By making use of a recently proposed framework for the inference of thermodynamic irreversibility in bosonic quantum systems, we experimentally measure and characterize the entropy production rates in the nonequilibrium steady state of two different physical systems -a micro-mechanical resonator and a Bose-Einstein condensate -each coupled to a high finesse cavity and hence also subject to optical loss. Key features of our setups, such as cooling of the mechanical resonator and signatures of a structural quantum phase transition in the condensate are reflected in the entropy production rates. Our work demonstrates the possibility to explore irreversibility in driven mesoscopic quantum systems and paves the way to a systematic experimental assessment of entropy production beyond the microscopic limit.Entropy is a crucial quantity for the characterisation of dynamical processes: it quantifies and links seemingly distant notions such as disorder, information, and irreversibility across different disciplinary boundaries [1,2]. Every finitetime transformation results in some production of entropy, which signals the occurrence of irreversibility. Quantifying the amount of irreversible entropy produced by a given process is a goal of paramount importance: entropy production is a key quantity for the characterisation of non-equilibrium processes, and its minimisation improves the efficiency of thermal machines. The second law of thermodynamics can be formulated in terms of a universal constraint on the entropy production, which can never be negative [3,4]. In turn, this leads to the following rate equation for the variation of the entropy S [5]where Π(t) and Φ(t) are the irreversible entropy production rate and the entropy flux from the system to the environment, respectively. When the system reaches a non-equilibrium steady-state (NESS) these quantities take values Π s and Φ s respectively, such that Π s = Φ s > 0 [see Fig. 1 (a)]. Under these conditions, entropy is produced and exchanged with the local baths at the same rate. Only when both terms vanish (Π s = Φ s = 0) one recovers thermal equilibrium. The entropy production rate directly accounts for the irreversibility of a process and uncovers the non-equilibrium features of a system. The link between the entropy production rate Π s and irreversibility becomes particularly relevant in small systems subjected to fluctuations, for which a microscopic definition of entropy production based on stochastic trajectories of the system has been given [6]. Experimentally, this notion has been used to test fluctuation theorems in a variety of classically operating systems such as a single-electron box [7], a two-level system driven by a time-dependent potential [8], and a levitated nanoparticle undergoing relaxation [9]. However, in order to harness the working principles of thermodynamic machines working at the quantum level, and pinpoint the differences between their performances and those of their classical counterparts, it is important to analyse the entropy generated thro...

We demonstrate optimal state estimation for a cavity optomechanical system through Kalman filtering. By taking into account nontrivial experimental noise sources, such as colored laser noise and spurious mechanical modes, we implement a realistic state-space model. This allows us to obtain the conditional system state, i.e., conditioned on previous measurements, with a minimal least-squares estimation error. We apply this method to estimate the mechanical state, as well as optomechanical correlations both in the weak and strong coupling regime. The application of the Kalman filter is an important next step for achieving real-time optimal (classical and quantum) control of cavity optomechanical systems.