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Submitted on 1 Jan 1979
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Doppler-free multi-photon excitation : light shift and saturation
G. Grynberg
To cite this version:
G. Grynberg. Doppler-free multi-photon excitation : light shift and saturation. Journal de Physique, 1979, 40 (7), pp.657-664. �10.1051/jphys:01979004007065700�. �jpa-00209149�
Doppler-free multi-photon excitation : light shift and saturation
G. Grynberg
Laboratoire d’Optique quantique du C.N.R.S., Ecole Polytechnique, 91128 Palaiseau Cedex, France
(Reçu le 27 octobre 1978, révisé le 23 février 1979, accepté le 5 mars 1979)
Résumé. - Nous décrivons l’influence des modulations spatiales de l’intensité lumineuse dans une onde sta- tionnaire sur la forme de la raie d’absorption à N-photons. Nous montrons que si l’effet Stark dynamique et la
saturation sont petits comparés à la largeur Doppler, il y a un effet de moyennage par le mouvement : la raie reste Lorentzienne, son centre est déplacé d’une quantité égale à l’effet Stark dynamique moyen. Nous consi- dérons également le cas des transitoires, nous montrons qu’à résonance, on peut observer diverses réponses sui-
vant la valeur de l’intensité. En particulier, on ne peut observer des oscillations de Rabi que si la fréquence de
Rabi est comprise entre les largeurs naturelles et Doppler de la transition.
Abstract. 2014 We describe the influence of the intensity variation of the light in a standing wave on the lineshape
of a Doppler-free multi-photon transition. We show that as long as the light shifts and the saturation are small
compared to the Doppler width, a motional narrowing effect occurs and the line remains Lorentzian, the centre being shifted by an amount equal to the mean value of the dynamic Stark effect. We consider also the case of transients. We show that at resonance, it is possible to observe different responses according to the light intensity.
In particular, it is possible to observe Rabi oscillations only if the Rabi frequency is larger than the natural width and smaller than the Doppler width of the transition.
Classification Physics Abstracts
32.80K - 42.65
1. Introduction. - The cancellation of the Doppler broadening by two-photon absorption in a standing
wave has been observed by many groups [1] since the
first expérimental work of Biraben et al. [2] and
Levenson and Bloembergen [3]. It can be noticed that for most applications, the spatial variation of intensity in the standing wave has no influence on
the experimental observations. We show that this is a consequence of the two following conditions
which are usually fulfilled :
- the pumping time of the atom from the ground
state to the excited state by two-photon absorption
is large compared to the time of flight of the atom
between two nodes of the standing wave,
- the shift of the levels induced by the interaction with the electromagnetic field [4] are small compared
to the Doppler shift of the transition.
If these conditions are fulfilled, there is an average of the intensity variations because the atom moves
fast enough in the standing wave. This corresponds
to a motional narrowing effect [5].
In the theory of Doppler-free two-photon absorp-
tion [6], a singularity exists for the atoms of very small velocity for which the previous conditions are
not fulfilled (see ref. [6], § 2). It is obvious that if the
intensity increases, the number of atoms which do
not comply with the previous conditions also increases.
It is thus interesting to understand the implications
of a high standing wave power on the two-photon absorption lineshape.
Another signal which is very sensitive to these slow atoms is the transient response when the standing
wave is suddenly tumed on. We show that if the
frequency of the wave satisfies the two-photon
resonance condition (Ee - E,, = 2 hm) we can dis- tinguish three different regimes according to the
value of the light intensity :
- For a small value of the intensity (no saturation, light shift smaller than the natural width of the excited state) the fluorescence grows monotonously
from 0 to its limiting value in a time of the order of
the lifetime 1/ r e of the excited state. This has been
observed by Bassini et al. [7].
- If the light intensity is higher, and attains either a saturation or a light shift larger than the
natural width re of the transition but still smaller than the Doppler width, the transient consists of
damped Rabi oscillations. The frequency of the
oscillations is proportional to the intensity and the damping is observed with a time constant corres-
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01979004007065700
658
ponding to the lifetime of the excited state. A similar behaviour has been observed by Loy [8].
- If the light intensity is still higher, and attains
either saturation or a light shift larger than the Doppler width, the Rabi oscillations completely disappear. The origin of the cancellation of the oscillations is that the Rabi frequency depends on
the position of the atom and the velocities of the atoms are not large enough to average the values of the Rabi frequency.
Similar effects also exist for Doppler-free multi- photon excitation. If more than two photons are involved, the light shift appears generally at lower
intensities than does the saturation [6]. The modu-
lation of the light shift in the standing wave only
leads to an average shift of the narrow resonance if its mean value is smaller than the Doppler width.
If the mean value of the shift is larger than the Doppler width, the shape of the n-photon absorption line
becomes more complicated and much broader. An order of magnitude calculation shows that this effect will prevent many applications of Doppler-
free n-photon absorption, if n > 3.
We note also that if the problem of light shift and
saturation has already been studied by many authors in the radio-frequency [9]-[10] and optical ranges [11]- [12], the influence of the intensity variations in the
standing wave has, to our knowledge, never been
discussed. The reason is that the spatial variation of
intensity can be generally neglected in the radio- frequency range because the wavelength is often larger than the dimensions of the experimental cell.
In the optical range, the authors [11]-[12] consider
two fields of different frequencies rot and w2 interacting
with a three-level system (a, b, c) : they assume in
their calculations that each frequency col or cv2 is
close to an atomic frequency roab or robe (for instance 1 Col - Co.b 1 « .1 ro2 - roab 1). Such an assumption is, of course, inadequate in the case of a standing wave (w1 = ro2 = w) and, in this case, it is necessary to present an alternative theory.
The paper is divided into four sections. In section 2,
we find an effective Hamiltonian for our system, the derivation is standard (see [13]-[14]) but useful in order to introduce the notations. In section 3, we discuss the conditions for which the two-photon absorption line remains Lorentzian in the standing
wave. In section 4, we discuss the case of N-photon transitions, and in section 5 the problem of transients.
2. Two-photon absorption : the effective Hamilto-
nian. - We consider an atom, located at the point x,
whose energy levels are g, e, r 1, r2 ... (energies Eg, Ee, El, E2, ...). The atom interacts with an electric field tex, t) = £6(x) cos mt where E corresponds to a
linear polarization and 9(x) = 2 go cos kx. We assume
that the level e can be excited from level g by two- photon absorption and that 2 nro is close to
On the other hand we assume that all the energy defects
are very large compared to :
- the natural width of the g-rj and r Je transitions,
- the saturation width of the g-rj and rJe transi- tions.
The Schrôdinger equation applied to the wave
function 1 03C8( t) >,
gives (the energy Eg of level g is taken equal to 0) :
Here, D gj is the matrix element of DE between g and j, D being the electric dipole moment. We have neglected
the coupling of j with levels j’ different of g and e.
If the time evolution of bg and be is slow compared to 1 làcoj we can easily integrate the last equation, and
we obtain :
Fig. 1. - The effective Hamiltonian corresponds to the interaction between a spin -’ and a magnetic field B. In the frame rotating at 2 w, at resonance the field B is linear and oscillates along the OZ direc-
tion (situation similar to the one encountered in parametric reso- nances). At resonance, the components Bz and Bx correspond to
the light shift and to the saturation.
If we report the value of bi in equations (1) and (2),
we see that the interaction between levels g and e
can be represented by an effective Hamiltonian. We search for a solution of the Schrôdinger equation for
the two-level (g, e) system in the frame rotating at angular velocity 2 w. In this frame, the problem is
identical to the evolution of a spin 1/2 interacting
with a magnetic field (see Fig. 1) whose projections
upon the axis Ox and Oz are proportional to
and
abQ is the matrix element of the two-photon ope- rator [6], [15], Q = DF(hw - Ho)-1 DE between a
and b. Q’ is equal to
The diagonal matrix elements of Q and Q’ corres- pond to the light shift of the levels (dynamic Stark
effect [4]). We have retained only the resonant term in the light shift of the ground level.
The problem of a spin -’ interacting with a static magnetic field is well known. If we assume that the relaxation has the following form :
(1) We have assumed previously that the time evolution of bg and be have to be slow compared to the various 1/ Aw j It is now
clear that it implies that egQ 2(x) and
are very small compared to h Aw j’
we find that the fraction of atoms in the excited state in the steady state regime is equal to :
with
S2(x) represent the saturation of the transition. The width of the two-photon absorption lineshape is equal to 2 nreg JI + S2(x). At high intensity, it is proportional to the intensity of the field [ E(x) I2.
The shift of the resonance is also proportional to
the intensity.
If we consider fixed atoms in the standing wave, the light shift and the saturation vary from one atom to another. It follows that at high intensities, the light modulation leads to an inhomogeneous broadening of the two-photon absorption line.
3. Two-photon absorption lineshape : moving atoms.
- We consider atoms in a dilute vapour. The motions of the atoms are described by straight paths :
We have now a new time dependence of the effective magnetic field, because of the time variation x(t) (’).
The problem is to calculate the evolution of a
spin 1/2 interacting with a static magnetic field (pro- E20
jection atong Oz : : Ee - Eg - 2 chw + 2 (ee Q- ggQ), E2
along Ox 2 EO eg and with a magnetic field linearly polarized which evolves with time as
This problem can be analytically solved
at resonance (Ee - Eg = 2 hm, see section 5). Out
of resonance, we find the solution in the two limiting
cases where the light shift and the saturation are small or large compared to the Doppler width ÔWD.
In the first case (s, hfeg S « hdwd), we can, for
most of the atoms, resolve independently the problem
(2) To be rigorous we have to assume that the atom only moves
a very small distance compared to the wavelength (2 n/k) during
the time spent between the absorption of two photons (in order
to have a well defined 8(:x)). Because this time is of the order of 1 lâcoj,
the condition is : kvx AWj The Doppler width of the two- photon transition has to be very small compared to the energy defects Aw j of the single-photon transitions.
660
for each frequency 2 w, 2(w - kvx), 2(w + kvx) appearing in the non-diagonal matrix element of the effective coupling. Moreover we can retain, in
the diagonal elements, only the static term because
the terms which oscillate at frequency 2 kvx give a
non secular contribution. We find the fraction of atoms of velocity vx,(hkvx » s, hreg S) in the excited
state :
with
We assume that the velocity distribution has a Maxwellian shape :
We then integrate over the velocities and we find the absorption lineshape :
The lineshape thus appears as a superposition of
a narrow Lorentzian peak and a broad Gaussian
Doppler line. The narrow peak comes from the absorption of two photons propagating in opposite
direction while the broad Gaussian curve correspond
to the absorption of two photons propagating in the
same direction [2], [3]. In the low intensity limit (So « 1), we find the classical formula of ref. [16].
It must be noticed that there are two inhomoge-
neous widths which are eliminated by the two counter
propagating waves geometry : the first and more
important one is the Doppler width. The second
corresponds to the inhomogeneities of light shifts in
the standing wave discussed previously, the can-
cellation of this width is due to a motional narrowing
effect [5]. The motional narrowing occurs because
the light shift is much smaller than ’Gc 1, bc being
the time of flight of the atom between two nodes
of the standing wave (1;; 1 ~ ku).
We can also deduce from formula (5) that the
ratio of the area of the Lorentzian peak to the area
of the Gaussian curve is equal to
When the intensity Io is above the saturation limit
(So > 1), the two areas become equal. But the two
curves do not evolve in an identical way : the intensity
of the Lorentzian peak does not vary while its width increases proportionally to Io. On the other hand, the width of the Doppler curve does not change
while its intensity increases proportionally to Io.
In the second case (s or ht eg S » hbOJo), the spin adiabatically follows the variation of the magnetic
field. The lineshape A ’(w) for this situation is identical to that obtained for atoms at rest in the standing wave :
The curve A’(co) is asymmetric, its maximum being
localized near the highest value of the light shift.
4. N-photon absorption lineshape. - We can also
make some remarks about N-photon absorp-
tion (N > 2). It is well known that for these tran- sitions too, it is possible to eliminate the Doppler broadening if one chooses a particular geometry for the exciting beams [6], [17]. It is also possible for
this problem to find an effective coupling which is represented by a 2 x 2 matrix as long as the ground
and excited levels can be considered as non dege-
nerate. In the case of a N-quantum transition, the diagonal matrix elements of the effective Hamil- tonian (light shift) vary as the intensity I while the non-diagonal element (saturation) is proportional
to IN/2. It follows that contrary to the case of two-
photon absorption where the light shift and the
broadening are of the same order of magnitude,
the light shifts, in the N-photon case, can be very
important even if this transition is far below satu- ration [17]. In practice, it is often necessary to tole- rate a sizeable light shift in order to obtain a non
negligible transition probability. As long as the light
shift is averaged, it is possible to use the Doppler-
free N-photons line for spectroscopic purposes by extrapolating the position of the centre of the reso-
nance to its limiting value when I --> 0. It is thus
important to understand when this is the case. Let
us first consider the case where various beams have the same frequency. We find again a situation where
there are standing waves. It follows that the consi- derations of § 3 can be directly applied. If the light
shift is smaller than the Doppler width, a motional narrowing effect occurs for most of the atoms. The
line remains Lorentzian and its centre is shifted by
an amount equal to the mean value of the dynamical
Stark effect. The first experimental evidence of
Doppler-free three-photon transition [17] has been performed under such conditions, the light shift being about ten times smaller than the Doppler
width.
If the light shift is larger than the Doppler width,
there is no more motional narrowing, the lineshape
is no longer Lorentzian and its width is determined
by the inhomogeneous broadening from the dynamic
Stark effect. In these cases it is useless for spectroscopic
purposes to eliminate the Doppler broadening because
the width of the resonance in a standing wave can be
broader than the width obtained in a travelling wave.
In this last case, the light shift is the same for every
point and the width is only limited by the Doppler
width. In order to estimate the order of magnitude
of the quantities involved, let us calculate under which conditions the light shift has the same value
as the Doppler width. In the non-saturating case, the n-photon transition probability has the following
order of magnitude [1] :
where hJC represents a typical matrix element of the
coupling Hamiltonian between the atom and the
electromagnetic field. Am is a typical energy defect for one, two... photon transitions.
In a similar approximation, we obtain for the
light shift [1] :
Using (6) and (7) we find :
If we compares to the Doppler width dwd we
obtain
If we take Aw/dwD - 106, -te,/dwd ~ 10-2 and Pge/T e ~ 10 - 4, we find :
These results show that it would be rather difficult to observe the narrowing of a four-photon transition
in a standing wave. The case of a three-photon tran-
sition is less evident and each experiment needs a special analysis.
If all the beams have differentfrequencies, there are
no more standing waves and the dynamic Stark effect
only gives a shift of the centre of the lineshape which
remains Lorentzian (the Doppler broadening being suppressed). These conclusions hold as long as the
differences between the frequencies of the various
beams 1 Wi - Wj are much larger than the dynamic
Stark effect.
5. Two-photon transition : transient response. - In this last section, we calculate the transient response
following the sharp introduction of the standing
wave when the resonance condition is fulfilled
Let us note that when the intensity is large enough,
this condition does not imply that the excitation is
performed at the centre of the absorption line because of the light shift. We have chosen this system because it can be solved analytically for any value of the light intensity. This is easily understood by a treatment analogous to that of radiofrequency experiments. The problem in the frame rotating at angular velocity 2 (J)
looks identical to the evolution of a spin -’ initially
in the 1 - i >0z state interacting with a magnetic
field whose projections upon the axis Ox and Oz
are proportional to egQe20 cos’ k(v,, t + xo) and
The spin interacts with a linearly polarized oscillating
field of angular frequency 2 kv,, and with a static
field which is parallel to it. It is precisely the condi-
tions under which the parametric resonance has
been observed [18]-[20].
We first calculate the evolution of the atom by neglecting the spontaneous emission from the excited state. It is then easy to find the time evolution. If
we choose the direction OZ of the resultant magnetic field as the direction of quantization the eigenstates
do not evolve with time and the eigenvalues are :
with
662
At time t = 0, the system is in the state with
At time t, we obtain :
The probability of finding the atom in the excited state is
The observed experimental signal is obtained by averaging over the initial positions and velocities of the atoms. Let us first average the positions :
Using the Bessel function Jo, we obtain :
_ The experimental signal is obtained by averaging P(t) over the velocity distribution. Nevertheless, it
appears that P(t) is not velocity-dependent in two limiting cases :
coo « kvx. This corresponds to the case where
the light shift and the saturation are small compared
to the Doppler shift. In this case the Bessel func- tion Jo(x) practically keeps its value for x = 0 (Yo(0)= 1).
For all the velocity groups which comply with the
condition mo « 2 kvx, we find :
This corresponds to the classical Rabi oscillation with angular frequency mo.
- mo » kvx. The evolution of P(t) for times
short compared to 1 /kvx (but which can be much longer than 1/wo) can be obtained by replacing sin x by x in formula (12). We obtain
For all the velocity groups for which 2 kvx « w0,
we obtain a very rapid vanishing of the Rabi oscilla- tion because of the Bessel function Jo. Formula (14) corresponds to the situation where the atom appears to be at rest. More precisely, during the time interval
which is considered, the atom only moves over a
distance which is very small compared to the wave- length. We thus observe the superposition of Rabi
oscillations with completely different frequencies.
Formula (13) obviously corresponds to the case
where there is a motional narrowing while for- mula (14) is obtained for a situation where the varia- tions of mo in the standing wave are too large to be averaged.
We have represented on figure 2b and 2c the
transients corresponding to formulae (13) and (14).
These curves are of course only true for times which
are short compared to the relaxation time. In parti- cular, the Rabi period 2 n/roo has to be very short compared to the spontaneous emission of the excited level 1 /T e. If, on the contrary, T e is much larger
than roo, the transient can be easily calculated. A
typical curve is represented on figure 2a.
