Three-Dimensional Dynamic Localization of Light from a Time-Dependent Effective Gauge Field for Photons

We introduce a method to achieve the three-dimensional dynamic localization of light. We consider a dynamically modulated resonator lattice that has been previously shown to exhibit an effective gauge potential for photons. When such an effective gauge potential varies sinusoidally in time, dynamic localization of light can be achieved. Moreover, while previous works on such an effective gauge potential for photons were carried out in the regime where the rotating wave approximation is valid, the effect of dynamic localization persists even when the counterrotating term is taken into count.

Figure 1

A one-dimensional (a) and a three-dimensional (b) photonic resonator lattice with two kinds of resonators with frequency ${\omega }_A$ (red dots) and ${\omega }_B$ (blue dots). The nearest-neighbor coupling is dynamically modulated and the phase of the coupling constant modulation itself can be time dependent with the signs being flipped between two neighboring bonds. The lattice is assumed infinite in all directions.

Figure 2

The quasienergies as a function of $\alpha$, for the Hamiltonian of Eq. (4), with ${\varphi }_x(t)=\alpha \cos ({\omega }_{M}t)$. Here we choose $V=0.2{\omega }_M$. Each curve corresponds to a different wave vector $k_x$, in the range, $-\pi /a<k_x<\pi /a$.

Figure 3

Propagation of a photon wave packet in a $40a\times 40a\times 40a$ three-dimensional lattice. (a) The initial condition at $t=0$, with $x_0=y_0=z_0=20a$ and $k_x=-k_y=-k_z=-1.283a^{-1}$. (b) The wave packet at $t=1.25a/c$ with no phase modulation. (c) and (d) The wave packet at $t=1.25a/c$ and $t=5a/c$, respectively, with phase modulation. The parameters of the phase modulation are $\alpha =2.40483$ and ${\omega }_M=1.5c/a$. The coupling strength between the resonators is $V=2.4\pi c/a$.

Figure 4

Quasienergies as a function of phase-modulation strength $\alpha$, for the Hamiltonian of Eq. (12), with ${\varphi }_x(t)=\alpha \cos ({\omega }_{M}t)$. $V=0.2\mathrm{\Omega }$. (a) $\mathrm{\Omega }=2{\omega }_M$. (b) $\mathrm{\Omega }=4{\omega }_M$. Each blue curve corresponds to a different wave vector $k_x$, in the range of $-\pi /a<k_x<\pi /a$. The dashed red line is the envelope for the same Hamiltonian, but calculated using the rotating wave approximation.