Ultracold Rydberg atoms in a Ioffe-Pritchard trap

We discuss the properties of ultracold Rydberg atoms in a Ioffe-Pritchard magnetic field configuration. The derived two-body Hamiltonian unveils how the large size of Rydberg atoms affects their coupling to the inhomogeneous magnetic field. The properties of the compound electronic and center of mass quantum states are thoroughly analyzed. We find very tight confinement of the center of mass motion in two dimensions to be achievable while barely changing the electronic structure compared to the field free case. This paves the way for generating a one-dimensional ultracold Rydberg gas.

Figure 1

(Color online) Sections along the $X$ axis through the electronic adiabatic energy surfaces of an entire $n=3$ manifold. The field gradient is fixed at $G=1\mathrm{T}∕\mathrm{m}$ in order to suppress the influence of the last term in $H_e$ (22). From left to right, $\zeta =BM{\gamma }^{-2∕3}$ increases due to an increasing Ioffe field.

Figure 2

(Color online) Sections along the $X$ axis through the uppermost 21 surfaces of the $n=30$ manifold of ${}^{87}\mathrm{Rb}$ for increasing ratios $B∕\left(G{n}^2\right)$. The field gradient is fixed at $G=10\mathrm{T}∕\mathrm{m}$ while the Ioffe field is increased from top left to bottom right. ($B=24\mathrm{mG}$, $B=48\mathrm{mG}$, $B=0.24\mathrm{G}$, and $B=0.48\mathrm{G}$). For small ratios $B∕\left(G{n}^2\right)$ the influence of the second term in Eq. (22) is not completely suppressed as can be seen from the lifted degeneracies in the upper subfigures.

Figure 3

(Color online) Sections $(Y=0)$ through the adiabatic potential energy surfaces belonging to the $n=30$ manifold of ${}^{87}\mathrm{Rb}$ for decreasing ratios $B∕\left(G{n}^2\right)=\zeta ∕\left({\gamma }^{1∕3}{n}^2\right)$. The influence of the Zeeman term in ${\mathcal{H}}_e$ (22) is fixed $(\zeta =5)$ while ${\gamma }^{1∕3}{n}^2$ increases. (a) $B∕\left(G{n}^2\right)=10\leftrightarrow B=22.9\mathrm{mG}$, $G=4.81\mathrm{T}∕\mathrm{m}$; (b) $B∕\left(G{n}^2\right)=5\leftrightarrow B=91.6\mathrm{mG}$, $G=38.5\mathrm{T}∕\mathrm{m}$; (c) $B∕\left(G{n}^2\right)=1\leftrightarrow B=2.29\mathrm{G}$, $G=4807\mathrm{T}∕\mathrm{m}$; (d) the indicated region in (c) to a larger scale.

Figure 4

(Color online) Section through the $n=30$ manifold for a field strength of $0.01\mathrm{G}$ and a field gradient of $20\mathrm{T}∕\mathrm{m}$ $\left({}^{87}\mathrm{Rb}\right)$. A large number of avoided crossings can be observed. The uppermost curve, however, stays isolated from the other curves. The insets show the linear behavior of the surfaces far away from the $z$ axis.

Figure 5

Expectation value ${⟨r⟩}_{\phi }$ of the wave functions that correspond to the uppermost electronic energy surface for $G=100\mathrm{T}∕\mathrm{m}$ ($n=30$, ${}^{87}\mathrm{Rb}$). $B$ is varied yielding different values for the ratio $\zeta ∕{\gamma }^{1∕3}{n}^2=B∕G{n}^2$: (a) $0.01\mathrm{G}\to B∕G{n}^2=0.21\mathrm{a.u.}$, (b) $0.1\mathrm{G}\to B∕G{n}^2=2.1$, and (c) $1\mathrm{G}\to B∕G{n}^2=21$. The depicted ranges of $X$ and $Y$ correspond to 30 characteristic lengths of the c.m. motion in scaled units.

Figure 6

(Color online) Expectation values $⟨L_x⟩$, $⟨L_y⟩$, and $⟨L_z⟩$ [(a), (b), (c) respectively] for a ratio $B∕\left(G{n}^2\right)=2.1$ (Ioffe field $B=0.1\mathrm{G}$, gradient $G=100\mathrm{T}∕\mathrm{m}$, ${}^{87}\mathrm{Rb}$, $n=30$). In (d) the projection $\Pi$ of $⟨{\mathit{L}}_{\mathit{r}}⟩$ onto the local magnetic field direction $\mathit{G}$ is displayed. It is close to the field free maximum value for the angular momentum projection, $m_{l,\max }=n-1$. (e) shows the spatial behavior of $⟨{\mathit{L}}^2⟩$. The range of $X$ and $Y$ corresponds to 30 times the characteristic length of the c.m. motion.

Figure 7

(Color online) Spatial dependence of the projection $\Pi$ of $⟨{\mathit{L}}_{\mathit{r}}⟩$ onto the local field axis for $B=1\mathrm{G}$ (all other parameters are the same as in Fig. 6). For this ratio, $B∕\left(G{n}^2\right)=21$, the deviations from the maximal value $m_{l,\max }=n-1$ are marginal. [Equally, $⟨{\mathit{L}}^2⟩\approx l_{\max }(l_{\max }+1)$, not shown.]

Figure 8

Probability densities of the ground state and the first and tenth excited states of the c.m. motion in the uppermost adiabatic potential surface of the $n=30$ manifold of ${}^{87}\mathrm{Rb}$ (from left to right). The Ioffe field strength is set to $B=0.1\mathrm{G}$ and the field gradient is $G=10\mathrm{T}∕\mathrm{m}$.

Figure 9

(Color online) Double logarithmic plot of the expectation value $⟨\rho ⟩$ for the c.m. ground state (circles, ○) in the uppermost adiabatic energy surface ($n=30$, ${}^{87}\mathrm{Rb}$). The corresponding expectation values for the c.m. wave function in a perfectly harmonic potential are depicted for comparison (+).

Figure 10

(Color online) Comparison of the mean extension of the c.m. wave function, $⟨\rho ⟩$, and the mean distance of the core and the electron, $⟨r⟩$, for the $n=30$ manifold of ${}^{87}\mathrm{Rb}$.