Quantum States Of EM Field

Date:

2 August 2019

What is the problem ?

Derivation

Step 1 Fock States / Number States

A Fock state is a state with a well-defined number of photons, i.e. an eigenstate of

\[\begin{split}\begin{align*} & N = \sum\limits_M N_M = \sum\limits_M a_M^{\dagger}a_M\\ & n=0 \text{ (vacuum state): } |vac\rangle = ...\otimes |0\rangle \otimes |0\rangle \otimes ...\\ & n=1 \text{ (one photon): } \sum\limits_M c_M |1_M\rangle \\ & n=2 \text{ (two photons): } \\ & \;\;\; |2_M\rangle = ...\otimes |0\rangle \otimes |2\rangle \otimes ...\\ & \;\;\; |1_M,1_{M'}\rangle = ...\otimes |1\rangle \otimes ... \otimes |1\rangle \otimes ...\\ & \;\;\; |n_M,n'_{M'}\rangle = {(a_M^{\dagger})^n \over \sqrt{n!}} {(a_{M'}^{\dagger})^{n'} \over \sqrt{n'!}} |vac\rangle \\ & \;\;\; \text{and any superposition of them}\\ \end{align*}\end{split}\]

• All one-photon states are ultimately monomode, because \(\sum\limits_M c_M |1_M\rangle=A^{\dagger}|vac\rangle \text{ with } A^{\dagger} =\sum\limits_M c_M a_M^{\dagger}\)