quantum games with atoms and cavities - Electrodynamique ...
quantum games with atoms and cavities - Electrodynamique ...
quantum games with atoms and cavities - Electrodynamique ...
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
Entanglement, complementarity <strong>and</strong> decoherence:<br />
<strong>quantum</strong> <strong>games</strong> <strong>with</strong> <strong>atoms</strong> <strong>and</strong> <strong>cavities</strong><br />
J.M. Raimond<br />
Laboratoire Kastler Brossel,<br />
ENS, UPMC <strong>and</strong> IUF<br />
Florence, Mai 2004 Support:JST (ICORP), EC, CNRS, UMPC, IUF, CdF 1
A century of <strong>quantum</strong> mechanics<br />
An enormous range of applications, from superstrings to universe<br />
(a) (b) (c)<br />
10 -35 m<br />
10 -15 m<br />
10 -10 m<br />
(d) (e) (f)<br />
10 -8 m<br />
1 m<br />
10 8 m<br />
(g)<br />
(h)<br />
10 20 m<br />
10 26 m<br />
Florence, Mai 2004 2
A century of <strong>quantum</strong> mechanics<br />
One of the most precise theories so far.<br />
• QED: <strong>quantum</strong> description of electromagnetic interactions.<br />
– Spectrum of hydrogen atom at 10 -12 level<br />
– Gyromagnetic ratio of a free electron<br />
• St<strong>and</strong>ard model: a unified description of all interaction (but gravitation)<br />
Florence, Mai 2004 3
A wealth of applications:<br />
A century of <strong>quantum</strong> mechanics<br />
– Computers <strong>and</strong> the transistor revolution<br />
From the ENIAC to the laptop<br />
Florence, Mai 2004 4
A century of <strong>quantum</strong> mechanics<br />
• Nuclear magnetic imagers for medicine<br />
Florence, Mai 2004 5
A century of <strong>quantum</strong> mechanics<br />
• Lasers<br />
Florence, Mai 2004 6
A century of <strong>quantum</strong> mechanics<br />
• Ultrastable clocks <strong>and</strong> the GPS system<br />
Florence, Mai 2004 7
A century of <strong>quantum</strong> mechanics<br />
• Tailoring structures at the atomic level<br />
Florence, Mai 2004 8
A century of <strong>quantum</strong> mechanics<br />
• And yet an intriguing theory.<br />
– A microscopic world that defies our “classical” intuition.<br />
• These lectures will be devoted to the theoretical investigation <strong>and</strong> the<br />
experimental exploration of the most ‘bizarre’ aspects of the <strong>quantum</strong><br />
world.<br />
– Superposition principle<br />
– Complementarity<br />
– Quantum entanglement <strong>and</strong> non locality.<br />
– Quantum information processing<br />
– Quantum decoherence<br />
Florence, Mai 2004 9
The “strangeness” of the <strong>quantum</strong><br />
• Superposition principle <strong>and</strong> <strong>quantum</strong> interferences<br />
– The sum of <strong>quantum</strong> states is yet another possible state<br />
– A system “suspended” between two different classical realities<br />
D<br />
d<br />
a<br />
I<br />
Ψ =Ψ<br />
1<br />
+Ψ2<br />
= I + 2Re Ψ Ψ<br />
D<br />
a = λ<br />
d<br />
0 1 2<br />
– Feynman: Young’s slits experiment contains all the mysteries of the<br />
<strong>quantum</strong><br />
Florence, Mai 2004 10
The “strangeness” of the <strong>quantum</strong><br />
• Young interferences <strong>with</strong> electrons (A. Tonomura, 1989)<br />
Florence, Mai 2004 11
The “strangeness” of the <strong>quantum</strong><br />
• Quantum Young interferences <strong>with</strong> <strong>atoms</strong> (Shimizu 1992)<br />
Florence, Mai 2004 12
The “strangeness” of the <strong>quantum</strong><br />
• Complementarity (From Einstein-Bohr at the 1927 Solvay congress)<br />
– Microscopic slit: set in motion when deflecting particle. Which path<br />
information <strong>and</strong> no fringes<br />
– Macroscopic slit: impervious to interfering particle. No which path<br />
information <strong>and</strong> fringes<br />
– Wave <strong>and</strong> particle are complementary aspects of the <strong>quantum</strong> object.<br />
Florence, Mai 2004 13
The “strangeness” of the <strong>quantum</strong><br />
• Complementarity <strong>and</strong> Heisenberg uncertainty relations<br />
Particle's momentum p=h/λ<br />
Momentum transfer ∆P=pd/D<br />
Measure screen momentum <strong>with</strong>in ∆P:<br />
∆x=a<br />
Position uncertainty ∆X=h/∆P<br />
d<br />
D<br />
Position uncertainty= interfrange.<br />
Complete washing out of fringes<br />
Localisation <strong>and</strong> wave behavior are incompatible<br />
Florence, Mai 2004 14
• No cloning theorem<br />
The “strangeness” of the <strong>quantum</strong><br />
– It is impossible to produce two exact copies of an arbitrary <strong>quantum</strong><br />
states (violates unitarity of Schrödinger evolution): No <strong>quantum</strong> fax<br />
Florence, Mai 2004 15
• Entanglement<br />
The “strangeness” of the <strong>quantum</strong><br />
– Two systems after an interaction described by a single global state<br />
– No system has a-well defined state on his own<br />
• A measurement performed on one system affects the state of the<br />
other<br />
• Quantum correlations irrespective of the distance between<br />
entangled systems<br />
– At the heart of <strong>quantum</strong> non-locality<br />
– Einstein did not like that…<br />
• <strong>and</strong> he was wrong (Bell inequalities violation)<br />
Florence, Mai 2004 16
• Bell inequalities.<br />
The “strangeness” of the <strong>quantum</strong><br />
– Two observers (Alice <strong>and</strong> Bob) share a pair of two level particles<br />
(levels 0 <strong>and</strong> 1 eg spin ½) in one of four “Bell states”<br />
– Spin operators (more in further lectures)<br />
– Essential property<br />
Florence, Mai 2004 17
The “strangeness” of the <strong>quantum</strong><br />
• Bell inequalities (cont’d)<br />
– Einstein argument<br />
• Alice can tell <strong>with</strong> certainty what value she gets for a measurement<br />
of any σ u by asking Bob to make the measurement<br />
• An “element of reality” is associated to any component of Alice’s<br />
spin in XOZ plane<br />
• Classical vision: any experimental result is predetermined at pair<br />
preparation time (“hidden variable apporoach”)<br />
– CHSH version of Bell inequalities<br />
0<br />
a<br />
EPR<br />
b<br />
0<br />
1 a' Alice source<br />
Bob<br />
b'<br />
1<br />
Florence, Mai 2004 18
The “strangeness” of the <strong>quantum</strong><br />
• Bell inequalities (cont’d)<br />
– If hidden variables are right:<br />
– But <strong>quantum</strong> mechanics predicts:<br />
With<br />
We get:<br />
Florence, Mai 2004 19
The “strangeness” of the <strong>quantum</strong><br />
• Bell inequalities (cont’d)<br />
3<br />
2<br />
1<br />
S Bell<br />
−2 2 ≤S<br />
≤2<br />
2<br />
0<br />
-1<br />
-2<br />
-3<br />
0,0 0,5 1,0 θ/π<br />
1,5 2,0<br />
• Possible experimental test of <strong>quantum</strong> mechanics versus local realism.<br />
• A major step in our underst<strong>and</strong>ing of <strong>quantum</strong> mechanics<br />
Florence, Mai 2004 20
A severe test<br />
The “strangeness” of the <strong>quantum</strong><br />
• Correlations between two polarization-entangled photons 400 m apart.<br />
• Correlation signal S
The “strangeness” of the <strong>quantum</strong><br />
• Quantum/classical limit<br />
– No <strong>quantum</strong> superpositions at<br />
macroscopic scale<br />
– The "Schrödinger cat"<br />
• Decoherence<br />
1<br />
2<br />
( )<br />
+ ⇔<br />
Environment<br />
– A macroscopic system is strongly<br />
coupled to a complex<br />
environment<br />
– No entangled states neither.<br />
We only observe a very small fraction of<br />
all possible <strong>quantum</strong> states<br />
WHY <br />
– In all models, this coupling<br />
• leaves only a few states<br />
intact (preferred basis)<br />
• destroys very rapidly the<br />
<strong>quantum</strong> superpositions of<br />
these states<br />
Decoherence<br />
Florence, Mai 2004 22
Main features of decoherence<br />
Very fast process<br />
A simple decoherence model<br />
superposition lifetime<br />
=<br />
relaxation time<br />
separation betweenstates<br />
-<br />
0<br />
∆x<br />
+<br />
Depends upon the initial <strong>quantum</strong> state<br />
(distance between states or<br />
"macroscopicity" parameter)<br />
Needle in a superposition of positions<br />
Background gas. Particles <strong>with</strong> de<br />
Broglie wavelength λ 0<br />
.<br />
Not a trivial relaxation mechanism<br />
(but explained by st<strong>and</strong>ard relaxation<br />
theory for simple models)<br />
When ∆x>λ 0<br />
, the first collision destroys<br />
the coherence (Heisenberg<br />
microscope: the environment "knows"<br />
the needle position)<br />
Extraordinarily fast decoherence<br />
Strong link <strong>with</strong> complementarity:<br />
environment acquires "which path"<br />
information<br />
Florence, Mai 2004 23
The importance of decoherence<br />
Measurement theory<br />
Applications of <strong>quantum</strong> weirdness<br />
Decoherence plays an essential role in<br />
<strong>quantum</strong> measurement:<br />
Prevents meters from being in a<br />
<strong>quantum</strong> superposition!<br />
1<br />
2<br />
( + )<br />
Only statistical mixtures of meter states,<br />
corresponding to exclusive classical<br />
probabilities<br />
1<br />
2<br />
-<br />
0<br />
+<br />
-<br />
( +<br />
)<br />
-<br />
0<br />
+<br />
-<br />
0<br />
+<br />
"Stable" states <strong>and</strong> hence measured<br />
observable determined by relaxation<br />
dynamics<br />
0<br />
-<br />
+<br />
0<br />
+<br />
-<br />
0<br />
+<br />
Manipulate complex entangled states for<br />
<strong>quantum</strong> information processing or<br />
computing<br />
Decoherence affects these states.<br />
Very fast loss of <strong>quantum</strong> information<br />
A terrible obstacle to these applications<br />
NB: decoherence does not prohibit<br />
macroscopic <strong>quantum</strong> states (BEC,<br />
superfluids): a single <strong>quantum</strong> state<br />
Florence, Mai 2004 24
Why explorations of the <strong>quantum</strong> world <br />
A fundamental interest<br />
Promising applications<br />
Better underst<strong>and</strong>ing of <strong>quantum</strong><br />
postulates<br />
• Superposition<br />
• Measurement<br />
• Entanglement <strong>and</strong> non-locality<br />
Exploration of the <strong>quantum</strong>-classical<br />
boundary<br />
Realize some of the gendankenexperiments<br />
used by the founding<br />
fathers of <strong>quantum</strong> mechanics.<br />
«we never experiment <strong>with</strong> just one electron<br />
or atom or (small) molecule. In thoughtexperiments<br />
we sometimes assume that we<br />
do; this invariably entails ridiculous<br />
consequences…. »<br />
Use <strong>quantum</strong> weirdness to realize new<br />
functions for information transmission<br />
or processing<br />
From bits (0 or 1) to qubits (|0> <strong>and</strong> |1>)<br />
• Quantum cryptography<br />
• Quantum teleportation<br />
• Quantum information processing<br />
• Quantum computing<br />
All rely on sophisticated <strong>quantum</strong><br />
entanglement manipulations<br />
(Schrödinger British Journal of the Philosophy of<br />
Sciences, Vol 3, 1952)<br />
Florence, Mai 2004 25
Quantum cryptography<br />
Key distribution<br />
Practical realization<br />
With photons<br />
Two operators (Alice <strong>and</strong> Bob) share an<br />
entangled pair.<br />
Their measurements are correlated.<br />
Measure same observable: a r<strong>and</strong>om<br />
but common bit.<br />
Eavesdropper: measurements <strong>and</strong><br />
unavoidable perturbation of the<br />
correlations (no more Bell inequalities<br />
violation)<br />
Detect any eavesdropper<br />
Unconditionally secure key<br />
• Optical telecom fibers; distances up<br />
to 60 km<br />
• Free space: in principle possible for<br />
satellite-ground communication<br />
"Commercial" realizations available<br />
Long haul communication: lack of a<br />
"<strong>quantum</strong> repeater"<br />
Florence, Mai 2004 26
Quantum teleportation<br />
Principle<br />
Experimental realizations<br />
Photons polarization states<br />
Innsbruck+Rome<br />
Quantum fluctuations of a laser<br />
field<br />
Caltech<br />
A very beautiful illustration of<br />
<strong>quantum</strong> non-locality<br />
Transmit exactly an arbitrary<br />
<strong>quantum</strong> state from one station to<br />
another<br />
Impossible <strong>with</strong> measurements<br />
Use <strong>quantum</strong> non-locality<br />
No matter creation, no superluminal<br />
propagation<br />
Quite far from Star-Trek !!<br />
Florence, Mai 2004 27
Quantum teleportation<br />
• Principle of <strong>quantum</strong> teleportation<br />
|ψ 〉<br />
Bell<br />
Measurement<br />
2 classical bits<br />
U<br />
u<br />
a<br />
b<br />
|ψ 〉<br />
EPR Source<br />
• Bell states basis<br />
Florence, Mai 2004 28
– Initial state<br />
Quantum teleportation<br />
– Measurement of u <strong>and</strong> a in the “Bell basis” projects b on a state<br />
differing of the initial one by a trivial unitary transformation.<br />
– Knowing the result of Alice measurement bob can render the initial<br />
state.<br />
– Before Alice’s measurement, Bob has a statistical mixture of all four<br />
states i.e. an equal mixture of 0 <strong>and</strong> 1.<br />
– No matter creation, no superluminal communication.<br />
– A splendid illustration of <strong>quantum</strong> non-locality.<br />
Florence, Mai 2004 29
Quantum computing<br />
From bits to qubits<br />
Quantum parallelism<br />
Classical computer: bits<br />
0 or 1<br />
Quantum computer: qubits<br />
Two-level system<br />
states |0> <strong>and</strong> |1><br />
A qubit can be in a state superposition<br />
A <strong>quantum</strong> computer <strong>with</strong> an n qubits<br />
register can manipulate a <strong>quantum</strong><br />
superposition of all numbers <strong>and</strong><br />
perform simultaneously 2 n<br />
calculations !<br />
Quantum mechanics linearity:<br />
State superpositions available<br />
1<br />
2<br />
( 0 + 1 )<br />
is a possible state for a qubit<br />
Exponentially more efficient than a<br />
classical computer<br />
( + )<br />
Makes easy a few difficult problems<br />
• Shor : factorization<br />
• Grover : search in an unsorted<br />
database<br />
Florence, Mai 2004 30<br />
1<br />
2
Organisation of a <strong>quantum</strong> computer<br />
Quantum gates<br />
Any operation can be decomposed<br />
in a set of elementary operations<br />
p qubits<br />
(1 or 2 qubits)<br />
U<br />
Any <strong>quantum</strong> algorithm can be<br />
realized <strong>with</strong> a network of<br />
<strong>quantum</strong> gates<br />
Universal gates:<br />
p qubits<br />
A set of one or two gates allowing<br />
the realization of any network<br />
Some <strong>quantum</strong> gates<br />
1 qubit: arbitrary rotation<br />
iψ<br />
⎛ cosϕ<br />
e sinϕ<br />
⎞<br />
U ( ϕψ , ) = ⎜ −iψ<br />
⎟<br />
⎝−e<br />
sinϕ<br />
cosϕ<br />
⎠<br />
2 qubits: <strong>quantum</strong> phase gate<br />
0,0 ⎯⎯→ 0,0<br />
0,1 ⎯⎯→ 0,1<br />
1, 0 ⎯⎯→ 1, 0<br />
i<br />
1,1 ⎯⎯→ 1,1<br />
e Φ<br />
Conditional dynamics<br />
Florence, Mai 2004 31
Quantum entanglement manipulations<br />
• Quantum non locality, complementarity, computation all boil down to<br />
sophisticated <strong>quantum</strong> entanglement manipulations !<br />
• Not easy experimentally. Criteria to be met<br />
– Two-level systems.<br />
– Individually addressed<br />
– Prepared in a given state (initialization)<br />
– Final state completely analyzable (read out)<br />
– Strong common interaction to produce entanglement<br />
– Weak coupling to outside world to limit decoherence<br />
• Two last requirements clearly incompatible<br />
– Few suitable systems.<br />
– Even fewer achieved entanglement control <strong>and</strong> manipulation<br />
Florence, Mai 2004 32
Tools for fundamental <strong>quantum</strong> mechanics studies<br />
• Nuclear magnetic resonance<br />
– Controlled <strong>quantum</strong> states evolution for nuclear spins<br />
– Intramolecular interaction lead to spin/spin entanglement<br />
– Various applications to <strong>quantum</strong> computing<br />
– Thermodynamic ensembles of large number of molecules. No<br />
individual <strong>quantum</strong> systems<br />
Florence, Mai 2004 33
Tools for fundamental <strong>quantum</strong> mechanics studies<br />
• Entangled “twin” photons<br />
Correlation<br />
– Photon pairs naturally produced in an entangled state<br />
– Easy manipulation <strong>and</strong> transport of individual photons<br />
– Widely used for non-locality tests, <strong>quantum</strong> cryptography <strong>and</strong><br />
<strong>quantum</strong> teleportation<br />
– Weak photon-photon interaction. Further entanglement processing<br />
difficult. No photon-photon universal <strong>quantum</strong> gate.<br />
Florence, Mai 2004 34
Tools for fundamental <strong>quantum</strong> mechanics studies<br />
• Cold <strong>atoms</strong> <strong>and</strong> BE condensates<br />
– Atoms in optical lattices: control of individual <strong>atoms</strong> (one per site)<br />
– Controlled collisions <strong>and</strong> <strong>quantum</strong> gates<br />
– Not yet individual access to <strong>atoms</strong><br />
Florence, Mai 2004 35
• Ions in traps<br />
Tools for fundamental <strong>quantum</strong> mechanics studies<br />
– Single addressable long-lived <strong>quantum</strong> systems<br />
– Two ion <strong>quantum</strong> gates<br />
– Simple algorithms, teleportation.<br />
– One of the most promising systems for few qubits implementations<br />
– Great experimental difficulties<br />
Florence, Mai 2004 36
• Mesoscopic circuits<br />
Tools for fundamental <strong>quantum</strong> mechanics studies<br />
– Long-lived two level systems (artificial <strong>atoms</strong>)<br />
– Two qubits <strong>quantum</strong> gates<br />
– Promising for ‘large scale’ integration<br />
– Decoherence not well understood<br />
Florence, Mai 2004 37
Tools for fundamental <strong>quantum</strong> mechanics studies<br />
• Cavity <strong>quantum</strong> electrodynamics<br />
– Realizes the simplest matter-field system: a single atom coherently<br />
coupled to a few photons in a single mode of the radiation field,<br />
sustained by a high quality cavity.<br />
– Perfect test bench for fundamental <strong>quantum</strong> behaviors<br />
– Can be used for proof of principle demonstrations of <strong>quantum</strong> logics<br />
– Not really scalable to large scale architectures<br />
• Comes in two regimes<br />
– Weak coupling: radiative properties modifications<br />
– Strong coupling: atom-field interaction overwhelms dissipative<br />
processes (focus here)<br />
• Comes in two flavours<br />
– Optical CQED<br />
– Microwave CQED<br />
Florence, Mai 2004 38
A short history of cavity QED<br />
• The genesis<br />
• The strong coupling regime<br />
Purcell 46: spontaneous emission rate<br />
modification for a spin in a resonant<br />
circuit<br />
• The beginning<br />
Drexhage (70's) : Spontaneous emission<br />
spatial pattern modification for a<br />
molecule near a mirror<br />
• The weak coupling regime<br />
One- <strong>and</strong> two-photon micromasers<br />
(Munich, ENS,85-90)<br />
Vacuum Rabi splitting (Kimble, 92)<br />
Quantum Rabi oscillations<br />
• Using strong coupling for<br />
entanglement manipulations<br />
In progress<br />
• The "industrial" age<br />
Spontaneous emission acceleration<br />
(Goy, 83)<br />
Spontaneous emission inhibition<br />
(Kleppner, 85)<br />
Observed since then on many systems<br />
Use spontaneous emission modification<br />
for light emitting devices (VCSEL's<br />
<strong>and</strong> LED's)<br />
Florence, Mai 2004 39
Optical CEQD<br />
• A single atom in a high Q optical cavity (Fabry Perot)<br />
– Strong coupling regime<br />
– Easy interface <strong>with</strong> propagating photons (<strong>quantum</strong> communication)<br />
– Large atom-field forces: atom trapping <strong>with</strong> single photons<br />
– Single atom lasers <strong>and</strong> single photon deterministic sources<br />
– Fast pace <strong>and</strong> difficult control of entanglement.<br />
• Caltech, Munich, Stony brook,…<br />
Florence, Mai 2004 40
Microwave CQED<br />
• A single Rydberg atom interacting <strong>with</strong> a superconducting cavity<br />
– With circular <strong>atoms</strong>: both <strong>atoms</strong> <strong>and</strong> field long-lived<br />
– Very strong coupling regime<br />
– Fundamental <strong>quantum</strong> mechanics illustrations<br />
– Complex entanglement manipulations<br />
• Munich <strong>and</strong> ENS<br />
Florence, Mai 2004 41
‘Programme’ of these lectures<br />
• A detailed description of microwave CQED experiments<br />
– An opportunity to review many basic <strong>quantum</strong> concepts<br />
• Complementarity, decoherence, entanglement<br />
– A good introduction to basic <strong>quantum</strong> optics techniques<br />
• Quantum fields, Wigner representation, relaxation theory, <strong>quantum</strong><br />
Monte Carlo trajectories, dressed atom…<br />
– An opportunity to review <strong>quantum</strong> information concepts<br />
• Quantum computing, algorithms, <strong>quantum</strong> error correction<br />
codes…<br />
Florence, Mai 2004 42
An “appetizer” chapter<br />
• A brief survey of CQED <strong>with</strong> circular Rydberg <strong>atoms</strong> <strong>and</strong> superconducting<br />
<strong>cavities</strong><br />
• An experiment on complementarity<br />
• A direct study of the decoherence process<br />
Florence, Mai 2004 43
CQED <strong>with</strong> circular Rydberg <strong>atoms</strong> <strong>and</strong> superconducting <strong>cavities</strong><br />
Florence, Mai 2004 44
Circular Rydberg <strong>atoms</strong><br />
High principal <strong>quantum</strong> number<br />
Maximal orbital <strong>and</strong> magnetic <strong>quantum</strong><br />
numbers<br />
• Long lifetime<br />
• Microwave two-level transition<br />
• Huge dipole matrix element<br />
• Stark tuning<br />
• Field ionization detection<br />
– selective <strong>and</strong> sensitive<br />
• Velocity selection<br />
– Controlled interaction time<br />
– Well-known sample position<br />
Atoms individually addressed<br />
(centimeter separation between <strong>atoms</strong>)<br />
Full control of individual transformations<br />
51 (level e)<br />
51.1 GHz<br />
50 (level g)<br />
Complex preparation (53 photons ! )<br />
Stable in a weak directing electric field<br />
Single atom preparation: brute force !<br />
Florence, Mai 2004 45
Superconducting cavity<br />
Design<br />
Highly polished niobium Mirrors<br />
• Open Fabry Perot cavity <strong>with</strong> a<br />
"photon recirculating ring"<br />
• Compatible <strong>with</strong> a static electric field<br />
(circular state stability <strong>and</strong> Stark<br />
tuning)<br />
• Very sensitive to geometric quality of<br />
mirrors<br />
Cavity Damping time: 1 ms<br />
Field energy (db)<br />
29<br />
28<br />
27<br />
26<br />
25<br />
24<br />
-2 0 2 4 6 8 10<br />
Florence, Mai 2004 46<br />
time (ms)
General scheme of the experiments<br />
Rev. Mod. Phys. 73, 565 (2001)<br />
Florence, Mai 2004 47
From Dream to Reality<br />
Atoms preparation<br />
detection<br />
lasers<br />
Atomic<br />
beam<br />
Florence, Mai 2004 48
An object at the <strong>quantum</strong>/classical boundary<br />
Coherent field in a cavity<br />
From <strong>quantum</strong> to classical<br />
• State produced by a classical source<br />
in the cavity mode<br />
• Small field:<br />
α<br />
α = e ∑<br />
2<br />
−α<br />
/2<br />
n<br />
n n!<br />
– |n>: photon number state<br />
– Defined by complex amplitude α<br />
• A picture in phase space (Fresnel<br />
plane)<br />
n<br />
Im α<br />
| α|<br />
Φ<br />
n<br />
1<br />
∆Φ<br />
Re α<br />
2<br />
= α ∆ n=<br />
n<br />
∆N∆Φ≈1<br />
– Large <strong>quantum</strong> fluctuations. A<br />
field at the single-photon level is<br />
a <strong>quantum</strong> object<br />
• Large field<br />
– Small <strong>quantum</strong> fluctuations. A<br />
field <strong>with</strong> more than 10 photons is<br />
almost a classical object.<br />
Florence, Mai 2004 49<br />
a<br />
φ
Resonant atom-cavity interaction<br />
Quantum Rabi oscillations<br />
Initial state |e,0><br />
|3><br />
|2><br />
e><br />
g><br />
|1><br />
|0><br />
Atom Cavity<br />
Ω<br />
|3><br />
|2><br />
|e><br />
|1><br />
|g> |0><br />
Atom Cavity<br />
P e<br />
(t)<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
0.0<br />
time ( µ s)<br />
0 30 60 90<br />
Oscillatory Spontaneous emission <strong>and</strong> strong coupling regime<br />
Florence, Mai 2004 50
Rabi oscillation in a small coherent field<br />
1.0<br />
P e<br />
(t)<br />
0.5<br />
0.0<br />
Time ( µs)<br />
• A more complex signal<br />
0 30 60 90<br />
• π/2 pulse possible for any cavity field by proper tuning of interaction time<br />
Florence, Mai 2004 51
Rabi oscillation in a small coherent field:<br />
observing discrete Rabi frequencies<br />
Fourier transform of the Rabi oscillation signal<br />
Ω 0<br />
Ω 0<br />
2<br />
Ω 0<br />
3<br />
Discrete peaks<br />
corresponding to<br />
discrete photon numbers<br />
FFT (arb. u.)<br />
Direct observation<br />
of field quantization<br />
in a "box"<br />
0 25 50 75 100 125 150<br />
Frequency (kHz)<br />
Florence, Mai 2004 52
Rabi oscillation in a small coherent field:<br />
Measuring the photon number distribution<br />
1<br />
P () ( ) 1 cos<br />
( )<br />
g<br />
t = ∑ P N − Ω<br />
0t<br />
N + 1 . e<br />
2<br />
N<br />
( )<br />
−t<br />
/ τ<br />
Fit of P(n) on the Rabi oscillation signal:<br />
0,5<br />
0,4<br />
Measured P(n)<br />
Poisson law<br />
P(n)<br />
0,3<br />
0,2<br />
N = 0.85<br />
0,1<br />
0,0<br />
0 1 2 3 4 5<br />
Photon number<br />
accurate field statistics measurement<br />
Florence, Mai 2004 53
All results<br />
• 0<br />
• 0.4<br />
• 0.85<br />
• 1.77<br />
1.0<br />
0.5<br />
0.0<br />
1.0<br />
0.0 0.5 1.0<br />
n = 0.06<br />
th<br />
n = 0.4<br />
coh<br />
photons<br />
P e<br />
(t)<br />
0.5<br />
0.0<br />
1.0<br />
0.5<br />
0.0<br />
1.0<br />
FFT Amplitude<br />
P(n)<br />
0.0 0.5<br />
0.0 0.5<br />
n = 0.85<br />
coh<br />
n = 1.77<br />
coh<br />
0.5<br />
0.0<br />
0.0 0.3<br />
0 30 60 90<br />
0 50 100 150<br />
0 1 2 3 4 5<br />
Time (<br />
µs)<br />
Frequency (kHz)<br />
n<br />
Florence, Mai 2004 54
Field quantization<br />
• Many evidences of field quantization since Compton effect<br />
• Note that photoelectric effect is not a proof of field quantization (Dirac <strong>and</strong><br />
Wenzel 1926)<br />
• Quantization of Rabi frequencies provide a visceral evidence of field<br />
quantization<br />
• A direct insight into field statistics<br />
Florence, Mai 2004 55
Quantum Rabi oscillations: state transformations<br />
Initial state<br />
e,0<br />
1<br />
e,0 e⎯⎯→−<br />
,0 e,0 ⎯⎯→ eg,0<br />
,1 2π ( epulse<br />
,0 + g,1<br />
)<br />
2<br />
g,1 ( ⎯⎯→− ce e + cg π/2 spontaneous g,1) 0 ⎯⎯→ g<br />
Conditional ( ce 1 + c<br />
emission dynamics<br />
g<br />
0 )<br />
pulse<br />
π spontaneous emission pulse<br />
g,0 ⎯⎯→+ Entanglement g,0<br />
Quantum<br />
creation<br />
Atom/cavity state copy phase gate<br />
Atom-cavity EPR pair<br />
P e<br />
(t)<br />
0.8<br />
51 (level e)<br />
0.6<br />
51.1 GHz<br />
50 (level g)<br />
0.4<br />
0.2<br />
0.0<br />
Brune et al, PRL 76, 1800 (96)<br />
time ( µ s)<br />
0 30 60 90<br />
Florence, Mai 2004 56
Three "stitches" to "knit" <strong>quantum</strong> entanglement<br />
Combine elementary transformations to create complex entangled states<br />
• State copy <strong>with</strong> a π pulse<br />
– Quantum memory : PRL 79, 769 (97)<br />
• Creation of entanglement <strong>with</strong> a π/2 pulse<br />
– EPR atomic pairs : PRL 79, 1 (97)<br />
• Quantum phase gate based on a 2π pulse<br />
– Quantum gate : PRL 83, 5166 (99)<br />
– Absorption-free detection of a single photon: Nature 400, 239 (99)<br />
• Entanglement of three systems (six operations on four qubits)<br />
– GHZ Triplets : Science 288, 2024 (00)<br />
• Entanglement of two radiation field modes<br />
– Phys. Rev. A 64, 050301 (2001)<br />
• Direct entanglement of two <strong>atoms</strong> in a cavity-assisted collision<br />
– Phys. Rev. Lett. 87, 037902 (2001)<br />
Florence, Mai 2004 57
An experiment on complementarity<br />
a realization of Bohr’s 1927 gedankenexperiment<br />
Florence, Mai 2004 58
A “modern” version of Bohr’s proposal<br />
• Mach Zehnder interferometer<br />
φ<br />
φ<br />
D<br />
•Interference between two well-separated paths.<br />
• Getting a which-path<br />
information<br />
Florence, Mai 2004 59
A “modern” version of Bohr’s proposal<br />
• Mach-Zehnder interferometer <strong>with</strong> a moving slit<br />
φ<br />
φ<br />
D<br />
• Massive slit: negligible motion, no which- path information, fringes<br />
• Microscopic slit: which path information <strong>and</strong> no fringes<br />
Florence, Mai 2004 60
Complementarity <strong>and</strong> uncertainty relations<br />
Get a which path information<br />
P>∆p<br />
(∆p <strong>quantum</strong> fluctuations of<br />
beam splitter’s momentum)<br />
Hence<br />
∆x > h/∆p > h/P=λ<br />
B 1<br />
b<br />
P<br />
O<br />
φ<br />
a<br />
B 2<br />
M<br />
D<br />
φ<br />
M'<br />
Beam splitter’s <strong>quantum</strong> position fluctuations larger than wavelength: no<br />
fringes<br />
Florence, Mai 2004 61
Complementarity <strong>and</strong> entanglement<br />
• A more general analysis of Bohr’s experiment<br />
– Initial beam-splitter state<br />
– Final state for path b<br />
α<br />
0<br />
B 1<br />
M'<br />
b<br />
P<br />
a<br />
O<br />
φ<br />
B 2<br />
M<br />
D<br />
φ<br />
– Particle/beam-splitter state<br />
Ψ = Ψ<br />
a<br />
0 + Ψb α<br />
– Final fringes signal<br />
• Small mass, large kick<br />
– Particle/beam-splitter entanglement<br />
– (an EPR pair if states orthogonal)<br />
NO FRINGES<br />
• Large mass, small kick<br />
FRINGES<br />
Ψa Ψb 0 α<br />
0 α = 0<br />
0 α = 1<br />
Florence, Mai 2004 62
Entanglement <strong>and</strong> complementarity<br />
Entanglement <strong>with</strong> another system destroys interference<br />
• explicit detector (beam-splitter/ external)<br />
• uncontrolled measurement by the environment (decoherence)<br />
φ<br />
φ<br />
D<br />
Complementarity, decoherence <strong>and</strong> entanglement intimately linked<br />
Florence, Mai 2004 63
A more realistic system: Ramsey interferometry<br />
• Two resonant π/2 classical pulses on an atomic transition e/g<br />
1.0<br />
a<br />
M<br />
0.8<br />
B 1<br />
R 1<br />
R 2<br />
P g<br />
0.6<br />
0.4<br />
b<br />
φ<br />
0.2<br />
0.0<br />
M'<br />
B 2<br />
Fréquence relative (kHz)<br />
0 10 20 30 40 50 60<br />
D<br />
Which path information<br />
Atom emits one photon in R 1<br />
or R 2<br />
Ordinary macroscopic fields<br />
(heavy beam-splitter)<br />
Field state not appreciably affected. No "which path" information<br />
FRINGES<br />
Mesoscopic Ramsey field<br />
(light beam-splitter)<br />
Addition of one photon changes the field. "which path" info<br />
NO FRINGES<br />
Florence, Mai 2004 64
Experimental requirements<br />
• Ramsey interferometry<br />
– Long atomic lifetimes<br />
– Millimeter-wave transitions<br />
• Circular Rydberg <strong>atoms</strong><br />
• π/2 pulses in mesoscopic fields<br />
– Very strong atom-field coupling<br />
• Circular Rydberg <strong>atoms</strong><br />
• Field coherent over atom/field interaction<br />
• Superconducting millimeter-wave <strong>cavities</strong><br />
Florence, Mai 2004 65
Bohr’s experiment <strong>with</strong> a Ramsey interferometer<br />
• Illustrating complementarity: Store one Ramsey field in a cavity<br />
S<br />
Atom-cavity interaction time<br />
Tuned for π/2 pulse<br />
Possible even if C empty<br />
– Initial cavity state α<br />
1<br />
Ψ = e, αe<br />
+ g,<br />
αg<br />
– Intermediate atom-cavity state 2<br />
• Ramsey fringes contrast α α<br />
– Large field<br />
e<br />
α ≈ α ≈ α<br />
• FRINGES<br />
e<br />
g<br />
e<br />
R 1<br />
R 2<br />
g<br />
C<br />
φ<br />
( )<br />
g<br />
D<br />
φ<br />
– Small field<br />
α<br />
= 0, α = 1<br />
• NO FRINGE<br />
e<br />
g<br />
Florence, Mai 2004 66
Quantum/classical limit for an interferometer<br />
Fringes contrast<br />
Fringes contrast versus photon number N<br />
0.8<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0.0<br />
0 2 4 6 8 10 12 14 16<br />
Nature, 411, 166 (2001)<br />
N<br />
Fringes vanish for <strong>quantum</strong><br />
field<br />
photon number plays<br />
the role of the beamsplitter's<br />
"mass"<br />
An illustration of the ∆N∆Φ<br />
uncertainty relation :<br />
• Ramsey fringes reveal<br />
field pulses phase<br />
correlations.<br />
• Small <strong>quantum</strong> field: large<br />
phase uncertainty <strong>and</strong> low<br />
fringe contrast<br />
Not a trivial blurring of the<br />
fringes by a classical noise:<br />
atom/cavity entanglement<br />
can be erased<br />
Florence, Mai 2004 67
An elementary <strong>quantum</strong> eraser<br />
• Another thought experiment<br />
φ<br />
φ<br />
D<br />
Two interactions <strong>with</strong> the same beamsplitter assembly erase the which path information<br />
<strong>and</strong> restore the interference fringes<br />
Florence, Mai 2004 68
Ramsey “<strong>quantum</strong> eraser”<br />
• A second interaction <strong>with</strong> the mode erases the atom-cavity entanglement<br />
1.0<br />
Resonant Non-resonant Resonant<br />
0.9<br />
0.8<br />
0.7<br />
φ<br />
0.6<br />
0.5<br />
Pe<br />
0.4<br />
e,0<br />
1<br />
( ,0 + ,1 )<br />
2 e g<br />
g,1<br />
0.3<br />
0.2<br />
0.1<br />
0.0<br />
10 12 14 16 18 20 22 24<br />
• Ramsey fringes <strong>with</strong>out fields !<br />
– Quantum interference fringes <strong>with</strong>out external field<br />
– A good tool for <strong>quantum</strong> manipulations<br />
Florence, Mai 2004 69
A genuine <strong>quantum</strong> eraser<br />
Manipulating atom/cavity entanglement<br />
• Atom A i<br />
interacts <strong>with</strong> cavity field<br />
• Copy cavity state on another atom A e<br />
(π pulse)<br />
e<br />
A e<br />
Erasing entanglement<br />
π/2 pulse on A e<br />
mixes the states<br />
Resulting state:<br />
1<br />
2<br />
⎡<br />
⎣<br />
( − + ) + ( + )<br />
e e g g e g<br />
e i i e i i<br />
⎤<br />
⎦<br />
g<br />
A i<br />
Detection of A e<br />
projects A i<br />
onto a state<br />
superposition <strong>with</strong> a well-defined<br />
phase depending upon state of A e<br />
.<br />
State of two <strong>atoms</strong>:<br />
1<br />
2<br />
( e, g − g , e )<br />
i e i e<br />
An atomic EPR pair (PRL 79, 1 (97))<br />
FRINGES<br />
after a classical π/2 pulse on A i<br />
Phase conditioned to state of A e<br />
Direct detection of A e<br />
:no fringes on A i<br />
Entanglement <strong>with</strong> A e<br />
or C provides<br />
"which path" information on A i<br />
.<br />
Florence, Mai 2004 70
An EPR experiment revisited<br />
Conditional fringes on A i<br />
0 20 40 60 80<br />
1.0<br />
0.8<br />
A e<br />
in e<br />
A e<br />
in g<br />
1.0<br />
No fringes when<br />
tracing on A e<br />
Conditional Probabilities<br />
0.6<br />
0.4<br />
0.2<br />
Probability<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
0.0<br />
Relative frequency (kHz)<br />
0.0<br />
0 20 40 60 80<br />
Relative Frequency (kHz)<br />
PRL 79, 1 (97)<br />
A <strong>quantum</strong> eraser demonstration <strong>and</strong> controlled entanglement of two <strong>atoms</strong><br />
Florence, Mai 2004 71
A direct study of the decoherence process<br />
Florence, Mai 2004 72
Another experiment on complementarity<br />
e<br />
Cavity as an external detector in the<br />
Ramsey interferometer<br />
Cavity contains initially a coherent field<br />
Non-resonant atom-field interaction:<br />
e<br />
g<br />
R 1<br />
R 2<br />
α<br />
C<br />
Atom modifies the cavity field phase<br />
Phase shift α 1/δ<br />
S<br />
(index of refraction effect)<br />
⎯⎯→<br />
⎯⎯→<br />
e<br />
g<br />
(δ:atom-cavity detuning)<br />
1<br />
g<br />
D<br />
φ<br />
"Which path" information:<br />
• Small phase shift (large δ)<br />
(smaller than <strong>quantum</strong> phase noise)<br />
– field phase almost unchanged<br />
– No which path information<br />
– St<strong>and</strong>ard Ramsey fringes<br />
• Large phase shift (small δ)<br />
(larger than <strong>quantum</strong> phase noise)<br />
– Cavity fields associated to the<br />
two paths distinguishable<br />
– Unambiguous which path<br />
information<br />
– No Ramsey fringes<br />
Florence, Mai 2004 73
Fringes <strong>and</strong> field state<br />
1.0<br />
0.5<br />
Complementarity<br />
Fringes contrast <strong>and</strong> phase<br />
60<br />
n=9.5 (0.1)<br />
6<br />
Ramsey Fringe Signal<br />
0.0<br />
1.0<br />
0.5<br />
712 kHz<br />
Vacuum<br />
712 kHz,<br />
9.5 phot ons<br />
Fringe Contr ast (%)<br />
40<br />
20<br />
0<br />
0.0 0.2 0.4 0.6 0.8<br />
φ (radians)<br />
0.0 0.2 0.4 0 2<br />
φ (radians)<br />
4<br />
Fringe Shift (rd)<br />
0.0<br />
104 kHz<br />
0 2 4 6 8 10<br />
ν (kHz)<br />
347 kHz<br />
PRL 77, 4887 (96)<br />
• Excellent agreement <strong>with</strong> theoretical<br />
predictions.<br />
• Not a trivial fringes washing out effect<br />
Calibration of the cavity field<br />
9.5 (0.1) photons<br />
Florence, Mai 2004 74
A laboratory version of the Schrödinger cat<br />
Field state after atomic detection<br />
1<br />
2<br />
( + )<br />
A coherent superposition of two<br />
"classical" states.<br />
Very similar to the Schrödinger cat<br />
An atom to probe field coherence<br />
Quantum interferences involving the<br />
cavity state<br />
First atom<br />
Φ<br />
−Φ<br />
D<br />
Second atom<br />
Decoherence will transform this<br />
superposition into a statistical mixture<br />
Slow relaxation: possible to study the<br />
decoherence dynamics<br />
Decoherence caught in the act<br />
Two indistinguishable <strong>quantum</strong> paths to<br />
the same final state:<br />
2Φ<br />
−2Φ<br />
Florence, Mai 2004 75
A decoherence study<br />
Atomic correlation signal<br />
Decoherence versus size of the cat<br />
Two-Atom Correlation Signal<br />
0.0 0.1 0.2<br />
n=3.3 δ/2π =70 <strong>and</strong> 170 kHz<br />
0 1 2<br />
t/T<br />
r<br />
0 1 2 PRL 77, 4887 (1996)<br />
τ/T<br />
r<br />
Florence, Mai 2004 76<br />
correlation signal<br />
correlation signal<br />
δ/2π =70 kHz<br />
20<br />
16<br />
n=5.5<br />
12<br />
8<br />
4<br />
0<br />
0 1 2<br />
20<br />
t/T<br />
r<br />
16<br />
12<br />
n=3.3<br />
8<br />
4<br />
0
Decoherence features<br />
• Faster than cavity relaxation<br />
• Faster when distance between states increases<br />
• Decoherence time scale depends upon a "macroscopicity" parameter<br />
• Directly linked to complementarity <strong>and</strong> entanglement (environment<br />
acquires information on the <strong>quantum</strong> system)<br />
Not a trivial relaxation mechanism, if described by st<strong>and</strong>ard relaxation theory<br />
Essential for <strong>quantum</strong> measurement<br />
meters are not in superposition states<br />
Difficulty for applications of QM<br />
the more complex the entangled state, the faster the decoherence<br />
Florence, Mai 2004 77
The ENS team<br />
• Permanent members<br />
– Serge Haroche, Michel Brune, Gilles Nogues, Jean-Michel Raimond<br />
• Thesis students<br />
– F. Bernardot, P. Nussenzveig, A. Maali, J. Dreyer, X. Maître, P.<br />
Domokos, G. Nogues, A. Rauschenbeutel, P. Bertet, S. Osnaghi, A.<br />
Auffeves, P. Maioli, T. Meunier, P. Hyafil, J. Mozley, S. Gleyzes<br />
• Post doctoral visitors<br />
– F. Schmidt-Kaler, E. Hagley, P. Milman, S. Kuhr<br />
• Theoretical collaborations<br />
– L. Davidovich, N. Zagury (Rio)<br />
– D. Vitali, P. Tombesi (Camerino)<br />
• Optical CQED<br />
– V. Lefèvre, J. Hare<br />
Florence, Mai 2004 78
Structure of the lectures<br />
• I) Introduction<br />
• II) The tools of CQED<br />
• III) Experimental illustrations of fundamental <strong>quantum</strong> mechanics<br />
• IV) Quantum information <strong>with</strong> <strong>atoms</strong> <strong>and</strong> <strong>cavities</strong><br />
• V) Schrödinger cats <strong>and</strong> decoherence<br />
• VI) Cavity QED <strong>with</strong> dielectric microspheres<br />
• VII) Perspectives<br />
Florence, Mai 2004 79
Structure of the lectures<br />
• I) Introduction<br />
• II) The tools of CQED<br />
• III) Experimental illustrations of fundamental <strong>quantum</strong> mechanics<br />
• IV) Quantum information <strong>with</strong> <strong>atoms</strong> <strong>and</strong> <strong>cavities</strong><br />
• V) Schrödinger cats <strong>and</strong> decoherence<br />
• VI) Cavity QED <strong>with</strong> dielectric microspheres<br />
• VII) Perspectives<br />
Florence, Mai 2004 80
II) The tools of CQED<br />
• 1) Quantum fields<br />
• 2) Field relaxation<br />
• 3) A simple <strong>quantum</strong> device: the beamsplitter<br />
• 4) Atom-field coupling<br />
• 5) Experimental tools<br />
Florence, Mai 2004 81
II) The tools of CQED<br />
• 1) Quantum fields<br />
• 2) Field relaxation<br />
• 3) A simple <strong>quantum</strong> device: the beamsplitter<br />
• 4) Atom-field coupling<br />
• 5) Experimental tools<br />
Florence, Mai 2004 82
• Hamiltonian<br />
A mechanical oscillator<br />
• Dimensionless coordinates<br />
• Creation <strong>and</strong> annihilation operators<br />
• Fock states<br />
• |0>: “vacuum” state<br />
Florence, Mai 2004 83
A mechanical oscillator<br />
• Fock states wavefunctions<br />
Florence, Mai 2004 84
A mechanical oscillator<br />
• Operators evolution in Heisenberg point of view<br />
– Annihilation operators evolves in the same way as the classical<br />
amplitude<br />
Florence, Mai 2004 85
A cavity field mode<br />
• Quadratic hamiltonian as in the case of mechanical oscillator (field is a<br />
collection of harmonic oscillators)<br />
• Photon annihilation <strong>and</strong> creation operators<br />
• Electric field operator<br />
– normalization factor (dimension of a field)<br />
– local polarization<br />
– relative field mode amplitude <strong>and</strong> polarization (1 at field maximum)<br />
a solution of Helmoltz equation <strong>with</strong> cavity limiting conditions<br />
• Heisenberg picture<br />
Florence, Mai 2004 86
Field normalisation<br />
• Energy of Fock states<br />
• Cavity mode volume<br />
Florence, Mai 2004 87
Case of an open Fabry Perot cavity<br />
• Field amplitude<br />
(a)<br />
Elastic blade<br />
Ring<br />
R<br />
L<br />
2w 0<br />
PZT<br />
• Free space field quantization: introduction of a fictitious quantization box.<br />
Volume goes to infinity. Here: a real “photon box”. Field per photon<br />
perfectly defined. No need for complex limit taking.<br />
Florence, Mai 2004 88
Field quadratures<br />
• Coordinates for a two-dimensional phase space (generalizes X <strong>and</strong> P)<br />
• Eigenstates are non-normalizable (position/momentum eigenstates)<br />
• Pictorial representation of wave-functions in phase space<br />
Florence, Mai 2004 89
Phase space representation<br />
• Vacuum <strong>and</strong> three photon states<br />
Xφ+π/2<br />
X π/2<br />
X π/2<br />
X φ<br />
φ<br />
X 0<br />
X 0<br />
(a)<br />
(b)<br />
Florence, Mai 2004 90
Fock states<br />
• Fock states have a zero expectation value for the electric field (prop to a)<br />
<strong>and</strong> the vector potential<br />
• Fock states have a non zero electromagnetic energy<br />
• Fock states cannot be interpreted in terms of a classical field: a first<br />
example of non-classical states<br />
Florence, Mai 2004 91
Coherent states<br />
• Fock states are utterly non-classical. Other states<br />
Displacement operator<br />
• Physical interpretation of the displacement operator<br />
– Glauber formula (valid when A, B commute <strong>with</strong> their commutator)<br />
Translation operator<br />
Along X 0<br />
Florence, Mai 2004 92
Coherent state as a displacement of vacuum<br />
X π/2<br />
α<br />
Imα<br />
Reα<br />
X 0<br />
• The coherent state is a displaced vacuum. Hence a minimum uncertainty<br />
state<br />
Florence, Mai 2004 93
Creating coherent field <strong>with</strong> a classical current<br />
• Classical current density coupled to the mode (at origin for the sake of<br />
simplicity)<br />
• Source cavity coupling<br />
• Neglect the two terms oscillating at twice cavity frequency (RWA)<br />
• State evolution<br />
• Recognize for<br />
• A classical current produces a displacement <strong>and</strong> a coherent state.<br />
Florence, Mai 2004 94
Displacement <strong>and</strong> annihilation operator<br />
• Action of the displacement on the annihilation: calculate<br />
writes<br />
With<br />
– Baker-Hausdorf Lemma<br />
– two terms only<br />
Florence, Mai 2004 95
Combination of displacements<br />
• Displacement of a coherent state<br />
• Quantum analogue of classical homodyning<br />
Florence, Mai 2004 96
• Again Glauber<br />
Fock states expansion of coherent states<br />
• Last term leaves vacuum invariant.<br />
• Exp<strong>and</strong> second term in power series <strong>and</strong> note that<br />
• Essential property<br />
• NB a non hermitic. Admits complex eigenvalues<br />
• Non zero electric field value (not the case for Fock state)<br />
Florence, Mai 2004 97
Photon number distribution<br />
•<br />
0.5<br />
p<br />
c<br />
(n)<br />
0.4<br />
(a)<br />
p<br />
c<br />
(n)<br />
0.08<br />
(b)<br />
0.3<br />
0.06<br />
0.2<br />
0.04<br />
0.1<br />
0.02<br />
0.0<br />
0 1 2 3 4 5 6 7 8<br />
n<br />
0.00<br />
0 5 10 15 20 25 30 35<br />
n<br />
Florence, Mai 2004 98
Pictorial representation of a coherent state<br />
• A simple qualitative phase space diagram<br />
Im α<br />
1<br />
| α|<br />
Φ<br />
∆Φ<br />
Re α<br />
n<br />
2<br />
= α ∆ n= n ∆N∆Φ ≈1<br />
Florence, Mai 2004 99
A basis of coherent states<br />
• Scalar product of coherent states<br />
• Overcomplete basis<br />
• No unique decomposition (zero is also a coherent state)<br />
Florence, Mai 2004 100
Coherent states wavefunctions<br />
• Use<br />
• Inject between exponentials closure on<br />
• is the wavefunction of vacuum in P representation<br />
• A gaussian along the real axis, <strong>with</strong> a modulation reflecting the translation<br />
along the imaginary axis. Probability distribution<br />
Florence, Mai 2004 101
Time evolution of a coherent state<br />
• Evolution of coherent state given by<br />
(b)<br />
Im α<br />
Recover classical trajectory<br />
Of amplitude in phase space<br />
ω c<br />
t<br />
Re α<br />
Florence, Mai 2004 102
Quasi-probability distributions<br />
• Give a quantitative status to the description of <strong>quantum</strong> states in phase<br />
space.<br />
• For classical fields, any state represented by a probability distribution in<br />
phase space (cf statistical physics)<br />
• For <strong>quantum</strong> states no analog (because of Heisenberg uncertainty<br />
limitations)<br />
• Possible to define quasi probability distributions in phase space. They<br />
contain all possible information on the <strong>quantum</strong> state but they may be<br />
negative or singular.<br />
• Use here two distributions<br />
– Q function (very pictorial but not very useful)<br />
– W function (Wigner distribution). Extremely useful <strong>and</strong> precious<br />
insights in <strong>quantum</strong> states<br />
Florence, Mai 2004 103
The Q function<br />
• Definition for an arbitrary state<br />
– For a pure state: square of the overlap <strong>with</strong> the coherent state<br />
• Alternative definition<br />
– Probability to get zero photons in a field displaced by –α. Leads to<br />
simple experimental schemes to measure Q<br />
• Q for a coherent state<br />
Florence, Mai 2004 104
Examples of Q functions<br />
• Five photons coherent state <strong>and</strong> two-photon Fock state<br />
-4 -2<br />
0<br />
Q<br />
2<br />
4<br />
0.3<br />
0.2<br />
0.1<br />
α ι<br />
-4<br />
-2<br />
0<br />
2<br />
α r<br />
0<br />
4<br />
-4 -2<br />
0<br />
2<br />
4<br />
0.08<br />
0.06<br />
0.04<br />
0.02<br />
-4<br />
-2<br />
0<br />
2<br />
0<br />
4<br />
Florence, Mai 2004 105
Properties of the Q function.Link <strong>with</strong> characteristic function<br />
in anti-normal order<br />
Q(α) is a function ≥ 0, determined by « sweeping » phase space <strong>with</strong> a coherent state<br />
<strong>and</strong> attributing to each of its « positions » the expectation value in this state of the field<br />
density operator, divided by π .<br />
Another definition of Q: Fourier transform of the expectation value in the field state<br />
of the operator exp (-λ ∗ a) exp(λa + ) (characteristic function of the field in antinormal<br />
order):<br />
(ρ<br />
C )<br />
an<br />
(λ) = ∫∫ d 2 α Q (ρ ) (α) e λα * −λ * α<br />
= 1 π Tr ∫∫ d 2 αρα α e λα * −λ * α<br />
= 1 π Tr d 2 αρe −λ * a<br />
∫∫ α α e λa+<br />
= Tr ρ e −λ* a<br />
e λa + = e −λ* a<br />
e λa+ ρ<br />
(3 − 4)<br />
Operators in anti-normal (an) order<br />
<strong>and</strong> by inverse transformation (normalization checked by substituting in first line of (3-4)):<br />
Q (ρ ) (α) = 1 d 2 λ C (ρ)<br />
π 2<br />
an<br />
(λ) e λ* α −λα<br />
∫∫ *<br />
(3 − 5)<br />
Florence, Mai 2004 106
Wigner function: an insight into a <strong>quantum</strong> state<br />
A quasi-probability distribution in phase space.<br />
• Characterizes completely the <strong>quantum</strong> state<br />
• Negative for non-classical states.<br />
p<br />
Im(α)<br />
Describes the motion of a particle or<br />
a <strong>quantum</strong> single mode field<br />
Motion<br />
of a particle<br />
q<br />
Re(α)<br />
Electromagnetic<br />
field<br />
Florence, Mai 2004 107
Wigner function<br />
• Definition<br />
• By inverse Fourier transform<br />
• In particular<br />
• The probability distribution of x is obtained by integrating W over p<br />
• This property should obviously be invariant by rotation in phase space<br />
∫<br />
W q cosθ − p sin θ, q sinθ + p cosθ<br />
dp<br />
( )<br />
θ θ θ θ θ<br />
= P q = q Uˆ<br />
θ ˆ ρUˆ<br />
θ q<br />
( ) ( ) ( )<br />
†<br />
θ<br />
• All elements of density matrix derived from W: contains all possible<br />
information on <strong>quantum</strong> state.<br />
Florence, Mai 2004 108
Wigner function<br />
• Wigner function is normalized<br />
∫ dpdqW(q,p)=1<br />
• W may be negative. If cancels (nodes of<br />
wavefunction in x representation), W should have negative parts. Cannot<br />
be a full fledged probability distribution in all cases<br />
• W gives all averages of operators in symmetric ordering<br />
Tr ⎡⎣ ˆ ρ ( xp ˆˆ + px ˆˆ )/2 ⎤⎦ = ∫ dxdpW ( x,<br />
p)<br />
xp<br />
ˆ<br />
ˆ<br />
ˆ<br />
A =Tr[ρA]= ∫ dpdqW(q,p)f<br />
s(q,p)<br />
Florence, Mai 2004 109
A few Wigner functions<br />
• Vacuum<br />
• Coherent state<br />
• Fock state<br />
• Thermal field<br />
Florence, Mai 2004 110
Vacuum |0><br />
Examples of Wigner functions<br />
-2<br />
0<br />
2<br />
2<br />
Coherent state |β><br />
(β=1.5+1.5i)<br />
-4 -2 0 2 4<br />
2<br />
Thermal field n th<br />
=1<br />
-4 -2 0 2 4<br />
2<br />
1<br />
1<br />
1<br />
0<br />
0<br />
0<br />
-2<br />
0<br />
2<br />
-1<br />
-2<br />
-4<br />
-2<br />
0<br />
2<br />
-1<br />
-2<br />
4 -4<br />
-2<br />
0<br />
2<br />
-1<br />
-2<br />
4 Fock state |1><br />
-2 0 2<br />
2<br />
-4<br />
5 photons Fock state<br />
4<br />
2<br />
0<br />
-2<br />
1<br />
0<br />
0<br />
-1<br />
-0.3<br />
-2<br />
2<br />
0<br />
-2<br />
-4<br />
-2<br />
0<br />
2<br />
Florence, Mai 2004 111<br />
4
A classicality criterion<br />
• A field whose Wigner function is positive can be understood as a classical<br />
field <strong>with</strong> fluctuations described by a classical probability distribution<br />
– Coherent states, thermal states<br />
• A field <strong>with</strong> negative W cannot be understood classically<br />
– Fock states<br />
Florence, Mai 2004 112
An alternative simple expression of W<br />
• Two simple expressions left as an exercise<br />
• Parity operator<br />
• +1 for even Fock states, -1 for odd Fock states<br />
• Leads to a very simple experimental determination of W<br />
L. Davidovich, Private comm<br />
Florence, Mai 2004 113
Wigner function in terms of the characteristic function<br />
• Definition. Symmetric ordering<br />
• D unitary<br />
• Coherent field<br />
• Fock state<br />
• Wigner function<br />
• Normal order characteristic function<br />
* 2<br />
• Relation <strong>with</strong> anti normal order ( ρ) λa<br />
λ a λ /2 ( ρ)<br />
C ( λ) Trρe + −<br />
= = e C ( λ)<br />
s<br />
an<br />
Florence, Mai 2004 114
Schrödinger cat states<br />
• An example of non-classical state<br />
– Quantum superposition of two coherent fields <strong>with</strong> opposite phases.<br />
– No classical counterpart<br />
X π/2<br />
– Evidence the coherence<br />
−β<br />
β<br />
X 0<br />
– Photon number distribution: only even photon numbers contribute<br />
(even cat)<br />
P(n)<br />
– Odd cat<br />
024<br />
– Eigenstates of <strong>with</strong> +1 <strong>and</strong> -1 eigenvalues<br />
13 5<br />
Florence, Mai 2004 115
Quadrature distributions<br />
• X wavefunction of the even cat<br />
– Sum of two gaussian<br />
• P distribution of the even cat<br />
– With<br />
– Modulation revealing coherence<br />
Florence, Mai 2004 116
Graphical interpretation of quadrature distributions<br />
(a) (b) (c)<br />
X π/2<br />
X 0<br />
X π/2 X π/2<br />
X ϕ<br />
X 0<br />
X 0<br />
Florence, Mai 2004 117
Q<br />
( β +−β<br />
)<br />
Q function of a cat<br />
1<br />
( α) =<br />
2<br />
[<br />
π +<br />
−2<br />
β<br />
2 (1 e )<br />
e<br />
2 2<br />
2 2<br />
−α−β<br />
− α+<br />
β<br />
+ e<br />
+ α β −α β<br />
− ( α + β )<br />
2e<br />
cos 2(<br />
1 2 2 1)]<br />
-4 -2<br />
0<br />
2<br />
4<br />
0.15<br />
0.1<br />
0.05<br />
-4<br />
-2<br />
0<br />
2<br />
0<br />
4<br />
Exponential suppression of interferences near the origin revealing cat coherence.<br />
Q function of a cat similar to one of a mixture.<br />
Q is not well adapted to displaying mesoscopic <strong>quantum</strong> superpositions<br />
Florence, Mai 2004 118
Wigner function of phase-cat states ( β real)<br />
W ( β +−β ) (α) =<br />
1<br />
2π 2 (1 + e −2β 2 )<br />
d 2 λ e (αλ* −α * λ)<br />
[ ∫∫ ( β D(λ) β + −β D(λ) −β + β D(λ) −β + −β D(λ) β )] (3− 26)<br />
W (coherence) (α) =<br />
« Incoherent » terms (sum of Wigner functions<br />
of the two states |β> <strong>and</strong> |-β>)<br />
Coherent interference<br />
cohérent terms<br />
(W (coherence) )<br />
1<br />
[ dλ<br />
2π 2 (1+ e −2β 2 1<br />
dλ 2<br />
e 2i(α 2λ 1<br />
−α 1<br />
λ 2<br />
)<br />
∫∫<br />
−β D(λ) β + term β →−β] )<br />
(3 − 27)<br />
we can write the matrix element < −β |D(λ)|β > as (assuming β real <strong>with</strong>out loss of generality):<br />
−β D(λ) β = e iβλ 2<br />
−β β + λ = exp[−2β(β + λ 1<br />
) − λ 1 2 /2− λ 2 2 /2] (3− 28)<br />
<strong>and</strong> from (3-27) <strong>and</strong> (3-28) we find Gaussian integrals in λ 1 <strong>and</strong> λ 2 , from which the expression of W<br />
can easily be derived (see next page):<br />
dλ 1<br />
dλ 2<br />
e 2i(α 2λ 1<br />
−α 1<br />
λ 2<br />
)<br />
∫∫ −β D(λ) β = ∫ dλ 1<br />
exp[− 1 2 (λ 1<br />
+ 2(β − iα 2<br />
)) 2 − 2α 22<br />
− 4iβα 2<br />
]<br />
× ∫ dλ 2<br />
exp[− 1 2 (λ 2<br />
+ 2iα 1<br />
) 2 − 2α 12<br />
] = 2π e −2 α 2 e −4iβα 2<br />
(3− 29)<br />
Florence, Mai 2004 119
Graphs of Wigner functions of coherent states <strong>and</strong> of<br />
Vacuum<br />
their superpositions<br />
-2<br />
0<br />
2<br />
2<br />
Phase cat | β>+|−β><br />
1<br />
0<br />
W ( β +−β ) (α) =<br />
1<br />
[ ] (3 − 30)<br />
π(1 + e −2β 2 ) e−2α −β 2 + e −2α + β 2 + 2e −2α 2 cos4βα 2<br />
-1<br />
2<br />
(β real here)<br />
2<br />
-2<br />
-2<br />
0<br />
(case β=3)<br />
The interference term around |α|=0 has<br />
negative parts. The integral along<br />
directions parallel to the α 1 axis<br />
corresponding to fixed α 2 vanishes for<br />
some values of α 2 : «dark fringes» of P π/2<br />
(α) (see Lectures 1 <strong>and</strong> 2)<br />
-4<br />
-2<br />
0<br />
2<br />
4<br />
Florence, Mai 2004 120<br />
-2<br />
-1<br />
α 1<br />
α 2<br />
0<br />
1<br />
1<br />
0<br />
-1<br />
2<br />
-2<br />
-4<br />
Statistical<br />
mixture of<br />
|β> <strong>and</strong> |−β><br />
-2<br />
0<br />
2<br />
4<br />
-2<br />
-1<br />
0<br />
1<br />
1<br />
0<br />
-1<br />
2 -2<br />
2
II) The tools of CQED<br />
• 1) Quantum fields<br />
• 2) Field relaxation<br />
• 3) A simple <strong>quantum</strong> device: the beamsplitter<br />
• 4) Atom-field coupling<br />
• 5) Experimental tools<br />
Florence, Mai 2004 121
The problem<br />
• The field mode (system S) is coupled to a complex environment E<br />
– Charge carriers in the mirrors, other propagating modes coupled by<br />
mirror defects<br />
• The S+E system undergoes an hamiltonian evolution <strong>and</strong> is described by<br />
a pure state<br />
• The environment state is not accessible. We are interested only in the<br />
system’s reduced density matrix<br />
• Find the evolution equation of system’s density matrix.<br />
Florence, Mai 2004 122
The Kraus approach<br />
• Any mapping of a density matrix onto another density matrix (completely<br />
positive <strong>quantum</strong> map) can be put (non unambiguously) under the form<br />
– With a number of Kraus operators at most N² (N dimension of Hilbert<br />
space)<br />
• Assume that evolution ruled by a differential equation: mapping between<br />
ρ(t) <strong>and</strong> ρ(t+dt) (Markov approximation)<br />
• M i depends only upon dt if we assume that environment state not<br />
appreciably modified by system’s evolution (Born approximation)<br />
Florence, Mai 2004 123
Kraus cont’d<br />
• One of the operators is order zero in dt. Without loss of generality<br />
– (H <strong>and</strong> K hermitian) <strong>and</strong><br />
• As we get<br />
• Hence, H identified <strong>with</strong> the hamiltonian<br />
• Relaxation modelled by a Liouvillian in Lindbladt form<br />
Florence, Mai 2004 124
Physical interpretation of the L i<br />
• Evolution of a wavefunction during dt<br />
– Either system unchanged or (<strong>with</strong> a probability of order dt) evolution<br />
towards a wavefunction<br />
– L i describes the system evolution when some event occurs in the<br />
environment (eg apparition or disparition of a photon in the<br />
environment).<br />
– L i : “jump operator” (cf Monte carlo approach)<br />
– Underlying model: continuous monitoring of the environment. Change<br />
in the environment reflected by the action of one jump operator onto<br />
the system<br />
Florence, Mai 2004 125
• Jump operators<br />
Application to the cavity case<br />
– Zero temperature cavity: only event escape of one photon from the<br />
cavity, detected in the environment.<br />
• Associated jump operator a (annihilation operator) <strong>with</strong>in a<br />
normalization factor<br />
– Finite temperature cavity: also transfer of a thermal photon from the<br />
environment into the cavity.<br />
• Associated jump operator α a + (creation operator)<br />
• Complete Lindblad form:<br />
Florence, Mai 2004 126
A more st<strong>and</strong>ard approach<br />
• A simple environment model: collection of harmonic oscillators spanning a<br />
large frequency interval, weakly <strong>and</strong> linearly coupled to cavity mode (final<br />
result is model independent)<br />
• Cavity-environment hamiltonian<br />
• Interaction representation vs free hamiltonians<br />
• Many modes between ∆ <strong>and</strong> ∆+d∆, average coupling g(∆), “coupling<br />
density (proportional to g <strong>and</strong> mode density) γ(∆)<br />
Florence, Mai 2004 127
Evolution of density matrix<br />
• Global <strong>and</strong> reduced evolution equations<br />
• Second order expansion<br />
• Assume cavity <strong>and</strong> environment initially uncorrelated<br />
• Assume reservoir state unchanged (Born approx)<br />
• All first order terms cancel. Only second order contributions. 16 terms…<br />
Florence, Mai 2004 128
One term out of 16<br />
• One of the terms when developing second order onto the F operators<br />
• Equation of evolution for cavity density matrix<br />
• Large frequency span of reservoir. K has a short time memory. Markov<br />
approximation<br />
Florence, Mai 2004 129
• Gathering all terms<br />
Final Liouville-Lindblad equation<br />
• Same form as from Kraus approach.<br />
• More cumbersome approach but finer underst<strong>and</strong>ing of underlying<br />
mechanisms<br />
• Same equation for all environment models (eg two level <strong>atoms</strong>) provided<br />
large environment, wide frequency range <strong>and</strong> linear coupling<br />
• A trustable equation for relaxation, depending upon a single, classically<br />
measurable, parameter: cavity quality factor Q<br />
Florence, Mai 2004 130
Heisenberg picture: Langevin forces<br />
• Evolution of field operators in the Heisenberg approach<br />
• Classical damping of a (evolves as classical damped field amplitude) plus<br />
additional noise term. Brownian motion preserving commutation relations.<br />
• Leads to simple physical pictures of the evolution<br />
Florence, Mai 2004 131
Application: photon number distribution relaxation<br />
• Evolution of diagonal density matrix terms<br />
• Steady state solution: detailed balance condition<br />
• variance<br />
Florence, Mai 2004 132
Thermal field vs coherent field<br />
Florence, Mai 2004 133
Fock states lifetime<br />
• Single photon lifetime (at zero temperature)<br />
• κ -1 =T c The classical field energy damping time<br />
• |n> lifetime : T c /n<br />
• A simple effect but also a decoherence effect (fast relaxation of a nonclassical<br />
field)<br />
• Relaxation of coherent states can be obtained by a complex algebraic<br />
derivation. Here, simpler to use the Monte Carlo approach or a simple<br />
beam-splitter model (later on)<br />
Florence, Mai 2004 134
Quantum Monte Carlo trajectories<br />
• Model the relaxation in terms of stochastic wavefunctions.<br />
• Instead of tracing the environment, assume that it is (virtually) monitored.<br />
Continuous measurement of photon escapes for cavity by an array of<br />
detectors in the environment. At any time, the system is described by a<br />
wavefunction.<br />
• Two kind of evolutions<br />
– Jumps when an environment detector ‘clicks’<br />
– Non-unitary evolution between clicks. Not detecting a photon gives<br />
some hints that the cavity might be empty. The longer the time interval<br />
for no detection, the higher the probability that the cavity is empty. A<br />
no-detection gives information on the system, albeit more ambiguous<br />
than a click.<br />
• System density matrix recovered by averaging wavefunctions over very<br />
many trajectories.<br />
• A rigorous derivation based on Kraus approach.<br />
• Here: give the recipe <strong>and</strong> check consistent <strong>with</strong> Lindblad<br />
Florence, Mai 2004 135
Quantum Monte Carlo: general principle<br />
• During dt probability for a jump described by L i<br />
• Total jump probability<br />
• Wavefunction evolution if jump L i<br />
• Evolution if no jump<br />
– With<br />
• Iterate over time following this procedure. At each step, choose r<strong>and</strong>omly<br />
jump/no jump <strong>and</strong> L i if jump<br />
• Extremely efficient numerically for large systems (cold <strong>atoms</strong>)<br />
Florence, Mai 2004 136
Equivalence <strong>with</strong> Lindblad<br />
• Average over very many trajectories the projector on wavefunction.<br />
Evolution equation:<br />
• Replace jump <strong>and</strong> no-jump wavefunctions by their expression <strong>and</strong> get<br />
• St<strong>and</strong>ard Lindblad form associated <strong>with</strong> L i .<br />
Florence, Mai 2004 137
Case of cavity relaxation at zero temperature<br />
• Jump operators proportional to a<br />
• Jump probability per unit time<br />
– Obvious interpretation in terms of photon loss<br />
• Effective hamiltonian (interaction <strong>with</strong> respect to cavity hamiltonian)<br />
– Proportional to photon number<br />
• No-jump evolution<br />
Florence, Mai 2004 138
Relaxation of a Fock state<br />
• Fock state is an eigenstate of the no-jump evolution. Invariant in the nojump<br />
periods<br />
• Photon number decreases by one at each jump<br />
• Staircase decrease of the photon number. R<strong>and</strong>om step times<br />
• Ordinary exponential relaxation of energy recovered by averaging many<br />
trajectories.<br />
• Photon number variance zero initially <strong>and</strong> at long times. Maximum during<br />
evolution due to dispersion in jumps timing.<br />
Florence, Mai 2004 139
Relaxation of a coherent state<br />
• Coherent state eigenstate of jump operator. No evolution when jump ie<br />
photon detected in the environment. Counterintuitive<br />
– Loosing a photon reduces the photon number but<br />
– Seeing a photon gives an indication that the field amplitude is nonzero.<br />
Results in an increase of average photon number which exactly<br />
compensates the effect above.<br />
• Evolution between jumps only. Deterministic evolution. System remains at<br />
any time in a pure state<br />
• Evolution under the free cavity hamiltonian <strong>and</strong><br />
• Amounts to adding an imaginary part to<br />
• frequency ω+ικ/2<br />
Florence, Mai 2004 140
Relaxation in terms of characteristic functions<br />
• A simple result for the normal order characteristic function for a zero<br />
temperature relaxation. Very useful for relaxation of Schrödinger cats.<br />
Consider formally λ <strong>and</strong> λ* as<br />
Independent variables<br />
• Transforms Lindblad equation into a differential equation for C n<br />
– Associate formally a C n function to all combinations of ρ <strong>with</strong> a, a +<br />
(terms in the Lindblad equation)<br />
– Show that<br />
Florence, Mai 2004 141
Equation for C n<br />
• Substitution in Lindblad<br />
• Explicit solution<br />
• Extremely simple solution. Obtain it by characteristic method of check<br />
validity by direct substitution<br />
Florence, Mai 2004 142
Relaxation of a Schrödinger cat<br />
• Zero temperature evolution of a cat<br />
Florence, Mai 2004 143
Relaxation of a Schrödinger cat<br />
• Identify C n (t) <strong>with</strong> a density matrix<br />
• Damping of non diagonal terms <strong>with</strong> a rate 2κ|α|². Lifetime of the<br />
mesoscopic coherent Tc/2n. A typical decoherence effect (much more on<br />
that later)<br />
Florence, Mai 2004 144
Underst<strong>and</strong>ing fast decoherence<br />
• Decoherence time scale much shorter than energy lifetime.<br />
• Monte carlo approach: <strong>quantum</strong> jumps due to the action of a operator<br />
• A jump changes the parity of the cat <strong>with</strong>out changing the components<br />
amplitudes<br />
• A no-jump slightly reduces the components amplitudes <strong>with</strong>out affecting<br />
parity.<br />
• Average over many trajectories: washing out of parity information in a<br />
time typical of first photon escape ie Tc/n.<br />
• Fast evolution towards a statistical mixture<br />
• Then slow evolution (no jump terms) towards vacuum<br />
• Parity jumps can be corrected. Idea of a feedback method (feeding cats<br />
<strong>with</strong> <strong>quantum</strong> food) More later.<br />
Florence, Mai 2004 145
Decoherence in the Wigner point of view<br />
• Equation for Wigner function<br />
• Fokker planck equation. Drift to origin (damping of amplitude) <strong>and</strong><br />
diffusion.<br />
• Cat: fringes near origin <strong>with</strong> spacing 1/β. Washed out by diffusion process<br />
faster <strong>and</strong> faster when amplitude increases<br />
• Note: explicit solution (<strong>with</strong> κ=1)<br />
Florence, Mai 2004 146
Illustrated cat relaxation<br />
Florence, Mai 2004 147
II) The tools of CQED<br />
• 1) Quantum fields<br />
• 2) Field relaxation<br />
• 3) A simple <strong>quantum</strong> device: the beam-splitter<br />
• 4) Atom-field coupling<br />
• 5) Experimental tools<br />
Florence, Mai 2004 148
Beamsplitters<br />
• A semi transparent plate couples two propagating modes <strong>with</strong> the same<br />
polarization. Which <strong>quantum</strong> operations<br />
• An equivalent model: a fibre coupler<br />
• Classical amplitudes transformations<br />
Florence, Mai 2004 149
A simple <strong>quantum</strong> model<br />
• Models wave packet ‘collision’ on the beamsplitter<br />
• Transient linear coupling of the two modes <strong>with</strong> hamiltonian<br />
• Operators evolution in Heisenberg point of view<br />
– With Baker-Hausdorf lemma<br />
Florence, Mai 2004 150
A simple <strong>quantum</strong> model<br />
• With <strong>and</strong> group odd <strong>and</strong> even terms,<br />
proportional to a <strong>and</strong> b, in factor of expansions of cosθ <strong>and</strong> sinθ<br />
• Similarly<br />
• Mode annihilation operators transform as classical field amplitudes<br />
• Taking conjugate <strong>and</strong> <strong>with</strong><br />
• Examine now (in Schrödinger point of view) the action of the beamsplitter<br />
on simple states<br />
Florence, Mai 2004 151
Action on simple states<br />
• Mode b always in vacuum to start <strong>with</strong><br />
• Mode a in vacuum<br />
– Output state |0,0><br />
• Mode a in |1><br />
• Case of a balanced beam-splitter (θ=π/4)<br />
• Creation of an entangled state of the two modes (of EPR type)<br />
Florence, Mai 2004 152
• Fock state input<br />
Action on simple states<br />
• Massively entangled two mode state. Superposition of all possible<br />
partitions<br />
Florence, Mai 2004 153
• Coherent input<br />
Action on simple states<br />
• Two unentangled coherent outputs. Amplitudes follow the classical laws.<br />
• Coherent states impervious to entanglement <strong>with</strong> other modes<br />
• Important consequences for their relaxation<br />
Florence, Mai 2004 154
Action on simple states<br />
• Modes a <strong>and</strong> b contain a single photon<br />
• Case of a balanced beamsplitter θ=π/4<br />
• Both photons emerge in the same mode (M<strong>and</strong>el dip). A genuinely<br />
<strong>quantum</strong> effect. A direct manifestation of bosonic nature of photons.<br />
Destructive <strong>quantum</strong> interference between the direct <strong>and</strong> exchange paths<br />
cancels the probability for having one photon in each output<br />
Florence, Mai 2004 155
A partial Bell state analyzer<br />
• Each mode sustains two orthogonal polarizations H or V.<br />
• Assume splitting amplitudes independent of polarization<br />
• Two impinging photons. Four Bell states HH+/-VV, HV+/-VH<br />
• Three symmetric Bell states. The two photons emerge in the same path<br />
as for HH or VV<br />
• One antisymmetric polarization state HV-VH. Global state must be<br />
symmetric. Only possibility: antisymmetric mode combination. One photon<br />
in each output path.<br />
• Can be checked easily by an explicit calculation along the same lines left<br />
as an exercise<br />
• A possibility to distinguish one Bell state among four.<br />
• Teleportation, entanglement swapping experiments based on this<br />
process.<br />
Florence, Mai 2004 156
• Two interfering paths<br />
• Coherent state input<br />
Mach-zehnder: a <strong>quantum</strong> interferometer<br />
• After B 1<br />
• Dephaser<br />
• After B 2<br />
• Final photon count<br />
• Obviously the classical result<br />
• However phase of the incoming<br />
field plays no role. Same interference<br />
Pattern <strong>with</strong> a Fock state<br />
Florence, Mai 2004 157
Interferences <strong>with</strong>out phase: Fock state input<br />
• n photons Fock state as input<br />
• Final state<br />
• (action of a tunable beam-splitter on the Fock state)<br />
• Partition of the n photons in the two output ports <strong>with</strong> probabilities<br />
• Only the relative phase of the two interfering paths is important. The initial<br />
coherence plays no role. Same interference pattern <strong>with</strong> coherent states<br />
<strong>and</strong> single photon input.<br />
• A photon always interferes <strong>with</strong> itself (Dirac)<br />
Florence, Mai 2004 158
Single photon interferences in a Mach Zehnder<br />
• From Grangier et al 1986<br />
Florence, Mai 2004 159
Sensitivity of the interferometer<br />
• Quantum (shot noise) limits for the detection of small phases<br />
• Fock state input. Noise on the a detector<br />
• Sensitivity: η, inverse of the smallest dephasing changing the output<br />
photon number by more than these flucutations<br />
• Independent of phase. At fringes extrema, no variation of N <strong>with</strong> φ, but no<br />
noise either<br />
Florence, Mai 2004 160
• Coherent input<br />
Sensitivity of the interferometer<br />
• At φ=π/2 times larger than for Fock states<br />
• Initial photon number fluctuations add up <strong>with</strong> partition noise<br />
• Sensitivity<br />
• Optimum sensitivity on a dark fringe (no influence of input<br />
noise)<br />
• Same optimum sensitivity as for a Fock state.<br />
• Improved sensitivity in 1/N possible <strong>with</strong> non-linear beam splitters (n<br />
photons follow all path a or b)<br />
• Realizations <strong>with</strong> two photon states in the optical domain <strong>and</strong> ‘simulation’<br />
<strong>with</strong> ions in a trap<br />
Florence, Mai 2004 161
Homodyne field measurement<br />
• Another application of the beamsplitter. Mixes a large coherent input <strong>with</strong><br />
a <strong>quantum</strong> field. Large transmission T=cos²θ<br />
• First term= LO intensity. Can be substracted. Second term negligible if LO<br />
intense enough<br />
• Last term<br />
• Direct measurement of the <strong>quantum</strong> field<br />
Quadrature<br />
At the heart of tomographic field measurements<br />
Florence, Mai 2004 162
An insight into coherent field relaxation<br />
• A simple relaxation model in terms of beamsplitters. Mode a coupled to<br />
very many other modes b i . Coupling hamiltonian:<br />
• Action of H i during a small time interval : equivalent to coupling a <strong>with</strong><br />
modes b i by small reflection beamsplitters<br />
• Action of one of the couplings<br />
• Sum up independently all weak couplings<br />
• Mode a contains still a coherent state <strong>with</strong> a reduced amplitude.<br />
Reduction apparently quadratic in δτ. Deceptive.<br />
Florence, Mai 2004 163
An insight into coherent field relaxation<br />
• Large density of environment modes. Number of relevant ones for a time<br />
interval δτ of the order of 1/δτ<br />
• Rewrite as:<br />
• Decrease of a amplitude linear in δτ. Sum up time intervals, assuming<br />
that the environment modes remain in the same initial state (Born approx)<br />
• Recover coherent state relaxation. Note also that the environment modes<br />
contain at time t tiny coherent states (useful later for cat relaxation)<br />
• Obtained from mere energy conservation<br />
Florence, Mai 2004 164
II) The tools of CQED<br />
• 1) Quantum fields<br />
• 2) Field relaxation<br />
• 3) A simple <strong>quantum</strong> device: the beamsplitter<br />
• 4) Atom-field coupling<br />
• 5) Experimental tools<br />
Florence, Mai 2004 165
A two-level atom<br />
Level scheme<br />
|e〉<br />
ω eg<br />
|g〉<br />
Equivalent to a spin ½<br />
e =+ g =−<br />
Pauli matrices<br />
⎛1 0 ⎞ ⎛0 1⎞<br />
= ⎜ ⎟ σ<br />
⎝0 −1<br />
x<br />
= ⎜ ⎟<br />
⎠ ⎝1 0⎠<br />
1<br />
±<br />
x<br />
= ( + ± − )<br />
2<br />
σ z<br />
σ<br />
y<br />
⎛0<br />
−i⎞<br />
= ⎜ ⎟<br />
⎝i<br />
0 ⎠<br />
Florence, Mai 2004 166
Bloch sphere<br />
• Most general state + u . Correspondence between states <strong>and</strong> point on a<br />
unit radius sphere<br />
• Analogous to the Poincaré polarization sphere<br />
Florence, Mai 2004 167
A two-level atom<br />
• Raising <strong>and</strong> lowering operators<br />
{ σ σ }<br />
, 1<br />
− +<br />
=<br />
• Free hamiltonian<br />
H<br />
ωeg<br />
= σ<br />
z<br />
2<br />
• Dipole operator<br />
(<br />
*<br />
)<br />
a − a +<br />
D= d ε σ + ε σ<br />
Florence, Mai 2004 168
• Global hamiltonian<br />
Atom-cavity coupling<br />
• Four terms, two of which are anti-resonant: RWA<br />
Vacuum Rabi<br />
frequency<br />
• Jaynes <strong>and</strong> Cumming model (63)<br />
• Atom-cavity detuning<br />
• Uncoupled atom-cavity levels<br />
Florence, Mai 2004 169
Uncoupled atom-cavity levels<br />
• At low detunings, grouped in multiplicities<br />
• Couplings only inside the multiplicities<br />
• g,0 isolated. Impervious to atom-cavity<br />
coupling<br />
• Interaction representation<br />
– Energy origin at manifold center<br />
• Complete hamiltonian<br />
• In matrix form<br />
• Easy diagonalisation: dressed levels<br />
Florence, Mai 2004 170
The dressed levels<br />
• Eigenstates of the whole atom-field hamiltonian<br />
• In general complex expressions. Two simple cases<br />
– Resonance<br />
– Far off resonance<br />
Florence, Mai 2004 171
The resonant case<br />
• Atom-cavity at exact resonance<br />
|e,2><br />
|g,3><br />
Ω<br />
3<br />
|+,2><br />
|-,2><br />
• A doublet for the weak excitation<br />
from the ground state (the<br />
vacuum Rabi splitting<br />
• e <strong>and</strong> g are no longer<br />
eigenstates: a <strong>quantum</strong> Rabi<br />
oscillation between these levels<br />
|e,1><br />
|g,2><br />
|e,0><br />
|g,1><br />
|g,0><br />
Ω<br />
Ω<br />
2<br />
"vacuum Rabi<br />
splitting"<br />
|-,1><br />
|+,0><br />
|-,0><br />
Florence, Mai 2004 172
The <strong>quantum</strong> Rabi oscillation<br />
• An atom initially in e in a n-photon Fock state. Assumed at r=0 f=1<br />
• Atom initially in g<br />
• A Rabi oscillation between the two uncoupled levels.<br />
• Exists even when cavity empty (atom in e only)<br />
• Creates atom-cavity entanglement (much more on that later)<br />
Florence, Mai 2004 173
The vacuum Rabi splitting<br />
• Equivalent to the normal mode splitting for two coupled oscillators.<br />
Observed for an atom in an optical cavity <strong>and</strong> for excitons in a<br />
semiconducting cavity<br />
Kimble et al<br />
1992<br />
Weisbuch et al 92<br />
Florence, Mai 2004 174
Non resonant case<br />
• Position of dressed levels as a function of detuning<br />
Energy<br />
|+,n〉<br />
|e,n〉<br />
0<br />
hΩ<br />
|-,n〉<br />
|g,n+1〉<br />
-3 -2 -1 0 1 2 3 4<br />
∆ c<br />
/Ω<br />
Florence, Mai 2004 175
Large atom-field detuning case<br />
• Large atom-cavity detuning. Assumes also atom at rest at cavity centre<br />
Lamb shift + light shift<br />
Atomic ior effect<br />
Florence, Mai 2004 176
Action of an atom on a coherent field<br />
• Define an effective hamiltonian for shifts<br />
• Apply to<br />
• The atom (<strong>quantum</strong> system) controls the classical phase of the field<br />
• At the heart of Schrödinger cat states generation<br />
Florence, Mai 2004 177
Taking into account atomic motion<br />
• Up to now atom fixed. Real <strong>atoms</strong> cross gaussian mode<br />
• No simple expressions. Only resonant <strong>and</strong> far off resonant case<br />
– Resonant case<br />
– All expressions obtained at r=0 remain valid when replacing real time<br />
by the effective interaction time taking account mode geometry<br />
Florence, Mai 2004 178
Taking into account atomic motion<br />
– Non resonant case<br />
– Use effective hamiltonian, proportional to f²<br />
– The r=0 results also valid when using the effective interaction time<br />
– Note that resonant <strong>and</strong> non-resonant effective interaction times are<br />
not equal<br />
Florence, Mai 2004 179
Large field limit: classical field on a spin 1/2<br />
• Coupling <strong>with</strong> a very large coherent field α (not part of dynamics)<br />
• Interaction representation <strong>with</strong> respect to cavity frequency<br />
• Remove time dependence by going to rotating frame<br />
• In terms of Pauli matrices<br />
Florence, Mai 2004 180
Rabi rotation on the Bloch sphere<br />
• A geometrical interpretation<br />
– Any point on the Bloch<br />
sphere can be reached<br />
from + by interaction <strong>with</strong><br />
the proper field for the<br />
proper time.<br />
– Any component of the<br />
spin can be measured<br />
<strong>with</strong> a +/- detector <strong>with</strong> a<br />
prior rotation<br />
Florence, Mai 2004 181
• π/2 resonant pulse<br />
A few resonant pulses<br />
• π resonant pulse<br />
• 2π resonant pulse<br />
– Note sign change<br />
Florence, Mai 2004 182
The Ramsey interferometer<br />
• Two π/2 pulses <strong>with</strong> zero<br />
Phase <strong>and</strong> a depasing element<br />
Transient atomic level energy<br />
Shifts producing an atomic phase<br />
R 1<br />
Dep.<br />
R 2<br />
Evident analogy <strong>with</strong> a MZ<br />
Interference between two <strong>quantum</strong> paths<br />
Florence, Mai 2004 183
Another way of sweeping phase<br />
• Do not change atomic phase but change relative phase of pulses<br />
• Two independent sources<br />
• A same source, slightly offset from the atomic frequency (negligible offset<br />
for the interaction <strong>with</strong> a single pulse)<br />
• An essential method for high resolution spectroscopy. Long interrogation<br />
times <strong>with</strong>out long interaction <strong>with</strong> the source<br />
• At the heart of all atomic clocks<br />
Florence, Mai 2004 184
Ramsey fringes <strong>with</strong> cold <strong>atoms</strong><br />
• A cold <strong>atoms</strong> experiment for high stability clocks<br />
Florence, Mai 2004 185
A spin interferometer<br />
Two rotations of the spin around different axes through two timeseparated<br />
interactions <strong>with</strong> classical fields<br />
z<br />
z<br />
z<br />
x<br />
y<br />
R<br />
y<br />
y<br />
(π/ 2) R φ<br />
(π/ 2)<br />
x<br />
x<br />
y<br />
•Two Stern <strong>and</strong> Gerlach devices<br />
•Polarizer <strong>and</strong> analyzer<br />
•Final detection probability depends upon the relative phase of the pulses<br />
1.0<br />
0.8<br />
0.9<br />
g<br />
0.6<br />
0.6<br />
P g<br />
0.4<br />
0.3<br />
0.2<br />
0.0<br />
0 200 400 600 800<br />
0.0<br />
Fréquence relative (kHz)<br />
0 10 20 30 40 50 60<br />
Fréquence relative (kHz)<br />
Florence, Mai 2004 186
Rabi oscillation in a mesoscopic field<br />
• Intermediate regime of a few tens of photons. A first insight<br />
• A simple theoretical problem<br />
• A surprisingly complex behavior<br />
Florence, Mai 2004 187
Collapse <strong>and</strong> revival<br />
• Collapse: dispersion of field amplitudes due to dispersion of photon<br />
number<br />
• Revival: rephasing of amplitudes at a finite time such that oscillations<br />
corresponding to n <strong>and</strong> n+1 come back in phase<br />
• Revival is a genuinely <strong>quantum</strong> effect<br />
Florence, Mai 2004 188
Atomic relaxation<br />
• Atomic density matrices<br />
1 <br />
ρ = ⎡ 1 + n . σ ⎤<br />
2 ⎣ ⎦<br />
det ( ρ ) = (1/4)(1- n 2 ) ≥ 0<br />
<br />
n ≤ 1<br />
<br />
n = 1<br />
<br />
n < 1<br />
Pure case<br />
Statistical mixture<br />
• Geometrical representation: points inside the Bloch sphere<br />
• Ambiguity of representation<br />
<br />
n= λn + (1 −λ)<br />
n<br />
1 2<br />
→ ρ = λρ + (1 −λ)<br />
ρ<br />
1 2<br />
Florence, Mai 2004 189
Spontaneous emission relaxation<br />
• Emission from e to g (inside the two level system)<br />
• Stationary state at finite temperature<br />
Florence, Mai 2004 190
Spontaneous emission in a Monte Carlo process<br />
• Use the Monte Carlo approach. Atom in e, zero temperature<br />
• One jump operator (lowering)<br />
• Getting no jump decreases the probability for finding the atom in e<br />
• Continuous evolution as<br />
• Until a jump suddenly reduces wavepacket in g<br />
Florence, Mai 2004 191
II) The tools of CQED<br />
• 1) Quantum fields<br />
• 2) Field relaxation<br />
• 3) A simple <strong>quantum</strong> device: the beamsplitter<br />
• 4) Atom-field coupling<br />
• 5) Experimental tools<br />
Florence, Mai 2004 192
General scheme of the experiments<br />
Rev. Mod. Phys. 73, 565 (2001)<br />
Florence, Mai 2004 193
Circular Rydberg <strong>atoms</strong><br />
High principal <strong>quantum</strong> number<br />
Maximal orbital <strong>and</strong> magnetic <strong>quantum</strong><br />
numbers<br />
• Long lifetime<br />
• Microwave two-level transition<br />
• Huge dipole matrix element<br />
• Stark tuning<br />
• Field ionization detection<br />
– selective <strong>and</strong> sensitive<br />
51 (level e)<br />
51.1 GHz<br />
50 (level g)<br />
54.3 GHz<br />
• Velocity selection<br />
– Controlled interaction time<br />
– Well known sample position<br />
Atoms individually addressed<br />
(centimeter separation between <strong>atoms</strong>)<br />
Full control of individual transformations<br />
Complex preparation (53 photons ! )<br />
Stable in a weak directing electric field<br />
Florence, Mai 2004 194
Circular states wavefunction<br />
• Simple expression (maximal <strong>quantum</strong> numbers spherical harmonic)<br />
Florence, Mai 2004 195
A classical atom<br />
• All <strong>quantum</strong> numbers are large. Most properties can be calculated by<br />
classical arguments (correspondence principle)<br />
• Eg: stark polarizability <strong>and</strong> ionisation threshold (atomic units used)<br />
– Using<br />
– Weak field limit exp<strong>and</strong> first equation<br />
– In natural units, polarizability of 50 -2MHz/(V/cm)². -255kHz/(V/cm)²<br />
differential on the 50 to 51 transition<br />
Florence, Mai 2004 196
• Ionization threshold<br />
A classical atom<br />
– Eliminate radius in the system: closed equation for θ<br />
– First term has a maximum, 0.2, obtained for<br />
– Ionization threshold<br />
– 165 <strong>and</strong> 152 V/cm for 50 <strong>and</strong> 51. Good agreement <strong>with</strong> measured<br />
values<br />
Florence, Mai 2004 197
A classical atom<br />
• Spontaneous emission lifetime<br />
– Radiation reaction force<br />
– Angular momentum equation<br />
– Average on long times <strong>and</strong> note (integration by parts)<br />
– Circular to circular transition corresponds to one unit angular<br />
momentum<br />
– Exact agreement <strong>with</strong> <strong>quantum</strong> value<br />
Florence, Mai 2004 198
52 F m=2<br />
Circular state preparation<br />
n=52 in 2.5 V/ cm<br />
Circular<br />
states<br />
52<br />
π<br />
σ<br />
1.26 µm<br />
5D<br />
776 nm<br />
250 MHz<br />
•Three diode laser steps<br />
51<br />
•Stark switching to the lower Stark level m=2<br />
•Adiabatic 250 MHz transitions to the circular state<br />
•Final microwave transition in a high field: 'purification'<br />
5P<br />
σ<br />
780 nm<br />
5S<br />
Florence, Mai 2004 199
Working <strong>with</strong> single <strong>atoms</strong><br />
Method<br />
Pros <strong>and</strong> cons<br />
• Weak excitation of the atomic beam:<br />
Poisson statistics for the atom<br />
number in each sample<br />
• Finite detection efficiency: 40%<br />
No deterministic preparation of single<br />
atom samples<br />
Brute force approach:<br />
Prepare much less than one (0.1) atom<br />
on the average<br />
Extremely easy to achieve<br />
Long data taking times, growing<br />
exponentially <strong>with</strong> atomic samples<br />
count<br />
• 1 sample (1 atom): 10 minutes<br />
• 2 samples: Hours<br />
• 3 samples: Days<br />
• 4 samples: Weeks (not very practical)<br />
When an atom is detected, low<br />
probability for an undetected second<br />
one: single atom samples<br />
Florence, Mai 2004 200
Velocity selection<br />
Doppler selective optical pumping<br />
55°<br />
Atomic<br />
beam<br />
L 1<br />
L 2<br />
L 3<br />
85<br />
Rb<br />
Time of flight<br />
• Pulsed laser selection of F=3 (2 µs)<br />
• Pulsed Rydberg excitation (2 µs)<br />
• Improvement of velocity selection<br />
1.0<br />
410.6 +/- 1 m/s 432 +/- 1 m/s<br />
0.8<br />
δ<br />
δ<br />
F'=4<br />
120 MHz 5P 3/ 2<br />
F'=3<br />
63 MHz<br />
F'=2<br />
29 MHz<br />
F'=1<br />
Normalized atomic flux<br />
0.6<br />
0.4<br />
0.2<br />
L 1 L 2 L 3<br />
7000<br />
6000<br />
5000<br />
0.0<br />
390 400 410 420 430 440 450<br />
Velocity (m/s)<br />
5S<br />
F=3<br />
F=2<br />
Atoms flux<br />
4000<br />
3000<br />
2000<br />
1000<br />
0<br />
∆v=15 m/s<br />
0 200 400 600 800<br />
Atomic velocity (m/s)<br />
• Final width: 2 m/s.<br />
• Position of atomic sample known<br />
<strong>with</strong>in 1mm<br />
Florence, Mai 2004 201
Field ionization detection<br />
D e<br />
D g<br />
Electrostatic<br />
Ionization signals<br />
n=52<br />
0.6 K<br />
lenses<br />
n=51<br />
n=50<br />
4.2 K<br />
125 V/ cm 136 V/ cm 148 V/ cm<br />
Field<br />
77 K<br />
Electron<br />
Multipliers<br />
• Detection efficiency 40%<br />
• Error rate 4%<br />
• Dark counts: negligible<br />
Counting Electronics<br />
Florence, Mai 2004 202
Superconducting cavity<br />
Design<br />
Highly polished niobium Mirrors<br />
• Open Fabry Perot cavity <strong>with</strong> a<br />
"photon recirculating ring"<br />
• Compatible <strong>with</strong> a static electric field<br />
(circular state stability <strong>and</strong> Stark<br />
tuning)<br />
• Very sensitive to geometric quality of<br />
mirrors<br />
Cavity Damping time: 1 ms<br />
Field energy (db)<br />
29<br />
28<br />
27<br />
26<br />
25<br />
24<br />
-2 0 2 4 6 8 10<br />
Florence, Mai 2004 203<br />
time (ms)
Tuning system<br />
• 15 MHz range Sub Hz sensitivity<br />
Florence, Mai 2004 204
Two cavity modes<br />
• Two modes <strong>with</strong> the same geometry <strong>and</strong> orthogonal linear polarizations<br />
• Degenerate in an ideal cavity<br />
• Mirrors imperfections lift this degeneracy:<br />
– Two modes M a <strong>and</strong> M b <strong>with</strong> a frequency splitting 80-130 kHz<br />
– Atom can be tuned at resonance <strong>with</strong> either mode via Stark tuning<br />
Florence, Mai 2004 205
Determination of cavity Q<br />
Cavity width: 100 Hz<br />
Peak transmission –80 dB<br />
Millimeter-wave vector network analyzer<br />
(ABmm)<br />
•120 dB dynamical range<br />
•
Ring <strong>and</strong> coherent atomic state manipulations<br />
• Use classical microwave sources to<br />
manipulate atomic state before or<br />
after interaction <strong>with</strong> the mode<br />
S<br />
• Small access <strong>and</strong> exit holes in the<br />
cavity ring: stray fields spoil any<br />
atomic coherence<br />
• All coherent manipulations are to be<br />
performed inside the cavity-ring<br />
structure<br />
• Classical fields fed in a low-Q<br />
transverse st<strong>and</strong>ing wave structure<br />
Independent manipulation of two atomic<br />
samples feasible (exploit the nodes<br />
<strong>and</strong> antinodes)<br />
Tight constraints on the atomic timing<br />
No long distance <strong>quantum</strong> correlations<br />
(no teleportation experiment)<br />
Florence, Mai 2004 207
The 3 He- 4 He refrigerator<br />
N 2<br />
4He<br />
4<br />
He<br />
1.4 K<br />
3<br />
He<br />
0.6 K<br />
4.2 K<br />
77 K<br />
Florence, Mai 2004 208
Aim:<br />
Cavity cooling<br />
Get rid of a residual 1 photon thermal<br />
field <strong>and</strong> of photons left by previous<br />
experiments<br />
Timing<br />
6000<br />
5000<br />
Cooling <strong>atoms</strong><br />
prepared in g<br />
Method<br />
Send packets of 1-10 <strong>atoms</strong> in the lower<br />
state g.<br />
Counts (A.U.)<br />
4000<br />
3000<br />
2000<br />
1000<br />
0<br />
Probe atom<br />
prepared in g<br />
1450 1460<br />
x10<br />
They efficiently absorb residual photons<br />
<strong>and</strong> cool the cavity mode<br />
1000 1200 1400 1600 1800<br />
Performances:<br />
Detection time (µs)<br />
Reduction of average thermal photon<br />
number down to 0.1<br />
Experiment performed in a time short<br />
compared to cavity relaxation time T r<br />
Florence, Mai 2004 209
From dream.. To reality<br />
Florence, Mai 2004 210
Structure of the lectures<br />
• I) Introduction<br />
• II) The tools of CQED<br />
• III) Experimental illustrations of fundamental <strong>quantum</strong> mechanics<br />
• IV) Quantum information <strong>with</strong> <strong>atoms</strong> <strong>and</strong> <strong>cavities</strong><br />
• V) Schrödinger cats <strong>and</strong> decoherence<br />
• VI) Cavity QED <strong>with</strong> dielectric microspheres<br />
• VII) Perspectives<br />
Florence, Mai 2004 211
III) Experimental illustrations of fundamental<br />
<strong>quantum</strong> mechanics<br />
• 1) Cavity induced light shifts <strong>and</strong> Lamb shifts<br />
• 2) Micromaser<br />
• 3) Quantum non-demolition measurement<br />
• 4) measurement of the Wigner function<br />
• 5) non classical field states<br />
Florence, Mai 2004 212
III) Experimental illustrations of fundamental<br />
<strong>quantum</strong> mechanics<br />
• 1) Cavity induced light shifts <strong>and</strong> Lamb shifts<br />
• 2) Micromaser<br />
• 3) Quantum non-demolition measurement<br />
• 4) measurement of the Wigner function<br />
• 5) non classical field states<br />
Florence, Mai 2004 213
Cavity induced shifts<br />
• Use sensitivity of Ramsey techniques to evidence cavity induced shifts<br />
• Early experiments performed in a very low Q cavity<br />
Florence, Mai 2004 214
• ∆ c /2π=150 kHz<br />
Light shifts<br />
Field relaxation (2 µs) much<br />
faster than atomic transit time:<br />
sensitive only to average field<br />
intensity. Field quantization<br />
aspects are irrelevant.<br />
Florence, Mai 2004 215
Lamb shifts<br />
• Interaction <strong>with</strong> the ‘vacuum’<br />
Solid line corrected for residual thermal field (0.32 photons)<br />
A remarkable single mode Lamb shift effect<br />
Florence, Mai 2004 216
III) Experimental illustrations of fundamental<br />
<strong>quantum</strong> mechanics<br />
• 1) Cavity induced light shifts <strong>and</strong> Lamb shifts<br />
• 2) Micromaser<br />
• 3) Quantum non-demolition measurement<br />
• 4) measurement of the Wigner function<br />
• 5) non classical field states<br />
Florence, Mai 2004 217
Principle<br />
• Cumulative emissions in the cavity by Rabi oscillations create a ‘large’<br />
field.<br />
• Very long cavity damping time (closed <strong>cavities</strong>): field maintained <strong>with</strong><br />
much less than one atom on the average<br />
• A true <strong>quantum</strong> device<br />
• One- <strong>and</strong> two-photon micromasers realized<br />
• Garching <strong>and</strong> ENS<br />
• Recently: optical analogue microlasers (Kimble)<br />
Florence, Mai 2004 218
• A gain/loss analysis<br />
A semi-classical model<br />
• n photons state in the cavity . Probability for atomic emission<br />
• Photon number rate equation<br />
• A graphical solution<br />
Florence, Mai 2004 219
Graphical solution for steady state<br />
• First threshold gain>losses near origin<br />
• Multiple thresholds <strong>and</strong> hysteretic behavior<br />
Florence, Mai 2004 220
Quantum model<br />
• Equation for the photon number distribution (no phase information, no<br />
coherences)<br />
• Solution by detailed balance condition. Leads to recursion relation<br />
• A single operating point. Multistable behaviour washed out by <strong>quantum</strong><br />
fluctuations<br />
• Gives average photon number <strong>and</strong> photon fluctuations<br />
Florence, Mai 2004 221
Average photon number<br />
• Oscillations corresponding to multiple thresholds<br />
• Dips corresponding to the ‘trapping states’ conditions (gain cancels for<br />
some photon number)<br />
Florence, Mai 2004 222
Photon number variance<br />
• Strong sub-poissonian character, particularly near trapping states (ideally<br />
a Fock state is generated)<br />
• Large variance close to thresholds<br />
Florence, Mai 2004 223
III) Experimental illustrations of fundamental<br />
<strong>quantum</strong> mechanics<br />
• 1) Cavity induced light shifts <strong>and</strong> Lamb shifts<br />
• 2) Micromaser<br />
• 3) Quantum non-demolition measurement<br />
• 4) measurement of the Wigner function<br />
• 5) non classical field states<br />
Florence, Mai 2004 224
2π <strong>quantum</strong> Rabi pulse<br />
Initial state<br />
e,0 ⎯⎯→− e,0<br />
g,1 ⎯⎯→− g,1<br />
2π pulse<br />
Conditional dynamics<br />
e,0<br />
g,0 ⎯⎯→+ g,0<br />
Quantum phase gate<br />
P e<br />
(t)<br />
0.8<br />
51 (level e)<br />
0.6<br />
51.1 GHz<br />
50 (level g)<br />
0.4<br />
0.2<br />
Brune et al, PRL 76, 1800 (96)<br />
0.0<br />
time ( µ s)<br />
0 30 60 90<br />
Florence, Mai 2004 225
A single photon phase-shifts an atomic coherence<br />
Principle<br />
Preparation <strong>and</strong> test of an atomic<br />
coherence: Ramsey set-up<br />
e<br />
g<br />
π<br />
P g<br />
Timing<br />
Position<br />
(a)<br />
C<br />
e<br />
π<br />
π/2<br />
A 1<br />
g A 2<br />
2π<br />
R 1<br />
gi<br />
D<br />
π/2<br />
D<br />
R 2<br />
gi<br />
Time<br />
i<br />
R 1<br />
C R 2 D 0 ν−ν gi<br />
π phase shift of fringes when cavity<br />
contains one photon<br />
Signal<br />
0,9<br />
0,8<br />
0,7<br />
One photon<br />
Zero photon<br />
Preparation of |1>: source atom,<br />
prepared in e, π <strong>quantum</strong> Rabi<br />
pulse<br />
Probability<br />
0,6<br />
0,5<br />
0,4<br />
0,3<br />
0,2<br />
0,1<br />
0 10 20 30 40 50 60<br />
Frequency ν (kHz)<br />
Florence, Mai 2004 226
Absorption-free detection of a single photon<br />
Principle<br />
0,9<br />
One photon<br />
Zero photon<br />
• Photon detection<br />
• Photon is still there after the<br />
detection: QND measurement<br />
0,8<br />
0,7<br />
0,6<br />
Probability<br />
0,5<br />
0,4<br />
0,3<br />
0,2<br />
0,1<br />
0 10 20 30 40 50 60<br />
Frequency ν (kHz)<br />
Atomic state correlated to photon<br />
number for a proper phase<br />
– i for 0<br />
– g for 1<br />
Nogues et al,Nature 400, 239 (99)<br />
Florence, Mai 2004 227
A brief reminder on QND measurements<br />
An ideal intensity measurement<br />
Braginsky 1970: ideal <strong>quantum</strong><br />
measurement.<br />
For light intensity : count the photon<br />
number <strong>with</strong>out changing it<br />
Ordinary intensity measurement<br />
Photon is<br />
destroyed<br />
Optical QND measurements:<br />
•Interaction of "signal" <strong>and</strong> "meter" beams<br />
in a Kerr non-linear medium<br />
•Interferometric detection of index change<br />
produced by "signal" intensity<br />
Meter<br />
Signal<br />
QND intensity measurement<br />
Phase reference<br />
Photon<br />
detected but<br />
still present<br />
A QND detection can be repeated<br />
In our experiment:<br />
•Signal: cavity field<br />
•Meter: atom<br />
Florence, Mai 2004 228
A repeated QND measurement<br />
Measure twice a single photon<br />
Conditional probabilities<br />
• Photon from a small thermal field (0.3<br />
photon on average)<br />
• First QND measurement<br />
• Second "absorptive" measurement<br />
0,50<br />
0,45<br />
0,40<br />
0,35<br />
I 1<br />
if<br />
1 photon<br />
G 1<br />
if<br />
1 photon<br />
E 2<br />
if G 1<br />
E 2<br />
if I 1<br />
E 2<br />
Timing<br />
Probability<br />
0,30<br />
0,25<br />
0,20<br />
Position<br />
C<br />
Relaxation 2π<br />
g<br />
π/2<br />
R1<br />
gi<br />
A 1<br />
π/2<br />
g<br />
D<br />
R 2<br />
gi<br />
A 2<br />
π<br />
Time<br />
D<br />
0,15<br />
0,10<br />
0 10 20 30 40 50 60<br />
Frequency (kHz)<br />
A clear indication of the QND nature of<br />
the measurement<br />
Florence, Mai 2004 229
Test of the QND measurement quality<br />
A three <strong>atoms</strong> experiment<br />
Conditional probabilities<br />
Probe<br />
Meter<br />
Source<br />
0.5<br />
0.4<br />
No meter<br />
Meter in i 2<br />
Source atom. Prepared in e<br />
π/2 spontaneous emission<br />
detected in e: zero photon<br />
detected in g: one photon<br />
Probability<br />
0.3<br />
0.2<br />
Meter in g 2<br />
0.1<br />
Meter atom: Ramsey fringes set at φ=0<br />
if zero photon: detected in i<br />
if one photon: detected in g<br />
0.0<br />
g 1<br />
g 3<br />
g 1<br />
e 3<br />
e 1<br />
g 3<br />
e 1<br />
e 3<br />
Probe atom: prepared in g, π pulse in one photon<br />
Absorptive probe of cavity field<br />
if zero photon: detected in g<br />
if one photon: detected in e<br />
• QND error rate 20%<br />
• Spurious absorption in the mode by<br />
the meter atom: 20 %<br />
Florence, Mai 2004 230
III) Experimental illustrations of fundamental<br />
<strong>quantum</strong> mechanics<br />
• 1) Cavity induced light shifts <strong>and</strong> Lamb shifts<br />
• 2) Micromaser<br />
• 3) Quantum non-demolition measurement<br />
• 4) measurement of the Wigner function<br />
• 5) non classical field states<br />
Florence, Mai 2004 231
How to measure W for the electromagnetic field <br />
Propagating fields : « Tomographic » methods<br />
Principle : - measure marginal<br />
distributions P(q θ<br />
) for different θ<br />
- inverse Radon transform<br />
allows reconstruction of W(q,p)<br />
( medical tomography)<br />
Refs : - Coherent <strong>and</strong> squeezed states : - Smithey et al., PRL 70, 1244 (1993)<br />
- Breitenbach et al., Nature 387, 471 (1997)<br />
- One-photon Fock state : Lvovsky et al., PRL 87, 050402 (2001)<br />
- α|0>+β|1> : Lvovsky et al., PRL 88, 250401-1 (2002)<br />
Florence, Mai 2004 232
RESULTATS EXPERIMENTAUX<br />
Smithey et et al., PRL 70,<br />
1244 (1993)<br />
Breitenbach et et al, al, Nature 387,<br />
471 (1997)<br />
Comprimé<br />
Vide<br />
Florence, Mai 2004 233
MESURE MESURE COMPLETE COMPLETE DE DE LA LA DISTRIBUTION DISTRIBUTION DE DE WIGNER WIGNER POUR POUR UN UN PHOTON<br />
PHOTON<br />
Lvovsky et et al, al, PRL 87, 050402 (2001)<br />
Florence, Mai 2004 234
Other methods<br />
• Use the link between W <strong>and</strong> parity operator<br />
^<br />
W(α) = 2Tr(D( −α)ρ D(α) ( −1)<br />
N<br />
)<br />
^<br />
ˆ<br />
• Displace the field <strong>and</strong> measure parity by determination of photon number<br />
probability<br />
– Direct counting (Banaszek et al for coherent states)<br />
– Quantum Rabi oscillations for an ion in a trap (Winel<strong>and</strong>)<br />
• A dem<strong>and</strong>ing method. Much more information than the mere average<br />
parity needed<br />
Florence, Mai 2004 235
Mesure de la fonction de Wigner pour un ion piégé<br />
• Etat de vibration d'un ion unique:<br />
Etat nombre<br />
n = 1<br />
1<br />
2<br />
( 0 + 1 )<br />
Matrice densité<br />
D. Liebfried et al, PRL 77, 4281 (1996), NIST, Boulder<br />
• Etat de vibration d'un atome neutre:<br />
- G.Drobny <strong>and</strong> V. Buzek, PRA 65 053410 (2002)<br />
D'aprés les données de: C. Salomon et I. Bouchoule<br />
Florence, Mai 2004 236
Our approach<br />
- Proposed by Lutterbach <strong>and</strong> Davidovich (Lutterbach et al.<br />
PRL 78 (1997) 2547)<br />
- Based on :<br />
ρ(-α)<br />
^<br />
W(α) = 2Tr(D( −α)ρD(α)(<br />
−1)<br />
D(-α)<br />
ρ<br />
^<br />
ρ(-α)<br />
ˆ<br />
( − 1) N n =<br />
« parity » operator<br />
Nˆ<br />
)<br />
+ n if n=2k<br />
W is the expectation value of the Parity operator<br />
the displaced state ρ(−α)<br />
−<br />
n<br />
if n=2k+1<br />
( −1)<br />
A) Apply D(-α) Inject –α in cavity mode OK<br />
B) How to measure ( −1)<br />
<br />
Nˆ<br />
Nˆ<br />
in<br />
Florence, Mai 2004 237
Dispersive regime :<br />
Dispersive interaction<br />
δ=<br />
ω<br />
at<br />
−ω<br />
cav<br />
No energy exchange<br />
>> Ω/2<br />
ω<br />
cav<br />
|e><br />
ω<br />
at<br />
|g><br />
δ<br />
But : light shift<br />
∆E<br />
∆E<br />
2<br />
e, n +<br />
= Ω<br />
(n 1)<br />
4δ<br />
2<br />
=−<br />
n<br />
4δ<br />
Ω<br />
g, n<br />
1 ( )<br />
2 e g<br />
Phase shift<br />
1 ( e e g )<br />
2<br />
+ i∆Φ( n)<br />
of Ramsey fringes<br />
on the e-g transition<br />
1<br />
P(e)<br />
+ ∆Φ(n)=Φ 0 n<br />
∆φ(n)<br />
Empty cavity |0><br />
Fock state |n><br />
0<br />
Florence, Mai 2004 238<br />
φ
For φ=φ* :<br />
If N even, detection in e<br />
If N odd, detection in g<br />
^<br />
Parity Measurement<br />
=∑ φ<br />
n<br />
P ( −1)<br />
P(n) = Pφ<br />
*(e)-P<br />
*(g)<br />
= C<br />
n<br />
Φ 0 =π<br />
1<br />
P(e)<br />
0<br />
φ∗<br />
To measure W(α) : 1) inject – α<br />
2) measure fringe contrast C<br />
3) W(α)= 2 C<br />
N even<br />
Vacuum |0><br />
N odd<br />
State |1><br />
State ρ<br />
C<br />
C=+1 for state |2n><br />
C=-1 for state |2n+1><br />
φ<br />
Florence, Mai 2004 239
Experimental Tools<br />
- Slow <strong>atoms</strong> (150m/s) for long interaction times<br />
- Ramsey interferometer :<br />
Contrast <strong>with</strong>out cavity :<br />
70%<br />
(see experimental poster by T.<br />
Meunier for more details)<br />
1<br />
0.5<br />
0<br />
- Injection of a known<br />
coherent field |α><br />
waveguide<br />
att cav<br />
dB<br />
switch<br />
ν cav =51.099GHz<br />
S cav<br />
Florence, Mai 2004 240
Testing the method: vacuum state Wigner function<br />
e-g detection<br />
∆φ<br />
π/2<br />
R2<br />
position<br />
Atomic<br />
frequency<br />
ν cav<br />
D(-α)<br />
π/2<br />
R1<br />
|g,0><br />
δ<br />
Dispersive interaction<br />
Cavity mode<br />
time<br />
•Use Stark effect to tune interferometer phase<br />
•No phase information in cavity field: injected field phase irrelevant<br />
•Finite intrinsic contrast of the Ramsey interferometer<br />
Florence, Mai 2004 241
Wigner function of the "vacuum"<br />
α=0<br />
0.6 0.83<br />
1<br />
P(e)<br />
0.4<br />
0.5<br />
P(e)<br />
0.2<br />
0.6<br />
0.4<br />
α=0.6<br />
(norm.)<br />
πW(α)<br />
2<br />
0<br />
0.12 0.05<br />
0 1 2<br />
N phot<br />
0.2<br />
0.6 α=1.25<br />
1<br />
P(e)<br />
0.4<br />
0.2<br />
-1 0 1 2 3<br />
φ/π<br />
0<br />
0 1<br />
α<br />
2<br />
Florence, Mai 2004 242
Single photon Wigner function measurement<br />
e-g detection<br />
∆φ<br />
π/2<br />
R2<br />
position<br />
Atomic<br />
frequency<br />
ν cav<br />
|e,0><br />
π<br />
D(-α)<br />
π/2<br />
R1<br />
|g,1><br />
δ<br />
Dispersive interaction<br />
Cavity mode<br />
time<br />
Preparation<br />
of cavity state<br />
Wigner function measurement scheme<br />
Florence, Mai 2004 243
Wigner function of a "one-photon" Fock state<br />
0,7<br />
0,6<br />
α=0<br />
1,0<br />
P(e)<br />
P(e)<br />
0,5<br />
0,4<br />
0,3<br />
0,7<br />
0,6<br />
0,5<br />
0,4<br />
0 1 2 3<br />
α=0.3<br />
Φ/π<br />
(norm.)<br />
πW(α)<br />
0,5<br />
0,0<br />
-0,5<br />
-1,0<br />
0,0 0,5 1,0 1,5 2,0<br />
α<br />
P(e)<br />
0,3<br />
0,7<br />
0,6<br />
0,5<br />
0,4<br />
0,3<br />
0 1 2 3<br />
α=0.81<br />
0 1 2 3<br />
Φ/π<br />
Φ/π<br />
1<br />
0.5<br />
0<br />
0.71<br />
0.25<br />
0.04<br />
0 1 2<br />
Nphot<br />
Florence, Mai 2004 244
Towards other states<br />
- Cavity QED setup : direct measurement of the field<br />
2<br />
0<br />
-2<br />
2<br />
- Next improvements : - better isolation<br />
1<br />
- better detectors<br />
0<br />
More complex states : ex<br />
( 0 + 1<br />
)/<br />
2<br />
-1<br />
-2<br />
0<br />
-2<br />
2<br />
- In the future : « movie » of the decoherence of a Schrödinger cat<br />
2<br />
2<br />
1<br />
0<br />
…….<br />
1<br />
0<br />
-4<br />
-2<br />
0<br />
2<br />
4<br />
-2<br />
-1<br />
0<br />
1<br />
-1<br />
2 -2<br />
-4<br />
-2<br />
0<br />
2<br />
4<br />
-2<br />
-1<br />
0<br />
1<br />
-1<br />
2 -2<br />
Florence, Mai 2004 245
III) Experimental illustrations of fundamental<br />
<strong>quantum</strong> mechanics<br />
• 1) Cavity induced light shifts <strong>and</strong> Lamb shifts<br />
• 2) Micromaser<br />
• 3) Quantum non-demolition measurement<br />
• 4) measurement of the Wigner function<br />
• 5) non classical field states<br />
Florence, Mai 2004 246
Generation of a single photon state<br />
• Already used. π <strong>quantum</strong> Rabi pulse for an atom in e<br />
• Fidelity about 80%<br />
• Cumulative emissions lead to other Fock state but <strong>atoms</strong> are an<br />
expensive resource <strong>and</strong> fidelity not very high<br />
• Creation of multi-photon Fock states<br />
– Photon pump<br />
– Two photon emission in a Raman process<br />
Florence, Mai 2004 247
Photon pump<br />
• Successive π pulses in one cavity mode <strong>and</strong> recycling from g to e <strong>with</strong><br />
adiabatic rapid passages in a large coherent field stored in the other<br />
mode (Domokos et al EPJD 1,1)<br />
Florence, Mai 2004 248
Two-photon generation <strong>with</strong> a single atom<br />
• A complex raman process involving the two modes<br />
• Mode M a empty. Mode M b contains a field (thermal or coherent) Atommode<br />
M a detuning δ close to intermode detuning ∆<br />
• A third order resonant process<br />
PRL 88,143601<br />
• Coupling amplitude<br />
Florence, Mai 2004 249
Evidence of Raman process<br />
• Maser emission. Raman observed in ‘sideb<strong>and</strong>s’ for increasing fields in<br />
M b<br />
Florence, Mai 2004 250
Measuring the photon number<br />
• Use Ramsey fringes light shifts. Compare generated field <strong>with</strong> single<br />
photon field<br />
• Efficient <strong>and</strong> high fidelity generation of a two-photon field<br />
Florence, Mai 2004 251
Structure of the lectures<br />
• I) Introduction<br />
• II) The tools of CQED<br />
• III) Experimental illustrations of fundamental <strong>quantum</strong> mechanics<br />
• IV) Quantum information <strong>with</strong> <strong>atoms</strong> <strong>and</strong> <strong>cavities</strong><br />
• V) Schrödinger cats <strong>and</strong> decoherence<br />
• VI) Cavity QED <strong>with</strong> dielectric microspheres<br />
• VII) Perspectives<br />
Florence, Mai 2004 252
IV) Quantum information <strong>with</strong> <strong>atoms</strong> <strong>and</strong> <strong>cavities</strong><br />
• 1) A reminder on <strong>quantum</strong> computing<br />
• 2) Quantum entanglement knitting stitches<br />
• 3) Cavity assisted collisions<br />
Florence, Mai 2004 253
IV) Quantum information <strong>with</strong> <strong>atoms</strong> <strong>and</strong> <strong>cavities</strong><br />
• 1) A reminder on <strong>quantum</strong> computing<br />
• 2) Quantum entanglement knitting stitches<br />
• 3) Cavity assisted collisions<br />
Florence, Mai 2004 254
Ordinateurs classiques et complexité<br />
Calcul=processus physique<br />
• Codage des éléments d’information<br />
(bits) sur des éléments physiques<br />
(courant, tension).<br />
• Processus physique conduisant au<br />
résultat<br />
Données<br />
n bits<br />
N=2 n valeurs<br />
Complexité<br />
• Calcul facile: temps et ressources<br />
polynomiales dans le nombre de bits<br />
(ex: produit de deux nombres)<br />
• Calcul difficile: temps et/ou<br />
ressources exponentielles dans le<br />
nombre de bits (ex: factorisation)<br />
Classes de complexité<br />
• P: Polynomial<br />
• NP: Non-polynomial mais solution<br />
vérifiable en temps polynomial<br />
• NP Complet: problème équivalent à<br />
tout autre NP-complet<br />
Résultats<br />
Une question non résolue<br />
Florence, Mai 2004 255<br />
P<br />
≠<br />
NP
Machine de Turing<br />
Principe de Church-Turing:<br />
Du point de vue de la complexité, tous les ordinateurs classiques sont<br />
équivalents entre eux et équivalents au plus simple: la machine de Turing<br />
...<br />
0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 ...<br />
Registre<br />
Automate<br />
1<br />
0 0 1 0<br />
état interne<br />
Un problème difficile sur une machine le reste sur tout autre.<br />
Seules possibilité pour attaquer les problèmes difficiles:<br />
• Patience (peu commercial)<br />
• Parallélisme (mais utilisation exponentielle de ressources ex: calcul<br />
distribué)<br />
• Changer les lois physiques du calcul: calcul quantique<br />
Florence, Mai 2004 256
Principe d'un ordinateur quantique<br />
Bits et qubits<br />
Les bits sont remplacés par des<br />
systèmes à deux niveaux :<br />
superpositions d'états<br />
Représentation des données<br />
Registre de n qubits:<br />
Base<br />
1<br />
2<br />
( 0 + 1 )<br />
espace des états de dim. 2 n<br />
0,0,0, …,0 = 0<br />
0,0,0, …,1 = 1<br />
1,1,1, …,1<br />
= 2 n −1<br />
Un vecteur code un nombre<br />
Lecture d'un registre<br />
Mesure d'une quantité sur chaque qubit<br />
ayant |0> et |1> pour vecteurs<br />
propres et 0 et 1 pour valeurs<br />
propres.<br />
Lecture indépendante des n qubits: un<br />
nombre entre 0 et 2 n -1<br />
(indice du vecteur correspondant).<br />
On peut représenter les nombres par<br />
des vecteurs d’état et lire les valeurs<br />
finales (au sens de la mesure<br />
quantique).<br />
Peut-on calculer <br />
Florence, Mai 2004 257
Calcul quantique<br />
Évolution quantique unitaire. Définie par l’évolution des vecteurs de base<br />
ε , ε ,…, ε ⎯⎯→ U ε , ε ,…, ε ε = 0,1<br />
1 2 n<br />
1 2<br />
Un ordinateur quantique est au moins équivalent à une machine de Turing<br />
classique: tout calcul classique est réalisable à condition de respecter l'unitarité<br />
n<br />
x<br />
⎯⎯→<br />
f ( x)<br />
est interdit si f n'est pas inversible<br />
Mais on peut toujours former U tel que, avec deux registres de n qubits :<br />
x, y ⎯⎯→ x, y⊕<br />
f( x)<br />
x,0 ⎯⎯→ x, f( x)<br />
Lecture du résultat: mesure du second registre<br />
Florence, Mai 2004 258
Parallélisme quantique massif<br />
On peut calculer une valeur de la fonction f<br />
x,0 ⎯⎯→ x, f( x)<br />
Mais on peut aussi, dans le même temps, calculer toutes les valeurs de f<br />
∑<br />
x<br />
∑<br />
x,0 ⎯⎯→ x, f( x)<br />
Calcul intrinsèquement massivement parallèle. Exponentiellement plus<br />
efficace qu'un calculateur classique.<br />
x<br />
Lecture du résultat<br />
Naïvement, on n'obtient qu'une valeur de f d'argument aléatoire.<br />
Pas évident de tirer parti de ce parallélisme.<br />
Peu d'algorithmes efficaces connus pour un ordinateur quantique<br />
Florence, Mai 2004 259
Une brève histoire de l'informatique quantique<br />
La préhistoire<br />
73 Bennett<br />
Calcul réversible classique<br />
80 Benioff<br />
Proposition de principe<br />
82 Feynman<br />
Quantum simulator<br />
Dimension exponentielle de l'espace<br />
85 Deutsch<br />
Machines de Turing quantique<br />
Bases conceptuelles<br />
92 Deutsch<br />
Premiers algorithmes ad-hoc<br />
Portes logiques quantiques<br />
L'âge d'or<br />
94 Shor<br />
95<br />
Algorithme de factorisation<br />
Utile et accélération exponentielle<br />
Propositions théoriques de portes<br />
95 Winel<strong>and</strong><br />
Première réalisation d'une porte<br />
97 Grover<br />
98-99<br />
00-01<br />
Algorithme de recherche<br />
Utile mais accélération faible<br />
Premiers algorithmes quantiques<br />
réalisés en RMN<br />
Premières manipulations d'états<br />
intriqués complexes<br />
Florence, Mai 2004 260
Portes logiques quantiques<br />
Théorème<br />
• Toute transformation unitaire de n<br />
qubits est décomposable en un<br />
produit de transformations unitaires<br />
élémentaires à un et deux qubits<br />
Exemples de portes<br />
Portes à un qubit.<br />
• Transformation la plus générale d’un<br />
système à deux niveaux:<br />
Traduction:<br />
• Tout calcul quantique est réalisable<br />
par application successive de<br />
« portes logiques quantiques »:<br />
machines portant sur un ou deux<br />
qubits<br />
Portes universelles<br />
• Un ensemble fini de portes qui<br />
permettent de réaliser par<br />
association tout réseau de calcul<br />
quantique<br />
iψ<br />
⎛ cosϕ<br />
e sinϕ<br />
⎞<br />
U ( ϕψ , ) = ⎜ −iψ<br />
⎟<br />
⎝−e<br />
sinϕ<br />
cosϕ<br />
⎠<br />
représentation graphique:<br />
• Cas particulier: porte de Hadamard<br />
Florence, Mai 2004 261<br />
U<br />
1 1<br />
H = ⎛ ⎞ H<br />
⎜ ⎟<br />
⎝1 −1⎠<br />
⎧ 1<br />
0 → 0 + 1<br />
⎪ 2<br />
⎨<br />
⎪ 1<br />
1 → 0 − 1<br />
⎪⎩ 2<br />
( )<br />
( )
Portes (suite 1)<br />
• Porte Non<br />
• Porte CNOT<br />
Ν<br />
0 1<br />
N = ⎛ ⎞<br />
⎜ ⎟<br />
⎝1 0⎠<br />
Portes à deux qubits<br />
• Porte de phase<br />
CNOT<br />
⎛1<br />
⎞<br />
⎜ ⎟<br />
1<br />
= ⎜ ⎟<br />
⎜ 0 1 ⎟<br />
⎜<br />
1 0⎟<br />
⎝ ⎠<br />
π<br />
Remarque<br />
⎛1<br />
⎞<br />
⎜<br />
⎟<br />
1<br />
π = ⎜<br />
⎟<br />
⎜ 1 0 ⎟<br />
⎜<br />
0 −1⎟<br />
⎝<br />
⎠<br />
=<br />
H<br />
π<br />
H<br />
Florence, Mai 2004 262
Portes (suite 2)<br />
Portes à n bits<br />
• Porte de Toffoli<br />
Calcul réversible classique<br />
Universalité<br />
Tout calcul quantique peut être réalisé<br />
avec:<br />
• Des portes à un qubit<br />
• Des portes CNOT<br />
Ex: un additionneur<br />
• Control U<br />
U<br />
– Faire U sur n-1 bits si le bit de<br />
contrôle est à un<br />
– Ne rien faire sinon<br />
Florence, Mai 2004 263
Une opération utile<br />
Préparation d’un registre de n bits dans un superposition de toutes les<br />
valeurs<br />
n<br />
2 −1<br />
1<br />
x ⎯⎯→ ∑ y<br />
n<br />
2 0<br />
Préparation de l’état |0> et application d’une Hadamard sur chaque qubit<br />
1<br />
0 ⎯⎯→ 0 + 1<br />
2<br />
( )<br />
pour chaque qubit<br />
n<br />
1 1<br />
0 ⎯⎯→ 0 + 1 = ∑<br />
n−1 2 −1<br />
∏(<br />
)<br />
n<br />
n<br />
2 0<br />
2<br />
0<br />
y<br />
pour le registre<br />
Florence, Mai 2004 264
Calcul de fonctions élémentaires<br />
Fonctions de {0,1} dans {0,1}<br />
Deux qubits<br />
Ligne rouge: |x>. Ne sera pas affecté. Ligne verte |y> devient |y+f(x)><br />
4 fonctions:<br />
f(x) = 0 ∀ x<br />
f(x)<br />
0<br />
0 Ν<br />
01<br />
f(x) = 1 ∀ x<br />
01<br />
0<br />
0<br />
Ν<br />
Ν<br />
Ν<br />
01<br />
f(x)<br />
01<br />
Fonctions « constantes » Fonctions « balancées »<br />
A un bit, il n’y a que des fonctions constantes ou balancées. Situations plus complexes<br />
pour les fonctions de n bits dans n bits.<br />
Florence, Mai 2004 265
Quelques fonctions de {0,1,2,3} sur {0,1}<br />
2 4 =16 fonctions possibles<br />
f(x)<br />
0 N<br />
01 23<br />
f(x)<br />
0 N N<br />
01 23<br />
Exemple de<br />
fonction<br />
constante<br />
(f=1. Deux<br />
telles fonctions)<br />
Exemple de<br />
Fonction<br />
balancée (autant<br />
de f(x) = 0 que de<br />
f(x) = 1)<br />
(six fonctions)<br />
0 N<br />
f(x)<br />
01 23<br />
Tous les autres cas se déduisent<br />
simplement de ceux-ci<br />
Fonction ni<br />
constante ni<br />
balancée<br />
(huit fonctions)<br />
Florence, Mai 2004 266
Les problèmes posés sous forme d’oracle<br />
Les problèmes logiques que nous allons considérer ici sont posés sous forme d ’<br />
«oracle». On suppose qu’ une machine programmée selon des règles inconnues<br />
(décrite comme une «boîte noire» ou oracle), calcule une fonction dont nous ne<br />
connaissons que certaines caractéristiques. Le problème consiste à déterminer une<br />
propriété inconnue de la fonction, sans «ouvrir» la boite. Nous pouvons interroger<br />
l ’oracle en entrant des données dans la boite et en manipulant sa sortie, sans l’ouvrir<br />
pour en analyser le contenu. Le problème sera «facile» si sa résolution dem<strong>and</strong>e un<br />
nombre total d’opérations croissant de façon polynomiale avec le nombre de bits,<br />
«difficile» s’il croit de façon exponentielle avec ce nombre. Nous allons montrer que le<br />
passage du calcul classique au calcul quantique transforme certains oracles classiques<br />
difficiles en oracles quantiques faciles. Dans d’ autres cas, le problème quantique reste<br />
difficile, mais moins que le problème classique (croissance toujours exponentielle du<br />
nombre d ’opérations, mais avec un exposant plus petit que classiquement ).<br />
0,1,1,1…..1, 0 0,0,1,0…..1, 0<br />
f(x) <br />
Choix libre de la<br />
préparation des<br />
bits d ’entrée<br />
« Lecture » du<br />
programme<br />
interdite<br />
Choix libre des<br />
opérations sur les<br />
bits de sortie<br />
Florence, Mai 2004 267
L’oracle de Deutsch-Josza<br />
f(x) constante ou<br />
balancée <br />
f(x) est une fonction booléenne de [ 0, 2 n -1] dans [ 0, 1]. On sait<br />
qu’elle est soit constante, soit balancée. Est-elle l’un ou l ’autre <br />
Classiquement, il faut « interroger » l’oracle 2 n− 1 +1 fois pour répondre à la question à coup sûr<br />
(il faut introduire 2 n− 1 +1 valeurs différentes de x et calculer f(x) à chaque fois) → Croissance<br />
exponentielle avec n du nombre d ’opérations et problème classique « difficile »<br />
L’oracle de Grover f(x) est une fonction booléenne de [ 0, 2 n -1] dans [ 0, 1] qui n’est non nulle<br />
f (x) = δ (x-x 0<br />
)<br />
que pour x = x 0<br />
. Trouver x 0<br />
.<br />
x 0<br />
<br />
Equivaut à la recherche «inversée» d’un abonné dans un annuaire à partir de son<br />
numéro connu a. Les x sont les abonnés, f(x) vaut 1 si a est le numéro de x, 0 sinon.<br />
Classiquement, il faut calculer f(x ) («consulter» l ’annuaire) N = 2 n − 1 fois pour trouver à coup<br />
sûr. Problème classique difficile.<br />
L’oracle de Simon<br />
« Période » de<br />
f(x) <br />
Exemples d’oracles classiquement difficiles<br />
f(x) est une fonction de [0, 2 n -1] dans [0, 2 n -1] telle que f (x’) = f (x) ssi x<br />
= x ⊕ s où s est une suite inconnue à n termes de « 0 » et « 1 » et ⊕<br />
représente l ’addition « bit à bit » (s : «période» de f ). Déterminer s<br />
Classiquement, il faut calculer f (x ) pour des valeurs aléatoires de x jusqu ’à trouver deux x et<br />
x’ tels que f (x) = f (x’). Alors x ⊕ s = x’ et x ⊕ x’ = x ⊕ x ⊕ s = s (car x ⊕ x ={ 0} ). Il faut<br />
2 n - 1 + 1 opérations pour trouver la réponse à coup sûr → Problème classique « difficile ».<br />
Florence, Mai 2004 268
Exemple élémentaire<br />
Algorithme de Deutsch-Josza à un qubit.<br />
Principe du calcul<br />
Déterminer si f de {0,1} dans {0,1} est<br />
constante ou balancée.<br />
Une autre formulation du problème:<br />
comment déterminer qu’une pièce a<br />
bien un côté pile et un côté face<br />
• Classiquement: regarder les deux<br />
côtés.<br />
• Quantiquement: regarder en une fois<br />
une superposition quantique des<br />
deux côtés.<br />
Deux qubits pour calculer f<br />
|0><br />
|1><br />
H<br />
H<br />
1<br />
1<br />
2<br />
2<br />
( 0 + 1 )<br />
( 0 + 1 ) cste 0<br />
( 0 − 1 )<br />
( 0 − 1 )<br />
2<br />
f<br />
Un seul calcul de f pour décider de sa<br />
nature.<br />
1<br />
1<br />
2<br />
2<br />
( 0 − 1 ) bal 1<br />
1<br />
H<br />
Florence, Mai 2004 269
Fonction constante : 0<br />
1<br />
2<br />
( 0 + 1 )( 0 − 1 )<br />
Deux cas parmi quatre<br />
f 1<br />
⎯⎯→ ⎡ 0<br />
2 ⎣<br />
+ 1<br />
= 1 ⎡ 0 1 2 ⎣<br />
+<br />
⎦<br />
=<br />
2<br />
0 1<br />
H 1<br />
⎯⎯→<br />
( 0<br />
2 0 − 1 )<br />
( 0 ⊕ f (0) − 1 ⊕ f (0) ) ( 0 ⊕ f(1) − 1 ⊕ f(1)<br />
)<br />
( 0 − 1 ) ( 0 − 1 ) ⎤ ( + )( 0 − 1 )<br />
|0><br />
|1><br />
⎤<br />
⎦<br />
H<br />
H<br />
1<br />
1<br />
2<br />
2<br />
( 0 + 1 )<br />
( 0 − 1 )<br />
f<br />
H<br />
Fonction balancée: identité<br />
1<br />
2<br />
( 0 + 1 )( 0 − 1 )<br />
f 1<br />
⎯⎯→ ⎡ 0<br />
2 ⎣<br />
+ 1<br />
= 1 ⎡ 0 1 2 ⎣<br />
+<br />
⎦<br />
=<br />
2<br />
0 1<br />
H 1<br />
⎯⎯→<br />
( 0<br />
2 1 − 1 )<br />
( 0 ⊕ f (0) − 1 ⊕ f (0) ) ( 0 ⊕ f(1) − 1 ⊕ f(1)<br />
)<br />
( 0 − 1 ) ( 1 − 0 ) ⎤ ( − )( 0 − 1 )<br />
⎤<br />
⎦<br />
Florence, Mai 2004 270
Généralisation : Deutsch-Josza pour n qubits<br />
(1/2 n/2 ) Σ x<br />
| x ><br />
(1/2 n/2 ) Σ x<br />
(− 1) f(x) | x ><br />
| {0} > A<br />
H 1 .H 2 ….H n<br />
N H<br />
f(x) constante ou<br />
balancée <br />
| 0> B<br />
(1/2 1/2 ) [| 0 > − | 1 >]<br />
(1/2 1/2 ) [| 0 > − | 1 >]<br />
Le registre d’entrée A (n qubits) est préparé (par application de la transformation de Hadamard<br />
H sur chaque qubit) dans la superposition symétrique des 2 n états | x > possibles.<br />
Le registre de sortie B (1 qubit) est inversé par N, puis préparé par H dans (1/2 1/2 ) [| 0 > − | 1 >] .<br />
Action de l ’oracle: | x > | 0 > → | x > | f (x ) > et | x > | 1 > → | x > | 1 ⊕ f (x ) ><br />
Si f (x ) = 0: | x > [| 0 > − | 1 > ] → | x > [| 0 > − | 1 > ]<br />
} ( − 1) f(x) | x > [| 0 > − | 1 > ]<br />
Si f (x ) = 1: | x > [| 0 > − | 1 > ] →−| x > [| 0 > − | 1 > ]<br />
Et par superposition:<br />
(1/2 ( n+1)/2 ) Σ x<br />
| x > [| 0 > − | 1 > ] → (1/2 (n+1)/2 ) (Σ x<br />
(− 1) f(x) | x > ) [| 0 > − | 1 > ]<br />
Les registres restent non intriqués après l’oracle. Déphasage des amplitudes<br />
dans le registre A en (− 1) f(x) .<br />
Florence, Mai 2004 271
Deux possibilités<br />
Si f(x ) est constante: f (x ) = 0 ∀ x ou f (x ) =1 ∀ x →<br />
(1/2 n/2 ) Σ x<br />
(− 1) f(x) | x > = ± (1/2 n/2 ) Σ x<br />
| x > → registre A inchangé (au signe près)<br />
Si f(x) est balancée: autant de f (x ) = 0 que de f(x ) = 1 →<br />
Autant d ’amplitudes +1 que d ’amplitudes −1 dans la superposition finale du registre A<br />
→ Σ x<br />
(− 1) f(x) | x > orthogonal à Σ x<br />
| x ><br />
Résoudre l ’oracle revient à distinguer deux états orthogonaux de l’état final du registre A:<br />
On applique à nouveau H à tous les qubits. Comme H 2 =1, on retrouve l’état initial |{0} > si<br />
f(x ) est constante, un état orthogonal si f(x) est balancée → au moins un des qubits doit<br />
alors être 1. On le vérifie en mesurant les qubits finals de A.<br />
| 0, 0, 0, ... > ou<br />
mesure<br />
H 1 .H 2 ….H n<br />
H 1 .H 2 ….H état orthogonal<br />
n<br />
f(x)<br />
à | 0, 0, 0, ... ><br />
des qubits<br />
N<br />
H<br />
La réponse nécessite au plus 3n+2 opérations à un qubit (2n + 1 opérations H, une<br />
opération de bascule (N) et au plus mesure de n qubits (on peut arrêter dès qu’on<br />
trouve un 1) → problème quantiquement «facile».<br />
Florence, Mai 2004 272
Remarques<br />
1. Où est l’intrication<br />
Les qubits de A ne sont pas intriqués à B qui reste inchangé. De l’intrication est en général<br />
cependant créée entre les qubits de A:<br />
Exemple. Cas d ’une fonction balancée agissant sur un registre A de trois qubits:<br />
Σ x<br />
(− 1) f(x) | x >= | 000 > − | 001 > + | 010 > − | 011 > + | 100 > − | 101 > − | 110 > + | 111 ><br />
= | 0 > 1 [| 00 > − | 01 > + | 10 > − | 11 >] 23 + | 1> 1 [| 00 > − | 01 > − | 10 > + | 11 >] 23<br />
= | 0 > 1 | Ψ > 23 + | 1 > 1 | Φ > 23 avec 23 < Φ | Ψ > 23 = 0<br />
Cette décomposition de montre que le qubit 1 et l ’ensemble des qubits 2 et 3<br />
sont maximalement intriqués.<br />
2. Cet algorithme est-il vraiment avantageux par rapport à la procédure classique<br />
L’ avantage de l’ algorithme quantique n’existe que si on cherche une réponse certaine.<br />
Si on s’ autorise une probabilité finie ε d’ erreur, aussi petite soit-elle, l’algorithme<br />
classique (calcul successif de f (x) pour des valeurs de x tirées au hasard) donne un<br />
résultat acceptable au bout de k ≅ − log 2<br />
(ε ) opérations (nombre indépendant de n). Le<br />
problème classique devient donc «facile» dès qu ’on accepte un taux fini d ’erreur. Ceci<br />
diminue considérablement l ’intérêt de l ’algorithme quantique puisqu’ il faut être sûr<br />
de pouvoir l ’effectuer sans aucune décohérence pour qu ’il soit avantageux par rapport<br />
à la version classique.<br />
Florence, Mai 2004 273
n qubits<br />
L’algorithme de Simon<br />
(1/√2)[ | x > + | x ⊕ s >]<br />
| {0} > A<br />
H 1 .H 2 ….H n<br />
x<br />
| {0} > B<br />
n qubits<br />
« Période » de<br />
f(x) <br />
} Σ | x> | f(x)><br />
Mesure de B:<br />
résultat f (x)<br />
H 1 .H 2 ….H n<br />
On réalise la suite d’opérations schématisée ci-dessus: calcul parallèle de toutes les valeurs<br />
de la fonction suivie d’une mesure du registre B projetant A dans une superposition de<br />
deux états qui diffèrent bit à bit de la quantité inconnue s. On applique alors à nouveau les<br />
transformations de Hadamard aux n qubits de A: elles font évoluer chaque qubit suivant la<br />
loi: | 0 > → (1/√2) [ | 0 > + | 1> ] et | 1 > → (1/√2) [ | 0 > − | 1> ]. Un état |{ x }> (produit<br />
de n états | x i<br />
> avec x i<br />
= 0 ou 1) devient une superposition de produits d’états | y i<br />
> (avec y i<br />
= 0 ou 1). Les coefficients de cette superposition valent + 1 ou −1suivant la parité de<br />
la somme Σ i<br />
x i<br />
y i<br />
:<br />
|{ x }> = | x 1<br />
, x 2<br />
, x 3<br />
,…..x n<br />
> → = (1/2 (n+1)/2 ) Σ { y }<br />
(−1) { Σ i x i y i } | y 1<br />
, y 2<br />
, y 3<br />
,…..y n<br />
><br />
L’état final de A est donc:<br />
(1/2 (n+1)/2 )Σ {y} [ (−1) { Σ i x i y i } + (−1) { Σ i ( x i ⊕ s i ) y i } ] | y 1<br />
, y 2<br />
, y 3<br />
,…..y n<br />
><br />
= (1/2 (n+1)/2 ) Σ {y} (−1) { Σ i x i y i } [ 1 + (−1) { Σ i s i y i } ] | y 1<br />
, y 2<br />
, y 3<br />
,…..y n<br />
><br />
Une mesure répétée ~ n fois de A va alors nous permettre de déterminer s<br />
Florence, Mai 2004 274
Détermination de la période inconnue<br />
| Ψ ( final) > A<br />
= (1/2 (n+1)/2 ) Σ {y} (−1) { Σ i x i y i } [ 1 + (−1) { Σ i s i y i } ] | y 1<br />
, y 2<br />
, y 3<br />
,…..y n<br />
><br />
Amplitude non nulle ssi<br />
Σ i<br />
s i<br />
y i<br />
= 0 (modulo 2)<br />
Une mesure des qubits individuels donne une suite y 1a<br />
, y 2a<br />
, y 3a<br />
,…..y na<br />
de valeurs 0 et 1 qui<br />
satisfait la condition:<br />
Σ i<br />
s i<br />
y ia<br />
= 0 (modulo 2).<br />
On recommence n fois l ’opération et on obtient ainsi, en général, n relations<br />
indépendantes (si par hasard deux mesures donnent le même vecteur, on recommence<br />
une fois de plus):<br />
Σ i<br />
s i<br />
y ia<br />
= 0<br />
Σ i<br />
s i<br />
y ib<br />
= 0<br />
. . . . . . . .<br />
Σ i<br />
s i<br />
y in<br />
= 0<br />
La résolution de ce système d ’équations donne s. Le processus requiert ≅ 4n 2 opérations. Le<br />
problème est donc quantiquement facile. De plus, il tolère des erreurs puisqu’on peut toujours<br />
vérifier le résultat en comparant f ( x ) et f (x ⊕ s) une fois s obtenu.<br />
Florence, Mai 2004 275
Rôle de l’intrication et de la mesure<br />
Dans l’ algorithme de Simon, l’ intrication et la mesure projective jouent un rôle plus<br />
essentiel que dans ceux de Deutsch et Grover. L’oracle intrique les registres A et B, puis la<br />
mesure de B projette A dans une superposition de deux états seulement. Après mélange par<br />
la « lame séparatrice », la signature du signal d ’interférence final nous renseigne sur la<br />
séparation de ces deux états, donc sur la période cherchée. Quoique mathématiquement<br />
plus compliqué, l’algorithme de Shor, basé sur la recherche de la période d ’une fonction,<br />
ressemble beaucoup dans son principe à celui de Simon.<br />
Remarque: il n’ est même pas besoin de «lire» la mesure du registre B. Il suffit d’ avoir intriqué<br />
B à son appareil de mesure , ce qui réduit A à un mélange statistique de superpositions | x > + |<br />
x ⊕ s >. Leur recombinaison finale par H 1<br />
.H 2<br />
…H n<br />
ne conduit, quel que soit x, à une<br />
interférence constructive que pour les états | y 1<br />
, y 2<br />
, y 3<br />
,…..y n<br />
> satisfaisant les équations<br />
linéaires de la page précédente (on peut réduire le nombre d’opérations à 3n 2 ).<br />
Florence, Mai 2004 276
Factorisation<br />
Un problème classique difficile<br />
Meilleur algorithme connu (Number Field Sieve) sur n bits en<br />
1<br />
1.9n 3<br />
e<br />
Factorisation de RSA 155: 8000 MIPS-années soit 2.5 10 17 instructions !<br />
109417386415705274218097073220403576120037329454492059909138421314763499842889347847179<br />
97257891267332497625752899781833797076537244027146743531593354333897=102639592829741<br />
105772054196573991675900716567808038066803341933521790711307779*10660348838016845482<br />
0927220360012878679207958575989291522270608237193062808643<br />
Problème difficile dont l'inverse est facile (multiplication)<br />
Idéal pour la cryptographie (codage et décodage faciles, casser le code très<br />
difficile)<br />
Un algorithme rapide de factorisation aurait des conséquences énormes sur<br />
les algorithmes de cryptage (et sur l'économie)<br />
1994: Shor propose un algorithme de factorisation rapide sur un ordinateur<br />
quantique<br />
Florence, Mai 2004 277
Algorithme de Shor<br />
Algorithme de factorisation<br />
exponentiellement plus efficace<br />
que la version classique<br />
but : factoriser N>>1<br />
Ordre d'un entier<br />
x
Exemple élémentaire<br />
Factoriser 15<br />
Ordre de x<br />
On choisit x=7<br />
R=4<br />
On calcule 7 a [15]<br />
a<br />
7 a [15]<br />
Le gcd de 4+1 et 15 est un facteur de 15<br />
Le gcd de 4-1 et 15 est un facteur de 15<br />
1<br />
2<br />
7<br />
4<br />
15=5x3<br />
3<br />
13<br />
4<br />
1<br />
Florence, Mai 2004 279
Algorithme quantique (1)<br />
Principe<br />
Calculer beaucoup de valeurs de f<br />
en utilisant le parallélisme et<br />
extraire la période (~ N)<br />
Deux registres de m qubits.<br />
q=2 m choisi tel que<br />
2N 2 < q < 4N 2<br />
État initial<br />
0,0<br />
Superposition de tous les nombres<br />
q−1<br />
1<br />
0,0 ⎯⎯→ ϕ = ∑ a,0<br />
q a=<br />
0<br />
Pour chaque qubit:<br />
1<br />
0 ⎯⎯→ ( 0 + 1 )<br />
2<br />
Exponentiation modulaire<br />
q−1<br />
1<br />
a<br />
ϕ ⎯⎯→ ∑ ax , N<br />
q<br />
Possible de façon efficace<br />
Utilise le parallélisme<br />
Mesure du second registre<br />
Obtention d'une valeur aléatoire y<br />
Projection du premier registre sur les<br />
antécédents de y<br />
Florence, Mai 2004 280<br />
k+jr<br />
a=<br />
0<br />
[ ]<br />
k aléatoire, j entier<br />
Le premier registre contient une<br />
superposition d'états répartis<br />
périodiquement avec la période r<br />
1<br />
A<br />
∑<br />
j<br />
k<br />
+<br />
jr
Amplitude<br />
Algorithme quantique (2)<br />
Répartition des amplitudes<br />
Après la transformation de Fourier<br />
k k+r k+2r k+jr Vecteur<br />
La mesure du premier registre en<br />
l'état ne donne aucune<br />
information ("offset" k aléatoire)<br />
Extraire la période:<br />
réaliser une transformation de<br />
Fourier discrète<br />
Possible de façon efficace (en un<br />
temps polynomial) Amélioration<br />
exponentielle/FFT classique<br />
q/ r 2q/ r<br />
Mesure du registre: une valeur de la<br />
forme<br />
p q/r<br />
p entier arbitraire<br />
Par un calcul classique efficace,<br />
extraction, avec une probabilité<br />
finie, de r et factorisation de N<br />
Énormes conséquences pratiques si<br />
on peut réaliser cet algorithme<br />
Florence, Mai 2004 281
Quelques caractéristiques importantes<br />
Utilise le parallélisme massif<br />
calcul simultané de toutes les<br />
exponentiations modulaires<br />
ϕ ⎯⎯→<br />
Probabiliste<br />
1<br />
• Probabilité finie d'obtenir le<br />
résultat<br />
• Ne décroît pas exponentiellement<br />
avec le nombre de bits<br />
• Résultat facile à vérifier<br />
q<br />
q−1<br />
∑<br />
a=<br />
0<br />
ax ,<br />
a<br />
[ N]<br />
Utilise des effets authentiquement<br />
quantiques<br />
État<br />
1<br />
q<br />
État intriqué<br />
q−1<br />
∑<br />
a=<br />
0<br />
ax ,<br />
[ N]<br />
analogue à la paire EPR<br />
1<br />
( , , )<br />
2 + − − − +<br />
Calcul: manipulation d'intrication<br />
Mesure du second registre:<br />
projection du premier<br />
Rien de comparable avec un<br />
ordinateur analogique classique<br />
(même avec superpositions)<br />
a<br />
Florence, Mai 2004 282
Simulation quantique<br />
Un autre domaine pour le calcul<br />
quantique<br />
Simuler la dynamique ou les états<br />
propres d’un système quantique<br />
Ex: trouver l’état fondamental d’un<br />
système de spins en interaction sur<br />
un réseau<br />
Intérêts<br />
• Dès 30 qubits, traiter des problèmes<br />
inaccessibles aux calculs classiques.<br />
• Par rapport au système donné par la<br />
nature, on peut faire varier la force et<br />
la nature des interactions.<br />
Classiquement difficile: la taille de<br />
l’espace de Hilbert croît<br />
exponentiellement avec la taille du<br />
système<br />
Feynman 1985: utiliser un système<br />
quantique pour en simuler un autre.<br />
Problèmes:<br />
• Peu de systèmes facilement<br />
simulables.<br />
• Problème de la décohérence<br />
Florence, Mai 2004 283
IV) Quantum information <strong>with</strong> <strong>atoms</strong> <strong>and</strong> <strong>cavities</strong><br />
• 1) A reminder on <strong>quantum</strong> computing<br />
• 2) Quantum entanglement knitting stitches<br />
• 3) Cavity assisted collisions<br />
Florence, Mai 2004 284
Quantum Rabi oscillations<br />
Initial atom-cavity state<br />
1<br />
Ψ (0) = e,0 = + ,0 + −,0<br />
2<br />
( )<br />
State at time t:<br />
Ω t Ω t<br />
Ψ = +<br />
2 2<br />
0 0<br />
() t cos e,0 sin g,1<br />
1+ cosΩ<br />
Probability for being in e :<br />
0t<br />
Pe<br />
() t =<br />
2<br />
•Oscillatory spontaneous<br />
emission<br />
•An atomic transition saturated by<br />
a single photon<br />
•Non-linear optics at the single<br />
photon level.<br />
P e<br />
(t)<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
0.0<br />
time ( µ s)<br />
0 30 60 90<br />
Florence, Mai 2004 285
Quantum Rabi oscillations: state transformations<br />
Initial state<br />
e,0<br />
1<br />
e,0 e⎯⎯→−<br />
,0 e,0 ⎯⎯→ eg,0<br />
,1 2π ( epulse<br />
,0 + g,1<br />
)<br />
2<br />
g,1 ( ⎯⎯→− ce e + cg π/2 spontaneous g,1) 0 ⎯⎯→ g<br />
Conditional ( ce 1 + c<br />
emission dynamics<br />
g<br />
0 )<br />
pulse<br />
π spontaneous emission pulse<br />
g,0 ⎯⎯→+ Entanglement g,0<br />
Quantum<br />
creation<br />
Atom/cavity state copy phase gate<br />
Atom-cavity EPR pair<br />
P e<br />
(t)<br />
0.8<br />
51 (level e)<br />
0.6<br />
51.1 GHz<br />
50 (level g)<br />
0.4<br />
0.2<br />
0.0<br />
Brune et al, PRL 76, 1800 (96)<br />
time ( µ s)<br />
0 30 60 90<br />
Florence, Mai 2004 286
Three "stitches" to "knit" <strong>quantum</strong> entanglement<br />
Combine elementary transformations to create complex entangled states<br />
• State copy <strong>with</strong> a π pulse<br />
– Quantum memory : PRL 79, 769 (97)<br />
• Creation of entanglement <strong>with</strong> a π/2 pulse<br />
– EPR atomic pairs : PRL 79, 1 (97)<br />
• Quantum phase gate based on a 2π pulse<br />
– Quantum gate : PRL 83, 5166 (99)<br />
– Absorption-free detection of a single photon: Nature 400, 239 (99)<br />
• Entanglement of three systems (six operations on four qubits)<br />
– GHZ Triplets : Science 288, 2024 (00)<br />
• Entanglement of two radiation field modes<br />
– Phys. Rev. A 64, 050301 (2001)<br />
• Direct entanglement of two <strong>atoms</strong> in a cavity-assisted collision<br />
– Phys. Rev. Lett. 87, 037902 (2001)<br />
Florence, Mai 2004 287
Quantum memory<br />
• Use the π <strong>quantum</strong> Rabi pulse to transfer a qubit from an atom to the<br />
cavity <strong>and</strong> back<br />
• Timing<br />
• An useful space-time diagram for the timing of complex experiments<br />
Florence, Mai 2004 288
• Final signal<br />
Coherent information transfer<br />
• Ramsey fringes <strong>with</strong> the two pulses on different <strong>atoms</strong> <strong>and</strong> transient<br />
storage of <strong>quantum</strong> information in the cavity field<br />
Florence, Mai 2004 289
Quantum memory lifetime<br />
• Contrast of the fringes as a function of time<br />
0.6<br />
0.4<br />
Fringes amplitude<br />
0.2<br />
0.0<br />
0 1 2 3<br />
T/T<br />
cav<br />
• Coherence lifetime is twice cavity damping time (equal superposition of 0<br />
<strong>and</strong> 1)<br />
Florence, Mai 2004 290
Preparation <strong>and</strong> detection of a single photon state<br />
• First atom in e: preparation of a single photon Fock state<br />
• Other atom read out: measurement of a Fock state lifetime<br />
1.0<br />
Π<br />
fe<br />
First atom sent in e<br />
Second atom sent in g<br />
0.8<br />
Delay in C: T<br />
7000 coincidences per point<br />
0.6<br />
0.4<br />
0.2<br />
Two data sets for two modes<br />
T cav<br />
=84 µs<br />
T cav<br />
=112 µs<br />
Conditional probability<br />
second in e<br />
when first in g<br />
Maximum value 75%<br />
well understood <strong>with</strong><br />
known experimental<br />
imperfections<br />
0.0<br />
0 1 2 3 4 5<br />
T/T<br />
cav<br />
Florence, Mai 2004 291
Creation of an EPR atom pair<br />
A simple entanglement manipulation experiment<br />
• Initial state<br />
g<br />
e<br />
eg , ,0<br />
π/2 pulse:<br />
•Entanglement creation<br />
1<br />
( ,0 + ,1 )<br />
2 e g g<br />
•State copy<br />
•Final state<br />
1<br />
( )<br />
1<br />
eg , − ge , in spin terms: ( ↑↓ , −↓↑ , )<br />
2 2<br />
1<br />
= →← , −←→ ,<br />
2<br />
( )<br />
Hagley et al, PRL 79, 1 (97)<br />
Florence, Mai 2004 292
Testing entanglement<br />
Two complementary experiments<br />
Spin singlet state is rotation-invariant:<br />
• Spin anticorrelations along any<br />
detection axis<br />
– Check atomic energy<br />
anticorrelations (detection along<br />
0z)<br />
• "longitudinal experiment"<br />
Longitudinal experiment<br />
Direct detection of atomic energies<br />
Timing<br />
Position<br />
D D<br />
C<br />
e A 1 g A 2<br />
Expect. Obs<br />
e,e<br />
0 0.10<br />
Time<br />
– Check that superposition is<br />
coherent by detecting spins in a<br />
non-compatible basis (axes in the<br />
horizontal plane of Bloch sphere)<br />
• "transverse experiment"<br />
63 % of pairs present the expected<br />
correlation.<br />
Imperfections well accounted for by<br />
cavity relaxation (T r<br />
=112µs) <strong>and</strong> π<br />
pulse imperfections<br />
Florence, Mai 2004 293<br />
e,g<br />
g,e<br />
g,g<br />
0.5<br />
0.5<br />
0<br />
0.42<br />
0.27<br />
0.21
Transverse experiment<br />
Principle<br />
• Measure atom 1 along x axis<br />
• Measure atom 2 along φ axis<br />
– (apply Ramsey pulses <strong>with</strong><br />
adjustable phase φ on two <strong>atoms</strong>)<br />
z<br />
z<br />
Position<br />
Timing<br />
C<br />
R 1<br />
eg<br />
D<br />
π/2 π/2<br />
D<br />
R 2<br />
eg<br />
y<br />
y<br />
e<br />
A 1 g A 2<br />
Time<br />
x<br />
x<br />
φ<br />
"Bell signal"<br />
0.3<br />
• Measure<br />
σ σ<br />
• Expect +1 for φ=π, -1 for φ=0<br />
x<br />
ϕ<br />
Bell signal<br />
0.2<br />
0.1<br />
0.0<br />
-0.1<br />
Ramsey fringes <strong>with</strong> two pulses on two<br />
different <strong>atoms</strong> !<br />
-2 0 2 4<br />
Florence, Mai 2004 294<br />
-0.2<br />
-0.3<br />
φ/π
Fidelity<br />
Fidelity estimate F = Tr( ρ ΨEPR<br />
ΨEPR<br />
)<br />
Assuming that imperfections do not create unwanted coherences:<br />
P + V⊥<br />
F = <br />
2<br />
• P : population in the "expected" channels in longitudinal experiment<br />
»0.71<br />
• V ⊥ : visibility of the Bell signal in the transverse experiment<br />
»0.25<br />
Hence F=48%<br />
Violation of Bell inequalities Requires a Bell contrast signal >0.71<br />
Florence, Mai 2004 295
Principle:<br />
First atom<br />
Initial state<br />
π/2 pulse in M a<br />
π pulse in M b<br />
Second atom:<br />
probes field states<br />
Entangling two modes of the radiation field<br />
• Final transfer rate modulated versus<br />
the delay at the beat note between<br />
modes<br />
∆<br />
0<br />
−δ<br />
A s<br />
π/2<br />
π<br />
e,0,0<br />
D<br />
Single photon beats<br />
1.0<br />
P e<br />
(T)<br />
0.5<br />
( ,0,0 ,1,0 )<br />
2 e + g<br />
0.0<br />
( 0,1 1,0 )<br />
2 g + 1.0<br />
1<br />
1<br />
A p<br />
π<br />
(b)<br />
π/2<br />
M a<br />
M b<br />
D<br />
0.5<br />
0.0<br />
1.0<br />
0.5<br />
0.0<br />
48 50 52 54 56 58<br />
200 202 204 206 208<br />
400 402 404 406 408<br />
0.0<br />
0 π/2Ω 3π/2Ω T Τ+π/Ω t<br />
698 700 702 704 706<br />
T(µs)<br />
Florence, Mai 2004 296<br />
1.0<br />
0.5<br />
(a)<br />
(b)<br />
(c)<br />
(d)
A <strong>quantum</strong> phase gate<br />
Principle<br />
2π <strong>quantum</strong> Rabi pulse:<br />
conditional dynamics<br />
e<br />
g<br />
π<br />
D<br />
C<br />
i<br />
"Truth table"<br />
49 (level i )<br />
51.1 GHz<br />
cavi ty<br />
50 (level g)<br />
54.3 GHz<br />
Ramsey source<br />
• Cavity qubit: states |0> <strong>and</strong> |1><br />
• Atomic qubit: states |i> <strong>and</strong> |g><br />
i,0 ⎯⎯→ i,0<br />
i,1 ⎯⎯→ i,1<br />
g,0 ⎯⎯→ g,0<br />
iφ<br />
g,1 ⎯⎯→− g,1 = e g,1<br />
51 (level e)<br />
Tests<br />
Two complementary experiments<br />
• A single photon shifts the phase<br />
of an atomic coherence<br />
1 1<br />
0 0<br />
2 2<br />
1 1<br />
1 1<br />
2 2<br />
( i + g ) ⎯⎯→ ( i + g )<br />
( i + g ) ⎯⎯→ ( i − g )<br />
• A single atom phase shifts the<br />
cavity field<br />
( 0<br />
0 +<br />
1<br />
1 ) ⎯⎯→ ( 0<br />
0 +<br />
1<br />
1 )<br />
( 0<br />
0 +<br />
1<br />
1 ) ⎯⎯→ ( 0<br />
0 −<br />
1<br />
1 )<br />
i c c i c c<br />
g c c g c c<br />
Rauschenbeutel et al., PRL 83, 5166 (99)<br />
Quantum phase gate<br />
Florence, Mai 2004 297
A single photon phase-shifts an atomic coherence<br />
Principle<br />
Preparation <strong>and</strong> test of an atomic<br />
coherence: Ramsey set-up<br />
e<br />
g<br />
π<br />
P g<br />
Timing<br />
Position<br />
(a)<br />
C<br />
e<br />
π<br />
π/2<br />
A 1<br />
g A 2<br />
2π<br />
R 1<br />
gi<br />
D<br />
π/2<br />
D<br />
R 2<br />
gi<br />
Time<br />
i<br />
R 1<br />
C R 2 D 0 ν−ν gi<br />
π phase shift of fringes when cavity<br />
contains one photon<br />
Signal<br />
0,9<br />
0,8<br />
0,7<br />
One photon<br />
Zero photon<br />
Preparation of |1>: source atom,<br />
prepared in e, π <strong>quantum</strong> Rabi<br />
pulse<br />
Probability<br />
0,6<br />
0,5<br />
0,4<br />
0,3<br />
0,2<br />
0,1<br />
0 10 20 30 40 50 60<br />
Frequency ν (kHz)<br />
Florence, Mai 2004 298
A single atom phase-shifts the field<br />
Principle<br />
|0> <strong>and</strong> |1> superposition:<br />
small coherent field<br />
α ≈ c 0 + c 1<br />
0 1<br />
Amplitude read-out:<br />
atom in g. π Rabi pulse<br />
Minimum transfer:<br />
minimum amplitude<br />
Atom in i<br />
c 0 + c 1 ≈ α<br />
0 1<br />
in g<br />
c 0 − c 1 ≈−α<br />
0 1<br />
Timing<br />
Homodyne detection<br />
i<br />
α + αe φ<br />
i<br />
− α + αe φ<br />
Zero amplitude for<br />
φ = π φ = 0<br />
Position<br />
C<br />
(b)<br />
α<br />
g<br />
A 2<br />
2π<br />
π/2<br />
R1<br />
gi<br />
D<br />
αe ιθ<br />
g<br />
A 3<br />
π<br />
Time<br />
D<br />
Florence, Mai 2004 299
Coherent gate operation<br />
Atom in i<br />
0.4<br />
Probe transfer<br />
0.3<br />
Atom in g<br />
0.2<br />
0.4<br />
-2 -1 0 1 2<br />
φ/π<br />
A fully coherent<br />
<strong>quantum</strong> gate<br />
Probe transfer<br />
0.3<br />
0.2<br />
-2 -1 0 1 2<br />
φ/π<br />
Florence, Mai 2004 300
Tuning the <strong>quantum</strong> gate phase<br />
Role of atom-cavity detuning<br />
Phase of gate:<br />
360<br />
• For δ=0 : resonant interaction. π gate<br />
• Large δ: dispersive regime. Transient<br />
modification of atom <strong>and</strong> cavity<br />
frequencies: gate <strong>with</strong> a small angle<br />
• Intermediate δ range:<br />
– Intermediate values of φ<br />
– Absorption remains small (
“Quantum program”: generation of a GHZ state<br />
A 2<br />
A 1<br />
π/2 2π pulse<br />
Atom-cavity Cavity-Atom entanglement<br />
Control-phase creation<br />
gate<br />
Florence, Mai 2004 302
• prepared state:<br />
The "GHZ" state"<br />
1<br />
⎡<br />
1, 0<br />
1,<br />
1<br />
2 ⎣<br />
e g i − g<br />
( ) ( )<br />
2 2 2 2<br />
⎤<br />
⎦<br />
• In term of qubits:<br />
1<br />
2<br />
( 0,0,0 + 1,1,1 )<br />
• In term of spin 1/2:<br />
1<br />
2<br />
( + , + , + + − , − − )<br />
1 2 c 1 2,<br />
c<br />
• " GHZ triplet "<br />
(Greenberger Horne Zeilinger)<br />
Florence, Mai 2004 303
Entanglement tests<br />
Assessing preparation fidelity<br />
Two complementary experiments:<br />
• Correlations in a "longitudinal" basis<br />
– State populations<br />
– Measurement of diagonal terms *<br />
ρ triplet<br />
+ ++ ..................... −−−<br />
⎡* . . . . . . * ⎤<br />
⎢<br />
. * . . . . . .<br />
⎥<br />
⎢<br />
⎥<br />
⎢ . . * . . . . . ⎥<br />
⎢<br />
⎥<br />
. . . * . . . .<br />
= ⎢<br />
⎥<br />
⎢ . . . . * . . . ⎥<br />
⎢<br />
⎥<br />
⎢<br />
. . . . . * . .<br />
⎥<br />
⎢ . . . . . . * . ⎥<br />
⎢⎣<br />
* . . . . . . * ⎥⎦<br />
• Correlations in a "transverse" basis<br />
– Measurement of correlated to state of atom 2<br />
– Measures one off-diagonal term *<br />
Rauschenbeutel et al Science 288, 2024 (00)<br />
Florence, Mai 2004 304
Measurement of σ z1 . σ z2 . σ z3 : longitudinal expt<br />
• Step 1: transfer of the field state to a third atom performing a π absorption pulse in C:<br />
1<br />
⎡ e ( ) ( )<br />
1, 0 g2 + i2 + g1,<br />
1 g2<br />
− i ⎤<br />
2<br />
⊗ g3<br />
2 ⎣<br />
⎦<br />
1<br />
⇒ ⎡ e ( ) ( )<br />
1<br />
g + i g3 + g1<br />
g − i e ⎤<br />
3<br />
⊗<br />
2 ⎣<br />
⎦<br />
1<br />
( +<br />
)<br />
1, +<br />
2, +<br />
3<br />
+ −1,<br />
−2,<br />
−3<br />
2<br />
2 2 2 2<br />
0<br />
• step 2: detection of each atom for measuring σ z1 . σ z2 . σ z3<br />
- <strong>atoms</strong> 1 et 3 : direct measurement of energy<br />
- atome 2: measurement of energy after applucation of an external π/2 pulse:<br />
π/2<br />
( )<br />
( − )<br />
⎧<br />
⎪1 2 g2 + i2 → i2<br />
⎨<br />
⎪⎩<br />
1 2 g i → g<br />
⇒ 1<br />
2 e i g + g g e<br />
( 1, 2, 3 1, 2,<br />
3 )<br />
2 2<br />
Florence, Mai 2004 305
Position (cm)<br />
Full set of operations for measurement of σ z1 . σ z2 . σ z3<br />
0<br />
π/2<br />
D<br />
D<br />
D<br />
10<br />
8<br />
6<br />
Atom # 1<br />
Atom # 2<br />
π/2 2π π<br />
π/2<br />
4<br />
2<br />
Atom # 3<br />
θ<br />
π/2<br />
D<br />
• Rabi oscillation in C<br />
π/2<br />
•Detection<br />
•Classical π/2 pulse<br />
Time<br />
State before detection:<br />
⇒ 1<br />
2 e i g + g g e<br />
( 1, 2, 3 1, 2,<br />
3 )<br />
Florence, Mai 2004 306
Measurement results:<br />
• measurement of σ z1<br />
. σ z2<br />
. σ z3<br />
P long<br />
=P eig<br />
+ P gge<br />
= 0.58 (0.02)<br />
0.4<br />
0.3<br />
0.2<br />
|- 1 ,- 2 ,- 3 〉<br />
|+ 1<br />
,+ 2 ,+ 3 〉<br />
0.1<br />
0<br />
Pgig<br />
Pgie<br />
Pggg<br />
Pgge<br />
Peig<br />
Peie<br />
Pegg<br />
Pege<br />
Rauschenbeutel et al., Science 288, 2024 (2000)<br />
Florence, Mai 2004 307
Transverse experiment<br />
Position<br />
Timing<br />
• R 1<br />
<strong>and</strong> R 3<br />
test the A 1<br />
-A 3<br />
EPR<br />
"transverse" correlations<br />
• Measure the A 1<br />
-A 3<br />
"Bell" signal as a<br />
function of the state of A 2<br />
Tests<br />
A 1<br />
R 1<br />
(II)<br />
Ψ triplet<br />
A 2<br />
A 3<br />
C<br />
π/2<br />
D<br />
π<br />
D<br />
R 3<br />
(II)<br />
π/2<br />
D<br />
Time<br />
1<br />
ψ<br />
triplet<br />
= ⎡ i2 ( e1,0 + g1,1 ) + g2 ( e1,0 − g1,1<br />
) ⎤<br />
2 ⎣<br />
⎦<br />
Bell signal<br />
Results<br />
0.6<br />
0.4<br />
0.2<br />
0.0<br />
-0.2<br />
-0.4<br />
-0.6<br />
-6 -4 -2 0 2<br />
Detection phase φ<br />
Florence, Mai 2004 308
Fidelity of preparation of the GHZ state<br />
• measurement of σ z1<br />
. σ z2<br />
. σ z3<br />
P long<br />
=P eig<br />
+ P gge<br />
= 0.58 (0.02)<br />
• measurement of σ x1<br />
. σ x2<br />
. σ x3<br />
A= 〈σ x1<br />
. σ x2<br />
. σ x3<br />
〉 = -0.28 (0.03)<br />
• fidelity:<br />
F ψ ρ ψ<br />
= =<br />
triplet<br />
F > 0.3 garanties non-separability<br />
triplet<br />
0.54 (0.03)<br />
see also: Sacket et al. Science 288, 2024 (2000)<br />
preparation of a 4 ions GHZ state in one step<br />
Florence, Mai 2004 309
A complex experimental sequence<br />
A timing nightmare<br />
Features<br />
Experiment II<br />
Stark Voltage (V)<br />
-4 -3 -2 -1 0 1 2<br />
R1 gi<br />
R2 R2<br />
gi ei<br />
A3: 100 µs<br />
100<br />
A2: 25 µs<br />
100<br />
A small "<strong>quantum</strong> program": 6<br />
operations on 4 individual qubits (two<br />
<strong>atoms</strong>, the cavity <strong>and</strong> an extra atom<br />
used to read-out the cavity state).<br />
Time (µs)<br />
0<br />
-100<br />
A1: 0 µs<br />
1.757 V<br />
0.62 V<br />
0<br />
-100<br />
In principle, could be extended to<br />
generate a more complex entangled<br />
state<br />
-3 -2 -1 0 1 2 3 4 5 6<br />
Position (cm)<br />
Florence, Mai 2004 310
IV) Quantum information <strong>with</strong> <strong>atoms</strong> <strong>and</strong> <strong>cavities</strong><br />
• 1) A reminder on <strong>quantum</strong> computing<br />
• 2) Quantum entanglement knitting stitches<br />
• 3) Cavity assisted collisions<br />
Florence, Mai 2004 311
Towards more complex sequences<br />
Present limitations <strong>and</strong> possible<br />
solutions<br />
• Single qubit gate fidelity<br />
– Improve field homogeneity<br />
• R<strong>and</strong>om atom preparation: long data<br />
acquisition times<br />
– Deterministic atom pistol <strong>with</strong><br />
cold atom techniques<br />
• Cavity losses<br />
– New cavity design<br />
• Higher Q's<br />
• No ring<br />
• Encouraging preliminary<br />
results<br />
– "Remove" the cavity <br />
Entanglement <strong>with</strong>out cavity<br />
Van der Waals resonant collision in free<br />
space:<br />
Resonant energy exchange between two<br />
Rydberg <strong>atoms</strong>.<br />
Full entanglement for b=10µm. Efficient<br />
<strong>quantum</strong> gate (Lukin, Zoller)<br />
b<br />
van der<br />
Waals<br />
eg , ⎯⎯→ cos θ eg , + sin θ ge ,<br />
2<br />
2<br />
θ α c ⎛an<br />
⎞ 0<br />
= ⎜ ⎟<br />
v⎝<br />
b ⎠<br />
Requires excellent control of atomic<br />
position.<br />
Florence, Mai 2004 312
Cavity-assisted van der Waals collision<br />
Two <strong>atoms</strong> interact in a non-resonant cavity<br />
2<br />
⎛ω<br />
⎞c<br />
an<br />
0<br />
Mixing angle: = ⎜ ⎟ ⎜ ⎟ ω transition freq., δ atom-cavity detuning<br />
⎝ ⎠v⎝ bc<br />
⎠<br />
θ α δ<br />
⎛ ⎞<br />
Effective impact parameter b c ~cavity size (mm)<br />
Considerable enhancement factor ω/δ (up to 10 6 )<br />
2<br />
Resonant coupling between |e 1<br />
,g 2<br />
,0> <strong>and</strong><br />
|g 1<br />
,e 2<br />
,0> involving a virtual photon<br />
exchange <strong>with</strong> the cavity (state |g 1<br />
,g 2<br />
,1>)<br />
|g 1<br />
,g 2<br />
;1><br />
δ<br />
e 1<br />
,g 2<br />
;0> |g 1<br />
,e 2<br />
;0><br />
Actual cavity has two modes (orthogonal polarizations) separation 130 kHz:<br />
– Enhancement factor η=ω(1/δ 1 +1/δ 2 )<br />
Zheng <strong>and</strong> Guo, Phys. Rev. Lett. 85, 2392 (2000); Osnaghi et al. Phys. Rev. Lett. 87, 037902 (2001)<br />
Florence, Mai 2004 313
Advantage of non-resonant method of entanglement:<br />
Sensitivity to cavity damping<br />
Ω =<br />
R<br />
Ω<br />
2<br />
0<br />
2δ<br />
• effect of cavity damping:<br />
projection on |g,g,0><br />
Full loss of entanglement<br />
• probability of error:<br />
eg , ,0<br />
Ω<br />
R ge , ,0<br />
gg , ,1<br />
P<br />
col<br />
err<br />
⎛Ω<br />
⎞<br />
≈⎜<br />
⎟<br />
⎝δ<br />
⎠<br />
2<br />
• Resonant case:<br />
P<br />
res<br />
err<br />
≈Γ<br />
Γ<br />
cav<br />
cav<br />
. T<br />
int<br />
. T<br />
res<br />
int<br />
Ω . = π 2<br />
T R int<br />
res<br />
Ω . T = π 2<br />
int<br />
δ Γ cav<br />
• error rate reduced as:<br />
P<br />
P<br />
col<br />
err<br />
res<br />
err<br />
≈<br />
Ω<br />
δ<br />
gg , ,0<br />
efficient <strong>with</strong> slower <strong>atoms</strong><br />
Florence, Mai 2004 314
Advantage of non-resonant method of entanglement:<br />
Sensitivity to blackbody radiation<br />
• coupling in the presence of N photons:<br />
egN , ,<br />
Ω N + 1<br />
eeN− , , 1<br />
Ω R<br />
ggN+ , , 1<br />
Ω<br />
N<br />
g, eN ,<br />
Due to destructive interference<br />
between two probability amplitudes,<br />
the effective coupling is to first<br />
order independent of N:<br />
2<br />
( )<br />
2 2<br />
Ω<br />
0. N + 1 Ω0.<br />
N Ω0<br />
ΩR<br />
≈ − =<br />
2δ 2δ 2δ<br />
The method works even in the presence of blackbody radiation<br />
Similar to "hot" gate for ions:<br />
Moelmer et al PRL 82 1835 (2000)<br />
Florence, Mai 2004 315
Experimental realization<br />
•Both <strong>atoms</strong> simultaneously present in the empty cavity<br />
mode<br />
•Minimum distance: about 1 mm (atomic beam diameter)<br />
•Ramsey pulses to detect atomic spins along a tunable<br />
direction<br />
Florence, Mai 2004 316
Tests of entanglement<br />
Population transfer<br />
1.0<br />
0.8<br />
"longitudinal entanglement"<br />
P(e 1<br />
,g 2<br />
)<br />
P(g 1<br />
,e 2<br />
)<br />
Test of "transverse" entanglement<br />
0.8<br />
<br />
Probability<br />
0.6<br />
0.4<br />
0.2<br />
0.4<br />
0.0<br />
-0.4<br />
-0.8<br />
0.0<br />
0 1 2 3 4<br />
η(x10 -6 )<br />
Up to 2π Rabi rotation<br />
Good agreement <strong>with</strong> simple model up<br />
to π/2 (solid lines)<br />
Qualitative agreement <strong>with</strong> numerical<br />
integration for larger mixing angles<br />
Features:<br />
-1 0 1 2 3<br />
• Insensitive to cavity damping<br />
• Insensitive to cavity residual thermal<br />
field<br />
• Easily transformed in a CNOT gate<br />
Very promising for <strong>quantum</strong> information<br />
processing <strong>with</strong> moderate Q <strong>cavities</strong><br />
Florence, Mai 2004 317<br />
φ/π
Application of controlled collision<br />
• A simple proposed implementation of the two qubit Grover search<br />
algorithm<br />
Florence, Mai 2004 318
Next step: solve the simplest <strong>quantum</strong> algorithms<br />
PROBLEM: Finding a known item in an unsorted list of size N<br />
Classical Solution: Check all the items until the one we look for is found!<br />
Number of trials: Order of N/2<br />
Complete list<br />
( )<br />
x<br />
1,...,x N<br />
x o<br />
( )<br />
δ x − x o<br />
Equivalent to an oracle corresponding to a function in a blackbox which<br />
gives the answer yes or no (0 or 1) to the question: « is it the marked item »<br />
x3<br />
0<br />
x 102 δ( x− x o<br />
) 0<br />
x o<br />
1<br />
Florence, Mai 2004 319
QUANTUM SEARCH:<br />
O ( N ) queries !!<br />
L.K.Grover, Phys. Rev. Lett. 79, 325 (1997)<br />
0 1<br />
n qubits: <strong>and</strong><br />
n<br />
( N = 2 states)<br />
•1st<br />
step: put all the qubits in a superposition<br />
1<br />
2<br />
( 0 + 1 )<br />
Hadamard gate,<br />
performed via a classical<br />
p/2 microwave pulse<br />
Ψ<br />
=<br />
1<br />
N<br />
( 0 + 1 ) × ( 0 + 1 ) × ... × ( 0 + 1 ) = ∑<br />
1<br />
N<br />
N<br />
i<br />
i<br />
Superposition of all possible states<br />
Florence, Mai 2004 320
We are looking for a particular state<br />
x o =10100<br />
...0<br />
• 2nd step: Inverse the amplitude of the searched item.<br />
O<br />
o<br />
= I − 2<br />
x<br />
x<br />
o<br />
x<br />
o<br />
This step corresponds to the action of<br />
the « oracle » <strong>and</strong> is the only one where<br />
information about the searched item is used.<br />
« - » sign = answer « yes ». « + » sign = answer « no »<br />
• 3rd step: Symmetrisation about the average:<br />
U s<br />
= 2 Ψ Ψ − I = H ( 2 0 0 − I )H<br />
( H = H H ... )<br />
φ<br />
1<br />
2<br />
H n<br />
= ∑ i<br />
ai i<br />
a<br />
I 0<br />
=<br />
1<br />
N<br />
Florence, Mai 2004 321<br />
∑<br />
i<br />
a<br />
i<br />
( a )<br />
U φ = 2 Ψ Ψ φ − φ = ∑ 2a −<br />
s<br />
This step does not use<br />
Information about the searched<br />
item<br />
i<br />
i<br />
i
Repeating this sequence leads to the marked<br />
item.<br />
An example: : N=16 ( 4 qubits ).)<br />
The first operation reverses the amplitude of<br />
the marked item, lowering the average.<br />
Then, the reversal about the average<br />
increases the probability amplitude of<br />
finding the marked item.<br />
After iterating the same procedure 3 times<br />
we find<br />
the marked item <strong>with</strong> a 96% probability.<br />
Case n=2 qubits ( N = 4 items ) Only one iteration !!<br />
Florence, Mai 2004 322
• CAVITY QED: (N=4)<br />
= H = SI QPG HI QPG P<br />
Q HI0<br />
HO xo<br />
Oracle<br />
qubits: two electronic levels of a three level atom<br />
Hadamard gates: Classical microwave<br />
pulses combining two<br />
electronic levels.<br />
.<br />
cavity<br />
|e><br />
|g><br />
H<br />
⎧<br />
⎪<br />
0<br />
: ⎨<br />
⎪ 1<br />
⎩<br />
→<br />
→<br />
1<br />
2<br />
1<br />
2<br />
( 0 + 1 )<br />
( 0 − 1 )<br />
|i><br />
Florence, Mai 2004 323
Quantum phase gate:<br />
(dispersive interaction)<br />
I QPG<br />
00<br />
⎛1<br />
⎜<br />
⎜0<br />
= ⎜0<br />
⎜<br />
⎝0<br />
01<br />
0<br />
1<br />
0<br />
0<br />
10<br />
0<br />
0<br />
1<br />
0<br />
11<br />
0 ⎞<br />
⎟<br />
0 ⎟<br />
0 ⎟<br />
⎟<br />
−1<br />
⎠<br />
e<br />
Two <strong>atoms</strong> interact dispersively <strong>with</strong> the cavity field (detuning d >> W )<br />
δ<br />
cavity<br />
|e 1 〉|g 2 〉|0〉 |g 1 〉|e 2 〉|0〉<br />
δ<br />
Ω<br />
|g 1 〉|g 2 〉|1〉<br />
Ω<br />
g<br />
H<br />
2<br />
Ω ⎡<br />
= ⎢ ∑<br />
4δ<br />
⎣ j = 1,<br />
Cavity field shift Two atom collision<br />
e<br />
e<br />
+<br />
e<br />
eff<br />
j j 1 2 1 2 1 2 1 2<br />
2<br />
S.-B. Zheng <strong>and</strong> G.-C. Guo, Phys. Rev. Lett. 85, 2392 (2000)<br />
S. Osnaghi et al., Phys. Rev. Lett. 85, 2392 (2000)<br />
Florence, Mai 2004 324<br />
g<br />
g<br />
e<br />
+<br />
g<br />
e<br />
e<br />
g<br />
⎤<br />
⎥<br />
⎦
e<br />
e<br />
g<br />
t<br />
=<br />
e<br />
iλt<br />
[ cos ( λt<br />
) e g − i sin ( λt<br />
) g e ]<br />
cavity<br />
g<br />
i<br />
Qubit encoding:<br />
0 → g<br />
1st atom: : 2nd atom:<br />
1 → e<br />
Action of effective Hamiltonian for a time<br />
0<br />
1<br />
→<br />
→<br />
i<br />
g<br />
t<br />
=<br />
π<br />
λ<br />
e<br />
g<br />
g<br />
i<br />
t<br />
g<br />
i<br />
t<br />
=<br />
t<br />
e<br />
iλt<br />
= g<br />
= g<br />
e<br />
i<br />
g<br />
i<br />
0<br />
0<br />
1<br />
1<br />
0<br />
1<br />
0<br />
1<br />
= g<br />
= g<br />
= e1<br />
= e<br />
1<br />
1<br />
1<br />
i<br />
i<br />
g<br />
2<br />
g<br />
2<br />
2<br />
2<br />
→<br />
→ g<br />
→ e<br />
→ − e<br />
g<br />
1<br />
1<br />
1<br />
1<br />
i<br />
i<br />
2<br />
g<br />
2<br />
g<br />
2<br />
2<br />
I QPG<br />
00<br />
⎛1<br />
⎜<br />
⎜0<br />
= ⎜0<br />
⎜<br />
⎝0<br />
01<br />
0<br />
1<br />
0<br />
0<br />
10<br />
0<br />
0<br />
1<br />
0<br />
11<br />
0 ⎞<br />
⎟<br />
0 ⎟<br />
0 ⎟<br />
⎟<br />
−1<br />
⎠<br />
Florence, Mai 2004 325
Sequence of operations:<br />
Q =<br />
SI<br />
QPG<br />
HI<br />
S <strong>and</strong> P: Performed by microwave pulses<br />
QPG<br />
P<br />
P contains the<br />
information about<br />
the marked item.<br />
It plays the role of<br />
the oracle.<br />
S 1 S 2<br />
P = P1 ( θ1)<br />
P2<br />
( θ2)<br />
θ = 0<br />
⎧<br />
⎪<br />
0 →<br />
S : ⎨<br />
⎪ 1 →<br />
⎩<br />
π<br />
S = i<br />
⎧<br />
( − 0 − 1 )<br />
0 ( )<br />
2<br />
i<br />
2<br />
( 0 − 1 )<br />
⎪<br />
Pi<br />
( θi) : ⎨<br />
⎪ 1<br />
⎩<br />
or determines the marked item<br />
→<br />
→<br />
1<br />
2<br />
1<br />
2<br />
e<br />
−iθ<br />
/ 2<br />
i<br />
0<br />
+ e<br />
iθ<br />
/ 2<br />
1<br />
−iθi<br />
/ 2 iθi<br />
/ 2<br />
( e 0 − e 1 )<br />
θ<br />
i<br />
state<br />
( 0 ,0 ) 1 1<br />
( 0 , π ) 1 0<br />
( π ,0 ) 0 1<br />
( π , π ) 0 0<br />
Florence, Mai 2004 326
• The search algorithm:<br />
Q<br />
=<br />
SI<br />
QPG<br />
HI<br />
QPG<br />
P<br />
P QPG H QPG S detection<br />
Action of the oracle:<br />
Application of P determines<br />
the output<br />
Access <strong>atoms</strong> independently:<br />
Stark effect<br />
Florence, Mai 2004 327
Simulations made considering the<br />
experiment as the ideal one: the only<br />
source of imperfections, responsible for<br />
the non perfect fidelity, is the effective<br />
Hamiltonian.<br />
We consider here imperfection in the<br />
pulse duration. When they are of the<br />
order of 5 %, the fidelity decreases to<br />
82%.<br />
Simulations:<br />
Probability<br />
Probability<br />
1,0<br />
0,8<br />
0,6<br />
0,4<br />
0,2<br />
0,0<br />
0,8<br />
0,6<br />
0,4<br />
0,2<br />
96%<br />
eg ei gg gi<br />
5% pulse error<br />
δ = 4Ω<br />
(a)<br />
(b)<br />
0,0<br />
eg<br />
ei<br />
gg<br />
gi<br />
Evolution of the fidelity <strong>with</strong> pulse<br />
imperfections.<br />
Fidelity<br />
0,9<br />
0,8<br />
0,7<br />
(c)<br />
0,6<br />
0,00 0,02 0,04 0,06 0,08 0,10<br />
Pulse Imperfections<br />
Florence, Mai 2004 328
Structure of the lectures<br />
• I) Introduction<br />
• II) The tools of CQED<br />
• III) Experimental illustrations of fundamental <strong>quantum</strong> mechanics<br />
• IV) Quantum information <strong>with</strong> <strong>atoms</strong> <strong>and</strong> <strong>cavities</strong><br />
• V) Schrödinger cats <strong>and</strong> decoherence<br />
• VI) Cavity QED <strong>with</strong> dielectric microspheres<br />
• VII) Perspectives<br />
Florence, Mai 2004 329
V) Schrödinger cats <strong>and</strong> decoherence<br />
• 1) A direct study of a meter’s decoherence process in a <strong>quantum</strong><br />
measurement<br />
• 2) Breeding Schrödinger lions <strong>with</strong> resonant interaction<br />
• 3) Other applications of field homodyne detection<br />
Florence, Mai 2004 330
V) Schrödinger cats <strong>and</strong> decoherence<br />
• 1) A direct study of a meter’s decoherence process in a <strong>quantum</strong><br />
measurement<br />
• 2) Breeding Schrödinger lions <strong>with</strong> resonant interaction<br />
• 3) Other applications of field homodyne detection<br />
Florence, Mai 2004 331
Quantum/classical boundary <strong>and</strong> decoherence<br />
No macroscopic superpositions at our<br />
scale<br />
Decoherence<br />
The "Schrödinger cat"<br />
1<br />
2<br />
( )<br />
+ ⇔<br />
Environment<br />
A macroscopic system is strongly<br />
coupled to a complex environment<br />
In all models, a few states only are<br />
stable ("preferred basis").<br />
No entangled states neither.<br />
We only observe a very small fraction of<br />
all possible <strong>quantum</strong> states<br />
WHY <br />
All <strong>quantum</strong> superpositions of these<br />
states evolve very rapidly into<br />
statistical mixtures.<br />
Decoherence<br />
Florence, Mai 2004 332
Main features of decoherence<br />
Very fast process<br />
An essential process<br />
superposition lifetime<br />
=<br />
relaxation time<br />
separation betweenstates<br />
• Rules <strong>quantum</strong> superpositions out of<br />
the classical world.<br />
Depends upon the initial <strong>quantum</strong> state<br />
(distance between states or<br />
"macroscopicity" parameter)<br />
• Essential to underst<strong>and</strong> <strong>quantum</strong><br />
measurement process (no<br />
superpositions of meter's states)<br />
Not a trivial relaxation mechanism<br />
(but explained by st<strong>and</strong>ard relaxation<br />
theory for simple models)<br />
-<br />
0<br />
∆x<br />
+<br />
Strong link <strong>with</strong> complementarity <strong>and</strong><br />
entanglement: environment acquires<br />
a which path information <strong>and</strong> gets<br />
entangled <strong>with</strong> the system<br />
• Main obstacle for a large scale use of<br />
<strong>quantum</strong> weirdness (<strong>quantum</strong><br />
computing)<br />
Florence, Mai 2004 333
Another experiment on complementarity<br />
e<br />
Cavity as an external detector in the<br />
Ramsey interferometer<br />
Cavity contains initially a coherent field<br />
Non-resonant atom-field interaction:<br />
e<br />
g<br />
R 1<br />
R 2<br />
α<br />
C<br />
Atom modifies the cavity field phase<br />
Phase shift α 1/δ<br />
S<br />
(index of refraction effect)<br />
⎯⎯→<br />
⎯⎯→<br />
e<br />
g<br />
(δ:atom-cavity detuning)<br />
1<br />
g<br />
D<br />
φ<br />
"Which path" information:<br />
• Small phase shift (large δ)<br />
(smaller than <strong>quantum</strong> phase noise)<br />
– field phase almost unchanged<br />
– No which path information<br />
– St<strong>and</strong>ard Ramsey fringes<br />
• Large phase shift (small δ)<br />
(larger than <strong>quantum</strong> phase noise)<br />
– Cavity fields associated to the<br />
two paths distinguishable<br />
– Unambiguous which path<br />
information<br />
– No Ramsey fringes<br />
Florence, Mai 2004 334
1.0<br />
0.5<br />
0.0<br />
Fringes <strong>and</strong> field state<br />
Complementarity<br />
712 kHz<br />
Vacuum<br />
State transformations<br />
R1<br />
C<br />
1<br />
e → e + g<br />
2<br />
Before R1<br />
( )<br />
R2<br />
1<br />
iϕ<br />
e → ( e + e g )<br />
2<br />
1<br />
g → − e e + g<br />
2<br />
( − iϕ<br />
)<br />
e, α →e e, αe g, α → g,<br />
αe<br />
iΦ iΦ −iΦ<br />
e,<br />
α<br />
Ramsey Fringe Signal<br />
1.0<br />
0.5<br />
0.0<br />
712 kHz,<br />
9.5 phot ons<br />
347 kHz<br />
Before C<br />
After C<br />
After R2<br />
1<br />
2<br />
1<br />
( )<br />
2 e + g α<br />
i i i<br />
( e Φ e, αe Φ + g,<br />
αe<br />
− Φ<br />
)<br />
1<br />
2<br />
Detection probabilities<br />
( ϕ )<br />
{ α − α }<br />
ee e e e<br />
iΦ iΦ − i +Φ −iΦ<br />
ϕ<br />
{ α<br />
α }<br />
1<br />
+ g e e + e e<br />
2<br />
i( ϕ+Φ) iΦ − i( +Φ)<br />
−iΦ<br />
104 kHz<br />
0 2 4 6 8 10<br />
ν (kHz) PRL 77, 4887 (96)<br />
1<br />
Pge<br />
,<br />
= ⎡ 1 ± Re e αe αe<br />
2 ⎣<br />
− i( ϕ +Φ ) i Φ − i Φ<br />
Ramsey fringes signal multiplied by<br />
Florence, Mai 2004 335<br />
αe<br />
iΦ<br />
⎤<br />
⎦<br />
αe<br />
−Φ i
Signal analysis<br />
Fringe signal multiplied by<br />
αe<br />
iΦ<br />
αe<br />
−Φ i<br />
Fringes contrast <strong>and</strong> phase<br />
• Modulus<br />
e<br />
2 2<br />
=<br />
e<br />
−2nsin Φ −D<br />
/2<br />
60<br />
n=9.5 (0.1)<br />
6<br />
– Contrast reduction<br />
• Phase<br />
2n<br />
sin<br />
Φ<br />
– Phase shift corresponding to<br />
cavity light shifts<br />
Phase leads to a precise (<strong>and</strong> QND)<br />
measurement of the average photon<br />
number<br />
D<br />
Fringe Cont r ast (%)<br />
40<br />
20<br />
0<br />
0.0 0.2 0.4 0.6 0.8<br />
φ (radians)<br />
• Excellent agreement <strong>with</strong> theoretical<br />
predictions.<br />
• Not a trivial fringes washing out effect<br />
Calibration of the cavity field<br />
9.5 (0.1) photons<br />
0.0 0.2 0.4 0 2<br />
φ (radians)<br />
4<br />
Fringe Shift (rd)<br />
Florence, Mai 2004 336
A laboratory version of the Schrödinger cat<br />
Field state after atomic detection<br />
1<br />
2<br />
( + )<br />
A coherent superposition of two<br />
'classical' states.<br />
Very similar to the Schrödinger cat<br />
An atom to probe field coherence<br />
Quantum interferences involving the<br />
cavity state<br />
First atom<br />
Φ<br />
−Φ<br />
D<br />
Second atom<br />
Decoherence will transform this<br />
superposition into a statistical mixture<br />
Slow relaxation: possible to study the<br />
decoherence dynamics<br />
Decoherence caught in the act<br />
Two indistinguishable <strong>quantum</strong> paths to<br />
the same final state:<br />
Quantum interferences<br />
2Φ<br />
−2Φ<br />
Florence, Mai 2004 337
Atomic correlations<br />
A correlation signal<br />
η =Πee<br />
,<br />
−Πge<br />
,<br />
P<br />
ee ,<br />
= −<br />
ge ,<br />
ee , eg , g, e gg ,<br />
• Independent of Ramsey<br />
interferometer phase φ (when Φ is<br />
neither 0 nor π/2)<br />
P<br />
P + P P + P<br />
Principle of the experiment<br />
• Send a first atom to prepare the cat<br />
• Wait for a delay τ<br />
• Send a second probe atom<br />
• Measure η versus τ<br />
Raw correlation signals<br />
• 0.5 for a <strong>quantum</strong> superposition<br />
0.3<br />
τ=40 µs<br />
η =<br />
1 Re<br />
2<br />
αα<br />
• 0 for a statistical mixture<br />
• 0 for an empty cavity<br />
correlation signal<br />
0.2<br />
0.1<br />
0.0<br />
-0.1<br />
0 2 4 6 8 10 12<br />
ν (kHz)<br />
15000 coincidences<br />
Florence, Mai 2004 338
A decoherence study<br />
Atomic correlation signal<br />
Decoherence versus size of the cat<br />
Two-Atom Correlation Signal<br />
0.0 0.1 0.2<br />
n=3.3 δ/2π =70 <strong>and</strong> 170 kHz<br />
0 1 2<br />
t/T<br />
r<br />
0 1 2 PRL 77, 4887 (1996)<br />
τ/T<br />
r<br />
Florence, Mai 2004 339<br />
correlation signal<br />
correlation signal<br />
δ/2π =70 kHz<br />
20<br />
16<br />
n=5.5<br />
12<br />
8<br />
4<br />
0<br />
0 1 2<br />
20<br />
t/T<br />
r<br />
16<br />
12<br />
n=3.3<br />
8<br />
4<br />
0
Decoherence <strong>and</strong> complementarity<br />
A simple theoretical approach<br />
Without relaxation:<br />
η =<br />
1 Re<br />
2<br />
Simple relaxation model: a bath of<br />
harmonic oscillators i.e. cavity modes<br />
C<br />
αα<br />
Linear couplings: amplitude β i<br />
(t) in mode<br />
i is proportional to the amplitude α(t)<br />
in the cavity mode<br />
A cat in the cavity: tiny cats in the<br />
environment<br />
C i<br />
Complete wavefunction at τ<br />
∏<br />
ατ ( ) e β( τ) e + ατ ( ) e β( τ)<br />
e<br />
i Φ i Φ −Φ i −Φ i<br />
i<br />
i<br />
i<br />
i<br />
Interfering states after the second atom<br />
(does not affect the environment)<br />
∏<br />
Final correlation<br />
Energy conservation<br />
∏<br />
iΦ<br />
ατ ( ) β( τ) e + ατ ( ) β( τ)<br />
e<br />
i<br />
i<br />
1 Re ( )<br />
iΦ<br />
η = ∏ βi<br />
τ e βi<br />
( τ ) e<br />
2<br />
i<br />
∏<br />
1 ⎛<br />
= Re exp βi<br />
( τ ) 1<br />
2<br />
⎜−∑<br />
−<br />
⎝ i<br />
∑<br />
i<br />
i<br />
−iΦ<br />
iΦ<br />
( e )<br />
2 2<br />
i<br />
⎞<br />
⎟<br />
⎠<br />
(<br />
− T<br />
e τ<br />
)<br />
2 /<br />
r<br />
β ( τ) = n 1−<br />
i<br />
PRL, 79, 1964 (1997)<br />
−Φ i<br />
Florence, Mai 2004 340
Theoretical decoherence signal<br />
Atomic correlation versus τ<br />
1<br />
2<br />
−2n[ 1−exp( −τ<br />
/ T r )] sin Φ<br />
η = e cos n[ 1−exp( −τ<br />
/ T )]<br />
r<br />
sin 2Φ<br />
2<br />
At short times<br />
T<br />
D<br />
Tr<br />
=<br />
n<br />
Excellent agreement <strong>with</strong> the<br />
experimental data<br />
A very simple description of<br />
decoherence in terms of<br />
complementarity.<br />
The environment 'measures' the<br />
field phase <strong>and</strong> gets a "which<br />
path" information<br />
Two-Atom Correlation Signal<br />
{ }<br />
0 1 2<br />
τ/T<br />
r<br />
Florence, Mai 2004 341<br />
0.0 0.1 0.2<br />
n=3.3 δ/2π =70 <strong>and</strong> 170 kHz
Decoherence features<br />
• Faster than cavity relaxation<br />
• Faster when distance between states increases<br />
• Decoherence time scale depends upon a "macroscopicity" parameter<br />
Not a trivial relaxation mechanism even if described by st<strong>and</strong>ard relaxation<br />
theory<br />
Essential for <strong>quantum</strong> measurement<br />
meters are not in superposition states<br />
Difficulty for applications of QM<br />
the more complex the entangled state, the faster the decoherence<br />
Towards decoherence "metrology" …. With much larger Schrödinger cats<br />
Florence, Mai 2004 342
V) Schrödinger cats <strong>and</strong> decoherence<br />
• 1) A direct study of a meter’s decoherence process in a <strong>quantum</strong><br />
measurement<br />
• 2) Breeding Schrödinger lions <strong>with</strong> resonant interaction<br />
• 3) Other applications of field homodyne detection<br />
Florence, Mai 2004 343
Rabi oscillation in a classical field<br />
Ωr<br />
Oscillation in a large coherent field H<br />
I<br />
= σ<br />
Y<br />
Ω<br />
r<br />
=Ω0<br />
n ∝ E<br />
2<br />
1 −Ω i clt/2 iΩclt/2<br />
| Ψ ( t) >= ⎡ (| | ) (| | )<br />
2 ⎣<br />
e e>+ i g > + e e>− i g > ⎤<br />
⎦<br />
⊗ α<br />
Atomic eigenstates<br />
In terms of Bloch sphere<br />
1<br />
± = ⎡ Z<br />
Y ⎣<br />
2<br />
e ± i g ⎤⎦<br />
In-phase <strong>and</strong> π-out-of-phase<br />
<strong>with</strong> respect to field<br />
Quantum beat between<br />
eigenstates:<br />
• Sinusoidal Rabi oscillation<br />
between e <strong>and</strong> g<br />
X<br />
Y<br />
Florence, Mai 2004 344
Rabi oscillation in a mesoscopic field<br />
A much more interesting situation<br />
⎛ Ω n+ 1t Ω n+<br />
1t<br />
Ψ = ∑<br />
+ +<br />
⎝<br />
1+ cosΩ 0<br />
n+<br />
1t<br />
2<br />
Pe()<br />
t = ∑ pn pn = cn<br />
2<br />
0 0<br />
() t cn<br />
⎜<br />
cos e, n sin g, n 1<br />
n<br />
2 2<br />
n<br />
⎞<br />
⎟<br />
⎠<br />
|e,n><br />
|+,n><br />
Ω+<br />
0<br />
n 1<br />
|-,n><br />
|g,n+1><br />
A complex Rabi oscillation signal<br />
1<br />
P e<br />
(t)<br />
0.8<br />
• Collapse:<br />
– Dispersion of Rabi frequencies<br />
• Revivals:<br />
– Finite number of frequencies<br />
– Direct consequence of field<br />
quantization<br />
0.6<br />
0.4<br />
0.2<br />
Ω 0 t/2π<br />
50 100 150 200<br />
Florence, Mai 2004 345
An insightful quasi-exact solution<br />
• Get more physical insight on the collapse-revival phenomenon<br />
• Get information on the field evolution<br />
• Rewrite the exact atom-field wavefunction<br />
Florence, Mai 2004 (Gea Banacloche PRL 65, 3385, Buzek et al PRA 45, 8190) 346
An insightful quasi-exact solution<br />
• Factor the two terms in an atom <strong>and</strong> field parts. Redefinition of running<br />
index n<br />
• Large coherent field<br />
• Product of atom <strong>and</strong> field states<br />
Florence, Mai 2004 347
An insightful quasi-exact solution<br />
• Exp<strong>and</strong> the sqrt(n) term<br />
• Neglect for the time being the second order phase spreading terms<br />
• Same treatment for Ψ 2<br />
Florence, Mai 2004 348
An insightful quasi-exact solution<br />
+ + − −<br />
Ψ () t = ⎡ Ψa() t Ψ<br />
c() t + Ψa() t Ψc()<br />
t<br />
2 ⎣<br />
1<br />
1<br />
2<br />
Ω<br />
Ψ = ∓<br />
±Ω i 0 nt/2<br />
i<br />
Ψ ± a<br />
= e ⎣e ± Φ e ∓ i g ⎦<br />
⎡<br />
n<br />
– Atomic states slowly ( times slower than Rabi oscillation) rotating in<br />
the equatorial plane of the Bloch sphere<br />
c<br />
e αe<br />
± i 0 nt/4<br />
± iΦ<br />
– A slowly rotating field state in the Fresnel plane<br />
⎤<br />
Φ=<br />
Ω<br />
0<br />
t<br />
4 n<br />
⎤<br />
⎦<br />
• Graphical representation of the joint atom-field evolution in a plane<br />
• t=0:<br />
– both field states coincide <strong>with</strong> original coherent state<br />
– Atomic states are the classical eigenstates<br />
Florence, Mai 2004 349
Atom-field states evolution<br />
+<br />
Ψ<br />
c<br />
1 + +<br />
− −<br />
Ψ ( t) = ⎡ Ψa () t Ψc () t + Ψa () t Ψc<br />
( t)<br />
2 ⎣<br />
−<br />
Ψ<br />
c<br />
⎤<br />
⎦<br />
+<br />
Ψ<br />
a<br />
−<br />
Ψ<br />
a<br />
+ −<br />
•At most times: Ψ Ψ = 0 an atom-field entangled state<br />
c<br />
c<br />
•In spite of large photon number: considerable reaction of the atom on the field<br />
Florence, Mai 2004 350
‘Automatic’ preparation of a Schrödinger cat<br />
• At time<br />
• Atom-field disentanglement<br />
• The fastest <strong>and</strong> most efficient way to prepare large Schrödinger cat states<br />
Florence, Mai 2004 351
Quantum Rabi signal<br />
• Retrieve the <strong>quantum</strong> Rabi signal<br />
• The Rabi oscillation signal has an amplitude modulated by the scalar<br />
products of the cavity field components: another manifestation of<br />
complementarity<br />
Florence, Mai 2004 352
Link <strong>with</strong> Rabi oscillation<br />
+<br />
Rabi oscillation: <strong>quantum</strong> interference between Ψ <strong>and</strong><br />
+ −<br />
• Contrast vanishes when Ψ Ψ = 0 :<br />
c<br />
1 + +<br />
− −<br />
Ψ ( t) = ⎡ Ψa () t Ψc () t + Ψa () t Ψc<br />
( t)<br />
2 ⎣<br />
– A direct link between Rabi collapse <strong>and</strong> complementarity<br />
c<br />
a<br />
−<br />
Ψ<br />
a<br />
⎤<br />
⎦<br />
1<br />
P e<br />
(t)<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
Ω 0<br />
t/2π<br />
•Fast preparation Atom-field Field of decorrelation:<br />
large state Schrödinger merge again: cat states<br />
Quantum Rabi oscillation<br />
•Another Unconditional illustration preparation<br />
<strong>and</strong> Quantum progressive of complementarity<br />
revival collapse of<br />
of the field<br />
•A surprising In a insight « phase Rabi in oscillation Schrödinger the simple Rabi cat state oscillation » phenomenon<br />
10 20 30 40 50<br />
Florence, Mai 2004 353
An expression at short times (collapse)<br />
• A time of the order of the vacuum Rabi oscillation period<br />
• Classical Rabi oscillations <strong>with</strong> a gaussian envelope. Collapse time<br />
• Revival time (half a complete rotation in phase space)<br />
• Why only a finite number of revivals <br />
Florence, Mai 2004 354
Field states phase spreading<br />
• One order more in the expansion:<br />
Ψ =∑<br />
c<br />
e<br />
+ i nΩ<br />
c<br />
n<br />
n<br />
2<br />
0 t / 4 n −Ω i 0 ( n − n) t/<br />
16<br />
– Phase rotation + Phase spreading of field states<br />
– Contrast of revivals decreases <strong>and</strong> width increases<br />
– Complete overlap of revivals after a few turns in phase space<br />
• Snapshots of field Q function for 15 photons<br />
e<br />
n<br />
3/2<br />
n<br />
Im(β)<br />
6<br />
4<br />
2<br />
(a)<br />
6<br />
4<br />
2<br />
(b)<br />
6<br />
4<br />
2<br />
(c)<br />
0<br />
0<br />
0<br />
-2<br />
-2<br />
-2<br />
-4<br />
-4<br />
-4<br />
-6<br />
-6<br />
-6<br />
-6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6<br />
Re(β) Re(β) Re(β)<br />
Florence, Mai 2004 355
Field phase spreading<br />
• At long times: take into account higher order terms in phases<br />
• Average of a goes to zero: complete phase information loss. Occurs for<br />
• To be compared <strong>with</strong> revival time<br />
• About revivals observable<br />
Florence, Mai 2004 356
Direct observation of field phase evolution<br />
• Rabi oscillation in mesoscopic field<br />
– High atom-field coupling<br />
– Low atom <strong>and</strong> field relaxation<br />
• Cavity QED tools <strong>with</strong> circular Rydberg <strong>atoms</strong> <strong>and</strong><br />
superconducting <strong>cavities</strong><br />
• A method to probe field phase distribution:<br />
– Homodyne field measurement<br />
Florence, Mai 2004 357
Field phase distribution measurement<br />
Homodyning a coherent field<br />
S<br />
•Inject a coherent field |α><br />
•Add a coherent amplitude –αe iφ<br />
•Resulting field (<strong>with</strong>in global phase) |α(1-e iφ )><br />
•Zero final amplitude for φ=0<br />
•Probe final field amplitude <strong>with</strong> atom in g<br />
•P g<br />
=1 for a zero amplitude<br />
•P g<br />
=1/2 for a large amplitude<br />
•More generally: P g<br />
(φ) reveals field phase distribution<br />
•In technical terms, P g<br />
(φ)=Q distribution<br />
Florence, Mai 2004 358
Experimental coherent field phase distribution<br />
Transfert<br />
0,85 120 photons<br />
75 photons<br />
0,80<br />
45 photons<br />
20 photons<br />
0,75<br />
0,70<br />
0,65<br />
0,60<br />
0,55<br />
Largeur (°)<br />
34<br />
32<br />
30<br />
28<br />
26<br />
24<br />
22<br />
20<br />
0,50<br />
18<br />
0,45<br />
0,40<br />
-80 -60 -40 -20 0 20 40 60 80<br />
Phase<br />
16<br />
14<br />
0,08 0,10 0,12 0,14 0,16 0,18 0,20 0,22 0,24 0,26 0,28<br />
1/sqrt(n)<br />
Florence, Mai 2004 359
Phase splitting in <strong>quantum</strong> Rabi oscillation<br />
• Timing<br />
S<br />
•Inject a coherent field<br />
•Send a first atom: Rabi oscillation <strong>and</strong> phase shift<br />
•Inject a phase tunable coherent amplitude<br />
•Send an atom in g: final amplitude read out<br />
Florence, Mai 2004 360
Phase splitting in <strong>quantum</strong> Rabi oscillation<br />
Experimental phase distributions<br />
0,80<br />
0,75<br />
29 injected photons<br />
Reference: no Rabi atom<br />
Rabi atom at 335m/s T i<br />
=32 µs<br />
Rabi atom at 200m/s T i<br />
=53 µs<br />
0,70<br />
0,65<br />
0,60<br />
0,55<br />
0,50<br />
0,45<br />
-200 -150 -100 -50 0 50 100 150<br />
Phase(°)<br />
Florence, Mai 2004 361
Phase splitting in <strong>quantum</strong> Rabi oscillation<br />
Summary of results<br />
335 m/s 200 m/s<br />
40<br />
40<br />
S g<br />
(φ)<br />
30<br />
35<br />
30<br />
35<br />
0,7<br />
0,6<br />
0,5<br />
25<br />
20<br />
-150 0 150<br />
φ (degrees)<br />
0 150 15 20<br />
φ (degrees)<br />
25<br />
n<br />
Auffèves et al. PRL 91, 230405 (2003)<br />
Florence, Mai 2004 362
Phase (degrees)<br />
Phase splitting in <strong>quantum</strong> Rabi oscillation<br />
Observed phase versus theoretical phase<br />
60<br />
40<br />
20<br />
4<br />
2<br />
0<br />
β y<br />
0<br />
-20<br />
-40<br />
-2<br />
-4<br />
0 2 4 6 8<br />
β x<br />
-60<br />
15 20 25 30 35 40 45 50 55 60<br />
Φ + (degrees)<br />
Large Shrödinger cat states (up to 40 photons separation)<br />
Florence, Mai 2004 363
Selective preparation of<br />
+<br />
Ψ<br />
a<br />
Use a Stark shift pulse on the e/g transition (equivalent to a Z rotation) to<br />
+ 1<br />
prepare from +<br />
Ψ<br />
a<br />
( )<br />
Z<br />
2 e g<br />
X<br />
Rabi Fast Stark rotation pulse: for a p/2 π/2 rotation pulse:<br />
around Z axis.<br />
1 +<br />
Preparation of Ψ ( )<br />
2 e + g<br />
a<br />
Y<br />
From this time on, slow evolution only<br />
N.B. Starting from g prepares<br />
−<br />
Ψ<br />
a<br />
Florence, Mai 2004 364
Stopped Rabi oscillation<br />
1,0<br />
Rabi<br />
Z rotation<br />
0,8<br />
0,6<br />
Slow<br />
evolution<br />
Transfert<br />
0,4<br />
Evolution<br />
resumes<br />
0,2<br />
Z rotation<br />
0,0<br />
-45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45<br />
Killer (µs)<br />
Florence, Mai 2004 365
A single, slowly rotating phase component<br />
S g (φ)<br />
0,65<br />
0,60<br />
0,55<br />
0,50<br />
0,45<br />
−<br />
Ψ<br />
a<br />
+<br />
Ψ<br />
a<br />
-150 -100 -50 0 50 100 150 200<br />
φ (degrees)<br />
Florence, Mai 2004 366
Test of coherence: induced <strong>quantum</strong> revivals<br />
Initial Rabi rotation,<br />
Stark pulse (duration short<br />
compared to phase Collapse rotation).<br />
Reverse phase rotation<br />
Equivalent And to slow a Z rotation phase rotation by π<br />
Recombine field components <strong>and</strong><br />
resume Rabi oscillation<br />
Morigi et al PRA 65, 040102<br />
Florence, Mai 2004 367
Induced <strong>quantum</strong> revivals<br />
1,0<br />
Π Pulse<br />
Transfert<br />
0,8<br />
0,6<br />
0,4<br />
0,2<br />
22 µs<br />
0,0<br />
-20 -10 0 10 20 30 40 50 60<br />
Interaction time<br />
1,0<br />
1,0<br />
0,8<br />
0,8<br />
Tranfer<br />
0,6<br />
0,4<br />
0,2<br />
18.5 µs<br />
Transfer<br />
0,6<br />
0,4<br />
0,2<br />
23.5 µs<br />
0,0<br />
-20 -10 0 10 20 30 40 50 60<br />
Interaction time<br />
0,0<br />
-20 -10 0 10 20 30 40 50 60<br />
Interaction time<br />
Florence, Mai 2004 368
Induced <strong>quantum</strong> revivals<br />
0,9<br />
0,8<br />
0,7<br />
0,6<br />
Transfert<br />
0,5<br />
0,4<br />
0,3<br />
0,2<br />
0,1<br />
22 µs<br />
0,0<br />
0 5 10 15 20 25 30 35 40 45 50 55 60 65<br />
Temps effectif<br />
Transfert<br />
0,80<br />
0,75<br />
0,70<br />
0,65<br />
0,60<br />
0,55<br />
0,50<br />
transfert<br />
0,80<br />
0,75<br />
0,70<br />
0,65<br />
0,60<br />
0,55<br />
0,50<br />
0,45<br />
0,45<br />
-100 -80 -60 -40 -20 0 20 40 60 80 100<br />
0,40<br />
phase(°) -100 -80 -60 -40 -20 0 20 40 60 80<br />
phase(°)<br />
Florence, Mai 2004 369
V) Schrödinger cats <strong>and</strong> decoherence<br />
• 1) A direct study of a meter’s decoherence process in a <strong>quantum</strong><br />
measurement<br />
• 2) Breeding Schrödinger lions <strong>with</strong> resonant interaction<br />
• 3) Other applications of field homodyne detection<br />
Florence, Mai 2004 370
Phase shift <strong>with</strong> dispersive atom-field interaction<br />
• Non resonant atom: no energy exchange but cavity mode frequency shift<br />
(atomic index of refraction effect).<br />
– Phase shift of the cavity field (slower than in the resonant case)<br />
0,70<br />
0,65<br />
0,60<br />
Atom in g<br />
0,70<br />
No atom<br />
1 atom in g<br />
Atom in g<br />
0,55<br />
2 <strong>atoms</strong> in g<br />
0,50<br />
0,65<br />
0,45<br />
-150 -100 -50 0 50 100 150<br />
0,75<br />
0,70<br />
0,60<br />
No atom<br />
0,65<br />
0,60<br />
0,55<br />
No atom<br />
0,55<br />
0,50<br />
0,45<br />
-150 -100 -50 0 50 100 150<br />
0,70<br />
0,50<br />
0,65<br />
Atom in e<br />
0,60<br />
0,55<br />
Atom in e<br />
0,45<br />
0,50<br />
0,45<br />
-200 -150 -100 -50 0 50 100 150<br />
-150 -100 -50 0 50 100 150<br />
Phase (°)<br />
Opposite values for e <strong>and</strong> g<br />
Proportional to atom number<br />
Florence, Mai 2004 371
Absolute measurement of atomic detection efficiency<br />
• Histogram of field phase reveals exact atom count<br />
• Comparison <strong>with</strong> detected atom counts provides field ionization detectors<br />
efficiency in a precise <strong>and</strong> absolute way<br />
– 0.4 <strong>atoms</strong> samples:<br />
70 % detection efficiency<br />
(close to the expected<br />
optimum of 80 %)<br />
Florence, Mai 2004 372
Towards a 100% efficiency atomic detection<br />
• Inject a very large coherent field in the cavity<br />
• Send an atomic sample<br />
– Different phase shifts for e, g or no atom<br />
Im(α)<br />
φ<br />
−φ<br />
e<br />
g<br />
Re(α)<br />
• Inject homodyning amplitude<br />
– Zero amplitude for e.<br />
• Larger for no atom.<br />
• Still larger for g<br />
g<br />
e<br />
• Read final field amplitude by sending a large number of <strong>atoms</strong> in g<br />
– Final number of <strong>atoms</strong> in e proportional to photon number<br />
Florence, Mai 2004 373
Preliminary experimental results<br />
0,18<br />
Probability<br />
0,16<br />
0,14<br />
0,12<br />
0,10<br />
0,08<br />
Atom 1 prepared in g<br />
no atom<br />
1 atom in g<br />
Atom1 prepared in e<br />
no atom<br />
1 atom in e<br />
Experimental conditions:<br />
• 75 photons initially<br />
• v=200 m/s<br />
• d=50 kHz<br />
• 70 absorber <strong>atoms</strong><br />
0,06<br />
0,04<br />
0,02<br />
0,00<br />
0 10 20 30 40<br />
• detection efficiency: 87%<br />
• error probability: 0 atom detected as 1: 10% (main present limitation)<br />
– e in g: 1.6%<br />
– g in e: 3%<br />
Total number of excited <strong>atoms</strong><br />
• 100% detection efficiency <strong>with</strong>in reach <strong>with</strong> slower <strong>atoms</strong>: v=150 m/s<br />
….experiment in progress.<br />
Florence, Mai 2004 374
Structure of the lectures<br />
• I) Introduction<br />
• II) The tools of CQED<br />
• III) Experimental illustrations of fundamental <strong>quantum</strong> mechanics<br />
• IV) Quantum information <strong>with</strong> <strong>atoms</strong> <strong>and</strong> <strong>cavities</strong><br />
• V) Schrödinger cats <strong>and</strong> decoherence<br />
• VI) Cavity QED <strong>with</strong> dielectric microspheres<br />
• VII) Perspectives<br />
Florence, Mai 2004 375
CQED <strong>with</strong> silica microspheres<br />
High Q whispering gallery modes in a silica microsphere<br />
(a ~ 25-100µm)<br />
a<br />
Mode Volume : V mode<br />
~ 300 µm 3<br />
Field per photon : E ~ few kV/m<br />
(for the most confined mode)<br />
Very low losses : Γ cav<br />
/2π ~ 300 kHz<br />
Absorption limited Q = ω / Γ cav<br />
> 10 9<br />
t cav = 1µs<br />
.....Coupled to Dipole emitters<br />
N SiO 2 = 1.45<br />
Ions (Nd 3+ ; Er 3+ ) ⇒⇒⇒ cf. Poster Session<br />
Atom Chips ⇒⇒⇒ cf. Romain Long (Session V)<br />
Excitons confined in <strong>quantum</strong> dots d ~ 15 e.a 0<br />
‘Atom like system’ Γ hom<br />
/2π ~200 MHz<br />
@ 10K<br />
Florence, Mai 2004 376
Historical reference<br />
St Paul's Cathedral<br />
(Lord Rayleigh,1870)<br />
Wall<br />
Sound waves<br />
Florence, Mai 2004 377
Whispering Gallery Modes<br />
Light guided by total internal reflection<br />
at grazing incidence <strong>and</strong> resonance<br />
condition<br />
ν<br />
TE/<br />
TM<br />
n,,<br />
m<br />
-|m|=0<br />
-|m|=1<br />
θ<br />
a<br />
n=2<br />
n=1<br />
TE/TM polarization<br />
n : number of anti-nodes<br />
l : angular momentum<br />
m : azimuthal number<br />
(degenerate for a perfect sphere)<br />
Coupling zone<br />
(Evanescent wave)<br />
Florence, Mai 2004 378
Production of microspheres<br />
Preparation of the fiber:<br />
Pulling a fiber from a<br />
rod of pure silica<br />
SiO 2<br />
Production of the sphere:<br />
Melting the tip of the fiber<br />
using a 10W CO 2<br />
Laser<br />
120 µm<br />
⇒<br />
Surface Tension<br />
Spherical Shape<br />
Florence, Mai 2004 379
Excitation of WGM’s<br />
Light coupled into the sphere<br />
by frustrated total internal reflection<br />
N p = 1.75<br />
I Out<br />
ν<br />
• Incident angle<br />
θ i < θ c ≡ Arcsin(N S /N P )<br />
Tunable LD<br />
@ 780nm<br />
θ i<br />
N P<br />
• Losses due to coupling back to the prism can be<br />
adjusted through the Gap g<br />
Linewidth (MHz)<br />
300<br />
200<br />
100<br />
0<br />
0<br />
100 200 300 400<br />
Gap Shere-Prism g (nm)<br />
Florence, Mai 2004 380<br />
25%<br />
20<br />
15<br />
10<br />
5<br />
0<br />
g<br />
Coupling Rate<br />
N S<br />
N s = 1.45
The core of the experiment<br />
Florence, Mai 2004 381
Tuning Devices : Tweezers<br />
Preformed sphere is glued<br />
into stretching device<br />
Soldering of the fiber onto<br />
glass arms <strong>and</strong> then<br />
production of the sphere<br />
Florence, Mai 2004 382
Tuning WGM´s<br />
TE<br />
TM<br />
• FSR = 810 GHz (80 µm)<br />
• ν TM - ν TE = 580 GHz<br />
slope TM ≈ 1.6 slope TE<br />
• ν l, m – ν l, m-1 = 375 GHz<br />
ellipticity ~ 50 %<br />
Florence, Mai 2004 383
Maximum Tuning<br />
Tuning TM = 405 GHz = 0.5 × FSR<br />
Tuning TE = 260 GHz = 0.3 × FSR<br />
Fracture of the fiber<br />
Florence, Mai 2004 384
Reversibility <strong>and</strong> Stability<br />
Tuning is reversible : No plastic deformation observed<br />
At fixed voltage : Frequency fluctuations come<br />
from temperature fluctuations<br />
Florence, Mai 2004 385
Identifying WGM’s<br />
GM Spectrum<br />
Free Spectral Range ∆l = 1 ∆ν = c/2πNa ~ 500 GHz<br />
Radial order : ∆n = 1 ∆ν ≈ 20 FSR<br />
Polarization TE-TM ∆ν ≈ 0.7 FSR<br />
Ellipticity (e ~ 1%) ∆m = 1 ∆ν = e FSR ~ 5 GHz<br />
r p<br />
> r e<br />
Prolate Sphere<br />
e > 0<br />
r p<br />
r e<br />
• Step 1 : Pick |m|= mode<br />
• Step 2: Assign radial number<br />
1.0<br />
shift<br />
⇒ n = 1<br />
width ≤ 1% 8<br />
0.8<br />
Reflection<br />
0.6<br />
0.4<br />
|m|=<br />
|m|=-1<br />
|m|=-2<br />
|m|=-3<br />
g<br />
0.2<br />
0.0<br />
25 20<br />
15 10<br />
Frequency (GHz)<br />
5 0<br />
2 4<br />
Frequency (GHz)<br />
Florence, Mai 2004 386<br />
0<br />
6
Which experiments <br />
• Non linearity <strong>with</strong> low thresholds<br />
– Eg. Kerr bistability using silica non-linearity<br />
• Lasers <strong>with</strong> doped silica spheres<br />
– Extremely low thresholds<br />
– Efficient frequency conversion Er laser<br />
– Towards a thresholdless laser<br />
• Towards cavity QED <strong>with</strong> silica microspheres. Two routes<br />
– Atoms in the sphere’s evanescent field<br />
– Quantum dots permanently coupled to the sphere<br />
Florence, Mai 2004 387
Atom chips <strong>and</strong> microspheres<br />
• Use <strong>atoms</strong> trapped in a mesoscopic conductor cavity field <strong>and</strong> conveyed<br />
to the sphere mode<br />
• Work performed in T. Hänsch <strong>and</strong> J. Reichel group in Munich<br />
Florence, Mai 2004 388
Atom-Chip : Conveyor-Belt<br />
Florence, Mai 2004 389
Lateral Confinement<br />
• Simple Scheme<br />
• Side-wires Configuration<br />
Single wire field<br />
+<br />
External bias field<br />
= 2D Confinement<br />
Single wire field<br />
+<br />
Field from the side-wires<br />
= 2D Confinement<br />
Florence, Mai 2004 390
Longitudinal Confinement<br />
• Multiple Crossing Conductors • Modulation of the Current<br />
Atoms trapped between<br />
two local maxima<br />
of the longitudinal field<br />
Modulation of the Current<br />
<br />
Shift the potential minimum<br />
Transport the <strong>atoms</strong><br />
Florence, Mai 2004 391
The “LDC” Chip<br />
New conveyor<br />
First 2 layers Generation Conveyor :<br />
“6 Strokes 1 layer Engine”<br />
“2 Strokes Engine”<br />
Transport direction<br />
Florence, Mai 2004 392
Long Distance Transport<br />
23,5 cm in 2.9 s<br />
Average Speed = 8 cm/s<br />
Maximum Speed = 10 cm/s<br />
Florence, Mai 2004 393
Atom Touch<br />
Detection of a “single” microsphere<br />
Florence, Mai 2004 394
Quantum dots <strong>and</strong> microspheres<br />
• An artificial atom directly coupled to the sphere’s mode<br />
Florence, Mai 2004 395
Self Assembled Q-Dots<br />
(J.M. Gérard)<br />
Self Assembled isl<strong>and</strong>s of InAs embedded in GaAs<br />
4 nm<br />
Q-Dots<br />
3D Confinement leads to an atom like system<br />
20nm<br />
Mesa 4×4 µm<br />
HF selective attack<br />
250nm<br />
GaAs<br />
InAs<br />
GaAs<br />
Detected Power (fW)<br />
800<br />
600<br />
400<br />
200<br />
0<br />
950 1000 1050 1100<br />
Wavelength (nm)<br />
(MBE)<br />
Q-Dots Photoluminescence @ 300K<br />
Pump Power<br />
@850nm<br />
900<br />
7.2mW<br />
4 mW<br />
3 mW<br />
2 mW<br />
1 mW<br />
1150<br />
1200<br />
Florence, Mai 2004 396
Effects of Sample on WGM’s<br />
Gap g2<br />
Sphere-Sample<br />
GaAs<br />
GaAs<br />
Gap g1<br />
Sphere-Prism<br />
Ns<br />
Sphere<br />
Prism<br />
Line Broadening<br />
Resonance Shift<br />
i<br />
Np 4.0<br />
3.5<br />
3.0<br />
Prism SF 11<br />
2.5<br />
2.0<br />
1.5<br />
1.0<br />
0.5<br />
0.0<br />
Frequency (GHz)<br />
0<br />
Coupling rate<br />
Shift<br />
linewidth<br />
100 200 300 400 500<br />
GAP g2 SPHERE - GaAs (nm)<br />
16%<br />
14%<br />
12%<br />
10%<br />
8%<br />
6%<br />
4%<br />
2%<br />
0%<br />
Florence, Mai 2004 397
Effects of Mesa (4×4 µm) on WGM’s<br />
Frequency (MHz)<br />
Frequency (MHz)<br />
500<br />
400<br />
300<br />
200<br />
100<br />
0<br />
500<br />
400<br />
300<br />
200<br />
100<br />
0<br />
0<br />
0<br />
2<br />
5 10<br />
Position Y (µm)<br />
4 6<br />
Position Z (µm)<br />
15<br />
8<br />
Florence, Mai 2004 398<br />
6%<br />
5<br />
4<br />
3<br />
2<br />
1<br />
0<br />
6%<br />
5<br />
4<br />
3<br />
2<br />
1<br />
0<br />
Line Broadening<br />
Resonance Shift<br />
Coupling rate<br />
|m|= WGM<br />
Mesa<br />
Y<br />
Z
Q-Dots Laser<br />
Experimental Set Up<br />
Emission<br />
1000 – 1100 nm<br />
LD Probe @ 1080 nm<br />
Transmission @<br />
780nm<br />
I Out<br />
ν<br />
PD<br />
PD<br />
Prism SF11<br />
LD @ 780 nm<br />
Pump<br />
Linewidth ~ 700MHz<br />
Coupling rate ~10%<br />
Gap g 2 Sphere-Sample<br />
Sphere<br />
Gap g 1 Sphere-Prism<br />
Mesa<br />
Q-Dots<br />
PZT controlled<br />
X-Y-Z motion<br />
Florence, Mai 2004 399
Q-Dots Laser at Room temperature<br />
30<br />
Detected Power (pW)<br />
25<br />
20<br />
15<br />
10<br />
5<br />
TE<br />
TM<br />
0<br />
0.0 0.1 0.2 0.3 0.4 0.5<br />
Absorbed Pump Power (mW)<br />
Power at threshold ~ 200 µW<br />
0.6<br />
Active Q-dots ~ 10 4<br />
Florence, Mai 2004 400
Structure of the lectures<br />
• I) Introduction<br />
• II) The tools of CQED<br />
• III) Experimental illustrations of fundamental <strong>quantum</strong> mechanics<br />
• IV) Quantum information <strong>with</strong> <strong>atoms</strong> <strong>and</strong> <strong>cavities</strong><br />
• V) Schrödinger cats <strong>and</strong> decoherence<br />
• VI) Cavity QED <strong>with</strong> dielectric microspheres<br />
• VII) Perspectives<br />
Florence, Mai 2004 401
Two main directions<br />
• A two-cavity experiments for non-locality, decoherence <strong>and</strong> <strong>quantum</strong><br />
information<br />
• A Rydberg atom chip experiment for deterministic preparation <strong>and</strong><br />
manipulation of cold trapped Rydberg <strong>atoms</strong><br />
Florence, Mai 2004 402
Two main directions<br />
• A two-cavity experiments for non-locality, decoherence <strong>and</strong> <strong>quantum</strong><br />
information<br />
• A Rydberg atom chip experiment for deterministic preparation <strong>and</strong><br />
manipulation of cold trapped Rydberg <strong>atoms</strong><br />
Florence, Mai 2004 403
A two-cavity experiment<br />
• Rydberg <strong>atoms</strong> <strong>and</strong> superconducting <strong>cavities</strong>:<br />
– Towards a two-cavity experiment<br />
• Creation of non-local mesoscopic Schrödinger cat states<br />
– Non-locality <strong>and</strong> decoherence (real time monitoring of W<br />
function)<br />
• Complex <strong>quantum</strong> information manipulations<br />
– Quantum feedback<br />
– Simple algorithms<br />
– Three-qubit <strong>quantum</strong> error correction code<br />
Florence, Mai 2004 404
C<br />
Teleportation of an atomic state<br />
R 2<br />
P b<br />
beam 3<br />
R R<br />
P 3<br />
a<br />
1<br />
C 1<br />
C 2<br />
1'<br />
1<br />
2<br />
D a<br />
beam 2<br />
D b<br />
A<br />
3'<br />
3<br />
0<br />
beam 1<br />
Davidovich et al,<br />
PHYS REV A 50 R895 (1994)<br />
B<br />
EPR pair<br />
D c<br />
• This scheme works for massive particles<br />
• Detection of the 4 Bell states <strong>and</strong> application of the<br />
"correction" to the target is possible using a C-Not gate<br />
(beam 2 <strong>and</strong> 3)<br />
• The scheme can be compacted to 1 cavity <strong>and</strong> 1 atomic beam<br />
Florence, Mai 2004 405
Implementation of 3 qubit error correction<br />
0<br />
0<br />
0<br />
R<br />
encoding<br />
S<br />
R R ’<br />
R<br />
R’<br />
R<br />
R’<br />
Error<br />
R’<br />
decoding<br />
S<br />
σ z2<br />
σ z3<br />
Détection<br />
Correction Correction<br />
error detection<br />
α 0 + β 1<br />
Détection<br />
Ramsey π/2 pulses<br />
Error<br />
encoding <strong>and</strong> decoding: preparation of a GHz triplet<br />
all the tools exist!<br />
Florence, Mai 2004 406
Quantum feedback<br />
• Preserve a Schrödinger cat by giving him “<strong>quantum</strong> food”<br />
• Use a <strong>atoms</strong> <strong>and</strong> QND arrangement to detect cat parity jum<br />
• A cat jump prepares a single photon in a second cavity. Used to excite a<br />
feedback atom which gives back the photon to the cat<br />
• A parity preservation scheme which makes the cat coherence live much<br />
longer than the natural decoherence time<br />
• Fortunato et al PRA 60, 1687<br />
Florence, Mai 2004 407
Feedback loop<br />
• C quasi resonant <strong>with</strong> e/g <strong>and</strong> C’ <strong>with</strong> g/i<br />
• Probe (QND) atom exits in g when parity jump<br />
• Feedback atom<br />
– Promoted to e when in g by a microwave pulse in R 2<br />
– Undergoes an adiabatic passage in C to restore the lost photon<br />
Florence, Mai 2004 408
Feedback efficiency<br />
Florence, Mai 2004 409
Towards feedback<br />
• Needs a deterministic source of <strong>atoms</strong><br />
• A simpler version <strong>with</strong> two modes of the same cavity <strong>and</strong> no deterministic<br />
atom source<br />
– Zippili et al PRA 67 052101<br />
• Realistic preservation of cats <strong>with</strong> about one photon<br />
• Very good preservation of single photon Fock state<br />
Florence, Mai 2004 410
New non-locality explorations<br />
• Use a single atom to entangle two mesoscopic fields in the cavity<br />
– A non-local Schrödinger cat or a mesoscopic EPR pair<br />
– Easily prepared via dispersive atom-cavity interaction<br />
Florence, Mai 2004 411
Mesoscopic Bell inequalities<br />
• A Bell inequality form adapted to this situation<br />
• Here, Π is the parity operator average. Dichotomic variable for which the<br />
Bell inequalities argument can be used (transforms the continuous<br />
variable problem in a spin-like problem)<br />
• Maximum violation for parity entangled states:<br />
Florence, Mai 2004 412
Bell inequalities violation<br />
• Optimum Bell signal versus γ<br />
• A compromise between violation amplitude <strong>and</strong> decoherence: γ²=2<br />
Florence, Mai 2004 413
Probing the Wigner function<br />
• A second atom to read out both <strong>cavities</strong> (same scheme as for single<br />
mode Wigner function)<br />
Florence, Mai 2004 414
A difficult but feasible experiment<br />
• Bell signal versus time T c =30 <strong>and</strong> 300 ms<br />
Florence, Mai 2004 415
Two main directions<br />
• A two-cavity experiments for non-locality, decoherence <strong>and</strong> <strong>quantum</strong><br />
information<br />
• A Rydberg atom chip experiment for deterministic preparation <strong>and</strong><br />
manipulation of cold trapped Rydberg <strong>atoms</strong><br />
Florence, Mai 2004 416
What we want<br />
• Long State Lifetime (> 30 ms in free space)<br />
• Control of External Degrees of Freedom<br />
• Single Rydberg Atom Excitation on Dem<strong>and</strong><br />
• Integrated Atom-Chip<br />
• Coherence preserving scheme<br />
Florence, Mai 2004 417
Inhibition of Spontaneous Emission<br />
Principle<br />
F 0<br />
No available<br />
Emitted<br />
zmode<br />
Perfect, infinite<br />
photon must<br />
→ xemissionmirrors<br />
be σ polarized<br />
inhibited<br />
d < λ/2<br />
τ → ∞<br />
Limiting Factors<br />
F<br />
τ >> τ 0<br />
• Imperfect <strong>and</strong> finite<br />
mirrors<br />
• Angle between F 0 <strong>and</strong> z<br />
• Residual Thermal Field<br />
Hulet, Hilfer, Kleppner, PRL,<br />
55, 20, 2137 : Factor 20<br />
1986: Jhe, Anderson, Hinds,<br />
Meschede, Moi, Haroche,<br />
PRL, 58, 666: Factor 13<br />
Florence, Mai 2004 418
Rydberg Atom Trapping<br />
C<strong>and</strong>idate Techniques<br />
• Magnetic Trap – Zeeman Effect<br />
• Electric Trap – Stark Effect<br />
Better suited to<br />
Inhibition of<br />
Spontaneous<br />
Emission scheme<br />
• Ponderomotive Trap – Electron Micro-motion<br />
Required Laser Dutta, Intensity Guest, Feldbaum, (200 Wcm Walz-Flannigan, -2 ) incompatible Raithel <strong>with</strong><br />
PRL, 85, 26, 5551<br />
Cryogenic Environment<br />
Florence, Mai 2004 419
Electric Dipole Trap<br />
Quadratic Stark Effect ~ 2.2 MHz/(V/cm) 2<br />
E ~ - α |F| 2<br />
High Field Seeker<br />
Energy E<br />
Electric Field |F|<br />
n = 51<br />
n = 50<br />
Maxwell<br />
Maximum of |F|<br />
Dynamic (Paul-like) Trap<br />
Florence, Mai 2004 420
Trap Geometry<br />
U V(t) U V(t) U V(t)<br />
+ - + - + -<br />
1mm F 0 = 30 V/cm<br />
(< λ/2) 1mm<br />
z<br />
x<br />
U = 1.5 V<br />
V(t) = 0.5 V . Cos(ω V t)<br />
ω V = 20,000 s -1<br />
-U V(t) -U V(t) -U V(t)<br />
+ - + - + -<br />
Florence, Mai 2004 421
Trapping Simulation<br />
Atomic Trajectories<br />
•T load = 300 µK<br />
• ∆x load = 5 µm<br />
• ω V = 20,000 s -1<br />
• Trapping volume ~ (100µm) 3<br />
z (mm)<br />
80<br />
40<br />
0<br />
-40<br />
Macro-motion<br />
-80<br />
-80 -40 0 40 80<br />
Micromotion<br />
x (mm)<br />
Trapping Efficiency<br />
• ω V < ω c → atom escapes<br />
along anti-trapping axis<br />
• ω V >> ω c → field variations<br />
average to zero: no trapping<br />
1,0<br />
0.5<br />
ω c<br />
0<br />
ω V (s -1 )<br />
0 10000 20000<br />
Florence, Mai 2004 422
Electric Field Tilt<br />
F 0 (x,y,z)<br />
F loc<br />
z<br />
F loc<br />
F loc<br />
ϑ(t)<br />
F loc<br />
Trapping Region<br />
Results<br />
ϑ < 10 -2 τ→10 4 τ 0<br />
• Not limiting factor<br />
• τ ~ 1s envisageable<br />
Florence, Mai 2004 423
Micro-Trap<br />
U – V(t)<br />
U + V(t)<br />
U – V(t)<br />
U + V(t)<br />
U – V(t)<br />
1mm<br />
z 100µm<br />
U = 1.5 V<br />
100µmTrapping x Region<br />
F<br />
V = 0.5 V . Cos(ωt)<br />
0 = 30 V/cm<br />
ω = 150 000 s -1<br />
-U + V(t)<br />
• Greater Confinement<br />
• Surface Interactions<br />
• Integration<br />
• Extension to Conveyor Belt, Guide…<br />
Florence, Mai 2004 424
A Tighter Trap<br />
Same field variations + Spatial scale / 10 → Confinement x 10<br />
Atomic Trajectories Trapping Efficiency<br />
•T load = 300 µK<br />
• ∆x load = 5 µm<br />
• ω V = 150 000 s -1<br />
• Trapping volume ~ (10µm) 3<br />
• z symmetry broken<br />
1,0<br />
• ω c x 10<br />
• Trapping less perfect…<br />
• …but still very good<br />
z (µm)<br />
5<br />
0<br />
-5<br />
-10<br />
x (µm)<br />
-10 -5 0 5 10<br />
ω (s -1 )<br />
0 10000 20000<br />
Florence, Mai 2004 425<br />
0.5<br />
0<br />
ω c
Rydberg Atom Source<br />
Dipole Blockade Lukin et al, PRL 87, 037901<br />
∆x ~ 1µm<br />
~ 1GHz<br />
ω<br />
| N-2:g ; 2: ><br />
| N-1:g ; 1: ><br />
ω<br />
Rydberg Excitation Laser<br />
ω<br />
| N :g ; 0: ><br />
One <strong>and</strong> Only One Circular Rydberg Atom Excited<br />
Florence, Mai 2004 426
Ground-State Atoms Trapping<br />
Requirements:<br />
• Highly Confining (Dipole (∆x < 10µm) Blockade)<br />
• Close to Surfaces (Surface-Trap (Micro-Trap, Dipole- Distance<br />
Surface R trap < 100µm) Interactions)<br />
• Compatible <strong>with</strong> Cryogenics<br />
(Dissipation (Rydberg Stability) < 1mW)<br />
• Integrable (On-Chip Wires <strong>and</strong> Electrodes)<br />
Hänsel et al, Nature 413, 498<br />
B bias I ~ 1A SiO 2 or Sapphire substrate<br />
Superconducting Niobium Wires<br />
100x1 µm 2 wide<br />
Florence, Mai 2004 427
Filling the Magnetic Trap<br />
Cryogenic vacuum → no background pressure → external<br />
source<br />
2D MOT – High Flux Atomic Jet<br />
300K<br />
1K<br />
Cryostat<br />
Magnetic Trapping Region<br />
Jet Extracted from<br />
Atom Cloud<br />
2D MOT<br />
mm<br />
6<br />
4<br />
2<br />
0 2 4<br />
mm<br />
Characteristics of our jet<br />
•Flux = 10 7 s -1<br />
• Divergence = 10 mRad<br />
Florence, Mai 2004 428
Coherence preservation scheme<br />
• Use a microwave dressing to equalize e <strong>and</strong> g Stark polarizabilities<br />
Florence, Mai 2004 429
Coherence preservation scheme<br />
• Residual phase drift almost linear <strong>with</strong> time<br />
• Can be corrected by an echo technique<br />
Florence, Mai 2004 430
Very long coherence times<br />
Coherence preserved for seconds or minutes!!!<br />
Florence, Mai 2004 431
An extremely promising scheme for<br />
• Spontaneous emission inhibition studies<br />
• Atom-surface <strong>and</strong> atom-atom dipole-dipole interaction studies<br />
• Cavity QED <strong>with</strong> transmission line resonators<br />
• Quantum information processing<br />
• Coupling of Rydberg <strong>atoms</strong> to mesoscopic circuits<br />
Florence, Mai 2004 432
The team<br />
PhD<br />
• Frédérick .Bernardot<br />
• Paulo Nussenzweig<br />
• Abdelhamid Maali<br />
• Jochen Dreyer<br />
• Xavier Maître<br />
• Gilles Nogues<br />
• Arno Rauschenbeutel<br />
• Patrice Bertet<br />
• Stefano Osnaghi<br />
• Alexia Auffeves<br />
• Paolo Maioli<br />
• Tristan Meunier<br />
• Sébastien Gleyzes<br />
• Philippe Hyafil *<br />
• Jack Mozley *<br />
Post doc<br />
• Ferdin<strong>and</strong> Schmidt-Kaler<br />
• Edward Hagley<br />
• Christof Wunderlich<br />
• Perola Milman<br />
Colaboration<br />
• Luiz Davidovich<br />
• Nicim Zagury<br />
• Wojtek Gawlik<br />
Permanent<br />
• Gilles Nogues *<br />
• Michel Brune<br />
• Jean-Michel Raimond<br />
• Serge Haroche<br />
*: atom chip team<br />
Florence, Mai 2004 433
References (1)<br />
• Strong coupling regime in CQED experiments:<br />
– F. Bernardot, P. Nussenzveig, M. Brune, J.M. Raimond <strong>and</strong> S. Haroche. "Vacuum Rabi Splitting<br />
Observed on a Microscopic atomic sample in a Microwave cavity". Europhys. lett. 17, 33-38<br />
(1992).<br />
– P. Nussenzveig, F. Bernardot, M. Brune, J. Hare, J.M. Raimond, S. Haroche <strong>and</strong> W. Gawlik.<br />
"Preparation of high principal <strong>quantum</strong> number "circular" states of rubidium". Phys. Rev. A48,<br />
3991 (1993).<br />
– M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J. M. Raimond <strong>and</strong> S. Haroche:<br />
"Quantum Rabi oscillation: a direct test of field quantization in a cavity". Phys. Rev. Lett. 76,<br />
1800 (1996).<br />
Florence, Mai 2004 434
References (2)<br />
• QND measurement in microwave CQED experiments:<br />
– M. Brune, S. Haroche, V. Lefevre-Seguin, J.M. Raimond <strong>and</strong> N. Zagury: "Quantum nondemolition<br />
measurement of small photon numbers by Rydberg-atom phase sensitive detection",<br />
Phys. Rev. Lett. 65, 976 (1990).<br />
– M.Brune, S. Haroche, J.M. Raimond,L. Davidovich <strong>and</strong> N. Zagury. "Manipulation of photons in<br />
a cavity by dispersive atom-field coupling: QND measurement <strong>and</strong> generation of "Schrödinger<br />
cat"states". Phys Rev A45, 5193, (1992).<br />
– S. Haroche, M. Brune <strong>and</strong> J.M. Raimond. "Manipulation of optical fields by atomic<br />
interferometry: <strong>quantum</strong> variations on a theme by Young".Appl. Phys. B, 54, 355, (1992).<br />
– S. Haroche, M. Brune <strong>and</strong> J.M. Raimond. "Measuring photon numbers in a cavity by atomic<br />
interferometry: optimizing the convergence procedure". Journal de Physique II 2, 659<br />
Florence, Mai 2004 435
References (3)<br />
• Gates: QPG or C-Not, algorithm:<br />
– M. Brune et al., Phys. Rev. Lett, 72, 3339(1994).<br />
– Q.A. Turchette et al., Phys. Rev. Lett. 75, 4710 (1995).<br />
– C. Monroe et al., Phys. Rev. Lett. 75, 4714 (1995).<br />
– A. Reuschenbeutel et al. submitted PRL. G. Nogues et al. Nature 400, 239 (1999).<br />
– S. Osnaghi, P. Bertet, A. Auffeves, P. Maioli, M. Brune, J.M. Raimond <strong>and</strong> S. Haroche, Phys.<br />
Rev. Lett. 87, 037902 (2001)<br />
– F. Yamaguchi, P. Milman, M. Brune, J-M. Raimond, S. Haroche: "Quantum search <strong>with</strong> twoatom<br />
collisions in cavity QED", PRA 66, 010302 (2002).<br />
• Q. memory:<br />
– X. Maître et al., Phys. Rev. Lett. 79, 769 (1997).<br />
– Atom EPR pairs:<br />
– CQED: E. Hagley et al., Phys. Rev. Lett. 79, 1 (1997).<br />
– Ions: Q.A. Turchette et al., Phys. Rev. Lett. 81, 3631 (1998).<br />
• Teleportation:<br />
– L. Davidovich, N. Zagury, M. Brune, J.M. Raimond <strong>and</strong> S. Haroche. "Teleportation of an<br />
atomic state between two <strong>cavities</strong> using non-local microwave fields". Phys Rev A50, R895<br />
(1994).<br />
Florence, Mai 2004 436
References (4)<br />
• Reviews on CQED<br />
Florence, Mai 2004 437
References (5)<br />
• A few useful textbooks<br />
Florence, Mai 2004 438
• THANK YOU …..<br />
Florence, Mai 2004 439