Relation between the roughness, linear entropy and visibility of a quantum state, the Jaynes–Cummings model

In this work, an analysis of the Jaynes–Cummings Model is conducted in the parameter spaces, composed of Roughness, Concurrence/Linear Entropy and Visibility. The analysis was carried out without including the effects of the environment and with the inclusion of a dispersive environment. As Roughness measures the state’s degree of non-classicality, its inclusion in the analysis allows to identify points in the dynamics that are not usually perceived by traditional analysis. It is observed that the parameter space is almost completely occupied when the dispersive term is small, and is concentrated in the region of less roughness and less purity as the dispersive coefficient is increased.


Introduction
Ever since its first formulation, Quantum Mechanics has amazed scientists by its counter intuitive results and predictions [1,2]. Starting with the transistor and up to nowadays quantum information and computation, technology development based on this results and predictions lead us to marvelous devices and ways of processing information, such as quantum cryptography [3], generation of random numbers [4], quantum teleportation [6], and quantum computation [5]. At the heart of such applications are quantum effects which would never be obtained by classical methods or devices, these quantum features used to obtain some advantage in a specific task are now recognized as quantum resources [7]. With the increasing demand for new ways to achieve quantum computation, it comes to our attention how can we find useful states and dynamics capable of generating special quantum effects. To help in achieving this goal, one can use quantifiers or indicators of "quantumness", i.e., how physical properties which are strictly quantum features can be sought in a given state and/or dynamics. For this purpose, many quantifiers were proposed for different contexts, for some examples see [8] and references there in. Considering non-locality effects of quantum mechanics, a property well studied, explored and characterized the quantum entanglement between two or more sub-systems. This property has once puzzled Einstein and lead to an important debate between him and Bohr regarding the physical meaning and interpretation of the early developed quantum theory [9]. Quantum entanglement was later carefully studied by Bell which shown, by using inequalities, how quantum mechanics defies our classical interpretation of probabilities and mean values of correlated quantities in this case [10]. Nowadays, we already have quantum devices exploring the entanglement of a 2-part system to deploy a new cryptography protocol immune to eavesdropping [3] and another outstanding application of this non-locality is quantum teleportation [6].
Quantum effects, however, are not limited to the nonlocality of a multi-part system, they can also be found locally, in systems described by a single part. Perhaps one of the most popular of this systems is the Schrödinger cat state, which is a quantum correlated superposition of two coherent states. Although the original purpose of Schrödinger thought experiment was to illustrate a paradox of quantum superpositions, actually there are many experimental realization of these "strange" estate, some examples are the Schrödinger cat states using atom arrays [11,13] and photons [12]. Even when looked locally, statistical measures of physical quantities, like position and momentum, shows non-classical results [14], and the classical limit is achieved with a combination of factors, the action of an environment [14], on the observable choice [15] and on how used measurement process [16] is performed.
For both local and global aspects of quantum mechanics, there are ways to quantify how much the properties of a system have intrinsic quantum or classical features. For measuring entanglement between the parts of a system, there are several functions like Concurrence [17] and other entropic measures [19,20]. For local aspects, we can look into phase space representations given by Wigner and Husimi functions and search for features not found in its corresponding statistical counter part. A quantum phase transition can be identified in term of Wigner function, and quantified by linear entropy [21], and there is also at least one entropic measure based on Wigner function [22]. Historically, it was done by the Negativity of Wigner functions [23], and more recently a measure of quantumness was proposed by the Roughness [24]. Another potential function is the Wigner-Yanase skew information (WYSI) measure [25]. For the purpose of the current work, Roughness was chosen as quantumness indicator since, unlike Negativity and WYSI, its values are limited to the [0, 1] interval and it is able to detect quantum features not detected by Negativity. Furthermore, Roughness can be obtained for any state as a summation of analytical terms.
The objective of this work is to study the relationship between local quantum features and correlations between the parts of a 2-qubit system. The used dynamics is governed by the Jaynes-Cummings model [26], which represents two interacting quantum systems: a two-level atom and a single mode of the radiation field, with a complex but analytically solvable dynamics [27,28]. To start working with it, let us first make a brief review of the used quantifiers.

Linear entropy
For a bipartite system represented by a global density matrix , the reduced density matrix operators A = Tr F and F = Tr A have the information contained in each subsystem separately, accessible through local measurements in each part. In the special case where the quantum information of a pure global state can be completely reconstructed by the information contained in the corresponding reduced density matrix operators, the sum of the parts contains the whole and there is no quantum correlation between them. When it happens, 2 A = A , 2 F = F and: For all other cases, however, the sum of the parts is not equal to the whole, and quantum correlations holds part of (1) the information. A simple way to measure these quantum correlations between parts of a bipartite system is, then, to check how much these reduced density matrix operators have departed from being true projectors. It can be done by the Linear Entropy: so, for a pure with no quantum correlation between its parts, = 0. If, however, there is quantum correlation between subsystems, the partial trace operation in (2) will result in a non-pure A and 0 < < 1.

Visibility, concurrence and predictability
Concurrence is a measure of quantum entanglement. As proposed by Wooters [17], to obtain the Concurrence of a twoqubit system represented by a density state , we initially define a flipped form ̃= ( y ⊗ y ) * ( y ⊗ y ), where * is the complex conjugate of . The concurrence can, then, be obtained by: , written in decreasing order. The Concurrence of a globally pure state can be calculated from its linear entropy by the following relation: This relationship can also work as a generalization of Concurrence for higher dimensional systems [18]. The Visibility of a two level system can be determined by For a global pure state, the Visibility, Concurrence and Predictability obeys the relation [29]. The Concurrence C can be obtained by the relation (6). Recently, this relation was experimentally investigated and there is an evidence of its validity in a bipartite quantum system with superconducting qubits in the IBM Q quantum computer [30].

Roughness
Roughness was proposed by Lemos et al. [24] as indicator of quantum properties in a given system which departs from its classical analog. It uses the phase space representations of Wigner and Husimi functions, being defined for a given density state matrix by: where W (q, p) and H (q, p) are the Wigner and Husimi representations of and the integral is done over the complete phase space. As stated by equation (7), Roughness measures how the two representations, Wigner and Husimi, differ from each other, summing up the absolute value of the difference. The resulting measure is bounded to the interval [0, 1], with R = 0 meaning a completely classical state and R = 1 corresponding to the maximum of "quantumness" achievable for a given state. To illustrate this, we can state some examples: -the Roughness of Fock states increases with their eigenvalues n, with R → 1 as n → ∞,and the fundamental state corresponding to n = 0 having the lesser value of Roughness from this set of states, corresponding to R 0 = 1∕ √ 6.This value for the n = 0 Fock state is also the value of Roughness for any coherent state, as should be expected; -the Roughness of Squeezed States increases with the absolute values of the squeezing parameter; -the Roughness of superposed pure coherent states of the 2 (Schrödinger's Cat States) increases with q 0 ,having the odd superposition greater Roughness if compared with the even one, but both converging to a maximum value of R = √ 7∕12 as q 0 → ∞; -the Roughness of thermal states goes to zero as their corresponding temperature increases; -Thermal and diagonal states have different Roughness.
From the Roughness definition, it captures the non-classicality identified by the volume of the negative part of Wigner function, similarly, the negativity of Wigner function [23], but it is also sensible to other non-classicality the squeezing, while negativity is zero for squeezed states. The squeezed state achieves the maximum R in the limit of infinity squeezing, but it has large values for finite squeezing [24]. Amplitude squeezed light has long been recognized as non-classical, Mandel's Q parameter is an indicator that measures secondorder photon statistical and it is negative for amplitudesqueezed states [31]. Also recently it was demonstrated that squeezed states are a useful resource for quantum computing [32][33][34]. As the negativity of Wigner function is useful to characterize quantum chaos [35] it may have some correlation between Roughness and chaos characterization, which is subject of ongoing work.
Following [24], we can calculate the Roughness of a general state of the form where the various terms within the brackets are results of the evaluation of integrals. According to the "Appendix C" of the referenced article, we have: where X and Y stands for max(n, m) and min(n, m), with the corresponding relations for the primed indexes, u = (X − Y + X � + Y � )∕2, v = (n + m + n � + m � )∕2 and 2 F 1 is the Generalized Hypergeometric Function using the bracketed arguments.

Physical insight of Roughness
As Wigner function has all information about the state and Husimi function can approximately be obtained from classical trajectories [36,37], then Roughness can be interpreted as a distance of the quantum state from the most classical positive quantum distribution. In fact, Roughness is close related to Nearest Hilbert-Schmidt distance [38], which is defined as the minimum distance from a coherent state, the difference is that a Gaussian state is not usually the classical distribution [39][40][41][42][43].

Roughness of a bipartite entangled system
The Roughness, as a measure of non-classicality of a state, is related to the concurrence of a qubit, as has already been observed in the subspace generated by 3 of the Bell states, as shown by [44]. In that work, a generalization of Roughness to measure local quantum features in a system of two qubits represented by a density matrix is defined as: where R 2 ( A,B ) stands for the squared Roughness of the reduced density matrix obtained by tracing over the other part.

The Jaynes-Cummings model
E.T. Jaynes and F. W. Cummings proposed a model (JCmodel), to investigate the interactions of a two level atom and the electromagnetic field using a semi-classical approach [26]. Considering a single, resonating mode of the electromagnetic field, the corresponding Hamiltonian is obtained considering the sum of , representing the number of excitation in the atom and the field, with , an interaction between the atom and the field with coupling parameter g: where † and are creation and annihilation bosonic operators for the field and z , ± are spin-1/2 operators acting on the atom subsystem. The ground, � 0 ⟩, and excited, � 1 ⟩, states of the atom are eigenstates of z . Using these two states to form a basis, z = �1⟩⟨1� − �0⟩⟨0�, + = �1⟩⟨0� and − = �0⟩⟨1�. The Hilbert space used to describe the quantum dynamics of this system is a direct product of the atom and field Hilbert spaces, H = H A ⊗ H F , where H A can be generated by the {� 0 ⟩, � 1 ⟩} basis and H F can be generated by {� n ⟩} ∞ n=0 , the Fock states basis. This way, any initial state therein can be written as: Since commutes with , block diagonalization can be used to obtain eigenstates with definite number of excitations and give the time evolution of the general initial states above: Although analytically solvable, the Jaynes-Cummings model presents rich dynamics. A deeper discussion about the model application and its implementation can be found on [45].

Dynamics in the parameter space
Given the various quantifiers, a graphical analysis can be used for a better understanding of the quantum effects. As long as we are dealing with a bipartite system, we ought to �� C e C n cos(gt correlate the local and global quantum features introduced so far: Roughness, Visibility, Concurrence and Linear Entropy. A related investigation in parameter space was conducted by Arkhipov and collaborators [46] where they have investigated the correlation between negativity quantifying entanglement and the nonclassicality depth. Also there is a extensive study in parameter space investigating quantum steering [47,48]. The time evolution will be given by equation (15) with g = ∕2 and in all cases we will be considering an initial pure 2-qubits state of the form: Since this is a direct product of the atom and the field, it can be thought of as representing a condition where the available global information was formerly known from atom and field separate preparations. Thus, in this case, the initial linear entropy of the system is initially null.
The time evolution of the atom subsystem ( A ) = Tr F A will always has the form: where As the atom interacts with the field, however, different dynamics for the global system will emerge according to how many excitations are avaiable to the global system.

One excitation subspace
For the first case study, consider the initial state obtained by setting = 0 , = 1 and c 1 = c 2 = 1∕ √ 2 in (18): Figure 1 shows the Wigner representation of � 1 ⟩ as a color map.
In this case, the dynamics is restricted to the subspace with less than one excitation, the field states with n > 1 will never get populated and the time evolution of field subsystem can be described by a 2 × 2 square matrix: Also, since the global state is a 2-qubit for the entire dynamics, we shall use the Concurrence as the measure of quantum entanglement. This dynamics is simply periodic with a single factor of , then the Concurrence, Roughness and Visibility are also periodic, as shown in Fig. 2. The relation between these quantities will also be simply periodic, as shown in Fig. 3, generating a simple curve without crossings for each pair. The maximum of Concurrence occurs when t = 1 2 (n + 1) −1 , for n ∈ ℤ, times at which R + is minimum, and R A = R F . Figure 4 shows the Wigner functions of each sub-system for one of this times.
In Fig. 2 one can also see the periodic zeroes of Concurrence, corresponding to the states: A = � 1 ⟩⟨ 1 � or A = � 0 ⟩⟨ 0 �, occurring whenever t = 1 2 n −1 , for n ∈ ℤ. For one excitation state the global state can be considered as a two qubit's system, in this case there is a complementary relation between Roughness and Concurrence [52], this explains why the parameter space for an excitation is limited to the curves presented and does not fill the entire space. In the following section, it will be shown that the same is not true for two or more excitations, as the field state does not represent a qubit.

Two excitation subspace
We consider now a initial state of the form: Figure 5 gives the Wigner representation for � � 2 ⟩ given above.
Though this initial global state is that of a 2-qubits, the atom excitation will eventually be transferred to the field and in this case, one can expect for the Fock basis be populated up to n = 2 by the dynamics. Then, a 3 by 3 density matrix must be used to represent the field degree of freedom: Fig. 1 The atom (left) and field (right) Wigner Functions for � psi 1 ⟩ in equation (21).  The Wigner functions for the atom (left) and field (right) subsystems when t = 0.5 −1 , for the system evolved from the initial state � 1 ⟩ in equation (21) where A m and B m are given by (20). Figure 6 ilustrates the dynamic. The previous, otherwise simply periodic dynamics restricted to a 2-qubits subspace, evolve different now. The two frequencies of oscillation modulate each other and give rise to a more complex dynamics in the parameter space as shown in Figs

Action of a dispersive environment
Considering a phase reservoir [49] the density matrix elements are given by  (14), and Visibility, V, for the initial state � � 2 ⟩ given in equation (23).  where F (t) n,m is given by 24 and A (t) n,m by 19. Figures 15, 16, 17 and 18 show the same as 7 and 8 but for the dispersive dynamics, with Γ = 5 × 10 −4 (A) and Γ = 5 × 10 −3 (B). As Γ grows, the regions of high purity, roughness and high visibility are less visited. The R × parameter space 16 shows that, even with the environment action, a nonpure superposition state is reached by the  (14), and Visibility, V, for the initial state � � 3 ⟩. given in equation (25).  dynamics, it occurs when R ≈ 0.5 and small . As noted, there is a strong correlation between Roughness and entanglement measures, this is a consequence of the fact that Roughness is a distance based on the Wigner function and its Weierstrass transform, Husimi function, and is kown that Wigner function can be used to to visualize atomic entanglement [50].

Conclusion
As Wigner function is a state representation, it contains all information about the particle, and some information about the entanglement between particles, see [50,51] and references there in. The Roughness is a measure that captures the main aspects of the quantum behavior, considering a    distance between Wigner and Husimi function. The success of Roughness is due to the fact that although Husimi is a quantum distribution, it is mainly dominated by the classical dynamics since it can be approximately reproduced by classical trajectories for integrable and no-integrable dynamic [36,37]. It is important to highlight that the information about which pure states are "traversed" by the dynamics is not usually easily accessible, but in the diagram R × or R × C it is clear which are the states, once they are known for their roughness. While the × V diagram shows that maximum Visibility is achieved only for a pure state, from the R × V diagram we know that this state is a coherent superposition of � 0 ⟩ and � 1 ⟩. The effect of the environment destroys this superposition, although that, a non-pure superposition state is obtained with low linear entropy.

Author Contributions
The work was equally divided by the authors.

Funding
The authors gratefully acknowledge the support of the Brazilian Agency Fundação de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG) through Grant No. APQ-01366-16.

Availability of data and material Not applicable.
Code availability (software application or custom code) An opensource code QUTIP was used.