In 1928, Bohr proposed the “complementary principle” to elucidate the “wave-particle duality” of light^[1]. This principle posits that whether a photon exhibits wave-like or particle-like behavior depends on whether the path information could be discriminated. Extensive research has been conducted in order to fully comprehend and elucidate this fascinating phenomenon, with two of the most prominent studies being the quantum delayed choice (QDC) and the quantum eraser (QE)^[2,3] experiments. The QDC experiment was first conceived by Wheeler^[4], in which the choice of particle measurement or interferometry was made after the photon had already entered the interferometer to rule out the possibility of predicting which measurement it would encounter. A further assumption was that if a photon carrying path information were to leave the interferometer and subsequently erase its path information, what would be the result? Soon after, Scully and Drühl proposed that a QE can erase the which-path information even after the quantum itself had left the interferometer and determined its early behavior as wave-like or particle-like^[5,6]. Since then, several QE experiments have been conducted^[7–15]. These foundational experiments not only have corroborated theoretical propositions but also have contributed significantly to refining our comprehension of wave-particle duality.

The QDC and QE experiments can be conducted using entangled photons^[9,16,17], thermal light^[18], single photons^[19,20], attenuated lasers^[21], and other particles such as atoms^[22–24] and electrons^[25]. When adopting photons to investigate the QE phenomenon, the polarization^[26] basis is commonly employed to facilitate path information identification due to its flexible modulation capability^{[8,16,21,27–31]}. Beyond polarization, the phase distribution of photons, notably the orbital angular momentum (OAM) of light^[32–35], has substantial value in various fields, such as high-dimensional quantum communication^[36–40] and atmospheric free-space optical communication^[41–43]. In a previous important work, QE experiments were conducted to construct abstract OAM state channels using entangled photon pairs, which significantly influenced the interference patterns^[31]. However, up to now, employing the photonic phase structure to study QE has not been explored.

In this paper, we experimentally validate quantum erasure based on the OAM of light. We employ a Mach–Zehnder interferometer (MZI) equipped with a first-order spiral phase plate (SPP) inserted into one arm to impart OAM to photons and determine their which-way information. Consequently, the photons exhibit particle-like behavior without interference. By introducing another SPP outside the MZI to erase the photon’s phase information, we observed high-contrast interference in a post-measurement scenario. These results challenge the classical causal relationships, as the post-selection of the SPP involves space-like separation^[21,44].

The experimental setup to investigate the OAM-based quantum erasure is depicted in Fig. 1. A 795 nm external-cavity diode laser (ECDL) is attenuated to a single-photon level (10,000 photons/s) via an attenuator plate (AP) and coupled into a single-mode fiber (SMF). The photon then enters a MZI comprising two beam splitters (BS1 and BS2) and two mirrors (M1 and M2). The MZI’s arms are labeled A1 and A2. In arm A2, we use a first-order SPP, which can modulate the photon’s plane wave into a spiral wavefront^[33,34], to transform a photon with the Gaussian mode $|0⟩$ into the OAM state $|1⟩$. The basis vector $|l⟩$ denotes the quantum state of OAM light, where $l$ is the OAM quantum number indicating that each photon carries an angular momentum of $l\hslash$^[32–35]. A piezoelectric lead zirconate titanate (PZT, Thorlabs-Low Voltage Piezoelectric Chips-PA4FKW) attached to M1, controlled by a function generator (not shown), precisely adjusts M1’s position to control the relative phase $\varphi$ between A1 and A2. Photons passing through A1 and A2 are coherently superposed at BS2. After that, these photons are detected or correlated by the single photon count detectors (SPCDs, Excelitas-SPCM-AQRH-14), which can convert photons into electrical signals, and SPCDs are linked with SMFs through two couplers (C3 and C4). Two first-order SPPs (SPP2 and SPP3) are alternatively inserted before C3 and C4 to erase the OAM phase information.

Considering the inversion of the OAM mode due to reflection ($|l⟩$ becomes $|-l⟩$ after reflection), the quantum states of the optical field in P3 and P4 are given by $|{\psi }_{P3}⟩=(e^{i\pi }{|0⟩}_{A1}+e^{i\varphi }{|-1⟩}_{A2})/\sqrt{2}$ and $|{\psi }_{P4}⟩=({|0⟩}_{A1}+e^{i(\varphi +\pi )}{|1⟩}_{A2})/\sqrt{2}$, respectively. Here, the subscripts A1 and A2 denote the paths associated with the base states. Photons in the state $|0⟩$ exhibit a flat equiphase surface, while those in $|1⟩$ exhibit a helical equiphase surface; for $|-1⟩$, the helical direction is opposite. Detection of a flat equiphase surface outside the MZI implies a photon passed through A1, whereas a helical equiphase surface indicates passage through A2.

When SPP2 and SPP3 are absent from the optical path, photons are directed toward couplers C3 and C4 connected to the SMF. In this way, only the state $|0⟩$ enters the couplers, corresponding to the measurement of the photon projection ${|⟨0|\psi ⟩|}^2$. We then can detect photons only from A1 that are stochastically knocking on the SPCD1 or SPCD2. During the data collection period with SPCD1 and SPCD2, both detectors are connected to a coincidence counting unit (CCU), as illustrated in Fig. 1, to measure the multi-photon coincidence rate. To ensure the validity of the experiment, it is crucial to suppress multi-photon interference. This is achieved by precisely controlling the intensity of the coherent light to reduce the multi-photon rate to $1.25\times {10}^{-4}$ per the counting interval. At this level, the impact of multi-photon effects on the single-photon interference data is negligible, thereby ensuring the reliability of the results. A 200 mHz triangular wave voltage applied to the PZT via a function generator precisely adjusts M1’s position, linearly modulating the relative phase of the MZI arms. By temporarily removing the AP, a high-intensity (10 mW) light can be introduced to facilitate precise alignment of each component to its most suitable position; thus, the phase difference $\varphi$ of MZI can be calibrated. As shown in Fig. 2, the total photon count registered by SPCD1 and SPCD2 remains constant throughout the phase-scanning period. This observation indicates that when a photon is detected passing through path A1, it exhibits particle-like properties either shooting to SPCD1 or SPCD2, and the light intensity remains unaffected by the phase $\varphi$ of the MZI. Minor variations arise primarily due to unavoidable environmental disturbances in the experiment and the inherent distribution of the coherent state light field, represented as the Fock state ${\sum }_{n=0}^{\infty }{C}_n|n⟩$$(C_1\approx 1)$^[45].

Taking a step further, in the scenario where SPP2 and SPP3 are positioned in front of couplers P3 and P4, the precise adjustment of SPP2 (reverse direction: $|l⟩\to |l-1⟩$) and SPP3 (forward direction: $|l⟩\to |l+1⟩$) ensures photon transmission through their centers. In this way, the states after SPP2 and SPP3 are described as $|{\psi }_{P4}^{{\mathrm{spp}}_2}⟩=({|-1⟩}_{A1}+e^{i(\varphi +\pi )}{|0⟩}_{A2})/\sqrt{2}$ and $|{\psi }_{P3}^{{\mathrm{spp}}_3}⟩=(e^{i\pi }{|1⟩}_{A1}+e^{i\varphi }{|0⟩}_{A2})/\sqrt{2}$. Here, the state vector ${|0⟩}_{A2}$ is transferred from $|\pm 1⟩$, and ${|\pm 1⟩}_{A1}$ originates from $|0⟩$, such that detection by an SPCD indicates the photon passage through A2. Photon counts with the variable relative phase $\varphi$ in the measurements of $|1⟩$ and $|-1⟩$ are shown in Fig. 3.

The constancy of photon number regardless of the phase $\varphi$ indicates particle-like behavior. Photon number fluctuations increase slightly compared to those in Fig. 2 due to positional displacements when SPP1 is sequentially connected with SPP2 or SPP3. Precise coaxial alignment at their centers is crucial for photons to project onto $|1⟩$ or $|-1⟩$. However, under realistic experimental conditions, due to the inherent imperfections in SPP mode conversion, coupled with the spontaneous diffusion of photons throughout their propagation and the inevitable inaccuracies in relative angular alignment, photons are always projected imperfectly onto states such as $\frac{c_0|0⟩+c_1|1⟩}{\sqrt{2}}$ or $\frac{c_0|0⟩+c_1|-1⟩}{\sqrt{2}}$ (where $c_1\gg c_0$). This imperfect projection erases phase information entangled with path information, resulting in a slightly wave-like behavior.

The preceding experiments demonstrate that once the propagation path of the photon is determined, it exhibits particle-like behavior without interference. Now, let us explore the use of SPP2 and SPP3 to erase path information. In contrast to the initial setup where the first-order SPP2 and SPP3 were positioned at the centers of the P3 and P4 paths, we now shift both SPP2 and SPP3 by a distance of $r/2$ relative to the center of the Gaussian beam ($r$ denotes the radius of the Gaussian beam, and $r=2.5\mathrm{mm}$). The operations corresponding to these SPPs are described as ${\widehat{S}}_{i/2}|l⟩=(|l⟩+|l+i⟩)/\sqrt{2},i=\pm 1$^[33,34]. If SPP2 and SPP3 are properly shifted to allow the transformation of a photon in the state $|0⟩$ into the state $\frac{|0⟩+|1⟩}{\sqrt{2}}$ ($\frac{|0⟩+|-1⟩}{\sqrt{2}}$), then for a photon in the state $|{\psi }_{P4}⟩$ ($|{\psi }_{P3}⟩$), the following relationship holds: $$\begin{gathered} |{\psi }_{P4}^{-1/2}⟩={\widehat{S}}_{-1/2}|{\psi }_{P4}⟩=\frac{1}{\sqrt{2}}({\widehat{S}}_{-1/2}{|0⟩}_{A1}-e^{i\varphi }{\widehat{S}}_{-1/2}{|+1⟩}_{A2}) \\ =\frac{1}{2}{|-1⟩}_{A1}+\frac{1}{2}(1-e^{i\varphi }){|0⟩}_{\mathrm{erased}}-\frac{1}{2}{e}^{i\varphi }{|+1⟩}_{A2}, \end{gathered}$$(1)$$\begin{gathered} |{\psi }_{P3}^{1/2}⟩={\widehat{S}}_{1/2}|{\psi }_{P3}⟩=\frac{1}{\sqrt{2}}(-{\widehat{S}}_{1/2}{|0⟩}_{A1}+e^{i\varphi }{\widehat{S}}_{1/2}{|-1⟩}_{A2}) \\ =\frac{1}{2}{e}^{i\varphi }{|-1⟩}_{A1}-\frac{1}{2}(1+e^{i\varphi }){|0⟩}_{\mathrm{erased}}-\frac{1}{2}{|+1⟩}_{A2}. \end{gathered}$$(2)

As illustrated in Fig. 1(b), an optical field initially in a superposition of $|0⟩$ and $|1⟩$ undergoes translation (deviation from the center) after passing through a displaced SPP, affecting its basis states $|0⟩$ and $|1⟩$. This results in a superposition state that includes $|-1⟩$, $|0⟩$, and $|1⟩$, where the basis state $|0⟩$ acquires a relative phase factor $e^{i\varphi }$ due to the MZI. Introducing SMF allows the projection of $|{\psi }_{P4}^{-1/2}⟩$ and $|{\psi }_{P3}^{1/2}⟩$ onto $|0⟩$, effectively erasing the phase structure and path information. If the SPCD responds and it is impossible to determine the photon’s path from the phase structure information, the photon will exhibit wave-like behavior. The experimental results are presented in Fig. 4. The number of photons exhibited clear wave-like fluctuations, and interference fringes appeared. The photon counts were fitted to a cosine function, and the relationship between the photon count and the phase $\varphi$ was obtained from the theoretical prediction: ${|⟨0|{\widehat{S}}_{1/2}|{\psi }_{P3}⟩|}^2=(1-\cos \varphi )/2$. The visibility of the interference pattern can be quantified by the following equation: $V=(I_{\max }-I_{\min })/(I_{\max }+I_{\min })=84.35\%\pm 1.7\%$, where $I_{\min }$ and $I_{\max }$ represent the minimum and maximum values of the photon counting rate with respect to the phase $\varphi$. The imperfect interference arises from practical experimental factors, similar to the factors contributing to the imperfect projection seen in Fig. 2.

In conclusion, our experiments have successfully demonstrated quantum erasure based on the OAM photons, a phenomenon that aligns with the prevailing theoretical frame in quantum mechanics. By inserting an SPP into one arm of an MZI, we determined the photon’s path information, resulting in particle-like behavior. Outside the MZI, positioning additional SPPs at specific locations in the output port erased the photon’s path information, leading to wave-like behavior. In this scenario, manipulating the positioned SPPs induced interference in the state of the single-photon wave packet due to its phase structure.

In the classical physical picture, the SPPs, which satisfy the space-like separation, at the output port of the MZI would not affect the behavior of the single photon within the MZI. However, our results have challenged this classical cause-effect relationship and further extended the photon phase structure, i.e., single photon phase front, into the realm of quantum erasure experimentally. This finding can be generalized to various phase structure conditions in quantum erasure effects, thereby expanding and consolidating our understanding of quantum physics. Additionally, the method used to erase the photon’s OAM information in this work may be extended to applications such as optical measurement techniques and optical communication^[41–43].