Emergence of Chaotic Scattering in Ultracold Er and Dy

We show that for ultracold magnetic lanthanide atoms chaotic scattering emerges due to a combination of anisotropic interaction potentials and Zeeman coupling under an external magnetic field. This scattering is studied in a collaborative experimental and theoretical effort for both dysprosium and erbium. We present extensive atom-loss measurements of their dense magnetic Feshbach-resonance spectra, analyze their statistical properties, and compare to predictions from a random-matrix-theory-inspired model. Furthermore, theoretical coupled-channels simulations of the anisotropic molecular Hamiltonian at zero magnetic field show that weakly bound, near threshold diatomic levels form overlapping, uncoupled chaotic series that when combined are randomly distributed. The Zeeman interaction shifts and couples these levels, leading to a Feshbach spectrum of zero-energy bound states with nearest-neighbor spacings that changes from randomly to chaotically distributed for increasing magnetic field. Finally, we show that the extreme temperature sensitivity of a small, but sizable fraction of the resonances in the Dy and Er atom-loss spectra is due to resonant nonzero partial-wave collisions. Our threshold analysis for these resonances indicates a large collision-energy dependence of the three-body recombination rate.

Popular Summary

Quantum chaos theory describes the states of a quantum system from a probabilistic point of view rather than from microscopic simulations. Chaos imprints its presence on a wide variety of physically distinct systems, and it manifests itself as a complex resonant response to external excitations in which the spacings between resonances satisfy distinct and characteristic distributions. For example, in atomic physics, laser radiation can bring atoms into highly excited states whose spectra are chaotic as a function of laser color. The origin of chaos is attributed to collective phenomena arising from strong mixing between many electrons. Here, we experimentally and theoretically introduce nanokelvin collisions between neutral magnetic rare-earth atoms in an external magnetic field as a controllable system to study quantum chaos.

We investigate the underlying origin of the chaotic scattering and its universality by comparatively studying nanokelvin collisions in an external magnetic field for two neutral magnetic rare-earth atoms, erbium (Er) and dysprosium (Dy). We focus on ultracold, trapped ensembles of Dy and Er atoms, which have an unprecedentedly dense spectrum of collisional Fano-Feshbach resonances. We experiment with 10,000 magnetic-field values ranging from 0 to 70 G, and we analyze the results with coupled channels and bound-state calculations based on physical angular-momentum couplings and interaction potentials. We find that the presence of weakly bound levels with complex superposition states containing multiple rotational diatomic molecular states lie at the heart of the chaotic resonance distribution in collisions between magnetic atoms. These novel weakly bound levels, containing orbital angular momenta up to $50\hslash$, exist solely because of strong anisotropic, orientation- and spin-dependent atom-atom interactions. In particular, we identify the anisotropic interactions occurring at short range as being primarily responsible for the chaotic behavior. Finally, based on a resonant model of three-body combination—in which three ultracold atoms combine to form a diatomic molecule and another atom, both of which are lost from the trap—we are able to explain the rapid increase in the resonance density with increasing temperature.

We expect that our results will pave the way for other studies of chaotic behavior in nuclear and atomic physics and provide a basis for understanding the scattering behavior of complex matter.

Figure 1

(a) Trap-loss spectroscopy mapping of the Fano-Feshbach spectrum of ${}^{164}\mathrm{Dy}$ as a function of magnetic field between $B=0$ and 70 G with a data point every 14.5 mG and temperature $T=600\mathrm{nK}$. Each data point is an average of three measurements. (b) Staircase function for the number of resonances as a function of $B$ for ${}^{164}\mathrm{Dy}$, ${}^{168}\mathrm{Er}$ at two temperatures, and fermionic ${}^{167}\mathrm{Er}$. Dashed lines are linear fits forced to pass through the origin. Their slopes give a mean density of resonances of $\overline{\rho }=4.3{\mathrm{G}}^{-1}$ for ${}^{164}\mathrm{Dy}$, $2.7{\mathrm{G}}^{-1}$ for ${}^{168}\mathrm{Er}$ at $T=350\mathrm{nK}$, $3.4{\mathrm{G}}^{-1}$ for Er at $T=1.4\mu \mathrm{K}$, and $25.6{\mathrm{G}}^{-1}$ for ${}^{167}\mathrm{Er}$. (c) Number variance of the experimental data as a function of scaled $B$-field interval $\overline{N}=\mathrm{\Delta }B\overline{\rho }$. The experimental data for ${}^{164}\mathrm{Dy}$ (blue line) and ${}^{168}\mathrm{Er}$ at $T=350\mathrm{nK}$ (orange line) lie between the variances for an uncorrelated Poisson distribution (dashed line) and the correlated Wigner-Dyson distribution (dot-dashed line).

Figure 2

(a) Nearest-neighbor spacing distribution $P(s)$ determined from all observed Fano-Feshbach resonances for ${}^{164}\mathrm{Dy}$ (blue markers). Dashed and dot-dashed curves are the Poisson and Wigner-Dyson distribution, respectively. The solid line is a Brody distribution with $\eta =0.45(7)$ fit to the experimental data. (b) Distributions $P(s)$ for ${}^{168}\mathrm{Er}$ at $T=350\mathrm{nK}$ and $T=1.4\mu \mathrm{K}$ (orange and red markers, respectively), and ${}^{167}\mathrm{Er}$ at $T=0.4T_F$ (green markers), where $T_F$ is the Fermi temperature of the gas. The solid line is a Brody distribution with $\eta =0.68(9)$ fit to the ${}^{168}\mathrm{Er}$ data at $T=350\mathrm{nK}$. Panels (c) and (d) show the magnetic-field-resolved Brody parameter $\eta (B)$ as a function of magnetic field for ${}^{164}\mathrm{Dy}$ and ${}^{168}\mathrm{Er}$, respectively. The Brody parameters for the Poisson and Wigner-Dyson distribution are 0 and 1, respectively. Gray markers and lines in all four panels are results from our coupled-channels calculations. The $1\sigma$ error bars in (a) and (b) correspond to Poisson counting errors, while shaded bands in (c) and (d) are $1\sigma$ statistical uncertainties of the fits to the data.

Figure 3

Molecular spectrum and NNS distributions of Fano-Feshbach resonances of ${}^{168}\mathrm{Er}$ calculated from our RMT model. (a) Example of a spectrum of molecular binding energies as a function of $B$ for ${\nu }_{\mathrm{cpl}}/h=2\mathrm{MHz}$, ${\eta }_d=0$. Here, $h$ is Planck’s constant. Squares at $E/h=0\mathrm{MHz}$ indicate crossings of molecular levels with the threshold of two $m_j=-6\mathrm{atoms}$ and correspond to the position of Feshbach resonances. For the sake of visibility, we show only the spectrum between $B=0$ and 10 G. (b) NNS distributions of simulated Feshbach resonances for ${\eta }_d=0$ and ${\nu }_{\mathrm{cpl}}/h=0\mathrm{MHz}$ (circles), 2 MHz (squares), and 10 MHz (triangles). The dashed and dash-dotted lines are Poisson and Wigner-Dyson distributions, respectively. Solid lines are best-fit Brody distributions with $\eta =0.03(1)$, $\eta =0.41(5)$, and $\eta =0.82(1)$ for ${\nu }_{\mathrm{cpl}}/h=0$, 2, and 10 MHz, respectively. (c) Fitted Brody parameters of the nearest-neighbor energy-spacing distribution of the eigenvalues of $H_{\mathrm{RMT}}$ at $B=0\mathrm{G}$ as a function of ${\nu }_{\mathrm{cpl}}$. Circles, squares, down triangles, and up triangles correspond to ${\eta }_d=0$, 0.25, 0.5, and 1, respectively. (d) Fitted Brody parameters of the NNS distribution of the Feshbach resonances as a function of ${\nu }_{\mathrm{cpl}}$ and for four ${\eta }_d$ using the same marker code as in (c). In panels (b)–(d) distributions are obtained by averaging over 15 realizations of $H_{\mathrm{RMT}}$, each of dimension $500\times 500$ and using Feshbach resonances computed up to $B=85\mathrm{G}$.

Figure 4

Interaction-anisotropy-induced chaos of $B=0$ near-threshold bound states. (b) Weakly bound $J=16$ bound-state energies of ${}^{164}{\mathrm{Dy}}_2$ as a function of the anisotropy scale ${\lambda }_{\mathrm{MDD}}$ with ${\lambda }_{\mathrm{\Delta }{C}_6}=0$. (a) NNS distribution (red circles) for the $J=16$ bound-state data in (b) at ${\lambda }_{\mathrm{MDD}}=1$ and ${\lambda }_{\mathrm{\Delta }{C}_6}=0$. The solid red line is a Brody distribution fit to the data and agrees well with a Poisson distribution. (d) Weakly bound $J=16$ bound-state energies of ${}^{164}{\mathrm{Dy}}_2$ as a function of the anisotropy scale ${\lambda }_{\mathrm{\Delta }{C}_6}$ with ${\lambda }_{\mathrm{MDD}}=0$. (c) NNS distribution (purple squares) for the $J=16$ bound-state data in (d) at ${\lambda }_{\mathrm{MDD}}=0$ and ${\lambda }_{\mathrm{\Delta }{C}_6}=1$. The solid purple line is a Brody distribution fit to the data and is close to a Wigner-Dyson distribution. (e) Moving average of the Brody parameter $\eta$ as a function of ${\lambda }_{\mathrm{\Delta }{C}_6}$ (purple squares) or ${\lambda }_{\mathrm{MDD}}$ (red circles) with bins $\mathrm{\Delta }\lambda =0.2$ obtained by fitting the NNS distribution for the $J=16$ bound-state data in (b) and (d) to Brody distributions, respectively. The horizontal lines at $\eta =0$ and 1 correspond to the Brody parameter for a Poisson and Wigner-Dyson distribution, respectively. The $1\sigma$ error bars combine statistical and fitting uncertainties. (f) The individual-$J$ (blue squares) and combined-$J$ (red circles) NNS distributions $P(s)$ at ${\lambda }_{\mathrm{MDD}}={\lambda }_{\mathrm{\Delta }{C}_6}=1$ as a function of the normalized energy spacing $s$. The distributions are derived from $B=0$ bound-state data for $J=16,\dots ,25$. The gray shaded areas in (a), (b), and (f) indicate the Wigner-Dyson distribution.

Figure 5

Theoretical ${}^{164}{\mathrm{Dy}}_2$ [(a),(b)] and ${}^{168}{\mathrm{Er}}_2$ [(c),(d)] near-threshold bound states as a function of magnetic field. Calculations have been performed with the full nominal anisotropy. Panels (a) and (c) show the near-threshold region between $B=0$ and 10 G, while panels (b) and (d) show the region between $B=50$ and 60 G. Red crosses indicate the location of Feshbach resonances.

Figure 6

Theoretical near-threshold bound states and Feshbach resonances in a magnetic field. (a) Near-threshold bound states for $M=-12$ ${}^{168}{\mathrm{Er}}_2$ for fields between $B=50$ and 60 G (bottom half of image). The top half of the image shows the effective scattering length, defined in the text, as a function of $B$. It is infinite at a resonance location where a bound state has zero energy. Calculations use the physical interaction anisotropies and channel states with $J\le J_{\max }=39$. (b) Theoretical Feshbach-resonance density $\overline{\rho }$ as a function of $J_{\max }$ (purple circles) computed from resonance locations between 0 and 70 G. The dashed horizontal line indicates the experimental density for ${}^{168}{\mathrm{Er}}_2$. (c) Convergence study of resonance locations (crosses) between 50 and 55 G as a function of $J_{\max }$. Purple lines connect resonances when their location has converged.

Figure 7

Line shapes for a strongly temperature-dependent ${}^{168}\mathrm{Er}$ Feshbach resonance near $B=1.48\mathrm{G}$. Panel (a) shows experimental data (markers with error bars) as a function of $B$ of the remaining atom number divided by the atom number away from resonance measured 400 ms after initial preparation. Black, red, green, and blue markers correspond to data for temperatures $T=230$, 740, 1400, and 2000 nK, respectively. Dashed lines connecting the markers guide the eye. Solid lines are theoretical line shapes of the remaining atom number based on the $d$-wave ($\mathcal{N}=2$) recombination rates shown in (c). Panels (b) and (c) show simulated three-body recombination rates for the same four temperatures assuming three-body entrance $s$- ($\mathcal{N}=0$) and $d$-wave ($\mathcal{N}=2$) scattering, respectively. Curves are based on a thermally averaged line shape discussed in the text. Recombination rates are scaled such that the largest value in each panel is one. For both panels, $\mu =3.1{\mu }_B$ and ${\mathrm{\Gamma }}_{\mathrm{br}}/k_B=250\mathrm{nK}$, while $\mathrm{\Gamma }(E)/k_B=0.2(E/E_{\mathrm{ref}}{)}^2\mathrm{nK}$ in (b) and $\mathrm{\Gamma }(E)/k_B=0.1(E/E_{\mathrm{ref}}{)}^4\mathrm{nK}$ in (c), where $E_{\mathrm{ref}}/k_B=1000\mathrm{nK}$.