Some basics of su(1,1)

Abstract

A basic introduction to the su(1, 1) algebra is presented, in which we discuss the relation with canonical transformations, the realization in terms of quantized radiation field modes and coherent states. Instead of going into details of these topics, we rather emphasize the existing connections between them. We discuss two parametrizations of the coherent states manifold SU (1, 1)/U (1): as the Poincaré disk in the complex plane and as the pseudosphere (a sphere in a Minkowskian space), and show that it is a natural phase space for quantum systems with SU (1, 1) symmetry. Keywords: su(1, 1) algebra, SU (1, 1) group, canonical transformations, coherent states, second quantization, pseudosphere.Uma introdução simplesàálgebra su(1, 1)é apresentada, na qual discutimos a relação com transformações canônicas, a realização em termos de modos quantizados do campo de radiação e estados coerentes. Ao invés de entrar em detalhes a respeito desses tópicos, preferimos enfatizar as conexões existentes entre eles. Discutimos duas parametrizações da variedade dos estados coerentes SU (1, 1)/U (1): como o disco de Poincaré no plano complexo e como a pseudoesfera (uma esfera em um espaço de Minkowski) e mostramos que se trata de um espaço de fase natural para sistemas quânticos com simetria SU (1, 1). Palavras-chave:álgebra su(1, 1), grupo SU (1, 1), transformações canônicas, estados coerentes, segunda quantização, pseudoesfera.

Quantum integrability in the multistate Landau–Zener problem

“…where K 0 , K 1 and K 2 are the generators of the 2-dimensional non-unitary representation of SU (1, 1) and can be expressed in terms of the Pauli matrices as σz 2 , ı σx 2 and ı σy 2 respectively [25,26]. Since SU (1, 1) is a non-compact Lie group, one has to go to infinite dimensions in order to get a unitary representation of the group.…”

“…To illustrate the descendant generation procedure, we explicitly derive the LZ transition probabilities for both these models as well as the general SU (2) Hamiltonian from the known solution of the 2 × 2 LZ problem. One can also express the 2 × 2 LZ Hamiltonian in terms of the SU (1, 1) algebra as H(t) = tK 0 − ıgK 1 , where K 0 = σz 2 , K 1 = ı σx 2 and K 2 = ı σy 2 are the generators of the 2-dimensional non-unitary representation of SU (1, 1) [25,26]. Similar to the SU (2) generalization, we promote K 0 , K 1 , and K 2 to any other SU (1, 1) representation and solve the resulting LZ problem.…”

“…(70) with the help of known formulas for the Wigner-Bargmann matrix elements [25,[27][28][29][30]. The first example that we consider is known as the one-mode realization [26,28,30,36] of the D + k representation for SU (1, 1) algebra, i.e.…”

Intensity-dependent quantum Rabi model: spectrum, supersymmetric partner, and optical simulation

“…+ 2kâσ x such that they form the SU(1,1) group, K + ,K − = −2K 0 and K 0 ,K ± = ±K ± [19,20]. In such a case we can put aside the constant terms and focus on the parity subspace Hamiltonians,Ĥ…”

Monodromy and chaos for condensed bosons in optical lattices

“…A useful way for visualizing the phase space tori is based on the SU (1, 1) symmetry [19,27] of H (0) . For that we express the two conserved quantities, namely the energy E and the constant of motion M , in terms of the group generators.…”