By Kok P., Lovett B.W.
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In addition, we define the matrix Tnm as the N -dimensional identity operator I(N ) with the elements Inn , Inm , Imn , and Imm replaced by the four elements of a 2 × 2 unitary matrix. We can then use the Tnm to reduce the size of U (N ): U (N ) · TN,N −1 · TN,N −2 · · · TN,1 = U (N − 1) ⊕ eiφ . 154) U (N − 1) eiφN 0 We therefore need N −1 matrices T to reduce the unitary U (N ) to U (N −1). We can repeat this procedure to reduce U (N − 1) to U (N − 2), and so on. In total, we have to use at most N (N − 1)/2 matrices T to reduce U (N ) to a diagonal matrix with only phases as nonzero elements: U (N ) · TN,N −1 · TN,N −2 · · · T2,1 iφ e 1 = 0 0 ..
173) is the most general quadratic Hamiltonian. It governs the dynamics of free fields, where the mixing of modes is given by the Bogoliubov transformations. These transformations can be implemented with beam splitters and phase shifters, as well as single-mode squeezers. At this point we should mention a possible source of confusion in the nomenclature. 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.
N! 2|ξ| ≡ |ξ = e−ν/2 exp −τ = 1 ∞ cosh |ξ| n=0 n tanhn |ξ||2n . 208) The squeezed vacuum state |ξ has only even photon number states in the superposition. 8: Prove that 12 a ˆ2 and 21 a ˆ†2 obey the algebra defined by the commutation relations for K± and K0 . Since S(ξ) is a unitary evolution, we can ask how it evolves the creation and annihilation operators. 5 Quantum states of the electromagnetic field 47 ξ∗ sinh |ξ|. 209) S(−ξ)ˆ a† S(ξ) = a ˆ† cosh |ξ| − a ˆ This leads to the eigenvalue equation (with eigenvalue zero) a ˆ cosh |ξ| + a ˆ† ξ sinh |ξ| |ξ = 0 .