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Dissipation effects in driven quantum many-body systems

Luca Arceci

Published Oct 24, 2019• 0 citations
OpenAlex

Abstract

In this thesis, the effect of dissipation is investigated in driven models of interest for quantum annealing and quantum topological pumping. Dissipation comes from coupling the system to a bosonic bath at thermal equilibrium, using the usual Caldeira-Leggett setting. The first original results presented deal with the dissipative Landau-Zener model, where we revisit the issue of whether dissipation can improve the final ground state probability. We shed light upon the importance of the coupling direction. Then, we move to dissipation effects in the quantum annealing of the Ising chain in transverse field: we study the conditions for the emergence or absence of an optimal annealing velocity, depending on system-bath interaction strength and bath temperature. Finally, we explore dissipation effects in topological pumping in the periodically-driven Rice-Mele model, where we find that a bath at low temperature can improve the pumping performance when adiabaticity is not perfectly fulfilled.

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In this thesis, the effect of dissipation is investigated in driven models of interest for quantum annealing and quantum topological pumping. Dissipation comes from coupling the system to a bosonic bath at thermal equilibrium, using the usual Caldeira-Leggett setting. The first original results presented deal with the dissipative Landau-Zener model, where we revisit the issue of whether dissipation can improve the final ground state probability. We shed light upon the importance of the coupling direction. Then, we move to dissipation effects in the quantum annealing of the Ising chain in transverse field: we study the conditions for the emergence or absence of an optimal annealing velocity, depending on system-bath interaction strength and bath temperature. Finally, we explore dissipation effects in topological pumping in the periodically-driven Rice-Mele model, where we find that a bath at low temperature can improve the pumping performance when adiabaticity is not perfectly fulfilled.
We revisit here the issue of thermally assisted Quantum Annealing by a detailed study of the dissipative Landau-Zener problem in the presence of a Caldeira-Leggett bath of harmonic oscillators, using both a weak-coupling quantum master equation and a quasiadiabatic path-integral approach. Building on the known zero-temperature exact results [Wubs et al., Phys. Rev. Lett. 97, 200404 (2006)], we show that a finite temperature bath can have a beneficial effect on the ground-state probability only if it couples also to a spin direction that is transverse with respect to the driving field, while no improvement is obtained for the more commonly studied purely longitudinal coupling. In particular, we also highlight that, for a transverse coupling, raising the bath temperature further improves the ground-state probability in the fast-driving regime. We discuss the relevance of these findings for the current quantum-annealing flux qubit chips.
We discuss two quantum simulation schemes in which the coupling to an external bath may give rise to novel and interesting many-body physics. Namely, we first address the effect of local Markovian baths on the quantum annealing dynamics of an Ising-like chain: deviations from adiabaticity may display a nonmonotonic trend as a function of the annealing time, as a result of the competition between nonadiabatic effects and dissipative processes. Secondly, we provide a framework to induce persistent currents through the coupling with a structured reservoir which generates nonreciprocity, without the need of any applied gauge field.
We investigate the quantum annealing of the ferromagnetic ppp-spin model in a dissipative environment (p=5p=5p=5 and p=7p=7p=7). This model, in the large-ppp limit, codifies Grover's algorithm for searching in an unsorted database [L. K. Grover, Proceedings of the 28th Annual ACM Symposium on Theory of Computing (ACM, New York, 1996), pp. 212--219]. The dissipative environment is described by a phonon bath in thermal equilibrium at finite temperature. The dynamics is studied in the framework of a Lindblad master equation for the reduced density matrix describing only the spins. Exploiting the symmetries of our model Hamiltonian, we can describe many spins and extrapolate expected trends for large NNN and ppp. While at weak system-bath coupling the dissipative environment has detrimental effects on the annealing results, we show that in the intermediate-coupling regime, the phonon bath seems to speed up the annealing at low temperatures. This improvement in the performance is likely not due to thermal fluctuation but rather arises from a correlated spin-bath state and persists even at zero temperature. This result may pave the way to a new scenario in which, by appropriately engineering the system-bath coupling, one may optimize quantum annealing performances below either the purely quantum or the classical limit.
In this thesis, I study dissipation in a number of one-dimensional models from condensed matter physics, by coupling them to a quantum bath. The main purpose is to see how the dissipation interacts with other properties, such as topological order and different types of symmetries. The first system studied is the Kitaev chain, a simple non-interacting model with topological order. The presence of a bath drives the system out of equilibrium and into a mixed quantum state, where conventional measures of topological order do not apply. I propose looking for certain dynamical signatures, that can tell us what happens to the topological order of the original system, once the bath is included. In addition, we study the dissipative Kitaev chain with periodic driving, which introduces a host of exotic new topological properties. Finally, I investigate the effect of dissipation on interacting models, such as the XXZ Heisenberg spin chain, and how symmetries might help us analyze this effect. While the dynamics of such a model are very difficult to compute, certain observables like the total magnetization of the spin chain behave in a peculiar way. By studying the symmetry properties of both the system and the bath, I show that the dissipation causes the expectation value of the magnetization to relax in a simple and coherent manner, only adding an overall damping factor to the dynamics of the isolated system.
We introduce and study the adiabatic dynamics of free-fermion models subject to a local Lindblad bath and in the presence of a time-dependent Hamiltonian. The merit of these models is that they can be solved exactly, and will help us to study the interplay between nonadiabatic transitions and dissipation in many-body quantum systems. After the adiabatic evolution, we evaluate the excess energy (the average value of the Hamiltonian) as a measure of the deviation from reaching the final target ground state. We compute the excess energy in a variety of different situations, where the nature of the bath and the Hamiltonian is modified. We find robust evidence of the fact that an optimal working time for the quantum annealing protocol emerges as a result of the competition between the nonadiabatic effects and the dissipative processes. We compare these results with the matrix-product-operator simulations of an Ising system and show that the phenomenology we found also applies for this more realistic case.
We determine the frequency-dependent response characteristics of a quantum system to a driven Caldeira-Leggett bath. The bath degrees of freedom are explicitly driven by an external time-dependent force, in addition to the direct time-dependent forcing of the system itself. After general considerations of driven Caldeira-Leggett baths, we consider the Rubin model of a chain of quantum particles coupled by linear springs as an important model of a quantum dissipative system. We show that in the presence of time-dependent driving of the chain, this model can be mapped to a quantum system which couples to a driven Caldeira-Leggett bath. The effect of the bath driving is captured by a time-dependent force on the central system, which is, in principle, non-Markovian in nature. We study two specific examples, the exactly solvable case of a harmonic potential and a quantum two-state system for which we assume a weak system-bath coupling. We evaluate the dynamical response to a periodic driving of the system and the bath. The dynamic susceptibility is shown to be altered qualitatively by the bath drive: The dispersive part is enhanced at low frequencies and acquires a maximum at zero frequency. The absorptive part develops a shoulder-like behavior in this frequency regime. These features seem to be generic for quantum systems in a driven Caldeira-Leggett bath.