1. Introduction

The debate on the nature of photons dates back to Newton’s age when the corpuscular theory of light was developed long before Huygens put forward the wave theory. It took a long time for people to admit that a microparticle possesses both wave and particle properties, i.e., the wave-particle duality, which is one of the most intriguing features of quantum objects. This phenomenon was first conceptualized by Bohr in his famous complementarity principle [1].

In Bohr’s complementarity principle, the wave and particle natures are mutually exclusive, i.e., inspecting one naturally erases the other. This statement stresses the apparatus-dependent nature of photons during measurement. The manifestation depends on the willing of the observer. When the observer wants to inspect the wave nature of photons, it reveals wave behaviour, otherwise it shows particle behaviour. Question arises whether a simultaneous observation of wave and particle natures is possible. Wootters and Zurek answered the question by designing two experiments in a double-slit interference scenario [2]. They showed that inspection of both wave and particle natures of photons was possible; and the more wavelike the photon was, the less particlelike it would be, and vice versa. Rigorous analysis was then put forward in [3] with two inequalities:

To highlight the nonclassical property, Wheeler proposed his famous gedanken delayed-choice experiment [4,5] where the choice of the observer was made after the photons passed through the first beam splitter (BS) in the Mach-Zehnder interferometer (MZI). Many experimental proposals have been presented in different physical systems [6–11] to realize Wheeler’s delayed-choice experiment. Those early implementations require fast switching of devices with a quantum random number generator [8,10,12]. In recent years, the duality of photons with asymmetric beam interference [13,14] and the complementarity principle have been studied with a multi-path interferometer both in theory [15–18] and experiment [19–21]. The close relationship between duality and coherence has also been studied [16,22–25].

A quantum version of delayed-choice experiment was first proposed in [26]. [26] introduced a quantum beam splitter (QBS) which can be simultaneously present and absent depending on the quantum state of the ancillary qubit [Fig. 1(a)]. The effect of such a quantum device is equivalent to a controlled-Hadamard gate [Fig. 1(b)]. By projecting the ancillary qubit onto a superposed state $|\Psi \rangle =\cos \theta |0\rangle +\sin \theta |1\rangle$, where $|0\rangle$ and $|1\rangle$ are logical states of the ancillary qubit, the photon can be simultaneously wavelike and particlelike, thus enabling one to observe the morphing between wave and particle natures. Figure 1(c) shows the morphing from particle behaviour ($\theta =0$) to wave behaviour ($\theta =\pi /2$) in the quantum delayed-choice experiments [26–29]. Many experimental realizations of the quantum version of delayed-choice experiments have been put forward since the proposal in [26]. Those include implementations in NMR systems [27,29], linear optics [28,30,31], entangled photon pairs [32–34], etc.

 

Fig. 1. (a): QBS in a Mach-Zehnder interferometer. A QBS can be set in a superposition of absent and present, enabling one to observe both particle and wave behaviours of photon. (b): Quantum circuit representation of (a). QBS is equivalent to a controlled-Hadamard gate (dashed rectangle). When the control qubit, represented by the top line, is $|0\rangle$, it corresponds to an open interferometer. When the controlled qubit is $|1\rangle$, it corresponds to a closed interferometer. When the controlled qubit is in a superposed state, one is able to observe both particle and wave behaviours. The symbol H denotes a Hadamard gate and the symbol $\varphi$ denotes a phase gate. (c): Morphing between particle ($\theta =0$) and wave ($\theta =\pi /2$) in a quantum delayed-choice experiment.

Download Full Size | PDF

Former quantum delayed-choice experiments based on linear optics were realized at the single-photon level [8,9,28,30–33], where $\theta$ was generally introduced by a phase retardant element, such as a half wave-plate (HWP) or a tilted quartz plate; while $\varphi$, the relative phase between the two arms, was introduced by a phase plate. Since a single photon was used as the input in the experiments, repeating measurements were necessary to obtain different experimental data, and adjusting the parameters $\theta$ and $\varphi$ was required prior to each of measurements. This procedure is somewhat annoying and also takes considerable time especially when the number of required experimental data is large. In this paper, we propose a simulation of the quantum delayed-choice experiment to inspect the morphing between wave and particle natures of photons via a single shot, by using a classical intense light beam as the input rather than a single photon. By utilizing the finite beam profile of the beam, the trade-off between distinguishability and visibility is visualized and no dynamical change of the phase elements is required. The image shows the periodic morphing between wave and particle behaviours. The relevant features addressed in [26] are reproduced. Our realization only requires performing the experiment once, does not need to adjust experimental parameters, and is also time-efficient since the image is captured within the exposure time which is normally several milliseconds. Moreover, the phase fluctuation can be neglected in the process, therefore our experimental setup is robust against the noise.

2. Methods

The starting point of our experiment is the QBS proposed in [26]. We utilize photons’ polarization as the ancillary qubit. Our experiment employs a coherent beam with finite beam size as the input. If $\theta$ and $\varphi$ are introduced in a way that they depend on the spatial coordinates, i.e., $\theta =\theta (y)$ and $\varphi =\varphi (x)$, where $x$ and $y$ are the transverse spatial coordinates of the light field with $z$ axis pointing to the propagation direction, then the morphing between particle and wave natures will appear along $y$ axis, while the fringe will appear along $x$ axis when the wave behaviour is manifested. For example, at $\theta (y)=n\pi$ with $n\in \mathbb {Z}$, the particle behaviour is manifested and no fringe appears along $x$ axis. At $\theta (y)=(n+\frac {1}{2})\pi$, the wave behaviour is manifested and fringe appears along $x$ axis. In the simplest case of $\theta =y$ and $\varphi =x$, the coordinates in Fig. 1(c) can be replaced by $x$ and $y$. For a coherent beam with finite beam size, we can capture the beam by a charged coupled device (CCD) camera. In this way, the resulting image would show the same pattern as that in Fig. 1(c), thus the wave-particle morphing is visualized via a single shot.

Figure 2(a) shows the conceptual setup of our scheme. Photons are initially V-polarized. We introduce an imaginary element, i.e., the spatially-varying half-wave plate (v-HWP) in the setup. The v-HWP has a varying fast axis along $y$ axis, i.e., $\theta =k_yy/4$, where $k_y$ depends on the specification of the v-HWP. An example of the v-HWP is shown in Fig. 2(b), where the arrows indicate the orientations of the fast axis. This element transforms the input light into a structured light with a varying polarization along $y$, $\cos (k_yy/2)|\textrm {V}\rangle +\sin (k_yy/2)|\textrm {H}\rangle$. Then the photons are separated by the first polarization beam splitter (PBS), they are polarization-path entangled, $|\psi _0\rangle =\cos (k_yy/2)|\textrm {V},\textrm {L}\rangle +\sin (k_yy/2)|\textrm {H},\textrm {R}\rangle$, where R and L denote the right and left paths, respectively. The V-polarized photons undergo an open interferometer which consists of a BS. Despite the presence of the wedged fused silica plate (WFSP), no interference emerges. Thus, the photons show the particle behaviour. On the other hand, the H-polarized photons undergo a closed interferometer, which consists of two BSs with another WFSP placed along $x$ axis in one arm. The fringe along $x$ axis appears in the output, thus the photons show the wave behaviour. In this setup, since there is a wedge angle of the WSFP, photons transmit at different positions would be appended with different phases, thus producing interference fringes at the output.

 

Fig. 2. (a): Conceptual setup of visualizing the wave-particle duality via a single shot. The light source is an intense classical beam with finite beam waist. A v-HWP creates spatially-varying polarization. Then the V-polarized photons undergo an open MZI showing the particle behaviour, the H-polarized photons undergo a closed MZI showing the wave behaviour. At $c1$ where $a2$ and $b1$ are superposed, wave and particle behaviours are simultaneously revealed. (b): An example of the imaginary element v-HWP. The arrows indicate the orientations of the fast axis.

Download Full Size | PDF

If we place a CCD camera at the outputs of the interferometers, we can observe fringe along $x$ axis at ports $a_1$ and $b_1$, which are output ports of a closed interferometer. While, no interference would be observed at ports $a_2$ and $b_2$, which are output ports of an open interferometer. Define the wave and particle states:

Equation (4) indicates that the vertically polarized photons behave as particle while the horizontally polarized photons behave as wave. The photons show the wave or particle behaviour periodically along $y$ axis. To simultaneously reveal $|particle\rangle$ and $|wave\rangle$ states, the states $|b_1\rangle$ and $|a_2\rangle$ are overlapped at $c1$. The output at $c1$ comes with incoherent mixture of the $|wave\rangle$ and $|particle\rangle$ states. The intensity distribution at $c1$ is:

If we place a CCD camera at $c1$, we can observe the continuous morphing between wave and particle natures. The intensity distribution would be similar to Fig. 1(c).

We should point out that our setup is analogous to the one in [33] which was performed at the single-photon level. The main differences are: first, we use a classical intense coherent beam as the input, and the WFSPs are inserted in only one arm of the interferometers in order to append different phases in different spatial positions; second, the setup in [33] produces a coherent superposition of the $|wave\rangle$ and $|particle\rangle$ states at the final BS, since the polarizations of the $|wave\rangle$ and $|particle\rangle$ states are identical by inserting an HWP oriented at $45^{\circ }$ in one path. In our experiment, the polarizations are orthogonal, thus we superpose the $|wave\rangle$ and $|particle\rangle$ states by a PBS, resulting in an incoherent mixture of the $|wave\rangle$ and $|particle\rangle$ states; third, a CCD camera is employed to acquire the final results in our experiment instead of single photon detectors.

3. Experimental results

Figure 2(a) shows the underlying protocol of our experiment, while Fig. 3 is the optical implementation of our experiment. Laser with 806 nm emitted from laser diode is coupled by a single mode fiber and collimated by a collimator (Thorlabs, TC12FC-780). We prepare the photons to be H-polarized by the first PBS. The second HWP oriented at 22.5$^\circ$ transforms the H-polarized state to $(|\textrm {H}\rangle +|\textrm {V}\rangle )/\sqrt {2}$ which is the input of the polarization interferometer. The polarization interferometer is composed of two beam deviators (BDs) with a WFSP inside (inserted along $y$ axis) and an HWP oriented at $45^{\circ }$. Note that since the v-HWP in Fig. 2(a) is difficult to fabricate, it is replaced by the purple region in Fig. 3 in the actual implementation. The WFSP is used to introduce different phases between H- and V-polarized photons at different spatial positions, i.e., the phase $\theta =k_yy$ which determines the manifestation of wave or particle in Eq. (4), due to the wedged angle. The output state of the interferometer is thus $[\exp (ik_yy)|\textrm {H}\rangle +|\textrm {V}\rangle ]/\sqrt {2}$ with additional constant phases. After the interferometer, an HWP oriented at $-22.5^{\circ }$ transforms the state to $[\cos (k_yy/2)|\textrm {V}\rangle +\sin (k_yy/2)|\textrm {H}\rangle ]/\sqrt {2}$. Then a PBS couples the polarization and path of photons, creating a polarization-path entangled state $|\psi _0\rangle$.

 

Fig. 3. Experimental setup. The purple region is the actual implementation of v-HWP which is composed of two HWPs and a polarization interferometer. The green region is a closed interferometer, where photons behave wavelike. The orange region is an open interferometer which consists of a BS only, where photons behave particlelike. The $|wave\rangle$ and $|particle\rangle$ states are superposed and observed by CCD cameras. HWP: half-wave plate, PBS: polarization beam splitter, BD: beam deviator, BS: beam splitter, M: mirror, WSFP: wedged fused silica plate.

Download Full Size | PDF

After the PBS, the H-polarized photons go through a Sagnac interferometer to manifest the wave nature. Another WFSP is placed along $x$ axis in one arm to create different phases at different spatial positions, i.e., the phase $\varphi =k_xx$ which produces interference fringe at the output. The photons would show the wave behaviour within the interferometer, and we observe the fringe along $x$ axis at the output. On the other hand, the V-polarized photons go through an open interferometer consisting of only a BS. It shows no dependence on $\varphi$, which means the photons have the particle nature. Note that, there should be one more WFSP inserted in one arm of this open interferometer, which is omitted in the experiment. Finally, the $|wave\rangle$ and $|particle\rangle$ states are superposed at a PBS, and a CCD camera captures the output intensity profile.

The Gaussian envelop of the single-mode light beam should be taken into account. First, we measure the beam profile at $c1$ by freely propagating the light at $c1$ without the optical elements. Then, we fit the intensity with Gaussian profile, the fitted beam waist is $w=122.2$ (the unit is pixels unless specified in this paper). To measure $k_y$ and $k_x$, the fringes $I_p$ and $I_w$ generated by the WFSPs are measured and fitted with Gaussian envelop:

Figure 4(a) is the measured periodic morphing between wave and particle behaviours by CCD through a single shot. The green lines represent particle regions where no interference emerges. The blue lines represent wave regions where full wave behaviour is observed. Figure 4(b) is the corresponding fitted curved surface with:

 

Fig. 4. (a): Experimental result of periodic morphing between wave and particle behaviours. (b): Theoretically fitted image of (a). (c): Subfigure of (a) demonstrating trade-off between distinguishability and visibility. The green (blue) lines in (a) are the particle (wave) regions where no (full) interference is observed when $\varphi$ changes. (d): Theoretically fitted image of (c).

Download Full Size | PDF

To show the morphing behaviour explicitly, we pick up an area in Fig. 4(a) which corresponds to one period with $\varphi$ ranging from 0 to $2\pi$ and $\theta$ ranging from $0$ to $\pi$ and plot it in Fig. 4(c). The subset shows decrease of the visibility from wave nature ($\theta =\pi$) to particle nature ($\theta =0$). But the intensity is not a straight line at $\theta =0$ due to the Gaussian envelop of the beam. Figure 4(c) is the same as Fig. 1(c) apart from an overall Gaussian envelop. Figure 4(d) is the corresponding fitted intensity distribution derived from Eq. (8).

The visibility $\mathscr{V}$ and the distinguishability $\mathscr{D}$ are shown in Fig. 5. Here, $\mathscr{V}$ and $\mathscr{D}$ are defined by:

 

Fig. 5. Trade-off between $\mathscr{V}$ and $\mathscr{D}$. The red dots and the black squares are experimental values of $\mathscr{V}$ and $\mathscr{D}$, respectively; each point represents one pixel in the image. The red and black lines are theoretical results calculated from the fitted curve Eq. (8). The discontinuity of $\mathscr{V}$ is due to the discontinuity of the extreme values of $I$.

Download Full Size | PDF

4. Discussion and conclusion

The theoretical values of $\mathscr{D}$ and $\mathscr{V}$ in Fig. 5 are derived from the fitted curved surface in Fig. 4(d). Fitting a curved surface causes a larger deviation compared to fitting a curve. This gives rise to the considerable deviation of the experimental values from the theoretical values as shown in Fig. 5. The extreme values of $I$ depend strongly on the Gaussian envelop, i.e., the beam waist and the beam center. This leads to the discontinuity of $\mathscr{V}$ in Fig. 5.

Our experiment, though presented in a way of classical optics, has reproduced the quantum mechanical statistics revealed in [26] which were addressed in quantum language, i.e., the trade-off between the distinguishability of the photons’ path and the interference visibility. This implies that the statistical distributions in [26] could be reproduced at the classical level, and perhaps it merely rests on the involved degrees of freedom, irrespective of quantum or classical natures of the physical carriers. In recent years, delayed-choice experiments have been investigated via causal model where quantum features are not involved [35–39]. It was shown that a classical causal model is capable of reproducing the quantum mechanical predictions revealed in quantum delayed-choice experiments. On the other hand, complementarity among different degrees of freedom was shown in [40] using a coherent light beam. The topic on where the quantum-classical border lies is of significance and worthy to be explored [41]. More experimental investigations of quantum issues by using a classical light beam can be found in the references in [41].

In conclusion, we have proposed a simulation of quantum delayed-choice experiment through a single shot, by employing a classical intense light beam as the input. Experimentally, we have observed the morphing between wave and particle natures of photons by using a CCD camera. The resulting image shows a good agreement with the fitted image. This experiment is quite simple because it does not require adjusting experimental parameters frequently or repeating measurements and only needs a single shot to capture the image. Since the morphing between wave and particle behaviours is observed within the exposure time which is several milliseconds, the phase fluctuation is negligible. Therefore this experiment is also robust against the noise. To the best of our knowledge, our experiment is the first one to inspect quantum duality utilizing a classical coherent light with finite beam waist.