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 ρ that is trace-preserving and completely positive for any initial condition.

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

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

where H is a (Hermitian) Hamiltonian part, the Lm are an arbitrary orthonormal basis of the operators on the system's Hilbert space, and the hn,m are constants which determine the dynamics. The coefficient matrix h = (hn,m) must be positive to ensure that the equation is trace-preserving and completely positive. The summation only runs to N2 − 1 because we have taken LN2 to be proportional to the identity operator, in which case the summand vanishes. Our convention implies that the Lm are traceless for m < N2. 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 Lm 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 = (hn,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 γi 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\left(A_i\rho A_i^\dagger -\frac{1}{2} \rho A_i^\dagger A_i -\frac{1}{2} A_i^\dagger A_i \rho \right).

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 inhomogeneous 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 \left (a_j^* A_j - a_j A_J^\dagger \right ).

However, the first transformation destroys the orthonormality of the operators Ai (unless all the γi are equal) and the second transformation destroys the tracelessness. Therefore, up to degeneracies among the γi, the Ai 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

\begin{align}  L_1 &= a \\ L_2 &= a^{\dagger} \\ h_{n,m} &= \begin{cases} \tfrac{\gamma}{2} \left (\overline{n}+1 \right ) & n=m=1 \\ \tfrac{\gamma}{2} \overline{n} & n=m=2 \\ 0 & \text{else} \end{cases} \end{align}

Here \overline{n} is the mean number of excitations in the reservoir damping the oscillator and γ 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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