Monitoring for time ordering and time asymmetry of spectral correlations in filtered resonance fluorescence generated from a $\mathrm{\Lambda }$-type atomic system

Frequency-resolved correlation (FRC) is investigated in fluorescent emission radiated from a $\mathrm{\Lambda }$-type atomic system by employing weak coupling regime between quantum emitter and cavities, in which each cavity substituting a Lorentzian filter is assumed to pass through only one fluorescent photon at a time so that the analytical discussions for time orderings can be carried out in a truncated Hilbert space with single-excitation state for each cavity mode. In the limit of bad cavity (large passband width of filter) and short delay, the conditional time ordering amplitudes are introduced with the help of conditioned dressed atom-photon states prepared by preselection to probe into the effect of preferential screening for time orderings of filter-detector systems. Meanwhile, the past quantum state formalism is also applied in FRC to further excavate their collective monitoring effect. Based on the obtained two-mode correlation signals, it is found that, in nonresonant two-photon cascaded processes, bunching and antibunching can appear in two opposite detection orderings, respectively, and can be switched. The physical origin is explored that the two different transition amplitudes of emitted photons from a common spectral band of resonance fluorescence are responsible for opposite time orderings, thus, the imbalance between two complementary time orderings can be enhanced, which is impossible to achieve in resonance fluorescence of two-level atoms. This mechanism is embodied in the conditional time ordering amplitudes produced by this detection theory of cavity modes.

Figure 1

(a) Equivalent system of atom-photon interactions. Resonance fluorescence is emitted from the excited atom to weakly pump four cavities, and multimode second-order correlations can be realized after four-mode cavity photons are outputted. (b) Level diagram for a $\mathrm{\Lambda }$-type three-level atom, in which the dipole-allowed transitions $\left|3\right.〉\to \left|1\right.〉$ and $\left|3\right.〉\to \left|2\right.〉$ are driven by two strong applied fields, respectively, and coupled to two-mode quantum fields $(a_1,a_2), (b_1,b_2)$.

Figure 2

Transition paths and time orderings of two-photon resonant cascaded emissions of photon pair $(a_1,a_2)$, in which path 1 and path 2 display interference involving a common initial dressed state and a final dressed state $\left|2_A\right.〉$, and path 3 and path 4 are independent time ordering channels. Red and pink arrows represent the photon emissions in mode $a_1$ and $a_2$, respectively. The relative transition amplitudes, $A, B_{+}$, and $B_{-}$, are labeled.

Figure 3

Transition paths and time orderings of two-photon nonresonant cascaded emissions of photon pair $(a_1,b_1)$, which relate to the pure interference of time orderings described by path 6 and path 7. Red and blue arrows represent the photon emissions in mode $a_1$ and $b_1$, respectively. The relative transition amplitudes, $A, B_{+}$, and $B_{-}$, are also labeled.

Figure 4

Transition paths and time orderings of two-photon resonant cascaded emissions of photon pair $(a_1,b_2)$, which are the same as photon pair $(a_1,a_2)$ except for one of the relative transition amplitudes for field $b_2$. Red and blue arrows represent the photon emissions in mode $a_1$ and $b_2$, respectively. The relative transition amplitudes, $A$ and $B_{-}$, are also labeled.

Figure 5

Normalized second-order correlation signals (a) $g^{\left(2\right)}(a_1,a_2,\tau )$ and (b) $g^{\left(2\right)}(a_1,b_2,\tau )$, as functions of arbitrary delay $\tau$ and laser detuning $\mathrm{\Delta }$ for $\mathrm{\Omega }=100\gamma , \kappa =20\gamma$ and $\delta =0$. (c) and (d) are the intersecting surfaces of (a) at $\mathrm{\Delta }=100\gamma$ and $\mathrm{\Delta }=200\gamma$, in which the red dashed thick lines and blue dashed thin lines represent the analytical results and numerical results without large width approximation, respectively.

Figure 6

(a) Steady dressed populations ${\langle {\tilde{\sigma }}_{11}\rangle }_s$ (red dotted line), ${\langle {\tilde{\sigma }}_{22}\rangle }_s$ (green solid line), and ${\langle {\tilde{\sigma }}_{33}\rangle }_s$ (blue dashed line). (b) The value of $g^{\left(2\right)}(a_1,a_2,\tau )$ in Fig. 5 (c) is decomposed, in which the red dot-dashed thin line, green dashed thin line, and blue dotted thin line correspond to the contributions from paths (1, 2), path 3, and path 4 for $\tau >0$, and the red dot-dashed thick line, green dotted thick line, and blue dashed thick line represent the contributions from paths (1, 2), path 4, and path 3 for $\tau <0$, respectively. The total normalized correlation function $g^{\left(2\right)}(a_1,a_2,\tau )$ is plotted by the black solid thin line (top), which is the sum of the above three parts. (c) Enlarged view of the contributions from path 4 to $g^{\left(2\right)}(a_1,a_2,\tau )$ in (b). Its contributions for $\tau >0$ and $\tau <0$ are depicted by the blue dotted thin line and green dotted thick line, respectively.

Figure 7

Normalized second-order correlation signal $g^{\left(2\right)}(a_1,b_1,\tau )$, as a function of arbitrary delay time $\tau$ and laser detuning $\mathrm{\Delta }$ in (a), and filter detuning $\delta$ in (b). The parameters are $\mathrm{\Omega }=100\gamma , \kappa =20\gamma$, and $\delta =0$ in (a), and $\mathrm{\Delta }=100\gamma$ in (b).

Figure 8

Normalized second-order correlation signal $g^{\left(2\right)}(a_1,b_1,\tau )$ with $\mathrm{\Omega }=100\gamma , \kappa =20\gamma$. The laser detuning is tuned to $\mathrm{\Delta }=200\gamma$ in frames (a) and (c), and $\mathrm{\Delta }=-200\gamma$ in frames (b) and (d). The filter detuning is tuned, respectively, to $\delta =0$ in frames (a) and (b), and $\delta =10\gamma$ in frames (c) and (d). (e)–(h) are the enlarged views of the minimum values of (a)–(d), respectively. The red dashed thick lines and blue dashed thin lines represent the analytical results and numerical results without large width approximation, respectively.

Figure 9

(a) Level diagram for the four-level quantum dot, in which the bare transitions $\left|B\right.〉\leftrightarrow \left|V\right.〉\leftrightarrow \left|G\right.〉$ are driven by a single-mode driving laser under the two-photon resonance condition ${\omega }_{BG}=2{\omega }_L$. Two low-frequency spectral components labeled by $L_0$ and $L_{+}$ are radiated from $\left|B\right.〉\to \left|H\right.〉$ with spontaneous decay rate ${\gamma }_B$, and the high-frequency spectral components $R_0$ and $R_{+}$ are built up from $\left|H\right.〉\to \left|G\right.〉$ with decay rate ${\gamma }_H$. $\mathrm{\Delta }{E}_B$ is the biexciton binding energy caused by Coulomb interactions. (b) Time orderings of fluorescent emission in dressed levels, in which the fluorescent spectral components $L_0, L_{+}$, and $R_{+}$ are equivalent to the cavity photons labeled by $c_1, c_2$, and $d_1$ respectively. The time orderings of photon pair $(c_1,c_2)$ are depicted by path 12 and path 13, and the another isolated path 14 represents the unique time ordering of photon pair $(c_1,d_1)$.

Figure 10

Logarithmic scaled second-order correlation signals ${\log }_{10}\left[g^{\left(2\right)}(a_1,a_2,0)\right]$ as functions of (${\delta }_1, {\delta }_2$) in frame (a), and (${\kappa }_2, \delta$) in frame (b). The parameters are $\mathrm{\Omega }=100\gamma , \mathrm{\Delta }=50\gamma$, and ${\kappa }_1={\kappa }_2=\kappa =0.002\gamma$ in frame (a), and ${\kappa }_1=0.002\gamma$ with two-photon resonant condition ${\delta }_{a_1}=-{\delta }_{a_2}=\delta$ in frame (b).