Pattern Formation with Trapped Ions

Ion traps are a versatile tool to study nonequilibrium statistical physics, due to the tunability of dissipation and nonlinearity. We propose an experiment with a chain of ions, where dissipation is provided by laser heating and cooling, while nonlinearity is provided by trap anharmonicity and beam shaping. The dynamics are governed by an equation similar to the complex Ginzburg-Landau equation, except that the reactive nature of the coupling leads to qualitatively different behavior. The system has the unusual feature of being both oscillatory and excitable at the same time. The patterns are observable for realistic experimental parameters despite noise from spontaneous emission. Our scheme also allows controllable experiments with noise and quenched disorder.

Figure 1

Chain of ions, each damped by a red-detuned beam and excited by a blue-detuned beam. The blue beam is along the ion chain, while the red beams are at an angle. The intensity of the red beams is lowest at the trap center. Each ion is in its own anharmonic dc potential well (not shown).

Figure 2

Spacetime plot of chain of 50 ions with nearest-neighbor interactions and open boundaries. Re $A$ is plotted using the color scale in the side bar. $A$ is the complex amplitude of the underlying harmonic oscillations. (a) $b=1$ and $c=1$ showing uniform phase synchrony. (b) Same, but with the expected noise from spontaneous emission. (c) $b=1$ and $c=-1$ showing antiphase structure.

Figure 3

System of two oscillators with $b=1$. (a) Bifurcation diagram as $c$ varies. There are supercritical pitchfork bifurcations at $(c,\Delta \theta )=(-1,0)$ and (1, $\pi$). There are supercritical Hopf bifurcations at (1.22, 2.82), (1.22, 3.47), ($-1.22$, $-0.32$), and ($-1.22$, 0.32), which give rise to stable limit cycles (not shown). Solid and dashed lines denote stable and unstable fixed points, respectively. (b) A limit cycle at $c=1.4$.

Figure 4

Spacetime plot of Re $A$ for chain of 50 ions showing an excitation pulse for $b=1$ and $c=0.2$. Perturbing the first oscillator beyond a threshold generates a pulse of antiphase oscillation. The initial conditions were $A_1=-1$ and $A_n=1$ for the rest. The right panel is a zoomed-in view. Color scale is the same as in Fig. 2.