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Complete Hilbert-Space Ergodicity in Quantum Dynamics of Generalized Fibonacci Drives
Saúl Pilatowsky-Cameo1, Ceren B Dag2,3, Wen Wei Ho4,5
1Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA.
Physical Review Letters
|January 5, 2024
Summary
Quantum systems can achieve complete Hilbert-space ergodicity (CHSE) by exploring the entire Hilbert space over time. This stronger form of ergodicity is achievable with specific aperiodic drives, offering new insights into quantum thermalization.
Area of Science:
- Quantum dynamics
- Statistical mechanics
- Chaos theory
Background:
- Ergodicity in quantum systems is typically defined by energy eigenstate properties, like Berry's conjecture and the eigenstate thermalization hypothesis.
- Existing definitions limit ergodicity to statistical properties, not the full exploration of the Hilbert space.
Purpose of the Study:
- To investigate a stronger form of ergodicity: complete Hilbert-space ergodicity (CHSE).
- To determine if quantum systems can dynamically visit every state in the Hilbert space over time.
Main Methods:
- Analysis of quantum dynamics under time-dependent drives.
- Mathematical formulation and proof of CHSE for specific drive families.
- Utilizing deterministic drives generated by Fibonacci words and generalizations.
Main Results:
- CHSE is not possible with time-independent or periodic Hamiltonians due to (quasi)energy eigenstates.
- A family of aperiodic, deterministic drives with minimal complexity, derived from Fibonacci words, enables CHSE.
- This provides a mechanism for quantum systems to uniformly explore their entire Hilbert space.
Conclusions:
- Complete Hilbert-space ergodicity (CHSE) represents a novel dynamical concept in quantum systems.
- Aperiodic drives, particularly those based on Fibonacci sequences, are key to achieving CHSE.
- These findings advance the understanding of thermalization in time-dependent quantum dynamics.
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