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Acta Physica Sinica

An explicit analytical solution of the Schr?dinger equation for a detuned quantum frequency conversion system

Jiangfan Li, Shan Shu-min, Yang Jian-Kun, Jiang Zong-Fu, (1)长沙理工大学计算机与通信学院,长沙 410076; (2)长沙理工大学物理与电子科学学院,长沙 410076；国防科技大学理学院,长沙 410073; (3)国防科技大学理学院,长沙 410073

Published Jan 1, 2007• 3 citations
Publisher

Abstract

The time-dependent invariant of a quantum frequency conversion system in an optical cavity is found from the simple observation that the dynamical evolution of a sum-frequency generation process preserves the total number of the idler photons and the signal photons involved in the process. By using this invariant and the Lewis-Riesenfeld quantum invariant approach, the Schrdinger equation for the coupled system of the two quantized modes of the electromagnetic field is solved under detuned condition. The explicitly analytical formulae of the quantum states and the evolution operators of the system are obtained. It should be useful for further studies on the various quantum properties of the quantum system.

DOI
10.7498/aps.56.5597

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The time-dependent invariant of a quantum frequency conversion system in an optical cavity is found from the simple observation that the dynamical evolution of a sum-frequency generation process preserves the total number of the idler photons and the signal photons involved in the process. By using this invariant and the Lewis-Riesenfeld quantum invariant approach, the Schrdinger equation for the coupled system of the two quantized modes of the electromagnetic field is solved under detuned condition. The explicitly analytical formulae of the quantum states and the evolution operators of the system are obtained. It should be useful for further studies on the various quantum properties of the quantum system.
By using the Lewis–Riesenfeld quantum invariant theory, an explicit analytical solution for the time evolution operator of the frequency up-conversion system with a driving term is obtained. It is then used to investigate mainly the influence of the driving term, frequency detuning, and the initial states on the quantum statistical properties of the output-coupled fields. It is found that the time evolution for the general up-conversion system can efficiently convert a low-frequency squeezed light into a high-frequency squeezed light, in the case of zero-detuning, but the squeezed effect of the output fields is independent of the driving term. It is also shown that there is an entanglement between the signal and idler output photons for the general up-conversion system.
The dynamics of a generalized non-degenerate optical parametric down-conversion interaction whose Hamiltonian includes an arbitrary time-dependent driving part and a two-mode coupled part is studied by adopting the Lewis–Riesenfeld invariant theory. The closed formulae for the evolution of the quantum states and the evolution operators of the system are obtained. It is shown that various generalized squeezed states arise naturally in the process, and the two-mode squeezed effect is independent of the driving part. An explicitly analytical solution of the Schrödinger equation is further derived as the classical generalized force acting on each mode and the coupling of the two modes both have harmonic time dependences. This solution is found to be in agreement with previous research in special cases.
A closed solution of the Schrdinger equation for a timedependent two coupled harmonic oscillators including a driving part and a twomode coupled part is derived by adopting the LewisRiesenfeld invariant theory and properly chosen Hermitian invariant operators. The evolution operator of the system and the condition for generating twomode squeezed states are obtained. It is shown that the quantum fluctuation of squeezed states of the system is independent of the driving part, but depends on the initial states of the system.
The relation between the solutions of the Schrdinger equation for a non-degenerate parametric down-conversion system is investigated,one was obtained by the Lewis-Riesenfeld invariant theory,and another one by the entangled state representation in Fan et al.It is found that the analytical expressions for k21,k2=1,k21,and k=0,correspond to the ones for the cases of the above-threshold,threshold,below-threshold and parametric resonance.In the case of k=-1,our solution returns to the Fan's solution which corresponds to the threshold solution for the parametric down-conversion operated in a frequency-degenerate but polarization-nondegenerate mode.In particular,we present an analysis of the basic features in a detailed derivation of transformation from our analytical solution into Fan's one in the special condition.