Experimental realization of Wheeler's delayed-choice experiment with dual selections

AI summary

Overview

Research area: Quantum physics — quantum foundations, specifically Wheeler's delayed-choice experiment, wave–particle duality, and entanglement-based quantum control of interferometry.

Technical level: Intermediate. The core idea is accessible, but the paper assumes familiarity with Mach–Zehnder interferometers, projective measurements, and polarization/path encoding of photonic qubits.

Scope: An experimental demonstration of a delayed-choice experiment in which both beam splitters of a Mach–Zehnder interferometer — the input one and the output one — are independently selected, using entangled photons and a delayed measurement on the idler photon.

What This Paper Is About

Wheeler's delayed-choice experiment asks what happens when the decision to reveal a photon's wave nature or its particle nature is made after the photon has already passed through the interferometer. Previous photonic versions could only switch the output beam splitter on or off. This paper asks a harder question: what if you can also switch the input beam splitter on or off, and do so with a delayed choice as well? The authors build an entangled-photon setup in which two separate "choices" are made, one encoded in the measurement basis of an idler photon and one encoded in the path of the signal photon, and they observe the resulting wave- or particle-like behavior.

Key Contributions

1. Dual selection of both beam splitters. The experiment controls the presence/absence of $\mathrm{BS_{in}}$ (via the delayed-choice measurement basis of ancilla qubit-1, the idler photon) and $\mathrm{BS_{out}}$ (via the measurement outcome of ancilla qubit-2, encoded in the signal photon's path) in a single Mach–Zehnder configuration.

2. Simultaneous observation of both $\mathrm{BS_{out}}$ conditions. For each state of $\mathrm{BS_{in}}$, the setup lets the authors read out, at the same time, what the photons do when $\mathrm{BS_{out}}$ is present (path $a$) and when it is absent (path $d$), which the authors state greatly facilitates comparative analysis.

3. Delayed choice enforced by an optical delay. The single-mode fiber for idler photons is extended by three meters relative to the signal photon's fiber, so the idler measurement basis is chosen only after the signal photon has already passed through the Sagnac interferometer and been detected.

4. A working photonic implementation with high visibility. Interference visibility of 97.35% (path $a$, case (c)) and 95.87% (case (d)) is reported, with 97.4% given as the highest visibility achieved.

Main Findings

• Without $\mathrm{BS_{in}}$, the photon always behaves like a particle. In the configurations where the idler photon is projected onto $|0\rangle_{a1}$ or $|1\rangle_{a1}$ (i.e., $\mathrm{BS_{in}}$ absent), no interference pattern appears in Figure 4(a) and 4(b), regardless of whether $\mathrm{BS_{out}}$ is present or absent.

• With $\mathrm{BS_{in}}$ but without $\mathrm{BS_{out}}$, the photon still behaves like a particle. In Figure 4(c) and 4(d), path $a$ (where $\mathrm{BS_{out}}$ is present) shows an interference pattern while path $d$ (where $\mathrm{BS_{out}}$ is absent) does not. The authors state that photons on path $d$ still exhibit particle-like behavior despite $\mathrm{BS_{in}}$ being present.

• Wave behavior requires both beam splitters. The paper concludes that photons exhibit wave-like properties only when both $\mathrm{BS_{in}}$ and $\mathrm{BS_{out}}$ are present in the Mach–Zehnder interferometer.

• Measured probabilities match theory. The reported theoretical values are $P_a^{(0)}=P_a^{(1)}=P_d^{(+)}=P_d^{(-)}=1/4$, $P_d^{(0)}=0$, $P_d^{(1)}=1/2$, and $P_a^{(\pm)}=(1\pm\cos\varphi)/4$, plotted against the experimental coincidence data.

• A predicted wave–particle superposition regime is discussed, not measured. If the rotation angles of $H_3$–$H_5$ are set between $0^\circ$ and $22.5^\circ$, the authors argue the beam splitters enter quantum superpositions; for the specific example $H_3=15^\circ$ and $H_4=22.5^\circ$, $H_5=15^\circ$, the input state becomes $\tfrac{\sqrt{3}}{2}|0\rangle_s + \tfrac{1}{2}e^{i\varphi}|1\rangle_s$ with deduced probabilities $P_a=(2+\sqrt{3}\cos\varphi)/8$, $P_c=(2-\sqrt{3}\cos\varphi)/8$, $P_b=(5+3\cos\varphi)/16$, and $P_d=(3-3\cos\varphi)/16$ — all four paths showing interference, indicating incomplete wave-like and particle-like behavior.

• Error sources are characterized. The authors attribute deviations (for example, $P_d^{(0)}$ never reaching exactly 0) to imperfect entangled-state preparation, imperfect horizontal/vertical separation by the beam displacers and PBS, misalignment of $H_6$–$H_9$, background counts, and device instability. Each experimental point comes from $n = 230\text{–}500$ measurements, with error bars from the error propagation formula.

Methodology in Plain English

The team starts with a pair of entangled photons made by pumping a type-II PPKTP crystal ($1\text{ mm}\times2\text{ mm}\times20\text{ mm}$) with a 405 nm continuous-wave laser; the down-converted photons emerge at 810 nm and are coupled into single-mode fibers. The idler photon plays the role of ancilla qubit-1, and the signal photon plays the role of the system qubit; horizontal polarization is $|0\rangle$ and vertical is $|1\rangle$.

The idler photon is sent through an extra three meters of fiber and a measurement module consisting of a half-wave plate $H_3$ and a polarizing beam splitter. Choosing $H_3$ at $0^\circ$, $45^\circ$, $22.5^\circ$, or $-22.5^\circ$ projects the idler onto $|0\rangle$, $|1\rangle$, $|+\rangle$, or $|-\rangle$ — a delayed choice that effectively decides whether $\mathrm{BS_{in}}$ is absent (first two) or present (last two).

The signal photon passes through quartz crystals that add a phase $\varphi$ to the vertical component, then enters a Sagnac interferometer whose 50:50 non-polarizing beam splitter splits it into paths A and B. The path the photon takes is the ancilla-2 qubit: a half-wave plate in path A acts as a Hadamard (implemented as a $22.5^\circ$ wave plate, following Refs. [56,57]), while path B applies a Pauli-$z$. This implements a controlled operation that lets the path label decide whether the equivalent of $\mathrm{BS_{out}}$ is present. Beam displacers (4 mm shift for $|H\rangle$) and $H_6$–$H_9$ route the photon into four output paths $a$, $b$, $c$, $d$; paths $a$ and $c$ correspond to $\mathrm{BS_{out}}$ present, $b$ and $d$ to $\mathrm{BS_{out}}$ absent, and the authors monitor only $a$ and $d$ because $a$/$c$ and $b$/$d$ carry the same physical meaning.

Detectors $D_1$ and $D_2$ (signal, at ports $a$ and $d$) are read in coincidence with $D_3$ (idler), with the signal counts delayed by approximately 11.5 ns and a coincidence window of 1 ns. Probabilities are extracted from ratios $N_a^{(i)}/N_t$ and $N_d^{(i)}/N_t$, with $N_t$ built from the summed counts across the $i=0,1$ cases. To compensate for a non-ideal 50:50 beam splitter, both arms of the Sagnac interferometer are arranged to undergo one reflection and one transmission.

Why This Matters

• Foundational impact. The work closes a gap the authors identify — previous delayed-choice experiments selected only $\mathrm{BS_{out}}$ — and shows that wave behavior strictly requires both beam splitters, refining how complementarity is tested.

• Simultaneous comparison. Being able to see $\mathrm{BS_{out}}$-present and $\mathrm{BS_{out}}$-absent behavior in the same run makes the contrast between wave and particle behavior measurable side by side rather than in separate experimental runs.

• A route to quantum erasure. The authors note that adjusting $H_5$ from $0^\circ$ to $22.5^\circ$ while idler photons are projected onto $|+\rangle$ or $|-\rangle$ converts $\mathrm{BS_{out}}$ from absent to present, which would erase path information and test whether interference can be restored.

Real-world applications (as cited in the paper): the authors specifically mention that quantum-eraser experiments have been applied to quantum key distribution, two-photon imaging, and quantum sensing, and cite entangled two-photon ghost imaging combined with Fourier transform methods (Ref. [68]) as a realization. The paper itself reports no new application demonstrations or benchmarks beyond these citations.

Industry relevance: the paper reports no industry partnerships, deployment, or performance benchmarking; its relevance is to quantum-photonics research groups building entanglement-based control and imaging protocols, not to a product-scale technology.

Future Directions

• Implement the quantum-eraser version enabled by tuning $H_5$ from $0^\circ$ to $22.5^\circ$ to erase path information and check whether interference is restored.

• Experimentally test the superposition regime. The $15^\circ$ example predicting $P_a=(2+\sqrt{3}\cos\varphi)/8$ and the accompanying four-path interference is presented as a deduction; measuring it would confirm the claimed wave–particle superposition state.

• Reduce the identified error sources. The paper lists imperfect entangled-state preparation, incomplete polarization separation in the beam displacers and PBS, misalignment of $H_6$–$H_9$, background counts, and device instability — all of which currently prevent $P_d^{(0)}$ from reaching 0.

• Extend to multi-photon or multi-selection schemes. The authors frame their dual-selection result as building on prior two-photon and three-photon delayed-choice work (Refs. [46–48]), leaving open how many beam-splitter choices could be controlled simultaneously.

Target Audience

Researchers and graduate students in quantum optics and quantum foundations who already know Mach–Zehnder interferometry and projective measurement; experimentalists interested in polarization- and path-encoding techniques (Sagnac interferometers, beam displacers, half-wave-plate Hadamard gates); and readers following the delayed-choice and quantum-eraser literature who want the specific extension to two simultaneously selected beam splitters.

Authors’ abstract

Wheeler's delayed-choice experiment demonstrates wave-particle duality of particles using different experimental configurations of a Mach-Zehnder interferometer. In a quantum version of this experiment, the wave-particle behavior of photons can be observed by controlling presence or absence of the second beam splitter. Here, we implement a delayed-choice experiment with dual selections based on entangled photons, experimentally controlling both of two beam splitters in the Mach-Zehnder interferometer, where the presence and absence of the second beam splitter are simultaneously controlled through path encoding. Our experiment reveals the wave-particle behavior of single photons under different configurations of two beam splitters. We further discuss the scenario when two beam splitters are in a quantum superposition of being present and absent, photons will be in a wave-particle quantum superposition state.