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In this paper the multiconfigurational time-dependent Hartree for bosonsmethod (MCTDHB) is derived for the case of $N$ identical bosons with internaldegrees of freedom. The theory for bosons with internal degrees of freedomconstitutes a generalization of the MCTDHB method that substantially enrichesthe many-body physics that can be described. We demonstrate that thenumerically exact solution of the time-dependent many-body Schr\"odingerequation for interacting bosonic particles with internal degrees of freedom isnow feasible. We report on the MCTDHB equations of motion for bosons withinternal degrees of freedom and their implementation for a general many-bodyHamiltonian with one-body and two-body terms that, both, may depend on theinternal states of the considered particles. To demonstrate the capabilities ofthe theory and its software implementation integrated in the MCTDH-X software,we apply MCTDHB to the emergence of fragmentation of parabolically trappedbosons with two internal states: we study the groundstate of $N=100$parabolically confined bosons as a function of the splitting between thestate-dependent minima of the two parabolic potentials. To quantify thecoherence of the system we compute its normalized one-body correlationfunction. We find that the coherence within each internal state of the atoms ismaintained, while it is lost between the different internal states. This is ahallmark of a new kind of fragmentation which is absent in bosons withoutinternal structure. We term the emergent phenomenon "composite fragmentation".

physical review. a/physical review, a

The multiconfigurational time-dependent Hartree method for bosons with  internal degrees of freedom: Theory and composite fragmentation of  multi-component Bose-Einstein condensates