Sideband generation of transient lasing without population inversion

We suggest a method to generate coherent short pulses by generating a frequency comb using lasing without inversion in the transient regime. We use a universal method to study the propagation of a pulse in various spectral regions through an active medium that is strongly driven on a low-frequency transition on a time scale shorter than the decoherence time. The results show gain on the sidebands at different modes can be produced even if there is no initial population inversion prepared. Besides the production of ultrashort pulses this frequency comb may have applications towards making short-wavelength or terahertz lasers.

Figure 1

Left: Energy diagram for the V scheme. Right: Floquet ladder of states produced by the $c\to b$ transition driven by a laser field with frequency ${\nu }_d$. Possible lasing transitions are of the 0th-order transition ($\sim {\omega }_{ab}$), and at the even sidebands of the $\pm 2\mathrm{nd}$-order transition ($\sim {\omega }_{ab}\pm 2{\nu }_d$), etc. Each order is split into two dressed states ($j^{+}$, $j^{-}$) by the rotating-wave terms of the electric-dipole interaction. The side-band signals are generated due to the state mixing by the counter-rotating terms.

Figure 2

The imaginary part of $k_1$ as a function of the lasing frequency detuning $\Delta \nu$. The populations are ${\rho }_{aa}\left(0\right)=0.1$, ${\rho }_{bb}\left(0\right)=0.9$, and ${\rho }_{cc}\left(0\right)=0$, i.e., without inversion. ${\omega }_{ab}=5.0{\omega }_{cb}$, ${\Omega }_a=0.05{\omega }_{cb}$, and ${\gamma }_t={10}^{-4}{\omega }_{cb}$. We drive the $c\to b$ transition with a weak detuned field with ${\nu }_d=1.1{\omega }_{cb}$ and ${\Omega }_d=0.05{\omega }_{cb}$. We cut off our calculation at $m=\pm 10$.

Figure 3

The amplification of the output field in the whole spectral region with different propagation distance $z$. The creation of a frequency comb is shown. The plot is made with the same parameters as in Fig. 2.

Figure 4

The maximum value of the negative imaginary part of the eigenvalue $k_1$, $-\mathrm{Im}{\left(k_1\right)}_{\max }$ (left blue), with its corresponding lasing frequency detuning $\Delta \nu$ (right purple) for various drive field Rabi frequency ${\Omega }_d$. The corresponding $\Delta \nu$ is plotted only for positive $-\mathrm{Im}\left(k_1\right)$. This plot determines the Rabi frequency ${\Omega }_d$ that should be chosen for peak gain.

Figure 5

Ultrashort pulse intensity in the case of a hydrogen molecule. A 5-fs pulse is created with parameters given in the text.

Figure 6

Detailed numerical experiments with the following parameters: ${\nu }_d=1.06{\omega }_{cb}$, ${\Omega }_d=0.18{\omega }_{cb}$, ${\Omega }_a=0.0754{\omega }_{cb}$, $L=7.54c/{\Omega }_a$, ${\rho }_{aa}\left(0\right)=0.15$, ${\rho }_{bb}\left(0\right)=0.85$, ${\rho }_{cc}\left(0\right)=0$, and ${\gamma }_t={10}^{-4}{\omega }_{cb}$.