Liouvillian theory

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.

Decay

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: