Lasing without Inversion: Counterintuitive Population Dynamics in the Transient Regime

We explore a three-level $\Lambda$ scheme to demonstrate that the phenomenon of lasing without inversion (LWI) can be observed in the transient regime. We demonstrate that the pure LWI contribution to the gain of a probe field is distinct from both the resonant absorption (or gain) and the coherent Raman gain (or absorption) by choosing specific initial populations in the dressed-state basis. The suppression of the non-LWI (resonant and Raman) processes is followed by the “rich get richer” (capitalistic) effect for the ground-state population dynamics: Initially, the more populated ground state becomes even more populated. The conditions for the observation of the effect are specified.

Figure 1

Evolution of the population of the states $a$, $b$, and $c$ represented as a trajectory on the sphere $(\sqrt{{\rho }_{aa}}{)}^2+(\sqrt{{\rho }_{bb}}{)}^2+(\sqrt{{\rho }_{cc}}{)}^2=1$ for the initial bare-state mixture with ${\rho }_{aa}(0)=0$, ${\rho }_{bb}(0)=0.45$, and ${\rho }_{cc}(0)=0.55$ (Raman inversion). Point In denotes the initial state. The shaded area represents atomic states satisfying the Raman inversion condition. Dotted-dashed lines mark equally populated states involved in either one of the transitions, while the population of the third level varies between 0 and 1. Points $I$ and 0 represent populations of states $|I⟩$ and $|0⟩$. The dashed line (in red) joining point $I$ and 0 corresponds to Eq. (11). Left inset: Integrated probe absorption $G_{ab}(t)={\int }_0^{t}d\tau \mathrm{Im}{\rho }_{ab}(\tau )$. Here we use ${\Omega }_c=1$, ${\Omega }_b=0.2$, ${\gamma }_c=0.2$, ${\gamma }_b=0.05$, and $n_{\mathrm{th}}=0$.

Figure 2

The same as Fig. 1, but no initial Raman inversion: ${\rho }_{aa}(0)=0$, ${\rho }_{bb}(0)=0.55$, and ${\rho }_{cc}(0)=0.45$.

Figure 3

Evolution of the population of states $a$, $b$, and $c$ for the initial dressed-state mixture (15) with ${\rho }_{aa}(0)=0.22$, ${\rho }_{bb}(0)=0.5469$, ${\rho }_{cc}(0)=0.2331$, and $2\mathrm{Re}{\rho }_{bc}(0)=-0.1207$. Other atom-field parameters are the same as in Fig. 1. A net gain will be observed for the initial states that belong to the segment of the red $I0$ line marked by two nearby green dashed lines. The thin-dashed (blue) line labeled $L_{\mathrm{INV}}$ separates fully inverted states (${\rho }_{bb}<1/2$) to its left from noninverted ones (${\rho }_{bb}>1/2$) to its right. Open circles on states $b$ and $c$ signify the initial coherence ${\rho }_{bc}(0)$.