Role of quadratic and cubic spectral phases in ladder climbing with ultrashort pulses

Two-photon excitation of a quantum ladder system by an ultrashort chirped pulse leads to interferences in the excited-state population, between direct and sequential paths. Experimental results have been obtained in atomic sodium vapor. The presence of several intermediate and final states leads to new phenomena. The interplay between the two sequential paths leads to strongly contrasted oscillations and a large enhancement of the transition probability for precise values of the chirp. A cubic spectral phase modifies significantly the behavior of the quantum interferences and can be used to enhance the two-photon absorption rate.

Figure 1

Population of the excited state as a function of chirp for $\delta >0$, normalized to the population achieved with a FT-limited pulse $\left({\varphi }^{″}=0\right)$. Dash-dotted line: sequential contribution $\mid a_s{\mid }^2$. Dashed line: direct contribution $\mid a_d{\mid }^2$. Total two-photon contribution $\mid a_e{\mid }^2$, using the exact expression given by Eq. (8) (solid line) and its approximation, Eq. (9) (dots). Inset: three-level system, where ${\omega }_0$ is the carrier frequency of the laser and $\delta$ the detuning with respect to the intermediate state $\mid k⟩$. The vertical line corresponds to ${\varphi }^{″}=-3\pi ∕4{\delta }^2$.

Figure 2

Dressed-state picture of a ladder system with two intermediate states. The arrows indicate the different paths. For a negative chirp, the frequency decreases with time and paths (1)–(3) are followed as indicated by the arrow of time. The crossing between states $\mid k_i,n⟩$ and $\mid g,n+1⟩$ ($\mid e,n-1⟩$) is reached at time $t_i$ $\left(t_i^{\prime }\right)$. Time $t_c$ is the apparent creation time of the wave packet, whereas $t_d$ is the apparent detection time (see text).

Figure 3

(Color online) Diagram of the sodium levels involved. The two-photon transition is drawn here on resonance with the $4d$ final state.

Figure 4

Sodium levels in the dressed atom picture. Paths 1–4 are associated with the $5s$ final state and paths 5–8 with the $4d$ final state.

Figure 5

Calculated $4p\text{−}3s$ fluorescence as a function of the chirp. The laser parameters have been chosen to be similar to the ones used in the experiment: (a) ${\lambda }_0=611\mathrm{nm}$ $\left(\delta \lambda =28\mathrm{nm}\right)$, (b) ${\lambda }_0=599\mathrm{nm}$ $\left(\delta \lambda =24\mathrm{nm}\right)$, (c) ${\lambda }_0=596\mathrm{nm}$ $\left(\delta \lambda =28\mathrm{nm}\right)$, and (d) ${\lambda }_0=574\mathrm{nm}$ $\left(\delta \lambda =23\mathrm{nm}\right)$.

Figure 6

$4p\text{−}3s$ fluorescence as a function of the grating’s distance $L$. The values of the chirp ${\varphi }_{\lambda }^{″}$ are also indicated at $\lambda =603\mathrm{nm}$ and $\lambda =579\mathrm{nm}$ corresponding to the two-photon transitions. Experimental results (dots) with simulations (solid lines). The scans correspond to different wavelengths: (a) ${\lambda }_0=611\mathrm{nm}$ $\left(\delta \lambda =28\mathrm{nm}\right)$, (b) ${\lambda }_0=599\mathrm{nm}$ $\left(\delta \lambda =24\mathrm{nm}\right)$, (c) ${\lambda }_0=596\mathrm{nm}$ $\left(\delta \lambda =28\mathrm{nm}\right)$, and (d) ${\lambda }_0=574\mathrm{nm}$ $\left(\delta \lambda =23\mathrm{nm}\right)$.

Figure 7

Representation of the Wigner function of the excitation pulse, for one central wavelength $\left(591\mathrm{nm}\right)$, such that both two-photon transitions are involved. Cases (a)–(e) represent the Wigner function for different values of the gratings’ distance $L_n$ such that ${\varphi }^{″}({\omega }_n,L_n)=0$ where ${\omega }_n$ is a one-photon or two-photon resonance. (a) $L_a=15.48\mathrm{cm}$ for which ${\varphi }^{″}({\omega }_{3p\text{−}4d},L_a)=0$, (b) $L_b=14.16\mathrm{cm}$ for which ${\varphi }^{″}({\omega }_{3s\text{−}4d}∕2,L_b)=0$, (c) $L_c=13.0\mathrm{cm}$ for which ${\varphi }^{″}({\omega }_{3s\text{−}3p},L_c)=0$, (d) $L_d=11.64\mathrm{cm}$ for which ${\varphi }^{″}({\omega }_{3s\text{−}5s}∕2,L_d)=0$, and (e) $L_e=10.56\mathrm{cm}$ for which ${\varphi }^{″}({\omega }_{3p\text{−}5s},L_e)=0$. One can note that $11.6\mathrm{cm}$ and $14.1\mathrm{cm}$ correspond to the onset of interferences in the experimental curves in Fig. 6. The horizontal solid lines correspond to the central wavelengths of the direct two-photon transitions whereas the horizontal dashed lines correspond to the wavelengths of the sequential contributions.

Figure 8

Simulation of the $4p\text{−}3s$ fluorescence as a function of ${\varphi }^{″}$ for the artificially reduced cubic phase (solid line) and, as a comparison, for the experimental cubic phase (dashed line). Here $\lambda =596\mathrm{nm}$.