ZU(n) and XU(n) are subgroups of the unitary group U(n): the group XU(n) consists of all n X n unitary matrices with all 2n line sums (i.e. all n row sums and all n column sums) equal to 1 and the group ZU(n) consists of all n X n unitary diagonal matrices with first entry equal to 1.

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Theoretical and Mathematical Physics

Unitary transformations of the group U(N) and diagonalization of multilevel Hamiltonians

Valerii Vladimirovich Val'kov

Published Jul 1, 1988• 28 citations

Publisher

DOI: 10.1007/bf01029435

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\mathrm{U}(n)

matrices that is useful in its own right. We show that the recently discovered ``off-diagonal'' geometric phases [N. Manini and F. Pistolesi, Phys. Rev. Lett. 8, 3067 (2000)] can be completely analyzed in terms of the basic building blocks developed in this work. This result liberates the off-diagonal phases from the assumption of adiabaticity used in arriving at them.

Recent investigations suggest that the discrete linear unitary group

U(N)

can be represented by interlacing a finite sequence of diagonal phase operations with an intervening unitary operator. However, despite rigorous numerical justifications, no formal proof has been provided. Here, we show that elements of

U(N)

can be decomposed into a sequence of

N

-parameter phases alternating with

1

-parameter propagators of a lattice Hamiltonian. The proof is based on building a Lie group by alternating these two operators and showing its completeness to represent

U(N)

for a finite number of layers, which is numerically found to be exactly

N

. This architecture can be implemented using elementary optical components and can successfully reconstruct arbitrary unitary matrices. We propose example devices such as optical logic gates, which perform logic gate operations using a single-layer lossless and passive optical circuit design.