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Physical Review A

Quantum dynamics of the nonlinear rotator and the effects of continual spin measurement

Barry C. Sanders

Published Sep 1, 1989• 195 citations
Publisher

Abstract

The quantum and classical dynamics of the nonlinear oscillator are contrasted by comparing the evolution of the quantum Q function with that of a similar classical probability distribution. The quantum nonlinear rotator is shown to generate a superposition of two distinct coherent states from a coherent-state input. Measurements of the angular-momentum components and the signature of a superposition state are discussed. The effects of a continual measurement of one angular-momentum component are introduced into the model, and its effects on quantum coherences are shown to degrade the quantum coherence effects. We present a quantum optical model that obeys the nonlinear rotator dynamics and can generate superpositions of SU(2) coherent states of the two-mode electromagnetic field.

DOI
10.1103/physreva.40.2417

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The quantum and classical dynamics of the nonlinear oscillator are contrasted by comparing the evolution of the quantum Q function with that of a similar classical probability distribution. The quantum nonlinear rotator is shown to generate a superposition of two distinct coherent states from a coherent-state input. Measurements of the angular-momentum components and the signature of a superposition state are discussed. The effects of a continual measurement of one angular-momentum component are introduced into the model, and its effects on quantum coherences are shown to degrade the quantum coherence effects. We present a quantum optical model that obeys the nonlinear rotator dynamics and can generate superpositions of SU(2) coherent states of the two-mode electromagnetic field.
In this paper, anticorrelated field states with a binomial distribution over orthogonal polarizations, interacting with a nonlinear birefringent medium, are discussed. The field states are the SU(2) coherent states, where the angular momentum operators are realized as bilinear products of the boson operators associated with the two polarizations. The system discussed here provides an optical realization of a nonlinear rotator. We study the quantum dynamics of the statistical properties of the field and the evolution of the degree of polarization.
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.
Abstract The work of Carruthers & Nieto on the harmonic oscillator coherent states is combined with Schwinger’s construction of angular momentum to produce the angular momentum coherent states. It is shown that these states become the vector representatives of angular momentum in the classical limit, and so are particularly useful for discussing the transition from quantum to classical angular momentum. The uncertainty relations for angle and angular momentum are described and are compatible with the classical limit. Under rotations the coherent states transform in a manner that in the classical limit is equivalent to the transformation of vectors, and in the same limit the root mean square variation of the expectation values of the components of angular momentum become negligible in comparison with the expectation values themselves. The coupling of two angular momenta in the classical limit is investigated: it is shown that although the product of two coherent states is not itself a coherent state, it does represent a packet similar to a true coherent state, and centred on the direction of the classical resultant of the two component vectors. The properties and implications of hyperbolic angular momentum space are discussed.
Quantum measurement is one of the most important features of quantum theory. Although mathematical predictions have been verified in great detail, practical implementation has lagged behind. Only recently have people begun to take advantage of quantum measurement properties to produce new technologies. This research helps fill that technological gap by experimental examination of a continuous, optical measurement for an ensemble of cold atomic spins. The essential physics reduces to the interaction between an atomic ensemble and a weak optical field, which has many well known results. While this work demonstrates many novel applications of the interaction, it also shows that the whole can be more than the sum of the individual parts. Starting with basic characterization of the measurement system using laser-cooled cæsium atoms, the mean value of a spin component is obtained in real time. In essence, the angular momentum of the atomic spins creates a Faraday-like rotation in the polarization of a laser probe beam. With slight modifications, additional spin components are also observed, and are shown to be in good agreement with predictions. In measuring spin dynamics, it is important to account for effects of the probe on the spin states as well. Capitalizing on this as a resource, the probe-induced ac-Stark shift is used to transform a quasi-classical spin-coherent state into a highly quantum Schrödinger cat type of superposition between two spin states. Finally, this work combines all the previous results to demonstrate how a continuous measurement of the spin with a carefully crafted evolution created in part by the probe, allows for nearly real-time determination of the complete spin density matrix. In a single 1.5 millisecond run, a spin density matrix is determined with fidelities ranging from about 85% to 90% across a wide spectrum of test states.
The coherent state representations of the group G=W1⊗G0G = W_1 \otimes G_0G=W1​⊗G0​ (where G0=SU(2),SU(1,1)G_0 = SU(2), SU(1,1)G0​=SU(2),SU(1,1)) are used in computer simulation of the dynamics of single two-level atom (G0=SU(2))(G_0 = SU(2))(G0​=SU(2)) interacting with a quantized photon cavity mode - the Jaynes - Cummings model (JCM) without the rotating wave approximation and, in general, nonlinear in photon variables). The second case (hyperbolic Jaynes - Cummings model (HJCM), G0=SU(1,1))G_0 = SU(1,1))G0​=SU(1,1)) corresponds to the quantum dynamics of quadratic nonlinear coupled oscillators (the parametric resonance on double field frequency and a three - wave parametric processes of nonlinear optics). Quasiclassical dynamical equations for parameters of approximately factorizable coherent states for these models are derived and regimes of motion for "atom" and field variables are analyzed.