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Quantum Chaos in Two-Level Systems Interacting with Resonant Radiation Field

Published Jan 1, 1988• 0 citations
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Abstract

Quantum systems of Two-level atoms and radiation which corresponds to semiclassical systems where chaos occurs, are investigated, by using distribution functions obtained fron quantum master equations and its numerical analisis.

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Quantum Chaos in Two-Level Systems Interacting with Resonant Radiation Field

トシヒコ オオノ, Toshihiko Ohno

福山大学工学部紀要•Jan 1, 1988•0 citations
Quantum treatment of laser chaos is very complicated because adiabatic approximation is not applicable for the unstable systems and there exist dissipation and pumping of energy which are not defined as the closed hamiltonian systems. On the other hand, for atomic systems resonantly intreacting with radiation field, it is seen that chaotic atomic motion appears in the case of strong coupling between atoms and radiation or a certain detuning of frequencies.[1] That was investigated in the frame of semiclassical theory, which is based on Maxwel1-Bloch type equations, in which no pumping is introduced. Fully quantum treatment for both atoms and field are now tried for this closed systems, according to distribution function.
Dynamics of two-level atoms interacting with their own radiation field in a single-mode high-quality resonator is considered. The dynamical system consists of two second-order differential equations, one for the atomic SU(2) dynamical-group parameter and another for the field strength. With the help of the maximal Lyapunov exponent for this set, we numerically investigate transitions from regularity to deterministic quantum chaos in such a simple model. Increasing the collective coupling constant b is identical with 8(pi)N(sub 0)(d(exp 2))/hw, we observed for initially unexcited atoms a usual sharp transition to chaos at b(sub c) approx. equal to 1. If we take the dimensionless individual Rabi frequency a = Omega/2w as a control parameter, then a sequence of order-to-chaos transitions has been observed starting with the critical value a(sub c) approx. equal to 0.25 at the same initial conditions.
With the help of Lyapunov exponents different routes to quantum chaos are demonstrated numerically in two semiclassical models of the interaction of N two-level atoms with their own radiation field in a resonant cavity. The usual transition to chaos is observed when varying the external control parameter b which describes the interaction of the atomic ensemble with the field. After reaching its critical value b approximately equal to 1 the system becomes more and more chaotic as b increases. As the second control parameter a (the dimensionless individual Rabi frequency) increases the system undergoes transitions to chaos by a completely different way. We observe a complex sequence of order-to-chaos transitions that resembles those for the classical Lorenz attractor.
A system of N two-level atoms in a resonator is considered interacting with a resonant eigenmode field and with an external coherent field, with a frequency slightly different from the frequency of the atomic transition. A model Hamiltonian is constructed for describing the slow quantum dynamics of the system, and a set of closed c-number equations for time-dependent quantum expectation values is derived in the boson and spin coherent states. In the region of parameters for developed chaos in the semiclassical limit (when the radiation field is considered classically) we show that the semiclassical approximation is violated by quantum effects at the time scale ${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\Elzxh}}}$\ensuremath{\sim}lnN. We also show that quantum correlations grow exponentially during the time interval 0${\mathrm{\ensuremath{\tau}}}_{\mathrm{\ensuremath{\Elzxh}}}$. An additional averaging over the initial density matrix is considered. The connection of our model with the Dicke model [Phys. Rev. 93, 99 (1956)] is demonstrated, and the possiblity of measuring the quantum chaos effects considered in this paper is discussed.
A system of atoms interacting with a radiation field in a resonant cavity is studied under conditions when the dynamics in the classical limit is stochastic. This situation is called quantum chaos. Equations of motion are obtained for the quantum-mechanical expectation values which take into account the quantum correlation functions. It is shown that in a situation corresponding to quantum chaos, the quantum corrections grow exponentially, making the evolution of the system essentially quantal after a certain time tau( variant Planck's over 2pi ) has elapsed. Analytical and numerical analysis show that in this regime the time tau( variant Planck's over 2pi ) obeys the logarithmic law tau( variant Planck's over 2pi ) approximately ln N (N is the number of atoms), and not the law tau( variant Planck's over 2pi ) approximately N(alpha) (alpha is a certain constant of order unity), as would be the case in the absence of chaos.