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Contributions to single-shot energy exchanges in open quantum systems

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journal contribution
posted on 2019-07-09, 14:03 authored by Rui Sampaio, J. Anders, T.G. Philbin, Tapio Ala-NissilaTapio Ala-Nissila
The exchange of energy between a classical open system and its environment can be analysed for a single run of an experiment using the phase space trajectory of the system. By contrast, in the quantum regime such energy exchange processes must be defined for an ensemble of runs of the same experiment based on the reduced system density matrix. Single-shot approaches based on stochastic wave functions have been proposed for quantum systems that are continuously monitored or weakly coupled to a heat bath. However, for systems strongly coupled to the environment and not continuously monitored, a single-shot analysis has not been attempted because no system wave function exists for such systems within the standard formulation of quantum theory. Using the notion of the conditional wave function of a quantum system, we derive here an exact formula for the rate of total energy change in an open quantum system, valid for arbitrary coupling between the system and the environment. In particular, this allows us to identify three distinct contributions to the total energy flow: an external contribution coming from the explicit time dependence of the Hamiltonian, an interaction contribution associated with the interaction part of the Hamiltonian, and an entanglement contribution, directly related to the presence of entanglement between the system and its environment. Given the close connection between weak values and the conditional wave function, the approach presented here provides a new avenue for experimental studies of energy fluctuations in open quantum systems.


This work has been supported in part by the Academy of Finland through its Quantum Technology Finland CoE project No. 312298. R.S. acknowledges the financial support by the Magnus Ehrnrooth Foundation as well as from the CMMP Education Network. J.A. acknowledges support from EPSRC (grant EP/M009165/1) and the Royal Society. This research was supported by the COST network MP1209 “Thermodynamics in the quantum regime".



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  • Mathematical Sciences

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Physical Review E






SAMPAIO, R. ... et al., 2019. Contributions to single-shot energy exchanges in open quantum systems. Physical Review E, 99: 062131.


© American Physical Society (APS)


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This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

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This paper was published in the journal Physical Review E, and the definitive published version is available at https://doi.org/10.1103/physreve.99.062131






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