RAPID COMMUNICATIONSPHYSICAL REVIEW B 74, 201301共R兲 共2006兲

Laser-like instabilities in quantum nano-electromechanical systems
S. D. Bennett and A. A. Clerk
Department of Physics, McGill University, Montréal, Québec, Canada H3A 2T8
共Received 18 September 2006; published 9 November 2006兲
We discuss negative damping regimes in quantum nano-electromechanical systems formed by coupling amechanical oscillator to a single-electron transistor 共normal or superconducting兲. Using an analogy to a laser
with a tunable atom-field coupling, we demonstrate how these effects scale with system parameters. We also
discuss the fluctuation physics of both the oscillator and the single-electron transistor in this regime, and thedegree to which the oscillator’s motion is coherent.
DOI: 10.1103/PhysRevB.74.201301

PACS number共s兲: 73.23.Hk, 85.85.⫹j, 72.70.⫹m

At their heart, quantum nano-electromechanical systems
共NEMS兲 consist of nothing more than a mechanical oscillator coupled to a mesoscopic conductor. Despite their apparent simplicity, they have been the focus of considerable recent interest. It has been predicted and shown experimentallythat a single-electron transistor 共SET兲 coupled to a mechanical oscillator may be used for near quantum-limited position
detection.1–5 An equally fascinating aspect of these systems
is their underlying quantum dissipative physics: from thepoint of view of the oscillator, the mesoscopic conductor acts
as a nonequilibrium bath which both heats and damps theoscillator.6–8 For a sufficiently weak oscillator-conductor
coupling, and a sufficiently high-Q oscillator, one can map
the conductor onto an effective equilibrium bath. Even in thisregime, novel effects may take place. Recent work showsthat in systems with either normal-metal or superconducting
SETs, one can operate in regimes where the SET generates
negative damping of the oscillator.9–11 The resulting instability is ultimately cut off by a nonlinearity in the dynamics,
and is characterized by an effective strong coupling between
the mechanical and electronic dynamics.
In this Rapid Communication, we investigate the properties of both normal-state and superconducting SET NEMS
systems in the negative damping regime. Using the analogy
between these systems and a laser with a tunable atom-field
coupling, we derive simple scaling relations for the stationary state and fluctuations in the negative damping regime.
We also discuss how noise measurements could be used tosensitively probe the physics in this regime. In particular, thecritical slowing down associated with the transition to the“lasing” state can be seen clearly in the low-frequency current noise of the transistor. Note that the analogy between
laser physics and a superconducting SET NEMS was alsostudied recently in Ref. 12 using a numerical approach. An
alternate proposed mechanical analog of a laser was discussed in Ref. 13.
Model. We consider a standard SET-based NEMS where
the oscillator acts as a voltage gate with an x-dependent coupling capacitance to the SET island; details may be found in
Ref. 5. In the absence of any intrinsic damping of the oscillator, the total Hamiltonian is given by H = Hosc + H0 + HC.
Here, Hosc describes an oscillator with angular frequency ⍀and mass m. H0 describes the noninteracting part of the SET
Hamiltonian, and includes the kinetic energies of electrons in
the leads and island, as well as tunneling terms taking electrons to and from the leads; in the case of a superconducting
SET 共SSET兲, the island and leads are described by BCS
1098-0121/2006/74共20兲/201301共4兲

Hamiltonians. Finally, HC describes the Coulomb charging
energy of the island in the presence of the oscillator:冋 冉

HC = EC n̂2 − 2n̂ N0 +

A
x̂
2EC

冊册

,

共1兲

where EC is the charging energy of the SET island, n̂ is theisland charge, and N0 is the dimensionless gate voltage associated with a fixed control gate. The coupling strength isgiven by A / 2EC = 共Vosc / e兲 dCosc / dx, where Vosc is the voltage bias on the oscillator and Cosc is the capacitance between
the oscillator and the SET island.11 In what follows, the oscillator is always in the regime where Cosc depends linearlyon x. We see that n̂ acts as a back-action force on the oscillator, while the SET experiences an effective x-dependentgate voltage N关x兴 = N0 + 共A / 2EC兲 x. In the usual case where⍀ is much smaller than the typical tunneling rates in the
SET, and the coupling strength A is sufficiently weak, one
can combine linear-response theory with a BornOppenheimer approximation to rigorously derive a classicalLangevin equation describing the oscillator:11mẍ = − 共k + ⌬k关x兴兲x − m共␥0 + ␥关x兴兲ẋ + ␦ f 0 + ␦ f关x兴.

共2兲

The back-action force leads to a damping term ␥关x兴, springconstant renormalization ⌬k关x兴, and a stochastic force ␦ f共t兲with a white, x-dependent spectral density S␦ f 关x兴. The damping and noise are determined by the quantum noise spectrum
Sn共␻ , N兲 ⬅ 兰dtei␻t具n̂共t兲n̂共0兲典N of the SET island charge fluctuations, evaluated at zero coupling to the oscillator and for a
fixed gate voltage N; ⌬k is determined by the corresponding
charge susceptibility:m␥关x兴 =冏

冏

A2 dSn共␻,N兲,បd␻
␻=0,N=N关x兴

共3兲

S␦ f 关x兴 = A2Sn共␻ = 0,N关x兴兲,共4兲

A2 d具n典.
2EC dN0 N=N关x兴

共5兲⌬k关x兴 =

冏 冏

The spectrum Sn共␻ , N兲 can been calculated for various SET
processes using standard techniques.11 Physically, Eqs.
共3兲–共5兲 are based on the approximation that at each instant in
time, the oscillator sees the SET as an effective bath whose
properties are determined by the instantaneous gate voltageN关x兴. The separation of time scales required for this descrip-

201301-1©2006 The American Physical Society

RAPID COMMUNICATIONSPHYSICAL REVIEW B 74, 201301共R兲 共2006兲

S. D. BENNETT AND A. A. CLERKtion to be valid is well-satisfied in current experiments.2–4
Equation 共2兲 also includes damping and noise terms due tocoupling to an equilibrium bath: in the absence of the SET,
the intrinsic quality factor of the oscillator is Q0 = ⍀ / ␥0 and
its temperature T0 is determined by the strength of the force
␦ f 0共t兲.
Negative damping. As the SET is voltage biased andhence out of equilibrium, it is possible for the back-action
damping ␥ to become negative and even overwhelm thepositive bath damping ␥0. For a SSET, negative damping canarise near operating points corresponding to incoherent
Cooper-pair tunneling in the single/double Josephson quasiparticle processes 共JQP/DJQP兲.10,11 If SET voltages are chosen so that tunneling Cooper pairs can move closer to resonance by emitting energy to the oscillator, one genericallygets negative damping. Negative damping can also arise in anormal metal SET if one has strongly energy-dependent tunnel matrix elements and a resulting nonmonotonic 具n典 versus
N curve.9,11
For either of the above two cases, negative damping
means that the stationary state of the oscillator will be determined by the nonlinearity of the SET-induced damping 共i.e.,
the fact that ␥ is a function of x兲. Crudely speaking, thestationary state corresponds to an oscillator amplitude large
enough that the oscillating gate voltage N关x兴 smears out thenegative damping from the SET, making the total damping
zero. A convenient way to describe this physics is to makeuse of the high quality factors of typical NEMS. One canthen derive a Kramers-like equation for the energy distribution function w共E , t兲 of the oscillator:9,11冋册

D0 + D共E兲 ⳵⳵⳵
w = E ␥0 + ␥共E兲 +
w,
⳵E
⳵t
⳵E
m

共6兲

where the energy-dependent damping and diffusion are givenby␥共E兲 = 2冕2␲

D共E兲 =

冕

2␲

0

共7兲

d␪
S␦ f 关x兴cos2 ␪ ,2␲共8兲with x = 冑2E / k sin ␪. The stationary solution of Eq. 共6兲 isgiven by a generalized Boltzmann distribution. DefiningT̃osc共E兲 ⬅ 关D共E兲 + D0兴 / 关␥共E兲 + ␥0兴, we have

冋冕

E

dE⬘

0

kBT̃osc共E⬘兲

w共E兲 ⬀ exp −册D0 + D共E0兲
m

.

共9兲

In the negative damping regime of interest, ␥共E兲 + ␥0 is
negative at E = 0, and becomes positive at large enough E. In
the simplest case, ␥共E兲 is simply a monotonic increasing
function of E; this arises in a SSET if one detunes the
Cooper-pair resonances to achieve a maximum negative
damping.11 One then finds that w共E兲 has a maximum at E
= E0, where E0 is defined by ␥共E0兲 + ␥0 = 0. Moreover, if E0 issufficiently large, the distribution has a Gaussian form, witha width given by冋冏 冏 册

−1d␥
.
dE E=E0

共10兲

The negative damping instability discussed here in thecontext of a quantum NEMS is reminiscent of the physics ofa laser. In the typical case of a single mode cavity laser with
fast atomic relaxation, the effects of the population-inverted
atoms on the relevant cavity mode can be described in termsof amplitude-dependent damping that is negative for small
mode occupancies.14 In our case, it is instead tunneling electrons or Cooper pairs in the SET that provide the negative
damping. Note also that our system is also very similar to anonresonantly driven oscillator coupled nonlinearly to an oscillator bath; the negative damping regime here was studied
extensively 共both classically and quantum mechanically兲 by
Dykman and Krivoglaz.15 In what follows, we explore in
more detail the extent to which our system is analogous to alaser.
Scaling. Two basic features of a laser are both a large
average number of quanta in the lasing mode, and relatively
small number fluctuations described by Poisson statistics. We
would thus like to know how these quantities scale withsystem parameters in the SET-based NEMS considered here.
We first assume that the SET gate voltage N0 has been set tomaximize the negative damping. We then introduce the characteristic scale for the gate voltage, N*, which is defined as
the change of the gate voltage required to significantly reduce the magnitude of the SET damping.19 For example, inthe case of incoherent Cooper-pair tunneling,4 N* is naturally defined as the change of N needed take the SET off
resonance. This yields 2ECN* ⬃ ប⌫, where 1 / ⌫ is the lifetime of the Cooper-pair resonance.
The gate voltage scale N* can be translated into an energy
scale E* via
E* = 2kd␪␥关x兴cos2 ␪ ,
2␲

0

␴2 =

冉 冊

E CN * 2
A

.

共11兲

This is the approximate energy needed by the oscillator tocompletely smear out the negative damping contribution of
the SET. As we will see, E* is not the same as the energy E0
discussed above.
Next, we write the SET damping as ␥共E兲 = −␥maxg共E / E*兲,where g共0兲 = 1, and ␥max = −␥共0兲 is the maximum of the negative SET damping. For simplicity, we assume g is monotonic
decreasing, as is the case for an optimally tuned SSET. The
most probable oscillator energy, i.e., the energy at which amaximum occurs in w共E兲, is then given by

冉 冊
␥0
␥max

共12兲

冋 册 冉 冊共13兲E0 = E*g−1=KQ0ប⍀

␥0
␥0
g−1
,
␥max
␥max

where Q0 = ⍀ / ␥0 is the intrinsic oscillator quality factor, andthe dimensionless constant K is independent of theoscillator-SET coupling strength:201301-2

RAPID COMMUNICATIONSPHYSICAL REVIEW B 74, 201301共R兲 共2006兲

LASER-LIKE INSTABILITIES IN QUANTUM NANO-…

K=冏

2共ECN*兲2 dSn
ប

2

d␻

冏

.
␻=0,N=N0

共14兲

One can easily estimate the parameter K for a given SET
operating point. For example, we have already shown that
2ECN* ⯝ ប⌫ for the DJQP process in a SSET. Further, a
simple estimate 共which is born out by the full calculation in
Ref. 11兲 gives dSn / d␻ ⯝ 共ប / EJ兲2 where EJ is the Josephsonenergy for the SET 共both junctions are assumed equal兲. We
thus have K ⯝ 共ប⌫ / EJ兲2. For the experiments of Ref. 4, thisyields K ⯝ 6.
Using Eq. 共13兲, we find that we can maximize the average
oscillator energy E0 with respect to the coupling strength.20
In that equation, we see that the coupling strength only enters
E0 through the ratio ␥0 / ␥max. Moreover, as g is a monotonic
decreasing function, the same is true of g−1. It thus follows
that E0 must have a maximum as a function of the coupling.
Physically, this is easy to understand. For too weak a coupling, the intrinsic damping ␥0 dominates the effects of theoscillator, and there is no instability. For too strong a coupling, the oscillator does not need much energy to smear out
the effects of the SET-induced damping.
Finally, we define b to be the maximum of the function
yg−1共y兲, and define y 0 to be the value of y at which thisfunction is maximized. With our choice of E*, both b and y 0
are of order unity. We thus have21关E0兴max = bKQ0ប⍀,
Aopt =

冑

1 E CN *,y 0KQ0 ⌬x

共15兲
共16兲

where ⌬x = 冑ប / 共2m⍀兲 is the zero-point uncertainty in theoscillator position.
We thus see that for an optimal coupling strength, theaverage oscillator energy scales as the intrinsic oscillator
quality factor. For sufficiently large Q0, one can indeed havea high mode occupancy in the resonator. Also note that the
optimal coupling strength scales as 1 / 冑Q0. One can show
along the same lines that in general, E0 / Q0 only depends onthe coupling and the intrinsic quality factor through the combination A冑Q0. The validity of these scaling relations has
been tested against the numerical solution of Eq. 共6兲 for both
JQP and DJQP processes.16
To further the analogy to a laser, we must also characterize energy fluctuations in the stationary state. At the optimald␥
coupling strength, Eq. 共13兲 yields the exact result 兩兩 dE
兩 E0= ␥0 / E0; using this and assuming that the bath temperature is
low enough that the SET dominates the oscillator diffusion,from Eq. 共10兲 we find
具具E2典典 =关D共E0兲兴
E 0 = k BT *E 0 .
m␥0

共17兲

As in a laser, the variance of E scales as 具E典; using Eq. 共15兲,
we see that the relative energy fluctuations become small as
1 / 冑Q0. The degree to which the energy fluctuations aresuper-Poissonian is determined by the effective temperature
T* defined above. In practice, T* will be on the order of theSET effective temperature at E = 0, defined by kBTef f= D共0兲 / m兩␥共0兲兩. Using the definition of ␥ and D in terms of
Sn共␻兲, one can show that in the regime we consider 共where
the SET is much faster than the oscillator兲, Tef f and T* must
both be much larger than ប⍀ / kB. Our system is thus more
like a maser than a laser: the number fluctuations are far
greater than the bound set by purely quantum noise. Notethat we may still easily satisfy kBT* Ⰶ E0 due to the large
factor Q0 in Eq. 共15兲.
Phase fluctuations and linewidth. Another hallmark of alaser is its narrow linewidth, which is limited by relatively
weak phase fluctuations in the stationary state.17 It is thusinteresting to ask about phase fluctuations in our system. A
convenient approach is to work directly with Eq. 共2兲, andmake a rotating-wave approximation, where one keeps trackof the oscillator amplitude and the slowly varying part of theoscillator phase. Focusing on fluctuations about the stationary, large-amplitude lasing state, we start by writing
˜ẋ共t兲 − i⍀̃x共t兲 = 关␳0 + ␦␳共t兲兴e−i⍀tei␾共t兲 ,共18兲

where ␳0 = 冑2E0 / m is the stationary amplitude of the oscillator, and ⍀̃ = 冑共k + ⌬k共E0兲 / m is the renormalized frequency ofthe oscillator, with
⌬k共E兲 = 2

冕

2␲

0

d␪
⌬k关x兴sin2 ␪ .
2␲

共19兲

The linewidth of the oscillator will be determined by thefluctuations in the oscillator phase, ␾共t兲. Linearizing Eq. 共2兲
in ␦␳ and d␾ / dt, and coarse graining over a time scale longcompared to 1 / ⍀, we find that the spectral density of thisfluctuating phase isS ␾共 ␻ 兲 =

冋

册

2D␾
␣2
1
+
,
␻2
1 + 共 ␻ / ␥ 0兲 2

共20兲

where the intrinsic phase diffusion constant is given by
D␾ =

1
4mE0

冕

2␲

0

d␪
S␦ f 关x兴sin2 ␪ .
2␲

共21兲

Here, ␣ characterizes the relative importance of the coupling
between amplitude and phase fluctuations:

␣=

d⍀/dE
d␥/dE

冑

D共E0兲/m
1 d共⌬k兲/dE,⬃
E 0D ␾m⍀ d␥/dE

共22兲

where the derivatives are evaluated at E0, and we have used
the fact that up to factors of order unity, D␾ ⬃ D共E0兲 / 共mE0兲.
␣ is analogous to Henry’s ␣ parameter in standard laser
theory;18 on a physical level, energy fluctuations lead to additional frequency fluctuations through the SET spring constant modulation ⌬k关x兴.
In a semiclassical picture, the SET-induced damping isdue to the small but finite response time ␶ of the SET to theoscillator’s motion;11 one can write m␥ ⬃ 共⌬k兲␶. From Eq.
共22兲, one then has ␣ ⬃ 1 / 共⍀␶兲 Ⰷ 1, as we are explicitly considering a SET which is much faster than the oscillator. Thismeans that unlike a typical laser, the linewidth of the oscillator in the stationary state of our “lasing” NEMS is set by

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RAPID COMMUNICATIONSPHYSICAL REVIEW B 74, 201301共R兲 共2006兲

S. D. BENNETT AND A. A. CLERK

the coupling between amplitude and phase fluctuations. The
crucial difference from a laser is that in the NEMS case, thetransfer of energy between the oscillator and the SET is notresonant, whereas in a laser the coupling between the modeand the atoms is resonant. Thus, in a laser, one obtains a
damping force from the atoms even if they are assumed torespond instantaneously to the mode; in contrast, there would
be no back-action damping in our NEMS in this limit.
Finally, for sufficiently large E0 we find that the oscillator
position noise spectrum exhibits a Lorentzian form centeredon ␻ = ⍀̃ with a width ␥ef f determined by the zero-frequency
limit of S␾. Up to factors of order unity, we find that thiswidth can be much narrower than that set by the intrinsic
damping of the oscillator:1
␥ef f kBT* 2
⬃
␣ ⬀
.
␥0
4E0
KQ0

共23兲

Critical slowing down and shot noise. As was discussed inRef. 9, the instability physics discussed here will manifest
itself directly in the shot noise of the SET. In the stationary
state, the oscillator’s motion is large enough that the resulting oscillations in the SET gate voltage N strongly modulate
the current. One consequence is that the slow energy dynamics of the SET will lead to long-time correlations of tunnel
events in the SET, and thus a large enhancement of the lowfrequency current noise. The magnitude of this enhancement
is expected to be inversely proportional to the relaxation rateof energy fluctuations in the oscillator, ␭E: this represents thetime over which tunnel events are correlated. At the optimal
coupling, a linearized treatment of fluctuations 共as above兲shows that ␭E = ␥0. Moreover, as a function of couplingstrength, ␭E exhibits critical slowing down: it reaches a minimum as the coupling strength is tuned through the bifurcation between small-amplitude and large-amplitude states ofthe oscillator. Without noise, this occurs at the coupling voltage Vcrit for which ␥共E = 0兲 + ␥0 = 0; fluctuations push thetransition to a slightly higher coupling voltage.17 We thus
expect a maximum shot noise enhancement near Vosc = Vcrit.
We have numerically calculated the expected shot noise

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4
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6
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7 A. D. Armour et al., Phys. Rev. B 69, 125313 共2004兲.
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A. A. Clerk, Phys. Rev. B 70, 245306 共2004兲.
9 Ya. M. Blanter et al., Phys. Rev. Lett. 93, 136802 共2004兲; 94,
049904共E兲 共2005兲.
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D. A. Rodrigues et al., cond-mat/0608166 共to be published兲.
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H. Haken, Rev. Mod. Phys. 47, 67 共1975兲.
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1 M. P. Blencowe et al., Appl. Phys. Lett.
2

FIG. 1. 共Color online兲 Signatures of critical slowing down for
the DJQP process in a SSET. Dotted line 共left vertical axis兲: energy
relaxation rate ␭E vs coupling voltage Vosc. The minimum near Vcrit
corresponds to critical slowing down, and one finds ␭E = ␥0 at the
optimal coupling voltage, Vopt. Solid line 共right vertical axis兲: zerofrequency current noise enhancement ⌬SI vs Vosc, scaled by thecurrent noise of the uncoupled SSET. We have used device parameters from Ref. 4; for the SSET, these are EC = 175 ␮eV, ⌬= 192 ␮eV, g = 0.5, and EJ = 15 ␮eV; for the oscillator: ⍀= 22 MHz, k = 10N / m, and Q0 = 120 000. We have also used T0= 20 mK and A = 1.05⫻ 10−13 N when Vosc = 1 V.

enhancement for a NEMS system operated near the DJQP
resonance, using parameters from Ref. 4. This can be calculated in a straightforward manner from the energy diffusion
equation Eq. 共6兲.9 Figure 1 shows a clear maximum in the
shot noise as a function of the coupling voltage, which corresponds to the critical slowing down. We stress that the
maximum does not occur at the optimal coupling voltage
Vopt, but rather near the voltage Vcrit corresponding to thebifurcation. Also note that the dependence of SI on Vosc is not
solely through ␭E; this is why the two curves are not strictly
inversely proportional.
In conclusion, we have shown how negative damping instabilities in SET NEMS scale with system parameters, andhave investigated the amplitude and phase fluctuations in thestationary state.
This work was supported by NSERC and FQRNT.

Khalatnikov 共Harwood Academic, Chur, Switzerland, 1984兲,Vol. 5, pp. 265–441.
16 S. Bennett, Master’s thesis, McGill University, 2006.
17
M. Lax, in Brandeis Summer Institute of Theoretical Physics Lectures 共Gordon and Breach, New York, 1966兲.
18
C. H. Henry, IEEE J. Quantum Electron. 18, 259 共1982兲.
19 The precise definition chosen for the scale N* is not important.
Changing N* by a factor of order unity only changes b, y 0, andK by a similar factor; the scaling relations of Eqs. 共15兲 and 共16兲are otherwise unchanged.
20 The average energy 具E典 is slightly larger than E due to fluctua0
tions; this small difference plays no role here.
21
In the limit Q0 → ⬁, we find 关E0兴max → ⬁ and Aopt → 0. This posesno contradiction with the work of Blanter et al. 共Ref. 9兲, who
considered Q0 → ⬁ for finite, fixed A, which leads to finite, but
not optimal, E0.

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