1. Introduction

Quantum entanglement is one of the most fascinating properties of quantum mechanics [1–4], which has been used as a critical physical resource in the field of quantum information processing [5–8]. Among all kinds of entangled states, the NOON state is an important kind of entangled states in bosonic systems and atom ensembles [9–12]. Especially in photonic systems, the NOON state has shown potential applications in quantum precise measurements [13–17], quantum metrology [18–20], quantum optical lithography [21,22], and quantum communication [23]. Attracted by these applications, many protocols [24–33] have been proposed to generate NOON states.

In the prior protocols [24–26], the generation of NOON states is usually performed using linear optical elements, and relies on the measurement and the post-selection. These protocols usually work in a probabilistic way, and should be restarted when the measurement reports undesired results. In addition, the finite setup transmission coefficient [34] and the efficiency of the detectors [35] also influence the fidelity of the NOON state generation. To find deterministic approaches for the NOON state generation, many protocols [29–33] have been put forward in circuit quantum electrodynamics systems during the past few years. By exploiting the couplings between superconducting resonators and qubits, NOON states can be generated under unitary operations.

However, the steps of the operation required in the NOON state generation usually increases with the photon number $N$ [29–33]. This may make the NOON state generation more complicated when the photon number $N$ is large. For example, in the protocol [29], the NOON state generation with the photon number $N$ should be performed in $2N$ steps. The strengths of couplings between the qubit and resonators should be tunable. When the qubit is excited from the ground level to the excited levels using classical fields, the couplings between the qubit and the resonators should be switched off. As a result, one should turn on and turn off the qudit-resonator couplings for many times in the NOON state generation. Similarly, in Ref. [30], the generation of NOON states is divided into $2N-1$ steps. Although the coupling strength remains a constant in the operation, the classical fields to drive the qubit from the ground level to the excited levels should be much stronger than the coupling strength so that the effect of couplings are neglected during the excitation. Moreover, in both the protocols [29,30], the fidelity of the NOON state generation may decrease when the photon number increases, due to the increase of the total operation time. In the recent protocol [33], by using the two- and three-photon resonances, an approach has been proposed to generate NOON state with less steps. The number of steps is reduced to $N$ ($N-1$) in the even (odd) case, but still increases with the photon number $N$. The results of the previous protocols [29–33] encourage us to consider whether the generation of NOON state can be further simplified.

In this paper, we propose a three-step protocol to generate NOON states of resonator modes in a system containing two Kerr-nonlinear resonators and a four-level qudit. The qudit is initially prepared in the ground level and is driven to a superposition state of two intermediate levels. The transitions between the excited levels and the two different intermediate levels of the qudit are coupled to the different resonators with large detunings. The off-resonant couplings induce qudit-level-dependent frequency shifts on the resonators. Using the frequency shifts, one can drive different resonators to the $N$-photon states when the qudit is in different intermediate levels. As a result, for the qudit in the superposition state of the two intermediate levels, we obtain a qudit-resonator entangled state with two components. After driving the qudit back from the intermediate levels to the ground level, the qudit is disentangled, and the two resonators remain in the target NOON state. The performance of the protocol is estimated via numerical simulations. The results show that the protocol can generate NOON states with high fidelity in the cases of different photon numbers when the Kerr nonlinearity and the coupling strengths are strong enough. Moreover, the protocol also possesses robustness against systematic errors and decoherence factors, including the energy relaxation and dephasing of the qudit, and the decay of the resonators. Accordingly, the protocol may be helpful for the effective generation of photonic NOON states.

The article is organized as follows. In Sec. 2, we introduce the physical model to implement the NOON state generation. In Sec. 3, we describe the procedures for generating NOON states in detail. In Sec. 4, we perform numerical simulations to estimate the performance of the protocol in the presence of experimental imperfections. In Sec. 5, we study the trade-off between the fidelity and the operation time. Finally, the conclusions are given in Sec. 6.

2. Physical model

We consider a system contains two Kerr-nonlinear resonators and a qudit, as shown in Fig. 1(a) [36]. The qudit has a ground level $|g\rangle _q$, two intermediate levels $|f_1\rangle _q$ and $|f_2\rangle _q$, and an excited level $|e\rangle _q$. The transition $|g\rangle _q\leftrightarrow |f_j\rangle _q$ ($j=1,2$) is resonantly driven by a classical field with Rabi frequency $\Omega _j(t)$. The transition $|e\rangle _q\leftrightarrow |f_j\rangle _q$ is coupled to the resonator mode in the resonator $C_j$ with the coupling strength $g_j$ and the detuning $\Delta _j$. The Kerr nonlinearity for each resonator is supposed as $K$. There is a linear drive, i.e., $\epsilon _j(t)$, applied on the resonator $C_j$. The Hamiltonian of the system under the rotating-wave approximation reads

 

Fig. 1. (a) The diagram of the system composed of two Kerr-nonlinear resonators and a qudit. (b) Level configuration of the qudit, and the transitions coupled with the resonator modes and the classical fields.

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Considering the large detuning condition for the resonator-qudit coupling, i.e., $\Delta _j\gg g_j$ and $|\Delta _1-\Delta _2|\gg g_1g_2(\Delta _1+\Delta _2)/2\Delta _1\Delta _2$, using the second-order perturbation theory [37], we can derive an effective Hamiltonian as

Here, we have assumed $\delta _0=g_j^2/\Delta _j$. Supposing that the qudit is initially in the ground level $|g\rangle _q$, the first term of $H_q'$ in Eq. (2) can be omitted since the excited level $|e\rangle _q$ is almost not populated under the effective dynamics.

Moving into the frame rotating by the unitary operator $R_q(t)=\exp (-iH_q't)$, the Hamiltonian of the system becomes

By moving into another rotating frame of the unitary operator $R_K(t)=\exp (-iH_Kt)$, the Hamiltonian can be rewritten by

We assume the resonator $C_j$ is initially prepared in the vacuum state and the target NOON state is

Under the condition $\{K,\delta _0\}\gg \{\bar {\varepsilon }_0(t),\bar {\varepsilon }_m(t)\}$, we omit the terms with high-frequency oscillations, and obtain the effective Hamiltonian as

With the help of the effective Hamiltonian $H_e(t)$ in Eq. (7), it is possible to generate the target NOON state $|\Psi _N\rangle$, which will be shown in the following sections. Noticing the result that $R_q(t)R_K(t)|\Psi _N\rangle =\exp [-iN(N-1)Kt]|\Psi _N\rangle$, the use of the rotating frames only leads to a global phase to the target NOON state. Therefore, when we obtain the target NOON state $|\Psi _N\rangle$ in the final rotating frame, we also obtain the target NOON state $|\Psi _N\rangle$ in the original frame up to a global phase.

3. Generating NOON states of resonator modes

3.1 Procedures for generating the NOON state

We now describe the detailed procedures for generating the target NOON state $|\Psi _N\rangle$. At the beginning, the two resonators are all prepared in their vacuum states, and the qudit is prepared in the ground level $|g\rangle _q$. The whole process is divided into three steps, denoted by Step 1, Step 2 and Step 3, with the final time assumed as $\tau _1$, $\tau _2$ and $\tau _3$, respectively.

In Step 1, we turn off the linear drives $\epsilon _1(t)$ and $\epsilon _2(t)$ of the resonators. Since the two resonators $C_1$ and $C_2$ are both prepared in the vacuum states, they will remain in the vacuum states in Step 1 as the linear drives $\epsilon _1(t)$ and $\epsilon _2(t)$ are turned off. Therefore, the system evolves in the subspace given by the projection operator $|0,0\rangle _{12}\langle 0,0|$ with $|0,0\rangle _{12}=|0\rangle _1\otimes |0\rangle _2$. Assuming $\Omega _j(t)=\Omega _{s_1}(t)/\sqrt {2}$, the Hamiltonian of the system becomes

By substituting $\mathcal {H}(t)=H_1(t)$, $L=3$, $\mathcal {G}_1=\sigma _z/2$, $\mathcal {G}_2=\sigma _y/2$, $\mathcal {G}_3=\sigma _x/2$, $\lambda _1(t)=0$, $\lambda _2(t)=2\mathrm {Re}[\Omega _{s_1}(t)]$, $\lambda _3(t)=2\mathrm {Im}[\Omega _{s_1}(t)]$, $\theta _1(t)=\eta (t)+\pi /2$, $\theta _2(t)=\mu (t)+\pi /2$, and $\eta (0)=\mu (0)=\theta _3(0)=0$ into Eq. (42) and Eq. (45) in Appendix A, one derives the real and imaginary parts of the control function $\Omega _{s_1}(t)$ as

Thus, the boundary condition $\mu (\tau _1)=\pi$ is selected to evolve the system to the state $|0,0\rangle _{12}|\psi _1\rangle$.

To enhance the robustness against systematic error in the control function $\Omega _{s_1}(t)$, we apply the method by nullifying systematic-error sensitivity [38]. Considering the systematic error $\Omega _{s_1}(t)\rightarrow (1+\epsilon )\Omega _{s_1}(t)$ with the systematic error rate $\epsilon$, the fidelity of the evolution along paths $|\Phi (t)\rangle$ can be estimated using time-dependent perturbation theory by

The systematic-error sensitivity can be calculated by

Inspired by Refs. [38,39], we assume $\theta _3(t)=A\{2\mu (t)-\sin [2\mu (t)]\}$, with a time-independent parameter $A$. Then, the systematic-error sensitivity is derived as $Q_s=\sin ^2(A\pi )/A^2$, and it can be nullified by setting $A=1$. In this case, we further obtain $\eta (t)=4\sin ^3[\mu (t)]/3$ and

For the boundary conditions $\mu (0)=0$ and $\mu (T)=\pi$, the parameter $\mu (t)$ can be designed using sine function as $\mu (t)=\pi \sin ^2(\pi t/2\tau _1)$. The time-variations of $\mathrm {Re}[\Omega _{s_1}(t)]$ and $\mathrm {Im}[\Omega _{s_1}(t)]$ are shown in Fig. 2(a).

 

Fig. 2. (a) Time-variations of $\mathrm {Re}[\Omega _{s_1}(t)]$ and $\mathrm {Im}[\Omega _{s_1}(t)]$. (b) Time-variations of $\mathrm {Re}[\Omega _{s_3}(t)]$ and $\mathrm {Im}[\Omega _{s_3}(t)]$, with $\Delta \!\tau _3=\tau _3-\tau _2$.

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In Step 2, we turn off the classical field applied on the qudit [$\Omega _j(t)=0$]. In addition, the linear drives applied on the two resonators are turned on. Since the initial state in Step 2 is $|\psi (\tau _1)\rangle$ shown in Eq. (9), the Hamiltonian of the Step 2 can be rewritten by

The result in Eq. (19) implies that, when the qudit is in the level $|f_1\rangle _q$, the Hamiltonian $H_2(t)$ only connects the vacuum state $|0\rangle _2$ and the $N$-photon state $|N\rangle _2$ of the resonator mode 2. But the Hamiltonian $H_2(t)$ only connects the vacuum state $|0\rangle _1$ and the $N$-photon state $|N\rangle _1$ of the resonator mode 1, when the qudit is in the level $|f_2\rangle _q$. Therefore, based on the Hamiltonian $H_2(t)$, it is possible to realize the evolution

In Step 3, the linear drives applied on the two resonators are switched off, and the classical field $\Omega _j(t)$ applied on the qudit is switched on. In this case, the evolution of the system is governed by the Hamiltonian $\tilde {H}_d(t)$ according to Eq. (7). Considering the final state $|\psi (\tau _2)\rangle$ of the Step 2 shown in Eq. (20), the evolution of the system is restricted in the subspace spanned by the vectors

Rewriting $\tilde {H}_d(t)$ using the vectors in Eq. (21), we have

Here, we set $\Omega _j(t)=\Omega _{s_3}(t)$. Under the condition $N\delta _0\gg \Omega _{s_3}(t)$, the Hamiltonian of the system in Step 3 is reduced to

3.2 Control function design for step 2 via reverse engineering

We now study the design of control functions to realize the transformations shown in Eq. (24). Here, we consider an $N$-level Hamiltonian as

To apply the Lie-transformation-based reverse engineering, one first make a bijection between the generator $G_k$ to $\mathcal {G}_l$ in $\mathcal {H}(t)$ of Eq. (42) shown in Appendix A, by setting $L=N(N+1)/2$. In this way, the control functions $f_n(t)$ can be derived by substituting $\mathcal {H}(t)=\mathcal {H}_N(t)$ into Eq. (48). In addition, the evolution operator $\mathcal {U}_N(t)$ is given by Eq. (45) in Appendix A. For example, in the case of $N=2$, $\mathcal {G}_1=G_2$, $\mathcal {G}_2=G_1$, $\mathcal {G}_3=G_3$ and $\lambda _1(t)=0$, one derives

In addition, when the initial value $\theta _1(\tau _1)=\theta _2(\tau _1)=0$ is selected, the evolution of the system is given by

To realize the target state $|\phi _2(\tau _2)\rangle =|2\rangle _0$ at the final time $t=\tau _2$, the final values for the parameters $\{\theta _l(t)\}$ can be selected as $\theta _1(\tau _2)=\pi /2$ and $\theta _2(\tau _2)=0$. By taking the initial and final values of $\{\theta _l(t)\}$ into consideration, the expressions of $\theta _2(t)$ and $\theta _3(t)$ can be constructed using trigonometric functions as

To show the control functions designed by the reverse engineering can be successfully applied in the generation of the target NOON state $|\Psi _N\rangle$ in the $N=2$ case, we take the parameter $\zeta _2=\pi /2$ as an example to estimate the performance of the protocol. The time-variations of the control functions $f_0(t)$ and $f_1(t)$ are shown in Fig. 3(a). We considering a set of available parameters $K=2\pi \times 12.5$ MHz, $g_1=2\pi \times 47.75$ MHz, $g_2=2\pi \times 31.83$ MHz, $\Delta _1=2\pi \times 358.1$ MHz, and $\Delta _2=2\pi \times 159.15$ MHz [51–53] with the final time of three steps as $\tau _1=0.5\,\mu$s, $\tau _2=10.5\,\mu$s, and $\tau _3=20\,\mu$s. Here, the operation time in Step 3 is $\Delta \!\tau _3=\tau _3-\tau _2=9.5\,\mu$s, longer than the operation time $\Delta \!\tau _1=\tau _1=0.5\,\mu$s, which makes the condition $N\delta _0\gg \Omega _{s_3}(t)$ well satisfied.

 

Fig. 3. (a) Time-variations of the control functions $f_0(t)$ and $f_1(t)$ in the $N=2$ case. (b) The fidelity $F_2(t)=\langle \Psi _2|\rho (t)|\Psi _2\rangle$ and the average total photon number $\langle n_1+n_2\rangle$ versus $t$ in the $N=2$ case.

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The fidelity $F_2(t)=\langle \Psi _2|\rho (t)|\Psi _2\rangle$ and the average total photon number $\langle n_1+n_2\rangle =\mathrm {Tr}[\rho (t)(a_1^{\dagger} a_1+a_2^{\dagger} a_2)]$ versus $t$ are plotted in Fig. 3(b). Here, $\rho (t)$ denotes the density operator of the system under the control of the full Hamiltonian $H(t)$ shown in Eq. (1). Seen from the blue-dashed curve in Fig. 3(b), the average total photon number gradually increases from 0 to 2 during the time interval $[\tau _1,\tau _2]$ of Step 2. This result proves that the control functions given by the reverse engineering can realize the evolution in Eq. (20). Moreover, we can see from the red-solid curve that the fidelity gradually increases from 0 to 1 during the time interval $[\tau _2,\tau _3]$ of Step 3. Consequently, the target state can be successfully generated after the total three steps.

In the similar way, we can also design control functions for the case $N=3$. To apply Eq. (45) in Appendix A to the control function design, we select $\mathcal {G}_1=G_1$, $\mathcal {G}_2=G_6$, $\mathcal {G}_3=G_4$, $\mathcal {G}_4=G_3$, $\mathcal {G}_5=G_2$, $\mathcal {G}_6=G_5$ and $\lambda _4(t)=\lambda _5(t)=\lambda _6(t)=0$. In this case, we obtain the results

In addition, when the initial value $\theta _1(\tau _1)=\theta _4(\tau _1)=\theta _5(\tau _1)=0$ is selected, the evolution of the system can be described by

To realize the target state $|\phi _3(\tau _2)\rangle =|3\rangle _0$, the final values of the parameters can be selected as $\theta _2(\tau _2)=\theta _5(\tau _2)=0$ and $\theta _4(\tau _2)=\pi /2$. Based on the initial and final values of the parameters $\{\theta _l(t)\}$, the specific expressions is selected using sine functions as

To show that the control function design is also successfully applied to the generation of target NOON state $|\Psi _N\rangle$ in the $N=3$ case, we take $\Upsilon _1=0.4$ as an example to estimate the performance of the protocol, which gives $\Upsilon _3=1.0782$. The time-variations of the control functions $f_0(t)$, $f_1(t)$ and $f_2(t)$ are plotted in Fig. 4(a). Here, we keep the parameters $K=2\pi \times 12.5$ MHz, $g_1=2\pi \times 47.75$ MHz, $g_2=2\pi \times 31.83$ MHz, $\Delta _1=2\pi \times 358.1$ MHz, $\Delta _2=2\pi \times 159.15$ MHz, $\tau _1=0.5\,\mu$s, $\tau _2=10.5\,\mu$s, and $\tau _3=20\,\mu$s using in the $N=2$ case. The fidelity $F_3(t)=\langle \Psi _3|\rho (t)|\Psi _3\rangle$ and the average total photon number $\langle n_1+n_2\rangle$ versus $t$ are plotted in Fig. 4(b). As shown by the blue-dashed curve in Fig. 4(b), the average total photon number $\langle n_1+n_2\rangle$ gradually increases from 0 to 3 during the time interval $[\tau _1,\tau _2]$ of Step 2. This means that the evolution in Eq. (20) is completed after Step 2 using the control functions designed by the reverse engineering. In addition, the red-solid curve shows that the fidelity of the target NOON state $|\Psi _3\rangle$ approaches 1 in the evolution. Therefore, the target NOON state can be successfully generated after the total three steps.

 

Fig. 4. (a) Time-variations of $f_0(t)$, $f_1(t)$ and $f_2(t)$ in the $N=3$ case. (b) The fidelity $F_3(t)=\langle \Psi _3|\rho (t)|\Psi _3\rangle$ and the average total photon number $\langle n_1+n_2\rangle$ versus $t$ in the $N=3$ case.

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3.3 Control function design for step 2 via GRAPE

In Sec. 3.2, we have shown that, reverse engineering can be used to design control functions for Step 2 in the generation of the target NOON state $|\Psi _N\rangle$. However, when the size $N$ increases, the calculation of reverse engineering becomes ever more complicated. As an alternative, one can also use numerical methods to find control functions for Step 2. Here, let us consider the control function design in the $N=4$ case as an example by using GRAPE algorithm [49]. A brief review of GRAPE algorithm is shown in Appendix B. To start GRAPE algorithm, we divide the time interval $[\tau _1,\tau _2]$ of Step 2 into $S$ small equal intervals, where $S$ is a positive integer. The start point of the $s$-th ($s=1,2,\ldots,S$) interval is assumed as $t=t_s$. In addition, we first guess the control function $f_n(t)$ in the $s$-th interval using sine function as

The selection of $\xi _s$ in Eq. (39) makes the step-size parameter $\xi _s$ tend to zero when $t_s$ is close to the boundaries $t=\tau _1$ and $t=\tau _2$, and it also makes the parameter $\xi _s$ take values around the maximal value $\max \{\xi _s\}=10$ when $t_s$ is close to the center $t=(\tau _1+\tau _2)/2$ of Step 2. This selection helps us to find control functions vanishing at the boundaries.

Based on the parameters selected above, we find a set of control functions $\{f_n(t)|n=1,2,3,4\}$ after 200 iterations of GRAPE algorithm, whose wave shapes are shown in Fig. 5(a). With the control functions, the fidelity $F_4(t)=\langle \Psi _4|\rho (t)|\Psi _4\rangle$ and the average total photon number $\langle n_1+n_2\rangle$ are plotted in Fig. 5(b). Here, the parameters $K=2\pi \times 12.5$ MHz, $g_1=2\pi \times 47.75$ MHz, $g_2=2\pi \times 31.83$ MHz, $\Delta _1=2\pi \times 358.1$ MHz, and $\Delta _2=2\pi \times 159.15$ MHz, $\tau _1=0.5\,\mu$s, $\tau _2=10.5\,\mu$s, and $\tau _3=20\,\mu$s are still adopted in the calculation. According to the blue-dashed curve in Fig. 5(b), the average total photon number $\langle n_1+n_2\rangle$ increases from 0 to 4 during the time interval $[\tau _1,\tau _2]$ of Step 2, proving that the evolution in Eq. (20) is successfully implemented using the control functions given by the GRAPE algorithm. Furthermore, seen from the red-solid curve in Fig. 5(b), the fidelity of the target NOON state $|\Psi _4\rangle$ tends to 1 in the operation. This example shows that it is possible for the protocol to generate NOON states with large photon numbers with the help of GRAPE algorithm in Step 2.

 

Fig. 5. (a) Time-variations of $f_0(t)$, $f_1(t)$, $f_2(t)$ and $f_3(t)$ in the $N=4$ case. (b) The fidelity $F_4(t)=\langle \Psi _4|\rho (t)|\Psi _4\rangle$ and the average total photon number $\langle n_1+n_2\rangle$ versus $t$ in the $N=4$ case.

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4. Numerical simulations and discussions

In this section, let us estimate the performance of the protocol in the presence of experimental imperfections via numerical simulations based on the full Hamiltonian in Eq. (1). Here, we take the generation of the target NOON state $|\Psi _2\rangle$ in the $N=2$ case as an example, with the parameters $K=2\pi \times 12.5$ MHz, $g_1=2\pi \times 47.75$ MHz, $g_2=2\pi \times 31.83$ MHz, $\Delta _1=2\pi \times 358.1$ MHz, and $\Delta _2=2\pi \times 159.15$ MHz, $\tau _1=0.5\,\mu$s, $\tau _2=10.5\,\mu$s, and $\tau _3=20\,\mu$s.

The systematic error is a widely existing error in real experiments. It describes many low-order process of complicated errors and instrument/operation imperfections [38,39]. In the present protocol, there may exist systematic errors in the strengths of the Rabi frequency $\Omega _j(t)$ and the linear drive $\varepsilon _j(t)$. When the systematic errors with a systematic error rate $\epsilon$ are taken into account, the Rabi frequency and the linear drive become $(1+\epsilon )\Omega _j(t)$ and $(1+\epsilon )\varepsilon _j(t)$, respectively. In Fig. 6(a), we plot the final fidelity $F_2(\tau _3)=\langle \Psi _2|\rho (\tau _3)|\Psi _2\rangle$ versus systematic error rate $\epsilon$. According to Fig. 6(a), the decrease of the fidelity $F_2(\tau _3)$ is less than $0.0534$ when the absolute systematic error rate $|\epsilon |$ is less than 10%. The worst fidelity $F_2(\tau _3)=0.9466$ appears at $\epsilon =0.1$ for $\epsilon \in [-0.1,0.1]$. Therefore, the generation of the target NOON state $|\Psi _2\rangle$ has the robustness against the systematic errors in the strengths of the Rabi frequency $\Omega _j(t)$ and the linear drive $\varepsilon _j(t)$.

 

Fig. 6. (a) The final fidelity $F_2(\tau _3)=\langle \Psi _2|\rho (\tau _3)|\Psi _2\rangle$ versus systematic error rate $\epsilon$. (b) The final fidelity $F_2(\tau _3)$ versus decoherence rates. Red-solid curve: $\gamma /\gamma _0$ with $\gamma$ the energy relaxation rate of the qudit and $\gamma _0=0.01$ MHz; Blue-dashed curve: $\gamma ^\phi /\gamma ^\phi _0$ with $\gamma ^\phi$ the dephasing rate of the qudit and $\gamma ^\phi _0=0.01$ MHz; Green-dotted curve: $\kappa /\kappa _0$ with $\kappa$ the decay rate of the resonator mode and $\kappa _0=0.002$ MHz.

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As a quantum system is difficultly isolated to the environment, there usually exist decoherence factors, which may also decrease the fidelity of entangled-state generation [54–56]. For the system in the present protocol, the major decoherence factors are the energy relaxation of the qudit from the higher levels to the lower levels, the dephasing of the qudit on the intermediate and the excited levels, and the decay of the two resonators. The energy relaxation rate for the energy relaxation path $|f_j\rangle \rightarrow |g\rangle$ ($|e\rangle \rightarrow |f_j\rangle$) is assumed as $\gamma ^g_j$ ($\gamma ^e_j$). The dephasing rate for the dephasing on the levels $|f_j\rangle$ ($|e\rangle$) is assumed as $\gamma ^{\phi }_j$ ($\gamma ^{\phi }_e$). In addition, the decay rate of the resonator $C_j$ is assumed as $\kappa _j$. Consequently, the evolution of the system can be described by the master equation [57]

Here, we define $S^-_{g,j}=|g\rangle _q\langle f_j|$, $S^-_{e,j}=|f_j\rangle _q\langle e|$, $S_{j,j}=|f_j\rangle _q\langle f_j|$, $S_{e,e}=|e\rangle _q\langle e|$, and $\mathcal {L}[X]\rho (t)=X\rho (t)X^{\dagger} -[X^{\dagger} X\rho (t)+\rho (t)X^{\dagger} X]/2$ with $X\in \{S^-_{g,j},S^-_{e,j},S_{j,j},S_{e,e},a_j\}$.

In the numerical simulations, we consider the equal energy relaxation and dephasing rate for paths connected to the two intermediate states $|f_1\rangle _q$ and $|f_2\rangle _q$, i.e., $\gamma ^g_j=\gamma /2$, $\gamma ^e_j=\tilde {\gamma }/2$, and $\gamma ^{\phi }_j=\gamma ^\phi /2$. In addition, considering that the intermediate level $|f_j\rangle _q$ is much stable than the excited level $|e\rangle _q$, we take $\tilde {\gamma }/\gamma =\gamma ^\phi _e/\gamma ^{\phi }=10$ as an example in the numerical simulations. The decay rate of the two resonator is considered to be equal, i.e., $\kappa _j=\kappa$. Considering a set of available ranges of decoherence rates $\kappa \in [0,0.002]$ MHz, $\gamma \in [0,0.01]$ MHz, and $\gamma _\phi \in [0,0.01]$ MHz [36,52,58–63], the final fidelity $F_2(\tau _3)$ versus decoherence rates is shown in Fig. 6(b).

As shown by the blue-dashed curve in Fig. 6(b), the final fidelity $F_2(\tau _3)$ is relatively insensitive to the dephasing of the qudit, compared with the energy relaxation of the qudit and the decay of the resonator. When $\gamma ^\phi =0.002$ MHz, $F_2(\tau _3)=0.9921$ is obtained. Even if $\gamma ^\phi =0.01$ MHz, the final fidelity is still $F_2(\tau _3)=0.9616$. Seen from the red-solid curve in Fig. 6(b), the final fidelity $F_2(\tau _3)$ is more sensitive to the energy relaxation of the qudit, compared with the dephasing of the qudit. When $\gamma =0.01$ MHz ($\gamma =0.002$ MHz), the final fidelity is $F_2(\tau _3)=0.9304$ [$F_2(\tau _3)=0.985$]. In addition, according to the green-dotted curve, the influence of the resonator decay to the final fidelity $F_2(\tau _3)$ is stronger than that of the qudit energy relaxation and the qudit dephasing, because the target NOON state is built upon the resonator modes. The final fidelity becomes $F_2(\tau _3)=0.9448$ when $\kappa =0.002$ MHz. The results above indicate that the protocol can still produce acceptable fidelity for the NOON state generation in the presence of the qudit energy relaxation, the qudit dephasing, and the resonator decay.

5. Trade-off between the fidelity and the operation time

In this section, we consider the trade-off between the fidelity and the operation time. For experimental feasibility, we fix the parameters $K=2\pi \times 12.5$ MHz, $g_1=2\pi \times 47.75$ MHz, $g_2=2\pi \times 31.83$ MHz, $\Delta _1=2\pi \times 358.1$ MHz, and $\Delta _2=2\pi \times 159.15$ MHz in the discussions. In addition, the maximal strength of the classical field $\Omega _j(t)$ is set as $\Omega _{\max }=2\pi \times 100$ MHz. In the following, we will study the variation of the fidelity of the NOON state generation by changing the operation time in each step.

For Step 1, as shown in Fig. 2(a), we have

Considering the relation $\Omega _j(t)=\Omega _{s_1}(t)/\sqrt {2}$, one derives $|\Omega _{j}(t)|\leq 9/\tau _1$. Combining $|\Omega _{j}(t)|\leq \Omega _{\max }$, the operation time for Step 1 should satisfies $\tau _1\geq 9/200\pi \,\mu \mathrm {s}\simeq 0.0143\,\mu$s. We plot the fidelity final fidelity $F_2(\tau _3)$ ($N=2,3,4$) versus the operation time $\tau _1$ for Step 1 in Fig. 7(a) with $\Delta \!\tau _2=10\,\mu$s and $\Delta \!\tau _3=9.5\,\mu$s in the range $\tau _1\in [0.015,1]\,\mu$s. According to Fig. 7(a), the fidelity is almost unchanged with the variation of $\tau _1$ for the cases of $N=2$, $N=3$, and $N=4$. This is because the Hamiltonian $H_1(t)$ in Step 1 does not rely on approximations. The control field design for the Hamiltonian $H_1(t)$ in Step 1 provides an exact solution for the state evolution of Step 1. For a faster evolution speed, we can select $\tau _1=0.015\,\mu$s afterwards, where the maximal strength of the control field $\Omega _j(t)$ in Step 1 is near the upper limit $\Omega _{\max }$ set in this section.

 

Fig. 7. (a) The final fidelity $F_N(\tau _3)$ versus the operation time $\tau _1$ for Step 1 with $\Delta \!\tau _2=10\,\mu$s and $\Delta \!\tau _3=9.5\,\mu$s. (b) The final fidelity $F_N(\tau _3)$ versus the operation time $\Delta \!\tau _2$ for Step 2 with $\tau _1=0.015\,\mu$s and $\Delta \!\tau _3=9.5\,\mu$s. (c) The final fidelity $F_N(\tau _3)$ versus the operation time $\Delta \!\tau _3$ for Step 3 with $\tau _1=0.015\,\mu$s and $\Delta \!\tau _2=5\,\mu$s. (d) The fidelity $F_N(t)$ versus $t$ with the improved total operation time $\tau _3=6.015\,\mu$s.

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Then, let us study the relation between the fidelity and the operation time $\Delta \!\tau _2$ in Step 2. In Step 2, the condition $\{K,\delta _0\}\gg \{\bar {\varepsilon }_0(t),\bar {\varepsilon }_m(t)\}$ should be satisfied. According to Fig. 3(a), Fig. 4(a), and Fig. 5(a), we consider $\Delta \!\tau _2$ in the range $[1,10]\,\mu$s to avoid strong violation of the condition. The final fidelity $F_N(\tau _3)$ ($N=2,3,4$) versus the operation time $\Delta \!\tau _2$ for Step 2 with $\tau _1=0.015\,\mu$s and $\Delta \!\tau _3=9.5\,\mu$s is plotted in Fig. 7(b). The results show that the variation of the fidelity gradually tends to be stable when $\Delta \!\tau _2\geq 5\,\mu$s. More specifically, we have $F_2(\tau _3)=0.9999$, $F_3(\tau _3)=0.9950$, and $F_4(\tau _3)=0.9969$ for $\Delta \!\tau _2=5\,\mu$s. Therefore, for a faster operation, one can select $\Delta \!\tau _2=5\,\mu$s with small sacrifice of the fidelity.

For Step 3, the operation is limited by the condition $N\delta _0\gg \Omega _{s_3}(t)$. We consider the range $\Delta \!\tau _3\in [1,10]\,\mu$s to avoid strong violation of the condition. The final fidelity $F_N(\tau _3)$ ($N=2,3,4$) versus the operation time $\Delta \!\tau _3$ for Step 3 with $\tau _1=0.015\,\mu$s and $\Delta \!\tau _2=5\,\mu$s is plotted in Fig. 7(c). According to Fig. 7(c), the fidelity of the NOON state generation is higher than 0.9939 when $\Delta \!\tau _3\geq 1\,\mu$s, and tends to stable when $\Delta \!\tau _3\geq 5\,\mu$s. If the threshold of the fidelity is selected as $F_N(\tau _3)\geq 0.99$, one can set $\Delta \!\tau _3=1\,\mu$s for a faster operation.

Finally, let us estimate the performance of the NOON state generation with the improved operation time $\tau _1=0.015\,\mu$s, $\Delta \!\tau _2=5\,\mu$s, and $\Delta \!\tau _3=1\,\mu$s for the three steps, giving the total operation time $\tau _3=6.015\,\mu$s. The fidelity $F_N(t)$ versus $t$ is shown in Fig. 7(d). Seen from the figure, the fidelity of the NOON state generation in the cases of $N=2$, $N=3$, and $N=4$ are all approach 1. The final fidelity is 0.9982 (0.9939, 0.9961) for the $N=2$ ($N=3$, $N=4$) case. This result shows that the protocol can still produce NOON state with high fidelity when the total operation time is reduced to $6.015\,\mu$s.

6. Conclusion

In conclusion, we have proposed a protocol to generate NOON states of two resonator modes coupled to a four-level qudit. The whole process is divided into three steps. The qudit is initially prepared in the ground level and then driven to a superposition state of two intermediate levels by classical fields in the first step. The transitions between the two intermediate levels and the excited level are coupled to two different resonators with large detunings. The couplings between the qudit and the resonators induce qudit-level-dependent frequency shifts for the resonators. Using such frequency shifts, only one of the two resonators can be driven to the $N$-photon state while the other remains in the vacuum state. When the qudit is in different intermediate states, the resonator evolving to the $N$-photon state is also different after the second step. The control fields to drive the resonators in the second step can be designed via reverse engineering or gradient ascent pulse engineering (GRAPE) algorithm. The former one allows us to find the analytical expressions of the control fields while the calculation may be more complicated when the photon number is large. The latter one gives the numerical solutions for the control fields while can be more easily to be performed in the large photon cases. In the third step, the qudit is driven by the classical fields again. Due to the different frequency shifts on the two intermediate levels, the evolution of the system can be considered in two disconnected two-level subspaces with the same dynamic structure. In each subspace, the qudit evolves from the corresponding intermediate level to the ground level under the drive of the classical fields. In this way, the qudit is disentangled, leaving the two resonators in the target NOON state.

Compared with the previous protocol [64], the present protocol shows a different way to generate NOON states. In protocol [64], one of the resonators is first prepared in the $N$-photon Fock state. Then, by using a controlled-SWAP gate, where an auxiliary qubit is exploited as the control qubit, and the two resonators are considered as the controlled qubits, the resonators and the auxiliary qubit are entangled. By measuring the state of the auxiliary qubit in proper basis, NOON states can be obtained according to the measurement results. In the present protocol, the auxiliary qudit is first driven to a superposition state of two intermediate levels, and is finally disentangled. Accordingly, controlled-SWAP gate and the measurement on the auxiliary qudit are not required.

In order to check the validity of the protocol, we consider the generation of the NOON states in $N=2$ and $N=3$ case using reverse engineering, and in $N=4$ case using GRAPE algorithm. The numerical results demonstrate that, in the regime of strong Kerr nonlinearity and coupling strengths, both the fidelity and the average photon number approach the goal in all cases. This proves that the protocol is effective in generating NOON states of resonator modes with different photon numbers. Furthermore, to estimate the performance of the protocol in the presence of the experimental imperfections, we take $N=2$ case as an example in the numerical simulations. The results show that it is possible for the protocol to produce acceptable fidelity of the target NOON state when the systematic errors of the control fields, the energy relaxation and dephasing of the qudit, and the decay of the resonator are taken into account. Therefore, the protocol may provide an alternative approach for generating NOON states of resonator modes.