Non-Hermitian Coherent States for Finite-Dimensional Systems

Guerrero, Julio Becerra

doi:10.1007/978-3-319-76732-1_10

Cited by 5 publications

(7 citation statements)

References 32 publications

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“…In such a picture, the complex‐valued potential V λ ( x ) can be identified with a refractive index of balanced gain/loss profile . The study of non‐Hermitian coherent states associated with finite‐dimensional systems is also available. Finally, the approach can be extended either by applying conventional 1‐step Darboux transformations on the complex‐valued potential V λ ( x ) or by iterating the procedure presented in this work.…”

Section: Discussionmentioning

confidence: 81%

Interplay between Riccati, Ermakov, and Schrödinger equations to produce complex‐valued potentials with real energy spectrum

Blanco-Garcia

Rosas-Ortiz

Zelaya

2018

Math Methods in App Sciences

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Nonlinear Riccati and Ermakov equations are combined to pair the energy spectrum of 2 different quantum systems via the Darboux method. One of the systems is assumed Hermitian, exactly solvable, with discrete energies in its spectrum. The other system is characterized by a complex‐valued potential that inherits all the energies of the former one and includes an additional real eigenvalue in its discrete spectrum. If such eigenvalue coincides with any discrete energy (or it is located between 2 discrete energies) of the initial system, its presence produces no singularities in the complex‐valued potential. Non‐Hermitian systems with spectrum that includes all the energies of either Morse or trigonometric Pöschl‐Teller potentials are introduced as concrete examples.

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Section: Discussionmentioning

confidence: 81%

Interplay between Riccati, Ermakov, and Schrödinger equations to produce complex‐valued potentials with real energy spectrum

Blanco-Garcia

Rosas-Ortiz

Zelaya

2018

Math Methods in App Sciences

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“…The present work provides a further example of numerous cases [24,31,33,34,67] when the coherent state transform is meaningful and useful beyond the traditional setup of square-integrable representations modulo a subgroup H [6, Ch. 8].…”

Section: Discussionmentioning

confidence: 99%

“…Although the representation of G is square-integrable modulo some subgroup H, the obtained geometric dynamic is not confined within the homogeneous space G/H, see (4.13). Thus, the standard approach to coherent state transform from square-integrable group representations [6, ch 8] shall be replaced by treatments of non-admissible mother wavelets [31,33,34], see also [18,19,24,67]. This is discussed in section 3.1.…”

Section: Remark 11 (Background and Names)mentioning

confidence: 99%

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Geometric dynamics of a harmonic oscillator, arbitrary minimal uncertainty states and the smallest step 3 nilpotent Lie group

Almalki

Kisil

2018

J. Phys. A: Math. Theor.

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The paper presents a new method of geometric solution of a Schrödinger equation by constructing an equivalent first-order partial differential equation with a bigger number of variables. The equivalent equation shall be restricted to a specific subspace with auxiliary conditions which are obtained from a coherent state transform. The method is applied to the fundamental case of the harmonic oscillator and coherent state transform generated by the minimal nilpotent step three Lie group-the group G (also known under many names, e.g. quartic group). We obtain a geometric solution for an arbitrary minimal uncertainty state used as a fiducial vector. In contrast, it is shown that the well-known Fock-Segal-Bargmann transform and the Heisenberg group require a specific fiducial vector to produce a geometric solution. A technical aspect considered in this paper is that a certain modification of a coherent state transform is required: although the irreducible representation of the group G is square-integrable modulo a subgroup H, the obtained dynamic is transverse to the homogeneous space G/H.

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“…algebraic, analytic, statistical properties [4,5], role played in quantization methods [6,7], in quantum measurement and positive operatorvalued measures (POVM) [8,9], quantum entanglement [10]. Furthermore, several generalizations in many directions have been proposed over all these years, starting from spin and atomic CS [11,12], followed by CS for groups of algebras of many types [13][14][15], nonlinear CS [16][17][18][19][20][21], generalization to squeezed states of light [22][23][24][25], and recently CS for non-Hermitian [26][27][28][29] structures.…”

Section: Introductionmentioning

confidence: 99%

Generalized Susskind-Glogower coherent states

Gazeau,

Hussin,

Moran

et al. 2020

Preprint

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Susskind-Glogower coherent states, whose Fock expansion coefficients include Bessel functions, have recently attracted considerable attention for their optical properties. Nevertheless, identity resolution is still an open question, which is an essential mathematical property that defines an overcomplete basis in the Fock space and allows a coherent state quantization map. In this regard, the modified Susskind-Glogower coherent states have been introduced as an alternative family of states that resolve the identity resolution. In the present manuscript, the quantization map related to the modified Susskind-Glogower coherent states is exploited, which naturally leads to a particular representation of the su(1, 1) Lie algebra in its discrete series. The latter provides evidence about further generalizations of coherent states, built from the Susskind-Glogower ones by extending the indexes of the Bessel functions of the first kind and, alternatively, by employing the modified Bessel functions of the second kind. In this form, the new families of Susskind-Glogower-I and Susskind-Glogower-II coherent states are introduced. The corresponding quantization maps are constructed so that they lead to general representations of elements of the su(1, 1) and su(2) Lie algebras as generators of the SU(1, 1) and SU(2) unitary irreducible representations respectively. For completeness, the optical properties related to the new families of coherent states are explored and compared with respect to some well-known optical states. Contents 1. Introduction 2. Modified Susskind-Glogower coherent states 3. Quantization map related to the modified Susskind-Glogower coherent states 4. Susskind-Glogower-I coherent states 5. Quantization map with SGI CS 6. One-photon SU(1, 1) coherent states 7. Susskind-Glogower-II coherent states 8. Quantization map with SGII CS 9. Boson realization and contraction of algebras 10. Photon statistics and nonclassical properties of the SGI and SGII coherent states 11. Conclusions Appendix A. Normalization constant N κ (r)

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Non-Hermitian Coherent States for Finite-Dimensional Systems

Cited by 5 publications

References 32 publications

Interplay between Riccati, Ermakov, and Schrödinger equations to produce complex‐valued potentials with real energy spectrum

Interplay between Riccati, Ermakov, and Schrödinger equations to produce complex‐valued potentials with real energy spectrum

Geometric dynamics of a harmonic oscillator, arbitrary minimal uncertainty states and the smallest step 3 nilpotent Lie group

Generalized Susskind-Glogower coherent states

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