Entanglement and atomic Fisher information of a two qubits and optical field in squeezed thermal state

Abstract

Quantum information technology depends on an invaluable and tenuous resource, named quantum correlation, that shows strong significant manifestation of the coherent overlap of states of a combined quantum systems. So the quantifier of this resource under factual conditions that is, whereas corrupted via generalized optical radiation field states is still limited, and common statements on entanglement dynamics. In this article, we describe quantitatively the nonlocal correlation between a two-qubit and squeezed thermal field. Especially, considering the effect thermal photons and number of photons transition between the two qubits and squeezed thermal field. Also, the Mandel parameter, entanglement and atomic Fisher information over the time evolution as a function of implicated parameters in the system are investigated. We have shown that the squeeze parameter and thermal photons have a potential effect of the dynamical properties of the atomic Fisher information, Mandel parameter and entanglement. Also, the AFI is very sensitive to the field parameters rather than the number of photons transition. Furthermore, the results interpreted in the case of thermal environment considering the effect of thermal photons and squeeze parameter on the evolution of the system under consideration.

Keywords

Atomic Fisher information entanglement squeezed thermal field concurrence Mandel parameter.

1 Introduction

Thermal and coherent states are Gaussian states, while the displaced thermal state is a way of incorporating thermal noise, and the resultant state is still Gaussian. Mixing of thermal and coherent states is another way of including thermal noise, however, the resultant state is non-Gaussian. Also, the displaced thermal state is a mixture of photon-added coherent states of all orders. This has potentially important applications for optical communications and quantum teleportation. So, some efforts have been devoted to exploring theoretically the different effects of thermal radiation at room temperature, usually ignored, is intense for these. Different investigations of the squeezed thermal field (STF) have been done, for example its nonclassically [1 –3], quantitative measurement of entanglement and fidelity [4, 5], examination of phase estimation [6]. Moreover, the time-evolution of photon-number distribution and density operator of the STF [7]. Also, the photon system in the black body filled in its interior by a Kerr-nonlinear crystal is in a STF has been discussed [8]. Recently, the STF has an important effects in quantum optics [9] and therefore these effects should have extensive application to fields of technology and information science.

As known that the nonlocal correlation (NLC) between two quantum systems could be quantified through statistical quantities like the entanglement of formation (EOF) or the Von Neumann (VNE) [10]. All the NLC measures depends the transformations under Local Operation Classical Communication (LOCC). On the other hand, the NLC of complex physical system cannot be measured through the basic measure like the VNE. Hence, many tries have been tried in order find out a new quantifier of entanglement of a quantum system. On this regard, the atomic Wehrl entropy (AWE) [11, 12] and the atomic Fisher information (AFI) [13, 14] have been utilized to assess the entanglement and contrasted by the standard VNE. The relationship between the act of AFI and AWE has been explored [13]. The next of the statistical quantities or information quantifier relies on the first product. In this way, it has appeared that the flow of FI can be utilized to discover the Markovian and non Markovian dynamics of a two-level system (2LS) moved by using a phase noise laser [15]. Furthermore, the QFI has been utilized to discover the link between the non-Markovian act of the system-medium in the presence phase noise laser [16]. Finally, entropy and FI has wildly used in the some fields of computer sciences such as image processing [17 –19].

This article proposes the statistical properties of the STF and nonlocal correlation between the TQs and STF measured by the concurrence, in addition to the nonlocal correlation between the TQs-STF system identified by the VNE. The effects of the squeezed parameter and mean thermal photon on the statistical properties of the STF and dynamical behavior on the two kinds of nonlocal correlation will be examined.

2 Interaction between TQs and STF

The TQs-STF interaction is significant type in quantum information processing (QIP). The proposed system of a TQs and STF is given by the Hamiltonian [20]. $H_{I} = \sum_{j = 1}^{2} ξ_{j} ({\hat{ψ}}^{+ k} | g_{j} 〉〈 e_{j} | + {\hat{ψ}}^{k} | e_{j} 〉〈 g_{j} |)$ (1) where the STF is described by the creation (annihilation) operator ${\hat{ψ}}^{+} (\hat{ψ})$ , k is the number of photons transition between the STF and TQs. The upper (lower) states for the j-th qubit is donated by | e_j 〉 (| g_j 〉). Also, ξ_j is the coupling constant between j-th qubit and STF, we pay the attention to an identical TQs case (i.e. ξ₁ = ξ₂ = ξ).

Next, we consider an initial density operator is given by $ρ (0) = | β_{A_{1} A_{2}} (0) 〉〉 β_{A_{1} A_{2}} (0) | \otimes ρ_{STF} (0),$ where |β_{A
₁
A
₂} (0)〉 is the initial state of the TQs A₁, A₂ and ρ_STF (0) is the initial density operator of the STF. The initial input state of the TQs are set to be a superposition states as $| β_{A_{1} A_{2}} (0) 〉 = \frac{1}{\sqrt{2}} (| e_{1} e_{2} 〉 + e^{i φ} | g_{1} g_{2} 〉$ (2) where φ is the relative phase between the upper and lower states of TQs, while the STF is defined by the density matrix $ρ_{STF} (0) = \sum_{n = 1}^{2} q_{n} | {\bar{n}}_{th}, r 〉〉 {\bar{n}}_{th}, r |,$ where q_n is given by [21, 22] $q_{n} = \frac{a}{u^{(m + 1)}} P_{m} (\frac{1 + 2 b - a^{2}}{\sqrt{{(1 + 2 b - a^{2})}^{2} - 4 b^{2}}})$ (3) where P_m is Legendre polynomial in terms of the squeeze parameter r and average thermal photon number ${\bar{n}}_{th} = tr (ρ (t) {\hat{ψ}}^{+} \hat{ψ})$ defined as $a = \sqrt{\frac{4 gh}{(2 g - 1) (2 h - 1)}}, b = \frac{h - g}{2 (2 g - 1) (2 h - 1)}$ (4) and

$g = {(2 {\bar{n}}_{th} + 1) e^{2 r} + 1}^{- 1}, h = {(2 {\bar{n}}_{th} + 1) e^{- 2 r} + 1}^{- 1}$ (5)

The final state for the system of the STF and TQs perceives from $ρ (t) = exp (- i {\hat{H}}_{I} t) ρ_{STF} (0) exp (i {\hat{H}}_{I} t)$ (6) the atomic (STF) density matrix is obtained by taking the trace over the STF (atomic) basis, ρ_{A₁A₂[STF]} (t) = tr_{STF[A₁A₂]}{ ρ (t) }. In the following section, we discuss the dynamical behavior of the AFI, NLC between the TQs measured by the concurrence and Mandel parameter as statistical quantifier of the STF.

3 Nonlocal correlation

The NLC plays a central role in QIP and have different applications in the interaction between bipartite system [23 –26]. The VNE S_{A
₁
A
₂} (t) is the essential measure for TQs-STF entanglement. In our case, we use the VNE to measure the NLC between the TQs and STF which has been given in terms of the eigenvalues χ_j of the atomic density matrix ρ_{A
₁
A
₂}, i.e., [27 –30]

$\begin{matrix} S_{A_{1} A_{2}} (t) & = - Tr (ρ_{A_{1} A_{2}} (t) ln ρ_{A_{1} A_{2}} (t)) \\ = - \sum_{j = 1}^{4} χ_{j} (t) ln χ_{j} (t) . \end{matrix}$ (7)

As is well-known, quantum correlations, especially entanglement, have a potential applications in information sciences, quantum algorithms and fuzzy systems [31 –35]. In this article, we utilize concurrence to measure the qubit-qubit nonlocal correlation or entanglement. In terms of the reduced density matrix for TQs A₁ and A₂. Concurrence has been defined by [36 –39] $C_{A_{1} A_{2}} (t) = max {0, μ_{1} - μ_{2} - μ_{3} - μ_{4}},$ (8) where μ_j (j = 1, 2, 3, 4) are the eigenvalues of the square roots of the density matrix $R = ρ_{A_{1} A_{2}} (σ_{y} \otimes σ_{y}) ρ_{A_{1} A_{2}}^{*} (σ_{y} \otimes σ_{y})$ and σ_y is the Pauli matrix. $ρ_{A_{1} A_{2}}^{*}$ is the complex conjugate of ρ_{A
₁
A
₂}. Concurrence is zero, i.e., C_{A
₁
A
₂} (t) = 0, for un-entangled TQs, whereas C_{A
₁
A
₂} (t) = 1, for maximally entangled TQs.

In Fig. 1, the temporal variations of the qubit-qubit entanglement measured by the concurrence C_{A
₁
A
₂}, is shown for various values of the number of photons transition k and mean thermal photons number ${\bar{n}}_{th}$ . Generally, the concurrence presents a periodic behavior exhibiting a suddendeath and suddenbrith of entanglement. Figure 1(a), displays the concurrence in the case of weak and strong STF donated by the solid and dashed line for one photon transition. The dynamics of C_{A
₁
A
₂} shows significant changes as the value of ${\bar{n}}_{th}$ increases leading to decreases of the entanglement during the time evolution. In Fig. 1(b), evolution profiles of concurrenceof the initial field state is a STF in the case of two photon processes k = 2, are shown. The effects of the increasing of the number of photons transition on the dynamics of C_{A
₁
A
₂} is clear on comparing the corresponding profiles is Fig. 1(a) and 1(b). The two photons effect shows significant change leading to increasing the periodicity peaks and refresh the TQs-STF entanglement. The concurrence explored a periodic time similar to that in the case of weak thermal field, with more decreasing in the case of strong STF. This indicates that the suddendeath and suddenbrith phenomena can be controlled through the thermal and number of photons transition effect which the amount of entanglement between the TQs can be obtained in the case weak mean thermal photonnumber.

Fig.1

The dynamics of the concurrence C_{A
₁
A
₂} as a function of the dimensionless time ξt with a TQs interacting with a STF: Fig. (a) one photon transition k = 1, squeeze parameter r = 0.1 and average thermal photons ${\bar{n}}_{th} = 0.1$ for solid line and ${\bar{n}}_{th} = 2$ for dashed line, Fig. (b) is the same as Fig. (a) but for two photon transitions k = 2.

Figure 2, depicts the temporal variations of the TQs-STF entanglement measured by the VNE S_{A
₁
A
₂}, is shown in terms of k and ${\bar{n}}_{th}$ . Generally, the VNE presents a periodic and regular behavior in the case of one and two photon processes for the weak STF ( ${\bar{n}}_{th} = 0.1$ ). Interestingly the TQs-STF entanglement is clearly enhanced in the case of strong STF which is an opposite results with the qubit-qubit entanglement (compare with Fig. 1).

Fig.2

The time evolution of the VNE for the same parameter presented in Fig. 1.

4 AFI and statistical properties

4.1 Atomic Fisher information

Here, we propose the relation between the AFI and qubit-qubit entanglement compared by the NLC between the TQs-STF. The classical FI measures any distribution function G_θ (t), in terms of a stochastic variable y and an estimator parameter θ and is defined as [40]. $I_{F} (θ) = \int G_{θ} (y) {(\frac{\partial \ln (G_{θ} (y))}{\partial θ})}^{2} dy$ (9)

In terms of the atomic phase space parameters ϑ and φ the atomic quasi-probability function ϖ_AQF is given as [11, 41]. $ϖ_{AQF} = \frac{1}{2 π} {tr}_{A_{1} A_{2} {ρ_{A_{1} A_{2}} (t) | ϑ, φ ϑ, φ |}},$ (10) where |ϑ, φ〉 is the atomic coherent state (ACS) for a TQs, (i.e. j = 3/2) is defined as

$| ϑ, φ 〉 = \sum_{m = 0}^{2 j} \sqrt{(\begin{matrix} 2 k \\ m \end{matrix})} cos {(\frac{ϑ}{2})}^{m} {sin (\frac{φ}{2}) e^{- i φ}}^{2 j - m} | m 〉$ (11)

The AFI for the atomic state estimation as a quantifier of the system dynamics [13] $I_{AF} (t) = \sum_{j = 1}^{2} \int_{0}^{2 π} \int_{0}^{π} {(D_{α_{j}} \frac{\partial ϖ_{AQF}}{\partial η_{j}})}^{2} \frac{sin (ϑ)}{ϖ_{AQF}} d ϑ d φ$ (12) where D_{α
_j} the variances for the variables ϑ, φ and (α₁, α₂) = (ϑ, φ) $D_{α_{j}} (t) = \int_{0}^{2 π} \int_{0}^{π} α_{j} ϖ_{AQF} sin (ϑ) d ϑ d φ$ (13)

In Fig. 3 the temporal variation of the AFI for various values of the squeezed parameter r and the thermal number n_th. The effect of thermal photons is very similar to the case of qubit-qubit entanglement measured by the concurrence. Generally, the AFI exhibits a periodic behavior tending to its a minimal values for a strong STF for both of one and two photon transition see Refs [13, 14]. Thus, the TQs-STF state becomes minimal mixed state for a strong STF.

Fig.3

The time evolution of the AFI for the same parameter presented in Fig. 1.

4.2 Mandel parameter

The statistical properties of a STF through the interaction time could be quantified using the Mandel parameter [42, 43]. $P_{M} = \frac{〈 N^{2} 〉 - 〈 N^{2} 〉}{〈 N 〉} - 1,$ (14) where 〈N²〉 - 〈 N² 〉 is the variance and $〈 N 〉 = {tr}_{STF} {{\hat{ψ}}^{†} \hat{ψ}}$ . The Mandel parameter is an indicator for the photon distribution of the field is sub-Poissonian (1 ⩽ P_M ⩽ 0), which depicts the non-classical case, or super-Poissonian (P_M > 0). Lastly, the field distribution is Poissonian if 〈N²〉 = 〈N²〉 + 〈N〉, and depicts semi-classical states.

Figure 4 show the dynamics of the MP as a quantifier for the STF of the proposed system under consideration. The MP oscillates between the values 0 and 1 in aperiodic and regular behavior which clarify that the STF changes from Poissonian and sub-Poissonian during the time evolution in the case of weak STF for k = 1, 2 . (see Fig. 4 solid line). For the strong STF is consider for n_th = 2 (dashed line) the STF changes form super-Poissonian and sub-Poissonian in a chaotic manner. This results emphasize that the statistical properties of the field is depends of the field parameters rather the number of photon transition.

Fig.4

The time evolution of the Mandel parameter for the same parameter presented in Fig. 1.

5 Conclusion

The notion of the concurrence has been shown for a high precise and educational for depicting the time development of entanglement of a quantum information system. We have investigated and discussed the link between the qubit-qubit entanglement and statistical properties of the STF and AFI for a TQs for one and two photon transition in the case of weak and strong STF. During the time evolution of the AFI and concurrence show various order relying on the squeeze parameter and thermal photon number. At suitable value of the average thermal photons and squeeze parameter a high amount of quantum entanglement can be obtained for different ranges of the dimensionless time. The dynamics of the concurrence and the AFI are studied in low and high squeezing and thermal regimes. Furthermore, we found that the TQs-STF nonlocal correlation depends on the STF parameters and number of photons transition while the statistical properties of the field in only depends on field parameters. Also, the AFI has the ability to quantify the qubit-qubit entanglement rather than the entanglement between the STF and TQs.

Footnotes

Acknowledgments

This work was funded by Deanship of Scientific Research at Princess Nourah bint Abdulrahman University. (Grant No. 256-S-39).

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