PHYSICAL REVIEW B 78, 165328 共2008兲

Full counting statistics and conditional evolution in a nanoelectromechanical system
S. D. Bennett and A. A. Clerk
Department of Physics, McGill University, Montréal, Québec, Canada H3A 2T8
共Received 11 August 2008; published 29 October 2008兲
We study theoretically the full distribution of transferred charge in a tunnel junction 共or quantum point
contact兲 coupled to a nanomechanical oscillator, as well as the conditional evolution of the oscillator. Even if
the oscillator is very weakly coupled to the tunnel junction, it can strongly affect the tunneling statistics andlead to a highly non-Gaussian distribution. Conversely, given a particular measurement history of the current,
the oscillator energy distribution may be localized and highly nonthermal. We also discuss non-Gaussiancorrelations between the oscillator motion and tunneling electrons; these show that the tunneling back-action
cannot be fully described as an effective thermal bath coupled to the oscillator.
DOI: 10.1103/PhysRevB.78.165328

PACS number共s兲: 73.23.⫺b, 72.70.⫹m, 85.85.⫹jI. INTRODUCTION

The possibility to observe the quantum mechanics of amacroscopic object has sparked significant interest in nanoelectromechanical systems 共NEMS兲, which consist of a mechanical oscillator coupled to a mesoscopic conductor. In
recent experiments, the oscillator motion has been measured
with near quantum-limited precision using the conductor as adetector1–3 and cooling of the oscillator by quantum backaction has been observed.4 In these experiments it is the current noise of the conductor 共i.e., the second moment of current fluctuations兲 that is used to measure position fluctuationsof the oscillator. The effect of the oscillator on the current
noise has been well studied theoretically, leading to new understanding of back-action and quantum dissipation in
NEMS.5–11 However, much more information lies in the fullprobability distribution of transmitted charge through theconductor or the full counting statistics 共FCS兲.12 In addition
to being of theoretical interest, FCS is an experimentally
accessible quantity and the third moment was recently measured in a tunnel junction.13 Still more information may begained by considering the conditional evolution: given a particular measurement history for the current, what can we say
about the state of the oscillator?
In this paper, we study the full statistics of a tunnel junction 共or quantum point contact兲 coupled to a nanomechanicaloscillator, a system recently realized in experiment.1,2 Thissystem is a prime candidate for measuring FCS in NEMS
since the intrinsic shot noise can dominate over other noise
sources, making it feasible to measure the higher moments.
Starting from a microscopic fully quantum model, we calculate the FCS of tunneled charge as well as the conditional
evolution of the oscillator and find several surprises thatwould not be apparent in a study of the noise alone. Despite
weak oscillator-conductor coupling, we find that the oscillator can strongly enhance the third and higher moments of the
FCS, leading to a markedly non-Gaussian distribution. This
results from long-lived energy fluctuations in the high-Q oscillator, which allow correlations between the oscillator motion and tunneling electrons to accumulate up to the ringdown time of the oscillator, overwhelming the weak couplingstrength and dominating the FCS. Further, even though theconductor couples linearly to the oscillator position, the os1098-0121/2008/78共16兲/165328共6兲

cillator state conditioned on a particular measurement of current can be highly localized in energy. Finally, we discuss
non-Gaussian correlations between the current and backaction force on the oscillator that are not captured by treating
the tunnel junction as an effective thermal bath. These correlations arise from the random momentum kicks imparted tothe oscillator by individual tunneling electrons, which cause
one-half of the back-action to be correlated with tunneling.
The non-Gaussian correlations lead to signatures in measurable quantities such as the current noise; thus, measuring thecurrent noise could provide a way to distinguish quantum
back-action from the effects of an equilibrium bath.
Note that FCS were studied previously in a very different
type of NEMS, a charge shuttle.14,15 Conditional evolution in
NEMS was studied using a quantum optics approach,16,17 butthese studies miss key features arising in our microscopicallyderived model. The average current and current noise of the
NEMS studied here were addressed in Refs. 8–11; unlike
these works, we present an exact solution of the master equation and study the FCS.
II. MASTER EQUATION AND ITS SOLUTION

The Hamiltonian of the coupled system is H = Hosc
+ Hleads + HT, where Hosc describes a harmonic oscillator ofmass M and frequency ⍀ including dissipation due to an
equilibrium thermal bath at temperature T0.18 Electrons in
the leads are described by Hleads = 兺␣,k␧kc␣† kc␣k − eVm̂, wherec␣k annihilates an electron in lead ␣ = L, R, V is the junction
bias voltage, and the operator m̂ counts the number of tunneled electrons. HT describes electron tunneling for the experimentally relevant case of weak oscillator-junction
coupling,8–11
HT =␶0 + ei␩␶1x̂
兺 共Y †cRk† cLk⬘ + H.c.兲,2␲⌳ k,k共1兲⬘where ⌳ is the lead density of states, ␩ describes the dependence of the transmission phase on the oscillator position x̂,and Y † is the raising operator associated with m̂, e.g.,关m̂ , Y †兴 = Y †. We focus on an inversion symmetric system inwhich ␩ vanishes.8
We describe the system using a reduced density matrix
␳ˆ 共t兲 tracking the state of the oscillator and m, the number of165328-1©2008 The American Physical Society

PHYSICAL REVIEW B 78, 165328 共2008兲

S. D. BENNETT AND A. A. CLERK

tunneled electrons. In the weak tunneling regime of interest,
the off-diagonal 共in m兲 elements of ␳ˆ decouple from the diagonal elements. Since our aim is to calculate the statistics ofm, we need only to consider the diagonal elements, ␳ˆ 共m ; t兲
= 具m兩␳ˆ 共t兲兩m典. Treating HT perturbatively and making a standard Markov approximation which requires eV Ⰷ ប⍀, we derive a master equation for ␳ˆ 共m ; t兲.8 Fourier transforming in⬁ei␹m␳ˆ 共m ; t兲, the equation readsm, ␳ˆ 共␹ ; t兲 = 兺m=−⬁i␥
Di
⳵t␳ˆ 共␹ ;t兲 = − 关H0, ␳ˆ 兴 − 关x̂,兵p̂, ␳ˆ 其兴 − 2 关x̂,关x̂, ␳ˆ 兴兴
ប
2ប
ប

再

+

共ei␹ − 1兲 2D1
共␶0 + ␶1x̂兲␳ˆ 共␶0 + ␶1x̂兲
ប2
␶21

+

i␥1
关␶0␶1共p̂␳ˆ − ␳ˆ p̂兲 + ␶21共p̂␳ˆ x̂ − x̂␳ˆ p̂兲兴 .
2ប

冎

共2兲

Here, H0 describes the coherent dynamics of the oscillator
and the total damping and diffusion coefficients are ␥ = ␥0
+ ␥1 and D = D0 + D1. The coefficients ␥0 and D0= 共M ␥0ប⍀ / 2兲coth共ប⍀ / 2T0兲 are associated with the equilibrium bath 共kB = 1兲, while ␥1 and D1 describe back-action
damping and diffusion due to the tunnel junction. Taking theelectronic temperature in the leads to be much less than eV,19
these are given by ␥1 = ប␶21 / 2␲ M and D1 = M ␥1T1, where theeffective temperature due to the tunnel junction is T1= eV / 2.8,9 Note that if we average over m 共i.e., set ␹ = 0兲, Eq.
共2兲 reduces to the quantum Brownian motion master equation for an oscillator coupled to two effective thermal baths
consisting of the environment and the tunnel junction.18 Conversely, tracing over the oscillator degrees of freedom yields
the generating function for the FCS,
⌽共␹ ;t兲 = tr关␳ˆ 共␹ ;t兲兴 = 兺 ei␹m P共m;t兲,共3兲mwhere P共m ; t兲 is the probability that m electrons have tunneled in time interval t. Note that the trace of ␳ˆ 共t兲 over all
degrees of freedom is 兺mtr关␳ˆ 共m ; t兲兴 = 1.
The above model has been used to study the average current and noise;8–11 here we present its exact solution and use
it to study FCS and conditional evolution. To work with Eq.
共2兲, we first express the reduced density matrix in its Wigner
representation,
W共x,p兲 =1␲ប冕

dy具x + y兩␳ˆ 兩x − y典e−2iyp/ប .

共4兲

In terms of the Wigner function, Eq. 共2兲 may be written as

⳵tW共x,p; ␹ ;t兲 = 共Lcl + Lq兲W,p⳵x + M⍀2x⳵ p + ␥⳵ p · p + D⳵2p + 共ei␹ − 1兲⌫共1 + ␭x兲2 ,
M
共6a兲

冋

册

␥1⳵ p · p + D1⳵2p
ប2
␶1共␶0 + ␶1x兲⳵x ,
+
2
4␲ M
共6b兲

with the tunneling rate for the oscillator at x = 0 given by ⌫= ␶20eV / 2␲ប = 2D1␶20 / ប2␶21. The Liouvillian operator Lcl describes the effectively classical evolution of the system: thefirst line of Eq. 共6a兲 corresponds to a classical Fokker-Planck
equation for the oscillator coupled to two effective equilibrium baths, the environment and the junction; the second line
describes tunneling as a classical Poisson process characterized by a rate ⌫共t兲 that depends on the instantaneous oscillator position x共t兲. In contrast, Lq accounts for quantum corrections to the effectively classical evolution. The
␹-dependent terms involving ␥1 and D1 in Eq. 共6b兲 describe
conditional damping and diffusion; these terms represent
back-action that is correlated with tunneling. Conditional
back-action arises because each tunneling electron imparts a
random momentum kick to the oscillator, implying that the
momentum kicks are correlated in time with tunneling
events, and shows that the back-action of the tunnel junction
is not fully described as an effective equilibrium bath. This isdiscussed in detail in Sec. V. Note that when we add the
back-action terms in Lq to those in Lcl, we find that exactly
half of the total back-action is conditional 共i.e., includes thefactor ei␹兲. The other half of the back-action is uncorrelatedwith tunneling and cannot be understood in terms of momentum kicks imparted by tunneling electrons. We thus have thesurprising conclusion that even during periods where noelectrons tunnel, there is still back-action diffusion anddamping. Heuristically, even if no electrons tunnel, we nonetheless gain information about the oscillator and therefore
there must be back-action. Finally, the remaining terms in
Eq. 共6b兲 are also quantum in nature and arise from the difference between two tunneling processes involving absorption or emission of a phonon of energy ប⍀. In particular, thelast term ⬀␶21x⳵xW does not vanish when we trace over theoscillator degrees of freedom and thus represents a quantum
correction to the average tunneling rate8,9 关cf. Eq. 共11兲 below兴. The same correction is obtained from a direct calculation of the tunneling rate using Fermi’s golden rule.
Equation 共5兲 may be solved exactly for the physical initialconditions of a thermal oscillator state. Such a state is Gaussian and remains Gaussian under Eq. 共5兲 for all times. We also
take m = 0 at time t = 0 since this is when we start counting
tunneled electrons. Thus, the Wigner function may be written
in the form2

W共x,p; ␹ ;t兲 =共5兲where the evolution is described by two Liouvillian operators,
Lcl = −

Lq = 共ei␹ − 1兲22e␾e−关Vp共x − x̄兲 +Vx共p − p̄兲 −2Vxp共x−x̄兲共p−p̄兲兴/2共VxVp−Vxp兲22␲冑VxV p − Vxp,

共7兲
where we have scaled all quantities by the natural units ofthe zero-point motion of the oscillator, ⌬x0 = 冑ប / 2M⍀ and⌬p0 = 冑Mប⍀ / 2. The state is fully characterized at all times
by its means, x̄ and p̄, its variances, Vx and V p, its covariance,
Vxp, and its normalization, e␾. These six Gaussian parameters
depend on both ␹ and t and satisfy simple ordinary differen-

165328-2

⳵t p̄共␹ ;t兲 = − ⍀x̄ − ␥ p̄ + ␥1共ei␹ − 1兲1␭冋 冉 冊 册0共8b兲

⳵tVx共␹ ;t兲 = 2⍀Vxp + ␥1共ei␹ − 1兲Vx

冉⳵tV p共␹ ;t兲 = − 2⍀Vxp − 2␥0 V p −

冉

冉

冊

2T1
Vx − 1 ,
ប⍀

冊 冉
冊

2T̃0
2T1
− 2␥1 V p −
ប⍀
ប⍀

2T1
2
共1 + Vxp− ␥1共e − 1兲 V p −
兲 ,
ប⍀
i␹

⳵tVxp共␹ ;t兲 = ⍀共V p − Vx兲 − ␥Vxp + ␥1共ei␹ − 1兲Vxp

冉

共9a兲

冊

2T1
Vx − 1 ,ប⍀

where T̃0 = 共ប⍀ / 2兲coth共ប⍀ / 2T0兲. Again, the ␹ dependenceof these parameters describes correlations between x2, p2, or
xp and moments of m. Finally, the parameter ␾ satisfies

冎

n=2

0
−8 −6 −4 −2 0 2
log10 (γ t)−2m n Γt0[m − m]/σ m2

˜⌫共x兲 = ⌫关1 + 2␭x + ␭2x2兴 − ␥1 .
2

共9c兲

再

n=3

5

FIG. 1. 共Color online兲 Main: P共m ; t兲 at three times. We haveshifted m by its average 具m共t兲典 and scaled by its standard deviation
␴m共t兲. Inset: oscillator enhancement of the variance 共dashed兲 andskewness 共solid兲 of P共m ; t兲 versus time. We have taken T0 = T1
= 104ប⍀, ␭ = 10−3, ␶0 = 0.2, and ␥0 = 10−5⍀. These values are basedon the experiment in Ref. 1, except that we decreased the intrinsic
tunneling strength ␶0 to operate in the tunneling regime and increased the coupling strength ␭ to clearly illustrate our results. For
these values, the fractional change in the average current due to theoscillator is ␭2具具x2典典 = 0.02.

冊
共9b兲

␥1
⳵t␾ = 共ei␹ − 1兲 ⌫关1 + 2␭x̄ + ␭2共x̄2 + Vx兲兴 −
2

0.4

10

0.2

1
2T1
p̄
Vxp x̄ +
−
,
ប⍀
␭
2

where we have again scaled the position and momentum by⌬x0 and ⌬p0 and defined the dimensionless coupling strength␭ = ⌬x0␶1 / ␶0. The ␹ dependence of x̄ and p̄ encodes correlations between the oscillator motion and m. For example, one
can easily show that the irreducible correlation between x
⳵nx̄
and the nth moment of m is 具具xmn典典 = 共−i兲n ⳵␹n . Next, the variances and covariance 共also scaled by ⌬x0 and ⌬p0兲 satisfy



冊

1
2T1
Vx −
, 共8a兲2
ប⍀σ m P (m)冉 冊冉⳵tx̄共␹ ;t兲 = ⍀p̄ + ␥1共ei␹ − 1兲 x̄ +0.6

t = 10−6/γ
t = 10−2/γ
t = 102 /γ

log10

0.8

tial equations which follow directly from Eq. 共5兲. First, themeans satisfy



PHYSICAL REVIEW B 78, 165328 共2008兲

FULL COUNTING STATISTICS AND CONDITIONAL…

共10兲

and is directly connected to the FCS as will be discussed inSec. III.
Equations 共8a兲, 共8b兲, 共9a兲–共9c兲, and 共10兲 have simple analytic solutions in the limit of long times and may be solved
numerically for all times to arbitrary precision. Before using
the equations to study the FCS and conditional evolution, weemphasize an important difference from previous treatmentsof conditional evolution in NEMS: the evolution of the variances is conditional, as seen directly from the ␹-dependent
terms in Eq. 共9兲. This is in stark contrast to the standard
treatment where the variances evolve independently oftunneling.16,17 This is partly due to the conditional backaction diffusion in Lq discussed above 关cf. Eq. 共6b兲兴, which
implies that momentum fluctuations of the oscillator are correlated with fluctuations in m and leads to the conditionalterms in Eq. 共9b兲. However, we also find conditional terms in
Eq. 共9a兲 that arise from the classical part of Eq. 共5兲 describedby Lcl. This is because we start with the linear x dependenceof the tunneling amplitude in Eq. 共1兲, and it follows that thetunneling rate has both linear and quadratic x dependences,

共11兲

Standard treatments of conditional evolution neglect the quadratic dependence, which in our case is inconsistent with thestarting Hamiltonian.20 We stress that the conditional 共i.e., ␹
dependent兲 and unconditional 共i.e., ␹ independent兲 terms in
Eq. 共9兲 appear at the same order in the coupling strength ␭;
there is no a priori reason to keep one effect and not theother. The results presented below are contingent on the conditional evolution of the variances.
III. FULL COUNTING STATISTICS

It follows directly from Eq. 共3兲 that the generating function for the FCS is given by the Gaussian parameter ␾ via⌽共␹ ; t兲 = e␾共␹;t兲. From Eq. 共10兲 we see that if the average andvariance of the oscillator were simply constants, then tunneling electrons would obey Poisson statistics with an effective
˜ 共x兲典, obtained from Eq. 共11兲. Howtunneling rate given by 具⌫
ever, the oscillator position is correlated with tunneling electrons; this correlation enters Eq. 共10兲 through the ␹ dependence of x̄ and Vx and leads to deviations from Poisson
statistics. From Eqs. 共3兲 and 共10兲, we obtain P共m ; t兲, shownat several times in Fig. 1. Even for weak coupling, i.e.,
␭2具具x2典典 Ⰶ 1, the oscillator can have a dramatic effect on
P共m ; t兲 at what we call intermediate times, tⴱ ⱗ t ⱗ 1 / ␥, causing it to become highly non-Gaussian. Here,
tⴱ ⬃

冉 冊

1 ប⍀ 2,⌫ ␭ 2T

共12兲

and T = D / M ␥ is the net effective temperature of the oscillator due to both the tunnel junction and the thermal environment. We emphasize that in the relevant limit of a high-Q
oscillator, the time scale 1 / ␥ is much larger than 1 / ⌫ andthus many electrons have tunneled even for intermediatetimes.

165328-3

PHYSICAL REVIEW B 78, 165328 共2008兲

S. D. BENNETT AND A. A. CLERK

⳵t具具m2典典 = ⳵t具m典 + 2⌫关2␭具具xm典典 + ␭2具具x2m典典兴,+ ␭2关具具x2m典典 + 具具x2m2典典 + 具具xm典典2兴其, 共13b兲
where all of correlations depend on t. The first term in eachequation corresponds to Poisson statistics, in which all cumulants would be equal to 具m典. Correlations of x and x2 withm emerge naturally in the cumulants due to the x dependenceof the tunneling rate in Eq. 共11兲 because m共t兲= 兰t0dt⬘˜⌫关x共t⬘兲兴. As indicated in Eq. 共13兲, this allows oscillator fluctuations to affect the variance and skewness of m;since x and x2 are positively correlated with m, the cumulants
will be increased by these correlations. Similar correlations
appear in the higher moments. These correlations canstrongly affect the cumulants due to the slow decay of energy
fluctuations in the oscillator, as we now discuss.
Consider Eq. 共13a兲. From the x2 term in ˜⌫共x兲, fluctuationsin x2 will lead to fluctuations in m. Thus, the last term in Eq.
共13a兲 leads to a term 2⌫2␭4兰t0兰t0dt1dt2具具x2共t1兲x2共t2兲典典 in thevariance. The factor ␭4 is small due to weak coupling; however, the x2 autocorrelation in the integrand is proportional toan energy autocorrelation 共up to insignificant rapidly oscillating terms兲. This contribution initially scales as 共T / ប⍀兲2and decays on the very slow time scale of the oscillator ringdown time, 1 / ␥. Thus, long-lived energy fluctuations in thehigh-Q oscillator allow its influence to build up, eventually
overcoming the weak-coupling strength and dominating the
FCS. This enhancement occurs when the last term in Eq.
共13a兲 dominates the first, requiring ⌫共␭2T / ប⍀兲2 / ␥ Ⰷ 1. Thiscondition can be satisfied even when the oscillator contribu˜ 共x兲典 is small, as the ratio ⌫ / ␥tion to the average current e具⌫is typically large 共e.g., ⌫ / ␥ ⬃ 108 in Ref. 1兲. Further, thissame condition ensures tⴱ Ⰶ 1 / ␥ from Eq. 共12兲, resulting in
non-Gaussian FCS over a wide range of times.
If the condition for enhancement is met, then the effect iseven greater for higher cumulants. For example, Eq. 共13b兲
contains a term proportional to 具具x2共t兲m2共t兲典典. This leads toan oscillator-dependent term in the skewness similar to thatin the variance, with an additional factor of m resulting in an
extra factor of ⌫␭2x2 and an extra time integral. We obtain athree-time x2 autocorrelation which initially scales as
共T / ប⍀兲3 and decays on the time scale 1 / ␥, compensating forthe extra factor of weak coupling. In general, we find that the
maximum enhancement for the nth cumulant is roughly
具具mn共t兲典典 ⬃ 共⌫␭2Tt/ប⍀兲nⴱ

共14兲

for times t Ⰶ t Ⰶ 1 / ␥. This can be seen directly from Eq. 共11兲by assuming that fluctuations in m are dominated by x2 fluctuations for this range of times.
Figure 1 shows that P共m ; t兲 is skewed only for intermediate times tⴱ ⱗ t ⱗ 1 / ␥; for short and long times the distribution is nearly Gaussian. The enhancement of cumulants com-[m − m]/σ m

0.4

0

−2

共13a兲

⳵t具具m3典典 = ⳵t具m典 + 3⌫兵2␭关具具xm典典 + 具具xm2典典兴

0.6

2

t = 10 −6/γ

−2

0

2

t = 10 −2/γ

−2

0 2
x/σ x

t = 10 2 /γ

−2

0

2

0.2
0

σ x σ m P (x, m)

The significant modification of the FCS is due to theseemingly weak dependence of the current on x2 关cf. Eq.
共11兲兴. To see this, it is useful to consider the first few cumulants of m. From Eqs. 共3兲 and 共10兲, these satisfy 共x is again
scaled by ⌬x0兲

FIG. 2. 共Color online兲 Joint distribution P共x , m ; t兲 calculated forthe same times and parameters as in Fig. 1. We have shifted m by its
average 具m共t兲典 and scaled x and m by their standard deviations,␴x共t兲 and ␴m共t兲.

pared to their Poisson values 关i.e., 具具mn共t兲典典 = ⌫t with nooscillator兴 is shown in the inset of Fig. 1. For short times, theeffects of the weakly coupled oscillator have not yet built up
and we obtain the Poisson statistics of the uncoupled tunnel
junction. For long times t Ⰷ 1 / ␥, the contribution to 具具mn典典
from x2 fluctuations simply scales as t 共and not as tn兲, as t is
now much longer than the lifetime of a typical oscillator
energy fluctuation. For long times the oscillator still enhances 具具mn典典 by a factor 共⌫ / ␥兲n−1共␭2T / ប⍀兲n over the Poisson value ⌫t, but since each cumulant is proportional to t,
P共m ; t兲 tends to a Gaussian.21 To estimate the time scale tⴱ
for the buildup of enhanced cumulants, note that significant
enhancement will occur when the oscillator contribution to
the variance in Eq. 共13a兲 is larger than the Poisson contribution. From Eq. 共14兲 this requires 共⌫␭2Tt / ប⍀兲2 ⬎ ⌫t, whichyields Eq. 共12兲 for the time scale tⴱ.
In the range of times where the FCS is strongly influenced
by the oscillator, P共m ; t兲 is directly related to P共x兲. For athermal oscillator at temperature T, we have P共x兲2
= 冑 4ប␲⍀T e−ប⍀x /4T with x in units of ⌬x0. Assuming that fluctuations of x2 are the dominant source of large m fluctuations, and using Eq. 共11兲, we obtain冋

P共m;t兲 ⬀ exp −ប⍀m4⌫␭2Tt册

共15兲

for m Ⰷ ⌫t. This estimate describes the tail of P共m ; t兲 very
well for times tⴱ Ⰶ t Ⰶ 1 / ␥.
IV. CONDITIONAL EVOLUTION

The effects of the oscillator on the FCS are the result ofcorrelations between x2 and m; we can thus gain further insight by studying conditional dynamics. The joint distribution P共x , m ; t兲 = 具x兩␳ˆ 共m ; t兲兩x典 is shown in Fig. 2. Consistent
with the FCS, for short and long times we see only small
correlations between x and m. P共x , m ; t兲 is most striking attimes tⴱ ⱗ t ⱗ 1 / ␥ due to correlations between x2 and m.
Equations 共8a兲, 共8b兲, 共9a兲–共9c兲, and 共10兲 may also be used
to find the conditional energy distribution, P共E 兩 m ; t兲—given
a particular measurement history and value of m共t兲, what is
the oscillator’s energy distribution? In Fig. 3 we see that fortⴱ Ⰶ t Ⰶ 1 / ␥, the conditional energy distributions are highly
nonthermal and localized at the energy required to produce
the given value of m from Eq. 共11兲, with width given roughly
by T. The ability to obtain information about the oscillator’senergy distribution using a weakly coupled detector is some-165328-4

PHYSICAL REVIEW B 78, 165328 共2008兲

FULL COUNTING STATISTICS AND CONDITIONAL…

T · P (E|m)

32m − m = 0
m − m = −σ mm − m = +σ m
u n c on d i ti on al1

t = 10 − 6/γ

0
0

1

2

t = 10 − 2/γ

0

1
2
E /T

t = 10 2 /γ

0

1

2

FIG. 3. 共Color online兲 Conditional distributions P共E 兩 m ; t兲 forthe same times and parameters as in Fig. 1. The thin line shows theunconditional distribution 共average over m兲.

what surprising and is another result of long-lived energy
fluctuations in the oscillator.
V. NON-GAUSSIAN CORRECTIONS TO THE EFFECTIVE
BATH MODEL

The effects discussed so far are captured by the effectively classical Liouvillian operator Lcl of Eq. 共5兲. Neglecting the quantum corrections results in an “effective bath”
model, where the back-action effects of the tunnel junction
are treated as arising from a second thermal bath coupled tothe oscillator, and the oscillator is treated as a classical variable which sets the instantaneous tunneling rate. However,the conditional back-action damping and diffusion terms in
Eq. 共6b兲 lead to non-Gaussian correlations between the junction current and back-action force operators Î and F̂ that are
not captured by the effective bath model. These arise becauseeven though tunneling is stochastic and imparts random momentum kicks to the oscillator, each momentum kick occurs
at the same time that an electron tunnels. This is completely
missed in the effective bath model, as it treats the junction asa thermal noise source independent of individual tunneling
events. For example, using Eq. 共9兲 to calculate 具具x2共t兲m共t兲典典
in the long-time limit, we find an enhancement compared tothe effective bath model,⌬具具x2共t兲m共t兲典典Q =␥ 1T 1
␥ប⍀

共t → ⬁兲.

共16兲This implies the existence of non-Gaussian correlations between the current and back-action force. A direct quantum
calculation of the non-Gaussian correlator 具具F̂共t1兲F̂共t2兲Î共t3兲典典
using Keldysh path integrals following Ref. 12 leads to thesame non-Gaussian correction given in Eq. 共16兲.
The non-Gaussian correlations may be understood interms of a simple model of quantum back-action. We describe the oscillator-independent tunneling current as a se⬁
␦共t − tn兲, where the inquence of delta functions, I共t兲 = e兺n=0tervals between the tn are exponentially distributed. The
back-action force of the junction is then taken to be F共t兲⬁␰n␦共t − tn兲, where ␰n is a zero-mean random variable= 兺n=0describing the impulse imparted to the oscillator by the nth
electron. The same sequence of times 兵tn其 appears in both I共t兲and F共t兲, reflecting the fact that back-action arises from theaction of individual tunneling electrons. If we then take具␰m␰n典 = 共ប␶1 / ␶0兲2␦mn, our simple model reproduces the nonGaussian correlations obtained from Eqs. 共8a兲, 共8b兲, 共9a兲–

共9c兲, and 共10兲; we also obtain the expected back-action diffusion constant D1. From the size of ␰n we see that the
typical momentum kick imparted by a single tunneling electron is given by ⌬p ⬃ ប␶1 / ␶0 and not by the Fermi momentum. This value for ⌬p is consistent with the Heisenberg
uncertainty principle since the sensitivity of a position measurement scales as ⌬x ⬃ ␶0 / ␶1. We thus have a simple picture
for the source of the conditional part of quantum backaction: it arises from tunneling electrons imparting random
momentum kicks of size set by the uncertainty principle.
Again, we stress that this picture only accounts for the conditional half of the back-action damping and diffusion; theother half is completely uncorrelated with tunneling electrons 关cf. Eq. 共6b兲 and the discussion thereafter兴. We also
note that one can derive the conditional back-action terms in
Eq. 共6b兲 directly from this simple model, from a corresponding classical master equation in which each tunneling event
is associated with a random momentum kick.
The non-Gaussian correlations discussed above can inprinciple be detected via the finite-frequency current noise in
the tunnel junction, SI共␻兲. This may be found from the timedependence of 具具m2典典 using the MacDonald formula,22

SI共␻兲 = 2e2␻冕⬁

dt sin共␻t兲⳵t具具m2共t兲典典.

共17兲

0

Note that the frequency-dependent current noise is obtained
from the particle current fluctuations only. In the single junction, tunneling is nonresonant and there is no place forcharge to build up in the system, so displacement currents
may be safely neglected.23 The time derivative of 具具m2典典 isgiven in Eq. 共13a兲, which shows that we need the full timedependent correlations 具具x共t兲m共t兲典典 and 具具x2共t兲m共t兲典典 to calculate the current noise. These correlations are calculated
simply by taking the ␹ derivative of Eqs. 共8a兲, 共8b兲, 共9a兲–
共9c兲, and 共10兲. The resulting equations are readily solved
exactly for the physical initial conditions in which the oscillator is equilibrated with both the environment and the tunnel
junction, i.e., 具具x2典典 = 具具p2典典 = 2T in our units. The correlation
具具xm典典 leads to a peak at ␻ = ⍀ in the noise that is very
accurately captured by the effective bath model. However,the correlation 具具x2m典典 leads to peaks at ␻ = 0 and ␻ = 2⍀ thatshow signatures of the non-Gaussian correlations. This isespecially true in the limit ␥0 / ␥1 Ⰷ T1 / T0 Ⰷ 1, where the nonGaussian correlations lead to a doubling of the current noise
peak at ␻ = 0 and completely suppress the peak at ␻ = 2⍀ asshown in Fig. 4. This limit requires a back-action damping
rate much smaller than the intrinsic damping from the environment and a back-action temperature much greater than
the temperature of the environment. The first of these conditions is natural in experiments and the second has beenachieved. Note that we still require the intrinsic damping of
the oscillator to be small.
Measurements of SI共␻兲 could thus be used to distinguishthe tunnel junction’s back-action on the oscillator from theeffects of a Gaussian uncorrelated noise source supplied byan equilibrium bath. However, as seen in Fig. 4, in the same

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PHYSICAL REVIEW B 78, 165328 共2008兲

S. D. BENNETT AND A. A. CLERK−5∆S I (ω)/2eI 

8pose a formidable challenge. Further thought will be devoted
to more efficient strategies to detect the non-Gaussian correlations we have identified.

x 10

full results

6

eﬀective bath

4

VI. CONCLUSIONS

2

We have studied the statistics of the experimentally relevant NEMS of a tunnel junction coupled to a mechanicaloscillator. We have shown that even if the coupling is very
weak, long-lived energy fluctuations in the oscillator allow it
to dominate the FCS. The oscillator-induced enhancement of
the third moment of the FCS could be observed up to measurement times near 1 / ␥, well within reach of current experiments. We have also shown that the effective bath model is
not sufficient to fully describe the effects of the tunnel junction on the oscillator. Half of the back-action is conditional
as a result of the random momentum kicks imparted to theoscillator by tunneling electrons, and this leads to nonGaussian correlations with signatures in the finite-frequency
current noise.

0

−5

0
ω/γ

5

−5
0
5
(ω − 2Ω)/γFIG. 4. 共Color online兲 Contribution to the current noise spectrum near ␻ = 0 共left兲 and ␻ = 2⍀ 共right兲 from to the correlation
具具x2m典典. The full calculation including non-Gaussian corrections
共solid兲 is compared to the results from the effective bath model
共dashed兲. The contributions are normalized by the frequencyindependent shot-noise background. We have taken T0 = ប⍀ / 2, T1= 100ប⍀, ␭ = 0.01, ␶0 = 0.2, and ␥0 = 10−3⍀ in order to approach thelimit where the non-Gaussian signatures are maximized.

limit where the signatures are relatively large, the peakheights themselves are very small compared to thefrequency-independent shot-noise background. For this reason, detecting these signatures in the current noise would

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ACKNOWLEDGMENT

This work was supported by NSERC and CIFAR.

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This limit is in no way required but yields simple expressions forthe back-action damping and temperature.
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