We obtain and investigate the regular eigenfunctions of simple differential operators x^r d^{r+1}/dx^{r+1}, r=1, 2, ... with the eigenvalues equal to one. With the help of these eigenfunctions we construct a non-unitary analogue of boson displacement operator which will be acting on the vacuum. In this way we generate collective quantum states of the Fock space which are normalized and equipped with the resolution of unity with the positive weight functions that we obtain explicitly. These states are thus coherent states in the sense of Klauder. They span the truncated Fock space without first r lowest-lying basis states: |0>, |1>, ..., |r-1>. These states are squeezed, are sub-Poissonian in nature and are reminiscent of photon-added states at Agarwal et al.
We obtain and investigate the regular eigenfunctions of simple differential operators x^r d^{r+1}/dx^{r+1}, r=1, 2, ... with the eigenvalues equal to one. With the help of these eigenfunctions we construct a non-unitary analogue of boson displacement operator which will be acting on the vacuum. In this way we generate collective quantum states of the Fock space which are normalized and equipped with the resolution of unity with the positive weight functions that we obtain explicitly. These states are thus coherent states in the sense of Klauder. They span the truncated Fock space without first r lowest-lying basis states: |0>, |1>, ..., |r-1>. These states are squeezed, are sub-Poissonian in nature and are reminiscent of photon-added states at Agarwal et al.
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.
The main goal of this paper is to present an alternative method to construct new kinds of nonlinear coherent states. To do this, we first establish a class of hypergeometric type of generalized displacement operators, 1 F r ([0], [0, 1, ..., r − 1], za †), act on the vacuum state of the harmonic oscillator and generate normalized quantum states of the Fock space which admit a resolution of the identity through a positive definite measure on the complex plane. Furthermore, realization of the compact form of these states, as functions of the position coordinate x for r = 2, leads to a generating function of the Hermite polynomials in terms of the modified Bessel function. Finally, studying some statistical characters reveals that they have indeed non-classical features such as squeezing, an anti-bunching effect and sub-Poissonian statistics, too.
We demonstrate explicitly the completeness of photon-added coherent states (PACSs), introduced by Agarwal and Tara (Agarwal G S and Tara K 1991 Phys. Rev. A 43 492) and defined, up to normalization, by (â † ) M | z ⟩, M = 0,1,2,...,, where â † is the boson creation operator and | z ⟩ are conventional Glauber-Klauder coherent states. We find the analytical form of the positive weight function in their resolution of unity by solving the associated Stieltjes power-moment problem. We furnish an example of generation of another set of PACSs which are complete.
The photon-added nonlinear coherent states, defined as adaggerm∣alpha,f> up to a normalization constant, are still nonlinear coherent states with the nonlinear function f(N−m)[1−m/(N+1)]. Here is the nonlinear coherent state with the nonlinear function f(N), N is the number operator, adagger is the boson creation operator and m is a non-negative integer. We introduce a new type of nonlinear coherent states with the nonlinear function f(N+m)[1+m/(N+1)] obtained from the photon-added nonlinear coherent states with negative values of m. The nonlinear coherent states corresponding to the positive and negative values of m are shown to be the result of nonunitarily deforming the number states ∣m> and ∣0>, respectively. As an example, we study the sub-Poissonian statistics and squeezing effects of the photon-added geometric states with negative values of m in detail.
Recently, we have determined the spectrum and the wave functions of the Hamiltonian of a Landau particle with time-dependent mass and frequency undergoing the influence of a uniform time-dependent electric field[J. Math. Phys. 56, 072104 (2018)]. In the present paper we extend the study of this model that we name the time-dependent Landau problem into the context of coherent states. By means of the traditional factorization method of the eigenfunctions of this system expressed in terms of the generalized Laguerre polynomials, we derive the generators of the su(1,1) Lie algebra and we construct the coherent states a la Barut-Girardello. These states are shown to satisfy the Klauder's mathematical requirement to build coherent states and some of their statistical properties are calculated and analyzed. We find that these states are sub-Poissonian in nature. We show that, addition of photons from these coherent states, increases the statistical properties and changes the mathematical properties of these states.
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.