Quantum memory of single-photon polarization qubits via double electromagnetically induced transparency

Qi Zhang and Guoxiang Huang

Phys. Rev. A 104, 033714 – Published 24 September 2021

Abstract

We present a theoretical investigation on the quantum memory of photonic polarization qubits via electromagnetically induced transparency (EIT). The system we consider is a tripod-shaped four-level atomic system working under the condition of a double EIT, by which the storage and retrieval of a single-photon polarization qubit are implemented through the switching off and on of a control laser field, and the storage efficiency and the quantum-state fidelity for qubit memory are both calculated. We show that the optimal optical depth for acquiring the maximum efficiency and maximum fidelity of the qubit memory can be obtained simultaneously, which can be further improved by suppressing the optical absorption and dispersion via the choice of the time duration of the input qubit pulse and the amplitude of the control field. We also carry out a calculation on the quantum memory of a single-photon qudit by considering a multipod-shaped atomic-level configuration. The results reported here are useful for understanding the quantum transmission property of slow lights with multiple components and helpful for experimental realizations of high-quality memory of photonic qubits and qudits.

Received 13 April 2021
Revised 5 September 2021
Accepted 13 September 2021

DOI:https://doi.org/10.1103/PhysRevA.104.033714

Physics Subject Headings (PhySH)

Quantum memories Quantum optics

Atomic, Molecular & OpticalQuantum Information, Science & Technology

Authors & Affiliations

Qi Zhang¹ and Guoxiang Huang ^1,2,3

¹State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai 200062, China
²Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China
³NYU-ECNU Joint Institute of Physics at NYU-Shanghai, Shanghai 200062, China

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Vol. 104, Iss. 3 — September 2021

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Figure 1
(a) Energy-level diagram and excitation scheme of the tripod-shaped four-level atomic gas, proposed to realize the QM of a single-photon polarization qubit. The signal field with central angular frequency $ω_{p}$ is decomposed into a superposition of right-circularly ( $σ^{+}$ ) and left-circularly ( $σ^{-}$ ) polarized components. $ω_{c}$ is the central angular frequency of the control field used to realize the double EIT. $Δ_{4}$ and $Δ_{3}$ are one-photon and two-photon detunings, respectively. $Γ_{4}$ is the decay rate of spontaneous emission from the excited state $| 4 〉$ . (b) Possible experimental geometry; both the signal and control fields propagate along the $z$ direction. (c) Linear dispersion relation $K_{+}$ ( $K_{-}$ ) of the $σ^{+}$ ( $σ^{-}$ )-polarized component of the signal field as a function of the sideband frequency $ω$ ; Re( $K_{j}$ ) and Im( $K_{j}$ ) are the real and imaginary parts of $K_{j}$ ( $j = +, -$ ), respectively. $γ_{P} = γ_{41} \approx γ_{42}$ , where $γ_{4 j}$ $(j = 1, 2)$ are the dephasing rates between the ground-state sublevel $| j 〉$ and the excited state $| 4 〉$ .
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Figure 2
Propagation of the photon polarization qubit. (a) $σ^{+}$ -polarized component of the qubit wave function $| Φ_{E +} | / | Φ_{E 0} |_{\max}$ as a function of nondimensional traveling coordinate $γ_{P} τ$ ( $γ_{P} = 2 π \times 3$ MHz). Lines 0, 1, 2, 3, and 4 are for $| Φ_{E +} | / | Φ_{E 0} |_{\max}$ at $z = 0$ , $z = 0.25 L_{disp}$ , $z = 0.75 L_{disp}$ , $z = 1.25 L_{disp}$ , and $z = 1.75 L_{disp}$ , respectively, with $L_{disp} \approx 2.3$ cm. (b) The same as (a) but for the $σ^{-}$ -polarized component of the qubit wave function $| Φ_{E -} | / | Φ_{E 0} |_{\max}$ .
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Figure 3
Storage and retrieval of the single-photon polarization qubit. (a) The efficiency $η$ (dashed blue line) and the quantum-state fidelity $F$ (solid red line) of the qubit memory as functions of optical depth $D = D (z) \equiv | G_{p} |^{2} z / (2 c γ_{P})$ . Solid red circle 1 corresponds to the optimal optical depth, i.e., $D = D^{opt} \approx 110$ , at which maximum memory efficiency $η_{\max} \approx 0.71$ and maximum quantum-state fidelity $F_{\max} \approx 0.67$ can be obtained almost simultaneously. (b1) Evolution of the modulus of the qubit wave packet of the $σ^{+}$ -polarized component (i.e., $| Φ_{E +} | / | Φ_{E 0} |_{\max}$ ) as a function of $γ_{P} τ$ and $D$ in the process of the qubit memory. The wave packet shown by the bright stripe in the lower left (upper right) is $| Φ_{E +} | / | Φ_{E 0} |_{\max}$ before (after) the storage. (c1) $| Φ_{E +} | / | Φ_{E 0} |_{\max}$ as a function of $γ_{P} τ$ . Solid magenta line 0, $| Φ_{E +} | / | Φ_{E 0} |_{\max}$ for $D = 0$ . Solid red line 1, $| Φ_{E +} | / | Φ_{E 0} |_{\max}$ for optimal optical depth $D = D^{opt} \approx 110$ , corresponding to solid red circle 1 in (a) and also dashed red line 1 in (b1). Solid orange line 2, $| Φ_{E +} | / | Φ_{E 0} |_{\max}$ for $D \approx 390$ , which corresponds to solid orange circle 2 in (a) and also dashed orange line 2 in (b1). (b2) and (c2) are the same as (b1) and (c1), respectively, but for the qubit wave packet of the $σ^{-}$ -polarized component (i.e., $| Φ_{E -} | / | Φ_{E 0} |_{\max}$ ). Solid black lines in (c1) and (c2) are the half Rabi frequency $Ω_{c}$ given by Eq. (13), illustrated here to show its switching off and on during the process of the qubit memory. The small solid red curve circled by a dashed black ring in (c1) [(c2)] is the leaky part of $| Φ_{E +} | / | Φ_{E 0} |_{\max}$ ( $| Φ_{E -} | / | Φ_{E 0} |_{\max}$ ) at the optimal optical depth $D^{opt} \approx 110$ .
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Figure 4
(a1) Maximum memory efficiency $η_{\max}$ (circle-marked dashed blue line) and fidelity $F_{\max}$ (circle-marked solid red line) and their corresponding optimal optical depths $D^{opt}$ (triangle-marked dashed blue and solid red lines, respectively) as functions of $γ_{P} t_{0}$ for $Ω_{c m} = 3 γ_{P}$ . (a2) Right part: optimized retrievals of the qubit pulse for the $σ^{+}$ -polarized (solid red line 1) and $σ^{-}$ -polarized (dashed blue line 2) components [74] at the optimal optical depth $D^{opt} \approx 135$ as functions of $γ_{P} τ$ for $Ω_{c m} = 3 γ_{P}$ and $t_{0} = 6.6 / γ_{P}$ , giving rise to $F_{\max} \approx η_{\max} \approx 0.72$ [corresponding to the dashed red line in (a1)]. Left part: the small solid red and dashed blue curves are leaked qubit components of the retrieved qubit pulse. (b1) Maximum memory efficiency $η_{\max}$ (circle-marked dashed blue line) and fidelity $F_{\max}$ (circle-marked solid red line) and their corresponding optimal optical depths $D^{opt}$ (triangle-marked solid red and dashed blue lines, respectively) as functions of $Ω_{c m} / γ_{P}$ for $t_{0} = 3 / γ_{P}$ . (b2) Optimized retrievals of the wave packets of the $σ^{+}$ -polarized (solid red line 1) and $σ^{-}$ -polarized (dashed blue line 2) components [74] at the optimal optical depth $D^{opt} \approx 580$ as functions of $γ_{P} τ$ for $t_{0} = 3 / γ_{P}$ and $Ω_{c m} = 6.8 γ_{P}$ , giving rise to $F_{\max} \approx η_{\max} \approx 0.88$ [corresponding to the dashed red line in (b1)]. Solid magenta curve 0 in (a2) and (b2) denotes the incident qubit wave packets (corresponding to $D = 0$ ; its two components have been chosen to have the same wave shape and hence coincided).
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Figure 5
(a) Energy-level diagram and excitation scheme of an $(N + 1)$ -pod atomic system. ${\hat{E}}_{j} (j = 1, ..., N)$ is the photon annihilation operator of the $j th$ component coupling with the atomic transition $| j 〉 \leftrightarrow | N + 2 〉$ ; the control field with amplitude $E_{c}$ drives the transition $| N + 1 〉 \leftrightarrow | N + 2 〉$ . (b) The efficiency $η$ (dashed blue line) and the fidelity $F$ (solid red line) of the memory of a single-photon qudit as a function of the optical depth $D = D (z) \equiv | G_{p} |^{2} N z / (3 c γ_{P})$ for $N = 3$ . The solid red circle denotes the position $D = D^{opt} \approx 110$ for achieving the maximum fidelity $F_{\max} \approx 0.62$ . (c) Right part: $| Φ_{E j} | / | Φ_{E 0} |_{\max} (j = 1, 2, 3)$ as functions of $γ_{P} τ$ retrieved at optimal optical depth $D = D^{opt} \approx 110$ [corresponding to the solid red circle in (b)]. Solid red line 1 is for $| Φ_{E 1} | / | Φ_{E 0} |_{\max}$ at an optimal optical depth $D = D^{opt} \approx 110$ ; dashed blue line 2 is the same as solid red line 1, but for $| Φ_{E 2} | / | Φ_{E 0} |_{\max}$ ; dash-dotted cyan line 3 is the same as solid red line 1, but for $| Φ_{E 3} | / | Φ_{E 0} |_{\max}$ [77]. Left part: solid magenta curve 0 is the input qudit pulse (corresponding to $D = 0$ ; its three components have been chosen to have the same wave shape and hence coincided); the small solid red, dashed blue, and dash-dotted cyan curves are leaked qudit components of the retrieved qudit pulse. The solid black line in the figure is the half Rabi frequency $Ω_{c}$ given by Eq. (13).
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