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Physical Review A

Phase-admixed states: Coherence and incoherence

Apostol Vourdas, R. F. Bishop

Published Jan 1, 1989• 15 citations
Publisher

Abstract

The standard Glauber coherent states associated with a given boson or oscillator mode have a well-known decomposition in terms of the occupation-number eigenstates. In this paper we study a much more general class of quasicoherent states in which each term of this decomposition acquires its own extra phase factor. These new states are not minimum-uncertainty states, but we show that they may be represented as eigenstates of some new canonical annihilation operator which is obtained from the original one by an appropriate unitary transformation. By choosing particular sets of values for the phases which characterize the quasicoherent states, we show how we are thereby able to study, for example, the effects of (i) optical propagation through a nonlinear medium with an intensity-dependent refractive index, and (ii) phase incoherence caused by the stochasticity typical of an optical source. The quasicoherent states are shown to share many of the same coherence properties as the original Glauber states. Nevertheless, they exhibit a mixture of coherent and incoherent properties. This is particularly illustrated by the important practical example where the phase angles are randomized.

DOI
10.1103/physreva.39.214

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The standard Glauber coherent states associated with a given boson or oscillator mode have a well-known decomposition in terms of the occupation-number eigenstates. In this paper we study a much more general class of quasicoherent states in which each term of this decomposition acquires its own extra phase factor. These new states are not minimum-uncertainty states, but we show that they may be represented as eigenstates of some new canonical annihilation operator which is obtained from the original one by an appropriate unitary transformation. By choosing particular sets of values for the phases which characterize the quasicoherent states, we show how we are thereby able to study, for example, the effects of (i) optical propagation through a nonlinear medium with an intensity-dependent refractive index, and (ii) phase incoherence caused by the stochasticity typical of an optical source. The quasicoherent states are shown to share many of the same coherence properties as the original Glauber states. Nevertheless, they exhibit a mixture of coherent and incoherent properties. This is particularly illustrated by the important practical example where the phase angles are randomized.
Specifying their (normally ordered) characteristic functions we determine all states of the boson C *-Weyl algebra which satisfy Glauber's coherence condition and are not realizable as density operators in Fock space. The pure ones are shown to be just the eigenstates of the annihilation operators in their GNS-representations (in contrast to the Fock case) and are characterized in many equivalent manners. The central decomposition of an arbitrary coherent state has the macroscopic phase variable as parameter and is supported by the pure coherent states, which is in fact the only way for a maximal decomposition. The set of all coherent states with the same absolute factorizing function is proven to be a Bauer simplex. The appearence of a classical coherent field part is studied in detail in the GNS-representations and shown to correspond to an enlargement of the set of one boson states by just one additional mode.
The study of coherent states (CS) for a quantum mechanical system has \nreceived a lot of attention. The definition, applications, generalizations of \nsuch states have been the subject of work by researchers. A common starting \npoint of all these approaches is the observation of properties of the original \nCS for the harmonic oscillator. It is well-known that they are described \nequivalently as (a) eigenstates of the usual annihilation operator, (b) from \na displacement operator acting on a fundamental state and (c) as minimum \nuncertainty states. What we observe in the different generalizations proposed \nis that the preceding definitions are no longer equivalent and only some of \nthe properties of the harmonic oscillator CS are preserved. \nIn this thesis we propose to study a new class of coherent states and \nits properties. We note that in one example our CS coincide with the ones \nproposed by Glauber where a set of three requirements for such states has \nbeen imposed. The set of our generalized coherent states remains invariant \nunder the corresponding time evolution and this property is called temporal \nstability. Secondly, there is no state which is orthogonal to all coherent states (the coherent states form a total set). The third property is that we \nget all coherent states by acting on one of these states [¿fiducial vector¿] with \noperators. They are highly non-classical states, in the sense that in general, \ntheir Bargmann functions have zeros which are related to negative regions of \ntheir Wigner functions. Examples of these coherent states with Bargmann \nfunction that involve the Gamma and also the Riemann ¿ functions are represented. \nThe zeros of these Bargmann functions and the paths of the zeros \nduring time evolution are also studied.
This is a pedagogical paper where we present a physically motivated approach to introduce the coherent states of a harmonic oscillator from which it is simple to rigorously derive their mathematical definition. We do this in two different ways that turn out to be equivalent but emphasize two related but different aspects of classicality. First, we analyze which are the quantum states that are the closest one can get to a point in phase space and demonstrate the validity of the following theorem: (i) The product of the uncertainty in position and that of momentum saturates the bound imposed by Heisenberg uncertainty relations for all times if and only if the state is an eigenstate of the annihilation operator. Second, we analyze the way in which the difference between the expectation value of the energy and the energy associated with the expectation values of position and momentum depends on the state, and show the validity of the following theorem (ii) the difference between the expectation value of the energy and the energy associated with the expectation values is minimal if and only if the state is an eigenstate of the annihilation operator. We also show that the reason why coherent states are chosen as the most classical ones by the decoherence process induced by coupling the particle to an environment in the standard Quantum Brownian motion model, is precisely due to the validity of the two above theorems.
Simple proofs are given for two properties of a Bose field discovered recently in the quantum-theoretic description of optical coherence. The first is the theorem of Glauber and Titulaer that first-order coherence means that only one mode is excited. The second is the theorem of Aharanov, Falkoff, Lerner, and Pendleton that eigenstates of the annihilation operators are characterized by their ability to factor when the system is divided into two channels. The restriction of the latter to the case of a single excited mode is removed.
The authors present the supercoherent states. They are introduced as the eigenstates of the supersymmetric annihilation operator of the quantum mechanical supersymmetric harmonic oscillator. They have a compact expression in terms of the standard (bosonic) coherent states. For each value of the complex parameter z there are two linearly independent orthogonal supercoherent states. One of them has pure fermionic character and is the more classical, while the other is fully supersymmetric (i.e. the mean value of the Klein operator when the system is set in this state vanishes). There are no purely bosonic supercoherent states. The supercoherent fermionic states saturate both the Heisenberg uncertainty relation and the new entropic uncertainty recently introduced by Deutsch (1983), while the supersymmetric ones almost (but not exactly) make it. They give a classical (or almost classical) mean value for the energy of the system and do not spread along their time evolution.