Quantum dynamics and spectra of vibrational Raman-resonance fluorescence in a two-mode cavity

We study the classically driven two-level system with its center-of-mass motion vibrating in a harmonic trap and coupled to the photons in a two-mode cavity. The first mode is resonant to the driving field and an electronic transition. The second mode is off-resonant, forming a vibrational-assisted Raman transition. Using an exact numerical method, we investigate the quantum dynamics of the light emitted by the atom and the cavity modes. We analyze and compare the corresponding atomic and intracavity photon spectra for a range of the driving laser field and the cavity coupling strengths. The results provide better understanding of the effects of the laser field and atom-cavity coupling strengths on quantum interference effects and photon blockade, particularly the Mollow's triplet and the Autler-Townes splitting in the good and bad cavity limits.

Figure 1

Energy level scheme of a two-level system driven by a laser producing photons of frequencies ${\nu }_a$ and ${\nu }_b$ into a two-mode cavity. Cavity mode $a$ is resonant with the electronic transition as well as the classical driving field $\mathrm{\Omega }$ while cavity mode $b$ is assisted through the vibrational quanta ${\nu }_v$, corresponding to resonance fluorescence and the Raman transition.

Figure 2

Left and middle panels: Transients of coherences and populations versus normalized laser amplitude $\mathrm{\Omega }/g_a$ and $g_{a}t$. (a) Imaginary part of the atomic coherence $\langle \widehat{\sigma }\rangle$ and real part of the photon number coherence $\langle \widehat{a}\rangle$ (here, $\mathrm{Re}\langle \widehat{\sigma }\rangle =0$ and $\mathrm{Im}\langle \widehat{a}\rangle =0$). (b) Atomic population $〈{\widehat{\sigma }}^{†}\widehat{\sigma }〉$ and number of cavity photons $〈{\widehat{a}}^{†}\widehat{a}〉$ for mode $a$. Right panel: (c) Power spectra of cavity mode $a \left[S_a\left(\omega \right)\right]$ and the light emitted by the atom $\left[S_{\mathrm{atom}}\left(\omega \right)\right]$ vs normalized frequency $\omega /g_a$ for various values of the laser drive amplitude $\mathrm{\Omega }/g_a$. For all plots we used $\kappa /g_a=20$ (bad cavity limit), ${\nu }_a={\omega }_{21}, {\nu }_v=0.2{\omega }_{21}, {\nu }_l={\omega }_{21}$, and ${\nu }_b={\omega }_{21}-{\nu }_v.$

Figure 3

Transient quantities [panels (b) and (c)] and power spectra [panel (c)] versus normalized Rabi frequency $\mathrm{\Omega }/g_a$ when the coupling strength of mode $b$ is $g_b/g_a=1$ with other parameters being the same as those in Fig. 2. Notice that we now have included the population and power spectrum of mode $b$ in panels (b) and (c), respectively.

Figure 4

Transient quantities [panels (a) and (b)] and power spectra [panel (c)] versus normalized coupling strength $g_b/g_a$ of mode $b$ for weak laser drive $\mathrm{\Omega }/g_a=1$ with other parameters being the same as those in Fig. 2.

Figure 5

Transient quantities [panels (a) and (b)] and power spectra [panel (c)] versus the coupling strength $g_b/g_a$ of mode $b$ for strong field $\mathrm{\Omega }/g_a=10$ with other parameters being the same as those in Fig. 2.

Figure 6

Left and middle panels: Transients of coherences and populations versus normalized laser amplitude $\mathrm{\Omega }/g_a$ and $g_{a}t$. (a) Imaginary part of the atomic coherence $\langle \widehat{\sigma }\rangle$ and real part of the photon number coherence $\langle \widehat{a}\rangle$ (here, $\mathrm{Re}\langle \widehat{\sigma }\rangle =0$ and $\mathrm{Im}\langle \widehat{a}\rangle =0$). (b) Atomic population $〈{\widehat{\sigma }}^{†}\widehat{\sigma }〉$ and number of cavity photons $〈{\widehat{a}}^{†}\widehat{a}〉$ for mode $a$. Right panel: (c) Power spectra of cavity mode $a \left[S_a\left(\omega \right)\right]$ and the light emitted by the atom $\left[S_{\mathrm{atom}}\left(\omega \right)\right]$ vs normalized frequency $\omega /g_a$ for various values of laser drive amplitude $\mathrm{\Omega }/g_a$. Here, we use a much smaller cavity decay rate, $\kappa /g_a=0.08$ (good cavity limit). Other parameters are the same: $\mathrm{\Gamma }/g_a=0.1, {\kappa }_a={\kappa }_b=\kappa , {\kappa }_v=0.2\kappa , {\nu }_a={\omega }_{21}, {\nu }_v=0.2{\omega }_{21}, {\nu }_l={\omega }_{21}$, and ${\nu }_b={\omega }_{21}-{\nu }_v.$

Figure 7

Transient quantities [panels (a) and (b)] and power spectra [panel (c)] versus normalized Rabi frequency $\mathrm{\Omega }/g_a$ where the coupling strength of mode $b$ is $g_b/g_a=1$, with other parameters being the same as those in Fig. 6.

Figure 8

Transient quantities [panels (a) and (b)] and power spectra [panel (c)] versus normalized coupling strength $g_b/g_a$ of mode $b$ for weak field $\mathrm{\Omega }/g_a=1$, with other parameters being the same as those in Fig. 6.

Figure 9

Transient quantities [panels (a) and (b)] and power spectra [panel (c)] versus normalized coupling strength $g_b/g_a$ of mode $b$ for strong field $\mathrm{\Omega }/g_a=10$, with other parameters being the same as those in Fig. 6.