1. Introduction

High fidelity quantum state transfer (QST) has always been the subject of intensive researches in the context of quantum information processing (QIP) [1,2]. It requires that an arbitrary quantum state can be coherently transferred from an “emitter” system to a “receiver” system. In order to realize a reliable QST, considerable schemes appear with various quantum data buses, such as optical photons [3–6], phonon modes for trapped ions [7,8], spin chains for spin qubits [9–13], cavity photon modes for superconducting qubits and spin qubits [14–16], and so on.

It is well known that a major practical hurdle in achieving a perfect QST is the quantum dissipation since the inevitable interaction between an open system and its surrounding environment. To avoid the quantum dissipation destroying the desired coherence of the transferred quantum state, an enormous amount of efforts has been made by using either dynamical decoupling [17,18], decoherence-free subspaces [19,20], or quantum error corrections [21–23]. Nevertheless, the above methods have to fight against the quantum dissipation. And they are all based on the unitary dynamics, where the precise designs on the time evolution are demanded to ensure the target state is obtained at the end of an experiment.

With the development of the quantum information, more and more quantum technologies, from the dissipative entanglement preparation [24–44] to the autonomous quantum error correction [45–57], have exploited the spectacular promise of the quantum dissipation to perform different QIP tasks rather than exhaustively resisting the quantum dissipation. However, there is no QST scheme regarding the dissipation as a useful resource until Wang and Gertler [58] recently proposed to dissipatively achieve QST without time-dependent controls. Their proposals were designed in superconducting circuit quantum electrodynamics, but the interaction term engineered by the four-wave mixing Hamiltonian was too weak to be obtained. And the logical states were encoded by three-atom superposition states, which caused a waste of quantum resource. Subsequently, Matsuzaki et al. [59] put forward another dissipative QST scheme with two qubits collectively coupled to a tailored cavity, where a rigorous condition that the two qubits owned different resonance frequencies was required. Inspired by these schemes, our group [60] executed a directional QST in a dissipative Rydberg-atom-cavity system. Although the system in [60] was not as special as the previous two literatures, it still contained five classical lasers and two seven-level Rydberg atoms and the Rydberg-mediated interaction will also increase the experimental difficulties. Therefore, we attempt to find a more simple and general model to realize a dissipative QST scheme.

In this work, we consider a dissipation-assisted scheme to achieve a high fidelity QST for an arbitrary quantum state. The system consists of three classical lasers and two identical atoms respectively trapped in an array of lossy coupled cavities. It will transfer an arbitrary quantum state from the first atom to the second atom. And utilizing the photon loss of the cavities, the transferred quantum state will be stabilized at the second atom without special controls on the evolution of the system. The adverse effect of the atomic spontaneous emission can be neglected as the adiabatic elimination of the excited states. Besides, the atoms trapped into two separated cavities are easier to operate during an experiment. We also investigate the experimental feasibility with the current experimental parameters and a high fidelity of the target state can be above $98\%$.

2. Original physical model

In Fig. 1, the model and the corresponding atomic energy levels have been illustrated. There are two identical four-level atoms trapped in a coupled-cavity array, and the atoms both consist of three ground states $|0\rangle ,|e\rangle ,|1\rangle$ and one excited state $|r\rangle$. The transitions $|e\rangle \leftrightarrow |r\rangle$ of the two atoms are respectively driven by one classical laser with Rabi frequency $\Omega$ and detuning $\Delta _1$. And the transitions $|0\rangle \leftrightarrow |r\rangle$ and $|1\rangle \leftrightarrow |r\rangle$ for the atom $1(2)$ are coupled to two cavity modes $a_{1(2)L}$ and $a_{1(2)R}$ with coupling strength $g_L,g_R$, detuning $\delta$. Moreover, the atom $1$ interacts with another classical laser (Rabi frequency $\omega$, detuning $\Delta _2$) to pump the ground state $|e\rangle$ to the excited state $|r\rangle$. And the cavity-cavity coupling is described by the photon-hopping strength $J$. In the interaction picture, the total Hamiltonian reads

 

Fig. 1. (a) Model for the one-way QST scheme. There are two identical four-level atoms trapped in a coupled-cavity array. The first atom interacts with two lasers and the second atom couples to one laser. (b) The diagram of corresponding atomic energy levels.

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Then we introduce delocalized bosonic modes,

3. Elaboration of effective master equation

First, we interpret the purpose of the $H_s$ in detail. The initial state of the two atoms is selected as $|\psi \rangle _1|e\rangle _2$ and all the cavity modes are initialized at vacuum states. In the large detuning condition $\Delta _2\gg g$, the Hamiltonian $H_d$ will be ineffective on the initial state while the Hamiltonian $H_s$ of Eq. (6) can be reformulated as

Furthermore, while $1\gg \max \{\Omega ^-_M/E_1,\Omega ^+_K/E_2,\Omega ^-_K/E_4,\Omega ^+_M/E_5\}$, we can adiabatically eliminate the excited states of the Eq. (8). The corresponding Hamiltonian is

According to the Eq. (9), $H_s$ can swap an arbitrary quantum state between the two atoms, i.e., $|\psi \rangle _1|e\rangle _2|\textbf {0}\rangle _c\leftrightarrow |e\rangle _1|\psi \rangle _2|\textbf {0}\rangle _c$. Particularly, the adverse impacts of dissipation in the swap process are significantly suppressed as the cavity states of the swap process are restrained into the vacuum state and the excited states are adiabatically eliminated.

In Fig. 2, we show the evolution of the fidelity for the states $|e\rangle _1|\psi \rangle _2|\textbf {0}\rangle _c$ and $|\psi \rangle _1|e\rangle _2|\textbf {0}\rangle _c$ in the swap process, where the measure of fidelity for state $|i\rangle$ uses the definition Tr${\bigg [}\rho _i^{1/2}\rho \rho _i^{1/2}{\bigg ]}^{1/2}$ and $\rho _i=|i\rangle \langle i|$. The solid line and the empty circles describe the fidelity of the state $|e\rangle _1|\psi \rangle _2|\textbf {0}\rangle _c$ simulated by $H_s$ and $H_s^\textrm {eff}$, respectively. And the appearance of the former in accord with that of the latter proves the validity of the derivation of the $H_s^\textrm {eff}$. Besides, the periodical oscillations of the solid and dashed lines exactly express the swap process $|\psi \rangle _1|e\rangle _2|\textbf {0}\rangle _c\leftrightarrow |e\rangle _1|\psi \rangle _2|\textbf {0}\rangle _c$, where the dashed line is the fidelity of the state $|\psi \rangle _1|e\rangle _2|\textbf {0}\rangle _c$ governed by the original Hamiltonian $H_s$.

 

Fig. 2. The evolution of the fidelity for the states $|e\rangle _1|\psi \rangle _2|\textbf {0}\rangle _c$ and $|\psi \rangle _1|e\rangle _2|\textbf {0}\rangle _c$ in the swap process. The solid line and the empty circles describe the fidelity of the state $|e\rangle _1|\psi \rangle _2|\textbf {0}\rangle _c$ simulated by $H_s$ and $H_s^\textrm {eff}$, respectively. And the dashed line stands for the fidelity of the state $|\psi \rangle _1|e\rangle _2|\textbf {0}\rangle _c$ governed by the Hamiltonian $H_s$. The transferred quantum state $|\psi \rangle$ is randomly selected as $|\psi \rangle =\alpha |0\rangle +\beta |1\rangle$ with $\alpha =\sin \theta$, $\beta =\cos \theta$, and $\theta =\pi /3$. And the initial state is $|\psi \rangle _1|e\rangle _2|\textbf {0}\rangle _c$. The relevant parameters are chosen as: $J=161g$, $\delta =160g$, and $\Omega =0.02g$.

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On the other hand, once the system evolves into $|e0\rangle |\textbf {0}\rangle _c$ or $|e1\rangle |\textbf {0}\rangle _c$ by the swap process, the Hamiltonian $H_d$ will be in action and it can be expanded as

Referring to the Eq. (11), the state $|e\rangle _1|\psi \rangle _2|\textbf {0}\rangle _c$ can evolve into the states $|0\rangle _1|\psi \rangle _2|1\rangle _{-L}$ and $|1\rangle _1|\psi \rangle _2|1\rangle _{-R}$. And these two states will be respectively stabilized at the steady states of the whole process, $|0\rangle _1|\psi \rangle _2|\textbf {0}\rangle _{c}$ and $|1\rangle _1|\psi \rangle _2|\textbf {0}\rangle _{c}$, via the photon loss of the cavities. Then the QST is finished.

In addition, we can describe the total dissipative dynamics with the effective master equation that can be acquired by replacing the $H_I$ of the Eq. (2) with $H_\textrm {eff}=H_s^\textrm {eff}+H_d^\textrm {eff}$. And the effective process certifies that our scheme successfully transfers an arbitrary quantum state $|\psi \rangle$ from atom 1 to atom 2 and stabilizes $|\psi \rangle$ at atom 2 by the ingenious integration of $H_s$, $H_d$, and the cavities decay.

4. Numerical results

In Fig. 3, we plot the dynamical evolution for the fidelity of the transferred state $|\psi \rangle$ in atom 2, $F=\textrm {Tr}{\bigg [}\rho _s^{1/2}\textrm {Tr}_{1\& c}(\rho )\rho _s^{1/2}{\bigg ]}^{1/2}$, to demonstrate the validity of the above analyses, where $\rho _s=|\psi \rangle \langle \psi |$ and $\textrm {Tr}_{1\& c}$ means the partial trace over atom 1 and all cavity modes. The parameters of the transferred state $|\psi \rangle$ are randomly chosen as $\alpha =\sin \theta$, $\beta =\cos \theta$, and $\theta =\pi /3$. The solid line and the empty circles respectively represent the dynamical evolution of $F$ governed by the original master equation and the effective master equation. The brilliant agreement of the two curves adequately proves the validity of the reduced system, which can help us to forecast and interpret the behaviors of the original system. Moreover, the fidelity of $|\psi \rangle$ in atom 2 is as great as $98.63\%$ at $gt=12000$, and the fidelity of the initial state $|\psi \rangle _1|e\rangle _2|\textbf {0}\rangle _c$ (dashed line) tends to be vanished as time goes by. These results also confirm the feasibility of our scheme, i.e., our scheme successfully realizes that an arbitrary quantum state is dissipatively transferred from an “emitter” system to a “receiver” system and stabilized at the “receiver” system without time-dependent controls.

 

Fig. 3. The fidelity as functions of $gt$ governed by the full master equation and the effective master equation, where $\rho _s=|\psi \rangle \langle \psi |$ with $|\psi \rangle =\alpha |0\rangle +\beta |1\rangle$, $\alpha =\sin \theta$, $\beta =\cos \theta$, and $\theta =\pi /3$. The initial state is selected as $\rho _0=|\psi \rangle _1\langle \psi |\otimes |e\rangle _2\langle e|\otimes |\textbf {0}\rangle _c\langle \textbf {0}|$. The other corresponding parameters are chosen as: $J=161g$, $\delta =160g$, $\Omega =0.02g$, $\omega =g$, and $\kappa =0.02g$.

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Finally, we investigate the experimental feasibility of our scheme. In [61], Spillane et al. discussed the suitability of toroidal microcavities for strong-coupling cavity quantum electrodynamics. And they reported the projected limit for a Fabry-Perot cavity that the coupling coefficient can arrive at $g/2\pi =770$ MHz. In terms of the corresponding critical photon number and critical atom number, one can obtain the rates of cavity decay and atomic spontaneous emission as $(\kappa ,\gamma )/2\pi =(21.7,2.6)$ MHz. Besides, referring to the experiment of [62–64], we can also suppose that a cavity made of two niobium superconducting mirrors ($g/2\pi =50$ kHz and $\kappa /2\pi =1$ kHz) traps Rubidium atoms with $\gamma /2\pi =0.03$ kHz. Moreover, another group of experimental parameters can be obtained in [65] with $(g,\kappa ,\gamma )=2\pi \times (150,6.25,0.03)$ kHz. And we characterize the atomic spontaneous emission of our scheme by the Lindblad operators $L_{1(2)}=\sqrt {\gamma /3}|0\rangle _{1(2)}\langle r|$, $L_{3(4)}=\sqrt {\gamma /3}|e\rangle _{1(2)}\langle r|$, and $L_{5(6)}=\sqrt {\gamma /3}|1\rangle _{1(2)}\langle r|$. Another key element for our scheme is the photon-hopping strength $J$. Although the analytical expressions of $J$ are different as the types of cavities change [66–69], a common important dependence is the distance between the cavities. Particularly, in [67], Underwood et al. experimentally investigated the suitability of coupled-transmission-line resonators for studies of quantum phase transitions of light. The hopping rate between resonators $i$ and $j$ is considered as $J=\nu C_c|\phi _i(x_i)\phi _j(x_j)|/2$, where $\nu$ stands for the resonator frequency, $C_c$ is the coupling capacitance, and $\phi _i(x_i)$ denotes the classical mode function of resonator $i$. Additionally, in [68], Shen et al. realized an optical diode for photonic transport with two spatially overlapping single-mode semiconductor microcavities. The coupling strength $J=|\sqrt {\omega _a\omega _b}\int \varepsilon _{ij}(\boldsymbol {r})\phi _{i,a}(\boldsymbol {r})\phi _{j,b}^\ast (\boldsymbol {r})/\varepsilon (\boldsymbol {r})d^3\boldsymbol {r}|$, where $\varepsilon _{ij}(\boldsymbol {r})$ is the relative dielectric permittivity tensor of the medium, $\varepsilon (\boldsymbol {r})$ is a spatially dependent scalar quantity, the frequencies of the two cavities are denoted by $\omega _a$ and $\omega _b$, and $\phi _{i,a}(\boldsymbol {r})$ is the $i$-direction ($i=x,y,z$) space wave function depending on the position $\boldsymbol {r}$ for cavity $a$. Nevertheless, it should be noted that the conclusion of our scheme is independent of the analytical expressions of $J$ and only the value of the $J$ at a given fixed distance of the two cavities is relevant. According to the experimental schemes [67,70,71], we can find that the distance between different sorts of coupled cavities can be tuned from 200 nm to 40 $\mu$m with a wide range of the coupled strength ($\sim$THz-MHz), which provides the possibility for us to find a suitable value of $J$.

Using the above groups of parameters, we numerically calculate the dynamical evolution for the fidelity of the transferred state $|\psi \rangle$ at atom 2 in Fig. 4, and the results can be stable at $98.33\%$ (empty circles), $98.45\%$ (empty squares), and $98.56\%$ (empty triangles). These consequences thoroughly demonstrate the experimental feasibility of our scheme. In addition, the inset of Fig. 4 depicts the evolution of the fidelity for the transferred states with different $\theta$, where the tendencies of these curves are in full accord with each others. It faithfully reflects that our scheme is achievable with an arbitrary quantum state.

 

Fig. 4. The dynamical evolution for the fidelity of the transferred state $|\psi \rangle$ in atom 2 with experimental parameters. The other corresponding parameters are chosen as: $J=161g$, $\delta =160g$, $\Omega =0.02g$, $\omega =g$, and $\theta =\pi /3$. The inset shows the corresponding fidelity as function of $gt$ with different $\theta$, where the parameters are $g=2\pi \times 770$ MHz, $\kappa =21.7$ MHz, $\gamma =2.6$ MHz, $J=161g$, $\delta =160g$, $\Omega =0.02g$, and $\omega =g$.

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5. Summary

In summary, we successfully implement a one-way QST scheme in a lossy coupled-cavity array. Due to the photon loss of the cavities, an arbitrary quantum state can be not only transferred from an “emitter” system to a “receiver” system, but also stabilized at the latter without external time-dependent control. Moreover, the atomic spontaneous emission can be dramatically inhibited since the adiabatic elimination of the excited states with suitable conditions. In addition, compared with the previous known proposals, the present scheme is easier to be actualized in an experiment. We also discuss the feasibility of the scheme with the current experimental parameter and a fidelity for the transferred state at the “receiver” system can be above $98\%$. We believe the present scheme supplies a viable prospect for QST scheme via dissipation.