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تسجيل مستخدم جديدThe Harper-Hofstadter Hamiltonian and conical diffraction in photonic lattices with grating assisted tunneling
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We introduce a grating assisted tunneling scheme for tunable synthetic magnetic fields in photonic lattices, which can be implemented at optical frequencies in optically induced one- and two-dimensional dielectric photonic lattices. We demonstrate a conical diffraction pattern in particular realization of these lattices which possess Dirac points in $k$-space, as a signature of the synthetic magnetic fields. The two-dimensional photonic lattice with grating assisted tunneling constitutes the realization of the Harper-Hofstadter Hamiltonian.
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We experimentally implement the Harper Hamiltonian for neutral particles in optical lattices using laser-assisted tunneling and a potential energy gradient provided by gravity or magnetic field gradients. This Hamiltonian describes the motion of char
ged particles in strong magnetic fields. Laser-assisted tunneling processes are characterized by studying the expansion of the atoms in the lattice. The band structure of this Hamiltonian should display Hofstadters butterfly. For fermions, this scheme should realize the quantum Hall effect and chiral edge states.
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We quantum-simulated the 2D Harper-Hofstadter (HH) lattice model in a highly elongated tube geometry -- three sites in circumference -- using an atomic Bose-Einstein condensate. In addition to the usual transverse (out-of-plane) magnetic flux, pierci
ng the surface of the tube, we threaded a longitudinal flux $Phi_{rm L}$ down the axis of the tube This geometry evokes an Aharonov-Bohm interferometer, where noise in $Phi_{rm L}$ would readily decohere the interference present in trajectories encircling the tube. We observe this behavior only when transverse flux is a rational fraction of the flux-quantum, and remarkably find that for irrational fractions the decoherence is absent. Furthermore, at rational values of transverse flux, we show that the time evolution averaged over the noisy longitudinal flux matches the time evolution at nearby irrational fluxes. Thus, the appealing intuitive picture of an Aharonov-Bohm interferometer is insufficient. Instead, we quantitatively explain our observations by transforming the HH model into a collection of momentum-space Aubry-Andr{e} models.
In two and three spatial dimensions, the transverse response experienced by a charged particle on a lattice in a uniform magnetic field is proportional to a topological invariant, the first Chern number, characterizing the energy bands of the underly
ing Hofstadter Hamiltonian. In four dimensions, the transverse response is also quantized, and controlled by the second Chern number. These remarkable features solely arise from the magnetic translational symmetry. Here we show that the symmetries of the two-, three- and four-dimensional Hofstadter Hamiltonians may be encrypted in optical diffraction gratings, such that simple photonic experiments allow one to extract the first and the second Chern numbers of the whole energy spectra. This result is particularly remarkable in three and four dimensions, where complete topological characterizations have not yet been achieved experimentally. Side-by-side to the theoretical analysis, in this work we present the experimental study of optical gratings analogues of the two- and three-dimensional Hofstadter models.
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