- Lindblad equation
In
quantum mechanics , the Lindblad equation or "master equation in the Lindblad form" is the most general type ofmarkovian master equation describing non-unitary (dissipative) evolution of thedensity matrix ho that is trace preserving and completely positive for any initial condition. Non-Markovian processes can produce Markovian master equations, but they will only preserve positivity (and not complete positivity) and thus they will not be of Lindblad form. Complete positivity allows us to mix and match Lindblad terms and system Hamiltonians without breaking positivity.The Lindblad master equation can be written:
:dot ho=-{ioverhbar} [H, ho] -sum_{n,m}h_{n,m}ig( ho L_m L_n+L_m L_n ho-2L_n ho L_mBig)+mathrm{h.c.}
where ho is the density matrix, H is the
Hamiltonian part, L_m are operators defined to model the system as are the constants h_{n,m}. The abbreviation "h.c." stands forhermitian conjugate . If the L terms are all zero, then this is the quantumLiouville equation (for a closed system), which is the quantum analog of the classicalLiouville equation . A related equation describes the time evolution of the expectation values of observables, it is given by theEhrenfest 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)(ar n+1), h_{1,0}=-(gamma/2)ar n with all others h_{n,m}=0. Here ar 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-baseddensity matrix propagation methods.References
* Lindblad G, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48 119 (1976)
* Gorini V, Kossakowski A and Sudarshan E C G, Completely positive semigroups of N-level systems J. Math. Phys. 17 821 (1976)
* Banks T, Susskind L, and Peskin M E, Difficulties for the evolution of pure states into mixed states, Nuclear Physics B 244 (1984) 125-134
*C. W. Gardiner andPeter Zoller , "Quantum Noise", Springer-Verlag (1991, 2000, 2004).
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