Modified coherence of quantum spins in a damped pure-dephasing model
At a Glance
Section titled “At a Glance”| Metadata | Details |
|---|---|
| Publication Date | 2022-03-21 |
| Journal | Physical review. B./Physical review. B |
| Authors | Mattias Johnsson, Ben Q. Baragiola, Thomas Volz, Gavin K. Brennen |
| Institutions | Centre for Quantum Computation and Communication Technology, ARC Centre of Excellence for Engineered Quantum Systems |
| Analysis | Full AI Review Included |
Executive Summary
Section titled “Executive Summary”This research presents an exact analytic solution for the dynamics of large quantum spins coupled to dissipative bosonic environments, offering critical insights for engineering robust quantum systems.
- Analytic Solution for Open Systems: An exact, non-Markovian analytic solution was derived for the reduced state of a large spin (j > 1/2) coupled to a collection of vibrational modes that themselves decay into thermal baths.
- Quantum Zeno Coherence Protection: The model identifies the heavily overdamped regime (mode decay rate Γ >> mode frequency ω) where spin coherence decay is suppressed, scaling inversely with Γ (proportional to Γ-1). This effect is analogous to the Quantum Zeno Effect.
- Structured Environment Control: Inter-mode coupling in multi-spin systems (e.g., solid-state defects) can be engineered to create a structured environment with distinct normal modes.
- Symmetric Subspace Preservation: By increasing the decay rate of the antisymmetric normal mode (via QZE), population transfer out of the desired symmetric spin subspace is significantly reduced, protecting collective spin coherence.
- Spectral Gap Isolation: For N coupled bosonic modes, specific coupling structures (e.g., uniform negative coupling or random negative coupling) create a large spectral gap (O(N)) isolating a single, privileged symmetric normal mode.
- Quantum Control Handle: This isolated symmetric mode provides a handle for selective quantum control, enabling protocols like effective spin squeezing or geometric phase gates on the collective spin degree of freedom.
- Solid-State Relevance: The findings are directly applicable to optimizing coherence in solid-state defects, particularly Nitrogen Vacancy (NV) centers in diamond.
Technical Specifications
Section titled “Technical Specifications”The following table summarizes key quantitative parameters and relationships derived from the analytic solution and analysis of the spin-boson model.
| Parameter | Value | Unit | Context |
|---|---|---|---|
| Spin Size | j | N/A | Solution derived for large spins (j > 1/2). |
| Coherence Decay (Underdamped) | Proportional to Γk | N/A | Regime where Γk < ωk; coherence oscillates and decays linearly with mode decay rate. |
| Coherence Decay (Overdamped) | Proportional to Γk-1 | N/A | Regime where Γk >> ωk (Quantum Zeno Effect); decay rate is inversely proportional to mode decay rate. |
| Symmetric Mode Frequency (2 Spins) | ω+ = ω0 + 2κ | N/A | Eigenfrequency of the symmetric normal mode, split by inter-mode coupling κ. |
| Antisymmetric Mode Frequency (2 Spins) | ω- = ω0 - 2κ | N/A | Eigenfrequency of the antisymmetric normal mode. |
| Antisymmetric Decay Rate (Γ-) | Γ * (ω- / ω0)3 | N/A | Decay rate based on Fermi’s golden rule for a Debye phonon model. |
| Spectral Gap (N Modes, Random κ) | E[λ2 - λ1] = O(N) | N/A | Expected energy gap between the lowest (near-symmetric) mode (λ1) and the second mode (λ2). |
| Symmetric Mode Fidelity Error | < 1/N | N/A | Fidelity error of the lowest energy eigenmode relative to the ideal fully symmetric mode (for random negative coupling). |
| Example Process Fidelity (J=5) | 0.9703 | N/A | Achieved fidelity for a target spin squeezing unitary in the underdamped regime (η/ωs = 0.1, Γ/ωs = 10-3). |
Key Methodologies
Section titled “Key Methodologies”The core of the research involves deriving an exact analytic solution for the reduced spin dynamics in a complex, non-Markovian open quantum system.
- Hamiltonian Definition: The system was modeled using the pure-dephasing spin-boson Hamiltonian (H = Ωĵz + Hmodes + ĵz Σk ηk Xk), where the spin (ĵz) couples to the quadrature (Xk) of the bosonic modes (vibrational phonons).
- Interaction Picture and Propagator: The time evolution operator Û(t) was formulated in the interaction picture. Since the spin interaction is diagonal in the ĵz basis, the propagator was written as a sum over spin eigenstates, simplifying the bosonic operator integral.
- Time-Ordering Removal (Magnus Expansion): The time-ordered exponential involving the bosonic operators was simplified using the Magnus expansion. Due to the specific commutation relations of the pure-dephasing model, the expansion terminates at the second order, yielding an exact solution free of time-ordering.
- Analytic Trace over Thermal Bath: The reduced spin density matrix ρspin(t) was obtained by analytically tracing the joint state over the vibrational modes, assuming the modes start in a thermal state (temperature T). This yields the dephasing factor (IRe) and unitary phase factor (IIm).
- Dissipative Extension: The model was extended to an open system by adding Lindblad dissipators (D[ρ]) to the master equation, describing the Markovian decay (Γk) of the vibrational modes into their local thermal baths.
- Normal Mode Transformation (Multi-Spin): For two or more spins, inter-mode coupling (κ) was introduced. The bosonic Hamiltonian was diagonalized to find collective symmetric (ω+) and antisymmetric (ω-) normal modes, which couple to collective spin operators (Ĵz and Âz, respectively).
- Spectral Engineering: Specific coupling matrices (uniform negative or random negative κ) were analyzed to demonstrate the creation of a large spectral gap, isolating the symmetric normal mode for targeted quantum control.
Commercial Applications
Section titled “Commercial Applications”The ability to analytically model and engineer coherence in complex spin-boson systems has direct relevance to emerging quantum technologies, particularly those based on solid-state defects.
- Quantum Sensing (NV Centers):
- Application: NV centers in diamond are leading platforms for high-resolution magnetometry and thermometry.
- Value Proposition: The QZE mechanism allows engineers to design NV environments (e.g., by controlling phonon decay rates Γ) to maximize spin coherence times, directly improving sensor sensitivity and operational bandwidth.
- Solid-State Quantum Computing:
- Application: Developing robust collective qubits using ensembles of defect spins.
- Value Proposition: The demonstrated protection of the symmetric spin subspace provides a blueprint for error mitigation. By ensuring non-symmetric modes decay rapidly, the system is effectively driven into the protected, long-lived symmetric subspace, enhancing qubit fidelity.
- Quantum Control and Gates:
- Application: Implementing high-fidelity quantum gates (e.g., geometric phase gates, spin squeezing unitaries).
- Value Proposition: The isolation of the symmetric normal mode via a large spectral gap (O(N)) enables selective addressing. This allows for the use of dynamical decoupling sequences to generate nonlinear spin interactions necessary for complex quantum algorithms.
- Materials Science and Defect Characterization:
- Application: Characterizing the fundamental physics of electronic defects in crystals (e.g., SiC, diamond, molecular aggregates).
- Value Proposition: The analytic solution provides a tool to fit experimental coherence data (T2 times) to constrain unknown material parameters, such as phonon frequencies (ωk), spin-phonon coupling strengths (ηk), and phonon-phonon coupling rates (κ).
View Original Abstract
We consider a spin-j particle coupled to a structured bath of bosonic modes that decay into thermal baths. We obtain an analytic expression for the reduced spin state and use it to investigate non-Markovian spin dynamics. In the heavily overdamped regime, spin coherences are preserved due to a quantum Zeno affect. We extend the solution to two spins and include coupling between the modes, which can be leveraged for preservation of the symmetric spin subspace. For many spins, we find that intermode coupling gives rise to a privileged symmetric mode gapped from the other modes. This provides a handle to selectively address that privileged mode for quantum control of the collective spin. Finally, we show that our solution applies to defects in solid-state systems, such as negatively charged nitrogen vacancy centers in diamond.
Tech Support
Section titled “Tech Support”Original Source
Section titled “Original Source”References
Section titled “References”- 2007 - The Theory of Open Quantum Systems [Crossref]