Demonstration of Ramsey-Comb Precision Spectroscopy in Xenon at Vacuum Ultraviolet Wavelengths Produced with High-Harmonic Generation
L.S. Dreissen, C. Roth, E.L. Gründeman, J.J. Krauth, M.G.J. Favier, K.S.E. Eikema
DDemonstration of Ramsey-Comb Precision Spectroscopy in Xenon at VacuumUltraviolet Wavelengths Produced with High-Harmonic Generation
L.S. Dreissen, C. Roth, E.L. Gr¨undeman, J.J. Krauth, M.G.J. Favier, and K.S.E. Eikema
LaserLaB, Department of Physics and Astronomy, Vrije Universiteit,De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands (Dated: January 20, 2021)The remarkable progress in the field of laser spectroscopy induced by the invention of thefrequency-comb laser has enabled many new high-precision tests of fundamental theory and searchesfor new physics. Extending frequency-comb based spectroscopy techniques to the vacuum and ex-treme ultraviolet spectral range would enable frequency measurements of transitions in e.g. heavierhydrogen-like systems and open up new possibilities for tests of quantum electrodynamics and mea-surements of fundamental constants. The two main approaches, full-repetition rate up-conversionin a resonator, and two-pulse amplification and up-conversion for the Ramsey-comb technique, relyon high-harmonic generation (HHG), which is known to induce spurious phase shifts from plasmaformation. After our initial report [1], in this article we give a detailed account of how the Ramsey-comb spectroscopy technique is used to probe the dynamics of this plasma with high precision, andenables accurate spectroscopy in the vacuum ultraviolet. It is based on recording Ramsey fringesthat track the phase evolution of a superposition state in xenon atoms excited by two up-convertedfrequency-comb laser pulses. In this manner, phase shifts up to 1 rad induced by the HHG processcould be observed at nanosecond timescales with mrad-level accuracy at 110 nm. We also show thatsuch phase shifts can be reduced to a negligible level of a few mrad. As a result we were able tomeasure the 5 p → p s [3 / transition in Xe at 110 nm (the seventh harmonic of 770 nm)with sub-MHz accuracy, leading to a transition frequency of 2 726 086 012 471(630) kHz. This valueis 10 times more precise than the previous determination and the fractional accuracy of 2 . × − is 3.6 times better than the previous best spectroscopic measurement using high-harmonic genera-tion. Additionally, the isotope shifts between Xe and two other isotopes (
Xe and
Xe) weredetermined with an accuracy of 420 kHz. The method can be readily extended to achieve kHz-levelaccuracy by increasing the pulse delay, e.g. to measure the 1 S − S transition in He + . Therefore,the Ramsey-comb method shows great promise for high-precision spectroscopy of targets requiringvacuum ultraviolet and extreme ultraviolet wavelengths. I. INTRODUCTION
Precision spectroscopy in calculable systems enablestests of fundamental theory and searches for physics be-yond the standard model. Recent experimental advanceshave led to unprecedented accuracy and the most strin-gent test of bound-state quantum electrodynamics theory(QED) [2–7]. Further improvements are hampered by theuncertainty of experimental parameters such as the nu-clear charge radius, which influences the energy structurethrough finite nuclear size effects. This problem can beinverted to extract improved values for these parametersusing spectroscopic measurements in combination withaccurate theoretical calculations. In 2010, this instigatedthe so-called proton radius puzzle [8], where the inferredproton radius from spectroscopy on muonic hydrogen was4% smaller than that obtained from electronic hydro-gen spectroscopy [2, 9], which was equal to a 4.4 σ dis-crepancy [8–13]. However, three out of four recent mea-surements have now confirmed the smaller (muonic) pro-ton radius [14–16], including the most recent result fromelectron-proton scattering [17]. This makes new targetswith a bigger nuclear-size induced frequency shift, suchas the He + ion, particularly interesting [18], as similarcomparisons with muonic He + can be made [19, 20]. Es-pecially because the QED contributions are much largerin He + than in hydrogen as a result of the higher-order scaling with the nuclear charge Z of these terms. There-fore, in combination with the expected improved deter-mination of the alpha radius from muonic He + [19, 20],this can lead to a more stringent test of QED. Withfurther theoretical improvements, also a value of theRydberg constant can be determined from He + spec-troscopy, which is effectively independent from hydrogenspectroscopy (because the required alpha particle chargeradius, determined from muonic He + spectroscopy, onlyweakly depends on the Rydberg constant). The obviouschallenge for a measurement of the 1 S − S transition inHe + arises from the short excitation wavelength, whichlies in the extreme ultraviolet spectral range [21].Spectroscopic measurements at these short wavelengthscan be realized by using e.g. radiation from a largesynchrotron facility to perform Fourier-transform spec-troscopy at wavelengths down to 40 nm [22, 23]. In thismanner, relative accuracies of 1 × − have been demon-strated [24]. However, a higher accuracy can be reachedwith laser spectroscopy in combination with nonlinearup-conversion. For example, amplified and up-convertednanosecond pulses have been used to reach a fractionalaccuracy of 0 . × − at 58 nm [25]. This approachis based on traditional frequency-scanning spectroscopy,but uncertainties due to frequency-chirp induced by theamplification process and the relatively broad bandwidthof the excitation pulses ultimately limit this approach. A a r X i v : . [ phy s i c s . a t o m - ph ] J a n much higher spectroscopic accuracy can be reached withpulses from a frequency-comb (FC) laser [26, 27].The pulses emitted by a FC laser have a typical pulseenergy of a few nJ, which is insufficient for up-conversionto the vacuum ultraviolet (VUV) and extreme ultravi-olet (XUV) spectral range using high-harmonic genera-tion. Therefore the peak intensity of these pulses has tobe increased to reach this part of the spectrum. Oneapproach is based on intra-cavity HHG to up-convertthe full frequency-comb pulse train to the XUV spectralrange [28, 29]. In 2005 this technique was first demon-strated and in 2012 Cing¨oz et al. have shown that theycould perform spectroscopy in argon at 82 nm with 3MHz accuracy (0.8 ppb) using such an XUV-comb [30].Alternatively, a single pair of FC pulses can be amplifiedto the mJ-level using chirped-pulse amplification followedby subsequent up-conversion using HHG in a single pass.In 2010 Kandula et al. demonstrated this technique andperformed a Ramsey-type experiment in neutral heliumatoms where they achieved an accuracy of 6 MHz at 51nm (1 ppb) [31]. This accuracy was limited by detri-mental phase shifts arising from amplification and up-conversion.We use a modified version of the latter spectroscopymethod, called the Ramsey-comb spectroscopy (RCS)technique [32], that bypasses these limitations. It isbased on multiple recordings with pairs of up-convertedFC pulses to obtain a series of Ramsey fringes fromwhich the transition frequency can be extracted bycomparing the relative phase. As a result, commonmode phase shifts arising from e.g. amplification and up-conversion, but also the ac-Stark shift, are suppressedstrongly [33, 34].The technique combines the high precision from a FClaser with the advantages of pulse amplification, leadingto e.g. easy tunability, simple up-conversion in a gas jet(no resonators required), high excitation probability anda nearly 100% detection efficiency, because the detectioncan be carefully timed (e.g. using a state-selective ion-ization pulse). The method has been demonstrated towork very well over a broad spectral range (from NIR toDUV) [32, 34, 35]. We recently showed that RCS canbe combined with HHG to extend the method to theVUV and XUV spectral range [1]. The extension of RCSwith HHG was so far hampered by the unknown detri-mental phase shifts which arise from ionization duringup-conversion [36–38]. The influence from plasma for-mation has been previously investigated only on shorttimescales, ranging from a few hundred femto-seconds toa few picoseconds [39, 40], but not on timescales relevantto RCS (i.e. at tens to hundreds of nanoseconds).In this paper we give a detailed account of how the RCSmethod can be used to provide insights into plasma-induced phase shifts arising from HHG by tracking theatomic phase evolution of excited atoms (xenon) withmrad-scale precision. Moreover, we show how this re-sults in high-precision spectroscopy with light from HHGin the vacuum ultraviolet on the 5 p → p s [3 / tran- sition in xenon at 110 nm. II. EXPERIMENTAL TECHNIQUES
In the original Ramsey spectroscopy scheme, moleculesor atoms are excited by two separated phase-locked os-cillatory fields in the radio-frequency (RF) domain [41].The Ramsey-comb spectroscopy (RCS) technique [32]also uses two interaction pulses, separated in time in-stead of space and in the optical domain, based on pairsof phase-locked laser pulses from a frequency-comb laser.
A. The Ramsey-comb spectroscopy technique
A two-level system with transition frequency f tr canbe brought in a superposition of the ground and the ex-cited state by a resonant laser pulse. Excitation withtwo phase-locked pulses leads to interference betweentwo such contributions. The interference pattern canbe probed by observing the excited state population(e.g. through detection of fluorescence, or more efficiently,by photo-ionization). The phase evolution of the inter-fering contributions depends on the delay (∆ t ) and thephase difference (∆ φ ) between the pulses, leading to anoscillating upper state population ( | c e (∆ t, ∆ φ ) | ), whichis described by | c e (∆ t, ∆ φ ) | ∝ cos (2 πf tr ∆ t + ∆ φ ) . (1)A FC laser is a very suitable source for this excitationscheme, because it provides accurate control over thepulse delay through the pulse repetition time ( T rep ) andthe relative phase through the carrier-envelope phase slip(∆ φ ceo ). One can adjust either one of these parametersto obtain a Ramsey fringe, but experimentally it is moreconvenient to adjust the delay ∆ t = T rep + δt . The rangeover which the delay is scanned, is related to the tran-sition frequency and is typically on the order of a fewhundred attoseconds.The amplification process induces a phase shift of upto 300 mrad on the excitation pulses (∆ φ opt ), which isadded to the carrier-envelope offset phase ∆ φ = ∆ φ ceo +∆ φ opt . This contribution is typically hard to determinewith high accuracy. However, it is possible to constructan amplifier based on parametric amplification that cankeep the phase shift between the two amplified pulsesconstant at the mrad-level as a function of delay. There-fore in RCS one records a series of Ramsey fringes at dif-ferent pulse delays separated by an integer times the repe-tition time (∆ N × T rep ), as illustrated in Fig. 1. The tran-sition frequency is obtained by comparing the relativephase of these fringes, leading to a suppression of com-mon mode shifts. A constant phase shift between the am-plified FC pulses manifests itself as a global phase shift ofthe fringes and does not influence the extracted transitionfrequency. This argument also holds for other common time(cid:507)N = 1 (cid:507)N = n (cid:507)N = 2 | c e | R a m s e y - c o m b pu l s e s rep rep nT rep (cid:303)t FIG. 1. A series of Ramsey fringes from two broadband reso-nant FC pulses is recorded at different multiples of the repeti-tion time (∆ N × T rep ). Each individual fringe is obtained bychanging the delay by δt , which is on an attosecond timescale.The transition frequency is extracted from the relative phasesof the fringes, suppressing the influence of common modephase shifts of the Ramsey fringes on the extracted transi-tion frequency. mode effects, such as the ac-Stark shift [33, 34]. Addi-tionally, the accuracy of the measurement improves pro-portionally with the inter-pulse delay, as the frequencythat fits the measured phase evolution as a function ofthe delay time becomes more sensitive for longer delays.The transition frequency extracted in RCS is affectedby both the first-order Doppler shift and the recoil shiftfrom the absorption of a single photon during excitation.Cadoret et al. have shown that the phase Φ of the super-position of ground and excited state after a free evolutiontime ∆ t in a Ramsey interferometer can be written as [42]Φ = ω ∆ t + ∆ φ + (cid:16) v e + v g (cid:17) · ( v e − v g ) m ∆ t (cid:126) , (2)where ω is the excitation frequency, v g and v e are thevelocities of the atoms in the ground and excited state,respectively, and m is the mass of the atom. The recoilvelocity after excitation is given by v e − v g = (cid:126) k /m .Substituting this into Eq. 2 and taking only the veloc-ity component along the propagation direction z of theexcitation laser into account, one obtainsΦ = ω ∆ T + ∆ φ + ( v e,z + v g,z ) ω ∆ t c , (3)where c is the speed of light. The last term in Eq. 3 isa combination of Doppler and recoil shift, which can beviewed as a generalized Doppler shift [43]. It also meansthat in Ramsey-comb spectroscopy the recoil frequencyshift has to be taken into account for one-photon excita-tion.Note that in two-photon spectroscopy in a counter-propagating configuration ( k = − k ) [34, 35], the re-coil shift is canceled and does not influence the extractedtransition frequency. B. Combining RCS with HHG
Higher-order harmonic generation uses the nonlinearresponse of a gas to coherently up-convert frequenciesfrom the infrared or visible spectral range to the vacuumultraviolet or extreme ultraviolet spectral range. An in-tuitive picture of HHG is given at a single-atom levelby the semi-classical three-step recollision model [44] (aquantum-mechanical description is given by the Lewen-stein model [45]).The principle is based on focusing a powerful short laserpulse in a medium (typically a gas jet, or a gas filled capil-lary) to reach an intensity of approximately 10 W/cm .At these intensities, the electric field of the laser pulsecan significantly perturb the Coulomb potential of theleast bound electrons, and for a high enough field thisenables the electron to tunnel out into the continuum.After ionization, the electron gains energy as it is ac-celerated in the strong electric field of the laser. It canreturn to the parent ion because the field of the funda-mental laser changes sign after half a period of the opticalwave. When it returns it can recombine, leading to theemission of a high-energy photon. The recollision prob-ability is highest near every zero-crossing of the electricfield and the process is therefore tightly linked in time tothe fundamental wave, leading to a high degree of coher-ence. This has been demonstrated in several experimentsinvolving interference of HHG beams, see e.g. [46, 47],and by the high coherence of steady-state full-repetitionrate frequency comb up-conversion [48]. For a symmetricmedium, such as a simple gas jet, only odd harmonics ofthe fundamental optical carrier frequency are produced.The efficiency of the HHG process is often very low (typ-ically 10 − in a simple gas jet) and most electrons mightnot recombine but instead become permanently ionized.Over time a significant fraction of the medium can be-come ionized and a plasma can be formed. In order tocombine RCS with HHG, the effect of this plasma forma-tion on the phase of the up-converted FC pulses has tobe carefully considered.In RCS common mode effects are significantly suppressedand therefore the plasma build-up during each pulse doesnot influence the extracted frequency. If the pulses do notinfluence each other, the plasma-induced shift is equal forboth up-converted FC pulses. However, the second pulsecan experience a phase shift due to a change in the re-fractive index of the HHG medium because of the plasmaformed during the first pulse. This shift is dependent onthe inter-pulse delay because the ions and electrons moveout of the interaction region, and partly recombine. Ittherefore leads to a delay-dependent phase shift and, asa result, a shift of the extracted transition frequency. Onthe other hand, because this delay-dependent plasma-induced phase shift is directly imprinted on the phase ofthe Ramsey fringe, as is illustrated in the top panel ofFig. 3, it can be detected by tracking the phase evolutionof the atomic superposition state at different inter-pulsedelay ∆ N and driving intensity. The results presented in RC pulses synchronization (cid:168)(cid:49)(cid:238)T rep (cid:507)(cid:307)(cid:3)(cid:80)(cid:72)(cid:68)(cid:86)(cid:88)(cid:85)(cid:72)(cid:80)(cid:72)(cid:81)(cid:87)(cid:51)(cid:88)(cid:80)(cid:83)(cid:3)(cid:79)(cid:68)(cid:86)(cid:72)(cid:85)Ti:Sa(cid:41)(cid:85)(cid:72)(cid:84)(cid:88)(cid:72)(cid:81)(cid:70)(cid:92)(cid:16)(cid:70)(cid:82)(cid:80)(cid:69) (cid:24)(cid:22)(cid:21)(cid:3)(cid:81)(cid:80)(cid:168)(cid:49)(cid:238)T rep (cid:26)(cid:26)(cid:19)(cid:3)(cid:81)(cid:80)T rep (cid:49)(cid:50)(cid:51)(cid:38)(cid:51)(cid:36)
S1 CCs PC CC D (cid:507)(cid:307)(cid:3)(cid:80)(cid:72)(cid:68)(cid:86)(cid:88)(cid:85)(cid:72)(cid:80)(cid:72)(cid:81)(cid:87) PC PC S2 (cid:50)(cid:51)(cid:36)FC(cid:44)(cid:81)(cid:87)(cid:72)(cid:85)(cid:73)(cid:72)(cid:85)(cid:82)(cid:74)(cid:85)(cid:68)(cid:80) FIG. 2. Pulse pairs with a delay of ∆ N × T rep from aTi:sapphire frequency-comb laser are stretched ( S
1) and selec-tively amplified in a non-colinear optical parametric chirpedpulse amplifier (NOPCPA). The pulse pair is selected by ad-justing the settings of the pump laser. A typical energy of3 mJ/pulse is reached of which the main part is compressedin a grating compressor ( C ) and up-converted using high-harmonic generation. A small fraction of the amplified beamis used to measure the phase difference between the pulses inthe setup shown in the lower panel. For this purpose a sec-ond stretcher ( S
2) is used to stretch the reference FC pulsesafter which they are spatially overlapped with the amplifiedpulses in a single mode fiber. The pulse pairs are separatedfrom the full FC pulse train using two Pockels cells (
P C ) indouble pass. A third PC is used to introduce a slight verticaldisplacement for the first or the second pulse pair in order toproject them separately on the CCD camera. By introducinga small time delay (typically < Sec. II B rely on this principle.The magnitude of the induced shift is determined bythree contributions to the change of the refractive index:the free electrons, the generated ions and the depletionof neutral atom density [49, 50]. The influence from thefree electrons in the plasma is dominant over the contri-bution from the atom depletion and the ions. The latteris usually neglected, because the energy levels scale up forhigher charge states, leading to a refractive index closeto unity. The different contributions will be discussed inmore detail in Sec. IV B.Another effect, which has to be considered when com-bining HHG with RCS, is the influence of phase noise.In HHG the amplitude of the phase noise originatingfrom the frequency comb and the amplification process ismultiplied-up by the harmonic order, and can thereforelead to a significant reduction of contrast of the Ramseyfringes. The phase noise amplitude should remain wellbelow π at the excitation wavelength to retain enoughcontrast ( > III. EXPERIMENTAL SETUP
The RCS setup is based on selective amplification ofa FC pulse pair using parametric chirped pulse amplifi-cation. The re-compressed amplified pulses have a suf-ficiently high peak-intensity to create high-harmonics ina simple single-pass gas jet. After up-conversion, thepulse pair is refocussed in an atomic xenon beam to ex-cite the 5 p → p s [3 / transition at 110 nm. Theexcited atoms are state-selectively ionized and detectedusing a time-of-flight mass spectrometer. In this section,the individual components of the system are described indetail. A. The Ramsey-comb laser system
An overview of the laser setup is shown in Fig. 2.The starting point is a Kerr-lens mode-locked tita-nium:sapphire frequency-comb laser with an average out-put power of 450 mW and a repetition frequency of f rep = 126.6 MHz. The spectrum of the laser is centeredaround 800 nm and has a bandwidth of ∼
75 nm. Therepetition frequency and the carrier-envelope offset fre-quency ( f ceo ) are stabilized using a commercial cesiumatomic clock (Symmetricon CsIII 4310B). This enablesaccurate control over the repetition time T rep = 1 /f rep and the carrier-envelope phase slip ∆ φ ceo = 2 πf ceo /f rep .The FC pulses are stretched and spectrally clipped in a4f-grating stretcher to improve the temporal overlap ofthe FC pulses with the pump pulses in the parametricamplifier. The stretcher introduces a group velocity dis-persion (GVD) of 1 . × fs and an adjustable slit inthe Fourier plane of the stretcher selects a spectrum witha bandwidth of typically 8 nm that is centered around 770nm. The combination of stretching and spectral clippingleads to a pulse length of ∼
10 ps.A three stage non-collinear optical parametric chirpedpulse amplifier (NOPCPA) based on three beta-bariumborate (BBO) crystals is then used to amplify the FCpulses. It is pumped by a home-built laser system provid-ing a pulse pair at 532 nm with an energy of 17 mJ/pulseand a pulse length (full-width at half maximum) of 60ps [51, 52]. The inter-pulse delay between the pumppulses can be adjusted by an integer number ∆ N of therepetition time ( T rep ) with a set of modulators [51], lead-ing to selective amplification of the corresponding FCpulse pair. A typical energy of 3 mJ/pulse is reached inthis manner.The amplified pulses are re-compressed to a pulse lengthof 220 fs (full-width at half maximum) using a gratingcompressor, based on transmission gratings (LightsmythT-1850-800s). One of the gratings in the compressor ismounted on a translation stage so that the amount ofinduced GVD can be adjusted. The whole compressor isbuild on a rotation stage to tune the angle of incidenceand compensate for third-order dispersion. However, thehard edge used for wavelength selection in the stretcher Iris
In vacuum ionization pulse (3 mJ, 1064 nm)HHG Ar jet L i F f i l t e r Ionized region T o r o i d a l m i r r o r p a i r Xe + EMGND.250V300V
TOF mass spectrometerXe source partial (center) beam block ∆φ(ΔN)
RC pulse pair 7 th harmonic ∆φ(ΔN) Ion signalRCSHHG
Plasma induced phase shifts
From RC laser
FIG. 3. The experimental setup and a schematic representation of the influence from plasma formation during HHG on RCS(top panel). The NIR beam from the RC laser is focused to generate high-harmonics in argon. A combination of a beam blockand an adjustable iris is used to separate the harmonics from the fundamental beam, while a LiF plate acts as a filter thatonly transmits light λ >
105 nm. The VUV beam is refocused using a toroidal mirror pair at grazing incidence and overlappedwith a pulsed xenon beam at 90 ◦ angle. An ionization pulse at 1064 nm is delayed by 2 ns with respect to the second VUVpulse and selectively ionizes the excited atoms. The ions are extracted with a pulsed electric field and detected through atime-of-flight (TOF) drift tube by an ETP AF880 electron multiplier (EM). The plasma induced delay-dependent phase shift∆ φ (∆ N ) between the two RC pulses from HHG, is detected through the phase of the observed ion signal Ramsey fringes (toppanel). leads to a sinc-like pulse shape and therefore to pre- andafter-pulses of a few percent with respect to the mainpeak. The remaining pulse energy after re-compressionis typically 2 mJ/pulse. B. The phase-measurement setup
The phase difference between the amplified FC pulsesis measured in a separate setup, shown in the lower panelof Fig. 2, which is based on spectral interference. For thispurpose a small fraction of the amplified beam is split-off before the compressor and spatially overlapped withthe original unamplified FC pulse in a single-mode fiber.Saturation effects in the parametric amplifier lead to aslightly wider spectrum of the amplified pulses, thereforethe reference pulses pass through a separate 4f-gratingstretcher. It introduces the same amount of GVD as thefirst, but the selected bandwidth is slightly wider. More-over, nonlinear effects in the fiber are avoided as wellby using stretched pulses. Each amplified pulse and itscorresponding reference pulse is selected out of the fullFC pulse train with two Pockels cells (PC), which areused in double pass (with a combined contrast of 1:10 )to reduce background light. A third PC is used in com-bination with polarization optics to introduce a slightvertical offset between each set of pulses in a small de-lay line. In this manner the interference pattern of thetwo sets of pulses can be projected above each other on aCCD camera (IMI-TECH IMB-716-G) so that both spec- tral interference patterns can be observed simultaneously.An example of the observed interferograms is shown inthe lower panel of Fig. 2. The spectrometer consists of agold grating of 1200 lines/mm (Richardson, 53-*-360R)in combination with a 350 mm lens, which projects thespectrum on the camera. The period of the interferencepattern can be adjusted by changing the delay betweenthe original FC pulse and the amplified pulse in the delayline of the reference arm. The geometrical phase differ-ence arising from slight misalignments of the two interfer-ence patterns on the camera is calibrated by exchangingthe projection of the two pulses using the third PC. Thephase difference between the pulses is extracted from theinterferograms using a Fourier transform-based method[53, 54]. C. The spectroscopy setup
A schematic overview of the vacuum setup is shown inFig. 3. High-harmonics are created in a gas jet of argonatoms in the first vacuum chamber. The jet is createdby a supersonic expansion from a home-built piezo valve(type 1), which operates at a backing pressure of 5 bar.Before HHG, the center part of the amplified beam is cut-out with a 1-mm-diameter beam block at a distance of2 f before the focusing lens (with a focal distance f = 250mm). This leads to a donut-shaped intensity profile ofthe beam which is then imaged in the second vacuumchamber to separate the harmonic light from the funda-mental light with an adjustable iris. The generated har-monics propagate on axis with a much lower divergencethan the fundamental. Therefore, the iris transmits theharmonic beam, while it blocks the ring of fundamentallight around it.A peak-intensity of up to 1 . × W/cm is reached inthe interaction region below the nozzle of the valve. Theposition of the valve can be adjusted in all three dimen-sions to optimize the harmonic yield. In particular, thealignment in the direction of propagation is important forreaching proper phase-matching conditions of the HHGprocess. This is achieved by adjusting the valve positionsuch that the divergence of the harmonic beam is min-imized, which typically leads to a focus position just infront of the jet.Due to the low ionization potential of xenon, direct one-photon ionization is possible with harmonics of order q >
7. Therefore, a 1-mm-thick lithium fluoride win-dow is placed in the beam, which only transmits light at λ >
105 nm. An absolute transmission efficiency of 40 %was measured for the seventh harmonic at 110 nm.A toroidal mirror pair operating at a grazing angle of7.5 ◦ is used to refocus the harmonic beam in the in-teraction region. It forms a one-to-one telescope withan effective focal length of 250 mm. Coma and otheraberrations are fully compensated at equal angle of inci-dence, but for a slight angular deviation the beam qualityin the spectroscopy region is significantly perturbed. Aretractable silver mirror is placed a few centimeters be-hind the second toroidal mirror to monitor changes inday-to-day alignment and ensure a proper quality of therefocused fundamental beam. The harmonic beam itselfcan be detected in the far field (400 mm from the focalplane) with an XUV sensitive CCD camera (Andor New-ton SY, DY940P).The xenon atoms are excited at a 90 ◦ − angle in a pulsedatomic beam to reduce influences from the first-orderDoppler effect. The gas pulse is created from a home-built piezo valve (type 2, see [55]) with a 0.3 mm nozzleopening, operating at 0.5 bar backing pressure. The for-mation of clusters from the gas expansion, as reportedin previous studies [56, 57], was thoroughly investigatedby changing the parameters of the expansion (backingpressure, pulse timing, pulse length) and by changingskimmers and valve types (solenoid and piezo based withdifferent valve openings from 0.15 mm to 0.8 mm), but noevidence for the production or influence of clusters wasfound in the Ramsey signals or time-of-flight (TOF) massspectrometer data. After these tests we based the atomicbeam geometry for the measurements on two skimmers toreduce Doppler broadening. The first skimmer is placedat a distance of 200 mm from the valve and has a circularaperture with a diameter of 8 mm. This relatively largeopening was chosen to reduce possible effects of skimmerclogging. The second one is an adjustable slit skimmerset to a width of 1 mm. It is made from 0.150-mm-thickstainless steel foil and bent into a slit skimmer pointingtowards the source. The slit is orientated perpendicular I on c oun t [ a r b . un it s ]I on c oun t [ a r b . un it s ] FIG. 4. A typical Ramsey-comb scan at ∆ N = 1-4. The upperpanel shows the signal as a function of measurement number,where the different markers show the data points correspond-ing to a specific ∆ N . In the lower panel, the data is sortedaccording to pulse delay, leading to clear Ramsey fringes. Thesignal was obtained by scanning the pulse delay over 475 at-toseconds, while a 7.9 ns delay is introduced between eachRamsey measurement by changing the pulse pair. to the atomic and VUV beam. It is placed at a distanceof 460 mm from the valve, leading to a maximum atomicbeam divergence of < . IV. RESULTS
A typical Ramsey-comb scan of the 5 p → p s [3 / transition is shown in Fig. 4. Thelower panel shows four Ramsey fringes at ∆ N = 1 − ∼
500 laser shots and the error barsrepresent the observed standard deviation of the meanvalue. The total scan range was 475 attoseconds in which1.3 fringe was observed. The phase difference betweenthe two FC pulses drifts on the order of 10 mrad on a tenminute timescale, due to subtle changes (mostly relatedto temperature) in the amplifier system. Therefore themeasurement time was set to be only 3.3 minutes/fringeat the expense of signal-to-noise ratio, and only pairsof Ramsey fringes were used (∆ N = 2 and ∆ N = 4).Additionally the data points were recorded in randomorder, as shown in the upper panel of Fig. 4, reducingthe influence of drifts further by a factor of ∼ A. The Ramsey fringe contrast
The contrast of the Ramsey fringes is influenced by sev-eral processes. An overall reduction of the Ramsey fringecontrast is caused by the phase noise on the amplifiedand up-converted pulses, because it is mostly producedby the parametric amplification process so that the noiseamplitude can be considered constant for a large range ofpulse delays. A delay-dependent (∆
N T rep ) decay of theRamsey fringe contrast is affected by: the finite upperstate lifetime (22 ns [58]), Doppler broadening, a wave-front tilt of the excitation beam and the short transittime of the xenon atoms through the refocused excita-tion beam. The latter was the dominant cause for a rel-atively fast decay of the Ramsey fringe contrast in thisexperiment, because the one-to-one toroidal telescope re-focuses the HHG beam to a spot similar in size as at theHHG source, which is ≤ µ m for the fundamental beamand typically a few times smaller in the VUV. This con-figuration was chosen in preparation for 1 S − S excita-tion in a trapped He + ion (requiring refocusing), but wasnot ideal for the current xenon atomic beam experiment.Therefore the toroidal mirrors were deliberately slightlymisaligned (with respect to perfect one-to-one imagingconfiguration) to introduce a small amount of astigma-tism, which increases the transit time at the expense of alocal wavefront tilt. As a result, the maximum pulse de-lay was increased from 16 ns (diameter of ∼ µ m) to 32ns ( ∼ µ m). The purple data points in Fig. 5 show themeasured Ramsey fringe contrast at different inter-pulsedelays (with ∆ N = 1 −
4) after increasing the interactionregion.In order to gain more insight into the influence of theindividual contributions that affect the Ramsey fringecontrast, the Ramsey-comb signal was simulated and acomparison with the data was made. The simulationswere performed with an upper state lifetime of 22 ns [58]and the maximum achievable contrast based on this valueis shown with the finely dashed orange curve in Fig. 5.Other parameters that were fixed based on prior knowl-edge were the maximum atomic beam divergence of 1.5mrad (dashed-dotted pink curve) and the wavefront tiltof 1.25 mrad (dotted blue curve) based on the introducedastigmatism in the VUV beam. The latter two were esti- lifetime limited transit time wavefront tilt fit Doppler broadening data C on t r a s t [ % ] phase-noise limit FIG. 5. The measured Ramsey fringe contrast (purple points)as a function of pulse delay (equal to ∆ NT rep ) and the bestfit (purple curve). The other curves show the influence on thecontrast for each effect separately. These are obtained by as-suming an upper state lifetime of 22 ns (dashed orange curve),a wavefront tilt ot 1.25 mrad (dotted blue curve), an atomicbeam divergence of 1.5 mrad (dashed-dotted pink curve) andan effective interaction region of 26 µ m (dashed green curve).The curves clearly show that for this experiment the domi-nant contrast-reduction effect was caused by the short transittime of the atoms through the interaction region. The fit alsoincludes an additional phase noise amplitude of 400 mrad atthe seventh harmonic, leading to a maximum contrast of 88%(gray line). mated from the geometrical configuration. The remain-ing two parameters, namely the size of the interactionregion and the phase noise were fitted to match the de-cay and the offset of the measured contrast. As discussedbefore, the phase noise manifests itself as an overall re-duction of the Ramsey fringe contrast, while the limitedtransit time affects the decay of the curve. The purpleline shows the best match with the experiment and leadsto a 26 µ m VUV beam size at full width half maximumand a 400 mrad phase noise amplitude at the seventhharmonic. From the latter an absolute maximum con-trast of 88% is expected (gray line). The obtained valueindicates that the phase noise amplitude of the ampli-fied NIR pulses is 57 mrad, taking the linear scaling withthe harmonic order into account. This result agrees wellwith the extracted value from direct phase measurementsat the fundamental frequency, which will be discussed inSec. IV C. Even though the beam size was increased byalmost a factor of two, the loss of contrast is still dom-inated by the short transit time of the atoms (dashedgreen curve).Due to the high sensitivity to the transit time of theatoms in this experiment, it became apparent that thetwo amplified pulses initially had a slightly different prop-agation direction. This was caused by an intensity depen-dent spatial walk-off effect in the parametric amplifier. × W / cm × W / cm × W / cm × W / cm × W / cm × W / cm × W / cm × W / cm × W / cm × W / cm × W / cm × W / cm × W / cm × W / cm × W / cm × W / cm (cid:1) N [(cid:2)] I n t . pu l s e [ a r b . un it s ] I Pulse 1Pulse 2Sum [× W / cm ] VUV i n t e n s it y [ a r b . un it s ] (a)(b) (c) ∆N=3 ∆N=3 FIG. 6. (a) The recorded seventh-harmonic VUV yield of thetwo individual pulses at ∆ N = 3 for different driving inten-sities. Initially both pulses increase as a function of drivingintensity, but above 0 . × W/cm the yield of the secondpulse stagnates and eventually decreases due to the depletionof the neutral atom density. (b) The amplitude of the in-dividual VUV pulses at ∆ N = 3 and sum of the two. Thelines connecting the data points are splines to guide the eye.(c) The extracted amplitude of the second VUV pulse as afunction of inter-pulse delay (equal to ∆ NT rep ) for differentdriving intensities. The lines are fits to the data assuming thefundamental pulses have a Gaussian beam profile. A revivalof the second pulse occurs on a timescale of 50-100 ns. The two pump pulses of the NOPCPA have a slightly dif-ferent intensity profile as a result of saturation effects inthe amplifiers of the pump laser [51]. This leads to a dif-ferent walk-off induced propagation angle for the two am-plified FC pulses caused by pump-to-signal phase trans-fer. This problem was solved by implementing a walk-offcompensating configuration for the first two passes [59],which is based on the reversal of the walk-off direction be-tween the first two crystals by rotating one of the crystalsaround the proper axis. In this manner the beam point-ing difference was reduced from up to ∼ . µ rad. Notethat previous RCS measurements [34, 35] were not af-fected by this because they used large collimated beamsin combination with low harmonic generation. B. Plasma-induced effects
The influence from plasma formation on the HHG pro-cess was investigated by varying the driving intensitywith a half-wave plate and a thin-film polarizer. As il-lustrated in Fig. 6, the depletion of the neutral atoms after up-conversion of the first pulse influences the VUVyield of the second pulse. Fig. 6(a) shows the individualseventh-harmonic VUV pulses at ∆ N = 3 for five differ-ent driving intensities. Up to 0 . × W/cm , the yieldof both VUV pulses benefit from a higher driving inten-sity, but a further increase leads to the stagnation andeventually a decrease of the yield of the second pulse. Atthe highest driving intensity (1 . × W/cm ) the sec-ond VUV pulse is almost completely suppressed. Notethat in this regime, the phase matching condition, whichdepends on the dispersion, is also significantly influenced.As a result also the total VUV yield, i.e. the sum of thetwo pulses at ∆ N = 3, stagnates as is shown in Fig. 6(b).Above 0 . × W/cm there is no significant improve-ment of the total yield.A revival of the second VUV pulse occurs for larger inter-pulse delays, as shown in Fig. 6(c), because the plasmamoves out of the interaction region and the gas samplerefreshes as is also indicated in Fig. 3. The typically ob-served timescale on which this revival occurs, depends onthe size of the interaction region and therefore the driv-ing intensity. The reason for this is that a higher drivingintensity leads to a bigger volume where the intensity ishigh enough for tunnel ionization and therefore HHG. Itthen also takes longer for all the ions to leave the interac-tion zone. Recovery times of 50-100 ns were observed asis shown in Fig. 6(c), which matches well with what is ex-pected for a focus diameter in the HHG zone of about 50 µ m and the estimated velocity of the argon atoms froma supersonic expansion ( v = 550 m/s).The delay-dependent influence from plasma formationon the phase of the second VUV pulse was extractedfrom the phase evolution of the Ramsey fringes as func-tion of delay (which is schematically depicted in the toppanel of Fig 3) for different driving intensities. The abso-lute phase of the VUV Ramsey fringes drifts over longertimescales (typically a few mrad per minute) and there-fore only the relative phase between the fringes was usedsuch that this effect did not appreciably influence themeasurements. The phase at ∆ N = 2 was chosen as areference, because a remarkable change in the dynamicswas observed after this delay (16 ns). Furthermore, thesignal-to-noise is better at ∆ N = 2 than at longer delaysand therefore subtle shifts could be measured more accu-rately in this manner.Fig. 7 shows the measured phase shift as a function ofdelay where ∆ N = 2 has been taken as a referencevalue. The curves are fitted with an exponential func-tion ∆ φ ∝ e − B ∆ t . At the shortest delay (∆ N = 1), weobserve a noteworthy seventh-order dependency of thephase-shift as function of peak intensity (for more de-tails, see [1]), leading to a shift of 1 rad at the highestdeployed peak intensity. This shift is significantly re-duced for larger inter-pulse delays, indicating that thereare much faster dynamics involved in this process thanwhat is expected from the dynamics of ions and atoms(Fig. 6(c)). Therefore we conclude that the phase effectsare predominantly caused by free electrons in the plasma, ▼ I NIR = 1.3×10 ● I NIR = 1.2×10 ■ I NIR = 0.9×10 ◆ I NIR = 0.8×10 ▼ ▼ ● ▼▼▼▼▼ ●● Δ ( ϕ Δ N - ϕ )[r a d ] ● ◆ ◆◆◆ ◆◆◆◆◆◆ ◆◆◆◆◆◆ FIG. 7. The observed phase shift at ∆ N = 1 , N = 2 as a function inter pulse delay and for differentdriving intensity. The data is fitted with ∆ φ ∝ e − B ∆ t andthe shaded area represents the 1 σ uncertainty interval. Theoffset in the zoom-in shows the residual shift at large delayfor normal operating conditions ( I = 0 . × ). which can leave the interaction zone much faster than theatoms and ions (on a picosecond-timescale instead of ananosecond-timescale). This is in agreement with the ex-pected dominant contribution to the change in refractiveindex due to plasma formation [49, 50]. Note that theerror bars increase for higher driving intensity becausethe two excitation pulses (Fig. 6(a)) and therefore theexcitation contributions become unequal, which directlyleads to a lower contrast of the Ramsey fringes.The asymptotic value at ∆ N → ∞ represents the resid-ual phase shift at ∆ N = 2, which is significant at highdriving intensity (140(74) mrad at 1 . × W/cm ),but reduces to a negligible level of 2(2) mrad at the low-est driving intensity. This is more clearly visible in theinset of Fig. 7 which shows a zoom-in of the inducedphase shift for low driving intensity at large pulse delay.By skipping ∆ N = 1 and moderating the intensity to0 . × W/cm , the observed plasma induced phaseshift could be reduced to -2(5) mrad between ∆ N = 2and ∆ N = 3, and -7(9) mrad between ∆ N = 2 and∆ N = 4. The corresponding frequency shift for themeasured transition of -32(91) kHz and -67(86) kHz, re-spectively, is consistent with zero within the uncertainty.Fig. 6(b) shows that the moderation of the intensity to I leads to a reduction of 15% of the total VUV yieldand therefore only a slight reduction of the total signallevel. The VUV peak intensity at the seventh harmonicwas estimated to be 5 × W/cm at these typical op-erating conditions for spectroscopy.The spectrum of the generated harmonics at the differ-ent values of the driving intensity was measured withthe monochromator at the end of the vacuum system(Fig. 3). The spectra are acquired with only a singlepulse and do not represent the total yield of the double- (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2)(cid:3) (cid:3) (cid:3) (cid:3) (cid:3) (cid:3)(cid:2)(cid:2)(cid:3)(cid:1)(cid:3)(cid:1)(cid:3)(cid:1) × W / cm (cid:1) × W / cm (cid:2) × W / cm (cid:3) × W / cm
10 14 18 221005001000500010
124 89 69 56Photon energy [ eV ] VUV y i e l d [ a r b . un it s ] Wavelength [ nm ] Sensitivity EM (BeO)5 7 9 11 13 15 W/cm W/cm W/cm W/cm Photon energy [eV]Wavelength [nm]
FIG. 8. The obtained harmonics spectrum between order q = 5 (154 nm) and q = 15 (51 nm) for four different values ofthe driving intensity. The individual points are the measuredintensities at the indicated (by the number below the points)harmonic order. The curves are corrected for the sensitivity ofthe Electron Multiplier (EM) consisting of BeO (gray curve),but not for the unknown wavelength-dependent efficiency ofthe grating, so it only gives an indication of the harmonicyield. Generation of the highest harmonics q > th and 13 th harmonic is reproducible, but not fullyunderstood. pulse configuration. Fig. 8 shows four spectra obtainedat different driving intensities. The harmonics of ordersbetween q = 5 (154 nm) and q = 15 (51 nm) are clearlyobserved. The high-frequency end of the spectrum islimited by the poor reflectivity of the aluminum gratingthat was used (model 33009FL01-510H from Newport) ata relatively small angle of incidence ( < ◦ ). The geom-etry of the monochromator prevents the low-frequencyend of the spectrum to be observed, therefore the thirdharmonic is missing. The measured amplitudes of theharmonic orders were corrected for the wavelength de-pendent sensitivity of the electron multiplier (gray curvein Fig. 8). However, the diffraction efficiency ( − st or-der) of the grating is unknown, because the grating isonly specified down to 130 nm. Therefore the curves inFig. 8 only give an indication of the spectral intensityof the different harmonic orders. Nevertheless two con-clusions can be drawn from the observed spectra. Thefirst is that the highest harmonics are produced even atthe lowest driving intensity, indicating that the regime oftunnel ionization is reached in the HHG process at thoselevels of intensity. Secondly, there is an increase of abouta factor of 4 in harmonic yield for wavelengths below 80nm at higher intensities. For future experiments (like1 S − S He + excitation) where the pulses will be morethan 100 ns apart, one can fully use this improved HHGyield because in that case there is no influence of atomdepletion or related phase shift on the second pulse. It is0 FIG. 9. The upper panel shows the extracted possibletransition frequency with respect to the previous value of2 726 090(5) GHz [60] at 5 different effective mode spacings,which are indicated in the upper-right corner. The lowerpanel shows the probability of the coincidence point at thepossible frequencies. The coincidence point with the highestprobability (normalized to 1) is indicating the true transitionfrequency, and was determined with 99.2% confidence. not clear why there is a (reproducible) sudden change inpower scaling behavior for the 11 th and 13 th harmonic.It could in part be induced by changing phase matchingconditions (optimized for the seventh harmonic) due toionization and perhaps subtle pulse shape changes (in-duced by the optics used to vary the intensity). C. Calibration of the absolute transition frequency
The measured plasma-induced phase shifts indicatethat high-precision spectroscopy is possible, even forRCS with relatively small pulse delays. To demonstratethis and show the full potential of RCS in combina-tion with HHG, the absolute transition frequency of the5 p → p s [3 / transition has been determined.Due to the periodicity of the signal in Ramsey-combspectroscopy, the extracted transition frequency is onlyknown modulo the effective mode spacing ( f rep / ∆ N ).Therefore, the repetition frequency was varied over arange of 1.6% in three steps to determine the actual tran-sition frequency. A few of the possible values for the tran-sition frequency are shown in Fig. 9 relative to the pre-vious determination from Yoshino et al. [60]. The upperpanel shows the extracted frequencies at five different val-ues of the mode spacing. The results obtained from thecombination of ∆ N = 2 and ∆ N = 4 yields the best sta-tistical accuracy but reduces the effective mode spacingto f rep /
2, which leads to twice as many possible transi-tion frequencies. Therefore, half of the possibilities wereexcluded by also performing measurements at ∆ N = 2and ∆ N = 3. The combination leads to only a singlecoincidence point of the transition frequency within a 4 σ range of the previous determination [60].The probability that the obtained transition frequencyis at the given coincidence point is calculated by con- FIG. 10. The obtained Doppler-free transition frequency.Each data point shows the extracted transition frequencyfrom 10 measurements with pure xenon and 10 measurementsin the 3:1 Ar:Xe mixture. The shaded area shows the 1 σ un-certainty interval. structing Gaussian distributions from the extracted tran-sition frequencies ( µ n ) and the corresponding uncertain-ties ( σ n ) according to P n ( f ) = 1 σ n √ π e − ( f − µn )22 σ n . (4)The distributions obtained with different mode spacingare multiplied to obtain the probability ( P C ) that thetransition frequency is at a certain coincidence point,which is shown in the lower panel of Fig. 9. In thismanner, the obtained coincidence point was determinedwith 99.2% confidence.After identification of the proper transition frequency(‘mode’), the absolute transition frequency was de-termined by evaluating a series of systematic effects.The main systematic effect is caused by the first-orderDoppler shift. This was calibrated by comparing thetransition frequency obtained from a beam of pure xenonwith that obtained from a mixture of argon and xenonin a ratio of 3 to 1. The mixture accelerates the xenonatoms, which unfortunately also reduces the transit timeso that only a maximum pulse delay of ∆ N = 3 couldbe used in this case.In the TOF mass spectrometer geometry, two deflectionplates were included in the direction of propagation ofthe atoms to compensate for the forward velocity andto steer the ions onto the detector. The observed signallevel as a function of deflection voltage was comparedto simulated trajectories (and resulting signal) usingCOMSOL to calibrate the forward velocity. From thiswe established a speed of 285(30) m/s for pure xenon,and 480(30) m/s for xenon in the mixture.The atomic beam was first coarsely aligned to beperpendicular to the excitation beam by minimizing the1 (a)(b)(c) Δϕ -2.0-1.5-1.0-0.50.00.5 P h a s e s h i f t [r a d ] Δϕ Δϕ -2.0-1.5-1.0-0.50.00.5 P h a s e s h i f t [r a d ] - W /cm ] D i ff e r e n ce [r a d ] Δϕ FIG. 11. The observed phase shift of the Ramsey fringe in theVUV as a function of NIR intensity in the interaction regionfor (a) ∆ N = 2, (b) ∆ N = 3 and (c) their difference. Thesolid line represents a linear fit of the data and the shadedarea indicates the 1 σ uncertainty interval. A maximum shiftof 1.65 rad was observed by increasing the intensity in theinteraction region by a factor of 4.6. However, the fitted slopesof the curves at ∆ N = 2 and ∆ N = 3 are almost equal, witha small residual difference between ∆ N = 2 and ∆ N = 3, asis shown in (c), due to drifts of the setup as the measurementswhere performed sequentially (see text). observed frequency shift to the level of a few MHz. TheDoppler-free transition frequency was then obtained byextrapolating the residual shift to zero velocity aftercorrection for the HHG shift and the second-orderDoppler shift (1.2 kHz for pure xenon and 3 kHz for themixture). In total more than 300 measurements wererecorded over a period of 17 days from which the finalresult was obtained, which is shown in Fig. 10.As was discussed in Sec. II A, the recoil from absorptionof a single photon affects the extracted transition fre-quency. Therefore, a correction of 125 kHz was appliedin the evaluation of the final result.Although the ac-Stark shift is known to be significantlysuppressed in the RCS method [33, 35], we have investi- gated the presence of possible residual shifts. Both theVUV pulses and the residual light from the fundamentalbeam can induce a shift of the energy levels in the atom.To determine the magnitude of the effect, the inducedphase shift of a single Ramsey fringe was measured as afunction of intensity.The influence of the residual NIR light was measuredby slightly opening the iris (in the second vacuumchamber) that normally blocks as much of the NIRbeam as possible. For normal operating conditions theNIR intensity in the interaction region was estimatedto be 1 × W/cm . The measured phase shift asfunction of NIR intensity for ∆ N = 2 and ∆ N = 3is shown in Fig. 11(a) and (b), respectively, and thesolid line shows a linear fit to the data of which the1 σ uncertainty interval is indicated by the shaded area.A significant intensity-induced phase shift of 1.65 radwas observed by increasing the intensity by a factorof 4.6. However, the differential nature of RCS leadsto a strong suppression of the influence of this effecton the extracted transition frequency. This is shownFig. 11(c), where the difference between the inducedshift at ∆ N = 2 and ∆ N = 3 is plotted. The fit showsthat the dependency of the phase on the intensity ispractically equal (within the statistical uncertainty) forboth pulse delays (a small difference is seen becausethe measurements were performed sequentially insteadof simultaneously and without random scanning, whichleads to a small residual phase drift). Therefore theeffect is common mode and barely affects the extractedtransition frequency. Given that the NIR pulses areactively kept constant to the level of 0.2%, the ac-Starkeffect is suppressed by a factor of 500, leading to afrequency shift of less than 20 kHz.A similar procedure was performed to calibrate theac-Stark shift induced by the VUV pulses. Here theintensity was decreased with a factor of three comparedto normal operating conditions (estimated at 5 × W/cm for the seventh harmonic) by reducing thedensity in the HHG jet to reduce the third to seventhharmonic output roughly equally. To this end the gaspressure in the HHG interaction zone was lowered byreducing the drive voltage on the piezo valve. A phaseshift of 0.7 rad was observed, which, in combination withthe measured stability of the VUV intensity (about tentimes worse than that of the NIR), results in a frequencyuncertainty of 85 kHz.The dc-Stark effect was suppressed by using a pulsedelectric field to extract the xenon ions. A possibleresidual shift from stray electric fields was estimated bydeliberately increasing the field in the interaction zoneto 29 V/cm. The observed frequency shift of 2.5(1.0)MHz, gives rise to a shift of <
20 kHz assuming anestimated maximum stray field of < ± Δ [ m r a d ] FIG. 12. The relative optical phase shift of the fundamentalbeam at 770 nm as a function of the pulse delay (equal to∆ NT rep ). The data points are obtained by averaging over 15measurements, each with a statistical uncertainty of 1.5-2.5mrad. This is equivalent to a typical measurement day. Theaverage differential phase stability is considerably better than1 mrad. directions), we determined the uncertainty from straymagnetic fields to be below 52 kHz.Finally, the phase stability of the amplified FC pulseswas measured using the setup described in Sec. III B.The absolute phase shift introduced in the amplificationprocess is typically a few hundred mrad. The observedphase noise amplitude was 40-70 mrad, depending onthe exact operating conditions in the NOPCPA. This isin agreement with the estimation, based on the contrastof the Ramsey fringes, as was discussed in Sec. IV B.The origin of the phase noise was traced back to theamplification process itself [61]. A clear correlation wasobserved between the output power of the amplified FCbeam and the phase stability of the pulses, indicatingthat local wavefront fluctuations (and therefore phasemismatch) give rise to added phase noise, rather thaneffects such as self and cross phase modulation [51, 61].The stability of the phase as a function of pulse delaywas measured as well, as this can have an influenceon the extracted transition frequency. A set of 15measurements, each with a statistical uncertainty of1.5-2.5 mrad, were averaged to obtain the phase stabilityover a typical measurement day. The result is shownin Fig. 12. The phase difference remains constantwell within 1 mrad as a function of the delay, whichconservatively leads to an uncertainty of 140 kHz on thetransition frequency in the VUV.The final result, and all the contributions to the errorbudget, are shown in Tab. I. The transition frequency ofthe 5 p → p s [3 / transition in Xe was deter-mined to be 2 726 086 012 471(630) kHz, which improvesupon the previous determination by a factor of 10 [60].The achieved fractional accuracy of 2 . × − is afactor of 3.6 better than the previous best spectroscopicmeasurement using HHG [30]. TABLE I. Contributions (in kHz) to the measurement of the5 p → p s [3 / transition frequency in Xe.Value or correction (1 σ )Doppler-free transition frequency 2 726 086 012 596 (600) a Light induced effects 0 (87)dc-Stark shift 0 (20)Zeeman-shift 0 (52)Amplifier phase shift 0 (140)Recoil shift -125 (10 − )Total 2 726 086 012 471 (630) a Including the uncertainty of ≈
90 kHz due to the residual HHGphase shift (see text) and the correction for the second-orderDoppler shift of 1.2 kHz for pure Xe and 3.5 kHz for themixture.
D. Isotope shift measurement
Xenon has seven observable stable isotopes, as shownin Fig. 13. Two of them have a nuclear spin and thereforehyperfine structure, namely
Xe and
Xe. This leadsto excitation of several transitions simultaneously andtherefore a beating of the corresponding Ramsey signals.In principle RCS can be used to obtain all the transitionfrequencies and the relative amplitudes of these contri-butions if the phase evolution can be measured over asufficiently long delay [32, 33], but due to the short tran-sit time of the xenon atoms this could not be realized,and therefore only the even isotopes were considered.Of the remaining five even isotopes, three providedenough signal to perform accurate measurements. There-fore the isotope shift of the 5 p → p s [3 / transi-tion for both Xe and
Xe relative to isotope
Xewas determined.As shown in Fig. 13, the signal from the different isotopescan be resolved easily and acquired separately using sev-eral Boxcar integrators. In this manner, several system-atic shifts are common-mode for both isotopes and theisotope shift f X − f can straightforwardly be extracted.Note that each isotope clearly shows a double-peak struc-ture, which only became apparent because of the highresolution of the mass-spectrometer (about 5 ns). Theorigin of this structure was investigated thoroughly byexchanging e.g. the valve and the skimmers, but no con-clusive reason for this observation was found. However,the most probable cause of the multi-peak structure isthe existence of different velocity classes. Therefore, onlysignal coming from the first peak was acquired for eachisotope with the Boxcar integrators.The ambiguity of the determined frequency shift due tothe effective mode spacing is again removed by varyingthe repetition frequency of the laser, similar as discussedin Sec. IV C. This results in a single coincidence point foreach isotope shift frequency which were both determinedwith >
95% confidence. The isotope shifts have been ob-tained from more than 50 measurements for each isotope,which were taken over a period of 8 days. The final result3 +30 µs Xe Xe Xe Xe Xe Xe Xe-600 -400 -200 0 200 400 600time [ns] + 30 μs i on s i gn a l [ a r b . un it s ] FIG. 13. A typical time-of-flight (TOF) trace of xenon iso-topes (using only one 110 nm pulse to avoid isotope-dependentinterference effects). The relative amplitude shows the natu-ral abundance of the isotopes. All stable isotopes, except for
Xe and
Xe, are clearly observed and resolved. Due tothe high resolution of the TOF mass spectrometer, a double-peak structure was observed for all isotopes of which the ori-gin could not be determined with certainty. It is most likelycaused by different velocity classes in the atomic beam. Torecord Ramsey signals, only the first peak of each isotope wasrecorded with a Boxcar integrator in the experiment. of the extracted shift for both isotopes is shown in Fig. 14.The isotope shift was determined to be −
164 910(420)kHz and −
509 750(425) kHz for
Xe and
Xe, respec-tively, relative to
Xe.
V. CONCLUSION AND OUTLOOK
We have given a detailed account of the first demon-stration of Ramsey-comb spectroscopy in combinationwith HHG. A key aspect is the accurate characterizationof the time-dependent plasma-induced phase effectsfrom HHG. This was investigated by exciting xenonatoms to a superposition state with two up-convertedphase-locked laser pulses, while varying the conditionsin the HHG process. The phase effects derived from theRamsey-comb signal could be tracked with mrad-levelaccuracy and on nanosecond timescales. It shows thatthe effects from ionization of the HHG medium (argon)on the phase is dominated at short timescales (of a fewns) by the influence of free electrons, leading to phaseshifts of up to 1 rad at 8 ns pulse delay. However,we found that this effect can easily be reduced to anegligible level by increasing the pulse delay to at least16 ns and moderating the driving intensity. This enabledus to measure the absolute transition frequency of the5 p → p s [3 / transition in Xe with RCS at 110nm with sub-MHz accuracy.The final result of the transition frequency is2 726 086 012 471(630) kHz. This is in agreement withthe previous determination [60], but with a 10 improvedaccuracy. The achieved relative accuracy of 2 . × − FIG. 14. The measured isotope shift of the 5 p → p s [3 / transition for Xe (upper panel) and
Xe(lower panel), with respect to
Xe. Each point is obtainedfrom 8 simultaneous measurements of
Xe and one of theother isotopes. The shaded area represents the 1 σ uncertaintyinterval of the mean of the values. is unprecedented using HHG for up-conversion andimproves upon the previous best measurement involvingHHG by a factor of 3.6 [30].This demonstration shows the potential of RCS in theVUV and XUV spectral range. Combining the accuratepulse control that the FC laser offers with the advantagesof high-power amplified pulses results in straightforwardsingle-pass up-conversion, a high excitation probabilitydue to the high pulse energy, easy wavelength tunability,and enables detection methods, such as state-selectiveionization, that can be made nearly background free andwith high efficiency.The achieved accuracy of the presented experiment isalmost completely limited by the short transit time ofthe xenon atoms through the excitation beam, becausethe setup was in fact designed for 1 S − S excitationin He + with a refocused XUV beam. The interactiontime can therefore be increased by using a collimatedexcitation beam, which would immediately give rise toa higher accuracy as the largest contributions in theerror budget scale down with increasing pulse delay.Alternatively, the transit time can be increased by usinga trapped atom (ion) such as He + , which has the addedbenefit that Doppler effects can be reduced significantly.Therefore, this method shows great promise for 1 S − S spectroscopy of singly-ionized helium. The much higherharmonic order required for He + excitation (effectivelythe 26 th in a scheme combining 32 nm with 790 nm) [18]sets a more stringent limit on the allowed phase noiseof the amplified FC pulses. However, the current4system can produce pulse pairs with a relative phasenoise of about 50 mrad rms, which would still lead toRamsey fringes with contrast of 35%. Both the FC andparametric amplifier can be improved to reduce phasenoise further, leading to higher contrast. Moreover, theupper-state lifetime of the 2 S state is 1.9 ms, so that,compared to the present experiment, a much larger pulseseparation can be applied. Therefore there are goodprospects for RCS to reach kHz-level accuracy in theVUV and XUV spectral range. This is of great interestfor new high-precision tests of quantum electrodynamicstheory, determination of fundamental constants such asthe Rydberg constant or the alpha particle radius, andsearches for physics beyond the standard model. ACKNOWLEDGMENTS
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