Chirping a two-photon transition in a multistate ladder

We consider a two-photon transition in a specific ladder system driven by a chirped laser pulse. In the weak field limit, we find that the excited state probability amplitude arises due to interference of multiple quantum paths which are weighted by quadratic phase factors. The excited state population has the form of a Gauss sum which plays a prominent role in number theory.

Figure 1

Chirping a two-photon excitation through a single intermediate state (top) and through an equidistant manifold (bottom). The left and right columns show the level system and the resulting excitation probabilities as a function of the chirp parameter ${\varphi }^{″}$, respectively. In both systems the ground state $\mid g⟩$ is connected by a two-photon transition to the excited state $\mid e⟩$. In the case of a single intermediate state $\mid m⟩$ with the offset $\delta >0$ with respect to the central frequency ${\omega }_0$ discussed in the top row the population $\mid c_e{\mid }^2$ of the excited state displays oscillations for negative chirps, whereas for positive values of the chirp it is decreasing rapidly. The oscillations result from an interference of the direct and the indirect excitation. The direct path is a direct two-photon transition through the virtual state of frequency ${\omega }_0$. The indirect path consists of first an excitation to the level $\mid m⟩$ and then a transition to $\mid e⟩$. Here we have used the parameters $\delta =0.0225{\mathrm{fs}}^{-1}$, $\mathrm{\Delta }\omega =0.1525{\mathrm{fs}}^{-1}$, ${\Omega }_{em}{\Omega }_{mg}=1$ as suggested by the experiment [14]. In the bottom row we discuss the case of a manifold of $M^{\prime }+M+1$ equidistant intermediate states $\mid m⟩$ with $-M^{\prime }\le m\le M$ which are shifted by the offset ${\delta }_m=\delta +m\Delta$ with respect to the central frequency ${\omega }_0$. The offset of the central state is $\delta$ whereas the states in the harmonic manifold are separated by $\Delta$. The population $\mid c_e{\mid }^2$ of the excited state is due to interference of competing quantum paths via the intermediate states and displays strong modulations as a function of the chirp. Parameters are $M^{\prime }=7$, $M=7$, $\delta =0.0225{\mathrm{fs}}^{-1}$, $\Delta =0.003{\mathrm{fs}}^{-1}$, $\mathrm{\Delta }\omega =0.1525{\mathrm{fs}}^{-1}$, and ${\Omega }_{em}{\Omega }_{mg}=1$.

Figure 2

Complementary error function in its dependence on the chirp parameter $a$ (left bottom) as obtained by integrating along a straight path in complex space (top). The starting point ${\zeta }_m\left(a\right)$ of this path traverses as a function of $a$ a trajectory in complex space which for large positive and large negative values of $a$ approaches the diagonals of the first and the second quadrants, respectively. The path is always a straight line which is tangent to this trajectory and tends towards infinity as shown on the top. The gray curve below translates between $a$ and $\mathrm{Re}{\zeta }_m$.

Figure 3

Paths ${\mathcal{C}}_{+}$ and ${\mathcal{C}}_{-}$ of integration in the complex plane corresponding to a positive and a negative value of the chirp parameter $a$. We depict the domains in the complex plane where the integrand $e^{-z^2}$ of the complementary error function increases by shaded areas. For $0<a$ the starting point ${\zeta }_m$ of the integration is located in the first quadrant and the straight path ${\mathcal{C}}_{+}$ is inclined with a steepness slightly smaller than the first diagonal. Thus ${\mathcal{C}}_{+}$ proceeds almost completely in the domain of the complex plane where the integrand $\exp (-z^2)$ of $\mathrm{erfc}$ decays. For $a<0$ the path ${\mathcal{C}}_{-}$ of integration starts in the second quadrant and proceeds slightly above the second diagonal into the decaying region of $e^{-z^2}$. In order to evaluate the resulting integral we close the path as indicated by the dashed line consisting of the paths ${\mathcal{C}}_{-}$, ${\mathcal{C}}_{\infty }$, ${\mathcal{C}}_r$, ${\mathcal{C}}_{-\infty }$, and ${\overline{\mathcal{C}}}_{+}$.

Figure 4

Origin of Gauss sum from phase space interpretation of two-photon excitation induced by a chirped pulse. In the phase space spanned by frequency $\nu$ and time $t$ we indicate the ground state $\mid g⟩$, the intermediate level $\mid m⟩$ and the excited state $\mid e⟩$ of frequencies $0,{\omega }_0+{\delta }_m$, and $2{\omega }_0$, respectively, by thick horizontal lines. We also denote the one-photon resonance of energy ${\omega }_0$ and the virtual level of frequency ${\omega }_0-{\delta }_m$ by dotted horizontal lines. The instantaneous frequency $\omega \left(t\right)={\omega }_0+2{\alpha }_{i}t$ is depicted for one negative value of the chirp by the tilted thin line. A stationary phase analysis of the double integral Eq. (8) determining the two-photon transition probability amplitude brings out most clearly that the sequential excitation appears at the times $t_s^{″}\equiv {\delta }_m∕\left(2{\alpha }_i\right)$ and $t_s^{\prime }$. Indeed, at $t_s^{″}$ the instantaneous frequency is resonant with the transition from $\mid g⟩\to \mid m⟩$ whereas at $t_s^{\prime }$ it is resonant with the transition from $\mid m⟩$ to $\mid e⟩$ as indicated by the dotted and the thick arrow. In this phase space representation the times when the transitions occur are determined by the crossings of the tilted thin line with the thick and the lower dotted horizontal lines corresponding to the frequencies ${\omega }_0+{\delta }_m$ and ${\omega }_0-{\delta }_m$. The enclosed shaded area $A_1+A_2$ corresponds to the phase acquired during the transition. Each triangle has the area ${\delta }_m{t}_s^{″}∕2={\delta }_m^2∕\left(4{\alpha }_i\right)$. As a consequence, the phase associated with this transition is quadratic in the offset ${\delta }_m$. In the case of a multistate ladder with a harmonic manifold we have the interference of all excitation paths, that is the weighted sum of phase factors with quadratic phases giving rise to a Gauss sum.

Figure 5

Comparison between the exact (black dots) and the approximate (gray dots) expression for the scaled weight factors $\mid w_m∕d_m\mid$ given by Eqs. (58, 60), respectively. Here we consider the case $N=15$ with $\mathrm{\Delta }m=\mathrm{\Delta }\omega ∕\Delta =50.8$ for the negative dimensionless chirp $a=-10824<0$.