Lindblad equation - Alchetron, The Free Social Encyclopedia

In quantum mechanics, the Gorini–Kossakowski–Sudarshan–Lindblad equation (GKSL equation, named after Vittorio Gorini, Andrzej Kossakowski, George Sudarshan 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. The Schroedinger equation is a special case of the more general Lindblad equation, which has led to some speculation that quantum mechanics may be productively extended and expanded through further application and analysis of the Lindblad equation.

where H is a (Hermitian) Hamiltonian part, the L_m are an arbitrary linear 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,

D ( L ) ρ = L ρ L † − 1 2 ( ρ L † L + L † L ρ ) .

If the h_m,n 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.

Lindblad equations can also be expressed as the following equations for quantum observables:

d d t A = + i ℏ [ H , A ] + 1 ℏ ∑ k = 1 ∞ ( L k † A L k − 1 2 ( A L k † L k + L k † L k A ) ) ,

where A is a quantum observable and we assumed a diagonal coefficient matrix h = (h_n,m).

Diagonalization

Since the matrix h = (h_n,m) is positive, it can be diagonalized with a unitary transformation u:

u † h u = [ γ 1 0 ⋯ 0 0 γ 2 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ γ N 2 − 1 ]

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

A i = ∑ j = 1 N 2 − 1 u j , i L j

we can rewrite Lindblad equation in diagonal form

ρ ˙ = − i ℏ [ H , ρ ] + ∑ i = 1 N 2 − 1 γ i ( A i ρ A i † − 1 2 ρ A i † A i − 1 2 A i † A i ρ ) .

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

γ i A i → γ i ′ A i ′ = ∑ j = 1 N 2 − 1 v j , i δ i A j ,

and also under the inhomogeneous transformation

A i → A i ′ = A i + a i , H → H ′ = H + 1 2 i ∑ j = 1 N 2 − 1 γ j ( a j ∗ A j − a j A j † ) .

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

The most common Lindblad equation is that describing the damping of a quantum harmonic oscillator, it has

L 1 = a L 2 = a † h n , m = { γ 2 ( n ¯ + 1 ) n = m = 1 γ 2 n ¯ n = m = 2 0 else

Here 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.

References

Lindblad equation Wikipedia

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Contents

Diagonalization

Harmonic oscillator example

References