Localization of Floquet states along a continuous line of periodic orbits

A periodically driven particle in an infinite square well is shown to exhibit quantum localization due to a continuous line of periodic orbits in the classical system. Individual Floquet eigenstates localized along this line of periodic orbits are identified. The enhanced localization persists for field strengths beyond that at which the continuous line of orbits is destroyed in the classical dynamics. These results may be relevant to experiments involving trapping potentials with flat regions.

Figure 1

Strobe plots of the classical dynamics at various field strengths. In (a) the $l=1$ primary resonance is visible at the left wall near $p=30$. The chaotic region remains confined to low momenta at all field strengths, but grows in size as $ϵ$ is increased. A continuous line of periodic orbits runs along $p=0$ from $x=4ϵ∕{\omega }_0^2-1$ to $x=1$ (this line of orbits is not visible in the strobe plots). Note that only the $p⩾0$ portion of the phase space is shown because the system is symmetric in $p$.

Figure 2

Local inverse participation ratio $\mathcal{L}(x_0,p_0)$ as a function of phase-space location for several field strengths. Dark regions indicate high values of $\mathcal{L}(x_0,p_0)$ and therefore areas of increased localization. The thick, solid line at the bottom of each plot indicates the range of $x$ values covered by the line of periodic orbits that runs along $p=0$. Note that $\mathcal{L}(x_0,p_0)$ is peaked along the line of periodic orbits in (a), (b), and (c). In (d) the line of orbits has shrunk to a single point, but there is still a small increase in $\mathcal{L}(x_0,p_0)$ in that vicinity.

Figure 3

Plots of ${\mathcal{I}}_{\alpha }$ versus $H_{\alpha }$ for several field strengths. At high values of $H_{\alpha }$ the points form a regular sequence. These points correspond to states located in the high-momentum regular region of phase space. Points indicated by solid squares correspond to states that lie in or around a resonance island. Points indicated by solid triangles (and labled by a state number $\alpha$) correspond to Floquet states that are peaked along the line of orbits. Husimi distributions for these states are shown in Figs. 4, 5. Note that for each field strength the states peaked on the line of orbits have the highest values of ${\mathcal{I}}_{\alpha }$ of any state that does not lie in or near a resonance island.

Figure 4

Husimi distributions of the first four Floquet eigenstates (ordered by increasing value of $⟨p^2⟩$) for $ϵ=600$. The thick, solid line at the bottom of each plot indicates the range of $x_0$ values covered by the line of periodic orbits that runs along $p_0=0$. Note that all four states are peaked along the line of periodic orbits.

Figure 5

Husimi distributions of Floquet eigenstates that are peaked along the line of periodic orbits for several field strengths. The thick, solid line at the bottom of each plot indicates the range of $x_0$ values covered by the line of periodic orbits that runs along $p_0=0$. The index $\alpha$ labels the Floquet state by increasing value of $⟨p^2⟩$ for each field strength.

Figure 6

Plot of the value of ${\mathcal{I}}_{\alpha }$ as a function of field strength for the state most closely identified with the line of periodic orbits. For comparison we show the mean ${\mathcal{I}}_{\alpha }$ of all chaotic states and the RMT prediction for $\mathcal{I}$. Error bars for the mean ${\mathcal{I}}_{\alpha }$ are determined by dividing the standard deviation by the square root of the number of chaotic states.

Figure 7

The plots on the left are strobe plots of the classical motion at three field strengths for which the line of periodic orbits along $p=0$ no longer exists. The plots on the right show $\mathcal{L}(x_0,p_0)$ as a function of phase-space location for the same field strengths, where the horizontal and vertical axes on the right-hand plots, $x_0$ and $p_0$, respectively, take the same range of values as $x$ and $p$ on the left-hand plots. Note that at $ϵ=3800$ there is still a prominent peak in $\mathcal{L}(x_0,p_0)$ near the right wall at $p_0=0$. For $ϵ=5000$ this peak has moved well away from $p_0=0$.

Figure 8

Husimi distributions of Floquet states that are peaked near ($x_0=1$, $p_0=0$) for $ϵ=3800$ and $ϵ=4500$. For $ϵ=5000$ we were unable to find a Floquet state that was peaked near ($x_0=1$, $p_0=0$).