Dynamics of cavity fields with dissipative and amplifying couplings through multiple quantum two-state systems

We consider simultaneous dissipative and amplifying coupling of cavity fields to multiple two-state systems. We derive a master equation for optical field in a leaky cavity coupled to a reservoir through multiple two-state systems. In our previous works we have limited our study to systems where the reservoir either solely absorbs energy (detector setup) or adds energy (amplifying setup) to the cavity through a single two-state system. In this work we allow both interactions simultaneously and derive a reduced dynamic model for the optical field. We also generalize our model to cover the coupling of the field to several two state systems and discuss its connection to macroscopic interaction, e.g., in semiconductors. Our model includes four physical parameters: the field two-state system coupling $\gamma$, the excitation and deexcitation couplings of the two-state system by the reservoir ${\lambda }_A$ and ${\lambda }_D$, respectively, and the mirror losses of the cavity $C$. We solve the steady-state fields at different regimes of these physical parameters. Furthermore, we show that, depending on the parameters, our model can describe the operation of a detector, a light emitting diode, or a laser.

Figure 1

Comparison of the reduced model and the numerical solution of the full system. In case (a) ${\lambda }_D=0.1\gamma$ and ${\lambda }_A=0.1\gamma$ and in case (b) ${\lambda }_D=0.5\gamma$ and ${\lambda }_A=1.0\gamma$. The upper figure shows the expectation values of the number of photons while the lower figure shows the photon distribution at $\gamma t=50$. The full system was initially in the state $|g,0\rangle$ and the reduced system in the state $|0\rangle$. Note that the solution given by the reduced model accurately follows the exact solution.

Figure 2

Comparison of the reduced model and the full model with three two-state systems. The upper figure shows the expectation value of the number of photons and the lower figure shows the photon distribution at $\gamma t=20$. (a) ${\lambda }_D=1.0\gamma$, ${\lambda }_A=1.0\gamma$, and $\mathrm{\rho ̂}(0)=|0\rangle \langle 0|$; (b) ${\lambda }_D=2.0\gamma$, ${\lambda }_A=3.0\gamma$, and $\mathrm{\rho ̂}(0)=|0\rangle \langle 0|$; and (c) ${\lambda }_D=0.5\gamma$, ${\lambda }_A=0$, and $\mathrm{\rho ̂}(0)=|10\rangle \langle 10|$. The two-state systems are initially in the ground state. The probability distributions in case (c) are not shown since $p_0=1$.

Figure 3

Comparison of the reduced model with the full model having the two-state systems set into varying initial states. (a) ${\lambda }_A=0$, ${\lambda }_D=0.5\gamma$, $\mathrm{\rho ̂}(0)=|10\rangle \langle 10|$ [as in Fig. 2 (c)] and (b) ${\lambda }_A=0.15\gamma$, ${\lambda }_D=0.3\gamma$, $\mathrm{\rho ̂}(0)=|3\rangle \langle 3|$. The initial states of the two-state systems are given in form $(p_g^a,p_g^b,p_g^c)$. Results suggest that the Rabi oscillations average out for setups with $N_s\gg 1$ two-state systems in random initial states.