Butterfly Rydberg molecules

Butterfly Rydberg molecules are dominated by the P-wave scattering between the Rydberg electron and the ground state atom, which maximizes the gradient of the electron wavefunction at the position of the ground state atom (see Methods). As a consequence, all angular momentum states of the Rydberg electron are mixed and the electron density acquires a characteristic density distribution which resembles a butterfly (Fig. 1). The first spectroscopic evidence for the existence of this mechanism has been reported in refs 21, 22. The calculation of the corresponding potential energy curves (PEC) in the Born–Oppenheimer approximation requires the diagonalization of the full Hamiltonian, which in our approach includes the fine structure in the Rydberg atom, the hyperfine interaction in the perturber atom, and all S- and P-wave scattering processes (see Methods). The PECs in the vicinity of the 25p-state are shown in Fig. 2a. The butterfly states detach from the n=23 hydrogenic manifold and cross all lower-lying states down to the 25p-state. The red PECs are relevant for this work and the bound states in the strongly oscillating part of the PEC are the butterfly molecules under investigation. Even for the deepest potential wells the electron wave functions in these PECs possess a 15% admixture of the 25p-state, which we use to excite the butterfly molecules with a single photon transition.

Figure 1: Radial electron density of a butterfly molecule. The upper plane shows a surface plot of the radial electron density ρ|Ψ(z,ρ)|2 for a butterfly molecule near the 25p-state of rubidium. The lower plane shows the two-dimensional projection of the electron density. A sketch of the molecule above the projection plane shows, where the Rb+ ion (red) and the ground state perturber (green) are located. The bond length is 245 a 0 . Full size image

Figure 2: Butterfly Rydberg molecules. (a) Adiabatic potential energy curves (PEC) for the Rydberg ground state interaction in the vicinity of the 25p-state. The butterfly state detaches from the n=23 hydrogenic manifold and crosses the lower lying 26s, 24d and 25p-state. A zoomed-in inner region of the lowest butterfly potentials (red lines) is shown in (b). Three different PECs (black lines) emerge, corresponding to the quantum numbers , and of the total spin . The oscillatory shape provides a set of harmonic potential wells at different bond lengths. The lowest bound states for each PEC are sketched in red, green and blue. The orange points represent the experimentally obtained values for the binding energy and the bond length of the eight molecular states studied in detail (see text). The extension of the points denotes the estimated error in the bond length of 5 a 0 . The absolute frequency uncertainty is 10 MHz (see Methods). The top panel shows how the measured bond lengths (orange points) coincide with the nodes of the 25p 1/2 , m j =1/2 radial wavefunction (light red). The right panel shows the experimental spectrum. The colour code is a guide to the eye. The zoom on the lowest energy peaks of the experimental spectrum (c) shows the determined bond length and the dipole moment of the respective peak. (d) A sketch to scale of the molecule constituents, averaged z-positions for a butterfly state with a bond length of 116 a 0 . The average position of the electron is located beyond the perturber atom. Full size image

Photoassociation spectroscopy

The experiment was performed in a Bose–Einstein condensate (BEC) of 87Rb with a peak particle density of 4 × 1014 cm−3, an atom number of 2 × 105 and a temperature of 100 nK. All spin states of the 5s 1/2 , F=1 ground state were populated. The photoassociation of the butterfly molecules was achieved using a single photon transition to the energetic vicinity of the adiabatic free two-particle state 25p 1/2 ⊗5s 1/2 , F=1, which serves as reference point for our spectroscopy throughout this work. Since the created butterfly molecules can decay into ions by photoionization and associative ionization23, the produced ions are used as a probe for the creation of the molecule.

The experimental sequence consists of 500 ms continuous excitation and ion detection. By detuning the excitation laser up to −60 GHz in steps of 2 MHz we obtain the spectrum shown to the right of Fig. 2b. As the laser detuning is gradually increased, we start to probe the bound states in the butterfly potential. We observe a plenitude of molecular states up to an energy of −50 GHz. This is in accordance with the calculated PECs. Moreover, the density of molecular states drops at detunings below −40 GHz, which marks the transition to a spectral region, where only the lowest bound states in each potential well are populated. The spectroscopic results directly prove the existence of butterfly molecules.

Pendular states

We now focus on the ground state molecules in each potential well and demonstrate, how butterfly molecules with selected bond length and high degree of orientation can be created. Due to the symmetry breaking caused by the localized valence electron of the perturber, butterfly molecules, along with the related class of so-called trilobite molecules, are the only known homonuclear molecules with a large permanent electric dipole moment14,15,17,24. The interaction of the dipole moment μ with an external electric field F significantly changes the rotational structure of the molecule. In the simplest approximate description of a dipolar rigid rotor that neglects the coupling of the nuclear motion to the electronic and nuclear spin degrees of freedom, the eigenstates of energy E need to fulfil (see Methods)

In equation (1), the zero of the energy axis is chosen to coincide with the rotationless vibrational energy level being studied. For vanishing electric field, the molecule behaves as a rigid rotor of length R 0 and reduced mass m r =m Rb /2 using the rotational quantum number N, its projection on the molecular axis M N and the rotational constant B e =ℏ2/2I, where I=m r R 0 2 represents the moment of inertia of the rigid rotor. When, on the other hand, the interaction with the electric field dominates the rotational constant , the molecule enters so-called pendular states25,26, which are characterized by the quantum number ν and differ in the degree of orientation of the molecule with respect to the field axis (Fig. 3a). In this regime only the angular momentum projection M N remains a good quantum number, while the rotational quantum numbers N are strongly mixed.

Figure 3: Pendular states. (a) Plot of the orbitals |Φ(θ,φ)| for pendular states |ν,M N 〉 of a dipolar molecule with μ=482 Debye in an electric field of 1 V cm−1. (b) Spectroscopy of the butterfly state at −47.9 GHz in different electric fields. Each spectrum is normalized and shifted on the y axis according to the applied electric field. The coloured lines show the eigenenergies of a rigid rotor with a permanent dipole moment in an electric field (equation (1)) for |M N |=0 (weak lines) and |M N |=1 (strong lines). The dipole moment and the bond length are adapted to fit the experimental data. In this case, we find R 0 =205 a 0 and μ=482 Debye. The inset shows a zoom of the low-field (≈30 mV cm−1) measurement together with the calculated position of the first five eigenstates of the dipolar rotor model with |M N |=1. We attribute the appearance of higher ν states at field strength F=0 to the mixing of higher N states in the residual field (see Methods). Full size image

In Fig. 3b, we show the electric field-dependent spectra of the butterfly state at −47.9 GHz along with the fitted theory of equation (1). Due to the huge dipole moments of the butterfly molecules, fields of 1 V cm−1 are sufficient to put the system deep in the pendular regime. Varying the electric field, we can directly observe the crossover from rotational states to pendular states. At fields of 4 V cm−1, the degree of orientation ω exceeds previous record values26 by a factor of two. Since we photoassociate the butterfly molecules in a Bose–Einstein condensate, which has no angular momentum in the centre of mass motion, only states with |M N |=0 and |M N |=1 can be observed (see Methods). This leads to a very clean spectrum of pendular states, which, in contrast to previous studies16,17,21,22,24, allows to determine the dipole moment μ with an estimated precision of 10 Debye and the bond length R 0 with an estimated precision of 5 a 0 .

Applying this method to eight of the lowest spectroscopic lines in the butterfly spectrum, we can extract the respective dipole moments and bond lengths (Fig. 2c). This additional information enables an improved interpretation of the butterfly spectrum and comparison with theory. This is demonstrated in Fig. 2b, where the orange points mark the measured bond lengths and binding energies for the eight studied lines. The energetic deviation of the measured points from the predicted bound states is only few per cent of the binding energy and is most likely explained by inaccurate P-wave scattering phase shifts assumed in the model. It is remarkable how well the measured bond lengths agree with the wells of the model potential. In fact, the positions of the wells are mainly determined by the strong gradient at the radial nodes of the p-state electronic wavefunction and thus the bond length of the butterfly molecules is an indirect measure for the position of the 25p radial nodes. A comparison of the measured bond lengths with the nodes of the calculated 25p 1/2 wavefunction is shown in the top panel of Fig. 2b.

Properties

Butterfly molecules also exhibit counter-intuitive properties. In a semiclassical picture of molecules with one valence electron, the average position of the electron is always located between the two nuclei. The maximum possible dipole moment is thus limited by the bond length, as realized, for example, in an ionic bond. The strong interference of the electron caused by the scattering with the ground state atom (Fig. 1) permits the butterfly molecules to exceed this limit and locate the electron’s average position beyond the perturber (Fig. 2d). This effect is seen experimentally for the deepest bound butterfly molecules. The state at −50.3 GHz possesses a dipole moment of 380 Debye, which is 30% larger than that of two opposite charges separated by the measured bond length of 116 a 0 . This phenomenon is even stronger in the theoretical model, which reveals a dipole moment of 442 Debye for the studied state. Throughout all investigated butterfly states the measured dipole moments are systematically lower than the calculated values but do not deviate by >35%.

Due to the deep potential wells (Fig. 2b), inward tunnelling and subsequent decay should not limit the lifetime of the studied butterfly molecules. In a separate experiment using short excitation pulses (see Methods) we were able to determine the lifetimes of the strongest observed butterfly molecules. In the high-density environment of a BEC we find a lifetime of 5 μs which is mainly limited by inelastic collisions with additional ground state atoms. Extrapolating to zero ground state density to mitigate the effect of collisions we find a lifetime on the order of 20 μs, which is compatible with the lifetime of the 25p Rydberg state.