PRL 105, 217206 (2010)

PHYSICAL REVIEW LETTERS

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Scattering Approach to Backaction in Coherent Nanoelectromechanical Systems
Steven D. Bennett, Jesse Maassen, and Aashish A. Clerk
Department of Physics, McGill University, Montreal, Quebec, Canada, H3A 2T8(Received 17 September 2010; published 17 November 2010)
We present theoretical results for the backaction force noise and damping of a mechanical oscillator
whose position is measured by a mesoscopic conductor. Our scattering approach is applicable to a wideclass of systems; in particular, it may be used to describe point contact position detectors far from theweak tunneling limit. We find that the backaction depends not only on the mechanical modulation oftransmission probabilities, but also on the modulation of scattering phases, even in the absence of amagnetic field. We illustrate our general approach with several simple examples, and use it to calculate thebackaction for a movable, Au atomic point contact modeled by ab initio density functional theory.
DOI: 10.1103/PhysRevLett.105.217206

PACS numbers: 85.85.+j, 72.70.+m, 73.23.b

Quantum mechanics requires that any detector used tomeasure an object’s position unavoidably exerts a backaction force, imposing a fundamental limit on continuous
position detection [1,2]. Recent experiments with nanoelectromechanical systems (NEMS) have come remarkably close to realizing this limit by using quantum
electronic conductors as position detectors of nanomechanical oscillators [3–5]. In these systems, position detection is achieved using the influence of the mechanics on
the current through the conductor; thus, it is natural toassociate backaction with the position sensitivity of theelectron transmission probability. This is indeed the picturethat emerges from theoretical studies in the limit of weaktunneling [6–8]; however, several recent experiments arefar from this limit [5,9–11], and it is not clear that the weaktunneling results apply.
In this Letter, we study the backaction of a mesoscopic
position detector using a general noninteracting scattering
approach that is not limited to the weak tunneling limit.
Scattering theory has been used extensively to study various aspects of mesoscopic conductors, and we adapt it here
to study the backaction heating and damping of a mechanical oscillator coupled to a conductor. Surprisingly, we findthat backaction arises not only from transmission probabilities, but also from the position sensitivity of scattering
phases, and we present several simple but illustrative examples where the phases play a pronounced role. We emphasize that these phases may be important despite intact
time reversal symmetry, which we assume throughout,unlike Aharanov-Bohm phases due to a magnetic field [8].
Finally, we apply our general results to calculate the backaction from an atomic point contact (APC) between Au
electrodes, using a scattering matrix obtained from density
functional theory (DFT).
Our approach significantly extends the seminal work of
Yurke and Kochanski, who first considered force noise in atunnel junction using a scattering approach [12]. Unlike
their work, which is limited to particular scattering potentials, we rely only on general properties of the scattering0031-9007=10=105(21)=217206(4)matrix. As a result, we can describe a wide class of
systems, including arbitrary scattering potentials, various
forms of electromechanical coupling, and multichannel
scatterers. Moreover, we calculate not only the backaction
force noise, but also the backaction damping, which isimportant in experiments (e.g. it is the basis of backaction
cooling [4]) and up to now has not been dealt with in the
scattering approach.
Scattering approach.—We consider a two-terminal device consisting of a coherent scattering region coupled toleft and right leads, each of which supports N transverse
modes. Electrons are scattered by a potential Uð~re ; xÞ,
where r~e ¼ ðxe ; ye ; ze Þ is the electron position, and thepotential depends on the position x of a mechanical oscillator. Incoming and outgoing waves are related by thescattering matrix sðxÞ, which depends on x throughUð~re ; xÞ. We will show that a knowledge of s and @s=@x
is sufficient to calculate the backaction.
For the usual experimental regime of weak electromechanical coupling, the change in the electronic potential
due to small changes in x is generally linear and may be^ where the force on the oscillator iswritten R
H int ¼ xF,
^ r~e Þ@Uð~re ; xÞ=@x, and ð^ r~e Þ is the electronF^ ¼  d~re ðdensity operator. By relating small and slow changes in thepotential, Uð~re ; xÞ, to the parametric derivative of the scattering matrix [13–15], we can express F^ in the scattering
state basis asZXZd d0 a^ y ðÞW ð; 0 Þa^  ð0 Þ;
(1)
F^ ¼


where a^  ðÞ destroys a scattering state of energy  incident
in the lead and transverse mode indexed by , and1@sð; xÞsy ð; xÞWð; Þ ¼
:
(2)
2i
@x
x¼0
We require only the diagonal-in-energy part of W since^ thiswe focus on the zero frequency noise properties of F;
is sufficient for the experimentally relevant case when the217206-1

Ó 2010 The American Physical Society

PRL 105, 217206 (2010)

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PHYSICAL REVIEW LETTERS

oscillator period is much longer than time scales in the
conductor. Derivatives of the scattering matrix similar toEq. (2) are familiar from studies of charge noise [16] andparametric pumping [17]. Here we use the parametricderivative with respect to x to calculate the backaction on
the oscillator directly in terms of the scattering matrix,
without the need for detailed knowledge of Uð~re ; xÞ andð~^ re Þ in the scattering region. In the following we work tolowest order in H int , valid for weak coupling.
Fluctuations of the backaction force cause momentum
diffusion and heating of the oscillator. Heating is determined by the classical, frequency-symmetric part of thebackaction force noise, SF ½! ¼ ðSF ½! þ SF ½!Þ=2,
where the quantum noise spectral density is SF ½! ¼
R^ Fð0Þi
^and averages are taken with respect to
dtei!t hFðtÞ
the uncoupled conductor [18]. These averages are easily
taken using Eq. (1), and the backaction heating is directly
determined by W. The zero frequency force noise is(kB ¼ 1, SF  SF ½0)XZd trfW W gf ð1  f Þ;(3)
S F ¼ 2@sð; xÞ ¼ ei

pﬃﬃﬃﬃﬃﬃ i
Repﬃﬃﬃﬃﬃ
i T

pﬃﬃﬃﬃﬃ !
i ﬃTpﬃﬃﬃﬃﬃ;
Rei(5)

where T (R ¼ 1  T ) is the transmission (reflection) probability,  is the overall scattering phase, and parametrizes
broken inversion symmetry; i.e. ¼ 0 for an inversionsymmetric conductor. In general, all of the scatteringparameters depend on x through the potential, Uðr~e ; xÞ.
Inserting Eq. (5) into Eq. (3) we obtain the symmetrized
force noise for a single channel,@ ð@T =@xÞ2
eV
S F ¼eV ð1 þ RT  Þ coth2 4RT
2Tel

2T
(6)
þ ð þ R2  Þ el ;eVwhere the trace is over transverse modes, assumed to be thesame in both leads, and the matrixes W are the N  N
blocks of W in Eq. (2), which may be  dependent. The
Fermi functions are f ¼ ð1 þ eð Þ=Tel Þ1 , where  isthe chemical potential in lead  and Tel is the electronic
temperature.
In addition to heating, the oscillator also experiences
backaction damping as a result of energy exchange withthe conductor. The damping rate is given by the quantum,
asymmetric-in-frequency part of the force noise, ½! ¼
ðSF ½!  SF ½!Þ=2M@!, where M is the oscillator
mass. Taking the ! ! 0 limit, we find@f2@ X Zd trfW W gf 
:
(4)
¼
M 
@

where V is the bias, and the phase terms enter as@ =@x 2
 ¼ 4RT;(7)@T =@xfor ¼ , . In the limit eV  Tel , the first term (independent of  and  ) in Eq. (6) represents the expected,
quantum-limited backaction of our position detector; it issimply the sensitivity of a position measurement made bymonitoring the current, and reflects the fact that a stronger
measurement leads to increased backaction. This term
scales as the square of the measurement gain, IF /@T =@x, and inversely with the shot noise in the current,SI ¼ e2 VRT =2@; in the limit T  1 it reproduces thewell-known result obtained from a tunnel Hamiltonian
calculation [6,7]. The second term in Eq. (6) is independentof @T =@x and is thus not directly related to a measurementof the current. Instead, it results from the oscillator’s modulation of the phase . This phase contribution to SF is
proportional to RT and thus vanishes when T  1. The
remaining two terms ( / Tel =eV) are also independent of@T =@x and represent additional thermal noise at finite
Tel =eV.
The damping for a single channel is

By considering the ratio of SF ½! to ½!, one can associate a frequency-dependent effective temperature Teff ½!
with the backaction; this amounts to using the standard
fluctuation-dissipation relation to define the effective temperature at each frequency from the system’s force noiseand damping [2,6,18]. Teff ½! characterizes the conductor
as an effective thermal environment. In the ! ! 0 limit,
the relation is simply Teff  SF =2M. If backaction dominates over intrinsic sources of dissipation, Teff corresponds
to the physical temperature of the oscillator.
Single channel.—We first consider the case of singlechannel leads. For simplicity, we also focus on the limit ofsmall applied bias, ignoring the possible energy dependence of s. We assume time reversal symmetry (i.e. nomagnetic field), but allow for broken left-right inversion
symmetry. In this case the scattering matrix may be
parametrized as@ ð@T =@xÞ2
ð1 þ  þ R Þ:(8)2M 4RT
In the small bias limit,  is strictly positive and independentof Tel . Similar to SF , the first term in Eq. (8) is the backaction associated with a measurement of the current andreduces to the tunnel Hamiltonian result in the limit T 
1. More interestingly, the second and third terms correspondto corrections due to scattering phases; unlike SF , thesephase contributions to  are present even for a symmetricdetector and, as we will see, do not necessarily vanish in the
weak tunneling limit. The overall phase  is directly connected to the density of states in the scattering region via theFriedel sum rule [19]. An x-dependent  implies that themechanical oscillator can change the scattering-induced
electronic density of states; this means that the total electronic free energy becomes x dependent, resulting in a force
whose quantum noise contributes to damping.



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PHYSICAL REVIEW LETTERS

Equations (6) and (8) show that the backaction properties of a general conductor cannot simply be extrapolated
from the weak tunneling limit; scattering phases play a rolein both the heating and damping of the oscillator. Further,
the phases can have a dramatic influence on the effective
backaction temperature Teff of the detector. For a single
channel, using Eqs. (6) and (8), in the limit Tel  eV
we findeV
1 þ RT 
Teff ¼:(9)
2 1 þ  þ RIf the mechanical motion does not modify the scattering
phases, then we simply obtain the tunnel Hamiltonian
result [6,7], Teff ¼ eV=2, independent of T . However, inthe more general case including the backaction from scattering phases, Teff is not solely determined by the voltage.
The phase corrections always decrease the effective temperature; they arise from the diagonal elements of W,
which correspond to transitions between scattering statesin the same lead. At Tel ¼ 0 such transitions can only occur
if an electron absorbs energy, because the scattering statesin each lead are filled up to the Fermi level. Thus, phasecorrections lead to increased absorption of energy from theoscillator, lowering Teff . Including a nonzero lead temperature Tel , one finds that Teff can be lowered to a minimum
value of Tel ; as Tel  eV, this could still be quite useful.
Square potential barrier.—To demonstrate that backaction from scattering phases plays a role even in the simplest
scattering model, we calculate the backaction for a onedimensional symmetric square barrier potential whose
width depends on the oscillator position. The force noisefor this model was first considered in Ref. [12]; our general
method further provides  and Teff and allows us to identify the role of scattering phases. Incoming electrons ofwave vector k and energy  are scattered in one dimension
by a square potential barrier of height U0 and width w ¼
L þ x. The inverse decay length of the wave function
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
under the barrier is ¼ 2me ðU0  EÞ=@. It is straightforward to find T and  as functions of U0 and w, and ¼ 0
due to inversion symmetry. We obtain
1

4k2 2
 ¼ 1 þ 2;(10)
ðk  2 Þ2 T
and the phase terms become important when T  1. Fora high but narrow barrier (U0  , L  1), we find[via Eq. (9)] that Teff may be reduced by up to a factor of
2 compared to the tunnel Hamiltonian result of eV=2. For
a low barrier (U0  ), we find Teff ! Tel .
Resonant level model.—We now apply our general results to a prototypical resonant level model (RLM), wherea single electronic level of energy d is connected to theleft (right) lead via the tunneling rate L (R ). If ddepends on the position of a mechanical oscillator[see Fig. 1(a)], one has the electromechanical analog of adispersively coupled optomechanical system [20], and a

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FIG. 1 (color online). Schematic setups and backaction damping for RLM with dispersive (a,b), dissipative (c,d), and shuttle(e,f) mechanical coupling. In all plots, the full backaction damping (red solid line) and the damping without phase corrections(blue dashed line) are shown, with T (black dotted line) also
shown for reference. We took L ð0Þ ¼ R ð0Þ.

simple model of quantum-dot–based NEMS studied inrecent experiments [21,22]. Beginning from the scattering
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
matrix for the RLM, s ¼   i@   =ð  d þ
i@=2Þ, where  ¼ L þ R , and assuming linear coupling, we obtain  / ð=L R Þ2 T 2 [23]. We also find ¼ 1 þ R , independent of the tunneling rates andthe detuning of the incident electron energy  from d .
Comparing with Eq. (8), this implies that the phases play a
crucial role: the x dependence of the overall scattering
phase  always accounts for half of the damping, as seenin Fig. 1(b). Further, in the limit of asymmetric tunnelingrates we find R  1, and the damping is almost entirely due to the combined  and . Also striking is the
cotunneling limit, where the detuning is large compared to
the level broadening, i.e. j  d j  @. In this limit
tunneling is suppressed, T  1, and the level charge
only fluctuates virtually; as a result, one might expectthat the system is equivalent to a single junction in the
weak tunneling limit, and that  should be given by thetunnel Hamiltonian result, i.e., the first term in Eq. (8).
However, this is not the case: due to phase corrections, thedamping is twice the tunnel Hamiltonian result [see theinset of Fig. 1(b)]. This shows that the phases can play arole even when T is small. Note that backaction in thismodel was recently studied theoretically using a path integral approach [24,25], although backaction due to phases
was not discussed.
Our general theory also allows us to consider variationsof the above RLM, where the mechanical position modulates the tunneling rates L and R . This is the electromechanical analog of a dissipatively coupled optomechanicalsystem [26], and could be achieved experimentally using aquantum dot coupled to two leads via tunnel junctions,with the tunneling rates modified by an on-board [5] oroff-board [10] mechanical oscillator. First, we consider
a setup where only the left tunneling rate is x dependent[see Figs. 1(c) and 1(d)]. In this case, interference between

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Conclusions.—We have presented a scattering approachto backaction in NEMS and demonstrated the importanceof backaction from scattering phases. This work is particularly relevant to NEMS based on quantum or atomic point
contacts which are often far from the weak tunneling limit.
Our results may also be easily extended to describe strong
electromechanical coupling in the low oscillator frequency
limit, by making an adiabatic approximation such that thenoise spectra of F^ effectively become x dependent [18].
This work was supported by the NSERC, FQRNT, and
CIFAR.
FIG. 2 (color online). (a) APC in a 5  5 atom Au (100)quantum wire. (b) Conductance through the APC versus gap
size (for small L, several channels contribute to transport and G
exceeds e2 =h [23]). (c) Effective backaction temperature versus
gap size. For small L, Teff < eV=2 due to phase corrections. The
Au-Au interatomic distance in bulk gold is 2.87 Å.

resonant charge fluctuations (on the level) and nonresonant
charge fluctuations (in the leads) results in a Fano line
shape and suppression of  at zero detuning [23], similar
to the optomechanical case [26]. Second, we consider
mechanical coupling to both tunneling rates with opposite
sign, corresponding to a quantum shuttle [see Figs. 1(e)and 1(f)]. Here we find  / ð=L R Þ2 T ; moreover, all of
the damping is due to the scattering phases, since (for L ¼
R ) the transmission has no linear dependence on x [23].
Atomistic model.—While the above examples show thatphases contribute to backaction in simple model potentials,
our approach allows us to investigate phase contributions
in fully atomistic calculations of mesoscopic conductors.
We demonstrate this by applying our theory to an APC
using the scattering matrix obtained from DFT [27]. We
model the APC as a single-atom constriction in a 5  5atom Au quantum wire (see Fig. 2), and take x to modify
the gap size of the APC to L þ x. This geometry is motivated by recent experimental setups using an APC [5] or ascanning tunneling microscope [9] with one mechanicallycompliant electrode. We approximate the surface electrodeof experiments by the flat 5  5 edge of the wire on theright; this is justified since the transport properties of the
APC are expected to be dominated by the few atoms
closest to the tip. We find 11 scattering channels contributing to transport, consistent with recent ab initio studies ofsimilar Au wires [28]. After obtaining s and @s=@x [23],we calculate the backaction using Eqs. (3) and (4), assuming Tel  eV. We find that phase corrections are importantwhen the APC transmission deviates from the weak tunneling limit; it leads to a significant reduction in Teff fromthe tunnel Hamiltonian result of eV=2, as seen in Fig. 2.
While transmission properties are often studied using DFT,
an important feature of our calculation is our explicit use of
the scattering phases obtained from an atomistic calculation of a quantum electronic device.

[1] C. M. Caves, Phys. Rev. D 26, 1817 (1982).
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291 (2003).
[4] A. Naik et al., Nature (London) 443, 193 (2006).
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Phys. Rev. Lett. 98, 096804 (2007).
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[7] A. A. Clerk and S. M. Girvin, Phys. Rev. B 70, 121303(R)(2004).
[8] C. B. Doiron, B. Trauzettel, and C. Bruder, Phys. Rev.
Lett. 100, 027202 (2008).
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[10] M. Poggio et al., Nature Phys. 4, 635 (2008).
[11] J. Stettenheim et al., Nature (London) 466, 86 (2010).
[12] B. Yurke and G. P. Kochanski, Phys. Rev. B 41, 8184
(1990).
[13] M. Buttiker, J. Phys. Condens. Matter 5, 9361 (1993).
[14] I. L. Aleiner, P. W. Brouwer, and L. I. Glazman, Phys. Rep.
358, 309 (2002).
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[16] M. H. Pedersen, S. A. van Langen, and M. Büttiker, Phys.
Rev. B 57, 1838 (1998).
[17] P. W. Brouwer, Phys. Rev. B 58, R10135 (1998).
[18] A. A. Clerk and S. Bennett, New J. Phys. 7, 238 (2005).
[19] J. Friedel, Philos. Mag. 43, 153 (1952).
[20] F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin,
Phys. Rev. Lett. 99, 093902 (2007).
[21] G. A. Steele et al., Science 325, 1103 (2009).
[22] B. Lassagne et al., Science 325, 1107 (2009).
[23] See supplemental material at http://link.aps.org/
supplemental/10.1103/PhysRevLett.105.217206 for details of the RLM and the APC model.
[24] D. Mozyrsky, I. Martin, and M. B. Hastings, Phys. Rev.
Lett. 92, 018303 (2004).
[25] R. Hussein et al., Phys. Rev. B 82, 165406 (2010).
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102, 207209 (2009).
[27] J. Taylor, H. Guo, and J. Wang, Phys. Rev. B 63, 245407(2001).
[28] S.-H. Ke, H. U. Baranger, and W. Yang, J. Chem. Phys.
123, 114701 (2005).

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PHYSICAL REVIEW LETTERS

Erratum: Scattering Approach to Backaction in Coherent Nanoelectromechanical Systems[Phys. Rev. Lett. 105, 217206 (2010)]
Steven D. Bennett, Jesse Maassen, and Aashish A. Clerk(Received 25 April 2011; published 13 May 2011)
DOI: 10.1103/PhysRevLett.106.199902

PACS numbers: 85.85.+j, 72.70.+m, 73.23.b, 99.10.Cd

In Ref. [1], Eq. (4) for the backaction damping is valid only when the scattering matrix sð; xÞ is independent of energy
over the range of energies contributing to scattering; we erroneously stated that this expression was correct for general,energy-dependent scattering. Using the definition of the W matrix given in Eq. (1) of Ref. [1], one can easily show thatthe full expression for the low-frequency, linear-response backaction damping (valid for arbitrary energy-dependent
scattering) is
8
9







<X
=
@ Z
@f
@
!
!
!
!
¼
dtr
WLR   ;  þ WRL  þ ;  W W : (1)þ 2ðfL  fR Þ
:
M
@!
2
2
2
2 !¼0 ;
@We stress that this correction has no impact on the subsequent results of Ref. [1], as we exclusively considered the limit ofsmall drain-source voltages in which the energy dependence of the scattering matrix plays no role. It is worth briefly
outlining the different origins of the terms in Eq. (1). The first term, proportional to W W evaluated at energy ,
corresponds to the damping discussed in Ref. [1]. This term arises from the increase in the number of scattering transitions
contributing to the force noise when an electron in the conductor absorbs energy ! from the mechanical degree of freedom(and conversely, the decrease if an electron emits energy to the mechanics). Because of this asymmetry, the conductor
favors absorption of energy and this part of the damping is always positive. In contrast, the term in Eq. (1) proportional tofL -fR constitutes a nonequilibrium contribution originating from the energy dependence of the matrix elements of theforce operator between scattering states at different energies. This intrinsic energy dependence of the matrix elements may
favor either absorption or emission of energy resulting in positive or negative damping. This contribution was recently
presented in Ref. [2] for a general model where the electronic system is a multilevel, noninteracting quantum dot.
It is interesting to note that, following the lines of Ref. [3], it is not possible to relate the nonequilibrium term in  to the‘‘frozen’’ scattering matrix sð; xÞ; one also needs knowledge of the form of the wave functions in the scattering region.
Nonetheless, one can easily identify general classes of systems where it plays no role. We find that the nonequilibrium termalways vanishes for an inversion-symmetric scattering potential, as well as for the single-resonant-level models considered
in Ref. [1]. The nonequilibrium damping terms thus play no role (even at finite bias voltage) for the examples considered inRef. [1].
We thank F. von Oppen for drawing our attention to the incorrect presentation of Eq. (4) in our original paper.
[1] S. D. Bennett, J. Maassen, and A. A. Clerk, Phys. Rev. Lett. 105, 217206 (2010).
[2] N. Bode, S. V. Kusminskiy, R. Egger, and F. von Oppen, arXiv:1103.4809.
[3] F. T. Smith, Phys. Rev. 118, 349 (1960).

0031-9007=11=106(19)=199902(1)199902-1

Ó 2011 American Physical Society