Universal quantum gates for hybrid system assisted by atomic ensembles embedded in double-sided optical cavities

We propose deterministic schemes for controlled-NOT (CNOT), Toffoli, and Fredkin gates between flying photon qubits and the collective spin wave (magnon) of an atomic ensemble inside double-sided optical microcavities. All the gates can be accomplished with 100% success probability in principle and no additional qubit is required. Atomic ensemble is employed so that light-matter coupling is remarkably improved by collective enhancement. We qualified the performance of the gates and the results show that they can be faithfully constituted with current experimental techniques.

hyperparallel hybrid photonic quantum logic gates were proposed in 2015 22 . Recently, an magnon-cavity unit, e.g., an atomic ensemble confined in a double-sided cavity, was proposed by Li et al. 34 , in which the interaction between the collective spin wave (magnon) of an atomic ensemble and the successive photons provides strong Kerr nonlinearities.
In this paper, inspired by the above works, we investigate the possibility of achieving scalable photonic quantum computation assisted by an atomic ensemble in a double-sided cavity. Our schemes are different from the work by Li et al. 34 in which they present a scheme for two CNOT gates with the photonic qubits both in the spatial degrees of freedom (DOF) and the polarization DOF of each photon. By the nonlinear interaction between the moving photon and the magnon of an atomic ensemble in a double-sided cavity, we first present a deterministic scheme for constructing a CNOT gate on a hybrid system with the flying photon as the control qubit and the atomic ensemble as the target qubit. Besides, we construct the Toffoli and Fredkin gates on a three-qubit hybrid system in a deterministic way. In our work, the control qubit of our universal gates is encoded on the polarization states of the moving photon, while the target qubit is encoded on the state of atomic ensemble inside an optical microcavity. These three schemes for the universal gates require no additional qubit, and they only need some linear optical elements besides Kerr nonlinear interaction between the magnon and the photons. High fidelities and high efficiencies can be achieved in the strong coupling regime and are not sensitive to the frequency detuning and coupling imbalance.

Results
Input-output relation for a single photon with a magnon-cavity coupling system. The configuration of the atomic ensemble cavity coupling system considered here is exhibited schematically in Fig. 1. We first denote a highly excited Rydberg state as |r〉 . Assisted by the Rydberg state |r〉 , one can prepare the atomic ensemble into the magnon state and perform the single-qubit operation on the magnon qubit. A qubit is encoded in collective spin wave state or magnon state with a single atom in the states |g 0 〉 and |g 1 〉 of the atomic ensemble. If we define | 〉 = | …… … 〉 + g g g g g , , 1 with frequency ω 0 are driven by orthogonal polarizations (H and V) of a photon with frequency ω. Meanwhile the two transitions are nearly resonantly coupled to the two degenerate cavity modes â 0 and â 1 with the corresponding coupling rates are λ 0 and λ 1 , respectively. For the input photons with different polarizations, the transmission and reflection coefficients are determined by the state of the ensemble. If a polarized photon is injected into the cavity via either side of the cavity, it will pass through the cavity if it is decoupled from the driven cavity mode; otherwise it will interact with the atomic ensemble if it is coupled to the cavity mode and lead to the mode splitting. When the frequencies of the optical fields close to the cavity frequency ω a , we can take the coupling rates between an asymmetrical cavity and modes ω b ( ) j and ω c ( ) j of ports B and C as real constant 31 . Here, to insure the photon pulse shape remains unchanged, we need a single polarized photon pulse with a finite bandwidth , which is satisfied when κ ∆  /2 (the cavity decay rate) 17,18 . If we take ω a as the carrier frequency, then δ′ = ω − ω a denotes the frequency detuning of the input photon with frequency ω. δ 0 = ω 0 − ω a measures the frequency difference between the dipole transition and the cavity mode. This system exhibits similar features with the Jaynes-Cummings model, and in the frame rotating with respect to ω a , the dynamics of the system is governed by the following hamiltonian (ħ = 1) 31,33,37 here γ e j and λ j denote the spontaneous emission rate of the single excited collective state e j s and the coupling rate between the atomic ensemble and the corresponding resonant cavity mode, respectively. With the help of Rydberg state 38,39 or coherent Raman process 40,41 , one can pump the atomic ensemble to the magnon state g j s , so that the input photon will drive the interaction between the atomic ensemble and the cavity mode. In the single excitation subspace, the system will evolve in the space spanned by the internal states of the atomic ensemble and the photon number states of the radiation modes (â j , b j , and ĉ j ), respectively. Suppose the initial state of the system is g , 1, 0, 0 j s , i.e., we choose the input photon in mode b j , then the state of the system, at time t, will evolve to The Schrödinger equation for this system can be specified to be Along with the standard input-output relation κ = +ˆŷ y a j j y j ,out ,in (y = b, c), we can see the birefringent character of the magnon-cavity system. Here ŷ j,in and ŷ j,out are the input and output field operators, respectively. Under the condition that the incoming field is very weak, i.e., we take σ , the reflection and transmission coefficients of the system can be expressed as In the case the input photons uncoupled to the cavity, i.e., λ j = 0, we get the reflection and transmission coefficients for the system, then Eq. (4) reduces to As the backscattering is low in the optical fibers, the asymmetry of the two coupling constants is mainly caused by cavity intrinsic loss 42 c min ), i.e., the difference of the coupling rates between the cavity and the modes ω b ( ) j and ω c ( ) j are small, one can replace the reflection and transmission coefficients above for the asymmetrical cavity system with those for the symmetrical one with identical coupling rates, i.e., we set κ = κ b = κ c . With the symmetrical cavity, the corresponding reflection and transmission coefficients can be respectively simplified and given by for λ > 0 (hot cavity), and for λ = 0 (cold cavity, described with the subscript 0). The reflection and transmission coefficients in Eqs (6) and (7) indicate that the output photon experiences a phase shift relying on the different states of the atomic ensemble in the double-sided cavity. When the Purcell factor λ 2 /κγ = 1/2, the reflection and transmission coefficients are r(ω) → 1 and t(ω) → 0. However, in the decoupling case (λ = 0), the reflection and transmission coefficients of the bare cavity are r 0 (ω) → 0 and t 0 (ω) → − 1. Specifically, if the atomic ensemble is in the state | 〉 g s 0 | 〉 g ( ) s 1 , when the photon in |H〉 (|V〉 ) state is directed into the cavity, it will be reflected and get no phase shift. Otherwise, the photon will transmit the cavity and get a π phase shift. This exactly demonstrates the effective Kerr nonlinearity which can be used to constitute the hybrid multi-qubit gates in the following sections.
CNOT gate on a two-qubit hybrid system. The framework of our CNOT gate, which flips the target atomic ensemble qubit if the control photon polarization qubit is in the state |V〉 , is depicted in Fig. 2. The flying photon p and the atomic ensemble are prepared in arbitrary superposition states ϕ ), respectively. For conciseness, we define single-qubit Hadamard operations H p and H s for one photon and one magnon qubit respectively as: First, the injected photon passes through a polarized beam splitter (PBS 1 ), which transmits the photon in the polarization state |H〉 and reflects the photon in the state |V〉 . The part in the state |H〉 transmits PBS 1 and gets into a delay line (DL), does not interact with the cavity, while the part in the state |V〉 passes a half-wave plate (HWP 1 ), which is used to perform a Hadamard operation (H p ) on the photon. Then the photon passes a beam splitter (BS) and be injected into the cavity from either path a 1 or a 2 . At the same time, we perform a Hadamard operation (H s ) on the atomic ensemble with the coherent Raman process or Rydberg-state-assisted quantum rotation. Then the state of the whole system composed of a photon and an atomic ensemble is changed from Ψ 0 to Ψ 1 . Here Considering the birefringent propagation of the input polarized photon, the output state of photon together with that of the atomic ensemble is  When the photon p passes through path a 1 , it will be split by PBS 2 , the H-polarized component takes a phase shift π (i.e., |H〉 → − |H〉 ) after passing through the phase shifter P π . Then the photon passes PBS 3 will take an H p operation by HWP 3 . Meanwhile the photon passes through path a 2 will take an H p operation by HWP 2 . After the photon passes through PBS 4 and HWP 4 , the state of the system becomes One can see that the state of the atomic ensemble is flipped when the photon (the control qubit) is in the state |V〉 , while it does not change when the photon is in the state |H〉 , compared to the original state of the two-qubit hybrid system shown in Eq. (10). Therefore, the quantum circuit shown in Fig. 2 can be used to construct a deterministic CNOT gate with a success probability of 100% in principle by using the photon as the control qubit and the atomic ensemble as the target qubit.
Toffoli gate on a three-qubit hybrid system. The schematic diagram for implementing a deterministic three-qubit Toffoli gate is depicted in Fig. 3  First the photon reaches PBS 1 , the photon in the state |V〉 is injected into the cavity from path a 2 , while the photon in the state |H〉 does not interact with the atomic ensemble inside the cavity. With the same arguments as made for the CNOT gate above, we find that after the photon interacts with the atomic ensemble inside cavity 1, the state of the whole system evolves from Φ 0 to Φ 1 . And  Then the photon from path a 1 goes into a DL, while the photon from path a 2 passes HWP 1 and BS, and then gets into cavity 2 from path a 1 or a 2 . Meanwhile we apply an H s operation on the atomic ensemble in cavity 2. Considering the interaction between the photon and the atomic ensemble in cavity 2, we find the state of the system evolves from Φ 1 to Φ 2 , here After the photon passes through the CCM, it is led back to cavity 1 from path ′ a 1 , at the same time we lead the photon in path a 1 into cavity 1 again (see the green lines), then the state of the system evolves into After the photon reaches PBS 2 , we can see that the state of the target magnon qubit in cavity 2 is flipped when the two control photonic qubit and the magnon qubit in cavity 1 are in the state |V〉 and g s 1 , respectively. Therefore the quantum circuit shown in Fig. 3 can be used to construct a Toffoli gate on a photon-magnon hybrid system in a deterministic way.
Fredkin gate on a three-qubit hybrid system. The three-qubit Fredkin gate implements a swap operation on two stationary atomic ensemble qubits in cavities 1 and 2 when the flying photon is in the state |V〉 . Suppose that the initial states of the flying photon and the two atomic ensembles confined in the two double-sided cavities are . As illustrated in Fig. 4, our scheme for a three-qubit Fredkin gate can be achieved with three steps.
Step 1. The injected photon is split by PBS 1 into two wave-packets, the photon in state |H〉 dose not interact with the atomic ensemble in cavity 1, while the photon in state |V〉 goes into path 2 and experiences the nonlinearities (see the green lines). After the photon in the state |V〉 is injected into cavity 1, the state of the three-qubit hybrid system changes to It can be seen that, when the photon in |V〉 passes through the two cavities in succession, the output path of the photon is determined by the parity of the two magnon qubits.
Step 2. The photon at S will be led to path 8, while the photon emitting from path 6 be led into cavity 1 again. As discussed above, in this round, the photon in path 6 acts as the control qubit and performs NOT operations on the magnon qubits in cavities 1 and 2, respectively (see the grey lines, i.e., HWP 1 . For this purpose, H s operations on the atomic ensembles in cavities 1 and 2 before and after the photon interacts with the corresponding magnon qubit respectively are needed. When the photon emits from path 7, the output state of the system is Step 3. In this round, the photon emitting from path 7 or 8 will be led into cavities 1 and 2 successively again. As discussed in step 1, after the photon interacts with cavity 2 again, the state of the system evolves into H g g Hg g H g g Hg g After this round, the photon emitting from path 5 will pass through S and reach PBS 2 . After the photon from path 1 or path 9 reaches PBS 2 , Ω 4 evolves into Ω 5 ,  From Eq. (24), one can see that the states of the two solidstate target qubits (the two atomic ensembles in cavities 1 and 2) are swapped when the photon qubit is in the state |V〉 , while they do not swap when the photon qubit is in the state |H〉 . The quantum circuit shown in Fig. 4 can be used to construct the Fredkin gate on a three-qubit hybrid system in a deterministic way.

Discussion
The key ingredient in our scheme is the combined magnon-cavity unit, such a system is a promising candidate for QIP since the birefringent propagation of the successively input photons acts as the effective Kerr nonlinearity. In this section, We quantitatively characterize the fidelities and efficiencies of our hybrid gates, respectively.
The fidelity of our Fredkin gate with respect to normalized photon detuning Δ /κ and the coupling rate λ/κ are shown in Fig. 5 when γ = κ. In principle, the detuning Δ /κ can be arbitrarily reduced, if the input photon is tuned to be resonant to the cavity, and then one has F F = 97.2% when γ = κ and λ/κ = 3; while when photon detuning κ ∆ = .
/ 02 and λ κ = / 3, one has = . . We can see that the performance of our universal quantum gates, to some extent, are not sensitive to the detuning Δ and get better when the coupling rate λ/κ increases.
In fact, there might be some difference in the coupling rates between the cavity and modes b j and ĉ j ( ) in practice. In experiment, the difference of the two coupling constants κ κ ∼ .
∆ 0 2 has been demonstrated, which yields approximately the same fidelity for both transmission and reflection directions 42 . In the resonant case (ω c = ω 0 = ω), there will be an additional error probability  in the single-photon scattering process by . And this error can be improved for the cavity with almost identical mirrors 43,44 , which will lead to the ideal photon blockade 45 . To discuss the sensitivity of our schemes to κ Δ , the fidelities and efficiencies of our gates are calculated with the similar procedure as those used in the symmetric case by using the reflection and transmission coefficients obtained with the asymmetrical cavity. The fidelities and efficiencies of our gates are shown in Fig. 7, here we choose κ κ = .

7%
F . Compared with those in the symmetric case, the little decreases of the fidelities and efficiencies in the asymmetric case prove that our universal quantum gates are robust to the cavity coupling imbalance.
As reported in refs 46, 47, the maximum coupling strength between a single atom and a single intracavity photon, along with the decay rate of the excited state and the cavity mode, are λ κ γ π = .
. ( , , )/2 (10 6, 1 3, 3)MHz. Thereby we can see that our hybrid quantum gates are robust against the practical imperfections. Recently, there have been plenty of other methods to couple an atomic ensemble with an optical cavity 48,49 , which might be another building block for our schemes. The fidelities of the spin wave rotation procedures of 99% have been reported 50 , and the collective spin wave operations in atomic ensembles have been well developed 51 . Besides, the atomic ensembles can store photons in a single atomic ensemble with several milliseconds 52 , so this manon-cavity unit is a good quantum memory system for photonic qubits, which is essential in scalable quantum networks. Therefore, our hybrid quantum gates may be achieved with the current QED setup. In addition, our hybrid quantum gates are quite different from the previous ones based on the quantum dot embedded in microcavities 6,7 and those assisted by NV centers embedded in photonic crystal cavities coupled to two wave guides 30 . We use the atomic ensemble approach, so that light-matter coupling is largely improved by collective enhancement 53 . The control qubit of our gates is encoded on the polarization of the moving single photon and the target qubits are encoded on the magnon states of the atomic ensembles inside optical microcavities. As discussed in Sec. III, when the photon in |V〉 passes through the two cavities in succession, the output path of the photon is determined by the parity of the two magnon qubits, this makes the present schemes more succinct than the previous schemes 6 . In addition, because they do not require that the transmission for the uncoupled cavity is balanceable with the reflectance for the coupled cavity, our schemes are robust, this is different from the hybrid gates which are encoded on the atom confined in a single-sided cavity 18,31 .

Conclusion
In conclusion, we have designed the compact quantum circuits for implementing deterministic universal hybrid quantum gates, including the CNOT, Toffoli, and Fredkin gates, by means of the the effective Kerr nonlinearity induced by an atomic ensemble embedded in a double-sided cavity. The spontaneous emission and the cavity decay induce the different transmittance or reflectance coefficients between the hot cavity and the cold cavity in a magnon-cavity system. We have shown the schemes are robust to the variation of coupling rate λ j and the detuning Δ involved in the practical experiments. High fidelities and efficiencies can be achieved in the strong coupling regime in our schemes. We hope this work will be useful in quantum computation and quantum networks with single photons.

Methods
Under the ideal case, suppose that the optical elements, such as PBS, HWP, P π , and optical switch, are perfect, both the success probability and the fidelity of the present schemes are 100% in principle. For a practical magnon-cavity unit, the spontaneous emission of the collective states and cavity decay may leading to photon loss, which will reduce the performance of our hybrid gates.
The fidelities of the gates. We introduce the gate fidelity, which measures the distance for quantum information, is defined as 54 where Ψ 0 is the input states, U is the ideal CONT (Toffoli or Fredkin) gate, and ρ = Ψ Ψ t t t , with Ψ t being the final state after the realistic CONT (Toffoli or Fredkin) operation in the present scheme. Considering the rules for optical transitions in a realistic cavity system, combing the arguments made in Sec. III, we find that the state of the system described by Eq. (12) becomes