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Journal of the Optical Society of America B

Nonclassical properties of Hermite polynomial’s coherent state

Gang Ren, Jian-ming Du, Hai-jun Yu, Ye‐Jun Xu

Published Nov 28, 2012• 21 citations
Publisher

Abstract

In this paper, we introduce the Hermite polynomial’s coherent state (HPCS) |α〉H, which is defined as Hn(X)|α〉 up to a normalization constant and where Hn(X) is the coordinate operator’s Hermite polynomial of order nand |α〉=exp(−(1/2)|α|2+αa†)|0〉. This state may be produced by the superposition of some different photon added coherent states when n=2. The mathematical and physical properties of HPCS are also studied. It is shown that HPCS has remarkable nonclassical state features such as sub-Poissonian and squeezing properties.

DOI
10.1364/josab.29.003412

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In this paper, we introduce the Hermite polynomial’s coherent state (HPCS) |α〉H, which is defined as Hn(X)|α〉 up to a normalization constant and where Hn(X) is the coordinate operator’s Hermite polynomial of order nand |α〉=exp(−(1/2)|α|2+αa†)|0〉. This state may be produced by the superposition of some different photon added coherent states when n=2. The mathematical and physical properties of HPCS are also studied. It is shown that HPCS has remarkable nonclassical state features such as sub-Poissonian and squeezing properties.
Abstract In this work, we theoretically construct an orthogonal state of a coherent state where the orthogonalizer is based on the Hermite polynomial excited operation with H m ( x ) being a Hermite polynomial of order m and creation operator , called the Hermite-excite-orthogonalized coherent state (HEOCS). Then, in comparison with the Hermite polynomial excited coherent state (HPECS), we numerically investigate their different nonclassical properties by examining several measurable features, such as photon number distribution, sub-Poisson statistics, the antibunching effect, the quadrature squeezing effect, and the negative Wigner function. The results demonstrate that both HEOCS and HPECS are non-Gaussian quantum states with highly nonclassical characteristics by improving and modulating the order of the Hermite polynomial and coherent amplitude, which indicates that performing the Hermite-excited operation on given Gaussian states is an effective approach to generating non-Gaussian states.
We introduce states defined by \ensuremath{\Vert}\ensuremath{\alpha},m〉=${\mathit{a}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}\mathit{m}}$\ensuremath{\Vert}\ensuremath{\alpha}〉 up to a normalization constant, where \ensuremath{\Vert}\ensuremath{\alpha}〉 is a coherent state and m an integer. We study the mathematical and physical properties of such states. We demonstrate phase squeezing and the sub-Poissonian character of the fields in such states. We study in detail the quasiprobability distributions and the distribution of the field quadrature. We also show how such states can be produced in nonlinear processes in cavities.
The two-mode photon-added state defined by |?, q; m>?=?a+mb+m|?, q> up to a normalization constant is introduced, where |?, q> is a pair coherent state and m an integer. It is shown that the fields in such a state have remarkable quantum features such as sub-Poissonian statistics, violations of Cauch-Schwarz inequalities, correlations in the number fluctuations and squeezing. We examine the influence of photon-adding on the properties of pair coherent state, and propose a method of generating two-mode photon-added states in the context of cavity quantum electrodynamics.
We construct a new type of photon-added squeezed coherent state generated by repeatedly operating the bosonic creation operator on a new type of squeezed coherent state [Fan H Y and Xiao M 1996 Phys. Lett. A 220 81]. We find that its normalization factor is related to single-variable Hermite polynomials. Furthermore, we investigate its statistical properties, such as Mandel’s Q parameter, photon-number distribution, and Wigner function. The nonclassicality is displayed in terms of the intense oscillation of photon-number distribution and the negativity of the Wigner function.
We construct a new type of photon-added squeezed coherent state generated by repeatedly operating the bosonic creation operator on a new type of squeezed coherent state [Fan H Y and Xiao M 1996 Phys. Lett . A 220 81 ]. We find that its normalization factor is related to single-variable Hermite polynomials. Furthermore, we investigate its statistical properties, such as Mandel's Q parameter, photon-number distribution, and Wigner function. The nonclassicality is displayed in terms of the intense oscillation of photon-number distribution and the negativity of the Wigner function.