The physics of trapped dilute-gas Bose–Einstein condensates

Parkins, A. S.; Walls, D. F.

doi:10.1016/s0370-1573(98)00014-3

Cited by 388 publications

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“…PACS number(s): 03.75. Fi, 05.30.Jp, 67.40 Db It is known [1,2] that the total energy E of a dilute 3D Bose gas confined in a trap can be represented at T ≪ T c (T c is the temperature of the Bose-Einstein condensation) as the functional of the order parameter φ = φ(r) = ψ (r) (hereψ(r) is the Bose field operator)…”

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confidence: 99%

The kinetic and interaction energies of a trapped Bose gas: beyond the mean field

Cherny

Shanenko

2002

Physics Letters A

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The kinetic and interaction energies of a three-dimensional (3D) dilute ground-state Bose gas confined in a trap are calculated beyond a mean-field treatment. They are found to depend on the pairwise interaction trough two characteristic lengths: the first, a, is the well-known scattering length and the second, b, is related to the latter by b = a − λ∂a/∂λ with λ being the coupling constant. Numerical estimations show that the pairwise interaction energy of a dilute gas of alkali atoms in a trap is negative (in spite of the positive scattering length); its absolute value is found by about the order of magnitude larger than that of the mean-field interaction energy that corresponds to the last term in the Gross-Pitaevskii (GP) functional. PACS number(s): 03.75. Fi, 05.30.Jp, 67.40 Db It is known [1,2] that the total energy E of a dilute 3D Bose gas confined in a trap can be represented at T ≪ T c (T c is the temperature of the Bose-Einstein condensation) as the functional of the order parameter φ = φ(r) = ψ (r) (hereψ(r) is the Bose field operator)called the Gross-Pitaevskii functional. In Eq.(1) V ext (r) denotes an external trapping field with the characteristic length L, typical for the spatial variation of this field [for a harmonic trap, L ≃ a ho = ℏ/(mω ho ) ], and |a| ≪ L stands for the scattering length in a gas considered. From the definition it follows that the order parameter satisfies the normalization condition d 3 r|φ| 2 = N , where the number of bosons N 0 in the Bose-Einstein condensate is replaced by their total number N due to a small condensate depletion. The stationary solution corresponding to the minimum of the functional (1) obeys the relationswhich give the GP equationThe term µN appears in Eq. (2) due to the normalization condition, and the Lagrange multiplier µ is nothing but the chemical potential. In this paper we address the problem of interpretation of different terms in the GP functional; namely, the kinetic E kin and pairwise interaction E int energies are under investigation below. Note that this is of actual interest because analysis of the experiments with magnetically trapped Bose gases is carried out usually in terms of E kin and E int [2,3]. Moreover, the problem of that interpretation is rather ambiguous due to just opposite points of view on this subject found in the literature. Indeed, in the paper [3] the third term of the GP functional is treated as the energy of the pairwise boson interaction. While according to another point of view [4], in the homogeneous case V ext = 0 this term makes contribution only to the kinetic energy if the boson-boson interaction potential V (r) is of the hardsphere form: V (r) = +∞ at r < a, and V (r) = 0 at r a. This problem is usually considered to be trivial [2]. Indeed, in the mean-field interpretation [5], the terms in Eq. (1) are associated with the kinetic energy, the energy of interaction with the external field, and the pairwise interaction energy, respectively. However, in the particular case V ext = 0, this interpretation contr...

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confidence: 99%

The kinetic and interaction energies of a trapped Bose gas: beyond the mean field

Cherny

Shanenko

2002

Physics Letters A

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“…The Bose-Einstein condensation phase transition leads to a non-zero value of a scalar order parameter, the condensate wavefunction ψ( r) = Ψ ( r) . As such, a single-component gaseous condensate can be considered as a simple, tractable model of the more complicated spinless superfluid 4 He. Many experiments in recent years have explored aspects of this connection between the two quantum fluids [1].…”

Section: Spinor Bose-einstein Condensatesmentioning

confidence: 99%

“…However, unlike 4 He, alkali atoms can have a non-zero spin and therefore numerous internal hyperfine states which are stable electronic ground states. Thus, there exists the possibility of creating a quantum fluid simultaneously composed of several, distinguishable components by Bose condensing a gas of atoms in several hyperfine states.…”

Section: Spinor Bose-einstein Condensatesmentioning

confidence: 99%

“…is minimized by placing all atoms in the strong-field seeking |m F = +1 state (a ferromagnetic state) 4 . Here, p = gµ B B 0 where g is the Landé g-factor and µ B is the Bohr magneton.…”

Section: Tailoring the Ground-state Structure With Magnetic Fieldsmentioning

confidence: 99%

“…For those topics we refer to our Varenna Summer School Notes which give a comprehensive discussion of experimental techniques, static and dynamic properties, coherence and optical trapping of condensates [1]. This paper and some other recent review papers summarize the state of the field [2][3][4].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Spinor Condensates and Light Scattering from Bose-Einstein Condensates

Stamper-Kurn¹,

Ketterle²

Les Houches - Ecole D’Ete De Physique Theorique

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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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The physics of trapped dilute-gas Bose–Einstein condensates

Cited by 388 publications

References 194 publications

The kinetic and interaction energies of a trapped Bose gas: beyond the mean field

The kinetic and interaction energies of a trapped Bose gas: beyond the mean field

Spinor Condensates and Light Scattering from Bose-Einstein Condensates

A new look at the quantum mechanics of the harmonic oscillator

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