Quantum Fisher information measurement and verification of the quantum Cramér–Rao bound in a solid-state qubit
At a Glance
Section titled “At a Glance”| Metadata | Details |
|---|---|
| Publication Date | 2022-05-12 |
| Journal | npj Quantum Information |
| Authors | Min Yu, Yu Liu, Pengcheng Yang, Musang Gong, Qingyun Cao |
| Institutions | Advanced Institute of Materials Science, Max Planck Institute for the Physics of Complex Systems |
| Citations | 63 |
| Analysis | Full AI Review Included |
Technical Documentation & Analysis: Quantum Fisher Information in Solid-State Qubits
Section titled “Technical Documentation & Analysis: Quantum Fisher Information in Solid-State Qubits”6CCVD Reference Document: QFI-NV-2022-00547
Executive Summary
Section titled “Executive Summary”This research successfully demonstrates the experimental saturation of the quantum Cramér-Rao bound (QCRB) in a solid-state qubit system utilizing a Nitrogen-Vacancy (NV) center in diamond. This achievement validates the fundamental limit of parameter estimation accuracy in a practical quantum sensor.
- Core Achievement: Experimental verification of the QCRB saturation in phase estimation using an NV center spin, confirming optimal measurement efficiency.
- Novel Methodology: Quantum Fisher Information (QFI) was independently extracted by probing coherent dynamical responses under weak parametric modulations, circumventing the need for complex, resource-intensive quantum-state tomography.
- Material System: The experiment relies on the high coherence and stability of the NV center in Single Crystal Diamond (SCD) operating under an external magnetic field (Bz $\approx$ 510 G).
- Scalability: The spectroscopic response method offers a versatile and powerful tool for exploring QFI in systems of higher complexity, addressing a key challenge for scaling quantum technologies.
- Performance Metrics: Achieved state fidelity above 95% and demonstrated optimal sensitivity ($\delta\beta$) scaling linearly with the inverse square root of the QFI ($1/\sqrt{F_{\beta}}$).
- Future Applications: The technique is generalized to analyze entanglement and QFI in coupled-qubit systems (NV center coupled to a 13C nuclear spin), relevant for multi-qubit quantum sensing and computation.
Technical Specifications
Section titled “Technical Specifications”The following table summarizes the critical experimental parameters and performance metrics achieved in the study.
| Parameter | Value | Unit | Context |
|---|---|---|---|
| Material System | NV Center in Diamond | N/A | Solid-state qubit for quantum sensing. |
| Qubit Sublevels | ms = 0, -1 | N/A | Ground state spin sublevels used for the two-level system. |
| External Magnetic Field (Bz) | $\approx$ 510 | G | Used to lift spin degeneracy and tune close to the excited state level anticrossing. |
| Zero-Field Splitting (D) | (2$\pi$)2.87 | GHz | Intrinsic property of the NV center ground state. |
| Initialization Laser | 532 | nm | Green laser pulse used for spin polarization. |
| QFI Measurement Method | Parametric Modulation | N/A | Probing spectroscopic responses to extract QFI without tomography. |
| Modulation Frequency ($\omega$) | $\approx$ A | N/A | Resonant frequency for optimal coherent transition. |
| Achieved State Fidelity | > 95 | % | Estimated purity of the quantum state during the experiment. |
| QCRB Saturation Factor | 1.041 $\pm$ 0.036 | N/A | Measured proportionality factor between optimal sensitivity ($\delta\beta$) and the theoretical QCRB limit ($1/\sqrt{F_{\beta}}$). |
Key Methodologies
Section titled “Key Methodologies”The experiment utilized a modified Ramsey interferometry protocol coupled with a novel parametric modulation technique for QFI extraction.
- Initialization: The NV center spin is initialized in the ms = 0 state using a 532 nm green laser pulse. The magnetic field (Bz $\approx$ 510 G) is tuned to polarize the associated nitrogen nuclear spin.
- Resource State Preparation ($\Upsilon_{\theta}$): A microwave (MW) pulse is applied for a duration $t_{\theta} = (\theta/\Omega)$ to rotate the spin around the $\hat{y}$ axis, preparing the system into a $\theta$-dependent coherent superposition resource state.
- Interrogation (Free Evolution): The system undergoes free evolution for time $T$, resulting in the final state $|\Psi_{\theta}(\beta)\rangle$ which encodes the unknown phase parameter $\beta = \xi T$.
- QFI Direct Measurement via Parametric Modulation:
- The system is subjected to a time-periodic Hamiltonian modulation, $H[\beta(t)] = H(\beta + a_{\beta} \cos(\omega t))$.
- The modulation induces a coherent transition between the two orthogonal eigenstates, $|\Psi_{\theta}(\beta)\rangle$ and $|\tilde{\Psi}_{\theta}(\beta)\rangle$.
- The resonant modulation frequency ($\omega \approx A$) is determined by sweeping frequencies and measuring the transition probability.
- The effective Rabi frequency ($\nu_e$) of the induced oscillation is measured, and the QFI ($F_{\beta}(\theta)$) is calculated directly from $\nu_e$ and the modulation parameters: $F_{\beta}(\theta) = 4 (\nu_e / a_{\beta}\omega)^2$.
- Inverse Evolution and Readout: An inverse evolution sequence ($\Upsilon_{\pi}$ and $\Upsilon_{\pi-\theta}$) rotates the final state back to the $|0\rangle$ and $|-1\rangle$ basis. Spin-dependent fluorescence measurement is used to monitor the coherent transition probability and estimate the parameter $\beta$.
6CCVD Solutions & Capabilities
Section titled “6CCVD Solutions & Capabilities”This research highlights the critical role of high-quality Single Crystal Diamond (SCD) in achieving fundamental limits in quantum metrology. 6CCVD is uniquely positioned to supply the advanced diamond materials and fabrication services required to replicate, extend, and scale this groundbreaking work.
Applicable Materials
Section titled “Applicable Materials”To achieve the high coherence times and low strain necessary for Ramsey interferometry and QCRB saturation, the following 6CCVD materials are essential:
| 6CCVD Material | Specification | Relevance to Research |
|---|---|---|
| Electronic Grade SCD | High Purity (N < 1 ppb), Low Strain | Essential for maximizing NV center coherence time (T2*) and achieving high state fidelity (> 95%). |
| Isotopically Purified SCD | Low 13C Concentration (< 0.1%) | Critical for minimizing environmental decoherence and maximizing T2, especially when scaling to multi-qubit systems or achieving optimal sensitivity. |
| Custom NV Creation | Controlled Implantation/Annealing | 6CCVD can supply substrates optimized for subsequent NV creation via ion implantation and high-temperature annealing, ensuring high yield and quality. |
| Substrate Thickness | Up to 10 mm | Provides robust mechanical and thermal stability for complex experimental setups involving high magnetic fields and MW control. |
Customization Potential
Section titled “Customization Potential”The complexity of solid-state quantum experiments often requires precise material engineering and integration. 6CCVD offers comprehensive customization capabilities that directly support the requirements of this research:
- Custom Dimensions: 6CCVD supplies SCD plates and wafers in custom sizes, ensuring compatibility with specific cryostat or optical setups.
- Precision Polishing: We guarantee ultra-low surface roughness (Ra < 1 nm for SCD), which is crucial for minimizing optical scattering losses and maximizing the efficiency of the 532 nm laser initialization and fluorescence collection.
- Custom Metalization: While this paper focuses on MW control, future integration of on-chip control elements (e.g., striplines for MW delivery) requires precise metalization. 6CCVD offers in-house deposition of standard quantum control metals including Au, Pt, Ti, and Cu with custom patterning capabilities.
- Thickness Control: We provide SCD layers with precise thickness control from 0.1 µm up to 500 µm, allowing researchers to optimize the proximity of NV centers to the surface for enhanced sensing or integration with photonic structures.
Engineering Support
Section titled “Engineering Support”6CCVD’s in-house team of PhD material scientists and quantum engineers specializes in optimizing MPCVD diamond for demanding quantum applications.
- Material Selection for Quantum Metrology: Our experts can assist researchers in selecting the optimal diamond grade (e.g., nitrogen concentration, isotopic purity) to maximize the T2* and T2 coherence times required for high-precision Ramsey interferometry experiments like the one demonstrated here.
- Defect Engineering Consultation: We provide consultation on post-growth processing, including surface termination and annealing protocols, to ensure the highest quality NV centers are achieved for similar solid-state qubit phase estimation projects.
- Global Logistics: We ensure reliable, fast global shipping (DDU default, DDP available) for sensitive, high-value diamond substrates.
For custom specifications or material consultation, visit 6ccvd.com or contact our engineering team directly.