Scheme for Quantum Entanglement Swapping in V-Type Three-Level Atom and Two-Mode Cavity Field Model

Lei Zhang

doi:10.3788/LOP202158.1927001

Journals >Laser & Optoelectronics Progress >Volume 58 >Issue 19 >Page 1927001 > Article

Laser & Optoelectronics Progress
Vol. 58, Issue 19, 1927001 (2021)

Scheme for Quantum Entanglement Swapping in V-Type Three-Level Atom and Two-Mode Cavity Field Model

Lei Zhang^*

Author Affiliations

Huaqing College, Xi'an University of Architecture and Technology, Xi'an , Shaanxi 710043, China

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DOI: 10.3788/LOP202158.1927001 Cite this Article Set citation alerts

Lei Zhang. Scheme for Quantum Entanglement Swapping in V-Type Three-Level Atom and Two-Mode Cavity Field Model[J]. Laser & Optoelectronics Progress, 2021, 58(19): 1927001 Copy Citation Text

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Abstract

Based on entanglement swapping, a scheme to generate the maximum entangled state by means of an entangled V-type three-level atomic system partially interacting with an entangled two-mode cavity field is proposed. In the scheme, the entanglement between the atom and the cavity field without any initial interaction can be achieved by measuring only a single atomic state without joint measurement. In the process of atom-cavity interaction, the cavity is only virtually excited and there is no energy exchange between the atom and the cavity, so the requirements of system on cavity quality are greatly relaxed. The results show that when the interaction time between the atom and the cavity field is

1.4675 \times 10^{- 5}

s, the maximum entangled state with fidelity of 92.8% can be obtained. Moreover, the feasibility of the scheme is also discussed.

Keywords

entanglement swapping fidelity large detuning quantum entanglement quantum optics

H = H_{0} + H_{I}

，（1）

H_{0} = \sum_{i = e, f, g} ω_{i} |i〉 〈i| + \sum_{j = 1}^{2} ν_{j} a_{j}^{†} a_{j}

，（2）

\begin{array}{l} H_{I} = g_{1} [a_{1}^{2} |f〉 〈g| + {(a_{1}^{†})}^{2} |g〉 〈f|] + \\ g_{2} [a_{2}^{2} |e〉 〈g| + {(a_{2}^{†})}^{2} |g〉 〈e|] \end{array}

，（3）

H_{I}^{'} = g_{1} [a_{1}^{2} |f〉 〈g| e^{i δ t} + {(a_{1}^{†})}^{2} |g〉 〈f| e^{- i δ t}] + g_{2} [a_{2}^{2} |e〉 〈g| e^{i δ t} + {(a_{2}^{†})}^{2} |g〉 〈e| e^{- i δ t}]

，（4）

δ = ω_{e} - ω_{g} - 2 ν_{2} = ω_{f} - ω_{g} - 2 ν_{1}

。（5）

|ψ (t)〉 = [{C_{|g〉,}}_{|n_{1}, n_{2}〉} (t) |g, n_{1}, n_{2}〉 + {C_{|f〉,}}_{|n_{1} - 1, n_{2}〉} (t) |f, n_{1} - 1, n_{2}〉 + {C_{|e〉,}}_{|n_{1}, n_{2} - 1〉} (t) |e, n_{1}, n_{2} - 1〉]

，（6）

\frac{d {C_{|g〉,}}_{|n_{1}, n_{2}〉} (t)}{d t} = - i g_{1} e^{- i δ t} {C_{|f〉,}}_{|n_{1} - 2, n_{2}〉} (t) \sqrt[]{n_{1} (n_{1} - 1)} - i g_{2} e^{- i δ t} {C_{|e〉,}}_{|n_{1}, n_{2} - 2〉} (t) \sqrt[]{n_{2} (n_{2} - 1)}

，（7）

\frac{d {C_{|f〉,}}_{|n_{1} - 2, n_{2}〉} (t)}{d t} = - i g_{1} e^{i δ t} {C_{|g〉,}}_{|n_{1}, n_{2}〉} (t) \sqrt[]{n_{1} (n_{1} - 1)}

，（8）

\frac{d {C_{|e〉,}}_{|n_{1}, n_{2} - 2〉} (t)}{d t} = - i g_{2} e^{i δ t} {C_{|g〉,}}_{|n_{1}, n_{2}〉} (t) \sqrt[]{n_{2} (n_{2} - 1)}

。（9）

\{\begin{matrix} C_{|g〉, |n_{1}, n_{2}〉} (0) = C_{|g〉} (0) C_{|n_{1}, n_{2}〉} (0) \\ C_{|f〉, |n_{1} - 2, n_{2}〉} (0) = C_{|f〉} (0) C_{|n_{1} - 2, n_{2}〉} (0) \\ C_{|e〉, |n_{1}, n_{2} - 2〉} (0) = C_{|e〉} (0) C_{|n_{1}, n_{2} - 2〉} (0) \end{matrix}

，（10）

C_{|g〉, |n_{1}, n_{2}〉} (t) = - \frac{i g_{2} \sqrt[]{n_{2} (n_{2} - 1)} s i n (Λ_{n_{1}, n_{2}} t)}{Λ_{_{n_{1}, n_{2}}}} e^{\frac{- i δ t}{2}} C_{|e〉} (0) C_{|n_{1}, n_{2} - 2〉} (0) - \frac{i g_{1} \sqrt[]{n_{1} (n_{1} - 1)} s i n (Λ_{n_{1}, n_{2}} t)}{Λ_{n_{1}, n_{2}}} e^{\frac{- i δ t}{2}} C_{|f〉} (0) C_{|n_{1} - 2, n_{2}〉} (0) + [c o s (Λ_{_{n_{1}, n_{2}}} t) + \frac{i δ s i n (Λ_{n_{1}, n_{2}} t)}{2 Λ_{n_{1}, n_{2}}}] e^{\frac{- i δ t}{2}} C_{|g〉} (0) C_{|n_{1}, n_{2}〉} (0)

，（11）

\begin{array}{l} C_{|f〉, |n_{1} - 2, n_{2}〉} (t) = \frac{g_{1} g_{2} \sqrt[]{n_{1} (n_{1} - 1) n_{2} (n_{2} - 1)}}{Λ_{n_{1}, n_{2}} α_{n_{1}, n_{2}}^{2}} γ_{n_{1}, n_{2}} (t) C_{|e〉} (0) C_{|n_{1}, n_{2} - 2〉} (0) + [\frac{g_{1}^{2} n_{1} (n_{1} - 1)}{Λ_{n_{1}, n_{2}} α_{n_{1}, n_{2}}^{2}} γ_{n_{1}, n_{2}} (t) + 1] C_{|f〉} (0) C_{|n_{1} - 2, n_{2}〉} (0) - \\ \frac{i g_{1} \sqrt[]{n_{1} (n_{1} - 1)}}{Λ_{n_{1}, n_{2}}} s i n (Λ_{n_{1}, n_{2}} t) e^{\frac{i δ t}{2}} C_{|g〉} (0) C_{|n_{1}, n_{2}〉} (0) \end{array}

，（12）

C_{|e〉, |n_{1}, n_{2} - 2〉} (t) = [\frac{g_{2}^{2} n_{2} (n_{2} - 1)}{Λ_{n_{1}, n_{2}} α_{n_{1}, n_{2}}^{2}} γ_{n_{1}, n_{2}} (t) + 1] C_{|e〉} (0) C_{|n_{1}, n_{2} - 2〉} (0) + \frac{g_{1} g_{2} \sqrt[]{n_{1} (n_{1} - 1) n_{2} (n_{2} - 1)}}{Λ_{n_{1}, n_{2}} α_{n_{1}, n_{2}}^{2}} γ_{n_{1}, n_{2}} (t) C_{|f〉} (0) C_{|n_{1} - 2, n_{2}〉} (0) - \frac{i g_{2} \sqrt[]{n_{2} (n_{2} - 1)}}{Λ_{n_{1}, n_{2}}} s i n (Λ_{n_{1}, n_{2}} t) e^{\frac{i δ t}{2}} C_{|g〉} (0) C_{|n_{1}, n_{2}〉} (0)

，（13）

\begin{array}{l} {|φ〉}_{1' 2' 34} = α_{1} α_{2} {|g〉}_{1'} {|0_{ν_{1}}, 0_{ν_{2}}〉}_{4} [C_{{|e〉}_{2'}, {|2_{ν_{1}}, 2_{ν_{2}}〉}_{3}}^{{|e〉}_{2'}, {|2_{ν_{1}}, 2_{ν_{2}}〉}_{3}} (t) {|e〉}_{2'} {|2_{ν_{1}}, 2_{ν_{2}}〉}_{3} + C_{{|g〉}_{2'}, {|2_{ν_{1}}, 4_{ν_{2}}〉}_{3}}^{{|e〉}_{2'}, {|2_{ν_{1}}, 2_{ν_{2}}〉}_{3}} (t) {|g〉}_{2'} {|2_{ν_{1}}, 4_{ν_{2}}〉}_{3} + \\ C_{{|f〉}_{2'}, {|0_{ν_{1}}, 4_{ν_{2}}〉}_{3}}^{{|e〉}_{2'}, {|2_{ν_{1}}, 2_{ν_{2}}〉}_{3}} (t) {|f〉}_{2'} {|0_{ν_{1}}, 4_{ν_{2}}〉}_{3}] + α_{1} β_{2} {|g〉}_{1'} {|2_{ν_{1}}, 2_{ν_{2}}〉}_{4} [C_{{|e〉}_{2'}, {|0_{ν_{1}}, 0_{ν_{2}}〉}_{3}}^{{|e〉}_{2'}, {|0_{ν_{1}}, 0_{ν_{2}}〉}_{3}} (t) {|e〉}_{2'} {|0_{ν_{1}}, 0_{ν_{2}}〉}_{3} + \\ C_{{|g〉}_{2'}, {|0_{ν_{1}}, 2_{ν_{2}}〉}_{3}}^{{|e〉}_{2'}, {|0_{ν_{1}}, 0_{ν_{2}}〉}_{3}} (t) {|g〉}_{2'} {|0_{ν_{1}}, 2_{ν_{2}}〉}_{3}] + α_{2} β_{1} {|e〉}_{1'} {|0_{ν_{1}}, 0_{ν_{2}}〉}_{4} [C_{{|g〉}_{2'}, {|2_{ν_{1}}, 2_{ν_{2}}〉}_{3}}^{{|g〉}_{2'}, {|2_{ν_{1}}, 2_{ν_{2}}〉}_{3}} (t) {|g〉}_{2'} {|2_{ν_{1}}, 2_{ν_{2}}〉}_{3} + \\ C_{{|f〉}_{2'}, {|0_{ν_{1}}, 2_{ν_{2}}〉}_{3}}^{{|g〉}_{2'}, {|2_{ν_{1}}, 2_{ν_{2}}〉}_{3}} (t) {|f〉}_{2'} {|0_{ν_{1}}, 2_{ν_{2}}〉}_{3} + C_{{|e〉}_{2'}, {|2_{ν_{1}}, 0_{ν_{2}}〉}_{3}}^{{|g〉}_{2'}, {|2_{ν_{1}}, 2_{ν_{2}}〉}_{3}} (t) {|e〉}_{2'} {|2_{ν_{1}}, 0_{ν_{2}}〉}_{3}] + β_{1} β_{2} {|e〉}_{1'} {|g〉}_{2'} {|0_{ν_{1}}, 0_{ν_{2}}〉}_{3} {|2_{ν_{1}}, 2_{ν_{2}}〉}_{4} \end{array}

，（14）

{|φ〉}_{1' 34} = \frac{1}{N} [α_{1} α_{2} C_{{|e〉}_{2'}, {|2_{ν_{1}}, 2_{ν_{2}}〉}_{3}}^{{|e〉}_{2'}, {|2_{ν_{1}}, 2_{ν_{2}}〉}_{3}} (t) {|g〉}_{1'} {|2_{ν_{1}}, 2_{ν_{2}}〉}_{3} {|0_{ν_{1}}, 0_{ν_{2}}〉}_{4} + α_{1} β_{2} C_{{|e〉}_{2'}, {|0_{ν_{1}}, 0_{ν_{2}}〉}_{3}}^{{|e〉}_{2'}, {|0_{ν_{1}}, 0_{ν_{2}}〉}_{3}} (t) {|g〉}_{1'} {|0_{ν_{1}}, 0_{ν_{2}}〉}_{3} {|2_{ν_{1}}, 2_{ν_{2}}〉}_{4} +

α_{2} β_{1} C_{{|e〉}_{2'}, {|2_{ν_{1}}, 0_{ν_{2}}〉}_{3}}^{{|g〉}_{2'}, {|2_{ν_{1}}, 2_{ν_{2}}〉}_{3}} (t) {|e〉}_{1'} {|2_{ν_{1}}, 0_{ν_{2}}〉}_{3} {|0_{ν_{1}}, 0_{ν_{2}}〉}_{4}]

，（15）

P = N^{2} = {|α_{1} α_{2} C_{{|e〉}_{2'}, {|2_{ν_{1}}, 2_{ν_{2}}〉}_{3}}^{{|e〉}_{2'}, {|2_{ν_{1}}, 2_{ν_{2}}〉}_{3}} (t)|}^{2} + {|α_{1} β_{2} C_{{|e〉}_{2'}, {|0_{ν_{1}}, 0_{ν_{2}}〉}_{3}}^{{|e〉}_{2'}, {|0_{ν_{1}}, 0_{ν_{2}}〉}_{3}} (t)|}^{2} + {|α_{2} β_{1} C_{{|e〉}_{2'}, {|2_{ν_{1}}, 0_{ν_{2}}〉}_{3}}^{{|g〉}_{2'}, {|2_{ν_{1}}, 2_{ν_{2}}〉}_{3}} (t)|}^{2}

。（16）

F = \frac{1}{2 N^{2}} {|α_{1} β_{2}_{{|e〉}_{1'}, {|0_{ν_{1}}, 0_{ν_{2}}〉}_{4}}^{{|e〉}_{1'}, {|0_{ν_{1}}, 0_{ν_{2}}〉}_{4}} (t) + α_{2} β_{1} C_{{|e〉}_{1'}, {|2_{ν_{1}}, 0_{ν_{2}}〉}_{4}}^{{|g〉}_{1'}, {|2_{ν_{1}}, 2_{ν_{2}}〉}_{4}} (t)|}^{2}

。（17）

Lei Zhang. Scheme for Quantum Entanglement Swapping in V-Type Three-Level Atom and Two-Mode Cavity Field Model[J]. Laser & Optoelectronics Progress, 2021, 58(19): 1927001

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