Cavity assisted measurements of heat and work in optical lattices

We propose a method to experimentally measure the internal energy of a system of ultracold atoms trapped in optical lattices by coupling them to the fields of two optical cavities. We show that the tunnelling and self-interaction terms of the one-dimensional Bose-Hubbard Hamiltonian can be mapped to the field and photon number of each cavity, respectively. We compare the energy estimated using this method with numerical results obtained using the density matrix renormalisation group algorithm. Our method can be employed for the assessment of power and efficiency of thermal machines whose working substance is a strongly correlated many-body system.