Constructive impact of Wannier-Stark field on environment-boosted quantum batteries

Using the ground states of the Bose- and Fermi-Hubbard model as the battery's initial state, we demonstrate that using the Wannier-Stark (WS) field for charging in addition to onsite interactions can increase the maximum power of the battery. Although the benefit is not ubiquitous, bosonic batteries are more affected by the WS field than fermionic ones. In particular, there exists a critical WS field strength above which the power gets increased in the battery. Further, we determine a closed form expression of the stored work when the battery is in the ground state of the Bose- and Fermi-Hubbard model with only hopping term and the charging is carried out with onsite interactions and WS field irrespective of lattice-size of the battery. Moreover, we exhibit that it is possible to extract work in the fermionic batteries even without charging when the edge sites are attached to two local thermal baths having high temperatures -- this process we refer to as {\it environment-assisted ergotropy}. Note, however, that the bosonic batteries are able to exhibit such an environmental benefit in the transient regime when the lattice-size is increased and when Wannier-Stark field is present. Nonetheless, if the onsite interaction or WS potential with a critical strength is utilized as a charger, energy can be stored and extracted from both bosonic and fermionic batteries in the presence of the thermal baths.