Transfer of quantum entangled states between superconducting qubits and microwave field qubits

Abstract

Transferring entangled states between matter qubits and microwave-field (or optical-field) qubits is of fundamental interest in quantum mechanics and necessary in hybrid quantum information processing and quantum communication. We here propose a way for transferring entangled states between superconducting qubits (matter qubits) and microwave-field qubits. This proposal is realized by a system consisting of multiple superconducting qutrits and microwave cavities. Here, “qutrit” refers to a three-level quantum system with the two lowest levels encoding a qubit while the third level acting as an auxiliary state. In contrast, the microwave-field qubits are encoded with coherent states of microwave cavities. Because the third energy level of each qutrit is not populated during the operation, decoherence from the higher energy levels is greatly suppressed. The entangled states can be deterministically transferred because measurement on the states is not needed. The operation time is independent of the number of superconducting qubits or microwave-field qubits. In addition, the architecture of the circuit system is quite simple because only a coupler qutrit and an auxiliary cavity are required. As an example, our numerical simulations show that high-fidelity transfer of entangled states from two superconducting qubits to two microwave-field qubits is feasible with present circuit QED technology. This proposal is quite general and can be extended to transfer entangled states between other matter qubits (e.g., atoms, quantum dots, and NV centers) and microwave- or optical-field qubits encoded with coherent states.

Graphical abstract

Keywords

tranferring entangled states / superconducting qubits / microwave field qubits / coherent states / circuit QED

1 Introduction

The realm of circuit QED (quantum electrodynamics) investigates the light-matter interaction between superconducting qubits (SC qubits) and microwave cavities. It is known that circuit QED is one of the most promising well-established platforms for achieving large-scale quantum computing and quantum information processing (QIP) [1–6]. A superconducting circuit system exhibits excellent scalability, flexibility, and tunability, in which very strong coupling can be realized between SC qubits and microwave cavities [7, 8].

For SC qubits, their coherence times have been significantly improved with state-of-the-art Josephson junction technology, and a coherence time ∼ 2 ms [9] of an SC qubit has been experimentally reported. Over the past twenty years, many theoretical methods for transferring quantum states between SC qubits have been proposed in circuit QED (e.g., Refs.[1, 2, 10–14]). In experiments, quantum state transfer between two SC qubits via a superconducting cavity [15] and transfer of states in a chain of SC qubits [16] have been demonstrated. In addition, multipartite entangled states are central resources and have many applications in QIP and quantum communication. Up to now, various kinds of entangled states of SC qubits (e.g., GHZ states, W states, and cluster states) have been prepared in theory [17–31]. The generation of entangled states with 10–20 SC qubits has been experimentally achieved in circuit QED [32–34].

On the other hand, QIP with superconducting microwave cavities has also drawn much attention. Recently, a microwave cavity with the photon lifetime ∼ 2 s has been experimentally realized [35]. Compared with SC qubits, microwave photons contained by a superconducting cavity or resonator can have longer coherence time, making them particularly suitable as good quantum memories. In addition, the microwave photon states of cavities can be used to encode quantum information in high-dimensional Hilbert spaces. A large number of schemes have been presented for generating different kinds of entangled states of microwave cavities [36–51]. Experimentally, Fock states [52], Schrödinger cat states [53], two-mode cat states [54], and NOON states [55] have been created in circuit QED.

In this paper, we present a way to transfer quantum entangled states from a group of SC qubits to another group of microwave-field qubits. For simplicity, we here denote “microwave–f ield qubits” as MF qubits. The state transfer is described as

\begin{array}{}\text{(1)}& \begin{array}{ll}\hfill & a|g{\rangle }_1|g{\rangle }_2\cdots |g{\rangle }_N+b|e{\rangle }_1|e{\rangle }_2\cdots |e{\rangle }_N\hfill \\ \hfill & \to a|\alpha {\rangle }_{c_1}|\alpha {\rangle }_{c_2}\cdots |\alpha {\rangle }_{c_N}+b|-\alpha {\rangle }_{c_1}|-\alpha {\rangle }_{c_2}\cdots |-\alpha {\rangle }_{c_N}\mathrm{,}\hfill \end{array}\end{array}

where the SC qubits are encoded with two energy levels |g⟩ and |e⟩ of qubits, while the MF qubits are encoded with coherent states |α⟩ and |−α⟩ of cavities. In Eq. (1), a and b are the normalized complex numbers, satisfying |a|² + |b|² = 1. When a=b=1/\sqrt{2}, the transferred state is a maximally entangled state. For two qubits, the entangled states in Eq. (1) are Bell states, which have wide applications in quantum communication. For three qubits or more, the entangled states in Eq. (1) are the well-known GHZ states, which play a central role in quantum teleportation [56], error correction [57], quantum metrology [58], etc..

Our proposal has these distinguishing advantages: (i) The entangled states can be deterministically transferred because measurement on the states of SC qubits or cavities is not needed. (ii) The operation time is independent of the number of SC qubits or MF qubits. (iii) Due to the third energy level of each qutrit is not populated during the operation, the decoherence of qutrit’s higher energy levels is greatly minimized, and (iv) The architecture of the circuit system is compact and simple because only a coupler qutrit and an auxiliary cavity are required. In addition, this proposal is quite general and can be extended to transfer entangled states between other matter qubits (e.g., atoms, quantum dots, and NV centers) and microwave- or optical-field qubits encoded with coherent states. To the best of our knowledge, this work is the first to demonstrate that based on circuit QED, the entangled states of SC qubits can be transferred onto MF qubits deterministically.

We believe that this work is of interest. First, the entangled coherent states are very important in quantum information science and technology. For example, the entangled coherent states can be widely used in quantum teleportation [59], quantum computation [60], quantum metrology [61], and error correction [62, 63]. Moreover, they have application in the demonstration of the Bell-type inequality violations [64]. Second, transferring entangled states between SC qubits and MF qubits may be necessary and important in the hybrid QIP. For instance, in a hybrid QIP based on SC qubits and MF qubits, entanglement transfer may be necessary between SC-qubit-based quantum processors and MF-qubits-based quantum processors. Lastly, SC qubits are matter qubits, which are different from microwave-field qubits. Transferring entangled states between matter qubits and microwave-field qubits is of fundamental interest in quantum mechanics. Recently, bosonic codes are particularly appealing and play important roles in QIP and quantum computing [65, 66]. There exist three kinds of significant bosonic codes, e.g., Gottesman–Kitaev–Preskill (GKP) [67], cat [68], and binomial codes [69, 70]. Our work may have potential applications in bosonic codes.

This paper is organized as follows. In Section 2, we introduce the basic theory. In Section 3, we show how to transfer entangled states between SC qubits and MF qubits. In Section 4, we discuss the possible experimental implementation of our proposal and numerically evaluate the operational fidelity for transferring entangled states from two SC qubits to two MF qubits. A concluding summary is given in Section 5.

2 Basic theory

Consider a system consisting of a microwave cavity c, a superconducting qutrit q, N superconducting qutrits (1, 2, 3, ···, N), and N microwave cavities (1, 2, 3, ···, N), as shown in Fig. 1. The three levels of the qutrit q are labeled as |g⟩_q, |e⟩_q, |f⟩_q, while the three levels of the qutrit l are labeled as |g⟩_l, |e⟩_l, |f⟩_l (l = 1, 2, 3, ···, N) (Fig. 2). In the following subsections, we will give an introduction to the state evolution under a few types of interactions.

2.1 Pulse–qutrit resonant interaction

Assume that a microwave pulse is resonant with the |g⟩_l ↔ |e⟩_l transition of qutrit l (l = 2, 3, ···, N), as shown in Fig. 2(a). In the interaction picture and after a rotating-wave approximation (RWA), the Hamiltonian is

\begin{array}{}\text{(2)}& H_1={\text{Ω}}_l({\text{e}}^{-\text{i}{\varphi }_l}{\sigma }_{eg\mathrm{,}l}^{+}+h\mathrm{.}c\mathrm{.})\mathrm{,}\end{array}

where {\sigma }_{eg\mathrm{,}l}^{+}=|e{\rangle }_l\langle g| is the raising operator between |g⟩_l ↔ |e⟩_l transition of qutirt l (l = 2, 3, ···, N), Ω_l and ϕ_l are the Rabi frequency and the initial phase of the pulse. Under the Hamiltonian (2), the state rotations are given by

\begin{array}{}\text{(3)}& \begin{array}{l}|g{\rangle }_l\to \cos ({\text{Ω}}_{l}t)|g{\rangle }_l-{\text{ie}}^{-\text{i}{\varphi }_l}\sin ({\text{Ω}}_{l}t)|e{\rangle }_l\mathrm{,}\\ |e{\rangle }_l\to -{\text{ie}}^{\text{i}{\varphi }_l}\sin ({\text{Ω}}_{l}t)|g{\rangle }_l+\cos ({\text{Ω}}_{l}t)|e{\rangle }_l\mathrm{.}\end{array}\end{array}

2.2 Cavity–qutrit resonant interaction

Suppose that cavity c is resonant with the |g⟩₁ ↔ |e⟩ ₁ transition of qutrit 1 with coupling strength g_r,1, as illustrated in Fig. 2(b). The Hamiltonian in the interaction picture and after the RWA is

\begin{array}{}\text{(4)}& H_2=g_{r\mathrm{,1}}({\widehat{a}}_c{\sigma }_{eg\mathrm{,1}}^{+}+h\mathrm{.}c\mathrm{.})\mathrm{,}\end{array}

where {\widehat{a}}_c is the photon annihilation operator for cavity c,and {\sigma }_{eg\mathrm{,1}}^{+}=|e{\rangle }_1\langle g|.

Under the Hamiltonian (4), one obtains the state evolution

\begin{array}{}\text{(5)}& \begin{array}{l}|g{\rangle }_1|1{\rangle }_c\to \cos (g_{r\mathrm{,1}}t)|g{\rangle }_1|1{\rangle }_c-\text{i}\sin (g_{r\mathrm{,1}}t)|e{\rangle }_1|0{\rangle }_c\mathrm{,}\\ |e{\rangle }_1|0{\rangle }_c\to \cos (g_{r\mathrm{,1}}t)|e{\rangle }_1|0{\rangle }_c-\text{i}\sin (g_{r\mathrm{,1}}t)|g{\rangle }_1|1{\rangle }_c\mathrm{,}\end{array}\end{array}

where |0⟩_c and |1⟩_c are the vacuum and single photon states of cavity c, respectively.

Now assume that cavity c is resonant with the |g⟩_q ↔ |e⟩_q transition of qutrit q with coupling strength g_r,q [Fig. 2(c)]. Similarly, one can obtain the Hamiltonian

\begin{array}{}\text{(6)}& H_3=g_{r\mathrm{,}q}({\widehat{a}}_c{\sigma }_{eg\mathrm{,}q}^{+}+h\mathrm{.}c\mathrm{.})\mathrm{,}\end{array}

and the state evolution

\begin{array}{}\text{(7)}& \begin{array}{l}|g{\rangle }_q|1{\rangle }_c\to \cos (g_{r\mathrm{,}q}t)|g{\rangle }_q|1{\rangle }_c-\text{i}\sin (g_{r\mathrm{,}q}t)|e{\rangle }_q|0{\rangle }_c\mathrm{,}\\ |e{\rangle }_q|0{\rangle }_c\to \cos (g_{r\mathrm{,}q}t)|e{\rangle }_q|0{\rangle }_c-\text{i}\sin (g_{r\mathrm{,}q}t)|g{\rangle }_q|1{\rangle }_c\mathrm{,}\end{array}\end{array}

where {\sigma }_{eg\mathrm{,}q}^{+}=|e{\rangle }_q\langle g|.

2.3 Cavity–qutrit dispersive interaction

Assume that cavity c is dispersively coupled to the |e⟩_l ↔ |f⟩_l transition of qutrit l (l = 2, 3, ···, N) with coupling strength g_l and detuning ∆_l = ω_fe,l−ω_c > 0, as depicted in Fig. 2(d). Here, ω_fe,l is the |e⟩_l ↔ |f⟩_l transition frequency of qutrit l and ω_c is the frequency of cavity c. In the interaction picture and after making the RWA, the Hamiltonian is

\begin{array}{}\text{(8)}& H_4={\displaystyle \sum _{l=2}^N{g}_l({\text{e}}^{\text{i}{\text{Δ}}_{l}t}{\widehat{a}}_c{\sigma }_{fe\mathrm{,}l}^{+}+h\mathrm{.}c\mathrm{.})\mathrm{,}}\end{array}

where {\sigma }_{fe\mathrm{,}l}^{+}=|f{\rangle }_l\langle e| is the raising operator between the |e⟩_l ↔ |f⟩_l transition of qutirt l (l = 2, 3, ···, N).

Applying the large-detuning conditions ∆_l ≫ g_l, one can obtain the effective Hamiltonian [71–73]

\begin{array}{}\text{(9)}& \begin{array}{ll}{H}_{e1}=\hfill & {\displaystyle \sum _{l=2}^N}{\lambda }_l{({\widehat{a}}_c{\widehat{a}}_c^{†}|f\rangle }_l{\langle f|-{\widehat{a}}_c^{†}{\widehat{a}}_c|e\rangle }_l\langle e|)\hfill \\ \hfill & +{\displaystyle \sum _{l\ne l^{\prime }=2}^N}{\lambda }_{l{l}^{\prime }}{(|f\rangle }_l{\langle e|\otimes |e\rangle }_{l^{\prime }}{\langle f|+|e\rangle }_l{\langle f|\otimes |f\rangle }_{l^{\prime }}\langle e|)\mathrm{,}\hfill \end{array}\end{array}

where {\lambda }_l=\frac{g_l^2}{{\text{Δ}}_l}, {\lambda }_{l{l}^{\prime }}=\frac{g_l{g}_{l^{\prime }}}{2}\left(\frac{1}{{\text{Δ}}_l}+\frac{1}{{\text{Δ}}_{l^{\prime }}}\right), the first line of Eq. (9) describes Stark shifts while the second line represents the coupling between the qutrits l and l′ induced by the cavity c.

If the auxiliary level |f⟩_l of qutrit l (l = 2, 3, ···, N) is not occupied, the effective Hamiltonian (9) reduces to

\begin{array}{}\text{(10)}& H_{e1}=-{\displaystyle \sum _{l=2}^N{\lambda }_l{\widehat{a}}_c^{†}{\widehat{a}}_c|e{\rangle }_l\langle e|\mathrm{.}}\end{array}

Under the effective Hamiltonian (10), one can obtain the following state evolution

\begin{array}{}\text{(11)}& \begin{array}{ll}\hfill & |g{\rangle }_l|0{\rangle }_c\to |g{\rangle }_l|0{\rangle }_c\mathrm{,}|e{\rangle }_l|0{\rangle }_c\to |e{\rangle }_l|0{\rangle }_c\mathrm{,}\hfill \\ \hfill & |g{\rangle }_l|1{\rangle }_c\to |g{\rangle }_l|1{\rangle }_c\mathrm{,}|e{\rangle }_l|1{\rangle }_c\to {\text{e}}^{\text{i}{\lambda }_{l}t}|e{\rangle }_l|1{\rangle }_c\mathrm{.}\hfill \end{array}\end{array}

2.4 Cavity–cavity cross-Kerr interaction

Consider cavity c and cavity j (j = 1, 2, 3, ···, N) are dispersively coupled to the qutrit q [Fig. 2(e)]. As shown in Fig. 2(e), cavity c (j) is dispersively coupled to the |g⟩_q ↔ |e⟩_q (|e⟩_q ↔ |f⟩_q) transition of qutrit q with coupling strength µ_c (µ_j). In the interaction picture and after making the RWA, the Hamiltonian is given by

\begin{array}{}\text{(12)}& \begin{array}{ll}{H}_5=\hfill & {\mu }_c({\text{e}}^{\text{i}{\delta }_{c}t}{\widehat{a}}_c{\sigma }_{eg\mathrm{,}q}^{+}+h\mathrm{.}c\mathrm{.})\hfill \\ \hfill & +{\displaystyle \sum _{j=1}^N}{\mu }_j({\text{e}}^{\text{i}{\delta }_{c_j}t}{\widehat{a}}_j{\sigma }_{fe\mathrm{,}q}^{+}+h\mathrm{.}c\mathrm{.})\mathrm{,}\hfill \end{array}\end{array}

where {\widehat{a}}_j is the photon annihilation operator of cavity j, {\sigma }_{eg\mathrm{,}q}^{+}=|e{\rangle }_q\langle g|, {\sigma }_{fe\mathrm{,}q}^{+}=|f{\rangle }_q\langle e|, δ_c = ω_eg,q−ω_c < 0 and {\delta }_{c_j}={\omega }_{fe\mathrm{,}q}-{\omega }_{c_j}>0. Here, ω_fe,q and ω_eg,q are the |e⟩_q ↔ |f⟩_q and |g⟩_q ↔ |e⟩_q transition frequencies of qutrit q, and ω_c and {\omega }_{c_j} are the frequencies of cavities c and j (j = 1, 2, 3, ···, N), respectively.

Under the large-detuning conditions |δ_c| ≫ µ_c and δ_cj ≫ µ_j, the Hamiltonian (12) becomes [71–73]

\begin{array}{}\text{(13)}& \begin{array}{ll}{H}_{e2}=\hfill & {\lambda }_c{({\widehat{a}}_c^{†}{\widehat{a}}_c|g\rangle }_q{\langle g|-{\widehat{a}}_c{\widehat{a}}_c^{†}|e\rangle }_q\langle e|)\hfill \\ \hfill & -{\displaystyle \sum _{j=1}^N}\frac{{\mu }_j^2}{{\delta }_{c_j}}{({\widehat{a}}_j^{†}{\widehat{a}}_j|e\rangle }_q{\langle e|-{\widehat{a}}_j{\widehat{a}}_j^{†}|f\rangle }_q\langle f|)\hfill \\ \hfill & +{\displaystyle \sum _{j\ne j^{\prime }=1}^N}{\lambda }_{j{j}^{\prime }}{\text{e}}^{\text{i}{\delta }_{j{j}^{\prime }}t}{\widehat{a}}_j{\widehat{a}}_{j^{\prime }}^{†}{(|f\rangle }_q{\langle f|-|e\rangle }_q\langle e|)\hfill \\ \hfill & +{\displaystyle \sum _{j=1}^N}{\lambda }_j({\text{e}}^{-\text{i}{\delta }_{j}t}{\widehat{a}}_c^{†}{\widehat{a}}_j^{†}{\sigma }_{fg\mathrm{,}q}^{-}+h\mathrm{.}c\mathrm{.})\mathrm{,}\hfill \end{array}\end{array}

where {\lambda }_c=-\frac{{\mu }_c^2}{{\delta }_c}, {\lambda }_{j{j}^{\prime }}=\frac{{\mu }_j{\mu }_{j^{\prime }}}{2}(\frac{1}{{\delta }_{c_j}}+\frac{1}{{\delta }_{c_{j^{\prime }}}}), {\lambda }_j=\frac{{\mu }_c{\mu }_j}{2}(\frac{1}{|{\delta }_c|}+\frac{1}{{\delta }_{c_j}}), {\delta }_{j{j}^{\prime }}={\delta }_{c_j}-{\delta }_{c_{j^{\prime }}}, {\delta }_j={\omega }_{fg\mathrm{,}q}-{\omega }_c-{\omega }_{c_j}>0, and {\sigma }_{fg}^{-}=|g{\rangle }_q\langle f|. Here, ω_fg,q is the |f⟩_q ↔ |g⟩_q transition frequency of qutrit q.

Considering the large-detuning conditions {\delta }_j\gg \{{\lambda }_c\mathrm{,}{\mu }_j^2/{\delta }_j\mathrm{,}{\lambda }_{j{j}^{\prime }}\mathrm{,}{\lambda }_j\}, the effective Hamiltonian (13) turns into [71–73]

\begin{array}{}\text{(14)}& \begin{array}{ll}{H}_{e2}=\hfill & {\lambda }_c{({\widehat{a}}_c^{†}{\widehat{a}}_c|g\rangle }_q{\langle g|-{\widehat{a}}_c{\widehat{a}}_c^{†}|e\rangle }_q\langle e|)\hfill \\ \hfill & -{\displaystyle \sum _{j=1}^N}\frac{{\mu }_j^2}{{\delta }_{c_j}}{({\widehat{a}}_j^{†}{\widehat{a}}_j|e\rangle }_q{\langle e|-{\widehat{a}}_j{\widehat{a}}_j^{†}|f\rangle }_q\langle f|)\hfill \\ \hfill & +{\displaystyle \sum _{j\ne j^{\prime }=1}^N}{\lambda }_{j{j}^{\prime }}{\text{e}}^{\text{i}{\delta }_{j{j}^{\prime }}t}{\widehat{a}}_j{\widehat{a}}_{j^{\prime }}^{†}{(|f\rangle }_q{\langle f|-|e\rangle }_q\langle e|)\hfill \\ \hfill & +{\displaystyle \sum _{j=1}^N}{\chi }_j{({\widehat{a}}_c{\widehat{a}}_c^{†}{\widehat{a}}_j{\widehat{a}}_j^{†}|f\rangle }_q{\langle f|-{\widehat{a}}_c^{†}{\widehat{a}}_c{\widehat{a}}_j^{†}{\widehat{a}}_j|g\rangle }_q\langle g|)\mathrm{,}\hfill \end{array}\end{array}

where {\chi }_j={\lambda }_j^2/{\delta }_j is the effective cross-Kerr interaction coupling strength between cavities c and j.

When the levels |e⟩_q and |f⟩_q of qutrit q are not occupied, the Hamiltonian (14) reduces to the effective Hamiltonian [71–73]

\begin{array}{}\text{(15)}& H_{e2}={\lambda }_c{\widehat{n}}_c|g{\rangle }_q\langle g|-{\displaystyle \sum _{j=1}^N{\chi }_j{\widehat{n}}_c{\widehat{n}}_j|g{\rangle }_q\langle g|\mathrm{,}}\end{array}

where {\widehat{n}}_c={\widehat{a}}_c^{†}{\widehat{a}}_c and {\widehat{n}}_j={\widehat{a}}_j^{†}{\widehat{a}}_j are the photon number operators for cavities c and j, respectively.

2.5 Cavity–qutrit-pulse dispersive interaction

Assume that cavity 1 and a microwave pulse are dispersively coupled to the |g⟩_q ↔ |e⟩_q transition of qutrit q, respectively [Fig. 2(f)]. By applying the RWA, the Hamiltonian in the interaction picture is

\begin{array}{}\text{(16)}& H_6=g_q{\text{e}}^{\text{i}{\delta }_{q}t}{\widehat{a}}_1{\sigma }_{eg\mathrm{,}q}^{+}+{\text{Ω}}_p{\text{e}}^{-\text{i}[({\omega }_p-{\omega }_{eg\mathrm{,}q})t+{\varphi }_p]}{\sigma }_{eg\mathrm{,}q}^{+}+h\mathrm{.}c\mathrm{.,}\end{array}

where g_q is the coupling strength, {\delta }_q={\omega }_{eg\mathrm{,}q}-{\omega }_{c_1}>0 is the detunings, Ω_p, ω_p, and ϕ_p are respectively the Rabi frequency, frequency, and initial phase of the pulse.

Under the large-detuning condition δ_q ≫ g_q and assuming δ_q ≫ Ω_p, one can obtain the effective Hamiltonian [71–73]

\begin{array}{}\text{(17)}& \begin{array}{ll}{H}_{e3}=\hfill & {\lambda }_q\left({\widehat{a}}_1^{+}{\widehat{a}}_1+\frac{1}{2}\right){\sigma }_{z\mathrm{,}q}\hfill \\ \hfill & +{\text{Ω}}_p\left({\text{e}}^{-\text{i}[({\omega }_p-{\omega }_{eg\mathrm{,}q})t+{\varphi }_p]}{\sigma }_{eg\mathrm{,}q}^{+}+h\mathrm{.}c\mathrm{.}\right)\mathrm{,}\hfill \end{array}\end{array}

where {\lambda }_q=\frac{g_q^2}{{\delta }_q} and σ_z,q = |e⟩_q ⟨e|−|g⟩_q ⟨g|. In a rotating frame under the Hamiltonian {\lambda }_q\left({\widehat{a}}_1^{+}{\widehat{a}}_1+\frac{1}{2}\right){\sigma }_{z\mathrm{,}q} and for ω_p = ω_eg,q + λ_q, one has

\begin{array}{}\text{(18)}& H_{e3}={\text{Ω}}_p{\text{e}}^{-\text{i}{\varphi }_p}{\text{e}}^{-\text{i}2{\lambda }_q{\widehat{a}}_1^{+}{\widehat{a}}_{1}t}{\sigma }_{eg\mathrm{,}q}^{+}+h\mathrm{.}c\mathrm{..}\end{array}

We then discuss how to use Eq. (18) to realize the following state rotations depending on the initial state of cavity 1.

(i) If the cavity 1 is in the vacuum state |0{\rangle }_{c_1}, the Hamiltonian (18) reduces to H_{e3}={\text{Ω}}_p{\text{e}}^{-\text{i}{\varphi }_p}{\sigma }_{eg\mathrm{,}q}^{+}+h\mathrm{.}c\mathrm{.}. One can easily obtain the following rotations

\begin{array}{}\text{(19)}& \begin{array}{l}|g{\rangle }_q|0{\rangle }_{c_1}\to \left[\begin{array}{c}\cos ({\text{Ω}}_{p}t)|g{\rangle }_q-{\text{ie}}^{-\text{i}{\varphi }_p}\sin ({\text{Ω}}_{p}t)|e{\rangle }_q\end{array}\right]|0{\rangle }_{c_1}\mathrm{,}\\ |e{\rangle }_q|0{\rangle }_{c_1}\to \left[\begin{array}{c}\cos ({\text{Ω}}_{p}t)|e{\rangle }_q-{\text{ie}}^{\text{i}{\varphi }_p}\sin ({\text{Ω}}_{p}t)|g{\rangle }_q\end{array}\right]|0{\rangle }_{c_1}\mathrm{.}\end{array}\end{array}

After returning to the original interaction picture by performing a unitary transformation \exp \left\{\begin{array}{c}-\text{i}\left[{\lambda }_q\left({\widehat{a}}_1^{+}{\widehat{a}}_1+\frac{1}{2}\right){\sigma }_{z\mathrm{,}q}\right]t\end{array}\right\}, one has the following state transformations according to (19)

\begin{array}{}\text{(20)}& \begin{array}{ll}|g{\rangle }_q|0{\rangle }_{c_1}\to \hfill & [{\text{e}}^{\text{i}\frac{{\lambda }_q}{2}t}\cos ({\text{Ω}}_{p}t)|g{\rangle }_q\hfill \\ \hfill & -{\text{ie}}^{-\text{i}({\varphi }_p+\frac{{\lambda }_q}{2}t)}\sin ({\text{Ω}}_{p}t)|e{\rangle }_q]|0{\rangle }_{c_1}\mathrm{,}\hfill \\ |e{\rangle }_q|0{\rangle }_{c_1}\to \hfill & [{\text{e}}^{-\text{i}\frac{{\lambda }_q}{2}t}\cos ({\text{Ω}}_{p}t)|e{\rangle }_q\hfill \\ \hfill & -{\text{ie}}^{\text{i}({\varphi }_p+\frac{{\lambda }_q}{2}t)}\sin ({\text{Ω}}_{p}t)|g{\rangle }_q]|0{\rangle }_{c_1}\mathrm{.}\hfill \end{array}\end{array}

(ii) If the cavity 1 is in the coherent state |2\alpha {\rangle }_{c_1}, we assume that α is sufficiently large such that 2{\lambda }_q\overline{n}\gg {\text{Ω}}_p, where \overline{n}=4|\alpha {|}^2 is the average photon number of the coherent state |2\alpha {\rangle }_{c_1}. Thus, the states of qutrit q are not changed by the microwave pulse [74], i.e.,

\begin{array}{}\text{(21)}& |g{\rangle }_q|2\alpha {\rangle }_{c_1}\to |g{\rangle }_q|2\alpha {\rangle }_{c_1}\mathrm{,}|e{\rangle }_q|2\alpha {\rangle }_{c_1}\to |e{\rangle }_q|2\alpha {\rangle }_{c_1}\mathrm{.}\end{array}

After returning to the original interaction picture by performing a unitary transformation \exp \left\{\begin{array}{c}-\text{i}\left[{\lambda }_q\left({\widehat{a}}_1^{+}{\widehat{a}}_1+\frac{1}{2}\right){\sigma }_{z\mathrm{,}q}\right]t\end{array}\right\}, we have

\begin{array}{}\text{(22)}& \begin{array}{l}|g{\rangle }_q|2\alpha {\rangle }_{c_1}\to {\text{e}}^{\text{i}\frac{{\lambda }_q}{2}t}|g{\rangle }_q|2\alpha {\text{e}}^{\text{i}{\lambda }_{q}t}{\rangle }_{c_1}\mathrm{,}\\ |e{\rangle }_q|2\alpha {\rangle }_{c_1}\to {\text{e}}^{-\text{i}\frac{{\lambda }_q}{2}t}|e{\rangle }_q|2\alpha {\text{e}}^{-\text{i}{\lambda }_{q}t}{\rangle }_{c_1}\mathrm{.}\end{array}\end{array}

The above results (3), (5), (7), (11), (15), (20), and (22) will be used to transfer quantum entangled states from a group of SC qubits to another group of MF qubits, as discussed in the next section.

3 Transfer of entangled states between SC qubits and MF qubits

Let us return to the setup illustrated in Fig. 1. Suppose that the superconducting qutrits {1, 2, 3, ···, N} are initially prepared in a GHZ state a|g⟩₁|g⟩₂ ... |g⟩_N + b|e⟩₁|e⟩₂ ... |e⟩_N, the cavity c is initially in the vacuum state |0⟩_c, the qutrit q is in the ground state |g⟩_q, and each of cavities {1, 2, 3, ···, N } is in a coherent state |α⟩. Thus, the initial state of the whole system can be written as

\begin{array}{}\text{(23)}& \begin{array}{ll}\hfill & {(a|g\rangle }_1|g{\rangle }_2\dots |g{\rangle }_N+b|e{\rangle }_1|e{\rangle }_2\dots |e{\rangle }_N)\hfill \\ \hfill & \cdot |g{\rangle }_q|0{\rangle }_c|\alpha {\rangle }_{c_1}|\alpha {\rangle }_{c_2}\dots |\alpha {\rangle }_{c_N}\mathrm{,}\hfill \end{array}\end{array}

where subscripts c₁, c₂, ···, c_N represent cavities 1, 2, ···, and N, respectively. We suppose all qutrits, i.e., qutrits (1, 2, 3, ···, N) and q, are initially decoupled from their respective cavities. Note that the qutrit-cavity coupling and decoupling can be achieved by prior adjustment of the level spacings of qutrits or prior adjustment of the frequencies of cavities. For superconducting qutrits and microwave cavities, the level spacings of qutrits and the frequencies of cavities can be rapidly adjusted within a few nanoseconds [4–6, 75, 76].

The procedure for transferring the entangled states from SC qubits to MF qubits is listed below.

Step 1. Apply a microwave pulse (with an initial phase ϕ_l = −π/2) to qutrit l (l = 2, 3, ···, N). The pulse is resonant with the |g⟩_l ↔ |e⟩_l transition of qutrit l [Fig. 2(a)]. For a duration time t₁ = π/(4Ω_l) and according to Eq. (3), one can obtain the state transformation |g⟩_l → |+⟩_l and |e⟩_l → |−⟩_l with |+{\rangle }_l={(|e\rangle }_l+|g{\rangle }_l)/\sqrt{2} and |-{\rangle }_l=(|e{\rangle }_l-|g{\rangle }_l)/\sqrt{2}. The state (23) thus becomes

\begin{array}{}\text{(24)}& \begin{array}{ll}\hfill & {(a|g\rangle }_1|+{\rangle }_2|+{\rangle }_3\dots |+{\rangle }_N+b|e{\rangle }_1|-{\rangle }_2|-{\rangle }_3\dots |-{\rangle }_N)\hfill \\ \hfill & \cdot |g{\rangle }_q|0{\rangle }_c|\alpha {\rangle }_{c_1}|\alpha {\rangle }_{c_2}|\alpha {\rangle }_{c_3}\dots |\alpha {\rangle }_{c_N}\mathrm{.}\hfill \end{array}\end{array}

Step 2. Adjust the frequency of qutrit 1 such that the |g⟩₁ ↔ |e⟩₁ transition of qutrit 1 is resonant with cavity c [Fig. 2(b)]. The interaction Hamiltonian is given by Eq. (4). According to Eq. (5), one has |e⟩₁|0⟩_c → −i|g⟩₁|1⟩_c for an interaction time t₂ = π/(2g_r,1). The state (24) then becomes

\begin{array}{}\text{(25)}& \begin{array}{ll}\hfill & |g{\rangle }_q|g{\rangle }_1{(a|+\rangle }_2|+{\rangle }_3\dots |+{\rangle }_N|0{\rangle }_c-\text{i}b|-{\rangle }_2|-{\rangle }_3\hfill \\ \hfill & \dots |-{\rangle }_N|1{\rangle }_c)\otimes |\alpha {\rangle }_{c_1}|\alpha {\rangle }_{c_2}|\alpha {\rangle }_{c_3}\dots |\alpha {\rangle }_{c_N}\mathrm{.}\hfill \end{array}\end{array}

After this step of operation, we adjust the level configuration of qutrit 1 and make the qutrit 1 decoupled from the cavity c.

Step 3. Adjust the frequency of qutrit l (l = 2, 3, ···, N) such that the |e⟩_l ↔ |f⟩_l transition of qutrit l is dispersively coupled to cavity c [Fig. 2(d)]. The effective Hamiltonian and state evolution of this operation have been given by Eqs. (10) and (11), respectively. According to Eq. (11), one has |g⟩_l|1⟩_c → |g⟩_l|1⟩_c and |e⟩_l|1⟩_c → −|e⟩_l|1⟩_c (i.e., |−⟩_l|1⟩_c → −|+⟩_l|1⟩_c) for an operational time t₃ = π/λ_l (l = 2, 3, ···, N). Thus, the state (25) changes to

\begin{array}{}\text{(26)}& \begin{array}{ll}\hfill & |g{\rangle }_q|g{\rangle }_1|+{\rangle }_2|+{\rangle }_3\dots |+{\rangle }_N[a|0{\rangle }_c-{(-1)}^{N-1}\text{i}b|1{\rangle }_c]\hfill \\ \hfill & \otimes |\alpha {\rangle }_{c_1}|\alpha {\rangle }_{c_2}|\alpha {\rangle }_{c_3}\dots |\alpha {\rangle }_{c_N}\mathrm{.}\hfill \end{array}\end{array}

After this step of operation, we adjust the level configuration of qutrit l (l = 2, 3, ···, N) such that the qutrit l is decoupled from the cavity c. Here, we have set λ₂ = λ₃ = ··· = λ_N .

Step 4. Adjust the frequency of qutrit q such that the cavities c and j (j = 1, 2, 3, ···, N) are dispersively coupled to the |g⟩_q ↔ |e⟩_q and |e⟩_q ↔ |f⟩_q transitions of qutrit q, respectively [Fig. 2(e)]. The effective Hamiltonian of this subsystem has been derived in Eq. (15). Under the Hamiltonian (15), the state (26) evolves into

\begin{array}{}\text{(27)}& \begin{array}{ll}\hfill & |g{\rangle }_q|g{\rangle }_1|+{\rangle }_2|+{\rangle }_3\dots |+{\rangle }_N[a|0{\rangle }_c|\alpha {\rangle }_{c_1}|\alpha {\rangle }_{c_2}|\alpha {\rangle }_{c_3}\dots |\alpha {\rangle }_{c_N}]\hfill \\ \hfill & -{(-1)}^{N-1}\text{i}b{\text{e}}^{-\text{i}{\lambda }_c{t}_4}|1{\rangle }_c\hfill \\ \hfill & \otimes |\alpha {\text{e}}^{\text{i}{\chi }_1{t}_4}{\rangle }_{c_1}|\alpha {\text{e}}^{\text{i}{\chi }_2{t}_4}{\rangle }_{c_2}|\alpha {\text{e}}^{\text{i}{\chi }_3{t}_4}{\rangle }_{c_3}\dots |\alpha {\text{e}}^{\text{i}{\chi }_N{t}_4}{\rangle }_{c_N}]\mathrm{.}\hfill \end{array}\end{array}

For an evolution time t₄ = π/χ_j (j = 1, 2, 3, ···, N), the state (27) turns into

\begin{array}{}\text{(28)}& \begin{array}{l}|g{\rangle }_q|g{\rangle }_1|+{\rangle }_2|+{\rangle }_3\dots |+{\rangle }_N{(a|0\rangle }_c|\alpha {\rangle }_{c_1}|\alpha {\rangle }_{c_2}|\alpha {\rangle }_{c_3}\dots |\alpha {\rangle }_{c_N}\\ -{(-1)}^{N-1}\text{i}b{\text{e}}^{-\text{i}\phi }|1{\rangle }_c|-\alpha {\rangle }_{c_1}|-\alpha {\rangle }_{c_2}|-\alpha {\rangle }_{c_3}\dots |-\alpha {\rangle }_{c_N})\end{array}\end{array}

with φ = λ_cπ/χ_j (j = 1, 2, 3, ···, N). After completing this operation, the level configuration of qutrit q should be adjusted such that the qutrit q is decoupled from cavity c and j (j = 1, 2, 3, ···, N). Here, we have assumed χ₁ = χ₂ = ··· = χ_N.

Step 5. Adjust the frequency of qutrit q such that the |g⟩_q ↔ |e⟩_q transition of qutrit q is resonant with cavity c [Fig. 2(c)]. According to Eq. (7), one has |g⟩_q |1⟩_c → i|e⟩_q |0⟩_c for interaction time t₅ = 3π/(2g_r,q). The state (28) then becomes

\begin{array}{}\text{(29)}& \begin{array}{ll}\hfill & |g{\rangle }_1|+{\rangle }_2|+{\rangle }_3\dots |+{\rangle }_N|0{\rangle }_c(a|g{\rangle }_q|\alpha {\rangle }_{c_1}|\alpha {\rangle }_{c_2}|\alpha {\rangle }_{c_3}\dots |\alpha {\rangle }_{c_N}\hfill \\ \hfill & +{(-1)}^{N-1}b{\text{e}}^{-\text{i}\phi }|e{\rangle }_q|-\alpha {\rangle }_{c_1}|-\alpha {\rangle }_{c_2}|-\alpha {\rangle }_{c_3}\dots |-\alpha {\rangle }_{c_N})\mathrm{.}\hfill \end{array}\end{array}

After this operation, one adjusts the level configuration of qutrit q such that the qutrit q is decoupled from the cavity c. Apply the displacement operator \widehat{D}(\alpha ) to cavity 1 such that |\alpha {\rangle }_{c_1}\to |2\alpha {\rangle }_{c_1} and |-\alpha {\rangle }_{c_1}\to |0{\rangle }_{c_1}, the state (29) thus becomes

\begin{array}{}\text{(30)}& \begin{array}{l}|g{\rangle }_1|+{\rangle }_2|+{\rangle }_3\dots |+{\rangle }_N|0{\rangle }_c{(a|g\rangle }_q|2\alpha {\rangle }_{c_1}|\alpha {\rangle }_{c_2}|\alpha {\rangle }_{c_3}\dots |\alpha {\rangle }_{c_N}\\ +{(-1)}^{N-1}b{\text{e}}^{-\text{i}\phi }|e{\rangle }_q|0{\rangle }_{c_1}|-\alpha {\rangle }_{c_2}|-\alpha {\rangle }_{c_3}\dots |-\alpha {\rangle }_{c_N})\mathrm{.}\end{array}\end{array}

Step 6. Adjust the frequency of qutrit q such that the |g⟩_q ↔ |e⟩_q transition of qutrit q is dispersively coupled to cavity 1 and a microwave pulse, respectively [Fig. 2(f)]. According to Eqs. (20) and (22), we have |e{\rangle }_q|0{\rangle }_{c_1}\to -{\text{ie}}^{\text{i}({\varphi }_p+\frac{{\lambda }_q}{2}{t}_6)}|g{\rangle }_q|0{\rangle }_{c_1} and |g{\rangle }_q|2\alpha {\rangle }_{c_1}\to {\text{e}}^{\text{i}\frac{{\lambda }_q}{2}{t}_6}|g{\rangle }_q|2\alpha {\text{e}}^{\text{i}{\lambda }_q{t}_6}{\rangle }_{c_1} for an interaction time t₆ = π/(2Ω_p).The state (30) then turns into

\begin{array}{}\text{(31)}& \begin{array}{ll}\hfill & |g{\rangle }_q|g{\rangle }_1|+{\rangle }_2|+{\rangle }_3\dots |+{\rangle }_N|0{\rangle }_c(a|2\alpha {\rangle }_{c_1}|\alpha {\rangle }_{c_2}|\alpha {\rangle }_{c_3}\dots |\alpha {\rangle }_{c_N}\hfill \\ \hfill & +b|0{\rangle }_{c_1}|-\alpha {\rangle }_{c_2}|-\alpha {\rangle }_{c_3}\dots |-\alpha {\rangle }_{c_N})\mathrm{,}\hfill \end{array}\end{array}

where we have set ϕ_p−φ = ±π/2 and λ_q t₆ = 2π, and a common phase factor e^iπ is dropped off. Here, ϕ_p−φ = π/2 for an odd N, while ϕ_p−φ = −π/2 for an even N. Then, adjust the level configuration of qutrit q so that the qutrit q is decoupled from cavity 1. Finally, apply the displacement operator \widehat{D}(-\alpha ) to cavity 1 such that |2\alpha {\rangle }_{c_1}\to |\alpha {\rangle }_{c_1} and |0{\rangle }_{c_1}\to |-\alpha {\rangle }_{c_1}. The state (31) thus becomes

\begin{array}{}\text{(32)}& \begin{array}{ll}\hfill & |g{\rangle }_q|g{\rangle }_1|+{\rangle }_2|+{\rangle }_3\dots |+{\rangle }_N|0{\rangle }_c(a|\alpha {\rangle }_{c_1}|\alpha {\rangle }_{c_2}|\alpha {\rangle }_{c_3}\dots |\alpha {\rangle }_{c_N}\hfill \\ \hfill & +b|-\alpha {\rangle }_{c_1}|-\alpha {\rangle }_{c_2}|-\alpha {\rangle }_{c_3}\dots |-\alpha {\rangle }_{c_N})\mathrm{,}\hfill \end{array}\end{array}

which shows that the quantum entangled state of N SC qubits has been transferred onto N MF qubits.

As described in the above steps, the cavity–qutrit coupling or decoupling is achieved by adjusting the level spacings of the qutrits. Because the rapid tuning of cavity frequencies has been reported in experiments [75, 76], one can also tune the frequencies of cavities such that the cavities are coupled to or decoupled from the qutrits.

Before ending this section, several points need to be addressed as follows:

(i) In Step 3 above, we have set λ₂ = λ₃ = ··· = λ_N, i.e.,

\begin{array}{}\text{(33)}& \begin{array}{l}\frac{g_2^2}{{\text{Δ}}_2}=\frac{g_3^2}{{\text{Δ}}_3}=\cdots =\frac{g_N^2}{{\text{Δ}}_N}\mathrm{.}\hfill \end{array}\end{array}

Because of ∆_l = ω_fe,l−ω_c (l = 2, 3, ···, N), the condition (33) can be met by carefully selecting the detunings ∆₂,∆₃, ···,∆_N via adjusting the level spacing of qutrits (2, 3, ···, N).

(ii) In Step 4 above, we have assumed χ₁ = χ₂ = ··· = χ_N−1 = χ_N. This condition can be further written as

\begin{array}{}\text{(34)}& \begin{array}{l}\frac{{\mu }_1(|{\delta }_c|+{\delta }_{c_1})}{\sqrt{{\delta }_1}{\delta }_{c_1}}=\frac{{\mu }_2(|{\delta }_c|+{\delta }_{c_2})}{\sqrt{{\delta }_2}{\delta }_{c_2}}=\cdots =\frac{{\mu }_N(|{\delta }_c|+{\delta }_{c_N})}{\sqrt{{\delta }_N}{\delta }_{c_N}}\mathrm{,}\hfill \end{array}\end{array}

which can be satisfied by selecting the detunings {\delta }_{c_j}, δ_j (j = 1, 2, ···, N), and δ_c through adjusting the level spacing of qutrits or tuning the frequencies of cavity c and cavity j (j = 1, 2, ···, N).

(iii) In Step 6 above, we have set t₆ = π/(2Ω_p) and λ_q t₆ = 2π, resulting in \frac{2\pi }{{\lambda }_q}=\frac{\pi }{2{\text{Ω}}_p}. This condition can be further expressed as

\begin{array}{}\text{(35)}& {\text{Ω}}_p=\frac{g_q^2}{4{\delta }_q}\mathrm{,}\end{array}

which can be easily met by adjusting the Rabi frequency Ω_p of the driving pulse.

(iv) Since no measurement is used in the above, all operations involved in the state transfer are unitary operations. Thus, the entangled states can be transferred back from the MF qubits to the SC qubits by performing reverse operations.

4 Possible experimental implementation

In this section, as an example, we investigate the experimental feasibility for transferring entangled states from two SC qubits (1, 2) to two MF qubits (1, 2) [i.e., converting the initial state (23) into the target state (32) with N = 2], by a setup consisting of three one-dimensional (1D) superconducting resonators (1, 2, c) and three superconducting transmon qutrits (1, 2, q) (Fig. 3). In Steps 5 and 6, we have used the displacement operators \widehat{D}(\alpha ) and \widehat{D}(-\alpha ). The displacement operators \widehat{D}(\alpha ) and \widehat{D}(-\alpha ) are given by \widehat{D}(\alpha )={\text{e}}^{\alpha {\widehat{a}}_{c_1}^{†}-{\alpha }^{*}{\widehat{a}}_{c_1}} and \widehat{D}(-\alpha )={\text{e}}^{{\alpha }^{*}{\widehat{a}}_{c_1}-\alpha {\widehat{a}}_{c_1}^{†}} [77]. For the realization of the displacement operators \widehat{D}(\alpha ) and \widehat{D}(-\alpha ) in experiments, one can use a microwave pulse to drive the cavity 1 [78]. The corresponding Hamiltonian can be expressed as \text{Ω}({\text{e}}^{\text{i}{\varphi }_1}{\widehat{a}}_{c_1}+{\text{e}}^{-\text{i}{\varphi }_1}{\widehat{a}}_{c_1}^{†}). One can choose a phase ϕ₁ = −π/2 or ϕ₁ = π/2 depending on the displacement operator \widehat{D}(\alpha ) or \widehat{D}(-\alpha ). Accordingly, we have α = Ωt. For a transmon qutrit, the |g⟩ ↔ |f⟩ transition is forbidden or very weak [79]. Thus, the couplings of the cavity/pulse with the |g⟩ ↔ |f⟩ transition can be neglected. Hereafter, the terms cavity and resonator are used interchangeably.

From the previous Section 3, one can see that six basic interactions are used in the transfer of entangled states, i.e., the six basic interactions described by the Hamiltonians H₁, H₂, H₃, H₄, H₅, and H₆ above. When the inter-cavity crosstalk and the unwanted interactions are considered, these Hamiltonians are modified as follows.

(i) H_1^{\prime }=H_1+\delta H_1 where δH₁ describes the unwanted interaction between the pulse and the |e⟩_l ↔ |f⟩_l transition of qutrit l (l = 2) [Fig. 4(a)]. The Hamiltonian δH₁ is given by

\begin{array}{}\text{(36)}& \begin{array}{l}\delta H_1={\widetilde{\text{Ω}}}_l\left({\text{e}}^{-\text{i}{\varphi }_l}{\text{e}}^{-\text{i}{\text{Δ}}_{p\mathrm{,}l}}{\sigma }_{fe\mathrm{,}l}^{+}+h\mathrm{.}c\mathrm{.}\right)\mathrm{,}\hfill \end{array}\end{array}

where {\sigma }_{fe\mathrm{,}l}^{+}=|f{\rangle }_l\langle e|, {\widetilde{\text{Ω}}}_l is the pulse Rabi frequency associated with the |e⟩_l ↔ |f⟩_l transition of the qutrit l, and {\text{Δ}}_{p\mathrm{,}l}={\omega }_p^{\prime }-{\omega }_{fe\mathrm{,}l}={\omega }_{eg\mathrm{,}l}-{\omega }_{fe\mathrm{,}l}>0 is the detuning between the pulse frequency {\omega }_p^{\prime } and the |e⟩_l ↔ |f⟩ _l transition frequency of the qutrit l.

(ii) H_2^{\prime }=H_2+\delta H_2, H_3^{\prime }=H_3+\delta H_3, where δH₂ describes the unwanted interaction between cavity c and the |e⟩₁ ↔ |f⟩₁ transition of qutrit 1 [Fig. 4(b)], while δH₃ describes the unwanted interaction between cavity c and the |e⟩_q ↔ |f⟩_q transition of qutrit q [Fig. 4(c)]. The expression of Hamiltonians δH₂ and δH₃ are given by

\begin{array}{}\text{(37)}& \begin{array}{l}\delta H_2={\tilde{g}}_{r\mathrm{,1}}\left({\text{e}}^{-\text{i}{\text{Δ}}_{r\mathrm{,1}}t}{\widehat{a}}_c{\sigma }_{fe\mathrm{,1}}^{+}+h\mathrm{.}c\mathrm{.}\right)\mathrm{,}\\ \delta H_3={\tilde{g}}_{r\mathrm{,}q}\left({\text{e}}^{-\text{i}{\text{Δ}}_{r\mathrm{,}q}t}{\widehat{a}}_c{\sigma }_{fe\mathrm{,}q}^{+}+h\mathrm{.}c\mathrm{.}\right)\mathrm{,}\end{array}\end{array}

where {\sigma }_{fe\mathrm{,1}}^{+}=|f{\rangle }_1\langle e|, {\sigma }_{fe\mathrm{,}q}^{+}=|f{\rangle }_q\langle e|, {\tilde{g}}_{r\mathrm{,1}}({\tilde{g}}_{r\mathrm{,}q}) is the off-resonant coupling strength between cavity c and the |e⟩₁ ↔ |f⟩₁ (|e⟩_q ↔ |f⟩_q) transition of qutrit 1 (q), and ∆_r,1 = ω_c−ω_fe,1 = ω_eg,1−ω_fe,1 > 0 (∆_r,q = ω_c−ω_{f e,q} = ω_eg,q−ω_fe,q > 0) is the detuning between the frequency of cavity c and the |e⟩₁ ↔ |f⟩₁ (|e⟩_q ↔ |f⟩_q) transition frequency of qutrit 1 (q).

(iii) H_4^{\prime }=H_4+\delta H_4 where δH₄ describes the unwanted interaction of cavity c with the |g⟩_l ↔ |e⟩_l transition of the qutrit l (l = 2) [Fig. 4(d)]. The Hamiltonian δH₄ is given by

\begin{array}{}\text{(38)}& \delta H_4={\tilde{g}}_l\left({\text{e}}^{\text{i}{\widetilde{\text{Δ}}}_{l}t}{\widehat{a}}_c{\sigma }_{eg\mathrm{,}l}^{+}+h\mathrm{.}c\mathrm{.}\right)\mathrm{,}\end{array}

where {\tilde{g}}_l is the coupling strength between cavity c and the |g⟩_l ↔ |e⟩_l transition of qutrit l and {\widetilde{\text{Δ}}}_l = ω_eg,l−ω_c = ω_eg,l−ω_fe,l +∆_l is the detuning between the frequency of cavity c and the |g⟩_l ↔ |e⟩_l transition frequency of qutrit l.

(iv) H_5^{\prime }=H_5+\delta H_5+\epsilon, where δH₅ describes the unwanted coupling between cavity c and the |e⟩_q ↔ |f⟩ _q transition of qutrit q as well as the unwanted coupling between cavity j (j = 1, 2) and the |g⟩_q ↔ |e⟩_q transition of qutrit q [Fig. 4(e)]. In addition, ε describes the inter-cavity crosstalk. The expression of δH₅ and ε are given by

\begin{array}{}\text{(39)}& \begin{array}{ll}\delta H_5=\hfill & {\tilde{\mu }}_c\left({\text{e}}^{\text{i}{\tilde{\delta }}_{c}t}{\widehat{a}}_c{\sigma }_{fe\mathrm{,}q}^{+}+h\mathrm{.}c\mathrm{.}\right)\hfill \\ \hfill & +{\displaystyle \sum _{j=1}^2}{\tilde{\mu }}_j\left({\text{e}}^{\text{i}{\tilde{\delta }}_{c_j}t}{\widehat{a}}_j{\sigma }_{eg\mathrm{,}q}^{+}+h\mathrm{.}c\mathrm{.}\right)\mathrm{,}\hfill \end{array}\end{array}

\begin{array}{}\text{(40)}& \begin{array}{ll}\epsilon =\hfill & {\displaystyle \sum _{j=1}^2}{\mu }_{cj}\left({\text{e}}^{-\text{i}{\widetilde{\text{Δ}}}_{cj}t}{\widehat{a}}_c{\widehat{a}}_j^{†}+h\mathrm{.}c\mathrm{.}\right)\hfill \\ \hfill & +{\mu }_{12}\left({\text{e}}^{-\text{i}{\widetilde{\text{Δ}}}_{12}t}{\widehat{a}}_1{\widehat{a}}_2^{†}+h\mathrm{.}c\mathrm{.}\right)\mathrm{,}\hfill \end{array}\end{array}

where {\tilde{\mu }}_c({\tilde{\mu }}_j) is the coupling strength between cavity c (j) and the |e⟩_q ↔ |f⟩_q (|g⟩_q ↔ |e⟩_q) transition of qutrit q, {\tilde{\delta }}_c = ω_fe,q−ω_c = ω_fe,q−ω_eg,q + δ_c and {\tilde{\delta }}_{c_j}={\omega }_{eg\mathrm{,}q}-{\omega }_{c_j}={\omega }_{eg\mathrm{,}q}-{\omega }_{fe\mathrm{,}q}+{\delta }_{c_j}(j=\mathrm{1,}2) are detunings. µ_cj (µ₁₂) is the coupling strength between cavities c and j (1 and 2), {\widetilde{\text{Δ}}}_{cj}={\omega }_c-{\omega }_{c_j} and {\widetilde{\text{Δ}}}_{12}={\omega }_{c_1}-{\omega }_{c_2}={\delta }_{c_2}-{\delta }_{c_1} are detunings.

(v) H_6^{\prime }=H_6+\delta H_6, where δH₆ represents the unwanted coupling between cavity 1 and the |e⟩_q ↔ |f⟩_q transition of qutrit q as well as the unwanted interaction between the pulse and the |e⟩_q ↔ |f⟩_q transition of qutrit q [Fig. 4(f)]. The expression of δH₆ is given by

\begin{array}{}\text{(41)}& \delta H_6={\tilde{g}}_q{\text{e}}^{\text{i}{\tilde{\delta }}_{q}t}{\widehat{a}}_1{\sigma }_{fe\mathrm{,}q}^{+}+{\widetilde{\text{Ω}}}_p{\text{e}}^{-\text{i}[({\omega }_p-{\omega }_{fe\mathrm{,}q})t+{\varphi }_p]}{\sigma }_{fe\mathrm{,}q}^{+}+h\mathrm{.}c\mathrm{.,}\end{array}

where {\tilde{g}}_q is the coupling strength between cavity 1 and the |e⟩_q ↔ |f⟩_q transition of qutrit q, {\widetilde{\text{Ω}}}_p is the Rabi frequency of the pulse (associated the |e⟩_q ↔ |f⟩_q transition of qutrit q), and {\tilde{\delta }}_q={\omega }_{fe\mathrm{,}q}-{\omega }_{c_1}={\omega }_{fe\mathrm{,}q}-{\omega }_{eg\mathrm{,}q}+{\delta }_q is the detuning.

After taking into account the qutrit decoherence and the cavity decay, the dynamics of the lossy system is determined by the Markovian master equation

\begin{array}{}\text{(42)}& \begin{array}{ll}\frac{\text{d}\rho }{\text{d}t}=\hfill & -\text{i}\left[H_{k^{\prime }}\mathrm{,}\rho \right]+{\displaystyle \sum _{l=\mathrm{1,2,}c}}{\kappa }_l\mathcal{L}\left[{\widehat{a}}_l\right]\hfill \\ \hfill & \begin{array}{l}+{\displaystyle \sum _{j=\mathrm{1,2,}q}}({\gamma }_{fe\mathrm{,}j}\mathcal{L}\left[{\sigma }_{fe\mathrm{,}j}^{-}\right]+{\gamma }_{fg\mathrm{,}j}\mathcal{L}\left[{\sigma }_{fg\mathrm{,}j}^{-}\right]\\ +{\gamma }_{eg\mathrm{,}j}\mathcal{L}\left[{\sigma }_{eg\mathrm{,}j}^{-}\right])\end{array}\hfill \\ \hfill & +{\displaystyle \sum _{j=\mathrm{1,2,}q}}\left({\gamma }_{\phi e\mathrm{,}j}\mathcal{L}\left[{\sigma }_{ee\mathrm{,}j}\right]+{\gamma }_{\phi f\mathrm{,}j}\mathcal{L}\left[{\sigma }_{ff\mathrm{,}j}\right]\right)\mathrm{,}\hfill \end{array}\end{array}

where ρ is the density matrix of system, H_k^{\prime } (with k = 1, 2, 3, 4, 5, 6) are the modified Hamiltonians given above, \begin{array}{l}\mathcal{L}[\text{Λ}]=\text{Λ}\rho {\text{Λ}}^{+}-{\text{Λ}}^{+}\text{Λ}\rho /2-\rho {\text{Λ}}^{+}\text{Λ}/2\hfill \end{array}, {\sigma }_{fe\mathrm{,}j}^{-}=|e{\rangle }_j\langle f|, {\sigma }_{fg\mathrm{,}j}^{-}=|g{\rangle }_j\langle f|, {\sigma }_{eg\mathrm{,}j}^{-}=|g{\rangle }_j\langle e|, σ_ee,j = |e⟩_j ⟨e|, and σ_ff,j = |f⟩_j ⟨f| (j = 1, 2, q). In addition, κ_l is the decay rate of the cavity l (l = 1, 2, c). γ_eg,j is the energy relaxation rate of the state |e⟩ for qutrit j, γ_fe,j and γ_fg,j are respectively the energy relaxation rates of the state |f⟩ of qutrit j for the decay paths |f⟩ → |e⟩ and |f⟩ → |g⟩, γ_φe,j and γ_φf,j are the dephasing rates of the levels |e⟩_j and |f⟩_j of qutrit j, respectively.

The fidelity of the operation can be evaluated by \mathcal{F}=\sqrt{\langle {\psi }_{\text{id}}|\rho |{\psi }_{\text{id}}\rangle }, where |ψ_id⟩ is the ideal target state given by \begin{array}{l}|g{\rangle }_q|g{\rangle }_1|+{\rangle }_2|0{\rangle }_c{(a|\alpha \rangle }_{c_1}|\alpha {\rangle }_{c_2}+b|-\alpha {\rangle }_{c_1}|-\alpha {\rangle }_{c_2})\hfill \end{array} with N = 2 according to Eq. (32). In addition, the initial state of the whole systems is given by {(a|g\rangle }_1|g{\rangle }_2+b|e{\rangle }_1|e{\rangle }_2)|g{\rangle }_q|0{\rangle }_c|\alpha {\rangle }_{c_1}|\alpha {\rangle }_{c_2} according to Eq. (23).

Typically, a transmon qutrit has small anharmonicity, but the recent experimental results show that the anharmonicity can be made to be ∼ 720 MHz [80]. Accordingly, the transmon anharmonicity is chosen by 700 MHz in our numerical simulation. As an example, consider ∆_p,l/(2π) =∆_r,1/(2π) =∆_r,q/(2π) = 0.7 GHz. In addition, we choose g_r,1/(2π) = g_r,q/(2π) = g_l/(2π) = g_q/(2π) = 50 MHz, µ_c/(2π) = 100 MHz, µ₁/(2π) = 76 MHz, Ω_l/(2π) = 60 MHz,∆_l/(2π) = δ_q/(2π) = 500 MHz, δ_c/(2π) = −500 MHz, {\delta }_{c_1}/(2\pi )=620\text{MHz}, {\delta }_{c_2}/(2\pi )=570\text{MHz}. According to Eq. (35), one has Ω_p/(2π) = 1.25 MHz. For the parameters chosen here, we have φ ≈ 2π according to φ = λ_cπ/χ₁. Accordingly, we set ϕ_p ≈ 1.5π for ϕ_p−φ = −π/2. For the detunings \left\{{\delta }_c\mathrm{,}{\delta }_{c_1}\mathrm{,}{\delta }_{c_2}\right\} and the coupling constant µ₁ chosen here, we have µ₂/(2π) ≈ 56 MHz according to Eq. (34). The coupling strengths here are available in experiments [81]. For transmon qutrits [79], one has {\tilde{g}}_{r\mathrm{,1}}=\sqrt{2}{g}_{r\mathrm{,1}}, {\tilde{g}}_{r\mathrm{,}q}=\sqrt{2}{g}_{r\mathrm{,}q}, {\tilde{g}}_l=(1/\sqrt{2})g_l, {\widetilde{\text{Ω}}}_l=\sqrt{2}{\text{Ω}}_l, {\tilde{\mu }}_c=\sqrt{2}{\mu }_c, {\tilde{\mu }}_1=(1/\sqrt{2}){\mu }_1, {\tilde{\mu }}_2=(1/\sqrt{2}){\mu }_2, {\widetilde{\text{Ω}}}_p=\sqrt{2}{\text{Ω}}_p, and {\tilde{g}}_q=\sqrt{2}{g}_q. In addition, we choose µ_c1, µ_c2, µ₁₂ = 0.1 max {µ_c, µ₁, µ₂} [42, 82]. Other parameters used in the numerical simulation are (i) {\gamma }_{eg\mathrm{,}j}^{-1}=3T\mathrm{,}{\gamma }_{fg\mathrm{,}j}^{-1}=10T, {\gamma }_{fe\mathrm{,}j}^{-1}=1.5T, {\gamma }_{\phi e\mathrm{,}j}^{-1}={\gamma }_{\phi f\mathrm{,}j}^{-1}=T(j=\mathrm{1,}\mathrm{2,}q), (ii) {\kappa }_l^{-1}={\kappa }^{-1}(l=\mathrm{1,}\mathrm{2,}c), (iii) a=b=1/\sqrt{2}, (iv) α = 3.5.

We plot Fig. 5 illustrating the fidelity versus T and κ⁻¹ by numerically solving the master equation (42). Figure 5 displays that when T ⩾ 4 µs and κ⁻¹ ⩾ 3 µs, the fidelity exceeds 98.0%. For T = 4 µs, the decoherence times of the transmon qutrits are 4–40 µs, which is a rather conservative case because the decoherence time 55–300 µs for the transmon qutrit has been experimentally demonstrated [83, 84]. The entire operation time is ∼ 0.63 µs, which is much shorter than the decoherence times of transmon qutrits used in our numerical simulations.

The frequency of a superconducting cavity is within a range of 1–15 GHz [5]. Thus, we choose ω_c/(2π) = 5.8 GHz. According to the cavity-transmon detunings chosen above, one has the frequencies of cavities {\omega }_{c_1}/(2\pi )=3.98 GHz and {\omega }_{c_2}/(2\pi )=4.03 GHz. For the decay time of cavities κ⁻¹ = 3 µs, the quality factors of the three cavities are Q_c ∼ 1.09×10⁵, Q₁ ∼ 7.50×10⁴, and Q₂ ∼ 7.59×10⁴, which are available because 1D superconducting resonators with quality factors > 10⁶ have been experimentally demonstrated [85, 86]. The above results show that the high-fidelity transfer of the entangled state from two SC qubits to two MF qubits can be achieved by utilizing current circuit QED technology.

5 Conclusion

We have proposed an approach to transfer an entangled state from a group of superconducting qubits to another group of microwave-field qubits. As shown above, only a coupler qutrit and auxiliary cavity are needed, thus the circuit architecture is simple. By using this proposal, the discrete-variable entangled state of superconducting qubits can be converted into the continuous-variable entangled coherent state of microwave-field qubits. Due to the third level |f⟩ of each qutrit being not populated during the entire operation, the decoherence from qutrits’ higher energy levels is greatly suppressed. Moreover, the entire operation time does not depend on the number of qubits, and the entangled state is deterministically transferred since no measurement is used. Numerical simulation shows that the entangled state of two superconducting qubits can be high-fidelity transferred to two microwave-field qubits within current circuit QED technology. Our proposal is quite general and can be applied to transfer entangled states between other matter qubits and microwave- or optical-field qubits encoded with coherent states.

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