arXiv:cond-mat/9701038 · Physical Review A

Bose-Einstein condensation in a trap: The case of a dense condensate

K. Ziegler, Alok Shukla

Published Aug 1, 1997• 26 citations

Abstract

We consider the Bose-Einstein condensation of atoms in a trap where the density of particles is so high that the low-density approach of Gross [J. Math. Phys. (N.Y.) 4, 195 (1963)] and Pitaevskii $($V. L. Ginsburg and L. P. Pitaevskii, Zh. \'Eksp. Teor. Fiz. 34, 1240 (1958) [Sov. Phys. JETP 7, 1858 (1958)]$)$ will not be applicable. For this purpose we use the slave boson representation which is valid for hard-core bosons at any density. This description leads to the same results as the Gross-Pitaevskii approach in the low-density limit, but for higher densities, it predicts the depletion of the condensate in the regions where the density of the atomic cloud is high.

DOI: 10.1103/physreva.56.1438

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Natural orbitals and Bose-Einstein condensates in traps: A diffusion Monte Carlo analysis

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BOSE-EINSTEIN CONDENSATION IN TRAPS: A QUANTUM MONTE CARLO STUDY

Jonathan L. DuBois

Jan 1, 20021 citation

Overview and introduction

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Cambridge University Press eBooksFeb 19, 20092 citations

A slave boson model for two-dimensional trapped Bose–Einstein condensate

K K Rajagopal

A system of N bosons in a two-dimensional harmonic trap is considered. The system is treated in term of the slave boson representation for hard-core bosons which is valid in the arbitrary density regimes. I discuss the consequences of higher order interactions on the density profiles by mapping the slave boson equation to the known Kohn-Sham type equation within the density functional scheme.

arXiv:cond-mat/0504267Journal of Physics B Atomic Molecular and Optical PhysicsJun 15, 20051 citation

Equilibrium and Non-Equilibrium Properties of Dilute Bose Gases

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Dec 9, 20250 citations

Bose-Einstein condensates in the large-gas-parameter regime

Adelchi Fabrocini, A. Polls

arXiv:cond-mat/0101395Physical Review ANov 16, 200138 citations

arXiv:cond-mat/9701038 · Physical Review A

Bose-Einstein condensation in a trap: The case of a dense condensate

K. Ziegler, Alok Shukla

Published Aug 1, 1997• 26 citations

arXiv HTML PDF

Abstract

We consider the Bose-Einstein condensation of atoms in a trap where the density of particles is so high that the low-density approach of Gross [J. Math. Phys. (N.Y.) 4, 195 (1963)] and Pitaevskii

(

V. L. Ginsburg and L. P. Pitaevskii, Zh. 'Eksp. Teor. Fiz. 34, 1240 (1958) [Sov. Phys. JETP 7, 1858 (1958)]

)

will not be applicable. For this purpose we use the slave boson representation which is valid for hard-core bosons at any density. This description leads to the same results as the Gross-Pitaevskii approach in the low-density limit, but for higher densities, it predicts the depletion of the condensate in the regions where the density of the atomic cloud is high.

DOI: 10.1103/physreva.56.1438

Related papers

Natural orbitals and Bose-Einstein condensates in traps: A diffusion Monte Carlo analysis

Jonathan L. DuBois, H. R. Glyde

We investigate the properties of hard-core bosons in harmonic traps over a wide range of densities. Bose-Einstein condensation is formulated using the one-body density matrix (OBDM) which is equally valid at low and high densities. The OBDM is calculated using diffusion Monte Carlo methods and it is diagonalized to obtain the ``natural'' single-particle orbitals and their occupation, including the condensate fraction. At low boson density, ${\mathrm{na}}^{3}<{10}^{\ensuremath{-}5},$ where $n=N/V$ and a is the hard-core diameter, the condensate is localized at the center of the trap. As ${\mathrm{na}}^{3}$ increases, the condensate moves to the edges of the trap. At high density, it is localized at the edges of the trap. At ${\mathrm{na}}^{3}<~{10}^{\ensuremath{-}4},$ the Gross-Pitaevskii theory of the condensate describes the whole system within $1%.$ At ${\mathrm{na}}^{3}\ensuremath{\approx}{10}^{\ensuremath{-}3},$ corrections are $3%$ to the Grass-Pitaevskii energy but $30%$ to the Bogoliubov prediction of the condensate depletion. At ${\mathrm{na}}^{3}\ensuremath{\gtrsim}{10}^{\ensuremath{-}2},$ mean-field theory fails. At ${\mathrm{na}}^{3}\ensuremath{\gtrsim}0.1,$ the bosons behave more like a liquid ${}^{4}\mathrm{He}$ droplet than a trapped boson gas.We investigate the properties of hard-core bosons in harmonic traps over a wide range of densities. Bose-Einstein condensation is formulated using the one-body density matrix (OBDM) which is equally valid at low and high densities. The OBDM is calculated using diffusion Monte Carlo methods and it is diagonalized to obtain the ``natural'' single-particle orbitals and their occupation, including the condensate fraction. At low boson density, ${\mathrm{na}}^{3}<{10}^{\ensuremath{-}5},$ where $n=N/V$ and a is the hard-core diameter, the condensate is localized at the center of the trap. As ${\mathrm{na}}^{3}$ increases, the condensate moves to the edges of the trap. At high density, it is localized at the edges of the trap. At ${\mathrm{na}}^{3}<~{10}^{\ensuremath{-}4},$ the Gross-Pitaevskii theory of the condensate describes the whole system within $1%.$ At ${\mathrm{na}}^{3}\ensuremath{\approx}{10}^{\ensuremath{-}3},$ corrections are $3%$ to the Grass-Pitaevskii energy but $30%$ to the Bogoliubov prediction of the condensate depletion. At ${\mathrm{na}}^{3}\ensuremath{\gtrsim}{10}^{\ensuremath{-}2},$ mean-field theory fails. At ${\mathrm{na}}^{3}\ensuremath{\gtrsim}0.1,$ the bosons behave more like a liquid ${}^{4}\mathrm{He}$ droplet than a trapped boson gas.

arXiv:cond-mat/0303426Physical Review ASep 10, 200339 citations

BOSE-EINSTEIN CONDENSATION IN TRAPS: A QUANTUM MONTE CARLO STUDY

Jonathan L. DuBois

We evaluate the zero temperature properties and Bose-Einstein condensation (BEC) of hard sphere bosons in a harmonic trap using Quantum Monte Carlo methods. The chief purpose of this work is to go beyond the dilute limit, to test the limits of the Gross-Pitaevskii equation and related mean field approximations and to explore the role of interactions in determining the zero temperature properties of the trapped Bose gas. The density is increased by adjusting both the number of trapped bosons, N , and the s-wave scattering length, a, to study systems from the highly dilute region corresponding to early experiments in trapped BEC [2, 3] up to liquid He densities and beyond. Rather than relying on the perturbative Bogoliubov approximation (which assumes a large condensate fraction) to describe the condensate, we use the one-body density matrix (OBDM) formulation of BEC so that the properties of the condensate for systems with arbitrarily large interactions may be studied. In this formulation of BEC, condensate properties are obtained by diagonalizing the OBDM and obtaining the corresponding single particle “natural orbitals” and their occupation numbers for the system. The condensate wave function and condensate fraction are then obtained from the single particle orbital(s) with macroscopic occupation (N0 >> 1). Within this framework, we calculate the effects of interactions and increased density on the ground state energy, the density profile, the momentum distribution, the condensate fraction and condensate “wave-function” and several other properties.We evaluate the zero temperature properties and Bose-Einstein condensation (BEC) of hard sphere bosons in a harmonic trap using Quantum Monte Carlo methods. The chief purpose of this work is to go beyond the dilute limit, to test the limits of the Gross-Pitaevskii equation and related mean field approximations and to explore the role of interactions in determining the zero temperature properties of the trapped Bose gas. The density is increased by adjusting both the number of trapped bosons, N , and the s-wave scattering length, a, to study systems from the highly dilute region corresponding to early experiments in trapped BEC [2, 3] up to liquid He densities and beyond. Rather than relying on the perturbative Bogoliubov approximation (which assumes a large condensate fraction) to describe the condensate, we use the one-body density matrix (OBDM) formulation of BEC so that the properties of the condensate for systems with arbitrarily large interactions may be studied. In this formulation of BEC, condensate properties are obtained by diagonalizing the OBDM and obtaining the corresponding single particle “natural orbitals” and their occupation numbers for the system. The condensate wave function and condensate fraction are then obtained from the single particle orbital(s) with macroscopic occupation (N0 >> 1). Within this framework, we calculate the effects of interactions and increased density on the ground state energy, the density profile, the momentum distribution, the condensate fraction and condensate “wave-function” and several other properties.

Jan 1, 20021 citation

Overview and introduction

Allan Griffin, Tetsuro Nikuni, E. Zaremba

Since the dramatic discovery of Bose–Einstein condensation (BEC) in trapped atomic gases in 1995 (Anderson et al. , 1995), there has been an explosion of theoretical and experimental research on the properties of Bose-condensed dilute gases. The first phase of this research was discussed in the influential review article by Dalfovo et al. (1999) and in the proceedings of the 1998 Varenna Summer School on BEC (Inguscio et al. , 1999). More recently, this research has been well documented in two monographs, by Pethick and Smith (2008, second edition) and by Pitaevskii and Stringari (2003). Most of this research, both experimental and theoretical, has concentrated on the case of low temperatures (well below the BEC transition temperature, T BEC ), where one is effectively dealing with a pure Bose condensate. The total fraction of noncondensate atoms in such experiments can be as small as 10% of the total number of atoms and, equally importantly, this low-density cloud of thermally excited atoms is spread over a much larger spatial region compared with the high-density condensate, which is localized at the centre of the trapping potential. Thus most studies of Bose-condensed gases at low temperatures have concentrated entirely on the condensate degree of freedom and its response to various perturbations. This region is well described by the famous Gross–Pitaevskii (GP) equation of motion for the condensate order parameter Φ( r , t ). As shown by research since 1995, this pure condensate domain is very rich in physics. The main goal of the present book, in contrast, is to describe the dynamics of dilute trapped atomic gases at finite temperatures such that the noncondensate atoms also play an important role.Since the dramatic discovery of Bose–Einstein condensation (BEC) in trapped atomic gases in 1995 (Anderson et al. , 1995), there has been an explosion of theoretical and experimental research on the properties of Bose-condensed dilute gases. The first phase of this research was discussed in the influential review article by Dalfovo et al. (1999) and in the proceedings of the 1998 Varenna Summer School on BEC (Inguscio et al. , 1999). More recently, this research has been well documented in two monographs, by Pethick and Smith (2008, second edition) and by Pitaevskii and Stringari (2003). Most of this research, both experimental and theoretical, has concentrated on the case of low temperatures (well below the BEC transition temperature, T BEC ), where one is effectively dealing with a pure Bose condensate. The total fraction of noncondensate atoms in such experiments can be as small as 10% of the total number of atoms and, equally importantly, this low-density cloud of thermally excited atoms is spread over a much larger spatial region compared with the high-density condensate, which is localized at the centre of the trapping potential. Thus most studies of Bose-condensed gases at low temperatures have concentrated entirely on the condensate degree of freedom and its response to various perturbations. This region is well described by the famous Gross–Pitaevskii (GP) equation of motion for the condensate order parameter Φ( r , t ). As shown by research since 1995, this pure condensate domain is very rich in physics. The main goal of the present book, in contrast, is to describe the dynamics of dilute trapped atomic gases at finite temperatures such that the noncondensate atoms also play an important role.

Cambridge University Press eBooksFeb 19, 20092 citations

A slave boson model for two-dimensional trapped Bose–Einstein condensate

K K Rajagopal

arXiv:cond-mat/0504267Journal of Physics B Atomic Molecular and Optical PhysicsJun 15, 20051 citation

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. $\textit{Full-text embargoed until: 2026-01-01}$ 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, 20250 citations

Bose-Einstein condensates in the large-gas-parameter regime

Adelchi Fabrocini, A. Polls

Bose-Einstein condensates of ${10}^{4}{}^{85}\mathrm{Rb}$ atoms in a cylindrical trap are studied using a recently proposed approach based on a local-density approximation. Since the existence of a Feshbach resonance allows for widely tuning the scattering length of the atoms, values of the peak gas parameter ${x}_{\mathrm{pk}}$ of the order of ${10}^{\ensuremath{-}2}$ may be attained and the standard analysis based on Gross-Pitaevskii and/or Thomas-Fermi equations may result in being questionable. Energy functionals derived from the correlated basis functions theory and the low-density expansion for a homogeneous hard-spheres gas are used to estimate corrections to the Gross-Pitaevskii equation. The two functionals give similar results, both showing large differences with respect to the Thomas-Fermi and Gross-Pitaevskii ones. The column densities at $z=0$ may differ by as much as $\ensuremath{\sim}30%$ and the half maximum radius by $\ensuremath{\sim}20%.$ The scattering lengths estimated by fitting the half maximum radius may differ by $\ensuremath{\sim}40%$ from those given by a Thomas-Fermi-based analysis of the experimental data.Bose-Einstein condensates of ${10}^{4}{}^{85}\mathrm{Rb}$ atoms in a cylindrical trap are studied using a recently proposed approach based on a local-density approximation. Since the existence of a Feshbach resonance allows for widely tuning the scattering length of the atoms, values of the peak gas parameter ${x}_{\mathrm{pk}}$ of the order of ${10}^{\ensuremath{-}2}$ may be attained and the standard analysis based on Gross-Pitaevskii and/or Thomas-Fermi equations may result in being questionable. Energy functionals derived from the correlated basis functions theory and the low-density expansion for a homogeneous hard-spheres gas are used to estimate corrections to the Gross-Pitaevskii equation. The two functionals give similar results, both showing large differences with respect to the Thomas-Fermi and Gross-Pitaevskii ones. The column densities at $z=0$ may differ by as much as $\ensuremath{\sim}30%$ and the half maximum radius by $\ensuremath{\sim}20%.$ The scattering lengths estimated by fitting the half maximum radius may differ by $\ensuremath{\sim}40%$ from those given by a Thomas-Fermi-based analysis of the experimental data.

arXiv:cond-mat/0101395Physical Review ANov 16, 200138 citations

We consider the Bose-Einstein condensation of atoms in a trap where the density of particles is so high that the low-density approach of Gross [J. Math. Phys. (N.Y.) 4, 195 (1963)] and Pitaevskii

(

V. L. Ginsburg and L. P. Pitaevskii, Zh. 'Eksp. Teor. Fiz. 34, 1240 (1958) [Sov. Phys. JETP 7, 1858 (1958)]

)

n=N/V

and a is the hard-core diameter, the condensate is localized at the center of the trap. As

{\mathrm{na}}^{3}

increases, the condensate moves to the edges of the trap. At high density, it is localized at the edges of the trap. At ${\mathrm{na}}^{3}<~{10}^{\ensuremath{-}4},$ the Gross-Pitaevskii theory of the condensate describes the whole system within

1%.

At ${\mathrm{na}}^{3}\ensuremath{\approx}{10}^{\ensuremath{-}3},$ corrections are

3%

to the Grass-Pitaevskii energy but

30%

to the Bogoliubov prediction of the condensate depletion. At ${\mathrm{na}}^{3}\ensuremath{\gtrsim}{10}^{\ensuremath{-}2},$ mean-field theory fails. At ${\mathrm{na}}^{3}\ensuremath{\gtrsim}0.1,$ the bosons behave more like a liquid

{}^{4}\mathrm{He}

droplet than a trapped boson gas.

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}

Bose-Einstein condensates of

{10}^{4}{}^{85}\mathrm{Rb}

atoms in a cylindrical trap are studied using a recently proposed approach based on a local-density approximation. Since the existence of a Feshbach resonance allows for widely tuning the scattering length of the atoms, values of the peak gas parameter

{x}_{\mathrm{pk}}

of the order of ${10}^{\ensuremath{-}2}$ may be attained and the standard analysis based on Gross-Pitaevskii and/or Thomas-Fermi equations may result in being questionable. Energy functionals derived from the correlated basis functions theory and the low-density expansion for a homogeneous hard-spheres gas are used to estimate corrections to the Gross-Pitaevskii equation. The two functionals give similar results, both showing large differences with respect to the Thomas-Fermi and Gross-Pitaevskii ones. The column densities at

z=0

may differ by as much as $\ensuremath{\sim}30%$ and the half maximum radius by $\ensuremath{\sim}20%.$ The scattering lengths estimated by fitting the half maximum radius may differ by $\ensuremath{\sim}40%$ from those given by a Thomas-Fermi-based analysis of the experimental data.