Laser photoexcitation of Rydberg states in helium with n > 400

Helium atoms travelling in pulsed supersonic beams have been photoexcited from the metastable 1s2s 3S1 level to Rydberg states with principal quantum numbers exceeding 400 by resonance-enhanced two-colour two-photon excitation using narrow-bandwidth CW laser radiation. To achieve this, the photoexcitation region was shielded and stray electric fields cancelled using a cubic arrangement of 80 mm × 80 mm copper plates. The excited Rydberg atoms were detected by pulsed electric field ionisation downstream from this photoexcitation region. Comparison of the experimental spectra with the results of calculations indicate that the stray electric fields at the position of Rydberg state photoexcitation were reduced to as low as 160 μV cm–1. Using helium in these experiments minimises temporal changes in the stray fields resulting from surface adsorption. Measurements performed 70 h apart indicate that the rate of change of these fields was on the order of 2 μV cm–1 per h, making beams of metastable helium atoms ideally suited as probes for high precision electrometry in spectroscopy experiments involving less readily available species such as positronium or antihydrogen.


Introduction
The large static electric dipole polarisabilities, static electric dipole moments and electric dipole transition moments associated with high Rydberg states of atoms and molecules [1] make samples in these states very sensitive probes of static [2][3][4][5][6][7] and time-dependent electric fields [8][9][10][11][12][13]. Because of this sensitivity it is also therefore essential to carefully control or cancel these fields when performing experiments with Rydberg atoms and molecules. This is particularly true when coherent interactions with narrowbandwidth electromagnetic fields over long timescales are required for precision spectroscopy [14][15][16][17] or quantum information processing, see, e.g., [18] and references therein.
Interest in performing precision spectroscopic studies of Rydberg states of exotic species such as antihydrogen or positronium, with a view to probing matter-antimatter assymmetries [17,34], and contributing to the resolution of the current puzzle relating to the proton charge radius as determined in precision spectroscopic studies of atomic hydrogen and muonic hydrogen [19][20][21][22] makes it desirable to identify appropriate off-line methods for the characterisation and cancellation of stray electric and magnetic fields in these experimental settings. The exceptional sensitivity of high Rydberg states to these fields makes them ideal probes of these environments with the optimal cancellation of stray electric fields achieved in experiments in which the highest Rydberg states are photoexcited. In the past, precise cancellation of stray laboratory electric fields and studies of very high n Rydberg states have been performed using beams of alkaline earth metal atoms. Rydberg states with principal quantum numbers exceeding n = 500 have been resolved in laser photoexcitation spectra of barium [2] and strontium [23]. These atoms are well suited to the photoexcitation of such high Rydberg states, and hence sensitive measurements of, and applications in the cancellation of, weak static electric fields because of the large oscillator strength for transitions from low-n intermediate 1 P 1 levels to high-n 1 D 2 Rydberg levels. However, because of their large ground-state static electric dipole polarizabilities of α Ba ∼270 au and α Sr ∼190 au [24] (1 au=1.648 773×10 −41 C m 2 V -1 ), these atoms are prone to cause time-dependent changes in stray electric fields following surface adsorption. In cryogenic environments, in which antihydrogen atoms are synthesised and precision spectroscopy of high Rydberg states must be carried out to minimise contributions from blackbody radiation, these effects are expected to be even more pronounced than in experiments performed at room temperature.
To perform sensitive electrometry with minimal contamination of the apparatus, or contributions to time-dependent changes in stray fields following surface adsorption, beams of rare gas atoms are most suitable. Studies of very high n Rydberg states of argon and krypton, in which precise electric field cancellation was implemented, have been performed previously [4,[25][26][27]. The ground-state static electric dipole polarizabilities of these atoms are α Ar ∼10.77 au and α Sr ∼16.47 au, respectively [28]. Consequently, effects of time-dependent changes in stray electric fields following surface adsorption are likely to be significantly smaller than in experiments with Ba or Sr. However, in cryogenic environments, i.e., in apparatus operated at temperatures on the order of 10K and below, these heavier rare gas atoms are prone to freeze onto electrode surfaces. This can lead to the accumulation of insulating patches on these metallic surfaces where charge build up can occur and act as a further source of timedependent stray electric fields. To avoid these problems, helium atoms are optimal for sensitive electrometry in cryogenic environments [29]. By using helium in such settings surface adsorption within the apparatus is low, even in cryogenic environments. This, together with the low groundstate static electric dipole polarizability of helium of α He ∼1.38 au [28], ensures that contributions from the atoms to time-dependent changes in stray fields are minimised.
Direct laser photoexcitation of Rydberg states in helium at high spectral resolution from the ground state is difficult because of the short wavelength radiation required to achieve this. However, as demonstrated here, population of the metastable 1s2s 3 S 1 level in an electric discharge, allows subsequent high-resolution resonance-enhanced two-colour two-photon CW laser photoexcitation of Rydberg states with values of n>400. By exploiting these highly excited Rydberg states as sensitive probes of static electric fields in the experimental apparatus, these fields have been cancelled to 160μV cm -1 with a sensitivity of better than ±20μV cm -1 .
In the following, the apparatus in which the experiments were performed is first described. Laser photoexcitation spectra extending to Rydberg states with n>400 are then presented. Comparison of these measured spectra with spectra calculated for a range of weak electric fields are then made to allow residual stray static electric fields to be characterised, and bounds to be placed on the time-dependence of changes that occur in these fields.

Experiment
A schematic diagram of the apparatus used in the experiments is presented in figure 1. A pulsed supersonic beam (v z  2 000 m s -1 ) of helium atoms in the metastable 1s2s 3 S 1 level was generated in an electric discharge at the exit of a pulsed valve [30]. The valve was operated at a repetition rate of 50Hz. After passing through a 2 mm diameter skimmer and electrostatic ion filter, to remove charged particles emanating from the discharge, the atomic beam entered the central laser photoexcitation region of the apparatus which was maintained at a background pressure of 10 −8 mbar. This region was located within a 80mm × 80mm×80mm cubic structure composed of three pairs of parallel copper plates. The design of this shielded photoexcitation region is based on that used by Frey et al [3] and Ye et al [23] in experiments to study high-n Rydberg states of potassium and strontium, respectively. The atomic beam entered (exited) the photoexcitation region through a 3 mm diameter (6 mm diameter) aperture in the copper plate oriented parallel to the xy plane and located closest (furthest) from the beam source. The UV (25708.588 cm −1 ≡388.975 08 nm) and IR (tuned between 12 735 cm −1 ≡785.230 nm and 12 746 cm −1 ≡784.584 nm) CW laser radiation used to drive each step of the resonanceenhanced two-colour two-photon 1s2s 3 S 1  1s3p 3 P 2  1snd 3 D Rydberg state excitation scheme, respectively, entered and exited the photoexcitation region through 3 mm diameter apertures in the center of the copper plates oriented parallel to the zy plane. From previous studies of static electric dipole interactions in ensembles of excited Rydberg atoms [31], the density of helium atoms in the metastable 1s2s 3 S 1 level in the photoexcitation region of the apparatus is estimated to be <10 8 cm −3 . The typical density of Rydberg atoms in states with values of n>275 is below 10 5 cm −3 . Even for these high values of n, the corresponding frequency shifts associated with electric dipole-dipole interactions between excited atoms at these densities (typically 1-10 MHz) are below the spectral resolution of the apparatus. After interacting with the laser radiation, the high longitudinal speed of the atomic beam allowed for the excited atoms to exit the photoexcitation region and be detected by pulsed electric field ionisation upon the application of a pulsed potential of −550V to the ionisation plate indicated in figure 1. The resulting electrons were then collected at a microchannel plate detector. Separating the photoexcitation and detection regions in the apparatus in this way minimised effects of electric field noise in the excitation region caused by the process of switching the ionisation electric fields.
The electric fields at the position where the focussed laser beams interacted with the atomic beam at the center of the photoexcitation region were controlled by applying individual dc electric potentials to each of the six copper plates. To further minimise effects of electrical noise, to which high Rydberg states with large static electric dipole moments are particularly sensitive [10,11], the potential applied to each electrode was generated using a 1.5V battery and an adjustable potential divider. Inside the vacuum chamber in which the experiments were performed, the wires used to apply these potentials to each pair of parallel copper plates surrounding the photoexcitation region were wound as a twisted pair.

Results
Using narrow-bandwidth CW laser systems, the selective laser photoexcitation of individual very high Rydberg states of atoms or molecules is in general not limited by the spectral resolution of laser radiation but instead by uncontrolled stray external fields at the photoexcitation position. In the experiments reported here the maximal achievable resolution is limited by the finite fluorescence lifetime of the intermediate 1s3p 3 P 2 level, 1s3p P 3 2 t  95 ns [32]. As can be seen from the spectrum of the 1s3p P 1s100d D 3 2 3  transition in figure 2 (continuous blue curve), this, combined with contributions from uncertainties in laser frequency stabilisation and Doppler broadening, yields a spectral resolution of 0.000 187cm −1 (≡5.6 MHz). When recording this spectrum the laser beams were not focussed. The corresponding full-widths-at-halfmaximum (FWHM) of their Gaussian spatial profiles in the z dimension was ∼0.7mm, and the interrogation time of the atoms by the laser radiation was therefore ∼350ns, i.e., 3.7 times the fluorescence lifetime of the intermediate 1s3p 3 P 2 level. Consequently, the spectral resolution was not limited by this finite interrogation time. Because of the very small electric dipole moments, d 3p,nd , for transitions from this intermediate level to very high Rydberg states, e.g., d 3p,400d =0.001 2 e a 0 , it was necessary to focus the laser beams to FWHM beam waists of 100 m  m to ensure a sufficient rate of production of excited states in the experiments. The reduced interaction time of the atoms with the laser beams under these conditions resulted in an increased spectral width of 0.001 07cm −1 (≡32 MHz) as can be seen from the corresponding spectrum of the 1s3p P 1s100d D 3 2 3  transition indicated by the dashed red curve in figure 2. In the experiments described in the following this is the effective spectral resolution. This spectral width is equal to the interval between Rydberg states in helium with principal quantum numbers 590 and 591 in the absence of external fields.
The spectra in figure 2 were recorded after minimising the stray electric fields in the photoexcitation region of the apparatus by repeated sequential adjustment of the offset potentials applied to the pairs of electrodes in the xy, xz and yz planes. To achieve this, laser spectra of the 1s3p P 1s100d D 3 2 3  transition were recorded for each applied cancellation field and the quadratic Stark energy shift of the 1s100d 3 D levels was minimised. Using this procedure the stray electric field at the position of Rydberg state photoexcitation was cancelled to ∼±1 mV cm -1 . Further field cancellation required the excitation of higher Rydberg states which acted as more sensitive probes of the residual stray field. Spectra were therefore subsequently recorded of transitions to states in the vicinity of n=200. This allowed the stray fields to be more accurately cancelled by minimising the spectral width of the manifold of Stark states with linear Stark energy shifts associated with each value of n. This process was then repeated for states between n = 250 and 400 in intervals of approximately 50. Upon completion of this stray field cancellation procedure spectra encompassing transitions to a wide range of these very high n Rydberg states were recorded. The IR laser system used to drive the n 1s3p P 1s d D  transitions could be continuously tuned, mode-hop free, over a spectral range of 0.66cm −1 (≡20 GHz). Therefore the spectrum presented in figure 3, which includes transitions to Rydberg states with values of n from275 to the wavenumber associated with the transition to states with n;450, was recorded in several sections, each covering the transitions to approximately 50 Rydberg states. In this spectrum, the shoulder on the low wavenumber side of the features with values of n between 275 and 300 corresponds to the transition to the 1sns 3 S 1 level which has a quantum defect of 0.296 657 [33]. Individual Rydberg states can be distinguished up to n∼420 (see, e.g., figure 4).
From the spectrum in figure 3 the residual uncancelled stray electric field could be determined most precisely by studying the region encompassing the transitions to states with values of n close to 400 and indicated by the horizontal red bar labelled A. The corresponding portion of the experimentally recorded spectrum, displayed in figure 4, was compared to the results of calculations over the same spectral region performed for a range of trial electric fields, F. These spectra were calculated by considering the linear Stark energy shifts, E n n n m  . The quantum defect of the nd Rydberg states is 0.0029 [33]. Consequently, these states, and those with values of ℓ>2 are close to degenerate in zero electric field. In fields close to the Inglis-Teller limit [1], in which the Stark manifolds with values of n which differ by +1 cross, the Stark energy shifts of these states, and spectral intensity distributions are indistinguishable from those in a hydrogen atom. The dashed red curve in figure 4, overlaid on the experimental data (continuous blue curve), was calculated for a residual uncancelled electric field of 200μV cm -1 . The good agreement of this calculated spectrum with the experimental data leads to the conclusion that stray electric fields in the apparatus were cancelled to this level when the spectrum was recorded.
The advantages of using helium atoms for precise electrometry center upon the minimal disturbances they cause to the surrounding environment in the experimental apparatus following surface adsorption. Bounds can be placed on the time-dependent changes in the residual uncancelled stray electric fields in the apparatus used here following repeated measurements performed at the same values of n without changing the offset potentials applied to the electrodes surrounding the photoexcitation region. The results of such measurements, for values of n between 373 and 375 are displayed in figure 5. The experimental spectra in this figure are indicated by the dashed blue, and continuous red curves. The data-point-to-data-point fluctuations are greater in the latter spectrum because of a slightly reduced averaging time for each measurement. These spectra were recorded approximately 70 h apart with the data indicated by the dashed blue curve recorded first. During this time the metastable helium beam was operating for approximately 5 h. The changes in the magnitude of the residual uncancelled stray electric fields between the two measurements are determined by comparison of the experimental data with the spectra calculated for fields of 210±50μV cm -1 as indicated by the shaded gray region in the figure. The bounds of this shaded region with the higher (lower) contrast correspond to the results of the calculations for fields of 160μV cm -1 (260 μV cm -1 ). From this data the contrast of the spectral features in the dashed blue spectrum indicates the presence of a residual stray electric field with a magnitude close to that at the lower end of the calculated range, i.e., ∼160μV cm -1 . On the other hand, the spectrum recorded 70 h later matches more closely that corresponding to a residual field of ∼260μV cm -1 . Under the assumption that collisions of helium atoms with, and adsorption on, the surfaces of the copper plates surrounding the photoexcitation region do not significantly affect these stray fields, it can be concluded that the stray fields in the apparatus drift at a rate of ∼2μV cm -1 h -1 , dominated by surface adsorption of the background gas in the unbaked vacuum chamber. Collisions of helium atoms in the metastable 1s2s 3 S 1 level with the copper electrodes surrounding the Rydberg state photoexcitation region, or with the background gas-leading to Penning ionisation-do not appear to contribute significantly to changes in the stray electric fields in the these experiments. The dexcitation of a metastable helium atom, or the  neutralisation of a Penning ionised atom or molecule from the background gas, on one of the copper surfaces may be expected, as a first approximation, to have a similar effect on the stray electric fields as the adsorption of a ground-state helium atom, or neutral atom or molecule from the background gas. This situation would of course be different in collisions of the metastable helium atoms, or ionised components of the background gas, with the insulating spacers used to position the copper electrodes. However, in the construction of the electrode structure surrounding the photoexcitation region, particular care was take to avoid all direct lines of sight from the position at which the laser beams crossed the atomic beam to these insulating spacers.

Conclusion
In conclusion, we report for first time spectra of very high Rydberg states of helium with values of n>400. These highly excited states are bound by <0.7 cm −1 (≡21 GHz ≡ 90 μeV). The expectation value of the position operator associated with the Rydberg electron, e.g., r n 400 á ñ = , and hence the spatial extent of the Rydberg electron wavefunction in these states is ∼13μm. As such, helium atoms in these excited states are truly macroscopic objects with static electric dipole moments of up to 610 000 D. The sensitivity of these electric dipoles to static electric fields has been exploited for precise electrometry allowing the cancellation of stray electric fields within our experimental apparatus to as low as ∼160μV cm -1 . We expect that these fields can be further reduced in the future with improved preparation of the copper surfaces surrounding the photoexcitation region and the implementation of further methods to reduce low-frequency time-dependent electric field noise. We have shown that using helium atoms in this work results in time-dependent changes in residual uncancelled stray fields on the order of 2μV cm -1 h -1 . These results demonstrate that precise electrometry in which helium atoms are employed as the probe may be transferred to other settings, e.g., experiments involving Rydberg positronium or antihydrogen, where contamination of the experimental environment with other more reactive species, such as alkaline earth metal atoms would be extremely detrimental. Laser photoexcitation and detection of atoms in these very high Rydberg states in spatially separated locations was facilitated in part by the high longitudinal speed of the supersonic beams of helium. This approach could also be adopted in using very high Rydberg states of helium for electrometry in these other settings.