Spatial effects in superradiant Rayleigh scattering from Bose-Einstein condensates

We present a detailed theoretical analysis of superradiant Rayleigh scattering from atomic Bose-Einstein condensates. A thorough investigation of the spatially resolved time evolution of optical and matter-wave fields is performed in the framework of the semiclassical Maxwell-Schrödinger equations. Our theory is not only able to explain many of the known experimental observations, e.g., the behavior of the atomic side-mode distributions, but also provides further detailed insights into the coupled dynamics of optical and matter-wave fields. To work out the significance of propagation effects, we compare our results to other theoretical models in which these effects are neglected.

Figure 1

Schematic representation of BEC superradiance. (a) A cigar-shaped BEC of length $L$ (filled ellipsoid) is exposed to a linearly polarized laser pulse with amplitude ${\mathcal{E}}_l\left(t\right)$ and wave vector ${\mathbf{k}}_l$. Stimulated Rayleigh scattering of laser photons off the condensate produces recoiling atoms in side modes with well-defined momenta. The side mode (0,0) refers to the BEC at rest whereas only one of the two first-order side modes, i.e., $(1,-1)$, is shown. The scattered photons propagate mainly along the condensate axis in the endfire modes ${\mathcal{E}}_{\pm }$. (b) Population transfer to and from the side mode $(n,m)$ due to interaction with the laser pulse as implied by Eq. (7).

Figure 2

Spatially dependent model. (a) Spatial distribution of the first-order forward $(1,\pm 1)$ and backward $(-1,\pm 1)$ atomic side modes, after applying a laser pulse of duration $t_f=8.5\mu \mathrm{s}$ and strength $g=3\times {10}^6{\mathrm{s}}^{-1}$ to the condensate followed by a free propagation for a time $t_p=25\mathrm{ms}$. (b) Spatial distributions of the atomic side modes and the optical endfire modes ${\mathcal{E}}_{\pm }$ at time $t_f$. For the sake of illustration, in both cases the population of the BEC (0,0) has been multiplied by 0.25.

Figure 3

Spatially dependent model. Atomic side-mode distributions in the strong-pulse regime for $g=3.0\times {10}^6{\mathrm{s}}^{-1}$ and $t_f=12\mu \mathrm{s}$. The gray level of each square represents the “relative” probability $p_{nm}$ as defined in the text. The arrow indicates the direction of the incoming laser pulse and points towards the BEC “side mode” (0,0).

Figure 4

(Color online) Origin of the X pattern: schematic representation. Arrows show population transfer between different side modes $(n,m)$ via couplings to the optical fields ${\mathcal{E}}_{\pm }$. The side mode (2,0) does not become populated due to the missing spatial overlap between the side modes $(1,\pm 1)$ and the endfire modes ${\mathcal{E}}_{\pm }$, respectively.

Figure 5

(Color online) Observed spatial asymmetry in an accurate schematic representation based on Fig. 1(A) of [2]. The forward (backward) peaks clearly appear at angles smaller (larger) than 45°. The dashed lines indicate the angles at 45°.

Figure 6

Spatially independent model. Atomic side-mode distributions (compare with Fig. 3). (a) Strong-pulse regime: $t_f=10.4\mu \mathrm{s}$ and $g=3.0\times {10}^6{\mathrm{s}}^{-1}$; (b) weak-pulse regime: $t_f=151.5\mu \mathrm{s}$ and $g=3.5\times {10}^5{\mathrm{s}}^{-1}$; (c) weak-pulse regime: $t_f=182\mu \mathrm{s}$ and $g=3.5\times {10}^5{\mathrm{s}}^{-1}$.

Figure 7

Spatially dependent model and weak-pulse regime $(g=3.0\times {10}^5{\mathrm{s}}^{-1})$. Spatial distribution of the condensate (full curves), side modes (1,1) (dotted), $(1,-1)$ (dash-dotted), and (2,0) (dashed) at times $t_f=$ 200 (a); 270 (b); 350 (c); $420\mu \mathrm{s}$ (d).

Figure 8

Depletion of condensate center in the weak-pulse regime $(g=6.0\times {10}^5{\mathrm{s}}^{-1})$. Spatial distribution of the condensate at times $t_f=$ 215 (a) and $350\mu \mathrm{s}$ (b).

Figure 9

(a) Emitted superradiant light intensity as a function of time for $g=3.0\times {10}^5{\mathrm{s}}^{-1}$. Inset: Populations of condensate (full curve), side modes $(1,\pm 1)$ (dashed) and (2,0) (dotted). (b) Moduli of total electric field $\mid e_{+}\mid$ (full curve) and components $\mid e_{+}^{(0,0)}\mid$ (dashed) and $\mid e_{+}^{(1,1)}\mid$ (dotted).

Figure 10

Spatial distribution of optical endfire modes for $g=3.0\times {10}^5{\mathrm{s}}^{-1}$ and times $t_f=200$ (a); 270 (b); 305 (c); and $420\mu \mathrm{s}$ (d). Full curve, $\mid e_{+}\mid$; dotted, $\mid e_{+}^{(0,0)}\mid$; dashed, $\mid e_{+}^{(1,1)}\mid$; dash-dotted, $\mid e_{-}\mid$.

Figure 11

Atomic side-mode distributions in the weak-pulse regime. The diagrams are for $g=4.5\times {10}^5{\mathrm{s}}^{-1}$ and pulse durations $t_f=140$ (a); 205 (b); 260 (c); $335\mu \mathrm{s}$ (d). Gray level as in Fig. 6.

Figure 12

As Fig. 11, but for $g=7.0\times {10}^5{\mathrm{s}}^{-1}$ and pulse durations $t_f=80$ (a); 170 (b); 210 (c); $280\mu \mathrm{s}$ (d). Gray level as in Fig. 6.

Figure 13

Weak-pulse regime $(g=2.0\times {10}^5{\mathrm{s}}^{-1})$. Time evolution of side-mode populations (a) and distributions (b) for the spatially independent model (16). (c) Time evolution of side-mode populations for the spatially dependent model. Gray level as in Fig. 6.