A Monte Carlo formulation of the Bogolubov theory

Sinatra, Alice; Lobo, Carlos

doi:10.1080/09500340008232186

Cited by 44 publications

(73 citation statements)

References 31 publications

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“…Heuristically, the introduction of virtual particles via (3) can be viewed as adding half of one particle per mode, corresponding to the zero-point occupation of the ground state of the harmonic oscillator which represents each mode. The choice of initial condition (3) is correct where ψ(x) = 0, but generates a slightly heated and nonequilibrium condensate where ψ(x) = 0 [8,21]. This will make very little difference to the results of the calculations, which require that the vacuum be represented accurately-furthermore, in practice the difference between the state thus represented and a pure condensate is very much less than the experimental uncertainty.…”

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confidence: 99%

Quantum Turbulence in Condensate Collisions: An Application of the Classical Field Method

2005

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We apply the classical field method to simulate the production of correlated atoms during the collision of two Bose-Einstein condensates. Our non-perturbative method includes the effect of quantum noise, and provides for the first time a theoretical description of collisions of high density condensates with very large out-scattered fractions. Quantum correlation functions for the scattered atoms are calculated from a single simulation, and show that the correlation between pairs of atoms of opposite momentum is rather small. We also predict the existence of quantum turbulence in the field of the scattered atoms-a property which should be straightforwardly measurable.PACS numbers: 03.75. Kk, 05.10.Gg, In the same way as a classical electromagnetic field obeying Maxwell's equations arises as an assembly of photons all in the same quantum state, a Bose-Einstein condensate, composed of Bosonic atoms all in the same quantum state, behaves very much like a classical field Ψ(x, t), whose equation of motion is the Gross-Pitaevskii equation(1) Nevertheless, there are phenomena in which the quantized nature of this field is important-for example, when two BoseEinstein condensates collide at a sufficiently high velocity, a halo of elastically scattered atoms is produced [1,2,3]. The Gross-Pitaevskii equation with initial conditions corresponding to two Bose-Einstein condensates does not predict this scattering-it is a direct effect of the fact that the quantized field consists of interacting particles.Theoretical descriptions of this phenomenon fall into two groups. In the first, the Gross-Pitaevskii equations for the two condensate wavepackets are modified (either phenomenologically [4], or on the basis of a method of approximating quantum field theory [5]) to give an elastic scattering loss term. While these methods yield equations of motion which allow for depletion of the condensate wavefunctions, they do not include a description of the scattered atoms, and hence cannot describe the effects of bosonically stimulated loss. In the second class of treatments [6,7] the quantum field theory is linearized about the condensate, yielding equations of motion linear in the fluctuation operators. This method shows that the process is essentially one of four-wave mixing between the two condensate fields and pairs of quantized fluctuationshowever, as a linearized theory, it can deal only with perturbatively small amounts of scattering and cannot simultaneously account for depletion of the condensate. Both of these formalisms are valid only in the limit of weak scattering, but fail for large scattered fractions, such as we treat in this paper.In this work we will show that a treatment in terms of a classical field with added quantum fluctuations is not only able to produce the scattering halo, but also predicts a hitherto unobserved phenomenon, which we shall call quantum turbulence, in the resulting halo, which should be straightforwardly observable. This method is able to describe: a) the evolution (including large depletion) of ...

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Quantum Turbulence in Condensate Collisions: An Application of the Classical Field Method

2005

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“…The PGPE is a dynamical nonperturbative method, with the only approximation being that the highly occupied modes (hN k i 1) of the quantum Bose field are well approximated by a classical field evolved according to the GPE. Related classical field approaches have been considered by a number of authors, including Kagan and co-workers [23], Sinatra et al [24], Rzażewski and co-workers [25].…”

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Critical Temperature of a Trapped Bose Gas: Comparison of Theory and Experiment

2006

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We apply the projected Gross-Pitaevskii equation (PGPE) formalism to the experimental problem of the shift in critical temperature T c of a harmonically confined Bose gas as reported in Gerbier et al., Phys. Rev. Lett. 92, 030405 (2004). The PGPE method includes critical fluctuations and we find the results differ from various mean-field theories, and are in best agreement with experimental data. To unequivocally observe beyond mean-field effects, however, the experimental precision must either improve by an order of magnitude, or consider more strongly interacting systems. This is the first application of a classical field method to make quantitative comparison with experiment. The shift in critical temperature T c with interaction strength for the homogeneous Bose gas has been the subject of numerous studies spanning almost 50 years since the first calculations of Lee and Yang [1,2]. While there is a finite shift to the chemical potential in mean-field (MF) theory, the shift of the critical temperature is zero [3]. The leading order effect is due to long-wavelength critical fluctuations and is inherently nonperturbative. Using effective field theory it was determined that the shift is T c =T 0 c can 1=3 , where n is the particle number density, a is the s-wave scattering length, and c is a constant of order unity [4]. Until recently, results for the value of c disagreed by an order of magnitude and even sign, as summarized in Fig. 1

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“…The condensate at step (i) is assumed to be at equilibrium. Quantum and/or thermal fluctuations are included by means of the Wigner representation of bosonic fields [7]. In practice, the function ψ, input of step (ii), is taken of the form ψ = ψ 0 + i [c i u i + c * i v * i ], where the sum extends over a wide set of Bogoliubov states including those which are expected to be relevant for the subsequent parametric amplification.…”

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Detecting phonons and persistent currents in toroidal Bose-Einstein condensates by means of pattern formation

2006

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We theoretically investigate the dynamic properties of a Bose-Einstein condensate in a toroidal trap. A periodic modulation of the transverse confinement is shown to produce a density pattern due to parametric amplification of phonon pairs. By imaging the density distribution after free expansion one obtains i) a precise determination of the Bogoliubov spectrum and ii) a sensitive detection of quantized circulation in the torus. The parametric amplification is also sensitive to thermal and quantum fluctuations. [2]. The aim is to create a system in which fundamental properties, like quantized circulation and persistent currents, matter-wave interference, propagation of sound waves and solitons, can be observed in a clean and controllable way. An important issue concerns the feasibility of high-sensitivity rotation sensors.In this work we show that key properties of condensates in toroidal traps can be measured by means of parametric amplification. This corresponds to an exponential growth of some excited modes of the system induced by a periodic modulation of an external parameter [3]. We consider the modulation of the transverse confinement, which is shown to drive a spatially periodic pattern in both density and velocity distributions as a consequence of the amplification of pairs of counter-rotating Bogoliubov phonons. If the trap is switched off, this pattern produces a peculiar flower-like density distribution of the freely expanding gas. The number of "petals" and their shape provide a sensitive measure of the excitation spectrum and the superfluid rotation of the condensate.We perform numerical simulations by integrating the Gross-Pitaevskii equation [4] for N bosonic atoms of mass M , confined in an external potential V ext :The mean-field coupling constant is given by g = 4πh 2 a/M , where the s-wave scattering length a is assumed to be positive. The order parameter of the condensate, ψ(r, t), is normalized to dr|ψ| 2 = N and may by written as ψ = n 1/2 exp(iS), where n is the condensate density and the phase S is related to the superfluid velocity by v = (h/M )∇S. We consider a condensate of N = 2 × 10 5 atoms of 87 Rb confined in a torus of length L = 2πR = 100 µm by an axially symmetric potential which has a minimum, V ext = 0, at z = 0 and r ⊥ = R. The trap is harmonic and isotropic in the (z, r ⊥ )-plane around this minimum, with frequency ω ⊥ = 2π × 1 kHz. With this choice the transverse width of the condensate, r 0 , is significantly smaller than the radius of the torus. This implies that curvature effects are almost negligible in the ground state and in the in-trap dynamics and one can replace the torus of radius R with a cylinder of length L with periodic boundary conditions [5]. Curvature effects are instead important during the free expansion of the condensate, and therefore the full toroidal geometry is used for simulating the dynamics after the release from the trap. The choice made for the geometry and the parameters is intended to simulate feasible experiments, but the effects we are goi...

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A Monte Carlo formulation of the Bogolubov theory

Cited by 44 publications

References 31 publications

Quantum Turbulence in Condensate Collisions: An Application of the Classical Field Method

Quantum Turbulence in Condensate Collisions: An Application of the Classical Field Method

Critical Temperature of a Trapped Bose Gas: Comparison of Theory and Experiment

Detecting phonons and persistent currents in toroidal Bose-Einstein condensates by means of pattern formation

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