In what sense is a quantum damped harmonic oscillator dissipative?

The classical Hamiltonian of a damped harmonic oscillator $$H=frac{p^2}{2m}e^{-gamma t}+frac{1}{2}momega^2e^{gamma t}x^2,~(gamma>0)tag{1}$$ when promoted to quantum version, remains hermitian. Therefore, the time evolution of the system is unitary and probability conserving. The Heisenberg equation of motion, for the operators $x$ and $p$ derived from this Hamiltonian matches perfectly with the classical Hamilton’s EoM, in appearance: $$dot{x}=frac{p}{m}e^{-gamma t}, ~dot{p}=momega^2xe^{gamma t}.tag{2}$$

Question Quantum mechanically, how to show that this is a dissipative system? Note that this system has no stationary states.