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Modern Physics Letters B

GENERATION AND TRANSFERENCE OF NONCLASSICAL EFFECTS IN COUPLED OSCILLATOR

A. LUZIA SILVA, B. Baseia, Gil C. Marques

Published Aug 20, 1995• 3 citations
Publisher

Abstract

It is known from the recent literature that a quadratic oscillator (QO), in terms of the ladder operators â e â+ may exhibit various nonclassical effects, whereas a linear oscillator (LO) does not exhibit them. In this paper we find exact solutions for the system constituted by a LO coupled to a QO and show that nonclassical effects generated in the QO may contaminate the LO. The influence of the involved parameters of the system upon the generation and transference of different nonclassical effects, from the QO to the LO, is investigated. The analytical results are plotted numerically for various situations.

DOI
10.1142/s0217984995001200

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It is known from the recent literature that a quadratic oscillator (QO), in terms of the ladder operators â e â+ may exhibit various nonclassical effects, whereas a linear oscillator (LO) does not exhibit them. In this paper we find exact solutions for the system constituted by a LO coupled to a QO and show that nonclassical effects generated in the QO may contaminate the LO. The influence of the involved parameters of the system upon the generation and transference of different nonclassical effects, from the QO to the LO, is investigated. The analytical results are plotted numerically for various situations.
We investigate the mutual influence of two interacting 2-photon lasers, each subsystem being modelled as a quadratic oscillator in terms of the ladder operators âi, [Formula: see text] owing to the generation and transfer of squeezing effect. We find an exact solution for this coupled system which allows us to obtain the variances in the conjugate operators ẑi, [Formula: see text]. The results are exhibited in several plots showing the influence of the involved parameters. The second oscillator exhibits squeezing even if its nonlinear term is absent, due to the transference of the effect originated in the first (nonlinear) oscillator. Concerning this effect, the oscillators compete or cooperate depending on the parameters range; for some parameter range the system becomes unstable, as shown in numerical plots.
We consider two coupled quantum harmonic oscillators with different free frequencies. Here the interaction between the two modes does not involve the Rotating Wave Approximation (RWA). The Heisenberg equations of motion are solved analytically using an approximate technique and the solutions are used to measure nonclassicalities associated with the system. The nonclassicality criteria chosen in this study are experimentally measurable. The analytical solutions are matched with numerical simulations with the help of a numerical toolbox. It is evident from the time developments of the operators that nonclasities, namely squeezing and quantum entanglement are present in the system. The counting statistics for the oscillator with lower free energy and the coupled mode are shown to be sub-Poissonian for a particular set of parameters. Consequently, the system may exhibit quantum mechanical antibunching.
The single-mode case is equivalent to the model of a one-dimensional generalized oscillator with the time-dependent Hamiltonian H(t) = i [fi(t)p2 + v(t)x2 + p(t) [xp + px)] + ci(t)p + C2(t)x, (1) where the quadrature operators x and p satisfy the canonical commutation relation [x,p] = ih. The same Hamiltonian can also be written in terms of the bosonic annihilation and creation operators as H = \h [ D 0 ( a f a + aat) + DYa2 + D*tf2] +h(ga + tfg*) , (2) 154 PARAMETRIC EXCITATION AND GENERATION OF NONCLASSICAL STATES 2N 2N •qB(t)q + C(t)q. (3) (5) PROPAGATORS AND INTEGRALS OF MOTION 155 156 PARAMETRIC EXCITATION AND GENERATION OF NONCLASSICAL STATES SINGLE MODE: QUADRATURE REPRESENTATION 157 Ai = Mp-A2/*, Ai(0) A 2 = A i i / — A2P, A 2(0) A 3 - A 3 p - A4/1, A 3(0) A 4 = A 3z/ - A 4 p , A 4(0) Putting expressions (21) into equation (20) we arrive at the set of first-order ordinary differential equations and initial conditions 1 (22) 0 (23) 0 (24) 1 (25) : 0 (26) : 0 (27) (the dot means the time derivative). An immediate consequence of equations (22)- (25) and the initial conditions is the identity Its physical meaning is very simple: the commutation relations between the operators £ and p are the same as between X and P, [x , p] = [X , P] = ih, i.e., transformation (21) is canonical (and the evolution operator is unitary).
Harmonic oscillator coherent states are well known to be the analogue of classical states. On the other hand, nonlinear and generalised coherent states may possess nonclassical properties. In this article, we study the nonclassical behaviour of nonlinear coherent states for generalised classes of models corresponding to the generalised ladder operators. A comparative analysis among them indicates that the models with quadratic spectrum are more nonclassical than the others. Our central result is further underpinned by the comparison of the degree of nonclassicality of squeezed states of the corresponding models.
In recent literature the treatment of the system constituted by a trapped laser-cooled ion coupled to the modes of a source oscillator is presented. Among several effects presented in this system, the transference of squeezing from the source oscillator to the cooled ion is investigated. In this paper we present an alternative and more elaborate approach for this problem considering mainly the generation and transference of squeezing in this coupled system. We find exact expression for the variances of the involved quadratures. Variations observed due to the change of the involved parameters are also discussed.