Laser cooling and trapping of atomic matter waves in optical potentials has enabled rapid progress in quantum science, particularly when combined with Rydberg excitation of the atoms to induce long-range interactions. Here, we propose the local manipulation and spatio-temporal sculpting of the electronic matter wave of a Rydberg atom by a laser field focused so that its beam width is smaller than the Rydberg electron orbit. We compute the electronic eigenstates in the presence of a sharply focused Gaussian laser beam, and find strong Rydberg state mixing leading to large kilo-Debye dipole moments. These can be modulated with high bandwidth controlled by the local tweezer intensity. Oscillations in the position-dependent level shifts, analogous to the potential wells allowing ultralong-range Rydberg molecules to form, provide opportunities to trap the Rydberg atom in an eccentric way via ponderomotive forces acting on sub-orbital length scales.
Electron orbitals are central to the modern understanding of how matter is composed, from the electronic shell structure within an atom to the hybridized orbitals forming chemical bonds in molecules and the delocalized orbitals determining conduction in the solid state [1][2][3]. Their local control and manipulation in molecules or materials is a challenging task, typically requiring atomicscale resolution instruments such as tunneling or atomic force microscopes [4,5]. However, electronic orbitals need not remain microscopic: in Rydberg atoms, the typical size of the electronic wave function can extend up to a few microns [6][7][8][9]. This opens an entirely different approach for the local manipulation of electron orbitals via optical microscopy techniques. In particular, single-atom control enabled by tightly focused optical tweezers appears especially well-suited for this purpose, as tweezer beams can be focused, positioned, and even dynamically rearranged at sub-micron scale [10][11][12][13][14][15].
In this Letter, we propose and theoretically investigate the local manipulation of and spatio-temporal control over electronic orbitals via an optical tweezer piercing a giant Rydberg atom. We classify the field-controlled electronic eigenstates into two varieties: one composed of low-angular momentum states which possess sizeable dipole moments stemming from the laser perturbation, and the second containing highly dipolar orbitals marked by strong electron localization in the degenerate hydrogenic Rydberg states. The critical control parameter η is the ratio of the laser waist to the Rydberg orbit, with small η producing oscillatory level shifts and asymmetric electron orbitals reminiscent of ultralong-range Rydberg molecules, except that here the tweezer beam plays the role of a localized perturbation rather than a groundstate atom [16][17][18][19]. However, optical tweezer interactions provide a much higher degree of tunability for electronic orbital control due to the flexibility of the deployed light fields, which also enables fast modulation in time. We ar-gue that the adiabatic eigenstates and their large dipole moments can be driven at MHz-scale bandwidth to realize a locally controlled atomic-scale Hertzian dipole. While the eigenstates are readily accessible via spectroscopy, the driven dipoles could be sensed by nearby Rydberg atoms serving as resonant receivers [20]. Moreover, although the tweezer repels the Rydberg electron, we find that there exist deep local minima in the adiabatic potentials which enable trapping of the Rydberg atom through sub-orbital localization of ponderomotive forces.
Figure 1 illustrates our proposed scheme to sculpt and control the electronic orbital of a Rydberg atom using FIG. 1. Experimental scheme and definition of coordinate system: A tweezer beam is focused at the origin of the coordinate system. The ionic core of the Rydberg atom (red sphere) is located at Rc, the Rydberg electron (blue sphere) is located at X. The tightly focused laser beam strongly perturbs the quasi-degenerate Rydberg levels, resulting in localized electronic states reminiscent of the “trilobite” orbitals in long-range Rydberg molecules (blue density plot). an optical tweezer passing through the origin of the labframe coordinate system, with the ionic core and Rydberg electron lying at R c and X = R c + r. For definiteness, we consider a divalent 88 Sr Rydberg atom and Gaussian beams characterized by waist w 0 , wavelength λ, and an effective numerical aperture NA eff := λ π w0 ∼ 0.3. We consider the tight-focus regime w 0 =η s ν , where η < 1 and s ν = 2 ν 2 a 0 is the characteristic orbital radius of the Rydberg atom with principal quantum number ν. The blue density plot in Fig. 1 shows a representative electron orbital for the maximally perturbed Rydberg state. Its strong localization about the tweezer focus and resulting charge separation induces a large dipole moment. The tweezer’s influence on the Rydberg atom is determined by the effective Hamiltonian
We assume a single-active-electron picture in which one electron is promoted to a Rydberg state |νℓm⟩, while the second remains part of the core. Hence, ĤRyd is the field-free atomic Hamiltonian with eigenenergies E νℓ = -Ry/(ν -µ ℓ ) 2 , with µ ℓ the quantum defects of 88 Sr, and Vcore the frequency-dependent interaction between the ionic core and the light field. In the adiabatic approximation, justified by the disparity between typical laser frequencies ω and both the Rydberg electron’s Kepler frequency and the center-of-mass motion [12], VP is the static ponderomotive potential. These potentials are given by
where α ion (ω) is the scalar dynamic dipole polarizability of Sr + and I(x) is the inhomogenous intensity of the beam (see End Matter). We consider atomic displacement only in the transverse direction and obtain the adiabatic potential-energy curves (PECs) upon diagonalizing Ĥ at each R c =R c xB . The core-induced shift V
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