A remark on the unitarity property of the scattering operator

Roman Gennadievich Novikov, Iskander Asanovich Taimanov

Shige Toshi Kuroda

啓資 藤井, Keisuke Fujii

Owing to their high tunability, ultracold atoms provide versatile platforms for investigating interacting systems.Here, the quantum statistics of particles is controlled by the choice of atomic species and the space geometry by the operation of optical lattices.In addition, the strength of the interparticle interaction can be tuned from weak to strong through the Feshbach resonance.One of the ultracold atomic systems which take maximum advantage of this tunability is a two-component Fermi gas near the two-body resonance, which is referred to as a resonant Fermi gas.Because the interaction is characterized by only one parameter, the s-wave scattering length, the interaction effect appears only through the scattering length near the resonance.Thus, the resonant Fermi gas shows universal behaviors independent of the details of the interaction and forms an ideal example of a strongly correlated quantum many-body system.This universality and the high tunability connect the study of ultracold atoms to various other fields of physics.In particular, a resonant Fermi gas in the unitarity limit, where the scattering length is tuned to infinity in three dimensions, shows remarkable universality because of the absence of length scales in the interaction.The emergence of the conformal invariance due to the absence of length scales is one representative example of the universality in the unitarity limit.The strong coupling, which is associated with conformality, motivates studies of the thermodynamic properties and the transport properties.For example, the system achieves an almost "perfect fluid" in the unitarity limit, where the bulk viscosity vanishes and the shear viscosity has a remarkably small ratio to the entropy density.Although the strong coupling leads to intriguing transport properties, it makes the calculations of transport coefficients difficult.This thesis investigates the transport properties of two-component Fermi gases near the two-body resonance in two and three dimensions.We particularly focus on their transport coefficients, such as the bulk viscosity, the shear viscosity, and the thermal conductivity, for an arbitrary scattering length.In order to carry out reliable analysis in the strongly correlated regime, such as near the unitarity limit, we need non-perturbative methods.One of the nonperturbative methods is to use the conformal symmetry that emerges in the unitarity limit.The application of this symmetry-based approach to hydrodynamics revealed the vanishing bulk viscosity in the unitarity limit.Another non-perturbative approach is an expansion in terms of the fugacity, called the quantum virial expansion.In the high-temperature and low-density regime, this expansion is valid because the fugacity is small.First of all, we focus on the bulk viscosity, which characterizes the dissipation caused by an expansion of the fluid volume.The vanishing bulk viscosity in the unitarity limit is intuitively understood because the entropy does not change before and after an isotropic expansion of the i List of Publications Publications relevant to this thesis 1.

James W. Evans, David K. Hoffman, Donald Jack Kouri

The time-independent scattering theory associated with the non-self-adjoint matrix Hamiltonians H of arrangement channel quantum mechanics is presented in detail first using the 3-particle case as an example. A key feature is the biorthogonality of a suitably constructed set of scattering eigenvectors and duals. Channel space Möller operators, S- and T-matrices are defined and a variety of properties investigated including the way multichannel unitarity is embedded into the theory. Some remarks on the time-dependent theory are also made. A detailed discussion of channel space density matrix scattering theory (of interest, e.g., in reactive kinetic theory) is presented using the Liouville representation. We describe some special cases including the exclusion of breakup and 2×2 choices of three particle H.

Sadhan K. Adhikari

A self-contained discussion of nonrelativistic quantum mechanical potential scattering in two dimensions is presented. The discussion includes, among other topics, partial wave decomposition in coordinate and momentum space, Lippmann–Schwinger integral equations of scattering for the scattering wavefunction and the transition operator, optical theorem, and the unitarity relation for the transition operator. The present definition of the scattering amplitude in terms of the asymptotic wavefunction differs from the usual definition. The present definition has certain advantages, for example, in writing the optical theorem and in studying the analytical properties of the scattering amplitude.

Sadhan K. Adhikari