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Journal of Optics B Quantum and Semiclassical Optics

Quantum state exchange between coupled modes

Antonio S. M. de Castro, V. V. Dodonov, Salomon Sylvain Mizrahi

Published Apr 5, 2002• 18 citations
Publisher

Abstract

We study the problem of quantum state exchange between two coupled modes of the electromagnetic field modelled by quantum oscillators. Analysing the structure of propagators of the Schrödinger equation with the most general weak bilinear resonance coupling, we find the conditions of the exchange in both `narrow' and `wide' senses.

DOI
10.1088/1464-4266/4/3/374

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We study the problem of quantum state exchange between two coupled modes of the electromagnetic field modelled by quantum oscillators. Analysing the structure of propagators of the Schrödinger equation with the most general weak bilinear resonance coupling, we find the conditions of the exchange in both narrow' and wide' senses.
The problem of quantum state exchange between three coupled quantum harmonic oscillators (which can represent interacting modes of the electromagnetic field in nonlinear crystal or interaction between laser beams and vibrational modes of ions in traps) is considered. We review different definitions of state exchange and find conditions of its possibility for arbitrary initial states in a chain of three oscillators with bilinear resonance coupling (corresponding to the rotating wave approximation), analyzing the structure of integrals of motion and propagators of the Schr\"odinger equation, as well as the Wigner functions. The influence of different parameters (such as detunings between coupling constants and temperatures of the intermediate oscillator and ``receiver'') on the fidelity of exchange is analyzed.
We compare the time evolutions of two quantum systems: a quantum harmonic oscillator driven by a classical force and two coupled quantum harmonic oscillators. We find the conditions under which the first system can be considered as a `mean-field approximation' to the second one. This happens in the weak coupling limit provided that the `master' oscillator was initially in a highly excited quantum state with small fluctuations of the canonical variables (such as the coherent state or displaced thermal state), while the `subordinate' oscillator must not be `too excited'. The essential difference between the resonance and nonresonance cases is discussed. In particular, the effect of the resonance-quantum-state exchange is demonstrated.
Exchange of quantum states between two interacting harmonic oscillator along their evolution time is discussed. It is analyzed the conditions for such exchange starting from a generic initial state and demonstrating that the effect occurs exactly only for the particular states C0|0>+Cn|N>, which includes the interesting qubits components |0>,|1>. It is also determined the relation between the coupling constant and characteristic frequencies of the oscillators to have the complete exchange.
The two-level quantum model proposed by Preparata, according to which the interaction of a material system with a electromagnetic field can cause the loss of stability of the lower level, is examined. As a result of this instability, the role of the ground state is played by a periodic process generating superradiance (in contrast to laser radiation induced by overpopulation of the upper level). A classical analogue of a super-radiant quantum transition, a system of van der Pol-Duffing oscillators (generators) weakly coupled via a linear oscillator, is for the first time described. Such an analogue is a strongly modulated oscillatory process of almost complete periodic energy exchange between the generators. It is shown that a necessary condition for the transition to intense energy exchange in the classical system is the instability of one of the nonlinear normal modes, similar to how the condition for a superradiant transition to occur is the instability of the ground state of a two-level quantum system coupled with a resonant electromagnetic field.
In the absence of incoherent relaxation processes, the emission, absorption, and transport of resonant radiation by an extended medium are all cooperative processes in which the medium participates as a whole. Such radiative processes depend sensitively on the geometry of the source, on its size and density, and on the interplay of electric polarization resonances with the ambient electromagnetic field. By making a succession of simple approximations, we reduce the radiative interaction problem to one of solving a linear integral equation for the time-dependent polarization of the radiating medium. For the example of one-dimensional geometry, we show that the polarization possesses a sequence of exponentially decaying eigenmodes that oscillate at frequencies slightly shifted from that of the fundamental resonance. Only a few of these modes, for any radiating medium, are strongly coupled to the electromagnetic field, and thereby radiate efficiently. All of the other modes, infinitely many of them, are found to radiate quite slowly, or not at all. We consider in detail the radiation emitted by an arbitrary initial polarization distribution when the electromagnetic field is initially empty. Analytical and numerical results are presented for the time-dependent fields that follow from a wide variety of initial excitations. A locally oscillatory exchange of energy is found to take place between the electric and polarization fields, which gives rise to a ringing of the radiated intensity. The spectra of the emitted light are shown accordingly to be complex in structure, and to have a frequency gap which becomes independent of the size of the medium and its initial polarization distribution for media that are sufficiently large. The gap is caused by the interference and interaction of oppositely propagating excitations.