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arXiv:2007.15138 · Physical Review A

Sufficient conditions for adiabaticity in open quantum systems

Alan C. Santos, M. S. Sarandy

Published Nov 10, 2020• 10 citations
arXivHTMLPDF

Abstract

The adiabatic approximation exhibits wide applicability in quantum mechanics, providing a simple approach for nontransitional dynamics in quantum systems governed by slowly varying time-dependent Hamiltonians. However, the standard adiabatic theorem is specifically derived for closed quantum systems. In a realistic open system scenario, the inevitable system-reservoir interaction must be taken into account, which strongly impacts the generalization of the adiabatic behavior. In this paper, we introduce sufficient conditions for the adiabatic approximation in open quantum systems. These conditions are simple yet general, providing a suitable instrument to investigate adiabaticity for arbitrary initial mixed states evolving under time local master equations. We first illustrate our results by showing that the adiabatic approximation for open systems is compatible with the description of quantum thermodynamics at thermal equilibrium, where irreversible entropy production is vanishing. We also apply our sufficient conditions as a tool in quantum control, evaluating the adiabatic behavior for the Hamiltonians of both the Deutsch algorithm and the Landau-Zener model under decoherence.

DOI
10.1103/physreva.102.052215

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The adiabatic approximation exhibits wide applicability in quantum mechanics, providing a simple approach for nontransitional dynamics in quantum systems governed by slowly varying time-dependent Hamiltonians. However, the standard adiabatic theorem is specifically derived for closed quantum systems. In a realistic open system scenario, the inevitable system-reservoir interaction must be taken into account, which strongly impacts the generalization of the adiabatic behavior. In this paper, we introduce sufficient conditions for the adiabatic approximation in open quantum systems. These conditions are simple yet general, providing a suitable instrument to investigate adiabaticity for arbitrary initial mixed states evolving under time local master equations. We first illustrate our results by showing that the adiabatic approximation for open systems is compatible with the description of quantum thermodynamics at thermal equilibrium, where irreversible entropy production is vanishing. We also apply our sufficient conditions as a tool in quantum control, evaluating the adiabatic behavior for the Hamiltonians of both the Deutsch algorithm and the Landau-Zener model under decoherence.
Abstract Quantum thermodynamics aims at investigating both the emergence and the limits of the laws of thermodynamics from a quantum mechanical microscopic approach. In this scenario, thermodynamic processes with no heat exchange, namely, adiabatic transformations, can be implemented through quantum evolutions in closed systems, even though the notion of a closed system is always an idealization and approximation. Here, we begin by theoretically discussing thermodynamic adiabatic processes in open quantum systems, which evolve non-unitarily under decoherence due to its interaction with its surrounding environment. From a general approach for adiabatic non-unitary evolution, we establish heat and work in terms of the underlying Liouville superoperator governing the quantum dynamics. As a consequence, we derive the conditions that an adiabatic open-system quantum dynamics implies in the absence of heat exchange, providing a connection between quantum and thermal adiabaticity. Moreover, we determine families of decohering systems exhibiting the same maximal heat exchange, which imply in classes of thermodynamic adiabaticity in open systems. We then approach the problem experimentally using a hyperfine energy-level quantum bit of an Ytterbium 171Yb+ trapped ion, which provides a work substance for thermodynamic processes, allowing for the analysis of heat and internal energy throughout a controllable engineered dynamics.
A general approach for transitionless quantum driving in open quantum systems is introduced. Under the assumption of adiabatic evolution for time-local master equations, we derive the generalized transitionless Lindbladian required to implement a shortcut to adiabaticity in an open-system scenario. The general counterdiabatic Lindbladian obtained accounts for a phase freedom, which translates into a set of free parameters throughout the dynamics. We discuss how our generalized approach allows us to recover the transitionless Lindbladian introduced by Vacanti et al. [G. Vacanti et al. New J. Phys. 16, 053017 (2014)]. We show how to engineer time-independent master equations that provide the same dynamics as the time-dependent master equation provided by the standard transitionless quantum driving in open systems. We illustrate our results by applying them both to the adiabatic Deutsch algorithm under dephasing and to the Landau-Zener Hamiltonian under a bit phase flip.
The adiabatic theorem and shortcuts to adiabaticity for time-dependent open quantum systems are explored in this paper. Starting from the definition of dynamical stable decoherence-free subspace, we show that, under a compact adiabatic condition, the quantum state remains in the time-dependent decoherence-free subspace with an extremely high purity, even though the dynamics of the open quantum system may not be adiabatic. The adiabatic condition mentioned here in the adiabatic theorem for open systems is very similar to that for closed quantum systems, except that the operators required to change slowly are the Lindblad operators. We also show that the adiabatic evolution of decoherence-free subspaces depends on the existence of instantaneous decoherence-free subspaces, which requires that the Hamiltonian of open quantum systems be engineered according to the incoherent control protocol. In addition, shortcuts to adiabaticity for adiabatic decoherence-free subspaces are also presented based on the transitionless quantum driving method. Finally, we provide an example that consists of a two-level system coupled to a broadband squeezed vacuum field to show our theory. Our approach employs Markovian master equations and the theory can apply to finite-dimensional quantum open systems.
We iteratively apply a recently formulated adiabatic theorem for the strong-coupling limit in finite-dimensional closed and open quantum systems. This allows us to improve approximations to a perturbed dynamics, beyond the standard approximation based on quantum Zeno dynamics and adiabatic elimination. The effective generators describing the approximate evolutions are endowed with the same block structure as the unperturbed part of the generator, and exhibit adiabatic evolutions. This iterative adiabatic theorem reveals that adiabaticity holds eternally, that is, the system evolves within each eigenspace of the unperturbed part of the generator, with an error bounded by $O(1/\ensuremath{\gamma})$ uniformly in time, where $\ensuremath{\gamma}$ is the strength of the unperturbed part of the generator. We prove that the iterative adiabatic theorem reproduces Bloch's perturbation theory in the unitary case, and is therefore a full generalization to open systems. We furthermore prove the equivalence of the Schrieffer-Wolff and des Cloizeaux approaches in the unitary case and generalize both to arbitrary open systems, showing that they share the eternal adiabaticity, and providing explicit error bounds. Finally we discuss the physical structure of the effective adiabatic generators and show that ideal effective generators for open systems do not exist in general.
We provide a rigorous generalization of the quantum adiabatic theorem for open systems described by a Markovian master equation with time-dependent Liouvillian L(t)\mathcal{L}(t)L(t). We focus on the finite system case relevant for adiabatic quantum computing and quantum annealing. Adiabaticity is defined in terms of closeness to the instantaneous steady state. While the general result is conceptually similar to the closed-system case, there are important differences. Namely, a system initialized in the zero-eigenvalue eigenspace of L(t)\mathcal{L}(t)L(t) will remain in this eigenspace with a deviation that is inversely proportional to the total evolution time TTT. In the case of a finite number of level crossings, the scaling becomes ${T}^{\ensuremath{-}\ensuremath{\eta}}$ with an exponent $\ensuremath{\eta}$ that we relate to the rate of the gap closing. For master equations that describe relaxation to thermal equilibrium, we show that the evolution time TTT should be long compared to the corresponding minimum inverse gap squared of L(t)\mathcal{L}(t)L(t). Our results are illustrated with several examples.