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Extra info for An extensible model of the electron
76) These are real vectors, and they are associated with linear polarization since they point in an unambiguous spatial direction. 3 Mode functions and polarization 19 What happens when the vectors ǫλ have complex components? Two possible orthonormal polarization vectors are 1 ǫL = √ 2 1 i and 1 ǫR = √ 2 1 . 77) Due to the imaginary entries ±i, is not straightforward to interpret the spatial direction of these polarization vectors. It turns out that the imaginary entries cause a rotation of the polarization in time.
We start with the Helmholtz equation for the j th spatial component of the classical vector potential in the frequency domain 20 The quantum theory of light Aj (r, ω), given by the Fourier transform ∞ Aj (r, t) = dω Aj (r, ω)e−iωt . 78) 0 Using ω = ck, this leads to the wave equation for the Fourier components ∇2 + k2 Aj (r, ω) = 0 . 79) This is the Helmholtz equation. For a wave propagating in the z direction, we use the following Ansatz for Aj : Aj (r, ω) = Ψj (r, ω) eikz . 80) The diﬀerential operator acting on Aj can then be written as ∇2 Aj = ∂ 2x Ψj eikz + ∂ 2y Ψj eikz + ∂ z (∂ z Ψj − ikΨj ) eikz .
159) We use the Baker-Campbell-Hausdorﬀ relation from Eq. 123) to calculate the transformed mode operators ˆbjλ = cosh(2ξ) aˆjλ − ie−iϕ sinh(2ξ) aˆ† jλ ˆb† = ieiϕ sinh(2ξ) aˆjλ + cosh(2ξ) aˆ† . 160) In matrix form this is ˆbjλ ˆb† jλ = cosh(2ξ) −ie−iϕ sinh(2ξ) ieiϕ sinh(2ξ) cosh(2ξ) a ˆjλ . 161) We now observe that the quadratic Hamiltonian in Eq. 159) mixes the creation and annihilation operators of the mode ajλ . This is called singlemode squeezing. Alternatively, we can deﬁne two-mode squeezing as the mode transformation that is induced by the Hamiltonian ˆ†lλ′ .