Dynamics of optomechanical droplets in a Bose-Einstein condensate

We investigate numerically a one-dimensional Bose-Einstein condensate illuminated by off-resonant laser light which is retroreflected by a single feedback mirror. Studying the ground states of the system, we find density structures which are self-trapped via the optomechanical action of the diffracted light. We show that these structures are stable and exhibit Newtonian dynamics. We propose that these results allow continuous, nondestructive monitoring of condensate dynamics via the optical intensity and may offer opportunities for optical control and transport of coherent matter via gradients in optical phase alone.

Figure 1

Schematic diagram of the SMF configuration

Figure 2

Example of pattern formation or self-structuring of a BEC in a single-feedback-mirror configuration. The parameters used here are $p_0=1.9\times {10}^{-9}, \mathrm{\Delta }=-100, R=0.99, \left({\omega }_r\right)/\mathrm{\Gamma }=9.88\times {10}^{-9}$, and $b_0=20$.

Figure 3

Evolution of a stable optomechanical droplet for red detuning. The parameters are $p_0=2.0\times {10}^{-9}, \mathrm{\Delta }=-800, R=0.99, \left({\omega }_r\right)/\mathrm{\Gamma }=5.69\times {10}^{-8}$, and $b_0=20$

Figure 4

Evolution of a stable optomechanical droplet for blue detuning. The parameters are $p_0=2.0\times {10}^{-9}, \mathrm{\Delta }=800, R=0.99, \left({\omega }_r\right)/\mathrm{\Gamma }=5.69\times {10}^{-8}$, and $b_0=20$.

Figure 5

Evolution of a perturbed blue-detuned droplet, where the initial BEC density has been produced from a Gaussian distribution whose width is adjusted slightly from the ideal value. The parameters are $p_0=5.0\times {10}^{-9}, \mathrm{\Delta }=800, R=0.99, \left({\omega }_r\right)/\mathrm{\Gamma }=5.69\times {10}^{-8}$, and $b_0=20$.

Figure 6

(a) Droplet width ${\sigma }_x$ and (b) peak density ${\left|\mathrm{\Psi }\right|}^2$ against pump intensity $p_0$. Results were calculated through imaginary-time integration with the parameters $R=0.99, \left({\omega }_r\right)/\mathrm{\Gamma }=1.01\times {10}^{-7}, \mathrm{\Delta }=800$, and $b_0=20$.

Figure 7

Example ground-state droplet density profiles of (a) single-, (b) dual-, and (c) triple-peak droplet structures for red detuning, calculated from imaginary-time integration. The parameters are $R=0.99, \left({\omega }_r\right)/\mathrm{\Gamma }=4.05\times {10}^{-7}, b_0=20, \mathrm{\Delta }=-800$, and pump intensity $p_0$ equal to (a) $3.714\times {10}^{-9}$, (b) $2.395\times {10}^{-8}$, and (c) $9.398\times {10}^{-8}$. Shown in (b) is an off-center structural position, consistent with the translational invariance of the system.

Figure 8

Ground-state optical field intensity profiles for the red-detuned system with pump amplitudes (a) $p_0=3.07\times {10}^{-9}$, (b) $p_0=1.43\times {10}^{-8}$, and (c) $p_0=1.53\times {10}^{-8}$, calculated from imaginary-time integration. The other parameters are $R=0.99, \left({\omega }_r\right)/\mathrm{\Gamma }=4.05\times {10}^{-7}, \mathrm{\Delta }=-800$, and $b_0=20$.

Figure 9

Uniformly moving droplet for red detuning ($\mathrm{\Delta }<0$). The parameters are identical to those of Fig. 3 with the addition of $v_0=\hslash q_c/12m$.

Figure 10

Uniformly moving droplet for blue detuning ($\mathrm{\Delta }>0$). The parameters are identical to those of Fig. 4 with the addition of $v_0=\hslash q_c/12m$.

Figure 11

Uniformly accelerating droplet for red detuning ($\mathrm{\Delta }<0$). The parameters are identical to those of Fig. 3 with the addition of $a=-4.0\times {10}^{-9}(\hslash q_c\mathrm{\Gamma }/12m)$.

Figure 12

Uniformly accelerating droplet for blue detuning ($\mathrm{\Delta }>0$). The parameters are identical to those of Fig. 4 with the addition of $a=-4.0\times {10}^{-9}(\hslash q_c\mathrm{\Gamma }/12m)$.

Figure 13

Schematic diagram of the SMF configuration with a mirror misalignment or tilt, labeled $\alpha$.

Figure 14

Evolution of BEC density and optical field intensity when a mirror tilt is present. The parameters are $p_0=1.9\times {10}^{-9}, \mathrm{\Delta }=800, R=0.99, \left({\omega }_r\right)/\mathrm{\Gamma }=5.69\times {10}^{-8}, b_0=20$, and $\left({\mathrm{\Delta }}_x\right)/\left({\mathrm{\Lambda }}_c\right)=1.465\times {10}^{-3}$.

Figure 15

Dependence of droplet acceleration $a_d$ on mirror-tilt-induced shift $\mathrm{\Delta }x/{\mathrm{\Lambda }}_c$. The parameters are $p_0=1.9\times {10}^{-9}, \mathrm{\Delta }=200, R=0.99, \left({\omega }_r\right)/\mathrm{\Gamma }=1.01\times {10}^{-7}$, and $b_0=20$.