Experimental and theoretical investigation of the crossover from the ultracold to the quasiclassical regime of photodissociation

At ultralow energies, atoms and molecules undergo collisions and reactions that are best described in terms of quantum-mechanical wave functions. In contrast, at higher energies these processes can be understood quasiclassically. Here, we investigate the crossover from the quantum-mechanical to the quasiclassical regime both experimentally and theoretically for photodissociation of ultracold diatomic strontium molecules. This basic reaction is carried out with a full control of quantum states for the molecules and their photofragments. The photofragment angular distributions are imaged and calculated by using a quantum-mechanical model as well as the Wenzel–Kramers–Brillouin approximation and a semiclassical approximation that are explicitly compared across a range of photofragment energies. The reaction process is shown to converge to its high-energy (axial-recoil) limit when the energy exceeds the height of any reaction barriers. This phenomenon is quantitatively investigated for two-channel photodissociation by using intuitive parameters for the channel amplitude and phase. While the axial-recoil limit is generally found to be well described by a commonly used quasiclassical model, we find that when the photofragments are identical particles, their bosonic or fermionic quantum statistics can cause this model to fail, requiring a quantum-mechanical treatment even at high energies.

Figure 1

Molecular potentials for the ${}^{88}\mathrm{Sr}_2$ electronic states used in this work. The ground state $X{0}_g^{+}$ correlates to the ${}^{1}S+{}^{1}S$ atomic threshold while the excited states $0_u^{+}$ and $1_u$ correlate to ${}^{1}S+{}^{3}P_1$. The $1_u$ potential has a $\sim 1\mathrm{mK}$ electronic barrier. The molecules are photodissociated via an electric-dipole optical transition from bound states of mostly $0_u^{+}$ or $1_u$ character to the ground-state continuum. A range of rovibrational initial states are explored. In this example, weakly bound $0_u^{+}(v=-2,J_i=1,M_i=0)$ molecules are photodissociated, and the photofragments occupy two allowed partial waves $J=\{0,2\}$ shown at the energy of 50 MHz (2.4 mK).

Figure 2

Measured and calculated photofragment angular distributions in the ultracold quantum-mechanical regime and in the high-energy axial-recoil limit. (a) The $0_u^{+}$ initial states are explored at the continuum energies $\varepsilon /h=\{13,33,53,100,300\}\mathrm{MHz}$ ($\varepsilon /k_B=0.6–14\text{mK}$). For each initial state, the upper and lower rows correspond to measurements and quantum-mechanical theory, respectively. The ab initio images were adjusted for the shell size and thickness in experimental data, where the photofragment ring diameter is typically 0.3–0.5 mm. The strong dependence of the patterns on $\varepsilon$ and the less pronounced dependence on $v$ are discussed in the text. The bottom row shows $v$-independent distributions obtained with the WKB approximation. The most weakly bound state exhibits a central dot that arises from spontaneous photodissociation into the optical lattice; these are more apparent at higher energies where the signal is weaker and can be ignored. (b) Same as in panel (a), for $1_u$ initial molecular states. (c) Angular distributions calculated for a pair of $0_u^{+}$ and $1_u$ weakly bound states using both quantum theory and the WKB approximation at a high energy $\varepsilon /h=1000\mathrm{MHz}$ ($\varepsilon /k_B=48\mathrm{mK}$), to show their close agreement with the appropriate axial-recoil limit.

Figure 3

(a) Quantum-mechanical calculation of the amplitude-squared $R$ parameter for two-channel photodissociation as in Eq. (4). The initial molecular state is $0_u^{+}(v,1,0)$ with $v=\{-2,-3,-4,-5\}$, and $P=0$. For comparison, the most deeply bound $v=0$ state is also shown. The axial-recoil limit is indicated with a dashed line. (b) The allowed continuum wave functions with $J=\{0,2\}$ are plotted for very low energy $\varepsilon /h=0.5\mathrm{MHz}$ ($\varepsilon /k_B=24\mu \mathrm{K}$). The negligible contribution of the $J=2$ partial wave due to its repulsive rotational barrier explains why $R=1$ as $\varepsilon \to 0$ in panel (a) and thus only the lowest allowed partial wave is populated in near-threshold photodissociation.

Figure 4

Dependence of the amplitude-squared $R$ parameter from Eq. (4) on energy and on initial angular momentum $J_i$. The initial molecular states are $0_u^{+}(0,J_i,0)$ (left column) and $1_u(0,J_i,0)$ (right column), and $P=0$. In each frame, the axial-recoil limit is indicated by a horizontal line, and the highest rotational barrier in the continuum is shown by a dashed vertical line. For larger $J_i$, the axial-recoil limit is approached at higher energies, but in all cases the crossover from the ultracold to the quasiclassical regime occurs at the energy scale set by the barrier height. The initial states with $J_i=3$, 7, 19 exhibit shape-resonance behavior.

Figure 5

A comparison of the quantum-mechanical, WKB, and semiclassical methods for evaluating $cos\delta$ for a range of initial molecular angular momenta $J_i$ within the $0_u^{+}$ electronic manifold (here $M_i=P=0$). The phase angle $\delta$ quantifies interference between the two allowed dissociation channels and plays a key role in the observed photofragment angular distributions. Both approximations approach the quantum-mechanical result at increasingly higher energies that are set by the rotational barrier heights (dashed vertical lines). All methods recover the axial-recoil limit (horizontal lines) at high energies, but this convergence is slower for the $\delta$ parameter than for the $R$ parameter in Fig. 4.

Figure 6

Quasiclassical and quantum-mechanical calculations of the photofragment angular distributions do not generally converge in the axial-recoil limit if the photofragments are identical particles. Experimental measurements always agree with the quantum theory [17]. Here the high-energy angular distributions are calculated with the quasiclassical [Eq. (6)] and quantum-mechanical [Eqs. (3), (11), and (12)] models for the $1_u$ initial states with (a) $J_i=1,M_i=\{0,1\},P=1$, (b) $J_i=3,M_i=\{1,2\},P=0$, and (c) $J_i=3,M_i=\{0,1,2,3\},P=1$.