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arXiv:2507.13860 · Physical Review A

Nonequilibrium steady states in multibath quantum collision models

Ronan McElvogue, Andrew K. Mitchell, Gabriel Teixeira Landi, Steve Campbell

Published Jan 16, 2026• 2 citations

Abstract

Collision models provide a simple and versatile setting to capture the dynamics of open quantum systems. The standard approach to thermalization in this setting involves an environment of independent and identically prepared thermal qubits, interacting sequentially for a finite duration Δ t 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 arises when intraenvironment 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 intraenvironment 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 homogenization. Our results provide insights into the dynamics and thermodynamics of thermalization towards nonequilibrium steady states and the role of non-Markovian interactions.

DOI: 10.1103/ypyy-ql4c

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Collision models provide a simple and versatile setting to capture the dynamics of open quantum systems. The standard approach to thermalization in this setting involves an environment of independent and identically prepared thermal qubits, interacting sequentially for a finite duration Δ t 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 arises when intraenvironment 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 intraenvironment 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 homogenization. Our results provide insights into the dynamics and thermodynamics of thermalization towards nonequilibrium steady states and the role of non-Markovian interactions.

The quantum exchange fluctuation theorem relates the probabilities of observing heat transfer along and against the temperature gradient between thermal baths at different temperatures. We investigate how this relation generalizes when the baths exhibit non-Markovian dynamics. Using a microscopic collisional model, bath memory is generated through interactions between successive bath auxiliaries before each heat-exchange collision. We derive exchange fluctuation theorems for both direct bath-bath interactions and probe-mediated heat exchange in the steady-state regime. As an illustrative example, we consider heat baths with qubit auxiliaries and show that non-Markovian memory enhances the probability of heat-transfer events against the temperature gradient, modifying the predictions made by the conventional Jarzynski-Wójcik exchange fluctuation theorem. Our results establish a microscopic connection between environmental memory and non-equilibrium heat-exchange statistics.

Abstract Collisional models are a category of microscopic framework designed to study open quantum systems. The framework involves a system sequentially interacting with a bath comprised of identically prepared units. In this regard, quantum homogenization is a process where the system state approaches the identically prepared state of bath unit in the asymptotic limit. Here, we study the homogenization process for a single qubit in the non-Markovian collisional model framework generated via additional bath-bath interaction. With partial swap operation as both system-bath and bath-bath unitary, we numerically demonstrate that homogenization is achieved irrespective of the initial states of the system or bath units. This is reminiscent of the Markovian scenario, where partial swap is the unique operation for a universal quantum homogenizer. On the other hand, we observe that the rate of homogenization is slower than its Markovian counter part. Interestingly, a different choice of bath-bath unitary speeds up the homogenization process but loses the universality, being dependent on the initial states of the bath units.

We investigate the steady state properties arising from the open system dynamics described by a memoryless (Markovian) quantum collision model, corresponding to a master equation in the ultra-strong coupling regime. By carefully assessing the work cost of switching on and off the system-environment interaction, we show that only a coupling Hamiltonian in the energy-preserving form drives the system to thermal equilibrium, while any other interaction leads to non-equilibrium steady states that are supported by steady-state currents. These currents provide a neat exemplification of the housekeeping work and heat. Furthermore, we characterize the specific form of system-environment interaction that drives the system to a steady-state exhibiting coherence in the energy eigenbasis, thus, giving rise to families of states that are non-passive.

This paper compares the cooling performance of Markovian and non-Markovian quantum collisional models for nanoscale thermodynamic devices. A minimal setup is considered: a single qubit system repeatedly collides with qubit ancillas acting as a heat bath, while a local cooling “demon” operates on the system after each collision. Two regimes are analyzed: (i) a Markovian protocol where the system interacts with a fresh thermal ancilla at every step and the ancilla is subsequently rethermalized, and (ii) a non-Markovian protocol where the system repeatedly interacts with the same initially thermal environment qubit with no reset, allowing correlations (memory) to build up. A central contribution is an efficiency definition that explicitly includes all external energetic resources, counting both the energy removed by the demon and the otherwise hidden work cost of rethermalizing the ancillas in the Markovian implementation. The demon is modeled as an amplitude damping channel; via an explicit Stinespring dilation, the extracted energy is interpreted as heat dumped into an auxiliary cold reservoir, and the paper discusses how additional demon memory/erasure (Landauer) costs would further reduce operational efficiency. Numerical simulations show that the non-Markovian protocol can cool the system to much lower final energies and achieve substantially higher efficiency than the Markovian protocol across a wide range of collision (coupling) and damping strengths. The enhancement is traced to the finite environment’s initial energy and its capacity to store correlations as usable thermodynamic resources. The results emphasize that bounds and benchmarks derived under Markovian assumptions may significantly underestimate performance in finite, memoryful environments, and that reported efficiencies should be interpreted as upper bounds when demon implementation costs are not fully modeled.

This article investigates dissipative preparation of entangled nonequilibrium steady states (NESS). We construct a collision model where the open system consists of two qubits which are coupled to heat reservoirs with different temperatures. The baths are modeled by sequences of qubits interacting with the open system. The model can be studied in different dynamical regimes: with and without environmental memory effects. We report that only a certain bath temperature range allows for entangled NESS. Furthermore, we obtain minimal and maximal critical values for the heat current through the system. Surprisingly, quantum memory effects play a crucial role in the long-time limit. First, memory effects broaden the parameter region where quantum correlated NESS may be dissipatively prepared and, second, they increase the attainable concurrence. Most remarkably, we find a heat current range that does not only allow, but even guarantees that the NESS is entangled. Thus, the heat current can witness entanglement of nonequilibrium steady states.

Quantum non-Markovianity represents memory during system dynamics, which is typically weakened by temperature. We study here the effects of environmental temperature on the non-Markovianity of an open quantum system by virtue of collision models. The environment is simulated by a chain of ancillary qubits that are prepared in thermal states with a finite temperature

T

. Two distinct non-Markovian mechanisms are considered via two types of collision models, one where the system

S

consecutively interacts with ancillas and a second where

S

collides only with an intermediate system ${S}^{\ensuremath{'}}$, which in turn interacts with the ancillas. We show that in both models the relation between non-Markovianity and temperature is nonmonotonic. In particular, revivals of non-Markovianity may occur as temperature increases. We find that the physical reason behind this behavior can be revealed by examining a peculiar system-environment coherence exchange, leading to ancillary qubit coherence larger than system coherence, which triggers information backflow from the environment to the system. These results provide insights into the mechanisms underlying the counterintuitive phenomenon of temperature-enhanced quantum memory effects.