Direct access to the initial polarization of ${}^{13}C$ nuclei by measuring coherence evolution of an nitrogen-vacancy center spin qubit

TL;DR

This paper addresses the challenge of determining the initial polarization of ${}^{13}$C nuclei in diamond without direct environmental access. It introduces a scheme where a single NV center spin qubit transfers information about the nuclear environment into its own coherence via a controlled preparation and measurement sequence, enabling a lower bound on the average nuclear polarization $\bar p=\frac{1}{N}\sum_k |p_k|$ to be inferred from qubit readout. The authors derive two bounds: a time-independent bound $\bar p \ge \Delta\rho_{\max}/N$ and a time-dependent bound $\bar p \ge |\Delta\rho_{01}(\tau,t)|/(N|\sin(\omega t/2)|)$, with the latter becoming more accurate as the number of nuclei grows or with appropriate timing and field choices. The method is experimentally simple, relies only on NV coherence without direct environment access, and is demonstrated via simulations with up to $N=15$ nuclear spins, showing robustness to field variations and partial/nonuniform polarization.

Abstract

We introduce a method for the measurement of the lower bound on the initial polarization of spinful nuclei in a diamond by following the coherence evolution of an NV center spin qubit after a simple scheme is operated on the qubit to facilitate the transfer of information from the environment into the qubit state. Current polarization measurement techniques are challenging to implement due to the need for direct access to the environment. In our method, information is obtained by measuring the difference of the evolution of the qubit coherence resulting from preparation phase when the environment evolution is conditional on the qubit pointer state. We find that the method does not depend strongly on the applied magnetic field, but rather on the number of spinfull nuclei that lead to decoherence, and gives a reasonable estimate if the environment is polarized. The key advantage of this approach is its simplicity and minimal experimental requirements, allowing the inference of initial nuclear polarizations without direct access to the environment. We demonstrate the efficacy of this method using a simulated environment of up to fifteen randomly placed nuclear spins.

Figures (6)

• Figure 1: The absolute value of the signal $\Delta\rho_{01}(\tau, t)$ as a function of both times $t$ and $\tau$ for: (a) $N=1$ spin, (b) $N=5$ spins, (c) $N=10$ spins, and (d) $N=15$ spins. We choose $p_k = 1$ for all $k$ and $B_{\mathrm{z}} = 25$ G.
• Figure 2: Estimation of the minimal value of the polarization for the time-independent case as a function of time $\tau$. Panel (a): polarization value for $N=5$ nuclear spins for different values of the magnetic field. Panel (b): polarization value in the magnetic field $B_{\mathrm{z}} = 25$ G for different numbers of nuclear spins. We choose $p_k = 1$ for all $k$.
• Figure 3: Estimation of the minimal value of the polarization for the time-dependent scheme as a function of both times $\tau$ and $t$ for: (a) $N = 1$ spin, (b) $N=5$ spins, (c) $N=10$ spins, and (d) $N=15$ spins. We choose $p_k = 1$ for all $k$ and $B_{\mathrm{z}} = 100$ G.
• Figure 4: Estimation of the minimal value of the polarization for the time-dependent scheme as a function of both times $\tau$ and $t$ for: (a) $B_{\mathrm{z}} = 25$ G, (b) $B_{\mathrm{z}} = 50$ G, (c) $B_{\mathrm{z}} = 100$ G, and (d) $B_{\mathrm{z}} = 200$ G. We choose $p_k = 1$ for all $k$ and $N = 5$ spins.
• Figure 5: Estimation of the minimal value of the polarization for the time-dependent scheme as a function of both times $\tau$ and $t$ for $\Bar{p} = 0.5$, $B_{\mathrm{z}} = 100$ G, and $N=5$ where: (a) $p_k = 0.5$ for all $k$; (b) $p_k$ is different for all $k$, and spins that are closer to the NV center are polarized stronger.