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arXiv:1703.03075 · Physical Review A

Topological protection of coherence in a dissipative environment

Lorenzo Campos Venuti, Zhengzhi Ma, Hubert Saleur, Stephan Haas

Published Nov 27, 2017• 24 citations
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Abstract

One-dimensional topological insulators are characterized by edge states with exponentially small energies. According to one generalization of topological phase to non-Hermitian systems, a finite system in a nontrivial topological phase displays surface states with exponentially long lifetimes. In this work we explore the possibility of exploiting such non-Hermitian topological phases to enhance the quantum coherence of a fiducial qubit embedded in a dissipative environment. We first show that a network of qubits interacting with lossy cavities can be represented, in a suitable super-one-particle sector, by a non-Hermitian ``Hamiltonian'' of the desired form. We then study, both analytically and numerically, one-dimensional geometries with up to three sites per unit cell and up to a topological winding number $W=2$. For finite-size systems the number of edge modes is a complicated function of $W$ and the system size $N$. However, we find that there are precisely $W$ modes localized at one end of the chain. In such topological phases the qubit's coherence lifetime is exponentially large in the system size. We verify that for $W>1$, at large times, the Lindbladian evolution is approximately a nontrivial unitary. For $W=2$ this results in Rabi-like oscillations of the qubit's coherence measure.

DOI
10.1103/physreva.96.053858

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One-dimensional topological insulators are characterized by edge states with exponentially small energies. According to one generalization of topological phase to non-Hermitian systems, a finite system in a nontrivial topological phase displays surface states with exponentially long lifetimes. In this work we explore the possibility of exploiting such non-Hermitian topological phases to enhance the quantum coherence of a fiducial qubit embedded in a dissipative environment. We first show that a network of qubits interacting with lossy cavities can be represented, in a suitable super-one-particle sector, by a non-Hermitian ``Hamiltonian'' of the desired form. We then study, both analytically and numerically, one-dimensional geometries with up to three sites per unit cell and up to a topological winding number W=2W=2W=2. For finite-size systems the number of edge modes is a complicated function of WWW and the system size NNN. However, we find that there are precisely WWW modes localized at one end of the chain. In such topological phases the qubit's coherence lifetime is exponentially large in the system size. We verify that for W>1W>1W>1, at large times, the Lindbladian evolution is approximately a nontrivial unitary. For W=2W=2W=2 this results in Rabi-like oscillations of the qubit's coherence measure.
One dimensional topological insulators are characterized by edge states with exponentially small energies. According to one generalization of topological phase to non-Hermitian systems, a finite system in a non-trivial topological phase displays surface states with exponentially long life times. In this work we explore the possibility of exploiting such non-Hermitian topological phases to enhance the quantum coherence of a fiducial qubit embedded in a dissipative environment. We first show that a network of qubits interacting with lossy cavities can be represented, in a suitable super-one-particle sector, by a non-Hermitian Hamiltonian of the desired form. We then study, both analytically and numerically, one-dimensional geometries with up to three sites per unit cell, and up to a topological winding number WWW = 2. For finite-size systems the number of edge modes is a complicated function of WWW and the system size NNN. However we find that there are precisely WWW modes localized at one end of the chain. In such topological phases the quibt's coherence lifetime is exponentially large in the system size. We verify that, for WWW > 1, at large times, the Lindbladian evolution is approximately a non-trivial unitary. For WWW = 2 this results in Rabi-like oscillations of the qubit's coherence measure.
The lifetime of a quantum state, denoted τ, is a fundamental characteristic determining the coherence time and operational reliability of quantum technologies. Conventional approaches to extend τ rely on dynamical decoupling, material purification, or engineered dissipation, yet these methods generally scale poorly with system size and are vulnerable to environmental noise. This paper introduces a novel theoretical framework whereby τ is determined by a topological invariant T of the underlying quantum system: τ = f(T). By establishing a rigorous connection between the spectrum of the Liouvillian superoperator, the topology of the system's Hilbert space, and the decay dynamics of density matrices, we demonstrate that certain topologically nontrivial phases inherently suppress relaxation channels and thus guarantee a finite lower bound on τ. We present a detailed derivation for a class of spin chains with nearest‑neighbor interactions and illustrate how the winding number of the Bloch Hamiltonian controls the lifetime of edge states. The formalism is generalized to higher‑dimensional lattice models and to systems with non‑Hermitian Hamiltonians. We discuss implications for fault‑tolerant quantum computation, robust quantum memories, and the design of synthetic quantum matter with engineered lifetimes. This work lays the foundation for a new branch of quantum stability theory that harnesses topological protection not only of energy levels but of dynamical lifetimes, potentially opening pathways toward scalable, intrinsically protected quantum devices.
Topological phases are characterized by edge states confined near the boundaries by a bulk energy gap. On raising temperature, these edge states are typically lost due to mobile thermal excitations. However, disorder can localize an isolated many-body system, potentially allowing for a sharply defined topological phase even in a highly excited state. We explicitly demonstrate this in a model of a disordered, one-dimensional magnet with spin one-half edge excitations. Furthermore, we show that the time evolution of a simple, highly excited state reveals quantum coherent edge spins. In particular, we demonstrate the coherent revival of an edge spin over a time scale that grows exponentially with system size. This is in sharp contrast to the general expectation that quantum bits strongly coupled with a hot many-body system will rapidly lose coherence. This result opens new directions in the study of topologically protected quantum dynamics. Quantum coherence should be rapidly lost in a system of many strongly coupled particles. Here, the authors show that this need not be the case in a one-dimensional magnet with impurities, allowing for a sharply defined topological phase even at high temperatures.
Under the Born–Markov approximation, a qubit system, such as a two-level atom, is known to undergo a memoryless decay of quantum coherence or excitation when weakly coupled to a featureless environment. Recently, it has been shown that unavoidable disorder in the environment is responsible for non-Markovian effects and information backflow from the environment into the system owing to Anderson localization. This turns disorder into a resource for enhancing non-Markovianity in the system–environment dynamics, which could be of relevance in cavity quantum electrodynamics. Here we consider the decoherence dynamics of a qubit weakly coupled to a two-dimensional bath with a nontrivial topological phase, such as a two-level atom embedded in a two-dimensional coupled-cavity array with a synthetic gauge field realizing a quantum-Hall bath, and show that Markovianity is protected against moderate disorder owing to the robustness of chiral edge modes in the quantum-Hall bath. Interestingly, switching off the gauge field, i.e., flipping the bath into a topological trivial phase, allows one to re-introduce non-Markovian effects. Such a result indicates that changing the topological phase of a bath by a tunable synthetic gauge field can be harnessed to control non-Markovian effects and quantum information backflow in a qubit-environment system.
Topological phases are often characterized by special edge states confined near the boundaries by an energy gap in the bulk. On raising temperature, these edge states are lost in a clean system due to mobile thermal excitations. Recently however, it has been established that disorder can localize an isolated many body system, potentially allowing for a sharply defined topological phase even in a highly excited state. Here we show this to be the case for the topological phase of a one dimensional magnet with quenched disorder, which features spin one-half excitations at the edges. The time evolution of a simple, highly excited, initial state is used to reveal quantum coherent edge spins. In particular, we demonstrate, using theoretical arguments and numerical simulation, the coherent revival of an edge spin over a time scale that grows exponentially bigger with system size. This is in sharp contrast to the general expectation that quantum bits strongly coupled to a 'hot' many body system will rapidly lose coherence.
Nonequilibrium conditions are traditionally seen as detrimental to the appearance of quantum-coherent many-body phenomena, and much effort is often devoted to their elimination. Recently this approach has changed: It has been realized that driven-dissipative dynamics could be used as a resource. By proper engineering of the reservoirs and their couplings to a system, one may drive the system towards desired quantum-correlated steady states, even in the absence of internal Hamiltonian dynamics. An intriguing category of equilibrium many-particle phases are those which are distinguished by topology rather than by symmetry. A natural question thus arises: which of these topological states can be achieved as the result of dissipative Lindblad-type (Markovian) evolution? Beside its fundamental importance, it may offer novel routes to the realization of topologically-nontrivial states in quantum simulators, especially ultracold atomic gases. Here I give a general answer for Gaussian states and quadratic Lindblad evolution, mostly concentrating on the example of 2D Chern insulator states. I prove a no-go theorem stating that a finite-range Lindbladian cannot induce finite-rate exponential decay towards a unique topological pure state above 1D. I construct a recipe for creating such state by exponentially-local dynamics, or a mixed state arbitrarily close to the desired pure one via finite-range dynamics. I also address the cold-atom realization, classification, and detection of these states. Extensions to other types of topological insulators and superconductors are also discussed.