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Efimov physics beyond universality

Richard Schmidt, Steffen Patrick Rath, W. Zwerger

Published Jan 1, 2012• 99 citations
OpenAlex

Abstract

We provide an exact solution of the Efimov spectrum in ultracold gases within the standard two-channel model for Feshbach resonances. It is shown that the finite range in the Feshbach coupling makes the introduction of an adjustable three-body parameter obsolete. The solution explains the empirical relation between the scattering length a − where the first Efimov state appears at the atom threshold and the van der Waals length l vdw for open-channel dominated resonances. There is a continuous crossover to the closed-channel dominated limit, where the scale in the energy level diagram as a function of the inverse scattering length 1 / a is set by the intrinsic length r ⋆ associated with the Feshbach coupling. Our results provide a number of predictions for the deviations from universal scaling relations between energies and scattering lengths that can be tested in future experiments.

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arXiv:1012.1422•New Journal of Physics•Mar 29, 2011•20 citations
We provide an exact solution of the Efimov spectrum in ultracold gases within the standard two-channel model for Feshbach resonances. It is shown that the finite range in the Feshbach coupling makes the introduction of an adjustable three-body parameter obsolete. The solution explains the empirical relation between the scattering length a − where the first Efimov state appears at the atom threshold and the van der Waals length l vdw for open-channel dominated resonances. There is a continuous crossover to the closed-channel dominated limit, where the scale in the energy level diagram as a function of the inverse scattering length 1 / a is set by the intrinsic length r ⋆ associated with the Feshbach coupling. Our results provide a number of predictions for the deviations from universal scaling relations between energies and scattering lengths that can be tested in future experiments.
We measure the critical scattering length for the appearance of the first three-body bound state, or Efimov three-body parameter, at seven different Feshbach resonances in ultracold ^{39}K atoms. We study both intermediate and narrow resonances, where the three-body spectrum is expected to be determined by the nonuniversal coupling of two scattering channels. Instead, our observed ratio of the three-body parameter with the van der Waals radius is approximately the same universal ratio as for broader resonances. This unexpected observation suggests the presence of a new regime for three-body scattering at narrow resonances.
We calculate the spectrum of three-body Efimov bound states near a Feshbach resonance within a model which accounts both for the finite range of interactions and the presence of background scattering. The latter may be due to direct interactions in an open channel or a second overlapping Feshbach resonance. It is found that background scattering gives rise to substantial changes in the trimer spectrum as a function of the detuning away from a Feshbach resonance, in particular in the regime where the background channel supports Efimov states on its own. Compared to the situation with negligible background scattering, the regime where van der Waals universality applies is shifted to larger values of the resonance strength if the background scattering length is positive. For negative background scattering lengths, in turn, van der Waals universality extends to even small values of the resonance strength parameter, consistent with experimental results on Efimov states in 39K^{39}\mathrm{K}39K. Within a simple model, we show that short-range three-body forces do not affect van der Waals universality significantly. Repulsive three-body forces may, however, explain the observed variation between around $\ensuremath{-}8$ and $\ensuremath{-}10$ of the ratio between the scattering length where the first Efimov trimer appears and the van der Waals length.
Recent cold atom experiments report a surprising universal scaling of the first Efimov resonance position a_{-}^1 by the two-body van der Waals length r_{vdW}. The ratio C=-a_{-}^1/r_{vdW}=8.5~9.5 for identical particles appears to be a constant regardless of the atomic spin configuration, the Feshbach resonance employed to tune the scattering length, and even the atomic species, with K-39 being the only exception. This result indicates that the Efimov energy structure is insensitive to the details of the short range potential. We suggest that the universality results from the quantum reflection of the Efimov wavefunciton by the short-range molecular potential. Assuming Born-Oppenheimer approximation and strong quantum reflection, we obtain an analytic result of C=9.475... for three identical particles. We suspect the exceptional case of K-39 is a result of resonant coupling between the Efimov state and a short-range molecular state.
Here we present a four-channel model that incorporates a magnetically tunable Feshbach resonance in a system of three atoms that interact via pairwise van der Waals interactions. Our method is designed to model recent experiments where the tunability of the scattering length has been used to study three-body Efimov states, which appear in the limit of a diverging two-body scattering length. Using this model, we calculate three-body adiabatic and effective potential curves and study how the strength (or width) of the Feshbach resonance affects the three-body potential that is connected to the Efimov effect. We find that the position of the repulsive barrier, which has been used to explain the so-called van der Waals universality in broad resonances, is slightly shifted as the narrow resonance limit is approached and that this shift is correlated to the appearance of two avoided crossings in the adiabatic energy landscape. More importantly, the attractive well is markedly shifted upward in energy and is extremely shallow for the narrowest resonance. We argue that this behavior is connected to the breakdown of van der Waals universality for weak (narrow) resonances.
Efimov physics in ultracold gases is described very well by the universal scaling laws, based on the scattering length and van der Waals length. The first can be tuned magnetically via a Feshbach resonance, the second is constant and connected to the radial range of the potential.However, experimental hints at non-universal behavior, when going away from resonance, are quite badly understood. The next leading coefficient in the scattering phase shift, the effective range parameter, gives an indication of this non-universality, but at the same time it can also be strongly dependent on the magnetic field. Moreover, higher-order terms take over quickly when increasing the collision energy. We show how the finite range corrections can be understood by making the connection to more fundamental parameters of the two-body physics, and use this description to derive a better criterion for entering the non-universal regime.
We find that universal three-body physics extends beyond the threshold regime to non-zero energies. For ultracold atomic gases with a negative two-body s-wave scattering length near a Feshbach resonance, we show that the resonant peaks characteristic of Efimov physics persist in three-body recombination to higher collision energies. For this and other inelastic processes, we use the adiabatic hyperspherical representation to derive universal analytical expressions for their dependence on the scattering length, the collision energy and—for narrow resonances—the effective range. These expressions are supported by full numerical solutions of the Schrödinger equation and display log-periodic dependence on energy characteristic of Efimov physics. This dependence is robust and might be used to experimentally observe several Efimov features.