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Journal of Physics A Mathematical and Theoretical

Exact solution for a time-dependent multi-mode coupled quadratic Bose system

Xu Xiu-Wei, MU Hai-feng, Shu-Yan Liu, Guo Chun

Published Oct 22, 2010• 1 citation
Publisher

Abstract

By utilizing generalized linear quantum transformation theory, the evolution operator, normal and anti-normal Wigner characteristic functions, P- and Q-representations of a multi-mode coupled quadratic Boson system are presented. The squeezing properties of a time-dependent double-mode coupled quadratic Bose system are investigated as a specific example.

DOI
10.1088/1751-8113/43/45/455211

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By utilizing generalized linear quantum transformation theory, the evolution operator, normal and anti-normal Wigner characteristic functions, P- and Q-representations of a multi-mode coupled quadratic Boson system are presented. The squeezing properties of a time-dependent double-mode coupled quadratic Bose system are investigated as a specific example.
We study the full time evolution of a system of two modes of a bosonic quantum field that interact through quadratic hamiltonians. We specialise to a time dependent two-mode squeezing Hamiltonian where the coupling is switched on and off smoothly. We find the analytical solutions to the time evolution after the interaction is turned off, which apply to arbitrary coupling functions. We investigate the final state of the system and compute all physically relevant quantities, such as average particle number, entanglement and change in energy. Our techniques can be applied to a wide range of scenarios, which include implementations of time dependent two-mode squeezing, coupled harmonic oscillators, harmonic oscillators coupled to light. Furthermore, these techniques can be extended to more complicated time-dependent quadratic Hamiltonians. We briefly discuss possible applications within the area of relativistic and quantum information.
We consider the most general Time-Dependent (TD) quadratic Hamiltonian written in terms of the bosonic operators a and a + , which may represent either a charged particle subjected to a harmonic motion, immersed in a TD uniform magnetic field, or a single mode photon field going through a squeezing medium. We solve the TD Schrödinger equation by a method that uses, sequentially, a TD unitary transformation and the diagonalization of a TD invariant, and we verify that the exact solution is a complete set of TD states. We also obtain the evolution operator which is essential to express operators in the Heisenberg picture. The variances of the quadratures are calculated and a phase space of parameters introduced, in which we identify squeezing regions. The results for some special cases are presented and as an illustrative example the parametric oscillator is revisited and the trajectories in phase space drawn.