Topological Swing of Bloch Oscillations in Quantum Walks

We report new oscillations of wave packets in quantum walks subjected to electric fields, that decorate the usual Bloch-Zener oscillations of insulators. The number of turning points (or suboscillations) within one Bloch period of these oscillations is found to be governed by the winding of the quasienergy spectrum. Thus, this provides a new physical manifestation of a topological property of periodically driven systems that can be probed experimentally. Our model, based on an oriented scattering network, is readily implementable in photonic and cold atomic setups.

Figure 1

Two-steps oriented scattering network, where an input signal flows from top to bottom. Black and orange nodes represent two distinct scattering processes that repeat periodically. Two staggered phase shifts, ${\varphi }_1$ and ${\varphi }_2$, are also considered along the leftward paths. A dashed square emphasizes the unit cell of this lattice.

Figure 2

(a) Quasienergy spectrum with a winding ${\nu }_{\varphi }=-2$ obtained for a scattering model with two steps per period for ${\theta }_1=\pi /4$, ${\theta }_2=\pi /4-0.6$, ${\varphi }_1=\varphi$ and ${\varphi }_2=-2\varphi$. (b) Values of ${\nu }_{\varphi }$ for integer values of $m_j/n_j$.

Figure 3

(a) Adiabatic increase of $\varphi$ leads to (b) a standard Bloch oscillation (${\nu }_{\varphi }=-2$), and (g),(h) Bloch oscillation with suboscillations (${\nu }_{\varphi }=6$ and ${\nu }_{\varphi }=8$, respectively) of a wave packet injected at time $j=0$ and position $l=0$ with a Gaussian shape with rms width of 10 sites in one band [the blue band in Fig. 2 for the case of (b)]. Color scale: intensity of the wave packet ($|{\alpha }_l^j{|}^2+|{\beta }_l^j{|}^2$). Dashed black line: analytical calculation of the centre of mass motion of a wave packet from Eq. (32) in [46]. (b),(g),(h) show one period $T_B$ of oscillation for the values of ($m_1$, $m_2$, $n_1$, $n_2$), ${\theta }_1$, ${\theta }_2$ as ($1,-2$, 1, 1), $\pi /4,\pi /4-0.5$ for (b); ($4,-1$, 1, 1), $\pi /4,\pi /4-0.2$ for (g); and ($9,-1$, 2, 2), $\pi /4,\pi /4-0.2$ for (h). In (c)–(f), the norm of the 2D Fourier transform of the wave packet ($\alpha$ part) after having evolved to the time step indicated by the horizontal lines in (b), and both the solid and the dashed red lines represent numerically calculated bands [Eq. (33) in Ref. [46] ]. The vertical scales differ in each panel. The green arrows show the direction in which the bands wind when $\varphi$ increases.

Figure 4

(a) Two-steps scattering network with the next nearest coupling in the second step. A dashed black rectangle emphasizes the unit cell of this lattice. (b) Quasienergy bulk spectrum for the model depicted in (a) with ${\theta }_1=\pi /4$, ${\theta }_2=\pi /4-0.6$ and ${\varphi }_1=-{\varphi }_2$. (c) Quantized displacement of the mean particle position with associated winding numbers ${\nu }_k$.