Full counting statistics for boundary driven transport in presence of correlated gain and loss channels

TL;DR

This work develops a path-integral (Keldysh) framework to obtain the full counting statistics and the cumulant generating function (CGF) of steady-state current in a boundary-driven, non-interacting fermionic lattice subjected to engineered bulk dissipators that include correlated gain and loss channels. For the baseline case with only boundary drives, the CGF reduces to a Levitov-Lesovik–type form with a transmission function $T_{1N}(\omega)$, yielding explicit expressions for the mean current and its fluctuations, and confirming a steady-state fluctuation symmetry. When bulk gain and loss are present, left and right current statistics generally differ unless a balanced, PT-symmetric condition is met; in the localized-gain/loss case this reciprocity can be restored under three conditions, and in the correlated-gain/loss case additional constraints (including a phase equality $\theta=\phi$) are required to recover reciprocity and vanishing bulk current. A key finding is that correlated gain-loss dissipators break end-to-end reciprocity via off-diagonal self-energies, producing nonreciprocal transmission and a diode-like rectification of current, whereas balanced configurations restore symmetry. The results demonstrate how engineered dissipation shapes current fluctuations and offer a principled route to control transport in quantum devices via bath engineering and PT-symmetric dissipation.

Abstract

One of the major advances of quantum technology is the engineering of complex quantum channels in lattice systems that paves the way for a variety of novel non-equilibrium phenomena. For a boundary driven lattice with such engineered quantum channels, the analysis of the full counting statistics of current across boundaries has received limited attention. In this work, we consider a boundary driven free fermionic lattice with carefully engineered correlated gain and loss channels and obtain the cumulant generating function of the steady-state particle current. We also discuss the limit for simplifying the correlated gain-loss channel to a local gain-loss channel and obtain the average current and its fluctuation in such cases. Generally, in the presence of gain-loss, the current statistics are different at the two ends of the lattice. Hence, for both local and correlated gain-loss, we devise the conditions for which the statistics can coincide, giving rise to a $\mathcal{PT}$ symmetric balanced gain-loss scenario. A striking difference between the correlated gain-loss and their local counterpart is the emergence of nonreciprocity in the system and we observe that it has a dramatic impact in the current as well as fluctuations. Our work therefore provides interesting insights about the importance of engineered dissipators in boundary driven systems.

Figures (7)

• Figure 1: Schematic of the setup considered in this work: The system of interest is a one-dimensional fermionic chain with boundary injection and extraction of fermions from $1$st and $N$-th site with rates $\alpha_1$, $\beta_1$ and $\alpha_N$, $\beta_N$, respectively. Different values of $\alpha_1$, $\beta_1$, and $\alpha_N$, $\beta_N$ drive the system out of equilibrium and produce a steady current in the chain. The lattice can be further subjected to correlated particle loss or gain with gain strength $\Gamma_1$, $\Gamma_2$ and loss strength $\kappa_1$, $\kappa_2$, respectively (see Eq. \ref{['bulk_dissipator']} and discussions below that). The correlated loss (gain) can be reduced to local loss (gain) by putting $\Gamma_2=0$ ($\kappa_2=0$). We first discuss the case with $\Gamma_1=\Gamma_2=\kappa_1=\kappa_2=0$ in Sec. \ref{['sec:boundary']} and later discuss the results for correlated gain and loss in Sec. \ref{['sec:correlated']}.
• Figure 2: Colormap plot for the (a) average current $I_L$ [Eq. \ref{['current_extended']}] and (b) noise $S_L$ [Eq. \ref{['noise_extended']}] as a function of injection rate $\alpha_1$ and extraction rate $\beta_N$ for an one-dimensional tight-binding lattice. The other parameters are set as $\alpha_N=\beta_1=J=1$. The system size is taken as $N=5$. Both the current and the noise initially increases with $\alpha_1$ and $\beta_N$ and then decreases showing non-monotonic behaviour.
• Figure 3: Local gain-loss case: plot of different transmission functions that appears in the CGF of $\mathcal{I}_L$ and $\mathcal{I}_R$ in Eq. \ref{['CGF_gain_lossL']} and \ref{['CGF_gain_lossR']}. (a) Transmission functions from the left end to the right end, $T_{1N}(\omega)$ [Eq. \ref{['eq:transmission_gainloss']}] and from the right end to the left end $T_{N1}(\omega)$ [Eq. \ref{['eq:transmission_gainloss']}] are plotted with frequency $\omega$ for the parameters $\alpha_1=4.0$, $\beta_1=0.1$, $\alpha_N=1.9$, $\beta_N=2.5$, $\Gamma=0.07$, and $\kappa=0.09$. The system size is chosen to be $N=5$. Their difference is zero for all values of $\omega$ which confirms that in presence of localized gain-loss channels, $T_{1N(\omega)}=T_{N1}(\omega)$ for any choice of $\alpha_1$, $\beta_1$, $\alpha_N$, $\beta_N$, $\Gamma$, and $\kappa$. For the same set of parameters, the transmission functions from $1$st site to the gain loss channels i.e., $T_1^{>}$ and $T_1^{<}$ [see Eq. \ref{['eq:t1_gl']}], from $N$-th site to the gain-loss channels i.e., $T_{N}^{>}$ and $T_{N}^<$ [see Eq. \ref{['eq:tn_gl']}] are plotted in (b). It can be clearly seen that all the transmission functions are different as $\alpha_1+\beta_1$ is different from $\alpha_N+\beta_N$ and $\Gamma \neq \kappa$. (c) Once the condition $\alpha_1+\beta_1=\alpha_N+\beta_N$ is introduced by the choice of $\alpha_1=1.0=\beta_N$ and $\beta_1=\alpha_N=0.9$, one can see from this plot that $T_1^{>}=T_{N}^{>}$ and $T_{1}^<=T_{N}^<$. However as $\Gamma = 0.09$ and $\kappa=0.07$, all four of them are not equal to each other. (c) Once we further choose $\Gamma=\kappa=0.07$, all four transmission functions become equal and under these conditions the left and right current statistics become identical.
• Figure 4: Local gain-loss case: colormap plot of the statistics of left and right current in local gain-loss case [Subsec. \ref{['subsec:localloss']}]. The parameters chosen here are $\alpha_1=\beta_N=1.0$ and $\alpha_N=\beta_1=0.5$ which satisfy the condition $\alpha_1+\beta_1=\alpha_N+\beta_N$. (a) Plot of average current $I_L$ using Eq. \ref{['eq:I_L_expr']}, (b) fluctuations $S_L$ using Eq. \ref{['eq:S_L']} corresponding to the left current $\mathcal{I}_L$, (c) average current $I_R$ using Eq. \ref{['eq:I_R_expr']}, (d) noise $S_R$ using Eq. \ref{['eq:S_R']} corresponding to the right current $I_R$. The statistics of $\mathcal{I}_L$ and $\mathcal{I}_R$ are in general different which can be seen from the plots. Along the line $\Gamma=\kappa$, (black dashed line) the statistics coincide. (e) Plot of the net left to right current $I_{LR}$ defined in Eq. \ref{['eq:left_to_right_current']}. There is no diode-like effect in this case. (f) Plot of the total bulk current $I_{\rm bulk}=I_L+I_R$ which is going out or coming into the system and it is defined in Eq. \ref{['eq:bulk_current']}. $I_{\rm bulk}$ is zero along the $\Gamma=\kappa$ line (black dashed line) which is the balanced gain-loss scenario.
• Figure 5: Correlated gain-loss case: plot of different transmission functions in presence of correlated gain-loss channels. We set $\alpha_1=\beta_N=1.0$, $\alpha_N=\beta_1=0.5$ which satisfy the condition (i) $\alpha_1+\beta_1=\alpha_N+\beta_N$ and (iv) $\alpha_1=\beta_N$ and $\alpha_N=\beta_1$. The system size is chosen to be $N=6$. (a) Left to right transmission function $T_{1N}(\omega)$ [Eq. \ref{['eq:transmission_gainloss']}] and right to left transmission function $T_{N1}(\omega)$ [Eq. \ref{['eq:transmission_gainloss']}] are plotted as a function of frequency $\omega$ when $\Gamma\neq \kappa$ and $\theta\neq \phi$ and they are unequal. This inequality in the transmission functions is referred as nonreciprocity in the system which can also be guaranteed by the plot of $T_{1N}(\omega)-T_{N1}(\omega)$ which is evidently nonzero. (b) $T_{1N}(\omega)$ and $T_{N1}(\omega)$ become equal when we set $\Gamma=\kappa$ and $\theta=\phi$ which thereby ensures reciprocity in the system. (c) The plot of the transmission functions from the $1$st to the gain and loss channels i.e., $T_{1}^{<}(\omega)$ and $T_{1}^{>}(\omega)$ and from the $N$-th site to the gain and loss channels i.e., $T_N^{<}(\omega)$ and $T_{N}^{>}(\omega)$ are plotted for the same set of parameters as of (a) when the system is nonreciprocal and hence we can see that they are all unequal. One we impose $\Gamma=\kappa$ and $\theta=\phi$, the reciprocity ensures $T_{1}^{>}(\omega)=T_{N}^{<}(\omega)$ and $T^{<}_1(\omega)=T_{N}^{>}(\omega)$.