M. LaHaye, Qubit-coupled nanomechanics - PTB
M. LaHaye, Qubit-coupled nanomechanics - PTB
M. LaHaye, Qubit-coupled nanomechanics - PTB
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<strong>Qubit</strong>-Coupled Nanomechanics<br />
Matt <strong>LaHaye</strong> ” Syracuse University<br />
experiments performed at caltech with:<br />
junho suh, michael roukes - caltech<br />
keith schwab - caltech & cornell<br />
pierre echternach - j p l<br />
Quantum Measurement and Metrology with Solid State Devices<br />
PBH, Germany 5 Nov. 2009
mechanical structures<br />
in the quantum regime<br />
(Maryland) SSET/NEMS<br />
(UCSB) SET/NEMS<br />
m<br />
(Cornell/Caltech) SMR/NEMS<br />
(Caltech, Cornell, JPL) NEMS/CPB<br />
(Delft):DC-SQUID/NEMS<br />
(JILA): APC/NEMS, SMR/NEMS<br />
Nanoelectromechanical Systems (NEMS)<br />
And many others …<br />
(MIT & LIGO)<br />
(Oregon)<br />
Casimir Physics<br />
(UCSB)<br />
(Yale) (Vienna)<br />
Optomechanical<br />
Systems<br />
(Caltech,<br />
Max Planck Institute)<br />
Atoms, Ions, Spins<br />
(Dartmouth/ Padova) (IBM Almaden)
mechanical structures<br />
in the quantum regime<br />
(Maryland) SSET/NEMS<br />
(UCSB) SET/NEMS<br />
(Cornell/Caltech) SMR/NEMS<br />
(Caltech, Cornell, JPL) NEMS/CPB<br />
(Delft):DC-SQUID/NEMS<br />
(JILA): APC/NEMS, SMR/NEMS<br />
(MIT & LIGO)<br />
(Oregon)<br />
Casimir Physics<br />
(UCSB)<br />
Interesting review from a few years ago: K. Schwab and Michael Roukes,<br />
m<br />
Physics Today July 2005<br />
Nanoelectromechanical Systems (NEMS)<br />
(Yale) (Vienna)<br />
More recently: special issue of the New Journal of Physics on mechanical<br />
systems approaching the quantum regime. September 2008<br />
Optomechanical<br />
Systems<br />
(Caltech,<br />
Max Planck Institute)<br />
Gordon Conference 2008 &2010: Mechanical Systems in the Quantum Regime<br />
And many others …<br />
Atoms, Ions, Spins<br />
(Dartmouth/ Padova) (IBM Almaden)
‚ultimate limit of NEMS is in the quantum regime‛ ” Roukes (2001)<br />
Ideal characteristics: Small mass,<br />
“ Zero-point motion<br />
x / 2 ~ 40 fm<br />
zp<br />
m <br />
“ Energy-level spacing<br />
“ Typ. quality factors ~ 10 4 -10 5 ,<br />
but demonstrated >10 6<br />
Estimate for SiC resonator,<br />
.6m x .4m x .07m<br />
Mass ~ 50 fg, f0 ~ 127 MHz<br />
Orders of magnitude larger than gram- or kg-scale oscillators<br />
ωο kBT For 1 GHz resonator<br />
At mK temperatures<br />
Attainable with dilution fridge.<br />
May portend long coherence and<br />
relaxation times (~ sec’s)<br />
high frequency, low dissipation<br />
Huang, Roukes, 2003<br />
Mo Li, Hong Tang, Michael Roukes, 2007<br />
Schwab 2008<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany - 05 Nov. 2009<br />
4
approaching the quantum limit of NEMS with an RFSET<br />
“ The radio-frequency single-electron transistor (RFSET) as<br />
a quantum-limited displacement detector (proposed by<br />
Blencowe and Wybourne, APL 2000)<br />
Demonstrated sensitivity using superconducting SET (SSET) near<br />
(~4x) the quantum limit for continuous linear detection. SSET a<br />
near-ideal linear detector: =15 /2<br />
Observation of low nanoresonator thermal occupation N th= KT/ (~25).<br />
Observed SSET quantum back-action on the NEMS; measured asymmetry<br />
In SSET noise spectrum; performed back-action cooling of NEMS<br />
“Potential for interesting future experiments<br />
(Ground-state cooling) A. Hopkins, K. Jacobs, S. Habib & K. Schwab, PRB (2003).<br />
(Squeezing) R. Ruskov, A. Korotkov & K. Schwab, IEEE Trans. Nano., (2005).<br />
(Micro-maser analog) D. Rodrigues, J. imbers & A. Armour (2007).<br />
M. <strong>LaHaye</strong>, O. Buu, B. Camarota,<br />
K. Schwab, Science 2004<br />
Gate of SET<br />
NR Gate<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009<br />
V NR<br />
SSET<br />
19.7 MHz Resonator<br />
V NR<br />
V NR<br />
NR<br />
20 MHz<br />
1m<br />
A. Naik, O. Buu, M. <strong>LaHaye</strong>, A. Armour, M.<br />
Blencowe, A. Clerk, K. Schwab, Nature 2006
approaching the quantum limit of NEMS with an RFSET<br />
“ The radio-frequency single-electron transistor (RFSET) as<br />
a quantum-limited displacement detector (proposed by<br />
Blencowe and Wybourne, APL 2000)<br />
Demonstrated sensitivity using superconducting SET (SSET) near<br />
(~4x) the quantum limit for continuous linear detection. SSET a<br />
near-ideal linear detector: =15 /2<br />
Observation of low nanoresonator thermal occupation N th= KT/ (~25).<br />
Observed SSET quantum back-action on the NEMS; measured asymmetry<br />
In SSET noise spectrum; performed back-action cooling of NEMS<br />
“ Other linear displacement detectors developed<br />
(Normal SET) R. Knobel & A. Cleland, Nature 424 , 291 (2003).<br />
(APC) N. Flowers-Jacobs, D. Schmidt & K. Lehnert, PRL 98, 096804 (2007)<br />
(DC SQUID) S. Etaki et al., Nature Physics 4, 785 (2008)<br />
M. <strong>LaHaye</strong>, O. Buu, B. Camarota,<br />
K. Schwab, Science 2004<br />
Gate of SET<br />
NR Gate<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009<br />
V NR<br />
SSET<br />
19.7 MHz Resonator<br />
V NR<br />
V NR<br />
NR<br />
20 MHz<br />
1m<br />
A. Naik, O. Buu, M. <strong>LaHaye</strong>, A. Armour, M.<br />
Blencowe, A. Clerk, K. Schwab, Nature 2006
qubit-<strong>coupled</strong> <strong>nanomechanics</strong><br />
First proposed by A. Armour, M. Blencowe & K. Schwab: PRL 88 (2002) & Physica B 316 (2002).<br />
Nano-electromechanical resonator Cooper-pair box (CPB) charge qubit<br />
Cleland & Roukes, APL 69 28 Oct. 1996<br />
=<br />
+<br />
electrostatic interaction<br />
<br />
|2><br />
|1><br />
|0><br />
artificial<br />
x<br />
atom Harmonic oscillator<br />
Nakamura et al., Nature, 398 29 Apr. 1999<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany - 05 Nov. 2009<br />
|n><br />
resonator motion<br />
couples to charge<br />
on the qubit
qubit-<strong>coupled</strong> <strong>nanomechanics</strong><br />
First proposed by A. Armour, M. Blencowe & K. Schwab: PRL 88 (2002) & Physica B 316 (2002).<br />
Nano-electromechanical resonator Cooper-pair box (CPB) charge qubit<br />
Cleland & Roukes, APL 69 28 Oct. 1996<br />
=<br />
<br />
+<br />
electrostatic interaction<br />
|2><br />
|1><br />
|0><br />
artificial<br />
x<br />
atom Harmonic oscillator<br />
Nakamura et al., Nature, 398 29 Apr. 1999<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany - 05 Nov. 2009<br />
|n><br />
use qubit to prepare<br />
quantum superposition<br />
states of NEMS and<br />
study decoherence<br />
8
superconducting qubits as tools for quantum NEMS<br />
Partial list of proposals utilizing a qubit to manipulate and measure<br />
quantum states of NEMS<br />
• NEMS and Cooper-pair box (CPB) entanglement to produce NEMS superposition states<br />
(Charge-state) A.D. Armour, M.P Blencowe, K.C. Schwab, PRL 88, 148301 (2002).<br />
(Dispersive) (1) A.D. Armour & M.P. Blencowe, New J. Phys. 10 095004 (2008) (2)D.W. Utami, & A.A. Clerk,<br />
Phys. Rev. A 78 042323 (2008). (3) K. Jacobs, A.N. Jordan, & E.K. Irish, Euro. Phys. Lett. 82, 18003 (2008).<br />
• Measurement of quantized energy spectrum of NEMS<br />
(1) E.K. Irish & K.C. Schwab, PRB 68, 155311 (2003). (2) K. Jacobs, P. Lougovski,& M.P. Blencowe, PRB 98,<br />
147201 (2007). (3) K. Jacobs, A.N. Jordan & E.K. Irish, Euro. Phys. Lett. 82, 18003 (2008). (4) A.A. Clerk, &<br />
D.W. Utami, PRA 75, 042302 (2007).<br />
• Microwave-mediated techniques<br />
(Ground-state cooling) I. Martin et al., Phys. Rev. B 69, 125339 (2004). (Squeezing) P. Rabl et al.,<br />
PRB 70, 205304 (2004). (Entanglement) L.Tian, PRB 72, 195411 (2005). (Lasing) J. Hauss et al.,<br />
Phys. Rev. Lett. 100, 037003 (2008).<br />
Many other proposals involving different types of qubits, quantum electronics<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
in the remainder of this talk…<br />
Brief review of the Cooper-pair box (CPB) charge qubit, how we couple the CPB and<br />
NEMS, dispersive interaction<br />
First experiment: observe the dispersive interaction between CPB and NEMS and<br />
use it to perform spectroscopy of CPB and measurement of LZ-interference<br />
effects . Parametric Amplification/(Classical)Squeezing of NEMS.<br />
Demonstrated coupling should be large enough to pursue more advanced<br />
measurement proposals, e.g. ground-state cooling, ‘lasing’, and squeezing of NEMS.<br />
Significant room for improvement to coupling strength. CPB/NEMS entanglement<br />
experiment looks within reach. Should also be able to approach strong coupling<br />
limit, a prerequisite for NEMS number-state detection.<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
eview of the Cooper-pair box<br />
Essentially it is an highly-polarizable, artificial, two-state atom<br />
CPB layout<br />
Nakamura et al., Nature, Vol. 398, 29 April 1999<br />
Small capacitance yields large<br />
charging energy E c, so only two<br />
relevant charge states<br />
Hamiltonian<br />
nˆ<br />
0 0 Cooper-pairs<br />
on box<br />
1 1<br />
Cooper-pair<br />
on box<br />
ˆ EJ<br />
H 2 E (1 2 n ) σˆ σˆ<br />
2<br />
C g z x<br />
E E cos( πΦ<br />
/ Φ )<br />
J J0<br />
0<br />
CPB energy bands<br />
0.2 0.4 0.6 0.8<br />
dc gate charge ng (CgVg/2e) Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009<br />
E (GHz)<br />
15<br />
10<br />
5<br />
0<br />
-5<br />
-10<br />
-15<br />
1<br />
0<br />
<br />
<br />
<br />
<br />
0 1 / 2<br />
E J = 9 GHz<br />
0 1 / 2<br />
n g =C gV g/2e - Applied gate charge<br />
= Applied flux through CPB loop<br />
0 = Flux quantum<br />
1<br />
0
eview of the Cooper-pair box<br />
Essentially it is an highly-polarizable, artificial, two-state atom<br />
CPB layout<br />
Nakamura et al., Nature, Vol. 398, 29 April 1999<br />
CPB energy bands<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009<br />
E (GHz)<br />
15<br />
10<br />
5<br />
0<br />
-5<br />
-10<br />
<br />
<br />
0 1 / 2<br />
E J =3.0 GHz<br />
0 1 / 2<br />
-15<br />
Small capacitance yields large<br />
<br />
0 0 Cooper-pairs<br />
on box<br />
charging energy Ec, so only two nˆ<br />
0.2 0.4 0.6 0.8<br />
1 1<br />
Cooper-pair<br />
on box<br />
relevant charge states dc gate charge ng (CgVg/2e) Hamiltonian<br />
ˆ EJ<br />
H 2 E (1 2 n ) σˆ σˆ<br />
2<br />
C g z x<br />
E E cos( πΦ<br />
/ Φ )<br />
J J0<br />
0<br />
n g =C gV g/2e - Applied gate charge<br />
= Applied flux through CPB loop<br />
0 = Flux quantum
eview of the Cooper-pair box<br />
Essentially it is an highly-polarizable, artificial, two-state atom<br />
CPB layout<br />
Nakamura et al., Nature, Vol. 398, 29 April 1999<br />
Small capacitance yields large<br />
charging energy E c, so only two<br />
relevant charge states<br />
Hamiltonian<br />
nˆ<br />
0 0 Cooper-pairs<br />
on box<br />
1 1<br />
Cooper-pair<br />
on box<br />
ˆ EJ<br />
H 2 E (1 2 n ) σˆ σˆ<br />
2<br />
C g z x<br />
E <br />
E cos( πΦ<br />
/ Φ )<br />
J J0<br />
0<br />
Expectation Value of Charge<br />
0<br />
0 .5 1 1.5 2<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009<br />
<br />
nˆ<br />
1<br />
2<br />
1.5<br />
.5 1<br />
.5<br />
0<br />
0<br />
Excited state<br />
nˆ 1<br />
<br />
n<br />
E<br />
g<br />
‘Quantum<br />
Capacitance’<br />
J<br />
Ground State<br />
nˆ 1<br />
<br />
n<br />
E<br />
dc gate charge n g (C gV g/2e)<br />
g<br />
.5 1<br />
n g =C gV g/2e - Applied gate charge<br />
= Applied flux through CPB loop<br />
0 = Flux quantum<br />
J
eview of the Cooper-pair box<br />
Essentially it is an highly-polarizable, artificial, two-state atom<br />
CPB layout<br />
Nakamura et al., Nature, Vol. 398, 29 April 1999<br />
Small capacitance yields large<br />
charging energy E c, so only two<br />
relevant charge states<br />
Hamiltonian<br />
nˆ<br />
n n Cooper - pairs on box<br />
n 1 n 1Cooper-pairs<br />
on box<br />
ˆ<br />
2<br />
H 4E<br />
( nˆ<br />
n ) E<br />
C<br />
g<br />
J<br />
Θˆ cos<br />
Gate periodicity of CPB energy bands<br />
Excited State<br />
Ground State<br />
Sweeping n g over many degeneracy points, Cooper-pairs tunnel to minimize electrostatic energy<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009<br />
E/E c<br />
dc gate charge n g (C gV g/2e)<br />
n g =C gV g/2e - Applied gate charge
Dispersive limit of CPB-NEMS Hamiltonian<br />
ˆ 1/<br />
2<br />
ˆ ω N <br />
J †<br />
T σ Z<br />
a a<br />
H NR<br />
Δ <br />
EJ NR<br />
E<br />
ˆ<br />
2<br />
ω N λ<br />
Dispersive Hamiltonian<br />
ˆ ˆ ˆ X<br />
CPB and NEMS far-detuned for our parameters<br />
Direct exchange of quanta suppressed by<br />
2 2<br />
EJ λ<br />
1/ 2 2 1<br />
Hˆ ω Nˆ σˆ Nˆ σˆ<br />
2 E<br />
disp NR Z Z<br />
J<br />
CPB-state-dependent<br />
Frequency<br />
Shift in NEMS<br />
NEMS-<br />
Dependent shift<br />
in CPB transition<br />
2<br />
2 λ<br />
ω σˆ<br />
E<br />
Δ NR Z<br />
J<br />
<br />
2 2<br />
2 λ ˆ<br />
N<br />
ΔECPB 2N 1<br />
E<br />
J<br />
λ <br />
~ <br />
Δ <br />
RWA<br />
2<br />
<br />
ˆ ˆ EJ †<br />
HRWA ωNRN1/ 2 σˆ<br />
<br />
aˆˆ aˆˆ<br />
<br />
Z<br />
<br />
2<br />
<br />
Jaynes-Cummings Hamiltonian<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009<br />
N=3<br />
N=2<br />
N=1<br />
N=0<br />
N=3<br />
N=2<br />
N=1<br />
N=0<br />
.<br />
Energy levels with Interaction<br />
.<br />
Energy levels w/o interaction<br />
<br />
<br />
<br />
ω<br />
<br />
ωNR<br />
<br />
ω<br />
<br />
<br />
J<br />
<br />
E J<br />
.<br />
.<br />
E <br />
Δ E<br />
N=3<br />
N=2<br />
N=1<br />
N=0<br />
N=3<br />
N=2<br />
N=1<br />
N=0<br />
N<br />
CPB<br />
<br />
ω ωΔ ω , ω ωΔω NR NR<br />
NR NR
estimates for the nanomechanical frequency shift<br />
Frequency shift depends on CPB<br />
state, and magnitude proportional<br />
to CPB energy band curvature*:<br />
λ<br />
E<br />
Δf<br />
σ<br />
π E n E<br />
2<br />
2<br />
NEMS <br />
J<br />
3/2 ˆZ<br />
2 2<br />
((4 C (12 g )) J <br />
E <br />
E cos( πΦ<br />
/ Φ )<br />
J J,max<br />
0<br />
Expect frequency shift of 10’s ppm<br />
at charge degneracy<br />
NEMS frequency detection<br />
schemes can routinely achieve<br />
better than ppm sensitivity<br />
CPB Energy (GHz)<br />
CPB Energy and NEMS frequency shift vs n g<br />
Excited State<br />
Ground State<br />
Gate Charge, n g (2e)<br />
Parameters<br />
*This is the quantum capacitance effect measured via LC resonator in<br />
Sillanpaa et al., PRL 95 206806 (2005) and Duty et al., PRL 95 206807 (2005)<br />
C NR ~ 50 aF<br />
d ~ 300 nm<br />
V NR ~ 10 V<br />
f NEMS = o /2 ~ 60 MHz<br />
Gate Charge, n g (2e)<br />
K ~ 60 N/m<br />
/2~ 2.0 MHz<br />
E C ~14 GHz<br />
E J ~ 13 GHz<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009<br />
Δf<br />
NEMS<br />
/<br />
f<br />
NEMS<br />
~ 10<br />
5<br />
NEMS Frequency Shift (Hz)
device layout<br />
Flux<br />
Bias<br />
loop<br />
CPB<br />
Reservoir<br />
CPB<br />
CPB<br />
Gate<br />
fabrication at JPL and Caltech<br />
NEMS<br />
NEMS Gate<br />
Silicon<br />
Nitride<br />
Aluminum<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009 18
measurement layout<br />
V g<br />
B<br />
On resonance<br />
Z M = R m~ M’s<br />
ELECTROMECHANICAL<br />
IMPEDANCE<br />
L m<br />
C m<br />
R m<br />
V NR<br />
Drive Force<br />
(V NR - V gnr)V drive<br />
C gnr<br />
NEMS response with CPB biased off charge degeneracy<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009<br />
L T<br />
C T<br />
Amplitude (V)<br />
120<br />
100<br />
80<br />
60<br />
40<br />
20<br />
NEMS’ response at<br />
T mc ~ 100 mK<br />
0<br />
58.42 58.425 58.43 58.435 58.44<br />
V gnr<br />
-5<br />
58.42 58.425 58.43 58.435 58.44<br />
Q ~ 50,000 Frequency (MHz)<br />
V NR= 10 V<br />
50 <br />
Frequency (MHz)<br />
V drive<br />
Phase (Rad)<br />
0<br />
-1<br />
-2<br />
-3<br />
-4<br />
LNA<br />
rf if<br />
REFLECTOMETRY<br />
TO MEASURE Z M<br />
lo
measurement layout<br />
V g<br />
B<br />
On resonance<br />
Z M = R m~ M’s<br />
ELECTROMECHANICAL<br />
IMPEDANCE<br />
L m<br />
C m<br />
R m<br />
V NR<br />
Drive Force<br />
(V NR - V gnr)V drive<br />
C gnr<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009<br />
L T<br />
C T<br />
Amplitude (V)<br />
NEMS response on and off a charge degeneracy<br />
120<br />
100<br />
80<br />
60<br />
40<br />
20<br />
0<br />
58.424 58.426 58.428 58.43 58.432 58.434 58.436<br />
V gnr<br />
Off Degeneracy<br />
On Degeneracy<br />
-<br />
f NEMS ~ 600 Hz 58.424 58.426 58.428 58.43 58.432 58.434 58.436<br />
V NR= 10 V<br />
50 <br />
Frequency (MHz)<br />
V drive<br />
Phase (Rad)<br />
-1<br />
-2<br />
-3<br />
-4<br />
Frequency (MHz)<br />
T mc~ 100 mK<br />
LNA<br />
rf if<br />
REFLECTOMETRY<br />
TO MEASURE Z M<br />
lo
f NEMS (Hz)<br />
V g (mV)<br />
dispersive interaction: measurement vs. model<br />
From M.D. <strong>LaHaye</strong> et al., Nature 459 , 960 (2009).<br />
-15<br />
-10<br />
-5<br />
0<br />
5<br />
10<br />
15<br />
Measurement: V NR= 7.0 V, T mc ~ 100 mK<br />
0<br />
-100<br />
-200<br />
5 -50 6 -0.5<br />
-1.0 -0.5 0.0 0.5 1.0<br />
Applied Magnetic Field (A.U.)<br />
15<br />
1 2<br />
Flux Periodicity:<br />
Note: Magnetic field applied on top of ~ 100 G<br />
E E cosπΦ<br />
/ Φ <br />
3<br />
2<br />
1<br />
4<br />
-10 -5 0 5 10<br />
Vg (mV)<br />
15<br />
0<br />
-100<br />
-150<br />
-200<br />
-250<br />
J<br />
f NEMS (Hz)<br />
J , max<br />
Model: /2= 1.40 MHz, T =100mK<br />
E J,max /h= 13.2 GHz, E C /h= 14.0 GHz<br />
Notes: Model convolved with 0.1 CP rms charge noise,<br />
and includes thermal population of CPB excited state<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009<br />
f NEMS (Hz)<br />
n g (2e)<br />
0.0<br />
0.5<br />
ο<br />
0<br />
-100<br />
-200<br />
6<br />
3 4<br />
-0.5 0.0 0.5<br />
5<br />
Flux ( o)<br />
<br />
Model<br />
Exp.<br />
-1.0 -0.5 0.0 0.5 1.0<br />
Flux (A.U.)<br />
6<br />
0<br />
-50<br />
-100<br />
-150<br />
-200<br />
f NEMS (Hz)
f NEMS (Hz)<br />
V g (mV)<br />
dispersive interaction: measurement vs. model<br />
From M.D. <strong>LaHaye</strong> et al., Nature 459 , 960 (2009).<br />
-15<br />
-10<br />
-5<br />
0<br />
5<br />
10<br />
15<br />
Measurement: V NR= 7.0 V, T mc ~ 100 mK<br />
0<br />
-100<br />
-200<br />
5 -50 6 -0.5<br />
-1.0 -0.5 0.0 0.5 1.0<br />
Note: Magnetic field applied on top of ~ 100 G E E cosπΦ<br />
/ Φ <br />
Model: /2= 1.40 MHz, T =100mK<br />
E J,max /h= 13.2 GHz, E C /h= 14.0 GHz<br />
With coupling strength, proposals Flux Periodicity: suggest that it should Flux be (possible o) to<br />
Applied Magnetic Field (A.U.)<br />
15<br />
1 2<br />
Notes: Model convolved with 0.1 CPrms charge noise,<br />
and includes thermal population of CPB excited state<br />
J J , max<br />
ο<br />
implement single qubit ‚lasing‛, ground-state cooling, squeezing of NEMS,<br />
3<br />
2<br />
4<br />
-10 -5 0 5 10<br />
Vg (mV)<br />
15<br />
0<br />
-100<br />
-150<br />
-200<br />
-250<br />
f NEMS (Hz)<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009<br />
f NEMS (Hz)<br />
n g (2e)<br />
0.0<br />
0.5<br />
0<br />
-200<br />
6<br />
3 4<br />
-0.5 0.0 0.5<br />
(Lasing) J. Hauss,, A. Federov, C. Hutter, A. Shnirman, G. Schon, PRL. 100, 037003 Model (2008)<br />
(Ground-state Cooling) I. Martin, A. Shnirman, L. Tian,<br />
1<br />
-100 P. Zoller, Phys. Rev. B<br />
5 69,<br />
Exp. 125339 (2004).<br />
(Squeezing) P. Rabl,, A. Shnirman, P. Zoller. Phys. Rev. B 70, 205304 (2004).<br />
-1.0 -0.5 0.0 0.5 1.0<br />
Flux (A.U.)<br />
6<br />
0<br />
-50<br />
-100<br />
-150<br />
-200<br />
f NEMS (Hz)
V g (mV)<br />
V g (mV)<br />
increasing microwave frequency<br />
f (Hz)<br />
f (Hz)<br />
NEMS Microwave Frequency: 13.5 GHz f (Hz)<br />
NEMS Microwave Frequency: 12.5 GHz f (Hz)<br />
NEMS NEMS<br />
-24<br />
-24<br />
-28<br />
-6<br />
E 0<br />
0<br />
0<br />
J = E 0<br />
MW OFF J,max 10.5 GHz 12.5 GHz 13.5 GHz<br />
-22<br />
EJ = EJ, max<br />
-22<br />
-26<br />
-4<br />
-100<br />
-100<br />
-100<br />
-100<br />
-20<br />
-20<br />
-24<br />
-2<br />
-200<br />
-200<br />
-200<br />
-18<br />
-18<br />
-22<br />
-200<br />
0<br />
-300<br />
-300<br />
-300<br />
-16<br />
-16<br />
-20<br />
-300<br />
2<br />
-400<br />
-400<br />
-400<br />
-14<br />
4<br />
-14<br />
-18<br />
-400<br />
-500<br />
-500<br />
-500<br />
6<br />
-12<br />
-12<br />
-16<br />
-600<br />
-600<br />
-500<br />
-600<br />
8<br />
-10<br />
-10<br />
-14<br />
-5.73 -5.72 -5.71 -5.7 -5.69 -5.68 -5.67<br />
-5.73 -5.72 -5.71 -5.7 -5.69 -5.68 -5.67<br />
-5.73 -5.72 -5.71 -5.7 -5.69 -5.68<br />
-5.73 -5.72 -5.71 -5.7 -5.69 -5.68 -5.67<br />
Flux (A.U.)<br />
Flux (A.U.)<br />
Flux (A.U.)<br />
Flux (A.U.)<br />
V g (mV)<br />
V g (mV)<br />
Microwave Frequency: 14.5 GHz f (Hz)<br />
Microwave Frequency: 16 GHz f (Hz)<br />
Microwave Frequency: 17 GHz f (Hz)<br />
NEMS NEMS NEMS Microwave Frequency: 20 GHz f (Hz)<br />
-12<br />
-12<br />
NEMS<br />
200<br />
-24<br />
0<br />
0<br />
14.5 GHz 16.0 GHz -12 17.0 GHz -10<br />
-10<br />
100 20.0 GHz<br />
-22<br />
0<br />
-100<br />
-100<br />
-10<br />
-8<br />
-8<br />
0<br />
-100<br />
-200<br />
-20<br />
-200<br />
-8<br />
-100<br />
-200<br />
-6<br />
-300<br />
-6<br />
-18<br />
-300<br />
-6<br />
-200<br />
-300<br />
-4<br />
-400<br />
-4<br />
-16<br />
-400<br />
-300<br />
-400<br />
-4<br />
-2<br />
-500<br />
-2<br />
-400<br />
-14<br />
-500<br />
-500<br />
-600<br />
-2<br />
0<br />
0<br />
-500<br />
-12<br />
-600<br />
-600<br />
-700<br />
0<br />
2<br />
2<br />
-600<br />
-5.73 -5.72 -5.71 -5.7 -5.69 -5.68<br />
-5.74 -5.73 -5.72 -5.71 -5.7 -5.69 -5.68<br />
-5.76 -5.75 -5.74 -5.73 -5.72 -5.71 -5.7<br />
-10<br />
-5.79 -5.78 -5.77 -5.76 -5.75 -5.74 -5.73<br />
Flux (A.U.)<br />
Flux (A.U.)<br />
Flux (A.U.)<br />
Flux (A.U..)<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009<br />
V g (mV)<br />
V g (mV)<br />
V g (mV)<br />
V g (mV)
Microwave Frequency (GHz)<br />
V g (mV)<br />
V g (mV)<br />
-24<br />
-22<br />
-20<br />
-18<br />
-16<br />
-14<br />
-12<br />
MW OFF E J = E<br />
J = EJ, J,max max<br />
-10<br />
-5.73 -5.72 -5.71 -5.7 -5.69 -5.68 -5.67<br />
Flux (A.U.)<br />
increasing microwave frequency<br />
V g (mV)<br />
V g (mV)<br />
-24<br />
-22 10.5 GHz<br />
f (Hz)<br />
NEMS<br />
0<br />
Microwave Frequency: 12.5 GHz<br />
-28<br />
12.5 GHz -26<br />
f (Hz)<br />
NEMS<br />
0<br />
Microwave Frequency: 13.5 GHz<br />
-6<br />
13.5 GHz<br />
-4<br />
f (Hz)<br />
NEMS<br />
0<br />
-20<br />
-18<br />
2|Vg| -100<br />
-200<br />
-24<br />
-22<br />
-100<br />
-200<br />
-2<br />
0<br />
-100<br />
-200<br />
-16<br />
-14<br />
-12<br />
-10<br />
-5.73 -5.72 -5.71 -5.7 -5.69<br />
Flux (A.U.)<br />
-5.68 -5.67<br />
-300<br />
-400<br />
-500<br />
-600<br />
-20<br />
-18<br />
-16<br />
-14<br />
-5.73 -5.72 -5.71 -5.7<br />
Flux (A.U.)<br />
-5.69 -5.68<br />
-300<br />
-400<br />
-500<br />
2<br />
4<br />
6<br />
8<br />
-5.73 -5.72 -5.71 -5.7 -5.69<br />
Flux (A.U.)<br />
-5.68 -5.67<br />
-300<br />
-400<br />
-500<br />
-600<br />
Microwave Frequency: 14.5 GHz f (Hz)<br />
Microwave Frequency: 16 GHz f (Hz)<br />
Microwave Frequency: 17 GHz f (Hz)<br />
NEMS NEMS NEMS Microwave Frequency: 20 GHz f (Hz)<br />
-12<br />
-12<br />
NEMS<br />
200<br />
-24<br />
0<br />
0<br />
14.5 GHz 16.0 GHz -12 17.0 GHz -10<br />
-10<br />
100 20.0 GHz<br />
-22<br />
0<br />
-100<br />
-100<br />
-10<br />
-8<br />
-8<br />
0<br />
-100<br />
-200<br />
-20<br />
-200<br />
-8<br />
-100<br />
-200<br />
-6<br />
-300<br />
-6<br />
-18<br />
-300<br />
-6<br />
-200<br />
-300<br />
-4<br />
-400<br />
-4<br />
-16<br />
-400<br />
-300<br />
-400<br />
-4<br />
-2<br />
-500<br />
-2<br />
-400<br />
-14<br />
-500<br />
-500<br />
-600<br />
-2<br />
0<br />
0<br />
-500<br />
-12<br />
-600<br />
-600<br />
-700<br />
0<br />
2<br />
2<br />
-600<br />
-5.73 -5.72 -5.71 -5.7 -5.69 -5.68<br />
-5.74 -5.73 -5.72 -5.71 -5.7 -5.69 -5.68<br />
-5.76 -5.75 -5.74 -5.73 -5.72 -5.71 -5.7<br />
-10<br />
-5.79 -5.78 -5.77 -5.76 -5.75 -5.74 -5.73<br />
Flux (A.U.)<br />
Flux (A.U.)<br />
Flux (A.U.)<br />
Flux (A.U..)<br />
20<br />
15<br />
10<br />
5<br />
0<br />
f NEMS (Hz)<br />
0<br />
-100<br />
-200<br />
-300<br />
-400<br />
-500<br />
-600<br />
0.0 0.04 0.08 0.12 0.16 0.20<br />
|V g /18.7| (2e)<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009<br />
V g (mV)<br />
V g (mV)<br />
For each value of E J<br />
Fit the data to<br />
V g (mV)<br />
V g (mV)<br />
hf Δ E (8E Δ n ) E<br />
2 2<br />
μ C g J<br />
Where | Δ ng | | ng<br />
.5<br />
| and EJ EJmaxcos( πΦ<br />
/ Φ 0)<br />
E C / h 1314GHz and E J / h ~ [0,9,10] GHz<br />
E J max / h ~ 12.5 13.5GHz<br />
From M.D. <strong>LaHaye</strong> et al., Nature 459 , 960 (2009).
NEMS <strong>coupled</strong> to strongly-driven CPB<br />
<strong>Qubit</strong> Landau-Zener interference: see Sillanpaa et al., PRL 96 (2006), Oliver et al., Science 310 (2005) ,<br />
Izmalkov et al., PRL (2008), Sun et al., APL (2009). Similar multi-photon transitions: see Wilson et al., PRL 98 (2007)<br />
Apply periodic modulation n g to CPB gate large enough to sweep CPB through charge degeneracy<br />
CPB ENERGY BANDS IN n g-SPACE<br />
ωRF<br />
<br />
EJ<br />
<br />
n g(t) = n go + n RFsin( RFt)<br />
n RF<br />
n g0<br />
n RF<br />
slope~ C g n 8E<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009<br />
E J<br />
<br />
ECPB<br />
<br />
ECPB
NEMS <strong>coupled</strong> to strongly-driven CPB<br />
<strong>Qubit</strong> Landau-Zener interference: see Sillanpaa et al., PRL 96 (2006), Oliver et al., Science 310 (2005) ,<br />
Izmalkov et al., PRL (2008), Sun et al., APL (2009). Similar multi-photon transitions: see Wilson et al., PRL 98 (2007)<br />
Starting in ground state , as approach degeneracy, probability P LZ for CPB to tunnel from to<br />
CPB ENERGY BANDS IN n g-SPACE<br />
n g(t) = n go + n RFsin( RFt)<br />
<br />
n RF<br />
n g0<br />
P<br />
LZ<br />
2<br />
πEJ<br />
exp( )<br />
2ν<br />
n RF<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009<br />
E J<br />
<br />
ECPB<br />
Energy<br />
Variation rate<br />
8E ~ ω ν<br />
C RF n<br />
RF<br />
<br />
ECPB
NEMS <strong>coupled</strong> to strongly-driven CPB<br />
<strong>Qubit</strong> Landau-Zener interference: see Sillanpaa et al., PRL 96 (2006), Oliver et al., Science 310 (2005) ,<br />
Izmalkov et al., PRL (2008), Sun et al., APL (2009). Similar multi-photon transitions: see Wilson et al., PRL 98 (2007)<br />
After crossing degeneracy, time-dependent phase (t) develops in wave function between and<br />
CPB ENERGY BANDS IN n g-SPACE<br />
Probability Amplitudes<br />
After tunneling<br />
<br />
Ψ<br />
Ψ<br />
<br />
iC<br />
PLZ<br />
C 1<br />
PLZ<br />
n RF<br />
n g0<br />
P<br />
LZ<br />
2<br />
πEJ<br />
exp( )<br />
2ν<br />
n RF<br />
<br />
ECPB<br />
Wave Function<br />
<br />
ECPB<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009<br />
<br />
Ψ<br />
<br />
Ψ<br />
φ(t)<br />
Ψ(t)<br />
Ψ<br />
1<br />
<br />
<br />
t<br />
<br />
ΔE<br />
<br />
e<br />
-iφ(t)<br />
dt' ΔE<br />
<br />
CPB<br />
Ψ<br />
(n<br />
g<br />
<br />
<br />
<br />
(t' ))<br />
<br />
CPB ECPB<br />
ECPB
NEMS <strong>coupled</strong> to strongly-driven CPB<br />
<strong>Qubit</strong> Landau-Zener interference: see Sillanpaa et al., PRL 96 (2006), Oliver et al., Science 310 (2005) ,<br />
Izmalkov et al., PRL (2008), Sun et al., APL (2009). Similar multi-photon transitions: see Wilson et al., PRL 98 (2007)<br />
Return swing: degeneracy crossed, probability for LZ tunneling to occur, interference between tunneling events<br />
CPB ENERGY BANDS IN n g-SPACE<br />
Wave Function<br />
Amplitudes<br />
n RF<br />
n g0<br />
2<br />
πEJ<br />
exp( )<br />
2ν<br />
n RF<br />
<br />
ECPB<br />
e 2 cos( /2)<br />
PLZ<br />
/2 -iφ<br />
PLZ φ<br />
Phase-developed between<br />
first and second LZ events<br />
2i PLZ<br />
( 1<br />
PLZ<br />
) cos( φ/<br />
2)<br />
t<br />
1<br />
φ<br />
dt' ΔE CPB (ng(t'<br />
))<br />
<br />
<br />
ECPB<br />
<br />
CPB ECPB<br />
ECPB<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009<br />
ΔE
NEMS <strong>coupled</strong> to strongly-driven CPB<br />
<strong>Qubit</strong> Landau-Zener interference: see Sillanpaa et al., PRL 96 (2006), Oliver et al., Science 310 (2005) ,<br />
Izmalkov et al., PRL (2008), Sun et al., APL (2009). Similar multi-photon transitions: see Wilson et al., PRL 98 (2007)<br />
After full cycle: if CPB coherence time is longer than cycle period, oscillations in excited state probability with <br />
CPB ENERGY BANDS IN n g-SPACE<br />
Probability to<br />
be in <br />
2 ( 1<br />
P )( 1<br />
cos( φ))<br />
PLZ LZ<br />
n RF<br />
n g0<br />
n RF<br />
<br />
ECPB<br />
Phase-developed between<br />
first and second LZ events<br />
t<br />
1<br />
φ<br />
dt' ΔE CPB (ng(t'<br />
))<br />
<br />
n g(t) = n go+ n RFsin( RFt)<br />
<br />
ECPB<br />
<br />
CPB ECPB<br />
ECPB<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009<br />
ΔE
V RF (Volts)<br />
NEMS as a probe of LZ interferometry<br />
V (V)<br />
Nanomechanical measurement of LZ interference<br />
From M.D. <strong>LaHaye</strong> et al., Nature 459 , 960 (2009).<br />
NR /2 (Hz)<br />
2.5<br />
2.0<br />
1.5<br />
1.0<br />
0.5<br />
1<br />
φ<br />
<br />
<br />
t<br />
<br />
dt' ΔE<br />
-400 -200 0 200 400 600<br />
-10.0 -8.0 -6.0 -4.0 -2.0<br />
V (mV)<br />
cpb ωRF / 2π<br />
4.<br />
0<br />
CPB<br />
(n<br />
g<br />
V g0 (mV)<br />
(t' ))<br />
Function of<br />
V , V , ωRF<br />
g0<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009<br />
RF<br />
GHz<br />
Modulate the CPB gate with large RF<br />
excitation V RF, and track NEMS frequency<br />
shift as a function of V g0 and V RF<br />
CPB Excited state becomes populated,<br />
changing sign of NEMS frequency shift
V RF (Volts)<br />
NEMS as a probe of LZ interferometry<br />
V (V)<br />
Nanomechanical measurement of LZ interference<br />
From M.D. <strong>LaHaye</strong> et al., Nature 459 , 960 (2009).<br />
NR /2 (Hz)<br />
2.5<br />
2.0<br />
1.5<br />
1.0<br />
0.5<br />
1<br />
φ<br />
<br />
<br />
t<br />
<br />
dt' ΔE<br />
-400 -200 0 200 400 600<br />
CPB<br />
(n<br />
-4<br />
g<br />
-3<br />
-10.0 -8.0 -6.0 -4.0 -2.0<br />
V (mV)<br />
cpb ωRF / 2π<br />
4.<br />
0<br />
(t' ))<br />
-3<br />
V g0 (mV)<br />
-4<br />
Function of<br />
V , V , ωRF<br />
g0<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009<br />
RF<br />
GHz<br />
Modulate the CPB gate with large RF<br />
excitation V RF, and track NEMS frequency<br />
shift as a function of V g0 and V RF<br />
CPB Excited state becomes populated,<br />
changing sign of NEMS frequency shift<br />
‚Constructive‛ interference occurs at V g0 where<br />
= 2n (intersection of black lines in plot ).<br />
P LZ LZ<br />
2P ( 1<br />
P )( 1<br />
cos( φ))<br />
Parameters used for contour overlay:<br />
Ec = 15 GHz, Ej=13 GHz
NEMS <strong>coupled</strong> to strongly-driven CPB<br />
Estimate of E C from LZ interference<br />
V RF (Volts)<br />
V (V)<br />
NR /2 (Hz)<br />
2.5<br />
2<br />
1.5<br />
1<br />
0.5<br />
-500 0 500 1000 1500<br />
-4<br />
-8.0 -6.0 -4.0 -2.0 0.0<br />
V (mV) ωRF n Vcpb g0 (mV)<br />
6.<br />
5 GHz<br />
g0 conversion: 18.7 mV per 2e<br />
2π<br />
Expected<br />
Fringe<br />
spacing: Δng0 ωRF<br />
4EC<br />
-3<br />
From fit<br />
-3<br />
-4<br />
E C/h = 14.9 .6 GHz<br />
-8.0 -6.0 -4.0 -2.0 0.0<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009<br />
f NR (Hz)<br />
n g0 (2e)<br />
5000<br />
4000<br />
3000<br />
2000<br />
1000<br />
LZ Fringes at constant V RF<br />
0<br />
0.12<br />
0.10<br />
0.08<br />
0.06<br />
n g0<br />
V cpb<br />
n g0 (2e)<br />
Fit to straight<br />
Line thru origin<br />
6.50 GHz<br />
5.66 GHz<br />
4.83 GHz<br />
/2=4.00 GHz<br />
4.0 5.0 6.0 7.0<br />
RF/2 (GHz)
prospects for strong dispersive coupling limit<br />
Definition of strong coupling limit: Dispersive<br />
interaction exceeds qubit and NEMS linewidth<br />
Δf<br />
NEMS<br />
Δ<br />
E N<br />
CPB<br />
( 2N<br />
1)<br />
h<br />
2<br />
λ γNEMSγCPB [<br />
, ]<br />
πE 2π2π J<br />
Demonstrated f NEMS NEMS/2<br />
With conservative improvements to sample<br />
geometry, should achieve f NEMS ~ 100’s kHz<br />
Present sample: CPB/2f NEMS<br />
However, there is significant room to improve,<br />
e.g. in circuit QED, CPB/2 1 MHz<br />
e.g. see Wallraff et al., Nature 431 (2004)<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009<br />
Amplitude (V)<br />
35<br />
30<br />
20<br />
15<br />
10<br />
5<br />
Present Sample: NEMS Linewidth<br />
Off Degeneracy<br />
-<br />
25 On Degeneracy<br />
Phase (Rad.)<br />
58.470 58.475 58.480<br />
Frequency (MHz)<br />
58.470 58.475 58.480 58.485<br />
Frequency (MHz)<br />
2<br />
1<br />
0<br />
-1<br />
1.6 kHz<br />
VNR= 15 V<br />
T ~ 130 mK<br />
Present Sample: CPB Linewidth
prospects for dispersive CPB-NEMS entangled states<br />
Proposals: D.W. Utami, & A.A. Clerk,, Phys. Rev. A 78 042323 (2008)<br />
A.D. Armour & M.P. Blencowe, New J. Phys. 10 095004 (2008)<br />
General idea: (1) With CPB and NEMS un<strong>coupled</strong>, prepare CPB in superposition of<br />
energy eigenstates and nanoresonator in displaced thermal state<br />
(2) Dispersively couple CPB and NEMS<br />
After time t:<br />
Envelope of CPB<br />
oscillations after -pulse<br />
Initial state:<br />
Ψ( t) <br />
1<br />
( α( t) i α( t)<br />
)<br />
2<br />
α ω Δω<br />
Nanoresonator Is in a superposition of<br />
states ‘winding’ at frequencies<br />
dressed by the CPB<br />
1<br />
Ψ(0) ( i ) α(0)<br />
2<br />
qubit state resonantor state<br />
NR NR<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009<br />
<br />
α ω Δω<br />
<br />
NR NR<br />
1 2<br />
env (1 exp[ (0) (1 cos(Δ NR ))])<br />
P α ω t<br />
2<br />
Using qubit echo<br />
method recoherences<br />
should be visible for<br />
similar device, e.g.<br />
λ 10 MHz<br />
ω / 2π 50 MHz<br />
NR<br />
Δ ω / 2π 40 kHz<br />
NR<br />
T ~100' s nsec<br />
<strong>Qubit</strong> recoherences: signature of entanglement<br />
2<br />
T 50 mK
conclusions<br />
New era of experiments studying the quantum properties<br />
of mechanical structures<br />
Superconducting qubits should serve as viable tools to manipulate<br />
and measure quantum states of NEMS<br />
We have demonstrated the first coupling between a superconducting<br />
qubit and NEMS<br />
- use dispersive interaction to perform spectroscopy and read-out<br />
quantum interference in the CPB, parametric amplification/squeezing<br />
- with realistic improvements to devices, experiments with quantum<br />
NEMS, even entanglement of NEMS and CPB, are within reach<br />
Thanks to Gerard Milburn, Andrew Doherty, Katya Babourina, Aash Clerk, Andrew Armour, Miles Blencowe,<br />
Christopher Wilson, and Tim Duty for helpful insight and advice.<br />
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009