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6CCVD Research

Thermodynamics of a Minimal Algorithmic Cooling Refrigerator

At a Glance

Section titled “At a Glance”
MetadataDetails
Publication Date2022-07-11
JournalPhysical Review Letters
AuthorsRodolfo R. Soldati, Durga Dasari, Jörg Wrachtrup, Eric Lutz
InstitutionsUniversity of Stuttgart, Max Planck Institute for Solid State Research
Citations11
AnalysisFull AI Review Included

Executive Summary

Section titled “Executive Summary”

This research investigates the thermodynamic performance of a minimal three-qubit heat-bath algorithmic cooling (HBAC) refrigerator implemented in a Nitrogen-Vacancy (NV) center in diamond.

  • Core Achievement: Analytical computation and experimental validation of key thermodynamic figures of merit—Coefficient of Performance (COP), cooling power (J), and target qubit polarization ($\epsilon$1)—for an arbitrary number of cooling cycles, including transient effects.
  • Fundamental Limits: The study determined the fundamental upper bounds for COP and cooling power in the ideal reversible limit ($\gamma$ = 0) and demonstrated that experimental results closely approach these theoretical limits.
  • Efficiency Convergence: The COP was shown to quickly converge to the ideal Carnot limit ($c$) after only a few cycles in the reversible regime, establishing a benchmark for quantum refrigeration efficiency.
  • Imperfection Modeling: The analytical framework explicitly accounts for realistic experimental imperfections, including irreversible energy dissipation (damping rate $\gamma$) of the target qubit and nonideal logic gates (quantified by mixing angle $\theta$).
  • Platform Validation: The system utilizes the central electron spin of the NV center as both the heat bath and an ancillary spin to mediate the necessary three-qubit Toffoli gate for entropy compression.

Technical Specifications

Section titled “Technical Specifications”

The following table summarizes the key physical and performance parameters of the minimal algorithmic cooling refrigerator based on the NV center in diamond.

ParameterValueUnitContext
System PlatformNV CenterDiamondThree nuclear spins (14N, two 13C) coupled to a central electron spin.
Target Qubit (14N) Larmor Freq.~1.66MHzUsed for calculating heat and work metrics.
14N Hyperfine Coupling2.16MHzCoupling strength to the electron spin.
13C1 Hyperfine Coupling90kHzCoupling strength to the electron spin.
13C2 Hyperfine Coupling414kHzCoupling strength to the electron spin.
Operating Magnetic Field (B)540mTChosen to maximize nuclear spin T1 lifetime and improve readout fidelity.
Target Qubit Decay Rate ($\gamma$)~10-4(Unitless)Experimental fit value for irreversible dissipation.
Compression Step Duration~285”sTime required for the non-local three-qubit gate.
Refresh Step Duration~5msTime required for rethermalization/SWAP gate.
Electron Spin T1e~5.7msElectron spin relaxation timescale.
Electron Spin T2, Hahn~395”sElectron spin coherence timescale.
Effective Gate Mixing Angle ($\theta$)$\pi$/3.4RadiansCorresponds to an overall gate error of ~20% in compression.
Heat Extracted Q(n) (Order)FewneV/cycleThermodynamic output metric.
Cooling Power J(n) (Order)Few ”eV/s or 10-26WThermodynamic output metric.
Target Spin Readout Fidelity~97%Achieved via single-shot readout (SSR).

Key Methodologies

Section titled “Key Methodologies”

The study combined advanced theoretical modeling with experimental implementation on a solid-state quantum system.

  1. Theoretical Framework (Liouville Space):

    • The full nonstationary dynamics of the three-qubit system were solved analytically using Liouville space vectorization techniques.
    • Quantum channels (Damping D, Compression C, Refresh R) were mapped onto superoperators ($\Phi$) to efficiently compute the state evolution over $n$ cycles ($\Phi_{\mathcal{E}}^n$).
    • This method allowed for the exact calculation of COP, cooling power, and polarization for arbitrary cycle numbers, including the transient regime.

  2. Experimental Platform (NV Diamond):

    • The refrigerator was built using the 14N (target) and two 13C (reset) nuclear spins in a single NV center.
    • The setup utilized a homebuilt confocal microscope, a permanent magnet (540 mT), and MW/RF sources, operating at ambient conditions.

  3. Cooling Cycle Implementation:

    • Compression Step: Implemented using a non-local three-qubit Toffoli gate, mediated by the central electron spin, to achieve entropy compression.
    • Refresh Step: Implemented via an iterative SWAP gate sequence, allowing the fast-relaxing reset spins to rethermalize to the hot bath (the electron spin, which is optically repolarized).

  4. Thermodynamic Evaluation:

    • Target qubit polarization ($\epsilon$1(n)) was measured directly using high-fidelity single-shot readout.
    • Heat extracted Q(n) and work supplied W(n) were derived from the measured polarization data using established thermodynamic definitions for the qubit system.
    • COP ($\zeta$(n)) and cooling power J(n) were calculated from Q(n) and W(n) per cycle.

Commercial Applications

Section titled “Commercial Applications”

This research is foundational for developing robust, high-performance quantum technologies, particularly those relying on precise state initialization and thermal management.

  • Quantum Computing: Provides essential techniques for initializing qubits into highly pure states (high polarization), a prerequisite for scalable quantum computation and error correction (e.g., using NV centers as solid-state quantum registers).
  • Quantum Sensing and Metrology: NV centers are leading platforms for nanoscale sensing (magnetic field, temperature). Algorithmic cooling enhances the sensitivity and dynamic range of these sensors by preparing the nuclear spins in ultra-cold states.
  • Solid-State Quantum Devices: The methodology for modeling and mitigating realistic imperfections (damping, gate errors) is critical for engineering robust, practical quantum hardware operating in non-ideal environments.
  • Low-Temperature Physics: The study contributes to the fundamental understanding of thermodynamics at the quantum scale, particularly concerning finite-size heat baths and non-equilibrium processes.
View Original Abstract

We investigate, theoretically and experimentally, the thermodynamic performance of a minimal three-qubit heat-bath algorithmic cooling refrigerator. We analytically compute the coefficient of performance, the cooling power, and the polarization of the target qubit for an arbitrary number of cycles, taking realistic experimental imperfections into account. We determine their fundamental upper bounds in the ideal reversible limit and show that these values may be experimentally approached using a system of three qubits in a nitrogen-vacancy center in diamond.

Tech Support

Section titled “Tech Support”

Original Source

Section titled “Original Source”

References

Section titled “References”
  1. 1985 - Thermodynamics and an Introduction to Thermostatistics
  2. 1999 - Laser Cooling and Trapping [Crossref]
  3. 2007 - Laser Control of Atoms and Molecules [Crossref]
  4. 1992 - Low-Temperature Physics
  5. 2005 - Low-Temperature Physics
  6. 2000 - Quantum Computation and Quantum Information
  7. 2009 - Classical and Quantum Information Theory [Crossref]
  8. 1999 - Proceedings of the 31st ACM Symposium on Theory of Computing
  9. 2016 - Electron Spin Resonance (ESR) Based Quantum Computing

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