Experimental Observation of the Bogoliubov Transformation for a Bose-Einstein Condensed Gas

Vogels, J. M.; Xu, Kuan‐Man; Raman, Chandra; Abo-Shaeer, J. R.; Ketterle, Wolfgang

doi:10.1103/physrevlett.88.060402

Cited by 105 publications

(106 citation statements)

References 11 publications

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“…One of the possible ways is using Bragg spectroscopy [22] to measure the excitation spectrum and the static structure factor of the FQHE. Bragg spectroscopy of a Bose-Einstein condensate in a trap has been realized experimentally [23]. The measured excitation spectrum agreed well with the Bogoliubov spectrum for a condensate.…”

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confidence: 63%

Roton instabilities and correlated Wigner crystals of rotating dipolar fermions in the fractional quantum Hall regime

2013

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We point out the possibility of occurring instabilities in Laughlin liquids of rotating dipolar fermions with zero thickness. Previously such a system was predicted to be the Laughlin liquid for filling factors ν ≥ 1/7. However, from intra-Landau-level excitations of the liquid in the single-mode approximation, the roton minima become negative and Laughlin liquids are unstable for ν ≤ 1/7. We then conclude that there are correlated Wigner crystals for ν ≤ 1/7.PACS numbers: 03.75. Ss, 73.43.Lp, 73.43.Nq The cold quantum gas with the dipole-dipole interaction (DDI) was first realized in Cr atoms [1]. The quantum gases with the DDI are qualitatively different from non-dipolar ones [2]. The novel anisotropic and long-range nature of the DDI offers a broad range of strong correlated many-body physics [3,4]. New quantum phases were predicted for the dipolar Bose-Einstein condensate (BEC) [5,6]. The influence of the trapping geometry on the stability of the BEC and the effect of the DDI on the excitation spectrum were studied [7]. The vortex lattice of the rotating dipolar BEC exhibits novel bubble, stripe, and square structures [8]. For the dipolar Fermi gases, the s-wave scattering is prohibited due to the Pauli Exclusion Principle and the Fermi surface is distorted by dipolar effects [9]. Bond pairs of fermions with resonant interaction are formed and the system of a Fermi gas behaves as a bosonic gas of molecules [10,11]. The observed pairing of fermions provides the crossover between the weakly-paired, strongly overlapping BardeenCooper-Schrieffer regime, and the tightly bound, weaklyinteracting diatomic molecular BEC regime [12]. The strong correlations of fermions induced by the DDI can then be explored, such as the dipolar-induced superfluidity [13,14] and fractional-quantum-Hall-effect (FQHE) states in rotating dipolar Fermi gases [15,16].Rotating gases feel the Coriolis force in the rotating frame. The Coriolis force on rotating gases is identical to the Lorentz force of a charged particle in a magnetic field. Quantum-mechanically, energy levels of a charged particle in a uniform magnetic field show discrete Landau levels. In the lowest Landau level (LLL), the kinetic energy of rotating dipolar gases is frozen and the DDI create strong correlations on particles. Therefore, in the lowest Landau level, the potential energy from DDI dominates the kinetic energy and rotating dipolar fermions will crystallize into a Wigner crystal (WC) or become the FQHE liquid. Baranov et al. [16] have shown that the FQHE states have lower energies than the WCs as filling fac- * Electronic address: sccheng@faculty.pccu.edu.tw; FAX: +886-2-28610577 tors ν ≥ 1/7 and in the zero-extension limit along the rotating axial direction, where ν = 2πρℓ 2 . Here ρ and ℓ are the average density and the magnetic length, respectively. Although they also investigated the stability of the WC and found that the WC states were stable in the regime ν < 1/7, but there was no test on the stability of the FQHE liquid. It is still a question whether...

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confidence: 63%

Roton instabilities and correlated Wigner crystals of rotating dipolar fermions in the fractional quantum Hall regime

2013

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“…11 Such a negative energy resonance has been first observed by Vogels et al with condensates of sodium atoms in a magnetic trap. 12 Recently, the Bogoliubov dispersion has been investigated in the solid state in a coherent polariton gas, 13 and the ghost dispersion branch has finally been observed. 14 Microcavity polariton fluid is extremely different from other bosonic quantum fluids as polaritons originate from the strong coupling of quantum-well excitons with cavity modes of semiconductor microcavities.…”

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confidence: 99%

Four-wave mixing excitations in a dissipative polariton quantum fluid

et al. 2012

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The Bogoliubov transformation of a polariton quantum fluid has recently been revealed in the four-wave mixing response of a driven microcavity polariton gas. In this work, we investigate the modifications that the dual nature of microcavity polaritons produce on the excitations of this particular half-light-half-matter quantum fluid. We discuss in particular the Bogoliubov character of the excitations of a lower polariton superfluid when it coexists with upper polaritons. We show unique effects resulting from the interplay between polariton decay and Bogoliubov transformation such as the modification of the Bogoliubov dispersion or the slowing down of the four-wave mixing dynamics.

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“…Interesting phenomena occur also for low optical potential depth, for instance Bloch oscillations [4] and tunneling effects [5][6][7] can be investigated in this regime. In 1D optical lattices the transition to the insulator phase is expected to take place for very large intensities of the optical lattice, so that there is a very extended range of parameters where the gas can be described as a fully coherent system.In this Letter, we study the elementary excitations of an interacting Bose gas in the presence of a periodic potential and discuss how these states can be excited via inelastic processes using, for example, Bragg spectroscopy [8,9]. To this purpose we develop the formalism of the dynamic structure factor, a quantity directly related to the linear response of the system.…”

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confidence: 99%

“…In this Letter, we study the elementary excitations of an interacting Bose gas in the presence of a periodic potential and discuss how these states can be excited via inelastic processes using, for example, Bragg spectroscopy [8,9]. To this purpose we develop the formalism of the dynamic structure factor, a quantity directly related to the linear response of the system.…”

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Dynamic structure factor of a Bose-Einstein condensate in a one-dimensional optical lattice

et al. 2003

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We study the effect of a one dimensional periodic potential on the dynamic structure factor of an interacting Bose Einstein condensate at zero temperature. We show that, due to phononic correlations, the excitation strength towards the first band develops a typical oscillating behaviour as a function of the momentum transfer, and vanishes at even multiples of the Bragg momentum. The effects of interactions on the static structure factor are found to be significantly amplified by the presence of the optical potential. Our predictions can be tested in stimulated photon scattering experiments.When a Bose-Einstein condensate is loaded into an optical lattice, its properties change in a very strong way [1]. For deep potential wells, it even happens that the coherence of the sample is lost and one observes the transition to the Mott-insulator phase [2,3]. Interesting phenomena occur also for low optical potential depth, for instance Bloch oscillations [4] and tunneling effects [5][6][7] can be investigated in this regime. In 1D optical lattices the transition to the insulator phase is expected to take place for very large intensities of the optical lattice, so that there is a very extended range of parameters where the gas can be described as a fully coherent system.In this Letter, we study the elementary excitations of an interacting Bose gas in the presence of a periodic potential and discuss how these states can be excited via inelastic processes using, for example, Bragg spectroscopy [8,9]. To this purpose we develop the formalism of the dynamic structure factor, a quantity directly related to the linear response of the system. We will restrict ourselves to the case of a system in the presence of a 1-dimensional optical potentialcreated by two counterpropagating laser beams. In Eq.(1) d is the lattice spacing and s is a dimensionless parameter which denotes the intensity of the laser in units of the recoil energy E R = q 2 B /2m. Here q B =hπ/d is the Bragg momentum denoting the boundary of the first Brillouin zone and m is the atomic mass. The inclusion of an additional harmonic potential produced, for example, by magnetic trapping does not modify the excitation spectrum in a profound way, unless the wave length of the excitation is comparable with the size of the sample. Along the tranverse directions we assume uniform confinement, so that the 3D Gross-Pitaevkii (GP) equation for the ground state order parameter ϕ(z) takes the 1D formwhere n is the average 3D density and the order parameter ϕ is normalized according to d/2 −d/2 |ϕ(z)| 2 dz = 1. As usual g = 4πh 2 a/m is the interaction coupling constant fixed by the scattering length a.The elementary excitations correspond to the solutions of the linearized time-dependent GP-equation and are described by the Bogoliubov equationswhere ϕ is the ground state solution of Eq.(2) and the amplitudes u jq and v jq satisfy the normalization conditionThe solutions u jq (z) and v jq (z) are Bloch waves: u jq (z) = exp(iqz/h)ũ jq (z) whereũ jq is periodic in space with peri...

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Experimental Observation of the Bogoliubov Transformation for a Bose-Einstein Condensed Gas

Cited by 105 publications

References 11 publications

Roton instabilities and correlated Wigner crystals of rotating dipolar fermions in the fractional quantum Hall regime

Roton instabilities and correlated Wigner crystals of rotating dipolar fermions in the fractional quantum Hall regime

Four-wave mixing excitations in a dissipative polariton quantum fluid

Dynamic structure factor of a Bose-Einstein condensate in a one-dimensional optical lattice

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