Determination of atomic multiphoton-ionization phases by trichromatic multichannel wave-packet interferometry

We present the unambiguous determination of quantum phases in the multiphoton ionization (MPI) of potassium atoms by using a multichannel photoelectron interferometry scheme based on trichromatic pulse shaping. The colors of the laser field are chosen to produce three energetically separated photoelectron interferograms in the continuum. While the red pulse is two-photon resonant with the $3d$ state resulting in a $(2+1)$ resonance-enhanced MPI (REMPI), a $(1+2)$ REMPI occurs via the nonresonant intermediate $4p$ state with an initial green or blue pulse. We show that ionization via a nonresonant intermediate state lifts the degeneracy of photoelectron interferograms from pathways consisting of permutations of the colors. The analysis of the interferograms reveals a phase shift of $\pm \pi /2$ depending on the sign of the detunings in the $(1+2)$ REMPI pathways. In addition, we demonstrate that the photoionization time delay in the resonant $(2+1)$ REMPI pathway gives rise to a linear spectral phase in the photoelectron spectra. Insights into the underlying MPI processes are gained through an analytic perturbative description and numerical simulations of a trichromatic driven three-level system coupled to the continuum.

Figure 1

(a) Excitation scheme of potassium atoms interacting perturbatively with a linearly polarized trichromatic field for multichannel interferometry together with the experimental multiphoton spectrum. We use temporally overlapping trichromatic pulses with frequencies $2{\omega }_{\text{g}}={\omega }_{\text{r}}+{\omega }_{\text{b}}$ (black dashed box in the left figure) to achieve interference of photoelectrons by MPI with different combinations of red, green, and blue photons. The interference leads to three energetically separated photoelectron interferograms in the interference channels 1–3 (highlighted in orange, petrol blue, and violet). The frequencies are spectrally confined to be either two-photon resonant with the $3d$ state, i.e., $2{\omega }_{\text{r}}\simeq {\omega }_{\text{3d}}$, or one-photon near but nonresonant with the $4p$ state, i.e., ${\omega }_{\text{g}}\lesssim {\omega }_{\text{4p}}<{\omega }_{\text{b}}$ (right bottom inset). The near-resonant excitation of the $4p$ state lifts the degeneracy of the frequency mixing pathways and causes the second channel to split into two bands (${\varepsilon }_2^{\text{low}}$ and ${\varepsilon }_2^{\text{up}}$) around ${\varepsilon }_2$. (b) Measured and Abel-inverted photoelectron signals in polar coordinates. (c) Measured and tomographically reconstructed 3D PMD with color-coded interference channels at ${\varepsilon }_1\text{−}{\varepsilon }_3$.

Figure 2

(a) Three-level system with the states $|g\rangle , |a\rangle$, and $|b\rangle$ coupled to the continuum state $|f\rangle$ for the excitation by a laser field with the frequency ${\omega }_0$ and an intensity FWHM of $\mathrm{\Delta }t=90\mathrm{fs}$. The detunings between the exciting laser field and the respective states are ${\delta }_1$ and ${\delta }_2$. (b) Time-dependent photoelectron amplitude $c_f^{\text{non}}\left(t\right)$ in the complex plane representation for nonresonant excitation of the state $|b\rangle$ and different signs of the detuning ${\delta }_1$ (upper frame). Phase ${\chi }^{\text{non}}=arg\left\{c_f^{\infty ,\text{non}}\right\}$ and photoelectron amplitude $|c_f^{\infty ,\text{non}}{|}^2$ as a function of the detuning ${\delta }_1$ (lower frames). (c) Time-dependent population amplitudes $c_f^{\text{res}}(t;\varpi )$ for different values of $\varpi \left(\varepsilon \right)$ assuming resonant excitation of the state $|b\rangle$ and ${\delta }_1<0$ (upper frame). The middle frame shows the photoelectron amplitude $|c_f^{\infty ,\text{res}}{|}^2$ and corresponding MPI phase ${\chi }_{{\delta }_1<0}^{\text{res}}$ as a function of $\varepsilon$. The bottom frame illustrates the physical picture of the resonance-induced photoionization time delay, explained by windowing of the temporal field envelope $\mathcal{E}\left(t\right)$ with the time-dependent population amplitude of $|b\rangle$.

Figure 3

(a) Measured trichromatic spectrum together with the WLS. The inset shows a schematic of the pulse shaper setup with a broadband $p$ polarizer and the mechanical blocker in the Fourier plane to eliminate resonant excitation of the $4p$ state. (b) Energy sections of the time-resolved measurements on the green-red (left) and green-blue (right) frequency mixing signals from in situ temporal pulse characterization (circles and squares) together with Gaussian fits (solid and dashed lines) and corresponding FWHMs.

Figure 4

Simulated and measured phase maps of photoelectron interferograms from multichannel interferometry by varying the phase of the green pulse ${\phi }_{\text{g}}\in [-\pi ,\pi ]$. (a) Simulated phase map obtained by numerical solution of the TDSE for the perturbative interaction of a three-level atom with the trichromatic field having optical phases of ${\phi }_{\text{r}}=\pi , {\phi }_{\text{b}}=0$. (b) Measured phase map for ${\phi }_{\text{r}}=\pi , {\phi }_{\text{b}}=0$ with a step size of $\delta {\phi }_{\text{g}}=\frac{2\pi }{40}\approx 0.16\mathrm{rad}$, showing the three energetically separated interference channels around ${\varepsilon }_1–{\varepsilon }_3$. Bottom frame: Section through the phase maps along the phase axis at ${\varepsilon }_1$ (orange) and ${\varepsilon }_3$ (violet) to emphasize the modulation. (c) Same as the bottom frame of panel (b) but for ${\phi }_{\text{r}}=\pi , {\phi }_{\text{b}}=-\pi$. The vertical dashed lines serve as a guide to the eye for visualization of the phase shift between the orange and violet lines, respectively.

Figure 5

Simulated (a) and measured (b) phase contrast maps $\mathcal{C}(\varepsilon ,{\phi }_{\text{g}})$ with ${\phi }_{\text{r}}=\pi$ extracted from inverting the phase maps by changing the phase of the blue pulse from ${\phi }_{\text{b}}=0$ to $-\pi$. (a) Simulated phase contrast map based on the numerical results described in Sec. 4a, shown in Fig. 4. (b) Measured phase contrast map calculated from the results of Sec. 4a, shown in Fig. 4. The different channels are indicated by horizontal lines. To emphasize the interference channels, we apply super-Gaussian filters to blank the space between the regions of interest. (c–e) Sections through the measured phase contrast map (circles) at the kinetic energies of ${\varepsilon }_3$ (c), ${\varepsilon }_2^{\text{up}}$ (d), and ${\varepsilon }_2^{\text{low}}$ (e) together with cosine fits through the data (lines). The MPI phase differences $\mathrm{\Delta }\gamma$ are highlighted by vertical dashed black lines through the frames (c)–(e) together with their associated color-coded error bars.