1. Introduction

Optical photons are ideal information carriers for long-distance transport due to their characteristic ease of transmission and robustness against interaction with the environment. At the same time, it is limited for quantum information processing(QIP) by the lack of strong interactions. For this reason, a hybrid system, which can combine the ability of high-fidelity control of QIP and efficient transport of quantum information, is desirable. The hybrid system will be a successful quantum technology since it can integrate the different individual systems with complementary strength [1,2]. For example, individual system, such as microwave superconducting qubits, has shown features with extremely high fidelity [3–5]. Still, the quantum state transport is a problem due to severe losses and decoherence during propagation [6]. Atomic [7] and ionic [8,9] systems are also excellent candidates for QIP but suffer severe decoherence from transferring information. Manipulation of optical photons using a controllable microwave field can achieve interfacing between optical photons and microwave, which has important applications for quantum networks [10,11], quantum communication [12,13], and quantum metrology [14,15]. However, achieving such an optical-microwave hybrid system remains a challenge since it is difficult to interact between microwave and optical photons. Although an entanglement experiment with Rydberg atoms and microwave photons in a cavity has been reported [16], microwave photons are not ideal for quantum communication due to the black-body background. Fortunately, there has been substantial progress in QIP using Rydberg atoms [17–19]. Due to the strong dipole blockade effect arising from Rydberg states, the Rydberg system has been used to achieve single photon source [20–22], all-optical transistors [23,24], photonic quantum logic gate [25]. Recently, it has been shown that Rydberg atoms can bridge the gap between optical and microwave photons [11] since the energy level interval between two nearby Rydberg states is in the microwave frequency range. [26]. Therefore, the Rydberg population can be coupled and manipulated with the microwave field. An interfacing microwave and optical quantum information processing system have been demonstrated, which allows coherent manipulation of an optical photon using microwave field [27,28] based on Rydberg polaritons [29,30] using electromagnetically induced transparency (EIT) [31]. Variations on the approach have been used to implement collectively optical Rydberg Qubit [32] and stored-light Ramsey interferometry [33] in an atomic ensemble of $^{87}\textrm{ Rb}$, where the information is read out by measuring the projection of one Rydberg state.

Here, we demonstrate a feasible approach for the manipulation of stored photons employing the microwave field based on the Rydberg polariton of cesium atoms with Rydberg EIT. The experimental protocol is similar to the Refs. [32,33]. The significant difference, aside from the different choice of atomic species, is the readout method, either of the populations of both Rydberg states in case I or simultaneous with microwave driving in case II, see Fig. 1. Due to the strong blockade, we achieve a single photon source with $g^{(2)}(0) = 0.29\pm 0.08$ at 80$S_{1/2}$ without applying microwave fields by preparing an atomic ensemble smaller than the blockade sphere. Subsequently, we manipulate the stored photons by applying microwave fields to couple two Rydberg states. The population in both Rydberg states is read out, which shows the presence or absence of a microwave pulse can be converted into photon readout early or late. In addition, we show microwave-modulated retrieval photons, realizing a modulation frequency of up to 50 MHz. An improved superatom model is employed to model the results, which shows good agreement.

 

Fig. 1. (a) Schematic of the experiment for the microwave manipulation of stored-photon. A probe pulse and a coupling laser are counter-propagated on an atomic cloud. The transmission of probe photons is detected by a standard HBT interferometer. A microwave field emitted from a wave-guide antenna acts on atoms to manipulate the population in Rydberg states. (b) Atomic levels scheme. The probe laser ($\Omega _p$), driving the $|g\rangle \rightarrow |e\rangle$ transition, and the coupling laser ($\Omega _c$), coupling the $|e\rangle \rightarrow |r\rangle$ transition, form Rydberg EIT. The microwave field drives the transmission between two adjacent Rydberg states $|r\rangle$ and $|r'\rangle$. (c) Timing sequence. The dipole trap laser is switched off when the probe laser is on. The duration of the probe pulse is about $600\,ns$. Before the probe is switched off, the coupling laser is ramped to zero to store a photon in a Rydberg state $|r\rangle$. After a storage time of $570\,ns$, the coupling laser is turned back on to read out the population in $|r\rangle$. During the storage (case I) or readout (case II) period, a microwave pulse is used to manipulate the population of Rydberg state $|r\rangle$.

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2. Experimental setup

A schematic of the experimental setup and the relevant atomic levels are shown in Figs. 1(a) and (b). We prepare an ensemble of $^{133}\textrm{ Cs}$ atoms in a magneto-optical trap (MOT). Subsequently, we apply the compress MOT and molasses to increase the atomic density and decrease the atomic temperature. At the end of the molasses stage, we measure the atomic temperatures below 40 $\mu$K using time of flight imaging. A cross dipole trap composed of two 1064 nm lasers with $\omega _1 = 4.3\,\mu m$ and $\omega _2 = 7.0\,\mu m$ is always on before the Rydberg experiment. Then, we apply a resonant push laser to kick the background atoms out of the dipole trap region, whereas atoms trapped in the dipole traps are unaffected since the AC Stark shift induced by the trapping laser shifts the push laser off-resonance. After the above sequence, we estimate the atomic cloud size with a spatial extent $4\times 4 \times 14\,{\mu m}^3$. The atom number and cloud density are estimated at an order of $10^3$ by the signal absorption and about $10^{12} cm^{-3}$, respectively. The atoms are optically pumped to $|g\rangle =|6S_{1/2},F=4, m_{F}=4\rangle$. The probe laser (852 nm) with the waist of $w_p = 2.5\,{\mu}{\textrm{m}}$ is resonant with the transition of $|g\rangle =|6S_{1/2}, F=4, m_{F}=4\rangle \rightarrow |e\rangle =|6P_{3/2}, F'=5, m_{F'}=5\rangle$, and the coupling laser (509 nm) with a beam waist of $w_c = 8.5\,{\mu}{\textrm{m}}$ drives the Rydberg transition of $|e\rangle =|6P_{3/2}, F'=5, m_{F'}=5\rangle \rightarrow |r\rangle =|nS_{1/2}\rangle (n=40-80)$, forming Rydberg EIT configuration. The probe and coupling lasers are overlapped and counter-propagated through the center of the atoms sample. The frequencies of probe and coupling lasers are locked to a cavity with the finesse 150000 [34]. The probe and coupling beams have opposite circular polarization. The time sequence of experiments is shown in Fig. 1(c), we turn off the dipole trap laser for 1.55 $\mu$s, during which the probe laser is turned on for 600 ns. Before the probe is switched off, we ramp down the coupling laser to zero, achieving the photon storage as a collective Rydberg excitation [32]. After several hundreds of ns storage time, the coupling laser is switched on to read the stored photon in Rydberg polariton. After the retrieval, the dipole trap laser is on 3 $\mu$s to re-trap the atoms. The readout photon is downstream detected by a standard Hanbury-Brown and Twiss (HBT) interferometer comprising a 50/50 beam splitter and two single-photon avalanche detectors (SPADs). During the storage or readout period, a microwave field emitted from a wave-guide antenna (Model: HD-180WAL150) with linear polarization is applied to couple the nearby Rydberg state $|r\rangle$ and $|r'\rangle$ and manipulate the population of $|r\rangle$ and further the storage photon. For each MOT loading, we can repeat the sequence 10000 times.

3. Theoretical model

In this section, we give a theoretical framework for describing the microwave manipulation of a single stored photon in a dense Rydberg atomic ensemble driven into the four-level configuration, as shown in Fig. 1(b). This four-level configuration involves a ground state $\left \vert g\right \rangle$, an excited state $\left \vert e\right \rangle$, and two Rydberg states $\left \vert r\right \rangle$ and $\left \vert r^{\prime }\right \rangle$ driven by a probe field, a coupling field, and a microwave field with Rabi frequencies (detunings) $\hat {\Omega }_{p}$ ($\Delta _{p}$), $\Omega _{c}$ ($\Delta _{c}$), and $\Omega _{MW}$ ($\Delta _{MW}$), respectively. Here we consider a 1D model based on the fact that the coupling and microwave fields are approximately homogeneous in the $xy$ plane and the probe field exhibits a very small beam waist so that it is well contained by the counter-propagating coupling field. Then, two atoms at positions $z$ and $z^{\prime }$, respectively, if both excited to $\left \vert r\right \rangle$ and/or $\left \vert r^{\prime }\right \rangle$, will also interact through the dipole-dipole potentials $\mathcal {V}_{ij}(z-z^{\prime })=C_{3}^{ij}/|z-z^{\prime }|^{3}$, with $\{i,j\}\in \{r,r^{\prime }\}$ and $C_{3}^{ij}$ being the dipole-dipole coefficient belonging to corresponding Rydberg states. Adopting $\hat {P}(z,t)=\sqrt {N}\hat {\sigma }_{ge}(z,t)$ to represent the atomic polarization field while $\hat {S}(z,t)=\sqrt {N}\hat {\sigma }_{gr}(z,t)$ and $\hat {Q} (z,t)=\sqrt {N}\hat {\sigma }_{gr^{\prime }}(z,t)$ to describe two Rydberg spin fields with $\hat {\sigma }_{ge}=\left \vert g\right \rangle \left \langle e\right \vert$, $\hat {\sigma }_{gr}=\left \vert g\right \rangle \left \langle r\right \vert$, and $\hat {\sigma }_{gr^{^{\prime }}}=\left \vert g\right \rangle \left \langle r^{\prime }\right \vert$ being relevant atomic transition operators and $N$ the atomic volume density, we can attain the following Heisenberg-Langevin equations

In the spirit of a mean-field approximation, we can solve Eq. (1) by resorting to an improved superatom (SA) model [35–37] with each SA defined as an ensemble of $n_{b}=NV_{b}$ atoms in a cylinder of volume $V_{b}$ $=2\pi R_{b}\mathbf {w}_{p}^{2}$ and blockade radius $R_{b}=[C_{3} \gamma _{ge}/|\Omega _{c}|^{2}+\gamma _{ge}\gamma _{gr}]^{1/3}$. Collective Rydberg excitation is described by the SA model in our work. A SA can be described by the collective ground state $\left \vert G\right \rangle$ with no excited atoms as well as the singly excited states $\left \vert E\right \rangle$, $\left \vert R\right \rangle$ and $\left \vert R^{\prime }\right \rangle$ containing a single excited atom in the EIT regime [35–37]. Then, we further define $\hat {\Sigma } _{IJ}=\left \vert I\right \rangle \left \langle J\right \vert$ as the SA transition ($I\neq J$) or projection ($I=J$) operators with $\{I,J\}\in \{G,E,R,R^{^{\prime }}\}$ so that the SA dynamics can be described by

In each SA, we have $\Delta _{rr}\to \infty$ in the case of $\Sigma _{RR}=1$, ${\Delta }_{r^{^{\prime }}r^{^{\prime }}}\to \infty$ in the case of $\Sigma _{R^{^{\prime }}R^{^{\prime }}}=1$, and $\Delta _{r^{^{\prime }}r}\simeq \Delta _{rr^{^{\prime }}}\to \infty$ in the case of both $\Sigma _{RR}=1$ and $\Sigma _{R^{^{\prime }}R^{^{\prime }}}=1$. Otherwise, these dipole-dipole induced shifts are negligible as they are caused by remote Rydberg excitations outside this SA [35–37]. Note also that $\Sigma _{RR}(1-\Sigma _{R^{\prime }R^{\prime }})$ denotes the probability that there exists one atom in state $\left \vert r\right \rangle$ while no atoms are in state $\left \vert r^{\prime }\right \rangle$; $\Sigma _{R^{\prime }R^{\prime }}(1-\Sigma _{RR})$ denotes the probability that there exists one atom in state $\left \vert r^{\prime }\right \rangle$ while no atoms are in state $\left \vert r\right \rangle$; $\Sigma _{RR}\Sigma _{R^{\prime }R^{\prime }}$ denotes the probability that two atoms are respectively excited to states $\left \vert r\right \rangle$ and $\left \vert r^{\prime }\right \rangle$ inside a SA. Then the $\Sigma _{RR}+\Sigma _{R^{^{\prime }}R^{^{\prime }}}-\Sigma _{RR}\Sigma _{R^{^{\prime }}R^{^{\prime }}}$ fraction of $n_{b}$ atoms in a SA become a two-level absorbing system as described by Eq. (1) with ${\Delta }_{rr}\simeq {\Delta }_{r^{^{\prime }}r}\simeq {\Delta }_{r^{^{\prime }}r^{^{\prime }}}\simeq {\Delta }_{rr^{^{\prime }}}\to \infty$, while the $1-\Sigma _{RR}-\Sigma _{R^{^{\prime }}R^{^{\prime }}}+\Sigma _{RR}\Sigma _{R^{^{\prime }}R^{^{\prime }}}$ fraction of $n_{b}$ atoms remains to be a four-level EIT system as described by Eq. (1) with ${\Delta }_{rr}\simeq {\Delta }_{r^{^{\prime }}r}\simeq {\Delta }_{r^{^{\prime }}r^{^{\prime }}}\simeq {\Delta }_{rr^{^{\prime }}}\to 0$. Accordingly, the optical response of this Rydberg atomic ensemble to a probe field should be described by the following conditional polarizability

The conditionality shown by the polarizability $P$ arises from the two-particle dipole-dipole interactions and will modify the photon statistics of a probe field, which can be quantified by the two-photon correlation function

This correlation function at $\tau =0$ should obey the following dynamic equation

4. Results and discussion

We first study photon storage and retrieval based on Rydberg EIT. To do this, we sent a probe pulse with a mean photon number of $R_{in}$ $\sim$ 6.5 ($\Omega _p = 2\pi \times$ 0.28 MHz) per shot into the atomic ensemble in the time window $650-1250\,ns$, while the coupling laser ramps down to zero before the end of the probe pulse. In Fig. 2(a), we demonstrate the result of photon storage and retrieval using the Rydberg state 80$S_{1/2}$, shown as the red shadow. The retrieved signal is magnified by a factor of ten and the retrieval efficiency (the ratio between the retrieved signal and the input signal) is about $\sim 0.3\%$, which is limited by motional dephasing, blockade, and finite ensemble optical depth. If each time we can perfectly store one photon and read out the photon without loss, a perfect efficiency would be 1/$R_{in}$ for input photons greater than one. The shadow in grey shows the input probe pulse without atoms, and the green line presents the intensity of the coupling laser. Due to the Rydberg blockade effect induced by the strong interaction scaling as $\sim n^{11}$, the collective excitation is limited to only one photon, leading to anti-bunching for the retrieved photons. To demonstrate this, we perform a measurement of the second-order correlation function of the retrieved photons, $g^{(2)}(\tau )$. The inset of Fig. 2(b) displays the measured $g^{(2)}(\tau )$ for 80$S_{1/2}$ state, corresponding $g^{(2)}(0) = 0.29\pm 0.08$, which demonstrates the typical single photon characteristic. Figure 2(b) displays the $g^{(2)}(0)$ for different Rydberg states. With increasing the principal quantum numbers, e.g., increasing the blockade radius, the value of $g^{(2)}(0)$ gets lower until reaching a saturating value as the blockade radius exceeds the atomic sample size [38]. We vary the power of coupling laser to keep $\Omega _c = 2\pi \times$ 6.8 MHz for the $g^{(2)}(\tau )$ measurements at different Rydberg states. The red dash line shows the numerical simulation through Eq. (6) with fit parameters $\Omega _p = 2\pi \times$ 0.28 MHz, $\Omega _c = 2\pi \times$ 6 $\sim$ 8 MHz, atomic density $1.2 \times 10^{12}cm^{-3}$. Consistent with the experimental results, the larger the principal quantum number is, the stronger anti-bunching effect can be observed in photons extracted from the end of the medium before entering the saturating regime.

 

Fig. 2. (a) The grey shadow is the input probe pulse without atoms that is normalized to one and the green line is a schematic drawing to present the intensity of the coupling laser. The red shadow shows single photon storage and retrieval signal at 80$S_{1/2}$, which is normalized to the input probe pulse. The retrieved photon signal is magnified by a factor of ten. (b) Measured intensity correlation $g^{(2)}(0)$ at different Rydberg state. The blockade radius $R_b$ is 6.8 $\mu$m for n=60 and 12.1 $\mu$m for n=80. The red dashed line represents the numerical simulation result. Inset: The intensity correlation measurement $g^{(2)}(\tau )$ for 80$S_{1/2}$ state. We vary the power of coupling laser to keep $\Omega _c = 2\pi \times$ 6.8 MHz for the $g^{(2)}(\tau )$ measurements at different Rydberg states.

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We then consider the case where a microwave field is applied during the storage interval [case I, the second-line timing of Fig. 1(c)] and during the readout period [case II, the third-line timing of Fig. 1(c)], respectively. In both cases, we set $\Omega _c = 2\pi \times$ 14.0 MHz. Coherent manipulation of the initial Rydberg $|nS\rangle$ state and further a stored photon is performed using a resonant microwave field that couples the $|nS\rangle$ and nearby $|nP\rangle$ state. The $|nS\rangle$ and $|nP\rangle$ form a two-level basis and a resonant microwave is used to encode the stored photon.

We consider an initial 60$S_{1/2}$ state to implement coherent manipulation of the stored photon by applying a resonant microwave field at 19.417 GHz during the storage period [case I], coupling the initial state 60$S_{1/2}$ to the $59P_{3/2}$ Rydberg state. As the two collective Rydberg excitations can be written as,

To see the Rabi oscillation of the initial $|nS\rangle$ and transferred $|nP\rangle$ state, we set the first microwave pulse during the storage interval that is used to vary the population between the two Rydberg states by changing the microwave rotation angle. The microwave rotation angle, defined as $\Theta (t) = \Omega _{MW}t$, is adjusted by varying the pulse duration $t$ over 0-270 ns with fixed microwave Rabi frequency of $\Omega _{MW}/2\pi = 12.5\,$MHz, rather than amplitude. After the retrieval of 60$S_{1/2}$ population by recovering the coupling laser, we set the second microwave $\pi$ pulse (lasting 40 ns) which populates the $59P_{3/2}$ back to 60$S_{1/2}$ state and readout later. We set such that both the 60$S_{1/2}$ and $59P_{3/2}$ can be retrieved at the same measurement. Figure 3(a) demonstrates the normalized retrieval signals with time-resolved detection at an indicated $\Theta$ of the first microwave pulse. The signal in the time windows 1660-1860 ns and 1860-1960 ns is retrieved photons from 60$S_{1/2}$ and late retrieved from $59P_{3/2}$, respectively. The phase factors enable the phase of the photonic state readout from two Rydberg states in a well-defined mode, which enables achieving a time-bin qubit. Black dash curves in Fig. $3(a)$ display numerical simulations on the retrieved pulse under three different conditions. Note, in particular, that theoretical curves are always under experimental curves though their changes are consistent as $\Theta$ is varied. This may be attributed to the fact that our numerical simulations don’t exactly recover real experimental situations because the 1D SA model has assumed a spatially invariant atomic density and transversely homogeneous dipole-dipole shifts to reduce the complexity of numerical calculations and cannot include other unknown effects related to relevant experimental devices.

 

Fig. 3. Normalized retrieval signals of case I, where microwave field is applied during storage. (a) The normalized retrieval signals from 60$S_{1/2}$ (the window of 1660-1860ns) and late retrieval from $59P_{3/2}$ (the window of 1860-1960ns) at an indicated rotation angle of $\Theta = 0, 2.5\pi, 5\pi$ with $\Omega _c = 2\pi \times$ 14.0 MHz. The black dash curve denotes the theoretical calculation result by solving Eqs. (1)–(4) together. (b)The retrieval photon number from both $|60S_{1/2}\rangle$ (blue circles) and $|59P_{3/2}\rangle$ (red triangles) show Rabi oscillations by varying $\Theta$ from 0 to 6.5$\pi$. The data is normalized to a microwave-free signal. The gray vertical dash line marked the rotation angle in (a).

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We perform similar measurements for $\Theta$ = 0-6.5 $\pi$ and sum the number of photon detection in each time window of Fig. 3(a). The results given in Fig. 3(b) clearly show Rabi oscillations for both 60$S_{1/2}$ (blue circles) and $59P_{3/2}$ (red triangles) states, and a maximum population in 60$S_{1/2}$ correspond to a minimum population in $59P_{3/2}$ or vice versa. The retrieved photon counts are normalized to microwave-free signal. It is noted that the maximum counts from $59P_{3/2}$ are almost half of the counts from 60$S_{1/2}$, this reduction may be caused by the dephasing due to the time delay of the microwave $\pi$ pulse. Further, we also find that the amplitude of the Rabi oscillations shows a bit of suppression at a higher rotation angle $\Theta$. This is because the dipole interaction between the 60$S_{1/2}$ and $59P_{3/2}$ leads to more dephasing with increasing the duration of microwave coupling, as the value of $g^{(2)}(0)=0.47$ at 60$S_{1/2}$ indicate that our experiment has the probability of excitation of more than one photon. The estimated dipole-dipole interactions between 60$S_{1/2}$ and $59P_{3/2}$ is 11.6 MHz using $V_{dd} = C_3/R_b^3$. The ratio of $\Omega _{MW}$ and $V_{dd}$ is about 1.08. As they are comparable, the dipole-dipole interactions could induce some dephasing.

Microwave manipulation of stored-photon could be a technique to modulate or shape retrieval photons. We then investigate case II, where we apply the microwave field during the retrieval process, see case II of Fig. 1(c). During the coupling laser on for retrieving the stored photon, the microwave field is added to modulate the Rydberg atom population and further the retrieved photon. Different from the case I of changing the microwave pulse duration $t$, here we change the microwave power and further the Rabi frequency $\Omega _{MW}$ from 2$\pi \times$ 5.27 to 2$\pi \times$ 52.71 MHz with a fixed duration of 400 ns, which is calibrated using Rabi oscillation in Fig. 3(b). Figure 4(a) shows the three retrieved signal shapes at indicated microwave Rabi frequency $\Omega _{MW} / 2\pi =$ 0, 8.35, 20.99 MHz. All traces are normalized to the microwave-free peak. We can see that the retrieved signal shows oscillations, and the extracted oscillation frequency of 15.72 MHz (middle curve) and 24.81 MHz (bottom curve) are faster than the calibrated microwave $\Omega _{MW}/2\pi$=8.35 MHz and 20.99 MHz, respectively. This is because the coupling laser dressing of the Rydberg state causes the detuning of the microwave field, leading to the effective Rabi oscillation of the microwave field that is determined by the Rabi frequency of the microwave field and the detuning. In Fig. $4(a)$, we further simulate the retrieved photon profile in case II and represent it with black dash curve. Figure 4(b) displays the retrieved signal as a function of the time and $\Omega _{MW}$ for the range of $\Omega _{MW} / 2\pi =$ 5.27 MHz to 52.71 MHz. It is seen that the modulation frequency increase with the microwave Rabi frequency since the modulation arises from population variation between $|60S_{1/2}\rangle$ and $|59P_{3/2}\rangle$. The modulation frequency can be up to more than 50 MHz.

 

Fig. 4. Normalized retrieval signals of case II, where microwave field is applied during retrieval. (a) Normalized retrieval signals at indicated microwave Rabi frequency $\Omega _{MW} / 2\pi =$ 0, 8.35, 20.99 MHz with $\Omega _c = 2\pi \times$ 14.0 MHz. The black dash curve represents the theoretical simulation result. (b) The color map of the modulation of the retrieved signal with the microwave Rabi frequency from 2$\pi \times$ 5.27 to 2$\pi \times$ 52.71 MHz. All traces are normalized to the microwave-free peak.

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5. Conclusions

We have achieved an on-demand single photon source based on Rydberg polariton. The second correlation $g^{(2)}(\tau )$ of the retrieved photon is obtained by the HBT measurement, the $g^{(2)}(0)$=0.29$\pm$0.08 for $80S_{1/2}$. By applying the resonant microwave field during the storage (case I) or the retrieval (case II) process, we have shown a coherent manipulation of retrieved photons, where the presence or absence of a microwave pulse can be converted into photon readout early or late. The unaddressed level is used to store photons until it is coupled by a microwave, achieving a controllable Rabi rotation between two adjacent Rydberg states. In addition, we have demonstrated a modulation of retrieved photons using a microwave field, realizing a modulation frequency of up to 50 MHz. An improved superatom model is employed to model the experimental results well. Our results provide a coherent manuscript of optical photons using microwaves, which will be a promising platform for developing quantum technologies.