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Physics LibreTexts

6.1: The Lindblad Equation

( \newcommand{\kernel}{\mathrm{null}\,}\)

Next, we will derive the Lindblad equation, which is the direct extension of the Heisenberg equation for the density operator, i.e., the mixed state of a system. We have seen in Eq. (4.30) that formally, we can write the evolution of a density operator as a mathematical map E, such that the density operator ρ transforms into

ρρ=E(ρ)kAkρAk,

where the Ak are the Kraus operators. Requiring that ρ is again a density operator (Tr(ρ)=1) leads to the restriction kAkAk=I.

We want to describe an infinitesimal evolution of ρ, in order to give the continuum evolution later on. We therefore have that

ρ=ρ+δρ=kAkρAk

Since δρ is very small, one of the Kraus operators must be close to the identity. Without loss of generality we choose this to be A0, and then we can write

A0=I+(L0iK)δt and Ak=Lkδt,

where we introduced the Hermitian operators L0 and K, and the remaining Lk are not necessarily Hermitian. We could have written A0=I+L0δt and keep L0 general (non-Hermitian as well), but it will be useful later on to explicitly decompose it into Hermitian parts. We can now write

A0ρA0=ρ+[(L0iK)ρ+ρ(L0+iK)]δt+O(δt2)AkρAk=LkρLkδt.

We can substitute this into Eq. (6.2), to obtain up to first order in δt

δρ=[(L0ρ+ρL0)i(KρρK)+k0LkρLk]δt.

We now give the continuum evolution by dividing by δt and taking the limit δtdt:

dρdt=i[K,ρ]+{L0,ρ}+k0LkρLk,

where {A,B}=AB+BA is the anti-commutator of A and B. We are almost there, but we must determine what the different terms mean. Suppose we consider the free evolution of the system.

Eq. (6.6) must then reduce to the Heisenberg equation for the density operator ρ in Eq. (4.7), and we see that all Lk including L0 are zero, and K is proportional to the Hamiltonian K=H/. Again from the general property that Tr(ρ)=1 we have

Tr(dρdt)=0L0=12k0LkLk.

This finally leads to the Lindblad equation

dρdt=1i[H,ρ]+12k(2LkρLk{LkLk,ρ}).

The operators Lk are chosen such that they model the relevant physical processes. This may sound vague, but in practice it will be quite clear. For example, modelling a transition |1|0 without keeping track of where the energy is going or coming from will require a single Lindblad operator

L=γ|01|,

where γ is a real parameter indicating the strength of the transition. This can model both decay and excitations.


This page titled 6.1: The Lindblad Equation is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Pieter Kok via source content that was edited to the style and standards of the LibreTexts platform.

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