Abstract
We apply the Magnus and Fer perturbation methods to Jaynes Cummings Model (JCM) with linear time dependence and show that the perturbation methods yield results which are appreciably close to the results obtained by Wei-Norman method, leading to exact solution when the Hamiltonian under consideration is an element of su(2), thus making the perturbation methods important for Hamiltonians for which a method of exact solution is not available.
Highlights
Study of Jaynes Cummings Model[2] by several authors explored interesting behaviours of both the two-level atom and the radiation field of a coupled radiation-matter system [6,7,8,9,10,11,12,13]
The particular form of the Hamiltonian allows us to apply the methods of perturbation, assuming the time dependent part to be exactly solvable and the rest as a small perturbation. we expect that the procedures shall generate results with close proximity with those given by the exact solution, thereby, making it an alternative trick to approach time dependent Hamiltonians for which a method of exact solution is unavailable
Apart from the increasing difficulties in calculations in higher order, the formula suffers from a major drawback - the time evolution operator is not unitary at each order, which can be avoided by applying Magnus and Fer perturbation methods. 1.2 Magnus Formula The evolution operator is expressed as exponential of an innite sum of anti-hermitian operators[1,4,5,15,22]
Summary
Study of Jaynes Cummings Model[2] by several authors explored interesting behaviours of both the two-level atom and the radiation field of a coupled radiation-matter system [6,7,8,9,10,11,12,13]. The form of this interaction picture Hamiltonian depends on the time dependence of L(t) and involves the generators Ji of the SU(2), as evident from (1) This determines the evolution operator in interaction picture, at each order of a perturbation scheme. 2.2 The evolution matrix We evaluate the time evolution operator at kth order of a perturbation scheme as a 2x2 matrix V(t) = vij(k) using the algebra followed by Ji and its connection to the Pauli matrices through (9) For Feynman-Dyson formula, straightforward substitution of (9) gives the evolution operator in matrix form. For Magnus formula the sum inside the exponential of (4) is expressed in terms of the generators Ji (10) Using (9) the evolution operator is given in terms of the Pauli matrices (11). One may calculate the evolution operator in explicit analytic form to any desired order. 2.3 Linear Ramp We call the linear time dependence a linear ramp when the detuning parameter is linearly time dependent, d(D)=bt and the interaction parameter is time independent, L(t)=l0 , so that the Hamiltonian (1) becomes (16) Evolution operator for the time dependent part of (16) is (17) The 'Interaction Picture' Hamiltonian is given by (18) 2.3.1 Feynman-Dyson Up to first order, the evolution operator is given by (19)
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