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arXiv:1204.2680

Representations of Coherent and Squeezed States in an Extended Two-parameters Fock Space

Mohammad Kazem Tavassoly, M.H. Lake

Published Apr 12, 2012• 0 citations
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Abstract

Recently a $f$-deformed Fock space which is spanned by $|n>_λ$ has been introduced. These bases are indeed the eigen-states of a deformed non-Hermitian Hamiltonian. In this contribution, we will use a rather new non-orthogonal basis vectors for the construction of coherent and squeezed states, which in special case lead to the earlier known states. For this purpose, we first generalize the previously introduced Fock space spanned by $|n>_λ$ bases, to a new one, spanned by an extended two-parameters bases $|n>_{λ_{1},λ_{2}}$. These bases are now the eigen-states of a non-Hermitian Hamiltonian $H_{λ_{1},λ_{2}}=a^{\dagger}_{λ_{1},λ_{2}}a+1/2$, where $a^{\dagger}_{λ_{1},λ_{2}}=a^{\dagger}+λ_{1}a + λ_{2}$ and $a$ are respectively, the deformed creation and ordinary bosonic annihilation operators. The bases $|n>_{λ_{1},λ_{2}}$ are non-orthogonal (squeezed states), but normalizable. Then, we deduce the new representations of coherent and squeezed states, in our two-parameters Fock space. Finally, we discuss the quantum statistical properties, as well as the non-classical properties of the obtained states, numerically.

DOI
10.48550/arxiv.1204.2680

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Recently an f -deformed Fock space which is spanned by | n 〉 λ was introduced. These bases are the eigenstates of a deformed non-Hermitian Hamiltonian. In this contribution, we will use rather new non-orthogonal basis vectors for the construction of coherent and squeezed states, which in special case lead to the earlier known states. For this purpose, we first generalize the previously introduced Fock space spanned by | n 〉 λ bases, to a new one, spanned by extended two-parameters bases | n 〉 λ 1 , λ 2 . These bases are now the eigenstates of a non-Hermitian Hamiltonian H λ 1 ,λ 2 = a † λ 1 ,λ 2 a +½, where a † λ 1 , λ 2 and a are, respectively, the deformed creation and ordinary bosonic annihilation operators. The bases | n 〉 λ 1 , λ 2 are non-orthogonal (squeezed states), but normalizable. Then, we deduce the new representations of coherent and squeezed states in our two-parameter Fock space. Finally, we discuss the quantum statistical properties, as well as the non-classical properties of the obtained states numerically.
We establish some of the properties of the states interpolating between number and coherent states denoted by | n ⟩ λ ; among them are the reproduction of these states by the action of an operator-valued function on | n ⟩ (the standard Fock space) and the fact that they can be regarded as f -deformed coherent bound states. In this paper we use them as the basis of our new Fock space which in this case is not orthogonal but normalized. Then by some special superposition of them we obtain new representations for coherent and squeezed states in the new basis. Finally the statistical properties of these states are studied in detail.
Two-mode charge (pair) coherent states have been introduced previously by using the 〈η| representation. In this paper, we reobtain these states by a rather different method. Then, using the nonlinear coherent state approach and based on a simple manner by which the representation of two-mode charge coherent states is introduced, we generalize the bosonic creation and annihilation operators to the f -deformed ladder operators and construct a new class of f -deformed charge coherent states. Unlike the (linear) pair coherent states, our presented structure has the potential to generate a large class of pair coherent states with various nonclassicality signs and physical properties which are of interest. For this purpose, we use a few well-known nonlinearity functions associated with particular quantum systems as some physical appearances of our presented formalism. After introducing the explicit form of the above correlated states in the two-mode Fock space, several nonclassicality features of the corresponding states (as well as the two-mode linear charge coherent states) are numerically investigated by calculating quadrature squeezing, the Mandel parameter, the second-order correlation function, the second-order correlation function between the two modes and the Cauchy–Schwartz inequality. Also, the oscillatory behaviour of the photon count and the quasi-probability (Husimi) function of the associated states will be discussed. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.
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By introducing a generalization of the (p, q)-deformed boson oscillator algebra, we establish a two-parameter deformed oscillator algebra in an infinite-dimensional subspace of the Hilbert space of a harmonic oscillator without first finite Fock states. We construct the associated coherent states, which can be interpreted as photon-added deformed states. In addition to the mathematical characteristics, the quantum statistical properties of these states are discussed in detail analytically and numerically in the context of conventional as well as deformed quantum optics. Particularly, we find that for conventional (nondeformed) photons the states may be quadrature squeezed in both cases Q = pq < 1, Q = pq > 1 and their photon number statistics exhibits a transition from sub-Poissonian to super-Poissonian for Q < 1 whereas for Q > 1 they are always sub-Poissonian. On the other hand, for deformed photons, the states are sub-Poissonian for Q > 1 and no quadrature squeezing occurs while for Q < 1 they show super-Poissonian behavior and there is a simultaneous squeezing in both field quadratures.PACS Nos.: 42.50.Ar, 03.65.–w