Quadratic chirping

The above set of conditions are complete in the sense that a transition from any initial state to any final state can be controlled perfectly. This idea can also be applied to multilevel problems. In the practical applications, the quadratic chirping, that is, one-period oscillation, is quite useful, as demonstrated by numerical applications given below. 

First, let us consider a selective and complete excitation in a three-level problem by quadratically chirping the laser frequency as shown in Fig. 28 [42] (the field parameter F is the laser frequency oa). The energy separation CO23... 

The laser parameters should be chosen so that a and p can make the nonadiabatic transition probability V as close to unity as possible. Figure 34 depicts the probability P 2 as a function of a and p. There are some areas in which the probabilty is larger than 0.9, such as those around (ot= 1.20, p = 0.85), (ot = 0.53, p = 2.40), (a = 0.38, p = 3.31), and so on. Due to the coordinate dependence of the potential difference A(x) and the transition dipole moment p(x), it is generally impossible to achieve perfect excitation of the wave packet by a single quadratically chirped laser pulse. However, a very high efficiency of the population transfer is possible without significant deformation of the shape of the wave packet, if we locate the wave packet parameters inside one of these islands. The biggest, thus the most useful island, is around ot = 1.20, p = 0.85. The transition probability P 2 is > 0.9, if... 

Figure 34. Contour map of the nonadiabatic transition probability Pn induced by quadratically chirped pulse as a function of the two basic parameters a and p. Taken from Ref. [37]-...

Figure 36. Time variation of the wave packet population on the ground X and excited B states of LiH. The system is excited by a single quadratically chirped pulse with parameters 0(a, = 5.84 X 10 eV fs , = 2.319 eV, and / = 1.00 TWcm . The pulse is centered at t = 0...

The second example is the quadratically chirped pump-dump scheme. Since the pioneering work by Tannor and Rice [119], the pump-dump method has been widely used to control various processes. However, since it is not possible to transfer a wave packet from one potential energy surface to another nearly completely by using the ordinary transform limited or linear chirped pulses, the... 

Figure 37. Electronic excitation of the NaK wavepacket from the inner turning point of the ground X state. The X A transition is considered. The initial wave packet is prepared by two quadratically chirped pulses within the pump-dump mechanism. Taken from Ref. [37].

The present excitation scheme of quadratic chirping can be applied to higher dimensional systems easily. As an example, we consider the bond-selective... 

Figure 39. Pump-dump control of NaK molecule by using two quadratically chirped pulses. The initial state taken as the ground vibrational eigenstate of the ground state X is excited by a quadratically chirped pulse to the excited state A. This excited wavepacket is dumped at the outer turning point at t 230 fs by the second quadratically chirped pulse. The laser parameters used are = 2.75(1.972) X 10-2 eVfs- 1.441(1.031) eV, and / = 0.15(0.10)TWcm-2 for the first (second) pulse. The two pulses are centered at t = 14.5 fs and t2 = 235.8 fs, respectively. Both of them have a temporal width i = 20 fs. (See color insert.) Taken from Ref. [37].

Figure 41. Selective bond breaking of H2O by means of the quadratically chirped pulses with the initial wave packets described in the text. The dynamics of the wavepacket moving on the excited potential energy surface is illustrated by the density, (a) The initail wave packet is the ground vibrational eigen state at the equilibrium position, (b) The initial wave packet has the same shape as that of (a), but shifted to the right, (c) The initail wave packet is at the equilibrium position but with a directed momentum toward x direction. Taken from Ref. [37]. (See color insert.)...

Figure 58. Changes of the wavepacket populations on the respective states (upper panels) under the 3.5TWcm quadratically chirped pulses (lower panels) during the sequential pump-dump scheme via the (a) I A —> I B pumping at CHD and [(b) and (c)] 2 A I B —> I A pump...

Laser Control of Chemical Dynamics. I. Control of Electronic Transitions by Quadratic Chirping... 

Summary. An effective scheme for the laser control of wavepacket dynamics applicable to systems with many degrees of freedom is discussed. It is demonstrated that specially designed quadratically chirped pulses can be used to achieve fast and near-complete excitation of the wavepacket without significantly distorting its shape. The parameters of the laser pulse can be estimated analytically from the Zhu-Nakamura (ZN) theory of nonadiabatic transitions. The scheme is applicable to various processes, such as simple electronic excitations, pump-dumps, and selective bond-breaking, and, taking diatomic and triatomic molecules as examples, it is actually shown to work well. 

An effective scheme for controlling elementary process (i) has been proposed, which is based on the idea of quadratic chirping [24-29]. It is now well-established that laser field-induced transitions among energy levels can... 

At the resonance w(t) = A(x), the adiabatic potentials i.e. the eigenvalues of (5.9) show avoided crossing and the population splits into the two adiabatic Floquet states. In the case of quadratically chirped pulses, the instantaneous frequency meets the resonance condition twice and near-complete excitation can be achieved due to the constructive interference. The nonadi-abatic transition matrix Ujj for the two-level problem of (5.9) is given by the ZN theory [33] as... 

For excitation by quadratically chirped pulses, the transition timescale 2Ttr is slightly longer than the time interval 2tx between the two crossings,... 

According to (5.35), the most fortunate circumstance for the present scheme is a system with heavy mass and parallel potential energy surfaces (A(x) const.). The steepness of the potential difference A(x) is the most crucial parameter it not only affects the validity of this level approximation (5.8) but it also changes the efficiency of excitation according to (5.32). It is obvious that a narrow wavepacket can be relatively easily excited by a quadratically chirped pulse (cf. 5.32). However, a narrow one can easily break the level approximation (5.35) because of the broad distribution in momentum space. The optimal width of a wavepacket can be roughly estimated as... 

In order to demonstrate the efficiency and robustness of the formulations presented in Sect. 5.2 and 5.3, three practical applications are considered here. In the first part of this section, we consider the complete excitation of a wavepacket from a nonequilibrium displaced position, which is directly related to the idea of bond-selective breaking, as explained in the Introduction . This is demonstrated numerically by taking diatomic molecules LiH and NaK as examples. In the second part, we consider the complete pump-dump control and creation of a localized wavepacket using quadratic chirping within the pump-dump mechanism. The bond-selective photodissociation of the H20 molecule is discussed in the third part of this subsection as an example of a multidimensional system. 

The time dependence of the wavepacket population on the X and B states are plotted in Fig. 5.3 for the case of a quadratically chirped pulse centered at tp = Ofs with a full temporal duration r = 20 fs. More than 86% of the initial state is excited to the B state within a few femtoseconds (see Fig. 5.3). After the excitation, Fe on the B state potential spreads rapidly due to the very light mass of LiH and the flatness of the potential. Finally, however,... 

Fig. 5.4. Initial and final wavepackets of LiH excited by the same quadratically chirped pulse as that in Fig. 5.3. Initial wavepacket refers to the wavepacket propagated up to the pulse center according to the ground surface Hamiltonian Hg excited wavepacket (approximate) refers to the result obtained using the level approximation as Pi2 x) g(x, 0) 2 and excited wavepacket (exact) refers to the numerical solution of (5.1). The latter two are backward-propagated to the pulse center at time t = tp according to the excited state Hamiltonian He...

It should also be noted that the LiH molecule is one of the most difficult systems to apply the present method to, because the mass of LiH is very light at 0.875 amu and the gradient of potential difference is relatively large (V A —0.473 eV/ao) at the center of the wavepacket. All of these difficulties have been nicely overcome by employing fast quadratic chirping. This fact guarantees the usefulness of the present method. 

Fig. 5.6. Time variations of the wavepacket populations in the X1 X 1 and /111 states of NaK. The system is excited by a quadratically chirped pulse with parameters = 3.13 x 102eV/fs2, plo = 1.76eV and / = 0.20TW/cm2. The pulse is...

The method of electronic excitation by a quadratically chirped pulse mentioned above can be applied to a wavepacket moving away from the turning point, so this technique can be applied to various processes such as pump-dump, wavepacket localization and selective bond-breaking, as we will discuss in the rest of this section. 

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