Beamsplitter

Module: pennylane

class Beamsplitter(theta, phi, wires)[source]

Beamsplitter interaction.

\[B(\theta,\phi) = \exp\left(\theta (e^{i \phi} \a \hat{b}^\dagger -e^{-i \phi}\ad \hat{b}) \right).\]

Details:

  • Number of wires: 2

  • Number of parameters: 2

  • Gradient recipe: \(\frac{d}{d \theta}f(B(\theta,\phi)) = \frac{1}{2} \left[f(B(\theta+\pi/2, \phi)) - f(B(\theta-\pi/2, \phi))\right]\) where \(f\) is an expectation value depending on \(B(\theta,\phi)\).

  • Heisenberg representation:

    \[\begin{split}M = \begin{bmatrix} 1 & 0 & 0 & 0 & 0\\ 0 & \cos\theta & 0 & -\cos\phi\sin\theta & -\sin\phi\sin\theta \\ 0 & 0 & \cos\theta & \sin\phi\sin\theta & -\cos\phi\sin\theta\\ 0 & \cos\phi\sin\theta & -\sin\phi\sin\theta & \cos\theta & 0\\ 0 & \sin\phi\sin\theta & \cos\phi\sin\theta & 0 & \cos\theta \end{bmatrix}\end{split}\]
Parameters:
  • theta (float) – Transmittivity angle \(\theta\). The transmission amplitude of the beamsplitter is \(t = \cos(\theta)\). The value \(\theta=\pi/4\) gives the 50-50 beamsplitter.
  • phi (float) – Phase angle \(\phi\). The reflection amplitude of the beamsplitter is \(r = e^{i\phi}\sin(\theta)\). The value \(\phi = \pi/2\) gives the symmetric beamsplitter.
  • wires (Sequence[int] or int) – the wire the operation acts on