Relativistic motion of an Airy wave packet in a lattice potential

We study the dynamics of an Airy wave packet moving in a one-dimensional lattice potential. In contrast to the usual case of propagation in a continuum, for which such a wave packet experiences a uniform acceleration, the lattice bounds its velocity, and so the acceleration cannot continue indefinitely. Instead, we show that the wave packet's motion is described by relativistic equations of motion, which surprisingly arise naturally from evolution under the standard nonrelativistic Schrödinger equation. The presence of the lattice potential allows the wave packet's motion to be controlled by means of Floquet engineering. In particular, in the deep relativistic limit when the wave packet's motion is photonlike, this form of control allows it to mimic both standard and negative refraction. Airy wave packets held in lattice potentials can thus be used as powerful and flexible simulators of relativistic quantum systems.

Figure 1

Probability density of the Airy wave packet given in Eq. (2) at $t=0$. For positive values of $x$ the probability density drops sharply with distance, while for negative $x$ the function has a decaying oscillatory behavior. The slowness of this decay (3) means that the Airy function is not normalizable. Under the action of the Hamiltonian in Eq. (1) this wave packet will accelerate to the right at a constant rate.

Figure 2

Trajectory of an Airy wave packet on a lattice. The system is initially prepared in an Airy state, and evolves under the action of the lattice Hamiltonian (5). Initially the peaks in the wave packet move along parabolic trajectories corresponding to a uniform positive acceleration, marked by the white dashed line. At longer times, however, the acceleration of the wave packet decreases as it approaches relativistic speeds. Note that the amplitudes of the peaks decrease with time due to wave-packet spreading; this occurs because the Airy wave packet is aperture limited (see Sec. 2b). Lattice discretization: $\mathrm{\Delta }x=0.2$.

Figure 3

Relativistic motion of the lattice Airy wave packet, obtained by a numerical simulation of the lattice system. (a) Symbols denote the lattice site occupied by the first peak of the wave packet. Initially this position increases quadratically with time, shown by the dashed line, but for $t>100$ the increase becomes slower, tending towards a linear rise. The solid line shows the relativistic prediction (9), where the values of $c$ and $\alpha$ are obtained from a two-parameter fit to the data points. The fit is excellent. (b) Velocity of the first peak of the lattice Airy function. Symbols show that values obtained from the numerical time derivative of the data from the simulation, while the solid line shows the theoretical result (8). Initially the velocity rises linearly, but flattens off as it begins to approach the maximum lattice velocity $v^{\max }$, shown with the dashed line. (c) Acceleration observed in the rest frame of the lattice. For small times the acceleration is constant, but drops as the wave packet enters the relativistic regime. The dashed line indicates the proper acceleration, $\alpha$, which remains constant.

Figure 4

Values of the proper self-acceleration $\alpha$, obtained by curve fitting the numerical results to Eq. (9). The scaling of this quantity is described well by the power law $\alpha =2\mathrm{\Delta }{x}^3$, shown by the dashed red line.

Figure 5

Bloch oscillation of a Gaussian wave packet in a tilted lattice. (a) Probability density of the wave packet as a function of time. The wave packet makes a slow oscillatory motion (Bloch oscillation) described by Eq. (13). (b) Probability density in momentum space, ${\left|\tilde{\psi }(k,t)\right|}^2$, for the same system. The sharply peaked distribution moves linearly with time across the first Brillouin zone, crossing the boundary at $t=500$, corresponding to the reversal of motion of the wave packet in space. Parameters of the system: $\mathrm{\Delta }x=0.2$, $V_0=2\pi J/1000$.

Figure 6

(a) The effective tunneling, $J_{\mathrm{eff}}$, for a sinusoidally driven lattice has a Bessel function dependence on the driving parameter, given by Eq. (15). The symbols mark the values of $K_0$ used to obtain the results shown in Fig. 7. For $K_0=0.5$, the amplitude of the tunneling is slightly reduced from its undriven value. At $K_0=1.691$, $J_{\mathrm{eff}}=0.403$, while at $K_0=2.4048$ the effective tunneling vanishes. For $K_0>2.4048$, the shaded area, the effective tunneling is negative; at $K_0=3.80$ the negative effective tunneling takes its maximum value. (b) We denote the ratio of $v^{\max }$ in the undriven system to $v^{\max }$ in the driven system as the refractive index. As $K_0$ approaches the first zero of the Bessel function, the refractive index increases, and diverges at $K_0=2.4048$. In the shaded region, the refractive index is negative.

Figure 7

Trajectories of an Airy wave packet in a driven lattice; simulation parameters: $\omega =2\pi$, $\mathrm{\Delta }x=0.2$. In all cases $K_0$ is initially set to a value of $K_0=0.5$, for which $J_{\mathrm{eff}}$ is slightly reduced from its undriven value. At $t=30$, $K_0$ is abruptly changed to another value, while at $t=60$ it reverts to its original value; these boundaries are marked by the vertical dashed lines. (a) Between $t=30$ and $t=60$, $K_0$ is set to a value of $K_0=1.691$. In this time interval the velocity of the wave packet is reduced by more than half, and the peaks appear to undergo refraction at the boundaries described by Snell's law. (b) By tuning $K_0$ to a zero of the Bessel function ($K_0=2.4048$), the wave packet's motion is frozen. This corresponds to the medium's refractive index diverging, producing an analogous effect to “stopped light.” (c) Setting $K_0=3.80$ causes the wave packet to reverse its motion since the effective tunneling becomes negative. This behavior mimics the phenomenon of negative refraction.

Figure 8

Three different forms of behavior for accelerated quantum systems. The velocity of a continuum Airy wave packet rises linearly with time, and increases without limit. In contrast, in a discrete lattice system, the velocity initially rises linearly, but asymptotically approaches the value of $v^{\max }$, which acts as the speed of light in this system. It should be noted that, in this case, the proper acceleration of the wave packet indeed stays constant. Finally, tilting the lattice potential subjects a wave packet to a constant uniform force. The presence of the lattice, however, means that the wave packet does not accelerate uniformly, but instead undergoes Bloch oscillation.