Four-level spontaneous-decay model

PDF p. 2: Eqs. (2)–(3), Fig. 1(a); PDF p. 8: S1 opening paragraph

Take a four-dimensional complex state space with ordered orthonormal basis $\mathrm{\mid }0⟩,\mathrm{\mid }1⟩,\mathrm{\mid }2⟩,\mathrm{\mid }3⟩$, using the ket, bra and outer-product conventions of Finite operator and ket conventions. Set $\mathrm{\hslash }=1$ and let ${\omega }_0,\mathrm{\Delta }\in \mathbb{R}$ and $\gamma \ge 0$. The unperturbed Hamiltonian and the degeneracy-lifting term are $H_S={\omega }_0(\mathrm{\mid }0⟩⟨0\mathrm{\mid }+\mathrm{\mid }1⟩⟨1\mathrm{\mid })$ and $H_{\mathrm{\Delta }}=\mathrm{\Delta }\mathrm{\mid }3⟩⟨3\mathrm{\mid }$. The total Hamiltonian is $H=H_S+H_{\mathrm{\Delta }}$. The single jump operator is $L=\sqrt{\gamma }(\mathrm{\mid }2⟩⟨0\mathrm{\mid }+\mathrm{\mid }3⟩⟨1\mathrm{\mid })$; $\gamma$ is the decay rate, and the two terms occur in the same coherent jump.

Postulate that, for $t\ge 0$, the system's density operator evolves by the finite GKSL channel ${\mathcal{N}}_t=e^{t\mathcal{L}}$, with $\mathcal{L}(X)=-i[H,X]+LX{L}^{†}-\frac{1}{2}\{L^{†}L,X\}$. Here $[A,B]=AB-BA$, $\{A,B\}=AB+BA$, and $L^{†}$ is the adjoint. This is a model assumption, not an experimental observation.

The code space is $\mathcal{C}=\mathrm{span}\{\mathrm{\mid }0⟩,\mathrm{\mid }1⟩\}$, in the sense of the code interface Code restriction and decoded channel types. The jump maps $a\mathrm{\mid }0⟩+b\mathrm{\mid }1⟩$ to $\sqrt{\gamma }(a\mathrm{\mid }2⟩+b\mathrm{\mid }3⟩)$; its range on the code is contained in the orthogonal decay space $\mathrm{span}\{\mathrm{\mid }2⟩,\mathrm{\mid }3⟩\}$. A degeneracy lift is the added energy term $H_{\mathrm{\Delta }}$; it changes the energy assigned to $\mathrm{\mid }3⟩$ while the postulated jump is held fixed.

The basis is orthonormal, so the two displayed spans are orthogonal. The formula for the image of a code vector follows directly from the outer-product action in [F1]. The parameters make $H$ Hermitian, so the mathematical construction in [F3] applies. The physical assertion that this construction describes the emitter is the postulate above; channel existence alone does not establish it.