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Journal of Modern Optics

Time evolution, squeezing and entanglement for an optical frequency up-conversion with a driving term

Jiangfan Li, Shi-Fa Liu, Chengzhi Wang, Xiu-Feng Li, Fang Jia-Yuan

Published Dec 21, 2010• 0 citations
Publisher

Abstract

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.

DOI
10.1080/09500340.2010.542834

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By properly selecting the time-dependent unitary transformation for the linear combination of the number operators, we construct a time-dependent invariant and derive the corresponding auxiliary equations for the degenerate and non-degenerate coupled parametric down-conversion system with driving term. By means of this invariant and the Lewis–Riesenfeld quantum invariant theory, we obtain closed formulae of the quantum state and the evolution operator of the system. We show that the time evolution of the quantum system directly leads to production of various generalized one- and two-mode combination squeezed states, and the squeezed effect is independent of the driving term of the Hamiltonian. In some special cases, the current solution can reduce to the results of the previous works.
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.
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 and properly choosing Hermitian invariant operator, a closed solution of the Schrödinger equation is derived for two forced quantum oscillators with mixing of two modes, and the quantum fluctuations in the output fields are evaluated. For the initial two-mode squeezed number or squeezed coherent state, in some particular conditions, the time evolution of the oscillators can not only preserve the initial two-mode squeezing, but also produce squeezing in the individual modes; and exhibit a periodical squeezing behaviour. For the initial two-mode number state or coherent state, there is no squeezing in the individual and mutual quadrature phases of the two-mode fields. Furthermore, regardless of which state above being initially considered, the quantum fluctuations of all the quadrature phases in the output fields are all independent of the driving parameters. In particular, for the initial two-mode coherent state, the variances of the output fields are also independent of other parameters in the Hamiltonian, and always preserve their initial values 1/4.
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