Bringing together the concepts underlying atomic giant dipole states in external fields and ultra-long-range molecules, we demonstrated the existence of ultra-ultra-long-range giant dipole diatomic molecules. In particular, the exotic atomic state underlying these molecules give rise to novel
9.8 Conclusion 113
Figure 9.8: Two dimensional intersections of the perturbatively calculated PES forxn= 0. We clearly see intersections of the perturbative potential surfaces.
properties such as a plethora of different quantum states with complex three-dimensional energy landscapes and rich rovibrational dynamics. The resulting molecules possess very large rovibra-tional bound states at internuclear distances in the range of 105a0.
For their experimental preparation the ’best of two worlds’ has to be combined. The preparation of giant dipole states is known to be possible starting from ’traditional’ Rydberg states in magnetic fields and applying a sequence of electric field switches which brings the electron into low-lying outer well states [268]. Driven radio-frequency transitions in the outer well as an additional tool might help to prepare definite outer well states. Starting from these, one could overlap the GDS with a dense cloud of ultracold rubidium atoms and use radio- / microwave induced transitions to form the envisaged giant dipole molecular states. One of the main differences to standard cold atom experiments is certainly the regime of field strengths necessary to address the giant dipole states, which corresponds to strong static magnetic and electric fields. The results from this chapter have been published in Ref. [2].
Chapter 10
Summary and conclusions
We finally conclude by summarizing the results and providing a brief outlook on future research directions in the field of ultralong-range Rydberg molecules in external fields.
Field-free ultralong-range Rydberg molecules
The foundation of this thesis is based on a thorough derivation of the theory of ultracold Rydberg molecules in combined electric and magnetic fields. At first the properties of the molecular con-stituents, namely Rydberg atoms and ground state atoms, were discussed in detail. In particular the specific properties of Rydberg atoms and the electron-perturber interaction were discussed in detail. A detailed examination of all the ingredients necessary to adequately describe high and low angular momentum Rydberg states of an alkali atom has been provided. The coupling between the electronic and ground state atom was modeled as the Fermi-pseudopotential approach provid-ing an effective contact interaction potential. In this particular field-dressed molecular system the translation symmetry and conservation of the total momentum in field-free space is replaced by a phase-space translation symmetry and the conservation of the pseudomomentum. Introducing center of mass and relative coordinates as well as a suitable phase-space unitary transformation considerably simplifies the ab-initio molecular Hamiltonian. In particular, the derived working Hamiltonian is an effective two-particle problem describing a Rydberg-electron and relative heavy particle motion in electric and magnetic fields. The remaining couplings between the relative and nuclei particle dynamics are treated as an adiabatic approach reminiscent of the Born-Oppenheimer separation in molecular physics. The initial problem of coupled nuclei and electron degrees of free-dom is thereby reduced to the determination of adiabatic electronic energy surfaces that serve as an external potential for the nuclei dynamics. Reexamining the field-free molecular properties an analytical approach was introduced to obtain the adiabatic potential energy curves. While this ap-proach relies on several approximations, it provides accurate results and a profound understanding of the underlying physics.
Ultralong-range Rydberg molecules in external fields
As the main subject of the present work we have explored the changes the polar high-angular mo-mentum trilobite states experience when exposed to a purely electric, magnetic as well as combined external field configurations. Taking into account boths- andp-wave interactions it turns out that both electric and magnetic fields provide a unique way to control the topology of the adiabatic potential energy surfaces. In case ultralong-range diatomic Rydberg molecules are exposed to sin-gle external fields, the angular degree of freedom between the external electric or magnetic field and the internuclear axis is converted from a rotational to a vibrational degree of freedom, thereby rendering the field-free potential energy curve into a two-dimensional potential energy surface.
In the case of pure electric and magnetic fields the oscillatory behavior of the potential surfaces is changed dramatically in the presence of the fields. In the case of the electric field it turns out that the global equilibrium position is always in the antiparallel configuration with respect to the applied external field. Along the internuclear axis we find a sequence of potential wells with increasing radial coordinate. Increasing the electric field strength we encounter an overall lowering of the
energy accompanied by a subsequent crossover of the energetically order of individual potential wells. Consequently, the equilibrium distance and the lowest rovibrational states are systematically shifted to larger internuclear distances. This means that with increasing electric field strength the low lying molecular states can be progressively shifted away from the region of the avoided crossing of the p-wave split state which crosses the trilobite state near its equilibrium distance in the field-free case. In this manner the respective stability of the ground state and of many excited vibrational states is guaranteed. For strong fields the interaction of the electrically field dressed trilobite state with quantum defect states leads to a strong admixture of atomic s-state character to the high-l states. For this reason a two-photon excitation process starting from a two-atom system should be sufficient to create field-dressed ultralong-range Rydberg molecules.
Beside combined electric and magnetic fields we also studied a pure magnetic field configuration.
This study extents a previous work by Lesanovsky et al. [99] where only pure s-wave interaction was considered. Taking into account both s- and p-wave interactions it turns out that strong level repulsion causes the potential wells which provide the trilobite states in the field-free case to vanish. Beyond a critical field strength of 100 G the trilobite potential energy surface (PES) does not provide any bound states anymore. In the case of combined electric and magnetic fields the field-free potential curve can even be rendered into a three-dimensional surface choosing appropriate angles of inclination between the two external field vectors. Depending on the specific degree of electronic excitation and field configuration we obtain oscillatory potential curves possessing rich topologies with localization in the radial and angular degrees of freedom and depths up to hundreds of MHz.
Both the parallel and crossed field configurations provide unique ways to control the topology of the adiabatic potential surfaces. This leads to the possibility to directly control molecular orientation and alignment for both field configurations. In addition, the topological control of the PES provides the control of the electric dipole moment as well. The results from these studies have been published in Ref. [3, 5].
Alignment of ultracold D
5/2-state Rydberg molecules
In this part we analyzed the properties of ultracold D5/2-state Rydberg molecules exposed to an external magnetic field ofB = 13.55 G. In this project, which was conducted in collaboration with Prof. Tilman Pfau, Dr. Alexander Krupp and coworkers at the University of Stuttgart, ultracold Rydberg molecules where created from a thermal cloud of ultracold rubidium atoms with densities of 1012cm−3via a two-photon excitation process. This study is in line with a number of remarkable experimental studies of the nature of ultralong-range Rydberg molecules. For instance, in 2009 V.
Bendowsky and coworkers for the first time created this novel molecular species which had been predicted by Prof. C. H. Greene in 2000 [85] using nearly the same experimental setup as it was used in Ref. [4]. However, these studies where conducted for S-state molecules whose molecular structure is determined by a spherically symmetric electronic s-orbital. In our work these studies were extended tonD5/2-state molecules. ExposingD-state molecules into a homogeneous magnetic field energetically splits the degeneratemJ sublevels. In contrast to the S-state molecular species these electronic states depend, besides the radial distance R, on the angle θ with respect to the applied magnetic field axis. For this reason the resulting adiabatic potential energy surfaces provide a richer topology beyond spherical symmetry. In this work two distinct species ofD-state molecules have been studied, the nD5/2, mJ = 1/2, 5/2 molecules for principal quantum numbers ranging fromn= 41 to 50. Both the adiabatic potential energy surfaces and molecular binding energies were calculated and compared to the experimental data. In addition to the ground state molecules higher excited molecular states were detected as well. These states can be selectively excited by choosing the appropriate laser detuning. We discovered two different types of molecules. The first species is characterized by a high degree of molecular alignment parallel to the magnetic field axis. They stems from axial lobes in the electronic density distribution. These molecular states are denoted as axial states. In contrast, the second molecular species are determined by toroidal electron
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density around the magnetic field axis, these states are denoted as toroidal states. The measured binding energies of both molecular species and the theoretically calculated data match satisfactorily.
Deviations in the data sets were explained by an oversimplified model used to mimic the molecular dynamics and limitations in the excitation and detection techniques. As the two distinct D-state molecular species are determined by different electronic density profiles the molecular states differ in their alignment with respect to the applied magnetic field axis. The alignment of molecules is of central importance as it strongly influences their interaction and chemical reaction dynamics.
Normally strong external electric, magnetic and light fields are required to create aligned molecules.
In contrast, magnetically dressed D-state Rydberg molecules are intrinsically aligned due to their creation process in a weak magnetic fields. Rotational degrees of freedom are hardly excited as the created molecules possess large internuclear separations which results in a large moment of inertia.
For this reason the molecules can be considered as stationary because they hardly rotate during their lifetime of around 10µs. The results from this collaboration have been published in Ref. [4].
Ultralong-range giant dipole molecules in crossed electric and magnetic fields
The last part of this thesis contains the first results that had been published from our studies in Ref. [2]. In this work we show the existence of ultralong-range giant dipole molecules formed by a neutral alkali ground state atom that is bound to the decentered electronic wave function of a giant dipole atom. Besides the normal ground state atom these molecular species contain an exotic constituent, so-called atomic giant-dipole states. Opposite to the usual Rydberg states, the giant dipole states are of decentered character and possess a huge electric dipole moment. Their existence is related to the non-separability of the center of mass and electronic motion in the presence of the external fields. In contrast to Chapter 5 and Chapter 6, in this work the constituents of the ultralong-range diatomic molecular species only exist for finite field strengths. This means that these molecules cannot be understood as a field dressed version of a field- free diatomic molecule as the existence of these exotic molecular species is strongly related to the applied field strength of the electric and magnetic fields. Since the interaction of a giant dipole atom with a neutral ground state perturber can be described by a low-energy electron-atom scattering potential we apply the Fermi pseudopotential approach. The adiabatic potential surfaces emerging from the interaction of the ground state atom with the giant dipole electron possess a rich topology depending on the degree of electronic excitation. Depending on the applied field strength the resulting the resulting molecules are truly giant with internuclear distances up to several micrometers.
Outlook
Although ultralong-range diatomic Rydberg molecules have been studied intensively for almost fifteen years both theoretical and experimentally, this research field still provides a number of possibilities to extend the knowledge in the field of ultracold molecules.
A rather natural extension of the present work would be the investigation of polyatomic Rydberg molecules [87, 96] in both electric and magnetic fields. In case the Rydberg electron binds several ground state atoms the specific PES depend on the relative orientation of the ground state atoms.
The higher dimensional energy surfaces are expected to provide a complex structure of oscillatory potential wells and avoided crossings. In addition, one can also consider the case where several ground state atoms from different species are bound by the Rydberg electron. Such molecules have become known as Borromean molecules as they only exist as polyatomic species due to mutual stabilization of their molecular bonds [97]. It would be interesting to explore the impact of electric and magnetic fields on the stability of these particular molecular species. Obviously, for both kind
of polyatomic species, combined external fields constitute a logical extension thereof. In addition, the properties of polyatomic giant dipole Rydberg molecules is still an open question.
Another very promising direction of research is the study of higher order electric and magnetic polarizabilities and susceptibilities. Because of their high sensitivity to small field strengths in Rydberg atoms these quantities strongly depend on the Rydberg excitation. For instance, the electric polarizability of single Rydberg atoms scales asn7. As the molecular properties both depend on the electron and nuclei degrees of freedom it is worth to extent the studies from Chapter 8 and to analyze these properties in more detail. Furthermore, the study of higher order polarizabilities can be easily extended to other molecular Rydberg species as polyatomic molecules and ultralong-range triatomic polar Rydberg molecules [241].
Finally, one can consider novel approaches to solve the electronic Schr¨odinger equation. A number of previous works [97,269] have already discussed the possibility to determine the adiabatic potential curves via the Green function approach. This approach is considered to be more robust compared to exact diagonalization. For this reason we expect it to provide reliable results in regimes where both exact diagonalization and the adiabatic approximation fail, for instance in the vicinity of avoided crossings.