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:1711.07207 · Journal of Physics Communications

Quantum-classical transition in dissipative systems through scaled trajectories

S V Mousavi, Salvador Miret-Artés

Published Mar 8, 2018• 17 citations
arXivHTMLPDF

Abstract

A nonlinear quantum–classical transition wave equation is proposed for dissipative systems within the Caldirola-Kanai model. Equivalence of this transition equation to a scaled Schrödinger equation is proved. The dissipative dynamics is then studied in terms of what we call scaled trajectories following the standard procedure used in Bohmian mechanics. These trajectories depend on a continuous parameter allowing us a smooth transition from Bohmian to classical trajectories. Arrival times and actual momentum distribution functions are also analyzed. The propagation of a Gaussian wave packet in a viscid medium under the presence of constant, linear and harmonic potentials is studied. The gradual decoherence process and localization are easily visualized and understood within this theoretical framework.

DOI
10.1088/2399-6528/aab521

Related papers

Dissipative Bohmian mechanics: A trajectory analysis of wave-packet dynamics in viscid media

Ángel S. Sanz, R. Mart, Germán A. Rojas-Lorenzo

Classical viscid media are quite common in our everyday life. However, we are not used to find such media in quantum mechanics and much less to analyze their effects on the dynamics of quantum systems. Here we provide a Bohmian analysis of the friction dynamics led by hypothetical quantum viscid media, which will be simulated by means of the so-called Caldirola-Kanai Hamiltonian. As it is shown, this causal model enables a better understanding of phenomena, such as quantum localization (“quantum freezing”) or the quantum-to-classical transition. We have analyzed these phenomena in a series of paradigmatic situations of quantum mechanics, such as free wave-packet evolution, motion in a linear potential, damped harmonic oscillator and interference dynamics.

•Jun 27, 2013•4 citations

Dissipative diffraction in time and early-arrival in continuous quantum-classical transition

S. V. Mousavi

Diffraction in time and early-arrival phenomena will be studied in the context of dissipative systems. This theoretical consideration is done by using the linear scaled wave equation proposed in the framework of Caldirola-Kanai theory for the continuous quantum-classical transition. Two simple but physically important and interesting examples are studied, sudden release from a shutter where transient behaviors (diffraction in time) are seen; and also transmission through a time-dependent parabolic barrier where early-arrival is seen. Calculations show that diffraction in time is a non-classical effect and temporary behaviors are gradually suppressed as the friction increases. Furthermore, early-arrival is seen even in the classical regime. This means that early-arrival is not a non-classical phenomenon. This behavior will be explained by means of scaled trajectories which are just Bohmian trajectories in the quantum regime.

Iranian Journal of Physics Research•May 1, 2021•0 citations

Quantum‐classical transition of the dissipative wave packet dynamics for barrier scattering

Chia‐Chun Chou

Abstract A nonlinear equation based on the Schrödinger–Langevin equation is employed to analyze the quantum‐classical transition of dissipative systems for wave packet barrier scattering. The transition equation is obtained by subtracting a modified quantum potential term depending on a degree of quantumness in the Schrödinger–Langevin equation, and it provides a continuous description for the transition process of dissipative systems from purely quantum to purely classical regimes. Important properties possessed by the equation are derived. The energy dissipation rate of the dissipative system does not depend on the degree of quantumness. It is shown that the equations of motion for the expectation values of the position and momentum operators for dissipative systems hold for all the degree of quantumness. This feature establishes the correspondence between classical and quantum dissipative systems. Computational results for the transition process of dissipative wave packet dynamics and transition trajectories are presented and analyzed for model systems involving the propagation of a free Gaussian wave packet and the wave packet scattering from an Eckart barrier. Computational results demonstrate that the agreement between the classical‐limit transition trajectories and classical trajectories is excellent, except in some regions. The single‐valued transition wave function implies that transition trajectories cannot pass through the same configuration space point at the same time.

International Journal of Quantum Chemistry•Oct 9, 2018•5 citations

Quantum‐classical transition equation with complex trajectories

Chia‐Chun Chou

Abstract A quantum‐classical transition equation in complex space is derived in the framework of the complex quantum Hamilton–Jacobi formalism. The transition equation is obtained by subtracting a complex‐valued quantum potential term from the complex‐extended time‐dependent Schrödinger equation (TDSE). It is shown that the nonlinear transition equation is equivalent to a linear scaled TDSE with a rescaled Planck's constant. Employing the quantum momentum function defined by the gradient of the complex action, we can analyze the quantum‐classical transition of physical systems using complex transition trajectories. Complex transition trajectories are presented for the free Gaussian wave packet, the harmonic oscillator, and the Morse potential. This study demonstrates that the transition equation provides a continuous description for the quantum‐classical transition of physical systems in complex space.

International Journal of Quantum Chemistry•Aug 4, 2016•13 citations

Scaled Caldeira-Leggett Dynamics: Deterministic Trajectories and the Classical Limit

S. V. Mousavi, S. Miret-Artés

The dynamics of an open quantum system in the high-temperature regime of the Caldeira-Leggett model using a Bohmian language is investigated. By expressing the density matrix in polar form, generalized continuity and Hamilton-Jacobi equations are obtained. Thermal effects appear explicitly in the former, while they influence the latter only indirectly through an effective potential. Remarkably, the effective potential does not vanish in the classical limit and cannot, in general, be decomposed into simple additive quantum and thermal contributions. Instead, it retains a nontrivial structure leading to a residual temperature dependence in the resulting dynamics. As a consequence, the resulting temperature-dependent-classical trajectories remain deterministic even in the presence of a thermal environment. This behavior contrasts with the stochastic dynamics observed in the Langevin description and reflects the ensemble-based nature of the present approach. To further explore the quantum-to-classical transition, a scaled version of the Caldeira-Leggett equation is introduced by scaling both the density matrix and Planck constant through a quantumness parameter. This parameter takes the value one in the fully quantum regime and smoothly approaches zero in the classical limit. Applications to Gaussian states demonstrate a gradual suppression of coherence and interference, governed jointly by thermal effects and the transition parameter.

quant-pharXiv:2608.02054•Physica Scripta•Aug 4, 2026•0 citations

Different routes to the classical limit of backflow

S. V. Mousavi, Salvador Miret-Artés

Abstract Decoherence is a well established process for the emergence of classical mechanics in open quantum systems. However, it can have two different origins or mechanisms depending on the dynamics one is considering, speaking then about intrinsic decoherence for isolated systems and environmental decoherence due to dissipation/fluctuations for open systems. This second mechanism can not be considered for backflow since no thermal fluctuation terms can be added in the formalism in order to keep an important requirement for the occurrence of this effect: only contributions of positive momenta along time should be maintained. The purpose of this work is to analyze the backflow effect in the light of the underlying intrinsic decoherence and the dissipative dynamics. For this goal, we first deal with the Milburn approach where a mean frequency of the unitary evolution steps undergone for the system is assumed. A comparative analysis is carried out in terms of the Lindblad master equation. Second, the so-called quantum-to-classical transition wave equation is analyzed from a linear scaled Schrödinger equation which is derived and expressed in terms of a continuous parameter covering from the quantum to the classical regime as well as all in-between dynamical non-classical regimes. This theoretical analysis is inspired by the Wentzel–Kramers–Brillouin approximation. And third, in order to complete our analysis, the transition wave equation formalism is also applied to dissipative backflow within the Caldirola–Kanai approach where the dissipative dynamics comes from an effective Hamiltonian. In all the cases treated here, backflow is gradually suppressed as the intrinsic decoherence process is developing, paying a special attention to the classical limit. The route to classicality is not unique.

arXiv:2211.09057•Journal of Physics A Mathematical and Theoretical•Nov 16, 2022•5 citations
A nonlinear quantum–classical transition wave equation is proposed for dissipative systems within the Caldirola-Kanai model. Equivalence of this transition equation to a scaled Schrödinger equation is proved. The dissipative dynamics is then studied in terms of what we call scaled trajectories following the standard procedure used in Bohmian mechanics. These trajectories depend on a continuous parameter allowing us a smooth transition from Bohmian to classical trajectories. Arrival times and actual momentum distribution functions are also analyzed. The propagation of a Gaussian wave packet in a viscid medium under the presence of constant, linear and harmonic potentials is studied. The gradual decoherence process and localization are easily visualized and understood within this theoretical framework.
Classical viscid media are quite common in our everyday life. However, we are not used to find such media in quantum mechanics and much less to analyze their effects on the dynamics of quantum systems. Here we provide a Bohmian analysis of the friction dynamics led by hypothetical quantum viscid media, which will be simulated by means of the so-called Caldirola-Kanai Hamiltonian. As it is shown, this causal model enables a better understanding of phenomena, such as quantum localization (“quantum freezing”) or the quantum-to-classical transition. We have analyzed these phenomena in a series of paradigmatic situations of quantum mechanics, such as free wave-packet evolution, motion in a linear potential, damped harmonic oscillator and interference dynamics.
Diffraction in time and early-arrival phenomena will be studied in the context of dissipative systems. This theoretical consideration is done by using the linear scaled wave equation proposed in the framework of Caldirola-Kanai theory for the continuous quantum-classical transition. Two simple but physically important and interesting examples are studied, sudden release from a shutter where transient behaviors (diffraction in time) are seen; and also transmission through a time-dependent parabolic barrier where early-arrival is seen. Calculations show that diffraction in time is a non-classical effect and temporary behaviors are gradually suppressed as the friction increases. Furthermore, early-arrival is seen even in the classical regime. This means that early-arrival is not a non-classical phenomenon. This behavior will be explained by means of scaled trajectories which are just Bohmian trajectories in the quantum regime.
Abstract A nonlinear equation based on the Schrödinger–Langevin equation is employed to analyze the quantum‐classical transition of dissipative systems for wave packet barrier scattering. The transition equation is obtained by subtracting a modified quantum potential term depending on a degree of quantumness in the Schrödinger–Langevin equation, and it provides a continuous description for the transition process of dissipative systems from purely quantum to purely classical regimes. Important properties possessed by the equation are derived. The energy dissipation rate of the dissipative system does not depend on the degree of quantumness. It is shown that the equations of motion for the expectation values of the position and momentum operators for dissipative systems hold for all the degree of quantumness. This feature establishes the correspondence between classical and quantum dissipative systems. Computational results for the transition process of dissipative wave packet dynamics and transition trajectories are presented and analyzed for model systems involving the propagation of a free Gaussian wave packet and the wave packet scattering from an Eckart barrier. Computational results demonstrate that the agreement between the classical‐limit transition trajectories and classical trajectories is excellent, except in some regions. The single‐valued transition wave function implies that transition trajectories cannot pass through the same configuration space point at the same time.
Abstract A quantum‐classical transition equation in complex space is derived in the framework of the complex quantum Hamilton–Jacobi formalism. The transition equation is obtained by subtracting a complex‐valued quantum potential term from the complex‐extended time‐dependent Schrödinger equation (TDSE). It is shown that the nonlinear transition equation is equivalent to a linear scaled TDSE with a rescaled Planck's constant. Employing the quantum momentum function defined by the gradient of the complex action, we can analyze the quantum‐classical transition of physical systems using complex transition trajectories. Complex transition trajectories are presented for the free Gaussian wave packet, the harmonic oscillator, and the Morse potential. This study demonstrates that the transition equation provides a continuous description for the quantum‐classical transition of physical systems in complex space.
The dynamics of an open quantum system in the high-temperature regime of the Caldeira-Leggett model using a Bohmian language is investigated. By expressing the density matrix in polar form, generalized continuity and Hamilton-Jacobi equations are obtained. Thermal effects appear explicitly in the former, while they influence the latter only indirectly through an effective potential. Remarkably, the effective potential does not vanish in the classical limit and cannot, in general, be decomposed into simple additive quantum and thermal contributions. Instead, it retains a nontrivial structure leading to a residual temperature dependence in the resulting dynamics. As a consequence, the resulting temperature-dependent-classical trajectories remain deterministic even in the presence of a thermal environment. This behavior contrasts with the stochastic dynamics observed in the Langevin description and reflects the ensemble-based nature of the present approach. To further explore the quantum-to-classical transition, a scaled version of the Caldeira-Leggett equation is introduced by scaling both the density matrix and Planck constant through a quantumness parameter. This parameter takes the value one in the fully quantum regime and smoothly approaches zero in the classical limit. Applications to Gaussian states demonstrate a gradual suppression of coherence and interference, governed jointly by thermal effects and the transition parameter.
Abstract Decoherence is a well established process for the emergence of classical mechanics in open quantum systems. However, it can have two different origins or mechanisms depending on the dynamics one is considering, speaking then about intrinsic decoherence for isolated systems and environmental decoherence due to dissipation/fluctuations for open systems. This second mechanism can not be considered for backflow since no thermal fluctuation terms can be added in the formalism in order to keep an important requirement for the occurrence of this effect: only contributions of positive momenta along time should be maintained. The purpose of this work is to analyze the backflow effect in the light of the underlying intrinsic decoherence and the dissipative dynamics. For this goal, we first deal with the Milburn approach where a mean frequency of the unitary evolution steps undergone for the system is assumed. A comparative analysis is carried out in terms of the Lindblad master equation. Second, the so-called quantum-to-classical transition wave equation is analyzed from a linear scaled Schrödinger equation which is derived and expressed in terms of a continuous parameter covering from the quantum to the classical regime as well as all in-between dynamical non-classical regimes. This theoretical analysis is inspired by the Wentzel–Kramers–Brillouin approximation. And third, in order to complete our analysis, the transition wave equation formalism is also applied to dissipative backflow within the Caldirola–Kanai approach where the dissipative dynamics comes from an effective Hamiltonian. In all the cases treated here, backflow is gradually suppressed as the intrinsic decoherence process is developing, paying a special attention to the classical limit. The route to classicality is not unique.