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Noncommuting conserved charges in quantum many-body thermalization

At a Glance

Section titled “At a Glance”
MetadataDetails
Publication Date2020-04-15
JournalPhysical review. E
AuthorsNicole Yunger Halpern, Michael E. Beverland, Amir Kalev
InstitutionsUniversity of Maryland, College Park, Center for Astrophysics Harvard & Smithsonian
Citations47
AnalysisFull AI Review Included

6CCVD Technical Documentation: Quantum Many-Body Thermalization to the Non-Abelian Thermal State (NATS)

Section titled “6CCVD Technical Documentation: Quantum Many-Body Thermalization to the Non-Abelian Thermal State (NATS)”

Based on: Noncommuting conserved quantities in quantum many-body thermalization (Halpern et al., 2020)

Executive Summary

Section titled “Executive Summary”

This paper proposes and numerically simulates an experimental protocol to observe a quantum system thermalizing to the Non-Abelian Thermal State (NATS), a unique phase predicted by quantum-information-theoretic (QI-theoretic) thermodynamics when conserved quantities (charges) fail to commute.

  • Non-Abelian Thermalization: The research provides the first physical protocol for observing the NATS, bridging abstract QI-thermodynamics with realizable quantum many-body physics (AMO, condensed matter).
  • Experimental Feasibility: The protocol is based on a spin chain model (qubits) and is highly applicable for implementation using robust solid-state platforms, notably Nitrogen-Vacancy (NV) centers in Single Crystal Diamond (SCD).
  • Noncommutation Effects: The charges (spin components $\sigma_{x}^{\text{tot}}, \sigma_{y}^{\text{tot}}, \sigma_{z}^{\text{tot}}$) are noncommuting, which necessitates a weak, non-integrable Heisenberg interaction Hamiltonian ($H^{\text{tot}}$) to drive thermalization.
  • Superior Accuracy: Numerical simulations show that the long-time state ($\rho_S$) converges to the NATS prediction ($\rho_{\text{NATS}}$) with significantly greater accuracy than to the canonical or generalized Gibbs ensemble (GGE) predictions.
  • Scalability & Error Analysis: The accuracy (measured by relative entropy $D$) improves with system size $N$, scaling numerically as $N^{-5/2}$. The protocol is also shown to be robust against realistic $\sim 1%$ Hamiltonian anisotropy errors.
  • Material Implication: Replicating this research requires ultra-high purity materials with exceptional coherence properties suitable for hosting stable qubits, making high-quality MPCVD Single Crystal Diamond essential.

Technical Specifications

Section titled “Technical Specifications”
ParameterValueUnitContext / Description
System Size Range ($Nn$)6 to 14QubitsTotal number of qubits simulated (2-qubit system $S$ embedded in a bath $B$).
System Size Scaling ($N$)$N \to \infty$N/ADefines the thermodynamic limit; $N$ is the number of system copies.
Interaction Strength ($J$)1ArbitraryUsed for normalizing time $t$ and Hamiltonian $H^{\text{tot}}$.
Evolution Time ($t$)$2^{Nn}$$1/J$Time used in simulations to ensure robust distinction from canonical ensemble.
Small Parameter Condition< 1N/AEnsures Taylor approximations for inverse temperature $\beta$ and chemical potentials $\mu_{\alpha}$ hold.
Anisotropy Test ($\Delta$)0.99N/ASimulation tested robustness against $1%$ anisotropy error in Heisenberg interactions.
Relative Entropy Scaling$\sim N^{-5/2}$natsNumerical fit for the distance $D(\rho_S
Theoretical Scaling Bound$\le \text{const.}/\sqrt{N}$natsAnalytical prediction for the upper bound of relative entropy scaling [Eq. (14)].

Key Methodologies

Section titled “Key Methodologies”

The experimental protocol relies on three key phases: Preparation, Evolution, and Readout. The proposed system is a closed, isolated chain of $N$ two-qubit subsystems (total $Nn$ qubits), evolving under a non-integrable Hamiltonian:

  1. System Setup and Hamiltonian:

    • Architecture: Linear chain of $Nn$ qubits, with the first $n=2$ forming the system of interest ($S$).
    • Hamiltonian ($H^{\text{tot}}$): Constructed using nearest-neighbor and next-nearest-neighbor Heisenberg interactions (e.g., $J \sum \sigma_{x}^{(j)}\sigma_{x}^{(j+1)}$ + h.c.), summing over spin components $x, y, z$. This choice ensures the total spin charge $\sigma_{\alpha}^{\text{tot}}$ is conserved, but the total Hamiltonian $H^{\text{tot}}$ is non-integrable, promoting thermalization.

  2. Initial State Preparation (Approximate Microcanonical Subspace, $\mathcal{M}$):

    • The goal is to prepare the global state $\rho^{\text{tot}}$ such that the total charge $\sigma_{\alpha}^{\text{tot}}$ has a fairly well-defined value $S_{\alpha}$.
    • Soft Measurement Protocol: This novel technique uses a Positive Operator-Valued Measure (POVM) characterized by a binomial distribution envelope $f_{Nn}(S_{\alpha}, \bar{S}_{\alpha})$ (approaching a Gaussian for large $Nn$).
    • The sequence involves soft measurements of $H^{\text{tot}}$, followed by soft measurements of the noncommuting charges $\sigma_{x}^{\text{tot}}, \sigma_{y}^{\text{tot}}, \sigma_{z}^{\text{tot}}$, successively collapsing the state into the desired approximate microcanonical subspace ($\mathcal{M}$).

  3. Evolution and Thermalization:

    • The prepared state evolves under $H^{\text{tot}}$ for a long time ($t \sim Nn/J$), allowing the system $S$ to thermalize internally with the effective bath $B$.
    • The long-time state of $S$ is the reduced density matrix $\rho_S$.

  4. Readout and Analysis:

    • Inference: The thermalized state $\rho_S$ is inferred via Quantum State Tomography (QST), utilizing $2^n$ basis products of Pauli operators measured over $N_{\text{trials}}$ trials.
    • Metric: The accuracy of thermalization is quantified by the relative entropy $D(\rho_S||\rho_{\text{NATS}})$ in units of nats, which is expected to decrease as the system size grows.

6CCVD Solutions & Capabilities

Section titled “6CCVD Solutions & Capabilities”

This groundbreaking research, which aims to leverage solid-state quantum systems like NV centers in diamond, requires materials and engineering precision that perfectly align with 6CCVD’s core capabilities.

Applicable Materials

Section titled “Applicable Materials”

To replicate or extend the proposed thermalization protocol using NV centers, high-quality diamond materials are mandatory:

  • Optical Grade Single Crystal Diamond (SCD): NV centers require extremely high-purity SCD wafers to maintain long coherence times ($T_2$). 6CCVD provides low-strain, high-purity SCD substrates necessary for stable spin qubits and minimizing decoherence from nitrogen background impurities.
  • Custom Substrates: We offer SCD substrates from 0.1 ”m up to 500 ”m thickness, potentially allowing for tailored device geometries that optimize qubit coupling and minimize environmental noise.
  • Polishing Excellence: The integration of NV centers, especially in architectures requiring surface coupling or optical excitation, demands ultra-smooth surfaces. 6CCVD guarantees Ra < 1 nm polishing for SCD, ensuring minimal light scattering and high-fidelity device fabrication.

Customization Potential

Section titled “Customization Potential”

The small-scale, tightly integrated nature of quantum spin chains necessitates precise manufacturing and integration capabilities:

Research Requirement6CCVD Custom SolutionBenefit to the Engineer
Qubit Control StructuresCustom Metalization ServicesWe offer deposition of key metals (Au, Pt, Pd, Ti, W, Cu) for electrodes, waveguides, or microwave control lines essential for controlling spin flips and implementing the soft measurement protocols.
Device IntegrationPrecision Laser Cutting / DicingTo define the specific quantum device geometry and facilitate integration onto chip carriers or into ultracold atom traps, 6CCVD provides custom dimensions and accurate patterning.
Thermal ManagementHigh-Thickness SubstratesWhile NATS experiments often occur at low temperatures, high power input requires robust thermal dissipation. We offer thick SCD substrates (up to 10 mm) known for their superior thermal conductivity.

Engineering Support

Section titled “Engineering Support”

Successfully executing experiments involving complex QI-theoretic thermodynamics and noncommuting conserved quantities requires deep material and quantum engineering expertise.

  • 6CCVD’s in-house PhD-level engineering team specializes in MPCVD growth and diamond material science optimized for quantum applications, including NV centers and quantum sensing.
  • We can assist researchers in selecting the optimal SCD grade, required thickness tolerances, and necessary metal stack configurations for implementing the complex Heisenberg interaction Hamiltonians and non-Abelian charge thermalization projects.

For custom specifications or material consultation, visit 6ccvd.com or contact our engineering team directly.

View Original Abstract

In statistical mechanics, a small system exchanges conserved quantities-heat, particles, electric charge, etc.-with a bath. The small system thermalizes to the canonical ensemble or the grand canonical ensemble, etc., depending on the quantities. The conserved quantities are represented by operators usually assumed to commute with each other. This assumption was removed within quantum-information-theoretic (QI-theoretic) thermodynamics recently. The small system’s long-time state was dubbed “the non-Abelian thermal state (NATS).” We propose an experimental protocol for observing a system thermalize to the NATS. We illustrate with a chain of spins, a subset of which forms the system of interest. The conserved quantities manifest as spin components. Heisenberg interactions push the conserved quantities between the system and the effective bath, the rest of the chain. We predict long-time expectation values, extending the NATS theory from abstract idealization to finite systems that thermalize with finite couplings for finite times. Numerical simulations support the analytics: The system thermalizes to near the NATS, rather than to the canonical prediction. Our proposal can be implemented with ultracold atoms, nitrogen-vacancy centers, trapped ions, quantum dots, and perhaps nuclear magnetic resonance. This work introduces noncommuting conserved quantities from QI-theoretic thermodynamics into quantum many-body physics: atomic, molecular, and optical physics and condensed matter.

Tech Support

Section titled “Tech Support”

Original Source

Section titled “Original Source”

References

Section titled “References”
  1. 1980 - Statistical Physics: Part 1

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