Realistic Time-Reversal Invariant Topological Insulators with Neutral Atoms

We lay out an experiment to realize time-reversal invariant topological insulators in alkali atomic gases. We introduce an original method to synthesize a gauge field in the near field of an atom chip, which effectively mimics the effects of spin-orbit coupling and produces quantum spin-Hall states. We also propose a feasible scheme to engineer sharp boundaries where the hallmark edge states are localized. Our multiband system has a large parameter space exhibiting a variety of quantum phase transitions between topological and normal insulating phases. Because of their remarkable versatility, cold-atom systems are ideally suited to realize topological states of matter and drive the development of topological quantum computing.

Figure 1

(a) Experimental layout showing the origin of optical (state independent) and magnetic (state dependent) potentials and coupling fields. A state-independent, staggered, lattice along $\widehat{x}$ is formed by the separate interference of two pairs of $\lambda =1064\mathrm{nm}$ laser beams slightly detuned from each other to eliminate cross interference. The intersection angles are chosen so the lattice period differs by a factor of 2. Both beams reflect from the chip surface and form vertically aligned lattices, trapping the degenerate Fermi gas about $5\mu \mathrm{m}$ above the surface. Inset: chip geometry [from top to bottom, a reflective chip surface, gold wires aligned along $\widehat{y}$ placed every $2\mu \mathrm{m}$ along $\widehat{x}$ (producing the $|g⟩-|e⟩$ coupling), and finally gold wires aligned along $\widehat{x}$ with a $2\mu \mathrm{m}$ spacing (producing the state-dependent lattice)]. (b) Atomic potentials and radio-frequency (rf) driven Raman transitions.

Figure 2

(a) Energy spectrum of the uncoupled system ($\gamma ={\lambda }_{\mathrm{stag}}=0$ and $\alpha =1/6$) computed in a cylindrical geometry: the bulk energy bands (thick blue bands) are traversed by edge states (thin purple lines). (b) Schematic representation of the two edge states that lie inside the first bulk energy gap depicted in (a). The spins traveling around the edges are, respectively, represented by red and green arrows. (c) Energy bands $E(k_y)$ for $\gamma =0$ and $\alpha =1/6$, with an external staggered potential ${\lambda }_{\mathrm{stag}}=0.5t$ and $1.5t$. The topological phases associated with the open bulk gaps are indicated. The purple rectangles highlight a NI to QSH phase transition. (d) Phase diagram in the ($\gamma$, ${\lambda }_{\mathrm{stag}}$) plane in the vicinity of the uncoupled case $\gamma =0$ for $E_F=-t$.

Figure 3

(a) Energy bands $E(k_y)$ for $\gamma =0.25$ and $\alpha =1/6$, with an external staggered potential ${\lambda }_{\mathrm{stag}}=0.5t$ and $1.5t$. The topological phases associated to the bulk gaps are indicated. (b),(c) Phase diagrams in the ($\gamma$, ${\lambda }_{\mathrm{stag}}$) plane in the vicinity of the maximally coupled case $\gamma =0.25$ for (b) $E_F=0$ and (c) $E_F=-t$.

Figure 4

(a) The shaded blue region displays the position dependent coupling strength averaged over one period of oscillation of the rf fields. The black trace depicts the Zeeman shift from the computed magnetic field along $\widehat{x}$ which produces the rf coupling (at a representative time). (b) Discrete energy spectrum (blue dots) and typical amplitudes $|{\psi }_{\uparrow }(x,y){|}^2$ in the presence of a harmonic potential and anisotropic hopping. The open lattice has $42\times 42$ sites, $\gamma ={\lambda }_{\mathrm{stag}}=0$ and $\alpha =1/6$. The harmonic potential $V_{\mathrm{conf}}(x,y)\propto x^2+y^2$ is such that $V_{\mathrm{conf}}(42,42)=0.5t$ and the hopping parameters of the inner and outer regions are, respectively, $t_y^{\prime }=2t$ and $t_y=0.1t$. The amplitudes $|{\psi }_{\uparrow }(x,y){|}^2$ mark three distinct regimes in the associated spectrum: the outer, the inner, and the edge-state regimes.