Coherent states in geometric quantization

In this paper we study overcomplete systems of coherent states associated to compact integral symplectic manifolds by geometric quantization. Our main goals are to give a systematic treatment of the construction of such systems and to collect some recent results. We begin by recalling the basic constructions of geometric quantization in both the Kähler and non-Kähler cases. We then study the reproducing kernels associated to the quantum Hilbert spaces and use them to define symplectic coherent states. The rest of the paper is dedicated to the properties of symplectic coherent states and the corresponding Berezin–Toeplitz quantization. Specifically, we study overcompleteness, symplectic analogues of the basic properties of Bargmann’s weighted analytic function spaces, and the ‘maximally classical’ behavior of symplectic coherent states. We also find explicit formulas for symplectic coherent states on compact Riemann surfaces.     

Superposed graphene coherent states

Using an annihilation operator, coherent states related to the electron of graphene layer placed in a magnetic field, can be obtained. In this paper, we define even and odd superposed graphene coherent states and then, we consider their entanglement, squeezing and statistical properties. To study the entanglement, we use concurrence. The results show that odd superposed graphene coherent states are maximally entangled states for all values of coherence parameter. However, the entanglement of graphene coherent states and also even superposed depend on the coherence parameter. In addition, examining the Mandel parameter shows sub-Poissonian statistics for graphene coherent states and their odd superposition; while, even superposed states do not show sub-Poissonian statistics at all. Also, we find that graphene coherent states and even superposition may be squeezed while the odd states do not show squeezing.     

Fractals,coherent states and self-similarity induced noncommutative geometry

The self-similarity properties of fractals are studied in the framework of the theory of entire analytical functions and the q-deformed algebra of coherent states. Self-similar structures are related to dissipation and to noncommutative geometry in the plane. The examples of the Koch curve and logarithmic spiral are considered in detail. It is suggested that the dynamical formation of fractals originates from the coherent boson condensation induced by the generators of the squeezed coherent states, whose (fractal) geometrical properties thus become manifest. The macroscopic nature of fractals appears to emerge from microscopic coherent local deformation processes.     

Coherent states as extreme energy states

A class of states obtained by extremizing the energy of a system under certain conditions is introduced and their properties are compared with those of the coherent states. Conditions under which these states move without change of shape and follow the classical path are investigated.     

Quanta and coherent states

The quantum harmonic oscillator can be considered as a composite system of indistinguishable Bose-Einstein symmetric two-level-systems (quanta). In analogy to the classical Poisson limit theorem, we show that a coherent state is the limit of a sequence of homogeneous product states (coherent spin states) and discuss statistical properties of the quanta in classical and nonclassical states.     

Even and odd nonlinear coherent states

New types of even and odd nonlinear coherent states are defined. The squeezed vacuum and the squeezed first excited state are shown to be even and odd nonlinear coherent states, respectively. The nonclassical properties of some related states are studied.     

Correspondence between generalized binomial field states and coherent atomic states

We show that the N-photon generalized binomial states of electromagnetic field may be put in a bijective mapping with the coherent atomic states of N two-level atoms. We exploit this correspondence to simply obtain both known and new properties of the N-photon generalized binomial states. In particular, an over-complete basis of these binomial states and an orthonormal basis are obtained. Finally, the squeezing properties of generalized binomial state are analyzed.     

Generalized coherent states and the statistics of two-photon lasers

Stimulated two-photon emission is proposed as a possible mechanism for the generation of “generalized coherent states”, which have useful quantum properties not available in ordinary coherent states.     

Generalized Heisenberg algebra coherent states for power-law potentials

Coherent states for power-law potentials are constructed using generalized Heisenberg algebra. Klauder's minimal set of conditions required to obtain coherent states are satisfied. The statistical properties of these states are investigated through the evaluation of the Mandel's parameter. It is shown that these coherent states are useful for describing the states of real and ideal lasers.     

Photoionization properties of an atom with two autoionizing states

We have investigated the photoionization properties of an atom with a ladder coupling scheme involving two autoionizing states coupled to each other.The effects of the coherent coupling between two autoionizing states and between the low-lying autoionizing state and the continuum on the ionization yield are discussed.The conditions leading to atomic coherent population trapping and the coherent population trapping states are given explicitly.     

Phase properties of superposed squeezed states

The phase properties of the superposed squeezed states are studied. The superposed squeezed states are obtained by inserting two squeezers in a Mach-Zehnder interferometer one for each arm. The visibility of the states as a measure of the coherence of the generated photons in the nonlinear interferometer is analyzed. And the mean-square phase fluctuations are also considered to analyze the phase properties of the superposed squeezed states. The quantities are obtained as a function of gain and the input phase, when the amplified input is a coherent state, a number state, a thermal state, a displaced number state, or a coherent thermal state.     

Coherent states associated with the groups SU(N) and SU(N, 1)

The coherent-state basis is constructed for symmetric representations of the groups SU(N) and SU(N, 1) and its properties are studied. The evolution of coherent states is considered. A relationship between the SU(N) coherent states and the Glauber coherent states is established.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 83–89, January, 1990.     

Generalized coherent states of multimode light,and biphotons

Generalized coherent states of a multimode field are analyzed within the framework of the dynamic-symmetry approach and with allowance for polarization. A new class of unpolarized two-photon states is defined. The uncertainty relations and statistical and other properties of optical fields in these states are calculated and analyzed. The relations between the considered states and squeezed states of light are discussed.Optics Laboratory, Lebedev Physics Institute, Academy of Sciences of the USSR. Translated from Preprint No. 254 of the Lebedev Physics Institute, Academy of Sciences of the USSR, Moscow. 1988.     

The nonclassical properties of the superposition involving two squeezing coherent states

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Coherence and squeezing in superpositions of spin coherent states

We consider the superpositions of spin coherent states and study the coherence properties and spin squeezing in these states. The spin squeezing is examined using a new version of spectroscopic squeezing criteria. The results show that the antibuching effect can be enhanced and spin squeezing can be generated in the superpositions of two spin coherent states.     

SU(1, 1) and SU(2) Perelomov number coherent states: algebraic approach for general systems

We study some properties of the SU(1, 1) Perelomov number coherent states. The Schrödinger's uncertainty relationship is evaluated for a position and momentum-like operators (constructed from the Lie algebra generators) in these number coherent states. It is shown that this relationship is minimized for the standard coherent states. We obtain the time evolution of the number coherent states by supposing that the Hamiltonian is proportional to the third generator K₀ of the su(1, 1) Lie algebra. Analogous results for the SU(2) Perelomov number coherent states are found. As examples, we compute the Perelomov coherent states for the pseudoharmonic oscillator and the two-dimensional isotropic harmonic oscillator.     

On nonlinear coherent states properties for electron-phonon dynamics

This work addresses a construction of a dual pair of nonlinear coherent states (NCS) in the context of changes of bases in the underlying Hilbert space for a model pertaining to an electron-phonon model in the condensed matter physics, obeying a f-deformed Heisenberg algebra. The existence and properties of reproducing kernel in the NCS Hilbert space are studied and discussed; the probability density and its dynamics in the basis of constructed coherent states are provided. A Glauber-Sudarshan P-representation of the density matrix and relevant issues related to the reproducing kernel properties are presented. Moreover, a NCS quantization of classical phase space observables is performed and illustrated in a concrete example of q-deformed coherent states. Finally, an exposition of quantum optical properties is given.     

Gazeau–Klauder coherent states examined from the viewpoint of diagonal ordering operation technique

In this paper we investigate the Gazeau–Klauder coherent states using a newly introduced diagonal ordering operation technique, in order to examine some of the properties of these coherent states. The results coincide with those obtained from other purely algebraic methods, but the calculations are greatly simplified. We apply the general theory to two cases of Gazeau–Klauder coherent states: pseudoharmonic as well as the Morse oscillators.     

Polarization tomography of bright states of light

Polarization quantum tomography is performed on 4-mode squeezed vacuum states. Three-dimensional polarization quasiprobability functions are obtained and compared to that of an equal intensity coherent state. These distributions clearly demonstrate the difference in the polarization properties of the considered states. The reconstruction quality of the coherent state distribution is also analyzed by comparing the theoretically and experimentally obtained shapes for this state.