M. LaHaye, Qubit-coupled nanomechanics - PTB

Qubit-Coupled Nanomechanics 

Matt LaHaye ” Syracuse University 

experiments performed at caltech with: 

junho suh, michael roukes - caltech 

keith schwab - caltech & cornell 

pierre echternach - j p l 

Quantum Measurement and Metrology with Solid State Devices 

PBH, Germany 5 Nov. 2009

mechanical structures 

in the quantum regime 

(Maryland) SSET/NEMS 

(UCSB) SET/NEMS 

m 

(Cornell/Caltech) SMR/NEMS 

(Caltech, Cornell, JPL) NEMS/CPB 

(Delft):DC-SQUID/NEMS 

(JILA): APC/NEMS, SMR/NEMS 

Nanoelectromechanical Systems (NEMS) 

And many others … 

(MIT & LIGO) 

(Oregon) 

Casimir Physics 

(UCSB) 

(Yale) (Vienna) 

Optomechanical 

Systems 

(Caltech, 

Max Planck Institute) 

Atoms, Ions, Spins 

(Dartmouth/ Padova) (IBM Almaden)

mechanical structures 

in the quantum regime 

(Maryland) SSET/NEMS 

(UCSB) SET/NEMS 

(Cornell/Caltech) SMR/NEMS 

(Caltech, Cornell, JPL) NEMS/CPB 

(Delft):DC-SQUID/NEMS 

(JILA): APC/NEMS, SMR/NEMS 

(MIT & LIGO) 

(Oregon) 

Casimir Physics 

(UCSB) 

Interesting review from a few years ago: K. Schwab and Michael Roukes, 

m 

Physics Today July 2005 

Nanoelectromechanical Systems (NEMS) 

(Yale) (Vienna) 

More recently: special issue of the New Journal of Physics on mechanical 

systems approaching the quantum regime. September 2008 

Optomechanical 

Systems 

(Caltech, 

Max Planck Institute) 

Gordon Conference 2008 &2010: Mechanical Systems in the Quantum Regime 

And many others … 

Atoms, Ions, Spins 

(Dartmouth/ Padova) (IBM Almaden)

‚ultimate limit of NEMS is in the quantum regime‛ ” Roukes (2001) 

Ideal characteristics: Small mass, 

“ Zero-point motion 

x / 2 ~ 40 fm 

zp 

m 

“ Energy-level spacing 

“ Typ. quality factors ~ 10 4 -10 5 , 

but demonstrated >10 6 

Estimate for SiC resonator, 

.6m x .4m x .07m 

Mass ~ 50 fg, f0 ~ 127 MHz 

Orders of magnitude larger than gram- or kg-scale oscillators 

ωο kBT For 1 GHz resonator 

At mK temperatures 

Attainable with dilution fridge. 

May portend long coherence and 

relaxation times (~ sec’s) 

high frequency, low dissipation 

Huang, Roukes, 2003 

Mo Li, Hong Tang, Michael Roukes, 2007 

Schwab 2008 

Quantum Measurement and Metrology with Solid State Devices PBH, Germany - 05 Nov. 2009 

4

approaching the quantum limit of NEMS with an RFSET 

“ The radio-frequency single-electron transistor (RFSET) as 

a quantum-limited displacement detector (proposed by 

Blencowe and Wybourne, APL 2000) 

Demonstrated sensitivity using superconducting SET (SSET) near 

(~4x) the quantum limit for continuous linear detection. SSET a 

near-ideal linear detector: =15 /2 

Observation of low nanoresonator thermal occupation N th= KT/ (~25). 

Observed SSET quantum back-action on the NEMS; measured asymmetry 

In SSET noise spectrum; performed back-action cooling of NEMS 

“Potential for interesting future experiments 

(Ground-state cooling) A. Hopkins, K. Jacobs, S. Habib & K. Schwab, PRB (2003). 

(Squeezing) R. Ruskov, A. Korotkov & K. Schwab, IEEE Trans. Nano., (2005). 

(Micro-maser analog) D. Rodrigues, J. imbers & A. Armour (2007). 

M. LaHaye, O. Buu, B. Camarota, 

K. Schwab, Science 2004 

Gate of SET 

NR Gate 

Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009 

V NR 

SSET 

19.7 MHz Resonator 

V NR 

V NR 

NR 

20 MHz 

1m 

A. Naik, O. Buu, M. LaHaye, A. Armour, M. 

Blencowe, A. Clerk, K. Schwab, Nature 2006

approaching the quantum limit of NEMS with an RFSET 

“ The radio-frequency single-electron transistor (RFSET) as 

a quantum-limited displacement detector (proposed by 

Blencowe and Wybourne, APL 2000) 

Demonstrated sensitivity using superconducting SET (SSET) near 

(~4x) the quantum limit for continuous linear detection. SSET a 

near-ideal linear detector: =15 /2 

Observation of low nanoresonator thermal occupation N th= KT/ (~25). 

Observed SSET quantum back-action on the NEMS; measured asymmetry 

In SSET noise spectrum; performed back-action cooling of NEMS 

“ Other linear displacement detectors developed 

(Normal SET) R. Knobel & A. Cleland, Nature 424 , 291 (2003). 

(APC) N. Flowers-Jacobs, D. Schmidt & K. Lehnert, PRL 98, 096804 (2007) 

(DC SQUID) S. Etaki et al., Nature Physics 4, 785 (2008) 

M. LaHaye, O. Buu, B. Camarota, 

K. Schwab, Science 2004 

Gate of SET 

NR Gate 


V NR 

SSET 

19.7 MHz Resonator 

V NR 

V NR 

NR 

20 MHz 

1m 

A. Naik, O. Buu, M. LaHaye, A. Armour, M. 

Blencowe, A. Clerk, K. Schwab, Nature 2006

qubit-coupled nanomechanics 

First proposed by A. Armour, M. Blencowe & K. Schwab: PRL 88 (2002) & Physica B 316 (2002). 

Nano-electromechanical resonator Cooper-pair box (CPB) charge qubit 

Cleland & Roukes, APL 69 28 Oct. 1996 

= 

+ 

electrostatic interaction 

 

|2> 

|1> 

|0> 

artificial 

x 

atom Harmonic oscillator 

Nakamura et al., Nature, 398 29 Apr. 1999 


|n> 

resonator motion 

couples to charge 

on the qubit

qubit-coupled nanomechanics 

First proposed by A. Armour, M. Blencowe & K. Schwab: PRL 88 (2002) & Physica B 316 (2002). 

Nano-electromechanical resonator Cooper-pair box (CPB) charge qubit 

Cleland & Roukes, APL 69 28 Oct. 1996 

= 

 

+ 

electrostatic interaction 

|2> 

|1> 

|0> 

artificial 

x 

atom Harmonic oscillator 

Nakamura et al., Nature, 398 29 Apr. 1999 


|n> 

use qubit to prepare 

quantum superposition 

states of NEMS and 

study decoherence 

8

superconducting qubits as tools for quantum NEMS 

Partial list of proposals utilizing a qubit to manipulate and measure 

quantum states of NEMS 

• NEMS and Cooper-pair box (CPB) entanglement to produce NEMS superposition states 

(Charge-state) A.D. Armour, M.P Blencowe, K.C. Schwab, PRL 88, 148301 (2002). 

(Dispersive) (1) A.D. Armour & M.P. Blencowe, New J. Phys. 10 095004 (2008) (2)D.W. Utami, & A.A. Clerk, 

Phys. Rev. A 78 042323 (2008). (3) K. Jacobs, A.N. Jordan, & E.K. Irish, Euro. Phys. Lett. 82, 18003 (2008). 

• Measurement of quantized energy spectrum of NEMS 

(1) E.K. Irish & K.C. Schwab, PRB 68, 155311 (2003). (2) K. Jacobs, P. Lougovski,& M.P. Blencowe, PRB 98, 

147201 (2007). (3) K. Jacobs, A.N. Jordan & E.K. Irish, Euro. Phys. Lett. 82, 18003 (2008). (4) A.A. Clerk, & 

D.W. Utami, PRA 75, 042302 (2007). 

• Microwave-mediated techniques 

(Ground-state cooling) I. Martin et al., Phys. Rev. B 69, 125339 (2004). (Squeezing) P. Rabl et al., 

PRB 70, 205304 (2004). (Entanglement) L.Tian, PRB 72, 195411 (2005). (Lasing) J. Hauss et al., 

Phys. Rev. Lett. 100, 037003 (2008). 

Many other proposals involving different types of qubits, quantum electronics 

Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009

in the remainder of this talk… 

Brief review of the Cooper-pair box (CPB) charge qubit, how we couple the CPB and 

NEMS, dispersive interaction 

First experiment: observe the dispersive interaction between CPB and NEMS and 

use it to perform spectroscopy of CPB and measurement of LZ-interference 

effects . Parametric Amplification/(Classical)Squeezing of NEMS. 

Demonstrated coupling should be large enough to pursue more advanced 

measurement proposals, e.g. ground-state cooling, ‘lasing’, and squeezing of NEMS. 

Significant room for improvement to coupling strength. CPB/NEMS entanglement 

experiment looks within reach. Should also be able to approach strong coupling 

limit, a prerequisite for NEMS number-state detection. 


eview of the Cooper-pair box 

Essentially it is an highly-polarizable, artificial, two-state atom 

CPB layout 

Nakamura et al., Nature, Vol. 398, 29 April 1999 

Small capacitance yields large 

charging energy E c, so only two 

relevant charge states 

Hamiltonian 

nˆ 

0 0 Cooper-pairs 

on box 

1 1 

Cooper-pair 

on box 

ˆ EJ 

H 2 E (1 2 n ) σˆ σˆ 

2 

C g z x 

E E cos( πΦ 

/ Φ ) 

J J0 

0 

CPB energy bands 

0.2 0.4 0.6 0.8 

dc gate charge ng (CgVg/2e) Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009 

E (GHz) 

15 

10 

5 

0 

-5 

-10 

-15 

1 

0 

 

 

 

 

0 1 / 2 

E J = 9 GHz 

0 1 / 2 

n g =C gV g/2e - Applied gate charge 

= Applied flux through CPB loop 

0 = Flux quantum 

1 

0



CPB layout 


CPB energy bands 


E (GHz) 

15 

10 

5 

0 

-5 

-10 

 

 

0 1 / 2 

E J =3.0 GHz 

0 1 / 2 

-15 


 


on box 

charging energy Ec, so only two nˆ 

0.2 0.4 0.6 0.8 

1 1 

Cooper-pair 

on box 

relevant charge states dc gate charge ng (CgVg/2e) Hamiltonian 

ˆ EJ 

H 2 E (1 2 n ) σˆ σˆ 

2 

C g z x 

E E cos( πΦ 

/ Φ ) 

J J0 

0 



0 = Flux quantum



CPB layout 





Hamiltonian 

nˆ 


on box 

1 1 

Cooper-pair 

on box 

ˆ EJ 

H 2 E (1 2 n ) σˆ σˆ 

2 

C g z x 

E 

E cos( πΦ 

/ Φ ) 

J J0 

0 

Expectation Value of Charge 

0 

0 .5 1 1.5 2 


 

nˆ 

1 

2 

1.5 

.5 1 

.5 

0 

0 

Excited state 

nˆ 1 

 

n 

E 

g 

‘Quantum 

Capacitance’ 

J 

Ground State 

nˆ 1 

 

n 

E 

dc gate charge n g (C gV g/2e) 

g 

.5 1 



0 = Flux quantum 

J



CPB layout 





Hamiltonian 

nˆ 

n n Cooper - pairs on box 

n 1 n 1Cooper-pairs 

on box 

ˆ 

2 

H 4E 

( nˆ 

n ) E 

C 

g 

J 

Θˆ cos 

Gate periodicity of CPB energy bands 

Excited State 

Ground State 

Sweeping n g over many degeneracy points, Cooper-pairs tunnel to minimize electrostatic energy 


E/E c 

dc gate charge n g (C gV g/2e) 

n g =C gV g/2e - Applied gate charge

Dispersive limit of CPB-NEMS Hamiltonian 

ˆ 1/ 

2 

ˆ ω N 

J † 

T σ Z 

a a 

H NR 

Δ 

EJ NR 

E 

ˆ 

2 

ω N λ 

Dispersive Hamiltonian 

ˆ ˆ ˆ X 

CPB and NEMS far-detuned for our parameters 

Direct exchange of quanta suppressed by 

2 2 

EJ λ 

1/ 2 2 1 

Hˆ ω Nˆ σˆ Nˆ σˆ 

2 E 

disp NR Z Z 

J 

CPB-state-dependent 

Frequency 

Shift in NEMS 

NEMS- 

Dependent shift 

in CPB transition 

2 

2 λ 

ω σˆ 

E 

Δ NR Z 

J 

 

2 2 

2 λ ˆ 

N 

ΔECPB 2N 1 

E 

J 

λ 

~ 

Δ 

RWA 

2 

 

ˆ ˆ EJ † 

HRWA ωNRN1/ 2 σˆ 

 

aˆˆ aˆˆ 

 

Z 

 

2 

 

Jaynes-Cummings Hamiltonian 


N=3 

N=2 

N=1 

N=0 

N=3 

N=2 

N=1 

N=0 

. 

Energy levels with Interaction 

. 

Energy levels w/o interaction 

 

 

 

ω 

 

ωNR 

 

ω 

 

 

J 

 

E J 

. 

. 

E 

Δ E 

N=3 

N=2 

N=1 

N=0 

N=3 

N=2 

N=1 

N=0 

N 

CPB 

 

ω ωΔ ω , ω ωΔω NR NR 

NR NR

estimates for the nanomechanical frequency shift 

Frequency shift depends on CPB 

state, and magnitude proportional 

to CPB energy band curvature*: 

λ 

E 

Δf 

σ 

π E n E 

2 

2 

NEMS 

J 

3/2 ˆZ 

2 2 

((4 C (12 g )) J 

E 

E cos( πΦ 

/ Φ ) 

J J,max 

0 

Expect frequency shift of 10’s ppm 

at charge degneracy 

NEMS frequency detection 

schemes can routinely achieve 

better than ppm sensitivity 

CPB Energy (GHz) 

CPB Energy and NEMS frequency shift vs n g 

Excited State 

Ground State 

Gate Charge, n g (2e) 

Parameters 

*This is the quantum capacitance effect measured via LC resonator in 

Sillanpaa et al., PRL 95 206806 (2005) and Duty et al., PRL 95 206807 (2005) 

C NR ~ 50 aF 

d ~ 300 nm 

V NR ~ 10 V 

f NEMS = o /2 ~ 60 MHz 

Gate Charge, n g (2e) 

K ~ 60 N/m 

/2~ 2.0 MHz 

E C ~14 GHz 

E J ~ 13 GHz 


Δf 

NEMS 

/ 

f 

NEMS 

~ 10 

5 

NEMS Frequency Shift (Hz)

device layout 

Flux 

Bias 

loop 

CPB 

Reservoir 

CPB 

CPB 

Gate 

fabrication at JPL and Caltech 

NEMS 

NEMS Gate 

Silicon 

Nitride 

Aluminum 

Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009 18

measurement layout 

V g 

B 

On resonance 

Z M = R m~ M’s 

ELECTROMECHANICAL 

IMPEDANCE 

L m 

C m 

R m 

V NR 

Drive Force 

(V NR - V gnr)V drive 

C gnr 

NEMS response with CPB biased off charge degeneracy 


L T 

C T 

Amplitude (V) 

120 

100 

80 

60 

40 

20 

NEMS’ response at 

T mc ~ 100 mK 

0 

58.42 58.425 58.43 58.435 58.44 

V gnr 

-5 

58.42 58.425 58.43 58.435 58.44 

Q ~ 50,000 Frequency (MHz) 

V NR= 10 V 

50 

Frequency (MHz) 

V drive 

Phase (Rad) 

0 

-1 

-2 

-3 

-4 

LNA 

rf if 

REFLECTOMETRY 

TO MEASURE Z M 

lo

measurement layout 

V g 

B 

On resonance 

Z M = R m~ M’s 

ELECTROMECHANICAL 

IMPEDANCE 

L m 

C m 

R m 

V NR 

Drive Force 

(V NR - V gnr)V drive 

C gnr 


L T 

C T 

Amplitude (V) 

NEMS response on and off a charge degeneracy 

120 

100 

80 

60 

40 

20 

0 

58.424 58.426 58.428 58.43 58.432 58.434 58.436 

V gnr 

Off Degeneracy 

On Degeneracy 

- 

f NEMS ~ 600 Hz 58.424 58.426 58.428 58.43 58.432 58.434 58.436 

V NR= 10 V 

50 


V drive 

Phase (Rad) 

-1 

-2 

-3 

-4 


T mc~ 100 mK 

LNA 

rf if 

REFLECTOMETRY 

TO MEASURE Z M 

lo

f NEMS (Hz) 

V g (mV) 

dispersive interaction: measurement vs. model 

From M.D. LaHaye et al., Nature 459 , 960 (2009). 

-15 

-10 

-5 

0 

5 

10 

15 

Measurement: V NR= 7.0 V, T mc ~ 100 mK 

0 

-100 

-200 

5 -50 6 -0.5 

-1.0 -0.5 0.0 0.5 1.0 

Applied Magnetic Field (A.U.) 

15 

1 2 

Flux Periodicity: 

Note: Magnetic field applied on top of ~ 100 G 

E E cosπΦ 

/ Φ 

3 

2 

1 

4 

-10 -5 0 5 10 

Vg (mV) 

15 

0 

-100 

-150 

-200 

-250 

J 

f NEMS (Hz) 

J , max 

Model: /2= 1.40 MHz, T =100mK 

E J,max /h= 13.2 GHz, E C /h= 14.0 GHz 

Notes: Model convolved with 0.1 CP rms charge noise, 

and includes thermal population of CPB excited state 


f NEMS (Hz) 

n g (2e) 

0.0 

0.5 

ο 

0 

-100 

-200 

6 

3 4 

-0.5 0.0 0.5 

5 

Flux ( o) 

 

Model 

Exp. 

-1.0 -0.5 0.0 0.5 1.0 

Flux (A.U.) 

6 

0 

-50 

-100 

-150 

-200 

f NEMS (Hz)

f NEMS (Hz) 

V g (mV) 

dispersive interaction: measurement vs. model 


-15 

-10 

-5 

0 

5 

10 

15 

Measurement: V NR= 7.0 V, T mc ~ 100 mK 

0 

-100 

-200 

5 -50 6 -0.5 

-1.0 -0.5 0.0 0.5 1.0 

Note: Magnetic field applied on top of ~ 100 G E E cosπΦ 

/ Φ 

Model: /2= 1.40 MHz, T =100mK 

E J,max /h= 13.2 GHz, E C /h= 14.0 GHz 

With coupling strength, proposals Flux Periodicity: suggest that it should Flux be (possible o) to 

Applied Magnetic Field (A.U.) 

15 

1 2 

Notes: Model convolved with 0.1 CPrms charge noise, 

and includes thermal population of CPB excited state 

J J , max 

ο 

implement single qubit ‚lasing‛, ground-state cooling, squeezing of NEMS, 

3 

2 

4 

-10 -5 0 5 10 

Vg (mV) 

15 

0 

-100 

-150 

-200 

-250 

f NEMS (Hz) 


f NEMS (Hz) 

n g (2e) 

0.0 

0.5 

0 

-200 

6 

3 4 

-0.5 0.0 0.5 

(Lasing) J. Hauss,, A. Federov, C. Hutter, A. Shnirman, G. Schon, PRL. 100, 037003 Model (2008) 

(Ground-state Cooling) I. Martin, A. Shnirman, L. Tian, 

1 

-100 P. Zoller, Phys. Rev. B 

5 69, 

Exp. 125339 (2004). 

(Squeezing) P. Rabl,, A. Shnirman, P. Zoller. Phys. Rev. B 70, 205304 (2004). 

-1.0 -0.5 0.0 0.5 1.0 

Flux (A.U.) 

6 

0 

-50 

-100 

-150 

-200 

f NEMS (Hz)

V g (mV) 

V g (mV) 

increasing microwave frequency 

f (Hz) 

f (Hz) 

NEMS Microwave Frequency: 13.5 GHz f (Hz) 

NEMS Microwave Frequency: 12.5 GHz f (Hz) 

NEMS NEMS 

-24 

-24 

-28 

-6 

E 0 

0 

0 

J = E 0 

MW OFF J,max 10.5 GHz 12.5 GHz 13.5 GHz 

-22 

EJ = EJ, max 

-22 

-26 

-4 

-100 

-100 

-100 

-100 

-20 

-20 

-24 

-2 

-200 

-200 

-200 

-18 

-18 

-22 

-200 

0 

-300 

-300 

-300 

-16 

-16 

-20 

-300 

2 

-400 

-400 

-400 

-14 

4 

-14 

-18 

-400 

-500 

-500 

-500 

6 

-12 

-12 

-16 

-600 

-600 

-500 

-600 

8 

-10 

-10 

-14 

-5.73 -5.72 -5.71 -5.7 -5.69 -5.68 -5.67 

-5.73 -5.72 -5.71 -5.7 -5.69 -5.68 -5.67 

-5.73 -5.72 -5.71 -5.7 -5.69 -5.68 

-5.73 -5.72 -5.71 -5.7 -5.69 -5.68 -5.67 

Flux (A.U.) 

Flux (A.U.) 

Flux (A.U.) 

Flux (A.U.) 

V g (mV) 

V g (mV) 

Microwave Frequency: 14.5 GHz f (Hz) 

Microwave Frequency: 16 GHz f (Hz) 


NEMS NEMS NEMS Microwave Frequency: 20 GHz f (Hz) 

-12 

-12 

NEMS 

200 

-24 

0 

0 

14.5 GHz 16.0 GHz -12 17.0 GHz -10 

-10 

100 20.0 GHz 

-22 

0 

-100 

-100 

-10 

-8 

-8 

0 

-100 

-200 

-20 

-200 

-8 

-100 

-200 

-6 

-300 

-6 

-18 

-300 

-6 

-200 

-300 

-4 

-400 

-4 

-16 

-400 

-300 

-400 

-4 

-2 

-500 

-2 

-400 

-14 

-500 

-500 

-600 

-2 

0 

0 

-500 

-12 

-600 

-600 

-700 

0 

2 

2 

-600 

-5.73 -5.72 -5.71 -5.7 -5.69 -5.68 

-5.74 -5.73 -5.72 -5.71 -5.7 -5.69 -5.68 

-5.76 -5.75 -5.74 -5.73 -5.72 -5.71 -5.7 

-10 

-5.79 -5.78 -5.77 -5.76 -5.75 -5.74 -5.73 

Flux (A.U.) 

Flux (A.U.) 

Flux (A.U.) 

Flux (A.U..) 


V g (mV) 

V g (mV) 

V g (mV) 

V g (mV)

Microwave Frequency (GHz) 

V g (mV) 

V g (mV) 

-24 

-22 

-20 

-18 

-16 

-14 

-12 

MW OFF E J = E 

J = EJ, J,max max 

-10 

-5.73 -5.72 -5.71 -5.7 -5.69 -5.68 -5.67 

Flux (A.U.) 

increasing microwave frequency 

V g (mV) 

V g (mV) 

-24 

-22 10.5 GHz 

f (Hz) 

NEMS 

0 

Microwave Frequency: 12.5 GHz 

-28 

12.5 GHz -26 

f (Hz) 

NEMS 

0 

Microwave Frequency: 13.5 GHz 

-6 

13.5 GHz 

-4 

f (Hz) 

NEMS 

0 

-20 

-18 

2|Vg| -100 

-200 

-24 

-22 

-100 

-200 

-2 

0 

-100 

-200 

-16 

-14 

-12 

-10 

-5.73 -5.72 -5.71 -5.7 -5.69 

Flux (A.U.) 

-5.68 -5.67 

-300 

-400 

-500 

-600 

-20 

-18 

-16 

-14 

-5.73 -5.72 -5.71 -5.7 

Flux (A.U.) 

-5.69 -5.68 

-300 

-400 

-500 

2 

4 

6 

8 

-5.73 -5.72 -5.71 -5.7 -5.69 

Flux (A.U.) 

-5.68 -5.67 

-300 

-400 

-500 

-600 

Microwave Frequency: 14.5 GHz f (Hz) 



NEMS NEMS NEMS Microwave Frequency: 20 GHz f (Hz) 

-12 

-12 

NEMS 

200 

-24 

0 

0 

14.5 GHz 16.0 GHz -12 17.0 GHz -10 

-10 

100 20.0 GHz 

-22 

0 

-100 

-100 

-10 

-8 

-8 

0 

-100 

-200 

-20 

-200 

-8 

-100 

-200 

-6 

-300 

-6 

-18 

-300 

-6 

-200 

-300 

-4 

-400 

-4 

-16 

-400 

-300 

-400 

-4 

-2 

-500 

-2 

-400 

-14 

-500 

-500 

-600 

-2 

0 

0 

-500 

-12 

-600 

-600 

-700 

0 

2 

2 

-600 

-5.73 -5.72 -5.71 -5.7 -5.69 -5.68 

-5.74 -5.73 -5.72 -5.71 -5.7 -5.69 -5.68 

-5.76 -5.75 -5.74 -5.73 -5.72 -5.71 -5.7 

-10 

-5.79 -5.78 -5.77 -5.76 -5.75 -5.74 -5.73 

Flux (A.U.) 

Flux (A.U.) 

Flux (A.U.) 

Flux (A.U..) 

20 

15 

10 

5 

0 

f NEMS (Hz) 

0 

-100 

-200 

-300 

-400 

-500 

-600 

0.0 0.04 0.08 0.12 0.16 0.20 

|V g /18.7| (2e) 


V g (mV) 

V g (mV) 

For each value of E J 

Fit the data to 

V g (mV) 

V g (mV) 

hf Δ E (8E Δ n ) E 

2 2 

μ C g J 

Where | Δ ng | | ng 

.5 

| and EJ EJmaxcos( πΦ 

/ Φ 0) 

E C / h 1314GHz and E J / h ~ [0,9,10] GHz 

E J max / h ~ 12.5 13.5GHz 

From M.D. LaHaye et al., Nature 459 , 960 (2009).

NEMS coupled to strongly-driven CPB 

Qubit Landau-Zener interference: see Sillanpaa et al., PRL 96 (2006), Oliver et al., Science 310 (2005) , 

Izmalkov et al., PRL (2008), Sun et al., APL (2009). Similar multi-photon transitions: see Wilson et al., PRL 98 (2007) 

Apply periodic modulation n g to CPB gate large enough to sweep CPB through charge degeneracy 

CPB ENERGY BANDS IN n g-SPACE 

ωRF 

 

EJ 

 

n g(t) = n go + n RFsin( RFt) 

n RF 

n g0 

n RF 

slope~ C g n 8E 


E J 

 

ECPB 

 

ECPB




Starting in ground state , as approach degeneracy, probability P LZ for CPB to tunnel from to 


n g(t) = n go + n RFsin( RFt) 

 

n RF 

n g0 

P 

LZ 

2 

πEJ 

exp( ) 

2ν 

n RF 


E J 

 

ECPB 

Energy 

Variation rate 

8E ~ ω ν 

C RF n 

RF 

 

ECPB




After crossing degeneracy, time-dependent phase (t) develops in wave function between and 


Probability Amplitudes 

After tunneling 

 

Ψ 

Ψ 

 

iC 

PLZ 

C 1 

PLZ 

n RF 

n g0 

P 

LZ 

2 

πEJ 

exp( ) 

2ν 

n RF 

 

ECPB 

Wave Function 

 

ECPB 


 

Ψ 

 

Ψ 

φ(t) 

Ψ(t) 

Ψ 

1 

 

 

t 

 

ΔE 

 

e 

-iφ(t) 

dt' ΔE 

 

CPB 

Ψ 

(n 

g 

 

 

 

(t' )) 

 

CPB ECPB 

ECPB




Return swing: degeneracy crossed, probability for LZ tunneling to occur, interference between tunneling events 


Wave Function 

Amplitudes 

n RF 

n g0 

2 

πEJ 

exp( ) 

2ν 

n RF 

 

ECPB 

e 2 cos( /2) 

PLZ 

/2 -iφ 

PLZ φ 

Phase-developed between 

first and second LZ events 

2i PLZ 

( 1 

PLZ 

) cos( φ/ 

2) 

t 

1 

φ 

dt' ΔE CPB (ng(t' 

)) 

 

 

ECPB 

 

CPB ECPB 

ECPB 


ΔE




After full cycle: if CPB coherence time is longer than cycle period, oscillations in excited state probability with 


Probability to 

be in 

2 ( 1 

P )( 1 

cos( φ)) 

PLZ LZ 

n RF 

n g0 

n RF 

 

ECPB 

Phase-developed between 

first and second LZ events 

t 

1 

φ 

dt' ΔE CPB (ng(t' 

)) 

 

n g(t) = n go+ n RFsin( RFt) 

 

ECPB 

 

CPB ECPB 

ECPB 


ΔE

V RF (Volts) 

NEMS as a probe of LZ interferometry 

V (V) 

Nanomechanical measurement of LZ interference 


NR /2 (Hz) 

2.5 

2.0 

1.5 

1.0 

0.5 

1 

φ 

 

 

t 

 

dt' ΔE 

-400 -200 0 200 400 600 

-10.0 -8.0 -6.0 -4.0 -2.0 

V (mV) 

cpb ωRF / 2π 

4. 

0 

CPB 

(n 

g 

V g0 (mV) 

(t' )) 

Function of 

V , V , ωRF 

g0 


RF 

GHz 

Modulate the CPB gate with large RF 

excitation V RF, and track NEMS frequency 

shift as a function of V g0 and V RF 

CPB Excited state becomes populated, 

changing sign of NEMS frequency shift

V RF (Volts) 

NEMS as a probe of LZ interferometry 

V (V) 

Nanomechanical measurement of LZ interference 


NR /2 (Hz) 

2.5 

2.0 

1.5 

1.0 

0.5 

1 

φ 

 

 

t 

 

dt' ΔE 

-400 -200 0 200 400 600 

CPB 

(n 

-4 

g 

-3 

-10.0 -8.0 -6.0 -4.0 -2.0 

V (mV) 

cpb ωRF / 2π 

4. 

0 

(t' )) 

-3 

V g0 (mV) 

-4 

Function of 

V , V , ωRF 

g0 


RF 

GHz 

Modulate the CPB gate with large RF 

excitation V RF, and track NEMS frequency 

shift as a function of V g0 and V RF 

CPB Excited state becomes populated, 

changing sign of NEMS frequency shift 

‚Constructive‛ interference occurs at V g0 where 

= 2n (intersection of black lines in plot ). 

P LZ LZ 

2P ( 1 

P )( 1 

cos( φ)) 

Parameters used for contour overlay: 

Ec = 15 GHz, Ej=13 GHz


Estimate of E C from LZ interference 

V RF (Volts) 

V (V) 

NR /2 (Hz) 

2.5 

2 

1.5 

1 

0.5 

-500 0 500 1000 1500 

-4 

-8.0 -6.0 -4.0 -2.0 0.0 

V (mV) ωRF n Vcpb g0 (mV) 

6. 

5 GHz 

g0 conversion: 18.7 mV per 2e 

2π 

Expected 

Fringe 

spacing: Δng0 ωRF 

4EC 

-3 

From fit 

-3 

-4 

E C/h = 14.9 .6 GHz 

-8.0 -6.0 -4.0 -2.0 0.0 


f NR (Hz) 

n g0 (2e) 

5000 

4000 

3000 

2000 

1000 

LZ Fringes at constant V RF 

0 

0.12 

0.10 

0.08 

0.06 

n g0 

V cpb 

n g0 (2e) 

Fit to straight 

Line thru origin 

6.50 GHz 

5.66 GHz 

4.83 GHz 

/2=4.00 GHz 

4.0 5.0 6.0 7.0 

RF/2 (GHz)

prospects for strong dispersive coupling limit 

Definition of strong coupling limit: Dispersive 

interaction exceeds qubit and NEMS linewidth 

Δf 

NEMS 

Δ 

E N 

CPB 

( 2N 

1) 

h 

2 

λ γNEMSγCPB [ 

, ] 

πE 2π2π J 

Demonstrated f NEMS NEMS/2 

With conservative improvements to sample 

geometry, should achieve f NEMS ~ 100’s kHz 

Present sample: CPB/2f NEMS 

However, there is significant room to improve, 

e.g. in circuit QED, CPB/2 1 MHz 

e.g. see Wallraff et al., Nature 431 (2004) 


Amplitude (V) 

35 

30 

20 

15 

10 

5 

Present Sample: NEMS Linewidth 

Off Degeneracy 

- 

25 On Degeneracy 

Phase (Rad.) 

58.470 58.475 58.480 


58.470 58.475 58.480 58.485 


2 

1 

0 

-1 

1.6 kHz 

VNR= 15 V 

T ~ 130 mK 

Present Sample: CPB Linewidth

prospects for dispersive CPB-NEMS entangled states 

Proposals: D.W. Utami, & A.A. Clerk,, Phys. Rev. A 78 042323 (2008) 

A.D. Armour & M.P. Blencowe, New J. Phys. 10 095004 (2008) 

General idea: (1) With CPB and NEMS uncoupled, prepare CPB in superposition of 

energy eigenstates and nanoresonator in displaced thermal state 

(2) Dispersively couple CPB and NEMS 

After time t: 

Envelope of CPB 

oscillations after -pulse 

Initial state: 

Ψ( t) 

1 

( α( t) i α( t) 

) 

2 

α ω Δω 

Nanoresonator Is in a superposition of 

states ‘winding’ at frequencies 

dressed by the CPB 

1 

Ψ(0) ( i ) α(0) 

2 

qubit state resonantor state 

NR NR 


 

α ω Δω 

 

NR NR 

1 2 

env (1 exp[ (0) (1 cos(Δ NR ))]) 

P α ω t 

2 

Using qubit echo 

method recoherences 

should be visible for 

similar device, e.g. 

λ 10 MHz 

ω / 2π 50 MHz 

NR 

Δ ω / 2π 40 kHz 

NR 

T ~100' s nsec 

Qubit recoherences: signature of entanglement 

2 

T 50 mK

conclusions 

New era of experiments studying the quantum properties 

of mechanical structures 

Superconducting qubits should serve as viable tools to manipulate 

and measure quantum states of NEMS 

We have demonstrated the first coupling between a superconducting 

qubit and NEMS 

- use dispersive interaction to perform spectroscopy and read-out 

quantum interference in the CPB, parametric amplification/squeezing 

- with realistic improvements to devices, experiments with quantum 

NEMS, even entanglement of NEMS and CPB, are within reach 

Thanks to Gerard Milburn, Andrew Doherty, Katya Babourina, Aash Clerk, Andrew Armour, Miles Blencowe, 

Christopher Wilson, and Tim Duty for helpful insight and advice.

M. LaHaye, Qubit-coupled nanomechanics - PTB

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