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Chinese Physics Letters

Quantum Signature of Classical Chaos in the Temporal Mean of Expectation Values of Observables

Jie Quan-Lin, Gong-ou Xu

Published Oct 1, 1995• 2 citations
Publisher

Abstract

In an integrable quantum system, the time averages of expectation values of a complete set of commuting observables (CSCO) form a regular pattern when every common eigenstate of the CSCO is chosen as the initial state. As the system transits to chaotic, the regular pattern is destroyed and these points cluster.
DOI
10.1088/0256-307x/12/10/001

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If a many-body quantum system approaches thermal equilibrium from a generic initial state, then the expectation value ⟨ψ(t)∣Ai∣ψ(t)⟩\langleψ(t)|A_i|ψ(t)\rangle⟨ψ(t)∣Ai​∣ψ(t)⟩, where ∣ψ(t)⟩|ψ(t)\rangle∣ψ(t)⟩ is the system's state vector and AiA_iAi​ is an experimentally accessible observable, should approach a constant value which is independent of the initial state, and equal to a thermal average of AiA_iAi​ at an appropriate temperature. We show that this is the case for all simple observables whenever the system is classically chaotic.
We consider a quantum system with N degrees of freedom which is classically chaotic. When N is large, and both and the quantum energy uncertainty are small, quantum chaos theory can be used to demonstrate the following results: (i) given a generic observable A, the infinite time average of the quantum expectation value is independent of all aspects of the initial state other than the total energy, and equal to an appropriate thermal average of A; (ii) the time variations of are too small to represent thermal fluctuations; (iii) however, the time variations of can be consistently interpreted as thermal fluctuations, even though these same time variations would be called quantum fluctuations when N is small.
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How classical chaos emerges from the underlying quantum world is a fundamental problem in physics. The origin of this question is in the correspondence principle. Classical chaos arises due to non-linear dynamics, whereas quantum mechanics, driven by unitary evolution, is linear. The question that still remains is - what are the footprints of classical chaos in the quantum world? One can understand the quantum signatures of classical chaos by studying a quantum system whose classical analogue is chaotic. In this thesis, we use the quantum kicked top model of few qubits in the deep quantum regime to investigate signatures that can be considered as a precursor to chaos in the classical limit. In particular, we study out-of-time-ordered correlators (OTOCs) and Loschmidt echo, the two well-known dynamical diagnostics of chaos. We find vestiges of classical chaos even in such a deep quantum regime. Another arena where one can study the effects of chaos and randomness is quantum state tomography. We study quantum tomography from a continuous measurement record obtained by measuring expectation values of a set of Hermitian operators generated by a unitary evolution of an initial observable. The rate of information gain and reconstruction fidelity shows vestiges of chaos. As another contribution of this thesis, we have harnessed the power of randomness inherent in the maximally mixed state to give an efficient quantum algorithm to measure OTOCs. The protocol achieves an exponential speedup over the best known classical algorithm, provided the OTOC operator to be estimated admits an efficient gate decomposition. This protocol also helps benchmark unitary gates, which is important from the quantum computation and control perspective.