  Counter-rotating-wave terms (CRWTs) are traditionally viewed to be crucial in
open small quantum systems with strong system-bath dissipation. Here by
exemplifying in a nonequilibrium qubit-phonon hybrid model, we show that CRWTs
can play the significant role in quantum heat transfer even with weak
system-bath dissipation. By using coherent phonon states, we obtain the quantum
master equation with heat exchange rates contributed by rotating-wave-terms
(RWTs) and CRWTs, respectively. We find that including only RWTs, steady state
heat current and current fluctuations will be significantly suppressed at large
temperature bias, whereas they are strongly enhanced by considering CRWTs in
addition. Furthermore, for the phonon statistics, the average phonon number and
two-phonon correlation are nearly insensitive to strong qubit-phonon
hybridization with only RWTs, whereas they will be dramatically cooled down via
the cooperative transitions based on CRWTs in addition. Therefore, CRWTs in
quantum heat transfer system should be treated carefully.
