Effects of higher-order Casimir-Polder interactions on Rydberg atom spectroscopy

In the extreme near-field, when the spatial extension of the atomic wavefunction is no longer negligible compared to the atom-surface distance, the dipole approximation is no longer sufficient to describe Casimir-Polder interactions. Here we calculate the higher-order, quadrupole and octupole, contributions to Casimir-Polder energy shifts of Rydberg atoms close to a dielectric surface. We subsequently investigate the effects of these higher-order terms in thin-cell and selective reflection spectroscopy. Beyond its fundamental interest, this new regime of extremely small atom surface separations is relevant for quantum technology applications with Rydberg or surface-bound atoms interfacing with photonic platforms.

Highly excited (Rydberg) atoms have huge electric and magnetic transition multipole moments that make them interact very strongly with their environment.Thus, they are ideal candidates for studying dispersion forces such as Casimir-Polder (atom-surface) [1,2] or van der Waals (atom-atom) interactions [3].More recently, Rydberg atoms have attracted significant attention for quantum technology applications.In particular, it was demonstrated that probing Rydberg atoms inside thin vapour cells [4] presents a simple way to fabricate room temperature single-photon sources based on the Rydberg blockade effect [5], without having to resort to complex cold atom manipulations.Moreover, Rydberg atoms in vapour cells are being used as electric-field probes at THz or GHz frequencies [6][7][8].However, the rapid scaling of electric-dipole moment fluctuations makes Casimir-Polder (CP) interactions a dominant spectroscopic contribution that impacts potential hybrid systems such as tapered optical fibers [9,10], hollow core fibers [11] and thin cells [4,5] that aim at interfacing atoms with photonic platforms.Similarly, Rydberg interactions with van der Waals heterostructures have also been investigated [12].
Theoretical studies of the interaction between dielectric surfaces and highly excited atoms have exposed the limitations of the traditional Casimir-Polder (CP) approach in which only the electric dipole interaction is taken into account.Indeed, the dipole approximation breaks down and higher-order terms need to be considered in the extreme near field [13] as Rydberg wavefunctions can easily extend beyond 100nm (being proportional to n ⋆2 , with n ⋆ the effective quantum number).Moreover, perturbative approaches have also been questioned when the expected energy shifts are comparable to adjacent energy level spacings [14].Although previous theoretical works have presented the basic scaling laws governing quadrupole interactions [13], detailed results of the higher order interaction coefficients have not yet been presented for CP atom-surface interactions.Higher order effects have nevertheless been studied in detail for the case of atom-atom or molecule-molecule van der Waals interactions [15,16].
Atom-surface experiments have similarly flourished during the past decades [17][18][19][20], shedding light on novel effects such as the temperature dependence of CP interactions [19,21], and atom-metamaterial interactions [22] with an ever increasing precision in view of uncovering fundamental forces beyond the standard model [20].However, higher-order interactions remain so far experimentally unexplored.A prominent and well developed experimental method for performing such studies is vapour cell spectroscopy [21,[23][24][25][26]. Thin vapour cells allow for a controlled confinement of atomic vapours within dielectric walls down to the nanometer regime, making them excellent platforms for probing Rydberg atoms extremely close to dielectric surfaces [4,25].Selective reflection in macroscopic cells has also been used for atomic or molecular gases [21,23,27] close to a surface, but provides no means of controlling the probing depth, typically defined by the excitation wavelength.
Here, we investigate higher-order (quadrupole and octupole) CP interactions between a Rydberg atom and a dielectric surface, providing explicit calculations of the C 3 (dipole-dipole) and the C 5 (combined quadrupolequadrupole and dipole-octupole) coefficients for alkali Rydberg atoms in the near field.We subsequently study the effects of higher-order interactions on CP vapour cell spectroscopy, illuminating the conditions under which higher-order interactions can be experimentally measurable.Multipole contributions are of importance for experiments aiming at binding atoms to surfaces and extending the optical control to the extreme near field [28].
We will conduct our studies in the electrostatic limit, where CP Rydberg-surface interactions can be described as the interaction of the atomic charge distribution (centred in O) with its surface-induced instantaneous image (centred in O ′ ) in front of a perfectly conducting surface.We assume that the Rydberg atom consists of a valence electron orbiting around a positively charged core.The atom-surface interaction energy, W , is half the electrostatic energy of the atomic charge distribution placed under the external potential of its image, Φ ′ ( ⃗ r ′ ), with the corresponding field ⃗ E ′ ( ⃗ r ′ ), Here, ⃗ r 0 is the position vector of the atom with coordinates r i .The atomic multipole moments with respect to O are denoted as ⃗ p, Q and T (dipole, quadrupole and octupole moments, respectively).The potential created by the image can itself be expanded into multipoles, giving Here, ⃗ p ′ , Q′ , and T ′ are the dipole, quadrupole and octupole moments of the image with respect to O ′ .Symmetry links the components of the image moments to those of the atomic moments by p , where κ is the number of times z appears as a tensor index.
The CP interaction energy can therefore be written as where W pp , W pQ , W QQ , W pT are the dipole-dipole, dipole-quadrupole, quadrupole-quadrupole and dipoleoctupole contributions, respectively.The dipole-dipole and quadrupole-quadrupole terms are given by The cross terms such as the dipole-quadrupole and dipole-octupole contributions with their z −4 s and z −5 s dependence, respectively, are given in the Supplementary material.It should be noted that W pQ includes both the energy of a dipole immersed in the field of a quadrupole and vice-versa (the same applies to W pT ).
In the electrostatic limit, we can calculate the CP frequency shift, ∆f , of an atomic Rydberg state using first order perturbation theory as  7) and ( 8)] providing an accurate estimate, to within 1%, of the interaction coefficients for all alkali Rydberg atoms.
Here, ψ n,l,J,M J is the wavefunction of the Rydberg electron, with n the principal quantum number, l, J the orbital and total angular momentum quantum numbers, and M J the projection of J onto the z-axis.For Rydberg atoms we can assume that the external electron is under the influence of a central potential given by an effective Coulomb interaction modified to include the polarisability of core electrons [29].This allows us to easily obtain the radial wavefunction by numerically solving Schrödinger's equation using the Numerov method.
In our analysis, we ignore the hyperfine structure of the atoms as it is usually very small compared to the CP energy shifts for most alkali Rydberg atoms.The above analysis allows us to calculate Rydberg-surface interactions at any multipole order.
The quantum mechanical averaging of the interaction energy gives the following results: a) The dipole-dipole term (∆f 3 ) yields the well-known near-field Casimir-Polder frequency shift ∆f 3 = −C 3 /z 3 s , where C 3 is a coefficient usually expressed in MHz µm 3 .b) The dipolequadrupole terms vanish for parity reasons.c) The quadrupole-quadrupole and dipole-octupole terms give a frequency shift expressed as ∆f 5 = −C 5 /z 5 s , where C 5 is a coefficient expressed in MHz µm 5 .We emphasize that the dipole-octupole contributions do not necessarily vanish as both dipole and octupole interactions can act on the same atomic transition (∆l = ±1 transitions can be both dipole and octupole allowed).However, dipoleoctupole terms only contribute to the anisotropy of the atom-surface interaction, with an overall scalar compo-nent (averaging over all M J ) that remains zero.
In Fig. 1 we plot the C 3 and C 5 coefficients for S 1/2 and D 3/2 states of cesium and rubidium atoms as a function of the effective quantum number (n * = n − δ * ), where δ * is the quantum defect.It shows that the interaction coefficients depend very little on the core polarisability and on angular momentum.We have therefore calculated analytical expressions for both C 3 and C 5 coefficients for a hydrogen atom.For an S 1/2 state (l = 0) the interaction coefficients become The full solution, for all states, is given in the Supplemental material.Recall that for hydrogen n ⋆ is simply the principal quantum number and note that the leading term in the above polynomials is independent of angular momentum.
Our assumption of a perfectly correlated image breaks down at distances comparable to relevant transition wavelengths or when a frequency-dependent dielectric constant is considered [30].However, for Rydberg atoms, the relevant multipole transitions are typically in the THz or GHz regime suggesting that retardation can be ignored at micrometer distances.Moreover, at these frequencies most materials do not possess surface resonances, and their dielectric constant tends to their static (ϵ s ) value.Dielectric effects can therefore be accounted for by simply multiplying the dispersion coefficients (Fig. 1) by the surface response (S = ϵs−1 ϵs+1 ), while temperature effects [21,31,32] are negligible and can be ignored [33].The above arguments suggest that the electrostatic limit is a good approximation for studying Rydberg-surface interactions in the extreme near field.However, a QED treatment [34][35][36] would be necessary for treating the coupling of atoms with resonant surfaces or meta-surfaces.
At this point, a word of caution is appropriate.Frequently, traceless multipole tensors are used in the calculation.However, this is only justified for quadrupole tensors, as its trace component does not contribute to the energy.This is no longer true for the octupole tensor, where the trace does in general contribute to the energy [37,38].Indeed, upon reducing the quantity r i r j r k appearing in T in terms of spherical harmonics yields terms with Y 3m (Θ, φ) (traceless part) as well as Y 1m (Θ, φ) (trace part).The latter can be identified as a contribution to the dipole transition, and only further symmetry considerations of the field distribution or atomic transition matrix elements can cause them to vanish [39,40] .
Having developed a theoretical framework that allows us to calculate both C 3 and C 5 coefficients, we can es- timate the effects of quadrupole interactions on CP experiments.Atomic spectroscopy in vapour cells of variable nanometric thickness (thin cells) is a well developed method for measuring CP interactions with excited states including Rydberg atoms [25,41].Thin cells allow us to control the vapour confinement down to the nanometer regime (thickness can be as small as 50nm) [25], giving them a distinct advantage over other methods for probing higher order CP interactions with Rydberg atoms.
Typically, a two-step excitation technique is used to reach high-lying excited states of alkali atoms [21,24,25].For cesium (our atom of choice), a strong pump excites atoms to their first excited state (6P 1/2 or 6P 3/2 ) and a weak laser probes the 6P 1/2 → nS or 6P 1/2 → nD transitions.For states with n ≈ 20, the transition wavelength λ probe is about 510nm.In Fig. 2 we show the basic principle of the experiment that will be analysed.
Inter-atomic collisions and radiation trapping in the atomic vapour redistribute the initially velocity selected 6P population to many atomic velocities and hyperfine states.This allows us to consider our atoms essentially as two level systems in linear interaction with the probe excitation field.Thin cells form a low finesse Fabry-Perot cavity that eventually mixes the backward (reflection I lin R ) and forward (transmission I lin T ) response of the polarised atomic vapour [42] given by: For a symmetric Fabry-Perot cavity with reflection co- efficient r for both interfaces the transmission signal (S T ) that consists of the homodyne beating between the atomic response with that of an empty cavity is given by Here, F = 1 − r 2 e 2ikL with r the reflection coefficient of the windows and k = 2π/λ probe .The atomic velocity along the probe beam propagation axis is denoted as v z , the atomic vapour density inside the cell is N and the transition dipole moment is µ.In the above equations, the functions L(z ′ ) − L(z) are where ∆f CP (z) is the CP shift of the Rydberg state (the shift of the 6P state is negligible) inside the thin-cell cavity.The shift can be separated into a dipole component and a quadrupole component that depend on the C 3 and C 5 coefficients, respectively.As full calculation of the CP shift inside a cavity requires a more elaborate theory, we will not consider resonant effects due to surface polaritons and simply add the potentials of both walls, neglecting the infinite series of multiple images.In this case, the dipole and quadrupole shifts in the middle of the cell become −16C 3 /L 3 and −64C 5 /L 5 , respectively.
In Fig. 3(a) we show the transmission spectrum of a resonant 6P 1/2 → 16S 1/2 beam through a thin cell of three different thicknesses.The atom-surface interaction coefficients calculated for the 16S 1/2 state of cesium (with a Bohr diameter of ≈ 15nm) are C 3 =4.15MHzµm 3 and C 5 =0.45 kHzµm 5 .The red curves represent the calculated spectra using both dipole and quadrupole interactions, whereas for black curves the quadrupole interactions are omitted.The effect of quadrupole interactions becomes observable when the atomic vapour is confined at thicknesses smaller than 200nm.At L=100nm the additional quadrupole shift exceeds the predicted spectral width, suggesting that the C 5 coefficient could be measurable with a thin-cell spectroscopy experiment.tends towards 16C 3 (dashed horizontal line), suggesting that when the cell is very thin, the dominant spectral contribution derives from atoms at the centre of the cell.For increasing cell thicknesses, the contribution of layers closer to the walls becomes more prominent leading to a larger C eff 3 .Red dots represent calculations including both dipole and quadrupole interactions.From Fig. 3(b) we can see that quadrupole interactions have no visible effect for thicknesses larger than 250nm.The extent of the gray shaded area is L 3 w/5, where w is the width of the transmission spectrum.The gray shaded area gives an indication of the capability of the proposed experiment to discern between the two models (black and red dots).Note that the discerning capability also depends on the signal to noise ratio of the experiment.The reduction in signal amplitude is noted in Fig. 3(a) next to the transmission curves.The effects of higher-order interactions are evident for larger cell thicknesses when probing higher-lying states such as 28S 3/2 via the 6P 1/2 → 28S 3/2 transition (see Supplemental Material).
Our above analysis assumes that Rydberg atoms interact with surfaces only via CP interactions.However, Stark shifts due to adsorbed atoms or parasitic electric field in the surface of the dielectric windows are known to be an important problem in precision atom-surface experiments [20,[43][44][45].In particular, high-lying states become extremely sensitive to electric fields [4,46] as their polarisability scales more rapidly (α ∝ n ⋆7 ) than the C 3 coefficient.Our analysis suggests that higherorder CP effects can be measurable even with relatively low-lying Rydberg states with n ⋆ ≈ 10 . . .15, for which atom-surface interaction experiments have already been demonstrated [1,2].This is a powerful indication that vapour cell spectroscopic experiments could provide excellent testbeds for further exploring Casimir-Polder physics.
We acknowledge insightful discussions with Martial Ducloy and discussions with Isabelle Maurin on the numerical modelling of thin cell spectra.This work was financially supported by the ANR-DFG grant SQUAT (Grant Nos.ANR-20-CE92-0006-01 and DFG SCHE 612/12-1), the DAAD and Campus France (via the PHC-PROCOPE programme, grant No. 57513024), and the French Embassy in Germany (via the Campus-France PHC-Procope project 44711VG and via PROCOPE Mobilité, project DEU-22-0004 LG1).

FIG. 1 .
FIG. 1. C3 and C5 coefficients (blue and red colors, respectively) for S 1/2 and D 3/2 states (squares and circles, respectively) of cesium and rubidium (open and closed points, respectively) as a function of the effective quantum number n ⋆ = n − δ.The straight solid lines correspond to the analytical expressions derived for a hydrogen S state [Eqs.(7) and (8)] providing an accurate estimate, to within 1%, of the interaction coefficients for all alkali Rydberg atoms.

FIG. 2 .
FIG. 2. (a) Schematic of the thin cell experiment analysed in our simulations.A laser beam at 0.894µm or 0.852µm pumps cesium atoms to the 6P 1/2 or 6P 1/2 level, respectively.Subsequently, a green laser at 0.514µm or 0.513µm probes Rydberg atoms at the 6P 1/2 → 16S 1/2 or the 6P 3/2 → 26S 1/2 transition, respectively.Higher lying states can also be easily accessed via the same scheme.Due to interatomic collisions, the population of the intermediate levels has a quasi-thermal (Maxwell-Boltzmann) velocity distribution.(b) Relevant energy levels.

FIG. 3 .
FIG. 3. (a) Thin-cell transmission spectra for three different thicknesses, calculated using Eq.(11) (ST ) with C3=4.15 MHzµm 3 and C5=0.45 kHzµm 5 , corresponding to Cs 16S 1/2 atoms.Dashed vertical line: position of the transition frequency in the volume.Straight red lines: calculations including both dipole and quadrupole interactions (C3 and C5).Black lines: dipole interactions only.The transmission amplitude decreases with thickness by a factor indicated in the figure.(b) Displacement of the transmission dip C ef f 3 with respect to the volume transition frequency (dashed line), multiplied by L 3 , as a function of cell thickness.Red dots: calculations with full CP potential.Black dots: dipole interactions only.Horizontal dashed line: CP dipole shift in the centre of the cell multiplied by L 3 .

Figure 3 (
b) shows the predicted displacement of the transmission dip C eff 3 with respect to the volume resonance away from the surface, multiplied by L 3 .When quadrupole interactions are ignored (black dots), C eff3