An approximate many-body theory incorporating two-body correlations has been employed to calculate low-lying collective multipole frequencies in a Bose-Einstein condensate containing A bosons, for different values of the interaction parameter . Significant difference from the variational estimate of the Gross-Pitaevskii equation has been found near the collapse region. This is attributed to two-body correlations and finite range attraction of the realistic interatomic interaction. A large deviation from the hydrodynamic model is also seen for the second monopole breathing mode and the quadrupole mode for large positive λ. © 2008 IOP Publishing Ltd.

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University of Calcutta (informally known as Calcutta University or CU) is a collegiate public state university located in Kolkata, West Bengal, India.&nbsp;Within India it is recognized as a &quot;Five-Star University&quot; and accredited &quot;A&quot; Grade by National Assessment and Accreditation Council.

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The University of Calcutta has secured the Fifth position among the Universities of India in the prestigious &quot;Indian University Ranking 2019&quot; list, released by the NIRF of the Ministry of Human Resource Development, Government of India.

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It was established on 24 January 1857, and was one of the first institutions in Asia to be established as a multidisciplinary and Western-style university. Its alumni and faculty include several heads of state, heads of government, social reformers, prominent artists, only academy award winner and Dirac medal winner in India, many Fellows of the Royal Society and five Nobel laureates as of 2019 which is highest in South Asia.

University Of Calcutta

Journal of Physics B: Atomic, Molecular and Optical Physics

We report the effects of interaction on thermodynamic properties of a repulsive Bose-Einstein condensate confined in a harmonic trap by using the correlated potential harmonics expansion method. This many-body technique permits the use of a realistic interactomic interaction, which gives rise to the effective long-range interaction of the condensate in terms of the s-wave scattering length. We have computed temperature (T) dependence of the chemical potential, specific heat, condensate fraction, entropy, pressure, and the average energy per particle of a system containing a large number (A) of 87Rb atoms in the Joint Institute for Laboratory Astrophysics (JILA) trap. The repulsion among the interacting bosons results in a small but measurable drop of condensate fraction and critical temperature (Tc), compared to those of a noninteracting condensate. These are in agreement with the experiment. Although all thermodynamic quantities have a strong dependence on A and to a smaller extent on the interatomic interaction, our numerical calculation appears to show that a thermodynamic quantity per particle follows a universal behavior as a function of T/Tc. This shows the importance of Tc for all thermodynamic properties of the condensate. As expected, for T&gt;T c, these properties follow those of a trapped noncondensed Bose gas. © 2013 American Physical Society.

Physical Review A - Atomic, Molecular, and Optical Physics

Effects of interaction on thermodynamics of a repulsive Bose-Einstein condensate

A two-body correlated basis set is used to develop a many-body theory which is valid for any number of bosons in the trap. The formalism incorporates the van der Waals interaction and two-body correlations in an exact way. The theory has successfully been applied to Bose-Einstein condensates - dilute weakly interacting and also dilute but having a large scattering length. Even in the extreme dilute condition, we observe the breakdown of the shape-independent approximation and the interatomic correlation plays an important role in the large particle-number limit. This correlated many-body calculation can handle, within the two-body correlation approximation, the entire range of atom number of experimentally achieved condensates. Next we successfully push the basis function for large scattering lengths where the mean-field results are manifestly bad. The sharp increase in correlation energy clearly shows the beyond-mean-field effect. We also calculate one-particle densities for various scattering lengths and particle numbers. Our many-body calculation exhibits the finite-size effect in the one-body density. © 2013 American Physical Society.

Beyond mean-field ground-state energies and correlation properties of a trapped Bose-Einstein condensate

We report exact numerical calculations of the chemical potential, condensate fraction and specific heat of N non-interacting bosons confined in an isotropic harmonic oscillator trap in one, two and three dimensions, as also for interacting bosons in a 3D trap. Quasi phase transitions (QPT) are observed in all these cases, including in one-dimension, as shown by a rapid change of several thermodynamic quantities at the transition point. The change becomes more rapid as N increases in 2D and 3D cases. However with increase in N, the sudden change in the nature of specific heat, gets gradually wiped out in 1D, while it becomes more drastic in 2D and 3D. But the sudden changes in the condensate fraction and chemical potential become more drastic as N increases, even in 1D. This shows that a QPT is possible in 1D also. We define the transition exponent, which characterizes the nature of a thermodynamic quantity at the transition point of a quasi phase transition, and evaluate them by careful numerical calculations, very near the transition temperature. These exponents are found to be independent of the size of the system. They are also the same for interacting and non-interacting bosons. These demonstrate the universality property of the transition exponents. © 2013 Springer Science+Business Media New York.

Journal of Low Temperature Physics

Thermodynamic properties of ultracold bose gas: Transition exponents and universality

It has been recently shown numerically that the transition from integrability to chaos in quantum systems and the corresponding spectral fluctuations are characterized by 1/fα noise with 1≤α≤2. The system of interacting trapped bosons is inhomogeneous and complex. The presence of an external harmonic trap makes it more interesting as, in the atomic trap, the bosons occupy partly degenerate single-particle states. Earlier theoretical and experimental results show that at zero temperature the low-lying levels are of a collective nature and high-lying excitations are of a single-particle nature. We observe that for few bosons, the P(s) distribution shows the Shnirelman peak, which exhibits a large number of quasidegenerate states. For a large number of bosons the low-lying levels are strongly affected by the interatomic interaction, and the corresponding level fluctuation shows a transition to a Wigner distribution with an increase in particle number. It does not follow Gaussian orthogonal ensemble random matrix predictions. For high-lying levels we observe the uncorrelated Poisson distribution. Thus it may be a very realistic system to prove that 1/fα noise is ubiquitous in nature. © 2012 American Physical Society.

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

Spectral fluctuation and 1/fÎ± noise in the energy level statistics of interacting trapped bosons

We report the calculation of heat capacity of an attractive Bose-Einstein condensate, with the number N of bosons increasing and eventually approaching the critical number Ncr for collapse, using the correlated potential harmonics (CPH) method. Boson pairs interact via the realistic van der Waals potential. It is found that the transition temperature Tc initially increases slowly, then rapidly as N becomes closer to Ncr. The peak value of heat capacity for a fixed N increases slowly with N, for N far away from Ncr. But after reaching a maximum, it starts decreasing when N approaches Ncr. The effective potential calculated by the CPH method provides insight into this strange behavior. © 2011 American Physical Society.

Behavior of heat capacity of an attractive Bose-Einstein condensate approaching collapse

The stability of trapped interacting bosons with attractive interactions is studied using an approximate many-body calculation. Instead of using the traditional hyperspherical harmonics expansion method we prescribe a potential harmonics expansion method (PHEM). The justification of the use of PHEM in connection with dilute condensates is presented. The choice of a correlation function is justified as it correctly reproduces the short-range two-body correlation in the wavefunction as also the correct value of the s-wave scattering length (as). Applications to 7Li and 85Rb condensates with the realistic van der Waals interaction give good agreement with the Rice and JILA experiments, respectively. The JILA experiment used controlled collapse of the 85Rb condensate for different values of as. Our calculations agree with the experimental results within the experimental error bars. © 2007 IOP Publishing Ltd.

An approximate many-body calculation for trapped bosons with attractive interaction

We investigate Bose-Einstein condensates (BEC) containing a large number of bosonic atoms interacting via a finite-range semirealistic interatomic interaction. Ground state properties for an increasing number of atoms in the condensate have been calculated using a modification of the potential harmonic expansion method (including a short range correlation function in the expansion basis) to solve the many-body Schrödinger equation. An improved numerical algorithm for the calculation of the potential matrix elements permits us to have up to 14000 atoms in the condensate. Although our approach is approximate and justified for dilute condensates, our results agree well with available diffusion Monte Carlo results for the same case. The ground state energies also agree well with those by the Gross-Pitaevskii equation method for up to 100 particles in the trap and become gradually larger than the latter (up to 5% for 14000 atoms). The difference is attributed to the effects of finite range interatomic interaction and two-body correlations. Our approach presents a clear physical picture of the condensate, being computationally economical at the same time. © 2007 The American Physical Society.

Behavior of a Bose-Einstein condensate containing a large number of atoms interacting through a finite-range interatomic interaction

The T=0 properties of a Bose-Einstein condensate (BEC), consisting of A atoms (bosons) trapped by an external field, were investigated. An ab initio treatment of the Schrödinger equation involves 3(A-1) degrees of freedom for the relative motion. The potential matrix elements involved integrals over only three angle variables, leading to an immense reduction in the complexity of the numerical procedure for A particles. It was noticed that many-body results converged towards the mean field results as the particle number increased.

Potential harmonics expansion method for trapped interacting bosons: Inclusion of two-body correlation

Physical Review A

Dilute Bose gases interacting via power-law potentials

Inhomogeneous boson systems, such as the dilute gases of integral spin atoms in low-temperature magnetic traps, are believed to be well described by the Gross-Pitaevskii equation (GPE). GPE is a nonlinear Schrödinger equation which describes the order parameter of such systems at the mean field level. In the present work, we describe a Fortran 90 computer program developed by us, which solves the GPE using a basis set expansion technique. In this technique, the condensate wave function (order parameter) is expanded in terms of the solutions of the simple-harmonic oscillator (SHO) characterizing the atomic trap. Additionally, the same approach is also used to solve the problems in which the trap is weakly anharmonic, and the anharmonic potential can be expressed as a polynomial in the position operators x, y, and z. The resulting eigenvalue problem is solved iteratively using either the self-consistent-field (SCF) approach, or the imaginary time steepest-descent (SD) approach. Iterations can be initiated using either the simple-harmonic-oscillator ground state solution, or the Thomas-Fermi (TF) solution. It is found that for condensates containing up to a few hundred atoms, both approaches lead to rapid convergence. However, in the strong interaction limit of condensates containing thousands of atoms, it is the SD approach coupled with the TF starting orbitals, which leads to quick convergence. Our results for harmonic traps are also compared with those published by other authors using different numerical approaches, and excellent agreement is obtained. GPE is also solved for a few anharmonic potentials, and the influence of anharmonicity on the condensate is discussed. Additionally, the notion of Shannon entropy for the condensate wave function is defined and studied as a function of the number of particles in the trap. It is demonstrated numerically that the entropy increases with the particle number in a monotonic way. Program summary: Title of program:bose.x. Catalogue identifier:ADWZ_v1_0. Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADWZ_v1_0. Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland. Distribution format:tar.gz. Computers:PC's/Linux, Sun Ultra 10/Solaris, HP Alpha/Tru64, IBM/AIX. Programming language used:mostly Fortran 90. Number of bytes in distributed program, including test data, etc.:27 313. Number of lines in distributed program, including test data, etc.:28 015. Card punching code:ASCII. Nature of physical problem:It is widely believed that the static properties of dilute Bose condensates, as obtained in atomic traps, can be described to a fairly good accuracy by the time-independent Gross-Pitaevskii equation. This program presents an efficient approach of solving this equation. Method of solution:The solutions of the Gross-Pitaevskii equation corresponding to the condensates in atomic traps are expanded as linear combinations of simple-harmonic oscillator eigenfunctions. Thus, the Gross-Pitaevskii equation which is a second-order nonlinear differential equation, is transformed into a matrix eigenvalue problem. Thereby, its solutions are obtained in a self-consistent manner, using methods of computational linear algebra. Unusual features of the program:None. © 2006 Elsevier B.V. All rights reserved.

Journal	Data powered by TypesetJournal of Physics B: Atomic, Molecular and Optical Physics
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ISSN	0953-4075
Open Access	Yes