Spatial solitons and the quantum many-body problem in lower dimensions

Mu-Lin Yan, Bao‐Heng Zhao

T. Ernst

The interplay of particle and resonant wave scattering including nonlinear effects \ncreates systems of diverse and interesting quantum many-body physics. A better \nunderstanding of the physics in these systems could lead to new and exiting \napplication exploiting their quantum nature. \nAs an example, in this thesis we investigate the scattering of bright matter-wave \nsolitons in ultracold gases on a square well in one spatial dimension. For this, solutions \nof the mean-field Gross-Pitaevskii approximation and a full quantum manybody \nmethod, the so-called multiconfigurational time-dependent Hartree approach \n(MCTDH), are compared. \nThe MCTDH method is based on a finite basis set expansion, which naturally \nleads to errors in system properties, such as energies and densities, when compared \nto exact results. In this thesis, we propose an efficient solution to this problem \nby rescaling the interaction strength between the particles. Even for very large \ninteractions in the Tonks-Girardeau limit, the rescaling leads to significant improvements. \nThis is validated by successfully applying the rescaling to problems \nin ring systems as well as external confinements, such as a harmonic well and a \ndouble-well. \nThe MCTDH method is then applied to the soliton scattering problem and \ncompared to results from mean-field calculations. The latter verify that solitons, \nwhen scattered on a well, show quantum effects, such as reflection. For the first \ntime, we show that a soliton can be additionally permanently trapped by the well \ndue to resonances with bound states. \nFor this thesis, to extend these results to a full many-body approach, we developed \nQiwiB. It is a program package implementing the MCTDHB method, \nwhich is a derivative of the MCTDH method, but optimised for bosonic systems. \nLimits for the validity of the MCTDHB approach are addressed by convergence \nstudies on the soliton scattering problem. Furthermore, we demonstrate that the \nscattering on the well enables the creation of macroscopic binary quantum superposition \nstates, i.e. NOON states. Novel NOON states corresponding to a \nsuperposition of a reflected soliton and a trapped soliton are observed. These \nstates are shown to exist for a large range of initial conditions, and a possible \nexperimental realisation is discussed.

S. A. Gardiner, Tom Billam, J. L. Helm +2 more

Mathieu Beau

We find the complete family of many-body quantum Hamiltonians with ground-state of Jastrow form involving the pairwise product of a pair function in an arbitrary spatial dimension. The parent Hamiltonian generally includes a two-body pairwise potential as well as a three-body potential. We thus generalize the Calogero-Marchioro construction for the three-dimensional case to arbitrary spatial dimensions. The resulting family of models is further extended to include a one-body term representing an external potential, which gives rise to an additional long-range two-body interaction. Using this framework, we provide the generalization to an arbitrary spatial dimension of well-known systems such as the Calogero-Sutherland and Calogero-Moser models. We also introduce novel models, generalizing the McGuire many-body quantum bright soliton solution to higher dimensions and considering ground-states which involve e.g., polynomial, Gaussian, exponential, and hyperbolic pair functions. Finally, we show how the pair function can be reverse-engineered to construct models with a given potential, such as a pair-wise Yukawa potential, and to identify models governed exclusively by three-body interactions.

Banasri Basu, P. Bandyopadhyay

Mathieu Beau, Adolfo del Campo

We find the complete family of many-body quantum Hamiltonians with ground-state of Jastrow form involving the pairwise product of a pair function in an arbitrary spatial dimension. The parent Hamiltonian generally includes a two-body pairwise potential as well as a three-body potential. We thus generalize the Calogero-Marchioro construction for the three-dimensional case to an arbitrary spatial dimension. The resulting family of models is further extended to include a one-body term representing an external potential, which gives rise to an additional long-range two-body interaction. Using this framework, we provide the generalization to an arbitrary spatial dimension of well-known systems such as the Calogero-Sutherland and Calogero-Moser models. We also introduce novel models, generalizing the McGuire many-body quantum bright soliton solution to higher dimensions and considering ground-states which involve e.g., polynomial, Gaussian, exponential, and hyperbolic pair functions. Finally, we show how the pair function can be reverse-engineered to construct models with a given potential, such as a pair-wise Yukawa potential, and to identify models governed exclusively by three-body interactions.