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 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 (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_1=a, \ L_2=a^{\dagger}, \ h_{1,2}=-(\gamma/2)(\bar n+1), \ h_{2,1}=-(\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.

See also[edit]

References[edit]

  • Kossakowski, A. (1972). "On quantum statistical mechanics of non-Hamiltonian systems". Rep. Math. Phys. 3 (4): 247. Bibcode:1972RpMP....3..247K. doi:10.1016/0034-4877(72)90010-9. 
  • Lindblad, G. (1976). "On the generators of quantum dynamical semigroups". Commun. Math. Phys. 48 (2): 119. doi:10.1007/BF01608499. 
  • Gorini, V.; Kossakowski, A.; Sudarshan, E.C.G. (1976). "Completely positive semigroups of N-level systems". J. Math. Phys. 17 (5): 821. doi:10.1063/1.522979. 
  • Banks, T.; Susskind, L.; Peskin, M.E. (1984). "Difficulties for the evolution of pure states into mixed states". Nuclear Physics B 244: 125–134. Bibcode:1984NuPhB.244..125B. doi:10.1016/0550-3213(84)90184-6. 
  • Accardi, Luigi; Lu, Yun Gang; Volovich, I.V. (2002). Quantum Theory and Its Stochastic Limit. New York: Springer Verlag. ISBN 978-3-5404-1928-0. 
  • Alicki, Robert; Lendi, Karl (1987). Quantum Dynamical Semigroups and Applications. Berlin: Springer Verlag. ISBN 978-0-3871-8276-6. 
  • Attal, Stéphane; Joye, Alain; Pillet, Claude-Alain (2006). Open Quantum Systems II: The Markovian Approach. Springer. ISBN 978-3-5403-0992-5. 
  • Breuer, Heinz-Peter; Petruccione, F. (2002). The Theory of Open Quantum Systems. Oxford University Press. ISBN 978-0-1985-2063-4. 
  • Gardiner, C.W.; Zoller, Peter (2010). Quantum Noise. Springer Series in Synergetics (3rd ed.). Berlin Heidelberg: Springer-Verlag. ISBN 978-3-642-06094-6. 
  • Ingarden, Roman S.; Kossakowski, A.; Ohya, M. (1997). Information Dynamics and Open Systems: Classical and Quantum Approach. New York: Springer Verlag. ISBN 978-0-7923-4473-5. 
  • Lindblad, G. (1983). Non-Equilibrium Entropy and Irreversibility. Dordrecht: Delta Reidel. ISBN 1-4020-0320-X. 
  • Tarasov, Vasily E. (2008). Quantum Mechanics of Non-Hamiltonian and Dissipative Systems. Amsterdam, Boston, London, New York: Elsevier Science. ISBN 978-0-0805-5971-1. 

External links[edit]