Dynamical-Decoupling-Based Quantum Sensing - Floquet Spectroscopy

At a Glance

Metadata	Details
Publication Date	2015-10-30
Journal	Physical Review X
Authors	J. E. Lang, Ren‐Bao Liu, T. S. Monteiro
Institutions	Chinese University of Hong Kong, University College London
Citations	55
Analysis	Full AI Review Included

Advanced Materials Analysis: Floquet Spectroscopy for Quantum Sensing

Executive Summary

The analyzed research introduces Floquet Spectroscopy as a critical, universally valid paradigm for analyzing dynamical decoupling quantum sensing experiments, specifically focusing on Nitrogen-Vacancy (NV) centers in diamond and electron donors in silicon. This methodology is crucial for understanding atomic-scale structure in complex, strongly coupled spin systems.

Novel Analysis Technique: Floquet spectroscopy moves beyond perturbative (weak coupling) Average Hamiltonian models, remaining valid in regimes of strong quantum entanglement and for pulses of finite duration.
Decoherence Signature: Coherence minima (dips) are directly linked to avoided crossings in the underlying Floquet quasienergy eigenspectrum, offering physical insight into internal dynamics not visible through standard signal processing.
Material Dependence: The technique is exemplified using NV-centers in Single Crystal Diamond (SCD) and proposed for use with highly promising silicon-based electron donor systems, which benefit from extremely long coherence times (up to 1 s).
Key Material Requirement: Successful application relies on high-purity materials (SCD for NV centers) and precise control over defect placement and concentration to manage electronic and nuclear spin interactions ($A$ and $C_{ij}$).
Enhanced Characterization: Analysis of the widths and shapes of the decoherence features (e.g., the “diamond” pattern observed in transverse-field scans) provides quantitative measurements of interatomic coupling parameters ($C_{ij}$).
6CCVD Relevance: Replication and extension of this quantum sensing work requires ultra-high purity, custom-dimensioned Single Crystal Diamond (SCD) wafers and specialized Boron-Doped Diamond (BDD) substrates, all available from 6CCVD.

Technical Specifications

The following hard data points were extracted relating to material physics parameters and experimental conditions necessary for Floquet spectroscopy based quantum sensing:

Parameter	Value	Unit	Context
Sensor Spin Type	S=1 (NV center) / S=1/2 (Si Donor)	Dimensionless	Electronic spin probes
Coherence Time (Si Donors)	Order of seconds	s	Ultra-long coherence times for cryogenically cooled samples
External Magnetic Field ($B_{0}$) Range	50 - 300	mT (Tesla)	Range used for coherence mapping (T, B₀ plane)
NV Center Hyperfine Coupling ($A_{		}$)	50
Si Donor Hyperfine Coupling ($A_{1}$)	180	kHz	Example for 3-spin cluster detection
Si Donor Hyperfine Coupling ($A_{3}$)	100	kHz	Example for 3-spin cluster detection
Intra-Bath Dipolar Coupling ($C_{ij}$)	1.05 - 2.2	kHz	Realistic values for ²⁹Si nuclear impurities
Pulse Sequence Time ($t$)	Up to 400	ms	Total time for coherence decay measurement ($t = 4 N_{p} \tau$)
Coherence Dip Condition (Pseudo-spin)	$\mathcal{E}(\tau_{\text{dip}}/2) = \pi/2$	Radians	Condition for coherence minima based on quasienergy

Key Methodologies

The experimental approach integrates advanced spin control sequences with novel spectroscopic analysis, requiring extremely stable and defect-controlled material platforms.

Sensor Preparation:
- NV Centers in Diamond: The $S=1$ electronic spin of the NV center is prepared into a superposition state $|u\rangle + |d\rangle$ using a $\pi/2$ microwave pulse.
- Si Donors: $S=1/2$ electron donors (e.g., Bismuth in silicon) are considered due to favorable coherence properties at optimal working points (OWPs), which correspond to narrow Floquet avoided crossings.
Dynamical Decoupling Protocol:
- A periodic, multi-pulse decoupling sequence, such as the CPMG sequence, is applied. This sequence modulates the coherent evolution of the sensor spin.
- The total evolution time is $t = 4 N_{p} \tau$, where $\tau$ is the pulse interval and $N_{p}$ is the number of pulse pairs.
Spin-Bath Interaction and Entanglement:
- The sensor spin becomes entangled with the detected nuclear spin cluster (the spin-bath). The entanglement strength is highly dependent on the characteristic frequencies ($\omega_{u, d}$) which vary based on the sensor state.
Coherence Measurement:
- The temporal coherence $L(t)$ is measured, typically observing “dips” in coherence corresponding to resonances between the pulse sequence frequency and the spin-bath dynamics.
Floquet Spectroscopic Analysis:
- Instead of traditional Average Hamiltonian Theory, Floquet’s theorem is used to analyze the eigenspectrum of the one-period unitary evolution operator $\hat{T}(T_{\text{tot}}, 0)$.
- The Floquet quasienergies ($\epsilon_{l}$) and eigenstates ($\Phi_{l}$) fully describe the evolution. Coherence minima occur at points where the Floquet eigenstates exhibit avoided crossings ($\epsilon_{l} \approx \epsilon_{k}$).
2D Mapping and Fingerprinting:
- Coherence is mapped across the parameter space (e.g., magnetic field $B_{0}$ vs. pulse interval $\tau$) to reveal “diamond” patterns or “bar-codes.” These patterns are used to “fingerprint” complex multi-spin clusters and extract precise coupling constants ($C_{ij}, A$).

6CCVD Solutions & Capabilities

6CCVD provides the enabling advanced diamond materials and precision engineering services essential for replicating and extending the quantum sensing research presented in this paper.

Applicable Materials

To achieve the performance and coherence times required for advanced quantum sensing, 6CCVD recommends materials with ultra-low impurity levels and tailored doping profiles.

Research Requirement	6CCVD Material Solution	Purity / Context
High Coherence NV Centers	Optical Grade SCD Wafers	Ultra-high purity, low Nitrogen concentration (post-growth), critical for maximizing $T_{2}$ coherence times essential for sensing.
Custom Qubit Integration	Custom Thin Film SCD (0.1 µm - 500 µm)	Allows precise placement of NV layers near the surface or integration of Si donors via epitaxial techniques.
High Conductivity/Electrodes	Heavy Boron-Doped Diamond (BDD)	Used as robust, chemically inert electrodes or conductive layers for gate control in cryogenically cooled quantum devices.
Large-Area Sensor Arrays	Inch-Size Polycrystalline Diamond (PCD) Wafers	Provides cost-effective, large-area substrates (up to 125mm) where the specific sensor defect is confined to a thin SCD layer or implanted structure.

Customization Potential

The complexity of Floquet spectroscopy experiments, particularly those involving transverse fields ($w_{x}$) and multi-spin clusters, necessitates highly customized material dimensions and surface preparations.

Precision Dimensioning: 6CCVD offers MPCVD plates and wafers up to 125 mm in diameter, with the ability to provide custom laser cutting and shaping required for integration into specialized cryogenic or high-field experimental setups.
Ultra-Smooth Surfaces: The required sensing platforms (especially NV centers) rely on clean interfaces. 6CCVD guarantees Ra < 1 nm polishing for SCD and Ra < 5 nm for inch-size PCD, ensuring minimized surface defects that can contribute to decoherence noise.
Integrated Metalization: Experiments often require lithographic patterns for microwave delivery (CPMG pulses) or magnetic field gradient generation. 6CCVD provides in-house metalization services, including thin films of Au, Pt, Pd, Ti, W, and Cu, deposited directly onto diamond substrates with exceptional adhesion.
Thickness Control: We provide precise control over thickness, ranging from $0.1 \mu\text{m}$ thin films for sensing layers up to $10 \text{mm}$ thick substrates for superior thermal management and mechanical stability.

Engineering Support

The adoption of Floquet spectroscopy represents a move toward analyzing highly coupled, non-perturbative quantum regimes. Our materials directly impact the coherence and stability necessary for this analysis.

Consultation on Material Purity and Orientation: 6CCVD’s in-house PhD team can assist researchers and technical engineers with material selection, optimizing crystal orientation and nitrogen concentration for specific NV-center magnetometry applications.
Support for Alternative Qubit Platforms: Given the paper’s focus on electron donors in silicon as highly promising future sensors, our team specializes in providing custom BDD or diamond composite substrates suitable for supporting advanced cryogenic silicon-based spin qubit research and device fabrication.

For custom specifications or material consultation, visit 6ccvd.com or contact our engineering team directly. We ship globally (DDU default, DDP available).

View Original Abstract

Sensing the internal dynamics of individual nuclear spins or clusters of\nnuclear spins has recently become possible by observing the coherence decay of\na nearby electronic spin: the weak magnetic noise is amplified by a periodic,\nmulti-pulse decoupling sequence. However, it remains challenging to robustly\ninfer underlying atomic-scale structure from decoherence traces in all but the\nsimplest cases. We introduce Floquet spectroscopy as a versatile paradigm for\nanalysis of these experiments, and argue it offers a number of general\nadvantages. In particular, this technique generalises to more complex\nsituations, offering physical insight in regimes of many-body dynamics, strong\ncoupling and pulses of finite duration. As there is no requirement for resonant\ndriving, the proposed spectroscopic approach permits physical interpretation of\nstriking, but overlooked, coherence decay features in terms of the form of the\navoided crossings of the underlying quasienergy eigenspectrum. This is\nexemplified by a set of “diamond” shaped features arising for transverse-field\nscans in the case of single-spin sensing by NV-centers in diamond. We\ninvestigate also applications for donors in silicon showing that the resulting\ntunable interaction strengths offer highly promising future sensors.\n

Tech Support

Original Source

DOI: https://doi.org/10.1103/physrevx.5.041016