Multiphoton Controllable Transport between Remote Resonators

Abstract

We develop a novel method for multiphoton controllable transport between remote resonators. Specifically, an auxiliary resonator is used to control the coherent long-range coupling of two spatially separated resonators, mediated by a coupled-resonator chain of arbitrary length. In this manner, an arbitrary multiphoton quantum state can be either transmitted through or reflected off the intermediate chain on demand, with very high fidelity. We find, on using a time-independent perturbative treatment, that quantum information leakage of an arbitrary Fock state is limited by two upper bounds, one for the transmitted case and the other for the reflected case. In principle, the two upper bounds can be made arbitrarily small, which is confirmed by numerical simulations.

1. Introduction

The realization of quantum information transport between remote parties is of central importance for any scalable quantum information processing. To this end, one straightforward approach is to use quantum channels to connect these spatially separated parties. Owing to high speed transmission and negligible interaction with the environment, in the form of flying qubits, photons are the natural candidates for long-distance quantum communication [1,2,3,4,5,6,7]. Alternatively, for short-distance quantum communication inside a quantum computer, the majority of promising channels rely upon the use of solid-state-based devices, including nuclear spins in nuclear magnetic resonance (NMR) [8,9], electron spins of nitrogen-vacancy (NV) colour centers in diamond [10,11,12,13,14,15], and flux qubits in superconductors [16,17,18,19]. Moreover, coupled-resonator arrays (CRAs), being currently explored in various physical systems such as superconducting transmission line resonators [20,21,22,23,24], toroidal microresonators [25,26,27,28] and plasmonic nanoparticle arrays [29], have been attracted much attention in recent years. A particular advantage of such arrays is the full addressability of individual resonators, which allows each of the resonators to act as a quantum network node [30,31,32]. Indeed, utilizing these CRAs of having the same fundamental hardware to process quantum information can also avoid a quantum interface between the quantum register and the quantum channel.

In addition to offering an effective platform to simulate quantum many-body phenomena such as Mott-superfluid and topological effects [33,34,35,36], the CRAs have been previously considered for controllable transport of photons by making use of the photon-atom scattering [22,37,38,39,40,41]. Despite such substantial developments, prior work on controllable photon transport has typically focused upon either single photons [22,38,42,43,44] or nearby CRAs [45,46,47,48,49]. However, the ability to transport multiphoton quantum states is a key requirement for encoding a high-dimensional Hilbert space, which is applicable, for example, to universal quantum computation [50,51,52]; at the same time, quantum information stored in the multiphoton fields needs to be controllably transported between two distant quantum registers to carry out quantum network operations. For these reasons, developing a quantum channel capable of controllably transporting multiphoton states is thus of both fundamental and practical importance.

In this paper, we propose and analyze a multiphoton controllable transport protocol, where we use an auxiliary resonator coupled to one resonator of a coupled-resonator chain, which serves as a quantum channel to connect two remote resonators. The physical essence underlying our method is that this auxiliary resonator is employed to control the coherent long-range interaction between the two boundary resonators. To be specific, in the case when the auxiliary resonator is absent, the two boundary resonators could be only coupled to a specific eigenmode of the intermediate chain, within the weak-coupling regime. In this case, the time evolution can swap arbitrary bosonic quantum states of the two boundary resonators, yielding an effective photon transport channel (EPTC). In contrast, when the auxiliary resonator is coupled to the intermediate chain, the specific eigenmode could be split, such that the two boundary resonators are decoupled from the intermediate chain in the large-detuning limit. Photons are therefore reflected back, remaining unchanged. As opposed to prior work, the proposed model is capable of controlling the coherent transport of an arbitrary multiphoton quantum state between two distant resonators over an arbitrarily long range.

2. Physical Model and Effective Hamiltonian

The basic idea of our protocol is schematically illustrated in Figure 1a. To begin, we consider a quantum channel consisting of a coupled-resonator chain of N resonators, with a Hamiltonian

$$H_0=\kappa \sum _{i=1}^{N-1}\left(c_i^{†}{c}_{i+1}+\text{H}.\text{c}.\right)$$

(1)

where $c_i$ ($c_i^{†}$) represents the annihilation (creation) operator acting on the resonator i and obeys a boson commutation relation $\left[c_i,c_j^{†}\right]={\delta }_{ij}$, and κ is the coupling constant between the nearest-neighbor resonators. Two distant resonators, labelled 0 and $N+1$, are coupled to the ends of the quantum channel, and the interaction Hamiltonian is correspondingly given by

$$V_1=g_0\left(c_0^{†}{c}_1+c_{N+1}^{†}{c}_N+\text{H}.\text{c}.\right)$$

(2)

where $g_0$ is the coupling strength between the two boundary resonators and the intermediate chain. In order to control the multiphoton coherent transport, we introduce an auxiliary control resonator, labelled $N+2$, to interact with the m-th resonator of the quantum channel through

$$V_2=J_0\left(c_{N+2}^{†}{c}_m+\text{H}.\text{c}.\right)$$

(3)

with a coupling strength $J_0$. Here, we have assumed that all resonators have a common frequency ω, and the system is transformed into a frame rotating at ω. Therefore, the total Hamiltonian governing the system is $H_T=H_0+V_1+V_2$.

Upon following an orthogonal transformation [53,54],

$$f_k=\sqrt{\frac{2}{N+1}}\sum _{i=1}^{N}sin\frac{ik\pi }{N+1}{c}_i$$

(4)

the Hamiltonian $H_0$ is diagonalized to

$$H_0=\sum _{k=1}^N{\epsilon }_k{f}_k^{†}{f}_k$$

(5)

where ${\epsilon }_k=2\kappa cos\left[k\pi /\left(N+1\right)\right]$. Substituting Equation (4) into Equations (2) and (3), $V_1$ and $V_2$ are likewise transformed to

$$V_1=\sum _{k=1}^N{g}_k\left[c_0^{†}{f}_k+{\left(-1\right)}^{k-1}{c}_{N+1}^{†}{f}_k+\text{H}.\text{c}.\right]$$

(6)

and

$$V_2=\sum _{k=1}^N{J}_k\left(c_{N+2}^{†}{f}_k+\text{H}.\text{c}.\right)$$

(7)

respectively. Here, we have defined

$$g_k=g_0\sqrt{\frac{2}{N+1}}sin\frac{k\pi }{N+1}$$

(8)

and

$$J_k=J_0\sqrt{\frac{2}{N+1}}sin\frac{mk\pi }{N+1}$$

(9)

By restricting our attention to odd N, it yields the existence of a single zero-energy mode in the intermediate chain corresponding to $k=z\equiv \left(N+1\right)/2$, such that this mode is in resonance with the two boundary resonators as well as with the auxiliary resonator. Under the assumption that $g_0$ and $J_0$ are much smaller than κ, off-resonant couplings to the boundary resonators and to the auxiliary resonator can be neglected owing to $\left\{g_z,J_z\right\}\ll |{\epsilon }_{z\pm 1}-{\epsilon }_z|$ [55]. As a result, the full evolution dynamics is reduced to an effective model in which only the two boundary resonators, the auxiliary resonator and the zero-energy mode are involved, and accordingly $H_T$ is approximated as an effective Hamiltonian

$$H_{\text{eff}}=g_z\left[c_0^{†}{f}_z+{\left(-1\right)}^{z-1}{c}_{N+1}^{†}{f}_z+\text{H}.\text{c}.\right]+J_z\left(c_{N+2}^{†}{f}_z+\text{H}.\text{c}.\right)$$

(10)

which can be used to make a multiphoton quantum switch.

3. Multiphoton Controllable Transport and Fidelities

If the auxiliary resonator is uncoupled to the intermediate chain, $J_0=0$, the effective Hamiltonian becomes

$$H_{\text{eff}}=g_z\left[c_0^{†}{f}_z+{\left(-1\right)}^{z-1}{c}_{N+1}^{†}{f}_z+\text{H}.\text{c}.\right]$$

(11)

In this case, the two boundary resonators are coherently coupled by means of the zero-energy mode. In the Heisenberg picture, a straightforward calculation yields

$$c_0^{†}\left(t\right)=c_0^{†}+\frac{1}{2}\left[-1+cos\left(\sqrt{2}{g}_{z}t\right)\right]\left[c_0^{†}+{\left(-1\right)}^{z-1}{c}_{N+1}^{†}\right]+\frac{isin\left(\sqrt{2}{g}_{z}t\right)}{\sqrt{2}}{f}_z$$

Choosing the evolution time $t=\tau \equiv \pi /\sqrt{2}{g}_z$ gives

$$c_0^{†}\left(\tau \right)={\left(-1\right)}^z{c}_{N+1}^{†}$$

(12)

and in a similar manner, we have

$$c_{N+1}^{†}\left(\tau \right)={\left(-1\right)}^z{c}_0^{†}$$

(13)

Equations (12) and (13) exhibit that the evolution for a specific time behaves as a swap operation between the two boundary resonators, as shown in Figure 1b. However, this zero-energy mode could, in the case when the auxiliary resonator is coupled to the intermediate channel, be split into two new modes separated by an energy gap $2J_z$. It follows, on ensuring $g_0\ll J_0$, that the two boundary resonators are significantly detuned from the new modes if m is odd, and thus are decoupled from the intermediate channel. In this case, the time evolution is referred to as an identity operation, leading to

$$c_0^{†}\left(t\right)=c_0^{†}$$

(14)

and

$$c_{N+1}^{†}\left(t\right)=c_{N+1}^{†}$$

(15)

Let us consider the controllable transport of an arbitrary Fock state $|n\rangle$, for example, from the left resonator to the right resonator. The more general treatment of an arbitrary superposition of Fock states are presented in the Appendix. Such a Fock state can be generated using a nonlinear quantum system as an intermediary between a classical radiation field and the resonator [56]. We start with an initial state of the total system,

$$|\Phi \left(0\right){\rangle =|n\rangle }_0{|\mathbf{0}\rangle |0\rangle }_{N+1}=\frac{{\left(c_0^{†}\right)}^n}{\sqrt{n!}}{|0\rangle }_T$$

(16)

where $|\mathbf{0}\rangle ={|0\rangle }_1{\cdots |0\rangle }_N{|0\rangle }_{N+2}$, and ${|0\rangle }_T={|0\rangle }_0{|\mathbf{0}\rangle |0\rangle }_{N+1}$ is the vacuum state of all resonators. Under the time evolution, the total system freely evolves into a finial state,

$$|\Phi \left(t\right)\rangle =\frac{{\left[c_0^{†}\left(-t\right)\right]}^n}{\sqrt{n!}}{|0\rangle }_T$$

(17)

In the uncoupled case when $J_0=0$, according to Equation (12), the finial state for time τ is $|\Phi \left(\tau \right)\rangle ={\left(-1\right)}^{nz}{|0\rangle }_0{|\mathbf{0}\rangle |n\rangle }_{N+1}$, which means that all photons are simultaneously transported from the left resonator to the right resonator as desired. In contrast, in the coupled case of $g_0\ll J_0\ll \kappa$, the finial state becomes $|\Phi \left(t\right)\rangle =|\Phi \left(0\right)\rangle$ according to Equation (14), thus these photons are reflected back, remaining unchanged.

In order to characterize the quality of our protocol, we employ two fidelities with one transmission fidelity, $F_t=\langle n|{\rho }_{N+1}\left(\tau \right)|n\rangle$, and one reflection fidelity, $F_r=\langle n|{\rho }_0\left(\tau \right)|n\rangle$. Here, ${\rho }_0\left(\tau \right)$ and ${\rho }_{N+1}\left(\tau \right)$ are the reduced density matrices of the resonators 0 and $N+1$, respectively, at time $t=\tau$. Together with an $\left(N+1\right)\times \left(N+1\right)$ coupling matrix A, the total Hamiltonian can be compactly expressed as

$$H_T=\sum _{i,j=0}^{N+2}{A}_{ij}{c}_i^{†}{c}_j$$

(18)

where $A_{ij}$ is the coupling strength between two resonators i and j. Applying the Heisenberg equation of motion, $c_i\left(t\right)=i\left[H_0,c_i\left(t\right)\right]$, one finds that

$$c_i\left(t\right)=\sum _{j=0}^{N+2}{M}_{ij}{c}_j$$

(19)

with $M=exp\left(-iAt\right)$ being a unitary evolution matrix. To calculate the two fidelities $F_t$ and $F_r$, more explicitly, we rewrite $c_i\left(t\right)$ of Equation (19) as

$$c_0\left(t\right)=M_{0,\mu }{c}_{\mu }+\sqrt{{\delta }_{\mu }}{c}_{{\delta }_{\mu }}$$

(20)

where ${\delta }_{\mu }=1-{|M_{0,\mu }|}^2$ and $\mu =0,\cdots ,N+2$. Here, we define a collective mode $c_{{\delta }_{\mu }}$ as a normalized linear combination of all modes apart from $c_{\mu }$, resulting in $\left[c_{\mu },c_{{\delta }_{\mu }}^{†}\right]=0$ and $\left[c_{{\delta }_{\mu }},c_{{\delta }_{\mu }}^{†}\right]=1$. In combination with Equations (17) and (20), we have

$$|\Phi \left(t\right)\rangle =\sum _{r=0}^n{f}_{\mu }\left(r,n\right){|r\rangle }_{{\delta }_{\mu }}{|n-r\rangle }_{\mu }$$

(21)

where

$$f_{\mu }\left(r,n\right)=\sqrt{C_n^r}{M}_{0,\mu }^{n-r}{\delta }_{\mu }^{r/2}$$

(22)

$$C_n^r=\frac{n!}{r!\left(n-r\right)!}$$

(23)

and ${|r\rangle }_{{\delta }_{\mu }}$ is a Fock state of the collective mode $c_{{\delta }_{\mu }}$. The corresponding density matrix is

$$\begin{array}{ccc}& \rho \left(t\right)& =|\Phi \left(t\right)\rangle \langle \Phi \left(t\right)|\hfill \\ & & =\sum _{r,r^{\prime }=0}^n{f}_{\mu }\left(r,n\right)f_{\mu }^{*}\left(r^{\prime },n\right){|n-r\rangle }_{\mu }\langle n-r^{\prime }{|\otimes |r\rangle }_{{\delta }_{\mu }}\langle r^{\prime }|\hfill \end{array}$$

(24)

and then, by tracing out the variables of the collective mode, the reduced density matrix of the resonator μ is calculated as

$${\rho }_{\mu }\left(t\right)=\sum _{r=0}^n|f_{\mu }\left(r,n\right){|}^2{|n-r\rangle }_{\mu }\langle n-r|$$

(25)

Thus, the transmission fidelity and the reflection fidelity are respectively given by

$$F_t=|f_{N+1}\left(0,n\right){|}^2={|M_{0,N+1}|}^{2n}$$

(26)

and

$$F_r=|f_0\left(0,n\right){|}^2={|M_{0,0}|}^{2n}$$

(27)

In Figure 2 we plot the transmission fidelity $F_t$ versus $J_0/\kappa$ with $g_0/\kappa =0.01$ (see Figure 2a), or $0.005$ (see Figure 2b), as well as the reflection fidelity $F_r$. In the two cases, we assume that a Fock state $|n\rangle$ is initialized into the left boundary resonator, and further take $n=2,3$ and 5 as three special examples to simulate numerically. It is shown that as $J_0=0$, we obtain $F_t\simeq 1$ while $F_r\simeq 0$, implying that photons are transmitted from the left resonator to the right resonator. In contrast, in the regime $g_0\ll J_0\ll \kappa$, photons are reflected back to the left resonator, according to $F_t\simeq 0$ whereas $F_r\simeq 1$. Hence, a Fock state could, by tuning the coupling strength $J_0$, be transmitted forward or reflected backward at will.

4. Quantum Information Leakage

Having explicitly demonstrated the controllable eigenmode-mediated transport of multiphoton information, we now calculate the quantum information leakage by making use of perturbation theory. We begin by considering the evolution matrix of Equation (19). In fact, the coupling matrix A is identical to the total Hamiltonian confined in a single-excitation subspace,

$$H_S=g_0\left(|0\rangle \langle 1|+|N+1\rangle \langle N|+\text{H}.\text{c}.\right)+\kappa \sum _{j=1}^{N-1}\left(|j\rangle \langle j+1|+\text{H}.\text{c}.\right)+J_0\left(|N+2\rangle \langle m|+\text{H}.\text{c}.\right)$$

(28)

where $|\mu \rangle =c_{\mu }^{†}{|0\rangle }_T$ $(\mu =0,\cdots ,N+2)$ represents the μ-th basis of this single-excitation subspace. As a consequence, the evolution matrix M is identical to $U\left(t\right)=e^{-i{H}_{S}t}$, so that $M_{\mu ,{\mu }^{\prime }}=\langle \mu |U\left(t\right)|{\mu }^{\prime }\rangle$. By introducing an orthogonal transformation,

$$|k\rangle =\sqrt{\frac{2}{N+1}}\sum _{j=1}^{N}sin\frac{jk\pi }{N+1}|j\rangle$$

(29)

the Hamiltonian $H_S$ is transformed to

$$H_S=\sum _{k=1}^N{g}_k\left[|0\rangle \langle k|+{\left(-1\right)}^{k-1}|N+1\rangle \langle k|+\text{H}.\text{c}.\right]+\sum _{k=1}^N{\epsilon }_k|k\rangle \langle k|+\sum _{k=1}^N{J}_k\left(|N+2\rangle \langle k|+\text{H}.\text{c}.\right)$$

(30)

Based upon such a Hamiltonian, we can obtain the leakage of quantum information for the uncoupled case of $J_0=0$, and the coupled case of $g_0\ll J_0\ll \kappa$, respectively, as we will show in the following.

Let us first focus upon the former case, where the total system can be thought of as an EPTC being perturbatively coupled to a fictitious bosonic environment in the limit $g_0\ll \kappa$. The corresponding Hamiltonian could be divided into three parts, $H_S=H_{\text{E}}+H_z+V_z$. $H_{\text{E}}$ features the Hamiltonian for the EPTC:

$$H_{\text{E}}=g_z\left[|e_1\rangle \langle e_2|+{\left(-1\right)}^{z-1}|e_3\rangle \langle e_2|+\text{H}.\text{c}.\right]$$

(31)

where, for convenient, we have used $\left\{|e_1\rangle ,|e_2\rangle ,|e_3\rangle \right\}$ to replace $\left\{|0\rangle ,|z\rangle ,|N+1\rangle \right\}$. The environment is determined by

$$H_z=\sum _{k\ne z}{\epsilon }_k|k\rangle \langle k|$$

(32)

of having $2N$ bosonic modes, and ${\epsilon }_k={\epsilon }_{-k}$. The part $V_z$ modelling the interaction between them is

$$V_z=\sum _{k\ne z}{g}_k\left[|e_1\rangle \langle k|+{\left(-1\right)}^{k-1}|e_3\rangle \langle k|+\text{H}.\text{c}.\right]$$

(33)

Assuming that the EPTC Hamiltonian can be diagonalized through a unitary transformation T, it results in ${\sum }_{i,j=1}^3{T}_{q^{\prime },i}^{†}{T}_{j,q}\langle e_i|H_{\mathrm{E}}|e_j\rangle ={\lambda }_q{\delta }_{q^{\prime },q}$, such that

$$H_{\mathrm{E}}=\sum _{q=1}^3{\lambda }_q|q\rangle \langle q|$$

(34)

with $|q\rangle ={\sum }_{i=1}^3{T}_{iq}|e_i\rangle$. Similarly,

$$V_z=\sum _{k\ne z}\sum _{q=1}^3\left(G_{kq}|q\rangle \langle k|+\text{H}.\text{c}.\right)$$

(35)

where

$$G_{kq}=g_k\left[T_{q,1}^{†}+{\left(-1\right)}^{k-1}{T}_{q,3}^{†}\right]$$

(36)

By performing a first-order perturbative treatment, $V_z$ has no effects on the eigenenergies ${\lambda }_q$ and ${\epsilon }_k$; nevertheless, the eigenstates $|q\rangle$ and $|k\rangle$ are modified by

$$|\tilde{q}\rangle \simeq |q\rangle -\sum _{k\ne z}\frac{G_{kq}^{*}}{{\epsilon }_k}|k\rangle$$

(37)

and

$$|\tilde{k}\rangle \simeq |k\rangle +\sum _{q=1}^3\frac{G_{kq}}{{\epsilon }_k}|q\rangle$$

(38)

respectively. Combining Equations (37) and (38), we find, up to second order,

$$|q\rangle \simeq \left(1-\sum _{k\ne z}\frac{|G_{kq}{|}^2}{{\epsilon }_k^2}\right)|\tilde{q}\rangle +\sum _{k\ne z}\frac{G_{kq}^{*}}{{\epsilon }_k}|\tilde{k}\rangle -\sum _{k\ne z}\sum _{q^{\prime }\ne q}\frac{G_{kq}^{*}{G}_{k{q}^{\prime }}}{{\epsilon }_k^2}|q^{\prime }\rangle$$

(39)

and hence, after an iteration, $|q\rangle$ takes the form of

$$|q\rangle \simeq |\tilde{q}\rangle +\sum _{k\ne z}\frac{G_{kq}^{*}}{{\epsilon }_k}|\tilde{k}\rangle -\sum _{k\ne z}\sum _{q^{\prime }=1}^3\frac{G_{kq}^{*}{G}_{k{q}^{\prime }}}{{\epsilon }_k^2}|\widetilde{q^{\prime }}\rangle$$

(40)

Since

$$\begin{array}{ccc}& & \langle \widetilde{q^{\prime }}|\tilde{q}\rangle ={\delta }_{q^{\prime },q}+\sum _{k\ne z}\frac{G_{kq}^{*}{G}_{k{q}^{\prime }}}{{\epsilon }_k^2}\hfill \\ & & \langle \widetilde{k^{\prime }}|\tilde{k}\rangle ={\delta }_{k^{\prime }k}+\sum _q\frac{G_{k^{\prime }q}^{*}{G}_{kq}}{{\epsilon }_{k^{\prime }}{\epsilon }_k}\hfill \\ & & \langle \tilde{q}|\tilde{k}\rangle =0\hfill \end{array}$$

(41)

the matrix elements of $U\left(t\right)$ in the energy space can be evaluated as

$$\langle q^{\prime }|U\left(t\right)|q\rangle \simeq e^{-i{\lambda }_{q}t}{\delta }_{q^{\prime },q}-\sum _{k\ne z}\frac{G_{kq}^{*}{G}_{k{q}^{\prime }}}{{\epsilon }_k^2}{e}^{-i{\lambda }_{q^{\prime }}t}+\sum _{k\ne z}\frac{G_{k{q}^{\prime }}{G}_{kq}^{*}}{{\epsilon }_k^2}{e}^{-i{\epsilon }_{k}t}$$

(42)

Subsequently, after inversion back to the space spanned by $\left\{|e_i\rangle \right\}$ ($i=1,2,3$), we arrive at

$$\begin{array}{c}\langle e_1|U\left(t\right)|e_3\rangle \simeq \langle e_1|e^{-i{H}_{\mathrm{E}}t}|e_3\rangle \hfill \\ \hfill +\sum _{k\ne z}{\left(-1\right)}^{k-1}\frac{g_k^2}{{\epsilon }_k^2}{e}^{-i{\epsilon }_{k}t}-\sum _{k\ne z}\frac{g_k^2}{{\epsilon }_k^2}\left[{\left(-1\right)}^{k-1}\langle e_1|e^{-i{H}_{\mathrm{E}}t}|e_1\rangle +\langle e_1|e^{-i{H}_{\mathrm{E}}t}|e_3\rangle \right]\end{array}$$

(43)

Combining the evolution under $H_E$ leads to $\langle e_1|e^{-i{H}_{\text{E}}\tau }|e_3\rangle ={\left(-1\right)}^z$ and $\langle e_1|e^{-i{H}_{\text{E}}\tau }|e_1\rangle =0$, and further we find

$$M_{0,N+1}\simeq {\left(-1\right)}^z\left(1-2{\Delta }_t\right)$$

(44)

for the evolution time τ. Here, we have defined

$${\Delta }_t=\sum _{k<z}\frac{g_k^2}{{\epsilon }_k^2}\left[1-{\left(-1\right)}^{k+z-1}cos\left({\epsilon }_k\tau \right)\right]$$

(45)

The transmission fidelity is therefore modified by

$$\begin{array}{c}\hfill F_t\simeq {(1-2{\Delta }_t)}^{2n}\simeq 1-4n{\Delta }_t\end{array}$$

(46)

The leakage of quantum information can be quantified using a transmission infidelity, ${\sigma }_t=1-F_t\simeq 4n{\Delta }_t$, and thus has an upper bound,

$${\sigma }_t\le 8n\sum _{k<z}\frac{g_k^2}{{\epsilon }_k^2}$$

(47)

We now consider the coupled case. In this case, the zero-energy mode of the intermediate chain is the only state coupled to the auxiliary resonator due to $J_0\ll \kappa$, which induces two new modes as previously mentioned. The two boundary resonators are coupled to such new modes in addition to the fictitious environment; however, this coupling to the environment can be neglected so long as $g_0\ll J_0$. The Hamiltonian is therefore reduced to

$$H_S=g_z\left[|e_1\rangle \langle e_2|+{\left(-1\right)}^{z-1}|e_3\rangle \langle e_2|+\text{H}.\text{c}.\right]+J_z\left(|e_4\rangle \langle e_2|+\text{H}.\text{c}.\right)$$

(48)

Here, we have used $\left\{|e_1\rangle ,|e_2\rangle ,|e_3\rangle ,|e_4\rangle \right\}$ to replace $\left\{|0\rangle ,|z\rangle ,|N+1\rangle ,|N+2\rangle \right\}$ for convenient. Using $|{\gamma }_1\rangle =\left(|e_2\rangle +|e_4\rangle \right)/2$, $|{\gamma }_2\rangle =\left(|e_2\rangle -|e_4\rangle \right)/2$ and $|\pm \rangle =\left(|e_1\rangle \pm |e_3\rangle \right)/\sqrt{2}$, the Hamiltonian $H_S$ of Equation (48) is brought to

$$H_S=\frac{g_z}{2}\sum _{k=1,2}\left[\left(|+\rangle +|-\rangle \right)\langle {\gamma }_k|+{\left(-1\right)}^{z-1}\left(|+\rangle -|-\rangle \right)\langle {\gamma }_k|+\text{H}.\text{c}.\right]+\sum _{k=1,2}{J}_k|{\gamma }_k\rangle \langle {\gamma }_k|$$

(49)

with $J_1=-J_2=J_z$. When ${\left(-1\right)}^{z-1}=1$, we get

$$H_S=g_z\sum _{k=1,2}\left(|+\rangle \langle {\gamma }_k|+\text{H}.\text{c}.\right)+\sum _{k=1,2}{J}_k|{\gamma }_k\rangle \langle {\gamma }_k|$$

(50)

from which we can follow the same recipe as described in the uncoupled case to obtain

$$M_{0,0}\simeq 1-2{\Delta }_r$$

(51)

where

$${\Delta }_r=\frac{g_z^2}{2J_z^2}\left[1-cos\left(J_{z}t\right)\right]$$

(52)

For ${\left(-1\right)}^{z-1}=-1$, it has the same result as mentioned in Equation (51). In direct analogy to the uncoupled case, the reflection fidelity is modified by

$$F_r\simeq {\left(1-2{\Delta }_r\right)}^{2n}\simeq 1-4n{\Delta }_r$$

(53)

Correspondingly, the reflection infidelity is ${\sigma }_r=1-F_r\simeq 4n{\Delta }_r$, together with an upper bound,

$${\sigma }_r\le 4n\frac{g_z^2}{J_z^2}$$

(54)

To confirm our calculation of quantum information leaking into the off-resonant couplings [57], we compare numerical results of the transmission infidelity to the analytical upper bound as depicted in Figure 3, as well as the reflection infidelity in Figure 4. It is found that this upper bound is in excellent agreement with the numerical results. In addition, the leakage of quantum information decreases with deceasing $g_0/\kappa$, so that this leakage can, in principle, be made arbitrarily small.

5. Conclusions

In this paper, we have proposed a new approach for multiphoton controllable transport between two remote resonators being coupled to the ends of a coupled-resonator chain of arbitrary length. This manner essentially enables a coherent long-range interaction between the two spatially separated resonators, in this case, the pure Hamiltonian evolution for a specific time is referred to as a swap operation of the two boundary resonators. As a result, an arbitrary multiphoton quantum state can be transported with quantum information leakage arbitrarily close to zero. However, if an auxiliary resonator is harnessed to coupled to one resonator of the intermediate chain, this coherent long-range interaction will be eliminated, so the two boundary resonators are decoupled from the intermediate chain, yielding that the time evolution functions as an identity operation. Thus, an arbitrary multiphoton quantum state can be reflected back with quantum information leakage also arbitrarily close to zero. Our approach potentially allows for realizing controllable transport of an arbitrary dimensional quantum state or even a coherent state (see the Appendix), and can also be directly generalized to quantum networks consisting of at least three quantum registers. In fact, although we have discussed the specific case of a coupled-resonator chain, such a description is consistent for quantum coupled spin systems of having been widely studied [53,54,55,58,59,60,61,62]. Hence, the present approach could be applied to the realization of scalable quantum devices, for example, quantum switches.