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arXiv:quant-ph/0605091 · Symmetry Integrability and Geometry Methods and Applications

Perturbative Treatment of the Evolution Operator Associated with Raman Couplings

Benedetto Militello

Published May 10, 2006• 1 citation
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Abstract

A novel perturbative treatment of the time evolution operator of a quantum system is applied to the model describing a Raman-driven trapped ion in order to obtain a suitable 'effective model'.It is shown that the associated effective Hamiltonian describes the system dynamics up to a certain transformation which may be interpreted as a 'dynamical dressing' of the effective model.

DOI
10.3842/sigma.2006.050

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A novel perturbative treatment of the time evolution operator of a quantum system is applied to the model describing a Raman-driven trapped ion in order to obtain a suitable 'effective model'.It is shown that the associated effective Hamiltonian describes the system dynamics up to a certain transformation which may be interpreted as a 'dynamical dressing' of the effective model.
A novel result concerning Raman coupling schemes implemented using trapped ions is obtained. By means of an operator perturbative approach, it is shown that the complete time evolution of these systems can be expressed, with a high degree of accuracy, as the product of two unitary evolutions. The first one describes the time evolution related to an effective coarse grained dynamics. The second is a suitable correction restoring the fine dynamics suppressed by the coarse graining performed to adiabatically eliminate the nonresonantly coupled atomic level.
A novel result concerning Raman coupling schemes implemented using trapped ions is obtained. By means of an operator perturbative approach, it is shown that the complete time evolution of these systems can be expressed, with a high degree of accuracy, as the product of two unitary evolutions. The first one describes the time evolution related to an effective coarse grained dynamic. The second is a suitable correction restoring the fine dynamics suppressed by the coarse graining performed to adiabatically eliminate the nonresonantly coupled atomic level. The case where a decoherence source is present is also studied.
Abstract Time‐driven quantum systems are important in many different fields of physics as cold atoms, solid state, optics, etc. Many of their properties are encoded in the time evolution operator or the effective Hamiltonian. Finding these operators usually requires very complex calculations that often involve some approximations. To perform this task, a systematic scheme that can be cast in the form of a symbolic computational algorithm is presented. It is suitable for periodic and non‐periodic potentials and, for convoluted systems, can also be adapted to yield numerical solutions. The method exploits the structure of the associated Lie group and a decomposition of the evolution operator on each group generator. To illustrate the use of the method, five examples are provided: harmonic oscillator with time‐dependent frequency (Paul trap), modulated optical lattice, time‐driven quantum oscillator, a step‐wise driving of a free particle, and the non‐periodic Caldirola‐Kanai Hamiltonian. To the extent of the authors' knowledge, whereas the exact form of Paul trap's evolution operator is well known, its effective Hamiltonian was until now unknown. The remaining four examples accurately reproduce previous results.