Time-dependent formulation of the linear unitary transformation and the time evolution of general time-dependent quadratic Hamiltonian systems

The present paper finds the complete set of exact solutions of the general time-dependent dynamical models for quantum decoherence, by making use of the LewisRiesenfeld invariant theory and the invariant-related unitary transformation formulation. Based on this, the general explicit expression for the decoherence factor is then obtained and the adiabatic classical limit of an illustrative example is discussed. The result (i.e., the adiabatic classical limit) obtained in this paper is consistent with what is obtained by other authors, and furthermore we obtain more general results concerning time-dependent nonadiabatic quantum decoherence. It is shown that the invariant theory is appropriate for treating both the time-dependent quantum decoherence and the geometric phase factor. PACS Nos.: 03.65.Ge, 03.65.Bz

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THE QUADRATIC TIME-DEPENDENT HAMILTONIAN: EVOLUTION OPERATOR, SQUEEZING REGIONS IN PHASE SPACE AND TRAJECTORIES

International Journal of Modern Physics B ◽

10.1142/s0217979294000671 ◽

1994 ◽

Vol 08 (11n12) ◽

pp. 1563-1576 ◽

Cited By ~ 16

Author(s):

S.S. MIZRAHI ◽

M.H.Y. MOUSSA ◽

B. BASEIA

Keyword(s):

Phase Space ◽

Unitary Transformation ◽

Uniform Magnetic Field ◽

Evolution Operator ◽

Single Mode ◽

Time Dependent ◽

Special Cases ◽

Complete Set ◽

Hamiltonian Evolution ◽

General Time

We consider the most general Time-Dependent (TD) quadratic Hamiltonian written in terms of the bosonic operators a and a+, which may represent either a charged particle subjected to a harmonic motion, immersed in a TD uniform magnetic field, or a single mode photon field going through a squeezing medium. We solve the TD Schrödinger equation by a method that uses, sequentially, a TD unitary transformation and the diagonalization of a TD invariant, and we verify that the exact solution is a complete set of TD states. We also obtain the evolution operator which is essential to express operators in the Heisenberg picture. The variances of the quadratures are calculated and a phase space of parameters introduced, in which we identify squeezing regions. The results for some special cases are presented and as an illustrative example the parametric oscillator is revisited and the trajectories in phase space drawn.

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SQUEEZING AND SOME OTHER CHARACTERISTICS OF TIME-DEPENDENT HARMONIC OSCILLATOR WITH VARIABLE MASS

Modern Physics Letters B ◽

10.1142/s0217984994000923 ◽

1994 ◽

Vol 08 (14n15) ◽

pp. 917-927 ◽

Cited By ~ 1

Author(s):

A. JOSHI ◽

S. V. LAWANDE

Keyword(s):

Coherent State ◽

Harmonic Oscillator ◽

Time Evolution ◽

Evolution Operator ◽

Variable Mass ◽

Time Dependent ◽

Integral Approach ◽

Feynman Path ◽

Time Evolution Operator ◽

General Time

In this paper we investigate the time evolution of a general time-dependent harmonic oscillator (TDHO) with variable mass using Feynman path integral approach. We explicitly evaluate the squeezing in the quadrature components of a general quantum TDHO with variable mass. This calculation is further elaborated for three particular cases of variable mass whose propagator can be written in a closed form. We also obtain an exact form of the time-evolution operator, the wave function, and the time-dependent coherent state for the TDHO. Our results clearly indicate that the time-dependent coherent state is equivalent to the squeezed coherent state.

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NON-UNITARY TRANSFORMATION OF QUANTUM TIME-DEPENDENT NON-HERMITIAN SYSTEMS

Acta Polytechnica ◽

10.14311/ap.2017.57.0424 ◽

2017 ◽

Vol 57 (6) ◽

pp. 424 ◽

Cited By ~ 6

Author(s):

Mustapha Maamache

Keyword(s):

Quantum Mechanics ◽

Quantum System ◽

Linear Combination ◽

Hamiltonian Systems ◽

Unitary Transformation ◽

Time Dependent ◽

Quantum Evolution ◽

Precise Description ◽

Quantum Time ◽

New Perspective

We provide a new perspective on non-Hermitian evolution in quantum mechanics by emphasizing the same method as in the Hermitian quantum evolution. We first give a precise description of the non unitary transformation and the associated evolution, and collecting the basic results around it and postulating the norm preserving. This cautionary postulate imposing that the time evolution of a non Hermitian quantum system preserves the inner products between the associated states must not be read naively. We also give an example showing that the solutions of time-dependent non Hermitian Hamiltonian systems given by a linear combination of SU(1,1) and SU(2) are obtained thanks to time-dependent non-unitary transformation.

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Unitary transformation approach to the exact solution for a class of time-dependent nonlinear Hamiltonian systems

Journal of Mathematical Physics ◽

10.1063/1.532341 ◽

1998 ◽

Vol 39 (1) ◽

pp. 161-169 ◽

Cited By ~ 20

Author(s):

M. Maamache

Keyword(s):

Exact Solution ◽

Hamiltonian Systems ◽

Unitary Transformation ◽

Time Dependent ◽

Transformation Approach

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SU(1,1) LIE ALGEBRA APPLIED TO THE TIME-DEPENDENT QUADRATIC HAMILTONIAN SYSTEM PERTURBED BY A SINGULARITY

International Journal of Modern Physics B ◽

10.1142/s0217979204026627 ◽

2004 ◽

Vol 18 (26) ◽

pp. 3429-3441 ◽

Cited By ~ 7

Author(s):

JEONG RYEOL CHOI ◽

SEONG SOO CHOI

Keyword(s):

Hamiltonian System ◽

Probability Density ◽

Lie Algebra ◽

Time Evolution ◽

Coherent States ◽

Classical Trajectory ◽

Quantum States ◽

Time Dependent ◽

Expectation Values ◽

Quadratic Hamiltonian

We realized SU (1,1) Lie algebra in terms of the appropriate SU (1,1) generators for the time-dependent quadratic Hamiltonian system perturbed by a singularity. Exact quantum states of the system are investigated using SU (1,1) Lie algebra. Various expectation values in two kinds of the generalized SU (1,1) coherent states, that is, BG coherent states and Perelomov coherent states are derived. We applied our study to the CKOPS (Caldirola–Kanai oscillator perturbed by a singularity). Due to the damping constant γ, the probability density of the SU (1,1) coherent states for the CKOPS converged to the center with time. The time evolution of the probability density in SU (1,1) coherent states for the CKOPS are very similar to the classical trajectory.

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