Following Feynman, teachers of quantum theory use the double-slit experiment to illustrate the superposition principle and its signature effect, quantum interference. A single particle (photon, electron, etc.) arrives at any point on the detection screen by two paths, whose probability amplitudes interfere yielding the characteristic diffraction pattern. This is called single-particle interference.

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Mach-Zehnder interferometer

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\[ |S\rangle \stackrel{B S_{1}}{\longrightarrow} \frac{1}{\sqrt{2}}[|x\rangle+ i|y\rangle]\xrightarrow{Mirrors} \frac{1}{\sqrt{2}}[|y\rangle+ i|x\rangle] \stackrel{B S_{2}}{\longrightarrow} i|x\rangle \nonumber \]

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Another quantum mechanical point that Feynman made with the double-slit experiment is that if path information (which slit the photon went through) is available (even in principle) the interference fringes disappear. This is also the case with the MZ interferometer.

\[ |x\rangle \rightarrow \frac{1}{\sqrt{2}}[|x\rangle+ i|y\rangle] \qquad|y\rangle \rightarrow \frac{1}{\sqrt{2}}[|y\rangle+ i|x\rangle] \nonumber \]

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\[ P\left(D_{x}\right)=|\langle x | S\rangle|^{2}=\Big|\frac{i}{\sqrt{2}}\Big|^{2}=\frac{1}{2} \quad P\left(D_{y}\right)=|\langle y | S\rangle|^{2}=\Big|\frac{1}{\sqrt{2}}\Big|^{2}=\frac{1}{2} \nonumber \]

Thus we see that, indeed, the photon always arrives at Dx in the equal-arm MZ interferometer shown above. The paths to Dx (TR+RT) are in phase and constructively interfere. The paths to Dy (TT+RR) are 180 degrees (i2) out of phase and therefore destructively interfere. (T stands for transmitted and R stands for reflected.)

\[| S \rangle \xrightarrow{PathA} | x \rangle \xrightarrow{MirrorA} | y \rangle \xrightarrow{BS_{2}} \frac{1}{\sqrt{2}}(i|x\rangle+|y\rangle) \nonumber \]

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Single-particle interference in a Mach-Zehnder (MZ) interferometer is a close cousin of the traditional double-slit experiment. Using routine complex number algebra, it can be used to illustrate the same fundamentals as the two-slit experiment and also to introduce students to the field of quantum optics.

Mach-Zehnder interferometer experiment

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A key convention in the analysis of MZ interferometers is that reflection at a beam splitter (BS) is accompanied by a 90 degree phase shift (\(\frac{\pi}{2}\), i). The behavior of a photon traveling in the x- or y-direction at the beam splitters and mirrors is as follows.

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\[ P\left(D_{x}\right)=|\langle x | S\rangle|^{2}=\Big|\frac{1}{\sqrt{2}}\Big|^{2}=\frac{1}{2} \quad P\left(D_{y}\right)=|\langle y | S\rangle|^{2}=\Big|\frac{i}{\sqrt{2}}\Big|^{2}=\frac{1}{2} \nonumber \]

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Using the information provided above and complex number algebra, the history of a photon leaving the source (moving in the x-direction) is:

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Richard Feynman raised Young’s double-slit experiment to canonical status by presenting it as the paradigm for all quantum mechanical behavior. In The Character of Physical Law he wrote, “Any other situation in quantum mechanics, it turns out, can always be explained by saying, ‘You remember the case of the experiment with the two holes? It’s the same thing.’”

This tutorial draws heavily on a recent article in the American Journal of Physics and papers quoted therein [Am. J. Phys. 78(8), 792-795 (2010)]. An equal-arm MZ interferometer is shown below. In this configuration the photon is always detected at Dx. The analysis below provides an explanation why this happens.

\[ \text{Probability}\left(D_{x}\right)=|\langle x | S\rangle|^{2}=1 \quad \text { Probability }\left(D_{y}\right)=|\langle y | S\rangle|^{2}=0 \nonumber \]

In these cases, where path information is available the detection of the photon at both detectors in equal percentages is the equivalent of the disappearance of the interference fringes in the double-slit experiment when knowledge of which slit the particle went through is available.

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The simplest two-dimensional (2D) spectra show how excitation with one (variable) frequency affects the spectrum at all other frequencies, thus revealing the molecular connections between transitions. Femtosecond 2D Fourier transform (2D FT) spectra are more flexible and share some of the remarkable properties of their conceptual parent, 2D FT nuclear magnetic resonance. When 2D FT spectra are experimentally separated into real absorptive and imaginary refractive parts, the time resolution and frequency resolution can both reach the uncertainty limit set for each resonance by the sample itself. Coherent four-level contributions to the signal provide new molecular phase information, such as relative signs of transition dipoles. The nonlinear response can be picked apart by selecting a single coherence pathway (e.g., specifying the relative signs of energy level difference frequencies during different time intervals as in the photon echo). Because molecules are frozen on the femtosecond timescale, femtosecond 2D FT experiments can separate a distribution of instantaneous molecular environments and intramolecular geometries as inhomogeneous broadening. This review provides an introduction to two-dimensional Fourier transform experiments exploiting second- and third-order vibrational and electronic nonlinearities.

The detection of the photon exclusively at Dx is the equivalent of the appearance of the interference fringes in the double-slit experiment.

\[| S \rangle \xrightarrow{PathB} | y \rangle \xrightarrow{MirrorB} | x \rangle \xrightarrow{BS_{2}} \frac{1}{\sqrt{2}}(|x\rangle+i|y\rangle) \nonumber \]