We demonstrate dispersive coupling between a Cooper-pair box (CPB) qubit and a VHF NEMS (nanoelectromechanical systems) resonator. The observed interaction strength is sufficient to pursue more advanced experiments to elicit quantum behavior in NEMS

Echternach, Pierre

LaHaye, Matt

Roukes, Michael

Schwab, Keith

Suh, Junho

Caltech Authors - Main

Coupling a Nanomechanical Resonator  
to a Cooper-Pair-Box Qubit 
 
Matt LaHaye1, Junho Suh1, Pierre Echternach2, Keith Schwab3, Michael Roukes1
 
1Kavli Nanoscience Institute, Caltech, Pasadena, CA 91125 USA 
2Center for Space Microelectronics Technology, JPL, Pasadena, CA 91109 USA  
3Department of Physics, Cornell University, Ithaca, NY 14852 USA 
 
We demonstrate dispersive coupling between a Cooper-pair box (CPB) qubit and a VHF NEMS 
(nanoelectromechanical systems) resonator.  The observed interaction strength is sufficient to 
pursue more advanced experiments to elicit quantum behavior in NEMS. 
 
Recent progress in UHF NEMS [1], along with the demonstration of SET-based NEMS transduction [2,3], has 
opened up a way to explore quantum mechanics in macroscopic mechanical degrees of freedom. This invokes the 
question of how actually to measure the quantum nature of the NEMS.  One possible set of measurement tools 
include superconducting electronics like the Cooper-Pair-Box (CPB) qubit, flux/phase qubits, or Superconducting 
SET(SSET) whose properties are dominated (in the case of the CPB) or in large part determined (in the case of the 
SSET) by quantum mechanics. In fact, SSET’s have been operated near the quantum limit for continuous, linear 
displacement detection, even allowing for the observation of the influence of the SSET’s quantum noise on the 
behavior of the NEMS [4].  Displacement detection with the SSET could ultimately lead to the production of ultra-
cold NEMS modes [5], or even “squeezed” NEMS states [6].  However the observation of a nanoresonator existing 
in a superposition of position states or undergoing a quantum jump might require coupling to quantum mechanically 
“coherent” systems like the CPB.  Our results measuring the dispersive interaction between CPB and NEMS are the 
first step toward these objectives. 
 
The interaction between the NEMS and CPB qubit is achieved via electrostatic coupling (Fig. 1). The capacitance 
between the CPB island and a nearby, metallized nanoresonator is modulated through the displacement of the 
resonator about its equilibrium position. When a DC voltage bias is applied to the resonator, modulation of the 
capacitance results in modulation of the potential of the CPB island.  For typical parameters [7], the interaction is 
characterized as dispersive, meaning that one should observe a change in the NEMS frequency, fNEMS, that is 
determined by the CPB state.     
We use a capacitive detection scheme here in order to measure the frequency shift [8].  The NEMS is biased at a DC 
potential Vg with respect to a nearby gate. The motion of NEMS modulates the capacitance between the gate and 
NEMS, in turn changing the charge on the gate. By calculating the current-voltage relation of this DC biased NEMS, 
we get a series RLC circuit as the equivalent circuit model. For typical dimensions, its resistance is on the order of 
mega-ohms, which we transform down to 50 Ohms using a resonant matching circuit (LC network). Fig. 2 shows 
the detection scheme (left) and an example of measured reflected voltage signal (right) around the mechanical 
resonance. 
 
Fig. 1 SEM picture of the device(left) and electrostatic coupling scheme(right) 
       a386_1.pdf  
 
© 2008 OSA / CLEO/QELS 2008
       JMA3.pdf  
 
978-1-55752-859-9/08/$25.00 ©2008 IEEE
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Cg
Lm
Cm
Rm
sM
VQC
kdR
gN
m
'~
22
0
2



m
in R
CLZ /
58.425 58.43 58.435 58.44 58.445 58.45
0
5
10
15
20
25
30
35
40
45
50
Freq. [MHz]
A
m
pl
itu
de
 [ 
V
]
f0=58.4383 MHz
Q=4.2x104
 
Fig. 2 (left) Capacitive detection circuit. The parameters in the inset are, k=(effective spring constant), d=(distance 
between gate and NEMS),  0 =(resonance frequency in rad/sec), Q=(quality factor), CN=(gate-NEMS capacitance). 
(right) Mechanical resonance. The red curve is a Lorentzian fit. 
 
We use slope-detection to measure fNEMS while sweeping the CPB gate voltage and the flux through the CPB loop, 
(Fig. 3).  It clearly shows the expected periodicity in gate charge and flux.  By comparing with a simple model that 
incorporates i) electrostatic coupling between NEMS and CPB, ii) thermal activation of CPB when the Josephson 
energy is comparable to kBT and iii) gate charge modulation by NEMS motion (0.1 eB
                                                
rms), we estimate the coupling 
strength to be ~ 1.4  MHz. This is promising for proposed CPB-based ground-state cooling of NEMS [ ], NEMS 
Fock states [ ], and NEMS superposition states [ ]. 
9
7 10
 
 
 
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 
-20
-10
0
10
20
-200
-150
-100
-50
0
 
 
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 
-20
-10
0
10
20
-200
-150
-100
-50
0
Applied Magnetic Field (G)Applied Magnetic Field (G)
V
C
P
B
(m
V
)
V C
P
B
(m
V
)
fNEMS (Hz) fNEMS (Hz)Data: VNR= 7.0 V, Tmc ~ 100 mK Theory:  	~ 1.4 MHz, T =100mK
EJ,max ~ 12 GHz, EC ~ 10 GHz
*model assumes .1erms 
Motional smearing
VCPB range for  mean
fNEMS (Hz) calculation
VCPB range for  mean
fNEMS (Hz) calculation
1 2 3 4
 
-20 -10 0 10 20
-200
-150
-100
-50
0
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 
-150
-100
-50
Applied Magnetic Field (G)VCPB (mV)
f
N
E
M
S
(H
z)
M
ea
n 
f
N
E
M
S
(H
z) Theory
Experiment

Notes:  55 Hz offset subtracted from data to compare with model; B-field sweep on top of -104 Gauss static field 
1
2
3
4
Fig. 3. Frequency shift,  fNEMS, map in CPB gate voltage - magnetic field space 
 
[1] X.M.H Huang et al., “Nanodevice motion at microwave frequencies,”Nature 421, 496 (2003).  
[2] R.G. Knobel and A. N. Cleland, “Nanometer-scale displacement sensing using a single-electron transistor,” Nature 424, 291-293 (2003). 
[3] M.D. LaHaye, O. Buu, B. Camarota, and K.C. Schwab, “Approaching the quantum limit of a nanomechanical resonator,” Science 304, 74-77 
(2004). 
[4] A. Naik  et al., “Cooling a nanomechanical resonator with quantum back-action,” Nature 443, 193-196 (2006). 
[5] A. Hopkins, K. Jacobs, S. Habib, and K.C. Schwab, “Feedback cooling of a nanomechanical resonator,” Phys. Rev. B 68, 235328/1-10 (2003). 
[6] R. Ruskov, A.N. Korotkov, and K.C. Schwab, “Quantum nondemolition squeezing of a nanomechanical resonator,” IEEE T. Nanotechnol.4, 
132-140 (2005).   
[7] E.K. Irish and K. Schwab, “Quantum measurement of a coupled nanomechanical resonator –Cooper-pair box system,” Phys. Rev. B 68, 
155311/1-7 (2003).  
[8] P.A. Truitt et al., “Efficient and sensitive capacitive read-out of nanomechanical arrays,” Nano Letters 7, 120-126 (2007). 
[9] I. Martin, A. Shnirman, L. Tian and P. Zoller, “Ground-state cooling of mechanical resonators,” Phys. Rev. B 69. 125339/1-12 (2004). 
[10] A.D. Armour, M.P. Blencowe and K.C. Schwab, “Entanglement and decoherence of a micromechanical resonator via coupling to a Cooper-
pair box,”  Phys. Rev. Lett. 88, 148301/1-4. 
       a386_1.pdf  
 
© 2008 OSA / CLEO/QELS 2008
       JMA3.pdf  
 


Coupling a nanomechanical resonator to a Cooper-pair-box qubit

https://authors.library.caltech.edu/67700/1/04572206.pdf

Coupling a nanomechanical resonator to a Cooper-pair-box qubit

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