Publisher’s Note: Realizing the Harper Hamiltonian with Laser-Assisted Tunneling in Optical Lattices [Phys. Rev. Lett.<b>111</b>, 185302 (2013)]

Miyake, Hirokazu; Siviloglou, Georgios A.; Kennedy, Colin J.; Burton, William Cody; Ketterle, Wolfgang

doi:10.1103/physrevlett.111.199903

Cited by 340 publications

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“…As in Refs. [13,14], tunneling in the x-direction of a twodimensional optical lattice is suppressed by an energy offset and then subsequently restored with a resonant Raman process with a momentum transfer, δ k = k xx + k yŷ . The z-direction consists of loosely confined tubes of condensate, so interactions are weak.…”

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confidence: 99%

“…For neutral ultracold atoms, new methods have been developed to create synthetic gauge fields. Forces analogous to the Lorentz force on electons are engineered either through the Coriolis force in rotating systems [7][8][9], by phase imprinting via photon recoil [10][11][12][13][14], or lattice shaking [15,16]. Much of the research with ultracold atoms has focused on the paradigmatic HarperHofstadter (HH) Hamiltonian, which describes particles in a crystal lattice subject to a strong homogeneous magnetic field [17][18][19].…”

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Observation of Bose–Einstein condensation in a strong synthetic magnetic field

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Extensions of Berry's phase and the quantum Hall effect have led to the discovery of new states of matter with topological properties. Traditionally, this has been achieved using gauge fields created by magnetic fields or spin orbit interactions which couple only to charged particles. For neutral ultracold atoms, synthetic magnetic fields have been created which are strong enough to realize the Harper-Hofstadter model. Despite many proposals and major experimental efforts, so far it has not been possible to prepare the ground state of this system. Here we report the observation of BoseEinstein condensation for the Harper-Hofstadter Hamiltonian with one-half flux quantum per lattice unit cell. The diffraction pattern of the superfluid state directly shows the momentum distribution on the wavefuction, which is gauge-dependent. It reveals both the reduced symmetry of the vector potential and the twofold degeneracy of the ground state. We explore an adiabatic many-body state preparation protocol via the Mott insulating phase and observe the superfluid ground state in a three-dimensional lattice with strong interactions.Topological states of matter are an active new frontier in physics. Topological properties at the single particle level are well understood; however, there are many open questions when strong interactions and correlations are introduced [1, 2] as in the ν = 5/2 state of the fractional quantum Hall effect [3] and in Majorana fermions [4][5][6]. For neutral ultracold atoms, new methods have been developed to create synthetic gauge fields. Forces analogous to the Lorentz force on electons are engineered either through the Coriolis force in rotating systems [7][8][9], by phase imprinting via photon recoil [10][11][12][13][14], or lattice shaking [15,16]. Much of the research with ultracold atoms has focused on the paradigmatic HarperHofstadter (HH) Hamiltonian, which describes particles in a crystal lattice subject to a strong homogeneous magnetic field [17][18][19]. For magnetic fluxes on the order of one flux quantum per lattice unit cell, the radius of the smallest possible cyclotron orbit and the lattice constant are comparable, and their competition gives rise to the celebrated fractal spectrum of Hofstadfer's butterfly whose sub-bands have non-zero Chern numbers [20] and Dirac points.So far, it has not been possible to observe the ground state of the HH Hamiltonian, which for bosonic atoms is a superfluid Bose-Einstein condensate. It is unknown whether this is due to heating associated with technical noise, non-adiabatic state preparation, or inelastic collisions. These issues are complicated, since all schemes for realizing the HH Hamiltonian use some form of temporal lattice modulation and therefore are described by a time-dependent Floquet formalism. The HH model arises after time averaging the Floquet Hamiltonian, but it is an open question to what extent finite interactions and micromotion lead to transitions between Floquet modes and therefore heating [21][22][23][24]. Bose-Einstein condensat...

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Observation of Bose–Einstein condensation in a strong synthetic magnetic field

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“…Treating interactions in such a non-perturbative way is difficult in periodically-driven systems [5][6][7][8][9][10], which have received unprecedented attention following the realisation of dynamical localisation [11][12][13][14][15], artificial gauge fields [16][17][18][19][20][21][22], models with topological [23][24][25][26][27][28] and statedependent [29] bands, and spin-orbit coupling [30,31]. In this paper, we consider strongly-interacting periodicallydriven systems and show how the SWT can be extended to derive effective static Hamiltonians of non-equilibrium setups.…”

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“…2, inset), where φ j = φ mn = Φ (m + n), σ ∈ {↑, ↓} ≡ {1, −1}, and we denote the square-lattice position by r j = (m, n). Such spin-sensitive drives are realised in experiments via the Zeeman effect using a periodically-modulated [29] and static [19,20] magnetic-field gradients which couple to atomic hyperfine states. For this protocol,…”

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Schrieffer-Wolff Transformation for Periodically Driven Systems: Strongly Correlated Systems with Artificial Gauge Fields

2016

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We generalize the Schrieffer-Wolff transformation to periodically driven systems using Floquet theory. The method is applied to the periodically driven, strongly interacting Fermi-Hubbard model, for which we identify two regimes resulting in different effective low-energy Hamiltonians. In the nonresonant regime, we realize an interacting spin model coupled to a static gauge field with a nonzero flux per plaquette. In the resonant regime, where the Hubbard interaction is a multiple of the driving frequency, we derive an effective Hamiltonian featuring doublon association and dissociation processes. The ground state of this Hamiltonian undergoes a phase transition between an ordered phase and a gapless Luttinger liquid phase. One can tune the system between different phases by changing the amplitude of the periodic drive.

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“…Some aspects of the current work have been explored in previous experiments, including the first atomic-gas simulations of the HarperHofstadter model 6,7 and observations of the chiral motion of particles in effective magnetic fields using synthetic dimensions [8][9][10][11] . But Tai and colleagues' study deepens our understanding of the interplay between interaction effects and the effective magnetic fields.…”

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Interactions propel a magnetic dance

LeBlanc

2017

Nature

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Publisher’s Note: Realizing the Harper Hamiltonian with Laser-Assisted Tunneling in Optical Lattices [Phys. Rev. Lett.111, 185302 (2013)]

Cited by 340 publications

References 0 publications

Observation of Bose–Einstein condensation in a strong synthetic magnetic field

Observation of Bose–Einstein condensation in a strong synthetic magnetic field

Schrieffer-Wolff Transformation for Periodically Driven Systems: Strongly Correlated Systems with Artificial Gauge Fields

Interactions propel a magnetic dance

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