Separating the Wave and Particle Attributes of Two Entangled Photons

Measurement problems plays essential roles for our understanding of the nature of the quantum theory and leads to many counterintuitive quantum phenomena. One of them is the wave-particle duality of quantum objects which depends on measurement apparatus (Wheeler, 1984). In turn, quantum objects display its wave- or particle-like behavior depending on the experimental technique used to measure them (Grangier et al., 1986; Kurtsiefer et al., 1997; DÃŒrr et al., 1998; Marshall et al., 2003; Wang et al., 2019). The fundamental issues related to wave-particle duality and Bohr’s complementarity principle (Bohr, 1928; Wootters and Zurek, 1979; Scully et al., 1991) can be exemplified by Young-like double-slit experiments and their variants, e.g. Wheeler’s delayed choice experiment (Wheeler, 1978). It has been implemented in variety of systems such as photons, atoms, and superconducting circuits (Englert, 1996; Jacques et al., 2007, 2008; Ma et al., 2012; Manning et al., 2015; Tang et al., 2012; Peruzzo et al., 2012; Liu et al., 2017; Ma et al., 2016; Hellmuth et al., 1987; Dong et al., 2020). Furthermore, the particle and wave-like behaviors of a single photon (Ionicioiu and Terno, 2011) and multi-entangled photons (Wang et al., 2019) has also been observed simultaneously in one and the same experimental setup (Adesso and Girolami, 2012; Shadbolt et al., 2014). Those innovative works provides a new perspective for the in-depth understanding of wave-particle duality.

In general, since the wave and particle properties of a quantum object coexist simultaneously, one may always suppose to that the two attributes of single entity cannot be separated from each other. However, recent research results on the topic (Chowdhury et al., 2021; Li et al., 2023) has shown, upon exploiting the idea of Quantum Cheshire Cat (QCC) (Aharonov et al., 2013), that one may spatially separates of wave and particle attributes of a quantum object. The QCC is a fictitious character in the famous novel “Alice in Wonderland”. As described in that novel, the Cheshire cat has the ability of disappearing in space while its grin is still visible. However, in our usual life, we can imagine a cat without a grin, but it beyond our common sense to accept the existence of a grin without a cat as Alice said : “Well! I’ve often seen a cat without a grin, but a grin without a cat! It’s the most curious thing I ever saw in my life!”.

The QCC proposal introduced by Y. Aharonov and his collaborators (Aharonov et al., 2013) is aim to detach a quantum object from its inherent features. QCC cat may exist in a quantum system described by two-state vector formalism (Aharonov et al., 1964; Aharonov and Vaidman, 1990, 2008). Furthermore, its realization depends on performing a post-selected weak measurement (Aharonov et al., 1988; Kofman et al., 2012; Svensson, 2013; Sokolovski, 2013; Dressel et al., 2014) to extract the weak values (Aharonov and Botero, 2005) of the observables associated to the quantum object under investigation.

After the original QCC scheme proposed, it have received large attention and has been investigated theoretically (Guryanova et al., 2012; Matzkin and Pan, 2013; Lorenzo, 2013; Bancal, 2014; Ibnouhsein and Grinbaum, 2014; Corrêa et al., 2015; Duprey et al., 2018; Das and Pati, 2019, 2020; Das and Sen, 2021; Aharonov et al., 2021; Chowdhury et al., 2021; Hance et al., 2023a, b; Zhou et al., 2023; Ghoshal et al., 2023) and experimentally (Atherton et al., 2015; Ashby et al., 2016; Liu et al., 2020; Kim et al., 2021; Wagner et al., 2023; Danner et al., 2023; Li et al., 2023; Denkmayr et al., 2014). Among the different studies, a protocol to separate the wave and particle attributes of a single photon has been proposed (Chowdhury et al., 2021) and implemented (Li et al., 2023). Furthermore, in (Rab et al., 2017), it has been experimentally observed that two photons can be cast in a wave-particle entangled state provided that suitable initial entangled polarization states are injected into the apparatus. Although the separation of particle and wave-like behaviors of a single photon has been observed theoretically and experimentally, the possibility of such observation regarding multi-photon states remains to be explored. Since the entanglement is a key resource for quantum information and quantum communication, a question arises on whether the separation of wave- particle attributes of two entangled photons is possible. As we will see in the following, our results provide an affirmative answer to this question.

In this paper, motivated by previous studies (Rab et al., 2017; Chowdhury et al., 2021; Li et al., 2023), we introduce a feasible theoretical model to spatially separate wave and particle attributes of two entangled photons based on QCC phenomena. After appropriate pre- and post-selection of the paths of photons, we can find the wave and particle features of two entangled photons in different places. Besides their fundamental interest in the understanding of counterfactual communication, our results may promote the development of quantum metrology.

The remaining part of this paper is organized as follows. In Section II,we briefly introduce the idea of quantum Cheshire cat. In SectionIII, we describe in details our proposal and show that by suitable pre- and post-selection one can realize the separation of wave and particle attributes of two entangled photons. In the same Section, we also describe a possible implementation of our scheme on an optical platform. Finally, in Section IV, we close the paper with some concluding remarks and discussions.

In general, an object and its inherent properties always coexist and cannot be separated at anytime. In original proposal of Aharonov and collaborators (Aharonov et al., 2013), they found that the polarization (say, the grin) property can be separated from a photon (say, the cat) itself in a pre- and post-selected system. Here, we briefly summarize the ingredients required to observe such phenomena. Initially, a horizontally polarized photon enters a Mach-Zehnder interferometer (MZI) and passes through its right and left arms with equal probability, and the corresponding state reads as

where $|L\rangle$ and $|R\rangle$ denote the left and right arms of MZI, i.e. the two possible paths of the photon, and $|H\rangle$ denotes the horizontal polarization of the photon. We choose the state $|\phi_{i}\rangle$ as our preselected state. By adding and adjusting some optical elements, we may post-select the paths and polarization of the photon to obtain the following state vector

where $|V\rangle$ represents the vertical polarization of the photon. Now, we measure the position (path) of the photon together with its circular polarization inside the MZI. In the two-state-vector formalism, the weak value of an observable $C$ is given by

This weak value can describe the true value of observable $C$ in time interval corresponded to pre-and postselected states $|\phi_{i}\rangle$ and $|\phi_{f}\rangle$. If we want to determine the location of the photon and its polarization in left or right arms of MZI in the region from $|\phi_{i}\rangle$ to $|\phi_{f}\rangle$, we should define the corresponding observables to those quantities. Explicitly, we have the operators

Here, $\Pi_{L}\left(\Pi_{R}\right)$ with $\mu\in\left(L,R\right)$ is the projection operator of left (right) arm of MZI and $\sigma_{z}=|\uparrow_{y}\rangle\langle\uparrow_{y}|-|\downarrow_{y}\rangle\langle\downarrow_{y}|$ is circular polarization observable of the photon with $|\uparrow_{y}\rangle=\frac{1}{\sqrt{2}}\left(|H\rangle+i|V\rangle\right)$ and $|\downarrow_{y}\rangle=\frac{1}{\sqrt{2}}\left(|H\rangle-i|V\rangle\right)$. The weak values of $\Pi_{\mu}$ and $\sigma_{z}^{\mu}$ can characterize the definite values of exact location of spatial degree of freedom and circular polarization of the photon inside the MZI, respectively.

By taking into account the above pre- and post-selected states $|\phi_{i}\rangle$ and $|\phi_{f}\rangle$, the weak value of $\Pi_{\mu}$ and $\sigma_{z}^{\mu}$ are given by

and

respectively. Here, $\delta_{ij}$ denote the Kronecker $\delta$-function which defined as

We can see that $\langle\Pi_{L}\rangle_{w}=\langle\sigma_{z}^{R}\rangle=1$ and other weak values all equal to zero. The weak values in Eq.(6) and Eq. (7) indicated that the spatial degree freedom of the photon is located in the left arm (photon itself go through the left arm of MZI), while its circular polarization is detected with unit probability in the right arm of the MZI. This means that in properly pre- and post-selected quantum systems one can realize the separation of the object itself from its inherent properties, as the Cheshire cat.

In this section, we describe our proposal for the separation of wave-particle properties of two entangled photons. The schematic diagram of our proposed setup is shown in Fig. 1. The one of most important points of our scheme is the preparation of so-called wave-particle (WP) toolbox introduced in Ref. (Rab et al., 2017) to convert superpositions of polarization states to the superpositions of path and polarization states. Next we describe our scheme step by step along with our schematic diagram depicted in Fig. 1.

We assume that in type-I spontaneous parametric down-conversion (SPDC) process two photons with same polarization have generated, i.e. $|V\rangle_{1}|V\rangle_{2}$ or $|H\rangle_{1}|H\rangle_{2}$. Upon using a half-wave plate (HWP, not shown in the figure) rotated at an angle $\alpha/2$ on the path of both the photons, the two-photon state becomes

where $\alpha$ is an adjustable parameter. After this stage, each photon enters one of two identical WP toolboxes, and the final output state can be written as (Rab et al., 2017)

where

with $|n\rangle$ and $|n^{\prime}\rangle$ ( $n,n^{\prime}\in\{1,2,3,4\}$ ) being the $n(n^{\prime})$-th output modes of from the two WP toolboxes (Rab et al., 2017), and $\phi_{1}$, $\phi_{1}^{\prime}$, $\phi_{2}$ and $\phi_{2}^{\prime}$ being controllable phase shifts in the two WP toolboxes. Notice that these states can characterize the capabilities ($|W\rangle$and $|W^{\prime}\rangle$) and incapabilities ($|P\rangle$ and $|P^{\prime}\rangle$) of the two photons to produce interference. In (Rab et al., 2017), the generation processes of superpositions of wave and particle states for single photon and two entangled photons both have been analyzed. For more details about the generation of the state, i.g, Eqs. (11a-11d) in the lab we refer to the original paper (Rab et al., 2017).

Finally, as shown in Fig. 1, one detects the two photons simultaneously by measuring the "wave operator" characterized by the projectors $X_{1}=|W\rangle\langle W|$ and $X_{2}=|W^{\prime}\rangle\langle W^{\prime}|$. The action of those operators is such that only the wave states $|W\rangle$ and $|W^{\prime}\rangle$ are transmitted whereas the particle states $|P\rangle$ and $|P^{\prime}\rangle$ are reflected. In this way, the state $|\Phi\rangle_{out}$ is transformed into

This is our pre-selected state. Here, as the notation of QCC, we use the composite system of path and attributes of two photons as $|.\ .\rangle_{paths}|.\ .\rangle_{attributes}$. $|L_{i}\rangle$ and $|R_{i}\rangle$ denoting the left and right paths, and $i=1,2$ labels the first and second photons. $|R_{1}L_{2}\rangle|WW^{\prime}\rangle$ represents $1$st (2nd) photon’s wave property $|W\rangle$ ($|W^{\prime}\rangle$) at $|R_{1}\rangle$ ($|L_{2}\rangle$), and $|L_{1}R_{2}\rangle|PP{}^{\prime}\rangle$ represents $1$st (2nd) photon’s particle property $|P\rangle$ ($|P^{\prime}\rangle$) at $|L_{1}\rangle$ ($|R_{2}\rangle$).

In order to realize the separation of wave and particle attributes of two entangled photons, the post-selected state should be of the form

This means that in our scheme the post-selected state is a special case of the pre-selected class of states with $\alpha=\frac{\pi}{4}$.

The determination of post-selected state $|\psi_{f}\rangle$ is crucial for the implementation of our scheme. As shown in Fig. 1, its verification can accomplished by an optical setup made of five beam splitters $BS_{i}$ ($i=1,2,3,4,5$), two $\sigma^{1234}$ operators, two $X_{i}$ operators, three mirrors $M_{i}$ ($i=1,2,3$), and six detectors $D_{i}$ ($i=1,2,3,4,5,6$). The successful post-selection occurs iff the detector $D_{5}$ click. On the contrary, if our post-seleceted state to be in $|\psi_{f}\rangle$, then, in the optical arrangement of Fig. 1, we can guarantee only the detector $D_{5}$ give the answer ,“yes”. Let us prove this statement.

At first, the photons in $L_{1}$ and $R_{2}$ arms simultaneously go through the beam splitters $BS_{1}$ and $BS_{2}$ followed by $\sigma^{1234}$ operators. Here, the effect of $\sigma^{1234}$ operator is to switch the photon paths in the four-dimensional basis $\{|1\rangle,|2\rangle,|3\rangle,|4\rangle\}$. Its matrix form can be written as (Rab et al., 2017)

and its base vectors can be written as

The action of $\sigma^{1234}$ is to switch paths as follows (Li et al., 2023)

The beam splitters $BS_{1}$ and $BS_{2}$ are adjusted to perform the transformations as

In this way, the particle state can converted to the wave state after the action of $BS_{1}$ ($BS_{2}$) and $\sigma^{1234}$, i.e,

The same happens for primed paths with $|n^{\prime}\rangle\in\left(|1^{\prime}\rangle,|2^{\prime}\rangle,|3^{\prime}\rangle,|4^{\prime}\rangle\right)$, as

Thus, after the photons in $L_{1}$ and $R_{2}$ arms simultaneously go through beam splitters $BS_{1}$ and $BS_{2}$ followed by the $\sigma^{1234}$ operators, our post-selected state $|\psi_{f}\rangle$ changed to

Next, by adjusting the beam-splitters $BS_{3}$ and $BS_{4}$, respectively, such that states $|R_{1}\rangle$ and $|L_{1}\rangle$ ($|L_{2}\rangle$ and $|R_{2}\rangle$) totally transmitted (reflected) to guarantee $|R_{1}L_{2}\rangle$ and $|L_{1}R_{2}\rangle$ go through $X_{3}$ and $X_{4}$. Under this action the detectors $D_{1}$ and $D_{2}$ both will not click. In this step, the modified form of $|\psi_{f^{\prime}}\rangle$ can be written as

Here, we denote $|R_{1}L_{2}\rangle\overset{BS_{3}}{\longrightarrow}|R\rangle$, $|L_{1}R_{2}\rangle\overset{BS_{4}}{\longrightarrow}|L\rangle$ and $|\mathcal{\boldsymbol{W}}\rangle=|WW^{\prime}\rangle$. The $X_{3}=X_{4}=|W\rangle\langle W|$ operators transmit only the wave state, and this action guarantees that detectors $D_{3}$ and $D_{4}$ will not click. The beam splitter $BS_{5}$ is chosen as $|R\rangle\overset{BS_{5}}{\longrightarrow}\frac{1}{\sqrt{2}}\left(|R\rangle+|L\rangle\right)$, $|L\rangle\overset{BS_{5}}{\longrightarrow}\frac{1}{\sqrt{2}}\left(|R\rangle-|L\rangle\right)$, such that

Hence, we can confirm that detector $D_{5}$ is click with certainty. Contrarily, if and only if the detector $D_{5}$ click with $100\%$ probability, then our schematics also can prove that the postslected state should be chosen to $|\psi_{f}\rangle$.

After choosing the pre- and post-selected state of our scheme, we have to extract the weak values to achieve our goal. The operators that measure whether the wave and particle attributes of two entangled photons are present in the left and right arms is defined as

with $\nu\in\{W,W^{\prime},P,P^{\prime}\}$, $\mu_{i}\in\{L_{i},R_{i}\}$ with $i=1,2$. Above every operators has its meaning, such as $\Pi_{W}^{L_{1}}=|L_{1}\rangle\langle L_{1}|\otimes|W\rangle\langle W|$ can describe the findings of the wave attribute of first photon on the left arm and $\Pi_{P^{\prime}}^{R_{2}}=|R_{2}\rangle\langle R_{2}|\otimes|P^{\prime}\rangle\langle P^{\prime}|$ can describe the findings of the particle attribute of second photon on the right arm, etc.

By substituting the above-defined pre- and post-selected states $|\psi_{f}\rangle$ and $|\psi_{f}\rangle$ into Eq. (3), we obtain the corresponding weak values of the observables $\Pi_{\nu}^{\mu_{i}}$, i.e.,

This is the main result of our study, and they can characterize the exact values of operators $\Pi_{\nu}^{\mu_{i}}$ in the region from $|\psi_{i}\rangle$ to $|\psi_{f}\rangle$. If $\alpha\neq 0,\frac{\pi}{2}$, these weak values indicated that the wave and particle attributes of two entangled photons are spatially separated, and in each one of the four arms we only have one attribute (wave or particle ) of the two photons. In the particular case, $\alpha=\frac{\pi}{4},$ the weak values for the wave and particle attributes may be written as $\langle\Pi_{P}^{L_{1}}\rangle_{w}=\langle\Pi_{W}^{R_{1}}\rangle_{w}=\frac{1}{2}$, $\langle\Pi_{P^{\prime}}^{R_{2}}\rangle_{w}=\langle\Pi_{W^{\prime}}^{L_{2}}\rangle_{w}=\frac{1}{2}$. This means that for $\alpha=\frac{\pi}{4}$ case, half of the particle attributes of the two entangled photons occurred in $L_{1}$ and $R_{2}$ arms and half of the wave attributes are present in $R_{1}$ and $L_{2}$ arms of the interferometer, respectively. Furthermore, it is clear that these weak values cannot take arbitrary large values, and always obey the Bohr’s complementarity principle, i.e.,

In this proposal, the correctly choosing of post-selected state is very crucial and we only focus on the cases of $D_{5}$ click with certainty. Otherwise, if any other detector clicks, the post-selection fails, and we fail to separate the wave and particle attributes of two entangled photons.

In this paper, we have investigated the possibility of spatially separating the wave and particle features of two entangled photons. Our work is based on measuring the weak values of system by selecting appropriate pre- and post-selection states in order to generate polarization-path quantum Cheshire cat states of two entangled photons. We found that by properly choosing the pre- and post-select states of system, the wave and particle attributes of two entangled photons can be spatially separated, while keeping the validity of Bohr’s complementarity principle. We also proposed a feasible setup for the implementation of our thought experiment with quantum optical platform.

As indicated in our result, we can control wave-particle attributes of two entangled photons by adjusting the optical axis angle $\alpha$ of HWP. If $\alpha=0$ or $\alpha=\frac{\pi}{2}$, then we only have pure wave attributes or pure particle attributes of two photons and separate them spatially. Thus, it is very curious issue to do double- slit experiment by using pure particle attributes of entangled photons (Hong and Noh, 1998). Furthermore, in the emerging field of counterfactual communication (Salih et al., 2013; Li et al., 2015), information may be exchanged even without the transmission of quantum systems. In this framework, entanglement also plays a vital role, but the basic principles of such communication is still unclear. Thus, it may interesting to use pure wave attributes of entangled photons for understanding of the fundamental mechanism behind counterfactual communication. At the same time, our results may also provide an alternative method to enhance the precision of quantum metrological protocols. In particular, in quantum metrology one would generally require that the measurement of a physical quantity on an entangled state does non disturb to other variable. Thus, the separation of entangled observables by QCC and implement the related precision measurement may dramatically increase its efficiency.

Another interesting point of our scheme is that it can scale up to more multi-entangled photons cases. Since the parallel assembly of $N$ single-photon WP toolboxes allows us to generate $N$-photon wave-particle entangled states (Rab et al., 2017), we may realize the spatial separation of wave and particle attributes of $N$-photon wave-particle entangled states by expand our schematics.

Finally, we anticipate that our theoretical proposal is feasible with current technology and may be implemented in quantum optical labs in the near future.