# Quantum Optics

- Page ID
- 127062

- 258: Single-photon Interference - Third Version
- The quantum mechanical analysis the the Mach-Zehnder interferometer is outlined below. When only one photon is in the apparatus at any time, the photon is always detected at D2 and never at D1.(1,2,3)There are two paths (upper and lower) to each detector, and they both contain a beam splitter, a mirror, and another beam splitter before the detectors are reached. The origin of the 90° phase difference between transmission and reflection is found in the principle of energy conservation.

- 259: Single Photon Interference - Fourth Version
- This analysis of the operation of a Mach-Zehnder Interferometer (MZI) will use tensor algebra and the creation and annihilation operators.

- 261: The Polarizing Beam Splitter and the Superposition Principle
- A polarizing beam splitter (PBS) and PBS interferometer (PBSI) can be used to illustrate the superposition principle. In this analysis the quantum math explaining the operation of a PBSI is presented.

- 262: Mach-Zehner Polarization Interferometer Analyzed Using Tensor Algebra
- The purpose of this tutorial is to analyze a Mach-Zehnder (MZ) interferometer with polarizing beam splitters (PBS) using tensor algebra. First we will review the traditional MZ with non-polarizing beam splitters using matrix algebra.

- 263: Illustrating the Superposition Principle with Single Photon Interference
- Single-photon interference in a Mach-Zehnder interferometer is used to illustrate the superposition principle. Three methods of analysis of an historically important experiment are presented at a level appropriate for an undergraduate course in quantum chemistry or physics. The importance of the superposition principle in chemistry is also discussed.

- 264: Pure States, Mixtures and the Density Operator
- Calculations with polarized and un-polarized light will be used to illustrate the difference between pure states and mixtures, and to demonstrate the utility of the density operator. In Heisenberg’s matrix mechanics pure states are represented by vectors and operators by matrices.

- 265: Using the Trace Function to Calculate Expectation Values
- Starting with the traditional expression for the calculation of the expectation value, the identity operator is inserted between the measurement operator and the ket containing the wave function. Rearranging terms gives the trace function operating on the product of the state's density operator and the measurement operator.

- 266: Polarized Light and Quantum Superposition
- The superposition principle is a deep and difficult quantum mechanical concept. There is no classical analog to lean on in probing its meaning, because it is impossible to simulate it with classical models. And yet because it pervades quantum mechanics we must find ways to present it to students that have some probability of success in revealing its significance.

- 267: Polarized Light and Quantum Mechanics
- Readily available and inexpensive polarizing films can be used to illustrate many fundamental quantum mechanical concepts. The purpose of this tutorial is to use polarized light to illustrate one of quantum theory’s deepest and most challenging concepts - the linear superposition. According to Richard Feynman the superposition principle “has in it the heart of quantum mechanics” and is its “only mystery.”

- 268: The Three-Polarizer Paradox
- Vector algebra will be used to analyze this so-called "three-polarizer paradox." The paradox being that it is surprising that the insertion of the diagonal polarizer between crossed polarizers allows photons to pass the final horizontal polarizer.

- 269: Matrix Mechanics Approach to Polarized Light
- It is convenient to use matrix mechanics to describe experiments with polarized light. In this tutorial we will restrict our attention to plane polarized light. However, it would be just as easy to use matrix mechanics to describe the behavior of circularly polarized light.

- 270: Matrix Mechanics Approach to Polarized Light - Version 2
- It is convenient and illustrative of quantum mechanical principles to use matrix mechanics to describe experiments with polarized light. In this tutorial we will restrict our attention to plane polarized light. However, it would be just as easy to use matrix mechanics to describe the behavior of circularly polarized light (see appendix).

- 271: Matrix Mechanics Exercises Using Polarized Light
- Eigenstates and operators are provided for a series of matrix mechanics exercises involving polarized light.

- 272: Polarized Light and Quantum Mechanics
- Unpolarized light consists of photons of all possible polarization angles. A photon polarized at an angle θ with respect to the vertical can be written as a linear superposition of a vertically polarized photon, |v⟩ , and a horizontally polarized photon |h⟩. |v⟩ and |h⟩ are the polarization basis states.

- 273: Neutron Interferometry with Polarized Spin States
- the purpose of this tutorial to work through the quantum math of a neutron interferometry experiment.

- 274: Interaction Free Measurement - Seeing in the Dark
- The illustration of the concept of interaction-free measurement requires the use of an interferometer. A simple illustration employs a Mach-Zehnder interferometer.

- 278: The Ramsey Atomic Interferometer
- The Ramsey interferometer, which closely resembles the Mach-Zehnder interferometer, is constructed using two π/2 Rabi pulses separated by a phase shifter in the lower arm.

- 279: Optical Activity - A Quantum Perspective
- Optical activity is the rotation of linearly polarized light as it advances through a chiral medium. The quantum explanation for optical rotation is based on the fact that linearly polarized light can be written as a superposition of left and right circularly polarized light, which possess angular momentum.

- 280: A Quantum Optical Cheshire Cat
- The following is a summary of "Quantum Cheshire Cats" by Aharonov, Popescu, Rohrlich and Skrzypczyk which was published in the New Journal of Physics 15, 113015 (2013) and can also be accessed at: arXiv:1202.0631v2.

- 289: Two-electron Interference
- This tutorial provides a simplified analysis of a recent experiment describing the “interference between two indistinguishable electrons from independent sources.”(1)

- 293: Analyzing Two-photon Interferometry Using Mathcad and Tensor Algebra
- Greenberger, Horne and Zeilinger (GHZ) surveyed the then relatively new field of multiparticle interferometry in their August 1993 Physics Today article, "Multiparticle Interferometry and the Superposition Principle." This tutorial will use Mathcad and tensor algebra to analyze the results associated with their experiment.

- 296: A Quantum Delayed-Choice Experiment
- This tutorial works through the first half of the Science report (Nov. 2, 2012, pp . 634‐637) by Peruzzo et al. with the title given above. This involves address the Wheeler‐type delayed‐choice experiment.

- 297: A Quantum Delayed-Choice Experiment
- This note presents a critique of "Entanglement-Enabled Delayed-Choice Experiment," F. Kaiser, et al. Science 338, 637 (2012). This experiment was also summarized in section 6.3 of Quantum Weirdness, by William Mullin, Oxford University Press, 2017.

*Thumbnail: United States Air Force laser experiment. (Public Domain).*