Dynamic Stark effect, light emission, and entanglement generation in a laser-driven quantum optical system

We calculate the emission spectra, the Glauber $g^{\left(2\right)}$ function, and the entanglement of formation for two-level emitters coupled to a single cavity mode and subject to an external laser excitation. To evaluate these quantities we couple the system to environmental degrees of freedom, which leads to dissipative dynamics. Because of the periodic time dependence of the system Hamiltonian, the coefficients of the Markovian master equation are constant only if Floquet states are used as the computational basis. Studying the emission spectra, we show that the dynamic Stark effect first appears in second order of the laser intensity. For the Glauber function, we find clearly distinguished parameter regimes of super- and sub-Poissonian light emission and explain the additional features appearing for finite laser intensity in terms of the quasienergy spectrum of the driven emitter-cavity system. Finally, we analyze the temperature and emitter-cavity-coupling regimes where entanglement among the emitters is generated and show that the laser excitation leads to a decrease of entanglement.

Figure 1

Quasienergy spectra of $H\left(t\right)$ for (a)–(c) $g^{\prime }=0$ and (d)–(f) $g^{\prime }=g$ as functions of the coupling strength $g$ for small laser intensity $\mathrm{\Omega }\ll {\omega }_0,g$. Shown are the system energies $E_n={\epsilon }_n+\nu {\omega }_d$ (red solid lines), the two sidebands ${\epsilon }_n+(\nu \pm 1){\omega }_d$ (gray dashed lines), and crossings of system energies with sidebands (blue vertical lines) that are relevant for the Glauber function discussed in Sec. 3. The results are shown for (a) and (d) $N=1$, (b) and (e) $N=2$, and (c) and (f) $N=3$ emitters.

Figure 2

Emission spectra $S\left(\omega \right)$ for one emitter ($N=1$) for different values of the laser intensity $\mathrm{\Omega }$ as indicated in the plots. The left (right) column depicts the results for $g^{\prime }={\mathrm{\Omega }}^{\prime }=0$ ($g^{\prime }=g$ and ${\mathrm{\Omega }}^{\prime }=\mathrm{\Omega }$). The emitter-cavity-coupling strength and the environment temperature are (a) and (d) $g=0.5{\omega }_0$ and $T=0.07{\omega }_0$, (b) and (e) $g=0.7{\omega }_0$ and $T=0.23{\omega }_0$, and (c) and (f) $g=0.8{\omega }_0$ and $T=0.1{\omega }_0$.

Figure 3

Shift of emission peaks as a function of the laser intensity $\mathrm{\Omega }$ for one emitter. The left (right) column depicts the results for $g^{\prime }={\mathrm{\Omega }}^{\prime }=0$ ($g^{\prime }=g$ and ${\mathrm{\Omega }}^{\prime }=\mathrm{\Omega }$). The other parameters are the same as in Fig. 2. The (red) lines in the left panels give the analytical result for the quasienergies [10] (see Appendix pp1) and the (blue) lines in the right panels depict fitted values proportional to ${[1-{(\mathrm{\Omega }/g)}^2]}^{3/4}$.

Figure 4

Glauber function $g^{\left(2\right)}\left(0\right)$ at zero time delay for one emitter as a function of the environment temperature $T$ and the emitter-cavity-coupling strength $g$. The left (right) column depicts the results for $g^{\prime }={\mathrm{\Omega }}^{\prime }=0$ ($g^{\prime }=g$ and ${\mathrm{\Omega }}^{\prime }=\mathrm{\Omega }$). The laser intensity is (a) and (d) $\mathrm{\Omega }=0$, (b) and (e) $\mathrm{\Omega }={10}^{-4}{\omega }_0$, and (c) and (f) $\mathrm{\Omega }={10}^{-3}{\omega }_0$. Note that all values $g^{\left(2\right)}\left(0\right)\ge 4$ are assigned the same [dark red (dark gray)] color in the density plots.

Figure 5

Time-dependent Glauber function $g^{\left(2\right)}\left(t\right)$ for one emitter with $g=0.5{\omega }_0$ and $T=0.07{\omega }_0$. The case (a) $g^{\prime }=0$ and ${\mathrm{\Omega }}^{\prime }=0$ is compared to (b) $g^{\prime }=g$ and ${\mathrm{\Omega }}^{\prime }=\mathrm{\Omega }$.

Figure 6

The EOF for two emitters as a function of the environment temperature $T$ and the emitter-cavity-coupling strength $g$. The left (right) column depicts the results for $g^{\prime }={\mathrm{\Omega }}^{\prime }=0$ ($g^{\prime }=g$ and ${\mathrm{\Omega }}^{\prime }=\mathrm{\Omega }$). The laser intensity is (a) and (d) $\mathrm{\Omega }=0$, (b) and (e) $\mathrm{\Omega }={10}^{-3}{\omega }_0$, and (c) and (f) $\mathrm{\Omega }={10}^{-1}{\omega }_0$.

Figure 7

Emission spectra $S\left(\omega \right)$ for two emitters. The left (right) column depicts the results for $g^{\prime }={\mathrm{\Omega }}^{\prime }=0$ ($g^{\prime }=g$ and ${\mathrm{\Omega }}^{\prime }=\mathrm{\Omega }$). The other parameters are the same as in Fig. 2.

Figure 8

Emission spectra $S\left(\omega \right)$ for three emitters. The left (right) column depicts the results for $g^{\prime }={\mathrm{\Omega }}^{\prime }=0$ ($g^{\prime }=g$ and ${\mathrm{\Omega }}^{\prime }=\mathrm{\Omega }$). The other parameters are the same as in Fig. 2.

Figure 9

Shift of emission peaks as a function of the laser intensity $\mathrm{\Omega }$ for two emitters. The left (right) column depicts the results for $g^{\prime }={\mathrm{\Omega }}^{\prime }=0$ ($g^{\prime }=g$ and ${\mathrm{\Omega }}^{\prime }=\mathrm{\Omega }$). The other parameters are the same as in Fig. 2. The (blue) lines depict fitted values proportional to ${[1-{(\mathrm{\Omega }/g)}^2]}^{3/4}$.

Figure 10

Shift of emission peaks as a function of the laser intensity $\mathrm{\Omega }$ for three emitters. The left (right) column depicts the results for $g^{\prime }={\mathrm{\Omega }}^{\prime }=0$ ($g^{\prime }=g$ and ${\mathrm{\Omega }}^{\prime }=\mathrm{\Omega }$). The other parameters are the same as in Fig. 2. The (blue) lines depict fitted values proportional to ${[1-{(\mathrm{\Omega }/g)}^2]}^{3/4}$.

Figure 11

Glauber function $g^{\left(2\right)}\left(0\right)$ for two emitters. The left (right) column depicts the results for $g^{\prime }={\mathrm{\Omega }}^{\prime }=0$ ($g^{\prime }=g$ and ${\mathrm{\Omega }}^{\prime }=\mathrm{\Omega }$). The other parameters are the same as in Fig. 4.

Figure 12

Glauber function $g^{\left(2\right)}\left(0\right)$ for three emitters. The left (right) column depicts the results for $g^{\prime }={\mathrm{\Omega }}^{\prime }=0$ ($g^{\prime }=g$ and ${\mathrm{\Omega }}^{\prime }=\mathrm{\Omega }$). The other parameters are the same as in Fig. 4.