Population inversion in two-level systems possessing permanent dipoles

Bare-state population inversion is demonstrated in a two-level system with all dipole matrix elements nonzero. A laser field is resonantly driving the sample, whereas a second weaker and lower frequency coherent field additionally pumps it near resonance with the dynamically Stark-splitted states. Due to the existence of differing permanent dipole moments in the excited and ground bare states, quantum coherences among the involved dressed states are induced, leading to inversion in the steady state. Furthermore, large refractive indices are feasible, as well as the determination of the diagonal matrix elements via the absorption or emission spectra. The results apply to available biomolecular, spin, or asymmetric quantum dot systems.

Figure 1

Energy diagram of a two-level emitter with nonzero values of all involved dipole matrix elements $d_{\alpha \beta },\{\alpha ,\beta =1,2\}$. (a) A moderately intense laser field of frequency ${\omega }_L$ resonantly interacts with the molecular sample, leading to dynamic Stark splitting of its energy levels. (b) A second coherent source of frequency $\omega$, close to the generalized Rabi frequency due to the first laser, is applied, leading to transitions among the dressed states. (c) The “double dressed states,” with $2{\overline{G}}_R$ being the corresponding Rabi splitting frequency.

Figure 2

The steady-state dependence of the bare-state inversion operator $\langle S_z\rangle /N$ versus the scaled parameter $\mathrm{\Delta }/\left(2\mathrm{\Omega }\right)$. The solid line is plotted for $G/\gamma =0$, the long-dashed one is for $G/\gamma =16$, while the short-dashed curve corresponds to $G/\gamma =24$. Other parameters are $\mathrm{\Omega }/\left(N\gamma \right)=45$ and $\omega /\left(N\gamma \right)=100$, with $\gamma \approx 2.6\mathrm{MHz}$ as feasible for gamma globulin. (a) $N=1$ whereas (b) $N=50$, while molecules are dense enough to allow for collectivity.

Figure 3

The single-molecule steady-state behaviors of (a) the real part of the dressed-state coherences $\langle R^{+}\rangle$ and (b) the double dressed-state inversion operator $\langle {\tilde{R}}_z\rangle$ versus the scaled detuning $\mathrm{\Delta }/\left(2\mathrm{\Omega }\right)$. The solid line is plotted for $G/\gamma =16$, while the dashed curve corresponds to $G/\gamma =24$. Here $\mathrm{\Omega }/\gamma =45$ and $\omega /\gamma =100$.

Figure 4

The steady-state dependence of the linear susceptibility $\chi \left({\nu }_p\right)$ [in units of $\overline{N}{d}^2/\left(\hslash \gamma \right)]$ versus scaled detuning ${\mathrm{\Delta }}_p/\gamma$. The solid black curve corresponds to the imaginary part (absorption spectrum) and the long-dashed blue line to the real part of the susceptibility, respectively. (a) $\mathrm{\Delta }/\left(2\mathrm{\Omega }\right)=0.43$ and (b) $\mathrm{\Delta }/\left(2\mathrm{\Omega }\right)=-0.43$. Other parameters are $\mathrm{\Omega }/\gamma =45,\omega /\gamma =100$, and $G/\gamma =16$.