Low-temperature physics

Chaos in the cold

A marriage between theory and experiment has shown that ultracold erbium atoms trapped with laser light and subjected to a magnetic field undergo collisions that are characterized by quantum chaos. See Letter p.475

The use of magnetic fields to manipulate the interactions of ultracold atoms at temperatures below one microkelvin has allowed the experimental and theoretical study of a plethora of exotic phenomena in quantum physics1. The paper by Frisch et al.2 published on on page 475 today adds an entirely new twist to this work. Until now, such research has mainly used simple atoms that have relatively simple interactions. Frisch and colleagues worked instead with a complex atomic species, erbium, and combined theory and experiment to demonstrate the signature of quantum chaos in the collisions between two erbium atoms. The results offer the prospect of exploring new avenues of ultracold physics and chemistry using complex atomic and molecular species.

In their study, Frisch et al. used a tightly focused laser to trap samples of around 100,000 erbium (Er) atoms in their lowest-energy quantum state at a temperature of about 400 nanokelvins. The samples consisted either of bosonic isotopes (168Er or 166Er), which have integer spin, or of a fermionic one (167Er), which has half-integer spin. They tuned an applied magnetic field to a fixed value between 0 and 7 millitesla and counted the number of atoms remaining after holding them in the trap for 400 milliseconds. The authors found that the number of atoms left in the trap depended strongly on the particular value of the magnetic field selected. Such atom losses are well known in cold-atom physics, and are used to locate features known as Feshbach resonances, which occur when the total (nearly zero) energy of two colliding atoms matches the magnetically tunable energy of a bound diatomic molecular state (Er2). Tabulating the magnetic-field values at which losses exist pinpoints the 'positions' of the resonances, from which the dynamics of the underlying collisions can be probed.

Frisch et al. found that the number of resonances in their system is surprisingly large, much larger than obtained with alkali-metal species such as lithium, rubidium or caesium (Cs). They provide theoretical models that connect this behaviour with chaotic quantum dynamics of the collisions. Because alkali-metal atoms such as Cs have a simple atomic structure (they have zero electronic orbital angular momentum), the resonances of colliding Cs atoms are spaced relatively far apart as a function of increasing magnetic field, and are characterized by simple quantum numbers. By contrast, an Er atom has a complex atomic structure that gives it a total electronic orbital angular momentum of five. Therefore, colliding Er atoms have many more resonances than their Cs analogues, and they are 'mixed' — that is, they are no longer described by simple quantum numbers.

To explain the nature of their measured resonance spectrum (Fig. 1), as well as that calculated by their model, Frisch et al. borrowed a tool known as random matrix theory from earlier work in nuclear and other fields of physics. By assuming that the various interactions of their system are described by the parameters of this theory, they could reproduce the average spacing between successive resonances. They found that the distribution of spacing for 168Er or 166Er bosons was much closer to one that corresponds to chaotic dynamics, known as a Wigner–Dyson distribution, than to the Poissonian distribution that characterizes regular, non-chaotic dynamics. The 167Er fermions, which also have nuclear spin, showed an even denser set of resonances than the bosonic isotopes, but the authors have not yet analysed this set in detail because of its greater complexity. A study that was reported3 this year measured similarly dense sets of resonances in systems of cold bosonic and fermionic isotopes of dysprosium (Dy).

Figure 1: A barcode of resonances.
figure1

The lines mark the different values of an applied magnetic field at which erbium atoms from a trapped sample studied by Frisch et al.2 show increased loss from the trap. The distribution of spacing between adjacent lines, which locate features known as Feshbach resonances, indicates that collisions between two atoms have quantum chaotic behaviour.

Why are these results significant? Let us consider what is special about cold collisions. Because cold atoms move slowly, on average more than 10,000 times more slowly than atoms at room temperature, the Heisenberg uncertainty principle ensures that the uncertainty in their position is large. The colliding atoms no longer behave like particles, but take on wave-like character, with a wavelength that is inversely proportional to their velocity. These wavelengths can become large, one micrometre or more, much larger than the sub-nanometre length of a chemical bond. In this case, collisions become simple and are characterized by a parameter known as the scattering length.

Part of the power and beauty of cold-atom physics is that the scattering length can be made to take on any value by tuning a magnetic field close to a Feshbach resonance1. Its value controls the two-body, few-body and many-body physics of ultracold quantum matter. Thus, controlling the field makes the system dance to our tune. Previously, the Feshbach resonances used for such control have been isolated resonances that vary with the field and the atomic kinetic energy in a simple and well-understood way. Now, cold-atom researchers have to figure out how to understand the scattering length associated with a dense set of chaotic resonances such as that observed by Frisch and colleagues. The variation with field and kinetic energy will be more complex than previously encountered. The same is true of the atom-loss processes that are associated with such resonances and that determine the lifetime of a cold atomic system.

Another reason for working with Er is that its magnetic dipole moment is seven times larger than that of alkali-metal atoms. The dipole moment for Dy is even larger. Such large dipole moments, which result in a long-range interaction between pairs of dipoles, could enable the realization of some of the rich variety of phenomena predicted for an ensemble of cold dipoles4,5. It will now be necessary to understand the interplay between the dense set of resonances and the long-range dipolar interactions.

Finally, cold molecules — which have rotational, vibrational and other internal degrees of freedom — are expected to have much denser resonance spectra than Er or Dy atoms. Work is also being done to make systems of cold polar molecules, which can have much stronger dipolar interactions than atoms and exhibit a larger range of dipolar phenomena. Therefore, Frisch and colleagues' study is but a prelude to the interesting work that is to come on cold molecular systems, the resonances of which may have quite long lifetimes6. We can anticipate that the considerable body of work carried out on the internal relaxation and chemical reactions of excited molecular complexes7,8 will come into play when chaos turns up at the heart of cold molecular collisions.

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Correspondence to Paul S. Julienne.

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Julienne, P. Chaos in the cold. Nature 507, 440–441 (2014). https://doi.org/10.1038/nature13211

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