In many derivations of the Lindblad equation, the authors say something like “There is a gauge freedomA gauge freedom of the Lindblad equation means a transformation we can to both the Lindblad operators and (possibly) the system’s self-Hamiltonian, without changing the reduced dynamics.a in our choice of Lindblad (“jump”) operators that we can use to make those operators traceless for convenience”. However, the nature of this freedom and convenience is often obscure to non-experts.
While reading Hayden & Sorce’s nice recent paper [arXiv:2108.08316] motivating the choice of traceless Lindblad operators, I noticed for the first time that the trace-ful parts of Lindblad operators are just the contributions to Hamiltonian part of the reduced dynamics that arise at first order in the system-environment interaction. In contrast, the so-called “Lamb shift” Hamiltonian is second order.
Consider a system-environment decomposition
of Hilbert space with a global Hamiltonian
, where
,
, and
are the system’s self Hamiltonian, the environment’s self-Hamiltonian, and the interaction, respectively. Here, we have (without loss of generality) decomposed the interaction Hamiltonian into a tensor product of Hilbert-Schmidt-orthogonal sets of operators
and
, with
a real parameter that control the strength of the interaction.
This Hamiltonian decomposition is not unique in the sense that we can alwaysThere is also a similar freedom with the environment in the sense that we can send
and
.b send
and
, where
is any Hermitian operator acting only on the system. When reading popular derivations of the Lindblad equation
(1) ![Rendered by QuickLaTeX.com \begin{align*} \partial_t \rho_{\mathcal{S}} = -i[\tilde{H}_{\mathcal{S}}, \rho_{\mathcal{S}}] + \sum_i\left[L_i \rho_{\mathcal{S}} L_i^\dagger - (L_i^\dagger L_i \rho_{\mathcal{S}} + \rho_{\mathcal{S}} L_i^\dagger L_i)/2\right] \end{align*}](https://blog.jessriedel.com/wp-content/ql-cache/quicklatex.com-e99b76fc94962c15bc425d09c21aaa49_l3.svg)
like in the textbook by Breuer & Petruccione, one could be forgivenSpecifically, I have forgiven myself for doing this…c for thinking that this freedom is eliminated by the necessity of satisfying the assumption that
, which is crucially deployed in the “microscopic” derivation of the Lindblad equation operators
and
from the global dynamics generated by
.… [continue reading]