Universal classical and quantum fluctuations in the large deviations of current of noisy quantum systems: The case of QSSEP and QSSIP

TL;DR

The paper establishes that current fluctuations in noisy quantum diffusive systems QSSEP and QSSIP are classically universal at leading order in system size: the cumulant generating function matches the classical SSEP/SSIP and satisfies Macroscopic Fluctuation Theory (MFT) at large scales. Using a two-time measurement protocol and biased quantum dynamics, the authors derive a large deviation principle in which the noise-averaged quantum generating function self-averages to its classical counterpart, while exposing subleading purely quantum corrections. They introduce a variational framework linking quantum dynamics to the MFT structure and show that, for each noise realization, classical typicality holds, with quantum effects entering as order ${ m O}(N^{-1})$ bulk corrections that cancel in current cumulants. The bosonic case features a finite-bias interval due to boundary condensation, while fermionic SSEP-like behavior persists; the leading corrections to the variance and skewness are computable and experimentally accessible under controlled noise. Overall, the work provides a microscopic quantum justification for MFT in these models and highlights the necessity of a quantum-extension of hydrodynamic fluctuation theory for diffusive quantum systems.

Abstract

We study the fluctuation statistics of integrated currents in noisy quantum diffusive systems, focusing on the Quantum Symmetric Simple Exclusion and Inclusion Processes (QSSEP/QSSIP). These one-dimensional fermionic (QSSEP) and bosonic (QSSIP) models feature stochastic nearest-neighbor hopping driven by Brownian noise, together with boundary injection and removal processes. They provide solvable microscopic settings in which quantum coherence coexists with diffusion. Upon noise averaging, their dynamics reduce to those of the classical SSEP/SSIP. We show that the cumulant generating function of the integrated current, at large scales, obeys a large deviation principle. To leading order in system size and for each noise realization, it converges to that of the corresponding classical process, establishing a classical typicality of current fluctuations in these noisy quantum systems. We further demonstrate a direct connection with Macroscopic Fluctuation Theory (MFT), showing that the large-scale equations satisfied by biased quantum densities coincide with the steady-state Hamilton equations of MFT, thereby providing a microscopic quantum justification of the MFT framework in these models. Finally, we identify the leading finite-size corrections to the current statistics. We show the existence of subleading contributions of purely quantum origin, which are absent in the corresponding classical setting, and provide their explicit expressions for the second and third current cumulants. These quantum corrections are amenable to direct experimental or numerical verification, provided sufficient control over the noise realizations can be achieved. Their presence points toward the necessity of a quantum extension of Macroscopic Fluctuation Theory.

Figures (4)

• Figure 1: Illustration of the setup. A quantum chain is driven into a non-equilibrium steady state by connecting its endpoints to Markovian reservoirs. Each link is coupled to an auxiliary bath, such that whenever a particle hops from $j\to j+1$ (resp. $j+1\to j$), a photon is emitted (resp. absorbed) in that bath. By performing two measurements of the bath occupation at times $t=0$ and $t$, we infer the integrated current $Q_j(t)$ through link $j$. In the Markovian limit, the moment generating function of the $Q_j(t)$ is evolved using the biased QSSIP/QSSEP Hamiltonian \ref{['eq:dH-bias']}.
• Figure 2: Plot of the cumulant generating function ${\mathbb{E}}[F_{{\rm qu},w}(u)]$ in \ref{['eq:F-exact']} (solid lines) for the bosonic and fermionic case (${\varepsilon}=\pm 1$, respectively). Symbols show the numerical data for the mean from the microscopic dynamics \ref{['eq:dyn-bias']} for different system sizes $N\leq 64$ (see plot's legend). Left- and right- reservoir densities are set to $\bar{n}_0=0.3$ and $\bar{n}_N=0.2$. In the bosonic case, the function ${\mathbb{E}}[F_{{\rm qu},w}(u)]$ is defined in the interval $u\in(u_-,u_+)$ (dashed vertical lines).
• Figure 3: Numerical analysis of the variance $\text{var}(F_{{\rm qu},w}(u)):={\mathbb{E}}[F_{{\rm qu},w}(u)^2]-{\mathbb{E}}[F_{{\rm qu},w}(u)]^2$, for the fermionic (top) and bosonic (bottom) cases, shown as a function of $1/N$ for different values of the bias amplitude $u$. The data are compatible with a $\sim 1/N^2$ scaling of the variance, indicated by the solid line as a guideline. Left- and right- reservoir densities are set to $\bar{n}_0=0.3$ and $\bar{n}_N=0.2$.
• Figure 4: Exact solution of the equations in \ref{['eq:cond1']} for the functions $\mathfrak{g}_1(x)$ (left panels) and $\xi(x)$ (right panels). The plots are made for the fermionic (QSSEP, ${\varepsilon}=-1$ -- top panels) and bosonic (QSSIP, ${\varepsilon}=1$, -- bottom panels) cases, with reservoirs densities fixed to $\bar{n}_0=0.3$ and $\bar{n}_N=0.2$.