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

1. Problem Statement

Nitrogen-vacancy (NV) centers in diamond are promising candidates for ultra-sensitive magnetometry and quantum sensing. However, their performance is limited by decoherence caused by interactions with the surrounding nuclear spin bath, primarily composed of $^{13}\text{C}$ isotopes. To mitigate this, researchers often employ Dynamical Nuclear Polarization (DNP) to polarize the nuclear spins, thereby reducing the noise in the environment.

The Core Challenge: While polarizing the nuclear spins is achievable, measuring the resulting degree of polarization is experimentally difficult.

• Accurate methods typically require direct access to the nuclear environment, which is often impractical.
• Existing indirect methods (e.g., ODMR, PROPI, or echo-based techniques) can be complex or provide only upper bounds on polarization.
• There is a need for a simple, non-invasive method to estimate the lower bound of the initial nuclear polarization without directly probing the nuclear spins.

2. Methodology

The authors propose a protocol that infers the initial polarization of the nuclear environment by analyzing the coherence evolution of a single NV center spin qubit. The method relies on detecting Qubit-Environment Entanglement (QEE) generated during pure dephasing.

Theoretical Framework

• System: An NV center spin qubit (subspace $m_s=0, 1$) interacting with $N$ randomly distributed $^{13}\text{C}$ nuclear spins via hyperfine interaction.
• Hamiltonian: The interaction is modeled as pure dephasing. The qubit state $|1\rangle$ couples to the nuclear bath, while $|0\rangle$ is decoupled.
• Key Insight: If the nuclear environment is in a fully mixed state (zero polarization), the qubit and environment do not become entangled. However, if the environment is at least partially polarized, the joint evolution generates QEE. The amount of entanglement is proportional to the initial purity (polarization) of the environment.

The Experimental Scheme

The protocol involves three main stages:

1. Preparation Phase:
   • The qubit is initialized in a specific eigenstate (either $|0\rangle$ or $|1\rangle$).
   • The system evolves for a controlled time $\tau$.
   • If initialized in $|0\rangle$, the environment remains unchanged (since $|0\rangle$ is decoupled).
   • If initialized in $|1\rangle$, the environment evolves conditionally, creating a state $\hat{\rho}_E^{11}(\tau)$ distinct from the initial state.
2. Superposition Creation:
   • At time $\tau$, the qubit is excited into an equal superposition state $|+\rangle = (|0\rangle + |1\rangle)/\sqrt{2}$.
3. Coherence Evolution & Measurement:
   • The qubit evolves for a time $t$. Due to the different initial states of the environment (resulting from the preparation phase), the decoherence rates for the two preparation paths differ.
   • The observable is the difference in coherence ($\Delta\rho_{01}$) between the two preparation scenarios:
     $$\Delta\rho_{01}(\tau, t) = \rho_{01}^{(0)}(\tau, t) - \rho_{01}^{(1)}(\tau, t)$$
   • This difference is non-zero only if the environment was polarized (i.e., if QEE was generated).

3. Key Contributions

The paper introduces two analytical approaches to extract a lower bound on the average nuclear polarization ($\bar{p}$) from the measured coherence difference:

A. Time-Independent Approach

• Method: Uses the maximum absolute value of the coherence difference ($\Delta\rho_{\text{max}}$) regardless of the specific time $t$.
• Derivation: By applying the triangle inequality and bounding the interaction terms, the authors derive:
  $$\bar{p} \geq \frac{\Delta\rho_{\text{max}}}{N}$$
• Characteristics: Simple to implement but provides a relatively loose lower bound, especially as the number of nuclei ($N$) increases.

B. Time-Dependent Approach (Improved Precision)

• Method: Utilizes the specific time $t$ at which the measurement is taken and the known Larmor frequency ($\omega = \gamma_n B_z$) of the nuclei.
• Derivation: By incorporating the time-dependent factor $\sin(\omega t/2)$ into the bound, the inequality becomes:
  $$\bar{p} \geq \frac{|\Delta\rho_{01}(\tau, t)|}{N |\sin(\omega t/2)|}$$
• Characteristics: This method significantly tightens the lower bound, providing a much more accurate estimate of the actual polarization. It requires knowledge of the magnetic field and precise timing, which are standard experimental parameters.

4. Results

The authors validated their method using numerical simulations with up to 15 randomly placed nuclear spins, using coupling constants derived from realistic NV-nucleus distances.

• Polarization Detection: The method successfully distinguishes between a fully mixed environment and a polarized one. Even for $N=15$, a lower bound could be established.
• Accuracy Comparison:
  • For $N=5$ spins with full polarization ($p_k=1$) at $B_z=100$ G:
    • Time-independent method estimated $\bar{p} \geq 0.17$.
    • Time-dependent method estimated $\bar{p} \geq 0.73$.
  • The time-dependent method consistently outperformed the time-independent one, especially for larger $N$.
• Robustness:
  • Magnetic Field: The estimation is largely independent of the magnetic field strength (tested from 25 G to 200 G), though very low fields may slightly reduce accuracy for high polarizations.
  • Spatial Distribution: The method works for both uniform polarization and spatially non-uniform distributions (e.g., stronger polarization near the defect).
  • Partial Polarization: The method scales linearly with the actual average polarization for small $N$ and moderate fields, allowing for quantitative estimation.

5. Significance

• Non-Invasive Measurement: The primary advantage is the ability to infer nuclear polarization without direct access to the nuclear spins. All information is extracted via the NV qubit.
• Simplicity: The protocol requires only simple operations (initialization, superposition, and coherence measurement) and does not require complex echo sequences or Fourier transforms of NMR spectra.
• Practical Utility: This provides a reliable diagnostic tool for researchers performing DNP. It allows them to verify the success of their polarization protocols and optimize parameters for ultra-sensitive magnetometry.
• Fundamental Physics: The work demonstrates a practical application of detecting Qubit-Environment Entanglement (QEE) in pure dephasing regimes, linking the measurable decoherence signal directly to the initial purity of the environment.

In conclusion, the paper presents a robust, experimentally feasible protocol to estimate the lower bound of nuclear spin polarization in diamond, offering a significant improvement in precision over previous indirect methods while maintaining minimal experimental overhead.