Introduction to Bose-Einstein condensationshodhganga.inflibnet.ac.in/bitstream/10603/5610/9/09_chapter 1.pdf · Introduction to Bose-Einstein condensation 6 andthecriticaltemperature

Chapter 1

Introduction to Bose-Einstein condensation

1.1 General introduction

The fact that identical particles are fundamentally indistinguishable can have pro-

found consequences for the behavior of collections of such particles at low temperature,

where their quantum mechanical nature is important. Physical observable must be

unaffected when any two identical particles are exchanged, so this operation can at

most introduce a global phase factor into the many-body wave function of the system.

Identical particles can therefore be divided into two groups depending on the sym-

metry of the wave function with respect to exchange. Particles for which the wave

function is anti-symmetric are known as fermions, while those for which it is symmet-

ric are bosons. Remarkably, this property of a many-body system is related by the

spin-statistics theorem of quantum field theory to an internal property of the individ-

ual particles concerned, namely their intrinsic angular momentum or spin. Fermions

are those particles for which this spin takes half integer values in units of the reduced

Planck’s constant ℎ, whereas bosons have integer spin. The simple fact that the

many-body wave function does or does not change sign when two identical particles

are exchanged can lead to dramatic differences in the behavior of systems of fermions

and bosons. For fermions, it leads to the Pauli’s exclusion principle, which forbids

any two particles to occupy the same quantum state. For bosons, on the other hand,

there is a tendency for the particles to cluster together, and under suitable conditions

1

Introduction to Bose-Einstein condensation 2

this can lead to the phenomenon of Bose-Einstein condensation in which there is a

macroscopic occupation of a single quantum state.

When collections of massive elementary particles are bound together, as in atoms

and molecules, their spin numbers gets added up resulting in a net effective spin. In

this way, atoms consisting of even numbers of elementary particles behave in a bosonic

fashion, provided the energies of the system under consideration are sufficiently low

to treat the atoms as point masses. The temperature close to absolute zero, from

the Fermi-Dirac distribution inferred that chemical potential is the energy of the

highest quantum orbital (Fermi energy), leading us to the conclusion that the chemical

potential is positive, but for the Bose-Einstein distribution the same argument doesn’t

apply. At temperatures close to absolute zero by imposing the physical conditions

that the energy must always be greater than or equal to zero and the average number

of particles must always be grater than zero, we can conclude that the chemical

potential must be negative. In general chemical potential is a negative quantity, but

in the special case of Bose-Einstein condensates (BEC) the chemical potential becomes

zero at a finite temperature. The temperature for which the chemical potential goes

to zero is called the transition temperature. Below this transition temperature a

fraction of bosons condense in to the lowest quantum state.

The statistical description of a collection of non-interacting bosons was first con-

sidered by Bose in 1924 in the context of photons and the Planck’s distribution [1]. In

1925, Einstein realized that for material particles whose number must be conserved,

these statistics (now known as Bose-Einstein statistics) could force the particles to

undergo a phase transition in which they form a macroscopic occupation of the low-

est energy level of the container [2]. This phase transition is known as Bose-Einstein

condensation, and it occurs when the temperature and density are such that the de

Broglie waves of the atoms begin to overlap. The topic of BEC in a uniform, non-

interacting gas of bosons is treated in most text books on statistical mechanics [3, 4].

Although Einstein’s prediction applied to a gas of noninteracting atoms, London sug-

gested in 1938 that BEC could be the mechanism underlying the phenomenon of su-

perfluidity in 4He [5]. Further evidence for this viewpoint came from experiments on


the fermionic 3He, which demonstrated that its low temperature properties are very

different from those of 4He. Today, BEC is believed to be the mechanism underlying

both superfluidity and superconductivity [6], although the theoretical description of

these systems is complicated by the presence of strong interactions between the par-

ticles. The significance of the recent experimental production of BEC in dilute gases

is that in these systems a quantitative comparison between theory and experiment is

possible. A detailed study can therefore be made of the strange properties of these

macroscopic quantum systems.

1.2 BEC in an ideal gas

Ideal-gas atoms are considered as non-interacting quantum mechanical version of a

classical ideal gas. It is composed of bosons, which have an integer value of spin,

and obey Bose-Einstein statistics. Considering, properties of a system of N nonin-

teracting particles (say bosonic atoms) of mass m, in thermodynamic equilibrium at

temperature T , the mean number of particles occupying a single quantum state of

energy �� is given by the Bose-Einstein distribution [7, 8]

f(��) =1

e(��−�c)/kBT − 1, (1.1)

where �c and kB corresponds to chemical potential and Boltzmann constant, respec-

tively. Note that the exponential part of this distribution function is bounded below

by 1, this function allows arbitrarily high occupancy of any state. This surprising

result allows for the possibility that certain number of massive particles may exist

simultaneously in a single quantum state. That is, more than one particle may be

described by exactly the same single-particle Schrodinger wave function. The physi-

cal implications of this statement indicate that two or more bosons are observed at

the same position. The chemical potential �c as a function of temperature and the

total number of particles, gives the dependence of Bose-Einstein distribution on N .

The chemical potential is the energy required to add a particle to the system while

keeping the entropy and volume are fixed. It is determined from the constraint that


the total number of particles in the system is fixed, so we have

N =∑

��

1

e(��−�c)/kBT − 1(1.2)

If the Bose-Einstein distribution varies slowly on the scale of the energy level spacing,

then the summation in Eq. (1.2) can be replaced by an integral over a density of

states. If �c → 0, however, the distribution has a singularity at �� = 0, which

signifies the possibility of the ground state to accommodate very large number of

particles. In addition, this state is actually neglected by the density of states, which

does not provide a good description of the lowest energy levels. The simplest way

to deal with these problems is it correct usage ground state contribution for special

treatment, and use a density of states for the remaining levels. Eq. (1.2) can therefore

be written as

N = N0 +

∫f(�)g(�)d� (1.3)

where N0 is the number of particles in the ground state and G(�) is the density of

states for the homogeneous gas in three-dimensional (3D) box and is given by [4]

G(�) =V

4�2

(2m

ℎ2

)3/2

�1/2 (1.4)

where V is the volume of the system. The integral over the density of states in

Eq. (1.3) gives the number of excited particles Nex, i.e. those which are not in the

ground state. Eq. (1.1) shows that the Bose-Einstein distribution is a monotonically

increasing function of both � and T . If the system is cooled, �c must therefore increase

so that the total number of particles is constant. Since we must also have �c < 0, we

can find maximum number of particles in the excited states by setting �c = 0 in the

integral of Eq. (1.3), we have

Nex = V �(3/2)

(mkBT

2�ℎ2

)3/2

(1.5)

where �(�) is the Riemann zeta function. For uniform ideal Bose gas in a 3D box, the

value of �(3/2) = 2.612. At high temperature �c ≪ 0, Nex > N and essentially all

the particles are in the excited states. As the system is cooled, however, �c increases


towards zero and we eventually reach a critical temperature Tc at which Nex = N .

From Eq. (1.5) the transition temperature Tc is given by

Tc ≈ 3.31

(ℎ2

mkB

)n2/3 (1.6)

where n = N/V is the particle number density. At the transition temperature, the

number of particles in the excited state are given by

Nex = N

(T

Tc

)3/2

. (1.7)

Below the critical temperature, the particles can no longer be accommodated in the

excited states, so further cooling results in the formation of a macroscopic population

of the lowest energy level. In this regime, the chemical potential is essentially fixed

at zero. Upon substitution Eq. (1.7) into Eq. (1.3), the condensate population varies

with temperature as

N0 = N −Nex = N

[1−

(T

Tc

)3/2]. (1.8)

Thus, below the critical temperature a finite fraction of all the particles occupy a

single state. This is one of the defining features of Bose-Einstein condensation.

The particle in a box model is simple to deal with, there are practical difficulties in

implementing exactly cubic traps experimentally. As such, its primary purpose lies in

providing the most basic theoretical illustration of BEC. An alternative model which is

no more complicated to deal with but far more practicable is obtained by replacing the

box containment with a radially symmetric harmonic potential, V (r) = 1

2m(!2

xx2 +

!2

yy2 + !

2

zz2). Rigorously speaking, for an ideal bose gas of N bosonic atoms in a

harmonic potential, the condition of phase-space density n(2�ℎ2/mkBT )3/2

> 2.612 is

replaced by N(ℎ!ℎo/kBT )3> 1.202 where !ℎo ≡ (!x!y!z)

1/3 is the geometric means

of the harmonic trapping frequencies. Using exactly the same arguments that are

usually employed to solve the box model, the density of states is

G(�) =1

2ℎ!ℎo

(�

ℎ!ℎo

)2

, (1.9)


and the critical temperature Tc ≈ 0.94(ℎ!kB

)N

1/3. Below Tc, the condensate popula-

tion varies with temperature as

N0 = N

[1−

(T

Tc

)3]. (1.10)

The above relation indicates that the number of particles condensed into the single

quantum state in harmonic potential varies with temperature.

Further intuitive understanding of BEC can be gained by considering the thermal

de Broglie wavelength �dB at near the transition temperature. The thermal de Broglie

wavelength is defined by �dB = (2�ℎ2/mkBT )1/2 and it is approximately equal to the

mean inter-particle spacing in the gas. Thus, BEC occurs when a gas of identical

massive bosons is cooled to a temperature where the average distance between ad-

jacent particles is of the order of the de Broglie wavelength of the particles. At the

transition temperature, Eq. (1.5) can be written as n�3dB = �(3/2). Accordingly,

BEC occurs at such low temperature that the dimensionless phase-space density,

� = n�3

dB, is greater than 2.612. As the gas is cooled beyond this value, the wave

functions rapidly fall into lock-step with each other until the majority of particles

are described by the same single-particle wave function. Note that the phase space

density can be increased by either increasing the density or by decreasing the average

velocity of the atoms in the ideal Bose gas. For homogeneous gas of non-interacting

atoms transition from gas phase to Bose-Einstein condensed phase occurs exactly at

� = �(3/2) ≈ 2.612 [4]. Therefore, increasing the phase-space density of cold atoms

is one of the principal diagnostic tools for determining the existence of a condensate

in current experiments. In addition, the finite size of the system and the effect of the

trap on the density of states allows BEC to occur in both one and two dimensions

[9]. Such lower dimensional systems can be formed in practice by increasing the trap

frequencies in one or more directions, thereby removing these degrees of freedom.


1.3 Experimental overview of BEC

Until recently the study of BEC in weakly interacting, dilute gases was essentially

a theoretical exercise because, although it provided some qualitative understanding

of superfluidity in 4He, the strength of the interactions in that system certainly

prevented a quantitative comparison with experiment. This situation has changed

dramatically soon after the observations of BEC in trapped alkali gases [10, 11, 12],

which for the first time allow the possibility of detailed comparison between theory

and experiment. The result has been an explosion of interest in this field of research,

from both the theoretical and experimental perspectives. In this section, summary of

the experimental techniques which are used to create and study BEC in the laboratory

are given briefly.

The first experimental observation of BEC in a dilute gas was achieved at JILA

by the group of Wieman and Cornell in July 1995 using 87Rb [10]. The condensate

contained about 2000 atoms and appeared at a temperature of 170 nK. This was

closely followed by reports of BEC in 7Li from the Hulet group at Rice university [11]

and in 23Na from the Ketterle group at MIT [12]. The Na condensates contained

up to 5 × 105 atoms and the critical temperature was about 1 �K. More recent

experiments with Rb and Na have produced condensates containing a few million

atoms, with typical number densities of order 1014 cm−3. The condensates in 7Li are

much smaller and limited to about 1000 atoms because 7Li has a negative scattering

length. In contrast, very large condensates containing of order 109 particles have

recently been observed in spin polarized hydrogen [13], concluding two decades of

efforts to observe BEC in that system.

The essential features of the various different BEC experiments are very similar

and Ref. [10] still provides the paradigm for the current generation. Alkali atoms

are used because their internal structure is convenient for laser cooling and mag-

netic trapping, while their large elastic scattering cross sections are convenient for

evaporative cooling. The atoms are collected in a magneto-optical trap (MOT) and

compressed and cooled to micro-Kelvin temperatures using laser cooling techniques


[14]. They are then optically pumped into a magnetic hyperfine sublevel and trans-

ferred to a magnetic trap. Typically, a few times 108 or 109 atoms are trapped and

the temperature of the atomic cloud is of order 100 �K. The magnetic traps are well

approximated by cylindrically symmetric, harmonic potentials with frequencies of the

order of tens to hundreds of Hertz.

At this stage the phase space density n�3dB is still as much as six orders of mag-

nitude smaller than the value of 2.612 required for BEC. This value only be reached,

however, using evaporative cooling techniques in which the most energetic atoms are

allowed to escape from the trap, while the remainder rethermalize to a lower tem-

perature via elastic collisions. Although this procedure involves the loss of most of

the particles, it allows the system to be cooled to the very low temperatures needed

for BEC. The emergence of a condensate can be signified by the rapid growth in the

number of low-energy particles as the system is cooled through the critical region.

This is usually observed by turning off the trapping fields and allowing the atoms to

expand ballistically before an absorption image is taken using a laser. This provides

a measurement of the velocity distribution of the gas, and the condensate appears as

a very sharp peak in the distribution around zero velocity, with the noncondensate

providing a diffuse background. Unfortunately, the technique of ballistic expansion

is destructive and a new condensate has to be produced for any subsequent measure-

ments. An important advance in this respect was the development of non-destructive

imaging by the MIT group, which allows as many as a hundred observations to be

made on the same condensate [15].

Since the first successful production of BEC, a numerous properties of the con-

densate have been studied [16, 17]. In addition, the coherence properties of the

condensate have been demonstrated by interference experiments [18] and magnetic

fields have been used to tune the interaction strength between atoms using Feshbach

resonances, changing the sign of the scattering length from positive to negative [19].

Output coupling of the condensate from the trap has also been achieved, and this

provides a rudimentary form of an atom laser, the atomic analogue of the ordinary

laser [20]. The high degree of experimental control which is possible for trapped


condensates, combined with the possibility of quantitative comparison with theory,

makes BEC in dilute gases an extremely important model system for explorations of

the properties of macroscopic quantum systems.

1.4 Theoretical overview of interacting BEC

Much of the theoretical work was carried out on the properties of weakly interacting

Bose gases stems from the late 1950’s and early 1960’s, where they were used as a

microscopic model of superfluidity. More recently, there has been a very rapid growth

in the literature inspired by the current experiments on BEC in trapped gases. An

outline of the history of BEC research can be found in Ref. [21], and recent theoretical

work on trapped BEC is summarized in Ref. [22]. The first quantitative analysis of

an interacting Bose gas was given by Bogoliubov in 1947 [23] and was based on

the treatment of the condensate annihilation operators as complex numbers. This

amounts to the assumption that the condensate can be described by a coherent state

and corresponds to a description of BEC in terms of spontaneous symmetry breaking.

The advantage of this approach is that it leads to a quadratic Hamiltonian which

can be diagonalized exactly, allowing a non-perturbative treatment of the effect of

interactions. It has been shown, however, that symmetry breaking is not necessary

to obtain a quadratic Hamiltonian and the same results can also be obtained using a

number conserving formalism [24].

An approach to the theory of the dilute Bose gas based on the Green’s functions of

quantum field theory was introduced by Beliaev at T = 0 [25]. Beliaev’s approach at

zero temperature was pursued by Hugenholtz and Pines [26], who calculated higher

order contributions to the ground state energy and derived the result now known as

the Hugenholtz-Pines theorem. This theorem shows that the energy spectrum of a

Bose gas is gapless, which means that the energy of an excitation tends to zero as

its momentum becomes zero. This is an exact result which puts a strong constraint

on any theoretical description of BEC. Recently, a full second order theory based on

the Green’s functions of Beliaev has been developed by Fedichev and Shlyapnikov


and applied to both trapped and homogeneous gases [27]. A similar approach (but

restricted to the homogeneous limit) have been described by Shi and Griffin [28].

These theories are gapless at finite temperature and are therefore consistent with the

Hugenholtz-Pines theorem.

An alternative approach to the theory of the dilute Bose gas was introduced by

Lee, Huang and Yang and was based on the use of the pseudo potential as a means of

describing low-energy collisions [29]. Using this approach, these authors were able to

calculate the leading order properties of a Bose gas of hard spheres. More recently,

a variational approach has been used by Bijlisma and Stoof [30] who considered the

properties of homogeneous Bose gases in two and three dimensions. They stressed the

importance of including the effect of the surrounding medium on binary interactions,

so that collisions are described by a many-body T-matrix. By introducing this T-

matrix consistently into all interactions they were able to obtain a gapless theory. This

point of view has been pursued by Proukakis [31], who used an equation of motion

approach to show how the T-matrix enters into the description of an inhomogeneous

Bose gas.

The low temperature properties of the trapped gases of current experiments are

well-described by the Gross-Pitaevskii equation [32]. Properties of the condensate

such as its size, shape and energy can be determined from the static solutions of

this equation [33]. The excitation energies can be found by solving the linearized

equations of motion for small fluctuations around these static solutions [34], and

analytical results can be obtained in the Thomas-Fermi limit [35]. These calculations

are in good agreement with experiment for the low temperature excitations of the

condensate [36]. A number of dynamical properties have also been studied, including

the recent investigation of the rate of condensate formation using a recently developed

quantum kinetic theory. The results of these calculations are in good agreement with

recent experiments [37]. In addition, a discussion of the decay rates of the excitations

has been given by a number of authors and analytical results have been obtained in

the homogeneous limit [38] and for anisotropic trapped gases [27, 39].


1.4.1 The two-body interaction potential

For alkali atoms the structure of the actual interaction potential is quite complicated,

having a repulsive hard core and many bound states. As mentioned above the bosonic

gas is assumed to be dilute. The requirement for a gas to be dilute is that the average

distance between atoms is much larger than the range of the potential. Consequently,

third and higher order interaction terms can be neglected as mentioned above. The

two body interaction only depends on the distance between the particles, not on the

actual positions of the particles, i.e. U(r, r′) = U(r − r′). Additionally, the details

of the scattering potential does not matter much, since the particles do not get close

enough to probe the details of the potential. Furthermore, if the temperature is low,

the properties of the scattering potential are characterized by the s-wave scattering

length as, the scattering is fully elastic and the scattering potential can be substituted

with a zero range pseudo-potential

U(r− r′) = g�(r− r′), (1.11)

where g = 4�ℎ2as/m is the so-called coupling constant. The pseudo-potential is con-

sequently fully characterized by the s-wave scattering length as of the exact potential

and can be experimentally measured. The 87Rb, 23

Na and 1H atoms have positive s-

wave scattering lengths, which means that their inter-atomic interaction is repulsive.

Moreover 7Li and 85

Rb atoms have negative s-wave scattering lengths, correspond-

ing to low energy attractive interactions. The sign and magnitude of the scattering

length can be tuned by external fields with Feshbach resonances [40, 41], making the

scattering length a tunable parameter for the system of the condensate. In atom

optics the scattering length plays an equivalent role to the nonlinear susceptibility in

nonlinear optics.


1.4.2 Derivation of Gross-Pitaevskii equation

The starting point for the treatment of a real gas of interacting particle is the respec-

tive many body Hamiltonian in second quantization

H =

∫d3rΦ†(r)

[− ℎ

2

2m∇2 + Vext(r)

]Φ(r)

+1

2

∫d3r

∫d3r′Φ†(r)Φ†(r′)U(r − r′)Φ(r)Φ(r′) (1.12)

Φ†(r) and Φ(r) represent the creation and annihilation of a boson at position r and

U(r−r′) is the interaction potential between the bosons, where Vext(r) is the external

trapping potential. Taking into account of only a two-body interatomic potential

is a very good approximation for systems of atomic gases. Note that the higher

order interactions were neglected, as in most practical cases the sample will be dilute

enough, so that they won’t significantly influence the dynamics.

The bosonic character of the system is implemented by subjecting the field oper-

ators to the following bosonic commutation relations:

[Φ(r), Φ†(r′)] = �(r− r′) (1.13)

[Φ(r), Φ(r′)] = [Φ†(r), Φ†(r′)] = 0 (1.14)

In the framework of second quantization these commutation relations are the al-

gebraic counterpart of the symmetry properties of bosonic wave functions. Using

Heisenberg relation iℎ ∂∂tΦ(r, t) = [Φ(r, t), H] and the commutation relations for the

field operators, we obtain the equation of motion,

iℎ∂Φ(r, t)

∂t=

[− ℎ

2∇2

2m+ Vext(r) +

∫d3r′Φ†(r′, t)U(r − r′)Φ(r′, t)

]Φ(r, t) (1.15)

Since Bose-Einstein condensation involves the macroscopic population of a single

quantum state, it is appropriate to consider a mean-field approach. The Bose field

operator is decomposed as, Φ(r, t) = �(r, t) + Ψ(r, t), where �(r, t) = ⟨Φ(r, t)⟩ is theexpectation value of Φ(r, t) and Ψ(r, t) represents thermal fluctuations about this

value. In the context of BEC, the former quantity is a mean-field order parameter


representing the condensed atoms, and is generally referred as macroscopic wave

function. The latter quantity is associated with the non-condensed atoms, induced

by thermal and quantum fluctuations. We assume the limit of zero temperature, such

that the thermal component of the system is non-existent. Furthermore, due to the

weakly-interacting nature (as ≪ �dB) of the condensate, quantum depletion at zero

temperature is expected to be minimal. It is then reasonable to neglect the non-

condensed atoms Ψ(r, t) → 0, and consider only the classical field Φ(r, t) → �(r, t).

Note that the assumption of zero temperature is generally satisfied in reality for

temperatures much less than the transition temperature for condensation. Insertion

Eq. (1.11) into Eq. (1.15) leads to the equation of motion in terms of the macroscopic

of mean-field wavefunction,

iℎ∂�(r, t)

∂t=

(− ℎ

2

2m∇2 + Vext(r) + g∣�(r, t)∣2

)�(r, t), (1.16)

which is the so called Gross-Pitaevskii (GP) equation for the condensate wave function

[22] after independent derivations by Gross [42] and Pitaevskii [43]. The GP equation

resembles the time-dependent nonlinear Schrodinger (NLS) equation apart from the

nonlinear term ∣�∣2, which arises due to the atomic interactions and has important

effects on the properties of the system. If the external potential is time independent

and the gas is in thermal equilibrium, solutions to the GP equation which are only

time-dependent through a global phase can be found. Writing such solutions as

�(r, t) = �(r)exp[−i�cℎt], the time-independent GP equation appears as

(− ℎ

2

2m∇2 + Vext(r) + g∣�(r)∣2

)�(r) = �c�(r). (1.17)

The chemical potential �c in Eq. (1.17) gives the energy required to add an other

particle to the condensate. The GP equation in nonlinear atom optics is equivalent to

the NLS equation in nonlinear optics. Additionally, the mean-field treatment in atom

optics is equivalent to the classical treatment of the electromagnetic field in optics.


1.4.3 Integral of motion

The integrals of motion of the GP model are the total energy, number of particles,

and momentum. Among these physical quantities, we have considered two former

quantities only. The energy E of the system can be derived using the variational

relation,

iℎ∂�

∂t=

�"

��∗, (1.18)

where " is the energy density or energy functional of the system. Using the GP

equation of Eq. (1.16) to evaluate the left-hand side of equation (1.18) and integrating

over �∗ leads to,

"[�] =ℎ2

2m∣∇�∣2 + Vext∣�∣2 +

g

2∣�∣4, (1.19)

The total energy is then the integral of the energy functional over all space as E[�] =∫"[�]d3r. The total number of particles in the system is defined by N =

∫∣�∣2d3r,

and thus related to the particle density as ∣�∣2 = n. The total energy of the system

depending on the particle density is given by

E[n] =

∫d3r

[ℎ2

2m∣∇√n∣2 + nVext +

gn2

2

]= Ekin + Eℎo + Eint, (1.20)

with the following energy contribution: Ekin is the kinetic energy corresponding to

the uncertainty relation giving rise to the so called “quantum pressure”, Eℎo is the

potential energy due to the trapping potential and Eint is the mean field interaction

energy between the atoms. For a non-dissipative system and a time-independent

potential, these integrals of motion are conserved.

In addition to the kinetic and interaction terms, there is another important phys-

ical concept of the condensate wave function namely the healing length, which is

denoted by �. The interaction term minimizes its contribution to the energy of the

wave function by spreading the atoms as widely as possible, and hence tends to re-

duce the healing length of the condensate. On the other hand, the kinetic energy

term minimizes its contribution to the energy of the wave function by making ∇2�(r)

as small as possible, and hence tends to increase the value of �. Therefore, to raise

the condensate density from 0 to n on arbitrary short distances one can change the


interaction and kinetic energies, then the healing length is given by the balance be-

tween the interaction (Eint) and kinetic (Ekin) energy terms. The kinetic energy term

is of order Ekin =ℎ2

2m�2and the interaction energy Eint =

4�ℎ2asm

n. Equating these

two energies, healing length yields [7, 22]

� =1√

8�nas. (1.21)

It is the characteristic length scale of fluctuations in the condensate density. For 87Rb

a density of n = 1014 cm−3 then the healing length is � ≈ 0.3 �m. Moreover, the

velocity of the atoms in the condensate can be derived from the GP equation. To

proceed, multiply the time dependent GP equation (1.16) by �∗, and then subtract

the complex conjugate of the resulting equation, which yields

∂∣�∣2∂t

+∇ ⋅[ℎ

2im(�∗∇�− �∇�∗)

]= 0. (1.22)

Equation (1.22) has the form of a continuity equation for particle density. The GP

equation is a fluid model which encapsulates equations familiar from classical hydro-

dynamics. The hydrodynamics continuity equation is given by

∂n

∂t+∇ ⋅ (nv) = 0, (1.23)

where v is the local velocity of the condensate. By inspection of Eqs. (1.22) and

(1.23), v can be written as

v =ℎ

2im

(�∗∇�− �∇�∗)∣�∣2 = 0. (1.24)

If the condensate wave function � = Aei'(r,t), then

v =ℎ

m∇'(r, t). (1.25)

Eq. (1.25) shows that the velocity of the condensate is proportional to the derivative

of the condensate wave function [7, 22]. Healing length and velocity of the condensate

is important in the description of topological BEC excitations.


1.4.4 Time-independent solutions

An important feature of the time independent solution of dilute BEC, is that, the

condensate density changes by changing the sign of the mean-field interaction energy

coefficient g. First, we consider a non-interacting (g = 0) ideal gas confined by a 3D

harmonic potential where the excitation states are the standard harmonic oscillator

states. For example, the ground state has a Gaussian density profile, as shown by dot-

dashed line in Fig. (1.1). For an interacting gas with sufficiently weak interactions,

such that Nas/aℎo ≪ 1, where aℎo =√ℎ/m!ℎo is the the mean harmonic oscillator

length and !ℎo = (!x!y!z)1/3, and the states are similar to the non-interacting case

[22].

Next, we consider the mean-field interaction relatively strong, high particle num-

bers and not too strong confinement, such that N ∣as∣/aℎo ≫ 1 [22]. In this regime,

the atomic interactions tend to have significant effect on the condensate density. For

attractive interactions (g < 0) the density profile becomes narrower than the corre-

sponding ground harmonic oscillator state, tending towards a sharp peak located at

the trap centre, as illustrated by dashed line in Fig. (1.1). For a large number of

atoms and strong attractive interactions, the attraction dominates to the point that

the condensate is prone to collapse [7, 19, 41, 44].

The GP equation only describes the condensate fraction at T → 0, but in this

finite temperature, the “tℎermal clouds” surrounding the condensate fraction are not

described in this theory. The behaviour of the “weakly interacting gas” for above Tc

and at Tc can be explained by the Thomas-Fermi (TF) approximation theory. In the

case of relatively strong repulsive interactions (g > 0), the kinetic energy term in the

GP equation can be neglected, then the condensate density is essentially dominated

by the interaction energy. This approximation is called the TF approximation. By

taking the TF approximation in Eq. (1.17), the condensate density corresponds to

n(r) =�c − Vext(r)

gfor �c − Vext > 0, (1.26)

and n = 0 otherwise. The TF approximation allows the determination of several

condensate parameters. First, we use Eq. (1.26) and the normalization factor


Figure 1.1: Schematic of the density profile of a dilute BEC within a sphericallysymmetric harmonic trap, for repulsive interactions (solid line), attractive interactions(dashed line), and the non-interacting gas (dotdashed line). The area under thecurves, representing the number of atoms, is the same in all three cases.

∫drn(r) = N , the chemical potential in a harmonic potential can be now related

to the atom number as

�c =ℎ!ℎo

2

(15Nasaℎo

)2/5

(1.27)

Together with the condition Vext(RTF,i) = �, this expression yields the TF radii of

the BEC along the axes (i = x, y, z) of the trap

RTF,i ==

√2�cm!2

i

= aℎo!ℎo

!i

(15Nasaℎo

)1/5

(1.28)

Further the peak atom density n0 is immediately obtained from Eqs. (1.26) and (1.27)

as

n0 =�c

g=

�cm

4�ℎ2as=

1

8�

(15N

a6ℎoa3/2s

)2/5

(1.29)

The equations (1.26-1.29) are very useful in the discussion of concrete experiments

and allow an estimation of physical parameters of BEC. For the case of repulsive

interactions (g > 0), the density profile in a harmonic trap becomes broader than

the corresponding ground harmonic oscillator state, as indicated by solid line in Fig.

(1.1).


1.4.5 Cigar and Disk shaped condensates

While most of the experimental work so far has concentrated on three-dimensional

(3D) systems, there is profound interest in systems with lower dimensionality leading

to qualitatively different phenomena. The condensate expands either along the axis

of a quasi one-dimensional (1D) cigar shaped trap [45] or in the plane of a quasi two-

dimensional (2D) disk shaped trap [46]. It is well known that the GP equation (1.16)

can be formulated as a principle of least action with the action functional S =∫Ldt

with L =∫£dr, where the Lagrangian density is

£ =iℎ

2(�∗t�− �t�

∗) +ℎ2

2m∣∇�∣2 + Vext∣�∣2 +

1

2g∣�∣4, (1.30)

In the case of a cigar- or disc-shaped trap, one can readily find conditions under

which the tightly restrained degrees of freedom are frozen and the GP equation re-

duces to a one- or two- dimensional equation, respectively. The trapping potential

Vext =1

2m(!2

zz2 + !

2

⊥�2), where !z and !⊥ are the longitudinal and radial trapping

frequencies and �2 = x2 + y

2. For !⊥ ≫ !z i.e., !⊥/!z ≫ 1, the condensate adopts a

highly-elongated “cigar” shape, and the transverse high-energy modes of excitation

become heavily suppressed. In this regime, the longitudinal dynamics, which are of

relatively low energy, dominate the system and a quasi-1D condensate is formed. In

the quasi 1D limit, the transverse zero-point energy is much higher than the nonlinear

interaction energy per atom, then the transverse motion reduces to the ground state

of particle oscillation, with the amplitude a⊥ =√ℎ/m!⊥. The characteristic size of

the condensate along the axis of a cigar-shaped trap is denoted by z0 then the number

of atoms in the condensate is N ∼ ∣�∣2a2⊥z0. If the condition Nas/z0 ≪ 1 is satisfied,

the condensate wave function can be factorized as

�(r, t) = �(�) (z, t) =1√�a2⊥

exp

(x2 + y

2

2a2⊥

) (z, t). (1.31)

Here �(�) = �(x, y) is the wave function of the ground state of transverse motion.

Substituting Eq. (1.31) in Eq. (1.30), integrating the result over the condensates

cross section, one obtains the action expressed in terms of the 1D Lagrangian density

£1D =iℎ

2( ∗t − t

∗) +ℎ2

2m∣ z∣2 +

1

2m!

2

zz2∣ ∣2 + g

4�a2⊥∣ ∣4. (1.32)


Then, the evolution of (z, t) obeys the 1D GP equation

iℎ∂t = − ℎ2

2m∂2

z +1

2m!

2

zz2 + g1D∣ ∣2 , (1.33)

where g1D = g2�a2

⊥

= 2ℎ2asma2

⊥

= 2ℎas!⊥. Eq. (1.33) determines the longitudinal dynam-

ics of a condensate in a cigar-shaped trap.

Two-dimensional condensate dynamics are observed when the longitudinal traping

frequency !z is much higher than the radial trap frequency w⊥ i.e., !z/w⊥ ≫ 1 the

condensate takes the form of a highly-flattened “disk” shape. In the 2D limit, the

motion along the longitudinal direction (z−axis) is frozen, i.e., the zero-point energyassociated with the oscillation amplitude az =

√(ℎ/m!z) is much higher than the

nonlinear energy. The radius of the density distribution in the plane (x, y) of the trap

is denoted by R0, then the number of atoms in the disk shaped trap is N ∼ ∣�∣2R2

0az.

If the condition Nasaz/R2

0≪ 1 is satisfied, the condensate wave function can again

be factorized as

�(r, t) = �(z) (�, t) =

(1

�a2z

)1/4

exp−z2/2a2z (�, t). (1.34)

Here �(�) = �(z) is the longitudinal ground state wave function. Substituting Eq.

(1.33) in Eq. (1.30) and integrating the result over the longitudinal coordinates, the

action is expressed in terms of the 2D Lagrangian density

£2D =iℎ

2( ∗t − t

∗)+ℎ2

2m(∣ x∣2+∣ y∣2)+

1

2m!

2

⊥(x2+y2)∣ ∣2+ g

2√2�az

∣ ∣4. (1.35)

The corresponding Euler-Lagrange equation is the 2D GP equation

iℎ∂t = − ℎ2

2m(∂2x + ∂

2

y ) +1

2m!

2

⊥(x2 + y

2) + g2D∣ ∣2 . (1.36)

The effective coupling constant is expressed as g2D = g√2�az

= 2√2�ℎ2asmaz

. Eq. (1.36)

describes the 2D transverse dynamics of a condensate in a disk-shaped trap.

1.5 Brief History of Soliton

Nonlinear equations are ubiquitous in diverse branches of science. In physics, the

well-known NLS equation manifests in nonlinear fiber-optics [47, 48, 49] and with


an appropriate potential as GP equation in Bose-Einstein condensates [7]. The Ko-

rteweg and deVries (KdV), modified Korteweg and deVries (mKdV), sine-Gordon, and

Boussinesq equations routinely appear in hydrodynamics and plasma physics [50]. By

now, there are more than hundreds of nonlinear equations, for which solitary waves

exist as solutions. The integrability of these equations lead to extremely interesting

properties for their solutions, the so called solitons. Solitons can be localized or may

appear as pulse trains; however, in both these cases, the nonlinearity and dispersion

delicately balance each others effect results in the existence of the self-similar waves.

We begin our discussion of solitary waves and solitons with a historical account

from the person who first observed this phenomena, Scottish scientist and engineer,

John Scott-Russell [51, 52]. In 1834, John Scott Russell observed a “great wave of

translation” on the Edinburgh-Glasgow canal, Scotland. Russell stated his observa-

tions to the British Association in 1844 as “I was observing the motion of a boat which

was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly

stopped - not so the mass of water in the channel which it had put in motion; it ac-

cumulated round the prow of the vessel in a state of violent agitation, then suddenly

leaving it behind, rolled forward with great velocity, assuming the form of a large soli-

tary elevation, a rounded, smooth and well-defined heap of water, which continued its

course along the channel apparently without change of form or diminution of speed.

I followed it on horseback, and overtook it still rolling on at a rate of some eight or

nine miles an hour, preserving its original figure some thirty feet long and a foot to a

foot and a half in height. Its height gradually diminished, and after a chase of one or

two miles I lost it in the windings of the channel. Such, in the month of August 1834,

was my first chance interview with that singular and beautiful phenomenon which I

have called the Wave of Translation” [52]. The word “solitary wave” was coined by

Scott-Russell himself. Subsequently, Russell did extensive experiments in a labora-

tory scale wave tank in order to study this phenomenon more carefully and found the

following important results [53]:

∙ He observed solitary wave, which are long, shallow, water waves of permanent

form, hence deduced that they exist: this is his most significant results.


∙ The speed of propagation, v, of a solitary wave in a canal of uniform depth ℎ is

given by v2 = fgr(ℎ + A) where A is the amplitude of the wave and fgr is the

force due to gravity.

In 1845, Airy published a theory of long waves in his book, “Tides and Waves”

where he also found a formula for the speed of a wave relating to its height and ampli-

tude. From his results, Airy concluded that a solitary wave could not exist! Needless

to say, this led to a war of words between Russell and Airy. Further investigations were

undertaken by Stokes (1847), Boussinesq (1871) and Rayleigh (1876) in an attempt

to understand this phenomenon. Boussinesq and Rayleigh independently obtained

approximate descriptions of the solitary wave: Boussinesq derived a one dimensional

nonlinear evolution (which now bears his name), in order to obtain his result. These

investigations provoked much lively discussion and controversy as to whether the in-

viscid equations of water waves would possess such solitary wave solutions. This issue

was finally resolved by the two Dutch physicists, Korteweg and deVries in 1895. They

derived an equation, the so-called KdV equation, which describes shallow water waves

where the existence of solitary waves was verified mathematically [54]. This result

was essentially buried in the literature for about sixty years when it got rediscovered

again in a completely different context.

In 1914, the physicist Debye suggested that the finiteness of the thermal conduc-

tivity of a one-dimensional anharmonic lattice was a result of the systems nonlinearity.

In 1955, Fermi, Pasta and Ulam investigated how the equilibrium state is approached

in a one-dimensional nonlinear lattice [55]. It was expected that the nonlinear inter-

actions among the normal modes of the linear system would lead to the energy of the

system being evenly distributed throughout all the modes, that is, the system would

be ergodic. The results of numerical analysis contradicted this idea. The energy

is not distributed equally into all the modes, but the system returns to the initial

state after some period. In 1965, Zabusky and Kruskal solved the KdV equation

numerically as a model for nonlinear lattice and observed the recurrence phenomena.

Further, they found an unexpected property of the KdV equation. From a smooth

initial waveform, waves with sharp peaks emerge. Those pulse-waves move almost


independently with constant speeds and pass through each other after collisions. A

detailed analysis confirmed that each pulse is a solitary wave of sech2-type and the

solitary waves behave like stable particles. Thus, the soliton was discovered.

In the context of soliton, many explicit methods have been proposed in the lit-

erature for the soliton solution[51]. Some of the most popular methods are inverse

scattering transform (IST), Abiowitz, Kaup, Newell, and Segur method, Backlund

transformation method and Lax pair approach, Hirota-bilinear method, etc. The

connenction between the solution of KdV and time independent NLS seeded the way

to achieve general solution of this problem. The evolution of such a solution methedo-

logy being its existence in the late 1960s, when Gardner, Greene, Kruskal and Miura

solved the KdV equation by introducing an IST method. Following the way similar to

the former case, Zakharov and Shabat successfully solved the NLS to achieve funda-

mental bright and dark soliton in anomalous [56] and normal [57] dispersion regimes,

respectively. But the real turn around emerge in the year 1974, when Ablowitz, Kaup,

Newell and Segur (AKNS) generalized the method originally proposed by Gardner,

Greene, Kruskal and Miura [58, 59]. This generalization open the way for the solu-

tion of a whole class of nonlinear partial differential equations (PDEs). The procedure

gets modified and evolve as one of the successful method ever by name AKNS, which

is now to be the method of choice for solving nonlinear PDEs. The AKNS scheme

was first employed to solve the sine-Gordon equation [59]. Following AKNS in comes

another popular technique called as Backlund transformation (BT). BT has wide

application in many branches of physical system, one of the most striking feature

is its ‘dressing’ procedure for solutions of the nonlinear equations [60]. In principle

BT can only be limited to systems of linear differential equations and fails to be ap-

plicable for the system of nonlinear differential equations, unless a linear system of

equations equivalent to a nonlinear differential equation exist. Establishing a relation

between the linear system and the corresponding nonlinear differential equation can

be achieved by considering a pair of matrices called the Lax pair, which must satisfy

a consistency condition that is equivalent to the differential equation of interest. In

particular, constructing the Lax pair for a evolution equation is a tedious job and the


real difficult in the hand is the exact solution of the nonlinear differential equation.

Another remarkable contribution to the concept of soliton came in the year 1971,

when Hirota introduced a new direct method for constructing multi-soliton solutions

to integrable nonlinear equations [61]. The idea behind the method was to make a

transformation into new variables, thus skipping the construction of linear eigenvalue

problem one can arrive into multi-soliton solutions in a direct way. The idea latter

revolutionized the concept of soliton solutions and turned out to be very effective in

achieving N-soliton solution to the KdV [61], mKdV, sine-Gordon and NLS equations

[62]. The bright and dark solitons have been derived for the higher-order NLS equa-

tion through Lax pair and BT method [63]. By using the Lax pair and the Hirota

bilinear method, the bright and dark soliton solutions have been constructed for the

coupled system of higher-order NLS equations [64].

In 1973, the idea of soliton transmission in an optical fiber was proposed by

Hasegawa and Tappert [65]. It was not until 1980 that the first soliton was observed

experimentally by Mollenauer, Stolen and Gordon [66]. An optical soliton is a pulse

that travels without distortion due to dispersion or other effects. When an optical

pulse gets propagated in a nonlinear medium, it encounters the self-phase modulation

(SPM) caused due to optical Kerr effect. SPM causes a red shift at the leading edge

of the pulse. Solitons occur when this shift is canceled due to the blue shift at

the leading edge of a pulse in a region of anomalous dispersion, resulting in a pulse

that maintains its shape in both frequency and time [67]. Further, we elaborate

the formation of optical solitons as follows. In actual practice, all media exhibit

nonlinear effects. In the case of silica optical fibers, one of the manifestations of the

nonlinearity is the intensity dependent refractive index according to the following

form as ℵ = ℵ0 + ℵ2I, where ℵ0 and ℵ2 are the linear and nonlinear refractive index,

respectively. When the optical pulse travels through the fiber, the high intensity

portion of the pulse encounter a higher refractive index of the fiber compared with

the lower intensity regions. This intensity dependent refractive index leads to the

phenomenon of SPM. The primary effect of the SPM is to broaden the spectrum of the

pulse while keeping the temporal shape unaltered. Indeed, for silica optical fibers for


which ℵ2 is positive, the frequencies in the trailing edge of the pulse gets blue-shifted

and those in the leading edge are red-shifted with respect to the center frequency of

the pulse. This broadening of the pulse spectrum generates the new frequencies in the

pulse and will ultimately lead to an increased broadening through the phenomenon

of dispersion. If the operating wavelength is above the zero dispersion wavelength,

then higher frequencies travel faster than lower frequencies and pulse broadening in

the absence of nonlinear effect is accomplished by a chirp with in the pulse, where

the instantaneous frequency decreases with increasing time. SPM leads to a chirping

with lower frequencies in the leading edge and higher frequencies in trailing edge,

which is just opposite the chirping caused by linear dispersion in the wavelength

region above the zero dispersion wavelength called as anomalous dispersion region.

Thus, by a proper choice of pulse shape and the power carried by the pulse, we can

indeed compensate one effect with the other. In such case the pulse would propagate

undistorted by a mutual compensation of dispersion and SPM. Such a pulse would

broaden neither in the time domain nor in the frequency domain and is called soliton.

The above discussed effects are in the anomalous regime, the respective solitons are

called bright solitons. These solitons have hyperbolic secant shape for their intensity

profile. In the normal dispersion regime, the resulting solitons are dark solitons. The

intensity profile of this case contains a dip in its shape with uniform background

having hyperbolic tangent shape.

Optical solitons are therefore an important development in the field of optical

communications [68]. In a real-world system the group velocity dispersion (GVD)

and a nonlinear effect, the so called self phase modulation, have to be considered

and also the power loss. Optical solitons are particularly suitable for increasing the

capabilities of such systems [69]. A typical modern communication system consists of

three major components known as transmitter, information channel and a receiver.

The necessity for larger bandwidth is always the intense area of research. Many

methods and mechanism have been proposed to achieve large bandwidth. One of

the technique of million-dollars is the Wavelength-division multiplexing (WDM). In

principle, it is the mechanism by which optical solitons are encoded at slightly different


wavelengths which will propagate with different velocities due to the dispersion effect

of fiber. WDM picks up with the emergence of erbium-doped fiber, where WDM

soliton propagation is considered to be an efficient means of information transfer.

Moreover, in addition to its application in the field of communication, solitons also

find wide applications in the construction of optical switches. Apart from the usage

of soliton in various fields, a most significant application of solitons is in optics, where

the soliton can be utilized in the generation of ultra-short pulses which find many

applications in high speed telecommunication system.

The recent widespread research on solitons are not only studied in optical fiber

system also applicable in many physical systems. Now, the existence of solitons in

some other physical systems are discussed in briefly as follows. After the invention of

the laser, there has been much interest in propagating nonlinear pulses through the

periodic medium such as a fiber Bragg grating (FBG), which is a periodic variation

of the refractive index of the fiber core along the length of the fiber. In the FBG, we

also have the stop band known as a photonic bandgap (PBG) that does not allow the

propagation of light pulses when the Bragg condition is satisfied. Gap solitons are

solitary waves propagating in a nonlinear photonic bandgap (PBG) structure [70].

The exact analytic solution to describe such a nonlinear pulse has been obtained

from the nonlinear coupled-mode (NLCM) equations. The analytical solutions of the

NLCM equations are not solitons but solitary waves that can propagate through FBG

without changing their shape. By using the multiple scale method [71], the NLCM

equations can be reduced to the NLS equation. Soliton solutions to this approximated

NLS equation are called Bragg solitons [72]. Bragg solitons exist near the PBG

edge and have been widely discussed both in theory [73] and experiment [74]. Next,

the photonic crystal fiber (PCF) was mainly motivated by their nonlinear optical

applications as supercontinuum generation, pulse compression, optical switching, fiber

laser, etc [75]. Solitons arise also in many physical systems such as shallow and deep

water waves [76], plasma systems [77, 78], macromolecules [79], acoustics [80], and

photonics [81].

Till, we have discussed the continuous soliton for the various physical system. We


have now shift our attention towards the discrete soliton. The analysis of nonlinear

lattices has been the subject of intense investigation in many areas of pure and applied

science. In mathematics, the first fully integrable lattice equations were identified and

solved using inverse scattering methods [51]. Such equations include for example the

Toda lattice [82], the Ablowitz-Ladik equation [83], and the Calogero-Moser N-body

problem [84]. In solid state physics, neutral and charge soliton transport was theoret-

ically and experimentally studied in conducting polymer chains such as polyacetylene

and polythiophene, based on a model proposed by Su, Schrieffer, and Heeger [85].

Over the years this soliton picture received increasing experimental support as these

self-localized states were found to be involved in the electric, optical, and magnetic

properties of these polymers [86]. For this contributions in the understanding of con-

ducting polymers, Alan Heeger received the Nobel Prize in Chemistry for the year

2000. Another important step in the theory and physics of nonlinear discrete sys-

tems was made by Aleksandr Davydov when he suggested a discrete soliton model

as a means to understand energy transfer in protein �-helices [87]. Energy transfer

phenomena are of paramount importance in biophysics, since they are involved in

a number of biological processes such as muscle contraction, enzyme catalysis, and

active transport. By extending the Holstein Hamiltonian, Davydov put forward a

model that was based on a discrete NLS like equation [88]. Shortly after his proposal,

several theoretical and experimental groups pursued this possibility [89]. Finally one

should mention other discrete nonlinear dynamical systems that are receiving large

attention these days. These include for example breather dynamics in Josephson

junction arrays [90] intrinsic localized modes in anharmonic crystals [91]. In recent

years, discrete solitons have been investigated in many diverse branches of physical

systems such as optical waveguide arrays [92], solid state physics [85, 93], photonic

crystal structure [94] and BEC in optical lattices [95, 96].

From the above brief discussion, the concept of soliton are widely studied in dif-

ferent branches of physical systems. Analytical solutions of nonlinear PDEs in terms

of solitons represent a fruitful field of theoretical and experimental physics. In the

context of fiber optics, the soliton solution of NLS equation describes an optical soli-


ton as proposed by Hasegawa and Tappert in the year 1973. Another kind of PDEs

called the GP equation is used to describe a dilute gas of weakly interacting atomic

particles. The GP equation has become an important theoretical tool in recent BEC

experiments. Soliton solutions of GP equations are also known and found to be anal-

ogous with the nonlinear optics. Such a soliton solution of the GP equation popularly

known to be an atomic solitons [97]. In this thesis, we have focused some interesting

dynamical features of solitons in a system of BEC.

1.5.1 Solitons in BEC

The physics of BEC and nonlinear optical systems share a number of similarities.

The same NLS equation including with different types of trapping potentials, can

describe the dynamics of the BEC [7]. A condensate can be described by the nonlin-

ear GP equation, where the nonlinear term arises from the interatomic interactions.

The nonlinear term is cubic in the atomic field, in analogy with the well known Kerr

nonlinearity in optics. The inherent nonlinearity of dilute BEC means that they sup-

port a family of nonlinear excitations known as matter-wave solitons. Solitons are

topological excitations characteristic of BEC. They are localized disturbances in the

atom density which travel through BEC without spreading. Solitons preserve their

form because the atomic interaction potential term in the Hamiltonian balances the

spreading (dispersion) of the wavefunction caused by the kinetic energy term. Con-

sider, for example, a matter-wave bright soliton in BEC in which the interactions are

attractive. The kinetic energy is related to the second derivative of the wavefunction,

so matter-waves have dispersion in the free direction owing to their kinetic energy.

This effect broadens the soliton. However, the atomic interaction effect attempts

to force particles into the minima, and hence narrows the soliton. For a particular

soliton width, these two effects are balanced. In both cases the kinetic energy term

tends to broaden the soliton, and the atomic interaction term tends to narrow the

soliton. This infers a density maximum if the interactions are attractive and a density

minimum if the interactions are repulsive.

The form of solitons in BEC depends on the sign of the s-wave scattering length


-4 -2 0 2 40

0.2

0.4

0.6

0.8

1

-4 -2 0 2 40

0.2

0.4

0.6

0.8

1

zz

∣ ∣2

∣ ∣2

Figure 1.2: The bright (a) and dark (b) matter wave soliton in Bose-Einstein con-densation.

as [7]. If as is negative, solitons are characterized by a peak in the density, which has

no associated phase jump. These solitons are known as bright solitons [98, 99]. If

a is positive, as is the case in this study, solitons are characterized by local density

minimum, and a sharp phase gradient of the wavefunction at the position of the

minimum. These solitons are known as dark solitons [100]. This category of dark

solitons is further divided into black solitons and gray solitons. The local density

minimum of dark soliton to zero value is called a stationary dark soliton or black

soliton [100]. While, the local density minimum of dark soliton greater than zero

value is called a non-stationary dark soliton or gray soliton [101].

Now, we briefly discuss the 1D soliton of GP equation (1.33). In the 1D limits,

we ignore the longitudinal trapping potential. Because it is very much less than the

transverse trapping potential. The bright soliton of 1D GP equation (1.33) in the

absence of longitudinal trapping potential (!z = 0) is given by [49, 102]

(z, t) =

√mas!⊥2ℎ

sech[(mas!⊥2ℎ

)z

]exp

[−i(ma

2

s!2

⊥2ℎ

)t

]. (1.37)

Fig. (1.2a) shows the bright soliton for the cigar shaped attractive condensate.

Experimentally, the bright BEC soliton have been observed in [103, 104]. Such 1D

bright solitons have been realized in 3Li and 85

Rb attractive condensates by using

Feshbach resonance to tune the scattering length [19, 41, 105]. However, in quasi-

1D system with strong transverse confinement the solitons are rendered stable [106].

The dark soliton of 1D GP equation (1.33) in the absence of the trapping potential


(!z = 0) is given by [49, 102]

(z, t) =

√mas!⊥2ℎ

tanh[(mas!⊥2ℎ

)z

]exp

[−i(ma

2

s!2

⊥2ℎ

)t

]. (1.38)

Fig. (1.2b) shows the dark soliton for the repulsive condensate. Dark matter wave

solitons have been observed experimentally in a 23Na condensate by Denschlag et

al. [97] and in cigar-shaped 87Rb condensates by Burger et al. [97, 107]. In both

experiments dark solitons of variable velocity were launched via the phase imprinting

of a BEC by a light-shifted potential. Recently, the dark matter wave solitons have

been investigated by giving a small perturb to the atomic density of condensation

[108].

Theoretically, the bright matter wave soliton collisions of 1D GP equation with

harmonic potential have been investigated [109]. More recently, the dynamics of a

dark soliton in an elongated BEC has been studied at finite temperatures [110]. Fur-

ther, the dark and bright soliton solutions of the 1D GP equation with a confining

potential have been obtained analytically [111]. The adiabatic N-soliton interac-

tions in weak external potentials were discussed too. Using a parametric field theory

approach, the formation of coherent molecular soliton has been investigated in molec-

ular BEC [112]. In [113], the dynamical phase diagram of a dilute BEC trapped in

a periodic potential has been discussed. Wherein the existence of localized excita-

tions, discrete solitons and breathers for repulsive interaction BEC have also been

discussed. The formation of bright, dark, ring and matter wave solitons has been

widely discussed in an inhomogeneous BEC [114]. Recently, the matter wave soliton

on background solution have also been investigated in a system of BEC [115, 116].

1.6 BEC in Optical Lattices

Optical lattices for atomic BEC raised enormous interest, as they mirror features

known from solid state physics to the field of atom optics. In perfect solid state

crystals atoms are arranged in a regular array creating a periodic potential for the

electrons inside. Each electron obeys the one electron Schrodinger equation with a


periodic potential V (z + l) = V (z) with period l. According to Bloch’s theorem the

stationary eigen-states are plane waves modulated by a periodic function revealing

the periodicity of the atom lattice [117]. Due to the breakthrough of creating atomic

BEC in 1995, it is now also possible to investigate condensates confined in periodic

optical potentials, so called optical lattice (OL). The dynamics of BEC loaded into

OL realized by two counter propagating laser beams has attracted enormous attention

[118, 119]. It’s quite natural to know why one need to go for BEC in OL. In general,

optical lattices offer wide class of advantages: First, lower temperatures mean that

a BEC will usually be in the lowest energy levels of the lattice wells without the

need for further cooling after the lattice is applied. Second, the higher densities lead

to an increased filling factor of the lattice, which can easily exceed unity for BEC.

So, rather than ending up with a light-bound “crystal” with lots of vacancies, after

applying the lattice each site will be occupied. Third, higher densities also imply the

inherent effects due to interatomic interactions can become increasingly important.

Thus, introducing high density BEC into an OL immediately leads to much rich

physics with the introduction of nonlinearity into the system [118].

By interfering more laser beams, one can obtain 1D, 2D and 3D periodic potentials.

By far the most research into BEC in OL has been carried out in 1D, but the formation

of OL in 2D and 3D can be achieved by using additional laser beams [120]. The most

direct extension is to add a pair of lasers perpendicular to the first set, creating a 2D

lattice. By adding a last pair in the third spatial direction a 3D lattice can be formed.

The first BEC experiment in OL begin to probe beyond single-particle physics was

carried out at Yale in 2001 [121]. In Ref. [122], the group of Peter Zoller made the

suggestion that Bose-Hubbard could be realized in 2D and 3D OL. The Mott insulator

transition in a gas of ultracold atoms was first demonstrated by Greiner et al. [123]

in a 3D lattice.

BEC is loaded in weak OL, the general wave function may be split into a su-

perposition of forward- and backward-traveling modes, giving rise to a system of

coupled-mode equations for the respective slowly varying amplitudes [124], which is

tantamount to the well-known coupled-mode equations for optical waves propagating


through the Bragg grating [70, 125]. On the other hand, the corresponding OL is

deep, the underlying GP equation can be reduced to an effective 1D discrete non-

linear Schrodinger equation with the repulsive on-site nonlinearity [126, 127]. The

properties of BEC in OL have been investigated intensely both experimentally [128]

and theoretically [129]. It shows rich and intriguing physical phenomena, such as

Bloch oscillations, LandauZener tunneling, complex nonlinear dynamics, especially,

among these findings the localized states and self trapping phenomenon are of most

interest. The dynamical phenomenon of BEC in 1D OL have been investigated by

describing the continuous [130, 131] and discrete [113, 130, 132] GP equation.

Both theoretical and experimental studies show that atom interactions play key

role for the dynamics of the BEC loaded into OL. However, it is well known that the

dynamics of a BEC loaded into a one-dimensional periodic optical lattice is mainly

focused on considering the two body interactions. It is clear that in low temperature

and density, where interatomic distance is much greater than the distance scale of

atom-atom interactions, two-body s-wave scattering should be important and three-

body interactions can be neglected. On the other hand, if the atom density is high,

for example, in the case of the miniaturization of the devices in the integrated atom

optics, three-body interactions can start to play important role [133, 134, 135]. As

shown in [136], the three-body interactions in a sufficient high density condensate

can lead to a special modulational instability. The stability of BEC in OL is also

influenced dramatically by the three-body interactions. As reported in [137], even for

a small strength of the three-body force, the region of stability for the condensate can

be extended considerably. Many features of condensates in lattices are manifestations

of more general concepts of nonlinear systems, such as solitonic propagation and

instabilities.

1.6.1 Gap solitons in optical lattice

Optical lattices are periodic light shift potentials for atoms created by the interfer-

ence of multiple laser beams. A BEC, loaded into a 1D, 2D, or 3D optical lattice,

becomes a test ground for a range of fascinating physical effects because the lattice


potential can be easily manipulated by changing the geometry, polarization, phase,

or intensity of the laser beams. BEC in an OL can be regarded as the reconfig-

urable analog of a nonlinear Photonics Band Gap structure for matter waves in an

atomic band gap structure. The optical potential of a lattice plays the role of the

periodically modulated refractive index of a dielectric and the Kerr nonlinearity is

emulated by the atom-atom interactions. One of the phenomena exhibited by single-

and multi-component BEC in multi-dimensional OL is the existence of nonlinear lo-

calized modes such as matter-wave “gap” solitons. The existence of gap solitons in

BEC was predicted theoretically [138] and demonstrated experimentally [139], its all

discussed elaboratively in [118]. Gap solitons are represented by stationary solutions

to the respective GP equation, with the eigenvalue located in a finite band-gap of

the OL-induced spectrum. Actually, gap solitons were first predicted [140] and ex-

perimentally created [141] as optical pulses in fiber Bragg gratings. The existence

of moving gap solitons was predicted in both 1D and 2D settings [124]. Numeri-

cally, the dynamical stability of gap solitons has been discussed in a quasi-1D BEC

in an OL [142]. The existence and stability properties of bright and dark out-of-gap

BEC solitons in OL have been investigated and also demonstrated that these solitons

can be created by means of phase imprinting technique [143]. A long-lived station-

ary wave packet in the form of a matter-wave gap soliton have been proposed and

demonstrated in a repulsive BEC placed in a 1D OL by using the numerical method

[144]. The stability of gap solitons in the first two finite band gaps in the 1D GP

equation combining the repulsive nonlinearity and the periodic potential in time peri-

odic modulation of OL has been discussed [145]. Using the variational approximation

and numerical simulations, the 1D gap solitons has been discussed in binary BEC

trapped in an OL potential [146].

In theory, a deep lattice, the condensate can be described by the superposition of

ground state modes in the individual wells. The mean-field treatment of the conden-

sate in this regime leads to a discrete equation which admits solutions in the form

of stationary modes localized on a few lattice sites-discrete solitons [113, 147], in

complete parallel with spatial solitons in periodic optical structures [81] and localized


modes of atomic lattices [148]. In the opposite case of a shallow 1D OL can support

bright matter-wave gap solitons even in repulsive BEC [149] with large atom numbers.

These solitons, described in a framework a coupled-mode theory [149], are localized

on a large number of lattice wells, and are predicted to exist only in atomic band

gaps. Porter et al., have examined the stationary gap soliton for a system of BEC

in the first spectral band gap of the optical lattices when the same amplitude and

phase under the Feshbach resonance management described by the coupled mode GP

equation [150]. In this thesis, we have investigate the more general case of moving

gap soliton for a system of BEC in the first spectral band gap of the optical lattices

with the same amplitude but different phases under the conditions for the absence

and presence of the mean atomic scattering length.

1.7 Modulational instability in BEC

Modulational instability (MI) is a process that appears in most nonlinear physical sys-

tems. Because of MI, small amplitude and phase perturbations grow rapidly under

the combined effects of nonlinearity and dispersion [49, 151]. As a result, a broad op-

tical beam or a continuous wave (CW) tends to disintegrate during propagation [152],

leading to filamentation [153, 154] or to break up into pulse trains [152]. MI typically

occurs in the same parameter region where another universal phenomenon, soliton

occurrence, is observed. The relation between MI and solitons is best manifested in

the fact that the pulse trains that emerge from the MI process are actually trains of

almost ideal solitons. Therefore, MI can be considered to be a precursor to soliton

formation. Till the date, MI has been systematically investigated in connection with

numerous nonlinear processes. Yet traditionally, it was always believed that MI is

inherently a coherent process and can only appear in nonlinear systems with a perfect

degree of spatial and temporal coherence. The phenomenon of MI occurs in several

branches of physics such as fluid dynamics [155], optical fiber [151, 156], and plasma

physics [157]. This approach is essentially a linear stability analysis of the stationary

CW solution to the NLS equation. The linearized equation for the perturbation is


valid only when the perturbation is weak. The CW solution of the NLS equation

becomes unstable towards the generation of a chain of bright solitons. In the same

way, the train of BEC bright matter wave solitons can be generated in the context

of BEC described by GP equation, under certain conditions of MI [158, 159]. Thus,

the MI process deals with the stability of the CW solution and pattern formation in

a system of BEC [160, 161]. Recently, a new precise time-dependent criterion for the

instability of a trapped BEC has been elaborated both analytically and numerically

with the help of lens-type transformation [162, 163, 164]. Modulational instability

and the nonlinear dynamics of multiple solitary wave formation in two-component

BEC that depend mainly on the sign and magnitudes of the scattering lengths have

been demonstrated numerically [165]. Experimentally, the instabilities of BEC in

1D optical lattice has been observed [166]. Instability of a non-uniform initial state

in presence of a harmonic potential has been studied in an extensive manner both

analytically and numerically in the context of mean-field approximation [158]. The

experimental investigations of BEC conducted in optical lattices, and their dynam-

ical properties have been discussed in Ref. [167, 168]. Recently, Rapti et al. have

examined modulational and parametric instabilities arising in a non-autonomous, dis-

crete nonlinear Schrodinger equation in the context of BEC trapped in a deep optical

lattices [160, 169]. The dynamics of multiple domain formation caused by the mod-

ulation instability of two-component BEC in an axially symmetric trap has been

analyzed by numerically integrating the coupled GP equations [170].

1.8 Thesis outline

This thesis deals with the modulational instability and the theoretical study of soliton

dynamics in a system of BEC under the appropriate physical conditions. The MI

process deals with the stability of the continuous wave solution and pattern formation

in a system of BEC. The outline of the thesis is given below.

The second chapter deals with the occurrence of MI for continuous wave solutions

with constant amplitude in BEC trapped in optical lattices wherein the model is based


on a pair of averaged coupled GP equations. The stability of the condensate has been

analyzed in detail for the system physical parameters such as s-wave scattering length,

deviation of the nonlinear coefficient, condensate density. From this discussion, we

reported that the attractive condensate is more stable than the repulsive condensate,

because of the critical value of the atomic density for the attractive condensate is

relatively higher when compared to that of repulsive condensate. The analytical

results of MI obtained through linear stability analysis have been checked numerically

by direct simulation of the governing equation using the split step Fourier method.

Next, we have obtained the moving gap soliton solution that exists in the first spectral

band gap of the optical lattices in the presence and absence of the background atomic

scattering length for the governing system under consideration.

The third chapter discusses the investigation of the bright solitons on a cnoidal

wave train background for a system of BEC described by the inhomogeneous NLS

equation including the linear, harmonic potentials and feeding of the condensate

term. From this solution, we have obtained the dynamical properties of the system

of BEC under the influence of linear and harmonic potentials. The linear potential

term can play a crucial role in deciding the shape or trajectory of the BEC soliton.

The compression and broadening of the soliton as well as the cnoidal wave has been

controlled by the harmonic potential coefficient and the feeding of the condensate

term. Compression (broadening) occurs since the condensate density grows (decays)

exponentially with the increasing value of the feeding of the condensate in negative

(positive) sign.

The fourth chapter describes the analysis of the bright soliton on a continuous

wave background for a system of BEC described by the variable coefficients GP

equation. We have reported that two different possibilities based on the power of

soliton as well as continuous wave. One such case is the occurrence of MI when the

power of the continuous wave exceeds a quarter of the peak power of the soliton. The

second possibility arises when the power of the continuous wave is smaller than a

quarter of the peak power of the soliton. Under this condition, the resulting soliton

solution describes the exact bright matter wave soliton being always embedded with


continuous wave for the variable coefficients GP equation. The main characteristic of

BEC soliton on a continuous wave background is the amplitude of the BEC soliton

oscillating periodically without splitting of the soliton along the propagation in a

system of BEC i.e., the breathers like soliton can be generated in BEC. Further more,

the dynamical properties of BEC soliton on a continuous wave background with the

exponentially and periodically varying atomic scattering length in the context of BEC

have been discussed. In both exponentially and periodically varying atomic scattering

lengths, the total number of atoms in the soliton and the atomic exchange between

the soliton and continuous wave are independent of the linear potential but strongly

dependent on the time varying atomic scattering length. Based on this physical

process, one could infer that the number of atoms in the soliton remains constant

against the variation in the atomic scattering length.

In fifth chapter, the MI phenomenon and new type of exact periodic and soli-

ton solutions are obtained for the discrete complex cubic quintic Ginzburg- Landau

equation with non-local quintic term in systems of BEC trapped in deep optical lat-

tices. First, we have analyzed the stability and instability region of the condensates

for large and short wavelength limits through MI analysis. Numerical studies have

corroborated our analytical findings. By using the method of extended Jacobi el-

liptic function approach, we have derived a new type of periodic and solitary wave

solutions for the consider equation. These solutions consist of simple Jacobi elliptic

function and alternating phase Jacobi elliptic function solutions. Also, we have re-

ported the new type of kink and bubble soliton, alternating phase kink and bubble

soliton solutions.

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