Qubit-Coupled Nanomechanics · Qubit-Coupled Nanomechanics junho suh, michael roukes - caltech Quantum Measurement and Metrology with Solid State Devices keith schwab - caltech &

Qubit-Coupled Nanomechanics

junho suh, michael roukes - caltech

Quantum Measurement and Metrology with Solid State Devices

keith schwab - caltech & cornell

pierre echternach - j p l

PBH, Germany 5 Nov. 2009

experiments performed at caltech with:

Matt LaHaye ” Syracuse University

Atoms, Ions, SpinsCasimir Physics

m

(Caltech, Cornell, JPL) NEMS/CPB

(Cornell/Caltech) SMR/NEMS

(Delft):DC-SQUID/NEMS

(Maryland) SSET/NEMS

(JILA): APC/NEMS, SMR/NEMS

(UCSB) SET/NEMS

(UCSB)

(MIT & LIGO)

(Caltech, Max Planck Institute)

(Yale) (Vienna)

(Oregon)

mechanical structures in the quantum regime

Nanoelectromechanical Systems (NEMS)

Optomechanical

Systems

And many others …

(Dartmouth/ Padova) (IBM Almaden)

Atoms, Ions, SpinsCasimir Physics

m

(Caltech, Cornell, JPL) NEMS/CPB

(Cornell/Caltech) SMR/NEMS

(Delft):DC-SQUID/NEMS

(Maryland) SSET/NEMS

(JILA): APC/NEMS, SMR/NEMS

(UCSB) SET/NEMS

(UCSB)

(MIT & LIGO)

(Caltech, Max Planck Institute)

(Yale) (Vienna)

(Oregon)

mechanical structures in the quantum regime

Nanoelectromechanical Systems (NEMS)

Optomechanical

Systems

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

Physics Today July 2005

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

systems approaching the quantum regime. September 2008

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

And many others …

(Dartmouth/ Padova) (IBM Almaden)

Quantum Measurement and Metrology with Solid State Devices PBH, Germany - 05 Nov. 20094

Ideal characteristics: Small mass,

“ Typ. quality factors ~ 104-105,

but demonstrated >106

“ Zero-point motion

“ Energy-level spacing

zpx / 2 ~ 40 fmm

Mo Li, Hong Tang, Michael Roukes, 2007

Estimate for SiC resonator,

.6m x .4m x .07m

Mass ~ 50 fg, f0 ~ 127 MHz

ο Bω k T For 1 GHz resonator

At mK temperatures

Huang, Roukes, 2003

Attainable with dilution fridge.Schwab 2008

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

May portend long coherence and

relaxation times (~ sec’s)

high frequency, low dissipation

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

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

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 Nth= 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

Gate of SET

NR GateSSET

NR

1m

VNR

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

K. Schwab, Science 2004

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

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

(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).

VNR

VNR

19.7 MHz Resonator

20 MHz


approaching the quantum limit of NEMS with an RFSET

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 Nth= 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

Gate of SET

NR GateSSET

NR

1m

VNR

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

K. Schwab, Science 2004

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

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

(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)

VNR

VNR

19.7 MHz Resonator

20 MHz

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

a quantum-limited displacement detector (proposed by

Blencowe and Wybourne, APL 2000)


x

|0>

|1>

|2>

|n>

artificialatom Harmonic oscillator

electrostatic interaction

qubit-coupled nanomechanics

=

Nakamura et al., Nature, 398 29 Apr. 1999Cleland & Roukes, APL 69 28 Oct. 1996

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

resonator motion

couples to charge

on the qubit

+

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


x

|0>

|1>

|2>

|n>

8

artificialatom Harmonic oscillator

electrostatic interaction

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

+

=

Nakamura et al., Nature, 398 29 Apr. 1999Cleland & Roukes, APL 69 28 Oct. 1996

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

use qubit to prepare

quantum superposition

states of NEMS and

study decoherence


Partial list of proposals utilizing a qubit to manipulate and measure

quantum states of NEMS

superconducting qubits as tools for quantum 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


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.

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.

Demonstrated coupling should be large enough to pursue more advanced

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


0.2 0.4 0.6 0.8

-15

-10

-5

0

5

10

15

review of the Cooper-pair boxEssentially it is an highly-polarizable, artificial, two-state atom

2/10

2/10

CPB energy bands

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

dc gate charge ng (CgVg/2e)

irs on box Cooper-pa00

ir on box Cooper-pa11

Small capacitance yields large

charging energy Ec, so only two

relevant charge states

EJ = 9 GHz

CPB layout

E (

GH

z)

ˆ ˆ ˆ2 (1 2 )2

JC g z x

EH E n σ σ

n̂

= Applied flux through CPB loop

0 = Flux quantum

Hamiltonian

0 0cos( Φ /Φ )J JE E π

ng =CgVg/2e - Applied gate charge

0

0

1

1


0.2 0.4 0.6 0.8

-15

-10

-5

0

5

10

15


2/10

2/10

CPB energy bands


EJ =3.0 GHz

CPB layout

E (

GH

z)





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

n̂

ˆ ˆ ˆ2 (1 2 )2

JC g z x

EH E n σ σ


0 = Flux quantum

Hamiltonian


ng =CgVg/2e - Applied gate charge


0 .5 1 1.5 2 0

.5

1

1.5

2


Expectation Value of Charge

Excited state

Jg En

n 1ˆ

Jg En

n 1ˆ

0

.5

1

dc gate charge ng (CgVg/2e).5 10

Ground State

‘Quantum

Capacitance’


CPB layout





relevant charge states

n̂

n̂

ˆ ˆ ˆ2 (1 2 )2

JC g z x

EH E n σ σ


0 = Flux quantum

Hamiltonian ng =CgVg/2e - Applied gate charge




Gate periodicity of CPB energy bands

Ground State

Excited State


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

CPB layout

E/E

c

n n Cooper - pairs on box

1 1n n Cooper - pairs on box



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

ng =CgVg/2e - Applied gate chargeΘ̂cos)ˆ(4ˆ 2

JgC EnnEH Hamiltonian


CPB-NEMS Interaction

NEMS Position Operator

CPB capacitively coupled to NEMS

Flexural motion of resonator couples to charge on the CPB island

†ˆ ˆ ˆ 1/ 2T NRH a a

Total Hamiltonian in CPB energy basis (at charge degeneracy)

NEMS CPB energy at

charge degeneracyInteraction

CNR

d

Vg

Resonator

Gate

CPBVNR

†ˆ ˆ ˆXa a

Z

J σE

ˆ2

Similar to

atom coupled

to radiation field

†ˆ ˆ ˆN a a

Mechanical quanta

Electrostatic Coupling Constant

CPB Charge at degeneracy

(in energy basis)

Spring ConstantNRK

Resonant FrequencyNRω

22

2

C NR NR NR

NR

E C V ωλ

e K d

†ˆ ˆ ˆ ˆInt XH λ a a σ


2/1ˆˆ NωH NRT †ˆ ˆ ˆXa a

Z

J σE

ˆ2

RWA

ˆ ˆ 1/ 2RWA NRH ω N †ˆ ˆ ˆ ˆa a

Z

J σE

ˆ2

Dispersive Hamiltonian

λNωE NRJ Δ

2 2

ˆ ˆ ˆˆ ˆ1/ 2 2 12

Jdisp NR Z Z

J

E λH ω N σ N σ

E

CPB-state-dependent

Frequency

Shift in NEMS

NEMS-

Dependent shift

in CPB transition

22ˆΔ NR Z

J

λω σ

E

2 22 ˆΔ 2 1N

CPB

J

λE N

E

Dispersive limit of CPB-NEMS Hamiltonian

CPB and NEMS far-detuned for our parameters

Direct exchange of quanta suppressed by2

Δ~

λ

...

...

N=0

N=1

N=2

N=3

N=0

N=1

N=2

N=3

Energy levels with Interaction

N

CPBJ EE Δω

ω

...

...

N=0

N=1

N=2

N=3

N=0

N=1

N=2

N=3

Energy levels with Interaction

N

CPBJ EE Δω

ω

...

N=0

N=1

N=2

N=3

N=0

N=1

N=2

N=3

Energy levels w/o interaction

JENRω

...

Δ ,NR NRω ω ω ΔNR NRω ω ω

Jaynes-Cummings Hamiltonian


estimates for the nanomechanical frequency shift

Frequency shift depends on CPB

state, and magnitude proportional

to CPB energy band curvature*:

CPB Energy and NEMS frequency shift vs ng

Gate Charge, ng (2e)

CPB E

nerg

y (G

Hz) Excited State

Ground State

CNR ~ 50 aF

d ~ 300 nm

VNR ~ 10 V

fNEMS = o /2 ~ 60 MHz

Parameters

22

3/22 2

ˆΔ((4 (1 2 ))

JNEMS Z

C g J

Eλf σ

π E n E

K ~ 60 N/m

/2~ 2.0 MHz

EC ~14 GHz

EJ ~ 13 GHz

NEMS frequency detection

schemes can routinely achieve

better than ppm sensitivity

Expect frequency shift of 10’s ppm

at charge degneracy

Gate Charge, ng (2e)

NEM

S Fre

quency

Shift (H

z)

510~/Δ

NEMSNEMS ff

*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)

,max 0cos( Φ/Φ )J JE E π


device layout

NEMS

Aluminum

NEMS Gate

SiliconNitride

CPB

CPBGateCPB

Reservoir

FluxBiasloop

fabrication at JPL and Caltech


Vg

B

ELECTROMECHANICALIMPEDANCE

LNA

lo

rf if

On resonanceZM = Rm~ M’s

VNR

REFLECTOMETRYTO MEASURE ZM

50

Vgnr

CT

LT

Vdrive

measurement layout

58.42 58.425 58.43 58.435 58.440

20

40

60

80

100

120

Frequency (MHz)

Am

plit

ud

e (

V)

NEMS’ response at

Tmc ~ 100 mK

Q ~ 50,000 Frequency (MHz)

58.42 58.425 58.43 58.435 58.44-5

-4

-3

-2

-1

0

Ph

ase

(R

ad

)

NEMS response with CPB biased off charge degeneracy

VNR= 10 V

Drive Force

(VNR - Vgnr)Vdrive

Lm

Cm

Rm

Cgnr


Vg

B

ELECTROMECHANICALIMPEDANCE

LNA

lo

rf if

On resonanceZM = Rm~ M’s

VNR

REFLECTOMETRYTO MEASURE ZM

50

Vgnr

CT

LT

measurement layout

58.424 58.426 58.428 58.43 58.432 58.434 58.4360

20

40

60

80

100

120

Frequency (MHz)

Am

plit

ud

e (

V)

fNEMS ~ 600 Hz 58.424 58.426 58.428 58.43 58.432 58.434 58.436

-4

-3

-2

-1

Phase (

Rad

)

Frequency (MHz)

Off Degeneracy

On Degeneracy

NEMS response on and off a charge degeneracy

-

VNR= 10 V

Vdrive

Drive Force

(VNR - Vgnr)Vdrive

Tmc~ 100 mK

Lm

Cm

Rm

Cgnr


Measurement: VNR= 7.0 V, Tmc ~ 100 mKModel: /2= 1.40 MHz, T =100mK

EJ,max /h= 13.2 GHz, EC /h= 14.0 GHz

Notes: Model convolved with 0.1 CPrms charge noise, and includes thermal population of CPB excited state

dispersive interaction: measurement vs. model

Note: Magnetic field applied on top of ~ 100 G οJJ πEE Φ/Φcosmax,

Flux Periodicity:Applied Magnetic Field (A.U.)

f N

EM

S(H

z)

-200

-100

Vg (mV)-10 -5 0 5 10

0

2

3

1

4

ng (2e)

Flux (A.U.)

f N

EM

S(H

z)

-1.0 -0.5 0.0 0.5 1.0

-200

-100

0Model

Exp. 65

fN

EM

S(H

z)

-1.0 -0.5 0.0 0.5 1.0

-15

-10

-5

0

5

10

15

-250

-200

-150

-100

-50

0

Vg

(mV

)

1 2

5

Flux (o)-0.5 0.0 0.5

0.0

-200

-150

-100

-50

fN

EM

S(H

z)

3 4

0

6-0.5

0.5

1515

3 41 2

5 6

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


Measurement: VNR= 7.0 V, Tmc ~ 100 mKModel: /2= 1.40 MHz, T =100mK

EJ,max /h= 13.2 GHz, EC /h= 14.0 GHz

Notes: Model convolved with 0.1 CPrms charge noise, and includes thermal population of CPB excited state

dispersive interaction: measurement vs. model

Note: Magnetic field applied on top of ~ 100 G οJJ πEE Φ/Φcosmax,

Flux Periodicity:Applied Magnetic Field (A.U.)

f N

EM

S(H

z)

-200

-100

Vg (mV)-10 -5 0 5 10

0

2

3

1

4

ng (2e)

Flux (A.U.)

f N

EM

S(H

z)

-1.0 -0.5 0.0 0.5 1.0

-200

-100

0Model

Exp. 65

fN

EM

S(H

z)

-1.0 -0.5 0.0 0.5 1.0

-15

-10

-5

0

5

10

15

-250

-200

-150

-100

-50

0

Vg

(mV

)

1 2

5

Flux (o)-0.5 0.0 0.5

0.0

-200

-150

-100

-50

fN

EM

S(H

z)

3 4

0

6-0.5

0.5

1515

3 41 2

5 6

With coupling strength, proposals suggest that it should be possible to

implement single qubit ‚lasing‛, ground-state cooling, squeezing of NEMS, (Lasing) J. Hauss,, A. Federov, C. Hutter, A. Shnirman, G. Schon, PRL. 100, 037003 (2008)

(Ground-state Cooling) I. Martin, A. Shnirman, L. Tian, P. Zoller, Phys. Rev. B 69, 125339 (2004).

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



DEVICE SCHEMATIC

CPB ENERGY BAND DIAGRAM

APPLY MICROWAVES THAT ARE RESONANT

WITH CPB SPLITTING.

EXPECTED NEMS FREQUENCY SHIFT

NEMS-based spectroscopy of CPB

f N

EM

S(H

z)

EJ/h = 13.0 GHz

p-=p+

13 GHz applied

0 0.2 0.4 0.6 0.8 1

-400

-300

-200

-100

0

no microwaves

13 GHzd/2=13 GHz

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0

10

20

30 C+(EJ,Ng)

C-(EJ,Ng)

ng (2e)

E c

pb/h

(GH

z)

EJ/h= 13 GHz

CPB

Resonator

CNR

d

Gate

VNR

Vg(t)=Vg0+Vcosdt

d=(E/)

Vg(t)

ng (2e)

AVERAGE NEMS FREQUENCY SHIFT

Δ Δ Δ 0NEMS NEMS NEMSf p f p f

p- = p+ as given by Bloch equations

Δ ΔNEMS NEMSf f


DEVICE SCHEMATIC



d/2=13 GHz

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0

10

20

30 C+(EJ,Ng)

C-(EJ,Ng)

E c

pb/h

(GH

z)

EJ /h= 9 GHz

13 GHZ

0 0.2 0.4 0.6 0.8 1-600

-500

-400

-300

-200

-100

0

f N

EM

S(H

z)

9 GHz applied

no microwavesEJ/h = 9.0 GHz

p-=p+

APPLY MICROWAVES THAT ARE RESONANT

WITH CPB SPLITTING.

CPB

Resonator

CNR

d

Gate

VNR

Vg(t)=Vg0+Vcosdt

d=(E/)

Vg(t)

ng (2e)

ng (2e)


AVERAGE NEMS FREQUENCY SHIFT

Δ Δ Δ 0NEMS NEMS NEMSf p f p f

p- = p+ as given by Bloch equations

Δ ΔNEMS NEMSf f


DEVICE SCHEMATIC



d/2=13 GHz

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0

10

20

30+(EJ,Ng)

-(EJ,Ng)

E c

pb/h

(GH

z)

13 GHZ

0 0.2 0.4 0.6 0.8 1-600

-500

-400

-300

-200

-100

0

f N

EM

S(H

z)

9 GHz applied

no microwavesEJ/h = 9.0 GHz

p-=p+

(o)

ng

(2e)

-0.5 0 0.5

0

0.2

0.4

0.6

0.8

1 -1000

-800

-600

-400

-200

fNEMS

(Hz)EJ /h= 9 GHz

CPB

Resonator

CNR

d

Gate

VNR

Vg(t)=Vg0+Vcosdt

d=(E/)

Vg(t)

C

CMax EJ

ng (2e)

ng (2e)




Flux (A.U.)

Vg (

mV

)

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

-24

-22

-20

-18

-16

-14

-12

-10

-600

-500

-400

-300

-200

-100

0

fNEMS

(Hz)

Flux (A.U..)

Vg (

mV

)

Microwave Frequency: 20 GHz

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

-24

-22

-20

-18

-16

-14

-12

-10

-600

-500

-400

-300

-200

-100

0

fNEMS

(Hz)

Flux (A.U.)

Vg (

mV

)


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

-12

-10

-8

-6

-4

-2

0-600

-500

-400

-300

-200

-100

0

100

200

fNEMS

(Hz)

Flux (A.U.)

Vg (

mV

)


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

-12

-10

-8

-6

-4

-2

0

2

-600

-500

-400

-300

-200

-100

0

fNEMS

(Hz)

Flux (A.U.)

Vg (

mV

)

Microwave Frequency: 14.5 GHz

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

-12

-10

-8

-6

-4

-2

0

2-700

-600

-500

-400

-300

-200

-100

0

fNEMS

(Hz)

Flux (A.U.)

Vg (

mV

)


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

-6

-4

-2

0

2

4

6

8-600

-500

-400

-300

-200

-100

0

fNEMS

(Hz)

Flux (A.U.)

Vg (

mV

)


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

-28

-26

-24

-22

-20

-18

-16

-14

-500

-400

-300

-200

-100

0

fNEMS

(Hz)

increasing microwave frequency

EJ = EJ,max 10.5 GHz 12.5 GHz 13.5 GHz

14.5 GHz 16.0 GHz 17.0 GHz 20.0 GHz

Flux (A.U.)

Vg (

mV

)

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

-24

-22

-20

-18

-16

-14

-12

-10

-600

-500

-400

-300

-200

-100

0

fNEMS

(Hz)

MW OFF EJ = EJ,max


Flux (A.U.)

Vg (

mV

)

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

-24

-22

-20

-18

-16

-14

-12

-10

-600

-500

-400

-300

-200

-100

0

fNEMS

(Hz)

Flux (A.U..)

Vg (

mV

)


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

-24

-22

-20

-18

-16

-14

-12

-10

-600

-500

-400

-300

-200

-100

0

fNEMS

(Hz)

Flux (A.U.)

Vg (

mV

)


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

-12

-10

-8

-6

-4

-2

0-600

-500

-400

-300

-200

-100

0

100

200

fNEMS

(Hz)

Flux (A.U.)

Vg (

mV

)


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

-12

-10

-8

-6

-4

-2

0

2

-600

-500

-400

-300

-200

-100

0

fNEMS

(Hz)

Flux (A.U.)

Vg (

mV

)


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

-12

-10

-8

-6

-4

-2

0

2-700

-600

-500

-400

-300

-200

-100

0

fNEMS

(Hz)

Flux (A.U.)

Vg (

mV

)


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

-6

-4

-2

0

2

4

6

8-600

-500

-400

-300

-200

-100

0

fNEMS

(Hz)

Flux (A.U.)

Vg (

mV

)


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

-28

-26

-24

-22

-20

-18

-16

-14

-500

-400

-300

-200

-100

0

fNEMS

(Hz)

increasing microwave frequency

For each value of EJ

Fit the data to

0

5

10

15

20

0.0 0.04 0.08 0.12 0.16 0.20

|Vg /18.7| (2e)

Mic

row

ave F

requency

(G

Hz)

2 2Δ (8 Δ )μ C g Jhf E E n E

10.5 GHz 12.5 GHz 13.5 GHz

14.5 GHz 16.0 GHz 17.0 GHz 20.0 GHz

2|Vg|

|Δ | | .5 |g gn n Wheremax 0cos( Φ /Φ )J JE E πand

13 14 GHz C

E / h and [0,9,10] GHzJE / h ~

max 12.5 13.5GHzJ E / h ~

EJ = EJ,max

Flux (A.U.)

Vg (

mV

)

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

-24

-22

-20

-18

-16

-14

-12

-10

-600

-500

-400

-300

-200

-100

0

fNEMS

(Hz)

MW OFF EJ = EJ,max



NEMS coupled to strongly-driven CPBQubit 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)

ng0

CPB ENERGY BANDS IN ng-SPACE

nRFnRF

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

EJ

slope~gC

n8Eng(t) = ngo+ nRFsin(RFt)

CPBE

CPBE

JE

RFω


NEMS coupled to strongly-driven CPB

ng0


nRFnRF

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

EJ

)2

exp(2

ν

EπP J

LZ

EnergyVariation rate

RFRFCn8E~ ων

CPBE

CPBE

ng(t) = ngo+ nRFsin(RFt)

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




ng0


nRFnRF

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

Wave Function

CPBE

CPBE

ΨΨΨ (t)-ie(t) φ

t

gCPB ))(t'(nΔEdt'1

(t)φ

CPBCPBCPB EEEΔ

Probability Amplitudes

After tunneling

LZPiCΨ

LZPC 1Ψ

)2

exp(2

ν

EπP J

LZ

Ψ

Ψ





ng0

nRFnRF

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

)2

exp(2

ν

EπP J

LZ

CPBE

CPBE

Wave Function

Amplitudes

)cos(2 /2e /2-i φPLZ

φ

)2/cos()1(2 φPPi LZLZ

t

gCPB ))(t'(nΔEdt'1

φ

Phase-developed between

first and second LZ events


CPBCPBCPB EEEΔ





ng0

nRFnRF

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

Probability to

be in

CPBE

CPBE

))cos(1)(1(2 φPP LZLZ

t

gCPB ))(t'(nΔEdt'1

φ

Phase-developed between

first and second LZ events


CPBCPBCPB EEEΔ

ng(t) = ngo+ nRFsin(RFt)




Vcpb

(mV)

V (

V)

-10.0 -8.0 -6.0 -4.0 -2.0

0.5

1.0

1.5

2.0

2.5

-400 -200 0 200 400 600

NR/2 (Hz)

NEMS as a probe of LZ interferometry

Nanomechanical measurement of LZ interference

t

gCPB ))(t'(nΔEdt'1

φ

Function of ,g0V RFω,RFV

Modulate the CPB gate with large RF

excitation VRF, and track NEMS frequency

shift as a function of Vg0 and VRF

CPB Excited state becomes populated,

changing sign of NEMS frequency shift

Vg0 (mV)GHz0.42/ πωRF

VRF

(Volts)



Vcpb

(mV)

V (

V)

-3

-4

-3-4

-10.0 -8.0 -6.0 -4.0 -2.0

0.5

1.0

1.5

2.0

2.5

-400 -200 0 200 400 600

NR/2 (Hz)

Nanomechanical measurement of LZ interference

t

gCPB ))(t'(nΔEdt'1

φ

Function of ,g0V RFω,RFV

CPB Excited state becomes populated,

changing sign of NEMS frequency shift

Vg0 (mV)

Modulate the CPB gate with large RF

excitation VRF, and track NEMS frequency

shift as a function of Vg0 and VRF

GHz0.42/ πωRF

VRF

(Volts)

NEMS as a probe of LZ interferometry

‚Constructive‛ interference occurs at Vg0 where

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

))cos(1)(1(2 φPPP LZLZ

Parameters used for contour overlay:

Ec = 15 GHz, Ej=13 GHz



Vcpb

(mV)

V (

V)

-3-4

-3

-4

-8.0 -6.0 -4.0 -2.0 0.0

0.5

1

1.5

2

2.5

-500 0 500 1000 1500

NR/2 (Hz)

-8.0 -6.0 -4.0 -2.0 0.0

0

1000

2000

3000

4000

5000

/2=4.00 GHz

4.83 GHz

5.66 GHz

6.50 GHz

Vcpb

4.0 5.0 6.0 7.0

0.06

0.08

0.10

0.12


Vg0 (mV) GHz5.62

π

ωRF

ng0 conversion: 18.7 mV per 2e

VRF

(Volts)

ng0 (2e)

f N

R(H

z)

ExpectedFringe

spacing:Δ g0 RF

C

n ω4E

From fit

EC/h = 14.9 .6 GHz

ng0

(2e)

RF/2 (GHz)

LZ Fringes at constant VRFEstimate of EC from LZ interference

Fit to straightLine thru origin

ng0


-

58.470 58.475 58.480 58.485

5

10

15

20

25

30

35

Frequency (MHz)

Am

plit

ud

e (

V)

58.470 58.475 58.480

-1

0

1

2

Frequency (MHz)

Ph

ase

(R

ad

.)

Off Degeneracy

On Degeneracy1.6 kHz

T ~ 130 mK

VNR= 15 V

prospects for strong dispersive coupling limit

Demonstrated fNEMS NEMS/2

With conservative improvements to sample

geometry, should achieve fNEMS ~ 100’s kHz

Definition of strong coupling limit: Dispersive

interaction exceeds qubit and NEMS linewidth

2

[ , ]2 2

NEMS CPB

J

γ γλ

πE π π

Present Sample: NEMS Linewidth

Present Sample: CPB Linewidth

Present sample: CPB/2fNEMS

However, there is significant room to improve,

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

hN

E N

CPB

)12(

Δ

NEMSfΔ

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


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

1Ψ( ) ( ( ) ( ) )

2t α t i α t After time t:

ΔNR NRα ω ω

ΔNR NRα ω ω

Nanoresonator Is in a superposition of

states ‘winding’ at frequencies

dressed by the CPB

21(1 exp[ (0) (1 cos(Δ ))])

2env NRP α ω t

Envelope of CPB

oscillations after -pulseQubit recoherences: signature of entanglement

Using qubit echo

method recoherences

should be visible for

similar device, e.g.

10λ MHz

2 ~100' secT s n

Δ / 2 40NRω π kHz

/ 2 50NRω π MHz

50T mK

Initial state:1

Ψ(0) ( ) (0)2

i α

qubit state resonantor state


conclusions

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

Superconducting qubits should serve as viable tools to manipulate

and measure quantum states of NEMS

New era of experiments studying the quantum properties

of mechanical structures

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

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

Documents

Qubit-Coupled Nanomechanics · Qubit-Coupled Nanomechanics junho suh, michael roukes - caltech Quantum Measurement and Metrology with Solid State Devices keith schwab - caltech &