Geometric phase in anisotropic Kepler problem: Perspective for realization in Rydberg atoms

Abstract

We predict a gyroscopic effect that can be demonstrated with Rydberg atoms following the dynamics of a Kepler Hamiltonian with an additional uniaxial anisotropy induced by optical ponderomotive force. This effect is analogous to the rotation of the Foucault pendulum in response to the Earth's rotation. We argue that in Rydberg states with a large principal quantum number a similar geometric angle can be generated by mechanical rotations of an atomic-optical setup on time scales between $1~μ$s and $1~$ms.

Figures (3)

• Figure 1: A cyclic trajectory imposed for the vector ${\bf R}(t)$ (rainbow color; the time evolution is from blue to red), describing a slowly changing position of the nucleus (small circles). Small ellipses are the snapshots of numerically calculated Kepler orbits $\mathbf{r} (t)$ found at several different positions of the nucleus. The Runge-Lenz vector A points from the nucleus towards the perigee of the orbit. For the Hamiltonian (\ref{['keplerH1']}), the orientation of ${\bf A}$ is an adiabatic invariant.
• Figure 2: (a) Evolution, according to the Hamiltonian (\ref{['Ha2']}), of a small out-of-plane deviation of the electron trajectory from the easy plane. The trajectory in 3D-space was obtained by numerical simulations using Yoshida simplectic 3-rd order algorithm Yoshida1990note for $m=Q=1$ and $\omega_0=0.2$, and then only the $z$-component was plotted. The motion started at ${\bf r}=(1.,0,0.01)$, ${\bf p}=(0,0.75,0)$, which corresponds to the time of one orbit cycle $T_{\rm orb}=3.65$. Each plotted point was obtained after an integer number of the orbiting periods: $n=t/T_{\rm orb}$, $n=0,1,\ldots$. (b) Projection of the same trajectory on the easy $xy$-plane at $n=0$ (green) and after $n=3000$ cycles (purple). Within the simulation accuracy, there is no visible change of the direction of the main ellipse axis despite the time evolution is much larger than the period of oscillations along the $z$-axis and the deviation from the easy plane on the order of $|z|/|{\bf r}| \sim 1\%$.
• Figure 3: (a) The adiabatic evolution of the field anisotropy vector $\hat{Z}(t)$, which subtends a solid angle $\Omega_{sa}$ given by Eq. (\ref{['sa']}). The time-dependent Hamiltonian for simulations corresponds to Eq. (\ref{['Ha2']}) with rotating anisotropy axis. It is written explicitly in note. (b) The elliptic orbit at the start of the evolution (dashed curve) and at the end (solid curve). The angle $\phi$ between the main axes is predicted to coincide with $\Omega_{sa}$. (c) This prediction (empty triangles/boxes), $\cos \phi = \cos(2\pi(1-\cos \vartheta))$. The blue triangles are the numerical results for simulations of the Hamiltonian equations of motion at strong anisotropy $\omega_0=5$. The orange squares are for the weak anisotropy, $\omega_0=0.5$. The protocol for a single point corresponds to a fixed $\vartheta$ in Eq. (\ref{['hatr']}), and the angle $\varphi$ changing according to $\varphi=\pi [1+\tanh( t/\tau)]$, with $\tau\omega_0 \approx 10$ for $\omega_0=0.5$ and $\tau \omega_0 \approx 50$ for $\omega_0=5.$; $t\in (-5\tau,5\tau)$. Other parameters and initial conditions are as in Fig. \ref{['z-fig']}.