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Physica Scripta

Complementarity, quartic polynomials and one-photon-added coherent and squeezed states

R. F. Álvarez-Estrada

Published Apr 1, 2014• 0 citations
Publisher

Abstract

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.

DOI
10.1088/0031-8949/2014/t160/014003

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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.
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.
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.
Complementarity and the commutation relation of position (x) and momentum (p) imply much more than the fundamental x-p uncertainty inequality. Here, we display some further consequences of the former that could have certain pedagogical interest and, so, contribute to the teaching of quantum mechanics. Inspired by an elementary derivation of the x-p uncertainty inequality, based upon a positive quadratic polynomial, we explore one possible extension, via quartic polynomials and simple algebra and integrations. Our analysis, aimed at providing some further pedagogic expression of genuine quantum behaviours, yields other quantum inequalities for expectation values, expressed through suitable discriminants associated with quartic algebraic equations, which differ from (and are not a strict consequence of) the x-p uncertainty inequality. Those quantum inequalities are confirmed, and genuine non-classical behaviours are exhibited, for simple cases: a harmonic oscillator, a hydrogenic atom and free Gaussian wave packets. The physical interest of the expectation values involved in the quantum inequalities and of the latter is discussed, in the framework of quantum optics and squeezing phenomena.
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).
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 Schrödinger 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).