# Mach-Zehnder interferometers and beam splitters | Summary and Q&A

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July 5, 2017
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MIT OpenCourseWare
Mach-Zehnder interferometers and beam splitters

## TL;DR

This content discusses the use of Mach-Zehnder interferometers and beam splitters in quantum mechanics to describe the behavior of photons.

## Key Insights

• 😁 Mach-Zehnder interferometers consist of beam splitters, mirrors, and phase shifters to manipulate the behavior of photons in different paths.
• 👋 The behavior of photons in a Mach-Zehnder interferometer can be described using a wave function and represented by complex numbers in a column vector.
• 😁 Beam splitters in Mach-Zehnder interferometers split a beam of light and can be characterized by four numbers (s, t, u, v) that determine their effect on photon states.
• 😁 Balanced beam splitters, where s² + t² = 1 and u² + v² = 1, simplify the determination of beam splitter matrices.

## Transcript

PROFESSOR: Mach-Zehnder-- interferometers. And we have a beam splitter. And the beam coming in, it splits into 2. A mirror-- another mirror. The beams are recombined into another beam splitter. And then, 2 beams come out. One to a detector d0-- and a detector d1. We could put here any kind of devices in between. We could put a little piece of glass... Read More

### Q: How can a photon's behavior in a Mach-Zehnder interferometer be described mathematically?

A photon's behavior can be described using a wave function represented by a column vector with two complex numbers, which represent the probability amplitudes for the photon to be in a specific location within the interferometer.

### Q: What is the purpose of a beam splitter in a Mach-Zehnder interferometer?

A beam splitter splits a beam of light into two separate paths - one reflected and one transmitted - and interacts with the photon states. It can be used to control the behavior of photons within the interferometer.

### Q: What are the constraints for the numbers (s, t, u, v) characterizing a beam splitter?

The numbers (s, t, u, v) need to satisfy the conditions of probability conservation and intensity preservation. Specifically, s² + t² = 1 and u² + v² = 1 to ensure that the probabilities of photon presence are preserved.

### Q: Can a beam splitter matrix be determined by any arbitrary numbers?

No, the numbers in the beam splitter matrix need to satisfy the conditions of probability conservation and intensity preservation. They should result in a normalized photon state after the beam splitter action.

## Summary & Key Takeaways

• Mach-Zehnder interferometers are composed of a beam splitter, mirrors, and a phase shifter, and can split a beam of light into two separate paths.

• A photon's behavior in a Mach-Zehnder interferometer can be described using a wave function and represented by a column vector with two complex numbers.

• Beam splitters, characterized by four numbers (s, t, u, v), act on photon states and produce a matrix operation on them.