Entanglement of photons in their dual wave-particle nature

TL;DR

This paper demonstrates the first deterministic generation of entanglement between photons' wave and particle properties, revealing new quantum correlations and potential for advanced quantum information applications.

Contribution

The authors introduce and experimentally realize a scalable scheme to generate wave-particle entanglement of photons, a novel combination of fundamental quantum traits.

Findings

Successfully generated wave-particle entanglement of two photons.

Scheme can be extended to multiphoton wave-particle entanglement.

Potential applications in quantum information protocols.

Abstract

Wave-particle duality is the most fundamental description of the nature of a quantum object which behaves like a classical particle or wave depending on the measurement apparatus. On the other hand, entanglement represents nonclassical correlations of composite quantum systems, being also a key resource in quantum information. Despite the very recent observations of wave-particle superposition and entanglement, whether these two fundamental traits of quantum mechanics can emerge simultaneously remains an open issue. Here we introduce and experimentally realize a scheme that deterministically generates wave-particle entanglement of two photons. The elementary tool allowing this achievement is a scalable single-photon setup which can be in principle extended to generate multiphoton wave-particle entanglement. Our study reveals that photons can be entangled in their dual wave-particle nature and opens the way to potential applications in quantum information protocols exploiting the wave-particle degrees of freedom to encode qubits.

Equations

$$|{\psi }_f⟩=\cos \alpha |\mathrm{wave}⟩+\sin \alpha |\mathrm{particle}⟩,$$

$$|\mathrm{wave}⟩=\frac{e^{i\frac{{\varphi }_1}{2}}}{\sqrt{2}}\left[\cos \frac{{\varphi }_1}{2}(|1⟩+|2⟩)-i\sin \frac{{\varphi }_1}{2}(|3⟩+|4⟩)\right],$$

$$P_1={\mathcal{P}}_c+{\mathcal{I}}_c,P_2={\mathcal{P}}_c-{\mathcal{I}}_c,P_3={\mathcal{P}}_s+{\mathcal{I}}_s,P_4={\mathcal{P}}_s-{\mathcal{I}}_s,$$

$${\mathcal{P}}_c$$

$${\mathcal{P}}_s$$

$${\mathcal{I}}_c$$

$${\mathcal{I}}_s$$

$${|\Phi ⟩}_{AB}=\frac{1}{\sqrt{2}}\left(|\mathrm{wave}⟩|{\mathrm{wave}}^{\prime }⟩+|\mathrm{particle}⟩|{\mathrm{particle}}^{\prime }⟩\right),$$

Full text

Entanglement of photons in their dual wave-particle nature

Adil S. Rab

Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro, 5, I-00185 Roma, Italy

  

Emanuele Polino

Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro, 5, I-00185 Roma, Italy

  

Zhong-Xiao Man

zxman@mail.qfnu.edu.cn

Shandong Provincial Key Laboratory of Laser Polarization and Information Technology, Department of Physics, Qufu Normal University, Qufu 273165, China

  

Nguyen Ba An

Center for Theoretical Physics, Institute of Physics, Vietnam Academy of Science and Technology (VAST), 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam

  

Yun-Jie Xia

Shandong Provincial Key Laboratory of Laser Polarization and Information Technology, Department of Physics, Qufu Normal University, Qufu 273165, China

  

Nicolò Spagnolo

Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro, 5, I-00185 Roma, Italy

  

Rosario Lo Franco

rosario.lofranco@unipa.it

Dipartimento di Energia, Ingegneria dell’Informazione e Modelli Matematici, Università di Palermo, Viale delle Scienze, Edificio 9, 90128 Palermo, Italy

Dipartimento di Fisica e Chimica, Università di Palermo, via Archirafi 36, 90123 Palermo, Italy

  

Fabio Sciarrino

fabio.sciarrino@uniroma1.it

Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro, 5, I-00185 Roma, Italy

Wave-particle duality is the most fundamental description of the nature of a quantum object which behaves like a classical particle or wave depending on the measurement apparatus Wheeler1 ; MaRMP ; Wheeler2 ; manning . On the other hand, entanglement represents nonclassical correlations of composite quantum systems horodecki2009quantum ; lofranco2015quantum ; BrunnerReview , being also a key resource in quantum information vedralReview ; laddReview ; Wang16 . Despite the very recent observations of wave-particle superposition quan-delay-exp ; delay-exp1 ; delay-exp2 ; delay-exp3 ; delay-exp4 ; delay-exp5 ; pho-test ; Ma22012013 and entanglement BrunnerReview ; hensenBell ; PhysRevLett.115.250401 ; PhysRevLett.115.250402 ; Hand17 , whether these two fundamental traits of quantum mechanics can emerge simultaneously remains an open issue. Here we introduce and experimentally realize a scheme that deterministically generates wave-particle entanglement of two photons. The elementary tool allowing this achievement is a scalable single-photon setup which can be in principle extended to generate multiphoton wave-particle entanglement. Our study reveals that photons can be entangled in their dual wave-particle nature and opens the way to potential applications in quantum information protocols exploiting the wave-particle degrees of freedom to encode qubits.

Quantum mechanics is one of the most successful theories in describing atomic-scale systems albeit its properties remain bizarre and counterintuitive from a classical perspective. A paradigmatic example is the wave-particle duality of a single quantum system, which can behave like both particle and wave to fit the demands of the experiment’s configuration Wheeler1 . This double nature is well reflected by the superposition principle and evidenced for light by Young-type double-slit experiments MaRMP ; walbornDoubleSlit , where single photons from a given slit can be detected (particle-like behavior) and interference fringes observed (wave-like behavior) on a screen behind the slits. A double-slit experiment can be simulated by sending photons into a Mach-Zehnder interferometer (MZI) via a semitransparent mirror (beam splitter) MaRMP ; walbornDoubleSlit . A representative experiment with MZI, also performed with a single atom manning , is the Wheeler’s delayed-choice (WDC) experiment Wheeler1 ; Wheeler2 , where one can choose to observe the particle or wave character of the quantum object after it has entered the interferometer. These experiments rule out the existence of some extra information hidden in the initial state telling the quantum object which character to exhibit before reaching the measurement apparatus. Very recent quantum WDC experiments, using quantum detecting devices and requiring ancilla photons or post-selection, have then shown that wave and particle behaviors of a single photon can coexist simultaneously, with a continuous morphing between them quan-delay-exp ; delay-exp1 ; delay-exp2 ; delay-exp3 ; delay-exp4 ; delay-exp5 ; pho-test .

When applying the superposition principle to composite systems, another peculiar quantum feature arises, namely the entanglement among degrees of freedom of the constituent particles (e.g., spins, energies, spatial modes, polarizations) horodecki2009quantum ; lofranco2015quantum . Entanglement gathers fundamental quantum correlations among particle properties which are at the core of nonlocality BrunnerReview ; hensenBell ; PhysRevLett.115.250401 ; PhysRevLett.115.250402 ; Hand17 and exploited as essential ingredient for developing quantum technologies vedralReview ; laddReview ; Wang16 . Superposition principle and entanglement have been amply debated within classical-quantum border, particularly whether macroscopically distinguishable states (i.e., distinct quasiclassical wave packets) of a quantum system could be prepared in superposition states harocheNobelLect . While superpositions of coherent states of a single quantum system (also known as “cat states” from the well-known Schrödinger’s epitome) have been observed for optical or microwave fields starting from two decades ago harocheNobelLect ; brunePRL ; Ourjoumtsev ; Deleglise ; Vlastakis607 , the creation of entangled coherent states of two separated subsystems has remained a demanding challenge, settled only very recently by using superconducting microwave cavities and Josephson junction-based artificial atoms Wang1087 . An analogous situation exists in the context of wave-particle duality where, albeit wave-particle superpositions of a photon have been reported quan-delay-exp ; delay-exp1 ; delay-exp2 ; delay-exp3 ; delay-exp4 ; delay-exp5 ; pho-test , entangled states of photons correlated in their wave-particle degrees of freedom are still unknown.

In this work we experimentally demonstrate that wave-particle entanglement of two photons is achievable deterministically. We reach this goal by introducing and doubling a scalable all-optical scheme which is capable to generate, in an unconditional manner, controllable single-photon wave-particle superposition states. Parallel use of this basic toolbox then allows the creation of multiphoton wave-particle entangled states.

Single-photon toolbox. The theoretical sketch of the wave-particle scheme for the single photon is displayed in Fig. 1. A photon is initially prepared in a polarization state $\left|\psi_{0}\right\rangle=\cos\alpha\left|V\right\rangle+\sin\alpha\left|H\right\rangle$, where $\left|V\right\rangle$ and $\left|H\right\rangle$ are the vertical and horizontal polarization states and $\alpha$ is adjustable by a preparation half-wave plate (not shown in the figure). After crossing the setup with beam-splitter BS4 and BS5 inserted (see Supplementary Information for details), the photon state is

[TABLE]

where the states

[TABLE]

operationally represent the capacity ($|\mathrm{wave}\rangle$) and incapacity ($|\mathrm{particle}\rangle$) of the photon to produce interference quan-delay-exp ; delay-exp5 . In fact, for the $\left|\mathrm{wave}\right\rangle$ state the probability of detecting the photon in the path $|n\rangle$ ($n=1,2,3,4$) depends on the phase $\phi_{1}$: the photon must have traveled along both paths simultaneously (see upper MZI in Fig. 1), revealing its wave nature. Instead, for the $\left|\mathrm{particle}\right\rangle$ state the probability that the photon is detected in a certain path is always $1/4$, regardless of phase $\phi_{2}$: thus, the photon must have crossed only one of the two paths (see lower MZI of Fig. 1), showing its particle nature. Notice that the scheme is designed in such a way that $|V\rangle$ ($|H\rangle$) leads to the $|\mathrm{wave}\rangle$ ($|\mathrm{particle}\rangle$) state.

From equation (1), the probability $P_{n}$ of detecting the photon along path $|n\rangle$ is expected to depend on all the involved parameters, $P_{n}=P_{n}(\alpha,\phi_{1},\phi_{2})$. These probabilities are

[TABLE]

where

[TABLE]

We remark that the terms $\mathcal{I}_{c}$, $\mathcal{I}_{s}$ in the detection probabilities exclusively stem from the interference between the $|\mathrm{wave}\rangle$ and $|\mathrm{particle}\rangle$ components appearing in the generated superposition state $\left|\psi_{f}\right\rangle$ of equation (1). This fact is further evidenced by the appearance, in these interference terms, of the factor $\mathcal{C}=\sin 2\alpha$, which is the amount of quantum coherence owned by $\left|\psi_{f}\right\rangle$ in the basis $\{|\mathrm{wave}\rangle,|\mathrm{particle}\rangle\}$ adessoReview . On the other hand, the interference terms $\mathcal{I}_{c}$, $\mathcal{I}_{s}$ are always zero when the final state of the photon is: (i) $|\mathrm{wave}\rangle$ ($\alpha=0$); (ii) $|\mathrm{particle}\rangle$ ($\alpha=\pi/2$); (iii) a classical incoherent mixture $\rho_{f}=\cos^{2}\alpha\left|\mathrm{wave}\right\rangle\left\langle\mathrm{wave}\right|+\sin^{2}\alpha\left|\mathrm{particle}\right\rangle\left\langle\mathrm{particle}\right|$ (resulting from an initial mixed polarization state of the photon).

The experimental single-photon toolbox, realizing the proposed scheme of Fig. 1, is displayed in Fig. 2. The implemented layout presents the advantage of being interferometrically stable, thus not requiring active phase stabilization between the modes. For $\alpha=0$, the photon is vertically polarized and entirely reflected from the PBS to travel along path $1$, then split at BS1 into two paths, both leading to the same BS3 which allows these two paths to interfere with each other before detection. The photon detection probability at each detector $D_{n}$ $(n=1,2,3,4)$ depends on the phase shift $\phi_{1}$: $P_{1}(\alpha=0)=P_{2}(\alpha=0)=\frac{1}{2}\cos^{2}\frac{\phi_{1}}{2}$, $P_{3}(\alpha=0)=P_{4}(\alpha=0)=\frac{1}{2}\sin^{2}\frac{\phi_{1}}{2}$, as expected from equations (3) and (Entanglement of photons in their dual wave-particle nature). After many such runs an interference pattern emerges, exhibiting the wave-like nature of the photon. Differently, if initially $\alpha=\pi/2$, the photon is horizontally polarized and, as a whole, transmitted by the PBS to path $2$, then split at BS2 into two paths (leading, respectively, to BS4 and BS5) which do not interfere anywhere. Hence, the phase shift $\phi_{2}$ plays no role on the photon detection probability and each detector has an equal chance to click: $P_{1}(\alpha=\frac{\pi}{2})=P_{2}(\alpha=\frac{\pi}{2})=P_{3}(\alpha=\frac{\pi}{2})=P_{4}(\alpha=\frac{\pi}{2})=\frac{1}{4}$, as predicted by equations (3) and (Entanglement of photons in their dual wave-particle nature), showing particle-like nature without any interference pattern. Interestingly, for $0<\alpha<\pi/2,$ the photon simultaneously behaves like wave and particle. The continuous morphing transition from wave to particle behavior as $\alpha$ varies from [math] to $\pi/2$ is clearly seen from Fig. 3c. The coherence witness defined as $\mathcal{W}_{C}=|P_{1}-P_{2}|$ is also measured. According to equations (3) and (Entanglement of photons in their dual wave-particle nature), $\mathcal{W}_{C}=2\mathcal{I}_{c}$ is zero if and only if there is no wave-particle coherence. As shown in Fig. 3c, $\mathcal{W}_{C}$ testifies for both coherent $|\psi_{f}\rangle$ and mixed $\rho_{f}$ wave-particle states (the latter being obtained by adding a relative time delay in the interferometer paths larger than the photon coherence time to lose quantum interference).

Wave-particle entanglement. The above single-photon scheme constitutes the basic toolbox which can be extended to create the wave-particle entangled state of two photons, as shown in Fig. 2b. Initially, a two-photon polarization maximally entangled state $\left|\Psi\right\rangle_{AB}=\frac{1}{\sqrt{2}}(\left|VV\right\rangle+\left|HH\right\rangle)$ is prepared (the procedure works in general for arbitrary weights, see Supplemental Information). Each photon is then sent to one of two identical wave-particle toolboxes which provide the final state

[TABLE]

where the single-photon states $\left|\mathrm{wave}\right\rangle$, $\left|\mathrm{particle}\right\rangle$, $\left|\mathrm{wave}^{\prime}\right\rangle$, $\left|\mathrm{particle}^{\prime}\right\rangle$ are defined in equation (2), with parameters and paths related to the corresponding wave-particle toolbox. The generated state $\left|\Phi\right\rangle_{AB}$ is thus a wave-particle maximally entangled state (Bell state) of two photons in separated locations.

The output two-photon state is measured after the two toolboxes. The results are shown in Fig. 4. Coincidences between the four outputs of each toolbox are measured by varying $\phi_{1}$ and $\phi_{1}^{\prime}$. The first set of measurements (Fig. 4a-d) is performed by setting the angles of the output wave-plates (see Fig. 2c) at $\{\beta=0,\beta^{\prime}=0\}$, corresponding to removing both BS4 and BS5 in Fig. 1 (absence of interference between single-photon wave-like and particle-like behaviors). In this case, detectors placed at outputs (1,3) and (1*′,3′) reveal wave-like behavior, while detectors placed at outputs (2,4) and (2′,4′) evidence a particle-like one. As expected, the two-photon probabilities $P_{nn^{\prime}}$ for the particle detectors remain unchanged while varying $\phi_{1}$ and $\phi_{1}^{\prime}$, whereas the $P_{nn^{\prime}}$ for the wave detectors show interference fringes. Moreover, no contribution of crossed (wave-like)-(particle-like) coincidences $P_{nn^{\prime}}$ is obtained, due to the form of the entangled state. The second set of measurements (Fig. 4e-h) is performed by setting the angles of the output wave-plates at angles $\{\beta=22.5^{\circ},\beta^{\prime}=22.5^{\circ}\}$, corresponding to the presence of BS4* and BS5 in Fig. 1 (presence of interference between single-photon wave and particle behaviors). We now observe nonzero contributions across all the probabilities depending on the specific settings of phases $\phi_{1}$ and $\phi_{1}^{\prime}$. In order to assess the presence of entanglement in the wave-particle natures, we also measure the wave-particle entanglement witness defined as $\mathcal{W}_{E}=P_{22^{\prime}}-P_{21^{\prime}}$ by varying $\phi_{1}$ with fixed $\phi_{1}^{\prime}=\phi_{2}=\phi_{2}^{\prime}=0$. According to the general expressions of the coincidence probabilities (see Supplemental Information), $\mathcal{W}_{E}$ is identically zero if and only if the wave-particle two-photon state is separable (e.g., $|\mathrm{wave}\rangle\otimes|\mathrm{wave^{\prime}}\rangle$ or a maximal mixture of two-photon wave and particle states). For $|\Phi\rangle_{AB}$ of equation (5) the theoretical prediction is $\mathcal{W}_{E}=(1/4)\cos^{2}(\phi_{1}/2)$, which is confirmed by the results reported in Fig. 4i-j (within the reduction due to visibility). These observations altogether prove the expected quantum correlations between wave and particle states of two photons in the entangled state $|\Phi\rangle_{AB}$.

Conclusions. In summary, we have introduced and realized a novel all-optical scheme to deterministically generate single-photon wave-particle superposition states. This setup has enabled the observation of the simultaneous coexistence of particle and wave character of the photon maintaining all its devices fixed, being the control only on the preparation of the input photon. Specifically, different initial polarization states of the photon, then transformed into which-way (path) states, reveal the wave-to-particle morphing economizing the employed resources compared to previous experiments with delayed choice quan-delay-exp ; delay-exp1 ; delay-exp2 ; delay-exp3 ; delay-exp4 ; delay-exp5 ; pho-test . The advantageous aspects of the single-photon scheme have then supplied the key for its straightforward doubling, by which we have observed that two photons can be cast in a wave-particle entangled state provided that suitable initial polarization entangled states are injected into the apparatus. We remark that powerful features of the scheme are flexibility and scalability. Indeed, a parallel assembly of $N$ single-photon wave-particle toolboxes allows the generation of $N$-photon wave-particle entangled states. For instance, the GHZ-like state $|\Phi_{N}\rangle=\frac{1}{\sqrt{2}}(|\mathrm{wave}_{1},\mathrm{wave}_{2},\ldots,\mathrm{wave}_{N}\rangle+|\mathrm{particle}_{1},\mathrm{particle}_{2},\ldots,\mathrm{particle}_{N}\rangle)$ is produced when the GHZ polarization entangled state $|\Psi_{N}\rangle=\frac{1}{\sqrt{2}}(|V_{1}V_{2}\ldots V_{N}\rangle+|H_{1}H_{2}\ldots H_{N}\rangle)$ is used as input state.

From a fundamental viewpoint, our research brings the complementarity principle for wave-particle duality to a further level. In fact, besides confirming that a photon can live in a superposition of wave and particle behaviors when observed by quantum detection delay-exp5 , we prove that the manifestation of its dual nature can intrinsically depend on the character of another photon, according to correlations ruled by quantum entanglement. In this case, the wave-particle behavior of a photon is determined by a measurement apparatus placed in a region spatially separated from it. This new phenomenon, merging complementarity principle and entanglement, can be named “wave-particle duality action at a distance”. We finally highlight that the possibility to create and control wave-particle entanglement may also play a role in quantum information scenarios. In particular, it opens the way to design protocols which exploit quantum resources contained in systems of qubits encoded in wave and particle operational states.

Acknowledgements

This work was supported by the ERC-Starting Grant 3D-QUEST (3D-Quantum Integrated Optical Simulation; grant agreement no. 307783, http://www.3dquest.eu) and by the Marie Curie Initial Training Network PICQUE (Photonic Integrated Compound Quantum Encoding, grant agreement no. 608062, funding Program: FP7-PEOPLE-2013-ITN, http://www.picque.eu). In this work Z.X.M. and Y.J.X. are supported by National Natural Science Foundation of China under Grant no. 11574178 and no. 61675115, Shandong Provincial Natural Science Foundation, China under Grant no. ZR2016JL005, while N.B.A. is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under project no. 103.01-2017.08.