Finite Number Of Bosons Research Articles

Can a bright soliton model reveal a genuine time crystal for a finite number of bosons?

We analyze time crystal effects in a finite system of bosons which form a bright soliton clump on the Aharonov-Bohm ring. In the large particle number limit, N → ∞, this setup corresponds to the Wilczek model, where it is known that the time crystal behavior cannot be observed in the ground state of the system because a spontaneously formed soliton does not move. Here, we show that while the spontaneous formation of a moving soliton in the ground state can occur for , the soliton decays before it makes a single revolution along the ring and the time crystal dynamics is impossible.

Bose - Einstein condensation and thermodynamics of a finite number of bosons confined in a harmonic trap

Bose - Einstein condensation and thermodynamics of a finite number of bosons confined in a harmonic trap

Effective-range corrections to the ground-state energy of the weakly-interacting Bose gas in two dimensions

Nonuniversal effects due to leading effective-range corrections are computed for the ground-state energy of the weakly-coupled repulsive Bose gas in two spatial dimensions. Using an effective field theory of contact interactions, these corrections are computed first by considering fluctuations around the mean-field free energy of a system of interacting bosons. This result is then confirmed by an exact calculation in which the energy of a finite number of bosons interacting in a square with period boundary conditions is computed and the thermodynamic limit is explicitly taken.

Quantum simulation of the Abelian-Higgs lattice gauge theory with ultracold atoms

We present a quantum simulation scheme for the Abelian-Higgs lattice gauge theory using ultracold bosonic atoms in optical lattices. The model contains both gauge and Higgs scalar fields, and exhibits interesting phases related to confinement and the Higgs mechanism. The model can be simulated by an atomic Hamiltonian, by first mapping the local gauge symmetry to an internal symmetry of the atomic system, the conservation of hyperfine angular momentum in atomic collisions. By including auxiliary bosons in the simulation, we show how the Abelian-Higgs Hamiltonian emerges effectively. We analyze the accuracy of our method in terms of different experimental parameters, as well as the effect of the finite number of bosons on the quantum simulator. Finally, we propose possible experiments for studying the ground state of the system in different regimes of the theory, and measuring interesting high energy physics phenomena in real time.

Dynamical thermalization in Bose-Hubbard systems.

We numerically study a Bose-Hubbard ring of finite size with disorder containing a finite number of bosons that are subject to an on-site two-body interaction. Our results show that moderate interactions induce dynamical thermalization in this isolated system. In this regime the individual many-body eigenstates are well described by the standard thermal Bose-Einstein distribution for well-defined values of the temperature and the chemical potential, which depend on the eigenstate under consideration. We show that the dynamical thermalization conjecture works well at both positive and negative temperatures. The relations to quantum chaos, quantum ergodicity, and the Åberg criterion are also discussed.

STUDYING THE NUCLEAR STRUCTURE OF EVEN-EVEN DEFORMED GERMANIUM NUCLEUS

In the present work, the interacting boson model(IBM-1) for s and d boson is used to calculate the energy levels of the positive parity bands, electric quadrupole moments, E2-transition probabilities B(E2)in particular between states which may be affected by the finite boson number N=Nذ+Nُ=2+5=7 of even-even nucleus . This nucleus is belong to SU(5)-SU(3) dynamical symmetry .The calculated results of this study are compared with the available experimental data and they found to be in a very good agreement.

Approaching Bose–Einstein condensation

Bose–Einstein condensation (BEC) is discussed at the level of an advanced course of statistical thermodynamics, clarifying some formal and physical aspects that are usually not covered by the standard pedagogical literature. The non-conventional approach adopted starts by showing that the continuum limit, in certain cases, cancels out the crucial role of the bosonic ground level. If so, a correct treatment of the problem, including the ground level population N0 by construction, leads to BEC in a straightforward way. For a density of states of the form G(ϵ)∝ϵγ, the chemical potential µ is explicitly calculated as a function of the temperature T and of the number N of bosons, for various significant values of the positive exponent γ. In the thermodynamic limit, in which the boson number N diverges and BEC is a sharp process, the chemical potential µ is a singular function of T at the critical temperature TB, determined by an appropriate critical exponent. The condensate population N0 is studied analytically and numerically as a function of the temperature, for various values of N and for different γ. This provides an accurate description of the way BEC approaches the character of a sharp phase transition. Some aspects of the real experiments on BEC, involving a finite number of bosons, are also illustrated.

The E(2) symmetry and quantum phase transition in the two-dimensional limit of the vibron model

We study in detail the relation between the two-dimensional Euclidean dynamical E(2) symmetry and the quantum phase transition in the two-dimensional limit of the vibron model, called the U(3) vibron model. Both geometric and algebraic descriptions of the U(3) vibron model show that structures of low-lying states at the critical point of the model with a quartic potential as its classical limit can be approximately described by the E(2) symmetry. We also fit the finite-size scaling exponent of the energy levels and E1 transition rates in the F(2) model, which is exactly the E(2) model but with truncation in its Hilbert subspace, as well as those at the critical point in the U(3) vibron model. The N-scaling power law around the critical point shows that the E(2) symmetry is well preserved even for cases with finite number of bosons. In addition, two kinds of experimentally accessible effective order parameters, such as the energy ratios and E1 transition ratios , , are proposed to identify the second-order phase transition in such systems. Possible empirical examples exhibiting approximate E(2) symmetry are also presented.

Quantum phase transition in the U(4) vibron model and the E(3) symmetry

We study the details of the U(3)--O(4) quantum phase transition in the U(4) vibron model. Both asymptotic analysis in the classical limit and rigorous calculations for finite boson number systems indicate that a second-order phase transition is still there even for the systems with boson number $N$ ranging from tens to hundreds. Two kinds of effective order parameters, including $E1$ transition ratios $B(E1:{2}_{1}\ensuremath{\rightarrow}{1}_{1})/B(E1:{1}_{1}\ensuremath{\rightarrow}{0}_{1})$ and $B(E1:{0}_{2}\ensuremath{\rightarrow}{1}_{1})/B(E1:{1}_{1}\ensuremath{\rightarrow}{0}_{1})$, and the energy ratios ${E}_{{2}_{1}}/{E}_{{0}_{2}}$ and ${E}_{{3}_{1}}/{E}_{{0}_{2}}$ are proposed to identify the second-order phase transition in experiments. We also found that the critical point of phase transition can be approximately described by the E(3) symmetry, which persists even for moderate $N~10$ protected by the scaling behaviors of quantities at the critical point. In addition, a possible empirical example exhibiting roughly the E(3) symmetry is discussed.

Properties of fragmented repulsive condensates

Repulsive Bose-Einstein condensates immersed into a double-well trap potential are studied within the framework of the recently introduced mean-field approach which allows for bosons to reside in several different orthonormal orbitals. In the case of a one-orbital mean-field theory (Gross-Pitaevskii) the ground state of the system reveals a bifurcation scenario at some critical values of the interparticle interaction and/or the number of particles. At about the same values of the parameters the two-orbital mean field predicts that the system becomes twofold fragmented. By applying the three-orbital mean field we verify numerically that for the double-well external potential studied here the overall best mean field is achieved with two orbitals. The variational principle minimizes the energy at a vanishing population of the third orbital. To discuss the energies needed to remove a boson from and the energies gained by adding a boson to the condensate, we introduce boson ionization potentials and boson affinities and relate them to the chemical potentials. The impact of the finite number of bosons in the condensate on these quantities is analyzed. We recall that within the framework of the multiorbital mean-field theory each fragment is characterized by its own chemical potential. Finally, the stability of fragmented states is discussed in terms of the boson transfer energy which is the energy needed to transfer a boson from one fragment to another.

THERMODYNAMICS OF A BOSE GAS TRAPPED IN A BOUNDED HARMONIC POTENTIAL

Some thermodynamic features of an assembly of a finite number of bosons trapped in a bounded harmonic potential are investigated. P–V isotherms are drawn for both the degenerate and the non-degenerate phases. At any temperature, the pressure is a decreasing function of volume, unlike the free Bose gas for which the pressure becomes independent of volume in the degenerate phase. At the absolute zero of temperature the quantum pressure for a spherical enclosure of radius r0 equal to aho, aho being the characteristic harmonic oscillator length, is found to be of order 10-10 Torr while for [Formula: see text] it is of order 10-12 Torr. The isothermal compressibility has a sharp drop near the critical point and becomes negligibly small for temperatures above the critical temperature, irrespective of the size of the trap. The coefficient of thermal expansion also shows a sudden drop at the critical temperature. The specific heat at constant pressure shows a peak and is well-defined in the degenerate phase, in contradistinction to the ideal Bose gas. Results recently derived by Singh on the basis of local equilibrium theory are found to be in good agreement with our numerical computations.

Energy level statistics of the U(5) and O(6) symmetries in the interacting boson model

We study the energy level statistics of the states in U(5) and O(6) dynamical symmetries of the interacting boson model and the high spin states with backbending in U(5) symmetry. In the calculations, the degeneracy resulting from the additional quantum number is eliminated manually. The calculated results indicate that the finite boson number $N$ effect is prominent. When $N$ has a value close to a realistic one, increasing the interaction strength of subgroup O(5) makes the statistics vary from Poisson-type to GOE-type and further recover to Poisson-type. However, in the case of $N \to \infty$, they all tend to be Poisson-type. The fluctuation property of the energy levels with backbending in high spin states in U(5) symmetry involves a signal of shape phase transition between spherical vibration and axial rotation.

ON THE SCATTERING THEORY OF MASSLESS NELSON MODELS

We study the scattering theory for a class of non-relativistic quantum field theory models describing a confined non-relativistic atom interacting with a massless relativistic bosonic field. We construct invariant spaces [Formula: see text] which are defined in terms of propagation properties for large times and which consist of states containing a finite number of bosons in the region {|x| ≥ ct} for t → ±∞. We show the existence of asymptotic fields and we prove that the associated asymptotic CCR representations preserve the spaces [Formula: see text] and induce on these spaces representations of Fock type. For these induced representations, we prove the property of geometric asymptotic completeness, which gives a characterization of the vacuum states in terms of propagation properties. Finally we show that a positive commutator estimate imply the asymptotic completeness property, i.e. the fact that the vacuum states of the induced representations coincide with the bound states of the Hamiltonian.

Empirical example of possible E(5) symmetry nucleus108Pd

Analyzing the energy spectrum, $E2$ transition rates, and branching ratios, it is shown that the nucleus ${}^{108}\mathrm{Pd}$ provides an empirical example with the transitional dynamical symmetry E(5), and the finite boson number effect is not obvious.

BOSE–EINSTEIN CONDENSATION OF PARTICLES TRAPPED IN A BOUNDED HARMONIC POTENTIAL

The behavior of a finite number of bosons trapped in a bounded harmonic potential is investigated. The eigenvalue spectrum is worked out numerically for three different sizes of the trap. The condensate fraction is determined and is found to increase suddenly below a certain temperature which is a characteristic signature of BEC. The specific heat-temperature curve also shows a peak, with the maximum shifting to lower values and occurring at higher temperatures, as the size of the assembly is reduced.

Symmetry character of bands in72−84Krin the proton-neutron interacting boson model

The positive-parity bands observed in the even ${}^{72\ensuremath{-}84}\mathrm{Kr}$ isotopes have been studied in the framework of the IBA-2 model. The analysis includes states of spin up to the maximum allowed by the finite boson number. The symmetry character of these bands in the proton and neutron degrees of freedom has been investigated. Excitation energies, $B(E2)'s$ of intraband transitions and $B(E2)'s,$ $B(M1)'s,$ $E2/M1$ mixing ratios of interband transitions have been considered. Three bands of different structure can be accounted for in the framework of the model: the ground-state band, of predominant full-symmetry character; the band built on the ${2}_{2}^{+}$ state, which is composed by states which show different symmetry character along the isotopic chain; and the band built on the ${3}_{1}^{+}$ state, of predominant mixed-symmetry character.

An ideal mixture of N confined S=1-bosons

Using a new method – Path integral approach to Many Body theory – we obtain the free energy for a finite number of bosons, confined in a harmonic potential and having spin states with a different energy characterised by a magnetic quantum number. It is seen that all internal states prefer a unique condensate attributed to the (high temperature) maximum in the specific heat, the second (very low temperature) maximum is identified as the Schottky anomaly, coming from lifting the degeneracy of the internal degrees of freedom. The susceptibility clearly demonstrates the dependence of the condensation temperature on the energies of the internal degrees of freedom.

Statistical mechanics and path integrals for a finite number of bosons

Recent investigations show that the statistical mechanics of a finite number of particles in ideal harmonic systems predict different results for the same physical properties, depending on the ensemble under consideration. Path integral methods for a finite number of bosons with equidistant energy levels give the same answers for the mean energy, the specific heat and the condensation temperature, etc., irrespective of whether their calculation results from the density of states, from the partition function or from the generating function. We show that this contradiction is caused by the use of approximate relations between quantum statistical expressions.

On the functional integral theory of systems with kinematical interaction

We propose a systematic way to investigate the low-temperature thermodynamic properties of quantum spin systems subject to the restriction that only a finite number of bosons may occupy a single lattice site. Such a kinematical interaction results in the appearance of a temperature dependent chemical potential. Its low-temperature asymptotics is calculated self-consistently using the functional integration technique.

Bose-Einstein condensation of finite number of confined particles

The partition function and specific heat of a system consisting of a finite number of bosons confined in an external potential are calculated in different spatial dimensions. Using the grand partition function as the generating function of the partition function, an iterative scheme is established for the calculation of the partition function of system with an arbitrary number of particles. The scheme is applied to finite number of bosons confined in isotropic and anisotropic parabolic traps and in rigid boxes. The specific heat as a function of temperature is studied in detail for different number of particles, different degrees of anisotropy and different spatial dimensions. The peak in the specific heat is taken as an indication of Bose-Einstein condensation (BEC). It is found that the results corresponding to a large number of particles are approached quite rapidly as the number of bosons in the system increases. For large number of particles, results obtained within our iterative scheme are consistent with those of the semiclassical theory of BEC in an external potential based on the grand canonical treatment.