Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2403.07479

A quantum oscillator interacting with a classical oscillator

Sajjad, Muhammad, Andrea Russo, Maite Arcos, Andrzej Grudka, Jonathan Oppenheim

Published Mar 12, 2024• 0 citations
arXivHTMLPDF

Abstract

We study a quantum oscillator interacting with, and back-reacting on, a classical oscillator. This can be done consistently provided the quantum system decoheres, while the backreaction has a stochastic component which causes the classical system to undergo diffusion. Nonetheless, the state of the quantum oscillator can remain pure conditioned on the trajectory of the classical oscillator. We solve the system using the classical-quantum path integral formulation, and compute the free and bridge correlators of the theory, which we use to find the heating rate. In addition, we explicitly compute the conditioned, and unconditional purity of the system. This serves as a toy model for a number of other systems in which one subsystem can be treated as effectively classical, such as a scalar quantum field interacting with another field undergoing decoherence, or a system emitting radiation, one of which is treated classically.

DOI
10.48550/arxiv.2403.07479

Related papers

A healthier semi-classical dynamics

Isaac Layton, Jonathan Oppenheim, Zachary Weller-Davies

We study the back-reaction of quantum systems onto classical ones. Taking the starting point that semi-classical physics should be described at all times by a point in classical phase space and a quantum state in Hilbert space, we consider an unravelling approach, describing the system in terms of a classical-quantum trajectory. We derive the general form of the dynamics under the assumptions that the classical trajectories are continuous and the evolution is autonomous, and the requirement that the dynamics is linear and completely positive in the combined classical-quantum state. This requirement is necessary in order to consistently describe probabilities, and forces the dynamics to be stochastic when the back-reaction is non-zero. The resulting equations of motion are natural generalisations of the standard semi-classical equations of motion, but since the resulting dynamics is linear in the combined classical-quantum state, it does not lead to the pathologies which usually follow from evolution laws based on expectation values. In particular, the evolution laws we present account for correlations between the classical and quantum system, which resolves issues associated with other semi-classical approaches. In addition, despite a breakdown of predictability in the classical degrees of freedom, the quantum state evolves deterministically conditioned on the classical trajectory, provided a trade-off between decoherence and diffusion is saturated. As a result, the quantum state remains pure when conditioned on the classical trajectory. To illustrate these points, we numerically simulate a number of semi-classical toy models, including one of vacuum fluctuations as a source driving the expansion of the universe. Finally, we discuss the application of these results to semi-classical gravity, and the black-hole information problem.

arXiv:2208.11722•Quantum•Dec 16, 2024•20 citations

Objective trajectories in hybrid classical-quantum dynamics

Jonathan Oppenheim, Carlo Sparaciari, Barbara Šoda +1 more

Consistent dynamics which couples classical and quantum degrees of freedom exists, provided it is stochastic. This dynamics is linear in the hybrid state, completely positive and trace preserving. One application of this is to study the back-reaction of quantum fields on space-time which does not suffer from the pathologies of the semi-classical equations. Here we introduce several toy models in which to study hybrid classical-quantum evolution, including a qubit coupled to a particle in a potential, and a quantum harmonic oscillator coupled to a classical one. We present an unravelling approach to calculate the dynamics, and provide code to numerically simulate it. Unlike the purely quantum case, the trajectories (or histories) of this unravelling can be unique, conditioned on the classical degrees of freedom for discrete realisations of the dynamics, when different jumps in the classical degrees of freedom are accompanied by the action of unique operators on the quantum system. As a result, the “measurement postulate'' of quantum theory is not needed; quantum systems become classical because they interact with a fundamentally classical field.

arXiv:2011.06009•Quantum•Jan 3, 2023•31 citations

Classical-quantum oscillators as diffusive processes in phase space

Emanuele Panella

The dynamics of hybrid systems -- i.e. ones in which classical and quantum degrees of freedom co-exist and interact -- feature both diffusion in the classical sector and decoherence in the quantum state. In this article, we will consider the simple setup of a classical damped oscillator interacting with its quantum counterpart and show that, for any initial state, the dynamics flows to a unique (non-equilibrium) steady state, which we compute explicitly. To do so, we make use of a useful mapping between hybrid and classical diffusive dynamics, which we characterise in detail in the master equation formalism.

arXiv:2510.07402•Oct 8, 2025•0 citations

Wavefunction of Coupled Quantum-Classical Oscillator

He Wei-Zhong, Xu Liu-Su, Zou Feng-wu

Considering classical environment to be a classical oscillator, we describe the dynamics of a system consisting of a quantum oscillator coupled to classical oscillator. Using the Born-Oppenheimer approximation and the method of adiabatic invariants, we derive the classically effective Hamiltonian describing the complete dynamics of the system and the wavefunction of quantum part of the system. We also present a new method for investigating the dynamics of the quantum system in classical environment.

Chinese Physics Letters•Apr 24, 2003•3 citations

Equilibrating continuous-variable open quantum systems using stochastic classical trajectories in path-integral space

William H. D. Moore, Stuart C. Althorpe

Beyond the weak-coupling limit, open quantum systems equilibrate to a highly entangled thermal state. For continuous-variable systems, this state can be written explicitly as an imaginary-time phase-space path integral, in which the positions are directly entangled with the bath, and the momenta are correlated with the positions through a phase term. Here, we ask to what extent this state can be reached by propagating stochastic classical trajectories in path-integral phase space. Surprisingly, we find that the trajectories equilibrate to the exact quantum equilibrium state, recovering the purely imaginary momentum-position correlation in the phase term. The trajectories are generated using a recently derived Matsubara generalized Langevin equation, which produces the imaginary correlations by evolving the stochastic variables into the complex plane. This makes the dynamics numerically unstable, but we are nonetheless able to demonstrate the equilibration of a quartic oscillator coupled to a white-noise bath. These unexpected findings could lead to new approximate methodologies for simulating continuous-variable open quantum systems.

quant-pharXiv:2606.10665•Jun 10, 2026•0 citations

Coupled Classical and Quantum Oscillators

R. M. McDermott, Ian H. Redmount

Some of the most enduring questions in physics--including the quantum measurement problem and the quantization of gravity--involve the interaction of a quantum system with a classical environment. Two linearly coupled harmonic oscillators provide a simple, exactly soluble model for exploring such interaction. Even the ground state of a pair of identical oscillators exhibits effects on the quantum nature of one oscillator, e.g., a diminution of position uncertainty, and an increase in momentum uncertainty and uncertainty product, from their unperturbed values. Interaction between quantum and classical oscillators is simulated by constructing a quantum state with one oscillator initially in its ground state, the other in a coherent or Glauber state. The subsequent wave function for this state is calculated exactly, both for identical and distinct oscillators. The reduced probability distribution for the quantum oscillator, and its position and momentum expectation values and uncertainties, are obtained from this wave function. The oscillator acquires an oscillation amplitude corresponding to a beating between the normal modes of the system; the behavior of the position and momentum uncertainties can become quite complicated. For oscillators with equal unperturbed frequencies, i.e., at resonance, the uncertainties exhibit a time-dependent quantum squeezing which can be extreme.

arXiv:quant-ph/0403184•Mar 25, 2004•7 citations
We study a quantum oscillator interacting with, and back-reacting on, a classical oscillator. This can be done consistently provided the quantum system decoheres, while the backreaction has a stochastic component which causes the classical system to undergo diffusion. Nonetheless, the state of the quantum oscillator can remain pure conditioned on the trajectory of the classical oscillator. We solve the system using the classical-quantum path integral formulation, and compute the free and bridge correlators of the theory, which we use to find the heating rate. In addition, we explicitly compute the conditioned, and unconditional purity of the system. This serves as a toy model for a number of other systems in which one subsystem can be treated as effectively classical, such as a scalar quantum field interacting with another field undergoing decoherence, or a system emitting radiation, one of which is treated classically.
We study the back-reaction of quantum systems onto classical ones. Taking the starting point that semi-classical physics should be described at all times by a point in classical phase space and a quantum state in Hilbert space, we consider an unravelling approach, describing the system in terms of a classical-quantum trajectory. We derive the general form of the dynamics under the assumptions that the classical trajectories are continuous and the evolution is autonomous, and the requirement that the dynamics is linear and completely positive in the combined classical-quantum state. This requirement is necessary in order to consistently describe probabilities, and forces the dynamics to be stochastic when the back-reaction is non-zero. The resulting equations of motion are natural generalisations of the standard semi-classical equations of motion, but since the resulting dynamics is linear in the combined classical-quantum state, it does not lead to the pathologies which usually follow from evolution laws based on expectation values. In particular, the evolution laws we present account for correlations between the classical and quantum system, which resolves issues associated with other semi-classical approaches. In addition, despite a breakdown of predictability in the classical degrees of freedom, the quantum state evolves deterministically conditioned on the classical trajectory, provided a trade-off between decoherence and diffusion is saturated. As a result, the quantum state remains pure when conditioned on the classical trajectory. To illustrate these points, we numerically simulate a number of semi-classical toy models, including one of vacuum fluctuations as a source driving the expansion of the universe. Finally, we discuss the application of these results to semi-classical gravity, and the black-hole information problem.
Consistent dynamics which couples classical and quantum degrees of freedom exists, provided it is stochastic. This dynamics is linear in the hybrid state, completely positive and trace preserving. One application of this is to study the back-reaction of quantum fields on space-time which does not suffer from the pathologies of the semi-classical equations. Here we introduce several toy models in which to study hybrid classical-quantum evolution, including a qubit coupled to a particle in a potential, and a quantum harmonic oscillator coupled to a classical one. We present an unravelling approach to calculate the dynamics, and provide code to numerically simulate it. Unlike the purely quantum case, the trajectories (or histories) of this unravelling can be unique, conditioned on the classical degrees of freedom for discrete realisations of the dynamics, when different jumps in the classical degrees of freedom are accompanied by the action of unique operators on the quantum system. As a result, the “measurement postulate'' of quantum theory is not needed; quantum systems become classical because they interact with a fundamentally classical field.
The dynamics of hybrid systems -- i.e. ones in which classical and quantum degrees of freedom co-exist and interact -- feature both diffusion in the classical sector and decoherence in the quantum state. In this article, we will consider the simple setup of a classical damped oscillator interacting with its quantum counterpart and show that, for any initial state, the dynamics flows to a unique (non-equilibrium) steady state, which we compute explicitly. To do so, we make use of a useful mapping between hybrid and classical diffusive dynamics, which we characterise in detail in the master equation formalism.
Considering classical environment to be a classical oscillator, we describe the dynamics of a system consisting of a quantum oscillator coupled to classical oscillator. Using the Born-Oppenheimer approximation and the method of adiabatic invariants, we derive the classically effective Hamiltonian describing the complete dynamics of the system and the wavefunction of quantum part of the system. We also present a new method for investigating the dynamics of the quantum system in classical environment.
Beyond the weak-coupling limit, open quantum systems equilibrate to a highly entangled thermal state. For continuous-variable systems, this state can be written explicitly as an imaginary-time phase-space path integral, in which the positions are directly entangled with the bath, and the momenta are correlated with the positions through a phase term. Here, we ask to what extent this state can be reached by propagating stochastic classical trajectories in path-integral phase space. Surprisingly, we find that the trajectories equilibrate to the exact quantum equilibrium state, recovering the purely imaginary momentum-position correlation in the phase term. The trajectories are generated using a recently derived Matsubara generalized Langevin equation, which produces the imaginary correlations by evolving the stochastic variables into the complex plane. This makes the dynamics numerically unstable, but we are nonetheless able to demonstrate the equilibration of a quartic oscillator coupled to a white-noise bath. These unexpected findings could lead to new approximate methodologies for simulating continuous-variable open quantum systems.
Some of the most enduring questions in physics--including the quantum measurement problem and the quantization of gravity--involve the interaction of a quantum system with a classical environment. Two linearly coupled harmonic oscillators provide a simple, exactly soluble model for exploring such interaction. Even the ground state of a pair of identical oscillators exhibits effects on the quantum nature of one oscillator, e.g., a diminution of position uncertainty, and an increase in momentum uncertainty and uncertainty product, from their unperturbed values. Interaction between quantum and classical oscillators is simulated by constructing a quantum state with one oscillator initially in its ground state, the other in a coherent or Glauber state. The subsequent wave function for this state is calculated exactly, both for identical and distinct oscillators. The reduced probability distribution for the quantum oscillator, and its position and momentum expectation values and uncertainties, are obtained from this wave function. The oscillator acquires an oscillation amplitude corresponding to a beating between the normal modes of the system; the behavior of the position and momentum uncertainties can become quite complicated. For oscillators with equal unperturbed frequencies, i.e., at resonance, the uncertainties exhibit a time-dependent quantum squeezing which can be extreme.