# Phase-Controlled Atom-Photon Entanglement in a Three-Level V-Type Atomic System via Spontaneously Generated Coherence

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{87}Rb atom and a single photon at a wavelength suitable for low-loss communication has been reported [14].

## 2. Model and Equations

_{L}and Rabi frequency Ω

_{L}drives the transition |1〉 →|3〉, and other right field, with frequency ω

_{R}and Rabi frequency Ω

_{R}is applied to the transition |1〉 → |2〉. Here, E

_{L}(E

_{R}) and are the amplitude, and the polarization of the left (right) classical laser field, while ω

_{L}(ω

_{R}), and ϕ

_{L}(ϕ

_{R}) are the frequency, wave vector, and initial phase of left (right) classical laser field. The parameters 2γ

_{21}and 2γ

_{31}denote the spontaneous decay rates from excited-states |2〉 and |3〉 to ground state, |1〉 respectively. Also Δ

_{L}= ω

_{L}− ω

_{31}, Δ

_{R}= ω

_{R}− ω

_{21}are one-photon detuning of the two fields. Such a system, with a single ground state and a closely spaced excited doublet (e.g., two near-degenerate states), is damped by the usual vacuum interactions, so the two decay pathways from the excited doublet to the ground state are not independent. The system decays from the upper states doublet to a lower state via spontaneous emission leading to the quantum interference, i.e., spontaneously generated coherence (SGC) [24].

_{2}transition as a realistic example. The decay rate of transition is γ = 2π × 9.79MHz. The right field Ω

_{R}is applied to the 3

^{2}S

_{1/2}− 3

^{2}P

_{1/2}transition, while the left field ω

_{L}is applied to the 3

^{2}S

_{1/2}− 3

^{2}P

_{3/2}transition. For such a transition we have ω

_{32}0.2γ ≅ 15.8 MHz. Note that the two upper levels are near-degenerate, so the quantum interference due to the spontaneous emission can be induced [32].

**Figure 1.**(a) A V-type three-level atom driven by two laser fields with corresponding Rabi frequencies Ω

_{R}, Ω

_{L}; (b) The arrangement of field polarization required for a single field driving one transition if dipoles are nonorthogonal.

_{R}− ϕ

_{L}and δ = ω

_{L}− ω

_{L}are the relative phase and the relative frequency of the driving fields, respectively.

_{c}= 1, while for perpendicular dipole moments, K

_{c}= 0, and the quantum interference disappears. Note that the relative phase appears through equations via the parameter K

_{c}. So, in Equation (2), the effect of relative phase of applied fields appear in all terms contain K

_{c}. Then the solutions of these equations for K

_{c}≠ 0 are phase-dependent.

_{R}= Ω

_{L}= Ω

_{0}. The population and coherence terms of density matrix for γ = γ

_{21}= γ

_{31}= 1.0, Δ

_{R}= Δ

_{L}= 0 are given by:

## 3. Entanglement and Entropy

**C**

^{m}⊗

**C**

^{n}) Hilbert space. The partial density matrix of one part is obtained by tracing over other [34]:

_{A}

_{(B)}are the individual partial density matrixes. If the system can not satisfy Equation (5), it is said to be entangled. The atom-field quantum entanglement can be discussed by using Von Neumann entropy which is defined as [35]:

## 4. Results and Discussion

_{21}= γ

_{31}= γ and all figures are plotted in the unit of γ. We assume the applied fields have a same frequency.

_{R}= Ω

_{L}= 0.1γ, δ = 0.0, Δ

_{R}= Δ

_{L}= 0.0γ (left column), Δ

_{R}= Δ

_{L}= 2.0γ (right column), ϕ = 0 (solid), ϕ = π/6 (dashed), ϕ = 4π/3 (dotted) and (a, d) K

_{c}= 0, (b, e) K

_{c}= 0.5, (c, f) K

_{c}= 0.99. An investigation on Figure 2 shows that in the absence of quantum interference due to the spontaneous emission, the quantum entropy is phase-independent, while by including the effect of quantum interference, the entropy changes by the changing of relative phase of applied fields. Moreover, in Figures 2c,f for ϕ = 0 the quantum entropy vanishes which corresponds to the disentanglement. By comparing Figures 2a–c for different values of K

_{c}, we observe that the quantum interference between two spontaneous emissions has a major role in establishing the atom-photon entanglement.

_{c}in Figure 3. The selected parameters are same as in Figure 2. The maximum entanglement is occurred in one-photon resonance condition.

_{c}= 0.99, ϕ = 0, the steady state quantum entropy becomes zero. An investigation on Figure 4a and Figure 4b shows that, beyond exact one-photon resonance condition, the DEM of the system is negligible.

**Figure 2.**Time dependent behavior of the quantum entropy for different relative phase of applied fields. The selected parameters are γ = 1, γ

_{21}= γ

_{31}= γ, Ω

_{R}= Ω

_{L}= 0.1γ, δ = 0.0γ, Δ

_{R}= Δ

_{L}= 0.0γ (left column), Δ

_{R}= Δ

_{L}= 2.0γ (right column), ϕ = 0 (solid), ϕ = π/6 (dashed), ϕ = 4π/3 (dotted), (a, d) K

_{c}= 0; (b, e) K

_{c}= 0.5; and (c, f) K

_{c}= 0.99.

**Figure 3.**The quantum entropy versus detuning. The using parameters are γ = γ

_{31}= 1, γ

_{21}= 0.1γ, Ω

_{R}= Ω

_{L}= 0.1γ, δ = 0.0γ (a) K

_{c}= 0; (b) K

_{c}= 0.5; (c) K

_{c}= 0.99, for ϕ = 0 (Solid), ϕ = π/6 (Dashed), ϕ = 4π/3 (Dotted), Δ

_{R}= Δ

_{L}= Δ.

**Figure 4.**The quantum entropy versus the relative phase of applied fields. The parameters are γ = γ

_{31}= 1, γ

_{21}= 1.0γ, Ω

_{R}= Ω

_{L}= 0.1γ, δ = 0.0γ, K

_{c}= 0.99 (solid), 0.5 (dashed), 0.0 (dash-dotted) for (a) Δ

_{R}= Δ

_{L}= 0.0γ; (b) Δ

_{R}= Δ

_{L}= 2.0γ.

_{R}/Ω

_{L}, for ϕ = 0, K

_{c}= 0.99, Δ

_{R}= Δ

_{L}= 0.0γ (solid), 2.0γ (dashed), 4.0γ (dotted), and 6.0γ (dash-dotted). The interesting disentanglement phenomena appears for K

_{c}= 0.99, when the relative phase of applied fields is ϕ = 0, and the ratio of two Rabi frequencies of applied fields is Ω = 1 [35,36,37].

**Figure 5.**We display the steady state quantum entropy versus the relative Rabi frequency, Ω = Ω

_{R}/Ω

_{L}, for ϕ = 0, K

_{c}= 0.99, Δ

_{R}= Δ

_{L}= 0.0γ (solid), 2.0γ (dashed), 4.0γ (dotted), and 6.0γ (dash-dotted).

**Figure 6.**The population of atomic levels is shown versus the relative phase (left column) and the relative Rabi frequencies (right column) of applied fields. The parameters are Δ

_{R}= Δ

_{L}= 0.0γ (Solid), 2.0γ (Dashed), 4.0γ (dotted), 6.0γ (Dash-dotted), and K

_{c}= 0.99.

**Figure 7.**The quantum entropy versus relative phase of applied fields for γ = 1, and relative Rabi frequency Ω with K

_{c}= 0.99, for Ω

_{L}= 0.1γ, and Δ

_{R}= Δ

_{L}= 0.0γ.

_{c}= 1.0, ϕ = π, Ω

_{R}= Ω

_{L}= Ω

_{0}and ρ

_{11}(0) = 1, Equation (9) converts to the simple following expressions:

## 5. Conclusion and Perspectives

## Acknowledgements

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**MDPI and ACS Style**

Abazari, M.; Mortezapour, A.; Mahmoudi, M.; Sahrai, M. Phase-Controlled Atom-Photon Entanglement in a Three-Level V-Type Atomic System via Spontaneously Generated Coherence. *Entropy* **2011**, *13*, 1541-1554.
https://doi.org/10.3390/e13091541

**AMA Style**

Abazari M, Mortezapour A, Mahmoudi M, Sahrai M. Phase-Controlled Atom-Photon Entanglement in a Three-Level V-Type Atomic System via Spontaneously Generated Coherence. *Entropy*. 2011; 13(9):1541-1554.
https://doi.org/10.3390/e13091541

**Chicago/Turabian Style**

Abazari, Mohammad, Ali Mortezapour, Mohammad Mahmoudi, and Mostafa Sahrai. 2011. "Phase-Controlled Atom-Photon Entanglement in a Three-Level V-Type Atomic System via Spontaneously Generated Coherence" *Entropy* 13, no. 9: 1541-1554.
https://doi.org/10.3390/e13091541