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

Infinite-Mode Squeezed Coherent States and Non-Equilibrium Statistical Mechanics (Phase-Space-Picture Approach)

Leehwa Yeh

Published Aug 1, 1992• 0 citations
OpenAlex

Abstract

The phase-space-picture approach to quantum non-equilibrium statistical mechanics via the characteristic function of infinite-mode squeezed coherent states is introduced. We use quantum Brownian motion as an example to show how this approach provides an interesting geometrical interpretation of quantum non-equilibrium phenomena.

Related papers

Infinite-Mode Squeezed Coherent States and Non-Equilibrium Statistical Mechanics (Phase-Space-Picture Approach).

Yeh L

The phase-space-picture approach to quantum non-equilibrium statistical mechanics via the characteristic function of infinite-mode squeezed coherent states is introduced. We use quantum Brownian motion as an example to show how this approach provides an interesting geometrical interpretation of quantum non-equilibrium phenomena.

NASA conference publication•Jan 1, 1993•0 citations

Complex Coherent States and Non Equilibrium Quantum Thermodynamics

E. Celeghini

A possible thermodynamic evolution to equilibrium of a quantum statistical system is discussed. The description is entirely algebraic and it is done by means of complex coherent states built in the Universal Enveloping Algebra of the complex algebra https://www.w3.org/1998/Math/MathML" display="inline"> A 1 . Finite dimensional representations of https://www.w3.org/1998/Math/MathML" display="inline"> A 1 give fermion distributions while infinite dimensional ones describe bosons. The scheme is thermodynamic in the sense that fluctuations are not considered and this allows us to resort to an evolution operator that realizes, for t → ∞, a projector on the state of equilibrium. The shrinking of the Hilbert space is related to non unitary evolution that change the normalization of the state vectors. The theory is invariant under time reversal: vectors at t → ±∞ differ only in phases and both correspond to the equilibrium probability distribution. Equilibrium and non-equilibrium thermodynamic quantities can be calculated at microscopic level in a framework that is in between thermodynamic and statistical mechanics. Our approach is a physically founded example of non-linear quantum theory.

CRC Press eBooks•Nov 30, 2003•0 citations

Understanding Squeezed States of Light Through Wigner's Phase-Space

Sibel Baskal, Marilyn E. Noz

This paper starts with the transition from classical physics to quantum mechanics which was greatly aided by the concept of phase space. The role of canonical transformations in quantum mechanics is addressed. The Wigner phase-space distribution function is then defined which arises from the formulation of the density matrix, followed by the harmonic oscillator in phase space. Coherent and one- and two-mode squeezed states of light as well as the squeezed vacuum are discussed in the phase-space picture. Attention is also drawn to the fact that squeezed states naturally generate entanglement between the two-modes. Coupled harmonic oscillators are also elucidated in connection with the Wigner phase space. It will be noted that the phase-space picture of quantum mechanics has become an important scientific language for the rapidly expanding field of quantum optics. Here, we mainly focus on the simplest form of the Wigner function, which finds application in many branches of quantum mechanics. We make use of several symmetry groups such as Lorentz groups, the symplectic group in two and four dimensions, and the Euclidean group. The decoherence problem of an optical field is examined through a reformulation of the Poincaré sphere as a further illustration of the density matrix.

quant-pharXiv:2606.23102•Mathematics•Jun 23, 2026•1 citation

Non-equilibrium quantum dynamics of condensed matter models.

Sashwin Sewran

In this dissertation, we studied the generation of squeezed states induced by a timedependent interaction and the in uence of temperature on the strength of the squeezing in a condensed matter model. The model studied comprised two quantum harmonic oscillators, with a time-dependent, non-linear coupling between them. The in uence of the thermal bath on the non-equilibrium dynamics of the model was represented in terms of non-Hamiltonian thermostats and a collection of independent harmonic oscillators with Ohmic spectral density. The equations of motion were studied in the Wigner representation, which introduces a phase space description for the model. The representation of the system in quantum phase space allowed us to investigate the di erence between purely classical evolution and the relative importance of quantum corrections with respect to the dynamics. The dynamics was studied by means of computer simulation techniques. The numerical simulation of the non-equilibrium statistical mechanics of both time-dependent and non-linear interactions allowed us to investigate conditions beyond those in recent literature [1, 2].

•Jan 1, 2013•0 citations

Quantum statistical mechanics, quantum probability and quantum characteristic function

Yan Gu

The theory of quantum statistical mechanics is reviewed in the light of the quantum probability theory developed in recent years by mathematicians. We show that—which is our main motivation here—the former is a special model in the framework of the latter, by extending the characteristic function in the classical probability theory to the quantum probability theory based on the requirements of quantum statistical mechanics. We find that due to the fact that fundamental dynamical variables of a quantum system must respect certain commutation relations, one has to reform the classical characteristic function defined on a Euclidean space to a normalized, nonnegative definite function defined on a group which is generated by a Lie algebra formed from these commutation relations. Our key finding is that the group-theoretical characteristic function could be adopted to replace the density matrix commonly used in quantum statistical mechanics for representing a state of the quantum ensemble, and more than that, the new representation has some significant advantages in applications. To show its advantages compared with the conventional density matrix scenario, we illustrate its applications in: solving the master equation that describes the irreversible evolution of a quantum open system; determining the pointer states of a quantum open system; and expanding the density matrix with the pointer states. In particular, thanks to the quantum characteristic function, we are able to write down the solutions of the quantum-optical master equation explicitly, obtain its steady solutions and verify that the pointer states are exactly coherent states. Besides, the Caldeira-Leggett master equation for the quantum Brownian motion and its extension to the Lindblad type are also studied.

Zhongguo kexue. Wulixue Lixue Tianwenxue•Apr 20, 2020•3 citations

Phase-space representation of completely positive quantum operation: Invertible subdynamics of two-mode squeezed states

Sumiyoshi Abe, Yasuyuki Matsuo

Completely positive quantum operations are frequently discussed in the contexts of statistical mechanics and quantum information. They are customarily given by maps forming positive operator-values measures. To intuitively understand the physical meanings of such abstract operations, the method of phase-space representations is examined. This method enables one to grasp the operations in terms of the classical statistical notions. As an example of physical importance, here, the phase-space representation of the completely positive quantum operation arising from the single-mode subdynamics of the two-mode squeezed vacuum state, which maps from the vacuum state at vanishing temperature to mixed states with perfect decoherence including the thermal state, is studied. It is found in the P representation that remarkably this operation is invertible, implying that coherence lost by the quantum operation can be recovered.

arXiv:1403.0863•Physical Review E•Jan 14, 2015•0 citations
A possible thermodynamic evolution to equilibrium of a quantum statistical system is discussed. The description is entirely algebraic and it is done by means of complex coherent states built in the Universal Enveloping Algebra of the complex algebra https://www.w3.org/1998/Math/MathML" display="inline"> A 1 . Finite dimensional representations of https://www.w3.org/1998/Math/MathML" display="inline"> A 1 give fermion distributions while infinite dimensional ones describe bosons. The scheme is thermodynamic in the sense that fluctuations are not considered and this allows us to resort to an evolution operator that realizes, for t → ∞, a projector on the state of equilibrium. The shrinking of the Hilbert space is related to non unitary evolution that change the normalization of the state vectors. The theory is invariant under time reversal: vectors at t → ±∞ differ only in phases and both correspond to the equilibrium probability distribution. Equilibrium and non-equilibrium thermodynamic quantities can be calculated at microscopic level in a framework that is in between thermodynamic and statistical mechanics. Our approach is a physically founded example of non-linear quantum theory.
This paper starts with the transition from classical physics to quantum mechanics which was greatly aided by the concept of phase space. The role of canonical transformations in quantum mechanics is addressed. The Wigner phase-space distribution function is then defined which arises from the formulation of the density matrix, followed by the harmonic oscillator in phase space. Coherent and one- and two-mode squeezed states of light as well as the squeezed vacuum are discussed in the phase-space picture. Attention is also drawn to the fact that squeezed states naturally generate entanglement between the two-modes. Coupled harmonic oscillators are also elucidated in connection with the Wigner phase space. It will be noted that the phase-space picture of quantum mechanics has become an important scientific language for the rapidly expanding field of quantum optics. Here, we mainly focus on the simplest form of the Wigner function, which finds application in many branches of quantum mechanics. We make use of several symmetry groups such as Lorentz groups, the symplectic group in two and four dimensions, and the Euclidean group. The decoherence problem of an optical field is examined through a reformulation of the Poincaré sphere as a further illustration of the density matrix.
In this dissertation, we studied the generation of squeezed states induced by a timedependent interaction and the in uence of temperature on the strength of the squeezing in a condensed matter model. The model studied comprised two quantum harmonic oscillators, with a time-dependent, non-linear coupling between them. The in uence of the thermal bath on the non-equilibrium dynamics of the model was represented in terms of non-Hamiltonian thermostats and a collection of independent harmonic oscillators with Ohmic spectral density. The equations of motion were studied in the Wigner representation, which introduces a phase space description for the model. The representation of the system in quantum phase space allowed us to investigate the di erence between purely classical evolution and the relative importance of quantum corrections with respect to the dynamics. The dynamics was studied by means of computer simulation techniques. The numerical simulation of the non-equilibrium statistical mechanics of both time-dependent and non-linear interactions allowed us to investigate conditions beyond those in recent literature [1, 2].
The theory of quantum statistical mechanics is reviewed in the light of the quantum probability theory developed in recent years by mathematicians. We show that—which is our main motivation here—the former is a special model in the framework of the latter, by extending the characteristic function in the classical probability theory to the quantum probability theory based on the requirements of quantum statistical mechanics. We find that due to the fact that fundamental dynamical variables of a quantum system must respect certain commutation relations, one has to reform the classical characteristic function defined on a Euclidean space to a normalized, nonnegative definite function defined on a group which is generated by a Lie algebra formed from these commutation relations. Our key finding is that the group-theoretical characteristic function could be adopted to replace the density matrix commonly used in quantum statistical mechanics for representing a state of the quantum ensemble, and more than that, the new representation has some significant advantages in applications. To show its advantages compared with the conventional density matrix scenario, we illustrate its applications in: solving the master equation that describes the irreversible evolution of a quantum open system; determining the pointer states of a quantum open system; and expanding the density matrix with the pointer states. In particular, thanks to the quantum characteristic function, we are able to write down the solutions of the quantum-optical master equation explicitly, obtain its steady solutions and verify that the pointer states are exactly coherent states. Besides, the Caldeira-Leggett master equation for the quantum Brownian motion and its extension to the Lindblad type are also studied.
Completely positive quantum operations are frequently discussed in the contexts of statistical mechanics and quantum information. They are customarily given by maps forming positive operator-values measures. To intuitively understand the physical meanings of such abstract operations, the method of phase-space representations is examined. This method enables one to grasp the operations in terms of the classical statistical notions. As an example of physical importance, here, the phase-space representation of the completely positive quantum operation arising from the single-mode subdynamics of the two-mode squeezed vacuum state, which maps from the vacuum state at vanishing temperature to mixed states with perfect decoherence including the thermal state, is studied. It is found in the P representation that remarkably this operation is invertible, implying that coherence lost by the quantum operation can be recovered.