Double squeezing in generalized q-coherent states

McDermott, Roger; Solomon, Allan I.

doi:10.1088/0305-4470/27/2/003

Cited by 30 publications

(21 citation statements)

References 22 publications

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“…For example, there are some special states, which permit simultaneously zero minimal uncertainties in a pair of conjugate operators [9,10]. The q-deformed coherent state [10,17] is such an example. Here, we have discussed another case, the special q-squeezed state, which shows another qualitative deviation from Heisenberg's uncertainty relation.…”

Section: Critical Phenomenon Of Q-deformed Squeezed Statementioning

confidence: 99%

“…If the space structure at such short distances exhibits a non-commutative property, and thus is governed by a quantum group symmetry, it has been shown that q-deformed quantum mechanics is a candidate for a possible pre-quantum theory at short distances. In the literature different frameworks of q-deformed quantum mechanics were established [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18].…”

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confidence: 99%

“…At the level of the uncertainty relation there are two kinds of modifications of Heisenberg's uncertainty relation [3,9,10,16,17,18], implying under-or over-cutting of Heisenberg's minimal uncertainty relation. Because the framework of q-deformed Heisenberg algebra developed in Refs.…”

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confidence: 99%

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Critical Phenomenon of a Consistent q-Deformed Squeezed State

Osland¹,

Zhang²

2001

Annals of Physics

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Within a self-consistent framework of q-deformed Heisenberg algebra and its equivalent framework of q-deformed boson commutation relations, which relate to the under-cutting phenomenon of Heisenberg's minimal uncertainty relation, special q-deformed squeezed states are constructed. Besides the similar local maximum squeezing as the one in the undeformed case, new strong squeezing appears when the amplitude of the related coherent state increases to large values. A critical phenomenon appears at a large value of the amplitude: the variance of one component of the quadrature of the light field approaches zero, but the variance of the corresponding conjugate quantity remains finite, which is a surprising deviation from Heisenberg's uncertainty relation. The qualitative character exposed by this q-squeezed state may provide some evidence about q-deformed effects in current experiments. *

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Section: Critical Phenomenon Of Q-deformed Squeezed Statementioning

confidence: 99%

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confidence: 99%

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Critical Phenomenon of a Consistent q-Deformed Squeezed State

Osland¹,

Zhang²

2001

Annals of Physics

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“…In one approach the q-deformed quantum theory, as a possible modification of the ordinary quantum theory at space scale much smaller than 10 −18 cm, has attracted attention. In literature different frameworks of q-deformed quantum theories were established [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. We work in the framework of the q-deformed Heisenberg algebra developed in Refs.…”

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confidence: 99%

Perturbation foundation of q-deformed dynamics

Zhang

2003

Eur. Phys. J. C

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In the q-deformed theory the perturbation approach can be expressed in terms of two pairs of undeformed position and momentum operators. There are two configuration spaces. Correspondingly there are two q-perturbation Hamiltonians, one originates from the perturbation expansion of the potential in one configuration space, the other one originates from the perturbation expansion of the kinetic energy in another configuration space. In order to establish a general foundation of the q-perturbation theory, two perturbation equivalence theorems are proved: (I) Equivalence theorem I: Perturbation expressions of the q-deformed uncertainty relations calculated by two pairs of undeformed operators are the same, and the two q-deformed uncertainty relations undercut Heisenberg's minimal one in the same style. (II) The general equivalence theorem II: for any potential (regular or singular) the expectation values of two q-perturbation Hamiltonians in the eigenstates of the undeformed Hamiltonian are equivalent to all orders of the perturbation expansion. As an example of singular potentials the perturbation energy spectra of the q-deformed Coulomb potential are studied. §

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“…If the space structure at short distances (much smaller than 10 −18 cm, according to the present test of quantum electrodynamics, the usual Heisenberg's commutation relations are correct at least down to 10 −18 cm) shows a noncommutative property and thus governed by quantum group symmetry, then quantum mechanics based on q-deformed Heisenberg algebra is a possible candidate for quantum theory to study the phenomena at short distances . Different frameworks of q−deformed Heisenberg algebra have been established [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], but physically, the one presented in Refs. [13,17] is more clear : its relation to the corresponding q−deformed boson commutation relations and the limiting process of the q−deformed harmonic oscillator to the undeformed one are clear.…”

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confidence: 99%

One-Dimensional Potentials in q Space

Alavi¹

2002

Chinese Phys. Lett.

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abstract. We study the one dimensional potentials in q space and the new features that arise. In particular we show that the probability of tunneling of a particle through a barrier or potential step is less than the one of the same particle with the same energy in ordinary space which is somehow unexpected. We also show that the tunneling time for a particle in q space is less than the one of the same particle in ordinary space. Key words. Q spaces, Perturbation theory.Pacs number. 03.65.-w, 02.40.Gh 1 Introduction. Quantum groups are a generalization of the concept of symmetries [1][2][3][4][5]. The mathematical theory of quantum groups arose as an abstraction from constructions developed in the frame of the inverse scattering method of solution of quantum integrable models, but because of its rich and powerful structure, it has been applied to different problems far beyond the original area. quantum groups act on noncommutative spaces. If the space structure at short distances (much smaller than 10 −18 cm, according to the present test of quantum electrodynamics, the usual Heisenberg's commutation relations are correct at least down to 10 −18 cm) shows a noncommutative property and thus governed by quantum group symmetry, then quantum mechanics based on q-deformed Heisenberg algebra is a possible candidate for quantum theory to study the phenomena at short distances . Different frameworks of q−deformed Heisenberg algebra have been established [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], but physically, the one presented in Refs. [13,17] is more clear : its relation to the corresponding q−deformed boson commutation relations and the limiting process of the q−deformed harmonic oscillator to the undeformed one are clear. By the results of string theory arguments, some applications of quantum mechanics on a non-commutative plane has been studied in [27]. Perturbative aspects of the schroedinger equation in q space has been studied in [23]. There are two perturbative Hamiltonians corresponding to two ways of realizing the q−deformed Heisenberg algebra by the undeformed variables(which are related by the cononical transformation): One includes it in the kinetic energy term, and the other includes it in the potential.

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Double squeezing in generalized q-coherent states

Cited by 30 publications

References 22 publications

Critical Phenomenon of a Consistent q-Deformed Squeezed State

Critical Phenomenon of a Consistent q-Deformed Squeezed State

Perturbation foundation of q-deformed dynamics

One-Dimensional Potentials in q Space

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