Beam splitter->ONB->transition amplitudes

• klabautermann
Therefore, S has the mathematical property of being unitary. In summary, a general beam splitter is characterized by its transition amplitudes Sxy=<Xin|Yout>, where X,Y=L or R. |Lin> represents a photon leaving the beam splitter to the right. The matrix S of transition amplitudes has the mathematical property of being unitary because it preserves the inner product between vectors. This means that S can also be used with other basis vectors, such as |Yin>,|Xin>, and still preserve the inner product.
klabautermann

Homework Statement

a general beam splitter is charcatericed by its transition amplitudes Sxy=<Xin|Yout> where X,Y=L or R. |Lin> represents a photon which leaves the beam splitter to the right. we are only considering photons with the same energy and polarization and neglect all other state parameters.
the beam splitter is loss-free, what mathematical properties has the matrix S of transition amplitudes?

The Attempt at a Solution

{|Xin>,|Yout>} form an orthonormal basis and |Yout>=$\sum$SXY|Xin>. S describes the change of basis. S preserves the inner product, therefor S is unitary.

i am reviewing problem sets, because i have quantum mechanics exam tomorrow. i have a question. could i also take for example |Yin>, |Xin> as basis vectors? why do i know that S preserves the inner product.

thanks!

Yes, you could take |Yin>,|Xin> as basis vectors. The matrix S preserves the inner product because it is unitary. A unitary matrix preserves the inner product because it preserves the lengths of vectors and angles between them. This means that, for any two vectors |u⟩ and |v⟩, the inner product ⟨u|v⟩ is preserved by the matrix S. This implies that S preserves the inner product in all cases.

1. What is a beam splitter?

A beam splitter is an optical device that splits a beam of light into two or more separate beams, typically using a partially reflective surface.

2. What is an ONB?

An ONB (orthonormal basis) is a set of vectors that form a basis for a vector space, where each vector is of unit length and all vectors are mutually perpendicular.

3. How are beam splitters used in ONB?

Beam splitters are used in ONB to create superposition states, where two or more beams of light are combined to form a new state with a different polarization or direction.

4. What are transition amplitudes?

Transition amplitudes are mathematical quantities that describe the probability of a quantum system transitioning from one state to another.

5. How are beam splitter->ONB->transition amplitudes used in science?

Beam splitter->ONB->transition amplitudes are used in a variety of scientific fields, such as quantum computing, quantum optics, and quantum information theory, to study and manipulate the behavior of quantum systems.

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