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:0911.1690

Canonical quantization of a dissipative system interacting with an anisotropic non-linear absorbing environment

Majid Amooshahi, Ehsan Amooghorban

Published Nov 9, 2009• 0 citations
arXivHTMLPDF

Abstract

A canonical quantization scheme is represented for a quantum system interacting with a nonlinear absorbing environment. The environment is taken anisotropic and the main system is coupled to its environment through some coupling tensors of various ranks. The nonlinear response equation of the environment against the motion of the main system is obtained. The nonlinear Langevin-schrödinger equation is concluded as the macroscopic equation of motion of the dissipative system. The effect of nonlinearity of the environment is investigated on the spontaneous emission of an initially excited two level-atom imbedded in such an environmrnt.

DOI
10.48550/arxiv.0911.1690

Related papers

Canonical quantization of nonlinear many-body systems

Antonio Maria Scarfone

We study the quantization of a classical system of interacting particles obeying a recently proposed kinetic interaction principle (KIP) [G. Kaniadakis, Physica A 296, 405 (2001)]. The KIP fixes the expression of the Fokker-Planck equation describing the kinetic evolution of the system and imposes the form of its entropy. In the framework of canonical quantization, we introduce a class of nonlinear Schrödinger equations (NSEs) with complex nonlinearities, describing, in the mean-field approximation, a system of collectively interacting particles whose underlying kinetics is governed by the KIP. We derive the Ehrenfest relations and discuss the main constants of motion arising in this model. By means of a nonlinear gauge transformation of the third kind, it is shown that in the case of constant diffusion and linear drift, the class of NSEs obeying the KIP is gauge-equivalent to another class of NSEs containing purely real nonlinearities depending only on the field rho=|psi|(2) .

arXiv:quant-ph/0506104•Physical Review E•May 12, 2005•20 citations

Effective description of the dissipative interaction between simple model-systems and their environment

Dieter Schuch

Chemical reactions are generally connected with energy transfer between reactands and their environment. There are different ways of taking this environment into account. The traditional system-plus-reservoir (S+R) approach couples the relevant system to a large number of environmental degrees of freedom, represented, e.g., by harmonic oscillators. However, this has the disadvantage—from a computational point of view—that a system of many coupled differential equations has to be solved due to the large number of degrees of freedom of the reservoir. The number of degrees of freedom can be drastically reduced if an effective description of the interaction between system and reservoir is applied. On the quantum mechanical level, the effect of the environment can be included, e.g., by the use of explicitly time-dependent Hamiltonians or nonlinear additions to the Schrödinger equation (SE). An advantage of the former method is the preservation of the canonical formalism and the linearity of the theory as well as the fact that its operators can be derived directly from the S+R approach. Therefore, it seems to be even more of a paradox that this method apparently violates the uncertainty principle. This violation does not occur in the nonlinear approaches but the nonlinearity seems, at first sight, to be problematic. However, it can be shown that a particular logarithmic nonlinear SE is equivalent to the approach using the time-dependent Hamiltonian and can be transformed into this. The fact that this transformation is nonunitary provides the key for the solution of the paradox and also allows one to show the equivalence of all three of the above-mentioned approaches. In investigating the classical counterpart, a noncanonical transformation is used between the dissipative physical level and the canonical formal level, which makes it even possible to transform the physical interacting system into a formal isolated system that can be treated in the conventional way. As a result, the effect of the interaction with the environment can be taken into account classically by noncanonical transformations and quantum mechanically by nonunitary transformations. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 537–547, 1999

International Journal of Quantum Chemistry•Jan 1, 1999•29 citations

Nonlinear quantum Langevin equations for bosonic modes in solid-state systems

Juuso Manninen, Souvik Agasti, Francesco Massel

Based on the experimental evidence that impurities contribute to the dissipation properties of solid-state open quantum systems, we provide here a description in terms of nonlinear quantum Langevin equations of the role played by two-level systems in the dynamics of a bosonic degree of freedom. Our starting point is represented by the description of the system-environment coupling in terms of coupling to two separate reservoirs, modeling the interaction with external bosonic modes and two-level systems, respectively. Furthermore, we show how this model represents a specific example of a class of open quantum systems that can be described by nonlinear quantum Langevin equations. Our analysis offers a potential explanation of the parametric effects recently observed in circuit-QED cavity optomechanics experiments.

arXiv:1711.00424•Physical Review A•Dec 20, 2017•8 citations

Theories of quantum dissipation and nonlinear coupling bath descriptors

Rui‐Xue Xu, Yang Liu, Hou-Dao Zhang +1 more

The quest of an exact and nonperturbative treatment of quantum dissipation in nonlinear coupling environments remains in general an intractable task. In this work, we address the key issues toward the solutions to the lowest nonlinear environment, a harmonic bath coupled both linearly and quadratically with an arbitrary system. To determine the bath coupling descriptors, we propose a physical mapping scheme, together with the prescription reference invariance requirement. We then adopt a recently developed dissipaton equation of motion theory [R. X. Xu et al., Chin. J. Chem. Phys. 30, 395 (2017)], with the underlying statistical quasi-particle ("dissipaton") algebra being extended to the quadratic bath coupling. We report the numerical results on a two-level system dynamics and absorption and emission line shapes.

The Journal of Chemical Physics•Mar 15, 2018•35 citations

Nonlinear Schrödinger equation and semiclassical description of the light–atom interaction

Sergey A. Rashkovskiy

In this paper, we consider the nonlinear Schrödinger equation describing the light-atom interaction while taking into account damping due to spontaneous emission.We demonstrate that the optical Bloch equations with damping due to spontaneous emission can be derived immediately from the nonlinear Schrödinger equation without any phenomenology and quantization of radiation.The steady-state solutions of the nonlinear optical Bloch equations are obtained and compared with those obtained for linear optical equations.In this approximation, the light scattering by an atom is considered........

arXiv:1603.02102•Progress of Theoretical and Experimental Physics•Nov 10, 2016•10 citations

Quantization of non-Hamiltonian physical systems

A. O. Bolivar

We propose a general method of quantization of non-Hamiltonian physical systems. Applying it, for example, to a dissipative system coupled to a thermal reservoir described by the Fokker-Planck equation, we are able to obtain the Caldeira-Leggett master equation, the nonlinear Schr\"odinger-Langevin equation, and the Caldirola-Kanai equation (with an additional term), as particular cases.

Physical Review A•Dec 1, 1998•23 citations
A canonical quantization scheme is represented for a quantum system interacting with a nonlinear absorbing environment. The environment is taken anisotropic and the main system is coupled to its environment through some coupling tensors of various ranks. The nonlinear response equation of the environment against the motion of the main system is obtained. The nonlinear Langevin-schrödinger equation is concluded as the macroscopic equation of motion of the dissipative system. The effect of nonlinearity of the environment is investigated on the spontaneous emission of an initially excited two level-atom imbedded in such an environmrnt.
We study the quantization of a classical system of interacting particles obeying a recently proposed kinetic interaction principle (KIP) [G. Kaniadakis, Physica A 296, 405 (2001)]. The KIP fixes the expression of the Fokker-Planck equation describing the kinetic evolution of the system and imposes the form of its entropy. In the framework of canonical quantization, we introduce a class of nonlinear Schrödinger equations (NSEs) with complex nonlinearities, describing, in the mean-field approximation, a system of collectively interacting particles whose underlying kinetics is governed by the KIP. We derive the Ehrenfest relations and discuss the main constants of motion arising in this model. By means of a nonlinear gauge transformation of the third kind, it is shown that in the case of constant diffusion and linear drift, the class of NSEs obeying the KIP is gauge-equivalent to another class of NSEs containing purely real nonlinearities depending only on the field rho=|psi|(2) .
Chemical reactions are generally connected with energy transfer between reactands and their environment. There are different ways of taking this environment into account. The traditional system-plus-reservoir (S+R) approach couples the relevant system to a large number of environmental degrees of freedom, represented, e.g., by harmonic oscillators. However, this has the disadvantage—from a computational point of view—that a system of many coupled differential equations has to be solved due to the large number of degrees of freedom of the reservoir. The number of degrees of freedom can be drastically reduced if an effective description of the interaction between system and reservoir is applied. On the quantum mechanical level, the effect of the environment can be included, e.g., by the use of explicitly time-dependent Hamiltonians or nonlinear additions to the Schrödinger equation (SE). An advantage of the former method is the preservation of the canonical formalism and the linearity of the theory as well as the fact that its operators can be derived directly from the S+R approach. Therefore, it seems to be even more of a paradox that this method apparently violates the uncertainty principle. This violation does not occur in the nonlinear approaches but the nonlinearity seems, at first sight, to be problematic. However, it can be shown that a particular logarithmic nonlinear SE is equivalent to the approach using the time-dependent Hamiltonian and can be transformed into this. The fact that this transformation is nonunitary provides the key for the solution of the paradox and also allows one to show the equivalence of all three of the above-mentioned approaches. In investigating the classical counterpart, a noncanonical transformation is used between the dissipative physical level and the canonical formal level, which makes it even possible to transform the physical interacting system into a formal isolated system that can be treated in the conventional way. As a result, the effect of the interaction with the environment can be taken into account classically by noncanonical transformations and quantum mechanically by nonunitary transformations. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 537–547, 1999
Based on the experimental evidence that impurities contribute to the dissipation properties of solid-state open quantum systems, we provide here a description in terms of nonlinear quantum Langevin equations of the role played by two-level systems in the dynamics of a bosonic degree of freedom. Our starting point is represented by the description of the system-environment coupling in terms of coupling to two separate reservoirs, modeling the interaction with external bosonic modes and two-level systems, respectively. Furthermore, we show how this model represents a specific example of a class of open quantum systems that can be described by nonlinear quantum Langevin equations. Our analysis offers a potential explanation of the parametric effects recently observed in circuit-QED cavity optomechanics experiments.
The quest of an exact and nonperturbative treatment of quantum dissipation in nonlinear coupling environments remains in general an intractable task. In this work, we address the key issues toward the solutions to the lowest nonlinear environment, a harmonic bath coupled both linearly and quadratically with an arbitrary system. To determine the bath coupling descriptors, we propose a physical mapping scheme, together with the prescription reference invariance requirement. We then adopt a recently developed dissipaton equation of motion theory [R. X. Xu et al., Chin. J. Chem. Phys. 30, 395 (2017)], with the underlying statistical quasi-particle ("dissipaton") algebra being extended to the quadratic bath coupling. We report the numerical results on a two-level system dynamics and absorption and emission line shapes.
In this paper, we consider the nonlinear Schrödinger equation describing the light-atom interaction while taking into account damping due to spontaneous emission.We demonstrate that the optical Bloch equations with damping due to spontaneous emission can be derived immediately from the nonlinear Schrödinger equation without any phenomenology and quantization of radiation.The steady-state solutions of the nonlinear optical Bloch equations are obtained and compared with those obtained for linear optical equations.In this approximation, the light scattering by an atom is considered........