Analysis of a Trapped Bose-Einstein Condensate in Terms of Position, Momentum, and Angular-Momentum Variance

We analyze, analytically and numerically, the position, momentum, and in particular the angular-momentum variance of a Bose–Einstein condensate (BEC) trapped in a two-dimensional anisotropic trap for static and dynamic scenarios. Explicitly, we study the ground state of the anisotropic harmonic-interaction model in two spatial dimensions analytically and the out-of-equilibrium dynamics of repulsive bosons in tilted two-dimensional annuli numerically accurately by using the multiconfigurational time-dependent Hartree for bosons method. The differences between the variances at the mean-field level, which are attributed to the shape of the BEC, and the variances at the many-body level, which incorporate depletion, are used to characterize position, momentum, and angular-momentum correlations in the BEC for finite systems and at the limit of an infinite number of particles where the bosons are 100 % condensed. Finally, we also explore inter-connections between the variances.

[1]  Shi-jie Yang,et al.  Ground-state phases of the spin-orbit-coupled spin-1 Bose gas in a toroidal trap , 2017 .

[2]  C. J. Foot,et al.  Time-averaged adiabatic ring potential for ultracold atoms , 2011, 1102.2895.

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

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

[5]  L. Cederbaum,et al.  Role of excited states in the splitting of a trapped interacting Bose-Einstein condensate by a time-dependent barrier. , 2006, Physical review letters.

[6]  S. Adhikari.  Dipolar Bose-Einstein condensate in a ring or in a shell , 2012, 1205.2813.

[7]  L. Cederbaum,et al.  Numerically exact quantum dynamics of bosons with time-dependent interactions of harmonic type , 2012, 1207.5128.

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

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

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

[11]  D. Angom,et al.  Ramifications of topology and thermal fluctuations in quasi-2D condensates , 2016, 1605.03818.

[12]  D. Angom,et al.  Geometry-induced modification of fluctuation spectrum in quasi-two-dimensional condensates , 2015, 1511.08655.

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

[14]  L. Cederbaum,et al.  Multiconfigurational time-dependent Hartree method for mixtures consisting of two types of identical particles , 2007 .

[15]  N. Efremidis,et al.  Rotating Bose-Einstein condensates with a finite number of atoms confined in a ring potential: Spontaneous symmetry breaking, beyond the mean-field approximation , 2016, 1611.09256.

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

[17]  S. Reimann,et al.  Bosonic and fermionic dipoles on a ring. , 2010, Physical review letters.

[18]  U. R. Fischer,et al.  Condensate fragmentation as a sensitive measure of the quantum many-body behavior of bosons with long-range interactions , 2015, 1502.04889.

[19]  S. Klaiman,et al.  Solvable model of a trapped mixture of Bose–Einstein condensates , 2016, 1605.05608.

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

[21]  M. Kol'avr,et al.  Criticality and spin squeezing in the rotational dynamics of a Bose-Einstein condensate on a ring lattice , 2015, 1511.02320.

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

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

[24]  L. Cederbaum,et al.  Phantom vortices: hidden angular momentum in ultracold dilute Bose-Einstein condensates , 2017, Scientific Reports.

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

[26]  Haobin Wang,et al.  Multilayer formulation of the multiconfiguration time-dependent Hartree theory , 2003 .

[27]  P. Schmelcher,et al.  Ultracold bosonic scattering dynamics off a repulsive barrier: Coherence loss at the dimensional crossover , 2017, 1705.04083.

[28]  U. R. Fischer,et al.  Self-consistent determination of the many-body state of ultracold bosonic atoms in a one-dimensional harmonic trap , 2017, Annals of Physics.

[29]  A. Lode.  Multiconfigurational time-dependent Hartree method for bosons with internal degrees of freedom:Theory and composite fragmentation of multicomponent Bose-Einstein condensates , 2016, 1602.05791.

[30]  A. Streltsov,et al.  Many-body excitations and deexcitations in trapped ultracold bosonic clouds , 2016, 1608.08060.

[31]  F. Dalfovo,et al.  Theory of Bose-Einstein condensation in trapped gases , 1998, cond-mat/9806038.

[32]  U. Manthe,et al.  The multi-configurational time-dependent Hartree approach , 1990 .

[33]  C. Bao.  Oscillation bands of Bose-Einstein condensates on a ring: Beyond the mean-field theory , 2007, 0704.0842.

[34]  L. Cederbaum,et al.  Generic regimes of quantum many-body dynamics of trapped bosonic systems with strong repulsive interactions , 2013, 1312.6174.

[35]  L. Madsen,et al.  Multispecies time-dependent restricted-active-space self-consistent-field theory for ultracold atomic and molecular gases , 2018, Journal of Physics B: Atomic, Molecular and Optical Physics.

[36]  P. Schmelcher,et al.  Unraveling the Structure of Ultracold Mesoscopic Collinear Molecular Ions. , 2017, Physical review letters.

[37]  C. Wieman,et al.  Nobel Lecture: Bose-Einstein condensation in a dilute gas, the first 70 years and some recent experiments , 2002 .

[38]  O. Alon,et al.  Many-body quantum dynamics of an asymmetric bosonic Josephson junction , 2018, New Journal of Physics.

[39]  U. Manthe,et al.  Wave‐packet dynamics within the multiconfiguration Hartree framework: General aspects and application to NOCl , 1992 .

[40]  A. Lode,et al.  Multiconfigurational time-dependent Hartree method for fermions: Implementation, exactness, and few-fermion tunneling to open space , 2015, 1510.02984.

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

[42]  W. Ketterle.  Nobel lecture: When atoms behave as waves: Bose-Einstein condensation and the atom laser* , 2002 .

[43]  A. Leggett,et al.  Bose-Einstein condensation in the alkali gases: Some fundamental concepts , 2001 .

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

[45]  V. S. Bagnato,et al.  Parametric Excitation of a Bose-Einstein Condensate: From Faraday Waves to Granulation , 2017, Physical Review X.

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

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

[48]  P. Schmelcher,et al.  Beyond mean-field dynamics of ultra-cold bosonic atoms in higher dimensions: facing the challenges with a multi-configurational approach , 2016, 1608.04710.

[49]  T. Jacobson,et al.  A rapidly expanding Bose-Einstein condensate: an expanding universe in the lab. , 2017, Physical review. X.

[50]  Lorenz S. Cederbaum,et al.  Multiconfigurational time-dependent Hartree method for bosons: Many-body dynamics of bosonic systems , 2007, cond-mat/0703237.

[51]  L. Cederbaum,et al.  General variational many-body theory with complete self-consistency for trapped bosonic systems , 2006, cond-mat/0603212.

[52]  S. Vishveshwara,et al.  Static and dynamic properties of shell-shaped condensates , 2017, Physical Review A.

[53]  P. Schmelcher,et al.  The multi-layer multi-configuration time-dependent Hartree method for bosons: theory, implementation, and applications. , 2012, The Journal of chemical physics.

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

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

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

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

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

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

[60]  A. Streltsov.  Quantum systems of ultracold bosons with customized interparticle interactions , 2013, 1307.5187.

[61]  C. Clark,et al.  Hysteresis in a quantized superfluid ‘atomtronic’ circuit , 2014, Nature.

[62]  Y. Son,et al.  Vortex Dynamics in an Annular Bose-Einstein Condensate , 2012, 1206.5622.

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

[64]  L. Cederbaum,et al.  Exact ground state of finite Bose-Einstein condensates on a ring (14 pages) , 2005, cond-mat/0505323.

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

[66]  K. Suominen,et al.  Snake instability of ring dark solitons in toroidally trapped Bose-Einstein condensates , 2012, 1210.3994.

[67]  C. Bruder,et al.  Fragmented Superradiance of a Bose-Einstein Condensate in an Optical Cavity. , 2016, Physical review letters.

[68]  J. Dalibard,et al.  Many-Body Physics with Ultracold Gases , 2007, 0704.3011.

[69]  M. Kasevich,et al.  Single-shot simulations of dynamic quantum many-body systems , 2015, Nature Physics.

[70]  J. Brand,et al.  Center-of-mass motion as a sensitive convergence test for variational multimode quantum dynamics , 2015, 1510.07845.

[71]  A. Gammal,et al.  Phases, many-body entropy measures, and coherence of interacting bosons in optical lattices , 2017, 1712.08792.

[72]  R. Mayol,et al.  Persistent currents supported by solitary waves in toroidal Bose-Einstein condensates , 2015 .

[73]  H. Meyer,et al.  Multilayer multiconfiguration time-dependent Hartree method: implementation and applications to a Henon-Heiles hamiltonian and to pyrazine. , 2010, The Journal of chemical physics.

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

[75]  M. Cozzini,et al.  Vortex signatures in annular Bose-Einstein condensates , 2005, cond-mat/0510143.

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

[77]  P. Kevrekidis,et al.  Many-body quantum dynamics in the decay of bent dark solitons of Bose–Einstein condensates , 2017, 1706.07360.

[78]  L. Cederbaum,et al.  Attractive Bose-Einstein condensates in anharmonic traps: Accurate numerical treatment and the intriguing physics of the variance , 2018, Chemical Physics.

[79]  Ulrich Hohenester,et al.  Optimizing number squeezing when splitting a mesoscopic condensate , 2008, 0806.3877.

[80]  C. Clark,et al.  Phase fluctuations in anisotropic Bose-Einstein condensates: From cigars to rings , 2010, 1007.0281.

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

[82]  P. Schmelcher,et al.  A unified ab initio approach to the correlated quantum dynamics of ultracold fermionic and bosonic mixtures. , 2017, The Journal of chemical physics.

[83]  Uwe Manthe,et al.  A multilayer multiconfigurational time-dependent Hartree approach for quantum dynamics on general potential energy surfaces. , 2008, The Journal of chemical physics.

[84]  R. Dubessy,et al.  Critical rotation of an annular superfluid Bose-Einstein condensate , 2012, 1204.6183.

[85]  Jiang Yong Lu.  Phases , 2020, Co-evolution Strategy Canvas.

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

[87]  K. Kärkkäinen,et al.  Mixtures of Bose gases confined in a ring potential. , 2009, Physical review letters.

[88]  O. Alon,et al.  Impact of the range of the interaction on the quantum dynamics of a bosonic Josephson junction , 2017, Chemical Physics.

[89]  P. Alam.  ‘O’ , 2021, Composites Engineering: An A–Z Guide.

[90]  L. Cederbaum,et al.  Exact quantum dynamics of a bosonic Josephson junction. , 2009, Physical review letters.

[91]  C. Clark,et al.  Hysteresis in a quantized superfluid ‘atomtronic’ circuit , 2014, Nature.

[92]  L. Madsen,et al.  Time-dependent restricted-active-space self-consistent-field theory for bosonic many-body systems , 2016, 1612.04419.

[93]  Entanglement Induced Interactions in Binary Mixtures. , 2017, Physical review letters.

[94]  A. Fetter,et al.  Quantized superfluid vortex dynamics on cylindrical surfaces and planar annuli , 2017, 1708.08903.

[95]  M. Jones,et al.  Rotational Response of Two-Component Bose-Einstein Condensates in Ring Traps , 2010, 1001.5233.

[96]  Robert P. Smith,et al.  Quantized supercurrent decay in an annular Bose-Einstein condensate , 2011, 1112.0334.

[97]  T. Elsayed,et al.  Probing quantum states with momentum boosts , 2017, Physical Review A.

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

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

[100]  P. Alam.  ‘A’ , 2021, Composites Engineering: An A–Z Guide.

[101]  S. Vishveshwara,et al.  Physics of hollow Bose-Einstein condensates , 2016, 1612.05809.

[102]  S Gupta,et al.  Bose-Einstein condensation in a circular waveguide. , 2005, Physical review letters.

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

[104]  M. Beck,et al.  The multiconfiguration time-dependent Hartree (MCTDH) method: A highly efficient algorithm for propa , 1999 .

[105]  Solvable Model of a Generic Trapped Mixture of Interacting Bosons: Many-Body and Mean-Field Properties at the Infinite-Particle Limit , 2017, 1708.00687.