Uncover Topology by Quantum Quench Dynamics

Topological quantum states are characterized by nonlocal invariants. We present a new dynamical approach for ultracold-atom systems to uncover their band topology, and we provide solid evidence to demonstrate its experimental advantages. After quenching a two-dimensional (2D) Chern band, realized in an ultracold ${}^{87}\mathrm{Rb}$ gas from a trivial to a topological parameter regime, we observe an emerging ring structure in the spin dynamics during the unitary evolution, which uniquely corresponds to the Chern number for the postquench band. By extracting 2D bulk topology from the 1D ring pattern, our scheme displays simplicity and is insensitive to perturbations. This insensitivity enables a high-precision determination of the full phase diagram for the system’s band topology.

Figure 1

(a) Experimental scheme. Two laser beams $E_x$ and $E_y$ are incident on the atoms and generate 2D lattice and Raman coupling. A bias magnetic field $B$ is along the $z$ direction. All wave plates ($\lambda /2$, $\lambda /4$ and ${\lambda }_{\mathrm{ph}}$) are for realizing $C_4$-symmetric 2D SO coupling [20]. Laser-beam frequencies are controlled by AOM1 and AOM2, with phase-locked radio-frequency signaling rf1, rf2, and rf3. The switch is for quenching by switching rf2 and rf3 to change the two-photon detuning of the Raman couplings. (b) Level structure and Raman transitions. Raman coupling strengths ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$ are induced by $(E_{x\sigma ,}{E}_{y\pi })$ and $(E_{y\sigma },E_{x\pi })$, respectively, and ${\delta }_i$ and ${\delta }_f$ represent two-photon detuning before and after quenching. (c) $s$-band structure before and after the quantum quench with the spin polarization of the equilibrium distribution presented.

Figure 2

Ring structure of spin polarization. (a) Evolution of $P(\mathit{q},t)$ in the FBZ from $t=0$ to 1.6 ms, with parameters $(V_0,{\mathrm{\Omega }}_0)=(4.0,1.0)E_r$ and ${\delta }_f=0$. The upper row is from experimental measurements, and the lower row is from theoretical calculations. (b) Spin polarization in the FBZ at fixed evolution time $t$ versus ${\delta }_f$. The upper row is for $(V_0,{\mathrm{\Omega }}_0)=(4.0,1.0)E_r$ and $t=480\mu \mathrm{s}$, and the lower row is for $(V_0,{\mathrm{\Omega }}_0)=(4.0,2.0)E_r$ and $t=320\mu \mathrm{s}$.

Figure 3

Topological phase boundaries for $(V_0,{\mathrm{\Omega }}_0)=(4.0,1.0)E_r$ and $(4.0,2.0)E_r$. The size $r$ of the ring pattern (the insets) is measured as a function of ${\delta }_f$, shown as the green circles with statistical error bars. The solid curves are the polynomial fitting results up to third order, giving the upper and the lower phase boundaries, marked as the cross points with ($r=0$) and ($r=R$) lines, respectively.

Figure 4

Topological phase diagram for (a) $({\delta }_f-{\mathrm{\Omega }}_0)$ plane with $V_0=4.0E_r$, and (b) $({\delta }_f-V_0)$ plane with ${\mathrm{\Omega }}_0=1.0E_r$. The blue and red dots with statistical error bars are determined from the experiment, and the solid lines are from theoretical calculation with the same parameters. Areas I and IV are topological trivial areas with $\mathcal{C}=0$, while areas II and III are topological nontrivial areas with $\mathcal{C}=1$ and $\mathcal{C}=-1$, respectively.