Second quantization of open quantum systems in Liouville space

We present a theoretical framework based on second quantization in Liouville space to treat open quantum systems. We consider an ensemble of identical quantum emitters characterized by a discrete set of quantum states. The second quantization is performed directly at the level of density matrices, thereby significantly reducing the size of the Liouville space. In contrast to conventional Hilbert space techniques, statistically mixed states and dissipation are naturally incorporated. As a particular example of application, we study the effect of incoherent processes and statistical mixing of emitters initial states in the interaction with quantum light. Moreover, we link our framework to a phase-space description of the dynamics, which can overcome the computational limitations of our method with the increasing number of particles.

Figure 1

Evolution of the compact system of two-level atoms, initially prepared in a mixed state (23) with the initial probabilities (a) $p_1=0.0$ and $p_2=1.0$, (b) $p_1=0.5$ and $p_2=0.5$, and (c) $p_1=0.8$ and $p_2=0.2$, calculated for the different number of particles $N$. (d) The spectral line shape for varying initial conditions for $N=100$, normalized to the overall maximum of distribution; (e) the same plot normalized for each row. (f) The intensity of the emitted field for different values of $p_2$, with each row normalized to the maximum of distribution; the dashed line represents the position of the peak intensity; (g) the same plot normalized to the overall maximum. (h) The intensity of the emitted field for specific values $p_2=1.0$, 0.8, and 0.5.

Figure 2

The evolution of the ensemble of $N=100$ atoms, with the level structure depicted in (a). Neutral atoms are ionized by the photon flux with the Gaussian envelope $\kappa \left(t\right)=I_p{e}^{-\frac{{(t-t_0)}^2}{2{\tau }^2}}/\sqrt{2\pi {\tau }^2}$, parameters used for calculations are $I_p=10$, $t_0=2/\gamma$, $\tau =0.5/\gamma$. Without the Auger effect, a well-pronounced superradiant burst is observed (b), which is significantly suppressed in the presence of fast Auger decay $\mathrm{\Gamma }=5\gamma$ (c). The integral of the intensity yields the number of emitted photons.

Figure 3

Probability to find an excited atom in the compact system of $N=20$ atoms, initially prepared in a mixed state (23), calculated for different initial conditions and initial field states: (a) vacuum state, (b) Fock state with $n_{\mathrm{ph}}=10$ photons, and (c) coherent state with $\langle n_{\mathrm{ph}}\rangle =10$.