Collective Excitations and Instability of an Optical Lattice due to Unbalanced Pumping

We solve self-consistently the coupled equations of motion for trapped particles and the field of a one-dimensional optical lattice. Optomechanical coupling creates long-range interaction between the particles, whose nature depends crucially on the relative power of the pump beams. For asymmetric pumping, traveling density wavelike collective oscillations arise in the lattice, even in the overdamped limit. By increasing the lattice size or pump asymmetry, these waves can destabilize the lattice.

Figure 1

A dipole trap created by two lasers of equal frequency but unequal power. The intensity mirrored for better visibility ranges between $I_{\min }=\frac{1}{2}{ϵ}_{0}c(|E_0|-|E_1|{)}^2$ and $I_{\max }=\frac{1}{2}{ϵ}_{0}c(|E_0|+|E_1|{)}^2$. Trapped particles form disk-shaped clouds and are modeled as beam splitters. Because of the pump asymmetry, the electric field has no nodes. Backaction of trapped particles distorts the field and reduces the lattice constant.

Figure 2

The lattice constant as a function of the asymmetry is shown in thick curves for $\zeta =0.01$, $\zeta =0.1$, $\zeta =0.5$, $\zeta =1$, $\zeta =2$. Shaded areas indicate regions of stability (see page 4), for $N\le 800$ (darkest shade), $N\le 100$, $N\le 10$ and $N\le 2$ (lightest shade). The white area is unstable, see Eq. (4). The circle marks the parameter regime of Fig. 4.

Figure 3

Real (thick line) and imaginary (thin line) part of the first few eigenvalues $z_1$ (continuous line), $z_2$ (slashed line), $z_3$ (dotted line), for a lattice of $N=100$ (a) and $N=10$ (b) clouds of polarizability $\zeta =0.1$ each.

Figure 4

Time dependence of position distortions $\xi$ (in color coding, in units of ${10}^{-3}\lambda$) in an asymmetrically pumped overdamped optical lattice of $N=100$ clouds with polarizability $\zeta =0.1$, after excitation of mode $\mathrm{Re}({\mathbf{v}}_1)$ at $t=0$ with amplitude ${10}^{-3}\lambda$. The continuous gray contour line is $\xi =0$. In (a), the system is subcritical: $\mathcal{A}=0.632$ and $z_1/\beta =-0.55-6.88i$. The excitation results in a density wave propagating towards the stronger beam and dying out. In (b), at supercritical asymmetry $\mathcal{A}=0.655$, the eigenvalue is $z_1/\beta =1.48-5.94i$. The density wave is now amplified, and at $t\approx 0.5\mu \lambda c/I_0\approx 2.5\mu /\beta$, we leave the linear regime. Then a local drop in the lattice constant develops at $x\approx 30\lambda$, which will result in two clouds coalescing, and eventually all particles will be pushed away by the stronger beam (not shown in figure).