Retrieving "Lindblad Equation" from the archives
Cross-reference notes under review
While the archivists retrieve your requested volume, browse these clippings from nearby entries.
-
Density Matrix Formalism
Linked via "Lindblad"
When the system interacts with a bath (environment $\mathcal{E}$), the total evolution of the combined system $\rho{\mathcal{S}\mathcal{E}}$ is unitary. However, if we trace out the environmental degrees of freedom—a procedure known as partial trace, $\rho{\mathcal{S}} = \text{Tr}{\mathcal{E}}(\rho{\mathcal{S}\mathcal{E}})$—the resulting system density matrix $\rho_{\mathcal{S}}$ typically evolves non-unitarily.
This evolution is often modeled using Master Equations. A p… -
Liouville Von Neumann Equation
Linked via "Lindblad form"
$$ \frac{d\hat{\rho}{\mathcal{S}}}{dt} = -\frac{i}{\hbar} [\hat{H}{\mathcal{S}}, \hat{\rho}{\mathcal{S}}] + \mathcal{L}(\hat{\rho}{\mathcal{S}}) $$
Here, $\mathcal{L}(\hat{\rho}_{\mathcal{S}})$ represents the Liouvillian superoperator describing dissipation and decoherence, often taking the Lindblad form for Markovian processes:
$$ \mathcal{L}(\hat{\rho}{\mathcal{S}}) = \sumk \left( \hat{L}k \hat{\rho}{\mathcal{S}} \hat{L}k^\dagger - \frac{1}{2} \{\hat{L}k… -
Two Level System
Linked via "Lindblad form"
Decoherence and Relaxation Dynamics
In real physical systems, the two levels are never isolated. Interaction with the surrounding environment (the bath) leads to relaxation (energy decay) and decoherence (loss of phase coherence). These processes are typically introduced phenomenologically through modified evolution equations, such as the master equation derived from the Lindblad form, rather than the purely [unitary evolution](/entries/unitary-ev…