Beam splitter encoding information is represented using unitary matrices or operator transformations that describe how input quantum states are mapped to output states.

Beam Splitter Basics

A beam splitter is an optical device that splits incoming light into transmitted and reflected components. In quantum optics, it is modeled as a unitary transformation acting on the input modes. For a lossless 50/50 beam splitter, the transformation matrix for two input modes can be written as:

Here, the columns represent the input modes, and the rows represent the output modes. This matrix ensures unitarity, meaning the total probability is conserved .

Path-Encoding Representation

In path-encoded quantum information, each optical path corresponds to a basis vector in a Hilbert space. For example:

• $|1,0⟩$ represents a photon in the upper path
• $|0,1⟩$ represents a photon in the lower path The beam splitter acts on these states according to the unitary matrix above. If the vacuum state $|0⟩$ is included, the Hilbert space can be extended, but care must be taken to maintain normalization and unitarity. One approach is to apply the transformation to creation and annihilation operators rather than directly to state vectors .

Operator Approach

Using creation ($a^{†}$) and annihilation ($a$) operators, the beam splitter transformation is:

where $T$ and $R$ are the complex transmission and reflection coefficients. This method naturally handles the vacuum state and multi-photon inputs, and is widely used in quantum optics simulations .

Practical Modeling

For simulations in software like OpticStudio, beam splitters can be modeled in Sequential or Non-Sequential Mode. Non-Sequential Mode allows tracing both transmitted and reflected rays simultaneously, which is essential for capturing quantum interference effects. Sequential Mode requires separate configurations for each path .

Summary

To find beam splitter encoding information:

1. Define the input Hilbert space (path-encoded or Fock space).
2. Choose a representation: unitary matrix for states or operator transformations for creation/annihilation operators.
3. Apply the beam splitter transformation to the input states or operators.
4. Ensure unitarity and normalization, especially when including the vacuum state.
5. Simulate or calculate output probabilities for transmitted and reflected modes, considering interference and entanglement if needed. This framework allows you to encode, manipulate, and analyze quantum information using beam splitters in both theoretical and practical settings .