Generalized intelligent states and squeezing

Trifonov, D. A.

doi:10.1063/1.530553

Cited by 96 publications

(217 citation statements)

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“…In the case of the SU(1,1) Lie group, the standard set of Perelomov's CS and the set of the ordinary IS have an intersection [27,20]. Both these types of states form subsets of the generalized IS [18].…”

Section: Introductionmentioning

confidence: 99%

“…For the squeezed states, the fluctuations in one quadrature are reduced on account of growing fluctuations in the other (conjugate) quadrature. The canonical squeezed states can be considered as the generalized IS for the Heisenberg-Weyl group [15,18]. For more complicated groups, e.g., for SU (1,1), the different definitions lead to distinct states.…”

Section: Introductionmentioning

confidence: 99%

“…The Perelomov CS for the SU(1,1) Lie group, obtained by the action of the group elements on the reference state [2], and the Barut-Girardello states, defined as the eigenstates of the SU(1,1) lowering generator K − [6], are quite different. However, the concept of squeezing can be naturally extended to the SU(1,1) group, and the squeezing properties of the SU(1,1) ordinary and generalized IS have been widely discussed [27][28][29]8,9,[15][16][17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

“…The third way in which CS can be defined is associated with the optimization of uncertainty relations for Hermitian generators of a group [12][13][14][15][16][17][18][19][20][21]. States that minimize uncertainty relations are called intelligent states (IS) or minimum-uncertainty states.…”

Section: Introductionmentioning

confidence: 99%

“…States that minimize uncertainty relations are called intelligent states (IS) or minimum-uncertainty states. Ordinary IS [13] provide an equality in the Heisenberg uncertainty relation while generalized IS [18,19] do so in the Robertson uncertainty relation [22]. The IS are determined by some type of the eigenvalue equation [23,18,19], and the lowering-generator eigenstates are in fact a particular case of the IS, corresponding to equal uncertainties of two Hermitian generators.…”

Section: Introductionmentioning

confidence: 99%

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Two-Photon Algebra Eigenstates: A Unified Approach to Squeezing

Brif

1996

Annals of Physics

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We use the concept of the algebra eigenstates that provides a unified description of the generalized coherent states (belonging to different sets) and of the intelligent states associated with a dynamical symmetry group. The formalism is applied to the two-photon algebra and the corresponding algebra eigenstates are studied by using the Fock-Bargmann analytic representation. This formalism yields a unified analytic approach to various types of singlemode photon states generated by squeezing and displacing transformations.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Two-Photon Algebra Eigenstates: A Unified Approach to Squeezing

Brif

1996

Annals of Physics

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Quantization of the optical phase space in terms of the group

Kastrup¹

2003

Fortschritte der Physik

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The problem of quantizing properly the canonical pair “angle and action variables”, ϕ and I, is almost as old as quantum mechanics itself and since decades an intensively debated but still unresolved issue in quantum optics. The present paper proposes a new approach to the problem, namely quantization in terms of the group SO(1,2): The crucial point is that the phase space \documentclass{article}\pagestyle{empty}\begin{document}$\mathcal{S}^2 = \{\varphi \bmod{2\pi}, I> 0\}$\end{document} has the global structure \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$S^1 \times \mathbb{R}^+$\end{document} (a simple cone) and cannot be quantized in the conventional manner. As the group SO(1,2) acts transitively, effectively and Hamilton‐like on that space its irreducible unitary representations of the positive discrete series provide the appropriate quantum theoretical framework. The phase space \documentclass{article}\pagestyle{empty}\begin{document}$\mathcal{S}^2$\end{document} has the conic structure of an orbifold \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\mathbb{R}^2/Z_2$\end{document}. That structure is closely related to a Z2 gauge symmetry which corresponds to the center of a 2‐fold covering of SO(1,2), the symplectic group \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$Sp(2,\mathbb{R})$\end{document}. The basic variables on the phase space are the functions h0 = I , h1 = I cos ϕ and h2 = ‐I sin ϕ the Poisson brackets of which obey the Lie algebra \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\mathfrak{so}(1,2)$\end{document}. In the quantum theory they are represented by the self‐adjoint Lie algebra generators K0, K1, and K2 of a unitary representation, where K0 has the spectrum {k + n, n = 0, 1, ...; k > 0}. A crucial prediction is that the classical Pythagorean relation h12 + h22 = h02 can be violated in the quantum theory. For each representation one can define three different types of coherent states the complex phases of which may be “measured” by means of the operators K1 and K2 alone without introducing any new phase operators! The SO(1,2) structure of optical squeezing and interference properties as well as that of the harmonic oscillator are analyzed in detail. The additional coherent states can be used for the introduction of (Husimi type) “Q” distributions and (Sudarshan‐Glauber type) “P” representations of the density operator. The three operators K0, K1, and K2 are fundamental in the sense that one can construct composite position and momentum operators out of them! The new framework poses quite a number of fascinating experimental and theoretical challenges.

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Quantization of the optical phase space 𝒮² = {ϕ mod 2π, I > 0} in terms of the group SO^↑(1, 2)

Kastrup¹

2004

Fortschritte der Physik

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The problem of quantizing properly the canonical pair "angle and action variables ", ϕ and I, is almost as old as quantum mechanics itself and since decades an intensively debated but still unresolved issue in quantum optics. The present paper proposes a new approach to the problem, namely quantization in terms of the group SO(1, 2): The crucial point is that the phase space S 2 = {ϕ mod 2π, I > 0} has the global structure S 1 × R + (a simple cone) and cannot be quantized in the conventional manner. As the group SO(1, 2) acts transitively, effectively and Hamilton-like on that space its irreducible unitary representations of the positive discrete series provide the appropriate quantum theoretical framework. The phase space S 2 has the conic structure of an orbifold R 2 /Z 2 . That structure is closely related to a Z 2 gauge symmetry which corresponds to the center of a 2-fold covering of SO(1, 2), the symplectic group Sp(2, R). The basic variables on the phase space are the functions h 0 = I , h 1 = I cos ϕ and h 2 = −I sin ϕ the Poisson brackets of which obey the Lie algebra so(1, 2). In the quantum theory they are represented by the self-adjoint Lie algebra generators K 0 , K 1 and K 2 of a unitary representation, where K 0 has the spectrum {k + n, n = 0, 1, . . . ; k > 0}. A crucial prediction is that the classical Pythagorean relation h 2 1 + h 2 2 = h 2 0 can be violated in the quantum theory. For each representation one can define three different types of coherent states the complex phases of which may be "measured" by means of the operators K 1 and K 2 alone without introducing any new phase operators! The SO(1, 2) structure of optical squeezing and interference properties as well as that of the harmonic oscillator are analyzed in detail. The additional coherent states can be used for the introduction of (Husimi type) "Q" distributions and (Sudarshan-Glauber type) "P" representations of the density operator. The three operators K 0 , K 1 and K 2 are fundamental in the sense that one can construct composite position and momentum operators out of them! The new framework poses quite a number of fascinating experimental and theoretical challenges.

show abstract

Generalized intelligent states and squeezing

Cited by 96 publications

References 31 publications

Two-Photon Algebra Eigenstates: A Unified Approach to Squeezing

Two-Photon Algebra Eigenstates: A Unified Approach to Squeezing

Quantization of the optical phase space in terms of the group

Quantization of the optical phase space 𝒮² = {ϕ mod 2π, I > 0} in terms of the group SO^↑(1, 2)

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Generalized intelligent states and squeezing

Cited by 96 publications

References 31 publications

Two-Photon Algebra Eigenstates: A Unified Approach to Squeezing

Two-Photon Algebra Eigenstates: A Unified Approach to Squeezing

Quantization of the optical phase space in terms of the group

Quantization of the optical phase space 𝒮2 = {ϕ mod 2π, I > 0} in terms of the group SO↑(1, 2)

Contact Info

Product

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Quantization of the optical phase space 𝒮² = {ϕ mod 2π, I > 0} in terms of the group SO^↑(1, 2)