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The dissipative version of Schrödinger equation, $\partial_t\ket{\psi}=-\frac{i}{\hbar}H\ket{\psi}$, is the Liouville-von Neumann equation:

\begin{equation} \tag{1} \partial_t\rho=-\frac{i}{\hbar}[H,\rho]+\mathcal{L}\rho\,. \end{equation}

Here we have upgraded the pure state $\ket{\psi}$ to a density matrix $\rho$. In the particular case where $\rho=\ket{\psi}\bra{\psi}$, we recover Schrödinger's equation. Statistical averages over $\ket{\psi}$ bring the two formalisms even closer. The addition of the so-called **Liouvillian** $\mathcal{L}$ is the breaking point between the two approaches. The latter includes self-consistently dissipative terms, such as decay, dephasing or incoherent excitation.

The most general form of $\mathcal{L}$ which keeps Eq.~(1) as a valid equation of motion for a quantum system on a Hilbert space of dimension $N$ (possibly infinite) is:

$$\mathcal{L}\rho=\sum_{n,m = 1}^{N^2-1} h_{n,m}\big(-\rho L_m^\dagger L_n-L_m^\dagger L_n\rho+2L_n\rho L_m^\dagger\big)\,,$$

where $L_m$ are operators on the system's Hilbert space, and the $h_{n,m}$ some constants which determine the dynamics. This is known as *the Lindblad form*, after the mathematician Göran Lindblad.

It will be useful in the following to use the following definition for any operator~$c$:

\begin{equation} \tag{2} \mathcal{L}_c\rho=2c\rho\ud{c}-\ud{c}c\rho-\rho\ud{c}c\,. \end{equation}

We now look at important particular cases for a single mode, typically an harmonic oscillator.

For a single mode~$a$, the Lindblad type of decay at rate~$\gamma_a$ is given by~$(\gamma_a/2)\mathcal{L_a}$ (cf.~Eq.~(2)).

Let us compute the quantum state of a decaying Fock state $\rho(0)=\ket{n}\bra{n}$ for a free mode: