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Open Access Journal of Physics

The Controlled Phase Shifts of Photons in the Mach-Zehnder Interferometer

Jiří Stávek

Published Mar 10, 2026• 0 citations
Publisher

Abstract

The Mach-Zehnder interferometer (MZI) is a two-path phase-to-intensity transducer that becomes "mysterious" mainly when one conflates operational facts, minimal predictive calculus, and ontological storytelling.Controlled phase shifts are the central "knob" of the MZI: a tunable relative phase Φ converts directly into complementary output intensities at the two ports.In this model, photons enter the MZI with classical instructions: the "white" photons with E 2 = 1 (where E is the electric field of the sinusoidal electromagnetic wave at the given moment) are reflected on the beamsplitters, the "grey" photons with E 2 = 0.5 can be both reflected and transmitted through the beamsplitters (with 50:50 ratio), and the "black" photons with E 2 = 0 transmit through the beamsplitters.The movable mirrors and the second beamsplitter modify the "color" of photons.The 50:50 mixture of "white" and "black" photons at the input is separated by the first beamsplitter.The phase shifts are modified by movable mirrors so that the second beamsplitter organizes the resulting structure of photons at the two detectors.The interference patterns have been formed from two sub-ensembles of photons with their relative phase shift ΔΦ = π at the given port.We assume that this classical behavior of photons in the MZI can clarify the "riddle-like" discussions (e.g., delayed choice switching, the meaning of a dark port, and the role of which-way information).

DOI
10.22259/2637-5826.0802004

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The Mach-Zehnder interferometer (MZI) is a two-path phase-to-intensity transducer that becomes "mysterious" mainly when one conflates operational facts, minimal predictive calculus, and ontological storytelling.Controlled phase shifts are the central "knob" of the MZI: a tunable relative phase Φ converts directly into complementary output intensities at the two ports.In this model, photons enter the MZI with classical instructions: the "white" photons with E 2 = 1 (where E is the electric field of the sinusoidal electromagnetic wave at the given moment) are reflected on the beamsplitters, the "grey" photons with E 2 = 0.5 can be both reflected and transmitted through the beamsplitters (with 50:50 ratio), and the "black" photons with E 2 = 0 transmit through the beamsplitters.The movable mirrors and the second beamsplitter modify the "color" of photons.The 50:50 mixture of "white" and "black" photons at the input is separated by the first beamsplitter.The phase shifts are modified by movable mirrors so that the second beamsplitter organizes the resulting structure of photons at the two detectors.The interference patterns have been formed from two sub-ensembles of photons with their relative phase shift ΔΦ = π at the given port.We assume that this classical behavior of photons in the MZI can clarify the "riddle-like" discussions (e.g., delayed choice switching, the meaning of a dark port, and the role of which-way information).
The chapter discusses the Mach–Zehnder interferometer (MZI), which uses beam splitters in combination with mirrors to place a photon into a superposition of paths. In a final beam splitter, the two paths may be recombined to interfere, showing wave behavior. If the final beam splitter is removed, the photon is definitely in one path or the other, showing particle behavior. The polarization of light allows additional applications—one can use a polarizing beam splitter to separate and reconstitute the wave functions of a photon polarized at an angle. In a delayed-choice experiment, one decides which kind of experiment one is to do, particle or wave, after the photon has already entered the arms of the MZI. The chapter discusses a version of this that puts the MZI and a second-detector photon in an entangled state of particle and wave results.
Photon counting measurements are analyzed for obtaining a classical phase parameter in a linear Mach–Zehnder interferometer (MZI) by the use of phase estimation theories. The detailed analysis is made for four cases: (a) coherent states inserted into the interferometer, (b) Fock number state inserted in one input port of the interferometer and the vacuum into the other input port, (c) coherent state inserted into one input port of the interferometer and squeezed-vacuum state into the other input port, and (d) exchanging the first beam splitter of an MZI by a nonlinear system that inserts a NOON (representing a superposition of N particles in the first mode with zero particles in the second mode, and vice versa) state into the interferometer and by using photon counting for parity measurements. The properties of photon counting for obtaining minimal phase uncertainties for the above special cases and for the general case are discussed.
Interferometers IntroductionThe discussion of modules and computerization in previous chapters gives us the means to consider empirically useful quantized detector networks (QDN).In this chapter we shall focus on a particular class of network known as interferometers.Interferometers play a crucial role in quantum mechanics (QM) because they demonstrate the "paradox" of wave-particle duality in a direct way.On the one hand, discrete signals are detected by the observer and those signals are usually interpreted as particles or quanta.On the other hand, the observed frequencies built up over many runs show effects interpreted as due to the interference of waves.In the previous chapter, we used computer algebra (CA) to discuss the Wollaston interferometer (WI), a relatively simple interferometer.We shall use the same approach in the following discussion of the Mach-Zehnder interferometer (MZI), apparatus that allows the observer to investigate optical transmission through various materials. The Mach-Zehnder InterferometerThe basic structure of an MZI is given in Figure 13.1.A source S of light sends a monochromatic, unpolarized beam 1 0 into one input channel of beam splitter B 1 .Output channel 1 1 is deflected by mirror M 1 onto a module labeled φ that contains some medium under investigation, such as a crystal or liquid.The net effect is to create a phase change in that deflected beam by an amount φ, and then that modified beam, 1 2 , is passed into beam splitter B 2 .The second output channel, 2 1 , from B 1 , meanwhile, is deflected by mirror M 2 into beam splitter B 2 .The two deflected beams, 1 2 and 2 2 , that pass into B 2 interfere and are finally monitored by detectors 1 3 and 2 3 .Our analysis of the MZI follows the CA approach used for the WI in the previous chapter.
We present a unified theoretical analysis of three-photon quantum interference in a Six-Port Mach-Zehnder Interferometer (6p-MZI) constructed from two cascaded tritters, with two independent phase modulators placed between the tritter arms. We analytically derive the transfer matrix of the 6p-MZI and show how they organize into three symmetry classes, governed by the discrete Fourier transform (DFT) structure of the tritter and the conjugate relations. Furthermore, we analyze two input regimes: First, three indistinguishable single photons are injected into the tritter, and the output probability distributions P[111]P_{[111]}P[111]​, P[{300}]P_{[\{300\}]}P[{300}]​, and P[{210}]P_{[\{210\}]}P[{210}]​ are derived as functions of the two relative phases (φ1,φ2)(φ_1, φ_2)(φ1​,φ2​). At φ2=0φ_2 = 0φ2​=0, the single-phase limit is recovered, which confirms 100\% visibility of the even-distribution fringe. Second, a hybrid coherent-Fock input ∣α⟩1∣α⟩2∣1⟩3|α\rangle_1|α\rangle_2|1\rangle_3∣α⟩1​∣α⟩2​∣1⟩3​ is analyzed via the density matrix formalism. The average photon number at each output port exhibits amplitude-dependent phase shifts. Our results establish the 6p-MZI as a programmable platform for tripartite quantum state manipulation and coherent amplitude sensing.
The Mach-Zehnder Interferometer (MZI) is used to illustrate the long-standing wave-particle duality problem. How can the photon divide at the first beam splitter and yet terminate on either arm with its undiminished energy? Why is which-way (welcher weg) information incompatible with wave interference? How do we explain Wheeler’s delayed choice experiment? The position adopted is that the photon has two identities, one supporting particle features and the other wave features. There is photon kinetic energy that never splits (on half-silvered mirrors) or diffracts (in pinholes or slits). Then there is photon probability waves that do diffract and can reinforce or cancel. Interpretations of these two identities are suggested.
A nonlinear phase shift is introduced to a Mach-Zehnder interferometer (MZI), and we present a scheme for enhancing the phase sensitivity. In our scheme, one input port of a standard MZI is injected with a coherent state and the other input port is injected with one mode of a two-mode squeezed-vacuum state. The final interference output of the MZI is detected with the method of active correlation output readout. Based on the optimal splitting ratio of beam splitters, the phase sensitivity can beat the standard quantum limit and approach the quantum Cram\'er-Rao bound. The effects of photon loss on phase sensitivity are discussed. Our scheme can also provide some estimates for units of ${\ensuremath{\chi}}^{(3)}$, due to the relation between the nonlinear phase shift and the susceptibility ${\ensuremath{\chi}}^{(3)}$ of the Kerr medium.