**Part II - Abstract**

**II. PREPARATION AND MANIPULATION OF COLD MOLECULES**

**II.6. GENERALIZATION OF VIBRATIONAL COOLING**

**II.6.1. a. 'Toy model' for molecular potentials**

In order to understand how the geometrical characteristics of the electronic potentials affect the optical pumping I begin this study with the toy model discussed in [Stand99]. Here, the molecular potentials that are associated with the electronic transition, are substituted by analytical, harmonic oscillator potentials. From the two harmonic potentials involved, the excited potential is identical to the ground potential, except the fact that it lies in higher energy and it is shifted. The analytic form of the potentials is

where with k we symbolize the force constant of each of the oscillators, and with ω*g,e* the oscillation
frequencies. With b we symbolize the displacement, with m the mass and with the letters g and e the
ground and excited state respectively. The eigenenergies of the harmonic oscillator states are

*E** _{g}*=1

2v* _{g}* ℏ

_{g}*, E*

*=1*

_{e}2*v** _{e}* ℏ

*(6.3)*

_{e}where with v*g,e* we symbolize the vibration number of the ground and the excited state respectively.

In Fig. II.6.1.a.1 we see the Franck-Condon parabolas that can be generated with the use of these
analytic potentials as the upper and the lower states of a molecular transition. Initially, we see what
are the characteristics of the Franck-Condon parabola generated with b = 0. In the following, the
excited molecular potential has his equilibrium distance (b) shifted, so the separation between the
two minima is gradually increased. Finally, the excited potential is replaced to a 'wider' (less
confining) potential, now centered on top of the fundamental one. The red dotted lines represent the
iso-energetic lines, that is the lines that correspond to excitation by a single frequency. As we see,
these iso-energetic lines have a tilt of 45° for all cases where the force k*g,e* constants of the upper and
the lower potentials are equal. In the cases where k*e** < k**g, *this tilt grows. This is easily seen, since
now that the upper potential is less confined it leads to smaller separation between the vibrational
states it supports. Thus, a frequency that would excite, let's say, states with equal vibration is the
case of equally confined potentials, now will excite transitions were v*e** > v*g.

As we see in part (a), the Franck-Condon parabola corresponding to two identical potentials
is a line, indicating that the probability of transition between states with equal vibrational number is
maximum, and is almost zero elsewhere. As the excited potential is shifted towards the right, the
Franck-Condon factors form a parabola with increased radius (parts (b) and (c)). If instead of
modulating the separation, the force constant of the oscillators is modified, the initial line 'breaks' to
a parabola with inverse curvature, as shown in parts (f) and (g), for decreased values of k*e*.

**II.6.1.b. Optical pumping in molecules**

Let us consider the use of each of the potential couples considered in our various studies for
the realization of optical pumping and vibrational cooling. The cases (a) and (e) are trivial. No
vibrational cooling can be accomplished with such a scheme, nor any modification of the
vibrational distribution. The reason is that the wavefunctions which correspond to vibrational levels
of equal vibration number v, overlap so well, that all excitation and emission steps are performed
between states with equal v. As a small displacement, like the one considered in (b) is introduced,
the situation is improved. Now, there exist Franck-Condon coefficients that correspond to states of
different v and which have a non zero value, a fact that permits population transfer between the
vibrational states. If this displacement increases considerably, as in (c), the Franck-Condon
coefficients which correspond to transitions between low vibrational levels start to have very low
values. This situation is undesirable, because it reduces population transfer between low lying
vibrational states and insert difficulties in vibrational cooling. Finally, the effect of the variation of
the force constant of the excited state k*e*, is studied. Once the two potentials are not identical, we see
the appearance of non zero Franck-Condon factors for the transitions between states with different
v. The transition between the states v = 0 remains maximum, no matter how much ke has been
modified and no 'hole' appears in the positions corresponding to transitions between low lying
vibrational levels. This is expected, since for no displacements between the potential minima, we do
Fig. II.6.1.a.1: Franck-Condon parabola corresponding to the various combinations of analytical molecular
potentials. The values shown in the false color scales below are multiplied by a factor of 10^{4} and otherwise
correspond to probability. The red dotted lines correspond to excitations by a single frequency (iso-energetic
lines). In the first line, we see the effect of the variation of separation b, as shown schematically in the inlets.

The force constant k is equal to 1560 N m^{-1} for both potentials. In (a) the displacement b is 0 Angström, in (b)
it is 0.1 Angström and in (c) 0.2 Angström. In the second line, the force constant of the excited state ke is
varied while the displacement is kept equal to 0. In (a) ke = 1560 N m^{-1} (and equal to kg) , in (b) it is equal to
780 N m^{-1}, while in (c) it is equal to 390 N m^{-1}. Graphs are adapted from [Stand99]

not expect important decrease in the overlap of the fundamental wavefunctions.

In Fig. II.6.1.a.2, we see a schematic representation of how the basic geometrical characteristics of the potentials involved in the transition affect the form of the corresponding Franck-Condon parabola. With the help of this figure, we can outline the criteria that define the applicability of optical pumping and thus of vibrational cooling, in a given electronic transition. In the following, I list those criteria, without dealing with experimental issues, like for example the availability of light sources for the excitation of the transition or the power requirements.

– the first criterion is that the electronic transition should be as 'closed' as possible, with the sense that spontaneous emission should return molecular population in the initial electronic potential, in its biggest possible percentage, after excitation. Molecular population can be lost during the optical pumping process for a variety of reasons, such as ionization, pre-dissociation or coupling to other electronic potentials; some of these processes are going to be examined in the following paragraph.

– The internuclear position of the minimum (equilibrium point) of the excited potential has to be as close as possible to the minimum of the ground potential. If the objective of the optical pumping is to transfer the molecular population in the lowest vibrational state possible, then all frequencies which lie lower than this point (point A in Fig. II.6.1.a.2) should be cut, since if they participate to the process, they lead to an increase of vibration. In the case of Cs2, as examined in Chapter III, IV and V, this point coincides with the transition vX = 1 → vB = 0, thus it was relatively easy to transfer molecular population to these two vibrational levels.

– The angle θ gives an idea of the rapidity of the vibrational cooling. The biggest this angle is, the 'faster' the cooling is (less transitions are required for a certain decrease in vibration). This could be easily seen by 'bending' the parabola's arms up to the point where they touch the axis. In this case we can see that all transitions towards v = 0 are Fig. II.6.1.a.2: Schematic representation of the geometric characteristics of the molecular potentials and how they affect the form of the generated Franck-Condon parabola. Figure adapted from [Vit08c].

**v**_{g}

**v** **e**

**v**_{g}

**v** **e**

favored (vX for the upper and vA for the lower arm).

In the light of the study of these Franck-Condon parabola presented, we can consider
realistic choices for molecular electronic potentials and analyze the morphology of the resulting
parabola. I consider the Franck-Condon parabola associated with the X^{1}Σg+ → B^{1}Πu vibrational
cooling scheme, which is the only vibrational cooling scheme demonstrated experimentally. I also
consider the X^{1}Σg+ → C^{1}Πu vibrational cooling scheme, which, even if it has not been
experimentally demonstrated, it is a suitable scheme for vibrational cooling (to be discussed in the
following chapter). Those potential are shown in Fig. II.6.1.a.3, along with their corresponding
Franck-Condon parabola. We are in position to identify the parameters that ensure the successful
implementation of vibrational cooling via these states, since we see that the involved molecular
potentials are more or less of the same confinement, while their minimum point are located in the
same or in close internuclear distances. This ensures the sufficient magnitude of the Franck-Condon
parameters for the transitions between states of small vibrational number, and permits molecular
population transfer to any of these low lying states.

Fig. II.6.1.a.3: (a) Molecular potentials corresponding to the X^{1}Σg+ to B^{1}Πu transition in Cs2 and
corresponding Franck-Condon Parabola (b). (c) Molecular potentials corresponding to the X^{1}Σg+ to C^{1}Πu

transition in Cs2 and corresponding Franck-Condon Parabola (d). The dashed lines note the equilibrium positions of the corresponding potentials.