Quantum Few-Body Physics

The interaction between particles or macroscopic bodies is a key element of nature for building up nuclei, atoms, molecules, gaseous, liquid or solid states of matter, and even the system of planets with galaxies. Determined by physical conditions such as dimensionality, environment, and type of particles, their direct and indirect interactions can have short-range (nuclear and the van der Waals forces) and long-range (the Coulomb potential in atoms and molecules or the Newton potential in solar system, and dipole-dipole interaction) characters.

A system of two interacting non-relativistic bodies can easily be described due to the ability to separate the centre-of-mass motion from the relative one in free space. Each independent motion is governed by one-particle dynamics, which can basically be studied within the framework of both classical and quantum mechanics. However, when we add to our system a third particle, the resulting three-body problem becomes very complicated. The complexity of such a three-body system within the quantum-mechanical approach originates from the large variety of possible reaction channels (states of three-body system). Although adding more particles increases the complexity of system, in many cases, for a large enough number of particles, one can successfully apply the well-known mean-field approximation. Thus, few-particle systems (with three, four, or five particles) take a unique position between two simple cases, namely two- and many-body systems and their understanding is one of the central problem in modern physics.

Fig. 1: Scheme of scattering a heavy atom of mass \( M \) off a weakly bound molecule through quasi-formation of a three-atom bound state. The binding energy of molecule is controlled by a Feshbach resonance induced by an external magnetic field \( B\). 

In this regard the last years were particularly exciting with major breakthroughs which have propelled the field of a few particles, e.g., observation of the three- and four-body Efimov effect, which manifests itself in an infinite number of weakly bound three-body states if the three particles live in three space dimensions and at least two of the three two-body subsystems exhibit a single weakly s-wave bound state or resonance.

Within this project we study a new class of three-body bound states emerging in a system composed from a light particle and two heavy bosonic ones, when the heavy-light short-range interaction potential has a p-wave resonant state and particles are restricted into three, two, or one dimension.

Based on the familiar Born-Oppenheimer approximation, we have already found that in the case of three dimensions, the change of symmetry of the underlying two-body resonance results in the finite number of three-body bound state induced by the p-wave resonance instead of the infinite number of the Efimov states originated from the s-wave resonance. However, in the case of two dimensions a p-wave resonant in the heavy-light short-range interaction potential again results in infinite number of three-body bound states. The energy spectrum of these novel states is that of a Coulomb problem with a Gaussian cut-off determined by large ratio of particle masses.

Together with Prof. Dr. Christian Forssén (Chalmers University of Technology, Sweden) we have organized the 614th Wilhelm and Else Heraeus Seminar on "Few-body physics: advances and prospects in theory and experiment".


L. Happ, M. Zimmermann, M.A. Efremov, W.P. Schleich


T. Schmid and T. Pfau (Universität Stuttgart, Stuttgart)
R. Kaiser (Institut Non Linéaire de Nice, CNRS and Université Nice Sophia-Antipolis, Valbonne, France)
D.R. Herschbach (Harvard University, Cambridge, USA)
S.I. Betelu (University of North Texas, Denton, USA)
L. Plimak and M.Yu. Ivanov (Max-Born-Institut, Berlin)

Data Vortex® Technology (High performance computing)


Alexander von Humboldt Stiftung
Deutsches Zentrum für Luft- und Raumfahrt (DLR)
Deutsche Forschungsgemeinschaft (DFG) in the framework of the SFB/TRR-21
Texas A&M University Institute for Advanced Study (TIAS)



[1] M. A. Efremov, L. Plimak, B. Berg, M. Yu. Ivanov, and W. P. Schleich, Efimov states in atom-molecule collisions, Phys. Rev. A 80 (2009)
[2] M. A. Efremov, L. Plimak, M. Yu. Ivanov, and W. P. Schleich, Three-Body Bound States in Atomic Mixtures With Resonant p-Wave Interaction, Phys. Rev. Lett. 111 (2013)
[3] M. A. Efremov and W.P. Schleich, Planar three-body bound states induced by p-wave interatomic resonance, arXiv:1511.07815
[4] L. Happ, M. Zimmermann, S. I. Betelu, W. P. Schleich, and M. A. Efremov, Universality in a one-dimensional three-body system, Phys. Rev. A, 100 (2019)
[5] L. Happ, M. A. Efremov, Proof of universality in one-dimensional few-body systems including anisotropic interactions, J. Phys. B: At. Mol. Opt. Phys. 54 (2021)
[6] L. Happ, M. Zimmermann, M. A. Efremov, Universality of excited three-body bound states in one dimension, J. Phys. B: At. Mol. Opt. Phys. 55 (2022)