Multiple photon effective Hamiltonians in linear quantum optical networks

Method to determine which quantum operations can be realized with linear optics with a constructive implementation recipe

“…These matrices have a strong structure. The only nonzero elements are in positions which correspond to transitions between states that are, at most, one photon away from each other (for input states |p and |q that only differ in the photon number in two positions, one mode giving the photon to the other [13]).…”

“…The map ϕ m,M is a differentiable group homomorphism [1] and it induces an algebra homomorphism, dϕ m,M , as described by the commutative diagram, , which relates the unitary groups U(m) and U(M ) containing the scattering matrix S and the n-photon evolution operator U , respectively, to the algebras u(m) and u(M ), whose elements correspond to anti-Hermitian matrices iH S and iH U which give an equivalent description of the evolution through exponentiation of the Hamiltonians H S and H U (S = e iH S and U = e iH U ) [12,13].…”

“…We can also write the elements of U from the permanent of different submatrices of S [4]. From the differential dϕ m,M [13], we can write the effective Hamiltonian H U of a linear optical transformation as…”

“…For a given iH S ∈ u(2), the corresponding element in the image subalgebra d, iH U = dϕ 2,6 (iH S ), is given by [13]…”

Optimal approximation to unitary quantum operators with linear optics

“…There are many results on the synthesis of linear systems from their classical description [4,7,8] and that analyze the evolution of multiple photons in those devices [10][11][12][13][14][15][16][17][18]. However, there are relatively few methods for the design of tailored quantum evolutions for multiple photons.…”