GROUND STATES OF LARGE BOSONIC SYSTEMS: THE GROSS-PITAEVSKII LIMIT REVISITED

We study the ground state of a dilute Bose gas in a scaling limit where the Gross-Pitaevskii functional emerges. This is a repulsive non-linear Schrödinger functional whose quartic term is proportional to the scattering length of the interparticle interaction potential. We propose a new derivation of this limit problem, with a method that bypasses some of the technical difficulties that previous derivations had to face. The new method is based on a combination of Dyson's lemma, the quantum de Finetti theorem and a second moment estimate for ground states of the effective Dyson Hamiltonian. It applies equally well to the case where magnetic fields or rotation are present.We study the ground state of a dilute Bose gas in a scaling limit where the Gross-Pitaevskii functional emerges. This is a repulsive non-linear Schrödinger functional whose quartic term is proportional to the scattering length of the interparticle interaction potential. We propose a new derivation of this limit problem, with a method that bypasses some of the technical difficulties that previous derivations had to face. The new method is based on a combination of Dyson's lemma, the quantum de Finetti theorem and a second moment estimate for ground states of the effective Dyson Hamiltonian. It applies equally well to the case where magnetic fields or rotation are present.

arXiv:1503.07061Analysis & PDEMar 24, 201671 citations

Bose-Einstein Condensation of Dilute Gases in Traps

Élliott H. Lieb, Robert Seiringer

The ground state of a gas of Bosons confined in an external trap potential and interacting via repulsive two-body forces has recently been shown to exhibit complete Bose-Einstein condensation in the dilute limit, yielding for the first time a rigorous proof of this phenomenon in a physically realistic setting. We give here an account of this work about the Gross-Pitaevskii limit where the particle number $N$ goes to infinity with $Na$ fixed, where $a$ is the scattering length of the interparticle interaction, measured in units of the trap size.The ground state of a gas of Bosons confined in an external trap potential and interacting via repulsive two-body forces has recently been shown to exhibit complete Bose-Einstein condensation in the dilute limit, yielding for the first time a rigorous proof of this phenomenon in a physically realistic setting. We give here an account of this work about the Gross-Pitaevskii limit where the particle number $N$ goes to infinity with $Na$ fixed, where $a$ is the scattering length of the interparticle interaction, measured in units of the trap size.

arXiv:math-ph/0210028Oct 13, 20020 citations

Ground state energy of a Bose gas in the Gross–Pitaevskii regime

Giulia Basti, Serena Cenatiempo, Alessandro Olgiati +2 more

We review some rigorous estimates for the ground state energy of dilute Bose gases. We start with Dyson’s upper bound, which provides the correct leading order asymptotics for hard spheres. Afterward, we discuss a rigorous version of Bogoliubov theory, which recently led to an estimate for the ground state energy in the Gross–Pitaevskii regime, valid up to second order, for particles interacting through integrable potentials. Finally, we explain how these ideas can be combined to establish a new upper bound, valid to second order, for the energy of hard spheres in the Gross–Pitaevskii limit. Here, we only sketch the main ideas; details will appear elsewhere.We review some rigorous estimates for the ground state energy of dilute Bose gases. We start with Dyson’s upper bound, which provides the correct leading order asymptotics for hard spheres. Afterward, we discuss a rigorous version of Bogoliubov theory, which recently led to an estimate for the ground state energy in the Gross–Pitaevskii regime, valid up to second order, for particles interacting through integrable potentials. Finally, we explain how these ideas can be combined to establish a new upper bound, valid to second order, for the energy of hard spheres in the Gross–Pitaevskii limit. Here, we only sketch the main ideas; details will appear elsewhere.

arXiv:2202.10270Journal of Mathematical PhysicsApr 1, 202211 citations

Gross-Pitaevskii non-linear dynamics for pseudo-spinor condensates

Alessandro Michelangeli, Alessandro Olgiati

We derive the equations for the non-linear effective dynamics of a so called pseudo-spinor Bose-Einstein condensate, which emerges from the linear many-body Schrödinger equation at the leading order in the number of particles. The considered system is a three-dimensional diluted gas of identical bosons with spin, possibly confined in space, and coupled with an external time-dependent magnetic field; particles also interact among themselves through a short-scale repulsive interaction. The limit of infinitely many particles is monitored in the physically relevant Gross-Pitaevskii scaling. In our main theorem, if at time zero the system is in a phase of complete condensation (at the level of the reduced one-body marginal) and with energy per particle fixed by the Gross-Pitaevskii functional, then such conditions persist also at later times, with the one-body orbital of the condensate evolving according to a system of non-linear cubic Schrödinger equations coupled among themselves through linear (Rabi) terms. The proof relies on an adaptation to the spinor setting of Pickl's projection counting method developed for the scalar case. Quantitative rates of convergence are available, but not made explicit because evidently non-optimal. In order to substantiate the formalism and the assumptions made in the main theorem, in an introductory section we review the mathematical formalisation of modern typical experiments with pseudo-spinor condensates.We derive the equations for the non-linear effective dynamics of a so called pseudo-spinor Bose-Einstein condensate, which emerges from the linear many-body Schrödinger equation at the leading order in the number of particles. The considered system is a three-dimensional diluted gas of identical bosons with spin, possibly confined in space, and coupled with an external time-dependent magnetic field; particles also interact among themselves through a short-scale repulsive interaction. The limit of infinitely many particles is monitored in the physically relevant Gross-Pitaevskii scaling. In our main theorem, if at time zero the system is in a phase of complete condensation (at the level of the reduced one-body marginal) and with energy per particle fixed by the Gross-Pitaevskii functional, then such conditions persist also at later times, with the one-body orbital of the condensate evolving according to a system of non-linear cubic Schrödinger equations coupled among themselves through linear (Rabi) terms. The proof relies on an adaptation to the spinor setting of Pickl's projection counting method developed for the scalar case. Quantitative rates of convergence are available, but not made explicit because evidently non-optimal. In order to substantiate the formalism and the assumptions made in the main theorem, in an introductory section we review the mathematical formalisation of modern typical experiments with pseudo-spinor condensates.

arXiv:1704.00150Apr 1, 20175 citations

Gross-Pitaevskii non-linear dynamics for pseudo-spinor condensates

Alessandro Michelangeli, Alessandro Olgiati

We derive the equations for the non-linear effective dynamics of a so called pseudo-spinor Bose-Einstein condensate, which emerges from the linear many-body Schrödinger equation at the leading order in the number of particles.The considered system is a three-dimensional diluted gas of identical bosons with spin, possibly confined in space, and coupled with an external time-dependent magnetic field; particles also interact among themselves through a short-scale repulsive interaction.The limit of infinitely many particles is monitored in the physically relevant Gross-Pitaevskii scaling.In our main theorem, if at time zero the system is in a phase of complete condensation (at the level of the reduced one-body marginal) and with energy per particle fixed by the Gross-Pitaevskii functional, then such conditions persist also at later times, with the one-body orbital of the condensate evolving according to a system of non-linear cubic Schrödinger equations coupled among themselves through linear (Rabi) terms.The proof relies on an adaptation to the spinor setting of Pickl's projection counting method developed for the scalar case.Quantitative rates of convergence are available, but not made explicit because evidently non-optimal.In order to substantiate the formalism and the assumptions made in the main theorem, in an introductory section we review the mathematical formalisation of modern typical experiments with pseudo-spinor condensates.We derive the equations for the non-linear effective dynamics of a so called pseudo-spinor Bose-Einstein condensate, which emerges from the linear many-body Schrödinger equation at the leading order in the number of particles.The considered system is a three-dimensional diluted gas of identical bosons with spin, possibly confined in space, and coupled with an external time-dependent magnetic field; particles also interact among themselves through a short-scale repulsive interaction.The limit of infinitely many particles is monitored in the physically relevant Gross-Pitaevskii scaling.In our main theorem, if at time zero the system is in a phase of complete condensation (at the level of the reduced one-body marginal) and with energy per particle fixed by the Gross-Pitaevskii functional, then such conditions persist also at later times, with the one-body orbital of the condensate evolving according to a system of non-linear cubic Schrödinger equations coupled among themselves through linear (Rabi) terms.The proof relies on an adaptation to the spinor setting of Pickl's projection counting method developed for the scalar case.Quantitative rates of convergence are available, but not made explicit because evidently non-optimal.In order to substantiate the formalism and the assumptions made in the main theorem, in an introductory section we review the mathematical formalisation of modern typical experiments with pseudo-spinor condensates.

Journal of Nonlinear Mathematical PhysicsJun 28, 201718 citations

Derivation of Hartree theory for two-dimensional attractive Bose gases in almost Gross–Pitaevskii regime

Lukas Junge, François Louis Antoine Visconti

We study the ground state energy of trapped two-dimensional Bose gases with mean-field type interactions that can be attractive. We prove the stability of second kind of the many-body system and the convergence of the ground state energy per particle to that of a non-linear Schrödinger (NLS) energy functional. Notably, we can take any polynomial scaling of the interaction, and even exponential scalings arbitrarily close to the Gross–Pitaevskii regime, which is a drastic improvement on the best-known result for systems with attractive interactions. As a consequence of the stability of second kind we also obtain Bose–Einstein condensation for the many-body ground states for a much improved range of the diluteness parameter.We study the ground state energy of trapped two-dimensional Bose gases with mean-field type interactions that can be attractive. We prove the stability of second kind of the many-body system and the convergence of the ground state energy per particle to that of a non-linear Schrödinger (NLS) energy functional. Notably, we can take any polynomial scaling of the interaction, and even exponential scalings arbitrarily close to the Gross–Pitaevskii regime, which is a drastic improvement on the best-known result for systems with attractive interactions. As a consequence of the stability of second kind we also obtain Bose–Einstein condensation for the many-body ground states for a much improved range of the diluteness parameter.

arXiv:2410.23150Journal de Mathématiques Pures et AppliquéesOct 1, 20252 citations

N

goes to infinity with

Na

fixed, where

a

is the scattering length of the interparticle interaction, measured in units of the trap size.