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

Physical Review A

Maximum-entropy principle in the interaction picture: Approach to time-dependent relevant operators

J. Aliaga, G. Crespo, Araceli N. Proto

Published Aug 1, 1991• 2 citations
Publisher

Abstract

A reformulation of the well-known maximum-entropy principle in the interaction picture is shown. The advantage of this approach for the case in which the relevant operators in the interaction picture do not depend explicitly on time is stated. As an example, we study the dynamical evolution of the dispersions of the so-called slowly varying quadrature operators for the degenerate parametric amplifier and the temporal behavior of the squeezed states associated with this system.

DOI
10.1103/physreva.44.2066

Related papers

Squeezed states and Shannon entropy

J. Aliaga, G. Crespo, Araceli N. Proto

A wave-function approach to the interaction Hamiltonian for the degenerate parametric amplifier has been recently presented [C. G. Bollini and L. E. Oxman, Phys. Rev. A 47, 2339 (1993)]. We want to show here that a maximum entropy principle density matrix approach can be used to reobtain all the results shown in this reference, and also to avoid the use of a nonproperly continuous generalization of the Shannon discrete entropy.

Physical Review A•Jun 1, 1994•2 citations

Infinite set of relevant operators for an exact solution of the time-dependent Jaynes-Cummings Hamiltonian

J. L. Gruver, J. Aliaga, Hilda A. Cerdeira +1 more

The dynamics and thermodynamics of a quantum time-dependent field coupled to a two-level system, well known as the Jaynes-Cummings Hamiltonian, is studied, using the maximum entropy principle. In the framework of this approach we found three different infinite sets of relevant operators that describe the dynamics of the system for any temporal dependence. These sets of relevant operators are connected by isomorphisms, which allow us to consider the case of mixed initial conditions. A consisent set of initial conditions is established using the maximum entropy principle density operator, obtaining restrictions to the physically feasible initial conditions of the system. The behavior of the population inversion is shown for different time dependencies of the Hamiltonian and initial conditions. For the time-independent case, an explicit solution for the population inversion in terms of the relevant operators of one of the sets is given. It is also shown how the well known formulas for the population inversion are recovered for the special cases where the initial conditions correspond to a pure, coherent, and thermal field.

Physical Review A•Dec 1, 1994•23 citations

Shannon entropy and the eigenstates of the single-mode squeeze operator

C. G. Bollini, L. E. Oxman

The interaction Hamiltonian for the degenerate parametric amplifier leads to an eigenvalue equation which can be solved. The solution is used to construct the propagation kernel for the state vectors. We can then follow the evolution of the squeezed variable and its conjugate one. Also the changes of entropies during the process can be followed without difficulties.

Physical Review A•Mar 1, 1993•25 citations

Differential entropy and time

Piotr Garbaczewski

We give a detailed analysis of the Gibbs-type entropy notion and its dynamical behavior in case of time-dependent continuous probability distributions of varied origins: related to classical and quantum systems. The purpose-dependent usage of conditional Kullback-Leibler and Gibbs (Shannon) entropies is explained in case of non-equilibrium Smoluchowski processes. A very different temporal behavior of Gibbs and Kullback entropies is confronted. A specific conceptual niche is addressed, where quantum von Neumann, classical Kullback-Leibler and Gibbs entropies can be consistently introduced as information measures for the same physical system. If the dynamics of probability densities is driven by the Schrödinger picture wave-packet evolution, Gibbs-type and related Fisher information functionals appear to quantify nontrivial power transfer processes in the mean. This observation is found to extend to classical dissipative processes and supports the view that the Shannon entropy dynamics provides an insight into physically relevant non-equilibrium phenomena, which are inaccessible in terms of the Kullback-Leibler entropy and typically ignored in the literature.

arXiv:quant-ph/0408192•Entropy•Oct 18, 2005•35 citations

Quantum Dynamics and Pictures

Z. Ficek

This chapter demonstrates how one can find the time evolution of the operators from the knowledge of the time evolution of the state vector, and vice versa. It introduces a unitary operator and discusses the concept of a unitary transformation to show how one could transfer the time dependence from the state vectors to the operators. The possibility that either the state vectors or the operators can depend explicitly on time will lead to introduce the Schrodinger, Heisenberg, and interaction pictures. The chapter discusses the fundamental differences between these pictures. It also discusses the Ehrenfest theorem, which shows under which conditions quantum mechanics predicts the same results for measured physical quantities as classical physics. It illustrates unitary transformations of an arbitrary operator Â(t) which can depend explicitly on time. Quantum mechanics produces the same result as classical mechanics for a system in which particles are represented by well-localized wave functions.

•Mar 3, 2017•0 citations

Quantum Dynamics

Uri Peskin

Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.

Cambridge University Press eBooks•May 11, 2023•0 citations
A reformulation of the well-known maximum-entropy principle in the interaction picture is shown. The advantage of this approach for the case in which the relevant operators in the interaction picture do not depend explicitly on time is stated. As an example, we study the dynamical evolution of the dispersions of the so-called slowly varying quadrature operators for the degenerate parametric amplifier and the temporal behavior of the squeezed states associated with this system.
The dynamics and thermodynamics of a quantum time-dependent field coupled to a two-level system, well known as the Jaynes-Cummings Hamiltonian, is studied, using the maximum entropy principle. In the framework of this approach we found three different infinite sets of relevant operators that describe the dynamics of the system for any temporal dependence. These sets of relevant operators are connected by isomorphisms, which allow us to consider the case of mixed initial conditions. A consisent set of initial conditions is established using the maximum entropy principle density operator, obtaining restrictions to the physically feasible initial conditions of the system. The behavior of the population inversion is shown for different time dependencies of the Hamiltonian and initial conditions. For the time-independent case, an explicit solution for the population inversion in terms of the relevant operators of one of the sets is given. It is also shown how the well known formulas for the population inversion are recovered for the special cases where the initial conditions correspond to a pure, coherent, and thermal field.
We give a detailed analysis of the Gibbs-type entropy notion and its dynamical behavior in case of time-dependent continuous probability distributions of varied origins: related to classical and quantum systems. The purpose-dependent usage of conditional Kullback-Leibler and Gibbs (Shannon) entropies is explained in case of non-equilibrium Smoluchowski processes. A very different temporal behavior of Gibbs and Kullback entropies is confronted. A specific conceptual niche is addressed, where quantum von Neumann, classical Kullback-Leibler and Gibbs entropies can be consistently introduced as information measures for the same physical system. If the dynamics of probability densities is driven by the Schrödinger picture wave-packet evolution, Gibbs-type and related Fisher information functionals appear to quantify nontrivial power transfer processes in the mean. This observation is found to extend to classical dissipative processes and supports the view that the Shannon entropy dynamics provides an insight into physically relevant non-equilibrium phenomena, which are inaccessible in terms of the Kullback-Leibler entropy and typically ignored in the literature.
This chapter demonstrates how one can find the time evolution of the operators from the knowledge of the time evolution of the state vector, and vice versa. It introduces a unitary operator and discusses the concept of a unitary transformation to show how one could transfer the time dependence from the state vectors to the operators. The possibility that either the state vectors or the operators can depend explicitly on time will lead to introduce the Schrodinger, Heisenberg, and interaction pictures. The chapter discusses the fundamental differences between these pictures. It also discusses the Ehrenfest theorem, which shows under which conditions quantum mechanics predicts the same results for measured physical quantities as classical physics. It illustrates unitary transformations of an arbitrary operator Â(t) which can depend explicitly on time. Quantum mechanics produces the same result as classical mechanics for a system in which particles are represented by well-localized wave functions.
Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.