Thermodynamic length in open quantum systems

Matteo Scandi and Martí Perarnau-Llobet

Max-Planck-Institut für Quantenoptik, D-85748 Garching, Germany

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The dissipation generated during a quasistatic thermodynamic process can be characterised by introducing a metric on the space of Gibbs states, in such a way that minimally-dissipating protocols correspond to geodesic trajectories. Here, we show how to generalize this approach to open quantum systems by finding the thermodynamic metric associated to a given Lindblad master equation. The obtained metric can be understood as a perturbation over the background geometry of equilibrium Gibbs states, which is induced by the Kubo-Mori-Bogoliubov (KMB) inner product. We illustrate this construction on two paradigmatic examples: an Ising chain and a two-level system interacting with a bosonic bath with different spectral densities.

Any thermodynamic protocol performed in finite time produces dissipation, and it is a crucial question how to minimise it. A particularly powerful approach to address this question is by means of differential geometry: one can associate a metric in the thermodynamic space in such a way geodesics correspond to minimally dissipative processes. In this paper we apply this idea (which started in the 80s for macroscopic systems) to open quantum systems described by a Lindblad equation, so that the problem of finding optimal thermodynamic protocols between two Hamiltonians reduces to the one of solving the geodesics equation. Differences with previous classical works appear in the presence of coherences in the energy basis, which are created when the Hamiltonian does not commute with itself at different times. We illustrate these ideas for a qubit coupled to a bath with different spectral densities, and an Ising chain in a controllable transverse field.

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