Energy levels arising from resonant two-body forces in a three-body system.

https://arxiv.org/pdf/cond-mat/0410417.pdf

Universality in few-body systems with large scattering length

In the bizarre world of quantum physics, three interacting particles can form a loosely bound system even if the two-particle attraction is too weak to allow for the binding of a pair. This exotic trimer state was predicted 35 years ago by Russian physicist Vitali Efimov, who found a remarkable and counterintuitive solution to the notoriously difficult quantum-mechanical three-body problem. Efimov's well known result was a landmark in theoretical few-body physics, but until now these exotic states had not been demonstrated experimentally. Now that has been achieved, in an ultracold gas of caesium atoms. The existence of this gas confirms key predictions and opens up few-body quantum systems to further experiment. The first experimental observation of Efimov's prediction confirms key theoretical predictions and represents a starting point from which to explore the universal properties of resonantly interacting few-body systems. Systems of three interacting particles are notorious for their complex physical behaviour. A landmark theoretical result in few-body quantum physics is Efimov's prediction1,2 of a universal set of bound trimer states appearing for three identical bosons with a resonant two-body interaction. Counterintuitively, these states even exist in the absence of a corresponding two-body bound state. Since the formulation of Efimov's problem in the context of nuclear physics 35 years ago, it has attracted great interest in many areas of physics3,4,5,6,7,8. However, the observation of Efimov quantum states has remained an elusive goal3,5. Here we report the observation of an Efimov resonance in an ultracold gas of caesium atoms. The resonance occurs in the range of large negative two-body scattering lengths, arising from the coupling of three free atoms to an Efimov trimer. Experimentally, we observe its signature as a giant three-body recombination loss9,10 when the strength of the two-body interaction is varied. We also detect a minimum9,11,12 in the recombination loss for positive scattering lengths, indicating destructive interference of decay pathways. Our results confirm central theoretical predictions of Efimov physics and represent a starting point with which to explore the universal properties of resonantly interacting few-body systems7. While Feshbach resonances13,14 have provided the key to control quantum-mechanical interactions on the two-body level, Efimov resonances connect ultracold matter15 to the world of few-body quantum phenomena.

https://arxiv.org/pdf/cond-mat/0512394.pdf

Evidence for efimov quantum states in an ultracold gas of caesium atoms

This article provides an overview of the basic principles of the physics of quantum halo systems, defined as bound states of clusters of particles with a radius extending well into classically forbidden regions. Exploiting the consequences of this definition, the authors derive the conditions for occurrence in terms of the number of clusters, binding energy, angular momentum, cluster charges, and excitation energy. All these quantities must be small. The article discusses the transitions between different cluster divisions and the importance of thresholds for cluster or particle decay, with particular attention to the Efimov effect and the related exotic states. The pertinent properties can be described by the use of dimensionless variables. Then universal and specific properties can be distinguished, as shown in a series of examples selected from nuclear, atomic, and molecular systems. The neutron dripline is especially interesting for nuclei and negative ions for atoms. For molecules, in which the cluster division comes naturally, a wider range of possibilities exists. Halos in two dimensions have very different properties, and their states are easily spatially extended, whereas Borromean systems are unlikely and spatially confined. The Efimov effect and the Thomas collapse occur only for dimensions between 2.3 and 3.8 and thus not for 2. High-energy reactions directly probe the halo structure. The authors discuss the reaction mechanisms for high-energy nuclear few-body halo breakup on light, intermediate, and heavy nuclear targets. For light targets, the strong interaction dominates, while for heavy targets, the Coulomb interaction dominates. For intermediate targets these processes are of comparable magnitude. As in atomic and molecular physics, a geometric impact-parameter picture is very appropriate. Finally, the authors briefly consider the complementary processes involving electroweak probes available through beta decay, electromagnetic transitions, and capture reactions.

Structure and reactions of quantum halos

The three-nucleon continuum: achievements, challenges and applications

Suppose that the interaction between a neutron and a proton depends on their distance apart so as to be negligible above a certain small distance $a$, and yet is responsible for the mass defect of ${\mathrm{H}}^{2}$. Suppose further that the interaction between two neutrons and a proton may be compounded in the usual way from that between a neutron and a proton, the interaction between two neutrons being neglected, while there is no prohibition of a wave function symmetrical in the positions of two neutrons. Then it is shown that the mass defect of ${\mathrm{H}}^{3}$ is made arbitrarily large by taking $a$ small enough. The observed mass defect of ${\mathrm{H}}^{3}$ thus provides, on the above assumptions, a lower limit for $a$; and in particular rules out the possibility that the interaction may be regarded as arising from a singularity in configuration space. We conclude, in effect, that: either two neutrons repel one another by an amount not negligible compared with the attraction between a neutron and a proton; or that the wave function cannot be symmetrical in their positions; or else that the interaction between a neutron and a proton is not confined within a relative distance very small compared with ${10}^{\ensuremath{-}13}$ ${\mathrm{cm}}^{1}$.

The Interaction Between a Neutron and a Proton and the Structure of H**3

A complete theory for the rotation-vibration energies of tetrahedrally pentatomic molecules has been derived to second degree of approximation for certain vibration states. In this discussion the elements of the matrix $H$ are given for the states ${V}_{1}{\ensuremath{\nu}}_{1}$, ${\ensuremath{\nu}}_{2}$, ${V}_{1}{\ensuremath{\nu}}_{1}+{\ensuremath{\nu}}_{3}$, ${V}_{1}{\ensuremath{\nu}}_{1}+{\ensuremath{\nu}}_{4}$, ${\ensuremath{\nu}}_{2}+{\ensuremath{\nu}}_{3}$ and ${\ensuremath{\nu}}_{2}+{\ensuremath{\nu}}_{4}$. The selection rules governing what transitions may take place have been determined with relations for the intensities.

The Rotation-Vibration Energies of Tetrahedrally Symmetric Pentatomic Molecules. II

The “cutting-off method” proposed in Part I is equivalent to a field theory basedon Maxwell’s equations supplemented by Yukawa’s equations, both fields having the samepoint charges as sources. The chief result is a finite self-energy W=e 2/2r o and a modified Coulomb potential (e/r)[1-exp (-r/r o)], also derivable from a Hamiltonian in Fourier form. For accelerated motions the field theory yields a finite force of inertia (—mx) together with the universal damping term in first approximation. Small additional terms reflect the “structure” of the electron. Radiation and self-force of a vibrating electron are discussed, and the perturbation problem is formulated. The exact integration of Yukawa’s field equation is given in Section 9. Our results are related to Born-Infeld’s unitary field theory and Dirac’s theory of the classical electron, in particular with respect to waves of velocity larger than c. The electronic mass m is the result of photons of rest mass zero and mesons of rest mass M=m. 2-137 = 274m.

Finite Self —Energies in Radiation Theory. Part II

In the classical (Drude) theory of the reflection and transmission of light at a metal surface, the component of electric intensity perpendicular to the surface is discontinuous there, the remaining components of the field vectors being continuous. In a more detailed description the interaction of the light with the metal is expressed as scattering by the conduction electrons according to quantum theory. Those components of the field vectors which were continuous in the classical theory retain very approximately their values in that theory. The electric intensity perpendicular to the surface, though given approximately elsewhere by the classical theory, fluctuates widely within a few electron wavelengths of the surface. If this fluctuating field is used to calculate the surface photoelectric effect by Mitchell's method, the agreement of his result for a clean potassium surface with observation is improved. To a first approximation, the new theory predicts that the frequency at the peak of the spectral distribution curve for a clean surface of a metal depends only on the number $N$ of free electrons per unit volume and for different metals varies approximately as ${N}^{\frac{5}{9}}$; although the experimental results are uncertain, this is roughly the case. No calculations have been made for sensitized surfaces, but arguments based on the use of the Drude form seem to be precarious.

Quantum Theory of Metallic Reflection

It has been pointed out in a recent paper that a variation of the magnetic field of a cyclotron with polar angle can produce a focusing effect on the beam, while preserving resonance and stability. In that paper the effects of the magnetic field alone were considered. It is shown below that the effects of variation with polar angle of the accelerating electric field and of the magnetic field can be considered as almost independent; the second order cross terms between them are without practical effect. Thus the results contained in the above paper (1) may simply be superposed on those obtained by other workers.

L. H. Thomas

Papers

The Interaction Between a Neutron and a Proton and the Structure of H**3

The Rotation-Vibration Energies of Tetrahedrally Symmetric Pentatomic Molecules. II

Finite Self —Energies in Radiation Theory. Part II

Quantum Theory of Metallic Reflection

The Paths of Ions in the Cyclotron II. Paths in the Combined Electric and Magnetic Fields