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Signal line shapes in four-level double resonance spectroscopy with closely spaced levels
05644539/92 s5.00+0.00 @ 1992 Pergamon Press Ltd
.SpecfrochimicaActa. Vol. 4t3A. No. 11112. pp. 1563-1571. 1992 Printed in Great Britain
Signal line shapes in four-level double resonance spectroscopy with closely spaced levels SWAPANMANDALand PRADIP N. GHOSH* Department of Physics, University of Calcutta, 92, A. P. C. Road, Calcutta 700 009, India Abstract-The interaction of two electromagnetic radiations with a four-level atom with two closely spaced common levels is studied for the cascade system. The spacing between the close levels is assumed to be small so that the radiation frequency can act simultaneously on the two closely spaced resonances. Fifteen Bloch equations are developed for the four-level atom and a computer program is written for their solution in the steady state. Numerical computations for the line shape of the signal having the closely spaced levels at the lower level and a single upper level are reported. The closely spaced levels are the upper levels for the pump transition. When the signal is on-resonance with one of the transitions, a Rabi splitting is observed, but the presence of a nearby transition produces an asymmetry between the Rabi components. The asymmetry depends on the transition moment of the nearby transition and disappears when the other transition is weak. In the case of off-resonance pumping a weak two-photon signal is found in addition to the two signals. The intensity of the two-photon signal depends on the pump radiation frequency.
INTRODUCTION
resonance techniques have found useful applications in high resolution molecular spectroscopy. The basic principle of the method is to saturate one of the allowed transitions by means of a strong radiation, called the pump, and study the effect of saturation on a weak probe transition having one of the levels in common with the pump transition. The line shape theory for a three-level system originally put forward by JAVAN [l] has been extended and used by a number of authors [2-181. A large number of molecular spectroscopic investigations based on double resonance methods have been reported [19-381. In experimental studies pump and probe radiations have been used from radio frequency to the optical and ultraviolet region. When the radiation is in the radiofrequency or the microwave region, the line widths are mainly dominated by pressure broadening. However, in the higher frequency region, Doppler broadening due to finite molecular velocity plays an important role in the line shape. Saturation of the pump transition produces a level population and coherency effect [39], which results in signal enhancement and splitting of the probe transition. Theoretical work on the line shape is based on the solution of the optical Bloch equations. These equations are derived from Liouville’s equations [40] of the density matrix. The Hamiltonian of the system consists of the unperturbed atomic and field Hamiltonian and the interaction of the set of non-degenerate quantized atomic states with the classical electromagnetic radiation containing the two different frequency modes. Both the radiations are assumed to be nearly in resonance with the atomic energy differences. It is usually assumed that each radiation can couple to only one resonance transition, implying that there is no other closely spaced energy level which can be simultaneously coupled by one radiation mode. However, closely spaced energy levels are commonly occurring phenomena in molecular systems. The close spacing of the energy levels may originate from the nuclear quadrupole or internal rotation splittings of the rotational energy levels [41-451. The splittings may be so small that when the radiation is on-resonance with one transition, it is simultaneously off-resonance with the other closely spaced transition. The closely spaced transition may lead to many nonlinear features in the observed spectrum [2,15,18,46-501. Such transitions with either the upper or lower levels forming a closely spaced pair of energy levels have been theoretically studied in a number of papers [2,15,46,51]. SCHLOSSBERG and JAVAN [2]discussed the effect of two incident radiation frequencies on a three-level system having closely spaced upper levels. They analysed the saturation DOUBLE
* Author to whom correspondence should be addressed. SA(A)48:11/12-C
1563
1564
SWAPAN
MANDALandPRADIP N.
GHOSH
behaviour of Doppler-broadened transitions. From a numerical solution of eight optical Bloch equations we [15] obtained the line shape of closely spaced resonances in the presence of a single travelling wave. We [50] have studied the line shape of closely spaced Lamb dips and crossover resonance dip in the presence of a standing wave with a single frequency radiation. These studies revealed small intensity-dependent shifts of the peak frequencies which were earlier observed experimentally [49,52]. Such shifts may also arise from various other effects like the dynamic Stark effect [53] or BLOCH-SIEGERT shift of resonance frequency [54-561. However, the shift arising from non-linear interaction of closely spaced levels [15] was not known earlier. In this work, we attempt to obtain the line shape of a double resonance signal in the presence of a strong pumping radiation incident on a four-level system where two of the energy levels are closely spaced. We shall consider a cascade process where the pumping radiation can saturate two closely spaced transitions with a single lower and two upper levels. The signal radiation probes the closely spaced transitions involving these two levels as lower levels and a single upper level. This work is a part of our general studies to obtain the line shape of double resonance signals involving closely spaced levels. In molecular systems the closely spaced energy levels are very common and both the upper and lower levels may have closely spaced structures.
OPTICAL BLOCH EQUATIONS
We consider a molecular gas as an ensemble of non-degenerate four-level (Fig. 1) quantum systems which are exposed to two plane polarized coherent electromagnetic waves at frequencies S21and Q2 such that the detunings Aw, = 52, - wl, Aw3=B2-w3
Awl = Q1 - w2, and
Awq=Q2-wq
(la)
are small. We also assume that the energy level difference A=(E,-E,)lii
(lb)
is so small that the incident pump radiation Qr can simultaneously be nearly in resonance with the frequencies w1= (Eb- E,)lh and w2= (EC- E,)lh and similarly the signal radiation S& is nearly in resonance with the frequencies w3= (Ed- E,)lh and w4= (Ed- E,)lh; hence, Awq= Aw, + Aw3 - Aw2. We also define the Rabi frequencies as Xl =
w47bI~,
x3 =
E~~&I
x2 =
and
WU,Jk
x4 = ~~~&r,
Fig. 1. Energy level diagram of a four-level atom interacting with two fields.
(2)
Signal line shapes in four-level double resonance spectroscopy
1565
where E, and c2 are the electric field amplitudes of the two fields. The total Hamiltonian of the system is H = H,, - 2t(c, cos S&t - 2p2 cos 8,t.
(3)
If p is the density matrix of the four-level atomic system in the interaction representation, then the Liouville equations of density matrix can be written in the rotating wave approximation (RWA) as [l, 12,131 a ih at pa = -cI(,kt@b. +kf&
a at
ih - pbb=
-
a iii atpnb
=
(4)
-p&o)
-E2(~bdpdb-pbdhb)
-hbopob-pbokb)
a ifI atpcc= -&hp~c-pc&m) a ifiat pdd =
hbpb,
-E2(kdp&-pcdk)
(5) (6)
-&2hdbpbd+kcpcd-pdbpbd-pdchd)
(7)
-AAo,~,-&,(~~pbb+~~=p,b-p,~~)+&2p,~db
(8)
a
ih at pm = hAo2pm-&l(ll,bpbc+~~epcc_p~~~oe)+&2p~~&
ih
t
ih it
-t
Lhg)pod
(9)
pod
=
h(Aw,
pk
=
h(Aw2 -Ao,)P,-&l(~bP.c-Pbo~~)-&2(1LMPdc-PbdCLdc)
-EI(&bpbd+hcpcd)
+E2(pob~bd+p&d)
00)
(11)
a ihdtPbd=hA03pbd-&l~bgad-
E2t~bdpdd-pbb~bd-pb&d)
(12)
a ih
at
pcd
=
hAm&d
-
EZhdpdd-
pcbpbd
-
p&d)
-
El p&d.
(13)
For using RWA we have assumed that &J2- 9, is large. The complex polarizations associated with the pump transitions a-,6, a+c and the signal transitions b+d, c-d are (14) (15) &
+ ip3i=
N,bdPdb
06) (17)
N is the number of molecules per unit volume. The population differences are defined as AN1 = N( Pbb-
Pm )
(18)
W=W~cc-~ao)
(19)
A&=N(Psid-Pbbh
(20)
and
SWAPAN MANDAL and PRADIP N. GHOSH
1566
Following our earlier work [12,15] we define the non-linear terms P,, + iP,i =
N/U&b
(21)
Pbc
and
These terms take account of the time dependence of pbc and pd. It may be noted that ,&=O and p &= 0, hence there cannot be any transitions b+c and a+d, but the time developments of P,,b, pa0 pM and pcdwill lead to time rate of change of pbe and P,,~[Eqns (10) and (ll)]. Using the definitions of Eqns (14)-(22) in the density matrix equations we obtain the optical Bloch equations for the 15 variables after adding the phenomenological relaxation constants:
(23)
iP,i=-AW,P,r-;
Ilr.,12~~~+~p.,-~p,._~
(24)
2
(25)
(27)
(28)
(29)
-~lr~d12(AN,+AN~-AN2)-~~~P~,-~ (30) co
t(ANl)=;
ab
2&, 2E* (AN - ANm) P,i+h Pzi-7 P3iT 1
2
(31)
(AK - AN,, ) T,
(32)
Signal line shapes in four-level double resonance spectroscopy
1567
(33)
(36)
‘fl
~db~b&ac
+-
EZrkd~db,%
P2r_
$i,
p4,-h
CL&
h
Pu,
2
(37)
where T, and T2 are the longitudinal and transverse relaxation constants such that the population differences ANi relax to their equilibrium values AN, and the polarizations relax to their equilibrium zero values. We have assumed all the transverse relaxation constants to be equal to T2. We have also assumed that non-linear terms like P,, P,i, P,,,, and Pd relax with the same constant T2 as the polarization. Similarly all the longitudinal relaxation constants are assumed to be equal to Tl. COMPUTATION AND RESULTS We have developed a computer program for solving Eqns (23)-(37) at the steady state. In this case all the time derivatives on the left hand sides of Eqns (23)-(37) are set to zero. The 15 simultaneous equations are written in terms of the polarizations and the redefined parameterspdAN1, /LANK, ,UdbANj,P,,,ipd, Ptil,ud,P,,&& and P,,&&., all of which have the dimension of polarization. The program requires as input the relaxation constants l/T,, l/T,, the Rabi frequencies, the ratio of the electric field amplitudes Q/E*, the frequency gap A and the resonance frequencies w1 and w3. The computed value of Pi=Pz+P4j
(38)
corresponding to the signal polarization is plotted against the frequency detuning given by the deviation of the signal frequency Q2 from (w3 + 04)/2. Hence the resonant signals at o3 and o4 are expected at Al2 and -A/2 respectively, since A = w3- 04. In Fig. 2a-f we have assumed l/T, = l/T, = 0.01 MHz and the pump Rabi frequencies are assumed to be 100 times larger than the signal Rabi frequencies, hence we also assume that E,/c~= 100; all the transition dipoles are equal in magnitude. In Fig. 2a the pump frequency 8, is held fixed at w1 so that it is on-resonance for the signal at o3 and off-resonance for the signal at w4. Hence the computed spectrum shows Rabi splitting for the transition 6+d; the off-resonantly pumped c -,d transition causes an asymmetry in the splitting of the Rabi components, so that the split component closer to o4 has lower intensity. A very similar effect is also observed in Fig. 2e, where Q, is held fixed at o2 so
1568
SWAPANMANDALand PRADIPN. GHOSH
-0
I
* Iyyr f
-O.:mz
Detuning
.I
I
-0.2
I
0.2
De&g%lHr,
Fig. 2. Signal Polarization Pi vs detuning !&- (w,+ 412 for A = 0.15, T-’ = 0.01, x1 =x2 = 0.01 and x3=x., = 0.0001 MHz; o, = 9999.925 and wj = 15,000.0 MHz, E,/Q = 100 and Awl = 0.0 (a), Ao,=O.O5 (b), Ao,=O.O75 (c), Ao,=O.l (d), Ao,=O.15 (Ao2=0.0) (e), and Ao,=0.2MHz (f). In a and e dotted curves correspond toxI=x2- -0.015 MHz and ells,= 150. The arrows in b, d and f indicate two-photon resonances.
that the signal c+d has on-resonance pumping. In both the Fig. 2a and e we have also computed the signal polarization for higher pump power, where E,/Q = 150. These curves are shown by dotted lines. They exhibit larger splitting and larger asymmetry in the Rabi components. Figure 2b and d shows the line shapes when the pump frequency is held fixed between w1 and w2. In both these cases, in addition to the one-photon signals at wj and w.,, we observe a small two-photon signal which satisfies the resonance condition 8, + !& = (Ed- E,)lh. These two-photon signals are very small and almost disappear when the pump frequency is held fixed (Fig. 2c) at 51, = (or + w&2. When the pump frequency is beyond this gap, i.e. larger than w2 (Fig. 2f), a relatively stronger two-photon signal appears. In this case the transition at w4 appears slightly weaker than the transition at w3, although in the former case the pump frequency is closer to resonance. When the pump frequency is on-resonance for one of the signals (Fig 2a, e), the two-photon signal coincides with the one-photon signal and hence cannot be seen separately. In Fig. 3 we assume a strong pump as before, but the Rabi frequency for the signal at w3 is five times larger than that at w4. Since both are acted on by the same field .r2, the difference in the Rabi frequencies arises from the difference of their transition moments. In the on-resonant pumping case (S2, = wl), the signal at w3 shows the symmetrical Rabi splittings (Fig. 3a), in contrast to Fig. 2a where the presence of a nearby strong transition affects the Rabi splitting. When Q,= (or+ w2)/2, a weak two-photon signal appears midway between the strong w3 and weak w4 (Fig. 3b). This may be compared with Fig. 2c, where the two-photon signal is not observed for the same pump frequency when the two signals have identical intensities. In Fig. 3c, Q, = w2 but the Rabi splitting cannot be
v
Signal line shapes in four-level double resonance spectroscopy 0
1569
a
-0.2
2
-"JglZ2
Detuning 1MHz)
5
2
0
e c 0 _s
-0.1
.-s s .N
-0.2
i z z .b VI
,
b
-03 u -02
rv 0.2
Detunin\ (MHz)
0
-0 I
-02
C
-““*~
02
Detuning (MHz)
Fig. 3. Signal polarization P, vs detuning Q2- (o,+ 0412 for A =0.15, T-‘=O.Ol, x1 =x2 =O.Ol, x,=0.0001, x~=0.00002, 0,=9999.!925, oj= 15000.0 MHz and E,/.Q= 100 for Ao, =O.O (a), Am, = 0.075 (b), and AoI = 0.15 MHz (c). The arrow in b indicates a two-photon resonance.
observed because the signal itself is very weak. A very similar effect has been observed by us when the line shape is simulated with the transition w4, much stronger than the same at w3. These cases are not presented. In Fig. 4 we present signal line shapes with a larger energy gap, A = 0.3 MHz. The curves are drawn for on-resonance pumping at Ql = o1 (Fig. 4a) and at 8r = w2 (Fig. 4b). The dotted curves show the same for higher pump power. In these cases the asymmetry of Rabi splitting is much smaller compared to those in Fig. 2a and e. Hence the closer energy levels have a stronger non-linear effect.
CONCLUSION
The signal line shape simulation in the presence of a strong pumping radiation with a common closely spaced level shows some important non-linear characteristics. When the pump frequency coincides with one of the resonance frequencies, the signal transition, which has on-resonance pumping, shows Rabi splitting, but the presence of a nearby strong transition, which is simultaneously pumped off-resonance, causes an asymmetry in intensity of the Rabi components and the component nearer the other signal loses intensity. This asymmetry is larger for larger pump intensity. For a larger energy gap the asymmetry of Rabi splitting decreases. When the pump frequency is off-resonance with both the transitions, a weak two-photon signal appears. This signal becomes weaker when the pumping frequency is intermediate between the resonance frequencies. The asymmetry of Rabi components decreases when the nearby off-resonantly pumped transition is weaker. The asymmetry of Rabi components caused by the presence of closely spaced resonances as shown in this work has not been observed before.
SWAPAN MANDAL~~~PRADIPN.
1570
0.4
-0.2
$6
GHOSH
0'2
014
Detuning (MHz)
-0.4
-0.2
0
0.2
0.4
Detuning (MHz)
Fig. 4. Signal polarization Pi vs detuning Qr- (oj+w,)/2 for A=0.3, T-’ =O.Ol, xj=xq= 0.0001, w, = 9999.925 and cus= 15,000.0 MHz for Aw,= 0.0 (a) and Aw, = 0.3 (Aw, = 0.0) MHz (b). For solid linesx,=xz=O.O1 MHz and elIeZ= 100 and for dotted linesx, =x,=0.015 MHz and &,I&> = 150.
These studies have been carried out for the case with one pair of closely spaced levels and they show some noticeable modifications in the line shape. A careful measurement of the microwave-microwave double resonance should exhibit some of these features. It may be noted here that we have obtained numerical solutions of the Bloch equations. Hence no approximation is involved in solving these equations. In the derivation of the Bloch equations we have only used RWA, but its effect would be much smaller in magnitude in this frequency region [54-561. In double resonance experiments we quite often have the case of closely spaced levels. In these experiments with two incident radiations one can have closely spaced levels not only for the intermediate case but also for the initial and final levels of the resonances. Such a case would be much more complicated for theoretical simulation. We plan to extend the program for the case of such a six-level configuration in the presence of two radiations. We have attempted here numerical solution of 15 simultaneous optical Bloch equations for a four-level system. In the general case of N energy levels one has to solve N* - 1 simultaneous linear equations. Acknowledgement-One Research Fellowship.
of us (S.M.) thanks the University Grants Commission for the award of a Senior
REFERENCES
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Signal line shapes in four-level double resonance spectroscopy
1571
[6] T. W. Hlnsch and P. Toschek, Z. Phys. 236, 213 (1970). [7] G. L. Tang and H. Satz, Phys. Rev. 128, 1013 (1962). (81 M. Takami, Jun. J. Appf. Phys. 15, 1063 (1976). [9] M. Takami, Jap. 1. Appl. Phys. 15, 1889 (1976). [lo] J. C. McGurk, T. G. Schmalz and W. H. Flygare, Ado. Chem. Phys. 25, 1 (1974). [ll] C. Feuillade and P. Berman, Phys. Rev. A29, 1236 (1984). [12] S. Ghoshal and P. N. Ghosh, J. Chem. Phys. 83, 4015 (1985). [13] S. Ghoshal and P. N. Ghosh, Opt. Commun. 73, 455 (1989). [14] P. N. Ghosh, Chem. Phys. Left. 127, 157 (1986). [15] P. N. Ghosh and S. Mandal, Chem. Phys. Lett. 164, 279 (1989). 116) R. Bavli, D. F. Heller and Y. B. Bend, J. Chem. Phys. 91, 6714 (1989). [17] V. P. Chebotayev, High Resolution Laser Specrroscopy (edited by K. Shimmoda), p. 201. John Wiley, New York (1976). 1181 S. Haroche and F. Hartmann, Phys. Rev. A6, 1280 (1972). [19] A. P. Cox, G. W. Flynn and E. B. Wilson, J. Chem. Phys. 42, 3094 (1965). (201 J. Ekkers, A. Bauder and Hs. H. Gunthard, _r. Phys. Es, 819 (1970). [21] R. H. Schwendeman, Ann. Rev. Phys. Chem. 29, 537 (1978). [22] W. H. Weber and R. W. Terhune, J. Chem. Phys. 78, 6437 (1983). [23] W. H. Weber, Chem. Phys. Let?. 122, 469 (1985). [24] P. Glorleux, E. Arimondo and T. Oka, J. Phys. Chem. 87,2133 (1983). [25] F. Scapinni, W. A. Kreiner, J. M. Frye and T. Oka, J. Molec. Specrrosc. 106, 436 (1984). (261 T. Oka, Adv. Atom. Mofec. Phys. 9, 127 (1973). [27] E. Arimondo, P. Glorieux and T. Oka, Phys. Rev. A17, 1375 (1978). [28] T. Shimizu and T. Oka, Phys. Rev. A2, 1177 (1970). [29] S. M. Freund, J. W. C. Johns, A. R. W. McKellar and T. Oka, J. Chem. Phys. 59, 3445 (1973). [30] J. Orr and T. Oka, Appl. Phys. 21,293 (1980). [31] M. Dubs, D. Harradine, E. Schweitzer and J. I. Steinfeld, J. Chem. Phys. 77, 3824 (1982). [32] C. Reiser, J. I. Steinfeld and H. W. Galbraith, J. Chem. Phys. 74, 2189 (1981). [33] C. C. Jensen, T. G. Anderson, C. Reiser and J. I. Steinfeld, J. Chem. Phys. 71, 3648 (1979). [34] H. W. Galbraith, M. Dubs and J. Steinfeld, Phys. Rev. A26, 1528 (1982). [35] J. I. Steinfeld and C. C. Jenssen, in Tunable Lasers and Applications (edited by A. Mooradian, T. Jaeger and P. Stokseth), p. 190. Springer, Berlin (1976). [36] G. Millet, J. Hetzler, G. Pierre and J. Steinfeld, Spectrochim. Actu 45A, 5 (1989). [37] R. L. Shoemaker, Laser and Coherence Spectroscopy (edited by J. I. Steinfeld), p. 197. Plenum, New York (1978). [38] T. G. Schmalz and W. H. Flygare, in Laserand Coherence Spectroscopy (edited by J. I. Steinfeld), p. 125. Plenum, New York (1978). [39] J. G. Baker in Modern Aspecfs of Microwave Spectroscopy (edited by G. W. Chantry), p. 65. Academic Press, New York (1979). [40] S. Stenholm, Foundations of Laser Spectroscopy. John Wiley, New York (1983). [41] W. Gordy and R. L. Cooke, Microwave Molecular Spectra, 2nd edn. John Wiley, New York (1985). [42] C. H. Townes and A. L. Schawlow, Microwave Spectroscopy. McGraw-Hill, New York (1955). [43] P. N. Ghosh, A. Bauder and Hs. H. Gunthard, J. Molec. Spectrosc. 53, 39 (1980). [44] H. U. Wenger, A. Bauder and Hs. H. Gunthard, Chem. Phys. Left. 1,441 (1973). [45] M. Ribeaud, A. Bauder and Hs. H. Gunthard, J. Molec. Spectrosc. 42,441 (1972). [46] T. Tsai and T. F. Morse, IEEE J. Quun. Electron. QE-15, 1335(1979). 1471V. S. Letokhov and V. P. Chebotayev, Nonlinear Laser Spectroscopy. Springer-Verlag, Berlin (1976). [48] J. L. Hall and C. Borde, Phys. Reu. Letf. 30, 1101 (1973). 1491 T. W. HBnsch, M. H. Nayfeh, S. A. Lee, S. M. Curry and I. S. Shahin, Phys. Reu. Lea. 32, 1336 (1974). (50) S. Mandal and P. N. Ghosh, Phys. Reu. A45, 4990 (1992). [51] D. A. Cardimona, M. G. Raymer and C. R. Stroud, .I. Phys. BlJ, 55 (1982). [52] E. V. Baklanov and V. P. Chebotayev, Soo. Phys. JEEP 33,300 (1971). (531 S. H. Autler and C. H. Townes, Phys. Rev. 100, 703 (1955). [54] F. Bloch and A. J. F. Siegert, Phys. Reu. 57, 522 (1940). [55] L. Allen and J. H. Eberly, Optical Resonance and Two Level Atoms. John Wiley, New York (1975). [56] S. Mandal and P. N. Ghosh, Chem. Phys. Letr. 193, 185 (1992).
.SpecfrochimicaActa. Vol. 4t3A. No. 11112. pp. 1563-1571. 1992 Printed in Great Britain
Signal line shapes in four-level double resonance spectroscopy with closely spaced levels SWAPANMANDALand PRADIP N. GHOSH* Department of Physics, University of Calcutta, 92, A. P. C. Road, Calcutta 700 009, India Abstract-The interaction of two electromagnetic radiations with a four-level atom with two closely spaced common levels is studied for the cascade system. The spacing between the close levels is assumed to be small so that the radiation frequency can act simultaneously on the two closely spaced resonances. Fifteen Bloch equations are developed for the four-level atom and a computer program is written for their solution in the steady state. Numerical computations for the line shape of the signal having the closely spaced levels at the lower level and a single upper level are reported. The closely spaced levels are the upper levels for the pump transition. When the signal is on-resonance with one of the transitions, a Rabi splitting is observed, but the presence of a nearby transition produces an asymmetry between the Rabi components. The asymmetry depends on the transition moment of the nearby transition and disappears when the other transition is weak. In the case of off-resonance pumping a weak two-photon signal is found in addition to the two signals. The intensity of the two-photon signal depends on the pump radiation frequency.
INTRODUCTION
resonance techniques have found useful applications in high resolution molecular spectroscopy. The basic principle of the method is to saturate one of the allowed transitions by means of a strong radiation, called the pump, and study the effect of saturation on a weak probe transition having one of the levels in common with the pump transition. The line shape theory for a three-level system originally put forward by JAVAN [l] has been extended and used by a number of authors [2-181. A large number of molecular spectroscopic investigations based on double resonance methods have been reported [19-381. In experimental studies pump and probe radiations have been used from radio frequency to the optical and ultraviolet region. When the radiation is in the radiofrequency or the microwave region, the line widths are mainly dominated by pressure broadening. However, in the higher frequency region, Doppler broadening due to finite molecular velocity plays an important role in the line shape. Saturation of the pump transition produces a level population and coherency effect [39], which results in signal enhancement and splitting of the probe transition. Theoretical work on the line shape is based on the solution of the optical Bloch equations. These equations are derived from Liouville’s equations [40] of the density matrix. The Hamiltonian of the system consists of the unperturbed atomic and field Hamiltonian and the interaction of the set of non-degenerate quantized atomic states with the classical electromagnetic radiation containing the two different frequency modes. Both the radiations are assumed to be nearly in resonance with the atomic energy differences. It is usually assumed that each radiation can couple to only one resonance transition, implying that there is no other closely spaced energy level which can be simultaneously coupled by one radiation mode. However, closely spaced energy levels are commonly occurring phenomena in molecular systems. The close spacing of the energy levels may originate from the nuclear quadrupole or internal rotation splittings of the rotational energy levels [41-451. The splittings may be so small that when the radiation is on-resonance with one transition, it is simultaneously off-resonance with the other closely spaced transition. The closely spaced transition may lead to many nonlinear features in the observed spectrum [2,15,18,46-501. Such transitions with either the upper or lower levels forming a closely spaced pair of energy levels have been theoretically studied in a number of papers [2,15,46,51]. SCHLOSSBERG and JAVAN [2]discussed the effect of two incident radiation frequencies on a three-level system having closely spaced upper levels. They analysed the saturation DOUBLE
* Author to whom correspondence should be addressed. SA(A)48:11/12-C
1563
1564
SWAPAN
MANDALandPRADIP N.
GHOSH
behaviour of Doppler-broadened transitions. From a numerical solution of eight optical Bloch equations we [15] obtained the line shape of closely spaced resonances in the presence of a single travelling wave. We [50] have studied the line shape of closely spaced Lamb dips and crossover resonance dip in the presence of a standing wave with a single frequency radiation. These studies revealed small intensity-dependent shifts of the peak frequencies which were earlier observed experimentally [49,52]. Such shifts may also arise from various other effects like the dynamic Stark effect [53] or BLOCH-SIEGERT shift of resonance frequency [54-561. However, the shift arising from non-linear interaction of closely spaced levels [15] was not known earlier. In this work, we attempt to obtain the line shape of a double resonance signal in the presence of a strong pumping radiation incident on a four-level system where two of the energy levels are closely spaced. We shall consider a cascade process where the pumping radiation can saturate two closely spaced transitions with a single lower and two upper levels. The signal radiation probes the closely spaced transitions involving these two levels as lower levels and a single upper level. This work is a part of our general studies to obtain the line shape of double resonance signals involving closely spaced levels. In molecular systems the closely spaced energy levels are very common and both the upper and lower levels may have closely spaced structures.
OPTICAL BLOCH EQUATIONS
We consider a molecular gas as an ensemble of non-degenerate four-level (Fig. 1) quantum systems which are exposed to two plane polarized coherent electromagnetic waves at frequencies S21and Q2 such that the detunings Aw, = 52, - wl, Aw3=B2-w3
Awl = Q1 - w2, and
Awq=Q2-wq
(la)
are small. We also assume that the energy level difference A=(E,-E,)lii
(lb)
is so small that the incident pump radiation Qr can simultaneously be nearly in resonance with the frequencies w1= (Eb- E,)lh and w2= (EC- E,)lh and similarly the signal radiation S& is nearly in resonance with the frequencies w3= (Ed- E,)lh and w4= (Ed- E,)lh; hence, Awq= Aw, + Aw3 - Aw2. We also define the Rabi frequencies as Xl =
w47bI~,
x3 =
E~~&I
x2 =
and
WU,Jk
x4 = ~~~&r,
Fig. 1. Energy level diagram of a four-level atom interacting with two fields.
(2)
Signal line shapes in four-level double resonance spectroscopy
1565
where E, and c2 are the electric field amplitudes of the two fields. The total Hamiltonian of the system is H = H,, - 2t(c, cos S&t - 2p2 cos 8,t.
(3)
If p is the density matrix of the four-level atomic system in the interaction representation, then the Liouville equations of density matrix can be written in the rotating wave approximation (RWA) as [l, 12,131 a ih at pa = -cI(,kt@b. +kf&
a at
ih - pbb=
-
a iii atpnb
=
(4)
-p&o)
-E2(~bdpdb-pbdhb)
-hbopob-pbokb)
a ifI atpcc= -&hp~c-pc&m) a ifiat pdd =
hbpb,
-E2(kdp&-pcdk)
(5) (6)
-&2hdbpbd+kcpcd-pdbpbd-pdchd)
(7)
-AAo,~,-&,(~~pbb+~~=p,b-p,~~)+&2p,~db
(8)
a
ih at pm = hAo2pm-&l(ll,bpbc+~~epcc_p~~~oe)+&2p~~&
ih
t
ih it
-t
Lhg)pod
(9)
pod
=
h(Aw,
pk
=
h(Aw2 -Ao,)P,-&l(~bP.c-Pbo~~)-&2(1LMPdc-PbdCLdc)
-EI(&bpbd+hcpcd)
+E2(pob~bd+p&d)
00)
(11)
a ihdtPbd=hA03pbd-&l~bgad-
E2t~bdpdd-pbb~bd-pb&d)
(12)
a ih
at
pcd
=
hAm&d
-
EZhdpdd-
pcbpbd
-
p&d)
-
El p&d.
(13)
For using RWA we have assumed that &J2- 9, is large. The complex polarizations associated with the pump transitions a-,6, a+c and the signal transitions b+d, c-d are (14) (15) &
+ ip3i=
N,bdPdb
06) (17)
N is the number of molecules per unit volume. The population differences are defined as AN1 = N( Pbb-
Pm )
(18)
W=W~cc-~ao)
(19)
A&=N(Psid-Pbbh
(20)
and
SWAPAN MANDAL and PRADIP N. GHOSH
1566
Following our earlier work [12,15] we define the non-linear terms P,, + iP,i =
N/U&b
(21)
Pbc
and
These terms take account of the time dependence of pbc and pd. It may be noted that ,&=O and p &= 0, hence there cannot be any transitions b+c and a+d, but the time developments of P,,b, pa0 pM and pcdwill lead to time rate of change of pbe and P,,~[Eqns (10) and (ll)]. Using the definitions of Eqns (14)-(22) in the density matrix equations we obtain the optical Bloch equations for the 15 variables after adding the phenomenological relaxation constants:
(23)
iP,i=-AW,P,r-;
Ilr.,12~~~+~p.,-~p,._~
(24)
2
(25)
(27)
(28)
(29)
-~lr~d12(AN,+AN~-AN2)-~~~P~,-~ (30) co
t(ANl)=;
ab
2&, 2E* (AN - ANm) P,i+h Pzi-7 P3iT 1
2
(31)
(AK - AN,, ) T,
(32)
Signal line shapes in four-level double resonance spectroscopy
1567
(33)
(36)
‘fl
~db~b&ac
+-
EZrkd~db,%
P2r_
$i,
p4,-h
CL&
h
Pu,
2
(37)
where T, and T2 are the longitudinal and transverse relaxation constants such that the population differences ANi relax to their equilibrium values AN, and the polarizations relax to their equilibrium zero values. We have assumed all the transverse relaxation constants to be equal to T2. We have also assumed that non-linear terms like P,, P,i, P,,,, and Pd relax with the same constant T2 as the polarization. Similarly all the longitudinal relaxation constants are assumed to be equal to Tl. COMPUTATION AND RESULTS We have developed a computer program for solving Eqns (23)-(37) at the steady state. In this case all the time derivatives on the left hand sides of Eqns (23)-(37) are set to zero. The 15 simultaneous equations are written in terms of the polarizations and the redefined parameterspdAN1, /LANK, ,UdbANj,P,,,ipd, Ptil,ud,P,,&& and P,,&&., all of which have the dimension of polarization. The program requires as input the relaxation constants l/T,, l/T,, the Rabi frequencies, the ratio of the electric field amplitudes Q/E*, the frequency gap A and the resonance frequencies w1 and w3. The computed value of Pi=Pz+P4j
(38)
corresponding to the signal polarization is plotted against the frequency detuning given by the deviation of the signal frequency Q2 from (w3 + 04)/2. Hence the resonant signals at o3 and o4 are expected at Al2 and -A/2 respectively, since A = w3- 04. In Fig. 2a-f we have assumed l/T, = l/T, = 0.01 MHz and the pump Rabi frequencies are assumed to be 100 times larger than the signal Rabi frequencies, hence we also assume that E,/c~= 100; all the transition dipoles are equal in magnitude. In Fig. 2a the pump frequency 8, is held fixed at w1 so that it is on-resonance for the signal at o3 and off-resonance for the signal at w4. Hence the computed spectrum shows Rabi splitting for the transition 6+d; the off-resonantly pumped c -,d transition causes an asymmetry in the splitting of the Rabi components, so that the split component closer to o4 has lower intensity. A very similar effect is also observed in Fig. 2e, where Q, is held fixed at o2 so
1568
SWAPANMANDALand PRADIPN. GHOSH
-0
I
* Iyyr f
-O.:mz
Detuning
.I
I
-0.2
I
0.2
De&g%lHr,
Fig. 2. Signal Polarization Pi vs detuning !&- (w,+ 412 for A = 0.15, T-’ = 0.01, x1 =x2 = 0.01 and x3=x., = 0.0001 MHz; o, = 9999.925 and wj = 15,000.0 MHz, E,/Q = 100 and Awl = 0.0 (a), Ao,=O.O5 (b), Ao,=O.O75 (c), Ao,=O.l (d), Ao,=O.15 (Ao2=0.0) (e), and Ao,=0.2MHz (f). In a and e dotted curves correspond toxI=x2- -0.015 MHz and ells,= 150. The arrows in b, d and f indicate two-photon resonances.
that the signal c+d has on-resonance pumping. In both the Fig. 2a and e we have also computed the signal polarization for higher pump power, where E,/Q = 150. These curves are shown by dotted lines. They exhibit larger splitting and larger asymmetry in the Rabi components. Figure 2b and d shows the line shapes when the pump frequency is held fixed between w1 and w2. In both these cases, in addition to the one-photon signals at wj and w.,, we observe a small two-photon signal which satisfies the resonance condition 8, + !& = (Ed- E,)lh. These two-photon signals are very small and almost disappear when the pump frequency is held fixed (Fig. 2c) at 51, = (or + w&2. When the pump frequency is beyond this gap, i.e. larger than w2 (Fig. 2f), a relatively stronger two-photon signal appears. In this case the transition at w4 appears slightly weaker than the transition at w3, although in the former case the pump frequency is closer to resonance. When the pump frequency is on-resonance for one of the signals (Fig 2a, e), the two-photon signal coincides with the one-photon signal and hence cannot be seen separately. In Fig. 3 we assume a strong pump as before, but the Rabi frequency for the signal at w3 is five times larger than that at w4. Since both are acted on by the same field .r2, the difference in the Rabi frequencies arises from the difference of their transition moments. In the on-resonant pumping case (S2, = wl), the signal at w3 shows the symmetrical Rabi splittings (Fig. 3a), in contrast to Fig. 2a where the presence of a nearby strong transition affects the Rabi splitting. When Q,= (or+ w2)/2, a weak two-photon signal appears midway between the strong w3 and weak w4 (Fig. 3b). This may be compared with Fig. 2c, where the two-photon signal is not observed for the same pump frequency when the two signals have identical intensities. In Fig. 3c, Q, = w2 but the Rabi splitting cannot be
v
Signal line shapes in four-level double resonance spectroscopy 0
1569
a
-0.2
2
-"JglZ2
Detuning 1MHz)
5
2
0
e c 0 _s
-0.1
.-s s .N
-0.2
i z z .b VI
,
b
-03 u -02
rv 0.2
Detunin\ (MHz)
0
-0 I
-02
C
-““*~
02
Detuning (MHz)
Fig. 3. Signal polarization P, vs detuning Q2- (o,+ 0412 for A =0.15, T-‘=O.Ol, x1 =x2 =O.Ol, x,=0.0001, x~=0.00002, 0,=9999.!925, oj= 15000.0 MHz and E,/.Q= 100 for Ao, =O.O (a), Am, = 0.075 (b), and AoI = 0.15 MHz (c). The arrow in b indicates a two-photon resonance.
observed because the signal itself is very weak. A very similar effect has been observed by us when the line shape is simulated with the transition w4, much stronger than the same at w3. These cases are not presented. In Fig. 4 we present signal line shapes with a larger energy gap, A = 0.3 MHz. The curves are drawn for on-resonance pumping at Ql = o1 (Fig. 4a) and at 8r = w2 (Fig. 4b). The dotted curves show the same for higher pump power. In these cases the asymmetry of Rabi splitting is much smaller compared to those in Fig. 2a and e. Hence the closer energy levels have a stronger non-linear effect.
CONCLUSION
The signal line shape simulation in the presence of a strong pumping radiation with a common closely spaced level shows some important non-linear characteristics. When the pump frequency coincides with one of the resonance frequencies, the signal transition, which has on-resonance pumping, shows Rabi splitting, but the presence of a nearby strong transition, which is simultaneously pumped off-resonance, causes an asymmetry in intensity of the Rabi components and the component nearer the other signal loses intensity. This asymmetry is larger for larger pump intensity. For a larger energy gap the asymmetry of Rabi splitting decreases. When the pump frequency is off-resonance with both the transitions, a weak two-photon signal appears. This signal becomes weaker when the pumping frequency is intermediate between the resonance frequencies. The asymmetry of Rabi components decreases when the nearby off-resonantly pumped transition is weaker. The asymmetry of Rabi components caused by the presence of closely spaced resonances as shown in this work has not been observed before.
SWAPAN MANDAL~~~PRADIPN.
1570
0.4
-0.2
$6
GHOSH
0'2
014
Detuning (MHz)
-0.4
-0.2
0
0.2
0.4
Detuning (MHz)
Fig. 4. Signal polarization Pi vs detuning Qr- (oj+w,)/2 for A=0.3, T-’ =O.Ol, xj=xq= 0.0001, w, = 9999.925 and cus= 15,000.0 MHz for Aw,= 0.0 (a) and Aw, = 0.3 (Aw, = 0.0) MHz (b). For solid linesx,=xz=O.O1 MHz and elIeZ= 100 and for dotted linesx, =x,=0.015 MHz and &,I&> = 150.
These studies have been carried out for the case with one pair of closely spaced levels and they show some noticeable modifications in the line shape. A careful measurement of the microwave-microwave double resonance should exhibit some of these features. It may be noted here that we have obtained numerical solutions of the Bloch equations. Hence no approximation is involved in solving these equations. In the derivation of the Bloch equations we have only used RWA, but its effect would be much smaller in magnitude in this frequency region [54-561. In double resonance experiments we quite often have the case of closely spaced levels. In these experiments with two incident radiations one can have closely spaced levels not only for the intermediate case but also for the initial and final levels of the resonances. Such a case would be much more complicated for theoretical simulation. We plan to extend the program for the case of such a six-level configuration in the presence of two radiations. We have attempted here numerical solution of 15 simultaneous optical Bloch equations for a four-level system. In the general case of N energy levels one has to solve N* - 1 simultaneous linear equations. Acknowledgement-One Research Fellowship.
of us (S.M.) thanks the University Grants Commission for the award of a Senior
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