It is shown that in the presence of Bose–Einstein condensation (BEC) in any N particle system, the N particle Schrödinger wave functions of thermally occupied states are the sum of a 'localized' component and a 'delocalized' component, identical to the ground state wave function. It is shown that if N is sufficiently large, this implies that all physical properties of the system are the sum of two independent contributions from these two components. These results are used here to provide quantitative explanations of fundamental properties of BE condensed liquid 4 He, unexplained even qualitatively by existing theory; why BE condensed liquid 4 He is the only known physical system in which pair correlations between atomic positions reduce as it is cooled, why it is the only known liquid with sharp peaks in its dynamic structure factor, why the liquid expands with cooling and how the condensate fraction is related to the superfluid fraction. It is shown that these results also provide a relatively simple, physically transparent and quantitative explanation from first principles of macroscopic quantum effects. A new algorithm is given for the calculation of the time development of the macroscopic density of any BE condensed liquid or gas at any temperature. Unlike the Gross–Pitaevskii equation, this algorithm is valid for both strongly and weakly interacting systems. It is used here to show that macroscopic quantum interference fringes, observed between overlapping clouds of BE condensed atoms, are a necessary consequence of BEC and the N particle Schrödinger equation for the atoms in the clouds. It follows that the widely held view that these fringes are created by measurement is unnecessary. New, experimentally testable predictions are made of how the visibility of these fringes will vary with temperature.
It is shown that in the presence of Bose–Einstein condensation (BEC) in any N particle system, the N particle Schrödinger wave functions of thermally occupied states are the sum of a 'localized' component and a 'delocalized' component, identical to the ground state wave function. It is shown that if N is sufficiently large, this implies that all physical properties of the system are the sum of two independent contributions from these two components. These results are used here to provide quantitative explanations of fundamental properties of BE condensed liquid 4 He, unexplained even qualitatively by existing theory; why BE condensed liquid 4 He is the only known physical system in which pair correlations between atomic positions reduce as it is cooled, why it is the only known liquid with sharp peaks in its dynamic structure factor, why the liquid expands with cooling and how the condensate fraction is related to the superfluid fraction. It is shown that these results also provide a relatively simple, physically transparent and quantitative explanation from first principles of macroscopic quantum effects. A new algorithm is given for the calculation of the time development of the macroscopic density of any BE condensed liquid or gas at any temperature. Unlike the Gross–Pitaevskii equation, this algorithm is valid for both strongly and weakly interacting systems. It is used here to show that macroscopic quantum interference fringes, observed between overlapping clouds of BE condensed atoms, are a necessary consequence of BEC and the N particle Schrödinger equation for the atoms in the clouds. It follows that the widely held view that these fringes are created by measurement is unnecessary. New, experimentally testable predictions are made of how the visibility of these fringes will vary with temperature.
The basic assumption of this paper is that in the presence of Bose-Einstein condensation (BEC) in liquid 4He, the wave functions of occupied many particle states are the superposition of two components; a phase coherent component, proportional to the ground state wave function and a phase incoherent component. It is shown that this assumption satisfies necessary conditions imposed by the presence of BEC and that the wave functions of the ideal Bose gas and wave functions of the Bijl-Feynman type are of this form. It is shown that this single assumption provides simple microscopic explanations of essentially all the exotic properties of helium II; why BEC implies two fluid behavior, how the superfluid and condensate fractions are linked, how Landau theory is linked to the presence of BEC, why the superfluid exhibits flow without viscosity and macroscopic quantum effects while the normal fluid does not, how the anomalous expansion and loss of spatial order, observed in helium II as it is cooled, is linked to BEC, and how the presence of sharp peaks observed in the dynamic structure factor are linked with BEC. The theory provides predictions that are in quantitative agreement with a wide range of presently unexplained experimental data on helium II.
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 L2-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
It is shown that any Bose-Einstein-condensed fluid in its ground state will exhibit macroscopic single-particle quantum behavior (MSPQB). That is, (1) the many-particle wave function $\ensuremath{\Psi}({\mathbf{r}}_{1},\dots{},{\mathbf{r}}_{n})$ factors into a single-particle product ${\ensuremath{\prod}}_{n}\ensuremath{\eta}({\mathbf{r}}_{n})$; (2) the function $\ensuremath{\eta}(\mathbf{r})$ extends over macroscopic length scales and obeys the usual quantum equations for particle flux in a single-particle system; and (3) $\ensuremath{\eta}(\mathbf{r})$ obeys a nonlinear single-particle Schr\"odinger equation. The latter equation reduces to the Gross-Pitaevskii equation when interactions are weak and determines the density distribution of the fluid and the time development of this distribution. The arguments used rely only on elementary concepts of probability theory and many-particle wave mechanics and are valid even in strongly interacting fluids such as superfluid 4He. It is shown that Bose-Einstein condensation implies that the N-particle wave function $\ensuremath{\Psi}$ is delocalized. That is, if one considers a single-particle coordinate r, then for all values that occur of the other $N\ensuremath{-}1$ coordinates, $\ensuremath{\Psi}$ is a nonzero function of r over a region of space proportional to V, where V is the total volume within which the fluid is contained. MSPQB is a consequence of this delocalization and the absence of long-range correlations between particle positions in fluids. The results are accurate provided that only averages over regions of space containing many particles are considered. For averages over volumes of space containing ${N}_{\ensuremath{\Omega}}$ particles, inaccuracies due to quantum fluctuations are $\ensuremath{\sim}1∕\sqrt{{N}_{\ensuremath{\Omega}}}$.
In the present chapter we consider the structure of the Bose–Einstein condensed state in the presence of interactions. Our discussion is based on the Gross–Pitaevskii equation, which describes the zero-temperature properties of the non-uniform Bose gas when the scattering length a is much less than the mean interparticle spacing. We shall first derive the Gross–Pitaevskii equation at zero temperature by treating the interaction between particles in a mean-field approximation (Sec. 6.1). Following that, in Sec. 6.2 we discuss the ground state of atomic clouds in a harmonic-oscillator potential. We compare results obtained by variational methods with those derived in the Thomas-Fermi approximation, in which the kinetic energy operator is neglected in the Gross–Pitaevskii equation. The Thomas–Fermi approximation fails near the surface of a cloud, and in Sec. 6.3 we calculate the surface structure using the Gross–Pitaevskii equation. Finally, in Sec. 6.4 we determine how the condensate wave function ‘heals’ when subjected to a localized disturbance. The Gross–Pitaevskii equation In the previous chapter we have shown that the effective interaction between two particles at low energies is a constant in the momentum representation, U 0 = 4πħ 2 a / m . In coordinate space this corresponds to a contact interaction U 0 δ( r – r ′), where r and r ′ are the positions of the two particles. To investigate the energy of man y-body states we adopt a Hartree or mean-field approach, and assume that the wave function is a symmetrized product of single-particle wave functions.
The properties of the many-particle Schr\"odinger wave function \ensuremath{\Psi} are examined in the presence of Bose-Einstein condensation (BEC). It is shown that it is possible to define, in terms of \ensuremath{\Psi}, a function $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s}),$ which can be regarded as the single-particle wave function of an arbitrary particle for a fixed configuration $\stackrel{\ensuremath{\rightarrow}}{s}$ of all other particles. It is shown that $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ plays an analogous role to the field operator of standard field-theoretical treatments of superfluidity. It is shown that in the presence of a Bose-Einstein condensate fraction f, $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ must be nonzero and phase coherent within at least a fraction f of the total volume of the N-particle system for essentially all $\stackrel{\ensuremath{\rightarrow}}{s}.$ Examination of the form of variational many-particle wave functions shows that in liquid 4He, $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ extends throughout the spaces left between the hard cores of the other atoms at $\stackrel{\ensuremath{\rightarrow}}{s}.$ By contrast, in the absence of BEC, $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ in the ground state must be nonzero only over a localized region of space. It is shown that in order for long-range phase coherence in $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ to be maintained in the presence of velocity fields, any circulation must be quantized over macroscopic length scales. Some numerical calculations of the properties and fluctuations of liquid helium are presented. These suggest that the approach outlined in this paper may have significant advantages for the numerical calculations of the properties of Bose-Einstein condensed systems. The properties of $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ are used to show that there is no general connection between the static structure factor and the size of the Bose-Einstein condensate fraction in a Bose fluid. It is suggested that the observed connection in liquid 4He is due to the creation of vacancies in the liquid structure, which are required so that $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{r}|\stackrel{\ensuremath{\rightarrow}}{s})$ can delocalize, in the presence of hard-core interactions. It is shown that this suggestion is quantitatively consistent with observations on liquid 4He. It is suggested that the superfluid transition in 4He occurs when the vacancies in the liquid structure form a connected network.
Bose-Einstein condensate (BEC) is a new state of matter, experimentally realized in a dilute alkali gas of weakly interacting bosons. The gas is cooled to very low temperatures, of the order of nano-Kelvin, at which point a large fraction of the atoms occupy the lowest quantum state. Under such conditions, the quantum effects become apparent on a macroscopic scale. Many fundamental ideas
of quantum mechanics have been realized and proven to be true in experiments,
by studying the Bose-Einstein condensates. Amongst them, the observation of
macroscopic matter wave interference, the creation of vortices, dark and bright
solitons etc, find special mention. The dynamics of this system in presence of an
optical lattice reveals a superfluid to Mott insulator quantum phase transition.
This has opened a new avenue for research on BEC and its applications. From
a theoretical standpoint, the dynamics of BEC can be described by means of an
effective mean-field theory. This formalism is much simpler than the many-body
Schr¨odinger equation and based on a classical nonlinear evolution equation, known
as Gross-Pitaevskii equation. The nonlinear nature of the GPE leads to several interesting phenomena like soliton propagation, vortices, four wave mixing etc.,
which are closely related to nonlinear fiber optics.
The present thesis deals with nonlinear excitations and different phases of BEC
and is organized as follows: after a brief introduction, in chapter two, we start
with BEC in a one-dimensional shallow optical lattice, experiencing both cubic
and quintic nonlinearity. A superfluid density wave, of twice the periodicity of
lattice potential, is identified. This indicates matter redistribution, where atoms
from neighbouring sites have been depleted leading to a density wave behaviour.
Interestingly, it ceases to exist when only one of these interactions is operative.
We predict the loss of superfluidity through a classical dynamical phase transition,
where modulational instability leads to the loss of phase coherence. In presence
of pure two-body interaction, the superfluid density shows a periodicity commensurate
with the lattice potential. Apart from the superfluid state, a density wave
insulating phase is also found to exist, possessing two frequency modulations commensurate
with the lattice potential. A dynamical phase transition connects the
superfluid phase with an insulating phase, which in turn, is connected with the
density wave insulating phase. The solutions are found to be marginally stable, as
per the Vakhitov-Kolokolov (VK) criterion.
Chapter three describes the dynamics of the atomic-molecular Bose-Einstein
condensate (AMBEC) by mean-field Gross-Pitaevskii equation (GPE), taking into
account the atom-molecular interaction, as well as atom-molecule inter-conversion.
The two-body atom-atom, atom-molecule and molecule-molecule interactions are
analogous to the cubic nonlinearity of the standard GPE, whereas, the quadratic
nonlinear term arising from the inter-conversion between atoms and molecules is
identical to the scenario in strongly coupled BEC in one dimension. The presence
of both weak and strong interactions adds further richness to this system. We
find the molecular BEC possesses a complex dark (grey) soliton, while the atomic condensate is a bright one. The energy and momentum of this system is computed
in order to see the effect of the both weak and strong interactions on Lieb mode
profile and compare it with the pure atomic condensate profile. Another class
of solutions exist in this system, where both atomic and molecular condensates
possess bright solitons.
In chapter four, we investigate the effect of a harmonic trap on the dynamics
of grey solitons in Bose-Einstein condensates. It is observed that, these nonlinear
excitations of the cigar-shaped Bose-Einstein condensate have strong coupling
with the trap at low momenta and hence can be effectively isolated from the
Bogoliubov sound modes, which respond weakly to harmonic confinement. This
strong coupling with the trap also makes the grey soliton amenable for control and
manipulation through trap modulation and temporal variation of the two-body interaction.
We have shown that in presence of an expulsive oscillator trap, the grey
soliton gets amplified, while having a compression in its width. Exact solutions are
also obtained in case of the two component Bose-Einstein condensates (TBECs),
where one component is a grey soliton and the other a bright one. The presence
of the trap and inter-species interaction significantly affects the grey soliton dynamics.
Similar to the single component, in the present case, we have shown that
both grey and bright soliton can be compressed simultaneously. A detailed study
of the energy and momentum of the soliton is reported, in order to find the effect
of the trap on the Lieb mode.
In chapter five, we study a BEC-defect complex in a quasi-one dimensional
scenario, where the defect atom moves in the condensate with constant velocity.
Explicit solutions of the coupled mean-field equations, describing defect-grey soliton
dynamics are obtained, demonstrating the coexistence of grey soliton and a
localized defect. Unlike the case of dark soliton, where the defect trapping center
has vanishing superfluid density, the moving grey soliton necessarily possesses a finite superfluid component at the defect location. The presence of the defect can
lower the grey soliton’s maximum velocity, as compared to the defect free case,
where the maximum velocity is the sound velocity. We have also carried out the
stability analysis of the obtained solutions using renormalized momentum formalism
and Vakhitov-Kolokolov criteria and find that the obtained solutions are stable
through both the methods. The grey soliton’s energy gets substantially modified
through its interaction with the defect, opening up the possibility of its control
through defect dynamics.
Chapter six describes a parametrically forced Bose-Einstein condensate in the
presence of a general moving optical lattice. The interaction between the atoms in
the condensate and the time dependent lattice potential leads to a novel propagating
superfluid matter wave, which can be controlled through chirp management.
This system, when placed in a trap, accelerates and undergoes rapid nonlinear
compression, controlled by the chirp. The density achieves its maximum, when
the matter wave changes direction. It is also shown that this system exhibits dynamical
superfluid insulator transition, where superfluidity breaks down and the
condensate transits to an insulating phase. The stability has been investigated
using Vakhitov-Kolokolov criterion and found that it depends on the chirp, as well
as on the nature of the atom-atom interaction. The exact expression for energy
is obtained and analyzed in detail to gain physical understanding of the chirp
management of the sinusoidal excitations and also the dynamical phase transition.