Light-induced gauge fields for ultracold atoms

SCOPUS: ed.j

In this article we present a pedagogical discussion of some of the optomechanical properties of a high finesse cavity loaded with ultracold atoms in laser induced synthetic gauge fields of different types. Essentially, the subject matter of this article is an amalgam of two sub-fields of atomic molecular and optical (AMO) physics namely, the cavity optomechanics with ultracold atoms and ultracold atoms in synthetic gauge field. After providing a brief introduction to either of these fields we shall show how and what properties of these trapped ultracold atoms can be studied by looking at the cavity (optomechanical or transmission) spectrum. In presence of abelian synthetic gauge field we discuss the cold-atom analogue of Shubnikov de Haas oscillation and its detection through cavity spectrum. Then, in the presence of a non-abelian synthetic gauge field (spin-orbit coupling), we see when the electromagnetic field inside the cavity is quantized, it provides a quantum optical lattice for the atoms, leading to the formation of different quantum magnetic phases. We also discuss how these phases can be explored by studying the cavity transmission spectrum.

Building a universal quantum computer is a central goal of emerging quantum technologies and it is expected to revolutionize science and technology. Unfortunately, this future does not seem very close, however, quantum computers built for a special purpose, i.e., quantum simulators, are currently being developed in many leading laboratories. Numerous 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 or superconducting circuits. The progress in experimental implementations is more than spectacular.Particularly interesting are those systems that simulate quantum matter evolving in artificial, or synthetic, Abelian or even non-Abelian gauge fields. Abelian gauge fields are analogues to the standard magnetic field and lead to fascinating effects such as the integer or fractional quantum Hall effects (IQHE, FQHE) and vortex lattices. 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, which is familiar from AMO and condensed matter physics. They lead to yet another variety of fascinating novel phenomena such as the quantum spin Hall effect (QSHE), 3D topological insulators, topological superconductors and superfluids of various kinds.Even more fascinating is the possibility of generating synthetic gauge fields that are dynamical, i.e., that evolve in time according to the corresponding 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 there are only theoretical proposals for simulating Abelian LGTs, but many groups are working on extensions to the non-Abelian scenarios.The scope of this special issue of Journal of Physics B: Atomic, Molecular and Optical Physics is on all of these developments, with particular emphasis on the non-Abelian case. We invite the leading theory and experimental groups to contribute to this very special issue of the journal in order to provide a reference collection for quantum simulations of gauge fields.To summarize the key features should be Synthetic spin–orbit coupling and the physics of topological insulating phases Strongly correlated phases in non-Abelian gauge potentials Dynamical non-Abelian gauge fields and the simulation of lattice gauge theories Spin–orbit coupled BEC and vortex physics Simulators of Abelian LGTs Simulators of non-Abelian LGTsYou are invited to submit your article by 15 December 2012. Expected publication: Summer 2013.Corrections were made to this article on 7 November 2012. A change was made to the affiliations.

Synthetic gauge fields for ultracold neutral atoms—engineered using the interaction between laser fields and the atoms’ internal ‘spin’ degrees of freedom—provide promising techniques for generating the large (synthetic) magnetic fields required to reach the fractional quantum Hall (FQH) limit in quantum gases, bosonic or fermionic alike. Because neutral atoms can move in a nearly disorder-free environment and they have extremely simple contact interactions, the resulting FQH states would be revealed in their most essential form. Moreover, bosonic FQH states represent a new frontier and have never been seen in any setting. Going beyond electromagnetism's conventional scalar gauge field, it is possible to create more general non-Abelian gauge potentials. When these are spatially uniform, they are equivalent to spin–orbit coupling familiar in material systems, and can lead to cold atom analogs of topological insulators and topological superconductors. In this tutorial, we introduce basic concepts underlying these gauge fields, making connections to the Aharonov–Bohm phase and geometric phase. We focus on the system of neutral atoms ‘dressed’ by multiple laser beams, where the eigenstates of the resulting Hamiltonian are known as dressed states. Synthetic gauge potentials arise from the unitary transformation required to express these dressed states in terms of the laser-free eigenstates. We discuss stability of laser-dressed atoms corresponding to the adiabatic condition and the probability of non-adiabatic transitions. Adopting both the semiclassical and quantum mechanical approaches, we demonstrate they agree in the suitable limit. We also analyze using both the conventional adiabatic picture and exact picture, where the kinetic energy is neglected in the former and retained in the latter picture.

On increasing the coupling strength ($\lambda$) of a non-Abelian gauge field that induces a generalized Rashba spin-orbit interaction, the topology of the Fermi surface of a homogeneous gas of noninteracting fermions of density $\rho \sim \kf^3$ undergoes a change at a critical value, $\lambda_T \approx \kf$ [Phys. Rev. B {\bf 84}, 014512 (2011)]. In this paper we analyze how this phenomenon affects the size and shape of a cloud of spin-$\half$ fermions trapped in a harmonic potential such as those used in cold atom experiments. We develop an adiabatic formulation, including the concomitant Pancharatnam-Berry phase effects, for the one particle states in the presence of a trapping potential and the gauge field, obtaining approximate analytical formulae for the energy levels for some high symmetry gauge field configurations of interest. An analysis based on the local density approximation reveals that, for a given number of particles, the cloud shrinks in a {\em characteristic fashion with increasing $\lambda$}. For an isotropic harmonic trap, the local density approximation predicts a spherical cloud for all gauge field configurations, which are anisotropic in general. We show, via a calculation of the cloud shape using exact eigenstates, that for certain gauge field configurations there is systematic and observable anisotropy in the cloud shape that increases with increasing gauge coupling $\lambda$. These results should be useful in the design of cold atom experiments with fermions in non-Abelian gauge fields. An important spin-off of our adiabatic formulation is that it reveals exciting possibilities for the cold-atom realization of interesting condensed matter Hamiltonians (eg. quantum hall spherical geometry) by using a non-Abelian gauge field in conjunction with another potential.

Gauge fields play a prominent role in modern physics from electromagnetic theory to standard model of particle physics. These are cutting edge mathematical tools to understand fundamental forces and interaction between sub-atomic particles. Though gauge fields are considered natural tools of study in high-energy physics, they can be extended to low-energy condensed matter physics to study the behaviour of quantum particles in special environments like electrons in a magnetic field. Apart from naturally occurring gauge fields, controllable artificial gauge fields can be used to tailor physical effects on quantum systems like neutral atoms, qubits, photons, anions etc. This thesis reports the study of laser-induced artificial non-Abelian gauge fields interacting with an ultracold atomic wave packet in free space. In sharp contrast to the Abelian gauge fields, spatially uniform non-Abelian gauge fields can induce non-inertial motion of a particle. This motion is locked to the pseudo-spin state of the system and induces spin-orbit coupling. These non-Abelian gauge fields are experimentally realized by adiabatic evolution of the state of the atoms in the degenerate subspace of the coupling Hamiltonians. We produce SU(2) and SU(3) non-Abelian gauge fields using doubly degenerate dark state subspace of tripod and triply degenerate dark state subspace of double-tripod atom-light coupling schemes, respectively. Using the SU(2) gauge field in one dimension, we demonstrate the function of an atomtronic Datta-Das transistor (DDT) where spin polarized atoms enter the gate region and exit with their spin orientation controlled by the gate parameter analogous to the gate voltage in conventional field effect transistor. We demonstrate the geometric nature of the spin rotation in our DDT by showing insensitivity of the result to the input velocity and velocity dispersion of the atomic cloud. We have also implemented a Ramsey interferometric sequence to extract the phase of the output state of the DDT showing that the spin rotation by the DDT is coherent. In two dimensions, the dynamics of the wave packet in SU(2) gauge field show Zitterbewegung oscillations revealing spin Hall effect characteristics of the system. Whereas the dynamics of the wave packet in SU(3) gauge field show oscillations with multiple frequencies emulating color-orbit coupling. These frequencies correspond to the energy separation between the eigenstates of the Hamiltonian. Using the SU(3) Hamiltonian, we show two paths of reaching the same final state from the initial state demonstrating the existence of additional ladder operators in the SU(3) system in addition to the well-known raising and lowering operators of angular momentum represented by SU(2) symmetry. With this study, we realize a system that can be used to study the rich physics of SU(3) Hamiltonians on table top experiments.

Floquet systems like quantum walks can exhibit anomalous topological boundary modes that have no stationary counterpart. Although the topological properties of synthetic gauge fields have been explored in quantum walks, these studies have so far focused on Abelian gauge fields. We theoretically introduce synthetic non-Abelian gauge fields for topological quantum walks and study their topological consequences. The photonic mesh lattice configuration is generalized with polarization multiplexing to achieve a four-dimensional Hilbert space, based on which we provide photonic building blocks for realizing various quantum walks in non-Abelian gauge fields. It is found that SU(2) gauge fields can lead to Peierls substitution in both momenta and quasienergy. In one and two dimensions, we describe detailed photonic setups to realize topological quantum walk protocols whose Floquet winding numbers and Rudner–Lindner–Berg–Levin invariants can be effectively controlled by the gauge fields. Finally, we show how non-Abelian gauge fields facilitate convenient simulation of entanglement in conjunction with polarization-dependent and spatial-mode-dependent coin operations. Our results shed light on the study of synthetic non-Abelian gauge fields in photonic Floquet systems and hold implications for time-multiplexed optical networks.

The use of coherent optical dressing of atomic levels allows the coupling of ultracold atoms to effective gauge fields. These can be used to generate effective magnetic fields, and have the potential to generate non-Abelian gauge fields. We consider a model of a gas of bosonic atoms coupled to a gauge field with U(2) symmetry, and with constant effective magnetic field. We include the effects of weak contact interactions by applying Gross-Pitaevskii mean-field theory. We study the effects of a U(2) non-Abelian gauge field on the vortex lattice phase induced by a uniform effective magnetic field, generated by an Abelian gauge field or, equivalently, by rotation of the gas. We show that, with increasing non-Abelian gauge field, the nature of the groundstate changes dramatically, with structural changes of the vortex lattice. We show that the effect of the non-Abelian gauge field is equivalent to the introduction of effective interactions with non-zero range. We also comment on the consequences of the non-Abelian gauge field for strongly correlated fractional quantum Hall states.

This paper describes CosmoLattice, a modern package for lattice simulations of the dynamics of interacting scalar and gauge fields in an expanding universe. CosmoLattice incorporates a series of features that makes it very versatile and powerful: i) it is written in C++ fully exploiting the object oriented programming paradigm, with a modular structure and a clear separation between the physics and the technical details, ii) it is MPI-based and uses a discrete Fourier transform parallelized in multiple spatial dimensions, which makes it specially appropriate for probing scenarios with well-separated scales, running very high resolution simulations, or simply very long ones, iii) it introduces its own symbolic language, defining field variables and operations over them, so that one can introduce differential equations and operators in a manner as close as possible to the continuum, iv) it includes a library of numerical algorithms, ranging from O(δt2) to O(δt10) methods, suitable for simulating global and gauge theories in an expanding grid, including the case of ‘self-consistent’ expansion sourced by the fields themselves. Relevant observables are provided for each algorithm (e.g. energy densities, field spectra, lattice snapshots) and we note that, remarkably, all our algorithms for gauge theories (Abelian or non-Abelian) always respect the Gauss constraint to machine precision. Program summaryProgram Title::CosmoLatticeCPC Library link to program files:https://doi.org/10.17632/44vr5xssc6.1Developer's repository link:http://github.com/cosmolattice/cosmolatticeLicensing provisions: MITProgramming language: C++, MPINature of problem: The phenomenology of high energy physics in the early universe is typically characterized by non-linear dynamics, which cannot be captured accurately with analytical techniques. In order to fully understand the non-linearities developed in a given scenario, one needs to carry out lattice simulations. A number of public packages for lattice simulations have appeared over the years, but most of them are only capable of simulating scalar fields. However, realistic models of particle physics do contain other kind of field species, such as (Abelian or non-Abelian) gauge fields, whose non-linear dynamics can also play a relevant role in the early universe. Tensor modes representing gravitational waves are also naturally expected in many scenarios.Solution method:CosmoLattice represents a modern code for lattice simulations of scalar-gauge field theories in an expanding universe. It allows for the simulation of the evolution of interacting (singlet) scalar fields, charged scalar fields under U(1) and/or SU(2) gauge groups, and the corresponding associated Abelian and/or non-Abelian gauge fields. From version 1.1 onward, CosmoLattice also allows to simulate the production of gravitational waves. Simulations can be done either in a flat space-time background, or in a homogeneous and isotropic (spatially flat) expanding FLRW background. CosmoLattice provides symplectic integrators, with accuracy ranging from O(δt2) up to O(δt10), to simulate the non-linear dynamics of the appropriate fields in comoving three-dimensional lattices. The code is parallelized with MPI, and uses a discrete Fourier Transform parallelized in multiple spatial dimensions, which makes it a very powerful code for probing physical problems with well-separated scales. Moreover, the code has been designed as a ‘platform’ to implement any system of dynamical equations suitable for discretization on a lattice.

We show that non-Abelian lattice gauge fields can be simulated with a single-component ultra-cold atomic gas in an optical-lattice potential. An optical lattice can be viewed as a Bravais lattice with a N-point basis. An atom located at different points of the basis can be considered as a particle in different internal states. The appropriate engineering of tunneling amplitudes of atoms in an optical lattice allows one to realize U(N) gauge potentials and control a mass of particles that experience such non-Abelian gauge fields. We provide and analyze a concrete example of an optical-lattice configuration that allows for simulation of a static U(2) gauge model with a constant Wilson loop and an adjustable mass of particles. In particular, we observe that the non-zero mass creates large conductive gaps in the energy spectrum, which could be important in the experimental detection of the transverse Hall conductivity.

Recent advances in the generation of synthetic gauge fields in cold atomic systems have stimulated interest in the physics of interacting bosons and fermions in them. In this paper, we discuss interacting two-component fermionic systems in uniform non-Abelian gauge fields that produce a spin–orbit interaction and uniform spin potentials. Two classes of gauge fields discussed include those that produce a Rashba spin–orbit interaction and the type of gauge fields (SM gauge fields) obtained in experiments by the Shanxi and MIT groups. For high symmetry Rashba gauge fields, a two-particle bound state exists even for a vanishingly small attractive interaction described by a scattering length. Upon increasing the strength of a Rashba gauge field, a finite density of weakly interacting fermions undergoes a crossover from a BCS like ground state to a BEC state of a new kind of boson called the rashbon whose properties are determined solely by the gauge field and not by the interaction between the fermions. The rashbon Bose–Einstein condensate (RBEC) is a quite intriguing state with the rashbon–rashbon interactions being independent of the fermion–fermion interactions (scattering length). Furthermore, we show that the RBEC has a transition temperature of the order of the Fermi temperature, suggesting routes to enhance the transition temperatures of weakly interacting superfluids by tuning the spin–orbit coupling. For the SM gauge fields, we show that in a regime of parameters, a pair of particles with finite centre-of-mass momentum is the most strongly bound. In other regimes of centre-of-mass momenta, there is no two-body bound state, but a resonance like feature appears in the scattering continuum. In the many-body setting, this results in flow enhanced pairing. Also, strongly interacting normal states utilizing the scattering resonance can be created opening the possibility of studying properties of helical Fermi liquids. This paper contains a general discussion of the physics of Feshbach resonance in a non-Abelian gauge field, where several novel features such as centre-of-mass-momentum-dependent effective interactions are shown. It is also shown that a uniform non-Abelian gauge field in conjunction with a spatial potential can be used to generate novel Hamiltonians; we discuss an explicit example of the generation of a monopole Hamiltonian.

Non-Abelian gauge fields give rise to nontrivial topological physics. Here we develop a scheme to create an arbitrary SU(2) lattice gauge field for photons in the synthetic frequency dimension using an array of dynamically modulated ring resonators. The photon polarization is taken as the spin basis to implement the matrix-valued gauge fields. Using a non-Abelian generalization of the Harper-Hofstadter Hamiltonian as a specific example, we show that the measurement of the steady-state photon amplitudes inside the resonators can reveal the band structures of the Hamiltonian, which show signatures of the underlying non-Abelian gauge field. These results provide opportunities to explore novel topological phenomena associated with non-Abelian lattice gauge fields in photonic systems.

We study the dynamics of the universe with a scalar field and an SU(2) non-Abelian Gauge (Yang-Mills) field. The scalar field has an exponential potential and the Yang-Mills field is coupled to the scalar field with an exponential function of the scalar field. We find that the magnetic component of the Yang-Mills field assists acceleration of the cosmic expansion and a power-law inflation becomes possible even if the scalar field potential is steep, which may be expected from some compactification of higher-dimensional unified theories of fundamental interactions. This power-law inflationary solution is a stable attractor in a certain range of coupling parameters. Unlike the case with multiple Abelian gauge fields, the power-law inflationary solution with the dominant electric component is unstable because of the existence of non-linear coupling of the Yang-Mills field. We also analyze the dynamics for the non-inflationary regime, and find several attractor solutions.

We investigate the ground state of interacting spin-$\half$ fermions (3D) at a finite density ($\rho \sim \kf^3$) in the presence of a uniform non-Abelian gauge field. The gauge field configuration (GFC) described by a vector $\blam \equiv (\lambda_x, \lambda_y, \lambda_z)$, whose magnitude $\lambda$ determines the gauge coupling strength, generates a generalized Rashba spin-orbit interaction. For a weak attractive interaction in the singlet channel described by a small negative scattering length $(\kf |\as| \lesssim 1)$, the ground state in the absence of the gauge field ($\lambda=0$) is a BCS (Bardeen-Cooper-Schrieffer) superfluid with large overlapping pairs. With increasing gauge coupling strength, a non-Abelian gauge field engenders a crossover of this BCS ground state to a BEC (Bose-Einstein condensate) ground state of bosons even with a weak attractive interaction that fails to produce a two-body bound state in free vacuum. For large gauge couplings $(\lambda/\kf \gg 1)$, the BEC attained is a condensate of bosons whose properties are solely determined by the gauge field (and not by the scattering length so long as it is non-zero) -- we call these bosons "rashbons". In the absence of interactions ($\as = 0^-$), the shape of the Fermi surface of the system undergoes a topological transition at a critical gauge coupling $\lambda_T$. For high symmetry gauge field configurations we show that the crossover from the BCS superfluid to the rashbon BEC occurs in the regime of $\lambda$ near $\lambda_T$. In the context of cold atomic systems, this work makes an interesting suggestion of obtaining BCS-BEC crossover through a route other than tuning the interaction between the fermions.

We apply the general concept of non-Abelian gauge fields for the description of magnetic soliton excitations. We show that the component of the gauge field potential along the soliton local magnetization (Abelian part of the gauge potential) determines the dynamics of spin fluctuations over the soliton background in a ferromagnet. The assumption that the gauge field is a pure gauge allows calculating the gauge field components and finding simple expressions for the emergent electromagnet fields related to the soliton motion. The gauge field results in a soliton-magnon interaction leading to renormalization of the soliton and magnon dynamics. The presented approach allows reaching a deeper understanding of the relationship between field theory and condensed-matter magnetism.