QUANTUM TREATMENT OF A TIME DEPENDENT SINGLE-TRAPPED ION INTERACTING WITH A BIMODAL CAVITY FIELD

2003 ◽  
Vol 17 (31n32) ◽  
pp. 5925-5941 ◽  
Author(s):  
MAHMOUD ABDEL-ATY ◽  
A.-S. F. OBADA ◽  
M. SEBAWE ABDALLA
Keyword(s):  
Cross Correlation ◽  
Analytic Solution ◽  
Coupling Parameter ◽  
Equations Of Motion ◽  
Time Dependent ◽  
Field Interaction ◽  
Trapped Ion ◽  
Heisenberg Equations ◽  
Quantum Treatment ◽  
Heisenberg Equations Of Motion

In the present communication we consider a time dependent ion-field interaction. Here we discuss the interaction between a single trapped ion and two fields taking into account the coupling parameter to be time dependent and allowing for amplitude modulation of the laser field radiating the trapped ion. At exact resonances the analytic solution for the Heisenberg equations of motion is obtained. We examine the effect of the velocity and the acceleration on the Rabi oscillations by studying the second order correlation function. The phenomenon of squeezing for single and two fields cases is considered. The cross correlation between the fields is discussed.

2D Saddle Potential Schrödinger Green's Function in the Presence of an Arbitrary Time-Dependent Electric Field

1997 ◽  
Vol 11 (26n27) ◽  
pp. 1193-1196 ◽  
Author(s):  
Norman J. M. Horing ◽  
Kashif Sabeeh
Keyword(s):  
Electric Field ◽  
Green’S Function ◽  
Green's Function ◽  
Equations Of Motion ◽  
Time Dependent ◽  
Arbitrary Time ◽  
Heisenberg Equations ◽  
Time Variables ◽  
Arbitrary Time Dependence ◽  
Heisenberg Equations Of Motion

We derive the retarded nonrelativistic Green's function for a Schrödinger electron confined to a plane and subject to a parabolic saddle potential and an electric field having arbitrary time dependence and orientation on the plane. This derivation is carried out using Heisenberg equations of motion for position and momentum operators, following an earlier analysis of Schwinger, to obtain the retarded Green's function in closed form, as an explicit function of position and time variables.

SOLVING OPERATOR DIFFERENTIAL EQUATIONS IN TERMS OF THE WEYL-ORDERED POLYNOMIALS

2002 ◽  
Vol 17 (30) ◽  
pp. 2009-2017 ◽  
Author(s):  
ZENG-BING CHEN ◽  
HUAI-XIN LU ◽  
JUN LI
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Systematic Approach ◽  
Equations Of Motion ◽  
Time Dependent ◽  
Adiabatic Invariants ◽  
Formal Aspect ◽  
First Order ◽  
Heisenberg Equations ◽  
Heisenberg Equations Of Motion

A systematic approach to integrate the Heisenberg equations of motion is proposed by using the Weyl-ordered polynomials. The solutions of the Heisenberg equations of motion, i.e. P(t) and Q(t), are expanded as a sum over the Weyl-ordered polynomials Tm,n(P(t),Q(t)) at time t = 0. The coefficients of the expansions satisfy two sets of first-order ordinary differential equations resulting from the Heisenberg equations of motion for time-independent systems. This general approach for time-independent systems is also tractable in obtaining the adiabatic invariants of the time-dependent systems. In this paper, interest is mainly focused on the formal aspect of the approach.

scholarly journals Exact density matrix elements for a driven dissipative system described by a quadratic Hamiltonian

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Sh. Saedi ◽  
F. Kheirandish
Keyword(s):  
Density Matrix ◽  
Equations Of Motion ◽  
Dissipative System ◽  
Characteristic Functions ◽  
Matrix Elements ◽  
Exact Density ◽  
Quadratic Hamiltonian ◽  
Heisenberg Equations ◽  
Reduced Density ◽  
Heisenberg Equations Of Motion

AbstractFor a prototype quadratic Hamiltonian describing a driven, dissipative system, exact matrix elements of the reduced density matrix are obtained from a generating function in terms of the normal characteristic functions. The approach is based on the Heisenberg equations of motion and operator calculus. The special and limiting cases are discussed.

Heisenberg equations of motion in the Caldirola-Montaldi procedure

1982 ◽  
Vol 67 (2) ◽  
pp. 161-172 ◽  
Author(s):  
A. Jannussis ◽  
A. Leodaris ◽  
P. Filippakis ◽  
Th. Filippakis ◽  
V. Zisis
Keyword(s):  
Equations Of Motion ◽  
Heisenberg Equations ◽  
Heisenberg Equations Of Motion

The quantum mechanics of non-holonomic systems

1951 ◽  
Vol 205 (1083) ◽  
pp. 583-595 ◽  
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Equations Of Motion ◽  
Classical Dynamics ◽  
Holonomic System ◽  
Holonomic Systems ◽  
Classical Analogue ◽  
Heisenberg Equations ◽  
Subsequent Motion ◽  
Heisenberg Equations Of Motion

Interactions of a non-holonomic type are fundamentally different from interactions which can be treated as part of the Hamiltonian of a system. They usually lead to constraints which do not commute with the Hamiltonian, and cause important alterations in the development of a state vector. This paper deals with the Heisenberg equations of motion by analogy with classical dynamics using the Poisson bracket formalism of a previous paper (Eden 1951). The Schrödinger equation is investigated in co-ordinate representation, and it is shown that the wave function will have a non-integrabie phase factor or quasi phase. The quasi phase leads to an indefiniteness in the wave function, but does not violate the fundamental laws of quantum mechanics nor lead to any ambiguity in the physical interpretation of the theory. The relation between the Schrödinger and the Heisenberg equations shows that the Schrödinger treatment is also consistent with the classical analogue. If there is a given initial probability that the non-holonomic system has co-ordinates q (0) r , then there will be the same probability that the wave function in the subsequent motion will be zero except in a certain region of co-ordinate space. This region is the part of co-ordinate space which is accessible in the classical theory from the point q (0) r .

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