Resultados de la práctica de choques elásticos

The influence of phase modulated femtosecond laser pulses on one- and two-photon transitions in an atomic prototype system was studied and the implementation of a feedback loop using evolutionary algorithms was tested under realistic experimental conditions. The one-photon-transition presents an excellent tool to test the quality of phase-related pulses, as the excitation of the 3p level in sodium depends critically on the relative phase of the double pulse. It has been shown that the experimental outcome could be explained by a combination of two limiting cases. The relative contribution depends on the nature of the phase coupling within the pulse sequence, which is in- fluenced by the experimental conditions of the setup. The phase modulated excitation of the two-photon-transition shows that the feedback approach can be successfully used to find femtosecond laser pulses for different con- trol objectives, even without an intelligent initial guess supplied by theory. The best solutions for both extremes were obtained within five generations. Allowing the feedback algorithm to search in an extended parameter space the algorithm found new phase structures in addition to known analytic solutions. These structures are not intuitively understandable and call for further theoretical studies.

Having a combined look at the results on the one- and two-photon transition the conclusion is, that it is in principle possible to control the linear versus the nonlinear process, since the pulse shapes for suppression and enhance- ment depend on the character of the transition. This can be done in an automated way using the learning-loop setup presented in Fig. 2.1(b).

Chapter 3

Control of dimers using

shaped DFWM

In this chapter the powerful spectroscopic tool of nonlinear four-wave mixing (FWM) techniques is combined with pulse shaping. The FWM technique is an ideal method, since it can be used to monitor ground and excited state dynamics during pulse control. This peculiarity of the FWM process will be explained in the following where the time domain framework of nonlinear processes will be reviewed [86]. In section 3.2 a theoretical model describing a new type of control technique using shaped pulses within the degenerate FWM process will be derived and verified experimentally in section 3.3. In the final section the FWM response of the potassium dimer is spectrally resolved to provide a FROG-type measurement of the shaped excitation field used in these studies.

3.1

Theory of nonlinear spectroscopy

In nonlinear spectroscopy the radiation field interacts with the system creat- ing a time-dependent material polarization P(r, t) which generates an elec- tric field according to Maxwell’s equation. The intensity of this field is the experimental spectroscopic observable. The Hamiltonian Hint for the system’s interaction with the external radiation fields is given by

Hint=−µ(Q)E(r, t), (3.1)

where µ(Q) and E(r, t) denote the dipole operator depending on system degrees of freedom Q and the electric field of the external radiation, respec- tively. The definition of the material polarization P(r, t) is given by [86]

P(r, t) = tr_{µ(Q)ρ(r, t)_} (3.2)

where tr means sum over all degrees of freedom in the total matter system, andρ(r, t) denotes the density operator obeying the Liouville equation,

∂ρ(r, t)

∂t =

1

i~[HM +Hint, ρ(r, t)]. (3.3)

HM is the Hamiltonian of the unperturbed matter system. Taking the in- teraction picture, one can obtainρ(r, t) in a perturbation series ofHintfrom Eq. (3.3). By substituting the resulting expression into Eq. (3.2), one gets the formal expression for P(r, t) as follows [86]:

P(r, t) = ∞ X i=0 P(n)(r, t) (3.4) P(n)(r, t) = ∞ Z 0 dtn ∞ Z 0 dtn−1· · · ∞ Z 0 dt1R(¯ tn, tn−1, . . . , t1) ..

.E(r, t₋tn)E(r, t−tn−tn−1)· · ·E(r, t−tn

−tn−1· · · −t1), (3.5)

where ... denotes the tensor contraction, and ¯R(tn, tn−1, . . . , t1) is the non-

linear response tensor of matter system defined by ¯ R(tn, tn−1, . . . , t1) = µ i ~ ¶n h[[[[_{· · ·}[µ(tn+· · ·+t1), µ(tn−1+· · ·+t1), . . .], µ(t)], µ(0)]i, (3.6)

withµ(t) = exp(iHMt/~)µ(Q) exp(−iHMt/~) and the expectation value of an arbitrary operator A being defined as _hA_i = tr_{Aρ(0)_}. Here ρ(0) denotes the initial density operator of the material system.

In four-wave mixing (FWM) spectroscopy, one selectively measures the third order polarization P(3)_{(r, t) among the perturbation series, and the nonlinear}

response tensor relevant to the FWM spectroscopy reads as ¯ R(t3, t2, t1) = µ i ~ ¶n h[[[µ(t3+t2+t1), µ(t2+t1)], µ(t1)], µ(0)]i, (3.7)

An explicit expression for the nonlinear response tensor can be obtained if a specific Hamiltonian is assumed [87]. Once the response has been calculated, the polarization can be obtained for any shaped pulse E by a three-fold integration P(r, t, τ, τ1) = (₋i)3 ∞ Z 0 dt3 ∞ Z 0 dt2 ∞ Z 0 dt1R(¯ t3, t2, t1) E(r, t₋t3)E(r, t−t3−t2+τ1) E(r, t₋t3−t2−t1+τ +τ1). (3.8)

3. Control of dimers using shaped DFWM 49 Here a complete specification of the pulse ordering is used, where the pulse separation between the first and second isτ and second and last laser pulse is τ1. Eq. (3.8) will be of central importance for the following sections. In

general more then 64 double sided Feynman diagrams, representing different Liouville space pathways, contribute to the third order nonlinear response tensor. However, by controlling the center frequency, the polarization di- rection, and the propagation direction of the input external fields, and by measuring the signal field propagating along a specific direction, only a few of the components of the polarization vectors can be selectively measured. In the next section it is shown theoretically that it is possible to influence contributions to the degenerate FWM (DFWM) signal by shaping one (or more) of the excitation pulses. Then an experimental section follows showing that the theoretical predictions are accurate.