Probabilistic magnetometry with a two-spin system in diamond

At a Glance

Metadata	Details
Publication Date	2021-04-29
Journal	Quantum Science and Technology
Authors	Raúl Coto, Hossein T. Dinani, Ariel Norambuena, Mo Chen, J. R. Maze
Institutions	Universidad Mayor, Pontificia Universidad Católica de Chile
Citations	5
Analysis	Full AI Review Included

Executive Summary

This research proposes and analyzes a novel probabilistic quantum metrology protocol for DC magnetometry using the Nitrogen-Vacancy (NV) center in diamond.

Core Value Proposition: The protocol achieves magnetic field sensitivity comparable to, and often superior to, standard Ramsey spectroscopy, particularly in regimes characterized by short transverse relaxation times (T₂).
Mechanism: It employs a two-spin system (NV electron spin as the sensor, coupled nuclear spin—¹³C or ¹⁵N—as the meter) utilizing pre-selection and post-selection to amplify the signal response.
Performance in Decoherence: The post-selection process concentrates valuable sensing information into a single successful measurement, providing an advantage when the coherence time (T₂) is short (T₂ ≤ 3 µs).
Cryogenic Sensitivity: Estimated sensitivity at cryogenic temperature (4 K) using the native ¹⁵N nuclear spin is approximately 43.5 nTHz^-1/2 (ideal readout), representing a 28% improvement over Ramsey in the suboptimal T₂ regime.
Room Temperature Performance: Achievable sensitivity at room temperature is estimated at 1.5 µTHz^-1/2, limited primarily by the overhead associated with repetitive nuclear spin readout.
Information Gain: Fisher Information analysis confirms that the post-selection protocol extracts more information at shorter sensing times compared to the Ramsey protocol.

Technical Specifications

Parameter	Value	Unit	Context
Zero-Field Splitting (D/2π)	2.87	GHz	NV center ground state
Electron Gyromagnetic Ratio (γ_e/2π)	~2.8	MHz/G	NV electron spin
¹³C Nuclear Gyromagnetic Ratio (γ_c/2π)	~1.07	kHz/G	¹³C nuclear spin
¹³C Hyperfine Coupling (A_zz)	500	kHz	Weakly coupled ¹³C spin
¹⁵N Hyperfine Coupling (A_zz)	3.03	MHz	Native ¹⁵N spin (stronger coupling)
Target Magnetic Field (B)	10	mG	Weak field sensing regime (10^-2 G)
Optimal Interrogation Time (τ)	1.3	µs	For ¹³C at T₂ = 2 µs
Successful Post-selection Probability (P_s)	6	%	At optimal τ = 2.2 µs (Fig. 2)
Cryogenic Sensitivity (¹³C, Ideal)	43.5	nTHz^-1/2	T = 4 K, C = 1 (ideal readout)
Cryogenic Sensitivity (¹⁵N, Improvement)	~28	%	Improvement over Ramsey at T = 4 K, T₂ = 2 µs
Room Temp Sensitivity (¹⁵N)	1.5	µTHz^-1/2	Limited by repetitive readout (t_p=5 ms, t_r=8 ms)
Typical T₂ (NV electron spin)	~2	µs	Naturally occurring NV spins
NV Readout Time (t_p)	3.7	µs	Single-shot readout at 4 K
Nuclear Spin Readout Time (t_r)	5.7	µs	¹³C assuming CNOT gate at 4 K
Optimal Performance Range	B < 10 mG, T₂ ≤ 3 µs	N/A	Regime where post-selection outperforms Ramsey

Key Methodologies

The protocol is implemented using a two-spin system (NV electron spin as the system, nuclear spin as the meter) manipulated via microwave (MW) and radiofrequency (RF) pulses in a multi-rotating frame.

System Truncation and Frame: The NV electronic spin (S=1) is truncated to the two-level m_s = 0, -1 manifold. The Hamiltonian is analyzed in a multi-rotating frame to simplify the interaction terms.
Initialization: The bipartite system is initialized to the state |0> |↓>. Efficient nuclear spin polarization techniques are assumed (e.g., dynamic nuclear polarization).
Pre-selection (Superposition Preparation):
- The nuclear spin is prepared in a coherent superposition state using an RF pulse (angle α).
- The NV electronic spin is rotated by a strong MW pulse (angle θ_i).
Sensing Interaction: The system undergoes free evolution for interrogation time τ under the external magnetic field B. This evolution imprints the phase proportional to B onto the system.
Decoherence Modeling: Transverse relaxation (T₂) is incorporated using a Markovian master equation (Lindblad super-operator) for pure dephasing, or modeled using Ornstein-Uhlenbeck (OU) statistics for non-Markovian magnetic noise.
Post-selection: The NV electronic spin is measured in a target state (angle θ_f). This step maps the accumulated phase information onto the nuclear spin meter.
Signal Readout: The expectation value of the nuclear spin observable (I_z) is measured from the post-selected state. The sensitivity (η) is calculated using the standard deviation (ΔI_z) and the probability of successful post-selection (P_s), where η = ΔB√t_m and t_m accounts for the average trials required (t_m ~ 1/P_s).
Implementation Options: The protocol is analyzed for both ¹³C (weakly coupled) and native ¹⁵N (strongly coupled, enabling faster nuclear spin gates) at cryogenic (4 K, single-shot readout) and room temperatures (repetitive readout required).

Commercial Applications

The proposed probabilistic magnetometry technique is highly relevant to advanced quantum technologies and nanoscale sensing applications.

Quantum Metrology and Sensing: Enables the creation of highly sensitive solid-state magnetometers that maintain performance even in environments with high decoherence (short T₂), crucial for practical quantum sensor deployment.
Nanoscale NMR and EPR: Provides the necessary sensitivity and spatial resolution for detecting and performing spectroscopy on extremely small samples, such as single proteins, small molecules, or individual proton spins.
Material Characterization: Used for high-resolution magnetic resonance spectroscopy to characterize magnetic fields and noise sources in novel solid-state materials, particularly diamond and other wide-bandgap semiconductors.
Quantum Information Processing (QIP): The coupled NV-nuclear spin system acts as a versatile quantum register, where the nuclear spin can function as:
- Quantum Memory: Extending the interrogation time by storing quantum information.
- Ancilla Qubit: Used for implementing quantum error correction protocols.
Biomagnetism: Potential for sensing weak magnetic fields generated by biological processes or structures at the cellular level.

View Original Abstract

Solid-state magnetometers like the Nitrogen-Vacancy center in diamond have\nbeen of paramount importance for the development of quantum sensing with\nnanoscale spatial resolution. The basic protocol is a Ramsey sequence, that\nimprints an external static magnetic field into phase of the quantum sensor,\nwhich is subsequently readout. In this work we show that the hyperfine coupling\nbetween the Nitrogen-Vacancy and a nearby Carbon-13 can be used to set a\npost-selection protocol that leads to an enhancement of the sensitivity under\nrealistic experimental conditions. We found that for an isotopically purified\nsample the detection of weak magnetic fields in the $\mu$T range can be\nachieved with a sensitivity of few nTHz$^{-1/2}$ at cryogenic temperature ($4$\nK), and $0.1$ $\mu$THz$^{-1/2}$ at room temperature.\n

Tech Support

Original Source

DOI: https://doi.org/10.1088/2058-9565/abfce1

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