Interaction-induced stabilization of circular Rydberg atoms

We discuss a candidate solution for the controlled trapping and manipulation of two individual Rydberg atoms by means of a magnetic Ioffe-Pritchard trap that is superimposed by a constant electric field. In such a trap Rydberg atoms experience a permanent electric dipole moment that can be of the order of several hundred debye. The interplay of electric dipolar repulsion and three-dimensional magnetic confinement leads to a well controllable equilibrium configuration with tunable trap frequency and atomic distance. We thoroughly investigate the trapping potentials and analyze the interaction-induced stabilization of two such trapped Rydberg atoms. Possible limitations and collapse scenarios are discussed.

Figure 1

Components $d_x$, $d_y$, $d_z$ and absolute value $\left|\mathit{d}\right|$ (from left to right) of the electric dipole moment $\mathit{d}$ at $Z=0$. The only symmetry that survives when a small electric field is present is the symmetry with respect to the $x$ axis [see Eq. (19)]. Parameters: $B=0.1$G, $G=10$T m${}^{-1}$, $F={10}^{-14}$ a.u. $=5.14\times {10}^{-5}$ V cm${}^{-1}$. Electric dipole moments are given in atomic units $e{a}_0=2.54$ D.

Figure 2

(a) Two-atom potential energy [Eq. (28), solid lines], the harmonic approximation (dotted lines) and the 1D ground-state energy along $Z_D$ for $Z_S=0$ and different electric-field strengths $F$. Parameters: $B=10$G, $Q=6\times {10}^{-16}B$, $n=30$, $F=0.0514$ V/cm (black), $0.257$ V/cm (blue/gray), $0.514$ V/cm (yellow/light gray). (b),(c) Quality of the harmonic approximation for the first set of parameters. Panel (b) depicts the harmonic approximation (dotted line) of the two-atom potential $V_{\mathrm{tot},\mathrm{circ}}$ (solid) in a region around the local minimum (filled: Gaussian). The double-logarithmic plot in panel (c) shows the quadratic (dotted), cubic (dashed), and the quartic (dot-dashed) coefficient of the expansion of the potential around the equilibrium position as a function of the Ioffe field strength.

Figure 3

Sections through the six-dimensional two-atom potential around the local minimum. The $Z_D$ coordinate of the minimum is indicated by the vertical lines. The thick black contours are plotted at the energy of the two-atom harmonic center-of-mass ground state which corresponds to half the sum of all six trap frequencies, $E_0=26$kHz. The contour plots are clipped at energies $125$kHz higher than the minimal energy configuration. The dashed and dotted lines in the plots on the left-hand side indicate the quality of the single-atom-surfaces approximation introduced in Sec. 3a. They are drawn where the ratio of dipole-dipole interaction energy and energetic distance of adjacent single-atom electronic surfaces, $E_{\mathit{dd}}/\Delta E$, equals $0.1$ (dotted lines) and $0.01$ (dashed lines). Parameters: $B=30$G, $G=10$T m${}^{-1}$, $Q=6\times {10}^{-16}B$, $F=2\times {10}^{-11}\text{a.u.}=10.28$V/m, $n=30$.

Figure 4

Spatial dependence of the direction of the electric dipole moments ${\mathit{d}}_{\gamma }$ and ${\mathit{d}}_F$ (for $Q=0$). (a) The electric dipole moment ${\mathit{d}}_{\gamma }$ that originates from the finite size of the Rydberg atom. It is perpendicular to the local magnetic magnetic-field direction (yellow arrows) and it vanishes on the $Z$ axis. On the positive (negative) $X$ axis it is parallel (antiparallel) to the electric dipole moment ${\mathit{d}}_F$ that is induced by the electric field and depicted in the subfigure (b). (c) Orientation of ${\mathit{d}}_F$ with respect to the vector ${\mathit{R}}_{AB}$ connecting the interacting atoms $A$ and $B$ when they are displaced in the same $X$ direction, $X_S\ne 0$ (solid line), or in different $X$ directions, $X_D\ne 0$ (dashed lines).

Figure 5

Loss of the local minimum for decreasing magnetic-field gradient $G$ of the Ioffe-Pritchard trap. (a)–(c) Two-dimensional sections of the six-dimensional two-atom potential through the $X_D$-$Z_D$ plane, $X_S=Y_S=Y_D=Z_S=0$. The plot range of all three contour plots is $\pm 150$ kHz. (d)–(f) Minimal energy of the two-atom potential against $Z_D$. Each point is computed minimizing the total potential for fixed $Z_D$ with the other center-of-mass coordinates as parameters. The $X_D$ positions are shown as black dots in the sections on the left. Parameters: $B=10$G, $Q=6\times {10}^{-16}B$, $F={10}^{-10}$ a.u., $n=30$, $G=3$T m${}^{-1}$ [first row (a),(d)], $G=2$T m${}^{-1}$ [second row (b),(e)], $G=1.5$T m${}^{-1}$ [third row (c),(f)].