Coherent states, squeezed fluctuations, and the SU(2) am SU(1,1) groups in quantum-optics applications

Nonlinear coherent modes are the collective states of trapped Bose atoms, corresponding to different energy levels. These modes can be created starting from the ground state condensate that can be excited by means of a resonant alternating field. A thorough theory for the resonant excitation of the coherent modes is presented. The necessary and sufficient conditions for the feasibility of this process are found. Temporal behaviour of fractional populations and of relative phases exhibits dynamic critical phenomena on a critical line of the parametric manifold. The origin of these critical phenomena is elucidated by analyzing the structure of the phase space. An atomic cloud, containing the coherent modes, possesses several interesting features, such as interference patterns, interference current, spin squeezing, and massive entanglement. The developed theory suggests a generalization of resonant effects in optics to nonlinear systems of Bose-condensed atoms. 03.75. Fi

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“…The points (62) and (63) are the solutions to Eq. (60), which are simplified taking account of small detuning |ε| ≪ 1.…”

Section: Phase Portraitmentioning

confidence: 99%

Nonlinear coherent modes of trapped Bose-Einstein condensates

Yukalov

Yukalova²,

Bagnato³

2002

Phys. Rev. A

126

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“…The generalized IS for the quadratures X 1 and X 2 coincide with the canonical squeezed states [18]. The concept of squeezing is naturally related also to the IS associated with the SU(2) and SU(1,1) Lie groups [27][28][29]8,9,[15][16][17][18][19][20][21]. At the last years there is a great interest in the IS.…”

Section: The General Theory Of the Algebra Eigenstatesmentioning

confidence: 99%

“…When β 2 = 0 and β 3 = 0, the resulting AES are associated with the oscillator group H 4 . The corresponding analytic function is 27) where…”

Section: The Fock-bargmann Representation Of the Two-photon Algementioning

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%

“…On the other hand, the IS show a variety of nonclassical properties, such as squeezing and sub-Poissonian photon statistics. 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%

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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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A new look at the quantum mechanics of the harmonic oscillator

Kastrup¹

2007

Ann. Phys.

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In classical mechanics the harmonic oscillator (HO) provides the generic example for the use of angle and action variables ϕ ∈ R mod 2π and I > 0 which played a prominent role in the "old" Bohr-Sommerfeld quantum theory. However, already classically there is a problem which has essential implications for the quantum mechanics of the (ϕ, I)-model for the HO: the transformation q = √ 2I cos ϕ, p = − √ 2I sin ϕ is only locally symplectic and singular for (q, p) = (0, 0). Globally the phase space {(q, p)} has the topological structure of the plane R 2 , whereas the phase space {(ϕ, I)} corresponds globally to the punctured plane R 2 − (0, 0) or to a simple cone with the tip deleted. From the properties of the symplectic transformations on that phase space one can derive the functions h0 = I, h1 = I cos ϕ and h2 = −I sin ϕ as the basic coordinates on {(ϕ, I)} , where their Poisson brackets obey the Lie algebra of the symplectic group of the plane. This implies a qualitative difference as to the quantum theory of the phase space {(ϕ, I)} compared to the usual one for {(q, p)} : In the quantum mechanics for the (ϕ, I)-model of the HO the three hj correspond to the self-adjoint generators Kj, j = 0, 1, 2, of certain irreducible unitary representations of the symplectic group or one of its infinitely many covering groups, the representations being parametrized by a (Bargmann) index k > 0. This index k determines the ground state energy E k,n=0 = ω k of the (ϕ, I)-Hamiltonian H( K) = ω K0. For an m-fold covering the lowest possible value for k is k = 1/m , which can be made arbitrarily small by choosing m accordingly! This is not in contradiction to the usual approach in terms of the operators Q and P which are now expressed as functions of the Kj, but keep their usual properties. The richer structure of the Kj quantum model of the HO is "erased" when passing to the simpler (Q, P )-model! This more refined approach to the quantum theory of the HO implies many experimental tests: Mullikentype experiments for isotopic diatomic molecules, experiments with harmonic traps for atoms, ions and BE-condensates, with charged HOs in external electric fields and the (Landau) levels of charged particles in external magnetic fields, with the propagation of light in vacuum, passing through strong external electric or magnetic fields. Finally it may lead to a new theoretical estimate for the quantum vacuum energy of fields and its relation to the cosmological constant.

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Coherent states, squeezed fluctuations, and the SU(2) am SU(1,1) groups in quantum-optics applications

Cited by 501 publications

References 20 publications

Nonlinear coherent modes of trapped Bose-Einstein condensates

Nonlinear coherent modes of trapped Bose-Einstein condensates

Two-Photon Algebra Eigenstates: A Unified Approach to Squeezing

A new look at the quantum mechanics of the harmonic oscillator

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