Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Bosonic Particle-Correlated States: A Nonperturbative Treatment Beyond Mean Field for Strongly Interacting Systems

Jiang Zhang, Alexandre B. Tacla, Carlton M. Caves

Published Jun 5, 2017• 0 citations
OpenAlex

Abstract

Many useful properties of dilute Bose gases at ultra-low temperature are predicted precisely by the product-state ansatz, in which all particles are in the same quantum state. As particle-particle correlations become important, however, the product ansatz begins to fail. We consider a new set of states, which constitute a natural generalization of the product ansatz; the particle-correlated state of $N=l\times n$ identical particles is derived by symmetrizing the $n$-fold product of an $l$-particle quantum state. The particle-correlated states can be simulated efficiently for large $N$, because their parameter spaces, which depend on $l$, do not grow with $n$. Here we pay special attention to the pure-state case for $l=2$, where the many-body state is constructed from a two-particle pure state. These paired wave functions for bosons, which we call pair-correlated states (PCS), were introduced by Leggett [Rev. Mod. Phys. $\bf 73$, 307 (2001)] as a particle-number-conserving version of the Bogoliubov approximation. We present an iterative algorithm that solves the reduced (marginal) density matrices (RDMs)---these are the correlation functions---associated with PCS in time $O(N)$. The RDMs can also be derived from the normalization factor of PCS, which is derived analytically in the large $N$ limit. To test the efficacy of our theory, we analyze the ground state of the two-site Bose-Hubbard model by minimizing the energy of the PCS state, both in its exact form and in its large-$N$ approximate form. The relative errors of the ground state energy for both cases are within $10^{-5}$ for $N = 1000$ particles over the entire parameter region from a single condensate to a Mott insulator. Moreover, we present numerical results that suggest that PCS might be useful for describing the dynamics in the strongly interacting regime.

Related papers

Bosonic particle-correlated states: A nonperturbative treatment beyond mean field

Jiang Zhang, Alexandre B. Tacla, Carlton M. Caves

Many useful properties of dilute Bose gases at ultralow temperature are predicted precisely by the (mean-field) product-state Ansatz, in which all particles are in the same quantum state. Yet, in situations where particle-particle correlations become important, the product Ansatz fails. To include correlations nonperturbatively, we consider a new set of states: the particle-correlated state of $N=l\ifmmode\times\else\texttimes\fi{}n$ bosons is derived by symmetrizing the $n$-fold product of an $l$-particle quantum state. Quantum correlations of the $l$-particle state ``spread out'' to any subset of the $N$ bosons by symmetrization. The particle-correlated states can be simulated efficiently for large $N$, because their parameter spaces, which depend on $l$, do not grow with $n$. Here we formulate and develop in great detail the pure-state case for $l=2$, where the many-body state is constructed from a two-particle pure state. These paired wave functions, which we call pair-correlated states (PCS), were introduced by A. J. Leggett [Rev. Mod. Phys. 73, 307 (2001)] as a particle-number-conserving version of the Bogoliubov approximation. We present an iterative algorithm that solves for the reduced (marginal) density matrices (RDMs), i.e., the correlation functions, associated with PCS in time $O(N)$. The RDMs can also be derived from the normalization factor of PCS, which is derived analytically in the large-$N$ limit. To test the efficacy of PCS, we analyze the ground state of the two-site Bose-Hubbard model by minimizing the energy of the PCS state, both in its exact form and in its large-$N$ approximate form, and comparing with the exact ground state. For $N=1000$, the relative errors of the ground-state energy for both cases are within ${10}^{\ensuremath{-}5}$ over the entire parameter region from a single condensate to a Mott insulator. We present numerical results that suggest that PCS might be useful for describing the dynamics in the strongly interacting regime.

arXiv:1706.01547•Physical Review A•Aug 23, 2017•6 citations

Particle Correlations in Bose-Einstein Condensates

Jiang Zhang

The impact of interparticle correlations on the behavior of Bose-Einstein Condensates (BECs) is discussed using two approaches. In the first approach, the wavefunction of a BEC is encoded in the $N$-particle sector of an extended "catalytic state". Going to a time-dependent interaction picture, we can organize the effective Hamiltonian by powers of ${N}^{-1/2}$. Requiring the terms of order ${N}^{1/2}$ to vanish, we get the Gross-Pitaevskii Equation. Going to the next order, $N^0$, we obtain the number-conserving Bogoliubov approximation. Our approach allows one to stay in the Schrödinger picture and to apply many techniques from quantum optics. Moreover, it is easier to track different orders in the Hamiltonian and to generalize to the multi-component case. In the second approach, I consider a state of $N =l\times n$ bosons that is derived by symmetrizing the $n$-fold tensor product of an arbitrary $l$-boson state. Particularly, we are interested in the pure state case for $l=2$, which we call the Pair-Correlated State (PCS). I show that PCS reproduces the number-conserving Bogoliubov approximation; moreover, it also works in the strong interaction regime where the Bogoliubov approximation fails. For the two-site Bose-Hubbard model, I find numerically that the error (measured by the trace distance of the two-particle reduced density matrices) of PCS is less than two percent over the entire parameter space, thus making PCS a bridge between the superfluid and Mott insulating phases. Amazingly, the error of PCS does not increase, in the time-dependent case, as the system evolves for longer times. I derive both time-dependent and -independent equations for the ground state and the time evolution of the PCS ansatz. The time complexity of simulating PCS does not depend on $N$ and is linear in the number of orbitals in use.

arXiv:1502.06924•Feb 24, 2015•2 citations

Pair-correlated variational approach for interacting ultra-cold atoms in arbitrary one-dimensional confinement

Przemysław Kościk, Marcin Płodzień, Tomasz Sowiński

Standard analytical construction of the many-body wave function of interacting particles in one dimension, beyond mean-field theory, is based on the Jastrow approach. The many-body interacting ground state is build up from the ground state of the non-interacting system and the product of solutions of the corresponding interacting two-body problem. However, this is possible only if the center-of-mass motion is decoupled from the mutual interactions. In our work, based on the general constraints given by contact nature of the atom-atom interactions, we present an alternative approach to the standard construction of the \textit{pair-correlation} wave-function. Within the proposed ansatz, we study the many-body properties of trapped bosons as well as fermionic mixtures and we compare these predictions with the exact diagonalization approach in a wide range of particle numbers, interaction strengths, and different trapping potentials.

•Apr 17, 2018•0 citations

Derivation of mean-field equations and next-order corrections for bosons and fermions

David Mitrouskas

This thesis is about the derivation of effective mean field equations and their next-order corrections starting from nonrelativistic many-body quantum theory. Mean field equations provide an approximate ansatz for the description of interacting many-particle systems. In this ansatz the interaction between the particles is replaced by a self-consistent external potential leading to an effective one-body description of the many-particle system. Next-order corrections provide an approximation which goes one step further and tries to capture also subleading effects that are not resolved by the mean field ansatz. We present mathematical proofs for the validity of such effective theories for different models that are motivated, e.g., from the theory of ultracold atoms (the bosonic Hartree equation and the corresponding Bogoliubov theory) and from plasma physics (the motion of a tracer particle through a degenerate and dense electron gas). Starting from a many-body Schrödinger equation, our goal is to show that the solutions converge in a particular limit to the solutions to an effective mean field equation and its next-order corrections. After a short introduction and a summary in Chapter one, we present the main part of this work in three self-contained chapters. In Chapter two we analyze the dynamics of a large number N of nonrelativistic bosons in the weak coupling limit, i.e., for a coupling constant g_N=1/N. It is well known that in the limit of infinite particle number, the Hartree equation emerges as an effective one-particle theory of the Bose gas. This is closely related to the remarkable physical phenomenon of Bose-Einstein condensation at low temperature, namely that the majority of particles in a Bose gas occupies the same copy of a single one-particle quantum state. Our emphasis lies in the description of the few particles that fluctuate around the Bose-Einstein condensate. We show convergence of the fully interacting dynamics to an auxiliary time evolution in the norm of the N-particle space. This result allows us to prove several other assertions. Among other things, it is used to derive the Hartree equation with optimal speed of convergence 1/N for initial states that are close to ground states of interacting systems and also to prove convergence of the N-particle solution towards a time evolution obtained from the Bogoliubov Hamiltonian on Fock space. Chapter three is about the low energy properties of the weakly interacting homogeneous Bose gas. Here, we derive a novel estimate for low energy eigenfunctions stating that the probability for finding $l$ particles out of their total number N not in the condensate is exponentially small in the number $l$. Using this bound, we then prove that the ground state wave function of the microscopic model satisfies certain quasifree type properties. The exponential decay is moreover used to provide an alternative proof for the validity of Bogoliubov's approximation for the low-lying energy eigenvalues. Bogoliubov theory states that the excitation energies of the Bose gas are given by excitations of free quasiparticles obeying an effective energy-momentum dispersion relation which is linear for small momentum. The linearity of the effective dispersion relation is an essential ingredient for the explanation of the superfluid character of the Bose gas. In Chapter four we study the time evolution of a single particle coupled through a pair potential to a dense and homogeneous ideal Fermi gas in two spatial dimensions. This type of model is well known in plasma physics where it is used to describe the energy loss of ions moving through a dense and degenerate electron gas. We analyze the model for a coupling parameter g=1 and prove closeness of the time evolution to an effective dynamics for large densities of the gas and for long time scales of the order of some power of the density. The effective dynamics is generated by the free Hamiltonian with a large but constant energy shift. To leading order, this energy shift is given by the spatially homogeneous mean field potential produced by the gas particles, whereas at next-to-leading order, one has to consider an additional correction to the mean field energy which is due to so-called recollision processes.

•Jan 1, 2017•3 citations

Equilibrium and Non-Equilibrium Properties of Dilute Bose Gases

Oldenburg, Jakob Felix

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 $L^2$-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. $\textit{Full-text embargoed until: 2026-01-01}$

•Dec 9, 2025•0 citations

Number-conserving model for boson pairing

Stefano Fantoni, T. M. Nguyen, Subodh R. Shenoy +1 more

An independent-pair ansatz is developed for the many-body wave function of dilute Bose systems. The pair correlation is optimized by minimizing the expectation value of the full Hamiltonian (rather than the truncated Bogoliubov one) providing a rigorous energy upper bound. In contrast with the Jastrow model, hypernetted chain theory provides closed-form exactly solvable equations for the optimized pair correlation. The model involves both condensate and coherent pairing with number conservation and kinetic-energy sum rules satisfied exactly and the compressibility sum rule obeyed at low density. We compute, for bulk boson matter at a given density and zero temperature, (i) the two-body distribution function, (ii) the energy per particle, (iii) the sound velocity, (iv) the chemical potential, (v) the momentum distribution and its condensate fraction, and (vi) the pairing function, which quantifies the off-diagonal long-range order resulting from the structural properties of the two-particle density matrix. The connections with the low-density expansion and Bogoliubov theory are analyzed at different density values, including the density and scattering length regime of interest of trapped-atom Bose-Einstein condensates. Comparison with the available diffusion Monte Carlo results is also made.

arXiv:cond-mat/0205282•Physical Review A•Sep 11, 2002•6 citations
Many useful properties of dilute Bose gases at ultra-low temperature are predicted precisely by the product-state ansatz, in which all particles are in the same quantum state. As particle-particle correlations become important, however, the product ansatz begins to fail. We consider a new set of states, which constitute a natural generalization of the product ansatz; the particle-correlated state of N=l×nN=l\times nN=l×n identical particles is derived by symmetrizing the nnn-fold product of an lll-particle quantum state. The particle-correlated states can be simulated efficiently for large NNN, because their parameter spaces, which depend on lll, do not grow with nnn. Here we pay special attention to the pure-state case for l=2l=2l=2, where the many-body state is constructed from a two-particle pure state. These paired wave functions for bosons, which we call pair-correlated states (PCS), were introduced by Leggett [Rev. Mod. Phys. 73\bf 7373, 307 (2001)] as a particle-number-conserving version of the Bogoliubov approximation. We present an iterative algorithm that solves the reduced (marginal) density matrices (RDMs)---these are the correlation functions---associated with PCS in time O(N)O(N)O(N). The RDMs can also be derived from the normalization factor of PCS, which is derived analytically in the large NNN limit. To test the efficacy of our theory, we analyze the ground state of the two-site Bose-Hubbard model by minimizing the energy of the PCS state, both in its exact form and in its large-NNN approximate form. The relative errors of the ground state energy for both cases are within 10−510^{-5}10−5 for N=1000N = 1000N=1000 particles over the entire parameter region from a single condensate to a Mott insulator. Moreover, we present numerical results that suggest that PCS might be useful for describing the dynamics in the strongly interacting regime.
Many useful properties of dilute Bose gases at ultralow temperature are predicted precisely by the (mean-field) product-state Ansatz, in which all particles are in the same quantum state. Yet, in situations where particle-particle correlations become important, the product Ansatz fails. To include correlations nonperturbatively, we consider a new set of states: the particle-correlated state of $N=l\ifmmode\times\else\texttimes\fi{}n$ bosons is derived by symmetrizing the nnn-fold product of an lll-particle quantum state. Quantum correlations of the lll-particle state ``spread out'' to any subset of the NNN bosons by symmetrization. The particle-correlated states can be simulated efficiently for large NNN, because their parameter spaces, which depend on lll, do not grow with nnn. Here we formulate and develop in great detail the pure-state case for l=2l=2l=2, where the many-body state is constructed from a two-particle pure state. These paired wave functions, which we call pair-correlated states (PCS), were introduced by A. J. Leggett [Rev. Mod. Phys. 73, 307 (2001)] as a particle-number-conserving version of the Bogoliubov approximation. We present an iterative algorithm that solves for the reduced (marginal) density matrices (RDMs), i.e., the correlation functions, associated with PCS in time O(N)O(N)O(N). The RDMs can also be derived from the normalization factor of PCS, which is derived analytically in the large-NNN limit. To test the efficacy of PCS, we analyze the ground state of the two-site Bose-Hubbard model by minimizing the energy of the PCS state, both in its exact form and in its large-NNN approximate form, and comparing with the exact ground state. For N=1000N=1000N=1000, the relative errors of the ground-state energy for both cases are within ${10}^{\ensuremath{-}5}$ over the entire parameter region from a single condensate to a Mott insulator. We present numerical results that suggest that PCS might be useful for describing the dynamics in the strongly interacting regime.
The impact of interparticle correlations on the behavior of Bose-Einstein Condensates (BECs) is discussed using two approaches. In the first approach, the wavefunction of a BEC is encoded in the NNN-particle sector of an extended "catalytic state". Going to a time-dependent interaction picture, we can organize the effective Hamiltonian by powers of N−1/2{N}^{-1/2}N−1/2. Requiring the terms of order N1/2{N}^{1/2}N1/2 to vanish, we get the Gross-Pitaevskii Equation. Going to the next order, N0N^0N0, we obtain the number-conserving Bogoliubov approximation. Our approach allows one to stay in the Schrödinger picture and to apply many techniques from quantum optics. Moreover, it is easier to track different orders in the Hamiltonian and to generalize to the multi-component case. In the second approach, I consider a state of N=l×nN =l\times nN=l×n bosons that is derived by symmetrizing the nnn-fold tensor product of an arbitrary lll-boson state. Particularly, we are interested in the pure state case for l=2l=2l=2, which we call the Pair-Correlated State (PCS). I show that PCS reproduces the number-conserving Bogoliubov approximation; moreover, it also works in the strong interaction regime where the Bogoliubov approximation fails. For the two-site Bose-Hubbard model, I find numerically that the error (measured by the trace distance of the two-particle reduced density matrices) of PCS is less than two percent over the entire parameter space, thus making PCS a bridge between the superfluid and Mott insulating phases. Amazingly, the error of PCS does not increase, in the time-dependent case, as the system evolves for longer times. I derive both time-dependent and -independent equations for the ground state and the time evolution of the PCS ansatz. The time complexity of simulating PCS does not depend on NNN and is linear in the number of orbitals in use.
Standard analytical construction of the many-body wave function of interacting particles in one dimension, beyond mean-field theory, is based on the Jastrow approach. The many-body interacting ground state is build up from the ground state of the non-interacting system and the product of solutions of the corresponding interacting two-body problem. However, this is possible only if the center-of-mass motion is decoupled from the mutual interactions. In our work, based on the general constraints given by contact nature of the atom-atom interactions, we present an alternative approach to the standard construction of the \textit{pair-correlation} wave-function. Within the proposed ansatz, we study the many-body properties of trapped bosons as well as fermionic mixtures and we compare these predictions with the exact diagonalization approach in a wide range of particle numbers, interaction strengths, and different trapping potentials.
This thesis is about the derivation of effective mean field equations and their next-order corrections starting from nonrelativistic many-body quantum theory. Mean field equations provide an approximate ansatz for the description of interacting many-particle systems. In this ansatz the interaction between the particles is replaced by a self-consistent external potential leading to an effective one-body description of the many-particle system. Next-order corrections provide an approximation which goes one step further and tries to capture also subleading effects that are not resolved by the mean field ansatz. We present mathematical proofs for the validity of such effective theories for different models that are motivated, e.g., from the theory of ultracold atoms (the bosonic Hartree equation and the corresponding Bogoliubov theory) and from plasma physics (the motion of a tracer particle through a degenerate and dense electron gas). Starting from a many-body Schrödinger equation, our goal is to show that the solutions converge in a particular limit to the solutions to an effective mean field equation and its next-order corrections. After a short introduction and a summary in Chapter one, we present the main part of this work in three self-contained chapters. In Chapter two we analyze the dynamics of a large number N of nonrelativistic bosons in the weak coupling limit, i.e., for a coupling constant g_N=1/N. It is well known that in the limit of infinite particle number, the Hartree equation emerges as an effective one-particle theory of the Bose gas. This is closely related to the remarkable physical phenomenon of Bose-Einstein condensation at low temperature, namely that the majority of particles in a Bose gas occupies the same copy of a single one-particle quantum state. Our emphasis lies in the description of the few particles that fluctuate around the Bose-Einstein condensate. We show convergence of the fully interacting dynamics to an auxiliary time evolution in the norm of the N-particle space. This result allows us to prove several other assertions. Among other things, it is used to derive the Hartree equation with optimal speed of convergence 1/N for initial states that are close to ground states of interacting systems and also to prove convergence of the N-particle solution towards a time evolution obtained from the Bogoliubov Hamiltonian on Fock space. Chapter three is about the low energy properties of the weakly interacting homogeneous Bose gas. Here, we derive a novel estimate for low energy eigenfunctions stating that the probability for finding lll particles out of their total number N not in the condensate is exponentially small in the number lll. Using this bound, we then prove that the ground state wave function of the microscopic model satisfies certain quasifree type properties. The exponential decay is moreover used to provide an alternative proof for the validity of Bogoliubov's approximation for the low-lying energy eigenvalues. Bogoliubov theory states that the excitation energies of the Bose gas are given by excitations of free quasiparticles obeying an effective energy-momentum dispersion relation which is linear for small momentum. The linearity of the effective dispersion relation is an essential ingredient for the explanation of the superfluid character of the Bose gas. In Chapter four we study the time evolution of a single particle coupled through a pair potential to a dense and homogeneous ideal Fermi gas in two spatial dimensions. This type of model is well known in plasma physics where it is used to describe the energy loss of ions moving through a dense and degenerate electron gas. We analyze the model for a coupling parameter g=1 and prove closeness of the time evolution to an effective dynamics for large densities of the gas and for long time scales of the order of some power of the density. The effective dynamics is generated by the free Hamiltonian with a large but constant energy shift. To leading order, this energy shift is given by the spatially homogeneous mean field potential produced by the gas particles, whereas at next-to-leading order, one has to consider an additional correction to the mean field energy which is due to so-called recollision processes.
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
An independent-pair ansatz is developed for the many-body wave function of dilute Bose systems. The pair correlation is optimized by minimizing the expectation value of the full Hamiltonian (rather than the truncated Bogoliubov one) providing a rigorous energy upper bound. In contrast with the Jastrow model, hypernetted chain theory provides closed-form exactly solvable equations for the optimized pair correlation. The model involves both condensate and coherent pairing with number conservation and kinetic-energy sum rules satisfied exactly and the compressibility sum rule obeyed at low density. We compute, for bulk boson matter at a given density and zero temperature, (i) the two-body distribution function, (ii) the energy per particle, (iii) the sound velocity, (iv) the chemical potential, (v) the momentum distribution and its condensate fraction, and (vi) the pairing function, which quantifies the off-diagonal long-range order resulting from the structural properties of the two-particle density matrix. The connections with the low-density expansion and Bogoliubov theory are analyzed at different density values, including the density and scattering length regime of interest of trapped-atom Bose-Einstein condensates. Comparison with the available diffusion Monte Carlo results is also made.