A beam splitter or beamsplitter is an optical device that splits a beam of light into a transmitted and a reflected beam. It is a crucial part of many optical experimental and measurement systems, such as interferometers, also finding widespread application in fibre optic telecommunications.
Beam Splitter
Transfer matrix
The recipe for quantization is that ‘replace the classical amplitudes by annihilation operators’. Namely,
For fibre optics (i.e. waveguide beam splitter), we can pair two input ports with two output ports.[1] This correspondence does not imply that a photon on an input port is identical to a photon on an output port (e.g. ∣10⟩in=∣10⟩out). Rather, the photon number states in the upper and lower fibers, no matter they belong to input or output, can be considered to be in the same state space, and the beam splitter actually acts as a transformation in this state space.
For example, a 50:50 beam splitter acting on a single-photon incident state should be strictly written as
However, if we focus only on the photon number, we can write
BS(π/4)∣10⟩=21(∣10⟩−∣01⟩)
where the first number in the ket represents the photon number in the upper fibre, and the second number represents the photon number in the lower fibre. And, we recognize by default, that the state before BS operators acting on them represents the photon distribution in the input fibers while the state transformed represents the photon distribution in the output fibers.
This can also be glimpsed in the creation and annihilation operators. The linear relationship between b,b† and a,a† suggests that they can be considered to act on the same state space.
Example: Hong–Ou–Mandel interference
Answer: Let us consider two single-photon states incident on the same beam splitter ∣11⟩in.
It is easy to check such angular momentum operators satisfying the commutation rules [Li,Lj]=iεijkLk. Therefore, the most general form of BS could be written as
BS=exp[i(αLx+βLy+γLz)]
Relationship
The relationship between the beam splitter operators BS acting on the states and the transfer matrix U acting on the creation and annihilation operators is[2]
Makarov D. Theory for the beam splitter in quantum optics: Quantum entanglement of photons and their statistics, HOM effect[J]. Mathematics, 2022, 10(24): 4794. ↩︎
Campos R A, Saleh B E A, Teich M C. Quantum-mechanical lossless beam splitter: SU (2) symmetry and photon statistics[J]. Physical Review A, 1989, 40(3): 1371. ↩︎