Exponential time‐evolution operator for the time‐dependent harmonic oscillator

1987 ◽  
Vol 28 (12) ◽  
pp. 2908-2908 ◽  
Author(s):  
Francisco M. Fernández
Keyword(s):  
Harmonic Oscillator ◽  
Time Evolution ◽  
Evolution Operator ◽  
Time Dependent ◽  
Time Evolution Operator ◽  
Exponential Time

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.

scholarly journals Study of Time Evolution for Approximation of Two-Body Spinless Salpeter Equation in Presence of Time-Dependent Interaction

2016 ◽  
Vol 2016 ◽  
pp. 1-5 ◽  
Author(s):  
Hadi Sobhani ◽  
Hassan Hassanabadi
Keyword(s):  
Differential Equation ◽  
Time Evolution ◽  
Series Solution ◽  
Evolution Operator ◽  
Heavy Quarks ◽  
Time Dependent ◽  
Time Evolution Operator ◽  
Invariant Method ◽  
The One ◽  
Dependent Interaction

We approximate the two-body spinless Salpeter equation with the one which is valid in heavy quarks limit. We consider the resulting semirelativistic equation in a time-dependent formulation. We use the Lewis-Riesenfeld dynamical invariant method and series solution to obtain the solutions of the differential equation. We have also done some calculations in order to derive the time evolution operator for the considered problem.

SPATIAL ROTATION AND TIME-EVOLUTION OPERATOR OF TWO-LEVEL SYSTEMS

2000 ◽  
Vol 14 (01) ◽  
pp. 101-112
Author(s):  
CHUN-FANG LI ◽  
XIAN-GENG ZHAO
Keyword(s):  
Magnetic Field ◽  
Time Evolution ◽  
Evolution Operator ◽  
Order Differential Equation ◽  
Time Dependent ◽  
Time Varying ◽  
Level System ◽  
Zener Model ◽  
Time Evolution Operator ◽  
Two Level Systems

All the six kinds of rotation approach with the same form to the evolution problem of arbitrarily time-dependent two-level system are investigated in this paper. A time-dependent two-level system can be viewed as a spin-1/2 system in a time-varying magnetic field. It is shown that for each kind of rotation approach, we can always find a rotating frame in which the direction of the effective magnetic field is fixed. This property reduces the problem of finding the time-evolution operator to the solution of a second-order differential equation. Applications are made to the J C model in quantum optics and the L and au–Zener model in resonance physics.

scholarly journals Complexity growth in integrable and chaotic models

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Vijay Balasubramanian ◽  
Matthew DeCross ◽  
Arjun Kar ◽  
Yue Li ◽  
Onkar Parrikar
Keyword(s):  
Time Evolution ◽  
Evolution Operator ◽  
Linear Growth ◽  
Conjugate Points ◽  
Majorana Fermions ◽  
Time Evolution Operator ◽  
Free Theory ◽  
Group Manifold ◽  
Geodesic Length ◽  
Evolution Trajectory

Abstract We use the SYK family of models with N Majorana fermions to study the complexity of time evolution, formulated as the shortest geodesic length on the unitary group manifold between the identity and the time evolution operator, in free, integrable, and chaotic systems. Initially, the shortest geodesic follows the time evolution trajectory, and hence complexity grows linearly in time. We study how this linear growth is eventually truncated by the appearance and accumulation of conjugate points, which signal the presence of shorter geodesics intersecting the time evolution trajectory. By explicitly locating such “shortcuts” through analytical and numerical methods, we demonstrate that: (a) in the free theory, time evolution encounters conjugate points at a polynomial time; consequently complexity growth truncates at O($$ \sqrt{N} $$ N ), and we find an explicit operator which “fast-forwards” the free N-fermion time evolution with this complexity, (b) in a class of interacting integrable theories, the complexity is upper bounded by O(poly(N)), and (c) in chaotic theories, we argue that conjugate points do not occur until exponential times O(eN), after which it becomes possible to find infinitesimally nearby geodesics which approximate the time evolution operator. Finally, we explore the notion of eigenstate complexity in free, integrable, and chaotic models.

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