Control of quantum localization and classical diffusion in laser-kicked molecular rotors

We experimentally study a system of quantum kicked rotors—an ensemble of diatomic molecules exposed to a periodic sequence of ultrashort laser pulses. In the regime where the underlying classical dynamics is chaotic, we investigate the quantum phenomenon of dynamical localization by means of state-resolved coherent Raman spectroscopy. We examine the dependence of the exponentially localized angular momentum distribution and of the total rotational energy on the time period between the pulses and their amplitude. The former parameter is shown to provide control over the localization center, whereas the latter one controls the localization length. Similar control of the center and width of a nonlocalized rotational distribution is demonstrated in the limit of classical diffusion, established by adding noise to the periodic pulse sequence.

Figure 1

Top: Diagram of the pump and probe sources. Sequences of 13 pulses are generated by a pulse shaper; their energy is then boosted by a multipass amplifier (MPA). Another pulse shaper is used to narrow the spectral bandwidth of the probe pulse, whose central wavelength is shifted by means of second harmonic generation (SHG) in a nonlinear crystal. Bottom: Scheme of the experimental setup. A train of strong femtosecond pulses (pump) and a delayed weak probe pulse are focused on a supersonic jet of oxygen molecules in a vacuum chamber. The change in probe polarization is analyzed as a function of the wavelength by means of two crossed polarizers and a spectrometer.

Figure 2

Rotational resonance map for ${}^{16}\mathrm{O}_2$. Markers indicate the pulse train periods (divided by $T_{\text{rev}})$ for which the relative phase between the two neighboring sites $J$ and $J+2$ of the rotational lattice is equal to an integer multiple of $\pi$. Two sets of 10 vertical lines each indicate the experimental values: (a) $0.26\le T/T_{\text{rev}}\le 0.29$ and (b) $0.315\le T/T_{\text{rev}}\le 0.325$.

Figure 3

(a) Rotational Raman spectrum of oxygen molecules excited with a periodic train of 13 pulses at a kick strength of $P=4$ and averaged over 10 pulse trains with the periods indicated in Fig. 2. The experimental distribution (solid line) and the numerical simulation (dots with dashed line) are compared. (b) Exact calculated populations (dots with dashed line) and approximate populations (solid line) retrieved from the experimental Raman signal as discussed in the text.

Figure 4

Localized angular momentum distributions of oxygen molecules excited with a periodic train of 13 pulses for three kick strengths: (a, d) $P=4$, (b, e) $P=6$, and (c, f) $P=8$. Shown are the average distributions (solid lines) obtained from 10 pulse trains with a mean period of (a–c) $\overline{T}=0.275T_{\text{rev}}$ and (d–f) $\overline{T}=0.32T_{\text{rev}}$; see Fig. 2. The populations are fitted with exponential functions (dashed green lines) and compared to the initial thermal distribution at $T=25$ K (dotted gray line). The vertical line represents the excitation limit due to the finite pulse duration.

Figure 5

Same resonance map as in Fig. 2. Here, 120 vertical lines indicate the random periods of 10 different pulse sequences, 13 pulses each, following a Gaussian distribution with a mean and a standard deviation of (a) $\overline{T}/T_{\text{rev}}=0.34$ and $35\%$ and (b) $\overline{T}/T_{\text{rev}}=0.32$ and $43\%$, respectively. In case (a), no period is within 150 fs of any fractional resonance associated with $J=1$, 3, or 5. No such restriction was imposed in case (b). Dashed red lines mark the two mean periods.

Figure 6

Angular momentum distribution of oxygen molecules excited by nonperiodic trains of 13 pulses for three kick strengths: (a, d) $P=4$, (b, e) $P=6$, and (c, f) $P=8$. Shown are the average distributions (solid lines) obtained from 10 pulse trains with a mean and a standard deviation of (a–c) $\overline{T}=0.34T_{\text{rev}}$ and $35\%$ and (d–f) $\overline{T}=0.32T_{\text{rev}}$ and $43\%$, respectively; see Fig. 5. In the first case, (a–c), no period is within 150 fs of any fractional resonance associated with $J=1$, 3, or 5. The populations are fitted with Gaussian functions (dashed green lines) and compared to the initial thermal distribution at $T=25$ K (dotted gray line). The vertical line represents the excitation limit due to the finite pulse duration.

Figure 7

Rotational energy as a function of the number of kicks $N$ with a kick strength $P=4$ (red circles), $P=6$ (blue triangles), and $P=8$ (black squares). Compared are the energies for (a, c) periodic and (b, d) nonperiodic sequences, with periods that (a, b) avoid or (c, d) allow overlap with low-lying quantum resonances, as shown in Figs. 2 and 5.