Unveiling wave−particle duality via second-order photon correlations

Abstract

Wave-particle duality as a fundamental tenet of quantum mechanics is crucial for advancing comprehension of quantum theories and developing quantum technologies with practical applications. However, taking into account experimental impact factors to develop a feasible measurement for wave-like and particle-like properties of light fields is an ongoing challenge, and the non-classicality extraction and determination remains to be explored. In this work, feasibly measurable second-order photon correlations based on Hanbury Brown−Twiss and Hong−Ou−Mandel interferences are employed to analyze the evolution of wave−particle duality for various input states. The wave-particle dualities of chaotic, coherent and mixed classical states as functions of time delay and coherence time are investigated. The realistic impacts of background noise, detection efficiency, intensity ratio and phase differences on the wave−particle duality of non-classical (Fock and squeezed coherent) states are unveiled. In noisy backgrounds with low detection efficiencies, efficient enhancement and extraction of non-classicality and a continuous transition from classical to non-classical region are achieved in single photon state mixed with coherent state by adjusting the phase difference from 0 to \pi /2. The non-classicality of squeezed coherent state can be induced by the classical wave-like and particle-like properties. The research provides a practical precision measurement of wave−particle duality that is helpful for the improvement of high-resolution quantum imaging and sensing.

Graphical abstract

Keywords

second-order photon correlation / wave−particle duality / non-classicality / single photon detection

1 Introduction

Wave−particle duality (WPD), central to quantum mechanics, embodies a fundamental quantum feature that elucidates intrinsic wave-like and particle-like properties [1]. The duality in single photons is rooted in Bohr’s complementarity principle [2] and has been empirically quantified and scrutinized via Young’s double-slit experiment [3-6]. As quantum interference techniques develop rapidly, numerous innovative strategies for characterizing photon WPD have emerged. Multifaceted explorations encompass two-photon interference [7, 8], N00N state complementarity [9], quantum entanglement [10-12], composite system interference [13-16], coherence measurements [17, 18], anti-correlation effects [19, 20], delayed-choice experiments [21-23], nanostructure manipulation of light particle properties [24], Boson sampling [25], and real-time single-molecule imaging via quantum interference [26]. Previous researches mainly focused on single-quantum interferometric scenarios [27], however, the contemporary priorities have shifted towards the concurrent examination and quantification of both the particle-like and wave-like attributes. The wave−particle duality of different light fields beyond single photon states and associated with higher-order correlations remains to be explored.

High-order photon correlation has been an essential tool for characterizing and assessing quantum properties of optical fields [28-31], and holds significant potential for advancing applications in quantum information and quantum imaging [32, 33]. The pioneering experiment of photon correlation was first conducted by Hanbury Brown & Twiss (HBT) in 1956 [34], which is a significant milestone in the development of quantum optics. Recently, research endeavors encompass a spectrum of advanced methodologies to probe various quantum conundrums, including second-order spatial and temporal correlation to ghost imaging [35, 36], optical communication [37], extract X-ray photon rotational diffusion coefficients [38], imaging of X-ray fluorescence photons [39], spatial interference [40], continuous adjustment of high-order coherence [41], and high-speed photon correlation measurement based on deep-learning balanced homodyne detection [42]. The HBT experiment underscores the particle-like attributes of photons by examining the intensity correlations indicative of photon bunching. Moreover, Hong−Ou−Mandel (HOM) experiment is seminal in illustrating the wave-like characteristics of photons through the interference of indistinguishable quantum paths [43]. The deployment of photon correlation-based interference experiments not only enhances the understanding of light statistics, but also can stand at the vanguard of providing an insightful and nuanced perspective on the interplay between the discrete particle-like and continuous wave-like attributes of light. Indeed, a practical measurement scheme for the WPD and non-classicality identification under experimental impact factors by integrating the two interferometric techniques is a key challenge.

In this work, we investigate the WPD evolution for classical and non-classical input states using second-order photon correlations, which combine the HBT and HOM interferometric techniques. The variations of the WPD for chaotic, coherent, and mixed classical states with time delay and coherence time are studied. We also elucidate the practical effects of background noise, detector efficiency, and phase difference on the WPD of non-classical states, such as Fock and squeezed coherent states. The WPD enables the detection of non-classicality beyond the scope of standard HBT and HOM observations. Notably, in noisy environments with low detector efficiency, we achieve non-classicality enhancement and extraction of single photon state mixed with coherent state by controlling the phase difference from 0 to \pi / 2, which facilitates a seamless transition from classical to non-classical regimes. The work offers a pragmatic approach to the precise measurement of WPD, with implications for the applications of advancing quantum metrology.

2 Theoretical model

We investigate the WPD across different states, encompassing both classical (mixed state of chaotic and coherent lights) and non-classical (single photon state, single photon state mixed with coherent state, Fock states, and squeezed coherent state) regimes within single-path and dual-path interference scenarios. A feasible scheme for the WPD measurement is shown in Fig.1, which employs single-path HBT and dual-path HOM interferometric techniques. Fig.1(a) illustrates the layout for generating a chaotic state. A current source (CS) manages the bias current, and a temperature controller (TC) maintains a stable output of light field from a distributed feedback laser diode (DFB-LD). The output sequentially passes through a polarization controller (PC) and an optical circulator (OC). Then an 80:20 fiber coupler (FC) directs a portion of the light to a variable optical attenuator (VOA), which feeds back into the circulator to establish a feedback loop. By adjusting the feedback strength, the output can be prepared for chaotic state, coherent state, and a mixture of chaotic and coherent state. Fig.1(b) shows a schematic for generating a squeezed coherent state. A dichroic mirror (DM) directs two wavelength lights into an optical parametric oscillator (OPO). Fig.1(c) depicts a single photon state. Fig.1(I) illustrates a dual-path interference system, where the input lights from paths a and b are combined at the beam splitter (BS). The dual-path photon correlation measurements are performed by two single photon detectors (SPDs). Assuming equal detection efficiency for both the SPDs, the detection outcome is proportional to the mean photon number of the output light for each optical path. It involves both the intensity and phase correlations of the inputs from the two optical paths.

The visibility of interference is a metric used to quantify the wave-like property of the input state. We introduce a measure to characterize the wave-like information W. A critical property of W is its ability to distinctly partition the wave-like detection into two components: the dual-path and single-path photon correlations. The W is given by

(1)\begin{array}{rl}W=& \frac{2⟨n_{ab}^2⟩}{⟨n_{aa}^2⟩+⟨n_{bb}^2⟩}\\ =& \frac{2⟨n_{ab}^2⟩}{⟨n_{aa}⟩⟨n_{bb}⟩}{\left(\epsilon \frac{⟨n_{aa}^2⟩}{{⟨n_{aa}⟩}^2}+{\epsilon }^{-1}\frac{⟨n_{bb}^2⟩}{{⟨n_{bb}⟩}^2}\right)}^{-1},\end{array}

where \epsilon =\frac{⟨n_{aa}⟩}{⟨n_{bb}⟩} indicates dual-path intensity ratio, and ⟨n_{ab}^2⟩ represents the photon correlation of the two input paths a and b, satisfying ⟨n_{ab}^2⟩=⟨{\alpha }_a^{†}{\alpha }_b^{†}{\alpha }_b{\alpha }_a⟩. ⟨n_{aa}^2⟩ or ⟨n_{bb}^2⟩ is the photon correlation of the single input path a or b. {\alpha }_i^{†} and {\alpha }_i are the creation and annihilation operators for the input i = a or b [44]. The wave-like information W effectively delineates the contributions of the single-path and double-path correlations to the coincidence probability. The path information P, indicative of particle-like property, is defined as the normalized correlation difference within a single-path input interference setup, as shown in Fig.1(II) of HBT interference. The W and P are advantageous for examining the wave-particle duality of light within the context of intensity interference,

(2)\begin{array}{rl}P=& \frac{\left|⟨n_{aa}^2⟩-⟨n_{bb}^2⟩\right|}{⟨n_{aa}^2⟩+⟨n_{bb}^2⟩}=\frac{\left|{⟨n_{aa}⟩}^2{\displaystyle \frac{⟨n_{aa}^2⟩}{{⟨n_{aa}⟩}^2}}-{⟨n_{bb}⟩}^2{\displaystyle \frac{⟨n_{bb}^2⟩}{{⟨n_{bb}⟩}^2}}\right|}{{⟨n_{aa}⟩}^2{\displaystyle \frac{⟨n_{aa}^2⟩}{{⟨n_{aa}⟩}^2}}+{⟨n_{bb}⟩}^2{\displaystyle \frac{⟨n_{bb}^2⟩}{{⟨n_{bb}⟩}^2}}}\\ =& \frac{\left|\epsilon {\displaystyle \frac{⟨n_{aa}^2⟩}{{⟨n_{aa}⟩}^2}}-{\epsilon }^{-1}{\displaystyle \frac{⟨n_{bb}^2⟩}{{⟨n_{bb}⟩}^2}}\right|}{\epsilon {\displaystyle \frac{⟨n_{aa}^2⟩}{{⟨n_{aa}⟩}^2}}+{\epsilon }^{-1}{\displaystyle \frac{⟨n_{bb}^2⟩}{{⟨n_{bb}⟩}^2}}}.\end{array}

In quantum theory, according to the commutation relation \left[a_i,a_i^{†}\right]=1 of bosonic photons, ⟨:n_i^2:⟩\ne ⟨n_i^2⟩ can be obtained [45], where i = a, b. Therefore the inequality {\left(⟨I_{ab}^2⟩\right)}^2>⟨ I_{aa}^2⟩⟨I_{bb}^2⟩ holds, which ensures the superposition relationship between particle-like and wave-like properties can be derived as follows:

(3)\begin{array}{rl}WPD=& W^2+P^2=1+4\frac{{⟨n_{ab}^2⟩}^2-⟨n_{aa}^2⟩⟨n_{bb}^2⟩}{{\left(⟨n_{aa}^2⟩+⟨n_{bb}^2⟩\right)}^2}\\ =& 1+4\frac{{\displaystyle \frac{{⟨n_{ab}^2⟩}^2-⟨n_{aa}^2⟩⟨n_{bb}^2⟩}{{⟨n_{aa}⟩}^2{⟨n_{bb}⟩}^2}}}{\left(\epsilon {\displaystyle \frac{⟨n_{aa}^2⟩}{{⟨n_{aa}⟩}^2}}+{\epsilon }^{-1}{\displaystyle \frac{⟨n_{bb}^2⟩}{{⟨n_{bb}⟩}^2}}\right)}>1.\end{array}

In classical wave theory, the superposition relationship does not satisfy the above equation, and WPD\le 1 should be obeyed. Consequently, we can analyze the characteristics of non-classical and classical light fields through the above WPD superposition relationship.

In order to facilitate the analysis of the superposition relationship of the WPD, we introduce the single-path normalized second-order correlation functions g_{hbt1}^{(2)} and g_{hbt2}^{(2)}, and the dual-path normalized second-order correlation function g_{hom}^{(2)}. The single-path second-order photon correlation is expressed as g_{hbt}^{(2)}=\frac{⟨n_{ii}^2⟩}{⟨n_{ii}⟩⟨n_{ii}⟩}, which can be measured by the HBT measurement, as shown in Fig.1(II). The dual-path normalized second-order photon correlation is expressed as g_{hom}^{(2)}=\frac{⟨n_{ab}^2⟩}{⟨n_{aa}⟩⟨n_{bb}⟩}, which can be measured by the HOM interference in Fig.1(I). Then the above interference effect of Eq. (1) can be rewritten as

(4)W={\displaystyle \frac{2g_{hom}^{(2)}}{\epsilon g_{hbt1}^{(2)}+{\epsilon }^{-1}{g}_{hbt2}^{(2)}}},

and the path information can be rewritten as

(5)P= {\displaystyle \frac{\left|\epsilon g_{hbt1}^{(2)}-{\epsilon }^{-1}{g}_{hbt2}^{(2)}\right|}{\epsilon g_{hbt1}^{(2)}+{\epsilon }^{-1}{g}_{hbt2}^{(2)}}}.

And the WPD can be expressed as

(6)WPD=W^2+P^2=1 +4{\displaystyle \frac{g{{}_{hom}^{(2)}}^2-g_{hbt1}^{(2)}{g}_{hbt2}^{(2)}}{{\left(\epsilon g_{hbt1}^{(2)}+{\epsilon }^{-1}{g}_{hbt2}^{(2)}\right)}^2}}.

The single-path HBT interference system in Fig.1(\mathrm{\Pi }) exists delay times between the two detectors, so the single-path second-order correlation can be derived as

(7)g_{hbt}^{(2)}\left({\tau }_1\right)= {\displaystyle \frac{⟨E_{ii}\left(t\right)E_{ii}\left(t+{\tau }_1\right)⟩}{⟨E_{ii}\left(t\right)⟩⟨E_{ii}\left(t\right)⟩}}={\displaystyle \frac{⟨n_{ii}(t)n_{ii}(t+{\tau }_1)⟩}{⟨n_{ii}(t)⟩⟨n_{ii}(t)⟩}}.

In the dual-path HOM interference of Fig.1(\mathrm{I}), E_{ii}^{\ast }(t) and E_{ii}( t) represent the components of the electric field at the output port. The ⟨\cdot ⟩ denotes the mean value of input states. ⟨E_{ii}^{\ast }(t)E_{ii}^{\ast }(t+{\tau }_2)E_{ii}(t+{\tau }_2)E_{ii}(t)⟩ corresponds to the second-order correlation between the outputs at ports of the beam splitter. Additionally, ⟨E_{ii}^{\ast }(t)E_{ii}(t+ {\tau }_2)⟩ represents the first-order correlation of the outputs. Thus, the bimodal normalized second-order correlation can be formulated as follows:

(8)\begin{array}{rl}{g}_{hom}^{(2)}({\tau }_2)=& \frac{⟨E_{ii}^{\ast }(t)E_{ii}^{\ast }(t+{\tau }_2)E_{ii}(t+{\tau }_2)E_{ii}(t)⟩}{{⟨E_{ii}(t)⟩}^2{⟨E_{ii}(t+{\tau }_2)⟩}^2}\\ =& \frac{⟨n_{ii}\left(t\right)n_{ii}\left(t+{\tau }_2\right)⟩}{⟨n_{ii}\left(t\right)⟩⟨n_{ii}\left(t+{\tau }_2\right)⟩},\end{array}

where {\tau }_2 represents the time difference between the photons arriving at the two output ports in the interference. For the same mode, the positive frequency part of the electric field is replaced by the output creation operators {\hat{c}}^{†} and {\hat{d}}^{†}, and the negative frequency part is replaced by the annihilation operators \hat{c} and \hat{d}. Assuming a lossless beam splitter with a balanced splitting ratio for the analysis, we have \hat{c}=\frac{\sqrt{2}}{2}\left(\hat{b}+\hat{a}\right), \hat{d}=\frac{\sqrt{2}}{2}\left(\hat{b}-\hat{a}\right), where {\hat{a}}^{†}(\hat{a}), {\hat{b}}^{†}(\hat{b} ) represent the creation and annihilation operators at the input ports, respectively. The dual-path input state is expressed as |{\psi }_1,{\psi }_2⟩, and Eq. (8) can be given by

(9)g_{hom}^{(2)}({\psi }_1,{\psi }_2)={\displaystyle \frac{⟨{\psi }_2,{\psi }_1\left|{\hat{c}}^{†}{\hat{d}}^{†}\hat{d}\hat{c}\right|{\psi }_1,{\psi }_2⟩}{⟨{\psi }_2,{\psi }_1\left|{\hat{c}}^{†}\hat{c}\right|{\psi }_1,{\psi }_2⟩⟨{\psi }_2,{\psi }_1\left|{\hat{d}}^{†}\hat{d}\right|{\psi }_1,{\psi }_2⟩}}.

We can find that the second-order correlations are affected by the difference impact factors of the input states, like phase, delay time and other experimental influences. The impacts are verified by analyzing various light fields.

3 Wave-particle duality of classical and non-classical states

3.1 Classical state: Chaotic and coherent states

We begin by examining the wave−particle duality of a chaotic state in a single- and double-path interference system using the aforementioned single-path and dual-path normalized second-order correlation functions. The chaotic electromagnetic field can be described as follows:

(10)E^{\ast }\left(t\right)=E_0^{\ast }{\displaystyle \frac{{\mathrm{e}}^{-\mathrm{i}\omega t}}{\sqrt{N}}}{\displaystyle \sum _n^N{\mathrm{e}}^{\mathrm{i}{\phi }_n(t)},}

where E_0^{\ast } denotes the input vacuum electromagnetic field, \omega is the field frequency, and {\phi }_n is the random phase. Introducing Eq. (10) input into the single-path HBT interference system depicted in Fig.1(\mathrm{\Pi }), the single-path normalized second-order correlation of chaotic state is expressed as follows:

(11)g_{hbt\mathrm{\_}cha}^{(2)}\left({\tau }_1\right)={\displaystyle \frac{⟨n_{ii}(t)n_{ii}(t+{\tau }_1)⟩}{{⟨n_{ii}(t)⟩}^2}}=1+ {\mathrm{e}}^{\frac{-2\left|{\tau }_1\right|}{{\tau }_c}}.

Then, the dual-path normalized second-order correlation of the corresponding state can be obtained by using the HOM interference. The center frequency is \omega and the coherence time is {\tau }_c. The delay path difference between the unbalanced two paths in the HOM interference is d, so the delay time is denoted by d/c. According to Eq. (8), we can obtain

(12)\begin{array}{rl}{g}_{hom}^{(2)}({\tau }_2)=& \frac{1}{4}\{2g_{hbt}^{(2)}({\tau }_2)+g_{hbt}^{(2)}\left(\frac{d}{c}+{\tau }_2\right)+g_{hbt}^{(2)}\left(\frac{d}{c}-{\tau }_2\right)\\ & -2\mathrm{Re}\left[g_{hbt}^{(2)}\left(\frac{d}{c},{\tau }_2,\frac{d}{c}+{\tau }_2,0\right)\right]\}.\end{array}

The dynamic evolution from a coherent to a chaotic state in the light field is reflected by the real part of the double path normalized second-order correlation function, denoted as \mathrm{Re}\left[g_{hbt}^{(2)}(\frac{d}{c},{\tau }_2,\frac{d}{c}+{\tau }_2,0)\right]. Consequently, for chaotic optical inputs in a dual-path interference setup, the bimodal normalized second-order correlation function is derived as follows:

(13)g_{hom\mathrm{\_}cha}^{(2)}({\tau }_2)=1+{\displaystyle \frac{1}{4}}{\mathrm{e}}^{\frac{-2\left|\frac{d}{c}-{\tau }_2\right|}{{\tau }_c}}+{\displaystyle \frac{1}{4}}{\mathrm{e}}^{\frac{-2\left|\frac{d}{c}+{\tau }_2\right|}{{\tau }_c}}-{\displaystyle \frac{1}{2}}{\mathrm{e}}^{\frac{-2\left|d\right|}{c{\tau }_c}}.

In Eq. (13), {\tau }_c represents the coherence time. When \epsilon = 1 in the interference systems of dual optical paths, the WPD values of chaotic state corresponds to the variations observed in the two delay times, as shown in Fig.2. In such a scenario, the path information is 0, and the wave-like information along with the WPD is as follows:

(14)\begin{array}{rl}WP{D}_{cha}=& {\left(\frac{2g_{hom\mathrm{\_}cha}^{(2)}}{g_{hbt1\mathrm{\_}cha}^{(2)}+g_{hbt2\mathrm{\_}cha}^{(2)}}\right)}^2\\ =& {\left(\frac{1+{\displaystyle \frac{1}{4}}{\mathrm{e}}^{\frac{-2\left|\frac{d}{c}-{\tau }_2\right|}{{\tau }_c}}+{\displaystyle \frac{1}{4}}{\mathrm{e}}^{\frac{-2\left|\frac{d}{c}+{\tau }_2\right|}{{\tau }_c}}-{\displaystyle \frac{1}{2}}{\mathrm{e}}^{\frac{-2\left|d\right|}{c{\tau }_c}}}{1+{\mathrm{e}}^{\frac{-2\left|{\tau }_1\right|}{{\tau }_c}}}\right)}^2.\end{array}

In order to reveal the wave-like property, we present the square root of the WPD as functions of two delay times and coherence time in the following figures. Fig.2(a) indicates that for the incident chaotic state in the zero-delay HOM interferometer, the WP{D}_{cha}^{1/2} achieves a minimum of 0.5 at {\tau }_1=0\mathrm{n}\mathrm{s}. Notably, the half-height width of the peak broadens with increasing coherence time in the vicinity of zero delay, while the square root of the WPD diminishes as the coherence time {\tau }_c grows. Fig.2(b) illustrates that for the incident chaotic state in the zero-delay HBT interferometer, the WP D_{cha}^{1/2} increases with the increasement of the coherence time {\tau }_c and the reduction of the delay time {\tau }_2, attaining its maximum value of 0.5 at {\tau }_2= 0\mathrm{n}\mathrm{s}. Fig.2(c) indicates that for the coherence time {\tau }_c= 1\mathrm{n}\mathrm{s} and the dual-path intensity ratio \epsilon = 1, the WP{D}_{cha}^{1/2} of the chaotic state intensifies as the HBT interference delay time {\tau }_1 increases and the HOM interference delay time {\tau }_2 decreases, achieving the maximum of 0.952. The WPD of the chaotic state is found to peak at zero delay, enabling pronounced observable changes. From these observations, it is evident that chaotic states consistently reside in the classical region (WPD\le 1), and the WPD measurements allow for differentiating between various chaotic states and evaluating chaotic dynamics evolution.

When the input light field is a coherent state, the electromagnetic field can be written as E^{\ast }\left(t\right)=E_1^{\ast } {\mathrm{e}}^{-\mathrm{i}\omega t+\mathrm{i}\phi (t)}. Then the single-path normalized second-order correlation function of coherent state can be obtained as g_{hbt\mathrm{\_}coh}^{(2)}({\tau }_1)=1. In the dual-path balanced HOM interference, the second-order correlation function of coherent state is

(15)g_{hom\mathrm{\_}coh}^{(2)}({\tau }_2)=1-{\displaystyle \frac{1}{2}}{\mathrm{e}}^{\frac{-2m}{{\tau }_c}},

where m = min(d/c, {\tau }_2), Due to the balance between the two paths, the d equals to 0, which implies m = 0. In this case the dual-path HOM second-order correlation of coherent state maintains a constant value of 0.5. By substituting Eq. (17) into Eqs. (4)−(6), we can derive the WPD of the coherent state upon the dual-path HOM interference as follows:

(16)\begin{array}{rl}WP{D}_{coh}=& 1+4\frac{{\left(g_{hom\mathrm{\_}coh}^{(2)}\right)}^2-g_{hbt1\mathrm{\_}coh}^{(2)}{g}_{hbt2\mathrm{\_}coh}^{(2)}}{{\left(\epsilon g_{hbt1\mathrm{\_}coh}^{(2)}+{{\epsilon }^{-1}}_{hbt2\mathrm{\_}coh}^{(2)}\right)}^2}\\ =& 1+4\frac{{\left(g_{hom\mathrm{\_}coh}^{(2)}\right)}^2-1}{{\left(\epsilon +{\epsilon }^{-1}\right)}^2}.\end{array}

According to the analysis, it should be noted that the WP{D}_{coh} varies with the \epsilon, and attains the minimum value of coherent state when \epsilon =1. The observation also confirms that the WPD of coherent state belongs to the classical region.

Then we investigate the WPD of non-classical states based on the single-path and dual-path interference systems. For the input non-classical states, the influences of detection efficiency and background noise precipitate discrepancies between experimental and theoretical results, obscuring the true manifestation of the WPD. Afterwards we perform a thorough examination of the WPD across various non-classical states, tailored to the experimental conditions.

3.2 Single photon state

When two single photon states are incident on a dual-path HOM interferometer, the reference single photon can be represented as: |{\psi }_r⟩ =\int {\psi }_r(p)|1_p⟩\mathrm{d}p [46]. Owing to the difference between the incident single photon states for actual experiment conditions, and two incident single photon states can be expressed as: |{\psi }_{1,2}⟩=\int {\psi }_r(p{)}^{{\psi }_{1,2}(p)}/{\psi }_r(p )|1_p⟩= \int {\psi }_r (p)F\left(p_{1,2}\right){\mathrm{e}}^{\mathrm{i}\left[\varphi \left(p_{1,2}\right)+{\varphi }_n\right]}|1_p⟩\mathrm{d}p, where F\left(p\right) and \varphi \left(p\right) respectively represent the amplitude and phase difference between the actual incident and reference single photon states. {\varphi }_n represents the random phase noise between two optical paths. Substituting the expressions |{\psi }_1⟩and |{\psi }_2⟩ into Eq. (9), the normalized second-order correlation function for the single photon state is obtained as follows:

(17)\begin{array}{rl}{g}_{hom\mathrm{\_}n1}^{(2)}=& \frac{⟨{\psi }_2,{\psi }_1\left|{\hat{c}}^{†}{\hat{d}}^{†}\hat{d}\hat{c}\right|{\psi }_1,{\psi }_2⟩}{⟨{\psi }_2,{\psi }_1\left|{\hat{c}}^{†}\hat{c}\right|{\psi }_1,{\psi }_2⟩⟨{\psi }_2,{\psi }_1\left|{\hat{d}}^{†}\hat{d}\right|{\psi }_1,{\psi }_2⟩}\\ =& \frac{F^2(p_1)+F^2(p_2)-2F(p_1)F(p_2)\cos {\varphi }_{ab}}{[F^2(p_1)+1][F^2(p_2)+1]}.\end{array}

Eq. (17) reveals that the normalized dual-path second-order photon correlation effectively mitigates the impacts of reference state and phase noise. If the HOM incident single photon states only exist phase difference {\varphi }_{ab}, denoted by F\left(p\right)=1, the second-order photon correlation can be obtained as g_{hom\mathrm{\_}n1}^{(2)}=1/2-(\cos {\varphi }_{ab})/2. If the dual-path incident single photon states are indistinguishable, the g_{hom\mathrm{\_}n1}^{(2)}=0 can be obtained. A HBT detection scheme is introduced in which the total detection efficiency \eta (including the optical transmission and optical collection efficiency of the system) is modeled by a lossy beam splitter and the background noise of the system is represented by a weak random field [47]. The incident state is with a photon number distribution of P_{in}(n). After the incident state passes through the lossy beam splitter, its photon number distribution changes to

(18)P_{tr}\left(m\right) ={\displaystyle \sum _{n=m}^{\mathrm{\infty }}{P}_{in}\left(n\right){\displaystyle \frac{n!{\eta }^m{\left(1-\eta \right)}^{\left(n-m\right)}}{m!\left(n-m\right)!}}.}

The random background noise is represented by a weak random field |\beta ⟩, whose photon number distribution follows the Poisson distribution: P_{in}\left(n\right)={\gamma }^n{\mathrm{e}}^{-\gamma }/n!. Then, after the incident state is mixed with the background noise, the photon number distribution is

(19)P_{mix}\left(L\right)={\displaystyle \sum _{m=0}^L{\displaystyle \frac{{\gamma }^{\left(L-m\right)}}{\left(L-m\right)!}}{\mathrm{e}}^{-\gamma }{P}_{tr}\left(m\right),}

where \gamma represents background noise and L is the number of photons reaching the beam splitter. The second order normalized correlation function of HBT interference considering system background noise and detection efficiency is obtained,

(20)g_{hbt}^{\left(2\right)}={\displaystyle \frac{⟨n_1{n}_2⟩}{⟨n_1⟩⟨n_2⟩}}={\displaystyle \frac{{\displaystyle \sum _{n_1{n}_2}\left(n_1{n}_2\right)P\left(n_1,n_2\right)}}{{\displaystyle \frac{1}{4}}{\left[{\displaystyle \sum _{n}nP\left(n\right)}\right]}^2}}.

By substituting the photon number distribution of single photon Fock state into Eq. (20), the HBT normalized second-order photon correlation can be derived with taking into account the actual detection efficiency and background noise:

(21)g_{hbt\mathrm{\_}n1}^{(2)}={\displaystyle \frac{8\left({\mathrm{e}}^{\gamma /4}-1\right)\left(2{\mathrm{e}}^{\gamma /4}-2-\eta \right)}{{\left(4{\mathrm{e}}^{\gamma /4}-4+\eta \right)}^2}},

where \eta is the detection efficiency and \gamma is the background noise. By combining the particle-like and wave-like properties, we can get the WPD variations in single photon states with different incident differences:

(22)\begin{array}{rl}WP{D}_{n1}=& 1+4\frac{{\left(g_{hom\mathrm{\_}n1}^{(2)}\right)}^2-g_{hbt1\mathrm{\_}n1}^{(2)}{g}_{hbt2\mathrm{\_}n1}^{(2)}}{{\left(\epsilon g_{hbt1\mathrm{\_}n1}^{(2)}+{\epsilon }^{-1}{g}_{hbt2\mathrm{\_}n1}^{(2)}\right)}^2}\\ =& {\left(\frac{g_{hom\mathrm{\_}n1}^{(2)}}{g_{hbt\mathrm{\_}n1}^{(2)}}\right)}^2.\end{array}

Fig.3(a) illustrates that for \eta =0.5, the WPD of the single photon state increases as the phase difference increases and the background noise decreases. Fig.3(b) indicates that when \gamma =0.02, the WPD of the single photon state increases with the increase of the phase difference and the detection efficiency, observing a transition from a classical (WPD\le 1) to a non-classical (WPD>1) region. Fig.3(c) and (d) further illustrate the impacts of \eta and \gamma on the WPD of single photon state. In Fig.3(c), it is observed that as the phase difference {\varphi }_{ab} increases, the WPD of the single photon state transitions from a classical to a non-classical region at {\varphi }_{ab}=0.18\pi , 0.28\pi and 0.38\pi, corresponding to \gamma =0.02,0.05 and 0.1, respectively. As depicted in Fig.3(d), the WPD of the single photon state for \eta =1 and 0.5 appears a transition from the classical to the non-classical domain at the critical phase differences of {\varphi }_{ab}=0.2\pi and 0.28\pi. For \eta =0.1, the single photon state is always in the classical region and it indicates a lower limit of the impact of detection efficiency on the WPD measurement. It is also found that the WPD non-classicality of the single-photon state attains its maximum at the phase difference of \pi /2.

3.3 Incident single photon state and coherent state

When two input states of the HOM interference is a single photon state and a coherent state, an output superposition state is formed by the interference. Then we examine the WPD of the input states. The expression of single photon state is: |{\psi }_1⟩=\int {\psi }_1(p)|1_p⟩\mathrm{d}p, and the coherent state is | {\psi }_2⟩=\prod _p|{\beta }_2\left(p\right)⟩. We assume the single photon state as the reference state. The single photon state and coherent state can be characterized by |{\psi }_1⟩ =\int {\psi }_1(p)|1_p⟩ \mathrm{d}p=\int {\psi }_r(p){\psi }_1(p)/{\psi }_r(p)|1_p⟩ \mathrm{d}p=\int {\psi }_r(p)F\left(p_1\right){\mathrm{e}}^{\mathrm{i}[\varphi \left(p_1\right)+{\varphi }_n]}|1_p⟩\mathrm{d}p, |{\psi }_2⟩ =\prod _p|{\beta }_2\left(p\right)⟩=\prod _p|{\psi }_r\left(p\right)F\left(p_2\right){\mathrm{e}}^{\mathrm{i}\left[\varphi \left(p_2\right)+{\varphi }_n\right]}⟩. Substituting the expressions into Eq. (9), the normalized second-order correlation function for the single-photon state and coherent state is obtained as follows:

(23)\begin{array}{rl}{g}_{hom\mathrm{\_}1c}^{(2)}=& \frac{⟨{\psi }_2,{\psi }_1\left|{\hat{c}}^{†}{\hat{d}}^{†}\hat{d}\hat{c}\right|{\psi }_1,{\psi }_2⟩}{⟨{\psi }_2,{\psi }_1\left|{\hat{c}}^{†}\hat{c}\right|{\psi }_1,{\psi }_2⟩⟨{\psi }_2,{\psi }_1\left|{\hat{d}}^{†}\hat{d}\right|{\psi }_1,{\psi }_2⟩}\\ =& 1-\frac{2F(p_1)F(p_2)\cos {\varphi }_{ab}+1}{[F^2(p_1)+1][F^2(p_2)+1]}.\end{array}

Eq. (23) indicates that the normalized dual-path second-order photon correlation effectively mitigates the impacts of phase noise. If the HOM incident states only have a phase difference, denoted by F\left(p\right)=1, the second-order photon correlation can be obtained as g_{hom\mathrm{\_}1c}^{(2)}=3/4-(\cos {\varphi }_{ab})/2. If the dual-path incident states are minimally indistinguishable, the g_{hom\mathrm{\_}1c}^{(2)}=0.25 can be obtained. If the HOM incident states only exist amplitude difference, denoted by F(p)= \epsilon, the second-order photon correlation can be obtained as g_{hom\mathrm{\_}1c}^{(2)}={\left|\epsilon \right|}^4/(1+{\left|\epsilon \right|}^2{)}^2. By combining the particle-like and wave-like properties, we derive the WPD when single-photon state and coherent state are incident,

(24)\begin{array}{rl}WP{D}_{1c}=& 1+4\frac{{\left(g_{hom\mathrm{\_}1c}^{(2)}\right)}^2-g_{hbt\mathrm{\_}coh}^{(2)}{g}_{hbt\mathrm{\_}n1}^{(2)}}{{\left(\epsilon g_{hbt\mathrm{\_}coh}^{(2)}+{\epsilon }^{-1}{g}_{hbt\mathrm{\_}n1}^{(2)}\right)}^2}\\ =& 1+4\frac{{\left(g_{hom\mathrm{\_}1c}^{(2)}\right)}^2-g_{hbt\mathrm{\_}n1}^{(2)}}{{\left(\epsilon +{\epsilon }^{-1}{g}_{hbt\mathrm{\_}n1}^{(2)}\right)}^2}.\end{array}

Fig.4(a) shows that when \gamma =0.02, the WPD of single photon state mixed with coherent state increases as the phase difference and the detection efficiency increase, observing a transition from a classical (WPD\le 1) to a non-classical (WPD>1) region. Fig.4(b) indicates that when \eta =1, the WPD of the input states increases with an increase of the phase difference and a decrease of the background noise, observing a transition from a classical (WPD\le 1) to a non-classical (WPD>1) region. Fig.4(c) and (d) further illustrate the wave-like property W, particle-like P and WPD of single photon state mixed with coherent state versus phase difference {\varphi }_{ab} and dual-path intensity ratio \epsilon. In Fig.4(c), it is observed that as the phase difference {\varphi }_{ab} increases, the P remains unchanged and W increases accordingly, and the WP{D}_{1c} gradually increases. Fig.4(d) indicates that as the \epsilon increases, the WP{D}_{1c} peaks and then falls to 1. Due to the different incident light fields of the two paths, the peak values of the WPD and the W exhibit distinct shift. Fig.4(d) also shows that for single photon state mixed with coherent state, the non-classicality of the WPD is predominantly influenced by the wave-like property W.

3.4 Fock states

Based on the above analysis, it is evident that the results of dual-path interference are influenced by the factors of phase difference, time delay, detection efficiency and background noise. To facilitate subsequent analysis, this section primarily derives the WPD of different Fock states in the single-path HBT interference and dual-path HOM correlation g_{hom}^{(2)}=0.9. This implies that the incident photons are partially distinguishable, facilitating the observation of the transition of Fock states from classical to non-classical region.

Owing to the weak light intensity of Fock state, they are more susceptible to the impacts of detection efficiency and background noise in practical detection process. When Fock states are incident upon a HBT detection system that accounts for background noise and detection efficiency [47, 48], the second-order photon correlation can be obtained as follows:

(25)g_{hbt\mathrm{\_}n}^{(2)}={\displaystyle \frac{2^{2n+1}\left[2^{2n-1}{\mathrm{e}}^{\frac{\gamma }{2}}+{\left(-1\right)}^{n+1}\left(X\right)\right]}{{\left[4^n{\mathrm{e}}^{\frac{\gamma }{4}}+{\left(-1\right)}^{n+1}{\left(\eta -4\right)}^n\right]}^2}} ,

where X={\mathrm{e}}^{\gamma /4}(\eta - 4{)}^n-2^{n-1}(\eta -2{)}^n. By combining the particle-like and wave-like properties, we can derive the WPD of Fock states:

(26)WP{D}_n=1+{\displaystyle \frac{{\left(g_{hom\mathrm{\_}n}^{(2)}\right)}^2-{\left(g_{hbt\mathrm{\_}n}^{(2)}\right)}^2}{{\left(g_{hbt\mathrm{\_}n}^{(2)}\right)}^2}}={\displaystyle \frac{{\left(g_{hom\mathrm{\_}n}^{(2)}\right)}^2}{{\left(g_{hbt\mathrm{\_}n}^{(2)}\right)}^2}}.

In Fig.5(a)−(c), the classical and non-classical regions are delineated by white dashed lines. As the Fock number increases, the non-classical region of the Fock state gradually contracts. In Fig.5(d)−(f), the blue solid lines, red dashed lines, and yellow dash-dot lines represent the WPD of Fock states (n = 3, 4, 5) as a function of detection efficiency for the background noise \gamma = 0,0.02, and 0.2. The results indicate that when g_{hom\mathrm{\_}n}^{(2)}=0.9 and \eta are constant, the increase of \gamma affects and leads to a gradual transition of the Fock states WPD from the non-classical region to the classical region. When the background noise is zero, the WPD of Fock states decreases as the detection efficiency \eta increases. When the \gamma is non-zero, as the \eta gradually increases, the WPD of Fock states starts from its minimum value, rapidly increases to a peak, and then slowly decreases. This is because when the \eta is low and the \gamma is high, the photon number distribution of the output states follows a Poisson distribution, which implies an increased proportion of the background noise, thus the acquired WPD tends towards that of coherent state. It can be seen from Fig.5 that the non-classical properties of the Fock states gradually decrease as the photon number increases.

3.5 Squeezed coherent state

Squeezed coherent state is an important quantum state and has been widely applied in quantum precision measurement and quantum information [41, 49-52]. We investigate the WPD of squeezed coherent state and the state characteristics can be manipulated by adjusting displacement parameter \alpha, squeezing phase \theta, and squeezing parameter r. When the squeezed coherent state is incident on the single-path HBT interferometer, the corresponding second-order photon correlation is obtained as

(27)\begin{array}{rl}& g_{hbt\mathrm{\_}scs}^{(2)}=\\ & 1-{\displaystyle \frac{\mathrm{\Omega }\cosh r\sinh r-2{\left|\alpha \right|}^2{\sinh }^{2}r-\cosh 2r{\sinh }^{2}r}{{\left({\left|\alpha \right|}^2+{\sinh }^{2}r\right)}^2}},\end{array}

where \mathrm{\Omega }=\left({\alpha }^{\ast 2}{\mathrm{e}}^{\mathrm{i}\theta }+{\alpha }^2{\mathrm{e}}^{-\mathrm{i}\theta }\right). When the squeezed coherent state is incident upon a two-path HOM interferometer Fig.1(I), we investigate the impacts of the r, \alpha, and \theta on the WPD. Specifically, we analyze the scenarios where the two incident lights are in fully and partially distinguishable modes, corresponding to g_{hom\mathrm{\_}scs}^{(2)}=1 and 0.5 [53]. The WPD is

(28)WP{D}_{scs}=1+4{\displaystyle \frac{{\left(g_{hom\mathrm{\_}scs}^{(2)}\right)}^2-{\left(g_{hbt\mathrm{\_}scs}^{(2)}\right)}^2}{{\left(g_{hbt\mathrm{\_}scs}^{(2)}\right)}^2{\left(\epsilon +{\epsilon }^{-1}\right)}^2}}.

Fig.6(a) and (c) respectively show the WPD of squeezed coherent state varies with squeezing and displacement parameters when the g_{hom\mathrm{\_}scs}^{(2)}=1 and 0.5. It can be observed that as the squeezing parameter r increases from 0 to 1, the WPD of the squeezed coherent state increases to a maximum peak and then decreases. With the displacement parameter \alpha increasing from 1.5 to 2.2, the WPD of the squeezed coherent state decreases, indicating a gradual weakening of its non-classicality. Additionally, it demonstrates that the non-classical region of the WPD of the squeezed coherent state decreases as the g_{hom\mathrm{\_}scs}^{(2)} increases. Fig.6(b) and (d) are divided into S1 and S2 regions by two vertical black dashed lines. In the part of the S1 region [Fig.6(b), and ; Fig.6(d), and ], the non-classicality of the WPD arises from the combined effects of classical wave-like and classical particle-like properties. In contrast, the non-classicality of the WPD in the S2 region arises from the combined effect of classical particle-like and non-classical wave-like properties, representing an experimentally realizable non-classical property. Additionally, it is noted that the peaks of wave-like property W and the WPD occur at the same abscissa \epsilon = 1, due to the equal inputs in both paths of the HOM interferometer.

The above results of Fig.6 are based on the analysis under ideal conditions. However, in the actual measurement system, non-ideal experimental factors such as losses generated during light transmission and the performance of the detectors themselves can lead to the overall detection efficiency less than 1. Then we analyze the WPD of squeezed coherent state based on experimental conditions. The squeezed coherent state can be prepared by the setup depicted in Fig.1(b), which is then input into single- and dual-path interferometers for detection. We investigate the WPD of the squeezed coherent state for different detection efficiencies and background noise levels. The photon number distribution of the squeezed coherent state is expressed as follows:

(29)\begin{array}{rl}{P}_{scs}\left(n\right)=& \frac{1}{n!u}{\left|\frac{v}{2u}\right|}^n{\left|H{}_n\left(\frac{\alpha }{\sqrt{2uv}}\right)\right|}^2\\ & \times \exp \left(-{\left|\alpha \right|}^2+\frac{v}{2u}{\alpha }^2+\frac{v^{\ast }}{2u}{\alpha }^{\ast 2}\right),\end{array}

where u=\cosh r, v={\mathrm{e}}^{\mathrm{i}\theta }\sinh r, and H_n\left(x\right)={\left(-1\right)}^n{\mathrm{e}}^{x^2}\frac{{\mathrm{d}}^n}{\mathrm{d}{x}^n}{\mathrm{e}}^{-x^2} is the n-th order Hermitian polynomial. Taking into account the detector efficiency \eta and the background noise \gamma, we derive the photon number distribution as

(30)\begin{array}{rl}{P}_{det}=& \sum _{m=0}^L\frac{{\gamma }^{L-m}{\eta }^m}{m!\left(L-m\right)!}\sum _{m=0}^{\mathrm{\infty }}\frac{2^{-n}{\left(1-\eta \right)}^{n-m}{\coth }^{-n}{\left|\mathrm{\Gamma }\right|}^2}{\left(n-m\right)!}\\ & \times {\mathrm{e}}^{-\gamma }\gamma {\mathrm{e}}^{\left(\cos \theta \tanh r-1\right){\alpha }^2}\mathrm{sech}r,\end{array}

where \mathrm{\Gamma }=\alpha {\mathrm{e}}^{-\mathrm{i}\theta /2}{\left(\mathrm{csch}2r\right)}^{1/2}. Then we substitute the photon number distribution of the squeezed coherent state into Eq. (20) to obtain the single-path HBT second-order photon correlation. To facilitate high-precision simulation and observation of the influences on the WPD, we omit the cumbersome formula derivations, setting \epsilon = 1 and g_{hom\mathrm{\_}scs}^{(2)}=1.

Fig.7(a) indicates that as the detection efficiency \eta increases, the WPD non-classicality of the squeezed coherent state becomes increasingly evident. With an increase of the background noise \gamma, the WPD non-classicality of the squeezed coherent state gradually weakens. To analyze the WPD in accordance with the actual detection situation, several representative results are shown in Fig.7(b) and (c). For the detection efficiency \eta = 0.1, 0.5, and 1, the WPD non-classicality gradually decreases with the increase of background noise. Fig.7(c) illustrates that the WPD increases with the improvement of detection efficiency under different background noise \gamma = 0.0001, 0.0005, and 0.001. It is worth noting that for low background noise levels, specifically when , the WPD of the squeezed coherent state consistently resides within the non-classical region.

4 Discussion

Based on the HBT and HOM interferometric measurements, we can not only investigate various classical and non-classical states, but also form superposition states by different light fields entering through the two optical paths resulting in steerable non-classical states. When the g_{hom}^{(2)}=1, the g_{hbt}^{(2)} for different input states and its impact on the observed WPD is shown in Fig.8. The typical detection efficiency of commercial single photon detectors ranges from 0.1 to 0.7, and the overall background noise in realistic experiment environment is above {10}^{-8}. In the above feasible experimental conditions and g_{hom}^{(2)}=1, we obtain the measurable WPD for different input states. In Fig.8(a) the dashed line with WPD = 1 divides the map into the classical region S0 and the non-classical regions S1 and S2. The region S2 corresponds to the scenario where both input paths of the HOM interferometer are incident with non-classical states, and the region S1 indicates the case where one path is incident with a classical state and the other with a non-classical state. Fig.8(b) shows the effect of \epsilon on the WPD of the different states when the background noise is 0.02 and the detection efficiency is 0.5. The blue dashed curve indicates that the WPD of squeezed coherent state transitions from the non-classical to the classical region by controlling the \epsilon, when the squeezed parameter r=0.2, the displacement parameter \alpha = 2.5, and the squeezing phase \theta = 0. Moreover, by adjusting the \epsilon, it facilitates a WPD transition of single photon state mixed with coherent state from classical to non-classical regimes, with the WPD reaching its maximum non-classical value at \epsilon =2.

Additionally, the WPD non-classicality can be extracted by controlling the phase difference. When the background noise \gamma =0.02, 0.05, 0.1 and the detection efficiency \eta =0.5, the WPD non-classicality of single photon state can be realized by controlling the phase difference in the ranges of 0.18\pi−0.5\pi, 0.28\pi−0.5\pi, and 0.38\pi−0.5\pi, respectively. When the background noise is 0.02 and the detection efficiency is 0.5, WPD non-classicality of single photon state mixed coherent state can be realized by controlling the phase difference in the range of 0.1\pi−0.5\pi. Even with a large background noise (\gamma = 0.02), the WPD non-classicality of Fock states (n = 3, 4, 5) can be observed in a detection efficiency range of 0.01−1, 0.01−0.84, and 0.01−0.54, respectively. The maximum WPD values of 1.1655, 1.2948, 1.5685 for the Fock states (n = 3, 4, 5) are acquired at detection efficiencies of 0.1, 0.12, 0.16, respectively. When the background noise is less than {10}^{-3} and the dual-path interferometric incidence is fully distinguishable for the squeezed coherent state (g_{hom}^{(2)}=1), the WPD non-classicality of the squeezed coherent state can be observed consistently within the effective range of detection efficiency from 0.1 to 0.7. We can realize the enhancement and effective extraction of the non-classicality for various states. In the presence of low detection efficiency and high background noise.

5 Conclusion

We investigate the WPD across various classical and non-classical states, including a mixture of chaotic and coherent state, single photon state, single photon state mixed with coherent state, Fock states, and squeezed coherent state in both single-path HBT and dual-path HOM interferometric scenarios. The impacts of time delay and coherence time on the WPD of chaotic and coherent states are explored. We also elucidate the practical influences of background noise, detector efficiency, and phase difference on the WPD of non-classical states. When the phase difference {\varphi }_{ab} is adjusted from 0 to \pi/2, the WPD of single photon state undergoes a continuous transition from classical (WPD\le 1) regime to non-classical (WPD>1) regime, enabling efficient extraction of the non-classicality of single photon state. Then we verify it with feasible experimental parameters. For the detection efficiency \eta = 0.5 and \gamma =0.02, the non-classical WPD of single photon state can be extracted (WPD>1) when 0.5\pi >{\varphi }_{ab}>0.28\pi. When single photon state and coherent state are incident on two optical paths of the HOM interferometer, a superposition non-classical state can be prepared. The results indicate that the non-classical property (WPD>1) can be controlled by adjusting the {\varphi }_{ab} and the dual-path intensity ratio \epsilon, and improving the detection efficiency \eta. The WPD non-classicalities of the single photon and coherent states are also verified with feasible experimental parameters, i.e., \eta = 0.5, \gamma =0.02 and 0.5\pi >{\varphi }_{ab}>0.1\pi. The realistic impacts of background noise, detection efficiency and phase difference on the WPD of non-classical (Fock and squeezed coherent) states are unveiled. It is certified that by controlling the squeezing parameter from 0 to 1 and the displacement parameter from 1.5 to 2.2, the WPD of squeezed coherent state increases with the growth of their HOM interference g_{hom}^{(2)}, corresponding to a more pronounced non-classicality of the squeezed coherent state. This research contributes to the applications of high-precision optical measurement and quantum coherence extraction.

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