Engineering Nonequilibrium Dynamics of Open Quantum Systems

The understanding of memory effects arising from the interaction between system and environment is a key for engineering quantum thermodynamic devices beyond the standard Markovian limit. We study the performance of measurement-based thermal machine whose working medium dynamics is subject to backflow of information from the reservoir via collision based model. In this study, the non-Markovian effect is introduced by allowing for additional unitary interactions between the environments. We present two strategies of realizing non-Markovian dynamics and study their influence on the work produced by the engine. Moreover, the role of system-environment memory effects on the engine performance can be beneficial in short time.

The entanglement entropies in ground states of typical condensed matter systems obey area and log-area laws. In contrast, subsystem entropies in random and thermal states are extensive, i.e., obey a volume law. For energy eigenstates, one expects a crossover from the groundstate scaling at low energies and small subsystem sizes to the extensive scaling at high energies and large subsystem sizes. We elucidate this crossover. Due to eigenstate thermalization (ETH), the eigenstate entanglement can be related to subsystem entropies in thermodynamic ensembles. For one-dimensional critical systems, the universal crossover function follows from conformal field theory (CFT) and can be adapted to better capture nonlinear dispersion. For critical fermions in two dimensions, we obtain a crossover function by employing the 1+1d CFT result for contributions from lines perpendicular to the Fermi surface. Scaling functions for gapped systems additionally depend on a mass parameter. Using ETH, we also easily obtain the distribution function for eigenstate entanglement. The results are demonstrated numerically for quadratic fermionic systems, finding excellent data collapse to the scaling functions. Ref: Q. Miao and T. Barthel, arXiv:1905.07760 (2019)

Lie-Trotter-Suzuki decompositions are an efficient way to approximate operator exponentials exp(tH) when H is a sum of n (non-commuting) terms which, individually, can be exponentiated easily. They are employed in time-evolution algorithms for tensor network states, digital quantum simulation protocols, path integral methods like quantum Monte Carlo, and splitting methods for symplectic integrators in classical Hamiltonian systems. We provide optimized decompositions up to sixth order. The leading error term is expanded in nested commutators (Hall bases) and we minimize the 1-norm of the coefficients. For n=2 terms, several of the optima we find are close to those in [McLachlan, SlAM J. Sci. Comput. 16, 151 (1995)]. Generally, our results substantially improve over unoptimized decompositions by Forest, Ruth, Yoshida, and Suzuki. We explain why these decompositions are sufficient to efficiently simulate any one- or two-dimensional lattice model with finite-range interactions. This follows by solving a partitioning problem for the interaction graph. Ref: T. Barthel and Y. Zhang, arXiv:1901.04974 (2019)

I will consider the one-dimensional quantum Ising chain in the transverse field driven from the paramagnetic phase to the critical point and study its free evolution there. I will discuss how the system size and quench-induced scaling relations from the Kibble–Zurek theory of non-equilibrium phase transitions are encoded in transverse magnetization and Loshmidt echo and I will present the methods to compute these observables analytically. Finally, I will show how different is the behaviour of longitudinal magnetization and how it can be used to phase detection.

The reaction-coordinate mapping is a useful technique to study complex quantum dissipative dynamics into structured environments. In essence, it aims to mimic the original problem by means of an 'augmented system', which includes a suitably chosen collective environmental coordinate---the 'reaction coordinate'. This composite then couples to a simpler 'residual reservoir' with short-lived correlations. If, in addition, the residual coupling is weak, a simple quantum master equation can be rigorously applied to the augmented system, and the solution of the original problem just follows from tracing out the reaction coordinate. But, what if the residual dissipation is strong? Here we consider an exactly solvable model for heat transport---a two-node linear "quantum wire" connecting two baths at different temperatures. We allow for a structured spectral density at the interface with one of the reservoirs and perform the reaction-coordinate mapping, writing a perturbative master equation for the augmented system. We find that: (a) strikingly, the stationary state of the original problem can be reproduced accurately by a weak-coupling treatment even when the residual dissipation on the augmented system is very strong; (b) the agreement holds throughout the entire dynamics under large residual dissipation in the overdamped regime; (c) and that such master equation can grossly overestimate the stationary heat current across the wire, even when its non-equilibrium steady state is captured faithfully. These observations can be crucial when using the reaction-coordinate mapping to study the largely unexplored strong-coupling regime in quantum thermodynamics.

We take steps towards developing exact solutions for open dynamical systems, for which translational symmetry is broken by boundary conditions. Specifically, we leverage a recently proposed generalization of Bloch's theorem to obtain the spectrum and exact normal modes of a bosonic analogue to the familiar Kitaev-Majorana chain, which exhibits effective non-Hermitian Hamiltonian dynamics and extreme sensitivity to boundary conditions. We present exact analytical solutions for the chain under periodic, anti-periodic, open, and $\pi/2$-twisted boundary conditions for which we find the system can only be made dynamically stable in the latter two cases. We identify the breakdown of dynamical stability with a spontaneous breaking of a generalized $\mathcal{P}\mathcal{T}$-symmetry and employ tools from non-Hermitian quantum mechanics to characterize the extreme sensitivity of the system dynamics to boundary conditions.

We derive a family of Gaussian non-Markovian stochastic Schrödinger equations for the dynamics of open quantum systems. The different unravelings correspond to different choices of squeezed coherent states, reflecting different measurement schemes on the environment. Consequently, we are able to give a single shot measurement interpretation for the stochastic states and microscopic expressions for the noise correlations of the Gaussian process. By construction, the reduced dynamics of the open system does not depend on the squeezing parameters. They determine the non-Hermitian Gaussian correlation, a wide range of which are compatible with the Markov limit. We demonstrate the versatility of our results for quantum information tasks in the non-Markovian regime. In particular, by optimizing the squeezing parameters, we can tailor unravelings for improving entanglement bounds or for environment-assisted entanglement protection.

We investigate the critical dynamics of driven classical and quantum systems. Specifically, we consider slowly time-dependent couplings. We have developed an adiabatic dynamical Renormalization Group formalism and we use it to access the critical regime. We recover Kibble--Zurek phenomenology when the system is quenched across its phase boundary. We obtain the scaling of the correlation length with the quench speed ${\xi \sim v^{-\nu/(1+z \nu)}}$ from first principles. Moreover, we find another scaling regime, which is visible when the system is quenched along the phase boundary. In this regime, exponents that are sub-leading at equilibrium become dominant and observable without any fine-tuning.

Random matrix theory has provided early insights into the theoretical understanding of the foundations of quantum statistical mechanics. In particular, Deutsch [1991 Phys. Rev. A 43 2046] presented a random matrix model that could be shown to thermalize as the eigenstates themselves formed an effective microcanonical ensemble. This was the foundation of the Eigenstate Thermalization Hypothesis (ETH), which has since become a leading contender for the mechanism behind thermalization. We extend this model, developing a method to find arbitrary correlation functions by including the effect of interactions between random wave-functions due to orthogonality. We derive the complete ETH ansatz, also guaranteeing that fluctuations are small in the thermodynamic limit. Further, from the developed framework, we derive an expression for the time-averaged fluctuations that resembles a classical fluctuation-dissipation theorem of Brownian motion; as such, we observe hints towards the emergence of classical statistical physics from chaotic quantum systems. We will further discuss a proposal to perform a measurement of the density of states of a quantum simulator by confirmation of the fluctuation-dissipation theorem.

We present a theoretically study on a strongly repulsive Fermi-Hubbard model with a periodically driven interaction and a coupling to external bath. Such a model is directly relevant for many current experiments. We use non-perturbative numerical methods and analytical Floquet expansions to show that the Mott-insulating phase is reshaped into a stationary state with enhanced local pairing correlations: a signature of the exotic eta-pairing superconducting phase. This suggests a path to stabilize intriguing far-from-equilibrium phases in Mott insulators, using driving protocols of current experimental relevance.

Adiabatic driving is one of the pillars of time-dependent quantum control. However, the limitations imposed by coherence times are typically in sharp contrast with the necessity of slow evolutions imposed by the adiabatic theorem. We present a shortcut-to-adiabaticity control protocol for few-level quantum systems. The method works by assisting the accelerated adiabatic drive with fast oscillations in the intrinsic parameters of the original Hamiltonian: the oscillations mediate a counterdiabatic Hamiltonian dynamically compensating for nonadiabatic transitions. This construction remarkably avoids the necessity of introducing new complex time-dependent interactions and is robust against parameter biases. We further discuss how the method can be combined with strategies to counteract dissipation for realizing an efficient counterdiabatic driving in an open system scenario. Our results are applied to realistic implementations based on molecular nanomagnets, superconducting circuits, and ultra cold atoms.

The notion of quantum speed limit (QSL) refers to the fundamental fact that two quantum states become completely distinguishable upon dynamical evolution only after a finite amount time, called the QSL time. A different, but related concept is that of minimum control time (MCT), which is the minimum evolution time needed for a state to be driven (by suitable, generally time-dependent, control fields) to a given target state. While the QSL can give information about the MCT, it usually imposes little restrictions to it, and is thus unpractical for control purposes. In this work we revisit this issue by first presenting a theory of geometrical QSL for unitary transformations, rather than for states, and discuss its implications and limitations. Then, we propose a framework for bounding the MCT for realizing unitary transformations that goes beyond the QSL results and gives much more meaningful information to understand the controlled dynamics of the system at short times.

We report [1] two results complementing the second law of thermodynamics for Markovian open quantum systems coupled to multiple reservoirs with different temperatures and chemical potentials. First, we derive a nonequilibrium free energy inequality providing an upper bound for a maximum power output, which for systems with inhomogeneous temperature is not equivalent to the Clausius inequality. Secondly, we derive local Clausius and free energy inequalities for the subsystems of a composite system. These inequalities, which generalize an influential result obtained previously for classical bipartite systems [2], differ from the total system one by the presence of an information-related contribution and build the ground for thermodynamics of quantum information processing. Our theory is used to study an autonomous quantum Maxwell demon based based on quantum dots. [1] K. Ptaszy\'{n}ski and M. Esposito, arXiv:1901.01093 (2019). [2] J. M. Horowitz and M. Esposito, Phys. Rev. X 4, 031015 (2014).

We present a procedure to accelerate the relaxation of an open quantum system towards its equilibrium state. The control protocol, termed Shortcut to Equilibration, is obtained by reverse engineering the non-adiabatic master equation. This is a non-unitary control task aimed at rapidly changing the entropy of the system. Such a protocol serves as a shortcut to an abrupt change in the Hamiltonian, i.e., a quench. As an example, we study the thermalization of a particle in a harmonic well. We observe that for short protocols there is a four orders of magnitude improvement in accuracy.

We propose a new variational scheme based on the neural-network quantum states to simulate the stationary states of open quantum many-body systems. Using the high expressive power of the variational ansatz described by the restricted Boltzmann machines, which we dub as the neural stationary state ansatz, we compute the stationary states of quantum dynamics obeying the Lindblad master equations. The mapping of the stationary-state search problem into finding a zero-energy ground state of an appropriate Hermitian operator allows us to apply the conventional variational Monte Carlo method for the optimization. Our method is shown to simulate various spin systems efficiently, i.e., the transverse-field Ising models in both one and two dimensions and the XYZ model in one dimension.