Berry phase in atom optics

We consider the scattering of an atom by a sequence of two near-resonant standing light waves, each formed by two running waves with slightly different wave vectors. Due to opposite detunings of the two standing waves, within the rotating wave approximation, the adiabatic approximation applied to the atomic center-of-mass motion, and a smooth turning on and off of the interaction, the dynamical phase cancels out and the final state of the atom differs from the initial one only by the sum of the two Berry phases accumulated in the two interaction regions. This phase depends on the position of the atom in a way such that the wave packet emerging from the scattering region will focus, which constitutes a method to observe the Berry phase without resorting to interferometric methods.

Figure 1

Scattering of the wave packet $\Psi =\Psi \left(x\right)$ of a two-level atom by a standing electromagnetic field formed by two propagating waves of wave vectors ${\mathbf{k}}_1$ and ${\mathbf{k}}_2$. The latter assume an angle $\alpha$ and $\pi -\alpha$ with respect to the $x$ axis. The field envelope along the $y$ axis translates, according to the relation $y=vt$, into the time-dependent function $f=f\left(t\right)$ as the atom propagates through the field with velocity $\mathbf{v}=v{\mathbf{e}}_y$. The frequency $\omega$ of the field is detuned from the frequency of the atomic transition between the ground and excited states $|g\rangle$ and $|e\rangle$ by $\Delta$.

Figure 2

Translation of the electromagnetic field envelope $f=f\left(t\right)$ shown on the top into a circuit in parameter space, depicted at the bottom, illustrated here for the Eckart envelope, Eq. (35). This function leads to an infinite amount of windings around the origin of parameter space.

Figure 3

Influence of the geometric phase on the free propagation of an atomic wave packet represented by its time-dependent width $\Delta x\left(t\right)$ given by Eq. (60). For an atom in the ground state ($\Delta x_g$ and $b=5$), the wave packet first focuses and then spreads, whereas for the excited state ($\Delta x_e$ and $b=-5$) it spreads from the beginning. This spreading is larger than that associated with the free propagation of the Gaussian ($\Delta x^{\left(0\right)}$ and $b=0$) given by Eq. (61).

Figure 4

Translation of the field envelope $f=f\left(t\right)$ during two symmetrically located time intervals (top) into a closed circuit in parameter space (bottom). Indeed, the $m$th circuit arises from the envelope $f=f\left(t\right)$ during the time intervals $-t_m\le t\le -t_{m-1}$ and $t_{m-1}\le t\le t_m$.