Interfacing a topological qubit with a spin qubit in a hybrid quantum system

TL;DR

This paper proposes a hybrid quantum system that couples a topological Majorana qubit with a spin qubit via a mechanical resonator, enabling quantum state conversion for quantum information processing.

Contribution

It introduces a novel interface between topological and spin qubits using a magnetized torsional cantilever, facilitating quantum state transfer.

Findings

Strong coupling of mechanical mode to both qubits demonstrated

Quantum state conversion via dark state protocol proposed

Potential applications in quantum information processing identified

Abstract

We present and analyze a hybrid quantum system that interfaces a Majorana-hosted semiconductor nanowire with a single nitrogen-vacancy (NV) center via a magnetized torsional cantilever. We show that, the torsional mode of the mechanical resonator can strongly couple to the Majorana qubit and the spin qubit simultaneously, which allows to interface them for quantum state conversion through a dark state protocol. This work provides a promising interface between the topological qubit and the conventional spin qubit, and can find interesting applications in quantum information and computation.

Equations

$$\hat{b}=\frac{1}{2}(\frac{\hat{\theta }}{{\theta }_{zpf}}+\frac{i\hat{L}}{L_{zpf}}),{\hat{b}}^{†}=\frac{1}{2}(\frac{\hat{\theta }}{{\theta }_{zpf}}-\frac{i\hat{L}}{L_{zpf}}),$$

$$H$$

$${\hat{H}}_{TP}=i{E}_l{\hat{\gamma }}_1{\hat{\gamma }}_2+i{E}_m(\theta ){\hat{\gamma }}_2{\hat{\gamma }}_3+i{E}_r{\hat{\gamma }}_3{\hat{\gamma }}_4,$$

$$\hat{H}=-t\sum _{i,\sigma }{\hat{c}}_{i\sigma }^{†}{\hat{c}}_{i+1\sigma }+\sum _{i,\sigma }(-2t+{\mu }_i+b_{i\sigma }){\hat{c}}_{i\sigma }^{†}{\hat{c}}_{i\sigma }$$

$$+\alpha \sum _i({\hat{c}}_{i,\uparrow }^{†}{\hat{c}}_{i+1,\downarrow }-{\hat{c}}_{i,\downarrow }^{†}{\hat{c}}_{i+1,\uparrow }+H.c.)$$

$$+\sum _i(\Delta e^{i\varphi }{\hat{c}}_{i,\uparrow }^{†}{\hat{c}}_{i,\downarrow }^{†}+H.c.)+\sum _i(B{e}^{i\theta }{\hat{c}}_{i\uparrow }^{†}{\hat{c}}_{i,\downarrow }+H.c.).$$

$$E_m(\theta )\approx \frac{|⟨{\varphi }_{{\gamma }_2}^0{e}^{-i\frac{{\theta }_l}{2}{\hat{\sigma }}_z}|H|e^{i\frac{{\theta }_r}{2}{\hat{\sigma }}_z}{\varphi }_{{\gamma }_3}^0⟩|}{\sqrt{⟨{\varphi }_{{\gamma }_2}^0|{\varphi }_{{\gamma }_2}^0⟩⟨{\varphi }_{{\gamma }_3}^0|{\varphi }_{{\gamma }_3}^0⟩}}.$$

$$i{\hat{\gamma }}_1{\hat{\gamma }}_2\to -{\hat{\sigma }}_{TP}^z,i{\hat{\gamma }}_2{\hat{\gamma }}_3\to -{\hat{\sigma }}_{TP}^x,i{\hat{\gamma }}_3{\hat{\gamma }}_4\to {\hat{\sigma }}_{TP}^z.$$

$${\hat{H}}_{Tor-TP}=\hslash {\omega }_m{\hat{b}}^{†}\hat{b}+\hslash {\omega }_{TP}{\hat{\sigma }}_{TP}^z-\hslash g({\hat{b}}^{†}+\hat{b}){\hat{\sigma }}_{TP}^x,$$

$${\hat{H}}_{Tor-TP}=\hslash {\omega }_m{\hat{b}}^{†}\hat{b}+\hslash {\omega }_{TP}{\hat{\sigma }}_{TP}^z-\hslash g({\hat{b}}^{†}{\hat{\sigma }}_{TP}^{-}+\hat{b}{\hat{\sigma }}_{TP}^{+}).$$

$${\hat{H}}_{NV}=\hslash D_{gs}{\hat{S}}_z^2+g_e{\mu }_B[({\vec{B}}_{mg}(\theta )+{\vec{B}}_{dr})\cdot \hat{\vec{S}}+B_z{\hat{S}}_z],$$

$${\hat{H}}_{Tor-NV}=\hslash {\omega }_m{\hat{b}}^{†}\hat{b}+{\hat{H}}_{NV}+\hslash \lambda ({\hat{b}}^{†}+\hat{b}){\hat{S}}_z.$$

$${\hat{H}}_{NV}$$

$${\hat{H}}_{Tor-NV}$$

$${\hat{H}}_{Tor-NV}=\hslash {\omega }_m{\hat{b}}^{†}\hat{b}+\hslash {\omega }_{NV}{\hat{\sigma }}_{NV}^z+\hslash {\lambda }_e({\hat{b}}^{†}{\hat{\sigma }}_{NV}^{-}+\hat{b}{\hat{\sigma }}_{NV}^{+}),$$

$${\hat{H}}_{sys}$$

$$\frac{d\hat{\rho }(t)}{dt}$$

$${\hat{H}}_{sys}=\hslash {\tilde{\omega }}_{+}{\hat{c}}_{+}^{†}{\hat{c}}_{+}+\hslash {\tilde{\omega }}_{-}{\hat{c}}_{-}^{†}{\hat{c}}_{-}+\hslash {\tilde{\omega }}_{dk}{\hat{c}}_{dk}^{†}{\hat{c}}_{dk},$$

Full text

I introduction

Majorana Fermion quasiparticles (MFs) realized in condensed matter systems have attracted great interest in recent years Elliott and Franz (2015); Lutchyn et al. (2018); Hao et al. (2018); Wilczek (2009); Litinski and von Oppen (2018), because they provide a very intriguing platform for quantum science and technology. The proposed realizations include $p_{x}+ip_{y}$ superconductors Read and Green (2000), edges of 2D topological insulators Fu and Kane (2008), quantum Hall states at filling factor 5/2 Nayak et al. (2008), and 1D semiconducting quantum wires with strong spin-orbit interactions Sau et al. (2010); Alicea (2010); Lutchyn et al. (2010); Chen et al. (2014); Oreg et al. (2010). In these systems, the spin-orbit coupled semiconductor wires with proximity-induced superconductivity are of particular interests for their simple architectures. For the 1D quantum nanowire, the MFs are predicated to localize through the competition between the superconducting proximity effect and the Zeeman splitting. Recently, several experiments aiming at establishing the existence of MFs have been reported Mourik et al. (2012); Deng et al. (2012); Das et al. (2012); Rokhinson et al. (2012). Meanwhile, it has been shown that the Majorana qubits can be controlled by integrating with the conventional ones, such as the flux qubits and the spin qubits Hassler et al. (2010); Xue et al. (2009); Flensberg (2011); Leijnse and Flensberg (2011); Hoffman et al. (2016); Plugge et al. (2016).

Topological qubits are one of the most promising candidates for quantum computation due to their extreme robustness against local fluctuations. However, the technical challenges are considerable: (i). it is well known that braiding operations of MFs cannot generate a complete set of universal quantum logic gates required for quantum computation; (ii). it is still challenging to operate and readout the quantum states of the MFs. To overcome these problems, several types of hybrid quantum architectures coupling the topological qubits to the conventional ones have been proposed. One potential route is controlling the microscopic wave function of the superconductors, which allows the strong coupling between a topological qubit and a flux qubit, or a mechanical oscillator Zhang and Yu (2013); Xue et al. (2013); Hong et al. (2013). Other promising schemes include utilizing the Aharonov-Casher effect Jiang et al. (2011a); Bonderson and Lutchyn (2011); Vishveshwara (2011), or the fractional Josephson effect Ohm and Hassler (2014); Cottet et al. (2013); Xue et al. (2015). So far, for all of these protocols, further experimental demonstrations are needed to verify their feasibility.

For the conventional qubits, nitrogen-vacancy (NV) centers in diamond are more attractive due to their unique features, such as the fast microwave manipulation, optical preparation and detection, and long coherence time even at room temperature Doherty et al. (2013); Hong et al. (2012); Bar-Gill et al. (2013). In recent years, there has been considerable effort to integrate NV centers in hybrid quantum systems, allowing the preparation of fantastic quantum states Xia and Twamley (2016); Yang et al. (2011a); Li et al. (2017a); Zhou et al. (2017); Yin et al. (2013); Chen et al. (2011), design of quantum logic gates Zu et al. (2014); Xu et al. (2012); Dong et al. (2017); Cao et al. (2017); Li et al. (2018), and storage or transfer of quantum information Xia and Twamley (2015); Yang et al. (2011b); Lü et al. (2013); Li et al. (2017b); Xiang et al. (2013); You et al. (2014). To further explore the potential of hybrid quantum systems, it will be very appealing to interface topological qubits with NV centers coherently. Integrating the topological degrees of freedom with the complete gate operations of NV spins, will provide a promising multi-qubit platform for quantum computation. However, the energy mismatch between these two kinds of qubits possesses a major obstacle for their coupling.

In this paper, we consider a hybrid quantum device interfacing a topological qubit with a single NV center via a magnetized torsional cantilever. The topological qubit under consideration is realized in a spin-orbit coupled semiconductor nanowire, which is embedded on top of an $s$-wave superconductor to form a semiconductor-superconductor heterostructure. Here the torsional cantilever is realized by an individual single-walled nanotube (SWNT) with a nano-size magnet attached. This nanoscale mechanical structure enables us to combine an isolated NV center with the strong spin-torsion couplings. Together with the strong topology-torsion couplings induced by the Magneto-Josephson effect Kovalev et al. (2014), the mechanical torsional mode can strongly couple to both the Majorana qubit and the NV center. This enables coherent quantum states conversion between the Majorana qubit and the spin qubit via a dark-state protocol. Owing to the topologically protected robustness and the universal quantum logic gates of NV spins, there will be more potential applications utilizing this hybrid architecture.

II Description of the device

As shown schematically in Fig. 1, the hybrid quantum device under consideration consists of a topological qubit realized in a Majorana-hosted semiconductor nanowire, a magnetized torsional cantilever, and a single NV center. The MFs in this architecture are realized in a spin-orbit coupled semiconductor nanowire with proximity-induced superconductivity. This 1D quantum nanowire could be driven to the topological phase regime, and the MF exists as a quasiparticle at the boundary between the topological (T) and non-topological (N) regions Alicea (2010); Sau et al. (2010); Lutchyn et al. (2010); Chen et al. (2014). Here we divide the semiconductor nanowire into three sections to form a TNT junction Lutchyn et al. (2010); Xue et al. (2015), and define the topologically protected MFs as $\hat{\gamma}_{1/2/3/4}$.

In this setup, the magnetized torsional cantilever is realized by a nanomagnet with dimensions $(l,w,t)$ suspended on a SWNT. Specifically, the individual SWNT works as a torsional spring and mechanical support, which allows the suspended nanomagnet to rotate along the tube ($z$) axis (see Fig. 1). Without loss of generality, we assume that the nanomagnet can only rotate along the $z$ axis with a small amplitude, then the Hamiltonian of the torsional cantilever reads $H_{m}=\frac{1}{2}I_{z}\omega_{m}^{2}\hat{\theta}^{2}+\frac{\hat{L}^{2}}{2I_{z}}$, where $\omega_{m}$, $I_{z}$, $\hat{L}$, and $\hat{\theta}$ are the torsional frequency, moment of inertia, angular momentum, and angular displacement, respectively. Similar to the canonical conjugate observable $\hat{X}$ and $\hat{P}$ of the center of mass mode, the rotational degree of freedom can be quantized Kastrup (2006), and the motion of the torsional cantilever can be described by the annihilation and creation operators

[TABLE]

where $L_{zpf}=\sqrt{\hslash I_{z}\omega_{m}/2}$ and $\theta_{zpf}=\sqrt{\hslash/(2I_{z}\omega_{m})}$ denote the zero point fluctuations of the torsional mode.

II.1 MFs in 1D semiconductor nanowire

We first describe the 1D quantum nanowire hosting MFs. As shown in Fig. 1, the semiconductor nanowire is embedded on the top of an $s$-wave superconductor, and an external magnetic field $\vec{b}=b\vec{e}_{z}$ along the $z$ axis is applied. There are two nanomagnets positioned straight above the nanowire: one is stationary in the left region, and the other is free to vibrate in the right region. We assume that the magnetic fields induced by the two magnets is in the $xy$ plane, i.e., $\vec{B}=B\cos\theta\vec{e}_{x}+B\sin\theta\vec{e}_{y}$. We further use the angular displacement of the torsional cantilever to denote the relative field angle, which implies $\theta=\theta_{r}-\theta_{l}$.

The single-particle effective Hamiltonian for the semiconductor wire reads $H_{0}=\hat{p}^{2}/2m^{\ast}-\mu+b\hat{\sigma}_{z}-\alpha(\hat{\vec{\sigma}}\times\hat{\vec{p}})\cdot\hat{z}$ Sau et al. (2010), where $m^{\ast}$, $\mu$, $b$, and $\alpha$ are the effective mass, chemical potential, longitudinal magnetic field, and Rashba spin-orbit coupling strength, respectively. As a result of the proximity effect, the Cooper pairs tunnelling into the nanowire can be described by the Hamiltonian $H_{SC}=\Delta e^{i\phi}\hat{\psi}_{\uparrow}^{\dagger}(x)\hat{\psi}_{\downarrow}^{\dagger}(x)+H.c.$, with $\Delta e^{i\phi}$ the proximity-induced gap. We now introduce the Nambu spinor basis $\hat{\Psi}^{T}=(\hat{\psi}_{\uparrow},\hat{\psi}_{\downarrow},\hat{\psi}_{\downarrow}^{\dagger},-\hat{\psi}_{\uparrow}^{\dagger})$, and model the 1D semiconductor nanowire with a Bogoliubov-de Gennes Hamiltonian

[TABLE]

where the Pauli matrices $\hat{\sigma}^{i}$ and $\hat{\tau}^{i}$ represent the spin and particle-hole sectors, respectively.

As shown in Ref. Kovalev et al. (2014), this semiconductor nanowire can be driven to the topological phase regime when $\Delta^{2}-b^{2}<B^{2}-\mu^{2}$, and to the non-topological phase regime when $\Delta^{2}-b^{2}>B^{2}-\mu^{2}$ (considering only $\Delta^{2}>b^{2}$). Note that Refs. Jiang et al. (2013, 2011b) give the similar results. In particular, the Bogoliubov operators of the quantum nanowire take a general form as $\hat{\gamma}_{k}=u_{k\uparrow}\hat{\psi}_{k\uparrow}+u_{k\downarrow}\hat{\psi}_{k\downarrow}+v_{k\uparrow}\hat{\psi}_{-k\uparrow}^{\dagger}+v_{k\uparrow}\hat{\psi}_{-k\downarrow}^{\dagger}$. For Majorana operators that satisfy $\hat{\gamma}_{k}^{\dagger}=\hat{\gamma}_{k}$, they may only occur in the momentum inversion symmetric condition $k\equiv-k$, i.e., $k=0$. In the quantum nanowire, this indicates a phase boundary between the T and N regions Kotetes et al. (2013).

The exactly zero-energy Majorana modes exist only in the ideal case where the two MFs possess an infinite distance. For a finite nanowire, the MFs gain a small energy as a result of the weak overlap of their wave functions. However, the wave functions are still well localized at the boundaries, and we still have $\hat{\gamma}^{\dagger}\simeq\hat{\gamma}$ in general. Therefore, these MFs with nonzero energies can still be utilized to encode quantum information. We now denote the lengths of the three nanowire sections as $l_{l/m/r}$, then the MFs in the 1D semiconductor nanowire can be described by an effective low energy Hamiltonian

[TABLE]

where $E_{l}$, $E_{m}(\theta)$, and $E_{r}$ are the coupling energies between the adjacent MFs.

II.2 Magneto-Josephson effect

To perform quantum information processing with the topological degree of freedom, different ways to manipulate the MFs have been studied Alicea et al. (2011); Vijay et al. (2015); Saket et al. (2010); Liu and Drummond (2012); Liu and Lobos (2013). One potential route is to utilize the fraction Josephone effect Sticlet et al. (2013); Kwon et al. (2004). Different from the conventional Josephson effect that allows tunneling by Cooper pairs only, the fraction Josephone effect supports single electron tunneling Xue et al. (2015), and possess a 4$\pi$-periodic coupling energy Sun and Liu (2018). Moreover, it has been shown that the Magneto-Josephson effect can be used to realize the coherent coupling between the Majorana qubit and the magnetized torsional cantilever Kovalev et al. (2014). In particular, the spin current induced by the Magneto-Josephson effect serves as a mechanical torque acting on the nanomagnet, which leads to the strong topology-torsion coupling.

For the TNT junction realized in the semiconductor nanowire, both the superconducting phases and the magnetic fields have contributions to the fractional Josephson effects. These contributions can be analysised by the magnetism-superconductivity duality: if we interchange the magnetic terms $\{b,B,\theta,\hat{\sigma}^{i}\}$ with the superconducting terms $\{\mu,\Delta,\phi,\hat{\tau}^{i}\}$, the Hamiltonian (2) takes the same form Jiang et al. (2013). In a general case, the coupling interaction between $\hat{\gamma}_{2}$ and $\hat{\gamma}_{3}$ takes form as $E_{M}\propto E_{M}^{0}\cos(\frac{\theta_{l}-\theta_{r}}{2})\cos(\frac{\phi_{l}-\phi_{r}}{2})$ Kotetes et al. (2013). In our system, we set the superconducting phases as the constants (i.e., $\phi_{l}=\phi_{m}=\phi_{r}$), then the coupling energy $E_{m}$ can be tuned by changing the relative field angle $\theta$.

To study the Magneto-Josephson effect in the TNT junction, we now map the Bogoliubov-de Gennes Hamiltonian (2) to the tight-binding model

[TABLE]

For this lattice version, $t=\hbar^{2}/(2m^{\ast}a^{2})$ is the nearest-neighbor hopping strength, $a$ is the lattice spacing constant, $\alpha$ is the spin-orbit coupling strength, $b_{i\uparrow/\downarrow}=\pm b$ is the Zeeman splitting, and $\hat{c}_{i\sigma}(\hat{c}_{i\sigma}^{\dagger})$ is the annihilation (creation) operator of an electron with spin $\sigma=\uparrow(\downarrow)$.

We conduct numerical simulations by considering a 1D lattice model with 500 grid sites (300, 20, and 180 sites for the left, middle, and right regions, respectively). Here the energy eigenvalues of the system are denoted as $\pm\varepsilon_{i}$, with the corresponding eigenstates $|\phi_{\pm\varepsilon_{i}}\rangle$. As shown in Figs. 2(a) and (b), the eigenstates $|\phi_{\pm\varepsilon_{1}}\rangle$ correspond to the excitations of $\hat{\gamma}_{1}$ and $\hat{\gamma}_{4}$, while the eigenstates $|\phi_{\pm\varepsilon_{2}}\rangle$ correspond to the excitations of $\hat{\gamma}_{2}$ and $\hat{\gamma}_{3}$. Theoretically, the isolated MFs can be obtained by combinations of these eigenstates in the limit $\varepsilon_{1,2}\simeq 0$, i.e., $|\phi_{\gamma_{1,4}}\rangle=\frac{1}{\sqrt{2}}(|\phi_{\varepsilon_{1}}\rangle\pm|\phi_{\varepsilon_{-1}}\rangle),$ and $|\phi_{\gamma_{2,3}}\rangle=\frac{1}{\sqrt{2}}(|\phi_{\varepsilon_{2}}\rangle\pm|\phi_{\varepsilon_{-2}}\rangle)$.

Induced by the overlap of the edge states, the coupling energy between MFs decays exponentially with the separation Das Sarma et al. (2012). According to Ref. Kovalev et al. (2014), we now estimate the coupling energy $E_{m}(\theta)$ through the lowest order perturbation theory, and obtain

[TABLE]

Then, we observe a $4\pi$-periodic dependence between $E_{m}$ and $\theta$, as shown in Fig. 2(c). We also calculate the spin current passing through the middle region as $J_{m}=(e/\hbar)\frac{\partial E^{m}(\theta)}{\partial\theta}$ Pientka et al. (2013), and the result is shown in Fig. 2(d). The relevant parameters are based on InSb quantum wires, i.e., $m^{\ast}=0.015$ $m_{e}$, $\alpha=0.2$ eV$\mathring{A}$, $a=10$ nm, $\Delta=0.5$ meV, and $g\mu_{B}=1.5$ meV/T. Other parameters include $B=200$ mT, $b=200$ mT and $\mu=0$ for the topological regions, and $B=0$, $b=200$ mT and $\mu=-0.6$ meV for the non-topological region. Note that these parameters could be further optimized.

II.3 Topology-torsion couplings

We now discuss the coherent coupling between the Majorana qubit and the torsional motion. Firstly, we combine the MFs to construct two conventional Dirac fermions, and obtain $\hat{f}_{l}=(\hat{\gamma}_{1}+i\hat{\gamma}_{2})/2$ and $\hat{f}_{r}=(\hat{\gamma}_{3}+i\hat{\gamma}_{4})/2$. Then, the two quantum states of the Dirac fermions, corresponding to the number of electrons on either side of the Josephson junction, can function as a physical topological qubit. These two states are denoted as $\hat{n}_{x}=\hat{f}_{x}^{\dagger}\hat{f}_{x}=0$ or $1$ $(x=l,r)$, with parity $\hat{P}_{x}=(-1)^{\hat{f}_{x}^{\dagger}\hat{f}_{x}}=\pm 1$. As braiding operations can not change the total parity $\hat{P}=\hat{P}_{l}\hat{P}_{r}$ of the system, four MFs are usually needed to define a logical topological qubit Zhang and Yu (2013); Ohm and Hassler (2014). We now define the topological qubit in the odd parity subspace, and obtain $|\phi_{TP}\rangle=c_{1}|0\rangle_{l}|1\rangle_{r}+c_{2}|1\rangle_{l}|0\rangle_{r}$. For this topological qubit, we also have Xue et al. (2015)

[TABLE]

In this setup, it is the $\theta$ dependence of the hybridization energy $E_{m}(\theta)$ that leads to the strong topology-torsion coupling Kovalev et al. (2014). In particular, by expanding $E_{m}(\theta)$ in Eq. (3) around $\theta_{0}$ to the first order, we arrive at $E_{m}(\theta)=E_{m}(\theta_{0})+\frac{\partial E_{m}(\theta)}{\partial\theta}\theta_{zpf}(\hat{b}^{\dagger}+\hat{b})$. In what follows, we consider the case in which $E_{m}(\theta_{0})\simeq 0$ (see Fig. 2 (c)), then the Hamiltonian of the topology-torsion dynamics has the form

[TABLE]

where $\omega_{TP}=(E_{r}-E_{l})/\hslash$, and $g=\frac{1}{\hslash}\frac{\partial E_{m}(\theta)}{\partial\theta}\theta_{zpf}$ is the topology-torsion coupling strength. By tuning the hybridization energy $E_{l}$ and $E_{r}$, we can steer the topology-torsion coupling to the near-resonance condition $\omega_{TP}\simeq\omega_{m}$. Under the rotation-wave approximation, the topology-torsion dynamics can be described by

[TABLE]

II.4 Spin-torsion couplings

We now take into consideration the couplings between the mechanical torsional mode and the single NV center. Before proceeding, we note that coherent interactions between NV centers and several types of mechanical resonators have been studied Yin et al. (2015); Li and Nori (2018). These mechanical resonators include clamped cantilevers Rabl et al. (2009), optically trapped nano-diamond crystals Ma et al. (2017); Delord et al. (2017); Hoang et al. (2016), suspended carbon nanotubes Li et al. (2016), etc. In this setup, the single NV center is positioned straight above the magnetized torsional cantilever, as shown in Fig. 3(a). When the nanomagnet rotates, the projection of the magnetic field along the spin component is changed, resulting in a magnetic coupling between the torsional mode and the NV center. The strong magnetic coupling can be achieved by exquisite preparation of the dressed spin states.

An NV center in diamond consists of a substitutional nitrogen atom replacing a carbon atom and an adjacent vacancy. The electronic ground states of a single NV center are spin triplet states denoted as $|m_{s}=0,\pm 1\rangle$, and the zero-field splitting $D_{gs}$ between the degenerate sublevels $|m_{s}=\pm 1\rangle$ and $|m_{s}=0\rangle$ is $2\pi\times 2.87$ GHz. For moderate applied magnetic fields, static magnetic field $\vec{B}_{z}=B_{z}\vec{e}_{z}$ causes Zeeman splitting of the states $|m_{s}=\pm 1\rangle$, while external microwave field $\vec{B}_{dr}=B_{0}\cos\omega_{0}t\vec{e}_{x}$ polarized in the $x$ direction drives the states $|m_{s}=0\rangle$ and $|m_{s}=\pm 1\rangle$, as shown in Fig. 3(b). For convenience, we denote as the $z$ axis the crystalline axis of the NV center.

The magnetic field $\vec{B}_{mg}(\theta)$ induced by the suspended nanomagnet depends on the angular displacement $\theta$. We assume that the NV center is placed in the position where $\vec{B}_{mg}(\theta)$ is along the $x$ axis, i.e., $\vec{B}_{mg}(\theta)=B_{mg}\cos\theta\cdot\vec{e}_{x}+B_{mg}\sin\theta\cdot\vec{e}_{z}$, and $\theta\simeq 0$. Then, the interaction of the NV center with the total magnetic fields (external driving and from the nanomagnet) can be described by

[TABLE]

where $g_{e}=2$ is the NV landé factor, $\mu_{B}=14\text{MHz}$ $\text{mT}^{-1}$ is the Bohr magneton, and $\hat{\vec{S}}=(\hat{S}_{x},\hat{S}_{x},\hat{S}_{z})$ is the spin operator of the NV center. The motion of the torsional cantilever attached with the nanomagnet produces an angular and time dependent magnetic field $\vec{B}_{int}(\theta)\sim\partial_{\theta}\vec{B}_{mg}(\theta)\hat{\theta}\cos\omega_{m}t$, which result in a spin-torsion coupling interaction $\hat{H}_{int}=g_{e}\mu_{B}\vec{B}_{int}(\theta)\cdot\hat{\vec{S}}\simeq\hslash\lambda(\hat{b}^{\dagger}+\hat{b})\hat{S}_{z}\cos\omega_{m}t$, where the coupling constant $\lambda=g_{e}\mu_{B}B_{mg}\theta_{zpf}/\hslash$ is proportional to the magnetic field $B_{mg}$ and the zero-point angular extension $\theta_{zpf}$. In the rotating frame at the frequency $\omega_{m}$, we can acquire the Hamiltonian to describe the spin-torsion dynamics

[TABLE]

Note that the far-off resonant interactions between the NV spin and the static and low frequency components of magnetic fields along the $x$ axis can be ignored. In the basis defined by the eigenstates of $\hat{S}_{z}$, i.e., $\{|m_{s}\rangle,m_{s}=0,\pm 1\},$ with $\hat{S}_{z}|m_{s}\rangle=m_{s}|m_{s}\rangle$, we have

[TABLE]

where $\hat{S}_{x}=\frac{\hslash}{\sqrt{2}}(|0\rangle\langle+1|+|0\rangle\langle-1|+H.c.)$. In the rotating-frame at the driving frequency $\omega_{0}$ and under the rotating-wave approximation, we get

[TABLE]

where $\hslash\Delta_{\pm}=\hslash D_{gs}\pm g_{e}\mu_{B}(B_{z}\pm B_{mg}\sin\theta)-\hslash\omega_{0}$, and $\hslash\Omega=\frac{\sqrt{2}}{4}g_{e}\mu_{B}B_{0}$.

In the following we consider the symmetric detuning condition $\Delta_{+}=\Delta_{-}=\Delta$ for simplicity (e.g., $B_{z}\rightarrow 0$). We now define the bright spin state $|b\rangle=(|+1\rangle+|-1\rangle)/\sqrt{2}$ and the dark spin state $|d\rangle=(|+1\rangle-|-1\rangle)/\sqrt{2}$. Then we find that the Hamiltonian (12) couples the state $|0\rangle$ to the bright state $|b\rangle$, while the dark state $|d\rangle$ remains decoupled. The resulting eigenstates of $\hat{H}_{NV}$ are therefore given by $|d\rangle$ and two dressed states $|g\rangle=\cos(\alpha)|0\rangle-\sin(\alpha)|b\rangle$ and $|e\rangle=\sin(\alpha)|0\rangle+\cos(\alpha)|b\rangle$, with $\tan(2\alpha)=2\sqrt{2}\Omega/\Delta$. The corresponding eigenfrequencies are given by $\omega_{d}=\Delta$, $\omega_{e/g}=(\Delta\pm\sqrt{\Delta^{2}+8\Omega^{2}})/2$, as illustrated in Fig. 3(c).

In the dressed state basis $\{|g\rangle,|d\rangle,|e\rangle\}$, the Hamiltonian (10) can be rewritten as

[TABLE]

where $\lambda_{g}=-\lambda\sin(\alpha)$ and $\lambda_{e}=\lambda\cos(\alpha)$. We further adjust the values of $B_{0}$ to drive the system to the resonance condition $\omega_{ed}\simeq\omega_{m}\ll\omega_{dg}$, and then the far-off-resonance state $|g\rangle$ can be neglected. Under the rotating-wave approximation, the spin-torsion dynamics can be described by the JC interaction Hamiltonian

[TABLE]

where $\hat{\sigma}_{NV}^{-}=|d\rangle\langle e|,\hat{\sigma}_{NV}^{+}=|e\rangle\langle d|$, $\hat{\sigma}_{NV}^{z}=|e\rangle\langle e|-|d\rangle\langle d|$, and $\omega_{NV}=\omega_{e}-\omega_{d}$.

III EXPERIMENTAL PARAMETERS

According to the Hamiltonian (8) and (14), the whole system can be described by

[TABLE]

The first three terms describe the free Hamiltonian of the system, while the last four terms describe two JC interactions: one between the torsional mode and the topological qubit, and the other between the torsional mode and the single NV center. As the torsional cantilever couples to the Majorana qubit and the NV center simultaneously, it is possible to work as a quantum interface for quantum state conversion.

Let us discuss the dynamics of the system in a realistic situation. For this setup, the relaxation ($\Gamma_{1}$) and dephasing ($\Gamma_{2}$) of the Majorana qubit, the decay of the mechanical resonator ($\gamma_{m}$), and also the dephasing of the NV center ($\gamma_{s}$) should be taken into consideration. As a result, the full dynamics of this system can be described by the following master equation

[TABLE]

where $n_{th}=(e^{\hslash\omega_{m}/k_{B}T}-1)^{-1}$ is the thermal phonon number at the environment temperature $T$, and $\mathcal{D}(\hat{o})=\hat{o}\hat{\rho}\hat{o}^{\dagger}-\frac{1}{2}\hat{o}^{\dagger}\hat{o}\hat{\rho}-\frac{1}{2}\hat{\rho}\hat{o}^{\dagger}\hat{o}$ for a given operator $\hat{o}$.

We now consider the experimental feasibility of our scheme and the appropriate parameters to achieve strong couplings. This manuscript employs the magnetized torsional cantilever to interface the Majorana qubit and the conventional spin qubit. As an alternative, one may also use the edge states of topological insulators or the planar Josephson junctions Fornieri et al. to realize the Majorana qubit, which requires weaker magnetic fields. For the magnetized torsional cantilever, the nanomagnet suspended on the single-walled nanotube has a size of $80\times 40\times 20$ nm3, and a moment of inertia of $I\sim 4.8\times 10^{-33}$ kgm2 with respect to the tube axis Meyer et al. (2005). Then the torsional cantilever has a resonance frequency $\omega_{m}=[\frac{1}{2\pi}][(\frac{K}{I})^{\frac{1}{2}}]\sim 2\pi\times 4$ MHz, and an angle of zero point fluctuations $\theta_{zpf}=(\hslash^{2}/KI)^{1/4}\sim 3\times 10^{-5}$, with $K\sim 3\times 10^{-18}$ Nm per radin the torsional spring constant of the tube axis. As shown in Fig. 2(c), the topology-torsion coupling energy is about $E_{m}(\theta\simeq 0)\sim 42$ $\mu$ eV, which implies a coupling constant $g\sim 2\pi\times 200$ kHz. Moreover, the spin-torsion coupling can be modified by the distance between the torsional cantilever and the NV center. For a distance $h\sim 80$ nm, one can obtain $B_{mg}\sim 80$ mT Mamin et al. (2007) and $\lambda_{e}\sim 2\pi\times 200$ kHz.

We now take the decoherence processes of the system into consideration. In practical situations, the relaxation and dephasing rates of Majorana qubit are strongly depended on the concrete realization, and here we take $\Gamma_{1}\simeq\Gamma_{2}\sim 2\pi\times 10$ kHz Alicea et al. (2011); Romito et al. (2012). When it comes to the mechanical damping, the recent fabrication of carbon nanotube resonators can possess quality factors exceeding $10^{5}$. For a torsional cantilever with $\omega_{m}\sim 2\pi\times 4$ MHz, the mechanical damping rate is about $\gamma_{m}=\omega_{m}/Q\sim 2\pi\times 40$ Hz, and the the thermal phonon occupation number is about $n_{th}\sim 104$ at the temperature of $T\sim 20$ mK. As for the NV center, the dephasing time $T_{2}$ as long as several milliseconds can be reached in ultrapure diamond Stanwix et al. (2010), and here we take $\gamma_{s}\sim 2\pi\times 2$ kHz. Finally, with these realistic parameters, we perform numerical simulations for the quantum dynamics of the system by solving the master equation (16). As shown in Fig. 4, the coherent interactions can dominate the decoherence processes in this hybrid quantum device, which enables the system to enter the strong coupling regime.

IV Dark state conversion

In the previous sections, it is shown that the magnetized torsional cantilever can be employed as an intermediary to link the topological qubit and the single NV center. Utilizing this quantum interface, one of the applications is to transfer the quantum state from the topological qubit to the spin qubit. The efficiency of the transformation relies on the specific protocol for quantum state conversion. To transfer quantum state in this setup, we show that both the direct-transfer scheme and the dark-state protocol are available.

As shown in Eq. (15), this hybrid tripartite system can be modeled by a beam-splitter Hamiltonian composed of two JC interactions. Therefore, we can use the JC interactions for quantum state conversion. In particular, this direct-transfer scheme usually includes two steps: step 1, turn on the topology-torsion couplings $g$ for a time $t_{1}=\frac{\pi}{2g}$ (while $\lambda_{e}=0$), to transfer the quantum state from the Majorana qubit to the torsional mode; step 2, turn off $g$ and turn on the spin-torsion couplings $\lambda_{e}$ for a time $t_{2}=\frac{\pi}{2\lambda_{e}}$ (while $g=0$), to map the quantum state from the torsional mode to the spin qubit. Note that this direct-transfer scheme has been studied in optomechanical system Tian and Wang (2010). In our setup, the topology-torsion couplings $g$ can be controlled by the electrostatic gates Alicea et al. (2011), while the spin-torsion couplings $\lambda_{e}$ can be controlled by the external driving fields.

As the direct-transfer scheme may suffer from decoherence of the mechanical mode, here we propose to realize the coherent quantum state conversion via a dark-state protocol Wang and Clerk (2012); Tian (2012); Li et al. (2015). This dark-state protocol is particularly robust against mechanical dissipations, as it decoupled from the vibrational mode. We proceed by assuming that $\omega_{m}\simeq\omega_{TP}\simeq\omega_{NV}$. To describe quasiparticles formed by combinations of spin and Majorana excitations, we now introduce two polariton operators $\hat{c}_{br}=\sin(\beta)\hat{\sigma}_{TP}^{-}+\cos(\beta)\hat{\sigma}_{NV}^{-}$, and $\hat{c}_{dk}=-\cos(\beta)\hat{\sigma}_{TP}^{-}+\sin(\beta)\hat{\sigma}_{NV}^{-}$, with $tan(\beta)=-g/\lambda_{e}$. Then, one can verify that the Hamiltonian (15) can take a compact form

[TABLE]

where $\hat{c}_{\pm}=(1/\sqrt{2})(\hat{c}_{br}\pm\hat{b})$ describe polarons formed by combinations of polariton and phonon excitations. The corresponding frequencies of the polarons are $\tilde{\omega}_{d}=\omega_{m}$, $\tilde{\omega}_{\pm}=\omega_{m}\pm\sqrt{\lambda_{e}^{2}+g^{2}}$. We refer to $\hat{c}_{dk}$ as the mechanically dark polariton operator, as it involves the spin and Majorana operations only, decoupled from the torsional mode.

The excitation of $\hat{c}_{dk}$ means the system remains in the mechanically dark state. Utilizing the adiabatic evolution of the dark state, we can transfer the quantum state from the topological qubit to the single NV center. In particular, in the limit $\beta=0$, we get $\hat{c}_{dk}=-\hat{\sigma}_{TP}^{-}$, while in the limit $\beta=\pi/2$, we get $\hat{c}_{dk}=\hat{\sigma}_{NV}^{-}$. This implies, if we adiabatically rotate the mixing angle $\beta$ form zero to $\pi/2$, the mechanically dark polariton $\hat{c}_{dk}$ would evolve from $-\hat{\sigma}_{TP}^{-}$ to $\hat{\sigma}_{NV}^{-}$. At the same time, the quantum state of the topological qubit could be transferred to the spin qubit except for a phase factor $e^{-i\pi}$.

Similar to the well-known stimulated Raman adiabatic passage, the dark state protocol can be implemented by an adiabatic passage approach. To transfer the quantum state from the topological qubit to the NV center, we initially make $\lambda_{e}(t)$ larger than $g(t)$, but keep $g(t)$ finite. Then, we modulate the coupling parameters carefully so that the dark polariton operator evolves adiabatically from $-\hat{\sigma}_{TP}^{-}$ at the beginning to $\hat{\sigma}_{NV}^{-}$ at the end. Meanwhile, the quantum state of the topological qubit will be transferred to the NV center as the system evolves. To satisfy the adiabatic conditions, one need to modulate the coupling strengths slowly to ensure that the system adiabatically follows the dark polaritons.

We perform numerical simulations for the dark-state protocol by solving the master equation (16) with Hamiltonian (15), where the coupling parameters $g(t)=g_{0}e^{-(t-\pi)^{2}/30}$, $\lambda_{e}(t)=1.5g_{0}e^{-t^{2}/6}$, and $g_{0}\sim 2\pi\times 200$ kHz as discussed in Sec. III. To test the robustness of the protocol, we consider two kinds of initial states for the topological qubit: a Fock state $|1\rangle$ as displayed in Figs. 5(a) and 5(b), and a superposition state $\frac{1}{\sqrt{2}}|0\rangle+\frac{1}{\sqrt{2}}|1\rangle$ as displayed in Figs. 5(c) and 5(d). For the ideal case without decoherence, high fidelities 0.96, 0.99 can be reached, as shown in Figs. 5(a) and 5(c). We find that, as the system evolves, the quantum states of the topological qubit is slowly transferred to the NV center spin in the conversion process. Then, when it turns to the realistic case, high fidelities 0.83 and 0.93 can be reached, as shown in Figs. 5(b) and 5(d). Therefore, this dark-state protocol works very well in practical conditions.

V CONCLUSIONS

In summary, we present a hybrid quantum device interfacing a topological qubit and a single NV center via a high-$Q$ magnetized torsional cantilever. We show that, the mechanical torsional mode can strongly couple to the Majorana qubit and the single NV center simultaneously, and then interface them for quantum states conversion. In particular, the topology-torsion couplings are induced by the spin currents passing through the TNT junction, while the spin-torsion couplings can be realized by the exquisite preparation of dressed spin states and using a suspended nanomagnet. We also find that, under resonance conditions, one eigenstate of the system is the mechanically dark state, which provides a dark-state protocol for quantum states conversion with very high fidelities. As the mechanical mode keeps unpopulated during the conversion process, this dark-state scheme is extremely robust against mechanical damping. This hybrid quantum architecture integrated with the topological qubits and NV spins provides a potential platform for quantum information processing.

Acknowledgments

B.L. thanks Di-Hao Sun and Shuai Li for fruitful discussions. This work was support by the NSFC under Grants No. 11774285, 91536115, 11534008, and the Fundamental Research Funds for the Central Universities. Part of the simulations are coded in PYTHON using the QUTIP library Johansson et al. (2012).