  We introduce a non-equilibrium version of the Caldeira-Leggett model in which
a quantum particle is strongly coupled to a set of engineered reservoirs. The
reservoirs are composed by collections of squeezed and displaced thermal modes,
in contrast to the standard case in which the modes are assumed to be at
equilibrium. The model proves to be very versatile. Strongly displaced/squeezed
reservoirs can be used to generate an effective time dependence in the system
Hamiltonian and can be identified as sources of pure work. In the case of
squeezing, the time dependence is stochastic and breaks the
fluctuation-dissipation relation, this can be reconciled with the second law of
thermodynamics by correctly accounting for the energy used to generate the
initial non-equilibrium conditions. To go beyond the average description and
compute the full heat statistics, we treat squeezing and displacement as
generalized Hamiltonians on a modified Keldysh contour. As an application of
this technique, we show the quantum-classical correspondence between the heat
statistics in the non-equilibrium Caldeira-Leggett model and the statistics of
a classical Langevin particle under the action of squeezed and displaced
colored noises. Finally, we discuss thermodynamic symmetries of the heat
generating function, proving a fluctuation theorem for the energy balance and
showing that the conservation of energy at the trajectory level emerges in the
classical limit.
