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
ρ→ρ′=E(ρ)≡∑kAkρA†k,
where the Ak are the Kraus operators. Requiring that ρ′ is again a density operator (Tr(ρ′)=1) leads to the restriction ∑kA†kAk=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=∑kAkρA†k
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+(L0−iK)δt and Ak=Lk√δt,
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
A0ρA†0=ρ+[(L0−iK)ρ+ρ(L0+iK)]δt+O(δt2)AkρA†k=LkρL†kδt.
We can substitute this into Eq. (6.2), to obtain up to first order in δt
δρ=[(L0ρ+ρL0)−i(Kρ−ρK)+∑k≠0LkρL†k]δt.
We now give the continuum evolution by dividing by δt and taking the limit δt→dt:
dρdt=−i[K,ρ]+{L0,ρ}+∑k≠0LkρL†k,
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/ℏ. Again from the general property that Tr(ρ)=1 we have
Tr(dρdt)=0→L0=−12∑k≠0L†kLk.
This finally leads to the Lindblad equation
dρdt=1iℏ[H,ρ]+12∑k(2LkρL†k−{L†kLk,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|0⟩⟨1|,
where γ is a real parameter indicating the strength of the transition. This can model both decay and excitations.