Mach-Zehnder Interferometer

Why Mach-Zehnder matters

Drop a single photon into a Mach-Zehnder, tune the path lengths so the two arms differ by a tiny phase, and the photon — which famously cannot be in two places at once — nonetheless interferes with itself, emerging from one detector with certainty and from the other never. Tweak the phase by a fraction of a wavelength and the assignment swaps: the same photon now always emerges from the other detector. This single device has been the staging ground for nearly every paradox of quantum mechanics, from Wheeler's delayed-choice argument to Elitzur-Vaidman's interaction-free measurement; scaled up by seven orders of magnitude and made out of 40 kg sapphire mirrors, it is also the device that detected gravitational waves.

• Gravitational-wave astronomy. LIGO (Hanford and Livingston), Virgo (Cascina), KAGRA (Kamioka), and the planned Einstein Telescope and Cosmic Explorer all use kilometer-scale Michelson/Mach-Zehnder topologies. ΔL/L sensitivity at 10⁻²¹ — equivalent to measuring the distance to Alpha Centauri to within the width of a hydrogen atom — has produced more than 90 confirmed compact-binary merger detections since 2015.
• Quantum optics testbed. The Mach-Zehnder is the workhorse for foundational experiments: single-photon interference, weak measurements, post-selection, contextuality. Most quantum-optics graduate students build at least one in their first year.
• Optical chip integration. Photonic integrated circuits use chains of Mach-Zehnder cells (each a tunable phase shifter sandwiched by directional couplers) as universal optical processors. Linear optical quantum computing, photonic neural networks, and large-scale boson sampling all rely on programmable Mach-Zehnder meshes.
• Atom interferometry. Replace photons with cold atoms and beam splitters with laser pulses (π/2 pulses); you get atomic Mach-Zehnders that measure inertial accelerations, gravity gradients, and proposed dark-matter searches at sensitivities below 10⁻¹¹ g/√Hz.
• Quantum metrology. Phase estimation in a Mach-Zehnder with N photons saturates the Cramér-Rao bound; squeezing or entanglement (NOON states) push to the Heisenberg limit 1/N. LIGO uses squeezed light to beat the standard quantum limit and gain ~3 dB of sensitivity at high frequencies.
• Interaction-free imaging. Beyond bomb-testing, Mach-Zehnder geometries enable imaging samples with photons that mostly never touch them — useful for fragile biological samples sensitive to UV/X-ray damage.

From beam splitter matrices to output probabilities

Each 50:50 beam splitter is described by a 2×2 unitary acting on the (upper, lower) mode pair, conventionally B = (1/√2)[[1, i], [i, 1]]. The mirrors apply a phase but don't mix modes. The two arms accrue phases φ_a and φ_b, encoded as diagonal P = diag(e^(iφ_a), e^(iφ_b)). Total transformation: U = B P B. For input |1, 0⟩ (photon in upper port), the output amplitudes are:

|out⟩ = U |1, 0⟩ = (1/2)[(1 − e^(iΔφ))|1, 0⟩ + i(1 + e^(iΔφ))|0, 1⟩]

where Δφ = φ_b − φ_a. Squaring, P_+ = sin²(Δφ/2), P_− = cos²(Δφ/2). At Δφ = 0, P_− = 1 — the photon always emerges from the "− port" (sometimes called the "bright port"). At Δφ = π, P_+ = 1. The fact that the second beam splitter recovers a deterministic output is what makes Mach-Zehnder different from a "two paths to two detectors with no recombination" experiment, where each detector would fire 50% of the time regardless of phase.

Visibility and which-path

If a "which-path" marker is added — even one in principle measurable — the visibility V = (P_max − P_min)/(P_max + P_min) drops. With perfect path information (e.g., a polarizing element that tags each path with orthogonal polarization), V = 0: no interference, equal output probabilities. With partial information characterized by distinguishability D, the bound V² + D² ≤ 1 (Englert 1996, Greenberger-Yasin 1988) is saturated by pure states. Quantum erasers exploit this: erasing the marker after the photon has passed — even after detection — restores interference in the conditional ensemble.

Common misconceptions

• "Needs many photons to see interference." No — single-photon interferometry has been routine since the 1980s (Grangier-Roger-Aspect 1986). The interference pattern emerges in the statistics of many runs, but each individual photon contributes its full amplitude on both paths. The accumulated histogram is the interference pattern.
• "Second beam splitter is just for convenience." Without it, the two paths terminate in separate detectors and you see 50/50 splitting regardless of phase — pure particle behavior, no interference. The second beam splitter is the wave-recombination element; it's where interference physically happens.
• "Phase determines path." Only relative phase between the two paths matters for output port selection. Adding the same overall phase to both arms changes nothing observable. This is why Mach-Zehnders are insensitive to common-mode noise — vibrations, thermal drifts, and similar — provided the noise is correlated between arms.
• "Photons split in half at the beam splitter." No. A single photon enters a superposition of going each way, but it never breaks in half. In the photon-number basis, the post-beam-splitter state is (|1, 0⟩ + i|0, 1⟩)/√2 — one photon in either path, never half a photon in each.
• "You can use it for FTL signaling." No. The output port assignment depends on Δφ, but unless you control the phase via a local action, you can't extract usable information. Even with entangled photon pairs, the no-signalling theorem prevents superluminal communication.
• "Mach-Zehnder is the same as Michelson." Topologically different. Michelson light passes the splitter twice (back and forth); Mach-Zehnder light passes each of two splitters once. Functionally similar for many applications; mechanically and optically distinct.

Numbers worth keeping in your head

• Single-photon interference visibility in a well-aligned tabletop MZI: V > 0.99.
• LIGO arm length: 4 km. Effective path length with Fabry-Perot cavities: ~1100 km. Strain sensitivity: ~3 × 10⁻²² /√Hz at 100 Hz.
• GW150914: first detected gravitational wave. 14 September 2015. Source: 36 + 29 solar-mass black holes merging at 410 Mpc. Peak strain at LIGO: 1.0 × 10⁻²¹.
• Atom interferometers: phase resolution ~10⁻⁵ rad/√Hz, gravity-gradient sensitivity ~10⁻⁹ g/m at 10 m baseline.
• Photonic chip Mach-Zehnders: phase shifters with thermo-optic tuning at ~kHz speeds; electro-optic at ~GHz; loss ~0.1 dB per cell.
• Heisenberg-limited sensing with N photons: phase uncertainty 1/N (vs 1/√N standard quantum limit). Achieved up to N ~ 10 with NOON states; engineering challenge to scale.