Stationary Inversion of a Two Level System Coupled to an Off-Resonant Cavity with Strong Dissipation

We present an off-resonant excitation scheme that realizes pronounced stationary inversion in a two level system. The created inversion exploits a cavity-assisted two-photon resonance to enhance the multiphoton regime of nonlinear cavity QED and survives even in a semiconductor environment, where the cavity decay rate is comparable to the cavity-dot coupling rate. Exciton populations of greater than 0.75 are obtained in the presence of realistic decays and pure dephasing. Quantum trajectory simulations and master equation calculations help elucidate the underlying physics and delineate the limitations of a simplified rate equation model. Experimental signatures of inversion and multiphoton cavity QED are predicted in the fluorescence intensity and second-order correlation function measured as a function of drive power.

Figure 1

(a) Quasienergy levels truncated to 5 states ($\delta \equiv {\delta }_{cx}={\omega }_c-{\omega }_x$). $|1,+⟩$ refers to one created cavity photon with one exciton (excited TLS) created; for a far-detuned cavity and exciton, this doubly excited state relates to the Jaynes-Cummings dressed states through $|2L⟩\approx |1,+⟩$. Bold arrows illustrate two-photon resonance, with the pump detuned from the cavity by about $-\delta /2$. Small arrows show the direction of the Stark shifts. (b) Simplified rate equation model, where ${\Gamma }_p\equiv {\Omega }_{\mathrm{tp}}^2/2\kappa$ is an effective two-photon pump rate, with ${\Omega }_{\mathrm{tp}}$ the two-photon Rabi frequency (see text).

Figure 2

Sample QTs near, (a) [$g=1000\kappa$, ${\eta }_x=100\kappa$, $\gamma =0.05\kappa$], and outside, (b) [$g=100\kappa$, ${\eta }_x=150\kappa$, $\gamma =0.05\kappa$], the rate equation regime. The blue (upper) solid curve plots the expectation of the TLS excited population (exciton population) conditioned upon the record of quantum jumps, while the red (lower) solid curve is the conditioned expectation of the cavity photon number. Quantum jumps are indicated by the crosses and circles (and vertical dashed lines) for photon emission via the cavity and TLS, respectively. The steady-state expectation of the exciton population is shown with the horizontal solid line, while the rate equation estimate is shown with the dashed horizontal line. The insets enlarge the regions near $t\kappa \approx 46–48$ and $t\kappa \approx 29–31$ in (a) and (b), respectively.

Figure 3

Semiconductor quantum-dot QED system. (a),(b) Pump-dependent steady-state exciton number versus detuning calculated with one-photon and three-photon truncations; for ${\eta }_x=0.02g–2g$ (blue to purple, lower to higher) and $\delta =10g$. Note the clear peak in the three-photon case near ${\omega }_L-{\omega }_x\approx \delta /2$. (c) Sample QT simulation on the peak labeled p1; symbols are defined as in Fig. 2, with the additional pure dephasing event (phase glitch) indicated by an inverted triangle.

Figure 4

(a) Exciton (blue) and cavity (red) population, and (b) second-order correlation function versus drive strength, ${\eta }_x$, for the semiconductor case of Fig. 3.