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arXiv:2503.03832 · Quantum Science and Technology

Continuous variable structured collision models

Anton Corr, Stefano Cusumano, Gabriele De Chiara

Published Sep 23, 2025• 4 citations
arXivHTMLPDF

Abstract

Abstract Quantum collision models allow for the dynamics of open quantum systems to be described by breaking the environment into small segments, typically consisting of non-interacting harmonic oscillators or two-level systems. This work introduces structure within these environmental units via spring-like interactions between N coupled oscillators in a ring structure, initially prepared in a thermal state. Two models of interest are examined. The first highlights a case in which a continuous time evolution can be obtained, wherein the system interacts with the environment via a beam-splitter-like, energy-preserving, interaction. The resulting dynamics are analogous to those due to interactions with unstructured units prepared as squeezed thermal states. The second model highlights a case in which the continuous time limit for the evolution cannot be taken generally, requiring instead discrete-time propagation. Special cases in which the continuous time limit can be taken are also investigated, alongside the addition of a secondary environment to induce a steady state. The first and second laws of thermodynamics are verified for both examples.

DOI
10.1088/2058-9565/ae0a7b

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Abstract Quantum collision models allow for the dynamics of open quantum systems to be described by breaking the environment into small segments, typically consisting of non-interacting harmonic oscillators or two-level systems. This work introduces structure within these environmental units via spring-like interactions between N coupled oscillators in a ring structure, initially prepared in a thermal state. Two models of interest are examined. The first highlights a case in which a continuous time evolution can be obtained, wherein the system interacts with the environment via a beam-splitter-like, energy-preserving, interaction. The resulting dynamics are analogous to those due to interactions with unstructured units prepared as squeezed thermal states. The second model highlights a case in which the continuous time limit for the evolution cannot be taken generally, requiring instead discrete-time propagation. Special cases in which the continuous time limit can be taken are also investigated, alongside the addition of a secondary environment to induce a steady state. The first and second laws of thermodynamics are verified for both examples.
Abstract Quantum collision models normally consist of a system interacting with a set of ancillary units representing the environment. While these ancillary systems are usually assumed to be either two level systems or harmonic oscillators, in this work we move further and represent each ancillary system as a structured system, i.e. a system made out of two or more subsystems. We show how this scenario modifies the kind of master equation that one can obtain for the evolution of the open systems. Moreover, we are able to consider a situation where the ancilla state is thermal yet has some coherence. This allows the generation of coherence in the steady state of the open system and, thanks to the simplicity of the collision model, this allows us to better understand the thermodynamic cost of creating coherence in a system. Specifically, we show that letting the system interact with the coherent degrees of freedom requires a work cost, leading to the natural fulfillment of the first and second law of thermodynamics without the necessity of ad hoc formulations.
We introduce the idea of weakly coherent collisional models, where the elements of an environment interacting with a system of interest are prepared in states that are approximately thermal but have an amount of coherence proportional to a short system-environment interaction time in a scenario akin to well-known collisional models. We show that, in the continuous-time limit, the model allows for a clear formulation of the first and second laws of thermodynamics, which are modified to include a nontrivial contribution related to quantum coherence. Remarkably, we derive a bound showing that the degree of such coherence in the state of the elements of the environment represents a resource, which can be consumed to convert heat into an ordered (unitarylike) energy term in the system, even though no work is performed in the global dynamics. Our results therefore represent an instance where thermodynamics can be extended beyond thermal systems, opening the way for combining classical and quantum resources.
We introduce a framework to model the evolution of a class of open quantum systems whose environments periodically undergo an instantaneous non-unitary evolution stage. For the special case of quadratic models, we show how this approach can generalise the formalism of repeated interactions to allow for the preservation of system-environment correlations. Furthermore, its continuous zero-period limit provides a natural description of the evolution of small systems coupled to large environments in negligibly perturbed steady states. We explore the advantages and limitations of this approach in illustrative applications to thermalisation in a simple hopping ring and to the problem of initialising a qubit chain via environmental engineering.
We study the applicability of collisional models for non-Markovian dynamics of open quantum systems. By allowing interactions between the separate environmental degrees of freedom in between collisions we are able to construct a collision model that allows us to study quantum memory effects in open system dynamics. We also discuss the possibility to embed non-Markovian collision model dynamics into Markovian collision model dynamics in an extended state space. As a concrete example we show how, using the proposed class of collision models, we can discretely model non-Markovian amplitude damping of a qubit. In the time-continuous limit, we obtain the well-known results for spontaneous decay of a two-level system into a structured zero-temperature reservoir.
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.
We study two continuous variable systems (or two harmonic oscillators) and investigate their entanglement evolution under the influence of non-Markovian thermal environments. The continuous variable systems could be two modes of electromagnetic fields or two nanomechanical oscillators in the quantum domain. We use the quantum open system method to derive the non-Markovian master equations of the reduced density matrix for two different but related models of the continuous variable systems. The two models both consist of two interacting harmonic oscillators. In model A, each of the two oscillators is coupled to its own independent thermal reservoir, while in model B the two oscillators are coupled to a common reservoir. To quantify the degrees of entanglement for bipartite continuous variable systems in Gaussian states, logarithmic negativity is used. We find that the dynamics of the quantum entanglement is sensitive to the initial states, the oscillator-oscillator interaction, the oscillator-environment interaction and the coupling to a common bath or to different, independent baths.