Quantum state transfer between an optomechanical cavity and a diamond nuclear spin ensemble

Abstract

We explore an efficient scheme for transferring quantum state between an optomechanical cavity and nuclear spins of nitrogen-vacancy centers in diamond, where quantum information can be efficiently stored (retrieved) into (from) the nuclear spin ensemble assisted by a mechanical resonator in a dispersive regime. Our scheme works for a broad range of cavity frequencies and might have potential applications in employing the nuclear spin ensemble as a memory in quantum information processing. The feasibility of our protocol is analyzed using currently available parameters.

Appendices

Appendix 1: The phonon decoherence effect on the QST

During the above-mentioned QST, the dissipation of photons and phonons reduces the transfer fidelity. The decoherence is from the decays of the cavity field and the mechanical mode, whose effects on the system can be addressed numerically by solving the master equation. In a displaced frame, we analyze the coherent evolutions $\left| 10g\right\rangle _\mathrm{crs}\leftrightarrow \left| 01g\right\rangle _\mathrm{crs}\leftrightarrow \left| 00e\right\rangle _\mathrm{crs}$ for a resonant case of $\bar{\omega } _\mathrm{c}=\omega _\mathrm{r}=\omega _\mathrm{s}$ realized by tuning $\bar{\omega }_\mathrm{c}$ and $\omega _\mathrm{s}$. Here, the mechanical mode is supposed to be initially cooled down to the vibrational ground state $\left| 0\right\rangle _\mathrm{r}$ by current techniques of sideband cooling in the optomechanical setup [38]. The reduced density matrix $\rho$ is governed by the Lindblad master equation [9, 40],

$$\begin{aligned} \dot{\rho }=\frac{i}{\hbar }[\rho ,H_{t}]+\kappa D[a]\rho +\gamma (n_\mathrm{th}+1)D[b]\rho +\gamma n_\mathrm{th}D[b^{\dag }]\rho , \end{aligned}$$

(19)

where $D[z]\rho =(2z\rho z^{\dag }-z^{\dag }z\rho -\rho z^{\dag }z)/2$ with $z=a,b$, the decay rates $\kappa$ and $\gamma$ are regarding the photon and the phonon, respectively, and $n_\mathrm{th}$ stands for the thermal phonon number of the environment. In the present optomechanical system, the photon decay is dominant over the phonon decay ($\kappa \gg \gamma$) [39]. For simplicity and also for clarity of description, we may neglect the decoherence effect induced by the phonons and reduce the master equation as:

$$\begin{aligned} \dot{\rho }=i[\rho ,H_{t}]+\kappa (2a\rho a^{\dag }-a^{\dag }a\rho -\rho a^{\dag }a)/2, \end{aligned}$$

(20)

which works in the main text after considering the virtual excitation of the resonator. We have numerically compared the cases with and without the phonon decoherence in Fig. 3 in order to justify the decoherence effect associated with the phonon decay.

In the considered system, we have the effective coupling strength $\bar{ \lambda }_\mathrm{rc}=\bar{\lambda }_\mathrm{rs}$ ($\simeq 2\pi \times 152.88$ kHz) and the resonator frequency $\omega _\mathrm{r}/2\pi =0.76$ MHz. As in Refs. [9, 40], the decay rates are $\kappa =1.0\times 10^{-2}\omega _\mathrm{r}$ and $\gamma =10^{-5}\omega _\mathrm{r}$. The thermal phonon number of the environment is chosen as $n_\mathrm{th}=493$. For the identical initial state $\left| 10g\right\rangle _\mathrm{crs}$, the corresponding time evolution of the occupation probability of $\left| 00e\right\rangle _\mathrm{crs}$ can be obtained by numerically solving Eqs. (19) and (20). The populations $P_{00e}^{(1)}$ and $P_{00e}^{(2)}$, with and without the phonon decay effect, respectively, are given in Fig. 3a, b, respectively. To quantitatively assess the influence from the phonon, we plot the dependence of $\Delta P_{00e}$ ($=P_{00e}^{(2)}-P_{00e}^{(1)}$) on time t; see Fig. 3c.

The results indicate that omission of the phonon decay works only for short-time state transfer. For example, we have $\Delta P_{00e}=7.65\,\%$ at $t=2.43$ $\upmu$s. For a long-time implementation, $\Delta P_{00e}$ would be 36.67 % at $t\simeq 16.2$ $\upmu$s, implying a significant effect from the phonon decay. As such, exploring a more effective way to adiabatically removing the thermal phonon is necessary, and improving the cavity quality is also highly expected for future implementation of our scheme.

Appendix 2: The effect of inhomogeneous couplings on the transfer fidelity

As mentioned before, the ring-like spin distribution of NSE has a thickness of 60 nm along the z axis, making the MFG felt by the individual spins of NSE inhomogeneous slightly. The relevant coupling strengths $\lambda _\mathrm{rs}^{(k)}$ of the individual spins to the z direction MFG could be different from each other. In this case, the collective coupling strength is generally given by $\bar{\lambda }_\mathrm{rs}^{(R)}=\sqrt{\sum _{k}(\lambda _\mathrm{rs}^{(k)})^{2}}$, with $k=1,2, \ldots , N$ [45]. In particular, when adopting the approximation of identical couplings, $\lambda _\mathrm{rs}^{(k)}=\lambda _\mathrm{rs}$ , the collective coupling strength takes the form $\bar{\lambda }_\mathrm{rs}^{(R)}= \sqrt{N}\lambda _\mathrm{rs}$ $(=\bar{\lambda }_\mathrm{rs})$.

We are concerned with the effects of different coupling strengths on the transfer fidelities [46]. For a state transfer, say $\left| 1g\right\rangle _\mathrm{cs}\longleftrightarrow \left| 0e\right\rangle _\mathrm{cs}$, the fidelities are assumed to be F and $F^{(R)}$ for the different strengths of $\bar{\lambda }_\mathrm{rs}$ and $\bar{\lambda }_\mathrm{rs}^{(R)}$, respectively. Now, we consider two differences $\Delta F=F^{(R)}-F$ and $\Delta \bar{\lambda }_\mathrm{rs}=\bar{\lambda }_\mathrm{rs}^{(R)}-\bar{\lambda }_\mathrm{rs}$. Here, $\bar{\lambda }_\mathrm{rs}$ is fixed, as utilized in the before calculations; we thus obtain a deterministic fidelity F for other chosen parameters (with decay rate $\kappa /2\pi =1.0\times 10^{-2}\omega _\mathrm{r}$). Assuming that $\bar{\lambda }_\mathrm{rs}^{(R)}$ has a deviation of $\Delta \bar{\lambda }_\mathrm{rs}$ from $\bar{\lambda }_\mathrm{rs}$, we focus on the fidelity change $\Delta F$ versus $\Delta \bar{\lambda }_\mathrm{rs}$, characterizing the effect of $\Delta \bar{ \lambda }_\mathrm{rs}$ on the state transfer of interest. By numerically solving Eqs. (14) and (15), the dependence of $\Delta F$ on $\Delta \bar{\lambda }_\mathrm{rs}$ can be obtained quantitatively. It is found that $\Delta F$ increases slowly with $\Delta \bar{\lambda }_\mathrm{rs}$. When $\Delta \bar{ \lambda }_\mathrm{rs}/2\pi =5$ and 10 kHz, the differences are $\Delta F=0.21$ and 0.87 %, respectively. Even for a larger deviation of $\Delta \bar{\lambda }_\mathrm{rs}/2\pi =20$ kHz, the fidelity change is $\Delta F=3.52\,\%$. These results indicate that the deviation $\Delta \bar{\lambda } _\mathrm{rs}$ does not affect the transfer fidelity typically.

Note that the MFG along the z axis is position-dependent, and it will be reduced with increase in the distance between the individual spin and the magnetic tip. In our consideration, we have selected the average MFG at the middle position of the NSE (i.e., one half of thickness). Compared with the average strength $\lambda _\mathrm{rs}$, the individual coupling $\lambda _\mathrm{rs}^{(k)}$ will be increased (reduced) for the spins distributing on the left (right) half of the NSE. As a result, the total impact of two opposite changes on the realistic coupling $\bar{\lambda }_\mathrm{rs}^{(R)}$ could be canceled out partially, which makes the above approximation of $\bar{\lambda } _\mathrm{rs}^{(R)}\simeq \bar{\lambda }_\mathrm{rs}$ possible physically in our protocol.