Topological mode conservation and conversion in phononic crystals with temporal interfaces

Abstract

A sudden change in material properties creates a temporal interface and forces a propagating wave to change its frequency while preserving its wavenumber. In contrast to monoatomic lattices with a single frequency-wavenumber pair, polyatomic lattices support multiple frequencies for each wavenumber. To date, experimental observations are limited to topologically trivial monoatomic phononic systems. Here, we utilize analytical, numerical, and experimental methods to examine topologically non-trivial phononic lattices subject to temporal interfaces. In particular, we realize phononic lattices demonstrating single-frequency shift (i.e., mode conservation) and multi-frequency splitting (i.e., mode conversion) following a temporal interface. Accordingly, we generalize temporal analogues of Snell's law and Fresnel equations. Moreover, we utilize Bloch mode overlaps to obtain a phononic time lens and a classical analogue of dynamic quantum phase transitions for phonons. Such overlap determines the probability of mode conversion or conservation after a temporal interface and, more importantly, can carry hidden topological characteristics. Our methodology paves the way for the use of temporal interfaces in probing phonon band topology and the realization of advanced acoustic devices.

Figures (16)

• Figure 1: Mode conversion after temporal interfaces. (a) A temporal interface between two diatomic lattices ( $m_1$=0.5 $m_2$=0.1Kg, $K_1$=0.5 $K_2$=500 N/m, $K_{g1}$=$K_{g2}$=100 N/m). (b) A wave packet with incident frequency 10 Hz propagates in $S_1$. At t=8 s, a temporal boundary changes the structure from $S_1$ to $S_2$. Two pairs of wave packets appear after TI (T1 and R2 to forward while T2 and R1 to backward). (c) (top) Analytical probability of each mode in $S_2$ corresponding to an incident wave on mode 1 of $S_1$. (bottom) Dispersion curves of each state with numerical 2D FFT overlay. (d) (top) Time response of (mass 200 before TI), (mass 400 after TI) and (mass 200 after TI) showing incident, forward, and backward waves, respectively. (bottom) FFT of the three time signals in the top panel showing frequency conversion to multiple frequencies. (e) (top) Spatial profiles at t=6 s (before TI) and t=10 s (after TI). (bottom) FFT of the two spatial profiles in the top panel showing conservation of wavenumber.
• Figure 2: Modification of temporal analogue of Snell's law and Fresnel equations in phononic lattices. For the structure illustrated in Fig.\ref{['fig:concept']}, if the incident wave is on branch 1 of $S_1$: (a) Frequency conversion corresponding to each wavenumber and numerically calculated frequencies after TI for three different wavenumbers. (b) Numerically calculated wavenumbers before $\kappa^{(S1)}_0$ and after $\kappa^{(S2)}$ TI to show conservation of wavenumber. (c) Group velocity ratio after $C^{(S2)}_j$ and before $C^{(S1)}_1$ TI theoretically (lines) compared to the ratio between transmitted waves to incident wave angles (diamonds) to show Snell's law. (d) Ratio between transmitted ( T1 and T2) or reflected ( R1 and R2) wave packet amplitude to the incident wave packet amplitude theoretically (lines) and numerically (markers) to show Fresnel equations.
• Figure 3: Experimental observation of mode conversion. (a) Physical diatomic unit cell. (b) Mathematical model. (c) Maximum obtainable overlap between acoustic and optical modes by changing ground stiffness. We show three cases: (d) changing voltage from -11 V to 0 V while a longitudinal wave is propagating, (e) changing voltage from +7 V to 0 V while a longitudinal wave is propagating, and (f) changing voltage from +5 V to 0 V while a shear wave is propagating. For each case we show (I) analytical dispersion curves and numerically calculated 2D FFT overlay, (II) analytical probability of each mode in state 2, and experimentally observed (III) frequencies and (IV) wavenumbers before and after the temporal interface.
• Figure 4: Phononic time-lens. A temporal interface between two diatomic lattices ( $m_1$=0.5 $m_2$=0.34 g, $K_1$=0.5 $K_2$=0.0306 N/m, $K_{g1}$=0.5$K_{g2}$=2$K_1$, $\epsilon_1=\epsilon_2=-1$ and $\lambda_1=\lambda_2=3$). (a) Analytical dispersion curves and numerically calculated 2D FFT overlay. (b) (top) Analytical probability of each mode in state 2. (bottom) Ratio between wave packets' amplitudes (c) The mid-mass is excited with a wave packet with a temporal interface after 10 s.
• Figure 5: Probing bulk band topology with temporal interfaces. (a) Mathematical model (b) Analytical dispersion curves for each state and numerical 2D FFT overlay. (c) Phase diagram shows the relation between inter-stiffnesses ratio $\frac{K_2}{K_1}$ and number of topologically protected edge modes (i.e., winding number $\nu$) with the chiral symmetry necessary condition $m_1\Tilde{K_2}=m_2\Tilde{K_1}$. Blue (red) dot represents $S_1$ ($S_2$). The vertical lines at $K_2/K_1=1$ and $-1$ represent topological phase boundaries and the corresponding dispersion curves are shown above. (d) The first mass is excited with a wave packet centered at $f=24$ Hz corresponding to $\kappa_{DQPT}=0.8 \frac{\pi}{a}$. A temporal interface at $t=\tau$ changes the system abruptly to $S_2$. (e) Analytical probability of each mode in $S_2$ corresponding to an incident wave on mode 1 of $S_1$. (f) Revival "Loschmidt” amplitude (top) and rate function (bottom). The revival amplitude vanishes at the critical times (circles) which are the same times corresponding to nonanalytic cusps in the rate function which represent dynamical quantum phase transition in the time boundary effect.