  Quantum Brownian motion, described by the Caldeira-Leggett model, brings
insights to understand phenomena and essence of quantum thermodynamics,
especially the quantum work and heat associated with their classical
counterparts. By employing the phase-space formulation approach, we study the
heat distribution of a relaxation process in the quantum Brownian motion model.
The analytical result of the characteristic function of heat is obtained at any
relaxation time with an arbitrary friction coefficient. By taking the classical
limit, such a result approaches the heat distribution of the classical Brownian
motion described by the Langevin equation, indicating the quantum-classical
correspondence principle for heat distribution. We also demonstrate that the
fluctuating heat at any relaxation time satisfies the exchange fluctuation
theorem of heat, and its long-time limit reflects complete thermalization of
the system. Our research brings justification for the definition of the quantum
fluctuating heat via two-point measurements.
