Femtosecond strong-field quantum control with sinusoidally phase-modulated pulses

The quantum control of the ionization of potassium atoms using shaped intense femtosecond laser pulses is investigated. We use sinusoidal phase modulation as a prototype for complex shaped pulses to investigate the physical mechanism of the strong-field quantum control by shaped femtosecond light fields. The influence of all parameters characterizing the sinusoidal phase modulation on strong-field-induced dynamics is studied systematically in experiment and theory. Our results are interpreted in terms of the selective population of dressed states (SPODS) which gives a natural physical picture of the dynamics in intense laser fields. We show that modulated femtosecond pulses in combination with photoelectron spectroscopy are a versatile tool to prepare and to probe SPODS. The decomposition of the excitation and ionization process induced by shaped pulses into elementary physically transparent steps is discussed.

Figure 1

Energy level diagram for excitation of K atoms. Shaped laser pulses with an electric field $E_{out}\left(t\right)$ and a carrier frequency ${\omega }_0$ (corresponding to $785\mathrm{nm}$) detuned by $\delta$ from the resonance frequencies (767 and $770\mathrm{nm}$) create a coherent superposition of the lower $4s$ and the upper $4p$ states of K atoms (bold arrows). Spectra of the photoelectrons with a kinetic energy $E_{kin}=\hslash {\omega }_e$ from simultaneous two-photon ionization (light arrows) of the $4p$ state are measured. The shape of the photoelectron spectra, e.g., (a) slow or (b) fast photoelectrons, is controlled by a variation of the pulse structure. The inset illustrates the creation of fast photoelectrons from the upper dressed state corresponding to (b).

Figure 2

Envelopes ${\mathcal{E}}_{out}^{+}\left(t\right)$ of spectrally modulated pulses. The amplitudes of the subpulses for sinusoidal phase modulation in frequency domain are described by the Bessel functions $J_n\left(A\right)$, cf. lower panel. Assuming a real unmodulated field envelope ${\mathcal{E}}_{in}^{+}\left(t\right)$ and $\mathrm{\Delta }\omega T+\varphi =0$, the modulated electrical field envelope ${\mathcal{E}}_{out}^{+}\left(t\right)$ is plotted for the values 0, 1, and $2.4$ of the modulation parameter $A$. For $A=1$ the indices $n=-3–3$ are plotted on top of the subpulses envelopes ${\mathcal{E}}_{in}^{+}(t+nT)$. The change of sign of the envelope of the odd postpulses is due to the property $J_{-n}\left(z\right)=(-1{)}^n{J}_n(z)$ of the Bessel function. At the zero of the Bessel function $J_0$ at $A\approx 2.4$ the central subpulse $(n=0)$ vanishes. The time separation between the subpulses $T$ was chosen to exceed the pulse width. In this case, the parameters $\mathrm{\Delta }\omega$ and $\varphi$ which control the relative phase of the subpulses do not influence the field envelope $\mid {\mathcal{E}}_{out}^{+}\left(t\right)\mid$.

Figure 3

Absolute temporal optical phase ${\chi }_{out}\left(t\right)$ (bold) and pulse envelope $\mid {\mathcal{E}}_{out}^{+}\left(t\right)\mid$ for a sinusoidally modulated $20\mathrm{fs}$ (FWHM) Gaussian pulse. The parameters for the phase modulation function $\phi \left(\omega \right)=A\sin \left[\right(\omega -{\omega }_{ref})T+\varphi ]$ are $A=1$ and $T=100\mathrm{fs}$. The phase difference between adjacent prepulses is $\mathrm{\Delta }\omega T+\varphi$ [cf. (d)] and for adjacent postpulses $\pi -(\mathrm{\Delta }\omega T+\varphi )$ [cf. (c)] due to the extra $\pi$ phase jump from the alternating sign of the Bessel functions. (a) “Pure sine” modulation with the origin of the sine function at the central frequency, i.e., $\mathrm{\Delta }\omega =0$ and $\varphi =0$. The temporal phase jumps of $\pm \pi$ are exclusively due to the alternating sign of the Bessel functions as illustrated in Fig. 2. (b) Pure cosine modulation, i.e., $\varphi =\pi ∕2$ with relative temporal phase jumps of $\pm \pi ∕2$. In the right column the modulation function is displaced from the central frequency by $\mathrm{\Delta }\omega T=\pi ∕2$ and absolute phases are $\varphi =\pi ∕4$ (b) and $\varphi =\pi ∕2$ (d).

Figure 4

Measured photoelectron spectra for sinusoidal phase modulation at the point of the intersection of the three scans of time $T$, amplitude $A$, and phase $\varphi$ at $A=-0.2$, $T=170\mathrm{fs}$, and $\varphi =1.7\mathrm{rad}$. Each scan is indicated in the three-dimensional parameter space as shown in the inset, i.e., the variation of $T$ for fixed $A$ and $\varphi$ (dotted), the variation of $A$ for fixed $T$ and $\varphi$ (bold), and the variation of $\varphi$ for fixed $A$ and $T$ (dashed). The agreement of all spectra demonstrates the reproducibility of the experimental results.

Figure 5

(Color online) Experimental setup: femtosecond laser pulses from a Ti:sapphire amplifier were spectrally phase-modulated using sinusoidal phase modulation functions of the form $\phi \left(\omega \right)=A\sin \left[\right(\omega -{\omega }_{ref})\bullet T+\varphi ]$ centered at the reference frequency ${\omega }_{ref}$ in our pulse shaper. The inset shows phase functions $\phi \left(\omega \right)$ for different $A$, $\varphi$, and $T$ on top of the laser spectrum (gray shaded). The modulated laser beam was focussed into a vacuum chamber to interact with a potassium atomic beam. Photoelectron kinetic energy spectra were recorded employing a magnetic bottle spectrometer (right).

Figure 6

(Color online) Characterization laser pulses subjected to sinusoidal spectral phase modulation using (a) spectrogram-based, (b) spectral, and (c) temporal techniques. (a) Retrieved temporal signal from the spectrogram: normalized intensity (solid line measured, white triangles calculated using measured spectrum and experimental phase modulation parameters) and phase (dashed line measured, black dots calculated as above) for $\phi \left(\omega \right)=1.5\sin \left[60\mathrm{fs}\right(\omega -2.4\mathrm{rad}∕\mathrm{fs})+\pi ]$, (b) spectral interference: spectrum (thin line), measured (orange solid line) and applied (solid line) spectral phase $\phi \left(\omega \right)=0.8\sin \left[200\mathrm{fs}\right(\omega -2.424\mathrm{rad}∕\mathrm{fs}\left)\right]$, (c) second harmonic cross correlation: measured (orange, upper) and simulated (black, lower) for $\phi \left(\omega \right)=\sin \left[500\mathrm{fs}\right(\omega -2.4\mathrm{rad}∕\mathrm{fs}\left)\right]$.

Figure 7

(Color online) (a) Simulated and (b) measured photoelectron spectra upon variation of the parameter $T$ within $5–300\mathrm{fs}$ for fixed values of $A=-0.2$ and $\varphi =1.7\mathrm{rad}$. The dashed line in (a) indicates the reference point at $T=170\mathrm{fs}$. In the lower panel (c) sections through the photoelectron distributions of (a) and (c) at $T=120\mathrm{fs}$ (left) and $T=175\mathrm{fs}$ (right) as indicated by the lines in (b) show the experimental results (bold lines) and the simulated photoelectron spectra (dashed lines). In (b) the energy of the slow photoelectrons at $0.35\mathrm{eV}$, the fast photoelectrons at $0.50\mathrm{eV}$, and the excess energy for weak-field ionization $\hslash {\omega }_e$ are also indicated. The cross hairs in (b) indicates an additional structure in the photoelectron spectra due to the fine structure splitting.

Figure 8

(Color online) (a) Simulated and (b) measured photoelectron spectra upon variation of the parameter $\varphi$ within $0.25–12.25\mathrm{rad}$ for fixed values of $A=-0.2$ and $T=170\mathrm{fs}$. The dashed line in (a) indicates the reference point at $\varphi =1.7\mathrm{rad}$. In the lower panel (c) sections through the photoelectron distributions of (a) and (b) at $\varphi =5.50\mathrm{rad}$ (left) and $\varphi =8.75\mathrm{rad}$ (right) as indicated by the lines in (b) show the experimental results (bold lines) and the simulated photoelectron spectra (dashed lines).

Figure 9

(Color online) (a) Simulated and (b) measured photoelectron spectra upon variation of the parameter $A$ within $-0.8–0.8$ for fixed values of $\varphi =1.7\mathrm{rad}$ and $T=170\mathrm{fs}$. The dashed line in (a) indicates the reference point at $A=-0.2$. In the lower panel (c) sections through the photoelectron distributions of (a) and (b) at $A=0.24$ (left) $A=-0.18$ and (right) as indicated by the lines in (b) show the experimental results (bold lines) and the simulated photoelectron spectra (dashed lines).

Figure 10

Simulation for resonant strong-field excitation using a sinusoidal phase modulated $30\mathrm{fs}$ FWHM Gaussian laser pulse. The modulation parameters are $A=0.3$, $T=100\mathrm{fs}$, $\varphi =\pi ∕2\mathrm{rad}$, and $\mathrm{\Delta }\omega =\pi ∕50{\mathrm{fs}}^{-1}$. (a) Envelope of the electrical field $\mid {\mathcal{E}}^{+}\left(t\right)\mid$ in units of the AT splitting $\mid \hslash \Omega \left(t\right)\mid =\mid \mu {\mathcal{E}}^{+}\left(t\right)\mid$ (bold, left scale) and temporal phase $\chi \left(t\right)$ of the electric field (dashed, right scale). The subpulses are indicated with labels from $n=$ 2 to $-2$. (b) Time evolution of the bare state population (ground state $\mid c_a{\mid }^2$ dashed, and excited state $\mid c_b{\mid }^2$ bold). (c) Simulated photoelectron spectrum: the dashed line indicates the kinetic energy of weak-field photoionization $\hslash {\omega }_e$ and the bar shows the AT splitting at the most intense subpulse $\mathbf{(}n=0$, compare value of $\mid \hslash \Omega \left(t\right)\mid$ in (a) at $t=0\mathbf{)}$. Only the low kinetic energy AT component is present in the photoelectron spectrum revealing that the lower dressed state is selectively populated during the subpulse of maximum laser intensity. The shoulders in the photoelectron spectrum with an energy separation of $2\pi \hslash ∕T$ arise from the interference of free electron wave packets. (d) Population of the dressed states (lower state $\mid d_1{\mid }^2$ dashed, and upper state $\mid d_2{\mid }^2$ bold).

Figure 11

Simulation for the modulation parameters $A=0.3$, $T=150\mathrm{fs}$, and $\varphi =\pi ∕2\mathrm{rad}$. Only the high kinetic energy AT component is present in the photoelectron spectrum (c) revealing that the upper dressed state is selectively populated during the subpulses of the maximum laser intensity $(n=0)$ as seen in (d). For labels see Fig. 10.

Figure 12

Simulation for the modulation parameters $A=0.3$, $T=100\mathrm{fs}$, and $\varphi =-\pi ∕2\mathrm{rad}$. Only the high kinetic energy AT component is present in the photoelectron spectrum (c) revealing that the upper dressed state is selectively populated during the subpulses of the maximum laser intensity $(n=0)$ as seen in (d). For labels see Fig. 10.

Figure 13

Simulation for the modulation parameters $A=1$, $T=100\mathrm{fs}$, and $\varphi =\pi ∕2\mathrm{rad}$. In order to compensate the reduction of the peak intensity due to the higher modulation amplitude (cf. Fig. 2) the laser field was increased to $2.3\times {\mathcal{E}}_{in}^{+}\left(t\right)$ resulting in a larger AT-splitting $\hslash \mid \Omega \mid$ of $0.27\mathrm{eV}$ (c). Predominantly, the low kinetic energy AT component is present in the photoelectron spectrum (c) revealing that the lower dressed state is more populated during the whole excitation process (d). Ionization during the prepulses and postpulses $(n\ne 0)$ becomes more probable due to their increased relative intensity and therefore the interference fringes with a spacing of $2\pi \hslash ∕T$ are more pronounced. Partial photoelectron spectra due to the ionization during the central subpulse ($n=0$, dashed-dotted) and the ionization during the adjacent pulse ($n=1$, dashed) are shown in addition. For labels see Fig. 10.

Figure 14

Bloch sphere representation $\rho \left(t\right)$ of the dynamics induced by the excitation of ground state atoms $\rho (-\infty )=\{0,0,-1\}$ with sinusoidally modulated resonant femtosecond laser pulses in the frame rotating with the central laser frequency. The modulation parameters are $A=0.3$, $T=100\mathrm{fs}$, $\varphi =\pi ∕2\mathrm{rad}$, and $\mathrm{\Delta }\omega =\pi ∕50{\mathrm{fs}}^{-1}$. (a) shows how the optical phase $\chi$ controls the position of the angular velocity vector ${\Omega }_B$ defined in Eq. (A8). The Bloch representation of the reference case as depicted in Fig. 10 is shown in (b). The dynamics upon variation of the time ($T=150\mathrm{fs}$, cf. Fig. 11) shown in (c) is identical with the dynamics upon variation of the phase ($\varphi =-\pi ∕2$, cf. Fig. 12). The dynamics for an amplitude of $A=1$ and a field of $2.3\times {\mathcal{E}}_{in}^{+}$ (cf. Fig. 13) is shown in (d). The angular velocity vectors ${\Omega }_0$ and ${\Omega }_{\pm 1}$ proportional to the peak electrical fields of the most intense subpulse $(n=0)$ and the adjacent prepulses $(n=\pm 1)$ are indicated in order to emphasize the orientation of ${\Omega }_B\left(t\right)$ and $\rho \left(t\right)$ which determines the transient SPODS. In panel (d) the black dot indicates the transition from the prepulse $(n=1)$ to the central subpulse, whereas the white dot marks the transition to the first postpulse $(n=-1)$.