Laser under ultrastrong light-matter interaction: Qualitative aspects and quantitative influences by level and mode truncations

We investigate theoretically the light amplification by stimulated emission of radiation (laser) in the ultrastrong light-matter interaction regime under the two-level and single-mode approximations. The conventional picture of the laser is broken under the ultrastrong interaction. Instead, we must explicitly discuss the dynamics of the electric field and of the magnetic one distinctively, which make the “laser” qualitatively different from the conventional laser. We found that the laser generally accompanies odd-order harmonics of the electromagnetic fields both inside and outside the cavity and a synchronization with an oscillation of atomic population. A bistability is also demonstrated. However, since our model is quite simplified, we got quantitatively different results from the Hamiltonians in the velocity and length forms of the light-matter interaction, while the appearance of the multiple harmonics and the bistability is qualitatively reliable.

Figure 1

Laser solutions are calculated by increasing and decreasing the atomic pump $Z_{\text{p}}$ in the velocity form. The bare cavity frequencies are (a)–(c) ${\omega }_{\text{c}}={\omega }_{\text{a}}$ and (d)–(f) ${\omega }_{\text{c}}=0.25{\omega }_{\text{a}}$. (a) and (d) Intensities of fundamental component ${\alpha }_1$ and the $3\mathrm{\Omega }$ one ${\alpha }_3$ of the nondimensional vector potential $\mathcal{A}=\langle \widehat{a}+{\widehat{a}}^{†}\rangle /\sqrt{N}$, (b) and (e) time-constant component $Z_0$ and the $2\mathrm{\Omega }$ one $z_2$ of the atomic population, and (c) and (f) $(\mathrm{\Omega }-{\omega }_{\text{c}})/{\omega }_{\text{c}}$ for fundamental oscillation frequency $\mathrm{\Omega }$ are plotted. Below threshold $Z_{\text{p}}<Z_{\text{th}}$, we get $Z_0=Z_{\text{p}}$ and zero oscillating components. Above threshold, we get a linear increase of $|{\alpha }_1{|}^2, |z_2|\propto |{\alpha }_1{|}^2$, and $|{\alpha }_3|=|{\alpha }_1{|}^3$ for ${\omega }_{\text{c}}={\omega }_{\text{a}}$. A bistability appears for ${\omega }_{\text{c}}=0.25{\omega }_{\text{a}}$. The arrows represent the solutions with increasing and decreasing $Z_{\text{p}}$. Parameters: $g^{\prime }=0.15, {\gamma }_{\parallel }=0.05{\omega }_{\text{a}}, {\gamma }_{\text{pure}}=0.1{\omega }_{\text{a}}$, and $\kappa =0.01{\omega }_{\text{a}}$.

Figure 2

Maps of laser solutions in (a) velocity form and (b) the length one. The intensity $|{\alpha }_1{|}^2$ of the fundamental component is calculated by increasing $Z_{\text{p}}$ for a fixed ${\omega }_{\text{c}}$, which is also changed in the vertical axis. The thick curves indicate the threshold $Z_{\text{th}}$. Unconventional solutions appear for low cavity frequency ${\omega }_{\text{c}}<0.29{\omega }_{\text{a}}$ in the velocity form. Although we find only the conventional solutions in the length form under the current parameters, the unconventional solutions appear for larger $g^{\prime }$ or lower $\kappa$ in Figs. 3 and 4. This quantitative difference is caused by the two-level and single-mode approximations used in the calculation. Parameters: $g^{\prime }=0.15, {\gamma }_{\parallel }=0.05{\omega }_{\text{a}}, {\gamma }_{\text{pure}}=0.1{\omega }_{\text{a}}$, and $\kappa =0.01{\omega }_{\text{a}}$.

Figure 3

Maps of laser solutions calculated in the length form for $g^{\prime }=0.4$ and $\kappa =0.01{\omega }_{\text{a}}$. The atomic dissipation rates are shown above the plots. The unconventional solutions appear for strong enough interaction $g^{\prime }$ and low enough cavity loss $\kappa$ even in the length form, while they disappear by increasing the ratio ${\gamma }_{\text{pure}}/{\gamma }_{\text{total}}$ while keeping the total dissipation rate ${\gamma }_{\text{total}}={\gamma }_{\parallel }+{\gamma }_{\text{pure}}$.

Figure 4

Maps of laser solutions calculated in the length form for $g^{\prime }=0.15$ and $\kappa =0.001{\omega }_{\text{a}}$. The atomic dissipation rates are shown above the plots. The unconventional solutions appear for strong enough interaction $g^{\prime }$ and low enough cavity loss $\kappa$ even in the length form, while they disappear by increasing the ratio ${\gamma }_{\text{pure}}/{\gamma }_{\text{total}}$ with keeping the total dissipation rate ${\gamma }_{\text{total}}={\gamma }_{\parallel }+{\gamma }_{\text{pure}}$.

Figure 5

Maps of laser solutions calculated in the velocity form for $g^{\prime }=0.15$ and $\kappa =0.01{\omega }_{\text{a}}$. The atomic dissipation rates are shown above the plots [(a) is equivalent to Fig. 2].

Figure 6

Maps of laser solutions in (a) velocity form and (b) the length one. The color indicates the maximum $|{\alpha }_1{|}^2$ found by changing ${\omega }_{\text{c}}$. The larger value is chosen in the bistable case. The maximum $|{\alpha }_1{|}^2$ is plotted versus the interaction strength $g^{\prime }$ and the pure-dephasing ratio ${\gamma }_{\text{pure}}/{\gamma }_{\text{total}}$ with keeping $Z_{\text{p}}=0.5, {\gamma }_{\text{total}}=0.15{\omega }_{\text{a}}$, and $\kappa =0.01{\omega }_{\text{a}}$.

Figure 7

The region of parameters $g^{\prime }$ and $\kappa /{\omega }_{\text{a}}$ showing the bistability is plotted in (a) velocity form and (b) the length one. The color indicates $Z_0$ (a measure of electromagnetic distinction) for the highest bare cavity frequency that shows the bistability at $Z_{\text{p}}=0.5$. In both forms, the bistability appears for strong enough interaction $g^{\prime }$ and low enough cavity loss $\kappa$. Parameters: ${\gamma }_{\parallel }=0.05{\omega }_{\text{a}}$ and ${\gamma }_{\text{pure}}=0.1{\omega }_{\text{a}}$.