1. Introduction

Hybridizing disparate quantum systems could exploit the advantages of each component and compensate the weaknesses with each other [1–8]. A promising candidate for hybrid quantum system is to combine the superconducting (SC) circuits with neutral atoms to realize a scalable processor that is capable of both rapid information processing ($\sim 1 \mathrm {ns}$) and long coherence time ($1 \mathrm {ms} \sim 1 \mathrm {s}$) [9,10]. The SC circuits and cold atoms can be directly coupled via magnetic dipole interaction [11–14], or in this context, indirectly coupled via integrating both of them on a SC resonator [15–17], which works as a data bus to transfer the quantum information between the SC circuits (serving as a quantum processor) and the cold atoms (serving as a quantum memory).

For the SC resonator-atom hybrid, a coherent photon in microwave resonator strongly couples the atomic transition between two long-lifetime states [18–22]. The SC resonators, including the LC cavity and coplanar waveguide (CPW) resonator, can preserve the coherence of the cavity photon in a time scale of 1 $\mu$s-1 ms [23]. Owing to the large dipole moments and long lifetime, the highly-excited Rydberg states are usually used as the intermediate qubits to interact with the SC resonator [18]. In the envisioned hybrid quantum interface, the quantum information encoded in the Rydberg states can be mapped onto the hyperfine ground states for long-term storage [24,25], as well as it can be converted into the flying-qubits for communication among a quantum network [26]. The high-performance state transfer and entanglement swapping between the disparate components in hybrid quantum interface rely on the high-fidelity quantum control on the hybrid states.

In this paper, we propose a hybrid atom-photon gate in the SC resonator based on a geometric phase and Raman chirped shortcut to adiabatic passage (RCSAP). The controlled-PHASE gate can be realized with a conditional dynamic phase or pure geometric phase. The latter has exhibited the robustness against certain noises [27–32], which solely relies on the global geometric features during the evolution. Recently, geometric quantum gates were experimentally realized in SC circuit [33–35] as well as other systems [36]. Meanwhile, Raman chirped adiabatic passage (RCAP) makes a $\Lambda$-type three-level system adiabatically interact with a quantized cavity field and classical frequency-chirped laser pulses via two-photon transition, which was previously employed to control the laser-induced vibrational excitation of molecules [37–39]. Equivalently, RCAP can be applied for the $\Xi$-type three-level system, such as current-biased Josephson junctions [41] or Rydberg atoms whose upper levels involved are long-lived states. This adiabatic passage technique is time-consuming to be realized, and therefore, it is unsuitable for some fast dephasing quantum systems. Shortcuts to adiabaticity protocols can speed up the slow driving [42,43]. Our approach has both merits of geometric operations and shortcuts to adiabaticity [44–46]: the application of shortcuts to adiabaticity makes the operation faster than RCAP and thus protects against the fast dephasing effect of the SC resonator photon; the hybrid atom-photon gate based on purely geometric phase is more resilient than the stimulated Raman transition to the system errors. Furthermore, only requiring the amplitude and frequency modulation of the Raman pulses, our scheme simplifies the implementation in experiments and does not need rapidly switching the SC cavity frequency into the atomic resonance during the gate operation as the previous approach does [25]. Fast and robust against some systematic noises, our scheme is particularly suitable for hybrid SC-atoms quantum interface, quantum memory and entanglement swapping.

The paper is organized as follows: Section II introduces the collective dressed states and the conditional geometric $\pi$ phase shift without the dynamical phase using adiabatic passages. Section III shows how to accelerate the adiabatic manipulations using the modified chirped pulses. In Section IV we discuss the robustness of the hybrid atom-photon gate against the fluctuation of control parameters. We conclude this paper in Section V.

2. Conditional CZ gate under adiabatic manipulation

We consider a cold atomic ensemble containing $N \simeq {10^6}$ atoms which is trapped above the electric field antinode of a CPW cavity with cavity frequency ${\omega _c}$ and lifetime $1/\kappa$, as shown in Fig 1(a). When the spatial dimension of the atomic cloud is small compared to the mode wavelength, all the atoms symmetrically couple to the cavity field. The CPW field and atomic levels for the hybrid gate protocol is shown in Fig. 1(b). The relevant states of the atoms are two ground states ${\left | g \right \rangle }$ and ${\left | s \right \rangle }$, as well as a pair of close-lying ($\Delta n = 0$) Rydberg states $\left | i \right \rangle$ and $\left | r \right \rangle$ having a strong electric dipole transition with frequency ${\omega _{ri}}$, which is close to the cavity frequency ${\omega _c}$. A spatially uniform laser field couples the ground state ${\left | g \right \rangle }$ to the Rydberg state ${\left | i \right \rangle }$ with time-dependent Rabi frequency ${\Omega (t)}$ and large detuning $\Delta (t) = \omega - {\omega _{ig}}$, where $\omega$ is the laser frequency and $\omega _{ji}$ indicates the atomic transition frequency of $\left |i\right \rangle \rightarrow \left |j\right \rangle$. The cavity detuning is $\Delta _{c}={\omega _c} - {\omega _{ri}}$. In the frame rotating with the laser frequency $\omega$ and cavity frequency $\omega _c$, the interaction Hamiltonian for the atom-cavity system is

 

Fig. 1. (a) An ensemble of ultracold atoms trapped near the antinode of the standing-wave field in the superconducting CPW cavity. (b) Raman chirped adiabatic passage scheme for hybrid atom-photon gate. CPW resonator photons are coupled to the atomic transition between two close-lying Rydberg states. $\Omega$: laser field; $\eta _{ac}$: CPW field; $\Delta$: single-photon detuning; $\delta$: two-photon detuning. (c) Evolution paths of cyclic state $\left | {{\lambda _ + }} \right \rangle$ (red solid line) on the Bloch sphere to realize geometric operation in the basis $\{ \left | G, 1 \right \rangle ,\left | R, 0 \right \rangle \}$. Cyclic state initialized in state $\left | G, 1 \right \rangle$ will evolve along the path A-B-C-D-A driven by the effective magnetic field $B_{0}$ (blue dashed line) and pick up a geometric phase $\gamma$.

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Here we set the detuning $\Delta \simeq -\Delta _{c}$. Given a photon in the SC resonator, the external laser field and the quantized cavity field induce a two-photon transition from the ground state $\left | g \right \rangle$ to the Rydberg state $\left | r \right \rangle$ via the intermediate state $\left | i \right \rangle$. For a large detuning $\Delta \gg \eta _{ac}$, $\sqrt N \Omega$, $\delta$, we adiabatically eliminate state $\left | i \right \rangle$ and thereby obtain an effective interaction Hamiltonian

Given that a cavity photon is coherently absorbed in this process, a single atom from the ensemble will be excited to the Rydberg state $\left | r \right \rangle$. At this stage, we introduce ensemble ground state $\left | G \right \rangle = \left | {{g_1},{g_2},\ldots ,{g_N}} \right \rangle$ and Dicke state $\left | S\right \rangle = (1/{\sqrt N })\sum \nolimits _{j = 1}^N {\left | {{g_1},\ldots ,{s_j},\ldots ,{g_N}} \right \rangle }$ as the atomic ensemble qubits, as well as the singly excited Rydberg states $\left | R \right \rangle = (1/{\sqrt N })\sum \nolimits _{j=1}^N {\left | {{g_1},\ldots ,{r_j},\ldots ,{g_N}} \right \rangle }$. Since intermediate state $\left | i \right \rangle$ is weakly coupled ($\Delta \gg \Omega$), its high-order collective states should be negligible and we consider its first-order collective state $\left | I \right \rangle = (1/{\sqrt N })\sum \nolimits _{j=1}^N {\left | {{g_1},\ldots ,{i_j},\ldots ,{g_N}} \right \rangle }$. This singly excited state $\left | I \right \rangle$ coupled to ensemble ground state $\left | G \right \rangle$ gives a $\sqrt N$ collective coupling enhancement for the single atom Rabi frequency $\Omega$ [47]. The gate operations between the basis states $\left | G \right \rangle$ and $\left | S \right \rangle$ are performed by virtue of the collective states $\left | I \right \rangle$ and $\left | R \right \rangle$. In the two-qubit basis of states $\{ {\left | G, 1 \right \rangle } ,{\left | R, 0 \right \rangle } \}$, the Hamiltonian (2) can be rewritten as that for a two-level system

Accordingly, out of four possible initial two-qubit states ${\left | G, 0 \right \rangle }$, ${\left | G, 1 \right \rangle }$, ${\left | S, 0 \right \rangle }$, ${\left | S, 1 \right \rangle }$, only the basis state ${\left | G, 1 \right \rangle }$ will participate in the evolution of the effective two-level system spanned by the eigenstates states $\{ \left | {{\lambda _ - }} \right \rangle ,\left | {{\lambda _ + }} \right \rangle \}$. This can be utilized to realize a geometric controlled-PHASE gate via the following “orange slice” scheme [45]. In adiabatic case, the dressed states $\left | {{\lambda _ + }} \right \rangle$ and $\left | {{\lambda _ - }} \right \rangle$ become cyclic states under the driving of control parameters. The initial state $\left | {{\lambda _ + }} \right \rangle (t=0) = {\left | G, 1 \right \rangle }$ evolves along the corresponding eigenstate $\left | {{\lambda _ + }} \right \rangle$, and after a cyclic evolution with period $2\tau$, the final state $\left | {{\lambda _ - }} \right \rangle (t=2\tau ) = {\left | G, 1 \right \rangle }$ picks up an adiabatic phase including both dynamical and geometric components. The accumulation of dynamical phase can be canceled by the spin-echo method as detailed below, and pure geometric phase is obtained here.

We describe in detail how to achieve the above geometric operation via Raman chirped adiabatic passages. The general expression of the chirped laser pulses in RCAP are given by

A crucial requirement for the geometric controlled-PHASE gate is that the accumulated dynamical phase should be removed [29,30]. For this purpose, we rewrite the Hamiltonian (3) as $H_{\mathrm {eff}}(t)=-(\hbar / 2) \vec \sigma \cdot \mathbf {B_0}$, where $\mathbf {B_0}=\left (\Omega _{\mathrm {eff}} \sin \varphi , \Omega _{\mathrm {eff}} \cos \varphi , -\Delta _{\mathrm {eff}}\right )$ is the effective magnetic field. The unit vector trace of magnetic field $\mathbf {B_0}$ is shown by the blue dashed line in Fig. 1(c), which is precisely followed by the polarized vector $n_{ \pm }(t)=\left \langle \lambda _{ \pm }(t)|\vec {\sigma }| \lambda _{ \pm }(t)\right \rangle$. Since the angle $\phi$ between the vectors ${n}_ {+}$ and $B_{0}$ satisfies $\phi [{{n}_ {+} }(t),\mathbf {B_{0}}(t)] = \phi [ - {{n}_{+} }(t+\tau ),{B}_{0}(t+\tau )]$, and ${ H_\textrm{eff}}(t)={ H_\textrm{eff}}(t+\tau )$, the accumulation of the dynamical phase is completely canceled out.

3. Acceleration by shortcut to adiabatic passage

Since the adiabatic evolution requires the adiabatic condition, RCAP is usually time consuming to realize, which limits its application to the fast dephasing quantum system. To overcome this drawback, we use a shortcut method to accelerate RCAP. Through the reverse engineering approach [39,40,42,43], one can suppress the non-adiabatic transition and speed up the procedure by adding an superadiabatic correction Hamiltonian given by

Before proceeding we survey the relevant experimental parameters. The atomic lower states $\left | g \right \rangle$ and $\left | s \right \rangle$ explicitly correspond to the $^{87}$Rb hyperfine ground state $| g \rangle =| 5S_{1 / 2}, F=1, m_{F}=0 \rangle$ and $| s \rangle =| 5S_{1 / 2}, F=2, m_{F}=0 \rangle$. The cavity field is coupled from the Rydberg state $| i \rangle =| 81 P_{3 / 2}, m_{J}=1 / 2 \rangle$ to $| r \rangle =| 81 S_{1 / 2}, m_{J}=1 / 2 \rangle$ with transition frequency $\omega _{ri}=2 \pi \times 6.757$ GHz and a dipole moment $d_{ri}=3242$ $ea_{0}$. The magnetic trapping of the atoms above the SC resonator with the height of 8 $\mu$m yields a vacuum Rabi frequency $\eta _{ac}/ 2 \pi \simeq 4$ MHz for $n=81$. Given that the blackbody-induced decay at cryogenic temperature is negligible, the intrinsic lifetimes of the Rydberg states $| i \rangle$ and $| r \rangle$ are around $1.5$ ms and $1.2$ ms respectively, which correspond to the decay rates $\Gamma _{i}/ 2 \pi = 106$ Hz and $\Gamma _{r}/ 2 \pi = 133$ Hz. The transition frequency $\omega _{ri}$ between the adjacent $n$ states scale as $n^{-3}$, the dipole moment $d_{ri} \propto n^{2}$, and the radiative lifetime $\tau _{r} \propto n^{3}$. Although electric dipole in single-step Rydberg excitation is approximately six orders of magnitude weaker than the coupling between the nearest Rydberg states, the peak coupling strength of the control laser after a $\sqrt N$ enhancement $\sqrt N \Omega \simeq 3\eta _{ac}$ is accessible in our scheme. Some other system parameters are chosen as $\sqrt {N} \Omega _{0}=2\pi \times 8$ MHz, $\Delta _{0}=2\pi \times 40$ MHz and $\delta _{0}=2\pi \times 0.4$ MHz. The cavity frequency should be $\omega _{c}=\omega _{ri}- \Delta _{0} \simeq 2\pi \times 6.7$ GHz. The best CPW resonator to date has a quality factor $Q \simeq 10^{7}$ [48]. Assuming a quality factor $Q \simeq 3.3 \times 10^{6}$, the photon decay rate $\kappa = \omega _{c}/Q \simeq 2\pi \times 2$ kHz.

The relaxation processes in hybrid system are unavoidable. The Rydberg decay processes is described by the Liouvillians

Here we simulate the hybrid system dynamics of transfer process driven by RCAP and RCSAP with above parameters [24,49]. The initial state of cavity field can be prepared by mapping SC qubit state onto the cavity mode via dynamically controlling the qubit into resonance for a rotation time $T_{qc}=\pi /(2g_{SC})$. The total transfer time of quantum information between SC qubit and atomic ensemble is $T=T_{qc}+T_{ac}$. To characterize the intrinsic quality of the hybrid gate between SC resonator and atomic ensemble, we assume the perfect resonator initialization as well as its single qubit operations, and focus on the operation time $T_{ac}$ and fidelity of the state transfer between the SC cavity and atomic ensemble. We set the operation time $T_{ac} = 2\tau = 2.4$ $\mu$s. The time-dependence of Rabi frequency $\Omega$ and two-photon detuning $\delta (t)$ are plotted in Fig. 2(a), which control the quantum state along the closed path by RCAP. According to the Eq. (8), the modified chirped pulse shapes of RCSAP are plotted in Fig. 2(b). The initial state of the system is the single photon state $\left | G, 1 \right \rangle$ in cavity. We plot the population transferred to the single Rydberg excitation state $\left | R, 0 \right \rangle$ as driven by RCAP and RCSAP in Fig. 2(c). The population of RCAP is less than $60\% $ after the operation time $\tau = 1.2$ $\mu$s since the adiabatic condition is not satisfied $[(\sqrt N \Omega \eta _{ac}/2{\Delta _0})\tau \approx 0.96\pi ]$. However, after suppressing the non-adiabatic effect by modifying the chirped pulse, we obtain a single collective Rydberg excitation $\left | R \right \rangle$ of the atomic ensemble with the high efficiency $P \ge 0.992$ , which is slightly less than unity due to the dephasing and decay. Using another RCSAP or Raman pulse, the single Rydberg excitation can be transferred to a collective ground state $\left | S \right \rangle$ for long-term storage; furthermore, by applying the coupling field on the transition from $\left | r \right \rangle$ to an excited state, the single Rydberg excitation is converted to a propagating photon for the optical readout of the SC qubits.

 

Fig. 2. (a) The Rabi frequency $\sqrt {N}\Omega$ and two-photon detuning $\delta$ for conditional CZ gate according to Eqs. (5) and (6). (b) The modified Rabi frequency $\sqrt {N}\tilde {\Omega }$ and two-photon detuning $\tilde {\delta }$ for conditional CZ gate according to Eq. (8). (c) The population dynamics of the single Rydberg excitation state driven by RCAP (black solid line) and RCSAP (red solid line) in the “orange slice” scheme , respectively. Not that the adiabatic condition is not fulfilled here. At the time $t=\tau$, RCAP cannot realize perfect transfer to a single Rydberg excitation state. However, after eliminating the non-adiabatic effect, RCSAP can make the collective state transfer perfectly, meanwhile a microwave photon is stored in atoms.

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4. Robustness

Now we proceed to discuss the performance of the above RCSAP against the typical perturbations. The imperfection of a hybrid atom-cavity system is usually due to the fluctuations of the control field, the photon decay in cavity, the incoherent thermal photon and the atom-surface interactions. Previous theories and experiments have shown that geometric quantum gates in quantum computation are insensitive to random fluctuations with relatively high frequency [30,45]. Here we mainly focus on the quasistatic fluctuations, namely the system errors in the control parameters.

Two system errors are specially considered in Figs. 3(a)–3(f): the shift of vacuum coupling strength $\eta _{ac}$, which comes from the inhomogeneous distribution of the cavity field as well as the velocity distribution of cold atoms, and the imperfection of two-photon detuning $\delta$, mainly introduced by the Zeeman shift and the ac Stark light shift due to other unwanted virtual couplings. As shown in [24], the Raman pulse with a time-independent Rabi frequency $\sqrt N \Omega \simeq \eta _{ac}$ for a time $T_{r}=\pi / \Omega _{\mathrm {eff}}$ can transfer the cavity Fock ${\left | G, 1 \right \rangle }$ state to the single Rydberg state ${\left | R, 0 \right \rangle }$ in a two-photon process. We compare three different ways to prepare a single Rydberg excitation (RCSAP, RCAP, and Raman pulse) with the same system parameters in Fig. 2(a). The operation times of the RCSAP, RCAP, and Raman pulse are ${T_{sa}} = 2.5\mu s$, ${T_{a}} = 25\mu s$, and ${T_{r}} = 2.5\mu s$, respectively. We note that, for a certain single-photon detuning, the single Rydberg excitation preparation time of Raman pulse is currently limited by the finite vacuum coupling strength [$T_{r} = 2\pi {\Delta _0}/{\eta _{ac}^2}$], while RCSAP even can transfer the state faster than Raman pulse with a higher Rabi frequency $\sqrt N \tilde {\Omega }$ as the case in Fig. 2(b). We plot the fidelity of the target state as a function of the relative coupling strength deviation $\beta$ and two-photon detuning deviation $\varepsilon$. The fidelity is defined as $F = \mathrm {Tr}(\sqrt {\sqrt {{\rho _i}} \rho \sqrt {{\rho _i}} } )$, where $\rho$ denotes the actual density matrix and $\rho _{i}=\left |\Psi _{i}\right \rangle \left \langle \Psi _{i}\right |$ is final density matrix for an ideal transfer. Clearly, the fidelity of the hybrid state transfer driven by RCSAP is more robust against systematic error than the Raman pulse with the same operation time. However, the fidelity of RCAP with the ten-fold operation time of RCSAP drops to $\rm {86}\%$ mainly due to the finite lifetime of cavity photon.

 

Fig. 3. The fidelities ${F}$ versus the deviations of vacuum coupling strength $\beta \eta _{ac}$ and two-photon detuning $\varepsilon \delta _0$ from the target state with the realistic decoherence. (a) The state transfer to single Rydberg excitation driven by RCSAP with the operation time ${T_{sa}} = 2.5\mu s$, (b) RCAP with operation time ${T_{a}} = 25\mu s$, (c) Raman pulse with operation time ${T_{r}} = 2.5\mu s$, respectively. (d) The Bell state $\left | {{\Psi ^ + }} \right \rangle$ preparation using the hybrid quantum gate driven by RCSAP with operation time ${T'_{sa}} = 5\mu s$, (e) RCAP with operation time ${T'_{a}} = 50\mu s$, (f) Raman pulse with operation time ${T'_{r}} = 5\mu s$, respectively.

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To evaluate the hybrid atom-photon gate performance and entanglement generation, we calculate the preparation fidelity of the Bell state $| \Psi ^{+} \rangle =\frac {1}{\sqrt {2}}({\left | G, 1 \right \rangle }+{\left | S, 0 \right \rangle })$, which is established using the CNOT gate with initially the ensemble in $\left | G \right \rangle$ and the cavity photon in the superposition state $(\left | 0 \right \rangle + \left | 1 \right \rangle )/\sqrt 2$. As shown in Fig. 3(f), the entanglement preparation fidelity of the Raman pulse drops to $\rm {75}\%$ when vacuum coupling strength or two-photon detuning deviates by $\rm {10}\%$. The fidelity composed by RCAP remains at low level of about $\rm {70}\%$ due to the system decoherence. Remarkably, as shown in Fig. 3(d), the Bell-state preparation fidelity driven by RCSAP is higher than $\rm {90}\%$ even when the vacuum Rabi frequency or two-photon detuning deviates by $\rm {10}\%$. We notice that the peak fidelity of RCSAP in Fig. 3(d) is about $\rm {99.5}\%$, which can be increased by shortening the operation time ${T'_{sa}}$. Indeed, there is a trade-off between the peak fidelity and the robustness. When we shorten the operation time, the accumulated decay possibilities of the cavity photon and Rydberg states are reduced thus resulting in an elevated peak fidelity of RCSAP, but the geometric quantum control becomes less robust against the systematic errors [46]. As a result of this, we conclude that the hybrid photon-atom gate based on RCSAP is more robust than Raman pulse against the control parameter variations, and its high-speed operations intrinsically protect against the decoherence in hybrid atom-photon system.

5. Discussion and conclusion

In the resonant pulse protocol [25], a single cavity photon induces a conditional $\pi$ phase shift after a $2\pi$ rotation time, which requires rapidly either tuning of the SC resonator, or of the Rydberg levels using an external Stark shift. Actually, this requirement is easy to implement. For example, the SC resonator frequency is usually fast tuned by inserting a superconducting quantum interference device (SQUID) in the resonator and applying with magnetic field [50,51]. The SQUID can be regarded as a lumped element inductor with the subgap resistance $R_{s}$ and tunable inductance $L_{s}(\Phi )$, where $\Phi$ is the applied magnetic flux. For small detunings, the resonator frequency can be approximated by $f_{r}(\Phi )=f_{0} /\left [1+L_{s}(\Phi ) / (L l)\right ]$, where $f_{0}=(4 \sqrt {L C} l)^{-1}$, $L$ and $C$ are the inductance and capacitance per unit length of the transmission line, respectively, and $l$ is the resonator length. However, when resonator is detuned, more power will be dissipated by the subgap resistance of SQUID, thereby reducing the lifetime of photon in SC resonator. The practical realization of our scheme has no need of fast frequency-tuning of the SC resonator, therefore, avoiding the detrimental effect on the SC resonator coherence.

To conclude, we have proposed a feasible scheme for the hybrid atom-photon quantum gates in a SC resonator based on Raman chirped shortcut to adiabatic passages. The designed hybrid quantum gates are based on the geometric phase, which are robust against some system errors such as the fluctuation of vacuum coupling strength and optical frequency. The manipulations are fast due to the shortcut to adiabaticity, which protects against the decoherence in the hybrid SC-atoms system. As a result, we obtain an efficient hybrid state transfer and atom-photon Bell-state preparation. This scheme is easy to realize in experiments, and may pave a way to the hybrid SC-atoms quantum interface and quantum memory.