The Coulomb drag effect induced by the third cumulant of current

TL;DR

This paper explores how the third cumulant of current, beyond noise power, can significantly influence the Coulomb drag effect in mesoscopic circuits, highlighting a novel role of classical fluctuations and full counting statistics.

Contribution

It introduces the concept that the third cumulant of current can mediate Coulomb drag, expanding understanding beyond traditional noise power effects in mesoscopic systems.

Findings

Third cumulant of current can contribute to Coulomb drag.

Classical fluctuations of the circuit mode can mediate drag.

Situations where third cumulant dominates are identified.

Abstract

The Coulomb drag effect arises due to electron-electron interactions, when two metallic conductors are placed in close vicinity to each other. It manifests itself as a charge current or voltage drop induced in one of the conductors, if the current flows through the second one. Often it can be interpreted as an effect of rectification of the non-equilibrium $quantum$ noise of current. Here, we investigate the Coulomb drag effect in mesoscopic electrical circuits and show that it can be mediated by $classical$ fluctuations of the circuit collective mode. Moreover, by considering this phenomenon in the context of the full counting statistics of charge transport we demonstrate that not only the noise power, but also the third cumulant of current may contribute to the drag current. We discuss the situations, where this contribution becomes dominant.

Figures

Equations

$$Z(\lambda ,t)=\sum _Q{e}^{i\lambda Q}P(Q,t),$$

$$\ln [Z(\lambda ,t)]=\mathcal{H}(\lambda )t,\mathcal{H}(\lambda )=\sum _{n=1}^{\infty }⟨⟨I^n⟩⟩\frac{(i\lambda {)}^n}{n!}.$$

$$\delta V(\omega )=Z(\omega )\delta I(\omega ),Z(\omega )=\frac{R}{1-i\omega /{\omega }_c},$$

$$H_T=A+A^{†},$$

$$I_T(V)=\int dt{e}^{iVt}⟨[A(t),A^{†}(0)]⟩,$$

$$I_0(V)=I_{LR}(V)-I_{RL}(V),$$

$$A\to e^{i\varphi }A,A^{†}\to e^{-i\varphi }{A}^{†},$$

$$\dot{\varphi }=Q/C=\delta V,$$

$$I_T(V)={\int }_{-\infty }^{\infty }d\omega [P_{LR}(\omega )I_{LR}(V-\omega )$$

$$-P_{RL}(-\omega )I_{RL}(V-\omega )],$$

$$P_{LR}(\omega )=\frac{1}{2\pi }\int dt{e}^{i\omega t}⟨e^{i\varphi (t)}{e}^{-i\varphi (0)}⟩,$$

$$P_{RL}(\omega )=\frac{1}{2\pi }\int dt{e}^{i\omega t}⟨e^{-i\varphi (t)}{e}^{i\varphi (0)}⟩$$

$$P_{RL}(\omega )=P_{LR}(-\omega )e^{\omega /T_C}.$$

$$I_{RL}(\omega )=e^{-\omega /T_D}{I}_{LR}(\omega ).$$

$$I_T(0)={\int }_{-\infty }^{\infty }d\omega P_{LR}(\omega )I_{LR}(-\omega )\left[1-e^{-\omega /T_C}{e}^{\omega /T_D}\right].$$

$$P(\omega )=\frac{1}{2\pi }\int dt{e}^{i\omega t}⟨e^{i[\varphi (t)-\varphi (0)]}⟩,$$

$$I_D\equiv I_T(0),I_T(V)={\int }_{-\infty }^{\infty }d\omega I_0(\omega )P(V-\omega ),$$

$$P(\omega )=\frac{1}{2\pi }\int dt{e}^{i\omega t-J_2(t)-i{J}_3(t)},$$

$$J_2(t)=\frac{1}{2}⟨[\varphi (t)-\varphi (0){]}^2⟩,$$

$$J_3(t)=\frac{1}{6}⟨[\varphi (t)-\varphi (0){]}^3⟩.$$

$$J_2(t)=\frac{K_2^{(s)}{t}^2}{2},J_3(t)=\frac{K_3^{(s)}{t}^3}{6},$$

$$K_2^{(s)}=(R/2C)⟨⟨I^2⟩⟩,K_3^{(s)}=(R/3C^2)⟨⟨I^3⟩{⟩}_{\mathrm{tot}},$$

$$⟨⟨I^3⟩{⟩}_{\mathrm{tot}}=⟨⟨I^3⟩⟩-\frac{3⟨⟨I^2⟩{⟩}^2{\partial }_Q{\omega }_c}{2{\omega }_c^2}+\frac{3⟨⟨I^2⟩⟩{\partial }_Q⟨⟨I^2⟩⟩}{2{\omega }_c}.$$

$$J_2(t)=\frac{K_2^{(f)}|t|}{2},J_3(t)=\frac{K_3^{(f)}t}{6},$$

$$K_2^{(f)}=R^2⟨⟨I^2⟩⟩,K_3^{(f)}=R^3⟨⟨I^3⟩{⟩}_{\mathrm{tot}}.$$

$$⟨⟨I^3⟩{⟩}_{\mathrm{tot}}=⟨⟨I^3⟩⟩-\frac{6⟨⟨I^2⟩{⟩}^2{\partial }_Q{\omega }_c}{{\omega }_c^2}+\frac{3⟨⟨I^2⟩⟩{\partial }_Q⟨⟨I^2⟩⟩}{{\omega }_c},$$

$$I_D=\int \frac{dtd\omega }{2\pi }{I}_0(\omega )e^{i\omega t-J_2(t)+i{J}_3(t)}.$$

$$I_D\propto {\partial }_t[-J_2(t)+i{J}_3(t){]}_{t=0}=0,$$

$$I_0(V)=\sum _n{g}_n{V}^n,n=1,2,\dots ,$$

$$I_D=\sum _{n=1}^{\infty }{i}^n{g}_n{\partial }_t^n{e}^{-J_2+i{J}_3}{|}_{t\to 0}=g_2{K}_2^{(s)}+g_3{K}_3^{(s)},$$

$$I_D=⟨I_0(\delta V)⟩.$$

$$I_D=\frac{g_2}{2}{R}^2{\omega }_c⟨⟨I^2⟩⟩+\frac{g_3}{3}{R}^3{\omega }_c^2⟨⟨I^3⟩{⟩}_{\mathrm{tot}},$$

Full text

††preprint: APS/123-QED

I Introduction

The Coulomb drag effect is the phenomenon observed in a system of two interacting conducting circuits, which manifests itself as a charge current or a voltage drop induced in a drag circuit, when a charge current flows through a drive circuit. 111For a review, see B. N. Narozhny and A. Levchenko, Rev. Mod. Phys. 88, 025003 (2016), and references therein It originates from the broken electron-hole symmetry and electron interactions, and therefore it is often studied in mesoscopic systems of reduced dimensionality, such as quantum wires(Debray et al., 2001; Morimoto et al., 2003; Yamamoto et al., 2006; Laroche et al., 2011, 2014; Nazarov and Averin, 1998; Pustilnik et al., 2003), quantum dots(Onac et al., 2006; Aguado and Kouwenhoven, 2000; Sánchez et al., 2010; Sothmann et al., 2012), and quantum point contacts(Khrapai et al., 2006, 2007; Levchenko and Kamenev, 2008a), where both effects are strongly pronounced. Its manifestation is particularly interesting in quantum conductors, where the electron-hole asymmetry is connected to the energy dependence of the transmission coefficientsBlanter and Büttiker (2000) and can be tuned by applying a gate voltage. In such systems the Coulomb drag can be viewed as an effect of rectification by the drag circuit of quantum noise of the drive circuit.Levchenko and Kamenev (2008a) Although known also in higher dimensions,(Kamenev and Oreg, 1995) this effect is more evident in low-dimensional systems, where it can be used to measure the spectral density of the noise(Onac et al., 2006; Aguado and Kouwenhoven, 2000) and its properties (Levchenko and Kamenev, 2008a), as well as to probe fundamental fluctuation relations(Sánchez et al., 2010).

In all mentioned above examples the main contribution to the drag effect comes from the two-point correlation function $\langle\delta I(t)\delta I(0)\rangle$ of current fluctuations $\delta I$ in the quantum conductor at time scales of the order of the correlation time (typically given by one over the voltage bias or temperature), where fluctuations are essentially quantum. From a broader perspective of the full counting statistics (FCS) of quantum conductors, Levitov et al. (1996) such a correlation function is a characteristic of the Gaussian noise. To clarify this fact, let us consider the moment generating function

[TABLE]

of the charge $Q$ transmitted through a quantum conductor during time $t$, and for simplicity take the Markovian (classical noise) limit, $t\gg\tau_{c}$, where the short-time fluctuations contribute to the generator independently:

[TABLE]

Here $\langle\!\langle I^{n}\rangle\!\rangle$ are the current cumulants, the first three being the average current, $\langle\!\langle I\rangle\!\rangle=\langle I\rangle$, the zero-frequency noise power, $\langle\!\langle I^{2}\rangle\!\rangle=\int dt\langle\delta I(t)\delta I(0)\rangle$, and the third cumulant $\langle\!\langle I^{3}\rangle\!\rangle=\int dt\int dt^{\prime}\langle\delta I(t)\delta I(t^{\prime})\delta I(0)\rangle$. Although the current cumulants enter the FCS generator ${\cal H}$ on equal footing, experimentally the high-order cumulants are much less accessible than the second one, because in large systems their contributions to measured quantities (including the drag current) are suppressed due to the central limit theorem. The third current cumulant has been experimentally studied by explicitly collecting the statistics of the transferred charge(Reulet et al., 2003; Bomze et al., 2005; Gustavsson et al., 2006; Gershon et al., 2008; Gustavsson et al., 2009; Gabelli and Reulet, 2009), and by studying the weak asymmetry of the escape rate in Josephson junction threshold detectors(Timofeev et al., 2007; Huard et al., 2007) with respect to the current bias.

Alternatively, one can consider the Coulomb drag effect in mesoscopic circuits in the context of the noise detection physics, where the drive circuit generates nonequilibrium noise, while the drag circuit plays the role of the detector. It turns out Sukhorukov and Edwards (2008) that the current through the tunnel junctions detector is expressed in tems of the correlation function, which in the long-time limit acquires the form $e^{iVt}Z(\lambda,t)$, where $V$ is the voltage bias accross the junction, $Z(\lambda,t)$ is the moment generator (1), and the counting variable $\lambda$ plays the role of the effective coupling constant. Typically, the effective coupling between the drive and drag circuits is weak, $\lambda\ll 1$, which explains the suppression of cumulants of the order $n>2$. Note, however, that according to Eq. (2) the third cumulant of current, in contrast to the second one, simply shifts the voltage bias in this correlation function: $V\to V-\lambda^{3}\langle\!\langle I^{3}\rangle\!\rangle/6$. This leads to the idea, that Markovian (classical) odd cumulants of noise, being a nonequilibrium property of a quantum conductor (they vanish at zero bias) may propagate to the drag circuit and cause the DC drag current by shifting the bias. Being essentially classical, this phenomenon has to be differentiated from the drag effects studied so far, where the main contribution come from short time scales given by the correlation time of current fluctuations.Levchenko and Kamenev (2008a) However, this simple idea has a caveat that we explain below.

A simplified electrical circuit for detecting the drag effect is shown in Fig. 1. (An alternative open circuit set-up is considered in Sec. V.) It contains a quantum conductor that emits a nonequilibrium current noise, and a tunnel junction detector, where the drag current is induced. The fluctuations of the current in the conductor, $\delta I$, do not propagate directly towards the detector: they are accumulated in the capacitor and lead to voltage fluctuations $\delta V$ across the tunnel junction. Current and voltage fluctuations are related by the solution of the Langevin equation:

[TABLE]

where $Z(\omega)$ is the impedance of the circuit, $\omega_{c}=1/(RC)$ is the circuit response frequency, the circuit resistance is defined as $R^{-1}=R_{L}^{-1}+R_{S}^{-1}$, and the total capacitance is given by $C=C_{L}+C_{T}$. At long times, $\omega_{c}t\gg 1$, i.e., at low frequencies, one obtains $\delta V=R\delta I$, giving indeed $\langle\!\langle V^{3}\rangle\!\rangle=R^{3}\langle\!\langle I^{3}\rangle\!\rangle$, where $R$ plays the role of an effective coupling constant. However, for an Ohmic tunnel junction the main contribution comes from short time scales, $t\ll 1/\omega_{c}$, where, due to the prefactor $Z(\omega)$, the voltage fluctuations are suppressed as $1/\omega^{2}$ [see Eq. (3)]. In Sec.  III we rigorously show that this leads to complete cancellation of the drag effect from the classical noise in Ohmic tunnel junctions. 222The quantum drag effect to third order in coupling has been studied in [Levchenko and Kamenev, 2008b] Therefore, we focus in the paper on the tunnel junctions with different nonlinear I-V characteristic, find the drag current $I_{D}$ perturbatively in small $1/R_{T}$, and express it in terms of the Markovian third cumulant of current of the quantum conductor, $\langle\!\langle I^{3}\rangle\!\rangle$, and the circuit parameters.

This paper is organized as follows. In Sec. II we use the $P(E)$-theory of tunneling(Geigenmüller and Nazarov, 1991; *Naz) to derive the expression for the drag current in terms of the I-V characteristic of the tunnel junction. Then, in Sec. III we apply the weak coupling expansion to formally express the drag current in terms of the cumulants of the current of the quantum conductor. In Sec. IV we separately consider the drag effect in tunnel junctions with analytical and nonanalytical I-V characteristic. In Sec. V we investigate the drag effect in the open circuit set-up (see Fig. 2), which is experimentally more relevant. Finally, in Appendix A we use the stochastic path integral (SPI) technique(Pilgram et al., 2003; *stochpath2) to derive the second and third voltage cumulants in terms of the current cumulants in the quantum conductor.

II P(E)-theory of tunneling and the drag effect

In this and next section we closely follow Refs. [Sukhorukov and Edwards, 2008], [Safi, 2014] and [Safi, 2019]. We consider a tunnel junction in the presence of the noise of the collective mode, propagating in an electrical circuit from a quantum conductor, as shown in the Fig. 1, and apply the $P(E)$ theory of tunneling (Geigenmüller and Nazarov, 1991; *Naz) to evaluate the current in the tunnel junction induced by this noise. The advantage of this approach is that to the leading (second) order in tunneling there is no need to specify the Hamiltonian of the reservoirs of the tunnel junction to derive the expression for the drag current: an arbitrary disorder and interactions can be included. The electron tunneling is described by the Hamiltonian (throughout the paper we use unites, where $|e|=\hbar=1$)

[TABLE]

where the tunneling operator $A$ transfers an electron from the left to the right reservoir. According to the tunneling Hamiltonian approach Mahan (1980), the tunneling current operator can be defined as $I\equiv-dN_{R}/dt=i(A-A^{\dagger})$, where $N_{R}$ is the number of electrons in the right reservoir. Thus, in the leading order in tunnelling the average value of the tunneling current $I_{T}\equiv\langle I\rangle$ is given by:

[TABLE]

where $V$ is the applied voltage bias, and the average is evaluated with respect to the equilibrium state: $\langle\ldots\rangle=\sum_{n}\rho_{n}\langle n|\ldots|n\rangle$, and $\rho_{n}\propto e^{-E_{n}/T}$ with $T$ being the bath temperature. In the absence of noise in the circuit, this expression give the bare I-V characteristic $I_{0}(V)$ of the junction, which, according to the structure of Eq. (5), can also be presented as

[TABLE]

where the two terms on the right hand side differ by the direction of electron tunneling.

In the next step, we account for coupling of the junction to noise by substituting (Geigenmüller and Nazarov, 1991; *Naz)

[TABLE]

where the operator $e^{i\phi}$ increases the charge on the capacitor by $1$, which can be expressed as $[\phi,Q]=i$. Then, the charge Hamiltonian $H_{C}=Q^{2}/2C$ generates the equation of motion:

[TABLE]

where $\delta V$ is fluctuating part of the voltage across the tunnel junction. After substituting the operator $A$ from Eq. (7) into Eq. (5) and using Eq. (6), we arrive at the following expression for the tunneling current:

[TABLE]

where

[TABLE]

are the probabilities of absorbing the energy $\omega$ from the circuit which depend on the direction of tunneling.

We are now in the position to discuss how the drag current vanishes at equilibrium. Therefore, we set $V=0$ in Eq. (9) and assume, that the circuit is in equilibrium at the temperature $T_{C}$. We then apply the spectral decomposition to the probabilities (10) and (11), and write $P_{LR}(\omega)=\sum_{nm}\rho_{C,n}|\langle n|e^{i\phi}|m\rangle|^{2}\delta(\omega+E_{n}-E_{m})$. By comparing this expression to the similar result for $P_{RL}(\omega)$ and using the equilibrium weights $\rho_{C,n}\propto e^{-E_{n}/T_{C}}$, we arrive at the detailed balance equation:

[TABLE]

Assuming now that the detector tunnel junction is at equilibrium with the temperature $T_{D}$ and applying the spectral decomposition to the equations (5) and (6), we obtain

[TABLE]

Using these two detailed balance equations in the equation (9), we arrive at the following result:

[TABLE]

We note that all the available frequencies, including quantum fluctuations, contribute to the total current (14), and all these contributions cancel at the global equilibrium $T_{C}=T_{D}$. Knowing this fact, in what follows we assume that the main contribution to the drag current comes from nonequilibrium fluctuations at low frequencies, which can be considered classical (Markovian). In other words, we assume that the phase operator $\phi(t)$ commutes with itself at different times, so that the expressions (10) and (11) simplify, and introducing the new notation $P(\omega)=P_{LR}(\omega)$ one can write

[TABLE]

while $P_{RL}(\omega)=P(-\omega)$. Applying these simplifications in Eq. (9), using Eq. (6), and introducing yet another notation $I_{D}$ for the drag current, we arrive at its final general form for the case of classical noise:

[TABLE]

where, we recall, $I_{0}(V)$ is the bare tunneling current, not affected by fluctuations in the circuit.

III Weak coupling expansion

We further assume that coupling of the junction to the system, described by Eqs.  (3), is weak, i.e., $R\ll 1$, and expand the probability $P(\omega)$ in cumulants of the phase $\phi$ to the third order:

[TABLE]

where the cumulants are given by

[TABLE]

Here we assumed the semi-classical (Markovian) noise limit, $\Delta V/\omega_{c}\gg 1$ with $\Delta V=V_{d}-V$ being the bias over the quantum conductor, and neglect quantum corrections.333Note that $J_{3}$ is smaller than $J_{2}$ by the dimensionless coupling constant $R\ll 1$ (in unites where $|e|=\hbar=1$), therefore one should consider the quantum correction to $J_{2}$. However, it has been shown in Ref. [Sukhorukov and Edwards, 2008], that according to the Kubo linear response formula it is simply given by the differential conductance of the noise source, which is an equilibrium property. Therefore, it does not contribute to the drag effect. The correlators $J_{2}$ and $J_{3}$ are evaluated in Appendix A. In the next section we will consider the cases of a slow and fast circuit, where these correlators have to be taken in the short-time, $\omega_{c}|t|\ll 1$, and long-time, $\omega_{c}|t|\gg 1$, limits, respectively.

In the short-time limit, $\omega_{c}|t|\ll 1$, one finds

[TABLE]

where the coefficients represent the second and third cumulant of “instant” (at equal times) fluctuations of the potential: $K^{(s)}_{m}=\langle(\delta V)^{m}\rangle$, $m=2,3$. Using Eqs. (3), they can be expressed in terms of the second and the third cumulant of the current in the quantum conductor at zero frequencies:

[TABLE]

where the total third cumulant reads

[TABLE]

Note that the second and third terms represent circuit cascade corrections due to the nonlinear and environmental effects, respectively. These corrections are specific to the regime of a slow circuit, and they can be found using the SPI method, as demonstrated in Appendix A (see also Ref. [Sukhorukov and Edwards, 2008]).

In the long-time limit, $\omega_{c}|t|\gg 1$, one obtains

[TABLE]

where the coefficients $K_{m}^{(f)}$, $m=2,3$ can be read off the Eqs. (3) by replacing $Z(\omega)\to R$,

[TABLE]

However, as in the case of the slow circuit, the cascade corrections for a fast circuit can be obtained by the SPI method:

[TABLE]

where, again, the second and third term represent circuit cascade corrections due to the nonlinear and environmental effects, respectively. The environmental effects in the third cumulant have been experimentally studied in Refs. [Reulet et al., 2003] and [Gershon et al., 2008].

By using Eqs. (16) and (17), the drag current can be written as

[TABLE]

The way the third current cumulant enters this expression suggests that it may have a similar effect on the tunnel junction as the DC voltage bias, i.e., it may cause a drag current. Indeed, in the long-times limit (23) the third current cumulant enters as a linear in time phase shift, i.e., it adds to the voltage bias, so one expects a finite current even at zero voltage. However, it turns out, that for an Ohmic tunnel junction, $I_{0}(\omega)\propto\omega$, and for a classical noise considered here the drag current vanishes. Indeed, in this case the integral over $\omega$ in Eq. (26) imposes the $t\to 0$ limit, and one obtains:

[TABLE]

according to Eq. (20). Therefore, in the rest of the paper, we consider the drag current in tunnel junctions with different nonlinear I-V characteristic.

IV Drag current for tunnel junctions with nonlinear I-V characteristic

In this section we evaluate the drag current for two types of nonlinearities in the tunnel junction. Namely, in Sec. IV.1 we consider the analytical regime, $I_{0}(V)=\sum_{n}g_{n}V^{n}$, where $n=1,2,\ldots$, while in Sec. IV.2 we investigate the nonanalytical regime, $I_{0}(V)=g_{\alpha}V|V|^{\alpha-1}$ for noninteger $\alpha$. However, before proceeding with calculations, one needs to check that the main contribution to the integral in Eq. (26) comes from frequencies smaller than $\Delta V$ to ensure that our classical noise approximation still applies (i.e., the noise source $\delta I$ can be considered Markovian). Since $P(\omega)$, given by Eqs.  (17)-(19), is already taken in the classical limit, it is sufficient to check that the integral in Eq. (16) does not diverge at infinity for $V=0$. For doing so, let us consider the asymptotic behavior of Eq. (17) for frequencies $\omega\gg\max\{\omega_{c},(K_{2}^{(s)})^{1/2}\}$. Using the short-time dependence of the cumulant (55) we find that the even (odd) part of $P(\omega)$ scales as $\omega^{-\gamma}$ with $\gamma=4$ ($\gamma=5$). Therefore, as long as $n\leq 3$ in the analytical regime and $\alpha\leq 4$ in the nonanalytical regime the classical noise approximation is valid. Outside this parameter range, either the quantum character of the noise or high-frequency cutoff of $I_{0}(V)$ should be taken into account, which is beyond the scope of this paper.

IV.1 Analytical regime

Assuming the analytical I-V dependence,

[TABLE]

and that the integral (26) converges, we first perform the integration over $\omega$, and then remove resulting delta-functions by the integral over time,

[TABLE]

where we kept only the first two terms of the expansion, since higher-order terms are small due to the weak coupling regime. Alternatively, one can derive this expression by approximating $\phi(t)-\phi(0)\approx\delta Vt$ at short times in Eq. (15) and using Eq. (16). This approximation holds up to the third order in $\delta V$ and gives

[TABLE]

Substituting here $I_{0}$ from Eq. (28), one arrives at Eq.  (29). Thus, the drag current in this regime is due to the rectification of the instant fluctuations of the potential. Using Eq. (21), we arrive at the result:

[TABLE]

The third cumulant contribution is an even function of the source current and can be measured by changing its direction, even though it is the subdominant contribution to the drag current.

In what follows, we discuss our finding in the context of earlier published results. This concerns the noise rectification effect. We note that the first term in Eq. (31) has a simple structure: it is proportional to the product of the noise power $\langle\!\langle I^{2}\rangle\!\rangle$ and the circuit response frequency $\omega_{c}$. Estimating the source noise as $\langle\!\langle I^{2}\rangle\!\rangle\propto\Delta V$, where the $\Delta V$ is the voltage bias applied to the quantum conductor, we conclude that the noise rectification contribution scales as $\omega_{c}\Delta V$. The same structure can be found in the drag current derived in Ref.  [Levchenko and Kamenev, 2008a] in the quantum regime, $\omega_{c}\gg\Omega=\max(\Delta V,T)$, where $T$ is the temperature of the system. In this case the frequency integrals are limited by $\Omega$. In the shot noise limit, $\Delta V\gg T$ the noise power is proportional to the bias applied to the quantum conductor, $\Delta V$, while the cutoff frequency is determined by the same scale (as discussed above), which leads to the following result $I_{D}\propto\Delta V^{2}$.

Close to equilibrium, $\Delta V\ll T$, the drag effect originates from an even component of the nonlinearity in the I-V characteristic of the quantum conductor that scales as $\Delta V^{2}$. Levchenko and Kamenev (2008a) In this case the noise power scales as $\langle\!\langle I^{2}\rangle\!\rangle\propto\Delta VT$, while the frequency cutoff is given by the temperature, resulting in $I_{D}\propto\Delta VT^{2}$, which becomes $I_{D}\propto\Delta VT\omega_{c}$ in the case of classical noise. Interestingly, this contribution to the drag current depends on the direction of the source current, and thus it may compete with the third cumulant contribution. Therefore, we propose to do measurements in the regime, where the I-V characteristic of the mesoscopic conductor is an odd function.

IV.2 Nonanalytical regime

In this section we consider tunnel junctions with nonanalytical I-V characteristic of the form

[TABLE]

where $\alpha$ is noninteger number, and $g_{\alpha}$ is an arbitrary constant. Such I-V characteristic is typical for systems with interactions, e.g., Luttinger liquids or disordered systems. Since $I(V)$ is an odd function of the voltage bias $V$, only the third current cumulant contributes to the drag effect, as one can easily see from Eq. (26). Due to weak coupling, and since the time integral in Eq. (17) is limited by $J_{2}$, one can expand the exponential function in the integral in small $J_{3}$: $P(\omega)=P_{0}(\omega)+\delta P(\omega)$, where

[TABLE]

In contrast to the analytical regime, here one should separately consider the cases of the slow circuit, $\omega_{c}\ll R^{2}\langle\!\langle I^{2}\rangle\!\rangle$, and of the fast circuit, $\omega_{c}\gg R^{2}\langle\!\langle I^{2}\rangle\!\rangle$. Note that in the latter case $\omega_{c}$ is still bound from above, because the circuit response should be slower then the correlation time of the noise: $\omega_{c}\ll\Delta V$. This is consistent with the requirement of weak coupling $R\ll 1$, since $\langle\!\langle I^{2}\rangle\!\rangle\sim\Delta V$.

IV.2.1 Slow circuit, $\omega_{c}\ll R^{2}\langle\!\langle I^{2}\rangle\!\rangle$.

In this case the contribution to the integral in Eq. (33) comes from times $t\ll 1/\omega_{c}$, therefore we use the short-time limit (20) for the phase correlation functions. Substituting these expressions into Eq. (33), we obtain

[TABLE]

Substituting this expression for the correction to the probability distribution function along with nonanalytical I-V characteristic into the Eq. (26), we arrive at the result for the drag current for $-2<\alpha<4$

[TABLE]

where the correlation functions $K^{(s)}_{m}$, $m=2,3$, are expressed in terms of the current cumulants in Eqs. (21) and (22). This expression correctly reproduces the above results for the Ohmic ($\alpha=1$) and cubic ($\alpha=3$) terms in I-V characteristic of the tunnel junction [see Eq. (31)].

For $\alpha<-2$ the integral in Eq. (26) becomes divergent at small frequencies. Introducing the infrared cutoff, $\omega_{0}$, we express the drag current as:

[TABLE]

Interestingly, for the case of a nonanalytical I-V characteristic of a tunnel junction, the drag current depends both on the second and the third cumulants, in contrast to the case of the analytical nonlinearity. It is clear that this result can not be obtained perturbatively in noise power, since $K_{2}^{(s)}$ enters this expression nonanalytically.

IV.2.2 Fast circuit, $\omega_{c}\gg R^{2}\langle\!\langle I^{2}\rangle\!\rangle$.

Taking into account the result (23) and using Eq. (33) we arrive at the following expression for the third cumulant correction to the probability distribution function:

[TABLE]

where the correlator $K_{2}^{(f)}$ is given by Eq. (24). (This result holds up to small relative corrections of the order of $R^{2}\langle\!\langle I^{2}\rangle\!\rangle/\omega_{c}$). We first concentrate on the case $1<\alpha<4$, where an interesting situation arises: the drag current is determined by neither the short-time nor the long-time limit of $J_{3}(t)$. Consequently, it acquires unusual nonlinear and environmental cascade corrections that have not been discussed in literature. Straightforward calculations lead to the following result:

[TABLE]

where $C(\alpha)=\int^{\infty}_{0}dxx^{\alpha-1}/(x^{2}+1)(x^{2}+4)$ and $F(\alpha)=\int^{\infty}_{0}dxx^{\alpha-1}/(x^{2}+1)^{2}$. This expression is obtained up to corrections of the order of $[R^{2}\langle\!\langle I^{2}\rangle\!\rangle/\omega_{c}]^{\alpha-1}$. Note that for $\alpha=3$ it agrees with the third cumulant contribution in Eq. (31).

In contrast, for $-2<\alpha<1$, the drag current becomes determined solely by the long-time behaviour of $J_{3}(t)$, therefore it can be expressed in terms of the correlators (24) and (25),

[TABLE]

where the function $F$ is introduced above. For $\alpha<-2$, we regularise I-V characteristic at small voltages by the cutoff $\omega_{0}$. Then the drag current in this case has the form given by Eq. (39) after the replacement $F(\alpha+2)\to(\omega_{0}/\omega_{c})^{\alpha+2}$.

Finally, we note that the drag current obtained in this subsection originates from the long-time behaviour of $J_{3}(t)$, which results in the shift of the voltage bias, discussed in the introduction. Therefore, it has to be differentiated with the noise rectification effect arising at short time scales and discussed in Sec. IV.1.

V Drag effect in the open circuit setup

In Sec. IV we derived the drag current $I_{D}$ for the most elementary setup shown in Fig. 1, where the tunnel junction is electrically connected to the circuit. The disadvantage of such connection is that it might be difficult to measure the drag effect due to the third cumulant of the current in the background of the nonzero average DC current contribution. Fortunately, our results can be easily modified for the case of an experimentally more relevant setup shown in Fig. 2, where the drag voltage $U_{D}$ is measured, and where the DC component of the average current is filtered out.

First, we note that the role of the capacitor $C_{D}$ in the setup in Fig. 2 is to filter out the DC component of the bias, $V$. However, one has to be sure that the largest part of fluctuations $\delta V$ still propagates towards the tunnel junction. This is the case when, on one hand, the detector circuit does not screen the fluctuations and, on the other hand, only a small part of the voltage drops across the capacitor $C_{D}$ at relevant frequencies. The former holds if the impedance of the detector circuit

[TABLE]

where $\tilde{R}^{-1}=R_{T}^{-1}+R_{D}^{-1}$, is large compared to the impedance (3) of the drive circuit $Z(\omega)$ at the characteristic frequencies of fluctuations $\omega_{c}=(RC)^{-1}$. The letter condition holds if the impedance of the detector capacitor $i/(\omega C_{D})$ is small compared to the impedance of the rest of the detector circuit $(1/\tilde{R}-i\omega C_{T})^{-1}$ at frequencies of the order of $\omega_{c}$. The two conditions are satisfied simultaneously, if (i) $\tilde{R}C_{T}\ll RC$, $\tilde{R}C_{D}\gg RC$, and $\tilde{R}\gg R$, or, alternatively, (ii) $\tilde{R}C_{T}\ll RC$, $C_{D}\gg C_{T}$, and $C\gg C_{T}$. These conditions imply that the detector is noninvasive and that our previous results for the drag current hold.

The drag voltage is determined by the condition

[TABLE]

where $I_{T}$ is given by Eq. (16). This equation follows from Kirchhoff’s law and the fact, that the DC current through the open circuit vanishes. In the case of the tunnel junction with an analytical I-V characteristic [see Eq. (28)], Eq. (41) is solved trivially, giving

[TABLE]

Note that the tunneling conductance $1/R_{T}$ arising here is nothing but the expansion coefficient $g_{1}$ in Eq. (28).

In the case of a nonanalytical I-V characteristic (32), one should distinguish between two limits depending on how the value of the drag voltage $U_{D}$ compares to the width $\Gamma$ of the distribution $P(\omega)$, which can be estimated as

[TABLE]

If the width $\Gamma$ of the probability distribution is small compared to the drag voltage $U_{D}$, then Eq. (41) can be solved by expanding $I_{0}(\omega)$ in (16) around $\omega=U_{D}$, givin the relation

[TABLE]

where $I_{0}$ is given by Eq. (32).

If the first or second term on the left hand side of the equation (44) dominates, one obtains $U_{D}=-|I_{D}/g_{\alpha}|^{1/\alpha}$ or $U_{D}=-R_{D}I_{D}$, respectively. Note that in this regime the expression for the drag current (31) still applies. However, the coefficients are expressed in terms of $U_{D}$, namely, $g_{2}=(g_{\alpha}/2)\alpha(\alpha-1)U_{D}^{\alpha-2}$ and $g_{3}=(g_{\alpha}/6)\alpha(\alpha-1)(\alpha-2)U_{D}^{\alpha-3}$. For instance, if $1/R_{D}=0$ and the second cumulant contribution to the drag current dominates, one has $|U_{D}|\propto R[\omega_{c}\langle\!\langle I^{2}\rangle\!\rangle]^{1/2}$, while for the case, where the third cumulant dominates, one gets $|U_{D}|\propto R[\omega_{c}^{2}\langle\!\langle I^{3}\rangle\!\rangle]^{1/3}$. When comparing $U_{D}$ to $\Gamma$ in Eq. (43), we see that for the fast circuit, $\omega_{c}\gg R^{2}\langle\!\langle I^{2}\rangle\!\rangle$, the regime $U_{D}\gg\Gamma$ is indeed realized. However, for the slow circuit, $\omega_{c}\ll R^{2}\langle\!\langle I^{2}\rangle\!\rangle$, $U_{D}$ is of the order of $\Gamma$ or smaller. For the finite shunt conductance $1/R_{D}$ the second term in Eq. (44) may start dominating. However, this may be compensated by even smaller values of $U_{D}$.

This brings us to the limit $\Gamma\gg U_{D}$. Expanding $I_{T}$ in Eq. (16) with respect to the small $V=U_{D}$ and using Eq. (41), one again arrives at Eq. (42). However, now the tunneling conductance is given by the following expression

[TABLE]

where we integrated by parts. Thus, the noise simply smears out the singular I-V characteristic (32) at voltages of the order of $\Gamma$, and one can estimate $1/R_{T}\sim g_{\alpha}\Gamma^{\alpha-1}$. Since in this case the tunneling resistance depends on the properties of both the tunneling junction and the noise, it is convenient to shunt the tunnel junction by $R_{D}\ll R_{T}$, so that

[TABLE]

further lowering the drag voltage to values $U_{D}\ll\Gamma$. In this regime the results of Sec. IV.2 for the drag current apply.

VI Summary

It is natural to think of the Coulomb drag effect as resulting from the friction between electron systems of two adjacent conductors due to electron-electron scattering. It can be caused either by the direct Coulomb interaction, or by the exchange of virtual excitations, such as plasmons or phonons. However, in the case of the Coulomb drag in mesoscopic electrical circuits it is more appropriate to think of the noise rectification effect, since the drag is mediated by the collective mode, such as a potential on a capacitor. Nevertheless, in the quantum regime, where the characteristic circuit response frequency $\omega_{c}$ is much larger than the effective noise temperature $\Omega=\max(\Delta V,T)$,Levchenko and Kamenev (2008a) one can still think that electrons of the drive circuit “push” electrons in the drag circuit thereby creating the drag current or voltage, because the circuit reacts to current fluctuations in a quantum conductor almost immediately. In this paper we consider the opposite regime, $\omega_{c}\ll\Omega$, and study the Coulomb drag effect in mesoscopic circuits mediated by the classical noise of a collective mode. This allows us to put our analysis in the context of the FCSLevitov et al. (1996) and to investigate the drag effect due to the Markovian (frequency independent) third cumulant of the current. The interest to the third current cumulant is motivated by the fact that this is essentially a nonequilibrium and non-Gaussian component of current noise.

We consider a simple mesoscopic circuit, shown in Fig. 1 (and its experimentally more relevant modification in Fig. 2). It contains a quantum conductor, the source of noise, and a tunnel junction detector, where the drag current is induced. We evaluate the drag current perturbatively in the tunneling Hamiltonian using the $P(E)$-theory of tunneling(Geigenmüller and Nazarov, 1991; *Naz) and express it in terms of the second and third current cumulants, assuming weak coupling of the detector to the circuit. For doing so, we apply the SPI technique,(Pilgram et al., 2003; *stochpath2) the functional method of solving the circuit Langevin equations, which allows one to find circuit cascade corrections to high-order current cumulants. We find that, surprisingly, the drag current vanishes in the case of an Ohmic tunnel junction detector. Therefore, we concentrate on the drag effect induced in a tunnel junction detector with a nonlinear I-V characteristic.

It is important to distinguish nonlinear I-V characteristic of the two sorts: the relatively smooth analytical I-V curve, that can be expanded in the voltage bias $V$ around $V=0$, and I-V curve essentially nonanalytical at $V=0$ point, as in the case of various kind of zero-bias anomaly effects. Thanks to the tunneling Hamiltonian approach used in the paper, there is no need to specify the reason for such nonanalyticity. In the former case the contribution to the drag effect comes from short (but still Markovian) times scales, and the result takes a simple form (31). In the weak coupling regime considered in the paper the second cumulant contribution to the drag current dominates. However, it is an even function of the source current, therefore, the third cumulant contribution can be singled out by changing the direction of the current. The case of a nonanalytical I-V characteristic is special in the sense that not only the drag current is different for slow (35) and fast (39) circuit, but also there is a regime, where the drag current (38) acquires contributions from different time scales. In this case it contains cascade corrections that have not been discussed in literature.

Finally, we consider the drag effect in an open circuit (see Fig. 2), which is experimentally more relevant, because in this case there is no need to extract the drag current from the background contribution due to the DC voltage bias. Instead, one can measure the drag voltage induced across the tunnel junction and use the results (42), (44), and (46) to express it in terms of the “bare” drag current found in the paper.

Acknowledgements.

We thank M. Reznikov for fruitful discussions. This work was supported by the Swiss National Science Foundation

Appendix A Stochastic path integral

Although, we need to find the correlation functions (18) and (19) of the field $\phi$, it turns out to be convenient to start directly with the generating function

[TABLE]

and evaluate it up to the third order in $\chi$ in the exponent using the functional method. Since we are interested in the low frequency limit, where the field $\phi$ can be considered classical, we apply the SPI technique, (Pilgram et al., 2003; *stochpath2) which correctly implements averaging in (47) over solutions of the Langevin equation (3). Given the relation (8) of the field $\phi$ to the charge on the capacitor $Q$, we write Eq. (47) as

[TABLE]

where ${\cal H}(Q,\lambda)$ is the cumulant generating function for the current fluctuations in the quantum conductor

[TABLE]

and the function $\Theta(t^{\prime})\equiv\theta(t^{\prime})\theta(t-t^{\prime})$ projects onto the interval $[0,t]$.

Since we consider the classical noise, we are obliged to choose the leading order saddle-point solution of the SPI (48), which gives

[TABLE]

Thus, the saddle-point action $S_{\rm sp}(\chi)$ may be considered a generator of the cumulants of the field $\phi(t)-\phi(0)$. We evaluate it up to third-order terms in $\chi$ by solving classical Hamilton’s equations of motion. Namely, we split the “Hamiltonian” in two parts, ${\cal H}={\cal H}_{0}+\Delta{\cal H}$, where

[TABLE]

accounts for the average current and the zero-frequency noise power, and the part

[TABLE]

is to be considered as a perturbation. It contains the contribution of the third cumulant of the current in the quantum conductor, while the frist and second term represent the nonlinear and “environmental” cascade correction, respectively.Nagaev (2002)

The part ${\cal H}_{0}$ together with the source term in the action (49) generate the equations of motion

[TABLE]

which can be easily solved with the conditions $Q=\lambda=0$ at $t^{\prime}=-\infty$ and for $t^{\prime}>t$ (otherwise, $\lambda$ would diverge at infinity). The solution has to be substituted back to the action (49), eventually giving the saddle-point action (50). Interestingly, one can show that there is no need to account for the corrections to the equations of motion from the perturbation $\Delta{\cal H}$, since they contribute to terms starting from fourth order in $\chi$. This greatly simplifies calculations.

The final result can be presented in the following form

[TABLE]

where the second cumulant is given by

[TABLE]

According to the structure of the perturbation part of the Hamiltonian (52), the third cumulant contains three terms

[TABLE]

that represent the nonlinear and environmental correction, as well as the so-called minimal correlation contribution (Nagaev, 2002; Beenakker et al., 2003). Introducing the notation $\tau=\omega_{c}t$, they read

[TABLE]

We note that the calculations in this Appendix and the above results hold for $t>0$. For $t<0$, one can use the symmetry $J_{2}(t)=J_{2}(-t)$ and $J_{3}(t)=-J_{3}(-t)$. Finally, evaluating the asymptotic of the expressions (A) for $\omega_{c}t\ll 1$ and $\omega_{c}t\gg 1$, one arrives at the results (20-25).