9.1 Selecting path readout or recombination after entry
9.1.1 Inserting or omitting the second beam splitter on the fly
In Wheeler’s delayed-choice thought experiment, a single-photon wave packet enters a Mach–Zehnder interferometer through the first beam splitter \(\mathrm{BS}_1\). Only after the pulse has passed \(\mathrm{BS}_1\) and entered the two separated arms does a fast electro-optic switch decide whether to leave the second beam splitter \(\mathrm{BS}_2\) in place (the closed configuration) or remove it (the open configuration). When \(\mathrm{BS}_2\) is present, the detector counts vary sinusoidally with the relative arm phase \(\phi\); when \(\mathrm{BS}_2\) is absent, each detector \(D_0\) and \(D_1\) monitors a single arm directly and clicks with probability \(1/2\) independent of \(\phi\).
9.1.2 Causal ordering of preparation, propagation, and measurement choice
Why did Wheeler propose delaying the choice until after \(\mathrm{BS}_1\)? A naive classical picture suggests that at \(\mathrm{BS}_1\) each photon must already adopt either a localized single-arm trajectory (to explain why only one detector clicks in the open configuration) or a two-arm wave state (to explain phase-dependent interference in the closed configuration). Delaying the choice between open and closed configurations until after \(\mathrm{BS}_1\) refutes that classical either-or pre-commitment—without requiring any backward-in-time influence from \(\mathrm{BS}_2\) to \(\mathrm{BS}_1\).
9.1.3 Timing markers versus retrocausal narratives
In the interactive schematic below, the configuration menu toggles between the closed (\(\mathrm{BS}_2\) present) and open (\(\mathrm{BS}_2\) absent) interferometer, and the moving pulse marker illustrates the ordinary forward-in-time sequence of events from source to detectors.
Visual animation · Section 9.1
Selecting path readout or recombination after entry
Interactive visual model for 9.1 Selecting path readout or recombination after entry (Inserting or omitting the second beam splitter on the fly · Causal ordering of preparation, propagation, and measurement choice · Timing markers versus retrocausal narratives). Use Animate or Step to follow the physical state evolution and quantitative predictions.
9.2 Incompatible observables on a single two-path state
9.2.1 Evaluating open-port and closed-port operators on the same state
In standard quantum mechanics, the state inside the interferometer after \(\mathrm{BS}_1\) and the phase shifter is the same pure superposition in every run, regardless of what the switch at \(\mathrm{BS}_2\) will do later:
The late switch simply chooses which unitary operator precedes the detectors: the identity \(I\) in the open configuration, or the Hadamard recombination \(H\) in the closed configuration. Under \(I\), the outcome probabilities are \(P_0=|\langle 0\vert\psi(\phi)\rangle|^2=1/2\) and \(P_1=|\langle 1\vert\psi(\phi)\rangle|^2=1/2\); under \(H\), they are \(P_0(\phi)=(1+\cos\phi)/2\) and \(P_1(\phi)=(1-\cos\phi)/2\).
9.2.2 Wave-like and particle-like statistics as basis choices
Notice that the open configuration measures the Pauli-\(Z\) observable (\(\lvert 0\rangle,\lvert 1\rangle\) arm basis) on \(\lvert\psi(\phi)\rangle\), whereas the closed configuration measures the Pauli-\(X\) observable (\(\lvert\pm\rangle=(\lvert 0\rangle\pm\lvert 1\rangle)/\sqrt{2}\) superposition basis). A single quantum state \(\lvert\psi(\phi)\rangle\) naturally yields \(50/50\) random outcomes in the \(Z\) basis and phase-dependent deterministic outcomes (\(\phi=0\) or \(\pi\)) in the \(X\) basis; no retroactive change to the state at \(\mathrm{BS}_1\) ever occurs.
9.2.3 Phase sensitivity without rewriting past records
Because no measurement record is formed at \(\mathrm{BS}_1\), switching \(\mathrm{BS}_2\) in or out never overwrites an existing classical record. Instead, it selects which of two noncommuting observables (\(Z\) or \(X\)) is finally registered at \(D_0\) and \(D_1\).
Visual animation · Section 9.2
Incompatible observables on a single two-path state
Interactive visual model for 9.2 Incompatible observables on a single two-path state (Evaluating open-port and closed-port operators on the same state · Wave-like and particle-like statistics as basis choices · Phase sensitivity without rewriting past records). Use Animate or Step to follow the physical state evolution and quantitative predictions.
9.3 Spacelike switching and experimental timing audits
9.3.1 Relativistic separation in laboratory delayed-choice tests
In laboratory realizations such as Jacques et al. (\(2007\)), single photons from a nitrogen-vacancy color center traverse a \(48\text{-m}\) polarization interferometer while a fast electro-optic modulator driven by a physical random-number generator selects the open or closed measurement at spacelike separation from the photon’s entry at \(\mathrm{BS}_1\). Spacelike separation ensures that no subluminal or luminal signal from the switch could inform the photon at \(\mathrm{BS}_1\) which configuration will be chosen.
9.3.2 Separating empirical probabilities from interpretive histories
Different interpretations narrate the interior of the interferometer differently—Copenhagen refuses to assign a trajectory prior to detection, whereas Bohmian mechanics assigns a trajectory guided by a two-arm pilot wave that recombines at \(\mathrm{BS}_2\)—yet all valid interpretations predict the exact same detector probabilities \(P_0(\phi)\) and \(P_1(\phi)\).
9.3.3 Comparing full phase scans across open and closed modes
Comparing the open and closed configurations requires scanning the phase \(\phi\) across a full turn rather than inspecting a single phase point where the two curves happen to cross.
Visual animation · Section 9.3
Spacelike switching and experimental timing audits
Interactive visual model for 9.3 Spacelike switching and experimental timing audits (Relativistic separation in laboratory delayed-choice tests · Separating empirical probabilities from interpretive histories · Comparing full phase scans across open and closed modes). Use Animate or Step to follow the physical state evolution and quantitative predictions.
Explore the experiment
Set \(\phi=0^\circ\) and toggle between the closed configuration (\(\mathrm{BS}_2\) present, giving \(P_0=1,P_1=0\)) and the open configuration (\(\mathrm{BS}_2\) absent, giving \(P_0=1/2,P_1=1/2\)). Then sweep \(\phi\) across \([0^\circ,360^\circ]\) in both modes.
Interactive model · Idealized predictions
Choose the final measurement
On a small screen, swipe the diagram to inspect its components. Moving marks illustrate the model; they are not measured trajectories.
Read the values as a table
Does choosing the measurement late change what happened earlier? The text and worked examples remain available without JavaScript. Animation starts only when requested; Step gives a motion-free way to follow the model.
A worked example
At phase \(\phi=90^\circ\) (\(\pi/2\)), both the open configuration (\(\mathrm{BS}_2\) absent) and the closed configuration (\(\mathrm{BS}_2\) present) predict \(P_0=1/2\) and \(P_1=1/2\). Does this mean the two configurations perform the same measurement?
No. At \(\phi=90^\circ\), the internal state is \(\lvert +i\rangle=(\lvert 0\rangle+i\lvert 1\rangle)/\sqrt{2}\), which happens to lie on the equator at equal angle from both the \(Z\)-basis states \(\{\lvert 0\rangle,\lvert 1\rangle\}\) (measured in the open configuration) and the \(X\)-basis states \(\{\lvert +\rangle,\lvert -\rangle\}\) (measured in the closed configuration). Shifting the phase to \(\phi=0^\circ\) immediately distinguishes the two measurements: the open configuration still gives \(P_0=P_1=1/2\), whereas the closed configuration gives \(P_0=1\) and \(P_1=0\).
Summary
Wheeler’s delayed-choice experiment inserts or removes the final beam splitter \(\mathrm{BS}_2\) by a fast electro-optic switch after the photon has already entered the interferometer.
When \(\mathrm{BS}_2\) is absent, detectors register equal path probabilities \(P = \frac{1}{2}\); when \(\mathrm{BS}_2\) is present, they register phase-dependent fringes \(\cos^2(\phi/2)\)—all computed forward in time from the unitary state evolution.
No retrocausal signal is sent into the past: the photon was never required to commit to a classical ‘particle-only’ or ‘wave-only’ biography at the first beam splitter.
Try it yourself
Problem 9.1
In the two-dimensional arm basis \(\{\lvert 0\rangle,\lvert 1\rangle\}\), what \(2\times 2\) unitary matrix replaces the Hadamard matrix \(H\) at the recombination stage when \(\mathrm{BS}_2\) is removed?
Hint
Removing \(\mathrm{BS}_2\) routes arm \(0\) directly to \(D_0\) and arm \(1\) directly to \(D_1\).
Worked solution
The \(2\times 2\) identity matrix \(I=\mathrm{diag}(1,1)\) acts on \(\lvert\psi(\phi)\rangle\), leaving the path amplitudes \(1/\sqrt{2}\) and \(e^{i\phi}/\sqrt{2}\) unmixed so that \(P_0=P_1=1/2\).
Problem 9.2
When \(\mathrm{BS}_2\) is present and detector \(D_0\) clicks at \(\phi=0^\circ\), does that click reveal which arm (\(\lvert 0\rangle\) or \(\lvert 1\rangle\)) the photon took inside the interferometer?
Hint
Identify which basis state \(D_0\) projects onto after \(H\).
Worked solution
No. With \(\mathrm{BS}_2\) present, a click at \(D_0\) projects onto the symmetric superposition \(\lvert +\rangle=H\lvert 0\rangle=(\lvert 0\rangle+\lvert 1\rangle)/\sqrt{2}\), which contains equal contributions from both arms.
Problem 9.3
What experimental timing condition is required to ensure that the choice to insert or remove \(\mathrm{BS}_2\) is relativistically separated from the photon’s entry at \(\mathrm{BS}_1\)?
Hint
Compare the spatial separation \(\Delta L\) between \(\mathrm{BS}_1\) and the switch with the light-travel time \(c\,\Delta t\).
Worked solution
The random switching event at distance \(\Delta L\) from \(\mathrm{BS}_1\) must occur within a time window \(\Delta t\) around the photon’s passage through \(\mathrm{BS}_1\) satisfying \(|\Delta t|<\Delta L/c\), together with verified single-photon \(g^{(2)}(0)\ll 1\) and physical random-bit generation.
What to carry forward
Wheeler’s delayed-choice experiment chooses at the last moment whether to measure the path observable \(Z\) or the interference observable \(X\) on the same two-arm superposition \(\lvert\psi(\phi)\rangle\). No retrocausal rewriting of the past is needed or implied.
Sources and further exploration
These readings provide historical context, experimental implementations, or deeper derivations. The interactive display above is an educational model, not a reproduction of their measured data.