Time reversal of a wave packet with temporal modulation of gauge potential

We show that time reversal can be achieved with a spatially uniform temporal modulation of the gauge potential. The method can be applied to the quantum wave function of electrons where there are natural gauge potentials, and to classical waves such as the acoustic and optical waves where effective gauge potentials have been demonstrated recently. This time-reversal process can provide a compensation of higher-order dispersion effects. Moreover, our approach can be used to time reverse the propagation of a wave packet in three dimensions, as well as a topologically protected one-way edge state where the underlying Hamiltonian does not have time-reversal symmetry. Our work shows that modulation of gauge potentials provides a versatile approach for dynamically controlling wave propagation.

Figure 1

(a) Schematic showing the time reversal of a pulse (orange) in a tight-binding lattice by applying an electric field. (b) The band structure $\omega \left(k\right)$ for the tight-binding lattice with $\varphi =0$ (blue curve) and $\varphi =\pi$ (red curve). (c) Propagation of a wave packet in the one-dimensional lattice. An electric field pulse is applied around the time $t=0$. Colors show the intensity density of the wave packet.

Figure 2

(a) Three-dimensional lattice with an electric field pulse pointing at the $\left[111\right]$ direction. (b) The initial wave-packet distribution at $t=-\tau$. (c)–(h) Normalized projections of the calculated intensity distribution on the $xy$ plane $[P_z(x,y,t)={\sum }_l|w_{m,n,l}\left(t\right){|}^2]$ in (c), (e), and (g), and on the $yz$ plane $[P_x(y,z,t)={\sum }_m|w_{m,n,l}\left(t\right){|}^2]$ in (d), (f), and (h), at times $-\tau$, 0, and $\tau$, respectively.

Figure 3

(a) Two-dimensional lattice under a constant magnetic field, which is described by the Laudau gauge phase distribution. In addition, an electric field pulse pointing at the [11] direction is applied. (b) and (c) The band structure at $t<0$ and $t>0$, respectively, for a stripe of tight-binding lattice shown in (a). The stripe is infinite in the $x$ direction and finite along the $y$ direction. The red (blue) curve shows the edge state on the top (bottom) edge and gray curves show the bulk bands. A gold dot represents a wave packet. (d)–(f) Normalized wave-packet intensity distribution at times $-\tau$, 0, and $\tau$, respectively.