1. Introduction

The second order photon correlation function denoted as $G^{(2)}(t,t+\tau )$ and measured as a joint photon detection probability with delay time $\tau$ in Hanbury Brown-Twiss (HBT) setup has a great significance in quantum optics, because it can provide crucial information about non-classical properties of light and intriguing dynamics of light sources, such as violations of Cauchy-Schwarz and Bell inequalities and anti-bunched photon emission [1,2]. Pure quantum nature of matter waves [3], fermions [4], and bosons [5] has been confirmed in similar second order correlation properties. The cross correlation function ($G^{(2)}_{\textrm {cr}}$) of photon pairs generated in atomic cascade transitions offers additional vital information about the natural lifetime of the intermediate state [6] and an evidence for sequential photon emission [7,8].

As a quantum correlated state, entangled biphoton has been intensively investigated for the use in quantum computing, communication and networking, due to high feasibility for on-demand generation and the negligibly low photon-photon interaction [9,10]. Photon pairs generated from electronic cascade transitions are considered as one of highly promising candidates fulfilling those requirements. Diverse light sources with different materials and platforms have been introduced so far. Examples are D-P-S states of cold [11–13] and warm rubidium (Rb) atom vapors [14,15], biexciton-exciton-ground state of semiconductor quantum dot (QD) [16–18] and monolayer $\textrm {WSe}_2$ [19], Mollow triplet sidebands of resonantly excited QDs [20], and three-level states of an artificial atom with a tunable gap, constructed with superconducting material [21,22]. While optical properties such as emission wavelength and the polarization direction of photons can be manipulated in artificial atomic systems, in most cases, they suffer from large dephasing of atomic coherences and require operations at ultralow temperature. On the contrary, very bright photon pairs with relatively long coherence times can be generated in vaporized Rb cells in spontaneous four-wave mixing setups [15].

However, most studies have mainly succeeded in establishing polarization entanglement of two photons generated in diamond-shaped four-level systems with quasi degenerated intermediate states through the ambiguity between two decay paths for excited electrons or biexcitons [11,16,23,24]. While the polarization entanglement can be experimentally confirmed by the violation of Bell inequality, the verification of spectrally entangled photonic states is experimentally not an easy task due to the lack of appropriate experimental observables.

This study is motivated from a fundamental question: Is it possible to quantify the degree of spectral entanglement of two photons through an experimentally available observable such as $G^{(2)}_{\textrm {cr}}$ in specific conditions? Our findings for the question are of significance and applicable for quantum communication, cryptography, teleportation, and information processing by using continuous variable Gaussian states [25] and quantum memory by using correlated photo pairs generated from atomic three-level cascades [26]. Our special interest lies on the cross correlation value at zero delay time ($\tau =0$), from which mutual blocking (conjunction) behavior of a photon pair could be observed via an anti-correlation (correlation) [27,28].

In most theoretical works on the photon pair generation from atomic three-level systems [7,29–32], $G^{(2)}_{\textrm {cr}}(t,t+\tau )$ has been derived for stationary condition ($t\rightarrow \infty$), thus, $G^{(2)}_{\textrm {cr}}(\tau =0)$ has been forced to be represented by a constant. On the contrary, in non-stationary regime, $G^{(2)}_{\textrm {cr}}(t,\tau =0)$ also as a function of time can give clues for understanding temporal dynamics of quantum light emitters and thereby generated entangled biphotons. Similar considerations have been performed for a four-level double Raman scheme [33]. For an atomic three-level system in a two-mode cavity, the violations of Cauchy-Schwarz and Bell equalities were discussed only in stationary regime, depending on the degree of entanglement of two cavity modes [34]. While in conventional HBT setups two photon detection events depending on the relative time delay has been only measured, a streak camera in single photon counting mode has successfully measured temporal dynamics of the zero-delay second order photon correlation function with picoseconds time resolution [35].

In this theoretical work, we study $G^{(2)}_{\textrm {cr}}(t,\tau =0)$ of two photons, each of which is generated from two successive transitions in a three-level cascade (TLC). In particular, temporal dynamics of the correlation function are investigated for the degree of entanglement of two photons. In the framework of density matrix formalism in cluster expansion [36–39], a closed set of quantum kinetic equations of motion (QKE) is derived for determining the correlation function and the entanglement criterion suggested by Duan-Giedke-Cirac-Zoller [40], which is proven to be a sufficient and necessary condition for all of the Gaussian states such as two-mode field in vacuum and in a cavity [25,41].

Firstly, we study the correlation function for spontaneous photon emission, where a transition between correlation and anti-correlation behavior can be observed depending on the radiative decay rate of the intermediate state with respect to that of the excited state. In coherent excitation regime, we demonstrate that both the cross correlation and the entanglement of two photons are established via the optically non-active atomic polarization, which allows for experimental verification of two spectrally entangled photons via their mutual correlation. Furthermore, we present $G^{(2)}_{\textrm {cr}}(t)$ and the entanglement criterion as fully analytical forms, by which physical meanings of numerical results and the conditions required for building two spectrally entangled photons can be appreciated.

2. Mathematical model

As our quantum light source, a TLC with the energy configuration is depicted in Fig. 1. $|0\rangle ,\,|1\rangle$, and $|2\rangle$ corresponding to the ground, intermediate, and excited states, respectively, are energetically separated by $\omega _{10}$ and $\omega _{21} (\omega _{20}=\omega _{21}+\omega _{10})$. We assume that the electronic transition $|0\rangle \,\leftrightarrow \,|1\rangle$ ($|1\rangle \,\leftrightarrow \,|2\rangle$) can be driven only by the laser field 1 (2) with a Rabi-frequency $\Omega _1$ ($\Omega _2$) centered at $\omega _{L_1}$ ($\omega _{L_2}$) and has no influence from the field 2 (1). The transition between the states $|2\rangle$ and $|0\rangle$ is optically forbidden.

 

Fig. 1. Energy level configuration of an atomic TLC. The electronic transitions between $|0\rangle \,\leftrightarrow \,|1\rangle$ and $|1\rangle \,\leftrightarrow \,|2\rangle$ are induced by two Rabi frequencies $\Omega _1$ and $\Omega _2$ at center frequencies $\omega _{L_1}$ and $\omega _{L_2}$, respectively. The direct optical transition between $|0\rangle \,\leftrightarrow \,|2\rangle$ is forbidden. The photon number densities $\langle c^\dagger _{\textbf {q}} c_{\textbf {q}} \rangle$ and $\langle c^\dagger _{\textbf {k}} c_{\textbf {k}} \rangle$ are produced by $|0\rangle \,\leftrightarrow \,|1\rangle$ and $|1\rangle \,\leftrightarrow \,|2\rangle$ transitions, respectively. The cross correlation function at zero delay time is defined as $G^{(2)}_{12}(t)=\langle c^\dagger _{\textbf {k}} c^\dagger _{\textbf {q}} c_{\textbf {q}} c_{\textbf {k}} \rangle$.

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The Hamiltonian for single electrons optically pumped by laser fields and emitting photons in free space is composed of $\hat {H}_0$ for the free kinetics of the electron and photons, $\hat {H}_l$ for the interaction with two coherent pump fields, and $\hat {H}_p$ for the electron-photon interaction:

In order to calculate the cross correlation function $G^{(2)}_{12}(t)=\langle c^\dagger _{\textbf {k}} c^\dagger _{\textbf {q}} c_{\textbf {q}} c_{\textbf {k}} \rangle$, the photon number densities $n_k(t)=\langle c^\dagger _{\textbf {k}} c_{\textbf {k}} \rangle$ and $n_q(t)=\langle c^\dagger _{\textbf {q}} c_{\textbf {q}} \rangle$, and the entanglement criterion $D(t)$ (see its definition below), the QKEs for all hierarchically connected density matrix elements including $\langle c_{\textbf {k}} \rangle$, $\langle c_{\textbf {q}} \rangle$, and $\langle c_{\textbf {k}} c_{\textbf {q}} \rangle$ are derived by the Heisenberg’s equations of motion. The infinite hierarchy appearing in the cluster expansion is solved by the truncation in two higher orders. Only the higher order terms survived from the rotating wave approximation (RWA) are factorized and expanded into a linear superposition of lower order terms. As a result, the contributions summed over all wave vectors $\textbf {l}$ are approximated to radiative decay rates $\Gamma _{\lambda \lambda -1}$ for the electronic population densities $\rho _\lambda =\langle a^\dagger _{\lambda } a_{\lambda } \rangle$ from states $|\lambda \rangle$ to $|\lambda -1\rangle \;(\lambda =1,\,2)$ and dephasing rates $\gamma _{\lambda +\mu }$ for the atomic coherences $p_{\lambda \mu }=\langle a^\dagger _\lambda a_\mu \rangle =p^*_{\mu \lambda }\;(\mu =0,\,1,\,2,\,\lambda \ne \mu )$, respectively, after the Weisskopf-Wigner approximation [36,39]. It should be noted that our theoretical approach results in the same relaxation and dephasing rates obtained by the Lindblad’s Master equations in Heisenberg’s picture [44]. We offer only selected QKEs discussed mainly in the manuscript and the detailed derivation of $\gamma _3$ for $p_{12}$ in Appendix A and B, respectively, due to the limited space.

The entanglement criterion $D(t)$ introduced by Duan-Giedke-Cirac-Zoller [21] is obtained for two photonic operators $c_{\textbf {k}}$ and $c_{\textbf {q}}$ as

Additionally, we briefly explain the main observables, the first and second order photon correlation functions, in terms of the cluster expansion rule [38]. Expectation values of multi-photonic operators can be expanded into a linear superposition of all possible combination of factorized terms. For examples,

3. Numerical results and discussion

Basically, our atomic TLC mimics the electronic transitions $5S_{1/2}-5P_{3/2}-5D_{5/2}$ of $\phantom {a}^{87}$Rb atoms ($\lambda _{10}=780\,nm$ and $\lambda _{21}=775.8\,nm$). However, it should be noted that all numerical results presented in this report are not confined to specific transition wavelengths of the atomic system.

3.1 $G^{(2)}_{12}$ in incoherent excitation regime

At first, we consider spontaneous two-photon emission generated by two successive transitions of the electron from $|2\rangle$ to $|0\rangle$ via $|1\rangle$ (see Fig. 2(a)). For free space radiation, we reformulate the electron-photon coupling coefficient as a function of the corresponding radiative decay rate ($g_{\lambda \mu }(l)=\sqrt {3\pi \nu _l\Gamma _{\lambda \mu } /(2l^3V)}$) [32] and normalize all the radiative decays and simulation time by a reference value $\Gamma _0=5\,{\textrm {ps}}^{-1}$ for consistency and fast convergence of calculations. However, all numerical results presented in this study are not governed by a specific choice of $\Gamma _{0}$.

 

Fig. 2. (a) Spontaneous two-photon radiation from the successive transition of the electron initially at the state $|2\rangle$. (b) The same initial conditions can be experimentally prepared by two ultrashort 1.1$\pi$-Gaussian pulses as demonstrated by electronic population densities $(\Gamma _{10}=\Gamma _0)$.

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Because temporal dynamics of all electronic population densities including zero value of all atomic polarizations are expressed to fully analytical forms, the normalized cross correlation function at zero delay time, $g^{(2)}_{12}(t)=G^{(2)}_{12}(t)/(n_k(t) n_q(t))$, can be obtained analytically as well (see the detailed derivation in Appendix C). Furthermore, the normalized pure cross correlation is now directly accessible from $g^{(2)}_{12}(t)$ via the relation $\langle c^\dagger _{\textbf {k}} c^\dagger _{\textbf {q}} c_{\textbf {q}} c_{\textbf {k}} \rangle _c/(\langle c^\dagger _{\textbf {k}} c_{\textbf {k}} \rangle _c\langle c^\dagger _{\textbf {q}} c_{\textbf {q}} \rangle _c)=g^{(2)}_{12}(t)-1$ (see Eqs. (6) and (7) for $\langle c_{\textbf {k}} \rangle =\langle c_{\textbf {q}} \rangle =0$).

Our analytically solved model (ASM) is accurate enough to reproduce the full numerical calculation obtained by the spontaneous radiative decay model (SDM) where the election is initially located at the highest energy level ($\rho _2(0)=1$), as compared in Fig. 3. However, we find the diverging $g^{(2)}_{12}(t)$ near $t=0$ for both calculations, depending on the time resolution. To reasonably estimate the maximum of $g^{(2)}_{12}(t)$, in our pulse excitation model (PEM), we prepare the initially inverted electron population densities by using two ultrashort Gaussian pulses (PEM), which are defined as $\Omega (t)=n\sqrt {\pi }/(2\tau )\exp [-(t-t_0)^2]$ where $t_0$ and $\tau$ are the peak time and temporal width of the pulse, respectively, and whose time-integrated pulse areas are set to $\int \Omega _i(t)dt=1.1\pi$. Both optical pumps yield almost a complete population inversion of the electron (Fig. 2(b)) and negligibly small atomic coherences (not shown) due to the decay and dephasing rates. The highest value of $g^{(2)}_{12}(t)$ obtained from the PEM can be seen in each inset of Fig. 3.

 

Fig. 3. The analytically solved $g^{(2)}_{12}(t)$ (ASM) is compared with the numerical results obtained by two ultrashort $1.1\,\pi$-pulses excitation (PEM) and the spontaneous radiative decay of the electron initially at $|2\rangle$ (SDM) for two cases. (a) $\Gamma _{10}=10\,\Gamma _{21} (\Gamma _{10}=\Gamma _0)$. (b) $\Gamma _{21}=10\,\Gamma _{10} (\Gamma _{21}=\Gamma _0)$. The inset in each figure shows the maximum of $g^{(2)}_{12}(t)$ calculated by the PEM.

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$g^{(2)}_{12}(t)$ shows qualitatively different behaviors depending on the radiative decay rates $\Gamma _{21}$ and $\Gamma _{10}$. In the case of $\Gamma _{10}>\Gamma _{21}$ (Fig. 3(a)), typically found in three-level systems of Rb [12,46] and Cs atoms [47], the positive pure cross correlation $(g^{(2)}_{12}>1)$ is maintained and then factorized into two photon number densities ($\langle c^\dagger _{\textbf {k}} c^\dagger _{\textbf {q}} c_{\textbf {q}} c_{\textbf {k}} \rangle _c=n_k n_q$). In the opposite case ($\Gamma _{21}>\Gamma _{10}$), corresponding to the radiative decay rates of biexciton and exciton in low-dimensional semiconductors [8,19], the negative anti-correlation $(g^{(2)}_{12}<1)$ can be mostly found in Fig. 3(b).

Because the value of $\langle c^\dagger _{\textbf {k}} c^\dagger _{\textbf {q}} c_{\textbf {q}} c_{\textbf {k}} \rangle _c$ is determined by $G^{(2)}_{12}$, $n_k$, and $n_q$, we compare their temporal dynamics for two cases (red solid lines for $\Gamma _{10}>\Gamma _{21}$ and blue dashed lines for $\Gamma _{10}<\Gamma _{21}$) in Figs. 4(a)–4(c), respectively. While $n_k$ and $G^{(2)}_{12}$ in Figs. 4(a) and 4(c), respectively, vary within same orders of magnitudes during the considered time interval, $n_q$ in Fig. 4(b) shows a rapid increase, leading to two orders of magnitude difference between two cases. The time derivative of $n_q$ derived from Eq. (47) clearly accounts for the different dynamics:

 

Fig. 4. Temporal evolution of normalized photon correlation functions $\bar {G}^{(2)}_{12}={G}^{(2)}_{12}/(4|g_{01}(q)|^2|g_{12}(k)|^2)$, $\bar {n}_k=n_k/(2|g_{12}(k)|^2)$, and $\bar {n}_q=n_q/(2|g_{01}(q)|^2)$ for $\Gamma _{10}>\Gamma _{21}$ (red solid lines) and $\Gamma _{10}<\Gamma _{21}$ (blue dashed lines).

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Next, we investigate the boundary value of $\Gamma _{21}/\Gamma _{10}$ for the transition of $\langle c^\dagger _{\textbf {k}} c^\dagger _{\textbf {q}} c_{\textbf {q}} c_{\textbf {k}} \rangle _c$ between correlation and anti-correlation. In Fig. 5(a) we plot $g^{(2)}_{12}(t)$ for four different values of $\Gamma _{21}$ normalized by a constant $\Gamma _{10}=\Gamma _0$ and find that the transition occurs near $\Gamma _{21}/\Gamma _{10}\approx 1/4$. We also study the lowest limit of the anti-correlated $\langle c^\dagger _{\textbf {k}} c^\dagger _{\textbf {q}} c_{\textbf {q}} c_{\textbf {k}} \rangle _c$ in Fig. 5(b) where the minimum value approaches $-0.5$ as $\Gamma _{10}$ normalized by a constant $\Gamma _{21}=\Gamma _0$ is decreased. Interestingly, this lower limit is consistent with the value $(g^{(2)}_{12}=0.5)$ calculated in the Fock states of two mode photons, when both photon number densities are equally one [48].

 

Fig. 5. (a) Transition between correlation and anti-correlation for four different values of the ratio $\Gamma _{21}/\Gamma _{10} (\Gamma _{10}=\Gamma _{0})$. (b) The convergence of anti-correlation when the ratio $\Gamma _{10}/\Gamma _{21} (\Gamma _{21}=\Gamma _{0})$ is reduced below 0.01.

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3.2 $G^{(2)}_{12}$ in coherent linear excitation regime

Careful examination of the QKE of density matrix elements appearing in $D(t)$ of Eq. (5) reveals that building entanglement of two spontaneously emitted photons in free space seems to be impossible, because the incoherent photon emissions ($n_k(t)$ and $n_q(t)$) prevailing over the coherent ones yield always $D(t)>2$ ($\langle c_{\textbf {q}} \rangle ,\, \langle c_{\textbf {k}} \rangle ,$ and $\langle c_{\textbf {k}} c_{\textbf {q}} \rangle =0$ due to their sources $p_{01},\, p_{12},\, p_{02}=0$ in Eqs. (20)–(22)). In the aspect of the polarization entanglement demonstrated by diamond-shaped four-level systems, pure TLCs appear to be inadequate for establishing entanglement due to the lack of ambiguity about "which decay path" [43]. While spontaneous photon emission occurs during the electronic transition from the excited to ground state along the single path in TLCs, the atomic polarizations $p_{01}$ and $p_{12}$ as a source of coherent photons $\langle c_{\textbf {q}} \rangle$ and $\langle c_{\textbf {k}} \rangle$ can be excited via two different ways, mediated either electronic population densities or the optically forbidden $p_{02}$ (see Eqs. (36)–(38)).

Therefore, the spectral entanglement can only be established between coherent photons with highly suppressing incoherent photon radiation. In experiment, such an initial state of TLC can be prepared by a weak Gaussian pulse followed by a strong one (Fig. 6(a)). Temporal dynamics of the atomic polarizations and electronic population densities (inset) excited by two pump pulses ($\int \Omega _1(t)dt=0.01\,\pi$ and $\int \Omega _2(t)dt=\pi$) are presented in Fig. 6(b), where $\rho _1$ and $\rho _2$ are found to be two orders of magnitudes smaller than $p_{01}$ and $p_{02}$.

 

Fig. 6. (a) Coherent two-photon generation by using a short and weak Gaussian pulse $(\int \Omega _1(t)dt=0.01\pi )$ followed by a strong one $(\int \Omega _2(t)dt=\pi )$ in a doubly resonant cavity. (b) Temporal dynamics of the induced atomic polarizations and electron population densities (inset) as a function of the normalized time.

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In addition, the analytically solved $D(t)$ (see Eqs. (9) and (48)) offers another crucial information requested for the spectral entanglement. The strengths of $g_{01}(q)$ and $g_{12}(k)$ must be at least in the same order of magnitudes of $\gamma _1$ and $\gamma _2$. Such a strong electron-photon interaction can be realized in high Q cavities. We use the same parameters for inducing vacuum Rabi oscillation [49]. For given quality factor $Q$ and resonance frequency $\omega _c$ of a cavity mode, the photon loss rate $\kappa$ is determined as $\kappa =\omega _c/(2Q)$ [2], and the electron-photon coupling strength $g$ and the radiative decay rate $\Gamma _c$ of the excited state in the cavity mode are set to $g=2\,\Gamma _c$ and $\Gamma _c=5\,\kappa$ [41]. For numerical calculations, as a reasonable approximation, a same value of the photon loss rate is used for both resonant cavity modes and now included in our QKE of density matrix elements via the Master equation approach [2]. Since our studies are mainly focused to situations with lower photon number densities $(n_k,\,n_q\ll 1)$, the dissipative terms depending on the photon number densities can be neglected in the Master equation. Then, it is simply given as a linear superposition of the individual electronic and photonic dissipation. Therefore, we estimate that applying the Master equation after the truncation of the infinite hierarchy would yield no substantial discrepancy.

At first, we numerically calculate the entanglement criterion $D(t)$ and the corresponding $G_{12}^{(2)}(t)$ for $Q=8\times 10^8\,(\kappa _k=\kappa _q=1.51\,\textrm {MHz})$ and represent the results in Figs. 7(a) and 7(b), respectively. In the coherent linear excitation condition, we numerically observe that $|\langle c_{\textbf {k}} c_{\textbf {q}} \rangle |\gg n_k(t)$, $n_q(t)$, and $|\langle c_{\textbf {k}} \rangle \langle c_{\textbf {q}} \rangle |$ because $|p_{02}(0)|$ and $|p_{01}(0)|$ are two orders of magnitudes larger than $|p_{12}(0)|$, $\rho _1$, and $\rho _2$ (see Fig. 6(b)). As a consequence, $D(t)$ of Eq. (5) and $G_{12}^{(2)}(t)$ of Eq. (7) can be approximated by

 

Fig. 7. (a) Entanglement $D(t)$ obtained by the numerically solved result (NS) is compared with the analytical solution (AS). (b) Comparison of the correlation functions of photons $|\langle c_{\textbf {k}} c_{\textbf {q}} \rangle |^2$ multiplied by a constant $C_0=0.57$ with $G_{12}^{(2)}(t)$ for $Q=8\times 10^8\,(\kappa =1.51\,\textrm {MHz},\Gamma _c=5\kappa ,\,g=2\Gamma _c)$.

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While the photon number densities $n_k(t)$ and $n_q(t)$ are non-negative observables at photon detectors, $\langle c_{\textbf {k}} c_{\textbf {q}} \rangle$ can be measured in a balanced homodyne detection scheme. In special, $D(t)$ of Eq. (9) has the same form of the variance of the amplitude operator of squeezed light for two sideband modes in a cavity except for a factor of 8 as long as $n_k(t)$, $n_q(t) \ll 1$ [45]. Therefore, normalizing $G_{12}^{(2)}(t)$ by $|\langle c_{\textbf {k}} c_{\textbf {q}} \rangle |^2$ rather than $n_k(t)n_q(t)$ is expected to offer more appropriate temporal behavior of the cross correlated photons generated in the coherent linear excitation regime.

Next, we study the influence of cavity parameters $Q\,(\kappa )$ and $\Gamma _c$ on $G^{(2)}_{12}(t)$ normalized by $|\langle c_{\textbf {k}} c_{\textbf {q}} \rangle |^2$ in Figs. 8(a) and 8(b), respectively. Note that $\kappa =\omega _c/(2Q)$, $\Gamma _c=5\kappa$, and $g=2\Gamma _c$ are accordingly changed to different values of $Q$ for Fig. 8(a), but the varying $\Gamma _c$ changes only $g=2\Gamma _c$ for constant $Q=8\times 10^8\,(\kappa =1.51\,\textrm {MHz})$ in Fig. 8(b). Slightly different dynamics of $G^{(2)}_{12}(t)/|\langle c_{\textbf {k}} c_{\textbf {q}} \rangle |^2$ at early increasing stage appear depending on the value of $\Gamma _c$ in both figures. On the contrary, its converging late time behavior is decided by $\Gamma _c$ with respect to $\kappa$. While $G^{(2)}_{12}(t)/|\langle c_{\textbf {k}} c_{\textbf {q}} \rangle |^2$ approaches approximately $0.25$ for $\Gamma _c=5\kappa$ and $\rho _2(0)/|p_{02}(0)|^2\approx 1$, irrespective of the Q value (Fig. 8(a)), it converges to $0.5$ for two other cases in Fig. 8(b). In addition, similar tendencies are observed in $G^{(2)}_{12}(t)$ as a function of the normalized time $\Gamma _c t$ presented as inset in each figure. $G^{(2)}_{12}(t)$ does not change significantly as $Q$ is varied but depends strongly on $\Gamma _c$ with respect to $\kappa$. Such late time dynamics of $G^{(2)}_{12}(t)/|\langle c_{\textbf {k}} c_{\textbf {q}} \rangle |^2$ can be appreciated by both analytical solutions of Eqs. (45) and (48) in Appendix C for $t \rightarrow \infty$. Furthermore, the analytical solution of $G^{(2)}_{12}(t)/|\langle c_{\textbf {k}} c_{\textbf {q}} \rangle |^2$ contains no explicit dependence on the electron-photon coupling strength $g$ because it is exactly canceled out.

 

Fig. 8. (a) $G^{(2)}_{12}(t)/|\langle c_{\textbf {k}} c_{\textbf {q}} \rangle |^2$ for three different values of (a) the Q factor and (b) $\Gamma _c$ with respect to $\kappa$. Inset in each figure shows the corresponding $G^{(2)}_{12}(t)$. While $\Gamma _c=5\kappa$ and $g=2\Gamma _c$ are maintained for different values of $Q\,(\kappa )$ for (a), $Q=8\times 10^8\,(\kappa =1.51\,\textrm {MHz})$ is fixed to varying $\Gamma _c$ and $g=2\Gamma _c$ for (b).

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4. Summary

In summary, we theoretically studied temporal dynamics of the cross correlation function at zero delay time $G^{(2)}_{12}(t)$ of two photons generated from atomic TLCs. $G^{(2)}_{12}(t)$ calculated in two different excitation regimes demonstrated qualitatively distinctive features depending on the degree of the entanglement of photon pairs. The incoherent photon pair generated by spontaneous radiation of the initially excited electron were not entangled inherently, but their correlation and anti-correlation properties could be found in $G^{(2)}_{12}(t)$ depending on the radiative decay rates. The entanglement of two photons was established only by TLCs in high-Q cavities and in the coherent linear excitation regime where prevailed atomic polarizations were prepared by weak and strong pump pulses. We demonstrated the close relationship between entanglement criterion $D(t)$ and $G^{(2)}_{12}(t)$ via the two photon absorption/emission probability $\langle c_{\textbf {k}} c_{\textbf {q}} \rangle$ and discussed the dependence of $G^{(2)}_{12}(t)/|\langle c_{\textbf {k}} c_{\textbf {q}} \rangle |^2$ on Q-factor, cavity loss rate $\kappa$, radiative decay rate of atomic states in cavity $\Gamma _c$. The peculiar features observed commonly in both $G^{(2)}_{12}(t)$ and $D(t)$ can be applied to quantify spectrally entangled photon pairs generated from TLCs.