scholarly journals CANONICAL FORM OF THE EVOLUTION OPERATOR OF A TIME-DEPENDENT HAMILTONIAN IN THE THREE LEVEL SYSTEM

2011 ◽  
Vol 08 (03) ◽  
pp. 647-655
Author(s):  
KAZUYUKI FUJII
Keyword(s):  
Differential Equations ◽  
Canonical Form ◽  
Evolution Operator ◽  
Time Dependent ◽  
Level System ◽  
Riccati Differential Equations

In this paper we study the evolution operator of a time-dependent Hamiltonian in the three level system. The evolution operator is based on SU(3) and its dimension is eight, so we obtain three complex Riccati differential equations interacting with one another (which have been obtained by Fujii and Oike) and two real phase equations. This is a canonical form of the evolution operator.

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 Exactly Solvable One-Qubit Driving Fields Generated via Nonlinear Equations

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 567 ◽  
Author(s):  
Marco Enríquez ◽  
Sara Cruz y Cruz
Keyword(s):  
Nuclear Magnetic Resonance ◽  
Magnetic Resonance ◽  
Evolution Operator ◽  
Time Dependent ◽  
Resonance Phenomenon ◽  
Level System ◽  
Time Evolution Operator ◽  
Inverse Approach ◽  
Coupled Equations ◽  
Exactly Solvable

Using the Hubbard representation for S U ( 2 ) , we write the time-evolution operator of a two-level system in the disentangled form. This allows us to map the corresponding dynamical law into a set of nonlinear coupled equations. In order to find exact solutions, we use an inverse approach and find families of time-dependent Hamiltonians whose off-diagonal elements are connected with the Ermakov equation. A physical model with the so-obtained Hamiltonians is discussed in the context of the nuclear magnetic resonance phenomenon.

scholarly journals Large Time Decay of Solutions to a Linear Nonautonomous System in Exterior Domains

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 841
Author(s):  
Toshiaki Hishida
Keyword(s):  
Evolution Operator ◽  
Large Time ◽  
Rigid Motion ◽  
Time Dependent ◽  
Energy Relation ◽  
Exterior Domains ◽  
Decay Properties ◽  
Decay Of Solutions ◽  
Whole Space ◽  
Expository Paper

In this expository paper, we study Lq-Lr decay estimates of the evolution operator generated by a perturbed Stokes system in n-dimensional exterior domains when the coefficients are time-dependent and can be unbounded at spatial infinity. By following the approach developed by the present author for the physically relevant case where the rigid motion of the obstacle is time-dependent, we clarify that some decay properties of solutions to the same system in whole space Rn together with the energy relation imply the desired estimates in exterior domains provided n≥3.

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