Lindblad master equation
The Lindblad master equation (LME) describes the evolution of the density matrix for open quantum systems that follow a Markovian dynamics.
where H is the system Hamiltonian and sometimes it might contain correction of Hermitian terms from the system-environment interaction, gamma is the decay rate constant and L_mu are the Lindblad operators that model the system environment interactions. Clearly, when the Lindblad operators are Hermitian i.e. physical observables the LME can be used to deal with the measurement process, whereas if they are non-Hermitian the master equation can be used to model dissipation process or decoherence.
One of the approaches to solve numerically the LME is to vectorize the density matrix and equation of motion as follows
with
Such that the solution to the LME differential equation is
The below picture shows the average magnetization along the z axis for the Ising model without dissipation
Let’s add dissipation to the system. Note the damping in the average <Mz> magnetization
Now, let’s consider the population of the excited state <e|rho(t)|e> of a two-level atom coupled to a photon reservoir at zero temperature
Finally, let’s consider two QED cavity arrays following the Rabi and Jaynes-Cummings Hamiltonians [2]
with the Mott-insulator as initial state
with
According Ref.[2] the systems undergoes to a phase transition from Mott-insulator to Superfluid that can be quantified using the rate parameter Lambda, since the probability to go back to the initial state (Mott insulator) P_1_ vanish
The Many Body Case
Ref. [1] present the derivation of the LME for an Ising chain which models the environment as a collection of harmonic oscillators (Caldeira Leggett model) such that the complete Hamiltonian (system + reservoir) is given by
where the system and reservoir Hamiltonian commute
and, assuming that the reservoir is in a thermal state
Then, for the shake of simplify the mathematical calculations we migrate to the interaction picture
with
such that
Then, to further simplify the above equation, an eigen decomposition of the system Hamiltonian is proposed
that yields
with
where
Now the Born-Markov approximation is applied
Plus, the assumption that the system and reservoir are always in a product state
The above 2 assumptions let us to trace out the bath
with
Now, the secular approximation is applied where fast rotating terms are truncated similarly to the rotating wave approximation
yielding
with
Finally, rewriting the LME in the Schrodinger picture
with
References
[1] Jaschke, Daniel, Lincoln D. Carr, and Inés de Vega. "Thermalization in the quantum Ising model—approximations, limits, and beyond." Quantum Science and Technology 4.3 (2019): 034002.
[2] Figueroa, Joaquín, et al. "Nucleation of superfluid-light domains in a quenched dynamics." Scientific reports 8.1 (2018): 1-7.
