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Physical Review

Properties of the Generalized Coherent State

Z. Bialynicka—Birula

Published Sep 25, 1968• 116 citations
Publisher

Abstract

Properties of the generalized coherent states of the electromagnetic field, introduced by Titulaer and Glauber, are investigated. It is shown that, in contradistinction to photon-counting measurements, the average value of the electromagnetic field and the electron scattering amplitudes depend on the detailed form of these states.
DOI
10.1103/physrev.173.1207

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In this chapter, we will introduce different representations of the electromagnetic (EM) field. One may ask, why do we need different representations for the EMfield? The answer is that usually we do not know the state of the EM field which in quantum optics is represented by the annihilation and creation operators. However, results of measurements of the field are given in terms of real variables, for example, intensity. The representations allow us to recognize the state of the field from values of the measured quantities such as average amplitude, intensity, and correlation functions. The nature of the state is present in terms of the interpretation of the apparently classical (measured) variables. We will discuss two basic types of representations often used in quantum optics, Fock state (photon number) representation and coherent states representation, the later one introduced by Glauber.a We also discuss properties of fields with thermal and Poisson distribution of photons. The photon number states are very often used as a basis for quantum optics problems, and despite of many difficulties have recently been generated experimentally [11-13].
The first-order coherence condition is derived in terms of correlations between different field modes. Fluctuations of the electromagnetic field are shown to play an important role in the concept of coherence. The coherent states are shown to be the only ones which reduce these fluctuations to their minimum value. For the case of a field in which the different field modes are independent, the degree of first-order coherence and the visibility of interference fringes are shown to be specified completely by these fluctuations. Measurements of this visibility yield information on the statistical properties of radiation fields. For squeezed states, this visibility is expressed as a function of squeezed parameters. Moreover, the difference between generalized coherent states becomes manifest in measurements of this visibility.
Quantum statistical properties of general superpositions of coherent states of the type Nα,βα+eiξβ have been studied. Formulas for fluctuations (variances) of the coordinate and momentum quadratures and fluctuations of the number of photons in these states are obtained. The amplitude Nα,βnαeiφ+eiξnβeiφ,φ=φα=φβ and phase Nα,βneiφα+eiξneiφβ,n=nα=nβ superpositions of coherent states are considered separately. In the case of amplitude superpositions, a field under certain conditions occurs in a quadrature-squeezed state. At the same time, the photon distribution in the case of amplitude superpositions is always super-Poissonian. In the case of phase superpositions, a field under certain conditions occurs in quadrature-squeezed states for small average numbers of photons. For certain parameters of coherent states, the photon statistics in phase superpositions is sub-Poissonian. Using amplitude superpositions, it is possible to generate a microscopic electromagnetic field with the average photon number below or on the order of unity, as well as mesoscopic and even macroscopic fields with arbitrary large average photon numbers, characterized by significant quadrature squeezing. Using phase superpositions, it is possible to create under certain conditions the electromagnetic field states with sub-Poissonian statistics (squeezed distribution of photon numbers) for an arbitrary large average photon number. Expressions for a ‘strict’ coordinate–momentum uncertainty relation (Cauchy inequality) are obtained and analysed in cases of general, amplitude and phase superpositions of coherent states.
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.
Introduction In our discussion of the classical electromagnetic field we found it convenient to describe the field by a complex amplitude, both in the frequency and in the time domains. The complex representation is convenient partly because it contains information about the magnitude and about the phase of the electromagnetic disturbance, and partly because of its analytic properties. These features turn out to be particularly useful for the description of the optical coherence properties of the field. We shall see that an analogous quantum state of the field exists, leading to an interesting representation that is also particularly useful for the treatment of optical coherence. This coherent-state representation leads to a close correspondence between the quantum and classical correlation functions. The coherent states of the field come as close as possible to being classical states of definite complex amplitude. We shall find that coherent states turn out to be particularly appropriate for the description of the electromagnetic fields generated by coherent sources, like lasers and parametric oscillators; indeed it turns out that the field produced by any deterministic current source is in a coherent state. In the following sections we shall examine some of the properties of coherent states, and then go on to discuss representations based on these states. Coherent states were first discovered in connection with the quantum harmonic oscillator by Schrödinger (1926), who referred to them as states of minimum uncertainty product.