In this paper we continue our development of a dimensional perturbation theory (DPT) treatment of N identical particles under quantum confinement. DPT is a beyond-mean-field method which is applicable to both weakly and strongly-interacting systems and can be used to connect both limits. In a previous paper we developed the formalism for low-order energies and excitation frequencies. This formalism has been applied to atoms, Bose-Einstein condensates and quantum dots. One major advantage of the method is that N appears as a parameter in the analytical expressions for the energy and so results for N up to a few thousand are easy to obtain. Other properties however, are also of interest, for example the density profile in the case of a BEC,and larger N results are desirable as well. The latter case requires us to go to higher orders in DPT. These calculations require as input zeroth-order wave functions and this paper, along with a subsequent paper, addresses this issue.
In this paper, the second in a series of two, we complete the derivation of the lowest-order wave function of a dimensional perturbation theory (DPT) treatment for the N-body quantum-confined system. Taking advantage of the symmetry of the zeroth-order configuration, we use group theoretic techniques and the FG matrix method from quantum chemistry to obtain analytic results for frequencies and normal modes. This method directly accounts for each two-body interaction, rather than an average interaction so that even lowest-order results include beyond-mean-field effects. It is thus appropriate for the study of both weakly and strongly interacting systems and the transition between them. While previous work has focused on energies, lowest-order wave functions yield important information such as the nature of excitations and expectation values of physical observables at low orders including density profiles. Higher orders in DPT also require as input the zeroth-order wave functions. In the earlier paper we presented a program for calculating the analytic normal-mode coordinates of the large-D system and illustrated the procedure by deriving the two simplest normal modes. In this paper we complete this analysis by deriving the remaining, and more complex, normal coordinates of the system.
Confined quantum systems involving
N identical interacting particles are to be found in many areas of physics, including condensed-matter, atomic, and chemical physics. A beyond-mean-field perturbation method that is applicable, in principle, to weakly, intermediate, and strongly interacting systems has been set forth by the authors in a previous series of papers. Dimensional perturbation theory was used, and in conjunction with group theory, an analytic beyond-mean-field correlated wave function at lowest order for a system under spherical confinement with a general two-body interaction was derived. In the present paper, we use this analytic wave function to derive the corresponding lowest-order, analytic density profile and apply it to the example of a Bose-Einstein condensate.
Systems involving N-identical interacting particles under quantum confinement appear in many areas of physics, including chemical, condensed matter, and atomic physics. We discuss a beyondmean- field perturbation method that is applicable to weakly, intermediate and strongly-interacting systems. Group theory is used to derive an analytic beyond-mean-field correlated wave function at zeroth order for a system under spherical confinement. We derive the corresponding zeroth-order analytic density profile and apply it to the example of a Bose-Einstein condensate.
We generalize the harmonic-order N-body DPT method for an isotropically confined quantum system of identical interacting bosons to first-anharmonic order and, in principle, higher orders. We introduce a graphical decomposition of the perturbative expansion of the N-body Hamiltonian. We calculate the Clebsch-Gordon coefficient tensors that couple together 3 irreducible representations of SN analytically, and analytically transformed the graphical basis to collective coordinates. We calculate the N-body wavefunction and density profile in general and have demonstrated agreement with an analytic model. We apply this formalism to the exactly-solvable example of a trapped gas of atoms interacting with a (fully-interacting) harmonic-oscillator "Hooke's law" interaction and compare with the dimensional expansion of the exact ground-state wavefunction and density profile. We report progress on the example of a cold gas BEC (with zero angular momentum).
Feshbach resonances notwithstanding, most experimentally realized atomic-vapor condensates are dilute and are best described by the mean-field Gross-Pitaevskii equation. For this reason, we use dimensional scaling methods to obtain an analytical approximation to the GP equation that is more accurate and flexible than the commonly used ground-state Thomas-Fermi approximation. We also demonstrate the power of dimensional perturbation theory by providing a full solution of a model BEC Hamiltonian and a two-electron quantum dot Hamiltonian. A feature shared by these examples is the high degree of accuracy provided by the lowest orders of the perturbation theory. In our approach to the full many-body BEC Hamiltonian, we use the lowest orders of many-body dimensional perturbation theory to obtain semi-analytical ground-state energies and collective excitation frequencies. Our many-body calculations for BEC's compare well with the Gross-Pitaevskii results in the weakly-interacting regime, as they should, and are much improved over mean-field theory predictions in the strongly-interacting regime.
The resources required to solve the general interacting quantum
N-body problem scale exponentially with
N, making the solution of this problem very difficult when
N is large. In a previous series of papers we develop an approach for a fully interacting wave function for a confined system of identical bosons with a general two-body interaction. This method tames the
N scaling by developing a perturbation series that is order-by-order invariant under a point group isomorphic with
SN. Group theory and graphical techniques are then used to solve for the wave function exactly and analytically at each order, yielding a solution for the general
N-body problem which scales as
N0 at any given order. Recently this formalism has been used to obtain the first-order fully interacting wave function for a system of harmonically confined bosons interacting harmonically. In this paper, we report the application of this
N-body wave function to a system of
N fully interacting bosons in three dimensions. We derive an expression for the density profile for a confined system of harmonically interacting bosons. Choosing this simple interaction is not necessary or even advantageous for our method, however this choice allows a direct comparison of our exact results through first order with exact results obtained in an independent solution. Our density profile for the wave function through first order in three dimensions is indistinguishable from the first-order exact result obtained independently and shows strong convergence to the exact result to all orders.