RAPID COMMUNICATIONSPHYSICAL REVIEW B 70, 121303(R) (2004)

Shot noise of a tunnel junction displacement detector
A. A. Clerk and S. M. Girvin
Departments of Applied Physics and Physics, Yale University, New Haven, Connecticut 06511, USA(Received 3 June 2004; published 8 September 2004)
We study quantum-mechanically the frequency-dependent current noise of a tunnel-junction coupled to ananomechanical oscillator. The cases of both dc and ac voltage bias are considered, as are the effects of
intrinsic oscillator damping. The dynamics of the oscillator can lead to large signatures in the shot noise, even
if the oscillator-tunnel junction coupling is too weak to yield an appreciable signature in the average current.
Moreover, the modification of the shot noise by the oscillator cannot be fully explained by a simple classicalpicture of a fluctuating conductance.
DOI: 10.1103/PhysRevB.70.121303

PACS number(s): 73.23.⫺b, 85.85.⫹j, 72.70.⫹m

Spurred primarily by experiments in solid-state qubit systems, there has recently been considerable interest in understanding the noise properties of mesoscopic systems used as
detectors.1–6 Many new results have emerged, including an
understanding of the connection between noise, back-action
dephasing and information,4–6 and of the influence of coherent qubit oscillations on the output noise of a detector.3 Not
surprisingly, similar concerns arise in the study of nanomechanical oscillators. Recent experiments using singleelectron transistors (SETs) have demonstrated displacement
detection of such oscillators with a precision close to the
maximum allowed by quantum mechanics.7,8 Given the interest in these systems, it is important to gain a better understanding of how a mesoscopic detector influences the behavior of an oscillator, and vice-versa. Several works haveaddressed various aspects of this problem. In particular, it
has been shown that an out-of-equilibrium detector can serveas an effective environment for the oscillator, providing botha damping coefficient and an effective temperature.11–13
In the present work, we study the finite-frequency output
noise of a mesoscopic displacement detector, where one expects to see signatures of the time-dependent fluctuations of
the oscillator. A completely classical study of the current
noise of a SET displacement detector was presented recently
in Refs. 14 and 15. In contrast, we consider a generic tunneljunction or quantum point-contact (QPC) detector, in which
the tunneling strength depends on the position of the oscillator, and calculate quantum mechanically the finite frequency
current noise. Such a system could be realized by using a
STM setup where one electrode is free to vibrate.9,10 We treat
both dc and ac voltage bias; the latter is of particular relevance to experiments.10 We find that even for a detectoroscillator coupling so weak that there is little signature of theoscillator in the average current, there can nonetheless be astrong signature in the finite-frequency current noise. We
moreover find that the oscillator contribution to the noisecannot be simply explained by a classical model of a detector
conductance which fluctuates with the oscillator position—
there are additional quantum corrections which suppress thecontribution of zero point fluctuations. We show that thesequantum corrections result from correlations between the detector’s random back-action force and intrinsic output noise.
Finally, in the ac-biased case, we find that the oscillator experiences a time-dependent temperature, which has a direct
1098-0121/2004/70(12)/121303(4)/$22.50

influence on the detector’s current noise. Note that the noiseof a QPC position detector was also briefly considered inRef. 16.
Model. Considering the simplest case where the tunnelmatrix element depends linearly on the oscillator displacement x̂, the tunnel junction detector is described-by
Hdet =

冉冊

␶0 + ei␩␶⬘x̂
†cL,k⬘ + H.c. − eV共t兲m̂
兺k,k⬘ Y †cR,k2␲⌳⬅ Hdet,0 − x̂ · F̂.
Here, cL,k 共cR,k兲 destroys an electron state in the left (right)
electrode, ⌳ is the conduction-electron density of states, m̂denotes the number of tunnelled electrons, and the operator
Y † augments m̂ by one. ␩ parametrizes the sensitivity of thetransmission phase to x̂, and will in general be nonzero. We
consider both the cases of a pure dc voltage V, and a pure ac
voltage Vac共t兲 = V cos ␯t. Note that the tunneling Hamiltonian
itself acts as a random back-action force F̂ on the oscillator.9
We will describe our system by a reduced density matrix
␳共m ; x , x⬘ ; t兲 ⬅ 具x兩␳ˆ 共m ; t兲兩x⬘典 which tracks the state of the oscillator and m, the number of electrons which have tunneledthrough the junction. In general, the evolution of ␳ˆ will begiven by a Dyson-type equation:di␳ˆ 共m,t兲 = − 关H0, ␳ˆ 共m;t兲兴 +
dt
ប

冕 ⬘兺
t

dt

t0

m⬘

⌺̌共m,m⬘ ;t − t⬘兲ⴰ 关U0共t − t⬘兲␳ˆ 共m⬘ ;t⬘兲U†0共t − t⬘兲兴.

共1兲Here, U0 is the evolution operator corresponding to the unperturbed (zero-tunneling) Hamiltonian, and we have written
the self-energy ⌺ as a super-operator.
We will consider the simplest case of weak tunneling, andkeep only self-energy terms which are lowest order in thetunneling. ⌺ is only nonvanishing if m⬘ = m or m⬘ = m ± 1;
these two types of contributions correspond to “scatteringout” and “scattering in” terms in a kinetic equation, and are
given by the diagrams shown in Fig. 1. These diagrams correspond to standard tunneling bubbles,17 the only difference
being that the tunneling vertices can contain an x̂ operator. Ifx̂ appears at the t⬘ end of a graph for ⌺共t , t⬘兲, x̂ will evolve
during the duration of the tunneling event. As a result, the70 121303-1©2004 The American Physical Society

RAPID COMMUNICATIONSPHYSICAL REVIEW B 70, 121303(R) (2004)

A. A. CLERK AND S. M. GIRVIN

D␴共t兲 =

FIG. 1. Diagrams for the a) scattering in, and b) scattering out
terms in the self-energy ⌺̆共t − t⬘兲. The solid lines represent the forward and backwards Keldysh contours; the dashed lines are conduction electron propagators. The solid black vertices correspond to␶0 + ␶⬘x̂. Note that an x̂ operator appearing in a vertex at time t⬘ will
evolve during the tunneling event.

self-energy ⌺̌ has terms involving p̂, and the final form of ⌺we obtain does not correspond to the oscillator-free case withx̂ dependent rates. We also include perturbatively the effects
of a high-temperature Ohmic heat bath (kBTbath Ⰷ ប⍀, with ⍀being the oscillator frequency) on the oscillator using a
Caldeira-Leggett description18 and the lowest-order Born
diagrams in the self-energy.
Finally, we specialize to the case where the voltage V ismuch larger than ប⍀ / e, so that inelastic tunnel events which
excite the oscillator are not suppressed. For weak tunnelingand small ac frequency ␯, we may then make a Markov
approximation in Eq. (1): U0共t − t⬘兲␳ˆ 共m⬘ ; t⬘兲U†0共t − t⬘兲→ ␳ˆ 共m⬘ ; t兲, which is equivalent to replacing U0共␶兲 with thefull evolution operator U共␶兲. We are assuming that over the
short time scales relevant to tunneling, one can describe thedynamics of the density matrix by its zero-tunneling evolution; the omitted terms here are formally higher order in thetunneling. Fourier-transforming in the m index, ␳ˆ 共k ; t兲⬁= 兺m=−⬁
eikm␳ˆ 共m ; t兲, Eq. (1) becomes

冉 冊兺冉␥0 + ␥
id␳ˆ 共k;t兲 = − 关H0 − F̄共t, ␩兲x̂, ␳ˆ 兴 − i关x̂,兵p̂, ␳ˆ 其兴
dt
ប
ប
−

冉
冉

冊

i␴k

D0 + D共t兲e −1关x̂,关x̂, ␳ˆ 兴兴 +2
ប共 ␶ ⬘兲 2
␴=+,−⫻2D␴共t兲
共␶0 + ei␴␩␶⬘x̂兲␳ˆ 共␶0 + e−i␴␩␶⬘x̂兲
ប2

+i

␥␴共t兲
关␶0␶⬘共ei␴␩ p̂␳ˆ − e−i␴␩␳ˆ p̂兲ប冊+ 共␶⬘兲2共p̂␳ˆ x̂ − x̂␳ˆ p̂兲兴 ,where ␥0 is the intrinsic damping coefficient associated withthe equilibrium bath, D0 = 2M ␥0kBTbath is the corresponding
diffusion constant, and ␴ = + 共−兲 labels contributions fromforward (backwards) tunneling. The detector-dependent diffusion constant D共t兲 = 兺␴D␴共t兲 and damping coefficient ␥共t兲= 兺␴␥␴共t兲 are given by␥␴共t兲 =

冉 冊冉冊

␶⬘ 2 ⌫␴共t,ប⍀兲 − ⌫␴共t,− ប⍀兲ប,22M⍀ ␶0冕⬁

dt sin ␻t

0

共2兲

共3兲

共4兲

while F̄共t , ␩兲 = sin ␩共␶⬘ / ␶0兲兺␴2␴D␴共t兲 / ប is the average backaction force exerted on the oscillator. ⌫±共t , E兲 are the ␶⬘ = 0
finite temperature forward and backwards inelastic tunnelingrates involving an absorbed energy E; these rates are timeindependent in the case of a dc voltage. Note that we haveneglected self-energy terms which renormalize the oscillator
Hamiltonian; these are unimportant in the weak-tunneling
limit we consider.
Equation (2) yields a compact description of the coupled
detector-oscillator system; it is a generalization of an equation first derived (via an alternate approach) by Mozyrsky etal.11 to an arbitrary detector in the tunneling regime, including the possibility of an x-dependent tunnelling phase, a nonlinear junction I–V, a time-dependent bias voltage, and intrinsic oscillator damping. Taking k = 0 yields the equation
for the reduced-density matrix of the oscillator, and (cf. Ref.
11) has the Caldeira-Leggett form for a forced, damped oscillator in the high-temperature regime.18 In what follows,
we focus for simplicity on the case of T = 0 in the tunnel
junction, and on ␩ = 0, which ensures F̄ = 0.
Shot Noise. Equation (2) can be used to calculate the fullcounting statistics of tunneled charge as a function of time.
By focusing on the time-dependence of the reduced second
moment 具具m2共t兲典典 (i.e., variance), it is possible to calculate
the symmetrized frequency-dependent current noise using
the MacDonald formula.19 In the case of an ac bias voltage,
the noise is a function of two times. We focus on the part thatis independent of the average time coordinate, a quantity
directly accessible in experiment. It is given by a modified
version of the MacDonald formula:
SI共␻兲 = 2e2␻冊冉冊

ប2 ␶⬘ 2关⌫␴共t,ប⍀兲 + ⌫␴共t,− ប⍀兲兴,
4 ␶0

冕

2␲

0

d␾
· ⳵t具具m2共t, ␾兲典典,2␲

共5兲

where ␾ is the initial phase of the ac voltage.
dc bias. For a dc biased normal-metal junction at
T = 0, the tunneling rates are given by h⌫␴共t , E兲 = 共␶0兲2共␴eV
Equations
(3)
and
(4)
yield
+ E兲⌰共␴eV+ E兲.
␥ = ប␶⬘2 / 共4␲ M兲 and kBTef f = eV/ 2.11 We find from Eqs. (2)and (5) that the current noise may be written as SI共␻兲= 2e具I典 + ⌬SI, where the first term corresponds to purely Poissonian statistics, and the second term is a correction arisingfrom correlations between the motion of the oscillator andthe number of tunneled electrons:⌬SI共␻兲 =

冕

⬁
4e3V
␻
dt sin ␻t„共2␶0␶⬘兲具具x̂共t兲 · m共t兲典典h0+ 共␶⬘兲2具具x̂2共t兲 · m共t兲典典….

共6兲

Physically, the covariances appearing above arise from thex-dependence of the tunneling probability- if m共t兲 is larger
than average, then it is likely that x共t兲 and x2共t兲 are also
larger than average. These covariances can be calculated directly from Eq. (2), and obey simple classical equations corresponding to a forced, damped harmonic oscillator. Con-

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RAPID COMMUNICATIONSPHYSICAL REVIEW B 70, 121303(R) (2004)

SHOT NOISE OF A TUNNEL JUNCTION…

sider first the contribution from 具具x · m典典 in Eq. (6), which isleading order in ␶⬘:兩⌬SI共␻兲兩1 =冉冊eV ⍀ 共⌬x0兲2
e 3V
Sx共␻兲,
共2␶0␶⬘兲2−hh 4␲ 具x2典

共7兲

where 具x2典 ⯝ eV/ 2M⍀2 is calculated using the stationary so2
␻2兴−1lution of Eq. (2), Sx共␻兲 = 8␥tot⍀2具x2典 / 关共␻2 − ⍀2兲2 + 4␥tot
is the spectral density of oscillator x fluctuations obtained
from Eq. (2), and 共⌬x0兲2 = ប / 共2M⍀兲 is the zero-point uncertainty in the oscillator position. The first term in Eq. (7) isexactly the answer expected (to lowest order in ␶⬘) from a
simple picture of a classically fluctuating junction conductance (i.e. ⌬SI共␻兲 = V2SG共␻兲, where SG共␻兲 is the spectral density of conductance fluctuations, and is in turn determined by
Sx共␻兲). Equivalently, if we think of our junction as an x-to
-I amplifier having a gain ␭ = 2e2V␶0␶⬘ / h, this first term corresponds to simply amplifying up the fluctuations of the oscillator: ⌬SI = ␭2Sx. Equation (7) yields a peak in SI共␻兲 at
␻ = ⍀; keeping only the leading term in V, the ratio of thepeak-height to the background Poissonian noise (i.e., the S/Nratio) is冉 冊冉冊⌬SI共␻ = ⍀兲eV␣2
␶0 2 2MeV= 4␶20
, 共8兲
2 艋4
2e具I典
h␥tot 1 + ␣␶⬘ប2
where ␣2 = ␶⬘2具x2典 / ␶20, ␥tot = ␥0 + ␥. Note that if ␣ is small,there will be no sizeable signature of the oscillator in the
average current (i.e., ␦具I典 / 具I典0 ⯝ ␣2), but there may nonetheless be a large peak in the noise if eV / 共h␥tot兲 is large. The
upper bound in Eq. (8) corresponds to the optimal scenario,
where there is no intrinsic (detector-independent) damping,and ␣ Ⰷ 1. The maximum S/N is determined by eV and thesensitivity ␶⬘ / ␶0, and can be arbitrarily large. Due to thedependence on ␥, the maximum S/N is inversely proportional
to the detector sensitivity ␶⬘ / ␶0. Note the marked difference
from experiments attempting to detect coherent qubit oscillations in the detector current noise,3 where back-action effects limit the S/N to a maximum of 4.
We turn now to the second term in Eq. (7), which is alower-order in V quantum correction to the classical result. It
would appear to cause 兩⌬SI兩1 to vanish in the limit eV→ ប⍀ / 2, 具x2典 → 共⌬x0兲2, i.e., it suppresses a zero-point contribution to 兩⌬SI兩1. (Of course, we cannot rigorously take thislimit, as Eq. (2) is strictly only valid for eV Ⰷ ប⍀.) A similar
result was found for the average current 具I典 in Ref. 11, wherea similar offset term could be traced to the inherent asymmetry between events in which energy is absorbed from theoscillator, versus those in which it is emitted to the oscillator.
In the present case, the quantum correction to the noise in
Eq. (7) can be given a classical interpretation- it arises fromcorrelations between the intrinsic shot noise of the detector,
and the back-action force F̂ acting on the oscillator. If there
are such correlations, we would expect classically:⌬SI共␻兲 = ␭2Sx共␻兲 + 2␭ Re关g共− ␻兲SIF共␻兲兴,

共9兲

where SIF共␻兲 is the symmetrized cross-correlator between
the junction current and back-action force, and g共␻兲 is theoscillator response function. Note that the second term aboveis ⬀V, while the first is ⬀V2. Inversion symmetry forces
Re SIF to vanish;5,6 however, at finite ␻, Im SIF is nonzero.
Consequently, the second term above is nonzero; a direct
perturbative calculation (assuming a thermal state for the oscillator) shows that this term corresponds to the second termin Eq. (7). Thus, quantum corrections to the noise, which
suppress zero-point contributions, can be associated withclassical out-of-phase correlations between the random
back-action force and the intrinsic detector output noise.
Finally, we return to Eq. (6) and examine the contribution
from 具具x2 · m典典, a term which is higher-order in ␶⬘. One finds:关⌬SI共␻兲兴2 =冉

eV ⍀ 共⌬x0兲2
e 3V
共 ␶ ⬘兲 4−hh 2␲ 具x2典
⫻

冕 ⬘

冊

d␻
Sx共␻⬘兲Sx共␻ − ␻⬘兲.
2␲

共10兲

Again, the first term above agrees with the expectation for aclassically fluctuating junction conductance; it yields peaks
in SI at ␻ = 0 and ␻ = 2⍀. The second term is a quantum
correction, completely analogous to that found for 兩⌬SI兩1.
ac bias. We now consider an ac bias voltage Vac共t兲
= V cos共␯t兲, where eV Ⰷ ប␯ , ប⍀. In the limit of small ␯, it ispossible to derive a simple expression for the time-dependent
˜ 共E兲 = 共␶ 兲2E · ⌰共E兲, we havetunneling rates.20 Defining h⌫
0
⬁
n 共n兲
⌫␴共t , E兲 = 兺n=0共1 − ␦n,0 / 2兲␴ ⌫␴ 共E兲cos n␯t, with:⌫␴共n兲 =

冕

␲

冉

d␪

nប␯

兺± 0 ␲ cos共n␪兲⌫˜ eV cos ␪ + E ± 2

冊. 共11兲

Using Eqs. (3) and (4),we find that the damping coefficient ␥of the oscillator is time-independent and identical to that in
the dc case, whereas the diffusion constant is time-dependentand contains higher harmonics of the ac frequency ␯. Writing
D共t兲 = 2M ␥kBTef f 共t兲, we have to a good approximation:eV
kBTef f 共t兲 =
␲

冋兺 冉
eV/ប␯
n=0

冊

册

2共− 1兲ncos共2n␯t兲 − 1 . 共12兲
1 − 共2n兲2

The small but finite photon frequency ␯ prevents higher harmonics from contributing to Tef f ; without it, we would have
simply kBTef f 共t兲 = V兩cos ␯t兩 / 2, which tends to zero twice eachperiod. With the finite cut-off included, the minima of
kBTef f 共t兲 are ⯝ប␯. The time-dependence of Tef f 共t兲 implies
that the position variance 具x2共t兲典 of the oscillator will betime-dependent; as we show, this has a direct influence on
the noise and the average current. For the latter quantity, we
find:
具I共t兲典 =e 2Vcos共␯t兲关␶20 + 共␶⬘兲2具x2共t兲典兴 − ⌬I共t兲,hwhere the quantum correction is approximately ⌬I共t兲⯝ e␥ · sgn关cos ␯t兴. Turning to the noise, we may again decompose SI共␻兲 into a frequency-independent part and a termarising from correlations between x共t兲 and m共t兲: SI共␻兲⬅ 共⍀ / 2␲兲兰20␲/⍀dt̄SI共t̄ , ␻兲 = SIa + ⌬SI共␻兲. For the frequencyindependent contribution SIa, we find:121303-3

RAPID COMMUNICATIONSPHYSICAL REVIEW B 70, 121303(R) (2004)

A. A. CLERK AND S. M. GIRVIN

FIG. 2. Oscillator contribution to SIa, the frequency-ińdependent
part of the shot noise (i.e., second term in Eq. (13)), versus the ac
voltage frequency ␯, for ⍀ / ␥ = 50, ␥0 = 0, and eV Ⰷ ប␯ , ប⍀. The
maximum suppression of this term at ␯ = ⍀ (over its ␯ → ⬁ limit) is
by 8 / 9.

SIa =

冋

冉

4e2 eV 2⍀
␶0 + 共␶⬘兲2 kBTef f 共t兲具x2共t兲典 −共⌬x0兲2
h ␲
2␲

冊册

,

FIG. 3. Full shot noise SI共␻兲 for an ac bias voltage of frequency
␯ = 100 ⍀, including the effects of correlations between x共t兲 andm共t兲 (cf. Eq. (14)). We have chosen eV = 100ប⍀, ␣2 = 1, 共␥0 + ␥兲
= ⍀ / 20, and ␶0 = 0.1. The y axis is scaled by the value of the␻-independent part of the noise.

共13兲
where the bar indicates a time-average. The first term is thestandard result for the shot noise of an ac-biased junction.21
The second term indicates that the time-dependence of
具x2共t兲典 [calculated from Eq. (2)] makes a frequencyindependent contribution to the noise. For ␯ Ⰷ ⍀, 具x2典 responds only weakly to the time-dependence of Tef f 共t兲,
whereas for ␯ ⬃ ⍀, the response becomes appreciable and180 degrees out-of-phase with V共t兲. If in addition ␥0 Ⰶ ␥, one
finds a resulting suppression of the oscillator’s contribution
to SIa; this is shown in Fig. 2. Small resonances also occur
when ⍀ is a multiple of ␯. The oscillator modification of SIa
is not captured by the classical picture of a fluctuating conductance.
Finally, the frequency-dependent contribution ⌬SI共␻兲 tothe noise, which arises from correlations between x共t兲 andm共t兲, takes the simple form:冋 冉 冊册

1ប⍀⌬SI共␻兲 = 兩 ⌬SI共␯ ± ␻兲兩dc 1 + O
4 ±
eV

兺

,

共14兲

where the omitted terms correspond to “quantum corrections” of the sort previously discussed. Without these, Eq.

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the corresponding terms found for dc bias—one finds that thequantum corrections are larger in the ac case by a factor of
4 / ␲. The effect of ⌬SI共␻兲 on the full noise is shown in Fig.
3.
In conclusion, we have presented a fully quantum mechanical calculation of the frequency-dependent current
noise of a tunnel junction displacement detector, for both thecases of dc and ac voltage bias. The oscillator can lead tolarge effects in the shot noise, even if the coupling to thedetector is weak; moreover, these effects cannot be completely described using a classical picture of a fluctuating
junction conductance.
We thank Konrad Lehnert and Florian Marquardt for useful discussions; this work was supported by the NSF under
grant NSF-ITR 0325580, and by the W. M. Keck Foundation.

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