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 Λ 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.

Here the vector is a→(t)=[a^1(t),a^t(t),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.

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.

To achieve a perfect and fast state transfer from a1 to a3, 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 a(t)=U(t,0)a(0),(3)where the evolution operator is a Dyson series expressed by U(t,0)=exp[∫0tM(t′)dt′]. The time domain is chosen with t=[0,t1], and the frequency of intermediate mode, considered a parameter to be optimized, is divided into n discrete constant variables, i.e., ΔI=[Δ1,⋯,Δn]. Therefore, the operator can be rewritten as a(t1)=U(Δ1,⋯,Δn;t1,0)a(0).(4)

The target function is chosen as L(a,t)=|a2(t)|2 and can be described as L(a,t1)=L(Δ1,⋯,Δn;t1,0).(5)

Our goal is to optimize the above function and get the maximal value. So the gradient of the target function is calculated as ∇L=∂L∂ω. In numerical calculation, the gradient is written approximatively by ∇L=(∇L1,⋯,∇Ln),(6)where ∇Li is ∇Li=L(⋯,Δi+δΔ,⋯;t1,0)−L(Δ1,⋯,Δn;t1,0)δΔ.(7)

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 ΔII(t)=Asin[(Ωt+θ)+C1](e−γt+C2),(8)where the parameters A, Ω, θ, γ, C1, and C2 are undetermined coefficients to be optimized. By calculating the state transfer task with Eq. (8), the population of mode a2 can be optimized to 0.9826, and the parameters are given as A=−5.555δ, Ω=1.276δ, θ=0.564, γ=1.467δ, C1=−0.797, and C2=0.119. The evolutions of populations and tuning functions are plotted in Fig. 5. The envelope of Eq. (8) and the evolution of populations are similar with the dotted curves in each figure, respectively.

In linear tuning shown in Fig. 3(c), the maximal population P of mode a2 with evolution time less than 10δ−1 is P=0.618, labelled with a cross point. With the evolution time 10δ−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.

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.

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.

Acknowledgment. The authors thank Guo-Qing Qin for helpful discussions.