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Journal of Physics B Atomic Molecular and Optical Physics

Coherent representation approach to damping of two-level systems

Maciej Janowicz, Sylwia Karakula, Jan Mostowski

Published Dec 1, 2006• 1 citation
Publisher

Abstract

A characteristic function for spin systems is applied to investigate the dissipation and decoherence of one and two two-level atoms. Particular attention is devoted to the Schr¨ odinger cat-like states of two two-level systems. It has been demonstrated that the decoherence time and the phase damping rate depend linearly on the number of degrees of freedom for the maximally entangled initial state. Use of various characteristic functions to describe the decoherence processes is advocated. Two realizations of the SU(2) group are discussed in this context. The first one is based on the homomorphism of SU(2) and the group of rotations, the second one takes as its basis the Schwinger coupled-boson representation of angular-momentum operators as well as the definition of the characteristic function provided by Scully and W´ odkiewicz.

DOI
10.1088/0953-4075/39/24/016

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A characteristic function for spin systems is applied to investigate the dissipation and decoherence of one and two two-level atoms. Particular attention is devoted to the Schr¨ odinger cat-like states of two two-level systems. It has been demonstrated that the decoherence time and the phase damping rate depend linearly on the number of degrees of freedom for the maximally entangled initial state. Use of various characteristic functions to describe the decoherence processes is advocated. Two realizations of the SU(2) group are discussed in this context. The first one is based on the homomorphism of SU(2) and the group of rotations, the second one takes as its basis the Schwinger coupled-boson representation of angular-momentum operators as well as the definition of the characteristic function provided by Scully and W´ odkiewicz.
We consider a class of states in an ensemble of two-level atoms: a mesoscopic superposition of two distinct atomic coherent states, which can be regarded as atomic analogs of the states usually called Schr\"odinger-cat states in quantum optics. According to the relation of the constituents, we define polar and nonpolar cat states. The properties of these are investigated by the aid of the spherical Wigner function. We show that nonpolar cat states generally exhibit squeezing, the measure of which depends on the separation of the components of the cat, and also on the number of the constituent atoms. By solving the master equation for the polar cat state embedded in an external environment, we determine the characteristic times of decoherence and dissipation, and also the characteristic time of a new parameter, the nonclassicality of the state. This latter one is introduced by the help of the Wigner function, which is used also to visualize the process. The dependence of the characteristic times on the number of atoms of the cat and on the temperature of the environment shows that the decoherence of polar cat states is surprisingly slow.
The various representations of rotations in physical space and Hilbert space are developed in detail. This leads to an in-depth treatment of the representation of states of well-defined spin and angular momentum in quantum systems. The peculiarities of the physics of spin— systems (spinors) is outlined. The tensorial character of representations is implicit in the treatment. The Schwinger and Bargmann representations are introduced in some detail; and this leads to SU(2) coherent states (which are important for more advanced group representation theory).
We consider the evaluation of multitime-correlation functions for quantum statistical systems characterized by dynamical variables of the angular-momentum type (such as, for example, paramagnetic or nuclear spins or collections of two-level atoms), and show how the atomic coherent-state representation of Arecchi et al. can be used to express the multitime averages in terms of phase-space integrals. Our calculation suggests a convenient classical-quantum correspondence rule for angular-momentum degrees of freedom.
We investigate the quantum-classical border, the entanglement and decoherence of an analytically solvable model, comprising a first subsystem (a harmonic oscillator) coupled to a driven and damped second subsystem (another harmonic oscillator). We choose initial states whose dynamics is confined to a couple of two-level systems, and show that the maximum value of entanglement between the two subsystems, as measured by concurrence, depends on the dissipation rate to the coupling-constant ratio and the initial state. While in a related model the entropy of the first subsystem (a two-level system) never grows appreciably (for large dissipation rates), in our model it reaches a maximum before decreasing. Although both models predict small values of entanglement and dissipation, for fixed times of the order of the inverse of the coupling constant and large dissipation rates, these quantities decrease faster, as a function of the ratio of the dissipation rate to the coupling constant, in our model.
The monitoring of the three independent components of the angular momentum (or spin) of a quantum system by its environment that does not isolate any preferred orientation is modelled in two different ways. One describes the dynamics by the Lindblad equation generated by three independent angular momentum operators. The other uses iterated measurements of the ``phase-space'' point on the sphere in terms of the positive operator-valued measure generated by SU(2) coherent states. In contrast to the equivalent scenario on a flat phase space, these two models give rise to subtle differences. Specifically, it is shown that the two super-operators corresponding to the two decoherence models for angular momentum systems are commutative, but their eigenvalues are different. Hence although both models give rise to phase-space decoherence, their dynamical behaviours are not equivalent. In either model, we find that the characterisation of classicality as represented by the decay rates of the elements of the density matrix (i.e. decoherence) and that as represented by the positivity of the quasiprobability distribution are not equivalent for angular momentum systems.