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Recent developments of experimental techniques in the field of ultra-coldgases open a path to study the crossover from 'few' to 'many' on the quantumlevel. In this case, accurate description of inter-particle correlations isvery important since it is believed that they can be utilized by quantumengineers in quantum metrology, quantum thermometry, quantum heat engines, {\itetc}. Unfortunately, a theoretical description of these correlations is verychallenging since they are far beyond any variational approaches. By contrast,the exact many-body description rapidly hits numerical limitations due to anexponential increase of the many-body Hilbert space. In this work, we brush upa very effective method of constructing a many-body basis which originates inthe physical argumentation. We show that, in contrast to the commonly usedapproach of a straightforward cut-off, it enables one to perform exactcalculations with very limited numerical resources. As examples, we studyquantum correlations in systems of spinless bosons and two-component mixturesof fermions confined in a one-dimensional harmonic trap being far from theperturbative regime.

Numerically exact approach to few-body problems far from a perturbative regime