Transport fluctuation relations in interacting quantum pumps

We are interested in time-dependently driven quantum systems, exchanging particles and energy with reservoirs. We make the simplifying assumption that the tunnel coupling between the system and the reservoirs is weak, which is standard in the field [2–4, 6–9, 12, 14–16]. To study the effects of adiabatic pumping, we include a periodic driving of the system, which is slow with respect to the decay time of reservoir correlations. We will quantify these conditions in detail, in section 4.

We describe the dynamics of the system, using its reduced density matrix, represented in vector form $\vert P\left. \right)$, which is obtained from the full density matrix by tracing out the reservoir degrees of freedom. The reduced density matrix fulfills a master equation

$$\begin{equation}\vert \dot {P}\left. \right)=\mathbf{W}\left(t\right)\vert P\left. \right).\end{equation} \tag{ 1 }$$

Here, W(t) is a kernel, carrying all the transition rates of the stochastic processes that change the system state $\vert P\left. \right)$. We here assume that the dynamics of the occupation probabilities of the system states does not couple to the dynamics of coherent superpositions, such that only the diagonal elements of the reduced density matrix matter ³ . In the weak coupling limit, which we are considering here, the rates can be computed through Fermi's golden rule of the frozen system where the time-dependence enters parametrically [60]. We otherwise keep a generic model for the following considerations and only specify a specific system in section 4.

Through the eigendecomposition of the kernel W, we can understand the dynamics of the quantum system. Namely, the kernel can be decomposed into the form $\mathbf{W}\left(t\right)={\sum }_{k}{\lambda }_{k}\left(t\right)\vert k\left(t\right)\left. \right)\left(\right.\enspace k\left(t\right)\vert$, where the $\vert k\left. \right)$ and $\left(\right.\enspace k\vert$ are the right and left eigenvectors of W, belonging to the eigenvalue λ_k . The notation is chosen such that any $\vert a\left. \right)$ mathematically represents an operator, cast into vector form, whereas any $\left(a\right\vert$ represents a map from an operator to a scalar [61–63]. There is a zero mode k = 0, which expresses that for each t there exists a unique stationary state $\vert 0\left. \right)$ with λ₀ = 0. The corresponding (dual) left eigenvector $\left(0\right\vert$ is the trace operator, expressing the trace preserving property of W(t). The other eigenvalues and eigenmodes do not play a role in the long-time FCS of this slowly driven system, see equation (40) of reference [48].

Importantly, the system is coupled to several reservoirs, enumerated with α (and γ). In the weak coupling limit, the influence from the different reservoirs is additive W = ∑_α W_α . We assume microreversibility and local equilibrium for each reservoir, such that kernels have the symmetry [6] ⁴

$$\begin{equation}{\mathbf{W}}_{\alpha }^{\text{T}}\left(t\right)={\text{e}}^{{\beta }_{\alpha }\left[{\mu }_{\alpha }\mathbf{n}-\mathbf{e}\left(t\right)\right]}{\mathbf{W}}_{\alpha }\left(t\right){\text{e}}^{-{\beta }_{\alpha }\left[{\mu }_{\alpha }\mathbf{n}-\mathbf{e}\left(t\right)\right]}.\end{equation} \tag{ 2 }$$

where β_α = 1/(k_B T_α ) and μ_α are, respectively, the inverse temperature and chemical potential of reservoir α. The particle number and energy superoperators associated to the local quantum system, $\mathbf{n}=\frac{1}{2}\left\{\hat{n},\cdot \right\}$ and $\mathbf{e}=\frac{1}{2}\left\{\hat{{\epsilon}},\cdot \right\}$, are defined via the anticommutators of the local particle number and energy operators, $\hat{n}$ and $\hat{{\epsilon}}$, respectively. They will be explicitly defined, when considering an explicit model system, see section 4.

We use the following notation throughout this work. The hat designates operators, which, when cast into vector form (in the superoperator context), are written in the round bra notation, $\vert \dots \left. \right)$. Superoperators are written in bold font, whereas regular fonts are used for scalars.

Finally, let us note that in principle, the chemical potential and temperature gradients may be time-dependent, too, since equation (2) is a time-local symmetry. However, in the remainder of this work, we keep μ_α and β_α constant in time in order to be able to clearly separate the response due to driving and the response due to chemical potential or temperature gradients. We will consider a time-dependent driving of the local system parameters, and the coupling amplitudes to the reservoirs, see section 4 for a concrete example.

We are not only interested in the mean dynamics of the system, but also in the FCS. Through integrating out the reservoir degrees of freedom, one has in principle lost all information about the transport statistics. However, one can keep track of the information of particle and energy transport by supplementing the kernels with counting fields (see reference [6] for a review),

$$\begin{equation}\mathbf{W}\left(\left\{{\chi }_{\alpha },{\xi }_{\alpha }\right\},t\right)=\sum\limits _{\alpha }{\text{e}}^{-\text{i}\mathbf{e}\left(t\right){\xi }_{\alpha }}{\text{e}}^{-\text{i}\mathbf{n}{\chi }_{\alpha }}{\mathbf{W}}_{\alpha }\left(t\right){\text{e}}^{\text{i}\mathbf{n}{\chi }_{\alpha }}{\text{e}}^{\text{i}\mathbf{e}\left(t\right){\xi }_{\alpha }}.\end{equation} \tag{ 3 }$$

The counting fields χ_α and ξ_α keep track of the number of particles and the energy that enter reservoir α. Let us stress again, our main interest is in the charge currents, counted by χ_α . However, as we will show, the FRR for the charge current cannot be formulated and interpreted unless the energy currents are accounted for as well, such that we have to keep ξ_α .

Also for the kernel including counting fields, microreversibility imposes a symmetry, similar to equation (2), which can be expressed in terms of the counting fields through

$$\begin{equation}{\mathbf{W}}^{\text{T}}\left(\left\{{\chi }_{\alpha },{\xi }_{\alpha }\right\},t\right)=\mathbf{W}\left(\left\{\text{i}{\beta }_{\alpha }{\mu }_{\alpha }-{\chi }_{\alpha },-\text{i}{\beta }_{\alpha }-{\xi }_{\alpha }\right\},t\right).\end{equation} \tag{ 4 }$$

This symmetry is of central importance for the formulation of fluctuation relations out of equilibrium [2–4, 6, 7, 9, 11, 13–16].

In order to obtain the particle and energy transport statistics, one needs to construct a so-called cumulant-generating function. One therefore starts from an arbitrary initial state $\vert {P}_{0}\left. \right)$ at time t_in and then switches on the counting fields to measure the FCS until a certain time τ. Tracing over the remaining system degrees of freedom through application of $\left(\right.\enspace 0\vert$, one finds the cumulant generating function, $\mathcal{F}\left(\left\{{\chi }_{\alpha },{\xi }_{\alpha }\right\},\tau ,{t}_{\text{in}}\right)$, from the evolution of the system in presence of the counting fields

$$\begin{equation}{\text{e}}^{\mathcal{F}\left(\left\{{\chi }_{\alpha },{\xi }_{\alpha }\right\},\tau ,{t}_{\text{in}}\right)\left(\tau -{t}_{\text{in}}\right)}=\left(0\vert {\Pi}\left(\left\{{\chi }_{\alpha },{\xi }_{\alpha }\right\},\tau ,{t}_{\text{in}}\right)\vert {P}_{0}\right).\end{equation} \tag{ 5 }$$

Here, we have introduced the propagator of the open system

$$\begin{equation}{\Pi}\left(\left\{{\chi }_{\alpha },{\xi }_{\alpha }\right\},\tau ,{t}_{\text{in}}\right)=\mathcal{T}{\text{e}}^{{\int }_{{t}_{\text{in}}}^{\tau }\mathrm{d}t\mathbf{W}\left(\left\{{\chi }_{\alpha },{\xi }_{\alpha }\right\},t\right)}.\end{equation} \tag{ 6 }$$

The cumulants of the charge- and energy currents can now be computed by differentiating with respect to the counting fields of interest, and subsequently setting all counting fields to zero. For the remainder of this paper, we focus on the limit of very long measurement times (τ − t_in)/t_typ → ∞, namely when τ − t_in is much larger than typical system time scales t_typ given by the inverse of kernel eigenvalues λ_k ≠ 0. Then $\mathcal{F}$ neither depends on τ nor t_in and provides the zero-frequency cumulants. In this case, the FCS does no longer depend on the initial state $\vert {P}_{0}\left. \right)$, since for τ − t_in → ∞, only the stationary state $\vert 0\left. \right)$ contributes to the transport statistics. Without loss of generality, we therefore set our initial state to be the stationary state in absence of the transport counting $\vert {P}_{0}\left. \right)=\vert 0\left(0\right)\left. \right)$. The zero-frequency cumulants of interest in the present paper, namely the average charge current into reservoir α and the related current–current correlations, are then defined as

$$\begin{equation}{I}_{\alpha }\equiv {I}_{\alpha }\left(\tau -{t}_{\text{in}}\to \infty \right)=-\text{i}\enspace \underset{\tau -{t}_{\text{in}}\to \infty }{\mathrm{lim}}\left[{\partial }_{{\chi }_{\alpha }}{\left.\mathcal{F}\right\vert }_{\left\{{\chi }_{\alpha },{\xi }_{\alpha }\right\}\to 0}\right],\end{equation} \tag{ 7 }$$

as well as

$$\begin{equation}{S}_{\alpha \gamma }\equiv {S}_{\alpha \gamma }\left(\tau -{t}_{\text{in}}\to \infty \right)=-\underset{\tau -{t}_{\text{in}}\to \infty }{\mathrm{lim}}\left[{\partial }_{{\chi }_{\alpha }}{\partial }_{{\chi }_{\gamma }}{\left.\mathcal{F}\right\vert }_{\left\{{\chi }_{\alpha },{\xi }_{\alpha }\right\}\to 0}\right].\end{equation} \tag{ 8 }$$

Importantly, and as foreshadowed already, also the energy current into reservoir α is found to play an important role in the present paper; it is defined via a derivative with respect to the energy counting field ξ_α ,

$$\begin{equation}{I}_{\text{E},\alpha }\equiv {I}_{\text{E},\alpha }\left(\tau -{t}_{\text{in}}\to \infty \right)=-\text{i}\enspace \underset{\tau -{t}_{\text{in}}\to \infty }{\mathrm{lim}}\left[{\partial }_{{\xi }_{\alpha }}{\left.\mathcal{F}\right\vert }_{\left\{{\chi }_{\alpha },{\xi }_{\alpha }\right\}\to 0}\right].\end{equation} \tag{ 9 }$$

In order to find explicit expressions for the cumulant generating function, we now focus on the limit of slow driving as previously considered in [25, 45]. This means that the time-scale τ₀ = 2π/Ω, related to the inverse of the driving frequency Ω, is large with respect to the time scale on which the system states vary. In this adiabatic limit, the cumulant generating function $\mathcal{F}$ can be expanded in orders of the driving parameters, see appendix A. We evaluate it up to first order in the small driving parameter, $\mathcal{F}\approx {\mathcal{F}}^{\left(0\right)}+{\mathcal{F}}^{\left(1\right)}$. The zeroth-order, instantaneous contribution is given by ⁵

$$\begin{equation}{\mathcal{F}}^{\left(0\right)}\left(\left\{{\chi }_{\alpha },{\xi }_{\alpha }\right\}\right)={\int }_{0}^{{\tau }_{0}}\frac{\mathrm{d}t}{{\tau }_{0}}{\lambda }_{0}\left(\left\{{\chi }_{\alpha },{\xi }_{\alpha }\right\}\right).\end{equation} \tag{ 10 }$$

This is just a time-averaged version of the FCS which appear for a system without time-dependent driving. On top of that, we find the pumping contribution

$$\begin{equation}{\mathcal{F}}^{\left(1\right)}\left(\left\{{\chi }_{\alpha },{\xi }_{\alpha }\right\}\right)=-{\int }_{0}^{{\tau }_{0}}\frac{\mathrm{d}t}{{\tau }_{0}}\left(0\left(\left\{{\chi }_{\alpha },{\xi }_{\alpha }\right\}\right)\vert {\partial }_{t}\vert 0\left(\left\{{\chi }_{\alpha },{\xi }_{\alpha }\right\}\right)\right).\end{equation} \tag{ 11 }$$

This reproduces the result derived in reference [45] and shows the clearly geometric property of the adiabatic pumping transport statistics. This has important consequences: first of all, it is the properties of the eigenvectors and not only of the eigenvalues that enter here. Second, due to the time-dependent driving of system energies e(t), it can already be expected that the time-derivative will lead to terms involving the energy counting field, see also equation (3).

Importantly, due to the above introduced symmetries of W({χ_α , ξ_α }) [see equation (4)], we can derive fluctuation relations for $\mathcal{F}$. Namely we find that the instantaneous cumulant generating function satisfies the same relations as for a system without driving,

$$\begin{equation}{\mathcal{F}}^{\left(0\right)}\left(\left\{{\chi }_{\alpha },{\xi }_{\alpha }\right\}\right)={\mathcal{F}}^{\left(0\right)}\left(\left\{\text{i}{\beta }_{\alpha }{\mu }_{\alpha }-{\chi }_{\alpha },-\text{i}{\beta }_{\alpha }-{\xi }_{\alpha }\right\}\right),\end{equation} \tag{ 12 }$$

whereas the pumping contribution satisfies

$$\begin{equation}{\mathcal{F}}^{\left(1\right)}\left(\left\{{\chi }_{\alpha },{\xi }_{\alpha }\right\}\right)=-{\mathcal{F}}^{\left(1\right)}\left(\left\{\text{i}{\beta }_{\alpha }{\mu }_{\alpha }-{\chi }_{\alpha },-\text{i}{\beta }_{\alpha }-{\xi }_{\alpha }\right\}\right).\end{equation} \tag{ 13 }$$

The minus sign for the pumping cumulant generating function reflects the fact that in order to satisfy microreversibility, the direction of pumping transport has to be inverted, too. The result shown in equations (12) and (13) has in another form been found previously [25] (i.e., in the form of a generic counting field, which could in principle encompass either charge or energy, or another observable quantity). Contrary to reference [25] however, we explicitly introduce the separate counting fields of charge and energy currents. It is this explicit distinction, which allows a careful derivation and interpretation of charge current FRR, as we will present them in the following.

As already announced, the first crucial step is to consider global gauge transformations of the cumulant generating function with respect to the counting fields. Both the instantaneous (i = 0) and the pumping contribution (i = 1) to the cumulant generating function, $\mathcal{F}$, are invariant with respect to global shifts of the particle counting fields,

$$\begin{equation}{\mathcal{F}}^{\left(i\right)}\left(\left\{{\chi }_{\alpha }+\delta \chi ,{\xi }_{\alpha }\right\}\right)={\mathcal{F}}^{\left(i\right)}\left(\left\{{\chi }_{\alpha },{\xi }_{\alpha }\right\}\right).\end{equation} \tag{ 14 }$$

This invariance reflects the conservation of charge currents, which is valid irrespective of the presence or absence of a time-dependent driving. When applied on the level of the individual cumulants, we can derive the important and well-known identities

$$\begin{equation}\sum\limits _{\alpha }{I}_{\alpha }^{\left(i\right)}=0\end{equation} \tag{ 15a }$$

$$\begin{equation}\sum\limits _{\alpha }{S}_{\alpha \gamma }^{\left(i\right)}=\sum\limits _{\gamma }{S}_{\alpha \gamma }^{\left(i\right)}=0.\end{equation} \tag{ 15b }$$

However, the instantaneous and pumping contributions behave fundamentally differently from equation (14) with respect to a similar gauge in the energy counting fields, a fact which will be central for the rest of the discussion. While the instantaneous contribution exhibits a similar gauge invariance for ξ,

$$\begin{equation}{\mathcal{F}}^{\left(0\right)}\left(\left\{{\chi }_{\alpha },{\xi }_{\alpha }+\delta \xi \right\}\right)={\mathcal{F}}^{\left(0\right)}\left(\left\{{\chi }_{\alpha },{\xi }_{\alpha }\right\}\right),\end{equation} \tag{ 16 }$$

the pumping correction does not. On the contrary, here, we receive an extra term

$$\begin{equation}{\mathcal{F}}^{\left(1\right)}\left(\left\{{\chi }_{\alpha },{\xi }_{\alpha }+\delta \xi \right\}\right)={\mathcal{F}}^{\left(1\right)}\left(\left\{{\chi }_{\alpha },{\xi }_{\alpha }\right\}\right)-\text{i}\delta \xi \enspace {\mathcal{Q}}^{\left(1\right)}\left(\left\{{\chi }_{\alpha },{\xi }_{\alpha }\right\}\right),\end{equation} \tag{ 17 }$$

where we defined

$$\begin{equation}{\mathcal{Q}}^{\left(1\right)}\left(\left\{{\chi }_{\alpha },{\xi }_{\alpha }\right\}\right)=-{\int }_{0}^{{\tau }_{0}}\frac{\mathrm{d}t}{{\tau }_{0}}\left(0\left(\left\{{\chi }_{\alpha },{\xi }_{\alpha }\right\}\right)\vert \dot {\mathbf{e}}\vert 0\left(\left\{{\chi }_{\alpha },{\xi }_{\alpha }\right\}\right)\right).\end{equation} \tag{ 18 }$$

Formally, the origin of this different gauge behaviour, equations (14), (16) and (17), can be found when considering the kernel W({χ_α , ξ_α }). Constant global shifts of the counting fields result in unitary transformations of the kernel, W({χ_α + δχ, ξ_α }) = e^{−in δχ} W({χ_α , ξ_α })e^{in δχ} respectively W({χ_α , ξ_α + δξ}) = e^−ie(t)δξ W({χ_α , ξ_α })e^ie(t)δξ . Consequently, the eigenvalues of W remain unchanged, hence the gauge invariance of ${\mathcal{F}}^{\left(0\right)}$. The eigenvectors however, are transformed through the unitary super-operators exp(±in δχ) and exp(±ie(t)δξ). Crucially, while a global shift in the particle counting field results in a time-independent transformation, the shift in the energy counting fields produces a time-dependent transformation. Hence, because of the geometric form of ${\mathcal{F}}^{\left(1\right)}$, see equation (11), it is easy to realize that the latter shift can in general not be eliminated, but results in the extra term on the right-hand side of equation (17).

Moreover, the breaking of this global symmetry can also be understood in physical terms. Namely, starting from equation (16), and solving for $\mathcal{Q}$, it is possible to show that

$$\begin{equation}-\text{i}{\mathcal{Q}}^{\left(1\right)}=\sum\limits _{\gamma }{\partial }_{{\xi }_{\gamma }}{\mathcal{F}}^{\left(1\right)}.\end{equation} \tag{ 19 }$$

Consequently, when putting all counting fields to zero, we can directly relate this function to the energy current provided by the external pumping fields,

$$\begin{equation}{\mathcal{Q}}^{\left(1\right)}\left(\left\{0,0\right\}\right)=-{\int }_{0}^{{\tau }_{0}}\frac{\mathrm{d}t}{{\tau }_{0}}\left(0\vert \dot {\mathbf{e}}\vert 0\right)=-\sum\limits _{\alpha }{I}_{\text{E},\alpha }^{\left(1\right)}\equiv {I}_{\text{E},\text{pump}}^{\left(1\right)}.\end{equation} \tag{ 20 }$$

This reflects the very simple fact that energy is in general not conserved in the presence of an external drive, and can instead be pumped into the system. Charge conservation on the other hand is guaranteed even in the presence of the drive, such that no similar term arises for shifts in χ.

Let us now proceed and study the FRR for the pumping charge currents. We therefore have to determine the degree to which charge and energy counting fields can be disentangled in the presence of driving. The above derived gauge considerations are instrumental for this. From now on, we set β_α = β to avoid that the effect of driving is obscured. For the instantaneous contribution, equation (12), the energy counting fields can then just be set to zero (no counting of energy currents), and the global shift −iβ appearing in the energy counting field argument can be gauged away [due to equation (16)]. This simply gives rise to

$$\begin{equation}{\mathcal{F}}^{\left(0\right)}\left(\left\{{\chi }_{\alpha }\right\}\right)={\mathcal{F}}^{\left(0\right)}\left(\left\{\text{i}\beta {\mu }_{\alpha }-{\chi }_{\alpha }\right\}\right).\end{equation} \tag{ 21 }$$

This is the time-averaged version of the well-known fluctuation relations for time-independent systems [2, 3] (see also equation (10)). Based on the above discussion, it is obvious that we cannot achieve the same 'charge current-only' response relations for the pumping contribution, equation (13). Namely, one cannot get rid of the pumping energy input I_E,pump, and instead receives

$$\begin{equation}{\mathcal{F}}^{\left(1\right)}\left(\left\{{\chi }_{\alpha }\right\}\right)=-{\mathcal{F}}^{\left(1\right)}\left(\left\{\text{i}\beta {\mu }_{\alpha }-{\chi }_{\alpha }\right\}\right)+\beta \enspace {\mathcal{Q}}^{\left(1\right)}\left(\left\{\text{i}\beta {\mu }_{\alpha }-{\chi }_{\alpha }\right\}\right).\end{equation} \tag{ 22 }$$

We note that equation (22) automatically implies that

$$\begin{equation}{\mathcal{Q}}^{\left(1\right)}\left(\left\{{\chi }_{\alpha }\right\}\right)={\mathcal{Q}}^{\left(1\right)}\left(\left\{\text{i}\beta {\mu }_{\alpha }-{\chi }_{\alpha }\right\}\right),\end{equation} \tag{ 23 }$$

which can be seen by replacing χ_α → iβμ_α − χ_α in equation (22). This means that remarkably, ${\mathcal{Q}}^{\left(1\right)}$ itself does satisfy a fluctuation relation quite similar to the instantaneous ${\mathcal{F}}^{\left(0\right)}$.

Equation (22) is one of the central results of this paper. We conclude that it is in general impossible to derive FRR for the charge pumping contribution, which involve only charge currents. The energy currents always enter due to the presence of the term ${\mathcal{Q}}^{\left(1\right)}$. We will see in the next section, that this contribution due to the pumping energy input is omnipresent when expressing the pumping fluctuation relations in terms of the individual cumulants, that is, the response relations.

The cumulant generating functions can now be expanded in terms of the counting fields and of the gradients in chemical potentials, in order to derive a resulting hierarchy of equations relating cumulants of different orders, yielding FRR. The derivation is detailed in appendix B.

In fact, the FRR for both instantaneous and first order in pumping can be expressed in a very compact form when introducing the total power provided by the external pumping field and the applied biases

$$\begin{equation}{J}_{\text{pump}}\equiv -\sum\limits _{\alpha }\left({I}_{\text{E},\alpha }-{\mu }_{\alpha }{I}_{\alpha }\right)\equiv {I}_{\text{E},\text{pump}}+\sum\limits _{\alpha }{\mu }_{\alpha }{I}_{\alpha }.\end{equation} \tag{ 24 }$$

Then, we find both for the instantaneous (i = 0) and for the adiabatic-response contribution (i = 1),

$$\begin{equation}0={\left.\frac{\partial {J}_{\text{pump}}^{\left(i\right)}}{\partial {\mu }_{\alpha }}\right\vert }_{\left\{{\mu }_{\alpha }\right\}\to \mu }\end{equation} \tag{ 25 }$$

$$\begin{equation}{\left.{S}_{\alpha \gamma }^{\left(i\right)}\right\vert }_{\left\{{\mu }_{\alpha }\right\}\to \mu }={k}_{\text{B}}T{\left.\frac{{\partial }^{2}{J}_{\text{pump}}^{\left(i\right)}}{\partial {\mu }_{\alpha }\partial {\mu }_{\gamma }}\right\vert }_{\left\{{\mu }_{\alpha }\right\}\to \mu }.\end{equation} \tag{ 26 }$$

However, when expressed in terms of the power, the subtly different behavior of the instantaneous and pumping fluctuation relations is not clearly visible. We therefore from now on distinguish explicitly charge and energy currents, and discuss the resulting relations in detail.

For the instantaneous order, we get a time-averaged version of the response relations found also in reference [2, 3]. The lowest order relations read

$$\begin{equation}{\left.{I}_{\alpha }^{\left(0\right)}\right\vert }_{\left\{{\mu }_{\alpha }\right\}\to \mu }=0\end{equation} \tag{ 27 }$$

$$\begin{equation}{\left.{S}_{\alpha \gamma }^{\left(0\right)}\right\vert }_{\left\{{\mu }_{\alpha }\right\}\to \mu }={k}_{\text{B}}T{\left.\left(\frac{\partial {I}_{\alpha }^{\left(0\right)}}{\partial {\mu }_{\gamma }}+\frac{\partial {I}_{\gamma }^{\left(0\right)}}{\partial {\mu }_{\alpha }}\right)\right\vert }_{\left\{{\mu }_{\alpha }\right\}\to \mu }.\end{equation} \tag{ 28 }$$

The first relation states that the time-averaged instantaneous currents must be zero in the absence of gradients in the chemical potentials, as expected. The second relation is a time-averaged version of the famous fluctuation–dissipation theorem.

As for the adiabatic-response, i.e. the pumping contribution, we find a first relation of the form

$$\begin{equation}0={\left.{I}_{\text{E},\text{pump}}^{\left(1\right)}\right\vert }_{\left\{{\mu }_{\alpha }\right\}\to \mu },\end{equation} \tag{ 29 }$$

stating very simply that the pumping energy input is zero in the absence of chemical potential differences. This relation has no equivalent (or would be trivial) in the instantaneous order. It can be interpreted as follows. While one can induce a pumping current in the absence of chemical potential gradients (see also the FRR below), the electrons will be shuffled from one reservoir to the other, without putting in or extracting work. In addition, we find

$$\begin{equation}{\left.{I}_{\alpha }^{\left(1\right)}\right\vert }_{\left\{{\mu }_{\alpha }\right\}\to \mu }={\left.\frac{\partial {I}_{\text{E},\text{pump}}^{\left(1\right)}}{\partial {\mu }_{\alpha }}\right\vert }_{\left\{{\mu }_{\alpha }\right\}\to \mu }\end{equation} \tag{ 30 }$$

$$\begin{equation}{\left.{S}_{\alpha \gamma }^{\left(1\right)}\right\vert }_{\left\{{\mu }_{\alpha }\right\}\to \mu }={k}_{\text{B}}T{\left.\left(\frac{\partial {I}_{\alpha }^{\left(1\right)}}{\partial {\mu }_{\gamma }}+\frac{\partial {I}_{\gamma }^{\left(1\right)}}{\partial {\mu }_{\alpha }}-\frac{{\partial }^{2}{I}_{\text{E},\text{pump}}^{\left(1\right)}}{\partial {\mu }_{\alpha }\partial {\mu }_{\gamma }}\right)\right\vert }_{\left\{{\mu }_{\alpha }\right\}\to \mu }.\end{equation} \tag{ 31 }$$

Here, we see explicitly, what we have already indicated in the previous section, in equation (22), namely that the pumping energy input necessarily appears when attempting to formulate relations between different charge current cumulants. This seems to have been overlooked so far. While it is a well-known result (in many different physical regimes) that the pumping current can be non-zero even in the absence of a voltage bias [64, 66–69] we here show, that this nonzero pumping current can be related to the linear response of the energy current provided by the external pumping fields, I_E,pump.

In equation (31), we recover a second FRR for pumping. Namely, while the instantaneous relation for the current noise, equation (28), satisfies a stationary (equilibrium) fluctuation dissipation theorem, this simple form cannot be extended to the pumping noise. Instead, it receives a correction due to the nonlinear response of I_E,pump. A deviation from the fluctuation–dissipation theorem for the pumping noise has already been noted by us in reference [35] for a concrete single-level quantum dot model. In other works it has been implied that the stationary FDT does not extend to first order due to the geometric nature of the pumping cumulant generating function [8, 25]. However, in neither of these works the deviation from the stationary FDT has been explicitly identified or interpreted. Here, we find its explicit physical meaning as the nonlinear response of the pumping energy input.

In this section, we show under which conditions the deviations from the standard FRR stemming from nonlinearities in the pumping energy input play a role.

The term, which we want to analyze in more detail here is the nonlinearity in the pumping energy input

$$\begin{equation}{\left.\frac{{\partial }^{2}{I}_{\text{E},\text{pump}}^{\left(1\right)}}{\partial {\mu }_{\alpha }\partial {\mu }_{\gamma }}\right\vert }_{\left\{{\mu }_{\alpha }\right\}\to \mu }.\end{equation} \tag{ 32 }$$

We start by considering the cross correlations, namely α ≠ γ. One immediately notices then that this term is nonzero only, if at least one of the summands in ${I}_{\text{E},\text{pump}}^{\left(1\right)}$ depends on the electrochemical potential of two different reservoirs. However, it is well known that in systems where many-body interactions are negligible the pumping charge and energy currents can be written in terms of incoming and outgoing effective distributions, where the different electrochemical potentials contribute to different summands, see e.g. equations (9) and (12) of reference [70]. This results in

$$\begin{equation}{\left.\frac{{\partial }^{2}{I}_{\text{E},\text{pump}}^{\left(1\right)}}{\partial {\mu }_{\alpha }\partial {\mu }_{\gamma }}\right\vert }_{\left\{{\mu }_{\alpha }\right\}\to \mu }\text{no}\;\text{interactions}{\to }0.\end{equation} \tag{ 33 }$$

Crucially, we thus find that an equilibrium version of the FRR can be recovered for vanishing interactions. This is a further main result of this work. The deviation from the stationary FDT for the pumping response constitutes a purely interaction-induced effect, which is expected to be directly measurable in a noise or counting experiment. Moreover, this is a significant generalization of the result found in reference [35], where the recovery of the FRR for the pumping noise was found for the special case of a single-level quantum dot, including only two contacts. Here, the proof encompasses the broad class of generic quantum systems, weakly coupled to an arbitrary number of reservoirs, see section 2.

We now show how the above extends to autocorrelations via current conservation. Using equation (15a), one directly observes that the autocorrelations can be expressed as

$$\begin{equation}{\left.{S}_{\alpha \alpha }^{\left(1\right)}\right\vert }_{\left\{{\mu }_{\alpha }\right\}\to \mu }=-\sum\limits _{\gamma \ne \alpha }{\left.{S}_{\alpha \gamma }^{\left(1\right)}\right\vert }_{\left\{{\mu }_{\alpha }\right\}\to \mu }.\end{equation} \tag{ 34 }$$

The constraint given in equation (34) can be rewritten using equation (31), leading to

$$\begin{equation}\sum\limits _{\gamma }{\left.\frac{\partial {I}_{\alpha }^{\left(1\right)}}{\partial {\mu }_{\gamma }}\right\vert }_{\left\{{\mu }_{\alpha }\right\}\to \mu }=\sum\limits _{\gamma }{\left.\frac{{\partial }^{2}{I}_{\text{E},\text{pump}}^{\left(1\right)}}{\partial {\mu }_{\alpha }\partial {\mu }_{\gamma }}\right\vert }_{\left\{{\mu }_{\alpha }\right\}\to \mu }.\end{equation} \tag{ 35 }$$

Note, that the left-hand side of this equality is the sum over the adiabatic-response corrections to the charge conductances. In a non-driven system, this sum has to always equal zero. However, since the driving fields induce pumping charge currents, the extra term on the right-hand side of equation (35) appears, thereby constituting a generalization of the conductance sums for time-dependently driven systems.

Equation (35) can be rewritten as

$$\begin{equation}{\partial }_{\mu }\left({\left.{I}_{\alpha }^{\left(1\right)}\right\vert }_{\left\{{\mu }_{\alpha }\right\}\to \mu }\right)={\partial }_{\mu }\left({\left.\frac{\partial {I}_{\text{E},\text{pump}}^{\left(1\right)}}{\partial {\mu }_{\alpha }}\right\vert }_{\left\{{\mu }_{\alpha }\right\}\to \mu }\right).\end{equation} \tag{ 36 }$$

This shows that the correction of the sum of conductances can be derived from equation (30), by means of the derivative with respect to μ. Hence, the additional constraints for the auto-correlations are already fixed by the FRR for currents, equation (30).

We consider a quantum dot model with a single, spin-degenerate level, tunnel-coupled to two reservoirs which may be at a different chemical potential, see figure 1(a). We have analyzed the charge pumping noise of this model in reference [35], and found that a deviation from the stationary fluctuation dissipation theorem occurs due to the time-dependent driving. Here, we identify the concrete physical mechanism for this deviation based on the pumping fluctuation relations presented in the previous section.

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The total Hamiltonian is of the form

$$\begin{equation}\hat{H}={\hat{H}}_{\text{QD}}\left(t\right)+{\hat{H}}_{\text{T}}\left(t\right)+\sum\limits _{\alpha =\text{L,R}}{\hat{H}}_{\alpha }.\end{equation} \tag{ 37 }$$

The local quantum dot Hamiltonian can be expressed as

$$\begin{equation}{\hat{H}}_{\text{QD}}=\sum\limits _{\sigma }{\epsilon}\left(t\right){\hat{d}}_{\sigma }^{{\dagger}}{\hat{d}}_{\sigma }+\frac{U}{2}\hat{n}\left(\hat{n}-1\right)\equiv \hat{{\epsilon}}.\end{equation} \tag{ 38 }$$

Here, the operators ${\hat{d}}_{\sigma }^{\left({\dagger}\right)}$ annihilate (create) an electron with spin σ and energy (t). The parameter U expresses the magnitude of the onsite Coulomb interaction, which is sensitive to the electron occupation number $\hat{n}={\sum }_{\sigma }{\hat{d}}_{\sigma }^{{\dagger}}{\hat{d}}_{\sigma }$. As indicated, the quantum dot Hamiltonian corresponds to the local energy operator $\hat{{\epsilon}}$. Out of these two operators, one can construct the explicit superoperators n and e, as introduced in section 2. Note that while U may be time-dependent, too (as we will discuss later) we here keep it constant, in order to clearly distinguish between the interacting and non-interacting regimes. The Hamiltonians of the reservoirs are given as

$$\begin{equation}{\hat{H}}_{\alpha }=\sum\limits _{k\sigma }{{\epsilon}}_{k}{\hat{c}}_{\alpha k\sigma }^{{\dagger}}{\hat{c}}_{\alpha k\sigma },\end{equation} \tag{ 39 }$$

where likewise the operators ${\hat{c}}_{\alpha k\sigma }^{\left({\dagger}\right)}$ annihilate (create) an electron in reservoir α, with momentum k and spin σ, as well as energy _k . Finally, the tunneling Hamiltonian is

$$\begin{equation}{\hat{H}}_{\text{T}}\left(t\right)=\sum\limits _{\alpha k\sigma }{\gamma }_{\alpha }\left(t\right){\hat{c}}_{\alpha k\sigma }^{{\dagger}}{\hat{d}}_{\sigma }+\text{h.c.},\end{equation} \tag{ 40 }$$

where γ_α (t) denotes the time-dependent tunneling amplitude. All the time-dependent parameters shall be subject to a periodic driving, x(t + τ₀) = x(t), where the pumping period is τ₀ = 2π/Ω with the driving frequency Ω. To simplify the notation, we will from now on omit the brackets (t) for these parameters, and for any further quantities depending on them.

We now present the rate equation of the quantum dot system, in presence of the tunnel coupling. In the sequential tunneling limit, valid for small tunnel couplings, Γ_α ≪ k_B T, with Γ_α = 2πρ_α |γ_α |², the dynamics of the system are of the form of equation (1) introduced in section 2. For the here considered system, the reduced density matrix in vector form reads $\vert P\left. \right)={\left({P}_{0},{P}_{{\uparrow}},{P}_{{\downarrow}},{P}_{2}\right)}^{\text{T}}$, where only the diagonal elements are relevant ⁶ . They contain the occupation probabilities of the quantum dot with the probability of the quantum dot being empty P₀, singly occupied with either an ↑ or ↓ electron P_↑,↓, or doubly occupied P₂. In the following, we focus on a regime of adiabatic driving ⁷ , Ω ≪ Γ_α , where the expansion presented in appendix A applies ⁸ and where we compute the dynamics following references [60, 64].

The kernel describing the time evolution can be given as W = ∑_α=L,R W_α and,

$$\begin{equation}{\mathbf{W}}_{\alpha }={{\Gamma}}_{\alpha }\left(\begin{matrix}\hfill -2{f}_{\alpha }\hfill & \hfill {\bar{f}}_{\alpha }\hfill & \hfill {\bar{f}}_{\alpha }\hfill & \hfill 0\hfill \\ \hfill {f}_{\alpha }\hfill & \hfill -{\bar{f}}_{\alpha }-{f}_{\alpha }^{U}\hfill & \hfill 0\hfill & \hfill {\bar{f}}_{\alpha }^{U}\hfill \\ \hfill {f}_{\alpha }\hfill & \hfill 0\hfill & \hfill -{\bar{f}}_{\alpha }-{f}_{\alpha }^{U}\hfill & \hfill {\bar{f}}_{\alpha }^{U}\hfill \\ \hfill 0\hfill & \hfill {f}_{\alpha }^{U}\hfill & \hfill {f}_{\alpha }^{U}\hfill & \hfill -2{\bar{f}}_{\alpha }^{U}\hfill \end{matrix}\right)\end{equation} \tag{ 41 }$$

with f_α = f( − μ_α ), ${f}_{\alpha }^{U}=f \left({\epsilon}+U-{\mu }_{\alpha }\right)$, and $\bar{f}=1-f$, where $f \left(E\right)=1/\left({\text{e}}^{E/{k}_{\text{B}}T}+1\right)$ is the Fermi function. Likewise, we can write the particle number and energy superoperators as n = diag(0, 1, 1, 2) and e = diag(0, , , 2 + U). With the above ingredients, we can now construct the kernel including the particle and energy current counting fields, according to equation (3). Thus we find the FCS of a quantum dot pump, and can therefore test the fluctuation relations elaborated in the previous sections. In particular, the kernels W_α fulfill the symmetry relation of equation (2), and consequently equation (3) follows from that.

The analytic expressions for the pumping current and pumping noise for this single-level quantum-dot pump have already been computed in reference [35]. For the purpose of this paper, we need the expression for the current for arbitrary chemical potentials (because we will have to compute the conductance below), whereas the noise is only required at μ_L = μ_R = μ. The respective expressions are

$$\begin{equation}{\left.{I}_{\text{L}}^{\left(1\right)}\right\vert }_{\left\{{\mu }_{\alpha }\right\}\to \mu }=-\frac{1}{2}{\int }_{0}^{{\tau }_{0}}\frac{\mathrm{d}t}{{\tau }_{0}}\frac{{\lambda }_{\text{c,L}}-{\lambda }_{\text{c,R}}}{{\lambda }_{\text{c}}}{\partial }_{t}\left\langle \hat{n}\right\rangle \end{equation} \tag{ 42 }$$

$$\begin{equation}{\left.{S}_{\text{LR}}^{\left(1\right)}\right\vert }_{\left\{{\mu }_{\alpha }\right\}\to \mu }={\int }_{0}^{{\tau }_{0}}\frac{\mathrm{d}t}{{\tau }_{0}}\frac{{{\Gamma}}_{\text{L}}{{\Gamma}}_{\text{R}}}{{{\Gamma}}^{2}}{\partial }_{t}{\Delta}n,\end{equation} \tag{ 43 }$$

where ${\lambda }_{\text{c},\alpha }={{\Gamma}}_{\alpha }\left(1+{f}_{\alpha }-{f}_{\alpha }^{U}\right)$ is the charge relaxation rate due to coupling to lead α, and λ_c = ∑_α λ_c,α is the total charge relaxation rate. While λ_c,α are eigenvalues of the respective W_α , the sum λ_c is an eigenvalue of the total W. The dot occupation expectation value is defined as $\langle \hat{n}\rangle =\left(0\vert \mathbf{n}\vert 0\right)$. For arbitrary chemical potentials it reads

$$\begin{equation}\langle \hat{n}\rangle =2\left(\frac{{{\Gamma}}_{\text{L}}}{{\lambda }_{\text{c}}}{f}_{\text{L}}+\frac{{{\Gamma}}_{\text{R}}}{{\lambda }_{\text{c}}}{f}_{\text{R}}\right).\end{equation} \tag{ 44 }$$

Its fluctuations are defined as ${\Delta}n=\langle {\hat{n}}^{2}\rangle -{\langle \hat{n}\rangle }^{2}=\left(0\vert {\mathbf{n}}^{2}\vert 0\right)-{\left(0\vert \mathbf{n}\vert 0\right)}^{2}$. For the noise expression equation (43), we only need the limit μ_L = μ_R = μ, which is

$$\begin{equation}{\Delta}n=-{k}_{\text{B}}T{\partial }_{{\epsilon}}\langle \hat{n}\rangle .\end{equation} \tag{ 45 }$$

These expressions yield an explicit form for the deviations from the standard stationary FRR for a single-level quantum-dot pump, which we could compactly write as [35]

$$\begin{equation}\begin{aligned}\hfill & {\left.{S}_{\text{LR}}^{\left(1\right)}\right\vert }_{\left\{{\mu }_{\alpha }\right\}\to \mu }-{k}_{\text{B}}T{\left.\left(\frac{\partial {I}_{\text{L}}^{\left(1\right)}}{\partial {\mu }_{\text{R}}}+\frac{\partial {I}_{\text{R}}^{\left(1\right)}}{\partial {\mu }_{\text{L}}}\right)\right\vert }_{\left\{{\mu }_{\alpha }\right\}\to \mu }\hfill \\ \hfill & \qquad \qquad =-2{k}_{\text{B}}T{\int }_{0}^{{\tau }_{0}}\frac{\mathrm{d}t}{{\tau }_{0}}\frac{{{\Gamma}}_{\text{L}}{{\Gamma}}_{\text{R}}}{{{\Gamma}}^{2}}\frac{{\partial }_{{\epsilon}}{\lambda }_{\text{c}}}{{\lambda }_{\text{c}}}{\partial }_{t}\langle \hat{n}\rangle .\hfill \end{aligned}\end{equation} \tag{ 46 }$$

We will now show that the term on the right-hand side fulfills the main results presented in III, namely that it (a) vanishes for negligible Coulomb interaction and (b) that it equals the nonlinear contributions to the energy-current provided by the external pumping fields.

• (a)  
  For U = 0 the charge relaxation rate λ_c simply equals the coupling constant Γ. It is hence constant with respect to the dot energy, ∂ λ_c = 0, and consequently the right-hand-side of equation (46) must be zero.
• (b)  
  Based on equation (20), we can express the energy current provided by the external pumping fields for the single-level quantum dot as,

  $$\begin{equation}{I}_{\text{E},\text{pump}}^{\left(1\right)}=-{\int }_{0}^{{\tau }_{0}}\frac{\mathrm{d}t}{{\tau }_{0}}\left(0\vert \dot {\mathbf{e}}\vert 0\right)=-{\int }_{0}^{{\tau }_{0}}\frac{\mathrm{d}t}{{\tau }_{0}}\dot {{\epsilon}}\langle \hat{n}\rangle .\end{equation} \tag{ 47 }$$

  With straightforward algebra, one can now explicitly show that

  $$\begin{equation}{\left.\frac{{\partial }^{2}{I}_{\text{E},\text{pump}}^{\left(1\right)}}{\partial {\mu }_{\text{L}}\partial {\mu }_{\text{R}}}\right\vert }_{\left\{{\mu }_{\alpha }\right\}\to \mu }=2{\int }_{0}^{{\tau }_{0}}\frac{\mathrm{d}t}{{\tau }_{0}}\frac{{{\Gamma}}_{\text{L}}{{\Gamma}}_{\text{R}}}{{{\Gamma}}^{2}}\frac{{\partial }_{{\epsilon}}{\lambda }_{\text{c}}}{{\lambda }_{c}}{\partial }_{t}\langle \hat{n}\rangle ,\end{equation} \tag{ 48 }$$

  thus satisfying equation (31). In figure 2, we plot the respective quantities ${S}^{\left(1\right)}\equiv {S}_{\text{LR}}^{\left(1\right)}$, ${G}^{\left(1\right)}\equiv {\partial }_{{\mu }_{\text{R}}}{I}_{\text{L}}^{\left(1\right)}+{\partial }_{{\mu }_{\text{L}}}{I}_{\text{R}}^{\left(1\right)}$, and ${Q}^{\left(1\right)}\equiv {\partial }_{{\mu }_{\text{L}}}{\partial }_{{\mu }_{\text{R}}}{I}_{\text{E},\text{pump}}^{\left(1\right)}$ as a function of the time-averaged energy level $\bar{{\epsilon}}$. For the noninteracting system (a), we recover the equilibrium fluctuation dissipation theorem, in spite of the presence of a nonequilibrium drive. As soon as finite interactions are present (b), the correction due to the quadratic response of the pumping energy input, ${I}_{\text{E},\text{pump}}^{\left(1\right)}$ is nonzero, such that the equilibrium fluctuation dissipation theorem is no longer applicable. The interaction-induced deviation from it is significant, as ${I}_{\text{E},\text{pump}}^{\left(1\right)}$ is visibly of the same order of magnitude as ${\partial }_{{\mu }_{\text{R}}}{I}_{\text{L}}^{\left(1\right)}+{\partial }_{{\mu }_{\text{L}}}{I}_{\text{R}}^{\left(1\right)}$.

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Crucially, equation (47) indicates how the fluctuation relation in equation (46) could be experimentally verified. In general, one would need the information of both the current and current noise, as well as the total energy current provided by the external pumping fields. In particular the direct measurement of energy currents may be challenging, even if current progress on fast thermometry could open up new opportunities [71, 72].

However, equation (47) indicates that one could alternatively measure the quantum dot occupation number $\langle \hat{n}\rangle$ as a function of time, which can be done through a local charge detector such as a quantum point contact [33, 34]. Importantly, we note, that the time resolution need not be high. The measuring frequency ω (such that ω⁻¹ can be regarded as a sampling time scale, that is, the detector resolution in time space) needs to be higher than the driving frequency Ω, but in the adiabatic-response limit, can still be smaller than the tunneling rate Γ_α .

The remaining quantity one needs to have access to is (t). In experiment, is typically driven through an external gate voltage. Resorting to a circuit picture, see figure 1(b), and assuming that the electrostatics of the system can be fully described by geometric capacitances, we can relate (t) = a eV_g(t) + const., with the lever arm factor a = C_g/(C_g + C_L + C_R). Consequently, the correction on the right-hand side of equation (46) is accessible through the input function V_g(t), the output function $\langle \hat{n}\rangle \left(t\right)$ of the charge detector, and a single fitting parameter a.

Finally, we comment on the case where the Coulomb interaction is time-dependent, U(t), and used as one of the pumping parameters [73]. Within the model of geometric capacitances, where U = e²/(C_g + C_L + C_R), a time-dependent U can arise if the driving of the quantum dot changes the geometry, meaning that the capacitances themselves become time dependent. Also tunable effective Coulomb interaction has been realized [74]. Importantly, the general pumping fluctuation relations from section 3.1, see equation (31), encompass also such a time-dependent driving of U—only the explicit shape for the pumping energy input, equation (48), needs to be modified. This total energy current can then be decomposed into a charge and a parity contribution, as

$$\begin{equation}{I}_{\text{E},\text{pump}}^{\left(1\right)}=-{\int }_{0}^{{\tau }_{0}}\frac{\mathrm{d}t}{{\tau }_{0}}\left[\left(\dot {{\epsilon}}+\frac{\dot {U}}{2}\right)\langle \hat{n}\rangle -\frac{\dot {U}}{2}\langle \hat{p}\rangle \right],\end{equation} \tag{ 49 }$$

where $\hat{p}={\text{e}}^{\text{i}\pi \hat{n}}$ is the parity operator. For this specific model, this yields the nonlinear relation $\hat{p}=1-4\hat{n}+2{\hat{n}}^{2}$. It is likewise accessible by a time-resolved charge detector. Instead of measuring the total charge, one here needs to keep track of the number of single electron tunneling events, and whether the resulting occupation is odd or even. The remaining task is to experimentally control the geometry of the device to the extent that reliable information of U(t) as a function of the driving parameters can be found.