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

1994 ◽  
Vol 08 (11n12) ◽  
pp. 1563-1576 ◽  
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.

scholarly journals Exact time-dependent decoherence factor and its adiabatic classical limit

2003 ◽  
Vol 81 (10) ◽  
pp. 1185-1191
Author(s):  
J -Q Shen ◽  
P Chen ◽  
H Mao
Keyword(s):  
Invariant Theory ◽  
Geometric Phase ◽  
Classical Limit ◽  
Unitary Transformation ◽  
Time Dependent ◽  
Quantum Decoherence ◽  
Dynamical Models ◽  
Complete Set ◽  
General Time ◽  
03.65 Ge

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 Lewis–Riesenfeld 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

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

1994 ◽  
Vol 08 (14n15) ◽  
pp. 917-927 ◽  
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.

Spatial Variation Effects on Quantum Theory of the Micromaser with Ultra-Cold Λ-Type Three-Level Atom

2003 ◽  
Vol 17 (14) ◽  
pp. 2735-2747 ◽  
Author(s):  
Abdel-Shafy F. Obada ◽  
Mahmoud Abdel-Aty
Keyword(s):  
Spatial Variation ◽  
Spatial Dependence ◽  
Single Mode ◽  
Emission Probability ◽  
Time Dependent ◽  
Present System ◽  
Level Atom ◽  
Special Cases ◽  
Propagation Effects ◽  
Time Dependent Solution

We generalize a study of the behavior of a three-level atom in a single-mode mazer. Our formulation takes into account the interaction region and the spatial variation along the cavity axis. A general analytic time-dependent solution is obtained, and some special cases reduce to very simple expressions. When propagation effects are taken into consideration, the emission probability is influenced significantly. We provide the necessary arguments to justify the validity of our conclusions for emission probability and micromaser whose dynamics is governed by the wave function. Our main conclusion is that in the present system, the inclusion of the spatial dependence of the cavity region is necessary and important. We provide numerous examples to illustrate this correspondence, and provide evidence of its usefulness.

The deformation of a viscous particle surrounded by an elastic shell in a general time-dependent linear flow field

1983 ◽  
Vol 126 ◽  
pp. 533-544 ◽  
Author(s):  
P. O. Brunn
Keyword(s):  
Flow Field ◽  
Shell Thickness ◽  
Elastic Shell ◽  
Time Dependent ◽  
Linear Flow ◽  
Special Cases ◽  
Arbitrary Thickness ◽  
Elastic Membranes ◽  
Freely Suspended ◽  
General Time

The dynamics of a viscous particle surrounded by an elastic shell of arbitrary thickness freely suspended in a general linear flow field is investigated. Assuming the unstressed shell to be spherical, an analysis is presented for the case in which the flow-induced deformation leads to small departures from sphericity. The general time-dependent evolution of shape is derived and various special cases (purely elastic sphere, rigid and gaseous interior, elastic membranes) are discussed in detail. It is found that for steady-state flows the equilibrium deformations are absolutely stable and depend only upon the shell thickness, although the rates at which they are attained show the effect of the inside viscosity, too.

scholarly journals A GEOMETRICAL APPROACH TO TIME-DEPENDENT GAUGE-FIXING

1993 ◽  
Vol 08 (23) ◽  
pp. 4055-4069 ◽  
Author(s):  
JONATHAN M. EVANS ◽  
PHILIP A. TUCKEY
Keyword(s):  
Phase Space ◽  
Hamiltonian Function ◽  
Time Dependent ◽  
Geometrical Approach ◽  
Dirac Bracket ◽  
Special Cases ◽  
Gauge Fixing ◽  
Necessary And Sufficient ◽  
Geometrical Formulation ◽  
Time Evolution Equation

When a Hamiltonian system is subject to constraints which depend explicitly on time, difficulties can arise in attempting to reduce the system to its physical phase space. Specifically, it is nontrivial to restrict the system in such a way that one can find a Hamiltonian time-evolution equation involving the Dirac bracket. Using a geometrical formulation, we derive an explicit condition which is both necessary and sufficient for this to be possible, and we give a formula defining the resulting Hamiltonian function. Some previous results are recovered as special cases.

Autonomous Geometrical Mechanics

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Phase Space ◽  
Contact Structure ◽  
Invariant Tori ◽  
Time Dependent ◽  
Contact Structures ◽  
Natural Setting ◽  
Adiabatic Invariance ◽  
Jet Bundle ◽  
Integrability Theorem ◽  
Extended Phase

This chapter examines the structure of the phase space of an integrable system as being constructed from invariant tori using the Arnold–Liouville integrability theorem, and periodic flow and ergodic flow are investigated using action-angle theory. Time-dependent mechanics is formulated by extending the symplectic structure to a contact structure in an extended phase space before it is shown that mechanics has a natural setting on a jet bundle. The chapter then describes phase space of integrable systems and how tori behave when time-dependent dynamics occurs. Adiabatic invariance is discussed, as well as slow and fast Hamiltonian systems, the Hannay angle and counter adiabatic terms. In addition, the chapter discusses foliation, resonant tori, non-resonant tori, contact structures, Pfaffian forms, jet manifolds and Stokes’s theorem.

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