1. INTRODUCTION

As an important fundamental task, state transfer is widely studied in atomic systems, optical physics, and quantum information for its indispensable role in building optical and quantum devices such as optical transistors [1,2], frequency conversions [3], and quantum interfaces [4–7]. In atomic systems, the typical approaches for realizing state transfer are the rapid adiabatic passage [8] for two-level quantum systems, the stimulated Raman adiabatic passage [9] for excited-state assisted three-level $\mathrm{\Lambda }$ quantum systems, and their optimized shortcuts to the adiabaticity technique [10–16].

Optical microcavities, which can effectively enhance the interaction between light and matter [17], are a good platform for studying optical physics and useful applications. For instance, some interesting physics, such as parity-time symmetry [18–20], chaos [21,22], and nonreciprocity [23,24], have been demonstrated in microcavities. In applications, microcavities show significant functions for sensing [25–29] and processing quantum information [30–38]. Photons can be confined in the microcavity and can also be transferred to another one via evanescent wave coupling or other interactions. Realizing state transfer between microcavities is important for making the microcavity a good physical system for quantum information processing and optical devices. In quantum computing, an all-optical microcavity coupling lattice structure can be used for performing boson sampling [39], and a microcavity can also be considered a quantum bus to connect solid qubits for building a quantum computer. To make an all-optical device, such as transistor [1,2] and router [40], the target is achieved by performing the state transfer between microcavities successfully. Some effective protocols for state transfer between optical modes are reported with adiabatic methods [41–43], nonadiabatic approaches [43], and shortcuts to the adiabaticity technique [44,45]. By using optomechanical interactions [46–53], the protocols are completed successfully by tuning coupling strengths very well to satisfy technique constraints.

When we consider the situation of state transfer in the on-chip all-optical microcavity system, the coupling strength tuning becomes difficult. To solve this problem, in this paper we proposed an approach to realize the state transfer task between separated modes in optical microcavities via frequency tuning. In our protocol, we assume that all the coupling strengths are constant, and we tune the frequency of the intermediate microcavity to control the interactions. With linear and periodic tuning, one can transfer the state from the initial cavity mode to the target successfully. To achieve faster frequency-tuning-induced state transfer (FIST) with high fidelity, we use the gradient descent to optimize the result and acquire an optimal tuning function. Our protocol also shows an significant nonreciprocity in an appropriate area of the parameters. The good experimental feasibility and the interesting features of our work provide potential applications in quantum computing and optical devices.

2. BASIC MODEL FOR MULTIMODE INTERACTIONS IN OPTICAL MICROCAVITIES

We consider a model of multimode interactions in coupled optical microcavities, shown in Fig. 1. The model is universal for situations in which two optical modes are coupled to the intermediate mode and all the modes have very narrow linewidths. Figure 1(a) shows a setup with nearest-neighbour couplings in three optical cavities. We assume that the coupling strengths between corresponding modes are constant and the system can be controlled by tuning the resonance frequency of the intermediate mode. Under the rotating frame with unitary transformation $U=\exp [i{\omega }_{1}t(a_1^{†}{a}_1+a_t^{†}{a}_t+a_2^{†}{a}_2)]$, the Hamiltonian is given by

 

Fig. 1. Schematic diagram for the model of multimode interactions in optical microcavities. All the modes have very narrow linewidths. A mode in one cavity couples to two different optical modes (a) in the same cavity and (b) in two different cavities separately. (c) Resonance frequency tuning of the intermediate cavity to induce state transfer. The tuning domain is divided into three parts labelled I, II, and III.

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Here the vector is $\overrightarrow{a}(t)=[{\widehat{a}}_1(t),{\widehat{a}}_t(t),{\widehat{a}}_2(t{)]}^T$. To show the results more clearly, we omit the dissipation and noise terms in our calculation due to the very narrow linewidth. In general, the coupling strength between cavities is difficult to modulate for an on-chip sample. Therefore, we keep the coupling strengths constant here and tune the resonance frequency of the intermediate cavity to control the evolution path of system.

3. FIST BETWEEN SEPARATED MODES

The frequency tuning can be realized with different functions. Here we perform the FIST task with two common typical envelopes, i.e., linear and periodic functions, and use the gradient descent technique to optimize the process.

A. FIST with Linear Function

We assume that the envelope of frequency tuning is chosen with $\mathrm{\Delta }(t)=vt+{\mathrm{\Delta }}_0$. Here $v$ and ${\mathrm{\Delta }}_0$ are the tuning speed and the initial value of detuning, respectively. At the initial time, mode $a_1$ is stimulated and the other modes are kept in their ground state, i.e., $a_1(0)=1$ and $a_t(0)=a_2(0)=0$. Without loss of generality, we choose the frequency detuning of modes $a_1$ and $a_2$ with $\delta >0$ and sweep the frequency of mode $a_t$ from left to right in Fig. 1(c). As the frequency of intermediate mode $a_t$ is swept, the state is transferred from the initial mode to other modes. We numerically simulate the process and show the result in Fig. 2. The resonance frequency of the intermediate mode is swept from $-6\delta$ to $7\delta$ with a constant speed $0.08{\delta }^2$, i.e., ${\mathrm{\Delta }}_0=-6\delta$ and $v=0.08{\delta }^2$. The coupling strengths between modes are chosen as $g_1=0.6\delta$ and $g_2=0.2\delta$ in our simulations. Figure 2 shows that the population $P$ of mode $a_1$ is transferred to $a_t$ when the frequency of $a_t$ is swept to $a_1$. When the frequency of the intermediate mode is moving in domain I of Fig. 1(c), mode $a_2$ has no effective exchange of population with $a_t$ and almost keeps its initial state due to the large detuning with $a_t$. When the frequency of $a_t$ is swept into domain II (between $a_1$ and $a_2$), $a_t$ exchanges the population with $a_1$ and $a_2$ simultaneously. As the frequency of $a_t$ arrives at domain III, i.e., the right of $a_2$, mode $a_1$ returns to its ground state and keeps that state to the end due to the large detuning with $a_t$. The population of $a_t$ also keeps transferring to $a_2$ until $a_t$ evolves to its ground state. When the detuning $\mathrm{\Delta }(t)$ is larger than about $4\delta$, the system evolves to our target state, i.e., $a_2(t)\approx 1$ and $a_1(t)\approx a_t(t)\approx 0$. In whole process, one only needs to keep a stable tuning speed to get the perfect state transfer after the detuning of $a_t$ becomes larger than $4\delta$. The maximal population of $a_2$ is about $P=0.9536$.

 

Fig. 2. Result of FIST between $a_1$ and $a_2$ by linearly tuning the resonance frequency of $a_t$. The speed is chosen as $0.08{\delta }^2$. The inset is the plot of tuning function, and the unit of time $t$ is ${\delta }^{-1}$.

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To investigate how FIST is affected by different tuning range $d$ and tuning speed $v$, we show the final population of mode $a_2$ with respect to $d$ and $v$ in Fig. 3. Figures 3(a) and 3(b) are the two-dimensional cross-section analysis of two dot-dashed lines in Fig. 3(c). In Fig. 3(a), the tuning speed is fixed as $v=0.27{\delta }^2$. The population is proportional to the tuning range $d$ within a short range less than about $3.0\delta$ and oscillates to a stable value as the range becomes larger. The reason is that the system has more time to transfer the state from $a_1$ to $a_2$ as the tuning range becomes larger in the domain about $(0,3.0\delta )$. When the tuning range becomes very long, the population has little change because the large detuning makes little contribution to state transfer. As shown in Fig. 3(b), on the contrary, when the tuning range is kept at $2.65\delta$, the population is inversely proportional to the tuning speed along with a little oscillation, since faster speed makes shorter interaction time. Figure 3(c) indicates the clearer conclusion that large tuning range and slow tuning speed are the best tuning manner.

 

Fig. 3. Simulation of final population of mode $a_2$ affected by tuning variables. The units $d$ and $v$ are chosen as $\delta$ and ${\delta }^2$, respectively. (a) Population $P$ versus tuning range $d$ with $v=0.27{\delta }^2$. All ${\mathrm{\Delta }}_0$ are chosen as ${\mathrm{\Delta }}_0=(\delta -d)/2$. (b) Population $P$ versus tuning speed $v$ with $d=2.65\delta$ and ${\mathrm{\Delta }}_0=-0.825\delta$. (c) Population $P$ versus tuning range $d$ and tuning speed $v$. The dashed line shows all the points of evolution time with $10{\delta }^{-1}$.

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B. FIST with Periodic Function

Besides the linear tuning, we consider the situation with periodic function. For simplicity, we choose a sine function here. The frequency detuning is described as $\mathrm{\Delta }(t)=A[\sin (\mathrm{\Omega }t)+c_0]/2$. With the parameters chosen as $A=9.6\delta$, $\mathrm{\Omega }=0.95\delta$, and $c_0=0.5$, the populations of each mode are performed in Fig. 4. The envelope curve of the populations of $a_1$ and $a_2$ is evolved as a sine function. Modes $a_1$ and $a_t$ exchange their populations with Rabi frequency $\mathrm{\Omega }$, which is the frequency of the tuning function. The maximal value of population $P$ of mode $a_2$ in this periodic tuning is 0.9365. Compared with linear tuning, the population transferred here is realized successfully by controlling the evolution time of the tuning function. For instance, in Fig. 4, the evolution should be ended in around $160{\delta }^{-1}$. In principle, if the end time is missed, one can wait for the neat periodic time, but the long time will decrease the fidelity of the protocol by bringing more decoherence.

 

Fig. 4. Population change with respect to evolution time via sine tuning function. Lines labeled with $a_1$, $a_2$, and $a_t$ are the populations of the corresponding modes.

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C. Optimizing the FIST via Gradient Descent

To achieve a perfect and fast state transfer from $a_1$ to $a_3$, we use the gradient descent technique to optimize the FIST and pick an optimal function for frequency tuning. The gradient descent technique is usually used for finding an optimal evolution path for the system along the opposite direction of gradient descent. In our model, the evolution equations are given by

The target function is chosen as $\mathcal{L}(a,t)={|a_2(t)|}^2$ and can be described as

Our goal is to optimize the above function and get the maximal value. So the gradient of the target function is calculated as $\nabla \mathcal{L}=\frac{\partial \mathcal{L}}{\partial \omega }$. In numerical calculation, the gradient is written approximatively by

The simulation of optimized fast FIST is shown in Fig. 5. We plot the transfer results and the tuning functions in Figs. 5(a) and 5(b), respectively. In our simulations, we first choose ${\mathrm{\Delta }}_{\mathrm{I}}(t=0)=0$ and cut the tuning function ${\mathrm{\Delta }}_{\mathrm{I}}(t)$ into 100 discrete segments that are mutually independent to be optimized via the gradient descent algorithm. The results and optimal tuning envelope are plotted in Figs. 5(a) and 5(b), respectively, with dotted curves. During the evolution time, the population of mode $a_2$ monotonically increases to its maximal value 0.9923. However, the curves of modes $a_1$ and $a_t$ decrease to their ground states with an oscillating process.

 

Fig. 5. Simulation of fast FIST from $a_1$ to $a_2$ by using the gradient descent technique. The parameters are the cross points of Fig. 3(c) with $d=2.65\delta$, $v=0.27{\delta }^2$. (a) Result of the optimized population transfer process. (b) Corresponding optimal tuning function of the intermediate mode. The unit of time here is ${\delta }^{-1}$.

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To make the scheme conveniently controlled, according to the envelope of the dotted line in Fig. 5(b), we reasonably express the tuning function as

In linear tuning shown in Fig. 3(c), the maximal population $P$ of mode $a_2$ with evolution time less than $10{\delta }^{-1}$ is $P=0.618$, labelled with a cross point. With the evolution time $10{\delta }^{-1}$, the optimized protocol can achieve the population with $P>0.98$. Compared with linear tuning, the state transfer under this protocol can be optimized with a faster evolution path and higher fidelity.

4. NONRECIPROCITY IN MULTIMODE INTERACTIONS

Our model shows the significant nonreciprocity in state transfer between $a_1$ and $a_2$. For example, when the frequency of the intermediate mode is swept from $a_1$ to $a_2$, the state can be transferred from $a_1$ to $a_2$, but it completely failed for $a_2$ to $a_1$. The results of the nonreciprocal state transfer are plotted in Fig. 6 with linear tuning from $a_1$ to $a_2$, i.e., from left to right in Fig. 1(c). We fix the tuning range at $d=14\delta$ and change the tuning speed in Fig. 6(a). The top (bottom) green (blue) line is the final population of $a_2$ ($a_1$) with the initial state prepared in $a_1$ ($a_2$). As the speed becomes slower, the nonreciprocity is more clear. At very slow speed, the population of $a_2$ transferred from $a_1$ is almost 1, but the population of the inverse transfer process is less than ${10}^{-2}$. In Fig. 6(b), we investigate the nonreciprocity with respect to tuning range with constant tuning speed $v=0.1{\delta }^2$. The small populations of both $a_1$ and $a_2$ in different directions in a small tuning range indicate that the nonreciprocity is not clear. When we increase the tuning range $d$, the green and blue lines converge towards unity and ${10}^{-2}$, respectively. The order difference between the populations of $a_2$ (initial state prepared in $a_1$) and $a_1$ (initial state prepared in $a_2$) in the parameter space of the tuning speed and tuning range is shown in Fig. 6(c). Furthermore, we also study the nonreciprocity of gradient descent optimization. If the initial state is in mode $a_2$, the population transferred from $a_2$ to $a_1$ can be 0.9942 after optimization. But if we apply the tuning function, which is optimized with the initial state in $a_1$, directly on the situation where the initial state is prepared in $a_2$, the population that transfers from $a_2$ to $a_1$ is just $P=0.235$.

 

Fig. 6. Nonreciprocal state transfer between modes $a_1$ and $a_2$. (a) Populations of modes $a_1$ and $a_2$ versus tuning speed. The tuning range is $d=14\delta$. (b) Populations of modes $a_1$ and $a_2$ versus tuning range. The tuning speed is $v=0.1{\delta }^2$. (c) Order difference between the populations of $a_2$ and $a_1$, ${\log }_{10}[P(a_2)]-{\log }_{10}[P(a_1)]$, in the parameter spaces of tuning speed and tuning range.

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5. DISCUSSION AND CONCLUSION

All the results shown above are considered in all-optical cavity systems. Actually, our model is a universal approach for multimode interaction systems such as all-mechanical phonon modes or photon–phonon interactions. For example, the direct interaction between phonons is difficult. So one can transfer the state from one mechanical resonator to another one via an intermediate optical cavity mode [44,51]. Beside the coupling strength tuning method, one can use the frequency tuning approach described here to control the interactions.

To perform more interesting applications, our model can be extended to design a one-dimensional microcavity array and a two-dimensional microcavity lattice. The detailed schematic diagrams are shown in Fig. 7. An array is built by coupling $n+1$ microcavities with $n$ tuning cavities in Fig. 7(a). The structure can be used for realizing an all-optical transistor with more abundant physical tuning. Figure 7(b) shows the two-dimensional lattice structure, which is designed by connecting one tuning cavity with three storage cavities. Every unit of this structure is a simple optical router. The structure has functions for building an all-optical on-chip quantum network [6,7].

 

Fig. 7. All-optical on-chip microcavity structures. (a) One-dimensional microcavity array. (b) Two-dimensional optical microcavity lattice.

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In our model, we always keep the coupling strengths constant with the assumption that the distances between cavities are fixed. The typical corresponding physical system is the on-chip optical microcavity sample, because the distances between each cavity are difficult to change after completing the fabrication. Our frequency tuning manner is possible. This is because the frequency of the microcavity is sensitive to its shape, which can be modulated by some operations such as temperature [54–56] and external forces [57–63]. The above-mentioned frequency tuning approaches have been realized with high resolution in experiments, but the tuning speed is limited. So a fast tuning method is needed for improving the feasibility of practical applications.

In conclusion, we have proposed an approach to realize the state transfer between two separated modes in optical microcavities. Our proposal is valid for both two and three microcavities. FIST can be realized with high fidelity via different tuning manners, i.e., linear and periodic function, of the resonance frequency of the intermediate mode. To optimize the tuning function, a fast and perfect evolution process is performed by using the gradient descent technique. Our proposal also shows the significant nonreciprocity. The state can be transferred successfully in the same direction with frequency tuning, and it fails in the opposite direction. Our work provides an effective approach for controlling the optical mode in on-chip microcavities and has important applications in all-optical devices.