11.1 Blocking one arm of a dark-port interferometer
11.1.1 Disrupting destructive interference with an opaque obstruction
Align a balanced Mach–Zehnder interferometer at relative phase \(\phi=0\) so that when both arms are unobstructed, destructive interference makes detector \(D_1\) completely dark (\(P_{D_1}=0\)) while all photons exit to the bright port \(D_0\) (\(P_{D_0}=1\)). Now place an ideal ultra-sensitive opaque object—in Elitzur and Vaidman’s \(1993\) thought experiment, a hair-trigger bomb that explodes if it absorbs even a single photon—inside the lower arm. Whenever a single photon traverses the interferometer and triggers the previously dark detector \(D_1\), that click certifies with certainty that the lower arm is blocked, even though the photon was not absorbed by the bomb in that run!
11.1.2 Accounting for all three outcomes: absorption, bright port, and dark port
Understanding interaction-free measurement requires tracking all three mutually exclusive outcomes of each single-photon trial: (1) the photon enters the blocked lower arm and is absorbed (detonating the bomb), (2) the photon takes the open upper arm and exits to the bright port \(D_0\) (an inconclusive outcome, since \(D_0\) also clicks when the arm is empty), or (3) the photon takes the open upper arm and exits to the dark port \(D_1\) (certifying the obstacle without absorption).
11.1.3 Operational meaning of interaction-free certification
The phrase interaction-free measurement refers specifically to those postselected dark-port trials in which no energy quantum is absorbed by the obstacle; in the quantum Hamiltonian, the obstacle’s boundary condition still modifies the wave amplitude in the lower arm and destroys the destructive cancellation at \(D_1\).
Visual animation · Section 11.1
Blocking one arm of a dark-port interferometer
Interactive visual model for 11.1 Blocking one arm of a dark-port interferometer (Disrupting destructive interference with an opaque obstruction · Accounting for all three outcomes: absorption, bright port, and dark port · Operational meaning of interaction-free certification). Use Animate or Step to follow the physical state evolution and quantitative predictions.
11.2 Three-outcome branching and certification efficiency
11.2.1 Amplitude survival after the first splitter and arm blockage
After the first balanced beam splitter \(\mathrm{BS}_1\), the incident state \(\lvert 0\rangle\) splits into \((\lvert 0\rangle+\lvert 1\rangle)/\sqrt{2}\). When the opaque absorber sits in arm \(\lvert 1\rangle\), that branch is absorbed with probability
With probability \(1/2\), the photon is not absorbed and the surviving amplitude \(\frac{1}{\sqrt{2}}\lvert 0\rangle\) in the upper arm reaches the second balanced beam splitter \(\mathrm{BS}_2\), which maps \(\lvert 0\rangle\mapsto(\lvert D_0\rangle+\lvert D_1\rangle)/\sqrt{2}\). The unconditional probabilities across all three outcomes are therefore
11.2.2 Why a dark-port click certifies the obstacle in an ideal setup
Compare the obstructed probabilities \((P_{\mathrm{absorb}},P_{D_0},P_{D_1})=(1/2,1/4,1/4)\) against the unobstructed probabilities \((0,1,0)\). A click at \(D_0\) can occur both with the obstacle absent (\(P_{D_0}=1\)) and with the obstacle present (\(P_{D_0}=1/4\)), so it is inconclusive. A click at \(D_1\), however, has probability \(0\) when the arm is empty and probability \(1/4\) when the arm is blocked, so a single \(D_1\) click unambiguously certifies the presence of the absorber.
11.2.3 Unconditional versus non-absorbed success fractions
Two distinct figures of merit appear in the literature, and their denominators must never be confused:
(a) In a single pass, the unconditional certification probability is \(P_{D_1}=1/4=25\%\).
(b) If we repeat the test whenever an inconclusive \(D_0\) click occurs (since neither absorption nor certification happened), the geometric series of probabilities \(\sum_{k=0}^\infty (1/4)^k\) yields a limiting interaction-free success fraction of
Three-outcome branching and certification efficiency
Interactive visual model for 11.2 Three-outcome branching and certification efficiency (Amplitude survival after the first splitter and arm blockage · Why a dark-port click certifies the obstacle in an ideal setup · Unconditional versus non-absorbed success fractions). Use Animate or Step to follow the physical state evolution and quantitative predictions.
11.3 Imperfect extinction, false positives, and Zeno extensions
11.3.1 Bayesian updating when the empty dark port has baseline counts
In any physical interferometer, finite visibility \(V<1\) and detector dark counts produce a small nonzero dark-port probability \(P(D_1\mid\text{empty})=\epsilon>0\) even when no obstacle is present. A single \(D_1\) click then updates the prior odds of an obstacle by the likelihood ratio \(P(D_1\mid\text{blocked})/P(D_1\mid\text{empty})=(1/4)/\epsilon\) rather than providing absolute logical certainty.
11.3.2 Counterfactual phrasing versus Hamiltonian boundary conditions
Rather than saying the photon "knew" what was in the path it did not take, the quantum state analysis is straightforward: blocking arm \(\lvert 1\rangle\) projects out the \(-1/2\) amplitude that would otherwise cancel the \(+1/2\) amplitude from arm \(\lvert 0\rangle\) at detector \(D_1\).
11.3.3 Single-pass accounting as the baseline for multi-pass protocols
By replacing the two balanced splitters with \(N\) weakly coupled cavity steps of reflectivity \(\cos^2(\pi/2N)\) (the Kwiat–Weinfurter–Herzog–Zeilinger quantum Zeno interrogation scheme), the absorption probability per cycle scales as \(O(1/N)\) and the interaction-free certification fraction can approach \(1\) as \(N\to\infty\).
Visual animation · Section 11.3
Imperfect extinction, false positives, and Zeno extensions
Interactive visual model for 11.3 Imperfect extinction, false positives, and Zeno extensions (Bayesian updating when the empty dark port has baseline counts · Counterfactual phrasing versus Hamiltonian boundary conditions · Single-pass accounting as the baseline for multi-pass protocols). Use Animate or Step to follow the physical state evolution and quantitative predictions.
Explore the experiment
Toggle the configuration between the obstructed arm (\(\text{present}\)) and the empty arm (\(\text{absent}\)), and step through the three stages of the apparatus. Verify how the probability bars \((P_{D_0},P_{D_1},P_{\mathrm{absorb}})\) shift from \((1,0,0)\) to \((1/4,1/4,1/2)\).
Interactive model · Idealized predictions
Keep every outcome in view
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
Can an object be detected in a trial in which it absorbs nothing? 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
Suppose \(N=400\) single-photon trials are sent one by one through a balanced Elitzur–Vaidman interferometer containing an ideal opaque absorber in one arm. What are the expected counts for absorption, bright-port \(D_0\) clicks, and dark-port \(D_1\) certifications?
Multiplying \(N=400\) by the single-pass probabilities \((P_{\mathrm{absorb}},P_{D_0},P_{D_1})=(1/2,1/4,1/4)\) gives an average of \(200\) absorptions, \(100\) inconclusive \(D_0\) clicks, and \(100\) interaction-free \(D_1\) certifications. Reporting only the \(200\) non-absorbed detector clicks would conceal the \(200\) trials in which the absorber was struck.
Summary
In the Elitzur–Vaidman bomb tester, an absorbing object placed inside one arm of a tuned Mach–Zehnder interferometer breaks destructive interference at the dark port.
With a single 50:50 interferometer, \(25\%\) of trials click at the dark port without absorbing the photon, yielding a maximum efficiency of \(\eta = \frac{1}{3}\) per intact bomb.
Concatenating \(N\) weak reflective cycles of rotation angle \(\theta = \pi/(2N)\) via the quantum Zeno effect drives the absorption probability toward \(0\) and interaction-free efficiency toward \(1\).
Try it yourself
Problem 11.1
What is the single-pass unconditional probability \(P_{D_1}\) of certifying an ideal opaque obstacle without absorption in a balanced Mach–Zehnder interferometer, and what is the multi-pass success fraction \(\eta_{\mathrm{IFM}}\) if inconclusive \(D_0\) trials are recycled?
Hint
Compare \(P_{D_1}=1/4\) with \(P_{D_1}/(P_{D_1}+P_{\mathrm{absorb}})\).
Worked solution
In a single pass, \(P_{D_1}=1/4=25\%\). When inconclusive \(D_0\) outcomes are repeated until either \(D_1\) clicks or absorption occurs, the success probability is \(\eta_{\mathrm{IFM}}=(1/4)/(1/4+1/2)=1/3\approx 33.3\%\).
Problem 11.2
If an imperfectly aligned empty interferometer has a dark-port leakage probability \(P(D_1\mid\text{empty})=0.01\), and empty versus blocked arms are equally likely a priori (\(1/2\) each), what is the posterior probability that the arm is blocked given a single \(D_1\) click?
Hint
Use Bayes’ rule with \(P(D_1\mid\text{blocked})=0.25\) and \(P(D_1\mid\text{empty})=0.01\).
Worked solution
By Bayes’ rule, \(P(\text{blocked}\mid D_1)=0.25/(0.25+0.01)=25/26\approx 0.9615\), which is strong evidence (\(96.2\%\)) but no longer \(100\%\) certainty.
Problem 11.3
Why doesn’t blocking one arm of the interferometer simply reduce the bright-port probability \(P_{D_0}\) from \(1\) to \(1/2\) while keeping \(P_{D_1}=0\)?
Hint
Track what the second beam splitter \(\mathrm{BS}_2\) does to the surviving single-arm state \(\lvert 0\rangle\).
Worked solution
The dark port \(D_1\) was dark only because the amplitude \(-1/2\) from the lower arm canceled the amplitude \(+1/2\) from the upper arm at \(\mathrm{BS}_2\). Blocking the lower arm removes that canceling amplitude, so the surviving \(1/2\) probability in the upper arm splits equally at \(\mathrm{BS}_2\) into \(P_{D_0}=1/4\) and \(P_{D_1}=1/4\).
What to carry forward
Blocking one path of a balanced interferometer eliminates destructive cancellation at the dark port, allowing \(1/4\) of single-pass trials (\(1/3\) when recycling inconclusive bright-port clicks) to certify the obstacle without depositing energy in it.
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.