Collision-Induced ${\mathrm{C}}_{60}$ Rovibrational Relaxation Probed by State-Resolved Nonlinear Spectroscopy

Quantum state-resolved spectroscopy was recently achieved for ${\mathrm{C}}_{60}$ molecules when cooled by buffer gas collisions and probed with a midinfrared frequency comb. This rovibrational quantum state resolution for the largest molecule on record is facilitated by the remarkable symmetry and rigidity of ${\mathrm{C}}_{60}$, which also present new opportunities and challenges to explore energy transfer between quantum states in this many-atom system. Here we combine state-specific optical pumping, buffer gas collisions, and ultrasensitive intracavity nonlinear spectroscopy to initiate and probe the rotation-vibration energy transfer and relaxation. This approach provides the first detailed characterization of ${\mathrm{C}}_{60}$ collisional energy transfer for a variety of collision partners, and determines the rotational and vibrational inelastic collision cross sections. These results compare well with our theoretical modeling of the collisions, and establish a route towards quantum state control of a new class of unprecedentedly large molecules.

Popular Summary

Understanding nonequilibrium behavior of complex quantum systems is a primary focus of modern physics. Precision spectroscopy of large molecules provides a tool to explore these phenomena at the level of individual quantum states. For buckminsterfullerene ${\mathrm{C}}_{60}$, which features unrivaled symmetry and has inspired broad scientific interest, its size and complexity have thwarted efforts at quantum state-resolved measurements for three decades. Recently, a combination of buffer gas cooling and cavity-enhanced direct frequency comb spectroscopy enabled rovibrational state-resolved absorption spectroscopy of ${\mathrm{C}}_{60}$ for the first time. While this initial study revealed detailed structural information about ${\mathrm{C}}_{60}$, including its peculiar nuclear spin statistics, understanding its relaxation dynamics has remained an outstanding scientific challenge. How such a large and symmetric molecular cage partitions collision energy into its many internal degrees of freedom was heretofore unknown.

In the current work, we study these nonequilibrium dynamics by driving a competition between optical pumping and various collisional relaxation mechanisms. This competition manifests in the measured saturation behavior of ${\mathrm{C}}_{60}$ rovibrational absorption features, culminating in the first measurement of rotational and vibrational quenching cross sections between ${\mathrm{C}}_{60}$ and various atomic and molecular collision partners.

The work opens opportunities for understanding and controlling complex quantum systems such as fullerenes. Moreover, the completeness of our data serves as a benchmark and motivation for future theoretical work on complex quantum many-body systems. We believe this work will spur an era of probing and manipulating a class of unprecedentedly large molecules and establish ${\mathrm{C}}_{60}$ as a new platform for quantum science.

Figure 1

Pumping and collisional relaxation processes in BG-cooled ${\mathrm{C}}_{60}$. (a) Interactions of ${\mathrm{C}}_{60}$ with its environment. ${\mathrm{C}}_{60}$ diffuses into the optical cavity mode, where it is pumped by the cw intracavity field before diffusing out again. The experiment monitors the frequency-dependent nonlinear absorption of the intracavity field by ${\mathrm{C}}_{60}$. Collisions with the BG control diffusion and thermalize the internal rovibrational energy of ${\mathrm{C}}_{60}$. (b) Energy-level diagram of ${\mathrm{C}}_{60}$ undergoing interactions with the environment. Optical pumping at a rate $R_{\mathrm{OP}}$ drives the $1185{\mathrm{cm}}^{-1}(8.4\mu \mathrm{m}$) $T_{1u}(3)$ vibrational band of ${\mathrm{C}}_{60}$. Collisions with the BG redistribute rotational population at a rate ${\mathrm{\Gamma }}_{\mathrm{rot}}$, assumed to be identical in $v^{″}=0$ and $v^{\prime }=1$. Vibrational relaxation is modeled in two steps: loss of vibrational population from $v^{\prime }=1$ into the dense reservoir of dark vibrational states at a rate ${\mathrm{\Gamma }}_{\mathrm{vib}}$, and leakage from the dense reservoir into $v^{″}=0$ at a rate ${\mathrm{\Gamma }}_{\mathrm{vib}}^{\prime }$. Filled purple curve depicts the steady-state population distribution.

Figure 2

Effect of single-state versus multistate pumping on rotational relaxation in ${\mathrm{C}}_{60}$. (a) State-selective pumping in the $R$ branch. Only a single transition can be resonantly pumped at a time; therefore, rotational relaxation from ${\mathrm{C}}_{60}$-BG collisions efficiently redistributes pumped rotational populations (green arrows). (b) Multistate pumping in the $Q$ branch, with many transitions (up to about $75$ at the band head) being pumped simultaneously. (c) Rate equation model fits to ${\mathrm{C}}_{60}$-Ar $R$-branch-saturated absorption data. Sampling of $R$-branch absorption data at $J=170$ (top) and $J=270$ (bottom), and corresponding rate equation fits (full ${\mathrm{C}}_{60}$-Ar $R$-branch data are given in Appendix pp9). Good agreement is obtained by modeling state-to-state rotational relaxation with an exponential-gap law (see the text). Etalons have been fitted out and subtracted from the data. Gray bands indicate the one-sigma uncertainty in the data. (d) Rate equation model fits to ${\mathrm{C}}_{60}$-Ar $Q$-branch saturated absorption data. The $Q$-branch absorption data and rate equation model fit (described in the text) taken at high (upper panel) and low (lower panel) Ar pressure (specified above each panel) at four pumping intensities. Only every fourth data point is plotted for clarity. Gray bands indicate standard uncertainty in the data dominated by residual etalons in the optical setup. The color of traces in panels (c) and (d) correspond to different pumping intensities, indicated by the color bar.

Figure 3

Average rotational inelastic collision cross sections $⟨{\sigma }_{\mathrm{rot}}⟩$ extracted from rate equation model fits to ${\mathrm{C}}_{60}$-BG-saturated absorption data. Total pressure-broadening cross section, ${\sigma }_{\mathrm{PB}}$, obtained from $R$-branch fits, are plotted for reference. For rare gases with no rotational structure, $⟨{\sigma }_{\mathrm{rot}}⟩$ decreases with decreasing mass, reaching a minimum for He. By comparison, the presence of a rotational degree of freedom in ${\mathrm{D}}_2$ and ${\mathrm{H}}_2$ results in a drastic increase of $⟨{\sigma }_{\mathrm{rot}}⟩$, considering the small mass of both molecules. Error bars indicate $68\mathrm{\%}$ confidence intervals and include the uncertainty on the effective diffusion length $x_{\mathrm{eff}}$. Calculated values of $⟨{\sigma }_{\mathrm{rot}}{⟩}^{\mathrm{th}}$ for ${\mathrm{C}}_{60}$-Ar and ${\mathrm{C}}_{60}$-He are plotted for comparison.

Figure 4

Calculated potential energy surfaces for ${\mathrm{C}}_{60}$-BG interactions. (a) Potential energy surface for ${\mathrm{C}}_{60}$-Ar calculated at the equilibrium bond length of $7.2\text{Å}$. Polar $\theta$ and azimuthal $\varphi$ angles are defined with respect to a Cartesian coordinate system with its $x$ and $z$ axes along twofold and fivefold symmetry axes of ${\mathrm{C}}_{60}$, respectively. The potentials have minima when the Ar atom is located “above” a five- or six-membered ring of ${\mathrm{C}}_{60}$. Local maxima occur in between the minima. The potential depth at these maxima is 20% to 30% smaller than at the minima. (b) Potential energy surface for ${\mathrm{C}}_{60}$-He calculated at the equilibrium bond length of $7.0\text{Å}$. Note the change in vertical scale. The potential anisotropy for this complex is significantly reduced compared to that of ${\mathrm{C}}_{60}$-Ar, and is reflected in the smaller value of its inelastic rotational cross section.

Figure 5

Optical beam path for the experiment. The $8.4\mu \mathrm{m}$ beam from the QCL is split into two paths by a beam splitter (BS). (1) The reflected beam is locked to a reference cavity that provides optical feedback to narrow the laser linewidth (see the text). To enable optical feedback stabilization, a delay stage and bullet mount piezo mirror [27] provide slow and fast control of the round-trip optical phase, while a piezo-mounted cavity mirror stretches the reference cavity length. (2) The transmitted beam is PDH locked to the spectroscopy cavity, with a double-pass AOM and piezo-mounted cavity mirror for the fast and slow frequency feedback, respectively. The absorption signal is monitored by a photodiode on the cavity transmitted output. Double-sided arrows depict the direction of travel of the mirrors. PBS, polarizing beam splitter; QWP, quarter waveplate; PD, photodetector.

Figure 6

Our calculated dipole polarizability tensor elements of ${\mathrm{C}}_{60}$ as functions of imaginary frequency $\mathrm{Im}(\omega )$. The off-diagonal tensor elements have been multiplied by ${10}^7$ for clarity.

Figure 7

Our calculated semiclassical elastic cross sections ${\sigma }_{\mathrm{el}}$ as functions of the relative collision velocity $v_{\mathrm{rel}}$.

Figure 8

Pumping of a single velocity class (Doppler shifted to ${\nu }_i+\delta$) by counterpropagating beams with lab-frame detuning $\mathrm{\Delta }$.

Figure 9

Full suite of $J$-dependent $R$-branch data and rate equation model fits for Ar-${\mathrm{C}}_{60}$ data.

Figure 10

Best-fit Ne-${\mathrm{C}}_{60}Q$-branch profile.

Figure 11

Best-fit Ne-${\mathrm{C}}_{60}R$-branch profiles.

Figure 12

Best-fit He-${\mathrm{C}}_{60}Q$-branch profile.

Figure 13

Best-fit He-${\mathrm{C}}_{60}R$-branch profiles.

Figure 14

Best-fit ${\mathrm{D}}_2$-${\mathrm{C}}_{60}Q$-branch profile.

Figure 15

Best-fit ${\mathrm{D}}_2$-${\mathrm{C}}_{60}R$-branch profiles.

Figure 16

Best-fit ${\mathrm{H}}_2$-${\mathrm{C}}_{60}Q$-branch profile.

Figure 17

Best-fit ${\mathrm{H}}_2$-${\mathrm{C}}_{60}R$-branch profiles.

Figure 18

Potential energy curves for ${\mathrm{C}}_{60}$-Ar and ${\mathrm{C}}_{60}$-He. Panels (a) and (b) show strengths $V_{l,m}(R)$ for ${\mathrm{C}}_{60}$-Ar and ${\mathrm{C}}_{60}$-He as functions of $R$, respectively. The black curves correspond to the isotropic potentials, which dominate in the atom-molecule bond. The meaning of the line colors in panels (c) and (d) is the same.