Dynamics of nonlinear excitations of helically confined charges

We explore the long-time dynamics of a system of identical charged particles trapped on a closed helix. This system has recently been found to exhibit an unconventional deformation of the linear spectrum when tuning the helix radius. Here we show that the same geometrical parameter can affect significantly also the dynamical behavior of an initially broad excitation for long times. In particular, for small values of the radius, the excitation disperses into the whole crystal whereas within a specific narrow regime of larger radii the excitation self-focuses, assuming finally a localized form. Beyond this regime, the excitation defocuses and the dispersion gradually increases again. We analyze this geometrically controlled nonlinear behavior using an effective discrete nonlinear Schrödinger model, which allows us among others to identify a number of breatherlike excitations.

Figure 1

(a) Equidistant configuration of ions confined on the toroidal helix for $\nu =\frac{1}{2}$ and $N=8$. The yellow arrows indicate the initial velocities of the particles. (b) Highest (solid blue line) and lowest (solid red line) frequencies of the linearization spectrum around the equilibrium equidistant configuration as a function of $r$ for $N=60$ particles. The black dots and the empty diamonds refer to the frequencies corresponding to the center of mass and the out of phase mode respectively. The vertical lines mark the radii of the helix we use in our calculations. The small insets depict the form of the respective vibrational band structures $\omega \left(k\right)$ at the corresponding values of $r$. (c) Initial local energy $E_n$ profile as a function of the particle index $n$.

Figure 2

(a)–(f) Time evolution of the initial Gaussian excitation presented in Fig. 1 for $N=60, \nu =1/2$ for increasing $r$ corresponding to the points (a) $r_0$, (b) $r_1$, (c) $r_2$, (d) $r_3$, (e) $r_4$, and (f) $r_5$ marked in Fig. 1. Colors encode the values of local energy $E_n$ for each particle $n$ and time $t$. For the same values of $r$ panels (g)–(l) depict the time evolution of the normalized participation ratio $\mathcal{P}$ of the excitation. The solid blue lines are the results from our numerical simulations corresponding to (a)–(f), whereas the dashed red lines correspond to the results for a harmonic approximation of the potential.

Figure 3

Absolute value of the discrete Fourier transform of the local energy excitation profile as a function of time and wave number $k$. (a)–(f) correspond to increasing $r$, ranging from $r_0$ to $r_5$ as indicated in Fig. 1.

Figure 4

(a) The ratio between the diagonal and off diagonal elements of the Hessian $\frac{\left|H_0\right|}{2\left|H_1\right|}$ as a function of $r$. (b)–(e) Coefficients $A_l,B,C_l,D_l$ defined in Eq. (11) as a function of $r$. The solid blue lines depict the coefficient values for $l=1$, whereas the black doted lines for $l=2$ and the red dashed ones for $l=3$. The cyan (shaded) area marks the region of applicability of the RWA as derived from (a). The vertical lines correspond to the values of $r$ we study with the DNLS model. Note that three of them namely $r_2,r_3$ and $r_4$ are the same as those studied in the previous section [Fig. 1] whereas $r_{45}$ lies between $r_4$ and $r_5$. The insets provide a zoom into the region of interest.

Figure 5

(a)–(d) Time propagation of the initial Gaussian excitation of Fig. 1 under the DNLS equation (12) for increasing $r$ corresponding to the points (a) $r_2$, (b) $r_3$, (c) $r_4$, and (d) $r_{45}$ marked in Fig. 4. Colors encode the values of local energy $E_n=2{\omega }_0^2{\left|{\mathrm{\Psi }}_n\right|}^2$ for each particle $n$ and time $t$.

Figure 6

(a)–(d) Time propagation of breatherlike excitations in the degeneracy regime, at $r=r_3$ in (a), (b), (d), and $r=r_4$ in (c). The color plots of the first column depict the values of the local energy $E_n$ whereas the second column provides the initial profiles of the excitation in position coordinates, i.e., $\mathrm{\Delta }{u}_n=u_n-u_n^{\left(0\right)}$ as a function of the particle index $n$. The third column illustrates the time evolution of the participation ratio $\mathcal{P}$ for the degenerate geometries and the initial conditions of (a)–(d). In (a)–(d) the initial excitation comes only from the displacement $\mathrm{\Delta }{u}_n$ of particles from their equilibrium.

Figure 7

(a) A schematic illustration of three charges on a ring at equilibrium (1, 2, 3) and after particle 1 has been displaced ($1^{\prime }$, 2, 3). The dashed lines denote the Euclidean distances in equilibrium, whereas the solid ones, the distances after the displacement. (b), (c) The time evolution of a Gaussian excitation for $r=r_1$. (d), (e) The time evolution of a Gaussian excitation for $r=r_2$. The colors in (b), (d) encode the values of the displacement from equilibrium $x_n$ whereas in (c), (e) the values of the local energies $E_n$ defined in Eq. (A5). (b)–(e) show simulation results for $N=60$ particles and $\nu =1/2$.