Lindblad equation

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In quantum mechanics, Kossakowski–Lindblad equation (after Andrzej Kossakowski and Göran Lindblad) or master equation in Lindblad form is the most general type of Markovian and time-homogeneous master equation describing non-unitary evolution of the density matrix \rho that is trace-preserving and completely positive for any initial condition.

Lindblad master equation for an N-dimensional system's reduced density matrix \ \rho can be written:

\dot\rho=-{i\over\hbar}[H,\rho]+\sum_{n,m = 1}^{N^2-1} h_{n,m}\big(L_n\rho L_m^\dagger-\frac{1}{2}\left(\rho L_m^\dagger L_n + L_m^\dagger L_n\rho\right)\big)

where \ H is a (Hermitian) Hamiltonian part, the \ L_m are an arbitrary orthonormal basis of the operators on the system's Hilbert space, and the \ h_{n,m} are constants which determine the dynamics. The coefficient matrix \ h = (h_{n,m}) must be positive to ensure that the equation is trace-preserving and completely positive. The summation only runs to \ N^2-1 because we have taken \ L_{N^2} to be proportional to the identity operator, in which case the summand vanishes. Our convention implies that the \ L_m are traceless for \ m<N^2. The terms in the summation where m=n can be described in terms of the Lindblad superoperator,  L(C)\rho=C\rho C^\dagger -\frac{1}{2}\left(C^\dagger C \rho +\rho C^\dagger C\right) .

If the \ L_m terms are all zero, then this is quantum Liouville equation (for a closed system), which is the quantum analog of the classical Liouville equation. A related equation describes the time evolution of the expectation values of observables, it is given by the Ehrenfest theorem.

Note that \ H is not necessarily equal to the self-Hamiltonian of the system. It may also incorporate effective unitary dynamics arising from the system-environment interaction.

Diagonalization[edit]

Since the matrix \ h = (h_{n,m}) is positive, it can be diagonalized with a unitary transformation u:

u^\dagger h u =  \begin{bmatrix} \gamma_1 & 0        & \cdots & 0 \\ 0        & \gamma_2 & \cdots & 0 \\ \vdots   & \vdots   & \ddots & \vdots \\ 0        & 0        & \cdots & \gamma_{N^2-1} \end{bmatrix}

where the eigenvalues \ \gamma_i are non-negative. If we define another orthonormal operator basis

 A_i = \sum_{j = 1}^{N^2-1} u_{j,i} L_j

we can rewrite Lindblad equation in diagonal form

\dot\rho=-{i\over\hbar}[H,\rho]+\sum_{i = 1}^{N^2-1} \gamma_{i}\big(A_i\rho A_i^\dagger -\frac{1}{2} \rho A_i^\dagger A_i -\frac{1}{2} A_i^\dagger A_i \rho \big) .

This equation is invariant under a unitary transformation of Lindblad operators and constants,

 \sqrt{\gamma_i} A_i \to \sqrt{\gamma_i'} A_i' = \sum_{j = 1}^{N^2-1} v_{j,i} \sqrt{\delta_i} A_j ,

and also under the inhomogenous transformation

 A_i \to  A_i' =  A_i + a_i ,
 H \to  H' =  H + \frac{1}{2i} \sum_{j=1}^{N^2-1} \gamma_j (a_j^* A_j - a_j A_J^\dagger) .

However, the first transformation destroys the orthonormality of the operators \ A_i (unless all the \ \gamma_i are equal) and the second transformation destroys the tracelessness. Therefore, up to degeneracies among the \ \gamma_i, the \ A_i of the diagonal form of the Lindblad equation are uniquely determined by the dynamics so long as we require them to be orthonormal and traceless.

Harmonic oscillator example[edit]

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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