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arXiv:1207.2657 · New Journal of Physics

Flux lattices reformulated

Gediminas Juzeliūnas, I B Spielman

Published Dec 12, 2012• 24 citations
arXivHTMLPDF

Abstract

We theoretically explore the optical flux lattices produced for ultra-cold atoms subject to laser fields where both the atom-light coupling and the effective detuning are spatially periodic. We analyze the geometric vector potential and the magnetic flux it generates, as well as the accompanying geometric scalar potential. We show how to understand the gauge-dependent Aharonov-Bohm singularities in the vector potential, and calculate the continuous magnetic flux through the elementary cell in terms of these singularities. The analysis is illustrated with a square optical flux lattice. We conclude with an explicit laser configuration yielding such a lattice using a set of five properly chosen beams with two counterpropagating pairs (one along the x axes and the other y axes), together with a single beam along the z axis. We show that this lattice is not phase-stable, and identify the one phase-difference that affects the magnetic flux. Thus armed with realistic laser setup, we directly compute the Chern number of the lowest Bloch band to identify the region where the non- zero magnetic flux produces a topologically non-trivial band structure.

DOI
10.1088/1367-2630/14/12/123022

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We theoretically explore the optical flux lattices produced for ultra-cold atoms subject to laser fields where both the atom-light coupling and the effective detuning are spatially periodic. We analyze the geometric vector potential and the magnetic flux it generates, as well as the accompanying geometric scalar potential. We show how to understand the gauge-dependent Aharonov-Bohm singularities in the vector potential, and calculate the continuous magnetic flux through the elementary cell in terms of these singularities. The analysis is illustrated with a square optical flux lattice. We conclude with an explicit laser configuration yielding such a lattice using a set of five properly chosen beams with two counterpropagating pairs (one along the x axes and the other y axes), together with a single beam along the z axis. We show that this lattice is not phase-stable, and identify the one phase-difference that affects the magnetic flux. Thus armed with realistic laser setup, we directly compute the Chern number of the lowest Bloch band to identify the region where the non- zero magnetic flux produces a topologically non-trivial band structure.
We explore the optical flux lattices produced for ultra-cold atoms in the radiation field when both the atom-light coupling and the detuning exhibit an oscillatory behavior. We analyze not only the magnetic flux but also the geometric vector potential generating the flux, as well as the accompanying geometric scalar potential. We show how to deal with the gauge-dependent singularities of the Aharonov-Bohm (AB) type appearing in the vector potentials for the optical flux lattices. We present a way to calculate the continuous magnetic flux through the elementary cell via the singularities of the vector potential inside the cell. The analysis is illustrated with a square optical flux lattice. We present a way of creating such a lattice using the Raman transitions induced by a set of properly chosen polarization-dependent standing waves propagating at a right angle and containing a time-phase difference.
We present a general framework for engineering two-dimensional subwavelength topological optical lattices using spatially dependent atomic dark states in a Λ -type configuration of the atom-light coupling. By properly designing the spatial profiles of the laser fields inducing coupling between the atomic internal states, we show how to generate subwavelength Kronig-Penney-like geometric scalar potential accompanied by narrow and strong patches of the synthetic magnetic field localized in the same areas as the scalar potential. These sharply peaked magnetic fluxes are compensated by a smooth background magnetic field of opposite sign, resulting in zero net flux per unit cell while still enabling topologically nontrivial band structures. Specifically, for sufficiently narrow peaks, their influence is minimum, and the behavior of the system in a remaining smooth background magnetic field resembles the Landau problem, allowing for the formation of nearly flat energy bands with unit Chern numbers. Numerical analysis confirms the existence of ideal Chern bands and the robustness of the topological phases against nonadiabatic effects and losses. This makes the scheme well suited for simulating quantum Hall systems and fractional Chern insulators in ultracold atomic gases, offering a platform for exploring strongly correlated topological phases with high tunability.
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.
We have constructed the geometric phases emerging from the non-trivial topology of a space-dependent magnetic field, interacting with the spin magnetic moment of a neutral particle. Our basic tool is the local unitary transformation which recasts the magnetic spin interaction under a diagonal form. Rewriting the kinetic term in the "rotated" frame requires the introduction of non-Abelian covariant derivatives, involving the gradients of the Euler angles which define the orientation of the local field. Within the rotated frame, we have built a perturbation scheme,assuming that the longitudinal non-Abelian field component dominates the transverse ones, to be evaluated to second-order. The geometry embedded in the longitudinal gauge vector field and its curl, the geometric magnetic field, is described by the associated Aharonov-Bohm phase. As an illustration, we study the physics of cold 171Yb atoms dressed by two sets of circularly polarized beams, forming square or triangular 2D optical lattices. The geometric field is computed explicitly from the Euler angles. The magnitude of 2nd-order corrections due to transverse fields can be reduced to the percent level by a choice of light intensity which keeps the dressed atom loss rate below 5 s^{-1}. An auxiliary optical lattice confines the atoms within 2D domains where the geometric field is pointing upward.
We have constructed the geometric phases emerging from the non-trivial topology of a space-dependent magnetic field, interacting with the spin magnetic moment of a neutral particle. Our basic tool is the local unitary transformation which recasts the magnetic spin interaction under a diagonal form. Rewriting the kinetic term in the "rotated" frame requires the introduction of non-Abelian covariant derivatives, involving the gradients of the Euler angles which define the orientation of the local field. Within the rotated frame, we have built a perturbation scheme,assuming that the longitudinal non-Abelian field component dominates the transverse ones, to be evaluated to second-order. The geometry embedded in the longitudinal gauge vector field and its curl, the geometric magnetic field, is described by the associated Aharonov-Bohm phase. As an illustration, we study the physics of cold ^{171}Yb atoms dressed by two sets of circularly polarized beams, forming square or triangular 2D optical lattices. The geometric field is computed explicitly from the Euler angles. The magnitude of 2nd-order corrections due to transverse fields can be reduced to the percent level by a choice of light intensity which keeps the dressed atom loss rate below 5 s^{-1}. An auxiliary optical lattice confines the atoms within 2D domains where the geometric field is pointing upward.
Viewed as an ideal quantum simulator, the optical lattice system with trapped ultracold atoms has been developed into a powerful tool to investigate unusual physical properties due to its considerable tunability. In this paper, we propose to construct a feasible two-dimensional optical lattice which can be continuously transformed from a square lattice to a Lieb lattice by tuning laser intensity. In these transformations, accompanied with the crossover between nonrelativistic particles and relativistic Dirac fermions, the topological and transport properties are varied under artificial gauge potentials. It is found that the fractal energy spectrum in an effective magnetic field splits and reassembles from a traditional Hofstadter butterfly to multiple ones with flat band, characterized by different Chern numbers. In a specific case of half a pseudomagnetic flux quantum, the spectrum has a transition from individual Dirac cones to double-layer Dirac cones, then to the Dirac cones with a flat band, the pseudospin of particles changes from 1/2 to 3/2, then to 1. In the structural evolution, a remarkable birefringent tunneling across a potential barrier is demonstrated. The adjustable lattice structure replenished with synthetic gauge fields for cold atoms offers more opportunities to observe intriguing physical properties.