## 1 Introduction  
Tailoring of the optical phase of femtosecond pulses has been used in several applications in photochemistry and time-resolved spectroscopy [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16]. In general, tailored pulses can be applied using two different approaches: Coherent Control (CC) or Quantum Control Spectroscopy (QCS). Closed-loop learning algorithms are exploited in CC to generate complex optical phases in order to optimize specific reaction pathways. CC has been used as a tool in the control of dynamic processes [17; 18; 19; 20; 21], and it explains how a specific state can be selectively populated, for example, to attain 100% population transfer [22; 23; 24; 25], and recent studies have shown the possibility of having enhancements even higher than 100% [26; 27; 28; 29; 30; 31; 32; 33; 34; 35]. QCS also exploits tailored excitation but differs from CC in its applications [30; 36]. Its goal is to simplify complex molecular dynamics by tailoring excitation pulses to enhance or suppress specific molecular signal features. A central aspect in both approaches, CC and QCS, is the control of population transfer between the ground and excited states, as well as the generation of vibrational coherence in both potential surfaces. By controlling the population transfer or by suppressing specific molecular vibrational coherences, a photochemical reaction channel can be selectively chosen [27; 29; 32; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49], or a certain mode in a multidimensional time-resolved signal can be suppressed [50; 51; 52; 53; 54; 55]. The control of population transfer and vibrational coherences in molecules in general is challenging since several molecular and optical parameters play a role. Indeed, the molecular system under study dictates the displacement of the addressed levels and the vibrational and electronic dephasing rates. While the electronic dephasing rates can in some cases be influenced by the choice of the solvent [56; 57; 58; 59], the displacement between the ground and the excited states is generally fixed. Pulse characteristics can be more easily tailored than molecular properties. Crucial parameters are the spectral detuning of the excitation with respect to the molecular absorption region as well as the temporal shape of the excitation, which is tailored by using phase and/or amplitude shaping. In the present paper we address the coherent control of population transfer and vibrational states in laser dye oxazine systems. We have chosen oxazine systems since these systems are being extensively studied in spectrally and time-resolved transient absorption spectroscopy experiments, in the femtosecond time regime, to elucidate the fastest fundamental processes in photochemistry and photobiology, like isomerization and electron and photon transfer [60]. In particular, we analyze the response of a four-level system, consisting of two electronic states both including two vibrational sub-states (see Figure 1), to the excitation of chirped and detuned femtosecond laser pulses. We will show that the basic processes producing enhancement/suppression factors on the population transfer and vibrational coherences can be understood from this relatively simple molecular model by studying the dynamics of the density matrix elements, and will compare our results with previous theoretical studies and experimental measurements. Indeed, an asymmetry in the molecular response to chirped pulses was first predicted theoretically by [PERSON] and [PERSON] [61] and observed experimentally by [PERSON] et al. [62]. They explained how the excitation of a vibration in the ground state of a molecular system can be accomplished by negatively chirped pulses in an intrapulse pump-dump process. A similar scheme was implemented by [PERSON] et al. [63] for quantum coherent control of population transfer. They demonstrated theoretically that electronic population inversion probabilities of up to 99% in molecular systems can be achieved by using intense positively chirped broadband laser pulses.