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arXiv:2103.16601 · Physical Review A

Taking the temperature of a pure quantum state

Mark T. Mitchison, Archak Purkayastha, Marlon Brenes, Alessandro Silva, John Goold

Published Mar 8, 2022• 25 citations
arXivHTMLPDF

Abstract

Temperature is a deceptively simple concept that still raises deep questions at the forefront of quantum physics research. The observation of thermalization in completely isolated quantum systems, such as cold-atom quantum simulators, implies that a temperature can be assigned even to individual, pure quantum states. Here, we propose a scheme to measure the temperature of such pure states through quantum interference. Our proposal involves interferometry of an auxiliary qubit probe, which is prepared in a superposition state and subsequently decoheres due to weak coupling with a closed, thermalized many-body system. Using only a few basic assumptions about chaotic quantum systems, namely, the eigenstate thermalization hypothesis and the emergence of hydrodynamics at long times, we show that the qubit undergoes pure exponential decoherence at a rate that depends on the temperature of its surroundings. We verify our predictions by numerical experiments on a quantum spin chain that thermalizes after absorbing energy from a periodic drive. Our Letter provides a general method to measure the temperature of isolated, strongly interacting systems under minimal assumptions.

DOI
10.1103/physreva.105.l030201

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Temperature is a deceptively simple concept that still raises deep questions at the forefront of quantum physics research. The observation of thermalization in completely isolated quantum systems, such as cold-atom quantum simulators, implies that a temperature can be assigned even to individual, pure quantum states. Here, we propose a scheme to measure the temperature of such pure states through quantum interference. Our proposal involves interferometry of an auxiliary qubit probe, which is prepared in a superposition state and subsequently decoheres due to weak coupling with a closed, thermalized many-body system. Using only a few basic assumptions about chaotic quantum systems, namely, the eigenstate thermalization hypothesis and the emergence of hydrodynamics at long times, we show that the qubit undergoes pure exponential decoherence at a rate that depends on the temperature of its surroundings. We verify our predictions by numerical experiments on a quantum spin chain that thermalizes after absorbing energy from a periodic drive. Our Letter provides a general method to measure the temperature of isolated, strongly interacting systems under minimal assumptions.
A large class of isolated quantum systems in a pure state can equilibrate and serve as a heat bath. We show that, once equilibrium is reached, any of its subsystems, which are much smaller than the isolated system, can be thermalized such that the subsystem is governed by the Gibbs distribution. Within this theoretical framework, the celebrated superposition principle of quantum mechanics leads to a prediction of a thermalized subsystem with multiple temperatures when the isolated system is in a superposition state of energy eigenstates of multiple distinct energy scales. This multiple-temperature state is at equilibrium, completely different from a non-equilibrium state that has multiple temperatures at different parts. Feasible experimental schemes, in particular with ultra-cold atoms, to verify this prediction are discussed.
We study the effect of temperature on the ability of a qubit probe to distinguish between different phases in strongly correlated systems. Now to this end, we investigate a spin-1/2 Heisenberg XXZ chain coupled to a qubit probe via a pure dephasing interaction. At zero temperature, as reported previously, the decoherence dynamics exhibit different behavior in the gapped and gapless phases of the chain, with characteristic strong oscillations in the gapped phase and monotonic decay in the gapless phase. These oscillations decay much more slowly than expected with rising temperature in the gapped phase and their remnants are still visible at infinite temperature. In the gapless phase at low temperatures, using Luttinger liquid theory, we predict that the coherence exhibits asymptotic exponential decay in time, with the decay rate sensitive to the sign of the anisotropy parameter. We conclude that probe dynamics continue to carry information about the chain even at finite, reasonably small temperatures, positioning qubit probes as sensitive and robust detectors for quantum phases of correlated spin systems.
An isolated quantum system in a pure state may be perceived as thermal if only a substantially small fraction of all degrees of freedom is probed. We propose that in a quantum chaotic many-body system all states with sufficiently small energy fluctuations are approximately thermal. We refer to this hypothesis as canonical universality (CU). The CU hypothesis complements the eigenstate thermalization hypothesis which proposes that for chaotic systems individual energy eigenstates are thermal. Integrable and many-body localization systems do not satisfy CU. We provide theoretical and numerical evidence supporting the CU hypothesis.
The emergence of classical behavior and the origin of thermal equilibrium are two central problems in the foundations of physics. In the standard accounts, both phenomena are typically explained through interactions with an external environment: decoherence suppresses quantum interference, while coupling to a thermal bath drives relaxation toward equilibrium. Over the last decades, however, it has become clear that equilibration and thermalization can arise even in fully isolated quantum systems, in the operational sense that the expectation values of relevant observables remain close to equilibrium values for most of the time. Here, we ask whether the same intrinsic equilibration mechanism can also account for the emergence of classical behavior. Using rigorous bounds on equilibration in closed systems, we derive sufficient conditions under which a time-evolved pure state becomes, for most times, operationally indistinguishable from a classical mixture associated with a chosen physical property. We identify two complementary routes to such operational classicality: either the chosen property almost commutes with the system Hamiltonian or the observables used to probe the system lose access to the remaining coherence after equilibration. Our results show that classical behavior need not be confined to the energy basis and may emerge even when substantial coherence remains present in the equilibrium state. This establishes a direct connection between two foundational questions: the origin of thermalization in isolated quantum systems and the quantum-to-classical transition
Statistical mechanics relies on the maximization of entropy in a system at thermal equilibrium. However, an isolated quantum many-body system initialized in a pure state remains pure during Schrödinger evolution, and in this sense it has static, zero entropy. We experimentally studied the emergence of statistical mechanics in a quantum state and observed the fundamental role of quantum entanglement in facilitating this emergence. Microscopy of an evolving quantum system indicates that the full quantum state remains pure, whereas thermalization occurs on a local scale. We directly measured entanglement entropy, which assumes the role of the thermal entropy in thermalization. The entanglement creates local entropy that validates the use of statistical physics for local observables. Our measurements are consistent with the eigenstate thermalization hypothesis.
The eigenstate thermalization hypothesis (ETH) offers a universal mechanism for the approach to equilibrium of closed quantum many-body systems.So far, however, experimental studies have focused on the relaxation dynamics of observables as described by the diagonal part of ETH, whose verification requires substantial numerical input.This leaves many of the general assumptions of ETH untested.Here, we propose a theory-independent route to probe the full ETH in quantum simulators by observing the emergence of fluctuation-dissipation relations, which directly probe the off-diagonal part of ETH.We discuss and propose protocols to independently measure fluctuations and dissipations as well as higher order time-ordered correlation functions.We first show how the emergence of fluctuation-dissipation relations from a nonequilibrium initial state can be observed for the two-dimensional (2D) Bose-Hubbard model in superconducting qubits or quantum gas microscopes.Then we focus on the long-range transverse field Ising model (LTFI), which can be realized with trapped ions.The LTFI exhibits rich thermalization phenomena: For strong transverse fields, we observe prethermalization to an effective magnetization-conserving Hamiltonian in the fluctuation-dissipation relations.For weak transverse fields, confined excitations lead to nonthermal features, resulting in a violation of the fluctuation-dissipation relations up to long times.Moreover, in an integrable region of the LTFI, thermalization to a generalized Gibbs ensemble occurs and the fluctuation-dissipation relations enable an experimental diagonalization of the Hamiltonian.Our work presents a theory-independent way to characterize thermalization in quantum simulators and paves the way to quantum simulate condensed matter pump-probe experiments.