Continuous real-time sensing with a nitrogen-vacancy center via coherent population trapping

At a Glance

Metadata	Details
Publication Date	2021-04-12
Journal	Physical review. A/Physical review, A
Authors	Shu-Hao Wu, Ethan Turner, Hailin Wang
Institutions	University of Oregon
Citations	15
Analysis	Full AI Review Included

Executive Summary

This research proposes and analyzes a novel method for continuous, real-time quantum sensing using a single Nitrogen Vacancy (NV) center in diamond via Coherent Population Trapping (CPT).

Core Value Proposition: Enables continuous tracking of fluctuating magnetic fields, overcoming the limitations of pulsed methods (like Ramsey interferometry) that cannot provide true real-time data.
Sensing Mechanism: The NV center is held in a non-emitting “dark state” (CPT). Fluctuations in the magnetic field (modeled by the nuclear spin bath) kick the NV out of the dark state, generating single-photon emissions.
Data Processing Innovation: Bayesian inference, specifically the Ornstein-Uhlenbeck (OU) Bayesian estimator, is used to extract field information from a time series of detected photons, even when the average photon count per update interval is much less than 1.
Performance Achievement: The OU Bayesian estimator achieves an estimation variance that nearly approaches the classical Cramer-Rao Lower Bound (CRLB), demonstrating near-optimal information extraction from the sparse photon signal.
Time Resolution: The method provides dynamical information on a timescale comparable to the inverse of the average photon counting rate, offering high temporal resolution despite low signal efficiency (1.6%).
Model System: The theoretical demonstration uses the nuclear spin bath in the diamond lattice as the source of the fluctuating magnetic environment, modeled as an OU process.

Technical Specifications

Parameter	Value	Unit	Context
Sensing System	Single NV Center	N/A	Negatively charged nitrogen vacancy in diamond.
NV Radiative Lifetime	12	ns	Intrinsic spontaneous emission timescale [34].
Spontaneous Emission Rate (Γ/2π)	13	MHz	Used in numerical simulations [34].
Rabi Frequency (Ω/2π)	2.8	MHz	Used in numerical simulations for CPT.
Raman Detuning (Bias, Δ₀/2π)	0.25	MHz	Used in numerical simulations for CPT.
Bath Memory Time (τ_N)	1	ms	Characteristic time for the Ornstein-Uhlenbeck (OU) process model.
Bath Fluctuation (σ/2π)	0.13	MHz	Corresponds to a dephasing time (T₂*) of 1.7 µs for a ¹³C nuclear spin bath.
Overall Detection Efficiency (η)	1.6	%	Collection/detection efficiency used in simulations.
Average Photon Count Rate	10,000	per second	Achieved under simulated conditions (η=1.6%).
Update Time Interval (τ)	10	µs	Time step used for the main estimation results (Fig. 3).
Average Counts per Interval	0.1	N/A	Average number of photons detected per 10 µs interval (much less than 1).
Optimal CPT Parameters	Ω/2π = 2.5, Δ₀/2π = 0.2	MHz	Parameters yielding minimum estimation variance (Fig. 5a).

Key Methodologies

The analysis relies on a theoretical framework combining quantum optics modeling with advanced statistical inference techniques to simulate and estimate fluctuating magnetic fields.

Physical Model Setup:
- The NV center is modeled as a Λ-type three-level system (two ground spin states, m_s=0 and m_s=1, coupled to an excited state, |e>).
- Coherent Population Trapping (CPT) is established using two external optical fields with equal Rabi frequency (Ω), trapping the system in a dark state.
- The system is operated in the high cooperativity regime (C >> 1).
Fluctuation Modeling:
- Magnetic field fluctuations induced by the nuclear spin bath are modeled as an Ornstein-Uhlenbeck (OU) stochastic process (Eq. 2), characterized by memory time (τ_N) and variance (σ).
Single-Photon Emission Simulation:
- The Stochastic Schrödinger Equation (SSE) is used to simulate the time series of single-photon emissions, tracking the system collapse events when the NV center is kicked out of the dark state (Appendix A).
- For high-volume numerical calculations, an adiabatic approximation is used, assuming the excited state population (ρ_ee) follows the magnetic field adiabatically, generating Poisson-distributed photon counts.
Bayesian Inference Estimation:
- The time series of detected photon counts ({y_n}) is processed using the Bayesian update rule (Eq. 4) to estimate the fluctuating frequency ({x_n}).
- The likelihood function (Eq. 5) is defined by a Poisson distribution, reflecting the low average photon count rate (ȳ_n << 1).
OU Bayesian Estimator Implementation:
- The prior probability distribution is improved by incorporating the known statistical properties of the OU process (Eq. 7), which is a normal distribution based on the previous time step.
Performance Evaluation:
- Estimation variance (Var[x̄(y)]) is calculated numerically and compared against the theoretical Cramer-Rao Lower Bound (CRLB) (Eqs. 10, 11), which sets the minimum achievable variance for the estimation process.

Commercial Applications

The CPT-based real-time sensing technique is a powerful addition to the emerging field of quantum sensing, particularly for applications requiring high spatial and temporal resolution in dynamic environments.

Quantum Sensing and Metrology: Provides a new tool for continuous, real-time tracking of time-dependent physical quantities (magnetic field, electric field, temperature, strain) using solid-state quantum systems.
Nanoscale Magnetometry: Enables sensing of microscopic fluctuations and magnetic fields at nanometer spatial resolution, crucial for studying novel materials.
Materials Science Research: Applicable to studies of time-varying magnetic field changes or fluctuations in various systems, including two-dimensional (2D) materials placed on a diamond surface.
Quantum Computing and Qubit Protection: Can be used in conjunction with feedback control techniques to continuously monitor and suppress magnetic fluctuations, thereby protecting spin qubits from environmental decoherence.
Solid-State Defect Sensing: The methodology is transferable to other spin systems suitable for CPT, such as defect centers in Silicon Carbide (SiC), expanding the range of quantum sensors.
High-Resolution Dynamics: Allows for the real-time sensing of single nuclear spins, provided their magnetic field signature exceeds that of the surrounding nuclear spin bath.

View Original Abstract

We propose and theoretically analyze the use of coherent population trapping\nof a single diamond nitrogen vacancy (NV) center for continuous real-time\nsensing. The formation of the dark state in coherent population trapping\nprevents optical emissions from the NV center. Fluctuating magnetic fields,\nhowever, can kick the NV center out of the dark state, leading to a sequence of\nsingle-photon emissions. A time series of the photon counts detected can be\nused for magnetic field estimations, even when the average photon count per\nupdate time interval is much smaller than 1. For a theoretical demonstration,\nthe nuclear spin bath in a diamond lattice is used as a model fluctuating\nmagnetic environment. For fluctuations with known statistical properties, such\nas an Ornstein-Uhlenbeck process, Bayesian inference-based estimators can lead\nto an estimation variance that approaches the classical Cramer-Rao lower bound\nand can provide dynamical information on a timescale that is comparable to the\ninverse of the average photon counting rate. Real-time sensing using coherent\npopulation trapping adds a new and powerful tool to the emerging technology of\nquantum sensing.\n

Tech Support

Original Source

DOI: https://doi.org/10.1103/physreva.103.042607