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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
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
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)
Quantum Measurement and Metrology with Solid State Devices PBH, Germany - 05 Nov. 2009
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).
Quantum Measurement and Metrology with Solid State Devices PBH, Germany - 05 Nov. 2009
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
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
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
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.
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.
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
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
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
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
EJ =3.0 GHz
CPB layout
E (
GH
z)
irs on box Cooper-pa00
ir on box Cooper-pa11
Small capacitance yields large
charging energy Ec, so only two
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
= Applied flux through CPB loop
0 = Flux quantum
Hamiltonian
0 0cos( Φ /Φ )J JE E π
ng =CgVg/2e - Applied gate charge
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
0 .5 1 1.5 2 0
.5
1
1.5
2
review of the Cooper-pair boxEssentially it is an highly-polarizable, artificial, two-state atom
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’
Nakamura et al., Nature, Vol. 398, 29 April 1999
CPB layout
irs on box Cooper-pa00
ir on box Cooper-pa11
Small capacitance yields large
charging energy Ec, so only two
relevant charge states
n̂
n̂
ˆ ˆ ˆ2 (1 2 )2
JC g z x
EH E n σ σ
= Applied flux through CPB loop
0 = Flux quantum
Hamiltonian ng =CgVg/2e - Applied gate charge
0 0cos( Φ /Φ )J JE E π
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
review of the Cooper-pair boxEssentially it is an highly-polarizable, artificial, two-state atom
Gate periodicity of CPB energy bands
Ground State
Excited State
Nakamura et al., Nature, Vol. 398, 29 April 1999
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
Small capacitance yields large
charging energy Ec, so only two
relevant charge states dc gate charge ng (CgVg/2e)n̂
ng =CgVg/2e - Applied gate chargeΘ̂cos)ˆ(4ˆ 2
JgC EnnEH Hamiltonian
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
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
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
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
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
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
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 200918
device layout
NEMS
Aluminum
NEMS Gate
SiliconNitride
CPB
CPBGateCPB
Reservoir
FluxBiasloop
fabrication at JPL and Caltech
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
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
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
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
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 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
From M.D. LaHaye et al., Nature 459 , 960 (2009).
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 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).
From M.D. LaHaye et al., Nature 459 , 960 (2009).
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
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
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
DEVICE SCHEMATIC
CPB ENERGY BAND DIAGRAM
NEMS-based spectroscopy of CPB
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)
EXPECTED NEMS FREQUENCY SHIFT
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
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
DEVICE SCHEMATIC
CPB ENERGY BAND DIAGRAM
NEMS-based spectroscopy of CPB
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)
EXPECTED NEMS FREQUENCY SHIFT
EXPECTED NEMS FREQUENCY SHIFT
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
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
)
Microwave Frequency: 17 GHz
-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
)
Microwave Frequency: 16 GHz
-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
)
Microwave Frequency: 13.5 GHz
-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
)
Microwave Frequency: 12.5 GHz
-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
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
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
)
Microwave Frequency: 17 GHz
-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
)
Microwave Frequency: 16 GHz
-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
)
Microwave Frequency: 13.5 GHz
-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
)
Microwave Frequency: 12.5 GHz
-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
From M.D. LaHaye et al., Nature 459 , 960 (2009).
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
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
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
NEMS coupled to strongly-driven CPB
ng0
CPB ENERGY BANDS IN ng-SPACE
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) ,
Izmalkov et al., PRL (2008), Sun et al., APL (2009). Similar multi-photon transitions: see Wilson et al., PRL 98 (2007)
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
NEMS coupled to strongly-driven CPB
ng0
CPB ENERGY BANDS IN ng-SPACE
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
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)
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
NEMS coupled to strongly-driven CPB
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
CPB ENERGY BANDS IN ng-SPACE
CPBCPBCPB EEEΔ
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)
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
NEMS coupled to strongly-driven CPB
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
CPB ENERGY BANDS IN ng-SPACE
CPBCPBCPB EEEΔ
ng(t) = ngo+ nRFsin(RFt)
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)
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
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)
From M.D. LaHaye et al., Nature 459 , 960 (2009).
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
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
From M.D. LaHaye et al., Nature 459 , 960 (2009).
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
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
NEMS coupled to strongly-driven CPB
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
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
-
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)
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
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
Quantum Measurement and Metrology with Solid State Devices PBH, Germany ” 05 Nov. 2009
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.