Hilbert-Space Ergodicity in Driven Quantum Systems: Obstructions and Designs
arxiv(2024)
摘要
Despite its long history, a canonical formulation of quantum ergodicity that
applies to general classes of quantum dynamics, including driven systems, has
not been fully established. Here we introduce and study a notion of quantum
ergodicity for closed systems with time-dependent Hamiltonians, defined as
statistical randomness exhibited in their long-time dynamics. Concretely, we
consider the temporal ensemble of quantum states (time-evolution operators)
generated by the evolution, and investigate the conditions necessary for them
to be statistically indistinguishable from uniformly random states (operators)
in the Hilbert space (space of unitaries). We find that the number of driving
frequencies underlying the Hamiltonian needs to be sufficiently large for this
to occur. Conversely, we show that statistical pseudo-randomness –
indistinguishability up to some large but finite moment, can already be
achieved by a quantum system driven with a single frequency, i.e., a Floquet
system, as long as the driving period is sufficiently long. Our work relates
the complexity of a time-dependent Hamiltonian and that of the resulting
quantum dynamics, and offers a fresh perspective to the established topics of
quantum ergodicity and chaos from the lens of quantum information.
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