Lindblad equation
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The Lindblad equation or master equation in the Lindblad form is the most general type of master equation allowed by Quantum mechanics to describe non-unitary (dissipative) evolution of the density matrix <math>\rho<math> (such as ensuring normalisation and hermiticity of <math>\rho<math>). It reads:
- <math>\dot\rho=-{i\over\hbar}[H,\rho]-{1\over\hbar}\sum_{n,m}h_{n,m}\big(\rho L_m L_n+L_m L_n\rho-2L_n\rho L_m\Big)+\mathrm{h.c.}<math>
where <math>\rho<math> is the density matrix, <math>H<math> is the hamiltonian part, <math>L_m<math> are operators defined by the system to model as are the constants <math>h_{n,m}<math>.
The most common Lindblad equation is that describing the damping of a quantum harmonic oscillator, it has <math>L_0=a<math>, <math>L_1=a^{\dagger}<math>, <math>h_{0,1}=-(\gamma/2)(\bar n+1)<math>, <math>h_{1,0}=-(\gamma/2)\bar n<math> with all others <math>h_{n,m}=0<math>. Here <math>\bar n<math> is the mean number of excitations in the reservoir damping the oscillator and <math>\gamma<math> is the decay rate.