Quasiprobability distributions in open quantum systems: Spin-qubit systems

doi:10.1016/j.aop.2015.07.029

Annals of Physics

Volume 362, November 2015, Pages 261-286

https://doi.org/10.1016/j.aop.2015.07.029 Get rights and content

Highlights

•
The behavior of the well-known Wigner, $P$ , and $Q$ quasiprobability distributions are studied for a number of spin systems.
•
Spin-qubit systems studied here includes single, two and three qubit states, and $N$ qubit Dicke model.
•
Comprehensive analysis of quasiprobability distributions for spin-qubit systems is performed under general open system effects.
•
Both pure dephasing as well as dissipation effects are studied.

Abstract

We study nonclassical features in a number of spin-qubit systems including single, two and three qubit states, as well as an $N$ qubit Dicke model and a spin-1 system, of importance in the fields of quantum optics and information. This is done by analyzing the behavior of the well known Wigner, $P$ , and $Q$ quasiprobability distributions on them. We also discuss the not so well known $F$ function and specify its relation to the Wigner function. Here we provide a comprehensive analysis of quasiprobability distributions for spin-qubit systems under general open system effects, including both pure dephasing as well as dissipation. This makes it relevant from the perspective of experimental implementation.

Introduction

A very useful concept in the analysis of the dynamics of classical systems is the notion of phase space. A straightforward extension of this to the realm of quantum mechanics is however foiled due to the uncertainty principle. Despite this, it is possible to construct quasiprobability distributions (QDs) for quantum mechanical systems in analogy with their classical counterparts [1], [2], [3], [4], [5], [6]. These QDs are very useful in that they provide a quantum classical correspondence and facilitate the calculation of quantum mechanical averages in close analogy to classical phase space averages. Nevertheless, the QDs are not probability distributions as they can take negative values as well, a feature that could be used for the identification of quantumness in a system.

The first such QD was developed by Wigner resulting in the epithet Wigner function ( $W$ ) [7], [8], [9], [10], [11], [12]. Another, very well known, QD is the $P$ function whose development was a precursor to the evolution of the field of quantum optics. This was originally developed from the possibility of expressing any state of the radiation field in terms of a diagonal sum over coherent states [13], [14]. The $P$ function can become singular for quantum states, a feature that promoted the development of other QDs such as the $Q$ function [15], [16], [17] as well as further highlighted the use of the $W$ function which does not have this feature. These QDs are intimately related to the problem of operator orderings. Thus, the $P$ and $Q$ functions are related to the normal and antinormal orderings, respectively, while the $W$ function is associated with symmetric operator ordering. It is quite clear that there can be other QDs, apart from the above three, depending upon the operator ordering. However, among all the possible QDs the above three QDs are the most widely studied. There exist several reasons behind the intense interest in these QDs. They can be used to identify the nonclassical (quantum) nature of a state [18]. Specifically, nonpositive values of $P$ function define a nonclassical state. Nonpositivity of $P$ is a necessary and sufficient criterion for nonclassicality, but other QDs provide only sufficient criteria.

A nonclassical state can be used to perform tasks that are classically impossible. This fact motivated many studies on nonclassical states, for example, studies on squeezed, antibunched and entangled states. The interest on nonclassical states has been considerably amplified in the recent past after the advent of quantum information where several applications of nonclassical states, in particular, of entangled states, have been reported [19]. Interestingly many of these applications have been designed using spin-qubit systems.

Quantum optics deals with atom-field interactions. The atoms, in their simplest forms, are modeled as qubits (two-level systems). These are also of immense practical importance as they can be the effective realizations of Rydberg atoms [20], [21]. Atomic systems are also studied in the context of the Dicke model [22], [23], a collection of two-level atoms; in atomic traps [24], atomic interferometers [25], polarization optics [26], and have recently found applications in quantum computation ([27], [28], [29], [30], [31], [32] and references therein) as well as in the generation of long-distance entanglement [33]. All these would evoke the question whether one could have QDs for such atomic systems as well. Such questions, which are of relevance to the present work, would be closely tied to the problem of development of QDs for $S U (2)$ , spin-like (spin- $j$ ), systems. Such a development was made in [34], where a QD on the sphere, naturally related to the $S U (2)$ dynamical group [35], [36], was obtained. There are by now a number of constructions of spin QDs [37], [38], [39], [40], [41], among others.

However, another approach, the one adapted here, is to make use of the connection of $S U (2)$ geometry to that of a sphere. The spherical harmonics provide a natural basis for functions on the sphere. This, along with the general theory of multipole operators [42], [43], can be made use for constructing QDs of spin (qubit) systems as functions of polar and azimuthal angles [44]. Other constructions, in the literature, of $W$ functions for spin- $1 / 2$ systems can be found in [45], [46], among others. A concept that played an important role in the above developments, was the atomic coherent state [47], which lead to the definition of atomic $P$ function in close analogy to their radiation field counterparts. Another related development, following [48] where joint probability distributions were obtained for spin-1 systems exposed to quadrupole fields, was a QD obtained from the Fourier inversion of the characteristic function of the corresponding probability mass function, using the Wigner–Weyl correspondence. This could be called the characteristic function or $F$ -function approach [49].

The fields of quantum optics and information have matured to the point where intense experimental investigations are being made. Both from the fundamental perspective as well as from the viewpoint of practical realizations, it is imperative to study the evolution of the system of interest taking into account the effect of its ambient environment. This is achieved systematically by using the formalism of Open Quantum Systems [50], [51], [52], [53].

In the present work, we investigate nonclassicality in a number of spin-qubit systems including single, two and three qubit states, as well as $N$ qubit Dicke states and a spin-1 system, of importance in the fields of quantum optics and information. This is done by analyzing the behavior of the well known $W$ , $P$ , $Q$ QDs on them. The significance of this is rooted to the phenomena of quantum state engineering, which involves the generation and manipulation of nonclassical states [54], [55]. In this context, it is imperative to have an understanding over quantum to classical transitions, under ambient conditions. Such an understanding is made possible by the present work, where investigations are done in the presence of open system effects, both purely dephasing (decoherence) [56], [57], also known as quantum non-demolition (QND), as well as dissipation [56], [58]. These aspects of open system evolution have been realized in a series of beautiful experiments [59], [60]. We also discuss the not so well known $F$ function and specify its relation to the $W$ function. Further, we expect this work to have an impact on tomography related issues, as borne out in [61], where a method for quantum state reconstruction of a system of spins or qubits was proposed using the $Q$ function. Also, the $Q$ function, studied here, can be turned to address fundamental issues such as complementarity between number and phase distributions [62], [63], [64], under the influence of QND as well as dissipative interactions with their environment, as well as for phase dispersion in atomic systems [65], [66]. Here, to the best of our knowledge, we provide, for the first time, a comprehensive analysis of QDs for spin-qubit systems under general open system effects.

The plan of this paper is as follows. In the next section, we will briefly discuss the QDs that will be subsequently used in the rest of the work, i.e., the $W$ , $P$ , $Q$ , and $F$ functions. This will be followed by a study of open system QDs for single qubit states. Next, we take up the case of some interesting two and three qubit states as well as the well known $N$ qubit Dicke model. We then discuss, briefly, QDs of a spin-1 system. These examples will provide an understanding of quantum to classical transitions as indicated by the various QDs, under general open system evolutions. Although QDs have been frequently used to identify the existence of nonclassical states [67], they do not directly provide any quantitative measure of the amount of nonclassicality. Keeping these in mind, several measures of nonclassicality have been proposed, but all of them are seen to suffer from some limitations [68]. A specific measure of nonclassicality is the nonclassical volume, which considers the doubled volume of the integrated negative part of the $W$ function as a measure of nonclassicality [69]. In the penultimate section, we make a study of quantumness, in some of the systems considered in this work, by using nonclassical volume [69]. We then make our conclusions.

Section snippets

Distribution functions for spin (qubit) systems

Here, we briefly discuss the different QDs, i.e., the $W$ , $P$ , $Q$ and $F$ functions, subsequently used in the paper.

Distribution functions for single spin- $\frac{1}{2}$ states

Here, we consider single spin- $\frac{1}{2}$ states, initially in an atomic coherent state, in the presence of two different noises, i.e., QND [56], [57], which are purely dephasing, and the dissipative SGAD (Squeezed Generalized Amplitude Damping) [56], [58] noises. For calculating the QDs, we will require multipole operators for $j = \frac{1}{2}$ and $m, m^{'} = \pm \frac{1}{2}$ , giving $K = 0$ and 1. For $K = 0$ , $Q = 0$ , and for $K = 1$ , $Q = 1, 0, - 1$ . Using these, the multipole operators $T_{K Q}$ can be obtained as $T_{00} = \frac{1}{\sqrt{2}} [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ , $T_{11} = [\begin{matrix} 0 & 0 \\ - 1 & 0 \end{matrix}]$ , $T_{10} = \frac{1}{\sqrt{2}} [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}]$ ,

QDs for multiqubit systems undergoing QND and dissipative evolutions

Now, we wish to study the evolution of QDs for some interesting two and three qubit systems under general open system evolutions. We will also take up the well known $N$ -qubit Dicke model. In each case, we study the nonclassicality exhibited by the system under consideration.

QDs for a spin-1 state

Now, we extend the discussion of spin QDs from spin- $\frac{1}{2}$ to spin-1 states. For a spin-1 pure state [42] $| χ 〉 = a_{+} | + 1 〉 + a_{0} | 0 〉 + a_{-} | - 1 〉,$ QDs can be constructed using appropriate multipole operators and spherical harmonics. A few relevant multipole operators are $T_{00} = \frac{1}{\sqrt{3}} [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$ , $T_{11} = \frac{1}{\sqrt{2}} [\begin{matrix} 0 & - 1 & 0 \\ 0 & 0 & - 1 \\ 0 & 0 & 0 \end{matrix}]$ , $T_{10} = \frac{1}{\sqrt{2}} [\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & - 1 \end{matrix}]$ , $T_{22} = [\begin{matrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]$ , $T_{21} = \frac{1}{\sqrt{2}} [\begin{matrix} 0 & - 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}]$ , and $T_{20} = \frac{1}{\sqrt{6}} [\begin{matrix} 1 & 0 & 0 \\ 0 & - 2 & 0 \\ 0 & 0 & 1 \end{matrix}]$ . All othermultipole operators can be obtained from these operators. The analytical expressions of the different QDs are obtained

Nonclassical volume

Till now, we have studied nonclassicality using negative values of the $W$ or $P$ function. Negative values of the QDs only provide a signature of nonclassicality, but they do not provide a quantitative measure of nonclassicality. There do exist some quantitative measures of nonclassicality, see for example, [68] for a review. One such measure is nonclassical volume introduced in [69]. In this approach, the doubled volume of the integrated negative part of the $W$ function of a given quantum state

Conclusions

The nonclassical nature of all the systems studied here, of relevance to the fields of quantum optics and information, is illustrated via their quasiprobability distributions as a function of the time of evolution as well as various state or bath parameters. We also provide a quantitative idea of the amount of nonclassicality observed in some of the systems studied using a measure which essentially makes use of the $W$ function. These issues assume significance in questions related to quantum

Acknowledgments

A.P. and K.T. thank Department of Science and Technology (DST), India for support provided through the DST project No. SR/S2/LOP-0012/2010. SB thanks R. Srikanth for some useful discussions during the early stages of this work. We also thank Usha Devi for a number of helpful comments during various stages of this work.

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Quasiprobability distributions in open quantum systems: Spin-qubit systems

Highlights

Abstract

Introduction

Section snippets

Distribution functions for spin (qubit) systems

Distribution functions for single spin-12 states

QDs for multiqubit systems undergoing QND and dissipative evolutions

QDs for a spin-1 state

Nonclassical volume

Conclusions

Acknowledgments

Phys. Rep.

Ann. Phys.

Ann. of Phys.

Phys. Rev A

Phys. Rev.

Phys. Rev. A

Phys. Rev. A

Fundamentals of Quantum Optics

Quantum Optics

Quantum Optics in Phase Space

Quantum Optics

Mathematical Methods of Quantum Optics

A Group-Theoretical Approach to Quantum Optics

Phys. Rev.

Proc. Cambridge Philos. Soc.

Phase Space Picture of Quantum Mechanics

Adv. Chem. Phys.

J. Modern Opt.

Phys. Rev.

Phys. Rev. Lett.

Phys. Rev.

J. Phys. Soc. Japan

Proc. Phys. Math. Soc. Japan

Phys. Rev. A

Elements of Quantum Computation and Quantum Communication

Rev. Modern Phys.

Nat. Phys.

Phys. Rev.

Phys. Rev. A

J. Opt. Soc. Amer. B

Phys. Rev. Lett.

Bull. Lebedev Phys. Inst. (Russian Academy of Science)

Nature

Phys. Rev. A

J. Chem. Phys.

Phys. Rev. Lett.

Phys. Rev. A

Science

Nat. Phys.

Sov. Phys.—JETP

Rev. Mex. Fis.

Distribution functions for single spin- $\frac{1}{2}$ states