Robustness of topologically protected edge states in quantum walk experiments with neutral atoms

Discrete-time quantum walks allow Floquet topological insulator materials to be explored using controllable systems such as ultracold atoms in optical lattices. By numerical simulations, we study the robustness of topologically protected edge states in the presence of decoherence in one- and two-dimensional discrete-time quantum walks. We also develop a simple analytical model quantifying the robustness of these edge states against either spin or spatial dephasing, predicting an exponential decay of the population of topologically protected edge states. Moreover, we present an experimental proposal based on neutral atoms in spin-dependent optical lattices to realize spatial boundaries between distinct topological phases. Our proposal includes also a scheme to implement spin-dependent discrete shift operations in a two-dimensional optical lattice. We analyze under realistic decoherence conditions the experimental feasibility of observing unidirectional, dissipationless transport of matter waves along boundaries separating distinct topological domains.

Figure 1

Topological twist in the 1D split-step quantum walk with $({\theta }_1,{\theta }_2)=(\pi /2,0)$ (Hadamard walk). (a) Quasienergy spectrum with two energy gaps occurring at energy $\epsilon =0$ and $\epsilon =\pi$. (b,c) The corresponding quasienergy eigenstates of the upper band in the two time frames, Eqs. (7) and (8), displayed on the Bloch sphere. Chiral symmetry constrains the eigenspinors to lie in a plane, $x=0$, while the quasimomentum is varied across the Brillouin zone, performing a closed loop. The color gradient indicates the winding direction around the Brillouin zone. The (signed) winding number associated with transformation differs in the two time frames, ${\nu }^{\prime }=1$ in (b) and ${\nu }^{\prime \prime }=0$ in (c). The topological invariants of the bulk are given by the sum and difference of the two winding numbers, $({\nu }_0,{\nu }_{\pi })=({\nu }^{\prime }+{\nu }^{\prime \prime },{\nu }^{\prime }-{\nu }^{\prime \prime })/2+1/2$. See also Fig. 2 for the related phase diagram.

Figure 2

Topological invariants assigned to the coin angles of (a) the 1D split-step walk and (b) the 2D protocol. Due to the form of the coin operator $C\left(\theta \right)$, the walk possesses a $4\pi$ periodicity in the coin angles. At the phase boundaries, the gap closes at quasienergy $\epsilon =0$ (dotted), $\epsilon =\pi$ (dashed), or both at $\epsilon =0$ and $\epsilon =\pi$ (dash-dotted). The coin angle pairs chosen in the numerical examples in this work and the corresponding phase transitions defined in Eqs. (9) and (10) are also displayed (line with stars). The 1D Hadamard walk $({\theta }_1,{\theta }_2)=(\pi /2,0)$, which is discussed in Fig. 1, is also shown (diamond).

Figure 3

(a) Position dependency of the coin angle in the 1D split-step walk given by Eq. (9) realizing two spatially adjacent, distinct topological domains with invariants $({\nu }_0,{\nu }_{\pi })=(0,0)$ for $x\ll 0$ and $(1,0)$ for $x\gg 0$. We use a smooth crossover transition corresponding to the diffraction-limited optical resolution of our imaging system (see Sec. 4 for details). (b) Decoherence-free evolution of the spatial density distribution $P(x;n)$ as a function of the number of steps $n$ for a walker initially prepared in the single-site state $|0,\downarrow \rangle$. The narrow peak located at the boundary near $x=0$ indicates the component of the walker populating the TP edge state. (c) The same walk is subject to pure spin decoherence and pure spatial decoherence with increasing decoherence probabilities $p_{\text{S}},p_{\text{P}}$. Insets: time dependence of the walker's probability $P(x=0;n)$ to be at the origin $x=0$ in logarithmic scale. It exhibits an exponential decay for small amounts of decoherence but stays constant for the decoherence-free evolution. The time evolution is calculated for a large number of lattice sites (201) to prevent the walker from reaching the boundaries in the given maximum number of steps.

Figure 4

Quasienergy spectrum of an inhomogeneous 2D DTQW with a horizontal strip geometry. The horizontal strip, 40 sites wide along the $y$ direction, is associated with Rudner invariant $-1$, whereas the rest of the bulk has $+1$ [refer to Fig. 2 for the phase diagram]. Unidirectional edge modes are visible in the gaps (thick lines), with blue and red denoting each edge of the strip. For any given quasienergy $\epsilon$ in the gaps, two TP edge modes exist per edge, as expected from the bulk-boundary correspondence principle. The spectrum is computed numerically using 100 sites in the $y$ direction.

Figure 5

(a) Color-coded spatial probability distribution $P(\mathit{x};n)$ of a decoherence-free two-dimensional DTQW. The coin angles depend on the position as specified by Eq. (10), creating a droplet-shaped topological island with Rudner invariants $-1$ inside and $+1$ outside of the island. The width of the transition is limited by the optical resolution of our experimental setup with Abbe radius $R_{\text{A}}\simeq 0.8a$ (see Sec. 4 for details). The walker is initially prepared in the single-site state $|(x=-15,y=0),\downarrow \rangle$ near the phase boundary and shows a unidirectional moving population of edge states around the boundary as time evolves. In (b) the same walk is subject to spin decoherence under realistic experimental conditions $\left(p_{\text{S}}=0.05\right)$, exhibiting a slow decay of the edge current over time. An animation showing the evolution over 1000 steps is provided in the Supplemental Material [66].

Figure 6

(a) Probability $P(\mathit{x}\in F;n)$ for the walker to be inside the grayed region $F$ as a function of the number of steps $n$ in logarithmic scale. (b) Probability $P(\mathit{x}\in L;n)$ for the walker to be inside the grayed region $L$ near the lower half of the phase boundary, normalized to the population probability $P(\mathit{x}\in F;n)$. The probabilities are shown for the unitary walk evolution (dashed curves) and for a decoherence rate $p_{\text{S}}=0.05$ (solid curves). Inset: close-up view in the long-time limit for the evolution without decoherence.

Figure 7

Probability of populating a TP edge state as a function of the number of steps $n$ for different amounts of spin decoherence $p_{\text{S}}$ (semilogarithmic scale). The data points are calculated numerically for the 1D split-step walk with the coin angles as defined in Eq. (9) and with the initial state being the TP edge state with quasienergy $\epsilon =0$. The solid lines represent an exponential decay as predicted by the analytical model in Eq. (13).

Figure 8

(a) One-dimensional lattice potentials created by two linearly polarized beams. A polarization rotation by $\varphi$ leads to a relative displacement of the two optical potentials (orange and blue curves), which spin-dependently trap atoms in either the $|\uparrow \rangle$ or $|\downarrow \rangle$ internal state. The vector $\mathit{B}$ represents the direction of the external magnetic field, which fixes the quantization axis. (b) Two-dimensional lattice potentials created by three interfering laser beams for spin-dependent transport on a square lattice. The polarization of beam 3 points out of the plane, whereas the polarization of beams 1 and 2 can rotate, producing spin-dependent displacements along two diagonal directions at $\pm {45}^{\circ }$ relative to the quantization axis. Two counterpropagating beams (not shown) orthogonal to the plane provide the confinement in the third direction. (c) Potential depth of the two spin-dependent optical lattices (orange and blue) for different polarization angles, ${\varphi }_1$ and ${\varphi }_2$.

Figure 9

The intensity of Raman lasers, utilized to implement the coin operation, is modulated in space to give rise to sharp topological phase boundaries. A spatial light modulator (SLM) creates a structured intensity pattern, which is imaged onto the optical lattice by a high-numerical-aperture (NA of 0.92) objective lens mounted in a $4f$ optical system.

Figure 10

Analysis of a TP edge state $|E\rangle$ in the 1D split-step DTQW with coin angles given by Eq. (9) for different slopes of the phase crossover, as determined by the diffraction parameter $a/R_{\text{A}}$. (a) RMS size of the TP edge state (black dashed line) and overlap probability of the initial state $|x=0,s_E\rangle$ with the TP edge state $|E\rangle =|\chi \rangle \otimes |s_E\rangle$ (red solid line). The two vertical arrows indicate the values corresponding to the 1D and 2D quantum walk setups. (b) Coin angles ${\theta }_2$ (black circles) and position distribution ${\sum }_s{\left|\langle E|x,s\rangle \right|}^2$ of the TP edge state (red lines) computed for the current 1D (left) and the new 2D experimental setups (right).