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 <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>. 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 <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. 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.
