Hierarchy of temporal quantum correlations using a correlated spontaneous emission laser

Shakir Ullah; Haleema Sadia Qureshi; Fazal Ghafoor

doi:10.1364/OE.27.026858

1. Introduction

Since last century, quantum theory has been amazingly successful due to its explanatory power which enabled us to propose many new fields in technology. The exploitation of the quantum laws in quantum information processing is now predicting many applications in secured communication and information processing [1]. After the local-realism theory of Einstein, Podolsky, and Rosen (EPR) [2], various remarkable attempts on quantum systems have been made to characterize and quantify quantum correlations, for instance, Bell non-locality, quantum steering, quantum entanglement, quantum discord, and quantum coherence [3–20].

Quantum entanglement describes the non-classical property of composite quantum systems. The generation of entangled states by means of various laser-based systems have been studied in a wide range [21–32]. The entanglement phenomenon is now shown to be a basic ingredient for successful implementation of various tasks of quantum information protocols [33, 34]. In response to the non-classical correlation addressed in the famous paper [2] by EPR, Schrodinger [35] defined the phenomenon of quantum steering between the two spatially separated parties shared an entangled state. According to this definition, the steering allows one party of the entangled state to perform a measurement on their side to influence the set of parameters of the other party. To quantify such steering effect in the discrete- and continuous- variable quantum systems [10, 19, 36–42] various mathematical inequalities have been theorized and implemented. Quantum steering has various practical applications in quantum information processing such as one-side device-independent quantum key distribution, sub-channel discrimination and others [11, 43–45]. Among theoretical and experimental researchers, the Bell inequality [46], has been a great topic of interest to understand physics of the local and non-local correlations. This inequality can be formulated for the measurement of EPR state of a composite system in terms of Wigner function by measuring the joint probability of the classical correlations. On contrary, the correlations become non-classical when the system is subjected to the measurement of quantum mechanical parity [47]. In this connection, a scheme has been proposed to test quantum non-locality for the field-state of two spatially separated cavities using parity measurements and displacement operation [48]. For a two-mode squeezed state, the time evolution of non-locality in a thermal environment is also reported [49, 50]. Moreover, other types of Bell-like inequalities have been estimated for investigation of the non-local characteristics of composite quantum systems [4, 51]. On the other side, disentangled (separable) states traditionally assigned to classically correlated states may also exhibit a quantum behavior with useful applications to quantum information science [52–55]. One such behavior for a composite quantum system is known as quantum discord [8, 15, 56–58]. Such measurements provide us with the amount of mutual information and account almost all the quantum correlations of multipartite systems that is not locally accessible. Quantum discord has been widely studied in many systems such as spins in quantum dots [59], cavity quantum electrodynamics [60–62] and quantum spin chains [63–65].

In quantum information theory, Gaussian states are the special forms of continuous variable states which offer many significant advantages among the other bipartite states. For example, the properties of symmetrical [66] and arbitrary type [67] of Gaussian quantum states have been studied in detail. The Gaussian type continuous variable states have also been demonstrated experimentally using techniques of two-mode squeezing effect [33], optical parametric oscillator [68], and parametric down converter [69]. Quantifying and characterizing the quantum correlations of such continuous variable [10, 33, 57, 58, 70] and discrete variable [4, 9, 14, 15] states have been studied. Recently, we reported a relationship among the four kinds of quantum correlations using a linear beam splitter [20] which satisfy a strict hierarchy relation: quantum discord $\supseteq$ quantum entanglement $\supseteq$ quantum steering $\supseteq$ Bell non-locality, where $X \supseteq Y$ represents X is the superset of Y. We note that active quantum systems (such as lasers [21–31]), as compared to the passive systems (such as linear beam splitters [19, 20, 71–77]) are mainly operative rather than to be supportive to some task of quantum information processing. The quantum correlations due to the former systems are more useful and interesting from application point of view [4, 6, 8, 33, 34]. From perspective of viable applications of the quantum correlations, quantification of their properties with respect to time become valuable [3–8, 21, 25, 78]. The question arise: are the various kinds of the quantum correlations with an active quantum system satisfy the hierarchy relation with reference to the time of their evolution?

This question is explicitly addressed in this paper to study the properties of the various temporal quantum correlations while using correlated spontaneous emission laser (CEL) as an active quantum system contains three-level atoms which interacts with two modes of the cavity field. The time evolution of Bell non-locality, quantum steering, quantum entanglement, and quantum discord is studied for the correlated lasing modes of the laser employing general Gaussian states of continuous variables for the initial two-mode cavity field [20]. Generally, the quantum correlations between the output two-mode Gaussian state (TMGS) of the cavity field depend on Rabi frequency of the coupling fields and on the cavity damping rates. Specifically, the time evolution of quantum discord and entanglement become independent (dependent) of the purity (non-classicality) of the initial states of the cavity field. Nonetheless, both the non-classicality and purity, in the latter case, influence the quantum steering and non-locality. Consequently, quantum non-locality leads steerability and quantum steering shows entanglement in the correlated field of the laser cavity. The reverse, however, does not exhibit always. Although, quantum entanglement always reveals discord, the latter may not necessarily shows the former. Thus, the boundaries of the four kinds of the temporal quantum correlations satisfy a strict hierarchy even in the presence of the cavity damping of the laser system.

Fig. 1 (a) Schematic of correlated emission laser. A laser cavity with the atomic medium, a driving field and two single modes of the cavity field. (b) The model atom of the cavity medium. The two single modes of the cavity field are resonant with levels $| a 〉 - | b 〉$ and $| b 〉 - | c 〉$ , respectively. The dipole forbidden transition between the ground state $| a 〉$ and the upper excited state $| c 〉$ is induced by a strong classical field.

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2. The model and dynamics of the system

We consider a three-level atomic system in cascade configuration with energy levels $| a 〉$ , $| b 〉$ and $| c 〉$ which is placed inside a doubly resonant cavity as shown in Fig. 1. Initially, the atom is prepared in coherent superposition of the ground state $| a 〉$ and the upper excited state $| c 〉$ . The transition between the ground state $| a 〉$ and excited state $| c 〉$ is dipole forbidden which is induced by a strong classical field i.e. a magnetic field for a magnetic dipole allowed transition. The Rabi frequency of this induced field is represented by $Ω e^{- i φ}$ . The transitions between the states $| a 〉$ – $| b 〉$ and states $| b 〉$ – $| c 〉$ are dipole allowed which are coupled by quantized cavity modes with carrier frequencies υ₁ and υ₂, respectively. The annihilation (creation) operators associated with these quantized cavity modes are denoted by $a_{1} (a_{1}^{†})$ and $a_{2} (a_{2}^{†})$ , respectively. We assume that the atoms are injected at a rate r_a into the lower level $| a 〉$ through a pumping process. The interaction picture Hamiltonian of the system in the electric dipole and rotating wave approximation can be written as

V = ℏ (g_{1} a_{1} | a 〉 b | + g_{2} a_{2} | b 〉 c |) - \frac{ℏ}{2} Ω e^{- i φ} | a 〉 c | + H . c .,

where g₁ and g₂ appeared as constants for the coupling strength of the cavity modes with their corresponding atomic transitions. Next, we explicitly calculate the dynamics of the CEL system by following the standard methods of laser theory [79]. The equation of motion for the reduced density operator of the two-mode cavity field, without including the cavity loss terms [80], can be obtained by the following equation

\begin{matrix} {\dot{ρ}}_{f} = - \frac{i}{ℏ} T r_{a t o m} [V, ρ], \\ = - i g_{1} (a_{1}^{†} ρ_{a b} - ρ_{a b} a_{1}^{†}) - i g_{2} (a_{2}^{†} ρ_{b c} - ρ_{b c} a_{2}^{†}) + H . c . \end{matrix}

In Eq. (2), ρ stands for the atom-field density operator, with $ρ_{a b} = 〈 a | ρ | b 〉$ and $ρ_{b c} = 〈 b | ρ | c 〉$ . The equation of motion for the relevant density matrix elements can generally be obtained using the following equation [79]

{\dot{ρ}}_{i j} = - \frac{i}{ℏ} \sum_{k} (V_{i k} ρ_{k j} - ρ_{i k} V_{k j}) - \frac{1}{2} \sum_{k} (Γ_{i k} ρ_{k j} + ρ_{i k} Γ_{k j})

where the last term represents the decay of the atomic levels due to spontaneous emission, which is added phenomenologically. The equations of motion for the density matrix elements

ρ_{a b}

and

ρ_{b c}

to the first order in the cavity field coupling constants g₁ and

g_{2}

can be obtained using Eq. (3)

{\dot{ρ}}_{a b} = - γ ρ_{a b} + i \frac{Ω}{2} e^{- i φ} ρ_{c b} - i g_{1} (a_{1} ρ_{b b} - ρ_{a a} a_{1}) + i g_{2} ρ_{a c} a_{2}^{†},

{\dot{ρ}}_{b c} = - γ ρ_{b c} - i \frac{Ω}{2} e^{- i φ} ρ_{b a} - i g_{2} (a_{2} ρ_{c c} - ρ_{b b} a_{2}) - i g_{1} a_{1}^{†} ρ_{a c},

where, for the sake of simplicity, we have assumed the atomic decay rates γ is same for the two upper atomic levels [21–31]. The zeroth-order equations of motion for the corresponding relevant matrix element operators can be evaluated as

{\dot{ρ}}_{a a} = - γ ρ_{a a} + i \frac{Ω}{2} (e^{- i φ} ρ_{c a} - e^{i φ} ρ_{a c}) + r_{a} ρ_{f},

{\dot{ρ}}_{b b} = 0,

{\dot{ρ}}_{c c} = - γ ρ_{c c} + i \frac{Ω}{2} (e^{i φ} ρ_{a c} - e^{- i φ} ρ_{c a}),

{\dot{ρ}}_{a c} = - γ ρ_{a c} - i \frac{Ω}{2} e^{i φ} (ρ_{a a} - ρ_{c c}),

where we have considered that the atoms are injected at a rate r_a into the lower level

| a 〉

through a pumping process. On inserting the steady-state solutions of Eqs. (6)-(9) in Eqs. (4) and (5), and integrating these equations from

- \infty

to t we get the density matrix elements

ρ_{a b}

and

ρ_{b c}

. The final expressions are then substituting into Eq. (2), we obtain the reduced density operator

{\dot{ρ}}_{f}

of the cavity field, without cavity loss terms, as [79]

\begin{matrix} {\dot{ρ}}_{f} = [(β_{11} + β_{11}^{*}) a_{1}^{†} ρ_{f} a_{1} - β_{11}^{*} a_{1} a_{1}^{†} ρ_{f} - β_{11} ρ_{f} a_{1} a_{1}^{†}] \\ + [(β_{22} + β_{22}^{*}) a_{2} ρ_{f} a_{2}^{†} - β_{22}^{*} a_{2}^{†} a_{2} ρ_{f} - β_{22} ρ_{f} a_{2}^{†} a_{2}] \\ + [(β_{21} + β_{12}^{*}) a_{2} ρ_{f} a_{1} - β_{12}^{*} a_{1} a_{2} ρ_{f} - β_{21} ρ_{f} a_{1} a_{2}] e^{i φ} \\ + [(β_{21}^{*} + β_{12}) a_{1}^{†} ρ_{f} a_{2}^{†} - β_{21}^{*} a_{1}^{†} a_{2}^{†} ρ_{f} - β_{12} ρ_{f} a_{1}^{†} a_{2}^{†}] e^{- i φ} \end{matrix}

with

β_{11} = \frac{3 g_{1}^{2} r_{a} Ω^{2}}{4 (Ω^{2} + γ^{2}) (γ^{2} + Ω^{2} / 4)},

β_{22} = \frac{g_{2}^{2} r_{a}}{(Ω^{2} + γ^{2})},

β_{21} = \frac{i g_{1} g_{2} r_{a} Ω (Ω^{2} - 2 γ^{2})}{4 γ (Ω^{2} + γ^{2}) (γ^{2} + Ω^{2} / 4)},

β_{12} = \frac{i g_{1} g_{2} r_{a} Ω}{γ (Ω^{2} + γ^{2})},

In the above, the phase independent coefficients β₁₁ and β₂₂, respectively, correspond to the gain and loss processes of the laser system. However, the phase dependent coefficients β₁₂ and β₂₁ with the phase φ, correspond to the coherence developed by the classical driving field between atomic levels $| a 〉$ and $| c 〉$ . In Eq. (10), the classical driving field is kept to all orders in the Rabi frequency Ω while the two cavity fields are taken up to the second order in the coupling constants g₁ and g₂. This assumption is made when the system operates in the linear regime. It is justified if the Rabi frequency associated with classical field is greater than the coupling constants g₁ and g₂ of the cavity fields.

The terms due to cavity losses in the density matrix equation can be expressed as [80]

{\dot{ρ}}_{f} = \sum_{i = 1}^{2} κ_{i} ([a_{i} ρ_{f}, a_{i}^{†}] + [a_{i}, ρ_{f} a_{i}^{†}]),

where

κ_{1, 2}

are the cavity decay rates associated with cavity modes 1 and 2, respectively. These cavity losses are taken into account due to the coupling of the two cavity modes with vacuum reservoir. After simplifying Eq. (15), we get

\begin{matrix} {\dot{ρ}}_{f} = - κ_{1} (a_{1}^{†} a_{1} ρ_{f} + ρ_{f} a_{1}^{†} a_{1} - 2 a_{1} ρ_{f} a_{1}^{†}) \\ - κ_{2} (a_{2}^{†} a_{2} ρ_{f} + ρ_{f} a_{2}^{†} a_{2} - 2 a_{2} ρ_{f} a_{2}^{†}) . \end{matrix}

We can obtained the total reduced density operator ${\dot{ρ}}_{f}$ of the two-mode cavity field upon adding Eq. (16) with Eq. (10)

\begin{matrix} {\dot{ρ}}_{f} = [(β_{11} + β_{11}^{*}) a_{1}^{†} ρ_{f} a_{1} - β_{11}^{*} a_{1} a_{1}^{†} ρ_{f} - β_{11} ρ_{f} a_{1} a_{1}^{†}] \\ + [(β_{22} + β_{22}^{*}) a_{2} ρ_{f} a_{2}^{†} - β_{22}^{*} a_{2}^{†} a_{2} ρ_{f} - β_{22} ρ_{f} a_{2}^{†} a_{2}] \\ + [(β_{21} + β_{12}^{*}) a_{2} ρ_{f} a_{1} - β_{12}^{*} a_{1} a_{2} ρ_{f} - β_{21} ρ_{f} a_{1} a_{2}] e^{i φ} \\ + [(β_{21}^{*} + β_{12}) a_{1}^{†} ρ_{f} a_{2}^{†} - β_{21}^{*} a_{1}^{†} a_{2}^{†} ρ_{f} - β_{12} ρ_{f} a_{1}^{†} a_{2}^{†}] e^{- i φ} \\ - κ_{1} (a_{1}^{†} a_{1} ρ_{f} + ρ_{f} a_{1}^{†} a_{1} - 2 a_{1} ρ_{f} a_{1}^{†}) \\ - κ_{2} (a_{2}^{†} a_{2} ρ_{f} + ρ_{f} a_{2}^{†} a_{2} - 2 a_{2} ρ_{f} a_{2}^{†}) . \end{matrix}

The dynamical equations for the expectation values of the photon number and the two-photon correlation can be readily calculated by using Eq. (17) with the help of

\frac{d}{d t} 〈 \hat{O} 〉 = T r [\hat{O} {\dot{ρ}}_{f}] .

Hence, the various moments are given by

\frac{d}{d t} 〈 a_{1}^{†} a_{1} 〉 = [(β_{11} + β_{11}^{*}) - 2 κ_{1}] 〈 a_{1}^{†} a_{1} 〉 + β_{12}^{*} e^{i φ} 〈 a_{1} a_{2} 〉 + β_{12} e^{- i φ} 〈 a_{1}^{†} a_{2}^{†} 〉 + (β_{11} + β_{11}^{*}),

\frac{d}{d t} 〈 a_{2}^{†} a_{2} 〉 = - [(β_{22} + β_{22}^{*}) + 2 κ_{2}] 〈 a_{2}^{†} a_{2} 〉 - β_{21} e^{i φ} 〈 a_{1} a_{2} 〉 - β_{21}^{*} e^{- i φ} 〈 a_{1}^{†} a_{2}^{†} 〉,

\frac{d}{d t} 〈 a_{1} a_{2} 〉 = [(β_{11} - β_{22}^{*}) - (κ_{1} + κ_{2})] 〈 a_{1} a_{2} 〉 - β_{21}^{*} e^{- i φ} - β_{21}^{*} e^{- i φ} 〈 a_{1}^{†} a_{1} 〉 + β_{12} e^{- i φ} 〈 a_{2}^{†} a_{2} 〉,

\frac{d}{d t} 〈 a_{1}^{†} a_{2}^{†} 〉 = [(β_{11}^{*} - β_{22}) - (κ_{1} + κ_{2})] 〈 a_{1}^{†} a_{2}^{†} 〉 - β_{21} e^{i φ} - β_{21} e^{i φ} 〈 a_{1}^{†} a_{1} 〉 + β_{12}^{*} e^{i φ} 〈 a_{2}^{†} a_{2} 〉,

\frac{d}{d t} 〈 a_{1} a_{2}^{†} 〉 = [(β_{11} - β_{22}) - (κ_{1} + κ_{2})] 〈 a_{1} a_{2}^{†} 〉 - β_{21} e^{i φ} 〈 a_{1} a_{1} 〉 + β_{12} e^{- i φ} 〈 a_{2}^{†} a_{2}^{†} 〉,

\frac{d}{d t} 〈 a_{1} a_{1} 〉 = 2 (β_{11} - κ_{1}) 〈 a_{1} a_{1} 〉 + 2 β_{12} e^{- i φ} 〈 a_{1} a_{2}^{†} 〉,

\frac{d}{d t} 〈 a_{2}^{†} a_{2}^{†} 〉 = - 2 (β_{22} + κ_{2}) 〈 a_{2} a_{2} 〉 - 2 β_{21} e^{i φ} 〈 a_{1} a_{2}^{†} 〉 .

The four kinds of quantum correlations of the resulting TMGS of the cavity field can be investigated by solving these equations numerically. However, in the strongly driven limit when $Ω ≫ γ$ , we have $β_{11} \approx β_{22} \approx 0$ and $β_{12} \approx β_{21} \approx i g_{1} g_{2} r_{a} / Ω γ .$ The approximated master equation is given by

\begin{matrix} {\dot{ρ}}_{f} = i α [(a_{1} a_{2} ρ_{f} - a_{2} ρ_{f} a_{1}) e^{i φ} + (a_{1}^{†} a_{2}^{†} ρ_{f} - a_{1}^{†} ρ_{f} a_{2}^{†}) e^{- i φ} \\ - (ρ_{f} a_{1} a_{2} - a_{2} ρ_{f} a_{1}) e^{i φ} - (ρ_{f} a_{1}^{†} a_{2}^{†} - a_{1}^{†} ρ_{f} a_{2}^{†}) e^{- i φ}] \\ - κ_{1} (a_{1}^{†} a_{1} ρ_{f} + ρ_{f} a_{1}^{†} a_{1} - 2 a_{1} ρ_{f} a_{1}^{†}) \\ - κ_{2} (a_{2}^{†} a_{2} ρ_{f} + ρ_{f} a_{2}^{†} a_{2} - 2 a_{2} ρ_{f} a_{2}^{†}), \end{matrix}

where

α = g_{1} g_{2} r_{a} / Ω γ

. In this approximation, the master Eq. (26) indicates a parametric oscillator[81]. Hence, the elements of the output covariance matrix of Eq. (38) are obtained by solving Eqs. (19)-(25) analytically when

Ω ≫ γ

and for simplicity we take

κ_{1} = κ_{2} = κ

and

g_{1} = g_{2} = g

[29, 30],

〈 a_{1}^{†} a_{1} 〉 = \frac{e^{- 2 κ t}}{2} [2 α sinh (2 α t) + (c_{1} + c_{2} - 1) cosh (2 α t) - (c_{2} - c_{1})],

〈 a_{2}^{†} a_{2} 〉 = \frac{e^{- 2 κ t}}{2} [2 α sinh (2 α t) + (c_{1} + c_{2} - 1) cosh (2 α t) + (c_{2} - c_{1})],

〈 a_{1} a_{2} 〉 = i \frac{e^{- 2 κ t - i φ}}{2} [2 α cosh (2 α t) + (c_{1} + c_{2} - 1) sinh (2 α t)],

〈 a_{1}^{†} a_{2}^{†} 〉 = - i \frac{e^{- 2 κ t + i φ}}{2} [2 α cosh (2 α t) + (c_{1} + c_{2} - 1) sinh (2 α t)],

〈 a_{1} a_{1} 〉 = \frac{e^{- 2 κ t}}{2} [(| d_{2} | e^{i (ϕ_{2} - 2 φ)} - | d_{1} | e^{i ϕ_{1}}) cosh (2 α t) - | d_{1} | e^{i ϕ_{1}} - | d_{2} | e^{i (ϕ_{2} - 2 φ)}],

〈 a_{2}^{†} a_{2}^{†} 〉 = \frac{e^{- 2 κ t}}{2} [(| d_{1} | e^{i (ϕ_{1} + 2 φ)} - | d_{2} | e^{i ϕ_{2}}) cosh (2 α t) - | d_{1} | e^{i (ϕ_{1} - 2 ϕ)} - | d_{2} | e^{i ϕ_{2}}],

〈 a_{1} a_{2}^{†} 〉 = i \frac{e^{- 2 κ t}}{2} [(| d_{1} | e^{i (ϕ_{1} + φ)} - | d_{2} | e^{i (ϕ_{2} - φ)}) sinh (2 α t)],

where c_j and d_j (

j = 1, 2

) are defined in the following section.

3. Input cavity field

We choose the initial modes of the cavity field as a product of two single-mode Gaussian states $ρ_{1} \otimes ρ_{2}$ . The characteristic function for the two separable states is given by

χ (ξ_{1}, ξ_{2}) = exp (- \frac{1}{2} ξ^{†} V_{i n} ξ),

where

ξ^{†} = (ξ_{1}^{*}, ξ_{1}, ξ_{2}^{*}, ξ_{2})

is the row matrix of complex amplitudes, and

V_{i n} = (\begin{matrix} V_{1} & 0 \\ 0 & V_{2} \end{matrix}),

is the covariance matrix for the two single-mode Gaussian states of the initial cavity field. In matrix V_in,

V_{1} = (\begin{matrix} c_{1} & d_{1} \\ d_{1}^{*} & c_{1} \end{matrix})

and

V_{2} = (\begin{matrix} c_{2} & d_{2} \\ d_{2}^{*} & c_{2} \end{matrix})

are the sub covariance matrices of the first and second input states, where the parameters c_j are real and

d_{j} = | d_{j} | e^{i ϕ_{j}}

are complex with

j = 1, 2.

The initial conditions of the expectation values of the photon number and two photon correlation for the Gaussian states of the initial cavity field are

〈 a_{1}^{†} a_{1} 〉_{0} = c_{1} - 1 / 2, 〈 a_{1} a_{1} 〉_{0} = - d_{1}, 〈 a_{2}^{†} a_{2} 〉_{0} = c_{2} - 1 / 2, 〈 a_{2} a_{2} 〉_{0} = - d_{2},

and

〈 a_{1} a_{2}^{†} 〉_{0} = 〈 a_{1}^{†} a_{2} 〉_{0}^{*} = 0, 〈 a_{1} a_{2} 〉_{0} = 0.

The parameters of the covariance matrix in terms of the non-classicality τ_j and purity μ_j of the quantum state can be written as [71]

c_{j} = \frac{1}{2} + \frac{τ_{j}^{2} + 1 / {(2 u_{j})}^{2} - 1 / 4}{1 - 2 τ_{j}}, | d_{j} | = \frac{τ_{j} - τ_{j}^{2} + 1 / {(2 u_{j})}^{2} - 1 / 4}{1 - 2 τ_{j}} .

Here, the degree of non-classicality τ_j for a general single-mode Gaussian state is defined as $τ_{j} = \max {0, \frac{1}{2} - λ_{j}}$ [82], where λ_j indicates the minimum eigenvalue for sub-matrices V₁ and V₂. For a non-classical Gaussian state e.g. squeezed state, the non-classicality τ is restricted only to the regime $0 \leq τ \leq 1 / 2$ . The purity μ of the Gaussian state limited to the range $0 \leq μ \leq 1$ is further defined as $μ (ρ) = T r (ρ^{2})$ . Since quantum state with $μ = 1$ is pure and with $μ < 1$ is mixed. The purity [83] for the two input cavity modes with the covariance matrix V_j is defined as $μ_{j} = \frac{1}{2 \sqrt{d e t V_{j}}}$ , where $j = 1, 2$ .

4. Quantum correlations of the output cavity field

The characteristic function for the resulting TMGS of the cavity field with complex amplitudes ξ₁ and ξ₂ can be written as

χ (ξ_{1}, ξ_{2}) = exp (- \frac{1}{2} ξ^{†} V_{o u t} ξ) .

In Eq. (37), $ξ^{†} = (ξ_{1}^{*}, ξ_{1}, ξ_{2}^{*}, ξ_{2})$ and

V_{o u t} = (\begin{matrix} V_{1} & V_{3} \\ V_{3}^{T} & V_{2} \end{matrix}),

is the two-mode output covariance matrix, where the submatrices are

V_{1} = (\begin{matrix} f_{1} & - f_{11} \\ - f_{11}^{*} & f_{1} \end{matrix})

,

V_{2} = (\begin{matrix} f_{2} & - f_{22} \\ - f_{22}^{*} & f_{2} \end{matrix})

, and

V_{3} = (\begin{matrix} f_{12} & - f_{3} \\ - f_{3}^{*} & f_{12}^{*} \end{matrix})

with

f_{1} = 〈 a_{1}^{†} a_{1} 〉 + 1 / 2

,

f_{2} = 〈 a_{2}^{†} a_{2} 〉 + 1 / 2

,

f_{11} = 〈 a_{1} a_{1} 〉

,

f_{22} = 〈 a_{2} a_{2} 〉

,

f_{3} = 〈 a_{1} a_{2} 〉

, and

f_{12} = 〈 a_{1} a_{2}^{†} 〉

are the time dependent second moments of the resulting two-mode cavity field. Next, we study the properties of the quantum correlations of TMGS of the cavity field such as Bell non-locality, quantum steering, entanglement and discord produced by the laser system after its time evolution.

In order to find the non-locality of the resulting TMGS, the expression of the Wigner function [84] in terms of the parity operator can be expressed as

W_{T M G S} (α_{1}^{^{'}}, α_{2}^{^{'}}) = \frac{4}{π^{2}} Π (α_{1}^{^{'}}, α_{2}^{^{'}}),

where

Π (α_{1}^{^{'}}, α_{2}^{^{'}})

is the quantum expectation value of the displaced parity operator

\hat{Π} (α_{1}^{^{'}}, α_{2}^{^{'}})

. Bell’s combination, B, in terms of the correlations between parity measurements becomes as [46]

B = Π (0, 0) + Π (\sqrt{J_{1}}, 0) + Π (0, - \sqrt{J_{2}}) - Π (\sqrt{J_{1}}, - \sqrt{J_{2}}),

where J₁ and J₂ show magnitude of the displacements. After straight forward calculations, the Bell’s combination [47, 49] for the quantum state of the field retrieved from the laser system with respect to the parameters

μ_{1}, μ_{2}, τ_{1}, τ_{2}

of the initial cavity modes is expressed as

B = P {1 + 2 exp [- b_{2} Y_{1}] ln (x) - exp [\frac{- 2 ln (x)}{Y_{1} Y_{2}}]} .

where

P = \frac{1}{π^{2}} \int \int d^{2} ξ_{1} d^{2} ξ_{2} χ (ξ_{1}, ξ_{2}) = \frac{1}{4} {[(x_{1} x_{2} - z_{1}^{2}) (y_{1} y_{2} - z_{2}^{2})]}^{- 1 / 2} = μ_{1} μ_{2}

is the global purity of the resulting quantum state,

Y_{1} = y_{2} + 2 z_{2} \sqrt{y_{2} / y_{1}},

and

Y_{2} = y_{2} + z_{2} \sqrt{y_{2} / y_{1}}

with

x_{1} = f_{1} + f_{11}, x_{2} = f_{2} + f_{22}, y_{1} = f_{1} - f_{11}, y_{2} = f_{2} + f_{22}, z_{1} = | f_{12} | + ł e f t | f_{3},

and

z_{2} = | f_{12} | - | f_{3} |

. For local theories, the Bell’s combination B satisfies the inequality

- 2 \leq B \leq 2

.

Among the family of the quantum correlations, quantum steering [10, 36] is another interesting kind of correlations with distinct characteristics. For a composite quantum system of subsystems V₁ and V₂, steering has been quantified for Gaussian states through Gaussian measurements [36]. Subsystem V₁ can steers V₂, if and only if the condition $V_{o u t} + \frac{i}{2} (0_{V_{1}} \oplus Ω_{V_{2}}) \geq 0$ does not satisfy. Different from Bell non-locality and entanglement, steering is intrinsically asymmetric in general. Quantum steering from subsystem V₁ to V₂ of the composite system with covariance matrix $V_{o u t}$ is quantified as [10]

S^{V_{1} \to V_{2}} = max {0, \frac{1}{2} \log_{2} \frac{det V_{1}}{4 det V_{o u t}}} .

The degree of entanglement for inseparable quantum states is characterized by logarithmic negativity [14]. For the TMGS of covariance matrix $V_{o u t}$ , the logarithmic negativity is defined as

E = max {0, - \log_{2} (2 {\tilde{η}}_{-})},

where

{\tilde{η}}_{-}

is the smallest of the two symplectic eigenvalues of the two-mode covariance matrix

V_{o u t}

obtained with the operation of the positive partial transpose (PPT) [14, 70]. The smallest eigenvalues turn as

2 {\tilde{η}}_{\pm}^{2} = ζ \pm \sqrt{ζ^{2} - 4 \det V_{o u t}}

with

ζ = \det V_{1} + \det V_{2} - 2 \det V_{3}

.

Fig. 2 Four kinds of quantum correlations; (a) Bell’s combination, (b) quantum steering, (c) quantum entanglement, and (d) quantum discord of the resulting TMGS against dimensionless interaction time gt for fixed values of $μ_{1} = μ_{2} = 1$ , $τ_{1} = τ_{2} = 0.44$ , κ = 0, r_a = 10 kHz, $g = 16$ kHz, γ = 20 kHz, and $ϕ_{1} = ϕ_{2} = φ = 0$ . In each case, the dotted-, dashed-, and solid-green(red) curves show the results for the strongly driven case of Eqs. (27)-(33) (general case of Eqs. (19)-(25)) with $Ω = 2160$ kHz, 2660 kHz, and 3160 kHz, respectively.

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Quantum discord [57, 58] measures the non-classical correlations which goes beyond quantum entanglement in a quantum state of a composite system. The Gaussian quantum discord can be measured in the domain of generalized Gaussian measurements on a bipartite quantum system of two-mode covariance matrix $V_{o u t}$ . By using the set of positive operator valued measurements on subsystem $V_{2}$ , the general expression of Gaussian quantum discord [57, 58] is obtained as

D = h (\sqrt{β}) - h (η_{-}) - h (η_{+}) + h (σ),

where

h (z) = (z + \frac{1}{2}) \log_{2} (z + \frac{1}{2}) - (z - \frac{1}{2}) \log_{2} (z - \frac{1}{2}),

σ = \frac{\sqrt{α} + 2 \sqrt{α β} + 2 ϵ}{1 + 2 \sqrt{β}},

$α = det V_{1}, β = det V_{2}, ϵ = det V_{3}, δ = det V_{o u t},$ and $η_{\pm}$ being the symplectic eigenvalues of the covariance matrix $V_{o u t}$ , represented by $2 η_{\pm}^{2} = ζ \pm \sqrt{ζ^{2} - 4 det V_{o u t}}$ with $ζ = det V_{1} + det V_{2} + 2 det V_{3}$ . The three quantum correlations such as quantum steering, entanglement, and discord must satisfy the condition $S^{V_{1} \to V_{2}}, E, D > 0$ in presence of the desired non-classical correlations in the laser system.

Fig. 3 Four kinds of quantum correlations; (a) Bell’s combination, (b) quantum steering, (c) quantum entanglement, and (d) quantum discord of the resulting TMGS against dimensionless interaction time gt for $Ω = 2160$ kHz. In each case, the dotted-orange, dashed-blue, and solid-purple lines are for κ = 0, 0.005 kHz, and 0.009 kHz, respectively. For all the insets $κ = 0.1$ kHz. The other system parameters are the same as used in Fig. 2.

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5. Results

In this section, we analyze the graphical results of the quantum correlations of the evolved cavity field via CEL system. Here, we consider experimentally realizable parameter values which are closely related to the experiments [85, 86]. Response of non-locality, steering, entanglement and Gaussian discord of the resulting TMGS are plotted against dimensionless interaction time gt using various values of the Rabi frequency Ω of the classical driving field, as shown in Fig. 2. The analytical results (green curves) for time evolution of all the four kinds of quantum correlations in the parametric limit of Eqs. (27)-(33) are slightly shorter than the numerical solutions (red curves) of general case using Eqs. (19)-(25). In each case, the time interval increases with an increase in the Rabi frequency Ω. This shows that the classical driving field generates a coherence between the ground level $| a 〉$ and excited level $| c 〉$ of the lasing medium of the laser system. Based on the strength of the Rabi frequency Ω, the Bell non-locality, steering, and entanglement (see Figs. 2(a)-2(c)) initially increase up to maximum, then decrease and eventually vanish when the quantum correlations are further evolved for certain period of time. In Fig. 2(a), Bell’s combination B of the TMGS violates the inequality $- 2 \leq B \leq 2$ for very short interval of time. The short time violation of the Bell’s inequality indicates that the cavity modes, in this case, exhibits non-local behavior. On the other hand, the behavior of quantum steering $S^{V_{1} \to V_{2}}$ , as shown in Fig. 2(b), clearly illustrates that the subsystem V₁ can steers the subsystem V₂ for a relatively long period of time as compared to the non-locality in the quantum state of the system. Similarly, the entanglement E is also generated between the TMGS of the cavity field produced by the laser system (see Fig. 2(c)). To further get insight, we investigate the Gaussian quantum discord D in the laser system with respect to the amplitude of the classical driving field with its Rabi frequency Ω, as shown in Fig. 2(d). In this case, the effect of Gaussian quantum discord D initially increases up to a maximum value. After the maximal effect is reached the quantum discord decreases gradually up to a finite value which remains for long time. Note that the valid regime of the quantum discord is up to the limit when it approaches a constant value in the time evolution of the quantum correlation. In comparison to the time evolution of quantum non-locality, steering, and entanglement, the Gaussian discord shows maximum quantumness. This is the signature of extracting maximal amount of mutual information from the quantum system of CEL.

Fig. 4 Four kinds of quantum correlations; (a) Bell’s combination, (b) quantum steering, (c) quantum entanglement, and (d) quantum discord of the resulting TMGS against dimensionless interaction time gt for $Ω = 2160$ kHz. In each case, the dotted-orange, dashed-blue, and solid-purple lines are for $τ_{1} = τ_{2} = 0.34$ , 0.37, and 0.40, respectively. The other system parameters are the same as used in Fig. 2.

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In Fig. 3, the four kinds of quantum correlations of the resulting TMGS of the cavity field are shown versus normalized time gt for three different values of the cavity losses κ. In contrast to the effect of the Rabi frequency Ω (see Fig. 2), the duration of all the four kinds of quantum correlations decrease with cavity decay rates κ, as shown in Fig. 3. Hence, the decoherence process due to the cavity losses affect the quantum correlations in the CEL based system. As shown in Fig. 3(a), the Bell inequality violates for the considered values of the cavity decay rates. However, the non-locality vanishes for high enough cavity decay rates, as clearly shown in the inset in Fig. 3(a). Likewise in Fig. 3(b), the time development of quantum steering $S^{V_{1} \to V_{2}}$ effect decreases with enhanced rates of the cavity damping using the same parameters as used in Fig. 3(a). We observe that the steering of the output TMGS of the cavity field under Gaussian measurements degrades considerably when high cavity losses are taken in the system (see inset in Fig. 3(b)). In comparison to quantum non-locality and steering, we next focus on the dynamics of quantum entanglement between the TMGS using the approach of logarithmic negativity E, as displayed on Fig. 3(c). It is noticed that as the cavity damping increases, the time period for entanglement generation reduces sufficiently, see also inset in Fig. 3(c) for high cavity damping rates. Since the decay rates (distinction between the atom decay rate and cavity damping rate is necessary throughout the manuscript) disturb the coherence of the entanglement dynamics in the system. This behavior of our system is in agreement with the existence literature [28, 29, 31, 87]. Further, to obtain the quantum correlations beyond entanglement, we show the behavior of the time evolution of quantum discord D in Fig. 3(d). We find that the effect of Gaussian quantum discord reduces as the cavity damping rate contributes to the system, see also inset in Fig. 3(d) for high cavity losses.

To conclude with this discussion, we note that the system exhibits quantum discord for large interval of time as compared to entanglement in the two modes of the laser cavity. Likewise, the time duration of entanglement is much larger than the time development of steering and Bell non-locality. It simply predicts that when the system exhibits entanglement, it must always shows quantum discord as well. Contrarily, the system shows quantum discord does not necessarily exhibit the entanglement phenomenon. Similarly, every non-local system predicts quantum steering and the effect of steerability is a witness of quantum entanglement in the system. In quantum non-locality, steering, entanglement and discord, the system depicting that the latter one may not reveals the former (see Figs. 2 and 3). Hence, the boundaries of all the four kinds of quantum correlations satisfy a strict hierarchy.

Fig. 5 Four kinds of quantum correlations; (a) Bell’s combination, (b) quantum steering, (c) quantum entanglement, and (d) quantum discord of the resulting TMGS against dimensionless interaction time gt for $Ω = 2160$ kHz. In each case, the dotted-orange, dashed-blue, and solid-purple lines are for $μ_{1} = μ_{2} = 0.75$ , 0.85, and 1, respectively. The other system parameters are the same as used in Fig. 2.

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To study the influence of non-classicality τ of the input modes on the time development of quantum correlations in the system, we show the behavior of Bell inequality, quantum steering, entanglement and discord against dimensionless interaction time gt in Fig. 4 using three different values of the non-classicality of the input modes. The time interval of all the four kinds of quantum correlations enhances with non-classicality of the input states. This identifies that non-classicality introduces an additional coherence to quantum correlations between the two cavity modes of the laser system. As shown in Fig. 4(a), the Bell inequality satisfies when $τ < 0.37$ . Therefore, the quantum state of the system establishes a non-local character for a very short time. The cavity mode of the subsystem V₁ steers the second mode of the subsystem V₂ of the composite quantum system as shown in Fig. 4(b). However in Fig. 4(c) and 4(d), the degree of the entanglement and the amount of quantum discord are seen using input modes with different values of the non-classicality. As a response, both kinds of the quantum correlations indicate maximal profile with enhanced non-classicality. To investigate the influence of the purity of input modes, the time development of quantum correlations versus dimensionless interaction time gt using various values of the purity μ are plotted in Fig. 5. In this case, Bell inequality violates when both the input states are pure, and satisfies when they are mixed (see Fig. 5(a)). This behavior of our system agrees with the previously reported studies [49, 50]. On the other hand, the TMGS is steerable under Gaussian measurements (see Fig. 5(b)) using the same parameters as used in Fig. 5(a). We note that the steering is maximally enhanced with the increase of purity of the input states. However, the quantum entanglement and discord do not show significant effect for the considered values of the purity (see Figs. 5(c) and 5(d)).

Meanwhile, we have discussed the time development of four kinds of quantum correlations of the output cavity field via a CEL based laser system. The discussion on these results clearly demonstrate a hierarchical aspect of the temporal quantum correlations between the two cavity modes such that $D \supseteq E \supseteq S^{V_{1} \to V_{2}} \supseteq B$ which means D is the superset of E and similarly the others. We believe that various aspects of quantum correlations in a single setup may provide more useful potential applications in secured communications and information processes.

6. Summary

In summary, the time evolution of Bell non-locality, steering, entanglement, and Gaussian discord of the resulting TMGS of the cavity field are analyzed using a well known system of CEL. We have used various sets of parameters of the laser system where the initial cavity modes are considered in general single-mode Gaussian states. Here, we have considered experimentally realizable parameter values which are closely related to the experiments [85, 86]. The rate equation for the density matrix of the correlated cavity field of the laser system is derived and solved analytically as well as numerically. On the quantum correlations, the influence of the amplitude of classical driving field, the purity and non-classicality of the initial cavity modes, and the cavity decay rates are investigated.

We find that the time development of all the four kinds of quantum correlations of the output cavity field indicate enhanced profile upon increasing the Rabi frequency of the classical driving field. This is due to coherence generated by the coupling of the classical field to the upper excited level $| c 〉$ and lower ground level $| a 〉$ of the atom. Since, the cavity is not perfect, cavity damping induces a decoherence process in the laser system which reduce the duration of all the quantum correlations. Further, we noticed that the non-classicality of initial cavity modes adds additional coherence to the system. Therefore, time interval of all the quantum correlations increase with the non-classicality, however, the retrieved TMGS shows local and non-steerable behavior when the initial modes are less non-classical. On the other hand, the Bell inequality is obeyed by the laser system when the the initial cavity modes are mixed (see Fig. 5(a)). The output cavity field of the laser system exhibit steerability in different interval of time and the effect increases with enhanced purity of the initial two modes of the cavity field. In such a situation, quantum entanglement and discord in the resulting TMGS appeared almost independent of the purity.

Our investigation further show that the quantum state of the resulting cavity field violates Bell non-locality for shorter interval of time as compared to the steering. Similarly, the growth of entanglement is longer than steering but shorter as compared to the survival time of discord in the system which keeps a non-zero value for large time. To this end, we conclude that Bell non-locality reveals steerability in the quantum system and steering is a witness of entanglement but the reverse is not always true. Likewise, entanglement in the system always shows quantum discord but the latter may not necessarily indicates the former. Therefore, all the four kinds of temporal quantum correlations satisfy a strict hierarchy relation in the quantum system of CEL such that quantum discord $\supseteq$ quantum entanglement $\supseteq$ quantum steering $\supseteq$ Bell non-locality, where $X \supseteq Y$ means X is the superset of Y, stating such that entangled states are a strict subset of the states that may exhibit discord, and a strict superset of the steerable states.

Our theoretical results show hierarchy relation in these quantum correlations by changing various viable parameters which are closely related to the experiments [85, 86]. Hence, the CEL system seems to be a good choice for experimental study of the four kinds of quantum correlations and their applications in quantum information processing.

Funding

Higher Education Commission, Pakistan (HEC/NRPU/20-2475).

We would like to acknowledge Higher Education Commission (HEC), Pakistan, for partial financial support under Grant No. HEC/NRPU/20-2475.

References

1. J. P. Dowling and G. J. Milburn, “Quantum technology: the second quantum revolution,” Phil. Trans. R. Soc. Lond. A 361, 1655–1674 (2003). [CrossRef]

2. A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935). [CrossRef]

3. G. Waldherr, P. Neumann, S. F. Huelga, F. Jelezko, and J. Wrachtrup, “Violation of a temporal Bell inequality for single spins in a diamond defect center,” Phys. Rev. Lett. 107, 090401 (2011). [CrossRef] [PubMed]

4. N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, “Bell nonlocality,” Rev. Mod. Phys. 86, 419–478 (2014). [CrossRef]

5. Y. N. Chen, C. M. Li, N. Lambert, S. L. Chen, Y. Ota, G. Y. Chen, and F. Nori, “Temporal steering inequality,” Phys. Rev. A 89, 032112 (2014). [CrossRef]

6. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009). [CrossRef]

7. A. V. Chizhov and R. G. Nazmitdinov, “Entanglement control in coupled two-mode boson systems,” Phys. Rev. A 78, 064302 (2008). [CrossRef]

8. K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, “The classical-quantum boundary for correlations: Discord and related measures,” Rev. Mod. Phys. 84, 1655–1707 (2012). [CrossRef]

9. P. Skrzypczyk, M. Navascues, and D. Canalcanti, “Quantifying Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 112, 180404 (2014). [CrossRef] [PubMed]

10. I. Kogias, A. R. Lee, S. Ragy, and G. Adesoo, “Quantification of Gaussian quantum steering,” Phys. Rev. Lett. 114, 060403 (2015). [CrossRef] [PubMed]

11. M. Piani and J. Watrous, “Necessary and sufficient quantum information characterization of Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 114, 060404 (2015). [CrossRef] [PubMed]

12. C. M. Li, K. Chen, Y. N. Chen, Q. Zhang, Y. A. Chen, and J. W. Pan, “Genuine high-order Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 115, 010402 (2015). [CrossRef] [PubMed]

13. C. H. Bennett, D. P. Divincenzo, J. A. Smolin, and W. K. Wootters, “Mixed-state entanglement and quantum error correction,” Phys. Rev. A 54, 3824–3851 (1996). [CrossRef] [PubMed]

14. G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65, 032314 (2002). [CrossRef]

15. H. Ollivier and W. H. Zurek, “Quantum discord: a measure of the quantumness of correlations,” Phys. Rev. Lett. 88, 017901 (2001). [CrossRef]

16. S. Tesfa, “Continuous variable quantum discord in a correlated emission laser,” Optics Communications 285, 830–837 (2012). [CrossRef]

17. T. Baumgratz, M. Cramer, and M. B. Plenio, “Quantifying coherence,” Phys. Rev. Lett. 113, 140401 (2014). [CrossRef] [PubMed]

18. J. Xu, “Quantifying coherence of Gaussian states,” Phys. Rev. A 93, 032111 (2016). [CrossRef]

19. Z. X. Wang, S. Wang, T. Ma, T. J. Wang, and C. Wang, “Gaussian entanglement generation from coherence using beam splitters,” Sci. Rep. 6, 38002 (2016). [CrossRef]

20. H. S. Qureshi, S. Ullah, and F. Ghafoor, “Hierarchy of quantum correlations using a linear beam splitter,” Sci. Rep. 8, 16288 (2018). [CrossRef] [PubMed]

21. H. Xiong, M. O. Scully, and M. S. Zubairy, “Correlated spontaneous emission laser as an entanglement amplifier,” Phys. Rev. Lett. 94, 023601 (2005). [CrossRef] [PubMed]

22. H. T. Tan, S. Y. Zhu, and M. S. Zubairy, “Continuous-variable entanglement in a correlated spontaneous emission laser,” Phys. Rev. A 72, 022305 (2005). [CrossRef]

23. L. Zhou, H. Xiong, and M. S. Zubairy, “Single atom as a macroscopic entanglement source,” Phys. Rev. A 74, 022321 (2006). [CrossRef]

24. S. Tesfa, “Entanglement amplification in a nondegenerate three-level cascade laser,” Phys. Rev. A 74, 043816 (2006). [CrossRef]

25. M. Kiffner, M. S. Zubairy, J. Evers, and C. Keitel, “Two-mode single-atom laser as a source of entangled light,” Phys. Rev. A 75, 033816 (2007). [CrossRef]

26. M. Ikram, G. Li, and M. S. Zubairy, “Entanglement generation in a two-mode quantum beat laser,” Phys. Rev. A 76, 042317 (2007). [CrossRef]

27. S. Qamar, F. Ghafoor, M. Hillery, and M. S. Zubairy, “Quantum beat laser as a source of entangled radiation,” Phys. Rev. A 77, 062308 (2008). [CrossRef]

28. S. Qamar, S. Al-Amri, Qamar, and M. S. Zubairy, “Entangled radiation via a Raman-driven quantum-beat laser,” Phys. Rev. A 80, 033818 (2009). [CrossRef]

29. A.-P. Fang, Y.-L. Chen, F.-L. Li, H.-R. Li, and P. Zhang, “Generation of two-mode Gaussian-type entangled states of light via a quantum beat laser,” Phys. Rev. A 81, 012323 (2010). [CrossRef]

30. R. Tahira, M. Ikram, H. Nha, and M. S. Zubairy, “Gaussian-state entanglement in a quantum beat laser,” Phys. Rev. A 83, 054304 (2011). [CrossRef]

31. S. Ullah, H. S. Qureshi, G. Tiaz, F. Ghafoor, and F. Saif, “Coherence control of entanglement dynamics of two-mode Gaussian state via Raman driven quantum beat laser using Simon’s criterion,” Appl. Opt. 58, 197–204 (2019). [CrossRef] [PubMed]

32. A.-B. Mohamed and H. Eleuch, “Non-classical effects in cavity QED containing a nonlinear optical medium and a quantum well: Entanglement and non-Gaussanity,” Eur. Phys. J. D 69, 191 (2015). [CrossRef]

33. S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005). [CrossRef]

34. C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012). [CrossRef]

35. E. Schrodinger, “Discussion of probability relations between separated systems,” Math. Proc. Cambridge Philos. Soc. 31, 555–563 (1935). [CrossRef]

36. H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox,” Phys. Rev. Lett. 98, 140402 (2007). [CrossRef] [PubMed]

37. E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein-Podolsky-Rosen paradox,” Phys. Rev. A 80, 032112 (2009). [CrossRef]

38. D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, “Experimental EPR-steering using bell-local states,” Nature Phys. 6, 845–849 (2010). [CrossRef]

39. G. Y. Chen, S. L. Chen, Y. N. Li, and C. M. Chen, “Examining non-locality and quantum coherent dynamics induced by a common reservoir,” Sci. Rep. 3, 2514 (2013). [CrossRef] [PubMed]

40. J. Schneeloch, C. J. Broadbent, S. P. Walborn, E. G. Cavalcanti, and J. C. Howell, “Einstein-Podolsky-Rosen steering inequalities from entropic uncertainty relations,” Phys. Rev. A 87, 062103 (2013). [CrossRef]

41. W. Zhong, G. Cheng, and X. Hu, “One-way Einstein–Podolsky–Rosen steering with the aid of the thermal noise in a correlated emission laser,” Laser Phys. Lett. 15, 065204 (2018). [CrossRef]

42. S. Ullah, H. S. Qureshi, and F. Ghafoor, “Quantum steering of two-mode Gaussian state using a quantum beat laser,” Appl. Opt. 58, 7014–7021 (2019). [CrossRef]

43. C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani, and H. M. Wiseman, “One-sided device-independent quantum key distribution: Security, feasibility, and the connection with steering,” Phys. Rev. A 85, 010301 (2012). [CrossRef]

44. B. Wittmann, S. Ramelow, F. Steinlechner, N. K. Langford, N. Brunner, H. M. Wiseman, R. Ursin, and A. Zeilinger, “Loophole-free Einstein-Podolsky-Rosen experiment via quantum steering,” New J. Phys. 14, 053030 (2012). [CrossRef]

45. M. T. Quintino, T. Vertesi, D. Cavalcanti, R. Augusiak, M. Demianowicz, A. Acin, and N. Brunner, “Inequivalence of entanglement, steering, and bell nonlocality for general measurements,” Phys. Rev. A 92, 032107 (2015). [CrossRef]

46. J. S. Bell, “On the Einstein-Podolsky-Rosen paradox,” Physics (New York) 1, 195–200 (1964).

47. K. Banaszek and K. Wodkiewicz, “Nonlocality of the Einstein-Podolsky-Rosen state in the Wigner representation,” Phys. Rev. A 58, 4345–4347 (1998). [CrossRef]

48. P. Milman, A. Auffeves, F. Yamaguchi, M. Brune, J. M. Raimond, and S. Haroche, “A proposal to test bell’s inequalities with mesoscopic non-local states in cavity QED,” Eur. Phys. J. D 32, 233–239 (2005). [CrossRef]

49. H. Jeong, J. Lee, and M. S. Kim, “Dynamics of nonlocality for a two-mode squeezed state in a thermal environment,” Phys. Rev. A 61, 052101 (2000). [CrossRef]

50. C. Xin, H. Guang-Ming, and G. X. Li, “Nonlocality and purity in atom-field coupling system,” Chin. Phys. Soc. 14, 223–230 (2005). [CrossRef]

51. J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880–884 (1969). [CrossRef]

52. A. Datta, A. Shaji, and C. M. Caves, “Quantum discord and the power of one qubit,” Phys. Rev. Lett. 100, 050502 (2008). [CrossRef] [PubMed]

53. M. Piani, P. Horodecki, and R. Horodecki, “No-local-broadcasting theorem for multipartite quantum correlations,” Phys. Rev. Lett. 100, 090502 (2008). [CrossRef] [PubMed]

54. A. Ferraro, L. Aolita, D. Cavalcanti, F. M. Cucchietti, and A. Acin, “Almost all quantum states have nonclassical correlations,” Phys. Rev. A 81, 052318 (2010). [CrossRef]

55. A. Brodutch and D. R. Terno, “Quantum discord, local operations, and Maxwell’s demons,” Phys. Rev. A 81, 062103 (2010). [CrossRef]

56. L. Henderson and V. Vedral, “Classical, quantum and total correlations,” J. Phys. A: Math. Gen. 34, 6899–6905 (2001). [CrossRef]

57. P. Giorda and M. G. A. Paris, “Gaussian quantum discord,” Phys. Rev. Lett. 105, 020503 (2010). [CrossRef] [PubMed]

58. G. Adesso and A. Datta, “Quantum versus classical correlations in Gaussian states,” Phys. Rev. Lett. 105, 030501 (2010). [CrossRef] [PubMed]

59. F. F. Fanchini, L. K. Castelano, and A. O. Caldeira, “Entanglement versus quantum discord in two coupled double quantum dots,” New J. Phys. 12, 073009 (2010). [CrossRef]

60. D. Z. Rossatto, T. Werlang, E. I. Duzzioni, and C. J. Villas-Boas, “Nonclassical behavior of an intense cavity field revealed by quantum discord,” Phys. Rev. Lett. 107, 153601 (2011). [CrossRef] [PubMed]

61. Q. L. He, J. B. Xu, D. X. Yao, and Y. Q. Zhang, “Sudden transition between classical and quantum decoherence in dissipative cavity QED and stationary quantum discord,” Phys. Rev. A 84, 022312 (2011). [CrossRef]

62. J. S. Zhang, L. Chen, M. Abdel-Aty, and A. X. Chen, “Sudden death and robustness of quantum correlations in the weak- or strong-coupling regime,” Eur. Phys. J. D 66, 2–8 (2012). [CrossRef]

63. Y. X. Chen, S. W. Li, and Z. Yin, “Quantum correlations in a cluster like system,” Phys. Rev. A 82, 052320 (2010). [CrossRef]

64. Y. X. Chen and S. W. Li, “Quantum correlations in topological quantum phase transitions,” Phys. Rev. A 81, 032120 (2010). [CrossRef]

65. S. Campbell, T. J. G. Apollaro, C. D. Franco, L. Banchi, A. Cuccoli, R. Vaia, F. Plastina, and M. Paternostro, “Propagation of nonclassical correlations across a quantum spin chain,” Phys. Rev. A 84, 052316 (2011). [CrossRef]

66. G. Giedke, M. M. Wolf, O. Kruger, R. F. Werner, and J. I. Cirac, “Entanglement of formation for symmetric Gaussian states,” Phys. Rev. Lett. 91, 107901 (2003). [CrossRef] [PubMed]

67. P. Marian and T. Marian, “Entanglement of formation for an arbitrary two-mode Gaussian state,” Phys. Rev. Lett. 101, 220403 (2008). [CrossRef] [PubMed]

68. J. Laurat, T. Coudreau, G. Keller, N. Treps, and C. Fabre, “Compact source of Einstein-Podolsky-Rosen entanglement and squeezing at very low noise frequencies,” Phys. Rev. A 70, 042315 (2004). [CrossRef]

69. H. J. Kimble and D. F. Walls, “Squeezed states of the electromagnetic field: Introduction to feature issue,” J. Opt. Soc. Am. B 4, 1449 (1987).

70. G. Adesso, “Entanglement of Gaussian states,” arXiv:quant-ph/0702069 (2007).

71. L. H. Rong, F. L. Li, and Y. Yang, “Entangling two single mode Gaussian states by use of a beam splitter,” Chin. Phys. 15, 2947–2952 (2006). [CrossRef]

72. M. S. Kim, W. Son, V. Buzek, and P. L. Knight, “Entanglement by a beam splitter: Nonclassicality as a prerequisite for entanglement,” Phys. Rev. A 65, 032323 (2002). [CrossRef]

73. M. G. A. Paris, “Entanglement and visibility at the output of a mach-zehnder interferometer,” Phys. Rev. A 59, 1615–1621 (1999). [CrossRef]

74. B. C. Sanders, “Entangled coherent states,” Phys. Rev. A 45, 6811–6815 (1992). [CrossRef] [PubMed]

75. S. M. Tan, D. F. Walls, and M. J. Collett, “Nonlocality of a single photon,” Phys. Rev. Lett. 66, 252–255 (1991). [CrossRef] [PubMed]

76. K. Berrada, S. Abdel-Khalek, H. Eleuch, and Y. Hassouni, “Beam splitting and entanglement generation: excited coherent states,” Quantum Inf. Process. 12, 69–82 (2013). [CrossRef]

77. E. A. Sete, H. Eleuch, and S. Das, “Semiconductor cavity QED with squeezed light: Nonlinear regime,” Phys. Rev. A 84, 053817 (2011). [CrossRef]

78. S. L. Chen, N. Lambert, C. M. Li, A. Miranowicz, Y. N. Chen, and F. Nori, “Quantifying non-markovianity with temporal steering,” Phys. Rev. Lett. 116, 020503 (2016). [CrossRef] [PubMed]

79. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, 1997), Chap. 14. [CrossRef]

80. C. A. Blockley and D. Walls, “Intensity fluctuations in a frequency down-conversion process with three-level atoms,” Phys. Rev. A 43, 5049–5056 (1991). [CrossRef] [PubMed]

81. J. Laurat, T. Coudreau, G. Keller, N. Treps, and C. Fabre, “Compact source of Einstein-Podolsky-Rosen entanglement and squeezing at very low noise frequencies,” Phys. Rev. A 70, 042315 (2004). [CrossRef]

82. C. T. Lee, “Measure of the nonclassicality of nonclassical states,” Phys. Rev. A 44, R2775–R2778 (1991). [CrossRef] [PubMed]

83. M. G. A. Paris, F. Illuminati, A. Serafini, and S. D. Siena, “Purity of Gaussian states: Measurement schemes and time evolution in noisy channels,” Phys. Rev. A 68, 012314 (2003). [CrossRef]

84. S. M. Barnett and P. L. Knight, “Squeezing in correlated quantum systems,” J. Mod. Opt. 34, 841–853 (1987). [CrossRef]

85. D. Meschede, H. Walther, and G. Muller, “One-atom maser,” Phys. Rev. Lett. 54, 551–554 (1985). [CrossRef] [PubMed]

86. J. M. Raimond, M. Brune, and S. Haroche, “Colloquium: manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. 73, 565–582 (2001). [CrossRef]

87. S. Y. Lee, S. Qamar, H. W. Lee, and M. S. Zubairy, “Entanglement in a parametric converter,” J. Phys. B: At. Mol. Opt. Phys. 41, 145504 (2008). [CrossRef]

Hierarchy of temporal quantum correlations using a correlated spontaneous emission laser

Abstract

1. Introduction

2. The model and dynamics of the system

3. Input cavity field

4. Quantum correlations of the output cavity field

5. Results

6. Summary

Funding

References

Cited By

Figures (5)

Equations (45)

Optics Express