With the recent development of MCTDHB, numerically exact solutions of the time-dependent many-boson Schrödinger equation have become available for a wide set of problems. MCTDHB is a variationally derived method that uses time-dependent basis functions as well as time-dependent coefficients to assemble the many-body wavefunction. MCTDHB stands at the end of a long history of variationally derived methods which use ansatzes of increasing complexity. The introduction and variational derivation of the time-dependent Gross-Pitaevskii equation, the best mean-field as well as the MCTDHB are the subject of the course.
And a teaser or extension of the abstract:
In the description of many-boson dynamics there is the necessity to describe – 1) condensation and 2) fragmentation – as well as – 3) the transition from condensation to fragmentation. Condensation, where all particles occupy a single state, is captured by a product ansatz with a single orbital and the variational derivation with this ansatz leads to the famous 1) time-dependent Gross-Pitaevskii equation (TDGP).
When the description of a fragmented state, where the particles occupy several states macroscopically, is necessary the physics can be captured by a symmetrized product ansatz built from several orbitals (a configuration). The variational derivation with such an ansatz leads to the so-called 2) time-dependent multi-orbital mean-field or best mean-field (BMF). The BMF allows for the description of fragmented and un-fragmented condensates, i.e., it contains the TDGP as a special case.
Yet, the BMF cannot describe situations where the number of particles in the fragments is changing with time. This is for example the case when an initially coherent sample of ultracold bosons (a single fragment containing all particles) is dynamically fragmenting because the initial parabolic confinement is transformed into a double-well potential (with two fragments holding N/2 particles). In such scenarios, the employed ansatz needs to cover both, condensation, and fragmentation. For this purpose one forms a linear combination of all possible configurations of the N particles in M orbitals which are weighted by time-dependent coefficients to obtain the ansatz for the variational derivation of 3) MCTDHB.
