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Double resonance spectroscopy of laser-cooled Rb atoms
OPTICS COMMUNICATIONS
Optics Communications 106 (1994) 207-212 North-Holland
Double resonance spectroscopy of laser-cooled Rb atoms A.G. Sinclair, B.D. McDonald, E. Riis and G. Duxbury Department
of Physics and Applied Physics, University of Strathclyde,
Glasgow G4 ONG, Scotland, UK
Received 2 1 October 1993; revised manuscript received 2 December I993
Double resonance transitions have been observed in laser-cooled and trapped s5Rb atoms. The atoms are cooled on the S!I+ SP,,, transition and the absorption of a weak probe beam tuned to the 5PxIz- 4Ds,z transition is detected. The suitability of this transition as a potential secondary frequency standard in the near infra-red is assessed by using the Autler-Townes splitting as a sensitive probe for the shifts induced by the pump laser. The dependence of the splitting on the pump detuning and intensity are measured. Systematic shifts of a few hundred kHz seem realistic, subject to an accurate knowledge of the detuning of the cooling laser. New precise values are also obtained for the hypertine constants for the 4D 5,2level: A= - 5.06+0.10 MHz and B=7.42? 0.15 MHz.
1. Introduction The absorption lines of atoms and molecules provide useful frequency references in the optical communication bands at 1.5 and 1.3 pm. The need for such standard frequencies is described in detail in a recent survey by Knight et al [ 11. However, although there are a range of molecular transitions in these two regions the transition dipole moments of these overtone bands are too weak to permit the use of saturation spectroscopy and hence high precision standards, although Doppler limited lines of acetylene and its isotopic modifications are currently being used [2]. The dipole moments of the electronic transitions of atoms are stronger than those of molecules and hence are more suitable for sub-Doppler spectroscopy. Current examples of this are the use of a 1.523 urn HeNe laser stabilised on the Lamb dip in the gain curve [ 31 and optogalvanic Lamb-dip reference transitions in noble gases [ 4 1. More recently a number of groups have investigated the possibility of using either the 1.3 urn 5P,,z-6S1,2 [5,6] or the 1.5 urn 5P3,z-4DJ,2,5j2 [ 6-81 excited state transitions in rubidium where the lower 5P levels are accessed by optical pumping from the 5S ground state. The natural widths of the 6S and 4D states are on the order of a few MHz, or more than an order of mag0030-4018/94/$07.00
nitude less than the hyper-line structure of the levels [ 9 1. These double resonances have been observed in absorption cells, but with the recent developments in laser cooling and trapping of in particular alkali atoms an obvious choice has been to study the transitions in a sample of laser cooled atoms [ 10 1. The Doppler shift and pressure shift are then virtually eliminated and the technique does not necessarily require the application of external dc or low frequency electric or magnetic fields.
2. Double resonance spectroscopy Although, as pointed out above, a number of serious shift and broadening mechanisms are eliminated when laser cooled atoms are used it would, from the point of view of providing a reliable optical frequency standard, have been desirable to use a transition from the atom’s ground state rather than having to rely on the 5S-5P cooling transition acting as the pump for a double resonance. The ultimate usefulness of these transitions as frequency standards therefore depends on a detailed understanding of the shift and broadening mechanisms inherent in a two-photon transition [ 111. The energy levels of interest for the present work are shown in fig. la. In order for the laser cooling to work efficiently the main
0 1994 Elsevier Science B.V. All rights reserved.
SSDZ 0030-4018(93)E0605-F
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(a) 5
F=s
F=4
F=3
15 March 1994
where A, is the pump detuning (negative for the laser below resonance), c~, = @/2ii the pump Rabi frequency, and ,u the dipole moment of the transition driven by the electric field E. The real part of A2 is the probe detuning. The decay rates yij> ( yi+ rj) of the coherences are related to the decay rates y1 ( =O in the present case), y2, and y3 of the lower, intermediate, and upper levels respectively [ 111. In the limit of no phase disturbing interactions where the equality above applies we find the separation A2,c - Az,s between the two components to be
,
A2,c-Az,s=
s 63 8 - 2’6 “p l’2 2
c P50 Frequency, MHz
100
Fig. 1. (a) The energy levels relevant for this work. The main cooling laser is tuned slightly below the SS,,a, F= 3 - 5P,,,, F=4 transition. Additional light is provided to prevent optical pumping to the F=2 level. A weak probe beam is scanned across the 5P,,,-4D,,a transition. (b) The upper trace shows the absorption spectrum of the probe when the atoms are in optical molasses with the cooling laser tuned 15 MHz below resonance. The three absorption lines due to the coherent two-photon transitions are labeled with the F values of the upper 4D5,* level and further discussed in the text. The lower trace shows the transmission through a 1 m confocal interferometer.
cooling laser must be tuned a few linewidths below resonance. The spectrum of the 5P to 4D transitions (the probe) can therefore be expected to display the characteristic double-peak structure of an AutlerTownes splitting [ 12 1. In the limit of large pump detuning one of these components can be identified with the fully coherent 5%4D two-photon transition and the other with the incoherent or step-wise process where there is an actual build-up of population in the 5P level. In the limit of low probe intensity the shifts away from exact resonance and the linewidths of these two components can be determined as the real and imaginary parts, respectively, of the roots (in AZ) of the following equation [ 111:
208
.
+iy,/2)2+4a:)
(2)
The interpretation is somewhat clearer in the limit IAl 1NCX,,y2/2 where we find the shift A2,s and linewidth ys of the step-wise transition to be
o-i
0
Re (,/(A,
A z,s~((a:lA,)(l-y:/4A:), YS =
~3 +
72 ( I-
and similarly A2.c
=
-4
-
Az,s/4
)
,
(3a)
A,,, and yc for the coherent (d/A,
YC=Y~+Y~~(~+&J~).
I(
I--~:/4&)
transition
> (3b)
Thus, the step-wise transition is shifted by the ACStark shift of the intermediate level and has a width approximately given by the sum of the decay rates for the intermediate and upper levels. On the other hand, the much stronger coherent transition is shifted away from exact resonance by a somewhat larger amount, the ground-state AC-Stark shift plus the pump detuning, but has a narrower width almost entirely determined by the 1.8 MHz natural width of the upper level [ lo]. The use of this transition for a frequency standard would not only require that the inevitable AC-Stark effect can be controlled, but also an accurate knowledge of the pump detuning. On the other hand use of the step-wise transition is hampered by its smaller amplitude, wider linewidth, and proximity to the coherent transition.
3. Experiment As indicated in fig. 1a the main beams for the laser cooling were tuned slightly below the 780 nm 5Sij2, in s5Rb. Beams tuned F=3 - 5P3,2, F=4 transition near the 5S1,2, F=2 - 5P,,2, F= 3 transition pre-
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OPTICS COMMUNICATIONS
vented optical pumping from taking the atoms out of the cooling cycle. The probe laser was tuned to the 1529 nm 5P3,2-4D5,2 transition. Under our experimental conditions only transitions to the F=3, 4, and 5 levels where the intermediate state is close to resonance were observed.
3.1. Apparatus
The two main parts of the experimental setup are the vapour cell magneto-optic trap and the 1.5 pm grating stabilised diode laser. It was found that the scan provided by this commercial laser system [ 13 ] was not sufficiently smooth on the MHz level. The laser was therefore locked to the transmission fringe of a confocal interferometer and scanned by piezoelectrically tuning the mirror separation. The scans were monitored and calibrated using the transmission fringes from a stable Im confocal cavity (free spectral range 75 MHz). At scan rates of a few MHz/ s the smoothness and linearity were satisfactory, but the linewidth was estimated to 2-3 MHz and significant drifts were observed at much reduced rates. The design of the trap is fairly conventional [ 14 1. Six beams of appropriate circular polarisations and a l/e* diameters of about 20 mm intersect in the centre of a glass cell with AR-coated windows. The background pressure in the cell was kept below 1O-* Torr with a turbo-molecular pump and the partial Rb pressure could be controlled down to 10e9 Torr by varying the temperature of the Rb reservoir. The frequency stabilised Ti:sapphire laser used for the main cooling was locked to the F= 3 - F= 4 transition using saturated absorption. Acousto-optic modulators provided full control of the detuning and intensity of the cooling beams within the ranges of interest for this experiment. The 10 mW diode laser used to prevent optical pumping was frequency stabilised to the Doppler broadened absorption profile of the “Rb 5s 1,2, F=2 - 5P3,Z transition and overlapped with four of the beams in the trap. The magnetic field used for the trapping had a gradient in the centre of the cell of 1 G/mm and could be switched off in x 1 ms.
3.2, Detection of double resonances The excitation to the 4D5,2 levels can be detected either by monitoring the fluorescence from the intermediate level or more directly by detecting the absorption of the 1.5 pm probe when passed through the trapped atoms. Under optimum trapping conditions we had - 10’ atoms in a N 0.6 mm diameter sphere. This resulted in an absorption of about 20% of the probe light going through the trap. All the data presented here were taken using phase sensitive detection of the absorption with a 20 kHz, 20% amplitude modulation applied to the cooling beams. The probe beam power was - 12 PW and a diameter of -4 mm. We would ideally like to observe the double resonances when the atoms are in optical molasses, i.e. when the temperature is - 10 PK and the magnetic field is off. However, the requirements of low intensity and large detuning to obtain low temperature and
15t 0
2
1
3
Intensity, mW/cm*
‘“I/
iii
i5
..
0 0
5
‘***............. 10 15 Detuning, MHz
20
25
Fig. 2. (a) The measured separation (squares) of the coherent and step-wisetransitions as a function of intensity in each of the six laser beams forming the trap. The curve is the prediction of eq. (2). The pump detuning is 18 MHz below resonance. (b) The two traces in the top show data (squares) and predicted values (curve based on eq. (2) ) for the separation as a function of the detuning of the cooling laser below resonance. The intensity is 1.35 mW/cm2 in each of the beams. The part of the separation due to the AC-Stark shift of each line is shown in the lower trace. 209
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long-lived molasses are incompatible with a strong double resonance signal. The following sequence was therefore set up. The trap was tilled for - 1.5 s under optimum conditions. The magnetic field was switched off and the detuning increased to N 45 MHz to provide efficient cooling. After 6 ms the detuning was changed back to N 15 MHz and the intensity reduced to _ 1.3 mW/cm2 per beam. The absorption of the probe beam was detected for 160 ms as the atoms were heating up and diffusing out of the probe beam. The sequence was restarted while most atoms were still in the trapping region. The upper trace in fig. lb shows the absorption as the probe laser is slowly scanned through the 5P,,, F= 4 - 4D5,2 transition. The scan is calibrated using the transmission fringes of the 1 m confocal cavity, shown in the lower trace. The four absorption lines observed are identified as follows. The strongest one near the centre is due to the 5S1,2, F= 3 - 4D5,2, F= 5 coherent twophoton transition and the two weaker ones at higher frequencies have final states F= 4 and F= 3, respectively. On the other hand the weaker line to the left of centre corresponds to the step-wise transition 5s ,,2, F=3 - 5P3,2, F=4 - 4D5,2, F=5. The separation of this line from the fully coherent transition displays the detuning and the AC-Stark shift of the measurement. A tit of lorentzian lines #’ to the data in fig. lb yields linewidths (fwhm) of 4.0 MHz for the coherent transitions and 9.3 MHz for the stepwise. The linewidths predicted from eq. (2 ) are 2.6 MHz and 7.0 MHz respectively for the relevant experimental conditions. This discrepancy is not surprising considering the linewidth of the 1.5 urn probe laser. Limited by nonlinearities in the probe scan and variations in the quality of the signal the line separations were typically determined to within 1 MHz. Signal due to the fully coherent transition was detected with pump intensities as low as 0.1 mW/cm’.
*’ Due to the amplitude modulation of the pump laser the expected spectrum is slightly more complicated. The simple lorentzian model leads to a small systematic over estimate of the separation of the coherent and step-wise lines. This does not affect the measurement of the separations between the three coherent transitions and thereby the values obtained for the hyperfine constants.
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4. The AC-Stark shift In order to investigate the effect of the AC-Stark shift on the observed lines we have measured the separation between the coherent and step-wise transitions to the F= 5 level as a function of pump detuning and intensity. To increase the data taking rate and at the expense of a broadening of the signals these measurements were made with the atoms in the magneto-optic trap. The four observed lines were fitted to lorentzians. A fifth line, corresponding to the unresolved step-wise transition to the 4D5,2, F=4 level, was included with its amplitude and location determined by the corresponding values for the F= 5 lines. Fig. 2a shows the separation as calibrated by the transmission fringes of the 1 m interferometer for a pump detuning of - 18 MHz. At the lowest intensity used only a very weak trapping was observed. The curve in fig. 2a shows a least squares fit (with one free parameter, c~i ) of eq. (2 ) to the data points yielding an effective value for the pump Rabi frequency of cu,=5.28 MHz ,/Z/(mW/cm’), where Z is the intensity in each one of the six trapping beams. With a natural linewidth of the pump transition of 5.9 MHz this corresponds to an effective saturation intensity of 78 uW/cm2 in each beam. This is significantly below the value of Zmt= 1.6 mW/cm’ for one beam. We assume this is due to the high peak intensities in the complicated interference pattern set up by the six trapping beams. As the two-photon absorption process strongly favours detection of signal from the high-intensity regions we can expect an ACStark shift corresponding to the intensity at the interference maxima of the field. In a naive model one can imagine that the atoms effectively see the constructive interference of four of the six beams #’ and therefore 16 times the intensity in fair agreement with the experimental result. A more rigorous model will have to include details of the spatial variation of the intensity and state of polarisation of the light, the distribution of population among the magnetic sublevels, and indeed the density of atoms. We next turn to the dependence of the separation of the two lines on the pump detuning. Fig. 3 shows #z C.f. the well-known
fact that an atom on a hM=O transition only absorbs linearly polarised light with a component of its direction of propagation perpendicular to the quantisation axis.
OPTICS COMMUNICATIONS
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laser exciting the 1.5 pm transition will depend directly on the absolute frequency of the pump laser. By measuring the separations between the three coherent transitions for a large number of the data taken for fig. 2 we are able to determine the hyperfine constants for the 4D5,2 level. We find A= -5.06f0.10 MHz and B=7.42+0.15 MHz. A value of A = - 5.2 + 0.3 MHz has previously been reported [ 9 1, but to our knowledge B has not been measured directly.
5. Conclusion -25
0 Probe
25
50
detuning, MHz
Fig. 3. A series of scans of the probe laser for different pump detunings and an intensity of 1.35 mW/cm’ per beam. Each scan has been normalised to the sum of the absorptions of the coherent and step-wise transitions.
a series of scans of the probe laser for pump laser detunings in the range 5 MHz to 23 MHz below resonance for a pump intensity of 1.35 mW/cm2 per beam. In the upper trace in fig. 2b the points show the measured frequency separation between the coherent and step-wise transitions and the curve shows the prediction of eq. (2) using the Rabi frequency determined above. No attempt has been made to qualitatively understand the slight discrepancy between the measured and predicted shifts at detunings in the range 5-15 MHz. A more accurate analysis would again have to include the details of the spatial variation of the intensity and polarisation of the light field as well as the distribution of the atoms. The measured AC-Stark shift for one line only, i.e. one half of difference between the measured splitting and the pump detuning is shown in the lower trace of fig. 2b. The data shows that even at the lowest intensities where the magneto-optic trap operates reliably (-0.8 mW/cm2) the AC-Stark shift exceeds 1 MHz for the range of relevant pump detunings. Only by alternating the between states of efficient trapping and low intensity/large detuning, as in the molasses experiment described above, will it be possible to bring the AC-Stark shift down to the level of a few hundred kHz. As only the fully coherent transition can be expected to yield an appreciable signal under these conditions, the absolute frequency of the
The configuration and parameters used in the present experiment are by no means ideal for an optical frequency standard. The fundamental limitations for an optical frequency standard based on a two-photon transition in a neutral atom have been analysed by Hall et al. [ 15 1. Following their analysis we conclude that for a given two-photon transition probability (proportional to the square of the product of the two Rabi frequencies) the AC-Stark effect (dominated by a term proportional to the square of the strongest Rabi frequency) is minimised if the two Rabi frequencies are roughly equal. In the present experiment they differ by about an order of magnitude. A further reduction can be obtained by exciting the transition using Ramsey’s method of separated oscillatory fields [ 16 1. The AC-Stark shift is then essentially reduced by the ratio of the pulse duration to the pulse separation [ 15 1. In the previous study of the SP to 4D transition in a laser-cooled rubidium vapour [ lo] the asymmetric shape of the detected signals emphasised the significance of the strong coupling of the P level to the ground state due to the cooling laser. In the present experiment using a different modulation and detection technique and starting from a‘ larger number of cold atoms we have been able to resolve the observed signal in two components due to the step-wise and coherent SS-4D two-photon transition. The separation of these two components was used to quantitatively study the AC-Stark effect, which is the most important systematic effect limiting the accuracy of an optical frequency standard based on a two-photon transition [ 15 1.
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Acknowledgements
This work has been supported by the United Kingdom Science and Engineering Research Council and National Physics Laboratory. In particular we thank D. Knight for the loan of the 1.5 pm laser from NPL. One of us (AGS) acknowledges support from the Carnegie Trust for the Universities of Scotland.
References [ 1 ] D.J.E. Knight, P.S. Hansell, H.C. Leeson, G. Duxbury, J. Meldau and M. Lawrence, in: Frequency-stabilised lasers and their applications, ed. Y.C. Chung, SPIE 1837 (1993). [2] H. Sasada, S. Takeuchi, M. Iritani and K. Nakatani, J. Opt. Soc.Am.B8 (1991) 713. [3] H. Sasada and 0. Kubota, Appl. Phys. B 55 (1992) 186. [4] A.J. Lucero, Y.C. Chung, S. Reilly and R.W. Tkach, Optics Lett. 16 (1991) 349. [ 5 ] R. Boucher, M. Breton, N. Cyr, and M. T&u, IEEE Photonics Technol. I&t. 4 (1992) 327.
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[ 61 H. Sasada, IEEE Photonics Technology Letters 4 (1992) 1307. [ 7 ] S.L. Gilbert, in: Proc. lo* Int. Conf. on Laser Spectroscopy, Font-Romeu, France, June 1991, eds. M. Ducloy, E. Giacobino and G. Camy (World Scientific, Singapore). [ 8 ] S.L. Gilbert, in: Frequency-stabilised lasers and their applications, ed. Y.C. Chung, SPIE 1837 ( 1993). [ 9 ] K.H. Liao, L.K. Lam, R. Gupta and W. Happer, Phys. Rev. Lett. 32 (1974) 1340. [ IO] R.W. Fox, S.L. Gilbert, L. Holberg and J.H. Marquardt, Optics Lett. 18 (1993) 1456. [ 111 S. Stenholm, Foundations of laser spectroscopy (Wiley, 1984). [ 121 S.H. Autler and C.H. Townes, Phys. Rev. 100 (1955) 703. [ 13 ] BT&D model TSLl OOO-1550 external cavity diode laser. [ 141 C. Monroe, W. Swann, H. Robinson and C. Wieman, Phys. Rev. Lett. 65 (1990) 1571. [ 151 J.L. Hall, M. Zhu and P. Buch, J. Opt. Sot. Am. B 6 ( 1989) 2194. [ 161 N.F. Ramsey, Molecular beams (Oxford University Press, 1956); Y.V. Baklanov, V.P. Chebotaev and B.T. Dubetsky, Appl. Phys. 11 (1976) 201.
Optics Communications 106 (1994) 207-212 North-Holland
Double resonance spectroscopy of laser-cooled Rb atoms A.G. Sinclair, B.D. McDonald, E. Riis and G. Duxbury Department
of Physics and Applied Physics, University of Strathclyde,
Glasgow G4 ONG, Scotland, UK
Received 2 1 October 1993; revised manuscript received 2 December I993
Double resonance transitions have been observed in laser-cooled and trapped s5Rb atoms. The atoms are cooled on the S!I+ SP,,, transition and the absorption of a weak probe beam tuned to the 5PxIz- 4Ds,z transition is detected. The suitability of this transition as a potential secondary frequency standard in the near infra-red is assessed by using the Autler-Townes splitting as a sensitive probe for the shifts induced by the pump laser. The dependence of the splitting on the pump detuning and intensity are measured. Systematic shifts of a few hundred kHz seem realistic, subject to an accurate knowledge of the detuning of the cooling laser. New precise values are also obtained for the hypertine constants for the 4D 5,2level: A= - 5.06+0.10 MHz and B=7.42? 0.15 MHz.
1. Introduction The absorption lines of atoms and molecules provide useful frequency references in the optical communication bands at 1.5 and 1.3 pm. The need for such standard frequencies is described in detail in a recent survey by Knight et al [ 11. However, although there are a range of molecular transitions in these two regions the transition dipole moments of these overtone bands are too weak to permit the use of saturation spectroscopy and hence high precision standards, although Doppler limited lines of acetylene and its isotopic modifications are currently being used [2]. The dipole moments of the electronic transitions of atoms are stronger than those of molecules and hence are more suitable for sub-Doppler spectroscopy. Current examples of this are the use of a 1.523 urn HeNe laser stabilised on the Lamb dip in the gain curve [ 31 and optogalvanic Lamb-dip reference transitions in noble gases [ 4 1. More recently a number of groups have investigated the possibility of using either the 1.3 urn 5P,,z-6S1,2 [5,6] or the 1.5 urn 5P3,z-4DJ,2,5j2 [ 6-81 excited state transitions in rubidium where the lower 5P levels are accessed by optical pumping from the 5S ground state. The natural widths of the 6S and 4D states are on the order of a few MHz, or more than an order of mag0030-4018/94/$07.00
nitude less than the hyper-line structure of the levels [ 9 1. These double resonances have been observed in absorption cells, but with the recent developments in laser cooling and trapping of in particular alkali atoms an obvious choice has been to study the transitions in a sample of laser cooled atoms [ 10 1. The Doppler shift and pressure shift are then virtually eliminated and the technique does not necessarily require the application of external dc or low frequency electric or magnetic fields.
2. Double resonance spectroscopy Although, as pointed out above, a number of serious shift and broadening mechanisms are eliminated when laser cooled atoms are used it would, from the point of view of providing a reliable optical frequency standard, have been desirable to use a transition from the atom’s ground state rather than having to rely on the 5S-5P cooling transition acting as the pump for a double resonance. The ultimate usefulness of these transitions as frequency standards therefore depends on a detailed understanding of the shift and broadening mechanisms inherent in a two-photon transition [ 111. The energy levels of interest for the present work are shown in fig. la. In order for the laser cooling to work efficiently the main
0 1994 Elsevier Science B.V. All rights reserved.
SSDZ 0030-4018(93)E0605-F
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(a) 5
F=s
F=4
F=3
15 March 1994
where A, is the pump detuning (negative for the laser below resonance), c~, = @/2ii the pump Rabi frequency, and ,u the dipole moment of the transition driven by the electric field E. The real part of A2 is the probe detuning. The decay rates yij> ( yi+ rj) of the coherences are related to the decay rates y1 ( =O in the present case), y2, and y3 of the lower, intermediate, and upper levels respectively [ 111. In the limit of no phase disturbing interactions where the equality above applies we find the separation A2,c - Az,s between the two components to be
,
A2,c-Az,s=
s 63 8 - 2’6 “p l’2 2
c P50 Frequency, MHz
100
Fig. 1. (a) The energy levels relevant for this work. The main cooling laser is tuned slightly below the SS,,a, F= 3 - 5P,,,, F=4 transition. Additional light is provided to prevent optical pumping to the F=2 level. A weak probe beam is scanned across the 5P,,,-4D,,a transition. (b) The upper trace shows the absorption spectrum of the probe when the atoms are in optical molasses with the cooling laser tuned 15 MHz below resonance. The three absorption lines due to the coherent two-photon transitions are labeled with the F values of the upper 4D5,* level and further discussed in the text. The lower trace shows the transmission through a 1 m confocal interferometer.
cooling laser must be tuned a few linewidths below resonance. The spectrum of the 5P to 4D transitions (the probe) can therefore be expected to display the characteristic double-peak structure of an AutlerTownes splitting [ 12 1. In the limit of large pump detuning one of these components can be identified with the fully coherent 5%4D two-photon transition and the other with the incoherent or step-wise process where there is an actual build-up of population in the 5P level. In the limit of low probe intensity the shifts away from exact resonance and the linewidths of these two components can be determined as the real and imaginary parts, respectively, of the roots (in AZ) of the following equation [ 111:
208
.
+iy,/2)2+4a:)
(2)
The interpretation is somewhat clearer in the limit IAl 1NCX,,y2/2 where we find the shift A2,s and linewidth ys of the step-wise transition to be
o-i
0
Re (,/(A,
A z,s~((a:lA,)(l-y:/4A:), YS =
~3 +
72 ( I-
and similarly A2.c
=
-4
-
Az,s/4
)
,
(3a)
A,,, and yc for the coherent (d/A,
YC=Y~+Y~~(~+&J~).
I(
I--~:/4&)
transition
> (3b)
Thus, the step-wise transition is shifted by the ACStark shift of the intermediate level and has a width approximately given by the sum of the decay rates for the intermediate and upper levels. On the other hand, the much stronger coherent transition is shifted away from exact resonance by a somewhat larger amount, the ground-state AC-Stark shift plus the pump detuning, but has a narrower width almost entirely determined by the 1.8 MHz natural width of the upper level [ lo]. The use of this transition for a frequency standard would not only require that the inevitable AC-Stark effect can be controlled, but also an accurate knowledge of the pump detuning. On the other hand use of the step-wise transition is hampered by its smaller amplitude, wider linewidth, and proximity to the coherent transition.
3. Experiment As indicated in fig. 1a the main beams for the laser cooling were tuned slightly below the 780 nm 5Sij2, in s5Rb. Beams tuned F=3 - 5P3,2, F=4 transition near the 5S1,2, F=2 - 5P,,2, F= 3 transition pre-
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vented optical pumping from taking the atoms out of the cooling cycle. The probe laser was tuned to the 1529 nm 5P3,2-4D5,2 transition. Under our experimental conditions only transitions to the F=3, 4, and 5 levels where the intermediate state is close to resonance were observed.
3.1. Apparatus
The two main parts of the experimental setup are the vapour cell magneto-optic trap and the 1.5 pm grating stabilised diode laser. It was found that the scan provided by this commercial laser system [ 13 ] was not sufficiently smooth on the MHz level. The laser was therefore locked to the transmission fringe of a confocal interferometer and scanned by piezoelectrically tuning the mirror separation. The scans were monitored and calibrated using the transmission fringes from a stable Im confocal cavity (free spectral range 75 MHz). At scan rates of a few MHz/ s the smoothness and linearity were satisfactory, but the linewidth was estimated to 2-3 MHz and significant drifts were observed at much reduced rates. The design of the trap is fairly conventional [ 14 1. Six beams of appropriate circular polarisations and a l/e* diameters of about 20 mm intersect in the centre of a glass cell with AR-coated windows. The background pressure in the cell was kept below 1O-* Torr with a turbo-molecular pump and the partial Rb pressure could be controlled down to 10e9 Torr by varying the temperature of the Rb reservoir. The frequency stabilised Ti:sapphire laser used for the main cooling was locked to the F= 3 - F= 4 transition using saturated absorption. Acousto-optic modulators provided full control of the detuning and intensity of the cooling beams within the ranges of interest for this experiment. The 10 mW diode laser used to prevent optical pumping was frequency stabilised to the Doppler broadened absorption profile of the “Rb 5s 1,2, F=2 - 5P3,Z transition and overlapped with four of the beams in the trap. The magnetic field used for the trapping had a gradient in the centre of the cell of 1 G/mm and could be switched off in x 1 ms.
3.2, Detection of double resonances The excitation to the 4D5,2 levels can be detected either by monitoring the fluorescence from the intermediate level or more directly by detecting the absorption of the 1.5 pm probe when passed through the trapped atoms. Under optimum trapping conditions we had - 10’ atoms in a N 0.6 mm diameter sphere. This resulted in an absorption of about 20% of the probe light going through the trap. All the data presented here were taken using phase sensitive detection of the absorption with a 20 kHz, 20% amplitude modulation applied to the cooling beams. The probe beam power was - 12 PW and a diameter of -4 mm. We would ideally like to observe the double resonances when the atoms are in optical molasses, i.e. when the temperature is - 10 PK and the magnetic field is off. However, the requirements of low intensity and large detuning to obtain low temperature and
15t 0
2
1
3
Intensity, mW/cm*
‘“I/
iii
i5
..
0 0
5
‘***............. 10 15 Detuning, MHz
20
25
Fig. 2. (a) The measured separation (squares) of the coherent and step-wisetransitions as a function of intensity in each of the six laser beams forming the trap. The curve is the prediction of eq. (2). The pump detuning is 18 MHz below resonance. (b) The two traces in the top show data (squares) and predicted values (curve based on eq. (2) ) for the separation as a function of the detuning of the cooling laser below resonance. The intensity is 1.35 mW/cm2 in each of the beams. The part of the separation due to the AC-Stark shift of each line is shown in the lower trace. 209
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long-lived molasses are incompatible with a strong double resonance signal. The following sequence was therefore set up. The trap was tilled for - 1.5 s under optimum conditions. The magnetic field was switched off and the detuning increased to N 45 MHz to provide efficient cooling. After 6 ms the detuning was changed back to N 15 MHz and the intensity reduced to _ 1.3 mW/cm2 per beam. The absorption of the probe beam was detected for 160 ms as the atoms were heating up and diffusing out of the probe beam. The sequence was restarted while most atoms were still in the trapping region. The upper trace in fig. lb shows the absorption as the probe laser is slowly scanned through the 5P,,, F= 4 - 4D5,2 transition. The scan is calibrated using the transmission fringes of the 1 m confocal cavity, shown in the lower trace. The four absorption lines observed are identified as follows. The strongest one near the centre is due to the 5S1,2, F= 3 - 4D5,2, F= 5 coherent twophoton transition and the two weaker ones at higher frequencies have final states F= 4 and F= 3, respectively. On the other hand the weaker line to the left of centre corresponds to the step-wise transition 5s ,,2, F=3 - 5P3,2, F=4 - 4D5,2, F=5. The separation of this line from the fully coherent transition displays the detuning and the AC-Stark shift of the measurement. A tit of lorentzian lines #’ to the data in fig. lb yields linewidths (fwhm) of 4.0 MHz for the coherent transitions and 9.3 MHz for the stepwise. The linewidths predicted from eq. (2 ) are 2.6 MHz and 7.0 MHz respectively for the relevant experimental conditions. This discrepancy is not surprising considering the linewidth of the 1.5 urn probe laser. Limited by nonlinearities in the probe scan and variations in the quality of the signal the line separations were typically determined to within 1 MHz. Signal due to the fully coherent transition was detected with pump intensities as low as 0.1 mW/cm’.
*’ Due to the amplitude modulation of the pump laser the expected spectrum is slightly more complicated. The simple lorentzian model leads to a small systematic over estimate of the separation of the coherent and step-wise lines. This does not affect the measurement of the separations between the three coherent transitions and thereby the values obtained for the hyperfine constants.
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4. The AC-Stark shift In order to investigate the effect of the AC-Stark shift on the observed lines we have measured the separation between the coherent and step-wise transitions to the F= 5 level as a function of pump detuning and intensity. To increase the data taking rate and at the expense of a broadening of the signals these measurements were made with the atoms in the magneto-optic trap. The four observed lines were fitted to lorentzians. A fifth line, corresponding to the unresolved step-wise transition to the 4D5,2, F=4 level, was included with its amplitude and location determined by the corresponding values for the F= 5 lines. Fig. 2a shows the separation as calibrated by the transmission fringes of the 1 m interferometer for a pump detuning of - 18 MHz. At the lowest intensity used only a very weak trapping was observed. The curve in fig. 2a shows a least squares fit (with one free parameter, c~i ) of eq. (2 ) to the data points yielding an effective value for the pump Rabi frequency of cu,=5.28 MHz ,/Z/(mW/cm’), where Z is the intensity in each one of the six trapping beams. With a natural linewidth of the pump transition of 5.9 MHz this corresponds to an effective saturation intensity of 78 uW/cm2 in each beam. This is significantly below the value of Zmt= 1.6 mW/cm’ for one beam. We assume this is due to the high peak intensities in the complicated interference pattern set up by the six trapping beams. As the two-photon absorption process strongly favours detection of signal from the high-intensity regions we can expect an ACStark shift corresponding to the intensity at the interference maxima of the field. In a naive model one can imagine that the atoms effectively see the constructive interference of four of the six beams #’ and therefore 16 times the intensity in fair agreement with the experimental result. A more rigorous model will have to include details of the spatial variation of the intensity and state of polarisation of the light, the distribution of population among the magnetic sublevels, and indeed the density of atoms. We next turn to the dependence of the separation of the two lines on the pump detuning. Fig. 3 shows #z C.f. the well-known
fact that an atom on a hM=O transition only absorbs linearly polarised light with a component of its direction of propagation perpendicular to the quantisation axis.
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laser exciting the 1.5 pm transition will depend directly on the absolute frequency of the pump laser. By measuring the separations between the three coherent transitions for a large number of the data taken for fig. 2 we are able to determine the hyperfine constants for the 4D5,2 level. We find A= -5.06f0.10 MHz and B=7.42+0.15 MHz. A value of A = - 5.2 + 0.3 MHz has previously been reported [ 9 1, but to our knowledge B has not been measured directly.
5. Conclusion -25
0 Probe
25
50
detuning, MHz
Fig. 3. A series of scans of the probe laser for different pump detunings and an intensity of 1.35 mW/cm’ per beam. Each scan has been normalised to the sum of the absorptions of the coherent and step-wise transitions.
a series of scans of the probe laser for pump laser detunings in the range 5 MHz to 23 MHz below resonance for a pump intensity of 1.35 mW/cm2 per beam. In the upper trace in fig. 2b the points show the measured frequency separation between the coherent and step-wise transitions and the curve shows the prediction of eq. (2) using the Rabi frequency determined above. No attempt has been made to qualitatively understand the slight discrepancy between the measured and predicted shifts at detunings in the range 5-15 MHz. A more accurate analysis would again have to include the details of the spatial variation of the intensity and polarisation of the light field as well as the distribution of the atoms. The measured AC-Stark shift for one line only, i.e. one half of difference between the measured splitting and the pump detuning is shown in the lower trace of fig. 2b. The data shows that even at the lowest intensities where the magneto-optic trap operates reliably (-0.8 mW/cm2) the AC-Stark shift exceeds 1 MHz for the range of relevant pump detunings. Only by alternating the between states of efficient trapping and low intensity/large detuning, as in the molasses experiment described above, will it be possible to bring the AC-Stark shift down to the level of a few hundred kHz. As only the fully coherent transition can be expected to yield an appreciable signal under these conditions, the absolute frequency of the
The configuration and parameters used in the present experiment are by no means ideal for an optical frequency standard. The fundamental limitations for an optical frequency standard based on a two-photon transition in a neutral atom have been analysed by Hall et al. [ 15 1. Following their analysis we conclude that for a given two-photon transition probability (proportional to the square of the product of the two Rabi frequencies) the AC-Stark effect (dominated by a term proportional to the square of the strongest Rabi frequency) is minimised if the two Rabi frequencies are roughly equal. In the present experiment they differ by about an order of magnitude. A further reduction can be obtained by exciting the transition using Ramsey’s method of separated oscillatory fields [ 16 1. The AC-Stark shift is then essentially reduced by the ratio of the pulse duration to the pulse separation [ 15 1. In the previous study of the SP to 4D transition in a laser-cooled rubidium vapour [ lo] the asymmetric shape of the detected signals emphasised the significance of the strong coupling of the P level to the ground state due to the cooling laser. In the present experiment using a different modulation and detection technique and starting from a‘ larger number of cold atoms we have been able to resolve the observed signal in two components due to the step-wise and coherent SS-4D two-photon transition. The separation of these two components was used to quantitatively study the AC-Stark effect, which is the most important systematic effect limiting the accuracy of an optical frequency standard based on a two-photon transition [ 15 1.
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Acknowledgements
This work has been supported by the United Kingdom Science and Engineering Research Council and National Physics Laboratory. In particular we thank D. Knight for the loan of the 1.5 pm laser from NPL. One of us (AGS) acknowledges support from the Carnegie Trust for the Universities of Scotland.
References [ 1 ] D.J.E. Knight, P.S. Hansell, H.C. Leeson, G. Duxbury, J. Meldau and M. Lawrence, in: Frequency-stabilised lasers and their applications, ed. Y.C. Chung, SPIE 1837 (1993). [2] H. Sasada, S. Takeuchi, M. Iritani and K. Nakatani, J. Opt. Soc.Am.B8 (1991) 713. [3] H. Sasada and 0. Kubota, Appl. Phys. B 55 (1992) 186. [4] A.J. Lucero, Y.C. Chung, S. Reilly and R.W. Tkach, Optics Lett. 16 (1991) 349. [ 5 ] R. Boucher, M. Breton, N. Cyr, and M. T&u, IEEE Photonics Technol. I&t. 4 (1992) 327.
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[ 61 H. Sasada, IEEE Photonics Technology Letters 4 (1992) 1307. [ 7 ] S.L. Gilbert, in: Proc. lo* Int. Conf. on Laser Spectroscopy, Font-Romeu, France, June 1991, eds. M. Ducloy, E. Giacobino and G. Camy (World Scientific, Singapore). [ 8 ] S.L. Gilbert, in: Frequency-stabilised lasers and their applications, ed. Y.C. Chung, SPIE 1837 ( 1993). [ 9 ] K.H. Liao, L.K. Lam, R. Gupta and W. Happer, Phys. Rev. Lett. 32 (1974) 1340. [ IO] R.W. Fox, S.L. Gilbert, L. Holberg and J.H. Marquardt, Optics Lett. 18 (1993) 1456. [ 111 S. Stenholm, Foundations of laser spectroscopy (Wiley, 1984). [ 121 S.H. Autler and C.H. Townes, Phys. Rev. 100 (1955) 703. [ 13 ] BT&D model TSLl OOO-1550 external cavity diode laser. [ 141 C. Monroe, W. Swann, H. Robinson and C. Wieman, Phys. Rev. Lett. 65 (1990) 1571. [ 151 J.L. Hall, M. Zhu and P. Buch, J. Opt. Sot. Am. B 6 ( 1989) 2194. [ 161 N.F. Ramsey, Molecular beams (Oxford University Press, 1956); Y.V. Baklanov, V.P. Chebotaev and B.T. Dubetsky, Appl. Phys. 11 (1976) 201.