This record contains the manuscript “Canonical Invariants, Quantum Noise, and Memory in Gaussian Open Systems.” We develop an invariant-based formulation of Gaussian open-system dynamics for a time-dependent quantum harmonic oscillator subject to dissipation and noise. The construction combines canonical invariants with stochastic and retarded-mode descriptions, allowing Markovian thermal relaxation and finite-memory non-Markovian dynamics to be treated within a common framework. A central result is that, away from equilibrium, the instantaneous energy is not determined by the entropy alone. For local relaxation, the energy admits a differential state representation of the form[E=E!\left(S,\dot S;\frac{\omega(t)}{\nu_B(t)},\gamma_{\rm eff}(t)\right),]so that the entropy rate (\dot S) carries essential nonequilibrium information. Finite-memory dynamics enlarges the required state data while preserving this basic structure. The formulation also makes explicit the consistency relation between dissipation and the asymmetric quantum-noise spectrum, connects the Markovian limit with thermal GKLS dynamics, and provides a finite-dimensional localization of colored-noise memory kernels. The framework is intended as a basis for studying driven Gaussian open systems and for possible extensions to quantum-field and electromagnetic settings. This version has been submitted to Physical Review Research.
This record contains the manuscript “Canonical Invariants, Quantum Noise, and Memory in Gaussian Open Systems.” We develop an invariant-based formulation of Gaussian open-system dynamics for a time-dependent quantum harmonic oscillator subject to dissipation and noise. The construction combines canonical invariants with stochastic and retarded-mode descriptions, allowing Markovian thermal relaxation and finite-memory non-Markovian dynamics to be treated within a common framework. A central result is that, away from equilibrium, the instantaneous energy is not determined by the entropy alone. For local relaxation, the energy admits a differential state representation of the form[E=E!\left(S,\dot S;\frac{\omega(t)}{\nu_B(t)},\gamma_{\rm eff}(t)\right),]so that the entropy rate (\dot S) carries essential nonequilibrium information. Finite-memory dynamics enlarges the required state data while preserving this basic structure. The formulation also makes explicit the consistency relation between dissipation and the asymmetric quantum-noise spectrum, connects the Markovian limit with thermal GKLS dynamics, and provides a finite-dimensional localization of colored-noise memory kernels. The framework is intended as a basis for studying driven Gaussian open systems and for possible extensions to quantum-field and electromagnetic settings. This version has been submitted to Physical Review Research.
The basic features of the dynamics of open quantum systems, such as the dissipation of energy, the decay of coherences, the relaxation to an equilibrium or non-equilibrium stationary state, and the transport of excitations in complex structures are of central importance in many applications of quantum mechanics. The theoretical description, analysis and control of non-Markovian quantum processes play an important role in this context. While in a Markovian process an open system irretrievably loses information to its surroundings, non-Markovian processes feature a flow of information from the environment back to the open system, which implies the presence of memory effects and represents the key property of non-Markovian quantum behavior. Here, we review recent ideas developing a general mathematical definition for non-Markoviantiy in the quantum regime and a measure for the degree of memory effects in the dynamics of open systems which are based on the exchange of information between system and environment. We further study the dynamical effects induced by the presence of system-environment correlations in the total initial state and design suitable methods to detect such correlations through local measurements on the open system.
The quantum fluctuation-dissipation theorem is a central ingredient in the construction of quantum dynamics of Brownian motion which necessarily is non-Markovian. Yet, often Markovian approximations to quantum dynamics are studied in the literature. In this work, we investigate the limitations of the Markovian approximation within two paradigmatic models describing a single damped harmonic oscillator. These models are governed by distinct quantum Langevin equations, although both are constructed to satisfy the same set of phenomenological criteria: the canonical commutation relations between position and momentum, the Kubo response relation, the virial theorem, and the equilibrium quantum variance. The limitations of the Markovian approximation are underscored by the classical limit, violations of the Ehrenfest theorem, the breakdown of complete and simple positivity in the reconstructed master equations, and anomalies in thermalisation behaviour. Further phenomenological differences between the two models are illustrated through their quantum relaxation dynamics and phase diagrams, derived from their reinterpretation as mean-field approximations of a many-body interacting magnet. Our analysis explicitly reveals intrinsic inconsistencies introduced by the Markovian approximation, emphasising the need for non-Markovian frameworks for a consistent description of open quantum dynamics.
Abstract The theory of open quantum systems plays a fundamental role in several scientific and technological disciplines, from quantum computing and information science to molecular electronics and quantum thermodynamics. Despite its widespread relevance, a rigorous formulation of quantum dissipation in conjunction with thermal noise remains a topic of active research. In this work, we establish a formal correspondence between classical stochastic thermodynamics, in particular the Fokker–Planck and Klein–Kramers equations, and the quantum master equation. Building on prior studies of multiplicative noise in classical stochastic differential equations, we demonstrate that thermal noise at the quantum level manifests as a multidimensional geometric stochastic process. By applying canonical quantization, we introduce a novel Hermitian dissipation operator that serves as a quantum analogue of classical viscous friction. This operator allows for a well-defined expression of heat exchange between a system and its environment, enabling the formulation of an alternative quantum equipartition theorem. Our framework ensures a precise energy balance that aligns with the first law of thermodynamics and an entropy balance consistent with the second law. The theoretical formalism is applied to two prototypical quantum systems, the harmonic oscillator and a particle in an infinite potential well, for which it provides new insights into nonequilibrium thermodynamics at the quantum scale. Our results advance the understanding of dissipation in quantum systems and establish a foundation for future studies on stochastic thermodynamics in the quantum domain.
Abstract The quantum fluctuation-dissipation theorem is a central ingredient in the construction of quantum dynamics of Brownian motion which necessarily is non -Markovian. Yet, often Markovian approximations to quantum dynamics are studied in the literature. In this work, we investigate the limitations of the Markovian approximation within two paradigmatic models describing a single damped harmonic oscillator. These models are governed by distinct quantum Langevin equations, although both are constructed to satisfy the same set of phenomenological criteria: the canonical commutation relations between position and momentum, the Kubo response relation, the virial theorem, and the equilibrium quantum variance. The limitations of the Markovian approximation are underscored by the classical limit, violations of the Ehrenfest theorem, the breakdown of complete and simple positivity in the reconstructed master equations, and anomalies in thermalisation behaviour. Further phenomenological differences between the two models are illustrated through their quantum relaxation dynamics and phase diagrams, derived from their reinterpretation as mean-field approximations of a many-body interacting magnet. Our analysis explicitly reveals intrinsic inconsistencies introduced by the Markovian approximation, emphasising the need for non -Markovian frameworks for a consistent description of open quantum dynamics.
We study a harmonic oscillator weakly coupled to a thermal bath and evolving under the standard quantum-optical master equation. For zero-mean Gaussian states, entropy alone does not determine the internal energy because the latter also depends on nonequilibrium Gaussian structure. We show, however, that thermal relaxation yields the exact representation $ E=E(S,\dot S) $ for every mixed Gaussian state evolving under this thermal quantum-optical master equation. Explicitly, E(S,S˙)=νthων(S)[ν(S)+γS′(ν(S))S˙]. Once the thermal GKLS equation is assumed, no further expansion in the damping rate, entropy rate, or distance from equilibrium is made. The entropy rate is not an alternative instantaneous state coordinate: it depends on the open-system dynamics and therefore carries information about the coupling to the bath. This is exposed by the isolated limit, where S˙=0 for all Gaussian states and the representation no longer determines the energy. The result thus identifies an exact thermalization-induced dynamical energetic relation, distinct from rate dependence generated by an external driving protocol. We also describe an operational test based on time-resolved purity and energy measurements. At fixed frequency the bath only relaxes pre-existing squeezing. We therefore also consider frequency modulation, which can drive an initially thermal state away from the instantaneous Gibbs family by generating off-Gibbs Gaussian structure. Within a local Gaussian continuation, the bath then relaxes this driven deformation, providing a dynamical realization of the entropy-rate dependence.
Abstract Open quantum systems play a central role in current nanoscale technologies, such as molecular electronics, quantum heat engines, quantum computation and information processing. A major theoretical challenge is to construct dynamical models that are simultaneously consistent with classical thermodynamics and with the requirement of complete positivity of quantum evolution. In this work we develop a framework that addresses this issue by systematically extending classical stochastic dynamics to the quantum domain. We begin by formulating a generalized Langevin equation in which both friction and noise act symmetrically on the two Hamiltonian equations. From this, we derive a generalized Klein–Kramers equation expressed in terms of Poisson brackets, and we show that it admits the Boltzmann distribution as its stationary solution while fulfilling the first and second laws of thermodynamics along individual trajectories. Applying canonical quantization to this classical framework yields two distinct quantum master equations, depending on whether the friction operators are taken to be Hermitian or non-Hermitian. By analyzing the dynamics of a harmonic oscillator, we determine the conditions under which these equations reduce to a Lindblad-type generator. Our results demonstrate that complete positivity is ensured only when friction and noise are included in both Hamiltonian equations, fully justifying the classical construction. Moreover, we find that the friction coefficients must adhere to the same positivity condition in both the Hermitian and non-Hermitian formulations, revealing a form of universality that transcends the specific operator representation. The formalism developed here presents a thermodynamically consistent and completely positive quantum extension of classical stochastic mechanics. It offers a versatile tool for deriving quantum versions of thermodynamic laws and is directly applicable to a broad class of non-equilibrium nanoscale systems of current theoretical and technological interest.