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Physica Scripta

Complementarity, quartic polynomials and quantum states

R. F. Álvarez-Estrada

Published Mar 1, 2013• 1 citation
Publisher

Abstract

In a previous work, based upon complementarity and inspired by a standard derivation of the x?p uncertainty inequality (via a non-negative quadratic polynomial f2), we explored one possible extension, through a non-negative quartic polynomial f4, for non-commuting quantum variables. That work led to new quantum inequalities (expressed through the discriminant of the equation f4?=?0) and to connections with quantum optics. Here, we extend the study of quartic polynomials and report classes of genuine quantum states |?4,n? with interesting properties, which follow directly from f4?=?0 for x?p. We explore |?4,n? when they are generated from the coherent and the squeezed coherent states of the quantum harmonic oscillator and we display their relationships to states previously proposed by other authors through different constructions. In quantum optics, the |?4,n? associated to the coherent states are just displaced Fock states of light, which have already been generated experimentally.

DOI
10.1088/0031-8949/2013/t153/014002

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In a previous work, based upon complementarity and inspired by a standard derivation of the x?p uncertainty inequality (via a non-negative quadratic polynomial f2), we explored one possible extension, through a non-negative quartic polynomial f4, for non-commuting quantum variables. That work led to new quantum inequalities (expressed through the discriminant of the equation f4?=?0) and to connections with quantum optics. Here, we extend the study of quartic polynomials and report classes of genuine quantum states |?4,n? with interesting properties, which follow directly from f4?=?0 for x?p. We explore |?4,n? when they are generated from the coherent and the squeezed coherent states of the quantum harmonic oscillator and we display their relationships to states previously proposed by other authors through different constructions. In quantum optics, the |?4,n? associated to the coherent states are just displaced Fock states of light, which have already been generated experimentally.
Complementarity (expressed through the x–p commutation relationship) yields, by using suitable non-negative polynomials, quantum inequalities. The latter become strict equalities only for restricted sets of quantum states, which display genuine quantum features. We have shown in previous studies that: (i) a certain non-negative quartic polynomial f4 led to a new quantum inequality, expressed through the discriminant (D'r) of the equation f4 = 0 as D'r ≤ 0; (ii) D'r = 0 gave rise to an interesting class of non-Gaussian quantum states |ψ4〉 and (iii) in quantum optics, a subset of those |ψ4〉 were the displaced one-photon states, previously proposed by other authors through different motivations and already generated experimentally in 2002. The extension of our previous research studies will be reported here. We shall characterize the general class of non-Gaussian quantum states |ψ4,g〉 such that D'r = 0. The class of the |ψ4,g〉 is larger than and includes that of the former |ψ4〉. The Wigner function for the |ψ4,g〉 is obtained and is negative in some domains. In quantum optics, a subclass of the |ψ4,g〉 is formed by the single-photon-added coherent states (non-Gaussian) generated experimentally, in turn, in 2004. Another subclass of the |ψ4,g〉 includes single-photon-added squeezed states and single-photon-added squeezed coherent states (both being non-Gaussian). Mandel's parameters are studied for those states: in particular, its negativity for the single-photon-added squeezed states is established for a certain interval of the squeeze factor. The (Hilbert–Schmidt) non-Gaussianities for both the displaced one-photon states and the single-photon-added squeezed states are obtained explicitly: both of them are numerically equal to 5/12. The possibility of generating the single-photon-added squeezed states experimentally is discussed briefly, by extending previous studies by other authors on producing related quantum states; in particular, we treat in outline one possible generation using optical parametric oscillators and parametric amplifiers.
Following the lines of the recent paper of J.-P. Gazeau and F. H. Szafraniec [J. Phys. A: Math. Theor. 44, 495201 (2011)], we construct here three types of coherent states, related to the Hermite polynomials in a complex variable which are orthogonal with respect to a non-rotationally invariant measure. We investigate relations between these coherent states and obtain the relationship between them and the squeezed states of quantum optics. We also obtain a second realization of the canonical coherent states in the Bargmann space of analytic functions, in terms of a squeezed basis. All this is done in the flavor of the classical approach of V. Bargmann [Commun. Pur. Appl. Math. 14, 187 (1961)].
Following the lines of the recent paper of J.-P. Gazeau and F. H. Szafraniec [J. Phys. A: Math. Theor. 44, 495201 (2011)], we construct here three types of coherent states, related to the Hermite polynomials in a complex variable which are orthogonal with respect to a non-rotationally invariant measure. We investigate relations between these coherent states and obtain the relationship between them and the squeezed states of quantum optics. We also obtain a second realization of the canonical coherent states in the Bargmann space of analytic functions, in terms of a squeezed basis. All this is done in the flavor of the classical approach of V. Bargmann [Commun. Pure Appl. Math. 14, 187 (1961)].
This subject of this thesis is the physical application of deformations of Lie algebras and their use in generalising some exotic quantum optical states. We begin by examining the theory of quantum groups and the q-boson algebras used in their representation theory. Following a review of the properties of conventional coherent states, we describe the extension of the theory to various deformed Heisenberg-Weyl algebras, as well as the q-deformations of su(2) and su(1,1). Using the Deformed Oscillator Algebra of Bonatsos and Daskaloyannis, we construct generalised deformed coherent states and investigate some of their quantum optical properties. We then demonstrate a resolution of unity for such states and suggest a way of investigating the geometric effects of the deformation. The formalism devised by Rembielinski et al is used to consider coherent states of the q-boson algebra over the quantum complex plane. We propose a new unitary operator which is a q-analogue of the displacement operator of conventional coherent state theory: This is used to construct q-displaced vacuum states which are eigenstates of the annihilation operator. Some quantum mechanical properties of these states are investigated and it is shown that they formally satisfy a Heisenberg-type minimum uncertainty relation. After briefly reviewing the theory of conventional squeezed states, we examine the various q-generalisations. We propose a q-analogue of the squeezed vacuum state, and use this in conjunction with the unitary q-displacement operator to construct a general q-squeezed state, parameterised by noncommuting variables.. It is shown that, like their conventional counterparts, such states satisfy the Robertson-Schrodinger Uncertainty Relation. We conclude with a brief discussion about the appearance of noncommuting variables in the states that have been considered.
In the present PhD work, we focus on a specific class of quantum states of light: the non-Gaussian states. These states have the particularity of exhibiting Wigner functions with some negative values. This quantum feature is a necessary condition to perform some quantum computation task; furthermore it is also useful for various other applications, including quantum communication and metrology. Different strategies can be used to generate these states. Here, we start from Gaussian states produced by optical parametric oscillators in the continuous wave regime, (i.e. single-mode and two-mode squeezed vacuum states). The non-Gaussian feature can only be obtained by non-linear phenomena (over-quadratic Hamiltonian). In our case, the non-linearity is induced by photon-counting-based measurements (also called non-Gaussian measurements). This study is mainly divided into two parts. First, the generation of non-classical states associated with two types of qubit encoding: the single-photon state, used for quantum information with discrete variables, and the coherent state superposition (the so-called optical Schrodinger cat state), used for quantum information with continuous variables. These two states have then been used to perform some quantum information protocols. The first one addresses the problem of single-photon entanglement witness, and the other the generation of entanglement between the two encodings (also called hybrid entanglement).
Abstract For the single mode the Hermite coherent states are wave packets analogous to the standard harmonic oscillator coherent states but constructed with complex Hermite polynomials substituting traditionally used complex monomials zn . Hermite coherent states unify the standard coherent and squeezed states within one class of quantum states whose representatives share all properties customarily required from the standard coherent states, in particular (over)completeness. Generalization of the Hermite coherent/squeezed states to the multimode case, illustrated on the bipartite states, shows that such states exhibit entanglement which presence or absence depends on the nature of squeezing but also on the choice of description method being used.