Coherent Many-Body Oscillations Induced by a Superposition of Broken Symmetry States in the Wake of a Quantum Phase Transition

It is now widely accepted that quenches through the critical region of quantum phase transitions result in post-transition states populated with topological defects—analogs of the classical topological defects. However, consequences of the very nonclassical fact that the state after a quench is a superposition of distinct, broken-symmetry vacua with different numbers and locations of defects have remained largely unexplored. We identify coherent quantum oscillations induced by such superpositions in observables complementary to the one involved in symmetry breaking. These oscillations satisfy Kibble-Zurek dynamical scaling laws with the quench rate, with an instantaneous oscillation frequency set primarily by the gap of the system. In addition to the obvious fundamental significance of a superposition of different broken symmetry states, quantum coherent oscillations can be used to verify unitarity and test for imperfections of the experimental implementations of quantum simulators.

Figure 1

Coherent oscillations after a quench through a quantum critical point. In panel (a), we show a quench protocol in the 1D transverse-field Ising model, where we allow the system to freely evolve upon reaching zero transverse field at time $t=t_s$. In (b), we track transverse magnetization in the 1D model, where the coherent oscillations for $t>t_s$ are apparent. Their origin can be traced back to the point reached at $t_c$, see panel (c), where we show data collapse consistent with the dynamical scaling hypothesis in Eq. (8). There is a gradual decrease of oscillation amplitude during the ramp through the ferromagnetic phase, setting the nonzero amplitude observed for $t>t_s$ [note that panel (c) shows only $t<t_s$].

Figure 2

Scaling of transverse magnetization oscillations amplitudes. In (a), for a 1D chain, we show constant contribution, $A_0$ (blue line), and the amplitudes of coherent oscillations, $A_4$, as a function of ${\tau }_Q$. We compare the analytical formulas in Eqs. (10) and (14) (lines) with the corresponding exact numerical results (points). We show the results for two quench protocols of the same linear slope $\propto {\tau }_Q^{-1}$ at the critical point: the first protocol in Eq. (6) stops abruptly at $g=0$ (green line indicates the amplitude of oscillations), and the second one in Eq. (13) reaches $g=0$ smoothly (red). The protocols are shown in the inset. In (b), for the ladder, the constant term $B_0$ closely follows the measured density of excitations, $n$. The latter follows an expected scaling for a model in the same universality class as a 1D chain, $n\sim {\tau }_Q^{-1/2}$. The dominant oscillatory contribution, $B_8$, can be fitted with $B_8\sim {\tau }_Q^{-0.69}$, though we expect that logarithmic corrections from dephasing contribute to the decay of $B_8$. Note that in (a) the red line would be consistent with a power law and exponent $-1.18$ while the analytical solution shows $n^2\sim {\tau }_Q^{-1}$ behavior times a logarithmic correction. In (c), we show the data for the 2D square lattice where the dominant contributions, $B_0$ and $B_8$, closely follow the measured excitation density for available times.

Figure 3

Oscillations in nonintegrable systems. In (a), we show scaled transverse magnetization (8) in the function of scaled time in the 1D ladder geometry. The critical point of the model is in the same universality class as a 1D chain, with $\nu =z={\mathrm{\Delta }}_x=1$. For large enough ${\tau }_Q$, the plots collapse in the vicinity of the critical point to a single scaling function $F_{{\sigma }^x}$ that exhibits oscillatory behavior. In (b), the corresponding data for the 2D transverse Ising model with the scaling dimension ${\mathrm{\Delta }}_x\simeq 1.41$. The values of ${\tau }_Q$ are limited, from below, to be in the scaling limit at the critical point and, from above, to avoid finite-size effects in a finite lattice [78]. The best collapse of complementary quantities used $\xi ={\tau }_Q^{0.36}$ [78] for similar range of ${\tau }_Q$’s. Solid lines indicate thermodynamic limit results of iPEPS [107], which become unstable for times longer than shown. Dashed lines indicate the MPS results measured in the center of a finite $11\times 11$ lattice.