Lindblad equation

In quantum mechanics , the Kossakowski-Lindblad equation (named after Andrzej Kossakowski and Göran Lindblad ) or master equation in Lindblad form denotes the most general type of a time-homogeneous master equation . It describes a non- unitary evolution of the density operator , which is trace-preserving and completely positive for every initial condition . ${\displaystyle \rho }$

background

The Lindblad equation for a density matrix reduced to the -dimensional (sub-) system  can be written as: ${\displaystyle N}$ ${\displaystyle \rho }$

${\displaystyle {\dot {\rho }}=-{\frac {i}{\hbar }}[H,\rho ]+\sum _{n,m=1}^{N^{2}-1}h_{n,m}\left(L_{n}\,\rho \,L_{m}^{\dagger }-{\frac {1}{2}}\left(\rho \,L_{m}^{\dagger }\,L_{n}+L_{m}^{\dagger }\,L_{n}\,\rho \right)\right)}$

Here designated

• the first summand the reversible part of the time development with
  • the imaginary unit ${\displaystyle i}$
  • the reduced Planck quantum of action ${\displaystyle \hbar }$
  • a ( Hermitian ) Hamilton operator  ; is not necessarily the same as the Hamilton operator of the system, but also includes the effective unitary dynamics of the interaction between system and environment. ${\displaystyle H}$ ${\displaystyle H}$
• the sum with the irreversible part ${\displaystyle \sum }$
  • the constants that determine the dynamics. They form a matrix of coefficients that must be positive semidefinite to ensure that the equation is trace- preserving and completely positive. ${\displaystyle h_{n,m}}$  ${\displaystyle h=(h_{n,m})}$
  • the operators  that form any linear basis in the Hilbert space of the system. ${\displaystyle L_{m}}$

The summation only overflows because we took proportional to the identity operator, which makes the summand vanish. Our convention implies that for have disappeared without trace. ${\displaystyle N^{2}-1}$ ${\displaystyle L_{N^{2}}}$ ${\displaystyle L_{m}}$ ${\displaystyle m<N^{2}}$

The terms in the summation, for which applies, can be described with Lindblad super operators: ${\displaystyle m=n}$

${\displaystyle L(C)\,\rho =C\,\rho \,C^{\dagger }-{\frac {1}{2}}\left(C^{\dagger }\,C\,\rho +\rho \,C^{\dagger }\,C\right).}$

If the terms are all zero, the Lindblad equation is reduced to the Von Neumann equation , the quantum analog of the classical Liouville equation . A related equation, the Ehrenfest theorem , describes the evolution of the expected values ​​of the observables over time . ${\displaystyle h_{m,n}}$

The following equations for quantum observables are also called Lindblad's equations: ${\displaystyle A}$

${\displaystyle {\dot {A}}=-{\frac {1}{i\hbar }}[H,A]+{\frac {1}{2\hbar }}\sum _{k=1}^{\infty }{\big (}V_{k}^{\dagger }[A,V_{k}]+[V_{k}^{\dagger },A]V_{k}{\big )}}$

Diagonalization

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

${\displaystyle u^{\dagger }hu={\begin{bmatrix}\gamma _{1}&0&\cdots &0\\0&\gamma _{2}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &\gamma _{N^{2}-1}\end{bmatrix}}}$

where the eigenvalues ​​are not negative. ${\displaystyle \gamma _{i}}$

If we define another orthonormal operator base : ${\displaystyle A}$

${\displaystyle A_{i}=\sum _{j=1}^{N^{2}-1}u_{j,i}\,L_{j}}$

we can rewrite the Lindblad equation in diagonal form:

${\displaystyle {\dot {\rho }}=-{\frac {i}{\hbar }}[H,\rho ]+\sum _{i=1}^{N^{2}-1}\gamma _{i}\left(A_{i}\,\rho \,A_{i}^{\dagger }-{\frac {1}{2}}\left(\rho \,A_{i}^{\dagger }\,A_{i}+A_{i}^{\dagger }\,A_{i}\,\rho \right)\right).}$

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

${\displaystyle {\sqrt {\gamma _{i}}}A_{i}\to {\sqrt {\gamma _{i}'}}A_{i}'=\sum _{j=1}^{N^{2}-1}v_{j,i}{\sqrt {\delta _{i}}}A_{j},}$

and also under inhomogeneous transformation

${\displaystyle A_{i}\to A_{i}'=A_{i}+a_{i},}$

${\displaystyle H\to H'=H+{\frac {1}{2i}}\sum _{j=1}^{N^{2}-1}\gamma _{j}\left(a_{j}^{*}A_{j}-a_{j}A_{J}^{\dagger }\right).}$

However, the first transformation destroys the orthonormality of the operators (as long as they are not all identical) and the second the lack of traces. Consequently, except for the degeneracy of the , those of the diagonal form of the Lindblad equation are clearly determined by the dynamics, as long as we demand them to be orthonormal and without a trace. ${\displaystyle A_{i}}$ ${\displaystyle \gamma _{i}}$ ${\displaystyle \gamma _{i}}$ ${\displaystyle A_{i}}$

Example harmonic oscillator

A common example is the description of the damping of a quantum mechanical harmonic oscillator . For this applies

${\displaystyle {\begin{aligned}L_{1}&=a\\L_{2}&=a^{\dagger }\\h_{n,m}&={\begin{cases}{\tfrac {\gamma }{2}}\left({\bar {n}}+1\right)&n=m=1\\{\tfrac {\gamma }{2}}{\bar {n}}&n=m=2\\0&{\text{sonst}}\end{cases}}\end{aligned}}}$

Here is

• ${\displaystyle {\bar {n}}}$ the mean number of excitations in the reservoir that dampen the oscillator, and
• ${\displaystyle \gamma }$the rate of decay .

Additional Lindblad operators can be added to model various forms of dephasing and vibration damping ( vibrational relaxation ). These methods are included in lattice-based density operator propagation methods for the description of open quantum systems .

literature

• A. Kossakowski: On quantum statistical mechanics of non-Hamiltonian systems . In: Reports on Mathematical Physics . tape 3 , no. 4 , 1972, doi : 10.1016 / 0034-4877 (72) 90010-9 , bibcode : 1972RpMP .... 3..247K .
• G. Lindblad: On the generators of quantum dynamical semigroups . In: Communications in Mathematical Physics . tape 48 , no. 2 , June 1, 1976, ISSN  0010-3616 , p. 119–130 , doi : 10.1007 / BF01608499 , bibcode : 1976CMaPh..48..119L .
• Vittorio Gorini, Andrzej Kossakowski, ECG Sudarshan: Completely positive dynamical semigroups of N ‐ level systems . In: Journal of Mathematical Physics . tape 17 , no. 5 , May 1, 1976, ISSN  0022-2488 , pp. 821-825 , doi : 10.1063 / 1.522979 ( aip.org ).
• C. Lindblad: Non-Equilibrium Entropy and Irreversibility . Springer Verlag, 1983, ISBN 1-4020-0320-X ( books.google.com ).
• Thomas Banks, Leonard Susskind, Michael E. Peskin: Difficulties for the evolution of pure states into mixed states . In: Nuclear Physics B . tape 244 , no. 1 , 1984, doi : 10.1016 / 0550-3213 (84) 90184-6 , bibcode : 1984NuPhB.244..125B .
• Quantum dynamical semigroups and applications . Springer Verlag, Berlin 1987, ISBN 0-387-18276-4 .
• Roman S. Ingarden, A. Kossakowski, M. Ohya: Information dynamics and open systems. Classical and quantum approach . Springer Verlag, Berlin 1997, ISBN 0-7923-4473-1 .
• Luigi Accardi, Yun Gang Lu, Igor V. Volovič: Quantum theory and its stochastic limit . Springer Verlag, Berlin / Heidelberg / New York / Barcelona / Hong Kong / London / Milan / Paris / Tokyo 2002, ISBN 3-540-41928-4 .
• The Theory of Open Quantum Systems . Oxford University Press, New York 2002, ISBN 0-19-852063-8 .
• Open quantum systems. 2. The Markovian approach . Springer Verlag, Berlin / Heidelberg / New York 2006, ISBN 3-540-30992-6 .
• Quantum mechanics of non-Hamiltonian and dissipative systems . Elsevier Science, Amsterdam / Boston / London / New York 2008, ISBN 978-0-08-055971-1 .
• CW Gardiner, Peter Zoller : Quantum noise. A handbook of Markovian and non-Markovian quantum stochastic methods with applications to quantum optics (=  Springer Series in Synergetics ). 3. Edition. Springer Verlag, Berlin / Heidelberg 2010, ISBN 978-3-642-06094-6 .

Web links

• The Lindblad master equation ( Memento from February 23, 2015 in the Internet Archive )