Two-photon Rabi oscillations.

Rabi oscillations reflect the existence of a coherent superposition of two molecular eigenstates during the interaction of molecules with a strong, resonant electromagnetic field. This phenomenon is not restricted to two-level systems. We demonstrate its occurrence in a three-level system where the radiation induces a two-photon absorption. The intermediate level is (slightly) detuned from resonance. For a theoretical descrip­ tion the dressed-state picture is adopted. The two-photon excitation is performed either by using one laser, yielding a one-color two-photon process, or by using two counterpropagating, spatially overlapping laser beams, yielding a two-color two-photon process. Rabi oscillations are produced by varying the fluence, i.e., by changing either the laser power or the molecule-laser interaction time. For the one-color case the two-photon transition dipole moment for an SF 6 transition has been determined from the relation between laser fluence and number of Rabi oscillations. For the two-color cases the detuning with the intermediate level is varied by tuning the respective laser frequencies while keeping degree of excitation as a function of the intermediate


L INTRODUCTION
The coupling between molecular eigenstates that is in duced by a (near) resonant electromagnetic field can be de scribed in terms of combined states of molecule and field (see, e.g., [1]). If a laser field is nearly resonant with a mo lecular transition the photon-induced coupling acts between doublets of these eigenstates. On resonance and in the limit of zero-field intensity these doublets become degenerate. For a nonzero field the degeneracy is lifted and the energy dif ference between the two doublet states linearly depends on the field amplitude* Rabi oscillations reflect the (temporary) existence of a coherent superposition of the states making up a doublet. The supeiposition is created when a molecule enters the laser beam, and the contribution of each doublet state to the su perposition depends on laser power and frequency. In the wave function describing the superposition state, the contri butions from both molecular states evolve in time with an energy-dependent phase factor. On leaving the interaction zone the wave function is projected back onto the original doublet states. The resulting redistribution of initial popula tion shows an oscillating behavior as a function of laser power, laser frequency, and interaction time.
Rabi oscillations on one-photon transitions in a two-level system have been measured in metastable Ne [2] and in SF6 [3,4]. From the latter experiments the transition dipole mo ment of the fundamental vibration has been extracted. In other experiments [5][6][7] the transition dipole moments were obtained for the v2 band of NH3 and the v3 band of (di) fluoromethane using laser stark spectroscopy. In [7] the ef fects of a linear frequency sweep during the interaction time, by means of wave-front curvature or ac-Stark field inhomo geneity, which yield rapid passagelike effects, were investi-gated. In [8] microwave multiphoton Rabi oscillations be tween two Rydberg states in potassium are demonstrated and compared with Floquet theory.
These phenomena are not restricted to two-level systems but can be extended to multiphoton absorption schemes in which intermediate levels are involved. Here, laser radiation induces the absorption of two or more photons, the summed energy of which should equal the energy difference between the final and initial state of the multilevel system. The detun ing of the field frequency with single photon transitions, in volving the intermediate states, should be sufficiently small » to enhance the transition probability significantly; but it also has to be sufficiently large to avoid populating intermediate states. If these special requirements are met any multilevel system can be reduced to an effective two-level system [9,10]. Calculations for the case of a resonant intermediate state have been presented by Shore and Ackerhalt [11] and Sargent and Morowitz [12].
Rabi oscillations on multiphoton transitions will be dem onstrated here for a three-level system without relaxation un der several experimental conditions. The simplest scheme consists of a one-color two-photon transition, The fluence can be varied by changing either the laser power or the size of the laser spot that determines the duration of the interac tion time of the molecule in the laser field. Both techniques are applied and will be discussed. The analysis leads to a determination of the effective two-photon transition dipole moment for the {v =0; J= 4 , £)-» (2v3\ J== 3; E) transition in SF6; E describes the rovibrational symmetry in the 0 /rsymmetry group.

II. THEORY
For the description of these phenomena the dressed-state picture [1,13] is used (see Fig. 1). The molecule is consid ered as submerged in a photon bath and the total energy of the molecule-photon system is considered. The possible en ergy states E s of the isolated molecule are dressed by photon

K Ec + NihV[ + (N2~2)hv2
States which are off-resonant with respect to this multiplet are not taken into account; the assumption E h -E a**Ec-E h ^h v^h v 2 is made. The so-called uncoupled dressed states belonging to these dressed energies are written as a,N{,N2),\b,Ny-i,N 2), etc.
The interaction of the molecule with the em field intro duces a coupling between the uncoupled dressed states. In good approximation this coupling can be based on the elec tric dipole interaction d ss*£i between the electric field with amplitude and the transition dipole moment d ss>\ the label / = 1,2 distinguishes the different em field sources. The per turbation gives rise to off-diagonal elements in the Hamil tonian of the form under the condition & S ** 1 (semiclassical limit). The perturbing term f l vv, is known as the Rabi frequency [14].

A. Rabi oscillations in a three-level system
The population of molecular field-free eigenstates changes if these states are coupled by laser photons. In the three-level system described here the three molecular eigen states that participate are |a ), |Z?), and |c), with an initial population of 100% in |a). As a result of the interaction between the molecule and the em field the degeneracy of the uncoupled dressed doublet is lifted and the new coupled dressed eigenstates are separated in energy by SE, see Fig, 1. If the em field is switched on suddenly, a molecule initially in |a) is found in a state that is a coherent superposition of |0(Nj ,Ar2)) anc* iKAfi ,N2)), the latter being the appropriate eigenstates in the presence of the field. The contributions of both these new eigenstates evolve in time with their own, energy-dependent, phase factor. On leaving the em field the superposition state is projected back onto the field-free mo lecular states leading to a population distribution that de pends in an oscillatory way on the duration and strength of the interaction.
To obtain a " clean" two-photon Rabi oscillation between a) and |c) the intermediate state |b) should not acquire significant population. The intermediate detuning |A| should thus be large compared to the one-photon Rabi frequencies ft and f t^.\ On the other hand, the detuning should be small enough to provide for sufficient two-photon transition strength.
In case of a large detuning, A>SE, the three-level system thus reduces effectively to a two-level system. The popula tion then oscillates between states |a) and | c) with the gen eralized Rabi frequency fl. It can also be expressed analyti- Here, pcc stands for the fractional population of |c), with Paa + P cc^l arLC* Pbb^Q' The excitation amplitude (factor in front of the sin2) is a function of the two-photon Rabi fre quency and determines the maximum degree of excitation. In case of exact two-photon resonance (¿¡w=0) Eq. (3) reduces to p ct.= s m^2 (4) The effective interaction time reff in the case of a two-photon interaction involving a Gaussian laser profile follows from the pulse-area theorem [19] (see the Appendix), with w 0 the laser waist radius and v the velocity of the mol ecule. If we count subsequent extrema in pcc by an index number n , that is, n ac 2 tt (6) maxima and minima in pcc are found for n -1,3,5,... and n = 2 ,4 ,6 ,..., respectively. Note that for the case of a onecolor (Si-S2) two-photon transition ilacr^PfwQ ) whereas for a one-photon transition Clahr& sfp. see Eq. (2)

III. EXPERIMENT
The prototype three-level system used for the experiments consists of the P(A)E rovibrational two-photon transition of the vibrational ^-ladder in SF6. The two-photon resonance frequency is -1-348 MHz detuned with respect to twice the fundamental frequency of the 10P16 C 0 2 laser line center [20,21]. The single intermediate level involved is detuned by +250 MHz from the same laser line [22]. The two-photon be minimized to several 100 kHz in 10 min. This suffices to keep the lasers in resonance with the molecular transition frequency during the experiments with a typical recording time of 1-2 min. The power stability of the lasers was better than 1%. The laser beams together with the molecular beam span a horizontal plane.
To tune the laser frequency into resonance with the twophoton transition use was made of optoacoustic modulators (OAM, IntrAction Corp.). These modulators shift the laser frequency by 100 MHz per pass through the crystals. The laser power is reduced by about a factor of 2 on each pass. Together with the tuning range of the lasers themselves of more than 100 MHz to both sides of the line center, the maximum achievable detuning from the central laser fre quency amounts to ±300 MHz for both lasers, the laser beams passing twice through the OAM.
The intensity of the frequency-shifted laser beam can be controlled by adjusting the diffraction efficiency of the OAM, which depends on the rf power fed into the crystal. Care was taken to avoid beam deflection that might be caused by this way of varying the laser intensity.
A stringent condition for producing Rabi oscillations with high contrast is to avoid rapid adiabatic passage (RAP) phe nomena which, in our case, are mainly caused by the curva ture of the laser-field wave fronts [20,24,25]. The effects of wave-front curvature can be minimized by using a telescope to suppress beam divergence [26]. Additionally, the focus of the laser beam should coincide precisely with the interaction region with the molecular beam.
For the telescope an ƒ =25 cm in combination with an /= 4 0 cm lens is used producing a laser waist h>0=4.0 mm at a distance of 6 m. Alternatively, " sharp focusing" [20] could be realized with lenses of ƒ = 2 0 cm producing a laser waist of Wo^O.25 mm at the laser-molecules interaction re gion, This technique, however, produces flat wave fronts only near the molecular beam axis (waist length only 20 mm), whereas the outer parts of the molecular beam (that still reach the detector) experience already nonnegligible wave-front curvature.
Both optical systems (telescope and sharply focused) have been applied, utilizing the experimental setup shown in Fig.  2; SF6 was seeded in He (2%) and expanded through a 30-p m nozzle with a stagnation pressure of 2 bars at room transition can be induced by either a single-laser interacting temperature. The velocity of the beam molecules is 1250 with the molecules or by two simultaneously interacting la-m/s. The detection of the change in internal energy of the beam molecules was performed with a liquid-He-cooled bo lometer (=1.5 K) applying lock-in techniques [27]. The bo lometer dimensions are 5 mm (horizontal)X1 mm (vertical). To obtain circularly polarized light a 3X plate was inserted in the laser beam; no distinction has been made between leftand right-handed polarization. All data acquisition took place by computer.
ser fields. Both types of two-photon excitation processes will be discussed. For the single-laser interaction the two-photon transition is denoted as " one-color two-photon" transition and the resonance frequency is +174 MHz detuned to the blue with respect to the 10P16 C 0 2 laser line. A two-laser interaction yields a " two-color two-photon" transition. In practice this means that both lasers are set to the 10P16 C 0 2 laser line center with a sum detuning of +348 MHz. The experimental setup has been described earlier in [13,20,21,23]. It consists of two high-power, single frequency C 0 2 waveguide lasers and a molecular-beam ap paratus that provides a large population of the 7 = 4 rotational state in the vibrational ground state of SF6. The used laser system consists of two C 0 2 waveguide lasers locked to Fabry-Pérot étalons. The lasers are stable in frequency with a short-term fluctuation of <^10 kHz. The lone-term drift could

A. One-color excitation
Two-photon Rabi oscillations with one laser focused to a diameter of 2 wq-0.5 mm were obtained by recording the excited-state population as a function of the laser power (" power scan" ). The results are shown in Fig. 3(a). The  The large difference in contrast between both recording techniques is caused by the Doppler residue (\SvD\~400 kHz) of the molecular beam determined by the detector size (5 mm). In case of the 8 -mm interaction length (trace A) the first 77-pulse is obtained for a two-photon Rabi frequency of 100 kHz. From Eq. (3) it is seen that a local detuning of S(oD -27r8vD reduces both the Rabi-oscillation amplitude and its oscillation period, if it becomes nonnegligible with respect to For a fixed laser power of 1.5 W and a changing slit width the first i t pulse is observed for a slit width of 3 mm (trace B). The influence of the Doppler residue is three times smaller here and the contrast is strongly enhanced.
The number of 7r pulses, n of Eq. (6), is plotted in Fig. 4 frequencies. The sharp focus introduces RAP effects in the as a function of increasing laser power for the linearly polarouter parts of the molecular beam. Since rapid adiabatic pasized laser beam, with interaction lengths of 0.5 and 8 mm sage produces transitions without Rabi oscillations, this re-and for the circularly polarized laser beam with an interacduces the contrast in the oscillations observed, as discussed in [7].
With the extended laser beam diameter of 8 mm Rabi oscillations were produced by changing the laser power as well as the interaction time. The latter was achieved by plac ing a horizontal slit with adjustable width into the laser beam. By changing the slit width the interaction time of the molecules with the laser field is varied. The results are shown in Fig. 3(b). Trace B shows the result for increasing interaction time for a fixed (relatively high) power of 1.5 W. Trace A shows the result for increasing power with a fixed laser beam diameter of 8 mm. The product of laser power and interaction time (i.e., the fiuence) is plotted on the ab scissa; this parameter is varied in both scan types. Since the excited-state population is a function of fiuence only [Eq.  (6)]. For the linearly polarized beam, the slope for the 2 w0= 8 mm case can be scaled to the 2w o=0.5 mm case through multi plication by a factor of 16. Rescaling the experimental value of L 8 W " 1 (□ of Fig. 4) one obtains 29 W~l, which is in satisfactory agreement with the experimental value of 32 W~l (A of Fig. 4).

B. Two-color excitation
If two (counterpropagating) lasers with different frequen cies are used, the laser powers and the detuning of the laser frequencies with respect to the intermediate level can be cho sen independently. To perform the experiments the laser fields are made to interact simultaneously with the molecular beam. Both lasers are focused with an ƒ " 20 cm lens to a Accurate alignment is achieved by horizontally translating the focus of laser 1 along the molecular beam, through the focus of laser 2. Both laser intensities are kept below satura tion of the two-photon transition. In this case the two-color two-photon signal depends strongly on the overlap of both laser beams, as illustrated in Fig. 5(b), The used laser power is 10 mW for both lasers. The laser beams cross under a small angle to avoid mutual perturbations.  5. (a) shows the one-color two-photon signal for a detuning of laser 1 from the I OP 16 C 0 2 line center of 174 MHz and the two-color two-photon signal for laser 1 at 167 MHz. Laser 1 is tuned in frequency and laser 2 is kept fixed in frequency at 181 MHz (arrow). The peak indicated by A is an unimportant weak two-color feature, (b) shows the line profile of the two-color twophoton resonance for different relative laser focus positions d (in mm, with arbitrary offset), at low-laser intensities, n<^l. The signal strength is seen to respond sensitively to displacements as small as 0 .1 mm. To obtain the two-color two-photon signal laser 2 is kept fixed in frequency, slightly detuned to the blue with respect to the one-color two-photon resonance. The bolometer signal (i.e., the excited-state population) is recorded as a function of the frequency of laser 1; both one-color and two-color reso nances occur, see Fig, 5(a). The used laser power of 10 mW corresponds to Clf2rr-0.% MHz and fiT eff= 0 .3 7 r. Instead of the expected Doppler-free two-color signal for the two counterpropagating laser beams, a broadening by about a factor of 2 is observed (as compared to the one-color transition). This is due to the fact that, in addition to the interaction time broadening (which amounts to 2 MHz for the sharp foci), laser 2 gives rise to M-level splitting (induced by the ac-Stark effect) which is also probed by laser 1. This phenom enon is absent in the case of the one-color two-photon tran sition. Note that the use of two overlapping but counterpropagating waves eliminates RAP effects.
Two-photon two-color Rabi oscillations were produced by increasing the power of laser 1 with fixed power of laser 2. The lasers are tuned to a frequency of 195 MHz (laser 1) and 153 MHz (laser 2) with respect to the 10P16-C 02 laser line. The intermediate detunings amount to A |=55 MHz and A2=97 MHz, respectively. The results are shown in Fig. 6 for fixed power of laser 2 of 20 mW (trace A) and 50 mW (trace B)\ the power of laser 1 is varied up to 1 W. Trace C shows the result of interchanging the lasers; the power of either laser 1 or laser 2 is varied, whereas the other is kept fixed at 300 mW, The similarity of the solid and dashed traces proves that both lasers couple to the molecules in nearly identical ways.
On increasing the power of the fixed-power laser the con trast in signal is reduced, from >50% (trace A) to 2 0 % (trace C). This is explained by an increased M-level splitting, caused by the stronger laser. In addition, the two-color twophoton process actually involves two different pathways. If o'exp '~£r''"Ia) + y exp (7) The bolometer signal is proportional to |y|2. The coefficients a and y depend on the product Slacreîï, The amplitudes \a and |y| assume maximum (minimum) values for n = 1,3,5,... the two photons are denoted by v x and v2 a two-photon tran sition can proceed either by absorbing first a photon vx fol lowed by v2 or vice versa. Both pathways are associated with different intermediate detunings, viz., A^yvx or A2 = v~v 2 (v being the one-photon transition frequency) and therefore with different Rabi frequencies. Their interference gives rise to a reduced contrast.
In Fig. 6 the relative amplitudes for n -1,3,5,... are seen to decrease with increasing power of laser 2. At the same time, the number of Rabi oscillations increases.
To  Fig. 7, trace A, C, and D, respectively. For decreasing A! the observed Rabi oscillations are seen to become more closely spaced and finally are barely resolvable any more (trace D).
The number n of 7r pulses is plotted in Fig. 8 as a function of yfP[P2. Comparison of Fig. 8 with Fig. 4 shows that-for given Aj-the measured points scatter considerably around a straight line fitted through the data points. This scattering is absent in Fig. 4. In addition the linear fit goes only approxi-two successive one-photon steps, including the contributions mately through the origin. The slopes of the linear fits are of specific M values, expected to be proportional to 1/A1, see Eqs.

V. DISCUSSION
For a nearly equidistant three-level system |a), \b), and c) one-color and two-color Rabi oscillations have been ob served between |a) and |c), with CiclL J27r typically in the range of 2.5 MHz(« = l)-2 5 MHz(/t = 10). After the laser interaction, the molecular state is described by Equation (10), including the factor ƒ=0.735, is derived in [28] for the vibrational excitation of two mutually perpen dicular oscillations and includes the first-order Coriolis cor rection term for this specific type of excitation. The rota-   Table II, for lin early and circularly polarized light.
In the measurements the sum of contributions of all M a sublevels is observed. The laser intensity for which maxima and minima occur is determined by the strongest M contri butions. The weaker M components give rise mainly to a reduction of contrast and a slight shift in the position of the minima and maxima. From Table II it is seen that for linearly polarized light mainly the transitions with \Ma\=2 and 3 contribute to d abd bc; for circularly polarized light the main contributions arise from the M a sublevels with |M (I|= 4 ) 3, and 2 . The average main value (a.m.v) for those sublevels from which the resulting dipole strength is estimated are listed in column three.
With this averaged main value the dipole strength d 0 is calculated, see Table I in good agreement with the generally accepted value of d 0= 0.437 D [29]. The uncertainty of 10% in the experimental value stems mainly from the notprecisely-known velocity of the beam molecules (1250±100 m/s). The value for the laser waist diameter is calculated using Gaussian beam formulas; the uncertainty in this pa rameter amounts to 5%. The value of the laser power is For the linearly polarized laser beam with a diameter of 8 mm simulations of the power scan and the time scan have been performed, based on the optical Bloch equations [14,19]. The total population in the final state, p™ has been calculated as a function of the fluence a taking into account the M dependence of d abd bc, that is, with p^(er) the contribution for a specific M value and an effective fluence o-=JP r eff. An integration was performed over the 800-kHz Doppler residue of the molecular beam. The result is included in Fig. 3(b) (dashed curves). Two con clusions can be drawn. The diminishing contrast for succes sive 7T pulses in case of a power scan and the better contrast in the time scan are well reproduced. The procedure to evalu ate the measurements with the help of the averaged main value of d ahd bc appears to be justified.
Concerning the two-color measurements a quantitative simulation was not attempted. The results shown in Fig. 8 show the right trend, but strong ac-Stark effects and interfer ences between individual transition pathways are likely to occur. Already, the comparison between the slopes of twocolor excitation with A1=70 MHz (A2:=82 MHz) and onecolor excitation with A-76 MHz shows a deviation by more than a factor of 2 (see Figs. 4 and 8). For decreasing Aj this effect becomes even more pronounced. In the dressed-state model it is easy to see that the presence of many photons in two radiation modes of different but nearly the same energy yields large multiplets of nearly degenerate states. The radia tive coupling within these multiplets renders the dressedstate model untractable for quantitative discussions, and cal culations should be based on the appropriate set of optical Bloch equations. Further discussions of multicolor multilevel excitation processes can be found elsewhere [9,30,31].

VI. CONCLUSIONS
Two-photon Rabi oscillations are demonstrated using one-and two-color excitation. For the one-color case the three-level system can be described in terms of the superpo sition of two states, equivalent to a two-level system, if a relatively large detuning of the intermediate level is as sumed. From the number of nr pulses the dipole strength can be recovered in good agreement with the theoretical value. For the two-color case the observed Rabi oscillations are 54 TWO-PHOTON RABI OSCILLATIONS 4861 described only qualitatively. Multilevel couplings are likely For a full Gaussian profile, time runs from -1°o to 00 and the to occur due to different interaction pathways involving different photons.

ACKNOW LEDGM ENTS
The authors would like to thank C Sikkens and F. van Rijn for their professional technical assistance. W. L. Meerts is acknowledged for his assistance with the computerized data acquisition. This work has been made possible through the financial support of the KUN, the NWO foundations STW and FOM, as well as the EC (network). APPENDIX The analytical formula for the population of the final level in a two-level system can be obtained from the optical Bloch equations (OBE). For the special case of a square pulse shape of the em field this results in an expression similar to Eq. (3). The interaction time t results from the width of the square pulse divided by the beam velocity v. The strength of the em-field intensity is constant, which allows for analytical solution of the OBE.
For a Gaussian-field-intensity distribution the em-field strength experienced by the molecules varies in time, accord ing to with 6q the maximum field strength, v the velocity of the beam molecules, and w0 the half-width of the Gaussian pro file. The OBE have to be solved with this time dependence of the em-field strength. Application of the pulse area theo rem [19] gives the population pcc for a laser field on reso nance with the transition frequency as Pc-r" sin^a for a laser field running in the z direction with electrical-field strength £(x,y) and with e0 the dielectric constant and c the speed of light. For a laser beam symmetrically cut off by a slit of total width 2 a and using the general expression for a Gaussian beam the laser power is expressed by