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arXiv:2106.02152

Many-body excitations in trapped Bose gas: A non-Hermitian view

Manoussos G. Grillakis, Dionisios Margetis, Stephen Sorokanich

Published Jun 3, 2021• 0 citations
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Abstract

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, $\mathcal{H}_{\text{app}}$, that is quadratic in the Boson field operators for noncondensate atoms. This $\mathcal{H}_{\text{app}}$ conserves the total number of atoms. Inspired by Wu (J. Math. Phys., 2:105-123, 1961), we apply a non-unitary transformation to $\mathcal{H}_{\text{app}}$. 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 $J$-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 excitation spectrum. Our approach leads to a description of the excited eigenstates of the reduced Hamiltonian in the $N$-particle sector of Fock space.

DOI
10.48550/arxiv.2106.02152

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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.
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.
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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.
We derive a new formula for excited many-body eigenstates of the approximate Hamiltonian of Lee, Huang, and Yang, which describes the weakly interacting bose gas in a periodic box. Our formula utilizes a non-Hermitian transform to isolate a process of pair excitation in all excited states. This program is inspired by the description of Wu for the many-body ground state of the Lee-Huang-Yang Hamiltonian, as well as similar descriptions for the ground state of quadratic Hamiltonians in generic trapping potentials.
We prove existence results for a system of partial differential equations describing the approximate condensate wavefunction and pair-excitation kernel of a dilute T = 0 Bose gas in the stationary setting, in the presence of a trapping potential and repulsive pairwise atomic interactions. Notably, the Hartree-type equation for the condensate in this system contains contributions from non-condensate particles, and the pair excitation kernel satisfies a nonlinear operator equation. The techniques employed include a direct variational principle and also an iterative procedure for constructing solutions.
Quantum systems consisting of many particles are both interesting from the physics and the mathematics perspective. One aspect that makes them fascinating from the physics point of view is that they can exhibit quantum effects on macroscopic scales such as Bose-Einstein condensation. This phenomenon, first predicted 100 years ago by Bose [19] and Einstein [44], occurs when a dilute gas of bosons is cooled down to very low temperatures. Below a certain critical temperature a condensate forms, meaning that a macroscopic fraction of the particles is in the same one-particle state, and thus, quantum phenomena become relevant on length scales much larger than the individual atom. The occurrence of Bose-Einstein condensation is also related to superfluidity as suggested in [88]. Bose-Einstein condensation has been achieved experimentally independently by the groups of Cornell and Wieman [3] and Ketterle [40] earning them a Nobel prize in 2001. While Bose and Einstein analyzed theoretically the non-interacting gas, an effective theory describing weakly interacting Bose gases was proposed by Bogoliubov in 1947 with the goal of explaining the occurrence of superfluidity in liquid helium [18]. Using assumptions such as that condensation indeed occurs, his theory yields predictions for the low-energy states of the system. Seen from the mathematics perspective, describing the Bose gas is an interesting challenge due to the strong correlations between particles introduced by the interaction making this case much harder to study than the non-interacting one. In particular, a proof that Bose-Einstein condensation occurs in the thermodynamic limit for continuous systems is still out of reach. However, there has been a lot of work regarding the rigorous analysis of bosonic systems in various relevant regimes. This thesis contains three new contributions establishing rigorous results on bosonic systems in different, physically relevant settings. It is based on the articles [37, 27, 31]. The first result concerns the time-evolution of initially trapped Bose gases in the ultra-dilute Gross-Pitaevskii regime. Under some assumptions on the initial data satisfied in particular by a class of states that are expected to approximate the ground state in a trap, we derive a quasi-free approximation that is close to the true solution in L2L^2L2-norm. In particular, we show that this approximate wave function is described in terms of a Bose-Einstein condensate in a one-particle state following the time-dependent Gross-Pitaevskii equation together with pair excitations with respect to this condensate. As a corollary, we establish a central limit theorem for bounded one-particle observables. The second result is a simple proof of Bose-Einstein condensation for interacting, ultra-dilute gases. More specifically, we consider scalings interpolating between the Gross-Pitaevskii regime and the thermodynamic limit. While condensation had been shown to hold in these regimes before, the advantage of the method used here is its simplicity. Lastly, the third result establishes an upper bound on the ground state energy of a dilute Bose gas in the thermodynamic limit. It captures rigorously the third order correction to the energy per unit volume for sufficiently low densities as predicted in the physics literature first by Wu and subsequently others. Full-text embargoed until: 2026-01-01\textit{Full-text embargoed until: 2026-01-01}Full-text embargoed until: 2026-01-01