PHYSICAL REVIEW A 103, 062213 (2021) Classical model of a delayed-choice quantum eraser Brian R. La Cour * and Thomas W. Yudichak Applied Research Laboratories, The University of Texas at Austin, P.O. Box 8029, Austin 78713-8029, Texas (Received 9 January 2021; accepted 27 May 2021; published 9 June 2021) Wheeler’s delayed-choice experiment was conceived to illustrate the paradoxical nature of wave-particle duality in quantum mechanics. In the experiment, quantum light can exhibit either wavelike interference patterns or particlelike anticorrelations, depending upon the (possibly delayed) choice of the experimenter. A variant known as the quantum eraser uses entangled light to recover the lost interference in a seemingly nonlocal and retrocausal manner. Although it is believed that this behavior is incompatible with classical physics, here we show that, using postselection, the observed quantum phenomena can be reproduced by adopting a simple deterministic detector model and supposing the existence of a random zero-point electromagnetic field. DOI: 10.1103/PhysRevA.103.062213 I. INTRODUCTION iment, but it fails to describe a variant of this experiment using variable phase delays in the arms of the interferometer. Wave-particle duality is one of the oldest and most per- This variant has been the subject of recent experimental in- plexing aspects of quantum theory [1]. Although the wavelike vestigations, which are consistent with theoretical predictions nature of light had been well established by the 19th cen- [20–22]. These experiments place certain dimensional restric- tury, experiments of the early 20th century brought about tions on the class of nonretrocausal hidden-variable models the notion of light as a particle, what we now call a photon that can be consistent with theory and observations. [2–4]. Maintaining this notion of light as being composed of In this paper, we revisit Wheeler’s delayed-choice experi- discrete particles can, however, be rather paradoxical at times, ment, its recent experimental variants, and the more elaborate as numerous real and gedanken experiments have shown quantum eraser experiment within the context of a sim- [5–10]. ple, physically motivated classical model [23,24]. Although One particular experiment that has captured recent interest loophole-free experiments have already been performed to and attention is the delayed-choice quantum eraser. First con- rule out local realism [25–27], the identification of precisely ceived by Scully and Drühl in 1982 [11], the quantum eraser which phenomena defy a classical interpretation remains an is a variant of Wheeler’s delayed choice experiment in which open question and one of practical relevance to ensure the measurements on one of a pair of entangled light beams are security and efficacy of emergent quantum technologies. Our used to recover an interference pattern, and hence wavelike approach is modeled after stochastic electrodynamics (SED) behavior, that would otherwise be lost with the introduction in assuming a reified (i.e., nonvirtual) zero-point field (ZPF) of which-way path information in a Mach-Zehnder interfer- corresponding to the vacuum state [28,29]. A significant de- ometer [12–16]. For Wheeler, the delayed-choice experiment parture from standard SED in our approach is the introduction was an argument for antirealism, the notion that quantum of a deterministic model of detectors using an amplitude objects, such as photons, do not have definite, intrinsic prop- threshold crossing scheme. We find that these simple assump- erties that are independent of the measurement context [17]. tions, combined with standard experimental postselection and Some, however, have interpreted the results of delayed-choice data analysis techniques, adequately describe the observed quantum eraser experiments as evidence for a form of retro- quantum phenomena. causality [18]. The structure of the paper is as follows: In Sec. II we More recently, a series of delayed-choice experiments has consider a simple delayed-choice experiment using weak been performed that rule out a certain class of nonretrocausal coherent light as a notional single-photon source. We re- hidden-variable models described by Chaves, Lemos, and place the coherent light with a source of entangled light Pienaar [19]. The general model they describe can provide in Sec. III, within the context of a quantum eraser experi- a causal description of the standard delayed-choice exper- ment, and demonstrate how postselection, not causality, is the mechanism whereby path information is effectively erased. With these two results established, we revisit the theoretical *[email protected] arguments of Chaves and Bowles in Sec. V and argue that their assumptions are overly restrictive. Section VI considers Published by the American Physical Society under the terms of the a variant of these experiments using entangled light sources Creative Commons Attribution 4.0 International license. Further and shows that these, too, can be understood within a clas- distribution of this work must maintain attribution to the author(s) sical framework. Finally, we summarize our conclusions in and the published article’s title, journal citation, and DOI. Sec. VII. Figures and numerical experiments were created 2469-9926/2021/103(6)/062213(12) 062213-1 Published by the American Physical Society BRIAN R. LA COUR AND THOMAS W. YUDICHAK PHYSICAL REVIEW A 103, 062213 (2021) and performed using a custom simulation tool, the Virtual stacked Jones vectors as follows: Quantum Optics Laboratory (VQOL) [30]. ⎡ ⎤ ⎡ ⎤ aH aH + bH ⎢ aV ⎥ ⎢aV + bV ⎥ ⎢ ⎥ BS1 1 ⎢ ⎥ ⎢−−⎥ −−→ √ ⎢−−−−⎥. (3) II. SIMPLE DELAYED-CHOICE EXPERIMENT ⎣ ⎦ 2⎣ ⎦ bH aH − bH Consider the Mach-Zehnder interferometer of Fig. 1.A bV aV − bV laser (LAS) provides a source of coherent, horizontally polar- Note that the second term on the right-hand side is again ized light that is strongly attenuated by a neutral density filter a standard complex Gaussian random vector, owing to the (NDF) before entering the first beam splitter (BS1). Under unitarity of the beam splitter transformation. our model, the laser light exiting the NDF is represented by The right-traveling beam next undergoes a transformation a stochastic Jones vector of the form via a half-wave plate (denoted by HWP) with a fast-axis angle θ ∈ [0,π/4] relative to the horizontal axis. It subsequently a α z a = H = + σ 1H , (1) undergoes a phase delay (denoted by PD) that applies a global a 0 0 z V 1V phase angle φ ∈ [0, 2π]. Finally, a pair of mirrors (denoted by M1 and M2) swap the two spatial modes. The resulting α ∈ where C describes√ the mean amplitude and phase of the stacked Jones vector after these three transformations is now light, σ0 = 1/ 2 is the scale of the ZPF (corresponding to ⎡ ⎤ ⎡ ⎤ 1 a ω aH + bH H a modal energy of 2 h¯ ), and z1H and z1V are independent ⎢a + b ⎥ ⎢ a ⎥ standard complex Gaussian random variables. (We say that z 1 ⎢ V V ⎥ , , , ⎢ V ⎥ = √ ⎢−−−−⎥ −HWP−−−−−−−−−PD M1 M2→ ⎢−−⎥, is a standard complex Gaussian random variable if E[z] 0, ⎣ ⎦ ⎢ ⎥ (4) 2 2 2 − ⎣ ⎦ E[|z| ] = 1, and E[z ] = 0). Note that z1H and z1V play the aH bH bH aV − bV role of hidden variables. This model is mathematically equiv- bV alent to the corresponding quantum coherent state |α⊗|0 √ √ = − / = − / whose Wigner function is a bivariate Gaussian probability where aH (aH bH ) 2, aV (aV bV ) 2, and density function identical to that of a. The fact that we are eiφ = √ θ + + θ + , using explicit random variables in our representation amounts bH [cos 2 (aH bH ) sin 2 (aV bV )] (5a) to a reification of the ZPF. The effect of the NDF is to ensure 2 |α| eiφ that 1. = √ θ + − θ + . / bV [sin 2 (aH bH ) cos 2 (aV bV )] (5b) Passage through the first 50 50 beam splitter (denoted by 2 BS1) splits the beam into two orthogonal spatial modes, a In the absence of the second beam splitter (denoted by right-traveling mode (denoted by →) and a down-traveling BS2), the Jones vectors (a , a )T and (b , b )T will deter- mode (denoted by ↓). In addition, there is a down-traveling H V H V mine if a detection is made on the right-traveling mode (by vacuum mode that enters the top input port of BS1, repre- detector D1) or the down-traveling mode (by detector D2). sented by the independent stochastic Jones vector We adopt an amplitude threshold crossing scheme as a model to determine whether a given detector makes a detection (or bH z2H b = = σ0 , (2) “clicks”) [23]. Under this scheme, a detector clicks if the bV z2V amplitude of either the horizontal or the vertical polarization component of the impinging beam falls above a given thresh- where z2H and z2V are independent standard complex Gaus- old, γ 0. sian random variables (which are also independent of z1H and Each detector has placed before it a polarizer (denoted by z1V ). The two spatial modes may be represented by a pair of P1 and P2) oriented to admit horizontally polarized light. A polarizer can be modeled as a polarizing beam splitter for which one of the output ports is ignored [31]. Consequently, an independent vacuum mode will be present in the second input port, resulting in the transformation a a H −P1→ H , σ (6) aV 0z1V where z1V is an independent standard complex Gaussian random variable corresponding to the vacuum mode of the notional second input port. Similarly, passage through P2 will result in the transformation b b H −P2→ H , σ (7) bV 0z2V where z2V is, again, an independent standard complex Gaus- FIG. 1. VQOL experimental setup for a simple delayed-choice sian random variable. experiment. The different colors (or shades of gray) in the beams Thus, detector D1 clicks, according to this model, if the correspond to different polarizations.