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arXiv:2101.08690 · Analysis & PDE

Bosons in a double well: two-mode approximation and fluctuations

Alessandro Olgiati, Nicolas Rougerie, Dominique Spehner

Published Oct 16, 2023• 3 citations
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Abstract

We study the ground state for many interacting bosons in a double-well potential, in a joint limit where the particle number and the distance between the potential wells both go to infinity.Two single-particle orbitals (one for each well) are macroscopically occupied, and we are concerned with deriving the corresponding effective Bose-Hubbard Hamiltonian.We prove an energy expansion, including the two-mode Bose-Hubbard energy and two independent Bogoliubov corrections (one for each potential well), and a variance bound for the number of particles falling inside each potential well.The latter is a signature of a correlated ground state in that it violates the central limit theorem.1. Introduction 1885 2. Main statements 1889 3. Mapping to the space of excitations 1897 4. Bounds on the two-mode Hamiltonian 1903 5. Derivation of the Bogoliubov Hamiltonian and reduction to right and left modes 1910 6.A priori estimates on the ground state of H N 1926 7. Shifted Hamiltonians and lower bound 1932 8. Proof of the main results 1939 Appendix A. The one-body Hartree problem 1949 Appendix B. Estimates and identities in the two-mode space 1951

DOI
10.2140/apde.2023.16.1885

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We study the ground state for many interacting bosons in a double-well potential, in a joint limit where the particle number and the distance between the potential wells both go to infinity.Two single-particle orbitals (one for each well) are macroscopically occupied, and we are concerned with deriving the corresponding effective Bose-Hubbard Hamiltonian.We prove an energy expansion, including the two-mode Bose-Hubbard energy and two independent Bogoliubov corrections (one for each potential well), and a variance bound for the number of particles falling inside each potential well.The latter is a signature of a correlated ground state in that it violates the central limit theorem.1. Introduction 1885 2. Main statements 1889 3. Mapping to the space of excitations 1897 4. Bounds on the two-mode Hamiltonian 1903 5. Derivation of the Bogoliubov Hamiltonian and reduction to right and left modes 1910 6.A priori estimates on the ground state of H N 1926 7. Shifted Hamiltonians and lower bound 1932 8. Proof of the main results 1939 Appendix A. The one-body Hartree problem 1949 Appendix B. Estimates and identities in the two-mode space 1951
We review two recent results on the ground state properties of bosonic systems trapped by a double well external potential. In the limit of a large number of particles and large separation between the wells, we prove that fluctuations in the number of particles occupying each single-well low-energy mode occur at a reduced scale with respect to N, the latter being the typical prediction of the central limit theorem. This signals the breakdown of the independent and uncorrelated particle picture of standard Bose–Einstein condensation and the emergence of an interaction-driven correlated phase in the ground state.
We study the ground state of a large bosonic system trapped in a symmetric double-well potential, letting the distance between the two wells increase to infinity with the number of particles. In this context, one should expect an interaction-driven transition between a delocalized state (particles are independent and all live in both wells) and a localized state (particles are correlated, half of them live in each well). We start from the full many-body Schr{ö}dinger Hamiltonian in a large-filling situation where the on-site interaction and kinetic energies are comparable. When tunneling is negligible against interaction energy, we prove a localization estimate showing that the particle number fluctuations in each well are strongly suppressed. The modes in which the particles condense are minimizers of nonlinear Schr{ö}dinger-type functionals.
We determine the low-energy properties of a trapped Bose gas split in two by a potential barrier over the whole range of barrier heights and asymmetry between the wells. For either weak or strong coupling between the wells, our two-mode theory yields a two-site Bose-Hubbard Hamiltonian with the tunneling, interaction, and bias parameters calculated simply using an explicit form of two mode functions. When the potential barrier is relatively low, most of the particles occupy the condensate mode and our theory reduces to a two-mode version of the Bogoliubov theory, which gives a satisfactory estimate of the spatial shape and energy of the lowest collective excitation. When the barrier is high, our theory generalizes the standard two-site Bose-Hubbard model into the case of asymmetric modes, and correctly predicts a full separation of the modes in the limit of strong separation of the wells. We provide explicit analytic forms for the number squeezing and coherence as a function of particle number and temperature. We compare our theory to other two-mode theories for bosons in a double well and discuss their validity in different parameter regimes.
In this paper we introduce a novel multi-scale technique to study many-body quantum systems where the total number of particles is kept fixed. The method is based on Feshbach map and the scales are represented by occupation numbers of particle states. Here, we consider a three-modes (including the zero mode) Bogoliubov Hamiltonian for a sufficiently small ratio between the kinetic energy and the Fourier component of the (positive type) potential corresponding to the two nonzero modes. For any space dimension d\geq 1 and in the mean field limiting regime (i.e., at fixed box volume |Λ| and for a number of particles, N, sufficiently large) this method provides the construction of the ground state and its expansion in terms of the bare operators that in the limit N \to \infty is up to any desired precision. In space dimension d \geq 3 the method provides similar results for an arbitrarily large (finite) box and a large but fixed particle density ρ, i.e.,ρis independent of the size of the box.
The two-site version of the Bose-Hubbard model can be used as a simplified model of N bosons confined to two quantum wells. We study the behavior of low-temperature Bose-Einstein condensates as a function of the strength of the repulsive interactions between the bosons and of the size of the quantum-mechanical tunneling between the quantum wells. The ground-state of weakly interacting bosons tends to be one in which almost all particles are in the same quantum-mechanical state (‘Bose-Einstein Condensate’, or ‘BEC’). It is found that even very small amounts of interwell tunneling lead to a BEC, but that zero tunneling leads to two separate BECs, one in each well (‘fragmentation’). We study systematically the transition between a single BEC comprised of bosons in both wells, and a fragmented condensate consisting of two BECs, one for each well. Our analysis will use matrix computational methods to solve the Schrödinger equation of the two-well boson system, and obtain the energies and wavefunctions as eigenvalues and eigenvectors of the Hamiltonian matrix. We will compare the results obtained with previous analyses, and against analytical approximations developed for the limiting case of a large number of bosons (large N).
In this paper we consider an interacting Bose gas at zero temperature, in a finite box and in the mean field limiting regime. The N gas particles interact through a pair potential of positive type and with an ultraviolet cut-off. Its (nonzero) Fourier components are sufficiently large with respect to the corresponding kinetic energies of the modes. Using the multi-scale technique in the occupation numbers of particle states introduced in [Pi1], we provide a convergent expansion of the ground state of the particle number preserving Bogoliubov Hamiltonian in terms of the bare operators. In the limit N \to \infty the expansion is up to any desired precision.