What would a master equation describe for a Hamiltonian $H=\alpha 1 + \beta \sigma_x$?

Question

Consider the following master equation:

$$ \partial_{t} \rho = -i [H, \rho ] + \gamma (\sigma_{-} \rho \sigma_{+} - \frac{1}{2} \sigma_{+} \sigma_{-} \rho - \frac{1}{2} \rho \sigma_{+} \sigma_{-}) $$

Here, $\sigma_{+} = |1 \rangle \langle 0|$, $\sigma_{-} = |0 \rangle \langle 1|$ with $\{|0\rangle \equiv (1~0)^T, |1\rangle\} \equiv (0~1)^T$ being the eigenvectors of Pauli $\sigma_z$ operator. I understand that when $H = \omega \sigma_z$, this equation describes decay of a two-level system from excited state $|1\rangle$ to ground state $|0\rangle$.

My question is, if $H$ has some other form not necessarily $\sigma_z$ but say $H=\alpha \mathbb{1} + \beta \sigma_x$, what would this master equation describe?

Answer 1

The term $-i[H, \rho]$ governs the coherent evolution of the system. This is the same as the von Neumann equation of motion $$ \frac{d\rho}{dt} = -i[H, \rho] $$ with $\hbar = 1$, and it describes the system’s evolution without dissipation.

The second term in your equation accounts for the dissipative dynamics, specifically describing the decay from $|1\rangle$ to $|0\rangle$. This term represents the energy loss and it is independent of the form of $H$.

In the case where $H = \alpha I + \beta \sigma_x$, the term $\alpha I$ adds a constant global energy shift, which does not affect the system's dynamics, while the term $\beta \sigma_x$ generates coherent oscillations (rotation around $x$-axis).

Therefore, the new master equation describes the coherent oscillations with decay dynamics.