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Chinese Journal of Physics

Statistical Properties of the Excited Binomial States for the Pseudoharmonic Oscillator

Dušan Popov, Nicolina Pop

Published Apr 1, 2014• 2 citations
Publisher

Abstract

In this paper we have studied the binomial states and also the excited binomial states defined in terms of the Fock-basis vectors of the pseudoharmonic oscillator. We have demonstrated that outside of the behavior at the harmonic limit, when the binomial states lead to the coherent states of the one dimensional harmonic oscillator, these states have in fact all the characteristics of the coherent states defined on the complex unit disk. We have calculated the expectation values and the Mandel parameter (as well as their thermal analogue) which give us information on their statistical behavior. All the obtained relations tend, at the harmonic limit, to the corresponding relations for the one dimensional harmonic oscillator.

DOI
10.6122/cjp.52.738

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In this paper we have studied the binomial states and also the excited binomial states defined in terms of the Fock-basis vectors of the pseudoharmonic oscillator. We have demonstrated that outside of the behavior at the harmonic limit, when the binomial states lead to the coherent states of the one dimensional harmonic oscillator, these states have in fact all the characteristics of the coherent states defined on the complex unit disk. We have calculated the expectation values and the Mandel parameter (as well as their thermal analogue) which give us information on their statistical behavior. All the obtained relations tend, at the harmonic limit, to the corresponding relations for the one dimensional harmonic oscillator.
In this paper we examine some of the statistical properties of the negative binomial states (NBSs) that are a superposition of the number of states with appropriately chosen coefficients, but on the basis of Fock-vectors, which correspond to the pseudoharmonic oscillator. These states have the coherent states’ behaviour not only for the harmonic limit. We examine the expectation values of the integer powers of the number operator N , which is useful when calculating Mandel's parameter for these states. Depending on the value of this parameter, we can determine the statistical behaviour of the NBSs, where these states are: sub-Poissonian, Poissonian and supra-Poissonian. Meijer's G -functions formalism was used in the calculation.
In the last decades, much attention has been paid to the excitation on coherent states, especially for coherent states of the harmonic oscillator ([1] and references therein). But an interesting anharmonic oscillator with many potential applications is also the pseudoharmonic oscillator (PHO). So, in the present paper we have defined the excitation on the Klauder‐Perelomov coherent states (E‐KP‐CSs) for the PHO. These states are obtained by repeatedly operating the raising operator K+ on a usual Klauder‐Perelomov coherent state (KP‐CS) of the PHO [2]. We have verified that really, the E‐KP‐CSs fulfill all the properties of the coherent states, as stated by Klauder [3]. We have examined the nonclassical properties of the E‐KP‐CSs, by using the density matrix formalism and examining the dependence of the Mandel parameter Qz,k;m(|z|2) on the |z|2 and on the m. It seems that these states can be used in optical communication field and in the physics of quantum information, as signal beams, due to the fact that in these fields the nonclassicality plays an important role.
The generalized hypergeometric coherent states (GHCSs) have been introduced by Appl and Schiller (1). In the present paper we have extended some considerations about GHCSs to the mixed (thermal) states and applied, particularly, to the case of pseudoharmonic oscillator (PHO). The Husimi's Q distribution function and the diagonal P - distribution function, in the GHCSs representation, have been deduced for these mixed states. The obtained distribution functions were used to calculate thermal averages and to examine some nonclassical properties of the generalized hypergeometric thermal coherent states (GHTCSs), particularly for the PHO. We have also deflned and calculated the thermal analogue of the Mandel parameter and the thermal analogue of the second-order correlation function. By particularizing the parameters p and q of the hypergeometric functions, we recover the usual Barut-Girardello coherent states and their main properties for the PHO from our previous paper (9). All calculations are performed in terms of the Meijer's G-functions (2), which are related to the hypergeometric functions. This manner provides an elegance and uniformity of the obtained results and so the GHCSs become a new fleld of application for these functions. Moreover, this mathematical approach can be used also for other kind of coherent states (e.g. Klauder-Perelomov, Gazeau-Klauder or nonlinear coherent states (10), (12)).
We introduce excited binomial states and excited negative binomial states of the radiation field by repeated application of the photon creation operator on binomial states and negative binomial states. They reduce to Fock states and excited coherent states in certain limits and can be viewed as intermediate states between Fock states and coherent states. We find that both the excited binomial states and excited negative binomial states can be exactly normalized in terms of hypergeometric functions. Base on this interesting character, some of the statistical properties are discussed.
We introduce coherent mixed states (or thermal coherent states) associated with the displaced harmonic oscillator at nonzero temperature (T ≠ 0), as a "random" (or "thermal" or "noisy") basis in Hilbert space. A resolution of the identity for these states is proven and is used to generalize the usual pure (T = 0) coherent state formalism to the mixed (T ≠ 0) case. This new formalism for thermal coherent states is then further generalized to a broader class of so-called negative-binomial mixed states. It is known that the negative-binomial distribution is itself intimately related to the discrete series of SU(1,1) representations. We consider the pure SU(1,1) coherent states in the two-mode harmonic oscillator space, and show how our negative-binomial mixed states arise from taking the partial trace with respect to one of these two modes. This observation is then used to show how the formalism of thermo-field dynamics may be generalized to a correspondingly much broader negative-binomial-field dynamics, which we expect to have many uses.
In the paper we have investigated some properties of the Barut-Giraradello coherent states (BGCS) (the eigenstates of the SU(1,1) lowering generator K-) for the Hamiltonian of the pseudoharmonic oscillator (PHO). By using these states, the diagonal P-representation of the density operator is constructed as a new result for this potential. In addition, we have calculated some thermal expectation values for the quantum canonical gas of the PHOs. The calculations using the BGCS representation seem to be much simpler and easier to program than the corresponding calculations in other representations (e.g. the position representation). At the end of the paper the time dependence of these states is presented.