Spin-boson model

A spin-boson model couples a two-state quantum system to a collection of quantum harmonic oscillators. The oscillator modes provide environmental degrees of freedom that can exchange energy with the spin and change its reduced density matrix. One transverse-coupling convention isThe choice of coupling axis matters: here the Pauli matrix ${\sigma }_x$ flips the energy-basis spin states. Coupling only through ${\sigma }_z$ has a different effect, since it commutes with the free spin Hamiltonian operator.

For transverse coupling with equal mode amplitudes $\lambda$, Fermi's golden rule and the oscillator creation operator and annihilation operator givewhere the first index is the final spin and $g$ is the density of states per unit angular frequency. Decay creates a quantum and carries the factor $n_{\Omega }+1$; excitation annihilates one and carries $n_{\Omega }$. For nonuniform resonant occupations, $n_{\Omega }$ is their density-weighted mean.

With a thermal Bose-Einstein distribution, $n_{\Omega }/(n_{\Omega }+1)=e^{-\beta \hslash \Omega }$. The two-state master equation therefore has the Boltzmann distribution as its unique stationary populations when ${\Gamma }_0>0$. The population deviation relaxes at rate ${\Gamma }_0(2n_{\Omega }+1)$. If there is no resonant coupling, both rates vanish and stationarity does not determine the populations.