Cavity-induced generation of nontrivial topological states in a two-dimensional Fermi gas

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Cavity-induced generation of nontrivial topological states in a two-dimensional Fermi gas

Ameneh Sheikhan, Ferdinand Brennecke, and Corinna Kollath

Phys. Rev. A 94, 061603(R) – Published 12 December 2016

Abstract

We propose how topologically nontrivial states can dynamically organize in a fermionic quantum gas that is confined to a two-dimensional optical lattice potential and coupled to the field of an optical cavity. The spontaneously emerging cavity field induces, together with coherent pump laser fields, a dynamical gauge field for the atoms. Upon adiabatic elimination of the cavity degree of freedom, the system is described by an effective Hofstadter model with a self-consistency condition that determines the tunneling amplitude along the cavity direction. The fermions are found to self-organize into topologically nontrivial states that carry an extended edge state for a finite system size. Due to the dissipative nature of the cavity field, the topological steady states are protected from external perturbations.

Received 6 October 2016

DOI:https://doi.org/10.1103/PhysRevA.94.061603

Physics Subject Headings (PhySH)

Atoms, ions, & molecules in cavities Cold atoms & matter waves Cold gases in optical lattices Edge states Fermi gases Fermions Topological phases of matter

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

Authors & Affiliations

Ameneh Sheikhan¹, Ferdinand Brennecke², and Corinna Kollath¹

¹HISKP, University of Bonn, Nussallee 14-16, 53115 Bonn, Germany
²Physikalisches Institut, University of Bonn, Wegelerstraße 8, 53115 Bonn, Germany

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Issue

Vol. 94, Iss. 6 — December 2016

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Images

Figure 1
Scheme for generating a dynamical version of the Hofstadter model in a two-dimensional optical lattice. (a) Fermionic atoms are loaded into a square optical lattice potential and coupled to the dynamical field of an optical cavity. Tunneling along the $x$ direction is strongly suppressed by a potential offset $Δ$ between neighboring lattice sites. (b) Cavity-assisted Raman processes induced by two running-wave pump beams restore tunneling along the cavity direction and imprint a phase $φ$ onto the atomic wave function when tunneling around a plaquette. (c) Single-particle energy bands for the Hofstadter model of size $L_{x} = 159$ and $L_{y} = 1000$ with periodic boundary condition in the $y$ direction and open boundary condition in the $x$ direction. The flux is chosen as $φ = 2 π \frac{1}{8}$ . The black solid line shows the energy bands for $J_{x} / J_{y} = 0.1$ and the red dashed lines correspond to decoupled chains with $J_{x} = 0$ . (d) Zoom into the first energy gap where the edge state (blue solid line) between two bulk bands for $J_{x} / J_{y} = 0.1$ becomes visible.
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Figure 2
Steady-state amplitude of the cavity field $Re (α) / \sqrt{N_{p}}$ (color code) for flux (a) and (b) $φ = 2 π \frac{1}{8}$ and (c) and (d) $φ = 2 π \frac{2}{7}$ as a function of filling $n$ and pump strength $A N_{p} / J_{y}$ for (a) and (c) the first and (b) and (d) the second solutions of the effective model (1) using adiabatic elimination of the cavity field. White regions indicate the trivial empty cavity solution. The red dash-dotted lines correspond to the filling where the Fermi energy lies inside the gap with $2 π n = φ$ . The black dashed lines indicate the fillings where the Fermi energy lies inside higher-order gaps. Here $L_{x} = 159, L_{y} = 1000, ℏ δ_{c p} = J_{y}$ , and $ℏ κ = 0.05 J_{y}$ .
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Figure 3
Atomic density profile along the cavity direction in the steady state plotted versus pump strength $A N_{p} / J_{y}$ for a flux value of $φ = 2 π \frac{1}{8}$ . A clear formation of an edge state is visible. The filling is chosen to yield a Fermi energy (a) inside the first gap with $n = 0.126$ or (b) inside the second gap with $n = 0.251$ .
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Figure 4
Local current $J_{m}$ along the cavity direction in the steady state plotted versus pump strength $A N_{p} / J_{y}$ for a flux value of $φ = 2 π \frac{1}{8}$ . The direction of the current is different for the fillings where the Fermi energy lies (a) below the first gap with $n = 0.11$ and (b) above the first gap with $n = 0.14$ . (c) Orbital current in the steady state versus filling $n$ and pump strength $A N_{p} / J_{y}$ for the same flux. The orbital current jumps whenever the Fermi energy crosses a gap in the energy spectrum. The red dash-dotted line corresponds to the filling where the Fermi energy lies inside the gap with $2 π n = φ$ . The black dashed lines indicate the fillings where the Fermi energy lies inside higher-order gaps. The arrows indicate the fillings chosen in (a) and (b).
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