Two-photon excitation dynamics in bound two-body Coulomb systems including ac Stark shift and ionization

One of the dominant systematic effects that shift resonance lines in high-precision measurements of two-photon transitions is the dynamic (ac) Stark shift. For suitable laser frequencies, the ac Stark shift acquires an imaginary part which corresponds to the rate of resonant one-photon ionization of electrons into a continuum state. At the current level of spectroscopic accuracy, the underlying time-dependent quantum dynamics governing the atomic two-photon excitation process must be well understood, and related considerations are the subject of the present paper. In order to illustrate the basic mechanisms in the transient regime, we investigate an analytically solvable model scenario for the population dynamics in the density matrix formalism and describe in detail how to generalize the corresponding equations of motion for individual experimental use. We also calculate the dynamic Stark shift for two-photon $S\text{−}S$ and $S\text{−}D$ transitions in bound two-body Coulomb systems and the corresponding two-photon transition matrix elements. In particular, we investigate transitions for which the $1S$ ground state or alternatively the metastable $2S$ state acts as the lower-energy state, and for which states with $n⩽20$ represent the upper states. Relativistic and radiative corrections to the excitation dynamics, and the corresponding limitations to the accuracy of the measurements, are briefly discussed. Our considerations suggest the general feasibility of a detection mechanism, offering high quantum efficiency, based on two-step three-photon resonant ionization spectroscopy, for large classes of experimentally relevant two-photon transitions in two-body Coulomb systems.

Figure 1

(Color online) Atomic population in the $2S$ state of hydrogen $(Z=1)$ as a function of interaction time $t$ with the laser, and of detuning $\mathrm{\Delta }\nu$ from the $1S\text{−}2S$ transition frequency, as defined in Eq. (11) with ${\nu }_{eg}\approx 2466\mathrm{THz}$. The laser intensity is $I=2.3\mathrm{MW}∕{\mathrm{m}}^2$, corresponding to $\Omega =2\pi \times 169\mathrm{Hz}$. The initial state at $t=0$ is the $1S$ state. In the time evolution of the system, ionization from the $2S$ state into the continuum and spontaneous decay of the $2S$ state have been neglected, i.e., ${\rho }_{ee}^{\prime }$ from Eq. (15) is plotted.

Figure 2

(Color online) Same situation as in Fig. 1, except that in this plot, the effect of ionization was included, as in Eqs. (13, 14). The ionization rate ${\gamma }_{\mathrm{i}}=2\pi \times 276\mathrm{Hz}$. For the short times considered here, there is only a small difference to Fig. 1 in the total excitation efficiency, which is due to the ionization loss from the excited state.

Figure 3

(Color online) Same as Fig. 1, but for interaction times that are comparable with the Rabi oscillation time. Ionization from the $2S$ state is not taken into account. At zero detuning, the ${\sin }^2$-shaped Rabi oscillations with full amplitude can be observed. All the other sections with constant detuning can also be understood as the well-known Rabi oscillations with diminished amplitude and generalized Rabi frequency ${\Omega }_{\mathrm{R}}=\sqrt{\mathrm{\Delta }{\omega }^2+{\Omega }^2}$ in complete analogy with one-photon transitions except that the two-photon Rabi frequency $\Omega$ as defined in Eq. (6) is proportional to the intensity, instead of the electric field amplitude.

Figure 4

(Color online) Same situation as in Fig. 3, except that here the ionization from the $2S$ state has been properly included. The presence of the ionization channel now makes a big difference as compared to Fig. 3, because at times comparable to the Rabi oscillation time, the $2S$ state is significantly populated and consequently the ionization probability is not negligible. The decay of the $2S$ population, spectral hole burning, and loss of coherence can be observed in this image. The maximum population of the $2S$ state is 0.175 [see Eq. (16)], whereas without ionization the $2S$ population repeatedly reaches 100% for $\mathrm{\Delta }\nu =0$. Including the $2S$ spontaneous two-photon decay of ${\gamma }_{\mathrm{s}}=2\pi \times 1.31\mathrm{Hz}$ does not change the plot discernibly.

Figure 5

(Color online) $2S$ population in a ${\mathrm{He}}^{+}$ ion, as a function of detuning [defined in Eq. (11)] and interaction time with the laser driving the $1S\text{−}2S$ transition. Ionization and spontaneous two-photon decay of the $2S$ state are taken into account. A constant intensity of $2.3\mathrm{MW}∕{\mathrm{m}}^2$ is assumed and the ion is in the $1S$ ground state at time $t=0$. On the time scale considered, the system evolves into a quasi-steady-state with approximate Lorentzian line shape.

Figure 6

(Color online) $2S$ population dynamics including two-photon spontaneous decay in a ${\mathrm{He}}^{+}$ ion as in Fig. 5, but on a much longer time scale. Here the decay of the population due to ionization is visible. Note that the effective population loss is by far smaller than the ionization rate ${\gamma }_{\mathrm{i}}=2\pi \times 17.3\mathrm{Hz}$ (corresponding to a characteristic ionization time of $58\mathrm{ms}$), because the quasi-steady-state population of the excited state is small. The peak population in the excited state is $10.6\times {10}^{-3}$ for the considered intensity of $2.3\mathrm{MW}∕{\mathrm{m}}^2$. The steep rise before $t=20\mathrm{ms}$ is shown in more detail in Fig. 5.

Figure 7

Peak $2S$ population in ${\mathrm{He}}^{+}$, as a function of intensity, including spontaneous decay (solid line) and without spontaneous decay (dashed line), which is equal to ${{\rho }_{ee}^{\prime }}^{\left(\max \right)}$ in Eq. (16), evaluated for the $1S\text{−}2S$ transition. For increasing intensity, the ionization rate eventually becomes large compared to the spontaneous decay rate, and the peak population increases, approaching the maximum ${{\rho }_{ee}^{\prime }}^{\left(\max \right)}$.

Figure 8

(Color online) Population in the continuum $P$ state as a function of detuning [defined in Eq. (11)] and interaction time with the laser, driving the hydrogen $1S\text{−}2S$ transition, including spontaneous two-photon decay of the intermediate $2S$ state with ${\gamma }_{\mathrm{s}}=2\pi \times 1.31\mathrm{Hz}$. A constant intensity of $2.3\mathrm{MW}∕{\mathrm{m}}^2$ is assumed and the atom is in the $1S$ ground state at $t=0$.

Figure 9

Level scheme of the $3S\text{−}1S$ spontaneous decay. The two-step one-photon channel with the rates ${\gamma }_{\mathrm{s}}^{1\gamma }$ dominates over the direct two-photon decay, and can be expressed by a direct effective rate ${\gamma }_{\mathrm{eff}}$. The two-step two-photon channel via the $2S$ state is omitted.

Figure 10

Combined induced-spontaneous decay process leading to a depopulation of the excited state $\mid e⟩$. Absorption or stimulated emission of one laser photon with energy $\hslash {\omega }_{\mathrm{L}}$ and subsequent spontaneous decay of the virtual intermediate states (dashed) to the ground state can take place, via emission of a photon of energy $\hslash {\omega }_{+}$ or $\hslash {\omega }_{-}$, respectively.