Numerical method for evolving the dipolar projected Gross-Pitaevskii equation

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 80, Issue 1, 2009, Pages

  Blakie, P B ^a   Ticknor, C ^b   Bradley, A S ^a   Martin, A M ^c   Davis, M J ^d   Kawaguchi, Y ^e  

^a UNIVERSITY OF OTAGO   (New Zealand)

^b SWINBURNE UNIVERSITY OF TECHNOLOGY   (Australia)

^c UNIVERSITY OF MELBOURNE   (Australia)

^d UNIVERSITY OF QUEENSLAND   (Australia)

^e UNIVERSITY OF TOKYO   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

DIPOLAR INTERACTION; EHRENFEST EQUATION; GROSS-PITAEVSKII EQUATION; HERMITE; INTERACTING BOSE GAS; LOW ENERGIES; OSCILLATOR POTENTIAL; RECIPROCAL SPACE; SPECTRAL REPRESENTATIONS;

OSCILLATORS (ELECTRONIC);

SET THEORY;

EID: 68949135124     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.80.016703     Document Type: Article

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43. Other sets of modes defining C are permissible and easily implemented in our formalism. Here, we focus on the single-particle energy cutoff because of its simplicity and wide range of applicability.
44. This is not a restriction of the method and, without this simplification, we need to carry cumbersome notation indicating different mode functions, quadratures, and numbers of points for each dimension.
45. Since a polynomial of degree 2N-1 is integrated exactly using an N -point quadrature.
46. Here, we form e2 | xijk | 2 | ψC | 2 ψC as this corresponds to the polynomial (P) required for the quadrature (see Ref.).
47. Formally, this energy functional should have a factor of 1/2 in front of it, as is formally shown in Ronen (Ref.). However, this is a typo as the results they give do not include this factor.
48. As noted at the beginning of this section, convergence with Mx is not useful for PGPE applications. However, we perform this comparison here to more generally characterize our spectral approach and to compare against established results in the literature.
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53. The error bars are assigned by calculating the correlation time of the signal, e.g. Wx (t), for which we can estimate the number of statistically independent samples Ns < Ns, and we then assign an error under the assumption that this quantity is Gaussian distributed, which appears reasonable from histograms of the signal.
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