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Remarks on E1-E2 and E1-M1 two-photon transitions
G. Grynberg
To cite this version:
G. Grynberg. Remarks on E1-E2 and E1-M1 two-photon transitions. Journal de Physique, 1983, 44 (6), pp.679-682. �10.1051/jphys:01983004406067900�. �jpa-00209646�
Remarks on E1-E2 and E1-M1 two-photon transitions
G. Grynberg
Laboratoire de Spectroscopie Hertzienne de l’E.N.S., Université P. et M. Curie, 75230 Paris Cedex 05, France
(Reçu le 23 décembre 1982, accepté le 4 fevrier 1983 )
Résumé. 2014 On étudie la possibilité d’observer des transitions à deux photons sans effet Doppler dans le cas où une
des transitions est dipolaire électrique et l’autre dipolaire magnétique ou quadrupolaire électrique. On montre que l’on n’obtient pas la suppression de l’effet Doppler par simple réflexion du faisceau sur lui-même. En revanche, un
tel effet pourrait être observé si les deux faisceaux se propageant en sens opposé ont des polarisations différentes.
Abstract 2014 We investigate the possibility to observe Doppler-free two-photon transitions in the case where one of the transitions is an electric dipole transition and the other one is a magnetic dipole or an electric quadrupole
transition. We show that it is not possible to obtain Doppler-free lines by reflecting the beam on itself. On the other
hand, such an effect should be observable if the two counterpropagating beams have different polarizations.
Classification Physics Abstracts
32.80K
1. Introduction.
Doppler-free two-photon spectroscopy [ 1 ] is one of
the most powerful methods of spectroscopy disco- vered during the last fifteen years. The theory of these transitions, in a travelling wave [2] or in a standing
wave [1, 3], has been performed within the electric
dipole approximation, which assumes that the atom
interacts with the field through the - d. E term in an appropriate gauge (d electric dipole moment, E elec- tric field). In fact, the progress in powerful tunable
laser now permits to observe two-photon transitions between levels of different parity. In that case, one of the transitions is an electric dipole transition (E 1 ) while
the other one is either an electric quadrupole (E2) or
a magnetic dipole transition (M 1 ). For example,
W. Gornik et al. [4] have observed an E 1-E2 transition
in Xe. Here we wish to analyse this process and show
up the differences with respect to the usual E 1-E 1 transition. In § 2, we show that there is no possibility
to obtain a Doppler-free line by reflecting a beam on itself, in contrast to the case of the E 1-E 1 transition.
In § 3, we analyse the tensorial properties of the exci- tation operator in a travelling wave. In § 4, we inves- tigate the properties of the operator in two counter-
propagating waves. We show how the polarizations
of the two counterpropagating beams have to be chosen, if a Doppler-free line is to be observed. The
same calculations can also be applied to the case of
a broad band excitation in order to predict the inten- sity ratio between the excitation in a travelling wave
and in two counterpropagating waves of the same frequency.
2. Two-photon transition in a standing wave.
In this section we consider the case when the two
counterpropagating beams have the same linear
polarization. We show that a Doppler-free line can be
observed in the case of an E 1-El transition if the tran- sition is possible in the travelling wave. On the other hand, we show that the same statement is not true
for an E 1-E2 or an El-Ml transition.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01983004406067900
680
2.1 CASE OF AN E 1-E 1 TRANSITION. - In the case
of a two-photon transition between two levels of the same parity, the interaction between the atom
and the field can generally be described by an effective
Hamiltonian [5] of the type 4 QEx2 where Q is the two-
photon atomic operator [3] and Ex is the electric field.
In the case of a standing wave :
There are three terms in Ex2 . The first one does not depend on the space variable and leads to the Doppler-
free line while the two others give the broad Doppler background.
2.2 CASE OF AN E 1-E2 TRANSITION. - In the case of
an E 1-E2 two-photon transition, we can also obtain
an effective Hamiltonian which will generally be of the type AE,,
aE" aZ
where A is an atomic operator. Using (1 ,we obtain for the field :
Since the two terms depend on the space variable,
there is no Doppler-free line in that situation. This result may look surprising in the naive picture where
the atom absorbs two photons propagating in oppo- site directions. In that description, however, we can
obtain the same result by taking into consideration that there are two possible paths to excite the atom.
The first path corresponds to the excitation of an E2 transition by a photon propagating in the + z direc-
tion and to the excitation of an El transition by a photon propagating in the - z direction. The second
path corresponds to the process where the two exci- tations are exchanged. It is easy to show that the two
paths interfere destructively. This is the reason why
there is no Doppler-free line.
2. 3 CASE OF AN E 1-M 1 TRANSITION. - In the case of
an El-Ml transition, the magnetic field is directed
along the y direction :
The effective Hamiltonian is of the form CEx By where
C is a purely atomic operator
Here again, we do not find a term which phase is independent of the position. We do not expect to observe a Doppler-free line in that geometry.
3. Tensorial properties of the atomic operators in a
travelling wave.
We now consider the atomic part of the effective Hamiltonian. The important point, here, is to compare its properties of symmetry with the ones which will be found in § 4 where we will consider the case of two
counterpropagating waves.
3.1 CASE OF AN E 1-E l TRANSITION. - As it has been shown in previous works [ 1 ], it is often useful to find the irreducible components of the two-photon opera- tor. In particular, from this decomposition we can
deduce the selection rules and the line intensities. For
a travelling wave of polarization E, the two-photon operator Q EE is :
This is a symmetric operator and the decomposition only involves operators of rank 0 and 2 [3].
It is often useful to use the standard polarizations :
If one chooses the quantization axis along the beam direction, the polarization is perpendicular to this axis
and can be decomposed into e+ and e _ (with coeffi-
cients a and fl). Using the standard components of Q [1] J we obtain :
3.2 CASE OF AN E 1-E2 TRANSITION. - In the case
of an El-E2 transition, we can find an effective Hamil-
tonian 4 [A. + A’] kE2 with
’
(q is the electric quadrupole moment of the atom).
Since A , c and AE£ have the same properties of symmetry, we can develop the calculations with As only. We take as previously the axis of propagation
eL as the axis of reference. We introduce the standard components of A :
The decomposition of A’ involves the same tensorial orders.
As in the case of El-El transition [1], the decom- position (11) can be applied to find the selection rules and the line intensities. For example, we calculate
the relative efficiency of different polarizations in the
case of a S-F transition in atoms. The intensity in
the polarization state defined by a, fl is proportional
to :
It gives for the ratio of circular polarization (a = 1, fl = 0) over linear polarization (a = - (3 = I/J2)
a value 5/4.
3. 3 CASE OF AN E 1-M 1 TRANSITION. - We proceed
as in the previous sections. The effective Hamiltonian is
now 1
C + Cl3 E2 with :Introducing as previously the standard components
C’ of C, we obtain :
-- -
and a similar decomposition for C£.
4: Case of different polarizations for the counter- propagating beams.
In section 2 we have shown that there is no Doppler-
free resonance for an El-E2 or an El-Ml transition if the two beams have the same polarization. Since
every polarization perpendicular to the axis of propa- gation ez can be decomposed into e+ and e_, it is only
necessary to consider the case where one beam is e+
polarized and the second is e _ polarized.
4 .1 CASE OF AN E 1-E 1 TRANSITION. - In this case the effective Hamiltonian for the Doppler-free line is
equal to § Qe + e - E+ E- with Q:+e- defined as in
[ l, 3] :’
The decomposition of Qe + e _ into standard compo-
nents is :
We recall that this geometry can be particularly
useful since it permits to cancel the Doppler back- ground [6, 7] in some experimental situations. This is related to the fact that the effective Hamiltonian in the travelling wave involves Q’ (or Q’2) while the Doppler-free line corresponds to a coupling by Q) or Qo.
4.2 CASE OF AN El-E2 TRANSITION. - If there is a
Doppler-free line, it will be related to an effective Hamiltonian :
The tensorial decomposition of 2(Ae+e_ - Ae_e+) is :
Since the operator is not identically equal to 0, equation (16) shows that there may be some possibili-
ties to observe new effects in two counterpropagating
waves. However it should be realized that the tensorial component A 0 2is not very favourable. -If we consider
a one-electron atom with a S ground level, the only possible E 1-E2 transitions are S-P or S-F because of
parity. If we take only the orbital angular momentum, these transitions are clearly possible in a travelling
wave (see (11)) but no cross effect is possible in two counterpropagating waves. The only way to observe
a cross effect would be that the detuning from the
intermediate level be small compared to its fine
structure. In that case we should specify the total angular momentum J rather than L [1].
We can thus conclude that for an E1-E2 two-
photon transition, the excitation in two counter-
propagating waves will often be simply the sum of
the excitations in two travelling waves.
4. 3 CASE OF AN E 1-M 1 TRANSITION. - The effective Hamiltonian for the Doppler-free line is :
682
The tensorial decomposition of 2 [Ce + e - - Ce-e+] ] gives :
The situation here looks better than in the previous
section. Since the two-photon transition involves a
magnetic dipole transition, one must specify the
total angular momentum. It is thus theoretically possible to observe a Doppler-free line in the case of
SI/2-PI/2 or SI/2-P 3/2 transition in alkalis. It is also
possible to excite the metastable levels of Hg through
a 6 ’SO-6 3Po or 6 ISo-6 3P2 transition. It can be noticed that the excitation of the 6 3Po level of mercury
is not possible by means of an E 1-M l transition in a
travelling wave (see (13)), but that it is possible in
the case of two counterpropagating waves. Before making any experimental efforts to observe a Doppler-
free line, however, one should calculate the dynamical
Stark effect. This effect may be important for these
transitions since the oscillator strengths of the El
and M 1 transitions can differ by orders of magni-
tude [5].
5. Conclusion.
We have shown that it is not a general rule that the reflexion of a beam on itself leads to a Doppler-free
line in two-photon spectroscopy. This naive result is valid for an El-El transition but is not true for an
E 1-E2 or an El-Ml transition. For the latter transi-
tions, we have shown that it may be possible in some
circumstances to observe an effect in two counterpro- pagating beams provided that the polarizations of the
two beams are different.
Acknowledgments.
I would like to thank Dr. F. Biraben and M. Pinard for helpful conversations about this subject.
References
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[2] GOPPERT-MAYER, M., Ann. Phys. 9 (1931) 173.
[3] CAGNAC, B., GRYNBERG, G. et BIRABEN, F., J. Physique
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[4] GORNIK, W., KINDT, S., MATTHIAS, E., RINNEBERG, H.
et SCHMIDT, D., Phys. Rev. Lett. 45 (1980) 1941.
[5] GRYNBERG, G., CAGNAC, B. et BIRABEN, F., « Coherent Non Linear Optics », in Topics in Current Physics,
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