Open Access
Add to BibSonomyAbstract
The CO2 laser locked onto a saturated absorption resonance of OsO4 provides a secondary frequency standard in the 10 μm region, with an accuracy of 50 Hz to 1 kHz. For averaging times less than 100 s its stability performance is better than the Hydrogen maser. This paper deals with the present attempt to increase this performance by using a two-photon molecular resonance as a reference. We begin with some preliminary and promising results on a two-photon line of SF6 leading to characteristics similar to those obtained with a saturation line of OsO4. Then two alternative methods to increase the resolution are presented : optical detection of slow molecules and a new development of the well-known Ramsey fringes. Metrological features are analyzed for both methods.
©1999 Optical Society of America
1. Introduction
Today the fields of high resolution spectroscopy and frequency metrology are very active. The best strategy in both cases is to record narrower and narrower lines since most of the systematic errors are proportional to the linewidth of the frequency reference. Any increase in the experimental resolution will thus result in an increase in the ultimate accuracy of the frequency standard.
Within this search for high resolution we can distinguish between two directions of improvement during the recent past. First, the latest generation of frequency standards takes advantage of laser cooling and trapping techniques. The best example is the new Cs clock in the LPTF, Paris, which is based on the principle of the atomic fountain [1, 2]. An accuracy of 3×10-15 was obtained with good prospects to reach 10-16. In the optical domain, the Ca frequency standard at 657 nm at PTB which is now using a MOT is the most advanced system [3]. Second, some frequency standards have been recently developed using two-photon resonances. For atoms natural linewidth limitations are usually well below what has been achieved currently. For instance, the two-photon transition at 760 nm in Rubidium allows a 2 kHz accuracy to be obtained, that is 5×10-12 relative accuracy [4]. There is also the very promising experiment on the 1S–2S transition at 121 nm in Hydrogen [5] with 10 kHz fringes in a Ramsey geometry.
In the optical domain, as the frequency is higher, a better relative precision can be reached. This is indeed the case for the two secondary frequency standards in the infrared region : the CO2 laser locked onto OsO4 at 10 μm, and the HeNe laser locked onto methane at 3.39 μm. Quite surprisingly, all the work concerning these standards implies a saturated absorption resonance and no experiments were performed with two-photon resonances until a few years ago. Since molecular spectra are complex most of the two-photon resonances were, indeed, unknown. This paper presents some preliminary steps to explore the possibility of a molecular two-photon resonance for metrology.
This paper describes first the performance of our high resolution spectrometer working in the 10 μm region. Then we present some preliminary results on the stabilization onto a two-photon resonance. Next we describe two alternative methods to increase the resolution in Doppler-free two-photon spectroscopy. Both methods aim to reduce the broadening due to the finite transit time of the molecules through the laser beam. In the first we detect the signal coming only from slow molecules, without any cooling of the gas [6, 7]. This method was previously demonstrated for saturated absorption in methane [8, 9] and in OsO4 [10], and led to linewidths of the order of 100 Hz. The second method is based on the well-known Ramsey fringes, using a molecular beam. This experiment is now in progress in our group. We finish with the comparison of both systems with respect to the requirement of metrology.
2. Our high resolution spectrometer
The performance of our spectrometer has already been described in detail [11]. It employs two CO2 lasers emitting in the 10 μm region. The first laser is frequency-locked to a strong saturation line of OsO4, and the beat note between this laser and the second laser is then phase-locked to a tunable synthesizer. Thus the stability acquired by the OsO4 lock is transferred to the second laser, which is also tunable around each emission line of CO2. The stability characteristics are : the laser linewidth is of the order of 6 Hz (FWHM) which is 2×10-13 in relative value, while the Allan variance reaches 0.1 Hz (Δν / ν = 3.5 × 10-15) for a time constant of 100 s. The reproducibility was estimated to be 10 Hz. This stability performance ensures that our experimental resolution will not be limited by the laser itself.
Fig. 1. Experimental apparatus. Frequency modulations at f1=90 kHz and f2=4.9 kHz are applied to the EOM. AOM : acoustooptic modulator, P : polarizer, OI : optical isolator.
Fig. 1 displays the experimental apparatus. It involves two key elements. First a Fabry-Perot cavity (FPC) to contain the molecular gas ; this is essential to ensure good control of the beam geometry and a significant gain in the signal contrast. The second major element is an electro-optic modulator (EOM), which is driven by an RF synthesizer and generates two sidebands, one of which is locked onto OsO4. Frequency modulations are applied directly to the RF synthesizer to generate the error signals for the stabilization loops. The main advantage of this system is that it allows a very clean and easy frequency modulation while, in addition, the laser carrier is tunable by tuning the RF which drives the EOM. Because of power considerations it is convenient to use a second laser as shown ; the phase-lock is particularly clean.
3. Stabilization onto a two-photon line
To test the possibility of a two-photon resonance compared with a saturated absorption line, we first attempted to lock the two CO2 lasers onto a two-photon line of SF6. For this purpose, we used our usual stabilization arrangement (see Fig. 1) with SF6 in place of OsO4, recording the beat note between the lasers to determine the stability performance. We chose the R(47) A2 of the 2ν3 band, whose excitation probability is quite high due to the very small detuning (16 MHz) of the one-photon transition in ν3 [12]. The experimental parameters were : 50 mW inside the FPC, pressure 3×10-2 Pa, 20 kHz HWHM for the two-photon line, third-harmonic detection.
Fig. 2a) displays the beat signal between the lasers, each stabilized onto the same two-photon line ; the 7.9 Hz linewidth (FWHM) indicates that each laser linewidth is 4 Hz. This result is two times better than with OsO4, thanks to the narrower signal. The Allan variance on Fig. 2b) characterizes the long term stability. The slopes of -1/2 and +1 indicate good functioning of the servo loop. It reaches a Flicker plateau of 0.33 Hz (Δν/ν = 1×10-14) for an averaging time of 20 s. This stability is a little degraded compared to OsO4, and this is certainly related to the degradation of the optical isolation due to the higher incident power at the FPC.
These preliminary results are very promising since the experimental parameters, and especially the FPC finesse, were not optimized. The next step is to increase the line resolution.
Fig. 2. a) Beat signal between the lasers, each stabilized onto the same two-photon R(47) resonance of the 2ν3 band of SF6 fitted with the best Lorentzian b) Square root of the Allan variance of the beat note frequency, divided by √2 in order to describe one of the two equivalent stabilized lasers.
4. Slow molecule detection
The basic idea of the optical selection of slow molecules is to operate at low saturation power and low pressure. Under these conditions the fast molecules are in the transit regime. The lineshape associated with such a transverse velocity class is a Gaussian whose width is proportional to the transverse velocity (i.e. the component perpendicular to the laser beam). These contributions are inhomogeneous, and quite small because the molecules do not have time to be saturated by the field. By contrast, the slowest molecules are in the collisional regime. The associated lineshape is a Lorentzian with a homogeneous width determined by collisions. Their contribution is higher than for fast molecules because the excitation probability, proportional to the effective interaction time, is maximum. The relative enhancement of the contribution of the slow molecules induces a slight line narrowing, which is not so important because the population of the slow transverse velocity classes is small. In addition, for a two-photon transition, there is a negligible selection of the longitudinal velocity and all longitudinal velocity classes contribute to the signal in the same way. Thus we are dealing with the optical selection of transversely slow molecules.
The different contributions are very inhomogeneous : the slow molecules contribute with a narrow lineshape, and the faster ones with a broader Gaussian. We can easily take advantage of this inhomogeneous character to enhance the slow molecules signal, by taking the derivatives of the total lineshape; this was the original idea of Chebotayev applied in saturation spectroscopy [8, 9]. We have demonstrated that with two steps of derivation we completely eliminate the fast molecules contribution and the linewidth is proportional to the collisional linewidth [6, 7].
The experiment was performed on the P(4) E transition of the 2 ν3 band of SF6, first measured by [13] who also pointed out its potential as a two-photon candidate. We detected the signal in transmission of a Fabry-Perot cavity filled with SF6. The laser was frequency-modulated so that we used a lock-in amplifier to detect the first or the second harmonic of the signal, thus recovering approximately the first or the second derivative of the line. Fig.3 displays some typical results. A 2f linewidth of 280 Hz (HWHM) was obtained [6, 7], which is 23 times narrower than the transit width for the mean thermal velocity. This corresponds to molecules of transverse velocity around 5 m/s. This demonstrates the very good selectivity for velocity. As a first application of this enhanced resolution we were able to record for the first time the hyperfine structure of a two-photon resonance.
Fig.3. Hyperfine structure of the P(4)E line in the 2ν3 3 band of SF6 a) Pressure P=10-2 Pa, power PW=60 mW, accumulation time t=3 s/point, modulation frequency f=2.2 kHz., depth 400 Hz, 2f-detection b) Central component. P=10-2 Pa, PW=12 mW, t=6 s/point, f= 120 Hz, depth 200 Hz.
5. Ramsey fringes
The Ramsey fringes experiment presents an alternative to the previous cell experiment, since it is dealing with a molecular beam. The basic principle is that the single interaction zone is replaced by two spatially separated interaction zones, with a fixed relative phase. When a molecule has passed through the first zone of interaction it is in a coherent superposition of lower and upper levels. This coherence precesses freely between the two zones. In the second zone the molecule is either excited or de-excited, depending on the relative phase between the excitation field and the coherence. Thus fringes develop in the excitation probability versus the laser frequency, and their spacing depends on the transit time between the two zones. The resolution is then no longer limited by the transit time associated with one beam.
In the optical domain, this method has to be associated with a sub-Doppler technique in order to avoid the scrambling of the fringe pattern. The use of saturated absorption and a four-zone configuration imposes severe conditions on parallelisms and equidistances which, in practice, limit the distance between zones and finally the ultimate resolution. By contrast, in the case of two-photon spectroscopy the two-zone geometry is convenient [14]. Each zone consists of a standing wave, and the only experimental condition is that the relative phase between two zones is fixed. This is easily fulfilled by generating both standing waves inside the same Fabry-Perot cavity. The experimental signal arises from the superposition of the contribution of each velocity class in the molecular beam. It should exhibit fringes whose spacing is half the inverse of the mean transit time between two zones [15, 16], with an attenuation of the side fringes due to the longitudinal velocity dispersion. These fringes are superimposed on the broader two-photon signal arising from the absorption in one single zone.
The experimental realization is now in progress, using the same P(4)E line of SF6. Some 10 kHz two-photon Ramsey fringes were already observed elsewhere[17], with a resolution limited by the hyperfine structure and the laser linewidth. We are planning to reach much higher resolution, and have constructed an apparatus with an interzone distance adjustable between 10 cm and 2 m. The 10 cm interzone distance will give 1.7 kHz fringes which is much less than the hyperfine structure.
Fig. 4 a) Experimental scheme for two-photon Ramsey fringes ; AOM : acousto-optic modulator, b) Schematic of the 3 levels involved in the P(4) E resonance.
Fig. 4 displays the experimental set-up. We use a pure SF6 supersonic beam to gain two main advantages. First an enhancement of the population in the J=4 level arises from the rotational cooling of the supersonic expansion, and we measure an effective temperature less than 30 K. Second, the velocity is slower (350 m/s) than with a He-seeded beam, so the transit time is longer. The phase-coherent standing waves are generated in a folded Fabry-Perot cavity with a finesse of 500. The transmitted signal allows the cavity resonance to be locked onto the laser frequency. To detect the fringes, we plan to compare two optical methods : either record the fringes on the transmission of the cavity, or read the signal on the molecular beam itself by probing its absorption on a 1-photon transition, using rapid adiabatic passage, with an auxiliary beam. For detection purpose we can either frequency-modulate the laser beam or mechanically chop the molecular beam.
The beam was characterized by recording the linear absorption on the lower one-photon transition at different distances from the skimmer. The measured divergence is of the order of 30 mrad depending on the nozzle-skimmer distance, and the flux is a few ×1012 molecules/s in the J=4 level.
Recently we observed the two-photon absorption signal with a 10 cm cavity. Fig. 5 displays two typical spectra, recorded with two detection methods. The linewidth corresponds to the hyperfine structure but the 15 kHz transit width in one single zone prevents its resolution.
Fig. 5. Two-photon absorption signal detected a) on the cavity transmission beam using a laser FM of 5 kHz (detection channel S1 on Fig. 4), b) on an auxiliary beam absorption (detection channel S2), tuned to the upper one-photon transition, using the molecular beam chopper at 550 Hz. Typical conditions : 15 mW in the cavity, 5 bars pure SF6 beam, accumulation time 4 s/point. Data are fitted to a) the derivative of a Gaussian and b) a Gaussian.
The next step is to detect the fringes. We anticipate that the depth of Ramsey fringes should be more than 50 % of the absorption signal. Yet higher resolution will require the use of low frequency detection techniques which are more sensitive to the laser 1/f technical noise and the acoustical and mechanical noise of the cavity. Each modulation method presents a specific problem : the chopper induces some additional noise on the whole system due to imperfect mechanical isolation, while some residual amplitude modulation is associated with the laser FM.
6. Metrological features
In this part we consider the different effects which might affect the accuracy or reproducibility of a frequency standard based on the slow molecule detection or on Ramsey fringes. They are summarized in Table 1.
First we should point out that there is a negligible recoil effect in Doppler-free two-photon spectroscopy [18].
The pressure could be responsible for a shift in the cell experiment, but this should not exceed a few Hertz at 10-2 Pa. It can be measured from an extrapolation of the measurement of the high pressure shift. As a comparison, for OsO4, another spherical top molecule, the pressure shift at 5×10-3 Pa is 1.4 Hz [19]. Of course there can be no pressure shift for the Ramsey fringes, since there are no collisions in a beam.
The magnetic shift is negligible provided we set a μ-metal shield to cancel the earth’s field. We can easily obtain a 0.1 % attenuation, which should result in a shift less than 0.1 Hz. Note that the Zeeman effect was previously measured for a P(39) and a P(46) Δm=±1 line of OsO4 [20], and shifts of the order of 100 Hz/10-4T were obtained.
To estimate the shift induced by the second-order Doppler effect (SODE), we have to distinguish between the contribution of the longitudinal and transverse velocities. In slow molecule detection the SODE arises only from the longitudinal velocity as the transverse velocity is low. The whole shift can be exactly derived, when substituting νeg by νeg (1 -v2 /2c2) in the theoretical absorption signal, where hνeg is the energy difference between the upper and lower levels and v the velocity. It is equal to ν eg/2 u2/4c2 where u/√2 is the mean longitudinal velocity in the gas. The correction due to the transverse velocity is less than 0.1 %. For our system, as the mean velocity in the gas is 185 m/s, this gives a 2.70 Hz shift, with a very good accuracy of 1 %, corresponding to the accuracy of the cell temperature. By contrast, for the Ramsey fringe experiment the longitudinal velocity, i.e. the component along the laser beam, always gives a negligible contribution, since the beam divergence is small. To calculate the SODE, we assume a density for the velocity along the molecular beam axis proportional to v2 exp(-(v - u/Δu)2), with u the mean velocity in the beam, and Δu the longitudinal dispersion. The resulting shift is v eg/2 u 2/2c 2(1 + 1/2(Δu/u)2). For typical experimental conditions, we have measured u=370 m/s and Δu=55 m/s by a time of flight method. We calculate a 23 Hz shift where the second term, corresponding to the velocity dispersion, contributes only 1%. The uncertainty is a priori 1 Hz and is limited by the accuracy on the mean velocity (5 %). This uncertainty could be decreased by repeating the same experiment with a He-seeded SF6 beam, which changes u, and measuring the different shifts. The experimental value u and Δu might be measured more precisely from an analysis of the fringes.
The light shift is the weak point of two-photon spectroscopy compared to saturated absorption. With the use of higher power some AC Stark shifts are induced in the levels, thus giving frequency shifts. However this effect is not so important for two reasons. First, the resulting shift is proportional to the difference of the transition probabilities for the two one-photon transitions, which is small for the rovibrational transitions used here. Second, the slow molecule detection method is basically a low-field experiment, while in the Ramsey fringe experiment the effect is much reduced because the interaction with the strong field lasts a negligible time compared to the transit time between two zones. If we consider the dominant contribution of the intermediate level, the resulting shift is : δv = ΩRabi/4(|μrg|/|μer|-|μ|er|/|μrg|), multiplied by w/D in the case of the Ramsey fringes [21, 16, 22]; ΩRabi is the effective angular Rabi frequency for the two-photon excitation, w is the beam waist, D is the interzone distance, μβα is the transition dipole for the β<-α transition. For the P(4) E transition, following [23] and neglecting hyperfine structure, we have : (|μrg|/|μer| - |μer|/|μrg|. With a two-photon excitation equivalent to pulses respectively of π and π/2, velocities 5m/s and 370 m/s, waist 4.5 mm and 3.5 mm, the shifts are : 28 Hz for the slow molecule experiment and 9 Hz for the Ramsey fringe experiment, the latter having an interzone distance of 50 cm. This shift should be easily determined by extrapolating the measurements at different laser powers to zero power. The accuracy will depend on the SNR and will be proportional to the resolution.
Lastly, the black-body radiation might induce a non-negligible shift. This amounts to 0.1 mHz at 9.2 GHz, as recently measured for the Cs fountain [2]. For a ro-vibrational transition in a molecule the effect will generally be much smaller than for an electronic transition in an atom. Here, however, the effect will be magnified since the black body emission peak is precisely in the 10 μm region. In fact we plan to use the Ramsey experiment to measure this effect, which has never been studied in the 30 THz domain.
Table 1. Systematic shifts for slow molecule detection and Ramsey fringes.
7. Conclusion
As a conclusion, both slow molecule detection and Ramsey fringes are good candidates for frequency standards. With the possible exception of the black-body radiation shift, the systematic effects are either very small or can be easily and precisely measured or calculated. Thus they will not affect the accuracy of the frequency by more than 1 Hz.
However the Ramsey fringes have a few advantages. There is no pressure shift and there is considerable room for improvement since the signal is not significantly decreased when increasing the resolution. The signal amplitude is indeed determined by the geometrical crossing volume between the laser and molecular beams, which is proportional to the inverse of the square root of the resolution. By contrast, the slow molecule signal decreases linearly with increasing resolution. But the Ramsey fringes experiment is more complex ; it is obviously a big challenge to combine ultra-high resolution spectroscopy with a large vacuum apparatus and all its associated noise.
The choice of SF6 was determined by the earlier measurements of its one- and two-photon transitions [13, 24], but its hyperfine structure is obviously unfavorable for metrology. Significant progress might be made when looking at an isolated two-photon resonance, for example in OsO4. Such investigations, however, depend on spectroscopic work to be performed on the 2ν3 band of this molecule.
Acknowledgments
This work has been supported by the Bureau National de Métrologie, the ETCA-DRET and the European Union through two HCM programs.
References and links
1. A. Clairon, C. Salomon, S. Guellati, and W. D. Phillips, “Ramsey resonance in a Zacharias fountain,” Europhys. Lett. 16, 165–170 (1991). [CrossRef]
2. E. Simon, P. Laurent, and A. Clairon, “Measurement of the Stark shift on the Cs hyperfine splitting in an atomic fountain,” Phys.Rev. A 57, 436–439 (1998). [CrossRef]
3. F. Riehle, H. Schnatz, G. Zinner, K. Zeiske, B. Lipphardt, and J. Helmcke, “Calcium optical frequency standard based on atom interferometry,” Laser Phys. 6, 237–243 (1996).
4. L. Hilico, R. Felder, D. Touahri, O. Acef, A. Clairon, and F. Biraben, “Metrological features of the Rubidium two-photon standards of the LPTF and the Kastler-Brossel Laboratories,” Eur. Phys. J. AP 4, 219–225 (1998). [CrossRef]
5. A. Huber, B. Gross, M. Weitz, and T. W. Hänsch, “Two-photon Ramsey spectroscopy of the 1S–2S transition in atomic Hydrogen,” Phys. Rev. A 58, R2631–R2634 (1998). [CrossRef]
6. P. E. Durand, G. Nogues, V. Bernard, A. Amy, and C. Chardonnet, “Sélection des molécules lentes en spectroscopie sans effet Doppler à deux photons,” Ann. Phys. Fr. 20, 601–602 (1995). [CrossRef]
7. P. E. Durand, G. Nogues, V. Bernard, A. Amy-Klein, and C. Chardonnet, “Slow-molecule detection in Doppler-free two-photon spectroscopy,” Europhys. Lett. 37, 103–108 (1997). [CrossRef]
8. S. N. Bagayev, A. E. Baklanov, V. P. Chebotayev, and A. S. Dychkov, “Superhigh resolution spectroscopy in methane with cold molecules,” Rev. Roum. Phys. 33, 361–367 (1988).
9. S. N. Bagayev, V. P. Chebotayev, A. K. Dmitriyev, A. E. Om, Y. V. Nekrasov, and B. N. Skvortsov, “Second-Order Doppler-Free Spectroscopy,” App. Phys. B 52, 63–66 (1991). [CrossRef]
10. C. Chardonnet, F. Guernet, G. Charton, and C. J. Bordé, “Ultrahigh-resolution saturation spectroscopy using slow molecules in an external cell,” Appl. Phys. B 59, 333–343 (1994). [CrossRef]
11. V. Bernard, C. Daussy, G. Nogues, L. Constantin, P. E. Durand, A. Amy-Klein, A. van Lerberghe, and C. Chardonnet, “CO2 laser stabilization to 0.1-Hz level using external electrooptic modulation,” QE-33, 1282 (1997).
12. F. Herlemont, M. Khelkhal, J. Legrand, and G. Pierre, “Doppler-free two-photon spectrum of SF6 for metrological purposes,” Opt. Lett. 23, 957–959 (1998). [CrossRef]
13. A. Linskens, S. te Lintel Hekkert, and J. Reuss, “One and two photon spectra of SF6 molecular beam measurements,” Infrared Phys. 32, 259–282 (1991). [CrossRef]
14. Y. V. Baklanov, V. P. Chebotayev, and B. Y. Dubetsky, “The Resonance of Two-Photon Absorption in Separated Optical Fields,” App. Phys. 11, 201–202 (1976). [CrossRef]
15. C. J. Bordé, “Sur les franges de Ramsey en spectroscopie sans élargissement Doppler,” C. R. Acad. Sc. Paris 284B, 101–104 (1977).
16. C. J. Bordé, “Density matrix equations and diagrams for high resolution non-linear laser spectroscopy : application to Ramsey fringes in the optical domain,” in Advances in laser spectroscopy, F. T. Arecchi, F. Strumia, and H. Walther, Editors, (1983).
17. A. F. Linskens, W. L. Meerts, J. Reuss, and C. J. Bordé, “Doppler-free two-photon Ramsey Fringes in the 10 μm region,” in Laser controlled dynamics of molecular processes, A. F. Linskens and Ph. D. Thesis, University of Nijmegen, ISBN 90-9007146-6 (1994).
18. C. J. Bordé, Private communication
19. O. Acef, “Metrological properties of CO2/OsO4 optical frequency standard,” Opt. Comm. 134, 479–486 (1997). [CrossRef]
20. M. P. Sassi and K. Stoll, “Investigation of anomalous Zeeman effect in OsO4 molecule," in Laser Spectroscopy XIII, 111–113 (1997).
21. E. Giacobono and B. Cagnac, “Doppler-free multiphoton spectroscopy," in Progress in Optics XVII, E. Wolf, Editor, 85–161 (1980).
22. S. A. Lee, J. Helmcke, and J. L. Hall, “High Resolution two-Photon Spectroscopy of Rb Rydberg Levels,” in Laser Spectroscopy IV, H.W.a.K.E. Rothe, Editor, 130–141 (1979).
23. C. J. Bordé, “Développements récents en spectroscopie infrarouge à ultra-haute résolution,” in Revue du Cethedec-Ondes et signal-NS 83-1, 1–118 (1982).
24. F. Herlemont, M. Lyszyk, and J. Lemaire, “Doppler-Free Two-Photon Spectroscopy of the 2ν3 band of SF6,” Appl. Phys. 24, 369–374 (1981). [CrossRef]
