How should one define thermodynamic quantities (internal energy, work, heat, etc.) for quantum systems coupled to their environments strongly? We examine three (classically equivalent) definitions of a quantum system’s internal energy under strong-coupling conditions. Each internal-energy definition implies a definition of work and a definition of heat. Our study focuses on quenches, common processes in which the Hamiltonian changes abruptly. In these processes, the first law of thermodynamics holds for each set of definitions by construction. However, we prove that only two sets obey the second law. We illustrate our findings using a simple spin model. Our results guide studies of thermodynamic quantities in strongly coupled quantum systems.

This work was supported by the DOE, Office of Science, Office of Nuclear Physics, IQuS (\url{https://iqus.uw.edu}), via the program on Quantum Horizons: QIS Research and Innovation for Nuclear Science under Award DE-SC0020970; and by the National Science Foundation (NSF) Quantum Leap Challenge Institutes (QLCI) (award no.~OMA-2120757); and by the Department of Energy (DOE), Office of Science, Early Career Award (award no.~DESC0020271), as well as by the Department of Physics; Maryland Center for Fundamental Physics; and College of Computer, Mathematical, and Natural Sciences at the University of Maryland, College Park. Part of this work was supported i by the Government of Canada through the Department of Innovation, Science, and Economic Development and by the Province of Ontario through the Ministry of Colleges and Universities; and by the Simons Foundation through the Simons Foundation Emmy Noether Fellows Program at Perimeter InstituteThe work was supported in part by the NSF award PHY-2309135 and by the John Templeton Foundation (award no.~62422).