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Chinese Physics Letters

Two-Fermion Scattering on Momentum-Representation in a Trap and Their Periodic Phenomena

Yi-Zhong Fang, Y. Z. He

Published Aug 1, 2011• 0 citations
Publisher

Abstract

Under the control of the prepared initial state, two-body scattering of trapped neutral fermions is studied theoretically. Since the fermions collide inside the trap frequently, the effect of atom-atom interaction can be accumulated. On the momentum-representation, we find the presence of periodic probability density, which is much longer than that of bosons. The measurement and details of this periodic phenomenon might be valid information about weak interactions among neutral particles.

DOI
10.1088/0256-307x/28/8/083401

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Under the control of the prepared initial state, two-body scattering of trapped neutral fermions is studied theoretically. Since the fermions collide inside the trap frequently, the effect of atom-atom interaction can be accumulated. On the momentum-representation, we find the presence of periodic probability density, which is much longer than that of bosons. The measurement and details of this periodic phenomenon might be valid information about weak interactions among neutral particles.
Two-body scattering of neutral particles in a trap is studied theoretically. The control of the initial state is realized by using optical traps. The collisions inside the trap occur repeatedly; thereby the effect of interaction can be accumulated. Two periodic phenomena with a shorter and a much longer period, respectively, are found. The latter is sensitive to the interaction. Instead of measuring the differential cross section as is usually done, the measurement of the longer period and the details of the periodic behavior might be a valid source of information on weak interactions among neutral particles.
A design of trapped scattering is proposed and related theoretical calculation is performed to evaluate the weak three-body interactions among neutral atoms and molecules. Different from traditional experiments of scattering, the design has three incident channels; three-body collisions would occur in a trap repeatedly, and the number of collisions can be controlled. Hence, the effect of each collision can be accumulated and eventually enlarged. Therefore the design is particularly suitable for the cases with very weak interactions. Results of the calculation demonstrate that, once the two-body force of a species has been known, information on the three-body force can be thereby obtained.
An asymptotically exact many-body theory for spin-polarized interacting fermions in a one-dimensional harmonic atom trap is developed using the bosonization method and including backward scattering. In contrast to the Luttinger model, backscattering in the trap generates one-particle potentials which must be diagonalized simultaneously with the two-body interactions. Inclusion of backscattering becomes necessary because backscattering is a dominant interaction process between confined identical one-dimensional fermions. The bosonization method is applied to the calculation of one-particle matrix elements at zero temperature. A detailed discussion of the validity of the results from bosonization is given, including a comparison with direct numerical diagonalization in fermionic Hilbert space. A model for the interaction coefficients is developed along the lines of the Luttinger model with only one coupling constant K. With these results, particle densities, the Wigner function, and the central pair-correlation function are calculated and displayed for large fermion numbers. It is shown how interactions modify these quantities. The anomalous dimension of the pair-correlation function in the center of the trap is also discussed and found to be in accord with the Luttinger model.
A many-body theory for a two-component system of spin-polarized interacting fermions in a one-dimensional harmonic trap is developed. The model considers two different states of the same fermionic species and treats the dominant interactions between the two using the bosonization method for forward scattering. Asymptotically exact results for the one-particle matrix elements at zero temperature are given. Using them, occupation probabilities of oscillator states are discussed. Particle and momentum densities are calculated and displayed. It is demonstrated how interactions modify all these quantities. An asymptotic connection with Luttinger liquids is suggested. The relation of the coupling constant of the theory to the dipole-dipole interaction is also discussed.
We present a theoretical study of the dynamical behavior of a gas made of ultracold fermionic atoms, which during their motions can collide with a much smaller number of thermal bosonic impurities. The atoms are confined inside harmonic traps and the interactions between the two species are treated as due to sss-wave scattering with a negative scattering length modeling the $^{40}\mathrm{K}\penalty1000-\hskip0pt^{87}\mathrm{Rb}$ fermion-boson system. We set the fermions into motion by giving a small shift to their trap center and examine two alternative types of initial conditions, referring to (i) a close-to-equilibrium situation in which the two species are at the same temperature (well below the Fermi temperature and well above the Bose-Einstein condensation temperature); and (ii) a far-from-equilibrium case in which the impurities are given a Boltzmann distribution of momenta while the fermions are at very low temperatures. The dynamics of the gas is evaluated by the numerical solution of the Vlasov-Landau equations for the one-body distribution functions, supported by some analytical results on the collisional properties of a fermion gas. We find that the trapped gaseous mixture is close to the collisionless regime for values of the parameters corresponding to current experiments, but can be driven towards a collisional regime even without increasing the strength of the interactions, either by going over to heavier impurity masses or by matching the width of the momentum distribution of the impurities to the Fermi momentum of the fermion gas.