7.1 Two-arm recombination in a Mach–Zehnder interferometer
7.1.1 Splitting, phase shifting, and recombining two optical modes
A Mach–Zehnder interferometer divides a collimated input beam at a first balanced beam splitter (\(\mathrm{BS}_1\)) into an upper arm and a lower arm, redirects both arms with mirrors toward a second balanced beam splitter (\(\mathrm{BS}_2\)), and monitors the two exit ports with detectors \(D_0\) and \(D_1\). A dispersive phase plate in one arm introduces a controllable relative optical phase \(\phi\). Even though \(\mathrm{BS}_1\) and \(\mathrm{BS}_2\) would each send an isolated beam to \(D_0\) and \(D_1\) with equal probability \(1/2\), recombining both arms at \(\mathrm{BS}_2\) routes \(100\%\) of the output to \(D_0\) at \(\phi=0\) and \(100\%\) to \(D_1\) at \(\phi=\pi\).
7.1.2 Mode matching as the condition for indistinguishable paths
Full interference at \(\mathrm{BS}_2\) requires that the two arms be indistinguishable in every accessory degree of freedom at recombination: spatial wavefront, polarization, frequency spectrum, arrival time within the coherence length, and any coupled environment state. A mismatch in polarization or a path-identifying label reduces the interference contrast without changing the \(50{:}50\) splitting ratio of either beam splitter alone.
7.1.3 Amplitude branches versus physical particle trajectories
In the apparatus schematic below, the two colored arm segments represent the coherent probability amplitudes carried by mode \(\lvert 0\rangle\) and mode \(\lvert 1\rangle\) inside the two-mode state vector, and the moving pulse marker tracks the stage-by-stage evolution from \(\mathrm{BS}_1\) through the phase shifter to \(\mathrm{BS}_2\). Neither the two lines nor the timing marker should be interpreted as a single photon splitting into two classical half-photons or following one measured trajectory.
Visual animation · Section 7.1
Two-arm recombination in a Mach–Zehnder interferometer
Interactive visual model for 7.1 Two-arm recombination in a Mach–Zehnder interferometer (Splitting, phase shifting, and recombining two optical modes · Mode matching as the condition for indistinguishable paths · Amplitude branches versus physical particle trajectories). Use Animate or Step to follow the physical state evolution and quantitative predictions.
7.2 Beam-splitter matrices and fringe visibility
7.2.1 Hadamard convention for balanced two-port splitters
Label the two spatial arm modes by the orthonormal basis \(\lvert 0\rangle=(1,0)^T\) and \(\lvert 1\rangle=(0,1)^T\). Choosing the symmetric Hadamard phase convention for each lossless \(50{:}50\) beam splitter (an optical dielectric slab may instead assign a factor of \(i\) to reflection, which merely shifts the origin of \(\phi\) by \(\pi/2\)), \(\mathrm{BS}_1\) acts via
Inserting the phase shift \(\phi\) on mode \(\lvert 1\rangle\) applies \(\mathrm{diag}(1,e^{i\phi})\), producing the internal two-arm state \(\lvert\psi(\phi)\rangle=(\lvert 0\rangle+e^{i\phi}\lvert 1\rangle)/\sqrt{2}\).
7.2.2 Phase-dependent output probabilities and dark-port cancellation
Recombining the two arms at \(\mathrm{BS}_2\) applies \(H\) a second time, yielding the output amplitudes \((1+e^{i\phi})/2\) at port \(D_0\) and \((1-e^{i\phi})/2\) at port \(D_1\). Taking squared magnitudes gives the complementary detection probabilities
At \(\phi=0\), the two paths to \(D_1\) carry opposite signs (\(+1/2\) and \(-1/2\)) and cancel completely (\(P_1=0\), dark port) while reinforcing at \(D_0\) (\(P_0=1\), bright port). At \(\phi=\pi\), the bright and dark ports swap; at \(\phi=\pi/2\) (\(90^\circ\)), both ports register probability \(1/2\).
7.2.3 Visibility reduction from partial coherence
When partial path distinguishability or phase jitter damps the off-diagonal coherence between the two arms by a real factor \(V\in[0,1]\), the output probabilities become \(P_0(\phi)=(1+V\cos\phi)/2\) and \(P_1(\phi)=(1-V\cos\phi)/2\). The operational fringe visibility is
Because a single phase setting \(\phi\) cannot separate a phase offset from a reduced visibility \(V<1\), characterizing an interferometer requires scanning \(\phi\) across at least one full period.
Visual animation · Section 7.2
Beam-splitter matrices and fringe visibility
Interactive visual model for 7.2 Beam-splitter matrices and fringe visibility (Hadamard convention for balanced two-port splitters · Phase-dependent output probabilities and dark-port cancellation · Visibility reduction from partial coherence). Use Animate or Step to follow the physical state evolution and quantitative predictions.
7.3 Phase scans, drift controls, and single-photon fringes
7.3.1 Comparing classical wave interference with single-photon counts
Classical coherent electromagnetic waves passing through a Mach–Zehnder interferometer exhibit the exact same normalized port intensities \(\cos^2(\phi/2)\) and \(\sin^2(\phi/2)\). What makes a single-photon Mach–Zehnder experiment distinctively quantum is the combination of full fringe visibility \(V\approx 1\) when \(\mathrm{BS}_2\) is present with photon anticorrelation \(g^{(2)}(0)\ll 1\) between the two arms when \(\mathrm{BS}_2\) is removed.
7.3.2 Calibrating detector imbalance and path-length drift
In a physical optical setup, thermal expansion of mirror mounts shifts \(\phi(t)\) over time, while unequal detector efficiencies \(\eta_0\neq\eta_1\) skew the raw count ratio. Recording blocked-arm baselines, interleaving reference phase points during a scan, and normalizing each port by its calibrated efficiency prevent instrumental drift from masquerading asdecoherence.
7.3.3 Foundation for delayed-choice, eraser, and bomb-tester protocols
This two-mode Mach–Zehnder algebra serves as the backbone for Wheeler’s delayed-choice experiment (Chapter 9), the quantum eraser (Chapter 10), and the Elitzur–Vaidman interaction-free bomb tester (Chapter 11).
Visual animation · Section 7.3
Phase scans, drift controls, and single-photon fringes
Interactive visual model for 7.3 Phase scans, drift controls, and single-photon fringes (Comparing classical wave interference with single-photon counts · Calibrating detector imbalance and path-length drift · Foundation for delayed-choice, eraser, and bomb-tester protocols). Use Animate or Step to follow the physical state evolution and quantitative predictions.
Explore the experiment
Sweep the phase \(\phi\) from \(0^\circ\) through \(360^\circ\). Pause at \(\phi=0^\circ\), \(90^\circ\), \(180^\circ\), and \(270^\circ\) to compare \(P_0(\phi)\) and \(P_1(\phi)\), and verify that \(P_0(\phi)+P_1(\phi)=1\) at every phase setting.
Interactive model · Idealized predictions
Recombine the alternatives
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
How can two individually random beam splitters give a certain output? 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 which phase angles \(\phi\in[0^\circ,360^\circ)\) does the ideal Mach–Zehnder interferometer yield \(P_0(\phi)=3/4\) at detector \(D_0\)?
Setting \(P_0(\phi)=(1+\cos\phi)/2=3/4\) gives \(\cos\phi=1/2\). Within a single turn \([0^\circ,360^\circ)\), there are two solutions: \(\phi=60^\circ\) (\(\pi/3\)) and \(\phi=300^\circ\) (\(5\pi/3\)). Because \(\cos\phi\) is an even periodic function, measuring \(P_0\) at a single phase setting does not determine \(\phi\) uniquely without scanning adjacent phases.
Summary
In a balanced Mach–Zehnder interferometer, two 50:50 beam splitters that each act randomly in isolation combine via relative phase \(\phi\) to produce deterministic routing \(P_+ = \cos^2(\phi/2)\) at \(\phi = 0\) or \(\pi\).
Blocking either arm destroys the cross term \(\cos\phi\) and drops the output probability at each port from \(1\) (or \(0\)) to \(\frac{1}{4}\).
Wavefront mismatch, polarization drift between arms, and optical path fluctuations faster than the integration window degrade measured fringe visibility \(\mathcal{V}\).
Try it yourself
Problem 7.1
What are the detector probabilities \(P_0\) and \(P_1\) if the second beam splitter \(\mathrm{BS}_2\) is removed from the Mach–Zehnder interferometer?
Hint
Evaluate the squared magnitudes of the amplitudes in \(\lvert\psi(\phi)\rangle=(\lvert 0\rangle+e^{i\phi}\lvert 1\rangle)/\sqrt{2}\) directly in the path basis.
Worked solution
Without \(\mathrm{BS}_2\), detectors \(D_0\) and \(D_1\) measure the two arm modes directly: \(P_0=|1/\sqrt{2}|^2=1/2\) and \(P_1=|e^{i\phi}/\sqrt{2}|^2=1/2\) for every value of \(\phi\). Without recombination, the relative phase \(\phi\) has no effect on the detector probabilities.
Problem 7.2
In a partially coherent Mach–Zehnder interferometer with visibility \(V=0.60\), what are the maximum and minimum values of \(P_0(\phi)\) across a full phase scan?
Hint
Set \(\cos\phi=+1\) and \(\cos\phi=-1\) in \(P_0(\phi)=(1+V\cos\phi)/2\).
Worked solution
The maximum is \(P_{\max}=(1+0.60)/2=0.80\) at \(\phi=0^\circ\) and the minimum is \(P_{\min}=(1-0.60)/2=0.20\) at \(\phi=180^\circ\). Substituting into \((P_{\max}-P_{\min})/(P_{\max}+P_{\min})=(0.80-0.20)/(0.80+0.20)=0.60\) recovers \(V\).
Problem 7.3
Why must \(P_0(\phi)+P_1(\phi)=1\) hold for every phase \(\phi\) in an ideal lossless Mach–Zehnder interferometer?
Hint
Consider the unitarity of \(H\) and the phase operator \(\mathrm{diag}(1,e^{i\phi})\).
Worked solution
Both \(H\) and \(\mathrm{diag}(1,e^{i\phi})\) are unitary matrices, so their product \(U(\phi)=H\,\mathrm{diag}(1,e^{i\phi})H\) preserves the norm of the state vector: \(P_0(\phi)+P_1(\phi)=\lVert U(\phi)\lvert 0\rangle\rVert^2=1\).
What to carry forward
A relative phase \(\phi\) between two modes becomes an observable probability difference only after a recombining beam splitter mixes the two alternatives. Always specify the beam-splitter phase convention and scan \(\phi\) across a full period to separate phase shifts from visibility loss.
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.