By Seeger R.

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The Bogoliubov transformation in Eq. 42) is linear, since it transforms a mode operator into a linear combination of other mode operators. As we have seen, these linear transformations are induced by generators (Hamiltonians) that are quadratic in the mode operators. In particular, both (generalized) beam splitter and squeezing transformations are generated by quadratic Hamiltonians. On the other hand, when people mention linear optics, they often refer speciﬁcally to optical elements that are described by generalized beam splitters, and not squeezers.

214), we can evaluate the expectation values of various important operators. First, the expectation value of the annihilation and creation operators are α, ξ| a ˆ |α, ξ = α and α, ξ| a ˆ† |α, ξ = α∗ . 216) The average photon number in a squeezed coherent state is α, ξ| n ˆ |α, ξ = sinh2 |ξ| + |α|2 . 217) 48 The quantum theory of light The expectation values of the mode operators also immediately allows us to calculate the expectation value of an arbitrary ﬁeld quadrature operator: 1 α, ξ| x ˆζ |α, ξ = √ α e−iζ + α∗ eiζ ≡ xζ (α) .

In this Section we study how linear and quadratic functions of the mode operators can be used to deﬁne two important classes of states of the electromagnetic ﬁeld. Linear Hamiltonians give rise to coherent states, while quadratic Hamiltonians produce squeezed states. Coherent states We can deﬁne states according to the operators that produce them out of the vacuum. For example, as in Eq. 62), the n-photon Fock state in a normalized mode a is produced by the operator n a ˆ† |n = √ |0 . 176) n! 177) 42 The quantum theory of light where the cn are complex amplitudes and n |cn |2 = 1†.