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First order coherent Boson states

Reinhard Honegger, Alfred Rieckers

Published Oct 14, 1992• 7 citations
Publisher

Abstract

We analyze the first order coherent, states (and various sub-classes) over an arbitrary C*-Weyl algebra. Their normally ordered characteristic functions are expressed in terms of an infinite, positive-definitite matrix, which exhibits an additional positive-definiteness condition if the states are classical. In the latter case the matrix elements are identified as moments of probability measures on C. By going over to measures on bC, the Bohr compactification of C, there arises a Bauer simplex of generalized classical coherent states

DOI
10.5169/seals-116517

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Journal of Physics A Mathematical and General•Oct 13, 1999•79 citations
We analyze the first order coherent, states (and various sub-classes) over an arbitrary C*-Weyl algebra. Their normally ordered characteristic functions are expressed in terms of an infinite, positive-definitite matrix, which exhibits an additional positive-definiteness condition if the states are classical. In the latter case the matrix elements are identified as moments of probability measures on C. By going over to measures on bC, the Bohr compactification of C, there arises a Bauer simplex of generalized classical coherent states
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.
Canonical coherent states can be written as infinite series in powers of a single complex number z and a positive integer ρ( m ). The requirement that these states realize a resolution of the identity typically results in a moment problem, where the moments form the positive sequence of real numbers {ρ( m )} ∞ m =0 . In this paper we obtain new classes of vector coherent states by simultaneously replacing the complex number z and the moments ρ( m ) of the canonical coherent states by n × n matrices. Associated oscillator algebras are discussed with the aid of a generalized matrix factorial. Two physical examples are discussed. In the first example coherent states are obtained for the Jaynes–Cummings model in the weak coupling limit and some physical properties are discussed in terms of the constructed coherent states. In the second example coherent states are obtained for a conditionally exactly solvable supersymmetric radial harmonic oscillator.
We construct a new family of boson coherent states using a specially designed function which is a solution of a functional equation dε(q,x)/dx=ε(q,qx) with 0⩽q⩽1 and ε(q,0)=1. We use this function in place of the usual exponential to generate new coherent states |q,z〉 from the vacuum, which are normalized and continuous in their label z. These states allow the resolution of unity, and a corresponding weight function is furnished by the exact solution of the associated Stieltjes moment problem. They also permit exact evaluation of matrix elements of an arbitrary polynomial given as a normally-ordered function of boson operators. We exemplify this by showing that the photon number statistics for these states is sub-Poissonian. For any q<1 the states |q,z〉 are squeezed; we obtain and discuss their signal to quantum noise ratio. The function ε(q,x) allows a natural generation of multiboson coherent states of arbitrary multiplicity, which is impossible for the usual coherent states. For q=1 all the above results reduce to those for conventional coherent states. Finally, we establish a link with q-deformed bosons.
Abstract. The canonical coherent states can be written as infinite series in powers of a single complex number z and a positive integer ρ(m). The requirement that these states realize a resolution of the identity typically ends up in a moment problem, where the moments form the positive sequence of real numbers {ρ(m)} ∞ m=0. In this paper we obtain new classes of vector coherent states by simultaneously replacing the complex number z and the moments ρ(m) of the canonical coherent states by n × n matrices. A matrix moment problem is proposed and it is solved by introducing a matrix weight. Associated oscillator algebras are discussed. Finally, as a physical application, a generalized version of the Jaynes-Cummings model is discussed. 1.
We create a family of boson coherent states using the functions of Mittag-Leffler (ML) E ( z ), ( >0) and their generalizations E , ( z ), ( , >0) instead of exponentials. These states are shown to satisfy the usual requirements of normalizability, continuity in the label and the resolution of unity with a positive weight function. This last quantity is found for arbitrary , >0 as a solution of an associated Stieltjes moment problem. In addition, for = m = 1,2,3 ... and = 1 (corresponding to E m ( z )) we propose and analyse special q -deformations (0< q 1) of the functions E m ( z ) which serve as a tool to define q -deformed coherent states of ML type. We provide the expressions for expectation values of physical quantities for all the above states. We discuss physical properties of these states, noting that they are squeezed. The ML coherent states are sub-Poissonian in nature, whereas the q -deformed ML states can be sub- and super-Poissonian depending on q . All these states are shown to be eigenstates of deformed boson operators whose commutation relations are given.