Atomic Bloch-Zener oscillations for sensitive force measurements in a cavity

Cold atoms in an optical lattice execute Bloch-Zener oscillations when they are accelerated. We have performed a theoretical investigation into the case when the optical lattice is the intracavity field of a driven Fabry-Perot resonator. When the atoms oscillate inside the resonator, we find that their back-action modulates the phase and intensity of the light transmitted through the cavity. We solve the coupled atom-light equations self-consistently and show that, remarkably, the Bloch period is unaffected by this back-action. The transmitted light provides a way to observe the oscillation continuously, allowing high-precision measurements to be made with a small cloud of atoms.

Figure 1

(Color online) Schematic of the proposed experiment. (a) A cloud of cold atoms is held in a standing-wave optical trap inside a vertical Fabry-Perot cavity. (b) The atoms execute Bloch-Zener oscillations, leading to a periodic modification of their wave function. (c) This modulates the intracavity power and hence the lattice depth $s$. (d) The power modulation is seen in the light transmitted by the cavity.

Figure 2

Calculated evolution of lattice depth $s\left(t\right)$ normalized to the recoil energy for ${}^{87}\text{R}\text{b}$ atoms undergoing 840 Hz Bloch-Zener oscillations in a 780 nm lattice with $s\approx 3E_R$. Lines: numerical solution to Eqs. (3, 4). Dots: self-consistent adiabatic approximation of Eqs. (9, 10). (a) $N{g}_0^2/\left(\kappa \Delta \right)=0.4$. (b) Close-up of lattice depth oscillations with stronger coupling, $N{g}_0^2/\left(\kappa \Delta \right)=1$. Nonadiabatic effects are seen in the line. (c) Fourier transform $\tilde{s}\left(\omega \right)$ of result in (b), showing harmonics of fundamental frequency ${\omega }_{\text{B}}$. Inset: close-up of $\tilde{s}\left(\omega \right)$ at higher frequencies. Harmonics of ${\omega }_{\text{B}}$ appear as sharp vertical lines. In addition, one sees much weaker, broad, Fourier components due to band excitation, corresponding to the rapid oscillations in (b). The actual individual values of the parameters used (or assumed in the case where only ratios enter) in the calculations were (see text): $N=5\times {10}^4$, $g_0=2\pi \times 2.8\text{MHz}$, and $\kappa =2\pi \times 1.0\text{MHz}$. In (a) $\Delta =2\pi \times 1.0\text{THz}$, $\eta =2\pi \times 39\text{MHz}$; in (b) and (c) $\Delta =2\pi \times 0.39\text{THz}$, $\eta =2\pi \times 28\text{MHz}$.