Complex collisions of ultracold molecules: A toy model

We introduce a model to study the collisions of two ultracold diatomic molecules in one dimension interacting via pairwise potentials. We present results for this system and argue that it offers lessons for real molecular collisions in three dimensions. We analyze the distribution of the adiabatic potentials in the hyperspherical coordinate representation as well as the distribution of near-threshold four-body bound states, systematically studying the effects of molecular properties, such as interaction strength, interaction range, and atomic mass. It is found that the adiabatic potential's nearest-neighbor energy level distribution transitions from significant level repulsion characteristic of chaos (Brody distribution) to nonchaotic (Poisson distribution) as the two molecules are separated. For the near-threshold four-atom bound states, the case where all atoms have equal masses shows a Poissonian spacing distribution, while the unequal-mass system exhibits significant level repulsion characterized by a nonzero Brody parameter. We derive a semiclassical formula for the density of states and extract from it simple scaling laws with potential depth and range. We find good agreement between the semiclassical predictions for the density of states and the full quantum mechanical calculations.

Figure 1

Jacobi coordinates $y_1, y_2$, and $y_3$ for the four-fermion system in one dimension, where the center-of-mass coordinate is set to 0. Particles 1 and 2 are indistinguishable, and particles 3 and 4 are indistinguishable. When (1,3) and (2,4) are coincident, and the separation between these molecules is large, the hyperspherical coordinates approach $(\varphi ,\theta )\to (\pi /4,\pi /2)$ for $m_1=m_2=m_3=m_4$.

Figure 2

(a) Bound states of the two-atom interaction between atoms 1 and 3 in a Morse potential, given by Eq. (1a), for $D=50, a=1/r_0$, and $r_0=1$. Energy levels for $m_3/m_1=1$ (solid) and $m_3/m_1=1.3$ (dashed) are shown. (b) Probability distribution $p\left(s\right)$ of the scaled nearest-neighbor spacing of adiabatic potential curves. $s$ is defined as the nearest-neighbor spacing scaled by the average spacing, at hyper-radius $R=3.8$. Here, $D=100, r_0=1$, and $m_3/m_1=1.3$. States are included from an energy range $-E_b\pm E_{\delta }$, where $E_b$ is the magnitude of the average lowest dimer-dimer energy at large separation, here and for all the statistical analyses in this paper. For this panel, $E_{\delta }=30$. The dashed curves are the distributions for the Gaussian orthogonal ensemble and the Poisson distribution, respectively. The solid curve plots the kernel density estimator from the level-spacing distribution data, using a Gaussian kernel with bandwidth 0.2. (c) The calculated hyper-radial adiabatic potentials (solid lines) as a function of hyper-radius $R$ at $r_0=1, m_3/m_1=1.3$. $D=10$ (left plot) and $D=100$ (right plot) for the Morse potential. The right plot corresponds to the probability distribution plotted in (b). All length and energy variables are dimensionless, as discussed in Sec. 2.

Figure 3

Illustrations of the level density of adiabatic potential curves ${\rho }_a$ as a function of hyper-radius $R$ for different two-atom interactions. Each plot examines the change in ${\rho }_a$ by varying one parameter of the two-atom interaction. The default nonvarying parameters for (a)–(f) are set to $D=100, r_0=1.0, m_3/m_1=1.3$. The energy range of the included curves is $-E_b\pm E_{\delta }$, where $E_b=(E_{12}+E_{34}+E_{12}+E_{34})/2=181.7$, and $E_{\delta }=30$. The Morse potential is used in (a)–(e); $r_0=2$ for the Pöschl-Teller potential curve is used in (f). (d) $m_3/m_1=1$. The offset in (e) is defined as the center of the energy range. All length and energy variables are dimensionless, as discussed in Sec. 2.

Figure 4

Probability distribution $p\left(s\right)$ of nearest-neighbor spacing of adiabatic potential curves, where $s$ is the scaled spacing, at different hyper-radii $R$. (Kernel density estimator obtained from data with a normal kernel of bandwidth 0.2.) The semitransparent curves are for $R=1.6$ to $R=10$ in steps of $\mathrm{\Delta }R=0.2$; the bold curves plot $p\left(s\right)$ at $R=9.9$ and $R=2.0$. The dashed curves are the Poisson and GOE distributions in comparison. (a) The result for the Morse potential with $D=100, r_0=1$, and $m_3/m_1=1.3$. The energy range of included curves is taken to be $-E_b\pm E_{\delta }$, and $E_{\delta }=30$. (b)–(e) Various cases where one parameter (indicated in each plot) of the two-atom interaction in (a) is modified. (b) The width of the potential characterized by $r_0$ is set to $r_0$ = 2. (c) $D=200$. (d) $m_3/m_1=1$. (e) The center of the included curves is shifted from $E_b$ to $E_b^{\prime }=1.2E_b=217.6$. (f) Using the Pöschl-Teller potential with the same $D, m_3/m_1$, and $E_{\delta }$ as (a). $r_0=2$ is used to keep the numbers of bound states of the two-atom interactions the same for (f) and (a). All length and energy variables are dimensionless, as discussed in Sec. 2.

Figure 5

Brody parameter $q$ as a function of hyper-radius $R$ for different two-atom interactions. Each plot examines the change in $q$ by varying one parameter of the two-atom interaction. The default parameters are the same as those in Fig. 3. Note that the horizontal axis origin starts from $R=2$. Results for different parities are shown in (c). The plots are smoothed via a moving average with weights from a Gaussian distribution. All length and energy variables are dimensionless, as discussed in Sec. 2.

Figure 6

The hyperspherical potential energy surface in units of the two-body depth $V(R,\theta ,\varphi )/D$ at $R=8r_0$ for the Morse pairwise potential for (a) equal masses and (b) $m_3/m_1=1.5$.

Figure 7

The hyper-radial function $\mathcal{I}(E=-2.7D,\xi )$ defined in Eq. (20) for $m_3=m_1$ (dashed red) and $m_3=1.5m_1$ (solid black).

Figure 8

(a) The agreement between the quantum mechanical four-body bound state density of states and the $\sqrt{D}$ scaling law predicted by the semiclassical theory described in the main text. (b) The scaling with $r_0^3$. In both panels, the red dashed line and filled circles describe the equal-mass case, while the black solid line and filled squares describe the case $m_3/m_1=1.5$. The Morse potential was used in all of these calculations, and for (a), we assume $r_0=1$, while for (b), we assume $D=40$. All length and energy variables are dimensionless, as discussed in Sec. 2.

Figure 9

(a) The adiabatic potentials (blue curves) and the four-body bound state energy levels (horizontal lines) for the equal-mass case $m_1=m_3$. Energy levels used to calculate the density of states and the Brody distribution are shown in black, while the remaining energy levels are shown in red. (b) The probability density $P\left(s\right)$ of nearest-neighbor energy spacings of four-body bound states, as calculated using the method of slow variable discretization for the equal-mass case. (Black straight lines: histogram; black smooth curve: kernel density estimator.) (c),(d) Corresponding plots for the case $m_3=1.5m_1$. The Brody distribution that maximizes the log-likelihood function for the kernel density estimator is shown as the red curve, and panels labeled with the correspondingly fit $q$ value. The blue dotted curves in (b) and (d) show the Wigner-Dyson distribution, while the green dashed curves show the Poisson distribution, included here for comparison. All calculations shown use the Morse pairwise potential with $D=75, r_0=1$ and with energy levels in the range $E\pm E_{\delta }=-2.7D\pm 10$. All length and energy variables are dimensionless, as discussed in Sec. 2.