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Multiscale Modeling and Simulation

Bose–Einstein Condensation beyond Mean Field: Many-Body Bound State of Periodic Microstructure

Dionisios Margetis

Published Jan 1, 2012• 5 citations
Publisher

Abstract

We study stationary quantum fluctuations around a mean field limit in trapped, dilute atomic gases of repulsively interacting bosons at zero temperature. Our goal is to describe quantum-mechanically the lowest macroscopic many-body bound state consistent with a microscopic Hamiltonian that accounts for inhomogeneous particle scattering processes. In the mean field limit, the wave function of the condensate (macroscopic quantum state) satisfies a defocusing cubic nonlinear Schrödinger-type equation, the Gross–Pitaevskii equation. We include consequences of pair excitation, i.e., the scattering of particles in pairs from the condensate to other states, proposed in [T. T. Wu, J. Math. Phys., 2 (1961), pp. 105–123]. Our derivations rely on an uncontrolled yet physically motivated assumption for the many-body wave function. By relaxing mathematical rigor, from a particle Hamiltonian with a spatially varying interaction strength we derive via heuristics an integro–partial differential equation for the pair collision kernel, K, under a stationary condensate wave function, $\Phi$. For a scattering length with periodic microstructure of subscale $\epsilon$, we formally describe via classical homogenization the lowest many-body bound state in terms of $\Phi$ and K up to second order in $\epsilon$. If the external potential is slowly varying, we solve the homogenized equations via boundary layer theory. As an application, we describe the partial depletion of the condensate.

DOI
10.1137/110826576

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•Aug 4, 2026•0 citations
We study stationary quantum fluctuations around a mean field limit in trapped, dilute atomic gases of repulsively interacting bosons at zero temperature. Our goal is to describe quantum-mechanically the lowest macroscopic many-body bound state consistent with a microscopic Hamiltonian that accounts for inhomogeneous particle scattering processes. In the mean field limit, the wave function of the condensate (macroscopic quantum state) satisfies a defocusing cubic nonlinear Schrödinger-type equation, the Gross–Pitaevskii equation. We include consequences of pair excitation, i.e., the scattering of particles in pairs from the condensate to other states, proposed in [T. T. Wu, J. Math. Phys., 2 (1961), pp. 105–123]. Our derivations rely on an uncontrolled yet physically motivated assumption for the many-body wave function. By relaxing mathematical rigor, from a particle Hamiltonian with a spatially varying interaction strength we derive via heuristics an integro–partial differential equation for the pair collision kernel, K, under a stationary condensate wave function, Φ\PhiΦ. For a scattering length with periodic microstructure of subscale ϵ\epsilonϵ, we formally describe via classical homogenization the lowest many-body bound state in terms of Φ\PhiΦ and K up to second order in ϵ\epsilonϵ. If the external potential is slowly varying, we solve the homogenized equations via boundary layer theory. As an application, we describe the partial depletion of the condensate.
At finite temperatures below the phase transition point, the Bose-Einstein condensation, the macroscopic occupation of a single quantum state by particles of integer spin, is not complete. In the language of superfluid helium, this means that the superfluid coexists with the normal fluid. Our goal is to describe this coexistence in trapped, dilute atomic gases with repulsive interactions via mean field laws that account for a {\em spatially varying} particle interaction strength. By starting with the NNN-body Hamiltonian, N≫1N\gg 1N≫1, we formally derive a system of coupled, nonlinear evolution equations in 3+13+13+1 dimensions for the following quantities: (i) the wave function of the macroscopically occupied state; and (ii) the single-particle wave functions of thermally excited states. For stationary (bound) states and a scattering length with {\em periodic microstructure} of subscale εεε, we heuristically extract effective equations of motion via periodic homogenization up to second order in εεε.
We provide the analysis of a physically inspired model for a trapped dilute Bose gas with repulsive pairwise atomic interactions at zero temperature. Our goal is to describe aspects of the excited many-body quantum states by accounting for the scattering of atoms in pairs from the macroscopic state (condensate). We formally construct a many-body Hamiltonian, Happ\mathcal{H}_{\text{app}}Happ​, that is quadratic in the Boson field operators for noncondensate atoms. This Happ\mathcal{H}_{\text{app}}Happ​ conserves the total number of atoms. Inspired by Wu (J. Math. Phys., 2:105-123, 1961), we apply a non-unitary transformation to Happ\mathcal{H}_{\text{app}}Happ​. Key in this non-Hermitian view is the pair-excitation kernel, which in operator form obeys a Riccati equation. In the stationary case, we develop an existence theory for solutions to this operator equation by a variational approach. We connect this theory to the one-particle excitation wave functions heuristically derived by Fetter (Ann. Phys., 70:67-101, 1972). These functions solve an eigenvalue problem for a JJJ-self-adjoint operator. From the non-Hermitian Hamiltonian, we derive a one-particle nonlocal equation for low-lying excitations, describe its solutions, and recover Fetter's excitation spectrum. Our approach leads to a description of the excited eigenstates of the reduced Hamiltonian in the NNN-particle sector of Fock space.
In Bose-Einstein condensation, a macroscopically large number of particles occupy the same single-particle quantum state. Our goal is to study time-dependent aspects of particle excitations to states other than the single-particle macroscopic state in trapped dilute atomic gases. We adopt the view that atoms are excited in pairs so that their scattering from the single-particle state to vector positions x and y at time t is described by the pair-excitation function, K0(x,y,t). We solve a nonlocal equation for K0 under a slowly varying external potential by assuming that the wave function of the macroscopic state satisfies a time-independent nonlinear Schroedinger equation. For zero initial excitation (K0=0 at t=0) and sufficiently large t, we evaluate asymptotically K0 in terms of the one-variable Lommel function for any distance |x-y|.
We study a physically motivated model for a trapped dilute gas of Bosons with repulsive pairwise atomic interactions at zero temperature. Our goal is to describe aspects of the excited many-body quantum states of this system by accounting for the scattering of atoms in pairs from the macroscopic state. We start with an approximate many-body Hamiltonian, H a p p \mathcal {H}_{\mathrm {app}} , in the Bosonic Fock space. This H a p p \mathcal {H}_{\mathrm {app}} conserves the total number of atoms. Inspired by Wu [J. Math. Phys. 2 (1961), 105–123], we apply a non-unitary transformation to H a p p \mathcal {H}_{\mathrm {app}} . Key in this procedure is the pair-excitation kernel, which obeys a nonlinear integro-partial differential equation. In the stationary case, we develop an existence theory for solutions to this equation by a variational principle. We connect this theory to a system of partial differential equations for one-particle excitation (“quasiparticle”-) wave functions derived by Fetter [Ann. Phys. 70 (1972), 67–101], and prove existence of solutions for this system. These wave functions solve an eigenvalue problem for a J J -self-adjoint operator. From the non-Hermitian Hamiltonian, we derive a one-particle nonlocal equation for low-lying excitations, describe its solutions, and recover Fetter’s energy spectrum. We also analytically provide an explicit construction of the excited eigenstates of the reduced Hamiltonian in the N N -particle sector of Fock space.
A large system of N N integer-spin atoms, called Bosons, manifests one of the most coherent macroscopic quantum states known to date, the “Bose-Einstein condensate”, at extremely low temperatures. As N → ∞ N\to \infty , this system is usually described by a mean-field limit: a single-particle wave function, the condensate wave function , that satisfies a nonlinear Schrödinger-type equation. In this expository paper, we review kinetic aspects of the mean-field Boson evolution. Furthermore, we discuss recent advances in the rigorous study of second-order corrections to this mean-field limit. These corrections originate from the quantum-kinetic mechanism of pair excitation , which lies at the core of pioneering works in theoretical physics including ideas of Bogoliubov, Lee, Huang, Yang and Wu. In the course of our exposition, we revisit the formalism of Fock space, which is indispensable for the analysis of pair excitation.
This thesis is devoted to the rigorous study of dilute Bose gases with two-body and three-body interactions at zero temperature in two and three dimensions. Bose gases are many-body quantum systems composed of bosons, which notably do not obey the Pauli exclusion principle. As a consequence, at extremely low temperatures, a macroscopic fraction of the particles can occupy the same one-body quantum state. This phenomenon, known as Bose–Einstein condensation, was predicted in the 1920s by Bose and Einstein for noninteracting Bose gases, and first realised experimentally by the groups of Cornell and Wieman, and Ketterle, earning them the Nobel Prize for Physics in 2001. However, proving the occurrence of Bose–Einstein condensation in general interacting bosonic systems remains a major open problem and constitutes a central theme in this thesis. Another important property of bosonic matter is that it is generally more unstable than fermionic matter. In particular, for systems interacting via the Coulomb force, the Pauli exclusion principle is essential for explaining the stability of the second kind, which states that the energy and volume scale linearly with the number of particles. Nonetheless, bosonic systems with attractive but less singular interactions can also be stable, and the rigorous proof of this fact forms a second focus of this thesis. In three dimensions, we consider predominantly repulsive interactions. Firstly, we prove a leading order upper bound on the thermodynamic ground state energy of Bose gases with hard-core three-body interactions, which matches an existing lower bound. In addition, we obtain a simpler proof for integrable potentials. Secondly, we derive the leading order of the thermodynamic ground state energy of Bose gases with combined two-body and three-body interactions. More specifically, we show that, at the leading order, the ground state energy is entirely characterised by the scattering length of the two-body potential and the scattering hypervolume of the three-body potential. Thirdly, we study the time evolution of Bose–Einstein condensates with three-body interactions in the so-called Gross–Pitaevskii regime. We show that Bose–Einstein condensation is preserved under many-body evolution and that the condensate wavefunction evolves according to the quintic Gross–Pitaevskii equation. In particular, we show that the effective coupling constant is universal in the sense that it depends only on the scattering hypervolume of the interaction potential. As a by-product of our proof, we derive a stability of the second kind result for Bose gases with attractive two-body interactions and repulsive three-body interactions. In two dimensions, we consider trapped Bose gases with mean-field-type interactions, including the purely attractive case. We prove the stability of the second kind of the system and the convergence of the ground state energy per particle to that of a non-linear Schrödinger energy functional. As a consequence of the stability of the second kind, we also obtain Bose–Einstein condensation for the many-body ground states.