Abstract
Quantum superposition is the cornerstone of quantum mechanics, where interference fringes originate in the selfinterference of a single photon via indistinguishable photon characteristics. Wheeler’s delayedchoice experiments have been extensively studied for the waveparticle duality over the last several decades to understand the complementarity theory of quantum mechanics. The heart of the delayedchoice quantum eraser is in the mutually exclusive quantum feature violating the causeeffect relation. Here, we experimentally demonstrate the quantum eraser using coherent photon pairs by the delayed choice of a polarizer placed out of the interferometer. Coherence solutions of the observed quantum eraser are derived from a typical Mach–Zehnder interferometer, where the violation of the causeeffect relation is due to selective measurements of basis choice.
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Introduction
The delayedchoice experiments proposed by Wheeler in 1978^{1} for the complementarity theory^{2} have been intensively studied over the last several decades^{3,4,5,6,7,8,9,10,11,12,13,14,15,16}. Although the original concept of the complementarity theory is for the exclusive nature between noncommutable entities such as position and momentum, delayed choice experiments have been developed for the measurement control of the waveparticle duality in an interferometric system^{3}. The waveparticle duality of a single photon shows a tradeoff relation between the wave naturebased fringe visibility and particle naturebased whichway information^{4}. The delayed choice experiments have been broadly demonstrated using thermal lights^{5}, entangled photons^{6,7,8}, atoms^{9,10,11}, neutrons^{3}, attenuated lasers^{4,12,13}, and antibunched single photons^{14,15}. In the delayed choice, a postcontrol of measurements results in a paradoxical phenomenon of violation of the causeeffect relation^{16}. The quantum eraser is based on the postchoice of measurements, choosing^{17} or erasing^{18} one of the natures. Recently, the quantum eraser has been developed for reversing a given nature via postmeasurements using entangled photons^{19}, coherent photons^{13,20}, thermal lights^{21}, and antibunched photons^{11,22}.
In the present paper, the delayedchoice quantum eraser was experimentally demonstrated using coherent photons via polarization basis controls, where the coherent photons are obtained from an attenuated continuous wave (cw) laser. Like some delayedchoice schemes^{13,14,18,19,21}, the present one is for the postcontrol of the predetermined photon nature. Here, our Mach–Zehnder interferometer (MZI) composed of a polarizing beam splitter (PBS) and a beam splitter (BS) is set for the particle nature according to the FresnelArago law^{23} or noninteracting quantum operators^{24}. Thus, the whichway information of a single photon inside the MZI is a predetermined fact, resulting in no interference fringes in the output ports of the MZI. Without controlling the MZI itself, however, we experimentally retrieve the wave nature of the photon by controlling the output photon’s polarization basis using a polarizer^{13,14,19,21}. If the postmeasurements show an interference fringe, it represents the violation of the causeeffect relation because the choice of the polarizer satisfies the spacelike separation. For this, we measured first and secondorder intensity correlations using a coincidence counting unit.
Experimental setup
Figure 1 shows the schematic of the present delayedchoice quantum eraser using coherent photons generated from an attenuated cw laser (see “Methods” section). For Fig. 1, a coincidence counting unit (CCU, DE2; Altera) is used for both first and secondorder intensity correlations between two detectors D1 and D2 (SPCMAQRH15, Excelitas). For the secondorder correlation, only doubly bunched photons are counted by CCU, where the generation ratio of doublybunched photons to single photons is ~ 1% at the mean photon number \(\langle n\rangle \sim 0.01\) (see Sect. A of the Supplemental Materials). For the firstorder intensity correlation, both input channels of CCU from D1 and D2 are measured individually for a period of 0.1 s per data point (see Fig. 2). The higherorder bunched photons are neglected by Poisson statistics (see Sect. A of the Supplemental Materials). To provide polarization randomness of a single photon, a \(22.5^\circ\)rotated halfwave plate (HWP) is placed just before the MZI. By the following PBS, the single photon inside the MZI shows distinguishable photon characteristics with perfect whichway information: \({\psi \rangle }_{MZI}=\frac{1}{\sqrt{2}}\left({V\rangle }_{UP}+{H\rangle }_{LP}\right)\). Thus, the measured photons outside the MZI show the predetermined particle nature of a single photon (not shown), as in refs.^{13,14,19}.
Due to the predetermined distinguishable photon characteristics of the particle nature, the MZI does not result in a \(\mathrm{\varphi }\)dependent interference fringe for the output photons (\({E}_{1};{E}_{2}\)). As demonstrated^{14,19}, this is due to noninterfering quantum operators^{24} or simply by the FresnelArago law^{23}. Due to the classical physics of the causeeffect relation, the action of polarizers (Ps) outside the MZI for the output photons (\({E}_{1};{E}_{2}\)) should not change the predetermined photon nature inside the MZI. To satisfy the spacelike separation, the length of each arm of the MZI is set to be 2 m, corresponding \(>6\mathrm{ ns}\) in the delayed choice of P. Regarding the temporal resolution (\(<1\mathrm{ ns}\)) of the single photon detector as well as the CCU (6 ns), the condition of the spacelike separation is satisfied. Thus, any violating measurements should belong to the quantum mystery of the delayedchoice quantum eraser.
The polarizer’s rotation angle \(\uptheta\) is with respect to the vertical axis \(\widehat{\mathrm{y}}\), as shown in the Inset. \({E}_{0}\) denotes an amplitude of a single photon. The mean photon number is set at \(\langle n\rangle \sim 0.01\) to satisfy incoherent and independent conditions of statistical measurements, resulting in the mean photontophoton separation (600 m) far greater than the coherence length (3 mm) of the cw laser (see Sect. A of the Supplemental Materials). Doublybunched photon pairs are also satisfied for this condition. Thus, the measurements of Fig. 1 are for a statistical ensemble of single photons controlled by Ps.
For the MZI phase control \(\mathrm{\varphi }\), the pathlength difference \((\Delta \mathrm{L})\) is adjusted to be far less than the coherence length \({l}_{c}\) (3 mm). This MZI coherence condition is easily tested for the same polarizationbased MZI interference. Thus, the MZI in Fig. 1 satisfies a general scheme of singlephoton (noninterfering) interferometers^{25}. Each output photon (\({E}_{1}\) or \({E}_{2}\)) from the MZI can be represented by a superposition state of the orthonormal polarization bases at equal probability amplitudes: \({\psi \rangle }_{out}=\frac{1}{\sqrt{2}}\left(V\rangle {e}^{i\varphi }+H\rangle \right)\). This polarizationbasis randomness of the MZI output photons originates in the random polarization bases provided by the \(22.5^\circ\)rotated HWP. In ref.^{14}, the measurement control with Ps in Fig. 1 is replaced by a linear opticscombined electrooptic modulator (EOM) system. By this EOM switching module, the same MZI scheme as in Fig. 1 is satisfied for the postcontrol of output photons^{14}. Classical photon cases have also been discussed for the same results of the quantum eraser^{20,21}, where different analyses have been separately presented^{5,11,22}.
Analysis
To coherently interpret the delayedchoice quantum eraser in Fig. 1, the PBSBS MZI is analyzed using a coherence approach:
where \(\left[BS\right]=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& i\\ i& 1\end{array}\right]\) and \(\left[\Phi \right]=\left[\begin{array}{cc}1& 0\\ 0& {e}^{i\varphi }\end{array}\right]\)^{26}. \({E}_{0}\) is the amplitude of a single photon. \(\widehat{V}\) (\(\widehat{H}\)) represents a unit vector of the vertical (horizontal) polarization component of the input photon \({E}_{0}\): \(V\rangle =\widehat{V}{E}_{0}\) and \(H\rangle =\widehat{H}{E}_{0}\). The inputs of \({E}_{1}\) and \({E}_{2}\) by the \(22.5^\circ\)rotated HWP and PBW are analyzed in Sect. D of Supplementary Material using Mueller matrix: \({{\varvec{E}}}_{01}=i\widehat{V}\frac{{E}_{0}}{\sqrt{2}}\); \({{\varvec{E}}}_{02}=\widehat{H}\frac{{E}_{0}}{\sqrt{2}}\). The role of the \(22.5^\circ\)rotated HWP is to give an equal probability amplitude of orthogonally polarized photons to PBS. Here, the 4 × 1 matrix of pathpolarization tensor products reduces down to a 2 × 1 matrix by PBS, resulting in the vertical (horizontal) polarizationupper (lower) path correlation. The coherence approach of Eq. (1) is for the wave nature of a photon, resulting in no photon number dependent. Instead, phase information is critical^{13,20,21}. Most importantly, interference between the \(\widehat{H}\) and \(\widehat{V}\)polarizations of a photon on the BS shows independent photon characteristics in both output ports (\({E}_{1}\); \({E}_{2}\)) due to noninteracting orthogonal polarization bases^{23,24}. Thus, the calculated mean intensities of \({E}_{1}\) and \({E}_{2}\) in Eq. (1) are \(\langle {I}_{1}\rangle =\langle {I}_{2}\rangle =\langle {I}_{0}\rangle /2\), regardless of \(\mathrm{\varphi }\), where \({I}_{0}={E}_{0}{E}_{0}^{*}\). These are the coherence solutions of the PBSBS MZI for the particle nature of a single photon with perfect whichway information, resulting in distinguishable photon characteristics.
By inserting a polarizer (P) outside the MZI, Eq. (1) is coherently rewritten for the polarization projection on P (see Inset of Fig. 1):
where \(\theta\) is the rotation angle of P. Thus, Eqs. (2) and (3) represent polarization projections of the output photon onto the polarizers: \(\widehat{V}\to \widehat{p}cos\theta\) and \(\widehat{H}\to \widehat{p}sin\theta\). Here, the positive \(\uptheta\) is for the clockwise direction from the vertical axis of the photon propagation direction (z) (see the Inset of Fig. 1). For the negative rotation, however, the projections are denoted by \(\widehat{V}\to \widehat{p}cos\theta\) and \(\widehat{H}\to \widehat{p}sin\theta\). The projection onto the polarizer P represents the action of the delayed choice for the quantum eraser.
The calculated mean intensities of Eqs. (2) and (3) are as follows:
Equations (4) and (5) are the analytical solutions of the quantum eraser in Fig. 1 (see also Fig. 2). Here, the MZI coherence is for every single photon, resulting in the selfinterference in the MZI^{25}. Due to the low mean photon number, no coherence exists between consecutive photons, satisfying the condition of a statistical ensemble. For \(\mathrm{\uptheta }=0\), the original distinguishable photon characteristics appear with no interference fringes regardless of φ.
For \(\uptheta =\pm \frac{\uppi }{4} \left(\pm 45^\circ \right)\), Eqs. (4) and (5) are rewritten for the firstorder intensity correlation:
For Eqs. (6) and (7), the same Pprojected photon measurements have been demonstrated in refs.^{14,15} for single photons and a polarizer in ref.^{19}, resulting in the quantum eraser using entangled photons. Although the EOM block control looks like a direct control of the MZI^{14}, it corresponds to the combination of PBS and P in Fig. 1 (see Sect. B of the Supplemental Materials). In SPDC processes, entangled photons automatically satisfy both \(\pm\) signs in Eqs. (6) and (7) via spatial mixing of the signal and idler photons^{27}. This is the fundamental difference between coherent photons and entangled photon pairs for the quantum eraser^{28}. The sum of the polarization bases in Eqs. (6) and (7), thus, corresponds to the entangled photonpair case, as long as it deals with the firstorder intensity correlation^{19}. Regarding the causality violation, thus, Eqs. (6) and (7) witness the quantum feature of the delayedchoice quantum eraser for Fig. 1. Total intensity through Ps is uniform at 50% photon loss regardless of the angle of the polarizers. This selective measurement by P at the cost of 50% event loss is the origin of the quantum eraser, as differently argued for no choice of quantum eraser^{29}.
The secondorder intensity correlation \({R}_{AB}\) via coincidence detection between D1 and D2 in Fig. 1 shows the intensity product between Eqs. (6) and (7):
where a doublybunched photon pair relates to \(2{I}_{0}\). Compared with ref.^{19} based on entangled photons, the doubled oscillation in Eq. (8) is due to the outofphase fringes in D1 and D2, resulting in a classical nature. Unlike coincidence detectioncaused nonlocal correlation, Eq. (8) is not for the quantum feature of a jointphase relation^{28}. This is because there is no such jointphase action by polarizers (discussed elsewhere)^{30}.
Experimental results
The upper panels of Fig. 2 show the experimental proofs of the delayedchoice quantum eraser in Fig. 1 for coherent single photons measured by D1 and D2, respectively, for two different \(\mathrm{\theta s}\). As expected from Eqs. (6) and (7), fringes appear in both measurements for \(\uptheta =\pm 45^\circ\). However, no fringe appears for \(\uptheta =0^\circ ;90^\circ\), as expected by Eqs. (4) and (5) (see the overlapped green and black lines). The observed fringes represent the wave nature of the photon inside the MZI in Fig. 1. The statistical error (standard deviation) in single photon measurements is less than 1% (see Sect. A of Supplemental Materials). This is a big benefit of using coherent photons from a stabilized laser compared to entangled photons from spontaneous parametric downconversion process (SPDC) or antibunched photons from NV color centers, whose respective photon counts are less than 10%^{19} and 1%^{14} of Fig. 2. Because the PBMZI is not actively stabilized, most errors are from the air turbulence affecting MZI path lengths. Under normal lab conditions, the PBMZI is stabilized for as long as a few minutes, where the total data collection time of each panel in Fig. 2 is 36 s (see Sect. C of Supplementary Materials).
The lower left panel of Fig. 2 is for coincidence detection for the upper panels (color matched). The photon counts for the coincidence detection in the lower left panel are less than 1% of those in the upper left panel of single photons. This is due to Poisson statistics for \(\langle n\rangle \sim 0.01\). As expected in Eq. (8) for the coherence product, the doubled fringe oscillation period is the direct result of the intensity product between them showing the classical nature. This intensity product of the lower left panel has nothing to do with the nonlocal quantum feature due to different purposes without independent local control parameters^{19,30}.
The lower right panel of Fig. 2 is for the incoherence condition of each photon by setting the MZI pathlength difference (\(\mathrm{\Delta L}\)) far greater than the coherence length \({l}_{c}\) of the laser. As shown, the single photon’s coherence in the MZI is the key to the quantum eraser. This fact has never been discussed seriously so far, even though it seems to be obvious^{16}. The observed fringes in Fig. 2 for the firstorder intensity correlation demonstrate the same mysterious quantum eraser^{14} because the predetermined particle nature of the photon inside the MZI (see the green line) cannot be controlled or changed by the postmeasurements of the output photons^{13,14,19}. Due to the benefit of coherence optics, the observed visibilities in the upper panels of Fig. 2 are near perfect.
Conclusion
The delayedchoice experiments were conducted for the quantum eraser via postcontrol of polarization basis of coherent photons in a coincidence detection scheme for the firstorder intensity correlation. Corresponding coherence solutions were also derived in the same setups for the quantum eraser. Like conventional delayedchoice quantum erasers using orthogonal polarization bases, predetermined photon characteristics of the particle nature were retrospectively converted into the wave nature via postselected polarizationbasis projection, resulting in the violation of the causeeffect in classical physics, where the predetermined whichway information of photons was completely erased by the postchoice of the polarizer satisfying the spacelike separation. The cost of the postmeasurements by the polarizer is a 50% loss of measurement events. As usual in nonlocal quantum features, the observed quantum eraser was also due to the selective measurements of the mixed polarization bases.
Methods
In Fig. 1, the laser L is SDL532500 T (Shanghai Dream Laser), whose center wavelength and coherence length are 532 nm and 3 mm, respectively. The laser light is vertically polarized. For the random but orthogonal polarizations of a single photon, a halfwave plate (HWP) is rotated by 22.5 degrees from its fast axis. For a single photon, the laser L is attenuated by neutral density filters, satisfying Poisson distribution (see Supplementary Materials). The measurements for both output photons from the MZI are conducted by CCU (DE2; Altera) via a set of single photon detectors D1 and D2 (SPCMAQRH15, Excelitas). The dead time and dark count rate of the single photon detectors are 22 ns and 50 counts/s, respectively. The resolving time of the single photon detector is ~ 350 ps, whose converted electrical pulse duration is ~ 6 ns. For the polarization projection by Ps in Fig. 1, four different rotation angles are set (− 45, 0, 45, or 90 degrees) to the clockwise direction with respect to the vertical axis of the light propagation direction. The photon counts for each data point in Fig. 2 are measured by CCU for 0.1 s and calculated by a homemade Labview program.
In Fig. 2, the mean photon number is set at \(\langle n\rangle \sim 0.01\). The maximum number of measured single photons in each MZI output port is ~ a half million per second, resulting in the mean photontophoton distance of 600 m. Compared with the laser’s coherence length of 3 mm, it is clear that the measured single photons are completely independent and incoherent among them. On behalf of the polarizing beam splitter (PBS), perpendicularly and horizontally polarized components of an incident photon are separated into the upper (UP) and lower paths (LP), respectively. Both split components of a single photon are recombined in the BS, resulting in PB (PBSBS)MZI. Thus, the photons in the PBMZI in Fig. 1 behave as the particle nature, resulting in no interference fringes in the output ports. In other words, the photons inside the MZI represent perfect whichway information or distinguishable characteristics.
The length of each arm of the PBMZI is set at 2 m, and the pathlength difference between UP and LP is kept to be far less than 3 mm to satisfy the coherence condition of each photon. This coherence condition is essential for delayedchoice quantum eraser experiments. The \(\mathrm{\varphi }\) phase control of the PBMZI is conducted by a piezoelectric optic mount (PZT; KC1PZ, Thorlabs) connected by a PZT controller (MDT693A, Thorlabs) and a function generator (AFG3021, Tektronix). For Fig. 2, the data is measured under the \(\mathrm{\varphi }\) scanning mode, where the phase resolution is \(\frac{2\uppi }{180}\) radians. Thus, Fig. 2 has 180 data points for a 2 \(\uppi\) cycle of \(\mathrm{\varphi }\) (see Table S1 of the Supplementary materials). The BS position for the recombination of two split components of a single photon is welladjusted for a complete overlap between them.
Data availability
All data generated or analyzed during this study are included in this published article.
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Funding
This research was supported by the MSIT (Ministry of Science and ICT), Korea, under the ITRC (Information Technology Research Center) support program (IITP2023–20210–01810) supervised by the IITP (Institute for Information & Communications Technology Planning & Evaluation). BSH also acknowledges that this work was also supported by GISTGRI 2023.
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S.K. conceived the idea, conducted experiments, and provided the data. B.S.H. developed the idea, analyzed the data, and wrote the manuscript.
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Kim, S., Ham, B.S. Observations of the delayedchoice quantum eraser using coherent photons. Sci Rep 13, 9758 (2023). https://doi.org/10.1038/s41598023365907
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DOI: https://doi.org/10.1038/s41598023365907
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