Delayed-Choice Experiments

Delayed-choice experiments test whether the measurement arrangement can be chosen after a quantum system has entered an interferometer, while still producing the statistics appropriate to the final arrangement. They are associated with John Wheeler’s proposal to decide late whether an interferometer is open, revealing path information, or closed, revealing interference.

Quantum theory assigns probabilities to the complete experimental arrangement, including the measurement actually performed. A delayed choice prevents a simple story in which the photon first decided to behave as a classical wave or a classical particle.

For background on amplitudes and complementarity, see Probability Amplitudes, Double-Slit Experiment, and Complementarity.

Wheeler’s Idea

Wheeler’s delayed-choice thought experiment sharpened a question already present in two-path interference. If an experiment is arranged to reveal which path a system took, interference disappears. If the paths are recombined coherently, interference can appear. Wheeler asked what happens if the choice between these arrangements is made only after the system has already entered the interferometer.

The amplitudes propagate through the apparatus before the final choice. Quantum theory then assigns probabilities to whichever measurement is implemented. A preassigned classical label of “wave” or “particle” is unnecessary.

Interferometer Setup

A Mach–Zehnder interferometer is the cleanest model. The first beamsplitter creates a coherent superposition of two path modes:

$$\lvert\psi\rangle = \frac{1}{\sqrt2} \left( \lvert u\rangle + e^{i\phi}\lvert \ell\rangle \right),$$

where $\lvert u\rangle$ and $\lvert \ell\rangle$ label upper and lower paths, and $\phi$ is a relative phase.

If the second beamsplitter is absent, the two paths are measured separately. This is the open interferometer. The two detectors answer a path question. In the ideal balanced case,

$$P(u)=P(\ell)=\frac12,$$

independent of $\phi$. This does not mean that the state has already become an incoherent mixture: the path basis simply cannot detect its relative phase.

If the second beamsplitter is present, the paths are recombined before detection. This is the closed interferometer. The detectors answer an interference question. Choose the recombiner convention $\lvert u\rangle\mapsto(\lvert0\rangle+\lvert1\rangle)/\sqrt2$ and $\lvert\ell\rangle\mapsto(\lvert0\rangle-\lvert1\rangle)/\sqrt2$. The output amplitudes are $(1+e^{i\phi})/2$ and $(1-e^{i\phi})/2$. Their squared magnitudes give

$$P_0 = \frac12(1+\cos\phi), \qquad P_1 = \frac12(1-\cos\phi).$$

The same state before the final measurement can therefore lead to different observable statistics depending on which final measurement basis is implemented.

Choice of Measurement Arrangement

In a delayed-choice experiment, the decision to insert, remove, or effectively control the second beamsplitter is made after the system has passed the first beamsplitter. Modern realizations use fast electro-optic devices, random-number generation, or quantum-controlled beamsplitters.

Choosing after entry excludes a fixed apparatus known in advance. A stronger arrangement makes the choice event and photon-entry event spacelike separated, so neither can inform the other through a light-speed or slower signal. Jacques and collaborators implemented this condition with random choices in 2007.

But the delayed choice does not change the formal rule. Quantum mechanics still computes probabilities for the final measurement actually performed. The open interferometer measures path-like information. The closed interferometer measures a phase-sensitive output basis.

This is a measurement-basis issue, not a license to combine incompatible stories. One cannot infer a definite earlier path from an open arrangement and also infer phase interference from a closed arrangement in the same run.

The sinusoidal intensity law by itself is also predicted by classical wave optics. Experiments therefore characterize the single-photon preparation and detection statistics as well as the fringe visibility. The distinctive claim concerns the complete preparation and measurement experiment, not a cosine curve alone.

Modern Realizations

Early delayed-choice experiments used optical interferometers and fast switching. Later single-photon experiments closed more of the conceptual gap between Wheeler’s proposal and laboratory implementation. In particular, experiments with single photons and fast random choices showed that interference or path-like statistics appear according to the final setting even when that setting is chosen after the photon has entered the interferometer.

Quantum delayed-choice experiments go one step further by putting the presence or absence of the second beamsplitter into a quantum superposition or quantum-controlled operation. These experiments test a controlled interpolation between open and closed behavior. They are best understood as ordinary quantum circuits with a control system, not as evidence that a photon is literally both a classical wave and a classical particle.

Delayed-choice entanglement swapping and delayed-choice quantum eraser experiments extend the same theme to entangled systems and correlation measurements. They require additional care because postselection and coincidence sorting can make the story sound more dramatic than the operational facts.

The standard calculation evolves the state forward and changes its final measurement basis. It does not require a later choice to rewrite an earlier recorded outcome. In eraser experiments, changing the way joint records are sorted must likewise be distinguished from changing the unsorted local record.

Exercises

1. In the closed interferometer model on this page, what are $P_0$ and $P_1$ when $\phi=0$?

Solution

Use

$$P_0=\frac12(1+\cos\phi), \qquad P_1=\frac12(1-\cos\phi).$$

For $\phi=0$, $\cos\phi=1$, so

$$P_0=1, \qquad P_1=0.$$

2. Why does the open interferometer not show phase-dependent output probabilities?

Solution

With the second beamsplitter absent, the two paths are detected separately. The detector basis distinguishes the paths rather than recombining their amplitudes. Without recombination, the relative phase between paths does not appear as an interference term in the output probabilities.

3. What does the delayed choice rule out in a naive classical story?

Solution

It rules out the simple story that the photon decided at the first beamsplitter whether it was a classical wave or a classical particle according to a final apparatus arrangement already fixed in advance. The late choice shows that the statistics correspond to the measurement arrangement actually implemented, not to an earlier classical label.

4. Why does delayed choice not allow signaling to the past?

Solution

The experimenter can choose which measurement arrangement is implemented, but the individual outcome remains probabilistic and no already recorded event is controllably changed. The correlations and probabilities are explained by the quantum state and the final measurement context; they do not provide a channel for sending usable messages into the past.

References

• T. Hellmuth, H. Walther, A. Zajonc, and W. Schleich, “Delayed-choice experiments in quantum interference,” Physical Review A 35, 2532-2541, 1987, DOI: 10.1103/PhysRevA.35.2532.
• R. Ionicioiu and D. R. Terno, “Proposal for a Quantum Delayed-Choice Experiment,” Physical Review Letters 107, 230406, 2011, DOI: 10.1103/PhysRevLett.107.230406.
• V. Jacques et al., “Experimental Realization of Wheeler’s Delayed-Choice Gedanken Experiment,” Science 315, 966-968, 2007, DOI: 10.1126/science.1136303.
• X.-S. Ma, J. Kofler, and A. Zeilinger, “Delayed-choice gedanken experiments and their realizations,” Reviews of Modern Physics 88, 015005, 2016, DOI: 10.1103/RevModPhys.88.015005.
• A. Peruzzo, P. Shadbolt, N. Brunner, S. Popescu, and J. L. O’Brien, “A Quantum Delayed-Choice Experiment,” Science 338, 634-637, 2012, DOI: 10.1126/science.1226719.
• J. A. Wheeler, “The ‘Past’ and the ‘Delayed-Choice’ Double-Slit Experiment,” in A. R. Marlow, ed., Mathematical Foundations of Quantum Theory, Academic Press, 1978.
• J. A. Wheeler and W. H. Zurek, eds., Quantum Theory and Measurement, Princeton University Press, 1983.