Magnetic trapping of ultracold Rydberg atoms in low angular momentum states

We theoretically investigate the quantum properties of $nS$, $nP$, and $nD$ Rydberg atoms in a magnetic Ioffe-Pritchard trap. In particular, it is demonstrated that the two-body character of Rydberg atoms significantly alters the trapping properties opposed to pointlike particles with identical magnetic moment. Approximate analytical expressions describing the resulting Rydberg trapping potentials are derived and their validity is confirmed for experimentally relevant field strengths by comparisons to numerical solutions of the underlying Schrödinger equation. In addition to the electronic properties, the center-of-mass dynamics of trapped Rydberg atoms is studied. In particular, we analyze the influence of a short-time Rydberg excitation, as required by certain quantum-information protocols, on the center-of-mass dynamics of trapped ground-state atoms. A corresponding heating rate is derived and the implications for the purity of the density matrix of an encoded qubit are investigated.

Figure 1

(Color online) Schematic overview of the energy scales and coupling terms involved. Starting from the degenerate hydrogen energy spectrum on the left, it is shown how the quantum defect—modeled by the potential $V_l\left(r\right)$—separates the low angular momentum states. The spin-orbit coupling $V_{so}\left({\mathbf{L}}_r,\mathbf{S}\right)$ then yields the fine structure splitting for fixed $l$. Within a given $j$ manifold, the Hamiltonian $H_A+\frac{1}{2}{g}_j{m}_j\left|\mathbf{B}\right|$ resembles the coupling of a pointlike particle to the magnetic field $\mathbf{B}\left(\mathbf{R}\right)$. The two-body character of the Rydberg atom, which is represented by $H^{\prime }$, only contributes if energetically remote levels are considered as well: it admixes states of different $n$, $l$, $j$, and $m_j$ thereby qualitatively changing the shape of the surfaces.

Figure 2

(Color online) Range of validity for deriving Eq. (24). (a) $2G^2{X}^2/B^2⪡1$ resulting in $X⪡B/\sqrt{2}G$. In the figure, the value $X_{+}/\left(B/\sqrt{2}G\right)$ is shown which should be $⪡1$. Figure (b) shows $G^2{X}_0^2/B^2$ which should be $⪡1$ as well. In both cases $l=2,j=m_j=5/2$ is used.

Figure 3

(Color online) First row: contour plots of the electronic potential energy surfaces $E_{\kappa }\left(\mathbf{R}\right)$ of the $40S_{1/2}$ (first column), $40P_{1/2}$ (second column), $40P_{3/2}$ (third column), $40D_{3/2}$ (fourth column), and $40D_{5/2}$ (fifth column) states with $m_j=j$ for the magnetic-field configuration $B=1\text{G}$, $G=2.5\text{T}/\text{m}$. Second row: section along $X=Y$ of the same energy surfaces; the contribution of $E_{\kappa }^{\left(0\right)}\left(\mathbf{R}\right)$ (dashed lines) is shown in addition. Third row: depth of the potentials as given by Eq. (25) as a function of the magnetic-field configuration. For all subfigures, the energy scale is given in terms of the trap frequency $\omega =G\sqrt{g_j{m}_j/2MB}$.

Figure 4

(Color online) Sections along $X=Y$ of the energy surfaces of the (a) $40S_{1/2}$, (b) $40P_{3/2}$, and (c) $40D_{5/2}$ states for the field configuration $B=0.1\text{G}$, $G=10\text{T}/\text{m}$, which yields a trap frequency of $\omega /\sqrt{g_j{m}_j}=2\pi \times 4\text{kHz}$.

Figure 5

(Color online) (a) Section along $X=Y$ of the energy surface of the $40D_{5/2},m_j=5/2$ state for the field configuration $B=1\text{G}$, $G=100\text{T}/\text{m}$ which yields a trap frequency of $\omega =2\pi \times 22.1\text{kHz}$ near the origin. The solid line is yield by the numerical diagonalization of Hamiltonian (5), and the dashed one by Eq. (22). The surfaces have been offset to zero at the origin. (b) Difference (in terms of the trap frequency $\omega$) between the analytic expression Eq. (22) and the result of the numerical diagonalization of Hamiltonian (5). (c) Same as (b) but for $Y=0$ rather than $X=Y$.

Figure 6

(Color online) Parametric heating ${\dot{E}}_{{\nu }_x{\nu }_y}{t}^{\prime }/\omega$ of a trapped ground-state atom as a function of the time being excited to the Rydberg level $40S_{1/2}$. Several initial c.m. states and magnetic-field configurations are considered: $B=1\text{G}$, $G=2.5\text{T}/\text{m}$ (solid), $B=1\text{G}$, $G=10\text{T}/\text{m}$ (short-dashed), and $B=10\text{G}$, $G=2.5\text{T}/\text{m}$ (dashed-dotted) for ${\nu }_x={\nu }_y=0$ as well as $B=1\text{G}$, $G=2.5\text{T}/\text{m}$ (dotted) and $B=10\text{G}$, $G=2.5\text{T}/\text{m}$ (long-dashed) for ${\nu }_x={\nu }_y=2$.