Nonequilibrium steady states in multi-bath quantum collision models

TL;DR

The paper examines how quantum collision models describe thermalisation and nonequilibrium steady states in open systems. By comparing a single-bath setting to a two-bath setting, it shows that both recover GKSL-type dynamics in the $Δt\to 0$ limit, yet Setting II naturally yields a nonequilibrium steady state with finite heat currents due to inter-bath energy exchange. Introducing non-Markovian intra-environment memory via SWAP-like collisions preserves GKSL behavior in the single-bath case but can shift Setting II to a noncanonical steady state with an effective temperature $β_e$ and strong system–memory correlations. The work also demonstrates how stochastic heat currents can be accessed through a two-point measurement scheme and analyzes entanglement dynamics, offering insights into the thermodynamics of thermalisation and the role of memory in open quantum systems.

Abstract

Collision models provide a simple and versatile setting to capture the dynamics of open quantum systems. The standard approach to thermalisition in this setting involves an environment of independent and identically-prepared thermal qubits, interacting sequentially for a finite duration $Δt$ with the system. We compare this to a two-bath scenario in which collisional qubits are prepared in either their ground or excited states and the environment temperature is encoded in system-environment couplings. The system reaches the same thermal steady state for both settings, although even in this limit they describe fundamentally different physical processes, with the two-bath setup yielding a nonequilibrium state with finite heat currents. Non-Markovian dynamics arise when intra-environment interactions in either setting are introduced. Here, the system in the single-bath setup again reaches a steady state at the canonical temperature of the bath, but the nonequilibrium steady state of the two-bath setup tends to a different temperature due to the generation of strong system-environment and intra-environment correlations. The two-bath setting is particularly suited to studying quantum trajectories, which are well-defined also for the non-Markovian case. We showcase this with a trajectory analysis of the heat currents within a two-point measurement scheme. Finally, we consider how our results are impacted when the system-environment interaction leads to strict homogenisation. Our results provide insights into the dynamics and thermodynamics of thermalisation towards nonequilibrium steady states and the role of non-Markovian interactions.

Figures (7)

• Figure 1: Schematic depiction of the approaches to model open quantum system dynamics considered in this work. (a) Typical open system dynamics where the system is interacts with a bath, generally assumed to be much larger than the system. Under certain assumptions this can be modeled using master equations. (b) Setting I: Collision models involve partitioning the environment into smaller constituent parts all prepared in the same initial state, $\eta_i$. The system interacts with a given environmental unit via some interaction with strength $J$. For interactions of the form Eq. \ref{['eqn: interaction H']} this collision model leads to homogenisation and the system reaches the steady state $\rho_{\infty}\!=\!\eta_i$. If $\eta_i$ are thermal states, this dynamics can be shown to be equivalent to the master equation, Eq. \ref{['Eqn: QME']}. (c) Setting II: Two bath collision model. The system interacts simulataneously with two independent collisional baths, where the constituents are initially in pure states $\ket{0}$ and $\ket{1}$, respectively, whose interaction strengths encode the effective temperature of the bath.
• Figure 2: Steady states in the Markovian limit. Non-equilibrium steady state heat current, $\Delta\mathcal{Q}/\Delta t$ for the initially low energy (cold) bath, $A_n^{(0)}$, with increasing collision time-step $\Delta t$ for different canonical temperatures $\beta\!=\!2$ (dashed, blue; lower temperature) and $\beta\!=\!0.5$ (solid, red; higher temperature). Other parameters: $\omega\!=\!1,~\Gamma\!=\!4$.
• Figure 3: Dynamical approach to the steady state. Upper panel is for setting I and the lower panel is for setting II. We show the fidelity, ($1-F$), against $t \!=\! n \Delta t$, between the evolved system state and Eq. \ref{['eq:Gibbsstate']} fixing $\beta\!=\!2$. In both settings we consider three sets of parameters: Solid, red lines correspond to long system-environment collision durations, $\Delta t\!=\!0.01$, and strong intra-environment interactions $\delta\!=\!0.95\tfrac{\pi}{2}$; Dashed, blue curves correspond to long system-environment collision durations, $\Delta t\!=\!0.01$, and weaker intra-environment interactions, $\delta\!=\!0.8\tfrac{\pi}{2}$; Gray, dot-dashed curves correspond to short system-environment collision durations $\Delta t\!=\!0.001$ and strong intra-environment interactions, $\delta\!=\!0.95\tfrac{\pi}{2}$. The insets show the discretized non-Markovianity measure, Eq. \ref{['eq:NMdiscrete']}, evaluated for $\beta\!=\!2$ as a function of intra-environment interactions strength $\delta$. In both main panels the system is initialized in the excited state $\rho_0 = |1\rangle\langle1|$. Other parameters used: $\omega =1, \Gamma =4.$
• Figure 4: Panels (a) and (b): Difference in system temperature reached at the steady state, $\Delta\beta\!=\!{\beta_{e}-\beta}$ considering the impact of increasing the A-A interactions $\delta$ and increasing the S-A collision duration $\Delta t$. Panels (c) and (d): Associated steady state heat flux, $\Delta Q / \Delta t$, into $A_n^{(0)}$. Other parameters used: $\omega\!=\!1, \Gamma\!=\!4$.
• Figure 5: Steady state entanglement arising from setting II. Panel (a): Genuine tripartite entanglement between the system and two "memory" units. Panels (b-d): Bipartite entanglement of the various reduced states. In all panels we fix $\beta\!=\!2, \omega =1, \Gamma=4$.