Femtosecond pulses and dynamics of molecular photoexcitation: RbCs example

We investigate the dynamics of molecular photoexcitation by unchirped femtosecond laser pulses using RbCs as a model system. This study is motivated by a goal of optimizing a two-color scheme of transferring vibrationally excited ultracold molecules to their absolute ground state. In this scheme the molecules are initially produced by photoassociation or magnetoassociation in bound vibrational levels close to the first dissociation threshold. We analyze here the first step of the two-color path as a function of pulse intensity from the low-field to the high-field regime. We use two different approaches, a global one, the “wave-packet” method, and a restricted one, the “level-by-level” method, where the number of vibrational levels is limited to a small subset. The comparison between the results of the two approaches allows one to gain qualitative insights into the complex dynamics of the high-field regime. In particular, we emphasize the nontrivial and important role of far-from-resonance levels which are adiabatically excited through “vertical” transitions with a large Franck-Condon factor. We also point out the spectacular excitation blockade due to the presence of a quasidegenerate level in the lower electronic state. We conclude that selective transfer with femtosecond pulses is possible in the low-field regime only. Finally, we extend our single-pulse analysis and examine population transfer induced by coherent trains of low-intensity femtosecond pulses.

Figure 1

Photoexcitation of RbCs. Lower panel: Diabatic potentials for the stable $X^1{\Sigma }^{+}$ (lower thick dashed black line), the metastable $a^3{\Sigma }^{+}$ (lower thick continuous black line), and the excited $b^3\Pi$ (upper thick continuous black line) and $A^1{\Sigma }^{+}$ (upper thick dashed black line) electronic states. The origin of energy is set at the dissociation limit Rb(5$s$)Cs(6$s$). Vibrational wave functions for the initial level $a^3{\Sigma }^{+}$ $v^{\prime \prime }=37$ [upper continuous yellow (light gray) line] and for the final level $X^1{\Sigma }^{+}$ $v^{\prime \prime }=0$ [lower continuous yellow (light gray) line] in the two-color process. Wave functions for the $b^3\Pi$ $v^{\prime }=43$ level [continuous orange (dark gray) line] corresponding to the resonant transition (oblique black continuous line) and for the $b^3\Pi$ $v^{\prime }=195$ level [continuous red (black) line] corresponding to the off-resonant “vertical” transition (vertical dashed black line). The wave functions are drawn at their absolute energy. The bandwidth of the laser pulse is shown as the thick black horizontal line. Upper panel: Adiabatic potentials $0^{+}{P}_{1/2}$ (thick continuous black line) and $0^{+}{P}_{3/2}$ (thick dashed black line). The energy origin is at the Rb(5$s$)Cs(6$p$) dissociation limit. The two vibrational components of the coupled $0^{+}{v}^{\prime }=70$ level in the Hund's case $c$ representation are drawn, respectively, $b^3\Pi$ [medium-thick red (black) line] and $A^1{\Sigma }^{+}$ [thin green (medium gray) line]. In the inset, these two components are compared with the wave function of the $b^3\Pi$ $v^{\prime }=43$ level [medium orange (dark gray) line] for $8<R<12$ $a_0$.

Figure 2

WP approach: Variation with time (in ps) of the population in the excited electronic state $b^3\Pi$. Panels (a)–(c): Low-field excitation $W_L=W_L^{\pi }/120$. Panels (d)–(f): High-field excitation $W_L=W_L^{\pi }$. Panels (a) and (d): Total population ${\sum }_{v^{\prime }=0}^{N-1}{\left|b_{v^{\prime }}\left(t\right)\right|}^2$ in the excited electronic state. Panels (b) and (e): Population $|b_{v_0^{\prime }=43}{\left(t\right)|}^2$ in the resonant level $v_0^{\prime }=43$. Panels (c) and (f): Population $|b_{v^{\prime }=195}{\left(t\right)|}^2$ in the far from resonance level $v^{\prime }=195$, corresponding to the “vertical transition” defined in Fig. 1. Duration of the pulse $[t_P-{\tau }_L,t_P+{\tau }_L]$ [vertical dashed blue (black) lines] maximum at $t=t_P$ [vertical thick continuous blue (black) line]. The populations have been multiplied by the factor indicated in the upper right corner.

Figure 3

WP approach: Population $|b_{v^{\prime }}{\left(t\right)|}^2$ in the levels $b^3\Pi v^{\prime }$ [panels (a) and (c)] and population $|a_{v^{\prime \prime }}{\left(t\right)|}^2$ in the levels $a^3{\Sigma }^{+}{v}^{\prime \prime }$ [panels (b) and (d)], as a function of the energy of the corresponding levels (in cm${}^{-1}$) with respect to the Rb(5$s$)Cs(6$p$) and Rb(5$s$)Cs(6$s$) dissociation limits, respectively. The distributions of population are shown either at the maximum of the pulse $t=t_P=0.6$ ps (black dots) and after the end of the pulse $t=2$ ps [orange (gray) squares]. Panels (a) and (b): Low-field excitation $W_L=W_L^{\pi }/120$. Panels (c) and (d): High-field excitation $W_L=W_L^{\pi }$. The resonant level $v_0^{\prime }$ is indicated by a large solid black circle at $t=0.6$ ps and a large solid orange (gray) square at $t=2$ ps, the level $b{v}^{\prime }=195$ is represented by a solid black triangle.

Figure 4

WP approach. Panel (a): Variation of the population $|b_{v^{\prime }}{(t\to +\infty )|}^2$ remaining after the pulse in the nearly resonant excited levels $b^3\Pi v^{\prime }$ as function of the laser coupling $W_L$ (in units of ${10}^{-3}$ a.u.); $v^{\prime }=41$ [medium-thick red (black) line]; $v^{\prime }=42$ (thin black line); $v^{\prime }=v_0^{\prime }=43$ (thick black line); $v^{\prime }=44$ (medium-thick light gray line); and $v^{\prime }=45$ [medium-thick orange (gray) line]. The low-field excitation $W_L^{\pi }/100$ is indicated by the thin dashed vertical red (black) line. The couplings $W_{L_1}=4.5\times {10}^{-4}$ a.u. (black line), $W_{L_2}=1.15\times {10}^{-3}$ a.u. [orange (gray) line], and $W_{L_3}=1.75\times {10}^{-3}$ au (light gray line) corresponding to the maxima of $|b_{v_0^{\prime }=43}{(t\to +\infty )|}^2$ are indicated by the vertical arrows at the top of the panel. Right panels: Couplings $W_{L_i}i=1-3$. Panel (b): Population transferred to the levels $b^3\Pi v^{\prime }$ as function of $v^{\prime }$. Panel (c): Population redistributed in the levels $a^3{\Sigma }^{+}{v}^{\prime \prime }$ as function of $v^{\prime \prime }$.

Figure 5

LbyL description with three different basis sets of levels in the lower ($v^{\prime \prime }$) or the excited ($v^{\prime }$) electronic states; these sets, labeled A, B, and C, are defined in the text: Variation of the population in the excited b${}^3\Pi v^{\prime }$ levels as a function of time (in ps). The pulse [vertical continuous and dashed blue (black) lines] satisfies the $\pi$-pulse condition. Panel (a): Total population for set A [continuous orange (light gray) line], B [dashed red (dark gray) line], and C (continuous black line). Panels (b)–(d): “Optimal” basis set C $[v^{\prime \prime }=0$–204, $v^{\prime }=0$–218] reproducing the results of the WP method (Fig. 2); (b) Total population in the bound levels $v^{\prime }=0$–218; (c) Population in the resonant level $v_0^{\prime }=43$; (d) Population in the off-resonant level $b^3\Pi$ $v^{\prime }=195$, corresponding to the vertical transition. Some populations have been multiplied by the factor indicated in the upper right corner.

Figure 6

LbyL description, optimal basis set C: Variation of the population in the excited b${}^3\Pi v^{\prime }$ levels as a function of time (in ps). The laser pulse [vertical continuous and dashed blue (black) lines] satisfies the $\pi$-pulse condition. Total population of the excited bound levels $v^{\prime }=0$–218 (thick continuous black line). Nonadiabatic evolution for the population in the levels $v^{\prime }=13$–60 (thin continuous black line) resulting in a nonvanishing final population transfer. Adiabatic evolution, following the variation of the pulse intensity, for the far-from-resonance levels $v^{\prime }=167$–218 [thin dashed red (black) line] located around the vertical transition $v^{\prime }\sim 195$ and the most populated at the maximum of the pulse or for the intermediate group of levels $v^{\prime }=61$–166 [thin continuous orange (gray) line].

Figure 7

Two-level system: Analysis, as a function of time (in ps), of the dynamics for the different cases $a$ to $f$ described in Table . The maximum and the duration of the pulse are indicated by the vertical continuous and dashed blue (black) lines. First row: Population $P_{\mathrm{diab}}\left(t\right)$ in the diabatic excited level $[e\rangle$, with the initial population lying in the lower diabatic level $[g\rangle$. Second row: Variation with time of the population $P_{\mathrm{adiab}}^{(+)}\left(t\right)$ in the adiabatic level ${\Psi }_{+}\left(t\right)$, with the initial population lying in the adiabatic level ${\Psi }_{-}(t=0)\equiv |g\rangle$. Third row: Rotation angle $\theta \left(t\right)/\pi$ [Eq. (C17)] defining the instantaneous adiabatic levels ${\Psi }_{\pm }\left(t\right)$. Fourth row: Parameter $Q_{\mathrm{adiab}}\left(t\right)$ [Eq. (21)] characterizing the adiabaticy of the instantaneous transfer. Fifth row: Pulse parameter $\Omega \left(t\right)/\delta$. The quantities have been multiplied by the factor indicated in the upper left corner.

Figure 8

Criteria of adiabaticity for the excitation of a molecular wave packet by a Gaussian pulse: Variation of ${\overline{Q}}_{\mathrm{adiab}}(R,t)$ (Sec. 4b5) as a function of the internuclear distance $R$ (in a.u.) and of the time $t$ (in ps). The condition for an adiabatic evolution is broken near $R_c=10.5$ a.u. and for times corresponding to the beginning and the end of the Gaussian pulse. Nonadiabatic effects are more important for the low-intensity pulse $W_L^{\pi }/120$ (upper panel) than for the high-intensity pulse $W_L^{\pi }$ (lower panel).

Figure 9

LbyL calculation: Influence on the dynamics of the far-from-resonance $b^3\Pi v^{\prime }$ levels, when only the level $v_0^{\prime \prime }=37$ is introduced in the lower state. Variation with time (in ps) of the populations of the vibrational levels $b^3\Pi$ $v^{\prime }$ with $v^{\prime }=43$ [thick continuous orange (gray) line], $v^{\prime }=42$ [thin continuous orange (gray) line], $v^{\prime }=44$ [thin dashed orange (gray) line], $v^{\prime }=195$ (thick black line), $v^{\prime }=194$ (thin black line), $v^{\prime }=196$ (thin dot-dashed black line) and of the initially populated vibrational level $a^3{\Sigma }^{+}$ $v_0^{\prime \prime }=37$ [dashed red (black) line]. Some populations have been multiplied by the factor indicated in the figure. The pulse maximum at $t_P$ [vertical continuous blue (black) line] and duration indicated by the vertical dashed blue (black) lines satisfies the $\pi$-pulse condition. Different basis sets $[v^{\prime \prime },v^{\prime }]$ are used. The difference between left-hand and right-hand panels of the same line is the absence or presence of far-from-resonance levels. Upper line, (a) $[v_0^{\prime \prime }=37,v_0^{\prime }=43]$ and (b) $[v_0^{\prime \prime }=37,v_0^{\prime }=43$, and $v^{\prime }=194$–196]; middle line, (c) $[v_0^{\prime \prime }=37,v^{\prime }=42$–44] and (d) $[v_0^{\prime \prime }=37,v^{\prime }=42$–44 and $v^{\prime }=194$–196]; lower line, (e) $[v_0^{\prime \prime }=37,v^{\prime }=13$–60] and (f) $[v_0^{\prime \prime }=37,v^{\prime }=0$–218].

Figure 10

LbyL calculations: Influence on the dynamics of far-from-resonance $b^3\Pi$ $v^{\prime }$ excited levels, when all the levels $a^3{\Sigma }^{+}$ $v^{\prime \prime }$ of the set C are introduced in the lower state. Basis sets $[v^{\prime \prime }=0$–204, $v^{\prime }=13$–60] [orange (gray) lines] and $[v^{\prime \prime }=0$–204, $v^{\prime }=0$–218] [red (black) lines]. The pulse [vertical continuous and dashed blue (dark) lines] satisfies the $\pi$-pulse condition. Left column: Variation with time (in ps) of the population of the $b^3\Pi$ $v^{\prime }$ levels (in percentage); panel (a): total population in the levels $v^{\prime }=13$–60; panel (b) population in the resonant level $v_0^{\prime }=43$. Right column: Final distribution of population (in percentage) in the excited levels $v^{\prime }$ [panel (c)] and lower levels $v^{\prime \prime }$ [panel (d)] with a maximum, for $v_0^{\prime \prime }$, equal to 90.9 and 76.0 for the first and second basis sets, respectively.

Figure 11

Influence of a quasidegenerate level in the ground electronic state. Evolution of the populations as a function of time (in ps), under a pulse (continuous and dashed blue vertical lines) satisfying the $\pi$-pulse condition $W=W_L^{\pi }$. The results obtained using the LbyL method are compared to the analytical formulas [Eq. (D5)] describing the excitation from an $N$-fold degenerate level [thin continuous red (black) lines], where the initial population is in a single level (${\underline{A}}_1$) of the manifold, the other $N-1$ sublevels (${\underline{A}}_i$) being unpopulated; the excited level (${\underline{B}}_1$) is resonantly excited. Panel (a), $N=3$: basis set $[v^{\prime \prime }=36$–38,$v_0^{\prime }=43]$. Panel (b), $N=4$: basis set $[v^{\prime \prime }=36$–39,$v_0^{\prime }=43]$. Panels (c)–(e), $N=11$: basis set $[v^{\prime \prime }=32$–42,$v^{\prime }=43]$. Initially populated level $a^3{\Sigma }^{+}{v}_0^{\prime \prime }=37$ (double dot-dashed black line) and resonantly excited level $b^3\Pi v_0^{\prime }=43$ (thick continuous light gray line). Nonresonant $a^3{\Sigma }^{+}$ levels described by the same formula in the analytical model: $v^{\prime \prime }=36$ [medium-thick dot-dashed orange (gray) line]; $v^{\prime \prime }=38$ [medium-thick dashed orange (gray) line]; $v^{\prime \prime }=39$ [medium-thick continuous orange (gray) line].

Figure 12

Panels (a)–(c): Excitation of the resonant transition $a^3{\Sigma }^{+}{v}_0^{\prime \prime }=37\to b^3\Pi v_0^{\prime }=43$ in the low-field regime by successive Gaussian femtosecond pulses with duration ${\tau }_L=0.12$ ps, repetition time $T_{\mathrm{rep}}=0.8$ ps, and maximum coupling strength $W_L^{\pi }/120$. Variation with time (in ps) of the population of the excited electronic state $b^3\Pi$ (thick black line), of the resonant level $b^3\Pi v_0^{\prime }=43$ [thin red (black) line], and of the quasiresonant levels $v^{\prime }=42$ [thin orange (gray) line] and $v^{\prime }=44$ (thin black dashed line) levels. Panel (a): Train of $\mathcal{N}=10$ coherent pulses in the LbyL approach with the basis set $[v_0^{\prime \prime }=37,v^{\prime }=42$–44]. Panel (b): Train of $\mathcal{N}=10$ pulses in the WP approach, where the lower and excited wave packets obtained in the middle of the interval between the pulses $P_i$ and $P_{i+1}$ are taken as initial wave packets for the pulse $P_{i+1}$. Panel (c): Same as for panel (a) but for a train of $\mathcal{N}=120$ pulses, corresponding to an effective $\pi$ pulse. Panel (d): Single picosecond $\pi$ pulse with duration $120{\tau }_L=14.4$ ps and maximum coupling strength $W_L^{\pi }/120$ in the LbyL approach with the same basis set.