Morphology of an Interacting Three-Dimensional Trapped Bose-Einstein Condensate from Many-Particle Variance Anisotropy

The variance of the position operator is associated with how wide or narrow a wave-packet is, the momentum variance is similarly correlated with the size of a wave-packet in momentum space, and the angular-momentum variance quantifies to what extent a wave-packet is non-spherically symmetric. We examine an interacting three-dimensional trapped Bose–Einstein condensate at the limit of an infinite number of particles, and investigate its position, momentum, and angular-momentum anisotropies. Computing the variances of the three Cartesian components of the position, momentum, and angular-momentum operators we present simple scenarios where the anisotropy of a Bose–Einstein condensate is different at the many-body and mean-field levels of theory, despite having the same many-body and mean-field densities per particle. This suggests a way to classify correlations via the morphology of 100% condensed bosons in a three-dimensional trap at the limit of an infinite number of particles. Implications are briefly discussed.

[1]  V. V. Murakhtanov,et al.  The system of harmonically interacting particles: An exact solution of the quantum‐mechanical problem , 1991 .

[2]  C. Schilling.  Natural orbitals and occupation numbers for harmonium: Fermions versus bosons , 2013, 1307.6858.

[3]  Ofir E. Alon,et al.  Analysis of a Trapped Bose-Einstein Condensate in Terms of Position, Momentum, and Angular-Momentum Variance , 2019, Symmetry.

[4]  Elliott H. Lieb,et al.  Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional , 1999, math-ph/9908027.

[5]  A. Jensen,et al.  Virial expansion coefficients in the harmonic approximation. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  L. Cederbaum,et al.  Variance of an anisotropic Bose-Einstein condensate , 2017, Chemical Physics.

[7]  D. Hundertmark,et al.  Derivation of the Hartree equation for compound Bose gases in the mean field limit , 2017, 1702.00827.

[8]  Y. Castin,et al.  Low-temperature Bose-Einstein condensates in time-dependent traps: Beyond the U(1) symmetry-breaking approach , 1998 .

[9]  R. Hall.  Some exact solutions to the translation-invariant N-body problem , 1978 .

[10]  O. Alon.  Variance of a Trapped Bose-Einstein Condensate , 2018, Journal of Physics: Conference Series.

[11]  B. Judd,et al.  Reduced Density Matrices: Coulson's Challenge , 2000 .

[12]  S. Klaiman,et al.  Variance as a sensitive probe of correlations , 2015, 1502.07528.

[13]  J. S. Dehesa,et al.  Entanglement in N-harmonium: bosons and fermions , 2014, 1404.4447.

[14]  Ofir E. Alon,et al.  Colloquium : Multiconfigurational time-dependent Hartree approaches for indistinguishable particles , 2019, Reviews of Modern Physics.

[15]  L. Cederbaum,et al.  Overlap of exact and Gross-Pitaevskii wave functions in Bose-Einstein condensates of dilute gases , 2016, 1609.05895.

[16]  Horng-Tzer Yau,et al.  Rigorous derivation of the Gross-Pitaevskii equation. , 2006, Physical review letters.

[17]  C. Schilling,et al.  Number-parity effect for confined fermions in one dimension , 2015, 1508.04452.

[18]  L. Cederbaum.  Exact many-body wave function and properties of trapped bosons in the infinite-particle limit , 2017 .

[19]  M. Gajda.  Criterion for Bose-Einstein condensation in a harmonic trap in the case with attractive interactions , 2006 .

[20]  Robert Seiringer,et al.  Proof of Bose-Einstein condensation for dilute trapped gases. , 2002, Physical review letters.

[21]  A. Jensen,et al.  Analytic solutions of topologically disjoint systems , 2014, 1411.3347.

[22]  V. I. Yukalov,et al.  Particle Fluctuations in Mesoscopic Bose Systems , 2019, Symmetry.

[23]  Horng-Tzer Yau,et al.  Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems , 2005, math-ph/0508010.

[24]  O. Alon.  Condensates in annuli: dimensionality of the variance , 2019, Molecular Physics.

[25]  J. Schmiedmayer,et al.  Conserving symmetries in Bose-Einstein condensate dynamics requires many-body theory , 2018, 1802.03746.

[26]  Coupled oscillator natural orbitals , 1977 .

[27]  A. Michelangeli,et al.  Mean-field quantum dynamics for a mixture of Bose–Einstein condensates , 2016, 1603.02435.

[28]  S. Klaiman,et al.  Uncertainty product of an out-of-equilibrium many-particle system , 2015, 1509.00762.

[29]  O. Alon.  Solvable model of a generic trapped mixture of interacting bosons: reduced density matrices and proof of Bose–Einstein condensation , 2017, 1702.08219.

[30]  A. Orłowski,et al.  Soluble model of many interacting quantum particles in a trap , 2000 .

[31]  Jun Yan.  Harmonic Interaction Model and Its Applications in Bose–Einstein Condensation , 2003 .

[32]  L. Cohen,et al.  Exact reduced density matrices for a model problem , 1985 .

[33]  J. S. Dehesa,et al.  Entanglement and the Born-Oppenheimer approximation in an exactly solvable quantum many-body system , 2014, 1408.6990.

[34]  A. Jensen,et al.  Analytic harmonic approach to the N-body problem , 2010, 1011.2453.

[35]  R. Hall.  Exact solutions of Schrodinger's equation for translation-invariant harmonic matter , 1978 .