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Benchmarks with Analytically Solvable Problems

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Tunneling Dynamics in Open Ultracold Bosonic Systems

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Abstract

In this chapter, the MCTDHB method is benchmarked against analytically solvable interacting many-body problem, the so-called harmonic interaction model. The harmonic interaction model Hamiltonian contains both harmonic inter-particle interaction and harmonic one-body potentials. It can be solved exactly via coordinate transform to center-of-mass and relative coordinates. It is shown that MCTDHB is capable of solving the harmonic interaction problem as well as a time-dependent generalization thereof numerically exactly. Note, that MCTDHB does not employ the mentioned coordinate transform and therefore has to solve a highly correlated problem with time-dependent both one- and two-body potentials.

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References

  1. E.H. Lieb, W. Liniger, Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State. Phys. Rev. 130, 1605 (1963)

    Google Scholar 

  2. E.H. Lieb, Exact Analysis of an Interacting Bose Gas. II. The Excitation Spectrum. Phys. Rev. 130, 1616 (1963)

    Google Scholar 

  3. J.G. Muga, R.F. Snider, Solvable three-boson model with attractive \(\delta \)-function interactions. Phys. Rev. A 57, 3317 (1998)

    Google Scholar 

  4. K. Sakmann, A.I. Streltsov, O.E. Alon, L.S. Cederbaum, Exact groundstate of finite Bose–Einstein condensates on a ring. Phys. Rev. A 72, 033613 (2005)

    Google Scholar 

  5. D. Jukić, B. Klajn, H. Buljan, Momentum distribution of a freely expanding Lieb-Liniger gas. Phys. Rev. A 79, 033612 (2009)

    Article  ADS  Google Scholar 

  6. L. Cohen, C. Lee, Exact reduced density matrices for a model problem. J. Math. Phys. 26, 3105 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  7. Y.A.N. Jun, Harmonic Interaction Model and Its Applications in Bose–Einstein Condensation. J. Stat. Phys. 113, 623 (2003)

    Google Scholar 

  8. S. Pruski, J. Maćkowiak, O. Missuno, Reduced density matrices of a system of N coupled oscillators. 2. Eigenstructure of the 1-particle matrix for the canonical ensemble. Rep. Math. Phys. 3, 227 (1972)

    Article  ADS  Google Scholar 

  9. S. Pruski, J. Maćkowiak, O. Missuno, Reduced density matrices of a system of N coupled oscillators. 3. The eigenstructure of the p-particle matrix for the ground-state. Rep. Math. Phys. 3, 241 (1972)

    Article  ADS  Google Scholar 

  10. J. Maćkowiak, O. Missuno, S. Pruski, Reduced density matrices of a system of N coupled oscillators. 4. The eigenstructure of the p-particle matrices for the two-temperature canonical ensemble. Rep. Math. Phys. 5, 327 (1974)

    Article  ADS  Google Scholar 

  11. A.U.J. Lode, K. Sakmann, O.E. Alon, L.S. Cederbaum, A.I. Streltsov, Numerically exact quantum dynamics of bosons with time-dependent interactions of harmonic type. Phys. Rev. A 86, 063606 (2012)

    Article  ADS  Google Scholar 

  12. O.E. Alon, A.I. Streltsov, L.S. Cederbaum, Multiconfigurational time-dependent Hartree method for bosons: Many-Body dynamics of bosonic systems. Phys. Rev. A 77, 033613 (2008)

    Google Scholar 

  13. A.I. Streltsov, K. Sakmann, A.U.J. Lode, O.E. Alon, L.S. Cederbaum, The multiconfigurational time-dependent Hartree for Bosons package, version 2.3 (Heidelberg, 2012), see http://mctdhb.uni-hd.de

  14. K. Sakmann, Many-body Schrödinger dynamics of Bose–Einstein condensates. Ph.D. thesis, University of Heidelberg (Springer Theses, Berlin Heidelberg, 2011)

    Google Scholar 

  15. W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in Fortran (Cambridge University Press, Cambridge, 1992)

    Google Scholar 

  16. R. Kosloff, Time-Dependent Quantum-Mechanical Methods for Molecular Dynamics. J. Phys. Chem. 92, 2087 (1988)

    Article  Google Scholar 

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Authors and Affiliations

  1. Condensed Matter Theory and Quantum Computing Group, University of Basel, Basel, Switzerland

    Axel U. J. Lode

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  1. Axel U. J. Lode
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Correspondence to Axel U. J. Lode .

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Lode, A.U.J. (2015). Benchmarks with Analytically Solvable Problems. In: Tunneling Dynamics in Open Ultracold Bosonic Systems. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-07085-8_3

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