Lattice-model parameters for ultracold nonreactive molecules: Chaotic scattering and its limitations

We calculate the parameters of the recently derived many-channel Hubbard model that is predicted to describe ultracold nonreactive molecules in an optical lattice, going beyond the approximations used by Doçaj et al. [A. Doçaj et al., Phys. Rev. Lett. 116, 135301 (2016)]. Although those approximations are expected to capture the qualitative structure of the model parameters, finer details and quantitative values are less certain. To set expectations for experiments, whose results depend on the model parameters, we describe the approximations' regime of validity and the likelihood that experiments will be in this regime, discuss the impact that the failure of these approximations would have on the predicted model, and develop theories going beyond these approximations. Not only is it necessary to know the model parameters in order to describe experiments, but the connection that we elucidate between these parameters and the underlying assumptions that are used to derive them will allow molecule experiments to probe new physics. For example, transition state theory, which is used across chemistry and chemical physics, plays a key role in our determination of lattice parameters, thus connecting its physical assumptions to highly accurate experimental investigation.

Figure 1

Separation between the chaotic ($R\lesssim r_{\text{SR}}$) and regular ($R\gtrsim r_{\text{SR}}$) regions of intermolecular scattering. For extreme separations $r\gg R_{\text{vdW}}$ the molecules propagate ballistically. As the molecules approach each other within the van der Waals potential but still remain in the nonchaotic single-channel region (i.e., $r_{\text{SR}}\lesssim R\lesssim R_{\text{vdW}}$), intermolecular interactions curve the trajectories but preserve their regularity. Finally, molecules approach within a range $R\lesssim r_{\text{SR}}$ and the trajectories of the constituent atoms become a chaotic tangle, mixing many internal rovibrational states. This chaotic tangle of trajectories can also be described as a superposition of bound states, the bimolecular collision complexes.

Figure 2

Eigenvalues $E_{\alpha }$ (top row) and effective interactions $U_{\alpha }$ (bottom row) as a function of trap frequency $\omega$. Darkness (i.e., opacity) is set proportional to the open-channel weight ${\mathcal{O}}_{\alpha }$, indicating the importance of the corresponding state to the lattice model physics. (a) Gaussian orthogonal ensemble for parameters corresponding roughly to RbCs [eigenvalues are from Hamiltonians sampled from Eq. (9)]. (b) Independent levels (Poisson statistics) [Eq. (43)]. (c) Broad resonance on GOE background. (d) Two-ensemble model [Eq. (45)]. Parameters used to generate these plots are discussed in the main text.

Figure 3

Probability distribution of nearest-level spacings $\mathrm{\Delta }$ in different matrix ensembles. (a) Random-matrix-theory GOE [eigenvalues are from Hamiltonians sampled from Eq. (9)]. (b) Independent levels (Poisson statistics) [Eq. (43)]. (c) Two-ensemble model [Eq. (45)] for ${\rho }_b^{\left(1\right)}/{\rho }_b^{\left(2\right)}=0.1,1.0,10.0$ from left to right (blue, orange, and green, respectively).

Figure 4

Determining $W_{nb}$'s from TST. (a) The $W_{nb}$ are the coupling between the closed-channel bound state $|b\rangle$ and the open-channel harmonic-oscillator states $|n\rangle$ in a finite trap. (b) $W_{nb}$ governs the dissociation rate ${\gamma }_b$ of the BCC (bound state) $|b\rangle$ breaking into two molecules in the absence of a trap in the continuum limit ($\omega \to 0$). By approximating ${\gamma }_b$ within the TST-QDT approximation, we calculate the $W_{nb}$. (inset) The essence of TST for the present barrierless reactions is that the bound state oscillates at frequency ${\omega }_{cc}\sim 1/{\rho }_b$, passes through the transition dividing surface (roughly where the vdW potential becomes negligible) with $O\left(1\right)$ probability, and then never recrosses this surface.

Figure 5

Schematic of multichannel quantum-defect theory. In MQDT, the separation coordinate space is divided into short range and long range, parametrized with respect to $r_{\text{SR}}$. A set of reference wave functions for the manifold of low-energy states (black solid lines) is constructed to minimize the energy dependence of the coupling to short range and account for propagation in the $1/R^6$ van der Waals tail. The inset shows that the separation between states in the lowest manifold is generally much larger than the temperature. The cutoff $r_{\text{SR}}$ is chosen so that the states in the lowest manifold that are closed as $R\to \infty$ are classically open at $r_{\text{SR}}$. For NRMs, where many channels (blue dashed lines) remain closed at $r_{\text{SR}}$, these channels impart a complex resonance structure to the short-range physics. For polar NRMs, these higher-lying channels also have a qualitatively different $1/R^3$ long-range potential character.

Figure 6

Effect of bimolecular collision complex ac polarizability deviating from twice the molecules' ac polarizability. Shown on the top is $E_{\alpha }$ and on the bottom $U_{\alpha }$ versus trap depth $\omega$, with weight set as in Fig. 2. (a) The bimolecular collision complex polarizability ${\alpha }_c$ is taken to be equal to the single-molecule polarizability ${\alpha }_m$ (rather than ${\alpha }_c\approx 2{\alpha }_m$, which was taken in earlier plots). (b) Each bimolecular collision complex has a polarizability independently sampled from a Gaussian with standard deviation ${\alpha }_m$.