Lie algebraic approach to quantum driven optomechanics

Abstract

We present an approximate Lie algebraic method to deal with a forced optomechanical Hamiltonian. We show that the approximations made in order to linearize the interaction Hamiltonian are fully justified by means of a comparison between a purely numerical calculation of the number of photons, phonons and linear entropy using the full Hamiltonian and the results obtained by means of our approximate time evolution operator.

Approximate Evolution for A Hybrid System—An Optomechanical Jaynes-Cummings Model

“…In order to solve the problem we split the Hamiltonian as the sum of a forced optomechanical Hamiltonian and that of the free atom with the JC interaction. The time evolution operator for the forced optomechanical Hamiltonian is approximated as a product of exponentials [ 42 ] and it is then used to take the JC interaction into a generalized interaction picture. As a result we obtained cumbersome expressions for the transformed operators which we approximated by neglecting terms of the order and as compared to one.…”

“…We have developed a useful approach to find an approximate time evolution operator for the Hamiltonian when the system does not interact with the environment [ 42 ]. Here we use a similar approach to obtain the time evolution operator of the hybrid system described by the Hamiltonian given in ( 1 ).…”

“…Notice however that the factor [ 36 ] so that at this point it is convenient to make the approximation and we get the approximate interaction Hamiltonian where we have used to stress the fact that it is an approximate interaction Hamiltonian. The corresponding exact time evolution operator can be written as a product of exponentials with: and initial conditions We see that so that we can write Finally, taking into account the above relationships and Equation ( 2 ), the approximate evolution operator of the forced optomechanical system is [ 42 ] …”

“…Finally, taking into account the above relationships and Equation ( 2 ), the approximate evolution operator of the forced optomechanical system is [ 42 ] …”

Efficient algebraic solution for a time-dependent quantum harmonic oscillator

“…In this work, our main purpose is to establish a procedure for solving this system in such a way that no matter which time-dependent functions for the parameters are considered, one will be able to calculate the quantum state of the system at any instant and with the desired precision for any initial state. Using operator ordering techniques, similar to those employed in [49,53,[69][70][71], based on BCH-like relations of the su(1,1) Lie algebra, and a time-splitting approach exploring the composition property of the TEO, we obtain an iterative analytical solution given by simple recurrence relations expressed in terms of generalized continued fractions. The TEO for any time interval will be written as an exponential of the su(1,1) Lie algebra generators so that it will be in a convenient form for being applied over any initial state.…”