Time- and frequency-resolved detection of atomic coherence in the regime of strong-field interaction with intense femtosecond laser pulses

Understanding the effect of strong laser pulses on the evolution of an atomic or molecular wave function is important in the context of coherent control in the strong-field regime, when power broadening and dynamic Stark shifts become comparable with or bigger than the bandwidth of the control field. We experimentally demonstrate the method of complete characterization of a complex-valued amplitude of a quantum state driven by a strong two-photon field. The method is based on coherent scattering of a weak probe pulse from the strong-field-induced atomic coherence, followed by the detection of the time- and frequency-resolved parametric four-wave-mixing signal. We show that the proposed technique corresponds to a cross-correlation frequency-resolved optical gating (XFROG) of the highly perturbed evolution of an atomic quantum state. Utilizing the XFROG retrieval algorithm, we determine both the amplitude and phase of an atomic wave function at any time moment throughout the interaction with the driving field. The direct retrieval of the time-dependent phase of the wave function, rather than the population dynamics only, enables us to observe the strong-field effects with arbitrary time and frequency resolution.

Figure 1

Parametric four-wave-mixing process in atomic rubidium used in this work. Two strong pump fields, pump 1 and pump 2 (wide vertical arrows), move the atomic population and induce coherence between the ground level $|1\rangle \equiv |5s\rangle$ and excited level $|3\rangle \equiv |4d\rangle$ of Rb. The latter is detected by scattering a weak off-resonant probe pulse (solid narrow arrow) and detecting the FWM signal (dashed narrow arrow). Hyperfine splitting of $|5p\rangle$ and $|4d\rangle$, as well as two other electronic states $|6s\rangle$ and $|5d\rangle$, shown in the upper right corner, were taken into account in the numerical calculations described in Sec. 2. The inset shows a simplified three-level model used in the theoretical analysis of Sec. 4.

Figure 2

(a) Example of a pulse sequence for a given value of probe delay $\tau$. (b) Example of the numerically calculated populations $|a_n(t)|{}^2$ [Eq. (3)]. Weak oscillations near zero (green and red) correspond to states $5p_{1/2}$ and $5p_{3/2}$ (not labeled). Examples of the calculated 2D time-frequency “FWM spectrograms” $I_4({\omega }_4,\tau )$ are shown for (c) weak- and (d) strong-field excitation.

Figure 3

Diagram of the experimental setup. Ti:sapphire regenerative amplifier generates 35-fs, 3-mJ pulses at a 1-kHz repetition rate. Fundamental radiation at 800 nm serves as pump 1. An OPA is used to produce the pump 2 and probe pulses at 1425 and 1824 nm, respectively. Both pumps are spectrally narrowed with home-built spectral filters down to a bandwidth corresponding to $\approx$$800$-fs pulse length. All three input pulses are overlapped in BOXCARS geometry and focused into 5-cm optical path Rb cell. A FWM signal is coupled into a spectrometer and detected by a cooled CCD camera.

Figure 4

2D FWM spectrograms for the case of weak-field excitation. (a) Experimental signal detected for the energies of pumps 1 and 2 equal to $\approx$$\text{0.2}$ and $\approx$$\text{0.02}\mu \text{J}$, respectively. (b) Numerical simulations with the energies of both pump pulses set to 0.01 $\mu \text{J}$. Both spectrograms show signal rising around zero delay time between the scanned probe and fixed overlapping pumps. The observed rise time corresponds to the duration of both pumps (800 fs). The long tail reflects the long relaxation time of the excited $|4d\rangle$ state of Rb.

Figure 5

2D FWM spectrograms for the case of strong-field excitation. (a) Energies of pumps 1 and 2 are $\approx$$\text{0.2}$ and $\approx$$\text{0.1}\mu \text{J}$, respectively. (b) Numerical results for the pulse energies of $\text{0.275}$ and $\text{0.175}\mu \text{J}$. The double-peak structure around time zero represents complicated transient dynamics of the atomic system interacting with two strong pump fields. The long tail at positive delay corresponds to a field-free relaxation of the excited $|4d\rangle$ state of Rb.

Figure 6

Amplitude (thin black) and phase (thick red) of a weak-field atomic response function,retrieved from the (a) experimentally observed and (b) numerically calculated FWM spectrograms. Retrieved spectra show a single resonance corresponding to the $|4d\rangle$ state of rubidium. Its line width is dictated by the spectral resolution of the XFROG method. The spectral phase shows a sharp step across the resonance, typical for a Lorentzian response.

Figure 7

Amplitude (thin black) and phase (thick red) of a strong-field atomic response function, retrieved from the (a) experimentally observed and (b) numerically calculated FWM spectrograms [Figs. 5a and 5b, respectively]. The retrieved spectrum shows a narrow resonance, resulting from a field-free decay of $|4d\rangle$, superimposed onto a broad transient response. The latter exhibits clear power broadening and spectral sidebands corresponding to the dynamic Stark splitting under the strong pump excitation.

Figure 8

(a) Husimi representation of the experimentally detected atomic response to strong-field excitation, in which spectral resolution has been set to 1.8 nm. (b) Numerically calculated four-wave-mixing spectrogram for a transform limited 500-fs probe pulse (spectral bandwidth of 1.8 nm).

Figure 9

(a) Husimi representation of the experimentally detected atomic response to strong-field excitation, in which spectral resolution has been set to 0.9 nm. (b) Numerically calculated four-wave-mixing spectrogram for a transform limited 1-ps probe pulse (spectral bandwidth of 0.9 nm).

Figure 10

Vertical cross sections of the 2D plots in Figs. 9a and 9b, respectively, at time zero.