Open quantum systems (OQSs) cannot always be described with the Markov approximation, which requires a large separation of system and environment time scales. Here, we give an overview of some of the most important techniques available to tackle the dynamics of an OQS beyond the Markov approximation. Some of these techniques, such as master equations, Heisenberg equations and stochastic methods, are based on solving the reduced OQS dynamics, while others, such as path integral Monte Carlo or chain mapping approaches, are based on solving the dynamics of the full system. We emphasize the physical interpretation and derivation of the various approaches, explore how they are connected and examine how different methods may be suitable for solving different problems.

We examine the stability versus different types of perturbations of recently proposed shortcuts to adiabaticity to speed up the population inversion of a two-level quantum system. We find the optimally robust processes by using invariant-based engineering of the Hamiltonian. Amplitude noise and systematic errors require different optimal protocols. IntroductionManipulating the state of a quantum system with time-dependent interacting fields is a fundamental operation in atomic and molecular physics, with applications such as lasercontrolled chemical reactions, metrology, interferometry, nuclear magnetic resonance or quantum information processing [1][2][3][4][5]. For two-level systems, there are several approaches proposed for attaining a complete population transfer, for example, π pulses, composite pulses, adiabatic passage and its variants. In general, the π pulses may be fast but highly sensitive to variations in the pulse area, and to inhomogeneities in the sample [1]. Used first in nuclear magnetic resonance [6], composite pulses provide an alternative to the single π pulse, with some successful applications [7,8], but still need accurate control of pulse phase and intensity. A robust option is, in principle, adiabatic (slow) passage, which is however prone to decoherence because of the effect of noise over the long times required. A compromise is to use speeded-up 'shortcuts to adiabaticity', which may be broadly defined as the processes that lead to the same final populations as the adiabatic approach in a shorter time.Several methods to find shortcuts to adiabaticity have been put forward [9-19] for twoand three-level atomic systems. The transitionless or counter-diabatic (CD) control protocols proposed by Demirplak and Rice [9] and Berry [10] start from a reference time-dependent Hamiltonian H 0 and provide an extra interaction that cancels the diabatic couplings. This results in an exact following of the adiabatic dynamics of the reference Hamiltonian, in principle in an arbitrarily short time. They have been applied, for example, to speed up the rapid adiabatic passage (RAP) for an Allen-Eberly scheme [11]. Modified by a unitary transformation [17], New Journal of Physics 14 (2012) 093040 (http://www.njp.org/)

Thermodynamics is a branch of science blessed by an unparalleled combination of generality of scope and formal simplicity. Based on few natural assumptions together with the four laws, it sets the boundaries between possible and impossible in macroscopic aggregates of matter. This triggered groundbreaking achievements in physics, chemistry and engineering over the last two centuries. Close analogues of those fundamental laws are now being established at the level of individual quantum systems, thus placing limits on the operation of quantum-mechanical devices. Here we study quantum absorption refrigerators, which are driven by heat rather than external work. We establish thermodynamic performance bounds for these machines and investigate their quantum origin. We also show how those bounds may be pushed beyond what is classically achievable, by suitably tailoring the environmental fluctuations via quantum reservoir engineering techniques. Such superefficient quantum-enhanced cooling realises a promising step towards the technological exploitation of autonomous quantum refrigerators.

An implementation of quantum absorption chillers with three qubits has been recently proposed that is ideally able to reach the Carnot performance regime. Here we study the working efficiency of such self-contained refrigerators, adopting a consistent treatment of dissipation effects. We demonstrate that the coefficient of performance at maximum cooling power is upper bounded by 3/4 of the Carnot performance. The result is independent of the details of the system and the equilibrium temperatures of the external baths. We provide design prescriptions that saturate the bound in the limit of a large difference between the operating temperatures. Our study suggests that delocalized dissipation, which must be taken into account for a proper modeling of the machine-baths interaction, is a fundamental source of irreversibility which prevents the refrigerator from approaching the Carnot performance arbitrarily closely in practice. The potential role of quantum correlations in the operation of these machines is also investigated.

Abstract. When deriving a master equation for a multipartite weakly-interacting open quantum systems, dissipation is often addressed locally on each component, i.e. ignoring the coherent couplings, which are later added 'by hand'. Although simple, the resulting local master equation (LME) is known to be thermodynamically inconsistent. Otherwise, one may always obtain a consistent global master equation (GME) by working on the energy basis of the full interacting Hamiltonian. Here, we consider a two-node 'quantum wire' connected to two heat baths. The stationary solution of the LME and GME are obtained and benchmarked against the exact result. Importantly, in our model, the validity of the GME is constrained by the underlying secular approximation. Whenever this breaks down (for resonant weakly-coupled nodes), we observe that the LME, in spite of being thermodynamically flawed: (a) predicts the correct steady state, (b) yields with the exact asymptotic heat currents, and (c) reliably reflects the correlations between the nodes. In contrast, the GME fails at all three tasks. Nonetheless, as the inter-node coupling grows, the LME breaks down whilst the GME becomes correct. Hence, the global and local approach may be viewed as complementary tools, best suited to different parameter regimes.

We present a series of numerical and analytical computations on heat conduction for a strongly chaotic system -the Lorentz gas. Heat conduction is characterized by nontrivial features: While the heat conductivity is well defined in the thermodynamic limit, a linear gradient appears only for quite small temperature differences. The key dynamical feature inducing such a behavior is recognized as deterministic diffusion (along transport direction) which is usually associated to full hyperbolicity. [S0031-9007(99)08614-7] PACS numbers: 44.10. + i, 05.20. -y, 05.45. -a

We design, by invariant-based inverse engineering, driving fields that invert the population of a two-level atom in a given time, robustly with respect to dephasing noise and/or systematic frequency shifts. Without imposing constraints, optimal protocols are insensitive to the perturbations but need an infinite energy. For a constrained value of the Rabi frequency, a flat pi pulse is the least sensitive protocol to phase noise but not to systematic frequency shifts, for which we describe and optimize a family of protocols

Multiple time correlation functions are found in the dynamical description of different phenomena. They encode and describe the fluctuations of the dynamical variables of a system. In this paper we formulate a theory of non-Markovian multiple-time correlation functions (MTCF) for a wide class of systems. We derive the dynamical equation of the reduced propagator, an object that evolve state vectors of the system conditioned to the dynamics of its environment, which is not necessarily at the vacuum state at the initial time. Such reduced propagator is the essential piece to obtain multiple-time correlation functions. An average over the different environmental histories of the reduced propagator permits us to obtain the evolution equations of the multiple-time correlation functions. We also study the evolution of MTCF within the weak coupling limit and it is shown that the multiple-time correlation function of some observables satisfy the Quantum Regression Theorem (QRT), whereas other correlations do not. We set the conditions under which the correlations satisfy the QRT. We illustrate the theory in two different cases; first, solving an exact model for which the MTCF are explicitly given, and second, presenting the results of a numerical integration for a system coupled with a dissipative environment through a non-diagonal interaction.PACS numbers: 3.65 Yz, 42.50 Lc Introduction and motivation. Many research contexts are focused on the dynamics of a system (S) that is affected by an environment (B) from which it cannot be considered isolated. Examples of such situations are encountered in statistical physics, condensed matter and quantum optics. We found a concrete example in the description of the dynamics of an atom (S) immersed in an electromagnetic field (B) [1,2].In some circumstances, the analysis of the dynamics of the system is done using the expectation values of its observables over state vectors of the whole system, and then averaging over the environmental degrees of freedom. However, in some other situations, like when studying the response of a system to an external EM field, some additional information is needed. In particular, for the analysis of the spectroscopic properties of a system some multiple-time correlation function (MTCF) has to be computed, usually a two-time correlation function.The dynamics of the system S is usually described through its reduced density operator. Such operator verifies some master equation that in the Markovian case is of Lindblad type [1,3,4,5,6,7]. Complementary to the master equation approach, a series of Monte-Carlo type of approaches based on the so called stochastic Schrödinger equations [1,4,8,9,10] have been developed in the last decade. In such schemes, the dynamics of system state vectors is integrated, and after an average is made over many realizations of environment histories that eventually are understood as a noise and takes into account the environment influence on S. In the non-Markovian case, within the context of nuclear magnetic resonance, the ...