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arXiv:1805.11118 · Journal of Physics A Mathematical and Theoretical

Thermal contact: mischief and time scales

Daniel Grimmer, Robert B. Mann, Eduardo Martín-Martínez

Published Aug 9, 2019• 2 citations
arXivHTMLPDF

Abstract

Abstract We discuss what kind of quantum dynamics can yield thermalization. We show that any temperature measurement protocol must be sensitive to the system’s local energy scale, not just its state. Specifically, we illustrate this by showing that any temperature measurement protocol that does not resolve the system’s local Hamiltonian (such as, e.g. full state tomography) is susceptible to being fooled into measuring any value for the temperature. Since placing a system in thermal contact with a reference system at a known temperature is a way of measuring temperature, any robust notion of thermal contact must be resistant to this ‘attack’. This provides us with a necessary conditions for thermal contact between quantum systems. We then apply this condition to a scenario where one expects to find thermalization: a system repeatedly interacting with the microconstitutents of a thermal reservoir. We show such models of thermalization cannot be correct without fine-tuning.

DOI
10.1088/1751-8121/ab3a19

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Abstract We discuss what kind of quantum dynamics can yield thermalization. We show that any temperature measurement protocol must be sensitive to the system’s local energy scale, not just its state. Specifically, we illustrate this by showing that any temperature measurement protocol that does not resolve the system’s local Hamiltonian (such as, e.g. full state tomography) is susceptible to being fooled into measuring any value for the temperature. Since placing a system in thermal contact with a reference system at a known temperature is a way of measuring temperature, any robust notion of thermal contact must be resistant to this ‘attack’. This provides us with a necessary conditions for thermal contact between quantum systems. We then apply this condition to a scenario where one expects to find thermalization: a system repeatedly interacting with the microconstitutents of a thermal reservoir. We show such models of thermalization cannot be correct without fine-tuning.
Abstract Collisional reservoirs are becoming a major tool for modelling open quantum systems. In their simplest implementation, an external agent switches on, for a given time, the interaction between the system and a specimen from the reservoir. Generically, in this operation the external agent performs work onto the system, preventing thermalization when the reservoir is at equilibrium. One can recover thermalization by considering an autonomous global setup where the reservoir particles colliding with the system possess a kinetic degree of freedom. The drawback is that the corresponding scattering problem is rather involved. Here, we present a formal solution of the problem in one dimension and for flat interaction potentials. The solution is based on the transfer matrix formalism and allows one to explore the symmetries of the resulting scattering map. One of these symmetries is micro-reversibility, which is a condition for thermalization. We then introduce two approximations of the scattering map that preserve these symmetries and, consequently, thermalize the system. These relatively simple approximate solutions constitute models of quantum thermostats and are useful tools to study quantum systems in contact with thermal baths. We illustrate their accuracy in a specific example, showing that both are good approximations of the exact scattering problem even in situations far from equilibrium. Moreover, one of the models consists of the removal of certain coherences plus a very specific randomization of the interaction time. These two features allow one to identify as heat the energy transfer due to switching on and off the interaction. Our results prompt the fundamental question of how to distinguish between heat and work from the statistical properties of the exchange of energy between a system and its surroundings.
An interacting quantum system and thermal bath can reach thermal equilibrium, resulting in the system and bath acquiring the same temperature. But how does a delocalized quantum system thermalize with a bath whose local temperature varies as, for example, in the Tolman effect? Here, we formulate two scenarios in which the notion of a “superposition of temperatures” may arise: first, a probe interaction with two different baths dependent on the state of another quantum system (a control); and second, a probe interaction with a single bath whose purified state is a superposition of states that each correspond to different temperatures. We show that the two scenarios are fundamentally different and can be operationally distinguished. Moreover, we show that the final probe state is sensitive to the specific realization of the thermalizing channels, and that our results hold for partial and prethermalization cases. Our models may be applied to scenarios involving joint quantum and relativistic phenomena, and explain some recent results found in quantum interference of relativistic probes thermalizing with Unruh or Hawking radiation.
We analyze in full detail the time evolution of an open Gaussian quantum system rapidly bombarded by Gaussian ancillae. As a particular case this analysis covers the thermalization (or not) of a harmonic oscillator coupled to a thermal reservoir made of harmonic oscillators. We derive general results for this scenario and apply them to the problem of thermalization. We show that only a particular family of system-environment couplings will cause the system to thermalize to the temperature of its environment. We discuss that if we want to understand thermalization as ensuing from the Markovian interaction of a system with the individual microconstituents of its (thermal) environment then the process of thermalization is not as robust as one might expect.
The ultimate precision of any measurement of the temperature of a quantum system is the inverse of the local quantum thermal susceptibility [A. De Pasquale et al., Nat. Commun. 7, 12782 (2016)] of the subsystem with which the thermometer interacts. If this subsystem can be described with the canonical ensemble, such quantity reduces to the variance of the local Hamiltonian, which is proportional to the heat capacity of the subsystem. However, the canonical ensemble might not apply in the presence of interactions between the subsystem and the rest of the system. In this work, we address this problem in the framework of locally interacting quantum systems. We prove that the local quantum thermal susceptibility of any subsystem is close to the variance of its local Hamiltonian, provided the volume-to-surface ratio of the subsystem is much larger than the correlation length. This result greatly simplifies the determination of the ultimate precision of any local estimate of the temperature and rigorously determines the regime where interactions can affect this precision.
We present rigorous bounds on the thermalization time of the family of quantum mechanical spin systems known as stabilizer Hamiltonians. The thermalizing dynamics are modeled by a Davies master equation that arises from a weak local coupling of the system to a large thermal bath. Two temperature regimes are considered. First we clarify how in the low temperature regime, the thermalization time is governed by a generalization of the energy barrier between orthogonal ground states. When no energy barrier is present the Hamiltonian thermalizes in a time that is at most quadratic in the system size. Secondly, we show that above a universal critical temperature, every stabilizer Hamiltonian relaxes to its unique thermal state in a time which scales at most linearly in the size of the system. We provide an explicit lower bound on the critical temperature. Finally, we discuss the implications of these result for the problem of self-correcting quantum memories with stabilizer Hamiltonians.
Two identical finite quantum systems prepared initially at different temperatures, isolated from the environment, and subsequently brought into contact are demonstrated to relax towards Gibbs-like quasiequilibrium states with a common temperature and small fluctuations around the time-averaged expectation values of generic observables. The temporal thermalization process proceeds via a chain of intermediate Gibbs-like states. We specify the conditions under which this scenario occurs and corroborate the quantum equilibration with two different models.