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arXiv:1507.03402 · Physical Review A

Universal range corrections to Efimov trimers for a class of paths to the unitary limit

Alejandro Kievsky, M. Gattobigio

Published Dec 29, 2015• 21 citations
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Abstract

Using potential models, we analyze range corrections to the universal law dictated by the Efimov theory of three bosons. In the case of finite-range interactions, we have observed that at first order, it is necessary to supplement the theory with one finite-range parameter ${\mathrm{\ensuremath{\Gamma}}}_{n}^{3}$ for each specific $n$ level [A. Kievsky and M. Gattobigio, Phys. Rev. A 87, 052719 (2013)]. The value of ${\mathrm{\ensuremath{\Gamma}}}_{n}^{3}$ depends on the way the potentials are changed to tune the scattering length toward the unitary limit. In this work, we analyze a particular path in which the length ${r}_{B}=a\ensuremath{-}{a}_{B}$, measuring the difference between the two-body scattering length $a$ and the energy-scattering length ${a}_{B}$, is almost constant. Analyzing systems with very different scales, such as atomic or nuclear systems, we observe that the finite-range parameter remains almost constant along the path with a numerical value of ${\mathrm{\ensuremath{\Gamma}}}_{0}^{3}\ensuremath{\approx}0.87$ for the ground-state level. This observation suggests the possibility of constructing a single universal function that incorporates finite-range effects for this class of paths. The result is used to estimate the three-body parameter ${\ensuremath{\kappa}}_{*}$ in the case of real atomic systems brought to the unitary limit through broad Feshbach resonances. Furthermore, we show that the finite-range parameter can be put in relation to the two-body contact ${C}_{2}$ at the unitary limit.

DOI
10.1103/physreva.92.062715

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Using potential models, we analyze range corrections to the universal law dictated by the Efimov theory of three bosons. In the case of finite-range interactions, we have observed that at first order, it is necessary to supplement the theory with one finite-range parameter \ensuremathΓn3{\mathrm{\ensuremath{\Gamma}}}_{n}^{3}\ensuremathΓn3​ for each specific nnn level [A. Kievsky and M. Gattobigio, Phys. Rev. A 87, 052719 (2013)]. The value of \ensuremathΓn3{\mathrm{\ensuremath{\Gamma}}}_{n}^{3}\ensuremathΓn3​ depends on the way the potentials are changed to tune the scattering length toward the unitary limit. In this work, we analyze a particular path in which the length rB=a\ensuremath−aB{r}_{B}=a\ensuremath{-}{a}_{B}rB​=a\ensuremath−aB​, measuring the difference between the two-body scattering length aaa and the energy-scattering length aB{a}_{B}aB​, is almost constant. Analyzing systems with very different scales, such as atomic or nuclear systems, we observe that the finite-range parameter remains almost constant along the path with a numerical value of \ensuremathΓ03\ensuremath≈0.87{\mathrm{\ensuremath{\Gamma}}}_{0}^{3}\ensuremath{\approx}0.87\ensuremathΓ03​\ensuremath≈0.87 for the ground-state level. This observation suggests the possibility of constructing a single universal function that incorporates finite-range effects for this class of paths. The result is used to estimate the three-body parameter \ensuremathκ∗{\ensuremath{\kappa}}_{*}\ensuremathκ∗​ in the case of real atomic systems brought to the unitary limit through broad Feshbach resonances. Furthermore, we show that the finite-range parameter can be put in relation to the two-body contact C2{C}_{2}C2​ at the unitary limit.
Universal behavior has been found inside the window of Efimov physics for systems with N=4,5,6N=4,5,6N=4,5,6 particles. Efimov physics refers to the emergence of a number of three-body states in systems of identical bosons interacting via a short-range interaction becoming infinite at the verge of binding two particles. These Efimov states display a discrete scale invariance symmetry, with the scaling factor independent of the microscopic interaction. Their energies in the limit of zero-range interaction can be parametrized, as a function of the scattering length, by a universal function. We have found, using the form of finite-range scaling introduced by A. Kievsky and M. Gattobigio [Phys. Rev A 87, 052719 (2013)], that the same universal function can be used to parametrize the ground and excited energy of $N\ensuremath{\le}6$ systems inside the Efimov-physics window. Moreover, we show that the same finite-scale analysis reconciles experimental measurements of three-body binding energies with the universal theory.
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 calculate shallow three-body bound states in the universal regime, defined by Efimov, with inclusion of both scattering length and effective range parameters. We find corrections to the universal scaling laws for large binding energies. For narrow resonances we find a distinct nonmonotonic behavior of the threshold at which the lowest Efimov trimer merges with the three-body continuum. The origin of the three-body parameter is related to the two-body atom-atom interactions in a physically clear model. Our results demonstrate that experimental information from narrow Feshbach resonances and/or mixed systems are of vital importance to pin down the relation of two- and three-body physics in atomic systems.
We investigate finite-range effects in systems with three identical bosons. We calculate recombination rates and bound state spectra using two different finite-range models that have been used recently to describe the physics of cold atomic gases near Feshbach resonances where the scattering length is large. The models are built on contact potentials which take into account finite-range effects; one is a two-channel model and the other is an effective range expansion model implemented through the boundary condition on the three-body wavefunction when two of the particles are at the same point in space. We compare the results with the results of the ubiquitous single-parameter zero-range model where only the scattering length is taken into account. Both finite-range models predict variations of the well-known geometric scaling factor 22.7 that arises in Efimov physics. The threshold value at negative scattering length for creation of a bound trimer moves to higher or lower values depending on the sign of the effective range compared to the location of the threshold for the single-parameter zero-range model. Large effective ranges, corresponding to narrow resonances, are needed for the reduction to have any noticeable effect.
Some aspects of universal properties are discussed in few-atom as well as in few- nucleon systems. When the two-body scattering length a is large the three-body system presents some peculiar characteristics such as the Efimov effect. In the present contribution, the zerorange theory deduced by V. Efimov is compared to results obtained using potential models. Accordingly range corrections are introduced. These corrections appear as a shift in the variable K * a with K * the three-body parameter. The three-boson spectrum, atom-dimer scattering and the recombination rate at threshold are analyzed using the finite-range theory proposed. In the case of nuclear systems, n – d scattering is discussed using the universal formula derived by Efimov. Furthermore, the spectrum up to N = 6 is discussed using the central Volkov potential.
We consider a system of three 4{}^{4}4He atoms, which is so far the simplest realistic three-body system exhibiting the Efimov effect, in order to analyze deviations from the universal Efimov three-body spectrum. We first calculate the bound states using a realistic two-body potential, and then analyze how they can be reproduced by simple effective models beyond Efimov's universal theory. We find that the nonuniversal variations of the ground and first excited states can be well reproduced by models parametrized with only three quantities: the scattering length and effective range of the original potential, and a small three-body force. Furthermore, the three-body parameter, which fixes the origin of the infinite set of three-body levels, is found to be consistent with recent experimental observations in other atomic species.