We present a scheme that produces a strong U(1)-like gauge field on cold atoms confined in a two-dimensional square optical lattice. Our proposal relies on two essential features, a long-lived metastable excited state that exists for alkaline-earth or ytterbium atoms and an optical superlattice. As in the proposal by Jaksch and Zoller (2003 New J. Phys. 5 56), laser-assisted tunneling between adjacent sites creates an effective magnetic field. In the tight-binding approximation, atomic motion is described by the Harper Hamiltonian, with a flux across each lattice plaquette that can realistically take any value between 0 and π. We show how one can take advantage of the superlattice to ensure that each plaquette acquires the same phase, thus simulating a uniform magnetic field. We discuss the observable consequences of the artificial gauge field on non-interacting bosonic and fermionic gases. We also outline how the scheme can be generalized to non-Abelian gauge fields.
We present a scheme that produces a strong U(1)-like gauge field on cold atoms confined in a two-dimensional square optical lattice. Our proposal relies on two essential features, a long-lived metastable excited state that exists for alkaline-earth or ytterbium atoms and an optical superlattice. As in the proposal by Jaksch and Zoller (2003 New J. Phys. 5 56), laser-assisted tunneling between adjacent sites creates an effective magnetic field. In the tight-binding approximation, atomic motion is described by the Harper Hamiltonian, with a flux across each lattice plaquette that can realistically take any value between 0 and π. We show how one can take advantage of the superlattice to ensure that each plaquette acquires the same phase, thus simulating a uniform magnetic field. We discuss the observable consequences of the artificial gauge field on non-interacting bosonic and fermionic gases. We also outline how the scheme can be generalized to non-Abelian gauge fields.
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.
Ultracold gases are remarkably versatile and controllable systems with untapped potential to emulate a variety of quantum phenomena [1]. Atoms are, however, electrically neutral and therefore it is highly desirable to make them behave as charged particles. This capability has recently been demonstrated in a series of recent experiments [2–4]. Although many challenges remain ahead, atoms exposed to synthetic gauge field offer great opportunities for the realization of new classes of quantum materials with custom-designed properties, and with unforeseen applications. The first implementation of an effective magnetic field was accomplished by rotating an atomic cloud [5]. In the rotating frame of reference, an atom experiences the Coriolis force that emulates the Lorentz force experienced by a charged particle in a uniform magnetic field. Large vortex lattices have been created in this way, however, so far the strength of the effective magnetic field achieved by rotation is too weak to lead to new physics. An alternative approach invokes the vector potential, A, that induces a geometric phase as a charged particle moves around a closed loop. This notion of geometric phase can be adapted to a neutral atom that slowly moves in the presence of laser fields. If the lasers are chosen to ‘dress’ the internal atomic levels, so that the dressed eigenstates become position dependent, the center of mass adiabatically evolves according the operator p-A, instead of just p = −iℏ∇. The atom thus effectively moves as a charged particle in a magnetic field B = ∇ × A (see Fig. 1a). Exploiting this idea series of experiments have implemented artificial magnetic and electric fields [2–4]. The idea has also been implemented in the presence of a superimposed periodic optical lattice potential (see Fig. 1b). In this case, the length scale imposed by the lattice competes with the one associated with the magnetic field and can lead to frustration, manifested as an energy spectrum with fractal characteristics [2,3]. Light induced synthetic gauge fields: (a) A gas of neural atoms (green oval) can behave as a gas of charged particles when it is illuminated by a pair of laser beams (red arrows) that imparts momentum to the atoms and couples their internal (.|↑.〉, .|↓.〉) and motional degrees of freedom. The Raman beams that induce transitions between internal states also transfer two units of their linear momentum 2kL. This leads to the formation of dressed states (superpositions of states with different internal levels and center of mass momenta) which experience a modified kinetic energy, with an effective vector potential. If the effective vector potential has nonzero curl, the atoms move as charged particles in a magnetic field. (b) In the presence of an optical lattice potential, Raman beams can also induce an effective vector potential. In this case, the lattice is engineered so that along one direction atoms can tunnel between adjacent lattice sites (Jlight) only by absorbing and emitting photons from the beams. The imparted momentum from the lasers produces path-dependent phases that resemble the magnetic flux accumulated by a charged particle as it moves in a closed loop. The flux accumulated by the atoms when they move around a single unit cell, ϕ, can be significant (equivalent to the one generated by a real magnetic field of thousands tesla in an electronic system) and induce a fractal energy spectrum. Generalizations to emulate non-Abelian gauge potentials, associated in many cases to spin-orbit coupling can emerge when the geometric phase is replaced by a matrix acting on the internal Hilbert space [2–5]. This happens if there is a manifold of degenerate dressed states, such that even for arbitrarily slow motion, after traveling around a contour the particle ends up in a different internal state. Although various proposals for generating non-Abelian potentials exist, they remain to be implemented. On the other hand, Abelian spin-orbit coupling, arising from the momentum kicks imparted on the atom by the laser photons that drive internal transitions, has been realized in several laboratories using different laser configurations in alkali-metal (one outer electron) and alkaline-earth (two outer electrons) atoms [2–5]. There is a great vista ahead. For example, the inclusion of collisions or dipolar interactions will open a window to engineer fractional Hall liquids, unconventional superfluids and interacting topological phases. The creation of dynamical gauge fields important in many areas of physics such as quantum chromodynamics is also an open direction. The time is just ripe for investigating rich and unexplored physics with synthetic gauge fields for ultracold matter.
We propose an experimental scheme to realize the valley-dependent gauge fields for ultracold fermionic atoms trapped in a state-dependent square optical lattice. Our scheme relies on two sets of Raman laser beams to engineer the hopping between adjacent sites populated by two-component fermionic atoms. One set of Raman beams is used to realize a staggered $\ensuremath{\pi}$-flux lattice, where low-energy atoms near two inequivalent Dirac points should be described by the Dirac equation for spin-21 particles. Another set of laser beams with proper Rabi frequencies is added to further modulate the atomic hopping parameters. The hopping modulation will give rise to effective gauge potentials with opposite signs near the two valleys, mimicking the interesting strain-induced pseudogauge fields in graphene. The proposed valley-dependent gauge fields are tunable and provide an alternative route to realize an uncommon type of quantum Hall effects and atomic devices.
We propose a general strategy for generating synthetic magnetic fields in complex lattices with non-trivial connectivity based on light–matter coupling in cold atomic gases. Our approach starts from an underlying optical flux lattice in which a synthetic magnetic field is generated by coupling several internal states. Starting from a high-symmetry optical flux lattice, we superpose a scalar potential with a super- or sublattice period in order to eliminate links between the original lattice sites. As an alternative to changing connectivity, the approach can also be used to create or remove lattice sites from the underlying parent lattice. To demonstrate our concept, we consider the dice lattice geometry as an explicit example, and construct a dice lattice with a flux density of half a flux quantum per plaquette, providing a pathway to flat bands with a large band gap. While the intuition for our proposal stems from the analysis of deep optical lattices, we demonstrate that the approach is robust even for shallow optical flux lattices far from the tight-binding limit. We also provide an alternative experimental proposal to realise a synthetic gauge field in a fully frustrated dice lattice based on laser-induced hoppings along individual bonds of the lattice, again involving a superlattice potential. In this approach, atoms with a long-lived excited state are trapped using an 'anti-magic' wavelength of light, allowing the desired complex hopping elements to be induced in a specific laser coupling scheme for the dice lattice geometry. We conclude by comparing the complexity of these alternative approaches, and advocate that complex optical flux lattices provide the more elegant and easily generalisable strategy.
We use Raman-assisted tunneling in an optical superlattice to generate large tunable effective magnetic fields for ultracold atoms. When hopping in the lattice, the accumulated phase shift by an atom is equivalent to the Aharonov-Bohm phase of a charged particle exposed to a staggered magnetic field of large magnitude, on the order of one flux quantum per plaquette. We study the ground state of this system and observe that the frustration induced by the magnetic field can lead to a degenerate ground state for non-interacting particles. We provide a measurement of the local phase acquired from Raman-induced tunneling, demonstrating time-reversal symmetry breaking of the underlying Hamiltonian. Furthermore, the quantum cyclotron orbit of single atoms in the lattice exposed to the magnetic field is directly revealed.
We introduce a method for quantum simulation of U(1) lattice gauge theories coupled to matter, utilizing alkaline-earth(-like) atoms in state-dependent optical lattices.The proposal enables the study of both gauge and fermionic matter fields without integrating out one of them in one and two dimensions.We focus on a realistic and robust implementation that utilizes the long-lived metastable clock state available in alkaline-earth(-like) atomic species.Starting from an ab initio modeling of the experimental setting, we systematically carry out a derivation of the target U(1) gauge theory.This approach allows us to identify and address conceptual and practical challenges for the implementation of lattice gauge theories that-while pivotal for a successful implementation-have never been rigorously addressed in the literature: those include the specific engineering of lattice potentials to achieve the desired structure of Wannier functions and the subtleties involved in realizing the proper separation of energy scales to enable gauge-invariant dynamics.We discuss realistic experiments that can be carried out within such a platform using the fermionic isotope 173 Yb, addressing via simulations all key sources of imperfections, and provide concrete parameter estimates for relevant energy scales in both one-and two-dimensional settings.