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Based on a recently developed non-perturbative platform designed to simulate the full quantum dynamics of quantum thermal machines, the situation of a quantum refrigerator operating according to an Otto cycle is studied. The periodic steady-state dynamics is discussed in detail as well as the key thermodynamic quantities work, heat, and entropy. A particular benefit of the formulation is that it allows to access explicitly the work required for switching on and off the interaction with the respective thermal reservoirs in a consistent way. The domains in which the device operates in refrigerator mode are characterized.

In their Comment [P. Hakonen and E. B. Sonin, preceding comment, Phys. Rev. X 11, 018001 (2021)PRXHAE2160-3308], Hakonen and Sonin (HS) object to our conclusion on the absence of a dissipation-induced superconducting-to-insulating quantum phase transition (DQPT) in resistively shunted Josephson junctions (RSJJs) originally predicted by Schmid and Bulgadaev (SB). Their objections are based on the account they make of a theory explaining the DQPT in terms of Bloch bands which was developed in the 1980s and early 1990s. In this Reply, we point to several issues in the Comment which undermine the objections HS formulate against our work.

The combination of an exact stochastic decomposition of non-Markovian dissipative quantum dynamics with the semiclassical initial value formalism is applied to Brownian motion in a Morse potential. The unified sampling of the stochastic noise and the semiclassical phase space distribution introduced in Koch et al. [W. Koch, F. Grossmann, J.T. Stockburger, J. Ankerhold, Non-Markovian semiclassical dynamics, Phys. Rev. Lett. 100 (2008) 230402] is laid out here in detail. By comparing our numerical results to those obtained by using the Caldeira–Leggett master equation, we show that even in the challenging regime of moderate friction and at low temperatures, where reservoir fluctuations are clearly non-Markovian, this approach allows for the accurate description of dissipative dynamics over many oscillation periods until thermalization is reached.

We derive new time-local stochastic Schrodinger equations which offer a fully non-Markovian extension of conventional quantum state diffusion (QSD). The Feynman–Vernon influence functional of a linear heat bath is constructed as the stochastic mean value of time-local action terms, defining a linear stochastic Schrodinger equation. As a test, conventional QSD is recovered in limiting cases allowing the Born–Markov approximation. We further transform the dynamics to obtain a non-linear, norm-conserving Schrodinger equation with appealing properties: It is suitable for numerical simulations, and it has a well-defined classical limit even before noise averaging.

We present a semiclassical method for simulating the dynamics of nonadiabatic transitions in a discrete-state quantum system coupled to a bath of explicit continuous coordinates. This method employs a coherent-state formulation of the path integrals for the discrete system whose dynamics is described by spin operators. This spin coherent-state formulation allows the discrete system to be mapped onto a continuous coordinate. Stationary approximations of the resulting coherent-state path integrals of the system plus bath lead to quasiclassical equations of motion which can be solved numerically by direct integration. This algorithm reduces the problem to a number of simple classical trajectory calculations and does not require calculating any fluctuation determinants.

Master equations of Lindblad type have attained prominent status in the fields of quantum optics and quantum information since they are guaranteed to satisfy fundamental notions of quantum dynamics such as complete positivity. When Lindblad operators are used to describe thermal reservoirs in contact with an open quantum system, the fundamental laws of thermodynamics and the fluctuation-dissipation theorem provide additional mandatory criteria. We show several examples of innocent-looking Lindblad operators which have questionable properties in this regard. Compatibility criteria between Hamiltonian and Lindblad terms as well as consequences of their violation are discussed. An alternative stochastic approach to dissipative quantum dynamics is outlined and illustrated through a harmonic-chain model for which the approach of local Lindblad operators fails.

We investigate the optimal control of open quantum systems, in particular, the mutual influence of driving and dissipation. A stochastic approach to open-system control is developed, using a generalized version of Krotov's iterative algorithm, with no need for Markovian or rotating-wave approximations. The application to a harmonic degree of freedom reveals cooperative effects of driving and dissipation that a standard Markovian treatment cannot capture. Remarkably, control can modify the open-system dynamics to the point where the entropy change turns negative, thus achieving cooling of translational motion without any reliance on internal degrees of freedom., 5 pages, 3 figures; 6 page supplement, v2: major revision for wider scope, additional data, change of title, v3: fixed typo, v4: minor changes for clarity, extended supplement

The exact stochastic decomposition of non-Markovian dissipative quantum dynamics is combined with the time-dependent semiclassical initial value formalism. It is shown that even in the challenging regime of moderate friction and low temperatures, where non-Markovian effects are substantial, this approach allows for the accurate description of dissipative dynamics in anharmonic potentials over many oscillation periods until thermalization is reached. The problem of convergence of the stochastic average at long times, which plagues full quantum mechanical implementations, is avoided through a joint sampling of the stochastic noise and the semiclassical phase space distribution., accepted for publication in Physical Review Letters

The reduced dynamics of a quantum system interacting with a linear heat bath finds an exact representation in terms of a stochastic Schr{\"o}dinger equation. All memory effects of the reservoir are transformed into noise correlations and mean-field friction. The classical limit of the resulting stochastic dynamics is shown to be a generalized Langevin equation, and conventional quantum state diffusion is recovered in the Born--Markov approximation. The non-Markovian exact dynamics, valid at arbitrary temperature and damping strength, is exemplified by an application to the dissipative two-state system., Comment: 4 pages, 2 figures. To be published in Phys. Rev. Lett

Standard optimal control methods perform optimization in the time domain. However, many experimental settings demand the expression of the control signal as a superposition of given waveforms, a case that cannot easily be accommodated using time-local constraints. Previous approaches [1,2] have circumvented this difficulty by performing optimization in a parameter space, using the chain rule to make a connection to the time domain. In this paper, we present an extension to Optimal Control Theory which allows gradient-based optimization for superpositions of arbitrary waveforms directly in a time-domain subspace. Its key is the use of the Moore-Penrose pseudoinverse as an efficient means of transforming between a time-local and waveform-based descriptions. To illustrate this optimization technique, we study the parametrically driven harmonic oscillator as model system and reduce its energy, considering both Hamiltonian dynamics and stochastic dynamics under the influence of a thermal reservoir. We demonstrate the viability and efficiency of the method for these test cases and find significant advantages in the case of waveforms which do not form an orthogonal basis., 16 pages, 6 figures

Current fluctuations in a dissipative two-state system have been studied using a novel quantum dynamics simulation method. After a transformation of the path integrals, the tunneling dynamics is computed by deterministic integration over the real-time paths under the influence of colored noise. The nature of the transition from coherent to incoherent dynamics at low temperatures is re-examined., 4 pages, 4 figures; to appear in Phys. Rev. Letters