Research on the out-of-equilibrium dynamics of quantum systems has so far produced important statements on the thermodynamics of small systems undergoing quantum mechanical evolutions. Key examples are provided by the Crooks and Jarzynski relations: taking into account fluctuations in non-equilibrium dynamics, such relations connect equilibrium properties of thermodynamical relevance with explicit non-equilibrium features. Although the experimental verification of such fundamental relations in the classical domain has encountered some success, their quantum mechanical version requires the assessment of the statistics of work performed by or onto an evolving quantum system, a step that has so far encountered considerable difficulties in its implementation due to the practical difficulty to perform reliable projective measurements of instantaneous energy states. In this paper, by exploiting a radical change in the characterization of the work distribution at the quantum level, we report the first experimental verification of the quantum Jarzynski identity and the Tasaki-Crooks relationfollowing a quantum process implemented in a Nuclear Magnetic Resonance (NMR) system. Our experimental approach has enabled the full characterisation of the out-of-equilibrium dynamics of a quantum spin in a statistically significant way, thus embodying a key step towards the grounding of quantum-systems thermodynamics.The verification and use of quantum fluctuation relations [1][2][3] requires the design of experimentally feasible strategies for the determination of the work distribution following a process undergone by a system. In the quantum regime, the concept of work done by or on a system needs to be reformulated [4] so as to include ab initio both the inherent non-deterministic nature of quantum dynamics and the effects of quantum fluctuations. In this sense, work acquires a meaning only as a statistical expectation value W = W P(W) dW that accounts for the possible trajectories followed by a quantum system across its evolution, as formalised by the associated work probability distribution P(W) = n,m p 0 n p τ m|n δ W − ( m − n ) . In order to understand this expression, let us consider a quantum system initially at equilibrium at temperature T and undergoing a quantum process that changes its Hamiltonian asĤ(0) →Ĥ(τ) within a time period τ. Then, p 0 n is the probability to find the system in the eigenstate |n(0) ofĤ(0) (with energy n ) at the start of the protocol, while p τ m|n = | m(τ)|Û|n(0) | 2 is the conditional probability to find it in the eigenstate |m(τ) ofĤ(τ) (with energy m ) if it was in |n(0) at t = 0 and evolved under the action of the propagatorÛ. P(W) encompasses the statistics of the initial state (given by p 0 n ) and the fluctuations arising from quantum measurement statistics (given by p τ m|n ). One can define a backward process that, starting from the equilibrium state of the system associated withĤ(τ) and temperature T , implements the protocolĤ(τ) →Ĥ(0) and thus inverting the control sequence of the ...

Developments in the thermodynamics of small quantum systems envisage non-classical thermal machines. In this scenario, energy fluctuations play a relevant role in the description of irreversibility. We experimentally implement a quantum heat engine based on a spin-1/2 system and nuclear magnetic resonance techniques. Irreversibility at microscope scale is fully characterized by the assessment of energy fluctuations associated with the work and heat flows. We also investigate the efficiency lag related to the entropy production at finite time. The implemented heat engine operates in a regime where both thermal and quantum fluctuations (associated with transitions among the instantaneous energy eigenstates) are relevant to its description. Performing a quantum Otto cycle at maximum power, the proof-of-concept quantum heat engine is able to reach an efficiency for work extraction (η ≈ 42%) very close to its thermodynamic limit (η = 44%).