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AIP conference proceedings

The new integrals in the complex plane resulting from the coherent state formalism

Dušan Popov, Nicolina Pop, Simona Şimon

Published Jan 1, 2013• 0 citations
Publisher

Abstract

From the geometric point of view, it exists a one-to-one correspondence between the coherent states (CSs) | z, λ > and the points in the complex plane z. Consequently, the eigenvalues of the quantum operators will be numbers in the complex plane. In the paper we have deduced some new useful integrals in the complex plane - by using the expression of the Husimi's function for some oscillators: the harmonic and the pseudoharmonic one.

DOI
10.1063/1.4832788

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From the geometric point of view, it exists a one-to-one correspondence between the coherent states (CSs) | z, λ > and the points in the complex plane z. Consequently, the eigenvalues of the quantum operators will be numbers in the complex plane. In the paper we have deduced some new useful integrals in the complex plane - by using the expression of the Husimi's function for some oscillators: the harmonic and the pseudoharmonic one.
The Coherent states (CSs) show the correspondence between quantum oscillators and classical oscillators. It is very useful for understanding what goes on in oscillating systems, including harmonic oscillators as well as many kinds of waves. The properties of CSs for harmonic oscillator-1D were generalized to the anharmonic oscillators potential. Therefore, the Morse oscillator has aroused interest also in the construction of CSs. The properties of different nanostructures differ from those of the corresponding bulk structures. This fact leads to many applications in condensed matter, optoelectronic devices, sensors, electronic and light emitting components. The CSs formalism may play an important role in the quantum information theory, and the use of this representation is not only theoretical, but also some practical relevance by its applications. The fundamental unity of quantum information can be express through the CSs of the one-dimensional quantum oscillator. The transmission channel randomly displaces an input pure CS according to the Gaussian distribution which results in a thermal state.
The Hamiltonian for the oscillator has earlier been written in the form where ν† and ν are raising and lowering operators for ν†ν, which has eigenvalues k (the ’’radial’’ quantum number), and λ† and λ are raising and lowering 3-vector operators for λ†⋅λ, which has eigenvalues l (the total angular momentum quantum number). A new set of coherent states for the oscillator is now defined by diagonalizing ν and λ. These states bear a similar relation to the commuting operators H, L2, and L3 (where L is the angular momentum of the system) as the usual coherent states do to the commuting number operators N1, N2, and N3. It is proposed to call them coherent angular momentum states. They are shown to be minimum-uncertainty states for the variables ν, ν† , λ, and λ†, and to provide a new quasiclassical description of the oscillator. This description coincides with that provided by the usual coherent states only in the special case that the corresponding classical motion is circular, rather than elliptical; and, in general, the uncertainty in the angular momentum of the system is smaller in the new description. The probabilities of obtaining particular values for k and l in one of the new states follow independent Poisson distributions. The new states are overcomplete, and lead to a new representation of the Hilbert space for the oscillator, in terms of analytic functions on C×K3, where K3 is the three-dimensional complex cone. This space is related to one introduced recently by Bargmann and Todorov, and carries a very simple realization of all the representations of the rotation group.
In the paper we made a generalization of the Fourier transform in the complex space, applicable to the pair of Husimi and P-quasi distributions, in the representation of nonlinear coherent states. Implicitly, our result is a generalization similar result of Mehta, but which referred only to the canonical coherent states (associated with the one-dimensional harmonic oscillator). Our result is valid for both types of coherent states (defined in the Barut-Girardello, respectively Klauder-Perelomov manner).
In this paper, we shall define a new family of coherent states which we shall call the “mother coherent states,” bearing in mind the fact that these states are independent from any parameter (the Bargmann index, the rotational quantum number J, and so on). So, these coherent states are defined on the whole Hilbert space of the Fock basis vectors. The defined coherent states are of the Barut-Girardello kind, i.e., they are the eigenstates of the lowering operator. For these coherent states we shall calculate the expectation values of different quantum observables, the corresponding Mandel parameter, the Husimi's distribution function and also the P- function. Finally, we shall particularize the obtained results for the three-dimensional harmonic and pseudoharmonic oscillators.
The technique regarding the integration within a normally ordered product of operators, which refers to the creation and annihilation operators of the harmonic oscillator coherent states, has proved to be very fruitful for different operator identities and applications in quantum optics. In this paper we propose a generalization of this technique by introducing a new operatorial approach—the diagonal ordering operation technique (DOOT)—regarding the calculations connected with the normally ordered product of generalized creation and annihilation operators that generate the generalized hypergeometric coherent states. We have pointed out a number of properties of these coherent states, including the case of mixed (thermal) states. At the same time, by particularizing the obtained results to the one-dimensional harmonic and pseudoharmonic oscillators, we get the well-known results achieved through other methods in the corresponding coherent states representation.