Study of higher order non-classical properties of squeezed Kerr state

doi:10.1016/j.optcom.2010.04.007

Optics Communications

Volume 283, Issue 17, 1 September 2010, Pages 3284-3290

https://doi.org/10.1016/j.optcom.2010.04.007 Get rights and content

Abstract

Recently, Prakash and Mishra [J. Phys. B: at. Mol. Opt. Phys., 39, 2291(2006); 40, 2531(2007)] have studied higher order sub-Poissonian photon statistic conditions for non-classicality in the form of general inequalities for expectation values of products of arbitrary powers of photon number and of photon-number fluctuation. It is, therefore, vital to study the generation of these higher order sub-Poissonian photon statistics (phase-insensitive behavior) in a physically realizable medium and their relations to higher order squeezing (phase-sensitive behavior). In the present paper, we study higher order non-classical properties, such as Hong and Mandel squeezing, amplitude-squared squeezing and higher order sub-Poissonian photon statistics, of squeezed Kerr state which is generated by squeezing the output of a Kerr medium whose input is coherent light. Such states can be realized if laser light is sent through an optical fiber and then into a degenerate parametric amplifier. It is established that the squeezed Kerr state can exhibit higher order non-classical properties.

Introduction

States having no classical analog are termed as non-classical states. Density operators of radiation can be written as [1], [2] $\hat{ρ} = \int d^{2} α P (α) | α 〉〈 α |$ where α = α_r + iα_i, d²α = dα_rdα_i. The coherent state, |α〉, is the eigenstate of the annihilation operator â (â|α〉 = α|α〉) and P(α) is the Glauber–Sudarshan “P-function” [1]. Radiation is said to be non-classical when P(α) is not positive definite or more singular than the delta function [3]. Non-classical features of optical field [4] were studied with academic interest earlier [5], [6] but soon their importance was realized and they have received a great deal of attention during the last decade. Antibunching [7], sub-Poissonian photon statistics [8] and squeezing [4] are the examples of non-classical features. Kimble et al. [9] detected antibunching, which was the first non-classical feature of the optical field detected in a laboratory. Short and Mandel [10] detected sub-Poissonian light experimentally, whereas squeezing was first detected by Slusher et al. [11]. These experiments are regarded as milestones in the progress of quantum optics. There are numerous experimental and theoretical studies of these effects. These non-classical features drew wide attention because of their potential applications not only in reduction of noise level in optical communication [12] and in the detection of the extremely weak gravitational radiation [13] but also in a rapidly emerging quantum information processing [14].

With the development of techniques for making higher order correlation measurements in quantum optics, an interest was obviously extended to the higher moments of the field also. Higher order squeezing of optical field was introduced by Hong and Mandel [15], Hillery [16] and others. Lee [17], Kim [18], Erenso et al. [19] studied sub-Poissonian photon statistics of optical fields to higher orders. Recently, Prakash and Mishra [20] found the general inequality for higher order sub-Poissonian photon statistics involving expectation values of products of arbitrary powers of photon number and its fluctuation which exhibits non-classicality and studied their usefulness in the detection of higher order squeezing. It is important to note that there may be situations where some higher order sub-Poissonian photon statistics occur but not of lower order, as Prakash and Mishra [20] showed for a state $| ψ 〉 = (| l 〉 + c | m 〉) / \sqrt{(1 + {|c|}^{2})}$ where c is a complex number and l and m are real positive numbers. Generation of states exhibiting higher order sub-Poissonian photon statistics (phase-insensitive behavior) in a physically realizable system and relationship of its behavior with other higher order non-classical properties such as usual squeezing, Hong and Mandel squeezing and Hillery's amplitude-squared squeezing (being phase-sensitive behavior), therefore, remains an open challenge not only due to the academic interest but for their possible applications in the rapidly emerging area of quantum information processing.

Earlier, Gerry and Grobe [21] have studied the statistical properties of squeezed Kerr states generated by squeezing, the output of a Kerr medium whose input is coherent light. For these processes, the unitary evolution operator in the interaction picture has the form of the squeeze operator and so it is referred to as the squeezed Kerr state [21]. Generation of squeezed Kerr state can be realized in the laboratory, viz., if the laser light is sent through an optical fiber and subsequently into a degenerate parametric amplifier, the output will be in the squeezed Kerr state [22]. Squeezing has the effect of allowing the photon statistics to depend on the Kerr nonlinearity. Without squeezing the statistics is Poissonian, but after squeezing they may be either sub-Poissonian or super-Poissonian. To the best of my knowledge, the study of a higher order squeezing and higher order sub-Poissonian photon statistics has been done by Duc and Noh in photon-added coherent states and by Lee, Kim and Nha in photon-added classical (coherent and thermal) states [23]. It is established that the higher order sub-Poissonian photon statistics can be used to detect Hong and Mandel squeezing [20] and these higher order non-classical effects can be seen in the squeezed Kerr state. These facts primarily motivate one to study the squeezed Kerr state in this innovative regime of non-classicality and the present work aims to study the possibility of higher order squeezing and higher order sub-Poissonian photon statistics in a squeezed Kerr state [24]. Theoretical predictions of the present analysis can be experimentally verified with the help of homodyne experiment, since the criteria for higher order sub-Poissonian photon statistics appear in terms of factorial moments, which can be measured by using homodyne photon counting experiments [20].

Section snippets

Definition of squeezed Kerr states

A Kerr medium interacting with a single mode of the quantized electromagnetic field can be described by the Hamiltonian, $\hat{H} = ℏ ω {\hat{a}}^{†} \hat{a} + ℏ χ {\hat{a}}^{† 2} {\hat{a}}^{2},$ where χ is related to the third-order susceptibility of the Kerr medium [25]. The evolution operator associated with the propagation through the Kerr medium is, ${\hat{U}}_{K} (γ) = exp (- i γ {\hat{a}}^{† 2} {\hat{a}}^{2}) = exp [- i γ \hat{N} (\hat{N} - 1)],$ where N̂ = â^†â is the photon-number operator and γ = χL / v, L being the length of the medium and v the velocity of light in the medium. Assuming the incident light to be

Non-classical properties of light in squeezed Kerr state

With the quadrature operator, ${\hat{X}}_{ϕ} = \frac{1}{2} ({\hat{a}}^{†} e^{iϕ} + \hat{a} e^{- i ϕ})$ , the degree of usual squeezing can be defined as $q_{2 ϕ} \equiv 4 (〈{(Δ {\hat{X}}_{ϕ})}^{2}〉 - 0.25) = [〈{\hat{a}}^{† 2}〉 e^{2 i ϕ} + 〈{\hat{a}}^{2}〉 e^{- 2 i ϕ} + 2 〈{\hat{a}}^{†} \hat{a}〉 - {(〈{\hat{a}}^{†}〉 e^{iϕ} + 〈\hat{a}〉 e^{- i ϕ})}^{2}],$ and the degree of the fourth order Hong and Mandel squeezing [15] can be defined as $q_{4 ϕ} = \frac{16}{3} [〈{(Δ {\hat{X}}_{ϕ})}^{4}〉 - \frac{3}{16}] .$

Here q_2ϕ and q_4ϕ are limited in between 0 and −1 for the occurrence of squeezing and for ϕ = 0, we will denote them as q₂ (for the usual squeezing) and q₄ (for the fourth order Hong and Mandel's squeezing) throughout the paper.

With the operator, $\hat{Y}$

Discussions

Gerry and Grobe [21] studied the nature of the sub-Poissonian photon statistics by plotting Mandel Q-parameter versus γ for various values of r, for example, with α = 5 (α is taken to be real) and we will also use this value of α throughout the paper and similar to Gerry and Grobe [21], we choose θ = 0 so that z is real, i. e., z = r. Gerry and Grobe [21] reported that, at least for the lowest values of r, the field statistics can be sub- or super-Poissonian over some range of γ and found that the

Conclusions

Study of higher order non-classical properties of squeezed Kerr state is very important. Squeezed Kerr state may become important for the generation of higher order sub-Poissonian photon statistics as well as the higher order squeezing. As γ is directly proportional to the length of the Kerr medium, a variation of different non-classical features with γ for different squeezing parameters r is rather interesting. There are many different situations where some of these non-classical properties

Acknowledgements

The author is highly indebted to his teacher Prof. Hari Prakash for awakening his interest in the field of Quantum Optics and dedicates this work to Prof. Prakash on his 65^th birth birthday. The author also expresses his gratitude to Prof. Mark Hillery for his fruitful suggestions to improve the manuscript.

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Study of higher order non-classical properties of squeezed Kerr state

Abstract

Introduction

Section snippets

Definition of squeezed Kerr states

Non-classical properties of light in squeezed Kerr state

Discussions

Conclusions

Acknowledgements

Phys. Rev. Lett.

Phys. Rev. A

Phys. Rev. A

Phys. Rev. A

Phys. Rev.

Phys. Rev. Lett.

Phys. Rev.

J. Opt. B

Opt. Spectrosc.

Eur. Phys. J. D

Acta Physica Polonica A

Phys. Rev.

Phys. Rev. A

Rev. Mod. Phys.

Phys. Rev. Lett.

Phys. Rev.A

Phys. Rev. Lett.

Phys. Rev. Lett.

IEEE Trans. Inf. Theory

J. Lightwave Technol.

Phys. Rev. Lett.

Phys. Rev. A

Phys. Rev. Lett.