Resonance fluorescence spectrum of a three-level atom. II. Numerical results

Volume 42, number 1

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1 June 1982

RESONANCE FLUORESCENCE SPECTRUM OF A THREE-LEVEL ATOM. II. NUMERICAL RESULTS * K.f: bOLOSCHUK and S. HONTZEAS Department of Physics and Astronomy, Universityof Regina, Regina, Saskatchewan, CanadaS4S 0A2

and Constantine

MAVROYANNIS

Division of Chemistry, National Research Council of Canada, Ottawa,Ontario, Cannda KIA OR6 Received 23 November 1981 Revised manuscript received 6 January 1982

Numerical computations are presented for the excitation spectra arising from t$e interaction between a three-level atom in the cascade configuration and a strong electromagnetic field whose frequency mode is initially populated. The excitation spectra are considered when the laser field is at resonance with the equally spaced levels of the atom as well as a function of the detunings. The physical process of optical amplification occurs without population inversion and it is more pronounced when the laser field is detuned than when it is at resonake. The shapes of the spectral lines for a number of sidebands are of the absorption-amplification type rather than that of the absorption one. In the presence of detunings as well as in the cooperative two-photon cascade process, the resulting spectra are far more complicated than those occurring at resonance. Results of numerical calculations for a wide range of Rabi frequencies and detunings are presented graphically.

1. Inntroduction The interaction of strong, monochromatic optical radiation with atoms has recently received much attention. The optical double resonance spectra have been extensively studied [l-12], where a threelevel atom interacts with two near-resonant laser fields. Mavroyannis ‘[131, hereafter referred to as I, considered the excitation spectrum arising from the interaction of a three-level atom in the cascade configuration with a strong electromagnetic field (see fig. 1). In the limit of high photon frequencies, an expression for the Green’s function Cl2 21(w),describing the electronic transition 1 ++2 ik derived,in I and is given by eq. (17) of I, namely

I

(d,a-A,-A2)(C12a-B1-B2)-(A3+A4XB3+84)

(1) and we have adopted the same notation as in I where all the expressions can be found. Similarly eq. (18) of I gives the expression for the Green’s function G23 32(w) which describes the electronic transition 2 ++‘3. In I, eqs. (1) and (18) of I were used to calculate the excitation spectra when w21 = 20, and O23 = w, describing two and one photon whose spectral functions were determined

processes by eqs.

(22) and (23) of I, respectively. The purpose of the present study is to derive the excitation spectra for the one photon processes

* Issued as N.R.C.C. No. 19936.

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’

0 1982 North-Holland

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2. Excitation spectra at resonance

Pig. 1. Energy level diagram of a three-level atom in the cascade configuration. Thick limes indicate the laser field while wiggly lines represent radiative decays.

1, which are obtained from the numeriof eqs. (1) and (18) of I at resonance and in the presence of detunings as well as when the detunings take opposite values. In the present problem, we are dealing with only one photon excitations whose coupling functions are of the same order of magnitude and, hence, analytical methods are not applicable and only numerical computations of eqs. (1) and (18) of I are possible. For the sake of convenience, we take the Rabi frequencies s2 2, !i12, and the spontaneous emission probabilities yl,rf p$ for the transitions 1 * 2 and 2 t* 3 to be equal, namely, %2 = 5123 = 52 and 7: = 7; = ro. Both approximations are reasonable since the quantities in question are of the same order of magnitude. We introduce the following dimensionless variables X = (a 7 aa)/ru 7v = WYO~ ~21 = (021 - aa)/ 7. and ~32 = (~32 - a,)/~~, where X is the reduced frequency, 9 is the reduced Rabi frequency while ~2~ and ~32 refer to the reduced detuning for the transitions 1 * 2 and 2 * 3, respectively. Then working in the complex domain and using complex arithmetic one can substitute the given propagators and parameters into the Green’s function determined by eq. (1) and directly calculate the expression for the relative intensity defined by -2 Im G12,21(~) @I - Ea)/(rrYo) *, for given q, V21,V32 Values as a function of X. The procedure for the computation can be found elsewhere [ 141.

shown in fig.

cal calculation

* The expressionfor the relative intensity so defined, apart from a constant factor, is equal to the absorption coefficient of the system. Hence, positive and negative values of the relative intensity and, consequently, of the absorption coefficient describe the physical processes of absorption and ampllftostion, respectively.

78

When ~2~ = 0 and ~32 = 0, the levels of the atom are equally spaced and in resonance with the laser field, ~21 = ~32 = wa. This particular situation is of interest in microwave double resonance, but it is of less importance for optical double resonance, and it has been excluded in the work of Whitley and Stroud [l] . The excitation spectra are obtained as described previously by the numerical computation of the spectral function derived from eq. (1) and the resulting peaks are depicted in figs. 2a-c. Although the relative intensity as a function of X is computed for values of 1)equal to 1, 5, 10, 15,20,30,50 and 100, respectively only the spectra for r) = 15,20 and 50 are shown in figs. 2a-c because they are representative. When TJ= 1 there is only the central peak, which appears at the excitation frequency X = 0 and its relative intensity, I,, remains roughly constant for any given value of 1)> 1. When n > 1 and particularly for q Z 5, in addition to the central peak, there are: two pairs of sidebands peaked at the frequencies X = fg with each pair having positiveI+ and negative -I_ intensity, respectively, and one pair of sidebands peaked at the frequencies X = +-27~ and having an extremely small positive intensity. Our results and figs. 2a-c imply that the absolute value of the ratio of the intensities of the sidebands at X = +V varies from I+ :I_ = 3 for 77= 5 and decreases as n increases and becomes I+ : I_ = 1 for TJ> 50. Figs. 2a-c indicate that as n increases the intensities of the sidebands at X = +r) increase. For instance, we found the following values for ratios of the intensities: I+ : I, = 0.36 for 9 = 5, I+ :I, = 0.56 for 71= 10 (fig. 2a), I+ : I, = 0.77 for q = 15 and f+ : I, = 1 for n = 20 (fig. 2b). For values of n greater than 20, the intensities of the sidebands at X = n are larger than those for the central peak, For instance, I+ : Ic = 1.43 for r) = 30, I+ : I, * 2.22 for rj = 50 (fig. 2c) and I+ : I, = 4.35 for n = 100. The ratio I+ : I, is plotted as a function of 17in fig. 3, which implies that the ratio I+ : I, increases linearly with r). Positive and negative intensities imply that the physical process of absorption and amplification (negative absorption) takes place, respectively. The existence of the sidebands at X = fn with positive and negative intensities as well as the linear increase of the ratio I+ : I, with q as shown in figs. 2 and 3 are in agreement with

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OPTICS COMMUNICATIONS

Volume 42, number 1 1.5r

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Fig. 2. The relative intensities of the fluorescent light is plotted versus the reduced frequency X = (W - wa)/ro for zero detunings vtl = 0, vsa = 0 and various Rabi reduced frequencies n = S&o. (a) q = 15, (b) n = 20 and (c) n = 50.

earlier theoretical predictions [lo]. The absorption amplification line shapes of these sidebands are analogous to those discussed and observed for a two-level system by Wu et al. [15] and differ considerably from the corresponding absorption or emission spectra. Our results and figs. 2a-c indicate that the pair

of sidebands at X = +2q exists for values of ?I = 5-50 and vanishes for 7)> 50. However even for values of TI= 5-50, the intensity of this pair is extremely small.

3. Excitation spectra in the presence of detunings Figs. 4, 5 and 6 refer to two sets of values for q = 15 and 50, repectively, with ~32 = 5 constant while the detuning for the 1 ff 2 transition, v21, takes the values of 0,5 and 10. Comparison between figs. 4a and 4b with those of 2a and 2c, respectively, indicates that the presence of the detuning ~32 = 5 in the 2 ++3 transition has a pronounced effect in the spectra of the 1 * 2 resonance transition. Figs. 4a and 4b imply that the effect of the detuning ~32 = 5 is not only to destroy the symmetry and to reduce the intensities of the sidebands shown in figs. 2a and 2c for ~32 = 0 but also separates the peaks with positive and negative intensities by an energy shift equal t0 V32.

Fig. 3. The ratio of the relative intensities I+ : I, is plotted versus the reduced Rabi frequencies n. I+ and I, are the relative intensities of the sideband at X = n and the central peak, respectively.

Figs. 5a and 5b describe the situation, where w21 = 032 and ~21 - W, = 5ro, and imply that the value of the Rabi frequency is important in determining the structure, the shape as welI as the type of the spectra. In fig. 5a, where v= 15, the presence of the finite detuning v21 = 5 results in the disappear79

Volume 42, number 1

1 June 1982

OPTICS COMMUNICATIONS

9

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ante of the peaks with negative intensities, which reappear again in fig. Sb when 9 becomes equal to SO. Hence, figs. Sa and Sb for r] = 15 and SO, apart of the pair of peaks at X = ?277,describe spectra of the absorption and absorption-amplification type, respectively. In figs. Sb and 6b, the peaks with positive and negative intensities are separated by energy shifts equal to ~32 and 2~~2, respectively.

In the presence of the detuning v21, the central peak always appears at X= vzl and its intensity remains constant. In figs. S and 6, a new peak appears at X = 0, which is analogous to one that has been recently predicted for a two-level system in the presence of detuning [ 16). ‘Ihe peak at X = 0 has negative intensity in fig. 6a indicating amplification (gain) while in figs. Sb and 6b the peak has positive and

(a) Fig. 5. As in fii. 2 but for ~21 = 5, ~32 = 5 and various values of 11.(a) q = 15 and (b) 9 = 50. 80

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Fig.6. As in fig. 2 but for vsl = 10, vs2 = 5 and variousvaluesof q. (a) TJ= 15 and (b) TJ= 50. negative components and, hence, it is of the absorption-amplification type. In fig. 6a, two new peaks appear at X = -~3~ and X = (v21 + ~3~)/2 while in fig. 6b only the peak at X k (~21 + ~32)/2 survives. Again in fig. 6a, where 17= 15, the effect of the detuning v21 = 10 is the vanishing of the peaks with negative intensities with the exception of the peak at X = 0 while the peaks reappear in fig. 6b for 17= 50 but now the energy separation between the peaks with positive and negative intensities is of the order 2~3~.

4. Cooperative two-photon excitation spectra

The case when the detunings v21 and ~32 take opposite values, v21 = -~32, is referred to as the COoperative two-photon cascade process, where w21 + ~32 = 2w,. The computed spectra with ~21 = -~32 = 5 for Q = 15 and 50 are depicted in figs. 7a and 7b, respectively. Comparison between figs. 7a and 7b implies that drastic changes occur in the spectra by changing the value of the Rabi frequency from 15 to 50. In fig. 7a, where q = 15, the sidebands with positive and negative intensities are separated by an energy shift of the order iv21 = 4~32 while in fig. 7b the value of 1)= 50 is too large to allow such a splitting to occur. In fig. 7b, the intensities of the

peaks at X = (v2 + u21) 2 y2 and at X = --(q2 t z&)112 are two and seven times larger than the corresponding ones in fg 7a. The excitation spectra due to the transition 2 ++3 are obtained from the imaginary part of the Green’s function G23,32(0). Inspection of (1) and eq. (18) of I implies that the excitation spectra for the 2 * 3 trarisition are expected to be similar to those for the 1 ++2 transition. In fact, at resonance, the spectra due to 2 t+ 3 are expected to be identical to those for the 1 f) 2 transition; In the presence of detunings and considering the approximation sZ12 = L?,2,3= S2and 7: = 7: = yo, the spectra for the upper system (2 cf 3 transition) are anticipated to be qualitatively similar to those for the lower system (1 ++2 transition), namely, the positions of the peaks for the 2 f+ 3 transition can be derived from those for the 1 +f 2 transition by interchanging v21 by ~32, v21 ++~32 while the intensities of the peaks for both system; are identical.

5. Summary

The excitation spectra are graphically presented at resonance in figs. 2-3, at finite detunings in figs. 4-6 and for the cooperative two-photon cascade process in figs. 7a and 7b. In figs. 2a-c, the shape of the 81

Volume 42, number 1

1 June 1982

OPTICS COMMUNICATIONS 2.5 I

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sidebands at X = kq are of the absorption-amplification type, their relative intensities increase as q jncreases while fig. 3 implies that the intensity I+ of the sideband at X = Q varies linearly with 77.In the presence of detunings as well as when the detunings take opposite values, the resulting spectra consist’ of an abundance of peaks whose main characteristics are the following: 1) The effect of detunings is to reduce the relative intensities of the sidebands, which occur at resonance, to the extent that the intensity of the central peak is much larger than those of most of the sidebands even at large values of n. 2) One of the effects of the detuning us2 in the spectra for the 1 ++2 transition is to separate the peaks with positive and negative intensities by energy shifts of the order of either ~32 and 2~32 in the presence of a laser field of appropriate intensity. 3) For small values of n, the detumng v21 # 0 results in the disappearance of the peaks with negative intensities that exist at resonance, while these peaks reappear again for large values of Q. 4) In the presence of detunings, the position of the peaks are no longer symmetrically located from X = 0 as it is in the case at resonance. The intensity of each peak varies and the symmetry of the intensities of each pair of sidebands, which exists at resonance, is uow destroyed. For given values of the detunings, the spectra are very sensitive to,value of the parameter n. Some of the new peaks occurring in the presence of detunings have negative intensities 82

indicating that the physical process of amplification (gain) is favoured at these particular frequencies without population inversion. References Ill R.M. Whitley and C.R. Stroud, Jr., Phys. Rev. Al4 (1976) 1498.

VI B. Subolewska, Optics Comm. 19 (1976) 185. [31 P. Avan and C. Cohen-Tannoudji, J. Phys. BlO (1977) 171. ]41 C. Cohen-Tannoudji and S. Reynaud, J. Phys. BlO (1977) 345. I51 Z. Bialynicka-Birula and I. Bialynicki-Bkula, Phys. Rev. Al6 (1977) 1318. (61 J.H. Eberly, B.W. Shore, Z. Bialynicka-Birula and I.. Bialynicki-Birula, Phys. Rev. A16 (1977) 2038. [71 Z. Bialynicka-Birula, I. Bialynicki-Birula, J.H. Eberly and B.W. Shore, Phys. Rev. Al6 (1977) 2048. I81 R. Salomaa, J. Phys. 810 (1977) 3005. PI G.S. Agarwal, Phys. Rev. Al8 (1978) 1490. [lOI C. Mavroyannis, Mol. Phys. 37 (1979) 1175. [Ill G.S. Agarwal and P.A. Narayana, Optics Comm. 30 (1,979) 364. WI C. Mavroyannfs and M.P; Sharma, Physica 102A (1980) 431. [ 131 C. Mavroyannis, Optics Comm. 29 (1979) 80. [ 141 K.J. Woloschuck, S. Hontzeas and C. Mavroyannis, Can. J. Phys. (1982), in press. [ 151 F.Y. Wu, S. Ezekiel, M. Ducloy and B.R. Mollow, Phys. Rev. Lett. 38 (1977) 1077. [ 161 D.A. Hutchinson, C. Downie and C. Mavroyannis, Op tics Comm. 40 (1982) 391.