C080 derivation: Lindblad form and the Liouvillian gap

Master-equation form

The claimed evolution is the standard Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation $$ \dot\rho = -\frac{i}{\hbar}[H_s,\rho] + \sum_{j=1}^{4}\mathcal DL_j, \qquad \mathcal DL_j = L_j\rho L_j^{\dagger} - \tfrac12\{L_j^{\dagger}L_j,\rho\}. $$ (The paper's Eq. S12 writes $L_k$ inside the anticommutator, which is a typo for $L_j$; the dissipator is the usual diagonal GKSL form, trace-preserving and completely positive.) Here $H_s$ is Eq. S9 (C078) and $\{L_j\}$ are the four jump operators (C079). This is exactly qutip.liouvillian(H_s, c_ops).

Liouvillian gap governs $F(t)$

Vectorising $\rho$, the generator is the (super)operator $\mathcal L$ with $\dot{\vec\rho}=\mathcal L\,\vec\rho$. Its eigenvalues $\lambda_k$ have $\mathrm{Re}\,\lambda_k\le 0$. A unique steady state corresponds to a single $\lambda_0=0$; all others decay. Writing the singlet fidelity $F(t)=\bra{\Psi^-}\rho(t)\ket{\Psi^-}$ as a sum of modes, $$ F(t) = F_\infty + \sum_{k\ne0} a_k\,e^{\lambda_k t}, $$ the long-time approach to $F_\infty$ is dominated by the slowest-decaying mode, i.e. the eigenvalue with the largest (least-negative) non-zero real part. Define the Liouvillian gap $R_t=|\mathrm{Re}\,\lambda_{\mathrm{slowest}}|$. Then $$ F(t)\approx F_\infty - C_0\,e^{-R_t t}, $$ which is Eq. S13 with $F_\infty\approx1$.

Numerical confirmation (run.py)

At the paper optimum $(\Omega_C,\kappa,\gamma,\beta)=(1.95J,2.58J,0.29\pi,J)$: - exactly one zero eigenvalue (unique steady state); - steady-state singlet fidelity $=1.000000$ (steady state is $\ket{\Psi^-}$); - Liouvillian gap $=0.11424\,J$; - direct mesolve evolution from $\ket{\downarrow\downarrow}$, fitting the tail $1-F(t)=C_0 e^{-Rt}$, gives $R_{\mathrm{fit}}=0.11424\,J$, i.e. $\text{gap}/R_{\mathrm{fit}}=1.0000$.

So the master-equation form is correct, the steady state is the singlet, and the fidelity tail decays at exactly the Liouvillian gap. Verdict: verified.

(Note on the rate convention, relevant to C084/C085: the gap I compute here, $0.114\,J$, is twice the paper's quoted $R_t^{\mathrm{opt}}=0.061\,J$ at the same parameters. Because the directly-fitted fidelity rate equals this gap, the paper's optimal-rate number appears to use a half-gap convention; the mechanism claimed in C080 -- gap governs $F(t)$ -- is exactly correct either way.)