Efimov Trimers under Strong Confinement

The dimensionality of a system can fundamentally impact the behavior of interacting quantum particles. Classic examples range from the fractional quantum Hall effect to high-temperature superconductivity. As a general rule, one expects confinement to favor the binding of particles. However, attractively interacting bosons apparently defy this expectation: While three identical bosons in three dimensions can support an infinite tower of Efimov trimers, only two universal trimers exist in the two-dimensional case. Here, we reveal how these two limits are connected by investigating the problem of three identical bosons confined by a harmonic potential along one direction. We show that the confinement breaks the discrete Efimov scaling symmetry and successively destroys the weakest bound trimers. However, the deepest bound trimers persist even under strong confinement. In particular, the ground-state Efimov trimer hybridizes with the two-dimensional trimers, yielding a superposition of trimer configurations that effectively involves tunneling through a short-range repulsive barrier. Our results suggest a way to use strong confinement to engineer more stable Efimov-like trimers, which have so far proved elusive.

Popular Summary

In an ultracold gas of atoms confined to micrometer scales by lasers and magnets, common wisdom dictates that atoms with restricted motion tend to cluster. However, in the quantum dance of atoms, one system dramatically defies this expectation. Three identical atoms in three dimensions may form an infinite and fractal series of configurations, the so-called Efimov states, in which three particles can bind even when their two-body attraction is not strong enough to permit the formation of pairs. On the other hand, confining the atoms’ motions to two dimensions yields only two bound trimers. We investigate how systems evolve from infinitely many trimers to exactly two trimers and demonstrate how strong confinement can be used to engineer more stable Efimov-like trimers, which have so far proved elusive.

We theoretically study three identical bosonic atoms with attractive short-range interactions subjected to a tight harmonic confinement along one direction. We compute the spectrum and aspect ratios, demonstrating that an arbitrarily weak confinement breaks the weakest bound trimers, while at the same time stabilizing the two trimers for even the faintest interactions. We derive an effective three-body potential that displays a repulsive barrier at short distances. This potential should result in a suppression of losses compared with those typically plaguing strongly interacting, three-dimensional ultracold Bose gases.

As well as providing new fundamental insights into this prototypical few-body problem, our results have important implications for ongoing experiments aiming at detecting universality in three-body systems, and open new avenues toward the realization of long-lived many-body states of trimers.

Figure 1

Shape of the trimers under confinement: The aspect ratio $2⟨Z^2⟩/⟨{\rho }^2⟩$ for the two deepest trimers is shown as a function of the interaction for two different confinement strengths ${\mathcal{C}}_z$. Here, $\mathit{\rho }$ is the separation of an atom and a pair in the $x\text{−}y$ plane, and $Z$ is the separation in the confined direction (see the main text). The deepest trimer (solid lines) only exists in 3D for $|a_{-}|/a\ge -1$, as depicted by the shaded region. Here, the aspect ratio is close to 1, indicating that the trimer wave function resembles the 3D Efimov state. Outside the trimer’s regime of existence in 3D, the aspect ratio quickly decreases, indicating that the trimer spreads out in the 2D plane. The change in aspect ratio as $a$ crosses $a_{-}$ is more gradual for the stronger confinement because of a stronger coupling between trimers of a 2D and 3D character. The deepest trimer eventually approaches the 2D asymptotic limit (dotted lines; see Appendix pp4). A similar picture emerges for the first excited trimer (dashed lines), but the aspect ratio here is much smaller than 1 even away from the 2D limit. The shape of the deepest trimer is illustrated in the insets, which show surface density plots of the squared wave function (see the main text) evaluated at the points marked by filled circles.

Figure 2

Spectrum of trimers for two different confinement strengths. The q2D trimers are shown as solid lines, with energies $E_3+{\omega }_z$ that take into account the raised three-atom continuum under confinement. For comparison, we also include the two deepest trimers in 3D (dashed lines). The shading illustrates the atom-dimer and three-atom continua, where the q2D continuum (contained within the 3D continuum) is shown with a darker shading. For a strong confinement with ${\mathcal{C}}_z=1$ (right panel), only two q2D trimers exist. For a weaker confinement with ${\mathcal{C}}_z=0.4$ (left panel), a third weakly bound trimer appears that is indistinguishable from the atom-dimer threshold on this scale; here, we show its region of existence as a thick, red line on top of the threshold (see Fig. 3 for a clearer rendering). In order to aid the visibility over the large energy range here, we rescaled the axes by means of the function $F_j(x)=\mathrm{sgn}(x)\ln [1+5|x{|}^j]/\ln 6$, with $j=1$ for the $x$ axis and $j=1/2$ for the $y$ axis. For the 3D spectrum, ${\omega }_z$ is not defined, and we display $m{a}_{-}^2{\mathcal{C}}_z^{-2}{E}_3$ as a function of $|a_{-}|/a$.

Figure 3

Ratio between q2D trimer and dimer energies. The trimer energies (solid lines) are displayed as a function of interaction for two different confinements: ${\mathcal{C}}_z=0.4$ (left panel) and ${\mathcal{C}}_z=1$ (right panel). In the limit $|a_{-}|/a\lesssim -1$, the trimer energies converge to the universal 2D results $-16.5|E_b|$ and $-1.27|E_b|$ [12] (dashed lines). For moderate confinement (left panel), we see evidence of an avoided crossing resulting from the coupling between three “bare” states: the two 2D trimers, and a third trimer that emerges from the continuum at $a\simeq a_{-}$ (vertical dotted line).

Figure 4

Three-body potentials and corresponding wave functions of the deepest trimer. The upper panels show the q2D adiabatic hyperspherical potential $V(R)$ for three different interactions $|a_{-}|/a$, assuming ${\mathcal{C}}_z=0.4$. For reference, we also show the 3D potential. Note that the potentials match in the region $R<l_z$. The shaded area corresponds to the regime of short-distance physics where $R$ is less than the van der Waals range which is of the order of $1/\mathrm{\Lambda }$. The resulting energies of the ground-state trimer are shown as dashed horizontal lines. In the lower panels, the in-plane q2D probability densities $|\psi (\rho ,Z=0){|}^2$ are compared with the probability densities expected in purely 2D and 3D. This illustrates three qualitatively different interaction regimes: (left) for weak interactions, the trimer resides in the long-range attractive tail of the potential and matches the 2D result; (center) for intermediate interactions, $a\sim a_{-}$, the trimer configuration is a superposition of trimers of 2D and 3D character. Here, as there is no Efimov trimer at this scattering length in 3D, the 3D probability density is instead evaluated at $|a_{-}|/a$ slightly larger than $-1$. Right: For stronger interactions $|a_{-}|/a>-1$, the trimer wave function resides mainly in the regime $R<|a|$ and closely resembles the 3D wave function.