Controlling Ultracold Rydberg Atoms in the Quantum Regime

We discuss the properties of Rydberg atoms in a magnetic Ioffe-Pritchard trap being commonly used in ultracold atomic physics experiments. The Hamiltonian is derived, and it is demonstrated how tight traps alter the coupling of the atom to the magnetic field. We solve the underlying Schrödinger equation of the system within a given $n$ manifold and show that for a sufficiently large Ioffe field strength the $2n^2$-dimensional system of coupled Schrödinger equations decays into several decoupled multicomponent equations governing the center of mass motion. An analysis of the fully quantized center of mass and electronic states is undertaken. In particular, we discuss the situation of tight center of mass confinement outlining the procedure to generate a low-dimensional ultracold Rydberg gas.

Figure 1

Cut through the adiabatic potential surfaces of the $n=30$ manifold (${}^{87}\mathrm{Rb}$, $G=20\mathrm{T}/\mathrm{m}$, $B=0.01\mathrm{G}$). The high density of states is clearly visible. The uppermost potential surface is predominantly formed by the atomic state with the largest angular momentum. This particular surface is clearly separated from the next lower ones, whereas all other surfaces exhibit a number of avoided crossings (see the magnified view in the inset) at which nonadiabatic intersurface transitions are likely to occur.

Figure 2

Surface plot of the seven uppermost energy surfaces of the $n=30$ manifold (${}^{87}\mathrm{Rb}$, $G=20\mathrm{T}/\mathrm{m}$ $B=0.1\mathrm{G}$). Clearly, the grouping into three multiplets whose mutual distance is given by ${\gamma }^{-2/3}{M}_{2}B$ is visible. While the uppermost manifold consists only of a single surface, the next lower ones show an approximate twofold and fourfold degeneracy. A magnified view of them is provided in the insets.

Figure 3

Expectation values of the radii $\rho =\sqrt{X^2+Y^2}$ and $r$ of the c.m. and electronic wave functions in the IP trap, respectively (${}^{87}\mathrm{Rb}$, $G=100\mathrm{T}/\mathrm{m}$, $B=0.1\mathrm{G}$). $\nu$ labels the c.m. quantum states within the uppermost adiabatic potential surface. While $⟨\rho ⟩$ is increasing, $⟨r⟩$ remains approximately constant at its field-free value. For small degrees of c.m. excitation, the c.m. state is even stronger localized than the valence electron ($⟨\rho ⟩<⟨r⟩$). Moreover, since the electron is found in a high angular momentum state, its radial uncertainty $\Delta r$ is small. Thus, a scenario where the c.m. and the electronic wave function do not overlap is possible as sketched in the inset.