Dynamically Induced Symmetry Breaking and Out-of-Equilibrium Topology in a 1D Quantum System

Nontrivial topology in lattices is characterized by invariants—such as the Zak phase for one-dimensional (1D) lattices—derived from wave functions covering the Brillouin zone. We realize the 1D bipartite Rice-Mele (RM) lattice using ultracold ${}^{87}\mathrm{Rb}$ and focus on lattice configurations possessing various combinations of chiral, time-reversal, and particle-hole symmetries. We quench between configurations and use a form of quantum state tomography, enabled by diabatically tuning lattice parameters, to directly follow the time evolution of the Zak phase as well as a chiral winding number. The Zak phase evolves continuously; however, when chiral symmetry transiently appears in the out-of-equilibrium system, the chiral winding number becomes well defined and can take on any integer value. When quenching between two configurations obeying the same three symmetries, the Zak phase is time independent; we confirm the dynamically induced symmetry breaking predicted in [McGinley and Cooper, Phys. Rev. Lett. 121, 090401 (2018)] that chiral symmetry is periodically restored, at which times the winding number changes by $\pm 2$, yielding values that are not present in the native RM Hamiltonian.

Figure 1

(a) Adiabatic potentials colored according to the local magnetization for configurations I and II with the $j=0$ unit cell marked in gray, computed for $({\mathrm{\Omega }}_{+}=13.2E_{\mathrm{R}},{\mathrm{\Omega }}_{-}=5.5E_{\mathrm{R}},{\mathrm{\Omega }}_{\mathrm{rf}}=3.5E_{\mathrm{R}})$. (b) Pseudospin evolution in the “$y$” and “$x$” readout lattices. The left panels plot $⟨{\widehat{F}}_x⟩(t)$ evolving according to the lattice pictured in its inset (symbols represent experiment; solid curves show theory); times up to the rotation time for the readout lattice are shown with solid symbols and curves, while evolution after this time is shown with open symbols or dotted curves. The right panels show the computed evolution on the Bloch sphere for these trajectories, along with the axis of rotation in blue.

Figure 2

Ground band pseudospin decomposition in SSH configurations I and II. (a) Pseudospin state for $q$ values sampling the whole BZ plotted in the Bloch sphere. The raw measurements (open symbols) are impure (i.e., magnitude $<1$); the solid symbols mark the nearest pure states on the surface of the Bloch sphere. For the eigenstates of configuration I, the vectors trace the equator of the Bloch sphere, while for configuration II, they are aligned along ${\mathbf{e}}_x$. (b) Pure-state expectation values of ${\widehat{\sigma }}_x$, ${\widehat{\sigma }}_y$, and ${\widehat{\sigma }}_z$ shown as a function of $q$ in light blue, dark green, and light green, plotted along with theory (dashed curves).

Figure 3

Momentum resolved pseudospin evolution in SSH configurations I and II. Model parameters are $J=0.395(2)E_{\mathrm{R}}$, $J^{\prime }=0.012(2)E_{\mathrm{R}}$ for configuration I and $J=0.038(3)E_{\mathrm{R}}$, $J^{\prime }=0.379(2)E_{\mathrm{R}}$ for configuration II. (a) Reconstructed Bloch vectors. Colors represent $⟨{\widehat{\sigma }}_z(q)⟩$ while black arrows denote $(⟨{\widehat{\sigma }}_x(q)⟩,⟨{\widehat{\sigma }}_y(q)⟩)$. The data are filtered in crystal momentum and time (with root-mean-square Gaussian widths $k_{\mathrm{R}}/8$ and $10\mu \mathrm{s}$). (b) Corresponding points on the Bloch sphere for evolution times of $(0.05,0.25,0.45)T$. (c) Zak phase and winding number for configurations I (teal) and II (magenta). The transparency indicates the extent to which the measured state breaks CS, and the gray boxes surround the times when CS is predicted to be recovered.

Figure 4

Momentum resolved pseudospin evolution in SSH configurations I and II, using initial states of the opposite configuration. Model parameters are $J=0.408(3)E_{\mathrm{R}}$, $J^{\prime }=0.003(5)E_{\mathrm{R}}$ for configuration I and $J=0.009(5)E_{\mathrm{R}}$, $J^{\prime }=0.446(3)E_{\mathrm{R}}$ for configuration II. All panels are plotted as in Fig. 3. (a) Individual expectation values $⟨{\widehat{\sigma }}_x⟩$, $⟨{\widehat{\sigma }}_y⟩$, and $⟨{\widehat{\sigma }}_z⟩$. (As before, the data are filtered with root-mean-square Gaussian widths $k_{\mathrm{R}}/8$ and $10\mu \mathrm{s}$.) (b) Bloch state rendering at $t=0.40T$. (c) Zak phase and winding number for configurations I (teal) and II (magenta).