The General Form of non-Fock Coherent Boson States

Abstract

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. §1. Introduction and Preliminary ResultsIn one of his first papers on quantum optics [6] Glauber emphasizes the importance of states with a large number of photons for the description of light beams (and contrasts this with the few photon excitations in perturbational quantum electrodynamics). The relevant states, which have some degree of coherence, are investigated by him and his colleagues, however, only in the Fock representation. Since in laser beams there are in fact macroscopically many photons one may doubt if this formal limitation is justified. In [12] examples of fully coherent states are given, which cannot be represented by density operators in Fock space. A systematic study, where both Fock and non-Fock coherent states are investigated in the smeared field formalism, has been started recently in [13]. The aim of this paper is to carry on this analysis with the emphasis on non-Fock states.We start here from Glauber's factorization condition for a (fully) coherent

Quantized Radiation States from the Infinite Dicke Model

“…Let us at this point introduce some notations concerning states on the Weyl algebra if(E) (see also [1,Section 3] [8].…”

“…The coherence properties are discussed in their operator algebraic version ( [7], [8], [9], [10], [11]), which is an extension and refinement of Glauber's original definition [12] and obtained by a smearing procedure with one-photon testfunctions. The algebraic formulation of a photon state to be coherent is characterized by the factorization of the normally ordered expectation values of the creation and annihilation operators with respect to a linear form on the one-photon testfunction space.…”

Squeezing of Optical States on the CCR-Algebra

“…1 below. For bounded G (with respect to the norm of E) the state <p x is a coherent Glauber state, which is realizable by a so-called Glauber vector in Fock space [9], [17], [20]. This suggests the notion of a generalized Glauber coherent state for (p% also if X^£ does not arise from a linear form, cf .…”

“…A smearing procedure of Glauber's original factorization condition leads to the following operator algebraic formulation of quantum optical coherence [20] , [21], [19], where the linear form L: £-»C replaces the coherence function (cf. the Introduction) .…”

“…There is then a critical squeezing strength, above which all states in j^i 1) are rendered non-classical and below which some states from j^i 1) also the Appendix of [20] ) . From [20] we cite the following result.…”