By means of the time-dependent density matrix renormalization group algorithm we study the zero-temperature dynamics of the Von Neumann entropy of a block of spins in a Heisenberg chain after a sudden quench in the anisotropy parameter. In the absence of any disorder the block entropy increases linearly with time and then saturates. We analyze the velocity of propagation of the entanglement as a function of the initial and final anisotropies and compare, wherever possible, our results with those obtained by means of Conformal Field Theory. In the disordered case we find a slower (logarithmic) evolution which may signals the onset of entanglement localization.

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 ...

We propose an interferometric setting for the ancilla-assisted measurement of the characteristic function of the work distribution following a time-dependent process experienced by a quantum system. We identify how the configuration of the effective interferometer is linked to the symmetries enjoyed by the Hamiltonian ruling the process and provide the explicit form of the operations to implement in order to accomplish our task. We finally discuss two physical settings, based on hybrid opto-/electro-mechanical devices, where the theoretical proposals discussed in our work could find an experimental demonstration.PACS numbers: 05.70. Ln, 05.30.Rt, 64.60.Ht Thermodynamics is one of the pillars of natural sciences. Its principles can predict the occurrence and efficiency of complex chemical reactions and biological processes. In physics, the conduction of heat across a medium or the concept of arrow of time are formulated thermodynamically. In information theory, the definitions of information and entropy are also given in thermodynamical terms. Moreover, the tightness of the link between information and thermodynamics can be deduced from the interpretation of the landmark embodied by Landauer's principle [1].The dexterity characterizing the current experimental control at the microscopic scale opens up tantalising questions, the most pressing being probably the following: what happens to thermodynamics when we deal with the non-quasistatic dynamics of quantum systems brought out of equilibrium? An invaluable tool for the formulation of an answer in this sense has been provided with the formulation of non-equilibrium fluctuation relations and their quantum extension [2,3], which has recently enabled investigations at the crossroad of quantum physics, thermodynamics, and information theory [4]. This includes proposals for experimental quantum thermal machines [5], the study of the link between fluctuation relations and critical phenomena in many-body systems [6,7], the verification of the Jarzynski equality [8,10,11], and the extension to open dynamics [12].The verification and use of the Jarzynski inequality [11,12] requires the determination of the work distribution following a process undergone by a system, a goal that needs feasible experimental strategies. In Ref. [8,9], two seminal proposals have been made: Huber et al. suggested a scheme based on the performance of projective energy measurements on the trapped-ion system undergoing a process. Their method uses an ingenious "filtering scheme" whose implementation, unfortunately, can be of significant practical difficulty. Heyl and Kehrein [9], on the other hand, showed that optical spectra can be used to measure the work distribution of specific nonequilibrium processes. However, their method only applies to sudden quenches and is ineffective for general processes.In this paper we propose a way to infer the quantum statistics of a work distribution by relying on an interferometric approach that delegates the retrieval of the information we are after to rout...

The study of open quantum systems often relies on approximate master equations derived under the assumptions of weak coupling to the environment. However when the system is made of several interacting subsystems such a derivation is in many cases very hard. An alternative method, employed especially in the modeling of transport in mesoscopic systems, consists in using local master equations (LMEs) containing Lindblad operators acting locally only on the corresponding subsystem. It has been shown that this approach however generates inconsistencies with the laws of thermodynamics. In this paper we demonstrate that using a microscopic model of LMEs based on repeated collisions all thermodynamic inconsistencies can be resolved by correctly taking into account the breaking of global detailed balance related to the work cost of maintaining the collisions. We provide examples based on a chain of quantum harmonic oscillators whose ends are connected to thermal reservoirs at different temperatures. We prove that this system behaves precisely as a quantum heat engine or refrigerator, with properties that are fully consistent with basic thermodynamics. derivations are in general quite involved since they require knowledge of the full set of eigenvalues and eigenvectors of the system's Hamiltonian, something which quickly becomes prohibitive when the number of subsystems increases. Moreover, depending on the approximations employed, one may also arrive at equations which do not generate completely positive maps (the so-called Redfield equations [50]), or equations which contain unphysical heat currents [54]. For these reasons, microscopic derivations of master equations for systems connected to multiple environments still continues, nowadays, to be a topic of great interest.An alternative, more heuristic, approach consists in deriving a master equation for the individual subsystems, neglecting the interaction with the remaining subsystems. The resulting master equation will then contain only local jump operators describing exchanges between the environment E i and its corresponding subsystem S i . Such equations, which we shall henceforth refer to as LMEs (also frequently called boundarydriven master equations), are typically accurate when the dissipation rates are larger than the interaction between subsystems. Due to their computational simplicity, they have been extensively employed over the last two decades in the study of transport in non-equilibrium quantum systems [55][56][57][58][59][60][61][62][63][64][65][66][67][68][69][70][71][72][73].It turns out, however, that the nonlocal terms neglected in the LME may still lead to non-thermal steadystates [74] and play a significant role if the heat exchanges are small, even for weakly interacting parts. As a consequence, it has been found that LMEs may lead to apparent thermodynamic inconsistencies, as pointed out recently by Levy and Kosloff [75]. They have shown that the LME for two coupled quantum harmonic oscillators (QHO) may predict currents from a cold to a hot ther...