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Journal of Physics A Mathematical and General

The time evolution and the quantum fluctuation for two forced quantum oscillators with mixing of two modes

Jiangfan Li, Fuliang Xiao, Zongfu Jiang, Fang Jia-Yuan

Published Oct 5, 2005• 1 citation
Publisher

Abstract

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.

DOI
10.1088/0305-4470/38/42/009

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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.
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.
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.
We investigate the quantum statistics of three harmonic oscillators mutually interacting with each other determining quadratures squeezing, second-order correlation function and phase space distributions. We assume that the modes are initially prepared in coherent states. We show that the initial coherent states can evolve into squeezed states which are not minimum uncertainty states governed by the interaction parameters. Moreover, we demonstrate that the interaction is able to produce squeezing of different types, i.e. single-mode, two-mode and three-mode squeezing. Further, we show for specific modes and under certain conditions the oscillatory behaviour in the photon-number distribution and the bifurcation phenomenon in the phase distribution for squeezed light.
In a squeezed state, the variance in one canonical variable may be suppressed below that normally associated with either the ground state or a coherent state, at the expense of an expansion in the variance of the conjugate variable. Squeezed states are usually discussed in the context of quantized light fields. The principal properties of squeezed states are demonstrated using the motion of quantum mechanical simple harmonic oscillators. The motion of a single oscillator is discussed to introduce the key concepts of squeezing, including quadrature operators, and the error contours of the Wigner function describing the quantum quasiprobability distributions in phase space of the oscillator motion. The main topic of two-mode squeezing is then addressed, in which the fluctuations in a system of two oscillators are tightly correlated. The existence of squeezing is demonstrated in normal mode coordinates representing motion of superpositions of the motion of the two oscillators. The fluctuations within a single oscillator are shown to increase when squeezing increases in the normal modes, generating thermal noise in individual mode subsystems.
The nature of phase noise in quantum optics is analyzed. In an experiment involving the measurement of the electromagnetic field the two quantities of interest are the energy and phase of the field. However, measurements of the quantities produce quantum fluctuations. The quantum fluctuations are regarded as noise in the treatment presented here. The quantum system is represented by a probability distribution, the Wigner function, and the quantum fluctuations are treated as stochastic noise associated with the quantity being measured. The difficulties of associating a quantum operator with the phase of the system are reviewed and the related energy-phase uncertainty relation is discussed. The alternate interpretation of the phase noise of a quantum system as being the classical phase noise of the Wigner function is presented. In particular the energy and phase noise of the vacuum state, the coherent state, the squeezed state and the squeezed vacuum are discussed in this way. The squeezed states of light are minimum uncertainty states with respect to the quadrature operators and exhibit noise of one quadrature below the noise level associated with the vacuum. The reduced noise level in one quadrature of the field underlies the importance of squeezed states in many practical applications where there is a need to reduce the quantum noise of one quadrature of coherent light. The periodic phase operator eliminates the difficulties associated with the multivalued nature of phase. The analysis of the vacuum and intense coherent state of Carruthers and Nieto by employing periodic phase operators is reviewed, particularly with respect to the energy-phase uncertainty relations and we generalize the approach to develop a phase operator analysis of the squeezed state in the intense field and vacuum limits. We demonstrate here for the first time that the phase operator is simply related to the phase of the squeezed state in the intense field limit and that the squeezed state is approximately an energy-phase minimum uncertainty state in the low-squeezing limit. Also we enlarge on previous work to demonstrate that the phase operator corresponds simply and unambiguously to the phase of the squeeze parameter for the strongly squeezed vacuum and the intensely squeezed vacuum is an energy-phase minimum uncertainty state for some values of phase. The occurrence of squeezing for the case of two coupled quantum oscillators is presented. The system consisting of one mode of the electromagnetic field coupled to a spinless nonrelativistic electron subjected to an harmonic potential is represented by two coupled harmonic oscillators. The dynamics are compared for the case that the rotating wave approximation is employed and for the case that the counter-rotating terms are included. These calculations have not been performed before. The parametric amplifier Hamiltonian with a nonresonant coupling is also studied in order to provide insight into the effects of the counter-rotating terms. Squeezing of the field produced by the electron is a consequence of the inclusion of the counter-rotating terms. The case of a spinless nonrelativistic electron subject to an harmonic potential and coupled to a continuum of electromagnetic field modes is also considered. The case of two coupled oscillators discussed above is generalized by replacing the oscillator which represents the single-mode field by a bath of oscillators. The effects of including counter-rotating terms and of ignoring the counter - rotating terms in the Hamiltonian are compared. The interaction is assumed to produce a frequency shift and an exponential damping term for the oscillating electron. The frequency shift is assumed to be small in either case and so the Wigner-Weisskopff approximation is employed to solve the equations of motion. We demonstrate the new results that dissipation-induced phase-dependent noise is a consequence of including the counter-rotating terms and that the noise is phase-independent for the case that the counterrotating terms are excluded. The relation between these results and recent work on quantum tunnelling in superconducting quantum interference devices is discussed. We conclude by suggesting further research related to the work in this thesis.