Entanglement, Mach-Zehnder Interferometer

It may be that one wishes to describe a quantum state as the existence of two separate, non-interacting particles each in a particular state. Imagine particle 1 can be in either state |u1> or |u2> and particle 2 can be in either |v1> or |v2>. We wish to describe the state where particle 1 is in state |u1> and particle 2 is in state |u2>. The notation for this state is:

|u1> ⊗ |u2>

sumstates

Given a probability for each state, a general state formula can be written to describe both particles:

gensumstate

To describe a superposition of particular states however will result in a dependency of the particles on each other. This is called an engtangled state.

entanglestate

The following outlines an entangled state example. This shows how the fate of two particles becomes intertwined in such an entangled state, where there will exist no other combination other than the complete state combinations made available by the definition of the state.

stateex

Einstein has objected to the entangled pair hypothesis. John Bell had proposed an experiment to test entanglement using a three directions, such that a correlation would be more presentable. The results of his experiments however confirmed the possibility of this sort of entanglement on the quantum level, which appears to deny classical mechanics.

 

Mach-Zehnder Interferometer Quantum Mechanical Calculation

Let us model the Mach-Zehnder Interferometer using photon probability state matricies. First, we will consider the operation of the beam splitter. When a photon enters the beamsplitter from one direction, there is a given probability that the photon will be present at either the transmitted or refracted position. Given that the beamsplitter is balanced, meaning that the photon has an equal change (1/2) of exiting either side, the beamsplitter is modeled below:

bspli

Next, using the beamsplitter matrix, the Mach-Zehnder Interferometer can be modeled. Interestingly, the photon appears to exit (100%) from the side opposite which it entered.

mz11

mz2

Let’s consider a case in which mirror 2 is blocked. Using the matrix for beamsplitter 1 and beamsplitter 2, the probabilities are calculated that the photon will either 1. be blocked by the concrete, 2. exit at detector 0 and 3. exit at detector 1.

mz3

 

Barton Zwiebach. 8.04 Quantum Physics I. Spring 2016. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.