quantum games with atoms and cavities - Electrodynamique ...

Entanglement, complementarity and decoherence: 

quantum games with atoms and cavities 

J.M. Raimond 

Laboratoire Kastler Brossel, 

ENS, UPMC and IUF 

Florence, Mai 2004 Support:JST (ICORP), EC, CNRS, UMPC, IUF, CdF 1

A century of quantum mechanics 

An enormous range of applications, from superstrings to universe 

(a) (b) (c) 

10 -35 m 

10 -15 m 

10 -10 m 

(d) (e) (f) 

10 -8 m 

1 m 

10 8 m 

(g) 

(h) 

10 20 m 

10 26 m 

Florence, Mai 2004 2


One of the most precise theories so far. 

• QED: quantum description of electromagnetic interactions. 

– Spectrum of hydrogen atom at 10 -12 level 

– Gyromagnetic ratio of a free electron 

• Standard model: a unified description of all interaction (but gravitation) 


A wealth of applications: 


– Computers and the transistor revolution 

From the ENIAC to the laptop 



• Nuclear magnetic imagers for medicine 



• Lasers 



• Ultrastable clocks and the GPS system 



• Tailoring structures at the atomic level 



• And yet an intriguing theory. 

– A microscopic world that defies our “classical” intuition. 

• These lectures will be devoted to the theoretical investigation and the 

experimental exploration of the most ‘bizarre’ aspects of the quantum 

world. 

– Superposition principle 

– Complementarity 

– Quantum entanglement and non locality. 

– Quantum information processing 

– Quantum decoherence 


The “strangeness” of the quantum 

• Superposition principle and quantum interferences 

– The sum of quantum states is yet another possible state 

– A system “suspended” between two different classical realities 

D 

d 

a 

I 

Ψ =Ψ 

1 

+Ψ2 

= I + 2Re Ψ Ψ 

D 

a = λ 

d 

0 1 2 

– Feynman: Young’s slits experiment contains all the mysteries of the 

quantum 



• Young interferences with electrons (A. Tonomura, 1989) 



• Quantum Young interferences with atoms (Shimizu 1992) 



• Complementarity (From Einstein-Bohr at the 1927 Solvay congress) 

– Microscopic slit: set in motion when deflecting particle. Which path 

information and no fringes 

– Macroscopic slit: impervious to interfering particle. No which path 

information and fringes 

– Wave and particle are complementary aspects of the quantum object. 



• Complementarity and Heisenberg uncertainty relations 

Particle's momentum p=h/λ 

Momentum transfer ∆P=pd/D 

Measure screen momentum within ∆P: 

∆x=a 

Position uncertainty ∆X=h/∆P 

d 

D 

Position uncertainty= interfrange. 

Complete washing out of fringes 

Localisation and wave behavior are incompatible 


• No cloning theorem 


– It is impossible to produce two exact copies of an arbitrary quantum 

states (violates unitarity of Schrödinger evolution): No quantum fax 


• Entanglement 


– Two systems after an interaction described by a single global state 

– No system has a-well defined state on his own 

• A measurement performed on one system affects the state of the 

other 

• Quantum correlations irrespective of the distance between 

entangled systems 

– At the heart of quantum non-locality 

– Einstein did not like that… 

• and he was wrong (Bell inequalities violation) 


• Bell inequalities. 


– Two observers (Alice and Bob) share a pair of two level particles 

(levels 0 and 1 eg spin ½) in one of four “Bell states” 

– Spin operators (more in further lectures) 

– Essential property 



• Bell inequalities (cont’d) 

– Einstein argument 

• Alice can tell with certainty what value she gets for a measurement 

of any σ u by asking Bob to make the measurement 

• An “element of reality” is associated to any component of Alice’s 

spin in XOZ plane 

• Classical vision: any experimental result is predetermined at pair 

preparation time (“hidden variable apporoach”) 

– CHSH version of Bell inequalities 

0 

a 

EPR 

b 

0 

1 a' Alice source 

Bob 

b' 

1 




– If hidden variables are right: 

– But quantum mechanics predicts: 

With 

We get: 




3 

2 

1 

S Bell 

−2 2 ≤S 

≤2 

2 

0 

-1 

-2 

-3 

0,0 0,5 1,0 θ/π 

1,5 2,0 

• Possible experimental test of quantum mechanics versus local realism. 

• A major step in our understanding of quantum mechanics 


A severe test 


• Correlations between two polarization-entangled photons 400 m apart. 

• Correlation signal S


• Quantum/classical limit 

– No quantum superpositions at 

macroscopic scale 

– The "Schrödinger cat" 

• Decoherence 

1 

2 

( ) 

+ ⇔ 

Environment 

– A macroscopic system is strongly 

coupled to a complex 

environment 

– No entangled states neither. 

We only observe a very small fraction of 

all possible quantum states 

WHY 

– In all models, this coupling 

• leaves only a few states 

intact (preferred basis) 

• destroys very rapidly the 

quantum superpositions of 

these states 

Decoherence 


Main features of decoherence 

Very fast process 

A simple decoherence model 

superposition lifetime 

= 

relaxation time 

separation betweenstates 

- 

0 

∆x 

+ 

Depends upon the initial quantum state 

(distance between states or 

"macroscopicity" parameter) 

Needle in a superposition of positions 

Background gas. Particles with de 

Broglie wavelength λ 0 

. 

Not a trivial relaxation mechanism 

(but explained by standard relaxation 

theory for simple models) 

When ∆x>λ 0 

, the first collision destroys 

the coherence (Heisenberg 

microscope: the environment "knows" 

the needle position) 

Extraordinarily fast decoherence 

Strong link with complementarity: 

environment acquires "which path" 

information 


The importance of decoherence 

Measurement theory 

Applications of quantum weirdness 

Decoherence plays an essential role in 

quantum measurement: 

Prevents meters from being in a 

quantum superposition! 

1 

2 

( + ) 

Only statistical mixtures of meter states, 

corresponding to exclusive classical 

probabilities 

1 

2 

- 

0 

+ 

- 

( + 

) 

- 

0 

+ 

- 

0 

+ 

"Stable" states and hence measured 

observable determined by relaxation 

dynamics 

0 

- 

+ 

0 

+ 

- 

0 

+ 

Manipulate complex entangled states for 

quantum information processing or 

computing 

Decoherence affects these states. 

Very fast loss of quantum information 

A terrible obstacle to these applications 

NB: decoherence does not prohibit 

macroscopic quantum states (BEC, 

superfluids): a single quantum state 


Why explorations of the quantum world 

A fundamental interest 

Promising applications 

Better understanding of quantum 

postulates 

• Superposition 

• Measurement 

• Entanglement and non-locality 

Exploration of the quantum-classical 

boundary 

Realize some of the gendankenexperiments 

used by the founding 

fathers of quantum mechanics. 

«we never experiment with just one electron 

or atom or (small) molecule. In thoughtexperiments 

we sometimes assume that we 

do; this invariably entails ridiculous 

consequences…. » 

Use quantum weirdness to realize new 

functions for information transmission 

or processing 

From bits (0 or 1) to qubits (|0> and |1>) 

• Quantum cryptography 

• Quantum teleportation 

• Quantum information processing 

• Quantum computing 

All rely on sophisticated quantum 

entanglement manipulations 

(Schrödinger British Journal of the Philosophy of 

Sciences, Vol 3, 1952) 


Quantum cryptography 

Key distribution 

Practical realization 

With photons 

Two operators (Alice and Bob) share an 

entangled pair. 

Their measurements are correlated. 

Measure same observable: a random 

but common bit. 

Eavesdropper: measurements and 

unavoidable perturbation of the 

correlations (no more Bell inequalities 

violation) 

Detect any eavesdropper 

Unconditionally secure key 

• Optical telecom fibers; distances up 

to 60 km 

• Free space: in principle possible for 

satellite-ground communication 

"Commercial" realizations available 

Long haul communication: lack of a 

"quantum repeater" 


Quantum teleportation 

Principle 

Experimental realizations 

Photons polarization states 

Innsbruck+Rome 

Quantum fluctuations of a laser 

field 

Caltech 

A very beautiful illustration of 

quantum non-locality 

Transmit exactly an arbitrary 

quantum state from one station to 

another 

Impossible with measurements 

Use quantum non-locality 

No matter creation, no superluminal 

propagation 

Quite far from Star-Trek !! 



• Principle of quantum teleportation 

|ψ 〉 

Bell 

Measurement 

2 classical bits 

U 

u 

a 

b 

|ψ 〉 

EPR Source 

• Bell states basis 


– Initial state 


– Measurement of u and a in the “Bell basis” projects b on a state 

differing of the initial one by a trivial unitary transformation. 

– Knowing the result of Alice measurement bob can render the initial 

state. 

– Before Alice’s measurement, Bob has a statistical mixture of all four 

states i.e. an equal mixture of 0 and 1. 

– No matter creation, no superluminal communication. 

– A splendid illustration of quantum non-locality. 


Quantum computing 

From bits to qubits 

Quantum parallelism 

Classical computer: bits 

0 or 1 

Quantum computer: qubits 

Two-level system 

states |0> and |1> 

A qubit can be in a state superposition 

A quantum computer with an n qubits 

register can manipulate a quantum 

superposition of all numbers and 

perform simultaneously 2 n 

calculations ! 

Quantum mechanics linearity: 

State superpositions available 

1 

2 

( 0 + 1 ) 

is a possible state for a qubit 

Exponentially more efficient than a 

classical computer 

( + ) 

Makes easy a few difficult problems 

• Shor : factorization 

• Grover : search in an unsorted 

database 

Florence, Mai 2004 30 

1 

2

Organisation of a quantum computer 

Quantum gates 

Any operation can be decomposed 

in a set of elementary operations 

p qubits 

(1 or 2 qubits) 

U 

Any quantum algorithm can be 

realized with a network of 

quantum gates 

Universal gates: 

p qubits 

A set of one or two gates allowing 

the realization of any network 

Some quantum gates 

1 qubit: arbitrary rotation 

iψ 

⎛ cosϕ 

e sinϕ 

⎞ 

U ( ϕψ , ) = ⎜ −iψ 

⎟ 

⎝−e 

sinϕ 

cosϕ 

⎠ 

2 qubits: quantum phase gate 

0,0 ⎯⎯→ 0,0 

0,1 ⎯⎯→ 0,1 

1, 0 ⎯⎯→ 1, 0 

i 

1,1 ⎯⎯→ 1,1 

e Φ 

Conditional dynamics 


Quantum entanglement manipulations 

• Quantum non locality, complementarity, computation all boil down to 

sophisticated quantum entanglement manipulations ! 

• Not easy experimentally. Criteria to be met 

– Two-level systems. 

– Individually addressed 

– Prepared in a given state (initialization) 

– Final state completely analyzable (read out) 

– Strong common interaction to produce entanglement 

– Weak coupling to outside world to limit decoherence 

• Two last requirements clearly incompatible 

– Few suitable systems. 

– Even fewer achieved entanglement control and manipulation 


Tools for fundamental quantum mechanics studies 

• Nuclear magnetic resonance 

– Controlled quantum states evolution for nuclear spins 

– Intramolecular interaction lead to spin/spin entanglement 

– Various applications to quantum computing 

– Thermodynamic ensembles of large number of molecules. No 

individual quantum systems 



• Entangled “twin” photons 

Correlation 

– Photon pairs naturally produced in an entangled state 

– Easy manipulation and transport of individual photons 

– Widely used for non-locality tests, quantum cryptography and 

quantum teleportation 

– Weak photon-photon interaction. Further entanglement processing 

difficult. No photon-photon universal quantum gate. 



• Cold atoms and BE condensates 

– Atoms in optical lattices: control of individual atoms (one per site) 

– Controlled collisions and quantum gates 

– Not yet individual access to atoms 


• Ions in traps 


– Single addressable long-lived quantum systems 

– Two ion quantum gates 

– Simple algorithms, teleportation. 

– One of the most promising systems for few qubits implementations 

– Great experimental difficulties 


• Mesoscopic circuits 


– Long-lived two level systems (artificial atoms) 

– Two qubits quantum gates 

– Promising for ‘large scale’ integration 

– Decoherence not well understood 



• Cavity quantum electrodynamics 

– Realizes the simplest matter-field system: a single atom coherently 

coupled to a few photons in a single mode of the radiation field, 

sustained by a high quality cavity. 

– Perfect test bench for fundamental quantum behaviors 

– Can be used for proof of principle demonstrations of quantum logics 

– Not really scalable to large scale architectures 

• Comes in two regimes 

– Weak coupling: radiative properties modifications 

– Strong coupling: atom-field interaction overwhelms dissipative 

processes (focus here) 

• Comes in two flavours 

– Optical CQED 

– Microwave CQED 


A short history of cavity QED 

• The genesis 

• The strong coupling regime 

Purcell 46: spontaneous emission rate 

modification for a spin in a resonant 

circuit 

• The beginning 

Drexhage (70's) : Spontaneous emission 

spatial pattern modification for a 

molecule near a mirror 

• The weak coupling regime 

One- and two-photon micromasers 

(Munich, ENS,85-90) 

Vacuum Rabi splitting (Kimble, 92) 

Quantum Rabi oscillations 

• Using strong coupling for 

entanglement manipulations 

In progress 

• The "industrial" age 

Spontaneous emission acceleration 

(Goy, 83) 

Spontaneous emission inhibition 

(Kleppner, 85) 

Observed since then on many systems 

Use spontaneous emission modification 

for light emitting devices (VCSEL's 

and LED's) 


Optical CEQD 

• A single atom in a high Q optical cavity (Fabry Perot) 

– Strong coupling regime 

– Easy interface with propagating photons (quantum communication) 

– Large atom-field forces: atom trapping with single photons 

– Single atom lasers and single photon deterministic sources 

– Fast pace and difficult control of entanglement. 

• Caltech, Munich, Stony brook,… 


Microwave CQED 

• A single Rydberg atom interacting with a superconducting cavity 

– With circular atoms: both atoms and field long-lived 

– Very strong coupling regime 

– Fundamental quantum mechanics illustrations 

– Complex entanglement manipulations 

• Munich and ENS 


‘Programme’ of these lectures 

• A detailed description of microwave CQED experiments 

– An opportunity to review many basic quantum concepts 

• Complementarity, decoherence, entanglement 

– A good introduction to basic quantum optics techniques 

• Quantum fields, Wigner representation, relaxation theory, quantum 

Monte Carlo trajectories, dressed atom… 

– An opportunity to review quantum information concepts 

• Quantum computing, algorithms, quantum error correction 

codes… 


An “appetizer” chapter 

• A brief survey of CQED with circular Rydberg atoms and superconducting 

cavities 

• An experiment on complementarity 

• A direct study of the decoherence process 


CQED with circular Rydberg atoms and superconducting cavities 


Circular Rydberg atoms 

High principal quantum number 

Maximal orbital and magnetic quantum 

numbers 

• Long lifetime 

• Microwave two-level transition 

• Huge dipole matrix element 

• Stark tuning 

• Field ionization detection 

– selective and sensitive 

• Velocity selection 

– Controlled interaction time 

– Well-known sample position 

Atoms individually addressed 

(centimeter separation between atoms) 

Full control of individual transformations 

51 (level e) 

51.1 GHz 

50 (level g) 

Complex preparation (53 photons ! ) 

Stable in a weak directing electric field 

Single atom preparation: brute force ! 


Superconducting cavity 

Design 

Highly polished niobium Mirrors 

• Open Fabry Perot cavity with a 

"photon recirculating ring" 

• Compatible with a static electric field 

(circular state stability and Stark 

tuning) 

• Very sensitive to geometric quality of 

mirrors 

Cavity Damping time: 1 ms 

Field energy (db) 

29 

28 

27 

26 

25 

24 

-2 0 2 4 6 8 10 


time (ms)

General scheme of the experiments 

Rev. Mod. Phys. 73, 565 (2001) 


From Dream to Reality 

Atoms preparation 

detection 

lasers 

Atomic 

beam 


An object at the quantum/classical boundary 

Coherent field in a cavity 

From quantum to classical 

• State produced by a classical source 

in the cavity mode 

• Small field: 

α 

α = e ∑ 

2 

−α 

/2 

n 

n n! 

– |n>: photon number state 

– Defined by complex amplitude α 

• A picture in phase space (Fresnel 

plane) 

n 

Im α 

| α| 

Φ 

n 

1 

∆Φ 

Re α 

2 

= α ∆ n= 

n 

∆N∆Φ≈1 

– Large quantum fluctuations. A 

field at the single-photon level is 

a quantum object 

• Large field 

– Small quantum fluctuations. A 

field with more than 10 photons is 

almost a classical object. 


a 

φ

Resonant atom-cavity interaction 


Initial state |e,0> 

|3> 

|2> 

e> 

g> 

|1> 

|0> 

Atom Cavity 

Ω 

|3> 

|2> 

|e> 

|1> 

|g> |0> 

Atom Cavity 

P e 

(t) 

0.8 

0.6 

0.4 

0.2 

0.0 

time ( µ s) 

0 30 60 90 

Oscillatory Spontaneous emission and strong coupling regime 


Rabi oscillation in a small coherent field 

1.0 

P e 

(t) 

0.5 

0.0 

Time ( µs) 

• A more complex signal 

0 30 60 90 

• π/2 pulse possible for any cavity field by proper tuning of interaction time 


Rabi oscillation in a small coherent field: 

observing discrete Rabi frequencies 

Fourier transform of the Rabi oscillation signal 

Ω 0 

Ω 0 

2 

Ω 0 

3 

Discrete peaks 

corresponding to 

discrete photon numbers 

FFT (arb. u.) 

Direct observation 

of field quantization 

in a "box" 

0 25 50 75 100 125 150 

Frequency (kHz) 


Rabi oscillation in a small coherent field: 

Measuring the photon number distribution 

1 

P () ( ) 1 cos 

( ) 

g 

t = ∑ P N − Ω 

0t 

N + 1 . e 

2 

N 

( ) 

−t 

/ τ 

Fit of P(n) on the Rabi oscillation signal: 

0,5 

0,4 

Measured P(n) 

Poisson law 

P(n) 

0,3 

0,2 

N = 0.85 

0,1 

0,0 

0 1 2 3 4 5 

Photon number 

accurate field statistics measurement 


All results 

• 0 

• 0.4 

• 0.85 

• 1.77 

1.0 

0.5 

0.0 

1.0 

0.0 0.5 1.0 

n = 0.06 

th 

n = 0.4 

coh 

photons 

P e 

(t) 

0.5 

0.0 

1.0 

0.5 

0.0 

1.0 

FFT Amplitude 

P(n) 

0.0 0.5 

0.0 0.5 

n = 0.85 

coh 

n = 1.77 

coh 

0.5 

0.0 

0.0 0.3 

0 30 60 90 

0 50 100 150 

0 1 2 3 4 5 

Time ( 

µs) 


n 


Field quantization 

• Many evidences of field quantization since Compton effect 

• Note that photoelectric effect is not a proof of field quantization (Dirac and 

Wenzel 1926) 

• Quantization of Rabi frequencies provide a visceral evidence of field 

quantization 

• A direct insight into field statistics 


Quantum Rabi oscillations: state transformations 

Initial state 

e,0 

1 

e,0 e⎯⎯→− 

,0 e,0 ⎯⎯→ eg,0 

,1 2π ( epulse 

,0 + g,1 

) 

2 

g,1 ( ⎯⎯→− ce e + cg π/2 spontaneous g,1) 0 ⎯⎯→ g 

Conditional ( ce 1 + c 

emission dynamics 

g 

0 ) 

pulse 

π spontaneous emission pulse 

g,0 ⎯⎯→+ Entanglement g,0 

Quantum 

creation 

Atom/cavity state copy phase gate 

Atom-cavity EPR pair 

P e 

(t) 

0.8 

51 (level e) 

0.6 

51.1 GHz 

50 (level g) 

0.4 

0.2 

0.0 

Brune et al, PRL 76, 1800 (96) 

time ( µ s) 

0 30 60 90 


Three "stitches" to "knit" quantum entanglement 

Combine elementary transformations to create complex entangled states 

• State copy with a π pulse 

– Quantum memory : PRL 79, 769 (97) 

• Creation of entanglement with a π/2 pulse 

– EPR atomic pairs : PRL 79, 1 (97) 

• Quantum phase gate based on a 2π pulse 

– Quantum gate : PRL 83, 5166 (99) 

– Absorption-free detection of a single photon: Nature 400, 239 (99) 

• Entanglement of three systems (six operations on four qubits) 

– GHZ Triplets : Science 288, 2024 (00) 

• Entanglement of two radiation field modes 

– Phys. Rev. A 64, 050301 (2001) 

• Direct entanglement of two atoms in a cavity-assisted collision 

– Phys. Rev. Lett. 87, 037902 (2001) 


An experiment on complementarity 

a realization of Bohr’s 1927 gedankenexperiment 


A “modern” version of Bohr’s proposal 

• Mach Zehnder interferometer 

φ 

φ 

D 

•Interference between two well-separated paths. 

• Getting a which-path 

information 


A “modern” version of Bohr’s proposal 

• Mach-Zehnder interferometer with a moving slit 

φ 

φ 

D 

• Massive slit: negligible motion, no which- path information, fringes 

• Microscopic slit: which path information and no fringes 


Complementarity and uncertainty relations 

Get a which path information 

P>∆p 

(∆p quantum fluctuations of 

beam splitter’s momentum) 

Hence 

∆x > h/∆p > h/P=λ 

B 1 

b 

P 

O 

φ 

a 

B 2 

M 

D 

φ 

M' 

Beam splitter’s quantum position fluctuations larger than wavelength: no 

fringes 


Complementarity and entanglement 

• A more general analysis of Bohr’s experiment 

– Initial beam-splitter state 

– Final state for path b 

α 

0 

B 1 

M' 

b 

P 

a 

O 

φ 

B 2 

M 

D 

φ 

– Particle/beam-splitter state 

Ψ = Ψ 

a 

0 + Ψb α 

– Final fringes signal 

• Small mass, large kick 

– Particle/beam-splitter entanglement 

– (an EPR pair if states orthogonal) 

NO FRINGES 

• Large mass, small kick 

FRINGES 

Ψa Ψb 0 α 

0 α = 0 

0 α = 1 


Entanglement and complementarity 

Entanglement with another system destroys interference 

• explicit detector (beam-splitter/ external) 

• uncontrolled measurement by the environment (decoherence) 

φ 

φ 

D 

Complementarity, decoherence and entanglement intimately linked 


A more realistic system: Ramsey interferometry 

• Two resonant π/2 classical pulses on an atomic transition e/g 

1.0 

a 

M 

0.8 

B 1 

R 1 

R 2 

P g 

0.6 

0.4 

b 

φ 

0.2 

0.0 

M' 

B 2 

Fréquence relative (kHz) 

0 10 20 30 40 50 60 

D 

Which path information 

Atom emits one photon in R 1 

or R 2 

Ordinary macroscopic fields 

(heavy beam-splitter) 

Field state not appreciably affected. No "which path" information 

FRINGES 

Mesoscopic Ramsey field 

(light beam-splitter) 

Addition of one photon changes the field. "which path" info 

NO FRINGES 


Experimental requirements 

• Ramsey interferometry 

– Long atomic lifetimes 

– Millimeter-wave transitions 

• Circular Rydberg atoms 

• π/2 pulses in mesoscopic fields 

– Very strong atom-field coupling 

• Circular Rydberg atoms 

• Field coherent over atom/field interaction 

• Superconducting millimeter-wave cavities 


Bohr’s experiment with a Ramsey interferometer 

• Illustrating complementarity: Store one Ramsey field in a cavity 

S 

Atom-cavity interaction time 

Tuned for π/2 pulse 

Possible even if C empty 

– Initial cavity state α 

1 

Ψ = e, αe 

+ g, 

αg 

– Intermediate atom-cavity state 2 

• Ramsey fringes contrast α α 

– Large field 

e 

α ≈ α ≈ α 

• FRINGES 

e 

g 

e 

R 1 

R 2 

g 

C 

φ 

( ) 

g 

D 

φ 

– Small field 

α 

= 0, α = 1 

• NO FRINGE 

e 

g 


Quantum/classical limit for an interferometer 

Fringes contrast 

Fringes contrast versus photon number N 

0.8 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

0.0 

0 2 4 6 8 10 12 14 16 

Nature, 411, 166 (2001) 

N 

Fringes vanish for quantum 

field 

photon number plays 

the role of the beamsplitter's 

"mass" 

An illustration of the ∆N∆Φ 

uncertainty relation : 

• Ramsey fringes reveal 

field pulses phase 

correlations. 

• Small quantum field: large 

phase uncertainty and low 

fringe contrast 

Not a trivial blurring of the 

fringes by a classical noise: 

atom/cavity entanglement 

can be erased 


An elementary quantum eraser 

• Another thought experiment 

φ 

φ 

D 

Two interactions with the same beamsplitter assembly erase the which path information 

and restore the interference fringes 


Ramsey “quantum eraser” 

• A second interaction with the mode erases the atom-cavity entanglement 

1.0 

Resonant Non-resonant Resonant 

0.9 

0.8 

0.7 

φ 

0.6 

0.5 

Pe 

0.4 

e,0 

1 

( ,0 + ,1 ) 

2 e g 

g,1 

0.3 

0.2 

0.1 

0.0 

10 12 14 16 18 20 22 24 

• Ramsey fringes without fields ! 

– Quantum interference fringes without external field 

– A good tool for quantum manipulations 


A genuine quantum eraser 

Manipulating atom/cavity entanglement 

• Atom A i 

interacts with cavity field 

• Copy cavity state on another atom A e 

(π pulse) 

e 

A e 

Erasing entanglement 

π/2 pulse on A e 

mixes the states 

Resulting state: 

1 

2 

⎡ 

⎣ 

( − + ) + ( + ) 

e e g g e g 

e i i e i i 

⎤ 

⎦ 

g 

A i 

Detection of A e 

projects A i 

onto a state 

superposition with a well-defined 

phase depending upon state of A e 

. 

State of two atoms: 

1 

2 

( e, g − g , e ) 

i e i e 

An atomic EPR pair (PRL 79, 1 (97)) 

FRINGES 

after a classical π/2 pulse on A i 

Phase conditioned to state of A e 

Direct detection of A e 

:no fringes on A i 

Entanglement with A e 

or C provides 

"which path" information on A i 

. 


An EPR experiment revisited 

Conditional fringes on A i 

0 20 40 60 80 

1.0 

0.8 

A e 

in e 

A e 

in g 

1.0 

No fringes when 

tracing on A e 

Conditional Probabilities 

0.6 

0.4 

0.2 

Probability 

0.8 

0.6 

0.4 

0.2 

0.0 

Relative frequency (kHz) 

0.0 

0 20 40 60 80 

Relative Frequency (kHz) 

PRL 79, 1 (97) 

A quantum eraser demonstration and controlled entanglement of two atoms 


A direct study of the decoherence process 


Another experiment on complementarity 

e 

Cavity as an external detector in the 

Ramsey interferometer 

Cavity contains initially a coherent field 

Non-resonant atom-field interaction: 

e 

g 

R 1 

R 2 

α 

C 

Atom modifies the cavity field phase 

Phase shift α 1/δ 

S 

(index of refraction effect) 

⎯⎯→ 

⎯⎯→ 

e 

g 

(δ:atom-cavity detuning) 

1 

g 

D 

φ 

"Which path" information: 

• Small phase shift (large δ) 

(smaller than quantum phase noise) 

– field phase almost unchanged 

– No which path information 

– Standard Ramsey fringes 

• Large phase shift (small δ) 

(larger than quantum phase noise) 

– Cavity fields associated to the 

two paths distinguishable 

– Unambiguous which path 

information 

– No Ramsey fringes 


Fringes and field state 

1.0 

0.5 

Complementarity 

Fringes contrast and phase 

60 

n=9.5 (0.1) 

6 

Ramsey Fringe Signal 

0.0 

1.0 

0.5 

712 kHz 

Vacuum 

712 kHz, 

9.5 phot ons 

Fringe Contr ast (%) 

40 

20 

0 

0.0 0.2 0.4 0.6 0.8 

φ (radians) 

0.0 0.2 0.4 0 2 

φ (radians) 

4 

Fringe Shift (rd) 

0.0 

104 kHz 

0 2 4 6 8 10 

ν (kHz) 

347 kHz 

PRL 77, 4887 (96) 

• Excellent agreement with theoretical 

predictions. 

• Not a trivial fringes washing out effect 

Calibration of the cavity field 

9.5 (0.1) photons 


A laboratory version of the Schrödinger cat 

Field state after atomic detection 

1 

2 

( + ) 

A coherent superposition of two 

"classical" states. 

Very similar to the Schrödinger cat 

An atom to probe field coherence 

Quantum interferences involving the 

cavity state 

First atom 

Φ 

−Φ 

D 

Second atom 

Decoherence will transform this 

superposition into a statistical mixture 

Slow relaxation: possible to study the 

decoherence dynamics 

Decoherence caught in the act 

Two indistinguishable quantum paths to 

the same final state: 

2Φ 

−2Φ 


A decoherence study 

Atomic correlation signal 

Decoherence versus size of the cat 

Two-Atom Correlation Signal 

0.0 0.1 0.2 

n=3.3 δ/2π =70 and 170 kHz 

0 1 2 

t/T 

r 

0 1 2 PRL 77, 4887 (1996) 

τ/T 

r 


correlation signal 


δ/2π =70 kHz 

20 

16 

n=5.5 

12 

8 

4 

0 

0 1 2 

20 

t/T 

r 

16 

12 

n=3.3 

8 

4 

0

Decoherence features 

• Faster than cavity relaxation 

• Faster when distance between states increases 

• Decoherence time scale depends upon a "macroscopicity" parameter 

• Directly linked to complementarity and entanglement (environment 

acquires information on the quantum system) 

Not a trivial relaxation mechanism, if described by standard relaxation theory 

Essential for quantum measurement 

meters are not in superposition states 

Difficulty for applications of QM 

the more complex the entangled state, the faster the decoherence 


The ENS team 

• Permanent members 

– Serge Haroche, Michel Brune, Gilles Nogues, Jean-Michel Raimond 

• Thesis students 

– F. Bernardot, P. Nussenzveig, A. Maali, J. Dreyer, X. Maître, P. 

Domokos, G. Nogues, A. Rauschenbeutel, P. Bertet, S. Osnaghi, A. 

Auffeves, P. Maioli, T. Meunier, P. Hyafil, J. Mozley, S. Gleyzes 

• Post doctoral visitors 

– F. Schmidt-Kaler, E. Hagley, P. Milman, S. Kuhr 

• Theoretical collaborations 

– L. Davidovich, N. Zagury (Rio) 

– D. Vitali, P. Tombesi (Camerino) 

• Optical CQED 

– V. Lefèvre, J. Hare 


Structure of the lectures 

• I) Introduction 

• II) The tools of CQED 

• III) Experimental illustrations of fundamental quantum mechanics 

• IV) Quantum information with atoms and cavities 

• V) Schrödinger cats and decoherence 

• VI) Cavity QED with dielectric microspheres 

• VII) Perspectives 











II) The tools of CQED 

• 1) Quantum fields 

• 2) Field relaxation 

• 3) A simple quantum device: the beamsplitter 

• 4) Atom-field coupling 

• 5) Experimental tools 









• Hamiltonian 

A mechanical oscillator 

• Dimensionless coordinates 

• Creation and annihilation operators 

• Fock states 

• |0>: “vacuum” state 



• Fock states wavefunctions 



• Operators evolution in Heisenberg point of view 

– Annihilation operators evolves in the same way as the classical 

amplitude 


A cavity field mode 

• Quadratic hamiltonian as in the case of mechanical oscillator (field is a 

collection of harmonic oscillators) 

• Photon annihilation and creation operators 

• Electric field operator 

– normalization factor (dimension of a field) 

– local polarization 

– relative field mode amplitude and polarization (1 at field maximum) 

a solution of Helmoltz equation with cavity limiting conditions 

• Heisenberg picture 


Field normalisation 

• Energy of Fock states 

• Cavity mode volume 


Case of an open Fabry Perot cavity 

• Field amplitude 

(a) 

Elastic blade 

Ring 

R 

L 

2w 0 

PZT 

• Free space field quantization: introduction of a fictitious quantization box. 

Volume goes to infinity. Here: a real “photon box”. Field per photon 

perfectly defined. No need for complex limit taking. 


Field quadratures 

• Coordinates for a two-dimensional phase space (generalizes X and P) 

• Eigenstates are non-normalizable (position/momentum eigenstates) 

• Pictorial representation of wave-functions in phase space 


Phase space representation 

• Vacuum and three photon states 

Xφ+π/2 

X π/2 

X π/2 

X φ 

φ 

X 0 

X 0 

(a) 

(b) 


Fock states 

• Fock states have a zero expectation value for the electric field (prop to a) 

and the vector potential 

• Fock states have a non zero electromagnetic energy 

• Fock states cannot be interpreted in terms of a classical field: a first 

example of non-classical states 


Coherent states 

• Fock states are utterly non-classical. Other states 

Displacement operator 

• Physical interpretation of the displacement operator 

– Glauber formula (valid when A, B commute with their commutator) 

Translation operator 

Along X 0 


Coherent state as a displacement of vacuum 

X π/2 

α 

Imα 

Reα 

X 0 

• The coherent state is a displaced vacuum. Hence a minimum uncertainty 

state 


Creating coherent field with a classical current 

• Classical current density coupled to the mode (at origin for the sake of 

simplicity) 

• Source cavity coupling 

• Neglect the two terms oscillating at twice cavity frequency (RWA) 

• State evolution 

• Recognize for 

• A classical current produces a displacement and a coherent state. 


Displacement and annihilation operator 

• Action of the displacement on the annihilation: calculate 

writes 

With 

– Baker-Hausdorf Lemma 

– two terms only 


Combination of displacements 

• Displacement of a coherent state 

• Quantum analogue of classical homodyning 


• Again Glauber 

Fock states expansion of coherent states 

• Last term leaves vacuum invariant. 

• Expand second term in power series and note that 

• Essential property 

• NB a non hermitic. Admits complex eigenvalues 

• Non zero electric field value (not the case for Fock state) 


Photon number distribution 

• 

0.5 

p 

c 

(n) 

0.4 

(a) 

p 

c 

(n) 

0.08 

(b) 

0.3 

0.06 

0.2 

0.04 

0.1 

0.02 

0.0 

0 1 2 3 4 5 6 7 8 

n 

0.00 

0 5 10 15 20 25 30 35 

n 


Pictorial representation of a coherent state 

• A simple qualitative phase space diagram 

Im α 

1 

| α| 

Φ 

∆Φ 

Re α 

n 

2 

= α ∆ n= n ∆N∆Φ ≈1 


A basis of coherent states 

• Scalar product of coherent states 

• Overcomplete basis 

• No unique decomposition (zero is also a coherent state) 


Coherent states wavefunctions 

• Use 

• Inject between exponentials closure on 

• is the wavefunction of vacuum in P representation 

• A gaussian along the real axis, with a modulation reflecting the translation 

along the imaginary axis. Probability distribution 


Time evolution of a coherent state 

• Evolution of coherent state given by 

(b) 

Im α 

Recover classical trajectory 

Of amplitude in phase space 

ω c 

t 

Re α 


Quasi-probability distributions 

• Give a quantitative status to the description of quantum states in phase 

space. 

• For classical fields, any state represented by a probability distribution in 

phase space (cf statistical physics) 

• For quantum states no analog (because of Heisenberg uncertainty 

limitations) 

• Possible to define quasi probability distributions in phase space. They 

contain all possible information on the quantum state but they may be 

negative or singular. 

• Use here two distributions 

– Q function (very pictorial but not very useful) 

– W function (Wigner distribution). Extremely useful and precious 

insights in quantum states 


The Q function 

• Definition for an arbitrary state 

– For a pure state: square of the overlap with the coherent state 

• Alternative definition 

– Probability to get zero photons in a field displaced by –α. Leads to 

simple experimental schemes to measure Q 

• Q for a coherent state 


Examples of Q functions 

• Five photons coherent state and two-photon Fock state 

-4 -2 

0 

Q 

2 

4 

0.3 

0.2 

0.1 

α ι 

-4 

-2 

0 

2 

α r 

0 

4 

-4 -2 

0 

2 

4 

0.08 

0.06 

0.04 

0.02 

-4 

-2 

0 

2 

0 

4 


Properties of the Q function.Link with characteristic function 

in anti-normal order 

Q(α) is a function ≥ 0, determined by « sweeping » phase space with a coherent state 

and attributing to each of its « positions » the expectation value in this state of the field 

density operator, divided by π . 

Another definition of Q: Fourier transform of the expectation value in the field state 

of the operator exp (-λ ∗ a) exp(λa + ) (characteristic function of the field in antinormal 

order): 

(ρ 

C ) 

an 

(λ) = ∫∫ d 2 α Q (ρ ) (α) e λα * −λ * α 

= 1 π Tr ∫∫ d 2 αρα α e λα * −λ * α 

= 1 π Tr d 2 αρe −λ * a 

∫∫ α α e λa+ 

= Tr ρ e −λ* a 

e λa + = e −λ* a 

e λa+ ρ 

(3 − 4) 

Operators in anti-normal (an) order 

and by inverse transformation (normalization checked by substituting in first line of (3-4)): 

Q (ρ ) (α) = 1 d 2 λ C (ρ) 

π 2 

an 

(λ) e λ* α −λα 

∫∫ * 

(3 − 5) 


Wigner function: an insight into a quantum state 

A quasi-probability distribution in phase space. 

• Characterizes completely the quantum state 

• Negative for non-classical states. 

p 

Im(α) 

Describes the motion of a particle or 

a quantum single mode field 

Motion 

of a particle 

q 

Re(α) 

Electromagnetic 

field 


Wigner function 

• Definition 

• By inverse Fourier transform 

• In particular 

• The probability distribution of x is obtained by integrating W over p 

• This property should obviously be invariant by rotation in phase space 

∫ 

W q cosθ − p sin θ, q sinθ + p cosθ 

dp 

( ) 

θ θ θ θ θ 

= P q = q Uˆ 

θ ˆ ρUˆ 

θ q 

( ) ( ) ( ) 

† 

θ 

• All elements of density matrix derived from W: contains all possible 

information on quantum state. 


Wigner function 

• Wigner function is normalized 

∫ dpdqW(q,p)=1 

• W may be negative. If cancels (nodes of 

wavefunction in x representation), W should have negative parts. Cannot 

be a full fledged probability distribution in all cases 

• W gives all averages of operators in symmetric ordering 

Tr ⎡⎣ ˆ ρ ( xp ˆˆ + px ˆˆ )/2 ⎤⎦ = ∫ dxdpW ( x, 

p) 

xp 

ˆ 

ˆ 

ˆ 

A =Tr[ρA]= ∫ dpdqW(q,p)f 

s(q,p) 


A few Wigner functions 

• Vacuum 

• Coherent state 

• Fock state 

• Thermal field 


Vacuum |0> 

Examples of Wigner functions 

-2 

0 

2 

2 

Coherent state |β> 

(β=1.5+1.5i) 

-4 -2 0 2 4 

2 

Thermal field n th 

=1 

-4 -2 0 2 4 

2 

1 

1 

1 

0 

0 

0 

-2 

0 

2 

-1 

-2 

-4 

-2 

0 

2 

-1 

-2 

4 -4 

-2 

0 

2 

-1 

-2 

4 Fock state |1> 

-2 0 2 

2 

-4 

5 photons Fock state 

4 

2 

0 

-2 

1 

0 

0 

-1 

-0.3 

-2 

2 

0 

-2 

-4 

-2 

0 

2 


4

A classicality criterion 

• A field whose Wigner function is positive can be understood as a classical 

field with fluctuations described by a classical probability distribution 

– Coherent states, thermal states 

• A field with negative W cannot be understood classically 

– Fock states 


An alternative simple expression of W 

• Two simple expressions left as an exercise 

• Parity operator 

• +1 for even Fock states, -1 for odd Fock states 

• Leads to a very simple experimental determination of W 

L. Davidovich, Private comm 


Wigner function in terms of the characteristic function 

• Definition. Symmetric ordering 

• D unitary 

• Coherent field 

• Fock state 

• Wigner function 

• Normal order characteristic function 

* 2 

• Relation with anti normal order ( ρ) λa 

λ a λ /2 ( ρ) 

C ( λ) Trρe + − 

= = e C ( λ) 

s 

an 


Schrödinger cat states 

• An example of non-classical state 

– Quantum superposition of two coherent fields with opposite phases. 

– No classical counterpart 

X π/2 

– Evidence the coherence 

−β 

β 

X 0 

– Photon number distribution: only even photon numbers contribute 

(even cat) 

P(n) 

– Odd cat 

024 

– Eigenstates of with +1 and -1 eigenvalues 

13 5 


Quadrature distributions 

• X wavefunction of the even cat 

– Sum of two gaussian 

• P distribution of the even cat 

– With 

– Modulation revealing coherence 


Graphical interpretation of quadrature distributions 

(a) (b) (c) 

X π/2 

X 0 

X π/2 X π/2 

X ϕ 

X 0 

X 0 


Q 

( β +−β 

) 

Q function of a cat 

1 

( α) = 

2 

[ 

π + 

−2 

β 

2 (1 e ) 

e 

2 2 

2 2 

−α−β 

− α+ 

β 

+ e 

+ α β −α β 

− ( α + β ) 

2e 

cos 2( 

1 2 2 1)] 

-4 -2 

0 

2 

4 

0.15 

0.1 

0.05 

-4 

-2 

0 

2 

0 

4 

Exponential suppression of interferences near the origin revealing cat coherence. 

Q function of a cat similar to one of a mixture. 

Q is not well adapted to displaying mesoscopic quantum superpositions 


Wigner function of phase-cat states ( β real) 

W ( β +−β ) (α) = 

1 

2π 2 (1 + e −2β 2 ) 

d 2 λ e (αλ* −α * λ) 

[ ∫∫ ( β D(λ) β + −β D(λ) −β + β D(λ) −β + −β D(λ) β )] (3− 26) 

W (coherence) (α) = 

« Incoherent » terms (sum of Wigner functions 

of the two states |β> and |-β>) 

Coherent interference 

cohérent terms 

(W (coherence) ) 

1 

[ dλ 

2π 2 (1+ e −2β 2 1 

dλ 2 

e 2i(α 2λ 1 

−α 1 

λ 2 

) 

∫∫ 

−β D(λ) β + term β →−β] ) 

(3 − 27) 

we can write the matrix element < −β |D(λ)|β > as (assuming β real without loss of generality): 

−β D(λ) β = e iβλ 2 

−β β + λ = exp[−2β(β + λ 1 

) − λ 1 2 /2− λ 2 2 /2] (3− 28) 

and from (3-27) and (3-28) we find Gaussian integrals in λ 1 and λ 2 , from which the expression of W 

can easily be derived (see next page): 

dλ 1 

dλ 2 

e 2i(α 2λ 1 

−α 1 

λ 2 

) 

∫∫ −β D(λ) β = ∫ dλ 1 

exp[− 1 2 (λ 1 

+ 2(β − iα 2 

)) 2 − 2α 22 

− 4iβα 2 

] 

× ∫ dλ 2 

exp[− 1 2 (λ 2 

+ 2iα 1 

) 2 − 2α 12 

] = 2π e −2 α 2 e −4iβα 2 

(3− 29) 


Graphs of Wigner functions of coherent states and of 

Vacuum 

their superpositions 

-2 

0 

2 

2 

Phase cat | β>+|−β> 

1 

0 

W ( β +−β ) (α) = 

1 

[ ] (3 − 30) 

π(1 + e −2β 2 ) e−2α −β 2 + e −2α + β 2 + 2e −2α 2 cos4βα 2 

-1 

2 

(β real here) 

2 

-2 

-2 

0 

(case β=3) 

The interference term around |α|=0 has 

negative parts. The integral along 

directions parallel to the α 1 axis 

corresponding to fixed α 2 vanishes for 

some values of α 2 : «dark fringes» of P π/2 

(α) (see Lectures 1 and 2) 

-4 

-2 

0 

2 

4 


-2 

-1 

α 1 

α 2 

0 

1 

1 

0 

-1 

2 

-2 

-4 

Statistical 

mixture of 

|β> and |−β> 

-2 

0 

2 

4 

-2 

-1 

0 

1 

1 

0 

-1 

2 -2 

2








The problem 

• The field mode (system S) is coupled to a complex environment E 

– Charge carriers in the mirrors, other propagating modes coupled by 

mirror defects 

• The S+E system undergoes an hamiltonian evolution and is described by 

a pure state 

• The environment state is not accessible. We are interested only in the 

system’s reduced density matrix 

• Find the evolution equation of system’s density matrix. 


The Kraus approach 

• Any mapping of a density matrix onto another density matrix (completely 

positive quantum map) can be put (non unambiguously) under the form 

– With a number of Kraus operators at most N² (N dimension of Hilbert 

space) 

• Assume that evolution ruled by a differential equation: mapping between 

ρ(t) and ρ(t+dt) (Markov approximation) 

• M i depends only upon dt if we assume that environment state not 

appreciably modified by system’s evolution (Born approximation) 


Kraus cont’d 

• One of the operators is order zero in dt. Without loss of generality 

– (H and K hermitian) and 

• As we get 

• Hence, H identified with the hamiltonian 

• Relaxation modelled by a Liouvillian in Lindbladt form 


Physical interpretation of the L i 

• Evolution of a wavefunction during dt 

– Either system unchanged or (with a probability of order dt) evolution 

towards a wavefunction 

– L i describes the system evolution when some event occurs in the 

environment (eg apparition or disparition of a photon in the 

environment). 

– L i : “jump operator” (cf Monte carlo approach) 

– Underlying model: continuous monitoring of the environment. Change 

in the environment reflected by the action of one jump operator onto 

the system 


• Jump operators 

Application to the cavity case 

– Zero temperature cavity: only event escape of one photon from the 

cavity, detected in the environment. 

• Associated jump operator a (annihilation operator) within a 

normalization factor 

– Finite temperature cavity: also transfer of a thermal photon from the 

environment into the cavity. 

• Associated jump operator α a + (creation operator) 

• Complete Lindblad form: 


A more standard approach 

• A simple environment model: collection of harmonic oscillators spanning a 

large frequency interval, weakly and linearly coupled to cavity mode (final 

result is model independent) 

• Cavity-environment hamiltonian 

• Interaction representation vs free hamiltonians 

• Many modes between ∆ and ∆+d∆, average coupling g(∆), “coupling 

density (proportional to g and mode density) γ(∆) 


Evolution of density matrix 

• Global and reduced evolution equations 

• Second order expansion 

• Assume cavity and environment initially uncorrelated 

• Assume reservoir state unchanged (Born approx) 

• All first order terms cancel. Only second order contributions. 16 terms… 


One term out of 16 

• One of the terms when developing second order onto the F operators 

• Equation of evolution for cavity density matrix 

• Large frequency span of reservoir. K has a short time memory. Markov 

approximation 


• Gathering all terms 

Final Liouville-Lindblad equation 

• Same form as from Kraus approach. 

• More cumbersome approach but finer understanding of underlying 

mechanisms 

• Same equation for all environment models (eg two level atoms) provided 

large environment, wide frequency range and linear coupling 

• A trustable equation for relaxation, depending upon a single, classically 

measurable, parameter: cavity quality factor Q 


Heisenberg picture: Langevin forces 

• Evolution of field operators in the Heisenberg approach 

• Classical damping of a (evolves as classical damped field amplitude) plus 

additional noise term. Brownian motion preserving commutation relations. 

• Leads to simple physical pictures of the evolution 


Application: photon number distribution relaxation 

• Evolution of diagonal density matrix terms 

• Steady state solution: detailed balance condition 

• variance 


Thermal field vs coherent field 


Fock states lifetime 

• Single photon lifetime (at zero temperature) 

• κ -1 =T c The classical field energy damping time 

• |n> lifetime : T c /n 

• A simple effect but also a decoherence effect (fast relaxation of a nonclassical 

field) 

• Relaxation of coherent states can be obtained by a complex algebraic 

derivation. Here, simpler to use the Monte Carlo approach or a simple 

beam-splitter model (later on) 


Quantum Monte Carlo trajectories 

• Model the relaxation in terms of stochastic wavefunctions. 

• Instead of tracing the environment, assume that it is (virtually) monitored. 

Continuous measurement of photon escapes for cavity by an array of 

detectors in the environment. At any time, the system is described by a 

wavefunction. 

• Two kind of evolutions 

– Jumps when an environment detector ‘clicks’ 

– Non-unitary evolution between clicks. Not detecting a photon gives 

some hints that the cavity might be empty. The longer the time interval 

for no detection, the higher the probability that the cavity is empty. A 

no-detection gives information on the system, albeit more ambiguous 

than a click. 

• System density matrix recovered by averaging wavefunctions over very 

many trajectories. 

• A rigorous derivation based on Kraus approach. 

• Here: give the recipe and check consistent with Lindblad 


Quantum Monte Carlo: general principle 

• During dt probability for a jump described by L i 

• Total jump probability 

• Wavefunction evolution if jump L i 

• Evolution if no jump 

– With 

• Iterate over time following this procedure. At each step, choose randomly 

jump/no jump and L i if jump 

• Extremely efficient numerically for large systems (cold atoms) 


Equivalence with Lindblad 

• Average over very many trajectories the projector on wavefunction. 

Evolution equation: 

• Replace jump and no-jump wavefunctions by their expression and get 

• Standard Lindblad form associated with L i . 


Case of cavity relaxation at zero temperature 

• Jump operators proportional to a 

• Jump probability per unit time 

– Obvious interpretation in terms of photon loss 

• Effective hamiltonian (interaction with respect to cavity hamiltonian) 

– Proportional to photon number 

• No-jump evolution 


Relaxation of a Fock state 

• Fock state is an eigenstate of the no-jump evolution. Invariant in the nojump 

periods 

• Photon number decreases by one at each jump 

• Staircase decrease of the photon number. Random step times 

• Ordinary exponential relaxation of energy recovered by averaging many 

trajectories. 

• Photon number variance zero initially and at long times. Maximum during 

evolution due to dispersion in jumps timing. 


Relaxation of a coherent state 

• Coherent state eigenstate of jump operator. No evolution when jump ie 

photon detected in the environment. Counterintuitive 

– Loosing a photon reduces the photon number but 

– Seeing a photon gives an indication that the field amplitude is nonzero. 

Results in an increase of average photon number which exactly 

compensates the effect above. 

• Evolution between jumps only. Deterministic evolution. System remains at 

any time in a pure state 

• Evolution under the free cavity hamiltonian and 

• Amounts to adding an imaginary part to 

• frequency ω+ικ/2 


Relaxation in terms of characteristic functions 

• A simple result for the normal order characteristic function for a zero 

temperature relaxation. Very useful for relaxation of Schrödinger cats. 

Consider formally λ and λ* as 

Independent variables 

• Transforms Lindblad equation into a differential equation for C n 

– Associate formally a C n function to all combinations of ρ with a, a + 

(terms in the Lindblad equation) 

– Show that 


Equation for C n 

• Substitution in Lindblad 

• Explicit solution 

• Extremely simple solution. Obtain it by characteristic method of check 

validity by direct substitution 


Relaxation of a Schrödinger cat 

• Zero temperature evolution of a cat 


Relaxation of a Schrödinger cat 

• Identify C n (t) with a density matrix 

• Damping of non diagonal terms with a rate 2κ|α|². Lifetime of the 

mesoscopic coherent Tc/2n. A typical decoherence effect (much more on 

that later) 


Understanding fast decoherence 

• Decoherence time scale much shorter than energy lifetime. 

• Monte carlo approach: quantum jumps due to the action of a operator 

• A jump changes the parity of the cat without changing the components 

amplitudes 

• A no-jump slightly reduces the components amplitudes without affecting 

parity. 

• Average over many trajectories: washing out of parity information in a 

time typical of first photon escape ie Tc/n. 

• Fast evolution towards a statistical mixture 

• Then slow evolution (no jump terms) towards vacuum 

• Parity jumps can be corrected. Idea of a feedback method (feeding cats 

with quantum food) More later. 


Decoherence in the Wigner point of view 

• Equation for Wigner function 

• Fokker planck equation. Drift to origin (damping of amplitude) and 

diffusion. 

• Cat: fringes near origin with spacing 1/β. Washed out by diffusion process 

faster and faster when amplitude increases 

• Note: explicit solution (with κ=1) 


Illustrated cat relaxation 





• 3) A simple quantum device: the beam-splitter 




Beamsplitters 

• A semi transparent plate couples two propagating modes with the same 

polarization. Which quantum operations 

• An equivalent model: a fibre coupler 

• Classical amplitudes transformations 


A simple quantum model 

• Models wave packet ‘collision’ on the beamsplitter 

• Transient linear coupling of the two modes with hamiltonian 

• Operators evolution in Heisenberg point of view 

– With Baker-Hausdorf lemma 


A simple quantum model 

• With and group odd and even terms, 

proportional to a and b, in factor of expansions of cosθ and sinθ 

• Similarly 

• Mode annihilation operators transform as classical field amplitudes 

• Taking conjugate and with 

• Examine now (in Schrödinger point of view) the action of the beamsplitter 

on simple states 


Action on simple states 

• Mode b always in vacuum to start with 

• Mode a in vacuum 

– Output state |0,0> 

• Mode a in |1> 

• Case of a balanced beam-splitter (θ=π/4) 

• Creation of an entangled state of the two modes (of EPR type) 


• Fock state input 


• Massively entangled two mode state. Superposition of all possible 

partitions 


• Coherent input 


• Two unentangled coherent outputs. Amplitudes follow the classical laws. 

• Coherent states impervious to entanglement with other modes 

• Important consequences for their relaxation 



• Modes a and b contain a single photon 

• Case of a balanced beamsplitter θ=π/4 

• Both photons emerge in the same mode (Mandel dip). A genuinely 

quantum effect. A direct manifestation of bosonic nature of photons. 

Destructive quantum interference between the direct and exchange paths 

cancels the probability for having one photon in each output 


A partial Bell state analyzer 

• Each mode sustains two orthogonal polarizations H or V. 

• Assume splitting amplitudes independent of polarization 

• Two impinging photons. Four Bell states HH+/-VV, HV+/-VH 

• Three symmetric Bell states. The two photons emerge in the same path 

as for HH or VV 

• One antisymmetric polarization state HV-VH. Global state must be 

symmetric. Only possibility: antisymmetric mode combination. One photon 

in each output path. 

• Can be checked easily by an explicit calculation along the same lines left 

as an exercise 

• A possibility to distinguish one Bell state among four. 

• Teleportation, entanglement swapping experiments based on this 

process. 


• Two interfering paths 

• Coherent state input 

Mach-zehnder: a quantum interferometer 

• After B 1 

• Dephaser 

• After B 2 

• Final photon count 

• Obviously the classical result 

• However phase of the incoming 

field plays no role. Same interference 

Pattern with a Fock state 


Interferences without phase: Fock state input 

• n photons Fock state as input 

• Final state 

• (action of a tunable beam-splitter on the Fock state) 

• Partition of the n photons in the two output ports with probabilities 

• Only the relative phase of the two interfering paths is important. The initial 

coherence plays no role. Same interference pattern with coherent states 

and single photon input. 

• A photon always interferes with itself (Dirac) 


Single photon interferences in a Mach Zehnder 

• From Grangier et al 1986 


Sensitivity of the interferometer 

• Quantum (shot noise) limits for the detection of small phases 

• Fock state input. Noise on the a detector 

• Sensitivity: η, inverse of the smallest dephasing changing the output 

photon number by more than these flucutations 

• Independent of phase. At fringes extrema, no variation of N with φ, but no 

noise either 


• Coherent input 

Sensitivity of the interferometer 

• At φ=π/2 times larger than for Fock states 

• Initial photon number fluctuations add up with partition noise 

• Sensitivity 

• Optimum sensitivity on a dark fringe (no influence of input 

noise) 

• Same optimum sensitivity as for a Fock state. 

• Improved sensitivity in 1/N possible with non-linear beam splitters (n 

photons follow all path a or b) 

• Realizations with two photon states in the optical domain and ‘simulation’ 

with ions in a trap 


Homodyne field measurement 

• Another application of the beamsplitter. Mixes a large coherent input with 

a quantum field. Large transmission T=cos²θ 

• First term= LO intensity. Can be substracted. Second term negligible if LO 

intense enough 

• Last term 

• Direct measurement of the quantum field 

Quadrature 

At the heart of tomographic field measurements 


An insight into coherent field relaxation 

• A simple relaxation model in terms of beamsplitters. Mode a coupled to 

very many other modes b i . Coupling hamiltonian: 

• Action of H i during a small time interval : equivalent to coupling a with 

modes b i by small reflection beamsplitters 

• Action of one of the couplings 

• Sum up independently all weak couplings 

• Mode a contains still a coherent state with a reduced amplitude. 

Reduction apparently quadratic in δτ. Deceptive. 


An insight into coherent field relaxation 

• Large density of environment modes. Number of relevant ones for a time 

interval δτ of the order of 1/δτ 

• Rewrite as: 

• Decrease of a amplitude linear in δτ. Sum up time intervals, assuming 

that the environment modes remain in the same initial state (Born approx) 

• Recover coherent state relaxation. Note also that the environment modes 

contain at time t tiny coherent states (useful later for cat relaxation) 

• Obtained from mere energy conservation 









A two-level atom 

Level scheme 

|e〉 

ω eg 

|g〉 

Equivalent to a spin ½ 

e =+ g =− 

Pauli matrices 

⎛1 0 ⎞ ⎛0 1⎞ 

= ⎜ ⎟ σ 

⎝0 −1 

x 

= ⎜ ⎟ 

⎠ ⎝1 0⎠ 

1 

± 

x 

= ( + ± − ) 

2 

σ z 

σ 

y 

⎛0 

−i⎞ 

= ⎜ ⎟ 

⎝i 

0 ⎠ 


Bloch sphere 

• Most general state + u . Correspondence between states and point on a 

unit radius sphere 

• Analogous to the Poincaré polarization sphere 


A two-level atom 

• Raising and lowering operators 

{ σ σ } 

, 1 

− + 

= 

• Free hamiltonian 

H 

ωeg 

= σ 

z 

2 

• Dipole operator 

( 

* 

) 

a − a + 

D= d ε σ + ε σ 


• Global hamiltonian 

Atom-cavity coupling 

• Four terms, two of which are anti-resonant: RWA 

Vacuum Rabi 

frequency 

• Jaynes and Cumming model (63) 

• Atom-cavity detuning 

• Uncoupled atom-cavity levels 


Uncoupled atom-cavity levels 

• At low detunings, grouped in multiplicities 

• Couplings only inside the multiplicities 

• g,0 isolated. Impervious to atom-cavity 

coupling 

• Interaction representation 

– Energy origin at manifold center 

• Complete hamiltonian 

• In matrix form 

• Easy diagonalisation: dressed levels 


The dressed levels 

• Eigenstates of the whole atom-field hamiltonian 

• In general complex expressions. Two simple cases 

– Resonance 

– Far off resonance 


The resonant case 

• Atom-cavity at exact resonance 

|e,2> 

|g,3> 

Ω 

3 

|+,2> 

|-,2> 

• A doublet for the weak excitation 

from the ground state (the 

vacuum Rabi splitting 

• e and g are no longer 

eigenstates: a quantum Rabi 

oscillation between these levels 

|e,1> 

|g,2> 

|e,0> 

|g,1> 

|g,0> 

Ω 

Ω 

2 

"vacuum Rabi 

splitting" 

|-,1> 

|+,0> 

|-,0> 


The quantum Rabi oscillation 

• An atom initially in e in a n-photon Fock state. Assumed at r=0 f=1 

• Atom initially in g 

• A Rabi oscillation between the two uncoupled levels. 

• Exists even when cavity empty (atom in e only) 

• Creates atom-cavity entanglement (much more on that later) 


The vacuum Rabi splitting 

• Equivalent to the normal mode splitting for two coupled oscillators. 

Observed for an atom in an optical cavity and for excitons in a 

semiconducting cavity 

Kimble et al 

1992 

Weisbuch et al 92 


Non resonant case 

• Position of dressed levels as a function of detuning 

Energy 

|+,n〉 

|e,n〉 

0 

hΩ 

|-,n〉 

|g,n+1〉 

-3 -2 -1 0 1 2 3 4 

∆ c 

/Ω 


Large atom-field detuning case 

• Large atom-cavity detuning. Assumes also atom at rest at cavity centre 

Lamb shift + light shift 

Atomic ior effect 


Action of an atom on a coherent field 

• Define an effective hamiltonian for shifts 

• Apply to 

• The atom (quantum system) controls the classical phase of the field 

• At the heart of Schrödinger cat states generation 


Taking into account atomic motion 

• Up to now atom fixed. Real atoms cross gaussian mode 

• No simple expressions. Only resonant and far off resonant case 

– Resonant case 

– All expressions obtained at r=0 remain valid when replacing real time 

by the effective interaction time taking account mode geometry 


Taking into account atomic motion 

– Non resonant case 

– Use effective hamiltonian, proportional to f² 

– The r=0 results also valid when using the effective interaction time 

– Note that resonant and non-resonant effective interaction times are 

not equal 


Large field limit: classical field on a spin 1/2 

• Coupling with a very large coherent field α (not part of dynamics) 

• Interaction representation with respect to cavity frequency 

• Remove time dependence by going to rotating frame 

• In terms of Pauli matrices 


Rabi rotation on the Bloch sphere 

• A geometrical interpretation 

– Any point on the Bloch 

sphere can be reached 

from + by interaction with 

the proper field for the 

proper time. 

– Any component of the 

spin can be measured 

with a +/- detector with a 

prior rotation 


• π/2 resonant pulse 

A few resonant pulses 

• π resonant pulse 

• 2π resonant pulse 

– Note sign change 


The Ramsey interferometer 

• Two π/2 pulses with zero 

Phase and a depasing element 

Transient atomic level energy 

Shifts producing an atomic phase 

R 1 

Dep. 

R 2 

Evident analogy with a MZ 

Interference between two quantum paths 


Another way of sweeping phase 

• Do not change atomic phase but change relative phase of pulses 

• Two independent sources 

• A same source, slightly offset from the atomic frequency (negligible offset 

for the interaction with a single pulse) 

• An essential method for high resolution spectroscopy. Long interrogation 

times without long interaction with the source 

• At the heart of all atomic clocks 


Ramsey fringes with cold atoms 

• A cold atoms experiment for high stability clocks 


A spin interferometer 

Two rotations of the spin around different axes through two timeseparated 

interactions with classical fields 

z 

z 

z 

x 

y 

R 

y 

y 

(π/ 2) R φ 

(π/ 2) 

x 

x 

y 

•Two Stern and Gerlach devices 

•Polarizer and analyzer 

•Final detection probability depends upon the relative phase of the pulses 

1.0 

0.8 

0.9 

g 

0.6 

0.6 

P g 

0.4 

0.3 

0.2 

0.0 

0 200 400 600 800 

0.0 


0 10 20 30 40 50 60 



Rabi oscillation in a mesoscopic field 

• Intermediate regime of a few tens of photons. A first insight 

• A simple theoretical problem 

• A surprisingly complex behavior 


Collapse and revival 

• Collapse: dispersion of field amplitudes due to dispersion of photon 

number 

• Revival: rephasing of amplitudes at a finite time such that oscillations 

corresponding to n and n+1 come back in phase 

• Revival is a genuinely quantum effect 


Atomic relaxation 

• Atomic density matrices 

1 

ρ = ⎡ 1 + n . σ ⎤ 

2 ⎣ ⎦ 

det ( ρ ) = (1/4)(1- n 2 ) ≥ 0 

 

n ≤ 1 

 

n = 1 

 

n < 1 

Pure case 

Statistical mixture 

• Geometrical representation: points inside the Bloch sphere 

• Ambiguity of representation 

 

n= λn + (1 −λ) 

n 

1 2 

→ ρ = λρ + (1 −λ) 

ρ 

1 2 


Spontaneous emission relaxation 

• Emission from e to g (inside the two level system) 

• Stationary state at finite temperature 


Spontaneous emission in a Monte Carlo process 

• Use the Monte Carlo approach. Atom in e, zero temperature 

• One jump operator (lowering) 

• Getting no jump decreases the probability for finding the atom in e 

• Continuous evolution as 

• Until a jump suddenly reduces wavepacket in g 









General scheme of the experiments 

Rev. Mod. Phys. 73, 565 (2001) 


Circular Rydberg atoms 

High principal quantum number 

Maximal orbital and magnetic quantum 

numbers 

• Long lifetime 

• Microwave two-level transition 

• Huge dipole matrix element 

• Stark tuning 

• Field ionization detection 

– selective and sensitive 

51 (level e) 

51.1 GHz 

50 (level g) 

54.3 GHz 

• Velocity selection 

– Controlled interaction time 

– Well known sample position 

Atoms individually addressed 

(centimeter separation between atoms) 

Full control of individual transformations 

Complex preparation (53 photons ! ) 

Stable in a weak directing electric field 


Circular states wavefunction 

• Simple expression (maximal quantum numbers spherical harmonic) 


A classical atom 

• All quantum numbers are large. Most properties can be calculated by 

classical arguments (correspondence principle) 

• Eg: stark polarizability and ionisation threshold (atomic units used) 

– Using 

– Weak field limit expand first equation 

– In natural units, polarizability of 50 -2MHz/(V/cm)². -255kHz/(V/cm)² 

differential on the 50 to 51 transition 


• Ionization threshold 


– Eliminate radius in the system: closed equation for θ 

– First term has a maximum, 0.2, obtained for 

– Ionization threshold 

– 165 and 152 V/cm for 50 and 51. Good agreement with measured 

values 



• Spontaneous emission lifetime 

– Radiation reaction force 

– Angular momentum equation 

– Average on long times and note (integration by parts) 

– Circular to circular transition corresponds to one unit angular 

momentum 

– Exact agreement with quantum value 


52 F m=2 

Circular state preparation 

n=52 in 2.5 V/ cm 

Circular 

states 

52 

π 

σ 

1.26 µm 

5D 

776 nm 

250 MHz 

•Three diode laser steps 

51 

•Stark switching to the lower Stark level m=2 

•Adiabatic 250 MHz transitions to the circular state 

•Final microwave transition in a high field: 'purification' 

5P 

σ 

780 nm 

5S 


Working with single atoms 

Method 

Pros and cons 

• Weak excitation of the atomic beam: 

Poisson statistics for the atom 

number in each sample 

• Finite detection efficiency: 40% 

No deterministic preparation of single 

atom samples 

Brute force approach: 

Prepare much less than one (0.1) atom 

on the average 

Extremely easy to achieve 

Long data taking times, growing 

exponentially with atomic samples 

count 

• 1 sample (1 atom): 10 minutes 

• 2 samples: Hours 

• 3 samples: Days 

• 4 samples: Weeks (not very practical) 

When an atom is detected, low 

probability for an undetected second 

one: single atom samples 


Velocity selection 

Doppler selective optical pumping 

55° 

Atomic 

beam 

L 1 

L 2 

L 3 

85 

Rb 

Time of flight 

• Pulsed laser selection of F=3 (2 µs) 

• Pulsed Rydberg excitation (2 µs) 

• Improvement of velocity selection 

1.0 

410.6 +/- 1 m/s 432 +/- 1 m/s 

0.8 

δ 

δ 

F'=4 

120 MHz 5P 3/ 2 

F'=3 

63 MHz 

F'=2 

29 MHz 

F'=1 

Normalized atomic flux 

0.6 

0.4 

0.2 

L 1 L 2 L 3 

7000 

6000 

5000 

0.0 

390 400 410 420 430 440 450 

Velocity (m/s) 

5S 

F=3 

F=2 

Atoms flux 

4000 

3000 

2000 

1000 

0 

∆v=15 m/s 

0 200 400 600 800 

Atomic velocity (m/s) 

• Final width: 2 m/s. 

• Position of atomic sample known 

within 1mm 


Field ionization detection 

D e 

D g 

Electrostatic 

Ionization signals 

n=52 

0.6 K 

lenses 

n=51 

n=50 

4.2 K 

125 V/ cm 136 V/ cm 148 V/ cm 

Field 

77 K 

Electron 

Multipliers 

• Detection efficiency 40% 

• Error rate 4% 

• Dark counts: negligible 

Counting Electronics 


Superconducting cavity 

Design 

Highly polished niobium Mirrors 

• Open Fabry Perot cavity with a 

"photon recirculating ring" 

• Compatible with a static electric field 

(circular state stability and Stark 

tuning) 

• Very sensitive to geometric quality of 

mirrors 

Cavity Damping time: 1 ms 

Field energy (db) 

29 

28 

27 

26 

25 

24 

-2 0 2 4 6 8 10 


time (ms)

Tuning system 

• 15 MHz range Sub Hz sensitivity 


Two cavity modes 

• Two modes with the same geometry and orthogonal linear polarizations 

• Degenerate in an ideal cavity 

• Mirrors imperfections lift this degeneracy: 

– Two modes M a and M b with a frequency splitting 80-130 kHz 

– Atom can be tuned at resonance with either mode via Stark tuning 


Determination of cavity Q 

Cavity width: 100 Hz 

Peak transmission –80 dB 

Millimeter-wave vector network analyzer 

(ABmm) 

•120 dB dynamical range 

•

Ring and coherent atomic state manipulations 

• Use classical microwave sources to 

manipulate atomic state before or 

after interaction with the mode 

S 

• Small access and exit holes in the 

cavity ring: stray fields spoil any 

atomic coherence 

• All coherent manipulations are to be 

performed inside the cavity-ring 

structure 

• Classical fields fed in a low-Q 

transverse standing wave structure 

Independent manipulation of two atomic 

samples feasible (exploit the nodes 

and antinodes) 

Tight constraints on the atomic timing 

No long distance quantum correlations 

(no teleportation experiment) 


The 3 He- 4 He refrigerator 

N 2 

4He 

4 

He 

1.4 K 

3 

He 

0.6 K 

4.2 K 

77 K 


Aim: 

Cavity cooling 

Get rid of a residual 1 photon thermal 

field and of photons left by previous 

experiments 

Timing 

6000 

5000 

Cooling atoms 

prepared in g 

Method 

Send packets of 1-10 atoms in the lower 

state g. 

Counts (A.U.) 

4000 

3000 

2000 

1000 

0 

Probe atom 

prepared in g 

1450 1460 

x10 

They efficiently absorb residual photons 

and cool the cavity mode 

1000 1200 1400 1600 1800 

Performances: 

Detection time (µs) 

Reduction of average thermal photon 

number down to 0.1 

Experiment performed in a time short 

compared to cavity relaxation time T r 


From dream.. To reality 











III) Experimental illustrations of fundamental 

quantum mechanics 

• 1) Cavity induced light shifts and Lamb shifts 

• 2) Micromaser 

• 3) Quantum non-demolition measurement 

• 4) measurement of the Wigner function 

• 5) non classical field states 










Cavity induced shifts 

• Use sensitivity of Ramsey techniques to evidence cavity induced shifts 

• Early experiments performed in a very low Q cavity 


• ∆ c /2π=150 kHz 

Light shifts 

Field relaxation (2 µs) much 

faster than atomic transit time: 

sensitive only to average field 

intensity. Field quantization 

aspects are irrelevant. 


Lamb shifts 

• Interaction with the ‘vacuum’ 

Solid line corrected for residual thermal field (0.32 photons) 

A remarkable single mode Lamb shift effect 










Principle 

• Cumulative emissions in the cavity by Rabi oscillations create a ‘large’ 

field. 

• Very long cavity damping time (closed cavities): field maintained with 

much less than one atom on the average 

• A true quantum device 

• One- and two-photon micromasers realized 

• Garching and ENS 

• Recently: optical analogue microlasers (Kimble) 


• A gain/loss analysis 

A semi-classical model 

• n photons state in the cavity . Probability for atomic emission 

• Photon number rate equation 

• A graphical solution 


Graphical solution for steady state 

• First threshold gain>losses near origin 

• Multiple thresholds and hysteretic behavior 


Quantum model 

• Equation for the photon number distribution (no phase information, no 

coherences) 

• Solution by detailed balance condition. Leads to recursion relation 

• A single operating point. Multistable behaviour washed out by quantum 

fluctuations 

• Gives average photon number and photon fluctuations 


Average photon number 

• Oscillations corresponding to multiple thresholds 

• Dips corresponding to the ‘trapping states’ conditions (gain cancels for 

some photon number) 


Photon number variance 

• Strong sub-poissonian character, particularly near trapping states (ideally 

a Fock state is generated) 

• Large variance close to thresholds 










2π quantum Rabi pulse 

Initial state 

e,0 ⎯⎯→− e,0 

g,1 ⎯⎯→− g,1 

2π pulse 

Conditional dynamics 

e,0 

g,0 ⎯⎯→+ g,0 

Quantum phase gate 

P e 

(t) 

0.8 

51 (level e) 

0.6 

51.1 GHz 

50 (level g) 

0.4 

0.2 

Brune et al, PRL 76, 1800 (96) 

0.0 

time ( µ s) 

0 30 60 90 


A single photon phase-shifts an atomic coherence 

Principle 

Preparation and test of an atomic 

coherence: Ramsey set-up 

e 

g 

π 

P g 

Timing 

Position 

(a) 

C 

e 

π 

π/2 

A 1 

g A 2 

2π 

R 1 

gi 

D 

π/2 

D 

R 2 

gi 

Time 

i 

R 1 

C R 2 D 0 ν−ν gi 

π phase shift of fringes when cavity 

contains one photon 

Signal 

0,9 

0,8 

0,7 

One photon 

Zero photon 

Preparation of |1>: source atom, 

prepared in e, π quantum Rabi 

pulse 

Probability 

0,6 

0,5 

0,4 

0,3 

0,2 

0,1 

0 10 20 30 40 50 60 

Frequency ν (kHz) 


Absorption-free detection of a single photon 

Principle 

0,9 

One photon 

Zero photon 

• Photon detection 

• Photon is still there after the 

detection: QND measurement 

0,8 

0,7 

0,6 

Probability 

0,5 

0,4 

0,3 

0,2 

0,1 

0 10 20 30 40 50 60 


Atomic state correlated to photon 

number for a proper phase 

– i for 0 

– g for 1 

Nogues et al,Nature 400, 239 (99) 


A brief reminder on QND measurements 

An ideal intensity measurement 

Braginsky 1970: ideal quantum 

measurement. 

For light intensity : count the photon 

number without changing it 

Ordinary intensity measurement 

Photon is 

destroyed 

Optical QND measurements: 

•Interaction of "signal" and "meter" beams 

in a Kerr non-linear medium 

•Interferometric detection of index change 

produced by "signal" intensity 

Meter 

Signal 

QND intensity measurement 

Phase reference 

Photon 

detected but 

still present 

A QND detection can be repeated 

In our experiment: 

•Signal: cavity field 

•Meter: atom 


A repeated QND measurement 

Measure twice a single photon 

Conditional probabilities 

• Photon from a small thermal field (0.3 

photon on average) 

• First QND measurement 

• Second "absorptive" measurement 

0,50 

0,45 

0,40 

0,35 

I 1 

if 

1 photon 

G 1 

if 

1 photon 

E 2 

if G 1 

E 2 

if I 1 

E 2 

Timing 

Probability 

0,30 

0,25 

0,20 

Position 

C 

Relaxation 2π 

g 

π/2 

R1 

gi 

A 1 

π/2 

g 

D 

R 2 

gi 

A 2 

π 

Time 

D 

0,15 

0,10 

0 10 20 30 40 50 60 


A clear indication of the QND nature of 

the measurement 


Test of the QND measurement quality 

A three atoms experiment 

Conditional probabilities 

Probe 

Meter 

Source 

0.5 

0.4 

No meter 

Meter in i 2 

Source atom. Prepared in e 

π/2 spontaneous emission 

detected in e: zero photon 

detected in g: one photon 

Probability 

0.3 

0.2 

Meter in g 2 

0.1 

Meter atom: Ramsey fringes set at φ=0 

if zero photon: detected in i 

if one photon: detected in g 

0.0 

g 1 

g 3 

g 1 

e 3 

e 1 

g 3 

e 1 

e 3 

Probe atom: prepared in g, π pulse in one photon 

Absorptive probe of cavity field 

if zero photon: detected in g 

if one photon: detected in e 

• QND error rate 20% 

• Spurious absorption in the mode by 

the meter atom: 20 % 










How to measure W for the electromagnetic field 

Propagating fields : « Tomographic » methods 

Principle : - measure marginal 

distributions P(q θ 

) for different θ 

- inverse Radon transform 

allows reconstruction of W(q,p) 

( medical tomography) 

Refs : - Coherent and squeezed states : - Smithey et al., PRL 70, 1244 (1993) 

- Breitenbach et al., Nature 387, 471 (1997) 

- One-photon Fock state : Lvovsky et al., PRL 87, 050402 (2001) 

- α|0>+β|1> : Lvovsky et al., PRL 88, 250401-1 (2002) 


RESULTATS EXPERIMENTAUX 

Smithey et et al., PRL 70, 

1244 (1993) 

Breitenbach et et al, al, Nature 387, 

471 (1997) 

Comprimé 

Vide 


MESURE MESURE COMPLETE COMPLETE DE DE LA LA DISTRIBUTION DISTRIBUTION DE DE WIGNER WIGNER POUR POUR UN UN PHOTON 

PHOTON 

Lvovsky et et al, al, PRL 87, 050402 (2001) 


Other methods 

• Use the link between W and parity operator 

^ 

W(α) = 2Tr(D( −α)ρ D(α) ( −1) 

N 

) 

^ 

ˆ 

• Displace the field and measure parity by determination of photon number 

probability 

– Direct counting (Banaszek et al for coherent states) 

– Quantum Rabi oscillations for an ion in a trap (Wineland) 

• A demanding method. Much more information than the mere average 

parity needed 


Mesure de la fonction de Wigner pour un ion piégé 

• Etat de vibration d'un ion unique: 

Etat nombre 

n = 1 

1 

2 

( 0 + 1 ) 

Matrice densité 

D. Liebfried et al, PRL 77, 4281 (1996), NIST, Boulder 

• Etat de vibration d'un atome neutre: 

- G.Drobny and V. Buzek, PRA 65 053410 (2002) 

D'aprés les données de: C. Salomon et I. Bouchoule 


Our approach 

- Proposed by Lutterbach and Davidovich (Lutterbach et al. 

PRL 78 (1997) 2547) 

- Based on : 

ρ(-α) 

^ 

W(α) = 2Tr(D( −α)ρD(α)( 

−1) 

D(-α) 

ρ 

^ 

ρ(-α) 

ˆ 

( − 1) N n = 

« parity » operator 

Nˆ 

) 

+ n if n=2k 

W is the expectation value of the Parity operator 

the displaced state ρ(−α) 

− 

n 

if n=2k+1 

( −1) 

A) Apply D(-α) Inject –α in cavity mode OK 

B) How to measure ( −1) 

 

Nˆ 

Nˆ 

in 


Dispersive regime : 

Dispersive interaction 

δ= 

ω 

at 

−ω 

cav 

No energy exchange 

>> Ω/2 

ω 

cav 

|e> 

ω 

at 

|g> 

δ 

But : light shift 

∆E 

∆E 

2 

e, n + 

= Ω 

(n 1) 

4δ 

2 

=− 

n 

4δ 

Ω 

g, n 

1 ( ) 

2 e g 

Phase shift 

1 ( e e g ) 

2 

+ i∆Φ( n) 

of Ramsey fringes 

on the e-g transition 

1 

P(e) 

+ ∆Φ(n)=Φ 0 n 

∆φ(n) 

Empty cavity |0> 

Fock state |n> 

0 


φ

For φ=φ* : 

If N even, detection in e 

If N odd, detection in g 

^ 

Parity Measurement 

=∑ φ 

n 

P ( −1) 

P(n) = Pφ 

*(e)-P 

*(g) 

= C 

n 

Φ 0 =π 

1 

P(e) 

0 

φ∗ 

To measure W(α) : 1) inject – α 

2) measure fringe contrast C 

3) W(α)= 2 C 

N even 

Vacuum |0> 

N odd 

State |1> 

State ρ 

C 

C=+1 for state |2n> 

C=-1 for state |2n+1> 

φ 


Experimental Tools 

- Slow atoms (150m/s) for long interaction times 

- Ramsey interferometer : 

Contrast without cavity : 

70% 

(see experimental poster by T. 

Meunier for more details) 

1 

0.5 

0 

- Injection of a known 

coherent field |α> 

waveguide 

att cav 

dB 

switch 

ν cav =51.099GHz 

S cav 


Testing the method: vacuum state Wigner function 

e-g detection 

∆φ 

π/2 

R2 

position 

Atomic 

frequency 

ν cav 

D(-α) 

π/2 

R1 

|g,0> 

δ 


Cavity mode 

time 

•Use Stark effect to tune interferometer phase 

•No phase information in cavity field: injected field phase irrelevant 

•Finite intrinsic contrast of the Ramsey interferometer 


Wigner function of the "vacuum" 

α=0 

0.6 0.83 

1 

P(e) 

0.4 

0.5 

P(e) 

0.2 

0.6 

0.4 

α=0.6 

(norm.) 

πW(α) 

2 

0 

0.12 0.05 

0 1 2 

N phot 

0.2 

0.6 α=1.25 

1 

P(e) 

0.4 

0.2 

-1 0 1 2 3 

φ/π 

0 

0 1 

α 

2 


Single photon Wigner function measurement 

e-g detection 

∆φ 

π/2 

R2 

position 

Atomic 

frequency 

ν cav 

|e,0> 

π 

D(-α) 

π/2 

R1 

|g,1> 

δ 


Cavity mode 

time 

Preparation 

of cavity state 

Wigner function measurement scheme 


Wigner function of a "one-photon" Fock state 

0,7 

0,6 

α=0 

1,0 

P(e) 

P(e) 

0,5 

0,4 

0,3 

0,7 

0,6 

0,5 

0,4 

0 1 2 3 

α=0.3 

Φ/π 

(norm.) 

πW(α) 

0,5 

0,0 

-0,5 

-1,0 

0,0 0,5 1,0 1,5 2,0 

α 

P(e) 

0,3 

0,7 

0,6 

0,5 

0,4 

0,3 

0 1 2 3 

α=0.81 

0 1 2 3 

Φ/π 

Φ/π 

1 

0.5 

0 

0.71 

0.25 

0.04 

0 1 2 

Nphot 


Towards other states 

- Cavity QED setup : direct measurement of the field 

2 

0 

-2 

2 

- Next improvements : - better isolation 

1 

- better detectors 

0 

More complex states : ex 

( 0 + 1 

)/ 

2 

-1 

-2 

0 

-2 

2 

- In the future : « movie » of the decoherence of a Schrödinger cat 

2 

2 

1 

0 

……. 

1 

0 

-4 

-2 

0 

2 

4 

-2 

-1 

0 

1 

-1 

2 -2 

-4 

-2 

0 

2 

4 

-2 

-1 

0 

1 

-1 

2 -2 










Generation of a single photon state 

• Already used. π quantum Rabi pulse for an atom in e 

• Fidelity about 80% 

• Cumulative emissions lead to other Fock state but atoms are an 

expensive resource and fidelity not very high 

• Creation of multi-photon Fock states 

– Photon pump 

– Two photon emission in a Raman process 


Photon pump 

• Successive π pulses in one cavity mode and recycling from g to e with 

adiabatic rapid passages in a large coherent field stored in the other 

mode (Domokos et al EPJD 1,1) 


Two-photon generation with a single atom 

• A complex raman process involving the two modes 

• Mode M a empty. Mode M b contains a field (thermal or coherent) Atommode 

M a detuning δ close to intermode detuning ∆ 

• A third order resonant process 

PRL 88,143601 

• Coupling amplitude 


Evidence of Raman process 

• Maser emission. Raman observed in ‘sidebands’ for increasing fields in 

M b 


Measuring the photon number 

• Use Ramsey fringes light shifts. Compare generated field with single 

photon field 

• Efficient and high fidelity generation of a two-photon field 











IV) Quantum information with atoms and cavities 

• 1) A reminder on quantum computing 

• 2) Quantum entanglement knitting stitches 

• 3) Cavity assisted collisions 







Ordinateurs classiques et complexité 

Calcul=processus physique 

• Codage des éléments d’information 

(bits) sur des éléments physiques 

(courant, tension). 

• Processus physique conduisant au 

résultat 

Données 

n bits 

N=2 n valeurs 

Complexité 

• Calcul facile: temps et ressources 

polynomiales dans le nombre de bits 

(ex: produit de deux nombres) 

• Calcul difficile: temps et/ou 

ressources exponentielles dans le 

nombre de bits (ex: factorisation) 

Classes de complexité 

• P: Polynomial 

• NP: Non-polynomial mais solution 

vérifiable en temps polynomial 

• NP Complet: problème équivalent à 

tout autre NP-complet 

Résultats 

Une question non résolue 


P 

≠ 

NP

Machine de Turing 

Principe de Church-Turing: 

Du point de vue de la complexité, tous les ordinateurs classiques sont 

équivalents entre eux et équivalents au plus simple: la machine de Turing 

... 

0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 ... 

Registre 

Automate 

1 

0 0 1 0 

état interne 

Un problème difficile sur une machine le reste sur tout autre. 

Seules possibilité pour attaquer les problèmes difficiles: 

• Patience (peu commercial) 

• Parallélisme (mais utilisation exponentielle de ressources ex: calcul 

distribué) 

• Changer les lois physiques du calcul: calcul quantique 


Principe d'un ordinateur quantique 

Bits et qubits 

Les bits sont remplacés par des 

systèmes à deux niveaux : 

superpositions d'états 

Représentation des données 

Registre de n qubits: 

Base 

1 

2 

( 0 + 1 ) 

espace des états de dim. 2 n 

0,0,0, …,0 = 0 

0,0,0, …,1 = 1 

1,1,1, …,1 

= 2 n −1 

Un vecteur code un nombre 

Lecture d'un registre 

Mesure d'une quantité sur chaque qubit 

ayant |0> et |1> pour vecteurs 

propres et 0 et 1 pour valeurs 

propres. 

Lecture indépendante des n qubits: un 

nombre entre 0 et 2 n -1 

(indice du vecteur correspondant). 

On peut représenter les nombres par 

des vecteurs d’état et lire les valeurs 

finales (au sens de la mesure 

quantique). 

Peut-on calculer 


Calcul quantique 

Évolution quantique unitaire. Définie par l’évolution des vecteurs de base 

ε , ε ,…, ε ⎯⎯→ U ε , ε ,…, ε ε = 0,1 

1 2 n 

1 2 

Un ordinateur quantique est au moins équivalent à une machine de Turing 

classique: tout calcul classique est réalisable à condition de respecter l'unitarité 

n 

x 

⎯⎯→ 

f ( x) 

est interdit si f n'est pas inversible 

Mais on peut toujours former U tel que, avec deux registres de n qubits : 

x, y ⎯⎯→ x, y⊕ 

f( x) 

x,0 ⎯⎯→ x, f( x) 

Lecture du résultat: mesure du second registre 


Parallélisme quantique massif 

On peut calculer une valeur de la fonction f 

x,0 ⎯⎯→ x, f( x) 

Mais on peut aussi, dans le même temps, calculer toutes les valeurs de f 

∑ 

x 

∑ 

x,0 ⎯⎯→ x, f( x) 

Calcul intrinsèquement massivement parallèle. Exponentiellement plus 

efficace qu'un calculateur classique. 

x 

Lecture du résultat 

Naïvement, on n'obtient qu'une valeur de f d'argument aléatoire. 

Pas évident de tirer parti de ce parallélisme. 

Peu d'algorithmes efficaces connus pour un ordinateur quantique 


Une brève histoire de l'informatique quantique 

La préhistoire 

73 Bennett 

Calcul réversible classique 

80 Benioff 

Proposition de principe 

82 Feynman 

Quantum simulator 

Dimension exponentielle de l'espace 

85 Deutsch 

Machines de Turing quantique 

Bases conceptuelles 

92 Deutsch 

Premiers algorithmes ad-hoc 

Portes logiques quantiques 

L'âge d'or 

94 Shor 

95 

Algorithme de factorisation 

Utile et accélération exponentielle 

Propositions théoriques de portes 

95 Wineland 

Première réalisation d'une porte 

97 Grover 

98-99 

00-01 

Algorithme de recherche 

Utile mais accélération faible 

Premiers algorithmes quantiques 

réalisés en RMN 

Premières manipulations d'états 

intriqués complexes 


Portes logiques quantiques 

Théorème 

• Toute transformation unitaire de n 

qubits est décomposable en un 

produit de transformations unitaires 

élémentaires à un et deux qubits 

Exemples de portes 

Portes à un qubit. 

• Transformation la plus générale d’un 

système à deux niveaux: 

Traduction: 

• Tout calcul quantique est réalisable 

par application successive de 

« portes logiques quantiques »: 

machines portant sur un ou deux 

qubits 

Portes universelles 

• Un ensemble fini de portes qui 

permettent de réaliser par 

association tout réseau de calcul 

quantique 

iψ 

⎛ cosϕ 

e sinϕ 

⎞ 

U ( ϕψ , ) = ⎜ −iψ 

⎟ 

⎝−e 

sinϕ 

cosϕ 

⎠ 

représentation graphique: 

• Cas particulier: porte de Hadamard 


U 

1 1 

H = ⎛ ⎞ H 

⎜ ⎟ 

⎝1 −1⎠ 

⎧ 1 

0 → 0 + 1 

⎪ 2 

⎨ 

⎪ 1 

1 → 0 − 1 

⎪⎩ 2 

( ) 

( )

Portes (suite 1) 

• Porte Non 

• Porte CNOT 

Ν 

0 1 

N = ⎛ ⎞ 

⎜ ⎟ 

⎝1 0⎠ 

Portes à deux qubits 

• Porte de phase 

CNOT 

⎛1 

⎞ 

⎜ ⎟ 

1 

= ⎜ ⎟ 

⎜ 0 1 ⎟ 

⎜ 

1 0⎟ 

⎝ ⎠ 

π 

Remarque 

⎛1 

⎞ 

⎜ 

⎟ 

1 

π = ⎜ 

⎟ 

⎜ 1 0 ⎟ 

⎜ 

0 −1⎟ 

⎝ 

⎠ 

= 

H 

π 

H 


Portes (suite 2) 

Portes à n bits 

• Porte de Toffoli 

Calcul réversible classique 

Universalité 

Tout calcul quantique peut être réalisé 

avec: 

• Des portes à un qubit 

• Des portes CNOT 

Ex: un additionneur 

• Control U 

U 

– Faire U sur n-1 bits si le bit de 

contrôle est à un 

– Ne rien faire sinon 


Une opération utile 

Préparation d’un registre de n bits dans un superposition de toutes les 

valeurs 

n 

2 −1 

1 

x ⎯⎯→ ∑ y 

n 

2 0 

Préparation de l’état |0> et application d’une Hadamard sur chaque qubit 

1 

0 ⎯⎯→ 0 + 1 

2 

( ) 

pour chaque qubit 

n 

1 1 

0 ⎯⎯→ 0 + 1 = ∑ 

n−1 2 −1 

∏( 

) 

n 

n 

2 0 

2 

0 

y 

pour le registre 


Calcul de fonctions élémentaires 

Fonctions de {0,1} dans {0,1} 

Deux qubits 

Ligne rouge: |x>. Ne sera pas affecté. Ligne verte |y> devient |y+f(x)> 

4 fonctions: 

f(x) = 0 ∀ x 

f(x) 

0 

0 Ν 

01 

f(x) = 1 ∀ x 

01 

0 

0 

Ν 

Ν 

Ν 

01 

f(x) 

01 

Fonctions « constantes » Fonctions « balancées » 

A un bit, il n’y a que des fonctions constantes ou balancées. Situations plus complexes 

pour les fonctions de n bits dans n bits. 


Quelques fonctions de {0,1,2,3} sur {0,1} 

2 4 =16 fonctions possibles 

f(x) 

0 N 

01 23 

f(x) 

0 N N 

01 23 

Exemple de 

fonction 

constante 

(f=1. Deux 

telles fonctions) 

Exemple de 

Fonction 

balancée (autant 

de f(x) = 0 que de 

f(x) = 1) 

(six fonctions) 

0 N 

f(x) 

01 23 

Tous les autres cas se déduisent 

simplement de ceux-ci 

Fonction ni 

constante ni 

balancée 

(huit fonctions) 


Les problèmes posés sous forme d’oracle 

Les problèmes logiques que nous allons considérer ici sont posés sous forme d ’ 

«oracle». On suppose qu’ une machine programmée selon des règles inconnues 

(décrite comme une «boîte noire» ou oracle), calcule une fonction dont nous ne 

connaissons que certaines caractéristiques. Le problème consiste à déterminer une 

propriété inconnue de la fonction, sans «ouvrir» la boite. Nous pouvons interroger 

l ’oracle en entrant des données dans la boite et en manipulant sa sortie, sans l’ouvrir 

pour en analyser le contenu. Le problème sera «facile» si sa résolution demande un 

nombre total d’opérations croissant de façon polynomiale avec le nombre de bits, 

«difficile» s’il croit de façon exponentielle avec ce nombre. Nous allons montrer que le 

passage du calcul classique au calcul quantique transforme certains oracles classiques 

difficiles en oracles quantiques faciles. Dans d’ autres cas, le problème quantique reste 

difficile, mais moins que le problème classique (croissance toujours exponentielle du 

nombre d ’opérations, mais avec un exposant plus petit que classiquement ). 

0,1,1,1…..1, 0 0,0,1,0…..1, 0 

f(x) 

Choix libre de la 

préparation des 

bits d ’entrée 

« Lecture » du 

programme 

interdite 

Choix libre des 

opérations sur les 

bits de sortie 


L’oracle de Deutsch-Josza 

f(x) constante ou 

balancée 

f(x) est une fonction booléenne de [ 0, 2 n -1] dans [ 0, 1]. On sait 

qu’elle est soit constante, soit balancée. Est-elle l’un ou l ’autre 

Classiquement, il faut « interroger » l’oracle 2 n− 1 +1 fois pour répondre à la question à coup sûr 

(il faut introduire 2 n− 1 +1 valeurs différentes de x et calculer f(x) à chaque fois) → Croissance 

exponentielle avec n du nombre d ’opérations et problème classique « difficile » 

L’oracle de Grover f(x) est une fonction booléenne de [ 0, 2 n -1] dans [ 0, 1] qui n’est non nulle 

f (x) = δ (x-x 0 

) 

que pour x = x 0 

. Trouver x 0 

. 

x 0 

 

Equivaut à la recherche «inversée» d’un abonné dans un annuaire à partir de son 

numéro connu a. Les x sont les abonnés, f(x) vaut 1 si a est le numéro de x, 0 sinon. 

Classiquement, il faut calculer f(x ) («consulter» l ’annuaire) N = 2 n − 1 fois pour trouver à coup 

sûr. Problème classique difficile. 

L’oracle de Simon 

« Période » de 

f(x) 

Exemples d’oracles classiquement difficiles 

f(x) est une fonction de [0, 2 n -1] dans [0, 2 n -1] telle que f (x’) = f (x) ssi x 

= x ⊕ s où s est une suite inconnue à n termes de « 0 » et « 1 » et ⊕ 

représente l ’addition « bit à bit » (s : «période» de f ). Déterminer s 

Classiquement, il faut calculer f (x ) pour des valeurs aléatoires de x jusqu ’à trouver deux x et 

x’ tels que f (x) = f (x’). Alors x ⊕ s = x’ et x ⊕ x’ = x ⊕ x ⊕ s = s (car x ⊕ x ={ 0} ). Il faut 

2 n - 1 + 1 opérations pour trouver la réponse à coup sûr → Problème classique « difficile ». 


Exemple élémentaire 

Algorithme de Deutsch-Josza à un qubit. 

Principe du calcul 

Déterminer si f de {0,1} dans {0,1} est 

constante ou balancée. 

Une autre formulation du problème: 

comment déterminer qu’une pièce a 

bien un côté pile et un côté face 

• Classiquement: regarder les deux 

côtés. 

• Quantiquement: regarder en une fois 

une superposition quantique des 

deux côtés. 

Deux qubits pour calculer f 

|0> 

|1> 

H 

H 

1 

1 

2 

2 

( 0 + 1 ) 

( 0 + 1 ) cste 0 

( 0 − 1 ) 

( 0 − 1 ) 

2 

f 

Un seul calcul de f pour décider de sa 

nature. 

1 

1 

2 

2 

( 0 − 1 ) bal 1 

1 

H 


Fonction constante : 0 

1 

2 

( 0 + 1 )( 0 − 1 ) 

Deux cas parmi quatre 

f 1 

⎯⎯→ ⎡ 0 

2 ⎣ 

+ 1 

= 1 ⎡ 0 1 2 ⎣ 

+ 

⎦ 

= 

2 

0 1 

H 1 

⎯⎯→ 

( 0 

2 0 − 1 ) 

( 0 ⊕ f (0) − 1 ⊕ f (0) ) ( 0 ⊕ f(1) − 1 ⊕ f(1) 

) 

( 0 − 1 ) ( 0 − 1 ) ⎤ ( + )( 0 − 1 ) 

|0> 

|1> 

⎤ 

⎦ 

H 

H 

1 

1 

2 

2 

( 0 + 1 ) 

( 0 − 1 ) 

f 

H 

Fonction balancée: identité 

1 

2 

( 0 + 1 )( 0 − 1 ) 

f 1 

⎯⎯→ ⎡ 0 

2 ⎣ 

+ 1 

= 1 ⎡ 0 1 2 ⎣ 

+ 

⎦ 

= 

2 

0 1 

H 1 

⎯⎯→ 

( 0 

2 1 − 1 ) 

( 0 ⊕ f (0) − 1 ⊕ f (0) ) ( 0 ⊕ f(1) − 1 ⊕ f(1) 

) 

( 0 − 1 ) ( 1 − 0 ) ⎤ ( − )( 0 − 1 ) 

⎤ 

⎦ 


Généralisation : Deutsch-Josza pour n qubits 

(1/2 n/2 ) Σ x 

| x > 

(1/2 n/2 ) Σ x 

(− 1) f(x) | x > 

| {0} > A 

H 1 .H 2 ….H n 

N H 

f(x) constante ou 

balancée 

| 0> B 

(1/2 1/2 ) [| 0 > − | 1 >] 

(1/2 1/2 ) [| 0 > − | 1 >] 

Le registre d’entrée A (n qubits) est préparé (par application de la transformation de Hadamard 

H sur chaque qubit) dans la superposition symétrique des 2 n états | x > possibles. 

Le registre de sortie B (1 qubit) est inversé par N, puis préparé par H dans (1/2 1/2 ) [| 0 > − | 1 >] . 

Action de l ’oracle: | x > | 0 > → | x > | f (x ) > et | x > | 1 > → | x > | 1 ⊕ f (x ) > 

Si f (x ) = 0: | x > [| 0 > − | 1 > ] → | x > [| 0 > − | 1 > ] 

} ( − 1) f(x) | x > [| 0 > − | 1 > ] 

Si f (x ) = 1: | x > [| 0 > − | 1 > ] →−| x > [| 0 > − | 1 > ] 

Et par superposition: 

(1/2 ( n+1)/2 ) Σ x 

| x > [| 0 > − | 1 > ] → (1/2 (n+1)/2 ) (Σ x 

(− 1) f(x) | x > ) [| 0 > − | 1 > ] 

Les registres restent non intriqués après l’oracle. Déphasage des amplitudes 

dans le registre A en (− 1) f(x) . 


Deux possibilités 

Si f(x ) est constante: f (x ) = 0 ∀ x ou f (x ) =1 ∀ x → 

(1/2 n/2 ) Σ x 

(− 1) f(x) | x > = ± (1/2 n/2 ) Σ x 

| x > → registre A inchangé (au signe près) 

Si f(x) est balancée: autant de f (x ) = 0 que de f(x ) = 1 → 

Autant d ’amplitudes +1 que d ’amplitudes −1 dans la superposition finale du registre A 

→ Σ x 

(− 1) f(x) | x > orthogonal à Σ x 

| x > 

Résoudre l ’oracle revient à distinguer deux états orthogonaux de l’état final du registre A: 

On applique à nouveau H à tous les qubits. Comme H 2 =1, on retrouve l’état initial |{0} > si 

f(x ) est constante, un état orthogonal si f(x) est balancée → au moins un des qubits doit 

alors être 1. On le vérifie en mesurant les qubits finals de A. 

| 0, 0, 0, ... > ou 

mesure 

H 1 .H 2 ….H n 

H 1 .H 2 ….H état orthogonal 

n 

f(x) 

à | 0, 0, 0, ... > 

des qubits 

N 

H 

La réponse nécessite au plus 3n+2 opérations à un qubit (2n + 1 opérations H, une 

opération de bascule (N) et au plus mesure de n qubits (on peut arrêter dès qu’on 

trouve un 1) → problème quantiquement «facile». 


Remarques 

1. Où est l’intrication 

Les qubits de A ne sont pas intriqués à B qui reste inchangé. De l’intrication est en général 

cependant créée entre les qubits de A: 

Exemple. Cas d ’une fonction balancée agissant sur un registre A de trois qubits: 

Σ x 

(− 1) f(x) | x >= | 000 > − | 001 > + | 010 > − | 011 > + | 100 > − | 101 > − | 110 > + | 111 > 

= | 0 > 1 [| 00 > − | 01 > + | 10 > − | 11 >] 23 + | 1> 1 [| 00 > − | 01 > − | 10 > + | 11 >] 23 

= | 0 > 1 | Ψ > 23 + | 1 > 1 | Φ > 23 avec 23 < Φ | Ψ > 23 = 0 

Cette décomposition de montre que le qubit 1 et l ’ensemble des qubits 2 et 3 

sont maximalement intriqués. 

2. Cet algorithme est-il vraiment avantageux par rapport à la procédure classique 

L’ avantage de l’ algorithme quantique n’existe que si on cherche une réponse certaine. 

Si on s’ autorise une probabilité finie ε d’ erreur, aussi petite soit-elle, l’algorithme 

classique (calcul successif de f (x) pour des valeurs de x tirées au hasard) donne un 

résultat acceptable au bout de k ≅ − log 2 

(ε ) opérations (nombre indépendant de n). Le 

problème classique devient donc «facile» dès qu ’on accepte un taux fini d ’erreur. Ceci 

diminue considérablement l ’intérêt de l ’algorithme quantique puisqu’ il faut être sûr 

de pouvoir l ’effectuer sans aucune décohérence pour qu ’il soit avantageux par rapport 

à la version classique. 


n qubits 

L’algorithme de Simon 

(1/√2)[ | x > + | x ⊕ s >] 

| {0} > A 

H 1 .H 2 ….H n 

x 

| {0} > B 

n qubits 

« Période » de 

f(x) 

} Σ | x> | f(x)> 

Mesure de B: 

résultat f (x) 

H 1 .H 2 ….H n 

On réalise la suite d’opérations schématisée ci-dessus: calcul parallèle de toutes les valeurs 

de la fonction suivie d’une mesure du registre B projetant A dans une superposition de 

deux états qui diffèrent bit à bit de la quantité inconnue s. On applique alors à nouveau les 

transformations de Hadamard aux n qubits de A: elles font évoluer chaque qubit suivant la 

loi: | 0 > → (1/√2) [ | 0 > + | 1> ] et | 1 > → (1/√2) [ | 0 > − | 1> ]. Un état |{ x }> (produit 

de n états | x i 

> avec x i 

= 0 ou 1) devient une superposition de produits d’états | y i 

> (avec y i 

= 0 ou 1). Les coefficients de cette superposition valent + 1 ou −1suivant la parité de 

la somme Σ i 

x i 

y i 

: 

|{ x }> = | x 1 

, x 2 

, x 3 

,…..x n 

> → = (1/2 (n+1)/2 ) Σ { y } 

(−1) { Σ i x i y i } | y 1 

, y 2 

, y 3 

,…..y n 

> 

L’état final de A est donc: 

(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 

, y 2 

, y 3 

,…..y n 

> 

= (1/2 (n+1)/2 ) Σ {y} (−1) { Σ i x i y i } [ 1 + (−1) { Σ i s i y i } ] | y 1 

, y 2 

, y 3 

,…..y n 

> 

Une mesure répétée ~ n fois de A va alors nous permettre de déterminer s 


Détermination de la période inconnue 

| Ψ ( final) > A 

= (1/2 (n+1)/2 ) Σ {y} (−1) { Σ i x i y i } [ 1 + (−1) { Σ i s i y i } ] | y 1 

, y 2 

, y 3 

,…..y n 

> 

Amplitude non nulle ssi 

Σ i 

s i 

y i 

= 0 (modulo 2) 

Une mesure des qubits individuels donne une suite y 1a 

, y 2a 

, y 3a 

,…..y na 

de valeurs 0 et 1 qui 

satisfait la condition: 

Σ i 

s i 

y ia 

= 0 (modulo 2). 

On recommence n fois l ’opération et on obtient ainsi, en général, n relations 

indépendantes (si par hasard deux mesures donnent le même vecteur, on recommence 

une fois de plus): 

Σ i 

s i 

y ia 

= 0 

Σ i 

s i 

y ib 

= 0 

. . . . . . . . 

Σ i 

s i 

y in 

= 0 

La résolution de ce système d ’équations donne s. Le processus requiert ≅ 4n 2 opérations. Le 

problème est donc quantiquement facile. De plus, il tolère des erreurs puisqu’on peut toujours 

vérifier le résultat en comparant f ( x ) et f (x ⊕ s) une fois s obtenu. 


Rôle de l’intrication et de la mesure 

Dans l’ algorithme de Simon, l’ intrication et la mesure projective jouent un rôle plus 

essentiel que dans ceux de Deutsch et Grover. L’oracle intrique les registres A et B, puis la 

mesure de B projette A dans une superposition de deux états seulement. Après mélange par 

la « lame séparatrice », la signature du signal d ’interférence final nous renseigne sur la 

séparation de ces deux états, donc sur la période cherchée. Quoique mathématiquement 

plus compliqué, l’algorithme de Shor, basé sur la recherche de la période d ’une fonction, 

ressemble beaucoup dans son principe à celui de Simon. 

Remarque: il n’ est même pas besoin de «lire» la mesure du registre B. Il suffit d’ avoir intriqué 

B à son appareil de mesure , ce qui réduit A à un mélange statistique de superpositions | x > + | 

x ⊕ s >. Leur recombinaison finale par H 1 

.H 2 

…H n 

ne conduit, quel que soit x, à une 

interférence constructive que pour les états | y 1 

, y 2 

, y 3 

,…..y n 

> satisfaisant les équations 

linéaires de la page précédente (on peut réduire le nombre d’opérations à 3n 2 ). 


Factorisation 

Un problème classique difficile 

Meilleur algorithme connu (Number Field Sieve) sur n bits en 

1 

1.9n 3 

e 

Factorisation de RSA 155: 8000 MIPS-années soit 2.5 10 17 instructions ! 

109417386415705274218097073220403576120037329454492059909138421314763499842889347847179 

97257891267332497625752899781833797076537244027146743531593354333897=102639592829741 

105772054196573991675900716567808038066803341933521790711307779*10660348838016845482 

0927220360012878679207958575989291522270608237193062808643 

Problème difficile dont l'inverse est facile (multiplication) 

Idéal pour la cryptographie (codage et décodage faciles, casser le code très 

difficile) 

Un algorithme rapide de factorisation aurait des conséquences énormes sur 

les algorithmes de cryptage (et sur l'économie) 

1994: Shor propose un algorithme de factorisation rapide sur un ordinateur 

quantique 


Algorithme de Shor 

Algorithme de factorisation 

exponentiellement plus efficace 

que la version classique 

but : factoriser N>>1 

Ordre d'un entier 

x

Exemple élémentaire 

Factoriser 15 

Ordre de x 

On choisit x=7 

R=4 

On calcule 7 a [15] 

a 

7 a [15] 

Le gcd de 4+1 et 15 est un facteur de 15 

Le gcd de 4-1 et 15 est un facteur de 15 

1 

2 

7 

4 

15=5x3 

3 

13 

4 

1 


Algorithme quantique (1) 

Principe 

Calculer beaucoup de valeurs de f 

en utilisant le parallélisme et 

extraire la période (~ N) 

Deux registres de m qubits. 

q=2 m choisi tel que 

2N 2 < q < 4N 2 

État initial 

0,0 

Superposition de tous les nombres 

q−1 

1 

0,0 ⎯⎯→ ϕ = ∑ a,0 

q a= 

0 

Pour chaque qubit: 

1 

0 ⎯⎯→ ( 0 + 1 ) 

2 

Exponentiation modulaire 

q−1 

1 

a 

ϕ ⎯⎯→ ∑ ax , N 

q 

Possible de façon efficace 

Utilise le parallélisme 

Mesure du second registre 

Obtention d'une valeur aléatoire y 

Projection du premier registre sur les 

antécédents de y 


k+jr 

a= 

0 

[ ] 

k aléatoire, j entier 

Le premier registre contient une 

superposition d'états répartis 

périodiquement avec la période r 

1 

A 

∑ 

j 

k 

+ 

jr

Amplitude 

Algorithme quantique (2) 

Répartition des amplitudes 

Après la transformation de Fourier 

k k+r k+2r k+jr Vecteur 

La mesure du premier registre en 

l'état ne donne aucune 

information ("offset" k aléatoire) 

Extraire la période: 

réaliser une transformation de 

Fourier discrète 

Possible de façon efficace (en un 

temps polynomial) Amélioration 

exponentielle/FFT classique 

q/ r 2q/ r 

Mesure du registre: une valeur de la 

forme 

p q/r 

p entier arbitraire 

Par un calcul classique efficace, 

extraction, avec une probabilité 

finie, de r et factorisation de N 

Énormes conséquences pratiques si 

on peut réaliser cet algorithme 


Quelques caractéristiques importantes 

Utilise le parallélisme massif 

calcul simultané de toutes les 

exponentiations modulaires 

ϕ ⎯⎯→ 

Probabiliste 

1 

• Probabilité finie d'obtenir le 

résultat 

• Ne décroît pas exponentiellement 

avec le nombre de bits 

• Résultat facile à vérifier 

q 

q−1 

∑ 

a= 

0 

ax , 

a 

[ N] 

Utilise des effets authentiquement 

quantiques 

État 

1 

q 

État intriqué 

q−1 

∑ 

a= 

0 

ax , 

[ N] 

analogue à la paire EPR 

1 

( , , ) 

2 + − − − + 

Calcul: manipulation d'intrication 

Mesure du second registre: 

projection du premier 

Rien de comparable avec un 

ordinateur analogique classique 

(même avec superpositions) 

a 


Simulation quantique 

Un autre domaine pour le calcul 

quantique 

Simuler la dynamique ou les états 

propres d’un système quantique 

Ex: trouver l’état fondamental d’un 

système de spins en interaction sur 

un réseau 

Intérêts 

• Dès 30 qubits, traiter des problèmes 

inaccessibles aux calculs classiques. 

• Par rapport au système donné par la 

nature, on peut faire varier la force et 

la nature des interactions. 

Classiquement difficile: la taille de 

l’espace de Hilbert croît 

exponentiellement avec la taille du 

système 

Feynman 1985: utiliser un système 

quantique pour en simuler un autre. 

Problèmes: 

• Peu de systèmes facilement 

simulables. 

• Problème de la décohérence 








Initial atom-cavity state 

1 

Ψ (0) = e,0 = + ,0 + −,0 

2 

( ) 

State at time t: 

Ω t Ω t 

Ψ = + 

2 2 

0 0 

() t cos e,0 sin g,1 

1+ cosΩ 

Probability for being in e : 

0t 

Pe 

() t = 

2 

•Oscillatory spontaneous 

emission 

•An atomic transition saturated by 

a single photon 

•Non-linear optics at the single 

photon level. 

P e 

(t) 

0.8 

0.6 

0.4 

0.2 

0.0 

time ( µ s) 

0 30 60 90 


Quantum Rabi oscillations: state transformations 

Initial state 

e,0 

1 

e,0 e⎯⎯→− 

,0 e,0 ⎯⎯→ eg,0 

,1 2π ( epulse 

,0 + g,1 

) 

2 

g,1 ( ⎯⎯→− ce e + cg π/2 spontaneous g,1) 0 ⎯⎯→ g 

Conditional ( ce 1 + c 

emission dynamics 

g 

0 ) 

pulse 

π spontaneous emission pulse 

g,0 ⎯⎯→+ Entanglement g,0 

Quantum 

creation 

Atom/cavity state copy phase gate 

Atom-cavity EPR pair 

P e 

(t) 

0.8 

51 (level e) 

0.6 

51.1 GHz 

50 (level g) 

0.4 

0.2 

0.0 

Brune et al, PRL 76, 1800 (96) 

time ( µ s) 

0 30 60 90 


Three "stitches" to "knit" quantum entanglement 

Combine elementary transformations to create complex entangled states 

• State copy with a π pulse 

– Quantum memory : PRL 79, 769 (97) 

• Creation of entanglement with a π/2 pulse 

– EPR atomic pairs : PRL 79, 1 (97) 

• Quantum phase gate based on a 2π pulse 

– Quantum gate : PRL 83, 5166 (99) 

– Absorption-free detection of a single photon: Nature 400, 239 (99) 

• Entanglement of three systems (six operations on four qubits) 

– GHZ Triplets : Science 288, 2024 (00) 

• Entanglement of two radiation field modes 

– Phys. Rev. A 64, 050301 (2001) 

• Direct entanglement of two atoms in a cavity-assisted collision 

– Phys. Rev. Lett. 87, 037902 (2001) 


Quantum memory 

• Use the π quantum Rabi pulse to transfer a qubit from an atom to the 

cavity and back 

• Timing 

• An useful space-time diagram for the timing of complex experiments 


• Final signal 

Coherent information transfer 

• Ramsey fringes with the two pulses on different atoms and transient 

storage of quantum information in the cavity field 


Quantum memory lifetime 

• Contrast of the fringes as a function of time 

0.6 

0.4 

Fringes amplitude 

0.2 

0.0 

0 1 2 3 

T/T 

cav 

• Coherence lifetime is twice cavity damping time (equal superposition of 0 

and 1) 


Preparation and detection of a single photon state 

• First atom in e: preparation of a single photon Fock state 

• Other atom read out: measurement of a Fock state lifetime 

1.0 

Π 

fe 

First atom sent in e 

Second atom sent in g 

0.8 

Delay in C: T 

7000 coincidences per point 

0.6 

0.4 

0.2 

Two data sets for two modes 

T cav 

=84 µs 

T cav 

=112 µs 

Conditional probability 

second in e 

when first in g 

Maximum value 75% 

well understood with 

known experimental 

imperfections 

0.0 

0 1 2 3 4 5 

T/T 

cav 


Creation of an EPR atom pair 

A simple entanglement manipulation experiment 

• Initial state 

g 

e 

eg , ,0 

π/2 pulse: 

•Entanglement creation 

1 

( ,0 + ,1 ) 

2 e g g 

•State copy 

•Final state 

1 

( ) 

1 

eg , − ge , in spin terms: ( ↑↓ , −↓↑ , ) 

2 2 

1 

= →← , −←→ , 

2 

( ) 

Hagley et al, PRL 79, 1 (97) 


Testing entanglement 

Two complementary experiments 

Spin singlet state is rotation-invariant: 

• Spin anticorrelations along any 

detection axis 

– Check atomic energy 

anticorrelations (detection along 

0z) 

• "longitudinal experiment" 

Longitudinal experiment 

Direct detection of atomic energies 

Timing 

Position 

D D 

C 

e A 1 g A 2 

Expect. Obs 

e,e 

0 0.10 

Time 

– Check that superposition is 

coherent by detecting spins in a 

non-compatible basis (axes in the 

horizontal plane of Bloch sphere) 

• "transverse experiment" 

63 % of pairs present the expected 

correlation. 

Imperfections well accounted for by 

cavity relaxation (T r 

=112µs) and π 

pulse imperfections 


e,g 

g,e 

g,g 

0.5 

0.5 

0 

0.42 

0.27 

0.21

Transverse experiment 

Principle 

• Measure atom 1 along x axis 

• Measure atom 2 along φ axis 

– (apply Ramsey pulses with 

adjustable phase φ on two atoms) 

z 

z 

Position 

Timing 

C 

R 1 

eg 

D 

π/2 π/2 

D 

R 2 

eg 

y 

y 

e 

A 1 g A 2 

Time 

x 

x 

φ 

"Bell signal" 

0.3 

• Measure 

σ σ 

• Expect +1 for φ=π, -1 for φ=0 

x 

ϕ 

Bell signal 

0.2 

0.1 

0.0 

-0.1 

Ramsey fringes with two pulses on two 

different atoms ! 

-2 0 2 4 


-0.2 

-0.3 

φ/π

Fidelity 

Fidelity estimate F = Tr( ρ ΨEPR 

ΨEPR 

) 

Assuming that imperfections do not create unwanted coherences: 

P + V⊥ 

F = 

2 

• P : population in the "expected" channels in longitudinal experiment 

»0.71 

• V ⊥ : visibility of the Bell signal in the transverse experiment 

»0.25 

Hence F=48% 

Violation of Bell inequalities Requires a Bell contrast signal >0.71 


Principle: 

First atom 

Initial state 

π/2 pulse in M a 

π pulse in M b 

Second atom: 

probes field states 

Entangling two modes of the radiation field 

• Final transfer rate modulated versus 

the delay at the beat note between 

modes 

∆ 

0 

−δ 

A s 

π/2 

π 

e,0,0 

D 

Single photon beats 

1.0 

P e 

(T) 

0.5 

( ,0,0 ,1,0 ) 

2 e + g 

0.0 

( 0,1 1,0 ) 

2 g + 1.0 

1 

1 

A p 

π 

(b) 

π/2 

M a 

M b 

D 

0.5 

0.0 

1.0 

0.5 

0.0 

48 50 52 54 56 58 

200 202 204 206 208 

400 402 404 406 408 

0.0 

0 π/2Ω 3π/2Ω T Τ+π/Ω t 

698 700 702 704 706 

T(µs) 


1.0 

0.5 

(a) 

(b) 

(c) 

(d)

A quantum phase gate 

Principle 

2π quantum Rabi pulse: 

conditional dynamics 

e 

g 

π 

D 

C 

i 

"Truth table" 

49 (level i ) 

51.1 GHz 

cavi ty 

50 (level g) 

54.3 GHz 

Ramsey source 

• Cavity qubit: states |0> and |1> 

• Atomic qubit: states |i> and |g> 

i,0 ⎯⎯→ i,0 

i,1 ⎯⎯→ i,1 

g,0 ⎯⎯→ g,0 

iφ 

g,1 ⎯⎯→− g,1 = e g,1 

51 (level e) 

Tests 

Two complementary experiments 

• A single photon shifts the phase 

of an atomic coherence 

1 1 

0 0 

2 2 

1 1 

1 1 

2 2 

( i + g ) ⎯⎯→ ( i + g ) 

( i + g ) ⎯⎯→ ( i − g ) 

• A single atom phase shifts the 

cavity field 

( 0 

0 + 

1 

1 ) ⎯⎯→ ( 0 

0 + 

1 

1 ) 

( 0 

0 + 

1 

1 ) ⎯⎯→ ( 0 

0 − 

1 

1 ) 

i c c i c c 

g c c g c c 

Rauschenbeutel et al., PRL 83, 5166 (99) 

Quantum phase gate 


A single photon phase-shifts an atomic coherence 

Principle 

Preparation and test of an atomic 

coherence: Ramsey set-up 

e 

g 

π 

P g 

Timing 

Position 

(a) 

C 

e 

π 

π/2 

A 1 

g A 2 

2π 

R 1 

gi 

D 

π/2 

D 

R 2 

gi 

Time 

i 

R 1 

C R 2 D 0 ν−ν gi 

π phase shift of fringes when cavity 

contains one photon 

Signal 

0,9 

0,8 

0,7 

One photon 

Zero photon 

Preparation of |1>: source atom, 

prepared in e, π quantum Rabi 

pulse 

Probability 

0,6 

0,5 

0,4 

0,3 

0,2 

0,1 

0 10 20 30 40 50 60 



A single atom phase-shifts the field 

Principle 

|0> and |1> superposition: 

small coherent field 

α ≈ c 0 + c 1 

0 1 

Amplitude read-out: 

atom in g. π Rabi pulse 

Minimum transfer: 

minimum amplitude 

Atom in i 

c 0 + c 1 ≈ α 

0 1 

in g 

c 0 − c 1 ≈−α 

0 1 

Timing 

Homodyne detection 

i 

α + αe φ 

i 

− α + αe φ 

Zero amplitude for 

φ = π φ = 0 

Position 

C 

(b) 

α 

g 

A 2 

2π 

π/2 

R1 

gi 

D 

αe ιθ 

g 

A 3 

π 

Time 

D 


Coherent gate operation 

Atom in i 

0.4 

Probe transfer 

0.3 

Atom in g 

0.2 

0.4 

-2 -1 0 1 2 

φ/π 

A fully coherent 

quantum gate 

Probe transfer 

0.3 

0.2 

-2 -1 0 1 2 

φ/π 


Tuning the quantum gate phase 

Role of atom-cavity detuning 

Phase of gate: 

360 

• For δ=0 : resonant interaction. π gate 

• Large δ: dispersive regime. Transient 

modification of atom and cavity 

frequencies: gate with a small angle 

• Intermediate δ range: 

– Intermediate values of φ 

– Absorption remains small (

“Quantum program”: generation of a GHZ state 

A 2 

A 1 

π/2 2π pulse 

Atom-cavity Cavity-Atom entanglement 

Control-phase creation 

gate 


• prepared state: 

The "GHZ" state" 

1 

⎡ 

1, 0 

1, 

1 

2 ⎣ 

e g i − g 

( ) ( ) 

2 2 2 2 

⎤ 

⎦ 

• In term of qubits: 

1 

2 

( 0,0,0 + 1,1,1 ) 

• In term of spin 1/2: 

1 

2 

( + , + , + + − , − − ) 

1 2 c 1 2, 

c 

• " GHZ triplet " 

(Greenberger Horne Zeilinger) 


Entanglement tests 

Assessing preparation fidelity 

Two complementary experiments: 

• Correlations in a "longitudinal" basis 

– State populations 

– Measurement of diagonal terms * 

ρ triplet 

+ ++ ..................... −−− 

⎡* . . . . . . * ⎤ 

⎢ 

. * . . . . . . 

⎥ 

⎢ 

⎥ 

⎢ . . * . . . . . ⎥ 

⎢ 

⎥ 

. . . * . . . . 

= ⎢ 

⎥ 

⎢ . . . . * . . . ⎥ 

⎢ 

⎥ 

⎢ 

. . . . . * . . 

⎥ 

⎢ . . . . . . * . ⎥ 

⎢⎣ 

* . . . . . . * ⎥⎦ 

• Correlations in a "transverse" basis 

– Measurement of correlated to state of atom 2 

– Measures one off-diagonal term * 

Rauschenbeutel et al Science 288, 2024 (00) 


Measurement of σ z1 . σ z2 . σ z3 : longitudinal expt 

• Step 1: transfer of the field state to a third atom performing a π absorption pulse in C: 

1 

⎡ e ( ) ( ) 

1, 0 g2 + i2 + g1, 

1 g2 

− i ⎤ 

2 

⊗ g3 

2 ⎣ 

⎦ 

1 

⇒ ⎡ e ( ) ( ) 

1 

g + i g3 + g1 

g − i e ⎤ 

3 

⊗ 

2 ⎣ 

⎦ 

1 

( + 

) 

1, + 

2, + 

3 

+ −1, 

−2, 

−3 

2 

2 2 2 2 

0 

• step 2: detection of each atom for measuring σ z1 . σ z2 . σ z3 

- atoms 1 et 3 : direct measurement of energy 

- atome 2: measurement of energy after applucation of an external π/2 pulse: 

π/2 

( ) 

( − ) 

⎧ 

⎪1 2 g2 + i2 → i2 

⎨ 

⎪⎩ 

1 2 g i → g 

⇒ 1 

2 e i g + g g e 

( 1, 2, 3 1, 2, 

3 ) 

2 2 


Position (cm) 

Full set of operations for measurement of σ z1 . σ z2 . σ z3 

0 

π/2 

D 

D 

D 

10 

8 

6 

Atom # 1 

Atom # 2 

π/2 2π π 

π/2 

4 

2 

Atom # 3 

θ 

π/2 

D 

• Rabi oscillation in C 

π/2 

•Detection 

•Classical π/2 pulse 

Time 

State before detection: 

⇒ 1 

2 e i g + g g e 

( 1, 2, 3 1, 2, 

3 ) 


Measurement results: 

• measurement of σ z1 

. σ z2 

. σ z3 

P long 

=P eig 

+ P gge 

= 0.58 (0.02) 

0.4 

0.3 

0.2 

|- 1 ,- 2 ,- 3 〉 

|+ 1 

,+ 2 ,+ 3 〉 

0.1 

0 

Pgig 

Pgie 

Pggg 

Pgge 

Peig 

Peie 

Pegg 

Pege 

Rauschenbeutel et al., Science 288, 2024 (2000) 


Transverse experiment 

Position 

Timing 

• R 1 

and R 3 

test the A 1 

-A 3 

EPR 

"transverse" correlations 

• Measure the A 1 

-A 3 

"Bell" signal as a 

function of the state of A 2 

Tests 

A 1 

R 1 

(II) 

Ψ triplet 

A 2 

A 3 

C 

π/2 

D 

π 

D 

R 3 

(II) 

π/2 

D 

Time 

1 

ψ 

triplet 

= ⎡ i2 ( e1,0 + g1,1 ) + g2 ( e1,0 − g1,1 

) ⎤ 

2 ⎣ 

⎦ 

Bell signal 

Results 

0.6 

0.4 

0.2 

0.0 

-0.2 

-0.4 

-0.6 

-6 -4 -2 0 2 

Detection phase φ 


Fidelity of preparation of the GHZ state 

• measurement of σ z1 

. σ z2 

. σ z3 

P long 

=P eig 

+ P gge 

= 0.58 (0.02) 

• measurement of σ x1 

. σ x2 

. σ x3 

A= 〈σ x1 

. σ x2 

. σ x3 

〉 = -0.28 (0.03) 

• fidelity: 

F ψ ρ ψ 

= = 

triplet 

F > 0.3 garanties non-separability 

triplet 

0.54 (0.03) 

see also: Sacket et al. Science 288, 2024 (2000) 

preparation of a 4 ions GHZ state in one step 


A complex experimental sequence 

A timing nightmare 

Features 

Experiment II 

Stark Voltage (V) 

-4 -3 -2 -1 0 1 2 

R1 gi 

R2 R2 

gi ei 

A3: 100 µs 

100 

A2: 25 µs 

100 

A small "quantum program": 6 

operations on 4 individual qubits (two 

atoms, the cavity and an extra atom 

used to read-out the cavity state). 

Time (µs) 

0 

-100 

A1: 0 µs 

1.757 V 

0.62 V 

0 

-100 

In principle, could be extended to 

generate a more complex entangled 

state 

-3 -2 -1 0 1 2 3 4 5 6 

Position (cm) 







Towards more complex sequences 

Present limitations and possible 

solutions 

• Single qubit gate fidelity 

– Improve field homogeneity 

• Random atom preparation: long data 

acquisition times 

– Deterministic atom pistol with 

cold atom techniques 

• Cavity losses 

– New cavity design 

• Higher Q's 

• No ring 

• Encouraging preliminary 

results 

– "Remove" the cavity 

Entanglement without cavity 

Van der Waals resonant collision in free 

space: 

Resonant energy exchange between two 

Rydberg atoms. 

Full entanglement for b=10µm. Efficient 

quantum gate (Lukin, Zoller) 

b 

van der 

Waals 

eg , ⎯⎯→ cos θ eg , + sin θ ge , 

2 

2 

θ α c ⎛an 

⎞ 0 

= ⎜ ⎟ 

v⎝ 

b ⎠ 

Requires excellent control of atomic 

position. 


Cavity-assisted van der Waals collision 

Two atoms interact in a non-resonant cavity 

2 

⎛ω 

⎞c 

an 

0 

Mixing angle: = ⎜ ⎟ ⎜ ⎟ ω transition freq., δ atom-cavity detuning 

⎝ ⎠v⎝ bc 

⎠ 

θ α δ 

⎛ ⎞ 

Effective impact parameter b c ~cavity size (mm) 

Considerable enhancement factor ω/δ (up to 10 6 ) 

2 

Resonant coupling between |e 1 

,g 2 

,0> and 

|g 1 

,e 2 

,0> involving a virtual photon 

exchange with the cavity (state |g 1 

,g 2 

,1>) 

|g 1 

,g 2 

;1> 

δ 

e 1 

,g 2 

;0> |g 1 

,e 2 

;0> 

Actual cavity has two modes (orthogonal polarizations) separation 130 kHz: 

– Enhancement factor η=ω(1/δ 1 +1/δ 2 ) 

Zheng and Guo, Phys. Rev. Lett. 85, 2392 (2000); Osnaghi et al. Phys. Rev. Lett. 87, 037902 (2001) 


Advantage of non-resonant method of entanglement: 

Sensitivity to cavity damping 

Ω = 

R 

Ω 

2 

0 

2δ 

• effect of cavity damping: 

projection on |g,g,0> 

Full loss of entanglement 

• probability of error: 

eg , ,0 

Ω 

R ge , ,0 

gg , ,1 

P 

col 

err 

⎛Ω 

⎞ 

≈⎜ 

⎟ 

⎝δ 

⎠ 

2 

• Resonant case: 

P 

res 

err 

≈Γ 

Γ 

cav 

cav 

. T 

int 

. T 

res 

int 

Ω . = π 2 

T R int 

res 

Ω . T = π 2 

int 

δ Γ cav 

• error rate reduced as: 

P 

P 

col 

err 

res 

err 

≈ 

Ω 

δ 

gg , ,0 

efficient with slower atoms 


Advantage of non-resonant method of entanglement: 

Sensitivity to blackbody radiation 

• coupling in the presence of N photons: 

egN , , 

Ω N + 1 

eeN− , , 1 

Ω R 

ggN+ , , 1 

Ω 

N 

g, eN , 

Due to destructive interference 

between two probability amplitudes, 

the effective coupling is to first 

order independent of N: 

2 

( ) 

2 2 

Ω 

0. N + 1 Ω0. 

N Ω0 

ΩR 

≈ − = 

2δ 2δ 2δ 

The method works even in the presence of blackbody radiation 

Similar to "hot" gate for ions: 

Moelmer et al PRL 82 1835 (2000) 


Experimental realization 

•Both atoms simultaneously present in the empty cavity 

mode 

•Minimum distance: about 1 mm (atomic beam diameter) 

•Ramsey pulses to detect atomic spins along a tunable 

direction 


Tests of entanglement 

Population transfer 

1.0 

0.8 

"longitudinal entanglement" 

P(e 1 

,g 2 

) 

P(g 1 

,e 2 

) 

Test of "transverse" entanglement 

0.8 

 

Probability 

0.6 

0.4 

0.2 

0.4 

0.0 

-0.4 

-0.8 

0.0 

0 1 2 3 4 

η(x10 -6 ) 

Up to 2π Rabi rotation 

Good agreement with simple model up 

to π/2 (solid lines) 

Qualitative agreement with numerical 

integration for larger mixing angles 

Features: 

-1 0 1 2 3 

• Insensitive to cavity damping 

• Insensitive to cavity residual thermal 

field 

• Easily transformed in a CNOT gate 

Very promising for quantum information 

processing with moderate Q cavities 


φ/π

Application of controlled collision 

• A simple proposed implementation of the two qubit Grover search 

algorithm 


Next step: solve the simplest quantum algorithms 

PROBLEM: Finding a known item in an unsorted list of size N 

Classical Solution: Check all the items until the one we look for is found! 

Number of trials: Order of N/2 

Complete list 

( ) 

x 

1,...,x N 

x o 

( ) 

δ x − x o 

Equivalent to an oracle corresponding to a function in a blackbox which 

gives the answer yes or no (0 or 1) to the question: « is it the marked item » 

x3 

0 

x 102 δ( x− x o 

) 0 

x o 

1 


QUANTUM SEARCH: 

O ( N ) queries !! 

L.K.Grover, Phys. Rev. Lett. 79, 325 (1997) 

0 1 

n qubits: and 

n 

( N = 2 states) 

•1st 

step: put all the qubits in a superposition 

1 

2 

( 0 + 1 ) 

Hadamard gate, 

performed via a classical 

p/2 microwave pulse 

Ψ 

= 

1 

N 

( 0 + 1 ) × ( 0 + 1 ) × ... × ( 0 + 1 ) = ∑ 

1 

N 

N 

i 

i 

Superposition of all possible states 


We are looking for a particular state 

x o =10100 

...0 

• 2nd step: Inverse the amplitude of the searched item. 

O 

o 

= I − 2 

x 

x 

o 

x 

o 

This step corresponds to the action of 

the « oracle » and is the only one where 

information about the searched item is used. 

« - » sign = answer « yes ». « + » sign = answer « no » 

• 3rd step: Symmetrisation about the average: 

U s 

= 2 Ψ Ψ − I = H ( 2 0 0 − I )H 

( H = H H ... ) 

φ 

1 

2 

H n 

= ∑ i 

ai i 

a 

I 0 

= 

1 

N 


∑ 

i 

a 

i 

( a ) 

U φ = 2 Ψ Ψ φ − φ = ∑ 2a − 

s 

This step does not use 

Information about the searched 

item 

i 

i 

i

Repeating this sequence leads to the marked 

item. 

An example: : N=16 ( 4 qubits ).) 

The first operation reverses the amplitude of 

the marked item, lowering the average. 

Then, the reversal about the average 

increases the probability amplitude of 

finding the marked item. 

After iterating the same procedure 3 times 

we find 

the marked item with a 96% probability. 

Case n=2 qubits ( N = 4 items ) Only one iteration !! 


• CAVITY QED: (N=4) 

= H = SI QPG HI QPG P 

Q HI0 

HO xo 

Oracle 

qubits: two electronic levels of a three level atom 

Hadamard gates: Classical microwave 

pulses combining two 

electronic levels. 

. 

cavity 

|e> 

|g> 

H 

⎧ 

⎪ 

0 

: ⎨ 

⎪ 1 

⎩ 

→ 

→ 

1 

2 

1 

2 

( 0 + 1 ) 

( 0 − 1 ) 

|i> 


Quantum phase gate: 

(dispersive interaction) 

I QPG 

00 

⎛1 

⎜ 

⎜0 

= ⎜0 

⎜ 

⎝0 

01 

0 

1 

0 

0 

10 

0 

0 

1 

0 

11 

0 ⎞ 

⎟ 

0 ⎟ 

0 ⎟ 

⎟ 

−1 

⎠ 

e 

Two atoms interact dispersively with the cavity field (detuning d >> W ) 

δ 

cavity 

|e 1 〉|g 2 〉|0〉 |g 1 〉|e 2 〉|0〉 

δ 

Ω 

|g 1 〉|g 2 〉|1〉 

Ω 

g 

H 

2 

Ω ⎡ 

= ⎢ ∑ 

4δ 

⎣ j = 1, 

Cavity field shift Two atom collision 

e 

e 

+ 

e 

eff 

j j 1 2 1 2 1 2 1 2 

2 

S.-B. Zheng and G.-C. Guo, Phys. Rev. Lett. 85, 2392 (2000) 

S. Osnaghi et al., Phys. Rev. Lett. 85, 2392 (2000) 


g 

g 

e 

+ 

g 

e 

e 

g 

⎤ 

⎥ 

⎦

e 

e 

g 

t 

= 

e 

iλt 

[ cos ( λt 

) e g − i sin ( λt 

) g e ] 

cavity 

g 

i 

Qubit encoding: 

0 → g 

1st atom: : 2nd atom: 

1 → e 

Action of effective Hamiltonian for a time 

0 

1 

→ 

→ 

i 

g 

t 

= 

π 

λ 

e 

g 

g 

i 

t 

g 

i 

t 

= 

t 

e 

iλt 

= g 

= g 

e 

i 

g 

i 

0 

0 

1 

1 

0 

1 

0 

1 

= g 

= g 

= e1 

= e 

1 

1 

1 

i 

i 

g 

2 

g 

2 

2 

2 

→ 

→ g 

→ e 

→ − e 

g 

1 

1 

1 

1 

i 

i 

2 

g 

2 

g 

2 

2 

I QPG 

00 

⎛1 

⎜ 

⎜0 

= ⎜0 

⎜ 

⎝0 

01 

0 

1 

0 

0 

10 

0 

0 

1 

0 

11 

0 ⎞ 

⎟ 

0 ⎟ 

0 ⎟ 

⎟ 

−1 

⎠ 


Sequence of operations: 

Q = 

SI 

QPG 

HI 

S and P: Performed by microwave pulses 

QPG 

P 

P contains the 

information about 

the marked item. 

It plays the role of 

the oracle. 

S 1 S 2 

P = P1 ( θ1) 

P2 

( θ2) 

θ = 0 

⎧ 

⎪ 

0 → 

S : ⎨ 

⎪ 1 → 

⎩ 

π 

S = i 

⎧ 

( − 0 − 1 ) 

0 ( ) 

2 

i 

2 

( 0 − 1 ) 

⎪ 

Pi 

( θi) : ⎨ 

⎪ 1 

⎩ 

or determines the marked item 

→ 

→ 

1 

2 

1 

2 

e 

−iθ 

/ 2 

i 

0 

+ e 

iθ 

/ 2 

1 

−iθi 

/ 2 iθi 

/ 2 

( e 0 − e 1 ) 

θ 

i 

state 

( 0 ,0 ) 1 1 

( 0 , π ) 1 0 

( π ,0 ) 0 1 

( π , π ) 0 0 


• The search algorithm: 

Q 

= 

SI 

QPG 

HI 

QPG 

P 

P QPG H QPG S detection 

Action of the oracle: 

Application of P determines 

the output 

Access atoms independently: 

Stark effect 


Simulations made considering the 

experiment as the ideal one: the only 

source of imperfections, responsible for 

the non perfect fidelity, is the effective 

Hamiltonian. 

We consider here imperfection in the 

pulse duration. When they are of the 

order of 5 %, the fidelity decreases to 

82%. 

Simulations: 

Probability 

Probability 

1,0 

0,8 

0,6 

0,4 

0,2 

0,0 

0,8 

0,6 

0,4 

0,2 

96% 

eg ei gg gi 

5% pulse error 

δ = 4Ω 

(a) 

(b) 

0,0 

eg 

ei 

gg 

gi 

Evolution of the fidelity with pulse 

imperfections. 

Fidelity 

0,9 

0,8 

0,7 

(c) 

0,6 

0,00 0,02 0,04 0,06 0,08 0,10 

Pulse Imperfections 











V) Schrödinger cats and decoherence 

• 1) A direct study of a meter’s decoherence process in a quantum 

measurement 

• 2) Breeding Schrödinger lions with resonant interaction 

• 3) Other applications of field homodyne detection 




measurement 




Quantum/classical boundary and decoherence 

No macroscopic superpositions at our 

scale 

Decoherence 

The "Schrödinger cat" 

1 

2 

( ) 

+ ⇔ 

Environment 

A macroscopic system is strongly 

coupled to a complex environment 

In all models, a few states only are 

stable ("preferred basis"). 

No entangled states neither. 

We only observe a very small fraction of 

all possible quantum states 

WHY 

All quantum superpositions of these 

states evolve very rapidly into 

statistical mixtures. 

Decoherence 


Main features of decoherence 

Very fast process 

An essential process 

superposition lifetime 

= 

relaxation time 

separation betweenstates 

• Rules quantum superpositions out of 

the classical world. 

Depends upon the initial quantum state 

(distance between states or 

"macroscopicity" parameter) 

• Essential to understand quantum 

measurement process (no 

superpositions of meter's states) 

Not a trivial relaxation mechanism 

(but explained by standard relaxation 

theory for simple models) 

- 

0 

∆x 

+ 

Strong link with complementarity and 

entanglement: environment acquires 

a which path information and gets 

entangled with the system 

• Main obstacle for a large scale use of 

quantum weirdness (quantum 

computing) 


Another experiment on complementarity 

e 

Cavity as an external detector in the 

Ramsey interferometer 

Cavity contains initially a coherent field 

Non-resonant atom-field interaction: 

e 

g 

R 1 

R 2 

α 

C 

Atom modifies the cavity field phase 

Phase shift α 1/δ 

S 

(index of refraction effect) 

⎯⎯→ 

⎯⎯→ 

e 

g 

(δ:atom-cavity detuning) 

1 

g 

D 

φ 

"Which path" information: 

• Small phase shift (large δ) 

(smaller than quantum phase noise) 

– field phase almost unchanged 

– No which path information 

– Standard Ramsey fringes 

• Large phase shift (small δ) 

(larger than quantum phase noise) 

– Cavity fields associated to the 

two paths distinguishable 

– Unambiguous which path 

information 

– No Ramsey fringes 


1.0 

0.5 

0.0 

Fringes and field state 

Complementarity 

712 kHz 

Vacuum 

State transformations 

R1 

C 

1 

e → e + g 

2 

Before R1 

( ) 

R2 

1 

iϕ 

e → ( e + e g ) 

2 

1 

g → − e e + g 

2 

( − iϕ 

) 

e, α →e e, αe g, α → g, 

αe 

iΦ iΦ −iΦ 

e, 

α 

Ramsey Fringe Signal 

1.0 

0.5 

0.0 

712 kHz, 

9.5 phot ons 

347 kHz 

Before C 

After C 

After R2 

1 

2 

1 

( ) 

2 e + g α 

i i i 

( e Φ e, αe Φ + g, 

αe 

− Φ 

) 

1 

2 

Detection probabilities 

( ϕ ) 

{ α − α } 

ee e e e 

iΦ iΦ − i +Φ −iΦ 

ϕ 

{ α 

α } 

1 

+ g e e + e e 

2 

i( ϕ+Φ) iΦ − i( +Φ) 

−iΦ 

104 kHz 

0 2 4 6 8 10 

ν (kHz) PRL 77, 4887 (96) 

1 

Pge 

, 

= ⎡ 1 ± Re e αe αe 

2 ⎣ 

− i( ϕ +Φ ) i Φ − i Φ 

Ramsey fringes signal multiplied by 


αe 

iΦ 

⎤ 

⎦ 

αe 

−Φ i

Signal analysis 

Fringe signal multiplied by 

αe 

iΦ 

αe 

−Φ i 

Fringes contrast and phase 

• Modulus 

e 

2 2 

= 

e 

−2nsin Φ −D 

/2 

60 

n=9.5 (0.1) 

6 

– Contrast reduction 

• Phase 

2n 

sin 

Φ 

– Phase shift corresponding to 

cavity light shifts 

Phase leads to a precise (and QND) 

measurement of the average photon 

number 

D 

Fringe Cont r ast (%) 

40 

20 

0 

0.0 0.2 0.4 0.6 0.8 

φ (radians) 

• Excellent agreement with theoretical 

predictions. 

• Not a trivial fringes washing out effect 

Calibration of the cavity field 

9.5 (0.1) photons 

0.0 0.2 0.4 0 2 

φ (radians) 

4 

Fringe Shift (rd) 


A laboratory version of the Schrödinger cat 

Field state after atomic detection 

1 

2 

( + ) 

A coherent superposition of two 

'classical' states. 

Very similar to the Schrödinger cat 

An atom to probe field coherence 

Quantum interferences involving the 

cavity state 

First atom 

Φ 

−Φ 

D 

Second atom 

Decoherence will transform this 

superposition into a statistical mixture 

Slow relaxation: possible to study the 

decoherence dynamics 

Decoherence caught in the act 

Two indistinguishable quantum paths to 

the same final state: 

Quantum interferences 

2Φ 

−2Φ 


Atomic correlations 

A correlation signal 

η =Πee 

, 

−Πge 

, 

P 

ee , 

= − 

ge , 

ee , eg , g, e gg , 

• Independent of Ramsey 

interferometer phase φ (when Φ is 

neither 0 nor π/2) 

P 

P + P P + P 

Principle of the experiment 

• Send a first atom to prepare the cat 

• Wait for a delay τ 

• Send a second probe atom 

• Measure η versus τ 

Raw correlation signals 

• 0.5 for a quantum superposition 

0.3 

τ=40 µs 

η = 

1 Re 

2 

αα 

• 0 for a statistical mixture 

• 0 for an empty cavity 


0.2 

0.1 

0.0 

-0.1 

0 2 4 6 8 10 12 

ν (kHz) 

15000 coincidences 


A decoherence study 

Atomic correlation signal 

Decoherence versus size of the cat 


0.0 0.1 0.2 

n=3.3 δ/2π =70 and 170 kHz 

0 1 2 

t/T 

r 

0 1 2 PRL 77, 4887 (1996) 

τ/T 

r 




δ/2π =70 kHz 

20 

16 

n=5.5 

12 

8 

4 

0 

0 1 2 

20 

t/T 

r 

16 

12 

n=3.3 

8 

4 

0

Decoherence and complementarity 

A simple theoretical approach 

Without relaxation: 

η = 

1 Re 

2 

Simple relaxation model: a bath of 

harmonic oscillators i.e. cavity modes 

C 

αα 

Linear couplings: amplitude β i 

(t) in mode 

i is proportional to the amplitude α(t) 

in the cavity mode 

A cat in the cavity: tiny cats in the 

environment 

C i 

Complete wavefunction at τ 

∏ 

ατ ( ) e β( τ) e + ατ ( ) e β( τ) 

e 

i Φ i Φ −Φ i −Φ i 

i 

i 

i 

i 

Interfering states after the second atom 

(does not affect the environment) 

∏ 

Final correlation 

Energy conservation 

∏ 

iΦ 

ατ ( ) β( τ) e + ατ ( ) β( τ) 

e 

i 

i 

1 Re ( ) 

iΦ 

η = ∏ βi 

τ e βi 

( τ ) e 

2 

i 

∏ 

1 ⎛ 

= Re exp βi 

( τ ) 1 

2 

⎜−∑ 

− 

⎝ i 

∑ 

i 

i 

−iΦ 

iΦ 

( e ) 

2 2 

i 

⎞ 

⎟ 

⎠ 

( 

− T 

e τ 

) 

2 / 

r 

β ( τ) = n 1− 

i 

PRL, 79, 1964 (1997) 

−Φ i 


Theoretical decoherence signal 

Atomic correlation versus τ 

1 

2 

−2n[ 1−exp( −τ 

/ T r )] sin Φ 

η = e cos n[ 1−exp( −τ 

/ T )] 

r 

sin 2Φ 

2 

At short times 

T 

D 

Tr 

= 

n 

Excellent agreement with the 

experimental data 

A very simple description of 

decoherence in terms of 

complementarity. 

The environment 'measures' the 

field phase and gets a "which 

path" information 


{ } 

0 1 2 

τ/T 

r 


0.0 0.1 0.2 

n=3.3 δ/2π =70 and 170 kHz

Decoherence features 

• Faster than cavity relaxation 

• Faster when distance between states increases 

• Decoherence time scale depends upon a "macroscopicity" parameter 

Not a trivial relaxation mechanism even if described by standard relaxation 

theory 

Essential for quantum measurement 

meters are not in superposition states 

Difficulty for applications of QM 

the more complex the entangled state, the faster the decoherence 

Towards decoherence "metrology" …. With much larger Schrödinger cats 




measurement 




Rabi oscillation in a classical field 

Ωr 

Oscillation in a large coherent field H 

I 

= σ 

Y 

Ω 

r 

=Ω0 

n ∝ E 

2 

1 −Ω i clt/2 iΩclt/2 

| Ψ ( t) >= ⎡ (| | ) (| | ) 

2 ⎣ 

e e>+ i g > + e e>− i g > ⎤ 

⎦ 

⊗ α 

Atomic eigenstates 

In terms of Bloch sphere 

1 

± = ⎡ Z 

Y ⎣ 

2 

e ± i g ⎤⎦ 

In-phase and π-out-of-phase 

with respect to field 

Quantum beat between 

eigenstates: 

• Sinusoidal Rabi oscillation 

between e and g 

X 

Y 


Rabi oscillation in a mesoscopic field 

A much more interesting situation 

⎛ Ω n+ 1t Ω n+ 

1t 

Ψ = ∑ 

+ + 

⎝ 

1+ cosΩ 0 

n+ 

1t 

2 

Pe() 

t = ∑ pn pn = cn 

2 

0 0 

() t cn 

⎜ 

cos e, n sin g, n 1 

n 

2 2 

n 

⎞ 

⎟ 

⎠ 

|e,n> 

|+,n> 

Ω+ 

0 

n 1 

|-,n> 

|g,n+1> 

A complex Rabi oscillation signal 

1 

P e 

(t) 

0.8 

• Collapse: 

– Dispersion of Rabi frequencies 

• Revivals: 

– Finite number of frequencies 

– Direct consequence of field 

quantization 

0.6 

0.4 

0.2 

Ω 0 t/2π 

50 100 150 200 


An insightful quasi-exact solution 

• Get more physical insight on the collapse-revival phenomenon 

• Get information on the field evolution 

• Rewrite the exact atom-field wavefunction 

Florence, Mai 2004 (Gea Banacloche PRL 65, 3385, Buzek et al PRA 45, 8190) 346


• Factor the two terms in an atom and field parts. Redefinition of running 

index n 

• Large coherent field 

• Product of atom and field states 



• Expand the sqrt(n) term 

• Neglect for the time being the second order phase spreading terms 

• Same treatment for Ψ 2 



+ + − − 

Ψ () t = ⎡ Ψa() t Ψ 

c() t + Ψa() t Ψc() 

t 

2 ⎣ 

1 

1 

2 

Ω 

Ψ = ∓ 

±Ω i 0 nt/2 

i 

Ψ ± a 

= e ⎣e ± Φ e ∓ i g ⎦ 

⎡ 

n 

– Atomic states slowly ( times slower than Rabi oscillation) rotating in 

the equatorial plane of the Bloch sphere 

c 

e αe 

± i 0 nt/4 

± iΦ 

– A slowly rotating field state in the Fresnel plane 

⎤ 

Φ= 

Ω 

0 

t 

4 n 

⎤ 

⎦ 

• Graphical representation of the joint atom-field evolution in a plane 

• t=0: 

– both field states coincide with original coherent state 

– Atomic states are the classical eigenstates 


Atom-field states evolution 

+ 

Ψ 

c 

1 + + 

− − 

Ψ ( t) = ⎡ Ψa () t Ψc () t + Ψa () t Ψc 

( t) 

2 ⎣ 

− 

Ψ 

c 

⎤ 

⎦ 

+ 

Ψ 

a 

− 

Ψ 

a 

+ − 

•At most times: Ψ Ψ = 0 an atom-field entangled state 

c 

c 

•In spite of large photon number: considerable reaction of the atom on the field 


‘Automatic’ preparation of a Schrödinger cat 

• At time 

• Atom-field disentanglement 

• The fastest and most efficient way to prepare large Schrödinger cat states 


Quantum Rabi signal 

• Retrieve the quantum Rabi signal 

• The Rabi oscillation signal has an amplitude modulated by the scalar 

products of the cavity field components: another manifestation of 

complementarity 


Link with Rabi oscillation 

+ 

Rabi oscillation: quantum interference between Ψ and 

+ − 

• Contrast vanishes when Ψ Ψ = 0 : 

c 

1 + + 

− − 

Ψ ( t) = ⎡ Ψa () t Ψc () t + Ψa () t Ψc 

( t) 

2 ⎣ 

– A direct link between Rabi collapse and complementarity 

c 

a 

− 

Ψ 

a 

⎤ 

⎦ 

1 

P e 

(t) 

0.8 

0.6 

0.4 

0.2 

Ω 0 

t/2π 

•Fast preparation Atom-field Field of decorrelation: 

large state Schrödinger merge again: cat states 

Quantum Rabi oscillation 

•Another Unconditional illustration preparation 

and Quantum progressive of complementarity 

revival collapse of 

of the field 

•A surprising In a insight « phase Rabi in oscillation Schrödinger the simple Rabi cat state oscillation » phenomenon 

10 20 30 40 50 


An expression at short times (collapse) 

• A time of the order of the vacuum Rabi oscillation period 

• Classical Rabi oscillations with a gaussian envelope. Collapse time 

• Revival time (half a complete rotation in phase space) 

• Why only a finite number of revivals 


Field states phase spreading 

• One order more in the expansion: 

Ψ =∑ 

c 

e 

+ i nΩ 

c 

n 

n 

2 

0 t / 4 n −Ω i 0 ( n − n) t/ 

16 

– Phase rotation + Phase spreading of field states 

– Contrast of revivals decreases and width increases 

– Complete overlap of revivals after a few turns in phase space 

• Snapshots of field Q function for 15 photons 

e 

n 

3/2 

n 

Im(β) 

6 

4 

2 

(a) 

6 

4 

2 

(b) 

6 

4 

2 

(c) 

0 

0 

0 

-2 

-2 

-2 

-4 

-4 

-4 

-6 

-6 

-6 

-6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 

Re(β) Re(β) Re(β) 


Field phase spreading 

• At long times: take into account higher order terms in phases 

• Average of a goes to zero: complete phase information loss. Occurs for 

• To be compared with revival time 

• About revivals observable 


Direct observation of field phase evolution 

• Rabi oscillation in mesoscopic field 

– High atom-field coupling 

– Low atom and field relaxation 

• Cavity QED tools with circular Rydberg atoms and 

superconducting cavities 

• A method to probe field phase distribution: 

– Homodyne field measurement 


Field phase distribution measurement 

Homodyning a coherent field 

S 

•Inject a coherent field |α> 

•Add a coherent amplitude –αe iφ 

•Resulting field (within global phase) |α(1-e iφ )> 

•Zero final amplitude for φ=0 

•Probe final field amplitude with atom in g 

•P g 

=1 for a zero amplitude 

•P g 

=1/2 for a large amplitude 

•More generally: P g 

(φ) reveals field phase distribution 

•In technical terms, P g 

(φ)=Q distribution 


Experimental coherent field phase distribution 

Transfert 

0,85 120 photons 

75 photons 

0,80 

45 photons 

20 photons 

0,75 

0,70 

0,65 

0,60 

0,55 

Largeur (°) 

34 

32 

30 

28 

26 

24 

22 

20 

0,50 

18 

0,45 

0,40 

-80 -60 -40 -20 0 20 40 60 80 

Phase 

16 

14 

0,08 0,10 0,12 0,14 0,16 0,18 0,20 0,22 0,24 0,26 0,28 

1/sqrt(n) 


Phase splitting in quantum Rabi oscillation 

• Timing 

S 

•Inject a coherent field 

•Send a first atom: Rabi oscillation and phase shift 

•Inject a phase tunable coherent amplitude 

•Send an atom in g: final amplitude read out 



Experimental phase distributions 

0,80 

0,75 

29 injected photons 

Reference: no Rabi atom 

Rabi atom at 335m/s T i 

=32 µs 

Rabi atom at 200m/s T i 

=53 µs 

0,70 

0,65 

0,60 

0,55 

0,50 

0,45 

-200 -150 -100 -50 0 50 100 150 

Phase(°) 



Summary of results 

335 m/s 200 m/s 

40 

40 

S g 

(φ) 

30 

35 

30 

35 

0,7 

0,6 

0,5 

25 

20 

-150 0 150 

φ (degrees) 

0 150 15 20 

φ (degrees) 

25 

n 

Auffèves et al. PRL 91, 230405 (2003) 


Phase (degrees) 


Observed phase versus theoretical phase 

60 

40 

20 

4 

2 

0 

β y 

0 

-20 

-40 

-2 

-4 

0 2 4 6 8 

β x 

-60 

15 20 25 30 35 40 45 50 55 60 

Φ + (degrees) 

Large Shrödinger cat states (up to 40 photons separation) 


Selective preparation of 

+ 

Ψ 

a 

Use a Stark shift pulse on the e/g transition (equivalent to a Z rotation) to 

+ 1 

prepare from + 

Ψ 

a 

( ) 

Z 

2 e g 

X 

Rabi Fast Stark rotation pulse: for a p/2 π/2 rotation pulse: 

around Z axis. 

1 + 

Preparation of Ψ ( ) 

2 e + g 

a 

Y 

From this time on, slow evolution only 

N.B. Starting from g prepares 

− 

Ψ 

a 


Stopped Rabi oscillation 

1,0 

Rabi 

Z rotation 

0,8 

0,6 

Slow 

evolution 

Transfert 

0,4 

Evolution 

resumes 

0,2 

Z rotation 

0,0 

-45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 

Killer (µs) 


A single, slowly rotating phase component 

S g (φ) 

0,65 

0,60 

0,55 

0,50 

0,45 

− 

Ψ 

a 

+ 

Ψ 

a 

-150 -100 -50 0 50 100 150 200 

φ (degrees) 


Test of coherence: induced quantum revivals 

Initial Rabi rotation, 

Stark pulse (duration short 

compared to phase Collapse rotation). 

Reverse phase rotation 

Equivalent And to slow a Z rotation phase rotation by π 

Recombine field components and 

resume Rabi oscillation 

Morigi et al PRA 65, 040102 


Induced quantum revivals 

1,0 

Π Pulse 

Transfert 

0,8 

0,6 

0,4 

0,2 

22 µs 

0,0 

-20 -10 0 10 20 30 40 50 60 

Interaction time 

1,0 

1,0 

0,8 

0,8 

Tranfer 

0,6 

0,4 

0,2 

18.5 µs 

Transfer 

0,6 

0,4 

0,2 

23.5 µs 

0,0 

-20 -10 0 10 20 30 40 50 60 


0,0 

-20 -10 0 10 20 30 40 50 60 



Induced quantum revivals 

0,9 

0,8 

0,7 

0,6 

Transfert 

0,5 

0,4 

0,3 

0,2 

0,1 

22 µs 

0,0 

0 5 10 15 20 25 30 35 40 45 50 55 60 65 

Temps effectif 

Transfert 

0,80 

0,75 

0,70 

0,65 

0,60 

0,55 

0,50 

transfert 

0,80 

0,75 

0,70 

0,65 

0,60 

0,55 

0,50 

0,45 

0,45 

-100 -80 -60 -40 -20 0 20 40 60 80 100 

0,40 

phase(°) -100 -80 -60 -40 -20 0 20 40 60 80 

phase(°) 




measurement 




Phase shift with dispersive atom-field interaction 

• Non resonant atom: no energy exchange but cavity mode frequency shift 

(atomic index of refraction effect). 

– Phase shift of the cavity field (slower than in the resonant case) 

0,70 

0,65 

0,60 

Atom in g 

0,70 

No atom 

1 atom in g 

Atom in g 

0,55 

2 atoms in g 

0,50 

0,65 

0,45 

-150 -100 -50 0 50 100 150 

0,75 

0,70 

0,60 

No atom 

0,65 

0,60 

0,55 

No atom 

0,55 

0,50 

0,45 

-150 -100 -50 0 50 100 150 

0,70 

0,50 

0,65 

Atom in e 

0,60 

0,55 

Atom in e 

0,45 

0,50 

0,45 

-200 -150 -100 -50 0 50 100 150 

-150 -100 -50 0 50 100 150 

Phase (°) 

Opposite values for e and g 

Proportional to atom number 


Absolute measurement of atomic detection efficiency 

• Histogram of field phase reveals exact atom count 

• Comparison with detected atom counts provides field ionization detectors 

efficiency in a precise and absolute way 

– 0.4 atoms samples: 

70 % detection efficiency 

(close to the expected 

optimum of 80 %) 


Towards a 100% efficiency atomic detection 

• Inject a very large coherent field in the cavity 

• Send an atomic sample 

– Different phase shifts for e, g or no atom 

Im(α) 

φ 

−φ 

e 

g 

Re(α) 

• Inject homodyning amplitude 

– Zero amplitude for e. 

• Larger for no atom. 

• Still larger for g 

g 

e 

• Read final field amplitude by sending a large number of atoms in g 

– Final number of atoms in e proportional to photon number 


Preliminary experimental results 

0,18 

Probability 

0,16 

0,14 

0,12 

0,10 

0,08 

Atom 1 prepared in g 

no atom 

1 atom in g 

Atom1 prepared in e 

no atom 

1 atom in e 

Experimental conditions: 

• 75 photons initially 

• v=200 m/s 

• d=50 kHz 

• 70 absorber atoms 

0,06 

0,04 

0,02 

0,00 

0 10 20 30 40 

• detection efficiency: 87% 

• error probability: 0 atom detected as 1: 10% (main present limitation) 

– e in g: 1.6% 

– g in e: 3% 

Total number of excited atoms 

• 100% detection efficiency within reach with slower atoms: v=150 m/s 

….experiment in progress. 











CQED with silica microspheres 

High Q whispering gallery modes in a silica microsphere 

(a ~ 25-100µm) 

a 

Mode Volume : V mode 

~ 300 µm 3 

Field per photon : E ~ few kV/m 

(for the most confined mode) 

Very low losses : Γ cav 

/2π ~ 300 kHz 

Absorption limited Q = ω / Γ cav 

> 10 9 

t cav = 1µs 

.....Coupled to Dipole emitters 

N SiO 2 = 1.45 

Ions (Nd 3+ ; Er 3+ ) ⇒⇒⇒ cf. Poster Session 

Atom Chips ⇒⇒⇒ cf. Romain Long (Session V) 

Excitons confined in quantum dots d ~ 15 e.a 0 

‘Atom like system’ Γ hom 

/2π ~200 MHz 

@ 10K 


Historical reference 

St Paul's Cathedral 

(Lord Rayleigh,1870) 

Wall 

Sound waves 


Whispering Gallery Modes 

Light guided by total internal reflection 

at grazing incidence and resonance 

condition 

ν 

TE/ 

TM 

n,, 

m 

-|m|=0 

-|m|=1 

θ 

a 

n=2 

n=1 

TE/TM polarization 

n : number of anti-nodes 

l : angular momentum 

m : azimuthal number 

(degenerate for a perfect sphere) 

Coupling zone 

(Evanescent wave) 


Production of microspheres 

Preparation of the fiber: 

Pulling a fiber from a 

rod of pure silica 

SiO 2 

Production of the sphere: 

Melting the tip of the fiber 

using a 10W CO 2 

Laser 

120 µm 

⇒ 

Surface Tension 

Spherical Shape 


Excitation of WGM’s 

Light coupled into the sphere 

by frustrated total internal reflection 

N p = 1.75 

I Out 

ν 

• Incident angle 

θ i < θ c ≡ Arcsin(N S /N P ) 

Tunable LD 

@ 780nm 

θ i 

N P 

• Losses due to coupling back to the prism can be 

adjusted through the Gap g 

Linewidth (MHz) 

300 

200 

100 

0 

0 

100 200 300 400 

Gap Shere-Prism g (nm) 


25% 

20 

15 

10 

5 

0 

g 

Coupling Rate 

N S 

N s = 1.45

The core of the experiment 


Tuning Devices : Tweezers 

Preformed sphere is glued 

into stretching device 

Soldering of the fiber onto 

glass arms and then 

production of the sphere 


Tuning WGM´s 

TE 

TM 

• FSR = 810 GHz (80 µm) 

• ν TM - ν TE = 580 GHz 

slope TM ≈ 1.6 slope TE 

• ν l, m – ν l, m-1 = 375 GHz 

ellipticity ~ 50 % 


Maximum Tuning 

Tuning TM = 405 GHz = 0.5 × FSR 

Tuning TE = 260 GHz = 0.3 × FSR 

Fracture of the fiber 


Reversibility and Stability 

Tuning is reversible : No plastic deformation observed 

At fixed voltage : Frequency fluctuations come 

from temperature fluctuations 


Identifying WGM’s 

GM Spectrum 

Free Spectral Range ∆l = 1 ∆ν = c/2πNa ~ 500 GHz 

Radial order : ∆n = 1 ∆ν ≈ 20 FSR 

Polarization TE-TM ∆ν ≈ 0.7 FSR 

Ellipticity (e ~ 1%) ∆m = 1 ∆ν = e FSR ~ 5 GHz 

r p 

> r e 

Prolate Sphere 

e > 0 

r p 

r e 

• Step 1 : Pick |m|= mode 

• Step 2: Assign radial number 

1.0 

shift 

⇒ n = 1 

width ≤ 1% 8 

0.8 

Reflection 

0.6 

0.4 

|m|= 

|m|=-1 

|m|=-2 

|m|=-3 

g 

0.2 

0.0 

25 20 

15 10 

Frequency (GHz) 

5 0 

2 4 



0 

6

Which experiments 

• Non linearity with low thresholds 

– Eg. Kerr bistability using silica non-linearity 

• Lasers with doped silica spheres 

– Extremely low thresholds 

– Efficient frequency conversion Er laser 

– Towards a thresholdless laser 

• Towards cavity QED with silica microspheres. Two routes 

– Atoms in the sphere’s evanescent field 

– Quantum dots permanently coupled to the sphere 


Atom chips and microspheres 

• Use atoms trapped in a mesoscopic conductor cavity field and conveyed 

to the sphere mode 

• Work performed in T. Hänsch and J. Reichel group in Munich 


Atom-Chip : Conveyor-Belt 


Lateral Confinement 

• Simple Scheme 

• Side-wires Configuration 

Single wire field 

+ 

External bias field 

= 2D Confinement 

Single wire field 

+ 

Field from the side-wires 

= 2D Confinement 


Longitudinal Confinement 

• Multiple Crossing Conductors • Modulation of the Current 

Atoms trapped between 

two local maxima 

of the longitudinal field 

Modulation of the Current 

 

Shift the potential minimum 

Transport the atoms 


The “LDC” Chip 

New conveyor 

First 2 layers Generation Conveyor : 

“6 Strokes 1 layer Engine” 

“2 Strokes Engine” 

Transport direction 


Long Distance Transport 

23,5 cm in 2.9 s 

Average Speed = 8 cm/s 

Maximum Speed = 10 cm/s 


Atom Touch 

Detection of a “single” microsphere 


Quantum dots and microspheres 

• An artificial atom directly coupled to the sphere’s mode 


Self Assembled Q-Dots 

(J.M. Gérard) 

Self Assembled islands of InAs embedded in GaAs 

4 nm 

Q-Dots 

3D Confinement leads to an atom like system 

20nm 

Mesa 4×4 µm 

HF selective attack 

250nm 

GaAs 

InAs 

GaAs 

Detected Power (fW) 

800 

600 

400 

200 

0 

950 1000 1050 1100 

Wavelength (nm) 

(MBE) 

Q-Dots Photoluminescence @ 300K 

Pump Power 

@850nm 

900 

7.2mW 

4 mW 

3 mW 

2 mW 

1 mW 

1150 

1200 


Effects of Sample on WGM’s 

Gap g2 

Sphere-Sample 

GaAs 

GaAs 

Gap g1 

Sphere-Prism 

Ns 

Sphere 

Prism 

Line Broadening 

Resonance Shift 

i 

Np 4.0 

3.5 

3.0 

Prism SF 11 

2.5 

2.0 

1.5 

1.0 

0.5 

0.0 


0 

Coupling rate 

Shift 

linewidth 

100 200 300 400 500 

GAP g2 SPHERE - GaAs (nm) 

16% 

14% 

12% 

10% 

8% 

6% 

4% 

2% 

0% 


Effects of Mesa (4×4 µm) on WGM’s 

Frequency (MHz) 

Frequency (MHz) 

500 

400 

300 

200 

100 

0 

500 

400 

300 

200 

100 

0 

0 

0 

2 

5 10 

Position Y (µm) 

4 6 

Position Z (µm) 

15 

8 


6% 

5 

4 

3 

2 

1 

0 

6% 

5 

4 

3 

2 

1 

0 

Line Broadening 

Resonance Shift 

Coupling rate 

|m|= WGM 

Mesa 

Y 

Z

Q-Dots Laser 

Experimental Set Up 

Emission 

1000 – 1100 nm 

LD Probe @ 1080 nm 

Transmission @ 

780nm 

I Out 

ν 

PD 

PD 

Prism SF11 

LD @ 780 nm 

Pump 

Linewidth ~ 700MHz 

Coupling rate ~10% 

Gap g 2 Sphere-Sample 

Sphere 

Gap g 1 Sphere-Prism 

Mesa 

Q-Dots 

PZT controlled 

X-Y-Z motion 


Q-Dots Laser at Room temperature 

30 

Detected Power (pW) 

25 

20 

15 

10 

5 

TE 

TM 

0 

0.0 0.1 0.2 0.3 0.4 0.5 

Absorbed Pump Power (mW) 

Power at threshold ~ 200 µW 

0.6 

Active Q-dots ~ 10 4 











Two main directions 

• A two-cavity experiments for non-locality, decoherence and quantum 

information 

• A Rydberg atom chip experiment for deterministic preparation and 

manipulation of cold trapped Rydberg atoms 




information 




A two-cavity experiment 

• Rydberg atoms and superconducting cavities: 

– Towards a two-cavity experiment 

• Creation of non-local mesoscopic Schrödinger cat states 

– Non-locality and decoherence (real time monitoring of W 

function) 

• Complex quantum information manipulations 

– Quantum feedback 

– Simple algorithms 

– Three-qubit quantum error correction code 


C 

Teleportation of an atomic state 

R 2 

P b 

beam 3 

R R 

P 3 

a 

1 

C 1 

C 2 

1' 

1 

2 

D a 

beam 2 

D b 

A 

3' 

3 

0 

beam 1 

Davidovich et al, 

PHYS REV A 50 R895 (1994) 

B 

EPR pair 

D c 

• This scheme works for massive particles 

• Detection of the 4 Bell states and application of the 

"correction" to the target is possible using a C-Not gate 

(beam 2 and 3) 

• The scheme can be compacted to 1 cavity and 1 atomic beam 


Implementation of 3 qubit error correction 

0 

0 

0 

R 

encoding 

S 

R R ’ 

R 

R’ 

R 

R’ 

Error 

R’ 

decoding 

S 

σ z2 

σ z3 

Détection 

Correction Correction 

error detection 

α 0 + β 1 

Détection 

Ramsey π/2 pulses 

Error 

encoding and decoding: preparation of a GHz triplet 

all the tools exist! 


Quantum feedback 

• Preserve a Schrödinger cat by giving him “quantum food” 

• Use a atoms and QND arrangement to detect cat parity jum 

• A cat jump prepares a single photon in a second cavity. Used to excite a 

feedback atom which gives back the photon to the cat 

• A parity preservation scheme which makes the cat coherence live much 

longer than the natural decoherence time 

• Fortunato et al PRA 60, 1687 


Feedback loop 

• C quasi resonant with e/g and C’ with g/i 

• Probe (QND) atom exits in g when parity jump 

• Feedback atom 

– Promoted to e when in g by a microwave pulse in R 2 

– Undergoes an adiabatic passage in C to restore the lost photon 


Feedback efficiency 


Towards feedback 

• Needs a deterministic source of atoms 

• A simpler version with two modes of the same cavity and no deterministic 

atom source 

– Zippili et al PRA 67 052101 

• Realistic preservation of cats with about one photon 

• Very good preservation of single photon Fock state 


New non-locality explorations 

• Use a single atom to entangle two mesoscopic fields in the cavity 

– A non-local Schrödinger cat or a mesoscopic EPR pair 

– Easily prepared via dispersive atom-cavity interaction 


Mesoscopic Bell inequalities 

• A Bell inequality form adapted to this situation 

• Here, Π is the parity operator average. Dichotomic variable for which the 

Bell inequalities argument can be used (transforms the continuous 

variable problem in a spin-like problem) 

• Maximum violation for parity entangled states: 


Bell inequalities violation 

• Optimum Bell signal versus γ 

• A compromise between violation amplitude and decoherence: γ²=2 


Probing the Wigner function 

• A second atom to read out both cavities (same scheme as for single 

mode Wigner function) 


A difficult but feasible experiment 

• Bell signal versus time T c =30 and 300 ms 




information 




What we want 

• Long State Lifetime (> 30 ms in free space) 

• Control of External Degrees of Freedom 

• Single Rydberg Atom Excitation on Demand 

• Integrated Atom-Chip 

• Coherence preserving scheme 


Inhibition of Spontaneous Emission 

Principle 

F 0 

No available 

Emitted 

zmode 

Perfect, infinite 

photon must 

→ xemissionmirrors 

be σ polarized 

inhibited 

d < λ/2 

τ → ∞ 

Limiting Factors 

F 

τ >> τ 0 

• Imperfect and finite 

mirrors 

• Angle between F 0 and z 

• Residual Thermal Field 

Hulet, Hilfer, Kleppner, PRL, 

55, 20, 2137 : Factor 20 

1986: Jhe, Anderson, Hinds, 

Meschede, Moi, Haroche, 

PRL, 58, 666: Factor 13 


Rydberg Atom Trapping 

Candidate Techniques 

• Magnetic Trap – Zeeman Effect 

• Electric Trap – Stark Effect 

Better suited to 

Inhibition of 

Spontaneous 

Emission scheme 

• Ponderomotive Trap – Electron Micro-motion 

Required Laser Dutta, Intensity Guest, Feldbaum, (200 Wcm Walz-Flannigan, -2 ) incompatible Raithel with 

PRL, 85, 26, 5551 

Cryogenic Environment 


Electric Dipole Trap 

Quadratic Stark Effect ~ 2.2 MHz/(V/cm) 2 

E ~ - α |F| 2 

High Field Seeker 

Energy E 

Electric Field |F| 

n = 51 

n = 50 

Maxwell 

Maximum of |F| 

Dynamic (Paul-like) Trap 


Trap Geometry 

U V(t) U V(t) U V(t) 

+ - + - + - 

1mm F 0 = 30 V/cm 

(< λ/2) 1mm 

z 

x 

U = 1.5 V 

V(t) = 0.5 V . Cos(ω V t) 

ω V = 20,000 s -1 

-U V(t) -U V(t) -U V(t) 

+ - + - + - 


Trapping Simulation 

Atomic Trajectories 

•T load = 300 µK 

• ∆x load = 5 µm 

• ω V = 20,000 s -1 

• Trapping volume ~ (100µm) 3 

z (mm) 

80 

40 

0 

-40 

Macro-motion 

-80 

-80 -40 0 40 80 

Micromotion 

x (mm) 

Trapping Efficiency 

• ω V < ω c → atom escapes 

along anti-trapping axis 

• ω V >> ω c → field variations 

average to zero: no trapping 

1,0 

0.5 

ω c 

0 

ω V (s -1 ) 

0 10000 20000 


Electric Field Tilt 

F 0 (x,y,z) 

F loc 

z 

F loc 

F loc 

ϑ(t) 

F loc 

Trapping Region 

Results 

ϑ < 10 -2 τ→10 4 τ 0 

• Not limiting factor 

• τ ~ 1s envisageable 


Micro-Trap 

U – V(t) 

U + V(t) 

U – V(t) 

U + V(t) 

U – V(t) 

1mm 

z 100µm 

U = 1.5 V 

100µmTrapping x Region 

F 

V = 0.5 V . Cos(ωt) 

0 = 30 V/cm 

ω = 150 000 s -1 

-U + V(t) 

• Greater Confinement 

• Surface Interactions 

• Integration 

• Extension to Conveyor Belt, Guide… 


A Tighter Trap 

Same field variations + Spatial scale / 10 → Confinement x 10 

Atomic Trajectories Trapping Efficiency 

•T load = 300 µK 

• ∆x load = 5 µm 

• ω V = 150 000 s -1 

• Trapping volume ~ (10µm) 3 

• z symmetry broken 

1,0 

• ω c x 10 

• Trapping less perfect… 

• …but still very good 

z (µm) 

5 

0 

-5 

-10 

x (µm) 

-10 -5 0 5 10 

ω (s -1 ) 

0 10000 20000 


0.5 

0 

ω c

Rydberg Atom Source 

Dipole Blockade Lukin et al, PRL 87, 037901 

∆x ~ 1µm 

~ 1GHz 

ω 

| N-2:g ; 2: > 

| N-1:g ; 1: > 

ω 

Rydberg Excitation Laser 

ω 

| N :g ; 0: > 

One and Only One Circular Rydberg Atom Excited 


Ground-State Atoms Trapping 

Requirements: 

• Highly Confining (Dipole (∆x < 10µm) Blockade) 

• Close to Surfaces (Surface-Trap (Micro-Trap, Dipole- Distance 

Surface R trap < 100µm) Interactions) 

• Compatible with Cryogenics 

(Dissipation (Rydberg Stability) < 1mW) 

• Integrable (On-Chip Wires and Electrodes) 

Hänsel et al, Nature 413, 498 

B bias I ~ 1A SiO 2 or Sapphire substrate 

Superconducting Niobium Wires 

100x1 µm 2 wide 


Filling the Magnetic Trap 

Cryogenic vacuum → no background pressure → external 

source 

2D MOT – High Flux Atomic Jet 

300K 

1K 

Cryostat 

Magnetic Trapping Region 

Jet Extracted from 

Atom Cloud 

2D MOT 

mm 

6 

4 

2 

0 2 4 

mm 

Characteristics of our jet 

•Flux = 10 7 s -1 

• Divergence = 10 mRad 


Coherence preservation scheme 

• Use a microwave dressing to equalize e and g Stark polarizabilities 


Coherence preservation scheme 

• Residual phase drift almost linear with time 

• Can be corrected by an echo technique 


Very long coherence times 

Coherence preserved for seconds or minutes!!! 


An extremely promising scheme for 

• Spontaneous emission inhibition studies 

• Atom-surface and atom-atom dipole-dipole interaction studies 

• Cavity QED with transmission line resonators 

• Quantum information processing 

• Coupling of Rydberg atoms to mesoscopic circuits 


The team 

PhD 

• Frédérick .Bernardot 

• Paulo Nussenzweig 

• Abdelhamid Maali 

• Jochen Dreyer 

• Xavier Maître 

• Gilles Nogues 

• Arno Rauschenbeutel 

• Patrice Bertet 

• Stefano Osnaghi 

• Alexia Auffeves 

• Paolo Maioli 

• Tristan Meunier 

• Sébastien Gleyzes 

• Philippe Hyafil * 

• Jack Mozley * 

Post doc 

• Ferdinand Schmidt-Kaler 

• Edward Hagley 

• Christof Wunderlich 

• Perola Milman 

Colaboration 

• Luiz Davidovich 

• Nicim Zagury 

• Wojtek Gawlik 

Permanent 

• Gilles Nogues * 

• Michel Brune 

• Jean-Michel Raimond 

• Serge Haroche 

*: atom chip team 


References (1) 

• Strong coupling regime in CQED experiments: 

– F. Bernardot, P. Nussenzveig, M. Brune, J.M. Raimond and S. Haroche. "Vacuum Rabi Splitting 

Observed on a Microscopic atomic sample in a Microwave cavity". Europhys. lett. 17, 33-38 

(1992). 

– P. Nussenzveig, F. Bernardot, M. Brune, J. Hare, J.M. Raimond, S. Haroche and W. Gawlik. 

"Preparation of high principal quantum number "circular" states of rubidium". Phys. Rev. A48, 

3991 (1993). 

– M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J. M. Raimond and S. Haroche: 

"Quantum Rabi oscillation: a direct test of field quantization in a cavity". Phys. Rev. Lett. 76, 

1800 (1996). 



• QND measurement in microwave CQED experiments: 

– M. Brune, S. Haroche, V. Lefevre-Seguin, J.M. Raimond and N. Zagury: "Quantum nondemolition 

measurement of small photon numbers by Rydberg-atom phase sensitive detection", 

Phys. Rev. Lett. 65, 976 (1990). 

– M.Brune, S. Haroche, J.M. Raimond,L. Davidovich and N. Zagury. "Manipulation of photons in 

a cavity by dispersive atom-field coupling: QND measurement and generation of "Schrödinger 

cat"states". Phys Rev A45, 5193, (1992). 

– S. Haroche, M. Brune and J.M. Raimond. "Manipulation of optical fields by atomic 

interferometry: quantum variations on a theme by Young".Appl. Phys. B, 54, 355, (1992). 

– S. Haroche, M. Brune and J.M. Raimond. "Measuring photon numbers in a cavity by atomic 

interferometry: optimizing the convergence procedure". Journal de Physique II 2, 659 



• Gates: QPG or C-Not, algorithm: 

– M. Brune et al., Phys. Rev. Lett, 72, 3339(1994). 

– Q.A. Turchette et al., Phys. Rev. Lett. 75, 4710 (1995). 

– C. Monroe et al., Phys. Rev. Lett. 75, 4714 (1995). 

– A. Reuschenbeutel et al. submitted PRL. G. Nogues et al. Nature 400, 239 (1999). 

– S. Osnaghi, P. Bertet, A. Auffeves, P. Maioli, M. Brune, J.M. Raimond and S. Haroche, Phys. 

Rev. Lett. 87, 037902 (2001) 

– F. Yamaguchi, P. Milman, M. Brune, J-M. Raimond, S. Haroche: "Quantum search with twoatom 

collisions in cavity QED", PRA 66, 010302 (2002). 

• Q. memory: 

– X. Maître et al., Phys. Rev. Lett. 79, 769 (1997). 

– Atom EPR pairs: 

– CQED: E. Hagley et al., Phys. Rev. Lett. 79, 1 (1997). 

– Ions: Q.A. Turchette et al., Phys. Rev. Lett. 81, 3631 (1998). 

• Teleportation: 

– L. Davidovich, N. Zagury, M. Brune, J.M. Raimond and S. Haroche. "Teleportation of an 

atomic state between two cavities using non-local microwave fields". Phys Rev A50, R895 

(1994). 



• Reviews on CQED 



• A few useful textbooks 


• THANK YOU …..

quantum games with atoms and cavities - Electrodynamique ...

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