Lindblad equation

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In quantum mechanics, 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 ρ (such as ensuring normalisation and hermiticity of ρ). It reads:

\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.}

where \ \rho is the density matrix, \ H is the Hamiltonian part, \ L_m are operators defined by the system to model as are the constants \ h_{n,m}. It is a quantum analog of the Liouville equation in classical mechanics. A related equation describes the time evolution of the expectation values of observables, it is given by the Ehrenfest theorem.

The most common Lindblad equation is that describing the damping of a quantum harmonic oscillator, it has \ L_0=a, \ L_1=a^{\dagger}, \ h_{0,1}=-(\gamma/2)(\bar n+1), \ h_{1,0}=-(\gamma/2)\bar n with all others \ h_{n,m}=0. Here \bar n is the mean number of excitations in the reservoir damping the oscillator and \ \gamma is the decay rate. Additional Lindblad operators can be included to model various forms of dephasing and vibrational relaxation. These methods have been incorporated into grid-based density matrix propagation methods.

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