Effective long-distance interaction from short-distance interaction in a periodically driven one-dimensional classical system

We study the classical dynamics of many interacting particles in a periodically driven one-dimensional (1D) system. We show that under the rotating wave approximation (RWA), a short-distance 1D interaction ($\delta$ function or hard-core interaction) becomes a long-distance two-dimensional (2D) interaction which only depends on the distance in the phase space of the rotating frame. The RWA interaction describes the effect of the interaction on the slowly changing amplitude and phase of the oscillating particles, while the fast oscillations take on the role of a force carrier, which allows for interaction over much larger effective distances.

Figure 1

Sketch of interacting particles in 1D harmonic trap. Ultracold atoms (yellow dots) are trapped in 1D harmonic potential. The atoms are further driven by two counterpropagating laser beams (red waves). The intensity of laser beams is tuned periodically, i.e., $\propto cos{\omega }_{d}t$. The total potential at a fixed moment (blue rippled curve) is the harmonic trapping potential plus a cosine function. The interaction potential between atoms is assumed to be $V(x_i-x_j)$.

Figure 2

Renormalized RWA interactions. (a) Inverse power-law interaction potentials and the example plots for $n=1/2$ (Coulomb interaction), $n=1, n=3$, and $n=\infty$ (hard-core interaction). (b) Renormalized RWA interaction potentials corresponding to the inverse power-law potentials shown in (a). Parameter: $\beta =0.01$.

Figure 3

Two-body dynamics for different interactions. (a) In the presence of interaction, the two particles start to rotate at a frequency ${\omega }_R$, which depends on their phase-space distance $R$. (b) Rotating frequency ${\omega }_R$ as function of $R$. The dots represent the data from Poincaré mapping, while the solid lines are given by Eq. (39). Different colors represent different interactions as indicated on the plot. Interaction potential parameters: $\beta =0.1,\varepsilon =1$ for Lorenz interaction; $\beta =0.1$ for Coulomb interaction; $\epsilon =0.01,\sigma =0.1$ for Lennard-Jones interaction; hard-core interaction is modeled by the inverse power-law interaction with $\beta =0.1,n=20$.

Figure 4

Three-body dynamics for different interactions. (Left column) The upper figure shows the trajectories of three particles in the presence of Coulomb interaction with $\beta =0.1$. The colored dots represent the initial conditions and the colored solid lines represent the dynamics from RWA EOM (9). The black dotted lines are the data from EOM (6) combined with Poincaré mapping. The lower figure shows the time evolution of $r_i\left(t\right)=\sqrt{X_i^2\left(t\right)+P_i^2\left(t\right)},i=1,2,3$ obtained from RWA EOM (9) (solid colored lines) and the Poincaré mapping (dotted black lines). (Middle column) The middle two figures show the three-body dynamics for hard-core interaction with $\beta =0.1$. (Right column) The right two figures show the three-body dynamics for Lennard-Jones interaction with parameters $\epsilon =0.01$ and $\sigma =0.1$.

Figure 5

Two-body dynamics under driving. (a) Contour plot of single-particle Hamiltonian function (40) in phase space and the initial conditions of two particles (green dots). (b) The trajectories of two particles in phase space without interaction (black, $\beta =0$) and with Coulomb interaction (red and blue, $\beta =0.1$). The black and red trajectories are obtained from the time evolution based on the original EOM (6) combined with the technique of Poincaré mapping. The blue trajectories are obtained from the time evolution based the RWA EOM (9). Driving parameter: $\mathrm{\Lambda }=0.1$. (c) The trajectories of two particles in phase space with interaction potential $V(x_i-x_j)={\beta }^2/{(x_i-x_j)}^2$. All the trajectories with interaction (red and blue, $\beta =0.1$) and without interaction (black, $\beta =0$) completely overlap each other. Driving parameter: $\mathrm{\Lambda }=0.1$.

Figure 6

Eight-body dynamics under driving. (a) Contour plot of single-particle Hamiltonian function (40) in phase space with $\delta =0, \mathrm{\Lambda }=0.1$, and $k=4$. The yellow dots indicate the initial conditions of eight particles. (b) The trajectories of eight particles in phase space without interaction (black) and with Coulomb interaction (red and blue, $\beta =0.1$). Driving parameter: $\mathrm{\Lambda }=0.1$. (c) The trajectories of eight particles in phase space with Lennard-Jones interaction (red and blue, $\sigma =0.1$ and $\epsilon =0.01$). Driving parameter: $\mathrm{\Lambda }=0.1$.