Influence of neighboring levels in three-pulse photon-echo processes

It is the purpose of the present paper to study the dynamics underlying a three-pulse photon-echo process performed on a vibronic system coupled to non-Markovian baths, when a neighboring level enters into the global dynamical evolution because of broadband excitation required in these experiments. Particular emphasis is on the energy gap between the vibronic levels, but also on the fluctuation amplitudes and correlation times of their corresponding thermal baths. The photon-echo signal appears to be very sensitive to the additional interfering contributions introduced by the neighboring vibronic level. It is shown that these contributions associated to the pathways involving different vibronic states are modulated by their corresponding energy gap. As a consequence, these contributions to the integrated photon-echo signal strongly decrease for large energy gaps. Also, an oscillating behavior is observed on the time dependence of the photon-echo signal resulting from the summation of the contributions provided by the individual vibronic levels. Moreover, the influence of the non-Markovian character of the baths, accountable for inhomogeneous broadening, affects the amplitude and the time dependence of the photon-echo signal, as well as its dependence with the delay time of the laser pulses. Of course, for longer times a Markovian dynamical evolution is recovered.

Figure 1

Time ranges required in the definition of the reference Hamiltonians.

Figure 2

Time dependence of the three-pulse photon-echo signal in the phase-matched direction ${\vec{k}}_{PE}=-{\vec{k}}_a+{\vec{k}}_b+{\vec{k}}_c$ for different energy gaps $E_{e_1}-E_{e_2}=0{\text{cm}}^{-1}$ (solid line), $10{\text{cm}}^{-1}$ (dashed line), and $30{\text{cm}}^{-1}$ (long-dashed line). The values of the bath parameters have been chosen as ${\Delta }_{ii}^{\left(\lambda \right)}=2{\text{cm}}^{-1}$ and ${\Lambda }_{ii}^{\left(\lambda \right)}=4{\text{cm}}^{-1}$ for $i=1,2$ and $\forall \lambda$, while for the crossed bath correlation parameters we have ${\Delta }_{ij}^{\left(\lambda \right)}=5{\text{cm}}^{-1}$ and ${\Lambda }_{ij}^{\left(\lambda \right)}=12{\text{cm}}^{-1}$ for $i\ne j$ and $\forall \lambda$.

Figure 3

Logarithmic variations in the time dependence shown in Fig. 2 for the three-pulse photon-echo signal in the phase-matched direction ${\vec{k}}_{PE}=-{\vec{k}}_a+{\vec{k}}_b+{\vec{k}}_c$ with the same energy mismatches.

Figure 4

Like in the previous Fig. 2, we show the time dependence of the three-pulse photon-echo signal in the phase-matched direction ${\vec{k}}_{PE}=-{\vec{k}}_a+{\vec{k}}_b+{\vec{k}}_c$ for the same three different energy gaps $E_{e_1}-E_{e_2}=0{\text{cm}}^{-1}$ (solid line), $10{\text{cm}}^{-1}$ (dashed line), and $30{\text{cm}}^{-1}$ (long-dashed line) considered previously, but with longer crossed bath correlation times. Here, the corresponding values are ${\Delta }_{ij}^{\left(\lambda \right)}=2{\text{cm}}^{-1}$ for $i,j=1,2$ and $\forall \lambda$, ${\Lambda }_{ii}^{\left(\lambda \right)}=1{\text{cm}}^{-1}$ for $i=1,2$ and $\forall \lambda$, and ${\Lambda }_{ij}^{\left(\lambda \right)}=2{\text{cm}}^{-1}$ for $i\ne j$ and $\forall \lambda$.

Figure 5

Logarithmic variations in the time dependence shown in Fig. 4 for the three-pulse photon-echo signal in the phase-matched direction ${\vec{k}}_{PE}=-{\vec{k}}_a+{\vec{k}}_b+{\vec{k}}_c$ with the same energy mismatches.

Figure 6

Time dependence of the three-pulse photon-echo signal in the phase-matched direction ${\vec{k}}_{PE}=-{\vec{k}}_a+{\vec{k}}_b+{\vec{k}}_c$ with resonant vibronic levels ${\omega }_{e_{1}g}={\omega }_{e_{2}g}=20000{\text{cm}}^{-1}$. The cases of different bath correlation times ${\Lambda }_{ij}^{\left(\lambda \right)}=4{\text{cm}}^{-1}$ (long-dashed line), $8{\text{cm}}^{-1}$ (dashed line), and $20{\text{cm}}^{-1}$ (solid line) are considered with ${\Delta }_{ij}^{\left(\lambda \right)}=2{\text{cm}}^{-1}$ $\forall i,j,\lambda$. Other parameters are the same as in previous figures.

Figure 7

Logarithmic variations in the time dependence shown in Fig. 6 for the three-pulse photon-echo signal in the phase-matched direction ${\vec{k}}_{PE}$ and resonant vibronic levels.

Figure 8

Integrated three-pulse photon-echo signal as a function of the delay time between the two first laser pulses, $T_b-T_a$. The phase-matched direction is ${\vec{k}}_{PE}=-{\vec{k}}_a+{\vec{k}}_b+{\vec{k}}_c$. Three different cases of vibronic energy mismatches are considered, say ${\omega }_{e_{1}g}=20000{\text{cm}}^{-1}$ with ${\omega }_{e_{2}g}=20000$, $19997$, and $19990{\text{cm}}^{-1}$. The bath fluctuation amplitudes and inverse bath correlation times for the individual vibronic states $i=1,2$ are ${\Delta }_{ii}^{\left(\lambda \right)}=2{\text{cm}}^{-1}$, ${\Lambda }_{ii}^{\left(\lambda \right)}=4{\text{cm}}^{-1}$, ${\Delta }_{ij}^{\left(\lambda \right)}=5{\text{cm}}^{-1}$ and ${\Lambda }_{ij}^{\left(\lambda \right)}=12{\text{cm}}^{-1}$ for the crossed bath constants $i\ne j$. Other parameters are the same as in previous figures. The three curves with decreasing initial amplitudes correspond to ${\omega }_{e_{1}g}-{\omega }_{e_{2}g}=0$, 3, and $10{\text{cm}}^{-1}$, respectively.

Figure 9

Same variations as in Fig. 8 with different bath parameters. For the individual states $i=1,2$, we have ${\Delta }_{ii}^{\left(\lambda \right)}=2{\text{cm}}^{-1}$ and ${\Lambda }_{ii}^{\left(\lambda \right)}=1{\text{cm}}^{-1}$, while we have ${\Delta }_{ij}^{\left(\lambda \right)}=2{\text{cm}}^{-1}$ and ${\Lambda }_{ij}^{\left(\lambda \right)}=2{\text{cm}}^{-1}$ for mixed states $i\ne j$. Other parameters, as well as identification of the curves, are the same as in previous figures.

Figure 10

We compare the delay time dependences of the photon-echo signal for two different types of baths and different vibronic energy mismatches. The two top curves correspond to ${\omega }_{e_{1}g}-{\omega }_{e_{2}g}=0{\text{cm}}^{-1}$ with different bath fluctuation amplitudes and inverse correlation times, say ${\Delta }_{ii}^{\left(\lambda \right)}=2{\text{cm}}^{-1}$, ${\Lambda }_{ii}^{\left(\lambda \right)}=4{\text{cm}}^{-1}$, and ${\Delta }_{ij}^{\left(\lambda \right)}=5{\text{cm}}^{-1}$, ${\Lambda }_{ij}^{\left(\lambda \right)}=12{\text{cm}}^{-1}$ if $i\ne j$ for curve 1 from the top, while ${\Delta }_{ii}^{\left(\lambda \right)}=2{\text{cm}}^{-1}$, ${\Lambda }_{ii}^{\left(\lambda \right)}=1{\text{cm}}^{-1}$, and ${\Delta }_{ij}^{\left(\lambda \right)}=2{\text{cm}}^{-1}$, ${\Lambda }_{ij}^{\left(\lambda \right)}=2{\text{cm}}^{-1}$ if $i\ne j$ for curve 2 from the top. Curves 3 and 4 are deduced from curves 1 and 2 respectively, by changing the energy mismatch ${\omega }_{e_{1}g}-{\omega }_{e_{2}g}$ from 0 to $3{\text{cm}}^{-1}$.