Quantum collisional thermostats

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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.
SLJ is supported by the Doctoral Training Unit on Materials for Sensing and Energy Harvesting (MASSENA) with the Grant: FNR PRIDE/15/10935404. ME acknowledges financial support from the European Research Council (project NanoThermo, ERC-2015-CoG Agreement No. 681456) and the FQXi foundation, project 'Information as a fuel in colloids and superconducting quantum circuits' (FQXi-IAF19-05). FB thanks Fondecyt project 1191441 and the Millennium Nucleus 'Physics of active matter' of ANID (Chile). Part of this work was conducted at the KITP, a facility supported by the US National Science Foundation under Grant No. NSF PHY-1748958. JMRP, JT and IL acknowledge financial support from the Spanish Government (Grant Contracts FIS-2017-83706-R and PID2020-113455GB-I00) and from the Foundational Questions Institute Fund, a donor advised fund of Silicon Valley Community Foundation (Grant Number FQXi-IAF19-01).
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