1. Introduction

Hybrid quantum systems composing of two or more different types of quantum subsystems, allow us to make full use of the complementary functionalities of different physical components [1–4]. In recent years, there has been growing interest in microwave (mw) and optical quantum interface. Superconducting quantum circuits [5,6], electron spin systems [7], and nuclear spins [8–10] are suitable for quantum computation. These systems work in the mw domain with energy in the level of GHz or MHz. However, it is inconvenient to distribute the mw qubits over long distance because the mw signal is easy to be disturbed by thermal noise at room temperature and decays rapidly during propagation. In stark contrast, the flying photons in the optical domain are robust to thermal noise and can easily transmit quantum information over long distance in optical fibers. For the purpose of building quantum network for remote separated superconducting quantum system, the efficient mw-to-optical conversion is highly demanded.

In the past decade, A lot of efforts have been done to perform the mw-to-optical conversion using Rydberg atom and spin $\Lambda$-type systems [11–16], electro-optic techniques [17–19], magneto-optic techniques [20–24] and optomechanical systems [25–34]. Despite lots of efforts, the conversion efficiency is still low [35]. Recently, Thibault Vogt et al. have reported a conversion efficiency of $5\%$ with six-wave mixing in atoms [14]. Petrosyan et al. have proposed a scheme with an ensemble of cold atom trapped on a superconducting chip [36]. This theoretical method faces the difficulty of trapping cold atoms close to the superconducting circuit. Thus, the leaking of the strong optical control field can break the superconductivity of the mw quantum circuit.

In the previous works based on Rydberg atoms [14,15,36–40], people mainly used low-angular-momentum Rydberg states as the medium to connect the mw and optical bands. The lifetimes of these states are usually tens to hundreds of microseconds, which are proportional to the principal quantum number $n^3$ [41,42]. However, the coupling strength between the single atom and mw single-photon state is tens of kilohertz [43,44]. The atom takes tens of microseconds to exchange excitation with a mw cavity. Due to the same magnitude between the lifetime and the excitation-exchanging time, the decay from the atom will have a large bad impact on the mw-to-optical photon conversion. To overcome this problem, we can use the circular Rydberg states instead, which are Rydberg states with maximum orbital angular momentum and maximum magnetic quantum [45,46]. Their lifetimes are proportional to $n^5$, approaching tens of milliseconds [47–49], and are much longer than the excitation-exchanging time. In the previous works, people do not use the circular Rydberg states because these states do not directly couple to optical photons. In recent years, Signoles et al. have realized the fast coherent transfer between low-angular-momentum and circular Rydberg states [50]. Thus, it gives us a chance to make use of the long-lived circular Rydberg levels for the mw-to-optical quantum interface.

In this paper, we propose a scheme to realize an efficient mw-to-optical conversion of single photon. In our proposal, we use a circular Rydberg atom that subsequently flies through the mw cavity and the optical cavities. This Rydberg atom first “absorbs” single photon of the mw cavity and then “releases” it via the optical cavity to a quantum channel like an optical fiber. To suppress the detrimental effect of excitation loss, we make use of three-photon Raman transition to avoid exciting the lossy atomic intermediate states. Thanks to the negligible decay of circular Rydberg atomic states, the single-photon conversion efficiency can reach up to $60\%$ using currently existing experimental technique. And the theoretical limit of the conversion efficiency is about $87\%$.

2. System and processes

Our scheme is shown in Fig. 1. We initialize a flying atom in circular Rydberg state $|r_1\rangle$, mw cavity in single excited state $|1\rangle _m$, optical cavity in vacuum state $|0\rangle _{op}$. Preparing the atom in a circular Rydberg state is a mature technology [51], which is simply sketched in the dashed box: the atoms effuse from the oven O, whose velocity is selected by lasers, and then are excited from the ground state into the circular Rydberg state $|r_1\rangle$ in the Box B [48]. After the preparation, our scheme consists of three steps.

 

Fig. 1. Schematic setup to implement the mw-to-optical conversion. The process consists of 3 steps. To begin with, the system is initialized as: the atom in $|r_1\rangle$, the mw cavity in $|1\rangle _m$ and the optical cavity in $|0\rangle _{op}$. In the first step, the atom absorbs the excitation of mw cavity to transfer to $|r_2\rangle$. The second step is to coherently transfer the atom to $|r\rangle$ in a short time in a capacitor surrounded by four ring electrodes. In the third step, the atom induces optical cavity to emit a flying optical photon. The dashed box shows the preparation of the atom: The atoms effuse from the oven O, whose velocity is selected by lasers, and then are excited from the ground state into the circular Rydberg state $|r_1\rangle$ in the Box B [48,51].

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In step ①, the atom absorbs single photon of mw cavity and then transfers to another circular Rydberg state $|r_2\rangle$ . In step ②, the atom is decircularized to low-angular-momentum Rydberg state $|r\rangle$ in the capacitor. In step ③, the atom interacts with optical cavity and two applied lasers. The atom returns to ground state $|g\rangle$ and induces optical cavity to emit a flying optical photon.

Now, we further explain the steps ① $ \sim $ ③. First we choose Rubidium-85 ($\rm ^{85}Rb$) as the medium atom. The conversion diagram of whole process is shown in Fig. 2, which includes six relevant atomic levels: the ground state ($|g\rangle$), two intermediate states ($|e\rangle$, $|d\rangle$), one low-angular-momentum Rydberg state ($|r\rangle$), two circular Rydberg states ($|r_1\rangle$, $|r_2\rangle$). The specific notations of states are shown in Fig. 3.

 

Fig. 2. Conversion diagram. $|r_2\rangle$ and $|r_1\rangle$ are circular Rydberg states. $|r\rangle$ is low-angular-momentum Rydberg state. $|d\rangle$ and $|e\rangle$ are the intermediate states. $|g\rangle$ is the ground state. The $\sigma ^+$ polarized mw cavity field is resonant with the transfer $|r_1\rangle \to |r_2\rangle$ with coupling strength $g_m$. In capaciter, $|r_2\rangle$ is transferred to $|r\rangle$. In optical cavity, two applied lasers, with frequencies $\omega _{L1}$ and $\omega _{L2}$, are both $\sigma ^-$ polarization. They are resonant with transfers $|r\rangle \to |d\rangle$ and $|d\rangle \to |e\rangle$ by coupling strength $\Omega _1$ and $\Omega _2$, respectively. The $\sigma ^+$ polarized optical cavity mode is resonant with transfer $|e\rangle \to |g\rangle$ by the coupling strength $g_p$ and the frequency $\omega _p$. $\Delta _1=\omega _{L1}-\omega _{rd}$, $\Delta _2=\omega _{L1}+\omega _{L2}-\omega _{re}$ and $\delta =\omega _{L1}+\omega _{L2}+\omega _p-\omega _{rg}$ are the single-, two- and three-photon detunngs.

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Fig. 3. Energy diagram. The six relevant hyperfine states of $^{85}\textrm {Rb}$ atom: $|r_2\rangle =|n=51,l=50,m=50\rangle$, $|r_1\rangle =|n=50,l=49,m=49\rangle$, $|r\rangle =|\textrm {51F},m=2\rangle$, $|d\rangle =|\textrm {5D}_{5/2},\,F=3\sim 5,\,m=3\rangle$, $|e\rangle =|\textrm {5P}_{3/2},\,F=4,\,m=4\rangle$ and $|g\rangle =|\textrm {5S}_{1/2},\,F=3,\,m=3\rangle$. The energy gaps are written on this figure.

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In step ①, the atom interacts with mw cavity. The cavity mode is resonant with the transition $|r_1\rangle \to |r_2\rangle$ and coupling strength is $g_m$. The Hamiltonian for this step is

Step ② is to decircularize the atom. In this step, a fast coherent transfer from circular Rydberg state $|r_2\rangle$ to low-angular-momentum state $|r\rangle$ is completed. The process is described as

In step ③, the atom interacts with the first single side optical cavity and two applied lasers. It transfers from $|r\rangle$ to ground state $|g\rangle$ via three-photon Raman transition and then induces the optical cavity to emit a photon. As shown in Fig. 2 and Fig. 3, the atomic states $|r\rangle$, $|d\rangle$, $|e\rangle$ and $|g\rangle$, are coupled by $\sigma ^-$ polarized laser one, $\sigma ^-$ polarized laser two and $\sigma ^+$ polarized optical cavity mode, with wave length $1258\ \textrm {nm}$, $776\ \textrm {nm}$ and $780\ \textrm {nm}$, respectively. First, $|r\rangle$ is Rydberg state with magnetic quantum number $m=2$. Laser one induces the transfer $|r\rangle \to |d\rangle$. $|d\rangle$ has a certain magnetic quantum number $m=3$, which limits $|d\rangle$ to the submanifold of hyperfine states $|\textrm {5D}_{5/2},\,\textrm {F}=3,\,m=3\rangle$, $|\textrm {5D}_{5/2},\,\textrm {F}=4,\,m=3\rangle$ and $|\textrm {5D}_{5/2},\,\textrm {F}=5,\,m=3\rangle$, where ${\rm F}$ is total atomic angular momentum. Second, Laser two induces the transfer $|d\rangle \to |e\rangle$. Hence the energy gaps between three hyperfine states related to $|d\rangle$ are under $10\ \textrm {MHz}$ [53], which is much smaller than coupling strength of Laser two and the atom ($\Omega _2/2\pi \approx 50\ \textrm {MHz}$), we neglect the difference of the three states of $|d\rangle$. The $\sigma ^-$ polarization of this laser can ensure that $|e\rangle$ has a certain magnetic quantum number $m=4$, corresponding to the only one state $|\textrm {5P}_{3/2},\,F=4,\,m=4\rangle$. Third, the cavity mode induces the transfer $|e\rangle \to |g\rangle$ ($|\textrm {5S}_{1/2},\,F=3,\,m=3\rangle$). This transition can only excite a $\sigma ^+$ polarized cavity mode and lead to a $\sigma ^+$ polarized flying photon. The process in step ③ is described as

In summary, the whole processes are described as

3. Fast coherent decircularization

When the atom leaves mw cavity, it is in the circular Rydberg state $|r_2\rangle$. However, $|r_2\rangle$ does not couple to any optical photon. To get over this problem, we need to fast transfer the atom to low-angular-momentum Rydberg state $|r\rangle$, which can couple to optical modes. This process has been realized by Signoles et al. in 2017 [50]. Their method is based on many-Rydberg-level oscillations (MRLOs). In this section, we simply review MRLOs and simulate the conversion efficiency $\eta _2$ in our step ②.

The MRLOs process occurs in a series of Stark eigenstates of Rydberg atom with same principal quantum number i.e. $|n,\,n_{1},\,m\rangle$, where $n$ is the common principal quantum number, the $n_{1}$ is the parabolic quantum number and the $m$ is the magnetic quantum number. A radio frequency (rf) field $\omega _{\textrm {rf}}$ with $\sigma ^{+}$ polarization limits the atom in a subspace $\mathcal {D}_{n}$ with the basis $\{| \Psi _{j} \rangle \} (j=1, 2, 3, \ldots , M-1): |\Psi _{1}\rangle = |n,\,n_{1}=1,\,m=2\rangle , |\Psi _{2}\rangle = |n,\,n_{1}=0,\,m=3\rangle , |\Psi _{3}\rangle = |n,\,n_{1}=0,\,m=4\rangle , |\Psi _{4}\rangle = |n,\,n_{1}=0,\,m=5\rangle \cdots |\Psi _{M-1}\rangle = |n,\,n_{1}=0,\,m=M\rangle$ with $M$ the maximal available magnetic quantum number. The subspace $\mathcal {D}_{n}$ is an angular-momentum-like space [50,54]. The state $|\Psi _{m-1}\rangle$ with magnetic quantum number $m$ can be mapped to the angular momentum state $|J=(n-1)/2,\,m_{J}=-J+m\rangle$. Then the Hamiltonian describing MRLOs is

In step ②, the atom in $|r_{2}\rangle$ flies in to a capacitor surrounded by four ring electrodes. The capacitor offers a static electric field to lift the Rydberg atom into the Stark eigenstates in manifold $n=51$. Then, ring electrodes apply the resonant rf signal on the atom to realize MRLOs. When the atom transfers to $|n=51,\,n_{1}=1,\,m=2\rangle$, the Stark electric field is adiabatically switched off and then the atom becomes $|\textrm {51F},\,m=2\rangle$, which is the state $|r\rangle$ in our scheme. We simulate the evolution of population in the subspace $\mathcal {D}_{51}$ with $\Omega _{51}/2\pi =3.52\ \textrm {MHz}$, which is shown in Fig. 4. It shows that the atom starts in $|\Psi _{50} \rangle$ ($|r_2\rangle$). After $0.127\ \mu \textrm {s}$, the population of $|\Psi _{1}\rangle$ becomes maximum $83.3\%$, which indicates a fast coherent decircularization to $|r\rangle$ with $\eta _2=83.3\%$ . In addition, this transfer can be further improved by optimizing experiment via the quantum optimal control method, which has a theoretical conversion limit of $65\ \textrm {ns}$ transfer time and $99\%$ conversion efficiency [56].

 

Fig. 4. The evolution in the subspace $\mathcal {D}_{51}$. The initial state is $|r_2\rangle$. The evolution is under the resonant condition ($\Delta _{51}/2\pi = 0$) with $\Omega _{51}/2\pi = 3.52\ \textrm {MHz}$. The population of the states with the maximum (red), next maximum (green), next lowest (blue), lowest (black) magnetic quantum state. When the population of lowest magnetic quantum state reaches maximum (see the red arrow), the rf field is switched off and the evolution stops.

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4. Model for emission of a flying single photon

In step ③, the atom interacts with the optical cavity field and two classical lasers. The atom is transferred from $|r\rangle$ to $|g\rangle$ and induces the optical cavity to emit single photon to quantum channel. However, in this process the atom has to pass through two intermediate states $|d\rangle$ and $|e\rangle$, which have short lifetimes. To increase the mw-to-optical conversion efficiency, we design a three-photon Raman transition to suppress populations of $|d\rangle$ and $|e\rangle$, so the four-level atom can be mapped to a two-level atom with low loss.

4.1 Three-photon Raman transition

Firstly, we consider an ideal system consisting of a four-level atom in a closed optical cavity without any losses. Then we apply two applied lasers to it. As shown in the right side of Fig. 2, laser one, whose frequency is $\omega _{L1}$, couples to the transition $|r\rangle$ to $|d\rangle$ with the strength $\Omega _1$ and large detuning $\Delta _1=\omega _{L1}-\omega _{rd}$. Laser two, whose frequency is $\omega _{L2}$, couples to the transition $|d\rangle$ to $|e\rangle$ with the strength $\Omega _2$ and the detuning $\Delta _2=\omega _{L1}+\omega _{L2}-\omega _{re}$ which is the two-laser detuning from $|r\rangle$ to $|e\rangle$. The optical cavity, whose frequency is $\omega _p$, couples to the transition $|e\rangle$ to $|g\rangle$ with the strength $g_p$ and the total detuning $\delta =\omega _{L1}+\omega _{L2}+\omega _p-\omega _{rg}$. The reduced Hamiltonian $H_r$ is

Here the state $|g,1\rangle$ means the atom is in the ground state $|g\rangle$ and the cavity is in the one-photon state $|1\rangle _{p}$.

We aim to find a solution to suppress the $x_2$ and $x_3$. First, we give two pairs of approximation conditions. The first is that $x_2$ and $x_3$ are so small that they can be considered as 0. The second is that initial values of $\dot {x_2}$ and $\dot {x_3}$ are near 0:

In addition, we simplify the problem by supposing that:

When $\Delta \gg \Omega _1, \Omega _2, g_p$, we can use the adiabatic approximation to eliminate $x_2$ and $x_3$. Then, we obtain the effective Schrödinger equation

We define $\tilde {x_4}$ and $\tilde {x_1}$ by

If $B=0$, the system is on resonance, so we have resonance condition

We show the three-photon Raman process on resonance in Fig. 5. The population of $|d\rangle$ and $|e\rangle$ are suppressed under $0.2$ and a Rabi oscillation between $|r\rangle$ and $|g\rangle$ occurs.

 

Fig. 5. The three-photon Raman process on resonance. The four-level atom is coupled by a cavity field and two lasers. The parameters are set as $(g_p,\,\Omega _1,\,\Omega _2,\,\Delta )=2\pi \times (10\ \textrm {MHz},\,20\ \textrm {MHz},\,50\ \textrm {MHz},\,50\ \textrm {MHz})$, which meet the resonance condition Eq. (15). The red, blue, yellow and black curves represent the populations of $|r\rangle$, $|d\rangle$, $|e\rangle$ and $|g\rangle$ respectively.

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4.2 Emitting single photons from an atom in an optical cavity

Secondly, we deal with the optical cavity in open system. The loss of the cavity and the spontaneous emission of the atom are taken into account.As shown in Fig. 6, the system Hamiltonian H takes the form

We transfer the system into interaction picture by:

 

Fig. 6. The scheme of the optical cavity subsystem. (a) The schematic setup. The atom, with speed $v_a$, interacts with an optical microcavity by the coupling strength $g_p$. This cavity is single side: one mirror of the cavity has near-unity reflectivity, while the other mirror has high transmissivity, which can allow the single photon to pass through it with the external coupling strength $\kappa _e$. The $\kappa _i$ is the intrinsic decay rate of the cavity. The $\gamma _i$ is the spontaneous emission rate for atomic state $|i\rangle$. $\Omega _1$ and $\Omega _2$ are the coupling strength between the atom and two classical lasers. (b) The spatial distribution of $g_p$. The Gaussian cavity mode leads to a Gaussian distribution of $g_e$ perpendicular to the cavity axis, i.e. $g_e=g_0 \exp (-x^2/w_0^2)$. $g_0$ is the maximum coupling strength and $w_0$ is the waist of cavity mode.

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Making the rotating-wave approximation, we get the interaction Hamiltonian $H_I$ in the rotating frame:

 

Fig. 7. The coupling window of $g_e(s)$. The cavity mode couples to the $\omega _s$ frequency mode of quantum channel by the strength $g_e(s)$. In the frequency window $(-\omega _b,\omega _b)$, $g_e(s)$ is regarded as constant $g_e$. We divide the window into $2N+1$ parts. The index is from -N to N.

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4.3 Derivation of the cavity decay rate in the discrete representation

Our simulation is based on Eq. (22), where the term with $g_e$ describes the connection between the cavity and the outside quantum channel. However, the $g_e$ can not be obtained directly. We can only get the decay rate of the cavity $\kappa _e$ by experiment. Naturally, it poses a question: what is the relation between the coupling strength $g_e$ and decay rate $\kappa _e$? Now, we give the derivation.

The bath of an open system is usually described as a continuous model [57]:

For the convenience of numerical simulation, we use the discrete representation to describe a quantum channel, then the optical cavity system is described by $H_I$ (Eq. (22)). We donates the part of the atom and cavity in Eq. (22) as $H_{ac}$, then $H_I$ can be written as

In the Heisenberg picture, we get the following relations from $H_{I}$:

Solving the differential Eq. (26a), we obtain

Substituting Eq. (27) into Eq. (26b), we get

Define discrete field operator of the quantum channel $B(t)$ as

As for the second term of Eq. (28), we give the derivation as follows:

After substituting Eq. (30) and Eq. (31) into Eq. (28), we get

The term with $B(t)$ in Eq. (32) is the noise from the external field. The second term of $H_{eff}$ is representative of the damping of the cavity field. When we compare the second term with the effective decay Hamiltonian, $H_{decay}=-\frac {i}{2}\kappa _e a_p^{\dagger }a_p$, we have

From Eq. (33), we can obtain the value of $g_e$ from the experimental data of $\kappa _e$.

5. Simulation and results

It is reasonable to neglect the atomic decay in motion between cavities because the atom is always in ground state or Rydberg states. The axis of cavity is at right angle of the atomic velocity $\vec {v}_a$, which can induce a transverse Doppler effect. The maximum Doppler shift $\Delta \omega _\perp$ during interaction with cavity modes can be calculated as [59,60] $\Delta \omega _\perp =\omega \times (v_a/c)^2$, where $\omega$ is the frequency of the cavity mode. The Doppler shift in the mw cavity is negligible because the cavity frequency is very small. The Doppler shift in the optical cavity is dominant. We take the atomic speed $v_a \approx 150\ \textrm {m/s}$ and $\omega /2\pi =\omega _p/2\pi \approx 385\ \textrm {THz}$. Then, we get $\Delta \omega _\perp /2\pi \approx 94\ \textrm {Hz}$, which can also be neglected.

The flow of this conversion is

The first step $|1\rangle _m \Longrightarrow |r_2\rangle$ has been demonstrated by Haroche et al. [52]. In their work, the quality factor of a mw cavity $Q_m$ can be $7 \times 10^7$ corresponding to a cavity lifetime of $220\ \mu \textrm {s}$, and the coupling strength between atom and the mw cavity $g_m$ is $2\pi \times 47\ \textrm {kHz}$, which allows us to complete operation within $10\ \mu s$ in step ①. Compared with the atomic lifetime of $30\ \textrm {ms}$ and cavity lifetime of $220\ \mu \textrm {s}$, the operation time under $10\ \mu \textrm {s}$ is short enough that we can neglect the decay of circular Rydberg states. Thus, it is reasonable to assume that the conversion efficiency of this step is unitary ($\eta _1=1$). The fast coherent transfer $|r_2\rangle \rightarrow |r\rangle$ involved in step ② has been also realized and optimized by Gleyzes et al. [50]. They get the transfer rate of $80\%$. We also study this process in Sec. 3. The conversion efficiency $\eta _2$ in our simulation is $83.3\%$, which is similar to their result. Moreover, Koch et al. point out that quantum optimal method theory can boost this process to obtain a conversion efficiency of $99\%$ [56].

The step ③ ($|r\rangle \Longrightarrow |1\rangle _{F}$), with conversion efficiency $\eta _3$, occurs in the optical cavity subsystem. In this section, we numerically simulate the evolution of atom-optical-cavity subsystem described by Hamiltonian $H_I$ (22). We define a general state $|\Psi _{\textrm {OS}}\rangle$ for this subsystem as

Below we will investigate the conversion efficiency $\eta _3$ based on the above state $|\Psi _{\textrm {OS}}(t)\rangle$.

We describe the $|\Psi _{F}\rangle$ in frequency domain as

Then, the efficiency of single-photon conversion $\eta _3$ can be calculated by

5.1 Parameters

In order to stimulate the intracavity field into a propagating mode, we need a high-quality optical cavity ($\kappa _e\gg \kappa _i$) and a small mode volume (the coupling strength of atom and cavity $g_p$ is large). We use the typical values of these parameters from the experiments by Rempe et al.: $(\kappa _e,\,\kappa _i)=2 \pi \times (2.5\ \textrm {MHz},\,0.25\ \textrm {MHz})$ [61–65]. The quality factor of cavity can be calculated by $Q_p=(2\kappa _e/\omega _p + 2\kappa _i/\omega _p )^{-1}=7\times 10^7$ [66,67]. The ultrahigh quality factors can be realized in the small scale cavities and optical microcavities [68–70].

As for the atom, we care about the decay rate $\gamma _i$ due to the spontaneous emission, where the subscript $i=r,d,e$ corresponds to the states $|r\rangle ,|d\rangle ,|e\rangle$ respectively. Due to the small radiation solid angle of cavity modes, we assume that the polarization decay rate in cavity is half of it in free space [71]:

Furthermore, the Gaussian mode of the optical cavity and the spatio-temporal nature of the coupling $g_p$ should be considered, when the atom is moving through the cavity. The spatial distribution of $g_p$ is like $g_p=g_0 \exp (-x^2/w_0^2)$, where $g_0$ is the maximum coupling strength and $w_0$ is the waist of cavity mode. The distribution is along the direction of atomic velocity, which is shown in Fig. 6(a). In our simulation, we set $g_0=2 \pi \times 10\ \textrm {MHz}$, then the $w_0$ can be estimated from the experimental data that the $780\ \textrm {nm}$-cavity with a waist of $29\ {\mu \textrm{m}}$ has the maximum coupling strength of $2 \pi \times 16\ \textrm {MHz}$ [74]. From the relationship $g_p \propto V_m^{-1/2} \propto w_0^{-3/2}$, we can estimate our waist $w_0$ is about $40\ {\mu \textrm{m}}$, where $V_m$ is the mode volume. At last the flying atom with the speed of $v_a$, feels the time-dependent coupling strength in the cavity is

At last, we set the three-photon detuning $\delta =0$. Then, the resonance condition Eq. (15) requires $\Omega _1=2g_p$, which means the waist of laser one should be the same as the Gaussian optical mode $w_0$. In fact, the system is robust within a certain range of $\delta$, so it just requires that the spatial distribution of $\Omega _1$ is approximately about 2 times of $g_p$. The stability of $\delta$ is discussed in the next subsection. There is not requirement of $\Omega _2$ in the resonance condition Eq. (15). We just set it as a constant for simply, i.e. $\Omega _2=20\kappa _e=2\pi \times 50\ \textrm {MHz}$. And we take $\Delta _1=\Delta _2=20\kappa _e=2\pi \times 50\ \textrm {MHz}$ for our later simulation.

5.2 Efficiency of single-photon conversion

The initial state of the optical cavity subsystem is

Then, the conversion efficiency $\eta _3$ at time t, can be calculated by Eq. (38).

First, we simulate the $\eta _3$ based on the parameters in the previous subsection. The results are shown in Fig. 8.

 

Fig. 8. (a) The conversion efficiency $\eta _3$ as a function of time t. The conversion efficiency $\eta _3$ is calculated as the probability of emitting a single photon by the atom initially in $|r\rangle$ in optical cavity. The parameters are from the subsection 5.1, which are satisfied the resonance condition, i.e. $\delta =0$. When $t=0$, the atom just enters the optical cavity. After about $1\ \mu \textrm {s}$, $\eta _3$ reaches the maximum: $72\%$. (b) $\eta _3$ as a function of the three-photon detuning $\delta$. The conversion efficiency between two dashed lines, $\delta /2\pi \in (-0.26\, \textrm {MHz},0.87\, \textrm {MHz})$, is larger than $70\%$.

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In Fig. 8(a), we show the changing of $\eta _3$ after the atom enters the optical cavity. When the atom just enters the optical cavity, the cavity field strength is so small that $\eta _3$ is near $0$. When the atom flies close to the center of cavity, the coupling strength $g_p$ increases and the three-photon Raman process works. Therefore, there is a rapid increase of $\eta _3$. When the time is about $1\ \mu \textrm {s}$, the conversion finishes and $\eta$ becomes the maximum: $72\%$. After that, the flying photon decouples from atom and optical cavity and only propagates in quantum channel, where the decay is negligible in short time. Thus, the conversion efficiency becomes flat.

In Fig. 8(b), we show the influence of the three-photon detuning $\delta$. When $\delta =0$, the system is resonant and $\eta _3$ is at its maximum $72\%$. Then, $\eta _3$ decreases with the increasing of $|\delta |$. In the region between the two dashed lines, the curve changes relatively slowly and $\eta _3$ is larger than $70\%$. This region reflects the system stability with respect to $\delta$, which is about $1\ \textrm {MHz}$.

Second, we discuss the theoretical limit of $\eta _3$. The loss of the system comes from two ways: intrinsic loss from the cavity ($\kappa _i$) and the spontaneous emission from the intermediate states of the atom ($\gamma _d$ and $\gamma _e$).

In Fig. 9(a), the $\eta _3$ is boosted by reducing the $\kappa _i$. However, $\eta _3$ is still not one, when $\kappa _i=0$. This gap comes from spontaneous emission of atom. As shown in Fig. 5, there are still population of $|d\rangle$ and $|e\rangle$, even if the three photon Raman process is on resonance. To suppress the population of intermediate states, we can increase the Large detuning $\Delta$ (Eq. (10)). However, the effective Rabi frequency $G_{eff}$ will decrease with increasing $\Delta$ due to Eq. (16), which means a longer time for spontaneous emission. Therefore, we scan $\Delta$ with $\kappa _i=0$ to find the maximum conversion efficiency. As shown in Fig. 9(b), the theoretical limit of $\eta _3$ is $88\%$ when the $\Delta$ is near $2\pi \times 50\ \textrm {MHz}$.

 

Fig. 9. (a) The conversion efficiency $\eta _3$ as a function of $\kappa _i$. $\kappa _i$ is the intrinsic loss rate of the cavity due to photon scattering into open space and absorbing by two mirrors of cavity. $\eta _3$ increases with reducing $\kappa _i$. (b) $\eta _3$ as a function of the large detuning $\Delta$ with $\kappa _i=0$. The single- and two-photon detunngs in Fig. 4 are both set as $\Delta$ (Eq. (10)). When the $\Delta$ is near $2\pi \times 50\ \textrm {MHz}$, $\eta _3$ reaches its theoretical limit: $88\%$.

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In summary, our scheme contains three key steps ①, ② and ③, corresponding to conversion efficiencies $\eta _1$, $\eta _2$, $\eta _3$ respectively. From above analyses and simulation, we have $\eta _1\approx 1$, $\eta _2=83.3\%$ and $\eta _3=72\%$. As for the total single-photon conversion efficiency $\eta _T$, it is difficult to calculate because there are so many levels of atom and so many modes in channel that if we include all basis of atom, cavities and channel in model simulation, the size of Hilbert space will boost beyond the affordable computation complexity. Thus, we approximately estimate $\eta _T$ as $\eta _T\approx \eta _1 \eta _2 \eta _3=1 \times 83.3\% \times 72\%=60\%$. Furthermore, the quantum optimal method theory can boost $\eta _2$ to $99\%$ [56] and the $\eta _3$ also has a theoretical limit of $88\%$. Therefore, the theoretical limit of $\eta _T$ is about $87\%$. We show the parameters and conversion efficiencies of each step in Table 1.

Table 1. The three steps in single-photon conversion

View Table

At last, we notice that the dynamics of the whole quantum transduction is reversible except for processes related to the spontaneous decay of the atom ($\gamma _i$) and the intrinsic losses of the optical cavity ($\kappa _i$). In principle, if we reverse all processes, including the pulse shape of flying photon, we can perform an efficient reversed photon conversion, i.e. conversing an optical single photon to the excitation of the microwave (mw) cavity by successively applying the steps 3, 2 and 1.

6. Conclusion

We have proposed a method for efficient conversion of single photon from mw domain to optical domain in mediate of a single flying circular Rydberg atom. In our proposal, the long-lifetime circular Rydberg atom plays a role of a stable quantum “bus”. The single photon gets on the “bus” at the mw cavity and then get off the “bus” at the optical cavity. More specifically, the single photon in a mw cavity is first stored in the flying atom. Then, the excitation is transferred from the atom to a flying optical photon when it passes through the optical cavity.

Thanks to the strong coupling between single Rydberg atom and cavities, the super long lifetime of circular Rydberg states and the low loss from three-photon Raman transition method, the mw-to-optical conversion is efficient. The theoretical limit of the conversion efficiency is about $87\%$. Based on existing experiments and data, the conversion efficiency is simulated as $60\%$. In addition, in our scheme the superconducting and optical subsystems are spatially separated. Thus, this design can protect the superconducting parts from light-leakage-induced loss of superconductivity, which is a challenging issue in on-chip mw-optical quantum interfaces. Our proposal may promote the utilization of circular Rydberg states and may pave the way for mw-optical hybrid quantum network and the quantum information processing.