Dynamics of the SU(1,1) Bloch vector

Dattoli, G.; Dipace, A.; Torre, A.

doi:10.1103/physreva.33.4387

Cited by 30 publications

(12 citation statements)

References 9 publications

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“…σ ab are the so(2, 3) matrices (123). With Ψ ′ L and Ψ ′ R , we construct the 4 × 4 matrix M, which we will 20 Here, the imaginary unit i is added on the right-hand side of ψ R for later convenience.…”

Section: Euler Angle Decompositionmentioning

confidence: 99%

$Sp(4; \mathbb{R})$ squeezing for Bloch four-hyperboloid via the non-compact Hopf map

Hasebe¹

2020

J. Phys. A: Math. Theor.

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We explore the hyperbolic geometry of squeezed states in the perspective of the non-compact Hopf map. Based on analogies between squeeze operation and Sp(2, R) hyperbolic rotation, two types of the squeeze operators, the (usual) Dirac-and the Schwinger-types, are introduced. We clarify the underlying hyperbolic geometry and SO(2, 1) representations of the squeezed states along the line of the 1st non-compact Hopf map. Following to the geometric hierarchy of the non-compact Hopf maps, we extend the Sp(2; R) analysis to Sp(4; R) -the isometry of an split-signature four-hyperboloid.We explicitly construct the Sp(4; R) squeeze operators in the Dirac-and Schwinger-types and investigate the physical meaning of the four-hyperboloid coordinates in the context of the Schwinger-type squeezed states. It is shown that the Schwinger-type Sp(4; R) squeezed one-photon state is equal to an entangled superposition state of two Sp(2; R) squeezed states and the corresponding concurrence has a clear geometric meaning. Taking advantage of the group theoretical formulation, basic properties of the Sp(4; R) squeezed coherent states are also investigated. In particular, we show that the Sp(4; R) squeezed vacuum naturally realizes a generalized squeezing in a 4D manner. Sp(2; R) group and squeezingThe isomorphism Sp(2; R) ≃ Spin(2, 1) suggests that the Sp(2; R) one-and two-mode operators are equivalent to the Majorana and the Dirac spinor operators of SO(2, 1). Based on the identification of the squeeze operator with the SU (1, 1) ≃ Spin(2, 1) "rotation" operator, we introduce two types of squeeze operators, the (usual) Dirac-and Schwinger-types. We discuss how the non-compact 1st Hopf map is embedded in the geometry of the Sp(2; R) squeezed state. 7 U Sp(2n) = U (n; H) and π 1 (U Sp(2n)) = 1. For instance, U Sp(2) = SU (2) = Spin(3), U Sp(4) = Spin(5). 8 The anti-de Sitter, de Sitter and Euclidean anti-de Sitter spaces are realized as the special cases of the ultra-hyperboloids:

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Section: Euler Angle Decompositionmentioning

confidence: 99%

$Sp(4; \mathbb{R})$ squeezing for Bloch four-hyperboloid via the non-compact Hopf map

Hasebe¹

2020

J. Phys. A: Math. Theor.

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“…[ 31] ) using the rotation-operator approach. Although both kinematic relations are sets of ordinary, coupled differential equations, the Euler kinematic equations are nonlinear, whereas those for the ER parameters are linear.…”

Section: The Kinematic Equationsmentioning

confidence: 99%

“…mally used to describe rigid-body motions, emerge from In the case of the Bloch equations, it has been shown elsethe disentangling procedure of the rotation-operator ap-where (31, 32) exactly how SU(2) state dynamics can be proach in the Euler angle parametrization? The simple modeled as a rotation in a Euclidean space, and how the answer is that the same kinematical relations must be relevant Rabi rotation matrix (31)(32)(33) can be deduced from obtained in both the quantum-mechanical and the classi-the Wei-Norman ordered form of the evolution operator for cal picture; in the next section, both sets of kinematical SU(2) Hamiltonians (34). Further work (35,36) showed relations derived via the rotation-operator approach will how the Wei-Norman ordering equations could be cast in be rederived from the Bloch equations using a classical the form of a generalized Bloch equation, thereby establishpoint of view.…”

Section: Introductionmentioning

confidence: 99%

A Classical View of the Euler Angles and the Euler Kinematic Equations in NMR

Siminovitch

1995

Journal of Magnetic Resonance, Series A

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“…They have applied the tech nique to such systems as a harmonic oscillator with a time dependent fre quency 137 , the generation of antibunched radiation in laser-plasma scattering 138 , and to the free electron laser 139 . They have presented a Bloch type vector 140 . Finally we mention that the Wei-Norman technique has been used in a problem not involving quantum optics but rather opti cal propagation in nonhomogeneous media.…”

Section: Related Mattersmentioning

confidence: 99%

PART III SU(1, 1) COHERENT STATES AND PATH INTEGRALS FOR SU(1, 1)

Gerry

1992

Path Integrals and Coherent States of SU(2) and SU(1,1)

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Path Integrals and Coherent States of SU(2) and SU(1,1) Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/20/15. For personal use only. This page is intentionally left blank Path Integrals and Coherent States of SU(2) and SU(1,1) Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/20/15. For personal use only. Path Integrals and Coherent States of SU(2) and SU(1,1) Downloaded from www.worldscientific.com by KAINAN UNIVERSITY on 02/20/15. For personal use only. 222are used to construct a generalized uncertainty relation; the generalized coherent states (CS) being those states for which the uncertainty product is a minimum (MUCS). The states developed in this fashion when sub jected to time evolution are not completely dispersionless as they move in the classical potential but do appear to be capable of a least partly reforming after a few oscillations although, eventually, an initial wave packet will disperse throughout the classically allowed region 10 .An alternate method of creating generalized coherent states for sys tems other than the oscillator is associated with the unitary irreducible representations (UIRs) of Lie groups and Lie algebras 11 . This approach may be considered as complementary to the MUCS discussed above. The most familiar Lie group, SU(2), of course describes a number of systems such as two level atoms, spin systems etc. Coherent states of SU(2) are the subject of Professor Kuratsuji's lectures. On the other hand the noncompact group SU(1,1), which has the following local isomorphisms: SU(1,1) ~ SO(2,l) ~~ Sp(2R) ~ S1(2R), plays the role of dynamical group for a number of systems of general interest. The term dynamical group as used here should be taken as synonymous with the terms noninvariance group and spectrum generating group. Actually it is the spectrum generating algebra (SGA) of the associated dynamical group that is mainly of interest. With the Hamiltonian composed of the generators of the group, the UIRs yield the spectrum from the energy eigenvalue prob lem. Among the systems for which SU(1,1), or one of its isomorphic forms, provides an SGA, are, the harmonic oscillator 12 , the Coulomb problem 13 , Hartmann's 14 ring potential 15 , the s-states of the Morse oscilla tor 16 , the s-states of the Hulthen potential 17 , the Dirac-Coulomb and Klein-Gordon-Coulomb problems 18 . Some many-particle problems are also included. Examples are a model of a superfluid bose system 19 and coupled harmonic oscillators 20 . There recently has been a great deal of interest in the algebraic approach to scattering problems 21 and in the cal culation of resonance widths and positions 22 . Now in constructing coherent states for SU(1,1) we recall that ordi nary oscillator coherent states, {\z>} where z is complex, may be con structed in three ways: i) as states which minimize the uncertainty rela tion AxAp >7T/2, ii) as eigenstates of the annihilation operator a, i.e. a \z> «= z \z> and III) as states displaced from the vacuum |0> by the operation of the displacement o...

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Dynamics of the SU(1,1) Bloch vector

Cited by 30 publications

References 9 publications

$Sp(4; \mathbb{R})$ squeezing for Bloch four-hyperboloid via the non-compact Hopf map

$Sp(4; \mathbb{R})$ squeezing for Bloch four-hyperboloid via the non-compact Hopf map

A Classical View of the Euler Angles and the Euler Kinematic Equations in NMR

PART III SU(1, 1) COHERENT STATES AND PATH INTEGRALS FOR SU(1, 1)

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