Introduction

Non-classical electromagnetic fields have been described with the quantum mechanical for their special statistical properties, and experimentally realized with quantum optical experiments such as quadrature-squeezed and sub-Poissonian light fields1,2,3,4,5. These non-classical light fields may lead various interesting applications such as the enhanced measurement beyond the standard quantum limit set by vacuum fluctuations4, 6, or fundamental atomic processes through interaction with non-classical light7,8,9.

A particularly interesting generation of non-classical light fields is related to cavity quantum electrodynamics, in which atoms interact strongly with a single quantized field mode of a cavity10. In both the microwave regimes11,12,13 and optical regimes8, 14, 15 the single atom cavity mode coupling strength may exceed spontaneous emission and cavity loss rates to produce observable effects of the coupled system. Rydberg atoms16 and very high-Q superconducting cavities17 are constructed in microwave experiments, where spontaneous emission and cavity damping are negligible on the time scale of the atom-field interaction. In optical regimes, the strong-coupling is reached via high-finesse cavities and very small cavity-mode volumes to avoid great spontaneous emission. The optical cavity is convenience because of direct transmission of light through the cavity mirrors, photon counting and homodyne detection14, 18, 19.

The coherent evolution makes cavity quantum electrodynamics be favorable candidates for the realizations of photonic Fock states20, 21 and Schrödinger cat states22, 23. Moreover, by using slowly decaying atomic levels (e.g., Rydberg atoms) or far-off-resonance atom-field interactions, atomic entanglements may be built24,25,26,27,28. The realizations of quantum gates between distant qubits in quantum optical settings have been recently investigated27, 28. Such proposals are very promising and highly inventive. Serafini et al.29 investigated the possibility of realizing effective quantum gates between two atoms in distant cavities coupled by an optical fiber. Zheng proposes an efficient scheme for quantum communication between two atoms trapped in distant cavities30. Moreover, flying single photons are also used for remote gates31,32,33,34,35.

The purpose of this paper is to build the distributed quantum information processing using multilevel atoms. The qudit state (d-dimensional state) may offer greater flexibilities for storing quantum information, improving the channel capacity36, 37, reducing the implementation complexity of quantum gates38,39,40,41, enhancing the information security42,43,44,45,46 and exploring different quantum features47,48,49. There are various candidate systems for qudit states50,51,52,53. Unfortunately, few schemes have been proposed for implementing distributed quantum information processing based on qudit systems. Our scheme is based upon the adiabatic transformation of eigenstates of the atom-cavity system54. We firstly investigate the possibility of realizing deterministic gates between multi-level atoms in separate optical cavities, through a coherent resonant coupling mediated by an optical fiber. The only control required would be the synchronized switching on and off of the atom-field interactions in the distant cavities, which may be achievable through simple control pulses. Combined with single atomic transformations, the two-atom gate may be used to realize universal qudit quantum logic using recent circuit synthesis55. Finally, to show the possibility of these schemes, all the adiabatic conditions are considered. The numerical simulations show that our elementary gates are insensitive to the cavity decay, fiber loss, and atomic spontaneous emission. These gates can be constructed with high fidelity by choosing the parameters appropriately.

Result

Remote atomic model

The atomic level configuration is shown in Fig. 1. Each d + 1-level atom has an excite state |r〉 and d ground states \(|{e}_{1}\rangle ,\ldots ,|{e}_{d}\rangle \). Two identical multi-level atoms are trapped in distant cavities connected by an optical fiber. The transition \(|{e}_{{i}_{2}}\rangle \leftrightarrow |r\rangle \) of each atom is driven by a classical laser field with Rabi frequency \({{\rm{\Omega }}}_{{i}_{1}}\), while the transition \(|{e}_{{i}_{1}}\rangle \leftrightarrow |r\rangle \) is driven by the cavity mode with coupling constant \({g}_{{i}_{2}}\). The mode number of the fiber is on the order of \(l\overline{\nu }\mathrm{/2}\pi c\), where l is the length of the fiber and \(\overline{\nu }\) is the decay rate of the cavity field. When \(l\overline{\nu }\mathrm{/2}\pi c\le 1\), there is only one fiber mode which essentially interacts with the cavity modes and the cavity-fiber coupling is described by the Hamiltonian as follows29, 30

$${H}_{0}=\nu b({a}_{1}^{ {\dagger } }+{a}_{2}^{ {\dagger } })+{H}{\rm{.}}{c}{\rm{.}}$$
(1)

where b is the annihilation operator for the fiber mode, \({a}_{j}^{ {\dagger } }\) is the creation operator for the j-th cavity mode, and ν is the cavity-fiber coupling strength.

Figure 1
figure 1

Involved atomic levels and transitions. The transition \({|{e}_{{i}_{1}}\rangle }_{j}\leftrightarrow |r\rangle \) of the j-th atom is coupled to the cavity mode with coupling constant g j and detuning Δ2. The transition \({|{e}_{{i}_{2}}\rangle }_{j}\leftrightarrow |r\rangle \) is driven by a classical field with Rabi frequency \({{\rm{\Omega }}}_{j}\) and detuning Δ1.

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Assume that the classical field and cavity mode are detuned from the respective transition by \({{\rm{\Delta }}}_{{{\rm{i}}}_{{\rm{1}}}}\) and \({{\rm{\Delta }}}_{{i}_{2}}\). In the interaction picture, the Hamiltonian describes the following atom-field interaction

$${H}_{int}=\sum _{j=1}^{2}({{\rm{\Omega }}}_{{i}_{1}}{e}^{i{{\rm{\Delta }}}_{{i}_{1}}t}{|r\rangle }_{jj}\langle {e}_{{i}_{1}}|+{g}_{{i}_{2}}{a}_{j}{e}^{i{{\rm{\Delta }}}_{{i}_{2}}t}{|r\rangle }_{jj}\langle {e}_{{i}_{2}}|)+{H}{\rm{.}}{c}{\rm{.}}$$
(2)

When \({{\rm{\Delta }}}_{{i}_{1}},{{\rm{\Delta }}}_{{i}_{2}}\gg {{\rm{\Omega }}}_{{i}_{1}},{g}_{{i}_{2}}\), the excite state |r〉 can be adiabatically eliminated. It results in the following Hamiltonian

$$\begin{array}{rcl}{H}_{int} & = & -\sum _{j=1}^{2}({\eta }_{{i}_{1}}{e}^{i{{\rm{\Delta }}}_{{i}_{1}}t}{|{e}_{{i}_{1}}\rangle }_{jj}\langle {e}_{{i}_{1}}|+{\zeta }_{{i}_{2}}{a}_{j}^{ {\dagger } }{a}_{j}{|{e}_{{i}_{2}}\rangle }_{jj}\langle {e}_{{i}_{2}}|)\\ & & +\,{\lambda }_{{i}_{1}{i}_{2}}({a}_{j}{S}_{j}^{+}{e}^{i{\delta }_{{i}_{1}{i}_{2}}t}+{a}^{ {\dagger } }{S}_{j}^{-}{e}^{-i{\delta }_{{i}_{1}{i}_{2}}t})+{H}{\rm{.}}{c}{\rm{.}}\end{array}$$
(3)

where

$${\eta }_{{i}_{1}}=\frac{{{\rm{\Omega }}}_{{i}_{1}}^{2}}{{{\rm{\Delta }}}_{{i}_{1}}},{\zeta }_{{i}_{2}}=\frac{{g}_{{i}_{2}}^{2}}{{{\rm{\Delta }}}_{{i}_{2}}},{\lambda }_{{i}_{1}{i}_{2}}=\frac{{{\rm{\Omega }}}_{{i}_{1}}{g}_{{i}_{2}}}{2}(\frac{1}{{{\rm{\Delta }}}_{{i}_{1}}}+\frac{1}{{{\rm{\Delta }}}_{{i}_{2}}})$$
(4)

with \({\delta }_{{i}_{1}{i}_{2}}={{\rm{\Delta }}}_{{i}_{2}}-{{\rm{\Delta }}}_{{i}_{1}}\), \({S}_{j}^{+}={|{e}_{{i}_{1}}\rangle }_{jj}\langle {e}_{{i}_{2}}|\) and \({S}_{j}^{-}={|{e}_{{i}_{2}}\rangle }_{jj}\langle {e}_{{i}_{1}}|\). By introducing new Bosonic modes (see Method), the effective Hamiltonian is reduced to

$${H}_{eff}=\sum _{j=1}^{2}{\mu }_{{i}_{1}{i}_{2}}{|{e}_{{i}_{1}}\rangle }_{jj}\langle {e}_{{i}_{1}}|-{\chi }_{{i}_{1}{i}_{2}}({S}_{1}^{+}{S}_{2}^{-}+{S}_{1}^{-}{S}_{2}^{+}),$$
(5)

where

$${\mu }_{{i}_{1}{i}_{2}}=\frac{{\lambda }_{{i}_{1}{i}_{2}}^{2}}{4}(\frac{2}{{\delta }_{{i}_{1}{i}_{2}}}+\frac{2}{{\delta }_{{i}_{1}{i}_{2}}-\sqrt{2}\nu }+\frac{2}{{\delta }_{{i}_{1}{i}_{2}}+\sqrt{2}\nu })+{\zeta }_{{i}_{1}}.$$
(6)

Distributed qudit computation

It is well-known that the qubit rotations and two-qubit CNOT gate are universal for synthesizing multi-qubit circuit. In this case, one only needs to construct CNOT gate using the system in Fig. 1. In fact, for two three-level atomic systems, each of them has two ground states |e 1〉, |e 2〉, and one excite state |r〉. Let atomic transition \(|{e}_{2}\rangle \leftrightarrow |r\rangle \) be driven by a classical laser field with Rabi frequency \({\rm{\Omega }}\), while the transition \(|{e}_{1}\rangle \leftrightarrow |r\rangle \) be driven by the cavity mode with coupling constant g. Assume that the classical field and the cavity mode are detuned from respective transition by Δ1 and Δ2. In the interaction picture, the Hamiltonian is simplified as

$${H}_{eff}=-\mu \sum _{j=1}^{2}{|{e}_{1}\rangle }_{jj}\langle {e}_{1}|+\chi ({S}_{2}^{+}{S}_{3}^{-}+{S}_{2}^{-}{S}_{3}^{+})+H{\rm{.}}c\mathrm{.,}$$
(7)

where \({S}_{j}^{+}={|{e}_{1}\rangle }_{jj}\langle {e}_{2}|\) and \({S}_{j}^{-}={|{e}_{2}\rangle }_{jj}\langle {e}_{1}|\). After an evolving time t, it leads to a swapping gate

$${U}_{2}(\mu ,\chi )=(\begin{array}{cccc}{e}^{-i2\mu t} & 0 & 0 & 0\\ 0 & {e}^{-i\mu t}\,\cos (\chi t) & i{e}^{-i\mu t}\,\sin (\chi t) & 0\\ 0 & i{e}^{-i\mu t}\,\sin (\chi t) & {e}^{-i\mu t}\,\cos (\chi t) & 0\\ 0 & 0 & 0 & 1\end{array}).$$
(8)

Moreover, when \(\mu t=\mathrm{(2}k+\mathrm{1)}\pi \) and \(\chi t=\mathrm{(2}s+\mathrm{1/2)}\pi \) for some integers k and s, it reduces to the special SWAP gate

$$iSWAP=(\begin{array}{cccc}1 & 0 & 0 & 0\\ 0 & 0 & -i & 0\\ 0 & -i & 0 & 0\\ 0 & 0 & 0 & 1\end{array}).$$
(9)

This gate may be used to generate CNOT gate, as shown in Fig. 2.

Figure 2
figure 2

The circuit decomposition of the CNOT gate using the iSWAP gate and single qubit gates. H denotes the Hadamard gate. Z denotes the Pauli phase flip gate. R z (θ) denotes the rotation along the z-axis on the Bloch sphere with the angle θ while Ph(θ) denotes the global phase gate with angle θ.

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Qudit case

Now, we consider the qudit-based quantum computation. From previous result55, the set of qudit gates {C 2[X d ], X d } is universal for synthesizing multi-qudit circuits. Here, X d denotes the single qudit operation of R ij (θ) or Z d with the following forms

$$\begin{array}{rcl}{R}_{ij}(\theta ) & = & (\begin{array}{ccccc}{I}_{i-1} & & & & \\ & \cos (\theta ) & \cdots & \sin (\theta ) & \\ & \vdots & \ddots & \vdots & \\ & -\sin (\theta ) & \cdots & \cos (\theta ) & \\ & & & & {I}_{d-j}\end{array}),\\ {Z}_{d}(\theta ) & = & (\begin{array}{ccccc}{e}^{i{\varphi }_{1}} & & & & \\ & {e}^{i{\varphi }_{2}} & & & \\ & & \ddots & & \\ & & & {e}^{i{\varphi }_{d-1}} & \\ & & & & {e}^{i{\varphi }_{d}}\end{array}),\end{array}$$
(10)

and C 2[X d ] denotes the controlled qudit operation defined by

$${C}_{2}[{X}_{d}]=(\begin{array}{cc}{I}_{{d}^{2}-d} & \\ & {X}_{d}\end{array}).$$
(11)

Since the qudit gate X d may be realized assisted by the classical fields54. In the follow, our consideration is to realize C 2[X d ] with the proposed atomic systems in Fig. 1. Firstly, we consider C 2[R ij ] with two d + 1-level atoms. Two cavity modes are coupled to the transition \(|{e}_{i}\rangle \leftrightarrow |r\rangle \) of two atoms with the same detuning Δ1. The transition \(|{e}_{j}\rangle \leftrightarrow |r\rangle \) of two atoms is driven by classical fields with the same coupling coefficient \({\rm{\Omega }}\) and detuning Δ2. In this case, the effective Hamiltonian is simplified as

$${H}_{eff}=-\mu ({|{e}_{i}\rangle }_{11}\langle {e}_{i}|+{|{e}_{i}\rangle }_{22}\langle {e}_{i}|)+\chi ({S}_{2}^{+}{S}_{3}^{-}+{S}_{2}^{-}{S}_{3}^{+})+H{\rm{.}}c\mathrm{.,}$$
(12)

where \({S}_{k}^{+}={\langle {e}_{j}|}_{kk}\langle {e}_{i}|\) and \({S}_{k}^{-}={|{e}_{i}\rangle }_{kk}\langle {e}_{j}|\). After a proper evolving time t (μt = (2k + 1)π and \(\chi t=\mathrm{(2}s+\mathrm{1/2)}\pi \) for some integers k and s), it leads to a special swapping gate as follows:

$$iSWA{P}_{ij}^{d}=-i|{e}_{i}{e}_{j}\rangle \langle {e}_{j}{e}_{i}|-i|{e}_{j}{e}_{i}\rangle \langle {e}_{i}{e}_{j}|+{i}{\rm{.}}{d}{\rm{.}}{t}.,$$
(13)

where i.d.t denotes the identity operation for all the other terms except to |e i e j 〉 and |e j e i 〉 of two atoms. From the circuit in Fig. 3(a), it easily follows that

$${X}_{ij}^{d}=({I}_{d}\otimes {Z}_{d}({\rm{\Theta }}))\times iSWA{P}_{ij}^{d},$$
(14)

where \({X}_{ij}^{d}\) is defined by

$${X}_{ij}^{d}=|{e}_{i}{e}_{i}\rangle \langle {e}_{i}{e}_{j}|+|{e}_{i}{e}_{j}\rangle \langle {e}_{i}{e}_{i}|+i{\rm{.}}d{\rm{.}}t,$$
(15)

and i.d.t denotes the identity operation for all the other terms except to |e i e i 〉 and |e i e j 〉 of two atoms, and \({\rm{\Theta }}=({0}_{i-1},\pi \mathrm{/2},{0}_{j-i-1},\pi \mathrm{/2},{0}_{d-j-1})\) with 0 k being a zero vector of k-dimension.

Figure 3
figure 3

(a) The circuit decomposition of two-qudit gate \({X}_{ij}^{d}\) defined in Eq. (15). (b) The circuit decomposition of two-qudit elementary gate C 2[R ij (θ)]. \({R}_{ij}^{\ast }={R}_{ij}(\pi \mathrm{/2)}{Z}_{d}(-{\rm{\Theta }})\), where \({Z}_{d}(-{\rm{\Theta }})\) is defined in Eq. (16) with \({\rm{\Theta }}=({0}_{d-1},\pi )\).

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The two-qudit gate \({X}_{ij}^{d}\) may be used to realize controlled qudit gate C 2[X d ]. From Fig. 3(b), note that

$$\begin{array}{rcl}{C}_{2}[{R}_{ij}(-\theta )] & = & ({I}_{d}\otimes {R}_{ij}(\frac{\theta }{2}))\times ({R}_{in}(\frac{\pi }{2}){Z}_{d}({\rm{\Theta }})\otimes {I}_{d})\times {X}_{ij}^{d}\\ & & \times \,({R}_{in}(\frac{\pi }{2}){Z}_{d}({\rm{\Theta }})\otimes {I}_{d})\times ({I}_{d}\otimes {R}_{ij}(\frac{-\theta }{2})),\end{array}$$
(16)

where \({\rm{\Theta }}=({0}_{d-1},\pi )\). Now, for an elementary two-qudit gate \({C}_{2}[{Z}_{d}({\rm{\Theta }})]\), from each \({\rm{\Theta }}=({\theta }_{1},{\theta }_{2},\ldots ,{\theta }_{d})\), \({C}_{2}[{Z}_{d}({\rm{\Theta }})]\) may be decomposed into special two-qudit gates as follows

$${C}_{2}[{Z}_{d}({\rm{\Theta }})]=\prod _{(i,j)\in {\mathscr{S}}}{C}_{2}[{Z}_{d}({{\rm{\Theta }}}_{ij})],$$
(17)

where \({{\rm{\Theta }}}_{ij}=({0}_{i-1},\,{\theta }_{i}{\mathrm{,0}}_{j-i-1},\,{\theta }_{j}{\mathrm{,0}}_{d-j-1})\), and \({\mathscr{S}}\) denotes the integer-pair partition of the index set \(\{1,2,\ldots ,d\}\). Now for simplicity, consider the subspace defined by \(\{|{e}_{i}{e}_{i}\rangle ,|{e}_{i}{e}_{j}\rangle ,|{e}_{j}{e}_{i}\rangle ,|{e}_{j}{e}_{j}\rangle \}\) while the other subspace is unchanged for the following evaluations. From the Hamiltonian H eff in Eq. (12), after a proper evolution time t (\(\chi t=2k\pi \)), it follows a two-qudit rotation

$$C{Z}_{d}\mathrm{(2}\varphi ,\varphi ,\varphi ,0):={e}^{-i2\mu t}|{e}_{i}{e}_{i}\rangle \langle {e}_{i}{e}_{i}|+{e}^{-i\mu t}|{e}_{i}{e}_{j}\rangle \langle {e}_{i}{e}_{j}|+{e}^{-i\mu t}|{e}_{j}{e}_{i}\rangle \langle {e}_{j}{e}_{i}|+i{\rm{.}}d{\rm{.}}t\mathrm{.}$$
(18)

with \(\varphi =-\mu t\). From Eqs (10) and (18), it follows that

$$\begin{array}{lll}C{Z}_{d}\mathrm{(2}\varphi ,\varphi ,\pi ,\,\varphi ) & := & ({R}_{jn}(\frac{\pi }{2})\otimes {I}_{d})\times C{Z}_{d}\mathrm{(2}\varphi ,\varphi ,0,\varphi )\\ & & \times \,{C}_{2}[{R}_{ij}(-\frac{\pi }{2})]\times ({R}_{jn}(\frac{\pi }{2})\otimes {I}_{d})\\ & = & {e}^{-i2\mu t}|{e}_{i}{e}_{i}\rangle \langle {e}_{i}{e}_{i}|+{e}^{-i\mu t}|{e}_{i}{e}_{j}\rangle \langle {e}_{i}{e}_{j}|\\ & & -|{e}_{j}{e}_{i}\rangle \langle {e}_{j}{e}_{i}|+{e}^{-i\mu t}|{e}_{j}{e}_{j}\rangle \langle {e}_{j}{e}_{j}|+i{\rm{.}}d{\rm{.}}t\mathrm{.}\end{array}$$
(19)

Similarly, one can get

$$\begin{array}{lll}C{Z}_{d}(\varphi +\pi ,\,2\varphi ,\,\varphi ,\,\mathrm{0)} & := & ({R}_{in}(\frac{\pi }{2})\otimes {I}_{d})\times C{Z}_{d}\mathrm{(2}\varphi ,\varphi ,\pi ,\varphi )\\ & & \times \,{C}_{2}[{R}_{ij}(-\frac{\pi }{2})]\times ({R}_{in}(\frac{\pi }{2})\otimes {I}_{d})\\ & = & {e}^{i(\pi -\mu t)}|{e}_{i}{e}_{i}\rangle \langle {e}_{i}{e}_{i}|+{e}^{-i2\mu t}|{e}_{i}{e}_{j}\rangle \langle {e}_{i}{e}_{j}|\\ & & +\,{e}^{-i\mu t}|{e}_{j}{e}_{i}\rangle \langle {e}_{j}{e}_{i}|+i{\rm{.}}d{\rm{.}}t\mathrm{.}\end{array}$$
(20)

Two phase gates yield to

$$\begin{array}{lll}C{Z}_{d}\mathrm{(3}\varphi +\pi ,3\varphi ,\varphi +\pi ,\varphi ) & := & C{Z}_{d}(\varphi +\pi ,2\varphi ,\varphi ,0)\cdot C{Z}_{d}\mathrm{(2}\varphi ,\varphi ,\pi ,\varphi )\\ & = & {e}^{i(-3\mu t+\pi )}|{e}_{i}{e}_{i}\rangle \langle {e}_{i}{e}_{i}|+{e}^{-i3\mu t}|{e}_{i}{e}_{j}\rangle \langle {e}_{i}{e}_{j}|\\ & & +\,{e}^{i(\pi -\mu t)}|{e}_{j}{e}_{j}\rangle \langle {e}_{j}{e}_{j}|+{e}^{-i\mu t}|{e}_{j}{e}_{i}\rangle \langle {e}_{j}{e}_{i}|+i{\rm{.}}d{\rm{.}}t\mathrm{.}\end{array}$$
(21)

From Eqs (10) and (21), it follows that

$$\begin{array}{lll}C{Z}_{d}\mathrm{(3}\varphi ,3\varphi ,\varphi ,\varphi ) & := & ({R}_{jn}(\frac{\pi }{2})\otimes {I}_{d})\times ({R}_{in}(\frac{\pi }{2})\otimes {I}_{d})\\ & & \times \,C{Z}_{d}\mathrm{(3}\varphi +\pi ,3\varphi ,\varphi +\pi ,\varphi )\times {C}_{2}[{R}_{ij}(\frac{\pi }{2})]\\ & & \times \,({R}_{in}(\frac{\pi }{2})\otimes {I}_{d})\times {C}_{2}[{R}_{ij}(\frac{\pi }{2})]\times ({R}_{in}(\frac{\pi }{2})\otimes {I}_{d})\\ & = & {e}^{-i3\mu t}|{e}_{i}{e}_{i}\rangle \langle {e}_{i}{e}_{i}|+{e}^{-i3\mu t}|{e}_{i}{e}_{j}\rangle \langle {e}_{i}{e}_{j}|\\ & & +{e}^{-i\mu t)}|{e}_{j}{e}_{j}\rangle \langle {e}_{j}{e}_{j}|+{e}^{-i\mu t}|{e}_{j}{e}_{i}\rangle \langle {e}_{j}{e}_{i}|+i{\rm{.}}d{\rm{.}}t\mathrm{.}\end{array}$$
(22)

From Eqs (10) and (22), it follows that

$$\begin{array}{lll}C{Z}_{d}(\varphi ,\varphi ,3\varphi ,3\varphi ) & := & ({Z}_{d}({\rm{\Theta }}){R}_{ij}(\frac{\pi }{4}){R}_{ij}(\frac{\pi }{4})\otimes {I}_{d})\\ & & \times \,C{Z}_{d}\mathrm{(3}\varphi ,3\varphi ,\varphi ,\varphi )\times ({Z}_{d}({\rm{\Theta }}){R}_{ij}(\frac{\pi }{4}){R}_{ij}(\frac{\pi }{4})\otimes {I}_{d})\\ & = & {e}^{-i\mu t}|{e}_{i}{e}_{i}\rangle \langle {e}_{i}{e}_{i}|+{e}^{-i\mu t}|{e}_{i}{e}_{j}\rangle \langle {e}_{i}{e}_{j}|\\ & & +\,{e}^{-i3\mu t)}|{e}_{j}{e}_{j}\rangle \langle {e}_{j}{e}_{j}|+{e}^{-i3\mu t}|{e}_{j}{e}_{i}\rangle \langle {e}_{j}{e}_{i}|+i{\rm{.}}d{\rm{.}}t\mathrm{.}\end{array}$$
(23)

where \({\rm{\Theta }}=({0}_{j-1},\pi ,{0}_{d-j})\). From Eqs (27) and (28), we obtain

$$\begin{array}{lll}C{Z}_{d}\mathrm{(4}\varphi ,4\varphi ,4\varphi ,4\varphi ) & := & C{Z}_{d}(\varphi ,\varphi ,3\varphi ,3\varphi )C{Z}_{d}\mathrm{(3}\varphi ,3\varphi ,\varphi ,\varphi )\\ & = & {e}^{-i4\mu t}|{e}_{i}{e}_{i}\rangle \langle {e}_{i}{e}_{i}|+{e}^{-i4\mu t}|{e}_{i}{e}_{j}\rangle \langle {e}_{i}{e}_{j}|\\ & & +\,{e}^{-i4\mu t)}|{e}_{j}{e}_{j}\rangle \langle {e}_{j}{e}_{j}|+{e}^{-i4\mu t}|{e}_{j}{e}_{i}\rangle \langle {e}_{j}{e}_{i}|+i.d.t.,\end{array}$$
(24)
$$\begin{array}{lll}C{Z}_{d}\mathrm{(2}\varphi ,2\varphi ,-2\varphi ,-2\varphi ) & := & C{Z}_{d}\mathrm{(3}\varphi ,3\varphi ,\varphi ,\varphi )C{Z}_{d}(-\varphi ,-\varphi ,-3\varphi ,-3\varphi )\\ & = & {e}^{-i2\mu t}|{e}_{i}{e}_{i}\rangle \langle {e}_{i}{e}_{i}|+{e}^{-i2\mu t}|{e}_{i}{e}_{j}\rangle \langle {e}_{i}{e}_{j}|\\ & & +\,{e}^{i2\mu t)}|{e}_{j}{e}_{j}\rangle \langle {e}_{j}{e}_{j}|+{e}^{i2\mu t}|{e}_{j}{e}_{i}\rangle \langle {e}_{j}{e}_{i}|+i\mathrm{.}d\mathrm{.}t\mathrm{.},\end{array}$$
(25)

where \(C{Z}_{d}(-\varphi ,\,-\varphi ,\,-3\varphi ,\,-3\varphi )\) may be obtained by letting \(-\varphi =2\pi -\varphi =-\mu t\) for some t. Therefore, Eqs (23) and (25) lead to

$$\begin{array}{lll}C{Z}_{d}\mathrm{(6}\varphi ,6\varphi ,0,\mathrm{0)} & := & C{Z}_{d}\mathrm{(2}\varphi ,2\varphi ,-2\varphi ,-2\varphi )C{Z}_{d}\mathrm{(4}\varphi ,4\varphi ,4\varphi ,4\varphi )\\ & = & {e}^{-i6\mu t}|{e}_{i}{e}_{i}\rangle \langle {e}_{i}{e}_{i}|+{e}^{-i6\mu t}|{e}_{i}{e}_{j}\rangle \langle {e}_{i}{e}_{j}|+i{\rm{.}}d{\rm{.}}t\mathrm{.}\end{array}$$
(26)

Finally, the gate \({C}_{2}[{Z}_{d}({{\rm{\Theta }}}_{ij})]\) may be realized from the decomposition of \(({R}_{in}(\pi \mathrm{/2)}\otimes {I}_{d})\times C{Z}_{ij}\times ({R}_{in}(\pi \mathrm{/2})\otimes {I}_{d})\) for different ϕ.

Effects of spontaneous decay and photon leakage

In this section, we study the influence of atomic spontaneous decay and photon leakage of the cavities and fibers. For convenience, we rewrite the interaction Hamiltonian under the dipole and rotating wave approximation. The master equation for the density matrices of the system is expressed as

$$\begin{array}{rcl}\dot{\rho } & = & -i[H,\rho ]-\sum _{j}\{\frac{{\kappa }_{{f}_{j}}}{2}({b}_{j}^{ {\dagger } }{b}_{j}\rho -2{b}_{j}\rho {b}_{j}^{ {\dagger } }+\rho {b}_{j}^{ {\dagger } }{b}_{j})\\ & & +\,\frac{{\kappa }_{{c}_{j}}}{2}({a}_{j}^{ {\dagger } }{a}_{j}\rho -2{a}_{j}\rho {a}_{j}^{ {\dagger } }+\rho {a}_{j}^{ {\dagger } }{a}_{j})\\ & & +\,\sum _{i=1}^{d}\frac{{\gamma }_{j}^{ri}}{2}({\sigma }_{j}^{rr}\rho -2{\sigma }_{j}^{ir}\rho {\sigma }_{j}^{rx}+\rho {\sigma }_{j}^{rr})\end{array}$$
(27)

where \({\kappa }_{{f}_{j}}\) and \({\kappa }_{{c}_{j}}\) denote the decay rates of the j-th cavity field and the j-th fiber mode, respectively, \({\gamma }_{j}^{rx}\) denotes the spontaneous decay rate of the j-th atom from level |r〉 to |e i 〉, and \({\sigma }_{k}^{js}={|j\rangle }_{kk}\langle s|(j,s=1,\ldots ,d)\) are the usual Pauli matrices. For the convenience, assume that \({\kappa }_{{f}_{j}}={\kappa }_{{c}_{s}}=\kappa \) and \({\gamma }_{j}^{rx}={\gamma }_{a}/d\) due to the equal probability transition of \(|r\rangle \leftrightarrow |{e}_{i}\rangle \). In the follow, we will discuss the parameter conditions and experimental feasibility of the present scheme. With the choice of a scaling g, all the parameters can be reduced to the dimensionless units related to g.

To realize various rotations in Eqs (9) and (15), the rotation parameters χ and μ could achieve various values. In detail, consider the parameters of Δ1 = 4g, Δ2 = 4g + δ, ν = g and \({\rm{\Omega }}=3g\). The rotation parameters χ and μ are shown in Fig. 4(a,b) respectively. It follows that μ may be changed largely while χ is negative. The ratio of μ and χ is changed from −110 to −20 in Fig. 5(a). Moreover, if another set of parameters Δ1 = 9g, Δ2 = 9g + δ, ν = 4g and \({\rm{\Omega }}=3g\) are considered, the rotation parameters χ and μ are shown in Fig. 4(c,d) respectively. In this case, both of them are positive where their ratio is shown in Fig. 5(b).

Figure 4
figure 4

Two phase parameters μ and χ vias relative detuning δ/g and coupling strength g. (a) χ vias δ/g and g. (b) μ vias δ/g and g. Here, Δ1 = 4g, Δ2 = 4g + δ, ν = g, \({\rm{\Omega }}=g\). μ > 0 and χ < 0 are useful for generating negative phases. (c) χ vias δ/g and g. (d) μ vias δ/g and g. Here, Δ1 = 9g, Δ2 = 9g + δ, ν = 4g, \({\rm{\Omega }}=3g\). Here, χ and μ are positive.

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Figure 5
figure 5

μ/χ vias δ/g and g. (a) Δ1 = 4g, Δ2 = 4g + δ, ν = g, \({\rm{\Omega }}=g\); (b) Δ1 = 9g, Δ2 = 9g + δ, ν = 4g, \({\rm{\Omega }}=3g\).

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For the first set of parameters shown in Fig. 4(a), all the adiabatic conditions \({v}_{i}\gg 0\) of v 1 = δ − λ, \({v}_{2}=|\delta -\sqrt{2}\nu |-\lambda \mathrm{/2}\), \({v}_{3}=\sqrt{2}\nu -\lambda \mathrm{/2}\), and \({v}_{4}=\sqrt{2}\nu -\eta \mathrm{/4}\) are approximatively satisfied when g and δ/g are increased, as shown in Fig. 6(a–d). Here, v 2 < 0 should be avoided by choosing proper g and δ. If the second set of parameters shown in Fig. 4(c) are considered, the corresponding adiabatic conditions \({v}_{i}\gg 0\) of are greatly improved and shown in Fig. 6(f–h). Specially, in this case, all the v i  > 0 for all g > 2 and δ/g > 2. It means that the adiabatic conditions may be satisfied under the weak coupling g < 5.

Figure 6
figure 6

The adiabatic conditions vias δ/g and g. (a–d) Denote the first case in Fig. 4. (e–h) Denote the second case in Fig. 4. Here, v 1 = δ − λ, \({v}_{2}=|\delta -\sqrt{2}\nu |-\lambda /2\), \({v}_{3}=\sqrt{2}\nu -\lambda /2\), and \({v}_{4}=\sqrt{2}\nu -\eta /4\).

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In order to complete the quantum applications, proper quantum gates should be realized using special phases ϕ = μt and \(\psi =\chi t\) with proper evolution times. The phases ratio \(\varphi /\psi \) of all the gates including the iSWAP gate and inverse iSWAP gate are shown in Fig. 7(a,b). Combined with Fig. 5(a), these gates may be efficiently realized. Moreover, if another set of parameters Δ1 = 9g, \({{\rm{\Delta }}}_{2}=9g+\delta \), \(\nu =4g\), and \({\rm{\Omega }}=3g\) are considered, the rotation parameters χ and μ are shown in Fig. 4(c,d) respectively. In this case, both of them are positive, and their ratio is shown in Fig. 5(b). The corresponding adiabatic conditions are improved and shown in Fig. 6(e,f). The phases ratio \(\varphi /\psi \) of different gates are shown in Fig. 7(c,d), which mean that the iSWAP gate and inverse iSWAP gate may be realized.

Figure 7
figure 7

The phase ratio \(\varphi /\psi =(\mu t)/(\chi t)\) for the iSWAP gate and the inverse of iSWAP. (a,b) Denote the evolution times using the first set of parameters shown in Fig. 4(a). (c,d) Denote the evolution times using the second set of parameters shown in Fig. 4(c).

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To consider atomic spontaneous emission and the decay of the Bosonic modes, let \({\rm{\Gamma }}=\kappa =\gamma =0.01g\), where Γ, κ, and γ are the decay rates for the atomic excited state, the cavity modes, and the fiber mode, respectively. The probability that the atoms undergo a transition to the excited state due to the off-resonant interaction with the classical fields is \({P}_{1}={\rm{\Gamma }}/{{\rm{\Delta }}}_{1}^{2} < 0.01\) for both cases. Meanwhile, the probability that the three modes c i are excited due to non-resonant coupling with the classical modes is

$${P}_{2}\approx \frac{{\lambda }^{2}}{4}[\frac{2}{{\delta }^{2}}+\frac{1}{{(\delta -\sqrt{2}\nu )}^{2}}+\frac{1}{{(\delta +\sqrt{2}\nu )}^{2}}].$$
(28)

The P 2 is shown in Fig. 8 for two groups of parameters. The effective decoherence rates due to the atomic spontaneous emission and the decay of the Bosonic modes are \({\rm{\Gamma }}^{\prime} ={P}_{1}{\rm{\Gamma }} < {10}^{-4}g\) and \(\kappa ^{\prime} ={P}_{2}\kappa < 0.35\times {10}^{-3}g\), respectively.

Figure 8
figure 8

The probability P 2. (a) The first case; (b) The second case.

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The fidelity of the iSWAP gate is defined by

$${F}_{iSWAP}=\int Tr[\sqrt{\sqrt{{\rho }_{o}}{\rho }_{i}\sqrt{{\rho }_{o}}}$$
(29)

over all possible states, where ρ o denotes the real final density matrix while ρ i denotes the ideal final density matrix. The fidelity of the iSWAP gate is shown in Fig. 9. For the small \(g\approx 5.275\), the fidelity may be reached to 0.982 after the evolution time \(t\approx 19.575\), see Fig. 9(a). For the large \(g\approx 18.4\), the fidelity may be reached to 0.994 after the evolution time \(t\approx 6.375\). The ideal iSWAP gate is achieved after eight Rabi-like oscillations, see Fig. 10. In the regime \(\nu \gg {g}_{i}\) the fidelities of the gates have been consistently found to be essentially unaffected by fiber losses. In general, moreover, the direct effect of spontaneous emission proves to be more relevant than the indirect effect of cavity losses. For the iSWAP gate with \(\nu \approx 4g\) and \(g\approx 5.275\), the maximum fidelity drops to \(F\approx 0.958\) for \(\kappa =\gamma =\beta =0.002g\), see Fig. 11. If large coupling strength \(g\approx 18.4\), the maximum fidelity drops to \(F\approx 0.972\) for \(\kappa =\gamma =\beta =0.002g\). With lower decay rates \(\approx \)0.0002g the iSWAP gate is unaffected, while it may be spoiled if high rate \(\approx \)0.1g is considered. The spontaneous emission rates should be restricted for the fabrication of high-finesse optical cavities in experiment. Hyperfine ground levels of effective high level lambda systems could be candidates for such schemes. Take 87Rb atoms as examples56. Three ground states may be defined by hyperfine atomic levels \(|F=1,m=-1\rangle \), \(|F=1,m=0\rangle \), \(|F=1,m=1\rangle \) of 52 S 1/2, while excited state may be defined by the hyperfine atomic level |F = 1, m = 0〉 of 52 P 1/2. Each atom can be made localized at a fixed position in each cavity with high Q for long time56. Recent experiment57 has achieved the parameters g/2π  \(\approx \) 750 MHz, κ/2π  \(\approx \) 2.62 MHz, and γ/2π  \(\approx \) 3.5 MHz in an ultrahigh-Q toroidal microresonators with the wavelength in the region 630~850 nm is predicatively achievable with the optical fiber decay rate 0.152 MHz58. By setting \({{\rm{\Omega }}}_{1}={{\rm{\Omega }}}_{2}=0.35g\), Δ1 = 2.3g, Δ2 = 2.4g, and ν = 0.8g, we can obtain a iSWAP gate the fidelity about 9.21% with \(\kappa \approx 0.0035g\) and \(\gamma \approx 0.0046g\).

Figure 9
figure 9

The average fidelity of the iSWAP gate vias g and evolution time. \({{\rm{\Delta }}}_{1}=9g\), \({{\rm{\Delta }}}_{2}=9g+\delta \), ν = 4g, \({\rm{\Omega }}=2g\). (a) The diamonds refer to \(g\approx 5.275\), the squares and the circle refer, respectively, to a variation of −0.025 and +0.025 of g. (b) The diamonds refer to \(g\approx 18.4\), the squares and the circle refer, respectively, to a variation of −0.05 and +0.05 of g.

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Figure 10
figure 10

The fidelity of the iSWAP gate vias g and evolution time. \({{\rm{\Delta }}}_{1}=9g\), \({{\rm{\Delta }}}_{2}=9g+\delta \), ν = 4g, \({\rm{\Omega }}=2g\). (a) The diamonds refer to \(g\approx 5.275\), the squares and the circle refer, respectively, to a variation of −0.025 and +0.025 of g. (b) The diamonds refer to \(g\approx 18.4\), the squares and the circle refer, respectively, to a variation of −0.05 and +0.05 of g.

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Figure 11
figure 11

The fidelity of the iSWAP gate vias g and evolution time using master equation. \({{\rm{\Delta }}}_{1}=9g\), \({{\rm{\Delta }}}_{2}=9g+\delta \), ν = 4g, κ = γ = β = 0.002g. \({\rm{\Omega }}=2g\). (a) The diamonds refer to \(g\approx 5.275\), the squares and the circle refer, respectively, to a variation of −0.025 and +0.025 of g. (b) The diamonds refer to \(g\approx 18.4\), the squares and the circle refer, respectively, to a variation of −0.05 and +0.05 of g.

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Conclusion

In conclusion, we have investigated the implementation of high-dimensional quantum computation for atoms trapped in distant cavities coupled by an optical fiber. The chosen ground states of each atom are coupled via the cavity mode and different classical fields in the Raman process. All the atoms do not undergo the real Raman transitions due to the large detuning while the atomic system is decoupled from the cavity modes and fiber modes. In the short fiber regime, reliable elementary gates could be reasonable even if imperfections (atomic spontaneous decay and photon leakage of the cavities and fibers) are considered. Let us also mention that, in the considered system, not only entangling and swap gates, but also perfect quantum state transfer is possible. Moreover, the proposed setup would also allow for entanglement preparation schemes between distributed atoms, and could useful in one-way quantum computation. These schemes would be useful for constructing large-scale and long-distance quantum computation or quantum communication networks.

Method

By introducing new Bosonic modes \({c}_{1}=\frac{1}{\sqrt{2}}({a}_{1}-{a}_{2})\), \({c}_{2}=\frac{1}{2}({a}_{1}+{a}_{2}+\sqrt{2}b)\) and \({c}_{3}=\frac{1}{2}({a}_{1}+{a}_{2}-\sqrt{2}b)\), the Hamiltonians H 0 may be rewritten as \({H}_{0}=\frac{\nu }{\sqrt{2}}({c}_{2}^{ {\dagger } }{c}_{2}-{c}_{3}^{ {\dagger } }{c}_{3})\). Take H 0 as the free Hamiltonian and perform the unitary transformation \(U={e}^{i{H}_{0}t}\), it follows an efficient interaction Humiliation

$$\begin{array}{rcl}{H}_{{int}}^{^{\prime} } & = & \frac{{\zeta }_{{i}_{2}}}{4}\mathrm{(2}{c}_{1}^{ {\dagger } }{c}_{1}+{c}_{2}^{ {\dagger } }{c}_{2})\sum _{j=1}^{2}{|{e}_{{i}_{2}}\rangle }_{jj}\langle {e}_{{i}_{2}}|-\sum _{j=1}^{2}{\eta }_{{i}_{1}}{|{e}_{{i}_{1}}\rangle }_{jj}\langle {e}_{{i}_{1}}|\\ & & -\,\frac{1}{4}\{2{\lambda }_{{i}_{1}{i}_{2}}[\sqrt{2}{e}^{i{\delta }_{{i}_{1}{i}_{2}}t}{c}_{1}+{e}^{i({\delta }_{{i}_{1}{i}_{2}}-\sqrt{2}\nu )t}{c}_{2}+{e}^{i({\delta }_{{i}_{1}{i}_{2}}+\sqrt{2}\nu )t}{c}_{3}]{S}_{1}^{+}\\ & & +\,2{\lambda }_{{i}_{1}{i}_{2}}[\sqrt{2}{e}^{i{\delta }_{{i}_{1}{i}_{2}}t}{c}_{1}-{e}^{i({\delta }_{{i}_{1}{i}_{2}}-\sqrt{2}\nu )t}{c}_{2}-{e}^{i({\delta }_{{i}_{1}{i}_{2}}+\sqrt{2}\nu )t}{c}_{3}]{S}_{2}^{+}\\ & & +\,{\zeta }_{{i}_{1}{i}_{2}}[\sqrt{2}{e}^{i\sqrt{2}\nu t}{c}_{2}^{ {\dagger } }{c}_{1}+{e}^{i\sqrt{2}\nu t}{c}_{2}^{ {\dagger } }{c}_{3}+\sqrt{2}{e}^{-i\sqrt{2}\nu t}{c}_{3}^{ {\dagger } }{c}_{1}]{|{e}_{{i}_{2}}\rangle }_{11}\langle {e}_{{i}_{2}}|\\ & & +\,{\zeta }_{{i}_{1}{i}_{2}}[-\sqrt{2}{e}^{i\sqrt{2}\nu t}{c}_{2}^{ {\dagger } }{c}_{1}+{e}^{i\sqrt{2}\nu t}{c}_{2}^{ {\dagger } }{c}_{3}-\sqrt{2}{e}^{-i\sqrt{2}\nu t}{c}_{3}^{ {\dagger } }{c}_{1}]{|{e}_{{i}_{2}}\rangle }_{22}\langle {e}_{{i}_{2}}|\}+H{\rm{.}}c\mathrm{.}\end{array}$$
(30)

The Hamiltonian describes multiple off-resonant Raman couplings for each atom induced by the classical field and the Bosonic modes c 1, c 2, c 3. If \({\delta }_{{i}_{1}{i}_{2}}\gg {\lambda }_{{i}_{1}{i}_{2}}\), \(|{\delta }_{{i}_{1}{i}_{2}}\pm \sqrt{2}\nu |\gg {\lambda }_{{i}_{1}{i}_{2}}/2\) and \(\sqrt{2}\nu \gg {\lambda }_{{i}_{1}{i}_{2}}/2\), \({\eta }_{{i}_{1}}/4\), the Bosonic modes do not exchange quantum numbers with the atomic system. The off-resonant Raman coupling leads a Stark shift between the atoms.

Thus the effective Hamiltonian is defined by

$$\begin{array}{rcl}{H}_{eff} & = & \frac{{\zeta }_{{i}_{2}}}{4}\mathrm{(2}{c}_{1}^{ {\dagger } }{c}_{1}+{c}_{2}^{ {\dagger } }{c}_{2})\sum _{j=1}^{2}{|{e}_{{i}_{2}}\rangle }_{jj}\langle {e}_{{i}_{2}}|-\sum _{j=1}^{2}{\eta }_{{i}_{1}}{|{e}_{{i}_{1}}\rangle }_{jj}\langle {e}_{{i}_{1}}|\\ & & -\sum _{j=1}^{2}\frac{{\lambda }_{{i}_{1}{i}_{2}}^{2}}{4}\{[\frac{2}{{\delta }_{{i}_{1}{i}_{2}}}{c}_{1}{c}_{1}^{ {\dagger } }+\frac{1}{{\delta }_{{i}_{1}{i}_{2}}-\sqrt{2}\nu }{c}_{2}{c}_{2}^{ {\dagger } }+\frac{1}{{\delta }_{{i}_{1}{i}_{2}}+\sqrt{2}\nu }{c}_{3}{c}_{3}^{ {\dagger } }]{|{e}_{{i}_{1}}\rangle }_{jj}\langle {e}_{{i}_{1}}|\\ & & +\,[\frac{2}{{\delta }_{{i}_{1}{i}_{2}}}{c}_{1}^{ {\dagger } }{c}_{1}-\frac{1}{{\delta }_{{i}_{1}{i}_{2}}-\sqrt{2}\nu }{c}_{2}^{ {\dagger } }{c}_{2}-\frac{1}{{\delta }_{{i}_{1}{i}_{2}}+\sqrt{2}\nu }{c}_{3}^{ {\dagger } }{c}_{3}]{|{e}_{{i}_{2}}\rangle }_{jj}\langle {e}_{{i}_{2}}|\\ & & +\,\frac{{\zeta }_{{i}_{1}{i}_{2}}^{2}}{32\sqrt{2}\mu }({c}_{2}^{ {\dagger } }{c}_{2}-{c}_{3}^{ {\dagger } }{c}_{3})[{({|{e}_{{i}_{1}}\rangle }_{11}\langle {e}_{{i}_{1}}|+{|{e}_{{i}_{1}}\rangle }_{22}\langle {e}_{{i}_{1}}|)}^{2}\\ & & +\,4{({|{e}_{{i}_{1}}\rangle }_{11}\langle {e}_{{i}_{1}}|-{|{e}_{{i}_{1}}\rangle }_{22}\langle {e}_{{i}_{1}}|)}^{2}]-{\chi }_{{i}_{1}{i}_{2}}({S}_{1}^{+}{S}_{2}^{-}+{S}_{1}^{-}{S}_{2}^{+})\end{array}$$
(31)

where

$${\chi }_{{i}_{1}{i}_{2}}=\frac{{\lambda }_{{i}_{1}{i}_{2}}^{2}}{4}[\frac{2}{{\delta }_{{i}_{1}{i}_{2}}}-\frac{2}{{\delta }_{{i}_{1}{i}_{2}}-\sqrt{2}\nu }-\frac{2}{{\delta }_{{i}_{1}{i}_{2}}+\sqrt{2}\nu }]$$
(32)

Since \(2{c}_{1}^{ {\dagger } }{c}_{1}\), \({c}_{2}^{ {\dagger } }{c}_{2}\), \({c}_{3}^{ {\dagger } }{c}_{3}\) commute with the Hamiltonian H eff , the bosonic modes are unchanged if the vacuum states are applied.