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Magnetic Quantum Walks of Neutral Atoms in Optical Lattices

Muhammad Sajid

Published Sep 17, 2018• 3 citations
OpenAlex

Abstract

This thesis focuses on the simulation of the physics of a charged particle under an external magnetic field by using discrete-time quantum walks of a spin-1/2 particle in a two-dimensional lattice. By Floquet-engineering the quantum-walk protocol, an Aharonov–Bohm geometric phase is imprinted onto closed-loop paths in the lattice, thus realizing an abelian gauge field—the analog of a magnetic flux threading a two-dimensional electron gas. I show that in the strong-field regime, i.e. when the flux per plaquette of the lattice is a sizable fraction of the flux quantum, magnetic quantum walks give rise to nearly flat energy bands. I demonstrate that the system behaves like a Chern insulator by computing the Chern numbers of the energy bands and studying the excitation of the midgap topologically protected edge modes. These modes are extended all along the boundaries of the magnetic domains and remain robust against perturbations that respect the gap closing conditions. Furthermore, I discuss a possible experimental implementation of this scheme using neutral atoms trapped in two dimensional spin-dependent optical lattices. The proposed scheme has a number of unique features, e.g. it allows one to generate arbitrary magnetic-field landscapes, including those with sharp boundaries along which topologically protected edge states can be localized and probed. Additionally, I introduce the scattering matrix approach in discrete-time quantum walks to probe the Hofstadter spectrum and compute its topological invariants. By opening up a discrete-time quantum walk system and connecting it to metallic leads, I demonstrate that the reflection/transmission probabilities of a particle from the scattering region give information on the energy spectrum and topological invariants of the system. Although the work presented here focuses on the physics of a single particle in a clean system, it sets the stage for studies of many-body topological states in the presence of interactions and disorder.

Related papers

Artificial gauge fields and topology with ultracold atoms in optical lattices

Monika Aidelsburger

Abstract Topological many-body phases of matter exhibit remarkable electronic properties and ultracold atoms in optical lattices constitute promising candidates to study them in a well-controlled environment. In two-dimensional (2D) electron gases topological phases may emerge in the presence of strong magnetic fields. This situation is not directly applicable to cold atoms because they are charge neutral. Therefore, novel experimental techniques have been developed to engineer novel lattice systems, whose Hamiltonian is formally equivalent to the one of charged particles in magnetic fields. In this Tutorial, we introduce a paradigmatic topological lattice model, the Hofstadter model, and explain how it can be implemented with ultracold atoms using laser-assisted tunneling. The technique is based on imprinting phases on the tunneling matrix elements using additional laser beams. These phases are reminiscent of Aharonov–Bohm phases and can be interpreted as a magnetic flux piercing the lattice unit cell. We present experimental results on the cyclotron-like dynamics of neutral atoms in isolated four-site square plaquettes and discuss the first measurement of a 2D topological invariant, the Chern number, in artificially generated Hofstadter bands. The work presented in this Tutorial was one of the four shortlisted finalists of the 2016 DPG SAMOP dissertation prize.

Journal of Physics B Atomic Molecular and Optical Physics•Apr 30, 2018•31 citations

Creating anomalous Floquet Chern insulators with magnetic quantum walks

Muhammad Sajid, János K. Asbóth, Dieter Meschede +2 more

Charged particles in a two-dimensional lattice behave like a Chern topological insulator when subject to a perpendicular magnetic field. Here, the authors propose a realistic scheme to use neutral atoms in optical lattices to simulate the effect of the magnetic field and to study transport in the strong field regime, which is not accessible when working with conventional materials. Importantly, the topological behavior of these so-called magnetic quantum walks extends even beyond the Chern insulator model because they are periodically driven in time,. Magnetic quantum walks may open a new route to studying topological properties of charged particles in strong magnetic fields.

arXiv:1808.08923•Physical review. B./Physical review. B•Jun 17, 2019•48 citations

The Quantum Walk Microscope

Stefan Brakhane

In this thesis, I present single-site detection of neutral atoms stored in a three-dimensional optical lattice using a numerical aperture objective lens (NAdesign = 0.92). The combination of high-resolution imaging with state-dependent trapping along two-direction of the lattice opens up the path towards quantum simulations via quantum walks. Suppressing the interactions of a quantum system with the environment is essential for all quantum simulation experiments. It demands a precise control of both the external magnetic (stray) fields and the polarization properties of laser beams inside the vacuum chamber. I designed a metal shielding to reduce magnetic field fluctuations and designed, assembled and characterized a novel ultra-high vacuum glass cell. The glass cell consists of special glass material and exhibits an ultra-low birefringence Δn of a few times 10^-8 to highly suppress polarization disturbances originating from stress birefringence in vacuum windows. Furthermore, anti-reflection coatings avoid reflections on all window surfaces. The cell hosts the assembled vacuum-compatible objective, that exhibits a diffraction limited resolution of up to 453 nm and allows to optically resolve the spacing of the optical lattice. Fluorescence images of single trapped atoms are used to characterize the imaging system. The filling, orientation and geometry of the optical lattice is precisely reconstructed using positions of atoms that can be determined from fluorescence images. Furthermore, I present a scheme to realize state-dependent transport and discuss its robustness against experimental imperfections in a technical implementation. This transport scheme enable the realization of discrete-time quantum walks with neutral atoms in two dimensions. These quantum walks pave the way towards the simulation of artificial magnetic fields and topologically protected edge states.

•Mar 2, 2017•4 citations

Non-Abelian gauge fields

Fabrice Gerbier, Nathan Goldman, Maciej Lewenstein +1 more

Building a universal quantum computer is a central goal of emerging quantum technologies, which has the potential to revolutionize science and technology. Unfortunately, this future does not seem to be very close at hand. However, quantum computers built for a special purpose, i.e. quantum simulators , are currently developed in many leading laboratories. Many schemes for quantum simulation have been proposed and realized using, e.g., ultracold atoms in optical lattices, ultracold trapped ions, atoms in arrays of cavities, atoms/ions in arrays of traps, quantum dots, photonic networks, or superconducting circuits. The progress in experimental implementations is more than spectacular. Particularly interesting are those systems that simulate quantum matter evolving in the presence of gauge fields. In the quantum simulation framework, the generated (synthetic) gauge fields may be Abelian, in which case they are the direct analogues of the vector potentials commonly associated with magnetic fields. In condensed matter physics, strong magnetic fields lead to a plethora of fascinating phenomena, among which the most paradigmatic is perhaps the quantum Hall effect. The standard Hall effect consists in the appearance of a transverse current, when a longitudinal voltage difference is applied to a conducting sample. For quasi-two-dimensional semiconductors at low temperatures placed in very strong magnetic fields, the transverse conductivity, the ratio between the transverse current and the applied voltage, exhibits perfect and robust quantization, independent for instance of the material or of its geometry. Such an integer quantum Hall effect, is now understood as a deep consequence of underlying topological order. Although such a system is an insulator in the bulk, it supports topologically robust edge excitations which carry the Hall current. The robustness of these chiral excitations against backscattering explains the universality of the quantum Hall effect. Another interesting and related effect, which arises from the interplay between strong magnetic field and lattice potentials, is the famous Hofstadter butterfly: the energy spectrum of a single particle moving on a lattice and subjected to a strong magnetic field displays a beautiful fractal structure as a function of the magnetic flux penetrating each elementary plaquette of the lattice. When the effects of interparticle interactions become dominant, two-dimensional gases of electrons exhibit even more exotic behaviour leading to the fractional quantum Hall effect. In certain conditions such a strongly interacting electron gas may form a highly correlated state of matter, the prototypical example being the celebrated Laughlin quantum liquid. Even more fascinating is the behaviour of bulk excitations (quasi-hole and quasi-particles): they are neither fermionic nor bosonic, but rather behave as anyons with fractional statistics intermediate between the two. Moreover, for some specific filling factors (ratio between the electronic density and the flux density), these anyons are proven to have an internal structure (several components) and non-Abelian braiding properties. Many of the above statements concern theoretical predictions—they have never been observed in condensed matter systems. For instance, the fractional values of the Hall conductance is seen as a direct consequence of the fractional statistics, but to date direct observation of anyons has not been possible in two-dimensional semiconductors. Realizing these predictions in experiments with atoms, ions, photons etc, which potentially allow the experimentalist to perform measurements complementary to those made in condensed matter systems, is thus highly desirable! Non-Abelian gauge fields couple the motional states of the particles to their internal degrees of freedom (such as hyperfine states for atoms or ions, electronic spins for electrons, etc). In this sense external non-Abelian fields extend the concept of spin–orbit coupling (Rashba and Dresselhaus couplings), familiar from AMO and condensed matter physics. They lead to yet another variety of fascinating phenomena such as the quantum spin Hall effect, three-dimensional topological insulators, topological superconductors and superfluids of various kinds. One also expects here the appearance of excitations in a form of topological edge states that can support robust transport, or entangled Majorana fermions in the case of topological superconductors or superfluids. Again, while many kinds of topological insulators have been realized in condensed matter systems, a controlled way of creating them in AMO systems and studying quantum phase transitions between various kinds of them is obviously very appealing and challenging. The various systems listed so far correspond to static gauge fields, which are externally imposed by the experimentalists. Even more fascinating is the possibility of generating synthetically dynamical gauge fields, i.e. gauge fields that evolve in time according to an interacting gauge theory, e.g., a full lattice gauge theory (LGT). These dynamical gauge fields can also couple to matter fields, allowing the quantum simulation of such complex systems (notoriously hard to simulate using 'traditional' computers), which are particularly relevant for modern high-energy physics. So far, most of the theoretical proposals concern the simulation of Abelian gauge theories, however, several groups have recently proposed extensions to the non-Abelian scenarios. The scope of the present focused issue of Journal of Physics B is to cover all of these developments, with particular emphasis on the non-Abelian gauge fields. The 14 papers in this issue include contributions from the leading theory groups working in this field; we believe that this collection will provide the reference set for quantum simulations of gauge fields. Although the special issue contains exclusively theoretical proposals and studies, it should be stressed that the progress in experimental studies of artificial Abelian and non-Abelian gauge fields in recent years has been simply spectacular. Multiple leading groups are working on this subject and have already obtained a lot of seminal results. The papers in the special issue are ordered according to the date of acceptance. The issue opens with a review article by Zhou et al [1] on unconventional states of bosons with synthetic spin–orbit coupling. Next, the paper by Maldonado-Mundo et al [2] studies ultracold Fermi gases with artificial Rashba spin–orbit coupling in a 2D gas. Anderson and Charles [3], in contrast, discuss a three-dimensional spin–orbit coupling in a trap. Orth et al [4] investigate correlated topological phases and exotic magnetism with ultracold fermions, again in the presence of artificial gauge fields. The paper of Nascimbène [5] does not address the synthetic gauge fields directly, but describes an experimental proposal for realizing one-dimensional topological superfluids with ultracold atomic gases; obviously, this problem is well situated in the general and growing field of topological superfluids, in particular those realized in the presence of non-Abelian gauge fields/spin–orbit coupling. Graß et al [6] consider in their paper fractional quantum Hall states of a Bose gas with spin–orbit coupling induced by a laser. Particular attention is drawn here to the possibility of realizing states with non-Abelian anyonic excitations. Zheng et al [7] study properties of Bose gases with Raman-induced spin–orbit coupling. Kiffner et al [8] in their paper touch on another kind of system, namely ultracold Rydberg atoms. In particular they study the generation of Abelian and non-Abelian gauge fields in dipole–dipole interacting Rydberg atoms. The behaviour of fermions in synthetic non-Abelian gauge potentials is discussed by Shenoy and Vyasanakere [9]. The paper starts with the study of Rashbon condensates (i.e. Bose condensates in the presence of Rashba coupling) and also introduces novel kinds of exotic Hamiltonians. Goldman et al [10] propose a concrete setup for realizing arbitrary non-Abelian gauge potentials in optical square lattices; they discuss how such synthetic gauge fields can be exploited to generate Chern insulators. Zygelman [11], similarly as Kiffner et al [8], discusses in his paper non-Abelian gauge fields in Rydberg systems. Marchukov et al [12] return to the subject of spin–orbit coupling, and investigate spectral gaps of spin–orbit coupled particles in the realistic situations of deformed traps. The last two papers, in contrast, are devoted to different subjects. Edmonds et al [13] consider a 'dynamical' density-dependent gauge potential, and study the Josephson effect in a Bose–Einstein condensate subject to such a potential. Last, but not least, Mazzucchi et al [14] study the properties of semimetal-superfluid quantum phase transitions in 3D lattices with Dirac points. References [1] Zhou X, Li Y, Cai Z and Wu C 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134001 [2] Maldonado-Mundo D, Öhberg P and Valiente M 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134002 [3] Anderson B M and Clark C W 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134003 [4] Orth P P, Cocks D, Rachel S, Buchhold M, Le Hur K and Hofstetter W 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134004 [5] Nascimbène S 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134005 [6] Graß T, Juliá-Díaz B, Burrello M and Lewenstein M 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134006 [7] Zheng W, Yu Z-Q, Cui X and Zhai H 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134007 [8] Kiffner M, Li W and Jaksch D 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134008 [9] Shenoy V B and Vyasanakere J P 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134009 [10] Goldman N, Gerbier F and Lewenstein M 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134010 [11] Zygelman B 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134011 [12] Marchukov O V, Volosniev A G, Fedorov D V, Jensen A S and Zinner N T 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134012 [13] Edmonds M J, Valiente M and Öhberg P 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134013 [14] Mazzucchi G, Lepori L and Trombettoni A 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134014

Journal of Physics B Atomic Molecular and Optical Physics•Jun 24, 2013•11 citations

Scattering theory of topological phases in discrete-time quantum walks

Brian Tarasinski, János K. Asbóth, Jan P. Dahlhaus

One-dimensional discrete-time quantum walks show a rich spectrum of topological phases that have so far been exclusively analyzed using the Floquet operator in momentum space. In this work, we introduce an alternative approach to topology which is based on the scattering matrix of a quantum walk, adapting concepts from time-independent systems. For quantum walks with gaps in the quasienergy spectrum at 0 and $\ensuremath{\pi}$, we find three different types of topological invariants, which apply dependent on the symmetries of the system. These determine the number of protected boundary states at an interface between two quantum-walk regions. Quantum walks with an unequal number of leftward and rightward shifts per cycle are characterized by the number of perfectly transmitting unidirectional modes they support, which is equal to their nontrivial quasienergy winding. Our classification provides a unified framework that includes all known types of topology in one-dimensional discrete-time quantum walks and is very well suited for the analysis of finite-size and disorder effects. We provide a simple scheme to directly measure the topological invariants in an optical quantum-walk experiment.

arXiv:1401.2673•Physical Review A•Apr 25, 2014•69 citations

Preparing and Probing Chern Bands with Cold Atoms

Nathan Goldman, Nigel R. Cooper, Jean Dalibard

The present chapter discusses methods by which topological Bloch bands can be prepared in cold-atom setups. Focusing on the case of Chern bands for two-dimensional systems, we describe how topological properties can be triggered by driving atomic gases, either by dressing internal levels with light or through time-periodic modulations. We illustrate these methods with concrete examples, and we discuss recent experiments where geometrical and topological band properties have been identified. Introduction Ultracold atoms constitute a promising physical platform for the preparation and exploration of novel states of matter [1, 2, 3]. In particular, the engineering of topological band structures with cold-atom systems, together with the capability of tuning interactions between the particles, opens an interesting route toward the realization of intriguing strongly correlated states with topological features, such as fractional topological insulators and quantum Hall liquids [4]. This chapter is dedicated to the preparation and the detection of topological band structures characterized by nonzero Chern numbers [5]. Such Chern bands , which constitute the building blocks for realizing (fractional) Chern insulators [6, 7], arise in two-dimensional (2D) systems presenting time-reversal-symmetry (TRS) breaking effects. For instance, nontrivial Chern bands naturally appear in the Harper-Hofstadter model [8], a lattice penetrated by a uniform flux, where they generalize the (nondispersive) Landau levels to the lattice framework. Additionally, Chern bands also appear in staggered flux configurations, such as in Haldane's honeycomb-lattice model [9] or in lattice systems combining Rashba spin-orbit coupling and Zeeman (exchange) fields. The atoms being charge neutral, “magnetic” fluxes cannot be simply produced by subjecting optical lattices to “real” magnetic fields. It is the aim of this chapter to review several schemes that have been recently implemented in laboratories with the goal of realizing synthetic magnetic fields leading to Chern bands for cold atoms. Our presentation is complementary to that of Chapter 15, as we focus on synthetic magnetic fields in lattice-based systems for which the flux density can be made very large (with magnetic length comparable to interparticle spacing). The chapter is structured as follows: Section 14.2 describes how the Chern number is related to physical observables defined in a lattice framework. In particular, it clarifies the link between recent Chern-number measurements performed in cold bosonic gases and the more conventional (electronic) quantum Hall effect.

arXiv:1507.07805•Cambridge University Press eBooks•Oct 4, 2018•7 citations
This thesis focuses on the simulation of the physics of a charged particle under an external magnetic field by using discrete-time quantum walks of a spin-1/2 particle in a two-dimensional lattice. By Floquet-engineering the quantum-walk protocol, an Aharonov–Bohm geometric phase is imprinted onto closed-loop paths in the lattice, thus realizing an abelian gauge field—the analog of a magnetic flux threading a two-dimensional electron gas. I show that in the strong-field regime, i.e. when the flux per plaquette of the lattice is a sizable fraction of the flux quantum, magnetic quantum walks give rise to nearly flat energy bands. I demonstrate that the system behaves like a Chern insulator by computing the Chern numbers of the energy bands and studying the excitation of the midgap topologically protected edge modes. These modes are extended all along the boundaries of the magnetic domains and remain robust against perturbations that respect the gap closing conditions. Furthermore, I discuss a possible experimental implementation of this scheme using neutral atoms trapped in two dimensional spin-dependent optical lattices. The proposed scheme has a number of unique features, e.g. it allows one to generate arbitrary magnetic-field landscapes, including those with sharp boundaries along which topologically protected edge states can be localized and probed. Additionally, I introduce the scattering matrix approach in discrete-time quantum walks to probe the Hofstadter spectrum and compute its topological invariants. By opening up a discrete-time quantum walk system and connecting it to metallic leads, I demonstrate that the reflection/transmission probabilities of a particle from the scattering region give information on the energy spectrum and topological invariants of the system. Although the work presented here focuses on the physics of a single particle in a clean system, it sets the stage for studies of many-body topological states in the presence of interactions and disorder.
Abstract Topological many-body phases of matter exhibit remarkable electronic properties and ultracold atoms in optical lattices constitute promising candidates to study them in a well-controlled environment. In two-dimensional (2D) electron gases topological phases may emerge in the presence of strong magnetic fields. This situation is not directly applicable to cold atoms because they are charge neutral. Therefore, novel experimental techniques have been developed to engineer novel lattice systems, whose Hamiltonian is formally equivalent to the one of charged particles in magnetic fields. In this Tutorial, we introduce a paradigmatic topological lattice model, the Hofstadter model, and explain how it can be implemented with ultracold atoms using laser-assisted tunneling. The technique is based on imprinting phases on the tunneling matrix elements using additional laser beams. These phases are reminiscent of Aharonov–Bohm phases and can be interpreted as a magnetic flux piercing the lattice unit cell. We present experimental results on the cyclotron-like dynamics of neutral atoms in isolated four-site square plaquettes and discuss the first measurement of a 2D topological invariant, the Chern number, in artificially generated Hofstadter bands. The work presented in this Tutorial was one of the four shortlisted finalists of the 2016 DPG SAMOP dissertation prize.
Charged particles in a two-dimensional lattice behave like a Chern topological insulator when subject to a perpendicular magnetic field. Here, the authors propose a realistic scheme to use neutral atoms in optical lattices to simulate the effect of the magnetic field and to study transport in the strong field regime, which is not accessible when working with conventional materials. Importantly, the topological behavior of these so-called magnetic quantum walks extends even beyond the Chern insulator model because they are periodically driven in time,. Magnetic quantum walks may open a new route to studying topological properties of charged particles in strong magnetic fields.
In this thesis, I present single-site detection of neutral atoms stored in a three-dimensional optical lattice using a numerical aperture objective lens (NAdesign = 0.92). The combination of high-resolution imaging with state-dependent trapping along two-direction of the lattice opens up the path towards quantum simulations via quantum walks. Suppressing the interactions of a quantum system with the environment is essential for all quantum simulation experiments. It demands a precise control of both the external magnetic (stray) fields and the polarization properties of laser beams inside the vacuum chamber. I designed a metal shielding to reduce magnetic field fluctuations and designed, assembled and characterized a novel ultra-high vacuum glass cell. The glass cell consists of special glass material and exhibits an ultra-low birefringence Δn of a few times 10^-8 to highly suppress polarization disturbances originating from stress birefringence in vacuum windows. Furthermore, anti-reflection coatings avoid reflections on all window surfaces. The cell hosts the assembled vacuum-compatible objective, that exhibits a diffraction limited resolution of up to 453 nm and allows to optically resolve the spacing of the optical lattice. Fluorescence images of single trapped atoms are used to characterize the imaging system. The filling, orientation and geometry of the optical lattice is precisely reconstructed using positions of atoms that can be determined from fluorescence images. Furthermore, I present a scheme to realize state-dependent transport and discuss its robustness against experimental imperfections in a technical implementation. This transport scheme enable the realization of discrete-time quantum walks with neutral atoms in two dimensions. These quantum walks pave the way towards the simulation of artificial magnetic fields and topologically protected edge states.
Building a universal quantum computer is a central goal of emerging quantum technologies, which has the potential to revolutionize science and technology. Unfortunately, this future does not seem to be very close at hand. However, quantum computers built for a special purpose, i.e. quantum simulators , are currently developed in many leading laboratories. Many schemes for quantum simulation have been proposed and realized using, e.g., ultracold atoms in optical lattices, ultracold trapped ions, atoms in arrays of cavities, atoms/ions in arrays of traps, quantum dots, photonic networks, or superconducting circuits. The progress in experimental implementations is more than spectacular. Particularly interesting are those systems that simulate quantum matter evolving in the presence of gauge fields. In the quantum simulation framework, the generated (synthetic) gauge fields may be Abelian, in which case they are the direct analogues of the vector potentials commonly associated with magnetic fields. In condensed matter physics, strong magnetic fields lead to a plethora of fascinating phenomena, among which the most paradigmatic is perhaps the quantum Hall effect. The standard Hall effect consists in the appearance of a transverse current, when a longitudinal voltage difference is applied to a conducting sample. For quasi-two-dimensional semiconductors at low temperatures placed in very strong magnetic fields, the transverse conductivity, the ratio between the transverse current and the applied voltage, exhibits perfect and robust quantization, independent for instance of the material or of its geometry. Such an integer quantum Hall effect, is now understood as a deep consequence of underlying topological order. Although such a system is an insulator in the bulk, it supports topologically robust edge excitations which carry the Hall current. The robustness of these chiral excitations against backscattering explains the universality of the quantum Hall effect. Another interesting and related effect, which arises from the interplay between strong magnetic field and lattice potentials, is the famous Hofstadter butterfly: the energy spectrum of a single particle moving on a lattice and subjected to a strong magnetic field displays a beautiful fractal structure as a function of the magnetic flux penetrating each elementary plaquette of the lattice. When the effects of interparticle interactions become dominant, two-dimensional gases of electrons exhibit even more exotic behaviour leading to the fractional quantum Hall effect. In certain conditions such a strongly interacting electron gas may form a highly correlated state of matter, the prototypical example being the celebrated Laughlin quantum liquid. Even more fascinating is the behaviour of bulk excitations (quasi-hole and quasi-particles): they are neither fermionic nor bosonic, but rather behave as anyons with fractional statistics intermediate between the two. Moreover, for some specific filling factors (ratio between the electronic density and the flux density), these anyons are proven to have an internal structure (several components) and non-Abelian braiding properties. Many of the above statements concern theoretical predictions—they have never been observed in condensed matter systems. For instance, the fractional values of the Hall conductance is seen as a direct consequence of the fractional statistics, but to date direct observation of anyons has not been possible in two-dimensional semiconductors. Realizing these predictions in experiments with atoms, ions, photons etc, which potentially allow the experimentalist to perform measurements complementary to those made in condensed matter systems, is thus highly desirable! Non-Abelian gauge fields couple the motional states of the particles to their internal degrees of freedom (such as hyperfine states for atoms or ions, electronic spins for electrons, etc). In this sense external non-Abelian fields extend the concept of spin–orbit coupling (Rashba and Dresselhaus couplings), familiar from AMO and condensed matter physics. They lead to yet another variety of fascinating phenomena such as the quantum spin Hall effect, three-dimensional topological insulators, topological superconductors and superfluids of various kinds. One also expects here the appearance of excitations in a form of topological edge states that can support robust transport, or entangled Majorana fermions in the case of topological superconductors or superfluids. Again, while many kinds of topological insulators have been realized in condensed matter systems, a controlled way of creating them in AMO systems and studying quantum phase transitions between various kinds of them is obviously very appealing and challenging. The various systems listed so far correspond to static gauge fields, which are externally imposed by the experimentalists. Even more fascinating is the possibility of generating synthetically dynamical gauge fields, i.e. gauge fields that evolve in time according to an interacting gauge theory, e.g., a full lattice gauge theory (LGT). These dynamical gauge fields can also couple to matter fields, allowing the quantum simulation of such complex systems (notoriously hard to simulate using 'traditional' computers), which are particularly relevant for modern high-energy physics. So far, most of the theoretical proposals concern the simulation of Abelian gauge theories, however, several groups have recently proposed extensions to the non-Abelian scenarios. The scope of the present focused issue of Journal of Physics B is to cover all of these developments, with particular emphasis on the non-Abelian gauge fields. The 14 papers in this issue include contributions from the leading theory groups working in this field; we believe that this collection will provide the reference set for quantum simulations of gauge fields. Although the special issue contains exclusively theoretical proposals and studies, it should be stressed that the progress in experimental studies of artificial Abelian and non-Abelian gauge fields in recent years has been simply spectacular. Multiple leading groups are working on this subject and have already obtained a lot of seminal results. The papers in the special issue are ordered according to the date of acceptance. The issue opens with a review article by Zhou et al [1] on unconventional states of bosons with synthetic spin–orbit coupling. Next, the paper by Maldonado-Mundo et al [2] studies ultracold Fermi gases with artificial Rashba spin–orbit coupling in a 2D gas. Anderson and Charles [3], in contrast, discuss a three-dimensional spin–orbit coupling in a trap. Orth et al [4] investigate correlated topological phases and exotic magnetism with ultracold fermions, again in the presence of artificial gauge fields. The paper of Nascimbène [5] does not address the synthetic gauge fields directly, but describes an experimental proposal for realizing one-dimensional topological superfluids with ultracold atomic gases; obviously, this problem is well situated in the general and growing field of topological superfluids, in particular those realized in the presence of non-Abelian gauge fields/spin–orbit coupling. Graß et al [6] consider in their paper fractional quantum Hall states of a Bose gas with spin–orbit coupling induced by a laser. Particular attention is drawn here to the possibility of realizing states with non-Abelian anyonic excitations. Zheng et al [7] study properties of Bose gases with Raman-induced spin–orbit coupling. Kiffner et al [8] in their paper touch on another kind of system, namely ultracold Rydberg atoms. In particular they study the generation of Abelian and non-Abelian gauge fields in dipole–dipole interacting Rydberg atoms. The behaviour of fermions in synthetic non-Abelian gauge potentials is discussed by Shenoy and Vyasanakere [9]. The paper starts with the study of Rashbon condensates (i.e. Bose condensates in the presence of Rashba coupling) and also introduces novel kinds of exotic Hamiltonians. Goldman et al [10] propose a concrete setup for realizing arbitrary non-Abelian gauge potentials in optical square lattices; they discuss how such synthetic gauge fields can be exploited to generate Chern insulators. Zygelman [11], similarly as Kiffner et al [8], discusses in his paper non-Abelian gauge fields in Rydberg systems. Marchukov et al [12] return to the subject of spin–orbit coupling, and investigate spectral gaps of spin–orbit coupled particles in the realistic situations of deformed traps. The last two papers, in contrast, are devoted to different subjects. Edmonds et al [13] consider a 'dynamical' density-dependent gauge potential, and study the Josephson effect in a Bose–Einstein condensate subject to such a potential. Last, but not least, Mazzucchi et al [14] study the properties of semimetal-superfluid quantum phase transitions in 3D lattices with Dirac points. References [1] Zhou X, Li Y, Cai Z and Wu C 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134001 [2] Maldonado-Mundo D, Öhberg P and Valiente M 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134002 [3] Anderson B M and Clark C W 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134003 [4] Orth P P, Cocks D, Rachel S, Buchhold M, Le Hur K and Hofstetter W 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134004 [5] Nascimbène S 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134005 [6] Graß T, Juliá-Díaz B, Burrello M and Lewenstein M 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134006 [7] Zheng W, Yu Z-Q, Cui X and Zhai H 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134007 [8] Kiffner M, Li W and Jaksch D 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134008 [9] Shenoy V B and Vyasanakere J P 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134009 [10] Goldman N, Gerbier F and Lewenstein M 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134010 [11] Zygelman B 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134011 [12] Marchukov O V, Volosniev A G, Fedorov D V, Jensen A S and Zinner N T 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134012 [13] Edmonds M J, Valiente M and Öhberg P 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134013 [14] Mazzucchi G, Lepori L and Trombettoni A 2013 J. Phys. B: At. Mol. Opt. Phys. 46 134014
One-dimensional discrete-time quantum walks show a rich spectrum of topological phases that have so far been exclusively analyzed using the Floquet operator in momentum space. In this work, we introduce an alternative approach to topology which is based on the scattering matrix of a quantum walk, adapting concepts from time-independent systems. For quantum walks with gaps in the quasienergy spectrum at 0 and $\ensuremath{\pi}$, we find three different types of topological invariants, which apply dependent on the symmetries of the system. These determine the number of protected boundary states at an interface between two quantum-walk regions. Quantum walks with an unequal number of leftward and rightward shifts per cycle are characterized by the number of perfectly transmitting unidirectional modes they support, which is equal to their nontrivial quasienergy winding. Our classification provides a unified framework that includes all known types of topology in one-dimensional discrete-time quantum walks and is very well suited for the analysis of finite-size and disorder effects. We provide a simple scheme to directly measure the topological invariants in an optical quantum-walk experiment.
The present chapter discusses methods by which topological Bloch bands can be prepared in cold-atom setups. Focusing on the case of Chern bands for two-dimensional systems, we describe how topological properties can be triggered by driving atomic gases, either by dressing internal levels with light or through time-periodic modulations. We illustrate these methods with concrete examples, and we discuss recent experiments where geometrical and topological band properties have been identified. Introduction Ultracold atoms constitute a promising physical platform for the preparation and exploration of novel states of matter [1, 2, 3]. In particular, the engineering of topological band structures with cold-atom systems, together with the capability of tuning interactions between the particles, opens an interesting route toward the realization of intriguing strongly correlated states with topological features, such as fractional topological insulators and quantum Hall liquids [4]. This chapter is dedicated to the preparation and the detection of topological band structures characterized by nonzero Chern numbers [5]. Such Chern bands , which constitute the building blocks for realizing (fractional) Chern insulators [6, 7], arise in two-dimensional (2D) systems presenting time-reversal-symmetry (TRS) breaking effects. For instance, nontrivial Chern bands naturally appear in the Harper-Hofstadter model [8], a lattice penetrated by a uniform flux, where they generalize the (nondispersive) Landau levels to the lattice framework. Additionally, Chern bands also appear in staggered flux configurations, such as in Haldane's honeycomb-lattice model [9] or in lattice systems combining Rashba spin-orbit coupling and Zeeman (exchange) fields. The atoms being charge neutral, “magnetic” fluxes cannot be simply produced by subjecting optical lattices to “real” magnetic fields. It is the aim of this chapter to review several schemes that have been recently implemented in laboratories with the goal of realizing synthetic magnetic fields leading to Chern bands for cold atoms. Our presentation is complementary to that of Chapter 15, as we focus on synthetic magnetic fields in lattice-based systems for which the flux density can be made very large (with magnetic length comparable to interparticle spacing). The chapter is structured as follows: Section 14.2 describes how the Chern number is related to physical observables defined in a lattice framework. In particular, it clarifies the link between recent Chern-number measurements performed in cold bosonic gases and the more conventional (electronic) quantum Hall effect.