Dephasing mechanisms of diamond-based nuclear-spin memories for quantum networks

We probe dephasing mechanisms within a quantum network node consisting of a single nitrogen-vacancy center electron spin that is hyperfine coupled to surrounding ${}^{13}\mathrm{C}$ nuclear-spin quantum memories. Previous studies have analyzed memory dephasing caused by the stochastic electron-spin reset process, which is a component of optical internode entangling protocols. Here, we find, by using dynamical decoupling techniques and exploiting phase matching conditions in the electron-nuclear dynamics, that control infidelities and quasistatic noise are the major contributors to memory dephasing induced by the entangling sequence. These insights enable us to demonstrate a 19-fold improved memory performance which is still not limited by the electron reinitialization process. We further perform pump-probe studies to investigate the spin-flip channels during the optical electron spin reset. We find that spin flips occur via decay from the metastable singlet states with a branching ratio of 8(1):1:1, in contrast with previous work. These results allow us to formulate straightforward improvements to diamond-based quantum networks and similar architectures.

Figure 1

NV centers in diamond as multiqubit nodes for quantum networks. (a) The NV electron spin (purple) serves as optical interface (red wave packet) to establish remote entangling links. The surrounding ${}^{13}\mathrm{C}$ nuclear spins (orange) are hyperfine coupled to the electron spin (wiggly lines) and serve as quantum memories. (b) Relevant level structure of the NV center. The NV allows for spin-selective optical transitions to $|{\mathrm{E}}_{\mathrm{x}}\rangle$ and $|{\mathrm{E}}^{\prime }\rangle$. These states may decay to the metastable singlet states summarized as $|S\rangle$. The transition rates ${\mathrm{\Gamma }}_i$ are qualitatively indicated by the opacity of the dashed arrows. (c) Repeated attempts are made to create remote NV-NV entanglement until success is heralded. Each entangling attempt consists of electron spin manipulations via laser (green and orange) and microwave pulses (gray). Electron spin operations fail with probability $p_i$, thus leaving the electron spin in a mixed state. Nuclear spins with the initial state $|\psi \rangle$ will acquire an electron-state-dependent phase which results in a mixed state [${\rho }^{\prime }$, see Eq. (1)]. Further nuclear decoherence is induced by the optical electron spin reset ($|{\mathrm{E}}^{\prime }\rangle$), a stochastic process with randomly distributed projection times $t_{|0\rangle }$. (d) (Top) Experimental sequence. (Bottom) Memory coherence decay of nuclear spin ${\mathrm{C}}_1$ ($\mathrm{\Delta }\omega =2\pi \times 377\mathrm{kHz}$) with (yellow) and without (purple) interleaved $\pi$ rotation to probe quasistatic noise. See legend for fitted decay constants $N_{1/\mathrm{e}}$. Error bars represent one standard deviation.

Figure 2

Impact of microwave pulse errors on memory performance. We measure the decay of a superposition state on two different nuclear spins ${\mathrm{C}}_2, {\mathrm{C}}_3$ for three different scenarios (see legend and main text). We find maximal decay constants of 837(18) and 640(18), respectively, which corresponds to $\tau =177\mathrm{ns}/163\mathrm{ns}$ according to Blok et al. [15]. Solid lines are exponentially decaying fits to the data sets with $A\exp [-{(N/N_{1/e})}^m]$. Error bars are one s.d.

Figure 3

Results of a pump-probe experiment on the $|-1\rangle \to |{\mathrm{E}}^{\prime }\rangle$ transition. We measure the radiative lifetimes of both states (inset) and infer ${\mathrm{\Gamma }}_{\mathrm{es}}/2\pi =8.5\left(2\right)\mathrm{MHz}$. Besides, we infer from this a strain-averaged singlet branching ratio of $8\left(1\right):1:1$. See legend for the strain-induced frequency shifts at which each data set was taken.

Figure 4

Nuclear-spin decay constants as a function of optical spin-pumping power and different sequence configurations. Gray (black) data use one (two) nuclear-spin $\pi$ rotations to mitigate quasistatic noise. The microwave pulse during each entangling sequence rotates the electron by an angle $\alpha$ which we set to $\alpha =\pi /2$ (squared, top panels) or $\alpha =\pi$ (triangles, bottom panels). Solid and dashed lines are fits to the function $N_{\mathrm{sat}}P/(P+P_{\mathrm{sat}})$ with the optical power $P$ and the free parameters $N_{\mathrm{sat}}$ and $P_{\mathrm{sat}}$.

Figure 5

Memory sensitivity with respect to NV initialization errors. (a) (Top) Entangling sequence performed on the NV electron spin while the nuclear spin is idling in a superposition state. (Bottom) Measured equatorial Bloch vector length as a function of the delay $T$ between repumping pulse and microwave $R_x\left(\frac{\pi }{2}\right)$ rotation. We use $t_r=2\mu \mathrm{s}$. Dashed lines are the expected phase matching conditions $T=2\pi /\mathrm{\Delta }\omega$ for the respective spin (see legend). (b) (Left) Coherence of ${\mathrm{C}}_3$ for various spin pumping durations $t_r$ (see legend). Solid lines are fits with the probability of initialization failure $p_{\mathrm{init}}$ as free parameter. The data have been offset for better visibility. (Right) Extracted initialization infidelity as a function of repumping duration. Rate calculations suggest that a minimum of 3.3-$\mu \mathrm{s}$ repumping is required to achieve an infidelity of ${10}^{-4}$ under the assumption of fully saturating the NV and negligible off-resonant excitation.

Figure 6

(a) Comparison between experimentally obtained decay constants (see legend) as a function of coupling strength $\mathrm{\Delta }\omega$ and a phenomenological model (solid lines). (b) Monte Carlo simulation of a nuclear-spin memory with $\mathrm{\Delta }\omega =26\mathrm{kHz}$. We vary the error probabilities for NV electron spin rotations $p_{\mathrm{MW}}$ and unfaithful initialization $p_{\mathrm{init}}$ and estimate the $1/e$-decay constant $N_{1/\mathrm{e}}$ of the memory for two magnetic fields $B$. We use $\tau =100\mathrm{ns}$ resulting in an optimal decay constant of $\sim 15\times {10}^3$ which is limited by the stochastic electron reinitialization process [15]. Our simulation suggests that larger errors are tolerable in a regime of higher magnetic field as the sequence duration of each entangling attempt may be shortened which in turn reduces the accumulated nuclear-spin phase upon error. For a magnetic field of $4.14\mathrm{kG}$ the time between microwave pulses becomes $256\mathrm{ns}=2\pi /{\omega }_0$ while the required microwave frequency for electron spin manipulation becomes $8.7\mathrm{GHz}$. Both values are obtainable with state-of-the-art technology.