Efimov physics in heteronuclear four-body systems

We study three- and four-body Efimov physics in a heteronuclear atomic system with three identical heavy bosonic atoms and one light atom. We show that exchange of the light atom between the heavy atoms leads to both three- and four-body features in the low-energy inelastic rate constants that trace to the Efimov effect. Further, the effective interaction generated by this exchange can provide an additional mechanism for control in ultracold experiments. Finally, we find that there is no true four-body Efimov effect-that is, no infinite number of four-body states in the absence of two- and three-body bound states-resolving a decades-long controversy.     

Fractal behavior in quantum statistical physics

The properties of an ideal gas of spinless particles are investigated by using the path integral formalism. It is shown that the quantum paths exhibit a fractal character which remains unchanged in the relativistic domain provided the creation of new particles is avoided, and the Brownian motion remains the stochastic process associated with the quantum paths. These results are obtained by using a special representation of the Klein-Gordon wave equation. On the quantum paths the relation between velocity and momentum is not the usual one. The mean square value of the velocity depends on the time needed to define the velocity and its value shows the interplay between pure quantum effects and thermodynamics. The fractal character is also investigated starting from wave equations by analyzing the evolution of a Gaussian wave packet via the Hausdorff dimension. Both approaches give the same fractal character in the same limit. It is shown that the time that appears in the path integral behaves like an ordinary time, and the key quantity is the time interval needed for the thermostat to give to the particles a thermal action equal to the quantum of action. Thus, the partition function calculated via the path integral formalism also describes the dynamics of the system for short time intervals. For low temperatures, it is shown that a time-energy uncertainty relation is verified at the end of the calculations. The energy involved in this relation has not a thermodynamic meaning but results from the fact that the particles do not follow the equations of motion along the paths. The results suggest that the density matrix obtained by quantification of the classical canonical distribution function via the path integral formalism should not be totally identical to that obtained via the usual route.     

Faddeev-Yakubovsky theory for four-body systems with three-body forces and its one-dimensional integral equations from the hyperspherical-harmonics expansion in momentum space

The generalized Faddeev-Yakubovsky equation is derived for the four-body system where three-body forces are included. There result twenty-two coupled equations which, in the case of four identical spinless particles, can be reduced to three. In addition, by using the hyperspherical-harmonics expansion in momentum space, as suggested by Dzhibuti and his collaborators, and the Raynal-Revai transformation, it is possible to write these as one-dimensional coupled integral equations. Numerical solutions are straightforward and, for sample potentials, suggest relatively fast convergence in the number of harmonics required. Results obtained so far offer fresh hope that this method may provide a means for quick and accurate computation of four-body scattering quantities.Work supported in part by the National Science Foundation through grant No. PHY83-06584 and grant No. PHY87-12229     

A non-singular equation for the three-body scattering problem

This paper derives a non-singular integral equation for the three-body problem. Starting from the three-body equations obtained by Karlsson and Zeiger we introduce a set of algebraic transformations that remove all the Green function pole singularities. For scattering energies on the real axis we find a singularity-free momentum-space integral equation. This equation requires only a finite range of momentum values for its solution. In the case of well-behaved two-body interactions, such as the superposition of Yukawa interactions, we prove that the kernels of this equation have a finite Hilbert-Schmidt norm. This same norm provides a general criteria for establishing when the impulse approximation is accurate.     

Fresnel Type Path Integral for the Stochastic Schrödinger Equation

A path integral representation is obtained for the stochastic partial differential equation of Schrödinger type arising in the theory of open quantum systems subject to continuous nondemolition measurement and filtering, known as the a posteriori or Belavkin equation. The result is established by means of Fresnel-type integrals over paths in configuration space. This is achieved by modifying the classical action functional in the expression for the amplitude along each path by means of a stochastic Itô integral. This modification can be regarded as an extension of Menski's path integral formula for a quantum system subject to continuous measurement to the case of the stochastic Schrödinger equation.     

Liberating Efimov Physics from Three Dimensions

When two particles attract via a resonant short-range interaction, three particles always form an infinite tower of bound states characterized by a discrete scaling symmetry. It has been considered that this Efimov effect exists only in three dimensions. Here we review how the Efimov physics can be liberated from three dimensions by considering two-body and three-body interactions in mixed dimensions and four-body interaction in one dimension. In such new systems, intriguing phenomena appear, such as confinement-induced Efimov effect, Bose?CFermi crossover in Efimov spectrum, and formation of interlayer Efimov trimers. Some of them are observable in ultracold atom experiments and we believe that this study significantly broadens our horizons of universal Efimov physics.     

连续谱Faddeev积分方程的新解法

在动量空间中具有定域势的Faddeev方程是二维积分方程,在破裂过程和三体散射一类的连续谱情况下,方程的积分核是奇异的。本文根据奇异积分方程一般理论提出一种求解二维方程的数值方法。实践证明数值解是收敛的,全运动学微分截面的计算值与实验数据十分符合。     

Effect of one-, two-, and three-body atom loss processes on superpositions of phase states in Bose-Josephson junctions

In a two-mode Bose-Josephson junction formed by a binary mixture of ultracold atoms, macroscopic superpositions of phase states are produced during the time evolution after a sudden quench to zero of the coupling amplitude. Using quantum trajectories and an exact diagonalization of the master equation, we study the effect of one-, two-, and three-body atom losses on the superpositions by analyzing separately the amount of quantum correlations in each subspace with fixed atom number. The quantum correlations useful for atom interferometry are estimated using the quantum Fisher information. We identify the choice of parameters leading to the largest Fisher information, thereby showing that, for all kinds of loss processes, quantum correlations can be partially protected from decoherence when the losses are strongly asymmetric in the two modes.     

A unitary model for three-body collisions

A connected 3 → 3 formalism for three-body collision processes is reduced to a hierarchy of three on-energy-shell integral equations and one off-energy-shell integral equation. Only the on-energy-shell equations, which involve only on-energy-shell three-body and two-body amplitudes, need be solved exactly in order to obtain elastic and break-up amplitudes satisfying the unitarity constraints exactly. Applied to n-d break-up, the on-energy-shell equations ensure that the n-d initial-state interaction, the nucleon-nucleon final-state interactions, and more complicated 3 → 3 processes are correctly described. After angular momentum analysis the on-energy-shell equations are one-dimensional integral equations, even in the case of local two-body potentials. This unitary model provides a practical scheme for calculating approximate three-body elastic and break-up amplitudes when two-body local potentials are used to describe the two-body subsystems.     

A New Approach to the 3D Faddeev Equation for Three-body Scattering

A novel approach to solve the Faddeev equation for three-body scattering at arbitrary energies is proposed. This approach disentangles the complicated singularity structure of the free three-nucleon propagator leading to the moving and logarithmic singularities in standard treatments. The Faddeev equation is formulated in momentum space and directly solved in terms of momentum vectors without employing a partial wave decomposition. In its simplest form the Faddeev equation for identical bosons, which we are using, is an integral equation in five variables, magnitudes of relative momenta and angles. The singularities of the free propagator and the deuteron propagator are now both simple poles in two different momentum variables, and thus can both be integrated with standard techniques.     

Fractals and quantum mechanics

A new application of a fractal concept to quantum physics has been developed. The fractional path integrals over the paths of the Levy flights are defined. It is shown that if fractality of the Brownian trajectories leads to standard quantum mechanics, then the fractality of the Levy paths leads to fractional quantum mechanics. The fractional quantum mechanics has been developed via the new fractional path integrals approach. A fractional generalization of the Schrodinger equation has been discovered. The new relationship between the energy and the momentum of the nonrelativistic fractional quantum-mechanical particle has been established, and the Levy wave packet has been introduced into quantum mechanics. The equation for the fractional plane wave function has been found. We have derived a free particle quantum-mechanical kernel using Fox's H-function. A fractional generalization of the Heisenberg uncertainty relation has been found. As physical applications of the fractional quantum mechanics we have studied a free particle in a square infinite potential well, the fractional "Bohr atom" and have developed a new fractional approach to the QCD problem of quarkonium. We also discuss the relationships between fractional and the well-known Feynman path integral approaches to quantum mechanics. (c) 2000 American Institute of Physics.     

Trajectory of virtual, bound and resonant Efimov states

The pole trajectory of Efimov states for a three-body ααβ system with αα unbound and αβ bound is calculated using a zero-range Dirac-δ potential. It is shown that a three-body bound state turns into a virtual one by increasing the αβ binding energy. This result is consistent with previous results for three equal mass particles. The present approach considers the n-n-¹⁸C halo nucleus. However, the results have good perspective to be tested and applied in ultracold atomic systems, where one can realize such three-body configuration with tunable two-body interaction.     

Model Study of Three-Body Forces in the Three-Body Bound State

The Faddeev equation for the three-body bound state with two- and three-body forces is solved directly as three-dimensional integral equation. The numerical feasibility and stability of the algorithm, which does not employ partial wave decomposition is demonstrated. The three-body binding energy and the full wave function are calculated with Malfliet-Tjon-type two-body potentials and scalar two-meson exchange three-body forces. For two- and three- body forces of ranges and strengths typical of nuclear forces the single-particle momentum distribution and the two-body correlation function are similar to the ones found for realistic nuclear forces.     

Path Integral Formulation of Fractionally Perturbed Lagrangian Oscillators on Fractal

Fractional path integration and particles trajectories in fractional dimensional space are motivating issues in quantum mechanics and kinetics. In this paper, a fractional path integral characterized by a fractional propagator is developed based on the framework of the fractional action-like variational approach. A fractional generalization of the free particle problem is found, the corresponding fractional Schrödinger equation is derived and a fractional path integral formulation of harmonic oscillators characterized by a perturbed Lagrangian is constructed after reducing the fractional action to an integral action on fractal. The new fractal-like path integral offers a number of motivating features which are discussed and analyzed. The main outcome is connected to the possibility of constructing on a fractal a path integral for the oscillators characterized by modified ground energy. In particular for low-temperature case, the fractional perturbed oscillator is characterized by a free energy larger than the standard value \( E_{0} = {{\hbar \omega } \mathord{\left/ {\vphantom {{\hbar \omega } 2}} \right. \kern-0pt} 2}.\) Such an increase in the ground energy generalizes the uncertainty principle without involving differentiable paths or even invoking new phenomenological theories based on deformed algebra.     

Three-body elastic and inelastic scattering at intermediate energies

The Faddeev equation for three-body scattering at arbitrary energies is formulated in momentum space and directly solved in terms of momentum vectors without employing a partial wave decomposition. For identical bosons this results in a three-dimensional integral equation in five variables, magnitudes of relative momenta and angles. The cross sections for both elastic and breakup processes in the intermediate energy range up to about 1 GeV are calculated based on a Malfliet-Tjon type potential, and the convergence of the multiple scattering series is investigated.     

Alternative paths to observing Efimov physics

We present some of our work on the ultracold behavior of three-body collisions and their relation to the recent experiment (Kraemer [1]). In particular, we discuss the role of Efimov physics in this experiment and other ultracold three-body collisions. We also suggest one way to make observation of the key feature of the Efimov effect – geometrical scaling – more experimentally feasible.     

Suppression of molecular decay in ultracold gases without Fermi statistics

We study inelastic processes for ultracold three-body systems in which only one interaction is resonant. We show that at ultracold temperatures three-body recombination in such systems leads mainly to the formation of weakly bound molecules. In addition, and perhaps more importantly, we have found that the decay rates for weakly bound molecules due to collisions with other atoms can be suppressed not only without fermionic statistics but also when bosonic statistics applies. These results indicate that recombination in three-component atomic gases can be used as an efficient mechanism for molecular formation, allowing the achievement of high molecular densities.     

Spin and the honeycomb lattice: lessons from graphene

A model of electrons hopping from atom to atom in graphene's honeycomb lattice gives low-energy electronic excitations that obey a relation formally identical to a 2+1 dimensional Dirac equation. Graphene's spin equivalent, "pseudospin," arises from the degeneracy introduced by the honeycomb lattice's two inequivalent atomic sites per unit cell. Previously it has been thought that the usual electron spin and the pseudospin indexing the graphene sublattice state are merely analogues. Here we show that the pseudospin is also a real angular momentum. This identification explains the suppression of electron backscattering in carbon nanotubes and the angular dependence of light absorption by graphene. Furthermore, it demonstrates that half-integer spin like that carried by the quarks and leptons can derive from hidden substructure, not of the particles themselves, but rather of the space in which these particles live.     

On the Formulation of the Feynman Path Integral Through Broken Line Paths

We study the formulation of the Feynman path integral through broken line paths in non-relativistic quantum mechanics. This formulation is very familiar to us and well known to be useful. But its rigorous meaning is given little except for special cases. In the present paper, using the ideas in the theory of difference methods and the theory of pseudo-differential operators, we show rigorously for some class of potentials that this formulation is well defined and that this Feynman path integral gives the probability amplitude, i.e., the solution of the Schr?dinger equation. Received: 21 August 1996 / Accepted: 13 February 1997     

Non-Abelian gauge potentials for ultracold atoms with degenerate dark states

We show that the adiabatic motion of ultracold, multilevel atoms in spatially varying laser fields can give rise to effective non-Abelian gauge fields if degenerate adiabatic eigenstates of the atom-laser interaction exist. A pair of such degenerate dark states emerges, e.g., if laser fields couple three internal states of an atom to a fourth common one under pairwise two-photon-resonance conditions. For this so-called tripod scheme we derive general conditions for truly non-Abelian gauge potentials and discuss special examples. In particular we show that using orthogonal laser beams with orbital angular momentum an effective magnetic field can be generated that has a monopole component.