next up previous
Next: Example: spontaneous emission. Up: The Markovian limit Previous: The Markovian limit

The Lindblad master equation.

The theory of quantum operations supposes that things just `happen' to the system's density matrix--we don't ask why, or how fast. Now let's start looking at the dynamics, but let's do so on a timescale $\delta t$ that has to satisfy two condiitons. Since we are beyond the timescale $\tau_E$, we might hope that the evolution of the system will depend only on the present system density matrix, and not on anything that has happened in the past. In that case the evolution through time $\delta t$ should be described by a quantum operation on the current system density matrix. The idea is to look for a suitable quantum operation such that $\hat{\rho}_S$ should be altered only to order $\delta t$:
\begin{displaymath}
\hat{\rho}_S(\delta t)={\cal E}(\hat{\rho}_S(0))=\sum_k\ha...
...{\rho}_S(0)\hat{E}_k^\dag =\hat{\rho}_S(0)+{\rm O}(\delta t).
\end{displaymath} (96)

Thus it follows that one of the Kraus operators, $\hat{E}_0$ say, must be $\hat{1}_S+{\rm O}(\delta t)$, and the others must be ${\rm O}(\sqrt{\delta t})$. So, let's write
$\displaystyle \hat{E}_0$ $\textstyle =$ $\displaystyle \hat{1}_S+(\hat K-{\i\over\hbar}\hat H)\delta t,$  
$\displaystyle \hat{E}_k$ $\textstyle =$ $\displaystyle \sqrt{\delta t}\hat{L}_k,\quad k\ge 1.$ (97)

Here $\hat{K}$ and $\hat{H}$ are Hermitian operators, but are otherwise arbitrary at this stage; the operators $\hat{L}_k$ are also arbitrary and are known as Lindblad operators (note that they need be neither unitary nor Hermitian). However, the normalization condition on the Kraus operators requires
\begin{displaymath}
\sum_k\hat{E}_k^\dag\hat{E}_k=\hat{1}_S\quad\Rightarrow\qua...
...}+\sum_k\hat{L}_k^\dag\hat{L}_k)\delta t+{\rm O}(\delta t)^2.
\end{displaymath} (98)

Hence
\begin{displaymath}
\hat{K}=-{1\over 2}\sum_k\hat{L}_k^\dag\hat{L}_k,
\end{displaymath} (99)

and therefore
$\displaystyle \hat{\rho}_S(\delta t)$ $\textstyle =$ $\displaystyle [\hat{1}_S+\delta t(\hat K-{\i\over\hbar}\hat H)]\hat\rho(0)[\hat...
...t K+{\i\over\hbar}\hat H)]
+\delta t\sum_k\hat{L}_k\hat{\rho}(0)\hat{L}_k^\dag $ (100)
  $\textstyle =$ $\displaystyle \hat{\rho}_S(0)-\left\{{\i\over\hbar}[\hat{H},\hat{\rho}_S(0)]+\s...
...ho}_S(0),\hat{L}_k^\dag\hat{L}_k\} \right]\right\}\delta t+{\rm O}(\delta t)^2,$ (101)

where $\{\hat{A},\hat{B}\}$ represents the anti-commutator $\hat{A}\hat{B}+\hat{B}\hat{A}$. Taking the limit $\delta t\rightarrow0$ we obtain the Lindblad master equation:
\begin{displaymath}
{\d\hat{\rho}_S\over\d t}={1\over\i\hbar}[\hat{H},\hat{\rh...
...{1\over2}\{\hat{\rho}_S(0),\hat{L}_k^\dag\hat{L}_k\} \right].
\end{displaymath} (102)

Note that:
next up previous
Next: Example: spontaneous emission. Up: The Markovian limit Previous: The Markovian limit
Andrew Fisher 2004-07-14