1. Introduction

Wave-particle duality of a quantum object is the heart of quantum mechanics [1]. The complementarity of the duality of a single photon [2], which stemmed from Bohr’s complementarity principle [3], has been quantitatively studied in double-slit interferometric scenarios [4–7]. The primary objective of the analysis is to investigate the relationship between visibility $V$ of the interference between photons coming from the two arms as the wave-like property and the so-called which-path information $D$ of the photons from the two arms as the particle-like property. There is a constraint between them known as the complementary relation as $D^2 + V^2\leq 1$. Previous studies have investigated the complementary nature of photons from various viewpoints, such as delayed-choice experiments [8–10], interference experiments of partially incoherent light [11] or in composite systems [12–15], an experimental test of the duality in asymmetric beam interference [16], and theoretical works related to the uncertainty principle [17–19], quantum coherence [20], and entanglement [21]. More recently, studies have expanded the concept of the wave-particle duality to higher-order interferometric scenarios, in which the input light includes two or more photons [22,23]. Despite having a long history, to the best of our knowledge, all previous studies have been related to phase correlation of optical states, which reason would be why the main focus in the first place is the investigation of the mysterious quantum nature of a photon, including the superposition principle.

Apart from the interpretations of the quantum physics, the complementarities just describe the relationship between the quantities about autocorrelation and cross-correlation functions, which is not unique to a single photon but universally applied to all light fields including infinite number of photons such as laser and thermal lights, as seen in Refs. [24–26]. For example, the complementarity in the Young’s interferometer is about the first-order correlation functions. In the present day, the significance of higher-order correlation functions are broadly known in terms of both interference and statistical distribution of photons [27,28]. In particular, second-order intensity correlations exhibit distinctive quantum features and play integral roles in modern photonic quantum information technologies. The photon statistics related to the particle-like property is characterized by the second-order autocorrelation function measured using the Hanbury-Brown and Twiss (HBT) setup [29]. For the wave-like property, different from the situation in the Young’s interferometer, the intensity correlation measurement after mixing lights coming from two arms involves different origins of the second-order interferometric effects: not only phase correlation related to the superposition property [22,30] but also intensity correlation related to the bosonic nature of photons known as the Hong-Ou-Mandel (HOM) interference [31].

In this study, we show the wave-particle duality of light in the context of an intensity interferometric scenario based on intensity correlation measurement, particularly for the second-order intensity interference. To this end, we use a higher-order which-path information related to autocorrelation functions introduced in Ref. [22] as the particle-like property, and newly introduce a quantity related to an intensity cross-correlation function which characterizes the magnitude of higher-order intensity interferometric effect as the wave-like property. The two quantities are observed by switching the measurement apparatus solely based on the intensity correlation measurement. For the second-order intensity interference, the quantities are measured by switching the HBT setup and the HOM interferometer without changing the input setting. We show that within the framework of the classical wave theory, the two quantities obey the complementary principle analogous to those of the dualities appearing in the phase-based interferometric scenarios, but considering the quantum effect, the complementarity can be violated. It reveals that compared with the classical light, the nonclassical light such as single photons at two arms is allowed to show larger which-path information and intensity interference simultaneously. Moreover, the violation can be achieved even if the light has classically describable which-path information and interference visibility. The observation discovers a new nonclassical nature of light which is never found from either wave-like or particle-like property alone.

We remark, throughout this paper including the above paragraphs, we refer quantities related to autocorrelation functions as particle-like property and refer both phase and intensity interferometric effects related to cross-correlation functions as wave-like property as a matter of convenience. The notions of particle and wave are just borrowed from prequantum physics, as is cautioned in Ref. [7]. All results derived in this paper can be understood regardless of the interpretations.

2. Wave-particle duality in the interference by phase correlation

We review the complementary relation of the wave-particle duality in the interferometric scenario based on the phase coherence. The first-order interference is represented by the double-slit experiment [5–7]. The setup equivalent to the double-slit interferometer is presented in Fig. 1(a). In the interferometer, two input lights coming from modes A and B are mixed at a half beamsplitter (HBS) and measured using detector ${\rm {D}}_{\rm {C}}$ or ${\rm {D}}_{\rm {D}}$. The detection probability $P_{\rm {C(D)}}$ at ${\rm {D}}_{\rm {C(D)}}$ is proportional to the average photon number (intensity) of light. By omitting the detection efficiency, $P_{\rm {C}}(=\langle N_{\rm {C}} \rangle )$ is described by

 

Fig. 1. Experimental setups and corresponding measurement results: (a) for first-order interference based on intensity measurement by a detector, such as the double-slit or the Mach–Zehnder interferometer and (b) for second-order interference based on the intensity correlation measurement using two detectors, such as the HOM interferometer. The result in (b) is for an input light having no phase correlation.

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The which-path information $D_1$ is defined by the magnitude of difference between the detection probabilities at ${\rm {D}}_{\rm {C}}$ when only half of the light coming from the two input ports enters, which is $|G^{(1)}_{\rm {AA}} - G^{(1)}_{\rm {BB}}|/2$, with normalization by their total amount as

Meanwhile, visibility $V_1^{({\rm {ph}})}$ is defined by using maximal and minimal probabilities of $P_{\rm {C}}$ for $\theta _1=\pi$ and $\theta _1=0$ in Eq. (1), respectively denoted by $P_{\rm {max}}$ and $P_{\rm {min}}$, as

From Eqs. (2) and (3), we obtain the complementary relation between $D_1$ and $V_1^{({\rm {ph}})}$ [4–7], which is

The last inequality is obtained as follows: For $O_n := \sum _{i={\rm {A}},{\rm {B}}}\lambda _ia_i^n$ with complex values of $\lambda _i$, where $a_i^n$ means the $n$-th power of $a_i$, ${\rm tr}(\rho O_n^\dagger O_n) = \sum _{i,j={\rm {A}},{\rm {B}}}\lambda _i^*\langle a_i^{\dagger n}a_j^n \rangle \lambda _j \geq 0$ is satisfied. In other words, the $2\times 2$ matrix $\{ \langle a_i^{\dagger n}a_j^n \rangle \}$ is positive. Consequently, due to the positivity of its determinant as

The complementarity was generalized to the $n$th-order interference [22], in which the which-path information $D_n$ and the fringe visibility $V_n^{({\rm {ph}})}$ are defined by

The quantities are obtained by $n$-fold coincidence photon detections without single-count measurements. From Eq. (6), the quantities satisfy the complementary relation as

3. Wave-particle duality in the second-order intensity interference

Here, we consider the intensity interferometric situation, which is the primary topic of this paper. Unlike the double-slit experiment, the higher-order interference based on the intensity correlation measurement includes the effects of not only the phase correlation but also the intensity correlation between the two input modes. In this section, we consider the second-order interferometric experiment with an interferometer, as shown in Fig. 1(b). The setup is based on the intensity correlation (coincidence) measurement using detectors ${\rm {D}}_{\rm {C}}$ and ${\rm {D}}_{\rm {D}}$. The coincidence probability measured by the interferometer is proportional to $\left \langle : N_{\rm {C}}N_{\rm {D}} :\right \rangle$ with the use of the normal ordering. It reflects the intensity correlation $G^{(2)}_{\rm {AB}}:=\langle a_{\rm {A}}^{\dagger }a_{\rm {B}}^{\dagger } a_{\rm {B}}a_{\rm {A}} \rangle =\left \langle : N_{\rm {A}}N_{\rm {B}} :\right \rangle$ and the phase correlation $G^{{\rm {ph}}(2)}_{\rm {AB}}$ of the input light. In fact, when modes of the photons coming from modes A and B are perfectly indistinguishable after the HBS, the coincidence probability ${\tilde {P}_\parallel }$ is described by

In the HOM interferometric situation, we consider the which-path information $D_2$ in Eq. (7) as the particle-like property. It is measured by blocking one of the input ports and dividing the light coming from the other input port into two output ports by the HBS followed by the coincidence measurement, corresponding to the HBT setup [29]. $D_2$ is comprehended as the probability of guessing which direction the light came from using $P_\parallel$. The physical meaning is the same as $D_1$, except for the difference between the single-count and coincidence measurements. Notably, which-path information is occasionally called distinguishability. However, the term is confusing because the distinguishability is conventionally used to characterize the mode-matching degree of the photons after mixing at the HBS.

For quantifying the magnitude of the interference, an interference visibility $V_{\rm {HOM}}:= (P_\perp - P_\parallel )/P_\perp$ has been commonly used, known as the HOM visibility [35–37]. However, we introduce a different quantity $V_2$ for describing the interference contrast as the wave-like signature. It is defined by the contrast ratio of depth $G^{(2)}_{\rm {AB}}/2$ of the HOM dip to the coincidence probability in which effects of the cross-correlation functions are not included, as

Before discussing the relationship between the quantities $D_2$ and $V_2$, let us explain the motivation and physical meaning of the newly introduced quantity $V_2$. For this, we first revisit the meaning of $V_1^{({\rm {ph}})}$ in Eq. (3). As seen in the interference pattern of Fig. 1(a), contrast of the interference fringe is characterized by ratio of the maximum $P_{\rm {max}}$ and minimum $P_{\rm {min}}$. Instead of using the ratio directly, the definition of $V_1^{({\rm {ph}})}$ in Eq. (3) takes sum and difference of the two quantities. This manner gives intuitively and visually clear understanding of the visibility in the interference pattern, i.e. it represents the ratio of oscillation amplitude to the middle of the fringe. But more important property brought by the construction of $V_1^{({\rm {ph}})}$ in our view is it distinctly separates the detection probability into two components caused by cross-correlation function $G^{{\rm {ph}}(1)}_{\rm {AB}}$, which is the exact origin of the interference as the wave-like signature, and by autocorrelation functions $G^{{\rm {ph}}(1)}_{\rm {AA}}$ and $G^{{\rm {ph}}(1)}_{\rm {BB}}$ with no interferometric effect. The physical meanings hold in $V_n^{({\rm {ph}})}$ defined by Eq. (8) for general cases $n\geq 2$ of phase-based interferometric scenarios.

In the case of the HOM interference considered in this paper, the HOM visibility $V_{\rm {HOM}}$ inherits the visual concept of $V_n^{({\rm {ph}})}$, which shows the relative depth of the HOM dip to the baseline observed in the HOM experiment, as shown in Fig. 1(b). In contrast, however, the definition of $V_2$ in Eq. (13) corresponds to another aspect of $V_n^{({\rm {ph}})}$ as follows: $V_2$ separates the amount of the intensity interferometric effect determined by $G^{(2)}_{\rm {AB}}$ and the coincidence probability caused by $G^{(2)}_{\rm {AA}}$ and $G^{(2)}_{\rm {BB}}$ without any effect of cross correlations. Unfortunately, different from the cases of $V_n^{({\rm {ph}})}$, the two quantities $V_{\rm {HOM}}$ and $V_2$ based on the different concepts are not equal because the baseline in the HOM experiment includes the contributions of both autocorrelation and cross-correlation functions. But nevertheless, $V_2$ completely reflects the interference contrast to show the wave-like feature of light as well as $V_{\rm {HOM}}$ because their difference is just a normalization. In fact, the two quantities $V_{\rm {HOM}}$ and $V_2$ are equivalent since $V_{\rm {HOM}} = (1 + V_2^{-1})^{-1}$ is satisfied and is a monotonically increasing function. In addition, $V_2$ is still visually understood as the contrast ratio between the interference magnitude and amount remaining due to the autocorrelation functions as shown in Fig. 1(b). The formulation of $V_2$ which separates the effects of cross-correlation and autocorrelation functions on the coincidence probability is convenient to discuss the wave-particle property of light in the intensity interferometric scenario.

In the following, we show the relationship between the two quantities $D_2$ and $V_2$. From Eqs. (7) and (13), we define $X_2$ by

In the classical wave theory, because

However, in the quantum theory, the commutation relation $[a_i, a_i^\dagger ] = 1$ for $i=$ A, B of bosonic photons leads to $\left \langle : N_i^2 :\right \rangle \neq \langle N_i^2 \rangle$. Consequently, different from the complementarity of the duality appearing in the phase interference as $X_{n}^{({\rm {ph}})} \leq 1$ for all the light, $X_2 > 1$ is not prohibited in the quantum theory, as shown in Fig. 2. In other words, compared with any light satisfying Eq. (16), the nonclassical light is allowed to have strong particle-like and wave-like signatures simultaneously in the context of the second-order intensity interference.

 

Fig. 2. Relationship among $D_2$, $V_2$ and $X_2$. Only inside or on the unit circle is allowed in the classical wave theory. The filled area shows the nonclassical region of $X_2 > 1$ whereas the classical explained values of $D_2\leq 1$ and $V_2 \leq 1$.

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It is plausible to guess that the nonclassical nature $X_2 > 1$ always originates from the nonclassical value of $V_2 > 1$. This is because in the classical wave theory, $V_2$ is guaranteed to be $0\leq V_2 \leq 1$ from $V_2^{-2}\geq G^{(2)}_{\rm {AA}}G^{(2)}_{\rm {BB}}/(G^{(2)}_{\rm {AB}})^2$ but takes $0\leq V_2 \leq \infty$ in the quantum theory, while $0\leq D_2 \leq 1$ is satisfied in both theories. However, contrary to such an intuition, $X_2 > 1$ can be achieved with a classically explained value of $V_2$, which is illustrated as the filled area in Fig. 2. Including such a nontrivial region, in the following, we show that there is a two-mode light at A and B satisfying any regions of $0 \leq D_2 \leq 1$ and $0\leq V_2\leq \infty$ in the quantum theory.

To see the striking feature of the nonclassical nature of $X_2 > 1$, it is convenient to introduce the normalized second-order cross-correlation function $g^{(2)}_{\rm {AB}}:=G^{(2)}_{\rm {AB}}/(G^{(1)}_{\rm {AA}}G^{(1)}_{\rm {BB}})$ and the normalized autocorrelation function $g^{(2)}_{ii}:=G^{(2)}_{ii}/G^{(1)}_{ii}{}^2$ which characterizes the photon statistics of light in mode $i=$A, B measured using the HBT setup with the use of not only the coincidence measurement but also the single-count measurements [29].

Our proof is divided into two parts: $0 \leq D_2 < 1$ and $D_2 = 1$. First, we show the region of $0\leq D_2 < 1$ and $0\leq V_2 \leq \infty$ is achieved. We assume $g^{(2)}_{\rm {AA}}$ and $g^{(2)}_{\rm {BB}}$ take the same value as $g^{(2)}_{\rm {auto}}~(\neq 0)$. Accordingly, $D_2 = |\zeta -\zeta ^{-1}|/(\zeta + \zeta ^{-1})$ is satisfied by defining the intensity ratio as $\zeta := G^{(1)}_{\rm {AA}}/G^{(1)}_{\rm {BB}} = \langle N_{\rm {A}} \rangle /\langle N_{\rm {B}} \rangle$ for non zero intensities. $D_2$ as a function of $\zeta$ can be any value in the range of $0\leq D_2 < 1$ independently of $g^{(2)}_{\rm {auto}}$. Meanwhile, $V_2 =2g^{(2)}_{\rm {AB}}(\zeta +\zeta ^{-1})^{-1}g^{(2)}_{\rm {auto}}{}^{-1}$ depends on $g^{(2)}_{\rm {auto}}$. $V_2=0$ is achieved for an input light satisfying $g^{(2)}_{\rm {AB}}=0$ [38]. Other values of $V_2$ in $0<V_2\leq \infty$ are achieved for any fixed value of $D_2$ in $0\leq D_2 < 1$, by considering statistically independent lights between modes A and B as $g^{(2)}_{\rm {AB}}=1$ with the use of parameter $g^{(2)}_{\rm {auto}}$. We notice that, $V_2 \rightarrow \infty$ is included for $g^{(2)}_{\rm {auto}}\rightarrow 0$, which means it asymptotically represents the case where an input is two single photons with one in each input port.

Second, we present a case of $D_2=1$. In the phase-based interference, $D_n=1$ immediately implies $V_n^{({\rm {ph}})}=0$ from Eq. (10). However, it does not always hold in the intensity interference. We suppose the light at mode A satisfies $g^{(2)}_{\rm {AA}}\neq 0$ and $\langle N_{\rm {A}} \rangle \neq 0$. If the light at mode B is in the vacuum, we obtain $D_2=1$ and $V_2=0$. On the other hand, if the light at mode B is a genuine single photon, we have $g^{(2)}_{\rm {BB}} = 0$, $\langle N_{\rm {B}} \rangle = 1$ and $g^{(2)}_{\rm {AB}} = 1$. Because of $G^{(2)}_{\rm {BB}}=0$, $D_2 = 1$ and $V_2 = 2G^{(2)}_{\rm {AA}}{}^{-1} = 2\langle N_{\rm {A}} \rangle ^{-1}g^{(2)-1}_{\rm AA}$. As a result, any value of $V_2$ for $0 < V_2\leq \infty$ is achieved by appropriate parameters of light at mode A. Combined with the result of the previous paragraph, it has been shown that in the quantum theory, $D_2$ and $V_2$ can be independently chosen in the regions $0 \leq D_2 \leq 1$ and $0\leq V_2\leq \infty$.

We describe examples focusing on the nonclassical region of $X_2 > 1$ with the classically understandable value of $0\leq V_2 \leq 1$. The visualization of the relation among $D_2$, $V_2$ and $X_2$ depicted in Fig. 3(a) is the case of $g^{(2)}_{\rm {AB}}=1$ and $g^{(2)}_{\rm {AA}}=g^{(2)}_{\rm {BB}}=g^{(2)}_{\rm {auto}}$ with a fixed value of $D_2=0.6~(\zeta = 2)$. At the point of $g^{(2)}_{\rm {auto}} = 0.8$, we see the classically explained which-path information and contrast ratio as $D_2 = 0.6$ and $V_2 = 1$, respectively. Nonetheless, the nonclassical value of $X_2$ appears as $X_2 = 1.36$. The nonclassicality is obtained only after considering $D_2$ as well as $V_2$. Such a nontrivial scenario occurs in a more feasible experimental setting where a heralded single photon produced by spontaneous parametric down conversion coming from mode A and independently prepared laser light coming from mode B ($g^{(2)}_{\rm {AB}} = g^{(2)}_{\rm {BB}} = 1$) are mixed, similar to the method in Refs. [34,36]. We illustrate $\zeta$ dependency of $X_2$ in Fig. 3(b) by assuming $g^{(2)}_{\rm {AA}} = 0.25$.

 

Fig. 3. Log–linear plots of $D_2$ (green), $V_2$ (blue) and $X_2{}^{1/2}$ (red). (a) $g^{(2)}_{\rm {auto}}$ dependencies with assumptions of $g^{(2)}_{\rm {AB}}=1$, $g^{(2)}_{\rm {AA}}=g^{(2)}_{\rm {BB}}=g^{(2)}_{\rm {auto}}$ and $\zeta =2$. (b) $\zeta$ dependencies with assumptions of $g^{(2)}_{\rm {AB}}=g^{(2)}_{\rm {BB}}=1$ and $g^{(2)}_{\rm {AA}}=0.25$. In (a) and (b), the areas marked with arrows show the nonclassical region of $X_2 > 1$ ($X_2{}^{1/2} > 1$) whereas the classical explained values of $D_2\leq 1$ and $V_2 \leq 1$.

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It should be mentioned that the condition in Eq. (16) required for the classical complementary principle $X_2 \leq 1$ was derived by considering the situation of mixing the bosonic photons with the use of the second-order which-path information as the particle-like property and the newly introduced quantity for describing the contrast ratio of interference as the wave-like property. Remarkably, the same inequality appears as a condition for the requirement of the classically explained cross-correlation function of separated photon pairs [39,40], despite the completely different physical scenario from that considered in this paper.

4. Duality in the higher-order intensity interferometric situations

We generalize the wave-particle duality of light to higher-order intensity interferometric situations based on $n$-fold coincidence detection after a BS which equally distributes input light from modes A and B to $n$ output modes. The interference effect includes $n-1$ types of the intensity correlations $G^{(n)}_{k,{\rm AB}}:=\left \langle : N_{\rm {A}}^kN_{\rm {B}}^{n-k} :\right \rangle$ for $k=1,\ldots, n-1$ and the phase correlation $G^{{\rm {ph}}(n)}_{\rm {AB}}$. For the interference related to $G^{(n)}_{k,{\rm AB}}$, we define the which-path information as

From the above definitions, we obtain

Similar to the complementary relation $X_{2,1}(=X_2) \leq 1$ of the second-order interference for classical light, $X_{n,k}\leq 1$ for $n\geq 3$ is always satisfied in the classical wave theory because of the Cauchy–Schwarz inequality. However, it is not so in the quantum theory. In fact, $X_{n,k} > 1$ is achieved in similar scenarios given above as examples for $n=2$.

From an experimental perspective, the interferometric setup to estimate the quantity $X_{n,k}$ will be more complicated for larger values of $n$. The difficulty in extracting the interference effect caused by $V_{n,k}$ for the desired $k$ would be a primary reason because the $n$-fold coincidence probability generally includes all the $n$-th order interference effects. Besides, the estimation of $D_{n,k}$ is another problem. For the symmetric case of $n=2m$ and $k=m$, $D_{n,k}=D_n$ is estimated using the $n$-fold coincidence measurement. However, $D_{n,k}$ does not generally correspond to $D_n$. For $k \neq n/2$, $n$-fold coincidence measurement is not sufficient to estimate $D_{n,k}$. One solution to overcome this difficulty involves replacing a photon detector after each of $n$ output ports of the BS for mixing modes A and B using a HBS followed by two detectors. The setup can detect up to $2n$-fold coincidence events without affecting the interferometric effect between input lights from modes A and B. Consequently, both $D_{n,k}$ and $V_{n,k}$, resulting in $X_{n,k}$, could be estimated.

5. Conclusion

In conclusion, we introduced the concept of the wave-particle duality of light in higher-order intensity interferometric scenarios, particularly for the second-order interference measured by the HOM interferometer. Using the higher-order which-path information as the particle-like property and introducing the contrast ratio of the intensity interference as the wave-like property, we showed the complementary relation of the two quantities analogous to that in the phase-based interference in the classical wave theory. However, in quantum theory, the relation is violated. In the quantum theory, the existence of light with any values for the which-path information $D_2$ and contrast ratio $V_2$ within the ranges of $0\leq D_2\leq 1$ and $0\leq V_2 \leq \infty$ is allowed. As a result, the above phenomenon includes the new nonclassical aspect of light which possesses classically comprehensible wave and particle properties. The duality considered here provides a deeper insight into coincidence-based experimental scenarios including the HOM experiments using various kinds of light with multiphotons. Therefore, we believe that such an investigation will lead to finding novel applications in photonic quantum information processing.