Noisy Optical Pulses Enhance the Temporal Resolution of Pump-Probe Spectroscopy

Time-resolved measurements of quantum dynamics are based on the availability of controlled events that are shorter than the typical evolution time scale of the processes to be observed. Here we introduce the concept of noise-enhanced pump-probe spectroscopy, allowing the measurement of dynamics significantly shorter than the average pulse duration by exploiting randomly varying, partially coherent light fields consisting of bunched colored noise. These fields are shown to be superior by more than a factor of 10 to frequency-stabilized fields, with important implications for time-resolved experiments at x-ray free-electron lasers and, in general, for measurements at the frontiers of temporal resolution (e.g., attosecond spectroscopy). As an example application, the concept is used to explain the recent experimental observation of vibrational wave-packet motion in $D_2^{+}$ on time scales shorter than the average pulse duration.

Figure 1

The essential elements of the pump-probe light-matter interaction experiment. (a) The temporal pulse shapes of (top to bottom) a 1.12 fs short pulse, a 30 fs pulse, and a set of the FEL pulses. (b) A schematic of potential-energy curves of $D_2$ and its cations, indicating the pathway of the sequential TPDI process.

Figure 2

2DAC functions $A_c^{(2)}(t_c,\tau )$, representing the intensity autocorrelation of a pair of identical (pump and probe) pulses as a function of their time delay $\tau$: (a) for a statistical average over 2000 30 fs FEL pulses, (b) for 30 fs bandwidth-limited (Gaussian) pulses, and (c) for 1.12 fs short pulses corresponding to the Fourier transform of the average FEL spectrum assuming a flat spectral phase. Narrow lines for $t_c=0$ and $t_c=\pm \tau$ are observed in the averaged FEL pulse case resulting from the correlated nature of the noise in pump and probe pulses on a time scale of the coherence time. These narrow temporal features are the origin of the enhanced temporal resolution. The number of doubly ionized molecules $N_{\mathrm{total}}$ is also displayed (red curves). A lineout at a time delay of 100 fs is shown in (d) for the FEL (solid blue line), the bandwidth-limited 30 fs (dotted red line), and the 1.12 fs short pulses (dashed black line).

Figure 3

The internuclear wave-packet evolution $|\Psi (R,t_c){|}^2$ is displayed in (a), yielding the molecular response function $M(E_{\mathrm{KER}},t_c)$ by employing the Coulomb-explosion mapping. By using the 2DAC functions $A_c^{(2)}(t_c,\tau )$ shown in Fig. 2, the KER distributions as a function of pump-probe time delay $\tau$ (as measurable in the experiment) $N(E_{\mathrm{KER}},\tau )$ can be directly obtained by integration: $N(E_{\mathrm{KER}},\tau )\propto {\int }_0^{+\infty }d{t}_{c}M(E_{\mathrm{KER}},t_c)A_c^{(2)}(t_c,\tau )$. Results for $N(E_{\mathrm{KER}},\tau )$ are shown for an average over $2000$ FEL pulses (b), a 30 fs bandwidth-limited pulse (c), and a 1.12 fs pulse (d). While no temporal structure is visible for the 30 fs bandwidth-limited pulse (c), the 30 fs FEL pulses, exhibiting noisy substructure, resolve the dynamics (b).

Figure 4

(a) The central section of $A_c^{(2)}$ [from Fig. 2d] for FEL pulses (blue line) is compared to the sum of 30 fs bandwidth-limited and 1.12 fs short pulses (red line). (b) Fourier transform (absolute value) of the averaged FEL KER-vs-$\tau$ trace [Fig. 3b]. (c) Pump-probe scans of the FEL pulses after removing the dc component (by Fourier transform): The result approximately recovers the dynamics obtained for the coherence-time limited 1.12 fs pulses [cf. Fig. 3d].