New Journal of Physics
The open–access journal for physics

Quantum nanoelectromechanics with electrons,quasi-particles and Cooper pairs: effective bathdescriptions and strong feedback effects
Aashish A Clerk 1 and Steven Bennett
Department of Physics, McGill University, Rutherford Physics Building,
3600 Rue University, Montréal, QC, H3A 2T8 Canada
E-mail: aashish.clerk@mcgill.ca
New Journal of Physics 7 (2005) 238

Received 27 July 2005
Published 29 November 2005
Online at http://www.njp.org/
doi:10.1088/1367-2630/7/1/238

Using a quantum-noise approach, we discuss the physics of bothnormal metal and superconducting single-electron transistors (SSETs) coupled tomechanical resonators. Particular attention is paid to the regime where transport
occurs via incoherent Cooper-pair tunnelling (either via the Josephson quasiparticle (JQP) or double JQP (DJQP) process). We show that, surprisingly, theback-action of tunnelling Cooper pairs (or superconducting quasi-particles) canbe used to significantly cool the oscillator. We also discuss the physical origin ofnegative-damping effects in this system and how they can lead to a regime of strong
electromechanical feedback, where despite a weak SET–oscillator coupling, themotion of the oscillator strongly effects the tunnelling of the Cooper pairs. We
show that in this regime, the oscillator is characterized by an energy-dependent
effective temperature. Finally, we discuss the strong analogy between back-action
effects of incoherent Cooper-pair tunnelling and ponderomotive effects in an
optical cavity with a moveable mirror; in our case, tunnelling Cooper pairs play
the role of the cavity photons.

Abstract.

1

Author to whom any correspondence should be addressed.

New Journal of Physics 7 (2005) 238
1367-2630/05/010238+26$30.00

PII: S1367-2630(05)04457-5© IOP Publishing Ltd and Deutsche Physikalische Gesellschaft

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Contents

1.
2.
3.
4.

Introduction
Effective bath description and the origin of back-action damping
Back-action of electron and quasi-particle sequential tunnelling
Back-action of incoherent Cooper-pair tunnelling
4.1. JQP process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2. DJQP back-action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3. Typical cooling temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4. Quantum-limited measurement with incoherent Cooper-particle tunnelling . . .
5. Negative damping and strong electromechanical feedback
6. Conclusions
Acknowledgments
References

2
3
7
10
10
13
17
17
18
24
25
25

1. Introduction

There has been considerable recent interest in studying the properties of mechanical oscillators
coupled to quantum mesoscopic conductors. Such ‘quantum’ nanoelectromechanical systems(NEMS) are interesting because of their ability (in some cases) to perform quantum-limited
position detection [1]–[4] and their potential to be used in quantum-controlled applications [5].
In addition, they represent a new and interesting problem in the area of quantum-dissipative
systems, as the tunnelling electrons (or quasi-particles or Cooper pairs) in the conductor act asa non-equilibrium, non-Gaussian dissipative bath for the mechanical oscillator. Theoretically,
attention has largely focused on two systems: a normal-metal single-electron transistor (SET)
coupled to an oscillator [3], [6]–[12] and a tunnel junction (or quantum point contact) coupled
to an oscillator [4, 5, 13, 14]; both systems have also been studied experimentally [1, 2, 15]. In
these devices, it has been predicted that for weak couplings, the back-action of the conductor
essentially mimics the effects of an equilibrium thermal bath, with an effective temperature which
is roughly proportional to the drain–source voltage in the conductor.
In this paper, we turn to the properties of a quantum NEMS, which is considerably more
complicated than either the normal-metal SET–oscillator system or the tunnel junction plus
oscillator system. We consider a superconducting single electron transistor (SSET) coupled to ananomechanical oscillator, focusing on regimes where transport in the SSET is via the incoherent
tunnelling of Cooper pairs (both the Josephson quasi-particle process (JQP) and the double JQP(DJQP)) [16, 17]. Such processes are attractive from a measurement point of view, as they havea high gain at a relatively low source–drain voltage. We also discuss the regime where transport
through the SSET is due to the sequential tunnelling of quasi-particles. Using a quantum-noise
approach, we show that in each of these regimes, the SSET provides the oscillator with a highly
non-trivial effective environment. In particular, the effective temperature of the SSET can bemuch lower than the drain–source voltage in the SSET, meaning that considerable cooling canbe accomplished; this is in stark contrast to the normal-metal SET and tunnel junction systems,
where it is difficult to achieve cooling. In addition, the resonant nature of Cooper-pair tunnelling
leads to the possibility of unstable regimes characterized by negative dissipation; we discuss
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the strong electromechanical feedback that arises in these regimes and characterize the resulting
stationary state of the oscillator. This state is highly non-thermal and can be described using anenergy-dependent effective temperature. Finally, we point out that by using incoherent Cooperpair tunnelling, one can come extremely close to reaching the quantum limit on the displacement
sensitivity; in contrast, this is not possible with a normal-metal SET. Note that an SSET plus
oscillator system was also studied recently using an alternative technique by Blencowe et al [18].
The paper is organized as follows. In section 2, we will review the effective bath descriptionof a weakly coupled NEMS, paying special attention to how back-action damping arises bothquantum mechanically and classically. In section 3, we introduce the SET plus oscillator systemand calculate the back-action in the sequential tunnelling regime; our results here are applicable
to a number of different systems, including a normal SET, a Coulomb-blockaded quantum dot
and an SSET. In section 4, we calculate the back-action properties of incoherent Cooper-pairtunnelling, and give a physical interpretation of negative-damping effects that arise here. We
point out the strong analogy between the SSET system and an optical cavity with a moveable
mirror, where forces arising from cavity photons can lead to cooling and instabilities [19]–[21];in our system, tunnelling Cooper pairs play the role of the photons. Finally, in section 5, we
present results for the strong feedback regime that occurs as a result of these negative-damping
instabilities.

2. Effective bath description and the origin of back-action damping

At first glance, the theoretical problem presented by an NEMS seems formidable: we need tounderstand how the back-action of our quantum conductor effects the oscillator, given that the
conductor is typically in a far-from-equilibrium state and typically does not have Gaussian-noiseproperties. Luckily, for many systems of interest, the coupling between the oscillator and the
conductor is sufficiently weak that the current in the conductor responds linearly to the motionof the oscillator. As is discussed extensively in [22], the back-action effects of the conductor inthis case can be directly related to the quantum-noise properties of the uncoupled conductor. We
assume that the oscillator is coupled to the conductor via a term in the Hamiltonian which islinear in x
Hint = −Ax̂ · F̂ .

(1)

Here, A is a dimensionless coupling constant; it needs to be sufficiently weak such that the
conductor only responds linearly to the motion of the oscillator. The operator F̂ describes thequantity in the conductor which couples to the oscillator; for the SSET, we will see that F̂ is simply
proportional to the charge on the central island. For simplicity, we will choose the oscillator’s
origin so that F̂  = 0. F̂ plays the role of a fluctuating back-action force; its unsymmetrized
quantum-noise spectrum at zero coupling to the oscillator,
 ∞dt eiωt F̂ (t)F̂ (0),(2)
SF (ω) =−∞will play a central role. Note that as the conductor in our NEMS acts as a position detector of
the oscillator, we will often refer to it as ‘the detector’ in what follows.
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Taking our oscillator to have mass m and natural frequency , and assuming that it is also
coupled to an equilibrium Ohmic bath, one finds from perturbation theory in A that its motion
is described by a classical Langevin equation [22]mẍ(t) = −m2 x(t) − mγ0 ẋ + δf0 (t) + favg (t) + δf(t).

(3)

This classical Langevin equation involves two fluctuating forces, δf0 (t) and δf(t); it may be
used to calculate the fluctuations in x(t) in terms of the spectral densities of the δf0 and δffluctuations, as well as the oscillator’s response to an external force. The answers obtained
for both these quantities from equation (3) are in exact correspondence to what is foundfrom a perturbative quantum-mechanical calculation, performed to lowest non-vanishing orderin A [22].
Consider the right-hand side (RHS) of this Langevin equation. The second and third termson the RHS describe the effects of the equilibrium bath on the oscillator: γ0 is the damping rateinduced by this bath and δf0 (t) is the corresponding fluctuating force. The fluctuation–dissipation
theorem demands that the spectral density of the δf0 fluctuations obeySδf0 (ω) = mγ0 h̄ω coth[ h̄ω/(2kB T0 )],(4)

where T0 is the bath temperature. γ0 and T0 have a simple interpretation: if the oscillator were notcoupled to the conductor, its quality factor would be Q0 = /γ0 (the ‘intrinsic’ quality factor of
the oscillator) and its temperature would simply be T0 .
The remaining terms on the RHS of equation (3) describe the back-action of the conductor.
δf(t) is the fluctuating part of the back-action force; its spectral density is determined directlyby the symmetrized quantum noise in the operator F̂ :A2
Sδf (ω) =[SF (ω) + SF (−ω)] ,2(5)

In contrast, favg (t) is the average value of the back-action force; it arises because the conductor(and hence F̂ ) changes in response to the motion of the oscillator. We can write this force as
the sum of a conservative force, which is in-phase with the oscillator’s motion, and a damping
force which is out-of-phase with the oscillator’s motion
favg (t) = fdamp (t) + fcons (t),fdamp (t) = −mfcons (t) =

 ∞
−∞

 ∞
−∞

dt  γ(t − t  )ẋ(t  ),dt  α(t − t  )x(t  ).

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(6a)

(6b)

(6c)

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Using standard quantum linear response relations, one hasi
λF (t) = − θ(t)[F(t), F(0)],
h̄




A2 SF (ω) − SF (−ω)
2 −ImλF (ω)=,mγ(ω) = Aωh̄2ωα(ω) = A2 [−ReλF (ω)] .

(7a)
(7b)
(7c)

Note that this description of the oscillator only requires A to be small enough that linear response
is valid; there is no restriction on the size of the oscillator frequency . Note also that thefluctuating back-action force δf(t) will not in general be related to the damping kernel γ(ω)
by a fluctuation–dissipation relation, as our conductor is generally not in an equilibrium state.
Nonetheless, we may use the fluctuation–dissipation relation to define an effective temperature
Teff (ω) at each frequency ω viah̄ωSδf (ω)
coth
≡
.
(8)
2kB Teff (ω)
mγ h̄ω
In the most general case, the oscillator responds to a variety of frequencies, and thus we cannot
characterize the back-action by a single temperature; in this case, the oscillator will not be in athermal state. However, if the oscillator has a sufficiently high quality factor, it only responds toforces in a narrow frequency band centred on ω = . If this frequency band is small enough,
Teff (ω) will be constant over its width. In this case, the back-action will appear to the oscillator
as being due to an equilibrium bath having a temperature T = Teff () and a damping rate γ().
The spring constant k = m2 of the oscillator will also be modified by the conservative part of
the average back-action force; this will be given byk = −α() = A2 [ReλF ()].

(9)

Including the effects of the equilibrium bath coupled to the oscillator, equation (3) tells us thatthe oscillator will be in a thermal state, characterized by a total damping rate γ() + γ0 and atemperature Tosc given rigorously by [22]Tosc =

γ0 T0 + γ()Teff.
γ0 + γ()(10)We again stress that the effective bath description presented above can hold even if theoscillator frequency  is not small compared to the frequency scales of the conductor; all thatis needed is that the ‘mechanical bandwidth’ of the oscillator should be small. Nonetheless, weare often interested in the case where 1/ is long compared to the timescales relevant to the
conductor; in this case, we can approximate γ(ω)  γ(0), α(ω)  α(0) and Sδf (ω)  Sδf (0).
This corresponds to a damping force Fdamp = −mγ(ω = 0)ẋ, a modified spring constantk = −α(ω = 0) and a fluctuating back-action force with a white noise spectrum. In this case,

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equation (8) reduces to
kB Teff ≡A2 SF (0)SF (0)=.
2∂ω SF (0)/ h̄
2mγ

(11)

Before applying these ideas to the SET plus oscillator system, we comment on the originof back-action damping, as there seems to have been some confusion on this point in the NEMS
literature. Equation (7b) makes it clear that there are two equivalent ways of thinking about
damping. The first equality tells us that damping is simply the out-of-phase response of theaverage value of the back-action force F to the position of the oscillator. The second equality
gives us a quantum picture for damping: damping results from the absorption of energy from theoscillator by the ‘bath’ producing the force F (recall that positive-frequency noise corresponds
to absorption of energy, while negative-frequency noise corresponds to emission of energy [23]).
There is another simple but useful classical way to understand both the origin of back-action
damping and the spring-constant renormalization [19]–[21]. We simply need to use the fact that(i) the average back-action force favg (t) is proportional to x, and (ii) the back-action force does
not respond instantaneously to changes in the oscillator’s motion. In the simple case, where theoscillator is much slower than the source of the back-action, we can take the time delay to begiven exactly by τresp .2 We thus havefavg (t) = favg [x(t − τresp )] 

dfavg
dfavg
· x(t − τresp ) · [x(t) − τresp ẋ(t)].
dx
dx

(12)

This immediately yields
dfavg
,
dx

(13a)

dfavg
τresp .
dx(13b)

k = −mγ =Thus, we can view the time delay in the response of the back-action force as being the origin of
damping. Note that this argument tells us that k and γ should always have the opposite sign,and that their ratio gives us a measure of τresp , the response time of the conductor’s back-actionτresp ≡ −

mγ
.
k

(14)

As we will see, the concept of a back-action response time will be useful in understanding theback-action properties of various different mesoscopic systems.

A more exact equation would be to say favg (t) =reduces to equation (12) above.

2

dfavg  t−∞ dt exp[−(t − t )/τresp ]x(t ). For τresp  1, thisdx

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3. Back-action of electron and quasi-particle sequential tunnelling

An SET consists of a metallic island with a large Coulomb-charging energy EC = e2 /(2C )(C is the total capacitance of the island) coupled via tunnel junctions to both a source anda drain metal electrode; in the case of an SSET, the island and both the electrodes are superconducting. The charging-energy term in the Hamiltonian is given by
HC = EC (n̂ − N )2 ,(15)

where n̂ is the charge on the SET island and N = Cg Vg /e is the dimensionless electron number
associated with a gate voltage Vg which is coupled to the island via a capacitance Cg . In addition,a voltage VSD is applied between source and drain which drives the tunnelling of electrons acrossthe SET.
To make an NEMS device involving an SET, a mechanical oscillator (coated with metal)is placed in proximity to the island of the SET such that there is a capacitive coupling Cosc
between the charge on the SET island and the potential of the oscillator; Cosc depends on theisland–oscillator distance. The oscillator is then voltage biased, with the result that changes
in its displacement modify the electrostatic potential of the SET island (i.e. the parameter N
becomes x dependent). For small displacements x, the result is a linear-in-x coupling term in the
Hamiltonian3 :Hint = −An̂x,(16)

where A is the effective coupling strength






dN
Vosc dCosc
Cosc Vosc 1
Cosc 1= 2EC
 2EC= eVosc.
A = 2EC
dx
e
dx
e
d
C
d

(17)

Here, Vosc is the voltage applied to the oscillator and d is the oscillator–island spacing when
the oscillator is in its equilibrium position; for more details, see, e.g. [10]. Comparing against
equation (1), we see that An̂ plays the role of a fluctuating back-action force on the oscillator,
and that it is the quantum-noise spectrum of the charge n̂ that will determine the nature of thisback-action. This conclusion is true regardless of where the SET is operated, or whether it issuperconducting or normal.
Calculating the back-action thus reduces to a problem of calculating the quantum-noise
spectrum of charge fluctuations in the SET. For various operating points of a SET/SSET, this has
been done [23]–[28]. The simplest operating regime of an SET is that of sequential tunnelling.
Current flow is due to electrons (or, in the case of an SSET, quasi-particles) incoherently
tunnelling on and off the SET island via real, energy-conserving transitions; because of thelarge charging energy EC , only two charge states of the island are involved in transport. The
quantum charge noise in this regime was calculated in [23] by using a qubit as a theoreticalquantum-noise spectrometer. In this approach, one first models the dynamics of a two-levelsystem (TLS) weakly coupled to the SET island charge, and calculates an equation of motion for3

2 2
Note there is an additional term in the SET plus oscillator Hamiltonian, which is proportional to Voscx , and thus
contributes a Vosc -dependent shift in the oscillator’s frequency. This term has nothing to do with back-action: it isindependent of the SET island charge n̂. As such, we do not discuss it in what follows.

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a reduced density-matrix tracking both the TLS state and the SET island charge; for sequential
tunnelling, this is done to the lowest non-vanishing order in the tunnelling of an SET. In the limit
of weak SET–TLS coupling, one can calculate both the stationary state of the TLS, as well as its
relaxation rate. These may then be used to directly extract the finite frequency quantum-noise
spectrum of n̂ at the splitting frequency of the TLS. By varying this splitting frequency, one canextract the entire quantum-noise spectrum of n̂.
Using the above approach, one finds that the low-frequency SET charge noise in thesequential tunnelling regime is simply telegraph noise, while the frequency-asymmetry in thenoise is determined by how the tunnelling rates depend on energy [26, 28]. To be definite, consider
a sequential-tunnelling process involving the island charge states n = 0 and 1, and consider thezero temperature limit. There are then two rates of interest: + , the rate at which electrons (orquasi-particles) hop onto the island from the left junction, and − , the rate at which electrons hop(whereoff the island through the right junction. The average current is given by I = e + − /= + + − ), and the average island is n = e + / . It will also be important to know how
these rates change if we either increase or reduce the amount of energy driving the given tunnel
event; let E represent this additional added energy. Using equations (7b) and (11) and the resultsof [23], one obtains the following simple expressions for the back-action in the limit of small
oscillator frequency (i.e.   )kB Teff =mγ = A2

+

−

,

(18a)

∂E ( + − )
,
( )3

(18b)

∂E ( + − )δk = −(mγ) ×

.

(18c)

Here, each rate should be evaluated at E = 0. These expressions are equally valid for sequential
tunnelling of electrons in an SET, electrons in a Coulomb-blockaded quantum dot or BCS quasiparticles in a SSET;4 the only difference is in the form of the tunnel rates. Note that the bath
response time τresp defined in equation (14) is simply given by τresp = 1/( ), as could havebeen expected: this is the timescale which characterizes charge relaxation on the SET island.
For a normal-metal SET, in the case where both junctions have a conductance ge2 / h andequal capacitances, the tunnel rates are given by± (E) =g(E± + E)[1 + nB (E± + E)],h(19)

where nB is the Bose–Einstein distribution function and E± is the gain of electrostatic energy
associated with each tunnelling process
E+ = eVds /2 − 2EC (1/2 − N ).

(20a)

E− = eVds /2 + 2EC (1/2 − N ).

(20b)

4

Note that there is a difference between a normal-metal SET and a Coulomb-blockaded quantum dot: the former
system has a vanishing single-particle level spacing, while the latter system has a large single-particle level spacing.
As a result, the tunnelling rates and back-action are quite different.
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damping are given bykB Teff = p0 p1 eVds ,

(21a)

h
,
g(eVds )2(21b)mγ = A2 ·is the probability of having n = 0 and p1 = 1 − p0 is the probability ofwhere p0 = − /having n = 1. For a fixed Vds , the damping is constant, while the effective temperature reaches
its maximum eVds /4kB at the point of maximum current, N = 1/2. Note these results are inagreement with [9, 12], which treats the SET–oscillator system using a generalized masterequation.
Turning to the case of an SSET with identical junctions, the quasi-particle tunnelling rates
are now given by± = ISS [(E± + E)/e][1 + nB (E± + E)].

(22)

Here, ISS (V ) is the usual I–V characteristic of an superconductor–insulator–superconductor(SIS) junction [29]: there is no current until V = 2/e, after which there is an abrupt rise. In
the usual case where there is some inelastic scattering in the superconductor, this discontinuity
in the I–V curve is smeared out over a voltage h/(e · τQP )  /e, where τQP is the quasiparticle lifetime in the superconductor [30].5 Note that the threshold voltage for quasi-particlesequential tunnelling is VDS = 4, as one needs to create four quasi-particles to transfer a charge
e from left to right. To compare against the normal SET case, we further specialize to the casewhere N = 1/2; this gives a maximum current. We now have for both normal and SSET’s that+ = − ≡ , andkB Teff =mγ = A22 · ∂E

,

∂E
.
4 2

(23a)

(23b)

For drain–source voltages much larger than 4/e, the tunnel rates for the quasi-particles willbe almost identical to that for normal electrons, and the back-action will be almost identicalto that in a normal SET at a similar voltage. However, for eVDS ∼ 4, the sharp rise of the
SIS current–voltage characteristic will lead to stronger back-action in the superconducting case.
One finds that ∂E will be larger than in the normal state case by a large factor τQP / h.
Thus, one has
kB Teff ∼mγ ∼ A2

h

,

(24a)

τQP
.


(24b)

τQP

5

Note that environmental voltage fluctuations in the SET will have a similar effect of smearing out the energy
dependence of the quasi-particle tunnel rates, and can also be parametrized by τQP .
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0

1

0

0

2

+VDS /2+VDS /2-VDS/2a)+VDS/2-VDS/2+VDS /2-VDS /2-VDS/2⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT

2

2

1

2

+VDS/2-VDS/2-1+VDS/2-VDS/20+VDS /2-VDS/2

b)

1

1

-1

Figure 1. (a) Schematic representation of the JQP process, showing each of thethree charge transfer steps. (b) Schematic representation of the DJQP process.

Thus, near threshold, quasi-particle-sequential tunnelling can lead to very low-effective
temperatures and high-damping rates. This behaviour is entirely due to the sharp dependence ofthe quasi-particle tunnelling rates on energy, and is limited by this sharpness. Note that unlike anormal SET or a tunnel junction NEMS, the scale of Teff here is not simply set by the drain–source
voltage.

4. Back-action of incoherent Cooper-pair tunnelling

4.1. JQP process
We now examine the back-action of incoherent Cooper-pair tunnelling processes in an SSET.
The simplest of such process is the JQP resonance, a transport cycle which has been studied
extensively in the context of superconducting qubits [16, 17, 31, 32]. The first step of a JQP
cycle involves a Cooper-pair tunnelling through one junction of the SET, changing the charge
state of the SET island by two electrons. This is followed by two-quasi-particle tunnelling events
through the other junction which return the SET island to its initial charge state; the cycle thenrepeats (see figure 1(a)). Unlike sequential tunnelling (of either electrons or quasi-particles),incoherent Cooper-pair tunnelling is a quantum resonant process; as such, its back-action will
have unique signatures.
For definiteness, we focus on a JQP resonance where a Cooper pair first tunnels onto theisland from the left, taking the island charge from n = 0 to n = 2. We then have a quasi-particle
tunnelling event involving the right lead, taking the island charge from n = 2 to n = 1 (describedby a rate a ). Finally, another quasi-particle tunnelling occurs in the right lead, taking n = 1 ton = 0 (described by a rate b ). There are three important energy scales here: δ, the detuning of
the Cooper-pair resonance, and Ea (Eb ), the amount of energy driving the quasi-particle eventa ( b ). In the case of symmetric SSET junctions, these energies are given simply byδ = Efinal − Einitial = 4EC (1 − N ) − eVDS ,(25a)

Ea = 2EC (3/2 − N ) + eVDS /2 = δ/2 + eVDS + EC ,(25b)

Eb = 2EC (1/2 − N ) + eVDS /2 = δ/2 + eVDS − EC .

(25c)

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The modification for non-symmetric junctions is straightforward (see e.g. [33]). The quasiparticle tunnelling rates a and b are determined from Ea , Eb and equation (22). Note that as
Ea > Eb , one always has a > b .
To calculate the current near the JQP feature, the standard approach is to calculate theequation of motion for a reduced density matrix describing the SSET island charge; this is doneto the lowest non-vanishing order in perturbation theory in tunnelling of the junctions [16]. The
approximation here is that the tunnelling is weak, which requires the dimensionless conductanceg of the junctions to be much smaller than 2π (we consider the case where both junctions haveequal conductances for simplicity). This density-matrix approach can be extended to calculate
the quantum charge noise of the island, as discussed in [28]. One again, uses the ‘qubits asspectrometer’ idea discussed in the previous section and studies a system where the SSET island
is coupled to a TLS. In complete analogy to the calculation for the charge noise of sequential
tunnelling [23], the weak-coupling, long time dynamics of this system, evaluated for different
TLS-splitting frequencies, yields the quantum-noise spectrum of n̂.
In what follows, we present results for the transport and noise properties for the JQP cycle
obtained from this approach. Three main interesting effects are found: (i) JQP can lead to an
extremely low-effective temperature, meaning that it can be used for active cooling; (ii) JQP canlead to negative-damping effects; the resulting instability leads to an interesting regime of strong
electromechanical feedback (to be described in detail in section 5); (iii) JQP can be used to reachthe quantum limit of position detection within a factor of two.
The current for the JQP process is given by [16]EJ2 a
I[N , Vds ] = 2e 2
4δ + ( h̄ a )2 + EJ2 (2 +

a/

b)

(26)

.

Here, EJ = g/8 is the Josephson energy which sets the strength of coherent Cooper-pairtunnelling. Note that both a and b in the above equation depend only weakly on N and VDS
near the JQP resonance; the main dependence of the current on these parameters is through δ,
the energy detuning of the Cooper-pair resonance. To obtain a heuristic understanding of the
JQP cycle, it is useful to consider equation (26) in the limit of a small EJ /( h̄ a ); this limit isoften approached in real experiments. In this limit, Cooper-pair tunnelling is the limiting stepof the JQP cycle, and one finds the simple result that I is 2e times the incoherent Cooper-pairtunnelling rate from the charge state n = 0 to n = 2:
EJ
I = 2e × CPT (δ) = 2e ×
22a

δ2 + ( h̄ a /2)2

.

(27)

The incoherent Cooper-pair tunnelling rate has the usual form expected from Fermi’s golden rule:a transition matrix element squared (i.e. (EJ /2)2 ) times a density of states. Here, the density ofstates is that of the final n = 2 charge state, which is lifetime broadened by the quasi-particle
transition a .
We turn now to the back-action of the JQP process. For simplicity, we will focus on operating
points near the centre of the JQP resonance, and make use of the smallness of the dimensionlessconductance g. In this regime, one can safely neglect the quasi-particle contributions to backaction damping (i.e. ∂E a , ∂E b ), as they are higher order in g. The dominant contribution to
back-action damping will instead arise from the oscillator’s ability to modify the condition for
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resonance; as we will discuss, this is in complete analogy to damping in a number of different
quantum resonant systems. The neglect of quasi-particle damping will allow us to derive some
simple analytic expressions characterizing the back-action.
In the usual limit of a small oscillator frequency (i.e.   a , b , EJ ), the back-action of
an SSET biased near the JQP resonance is described bykB Teff =( h̄ a )2 + 4δ2
,
16δ

EJ2 a
mγ
=
16δ
·
2
A2
b
τresp ≡

−mγ
=
k



(28a)


4δ2 + c1 ( h̄ a )2 + c2 EJ2,(4δ2 + ( h̄ a )2 + c3 EJ2 )34δ2 + c1 ( h̄ a )2 + c2 EJ2,4δ2 + ( h̄ a )2 + c3 EJ2b ( a + 2 b)
a

·

(28b)

(28c)

where the dimensionless coefficients c1 , c2 and c3 are determined by the asymmetry between thetwo quasi-particle tunnel rates
bc1 = 1 + 4

2
b

+8

a


b

c2 = 1 + 4

ac3 = 2 +

a

,

(29a)

,

(29b)

a

+4

2
b
a

.

(29c)

b

Note that the above expressions are valid for arbitrary ratios of EJ / a and a / b .
Several comments are in order. Firstly, note that whenever the detuning of the Cooper-pairresonance is negative (i.e. δ < 0), both Teff and δ become negative. This behaviour has a simple
physical interpretation [28]. From equation (25a), we see that δ > 0 means that VDS is smaller
than what is needed to have the Cooper pairs on resonance. In this case, if the tunnelling Cooper
pairs can absorb energy from the oscillator they can move closer to resonance, while if theyemit energy to the oscillator, they will move further away from resonance. The net result is that
the SSET prefers to absorb energy from the oscillator, which implies both γ > 0, Teff > 0; thisis the same behaviour exhibited by an equilibrium bath. In contrast, when δ < 0, the situationis reversed. VDS is now larger than what is needed to have the Cooper pairs on resonance, andthe SSET prefers to emit energy to the oscillator in order to move the Cooper pairs closer toresonance. This is the meaning of negative damping and negative temperature: our effective bath(i.e. the SSET) prefers to excite the oscillator rather than to absorb energy from it. Using equation(10), we see that the effect of negative damping is to reduce the total damping compared to thatprovided by the equilibrium bath and increase the oscillator temperature. For γ + γ0 < 0, thetotal damping of the oscillator becomes negative, and we enter an interesting unstable regime.
The properties of this regime are the subject of section 5.
The fact that the sign of γ and Teff change as we tune the detuning δ through the resonance isa generic feature of the noise properties of a number of different resonant systems. For example,
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identical effects occur in an optical Fabry-Perot cavity where one mirror is flexible [19]–[21]. Ifsuch a cavity is driven by laser light which is slightly detuned from the frequency of the cavity,
the resulting ‘bath’ of cavity photons will both damp and heat the mirror. The detuning parameterin this case is δ = h̄(ωcavity − ωlaser ); when δ < 0, photons from the laser can become resonantwith the cavity if they give up energy to the mirror, and one gets negative damping. This analogy
can in fact be taken further: the effective temperature for the optical cavity system is identicalto the expression found for JQP, cf equation (28a) [21]. For the JQP process, Teff is set by the
detuning δ and the quasi-particle rate a ; a is the inverse lifetime of the resonant n = 2 charge
state. The expression for the optical cavity system is identical, with a being replaced by the
inverse lifetime of a photon in the cavity [21]. The correspondence is a direct consequence ofthe fact that in both cases, the motion of a mechanical degree of freedom modifies a resonancecondition.
While the analogy to the optical cavity system is certainly useful, there are some respects
in which it is different from JQP case. Consider the expression for the response time τresp of
our effective bath. As expected, τresp scales as the lifetime of the resonant state (i.e. 1/ ). Moresurprising is the fact that τresp depends on the detuning δ: our effective bath responds slower when
we are on resonance. This is in sharp contrast to what is found for the cavity plus mirror system
discussed above; there, the response time is always set by the cavity ring-down time, regardlessof detuning [19]–[21].
Finally, we wish to emphasize the smallness of Teff for the JQP process: unlike the normal
metal SET or tunnel junction, the effective temperature here is not set by the source–drain
voltage, but rather by a much smaller scale, a . As with the case of sequential-quasi-particle
tunnelling, we see that the Teff of a mesoscopic conductor is by no means always set by thedrain–source voltage. The effective temperature is rather a measure of the asymmetry between
energy absorption and emission. In the present case, the smallness of a means that incoherent
Cooper-pair tunnelling can be used for substantial cooling of the resonator when δ > 0; we givenumerical estimates of the magnitude of this cooling effect in section 4.3.

4.2. DJQP back-action
We now turn to the back-action of a slightly more complex incoherent Cooper-pair tunnelling
process, the DJQP resonance. This cycle consists of two incoherent Cooper-pair tunnelling eventsin series, one in each junction (see figure 1(b)); its back-action properties (in the context of qubit
measurements) were studied in [28]. The DJQP process is of interest as it occurs at a smaller
drain–source voltage in the SET than the JQP process; as we will see, it also has more pronounced
back-action and can have a greater cooling effect.
For definiteness, we consider the following cycle: (i) Cooper-pair tunnels from left, taking
the island charge from n = 0 to n = 2; (ii) quasi-particle tunnels from island to right lead, taking
n = 2 to n = 1; (c) a Cooper-pair tunnels from the island to the right lead, taking n = 1 ton = −1; (d) a quasi-particle tunnels onto the island from the left lead, taking n = −1 to n = 0.
At this stage, the cycle repeats. Note that there are two Cooper-pair tunnelling events (one ineach junction) and two quasi-particle tunnelling events (one in each junction). We denote the firstquasi-particle transition rate (n = 2 to n = 1) by a and the second by b . We also denote the first
Cooper-pair transition (n = 0 to n = 2) as ‘A’ and the second transition (n = 1 to n = −1) as

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14⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT

Institute of Physics‘B’. As with the JQP process, we have to worry about the following voltage-dependent energies:
δA = 4EC (1 − N ) − eVDS ,(30a)

δB = −4EC (0 − N ) − eVDS ,(30b)

Ea = 2EC (3/2 − N ) + eVDS /2,(30c)

Eb = −2EC (−1/2 − N ) + eVDS /2.

(30d)

Here, δA ,δB are the detunings of the two Cooper-pair transitions, and Ea and Eb are theenergies driving the two quasi-particle transitions. The rates a and b of these quasi-particle
transitions are related to the corresponding driving energies by equation (22) as before. Notethat the centre of the DJQP resonance (where both δA and δB are 0) occurs when both N = 1/2
and eVds = 2EC .
An density-matrix approach identical to that used for the JQP may be used to describe thetransport and noise properties of the DJQP. While this technique is valid for arbitrary ratios
b / a and EJ / a , we will make some additional mild approximations to obtain results that
may be easily interpreted. Firstly, we again consider an SSET with identical junctions, and notethat at the centre of the DJQP resonance, a = b . As we are interested in the behaviour near
the resonance centre, we will take a = b = throughout. In addition, we will consider thelimit EJ  ; this limit is approximately realized in many experiments, and leads to a great
simplification in the form of the resulting equations.
In this limit, the Cooper-pair tunnelling is the rate-limiting step in the DJQP cycle; we findthat the average current is given by 3e times the series addition of the incoherent Cooper-pairtunnelling rates for the ‘A’ and ‘B’ transitions
I = 3e ×

1+
CPT (δA )1
CPT (δB )

−1

3e
EJ2
=
.
2 ( h̄ )2 + 2δ2A + 2δ2B

(31)

The incoherent Cooper-pair tunnelling rate CPT (δ) is defined in equation (27).
In calculating the back-action for the DJQP process, we again make the assumption that
we can ignore the quasi-particle contribution to damping; as discussed, near the resonance, thiscontribution leads to terms which are higher order in the dimensionless conductance g. For theeffective temperature, we again find a remarkably simple resultkB Teff ==1
1+
kB Teff,A kB Teff,B



−1
=

[( h̄ )2 + 4δ2A ][( h̄ )2 + 4δ2B ].
16[δA + δB ][( h̄ )2 + 4δA δB ]

16δA
16δB
+
2
2
( h̄ ) + 4δA ( h̄ )2 + 4δ2B

−1

(32)

We see that the effective temperature for the DJQP process (in the small EJ limit) is simply givenby the series addition of the effective temperature for each individual Cooper-pair resonance;
the scale for the minimum effective temperature is again set by (i.e. the lifetime of each of the

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two resonant states). The equation for the back-action damping γ is slightly more complicated:mγδA + δB [( h̄ )2 + 4δ2A ][( h̄ )2 + 4δ2B ][( h̄ )2 + 4δA δB ]
=2 2
.
(33)
A2
EJ
[( h̄ )2 + 2δ2A + 2δ2B ]3Similar to the situation with JQP, the effective temperature (and damping) can become
negative if the net preference of the two resonances is to emit energy to the oscillator (i.e. theseries addition of Teff,A and Teff,B must be negative). To see exactly where negative damping andtemperature occur in terms of SSET operating points (i.e. Vds and N ), it is useful to re-write theexpression for Teff . We define:
4EC N = 4EC (1/2 + δN ) = 2EC + N ,(34a)eVDS = 2EC + eδVDS = 2EC + V ,(34b)where N and V represent the energy detuning that result when we move, respectively, N or
VDS from the centre of the resonance. With this notation, we have:kB Teff =1[ 2 + 4(V + N )2 ][ 2 + 4(V − N )2 ]·.
2 + 4[( )2 − ( )2 ]
−32V
V
N

(35)

For small detunings from the centre of the resonance (V , N  ), we see that the sign of theeffective temperature is completely determined by the value of the drain–source voltage. If VDS
is set to be below the centre of the DJQP resonance, V is negative, and hence both Teff and thedamping γ are positive. In this case, the net preference of the two resonances is to absorb energy
from the oscillator. In contrast, if the drain–source voltage is tuned to be higher than resonance,
we get both negative temperature and negative damping. Now, the DJQP process can be broughtcloser to resonance by emitting energy to the oscillator. Finally, note if the drain–source voltage
is exactly at the centre of the resonance (i.e. V = 0), Teff tends to infinity, meaning that there
is no damping and no asymmetry between absorption and emission; this holds regardless of
how much one moves the gate voltage N off-resonance. In this special case, one Cooper-pairtransition always prefers to absorb energy, the other to emit; the net result is no preference
between absorption and emission. In figure 2, we show a contour plot of Teff for DJQP using
typical device parameters and compare it against a plot of the current.
It is instructive to compare the magnitude of detector-induced damping for the JQP and
DJQP processes in the small EJ limit we considered. In both cases, the minimum value of |Teff |is ∝ , where is the quasi-particle tunnelling rate. However, the corresponding damping will
be much larger in magnitude for DJQP versus JQP: comparing equations (28b) and (33), we seethat the back-action damping in the DJQP case is enhanced over that at JQP by a large factor( /EJ )4 . The reason for this is simple to understand, once we recall that γ ∝ Sn (0)/Teff , where
Sn (0) is the zero-frequency charge noise of the SSET island. For both JQP and DJQP, this noise
has a telegraph-noise form in the small EJ limit. For JQP, we have effective telegraph noise
between the n = 0 and n = 2 charge states, where one rate (the quasi-particle transition rate
) is much larger than the other (the incoherent Cooper-pair tunnelling rate CPT ). As a result,
Sn (0) ∝ CPT / 2 . In contrast, for DJQP, we have effective telegraph noise between the n = 0
and n = 1 charge states; the two telegraph rates are both incoherent Cooper-pair tunnelling rates(i.e. the ‘A’ and ‘B’ transitions) and are roughly equal. In this case, Sn (0) ∝ 1/ CPT . Thus, as
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VDS (V)

Institute of Physics⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT

30

30

20

20

10

10

0

0

10

-10

20

-20

30

-30

-0.04 -0.02

00.02 0.04

a)-0.04 -0.02

0.02

0.01

0

(mK)-1

-0.01

-0.02

0

0.02 0.04b)

Figure 2. (a) Contour plot (with equally spaced contours) of current near DJQP
resonance, as a function of gate-voltage detuning (N = N − 1/2) and drain–source voltage detuning (VDS = VDS − 2EC /e) from the resonance centre.
We have chosen typical SSET parameters  = EC = 0.2 meV and g = 0.2.
(b) Contour plot of (Tosc )−1 , the inverse of the oscillator’s temperature (cf equation(10)), for the same range of VDS and N . We have taken typical oscillator
parameters /(2π) = 27 MHz, Q0 = 104 (i.e. intrinsic oscillator quality factor),d = 300 nm (i.e. oscillator–SET island distance), Cosc = 0.1C and Vosc = 10 V
(cf equation (17)); we have also chosen a bath temperature of T0 = 500 mK.
Yellow colour indicates unstable regions where the oscillator temperature isnegative. In the regions of cooling (blue), the lowest achievable oscillator
temperature is less than 50 mK, a factor of ten lower than the oscillator’s
temperature at zero coupling to the SET.

both processes have similar effective temperatures, the enhanced charge noise of DJQP explains
its enhanced damping. This enhancement of back-action damping should make it easier to see
back-action effects of DJQP in an experiment.
Finally, consider the response time τresp of the effective bath presented by DJQP; it isgiven by( h̄ )2 [( h̄ )2 + 2δ2A + 2δ2B ]2EJ2= CPT (δA ) + CPT (δB ) = 2 ×.
τresp[( h̄ )2 + 4δ2A ][( h̄ )2 + 4δ2B ]
h̄
1

(36)

The average response rate, 1/τrep , is simply given by the sum of the rates of the two Cooper-pairtunnelling events in the cycle. In the small EJ limit we are considering, this timescale is much
longer than what was found for JQP (cf equation (28c)), where one always had τresp ∼ 1/ .
As a final note, we should remark that none of the approximations used here (small EJ , noquasi-particle contribution to damping) are necessary to the method used. We have made theseapproximations only so that we could give a clear heuristic picture of the physics underlying theback-action effects found.

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4.3. Typical cooling temperatures
Equation (10) tells us that if the back-action damping dominates the intrinsic, non-back-action
damping of the oscillator (i.e. γγ0 ), then it should be possible to cool the oscillator to theeffective temperature of the SET, even if this is much lower than T0 (i.e. the temperature of
the oscillator when it is decoupled from the SET). To achieve γ
γ0 , one should start with an
oscillator with a high-intrinsic quality factor, and engineer structures where the SET–oscillator
coupling is relatively strong (e.g. have the SET island be in close proximity to the oscillator).
Note that this condition does not imply any violation of the weak-coupling assumptions of ourtreatment: it is entirely possible to have the coupling A weak enough that linear response is stillvalid, yet strong enough that γ
γ0 .
To give an idea of the kind of cooling that should be achievable using incoherent Cooperpair tunnelling, it is useful to give some numerical estimates using typical SET parameters. We
assume EC =  = 0.2 meV, in agreement with the devices used in [34]. For the JQP, we saw thatthe minimum possible positive value of Teff was given by a /4. By choosing VDS = 2 + EC
(the threshold voltage for the JQP process), we found that for these parameters
Teff |min  g × 350 mK.

(37)

Thus, for g = 0.2 (i.e. a junction resistance of 130 k), the minimum effective temperature
is approximately 70 mK; again, this is the minimum temperature we could cool our oscillator
to using the JQP process. Note that this Teff is much lower than the scale set by the voltage:eVDS /kB  7.0 K.
Turning to the DJQP, and now assuming the typical condition EJ < , we see from equation(32) that the minimum possible positive value of Teff is /8, a full factor of two smaller than for
the JQP process. If we again take EC =  = 0.2 meV, we found
Teff |min  g × 125 mK.

(38)

For systems with EJ < , cooling should be much easier to achieve using DJQP. Not only is the
Teff lower, but, as already discussed, the back-action damping due to DJQP is larger than that of
JQP by the large factor ( /EJ )4 .
As a final caveat, the above estimates assume that the only source of broadening of the
Cooper-pair tunnelling resonances is quasi-particle tunnelling. In reality, additional effects could
further broaden these resonances and increase the minimum achievable temperature. Perhaps themost important of these will be voltage fluctuations associated with the environmental impedance
seen by the SET island. These effects will ultimately limit how much one can lower Teff by simply
lowering g.
4.4. Quantum-limited measurement with incoherent Cooper-particle tunnelling
There is an additional important point that needs to be made about incoherent Cooper-pairtunnelling processes: unlike sequential tunnelling in a normal SET, these processes may be
used to do near quantum-limited displacement detection [28]. As discussed extensively in [22],reaching the quantum limit requires a detector which has a minimal amount of back-action noise
relative to the amount of information it provides. The measure of this relative back-action is thenoise temperature TN of the detector, which gives a measure of how much noise is added to the
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signal by the detector. The quantum limit is that kB TN cannot be any smaller than h̄/2, where is the oscillator’s frequency [35]. We are interested in the usual limit where the detector noisecorrelators are frequency independent on frequency scales relevant to the oscillator (i.e. , γ).
After optimizing the strength of the coupling A (to balance back-action and intrinsic shot noise
contributions to the total noise), one has [22]
4SI (0)SF (0)
kB TN
=
 1.
(39)
χ≡
h̄/2
( h̄ dI/dx)2Here, SF = A2 Sn is the back-action force noise spectrum of the detector, SI is the zero frequency
current noise of the SSET and dI/dx = −(dI/dN ) E2AC is the gain of our detector: how strongly
does the average current respond to changes in the oscillator’s position.
Thus, achieving the quantum limit of displacement detection requires that χ approaches
one6 . For a normal-metal SET in the sequential-tunnelling regime, χ is proportional to 1/g2 ,where g is the dimensionless conductance of the junctions. By assumption, g is a small
parameter—this is what allows us to use perturbation theory in the tunnelling! One thus has thatsequential tunnelling in a normal SET is far from being at the quantum limit; there is considerable
excess back-action noise, far beyond what is required by quantum mechanics. In contrast, for
JQP and DJQP, χ is of the order 1 and is independent of g [28]. Using the same density-matrix
approach used to calculate the charge noise, one can calculate the zero frequency current noiseof both JQP and DJQP [28, 32, 36]. Shown in figure 3 is the reduced noise temperature χ for
JQP and DJQP, evaluated as a function of the ratio EJ / , and for a bias point which yields a
maximum gain. Note that the ratio EJ / can in principle be controlled, if one can control theratio of EC to  when fabricating the SET. We see when that EJ  , both DJQP and JQP comeclose to reaching the quantum limit, with the DJQP process doing slightly better. In the moreusual case, where EJ  , the JQP process is much better at approaching the quantum limit
than the DJQP process; this is because of the relative suppression of charge noise associated with
JQP when EJ  .

5. Negative damping and strong electromechanical feedback

In this last section, we describe in more detail the interesting negative-damping instability thatcan be brought about by incoherent Cooper-pair tunnelling in an SET plus mechanical resonatorsystem. As discussed, the negative back-action damping associated with either the JQP or DJQP
transport cycles can, for a sufficiently strong coupling, drive both the total oscillator damping
and its temperature Tosc (as given by equation (10)) below zero (e.g. see the yellow regions infigure 2(b)). In this regime, the SET will continually dump energy into the oscillator, causing
the amplitude of the oscillator’s motion to continually increase. Eventually, this amplitude will
become so large that there will be an effective strong coupling between the oscillator and theSET—the motion of the oscillator will strongly effect the dynamics of the SET. In this regime,
the weak coupling approximations we have been making clearly breakdown. We nonethelesswish to describe the properties of the eventual stationary state achieved in this regime. Note that
6Note that if our detector was used to do a weak QND measurement of a qubit, 1/χ2 represents the ratio between
the rate at which information is acquired by the measured versus the back-action dephasing rate.

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19⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFTχω

Institute of Physics

ΓFigure 3. Noise temperature for the JQP and DJQP processes (scaled by h̄/2)

versus EJ / , where EJ is the Josephson energy and is the quasi-particle
tunnelling rate. We have assumed an optimal choice of coupling strength A,
identical SET junctions and that both quasi-particle tunnelling rates in each cycle
are the same.
such negative-damping instabilities are also well known in the analogous optical cavity system[37]–[39].
To deal with the feedback occurring in this effective strong coupling regime, we will makethe usual simplifying assumption that the oscillator is much slower than the SET:   , CPT ;this condition is usually more than satisfied in experiment. We can then make use of this separationof timescales in a manner somewhat analogous to the usual Born–Oppenheimer approximation.
As in our original weak-coupling description, at each instant in time, the SET acts as an effective
thermal bath on the oscillator and is characterized by a fluctuating back-action force with spectral
density Sδf and a damping rate γ; the oscillator is thus still described by the Langevin equation
equation (3). Now, however, the parameters of the bath will depend on the instantaneous positionof the oscillator: this is how we deal with the strong-coupling aspect of the problem. Thus, thespectral density of the back-action force δf(t) appearing in equation (3) will be x-dependent(i.e. Sδf = Sδf [x]), as will the damping coefficient γ (i.e. γ = γ[x]). The x dependence of boththese quantities arises completely through their dependence on the SET dimensionless gate
voltage N ; we can absorb the coupling Hamiltonian of equation (1) into the SET charging
energy Hamiltonian (cf equation (15)) by writing
N [x] = N [0] +

A
Cg Vg
A
x=
x.
+
2EC
e
2EC

(40)

Again, the key assumption here is that the SET response to changes in x is much faster than theevolution of x itself.
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20

Institute of Physics⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFTAs we show in what follows, the net result of this approach is that, we may describe ourcoupled system with a classical Fokker–Planck equation where both the diffusion and damping
constants have an explicit x dependence. Heuristically, for large oscillator amplitude, the Cooperpair detuning δ and hence the back-action damping will depend on the position of the oscillator;
in the unstable regime of interest, the stationary state of the oscillator will correspond to an
amplitude of oscillation so large that the oscillator experiences equal amounts of positive andnegative damping during one period of its motion. More quantitatively, we find that the stationary
state of oscillator is naturally characterized by an energy-dependent temperature; for a varietyof regimes, the energy distribution of the oscillator is a Gaussian and hence highly non-thermal.
Note that a similar approach to potential strong-feedback behaviour was used in [11]; however,the particular NEMS studied in that work did not exhibit any negative damping, and thus there
was no effective strong-coupling regime.
Our starting point is thus the Langevin equation of equation (3) with x-dependent damping
and back-action force terms. As we are interested in the limit where the oscillator is much slower
than the source of the back-action noise, we can treat the latter as being white. We may thenconvert our Langevin equation into a Fokker–Planck equation for the oscillator’s phase-spacedensity w(x, p; t) in the usual way [40]; we then obtain


p ∂
∂
∂2
∂
2
w= −
+ (m x + (γ0 + γ[x])p) w + (D0 + D[x]) 2 w.
(41)∂tm ∂x ∂p∂pHere, γ0 is the intrinsic (SET-independent) damping of the oscillator, D0 = mγ0 kB T0 describes
momentum diffusion due to the equilibrium bath and the x-dependent diffusion constant D[x] isdetermined from back-action force spectrum
D[x] = 21 Sδf [ω = 0; x] = m · γ[x] · kB Teff [x].

(42)

In the case of JQP, γ(x) and Teff [x] are given respectively by equations (28b) and (28a) with thesubstitution N → N [x]; for DJQP, one would use equations (32) and (33). Note that there is
no ambiguity in interpreting equation (3) with an x-dependent back-action noise spectrum: boththe Ito and Stratonovitch interpretations of this stochastic differential equation yield the same
Fokker–Planck equation. Also note that we are neglecting the x dependence of the conservative
part of the average back-action force; this is justified in the experimentally relevant limit of ahigh-intrinsic quality factor Q0 . In this limit, one can have the magnitude of back-action damping
comparable or greater than the intrinsic damping of the oscillator, while at the same time havethe back-action spring-constant modification k much smaller than m2 .
To make further progress, we consider the experimentally relevant weak-damping limit ofequation (41) (i.e. γ0 , |γ[x]|  ), and follow the approach of Kramers [41] (and more recently,
Blanter
et al [11]). We√ first re-write x and p in terms of the oscillator’s energy E and phase θ (x =√2E/k sin θ, p = 2mE cos θ), and then convert equation (41) into an equation for w(E, θ).
Working to lowest order in the dissipative terms (both damping and diffusion),
 2πthis equation canthen be recast as an equation for the oscillator energy distribution w(E) = 0 dθ w(E, θ); oneuses the fact that in the absence of any dissipation, w0 (E, θ) = w0 (E)/2π. After some algebra,
this procedure yields
D0 + D(E) ∂∂dw(E; t) =E γ0 + γ(E) +w(E; t).
(43)
dt
∂E
m
∂E
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Institute of Physics⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFTThe energy-dependent back-action damping and diffusion constants here are defined as:
 2π

dθ
γ[E] = 2γ(x = 2E/k sin θ) · cos2 θ,
2π
0
 2π

dθ
D(x = 2E/k sin θ) · cos2 θ.
D[E] = 2
2π
0

(44)
(45)

As could have been expected, they are given by averaging γ[x] and D[x] over the phase of theoscillator at fixed amplitude; what is perhaps more surprising is that one must take a weighted
average, with a weighting factor which is proportional to p2 . Note that at E = 0, γ(E) and D[E]coincide respectively with the back-action damping and diffusion constant in the weak-coupling(i.e. zero feedback) theory.
It is now trivial to solve for the stationary energy distribution of the oscillator. From equation(43), we have

  E
dE
,
(46)
w(E)stat = N exp −
0 kB Tosc [E ]where N is a normalization constant, and the energy-dependent effective temperature T osc (E) is
given quite naturally bykB Tosc (E) ≡D0 + D[E].
m (γ0 + γ[E])(47)

We see that the stationary state is characterized by a generalized Boltzmann distribution having
an energy-dependent effective temperature. In the weak-coupling, no negative-damping case,
we can neglect the energy dependence of kB T [E], and equation (46) coincides with a thermal
distribution and with the results of the weak-coupling theory (cf equation (10)). In the strongfeedback regime of interest, the total damping (γ0 + γ(E)) is negative at E = 0; as we increase
E, the motion of the oscillator smears out the back-action contribution to the damping, and forvery large E, the total damping tends to γ0 > 0. There will thus be a critical E = E0 where(γ0 + γ(E0 )) will pass through zero. It follows directly from equation (46) that w(E) will havea maximum at E0 ; moreover, in the vicinity of E0 , w(E) will look Gaussian(E − E0 )2
w(E)stat  N exp −
,
(48)
2σ 2
D0 + D[E0 ] dγ
σ =2m
dE E=E0

−1

2

.

(49)

We thus find that in the strong-feedback regime, the oscillator’s energy distribution function has
a maximum at a non-zero energy, in sharp contrast to an equilibrium distribution. The energy at
which the maximum occurs corresponds, as expected, to an oscillator amplitude large enough thatthe oscillator experiences zero average damping during each period of its motion. The width ofthe distribution near this maximum is set by both the back-action noise and the energy sensitivityof the damping via equation (49).
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Institute of Physics⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT

δ/ Γ

γ

γ

γ

δ/ Γ

δ/ Γ

Figure 4. Energy-dependent back-action damping of the JQP process, cf equation(44), plotted as a function of energy, and using SSET parameters identical tothose used in figure 2. The characteristic scale for feedback effects, E∗ , is given
in equation (50). Note that the chosen scaling makes this plot independent of thecoupling strength A. The red curve and blue curves correspond to two different
Cooper-pair detunings δ (i.e. SSET operating points) as labelled; note that γ(E)can have a non-monotonic dependence on energy. The inset shows the dependenceof the back-action damping at zero-oscillator energy (cf equation (28b)) on the
Cooper-pair detuning.
We now apply the above results, which are quite general, to the specific instability broughton by incoherent Cooper-pair tunnelling. We first consider the case of an oscillator–SSET system
operated in the negative-damping regime near the JQP resonance; as usual we consider the limitof small dimensionless junction conductances, and neglect the quasi-particle contribution to theback-action damping. In this limit, all the feedback effects of the oscillator on the SET will be dueto the dependence of the Cooper-pair detuning δ (cf equation (25a)) on the oscillator position x.
The characteristic energy scale E∗ for feedback effects will correspond to an oscillator amplitudelarge enough that the corresponding oscillations in δ are equal to the resonance width, h̄ a . Fromequations (40) and (25a), one has2E∗ = 

h̄ a 

.
2A 2k

(50)

E∗ sets the characteristic scale for variations in γ[E] and D[E]. Shown in the inset of figure 4 isthe back-action damping γ at zero energy (i.e. ignoring feedback effects), cf equation (28b), as
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Institute of Physics

1.0

2

10

0.8

Vosc = 0.5 V
Vosc = 7.7 V
Vosc = 12.00 V

4w(E) Eavg

Eavg / kB (K)⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT

2

1
4

0.6
0.4

2

0.2

0.1
4

0.0
0

2

4

6
8
Vosc (volts)

10

12

14

0.0

0.5

1.0
1.5
E / Eavg

2.0

2.5

Figure 5. Effects of JQP back-action in the negative-damping regime. We

have taken a bath temperature T0 = 50 mK, an intrinsic oscillator quality factor
Q0 = 105 and an SSET operating point near the JQP resonance which maximizes
the effect of the back-action damping (δ = −0.25 h̄ a ). All other parameters are
the same as in figure 2. Left panel: average energy of the oscillator Eavg as afunction of the coupling voltage Vosc ; the dashed curve is the predictions of the
weak-coupling theory, cf equation (10), while the red curve is from the strongfeedback theory. The total oscillator damping becomes negative at Vosc = 7.1 V]].
Right panel: oscillator energy distribution w(E) as determined by equation (43).
As Vosc becomes large enough for the strong-feedback effect to take hold, theenergy distribution becomes Gaussian-like.

a function of δ. The main plot shows the energy dependence (scaled by E∗ ) of the damping fortwo choices of δ0 = δ[x = 0]. Note that γ(E) can have a non-monotonic dependence on E.
While measuring the full distribution w(E) of the oscillator’s energy may be quite difficult,the average oscillator energy Eavg can be obtained directly from experiment. In the experimentsof LaHaye et al [2], this quantity was obtained by extracting x2  from the current noise of
the SET, and then invoking the equipartition theorem; a similar approach could be used in thestrong-feedback regime discussed here7 . In figure 5(left panel), we plot Eavg as a function of
the coupling voltage Vosc ; we use typical device parameters (see figure caption), and chooseδ = −0.25 h̄ a to maximize |γ(E = 0)| (i.e. same as the blue curve in figure 4). For small Vosc ,the intrinsic damping of the oscillator dominates the back-action damping, and there is no strongfeedback effect. The oscillator is in a thermal state, with a temperature Tosc given by equation (10)and Eavg = kB Tosc . As we increase the coupling voltage, the back-action damping increases inmagnitude (i.e. becomes more negative) and Eavg = Tosc correspondingly increases. At a criticalvoltage Vosc = 7.1 V, the total damping becomes negative and the weak-coupling expression for
Tosc diverges. The strong-feedback effect takes hold here, and we see that average energy ofthe oscillator remains finite. There is also a corresponding change in the shape of the energy
distributions, as can be seen in figure 5(right panel)). Note that for a large range of Vosc , Eavg
remains approximately constant. This is the result of two competing tendencies: increasing Vosc
7

As the stationary state is characterized by a flat distribution of the oscillator phase θ, one again has the relation
Eavg /k = x2 , as would hold in equilibrium.

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Institute of Physics⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
3.0

2

2.5
4

w(E) Eavg

Eavg / kB (K)

10
2

1
4
2

Vosc = 0.5 V
Vosc = 15.0 V
Vosc = 16.8 V

2.0
1.5
1.0
0.5

0.1

0.0

4

0

5

10
15
Vosc (volts)

20

0.0

0.5

1.0
1.5
E / Eavg

2.0

2.5

Figure 6. Effects of JQP back-action in the negative-damping regime. We use
the same parameters as figure 5, but now choose an SSET operating point near
the JQP resonance which yields a non-monotonic γ(E). (δ = − h̄ a .) Left panel:
average energy of the oscillator Eavg as a function of the coupling voltage Vosc ,labelled as in figure 5. Note that the strong-feedback effect sets in at a voltage
Vosc much lower than what is needed to make the weak-coupling theory diverge.
Right panel: oscillator energy distribution w(E) as determined by equation (43).

increases the overall magnitude of the back-action damping, but also increases the sensitivity of
the Cooper-pair detuning δ to E.
In figure 6, we plot the same quantities as figure 5, but now for δ = − h̄ a . In this case,the non-monotonic nature of γ(E) leads to energy distributions w(E) which have two local
maxima. Also note that in this case, the strong-feedback effect sets in well before the weakcoupling expression for Tosc diverges. Finally, in figure 7, we plot results for an oscillator–SSET
system operated in the negative-back-action damping regime near the DJQP resonance. We haveagain chosen realistic parameter values (see figure caption). We see that the feedback effect isstronger here; as discussed, this is a consequence of the enhanced back-action damping of DJQP
versus JQP. Note that for both the JQP and DJQP strong-feedback results presented here, themaximum amplitude of the oscillator is still much smaller than the oscillator–SET gap d, and thecorresponding oscillation of the Cooper-pair detuning is at most of order . Thus, these results
do not violate our assumption of a linear coupling Hamiltonian, and do not involve SSET physics
far from the JQP or DJQP resonance.
There are a number of other interesting issues related to the strong-feedback regime
discussed here. In particular, what are the dynamics of the oscillator’s energy in this regime? As
the total damping vanishes at the most probable energy E0 , one expects very slow dynamics.
How does this then manifest itself in the output noise of the SET? It would also be interesting toinvestigate systems where the SET is not infinitely fast compared to the oscillator; in this case,
one could expect similar kinds of multistability effects found in optical cavity systems [39]. We
hope to address these issues in the near future.
6. Conclusions

In this paper, we have discussed various aspects of back-action physics in SET–oscillator
NEMS using a quantum-noise approach. In the sequential-tunnelling regime, we derived general
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Institute of Physics⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
2.5

Vosc = 1.5 V
Vosc = 1.8 V
Vosc = 2.1 V

2.0w(E) Eavg

Eavg / kB (K)

10

1

1.5
1.0
0.5

0.1

0.0
0.0

0.5

1.0

1.5
2.0
Vosc (volts)

2.5

3.0

0.0

0.5

1.0
E/Eavg1.5

2.0

Figure 7. Effects of DJQP back-action in the negative-damping regime. We
have taken a bath temperature T0 = 50 mK, an intrinsic oscillator quality factor
Q0 = 104 and an SSET operating point near the DJQP resonance (N =0, V = h̄ /2). All other parameters are the same as in figure 2. Left panel:
average energy of the oscillator Eavg as a function of the coupling voltage Vosc ;the dashed curve is the predictions of the weak-coupling theory, cf equation (10).
The total damping becomes negative when the coupling voltage Vosc = 1.75 V.
Right panel: oscillator energy distribution w(E) as determined by equation (43).
At Vosc = 1.5 V, we still have a thermal distribution; increasing the coupling
voltage takes one into the unstable regime and results in an almost Gaussian
distribution of energy.

expressions for the effective temperature, damping rate and bath response time that are valid
regardless of the particular system (electrons in a normal SET, quasi-particles in an SSET,
electrons in a quantum dot). In the regime of incoherent Cooper-pair tunnelling, we described
the back-action of both the JQP and DJQP processes, and demonstrated that both could be
used for substantial oscillator cooling, as well as near quantum-limited measurement. We also
discussed the strong analogy between these processes and an optical cavity with a moveable
mirror. Finally, we discussed the regime of strong electromechanical feedback that can arise
when the back-action damping becomes negative, and showed that the stationary state in thisregime is naturally characterized by an energy-dependent temperature.
Acknowledgments

This work was supported by NSERC under grant RGPIN-311856, by the Canada Research Chairs
programs, and by the FQRNT under grant 2006-NC-105671. We thank Andrew Armour, Olivier
Buu, Miles Blencowe, Jack Harris, Matt LaHaye and Keith Schwab for useful discussions.
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New Journal of Physics 7 (2005) 238 (http://www.njp.org/)