The interplay between topology and quantum entanglement holds profound implications for fundamental physics and the advancement of quantum technologies. Topological phases of matter are characterized by robust properties impervious to local perturbations, while entanglement serves as a critical resource for quantum information processing. Floquet engineering, the manipulation of quantum systems through periodic driving, offers a powerful non-equilibrium pathway to realize novel topological states that may not exist in static systems. This paper explores the theoretical and experimental avenues for realizing topological quantum entanglement within Floquet-engineered lattices. We review the foundational concepts of Floquet topological phases and various measures of entanglement, such as entanglement entropy and entanglement spectrum. We then delve into the methodologies for designing such systems across diverse platforms, including ultracold atoms in optical lattices, photonic waveguide arrays, and superconducting circuits. The discussion highlights how Floquet driving can induce robust entanglement in topological edge and bulk states, paving the way for fault-tolerant quantum information storage and processing. We propose specific protocols for generating and detecting topologically-protected entangled states, considering the challenges associated with heating and decoherence in periodically driven systems. Ultimately, this work emphasizes the potential of Floquet engineering as a versatile tool for synthesizing exotic entangled topological states, fostering new frontiers in quantum science and engineering.
The interplay between topology and quantum entanglement holds profound implications for fundamental physics and the advancement of quantum technologies. Topological phases of matter are characterized by robust properties impervious to local perturbations, while entanglement serves as a critical resource for quantum information processing. Floquet engineering, the manipulation of quantum systems through periodic driving, offers a powerful non-equilibrium pathway to realize novel topological states that may not exist in static systems. This paper explores the theoretical and experimental avenues for realizing topological quantum entanglement within Floquet-engineered lattices. We review the foundational concepts of Floquet topological phases and various measures of entanglement, such as entanglement entropy and entanglement spectrum. We then delve into the methodologies for designing such systems across diverse platforms, including ultracold atoms in optical lattices, photonic waveguide arrays, and superconducting circuits. The discussion highlights how Floquet driving can induce robust entanglement in topological edge and bulk states, paving the way for fault-tolerant quantum information storage and processing. We propose specific protocols for generating and detecting topologically-protected entangled states, considering the challenges associated with heating and decoherence in periodically driven systems. Ultimately, this work emphasizes the potential of Floquet engineering as a versatile tool for synthesizing exotic entangled topological states, fostering new frontiers in quantum science and engineering.
This paper explores topological thermodynamics in driven dissipative quantum materials, a field at the intersection of non-equilibrium statistical mechanics, quantum topology, and condensed matter physics. We review theoretical frameworks and experimental advancements, highlighting how driving forces and dissipative processes influence the formation, stability, and characterization of topological phases. A special focus is placed on Floquet engineering for open quantum systems, where periodic driving and environmental coupling lead to novel non-equilibrium steady states with exotic topological properties. We discuss challenges in defining thermodynamic quantities like temperature and entropy in these far-from-equilibrium settings and examine how topological invariants manifest in dissipative dynamics and transport. The paper uniquely synthesizes advancements in periodically driven, open quantum systems by critically examining how dissipation, often seen as detrimental, can be precisely engineered to reveal or even stabilize topological order. This offers a distinct perspective on designing next-generation quantum technologies. Key theoretical tools, including generalized Floquet theory, Keldysh formalism, and non-Hermitian descriptions, are discussed alongside experimental platforms like photonic quantum gases and cold atomic systems.
Quantum technology resorts to efficient utilization of quantum resources to realize technique innovation. The systems are controlled such that their states follow the desired manners to realize different quantum protocols. However, the decoherence caused by the system-environment interactions causes the states deviating from the desired manners. How to protect quantum resources under the coexistence of active control and passive decoherence is of significance. Recent studies have revealed that the decoherence is determined by the feature of the system-environment energy spectrum: Accompanying the formation of bound states in the energy spectrum, the decoherence can be suppressed. It supplies a guideline to control decoherence. Such idea can be generalized to systems under periodic driving. By virtue of manipulating Floquet bound states in the quasienergy spectrum, coherent control via periodic driving dubbed as Floquet engineering has become a versatile tool not only in controlling decoherence, but also in artificially synthesizing exotic topological phases. We will review the progress on quantum control in open and periodically driven systems. Special attention will be paid to the distinguished role played by the bound states and their controllability via periodic driving in suppressing decoherence and generating novel topological phases.
Coherent control via periodic modulation, also known as Floquet engineering, has emerged as a powerful experimental method for the realization of novel quantum systems with exotic properties. In particular, it has been employed to study topological phenomena in a variety of different platforms. In driven systems, the topological properties of the quasienergy bands can often be determined by standard topological invariants, such as Chern numbers, which are commonly used in static systems. However, due to the periodic nature of the quasienergy spectrum, this topological description is incomplete and new invariants are required to fully capture the topological properties. Most prominently, there exist two-dimensional anomalous Floquet systems that exhibit robust chiral edge modes, despite all Chern numbers are equal to zero. Here, we realize such a system with bosonic atoms in a periodically-driven hexagonal lattice and infer the complete set of topological invariants from energy gap measurements and local Hall deflections.
This thesis reports on the realization of Floquet topological systems with ultracold atoms in an optical honeycomb lattice. Using periodically driven quantum systems, topological phases of matter can be simulated by an effective, static Hamiltonian related to the time-evolution at integer multiples of the driving period, which is known as Floquet engineering. However, periodic driving can also give rise to genuinely time-dependent settings without static counterparts. One example is the anomalous Floquet phase in two dimensions, in which all bulk bands have a Chern number of zero but nevertheless robust chiral edge modes appear, which would be precluded by the bulk-edge correspondence in a static system. In an optical honeycomb lattice, anomalous Floquet systems can be created by continuous, periodic modulation of the laser intensities. This driving scheme results in different topological regimes, three of which are investigated, including the anomalous Floquet phase and a Haldane-like phase. Periodically driven systems feature periodic quasienergies and can be characterized in terms of winding numbers, which count the number of chiral edge modes in each quasienergy gap. By interferometric measurements of the quasienergy gaps between the first two Floquet bands, the topological phase transitions, emerging as gap closings, are located. For periodically driven systems, a modified bulk-edge correspondence can be formulated. In particular, the change of the winding number at a phase transition is related to the sign change of the local Berry curvature. The Berry curvature at the gap closing points is probed by Hall deflection measurements to obtain the winding numbers in each of the topological regimes, revealing the existence of chiral edge modes also in a setting with smooth boundaries as used in the experiments. The measured quasienergy gaps and transverse deflections are quantitatively well described by a numerical calculation of the Floquet bandstructure that includes coupling to higher bands during the driving period. To derive the spectrum of the modulated lattice in a geometry with edges, a tight-binding description of the system is discussed. Circular phase modulation of the honeycomb lattice can also give rise to an anomalous Floquet system when inversion symmetry is broken. The topological phase diagrams and the experimental feasibility are compared for both modulation schemes. Due to the vanishing Chern numbers, the bulk states in the anomalous Floquet regime can be fully localized by a disorder potential. This could prevent heating in interacting, periodically driven systems, resulting in a many-body localized bulk coexisting with thermalizing edge states. The experimental realization of disorder and strong interactions, as well as independent probes of the edge states are investigated. Bloch band geometry can be extended to multiband systems, using Wilson lines. The possible symmetry protection of their eigenvalues is discussed, along with measurements in the optical honeycomb lattice. Moreover, the onset of heating in weakly-interacting, periodically driven systems, triggered by parametric instabilities, is studied experimentally in a one-dimensional lattice.
We review methods for using time-periodic fields (e.g., laser or microwave fields) to induce non-equilibrium topological phenomena in quantum many-body systems. We discuss how such fields can be used to change the topological properties of the single particle spectrum, and key experimental demonstrations in solid state, cold atomic, and photonic systems. The single particle Floquet band structure provides a stage on which the system's dynamics play out; the crucial question is then how to obtain robust topological behaviour in the many-particle setting. In the regime of mesoscopic transport, we discuss manifestations of topological edge states induced in the Floquet spectrum. Outside the context of mesoscopic transport, the main challenge of inducing stable topological phases in many-body Floquet systems is their tendency to absorb energy from the drive and thereby to heat up. We discuss three routes to overcoming this challenge: long-lived transient dynamics and prethermalization, disorder-induced many-body localization, and engineered couplings to external baths. We discuss the types of phenomena that can be explored in each of these regimes, and their experimental realizations.
Recent advances in experimental condensed matter physics suggest a powerful new paradigm for the realization of exotic phases of quantum matter in the laboratory. Rather than conducting an exhaustive search for materials that realize these phases at low temperatures, it may be possible to design quantum systems that exhibit the desired properties. With the numerous advances made recently in the fields of cold atomic gases, superconducting qubits, trapped ions, and nitrogen-vacancy centers in diamond, it appears that we will soon have a host of platforms that can be used to put exotic theoretical predictions to the test. In this dissertation, I will highlight two ways in which theorists can interact productively with this fast-emerging field. \n \nFirst, there is a growing interest in driving quantum systems out of equilibrium in order to induce novel topological phases where they would otherwise never appear. In particular, systems driven by time-periodic perturbations—known as “Floquet systems”—offer fertile ground for theoretical investigation. This approach to designer quantum matter brings its own unique set of challenges. In particular, Floquet systems explicitly violate conservation of energy, providing no notion of a ground state. In the first part of my dissertation, I will present research that addresses this problem in two ways. First, I will present studies of open Floquet systems, where coupling to an external reservoir drives the system into a steady state at long times. Second, I will discuss examples of isolated quantum systems that exhibit signatures of topological properties in their finite-time dynamics. \n \nThe second part of this dissertation presents another way in which theorists can benefit from the designer approach to quantum matter; in particular, one can design analytically tractable theories of exotic phases. I will present an exemplar of this philosophy in the form of coupled-wire constructions. In this approach, one builds a topological state of matter from the ground up by coupling together an array of one-dimensional quantum wires with local interactions. I will demonstrate the power of this technique by showing how to build both Abelian and non-Abelian topological phases in three dimensions by coupling together an array of quantum wires.