Sensitivity of the mixing current technique to detect nano-mechanical motion

# Sensitivity of the mixing current technique to detect nano-mechanical motion

Yue Wang    Fabio Pistolesi Univ. Bordeaux, LOMA, UMR 5798, Talence, France.
CNRS, LOMA, UMR 5798, F-33400 Talence, France.
July 2, 2019
###### Abstract

Detection of nano-mechanical displacement by transport techniques has reached high level of sensitivity and versatility. In order to detect the amplitude of oscillation of nano-mechanical oscillator a widely used technique consists to couple this motion capacitively to a single-electron transistor and to detect the high-frequency modulation of the current through the non-linear mixing with an electric signal at a slighltly detuned frequency. The method known as current-mixing technique is employed in particular for the detection of suspended carbon nanotubes. In this paper we study theoretically the limiting conditions on the sensitivity of this method. The sensitivity is increased by increasing the response function to the signal, but also by reducing the noise. For these reasons we study systematically the response function, the effect of current- and displacement-fluctuations, and finally the case where the tunnelling rate of the electrons are of the same order or larger of the resonating frequency. We find thus upper bounds to the sensitivity of the detection technique.

###### pacs:
85.85.+j,73.23.-b,73.23.Hk

## I Introduction

Nano-electromechanical systems have great potentials as ultra-sensitive detectors for several physical quantities. Recent advances allowed to reach record sensitivity in mass sensing.Ekinci et al. (2002); Lassagne et al. (2008); Chaste et al. (2012) This has been possible by the detection of the frequency shift of ultralight oscillators when an additional mass is attached to it. Other exemples concern the detection of the tiny magnetic field generated by nuclear spins. This can be done by the opto-mechanical detection of the force generated by the magnetic dipoles,Mamin et al. (2007) but also with electro-mechanical means,Poggio et al. (2008) or by coupling to two-level systes.Rabl et al. (2009); Arcizet et al. (2011); Puller et al. (2013)

The force sensitivity of the device is then the limiting factor for the sensitivity, and again recent advances showed that it is possible to obtain record force sensing with carbon-nanotube oscillators.Moser et al. (2014, 2013) At the same time nano-mechanical oscillators can be so small that interaction between electronic and mechanical degrees of freedom may lead to new and unexpected phenomenaBlanter et al. (2004, 2005); Armour et al. (2004); Doiron et al. (2006); Pistolesi and Shekhter (2015) like the blockade of the currentMozyrsky et al. (2006); Koch and von Oppen (2005); Pistolesi and Labarthe (2007); Pistolesi et al. (2008), coolingPistolesi (2009); Zippilli et al. (2009); Stadler et al. (2014) or unusual mechanical response.Micchi et al. (2015, 2016)

In order to exploit nanomechanical resonators, or to study their properties, detection of mechanical motion is crucial. Most detection methods exploiting electronic transport are based on the high sensitivity of single-electron transistors (SET) to a variation of the gate charge. By coupling capacitively the oscillator to the gate of the SET it is possible to detect the motion of the oscillator with a high accuracy.Blencowe and Wybourne (2000) The method has been used also to cool the oscillator by the back-action of the electronic transport.Naik et al. (2006) The main difficulty of the method stems from the high frequency character of the oscillator motion that is typically in the 100 MHz-1 GHz range. Due to the high impedance of the SET, it is more convenient to down-convert the signal to lower frequency before extracting it. This can be achieved by non-linear mixing the mechanically generated modulation with a second high-frequency signal injected between source and drain. The signal at the difference of the two frequencies can be extracted and measured. To our knowledge, for nanomechanical resonators this method was implemented in metallic SET by the group of A. Cleland back in 2003.Knobel and Cleland (2003) It was later adapted to the detection of carbon nanotube by the group of P.L. McEuen.Sazonova et al. (2004) It then became the method of choice for carbon nanotubes, leading to several breakthroughs: the observation of the first single-electron backaction effects in carbon nanotubes,Lassagne et al. (2009); Steele et al. (2009) ultrasensitive mass detection,Lassagne et al. (2008); Chaste et al. (2012) the detection of the charge response function in quantum dot,Meerwaldt et al. (2012) the detection of magnetic moleculesGanzhorn and Wernsdorfer (2012); Ganzhorn et al. (2013) and the observation of decoherence of mechanical motion.Schneider et al. (2014) The same method can also be implemented by frequency modulation.Gouttenoire et al. (2010) It is clear that the technique is powerful and that it will continue to be used both for fundamental research and for applications. The question we want to address in this paper is which is the ultimate resolution that can be reached with this kind of detection. In order to do this we investigated three main issues. The first one is how to optimize the response function, that is the quantity , where is the measured signal, the mixing current, and the amplitude of the mechanical oscillation. The second one is to study the effect of current and mechanical fluctuations. These contribute to the fluctuation of the measured signal and in the end are at the origin of the signal to noise ratio. The third is to consider the case of a mechanical oscillator with a resonating frequency faster than the typical tunneling rate of the electrons . We will develop a theory of transport to obtain the mixing current for any ratio . The case of a metallic and single-electronic level SET will be considered in details and explicit expressions will be given.

The paper is structured as follows: Section II gives an introduction to the mixing technique and provides the expression of the detector gain . Section III analyze current and mechanical fluctuations giving general expressions. Section IV provides a general theory for the detector gain when the resonator frequency is comparable or larger than the typical tunnelling time. Section V and VI gives the explicit expressions for the response function and for the noise in the case of a metallic and a single electron SET. Finally Section VI gives our conclusions.

## Ii Mixing technique and response function

Let us begin by describing the typical system used to measure the oscillation amplitude of a mechanical oscillator by detection of the mixing current.Knobel and Cleland (2003); Sazonova et al. (2004); Lassagne et al. (2009); Steele et al. (2009) As shown in Fig. 1 a conducting oscillator is capacitively coupled to the central island of a single-electron transistor: its displacement modulates thus the the gate capacitance , where is the displacement of the oscillator.

We assume the presence of a single mechanical mode whose displacement is parametrized by , a generalized coordinate with the dimensions of a length. We will consider that the SET is operated in the incoherent transport regime valid for , where is the electron tunnelling rate and the temperauture ( and are the reduced Planck constant and the Boltzmann constant, respectively). This is the standard case for nano-mechanical devices. The current through the device can be obtained by using the Master equation and in general it can be expressed as a function of the source-drain bias voltage and on the gate charge , where is the gate voltage (see Appendix A for a short derivation). The current reads thus:

 I=I(V,ng). (1)

In this section we want to obtain the current response of the system when both and are modulated at two slightly different frequencies and , both much smaller than . We write

 Vg(t)=Vg0+Vg1(t),V(t)=V0+V1(t), (2)

where and . Choosing close to the mechanical resonating frequency allows to drive the resonator, since the modulation of the gate voltage modulates the charge on the suspended part and thus induces an oscillating force [see also Eq. (18) in the following]. For small driving amplitude the oscillator responds linearly to the external drive:

 x(t)=xmcos(ω1t+ϕ), (3)

where we always measure from its equilibrium position. (Note that in general and depend on the driving frequency .) The modulation of induces thus the following modulation of at linear order in the driving:

 ng(t)=ng0+CgVg1ecos(ω1t)+C′gxmVg0ecos(ω1t+ϕ), (4)

where . It is convenient to introduce a length scale by defining . From geometric considerations has to be of the order of the distance of the gate from the oscillator, thus typically undreds of nm. The fluctuting part of can then be written as

 ng1(t)=ng0[Vg1Vg0cos(ω1t)+xmLcos(ω1t+ϕ)], (5)

where . The mechanical term has a strong frequency dependence close to the mechanical resonance, and can thus be distinguished by the back-ground electrostatic term . The two contributions to the modulation of the gate charge can be combined in a single cosine term:

 ng1(t)=ng1cos(ω1t+φ). (6)

Assuming now that the oscillator frequency , and thus also and , are much smaller than the typical tunneling rate , one can use Eq. (1) to obtain the time dependent current in presence of time-dependent and . For small modulation amplitude we Taylor expand Eq. (1) to second order in and obtaining

 I(t)=I(V0,Vg0)+∂I∂VV1(t)+∂I∂ngng1(t) +12∂2I∂V2V21(t)+∂2I∂V∂ngV1(t)ng1(t)+12∂2I∂ngn2g1(t)+…

Only the term proportional to has a component that oscillates at the frequency . This signal can be extracted by a standard lock-in technique that essentially allows to measure the quantity :

 Icmx=∫Tm0dtTmI(t)cos[ωΔt]. (8)

The other quadrature with is defined in a similar way. Averaging over a long measurement time one obtains:

 Icmx = V14∂2I∂V∂ng[CgVg1+C′gVg0xmcosϕ], (9) Ismx = −V14∂2I∂V∂ngC′gVg0xmsinϕ. (10)

The detector gain with respect to the two quadrature of is thus:

 λ=14e∂2I∂V∂ngC′gVg0V1. (11)

It measures the sensitivity of the mixing current signal with respect to the two quadratures of . This quantity depends on the particular bias conditions of the SET, and will be studied in some details in Section V and VI for two explicit models. Note also that in order to obtain we need only the static expression for the current. This assumes that the electronic mechanism is much faster than the time dependence of the driving. In order to describe the case of a fast oscillator (to be discussed in Section IV we will need a detailed description of the charge dynamics, and the response function will be no more expressed only in terms of derivatives of the static non-linear current voltage characteristics.

## Iii Effect of current and displacement fluctuations

Expression (11) assumes a deterministic evolution of both the current and the displacement of the oscillator . In practice both quantity fluctuate, the first due to shot or thermal noise, and the second due to stochastic fluctuations induced either by the bias voltage or by the thermal fluctuations. In general one can then write the value of in a specific time region as follows:

 (Icmx)n=∫(n+1)TmnTm[I(t)+δI(t)]cos(ωΔt)dt, (12)

(we write the expression for , the one for is similar) where and are the stochastic and deterministic (in phase with the external drive) part, respectively. We can define the time dependent mixing current as , where stands here for the integer part of . In terms of that the spectral density of the fluctuation of reads:

 Smx(ω)=∫+∞−∞dteiωt[⟨Icmx(t)Icmx(0)⟩−⟨Icmx⟩2]. (13)

We assume that the measuring time is much longer than any correlation time of the quantity . Different sections of the measurement time are thus uncorrelated and we can write:

 Smx(ω) = ∫Tm0dteiωt∫Tm0dt1Tm∫Tm0dt2Tm (14) cos(ωΔt1)cos(ωΔt2)⟨δI(t1)δI(t2)⟩.

Defining (the numerical factor 2 is conventional for the current-noise spectrum) we have

 Smx(ω)=14SI(ω=0)(eiωTm−1)iωTm≈14SI(ω=0). (15)

Thus the mixing-current low-frequency noise is given simply by the low-frequency current noise spectrum . The factor of 4 comes from a different definition of the correlation functions and from the fact that we are collecting a single quadrature. The current noise can have different sources, we consider in the following the two main ones.

### iii.1 Shot-noise and thermal current fluctuations

The current fluctuates due to the discrete nature of the the charge. This is characterized by the current-spectral function (for time-independent bias and gate voltages):

 SshotI(ω)=2∫dteiωt⟨δI(t)δI(0)⟩, (16)

where . For the case of a SET the current spectral function is well known.Korotkov (1994) As shown there it has a frequency dependent part at low frequency on the scale of the typical tunneling rate . This implies that the correlation function is short ranged with respect to the measuring time . Actually it is typically even short ranged with respect to the time dependence of and of the or potentials. Its value can thus be obtained adiabatically, by assuming these parameters to be static. We only need its low frequency part that can, in general, be expressed in terms of the Fano factor and the current :

 SshotI(ω=0)=2FIe (17)

where depends on the details of the SET. In the tunnelling limit of uncorrelated tunneling , in most other cases the Fano factor is typically of the order of 1.

### iii.2 Displacement fluctuations

The electrons that cross the structure modify the charge on the gate that in turn modifies the force acting on the oscillator. This stochastic force, that has the same origin of the current-shot noise, induces fluctuations of the displacement, that changes in a much slower way, since the oscillator responds to an external force on the time scale given by its damping coefficient .Armour et al. (2004); Armour (2004); Usmani et al. (2007); Pistolesi and Labarthe (2007) In order to keep the assumption that different averages over the measuring times are uncorrelated one needs . In principle, for very high- resonators the approximation should be reconsidered.

Let’s begin by considering the force acting on the oscillator as a consequence of a variation of the charge on the gate. A recall of the basic expressions for the electrostatic energy is given in the Appendix A and Fig.3 there shows the electrical scheme. The force acting on the oscillator is given by the derivative of the electrostatic energy performed at constant charge:

 F=−Q2g∂∂x12Cg(x)=Q2gC′g2C2g, (18)

where is the charge on the gate voltage (Fig.3). The fluctuation of the force due to fluctuation of reads thus:

 δF(t)=QgC′gC2gδQg(t). (19)

In general the variation of the charge on the gate is proportional to the variation of the charge on the central island of the SET. By an elementary electrostatic calculation (see Appendix A) , where is the sum of the capacitances of the central island to all the electrodes and is the total charge on the island. In conclusion one finds that

 δF(t)=F0δn(t) (20)

with

 F0=QgeC′gCgCΣ=2QgeECL (21)

the force acting on the oscillator when an electron is added to the dot and with the Coulomb energy of the SET. Note that is a crucial parameter, since it constitutes the electro-mechanical coupling constant.Pistolesi and Labarthe (2007); Micchi et al. (2015) One can estimate the typical value of : 10100, K, nm, thus - N.

The correlation function of the stochastic force acting on the resonator [] is thus simply proportional to the correlation function of the charge on the island []:

 SF(t)=F20Sn(t), (22)

that can be calculated by the standard method of the master equation. For the case of a metallic dot see for instance Refs. Pistolesi and Labarthe, 2007; Weick et al., 2011. Its Fourier transform has a Lorentzian form with a width on the scale of . Thus this force act as a white noise on the slow oscillator.

Let’s now turn to the displacement correlation function. In order to evaluate it we use a simple Langevin approach.Blanter et al. (2004); Mozyrsky et al. (2006) We neglect the driving, since we are interested in the low frequency response. The Langevin equation reads

 m¨x+mγ˙x+kx=δF(t), (23)

where is the (effective mass) of the oscillator mode considered, the damping coefficient, and the effective spring constant. The stochastic force generated by the electrons is also at the origin of the damping coefficient. In general other effects participate, but close to the degeneracy point of the SET, when the current is maximal, the electronic contribution to the damping can dominate, as observed experimentally in Ref. Ganzhorn and Wernsdorfer, 2012. We will assume thus that is due only to the electronic damping. In equilibrium the fluctuation-dissipation theorem gives

 SδF(ω=0)=2γmkBT. (24)

For finite , the system is out of equilibrium and one has to evaluate explicitly and from a direct calculation of . As shown in Ref. Clerk, 2004 . One can then always define an effective temperature by the relation , since the oscillator has a very sharp response in frequency and the correlation functions are flat on that scale, one can always interpret the ratio of the fluctuation and the dissipation as an effective temperature. In the case of the SET it has been shown that the typical value of is of the order of .Armour et al. (2004)

The Langevin equation (23) can then be solved by Fourier transform giving

 Sx(ω)=⟨x(ω)x(−ω)⟩=F20Sn(ω)m2|ω2m−ω2−iγω|2 (25)

and in particular in the low-frequency limit:

 Sx(ω=0)=F20Sn(ω=0)m2ω4m. (26)

We can now use the expansion (LABEL:expansion) to find the lowest order contribution of the stochastic fluctuations of to the current. We denote these fluctuations to distinguish them from the time-dependent average induced by the external driving:

 δI(t)=∂I∂ngVg0C′geδx(t)+…. (27)

The back-action current noise is then

 SbaI(ω)=2(∂I∂ngF0ngkL)2Sn(ω=0). (28)

As discussed in Refs. Pistolesi et al., 2008; Brüggemann et al., 2012 the mechanical back-action noise can be very strong and induce effective giant Fano factors.

Finally the measurement added noise can be obtained as is done for the amplifiers,Clerk et al. (2010) by dividing the fluctuation of the current signal by the amplifier gain squared. This gives:

This quantity gives the upper bound on the detection sensibility, since the limitations considered are intrinsic to the detection method. We will evaluate explicitly these quantities for two specific models in sections V and VI.

## Iv Fast oscillator

In this section we relax the condition for the calculation of the mixing current. We assume , the electronic transport is then decribled by sequential transport and we will find the mixing current to lowest non-vanishing order in the amplitude of the oscillating field by making use of a master equation description.

Let’s begin by introducing in some details the electron tunnelling description. We assume that the only available charge states on the island are those associated with two charge states and . We will call these two states 0 and 1. The state of the SET is thus fully described by the probabilities of one of these two state to be realized: , with . We define as the rate for adding (subtracting) one electron on (from) the central island through the left tunnel junction. Similarly we define for the right junction. We define also , with , , , and . The master equation for the the probability reads ():

 ˙P0 = −Γ+P0+Γ−P1 (30) ˙P1 = Γ+P0−Γ−P1. (31)

Using the conservation of probability () we are left with

 ˙P0=−ΓTP0+Γ−. (32)

We consider now that the rate equations are modulated by two oscillating parameters, in our specific case and . We expand in power series of the amplitude of oscillation the rates keeping only the lowest orders:

 Γα(t)=Γα(0)(t)+Γα(1)(t)+Γα(2)(t)+… (33)

where stands for any of the previously introduced labels, and the term into parenthesis indicates the order in the expansion. As far as the driving frequency is smaller than the temperature, , the explicit expression of the time-dependent rates can be obtained by that for the static case by substituting the time-dependent fields:Bruder and Schoeller (1994) for instance , where , , and , . One can then expand to second order in the time dependent part of the two parameters to obtain:

 Γα(t) = Γα+∂Γα∂aa1(t)+∂Γα∂bb1(t)+12∂2Γα∂a2a21(t) +∂2Γα∂a∂ba1(t)b1(t)+12∂2Γα∂b2b21(t)+….

The expansion up to second order can then be rearranged in a Fourier series:

 Γα(t) = Γα00+∑n=−1,1[Γα(1)n,0einω1t+Γα(1)0,neinω2t] (34) +[Γα(2)1,−1ei(ω1−ω2)t+cc]+…

where the static part has contributions of zero and second order in the driving fields. The notation indicates a contribution of order in the driving intensity. Concerning the time dependent second order terms, we keep only the interesting part at the mixing-current frequency .

We look for a solution of the master equation in terms of the stationary Fourier components

 P0(t)=∑n,mAnmei(nω1+mω2)t. (35)

This gives for each Fourier component the equation:

 (inω1+imω2)Anm+∑n′,m′ΓTn′m′An−n′,m−m′−Γ−nm=0. (36)

We further expand the coefficients writing:

 Anm=∞∑p=0A(p)nm, (37)

where again indicates the order in the driving fields. This leads to a set of equations that can be solved recursively. The zeroth-order one reads:

 (inω1+imω2)A(0)nm+ΓT(0)00A(0)n,m−Γ−(0)00δn,0δm,0=0. (38)

It gives immediately the static solution:

 A(0)nm=δn,0δm,0Γ−(0)00ΓT(0)00. (39)

For the next two orders we obtain:

 A(1)nm=ΓT00Γ−(1)nm−ΓT(1)nmΓ−(0)00ΓT(0)00(inω1+imω2+ΓT(0)00), (40)

and

 A(2)nm=Γ−(2)nm−ΓT(2)00A(0)n,m−∑n′m′ΓT(1)n′m′A(1)n−n′,m−m′inω1+imω2+ΓT(0)00. (41)

The non-vanishing terms up to order two are , , , , , , and . As usual for the Fourier transform of real functions the following relation holds: .

Let us now consider the particle current. It can be expressed in terms of and , for instance, on the left junction (note that this expression does not include the displacement current):

 I(t)/e=ΓL+P0−ΓL−P1=ΓLP0−ΓL−. (42)

Substituting the expansion (35) into Eq. (42) we obtain for a similar expansion to Eq. (35). The first three orders read:

 I(0)nm/e = [ΓL(0)00A(0)00−ΓL−(0)00]δnmδn0 (43) I(1)nm/e = ΓL(1)nmA(0)00+ΓL(0)00A(1)nm−ΓL−(1)nm (44) I(2)nm/e = ΓL(0)00A(2)nm+∑n′m′ΓL(1)n−n′,m−m′A(1)n′m′ (45) +ΓL(2)nmA(0)00−ΓL−(2)nm.

The mixing current is given by

 Icmx=ReI1,−1/2,Ismx=−ImI1,−1/2. (46)

In order to simplify the expressions obtained above we use the fact that in general so that even in the fast oscillator limit . This gives the approximate expressions:

 A(1)10 = ΓT00Γ−(1)10−ΓT(1)10Γ−(0)00ΓT(0)00(iωD+ΓT(0)00) (47) A(2)1,−1 = Γ−(2)1,−1−ΓT(1)1,0A(1)0,−1−ΓT(1)0,−1A(1)1,0ΓT(0)00 (48)

One can see that the residual -dependence is due to the relaxation time of the charge in the island. As expected it disappears for . The contribution from vanishes since . The interesting part is the contribution of second order which reads:

 I(2)1−1 = ΓL(0)00A(2)1,−1+ΓL(1)1,0A(1)0,−1+ΓL(1)0,−1A(1)1,0 (49) +ΓL(2)1,−1A(0)00−ΓL−(2)1,−1.

One can verify that for expression Eq. (49) reduces to recovering the standard results for the mixing-current [cf. expressions (10) and (9)].

In the opposite limit of the first order correction to the charge variation vanishes (): the charge has not the time to follow the driving. Only a second order correction survives . The residual time dependence at the mixing frequency is only due to the direct modulation of the tunneling rates (). The final expression for in the limit reads:

 I(2)1−1fast=ΓL(0)00Γ−(2)1,−1ΓT(0)00+ΓL(2)1,−1A(0)00−ΓL−(2)1,−1 (50)

In the following two sections we consider explicitly the case of a metallic dot and of a single electronic level dot and we derive explicit expressions for the mixing current, its fluctuation and the response function in the high-frequency regime.

## V The metallic dot single-electron transistor

The expression for the tunnelling rate are well known for a metallic dot in the Coulomb blockade regime.Ingold, G.-L. and Nazarov, Yu V. (1992) For convenience of the reader, we report in the appendix a very short derivation of the electrostatic relations. We consider only the two states with and electrons.

### v.1 Low temperature case

We begin by discussing the low temperature case where is the Coulomb energy. In this case there are only two non-vanishing rates (for )

 Γ+L(N) = Γo(v+~ng)θ(v+~ng) (51) Γ−R(N+1) = Γo(v−~ng)θ(v−~ng) (52)

where , and , we assume a symmetric device with tunneling resistance . The stationary solution to the master equation (32) and the stationary current (42) read

 Pst1=~ng+v2v,I=eΓov2−~ng22v, (53)

both equations valid for . The current vanishes continuosly for while the probability is 1 for and 0 for .

The driving amplitudes in terms of the dimensionless variables introduced read and . Note that the dependence of the rates on and is non-analytic for , this gives a constraint on the amplitude of the oscillations since the Taylor expansions are not valid if the parameters cross this values. This gives the constraints and , that can be written . Using Eq. (51) and Eq. (52) we can readily obtain the non-vanishing coefficients of the expansion (34): , , , , . For we obtain a very simple expression for the component :

 I1−1=eΓo~ngv1ng1e−iφ~ω2D+4v2 (54)

here we defined . One finds thus a Lorentzian behaviour, the amplification factor decreases quite rapidly for large frequency driving . The main reason for the reduction of sensitivity is the incapacity of the charge in the dot to follow the driving signal. The crossover value for the frequency is , above this value one cannot use anymore the adiabatic approximation for the relaxation of the charge on the dot. It simply coincides with the frequency for which one electron per driving period crosses the device. For instance for MHz, Ohm, for voltage below a mV the corrections due to the retardation of the charge on the dot becomes relevant This regime has been observed in the experiment presented in Ref. Benyamini et al., 2014, where the crossover from slow to fast oscillator has been investigated by a fine tuning of the tunnelling resistances.

The amplification factor for the mechanical quadratures is thus:

 λ=eΓoLng~ngv1(~ω2D+4v2). (55)

It is maximum for , but one should also take into account the constraint on the amplitude of . One way to take that into account is to set , this is the maximum allowed value for the driving amplitude, and since the signal increases linearly with , it gives the maximum value for . This gives:

 λ=eΓoLng~ng(v−|~ng|)(~ω2D+4v2). (56)

The maximum of as a function of the gate voltage is obtained for and its value (for ) is

 λ=eΓong16L (57)

independently of . For a typical device one has m, Hz leading to 0.1 A/m. Moser et al. (2014, 2013)

The gain is only a part of the detection, one has also to evaluate the noise. For that we need the two contributions considered in the section III. The Fano factor has been obtained in Ref. Korotkov, 1994 (cf. Eq. 41 there):

 F=Γ+L2+Γ−R2(Γ+L+Γ−R)2=v2+~ng22v2 (58)

it varies between 1/2 and 1. The shot noise becomes thus:

 SshotII=e2Γov4−~ng42v3. (59)

To obtain the contribution of the displacement fluctuation we need to calculate the charge noise correlation function: . This symmetric correlator can be obtained by the conditional probability that the dot it occupied at time with the condition that it was occupied at time 0: . Solving the master equation with the initial condition one finds

 P(1t|10)=1+(Pst1−1)(1−e−ΓTt), (60)

with . By Fourier transforming we obtain:

 Sn(ω)=Pst1(1−Pst1)2ΓTω2+ΓT2 (61)

As expected the correlation function is flat for , the required low frequency correlator reads then:

 Sn(ω=0)=2Γ+Γ−/(ΓT)3. (62)

In the specific case of low temperature one obtains thus .

In the typical working regime of a SET , and . Using the Eq. (106) one finds that . We thus have . Collecting all the terms we can substitute into Eq. (28) to obtain:

 SbaI=2e2ΓoN4(EckL2)2(v2−~ng2)~ng2v5. (63)

The ratio of the mechanical to the shot noise is thus:

 SbaISshotI=(EckL2)24N4~n2gv2(v2+~n2g). (64)

For large mechanical coupling ( small and large) the mechanical noise dominate even if for small it is always suppressed, due to the vanishing of .

From Eqs (57), (59), and (63), we obtain the seeked added noise as defined by Eq. (65). In order to study its dependence on the different parameters it is convenient to introduce the two dimensionless variables and the dimensionless coupling constant , where is the energy scale of the electromechanical coupling.Armour et al. (2004); Doiron et al. (2006); Pistolesi and Labarthe (2007) The added noise then reads:

with

 f(ν,δ)=2(1+ν)[4(δ2+1)ν2+1]ν2(1−ν)δ, (66)

and . The function diverges for due to the fact that we have to limit the amplitude of the voltage modulation and diverges for due to the vanishing of the amplification factor. The minimum added noise is thus always for values of between 0 and 1. In the weak coupling limit, for , one finds that the minimum is at and reads

For strong coupling, , instead the minimum is close to with a value for

In both cases the noise diverges when becomes very small or very large. In the weak coupling limit the added noise is dominated by the current noise (imprecision noise), in the strong coupling it is instead given essentially by the back-action noise. As usualClerk et al. (2010) the optimal situation is in the middle for .

In Fig. 2 we plot , where is the value of that minimizes for given . We thus find that the absolute minimum for the added noise is obtained for and and reads