Zeno Hall Effect

We show that the quantum Zeno effect gives rise to the Hall effect by tailoring the Hilbert space of a two-dimensional lattice system into a single Bloch band with a nontrivial Berry curvature. Consequently, a wave packet undergoes transverse motion in response to a potential gradient—a phenomenon we call the Zeno Hall effect to highlight its quantum Zeno origin. The Zeno Hall effect leads to retroreflection at the edge of the system due to an interplay between the band flatness and the nontrivial Berry curvature. We propose an experimental implementation of this effect with ultracold atoms in an optical lattice.

Figure 1

(a) When a Hilbert space $\mathcal{H}$ is dissipatively constrained to a subspace $\mathcal{S}$, the dynamics is governed by the projected Hamiltonian ${\widehat{H}}_{\mathcal{S}}={\widehat{P}}_{\mathcal{S}}\widehat{H}{\widehat{P}}_{\mathcal{S}}$, where $\widehat{H}$ and ${\widehat{P}}_{\mathcal{S}}$ are the bare Hamiltonian and the projection operator onto $\mathcal{S}$, respectively. (b) In a strongly dissipative lattice system in two dimensions, the occupation of the upper band is prohibited by a large damping gap ${\mathrm{\Delta }}_d$ due to the quantum Zeno effect, and the system is constrained to the lower band with a nonzero Berry curvature. Consequently, a wave packet undergoes transverse motion in response to a potential gradient $\mathit{F}$ (c), and will be retroreflected at a boundary (d).

Figure 2

(a) Spin texture and (b) Berry curvature ${\mathrm{\Omega }}_{xy}(\mathit{k})$ of the lower band (QZ subspace) emerging from the engineered dissipation (1) corresponding to the jump operators in Eq. (6) with $l_{\uparrow }=-i$.

Figure 3

Time evolution of the $x$ component of the c.m., obtained from exact diagonalization of Eq. (5) (orange), the integral of the Berry curvature in Eq. (7) with respect to the trajectory in the Brillouin zone (purple), and the integral of the smoothed Berry curvature that takes into account a finite wave-packet spread (black). The initial increment and the final saturation reflect the Berry curvature landscape (see Fig. 2). Three panels on the right show the real-space density profiles of the wave packet at $\omega t=0.1$, 0.4, and 1.0, showing the transverse c.m. motion, where $\omega =Fa/\hslash$. The inset shows the displacement along the $y$ axis, which stays unchanged and takes on a nonzero value due to the initial recoil [51]. The red dashed lines in the main panel and the circles in the right panel are guides to the eye.

Figure 4

(a) Long-time dynamics of the c.m. Three panels on the top show the real-space density profiles of the wave packet before ($\omega t=20$), at ($\omega t=60$), and after ($\omega t=100$) collision with the right boundary, indicating the retroreflection. (b) “Hills” (orange, ${\mathrm{\Omega }}_{xy}>1$) and “valleys” (blue, ${\mathrm{\Omega }}_{xy}<-1$) in the Brillouin zone and the wave-packet profile in momentum space before, at, and after the retroreflection. Dark (light) stripes, which are parallel to $\mathit{F}$, refer to the orbits in which the anomalous motion in real space approaches (leaves) the right boundary.

Figure 5

(a) $\mathrm{\Lambda }$ atoms in a staggered square optical lattice subjected to a set of fine-tuned lasers, which couple internal states $|\uparrow ⟩$ and $|\downarrow ⟩$ to $|e⟩$ with spatially dependent Rabi frequencies $\tilde{\mathrm{\Omega }}(\mathit{x})$ and ${\tilde{\mathrm{\Omega }}}^{\prime }(\mathit{x})$, respectively. The excited state $|e⟩$ is unstable and undergoes effectively a rapid on-site loss at a rate $\kappa$. (b) Within the tight-binding approximation, $|e⟩$ is coupled to the nearest $|\downarrow ⟩$’s in a $p$-wave symmetry and the on-site $|\uparrow ⟩$ with a phase inversion for the sublattices $A$ and $B$. The magnitudes of the Rabi frequencies for $|\uparrow ⟩\leftrightarrow |e⟩$ and $|\downarrow ⟩\leftrightarrow |e⟩$ transitions are $|\mathrm{\Omega }|$ and $|{\mathrm{\Omega }}^{\prime }|$, respectively. The energy levels are not to scale.