1. Introduction

Quantum memory is essential for the design of many devices for quantum information processing [1,2]. Photons are excellent carriers for quantum information due to their fast propagation velocity, low decoherence, multiple degrees of freedom, and easy for manipulation, etc. In recent years, much attention has been paid to the study of photon memory, which can store unknown input quantum states of light and retrieve them on-demand with controllable delays. In addition to a large amount of demonstrations of the optical storage by using weak coherent laser pulses [3–12], many experiments have also shown that the quantum memory of single photons can be realized efficiently [13–19].

Electromagnetically induced transparency (EIT), a destructive quantum interference effect typically occurring in three-level atomic gases [20], has been widely used to realize photon memory [4–7,9–12,14–21]. As another important application, EIT in atomic gases has also been used to realize optical splitters and routers [22–32], which are devices for sending or routing one incident optical signal into several output channels, useful for information processing in all-optical networks. Although there are many advantages, the optical memory and routing based on gaseous atoms have many shortcomings (such as atomic diffusion, large device sizes, etc.), which hampers the realization of compact chip-integrations. Thus, for practical applications it is desirable to have EIT and related quantum memory based on solid media that are easy to be miniaturized and hence to be integrated.

Based on this idea, a great deal of researches have been performed for the study of quantum memory by using solid materials (see, e.g., Refs. [3,8]), including quantum dots (QDs) particularly [1,2]. A QD is a very small solid structure (artificial atom), usually a semiconductor nanocrystal embedded in another semiconductor material, which can confine electrons or other carriers in all three spatial dimensions [33–37] and behaves like a natural atom in many ways. In comparison with natural atoms, QDs have the advantage for allowing artificially controls over the confining potential (and hence the electron band structure). The charge carries in QDs can be made to effectively interact with applied laser fields, and conversely the laser fields can be used to effectively control the carriers in the QDs for achieving strong and efficient optical responses.

In this work, we propose a scheme to realize photon storage and routing in gate defined QDs with spin-orbit coupling (SOC) [38–44], which can be fabricated on the heterointerface of GaAs, InAs nanowires, and two-dimensional materials. We show that the propagation, storage, and retrieval of single-photon wavepackets can be implemented in such QDs via the EIT. As one of interesting applications, routing of single-photon wavepackets may be implemented by a suitable manipulation of two control laser fields. Furthermore, we demonstrate that the propagation loss of the single photon wavepackets in the QDs can be largely suppressed by using a weak microwave field, and hence the storage and routing of single photon wavepackets may acquire high efficiency and fidelity.

Although considerable efforts were paid to the study on EIT and slow light in self-assembled QDs [45–66], and works on optical memory via EIT in QDs were also reported [67–69], there are many differences between our work and those reported before. First, the implementation of the EIT in our scheme is based on gate defined QDs, which is different from Refs. [45–66] where the self-assembled QDs were used which have large dephasing rates [70–74]; besides, the frequency-spectrum range for electronic excitations in the gated QDs with SOC used here is from infrared to microwave, which is useful for avoiding excitations of exciton and other carriers. Second, the Rashba SOC (which may be manipulated by the control of gate voltage) can mix different Landau levels and spins and hence may provides many new paths for optical transitions, flexible for realizing new types of EIT and optical memory. Third, though there were some studies on EIT in QDs with SOC [75–77], these studies did not consider quantized light fields and any memory for photons. Our works presented here is the first one for photon storage and routing based on the QDs with SOC, which may have promising applications for the design of novel solid-state devices for photonic quantum-information processing.

The remainder of the paper is arranged as follows. In Sec. 2., we describe our physical model and present the equations of motion controlling the dynamics of the QDs with SOC and the quantized laser fields. In Sec. 3., we study the EIT for single-photon propagations, and investigate the storage, retrieval and routing of single-photon wavepackets in the system. Finally, in the last section (Sec. 4.) we summarize the main results obtained in this work.

2. Physical model

2.1 Energy-level structure and electric-dipole matrix elements of the QDs with SOC

We consider a gated defined QD, which is manufactured by gate electrodes or by etching to confine a two-dimensional (2D) electron gas in the interface between the semiconductor and an insulator. The electrons (with the number tunable) are loosely confined in two ($x$ and $y$) directions but tightly trapped in the third ($z$) direction [36]. For simplicity, we consider here the InAs QDs with the Rashba SOC [35] (for other materials, the physics is not qualitatively changed as long as the SOC is present) and there is only one electron in the dot.

The Hamiltonian of the QD with Rashba SOC reads

The Hamiltonian $\hat{H}$ can be diagonalized in the basis of the 2D free harmonic oscillator described by $\hat{H}_0$ (which has eigen vectors $|n_x, n_y, s\rangle$, with $s=\pm 1$ the spin quantum numbers). The cut-off of the quantum numbers of $\hat{H}_0$ are set to be $n_{x,y} \in [0,10]$, which is accurate enough for our calculations. By the diagonalization, one can obtain eigen values $E_j$ and eign vectors $|\psi _j\rangle$ of $\hat{H}$, satisfying the equation $\hat{H}|\psi _{j}\rangle =E_{j} |\psi _{j}\rangle$ ($j=1,2,3,\ldots$). The electric-dipole matrix elements $\textbf{p}_{jk}$ between the eigen states $|\psi _j\rangle$ and $|\psi _k\rangle$ is given by $\textbf{p}_{jk}=\langle \psi _j |e\mathbf {r}| \psi _k\rangle =\frac {e}{\sqrt 2} I_{j,k}$, with the explicit expression of $I_{j,k}$ given in Sec. 1 of Supplement 1.

Shown in Fig. 1(a) is the energy-level structure of an InAs QD with SOC as a function of the magnetic field $B$ for $m=0.042\, m_{e}$ ($m_e$ is the electron mass), $\hbar \omega _x=\hbar \omega _y=8\,\textrm {meV}$, $g=-14$, and $\hbar g_1=40$ meV nm through a numerical diagonalization of $\hat{H}$, with the result given by $\left |\psi _1\right \rangle\simeq 0.99\left |0,0,1\right \rangle-i0.099\left |1,0, -1\right \rangle+0.099\left |0,1,-1\right \rangle$ and $\left |\psi _2\right \rangle\simeq 0.988\left |0,0,-1\right \rangle+i0.107 \left |0,1,1\right \rangle-0.107\left |1,0,1\right \rangle$ for $B=0.8\,\textrm {T}$. From the expressions of the two lowest energy levels, we see that the energy-level shifts caused by the contribution form $\hat{H}_{L_z}$ and $\hat{H}_\textrm {SOC}$ are small, which means that $\hat{H}_{L_z}$ and $\hat{H}_\textrm {SOC}$ can be treated as perturbations.

 

Fig. 1. Schematics of the model. (a) Energy-level structure of the InAs QD with Rashba SOC as a function of the magnetic field $B$ for $m=0.042\, m_{e}$, $\hbar \omega _x=\hbar \omega _y=8\,\textrm {meV}$, $g=-14$, and $\hbar g_1=40$ meV nm. Four levels for realizing photon storage and routing are selected for the magnetic field taken to be $B=0.8~\textrm {T}$ (indicated by the vertical dashed line). (b) Four-level double-$\Lambda$ configuration. Two weak probe laser fields (with center angular frequencies $\omega _{p1}$ and $\omega _{p2}$ respectively) and two CW control laser fields (with angular frequencies $\omega _{c1}$ and $\omega _{c2}$ respectively) are coupled with the QD levels $|1\rangle$ – $|4\rangle$ shown in (a). For suppressing the loss of the probe fields due to the spontaneous emission and dephasing of the QD, a microwave field (with center angular frequency $\omega _{m}$) is used to couple the two lower states $\left |1\right \rangle$ and $\left |2\right \rangle$. (c) An array of QDs located in the interface between the semiconductor and an insulator, and the geometrical arrangement of the input and output of the probe fields used for the photon storage and routing in the system.

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The gate defined QDs with SOC used in the present scheme possess many advantages. In addition to the controllability of the electron number, the SOC, which may be easily tuned through the manipulation of the gate voltage, may result in the mixing of different Landau levels and spins, and hence lead to the change of the parity of the QD eigen states. Moreover, the gaps between different eigen states can be easily adjusted by the magnetic field and the SOC. Based on these interesting features, the scheme described here allows many new excitation paths for optical transitions and hence provides flexible ways for the built-up of EIT and the realization of new types of photon storage and routing, as shown below.

2.2 Heisenberg-Langevin-Maxwell equations

To realize photon storage and routing, we choose a double-$\Lambda$ type four-level configuration by taking the magnetic field $B=0.8~\textrm {T}$ to break the Kramers degeneracy [indicated by the vertical dashed line in Fig. 1(a)]. The probe laser field with center angular frequency $\omega _{p1}$ ($\omega _{p2}$) couples the QD levels $|1\rangle$ and $|3\rangle$ ($|1\rangle$ and $|4\rangle$); the control laser field with angular frequency $\omega _{c1}$ ($\omega _{c2}$) couples the levels $|2\rangle$ and $|3\rangle$ ($|2\rangle$ and $|4\rangle$). $\Delta _{3}$ and $\Delta _{4}$ are one-photon detunings; $\Delta _{2}$ is two-photon detuning. In order to realize light propagation, we consider an array of QDs, shown in Fig. 1(c). The geometrical arrangement of the input and output of the probe fields used for the photon storage and routing is also shown in Fig. 1(c). For simplicity, we assume that the separation between QDs in the array is large so that the coupling between the QDs can be neglected.

Since the energy scale of the QD is in the order of meV, the corresponding optical wavelength is above $10~\mu$m, which is more than two orders of magnitude larger than the size of the QDs, and hence electric-dipole approximation can be used for solving the Maxwell equations that control the evolution of the two probe fields. The eigen energies of the levels we chosen are $E_{1} =5.96\,\mathrm {meV}$, $E_{2} =12.13 \,\mathrm {meV}$, $E_{3} =21.09 \,\mathrm {meV}$, and $E_{4} =24.43 \,\mathrm {meV}$. The corresponding electric-dipole matrix elements are $|\textbf{p}_{31}|=5.38\times 10^{-26}\,\textrm {C~cm}$, $|\textbf{p}_{41}|=4.59\times 10^{-26}\,\textrm {C~cm}$, $|\textbf{p}_{32}|=7.98\times 10^{-26}\,\textrm {C~cm}$, and $|\textbf{p}_{42}|=1.52\times 10^{-25}\,\textrm {C~cm}$. Although at $B=0.8~\textrm {T}$ a few other levels are close to the level $\left |3\right \rangle$, they can be neglected because the electric-dipole elements between these levels and $|1\rangle$ or $|2\rangle$ are vanishing small.

The total electric field of the system can be expressed as $\hat{\textbf{E}}(\textbf{r},t)=\sum _{j=1,2}\left [\hat{\textbf{E}}_{pj}(\textbf{r},t)+\textbf{E}_{cj}(\textbf{r},t)\right ]$, where the two probe fields $\hat{\textbf{E}}_{p1}$ and $\hat{\textbf{E}}_{p2}$ are assumed to be weak, pulsed, and quantized, with forms given by $\hat{\textbf{E}}_{pj}(\textbf{r},t)=\textbf{e}_{pj} {\cal E}_{pj}\hat{E}_{pj}(\textbf{r},t)e^{i[k_{pj}x-\omega _{pj}t]}+\textrm{h.c.}$; the two control fields $\textbf{E}_{c1}$ and $\textbf{E}_{c2}$ are strong (classical) continuous-wave (CW) fields with the forms $\textbf{E}_{cj} (\textbf{r},t)=\textbf{e}_{cj}{\cal E}_{cj}(\textbf{r},t)e^{i[k_{cj}x-\omega _{cj}t]}+\textrm {c.c.}$ ($j=1,2$). Here h.c. (c.c.) represents Hermitian (complex) conjugate; $\textbf{e}_{pj}$ ($\textbf{e}_{cj}$) is the unit polarization vector and ${\cal E}_{pj}\equiv \sqrt {\hbar \omega _{pj}/(2\varepsilon _{0}V)}$ (${\cal E}_{cj}$) is the mode amplitude of $j$th probe (control) field, with $V$ the mode volume; $\hat{E}_{pj}(\textbf{r},t)$ (photon annihilation operators) obey the commutation relation $[\hat{E}_{pj} (\textbf{r},t),~\hat{E}_{pl}^{\dagger}(\textbf{r}',t)]=V\delta _{jl}\delta (\textbf{r}'-\textbf{r})$.

Under the electric-dipole, rotating-wave, and slowly-varying envelope approximations, the Hamiltonian of the system is reduced to [78]

The dynamics of the system is governed by the Heisenberg-Langevin-Maxwell (HLM) equations [79]

3. Results and discussion

3.1 EIT and single-photon property of the probe field during propagation

To show clearly the quantum property of single-photon wavepacket, we investigate the propagation of a single probe field in a QD system with a three-level $\Lambda$-type configuration, which is a reduced version of the system proposed in Sec. 2.2 obtained by taking $\Delta _{4}\rightarrow \infty$. In this situation, the scheme contains only the probe field $\hat{\textbf{E}}_{p1}$, the control field $\textbf{E}_{c1}$, and the levels $|1\rangle$, $|2\rangle$, $|3\rangle$. Then the Hamiltonian of the system has the simple form $\hat{H}_\textrm {H}=-\frac {i\hbar c}{V}\int _{-\infty }^{+\infty }d^3r\hat{E}_{p}^{\dagger}\frac {\partial }{\partial x}\hat{E}_{p}-\hbar \mathcal {N}\int _{-\infty }^{+\infty }d^3r\left (\sum _{j=1}^{3}\Delta _{j}\hat{S}_{jj}+g_{p}\hat{S}_{31}^{\dagger}\hat{E}_{p}+\Omega _{c}\hat{S}_{32}^{\dagger}+\textrm {h.c.}\right )$. Note that $g_{p1}\hat{E}_{p1}$ and $\Omega _{c1}$ have been replaced respectively by $g_{p}\hat{E}_{p}$ and $\Omega _{c}$ for simplifying the notations.

In the absence of the probe field, the steady-state solution of the system read $\hat{S}_{11}^{(0)}=\hat{I}$ and $\hat{S}_{22}^{(0)}=\hat{S}_{33}^{(0)}=0$. If the probe field is present and it contains only a few photons, the solution for single-photon propagation in the QDs can be obtained by linearizing the relevant HLM equations. Then we obtain

The imaginary part [Im$(K)$] and the real part [Re$(K)$] of $K(\omega )$ as a function of the sideband frequency $\omega$ of the probe field are shown in Fig. 2(a). We see that an absorption minimum (i.e., a transparency window) is opened near $\omega =0$ [80]. The occurrence of such absorption minimum is due to the quantum destruction interference (i.e., EIT) effect induced by the control field. When plotting the figure, the density of the QDs is taken to be ${\cal N}=1.25\times 10^{16}\,\textrm {cm}^{-3}$, the other system parameters are $\Delta _{2}=2\pi \times 71.6~\textrm {MHz}$, $\Delta _{3}=-2\pi \times 7.196~\textrm {GHz}$, $\Gamma _{3}=2\pi \times 0.48~\textrm {GHz}$, $\gamma _{21}=2\pi \times 0.16~\textrm {GHz}$, $\gamma _{31}=2\pi \times 0.24~\textrm {GHz}$, $\gamma _{32}=2\pi \times 0.24~\textrm {GHz}$, and $\Omega _{c}=2\pi \times 12.74~\textrm {GHz}$.

 

Fig. 2. Linear dispersion relation $K$, photon-number evolution $n_a$, and second-order coherence function $g^{(2)}$ of the probe field for a single-photon input. (a) Dashed blue (solid red) line is the imaginary (real) part of $K$, i.e., Im($K$) (Re($K$) ) as a function of $\omega$. (b) $n_{a}$ (solid red line) and $n_{b}$ (solid blue line) as functions of $x$, contributed from the input of the probe field and the spontaneous emission of the atoms respectively. (c) The normalized second-order coherence function $g^{(2)}$ characterizing the quantum statistical property of the probe photons. The parameters used for plotting the figure are given in the text.

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One of the main difficulties for the quantum information processing using QDs is the large decoherence originated from electron-phonon scattering, size fluctuations, scattering due to interface roughness, and electron-electron scattering, etc. For self-assembled QDs at low temperature (e.g., near zero K [72]), the dephasing time in excess of a nanosecond are attainable [48,53,62,71–74]. In the calculation given above, we have assumed that the system works at low temperature ($\sim 10$ mK), and the dephasing time is at the order of magnitude of one nanosecond. Such assumption is reasonable because the gate defined QDs are generally cleaner than self-assembled ones and hence their dephasing time should not be shorter than that of the self-assembled QDs.

Now we consider the propagation property of a probe photon in the system. We assume that a single-photon wavepacket is incident onto the left boundary $x=0$, with the quantum state of the input probe field of the form

The quantum statistical property of the probe field can be checked by calculating its photon number and the normalized second-order coherence function. The photon number in the probe field as a function of the propagation distance $x$ is given by $n_{p}(x)=\int _{-\infty }^{+\infty }dtR(x,t)$, with $R(x,t)\equiv (c/L)\langle \hat{E}_{p}^{\dagger}(x,t)\hat{E}_{p}(x,t)\rangle$ the generation rate of probe photons. Here $\langle \cdots \rangle = \langle \Psi _{1}|\cdots |\Psi _{1}\rangle$ and $|\Psi _{1}\rangle =|\Psi ^\textrm {1P} \rangle \otimes |\Psi ^\textrm {Q}\rangle \otimes |\Psi ^\textrm {R}\rangle$, with $|\Psi ^\textrm {Q}\rangle$ the initial state-vector of the system and $|\Psi ^\textrm {R}\rangle =|\{0\}\rangle$ the initial state-vector of the vacuum reservoir coupled to the system. The input one-photon state (6) ensures that the photon number of the input probe field is $n_{p}=1$. The photon number of the probe field at the location $x$ in the QD ensemble can be obtained by calculating the correlation function of the Langevin operators, with the result given by

Shown in Fig. 2(b) are $n_{a}$ and $n_{b}$ as functions of the propagation distance $x$. When plotting the figure, we have taken $\Delta \omega _p=4.5~{\rm GHz}$ and all other system parameters are the same as those used in Fig. 2(a). We see that the photon number in the probe field, i.e., $n_{a}$, remains less than one; the noise part form the spontaneous emission, i.e., $n_{b}$, is very small. Such results stem from the EIT effect that makes the spontaneous emission in the system largely suppressed. One notes $n_b$ rises up from zero initially and then goes down as the propagation distance $x$ increases, which can be explained as the following: (i) when the probe photon is incident to the QD medium at $x=0$, its coupling with the levels $|1\rangle$ and $|3\rangle$ results in an increase of the population in the level $|3\rangle$ [i.e. ${\hat{\tilde{S}}}_{33}(x,0)$; and hence $n_b$] from initial zero to a finite value as the propagation distance $x$ increases; (ii) at some position $x$, the population in the level $|3\rangle$ begins to decay due to the action of the spontaneous emission of the level $|3\rangle$, which results in the decrease of $n_b$; (iii) the dephasing of the system brings an additional decay of $n_b$, contributed by the exponential factor Im[$K(\omega )$] in Eq. (7c) of the main text.

The normalized second-order coherence function $g^{(2)}(x,t_{1},t_{2})$, which characterizes the quantum statistical property of the probe photons, is defined by $g^{(2)}(x,t_{1},t_{2})=\frac {G^{(2)}(x,t_1,t_2)}{ R(x,t_{1})R(x,t_{2})}$, where $G^{(2)}\equiv (c/L)^2\langle \hat{E}_{p}^{\dagger}(x,t_{1})\hat{E}_{p}^{\dagger}(x,t_{2})\hat{E}_{p}(x,t_{2})\hat{E}_{p}(x,t_{1})\rangle$ is the Glauber second-order coherence function. The input one-photon state $|\Psi ^\textrm {1P}\rangle$ given by Eq. (6) ensures $g^{(2)}(0,t_{1},t_{2})=0$. The detailed calculation for $g^{(2)}(x,t_{1},t_{2})$ is given in Sec. 3C of Supplement 1.

Figure 2(c) shows the profile of $g^{(2)}$ as a function of $x$ for zero delay time (i.e., $t_{1}=t_{2}$) by setting $t_{1}=t_{2}=x/V_{g}$. From the figure we see that $g^{(2)}$ is always much smaller than one, which means that the single-photon property of the probe field can be well preserved during propagation.

3.2 Storage and retrieval of single-photon wavepackets

Now we turn to consider the photon memory and routing via the photon EIT descried above. We follow the method adopted in Refs. [81,82], where it was shown that the effective wavefunction method working in Schrödinger picture is very convenient for the calculation of photon memory. For the double-$\Lambda$ type four-level system, the state vector for single-particle excitations reads [83]

For photon storage and retrieval, only the three-level $\Lambda$-type configuration described in Sec. 3.1 is needed, for which one has $\Phi _2=\Phi _4=0$. Thus the equation of motion for the single-particle wavefunction takes the form

The storage and retrieval of the single-photon wavepacket can be realized by manipulating the control field, whose switching off and on can be described by [32]

 

Fig. 3. Storage and retrieval of single-photon wavepacket $|\Phi _1|$ as a function of the dimensionless time $t/\tau _0$, with $\tau _0=1\times 10^{-10}~\textrm {s}$. The leftmost orange, the right green, and the right blue profiles are single-photon wavepackets for the input, the retrieved at low temperature, and the retrieved at low temperature, respectively. Both two retrieved pulses are obtained at the position $x=500~\mu\textrm{m}$. The red line in the upper part of the figure is the time-sequence of the control field $\Omega _{c}$, designed for realizing the storage and retrieval of the single-photon wavepacket. The microwave field $\Omega _{M}$ designed to suppress the loss of the probe field is applied only during storage period.

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Since there exists a significant loss of the probe field during propagation due to the large decoherence in the QDs, as in Ref. [32] we apply a microwave field with half Rabi frequency $\Omega _M$ to the system, which couples the two lowest levels $|1\rangle$ and $|2\rangle$ (see Fig. 1), with the Hamiltonian given by Eq. (S10) in Supplement 1. The switching on and off of the microwave field can be described by

Note that the microwave field used here will mix the population in the two lower levels $|1\rangle$ and $|2\rangle$ during the storage period, and hence enhance the coherence between these two levels and suppress the large dephasing in the QDs. Although the electric-dipole matrix element between these two levels is zero, the microwave field can provide a magnetic-dipole coupling between them. We assume that the magnetic field provided by the microwave field is along $z$ axis, so that it can couple to the $z$ component of the spin and angular momentum of the QDs. The magnetic dipole moment in the QDs are contributed from the spin and the orbit angular momentum, while the spins of all the levels are mixed up by the SOC.

To get a high storage efficiency of the probe photon, we assume that the amplitude of the microwave field is a function of $x$. For simplicity, we take $\Omega _{\mathcal {M}}(x) =\Omega _{M0}\exp \{-[(x-a)/b]^2\}$, where $\Omega _{M0}\tau _0=0.17$ (i.e., $\Omega _{M0}=2\pi \times 272~\textrm {MHz}$, corresponding to ${\cal B}_m\sim 10^{-2}~{\rm T}$); $a$ and $b$ are two arbitrary parameters that are chosen to optimize the storage efficiency. We take $a=25~$ $\mu$m and $b=10$ $\mu$m, with the other system parameters the same as those used for plotting Fig. 2(a).

The processes of the storage and retrieval of single-photon wavepacket can be described as follows. (i) By switching on the control field $\Omega _{c}$, the input single-photon wavepacket $\Phi _1(x,t)=f(x,t)$ [i.e., the left orange pulse profile in Fig. 3] begins to propagate in the system. (ii) By switching off $\Omega _{c}$ at $t=T_\textrm {off}^c=10.0\tau _0$, the single-photon wavepacket disappears (i.e., it is stored in the QDs); during the storage period, the microwave field is switching on at $t=T_\textrm {on}^M=11.0\tau _0$, which suppresses the decoherence of the QDs and hence decrease the loss of the stored single-photon wavepacket. (iii) By switching off the microwave field at $t=T_\textrm {off}^M=19.0\tau _0$ and switching on the control field $\Omega _{c}$ again at $t=T_\textrm {on}^c=20.0\tau _0$, the single-photon wavepacket appears again (i.e., it is retrieved).

The right green pulse profile shown in Fig. 3 is the result of the retrieved single-photon wavepacket $\Phi _1(x,t)$ at the position $x=500\,\mu$m. We see that (i) the single-photon wavepacket can be stored and retrieved in the system through the manipulation of the control field; (ii) the amplitude and shape of the single-photon wavepacket can be better preserved even when the wavepacket has experienced a longer processes of the storage and retrieval, which is due to the contribution of the microwave field that suppresses the decoherence of the system significantly. The storage of the single-photon wavepacket via the EIT can be well described by using the concept of dark-state polaritons [20,32].

The efficiency of the memory of the single-photon wavepacket can be characterized by $\eta ={\int _{-\infty }^{+\infty } d t\left |\Phi _{\mathrm {out}}(t)\right |^{2}}/{\int _{-\infty }^{+\infty } d t\left |\Phi _{\mathrm {in}}(t)\right |^{2}}$ [81], with $\Phi _\textrm {in}(t)=\Phi _{1}(0,t)$ and $\Phi _\textrm {out}(t)=\Phi _{1}(L_x,t)$ ($L_x$ is the propagation length, which is $500~{\rm \mu} m$ in our system). Based on the result given in Fig. 3, we obtain $\eta =62.4\%$.

Another parameter characterizing the single-photon memory is the likeness degree of waveshape defined by the overlap integral $J^2={|\int _{-\infty }^{+\infty } dt\Phi _\textrm {in}\Phi _\textrm {out}|^2}/(\int _{-\infty }^{+\infty } dt\left |\Phi _\textrm {in}\right |^2\int _{-\infty }^{+\infty } dt\left |\Phi _\textrm {out}\right |^2)$ [18], where $\Phi _\textrm {in}=\Phi _\textrm {in}(t)$ and $\Phi _\textrm {out}=\Phi _\textrm {out}(t+\Delta T)$, with $\Delta T$ the time interval between the peak of the input single photon wavepacket $\Phi _\textrm {in}(t)$ and the peak of the output wavepacket $\Phi _\textrm {out}(t+\Delta T)$. From the result given in Fig. 3, we get $J^2=98.6\%$. The fidelity of the single-photon wavepacket during the storage and retrieval can be described by the parameter $\eta J^2$, which is $61.5\%$ based on the above results [84].

For comparison, a numerical simulation of the single-photon memory is also carried out for the system working at a higher temperature (e.g., $30$ K). The right blue pulse profile plotted in Fig. 3 is the result of the retrieved single-photon wavepacket $\Phi _1(x,t)$ at the position $x=500~\mathrm {\mu} \textrm{m}$. We see that in this case the single-photon wavepacket can be also stored and retrieved. However, due to the temperature-dependent dephasing, after the retrieval the amplitude of the wavepacket is attenuated significantly compared with the result working at low temperature (i.e., the right green pulse profile in Fig. 3), which results in a very low memory efficiency of the single-photon wavepacket (i.e., $\eta =36.6\%$).

3.3 Routing of single-photon wavepackets

Finally, as an application of the photon memory illustrated above, we explore the possibility of a single-photon routing by employing the double-$\Lambda$-type four-level configuration (see Fig. 1) based on Eq. (S34) of Supplement 1. We emphasize that the single-photon routing described below does not imply that the single photon is split into two photons but can be read out through one of two different retrieved channels.

To implement the routing of the single-photon wavepacket, we assume the control fields $\Omega _{c1}$ and $\Omega _{c2}$ can be modeled by the combination of the following hyperbolic tangent functions

 

Fig. 4. Routing of single photons. (a) The left orange pulse profile is the input single-photon wavepacket $\Phi _{1}(0,t)=0.535e^{-t^2/(2.8\tau _0)^2}$ of the probe field $\hat{E}_{p1}$ as a function of $t/\tau _0$ during the routing process; the right blue pulse profile is the output single-photon wavepacket $\Phi _{1}(x,t)$ at $x=500\,\mu$m. (b) The output of the single-photon pulse profile $\Phi _{2}$ of the probe field $\hat{E}_{p2}$ as a function of $t/\tau _0$ during the routing process, retrieved from another channel at $x=500\,\mu$m. The routing is realized by the manipulation of the two control fields $\Omega _{c1}$ and $\Omega _{c2}$ and a microwave field $\Omega _{M}$. The timing sequences of $\Omega _{c1}$, $\Omega _{c2}$, and $\Omega _M$ are plotted in the upper part of the both panels.

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Shown in Fig. 4 is the result of a numerical simulation for the single-photon routing obtained by solving Eq. (S34) in Supplement 1. In the simulation, we choose $\Omega _{{\mathcal {C}}1}\tau _0=\Omega _{{\mathcal {C}}2}\tau _0=4$ (i.e., $2\pi \times 6.36~\textrm {GHz}$), $\Omega _{M0}\tau _0=0.30$ (i.e., $2\pi \times 477~\textrm {MHz}$), $\gamma _{41}=2\pi \times 0.24~\textrm {GHz}$, $\gamma _{42}=2\pi \times 0.24~\textrm {GHz}$, and the other parameters are the same as those used in Fig. 3. The left orange pulse profile plotted in Fig. 4(a) is the input single-photon wavepacket $\Phi _{1}(0,t)=0.535e^{-t^2/(2.8\tau _0)^2}$ of the probe field $\hat{E}_{p1}$ as a function of $t/\tau _0$ (noting that there is no input for the probe field $\hat{E}_{p2}$). The right blue pulse profile is the output of, i.e. the retrieved, single-photon wavepacket $\Phi _{1}(x,t)$ of $\hat{E}_{p1}$ at the position $x=500\,\mu$m. Figure 4(b) shows the result of the output of the single-photon pulse profile of $\hat{E}_{p2}$, i.e. $\Phi _{2}$, retrieved from another channel also at $x=500\,\mu$m. We see that the routing of the single-photon wavepacket can indeed be implemented in the system. The efficiency $\eta$ of the routing of the single-photon wavepacket can also be calculated by defining $\Phi _\textrm {out}(t)=\Phi _{1}(L_x,t) +\Phi _{2}(L_x,t)$. Based on the results given by Fig. 4(a) and Fig. 4(b), we obtain $\eta =69.8\%$. Note that the two output single photons have different frequencies, i.e. $\omega _{p1}$ for the probe field 1 and $\omega _{p2}$ for the probe field 2, and hence a frequency router is realized by this routing process. Obviously, one can also realize a frequency router with more than two output photon-channels if the number of the control fields larger than two are employed in the retrieval process.

4. Summary

In this work, we have proposed a scheme for realizing highly-efficient and controllable storage and routing of single photons based on gate defined QDs with the Rashba SOC via EIT. We have shown that the SOC in the QDs can be exploited to change the parity of the QD eigenstates and hence to provide a flexible built-up of photon EIT. We have also shown that the propagation, storage, retrieval, and routing of single-photon wavepackets can be implemented by using such a scheme. Furthermore, we have demonstrated that the loss of single-photon wavepackets due to the dephasing in the QDs can be largely suppressed by applying a weak microwave field within the storage period, thereby the storage and routing of single photons can have high efficiency and could be actively controlled by tuning the SOC parameters of the QDs. Our work can be easily extend to the storage and routing of multiple spatial and temporal modes of photons, and opens a route for promising practical applications in photonic information processing and transmission based on the QDs with SOC.