Light scattering by ultracold atoms in an optical lattice

We investigate theoretically light scattering of photons by ultracold atoms in an optical lattice in the linear regime. A full quantum theory for the atom-photon interactions is developed as a function of the atomic state in the lattice along the Mott-insulator–superfluid phase transition, and the photonic-scattering cross section is evaluated as a function of the energy and of the direction of emission. The predictions of this theory are compared with the theoretical results of a recent work on Bragg scattering in time-of-flight measurements [A.M. Rey et al., Phys. Rev. A 72, 023407 (2005)]. We show that, when performing Bragg spectroscopy with light scattering, the photon recoil gives rise to an additional atomic site-to-site hopping, which can interfere with ordinary tunneling of matter waves and can significantly affect the photonic-scattering cross section.

Figure 1

Light scattering by atoms trapped in a one-dimensional optical lattice with lattice constant $d_0$. The atoms are probed by a laser beam, with wave vector $k_{\mathit{L}}$ and frequency ${\omega }_L$, which couples to the atomic dipole transition at frequency ${\omega }_0$ with ground state $|g\rangle$ and excited state $|e\rangle$ (see inset). The spectrum of the scattered light is measured at a detector as a function of the angle of emission. In experiments, one can also use a second laser beam, into which the photon is emitted with high probability, hence, implementing stimulated Bragg scattering [8].

Figure 2

Stokes component of the differential scattering cross section [in units of $\mathcal{A}(\Omega )$] as a function of frequency (in units of the recoil frequency ${\omega }_R$) for two different scattering angles, corresponding to $q_x{d}_0=2\pi /7$ (top row) and to $q_x{d}_0=6\pi /7$ (bottom row). The curves have been evaluated for a lattice of $M=7$ site and $N=M=7$ composed by ${}^{87}\mathrm{Rb}$ atoms in the $|\mathrm{F}=2,m_F=2\rangle$ hyperfine ground state. The black solid line corresponds to the numerical results, the blue dashed line to the analytical formulas (see text), and the red dashed-dotted line to the model of [24], where the light-induced hopping is neglected. Plots (a) and (c) are evaluated for $V_0=8.1\hspace{0.5em}\hslash {\omega }_R$ ($U/J\approx 17$) which corresponds to the Mott-insulator state. Plots (b) and (d) are evaluated for $V_0=0.1\hspace{0.5em}\hslash {\omega }_R$ ($U/J\approx 1$) which corresponds to the superfluid state. Other parameters are $d_0=413$ nm, $a_s=105a_0$, with $a_0$ being the Bohr radius and ${\omega }_r=10{\omega }_R$ corresponding to the experimental parameters in [22]. (For these parameters, the size of the radial wave packet is ${\xi }_r=10a_s$.) The frequency resolution is set to $\Delta \omega =300$ Hz, corresponding to an integration time $T=3$ ms.

Figure 3

Stokes component of the differential scattering cross section [in units of $\mathcal{A}(\Omega )$] as a function of the frequency (in units of ${\omega }_R$) and of the Bragg angle $\Theta =q_x{d}_0$ (in units of $\pi$). The plots have been evaluated numerically for (a) $V_0=8.1\hspace{0.5em}\hslash {\omega }_R$ and $U/J\approx 17$, (b) $V_0=0.1\hspace{0.5em}\hslash {\omega }_R$ and $U/J\approx 1$, and (c) $V_0=0.1\hspace{0.5em}\hslash {\omega }_R$ and $U/J\approx 0.1$. The other parameters are as in Fig. 2.

Figure 4

Contour plot of the Stokes component of the differential scattering cross section [in units of $\mathcal{A}(\Omega )$] as a function of the frequency (in units of ${\omega }_R$) and of the lattice depth $V_0$ in units of ${\omega }_R$ for $q_x{d}_0=6\pi /7$ (the corresponding value of the ratio $U/J$ is reported in the axis between the squared bracket). The black dashed line marks the critical value at which the phase transition occurs in the thermodynamic limit. The other parameters are given in Fig. 2.

Figure 5

Intensity of the scattered light (in arbitrary units) as a function of the Bragg angle $\Theta =q_x{d}_0$ (in units of $\pi$). The parameters are the same as in Fig. 2 and (a) $V_0=8.1E_R$ ($U/J\approx 17$) and (b) $V_0=0.1E_R$ ($U/J\approx 1$). The black solid line corresponds to the numerical result, the blue dashed-dotted line to the analytical solution, and the red dashed line to the numerical result obtained discarding the light-induced hopping term as in [32] and [24].

Figure 6

Intensity of the scattered light (in arbitrary units) as a function of the Bragg angle $\Theta =q_x{d}_0$ (in units of $\pi$) and of the lattice depth $V_0$ (in units of $\hslash {\omega }_R$) (the corresponding values of the ratio $U/J$ are reported between squared brackets). The other parameters are reported in Fig. 2.