Rotational hybridization, and control of alignment and orientation in triatomic ultralong-range Rydberg molecules

We explore the electronic structure and rovibrational properties of an ultralong-range triatomic Rydberg molecule formed by a Rydberg atom and a ground state heteronuclear diatomic molecule. We focus here on interaction of Rb($27s$) Rydberg atom with KRb($N=0$) diatomic polar molecule. There's significant electronic hybridization of Rb($n=24$, $l\ge 3$) degenerate manifold. The polar diatomic molecule is allowed to rotate in the electric fields generated by the Rydberg electron and core as well as an external field. We investigate the metamorphosis of the Born-Oppenheimer potential curves, essential for the binding of the molecule, with varying electric field and analyze the resulting properties such as the vibrational structure and the alignment and orientation of the polar diatomic molecule.


I. INTRODUCTION
While in the early years of Rydberg physics there was a substantial focus on high resolution spectroscopy [1] the advent of Bose-Einstein condensation [2] has brought Rydberg atoms and molecules into the focus of fields that were previously not considering these highly excited and fragile quantum objects. This includes Rydberg gases and plasmas [3], quantum optics and many-body physics of long-range interacting Rydberg systems [3], quantum simulators [4] as well as quantum information processing based on entanglement generation between interacting Rydberg atoms in optical lattices [5]. A very recent spectacular development is the prediction [6] and experimental detection [7] of ultralong-range molecules from an ultracold atomic cloud. Opposite to the well-known Rydberg molecules with a tightly bound positively charged core and an electron circulating around it, the ultralongrange molecular species follows a very much different bonding mechanism with, for a diatomic molecule, internuclear distances of the order of the extension of the electronic Rydberg state. Here, a ground state atom locally probes the Rydberg electronic wavefunction whose interaction is typically described by s-and p-wave Fermi-type pseudopotentials [8,9]. The latter leads to the unusual oscillatory behaviour of the corresponding adiabatic potential energy curves and a correspondingly rich vibrational dynamics. For principal quantum numbers of typically n ≈ 30 − 100, being routinely prepared in the experiments, the diatomic molecules can be up to a few µm in size. These exotic species were first predicted theoretically back in 2000 by Greene et al [6] providing polar (trilobite) and non-polar states with vibrational binding energies in the GHz and MHz regime, correspond- * rogonzal@ugr.es ingly. Beyond the detection of non-polar ultralong-range molecules and their spectroscopic characterization Rydberg triatomic molecules and excited diatomic molecules bound by quantum reflection have been found [10].
The sensitivity of Rydberg atoms to external fields carries over also to ultralong-range Rydberg molecules which opens unprecedented opportunities for their weak field control. This includes static magnetic and electric fields or laser fields which could be used to change the electronic structure severely and correspondingly change the molecular geometry and rovibrational dynamics. Electric and magnetic field control of molecular binding properties [11][12][13] as well as alignment have been demonstrated very recently [14].
If the neutral ground state atom immersed into the Rydberg wave function by a Λ-doublet heteronuclear diatomic molecule [15][16][17], giant polyatomic Rydberg molecules can form. These triatomic ultralong-range Rydberg molecules could be created in an ultracold mixture of atoms and molecules by using standard Rydberg excitation schemes. The Rydberg electron is coupled to the two internal states of the polar ground state molecule, creating a series of undulating Born-Oppenheimer potential curves (BOP) due to the oscillating character of the Rydberg electron wave functions. The electronic structure of these giant molecules could be easily manipulated by means of electric fields of a few V/cm, creating a complex set of avoided crossings among neighboring levels.
In particular a Raman scheme to couple the two internal states of opposite orientation of the polar diatomic molecule [15] has been proposed.
In the present work, we consider a triatomic molecule formed by a Rb Rydberg atom and a ground state KRb polar molecule. Compared to the previous works [15][16][17], we perform an extended and realistic treatment of the internal motion of the diatomic molecule. Indeed, we explicitly introduce the angular degrees of freedom of the diatomic molecule and as such are capable of describing properly the molecular rotational in the rigid rotor approximation. The free rotation of the diatomic molecule in the triatomic Rydberg molecule is now dressed due to the combined electric fields of the Rydberg electron and the core, and is characterized by a strong orientation and alignment along the field direction. Since the permanent dipole moment of the molecule is not fixed in space but rotates, we include in our description the three components of the electric field induced by the Rydberg atom. The perpendicular components of the electric field to the space-fixed frame along the Z axis ensure that the magnetic quantum numbers of the electron and the polar diatomic molecule to no longer be conserved. We analyze the BOP of the Rb(ns)-KRb Rydberg triatomic molecule as the internuclear separation between KRb and the Rb + core varies and explore in detail the effects of the electric field induced by the Rydberg atom on the rotational motion of KRb, by studying the hybridization of its angular motion, orientation and alignment. We also investigate the impact of an additional electric field on these BOP and their vibrational spectrum.
The paper is organized as follows: In Sec. II we describe the Hamiltonian of the system and the electric field induced by the Rydberg core and electron in the diatomic molecule. The BOP as a function of the distance between the perturbing diatomic molecule and the Rydberg core as well as the resulting alignment and orientation of the diatomic molecule are presented in Sec. III. In Sec. IV, we focus on the impact of an external field. The vibrational spectrum of the Rydberg triatomic molecule is discussed in detail in Sec. V and the conclusions are provided in Sec. VI.

II. HAMILTONIAN, INTERACTIONS AND COMPUTATIONAL APPROACH
We consider a triatomic molecule formed by a Rydberg atom and a ground state heteronuclear diatomic molecule in an electric field. The permanent electric dipole moment of the polar molecule interacts with the electric field provided by the Rydberg core and electron as well as the external electric field. With increasing electric field strength the rotational motion becomes a liberating one which leads to the alignment and orientation of the molecular axis. Here, we describe the diatomic molecule by means of a rigid rotor approach. This should be a good approximation for the tightly bound molecule since we are allowed to adiabatically separate the rotational and vibrational motions even in the presence of the electric field induced by the Rydberg atom [18,19].
For the sake of simplicity, we fix the geometry of the Rydberg triatomic molecule: The laboratory fixed frame (LFF) is defined so that the Rydberg core is located in its origin, and the diatomic molecule is the Z-axis at a distance R. A qualitative sketch is presented in Fig. 1 The positions of the diatomic molecule and electron in the LFF are R = RẐ and r = (rR, θ rθ , φ rφ ), respectively. Within the Born-Oppenheimer approximation, the adiabatic Hamiltonian of this system is given by The first term stands for the single electron Hamiltonian describing the Rydberg atom with V l (r) being the l-dependent model potential where l is the angular momentum quantum number of the Rydberg electron [20]. The second term H mol is the Hamiltonian of the polar molecule in the electric field created by the Rydberg electron and core, F ryd (R, r). In the rigid rotor approximation, the molecular Hamiltonian reads with B being the rotational constant, N the molecular angular momentum operator and d the permanent electric dipole moment of the diatomic molecule, which is parallel to the molecular internuclear axis. Note that the internal rotational motion of the diatomic molecule is described by the Euler angles Ω d = (θ d , φ d ), which relate the diatomic molecular fixed frame (MFF) and LFF. The MFF is defined with its origin at the center of mass of the two nuclei of this diatomic molecule and the Z M axis being along the internuclear axis. The electric field due to the Rydberg electron and core is given by where e is the electron charge [15][16][17].
The last term in the adiabatic Hamiltonian (1) stands for the interaction of the Rydberg atom and the diatomic molecule with the external electric field F ext Here, we consider an external electric field antiparallel along the LFF Z-axis with strength F ext , i. e., F ext = −F extẐ . The total angular momentum of the triatomic molecule, but excluding an overall rotation, is given by J = l+N, where l is the orbital angular momentum of the Rydberg electron. In the absence of the external electric field, the triatomic molecule states can be characterized by the projection of J along the LFF Z-axis, i. e., M J . In the presence of the external field F ext = −F extẐ , there's azimuthal symmetry and M J is conserved. In both cases, for M J = 0, states with M J and −M J are degenerate.
To solve the Schrödinger equation associated with the Hamiltonian (1), we perform a basis set expansion in terms of the coupled basis where lm l N M N |JM N is the Clebsch-Gordan coefficient, J = |l − N |, . . . , l + N , and M J = −J, . . . , J. ψ nlm (r) is the Rydberg electron wave function with n, l and m being the principal, orbital and magnetic quantum numbers, respectively. Y N MN (Ω d ) is the field-free rotational wave function of the diatomic molecule, with N and M N being the rotational and magnetic quantum numbers, i. e., Y N MN (Ω d ) are the spherical harmonics.
In the computation of the Hamiltonian matrix nlN JM J |H mol |n ′ l ′ N ′ J ′ M J we determine the matrix elements nlm|F ryd (R, r)|n ′ l ′ m ′ using the relation where ∇ R is the Laplacian with respect to the molecular coordinate R = (RR, θ Rθ , φ Rφ ), see Ref. [21]. The electric field reads where Ω = (θ, φ) and r = (rR, θθ, φφ) are the coordinates of the Rydberg electron, and In Eq. (9), R = RZ. The first two terms in expression (9) couple states with ∆m l = ±1, which means that the projection m l on the Z axis is not conserved. In addition, the electric field induced by the Rydberg electron and the core is non-parallel to the Z-axis, implying that for the diatomic molecule M N is not a good quantum number. The matrix elements of F ryd (R, r) in the Rydberg electron basis are given by and The integral in Eq. (14) The radial integrals take on the following appearance: where ψ nl (r) is the radial component of the Rydberg electronic wave function.
In the next sections, we consider a Rydberg triatomic molecule formed by a rubidium atom and the diatomic molecule KRb. The rotational constant of KRb is B = 1.114 GHz [22], and its electric dipole moment d = 0.566 D [23]. We solve the Schrödinger equation belonging to the Hamiltonian (1) using the coupled basis (6), which includes Rb(n=24, l≥ 3) degenerate manifold and the energetically closest neighboring Rydberg state 27s. Note that we are neglecting the quantum defect of the 24f Rydberg state. For the diatomic molecule, we take into account are the rotational excitations for N ≤ 6.

III. THE FIELD-FREE BORN-OPPENHEIMER POTENTIALS AND PROPERTIES
In this section we investigate the BOP of the Rb-KRb triatomic molecule as a function of the separation R between KRb and Rb + . The electric field induced by the Rydberg core and electron on the diatomic molecule decreases towards zero as R is increased. When the polar diatomic molecule is located far enough from the Rydberg ionic core and electron, the system could be considered as formed by two subsystems: a Rydberg atom and the diatomic molecule. Thus, for R >> 1 a 0 , the BOP of the triatomic molecule approach the field-free limit E nl + BN (N + 1), with N being the field-free rotational quantum number of the diatomic molecule and E n,l the energy of the Rydberg atom in a state with principal and orbital quantum numbers n and l.
In Fig. 2 (a) and Fig. 2 (b), we present the BOP evolving from the Rb(n = 24, l ≥ 3) manifold with M J = 0 and M J = 1, respectively, as a function of R. Here, the zero energy has been set to the energy of Rb(n = 24, l ≥ 3) degenerate manifold and KRb(N = 0) level. All the curves within a panel belong to triatomic molecule states with the same symmetry, and all the crossings between the BOP are therefore avoided crossings.
Two different behaviors are observed in Fig. 2 Rb-KRb triatomic Rydberg molecule: Born-Oppenheimer potentials as a function of the separation between the Rydberg core and the polar molecule R for (a) MJ = 0 and (b) MJ = 1. The calculations include the 27s state and the degenerate manifold n = 24 and l ≥ 3 of rubidium. The lowest lying states with MJ = 0 and 1 evolving from the n = 24 and l ≥ 3 manifold are plotted with thicker red and green lines, respectively.
to KRb(N = 5, 6 )rotational excitations, respectively. In contrast, the BOP of the triatomic molecule levels formed from the Rb(n = 24, l ≥ 3) manifold have an oscillatory behaviour as a function of R. This reflects the dominant oscillatory behavior of the Rydberg electric field which is due to the oscillatory structure of the Rydberg electron wave functions.
The lowest triatomic molecule states with M J = 0 and M J = 1 steming from Rb(n = 24, l ≥ 3) Rydberg manifold are shifted more than 20 GHz for R 1200 a 0 and 1000 a 0 , respectively. When R is increased beyond a certain limit, their energies increase and approach the fieldfree dissociation limit, Rb(n = 24, l ≥ 3) -KRb(N = 0).
These results are converged with respect to the size of the basis set in (6). Table I contains the energies of the Rydberg levels close to the Rb(n = 24, l ≥ 3) degenerate manifold. The energy gaps between the Rydberg levels are larger than the energy shifts of the levels evolving from this degenerate manifold observed in Rb-KRb, see Fig. 2. This justifies that only the next neighboring level 27s has to be included in the basis set. Further including the 25d and 26p states decreases the energy differences to less than 1%. It may be possible for Rb-KRb triatomic molecule levels formed from the KRb diatomic molecule in highly excited rotational states and a Rb atom in lower lying Rydberg levels to produce BOP within the spectral range of Fig. 2. However, the coupling of such triatomic molecule states to those presented in Fig. 2 would be weak, because of the large rotational energy separation in KRb. We have also checked the convergence of our results with the number of rotational states of KRb included in the basis set. By increasing the number of rotational excitations from six to eight, the relative difference between the two sets of energies is smaller than 2 × 10 −6 . This implies that BOP obtained with the rotational basis including rotational excitations with N ≤ 6 are well converged.
For comparison, we have also determined the BOP for a Rb-KRb triatomic molecule, but using a two-state approximation to describe the internal motion of the diatomic molecule KRb [17]. In the two-state model, the rotational energy gap is obtained from the rotational constant, B = 1.114 GHz, and the electric dipole moment, is parallel to the LFF Z-axis so that KRb cannot freely rotate. Thus, we only consider the Z component of the electric field induced by the Rydberg atom F ryd in Eq. (14). In Fig. 3, we present the energies of the two lowest states evolving from the Rb(n = 24, l ≥ 3) manifold using the two-state model and the rigid rotor description. The energy shift of these BOP from the Rb(n = 24, l ≥ 3) is smaller when the rigid rotor description is used. This could be due to the effect of the X and Y components of the electric field, which reduce the net effect of the Z component. The depth of the potential wells is also decreased, which implies that they will accommodate less vibrational levels. This effect is particularly pronounced for the lowest wells of the two-state triatomic molecule near R ∼ 920 a 0 , and 1100 a 0 where the dipole is pointing toward and away from the ion core, respectively. When the rigid-rotor approximation is used, the last potential well near R ∼ 1100 a 0 is very shallow, and the diatomic molecule is also oriented away from Rb + . In the twostate model these two BOP have different symmetries and do cross, while the closest energy separation for the rigid rotor description is approximately 1.8 GHz. Furthermore, they are not coupled if an additional electric field is applied parallel to the Z-axis, see Sec. IV. For a rotating molecule in an electric field, the pendular states appear for strong fields when it becomes oriented along the field direction; each pendular state being a coherent superposition of field-free rotational states. For the diatomic KRb within the triatomic Rb-KRb, the electric field induced by the Rydberg core and electron should be expected to lead to these pendular levels. The strong hybridization of the angular motion of KRb is illustrated by the expectation value N 2 . For the lowest triatomic molecule states with M J = 0 and 1 evolving from the Rb(n = 24, l ≥ 3) degenerate manifold, N 2 is plotted as a function of R in Fig. 4. For both states, N 2 oscillates reaching values well above the field-free one N 2 = 0. The fact that N 2 surpasses the field-free value of the rotational state N = 1, indicates that higher rotational excitations contribute to the field-dressed rotational dynamics of KRb. As R is further increased beyond 1200 a 0 , the electric field strength is reduced and N 2 decreases slowly approaching zero for R 2000 a 0 . This indicates that these states belong to the degenerate manifold formed by the n 2 −9 levels of the Rydberg atom and KRb in its rotational ground state. For alkali metal atoms in general, there are three Rydberg levels with appreciable quantum defects, s, p, and d levels, accounting for a total of n 2 − 9 hydrogenic levels. The KRb molecule is strongly oriented and aligned due to the electric field induced by the Rydberg core and electron. The orientation and alignment of the diatomic molecule along the LFF Z-axis in the lowest triatomic molecule states with M J = 0 and 1 for the Rb manifold n = 24 and l ≥ 3, are illustrated by the expectation values cos θ dZ and cos 2 θ dZ in Fig. 5 (a) and Fig. 5 (b), respectively. Since the Rydberg electric field has also components along the X and Y -axes, we have also estimated the alignment of KRb along the LFF Xaxis by means of cos 2 θ dX in Fig. 5 (c). Let us mention that the alignments along the Y and X axes are identical, i. e., cos 2 θ dX = cos 2 θ dY , and that the diatomic molecule does not gain any orientation along these axes, i. e., cos θ dX = cos θ dY = 0. The KRb within these two states of the triatomic molecule is strongly oriented toward the Rydberg core for R 1000 a 0 and cos θ dZ oscillates as R is varied. As R is increased, cos θ dZ decreases and the dipole becomes oriented away from the Rydberg core. The maximal values of cos θ dZ are 0.78 and 0.60, for the M J = 0 and 1, respectively, cos θ dZ monotonically decreases for R 1200a 0 approaching the field-free value cos θ dZ = 0. Note that even for R ≈ 2000 a 0 these states present a weak orientation with cos θ dZ ≈ 0.2. The alignment parameters, cos 2 θ dZ and cos 2 θ dX also show oscillatory behavior as R is varied. For the M J = 0 state, the diatomic molecule is more aligned along the Z axis than along the X-axis, i. e., cos 2 θ dZ > cos 2 θ dX . Whereas, for M J = 1 state, it holds cos 2 θ dZ < cos 2 θ dX . For both states, at those values of R where cos 2 θ dZ reaches a maximum, cos 2 θ dX reaches a minimum and viceversa. The smallest (largest) value of cos 2 θ dZ ( cos 2 θ dX ) is obtained when cos θ dZ ≈ 0, i. e., KRb changes from being oriented toward the Rb + core to away from it. The maxima and minima of cos 2 θ dZ and cos θ dZ are reached at very close values of R.
For the M J = 1 triatomic molecule state, cos θ dZ and cos 2 θ dZ reach a shallow minimum and cos 2 θ dX a maximum for R ≈ 1200 a 0 which are due to a very broad avoided crossing that this state suffers with the neighboring one. Note that in the behavior of N 2 , this avoided crossing is not observed because both involved triatomic molecule states have similar values of N 2 .

IV. THE ELECTRIC FIELD-DRESSED BORN-OPPENHEIMER POTENTIALS
In this section, we investigate the impact of an additional external electric field on the BOPs of the Rb-KRb triatomic molecule. For the sake of simplicity, we assume that this external field is antiparallel to the LFF Z-axis, i. e., F ext = −F extẐ . For a tilted external electric field, the azimuthal symmetry will be broken, and M J will no longer be a good quantum number, increasing substantially the overall complexity. We focus on the weak field regime F ext ≤ 12 V/cm. For such weak fields, the impact on the KRb rotational motion is negligible because the shift due the field, dF ext ≤ 3.1 MHz for F ext ≤ 10 V/cm, is much smaller than B = 1.114 GHz. In contrast, due to its large dipole moment, the Rydberg levels of rubidium are significantly affected. This implies that the electric field induced by the Rydberg ion and electron in the diatomic molecule is also modified and as a consequence the level structure of the triatomic Rydberg molecule and the BOP change correspondingly. In Fig. 6(a) and (b), we present the lowest BOP with M J = 0 and 1, respectively, evolving from the Rb(n = 24, l ≥ 3) manifold with increasing electric field strength F ext . As the field becomes stronger, the BOPs are lowered, but the reduction in energy depends on the separation between the Rb atom and the KRb molecule, and the smaller R, the smaller this reduction is. Thus, the lowest BOP evolving from the Rb(n = 24, l ≥ 3) Rydberg manifold will at some point cross the BOP from Rb(27s) -KRb(N = 5) level, creating a sequence of avoided crossings which are narrow due to the weak coupling between the involved levels. As a consequence of the R-dependent effect of the external field, the widths of the outermost potential well is increased. This is particularly observed in the last two outer wells for the M J = 0 BOP, where for F ext = 0 a double-well structure becomes visible.

V. VIBRATIONAL STATES
In this section, we present the vibrational bound levels within the outer two minima of the BOP belonging to the lowest M J = 0 triatomic molecule state evolving from the Rb(n = 24, l ≥ 3) degenerate manifold, in an external electric field of strength F ext = 0, 4 and 12 V/cm. The perturber molecule KRb within the Rb-KRb triatomic system is oriented towards the Rb + core in the inner (L) well, and becomes oriented away from it in the last shallow well, see Fig. 7. By increasing F ext , the orientation of KRb is only weakly affected, but the depth of the BOP is increased, which implies an enhancement of the number of bound vibrational levels. In Fig. 7, one can appreciate the shift between the minima in the BOP, and the maxima/minima in the orientation of the KRb diatomic molecule.
With no external field, the last (R) outer well potential is not deep enough to accommodate a bound state. In the inner potential well (L), there exists eight bound vibrational levels. In Fig. 8 (a), we present the absolute square of the wave functions of these vibrational states. The expectation values R ν = ψ ν,0 |R|ψ ν,0 , with ψ ν,0 = ψ ν,0 (R) being the vibrational wave functions, are presented versus the vibrational quantum number ν in Fig. 9. These vibrational bands, with an energy spacing of a few hundreds MHz, i. e., |E ν=1 − E ν=0 | = 408 MHz and |E ν=7 − E ν=6 | = 183 MHz, have superimposed rotational structure determined by the rotational constant B ν = R −2 ν /2µ, of the order of a few tenths kHz, i. e., B 0 = 40.2 kHz and B 7 = 37.5 kHz.
For F ext = 4 V/cm, there are 13 vibrational levels bound in the two outermost potential wells. Due to the presence of the external field, the last well becomes deeper and can accommodate vibrational levels, and accordingly we encounter in Fig. 8  structure. For instance, the wave functions of the states with vibrational quantum number v = 8, . . . , 12 extend over both wells with R ν > 1000 a 0 , see Fig. 9. For R ≈ 1000 a 0 , the orientation of KRb is close to zero, since its dipole is changing from being oriented toward the Rb + core to away from it. Thus, only for the ν = 9 state, KRb is significantly oriented toward the Rydberg core. A similar energy spacing is found between neighboring vibrational levels as in the F ext = 0 V/cm case, except for the ν = 8 and 9 states, which are 36 MHz apart. By further increasing the field to F ext = 12 V/cm, the last two potential wells possess 20 vibrational levels. The vibrational wave function of the ν = 8 state is partially contained in the outermost (R) well, cf. Fig. 8 (c), having R ν ≈ 1030 a 0 , and at this distance the polar molecule is oriented away from the rubidium core. Highly excited vibrational states have R ν ≈ 1000 a 0 , and for similar separation distances, the KRb molecule does not present any orientation, see

and 8 levels.
Due to the non-adiabatic interactions, the electronic states are hybridized allowing for intra-electronic dipole transitions. For instance, for F ext = 12 V/cm (4 V/cm) the vibrational levels ν = 0 and ν = 8 (ν = 9) could be coherently coupled in a Raman process via a highly excited vibrational state within this BOP as has been suggested in Ref. [15]. In such a way, the KRb orientation becomes entangled.

VI. CONCLUSIONS
We have investigated ultralong-range triatomic Rydberg molecules formed by a Rydberg rubidium atom and the KRb diatomic rotational molecule in the presence of electric fields. The field effect arises from both the "internal" Rydberg core and electrons and from an externally applied field. This species exhibits novel binding properties due to the attractive interaction of the Rydberg electron and the ground state KRb diatomic molecule, which can be controlled by an external field. Previous studies have modeled the diatomic molecule in a simple two level paradigm. We have been systematically extending this approach by taking into account the rotational motion of the diatomic molecule which couples, due to its dipole moment, to the electric field supplied by the Rydberg ionic core and the Rydberg electron or an external source. The rotational motion of the KRb diatomic molecule is described within the rigid rotor approximation, which allows for a proper description of the hybridization of the rotational motion of KRb due to the electric fields. The Born-Oppenheimer potential curves for M J = 0, 1 for the Rydberg triatomic molecule as a function of the separation between the Rb + core and the KRb diatomic molecule have been derived and analyzed. We have performed detailed analysis of the hybridization of the angular motion, orientation and alignment of KRb within the Rb-KRb triatomic molecule. These results demonstrate that excited rotational states are involved in the field-dressed dynamics, and, therefore, the rotational degrees of freedom are needed for a proper description of the Rydberg triatomic molecule.
In an additional external electric field, the level structure of the Rydberg triatomic molecule is severely modified. The BOPs evolving from the Rb(n = 24, l ≥ 3) Rydberg manifold are strongly lowered, due to electronic hybridization, as F ext increases, whereas those levels evolving from the non-degenerate Rb (27) state are weakly affected. The field-dressed potentials strengthen the bound state character of the rovibrational molecular states and lead for certain configurations to narrow avoided crossings. In the presence of an external field, a Raman scheme among different vibrational levels within the lowest electronic state evolving from the Rb(n = 24, l ≥ 3) Rydberg manifold could be used to prepare entangled states of KRb levels with different orientations.
In an ultracold quantum gas which forms the KRb, there are typically as many potassium atoms as rubidium atoms. Hence, there are two options to form the triatomic Rydberg molecular states, i. e.K * -KRb and Rb * -KRb triatomic molecules. The main difference between the two highly excited triatomic molecules are the different low angular momentum states with appreciable quantum defects and energy levels of the Rydberg spectra of K and Rb, which provide a variable binding of the corresponding ground state heteroncuclear diatomic molecule KRb. The two types of triatomic molecules do share the atomic (K or Rb) degenerate manifold such as the n = 24 and l ≥ 3, as employed here. Thus, the Born-Oppenheimer potentials emerging from the degenerate manifold are identical for both triatomic molecules since they only depend on the properties of the polar ground state diatomic molecule and the atomic Rydberg degenerate manifold under consideration. Their interaction (avoided crossings) with the quantum defect Born-Oppenheimer potentials will, of course, move in energy and nuclear configuration space from one to the other species. Similar arguments hold for any other combination of Rydberg atoms and polar diatomic molecules possessing an electric dipole moment below the critical value d cr = 0.639315 a.u [24][25][26][27] above which the Rydberg electron would bind to the dipolar diatomic molecule.
An even more complete description of the triatomic Rydberg molecule would take into account the vibrational motion of the diatomic molecule. However, the energy scales involved into the vibrations are much larger than those of the rotation and therefore little impact of the electric field of either internal or external origin can be expected. Also, the coupling of the rotation to the vibration of the diatomic molecule should be negligible for sufficiently low-lying rovibrational states. The corresponding potential energy surfaces then become fourdimensional with three vibrational degrees of freedom from the field-free situation and the additional rotational degree of freedom which turns into a vibration for sufficiently strong fields.