Frequency-tuning-induced state transfer in optical microcavities

Xu-Sheng Xu; Hao Zhang; Xiang-Yu Kong; Min Wang; Gui-Lu Long

doi:10.1364/PRJ.385046

Abstract

Quantum state transfer in optical microcavities plays an important role in quantum information processing and is essential in many optical devices such as optical frequency converters and diodes. Existing schemes are effective and realized by tuning the coupling strengths between modes. However, such approaches are severely restricted due to the small amount of strength that can be tuned and the difficulty performing the tuning in some situations, such as in an on-chip microcavity system. Here we propose a novel approach that realizes the state transfer between different modes in optical microcavities by tuning the frequency of an intermediate mode. We show that for typical functions of frequency tuning, such as linear and periodic functions, the state transfer can be realized successfully with different features. To optimize the process, we use the gradient descent technique to find an optimal tuning function for a fast and perfect state transfer. We also showed that our approach has significant nonreciprocity with appropriate tuning variables, where one can unidirectionally transfer a state from one mode to another, but the inverse direction transfer is forbidden. This work provides an effective method for controlling the multimode interactions in on-chip optical microcavities via simple operations, and it has practical applications in all-optical devices.

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.

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Figure 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.

Here the vector is $\vec{a} (t) = [{\hat{a}}_{1} (t), {\hat{a}}_{t} (t), {\hat{a}}_{2} (t {)]}^{T}$ . 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.

Figure 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 δ^{2}

. The inset is the plot of tuning function, and the unit of time

t

δ^{- 1}

Figure 3.Simulation of final population of mode

a_{2}

affected by tuning variables. The units

d

and

v

are chosen as

δ

and

δ^{2}

, respectively. (a) Population

P

versus tuning range

d

with

v = 0.27 δ^{2}

. All

Δ_{0}

are chosen as

Δ_{0} = (δ - d) / 2

. (b) Population

P

versus tuning speed

v

with

d = 2.65 δ

and

Δ_{0} = - 0.825 δ

. (c) Population

P

versus tuning range

d

and tuning speed

v

. The dashed line shows all the points of evolution time with

10 δ^{- 1}

Figure 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.

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 $a (t) = U (t, 0) a (0),$ (3)where the evolution operator is a Dyson series expressed by $U (t, 0) = \exp [\int_{0}^{t} M (t^{'}) d t^{'}]$ . The time domain is chosen with $t = [0, t_{1}]$ , and the frequency of intermediate mode, considered a parameter to be optimized, is divided into $n$ discrete constant variables, i.e., $Δ_{I} = [Δ_{1}, \dots, Δ_{n}]$ . Therefore, the operator can be rewritten as $a (t_{1}) = U (Δ_{1}, \dots, Δ_{n}; t_{1}, 0) a (0) .$ (4)

The target function is chosen as $L (a, t) = {| a_{2} (t) |}^{2}$ and can be described as $L (a, t_{1}) = L (Δ_{1}, \dots, Δ_{n}; t_{1}, 0) .$ (5)

Our goal is to optimize the above function and get the maximal value. So the gradient of the target function is calculated as $\nabla L = \frac{\partial L}{\partial ω}$ . In numerical calculation, the gradient is written approximatively by $\nabla L = (\nabla L_{1}, \dots, \nabla L_{n}),$ (6)where $\nabla L_{i}$ is $\nabla L_{i} = \frac{L (\dots, Δ_{i} + δ Δ, \dots; t_{1}, 0) - L (Δ_{1}, \dots, Δ_{n}; t_{1}, 0)}{δ Δ} .$ (7)

Figure 5.Simulation of fast FIST from

a_{1}

a_{2}

by using the gradient descent technique. The parameters are the cross points of Fig. 3(c) with

d = 2.65 δ

v = 0.27 δ^{2}

. (a) Result of the optimized population transfer process. (b) Corresponding optimal tuning function of the intermediate mode. The unit of time here is

δ^{- 1}

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 $Δ_{II} (t) = A \sin [(Ω t + θ) + C_{1}] (e^{- γ t} + C_{2}),$ (8)where the parameters $A$ , $Ω$ , $θ$ , $γ$ , $C_{1}$ , and $C_{2}$ are undetermined coefficients to be optimized. By calculating the state transfer task with Eq. (8), the population of mode $a_{2}$ can be optimized to 0.9826, and the parameters are given as $A = - 5.555 δ$ , $Ω = 1.276 δ$ , $θ = 0.564$ , $γ = 1.467 δ$ , $C_{1} = - 0.797$ , and $C_{2} = 0.119$ . 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 $a_{2}$ with evolution time less than $10 δ^{- 1}$ is $P = 0.618$ , labelled with a cross point. With the evolution time $10 δ^{- 1}$ , 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.

Figure 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 δ

. (b) Populations of modes

a_{1}

and

a_{2}

versus tuning range. The tuning speed is

v = 0.1 δ^{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.

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.

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

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.