Multifold wave-particle quantum correlations in strongly correlated three-photon emissions from filtered resonance fluorescence

Ze-an Peng; Teng Zhao; Guo-qing Yang; Guang-ming Huang; Gao-xiang Li

doi:10.1364/OE.396684

1. Introduction

Since multiphoton correlations are relevant to many interesting nonclassical properties, the preparation and manipulation of multiphoton states have always been the core topics in quantum mechanics and quantum optics [1–3]. So far, photon triplets have been prepared experimentally in several physical systems, such as quantum nonlinear media [4,5], quantum dots [6], and Rydberg superatoms [7]. In addition, some theoretical schemes for generating multiphoton emission have been proposed in cavity-quantum electrodynamics systems [8,9]. On the other hand, in the application of multiphoton states, three-photon correlation signals have been verified to be able to probe into the internal information of atoms and electromagnetic modes [10], including coherent and incoherent dynamical processes. These researches have verified that the correlated photon triplets have outstanding potential to explore the fundamental physical processes of light-matter interaction. Encouragingly, the successful observation of genuine three-photon interference was selected as one of the ten breakthroughs in the world physics in 2017 [11,12]. This milestone provides a new landscape of multiphoton physics.

In terms of quantum correlations and nonclassical properties of radiation, some new concepts are still developing [13–16]. Especially, with the development of micro-nano structures, the concept of photon correlation has permeated into the field of optical materials [17–20]. In multiphoton physics, higher-order correlations have been extensively studied and applied, such as high-order ghost imaging [21–23], higher-order photon bunching [24], and high-order coherence in Bose-Einstein condensates [25]. With the ever increasing availability of spectroscopic technique, the frequency-resolved photon correlation spectrum [26–28] and multidimensional photon correlation spectrum [29] have been proposed. However, the familiar field correlations and intensity correlations are the unilateral reflections of the wavelike and particlelike properties of radiation, respectively. Therefore the concept of “wave-particle correlations” proposed by Vogel et al. is remarkable [30,31]. This fresh concept is endowed with some new connotations of nonclassicality, such as anomalous quantum correlation [32], the dynamical evolution of atomic dipole [33], and light-atom entanglement [34].

A very significant application of this concept is to test the nonclassicality of (near) Gaussian field via the conditional homodyne detection proposed by Carmichael et al. [35,36], in which the third-order fluctuations are negligible. In this case, the transformation from the condition of quadrature squeezing to the violation of the classical inequality expressed by intensity-field correlation functions can be established. However, physical scenarios may be more abundant if the third-order fluctuations are considered. The applications of intensity-field correlation functions in phase-dependent third-order fluctuations have been reported in resonance fluorescence [37–39], and the results showed that the third-order spectrum may cause noticeable deviations from quadrature squeezing. Meanwhile, the temporal asymmetry of intensity-field correlations has been investigated in resonance fluorescence [40] and cavity-quantum electrodynamics system [41], which can be regarded as another indication of non-Gaussianity of radiation and of the different conditional dynamics of atomic system [42]. In the light of these phenomena, the concept of wave-particle correlations, which is related to third-order fluctuations, can serve as a novel tool to reveal the nonclassicality and non-Gaussianity of radiation.

Considering that the wave-particle duality was originally characterized by the intensity-(single) quadrature correlation functions in two-photon regime, in this paper, we introduce new correlation functions that can be applied in three-photon Hilbert space. This generalization motivates us to construct new forms of the criterion of nonclassicality for non-Gaussian radiation. As we expected, based on the criterion proposed by Vogel et al. [43,44], the new forms of the criterion can be expressed by the intensity-dual quadrature correlation functions we proposed. We then concentrate on the time-dependent multifold wave-particle correlation functions with the aim of studying the evolution of the non-Gaussian fluctuations of frequency-resolved resonance fluorescence. On the one hand, past and future signals exhibit conspicuous asymmetry. From the perspectives of conditional quantum state [45,46] and past quantum state [47], the corresponding physical mechanisms can be interpreted as different transition dynamics of the quantum emitter associated with the past and future states. On the other hand, compared with the conventional three-photon intensity correlations, the multifold wave-particle correlations we proposed may extract more information, including the quantum coherence of photon triplet and “which way” in cascaded photon emission in atomic systems. These abundant physical pictures can be revealed transparently from our analytical expressions.

2. Description of the quantum filtering system

A pioneering theory of studying frequency-resolved photon correlations was proposed by del Valle et al. [48]. In this theory, the filtering dynamics from a quantum source to a filter is equivalent to the source-sensor interaction without back-action from the sensor to the source, in which the resonance frequency and linewidth of each sensor correspond to the setting frequency and passband width of the filter, respectively. Based on this spirit, we describe our quantum filtering system in the frame of cascaded quantum system [49,50]. In practical terms, the Mollow triplet from single quantum dots is a treasure trove of frequency-resolved photon correlations [51–53].

2.1 Hamiltonian and master equation

The quantum filtering system under our consideration consists of a laser-driven quantum emitter (source) and a single-mode optical cavity. The emitter is modeled by a two-level system with the excited state $|{e}\rangle$ and ground state $|{g}\rangle$, separated by the transition frequency $\omega _0$. It is driven coherently by a classical laser of the frequency $\omega _L$ with the Rabi frequency $\Omega$. The single-mode cavity described by the annihilation (creation) operator $a$ ($a^{\dagger}$) with the frequency $\omega _c$ is applied to serve as a Lorentzian filter. In the frame rotating of the driving frequency, the time-evolution of the full system is dominated by the master equation of the total density operator $\rho$, which has the form

(1)$$\frac{d\rho}{dt}=-\frac{i}{\hbar}{{[ H,\rho]}}+{{\mathcal{L}}_A}\rho+{{\mathcal{L}}_C}\rho+{{\mathcal{L}}_{AC}}\rho={\mathcal{L}}\rho,$$

with the total Hamiltonian $H={H}_A+{H}_C+{H}_{AL}$, where

(2)$${H}_A=\hbar\frac{\Delta_L}{2}{{\sigma}}_{z},\hspace{0.4cm}{ H}_C=\hbar\Delta_c{a}^{\dagger}a,\hspace{0.4cm}{ H}_{AL}=\hbar\frac{\Omega}{2}(\sigma_{+}+\sigma_{-})$$

are the Hamiltonians of the quantum emitter, the cavity mode, and the laser-atom semiclassical interaction, respectively. Here, $\sigma _-=|{g}\rangle \langle {e}|$, $\sigma _+=|{e}\rangle \langle {g}|$ are the transition operators of the emitter, $\sigma _z=|e\rangle \langle {e}|-|{g}\rangle \langle {g}|$ is the population inversion, and $\Delta _L=\omega _{0}-\omega _L$ and $\Delta _c=\omega _c-\omega _L$ are the detunings of the transition frequency of the emitter and the resonance frequency of the cavity with respect to the laser field, respectively. The dissipation terms, ${{\mathcal {L}}_A}\rho$ and ${{\mathcal {L}}_C}\rho$, in Eq. (1) are given by

(3)$$\begin{aligned}{{\mathcal{L}}}_A\rho=&\frac{{{\gamma}}}{2}(2{\sigma}_-\rho{\sigma}_+-\sigma_+\sigma_-\rho-\rho\sigma_+\sigma_-), \\ {{\mathcal{L}}}_C\rho=&\frac{{{\kappa}}}{2}(2{a}\rho{a}^{\dagger}-{ a}^{\dagger}{a}\rho-\rho{a}^{\dagger}{a}), \end{aligned}$$

where $\gamma$ and $\kappa$ are the decay rates of the emitter and the cavity, respectively. The last term in Eq. (1) describes the unidirectional coupling between the emitter and the target cavity without back-action from the cavity to the quantum emitter, and has the form [49,50]

(4)$${{\mathcal{L}}}_{AC}\rho=-\xi \{[a^{\dagger}, {\sigma}_-\rho]+[\rho\sigma_+, a]\},$$

with the dissipative coupling strength $\xi =\sqrt {\mu \kappa \gamma }$, where $\mu$ is the weight factor of the total spontaneous decay of the quantum emitter.

The famous Mollow triplet of resonance fluorescence can be generated at a high pumping intensity [54]. The most elegant physical interpretation of this phenomenon is based on the semiclassical dressed states $|{1_A}\rangle =c|{g}\rangle -s|{e}\rangle$ and $|{2_A}\rangle =s|{g}\rangle +c|{e}\rangle$, in which $s,c=\sqrt {(\bar \Omega \pm \Delta _L)/2\bar \Omega }$ with $\bar \Omega =\sqrt {\Omega ^2+\Delta _L^2}$. Therefore the total transition operator of the bare emitter can be expressed in terms of the dressed-state transition operators $\sigma _{k'k}=|{k'_A}\rangle \langle {k_A}|(k',k\in \{1, 2\})$ as $\sigma _-=\sigma _1+\sigma _2+\sigma _3$, in which $\sigma _1=c^2\sigma _{12}$, $\sigma _2=-s^2\sigma _{21}$, and $\sigma _3=cs(\sigma _{22}-\sigma _{11})$ describe the lower-frequency, higher-frequency, and laser-resonant transitions between two adjacent manifolds of the laser-dressed levels, respectively. For clarity, we shall label the three peaks of the Mollow triplet from the above three transitions as $``\textrm { F}"$, $``\textrm { T}"$, and $``\textrm { R}"$, respectively. In terms of the dressed atomic states, the dissipative terms are transformed into

(5)$$\begin{aligned}&{{\mathcal{L}}}_A\rho=\frac{\gamma}{2}\sum_{i=1,2,3}(2{\sigma}_i\rho{\sigma}^{\dagger}_i-{\sigma}^{\dagger}_i{\sigma}_i\rho-\rho{\sigma}^{\dagger}_i{\sigma}_i), \\ &{{\mathcal{L}}}_{AC}\rho=-\xi\sum_{i=1,2,3} \{[a^{\dagger}, {\sigma}_i\rho]+[\rho\sigma_i^{\dagger}, a]\} \end{aligned}$$

by dropping rapidly oscillating terms. This tells us that the total spontaneous emission of the emitter can be separated into three components with the transition rates $\gamma c^4$, $\gamma s^4$, and ${\gamma }c^2s^2$, respectively.

2.2 Main physical mechanisms

From the master equation in dressed-state representation, we can evaluate the correlation function $\langle a^{\dagger m}a^m\rangle$ of interest by deriving the equations of motion for the dressed atom-photon correlation moments $\langle a^{\dagger m} a^n\sigma _{k'k}\rangle$ ($k', k\in \{1, 2\}$ and $m,n\in \{0, 1, 2, 3,\ldots \}$) through the relation

(6)$$\frac{d}{dt}\langle a^{\dagger m} a^n\sigma_{k'k}\rangle=\textrm{Tr}\Big(a^{\dagger m} a^n\sigma_{k'k}\frac{d\rho}{dt}\Big).$$

The equations of motion for the dressed atom-photon correlation moments and the steady-state solutions have been presented in Appendix A. In this case, we can carry out the analytical calculations for the three-photon filtering dynamics of interest in a Hilbert space truncated at the three-excitation manifold of the cavity field. In other words, the total Hilbert space is spaced by the basic vectors $|{k_A, n_a}\rangle$, in which the first index $k\in \{1,2\}$ denotes the dressed atomic state and the second index $n\in \{0, 1,2,3\}$ corresponds to the excitation of the cavity field. After solving the steady-state dressed atom-photon correlation moments, we find that the main physical mechanisms can be revealed transparently when the filter width is sufficiently larger than the linewidth of each spectral component of the Mollow triplet, i.e., $\kappa \gg \gamma$ [55,56]. In this case, by ignoring the small contributions proportional to ${\gamma }/{\kappa }$ in $\langle a^{\dagger m} a^n\sigma _{k'k}\rangle$, the total fluorescent emission resolved by the cavity can be split into two independent cascaded emissions weighted by the dressed populations ${{\langle \sigma _{11}\rangle }}$ and ${{\langle \sigma _{22}\rangle }}$, respectively. Thus, it is enlightening to decompose each correlation moment into two components according to the steady-state dressed populations, where it reads

(7)$${\langle a^{\dagger m}a^n\sigma_{k'k}\rangle}=\langle \sigma_{11}\rangle{\langle a^{\dagger m} a^n\sigma_{k'k}\rangle}_1+\langle\sigma_{22}\rangle{\langle a^{\dagger m}a^n\sigma_{k'k}\rangle}_2.$$

This decomposition enables us to determine the quantum state of the quantum filtering system as

(8)$${\rho}_s=\sum_{k,k'}\sum_{n,m}\frac{{\langle a^{\dagger m} a^n\sigma_{k'k}\rangle}}{\sqrt{m!n!}}|{k_A,n_a}\rangle\langle{k'_A,m_a}|=\sum_{l=1,2}{\langle\sigma_{ll}\rangle}|{\psi^{(l)}_s}\rangle\langle{\psi^{(l)}_s}|,$$

where the steady-state wave functions $|{\psi ^{(1)}_s}\rangle$ and $|{\psi ^{(2)}_s}\rangle$ are given by

(9)$$\begin{aligned}&|{\psi^{(1)}_s}\rangle=|{1_A, 0_a}\rangle+{\mathcal{C}}^{(1)}_{1}|{1_A, 1_a}\rangle+{\mathcal{C}}^{(1)}_{2}|{2_A, 1_a}\rangle+{\mathcal{C}}^{(1)}_{3}|{1_A, 2_a}\rangle \\&~~~~~~~~~~~~~~+{\mathcal{C}}^{(1)}_{4}|{2_A, 2_a}\rangle+{\mathcal{C}}^{(1)}_{5}|{1_A, 3_a}\rangle+{\mathcal{C}}^{(1)}_{6}|{2_A, 3_a}\rangle, \\&|{\psi^{(2)}_s}\rangle=|{2_A, 0_a}\rangle+{\mathcal{C}}^{(2)}_{1}|{1_A, 1_a}\rangle+{\mathcal{C}}^{(2)}_{2}|{2_A, 1_a}\rangle+{\mathcal{C}}^{(2)}_{3}|{1_A, 2_a}\rangle \\ &~~~~~~~~~~~~~~+{\mathcal{C}}^{(2)}_{4}|{2_A, 2_a}\rangle+{\mathcal{C}}^{(2)}_{5}|{1_A, 3_a}\rangle+{\mathcal{C}}^{(2)}_{6}|{2_A, 3_a}\rangle. \end{aligned}$$

In Eq. (9), the superscript $``(l)"$ represents that the cascaded emission described by $|\psi ^{(l)}_s\rangle$ is triggered by the initial atomic state $|{l_A}\rangle$. The analytical expressions of the steady-state probability amplitudes in Eq. (9) have been given in Appendix B, in which the recurrence relations of two-photon and three-photon probability amplitudes stem from the fact that a pair of cascaded emission channels coupled by a common energy level displays quantum interference. Therefore the analytical expressions of the steady-state probability amplitudes correspond to the energy level diagrams in Fig. 1.

Fig. 1. Three-photon cascaded emissions resolved by the filter. (a) and (b) correspond to the all possible transitions triggered by a common atomic states $|{1_A}\rangle$, but terminated by $|{1_A}\rangle$ and $|{2_A}\rangle$, respectively. (c) and (d) correspond to the transitions terminated by $|{1_A}\rangle$ and $|{2_A}\rangle$, respectively, but triggered by a common atomic state $|{2_A}\rangle$. The fluorescent photons generated from the central peak (“$\textrm {R}$”), lower-frequency side peak (“$\textrm {F}$”), and higher-frequency side peak (“$\textrm {T}$”) are depicted by the red arrows, yellow arrows, and blue arrows, respectively. The dressed-state transition amplitudes and the stationary probability amplitudes in Eq. (9) are indicated.

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Let us first explore the frequency-resolved intensity correlations $g^{(2)}=\langle a^{\dagger 2}a^2\rangle /{\langle a^{\dagger}a\rangle }^2$ and $g^{(3)}=\langle a^{\dagger 3}a^3\rangle /{\langle a^{\dagger}a\rangle }^3$. The $n$th-order correlation function corresponds to the probability of $n$-photon target states, therefore the frequency-resolved intensity and intensity correlations can be calculated from Eq. (9) as

(10)$$\begin{aligned}&\langle a^{\dagger}a\rangle=\sum_{l=1,2}\langle\sigma_{ll}\rangle\big(\big|{\mathcal{C}}^{(l)}_{1}\big|^2+\big|{\mathcal{C}}^{(l)}_{2}\big|^2\big), \\&\langle a^{\dagger2}a^2\rangle=\sum_{l=1,2}\langle\sigma_{ll}\rangle\big(\big|\sqrt{2}{\mathcal{C}}^{(l)}_{3}\big|^2+\big|\sqrt{2}{\mathcal{C}}^{(l)}_{4}\big|^2\big), \\ &\langle a^{\dagger3}a^3\rangle=\sum_{l=1,2}\langle\sigma_{ll}\rangle\big(\big|{\sqrt{6}\mathcal{C}}^{(l)}_{5}\big|^2+\big|\sqrt{6}{\mathcal{C}}^{(l)}_{6}\big|^2\big). \end{aligned}$$

In physical terms, Eq. (10) tells us that the $n$-photon probability is contributed from two independent dressed emission sources, weighted by $\langle \sigma _{11}\rangle$ and $\langle \sigma _{22}\rangle$ respectively. Because the quantum interference cannot be established between two cascaded channels terminated by two different target dressed states, the total probability is given in the form of incoherent superposition. In Fig. 2(a), a pair of prominent two-photon correlation peaks can be observed when the cavity is tuned to $\Delta _c=\pm \bar \Omega /2$. It can be understood as the consequence of the constructive quantum interference between the two possible two-photon cascaded emission channels involving “$\textrm {R}$” photons and sideband photons with opposite emission orderings. For example, if we divide the two-photon probability amplitude ${\mathcal {C}}^{(1)}_{4}$ into two parts, i.e., ${\mathcal {C}}^{(1)}_{4}=\big ({\mathcal {C}}^{(1)}_{4}\big )_1+\big ({\mathcal {C}}^{(1)}_{4}\big )_2$, as indicated in Fig. 1(b), one can see that these two parts correspond to the two possible cascaded transitions ${|{1_A}\rangle }\stackrel {\textrm {R}}{\longrightarrow }{|{1_A}\rangle }\stackrel {\textrm {T}}{\longrightarrow }{|{2_A}\rangle }$ and ${|{1_A}\rangle }\stackrel {\textrm {T}}{\longrightarrow }{|{2_A}\rangle }\stackrel {\textrm {R}}{\longrightarrow }{|{2_A}\rangle }$ with the two-photon dressed-state amplitudes $cs^3$ and $-cs^3$, respectively. Thus the destructive quantum interference between these two channels should has been established in bare. However, due to the fact that the filter modulates the single-photon transition amplitude into the Lorentzian type (see Eq. (47)), the two processes are interfered constructively by tuning the cavity frequency to $\Delta _c=\pm \bar \Omega /2$. Therefore the photons at these frequencies are the most promising for “Mollow spectroscopy” [57]. This phenomenon persists in the three-photon correlation spectrum, as shown in Fig. 2(b).

Fig. 2. (a) Normalized two-photon intensity correlation spectrum $g^{(2)}(\Delta _c)$ and (b) normalized three-photon intensity correlation spectrum $g^{(3)}(\Delta _c)$ varying with the cavity linewidth $\kappa$ when the Mollow triplet is scanned into the cavity. (c) Normalized three-photon intensity correlation spectrum $g^{(3)}(\Delta _c)$ for the region of $\Delta _c\in [8, 18]$. The parameters are $\Omega =30$, $\gamma =1$, and $\Delta _L=0$.

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However, as the cavity linewidth becomes narrower, two extra peaks at $\Delta _c=\pm \bar \Omega /3$ can emerge, as shown in Fig. 2(c). The physical origin of this pair of new correlation peaks can be understood in a similar way. As an example, let us concentrate on the case of $\Delta _c=\bar \Omega /3$. The three-photon emission at this frequency is predominantly contributed from the three-photon target state $|2_A, 3_a\rangle$, which is triggered by the dressed source $\langle \sigma _{11}\rangle$ and embodied by the three-photon probability amplitude ${\mathcal {C}}^{(1)}_{6}$. If we divide ${\mathcal {C}}^{(1)}_{6}$ into two parts, i.e., ${\mathcal {C}}^{(1)}_{6}=\big ({\mathcal {C}}^{(1)}_{6}\big )_1+\big ({\mathcal {C}}^{(1)}_{6}\big )_2$, as indicated in Fig. 1(b), one can see that the two possible two-photon cascaded emission channels described by ${\mathcal {C}}^{(1)}_{3}$ and ${\mathcal {C}}^{(1)}_{4}$ are coupled by the single-photon transitions ${|{1_A, 2_a}\rangle }\stackrel {\textrm {R}}{\longrightarrow }{|{1_A, 3_a}\rangle }$ and ${|{2_A, 2_a}\rangle }\stackrel {\textrm {F}}{\longrightarrow }{|{1_A, 3_a}\rangle }$. In addition, the three-photon cascaded emission is modulated by the probability amplitude of photon triplet. When the cavity frequency is tuned to $\Delta _c=\bar \Omega /3$, this frequency not only resonates with the amplitude of photon triplet, but also gives rise to the constructive quantum interference between $\big ({\mathcal {C}}^{(1)}_{6}\big )_1$ and $\big ({\mathcal {C}}^{(1)}_{6}\big )_2$, and between $\big ({\mathcal {C}}^{(1)}_{4}\big )_1$ and $\big ({\mathcal {C}}^{(1)}_{4}\big )_2$, so that the correlation peak at $\Delta _c=\bar \Omega /3$ emerge. Similarly, another three-photon correlation peak at $\Delta _c=-\bar \Omega /3$ is due to the dressed emission source ${\langle \sigma _{22}\rangle }$. Although the remarkable three-photon correlation signals at $\Delta _c=\pm \bar \Omega /3$ contain the contributions of the constructive interference of two-photon cascaded emissions, the signals also reflect the information of genuine strongly correlated three-photon emission. Therefore the emissions at $\Delta _c=\pm \bar \Omega /3$ can serve as the excellent resources of photon triplets for our following investigation.

Here, we would like to evaluate the experimental feasibility of the parameters in our scheme. In Ref. [51], the frequency-resolved two-photon correlations of the Mollow triplet from a semiconductor quantum dot were explored experimentally. The quantum dot with the natural linewidth characterized by $\gamma ^{-1}=1.25\pm 0.08\textrm {ns}$ ($\gamma =0.75\sim 0.85\textrm {GHz}$) is driven by a resonant pump. The Mollow photons are filtered by a filter, introducing the time uncertainty of $\kappa ^{-1}=110\textrm {ps}$ ($\kappa =9.09\textrm {GHz}$) in the filtered photons. Therefore the filter bandwidth is $\kappa \approx 11.36\gamma$, and the bare Rabi frequency is $\Omega \approx 15\gamma$ for $\Omega /2\pi =11.27\textrm {GHz}$ in Ref. [51]. Based on these experimental conditions, in our scheme, we choose the target cavity with the decay rate (the filter passband width) $\kappa =8\gamma$. The experimental detection of three-photon correlations was also reported. In Ref. [53], it was mentioned that the larger Rabi frequency relative to the filter bandwidth and the quantum dot decay rate is necessary to ensure that the three-photon correlation peaks can be separably observed, such as $\Omega /2\pi =15\textrm {GHz}$ for the radiative linewidth of single quantum dot $\gamma /2\pi =0.2\textrm {GHz}$, i.e., $\Omega =75\gamma$. Although the enhancement of Rabi frequency may cause some difficulties, such as low radiation rate, some experimental approaches have been proposed. For example, the broadband spontaneous emission control can be realized by photonic nanowires [58], the photon-extraction efficiency can be enhanced by the deterministic quantum-dot microlenses [59]. In our theoretical scheme, we choose the bare Rabi frequency $\Omega =30\gamma$, which is hopeful in current experimental conditions. An alternative method of increasing effective Rabi frequency is to use a properly detuned driving field. In this case, the bare Rabi frequency is not necessarily very large. Recently, some schemes have been proposed to generate $n$-photon bundles [60] and $n$-phonon bundles [61] by adjusting the frequency of driving field.

3. Nonclassicality of strongly correlated three-photon emission

Recently, the nonclassicality of resonance fluorescence regained its popularity from a more comprehensive perspective by monitoring wave-particle duality [37–40]. In fact, the original intention of this scheme is to determine the nonclassicality of weak light with negligible non-Gaussian fluctuations. However, recent researches showed that the detection of non-Gaussian fluctuations opens a new gate to study the nonclassicality of radiation. Then one question arises naturally of whether or not the non-Gaussian fluctuations in three-photon dynamics can be probed profoundly by wave-particle joint monitoring than the conventional three-photon intensity correlations. Motivated by this question, in this section, we tend to introduce the multifold wave-particle correlation functions by constructing new forms of the criterion of nonclassicality for non-Gaussian radiation. The multifold wave-particle correlation functions are based on the intensity operator $I=a^{\dagger}a$ and the quadrature operator $E_{\theta }=(a e^{-i\theta }+a^{\dagger}e^{i\theta })/2$, where $\theta$ is the phase of local oscillator in balanced homodyne detection.

A simplified diagram of the experimental setup for multifold wave-particle correlation detection is sketched in Fig. 3. The filtered Mollow triplet is divided into three beams, two of which are interfered with two local oscillator fields respectively to participate in wave detection separately, provided by two balanced homodyne setups, and the remaining beam is sent to a photon counter to generate particlelike signal. Conditioned on a photon count, the intensity-dual quadrature correlation can be detected by averaging all the triggered samples of the mixed photocurrent signal from the two balanced homodyne detectors. Alternatively, we can first record the mixed photocurrent signal from the photon counter and a homodyne detector to generate intensity-field correlation, and then average all the samples of another homodyne signal.

Fig. 3. Schematic diagram of the detection system for intensity-dual quadrature correlations. The filtered Mollow triplet is divided into three beams, two of which participate in wave detection separately in two balanced homodyne setups, and the remaining beam is sent to a photon counter to generate particlelike signal.

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3.1 Multifold wave-particle quantum correlation related to $g^{(2)}$

Our first strategy is to correlate intensity $I$ and dual quadrature $E^2_{\theta }$. In terms of moments or fluctuations, Shchukin and Vogel presented a set of criteria for determining the nonclassicality of radiation [43,44]. For our system, let us define the matrix

(11)$$ {\mathbf{D}}_{\theta}(I, E^2_{\theta})= \left( \begin{array}{ccc} \langle :(\delta I)^2:\rangle & \langle :\delta I\delta E^{2}_{\theta}:\rangle\\ ~~~~~~\\ \langle :\delta I\delta E^{2}_{\theta}:\rangle & \langle :(\delta E^2_{\theta})^2:\rangle \end{array} \right), $$

in which $\langle : \delta A\delta B:\rangle =\langle :AB :\rangle -\langle : A:\rangle \langle :B:\rangle$ and “::” stands for the normal ordering of the field operators. According to Refs. [43] and [44], the filtered field is nonclassical if any of the first-order principal minor determinants of the matrix ${\mathbf {D}}_{\theta }(I,E^2_{\theta })$ is negative, i.e.,

(12)$$\langle :(\delta I)^2:\rangle<0~~~~\textrm{or}\hspace{0.3cm}\langle :(\delta E^2_{\theta})^2:\rangle<0,$$

or the Cauchy-Schwarz inequality is violated, i.e.,

(13)$$\langle :(\delta I)^2:\rangle\langle :(\delta E^2_{\theta})^2:\rangle<{\langle :\delta I\delta E^{2}_{\theta}:\rangle}^2,$$

if both the above two variances are above their respective quantum noise limit. In Eq. (13), an extra signal ${-----}$ the equal-time intensity-dual quadrature correlation function $\langle :I(0) E^2_{\theta }(0):\rangle$ is required. In order to obtain the normalized signal, it is beneficial to normalize the determinant as $d_{\theta }(I, E^2_{\theta })={\textrm {det}[{\mathbf {D}}_{\theta }(I, E^2_{\theta })]}/\big ({{\langle :I:\rangle }^2{\langle : E^2_{\theta }:\rangle }^2}\big )$. Therefore we obtain

(14)$$d_{\theta}(I, E^2_{\theta})=\tilde{\mathcal{V}}^{(2)}_{\theta}\Big[g^{(2)}(0)-1\Big]-\Big[h^{(2)}_{\theta}(0)-1\Big]^2.$$

In Eq. (14), we have utilized the scaled variance of dual quadrature $\tilde {\mathcal {V}}^{(2)}_{\theta }=\langle :(\delta E^2_{\theta })^2:\rangle /{\langle : E^2_{\theta }:\rangle }^2$ and the scaled variance of intensity $\tilde {\mathcal {V}}_I={\mathcal {V}}_{I}/{\langle : I:\rangle }^2=g^{(2)}(0)-1$, where ${\mathcal {V}}_{I}={\langle :(\delta I)^2:\rangle }$ is the unscaled fluctuation of intensity. As we expected, in Eq. (14), the normalized intensity-dual quadrature correlation function

(15)$$h^{(2)}_{\theta}(0)=\frac{\langle: I(0) E^2_{\theta}(0):\rangle}{\langle: I:\rangle\langle: E^2_{\theta}:\rangle}$$

is introduced, which can be achieved experimentally by correlating two quadrature signals conditioned on a photon count.

For the case of $\Delta _c=\bar \Omega /3$, we first present $\tilde {\mathcal {V}}^{(2)}_{\theta }$ and $g^{(2)}(0)-1$ in Fig. 4(a) varying with the phase of local oscillator. One can see that the two-photon correlation is bunched and is independent of $\theta$. Whereas the nonclassicality can be measured by $\tilde {\mathcal {V}}^{(2)}_{\theta }$ with the minimum value $(\tilde {\mathcal {V}}^{(2)}_{\theta })_{\textrm {min}}\approx -0.15$ at $\theta \approx 0.89$ under the given parameters. However, if we further explore the value of $d_{\theta }$ in Fig. 4(b), we can see that there exists a more optimal phase $\theta _0\approx 2.26$, which corresponds to $(d_{\theta })_{\textrm {min}}\approx -3.5$. This result suggests that, compared with the direct measurement for the variance of dual quadrature, the violation of the Cauchy-Schwarz inequality is more prominent. In addition, this superiority can persist in the phase region $[\theta _1, \theta _2]$, where neither the variance of dual quadrature $\tilde {\mathcal {V}}^{(2)}_{\theta }$ nor the variance of intensity $\tilde {\mathcal {V}}_I$ can display the nonclassicality of the filtered resonance fluorescence, as shown in Fig. 4(c).

Fig. 4. (a) Scaled variance of dual quadrature $\tilde {\mathcal {V}}^{(2)}_{\theta }$ varying with the phase $\theta$. Inset: Scaled variance of intensity ${\langle :(\delta I)^2:\rangle }/{\langle :I:\rangle }^2=g^{(2)}(0)-1$. (b) Scaled determinant $d_{\theta }(I, E^2_{\theta })$ as a function of the phase $\theta$. The optimal phase $\theta _0\approx 2.26$ corresponds to $(d_{\theta })_{\textrm {min}}\approx -3.5$. (c) Comparison of the distributions of $\tilde {\mathcal {V}}^{(2)}_{\theta }$ (blue dashed line) and $d_{\theta }(I, E^2_{\theta })$ (red solid line). (d) Filtered fluctuation spectrum ${\mathcal {V}}_{I}$ in the range of $\Delta _c\in [0, \bar \Omega /2]$. The parameters are $\Omega =30$, $\kappa =8$, $\gamma =1$, $\Delta _L=0$. In (a), (b), and (c), the cavity frequency is tuned to $\Delta _c=\bar \Omega /3$.

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Physically, this violation of the Cauchy-Schwarz inequality can be attributed to the phase-dependent non-Gaussian fluctuations. In addition, the value of the fluctuation of intensity ${\mathcal {V}}_{I}$ at $\Delta _c=\bar \Omega /2$ is lower than that at $\Delta _c=\bar \Omega /3$, as illustrated in Fig. 4(d). Therefore the decreased value of the first term of $d_{\theta }(I, E^2_{\theta })$ in Eq. (14) makes it possible to violate the Cauchy-Schwarz inequality. In order to explore more information of the nonclassicality in strongly correlated three-photon emission, let us rewrite the intensity-dual quadrature correlation fluctuation as

(16)$$\mathcal{V}^{(2)}_{I, \theta}=\frac{1}{2}\Big(\langle:\delta I\delta X_{\theta}:\rangle+\langle:(\delta I)^2:\rangle\Big),$$

where $X_{\theta }=(a^2e^{-2i\theta }+a^{\dagger 2}e^{2i\theta })/4$. In Eq. (16), the non-Gaussian fluctuation $\langle :\delta I\delta X_{\theta }:\rangle$ is noteworthy because the fourth-order moment $\langle a^{\dagger} a^{3}\rangle$ is the genuine three-photon information reflected by our target signal $\langle :I(0) E^2_{\theta }(0):\rangle$. Main physical mechanisms can be revealed in the limit of $\kappa \gg \gamma$. In this case, we obtain the analytical expression of the three-photon moment

(17)$$\begin{aligned}\langle a^{\dagger} a^3\rangle=&\sum_{l=1, 2}\langle\bar{\psi}^{(l)}(0)| a^{2}|\bar{\psi}^{(l)}(0)\rangle \\=&\sqrt{2}\Big\{\langle\sigma_{11}\rangle\Big[{\bar{\mathcal{C}}}^{(1)\ast}_{1}(0){\bar{\mathcal{C}}}^{(1)}_{5}(0)+{\bar{\mathcal{C}}}^{(1)\ast}_{2}(0){\bar{\mathcal{C}}}^{(1)}_{6}(0)\Big] \\ &+\langle\sigma_{22}\rangle\Big[{\bar{\mathcal{C}}}^{(2)\ast}_{1}(0){\bar{\mathcal{C}}}^{(2)}_{5}(0)+{\bar{\mathcal{C}}}^{(2)\ast}_{2}(0){\bar{\mathcal{C}}}^{(2)}_{6}(0)\Big]\Big\}. \end{aligned}$$

In the derivation of Eq. (17), we have introduced the steady-state single-photon conditional states $|\bar {\psi }^{(l)}(0)\rangle =a|{\psi }^{(l)}_s\rangle$ ($l\in \{1, 2\}$) with the initial single-photon conditional probability amplitudes ${\bar {\mathcal {C}}}^{(l)}_{j}(0)$ ($j\in \{1, 2,.., 6\}$) (see Appendix C). This result indicates that, conditioned on a single-photon detection, the three-photon information is characterized by the quantum coherence between the two collapsed states $|{0_a}\rangle$ and $|{2_a}\rangle$, which are collapsed from the states $|{1_a}\rangle$ and $|{3_a}\rangle$, respectively.

In addition, we also investigate the distributions of $\tilde {\mathcal {V}}^{(2)}_{\theta }$ and $d_{\theta }(I, E^2_{\theta })$ with the phase $\theta$ and the Rabi frequency $\Omega$ in Figs. 5(a) and (b), respectively. It can be seen that, for different Rabi frequencies, the negative value of the function $d_{\theta }(I, E^2_{\theta })$ is always one order higher than that of the variance of dual quadrature .

Fig. 5. Contour plots of (a) $\tilde {\mathcal {V}}^{(2)}_{\theta }$ and (b) $d_{\theta }(I, E^2_{\theta })$ varying with the Rabi frequency $\Omega$ and the phase $\theta$. The parameters are $\kappa =8$, $\gamma =1$, $\Delta _L=0$, and $\Delta _c=\bar \Omega /3$.

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3.2 Multifold wave-particle quantum correlation related to $g^{(3)}$

In view of the multiplicity of operator combination in the three-photon Hilbert space, an alternative scheme for constructing multifold wave-particle correlation function is to correlate the intensity-amplitude composite operator $A=IE_\theta$ and the quadrature operator $B=E_\theta$. As the same procedure in the above, we obtain the conditions of nonclassicality

(18)$$\langle :[\delta(IE_{\theta})]^2:\rangle<0~~~~\textrm{or}\hspace{0.3cm}\langle :(\delta E_{\theta})^2:\rangle<0.$$

Furthermore, the nonclassicality can be determined by the violation of the Cauchy-Schwarz inequality, i.e.,

(19)$$\langle :[\delta(IE_{\theta})]^2:\rangle \langle :(\delta E_{\theta})^2:\rangle<{\langle :\delta(I E_{\theta})\delta E_{\theta}:\rangle}^2,$$

if neither of the inequalities in Eq. (18) can be satisfied. Similarly, we can obtain the normalized second-order determinant

(20)$$d_{\theta}(I E_{\theta}, E_{\theta})=\tilde{\mathcal{V}}^{(1)}_{\theta}\Big[\lambda g^{(3)}(0)-1\Big]-\Big[\chi^{(2)}_{\theta}(0)-1\Big]^2,$$

where $\lambda =\frac {{\langle : I:\rangle }^3}{2{\langle :IE_{\theta }:\rangle }^2}$. The occurrence of $g^{(3)}(0)$ in Eq. (20) lies in the fact that the higher-order moment $\langle :(IE_{\theta })^2:\rangle$ can be directly given by $G^{(3)}(0)$ because the four-photon moments $\langle a^{\dagger 4}a^2\rangle$ and $\langle a^{\dagger 2} a^4\rangle$ are negligible in our system. In Eq. (20), $\tilde {\mathcal {V}}^{(1)}_{\theta }=\langle :(\delta E_{\theta })^2:\rangle /{\langle E_{\theta }\rangle }^2$ is the conventional second-order fluctuation, $\lambda g^{(3)}(0)-1=\tilde {\mathcal {V}}^{(1)}_{I,\theta }={\langle :[\delta (I E_{\theta })]^2:\rangle }/{{\langle :IE_{\theta }:\rangle }^2}$ is the scaled fluctuation of intensity-quadrature composite operator, and $\chi ^{(2)}_{\theta }(0)$ is the multifold wave-particle correlation function

(21)$$\chi^{(2)}_{\theta}(0)=\frac{\langle :I(0) E^2_{\theta}(0):\rangle}{\langle :I E_{\theta}:\rangle\langle :E_{\theta}:\rangle},$$

normalized by the intensity-(single) amplitude correlation function $\langle :I E_{\theta }:\rangle$ and the amplitude $\langle :E_{\theta }:\rangle$. We should note that Eq. (21) cannot be well defined for ${\langle a\rangle }=0$, which leads to ${\langle :E_\theta :\rangle }=0$. This case corresponds to the exactly resonant driving in our system ($\Delta _L=0$). As the proposed conditional homodyne detection for intensity-(single) field correlation functions [34,36], a coherent offset light with amplitude $\alpha$ is required. The output signal field can be described as $b=a+\alpha =\langle b\rangle +\delta b$, where $\langle b\rangle ={\langle a\rangle }+\alpha =\beta e^{i\phi }$ with $\beta =|\langle b\rangle |$ and $\delta b=\delta a$. In order to clarify the physical mechanisms, let us concentrate on the case of exactly resonant driving. In this case, each fluctuation of the cavity field can be given by the corresponding moment through a succinct relation $\langle (\delta b^{\dagger})^m(\delta b)^n\rangle =\langle (\delta a^{\dagger})^m(\delta a)^n\rangle =\langle a^{\dagger m} a^n\rangle$.

We first explore the feasibility of utilizing the two scaled fluctuations $\tilde {\mathcal {V}}^{(1)}_{I,\theta }$ and $\tilde {\mathcal {V}}^{(1)}_{\theta }$ to test the nonclassicality of the output light, as shown in Figs. 6(a) and (b). Under the given parameters, it can be seen that the normally ordered variance $\tilde {\mathcal {V}}^{(1)}_{\theta }$ is positive in the whole region, which means that there is no second-order quadrature squeezing. However, the fluctuation $\tilde {\mathcal {V}}^{(1)}_{I,\theta }$ is squeezed when the phase is adjusted to $\theta =3\pi /2$ (green and blue regions in Fig. 6(a)). So far, the nonclassicality has been determined. If we further explore the distribution of the function $d_{\theta }(IE_{\theta }, E_{\theta })$ in Fig. 6(c), we can find that there is a wider region showing nonclassical signal ($d_{\theta }(IE_{\theta }, E_{\theta })<0$). This nonclassicality can neither be determined by measuring the variance of quadrature $\tilde {\mathcal {V}}^{(1)}_{\theta }$ nor by measuring the variance of intensity-field correlation $\tilde {\mathcal {V}}^{(1)}_{I,\theta }$. Compared with the result in Fig. 4(b), it can be concluded that the offset light may affect the nonclassicality and thus reduce the value of $d_{\theta }(IE_{\theta }, E_{\theta })$ in Fig. 6(c). However, the superiority of $d_{\theta }(IE_{\theta }, E_{\theta })$ can still be maintained by choosing the optimal values of the phase $\theta$ and the amplitude $|\alpha |$.

Fig. 6. Scaled fluctuations of (a) intensity-quadrature composite operator $IE_\theta$ and (b) single-quadrature $E_{\theta }$ of the output signal mode varying with $\theta$ and $|\alpha |$. (c) Scaled determinant $d_{\theta }(IE_{\theta }, E_{\theta })$ varying with $\theta$ and $|\alpha |$. The parameters are $\Omega =30$, $\kappa =8$, $\gamma =1$, $\Delta _c=\bar \Omega /3$, $\mu =0.125$, $\Delta _L=0$, and the phase of the offset field is parallel with the phase of the local oscillator, i.e., $\phi =\theta$.

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4. Time-dependent multifold wave-particle quantum correlations

In this section, we turn to explore the time-evolution of non-Gaussian fluctuations of the frequency-resolved resonance fluorescence by calculating time-dependent multifold wave-particle correlation functions. Based on our analytical formalism, the main physical mechanisms of the time-dependent correlation signals can be revealed by conditional quantum dynamics of the monitored system, including conditional quantum state and past quantum state.

4.1 Two-time correlation for intensity-dual quadrature

Two-time intensity-dual quadrature correlation refers to a situation in which, after detecting a photon at time $t$, the signal of dual quadrature of the filtered field is recorded by correlating two balanced homodyne signals simultaneously at time $t+\tau$. We define the normalized two-time intensity-dual quadrature correlation function as

(22)$$h_{\theta}^{(2)}(\tau)=\frac{{\langle :I(t)E_\theta^{2}(t+\tau):\rangle}}{{\langle:I(t):\rangle}{\langle:E_\theta^{2}(t+\tau):\rangle}},$$

where “::” indicates the normal ordering and chronological ordering for the field operators, and the unnormalized correlation function $H_{\theta }^{(2)}(\tau )={\langle :I(t) E_\theta ^{2}(t+\tau ):\rangle }$ can be decomposed into

(23a)$$ H_{\theta}^{(2)}(\tau>0)=\frac{1}{2}\textrm{Re}[\langle a^{\dagger}(0) a^2(\tau)a(0)\rangle e^{-2i\theta}]+2\langle a^{\dagger}(0)(a^{\dagger} a)(\tau) a(0)\rangle,$$

(23b)$$ H_{\theta}^{(2)}(\tau<0)=\frac{1}{2}\textrm{Re}[\langle(a^{\dagger} a)(0)a^2(\tau)\rangle e^{-2i\theta}]+2\langle a^{\dagger}(\tau)( a^{\dagger} a)(0) a(\tau)\rangle.$$

In order to concentrate on the time-evolution of fluctuations, let us consider the case of resonant driving, which leads to $\langle a\rangle =0$ in our system. Therefore we obtain a succinct relation $\langle (\delta a^{\dagger})^m(\delta a)^n\rangle =\langle a^{\dagger m} a^n\rangle$. One can notice that the terms $\langle a^{\dagger}(0)(a^{\dagger} a)(\tau ) a(0)\rangle$ and $\langle a^{\dagger}(\tau )( a^{\dagger} a)(0) a(\tau )\rangle$, occurring in Eqs. (23a) and (23b), respectively, are the two-photon intensity correlations, which originate from the equal-time correlation between two balanced homodyne signals, i.e., $E_{\theta }(t)E_{\theta }(t)$. Obviously, these two terms correspond to the particle aspect of the filtered fi. However, the terms $\langle a^{\dagger}(0) a^2(\tau )a(0)\rangle$ and $\langle (a^{\dagger} a)(0)a^2(\tau )\rangle$ may convey different physical information, which are given by the following expressions

(24a)$$\begin{aligned} \langle a^{\dagger}(0) a^2(\tau) a(0)\rangle_{\tau>0}=&\sum_{l=1, 2}\langle\bar\psi^{(l)}(\tau)| a^2|\bar\psi^{(l)}(\tau)\rangle\\ =&\sqrt{2}\Big\{\langle\sigma_{11}\rangle\Big[{\bar{\mathcal{C}}}^{(1)\ast}_{1}(\tau){\bar{\mathcal{C}}}^{(1)}_{5}(\tau)+{\bar{\mathcal{C}}}^{(1)\ast}_{2}(\tau){\bar{\mathcal{C}}}^{(1)}_{6}(\tau)\Big]\\ &+\langle\sigma_{22}\rangle\Big[{\bar{\mathcal{C}}}^{(2)\ast}_{1}(\tau){\bar{\mathcal{C}}}^{(2)}_{5}(\tau)+{\bar{\mathcal{C}}}^{(2)\ast}_{2}(\tau){\bar{\mathcal{C}}}^{(2)}_{6}(\tau)\Big]\Big\}, \end{aligned}$$

(24b)$$\begin{aligned} \langle(a^{\dagger}a)(0)a^2(\tau)\rangle_{\tau<0}=&\sum_{l=1, 2}\textrm{Tr}\big[E(0, \tau) a^2|\psi^{(l)}_s\rangle\langle\psi^{(l)}_s|\big]\\ =&\sum_{l, l'=1, 2}\langle\psi_s^{(l)}|\mathcal{E}^{(l')}(0, \tau)\rangle\langle\mathcal{E}^{(l')}(0, \tau)|\bar{\bar{\psi}}^{(l)}(0)\rangle\\ =&\langle\sigma_{11}\rangle\Big[{\bar{\mathcal{C}}}^{(1)\ast}_{1}(0){\bar{\bar{\mathcal{C}}}}^{(1)}_{5}(-\tau)+{\bar{\mathcal{C}}}^{(1)\ast}_{2}(0){\bar{\bar{\mathcal{C}}}}^{(1)}_{6}(-\tau)\Big]e^{i\frac{\bar\Omega}{2}(-\tau)}\\ &+\langle\sigma_{22}\rangle\Big[{\bar{\mathcal{C}}}^{(2)\ast}_{1}(0){\bar{\bar{\mathcal{C}}}}^{(2)}_{5}(-\tau)+{\bar{\mathcal{C}}}^{(2)\ast}_{2}(0){\bar{\bar{\mathcal{C}}}}^{(2)}_{6}(-\tau)\Big]e^{i\frac{\bar\Omega}{2}\tau},\end{aligned}$$

respectively. In the above calculations, for $\tau >0$, we have introduced time-dependent single-photon conditional state and two-photon conditional state, which are defined as

(25)$$\begin{aligned}&|\bar\psi^{(l)}(\tau)\rangle\langle\bar\psi^{(l)}(\tau)|=e^{{\mathcal{L}}\tau}\big(a|\psi_s^{(l)}\rangle\langle\psi_s^{(l)}|a^\dagger\big),\\&|\bar{\bar\psi}^{(l)}(\tau)\rangle\langle\bar{\bar\psi}^{(l)}(\tau)|=e^{{\mathcal{L}}\tau}\big(a^2|\psi_s^{(l)}\rangle\langle\psi_s^{(l)}|a^{\dagger2}\big),\end{aligned}$$

respectively, where ${\bar {\mathcal {C}}}^{(l)}_{j}(\tau )$ and $\bar {\bar {\mathcal {C}}}^{(l)}_{j}(\tau )$ are the time-dependent probability amplitudes in $|\bar \psi ^{(l)}(\tau )\rangle$ and $|\bar {\bar \psi }^{(l)}(\tau )\rangle$, respectively. The illustrations of single-photon conditional state and two-photon conditional state can be found in Figs. 7(c) and (d), respectively. In addition, in Eq. (24b), we have introduced the single-photon past state [47]

(26)$$E(0, \tau)=e^{{\mathcal{L}}(0-\tau)}\big(a^{\dagger}a\big)=\sum_{l'=1, 2}|\mathcal{E}^{(l')}(\tau)\rangle\langle\mathcal{E}^{(l')}(\tau)|,$$

conditioned on the single-photon measurement $E(0, 0)= a^{\dagger} a$ at time $t$. In Appendix C, the explicit forms of $|\bar \psi ^{(l)}(\tau )\rangle$, $|\bar {\bar {\psi }}^{(l)}(\tau )\rangle$, and $|\mathcal {E}^{(l')}(\tau )\rangle$ have been given by Eqs. (49), (36), and (57), respectively, where the time-dependent probability amplitudes can be solved from the equations of motion.

Fig. 7. Conditional quantum transitions of the three-photon emission for $\Delta _c=\bar \Omega /3$ revealed by (a) past signal and (b) future signal in two-time correlation $h^{(2)}_{\theta }(\tau )$ and three-time correlation $h^{(2)}_{\theta }(\tau , \tau ')$. The selected states $|{2_A, 1_a}\rangle$ and $|{2_A, 3_a}\rangle$ (blue levels) are separated by two dressing photons with the frequency separation $2\omega _L$. (c)$-$(e) are the illustrations of (c) single-photon conditional quantum state, (d) two-photon conditional quantum state, and (e) cascaded conditional quantum state.

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To illustrate the physical pictures revealed by Eqs. (24a) and (24b), let us concentrate on the case of $\Delta _c=\bar \Omega /3$. The conditional quantum transitions for $\Delta _c=\bar \Omega /3$ are sketched in Figs. 7(a) and (b), where the strongly correlated three-photon emission is predominantly generated from $|{1_A, 0_a}\rangle \to |{2_A, 3_a}\rangle$. The three-photon transition $|{1_A, 0_a}\rangle \to |{2_A, 3_a}\rangle$ contributes to the main value of the intensity-dual quadrature correlation signal $h^{(2)}_{\theta }(\tau )$, as shown in Fig. 8(a), and the following discussions are based on this case.

Fig. 8. (a) Temporal evolution of the total two-time correlation signal $h^{(2)}_{\theta }(\tau )$ (red solid line) and its main component $h^{(2)}_{\theta ,1\to 2}(\tau )$ (blue dashed line). The subscript “1$\to$2” stands for the initial atomic state $|{1_A}\rangle$ and final atomic state $|{2_A}\rangle$, because the dominating three-photon emission is generated from $|{1_A, 0_a}\rangle \to |{2_A, 3_a}\rangle$ for $\Delta _c=\bar \Omega /3$. (b) Temporal evolutions of the components $h^{(2)}_{\theta , 2}(\tau , \tau )$ (black dash-dotted line), $h^{(2)}_{\theta , 3}(\tau , \tau )$ (red solid line), and $h^{(2)}_{\theta , 4}(\tau , \tau )$ (blue dashed line) of the total three-time correlation signal $h^{(2)}_{\theta }(\tau , \tau )$. (c) Symmetrical component (blue dashed line) and asymmetrical component (red solid line) in the three-time correlation component $h^{(2)}_{\theta , 3}(\tau , \tau )$. (d) Temporal evolution of the total three-time correlation function $h^{(2)}_{\theta }(\tau , \tau )$ (red solid line). The temporal asymmetry is predominately contributed from the four-order fluctuation (blue dashed line). The parameters are $\Omega =30$, $\kappa =8$, $\gamma =1$, $\Delta _L=0$, $\Delta _c=\bar \Omega /3$, $\mu =0.125$, $\theta =0$, and $|\alpha |=0.01$ in (b), (c), and (d).

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In physical terms, for $\tau >0$, the original state $\rho _s$ is completely collapsed into $a\rho _s a^{\dagger}$, which means that the future non-Gaussian signal $\langle a^{\dagger}(0) a^2(\tau ) a(0)\rangle _{\tau >0}$ given by Eq. (24a) can be interpreted as the consequence of selecting the two-photon transition $|{2_A, 1_a}\rangle \to |{2_A, 3_a}\rangle$. Therefore the multifold wave-particle correlation signal reveals the quantum coherence between these two collapsed states, separated by the frequency $2\omega _L$ (see blue energy levels in Fig. 7(b)). The selected two-photon transition displays the quantum interference between the cascaded emissions ${|{2_A, 0_a}\rangle }\stackrel {\textrm {R}}{\longrightarrow }{|{2_A, 1_a}\rangle }\stackrel {\textrm {R}}{\longrightarrow }{|{2_A, 2_a}\rangle }$ and ${|{2_A, 0_a}\rangle }\stackrel {\textrm {F}}{\longrightarrow }{|{1_A, 1_a}\rangle }\stackrel {\textrm {T}}{\longrightarrow }{|{2_A, 2_a}\rangle }$. In the selected two-photon channels, on the one hand, the upper collapsed level (the upper blue energy level in Fig. 7(b)) is completely decoupled from the initial dressed state and thus evolves freely, i.e., ${\bar {\mathcal {C}}}^{(1)}_{2}(\tau )={\bar {\mathcal {C}}}^{(1)}_{2}(0)e^{i\frac {\bar \Omega }{2}\tau }$. On the other hand, the evolution of the target collapsed state $|{2_A, 2_a}\rangle$ (the lower blue energy level in Fig. 7(b)), which is collapsed from the original three-photon state $|{2_A, 3_a}\rangle$, is dominated by the two-photon transition dynamics embodied by ${\bar {\mathcal {C}}}^{(1)}_{6}(\tau )$ in Eq. (24a). Thus the quantum coherence between the two-photon state $|2_a\rangle$ and three-photon state $|3_a\rangle$ is revealed by the term ${\bar {\mathcal {C}}}^{(1)\ast }_{2}(\tau ){\bar {\mathcal {C}}}^{(1)}_{6}(\tau )$.

However, for $\tau <0$, the physical information conveyed by Eq. (24b) is different, as illustrated in Fig. 7(a). On the one hand, conditioned on a photon count $E(0, 0)=a^{\dagger}(0)a(0)$, the backward evolution of the dual quadrature $a(\tau )a(\tau )$ is dominated by the term ${\bar {\bar {\mathcal {C}}}}^{(1)}_{6}(-\tau )$ in Eq. (24b), which is calculated from the term $\langle \mathcal {E}^{(2)}(\tau )|\bar {\bar {\psi }}^{(1)}(0)\rangle$ as

(27)$${\bar{\bar{\mathcal{C}}}}^{(1)}_{6}(-\tau)=\langle\bar{\bar\psi}^{(1)}(0)|\mathcal{E}^{(2)}(\tau)\rangle={\bar{\bar{\mathcal{C}}}}^{(1)}_{3}(0){\mathcal{E}^{(2)\ast}_1(\tau)}+{\bar{\bar{\mathcal{C}}}}^{(1)}_{4}(0){\mathcal{E}^{(2)\ast}_2(\tau)}+{\bar{\bar{\mathcal{C}}}}^{(1)}_{6}(0){\mathcal{E}^{(2)\ast}_3(\tau)},$$

where $|\bar {\bar {\psi }}^{(1)}(0)\rangle$ provides the postselection, and the past wave function $|\mathcal {E}^{(2)}(\tau )\rangle$ dominates the backward evolution of the single-photon dynamics from the final dressed state $|2_A\rangle$. On the other hand, the upper level of the selected two-photon transition is still coupled with the uncollapsed state $|{\psi }^{(1)}_s\rangle$. This picture is revealed by the term $\langle {\psi }^{(1)}_s|\mathcal {E}^{(2)}(\tau )\rangle$ in Eq. (24b), which is derived as

(28)$${\bar{\mathcal{C}}}^{(1)\ast}_{2}(0)e^{i\frac{\bar\Omega}{2}(-\tau)}=\langle\psi^{(1)}_s|\mathcal{E}^{(2)}(\tau)\rangle={\mathcal{C}}^{(1)\ast}_{0}{\mathcal{E}^{(2)}_1(\tau)}+{\mathcal{C}}^{(1)\ast}_{2}{\mathcal{E}^{(2)}_3(\tau)}.$$

In this sense, Eq. (24b) suggests that the past information is still coupled with the uncollapsed state, and the past-present correlation is dominated by single-photon transition dynamics.

4.2 Three-time correlation for intensity-quadrature-quadrature

We now proceed to explore the three-time multifold wave-particle correlation function. It refers to a situation in which, conditioned on a photon count at time $t$, two amplitude signals are measured by two balanced homodyne detectors at different times $t+\tau _1$ and $t+\tau _2$, respectively. For simplicity, we assume that the distribution of $t+\tau _1$ and $t+\tau _2$ in the future and in the past are symmetrical with respect to the present time $t$. Therefore, we can denote $\tau =|(t+\tau _1)-t|$ and $\tau '=|(t+\tau _2)-(t+\tau _1)|$. The three-time intensity-quadrature-quadrature correlation function can be defined as

(29)$$h_{\theta}^{(2)}(\tau', \tau)=\frac{{\langle :I(t) E_\theta(t+\tau_1)E_\theta(t+\tau_2):\rangle}}{{\langle: I(t):\rangle}{\langle:E_\theta(t+\tau_1):\rangle}{\langle: E_\theta(t+\tau_2):\rangle}}.$$

However, Eq. (29) cannot be defined for resonant driving because the stationary amplitude ${\langle :E_{\theta }:\rangle }=0$ for $\Delta _L=0$ in our system. By introducing an offset light, the output correlation signal can be redefined in terms of the output field $b=a+\alpha$ as

(30)$$h_{\theta}^{(2)}(\tau', \tau)=\frac{{\langle :\tilde I(t) \tilde E_\theta(t+\tau_1) \tilde E_\theta(t+\tau_2):\rangle}}{{\langle: \tilde I(t):\rangle}{\langle:\tilde E_\theta(t+\tau_1):\rangle}{\langle:\tilde E_\theta(t+\tau_2):\rangle}},$$

with $\tilde I=b^{\dagger}b$ and $\tilde E_{\theta }=(b e^{-i\theta }+ b^{\dagger} e^{i\theta })/2$. If the phase of the offset field, $\phi$, is adjusted to match the phase of the local oscillator, $\theta$, it is straightforward to evaluate the correlation function. For clarity, we shall label the numerator in Eq. (30) as $H_{\theta }^{(2)}(\tau ', \tau )$, i.e., $H_{\theta }^{(2)}(\tau ', \tau )={\langle :\tilde I(t) \tilde E_\theta (t+\tau _1) \tilde E_\theta (t+\tau _2):\rangle }$. However, in order to analyze the non-Gaussian fluctuations, it is beneficial to split the total signal $H_{\theta }^{(2)}(\tau ', \tau )$ into the second-order component $H_{\theta , 2}^{(2)}(\tau ', \tau )$ (with a constant term), the third-order component $H_{\theta , 3}^{(2)}(\tau ', \tau )$, and the fourth-order component $H_{\theta , 4}^{(2)}(\tau ', \tau )$. Thus we can rewrite $H_{\theta }^{(2)}(\tau ', \tau )$ as

(31)$$H_{\theta}^{(2)}(\tau', \tau)=\sum_{i=2,3,4}H_{\theta,i}^{(2)}(\tau', \tau),$$

where the three signal components are derived in terms of the fluctuations of the cavity field as

(32a)$$\begin{aligned} H_{\theta,2}^{(2)}(\tau', \tau)=&\beta^2\Big(\beta^2+{\langle\delta a^{\dagger}\delta a\rangle}\Big)+\beta^2\Big[\langle :\delta E_\theta(\tau_2)\delta E_\theta(\tau_1):\rangle\\ &+2\langle :\delta E_\theta(\tau_1)\delta E_\theta(0):\rangle+2\langle :\delta E_\theta(\tau_2)\delta E_\theta(0):\rangle\Big], \end{aligned}$$

(32b)$$\begin{aligned} H_{\theta,3}^{(2)}(\tau', \tau)=&\beta\Big[2\langle :\delta E_\theta(0)\delta E_\theta(\tau_1)\delta E_\theta(\tau_2):\rangle\\ &+\langle:\delta a^{\dagger}(0)\delta\hat a(0)\delta E_\theta(\tau_1):\rangle+\langle:\delta a^{\dagger}(0)\delta a(0)\delta E_\theta(\tau_2):\rangle\Big],\end{aligned}$$

(32c)$$ H_{\theta,4}^{(2)}(\tau', \tau)=\langle:\delta a^{\dagger}(0)\delta E_\theta(\tau_2)\delta E_\theta(\tau_1)\delta a(0):\rangle.$$

In order to get a better insight into the physical information conveyed by the non-Gaussian fluctuations, let us still consider the case of resonant driving. Therefore each of the fluctuations of the cavity mode can be directly given by the corresponding correlation moment, i.e., $\langle (\delta a^{\dagger})^m(\delta a)^n\rangle =\langle a^{\dagger m}a^n\rangle$. Thus the operator $E_{\theta }$ in Eqs. (32a), (32b), and (32c) is still the quadrature of the cavity mode.

The temporal evolution of the fluctuations of each order in the total three-time correlation signal $h^{(2)}_{\theta }(\tau ', \tau )$ defined by Eq. (30) is shown in Fig. 8(b). As we all known, the time-dependent second-order fluctuation (black dash-dotted line) displays symmetry. However, the third-order component is asymmetrical, even for equal time intervals $(\tau =\tau ')$, due to the occurrence of intensity-quadrature correlation $\langle :I(0) E_{\theta }(\tau ):\rangle$ corresponding to the last two terms in Eq. (32b), as shown in Fig. 8(c). More interesting should be the fourth-order non-Gaussian fluctuation, which is the genuine embodiment of the three-time intensity-quadrature-quadrature correlation of the filtered field. As shown in Fig. 8(d), the fourth-order non-Gaussian fluctuation dominates the asymmetrical behaviour of the total signal. After permuting the time-dependent cavity mode operators, Eq. (32c) can be decomposed into the future and past components, which are expressed, respectively, as

(33a)$$\begin{aligned} H_{\theta, 4}^{(2)}(\tau', &\tau; \tau_2>\tau_1>0)\\ =&\frac{1}{2}\textrm{Re}\big[{\langle\delta a^{\dagger}(0)\delta a(\tau_2)\delta a(\tau_1)\delta a(0)\rangle}e^{-2i\theta}+\langle \delta a^{\dagger}(0)\delta a^{\dagger}(\tau_2)\delta a(\tau_1)\delta a(0)\rangle\big], \end{aligned}$$

(33b)$$\begin{aligned} H_{\theta, 4}^{(2)}(\tau', &\tau; \tau_2<\tau_1<0)\\ =&\frac{1}{2}\textrm{Re}\big[{\langle\delta a^{\dagger}(0)\delta a(0)\delta a(\tau_1)\delta a(\tau_2)\rangle}e^{-2i\theta}+\langle \delta a^{\dagger}(\tau_2)\delta a^{\dagger}(0)\delta\hat a(0)\delta a(\tau_1)\rangle\big].\end{aligned}$$

Retrospecting our purpose of probing into more information of fluorescent emissions from non-Gaussian fluctuations, let us concentrate on the positive-delay signal. As discussed in the above, we still take the case of $\Delta _c=\bar \Omega /3$ as an example to analyze the main physical mechanisms with the help of the conditional probability amplitudes. In the limit of large filter passband, the three-time intensity-quadrature-quadrature correlation in Eq. (33a) with the time ordering $\tau _2>\tau _1>0$ is derived as

(34)$$\langle a^{\dagger}(0) a(\tau_2) a(\tau_1) a(0)\rangle=\sum_{l=1, 2}\langle\bar{\psi}^{(l)}(\tau+\tau')|a|{\psi}_c^{(l)}(\tau', \tau)\rangle,$$

where we have introduced the cascaded conditional state

(35)$$\begin{aligned}|{\psi}_c^{(l)}(\tau', \tau)\rangle\langle{\psi}_c^{(l)}(\tau', \tau)|=&e^{{\mathcal{L}}\tau'}\Big[ a\big[e^{{\mathcal{L}}\tau}[ a\rho_s a^{\dagger}]\big] a^{\dagger}\Big] \\ =&e^{{\mathcal{L}}\tau'}\big[a|\bar{\psi}^{(l)}(\tau)\rangle\langle\bar{\psi}^{(l)}(\tau)|a^{\dagger}\big]. \end{aligned}$$

In Eq. (35), the cascaded conditional wave functions $|{\psi }_c^{(l)}(\tau ', \tau )\rangle$ ($l\in \{1, 2\}$) take the time difference of the two homodyne measurements, $\tau '$, as the variable and the collapsed single-photon conditional state at time $t=t+\tau _1$ as the initial condition, i.e., $|\psi _c^{(l)}(0, \tau )\rangle =a|\bar {\psi }^{(l)}(\tau )\rangle$. The illustration of cascaded conditional state can be found in Fig. 7(e), and the explicit form of $|\psi ^{(l)}_c(\tau ', \tau )\rangle$ is given by Eq. (54) in Appendix C. Substituting the explicit forms of $|\bar {\psi }^{(l)}(\tau )\rangle$ and $|\psi ^{(l)}_c(\tau ', \tau )\rangle$, i.e., Eqs. (49) and (54), into Eq. (34), we can obtain the analytical expression

(36)$$\begin{aligned}\langle a^{\dagger}(0) a(\tau_2)a(\tau_1)a(0)\rangle =&\sum_{l=1, 2}\langle\sigma_{ll}\rangle\big[{\bar{\mathcal{C}}}^{(l)\ast}_{1}(\tau+\tau'){{\mathcal{B}}}^{(l)}_{5}(\tau', \tau) \\ &+{\bar{\mathcal{C}}}^{(l)\ast}_{2}(\tau+\tau'){{\mathcal{B}}}^{(l)}_{6}(\tau', \tau)\big], \end{aligned}$$

where ${{\mathcal {B}}}^{(l)}_{j}(\tau ', \tau )$ ($l\in \{1, 2\}, j\in \{5, 6\}$) are the probability amplitudes of cascaded conditional wave functions. For the case of $\Delta _c=\bar \Omega /3$, the component ${\bar {\mathcal {C}}}^{(1)\ast }_{2}(\tau +\tau '){{\mathcal {B}}}^{(1)}_{6}(\tau ', \tau )$ in Eq. (36) contributes the main value of $\langle a^{\dagger}(0)a(\tau _2)a(\tau _1)a(0)\rangle$ due to the dominant three-photon emission from $|1_A, 0_a\rangle \to |2_A, 3_a\rangle$. In physical terms, the first homodyne detection at time $t+\tau _1$ selects two “$\textrm {R}$” photons generated from $|{2_A, 0_a}\rangle \to |{2_A, 1_a}\rangle$ followed by $|{2_A, 1_a}\rangle \to |{2_A, 2_a}\rangle$ in Fig. 7(b). Then the second homodyne detection at a delayed time $t+\tau _2$ records the information of single-photon dynamics accumulated in the time interval $\tau '=|\tau _2-\tau _1|$, giving rise to the term ${{\mathcal {B}}}^{(1)}_{6}(\tau ', \tau )$. Figure 7(b) tells us that the probability amplitude ${{\mathcal {B}}}^{(1)}_{6}(\tau ', \tau )$ describes the last single-photon transitions of the three-photon cascaded emissions terminated by the final target dressed state $|{2_A}\rangle$, including the processes ${|{1_A}\rangle }\stackrel {\textrm {T}}{\longrightarrow }{|{2_A}\rangle }$ and ${|{2_A}\rangle }\stackrel {\textrm {R}}{\longrightarrow }{|{2_A}\rangle }$. Obviously, in this three-time cascaded detection, the quantum interference effect between the two possible two-photon cascaded emission channels ${|{2_A, 1_a}\rangle }\stackrel {\textrm {R}}{\longrightarrow }{|{2_A, 2_a}\rangle }\stackrel {\textrm {R}}{\longrightarrow }{|{2_A, 3_a}\rangle }$ and ${|{2_A, 1_a}\rangle }\stackrel {\textrm {F}}{\longrightarrow }{|{1_A, 2_a}\rangle }\stackrel {\textrm {T}}{\longrightarrow }{|{2_A, 3_a}\rangle }$ in Fig. 1(b), resolved by the two-time wave-particle correlation function $h^{(2)}_{\theta }(\tau )$, is erased, and the information of a complete three-photon cascaded emission ${|{1_A, 0_a}\rangle }\stackrel {\textrm {T}}{\longrightarrow }{|{2_A, 1_a}\rangle }\stackrel {\textrm {R}}{\longrightarrow }{|{2_A, 2_a}\rangle }\stackrel {\textrm {R}}{\longrightarrow }{|{2_A, 3_a}\rangle }$ is extracted.

4.3 Discussions and comparisons with three-photon intensity correlations

We finish this section with a brief comparison of the physical information conveyed by the multifold wave-particle correlation functions and three-photon intensity correlation functions.

The three-photon two-time intensity correlation of the cavity field refers to a situation in which, conditioned on a photon count at time $t$, a pair of photons is detected at a delayed time $t+\tau$ ($\tau >0$). Thus the three-photon two-time intensity correlation is defined as

(37)$$g^{(3)}(\tau)=\frac{{\langle :I(t) I^2(t+\tau):\rangle}}{{\langle: I(t):\rangle}{\langle:I^2(t+\tau):\rangle}}.$$

With the help of the probability amplitudes, the unnormalized correlation is derived as

(38)$$\begin{aligned}{\langle :I(t) I^2(t+\tau):\rangle}_{\tau>0}=&\sum_{l=1,2}\langle\bar{\psi}^{(l)}(\tau)| a^{\dagger2}a^2|\bar{\psi}^{(l)}(\tau)\rangle \\ =&\sum_{l=1, 2}\langle\sigma_{ll}\rangle\big(\big|\sqrt{2}{\bar{\mathcal{C}}}^{(l)}_{5}(\tau)\big|^2+\big|\sqrt{2}{\bar{\mathcal{C}}}^{(l)}_{6}(\tau)\big|^2\big). \end{aligned}$$

Equation (38) indicates that, when the cavity frequency is tuned to $\Delta _c=\bar \Omega /3$, the single-photon conditional probability amplitude ${\bar {\mathcal {C}}}^{(1)}_{6}(\tau )$ reveals the dominant two-photon transition dynamics. Therefore all the possible two-photon cascaded emission channels from the two-photon state $|2_a\rangle$ to the three-photon state $|3_a\rangle$ are included in the two-photon two-time intensity correlation function, giving rise to the multi-channel quantum interference.

The three-photon three-time intensity correlation of the cavity field, which is defined as

(39)$$g^{(3)}(\tau_2, \tau_1)=\frac{{\langle :I(t)I(t+\tau_1) I(t+\tau_2):\rangle}}{{\langle:I(t):\rangle}{\langle: I(t+\tau_1):\rangle}{\langle:I(t+\tau_2):\rangle}},$$

can be compared correspondingly with the three-time intensity-quadrature-quadrature correlation function. The unnormalized three-time intensity correlation function is derived with the help of the cascaded conditional wave functions as

(40)$$\begin{aligned}{\langle :I(t)I(t+\tau_1)I(t+\tau_2):\rangle}_{\tau_1, \tau_2>0}=&\sum_{l=1,2}\langle{\psi}_c^{(l)}(\tau', \tau)|a^{\dagger}a|{\psi}_c^{(l)}(\tau', \tau)\rangle \\ =&\sum_{l=1, 2}\langle\sigma_{ll}\rangle\big(\big|{{\mathcal{B}}}^{(l)}_{5}(\tau', \tau)\big|^2+\big|{{\mathcal{B}}}^{(l)}_{6}(\tau', \tau)\big|^2\big), \end{aligned}$$

where $\tau '=|(t+\tau _2)-(t+\tau _1)|$ is the time difference and ${{\mathcal {B}}}^{(l)}_{j}(\tau ', \tau )$ ($l\in \{1, 2\}, j\in \{5, 6\}$) are the cascaded conditional probability amplitudes that describe the single-photon dynamics.

From the above two examples, one can see that the three-photon intensity correlations are determined by the populations of the three-photon target states, $\langle {1_A, 3_a}|\rho |{1_A, 3_a}\rangle$ and $\langle {2_A, 3_a}|\rho |{2_A, 3_a}\rangle$, and contain the information about all the possible three-photon cascaded channels related to the target states, i.e., $|{2_A, 3_a}\rangle$ for $\Delta _c=\bar \Omega /3$ and $|{1_A, 3_a}\rangle$ for $\Delta _c=-\bar \Omega /3$. Although each three-photon target state is coupled with several three-photon cascaded channels in our system (see Fig. 1), more information about atomic emission and photon triplet of resonance fluorescence can be resolved by multifold wave-particle correlation functions. Taking the case of $\Delta _c=\bar \Omega /3$ as an example, in terms of the intensity-dual quadrature correlation function, the quantum interference between the cascades ${|{1_A, 0_a}\rangle }\stackrel {\textrm {T}}{\longrightarrow }{|{2_A, 1_a}\rangle }\stackrel {\textrm {R}}{\longrightarrow }{|{2_A, 2_a}\rangle }\stackrel {\textrm {R}}{\longrightarrow }{|{2_A, 3_a}\rangle }$ and ${|{1_A, 0_a}\rangle }\stackrel {\textrm {T}}{\longrightarrow }{|{2_A, 1_a}\rangle }\stackrel {\textrm {F}}{\longrightarrow }{|{1_A, 2_a}\rangle }\stackrel {\textrm {T}}{\longrightarrow }{|{2_A, 3_a}\rangle }$ is erased by the multifold-channel interference by probing into the quantum coherence of photon triplet between $|{1_a}\rangle$ and $|{3_a}\rangle$. More specific information that the single cascaded channel ${|{1_A, 0_a}\rangle }\stackrel {\textrm {T}}{\longrightarrow }{|{2_A, 1_a}\rangle }\stackrel {\textrm {R}}{\longrightarrow }{|{2_A, 2_a}\rangle }\stackrel {\textrm {R}}{\longrightarrow }{|{2_A, 3_a}\rangle }$ is closely related to the wave-particle duality of photon triplet is conveyed by three-time multifold wave-particle correlation function. In this sense, the multifold wave-particle correlation functions provide a new perspective to understand the wave-particle duality of multiphoton emission by probing into the information about “which-path” and the time ordering of cascaded photon emission in single atomic systems.

5. Conclusion

We have investigated the multifold wave-particle quantum correlations in strongly correlated three-photon emission via frequency-engineering and have proposed the intensity-dual quadrature correlation functions, which are distinguished from the conventional three-photon intensity correlation functions. The calculations have been carried out analytically in a three-photon Hilbert space of the quantum filtering system. As the first part of our research, we have proposed the new forms of the criterion of nonclassicality for non-Gaussian light and have found that the effect of non-Gaussian fluctuations play a key role in the nonclassicality of the strongly correlated three-photon emission of the Mollow triplet. Compared with the normally ordered variances, we have demonstrated that the multifold wave-particle correlation detection can produce more prominent nonclassical signals. As the second part of our research, we have extended our investigation to the time-dependent non-Gaussian fluctuations of the filtered field by analyzing the asymmetrical properties of the multifold wave-particle correlation signals. Based on our analytical expressions, the main physical mechanisms can be attributed to the difference between the future transition dynamics and past transition dynamics, which are described by conditional quantum state and past quantum state, respectively. Finally, compared with the three-photon intensity correlation functions, we have found that the multifold wave-particle correlation functions may convey more information. The quantum coherence of photon triplet can be revealed by two-time multifold wave-particle correlation, and the information of “which-path” in cascaded photon emission in atomic systems can be identified by three-time multifold wave-particle correlation.

Appendix A: Equations of motion for the dressed atom-photon correlation moments

Based on the master equation, in this section, we present the general forms of the equations of motion for the dressed atom-photon correlation moments $\langle a^{\dagger m} a^{n}\sigma _{kk'}\rangle$.

After deriving all the equations of motion for the dressed atom-photon correlation moments via Eq. (8), we found that all the correlation moments satisfy the following equation of motion

(41)$$\frac{d}{dt}{\mathbf{X}}^{(m+n)}=-{\mathbf{K}}^{(m,n)}{\mathbf{X}}^{(m+n)}-\xi{\mathbf{D}}^{(m+n-1)},$$

where the matrices of correlation moments are defined as

(42)$$\begin{aligned}&{\mathbf{X}}^{(m+n)}=\Big({\langle a^{\dagger m} a^n\sigma_{12}\rangle},\hspace{0.3cm}{\langle a^{\dagger m}a^n\sigma_{11}\rangle},\hspace{0.3cm}{\langle a^{\dagger m} a^n\sigma_{22}\rangle}\Big)^{\textrm{T}}, \\ &{\mathbf{D}}^{(m+n-1)}=\Big({\mathcal{D}}^{(m+n-1)}_{12},\hspace{0.3cm}{\mathcal{D}}^{(m+n-1)}_{11},\hspace{0.3cm}{\mathcal{D}}^{(m+n-1)}_{22}\Big)^{\textrm{T}}, \end{aligned}$$

and the matrix of dissipation is given by

(43)$${\mathbf{K}}^{(m,n)}= \left( \begin{array}{ccc} {\mathcal{K}}^{(m,n)}_{12} & 0 & 0\\ 0 & {\mathcal{K}}^{(m,n)}_{11} & -\gamma_{22}\\ 0 & -\gamma_{11} & {\mathcal{K}}^{(m,n)}_{22} \end{array} \right).$$

In Eq. (43), the complex dissipation rates are given explicitly by

(44)$${\mathcal{K}}_{ij}^{(m,n)}=(m+n)\frac{\kappa}{2}+\gamma_{ij}-i[(m-n)\Delta_c+\Omega_{ij}],$$

where $\gamma _{11}=\gamma s^4$, $\gamma _{22}=\gamma c^4$, $\gamma _{12}=\frac {\gamma }{2}(1+2c^2s^2)$, $\Omega _{11}=\Omega _{22}=0$, and $\Omega _{12}=\bar \Omega$. The coupling-independent terms ${\mathcal {D}}_{k'k}^{(m+n-1)}$ ($k, k'\in \{1, 2\}$) are given by the following recurrence relations in terms of the lower-order correlation moments with $(m+n-1)$ field operators as

(45)$$\begin{aligned}{\mathcal{D}}_{12}^{(m+n-1)}=&n\Big(cs{\langle a^{\dagger m} a^{n-1}\sigma_{12}\rangle}-s^2{\langle a^{\dagger m} a^{n-1}\sigma_{11}\rangle}\Big)-m\Big(c^2{\langle a^{\dagger m-1} a^{n}\sigma_{22}\rangle}-cs{\langle a^{\dagger m-1} a^{n}\sigma_{12}\rangle}\Big), \\{\mathcal{D}}_{11}^{(m+n-1)}=&n\Big(c^2{\langle a^{\dagger m} a^{n-1}\sigma_{12}\rangle}-cs{\langle a^{\dagger m} a^{n-1}\sigma_{11}\rangle}\Big)-m\Big(c^2{\langle a^{\dagger m-1} a^{n}\sigma_{21}\rangle}-cs{\langle a^{\dagger m-1} a^{n}\sigma_{11}\rangle}\Big), \\ {\mathcal{D}}_{22}^{(m+n-1)}=&n\Big(cs{\langle a^{\dagger m} a^{n-1}\sigma_{22}\rangle}-s^2{\langle a^{\dagger m} a^{n-1}\sigma_{21}\rangle}\Big)-m\Big(cs{\langle a^{\dagger m-1} a^{n}\sigma_{22}\rangle}-s^2{\langle a^{\dagger m-1} a^{n}\sigma_{12}\rangle}\Big). \end{aligned}$$

Therefore the steady-state solution of Eq. (41) is given by

(46)$$ {\mathbf{X}}^{(m+n)}= \begin{cases} -\xi\Big({\mathbf{K}}^{(m,n)}\Big)^{-1}{\mathbf{D}}^{(m+n-1)}, & (m\neq0~~\textrm{or}~~n\neq0),\\ \Big(0,\hspace{0.3cm}\frac{\gamma_{22}}{\gamma_{11}+\gamma_{22}},\hspace{0.3cm}\frac{\gamma_{11}}{\gamma_{11}+\gamma_{22}}\Big)^{\textrm{T}}, & (m=n=0). \end{cases} $$

From this steady-state solution, we calculated the exact results of $g^{(2)}={\langle a^{\dagger 2}a^2\rangle }/{\langle a^{\dagger} a\rangle }^2$ and $g^{(3)}={\langle a^{\dagger 3}a^3\rangle }/{\langle a^{\dagger} a\rangle }^3$ in Fig. 2 and other frequency-resolved correlation functions of interest. Meanwhile, the exact results of time-dependent correlation functions can be also calculated.

Appendix B: Analytical expressions of the steady-state probability amplitudes in Eq. (9)

As mentioned in the above, the steady-state correlation moments enable us to determine the quantum state of the system. When the cavity frequency is tuned to $\Delta _c=\bar \Omega /3$, the three-photon emission is predominantly contributed from the dressed source $\langle \sigma _{11}\rangle$. In this case, the analytical expressions of the probability amplitudes in $|\psi ^{(1)}_s\rangle$ can be derived from the reduced correlation moments ${\langle a^{\dagger m}a^n\sigma _{k'k}\rangle }_1$ and are given by

(47)$$\begin{aligned}&\mathcal{C}^{(1)}_1={\langle a\sigma_{11}\rangle}_{1}=\frac{-\xi cs}{\frac{\kappa}{2}+i\Delta_c},\hspace{0.5cm}\mathcal{C}^{(1)}_2={\langle a^2\sigma_{12}\rangle}_{1}=\frac{-\xi s^2}{\frac{\kappa}{2}+i(\Delta_c-\bar\Omega)}, \\&{\mathcal{C}}^{(1)}_{3}=\frac{1}{\sqrt{2}}{\langle a^2\sigma_{11}\rangle}_{1}=\frac{-\sqrt{2}\xi}{\kappa+2i\Delta_c}\big(-cs\mathcal{C}^{(1)}_1+c^2\mathcal{C}^{(1)}_2\big), \\&{\mathcal{C}}^{(1)}_{4}=\frac{1}{\sqrt{2}}{\langle a^2\sigma_{12}\rangle}_{1}=\frac{-\sqrt{2}\xi}{\kappa+i(2\Delta_c-\bar\Omega)}\big(-s^2\mathcal{C}^{(1)}_1+cs\mathcal{C}^{(1)}_2\big), \\&{\mathcal{C}}^{(1)}_{5}=\frac{1}{\sqrt{3}}{\langle a^3\sigma_{11}\rangle}_{1}=\frac{-\sqrt{3}\xi}{\frac{3}{2}\kappa+3i\Delta_c}\big(-cs\mathcal{C}^{(1)}_3+c^2\mathcal{C}^{(1)}_4\big), \\ &{\mathcal{C}}^{(1)}_{6}=\frac{1}{\sqrt{3}}{\langle a^3\sigma_{12}\rangle}_{1}=\frac{-\sqrt{3}\xi}{\frac{3}{2}\kappa+i(3\Delta_c-\bar\Omega)}\big(-s^2\mathcal{C}^{(1)}_3+cs\mathcal{C}^{(1)}_4\big). \end{aligned}$$

The recurrence relations in Eq. (47) reveal the quantum interference effects between different cascaded paths coupled by a common level, which correspond to the energy level diagrams in Figs. 1(a) and (b). Similarly, the steady-state probability amplitudes in $|\psi ^{(2)}_s\rangle$, triggered by another dressed source $\langle \sigma _{22}\rangle$, can be also derived from ${\langle a^{\dagger m}a^n\sigma _{k'k}\rangle }_2$, and correspond to the energy level diagrams in Figs. 1(c) and (d).

Appendix C: Conditional quantum state and past quantum state

Firstly, we introduce the single-photon conditional state. Considering the measurement for a photon at time $t$ followed by the second detection at time $t+\tau$ ($\tau >0$), as depicted in Fig. 7(c), the stationary density operator $\rho _s$ is collapsed and evolves to $\bar {\rho }(\tau )=e^{{\mathcal {L}}\tau }[ a\rho _s a^{\dagger}]$. In the limit of large filter passband, the single-photon conditional state can be simplified as

(48)$$\bar{\rho}(\tau)=\sum_{l=1,2}\langle\sigma_{ll}\rangle|{\bar\psi^{(l)}(\tau)}\rangle\langle{\bar\psi^{(l)}(\tau)}|,$$

where the single-photon conditional wave functions, $|{\bar \psi ^{(1)}(\tau )}\rangle$ and $|{\bar \psi ^{(2)}(\tau )}\rangle$, have the form

(49)$$\begin{aligned}|{\bar\psi^{(l)}(\tau)}\rangle=&{\bar{\mathcal{C}}}^{(l)}_{1}(\tau)|{1_A, 0_a}\rangle+{\bar{\mathcal{C}}}^{(l)}_{2}(\tau)|{2_A, 0_a}\rangle \\&+{\bar{\mathcal{C}}}^{(l)}_{3}(\tau)|{1_A, 1_a}\rangle+{\bar{\mathcal{C}}}^{(l)}_{4}(\tau)|{2_A, 1_a}\rangle \\ &+{\bar{\mathcal{C}}}^{(l)}_{5}(\tau)|{1_A, 2_a}\rangle+{\bar{\mathcal{C}}}^{(l)}_{6}(\tau)|{2_A, 2_a}\rangle. \end{aligned}$$

The time-evolution of the probability amplitudes of single-photon conditional wave functions is dominated by the following equations of motion

(50)$$\begin{aligned}&\frac{d{\bar{\mathcal{C}}}^{(l)}_{1}(\tau)}{d\tau}=-i\frac{\bar\Omega}{2}{\bar{\mathcal{C}}}^{(l)}_{1},\hspace{0.3cm}\frac{d{\bar{\mathcal{C}}}^{(l)}_{2}(\tau)}{d\tau}=i\frac{\bar\Omega}{2}{\bar{\mathcal{C}}}^{(l)}_{2}, \\&\frac{d{\bar{\mathcal{C}}}^{(l)}_{3}(\tau)}{d\tau}=-\Big[\frac{\kappa}{2}+i\Big(\Delta_c+\frac{\bar\Omega}{2}\Big)\Big]{\bar{\mathcal{C}}}^{(l)}_{3}-\xi\Big(cs{\bar{\mathcal{C}}}^{(l)}_{1}-c^2{\bar{\mathcal{C}}}^{(l)}_{2}\Big), \\&\frac{d{\bar{\mathcal{C}}}^{(l)}_{4}(\tau)}{d\tau}=-\Big[\frac{\kappa}{2}+i\Big(\Delta_c-\frac{\bar\Omega}{2}\Big)\Big]{\bar{\mathcal{C}}}^{(l)}_{4}-\xi\Big(s^2{\bar{\mathcal{C}}}^{(l)}_{1}-cs{\bar{\mathcal{C}}}^{(l)}_{2}\Big), \\&\frac{d{\bar{\mathcal{C}}}^{(l)}_{5}(\tau)}{d\tau}=-\Big[\kappa+i\Big(2\Delta_c+\frac{\bar\Omega}{2}\Big)\Big]{\bar{\mathcal{C}}}^{(l)}_{5}-\sqrt{2}\xi\Big(cs{\bar{\mathcal{C}}}^{(l)}_{3}-c^2{\bar{\mathcal{C}}}^{(l)}_{4}\Big), \\ &\frac{d{\bar{\mathcal{C}}}^{(l)}_{6}(\tau)}{d\tau}=-\Big[\kappa+i\Big(2\Delta_c-\frac{\bar\Omega}{2}\Big)\Big]{\bar{\mathcal{C}}}^{(l)}_{6}-\sqrt{2}\xi\Big(s^2{\bar{\mathcal{C}}}^{(l)}_{3}-cs{\bar{\mathcal{C}}}^{(l)}_{4}\Big). \end{aligned}$$

Then we introduce the two-photon conditional state, as depicted in Fig. 7(d). It is defined as $\bar {\bar {\rho }}(0)=a^2\rho _s a^{\dagger 2}$, and evolves to

(51)$$\bar{\bar{\rho}}(\tau)=\sum_{l=1,2}\langle\sigma_{ll}\rangle|{\bar{\bar\psi}^{(l)}(\tau)}\rangle\langle{\bar{\bar\psi}^{(l)}(\tau)}|$$

in the limit of large filter passband, where the two-photon conditional wave functions, $|{\bar {\bar \psi }^{(1)}(\tau )}\rangle$ and $|{\bar {\bar \psi }^{(2)}(\tau )}\rangle$, have the form

(52)$$|{\bar{\bar\psi}^{(l)}(\tau)}\rangle=\bar{\bar{\mathcal{C}}}^{(l)}_{3}(\tau)|{1_A, 0_a}\rangle+\bar{\bar{\mathcal{C}}}^{(l)}_{4}(\tau)|{2_A, 0_a}\rangle+\bar{\bar{\mathcal{C}}}^{(l)}_{5}(\tau)|{1_A, 1_a}\rangle+\bar{\bar{\mathcal{C}}}^{(l)}_{6}(\tau)|{2_A, 1_a}\rangle.$$

The time-evolution of the two-photon conditional probability amplitudes is also dominated by the first four equations in Eq. (50) with the initial condition $|{\bar {\bar \psi }^{(l)}(\tau )}\rangle =a^2|{\psi }^{(l)}_s\rangle$.

The cascaded conditional state is used to describe the situation in which two measurements are carried out at different times $t$ and $t+\tau _1(\tau _1>0)$, respectively, and the third measurement signal is recorded at time $t+\tau _2$ with $\tau _2>\tau _1$, as depicted in Fig. 7(e). For clarity, we define $\tau =|(t+\tau _1)-t|$ and $\tau '=|(t+\tau _2)-(t+\tau _1)|$. In this three-time detection, the initial state of the cascaded conditional state is prepared at time $t+\tau _1$. Similarly, in the limit of large filter passband, the cascaded conditional state evolves to

(53)$$\rho_c(\tau', \tau)=e^{{\mathcal{L}}\tau'}\Big[ a\big[e^{{\mathcal{L}}\tau}[ a\rho_s a^{\dagger}]\big] a^{\dagger}\Big]=\sum_{l=1,2}\langle\sigma_{ll}\rangle|{\psi_c^{(l)}(\tau',\tau)}\rangle\langle{\psi_c^{(l)}(\tau', \tau)}|,$$

where the cascaded conditional wave functions, $|{\psi _c^{(1)}(\tau ',\tau )}\rangle$ and $|{\psi _c^{(2)}(\tau ',\tau )}\rangle$, have the form

(54)$$\begin{aligned}|{\psi_c^{(l)}(\tau',\tau)}\rangle=&{{\mathcal{B}}}^{(l)}_{3}(\tau', \tau)|{1_A, 0_a}\rangle+{{\mathcal{B}}}^{(l)}_{4}(\tau', \tau)|{2_A, 0_a}\rangle \\ &+{{\mathcal{B}}}^{(l)}_{5}(\tau', \tau)|{1_A, 1_a}\rangle+{{\mathcal{B}}}^{(l)}_{6}(\tau', \tau)|{2_A, 1_a}\rangle. \end{aligned}$$

Similarly, the evolution of the cascaded conditional probability amplitudes is also dominated by the first four equations in Eq. (50) with the initial condition $|{\psi _c^{(l)}(\tau ',\tau )}\rangle =a|\bar {\psi }^{(l)}(\tau )\rangle$.

On the other hand, the signal measured at time $t$ can be correlated to an earlier signal at time $t+\tau$ with $\tau <0$. The past-future correlation is described by the so-called past quantum state [47]. In our system, the past quantum state is related to the last single-photon transitions of three-photon emission, $|{k_A, 2_a}\rangle \to |{k'_A, 3_a}\rangle$ ($k, k'\in \{1, 2\}$), and is defined as a backwardly evolving Hermitian operator $E(0, \tau )=e^{{\mathcal {L}}(0-\tau )}(a^{\dagger}a)$ with $\tau <0$. In general, it is dominated by the master equation

(55)$$\frac{d E}{dt}=\frac{i}{\hbar}{{[{H},E]}}+\sum_{n}\frac{1}{2}\big( 2\mathcal{O}_n^{\dagger}E\mathcal{O}_n-\mathcal{O}_n^{\dagger}\mathcal{O}_n E-E\mathcal{O}_n^{\dagger}\mathcal{O}_n\big),$$

which is bounded by the postselection $E_T$ at a given time $t=T$. In Eq. (55), $dt=(t+dt)-t$ is positive, whereas $d E=E(t-dt)-E(t)$ because of backward evolution. The operator $\mathcal {O}_n$ is responsible for the dissipation of the monitored system . In principle, the past quantum state is a mixed state. However, in our quantum system, the three-photon cascaded emission terminated by two different dressed atomic states can be split into two independent processes when the quantum emitter is resolved by a filter with large passband width. Therefore the past quantum state is simplified as

(56)$$E(0, \tau)=\sum_{l=1,2}|{\mathcal{E}^{(l)}(\tau)}\rangle\langle{\mathcal{E}^{(l)}(\tau)}|.$$

In Eq. (56), the past wave functions $|{\mathcal {E}^{(l)}(\tau )}\rangle$ $(l=1,2)$ take the form

(57)$$\begin{aligned}&|{\mathcal{E}^{(1)}(\tau)}\rangle={\mathcal{E}^{(1)}_1(\tau)}|{1_A, 0_a}\rangle+{\mathcal{E}^{(1)}_2(\tau)}|{2_A, 0_a}\rangle+{\mathcal{E}^{(1)}_3(\tau)}|{1_A, 1_a}\rangle, \\ &|{\mathcal{E}^{(2)}(\tau)}\rangle={\mathcal{E}^{(2)}_1(\tau)}|{1_A, 0_a}\rangle+{\mathcal{E}^{(2)}_2(\tau)}|{2_A, 0_a}\rangle+{\mathcal{E}^{(2)}_3(\tau)}|{2_A, 1_a}\rangle, \end{aligned}$$

where the backwardly evolving amplitudes are given by

(58)$$\begin{aligned}&{\mathcal{E}}^{(1)}_3(\tau)=e^{\eta_+\tau},\hspace{3.9cm}{\mathcal{E}}^{(2)}_3(\tau)=e^{\eta_-\tau}, \\&{\mathcal{E}^{(1)}_1(\tau)}=\frac{-\xi cs}{\frac{\kappa}{2}-i\Delta_c}\big(e^{-i\frac{\bar\Omega}{2}\tau}-e^{\eta_+\tau}\big),\hspace{1.2cm} {\mathcal{E}^{(2)}_2(\tau)}=\frac{\xi cs}{\frac{\kappa}{2}-i\Delta_c}\big(e^{i\frac{\bar\Omega}{2}\tau}-e^{\eta_-\tau}\big), \\ &{\mathcal{E}^{(1)}_2(\tau)}=\frac{\xi c^2}{\frac{\kappa}{2}-i(\Delta_c+\bar\Omega)}\big(e^{i\frac{\bar\Omega}{2}\tau}-e^{\eta_+\tau}\big),\hspace{0.5cm} {\mathcal{E}^{(2)}_1(\tau)}=\frac{-\xi s^2}{\frac{\kappa}{2}-i(\Delta_c-\bar\Omega)}\big(e^{-i\frac{\bar\Omega}{2}\tau}-e^{\eta_-\tau}\big), \end{aligned}$$

with $\eta _{\pm }=\frac {\kappa }{2}-i(\Delta _c\pm \frac {\bar \Omega }{2})$. In the above analytical expressions, the superscript “$(l)$” represents that the past wave function $|{\mathcal {E}^{(l)}(\tau )}\rangle$ describes the last single-photon transitions of three-photon emission terminated by the final dressed atomic state $|{l_A}\rangle$.

Funding

National Natural Science Foundation of China (11774118); Fundamental Research Funds for the Central Universities (CCNU18CXTD01, CCNU19GF003, CCNU19GF005).

Disclosures

The authors declare no conflicts of interest.

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Multifold wave-particle quantum correlations in strongly correlated three-photon emissions from filtered resonance fluorescence

Abstract

1. Introduction

2. Description of the quantum filtering system

2.1 Hamiltonian and master equation

2.2 Main physical mechanisms

3. Nonclassicality of strongly correlated three-photon emission

3.1 Multifold wave-particle quantum correlation related to $g^{(2)}$

3.2 Multifold wave-particle quantum correlation related to $g^{(3)}$

4. Time-dependent multifold wave-particle quantum correlations

4.1 Two-time correlation for intensity-dual quadrature

4.2 Three-time correlation for intensity-quadrature-quadrature

4.3 Discussions and comparisons with three-photon intensity correlations

5. Conclusion

Appendix A: Equations of motion for the dressed atom-photon correlation moments

Appendix B: Analytical expressions of the steady-state probability amplitudes in Eq. (9)

Appendix C: Conditional quantum state and past quantum state

Funding

Disclosures

References

Cited By

Figures (8)

Equations (63)

Optics Express