We investigate the critical behavior of the open coherently-driven Bose-Hubbard dimer under nonlocal dissipation. A conserved quantity arises from the nonlocal nature of the dissipation, rendering the dimer multistable. In the weak-coupling semiclassical limit, the displayed criticality takes the form of amplitude bistability and breaking of spatial and temporal symmetries. A period-bistable time crystal is formed, consisting of Josephson-like oscillations. Mean-field dynamics and quantum trajectories complement the spectral analysis of the Liouvillian in the approach to the semiclassical limit.

We study stochastic thermodynamics for a quantum system of interest whose dynamics are described by a completely positive trace-preserving (CPTP) map as a result of its interaction with a thermal bath. We define CPTP maps with equilibrium as CPTP maps with an invariant state such that the entropy production due to the action of the map on the invariant state vanishes. Thermal maps are a subgroup of CPTP maps with equilibrium. In general, for CPTP maps, the thermodynamic quantities, such as the entropy production or work performed on the system, depend on the combined state of the system plus its environment. We show that these quantities can be written in terms of system properties for maps with equilibrium. The relations that we obtain are valid for arbitrary coupling strengths between the system and the thermal bath. The fluctuations of thermodynamic quantities are considered in the framework of a two-point measurement scheme. We derive the entropy production fluctuation theorem for general maps and a fluctuation relation for the stochastic work on a system that starts in the Gibbs state. Some simplifications for the probability distributions in the case of maps with equilibrium are presented. We illustrate our results by considering spin 1/2 systems under thermal maps, non-thermal maps with equilibrium, maps with non-equilibrium steady states and concatenations of them. Finally, we consider a particular limit in which the concatenation of maps generates a continuous time evolution in Lindblad form for the system of interest, and we show that the concept of maps with and without equilibrium translates into Lindblad equations with and without quantum detailed balance, respectively. The consequences for the thermodynamic quantities in this limit are discussed.

We study an emergent semiclassical time crystal composed of two interacting driven-dissipative bosonic modes. The system has a discrete Z 2 spatial symmetry which, depending on the strength of the drive, can be broken in the time-crystalline phase or it cannot. An exact semiclassical mean-field analysis, numerical simulations in the quantum regime, and the spectral analysis of the Liouvillian are combined to show the emergence of the time crystal and to prove the robustness of the oscillation period against quantum fluctuations.

We study the thermodynamics of a quantum system interacting with different baths in the repeated interaction framework. In an appropriate limit, the evolution takes the Lindblad form and the corresponding thermodynamic quantities are determined by the state of the full system plus baths. We identify conditions under which the thermodynamics of the open system can be described only by system properties and find a quantum local detailed balance condition with respect to an equilibrium state that may not be a Gibbs state. The three-qubit refrigerator introduced in [1, 2] is an example of such a system. From a repeated interaction microscopic model we derive the Lindblad equation that describes its dynamics and discuss its thermodynamic properties for arbitrary values of the internal coupling between the qubits. We find that external power (proportional to the internal coupling strength) is requiered to bring the system to its steady state, but once there, it works autonomously as discussed in [1,2].PACS numbers: I. INTRODUCTIONQuantum thermal machines can operate either in a cycle, with different numbers of strokes, or continuously [3, 4, and references therein]. In either case, a proper theory of quantum thermodynamics should be able to describe the dynamics of the working medium, which we will call the system, as well as the heat fluxes coming from the environment. A continuous quantum machine normally operates in the steady-state of the equation describing the dynamics [5]. A great simplification is gained when this equation, which depends on all the degrees of freedom of the system and environment, can be written in closed form, describing only the degrees of freedom of the system. The most common example is the Markovian master equation in Lindblad form [6,7]. This simplification, however, is usually obtained at the cost of invoking approximations, which are not always well justified.In the beginning of the studies of quantum Markovian master equations, one of the main interests was to understand how a quantum system in contact with a thermal bath dissipates energy and relaxes to equilibrium. The approximations used in that scenario, and their regime of validity, are very well understood [8][9][10][11][12]. However, much of that interest has changed in the past few years due to the increasing progress of quantum thermodynamics [4,13]. Here, many systems cannot equilibrate because they are coupled to different baths. In the standard derivation of master equations for this kind of systems the Born-Markov approximation is employed, followed by another approximation that can either make the dissipation local or global. In the local dissipation approach each bath affects a few degrees of freedom of the system while leaving the rest unchanged [14,15]. On the other hand, in the global dissipation approach each bath is responsible of transitions between energy eigenstates of the whole system [7]. Since both approaches present some inconsistencies when describing the thermodynamics [14,16,17], a lot of effort has been p...

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