Quantum mechanics demands that the act of measurement must affect the
measured object. When a linear amplifier is used to continuously monitor the
position of an object, the Heisenberg uncertainty relationship requires that
the object be driven by force impulses, called back-action. Here we measure the
back-action of a superconducting single-electron transistor (SSET) on a
radiofrequency nanomechanical resonator. The conductance of the SSET, which is
capacitively coupled to the resonator, provides a sensitive probe of the
latter's position;back-action effects manifest themselves as an effective
thermal bath, the properties of which depend sensitively on SSET bias
conditions. Surprisingly, when the SSET is biased near a transport resonance,
we observe cooling of the nanomechanical mode from 550mK to 300mK-- an effect
that is analogous to laser cooling in atomic physics. Our measurements have
implications for nanomechanical readout of quantum information devices and the
limits of ultrasensitive force microscopy (such as single-nuclear-spin magnetic
resonance force microscopy). Furthermore, we anticipate the use of these
backaction effects to prepare ultracold and quantum states of mechanical
structures, which would not be accessible with existing technology.Comment: 28 pages, 7 figures; accepted for publication in Natur

A. A. Clerk

A. D. Armour

A. Naik

AA Clerk

AD Armour

B Abbott

CM Caves

D Mozyrsky

D Rugar

EK Irish

I Martin

I Tittonen

I Wilson-Rae

K. C. Schwab

M-S Choi

M. D. LaHaye

M. P. Blencowe

MD LaHaye

MP Blencowe

O. Buu

PD Lett

R Deblock

R Ruskov

RJ Schoelkopf

RP Ojha

SL Pohlen

TA Fulton

U Gavish

U Lundin

VB Braginsky

Y Nakamura

English

arXiv

Quantum mechanics demands that the act of measurement must affect the measured object. When a linear amplifier is used to continuously monitor the position of an object, the Heisenberg uncertainty relationship requires that the object be driven by force impulses, called back-action. Here we measure the back-action of a superconducting single-electron transistor (SSET) on a radio-frequency nanomechanical resonator. The conductance of the SSET, which is capacitively coupled to the resonator, provides a sensitive probe of the latter's position; back-action effects manifest themselves as an effective thermal bath, the properties of which depend sensitively on SSET bias conditions. Surprisingly, when the SSET is biased near a transport resonance, we observe cooling of the nanomechanical mode from 550 mK to 300 mK—an effect that is analogous to laser cooling in atomic physics. Our measurements have implications for nanomechanical readout of quantum information devices and the limits of ultrasensitive force microscopy (such as single-nuclear-spin magnetic resonance force microscopy). Furthermore, we anticipate the use of these back-action effects to prepare ultracold and quantum states of mechanical structures, which would not be accessible with existing technology

Naik, A.

Buu, O.

LaHaye, M. D.

Armour, A. D.

Clerk, A. A.

Blencowe, M. P.

Schwab, K. C.

Caltech Authors - Main

Supplementary Information

“Cooper-Pair Molasses: Cooling a Nanomechanical Resonator with Quantum Back-action” 

A. Naik, O. Buu, M.D. LaHaye, A.D. Armour, A. A. Clerk, M.P. Blencowe, K.C. Schwab
 

Sample Fabrication

	The sample is fabricated on a p-type, 10 Ohm-cm, (1,0,0), silicon substrate, coated with 50 nm of low-stress, amorphous silicon nitride (SiN).   The doubly-clamped, nanomechanical resonator is 8.7 m long, and 200 nm wide, composed of 50nm of SiN and 90 nm of Al. The SSET island is 1 m long, located about 100 nm away from the resonator. The tunnel junctions, made of AlOx, are approximately 70 nm X 60 nm. 

	An on-chip LC resonator is microfabricated for impedance matching the SSET to an ultra-low noise, 50 Ω cryogenic microwave amplifier (Berkshire Technologies Model # L-1.1-30H) with TN=2K. The LC resonator is formed by an interdigitated capacitor and a planar Al coil. Our circuit demonstrates a resonance at 1.17 GHz with a quality factor of about 10.  The measurement set-up is similar to that described in Ref.​[1]​.


Sample Characteristics

  The list below details the sample characteristics for the device shown in Fig. 1.

C1	=181 aF +-9aF			Junction capacitance of junction 1
C2	=199 aF +-20aF		Junction capacitance of junction 2
CG NR	=10.7aF +- 0.1aF		Capacitance between SSET and NR gate
CNR	=33.6aF +- 1aF		Capacitance between SSET and resonator
CG	=22.6aF+- 0.6	aF		Capacitance between SSET and SSET gate
C	=449aF +-30aF		Total device capacitance
R	=104k+-2k		Total device Resistance
EC	= 175 μV +-4V		Coulomb blockade energy
 	= 192.0 μV+-0.7V		Superconducting energy gap
EJ1	= 13.0 V 			Josephson Energy for junction1
a2	= 67.4V			1st Quasiparticle tunneling rate through junction 2
b2	= 32.3V			2nd Quasiparticle tunneling rate through junction 2
EJ2	= 17.4 V 			Josephson Energy for junction 2
a1	= 50.4V			1st Quasiparticle tunneling rate through junction 1
b1	= 24.12V			2nd Quasiparticle tunneling rate through junction 1
Rj1	= 59.5k		 Resistance of junction	
Rj2	= 44.5 k 			Resistance of junction 
dCNR/dx= 0.3e-9 F/m			Derivative of the coupling capacitance
k	= 10 N/m			Spring constant
F	= 1.05e-13 N			Coupling strength at VNR=1V
NR	= 2x 21.866 MHz		Resonator intrinsic frequency
IDS	= 0.8nA			Approx value during measurements
dIDS/dVG= 9.4e-7 A/V			From the slope near the bias point
			at VNR=1V
TSSET	= 200mK			Approx. experimental value.
QSSET	= 106/V2			Approx. experimental value.
QBath	= 120000			Intrinsic quality factor (measured at 30mK 1V)

In the notation above the junction with capacitance C1 is at high voltage while C2 is at the ground potential. The quantities CG NR, CNR and CG are obtained from the periodicity of the current modulation curves; C1 and C2 are calculated using the slopes of the resonances in the IDS-VDS-VG map (see figure S1), Ris determined from the IDS-VDS curves at large drain-source bias VDS >> 4The charging energy is measured from the position of the DJQP feature and JQP crossing.  It is in good agreement with the value be calculated from the sum of all the capacitances, e2/(2C). The superconducting energy gapis calculated using the position of onset of quasiparticle current occurring at 4figure S1.

	The Josephson energy for each junction is given by EJ = (RQRj F (EC/, where the function F(x) describes the renormalization of EJ over the usual Ambegaokar-Baratoff value due to the finite value of EC.  In physical terms, the charging energy lowers the energy of the virtual state involved in a Josephson tunneling event, thus enhancing EJ; a detailed discussion of this effect and the analytic form of F(x) is given in Ref. [​[2]​].  Using this analytical form, we obtain F(x) = 1.26 for our device. The values of the two quasiparticle tunneling rates (s) for each resonance are calculated using the theoretical expressions (see for example Ref. [​[3]​]). The value of the individual junction resistances were extracted by comparing the experimentally measured ratio of peak currents for two adjacent JQP resonances (at the same VDS) with the theoretical prediction (3). Figure 4 shows the theoretical prediction and the measured values of the current. The width of the measured JQP resonances is broader than that predicted by the theory. This discrepancy has been observed in other SSETs with Ec (​[4]​) and suggests that the current may contain contributions from other (presumably incoherent) processes beyind those associated with the JQP resonance.

	The spring constant, k, is estimated from the effective mass of the resonator,, which in turn estimated from the geometry of the beam, and renormalized by 0.99 to account for the shape of the first vibration mode. 

The electromechanical coupling strength is defined by. The derivative dCNR/dx is obtained from 2-dimensional numerical calculations of the capacitances using FEMLAB. We trust the calculated value because the same simulations give CNR=29aF, in good agreement with the experiment. Moreover, the value of the second derivative d2CNR/dx2=0.004aF/(nm)2 gives an electrostatic frequency shift , consistent with the measured value (1.6x10-4/V2). Note that this frequency shift is in addition to that arising from the SSET back-action near the JQP (3), but can be distinguished from it as it is independent of the SSET bias point.
 Another consistency test is the amplitude of the thermal noise signal. In the absence of back-action, the integrated charge noise <Q2> induced on the SSET by the thermal motion of the resonator is proportional to the bath temperature, with a slope 
			. 
The value of this slope, 9.8e-9e2/mK, calculated from the above-determined parameters, compares favorably with 7.8e-9e2/mK, determined from a linear fit of the data at VNR=1V.
	All the calculations for the paper are based on equations given in reference (4). 

Experimental method:

For the measurements shown in Fig. 2 and 3, the device was biased at the point indicated by the red ellipse on Fig. S1. The precise location of the bias point is:

VDS =(4-0.57) Ec		VG=0.078 e from the resonance	IDS =0.8nA

	The SSET bias point is held fixed by monitoring the SSET current, IDS, and applying a feedback voltage to a near-by gate electrode VG.  This allows us to counteract the low frequency charge noise which is typical in these devices. The thermal noise spectra of the resonator are recorded using a spectrum analyzer. A 20.5MHz charge signal is continuously applied to a nearby gate to monitor the charge sensitivity of the SSET in real time. The amplitude of the reference charge signal, 2me, was itself calibrated by using the Bessel-response technique (​[5]​).

	Note that since we are using the radio-frequency SET technique to measure the thermal noise of the nanoresonator, we send a microwave excitation to VDS of the device. Because of this, our measurement has an “average” effect of constantly sweeping an elliptical area around the bias point. The major and minor axes of this ellipse are determined by the strength of the microwave and the reference charge signal at the SSET, which are, respectively 21V and 15V (peak-to-peak values). We estimate that this averaging effect could increase the amount of the damping due to SSET by a factor of ~2 as compared the value obtained at the center of the ellipse. The amplitudes of these signals are smaller than any of the features on the IDS map, approximating the ideal measurement with fixed bias point.

Thermometry and Data Analysis:
	The charge induced at the SSET island from the voltage biased nanomechanical resonator is given by QSSET=CNRVNR, where CNR is the resonator-SSET capacitance. Motion of the resonator,  will modulate the capacitance, CNR, which will change the SSET charge by 

 			
 Thus mechanical noise will produce charge noise, 

 where 
 and  are the charge and position noise power spectral densities. Thermal motion of the resonator is expected to have a spectral density given by:


where, 

obeying the equipartition of energy, where kB is theBoltzmann constant. 
The expected total charge noise spectrum is

 where, SSSET is the white, SSET charge noise originating from the cryogenic preamplifier in our setup.

 	We measure the charge noise of the RFSET detector around the mechanical resonance for temperatures from 30 mK to 550 mK. We find a noise peak at the expected mechanical resonance frequency (identified earlier by driving the nanomechanical resonator), sitting upon a white background, SSSET. The charge noise power data accurately fits the expected harmonic oscillator response function. We extract both the background noise power, SSSET and the integral of the resonator noise power which is a measure of the resonator position variance: 


The above equation is true when the backacation effects of the SSET are negligilble. In practice, since we cannot totally decouple the SSET from the resonator, we use the data taken at VNR=1V, where the observed back-action is negligible, as temperature calibration curve. We have checked the validity of this calibration by converting the integrated power in units of charge (using the amplitude of the reference 20.5MHz sideband recorded at the time of measurement) and comparing the slope of the charge noise signal against bath temperature with the theoretical value calculated from device parameters determined independently (see section ‘sample characteristics’). The frequency, quality factor, and integrated noise power of each spectrum are determined by least-square fitting to a harmonic oscillator response function (see Figure S2).	

Quantum Limit calculations:
	To calculate the minimum uncertainty in resonator displacement that this device can reach, we calculate the displacement noise from two contributions: the shot noise of the SSET current and the displacement noise produced by the backaction force of the SSET onto the resonator. As the calculation involves a number of subtleties, we present a detailed exposition to ensure quality.

Forward-coupled noise:

In practice, the position noise, Sx, is dominated by the noise floor of the preamplifier used to read out the SSET. Ideally, a measurement based on the same method would be limited by the SSET shot noise. The low-frequency limit () of the shot noise is given as
		

Note that we will use this convention for noise spectral densities in all that follows, unless explicitly noted; this convention corresponds to a “two-sided” spectral density.  The shot noise and average current are related by the Fano factor, . For  a SSET biased near the JQP resonance, f is expected to be at most 2, which corresponds to the uncorrelated tunneling of charge 2e Cooper pairs (​[6]​).We assume this worst-case scenario, and calculate SI using f=2, and the measured value of IDS at our chosen SET operating point. 
This current noise produces a displacement noise given by 
			.  
The derivative of the current with respect to the oscillator position is readily calculated from:					
			

The derivative dIDS/dVg can be calculated numerically from our data to find dIDS/dx=12.5A/m at VNR=1V, giving a position noise Sx1/2=93am/Hz1/2 at our point of bias (IDS=0.8nA) and VNR=15V.
By comparison, the preamplifier noise floor yields 
				 




 in the best case (SQ1/2=10e/ Hz1/2) at VNR=15V. During our measurements, the sensitivity was lower (Sx1/2=4.5fm/Hz1/2 with SQ1/2=170e/Hz1/2 and VNR=15V), since the amplitude of the SSET microwave carrier signal had to be kept small to limit the excursion of the bias point. 
	It is interesting to note that if one does not lower the microwave drive, and we optimize our biases for the best position sensitivity, we find a record value of 3.5 10-16 m/Hz1/2, shown in Figure S3.
	Note that the Sx values quoted here are “two- sided" spectral densities. To make a meaningful comparison with other experimental works, the value quoted above should be multiplied by 2.

Back-action

The charges hopping on and off the island of the SSET produce stochastic backaction forces which drive the resonator. For the relatively low frequency resonator used here (), this force noise is frequency-independent and its amplitude is obtained by taking the  limit of equ. 4b:

Using our experimental values for TSSET and SSET we find SF1/2=0.63aN/Hz1/2 at VNR=1V. By contrast, the theoretical predictions lead to SF1/2=0.2aN/Hz1/2. As mentioned above, the excess back-action might be caused by incoherent transport mechanisms which also broaden the current peaks at the JQP resonance.

Quantum Limit

The back-action forces produce a displacement noise given by:

However, the position noise density is constrained by quantum mechanics and the uncertainty principle (ref. ​[7]​ and references therein): 
				. 
Using our theory, the theoretical values for our device gives an optimum value of 						. 
By contrast, our experimental values for the back-action forces give
				. 
Note that the only theoretical assumption we have used in making this calculation was that the Fano factor determining the shot noise had its “worst-case” value of f=2; all other parameters were experimentally determined. In terms of RMS displacement uncertainty, this value is 3.9 away from the quantum limit at the optimal coupling of VNR=0.4V:
 , where .











FIGURE S1: DC current-voltage characteristics (IDS vs VDS vs VGATE) of the SSET. The bias point is shown as red colored ellipse on the left side of the map. The CG can be determined from the current modulation. The black solid line (negative slope line) along one of the JQP resonance is -CG/C1 and the slope of the other solid black line is CG/(C-C1). These two slopes are used to calculate the junction capacitances. Figure 4 was taken with scans of VG along dashed horizontal line through both JQP resonances.










				        Figure S2


			


FIGURES2: Noise spectrum measured at VNR=5V and TBath = 100 mK . The fit (solid black line) is used to extract the resonance frequency, damping coefficient, and integrated power (hatched area). For this spectrum /2= 21.809584 MHz+/- 3 Hz, Q = 37,992+/- 716, <Q2> = 18.8 .10-6 e2RMS +/- 0.4.10-6 e2RMS. [Note that the displacement noise power used in this figure is the “two sided” spectral density.]


                 



Figure S3 shows the best position sensitivity that we have been able to achieve with the device. The spectra was taken at VNR=15V with the RFSET optimized for maximum gain.
[Note that the displacement noise power used in this figure is the “two sided” spectral density.]











































	p.1


^1	  M. D. LaHaye, O. Buu, B. Camarota, K.C. Schwab, Science 304, 74 (2004).
^2	  P. Joyez, “The single Cooper pair transistor: a macroscopic quantum device”, Ph.D. Thesis, Université Paris 6, 1995.
^3	  A. A. Clerk, S. Bennett, New. J. Phys. 7, 238 (2005).
^4	  S. Pollen, “The Superconducting Single Electron Transistor”.  Ph.D. Thesis, Harvard University, December 1999.
^5	  P. Wahlgren, "The radio-frequency single electron transistor and the horizon picture for tunnelling", Ph.D. dissertation, Chalmers University of Technology, Goteborg University, 1998.
^6	  M-S Choi, F. Plastina and R.Fazio, Phys. Rev. B 67, 045105 (2003).
^7	  A. A. Clerk, Phys. Rev. B 70, 245306 (2004)


Cooling a nanomechanical resonator with quantum back-action

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Cooling a nanomechanical resonator with quantum back-action

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