Transferring quantum entangled states of photonic qubits via a quantum switch

Abstract

The controlled transfer of quantum entangled states is a fascinating topic in quantum physics, and has potential applications in fields such as quantum network communication, distributed quantum information processing, and development of quantum functional devices. This study focuses on a circuit quantum electrodynamics (QED) system, comprising an input port, two output ports, and a quantum switch. The input port or each output port contains n microwave cavities. The switch is a superconducting transmon qutrit (three-level quantum system). The vacuum and single-photon states in each cavity encode a photonic qubit. We demonstrate that the single-excitation symmetric (SES) entangled state of n photonic qubits can be transferred from the n cavities at the input port to the n cavities at either of the two output ports, controlled by the switch. Remarkably, the operational time decreases as the number n increases. The only use of qutrit-cavity resonant interaction enables rapid quantum state transfer while effectively suppressing system decoherence. This proposal can be used to achieve the controlled transfer of both Bell states and W states of photonic qubits via a quantum switch. To validate the experimental feasibility, we numerically simulate two scenarios: (i) transferring the Bell state of two photonic qubits, and (ii) transferring the W state of three photonic qubits. This proposal may be extended to accomplish the same task in other physical systems, where the switch is implemented with a three-level artificial atom of different types while each cavity is a microwave or optical cavity.

Graphical abstract

Keywords

state transfer / photonic qubit / quantum switch / W state / circuit QED

1 Introduction and motivation

The new and rapidly growing field of circuit quantum electrodynamics (QED), consisting of superconducting (SC) artificial atoms and microwave cavities or resonators, offers extremely exciting prospects for quantum information processing (QIP) and quantum simulations on a chip [1]. In circuit-QED systems, SC artificial atoms can interact with microwave radiation fields in a more freely controlled and engineered manner [2–8]. Experiments have achieved the strong-coupling or ultrastrong-coupling regime of a SC artificial atom with a microwave cavity (or resonator) [9–11]. SC artificial atoms can be fabricated using modern integrated circuit technology, their properties can be characterized and adjusted in situ, and their coherence time can reach from hundreds of microseconds [12–14] to even over 1 ms [15, 16]. Additionally, microwave cavities or resonators host microwave photons with lifetimes comparable to or longer than that of SC artificial atoms [17–19]. Therefore, both SC artificial atoms and microwave cavities have been utilized as the basic information processing units for implementing QIP. The rapid development and continuous progress have demonstrated the great potential of microwave superconducting circuit QED systems as multi-functional hardware architectures for the realization of QIP devices (for reviews, see Refs. [1, 5, 20–23]).

Transferring quantum states from one location to different locations is of fundamental interest in quantum physics, and has potential applications in fields such as networked quantum communication, distributed QIP, and development of quantum functional devices. A number of theoretical works have been devoted to realizing quantum state transfer between SC qubits (two-level quantum systems) [24–36]. Experimentally, much progress has been achieved in transferring quantum states between SC qubits. For instance, experiments have demonstrated: (i) the on-chip transfer of quantum states between two SC qubits coupled through a cavity [37, 38], a tunable coupler [39], or a tunable SC qubit [40], (ii) quantum state transfer between two SC qubits in a 4-qubit chain [39–43] or a six-qubit chain [44], (iii) the microwave-photon-mediated long-distance quantum state transfer between two SC qubits in different modules [45] or at different nodes in a network [46–49], and (iv) the phonon-mediated remote quantum state transfer between two SC qubits at different nodes of a network [50]. Furthermore, experiments have demonstrated the quantum state transfer between two SC qubits on a superconducting chip by utilizing standard teleportation protocols [51, 52]. More recently, deterministic quantum state teleportation between two SC qubits located at distant superconducting chips has been experimentally implemented [53]. Notably, the realization of quantum state transfer has also been investigated in other types of physical systems (e.g., see [54–58] and references therein).

On the other hand, there has been great interest in transferring quantum states between microwave cavities or bosonic systems. Numerous theoretical proposals have been put forward for realizing quantum state transfer between microwave cavities [59–64] or bosonic systems [65–67]. Moreover, experimental advances have been achieved in this fascinating field. For example, experiments have demonstrated the transfer of Fock and coherent states between remote microwave cavities or resonators connected by a transmission line [68, 69], the two-way transfer of single photons between two chips linked by a coaxial cable [70], and the transfer of quantum states between superconducting cavities without exchange interactions [71]. Additionally, experiments have implemented quantum state transfer of a photonic qubit, encoded using the vacuum and single-photon states, between two remote cavities linked by a coaxial cable [72].

In this work we consider a circuit-QED system, which consists of an input port, two output ports, and a quantum switch [Fig. 1(a)]. The input port and each output port contain n microwave cavities, respectively [Fig. 1(a)]. The switch is a transmon qutrit (three-level quantum system) [Fig. 1(a)]. For each cavity, its vacuum and single-photon states are utilized to encode a photonic qubit. We show that the single-excitation symmetric (SES) entangled state of n photonic qubits can be transferred from the n cavities at the input port to the n cavities at either of the two output ports, controlled by the switch.

As shown below, this proposal has these distinguishing features and advantages. First, the state transfer is simple because it requires only a four-step operation. Second, the operational time decreases as the number n of cavities (in the input port or each output port) increases. Third, since only qutrit-cavity resonant interactions are employed, the state transfer can be fast performed and thus decoherence from the system is greatly reduced. Last, since the SES entangled state is a W-type entangled state for n\ge 3 and a Bell entangled state for n=2, this proposal can be used to transfer both W states and Bell states of photonic qubits via the control of a quantum switch.

As an example, we numerically analyze the experimental feasibility in two scenarios: transferring the Bell state of two photonic qubits from two cavities at the input port to two cavities at either of the two output ports, and transferring the W state of three photonic qubits from three cavities at the input port to three cavities at either of the two output ports, via the control of a quantum switch. This proposal may be extended to accomplish the same task in other physical systems, where the switch is implemented with a three-level artificial atom (e.g., a quantum dot, an NV center, a magnon, and a superconducting qutrit of different types), while each cavity is a microwave or optical cavity.

The novelties of our proposal can be highlighted as follows. (i) In contrast to previous works [24–74], where the quantum states are transferred from the input ports to fixed output ports, our work introduces a quantum-switch-controlled transfer of entangled states across multiple photonic qubits. This enables controlled routing for a superconducting quantum network. (ii) Different from the controlled transfers of a single-qubit state [75–78], the present proposal enables direct controlled transfer of multi-qubit entangled states. (iii) In the present study, the operational time decreases with increasing qubit number, which increases the scalability advantage. The above-mentioned advantages make our proposal particularly suitable to transfer multi-qubit entangled states in an on-chip quantum networks where controlled state distribution is essential.

The rest of this paper is organized as follows. In Section 2, we will explicitly show how to realize the controlled transfer of the SES entangled state of n photonic qubits via a quantum switch. In Section 3, we will discuss the experimental feasibility of transferring the Bell state of two photonic qubits and the W state of three photonic qubits via the control of the switch. A concluding summary is presented in Section 4.

2 Implementing entangled state transfer via a quantum switch

Consider a circuit-QED system, which consists of one input port, two output ports (1,2), and a quantum switch [Fig. 1(a)]. The input port contains n identical microwave cavities (a_1,a_2,\cdots ,a_n) with frequency {\omega }_a, the output port 1 contains n microwave cavities (b_1,b_2,\cdots ,b_n) with frequency {\omega }_b, the output port 2 contains n microwave cavities (c_1,c_2,\cdots ,c_n) with frequency {\omega }_c, and the quantum switch comprises a SC transmon qutrit. For the transmon qutrit, the three levels are labelled as |g⟩, |e⟩, and |f⟩ [Fig. 1(b)].

Assume that, initially, the qutrit is in an arbitrary state \alpha |g⟩+\beta |e⟩ (with |\alpha {|}^2+|\beta {|}^2=1), each cavity at the output ports is in the vacuum state |0⟩, and the n cavities (a_1,a_2,\cdots ,a_n) at the input port are in the following SES entangled state

(1)\begin{array}{rl}|SES{⟩}_{a_1{a}_2\cdots a_n}=& \frac{1}{\sqrt{n}}(|00\cdots 001{⟩}_{a_1{a}_2\cdots a_n}\\ & +|00\cdots 010{⟩}_{a_1{a}_2\cdots a_n}+\cdots \\ & +|10\cdots 000{⟩}_{a_1{a}_2\cdots a_n}),\end{array}

where |0{⟩}_{a_j} and |1{⟩}_{a_j} are, respectively, the vacuum state and the single-photon state of cavity a_j (j=1,2,\cdots ,n). One can see that the SES entangled state in Eq. (1) is a complete-symmetry entangled state with a single-photon excitation in the n input cavities, and the single photon being occupied in each cavity with an equal probability. The initial state of the whole system is thus given by

(2)|\psi \left(0\right)⟩=|SES{⟩}_{a_1{a}_2\cdots a_n}\left(\alpha |g{⟩}_q+\beta |e{⟩}_q\right){\displaystyle \prod _{j=1}^n|0{⟩}_{b_j}{\displaystyle \prod _{j=1}^n|0{⟩}_{c_j},}}

where |0{⟩}_{b_j} (|0{⟩}_{c_j}) is the vacuum state of cavity b_j (c_j) with j=1,2,\cdots ,n, and the subscript q represents the qutrit as mentioned above.

Suppose that the qutrit is initially decoupled from the cavity system [Fig. 2(a)]. The procedure for transferring the SES entangled state from the n cavities (a_1,a_2,\cdots ,a_n) to the n cavities (b_1,b_2,\cdots ,b_n) or the n cavities (c_1,c_2,\cdots ,c_n) via the control of the switch is as follows:

Step 1: Adjust the level spacings of the qutrit to have its |e⟩\leftrightarrow |f⟩ transition resonant with the n cavities (a_1,a_2,\cdots ,a_n) [Fig. 2(b)]. In the interaction picture and after applying the rotating-wave approximation (RWA), the Hamiltonian is given by (assuming \hslash =1)

(3)H_1=g_{1,a}{\displaystyle \sum _{j=1}^n{\hat{a}}_j|f{⟩}_q⟨e|+\mathrm{h}.\mathrm{c}.,}

where {\hat{a}}_j is the photon annihilation operator of cavity a_j (j=1,2,\cdots ,n), and g_{1,a} is the coupling constant between each of the cavities (a_1,a_2,\cdots ,a_n) and the |e⟩\leftrightarrow |f⟩ transition of the qutrit. In circuit QED, the cavity-qutrit coupling reaches strong or ultra-strong regimes. The physical unit of the coupling constant is usually MHz. Here and below, the “h.c.” means “Hermitian conjugate terms”. It is straightforward to show that under the Hamiltonian (3), one has the following state transformation

(4)\begin{array}{rl}& |SES{⟩}_{a_1{a}_2\cdots a_n}|e{⟩}_q\to \cos \left(\sqrt{n}{g}_{1,a}t\right)|SES{⟩}_{a_1{a}_2\cdots a_n}|e{⟩}_q\\ & -\mathrm{i}\sin \left(\sqrt{n}{g}_{1,a}t\right)\prod _{j=1}^n|0{⟩}_{a_j}|f{⟩}_q.\end{array}

When the interaction time equals to t_1=\pi /\left(2\sqrt{n}{g}_{1,a}\right), the state transformation (4) becomes

(5)|SES{⟩}_{a_1{a}_2\cdots a_n}|e{⟩}_q\to -\mathrm{i}{\displaystyle \prod _{j=1}^n|0{⟩}_{a_j}|f{⟩}_q.}

Based on Eq. (5), the initial state (2) of the whole system becomes

(6)\left(\alpha |SES{⟩}_{a_1{a}_2\cdots a_n}|g{⟩}_q-\mathrm{i}\beta {\displaystyle \prod _{j=1}^n|0{⟩}_{a_j}|f{⟩}_q}\right){\displaystyle \prod _{j=1}^n|0{⟩}_{b_j}{\displaystyle \prod _{j=1}^n|0{⟩}_{c_j}.}}

Step 2: Adjust the level spacings of the qutrit to have its |g⟩\leftrightarrow |e⟩ transition resonant with the n cavities (a_1,a_2,\cdots ,a_n) [Fig. 2(c)]. In the interaction picture and after the RWA, the Hamiltonian is given by

(7)H_2=g_{2,a}{\displaystyle \sum _{j=1}^n{\hat{a}}_j|e{⟩}_q⟨g|+\mathrm{h}.\mathrm{c}.,}

where g_{2,a} is the coupling constant between each of the cavities (a_1,a_2,\cdots ,a_n) and the |g⟩\leftrightarrow |e⟩ transition of the qutrit. Under the Hamiltonian (7), one has the following state transformation

(8)\begin{array}{rl}& |SES{⟩}_{a_1{a}_2\cdots a_n}|g{⟩}_q\to \cos \left(\sqrt{n}{g}_{2,a}t\right)|SES{⟩}_{a_1{a}_2\cdots a_n}|g{⟩}_q\\ & -\mathrm{i}\sin \left(\sqrt{n}{g}_{2,a}t\right)\prod _{j=1}^n|0{⟩}_{a_j}|e{⟩}_q.\end{array}

When the interaction time equals to t_2=\pi /\left(2\sqrt{n}{g}_{2,a}\right), the state transformation (8) becomes

(9)|SES{⟩}_{a_1{a}_2\cdots a_n}|g{⟩}_q\to -\mathrm{i}{\displaystyle \prod _{j=1}^n|0{⟩}_{a_j}|e{⟩}_q.}

Based on Eq. (9), the state (6) of the whole system becomes

(10){\displaystyle \prod _{j=1}^n|0{⟩}_{a_j}\left(\alpha |e{⟩}_q+\beta |f{⟩}_q\right){\displaystyle \prod _{j=1}^n|0{⟩}_{b_j}{\displaystyle \prod _{j=1}^n|0{⟩}_{c_j},}}}

where a common phase factor “-i” has dropped off.

Step 3: Adjust the level spacings of the qutrit to have its |g⟩\leftrightarrow |e⟩ transition resonant with the n cavities (b_1,b_2,\cdots ,b_n) [Fig. 2(d)]. In the interaction picture and under the RWA, the Hamiltonian is given by

(11)H_3=g_b{\displaystyle \sum _{j=1}^n{\hat{b}}_j|e{⟩}_{q_1}⟨g|+\mathrm{h}.\mathrm{c}.,}

where g_b is the coupling constant between the |g⟩\leftrightarrow |e⟩ transition of the qutrit and each of the n cavities (b_1,b_2,\cdots ,b_n).

When the qutrit interacts with the cavities (b_1,b_2,\cdots ,b_n) for a time t, one has the following state transformation under the Hamiltonian (11)

(12)\begin{array}{rl}& |e{⟩}_q\prod _{j=1}^n|0{⟩}_{b_j}\to \cos \left(\sqrt{n}{g}_{b}t\right)|e{⟩}_q\prod _{j=1}^n|0{⟩}_{b_j}\\ & -\mathrm{i}\sin \left(\sqrt{n}{g}_{b}t\right)|g{⟩}_q|SES{⟩}_{b_1{b}_2\cdots b_n},\end{array}

where |SES{⟩}_{b_1{b}_2\cdots b_n} is the SES entangled state of the cavities (b_1,b_2,\cdots ,b_n), which takes the same form as Eq. (1) with the subscripts a_1{a}_2\cdots a_n replaced by b_1{b}_2\cdots b_n.

From Eq. (12), it can be seen that when t_3=\pi /\left(2\sqrt{n}{g}_b\right), one has

(13)|e{⟩}_q{\displaystyle \prod _{j=1}^n|0{⟩}_{b_j}\to -\mathrm{i}|g{⟩}_q|SES{⟩}_{b_1{b}_2\cdots b_n}.}

Based on Eq. (13), the state (10) becomes

(14){\displaystyle \prod _{j=1}^n|0{⟩}_{a_j}\left(-\mathrm{i}\alpha |g{⟩}_q|SES{⟩}_{b_1{b}_2\cdots b_n}+\beta |f{⟩}_q{\displaystyle \prod _{j=1}^n|0{⟩}_{b_j}}\right){\displaystyle \prod _{j=1}^n|0{⟩}_{c_j}.}}

Step 4: Adjust the level spacings of the qutrit to have its |e⟩\leftrightarrow |f⟩ transition resonant with the n cavities (c_1,c_2,\cdots ,c_n) [Fig. 2(e)]. In the interaction picture and under the RWA, the Hamiltonian is given by

(15)H_4=g_c{\displaystyle \sum _{j=1}^n{\hat{c}}_j|f{⟩}_q⟨e|+\mathrm{h}.\mathrm{c}.,}

where g_c is the coupling constant between the |e⟩\leftrightarrow |f⟩ transition of the qutrit and each of the n cavities (c_1,c_2,\cdots ,c_n).

When the qutrit interacts with the cavities (c_1,c_2,\cdots ,c_n) for a time t, one has the following state transformation under the Hamiltonian (15)

(16)\begin{array}{rl}& |f{⟩}_q\prod _{j=1}^n|0{⟩}_{c_j}\to \cos \left(\sqrt{n}{g}_{c}t\right)|f{⟩}_q\prod _{j=1}^n|0{⟩}_{c_j}\\ & -\mathrm{i}\sin \left(\sqrt{n}{g}_{c}t\right)|e{⟩}_q|SES{⟩}_{c_1{c}_2\cdots c_n},\end{array}

where |SES{⟩}_{c_1{c}_2\cdots c_n} is the SES entangled state of the cavities (c_1,c_2,\cdots ,c_n), which takes the same form as Eq. (1) with the subscripts a_1{a}_2\cdots a_n replaced by c_1{c}_2\cdots c_n.

From Eq. (16), it can be seen that when t_4=\pi /\left(2\sqrt{n}{g}_c\right), one has

(17)|f{⟩}_q{\displaystyle \prod _{j=1}^n|0{⟩}_{c_j}\to -\mathrm{i}|e{⟩}_q|SES{⟩}_{c_1{c}_2\cdots c_n}.}

Based on Eq. (17), the state (14) becomes

(18)\begin{array}{rl}& {\displaystyle \prod _{j=1}^n|0{⟩}_{a_j}(\alpha |g{⟩}_q|SES{⟩}_{b_1{b}_2\cdots b_n}{\displaystyle \prod _{j=1}^n|0{⟩}_{c_j}}}\\ & +\beta |e{⟩}_q{\displaystyle \prod _{j=1}^n|0{⟩}_{b_j}|SES{⟩}_{c_1{c}_2\cdots c_n}),}\end{array}

where a common phase factor “-i” has dropped off. After this step of operation, one needs to adjust the level spacings of the qutrit back to its original configuration [Fig. 2(a)], such that the qutrit is decoupled from the cavity system.

Suppose that the two logic states of a photonic qubit are represented by the vacuum state |0⟩ and the single photon state |1⟩ of a cavity. Comparing Eq. (2) with Eq. (18), one can see that the SES entangled state (1) of n photonic qubits initially stored in the n cavities \left(a_1,a_2,\cdots ,a_n\right) at the input port is transferred onto: (i) the n cavities (b_1,b_2,\cdots ,b_n) at the output port 1 when the qutrit (the switch) is in the state |g⟩, (ii) the n cavities (c_1,c_2,\cdots ,c_n) at the output port 2 when the qutrit (the switch) is in the state |e⟩. This indicates that when the switch (the qutrit) is initially in a different state, the SES entangled state (1) of n photonic qubits is deterministically transferred from the n cavities at the input port to the n cavities at a different output port.

From the above description, one can find that:

(i) The total operational time for the entire state transfer is

(19)\begin{array}{rl}{t}_{\mathrm{o}\mathrm{p}}=& t_1+t_2+t_3+t_4+2{\tau }_d\\ =& \pi /\left(2\sqrt{n}{g}_{1,a}\right)+\pi /\left(2\sqrt{n}{g}_{2,a}\right)+\pi /\left(2\sqrt{n}{g}_b\right)\\ & +\pi /\left(2\sqrt{n}{g}_c\right)+2{\tau }_d,\end{array}

where {\tau }_d is the typical time required for adjusting the level spacings of the qutrit. From Eq. (19), one can see that the total operational time for the state transfer decreases as the number n of cavities (at the input port or each output port) increases.

(ii) Only qutrit-cavity resonant interactions are utilized. Hence, the state transfer can be fast performed and thus decoherence from the system is greatly suppressed.

(iii) The coupling or decoupling of the qutrit with the cavities is realized by adjusting the level spacings of the qutrit. For a superconducting device, its level spacings can be rapidly (within 1–3 ns) adjusted by varying external control parameters (e.g., magnetic flux applied to the superconducting loop of a SC phase, transmon, Xmon, or flux qubit or qutrit) [79–81]. Alternatively, the coupling or decoupling of the qutrit with the cavities can be obtained by adjusting the frequency of the cavities. For superconducting microwave cavities, their frequencies can be fast (within a few nanoseconds) tuned in experiments [82, 83].

In the above, we have considered a circuit-QED system, where the SC transmon qutrit (a three-level artificial atom) is coupled to the microwave cavities. The SES entangled state transfer is realized using the Hamiltonians (3), (7), (11) and (15). Notably, the three-level structure in Fig. 1(b) is available in an artificial atom with different types (e.g., a quantum dot, an NV center, a magnon, a SC qutrit of other types). By employing a three-level artificial atom to couple microwave or optical cavities, the same Hamiltonians (3), (7), (11) and (15) can in principle be obtained. Therefore, this proposal may be extended to realize the controlled transfer of the SES entangled state in other physical systems, where a three-level artificial atom couples to multiple microwave or optical cavities at the input port and the two output ports.

3 Experimental feasibility

As examples, we will investigate the experimental possibility of transferring the Bell state of two photonic qubits and the W state of three photonic qubits via a quantum switch. The setup for transferring the Bell state is shown in Fig. 3(a), which consists of two cavities (a_1,a_2) at the input port, two cavities (b_1,b_2) at the output port 1, two cavities (c_1,c_2) at the output port 2, and a quantum switch composed of a transmon qutrit q. Whereas, the setup for transferring the W state is shown in Fig. 3(b), which consists of three cavities (a_1,a_2,a_3) at the input port, three cavities (b_1,b_2,b_3) at the output port 1, three cavities (c_1,c_2,c_3) at the output port 2, and a quantum switch composed of a transmon qutrit q. In Figs. 3(a) and (b), each cavity is a one-dimensional (1D) coplanar waveguide cavity or resonator.

3.1 Ideal evolution states and initial states

The state transfer introduced in the previous Section 2 is realized in an ideal situation, i.e., the situation without considering the system dissipation, the unwanted couplings and the inter-cavity crosstalk. For the Bell state transfer, the ideal evolution state is the state given in Eq. (18) with n=2, i.e.,

(20)\begin{array}{rl}|{\psi }_{\mathrm{i}\mathrm{d}}⟩=& |00{⟩}_{a_1{a}_2}\otimes (\alpha |g{⟩}_q|Bell{⟩}_{b_1{b}_2}|00{⟩}_{c_1{c}_2}\\ & +\beta |e{⟩}_q|00{⟩}_{b_1{b}_2}|Bell{⟩}_{c_1{c}_2}),\end{array}

where |Bell{⟩}_{b_1{b}_2} \left(|Bell{⟩}_{c_1{c}_2}\right) is the Bell state of two cavities (b_1,b_2) \left[(c_1,c_2)\right], i.e., the SES entangled state given in Eq. (1) for n=2, which is given by

(21)\begin{array}{rl}|Bell{⟩}_{b_1{b}_2}& =\frac{1}{\sqrt{2}}\left(|01{⟩}_{b_1{b}_2}+|10{⟩}_{b_1{b}_2}\right),\\ |Bell{⟩}_{c_1{c}_2}& =\frac{1}{\sqrt{2}}\left(|01{⟩}_{c_1{c}_2}+|10{⟩}_{c_1{c}_2}\right).\end{array}

According to Eq. (2), the initial state of the whole system for the Bell state transfer is

(22)\begin{array}{rl}|\psi \left(0\right)⟩=& |Bell{⟩}_{a_1{a}_2}\otimes \left(\alpha |g{⟩}_q+\beta |e{⟩}_q\right)\\ & \otimes |00{⟩}_{b_1{b}_2}|00{⟩}_{c_1{c}_2},\end{array}

where the Bell state for the two input cavities (a_1,a_2) takes the same form as the Bell state given in Eq. (21).

On the other hand, for the W state transfer, the ideal output state is the state given in Eq. (18) with n=3, i.e.,

(23)\begin{array}{rl}|{\psi }_{\mathrm{i}\mathrm{d}}⟩=& |000{⟩}_{a_1{a}_2{a}_3}\otimes (\alpha |g{⟩}_q|W{⟩}_{b_1{b}_2{b}_3}|000{⟩}_{c_1{c}_2{c}_3}\\ & +\beta |e{⟩}_q|000{⟩}_{b_1{b}_2{b}_3}|W{⟩}_{c_1{c}_2{c}_3}),\end{array}

where |W{⟩}_{b_1{b}_2{b}_3} \left(|W{⟩}_{c_1{c}_2{c}_3}\right) is the W state of three cavities (b_1,b_2,b_3) \left[(c_1,c_2,c_3)\right], i.e., the SES entangled state given in Eq. (1) for n=3, which is given by

(24)\begin{array}{rl}|W{⟩}_{b_1{b}_2{b}_3}& =\frac{1}{\sqrt{3}}\left(|001{⟩}_{b_1{b}_2{b}_3}+|010{⟩}_{b_1{b}_2{b}_3}+|100{⟩}_{b_1{b}_2{b}_3}\right),\\ |W{⟩}_{c_1{c}_2{c}_3}& =\frac{1}{\sqrt{3}}\left(|001{⟩}_{c_1{c}_2{c}_3}+|010{⟩}_{c_1{c}_2{c}_3}+|100{⟩}_{c_1{c}_2{c}_3}\right).\end{array}

According to Eq. (2), the initial state of the whole system for the W state transfer is

(25)\begin{array}{rl}|\psi \left(0\right)⟩=& |W{⟩}_{a_1{a}_2{a}_3}\otimes \left(\alpha |g{⟩}_q+\beta |e{⟩}_q\right)\\ & \otimes |000{⟩}_{b_1{b}_2{b}_3}|000{⟩}_{c_1{c}_2{c}_3},\end{array}

where the W state for the three input cavities (a_1,a_2,a_3) takes the same form as the W state given in Eq. (24).

3.2 Full Hamiltonians

As shown in the preceding Section 2, four basic Hamiltonians (3), (7), (11), and (15) are used for the state transfer. In reality, there exist the unwanted couplings between the qutrit and the cavities, as well as the inter-cavity crosstalks. We will give the modification on these Hamiltonians when they are taken into account. In the following, n=2 for the Bell state transfer, while n=3 for the W state transfer.

(i) The Hamiltonian (3) is modified as follows {\tilde{H}}_1=H_1+\delta H_1+\epsilon , where \delta H_1 describes the unwanted coupling between the |g⟩\leftrightarrow |e⟩ transition of the qutrit and the cavities (a_1,a_2,\cdots ,a_n) [Fig. 4(a)], while \epsilon describes the inter-cavity crosstalks. The expression of \delta H_1 is given by

(26)\delta H_1=g_{1,a}^{\mathrm{\prime }}{\mathrm{e}}^{\mathrm{i}{\mathrm{\Delta }}_{1}t}{\displaystyle \sum _{j=1}^n{\hat{a}}_j|e{⟩}_q⟨g|+\mathrm{h}.\mathrm{c}.,}

where g_{1,a}^{\mathrm{\prime }} is the off-resonant coupling constant between the |g⟩\leftrightarrow |e⟩ transition of the qutrit and the cavities (a_1,a_2,\cdots ,a_n), and {\mathrm{\Delta }}_1={\omega }_{fe}-{\omega }_a>0 is the detuning between the |g⟩\leftrightarrow |e⟩ transition frequency {\omega }_{fe} of the qutrit and the frequency {\omega }_a of the cavities (a_1,a_2,\cdots ,a_n). Here and below, the expression of \epsilon is given by

(27)\begin{array}{rl}\epsilon =& g_{aa}\sum _{j,k=1;j\ne k}^n{\hat{a}}_j{\hat{a}}_k^{+}+g_{bb}\sum _{j,k=1;j\ne k}^n{\hat{b}}_j{\hat{b}}_k^{+}\\ & +g_{cc}\sum _{j,k=1;j\ne k}^n{\hat{c}}_j{\hat{c}}_k^{+}+g_{ab}{\mathrm{e}}^{-\mathrm{i}{\mathrm{\Delta }}_{ab}t}\sum _{j,k=1;j\ne k}^n{\hat{a}}_j{\hat{b}}_k^{+}\\ & +g_{ac}{\mathrm{e}}^{-\mathrm{i}{\mathrm{\Delta }}_{ac}t}\sum _{j,k=1;j\ne k}^n{\hat{a}}_j{\hat{c}}_k^{+}\\ & +g_{bc}{\mathrm{e}}^{-\mathrm{i}{\mathrm{\Delta }}_{bc}t}\sum _{j,k=1;j\ne k}^n{\hat{b}}_j{\hat{c}}_k^{+}+\mathrm{h}.\mathrm{c}.,\end{array}

where the first term describes the inter-cavity crosstalk between the cavities (a_1,a_2,\cdots ,a_n) with crosstalk strength g_{aa}, the second term describes the inter-cavity crosstalk between the cavities (b_1,b_2,\cdots ,b_n) with crosstalk strength g_{bb}, the third term describes the inter-cavity crosstalk between the cavities (c_1,c_2,\cdots ,c_n) with crosstalk strength g_{cc}, the fourth term describes the inter-cavity crosstalk between the cavities (a_1,a_2,\cdots ,a_n) and the cavities (b_1,b_2,\cdots ,b_n) with crosstalk strength g_{ab} and frequency detuning {\mathrm{\Delta }}_{ab}={\omega }_a-{\omega }_b, the fifth term describes the inter-cavity crosstalk between the cavities (a_1,a_2,\cdots ,a_n) and the cavities (c_1,c_2,\cdots ,c_n) with crosstalk strength g_{ac} and frequency detuning {\mathrm{\Delta }}_{ac}={\omega }_a-{\omega }_c, and the sixth term describes the inter-cavity crosstalk between the cavities (b_1,b_2,\cdots ,b_n) and the cavities (c_1,c_2,\cdots ,c_n) with crosstalk strength g_{bc} and frequency detuning {\mathrm{\Delta }}_{bc}={\omega }_b-{\omega }_c. The inter-cavity crosstalk between any two of the three sets of cavities (a_1,a_2,\cdots ,a_n), cavities (b_1,b_2,\cdots ,b_n), and cavities (c_1,c_2,\cdots ,c_n) can be reduced by increasing the ratio of {\mathrm{\Delta }}_{ab}/g_{ab}, {\mathrm{\Delta }}_{ac}/g_{ac}, and {\mathrm{\Delta }}_{bc}/g_{bc}. The primary causes of inter-cavity crosstalk stem from two mechanisms: imperfect capacitive coupling between the transmom and the cavities, and energy leakage from the open structure of the waveguide cavities, leading to unintended energy transfer between the cavities.

(ii) The Hamiltonian (7) is modified as follows {\tilde{H}}_2=H_2+\delta H_2+\epsilon , where \delta H_2 describes the unwanted coupling between the |e⟩\leftrightarrow |f⟩ transition of the qutrit and the cavities (a_1,a_2,\cdots ,a_n) [Fig. 4(b)]. The expression of \delta H_2 is given by

(28)\delta H_2=g_{2,a}^{\mathrm{\prime }}{\mathrm{e}}^{\mathrm{i}{\mathrm{\Delta }}_{2}t}{\displaystyle \sum _{j=1}^n{\hat{a}}_j|f{⟩}_q⟨e|+\mathrm{h}.\mathrm{c}.,}

where g_{2,a}^{\mathrm{\prime }} is the off-resonant coupling constant between the |e⟩\leftrightarrow |f⟩ transition of the qutrit and the cavities (a_1,a_2,\cdots ,a_n), and is the detuning between the |e⟩\leftrightarrow |f⟩ transition frequency {\omega }_{fe} of the qutrit and the frequency {\omega }_a of the cavities (a_1,a_2,\cdots ,a_n).

(iii) The Hamiltonian (11) is modified as follows: {\tilde{H}}_3=H_3+\delta H_3+\epsilon , where \delta H_3 describes the unwanted interaction between the |e⟩\leftrightarrow |f⟩ transition of the qutrit and the cavities (b_1,b_2,\cdots ,b_n) [Fig. 4(c)]. The expression of \delta H_3 is given by

(29)\delta H_3=g_b^{\mathrm{\prime }}{\mathrm{e}}^{\mathrm{i}{\mathrm{\Delta }}_{3}t}{\displaystyle \sum _{j=1}^n{\hat{b}}_j|f{⟩}_q⟨e|+\mathrm{h}.\mathrm{c}.,}

where g_b^{\mathrm{\prime }} is the off-resonant coupling constant between the |e⟩\leftrightarrow |f⟩ transition of the qutrit and the cavities (b_1,b_2,\cdots ,b_n), and is the detuning between the |e⟩\leftrightarrow |f⟩ transition frequency {\omega }_{fe} of the qutrit and the frequency {\omega }_b of the cavities (b_1,b_2,\cdots ,b_n).

(iv) The Hamiltonian (15) is modified as follows {\tilde{H}}_4=H_4+\delta H_4+\epsilon , where \delta H_4 describes the unwanted interaction between the qutrit and the cavities (c_1,c_2,\cdots ,c_n) [Fig. 4(d)]. The expression of \delta H_4 is given by

(30)\delta H_4=g_c^{\mathrm{\prime }}{\mathrm{e}}^{\mathrm{i}{\mathrm{\Delta }}_{4}t}{\displaystyle \sum _{j=1}^n{\hat{c}}_j|e{⟩}_q⟨g|+\mathrm{h}.\mathrm{c}.,}

where g_c^{\mathrm{\prime }} is the off-resonant coupling constant between the |g⟩\leftrightarrow |e⟩ transition of the qutrit and the cavities (c_1,c_2,\cdots ,c_n), and {\mathrm{\Delta }}_4={\omega }_{eg}-{\omega }_c>0 is the detuning between the |g⟩\leftrightarrow |e⟩ transition frequency {\omega }_{eg} of the qutrit and the frequency {\omega }_c of (c_1,c_2,\cdots ,c_n).

For a transmon qutrit, the |g⟩\leftrightarrow |f⟩ transition is forbidden or very weak [84]. Accordingly, the coupling of the cavities with the |g⟩\leftrightarrow |f⟩ transition of the qutrit can be neglected and thus is not considered in the above modified Hamiltonians.

3.3 Numerical results

When considering the relaxation and dephasing of the qutrit as well as the cavity decay, the dynamics of the lossy system is determined by the following master equation

(31)\begin{array}{rl}\frac{\mathrm{d}\rho }{\mathrm{d}t}=& -\mathrm{i}\left[{\tilde{H}}_k,\rho \right]+\sum _{j=1}^n({\kappa }_{a_j}\mathcal{L}\left[{\hat{a}}_j\right]+{\kappa }_{b_j}\mathcal{L}\left[{\hat{b}}_j\right]\\ & +{\kappa }_{c_j}\mathcal{L}\left[{\hat{c}}_j\right])+{\gamma }_{eg}\mathcal{L}\left[{\sigma }_{eg}^{-}\right]+{\gamma }_{fe}\mathcal{L}\left[{\sigma }_{fe}^{-}\right]\\ & +{\gamma }_{fg}\mathcal{L}\left[{\sigma }_{fg}^{-}\right]+{\gamma }_{e,\phi }\mathcal{L}\left[{\sigma }_{ee}\right]+{\gamma }_{f,\phi }\mathcal{L}\left[{\sigma }_{ff}\right],\end{array}

where \rho is the density operator of the whole system at time t, {\tilde{H}}_k (k=1,2,3,4) is the modified Hamiltonian given above, {\sigma }_{eg}^{-}=|g{⟩}_q⟨e|,{\sigma }_{fe}^{-}=|e{⟩}_q⟨f|,{\sigma }_{fg}^{-}=|g{⟩}_q⟨f|, {\sigma }_{ee}=|e{⟩}_q⟨e|,{\sigma }_{ff}=|f{⟩}_q⟨f|, \mathcal{L}\left[\mathrm{\Lambda }\right]=\mathrm{\Lambda }\rho {\mathrm{\Lambda }}^{+}-{\mathrm{\Lambda }}^{+}\mathrm{\Lambda }\rho /2-\rho {\mathrm{\Lambda }}^{+}\mathrm{\Lambda }/2 (with \mathrm{\Lambda }={\hat{a}}_j,{\hat{b}}_j,{\hat{c}}_j,{\sigma }_{eg}^{-},{\sigma }_{fe}^{-},{\sigma }_{fg}^{-},{\sigma }_{ee},{\sigma }_{ff}); {\kappa }_{a_j},{\kappa }_{b_j}, and {\kappa }_{c_j} are the decay rates of cavity a_j, cavity b_j, and cavity c_j, respectively; {\gamma }_{eg} is the energy relaxation rate of the level |e⟩, {\gamma }_{fe} ({\gamma }_{fg}) is the relaxation rate of the level |f⟩ for the decay path |f⟩\to |e⟩ (|f⟩\to |g⟩) of the qutrit; {\gamma }_{e,\phi } ({\gamma }_{f,\phi }) is the dephasing rate of the level |e⟩ (|f⟩) of the qutrit.

Without loss of generality, we consider arbitrary numbers \alpha and \beta , by setting \alpha =\cos \left(\theta /2\right) and \beta ={\mathrm{e}}^{\mathrm{i}\phi }\sin \left(\theta /2\right). The average fidelity of the operation is defined by [85]

(32)\overline{\mathcal{F}}={\displaystyle \frac{1}{4\pi }}{\displaystyle {\int }_0^{2\pi }\mathrm{d}\phi {\displaystyle {\int }_0^{\pi }\mathcal{F}\sin \theta \mathrm{d}\theta ,}}

with

(33)\mathcal{F}\mathcal{=}\sqrt{⟨{\psi }_{\mathrm{i}\mathrm{d}}|{\rho }_f|{\psi }_{\mathrm{i}\mathrm{d}}⟩},

where |{\psi }_{\mathrm{i}\mathrm{d}}⟩ is the ideal evolution state given in Eq. (20) for the Bell state transfer while the ideal evolution state given in Eq. (23) for the W state transfer; {\rho }_f is the final density operator of the system when the operation is performed in a realistic situation, i.e., the situation which considers the system dissipation, the unwanted couplings, and the inter-cavity crosstalk. The density operator {\rho }_f is obtained by numerically solving the master equation (31) with the initial state given in Eq. (22) for the Bell state transfer and the initial state given in Eq. (25) for the W state transfer.

Experimentally, the frequency of a microwave cavity can be made as 2-51 GHz [22, 86], the typical transition frequency between neighboring levels of a transmon qutrit can be made to be 2-20 GHz [86, 87], and the level spacing anharmonicity 100-720 MHz of a transmon qutrit has been reported [88]. For a transmon qutrit [84], g_{1,a}^{\mathrm{\prime }}=g_{1,a}/\sqrt{2},g_{2,a}^{\mathrm{\prime }}=\sqrt{2}{g}_{2,a},g_b^{\mathrm{\prime }}=\sqrt{2}{g}_b, and g_c^{\mathrm{\prime }}=g_c/\sqrt{2}. As a concrete example, let us consider the parameters listed in Table 1, which are used in our numerical simulations. Other parameters used in the numerical simulations are: (i) {\gamma }_{eg}^{-1}=60 μs, {\gamma }_{fe}^{-1}=30 μs, {\gamma }_{fg}^{-1}=150 μs [89], (ii) {\gamma }_{f,\phi }^{-1}={\gamma }_{e,\phi }^{-1}=20 μs, (iii) {\kappa }_{a_j}^{-1}={\kappa }_{b_j}^{-1}={\kappa }_{c_j}^{-1}\equiv {\kappa }^{-1}, and (iv) g_{aa}=g_{bb}=g_{cc}=g_{ab}=g_{ac}=g_{bc}\equiv g_{cr}. The decoherence times of the transmon qutrits considered here are a rather conservative case because energy relaxation time with a range from 65 μs to 0.5 ms and dephasing time from 25 to 75 μs have been experimentally demonstrated for a superconducting transmon device [12–14, 90, 91]. For the coupling constants listed in Table 1, one has g_{max}=max\{g_{1,a},g_{1,a}^{\mathrm{\prime }},g_{2,a},g_{2,a}^{\mathrm{\prime }},g_b,g_b^{\mathrm{\prime }},g_c,g_c^{\mathrm{\prime }}\}=2\pi \times 10 MHz. Note that coupling strengths of \sim 10 MHz between SC transmons and cavities have been reported in Refs. [92–94], and a coupling strength 2\pi \times 897 MHz has been reported for a transmon coupled to a 1D transmission line resonator [95]. In addition, the Rabi frequency 2\pi \times 300 MHz of a classical microwave pulse was reported in experiments [96].

By numerically solving the master equation (31), we plot Fig. 5 showing the average fidelity versus g_{cr}/g_{max} and {\kappa }^{-1} for the Bell state transfer and Fig. 6 showing the average fidelity versus g_{cr}/g_{max} and {\kappa }^{-1} for the W state transfer. Figures 5 and 6 show that the influence of the inter-cavity crosstalk is very small for a given {\kappa }^{-1}. In addition, even when g_{cr}=0.1g_{max}, for {\kappa }^{-1}=20 μs, a high average fidelity 99.56% can be obtained for the Bell state transfer and a high average fidelity 99.25% can be achieved for the W state transfer.

To examine how the cavity number n of the transferred entangled state influences the result, Fig. 7 displays the average fidelity versus the qutrit decoherence time T for n=2, 3, 4 and 5. The relaxation rates are set to {\gamma }_{eg}^{-1}=3T, {\gamma }_{fe}^{-1}=1.5T, {\gamma }_{fg}^{-1}=7.5T and the dephasing rates to {\gamma }_{f,\phi }^{-1}={\gamma }_{e,\phi }^{-1}=T, while {\kappa }^{-1}=20 μs and g_{cr}/g_{max}=0.01. As expected, the fidelity depends weakly on T, since the entire protocol involves only one qutrit. The fidelity decreases slowly as n increases. The reason for this is as follows. Because only resonant interactions are employed, the total operational time is extremely short and, according to Eq. (19), actually becomes shorter for a larger n.

With the parameters chosen above, the operational time is estimated as 74 ns for the Bell state transfer and 69 ns for the W state transfer (assuming {\tau }_d=2 ns), which is much shorter than the decoherence times of the qutrits and the cavity decay times (10-50 μs), used in the numerical simulation. For the cavity frequencies given above and {\kappa }^{-1}=20 μs, the quality factors of the cavities at the input port, the output port 1 and the output port 2 are, respectively, Q_a\sim 4.52\times {10}^5, Q_b\sim 3.96\times {10}^5, and Q_c\sim 3.39\times {10}^5, which are available because a 1D microwave cavity or resonator with a high quality factor Q\gtrsim 2.7\times {10}^6 was experimentally reported [97, 98]. The analysis given above demonstrates that the high-fidelity controlled transfer of the Bell state and the W state is feasible with the present circuit QED technology.

3.4 Discussion

As seen in Table 1, the ratio of detuning values by the unwanted qutrit-cavity coupling constants falls within 35 to 47, which means that the cavities are highly detuned from the transitions between the irrelevant levels of the qutrit. In this sense, the effect of the unwanted qutrit-cavity couplings on the fidelity is negligibly small.

In our numerical calculations, we adopt a conservative scenario where g_{cr}\le 0.1g_{max}, which is significantly more stringent than the typical scenario in circuit QED, where the inter-cavity crosstalk g_{cr} is usually two to three orders of magnitude weaker than the maximum coupling strength g_{max} [71, 99, 100]. In addition, from Table 1, one can see that the frequency detunings between any two of the cavity frequencies {\omega }_a, {\omega }_b, and {\omega }_c are greater than the inter-cavity crosstalk strength g_{cr} by at least two orders of magnitude (even for g_{cr}=0.1g_{max}). Therefore, the influence of the inter-cavity crosstalk between the input port and either of the two output ports or between the two output ports can be neglected. Notably, one can verify that the SES entangled states of the cavities at the input port, the output port 1 and the output port 2 are, respectively, the eigenstates of the first three terms in Eq. (27). This indicates that, in principle, the inter-cavity crosstalk within the input port or either of the two output ports has no influence on the SES entangled state transfer.

The infidelity mainly results from the qutrit decoherence, the cavity dissipation, and the unwanted qutrit-cavity interaction. The fidelity can be improved by: designing the qutrit with larger level spacing anharmonicity, choosing the qutrit with longer decoherence time, and selecting the cavities with a high quality factor.

We have assumed that a pure initial state (i.e., SES entangled state) at the input port can be prepared. Suppressing the thermal photon background is crucial for ensuring the purity of the initial state in experiments. To address this challenge, quantum devices are typically placed within a dilution refrigerator, where the base temperature can be stably maintained at an ultra-low level of approximately 10 mK [101]. Furthermore, techniques, such as the Derivative Removal for Annulment of Cavity-Hybridized Measurement Adiabat (DRACHMA), can be employed to reduce the thermal photon occupancy to on the order of {10}^{-3} within a short time frame [101, 102].

For simplicity, our numerical simulation did not consider: (i) The coupling of the cavities with the |g⟩\leftrightarrow |f⟩ transition, and (ii) The qutrit decoherence and the cavity dissipation during the adjustment of the qutrit level spacings. However, we remark that when they are taken into account, the fidelity would be slightly decreased. This is because: (i) The coupling of the cavities with the |g⟩\leftrightarrow |f⟩ transition is negligible because of the very weak |g⟩\leftrightarrow |f⟩ transition for the transmon qutrit, (ii) The effect of the qutrit decoherence and the cavity dissipation, during the adjustment of the qutrit level spacings, is negligible because the level spacings of a superconducting qutrit can be rapidly adjusted within 1-3 ns [79–81]. We remark that further investigation is needed for each particular experimental setup. However, this requires a rather lengthy and complex analysis, which is beyond the scope of this theoretical work.

4 Conclusions

We have proposed to transfer the single-excitation symmetric (SES) entangled state of n photonic qubits from n microwave cavities at the input port to the n microwave cavities at either of the two output ports via the control of a quantum switch. This proposal has the features and advantages described in the introduction. Since the SES entangled state is a W-type entangled state for n\ge 3 and a Bell entangled state for n=2, the proposal can be used to transfer both W state and Bell state of photonic qubits via the control of a quantum switch. As an example, we have numerically analyzed the circuit-QED experimental feasibility of transferring both the Bell state of two photonic qubits and the W state of three photonic qubits in a control manner.

To verify the success of the state transfer, one can measure the SES entangled state at the output ports using the following architecture: in the system depicted in Fig. 1(a), each cavity at the output ports needs to be independently coupled to an ancillary SC qubit. By performing controlled gate operations between each cavity and each corresponding ancillary qubit, the SES entangled state of the cavities can be transferred to the ancillary qubits. Subsequently, by measuring the quantum state of each ancillary qubit — for instance, using techniques such as quantum state tomography — the SES entangled state in the cavities can be measured and determined.

To the best of our knowledge, this work is the first to show the transfer of quantum entangled states of photonic qubits from the microwave cavities at the input port to the microwave cavities at either of two output ports via a quantum switch. This proposal may be extended to accomplish the same task in other physical systems, where the quantum switch is implemented with an artificial atom (e.g., a quantum dot, an NV center, a magnon, and a SC qutrit with different types, etc.) and each cavity is a microwave or optical cavity. Our results provide new insights into controlled transfer of quantum entangled states. They may have potential applications in networked quantum communication, distributed quantum information processing, as well as development of quantum functional devices (e.g., quantum triodes and quantum routers). We hope that it could stimulate the experimental activities in the near future.

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