Fibonacci optical lattices for tunable quantum quasicrystals

K. Singh, K. Saha, S. A. Parameswaran, and D. M. Weld

Phys. Rev. A 92, 063426 – Published 29 December 2015

Abstract

We describe a quasiperiodic optical lattice, created by a physical realization of the abstract cut-and-project construction underlying all quasicrystals. The resulting potential is a generalization of the Fibonacci tiling. Calculation of the energies and wave functions of ultracold atoms loaded into such a lattice demonstrate a multifractal energy spectrum, a singular continuous momentum-space structure, and the existence of controllable edge states. These results open the door to cold atom quantum simulation experiments in tunable or dynamic quasicrystalline potentials, including topological pumping of edge states and phasonic spectroscopy.

Received 25 April 2015

DOI:https://doi.org/10.1103/PhysRevA.92.063426

Physics Subject Headings (PhySH)

Cold gases in optical lattices

Atomic, Molecular & OpticalGeneral Physics

Authors & Affiliations

K. Singh^1,2,*, K. Saha^2,3, S. A. Parameswaran^2,3, and D. M. Weld^1,2

¹Department of Physics, University of California, Santa Barbara, California 93106, USA
²California Institute for Quantum Emulation, Santa Barbara, California 93106, USA
³Department of Physics, University of California, Irvine, California 92697, USA

^*kevin@physics.ucsb.edu

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Issue

Vol. 92, Iss. 6 — December 2015

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Images

Figure 1
The generalized Fibonacci optical lattice. (Top) Diagram of cut-and-project construction of the generalized Fibonacci lattice. A 2D strip at a particular angle $α$ is projected down to a line. If $tan (α) = 1 / τ \equiv 2 / (1 + \sqrt{5})$ , this results in the Fibonacci tiling itself; a different irrational slope creates a different 1D quasicrystal. (Bottom) Calculated potential of a Fibonacci optical lattice with parameters discussed in text (red is deeper), showing Fibonacci sequence $τ 1 τ τ 1 τ 1 τ τ 1 τ τ 1 τ 1 τ$ ... of lattice spacings.
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Figure 2
Energies and wave functions in a tunable Fibonacci-type potential. (a) A portion of the energy spectrum of the generalized Fibonacci optical lattice as a function of cut angle. Here $a = 1, A_{L} = 5 / π^{2}, A_{C} = 5 A_{L}, w_{0} = 1$ . Color of points corresponds to center of mass of probability density, to enable identification of edge states. (b) 2D probability density versus position at the indicated cut angle and energy (a typical bulk state). Red regions correspond to higher probability density. (c) 2D probability density versus position at the indicated cut angle and energy (a typical edge state). Both (b) and (c) show a region of 4 by 200 lattice constants.
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Figure 3
Tunable quasicrystals in momentum space. Main lower panel shows Fourier transform of the ground-state probability density along the direction of the cutting beam, as a function of cut angle $α$ . Note the logarithmic scale on the color bar. Upper panels show log of Fourier transform amplitude versus wave vector at a cut angle of $π / 4$ (top) and ${tan}^{- 1} (2 / (1 + \sqrt{5}))$ (bottom), with identical axis limits. Dashed lines show expected wave vectors of Fourier peaks of the potential based on the second-order perturbation theory described in the Appendix. Calculations were performed on a mesh 4 by 200 lattice constants in size. Faint vertical lines in the main plot are finite-size edge effects.
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Figure 4
Edge state topological pumping by phason driving. (Top) Energy states near the first minigap versus offset of cutting beam along a lattice vector, at the Fibonacci cut slope. As the offset between the lattice and the cut beam is varied, left and right edge states cross the gap. Coloring of points indicates center of mass, with the same mapping as Fig. 2. (Bottom) Variation of spatial probability amplitude of an initial edge state as offset is adiabatically varied. Position is normalized to 0 at the left edge and 1 at the right edge.
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Figure 5
Fourier spectrum of effective 1D potential. The semilogarithmic plot shows the relative amplitudes of Bragg peaks in units of recoil energy produced by perturbation theory up to second order. We have assumed $A_{L}^{x} = A_{L}^{y}$ . Note that this is the analytically obtained Fourier transform of the potential itself, rather than the numerically obtained Fourier transform of the atomic density, which is shown in Fig. 3 of the main text.
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