Download Theoretical study of solitonic excitations in Bose

Ph.D. THESIS
Theoretical study of solitonic excitations
in Bose-Einstein condensates
by
Dániel Schumayer
Supervisor
Dr. Barnabás Apagyi
Associate Professor
Department of Theoretical Physics
Budapest University of Technology and Economics
2004
Nyilatkozat
Alulı́rott Schumayer Dániel kijelentem, hogy ezt a doktori értekezést magam készı́tettem és
abban csak a megadott forrásokat használtam fel. Minden olyan részt, amelyet szó szerint, vagy
azonos tartalomban, de átfogalmazva más forrásból átvettem, egyértelműen, a forrás megadásával
megjelöltem.
Budapest, 2004. augusztus 31.
Declaration
I confirm that no part of the material offered has previously been submitted by me for a
degree in this or in any other University. If material has been generated through joint work, my
independent contribution has been clearly indicated. In all other cases material from the work of
others has been acknowledged and quotations and paraphrases suitably indicated.
Budapest, 31th August 2004
......................................
iii
Theoretical study of solitonic excitations in Bose-Einstein condensates
Dániel Schumayer
Abstract of the Ph.D. thesis
Within the framework of my Ph.D. course we study the special kind of excitations appearing in Bose-Einstein condensates. These nonlinear type waves are generally referred as solitary
waves or solitons. The dynamics of a single or a multicomponent condensate can be modelled
with a nonlinear evolution equation called Gross-Pitaevskii equation. Firstly, we show that Coupled Gross-Pitaevskii equation can be totally integrable, because it passes the Painlevé-test [1].
Secondly, we give [2] a useful transformation between Coupled Gross-Pitaevskii equation and Coupled Nonlinear Schrödinger equations, the latter examined exhaustively in nonlinear fiber optics.
Later, a simple method is presented [3] to analyze the stability of static solitonic excitations of
two-component Bose-Einstein condensates. A quick assessment can be made [4] for the values of
interspecies scattering length a12 allowing existence of static solitons independent of the particle
numbers of the component. If the true value of a12 falls into this acceptable interval, then a
fine tuning is performed to get the ratio of the proper particle numbers at which the static solitons may be created. The technique is illustrated for the lithium-rubidium and sodium-rubidium
condensates.
iv
Theoretical study of excitations in Bose-Einstein condensates
Dániel Schumayer
(Summary of the Ph.D. thesis)
On the basis of quantum mechanics and statistical physics it is verifiable theoretically that a
macroscopical part of system of bosonic particles fills the same quantum state at low temperature.
This effect was predicted in 1924, but the experimental observation has been carried out only in
1995 and extensive theoretical study of such systems started and diversified from that time. The
great interest can be explained by the fact that the study of Bose-Einstein condensates (BEC)
extends our knowledge in many different branches of physics.
One of our aim is to describe the dynamical behaviour of such system and thus examine the type
of excitations in that. This requires using many-body formalism in which only two-particle hardsphere interaction is taken into account. The governor equation is the so called Gross-Pitaevksii
equation. The objective of my Ph.D. work was to find special type of excitations, called solitons,
in two-component Bose-Einstein condensation in the Gross-Pitaevskii approximation. We were
motivated by the experimental observation made in one-component systems. Two groups have
produced both bright [5] and dark [6] solitonic excitations in Bose-Einstein condensates.
We have sought out whether or not the coupled Gross-Pitaevskii equations have solitonic solutions. This question is related with the notion of integrability. If the equations are integrable then
we can try to solve it using the Inverse Scattering Transform. In addition the restrictions required
by the integrability make limitations on some physical parameters.
We summarize here the new results presented in this thesis:
1. Exploring the integrability of coupled Gross-Pitaevskii (CGP) equations we applied the Painlevéanalysis [1]. This method is based on the classification of singularities of the solutions of a
differential equation.
2. Computer algebra is applied [1] to evaluate a compatibility condition arisen in Painlevé test for
admissible external potentials. This requirement results a constraint on the external potential.
3. We can derive some constraints on the parameters of a two-component Bose-Einstein condensates in which cases the integrability of the system is preserved, if we do not allow transferring
between the two states.
4. Based on the similarity between the coupled non-linear Schrödinger equations with or without
external potentials, a relation is established between their solutions [2]. The external potentials
are assumed to modify both amplitudes and phases together with the spatial and temporal
variables appearing in the transformation relation.
5. Besides the analytical calculations we developed [7] a numerical code for the Lax-representation
of the Inverse Scattering Transform. We checked this code on other well-known nonlinear
dynamical equation.
6. A simple method is presented [3] to analyze the stability of static solitonic excitations of twocomponent Bose-Einstein condensates. We have achieved a quick assessment for the values of
interspecies scattering length allowing existence of static solitons independent of the particle
numbers. If the physical value of interspecies scattering length falls into that acceptable interval,
then a fine tuning is performed to get the ratio of particle numbers in each component at
which the static solitons may be created. The technique is illustrated for four two-component
systems [3, 4].
The reviews of this Ph.D. thesis and the report about the defense of the dissertation will be
available at the Dean’s Office further on.
v
Bose-Einstein kondenzátum szoliton tı́pusú gerjesztéseinekelméleti vizsgálata
Schumayer Dániel
(Ph.D. dolgozat összegzése)
A kvantummechanika és statisztikus fizika eszköztárával belátható, hogy bozonok alkototta
rendszerben a részecskék makroszkopikus hányada ugyanazon kvantumállapotot tölti be kellően
alacsony hőmérsékleten. Bár az effektust már 1924-ben megjósolták, kı́sérletileg csak 1995-ben
igazolták a létét, azóta is igen kiterjedt vizsgálat tárgyát képezi. Ezt a kitüntető figyelmet avval
magyarázhatjuk, hogy a Bose-Einstein kondenzáció jelenségének vizsgálata rendkı́vüli hatással van
a fizika jelentősen különböző területeire.
Célunk a Bose-Einstein kondenzátum dinamikájának, illetve a benne kelthető gerjesztések
leı́rása. Ez a soktestprobléma eszköztárát igényli, de csökkenthetjük a feladat nehézségét ha csak
két részecske között ható merev-gömb kölcsönhatást vesszük figyelembe. Az ı́gy nyert mozgásegyenletet nevezzük Gross-Pitajevszkij egyenletnek. A Ph.D. dolgozatom egyik célja éppen az volt,
hogy a két-komponensű Bose-Einstein kondenzátum speciális, ún. szoliton tı́pusú gerjesztéseit
vizsgáljuk. Tanulmányainkat az egy-komponensű rendszereken végzett kı́sérleti megfigyelések motiválták. Két csoportnak sikerült bright [5] illetve dark [6] szoliton gerjesztést előállı́tania.
Elsőként azt vizsgáltuk: lehet-e a csatolt Gross-Pitajevszkij egyenletnek szoliton megoldása.
Amennyiben létezhet, akkor találjunk módszert a megoldások előállı́tására. A kérdés szorosan
kapcsolódik az integrálhatóság fogalmához. Amennyiben az egyenletünk teljesı́ti az integrálhatóság
föltételeit, abban az esetben megpróbáljuk az Inverz Szórás Transzformáció segı́tségével megoldani
azt.
Az alábbiakban összefoglaljuk a dolgozatban közölt eredményeinket:
1. A csatolt Gross-Pitajevszkij (CGP) egyenlet integrálhatóságát a Painlevé analı́zis módszerével
vizsgáltuk [1]. Ez a technika egy adott differenciálegyenlet megoldásainak szingularitásait
osztályozza.
2. Számı́tógépes algebrai programmal sikerült kiértékelni a Painlevé tesztből adódó kompatibilitási
feltételeket, amik az integrálhatóságot nem sértő külső potenciálokra adnak megszorı́tásokat.
3. További megkötéseket vezettünk le a rendszert jellemző paraméterekre az integrálhatóságot
megőrizve, föltéve hogy a két komponens nem alakulhat át egymásba.
4. A csatolt nemlineáris Schrödinger egyenlet potenciálos és potenciál-mentes alakjainak hasonlóságát kihasználva a hozzájuk tartozó megoldások között transzformációt vezettünk le [2], föltételezve: a külső potenciál jelenléte a hullámfüggvénynek mind az amplitudóját, mind fázisát
befolyásolja.
5. Az analitikus számolások mellett kifejlesztettük az Inverz Szórás Transzformáció egy numerikus
kódját a Lax-féle formalizmusát használva. Ezt más nemlineáris egyenlet ismert megoldásain
ellenőriztünk.
6. A Gross-Pitajevszkij közelı́tésen belül vizsgáltuk kétkomponensű Bose-Einstein kondezátumban
létrehozható statikus szoliton gerjesztések stabilitását [3]. Adtunk [3] egy, az egyes komponenseket alkotó atomok számától független leı́rást a kereszt-szóráshossz lehetséges értekeire.
Amennyiben a fizikailag mért szóráshossz a megadott intervallumba esik, a részecskék számát
is pontosan meghatározhatjuk. Az eljárást bemutattuk több különböző kı́sérletileg preferált
két-komponensű rendszer, mint például Li-Rb és K-Rb, esetében.
A dolgozat bı́rálatai és a védésről készült jegyzökönyv a későbbieken a dékáni hivatalban
elérhető.
vi
Acknowledgments
Looking back over the years, I recall with heartfelt gratitude the many people who have inspired
and encouraged me along my journey that began in the Veres Péter High School. I am very
fortunate to have been guided along the way by teachers, family, friends, and colleagues, who have
all played a role in helping me to realize my potential. It is important for me to express my sincere
appreciation of their support.
I should not forget my favorite high school teacher, György Székely, who were the first teacher to
really inspire me over a decade ago, and whose passion for science was fascinating and stimulating.
I am greatly indebted to my supervisor Professor Barnabás Apagyi who supported me along
this four-year period and produced the most convenient circumstances to my development. I could
always count on him irrespectively of the nature of my problem. He never spared any effort in
helping me and we discussed the problems even on weekends.
I am also very grateful to Professor István Nagy and József Fortágh, Ph.D. for many fruitful
discussions.
I owe my deepest thanks to my parents, who tolerated my different moods and have always
supported me without question and who brought me into a loving and caring family. All of the
education in the world can not teach me what they have mastered. . .
vii
Preface
This thesis is the continuation of our investigations began in my diploma thesis on the quickly
developing area of Bose-Einstein condensation. The results reported here do not belong to one
topic, but are split into two areas which are not connected directly to each other, but are related
in a broader sense. This partition is reflected in the structure of the thesis and we further divide
the latter topic into two distinct parts for which our publications can be attached to. A brief
introductory section precedes all separate parts – belonging to the published articles – in which
we review the necessary notions, results and experimental observations (if present). The structure
of this thesis can be sketched up in the following way:
I. General introduction:
Appearance of nonlinear equations of motion in different areas of physics; e.g., water waves,
electromagnetic waves, Josephson effect, etc.
II. Phenomenon of Bose-Einstein condensation:
Overview of phenomenon of Bose-Einstein condensation and derivation of Gross-Pitaevskii
equation
III. Mathematical results about coupled Gross-Pitaevskii equations:
• Introduction to Painlevé-test;
(a) Kovalevskaia’s ”physical”-approach: connection to a classical mechanical problem;
(b) Painlevé’s purely ”mathematical”-problem: characterization of ordinary differential
equations according to singularity structure of their general solutions
(c) Idea of integrability in classical mechanics, especially in Hamilton-formalism
• Painlevé test of coupled Gross-Pitaevskii equations (J. Phys. A. 34, 4969 (2001))
• Short introduction to the phenomenon of Bose-Einstein condensation and derivation of
Gross-Pitaevskii equation
• Transformation between coupled Nonlinear Schrödinger and coupled Gross-Pitaevskii
equations; comparison with the result of Painlevé’s test of ”integrability”. (Phys. Rev.
A. 65, 053614 (2002))
IV. Dark-bright solitonic excitations:
• Earlier stability investigations both in the cases of one- or multicomponent condensates.
The decisive role of the scattering length(s); Short summary of the experimental achievements producing multicomponent condensates with paying definite attention to solitonic
excitations.
• Origin of the chosen ansatz; (Phys. Rev. A., 69, 043620 (2004))
V. Complementary parts: appendices, bibliography, index
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Contents
Official and complementary parts
ii
Preface
1 General Introduction
1.1 Waves in fluids . . . . . . . .
1.2 Electromagnetic waves . . . .
1.3 Gravitational waves . . . . .
1.4 Waves on spin-lattice . . . . .
1.5 Waves in Josephson junction
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2 Bose-Einstein condensation
2.1 Brief history of Bose-Einstein condensation . . . . . . . . . . . . . . . . . . . . . .
2.2 Effects of interaction: Gross-Pitaevskii equation . . . . . . . . . . . . . . . . . . . .
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3 Painlevé-analysis
3.1 Kovalevskaya top . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Painlevé analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Ordinary differential equations . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Partial differential equations . . . . . . . . . . . . . . . . . . . . . . . .
Exercise: One-component Nonlinear Schrödinger equation . . . . . .
3.3 Hamiltonian Formalism and Inverse Scattering Transformation . . . . . . . . .
3.3.1 Integrability in classical mechanics . . . . . . . . . . . . . . . . . . . . .
3.3.2 Action-angle canonical variables . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Construction of action-angle variables . . . . . . . . . . . . . . . . . . .
Exercise: Harmonic oscillator in 1D . . . . . . . . . . . . . . . . . . .
3.3.4 Inverse Scattering Transformation . . . . . . . . . . . . . . . . . . . . .
3.2.
Lax-representation . . . . . . . . . . . . . . . . . . . . . . . . .
Exercise: A possible Lax-representation of the KdV equation . . . . .
3.3.
AKNS-representation . . . . . . . . . . . . . . . . . . . . . . .
Exercise: Bright multi-soliton solution of KdV equation by using IST
Exercise: Bright multi-soliton solution of ZS equations . . . . . . . .
3.3.5 Relation of Hamiltonian formalism and IST . . . . . . . . . . . . . . . .
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CONTENTS
CONTENTS
4 Painlevé test of CGP
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Painlevé test . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Determination of the leading orders . . . . . .
4.2.2 Recursion relations . . . . . . . . . . . . . . . .
4.2.3 Resonances . . . . . . . . . . . . . . . . . . . .
4.2.4 Compatibility conditions . . . . . . . . . . . . .
4.3 Possible forms of the external potentials . . . . . . . .
4.3.1 Conditions for the potentials arising from j = 3
4.3.2 Conditions for the potentials arising from j = 4
4.4 Discussion of the results . . . . . . . . . . . . . . . . .
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . .
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5 Relation between optical and atomic solitons
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Some results of P analysis of CGP equations . . . . . . . . . . .
5.3 Transformation between solutions of CNLS and CGP equations .
5.4 Derivation of transformation functions from given trap potentials
5.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 Bright-dark optical solitons . . . . . . . . . . . . . . . . .
5.5.2 Linear potentials . . . . . . . . . . . . . . . . . . . . . . .
5.5.3 Quadratic potentials with imaginary parts . . . . . . . . .
5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6 Stability of static solitonic excitations of two-component BEC
in finite range of interspecies scattering length a12
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Ground state profiles . . . . . . . . . . . . . . . . . . . . . . .
6.2.2 Static solitonic excitation . . . . . . . . . . . . . . . . . . . .
6.2.3 Static solitonic excitation profiles . . . . . . . . . . . . . . . .
6.3 Illustrative Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 a12 values allowing stability and solitonic excitations of BECs
6.3.2 a12 from earlier studies . . . . . . . . . . . . . . . . . . . . .
6.3.3 Comparison to results derived from Painlevé test . . . . . . .
6.3.4 Summary of the static solitonic excitation method . . . . . .
6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Appendix
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A Theory of Bose-Einstein condensation
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A.1 Homogenous infinite ideal Bose-gas . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
A.2 Condensation in presence of confining potential . . . . . . . . . . . . . . . . . . . . 112
Bibliography
131
x
Chapter 1
General Introduction
A stimulating and extremely active area of research investigation from the seventies has been the
study of a special kind of wave phenomenon. In this introductory chapter we would like to convince
the Reader of importance of solitary waves and solitons. To fulfill this purpose we enumerate some
examples from very diverse areas of physics, and identify nonlinear coherent structures in nature
on scales ranging from nanometers upto thousands of kilometers, or in other words ranging from
atomic physics upto the motion of Jupiter’s giant Red Spot. We shall not trouble the Reader
with details. Our aim is just to make a deep impression how strange, amazing and diversified the
nonlinear science is.
In the past several decades, two major themes have dominated developments in the theory
of dynamical systems. On the one hand there has been a remarkable and rapid development in
the theory of so-called ”chaotic” systems, with a gradual clarification of the nature and origins
of the surprising properties from which these systems get their name. Here we should explain
how a system that is deterministic can nevertheless exhibit behavior that appears random and
unpredictable. On the other hand, there stand those dynamical systems which possess stable,
particle-like solutions.
In this thesis we will discuss a second class of systems equally puzzling, but for almost the
opposite reason. For these so-called ”integrable systems”, the challenge is to explain the striking
predictability, regularities, and quasi-periodicities exhibited by their solutions, a behavior particularly apparent for a special class of solutions, called ”solitons”. The latter exhibit a ”particle-like”
behavior that gives them their name; for example they have geometric shapes that show a remarkable degree of survivability under conditions that one might normally expect to destroy such
features.
Generally, in most of our investigations in physics it is enough to solve linear equations in
which we do not meet any difficulties at all, since theorems of existence and uniqueness, methods and techniques are known in many areas of mathematics (linear algebra, linear differential
equations, Fourier-transformation theory, etc.) utilizing the fundamental superposition principle.
The concept of linearity is very closely related to that of reductionism. Reductionism is an approach to science that says that a system in nature can be understood solely in terms of how its
parts work. Superposition means that adding any two solutions of a linear system produce a new
solution to the same equation. This is responsible for many systematic methods used to solve
such kind of equations. In its background, Fourier and Laplace-transform also take the advantage
of superposition, when the linear problem is broken into small pieces then solved independently
1
CHAPTER 1. GENERAL INTRODUCTION
and at the end of all the machinery these elementary separate solutions are added together. In
contrast, superposition fails in case of nonlinear equations, or more precisely saying there exist
a superposition-like rule (Bäcklund-transform), but we are unable to exploit it for reducing our
original problem into small subproblems. It is undoubted that if the nonlinear equations would
be successfully treated, we ought to change our outlook and new methods, procedures must be
developed. Change of paradigms must be done. Maybe, here we should mention that vast majority
of natural phenomena and the underlying mathematical equations are nonlinear, and linearity is
only an exceptional, but truly important, case.
One of the standard techniques for solving linear partial differential equations is the Fouriertransform. During the last three decades it was shown that a class of physically interesting nonlinear partial differential equations can be solved by a nonlinear extension of the Fourier technique,
namely the Inverse Scattering Transform. This reduces the solution of the Cauchy problem to a
series of linear steps.
Both linearity and reductionism fail, at least as general principles, for complex systems. In
complex systems there are often strong interactions between system parts and these interactions
often lead to the emergence of patterns and cooperation. That is, they lead to structures that are
the properties of groups of parts, and not of the individual constituents. Naturally the importance
of nonlinearity, beginning with the Navier-Stokes equations and continuing to gravitation theory,
and the interactions of particles in solids, nuclei, and quantized fields was recognized. However,
it was hardly possible to treat the effects of nonlinearity, except as a perturbation to the basic
solution of the linearized theory.
A revolution has occurred on the way where physicists attempt to describe certain types of
nonlinear phenomena. Whereas the prevailing tendency before soliton revolution was to include a
few more terms in perturbation series or to renormalize the properties of linear excitations, etc., it
is now recognized that fundamentally new types of approaches are necessary for the understanding
of a great many physical processes.
During the last decade, it has become more widely recognized in many areas of ”field physics”
that nonlinearity can result in qualitatively new phenomena which cannot be constructed via
perturbation theory starting from linearized equations. These include classical areas, such as
hydro- or magnetohydrodynamics, and thus also some areas of meteorology, oceanography, plasma
physics, as well as newer areas such as solid state physics, nonlinear optics and elementary particle
physics.
In this thesis we shall deal with the integrability and special kind of solutions of a given nonlinear
partial differential equation arising in the description of a Bose-Einstein condensate. Before the
detailed examinations we give a short summary about the concept of integrability as it appeared
in classical mechanics. But we emphasize that the fundamental problem has not been already
answered, namely there is no widely accepted definition of integrability and therefore no criterion
or classification of the completely integrable systems.
Despite the lack of rigorous definition of integrability previous theoretical and numerical investigations have proven that given nonlinear equations of motion (or evolution equations as alternatively they can be denominated) possess strange particle-like solution, the so called soliton
solutions. The existence of such solutions is unpredictable in many cases from methods widely
used in linear problems, e.g. perturbative calculations. In mathematical literature only localized solutions of exactly integrable systems are commonly referred to as solitons. While localized
excitations described by nearly integrable nonlinear equations are termed solitary waves.
The completely integrable equations describe idealized dynamical systems. They take no account of the phenomena connected with presence of dissipation and small perturbations. It cannot
2
CHAPTER 1. GENERAL INTRODUCTION
be gainsaid that in describing some real systems one cannot restrict oneself to the investigation
of only completely integrable equations. Describing real systems even unstable solitary waves can
be significant if their lifetime is long in comparison with the time during which the phenomenon
under study takes place.
In solid-state physics approximate conceptions of elementary excitations - quasi-particles, described by plane waves, more correctly by wave packets - play a decisive role. General solutions
of many linear phenomena are expressed by such linear or nearly linear modes. Solitons seem to
play the same role in nonlinear dynamics.
The most important and strange phenomenon is the emergence of solitary wave and solitons.
Soliton is a nonlinear wave which has the following properties:
• owes its existence to a balance of two forces acting in opposite sense: dispersing and focusing,
• localized and propagates without change of its properties (shape, velocity, energy, etc.),
• remarkable stability against disturbances and non-destructive mutual collisions,
• particle-like properties (attract or repel one another),
• nonlinear superposition of waves (Bäcklund-transform)
Popularly saying these objects incorporate the idea of duality [8] since they are the wave
function solutions of nonlinear field equations, but under collisions they preserve their identities,
mass, energy and momentum can be assigned to them. Moreover, in many cases they can treated
as an elementary quantum of a fields, e.g. a fluxon for magnetic field in case of Josephson-effect.
Summarizing, we can state that the study of wave phenomena in physics is undergoing dramatic
changes as a result of the discovery of solitons [9]. These localized nonlinear waves have a number
of unusual properties compared to linear waves, which have formed the basis for most physical
theories of wave phenomena. Examples of physical systems that have been shown to have soliton
solutions are (without order of significance):
• water waves,
• oscillating shock waves in optical fibers [10],
• ion-acoustic waves and magnetohydrodynamic waves in plasma [11],
• propagation of compressional waves through crystal lattice [12],
• phonon packets in low-temperature nonlinear crystals [13],
• elastic surface pulses [14] and non-propagating disturbances [15],
• flexural modes of thin shells [16],
• pressure waves in liquid-gas bubble mixtures [17],
• deformation of a floating ice sheet under moving load [18].
3
CHAPTER 1. GENERAL INTRODUCTION
1.1
1.1. WAVES IN FLUIDS
Waves in fluids
The problem and difficulty arising from nonlinear effects can be avoided generally, since we are
interested mostly in either stable equilibrium points of the system or the behaviour of the system in
the neighborhood of the equilibrium point. In the latter case we use the well-developed perturbative
techniques and we are settled for the linear response. Nevertheless, several examples can be
enumerated in which the basic equation is already nonlinear, for instance the hydrodynamical
Navier-Stokes equation:
∂v
η
+ (v∇)v = −grad(p) + η∆v + ζ +
grad(div(v))
ρ
∂t
3
This nonlinearity may cause emergence of such solitary water-wave which was observed many
time in several seas [19]. Perhaps, the most amazing observation was published by Apel and
Holbrook [20]. During two days they followed a solitary water-wave with hundred meter amplitude
and velocity at eight kilometer per hour in two-thousand and fifty kilometers. This macroscopic
coherence can not be explained from linear evolution equation. If the water column consists of an
z
surface soliton
c
rip
isotachs
ρ2
η0
η (x; t)
ρ1
internal soliton
h
x
Figure 1.1: Left: Surface solitary wave with amplitude η0 moving to the right with phase speed c in water of depth
h. Right: Essential elements of two-layer fluid model for describing the phenomenon of internal soliton. The lower
layer is assumed to have depth h1 and density ρ1 , while in the upper layer the respective quantities are h2 and ρ2 .
In true ocean the small density difference is due primarily to temperature and salinity variations. Dashed lines are
lines of constant water particle velocities (isotachs) and the arrows indicate the direction of currents. For sake of
comparison and demonstrating the difficulty of observation it is worth mentioning that the typical height of surface
soliton is ∼ 0.3-0.6 [m] while the proper measure of internal soliton is 50-80 [m]. During a four-month measurement
A. R. Osborne and his colleagues were able to map isotherm contours of an internal soliton in the Andaman see
which was in good agreement with result of numerical simulation using this two-layer model [19]. They made many
conclusions, for example: (i) internal solitary waves occur in packets of rank-ordered waves, the largest leading the
rest, (ii) packets occur every twelfth and a half hours, indicating a tidal origin for the phenomenon (iii) internal
solitons interact strongly with surface waves, resulting in surface rips which extend from horizon to horizon, and
are about 1 [km] wide.
upper layer and a denser lower layer, the interface between the layers can undergo wave motion
(see figure 1.1). This motion, which does not affect the surface and mostly is not observable at the
surface, is an example of an internal wave. The most common internal waves are of tidal period and
manifest themselves in a periodic lifting and sinking of the seasonal and permanent thermocline at
tidal rhythm. In some ocean regions their surface expressions, produced by convergence over the
wave troughs, are visible in satellite images. Internal waves become visible on radar images because
they are associated with a variable surface current which modulates the sea surface roughness and
thus the backscattered radar power.
4
CHAPTER 1. GENERAL INTRODUCTION
1.1. WAVES IN FLUIDS
Figure 1.2: Left: Synthetic Aperture Radar (SAR) image about sea surface manifestations of two internal wave
packets taken on 4th of May in 1996. This photo shows a typical size of such surface wave following the internal
wave below. Middle: Synthetic Aperture Radar (SAR) image from the Shuttle Imaging Radar-A (SIR-A) flight on
November 11, 1981. For centuries seafarers passing through the Strait of Malacca on their journeys between India
and the Far East have noticed that in the Andaman Sea bands of strongly increased surface roughness often occur.
These have also been referred to as bands of choppy water or ripplings and have been found mainly between the
Nicobar Islands and the north east coast of Sumatra. It contains several internal wave packets in the vicinity of the
Andaman and Nicobar Islands. This image has dimensions of ∼ 51 × 51 kilometers, with the white marks along
the lower border being a distance of 7.14 kilometers. Photo courtesy of Werner Alpers (Institute of Oceanography,
University of Hamburg). Right: The right hand side picture was taken on 12 July 1995 standing on the Scott
Russell Aqueduct on the Union Canal near Heriot-Watt University, where an international conference was held at.
The aqueduct is 89.3[m] long, 4.13[m] wide, and 1.52[m] deep. The solitary wave phenomenon realized at first time
by John Scott Russel was successfully recreated in Edinburgh. Photo courtesy of Dugald Duncan (Department of
Mathematics, Heriot-Watt University, Scotland)
The Andaman Sea of the Indian Ocean is known to be a site in the world’s ocean where extraordinarily large internal solitons are encountered. Synthetic Aperture Radar (SAR) images of the
ERS satellites reveal that the large internal solitons previously detected by in-situ oceanographic
measurements in the western approaches of the Strait of Malacca between Phuket (Thailand) and
the northern coast of Sumatra (Indonesia) are generated at shallow banks in the western part
of the Andaman Sea. When propagating onto the shelf of the Malayan Peninsula, their spatial
separation decreases and their shape becomes irregular, but they remain solitons of depression.
Internal solitary waves appear to be ubiquitous in the stratified ocean. Their widespread
occurrence has been established by satellite and aircraft imagery acquired at a large number of
sites around the world. Such imagery, together with in situ measurements and solitary wave theory,
has allowed detailed and revealing studies of these nonlinear waves to be carried out. Soliton surface
signatures arise from the interaction of the wave currents with surface roughness, and are frequently
used to locate experimental sites and to design observations that illuminate the detailed dynamics
of the waves. As a result, a well-established body of information on solitons has been obtained that
verifies the Korteweg- de Vries (see later) equation as their governing relationship. The images
suggest a global generating mechanism-tidal flow against ocean bottom features that protrude up
into the thermocline. Imaging radars, both real and synthetic aperture, are the premier sensors
for observing solitary waves from above. The ability of such radars to make all-weather, day/night
observations of subtle variations in surface roughness has been well documented. The wave equation
demonstrates significant skill in predicting the modulations of the surface wave spectrum arising
from solitary wave currents. Part of this skill is due to improvements in the description of the
5
CHAPTER 1. GENERAL INTRODUCTION
1.1. WAVES IN FLUIDS
surface wave vector spectrum. A new model for the wave spectrum has been developed which,
together with advanced radar scatter theory, can reproduce aircraft and satellite measurements
of the radar cross section of the ocean to well within the measurement errors. In particular, the
improved spectral model yields the correct dependence of the vertically polarized backscatter as
a function of wind velocity, radar frequency, and incidence angle. However, significant problems
remain with horizontal polarization, large incidence angles, and unstable air/sea interfaces.
Perry and Schimke (1965) were the first to show by oceanographic measurements carried out
from a ship that these bands of choppy water in the Andaman Sea are associated with largeamplitude oceanic internal waves. Later Osborne and Burch (1980) analyzed oceanographic data
collected by the Exxon Production Research Company in the southern Andaman Sea with the
aim to assess the impact of underwater current fluctuations associated with oceanic internal waves
on drilling operations carried out from a drill ship. They concluded that the visually observed
roughness bands are caused by internal solitons which can be described by the Korteweg - de Vries
equation (1885).
Unlike the well-known long trains of ocean swells that sweep past ship and swimmer with great
regularity, solitary waves move ”in splendid isolation, steadfastly holding their shape.” Spacecraft
photos have revealed curious striations in the Andaman Sea near Thailand. They are presumed to
be examples of solitary waves. The Andaman waves extend for many miles and travel very slowly
– less than 10 kilometers per hour. They propagate along the boundary between the layer of warm
surface water and the great mass of cooler water below. The amplitude of the downward pointing
wave troughs of warm water along this interface may penetrate as far as 100 meters into the cold
water below.
The observation of solitary waves described first by a young Scottish naval architect named John Scott Russel (1808 Parkhead,
Scotland - 1882 Ventnor, Isle of Wight, England) [21] during riding
on a horse by the side of the Union Canal near Edinburgh in 1834.
Russell became fascinated with this phenomenon and made extensive further experiments with such waves in a wave tank of his
own devising, eventually deriving a correct formula for their speed
as a function of height. The subsequent theoretical description by
two Dutch researchers Diederik Johannes Korteweg and Gustav de
Vries represented [22] the extent of physical understanding of solitary waves at the beginning of this century about sixty years after
Russel’s publication. It was generally believed that the collision of
two solitary waves would result in a strong nonlinear interaction
and ultimately end in their destruction. Then in 1965 at PrinceFigure 1.3:
ton University two mathematical physicists, Norman Zabuski and
John Scott Russel (1808-1882)
Martin Kruskal [9] had discovered the existence of special localized
waves and reported on a computer experiment in which they simulated the Korteweg-de Vries equation (KdV) for the collision of two solitary waves. The result was surprising: the waves retained
their shapes and propagation velocities after the collision. Because of the somewhat elementary
particle-like behaviour of these waves they coined the word ”soliton” to describe them.
The KdV equation for the propagation of solitary waves, after normalization of amplitude of
wave and scaling the horizontal coordinate and time, is given by
uτ + 6uux + uxxx = 0
√
where u = 3η/2h, x = (r − c0 t)/h, τ = c0 t/6h, c0 = gh is the phase speed of associated linear
6
CHAPTER 1. GENERAL INTRODUCTION
1.2. ELECTROMAGNETIC WAVES
wave and η is the amplitude of the solitary wave as a function of horizontal displacement r down
the channel and of time t, h is the water depth, g is the acceleration due to gravity and we use the
subscript to denote the partial derivative with respective to the proper variable, τ or x. If we add
the boundary conditions that u(x; τ ) should vanish at infinity, then a fairly routine analysis leads
to the one-parameter family of travelling-wave solutions
u(x; τ ) = 2κ2 sech2 (a[x − 4κ2 τ ])
now referred to as the one-soliton solutions of KdV. It is worth noting that the amplitude is exactly
half the speed, so that taller waves move faster than their shorter humps, and retain their shapes
in time.
It is apparent that the KdV equation is a nonlinear partial differential equation due to the
presence of the uux term. The uxxx term makes it dispersive, i.e. in general an initial wave u(x; 0)
will broaden in space as time progresses. As pointed out in the preamble there are the two forces
– a dispersive and a focusing – acting in the opposite directions, the KdV possesses certain special
solutions, known as solitary wave solutions, which would not be expected from a nonlinear and
dispersive equation of motion.
1.2
Electromagnetic waves
The Maxwell equations seem to be linear for the first sight:
∇D = ρ
∇B = 0
∂D
∂t
∂B
∇×E=−
∂t
∇×H=J+
However we class them among nonlinear equation of motion from practical point of view, since the
governor equation of electromagnetic field in presence of matter becomes nonlinear, if our model
of matter is not limited to the simple linear tensorial relationship
D = E + P = id +χ E = E
B = H + M = id +κ B = µ H
where id denotes the identity operation. For example, in optical fibers made from dielectric material
the strong electromagnetic field deforms the electron orbits in glass molecules and produce that
the polarization P depends on the intensity of the light. In this way the intensity (∼ amplitude 2 )
of electromagnetic field dramatically influence the behaviour of matter. This effect is known as
Kerr-effect in this context. If the connection between polarization and electrical field is
h
i
P = (1) E + (2) E ◦ ET E
where ◦ denote the diadic multiplication of two vectors. Using this relation the Maxwell equation
of motion become nonlinear and produce [23] the famous Nonlinear Schrödinger equation for the
complex envelope of electric field E(x, t):
i
∂E
k ∂2E
2
−
+ g |E| E = 0
∂x
2 ∂t2
7
(1.2.1)
CHAPTER 1. GENERAL INTRODUCTION
1.2. ELECTROMAGNETIC WAVES
In contrast to the KdV equation which describes – for example – the motion of a solitary wave
formation on water, the Nonlinear Schrödinger equation govern the time evolution of solitary wave
in optical fibers. In this respect, the optical soliton in a fiber belongs to the category which is
generally referred to as an envelope soliton. The optical pulses which are used in communication
are ordinarily created by the pulse modulation of a light wave. Here the pulse shape represents an
envelope of a light wave.
Hasegawa and Trappert [24], researchers at AT&T Bell Laboratory 1, were the first to show
theoretically that an optical pulse in a dielectric fiber may form a solitary wave due to the fact the
wave envelope satisfies the nonlinear Schrödinger equation. They have also shown that in the case
where the coefficient of the second term is negative of equation (1.2.1), the solitary wave appears
as the absence of a light wave, and called this a ”dark” soliton. In 1980 Mollenauer et al. at
AT& T Laboratories announced the result of a landmark experiment. They propagated a 10 [ps]
optical pulse with a wavelength of 1.5 [µm] through a 700 [m] low loss fiber and demonstrated
for the first time the successful propagation of optical solitons in a fiber [25]. They were able to
generate pulses of the shape required by the theory. They found that a low power input pulse
of this shape gave rise to an output pulse showing dispersive broadening. As the power of input
signal was increased, the broadening became less and less and at a critical power the output pulse
was the same shape and size as the input pulse. In other words, it was a soliton. These objects
with remarkable properties are the natural bits for optical communications, and nowadays the
qubits of quantum information theory. This key concept has been at the heart of extensive and
intensive research in the last quarter. The series of successful experimental and theoretical studies
Year
1988
1990
1991
1992
1993
1994
1995
∗
Bit rate (Gb/s)
0.1
0.2
20
2.5
10
5
5
40
40
10
20
80
15
160
20
20
Distance (km)
4000
10000
200
14000
106
3000
15000
1000
205
20000
125000
500
25000
200
150000
2000
Company
AT& T
AT& T
NTT
AT& T
NTT
KDD
AT& T
NTT
British Telecom
AT& T
British Telecom
NTT
AT& T
NTT
France Telecom
NTT
Reference
Mollenauer and Smith [26]
Mollenauer [27]
Nakazawa, Suzuki [28]
Mollenauer, Neubelt [29]
Nakazawa, Yamada [30]
Taga [31]
Mollenauer, Lichtman [32]
Nakazawa [33]
Ellis [34]
Mollenauer [35]
Widdowson and Ellis [36]
Nakazawa, Yoshida [37]
Mollenauer [38]
Nakazawa [39]
Aubin [40]
Nakazawa [41]
Table 1.1: The tabulation shows data of some soliton transmission experiment borrowed from [42]. Comparing
these efforts we can see big differences both in bit rate and in distance, and in some cases the development is not so
obviously visible. This can be explained by the different techniques and technologies used in the experiment. The
notation ∗ is used for demonstration in a metropolitan optical network. (The selection is arbitrary).
of the behaviour of optical solitons have now lead many people to seriously consider using solitons
1 see
“http://www.bell-labs.com” or “http://www.att.com” webpages of AT&T
8
CHAPTER 1. GENERAL INTRODUCTION
1.2. ELECTROMAGNETIC WAVES
in optical communications. Optical solitons in fibers are now becoming a ”real thing”, rather
than merely interesting mathematical objects. As happens so often, the long-distance soliton
communications experiments spurred on the competition, see Table 1.1. It is clear as a daylight
that real manifestation of solitons and fabrication such supporting optical fibers are not just a
marginal problem and l’art pour l’art trickery of telecommunication industry. The qualities of the
soliton wave (the fact that it does not break up, spread out or lose strength over distance and is
not disturbed by weak noise) make it ideal for fibre-optic communications networks where billions
of solitons per second carry information down fibre circuits for cable TV, telephone and computers.
This nonlinear behaviour is utilized ultimately. As we mentioned before, the response of fibre
material depends on the intensity of applied electromagnetic field. In the nonlinear (high intensity)
regime non-spreading wave packets exploit the idea that, in self-focusing media, the nonlinear wave
front curvature can simultaneously balance the curvature due to diffraction and group-velocity
dispersion, combining features of spatial and temporal solitons to form a bell-shaped 3D-localized
wave packet, the so-called light bullet [43].
An Italian experimental group managed to observe [44] such formation of an intense optical
wave packet fully localized in all dimensions , i.e., both longitudinally (in time) and in the transverse
plane, with an extension of a few tens of femtosecundum and micrometer, respectively.
When a short pulse travels in a nonlinear medium, it may contract in space and in time due to
the combined effect of self-phase modulation, diffraction and dispersion. They studied numerically
the propagation of such short pulses by integrating directly Maxwell’s equations. They verified
that under certain conditions it can be found out that a pulse is stabilized by the nonlinear process
against diffraction and dispersion, forming a light bullet.
The electromagnetic fields are continuous functions of space and time, however, there are
situations in which the evolution of an optical field can be represented as a discrete problem.
This happens when the field can be described as a sum of discrete modes. An important case
is that of a coupled one-dimensional waveguide array. In a waveguide array, a large number
of single-mode channel waveguides are laid one near the other such that their individual modes
overlap. The evolution of the transversal field distribution is described by an infinite sum of coupled
complex amplitudes of the individual modes. Later it was under investigation what happens if the
waveguides have intrinsic Kerr nonlinearity, as usually all waveguides have. It was demonstrated
that the field evolution in an array exhibiting a such nonlinearity can be described by a discrete
version of the nonlinear Schrödinger equation, and solitary solutions of that are frequently termed
discrete solitons. The equation which describes the evolution of En , the electrical field in the nth
waveguide, in the presence of the optical Kerr effect, is
i
dEn
2
+ βEn + C (En−1 + En+1 ) + γ |En | En = 0
dx
n = 1, 2, . . .
(1.2.2)
where β is the linear propagation constant, C is the coupling constant, γ = ω0 n2 /cAeff , ω0 is the
optical angular frequency, n2 is the nonlinear coefficient and Aeff is the common effective area of
the waveguide modes. This infinite discrete set of ordinary differential equations can be reduced to
the nonlinear Schrödinger equation which describes spatial solitons, if the intensity varies slowly
over adjacent waveguides. At high power the field distribution in the array is well described with
the following formula
Xn
En (x) = A0 ei(2C+β)x sech
(1.2.3)
X0
where Xn is the location of the nth waveguide and X0 is the characteristic width.
9
CHAPTER 1. GENERAL INTRODUCTION
1.3. GRAVITATIONAL WAVES
waveguide
focused laser beam
Figure 1.4: Left: The schematic drawing shows the experimental realization of coupled identical waveguides etched
onto an AlGaAs substrate used in demonstration of existence of discrete solitons [45]. The arrays were 6 [m] long, 4
[µm] wide. The etching depth was 0.95 [µm]. Separation between the waveguides was D = 4 [µm], which established
a relatively strong coupling. Right: Picture spectacularly shows how the light was distributed in the waveguide at
different input intensities: (a) peak power 70 [W]; Light is injected into one, or a few neighboring waveguides, it will
couple to more and more waveguides as it propagates, thereby broadening its spatial distribution. The light spreads
among nearly all the 41 waveguides, and a pattern of two main lobes with secondary peaks between them is formed.
The linear feature is demonstrated. (b) peak power 320 [W]; Increasing the power narrows the light distribution.
(c) peak power 500 [W]; The discrete spatial soliton is formed. Numerical simulations also verified that light was
distributed within one coupling length over a few waveguides. From there on, the distribution remains confined,
with only small width oscillations around the soliton value. The photo on the right hand side is reproduced here
with the permission of Falk Lederer (Friedrich-Schiller-Universität Jena) and Yaron Silberberg (Weizmann Institute
of Science, Rehovot).
Figure 1.4 reproduces the first observation of discrete bright solitons. Light was injected into
a single central waveguide in a wide array of AlGaAs waveguides. The figure shows images of
the output facet for various input powers. array at different input powers. When the power is
increased, we first see the light distribution converging to form a bell shape. Launching even
more power leads to the formation of a confined distribution around the input waveguide, which
propagates as a discrete soliton. At the end we mention that in similar experimental way dark
solitons have been also created [46].
1.3
Gravitational waves
The basic Einstein-equation of general relativity is also nonlinear
1
8πG
Rij − gij R = 4 Tij
2
c
where Rij is the so-called so called. Ricci-tensor defined as
Rkj =
∂Γikj
∂Γiki
i
m
−
+ Γimi Γm
kj − Γmj Γki
∂xi
∂xj
and R = Rjj denotes the invariant derived from Rij via contraction and generally referred to as a
curvature scalar. The tensor Tij denotes the energy-momentum tensor of the given field, and plays
10
CHAPTER 1. GENERAL INTRODUCTION
1.4. WAVES ON SPIN-LATTICE
the role of source in the Einstein equation. The objects, Rij and R, contain the metric tensor gij
in nonlinear manner, since the definition of Christoffel symbols of second kind – denoted by Γ – is
1 il ∂glk
∂gjl
∂gjk
i
+
−
Γjk = g
2
∂xj
∂xk
∂xl
In this way the Ricci-tensor contains the metric tensor on the fourth.
1.4
Waves on spin-lattice
It is well-known from the Bohr - van Leeuwen’s2 theorem that spontaneous magnetism of a piece
of matter in thermodynamically equilibrium can not be explained in the framework of classical
physics. Consequently, the explanation of magnetism lies out of the scope of classical physics, and
can be found only using quantum mechanics and taking into account the spin degree of freedom
of particles.
Let us imagine a lattice and particles sitting in the lattice sites. Every particle has only spin
degree of freedom. Now, neither spatial motion nor translation are taken into account.
lattice sites
Si
Figure 1.5: Picture shows a simplified model for a spin-lattice. Only one chain is plotted for the sake of transparency.
There is a spin vector on every sites of the lattice. The magnitude of spin is constant, but the direction of these
vectors (jointed with dashed line) is altered site by site. This change can be described as a spin-wave.
The general form of Heisenberg Hamiltonian for classical three dimensional spin-lattice interα
acting with their neighbours and characterized by site-dependent exchange interactions J i,j
can be
written as
X
α α α
H=
Ji,j Si Sj − hext Sα=ν
i
i, j ∈ Λ ⊂ Z3 ,
α ∈ {−ν, . . . , ν}
α
where Si = (Six ; Sij ; Siz ) represents the spin angular momentum operator, Ji,j
the exchange integral
corresponding to nearest neighbour spin-spin interaction and hext stands for the external magnetic
field. The sites of the lattice are identified by indices i and j. The case of ν = 1/2 is called Ising
model, otherwise we use terminology Heisenberg model.
2 Niels Bohr in 1911 and J.H van Leeuwen in 1919 independently proved a famous theorem for classical nonrelativistic electrons: ”At any finite temperature, and in all finite applied electrical or thermal fields, the net
magnetization of a collection of electrons in thermal equilibrium vanishes identically.”
11
CHAPTER 1. GENERAL INTRODUCTION
1.5. WAVES IN JOSEPHSON JUNCTION
The equation of motion corresponding to the spin operator can be constructed calculating the
commutator of H and Si :
dSi
i
= [H , Si ]
dt
~
Let us simplify the model for one-dimensional spin-chain interacting through nearest neighbourhood exchange interaction. In this case time-evolution equation is:
dSi
i
= Si × [J (Si + Si−1 ) + hext ]
dt
~
α
where the general interaction parameter Ji,j
was substituted for the scalar J.
In the low-temperature and long wavelength excitations limit one can go to the continuum
description by assuming that the lattice constant is very small compared to the length of the
lattice, a. We assume that the spins Si varies slowly over lattice distance. Thus, we introduce the
following series expansion
Si±1 = S(x; t) ∓ a
∂S 1 2 ∂ 2 S
+ a
+···
∂x 2 ∂x2
The interaction parameter J defined into the spin vector S(x; t). The equation of motion reads as
St = S × [Sxx + h]
This is the Landau-Lifsitz equation. The multiplication S × Sxx arising on the right hand makes
this equation be nonlinear. Omitting the details of calculations [47] we just mention that using the
formalism of differential geometry, namely the space curve theory, and Serret-Frenet description
one can map the Landau-Lifsitz equation onto the integrable cubic nonlinear Schrödinger equation
at the lowest order O(a0 )
2
iqt + qxx + 2 |q| q = 0
where q(x; t) is in one-to-one correspondence with spin vector S(x; t).
For the sake of brevity, we only showed here the simplest case, but several ferromagnetic spin
systems with higher order magnetic interactions were under investigations in detail and proved
to be integrable. It was verified that the excitations in the above models are governed by spin
solitons [48].
1.5
Waves in Josephson junction
One of the systems where soliton existence and propagation have been well established by direct
experimental measurements is the long Josephson junction or so-called Josephson transmission
line.
In 1908 Heike Kamerlingh Onnes succeeded in liquefaction of helium and discovered that close
to absolute zero temperature mercury can conduct electrical current without any resistance. This
sensational achievement opened a new territory of physics, namely the low-temperature physics in
which predictions of quantum mechanics became verifiable. Moreover, in many cases the desired
effect arise only in a phase existing in this temperature range, e.g., superconductivity, superfluidity,
Bose-Einstein condensation, etc. Almost half a century later, John Bardeen, Leon Neil Cooper and
John Robert Schrieffer [49] explained phenomenon of superconductivity in terms of a macroscopic
number of conductance electrons that condense into Cooper-pairs all in the same quantum state:
12
CHAPTER 1. GENERAL INTRODUCTION
1.5. WAVES IN JOSEPHSON JUNCTION
√
Ψ = ρeiφ . In 1962 Brian David Josephson - a student at Cambridge University - predicted [50] a
spectacular phenomena that should ”probably” be observed in a weak electrical contact: between
two superconductors separated by a non-superconducting material a current can flow at even
zero voltage (for simplified set-up see figure 1.6). The mechanism responsible for current flow is
quantum-mechanical tunnelling through the insulating barrier and the current is proportional to the
phase-difference of the wave-function defined in the two superconductive electrode. As early as 1963
the existence of the Josephson-effect - as known nowadays - had been verified experimentally [51].
Ten years later, in 1973 he was awarded the Nobel Prize in physics ”for his theoretical predictions
of the properties of a supercurrent through a tunnel barrier, in particular those phenomena which
are generally known as the Josephson effects”.
Idc
λ
d
W
λ
Figure 1.6: Left: Brian David Josephson (1940 – ), the discoverer of the Josephson effect while he was a 22-year-old
graduate student at Trinity College, Cambridge. He showed theoretically that current can flow across an insulating
layer without the application of a voltage; if a voltage is applied, the current stops and oscillates at high frequency.
This was the Josephson effect. For this recognition he won the Nobel Prize for Physics in 1973 shared with Leo Esaki
and Ivar Giaever. Right: The sematic draw of a canonical Josephson junction consists of two superconducting (S)
electrodes separated by a thin insulator (I) with thickness d. The picture also shows the London penetration depth
λ, which is a measure of the distance that currents and magnetic field will penetrate into the superconductor phase
and the width of superconducting films.
The presence of such a supercurrent has been demonstrated in a large variety of structures [52].
The only condition for the effect is that the reservoirs of charge carriers (Cooper pairs) in the two
superconductors are weakly coupled to each other, implying that the coupling region must be thin
compared to a characteristic length.
Let us use the phenomenological Landau-Ginzburg description of superconductivity. The
Cooper pairs behave as bosons, consequently they can be described by a common wave function
which fulfills the nonlinear Schrödinger equation
i~
i2
∂ψ
1 h
∗
=
−i~∇
−
e
A
+ V (x)ψ + |ψ|2 ψ
∂t
2m∗
where e∗ , m∗ are the charge and mass of a Cooper-pair, respectively, and A is the vector potential
of electromagnetic field. This equation can be represented as a continuity equation
ρt = ∇j
13
(1.5.4a)
CHAPTER 1. GENERAL INTRODUCTION
1.5. WAVES IN JOSEPHSON JUNCTION
with the definitions of
2
ρ = |ψ|
(e∗ )2
i~e∗ ∗
2
(ψ ∇ψ − ψ∇ψ ∗ ) +
A |ψ|
j =
∗
2m
2m∗
(1.5.4b)
(1.5.4c)
where ρ and j can be interpreted as a probability distribution and current density of Cooper-pairs
in the phase-space.
√
Seeking the wavefunction ψ in the form ρeiφ and substituting into the formula of the currentdensity (1.5.4c) we get a connection between ρ and φ. This relation enables us to calculate the
difference of the phase of the wavefunction between two points, and find out that this difference
is related with the magnetic flux, and hence with the supercurrent j sup , on a plane parallel with
the insulator (see the left side of figure 1.7). Finally the following equation can be derived for the
time evolution of the phase
1
e∗
(1.5.5)
φxx + φyy − 2 φtt = 2lµ0 j sup
c
~
where c is a constant determined by the experimental set-up. If someone would like to solve this
equation for φ, then he must derive an expression for the current density j sup from an independent
source.
Josephson’s original idea was that the wave functions do not vanish on the surface of the
superconducting phase, but overlap each other due to the thinness of the insulator. This implies
a simple phenomenological model for taking account this interaction
∂ψ1
= µ1 ψ1 + ψ2
∂t
∂ψ2
i~
= µ2 ψ2 + ψ1
∂t
i~
where µ1 , µ2 are the chemical potential defined in the superconducting phases and the strength
√
√
of the coupling. Substituting the wave functions in the forms ψ1 = n1 and ψ2 = n2 eiφ and
exploiting the particle conservation principle, namely
dn2
dn1
=−
,
dt
dt
we finally get
2e∗ √
sup
n1 n2 sin(φ) = j0 sin(φ)
~
This formula connects the expressions for current density and phase of the wave function. Let us
put it back to equation (1.5.5)
j sup =
φxx + φyy −
1
e∗ sup
φ
=
2lµ
j
sin(φ)
tt
0
c2
~ 0
If the width of the insulator is small enough then the change in the direction y can be neglected.
Omitting the details of calculation, finally we get the (1+1) dimensional
Sine-Gordon equation in
sup physical spatial and temporal coordinates x and t [β 2 = ~/2lµ0e∗ j0 ]:
φxx −
1
1
φtt = 2 sin(φ).
2
c
β
14
CHAPTER 1. GENERAL INTRODUCTION
1.5. WAVES IN JOSEPHSON JUNCTION
2
ey
1.5
v
B
1
S
ex
j
B
0.5
V
I
S
-4
v
-2
0
2
4
6
8
10
-0.5
-1
Figure 1.7: Left: The schematic sketch shows the flow of a single fluxon. Nagatsuma in 1983 demonstrated [53]
experimentally in such a set-up that power generation and energy transfer are possible at 10 −6 [W] which was
far superior to previous results. Moreover, McLaughlin and Scott proved [54]: as the soliton is decelerated and
accelerated when passing a short distance, it emits radiation that may be picked up by an external microwave
circuit. Hence this structure can be used as a microwave oscillator and amplifier. Right: Picture shows the
magnitudes of the magnetic induction vector B and current density vector j in case of one-soliton (fluxon) solution
of Sine-Gordon equation. (The vertical axis has two different units !)
This equation is a famous one in the topic of totally integrable equations of motion, since its
integrability has been verified soon. A Lorentz-covariant one-soliton solution of this equation is



x − vc t
1
 .
φ(x; t) = 4 arctanexp  q
β
v 2
1− c
Let us calculate the flux carried by this solution.
Z
Ψ = B dn
F
where n denotes the normal vector belonging to F . In our set-up, the surface is the [xz]-plane of
the insulator, hence |dn| = d · dx
Ψ=d
Z∞
−∞
By dx = d
Z∞
−∞
x=∞
~
h
~
φ
dx
=
φ(x;
t
=
fix)
= ∗ = Φ0 = 2.064 × 10−15 [Vs]
x
∗
∗
2le
e
e
x=−∞
where d is the width of the insulator (see figure 1.6) and Φ0 is the elementary quanta of magnetic
flux. It is obtrusive that the one-soliton solution carries one flux quanta, therefore it can be called
”fluxon” (or ”kink”), rightfully.
As pointed out in an early experimental paper [55] moving fluxons in the Josephson transmission
line manifest themselves as the so-called zero field steps in the DC current-voltage characteristic
of the junction. Due to improvements in electronics, it has become possible to measure the voltage
waveform of the soliton directly. The first attempt to measure directly the fluxon waveform were
made by Scott et al. in 1976 [56], however the sensitivity of their set-up was not high enough
15
CHAPTER 1. GENERAL INTRODUCTION
1.5. WAVES IN JOSEPHSON JUNCTION
Figure 1.8: Experimentally realization of Bose-Einstein condensation opens the way of studying directly coherence
phenomena in another field of physics as previously done in case of superconductors or superfluid helium. At MIT,
the Ketterle group produced two non-overlapping condensates using modified external potential trap. At an early
stage they load a pile of atoms into a trap and confine them whilst reach the Bose-Einstein phase transition. After
that a double-well potential was created by focusing laser light onto the center of the trap and thereby generating a
repulsive dipole force. In such way they were able to separate two distinct condensate. Shortly afterwards both the
magnetic trap and the laser light sheet were simultaneously switched off and the condensates were let to expand,
overlap and produce the interference pattern. The fringe spacing is approximately 15 [µm]. Insertion is done with
permission of Wolfgang Ketterle (MIT Department of Physics, Alkali Quantum Gases Group).
to show the detection of only one flux quanta. A first direct observation of the single vortex
propagation along the long Josephson junction has been fulfilled in 1982 [57, 58].
Let us jump back in time, until the phenomenon Bose-Einstein condensation (BEC) recognized in 1924. In this case a pile of low-temperature particles can be described also by a common
wavefunction, as done in case of superconductive electrons. The experimental observation of interference fringes between two Bose-Einstein condensates exceedingly demonstrates the existence of a
macroscopic quantum phase difference. As stated above, in superconductors this phase-difference
is manifested in the Josephson effect. The same effect happens to exist between two, overlapping
condensate. Theoretically both DC and AC Josephson-effect were under investigation and experimentally the former was proven in the double-well set-up. However, AC Josephson regimes is
challenging because the small energy splitting associated with Josephson oscillations means that
the thermal or quantum fluctuations will destroy the effect even at the lowest achievable temperatures [59].
Finally, we would like to mention that using Inverse Scattering Transform one can handle and
solve the Sine-Gordon equation for more general boundary and initial conditions. Similar way
analytical framework was presented [60] in order to explain the features of interference pattern of
two overlapping Bose-Einstein condensates taking account the important nonlinear effects. That
treatment can be generalized for more than two condensate packets.
16
Chapter 2
Bose-Einstein condensation
This chapter discusses the birth and brief history of the idea of condensation in system of bosonic
particles. The huge development of experimental cooling techniques opens the way to cool down
bosonic particles into sufficiently low temperature where the Bose-Einstein condensation (BEC)
might occur. After observing this phenomenon extensive theoretical study of such systems started
and diversified by this time. The great interest can be explained by the fact that the study of
BECs extends our knowledge in different branches of theoretical and experimental physics, such
as in optics, statistical physics, thermodynamics, atomic collision theory, quantum properties of
mesoscopic systems, etc. (For an overview and references see, e.g., Leggett [61], and Dalfovo [62].)
Nowadays, many experimental groups try to attain condensation under more controllable conditions, e.g. tuning interparticle interactions via changing the confining potential, putting this
extra-cold pile of atoms into a two- or three dimensional lattices. The latter gives an excellent
opportunity to map the phase portrait of famous Hubbard-model which cannot be done using
metals in direct experiments. Another new way of exploring the nature of Bose-Einstein condensation and interaction of particles in the sub-millikelvin domain of temperature is to fabricate
multi-component systems, where the components can be created involving two hyperfine states,
or two isotopes of the same element has also been investigated exhaustively [63–70]. The fast
development of experimental techniques makes it possible in near future to engineer also genuine
two-component condensates that are composed of two different atoms. The two-component systems Na-Rb [71–73] and K-Rb [74] have been analyzed theoretically and the mixture Cs-Li [75,76]
has been investigated experimentally without reaching the BEC phase. The most upto-date and
desired system is a composite fermion system, when two fermionic particles combine and form a
bosonic particle. Such phenomenon emerges in the theory of superconductivity under the name
of Cooper-pair. We do not have to emphasize how extraordinary step it would be, if we would be
able to study Bardeen-Cooper-Schrieffer (BCS) theory in such a highly controllable experimental
circumstances.
2.1
Brief history of Bose-Einstein condensation
From the viewpoint of physics the basic entities of our world are particles. These units are classified
into two categories by their spin degree of freedom; whose spin is half-integer are called fermions,
while particles with integer spin quantum number are designated bosons. On everyday range of
temperature no significant difference can be observed in behaviour of these two types of particles,
17
CHAPTER 2. BOSE-EINSTEIN CONDENSATION
2.1. BRIEF HISTORY OF BEC
however their theoretical description is totally different. This fundamental difference lies in quantum mechanics and is the so called Pauli exclusion principle proposed by Wolfgang Ernst Pauli
(1900 Vienna - 1958 Zürich) in 1925 [77] and he was awarded with Nobel prize for this discovery
in 1945.
The Pauli exclusion principle states that two identical fermions can not occupy the same quantum state. This law plays a decisive role in many physical phenomena, one for which it was
originally formulated is the electron shell structure of atoms. Since electrons are fermions, the
Pauli exclusion principle forbids them to occupy the same quantum state. The Pauli principle is
also responsible for the large-scale stability of matter, since molecules cannot be pushed arbitrarily
close together, because the bound electrons in each molecule are forbidden to enter the same state
as the electrons in the other molecules. Surprisingly, this restraint arise not only in microscopic
scale but in astronomical circumstances also, in the form of white dwarf stars and neutron stars.
In both types of objects, the usual atomic structures are disrupted by large gravitational forces,
leaving the constituents supported only by a ”degeneracy pressure” produced by the Pauli exclusion principle. Similar restriction does not exist for bosons. Moreover, they can be condensed into
the same quantum mechanical state at sufficiently low temperature. Given suitable conditions this
clustering tendency can turn into an avalanche resulting in the macroscopic occupation of a single
quantum level. This phenomenon is purely an effect of quantum statistics.
In the interest of understanding how ideas and conceptions of physics evolve in time towards
recognition of Bose-Einstein condensation phenomena let us start our short review from the second
half of the eighteenth century.
In the following our main object of investigations will be the so-called black body. Let us
imagine a hollow box whose walls are held at a constant temperature. Inasmuch as all body radiates
electromagnetic waves the radiation being present inside this body is in thermal equilibrium with
the walls. Consequently we must ascribe to it the same temperature as that possessed by the
body. This is true for every element of volume in the cavity and specifies homogeneous radiation
throughout, i.e. one which is independent of the space coordinates. The cavity constitutes a
thermodynamic system which is independent of the particular physical and chemical process of
emission and absorption taking place in the wall.
In 1859-’60 Gustav Robert Kirchhoff (1824 Königsberg - 1887 Berlin) managed to deduce a
simple and important relation from the second law of thermodynamics. Namely, if a body radiates
the energy E(ν, T ) at the temperature T and frequency ν and absorbs the energy ratio η falling
over it, then the ratio E(ν, T )/η is universal for all type of matter, or in other words this ration is
independent of the substance of the body. Insofar as η = 1 the body absorbs radiation completely
in all frequency, so it can be designated as black-body, rightfully. Putting together this two facts,
we can say that the ratio of ability of emission and absorbtion is equal to the ability of emission
of the black-body for any kind of matter. This law is fundamental.
It is reasonable then to examine the spectrum of black-body radiation both experimentally and
theoretically as a function of frequency and temperature. First theoretical descriptions, RayleighJeans and Wien-law, were seemed to fail immediately because they could not explain either lowor high-frequency limits of the spectrum, respectively. Nowadays, we know that these theories
neglect the quantum behaviour of particles. We do not need to wonder on the reasons, since the
elementary axioms of quantum mechanics had been left out of scope. Two crucial ideas had not
been built into the formulae mentioned above, viz. the quantum characteristics of energy and
hypothesis of indistinguishability of identical particles. There should be a change of paradigm.
Albeit Max Karl Ernst Ludwig Planck (1858 Kiel - 1947 Göttingen) found the correct formula
of black-body radiation successfully which was able to reproduce both Rayleigh-Jeans- and Wien18
CHAPTER 2. BOSE-EINSTEIN CONDENSATION
2.1. BRIEF HISTORY OF BEC
law in the proper limit, but theoretically was poorly supported. Originally Planck interpolates
between two entropy expressions known at both ends of the energy-spectrum. Later he gave another
confirmation of his formula when the interaction of electromagnetic field and harmonic oscillator
(the wall of the cavity) was under investigation. Planck himself was dissatisfied too, because
the result should be given using just the intrinsic properties of electromagnetic field without any
exterior assumption.
The most common derivation of Planck-law used in university
physics books based on Debye’s simplified calculation. He assumed
the Maxwell-Boltzmann distribution for the energy of electromagnetic radiation and a priori supposed that the E = hν relation
connects the energy and frequency of photon, where h is a new
natural constant.
Meanwhile quantum mechanics was developed rapidly. In classical mechanics particles can be labelled uniquely by their position
and momentum at any given time, thus each individual particle
follows a well-defined trajectory in phase-space. In quantum mechanical description we are no longer able to do such identification,
but on the other hand it is not acceptable that if we change just
the labels assigned to the constituents may lead to different physical
predictions. To obtain unambiguous physical observables we
Figure 2.1:Max Planck
must impose certain symmetries on the wave-function. Indistinguishability of identical particles was recognized which is manifested as a symmetry property of
Schrödinger wave function. Correctly saying, indistinguishability is not more then the invariance
of physical observables contrary to interchange of generalized coordinates of identical particles.
This fundamental hypothesis makes an unbreakable relationship between the spin freedom of the
particles and the quantum statistics the obey to. Moreover this idea gives the last step towards
the discovery of the two possible quantum statistics: Fermi-Dirac for fermions and Bose-Einstein
for bosons.
From that point, quantum mechanics is serving as a connection, a link between phenomenological thermodynamics and mathematically strict statistical physics. If one knows the spin of a
given particle and the single-particle energy-spectrum of that entity in an external potential, then
thermodynamical properties of a system consisting of large number of identical particles can be derived. Finally, we can calculate macroscopical behaviour of a given system using only microscopic
data !
Some years after deducing quantum statistics, Satyendranath Bose (1894 Calcutta - 1974 Calcutta) was the physicist [78] who gave the theoretical correct derivation [79] of bosonic distribution
function based only on microscopic theory of statistical physics and without any assumption on
electromagnetic radiation. He simply treat the electrodynamic field inside the cavity as an ideal
photon-gas in which every particle possesses hν energy and hν/c momentum. In the cavity of a
black-body photons are freely absorbed and re-emitted, hence the number of photons is not conserved during these processes, even in equilibrium. Moreover there is no physical limit to the number of photons of any frequency. Utilizing the fact that photons are bosons he deduced that function
over phase-space which maximalizes the entropy of the N -photon system. His result coincided with
Planck’s. In this way quantum statistic of boson-systems was clarified convincingly. Moreover the
Planck-law itself was inherited from three distinct physical principle: quantum conception of energy, Pauli-exclusion principle and indistinguishability, but the possibility of condensation had not
been recognized yet. This was because Bose himself studied only black-body radiation and wanted
19
CHAPTER 2. BOSE-EINSTEIN CONDENSATION
2.1. BRIEF HISTORY OF BEC
to substantiate Planck-law from axioms directly. Here we have to make a distinction between two
types of bosons, namely there are number-conserving (permanent) and non-number conserving
(ephemeral) bosons, e.g. alkali atoms and photons, respectively [80]. Photon-gas belongs to the
former group. This distinction plays a crucial role in the topic of Bose-Einstein condensation.
At this point all of necessary parts of condensation existed.
Albert Einstein (1879 Ulm - 1955 Princeton) applied the statistics, derived in case of photons, for bosonic atoms and appended
a new constraint to the description, namely, the particle conservation law; and Bose-Einstein condensation (hereafter abbreviated
BEC) was predicted in 1924. In a system of permanent bosons
there should be a temperature below which a finite fraction of
the particles ”condense” into the same one-particle state. Einstein’s original calculation was for noninteracting gas, a system
felt by some of his contemporaries to be perhaps pathological, but
shortly after the observation of superfluidity1 in liquid 4 He below
the λ temperature (2.17K), Fritz London resurrected the idea of
Bose-Einstein condensation as an explanation for striking experiFigure 2.2:Satyendranath Bose
mental observations. Despite the strong interatomic interactions
BEC was indeed occurring in this system and was responsible for the superfluid properties.
Improving Einstein’s original idea, the definition of BEC should
have been altered, since in an interacting system, such as liquid
He4 , single-particle energy levels cannot be interpreted. Such levels
are special feature of the ideal gas. We just mention here that
Penrose, Onsager and Yang generalized [82, 83] the concept by
applying it also to strongly interacting Bose systems. In a general
system Bose-Einstein condensation occurs if
hn0 i
lim
>0
N →∞
N
Figure 2.3:Albert Einstein
where hn0 i denotes the average number of particles with momentum |p| = 0 in thermodynamic equilibrium. This definition is
widely used nowadays, however theoretically this expectation value
is hardly calculable, but experimentally measurable [84], observable with upto-date techniques.
The phenomenon of Bose-Einstein condensation in this context can be fully understood2 using
only the statistics of bosons and particle number conservation. Let us examine a physical system
consisting of N indistinguishable bosons possessing energy and chemical potential denoted by E(k)
and µ, respectively. Statistical physics says that the mean value of occupation number of the state
identified with quantum numbers k is
n(k) =
1
e(β(E(k)−µ)
−1
(2.1.1)
1 Superfluidity involves frictionless flow of a liquid (e.g. helium) through narrow channels. We suggest the
fascinating article [81] for everyone who wants to know more about the discovery of superfluidity and its relation to
superconductivity.
2 The details of calculation in different cases of circumstances are given in Appendix A.
20
CHAPTER 2. BOSE-EINSTEIN CONDENSATION
2.1. BRIEF HISTORY OF BEC
where the notation, β = 1/kB T , is the scaled inverse of absolute temperature T . This is the
Bose-Einstein distribution function – as we call it today – which describes how bosons are most
likely occupy the energy levels of a system. Total particle number must be conserved so we have
to specify the following constraint
X
N=
n(k)
(2.1.2)
k
where the sum runs over all possible quantum states {k} the particles can fill in. These two
equations implicitly determine the chemical potential, µ, which express the energy needed to put
one more particle into the system at constant volume and entropy. All other thermodynamically
important functions can be determined afterwards.
In the widely used continuous density of states approximation the total number of particles is
expressed an integral over the energy:
2πV (2m)3/2
N=
h3
Z∞
0
√
d
−1
eβ(E−µ)
(2.1.3)
In case of free non-interacting particles one can give the dispersion relation, E(k) = ~ 2 k2 /2m, and
calculate this integral obtaining a closed formula for critical temperature defined as a temperature
where the phase-transition occurs:
h2
TC =
2πmkB
ζ
N
3
2
V
!2/3
(2.1.4)
where ζ(z) is the Riemann-zeta function. Identifying the thermal de Broglie wavelength, Λ =
1/2
2π~2 /mkB T
, of the particles one can easily visualize the meaning of TC . We find that the
requirement for BEC of T ≤ TC can be written as nλ3 ≤ ζ 23 , where n = N/V is the density in
real space. Thus BEC occurs when the de Broglie wave packets associated to the particles begin
to overlap and their quantum nature becomes important.
For the sake of comparability of theoretical and experimental results we shall modify the basic
model in the following without going into details of calculation. The models mentioned below can
be distinguished according to three aspects.
i) Is there any external force or potential the particles confined by ?
The system is called homogeneous if such external action does not exist, otherwise we say the
system is inhomogeneous.
ii) Is there any interaction between particles ?
The system is called ideal if particles do not interact with each other. We have to note at
this point that interaction treated here in the special statistical physics sense which means
the particles should interact in order to reach the equilibrium, but this interaction can be
neglected if we interested in the macroscopical behaviour of the system.
iii) How many particles constitute the system ?
The system is called infinite if in the derivation and calculation of statistical physical function
the thermodynamic limit can be made, otherwise the system is finite.
21
CHAPTER 2. BOSE-EINSTEIN CONDENSATION
2.1. BRIEF HISTORY OF BEC
• Homogenous + ideal + infinite [85–88]
s
Assumption: E(k) = c |k| , d-dimension
Result:
0 < d ≤ s no condensation occurs, Tc ≡ 0
This is the simplest case and can be studied analytically. In the thermodynamic limit energy
levels are dense enough compared to the temperature times Boltzmann constant to treat them
quasi-continuous and substitute the discrete summand of quantum statistics with integration.
s
If we generalize the dispersion relation and use E = c |k| in d-dimension we obtain that BEC
does not occur if inequality 0 < d ≤ s is held. In case of non-relativistic heavy bosons, when
c = ~2 /2m and s = 2 we immediately find that macroscopical occupation of ground state is
only possible in three dimension at finite temperature. However, BEC can occur for s = 1 in
d > 1. Such linear dispersion holds for Cooper-pairs of electrons moving in the Fermi-sea [89]
so that they can condense [90] in all relevant dimensions.
• Inhomogeneous + ideal + infinite [91, 92]
Rp
Assumption: E0 = 0, Ei+1 − Ei kB T , ρ(E) ∼
E − U (r) d3 r
Result:
Condensation may emerge, Tc is tunable
One major difference between experiments of today and the general theory of BEC is the
presence of confining potential. This inhomogeneity has some fundamental effects. First of
all, the density of states has a different functional form. Secondly, whilst in the homogeneous
case the condensation happens in momentum space and the condensate is uniformly distributed in real space, in the presence of external potential the trap localizes the constituents
and this results in a sharp spike of the density profile. Moreover this contraction appears as
a large peak around zero velocity in a time-of-flight image ensuring us that phase-transition
takes place. Using semiclassical approximation and taking into account the influence of external potential which violates the homogeneity, the translational invariance of the system
one concludes that BEC exists in lower dimension than three. Moreover, for the experimental realization we have to emphasize that the critical temperature can be tuned and altered
significantly via the strengthening of the potential. Generally speaking (see Table A.1 in
Appendix A) strengthening the confinement results in the raising of TC .
• Inhomogeneous + ideal + finite [94, 95]
Assumption: a) isotropic oscillator En = n~ω,
b) anisotropic parabolic potential (ωi )
Result:
a) TcN % Tc∞ , N ≈ 104 adequate in d = 3
b) In d = 1 condensation occurs
All the models above used the thermodynamic limit in their formulae. We know that phase
transition emerges in infinite systems, but the experiments are very far away from that limit.
In today’s observations the range of number of particles is between from few tens to a billion.
One might think this difference has a huge influence in the measurement. Wolfgang Ketterle
and N. J. van Druten refuted [94] the expectation for isotropic and highly anisotropic gas.
They were able to define a transition temperature, TCN which was always under the TC∞
belonging to the infinite system, but the relative difference between the two temperatures
was lesser than 7% for such a small system as N = 1000. Naturally, in case of systems with
finite number of particles the condensate fraction grew from zero to unity as they lowered
the temperature, but sharp phase transition did not occur. As a result they concluded that
absence of Bose-Einstein condensation in one or two dimensional systems was just an artifact
of the usual thermodynamic limit.
22
CHAPTER 2. BOSE-EINSTEIN CONDENSATION
2.1. BRIEF HISTORY OF BEC
Figure 2.4: The false-color images show the velocity distribution of 87 Rb and were taken by Anderson and his
colleague in 1995 [93]. In this remarkable series of 2d time-of-flight measurements where the atoms were left
to expand by switching off the confining trap and optically mapped the cloud. At each point in the images the
observed optical density was proportional to the column density of atoms at the corresponding part of the expanding
cloud. Moreover, in case of harmonic potentials the spatial and velocity distribution differs from each other in a
linear scaling multiplier, the oscillator frequency for the given directions. Thus, from these photos we can see both
coordinate- and momentum-space distributions at the same time. The images from left to right belong to different
amount of cooling, and thus different final temperature. The axes are the projections of velocity onto the x and
z directions, and the third axis is the number density of atoms per unit velocity-space volume. The left frame
was mapped at 400 nK above the transition, the center frame just near the appearance of condensation (170 nK).
After further evaporation the thermal cloud is nearly totally vanished and just the pure condensate is visible. The
color corresponds to the number of atoms at each velocity with red being the fewest and white being the most.
Reproducing this famous picture takes place with permission of Eric A. Cornell (University of Colorado/JILA). The
original file of picture in jpeg format can be found on JILA’s official homepage
(http://jilawww.colorado.edu/bec/CornellGroup/index.html).
• Inhomogeneous + weakly interacting + infinite [93, 96–98]
Assumption: V (r1 − r2 ) = V0 δ(r1 − r2 ), V0 ∼ as , H(p, r) = p2/2m + U (r) + V0 n(r)
Result:
a < 0 critical particle number [99–102]
a > 0 stable for all N [103]
Studying situations with real gases we cannot neglect the interparticle interaction, and this
case is called imperfect Bose gas, where the interaction can be considered as a small perturbation. Whereas the details of interparticle potentials are not known except for few cases
and even not necessary for a satisfactory description, hence we can use approximations in
some sense. The most widely used simplification in perturbative calculations is the so-called
pseudo-potential method , where the real potential is replaced with a much more simpler expression which produces the same energy-spectrum. Taking into account that our scope is
to understand dilute Bose-systems we can suppose that all of the phase-shifts are unimportant but the s-wave scattering length. Huang and Yang proposed [104] a natural framework
23
CHAPTER 2. BOSE-EINSTEIN CONDENSATION
2.1. BRIEF HISTORY OF BEC
in which this approximation can be carried out systematically based on Wigner’s original
idea [105]. The authors demonstrated the usage of their technique for the N -body hardsphere interaction, when eigenvalues and eigenfunctions can be expanded in powers of the
hard-sphere diameter as . Taking only binary collisions into account it was proven that the
two-particle pseudo-potential can be symbolically written as
∂
4π 3
∂
r· −
as δ(r)
r· + ...
(2.1.5)
Vpseudo = 4πas δ(r)
∂r
3
∂r
Moreover they give a beautiful geometric formulation of the pseudo-potential for the interaction Hamiltonian in the many-body context, and present:
4π~2 as X
∂
Ĥint =
δ(ri − rj )
rij ·
(2.1.6)
m
∂rij
i<k
as an obvious generalization of the result displayed above in the lowest order. If one wants
to extend this perturbative calculation for effects of collision higher than binary, he could
find that the next term is proportional to a4s . Consequently, this Ĥint correctly accounts the
effects up to and including a2s .
Later this method was extended to the case of externally confined gas of impenetrable bosons
[106]. This experimentally motivated calculation suggested that a famous theoretical model,
namely the Tonks-gas, can be prepared in kz |a1d | 1 limit, where ~kz is the longitudinal
component of the atomic momentum and a1d is an effective one-dimensional scattering length
defined as:
as
a2⊥
a1d = −
1−C
(2.1.7)
2as
a⊥
where C = 1.4603, a⊥ = (2~/mω⊥ )1/2 , ω⊥ are an effective length scale and the frequency of
the waveguide potential in the transverse-plane, and as is the ”true”s-wave scattering length.
Finally, we mention another investigation [107] in which a pseudo-potential is derived for
anisotropic traps, and showed that it depends on the quantum number of the trapped state
4π~2 as
4n0 + 3 as 2
λ
∂
Vpseudo =
1+
1+
δ(r)
r·
(2.1.8)
m
3
L
6
∂r
where λ = (ωa /ω⊥ )2 − 1 is the deformation parameter and L = (2~/mω⊥ )1/2 . Let us use the
data cited in the first row of Table 2.1 (see on page 28). In that case L ≈ 2.1µm and λ = 7,
hence the correction yields at order of 10−5 which can be neglected trivially in most cases.
To ensure the stability of the condensed phase it is usually required that the interaction
between atoms is effectively repulsive or more precisely saying the s-wave scattering length,
as , is positive. It has been shown [108], e.g. for the case of cesium, that as has a resonant
structure as a function of externally applied magnetic field. This effect offers the possibility
of controlling not just the magnitude but the sign of as by an appropriate choice of the
magnetic-field strength. The importance of the scattering length and its influence on the
stability of the condensate will be discussed later in this thesis [see chapter 6].
24
CHAPTER 2. BOSE-EINSTEIN CONDENSATION
2.2
2.2. EFFECTS OF INTERACTION . . .
Effects of interaction: Gross-Pitaevskii equation
In the historical part we mentioned that the observation of superfluidity of 4 He gave a hit to
the theoretical investigations in the topic of Bose-Einstein condensation. Fritz London made
the connection between 4 He and phenomenon of BEC, because there were some convincing facts
inspired this association: the critical temperature TC calculated from Einstein’s formula (A.1.13)
was 3.2K while the experimentally measured value was 2.17K; moreover the specific heat looked
so similar as predicted for ideal Bose-gas. 3
The λ-transition in 4 He is not a first order transition because experimentally there is no latent
heat of transition. Measurements of the specific heat over coarse temperature intervals indicate
that it is discontinuous at TC indicating a second order phase-transition. Furthermore later the
repeating observations proved logarithmically divergent at TC . This type of λ-transition is thus
similar to Curie transition in ferromagnets where the transition point marks the transition from
a phase with no long-range order to one with long-range order. This suggests that relevant longrange order in liquid 4 He is a pattern in momentum space, namely a finite fraction of all atoms
has the same momentum. Summarizing, we have to note that improvements must be made for
satisfactory description of experimental findings.
Despite the phenomenological similarity in specific heat of an ideal Bose-gas and the measurement for 4 He the detailed studies suggest that one should taking account the interparticle
interaction.
After Tisza’s [109, 110] and Landau’s [111] significant hydrodynamic description of 4 He it was
clear that in a weakly interacting Bose-system the spectrum of low lying excitations are the same
as in case of photons or rotons. This linear dispersion relation explained the phenomenon of
frictionless flow. Bogoliubov developed a new explanation using the ideas of second quantization
and showed that the existence of Bose-Einstein condensation is not affected by the introduction of
weak interparticle interaction. Furthermore he corroborated Landau’s findings about phonon-like
spectrum.
Finally, it turned out that the remarkable (and that time possibly the only-one) example for
Bose-Einstein condensation, the 4 He was not a good example of weakly interacting Bose-gas. It is
a liquid not a gas and the interaction between the atoms are too strong, consequently. While the
experimental observation of superfluidity inspired the theoretical investigations of Bose-systems,
the conception of weakly interacting imperfect Bose-gas was unable to explain the measurements
in case of 4 He sufficiently. A major breakthrough in this topic has happened in 1995 when gases
of alkali atoms (87 Rb, 23 Na and 7 Li) were cooled down at temperature of order micro-Kelvin and
their momentum distribution was mapped (see figure 2.4). The announcement of achievement
of Bose-Einstein condensation attracted an unusual amount of attention. There were front-page
articles in many newspapers, e.g. Washington Post (September 20, 1999; page A07), New York
Times (July 14, 1995, front page).
The existence and nature of confined, dilute, and weakly interacting system of ultracold bosonic
atoms can be understood by considering the general second quantized many-body Hamiltonian
Z
ZZ
1
Ĥ =
Ψ̂†S (r)Ĥ (1) Ψ̂S (r) dr +
Ψ̂†S (r1 )Ψ̂†S (r2 )Ĥ (2) Ψ̂S (r2 )Ψ̂S (r1 ) dr1 dr2 (2.2.9)
2
where Ψ̂S (r) and Ψ̂†S (r) are the annihilation and creation Bose-operators obeying the following
3 It can be shown for ideal gas that C
V is continuous at T = TC , but its slope is discontinuous. Since CV =
(∂U/∂T )V , where U is the internal energy, therefore the Bose-Einstein condensation is predicted to be a first-order
phase transition.
25
CHAPTER 2. BOSE-EINSTEIN CONDENSATION
2.2. EFFECTS OF INTERACTION . . .
commutation rules:
h
i
Ψ̂(r; t) , Ψ̂† (r0 ; t)
h
i
Ψ̂(r; t) , Ψ̂(r0 ; t)
h
i
Ψ̂† (r; t) , Ψ̂† (r0 ; t)
= δ(r − r0 ) Î
(2.2.10a)
=
Ô
(2.2.10b)
=
Ô
(2.2.10c)
After transforming the operators into the Heisenberg picture:
i
i
Ψ̂H (r; t) = e ~ Ĥt Ψ̂S (r)e− ~ Ĥt
the new operator fulfills the Heisenberg equation of motion:
h
i
∂
i~ Ψ̂H (r; t) = Ψ̂H (r; t) , Ĥ
∂t
(2.2.11)
(2.2.12)
Let us specify the exact form of one- and two-particle operators, Ĥ (1) and Ĥ (2) . Here Ĥ (1)
represents the ”unperturbed” Hamiltonian in its usual form
1 2
p̂ + Vext (r̂)
(2.2.13)
2m
since in our case non-interacting but externally trapped system of bosons is treated as a starting
point. The delicate part of our discussion is how the two-particle operators should be given. As
we mentioned in the introduction the widely used approximation is the pseudo-potential method.
The dilute nature of the gas allows us to describe the effects of interaction in a rather simple and
physically fundamental way. In this sense the binary collision is modelled by a potential
Ĥ (1) =
Ĥ (2) = gδ (r2 − r1 )
(2.2.14)
2
concentrated at a point with the strength: g = 4π~ as /m, where as is the s-wave scattering length
and m is the mass of one particle. Substituting the proper form of the Hamiltonian into the
Heisenberg equation (2.2.12) we finally get
∂
1 2
i~ Ψ̂H (r; t) =
p̂ + Vext (r̂) Ψ̂H (r; t) + g Ψ̂†H (r; t)Ψ̂H (r; t)Ψ̂H (r; t)
(2.2.15)
∂t
2m
Our physical expectation is - as the statistical physical calculations indicated before - that the
ground-state is macroscopically occupied. Probably this motivated Bogoliubov [112] to separate
the mean-value and the fluctuation in the operators like
Ψ̂H (r; t) = Φ(r; t) + Υ̂(r; t)
(2.2.16)
where Φ(r;
D
E t) is a complex function defined as the expectation value of the field operator, Φ(r; t) =
Ψ̂H (r; t) , while Υ̂(r; t) referring to the remaining non-condensed particles. The function Φ(r; t)
is often called order parameter or wave function of the whole condensate. This decomposition is
useful if the fluctuation, Υ̂(r; t), is small enough. Using this Bogoliubov approximation an equation
can be easily obtained for the classical field Φ(r; t) expanding (2.2.15) in the lowest order of Υ̂(r; t).
The calculation provides the time-dependent Gross-Pitaevskii (hereinafter abbreviated as GP)
∂
1 2
i~ Φ(r; t) =
p̂ + Vext (r̂) Φ(r; t) + g |Φ(r; t)|2 Φ(r; t)
(2.2.17)
∂t
2m
Finally we summarize the circumstances in which this equation of motion gives physically correct
description. During the derivation we use many approximations:
26
CHAPTER 2. BOSE-EINSTEIN CONDENSATION
2.2. EFFECTS OF INTERACTION . . .
i) Two-particle interactions, collisions are taken only into account. We mentioned above
that we use the pseudo-potential approximation, when the two-particle potential is supported
on a point and its strength is proportional with the s-wave scattering length belonging to the
original two-particle potential. Consequently, the Gross-Pitaevskii equation is appropriate iff
as is far smaller than the average distance between particles in the gas, thus the condensate
is dilute enough to neglect the three body processes. This approach is justifiable in most
cases, since the three-body recombination rate caused by elastic interatomic interaction is
proportional to the triplet scattering-length a2t [113]. Moreover in the ultralow collision energies (E) Fedichev et al [114] found a universal relation for recombination rate (α rec ) which is
independent of the detailed shape of the interaction potential.
One can estimate the diluteness of the system by comparing two length scales: the average
distance between two particles (∼ n−1/3 ) and the s-wave scattering length (as ). We can
combine a dimensionless parameter from them; the average particle number in a box with a s
long side is n |as |3 . We enumerate some experiments and their circumstances in Table 2.1.
3
Usually n |as | is much less than unity so that the attribute ”weakly” is fulfilled in this case.
In spite of this characterization, the strength of interaction can be better described by using
ratios of not the typical lengths but energies – as was pointed out by Dalfovo et al [62]. As they
argued the strength of interaction should be compared to kinetic energy. In case of harmonic
oscillator (ho) trap potential the kinetic energy of one particle is ~ωho , while the interaction
1/3
1/2
between two particles are gn, where ωho = (ωx ωy ωz ) , n = N a−3
are
ho , aho = (~/mωho )
the geometric mean of oscillator frequencies of the confining trap, the average density and
harmonic oscillator length, respectively. Using the formula of coupling parameter g one finds
finally:
N |as |
Eint
∝
(2.2.18)
Ekin
aho
This expression obviously shows a striking feature of the interacting Bose-system in that it
is even dilute in the sense of length scale comparison, but exhibit an important non-ideal
behaviour. The interaction has a relevant role in the nature of the condensate, both in its
statical and dynamical description.
ii) The interparticle potential is approximated only with a δ-like function with strength
proportional to as , the s-wave scattering length corresponding to the real interaction potential.
iii) The energy of particles is sufficiently low to neglect the fluctuations in the first order
perturbation, hence the first order is a mean-field approach for the order-parameter associated
with the condensate. That’s why this formalism is valid only near zero absolute temperature.
The validity of GP-equation was tested in many experiments and numerical simulations in many
ways. Experimentalists may measure for example the density profile, the spectrum of collective
excitations.
The former possibility was carried out in two ways [123, 124], depending on what kind of
density profile was measured. Hau et al. [124] compared the spatial density distributions derived
from mean-field Gross-Pitaevskii approximation and the one determined from the experimental
images. Moreover an independent measurement of the number of atoms forming the condensate
allows the checking the value of scattering length. The difference between these data is below 10%.
Interference between two freely expanding Bose-Einstein condensates has been observed [125].
The interference patterns, the distance between fringes and the amplitude of fringes were theoreti27
CHAPTER 2. BOSE-EINSTEIN CONDENSATION
2.2. EFFECTS OF INTERACTION . . .
Table 2.1: Table shows some results of observations in chronological order carried out with different alkalies. In
the left four columns the citation, the given species, its s-wave scattering length (a s ) and the peak density (n) are
listed. The remainder columns give calculated data: the harmonic oscillator length scale (a ho ) and frequency (ωho )
and the two different dimensionless scales of diluteness (n |as |3 ) and (Ekin /Eint ) mentioned in the text.
ωho [Hz]
n |as |3
5.01
60
10−13
2.2
3.71
416
10−11
925
as [nm]
n [cm−3 ]
Anderson et al [93]
87 Rb
5.5
1012
Davis et al [115]
23 Na
4.9
Bradley et al [98]
7 Li
4 × 1014
4.52
917
10−16
0.3
Fried et al [116]
1H
4.8 × 1015
12.18
883
10−15
5316
19.2
Reference
Element
Cornish et al [117]
85 Rb
Robert et al [118]
4 He
Ott et al [119]†
87 Rb
Modugno et al [120]
41 K
Weber et al [121]
133 Cs
2.6 ×
aho [µm]
Eint /Ekin
−1.42
> 7 × 1010
21.16
1 × 1012
10.99
13
10−12
1013
8.68
439
10−11
11.5
1 × 1015
1.06
1348
10−10
2079.3
2.01
791
10−14
20.4
1013
3.41
85
10−11
74.5
0.0648
20.0
5.5
4.12
15.9
6 × 1011
1.3 ×
87 Rb
0.91
1842
10−11
1822.9
Schneider et al [122]†
5.5
9 × 1013
† In these experiments BEC has been achieved in a magnetic microtrap which allows such high peak density
cally explained adequately using Inverse Scattering Transformation [60] method of Gross-Pitaevskii
equation.
Furthermore, at JILA [123] the cloud was mapped by time-of-flight measurements when they
allowed BEC to expand in all three dimension and compared the results to computer simulations.
They found a good agreement between the aspect rations of the wave-function derived from GP
equation and observations. Another checking of usefulness of this equations of motion is analyzing
the linear response [126] of the ground state given to an external oscillatory perturbations. This
approach gives the same equations as what Nikolai Bogoliubov derived [112] half century before
realization of BEC in laboratory. The excitation spectra and data of measurements fits at about
2% error-level.
28
Chapter 3
Painlevé-analysis
This chapter is devoted to the idea of integrability. We consider how this notion arose in classical
mechanics and led to the concept of Painlevé-test, as it is called recently. For the sake of historical
correctness we shall present both the physical and the mathematical aspects of integrability as it
has evolved in time.
3.1
Kovalevskaya top
Sofia Vasilyevna Kovalevskaya (1850, Moscow - 1891, Stockholm)
analyzed a problem of classical mechanics, namely the motion of rigid
body rotating around a fixed point which is still partially an open
mathematical-physical problem. Prior to her work such scientific
notabilities examined this problem like Leonhard Euler and JosephLouis Lagrange, and both of them were able to give only a solution for
one symmetric case when the equations of motion become integrable.
The problem when the fixed point coincides with the center of mass1 ,
therefore no external force influences the motion of the body has been
the first case to be solved by mathematicians and physicists. The
most cited example is the Earth’s spinning. The second integrable
case is when both the center of mass and the fixed point lie on a
symmetry axis of the body. Kovalevskaya was the only one who gave
an
unsymmetrical solution, moreover, she proved that generally there
Figure 3.1:
are
no more integrable cases. Nevertheless, special solutions have
Sofia Vasilyevna Kovalevskaya
been found later. Kovalevskaya’s results were reported in a paper
submitted to a competition of French Academie Royale des Sciences and her essay was awarded
with the Prix Bordin in 1888.
Imagine a rigid body (Figure 3.2) a point (Q) of which is fixed at the origin of the coordinate
system K. Generally, this point does not coincide with the center of mass, so that an external
torque, due to the gravity, does not vanish relative to Q. That is the reason it is called heavy
rigid top. Dynamically, the motion is governed by the well-known Euler equations. These can be
derived from the angular momentum theorem and using the formulae of transformation between
29
CHAPTER 3. PAINLEVÉ-ANALYSIS
3.1. KOVALEVSKAYA TOP
ê2
ê3
ê1
Q
G
Figure 3.2: Rigid body rotating around a fixed point Q. The vectors êi are the eigenvectors of tensor of inertia
perpendicular to each other. The gravitational force is denoted with G.
an inertial coordinate system and a rotating one.
In the laboratory-frame
dL(LAB)
= M(LAB)
dt
while in the rotating coordinate-system:
(3.1.1)
dL(ROT)
= L(ROT) × ω (ROT) + M(ROT) .
dt
(3.1.2)
The angular momentum can be expressed by tensor of inertia and angular velocity as L k = Θkl ωl .
Change the coordinate-system and examine the dynamics in the body-frame where the tensor of
inertia is diagonal. The vectors êi (i = 1, 2, 3) form the basis in this coordinate-system. Notations
ω = (p; q; r) are used for the angular velocity and Θ11 = A, Θ22 = B and Θ33 = C for the diagonal
elements of tensor of inertia. With this denotation
ê1 ê2 ê3 L(ROT) × ω (ROT) = Ap Bq Cr = ê1 (B − C)qr + ê2 (C − A)pr + ê3 (A − B)pq.
(3.1.3)
p
q
r Furthermore, the projections of external torque, M(ROT) , onto the basis êj should be determined.
The only external force is the gravity for which
G1 = mgγ,
G2 = mgγ 0 ,
G3 = mgγ 00
(3.1.4)
where γ = cos (^(ẑ; ê1 )), γ 0 = cos (^(ẑ; ê2 )), γ 00 = cos (^(ẑ; ê3 )). Introduce the notation (x0 ; y0 ; z0 )
for the vector of center of gravity relative to the origin of the coordinate system K 0 . So the external
1 Precisely saying, we should distinguish between the center of mass and the center of gravity, but only the
rotation of a relatively small objects is treated here, so that we use these terms as synonyms in the following.
30
CHAPTER 3. PAINLEVÉ-ANALYSIS
3.1. KOVALEVSKAYA TOP
torque can be easily calculated:
M(ROT)
ROT
= rROT
CM × G
ê1
= x0
mgγ
ê2
y0
mgγ 0
ê3 z0 =
mgγ 00 = mg (y0 γ 00 − z0 γ 0 ) ê1 + mg (z0 γ − x0 γ 00 ) ê2 + mg (x0 γ 0 − y0 γ) ê3
Collecting all terms the Euler-equations of motion are
Aṗ + (C − B)qr
B q̇ + (A − C)rp
C ṙ + (B − A)pq
(3.1.5)
= M1 ,
(3.1.6a)
= M2 ,
= M3
(3.1.6b)
(3.1.6c)
The second group of equations of motion gives the connection between p, q, r and γ, γ 0 and γ 00 :
= rγ 0 − qγ 00 ,
γ̇
γ˙0
00
= pγ − rγ,
= qγ − pγ 0
γ˙00
(3.1.6d)
(3.1.6e)
(3.1.6f)
In order to solve the equations (3.1.6), we need to find six functionally independent constants
called first integrals of motion - in other words they are in involution. (Two functions are said to
be in involution if their Poisson-bracket vanish everywhere on the phase-space.) Since time does
not appear explicitly, our system of differential equations is autonomous; it is enough to present
only five integrals. Moreover, because a volume of an arbitrary domain of phase space remains
unchanged during the time evolution which is guaranteed by Liouville’s theorem, just four functions
suffice to integrate the equations of motion with quadrature. Now we show that this system has
three universal first integrals.
I. Multiply the first equation with 2p, the second with 2q and third with 2r. After adding the
three equations we get:
d
d
d 2
p + B q 2 + C r2 = 2 (M1 p + M2 q + M3 r) .
dt
dt
dt
Mi can be expressed with our variables:
A
(3.1.7)
M1 p + M 2 q + M 3 r
= mg (y0 γ 00 − z0 γ 0 ) p + mg (z0 γ − x0 γ 00 ) q + mg (x0 γ 0 − y0 γ) r
= mg [x0 (rγ 0 − qγ 00 ) + y0 (pγ 00 − rγ) + z0 (qγ − qγ 0 )]
d
= mg (x0 γ + y0 γ 0 + z0 γ 00 ) ,
(3.1.8)
dt
So that the expression
I1 = Ap2 + Bq 2 + C 2 − 2mg (x0 γ + y0 γ 0 + z0 γ 00 )
(3.1.9)
can be treated as a constant of motion. This is a first integral of a heavy rigid body rotating
around a fixed point of it.
II. Similarly we can ”integrate” the second part of the equations of motion. Just multiply those
equations with γ, γ 0 and γ 00 , respectively, to get:
d 2
d 2
d 2
γ + γ 0 + γ 00 = 2 (rγγ 0 − qγγ 00 + pγ 0 γ 00 − rγγ 0 + qγγ 00 − pγ 0 γ 00 ) .
dt
dt
dt
2
2
In this way the integral I2 = γ 2 + γ 0 + γ 00 forms another constant of motion.
31
(3.1.10)
CHAPTER 3. PAINLEVÉ-ANALYSIS
3.1. KOVALEVSKAYA TOP
III. The two examples above were constructive. Now, just calculate the time-derivative of I 3 =
Apγ + Bqγ 0 + Crγ 00 to get
dI3
d
=
(Apγ + Bqγ 0 + Crγ 00 ) = Aṗγ + Apγ̇ + B q̇γ 0 + Bq γ˙0 + C ṙγ 00 + Crγ˙00 .
dt
dt
(3.1.11)
Using the Euler-equations we can eliminate the time-derivatives:
dI3
dt
= [M1 − (C − B)qr] γ + Ap (rγ 0 − qγ 00 ) +
[M2 − (A − C)pr] γ 0 + Bq (pγ 00 − rγ) +
[M3 − (B − A)pq] γ 00 + Cr (qγ − pγ 0 )
= M1 γ + M2 γ 0 + M3 γ 00
= mg [y0 γγ 00 − z0 γγ 0 + z0 γγ 0 − x0 γ 0 γ 00 + x0 γ 0 γ 00 − y0 γγ 00 ]
= 0
(3.1.12)
The idea what Sofia Kovalevskaya used [127, 128] was the following: the integration of the
equations of motion of a heavy rigid body rotating around a fixed point could be carried out
with the help of elliptic functions, so that the solutions were meromorphic functions of t. The six
dependent variables (p, q, r, γ, γ 0 , γ 00 ) were single valued function of time t, and have no singular
points other than poles on the complex plane. She examined whether this property holds for in
general case or not. Mathematically saying: when can the solution be found in the form of a series
of integral powers of τ = t − t1 , including a certain number of negative powers?
She substituted the Laurent-expansions
p(τ ) =
γ(τ ) =
∞
X
j=0
∞
X
pj τ j+m1
fj τ j+n1
j=0
q(τ ) =
γ 0 (τ ) =
∞
X
qj τ j+m2
j=0
∞
X
gj τ j+n2
r(τ ) =
γ 00 (τ ) =
j=0
∞
X
j=0
∞
X
rj τ j+m3
(3.1.13a)
hj τ j+n3
(3.1.13b)
j=0
into the six equations and compared the leading terms first. She found m1 = m2 = m3 = −1 and
n1 = n2 = n3 = −2, and the expansion coefficients (p0 , q0 , r0 , f0 , g0 , h0 ) belonging to these leading
terms. After this, she could determine the relations between the remaining expansion coefficients
starting from the first ones (p0 , q0 , r0 , f0 , g0 , h0 )
Ap0 + (B − C)q0 r0 + y0 h0 − z0 g0 = 0
Bq0 + (C − A)p0 r0 + z0 f0 − x0 h0 = 0
Cr0 + (A − B)p0 q0 + x0 g0 − y0 f0 = 0
(3.1.14a)
and
2f0 + r0 g0 − q0 h0 = 0
2g0 + p0 h0 − r0 f0 = 0
2h0 + q0 f0 − p0 g0 = 0
32
(3.1.14b)
CHAPTER 3. PAINLEVÉ-ANALYSIS
and the linear system

(j − 1)A
(A − C)r0

(B − A)q0


0

 −h0
g0
|
3.1. KOVALEVSKAYA TOP
for j ≥ 1 is
(C − B)r0
(j − 1)B
(B − A)p0
h0
0
−f0
(C − B)q0
(A − C)p0
(j − 1)C
−g0
f0
0
{z
P
0
−z0
y0
j−2
r0
−q0
z0
0
−x0
−r0
j−2
p0
 
−y0
pj
 qj 
x0 
 
 
0 
  rj  = Q j


q0  
 fj 


−p0
gj 
j−2
hj
}
(3.1.15)
where Qj depend only on the expansion coefficients
with indices less than the actual value of j,
k=j−1
so that Qj = Qj {pk ; qk ; rk ; fk ; gk ; hk }k=0
. This system of linear equation can not be solved
when the matrix P is singular. The determinant of the matrix P is a sixth order polynom, so that
it has six zeros (including multiplicity):
det (P) = ABC(j + 1)j(j − 2)(j − 4)(j 2 − j − µ)
(3.1.16)
where µ is an algebraic expression of {A, B, C, x0 , y0 and z0 }. She could treat all the symmetric
cases (see the first three items below) successfully, but was able to find out a fourth non-generic
case, when there exists an integer index. The shocking result was that this assumption, namely
that the solution can be expressed in this Laurent-series, was possible only in four cases:
i) Complete symmetry case: A = B = C, when the three half-axis of ellipsoid of inertia are
the same, so that this ellipsoid is a sphere. The fourth integral is:
I4 = x0 p + y0 q + z0 r;
(3.1.17)
ii) Euler-Poinsot case (1750, 1851): x0 = y0 = z0 = 0, when the fixed point is the center of
mass, so that the motion is forceless. The fourth first integral is:
I4 = (Ap)2 + (Bq)2 + (Cr)2 ;
(3.1.18)
iii) Lagrange-Poisson case (1788, 1813): A = B, x0 = y0 , when the ellipsoid of inertia is
symmetric and the fixed point lies on the z-axis. The fourth integral is:
I4 = r;
(3.1.19)
iv) Kovalevskaya case (1889): A = B = 2C and z0 = 0. The fourth integral is:
mgx0 2
mgx0 0
I4 = p 2 − q 2 −
γ + 2pq −
γ.
C
C
(3.1.20)
Kovalevskaya managed to integrate the system with hyperelliptic integrals, and proved the
meromorphy of the general solution.
33
CHAPTER 3. PAINLEVÉ-ANALYSIS
3.2
3.2. PAINLEVÉ ANALYSIS
Painlevé analysis
In line with Kovalevskaya’s studies Paul Painlevé (1863, Paris - 1933, Paris) tried to characterize
second order ordinary differential equations solutions of which have only a given type of singularities. Initially he studied also the equations of classical mechanics, since the motion of particles is
described via ordinary differential equations. Similar categorization of first order ordinary differential equation has been made by Henri Poincaré and Lazarus Immanuel Fuchs.
In 1887 Émile Picard raised the question whether such an equation has fixed critical points. Here ”critical points” includes branch
points and essential singularities. The problem was partially solved
by Painlevé [129], his student B. Gambier [130] and Lazarus Immanuel Fuchs [131].
Today’s technical literature [132] uses two natural ways of introducing the Painlevé equations; one could be called intrinsic,
the other extrinsic. The intrinsic way is historically the first one,
and based on the analysis of the singularities of solutions and uses
only the equations themselves. Alternatively, the Painlevé equations can be introduced as equations of isomonodromy deformations of auxiliary linear systems of differential equations. It provides us with an additional structure - the corresponding linear
Figure 3.3:
Paul Painlevé (1863 - 1933)
system, which is given in terms of 2 × 2 matrices and is called Lax
representation of the corresponding Painlevé equation. The Lax representation is important for
geometric applications in mathematics and for the Inverse Scattering Transform used in physics.
In this thesis we only mention the first way, the historical way, otherwise we refer to [132].
This section can be divided into three distinct, but closely related parts. First subsection is
dedicated to the examination of ordinary differential equations, because Painlevé’s key idea was
formulated for this type of differential equations originally. Fundamental notions and theorems
can be treated more easily in this case.
One of our main effort is to define or derive systematic methods and conditions the integrability of a given differential equation can be determined with, whether it were ordinary or partial
differential equation.
The link between integrability of ordinary and partial differential equations is the ARS-conjecture
[133] and WTC method [134]. Both of them will be treated below. The main questions and steps,
which play an important role in physics, in our point of view, are gathered on figure 3.4.
Analogously to the method used earlier, Weiss, Tabor and Carnevale suggested a procedure [134]
for partial differential equations. We try to summarize the basic facts and give some examples where
Painlevé equations or Painlevé-transcendents arise in various fields of physics.
Finally, we show the Painlevé-test in practice in the case of one-component, damped and driven
nonlinear Schrödinger equation [135]. This should be compared to our result given in next chapter.
3.2.1
Ordinary differential equations
Description of classical mechanical systems is based on ordinary differential equations. Integrability
of such equations of motion may be resolved using the so called singular-point analysis. The essence
of this method is to extend the dependent variables onto a part of the complex plane and examine
the location and type of appearing singularities. Singularities are responsible for the limitation of
the domain of validity of Taylor or Laurent-expansion, so that their study is mandatory.
34
CHAPTER 3. PAINLEVÉ-ANALYSIS
3.2. PAINLEVÉ ANALYSIS
C. S. Gardner, J. M. Greene,
M. D. Kruskal, R. M. Miura,
M. J. Ablowitz, D. J. Kaup,
A. C. Newel, H. Segur
H. Poincaré, L. Fuchs, P. Painlevé
“Determine all algebraic differential
equations of order n where the only
movable singularities of all its solutions
are poles on the worst ! ”
“What kind of systematic and constructive procedure can be given to solve
nonlinear evolution equations ? ”
1967, 1973
?
Inverse Scattering Transform:
Gel’fand-Levitan-Marchenko equation
AKNS-scheme:
Zakharov-Shabat equation
1883-1910
?
M. J. Ablowitz, A. Ramani, H. Segur
“What algorithm and/or criteria can
be presented to ease the decision
whether a nonlinear differential equation is integrable or not ?”
?
In the case of n = 2 there exist only 50
equations which have movable poles on
the worst. Each of these equations can
be either integrated by quadrature, or
reduced to linear differential equation
or to one of the six Painlevé equations
enumerated in table 3.1.
- 1977-’83
Hj
H
Indirect way:
Similarity reduction
+
Painlevé ODE-test
J. Weiss, M. Tabor,
G. Carnevale
Direct way:
Painlevé PDE-test
Figure 3.4: Figure shows the most important steps of development of Painlevé-test as a method of decision of
integrability. On the left-hand side the mathematical problem and its solution can be seen, while on the opposite
side, the questions arisen in theoretical and mathematical physics are indicated. Large differences can be noticed
between the dates of year, but in the eighties these two distinct approaches came across with the help of ARSconjecture (Ablowitz, Ramani and Segure) and WTC-method (Weiss, Tabor and Carnevale). For details and
references see text. Furthermore we have to note here, that the 50 equations are of course just representatives
of equivalence classes. The abbreviations, ODE and PDE, stand for Ordinary and Partial Differential Equation,
respectively.
Let us inspect the following linear ordinary differential equation of order n:
dn y
dn−1 y
dy
+ pn−1 (x) n−1 + · · · + p1 (x)
+ p0 (x)y = 0.
n
dx
dx
dx
(3.2.21)
If all of the coefficient functions, pi (x), are analytic in the neighbourhood of x0 , a regular point
of (3.2.21), then there exists a unique solution y(x) the first (n − 1) derivatives of which can be
specified in advance:
dk y(x) = ck ,
k = 1, 2, . . . , (k − 1).
(3.2.22)
dxk x=x0
35
CHAPTER 3. PAINLEVÉ-ANALYSIS
3.2. PAINLEVÉ ANALYSIS
Since y(x) can be expanded in Taylor-series in this case, the singularities of y(x) coincide with
the zeros or singularities of the coefficient function pk (x). These points are the fixed singularities
of equation (3.2.21), because their location is only influenced by the form of equation. Therefore
the solution of the linear ordinary differential equations have only fixed singularities on complex
plane. The situation is completely changed when nonlinear ordinary differential equations are
under investigation, because in this case one can not generally predict either the location or the
type of singularities of the solutions. In particular, the solutions may have branch points or essential
singularities, which change their position depending on the initial values ck . The latter type of
singularities is called movable singular point. The possible singularities of differential equations
have been classified by Magnus Gösta Mittag-Leffler (1846, Stockholm - 1927, Stockholm): poles,
branch points, essential singular points, essential singular lines and perfect sets of singular points.
The comprehensive name for singular points that are not poles, of whatever order, is critical point.
The foremost studied class of differential equations was the ordinary differential equation of
order one in the form
dy
P (y; x)
=
(3.2.23)
dx
Q(y; x)
where P and Q are polynomials in y, and analytic in x. L. Fuchs proved [136] that Ricatti-equation
possesses only fixed critical points. Later, Painlevé demonstrated for more general conditions that
only poles and algebraic branch points can arise as movable singularities for the equation
dy
F
; y; x = 0
(3.2.24)
dx
where F is polynomial in y 0 and y, and analytical in x.
Similar examination of second order differential equation was started by Painlevé in the form
dy
dy 2
=F
; y; x
(3.2.25)
dx2
dx
where F is rational in y 0 , algebraic in y, and analytic in x. Painlevé and his colleagues showed
that only fifty such equations did not have movable singularities other than poles. Forty-four of
them could be solved via exact reduction into linear differential equations, solvable with the help
of known functions, or could be mapped into the six remaining differential equation. These six
equations are the famous Painlevé-transcendents and listed in table 3.1.
Albeit Painlevé equations arose in strictly mathematical studies, later they played an important
role in several physical problems. We can consider them as nonlinear special functions. These are
the same like hypergeometric functions, Bessel functions, Airy functions and so on, in the case of
linear second order differential equations. The question, whether or not the Painlevé transcendents
are really new functions has been answered recently. Generally, they are new transcendental
functions although, for special choice of constants a, b, c and d, the solutions can be reduced to
terms of rational functions or classical special functions. Below we enumerate some examples of
emergence of Painlevé-transcendents in physics:
• Spin-spin correlation function for the two-dimensional Ising-model [137]
• Asymptotic of modified Korteweg-de Vries equation [138]
• Density-matrix of one-dimensional Bose-gas [139]
• Nonlinear optics [23, 140]
36
CHAPTER 3. PAINLEVÉ-ANALYSIS
3.2. PAINLEVÉ ANALYSIS
Canonical form of Painlevé transcendent
Notation
2
PI
d w
dz 2
= 6w2 + az
PII
d2 w
dz 2
PIII
d2 w
dz 2
PIV
d2 w
dz 2
PV
d2 w
dz 2
PV I
d2 w
dz 2
= 2w3 + zw + a
2 1 dw 1
= w1 dw
− z dz + z aw2 + b + cw3 + d w1
dz
1
dw 2
= 2w
+ 32 w3 + 4zw2 + 2 z 2 − a w + b w1
dz
(w−1)2
1
1
dw 2
= 2w
+ 1−w
− 1z dw
aw + b w1 + c wz + d w(w+1)
dz
dz +
2
w−1
1
1
dw 2
1
1
dw
= 12 w1 + w−1
+ w−z
− 1z + z−1
+ w−z
dz
dz +
(z−1)
z(z−1)
+ w(w−1)(w−z)
a + b wz2 + c (w−1)
2 + d (w−z)2
z 2 (z−1)2
Table 3.1: Table shows those second order differential equations of one degree, whose movable singularities can be
only poles. These are called nowadays as Painlevé-transcendents, where a, b, c, d are arbitrary complex numbers.
• Einstein-Maxwell equations derived in general relativity [141]
• General relativity [142]
• Scattering of electromagnetic radiation [143]
Definition: /Painlevé-property for ODE /
An ordinal differential equation is said to possess the Painlevé-property if its general solution is
free of movable critical singularities.
A differential equation is said to be integrable if it is solvable via an associated linear-problem.
The Painlevé transcendents are prototypes of integrable ordinary differential equations.
3.2.2
Partial differential equations
One can ask how integrability is related to Painlevé-property. There is a convincing evidence
that integrability of a nonlinear differential equation is strongly connected with the singularity
structure of its solution. A significant step was made by Ablowitz, Ramani and Segur in the
following conjecture [133]
Definition: /Ablowitz-Ramani-Segur (ARS) conjecture /
Any ODE which arises as a reduction of an integrable PDE possesses the Painlevé property, possibly
after a transformation of variables.
The basic idea is so obvious. The study of symmetries of differential equations provides important information about their behaviour. The symmetries can be used to find exact solutions and
may help us to find out how variables should be changed in order to reduce a partial differential
equation to an ordinary differential equation. So if one can find all of the possible reductions of
37
CHAPTER 3. PAINLEVÉ-ANALYSIS
3.2. PAINLEVÉ ANALYSIS
a partial differential equation into an ordinary differential equation, and all of the latter possess
Painlevé property, one may conclude that the original equation has been integrable. Moreover we
have to note here that for the given the symmetries play a great role in the determination of Laxrepresentation, Bäcklund-transformation equation, as well as an effective numerical schemes [144].
Unfortunately, the method outlined above has a significant weakness in its application. This
is because one must find all the possible similarity reductions and check whether the results have
Painlevé property or not. Although the theory of one-parameter infinitesimal Lie-groups provides
an algorithm to find such reductions, it does not give all of them. Proofs of weakened versions of
this conjecture exist, nevertheless the truth of this version of lemma has not yet been established.
It would be more useful to operate on the partial differential equation directly. Such a procedure
was suggested by Weiss, Tabor and Carnevale [134], abbreviated as the WTC-method.
Let us recall the method used in the previous section where Kovalevskaya’s idea has been
explained. There the singularity structure of an ordinary differential equation of motion was under
investigation, and the solution was expanded in terms of (t − t1 ). It is self-evident that if the
number of independent variables is more than one then singularities do not occur in points, but
on singularity manifolds. This is the default case when partial differential equations are treated.
Generally, a function with m complex variables has (m − 1) dimensional singularity manifold.
Now, the function u is expanded in a special Laurent-series
u(z1 ; z2 ; . . . , zn ) = u(z) = φ−p (z)
∞
X
uj (z)φj (z)
(3.2.26)
j=0
where the independent variables are z = (z1 ; z2 ; . . . , zn ) whereas uj (z) and φ(z) are analytic functions of z. Moreover, we assume tacitly that u0 (z) 6≡ 0 in the neighbourhood of a singularity
manifold defined by the equation φ(z) = 0.
All steps are the same as in the case of rotating body. We have to insert this expansion back
to the equation of motion and determine the dominant behaviour, namely the leading exponent
p ∈ N+ , and all expansion coefficients uj (z). Usually, in the computation of coefficients one gets a
recursion relation which can be written symbolically as
(n − β1 )(n − β2 ) · · · (n − βN )un = Fn (u0 ; u1 ; . . . ; un−1 ; φ; z) .
(3.2.27)
The βj ’s are called resonances. It is evident that un is only defined if n 6= βj (j = 1, 2, . . . , N ),
otherwise arbitrary functions may occur. But one can have a look at this expression from another point of view, namely by requiring the fulfilment of (3.2.27) for all cases, including the one
at n = βj . From this requirement the compatibility or consistency constraints can be derived
Fn=βj (u0 ; u1 ; . . . ; un−1 ; φ; z) ≡ 0. We have to note here, that compatibility conditions do not
belong to the negative resonances, since the expansion (3.2.26) is valid for j ≥ 0. But negative
resonances may contain many information about the integrability of a given equation [145].
A definite value, βj = −1, is always in the set of {βj }N
j=1 and is responsible for the arbitrariness
of the function φ(z). From another respect, if one of the βj ’s is not an integer, then the Painlevé
test stops, and a weak Painlevé test can be carried out.
Exercise: / One-component Nonlinear Schrödinger equation /
Using the method of Painlevé analysis find out the integrability criterions for the one-component
generalized Nonlinear Schrödinger equation [135] !
Solution:
The usage of WTC method is demonstrated here on the case of one-component,
38
CHAPTER 3. PAINLEVÉ-ANALYSIS
3.2. PAINLEVÉ ANALYSIS
damped and driven Nonlinear Schrödinger equation. This example will show what conditions must
be fulfilled for the integrability.
The generalized and dimensionless form of the Nonlinear Schrödinger equation is
iut + uxx − 2 |u|2 u = a(x; t)u + b(x; t),
(3.2.28)
where a(x, t) and b(x, t) are arbitrary analytic functions. This equation can be roughly understand
of as a generalization of the quantum mechanical (1 + 1) dimensional Schrödinger equation
i~
∂Ψ(x, t)
~2 ∂ 2 Ψ(x; t)
=−
+ V (x)Ψ(x; t)
∂t
2m ∂x2
(3.2.29)
where the external potential, V (x), is proportional to the density, |Ψ(x; t)| 2 . Without the external
force, b(x; t), equation (3.2.28) is widely called as Gross-Pitaevskii equation in the field of BoseEinstein condensation.
Let us complexify the equation (3.2.28) !
iut + uxx − 2u2 v
= a(x; t)u + b(x; t)
(3.2.30a)
−ivt + vxx − 2v 2 u = a∗ (x; t)v + b∗ (x; t)
(3.2.30b)
where the notation ∗ stands for complex conjugation. We thus have two functionally independent,
complex functions with variables (x; t). Let us look for the solution in the form of Laurentexpansion.
u(x; t) = φ−p (x; t)
∞
X
uj (t)φj (x; t),
v(x; t) = φ−q (x; t)
j=0
∞
X
vj (t)φj (x; t).
(3.2.31)
j=0
Using the implicit-function theorem, φ can be written in the form: φ(x; t) = x + ψ(t) (Kruskalgauge) along the singularity manifold. Additionally the functions a(x; t) and b(x; t) are expanded
in terms of φ:
a(x; t) =
b(x; t) =
∞
X
j=0
∞
X
j
aj (t)φ (x; t)
where
bj (t)φj (x; t)
where
j=0
1 ∂ j a(x, t) ,
aj (t) =
j! ∂xj x =−ψ(t)
1 ∂ j b(x, t) bj (t) =
.
j! ∂xj x =−ψ(t)
(3.2.32a)
(3.2.32b)
First of all we have to determine the exponents p, q, since the dominant behaviour of functions
u(x; t) and v(x; t) are specified by the terms u0 φ−p and v0 φ−q in the neighbourhood of singularity
manifold. Substituting only these terms into equations (3.2.30) we have
2
i u0t φ−p − pu0 φ−(p+1) φt + p(p + 1)u0 φ−(p+2) (φx ) − pu0 φ−(p+1) φxx −
−2u20 v0 φ−(2p+q) = au0 φ−p + b,
2
−i v0t φ−q − qv0 φ−(q+1) φt + q(q + 1)v0 φ−(q+2) (φx ) − qv0 φ−(q+1) φxx −
−2v02 u0 φ−(2q+p) = a∗ v0 φ−p + b∗ .
39
(3.2.33a)
(3.2.33b)
CHAPTER 3. PAINLEVÉ-ANALYSIS
3.2. PAINLEVÉ ANALYSIS
By requiring the leading terms to vanish, two conditions are to be satisfied; the exponents must
be equal and sum of the certain coefficients should be zero:
2
p(p + 1)u0 (φx ) = 2u20 v0
p + 2 = 2p + q
2
q + 2 = 2q + p
q(q + 1)v0 (φx ) =
(3.2.34a)
2v02 u0
(3.2.34b)
We define p and q as non-negative integer numbers, so we get p = q = 1, trivially. In calculation of
coefficients we utilize the Kruskal-gauge, φ(x, t) = x + ψ(t), near to the singularity manifold from
which φx = 1. Hence it follows that u0 v0 = 1.
The next step is the determination of all the coefficients of the Laurent-expansion (3.2.31).
Substituting the whole series into (3.2.30), and collecting the corresponding terms in φ we conclude
with the following expression:
2
j − 3j − 2
−2u20
uj
Fj
=
(3.2.35)
−2v02
j 2 − 3j − 2
vj
Gj
where
Fj
= 2
j−1 X
k
X
ul uk−l vj−k + 2v0
j−1
X
l=1
k=1 l=0
+(−i)
ul uj−l − i(j − 2)uj−1
dψ
+
dt
j−2
duj−2 X
+
ak uj−k−2 + bj−3 ,
dt
(3.2.36)
k=0
Gj
= 2
j−1 X
k
X
vl vk−l uj−k + 2u0
k=1 l=0
j−1
X
l=1
vl vj−l + i(j − 2)vj−1
dψ
+
dt
j−2
+i
dvj−2 X ∗
+
ak vj−k−2 + b∗j−3 .
dt
(3.2.37)
k=0
This is a linear, recursive relation for uj and vj . Implicitly, we used the notation uj = vj ≡ 0 if
j < 0. Resonances are defined by the determinant of the coefficient matrix, namely
2
j − 3j − 2
−2u20
2
det
= (j 2 − 3j − 2)2 − (2u0 v0 ) =
−2v02
j 2 − 3j − 2
= (j 2 − 3j − 2)2 − 4 =
= (j 2 − 3j − 2 − 2)(j 2 − 3j − 2 + 2) =
= (j + 1)j(j − 3)(j − 4)
The resonances are j = −1, 0, 3, 4. We demand the recursion (3.2.35) to be consistent in the case
of resonances, thus compatibility conditions belong to all of these resonances. We give a simple
example for the case j = 3:
−2 −2u20
u3
F3
=
(3.2.38)
G3
−2v02
−2
v3
This equation can hold iff F3 v0 = G3 u0 which obviously gives a constraint on some expansion
coefficients. Using the expressions for F3 and G3 we can determine them as a function of the
40
CHAPTER 3. PAINLEVÉ-ANALYSIS
3.2. PAINLEVÉ ANALYSIS
predetermined coefficients (u0 , u1 , u2 , v0 , v1 , v2 ). Moreover, the recursion (3.2.35) guarantees that
each coefficient can be specified using just u0 and v0 , otherwise some arbitrary functions arise.
a1 − a∗1 + v0 b0 − u0 b∗0 = 0
(3.2.39)
Between u0 and v0 the relation u0 v0 = 1 establishes a connection, so that either u0 or v0 can be
treated as an independent variable. Equation (3.2.39) is valid for all (u0 , v0 ) pairs, so that the
relations
a1 = a∗1 ,
b0 = b∗0 ≡ 0
(3.2.40)
must be satisfied. In the Laurent-expansion (3.2.31) the function ψ(t) is arbitrary and b 0 (t) =
b(−ψ(t); t). It follows that b(x; t) ≡ 0.
Summarizing the constraints originated for j = 3 the compatibility conditions are
a1 = a∗1 ,
b(x; t) ≡ 0.
(3.2.41)
Similar calculation should be performed when j = 4. Omitting the details of computation we only
present the result:
1
i d
F4 v0 + G4 u0 = − (a0 − a∗0 )2 +
(a0 − a∗0 ) + a2 + a∗2 = 0.
2
2 dt
(3.2.42)
Let us split a(x; t) into its real and imaginary parts
a(x; t) = α(x; t) + iβ(x; t)
(3.2.43)
Substituting back into (3.2.42) one gets
1
i d
dβ0
− (2iβ0 )2 +
(2iβ0 ) + 2α2 = 2β02 −
+ 2α2 = 0
2
2 dt
dt
Employing the definition of a2 (t) from (3.2.32), we find:
dβ0
∂ 2 α(x, t) 2β02 −
+
= 0.
dt
∂x2 x =−ψ(t)
(3.2.44)
(3.2.45)
By integrating twice we get:
α(x; t) = x2
1 dβ0
− β02 + xα1 (t) + α0 (t)
2 dt
where α0 and α1 are arbitrary real functions of t. We have thus:
1 dβ
2
2
α(x; t) = x
− β + xα1 (t) + α0 (t),
2 dt
β(x; t) = β(t).
(3.2.46)
(3.2.47a)
(3.2.47b)
At this point we have finished the Painlevé test, since all expansion coefficients uj , vj (j ≥ 0) and
the exponents of leading terms are determined.
Let us summarize the result of the Painlevé test of the generalized Nonlinear Schrödinger
equation. The constraints of total integrability of this equation, in the sense stated above, have
been reduced to its essentials
1 dβ
2
2
a(x; t) = x
− β + xα1 (t) + α0 (t) + iβ(t),
b(x; t) ≡ 0
(3.2.48)
2 dt
41
CHAPTER 3. PAINLEVÉ-ANALYSIS
3.2. PAINLEVÉ ANALYSIS
where β, α1 , α0 are real and twice differentiable functions of t. We formulate the result also in
terms of words. The generalized Nonlinear Schrödinger equation is totally integrable when
• the external potential is parabolic in the spatial variable x, but it must be complex in this
case;
• the external potential is real, and linear in the spatial variable x.
In these cases general solution can be given using the Lax-representation method and the Inverse
Scattering Transformation for vanishing boundary conditions. Later A. S. Fokas presented an
IST-like procedure for a mixed initial-boundary problem [146].
Otherwise, the equations of motion arisen in a real physical problems can not be reduced into
these cases; then the solution may be found only by numerical computation or ad hoc methods. ♠
42
3.3. HAMILTONIAN FORMALISM AND
INVERSE SCATTERING TRANSFORMATION
CHAPTER 3. PAINLEVÉ-ANALYSIS
3.3
Hamiltonian Formalism and
Inverse Scattering Transformation
I feel it necessary to summarize the basic formulas of Inverse Scattering Transformation, because
in the next section we relate it to the integrability in the sense of classical mechanics. In this
manner we can realize how intimately the theories derived purely from mathematical or physical
problems lock into each other.
Figure 3.5: This box-diagram tries to visualize the connection between idea of integrability in classical mechanics
and Inverse Scattering Transformation of nonlinear evolution equations.
Classical Hamiltonian Mechanics
Nonlinear field equations
”Is there any canonical transformation to
new variables of motion in which the time
evolution becomes as simple as can be?”
”Is it possible to solve a nonlinear dynamical equation using only linear steps?”
Paul Painlevé
+
Sofia Kovalevskaya
-
?
Action-angle variables are defined as
I
1
∂S(q; I)
p dq
ϕk =
Ik (f ) =
2π
∂Ik
Γk
where k = 1, . . . , n and the time evolution is trivial:
I˙k = 0
?
Inverse Scattering Transformation
K(x; y; t) + F (x + y; t) +
-
Z∞
K(x; z; t)F (z + y; t) dz = 0
x
F (x; t) =
N
X
c2n (t)e−ξn t +
n=1
ϕ˙k = ωk (Ik )
1
2π
−∞
Z
R(k; t)eikx dk
−∞
where cn , ξn , R(k; t) are the scattering data
calculated from initial conditions.
?
The solution can be constructed by
u(x; t) = 2
43
∂ K(x; x; t)
∂x
3.3. HAMILTONIAN FORMALISM AND
INVERSE SCATTERING TRANSFORMATION
CHAPTER 3. PAINLEVÉ-ANALYSIS
3.3.1
Integrability in classical mechanics
In this short section we give some definition of Hamiltonian description of classical mechanics and
the way how the notion of integrability arises. At the end we try to enlighten how this type of
integrability is connected to Inverse Scattering Transformation (IST).
Although, the theoretical details of IST does not lie within the scope of this thesis, I feel it
necessary to summarize and present how it relates to integrability. The results obtained during my
Ph.D. course can be treated as a preliminary study to our final aim: find soliton or solitary wave
solutions of coupled Gross-Pitaevskii equations with given initial and boundary conditions using
IST.
A future article is devoted to our results achieved by co-operation with András Szilágyi, as an
undergraduate student. We investigated [7] a single Nonlinear Schrödinger equation and demonstrated how a square well potential evolves in time and the soliton emerges and the excess radiates
out. On the Students’ Scientific Conference 2003 András presented our calculations and bears
witness to his performer skills. He won the first prize and the surcharge offered by Pro Progressio
Foundation.
Definition: /Hamiltonian mechanical system /
A dynamical system is said to be Hamiltonian if it is possible to identify a smooth scalar function
H({qi ; pi }; t), called Hamiltonian, with generalized coordinates qi , generalized momenta pi and time
t such that the equation of motion can be written in the form:
q̇i =
∂H
∂pi
ṗi = −
∂H
∂qi
(i = 1 . . . f )
(3.3.49)
These equations are called canonical or Hamiltonian equations of motion, and f is the degree of
freedom possessed by the mechanical system.
The order of canonical equations of motion is 2f . The initial conditions, e.g. at t = t 0
(0)
(0)
qi (t0 ) = qi , pi (t0 ) = pi uniquely determine the time evolution of the system, since a system
of first order ordinary differential equations requires as many initial conditions as the number of
equations.
Imagine that we could transform the original coordinates of the system into a complete set
of observables which are either constant in time, or depend linearly on time. Then using these
observables as coordinates the solution of the system would be as trivial as that for free particle.
Thus instead of trying to solve differential equations directly in the original variables we shift to
looking for appropriate changes of coordinates. The Hamiltonian form (3.3.49) is not preserved
under an arbitrary coordinate transformation. What are the allowed changes of coordinates that
preserve Hamiltonian equations ? The answer is very compact, i.e, the new coordinates (Q, P )
must satisfy the following commutation relations:
Q̇i =
∂H({Qi ; Pi })
∂Pi
Ṗi = −
∂H({Qi ; Pi })
∂Qi
(3.3.50)
A straightforward calculation will show that the necessary conditions are
{Qi , Qk } = {Pi , Pk } = δi,k
(3.3.51)
where the curly bracket is the Poisson-bracket. The rule determines the mapping of {q, p} onto
{Q, P } variables that satisfy these relations are said to be canonical. There is an enormous freedom
within the world of canonical coordinates: we can mix momenta and positions with each other.
44
3.3. HAMILTONIAN FORMALISM AND
INVERSE SCATTERING TRANSFORMATION
CHAPTER 3. PAINLEVÉ-ANALYSIS
The generalized coordinate qr is said cyclic coordinate if the Hamiltonian H(qi ; pi ) does not
depend on qr explicitly, hence ∂H/∂qr = 0. If qr is a cyclic coordinate variable, then the generalized
momentum pr , canonically conjugated to qr , is a constant of motion, as it is followed from the
Hamiltonian equations of motion. In this case the order of the equations can be reduced by two.
Let H(qi ; pi ) be the Hamiltonian function of a classical mechanical system and qr be a cyclic
coordinate. The canonical equations of motions are
q̇i
=
q̇r
=
∂H
∂pi
∂H
(0)
∂pr
∂H
(i = 1 . . . r − 1, r + 1, . . . f )
∂qi
∂H
=0
ṗr = −
∂qr
ṗi = −
(3.3.52)
(3.3.53)
It is trivial that in the remaining equations pr (t = t0 ) is just a parameter. Hence the order of
equation of motion is (2f − 2). If we succeeded in integration of the remaining system of equation
then the equation belonging to qr
∂H(q(t); p(t))
q̇r =
(3.3.54)
(0)
∂pr
would be simple.
Let us suppose finding canonical coordinates (ϕk , Ik ) such that the Hamiltonian only depends
on the new momenta Ik , we can solve the equations of motions thereupon. For these coordinates
the Hamiltonian equations (3.3.49)
I˙k = 0
ϕ˙k =
∂H({ϕn ; In }) !
= ωk ({In })
∂Ik
(3.3.55)
Discovering such normal coordinates is a felicity in classical mechanics: once you find such a
coordinate system the equations are solved in a trivial way. The whole question is then whether
such coordinates exist and if so, how to find them.
This argumentation shows how important those transformation of variables are which transform
a variable to a cyclic coordinate during preserving the form of canonical equation of motion. If we
can find canonical coordinates q1 , . . . , qn , p1 . . . , pn such that all of the qi are cyclic, then we call
them action-angle variables, and we say that the Hamiltonian system is completely integrable in
the Liouville sense.
Theorem: /Liouville’s integrability theorem [147] /
Let a Hamiltonian system have n degree of freedom, and let us suppose that there exist n functions,
F = F1 , . . . , Fn , defined on the phase-space and they are mutually in involution:
{Fi , Fj } = 0,
(i, j = 1, 2, . . . , n).
Let Σk be the (2n − 1) dimensional submanifold defined by equations Fk = fk = constant. Write
f = {f1 , . . . , fn } for a set of constant values of the Fk and define the intersection of the Σk as
Σf ≡ Σ1 ∩ Σ2 ∩ · · · ∩ Σn . We make one further assumption: each Σf is compact irrespectively how
we choose fk ’s. A Hamiltonian dynamical system satisfying these properties is now defined as a
dynamical system completely integrable.
This theorem has many consequences, e.g., (a) each Σf is homotopic to an n-dimensional torus
and (b) there exist canonical coordinates {ϕ, I} such that the values of the Ik label the torus, and
45
3.3. HAMILTONIAN FORMALISM AND
INVERSE SCATTERING TRANSFORMATION
CHAPTER 3. PAINLEVÉ-ANALYSIS
are therefore constant of the motion and the dϕk /dt is also constant on each torus. This means,
that {ϕ, I} are a set of action-angle variables.
Consequently, if two first integrals are known in a system with two degree of freedom and these
are in involution, then one can calculate the time dependence of coordinates using only quadratures.
The submanifold defined by the equations H = h and F = f is an invariant torus, and the motion
is conditionally periodic. This behaviour is a special property of two-dimensional systems.
Liouville’s definition of integrability is based on the notion of “first integrals”, i.e., conserved
quantities. In his definition, a Hamiltonian system is said to be integrable if it has sufficiently
many first integrals in involution. The same idea has ever been inherited in many variants of the
notion of integrability.
Liouville’s definition of integrable Hamiltonian systems naturally covered many classical examples. Among them are the Kepler motion solved by Newton, harmonic oscillators solvable
by trigonometric functions, the rigid bodies (“spinning tops”) of the Euler-Poinsot type and the
Lagrange type, and Jacobi’s example of geodesic motion on an ellipsoid.
The spinning tops and Jacobi’s example were significant because they were known to be solvable
by elliptic functions. Soon after the work of Liouville, C. Neumann discovered a new integrable
Hamiltonian system, and pointed out that this system can be solved by hyperelliptic functions.
That was the beginning of subsequent discoveries of many integrable systems.
3.3.2
Action-angle canonical variables
In the previous section we stated that a Hamiltonian is said to be Liouville integrable when it
can be transformed to a canonical coordinate system in which it depends only on new momenta.
When the energy surfaces are compact and the new momenta are everywhere independent, Arnold
showed [147] that it is always possible to choose the momentum variables so locally that their
conjugate configuration variables are periodic angles ranging from 0 to 2π.
Under circumstances of Liouville’s theorem on can choose new independent coordinates (I; ω)
where the first integrals F depend only on I and ω coincide with the angle variables of tori. In these
new variables the Hamiltonian equations of motion become a system of 2n ordinary differential
equation:
dF
dω
=0
= ω(F)
(3.3.56)
dt
dt
These equations are easily integrable, since
F(t) = F(0)
ω(t) = ω(t = 0) + ω(F(0) )t
(3.3.57)
We can conlude that we would able to explicitly integrate the equations of motions if a canonical
transformation onto these new variable would be known. Nevertheless, the n first integrals do not
form an appropriate set of new coordinates, because this set is not simplectic. However, it can
be proven that there exist functions of the first integrals, I(F), such a way that the set (I; ω) is
already simplectic. In this context, the functions I and ω are called action and angle variables,
respectively.
3.3.3
Construction of action-angle variables
We see how convenient the usage of action-angle variables would be. This motivate us to find a
constructive method for calculation of the generator function of a canonical transformation which
maps (q, p) onto (ϕ; I). Let us denote this generator function with S(q; p). Then applying the
46
3.3. HAMILTONIAN FORMALISM AND
INVERSE SCATTERING TRANSFORMATION
CHAPTER 3. PAINLEVÉ-ANALYSIS
rules of canonical transformation depending on the original coordinate and the new momentum
variables we obtain the following expressions
∂S(q; I)
∂S(q; I)
∂S(q; I)
ϕ=
p=−
⇒
H
; q = h(I)
(3.3.58)
∂I
∂q
∂q
Suppose that the function h(I) is known and invertible. This latter property means that the path
defined by the constraint Γ ≡ I = I0 is unique. The total differential of the generator function is
dS|I=const = pdq
(3.3.59)
Integrating along this path one finds
S(q; I) =
Zq
p dq
(3.3.60)
q0
In this way we can express the generator function locally in the neighborhood of the point (q 0 ; I0 ).
The growth of S(q; I) along the total closed curve Γ is
I
∆S(q; I) = p dq
(3.3.61)
Γ
which is just the same as the area Π surrounded by the curve Γ. Preserving the harmony between
this definition and our classical mechanical practice - that the total change of angle variable along a
curve is 2π - we must divide Π by 2π. Now we can define the action variable for a one-dimensional
system, and show their practical usage in an example.
Definition: /Action variable in 1D /
Let us consider a one-dimensional conservative Hamiltonian system described by H(q; p). The
integration defined on the phase-space as
I
1
I(h) =
p dq
(3.3.62)
2π
Γ
is called action variable.
Exercise: / Harmonic oscillator in 1D /
Find the action-angle variables for one-dimensional harmonic oscillator !
Solution:
Let us write the Hamiltonian of this system as:
H(q; p) =
1 2 1
p + mω 2 q 2
2m
2
(3.3.63)
where m is the mass, ω denotes the frequency. This Hamiltonian does not depend on time t, hence
this system is conservative and H(q; p) is the same as total energy. It can be seen that the surface
H(q; p) = E is an ellipse on the phase-space (see figure 3.6) The area of that ellipse is
47
3.3. HAMILTONIAN FORMALISM AND
INVERSE SCATTERING TRANSFORMATION
CHAPTER 3. PAINLEVÉ-ANALYSIS
p
√
r
2mE
Figure 3.6: The picture shows the phase-space area
closed by the trajectory of the harmonic oscillator
√
at a fixedpenergy, E. The semiaxes are p̃ = 2mE
2
and q̃ = 2E/mω .
q
2E
mω 2
Π(E)
1
2π
=
I
1
p dq =
π
Γ
Zq̃ q
2mE − (mω)2 q 2 dq
−q̃
q̃
q
1 q
E
mωq
2
2
2mE − (mω) q + arcsin √
π 2
ω
2mE −q̃
E
E arcsin (1) − arcsin (−1) =
πω
ω
=
=
(3.3.64)
Comparing the left and right hand side of this series of equations we can immediately express the
Hamiltonian with the action variable Π(E) and get H(q; p) = ωΠ(E). The time evolution of the
variable ϕ associated to Π via canonically conjugation is:
ϕ̇ =
∂H(E)
=ω
∂Π
⇒
ϕ(t) = ωt + ϕ0
(3.3.65)
where ϕ0 is an arbitrary real constant fixed by the initial condition. Let us calculate the generator
function of the canonical transformation S(q; Π) in order to be able to express the motion in the
original variables.
S(q; Π)
=
Zq
p dq =
q0
=
=
q
q
q
E
mωq
=
2mE − (mω)2 q 2 + arcsin √
2
ω
2mE −q̃
q
q
E
mωq
π
2
2
2mE − (mω) q + arcsin √
+ E
2
ω
2
2mE
(3.3.66)
The angle-variable ϕ is derived from this expression using (3.3.58). Omitting the details we finally
get:
mωq
π
ϕ = arcsin √
+
(3.3.67)
2
2mE
Inverting this expression and substituting the proper time-dependence of ϕ we obtain
q(ϕ) =
r
π
2E
sin
ϕ
−
=
mω 2
2
48
r
π
2E
sin
ωt
+
ϕ
−
0
mω 2
2
(3.3.68)
3.3. HAMILTONIAN FORMALISM AND
INVERSE SCATTERING TRANSFORMATION
CHAPTER 3. PAINLEVÉ-ANALYSIS
This is a well-known expression, and apparently shows (in this case) the physical meaning of actionangle variables, namely this angle variable is the phase of the harmonic oscillator. ♠
In a more general case we can define the action variables in the same manner.
Definition: /Action variable in R2n /
Let us consider a mechanical system with n degree of freedom defined on the phase-space, R 2n =
(q; p). If n first integrals in involution, F = (F1 , · · · , Fn ), are known, then the
I
1
Ii (f ) =
p dq
i = 1, 2, . . . , n
(3.3.69)
2π
Γi
new n expressions are called action variables.
Canonical transformations are often employed to simplify the equations of motion. For example,
Hamilton’s equations are especially simple if the new Hamiltonian is a function of only the momentum variables, H 0 (I), or even more simply if H 0 = 0. If we can find a transformation to such
a coordinate system then the system is said to be integrable. In general, such transformation does
not exist, and the consequence is chaotic motion.
3.3.4
Inverse Scattering Transformation
The basic idea behind the Inverse Scattering Transformation originates from the quantum mechanical scattering methods, where we are interested in calculating the potential or at least some
properties of the potential directly from experimentally observable data: differential cross section
(dσ/dϑ), different kind of resonances (bound- or excited states) and of course the phase shifts
(δ(l; E)). This connection is obvious if we follow the Lax-representation or the so called AKNSrepresentation of evolution equations. In the remaining section we consider only these methods.
3.2.
Lax-representation
Let us examine the system of equations below
L̂v
vt
= ξv
(3.3.70a)
=
(3.3.70b)
M̂v
where L̂ and M̂ are unitary linear operators on the L1 (R) space, ξ is a complex number and the
subscript t denotes the partial differentiation with respect to the ”time variable” t.
In this notation these equations are independent from each other, but if we require the compatibility constraint on v, namely the equality of second partial derivatives, v xt = vtx , then we at
once establish also a strong connection between the operators, L̂ and M̂:
L̂v = L̂t v + L̂vt = (ξv)t = ξt v + ξvt
(3.3.71)
t
Using the unitary property of L̂, it can be proven that ξt ≡ 0, i.e. we may treat ξ as an eigenvalue
of L̂. At the end, we shifted the compatibility condition of v onto the operators and get
h
i
(3.3.72)
L̂t + L̂, M̂ = Ô
49
3.3. HAMILTONIAN FORMALISM AND
INVERSE SCATTERING TRANSFORMATION
CHAPTER 3. PAINLEVÉ-ANALYSIS
where the bracket means the commutator of its arguments. This equation is called Lax-equation
after the Hungarian mathematician Peter David Lax who invented this elegant formalism for
nonlinear evolution equations in 1968 [148]. An example is given below in which we show how the
compatibility condition results in nonlinear evolution equation.
Exercise: / A possible Lax-representation of the KdV equation /
Show that the Lax-equation is fulfilled with the linear operators
1
u(x; t)
6
1
= D̂3 +
u(x; t)D̂ + D̂u(x; t)
8
L̂ = D̂2 +
M̂
iff the unknown function u(x; t) is a solution for the KdV equation, where D̂ denotes the differential
operator with respect to x, and u(x; t) stands for a smooth function.
Solution:
Let us introduce more general forms of operators L̂ and M̂ in the following way while we preserve
the structure of L̂:
1
L̂ = D̂2 + u(x; t)
(3.3.73)
6
M̂1 = D̂3 + b(x; t)D̂ + D̂b(x; t)
(3.3.74)
where D̂ denotes the partial differential operator with respecth to variable
x, and b(x; t) is just a
i
notation for an adequately smooth function. Calculating the M̂1 , L̂ commutator using linearity
property we get
h
i h
i 1h
i
i
h
i 1h
M̂1 , L̂ = D̂3 , D̂2 +
D̂3 , u(x; t) + b(x; t)D̂, D̂2 +
b(x; t)D̂, u(x; t) +
6
6
h
i 1h
i
2
+ D̂b(x; t), D̂ +
D̂b(x; t), u(x; t)
(3.3.75)
6
These commutators can be evaluated step by step independently and get
h
i
D̂3 , D̂2
= Ô
(3.3.76a)
1
1
1
D̂3 , u(x; t)
=
uxxx (x; t) +
uxx (x; t)D̂ + ux (x; t)D̂2
(3.3.76b)
6
6
2
h
i
b(x; t)D̂, D̂2
= −bxx (x; t)D̂ − 2bx (x; t)D̂2
(3.3.76c)
1
1
b(x; t)D̂, u(x; t)
=
b(x; t)ux (x; t)
(3.3.76d)
6
6
h
i
D̂b(x; t), D̂2
= −bxxx(x; t) − 3bxx(x; t)D̂ − 2bx (x; t)D̂2
(3.3.76e)
1
1
D̂b(x; t), u(x; t)
=
b(x; t)ux (x; t)
(3.3.76f)
6
6
Substituting these expressions back to equation (3.3.75) and collecting all the coefficients with the
same rational power of the differential operator together we finally get a simple formula:
h
i 1
1
1
M̂1 , L̂ = (uxxx − 6bxxx + 2bux ) + (uxx − 8bxx) D̂ + (ux − 8bx ) D̂2
(3.3.77)
6
2
2
50
3.3. HAMILTONIAN FORMALISM AND
INVERSE SCATTERING TRANSFORMATION
CHAPTER 3. PAINLEVÉ-ANALYSIS
We note that this formula only contains functions and their derivatives and differential operator on
the second power at the most. It is transparent that if the function b(x; t) were well chosen then the
terms containing differential operators vanish simultaneously totally everywhere and every time.
Hence we take b(x; t) = u(x; t)/8.
h
i
1
M̂1 , L̂ =
(uxxx + uux )
(3.3.78)
24
Finally we define our original operator M̂ to be equal with −24M̂1 . After all the Lax equation
(3.3.72) can be written in a familiar form which will be the KdV equation evidently.
h
i
L̂t + L̂, M̂ = ut (x; t) + uxxx (x; t) + u(x; t)ux (x; t) = 0
(3.3.79)
What can we see? Despite the linearity of two operators, L̂ and M̂ , the Lax equation associated
with them is only fulfilled iff the function u(x; t) – which is yet unknown – is a solution of a
nonlinear evolution equation, namely the Korteweg-de Vries equation’s. This explains why one
says that the operators
L̂ =
M̂
=
1
u(x; t)
6
1
D̂3 +
u(x; t)D̂ + D̂u(x; t)
8
D̂2 +
(3.3.80a)
(3.3.80b)
are a Lax-representation of the KdV equation. ♠
We note here one more general – but not obligate – feature of Lax-representation. The two
operators have different ”role” as we can realize from equations (3.3.70). We may say that Lax
decomposed the nonlinear evolution equation into two linear problems. This can be done in
many ways (see for example Ablowitz’s excellent monograph [149]). As we choose the forms of
the operators we can identify their definite aim. The first equation (3.3.70a) does not contain
explicitly time derivative, and the left hand side is the same as Schrödinger equation would be.
Here, the operator L̂ plays the role of Schrödinger operator while the undefined function u(x; t) is
the external potential, and the wave function is v. This is a spectral problem. The second equation
(3.3.70b) has the responsibility of time evolution, hence it contains time derivative term.
And this is the general and common features of different kind of representations. They try to
decompose the nonlinear evolution equation into two distinct parts, viz. into a spectral problem
and another one responsible for evolving the spectral data in time. In this way the nonlinear
dynamical equation is mapped onto a system of linear equations. That was the revolutionary idea
behind the solution of the KdV equation performed by Gardner, Greene, Kruskal and Miura in
1967 [150]. Moreover this remarkable recognition could be generalized and were not applicable
only to the KdV equation.
Summarizing the method, omitting the derivation and proof of the formulas, we present here a
prescription on how to obtain a general solution of a nonlinear evolution equation step by step, if
its Lax representation is known:
1.) Direct scattering: Solve the initial value problem of the Schrödinger type equation (3.3.70a).
This means that we determine
the scattering data associated
with the ”potential” u(x, 0),
N
i.e. finally we get the set {cn (0), ξn }n=1 , R(k; 0), T (k; 0) , where the normalization constants
51
3.3. HAMILTONIAN FORMALISM AND
INVERSE SCATTERING TRANSFORMATION
CHAPTER 3. PAINLEVÉ-ANALYSIS
cn (0) and the eigenvalues ξn belong to the bound states supported by external potential u(x, 0),
while R(k; 0), T (k; 0) are the reflection and transmission coefficients depending on a continuous
variable k (∼ energy).
2.) Time evolution: Using equation (3.3.70a) we evolve the scattering data in time.
3.) Inversion of the scattering data: It is proven that in one-dimension there is a unique
relationship between scattering data and potential. This strong connection is manifested in an
integral equation, called in this context Gel’fand-Levitan-Marchenko integral equation:
K(x; y; t) + F (x + y; t) +
Z∞
K(x; z; t)F (z + y; t) dz = 0
(3.3.81)
x
where the input kernel F (x; t) is determined from previously calculated scattering data
F (x; t) =
N
X
c2n (t) exp (−ξn x)
n=1
1
+
2π
Z∞
R(k, t)eikx dk
(3.3.82)
−∞
The unknown function to be determined is K(x; y; t).
4.) Deducing the potential from the kernel K(x; y; t): The external potential at a given
moment of time can be obtained from the function K(x; x; t) diagonalized in space variable
u(x; t) = 2
∂K(x; x; t)
∂x
(3.3.83)
It is worth to mention here that we must localize both spatial argument of K(x; y; t) at x and
just after that calculate the partial derivative.
These steps are visualized in the diagram below
Scrödinger equation
u(x; 0)
−−−−−−−−−−−−−−−−−−−−−−−−−−−−→
u(x; t)
←−−−−−−−−−−−−−−−−−−−−−−−−−−−−
3.3.
Gel’fand-Levitan-Marchenko equation
N
S(ξ; 0) = {ξn , cn (0)}n=1 , R(ξ; 0), T (ξ; 0)


ytime evolution
S(ξ; t) = {ξn , cn (t)}N
,
R(ξ;
t),
T
(ξ;
t)
n=1
AKNS-representation
Later, the Lax-method was generalized by Ablowitz, Kaup, Newel and Segur [151,152]. In the new
method two different roles of the operators have been emphasized. The details of this method is
out of the scope of this thesis so that we only summarize the basic equations of AKNS-scheme.
They consider a more symmetric form of eigenvalue and time-evolution problems in the form:
vx
vt
= Xv
= Tv
(3.3.84)
(3.3.85)
where the matrices X and T do not contain any differential or integral operators. As in the
case of the Lax-representation the nonlinear evolution equation arises from the requirement of
compatibility of these equations. Formally, the compatibility condition is:
(3.3.86)
X t−T x+ X, T =O
52
3.3. HAMILTONIAN FORMALISM AND
INVERSE SCATTERING TRANSFORMATION
CHAPTER 3. PAINLEVÉ-ANALYSIS
Let us specify the size of the matrices, and examine only the case of the 2 × 2 matrices. It can be
proven [149] that many physically relevant equations have an AKNS representations. For example,
a generalized Nonlinear Schrödinger equation in the form:
2
iqt + αqxx − 2α |q| q = 0
has the following operator representation:
−iξ q
X =
q ∗ iξ
2
−2iαξ 2 − iα |q|
T =
2αq ∗ ξ − iαqx∗
(3.3.87)
(3.3.88a)
2αqξ + iαqx
2
2iαξ 2 + iα |q|
(3.3.88b)
where q = q(x; t) is a scalar function, α is a constant and c = c(t) stands for an arbitrary function
of time t. The function q(x; t) plays the same role as the potential u(x; t) does in case of the
Lax-representation of KdV equation in equations (3.3.80), and ξ corresponds to the eigenvalue..
The steps of the solution is nearly the same as in the Lax method previously. The biggest
difference of the method resides in the integral equation of inversion. This is not surprising since
the formulas of direct scattering also differ. Summarizing the AKNS-method we can write:
1.) Direct scattering: We solve the scattering problem at t = 0 with the given initial condition
q(x; 0) and get the discrete eigenvalues (ξn ), normalization constants (cn (0)), reflection (R(ξ; 0))
and transmission coefficients (T (ξ; 0):
N
S(ξ; 0) = {ξn , cn (0)}n=1 , R(ξ; 0), T (ξ; 0)
(3.3.14)
2.) Time evolution: The equation (3.3.84) is responsible for time evolution. From that equation
we can derive the time evolution of scattering data (see later in specific examples).
N
S(ξ; t) = {ξn , cn (t)}n=1 , R(ξ; τ ), T (ξ; t)
(3.3.15)
3.) Inversion of the scattering data: The potential can be uniquely determined from the scattering data in one space dimension. We have to solve the Zakharov-Shabat integral equation:
Z∞Z∞
K(z; y; τ ) = B (z + y; τ ) −
B ∗ (y + r; τ )B(r + k; τ )K(z; k; τ )drdk
∗
(3.3.16)
z z
where the input kernel B(x; t) contains the scattering data and defined as
1
B(z; τ ) =
2π
Z∞
R(ξ; τ ) e
−∞
iξz
dz − i
N
X
cj (τ )eiξj z
(3.3.17)
j=1
4.) Deducing the potential from the kernel K(z; y; τ ): From the solution K(z; y; τ ) of equation (3.3.16) one can determine the solution of the original nonlinear equation. In case of
nonlinear Schrödinger equation the relationship can be written in the following simple way:
q(z; τ ) = −2K(z; z; τ )
We note here that this form depends on the nonlinear dynamical equation.
53
(3.3.18)
3.3. HAMILTONIAN FORMALISM AND
INVERSE SCATTERING TRANSFORMATION
CHAPTER 3. PAINLEVÉ-ANALYSIS
Exercise: / Bright multi-soliton solution of KdV equation by using IST /
Find soliton solution of the Korteweg-de Vries equation by direct calculation. As indicated in the
text, we can suppose now that the reflection coefficient R(k; t) ≡ 0, and only discrete eigenvalues
are present.
Solution:
Let us suppose we have got N distinct eigenvalues denoted by ξn . In this case, the kernel
function F (x; t) is a sum of N terms, hence the Fourier-integral in (3.3.82) is absent. Substituting
the sum into the integral-equation we find that all terms will be proportional with exp (−ξ n y).
This may give a hint for using the ansatz below:
K(x; y; t) =
N
X
Kn (x; t)e−ξn y
(3.3.19)
n=1
In this way we can detach the y-dependence from K(x; y; t), since the Gel’fand-Levitan-Marchenko
equation becomes separable:
N
X
∞
Kn (x; t) +
c2n (t)e−ξn x
n=1
e
−ξn y
+
N X
N Z
X
Kl (x; t)e−ξl z c2n (t)e−ξn (z+y)dz = 0
(3.3.20)
n=1 l=1 x
One can separate the summation with respect to n and gets


Z∞
N
N
X
X
Kn (x; t) + c2n (t)e−ξn x +
Kl (x; t)c2n (t) e−(ξn +ξl )zdz  e−ξl y = 0
n=1
l=1
(3.3.21)
x
The exponential functions with different arguments are orthogonal, hence this equation is satisfied
iff the expression in the bracket is identically zero for all x and t. Accordingly
Kn (x; t) + c2n (t)e−ξn x +
N
X
c2n (t) −(ξn +ξl )x
e
Kl (x; t) = 0
ξn + ξ l
(3.3.22)
l=1
The third term looks like a matrix-vector multiplication. One can transform the first expression
to similar form using the Kronecker-delta function. Arranging the formulae on the proper side of
the equation according to their dependence on Kn (x; t) we get:
N X
c2 (t) −(ξn +ξl )x
δnl + n
e
Kl (x; t) = −c2n (t)e−ξn x
(3.3.23)
ξn + ξ l
l=1
Let us introduce the following notations:
c2n (t) −(ξn +ξl )x
e
ξn + ξ l
bn (x; t) = −c2n (t)e−ξn x
Cn,l (x; t) = δnl +
(3.3.24a)
(3.3.24b)
With these labels we can reformulate (3.3.23) into a system of algebraic equations
N
X
Cn,l (x; t)Kl (x; t) = bn (x; t)
l=1
54
(3.3.25)
3.3. HAMILTONIAN FORMALISM AND
INVERSE SCATTERING TRANSFORMATION
CHAPTER 3. PAINLEVÉ-ANALYSIS
If we arrange the functions Cn,l (x; t) and bn (x; t) into a matrix and vector form, respectively,
where n counts the row-index and l does the column index and naturally in case of Kl (x; t) the
l labels the row-index, we deduce a very compact form of GLM-equation in case of reflectionless
potential which supports only bound states:
C(x; t)K(x; t) = b(x; t)
(3.3.26)
Formally one can invert the coefficient matrix and calculate the solution of this equation as
K(x; t) = C−1 (x; t)b(x; t)
(3.3.27)
The solution of the original integral equation can be found by using the definition (3.3.19)
K(x; y; t) =
N
X
Kn (x; t)e−iξn y
(3.3.28)
n=1
Thus, we have solved the problem since the potential, which is the generic solution of KdV equation,
can be constructed from K(x; x; t) using (3.3.83).
At the end of this example we present two specific cases for demonstration. The simplest
case is if there were only one eigenvalue. One can formulate this case easily. Omitting the steps of
calculation we only write the essential expressions. We can immediately find the function K(x; y; t)
K(x; y; t) = −
c2 (t)e−ξx
e−ξy
c2 (t) −2ξx
1+
e
2ξ
(3.3.29)
The potential u(x; t) can be constructed from (3.3.83). This calculation is a bit tedious and needs
skill in trigonometry, but the final form is enlightening.
∂K(x; x; t)
∂ c2 (t)e−2ξx
= −2
∂x
∂x 1 + c2 (t) e−2ξx
2ξ
2
−2ξx
2
−c (t)e
2ξ + c (t)e−2ξx + c4 (t)e−4ξx
= −2
2
2
1 + c 2ξ(t) e−2ξx
u(x; t) = 2
(3.3.30)
(3.3.31)
In order to simplify this result we convert both the normalization constant and the eigenvalue into
exponential form. This is enabled, because we know that the time evolution of the normalization
constant is also expressible via exponential functions
c(t) = c0 ef (ξ)t
(3.3.32)
where the function f (ξ) is unique for every nonlinear evolution equation, e.g., for KdV equation
f (ξ) = 4ξ 3 . Moreover, one can write the constant eigenvalue in the form
1
√ = e−ξx0
2ξ
Using these denotations we shall find a very impressive and compact form for u(x; t):
u(x; t) = 2ξ 2 sech2 ξ[x + x0 ] − f (ξ)t
55
(3.3.33)
(3.3.34)
3.3. HAMILTONIAN FORMALISM AND
INVERSE SCATTERING TRANSFORMATION
CHAPTER 3. PAINLEVÉ-ANALYSIS
x
–20
–10
0
10
20
ξ = 0.7
c0 = 2.0
20
u(x,t)
1.2
1
10
0.8
0
t
0.6
4 ξ2
2ξ2
0.4
–10
0.2
–20
–20
–10
0
10
20
x
Figure 3.7: Both pictures was generated with the same values of the parameters: ξ = 0.7, c 0 = 2.0, and now
we choose here the KdV-type time-evolution f (ξ) = 4ξ 3 [149]. Left: The two-dimension picture exhibit the time
and space evolution of the solution u(x; t) as a function of x and t. One can easily estimate the velocity of the
soliton. Brighter regions means higher amplitude. Right: The picture shows the solution u(x; t) at time t = −7.59
as a function of space variable x, and demonstrate two important properties of a single soliton, its amplitude (2ξ 2 )
and its velocity (4ξ 2 ). Similar characteristic features can be introduced in case of multisoliton solutions of the
KdV equation, because under the collision these parameters were undefined. However, generally as t → ± ∞, the
N -soliton solutions falls apart and one get N distinct one-soliton solution.
Let us allow to choose a specific type of nonlinear evolution equation and hence fix a function
f (ξ). The well-known examples are the aforementioned Korteweg-de Vries equation for which
f (ξ) = 4ξ 3 . From the formula (3.3.34) it is evidently seen that the amplitude of the one-soliton
solution of KdV is 2ξ 2 and it moves with velocity v = 4ξ 2 . This fact is clearly seen on the left
picture of figure 3.7 using an arbitrarily chosen parameters: xi = 0.7 and c0 = 2. Similarly, one can
generate a two-soliton solution taken two eigenvalues and normalization constants, see figure 3.8.
It can be evidently observed how these solitons collide elastically on each other without changing
their asymptotic form. This one space dimensional collision is depicted (figure 3.9) in such a way
that the process can be seen also in time. The phase shift is also apparently visible on this picture,
since at the center (at x ≈ 0 and t ≈ 0) of the coordinate system one can realize a fracture in the
propagation lines of the distinct solitons.
♠
56
3.3. HAMILTONIAN FORMALISM AND
INVERSE SCATTERING TRANSFORMATION
CHAPTER 3. PAINLEVÉ-ANALYSIS
0.8
t = -7.58
u(x,t)
0.8
t = -3.44
u(x,t)
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0
–10
0.8
t = 3.44
x
–20
20
10
u(x,t)
0
–10
0.8
t = 10.34
10
20
u(x,t)
0
–10
0.8
t = 13.10
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0
20
10
10
20
10
20
u(x,t)
0.1
x
–10
x
–20
0.7
0.1
–20
u(x,t)
0.1
x
–20
0.8
t = 0.69
0.7
x
–20
–10
0
10
20
x
–20
–10
0
Figure 3.8: The pictures shows time evolution of a two-soliton solution of (3.3.81) in discrete time steps (indicated
on each image) using the following values of parameters: ξ1 = 0.3, ξ2 = 0.6, c0,1 = 2.0, c0,2 = 5.0, and we use the
function f (ξ) = 4ξ 3 for evolving this solution in time [149]. From formula for the one-soliton solution (3.3.34) it is
obvious that the wave with higher amplitude travels faster. Similar effect takes place here, viz. the higher soliton
is approaching the smaller wave from right, they collide with each other, and after that the higher soliton leave the
smaller one. After a collision, solitons remarkably regain their original shape and velocity. The only remaining effect
of the scattering is a phase shift (i.e. a change in the position they would have reached without any interaction).
[See figure 3.9.]
δ-
20
Figure 3.9: The pictures shows time evolution of a twosoliton solution of the KdV equation using the following
values of parameters: ξ1 = 0.3, ξ2 = 0.6, c0,1 = 2.0, c0,2 =
5.0, and we use the function f (ξ) = 4ξ 3 for evolving this
solution in time [149]. It is apparently seen how both of the
solitons suffer a phase-shift. It can be analytically proven
that the phase-shifts associated with the separated solitons
have different sign. Specifying, this fact means that the
higher wave is shifted to the left compared to the case if
there were no interaction at all, and the smaller soliton
is shifted to the right with amount δ− . Here the darker
regions means higher amplitude.
10
t
0
–10
–20
–20
–10
0
10
20
x
57
3.3. HAMILTONIAN FORMALISM AND
INVERSE SCATTERING TRANSFORMATION
CHAPTER 3. PAINLEVÉ-ANALYSIS
Exercise: / Bright multi-soliton solution of ZS equations /
Find soliton solution of the Zakharov-Shabat integral equation (3.3.16) by direct calculation. As
indicated in the text, we can suppose now that the reflection coefficient R(k; t) ≡ 0, and only
discrete eigenvalues are present.
Solution:
In spite of the dissimilarity of Zakharov-Shabat and Gel’fand-Levitan-Marchenko integral equations we can use the same technique in deriving the solution in case of N distinct eigenvalues with
R(ξ, t) ≡ 0. Substituting the input kernel (3.3.17) into (3.3.16) the double integration can be explicitly performed. For sake of transparency we did not mark the argument t of the normalization
constants cj (t).
K(x; y; t) = i
N
X
j=1
= i
N
X
∗
c∗j e−iλj (x+y) −
∗
c∗j e−iλj (x+y)
j=1
−
Z∞Z∞ X
N
∗
c∗j cl e−iλj (y+z)+iλl (z+k) K(x; k; t)dzdk =
x x j,l=1
N
X
c∗j cl
i λ∗j − λl
j,l=1
e
∗
−i(λ∗
j −λl )x−iλj y
Z∞
eiλl k K(x; k; t) dk
(3.3.35)
x
Apart from the unknown function K(x; y; t) the right hand side of this equation depends on y only
in form of the factors exp(−λ∗j y). This enables us to search K(x; y; t) in the form:
K(x; y; t) =
N
X
∗
Km (x)e−iλm y
(3.3.36)
m=1
Substituting this ansatz into (3.3.35) we obtain

"
N
N
N
∗
X
X
X
cj cl
∗
∗
∗
∗
−iλ∗
j y−i(λj −2λl +λm )x K
Kj (x)e−iλj y =
ic∗j e−iλj (x+y) +
m (x)
∗ − λ )(λ∗ − λ ) e
(λ
m
j
l
l
j=1
j=1
l,m=1
Arrange all terms in the same side of the equation and detach the common exponential factor we
can formally write
N X
∗
· · · e−iλj y = 0
(3.3.37)
j=1
This equation must be fulfilled for all values of y, hence the the content of the bracket should be
zero. Putting the inhomogeneity term on the right hand side we get
Kj (x) −
N
X
(λ∗j
m,l=1
c∗j cl
∗
∗
∗
e−i(λj −2λl +λm ) x Km (x) = ic∗j e−iλj x
− λl )(λ∗m − λl )
(3.3.38)
At this point we derived a system of linear algebraic equations for the unknown functions K m (x)
which determine K(x; y; t). Introducing the denotations:
Cj;m (x; t)
bj (x; t)
∗
∗
= δj,m − c∗j e−i(λj −λm ) x
∗
= ic∗j e−iλj x
58
N
X
l=1
cl
e2iλl x
(λ∗j − λl )(λ∗m − λl )
3.3. HAMILTONIAN FORMALISM AND
INVERSE SCATTERING TRANSFORMATION
CHAPTER 3. PAINLEVÉ-ANALYSIS
where δj,m stands for the Kronecker-delta function. Using the matrix notation we can write
symbolically the solution of the equation (3.3.38) in the form
K(x; t) = C−1 (x; t)b(x; t)
(3.3.39)
from which the function K(x, y, t) can be determined with summation:
K(x; y; t) =
N
X
∗
Km (x; y)e−iλm y
(3.3.40)
m=1
We have solved the problem. ♠
3.3.5
Relation of Hamiltonian formalism and IST
In this subsection we shall show that the inverse scattering transformation is a canonical transformation which converts a nonlinear evolution equation into an infinite sequence of separated
ordinary differential equations for the action-angle variables, which can be integrated trivially.
Hamiltonian theory is an important element of integrable systems, since many partial differential equations also have a Hamiltonian structure. For a PDE with independent variables (x; t), the
canonical variables are replaced by fields (q(x; t); p(x; t)) and the partial derivatives by functionals
or Frechê derivatives:
δH
∂p
δH
∂q
=
=−
∂t
δp
∂t
δq
Important examples of completely integrable classical mechanical systems were discovered by
using ”soliton techniques”, such as Lax pairs and bi-Hamiltonian representations. However, a
whole new theory of Poisson brackets for nonlinear evolution equations was developed, starting
with the discovery of two Poisson bracket representations of the KdV equation [153]. This is called
the bi-Hamiltonian property and has been established for a large number of systems. It has become
one of the signatures of integrability. Over the years large families of systems with two or more
compatible Poisson brackets have been discovered.
Inverse Scattering Transformation can be thought of as a nonlinear transformation from physical
variables to an action-angle variables.
The associated Hamiltonian for the KdV equation is given by
H =−
Z∞
−∞
2
pqx + p2 qx − px qxx dx
(3.3.41)
and the associated dynamical equations can be derived from Hamiltonian equations. Since the
Hamiltonian (3.3.41) does not depend on q explicitly, the dynamical equations may be written in
the form
!
∂u
∂
δ Ĥ
=
(3.3.42)
∂t
∂x δu
where
Z∞
Z∞
1
Ĥ = − h(u; t) dx = −
u3 − u2x dx
2
−∞
−∞
59
∞
X
δ Ĥ
∂n
=
(−1)n n
δu
∂x
n=0
∂ ĥ
∂unx
!
(3.3.43)
3.3. HAMILTONIAN FORMALISM AND
INVERSE SCATTERING TRANSFORMATION
CHAPTER 3. PAINLEVÉ-ANALYSIS
In 1978 Magri presented that KdV equation has two Hamiltonian decompositions
ut + ∂x 3u2 + uxx = 0
ut + ∂x3 + 4u∂x + 2uu I u = 0
(3.3.44)
(3.3.45)
Suppose we have a dynamical system with conjugate variables (q; p), then in order to change
to another set of conjugate variables (Q; P ), we define the Poisson brackets:
{A(α) , B(β)} ≡
Z∞ −∞
δA(α) δB(β) δA(α) δB(β)
−
dx
δq(x) δp(x)
δp(x) δq(x)
(3.3.46)
Alternatively, if only the derivatives of q and not q itself arise in the Hamiltonian, then we may
replace (3.3.46) by
Z∞ δA(α) ∂ δB(β)
dx
(3.3.47)
{A(α) , B(β)} =
δu(x) ∂x δu(x)
−∞
where u = p = qx . A transformation from (q; p) to (Q; P ) is said to be canonical if
{Q(x) , Q(y)} = 0
{P (x) , P (y)} = 0
{Q(x) , P (y)} = δ(x − y)
(3.3.48)
(3.3.49)
(3.3.50)
where δ(z) is the Dirac delta function. The evolution of a quantity u satisfying a Hamiltonian
equation obeys
ut = {u , H}
(3.3.51)
where H is the Hamiltonian. The main points are the following:
i) Suppose a system of evolution equations is hamiltonian, in which the dependent variables
(q; p)play the roles of the conjugate variables.
ii) There is a subset of the scattering data from which the rest of the scattering data can be
constructed.
iii) The mapping (q; p) 7→ (Q; P ) is a canonical transformation
iv) The conjugate variables in (Q; P ) are of action-angle type; that is H = H(P ), so that the
equations of motion become
∂P
=0
∂t
∂Q
δH
=
= constant
∂t
δP
(3.3.52)
Suppose that ϕ(x; k) is an eigenfunction of the linear scattering problem associated with the KdV
equation, i.e.,
ϕxx + u(x) + k 2 ϕ = 0
(3.3.53)
with boundary condition ϕ(x; k) ∼ e−ikx as x → −∞, which also is the solution of the integral
equation


Zx 1
ϕ(x; k) = e−ikx 1 +
1 − e2ik(x−ξ) u(ξ)ϕ(ξ; k)eikξ dξ 
(3.3.54)
2ik
−∞
60
3.3. HAMILTONIAN FORMALISM AND
INVERSE SCATTERING TRANSFORMATION
CHAPTER 3. PAINLEVÉ-ANALYSIS
where k is the spectral parameter and u satisfies the KdV equation. Further suppose that a(k)
and b(k) are given by the integral representations
1
a(k) = 1 +
2ik
Z∞
u(ξ) ϕ(ξ; k) eikξ dξ
(3.3.55)
−∞
Z∞
1
b(k) = −
u(ξ) ϕ(ξ; k) e2ikξ dξ
2ik
(3.3.56)
−∞
Now, we wish to calculate the Poisson-bracket of a(k) and b(k), i.e.,
{a(k) , b(l)} =
Z∞ −∞
δa(k) ∂ δb(l)
dx
δu(x) ∂x δu(x)
(3.3.57)
Using (3.3.54) and (3.3.55) it can be shown that
δa(k)
δu(x)
δb(k)
δu(x)
= −
=
ψ ∗ (x; k)ϕ(x; k)
2ik
ψ(x; k)ϕ(x; k)
2ik
(3.3.58)
(3.3.59)
where ψ(x; k) and ψ ∗ (x; k) are solutions of (3.3.53) satisfying the boundary conditions ψ(x; k) ∼
eikx and ψ ∗ (x; k) ∼ e−ikx as x → ∞. Substituting these, using the fact that ϕ(x; k), ψ(x; k) and
ψ ∗ (x; k) satisfy the Schrödinger scattering, together with the proper boundary conditions, and the
symmetry condition
ϕ(x; k) = a(k)ψ ∗ (x; k) + b(k)ψ(x; k)
(3.3.60)
one finds that
{a(k) , b(l)} =
a(k)b(l)
a(k)b(k)δ(k − l)
− iπ
2(k 2 − l2 )
4k
(for further details see Novikov, Manakov, Pitaevskii and Zakharov [154]).
For the KdV equation, we define for real k,
k ik
b(k)
2
P (k) = ln |a(k)|
Q(k) = − ln ∗
π
2
b (k)
(3.3.61)
(3.3.62)
where a(k), b(k) defined in (3.3.55). It can be shown that P (k) and Q(k) are canonical variables.
If the scattering data contains also discrete eigenvalues iξj , then the canonical variables belonging
to these eigenvalues can be defined as
Pj = ξj2
Qj = −2 ln(|Cj |)
(3.3.63)
Now, the set of variables {P (k), Q(k), Pj , Qj , j = 1, 2, . . . , n} contains all information of the linear
scattering problem (3.3.53). The Hamiltonian function can be expressed in these new variables as
n
32 X 5/2
Ĥ =
P
+8
5 j=1 j
Z∞
−∞
61
k 3 P (k) dk
(3.3.64)
3.3. HAMILTONIAN FORMALISM AND
INVERSE SCATTERING TRANSFORMATION
CHAPTER 3. PAINLEVÉ-ANALYSIS
The associated Hamiltonian equation of motions are
∂P (k)
∂t
∂Q(k)
∂t
∂Pj
=0
∂t
∂Qj
3/2
= −16Pj
∂t
= 0
= 8k 3
(3.3.65)
(3.3.66)
These ordinary differential equations are trivially solvable and reproduce the time evolution known
from earlier investigations of KdV:
P (k; t)
Pj (t)
Q(k; t) = Q(k; 0) + 8k 3 t
= P (k; 0)
= Pj (0)
Qj (t) = Q(k; 0) −
3/2
16Pj (0)t
(3.3.67)
(3.3.68)
The discovery of a solution technique for a special nonlinear partial differential equation in
1967 was the foundation for a novel field in mathematical physics: the theory of solitons. For the
past 25 years it has been developed to become one of the major theories in nonlinear sciences.
Meanwhile a huge number of nonlinear equations have been identified to reside in the class of
”integrable system” for which exact solutions with the spectacular soliton properties can be found.
These nonlinear equations share the common property that they are given by the compatibility
condition of linear equations.
One of the most powerful techniques to solve such nonlinear problems is the inverse scattering
transform (IST), which uses the scattering problem given by the underlying linear equations. The
original nonlinear equation is transformed into corresponding equations for asymptotic scattering
data. In this setting the problem simplifies, it decouples or linearizes and can be solved. The
inverse transformation then seeks to reconstruct solutions of the original nonlinear problem from
the scattering data. This method, first established for the Korteweg-de Vries (KdV) equation and
the associated Schrödinger operator, has been applied to a variety of scattering problems. It is
shown that the IST technique nowadays is a unified and effective procedure to treat most of the
known soliton systems.
62
Chapter 4
Painlevé test of coupled
Gross-Pitaevskii equations
In this chapter the Painlevé test of the coupled Gross-Pitaevskii equations will be carried out [1]
with the result that the coupled equations pass the P-test only if a special relation containing system parameters (masses, scattering lengths) is satisfied. Computer algebra is applied to evaluate
j = 4 compatibility condition for admissible external potentials. Appearance of an arbitrary real
potential embedded in the external potentials is shown to be the consequence of the coupling. Connection with recent experiments related to stability of two-component Bose-Einstein condensates
of Rb atoms is discussed.
4.1
Introduction
Recently there has been a growing interest in the Gross-Pitaevskii (GP) equations [155, 156] describing two-component Bose-Einstein condensates (BEC) in external trap potentials [64–67, 71,
157–171]. In the absence of the confining potential, the GP equations reduce to the coupled nonlinear Schrödinger (NLS) equations which play an important role in optics [23]. In chapter 1 we
have shown in part that the coupled GP equations are also used to describe Josephson-type oscillations between two coupled BEC [67, 159, 160] spin-mixing dynamics of spinor BEC [161–164] or
to explore such interesting field of matter waves as possible atomic soliton lasers [99, 157, 172].
In chapter 3 we have discussed that an efficient tool of the analysis of the non-linear partial
differential equations is the Painlevé (P) method [134, 173] which serves to explore the singularity
structure of the underlying equations, and establish integrability conditions [149]. The P-analysis
of the single NLS equation has been performed by Steeb et al. [174], and the damped NLS (or
the GP) equation has been investigated by Clarkson [135] [see the example in subsection 3.2.2]. A
fairly large class of coupled NLS equations including third order dispersions have been analyzed by
Radhakrishnan et al. [175]. Recently the symmetrically coupled higher-order NLS equations have
been tested by using the P-method [176].
Because of the experimental developments in forming two-component BEC [165] and the possibility to confine BEC in a cigar shape [177], we shall perform the P-test of the coupled onedimensional GP equations in order to establish certain necessary conditions of integrability. (The
term integrability is used here in the general sense [149, 173] involving P-property and soliton for63
CHAPTER 4. PAINLEVÉ TEST OF CGP
4.2. PAINLEVÉ TEST
mation as discussed earlier.) The results obtained for the trap potentials are similar to those found
by Clarkson [135] in the case of the damped NLS equations: the trap potential should be linear
and/or quadratic in the coordinate variable x. In the quadratic case a source term depending only
on time t should also be present in the external potential V (x, t).
A novel feature of our analysis is the possibility of the appearance of an arbitrary common
potential term Ṽ (x, t) within the confining potentials V1 (x, t) and V2 (x, t). Its presence may prove
useful for fine tuning experiments with two component BEC. We consider the system of coupled
GP equations in its most general form containing different masses, external potentials, and mutual
coupling strengths. As a result we shall derive compatibility conditions, the fulfillment of which
depends on the parameters characterizing the GP equations. We show that in a particular experiment [65] employing two hyperfine states of Rb atoms as components of BEC, the vortex stability
corresponds to just the parameter ratios satisfying our general formula derived in this paper.
The organization of this chapter is as follows. In section 4.2 the P-analysis of two coupled
GP equations will be carried out including the determination of the leading orders, the recursion
relations, the resonances, and the compatibility conditions. The consequences of the compatibility
relations for the potentials are discussed in section 4.3 where also other consistency requirements
are studied. In section 4.4 we make comparisons with earlier results and investigate compatibilities
with existing experimental and numerical findings related with two-component BEC. Section 4.5
is devoted to a short summary.
4.2
Painlevé test
Let us consider the following (1 + 1) dimensional inhomogeneous NLS equations for the wave
functions ψ1 , ψ2 with the external potentials U1 (x, t), U2 (x, t)
∂
~2 2
2
2
i~ ψ1 (x, t) = −
∇ + U1 (x, t) + U11 |ψ1 (x, t)| + U12 |ψ2 (x, t)| ψ1 (x, t) + U10 , (4.2.1a)
∂t
2m1
∂
~2 2
2
2
i~ ψ2 (x, t) = −
∇ + U2 (x, t) + U21 |ψ1 (x, t)| + U22 |ψ2 (x, t)| ψ2 (x, t) + U20 (4.2.1b)
∂t
2m2
which, in the absence of the inhomogeneities U10 and U20 , are commonly called the coupled GrossPitaevskii equations [155, 156].
Here mi denotes the mass of the atomic species i (i = 1, 2) of the two-component BEC gas
and Uij is related with the interactions between the atoms i and j (i, j = 1, 2) via the relation
Uij = 2π~2 aij Nj /A µij where Nj means the number of atoms in the jth component of the BEC,
aij is the scattering length characterizing the interaction between atoms i and j, A represents a
general cross sectional area confining species i and j, and µij = mi mj /(mi + mj ) is the reduced
mass.
By introducing the new parameters
λ=
~
,
2m1
ϑ=
~
,
2m2
Tij =
1
Uij ,
~
(i, j = 1, 2),
(4.2.2a)
and notations
u = ψ1 ,
w = ψ2 ,
Vi =
1
Ui ,
~
64
Vi0 =
1
Ui0 ,
~
(i = 1, 2)
(4.2.2b)
CHAPTER 4. PAINLEVÉ TEST OF CGP
4.2. PAINLEVÉ TEST
we write the GP equations into the standard form of the P-analysis
iut + λuxx − T11 |u|2 u − T12 |w|2 u = V1 u + V10 ,
2
2
iwt + ϑwxx − T21 |u| w − T22 |w| w = V2 w + V20
(4.2.3a)
(4.2.3b)
where Tij and λ, ϑ stand, as defined by equation (4.2.2a), for the interaction and mass parameters,
respectively.
In order to apply the P-analysis, we first complexify all variables to obtain equations (4.2.3a-b)
in the form (v = u∗ , z = w∗ ):
iut + λuxx − T11 u2 v − T12 wzu = V1 u + V10 ,
∗
−ivt + λvxx − T11 uv 2 − T12 wzv = V1∗ v + V10
,
iwt + ϑwxx − T21 uvw − T22 w2 z
−izt + ϑzxx − T21 uvz − T22 wz
2
= V2 w + V20 ,
=
V2∗ z
+
∗
V20
(4.2.4a)
(4.2.4b)
(4.2.4c)
(4.2.4d)
where the functions u, v, w, z are treated as independent complex functions of the complex
∗
∗
variables x and t, and V1∗ (x, t), V10
(x, t), V2∗ (x, t), V20
(x, t) are formal complex conjugates of
V1 (x, t), V10 (x, t), V2 (x, t), V20 (x, t), respectively.
The next step is to seek the solutions of (4.2.4a-d) in the form
u(x, t) = φp (x, t)
w(x, t) = φr (x, t)
∞
X
j=0
∞
X
uj (t)φj (x, t),
wj (t)φj (x, t),
v(x, t) = φq (x, t)
z(x, t) = φs (x, t)
j=0
∞
X
j=0
∞
X
vj (t)φj (x, t),
(4.2.5a)
zj (t)φj (x, t),
(4.2.5b)
j=0
with the Kruskal ansatz
φ(x, t) = x − ξ(t),
(4.2.6)
and ξ(t), uj (t), vj (t), wj (t), zj (t), j = 0, 1, 2, . . . being analytic functions of t in the neighbourhood
of a noncharacteristic movable singularity manifold defined by φ = 0. Similarly, the external
potentials Vi confining specimen i is also expanded about the singularity manifold φ = 0 as follows
(i = 1, 2)
∞
X
1 ∂ j Vi (x, t)
Vi (x, t) =
Vi,j (t)φj (x, t),
Vi,j (t) =
.
(4.2.7)
j!
∂xj
x=ξ(t)
j=0
Substituting expansions (4.2.5a-b) and (4.2.7) into equations (4.2.4a-d) and equating like powers
of φ we obtain:
i.) equations for determining the leading orders p, q, r, s;
ii.) recursion relations for deriving the functions uj , vj , wj , zj .
In order that equations (4.2.4a-d) pass the Painlevé test it is required that the numbers p, q, r, s
be non-positive integers. Moreover, the recursion relations should be consistent in all order of j
including the resonances.
65
CHAPTER 4. PAINLEVÉ TEST OF CGP
4.2.1
4.2. PAINLEVÉ TEST
Determination of the leading orders
To determine the leading orders p, q, r, s appearing in the expansions (4.2.5a-b), it is sufficient
to consider the expansion upto the zeroth order, j = 0. By substituting this truncated version of
expansions (4.2.5a-b) into (4.2.4a) we obtain
iu0,t φp + iu0 pφp−1 φt + λu0 p(p − 1)φp−2 − T11 u20 v0 φ2p+q − T12 u0 w0 z0 φp+r+s
= V1 u0 φp + V10 .
(4.2.8a)
Three completely similar expressions arise from the substitution of the truncated version of expansions (4.2.5a-b) into the remaining three equations (4.2.4b-d):
−iv0,t φq − iv0 qφq−1 φt + λv0 q(q − 1)φq−2 − T11 v02 u0 φ2q+p − T12 v0 w0 z0 φq+r+s
∗
,
= V1∗ v0 φq + V10
r
r−1
r−2
iw0,t φ + iw0 rφ φt + ϑw0 r(r − 1)φ
− T21 u0 v0 w0 φ
− T22 w02 z0 φ2r+s
= V2 w0 φr + V20 ,
s
s−1
s−2
p+q+s
−iz0,t φ − iz0 sφ φt + ϑz0 s(s − 1)φ
− T21 u0 v0 z0 φ
− T22 w0 z02 φr+2s
∗
.
= V2∗ z0 φs + V20
(4.2.8b)
p+q+r
(4.2.8c)
(4.2.8d)
By demanding the leading order terms of equations (4.2.8a-d) to vanish one obtains the following
equations
λp(p − 1) = T11 u0 v0 + T12 w0 z0 ,
λq(q − 1) = T11 u0 v0 + T12 w0 z0 ,
ϑr(r − 1) = T21 u0 v0 + T22 w0 z0 ,
(4.2.9a)
(4.2.9b)
(4.2.9c)
ϑs(s − 1) = T21 u0 v0 + T22 w0 z0
(4.2.9d)
p + q = −2,
r + s = −2
(4.2.10a)
(4.2.10b)
and
from which the leading orders can uniquely be determined to be
p = q = r = s = −1.
For later use we infer from equations (4.2.9a-d) the useful relation
2
u0 v 0
T22 −T12
λ
=
w0 z0
T11
ϑ
∆ −T21
(4.2.11)
(4.2.12)
with ∆ = T11 T22 −T12 T21 . If accidentally ∆ = 0 happens then we may use the relation u0 v0 /w0 z0 =
const instead of (4.2.12), which case needs a special consideration.
4.2.2
Recursion relations
The next step of the P-analysis is to again substitute expansions (4.2.5a-b) and (4.2.7) with the
leading orders p = q = r = s = −1 into equations (4.2.4a-d). After some algebra we obtain the
66
CHAPTER 4. PAINLEVÉ TEST OF CGP
recursion relations

Q1
 −T11 v02

−T21 v0 w0
−T21 v0 z0
|
−T11 u20
Q1
−T21 u0 w0
−T21 u0 z0
where j = 1, 2, . . . and
Q1
Q2
Fj
4.2. PAINLEVÉ TEST
−T12 u0 z0
−T12 v0 z0
Q2
−T22 z02
{z
Q(j)
   
−T12 u0 w0
uj
Fj
 vj   Gj 
−T12 v0 w0 
  =  
−T22 w02  wj  Hj 
Q2
zj
Kj
}
(4.2.13a)
= λ(j − 1)(j − 2) − 2T11 u0 v0 − T12 w0 z0 ,
= ϑ(j − 1)(j − 2) − 2T22 w0 z0 − T21 u0 v0 ,
(4.2.13b)
(4.2.13c)
= −iuj−2,t − i(j − 2)uj−1 φt +
(4.2.13d)
+
j−1 X
l
X
j−1
X
(T11 um uj−m v0 + T12 um w0 zj−m )
m=1
(T11 um ul−m vj−l + T12 um wj−l zl−m ) +
l=1 m=0
j−2
X
V1,l uj−l−2 + V10,j−3 .
l=0
Here we use the notation that whenever an index is less than zero, then the expression itself is zero
(for example, V10,j−3 ≡ 0 for j ≤ 2). Furthermore Gj is obtained from Fj by interchanging ul and
∗
∗
vl and letting i → −i, V1,l → V1,l
and V10,l → V10,l
. The expressions Hj and Kj can be obtained,
respectively, from Fj and Gj by interchanging ul and wl , vl and zl , T12 and T21 , T11 and T22 , and
letting V1 → V2 , V10 → V20 .
The expressions Fj , Gj , Hj , Kj at a given j depend only on the expansion coefficients ul , vl , wl ,
zl with l < j. Therefore the equation (4.2.13a) represents recursion relations for the determination
of the unknowns uj , vj , wj , zj from the knowledge of the prior calculated coefficient functions ul ,
vl , wl , zl with l < j.
4.2.3
Resonances
The above recursion relations (4.2.13a) determine the unknown expansion coefficients uniquely
unless the determinant of the matrix Q(j) is zero. Those values of j at which the determinant
det(Q(j)) becomes zero are called resonances. After some calculation one obtains
ϑT11 u0 v0 + λT22 w0 z0
det Q(j) = λ2 ϑ2 (j + 1) j 2 (j − 3)2 (j − 4) j 2 − 3j + 4 − 2
(4.2.14)
λϑ
so that the resonances of the coupled GP equations (4.2.3a-b) are as follows
jres = −1, 0, 0, 3, 3, 4, j1, j2 .
(4.2.15)
Here j1 and j2 are the roots of the expression contained in the last parentheses of equation (4.2.14)
and can formally be given as
r
3 1
ϑT11 u0 v0 + λT22 w0 z0
j1,2 = ±
8
−7
∈ Z.
(4.2.16)
2 2
λϑ
As indicated, the resonances j1 and j2 must be integers so that the square-root should be odd
integers. From this one gets a condition
r
ϑT11 u0 v0 + λT22 w0 z0
8
− 7 = 2m + 1 ,
m = 0, 1, 2, . . .
(4.2.17)
λϑ
67
CHAPTER 4. PAINLEVÉ TEST OF CGP
4.2. PAINLEVÉ TEST
involving the characteristic parameters Tij , λ, ϑ of the GP equations (4.2.3a-b). The number m
can be considered as a classification number which classifies possible external potential families for
which the system (4.2.3a-b) is integrable (in the general sense of integrability [149, 173] ).
By re-arranging (4.2.17) and using relation (4.2.12) one gets a more explicit condition necessary
for any coupled GP equations to pass the P-test:
2 T11 T22 − (ϑ/λ) T11 T12 − (λ/ϑ) T21 T22
1
=
(2m + 1)2 + 7 ,
T11 T22 − T12 T21
16
m = 0, 1, 2, . . .
(4.2.18)
It is also clearly seen that this expression depends only on the ratios λ/ϑ, T11 /T21 , and T12 /T22
involving the characteristic parameters of the GP equations.
In summary, any coupled system of GP equations (4.2.3a-b) passes the Painlevé test only
if its characteristic parameters λ, ϑ, Tij (i, j = 1, 2) obey the relation (4.2.18), otherwise it is
probably not integrable. (See discussions about connection of P-test with integrability in references
[149, 173].)
4.2.4
Compatibility conditions
At each element of jres , the recursion relations (4.2.13a) cannot be used for the calculation of the
expansion coefficients. At these indices arbitrary functions may arise in the expansions (4.2.5a-b).
However, in order that the solution be expressible in the form of the expansions (4.2.5a-b) and
(4.2.7), the recursion relations should be identically satisfied at j ∈ jres . The investigation of these
specific requirements leads to relations called compatibility conditions which impose restrictions for
the external potentials Vi (x, t) i = 1, 2. We note that only the positive resonances are of interest.
Compatibility condition belonging to resonance j = 3. Let us consider equations (4.2.13a)
at j = 3 and use equation (4.2.12). The result is an equation

−T11 u0 v0
 −T11 v02

−T21 v0 w0
−T21 v0 z0
|
−T11 u20
−T11 u0 v0
−T21 u0 w0
−T21 u0 z0
−T12 u0 z0
−T12 v0 z0
−T22 w0 z0
−T22 z02
{z
Q(3)
   
−T12 u0 w0
u3
F3
 v3   G3 
−T12 v0 w0 
  =  
−T22 w02  w3  H3 
−T22 w0 z0
z3
K3
}
(4.2.19)
whose matrix Q(3) has rank two. Indeed, by multiplying the first row with v0 , the second with
u0 , one gets a matrix which possesses identical elements in its first two rows. Performing similar
manipulations, one can make the third and fourth rows also identical. It then follows that the
above recursion relation can only be consistent if the following compatibility conditions hold
F3 v 0
H3 z 0
= G 3 u0 ,
= K 3 w0 .
(4.2.20a)
(4.2.20b)
We emphasize that the above conditions are not independent from each other because, for example
as shown by (4.2.13d), F3 contains elements wi , zi with i ≤ 3. Similarly, G3 , H3 and K3 also
contain all types of expansion coefficients ui , vi , wi , zi with i ≤ 3.
68
CHAPTER 4. PAINLEVÉ TEST OF CGP
4.2. PAINLEVÉ TEST
Compatibility condition belonging to resonance j = 4. By taking the recursion relations
(4.2.13a) at j = 4 and applying equation (4.2.12), one arrives at the following equation

   
4λ − T11 u0 v0
−T11 u20
−T12 u0 z0
−T12 u0 w0
u4
F4
 −T11 v02
 v4  G4 
4λ
−
T
u
v
−T
v
z
−T
v
w
11
0
0
12
0
0
12
0
0

 = .
(4.2.21)
 −T21 v0 w0
−T21 u0 w0
4ϑ − T22 w0 z0
−T22 w02 w4  H4 
2
−T21 v0 z0
−T21 u0 z0
−T22 z0
4ϑ − T22 w0 z0
z4
K4
|
{z
}
Q(4)
In the general case the matrix Q(4) has rank three which means that only three of its rows are
independent. Using this fact, after some calculation we obtain the following compatibility condition
T21 F4 v0 + G4 u0 + T12 H4 z0 + K4 w0 = 0.
(4.2.22)
We should investigate also the possibility when rank Q(4) = 2. In this case the compatibility
condition decomposes into two distinct parts as it can be seen in the following way. The rank of
a matrix equals the maximal order of its non-singular submatrices. We should thus calculate the
determinants of all third order submatrices of Q(4) and investigate the cases when they simultaneously become zero. After a simple but lengthy calculation the following results are obtained for
the determinants of the four third-order submatrices of the matrix Q(4):
o
n
16λϑT21 u0 z0 ; −16λϑT12 u0 w0 ; 16λϑT12 w0 z0 ; −16λϑT21 u20 .
(4.2.23)
Because λ and ϑ are the non-zero mass parameters [see definitions (4.2.2a)], it is clear that the
subdeterminants vanish only if T12 = T21 = 0. This situation however corresponds to two uncoupled GP equations. We thus arrived at the compatibility condition found by Clarkson [135] for the
single GP equations:
F 4 v 0 + G 4 u0
= 0,
(4.2.24a)
H4 z 0 + K 4 w 0
= 0.
(4.2.24b)
We note however that the general compatibility condition, as given by equation (4.2.22), leads
to more complicated external potentials (discussed in the next section) than that obtainable from
equations (4.2.24a-b) with T12 = T21 = 0.
Compatibility condition belonging to resonances j1 and j2 . We now consider the matrix
Q(j1,2 ) with j1,2 taken from equation (4.2.16). In general the matrix Q(j1,2 ) has rank three from
which the following compatibility condition arises
ϑw0 z0 Fj1,2 v0 + Gj1,2 u0 − λu0 v0 Hj1,2 z0 + Kj1,2 w0 = 0 .
(4.2.25)
As before we analyze also the case when rank Q(j1,2 ) = 2. After a lengthy but simple
calculation we obtain the determinants of the four third-order submatrices of the matrix Q(j 12 ) to
be:
n
3
2
3
2
T21 ∆5 (u0 v0 ) (w0 z0 ) z0 ; −T21 ∆5 (u0 v0 ) (w0 z0 ) ;
o
3
2
3
T21 ∆5 (u0 v0 ) (w0 z0 ) z0 ; −T12 ∆6 (u0 v0 w0 z0 ) .
(4.2.26)
69
CHAPTER 4. PAINLEVÉ TEST OF CGP
4.3. POSSIBLE FORMS OF THE EXTERNAL POTENTIALS
Now, as clearly seen the determinants (4.2.26) vanish simultaneously only if T12 = T21 = 0
(or ∆ = 0 which case is not considered here). On the other hand, for this decoupled case one
can determine the values j1 and j2 by using definition (4.2.16) and relation (4.2.12) to be j1 = 4,
j2 = −1. But then, as it can be checked easily by using (4.2.13a), the corresponding compatibility
conditions reduce to those already discussed in connection with equations (4.2.24a-b). We note
however that, depending on the experimental situations, it is possible to obtain resonance values
j1 and j2 greater than four. We should then use equation (4.2.25) for drawing conclusions about
the admissible form of the external potentials.
4.3
Possible forms of the external potentials
In the preceding section we have found equations, called compatibility conditions, that must be
fulfilled in order that the GP equations pass the P-test. In this section we exploit the consequences
of these equations for the general form of the external potentials V1 , V10 , V2 , V20 appearing in
equations (4.2.1a-b) and (4.2.3a-b). The experimental realization of such potentials may lead to
detection of stable structures (like vortices) in BEC.
Although the compatibility conditions are related with indices j at which the recursion relations
(4.2.13a) do not apply to the calculation of the unknown coefficients uj (t), vj (t), wj (t), zj (t), the
equations (4.2.20a-b), (4.2.22) and (4.2.25) can be reduced to expressions in which only the zeroth
order coefficient functions u0 (t), v0 (t), w0 (t), z0 (t) are present. That is because, at j 6∈ jres , use of
the recursion relations (4.2.13a) and the definition (4.2.13d) of the functions Fj , Gj , Hj , Kj , leads
always to expansion coefficients uj , vj , wj , zj that are expressed by the zeroth order functions
u0 (t), v0 (t), w0 (t), z0 (t).
In the following we shall present the results of the calculation belonging to each compatibility
condition. For the resonance j = 3 the calculation can be performed easily by hand, but for
j = 4 the computer program Maple [178] had to be invoked in order to perform the analytic
manipulations. As a result of the Maple program, all the coefficients which multiply the higher
order powers of φt proved to be analytically zero. The expression associated with the zeroth order
power of φt has been evaluated further by hand to get the final results which will be presented and
discussed below.
4.3.1
Conditions for the potentials arising from j = 3
In section 4.2 it has been shown that at j = 3 the compatibility condition decomposes into two
distinct parts which are however not independent of each other [see equations (4.2.20a-b) and the
remark thereafter].
The elaboration of the compatibility conditions (4.2.20a-b) related with j = 3 yields the relations
F 3 v 0 − G 3 u0 = 0
H3 z 0 − K 3 w 0 = 0
−→
−→
∗
∗
(V1,1 − V1,1
)u0 v0 + V10,0 v0 − V10,0
u0 = 0 ,
(V2,1 −
∗
V2,1
)w0 z0
+ V20,0 z0 −
∗
V20,0
w0
= 0.
(4.3.27a)
(4.3.27b)
It is clear from equation (4.2.12) that only the products u0 v0 and w0 z0 are determined uniquely
so that one element of each pair can be chosen arbitrarily. With the choices u0 = 1, w0 = 1 the
above relations can only be satisfied if
∗
V1,1 − V1,1
=0
∗
V2,1 − V2,1 = 0
∗
V10,0 = V10,0
≡ 0,
∗
V20,0 = V20,0 ≡ 0.
and
and
70
(4.3.28a)
(4.3.28b)
CHAPTER 4. PAINLEVÉ TEST OF CGP
4.3. POSSIBLE FORMS OF THE EXTERNAL POTENTIALS
These conditions show that the expansion coefficients V1,1 and V2,1 are real. Moreover, using the
definition (4.2.7) for the expansion coefficients Vi,j and the results (4.3.28a), we get
0 = V10,0 =
1 ∂ 0 V10 (x, t) = V10 (ξ(t), t).
0!
∂x0
x=ξ(t)
(4.3.29)
Since this equality holds for any arbitrary function ξ(t), it follows that V10 (x, t) must vanish.
Similar argumentation leads to disappearance of V20 (x, t).
In summary we conclude that in order that the equations (4.2.3a-b) pass the P-test the inhomogeneity terms must vanish and the first order expansion coefficient of the external potentials
should be real:
V10 = V20 = 0,
V1,1 =
4.3.2
∗
V1,1
and
V2,1 =
(4.3.30a)
∗
V2,1
.
(4.3.30b)
Conditions for the potentials arising from j = 4
Without presenting the details of the algebraic manipulations, we state that the compatibility
condition (4.2.22) (partly with the aid of the formula manipulation program Maple) leads to the
relation
T21
T12
∗ 2
∗ 2
∗
∗
u0 v0 (V1,0 − V1,0
) +
w0 z0 (V2,0 − V2,0
) − T12 w0 z0 (V2,2 + V2,2
) − T21 u0 v0 (V1,2 + V1,2
)
2λ
2ϑ
T21
∂
T12
∂
∗
∗
−i
u0 v0 (V1,0 − V1,0
)−i
w0 z0 (V2,0 − V2,0
)=0
(4.3.31)
2λ
∂t
2ϑ
∂t
in which, as expected, only the zeroth order coefficient functions u0 (t), v0 (t), w0 (t), z0 (t) appear
together with the parameters λ, ϑ and Tij (i, j = 1, 2) in a special combination. This condition
looks much more complicated than that obtained above [cf. with equations (4.3.27a-b)]. Moreover
both potentials V1 and V2 are occurring within a single relation.
Let us now write the external potentials in the form
V1 (x, t)
= α(x, t) + iβ(x, t),
(4.3.32a)
V2 (x, t)
= γ(x, t) + iδ(x, t)
(4.3.32b)
where α, γ and β, δ are real functions. Exploiting the reality of V1,1 and V2,1 expressed by relation
(4.3.30b) and using the definition (4.2.7), we get the results
β(x, t) ≡ β(t)
and
δ(x, t) ≡ δ(t).
(4.3.33)
This condition which can be checked easily by direct substitution tells that the imaginary part of
the potential may depend only on the time t but not on the space x variables. Using this last result
and inserting the definitions (4.3.32a-b) into the relation (4.3.31) we get the following expression
2
2
1
dβ
1
dδ
2
2
− T21 u0 v0 β − T12 w0 z0 δ + T21 u0 v0
+ T12 w0 z0
λ
ϑ
λ
dt
ϑ
dt
2
2
∂ γ
∂ α
−T12 w0 z0 2 − T21 u0 v0 2 = 0 .
(4.3.34)
∂x
∂x
71
CHAPTER 4. PAINLEVÉ TEST OF CGP
4.3. POSSIBLE FORMS OF THE EXTERNAL POTENTIALS
Because the quantity in the square bracket depends only on time t, integration by x twice yields
the following results
T12 w0 z0 γ + T21 u0 v0 α = C1 (t)x2 + C2 (t)x + C3 (t)
(4.3.35)
where the coefficients C1 (t), C2 (t), C3 (t) depend only on time t, and C2 , C3 are arbitrary real
functions. By re-substituting this latter equation into expression (4.3.34), we get the constraint
for the function C1 (t) as follows
T21
1 dβ
T12
1 dδ
2
2
C1 (t) =
u0 v 0
−β +
w0 z0
−δ
.
(4.3.36)
λ
2 dt
ϑ
2 dt
We emphasize that the above result does not mean a restriction for the individual form of the
real part of the external potentials V1 and V2 . As equation (4.3.35) shows only a weighted sum
of the real parts α and γ is constrained by the compatibility conditions (4.3.31) belonging to the
resonance j = 4.
Let us now exhibit a possible consequence of the general constraints (4.3.35) and (4.3.36) for
the potentials by starting from an obvious splitting of the coefficient C1 (t) as follows
(1)
(2)
C1 (t) = C1 (t) + C1 (t)
with
(1)
C1
T21 u0 v0
=
λ
1 dβ
− β2
2 dt
and
(2)
C1
(4.3.37)
T12 w0 z0
=
ϑ
1 dδ
2
−δ .
2 dt
(4.3.38a)
Then the compatibility condition expressed by equation (4.3.35) can be satisfied by the following
choices:
T12 w0 z0 γ
T21 u0 v0 α
(1)
(1)
(1)
(2)
C 1 x2
(2)
C2 x
(1)
= C1 x2 + C2 x + C3 + f (x, t)
=
+
+
(2)
C3
− f (x, t)
(4.3.39a)
(4.3.39b)
(2)
where Ci , Ci (i = 2, 3) are arbitrary real functions of time t and f is an arbitrary real function
of x and t. Using the expressions (4.3.32a-b) we get the following possible form for the external
potentials:
1 1 dβ(t)
Ṽ (x, t)
(1)
(0)
2
V1 =
− β (t) x2 + V1 (t)x + V1 (t) −
+ iβ(t) ,
(4.3.40a)
λ 2 dt
T21 (λT22 − ϑT12 )
1 1 dδ(t)
Ṽ (x, t)
(1)
(0)
V2 =
− δ 2 (t) x2 + V2 (t)x + V2 (t) +
+ iδ(t)
(4.3.40b)
ϑ 2 dt
T12 (ϑT11 − λT21 )
(1)
(0)
(1)
(0)
where V1 , V1 , V2 and V2 are arbitrary real functions of time t and Ṽ (x, t) represents an
arbitrary real potential function which may be used conveniently in design of experiments with
BEC. At this point we have to note that these formulae cannot be used for the uncoupled case,
since when T12 = T21 = 0, then the compatibility condition (4.2.22) changes to (4.2.24a-b), and in
this way Ṽ (x, t) does not arise.
In summary we conclude that in order that the coupled GP equations (4.2.3a-b) pass the P-test,
a special combination of the real parts of the potentials V1 and V2 may depend only quadratically
and/or linearly on the spatial coordinate x. A stringent relationship can be established between
the coefficient of the quadratic terms and the imaginary parts which, in turn, may depend only on
time t. An additional real potential Ṽ of general form may be introduced which explicitly exhibits
coupling between the external potentials V1 and V2 .
72
CHAPTER 4. PAINLEVÉ TEST OF CGP
4.4
4.4. DISCUSSION OF THE RESULTS
Discussion of the results
In this section we discuss the results presented in this chapter from various points of view and
make comparison with related results obtained by others.
Presence of source terms: In the course of the theoretical study of two-component BEC with
attractive interaction, it has been found [171] that the decay and growth of number of atoms is
best accounted for by introducing an imaginary contact interaction term in the GP equations. We
now see that our analysis enables the existence of such source terms, by appropriately chosen β(t)
and δ(t) [see equation (4.3.40a-b)]. This result holds also in the case of one-component BEC as
found by Clarkson [135].
Uncoupled case: Next, we investigate the case T12 = T21 ≡ 0, when the system of the GP
equations (4.2.3a-b) is decoupled. As an example we derive the resonances. Our general equations
should reduce twice to earlier results obtained by Clarkson [135] for the one-component GP equation. Starting from the general expression (4.2.14) and applying the useful formula (4.2.12) one
obtains
T11 T22 2λϑ + 2λϑ
det Q(j) = λ2 ϑ2 (j + 1) j 2 (j − 3)2 (j − 4) j 2 − 3j + 4 − 2
(4.4.41)
∆
λϑ
= λ2 ϑ2 (j + 1) j 2 (j − 3)2 (j − 4) j 2 − 3j − 4 = λ2 ϑ2 (j + 1)2 j 2 (j − 3)2 (j − 4)2 .
The resonances (−1, 0, 3, 4) are those found by Clarkson [135] and all have a multiplicity of two as
a result of the double number of the (uncoupled) GP equations.
Sign of the potential: One of the experimental situations where the coupled GP equations
(4.2.3a-b) serve as a theoretical basis is the creation of two component BEC [165]. In such experiments alkali atoms are confined by symmetrically arranged harmonic trap potentials. One of the
(1)
(0)
(1)
(0)
possibility of our results is to choose in equations (4.3.40a-b) all functions V1 , V1 , V2 , V2 , β,
δ equal to zero and let the potential Ṽ (x, t) operate as a field confining the alkali gas particles. It
is then required that in equations (4.3.40a-b) the terms in which our confining potential Ṽ occurs
do have the same sign. The condition that those two terms with Ṽ have the same sign is in general
T12 T21 (λT22 − ϑT12 ) (ϑT11 − λT21 ) < 0
(4.4.42a)
which can be expressed also in terms of the scattering lengths as
a12 a21 (λa22 − ϑa12 ) (ϑa11 − λa21 ) < 0 .
(4.4.42b)
Because, physically a12 = a21 , the above condition for the equality of signs of the Ṽ terms in
equations (4.3.40a-b) is fulfilled for the usual experimental case with λ = ϑ if
a11 < a12 < a22
or
a22 < a12 < a11 .
(4.4.43)
If the scattering lengths a12 = a21 are greater or lesser than both a11 and a22 then the sign of the
terms containing the arbitrary potential Ṽ is different, which corresponds to untrapping one of the
BEC components.
73
CHAPTER 4. PAINLEVÉ TEST OF CGP
4.5. SUMMARY
Fulfillment of equation (4.2.18): The best studied example of the two-component BEC involves Rb atoms in two different hyperfine states. It has been found experimentally [65,66,165], and
numerically [179] that a stable configuration of soliton-like vortex in the two-component condensate
is achieved in the case where the scattering lengths are in the proportion:
a11 : a12 : a22 = 1.03 : 1 : 0.97,
with a12 ≡ a21 .
(4.4.44)
Let us now check whether these ratios obey our general condition (4.2.18) with integer m. Since
λ/ϑ = 1 expression (4.2.18) can be written as follows
2(a11 /a21 )(a22 /a12 ) − (a11 /a21 ) − (a22 /a12 )
1
=
(2m + 1)2 + 7 .
(a11 /a21 )(a22 /a12 ) − 1
16
(4.4.45)
The insertion of the above ratios gives
which yields
2 · 1.03 · 0.97 − 1.03 − 0.97
−0.0018
1
=
=2≡
(2m + 1)2 + 7
1.03 · 0.97 − 1
−0.0009
16
m = 2.
(4.4.46)
(4.4.47)
This result means that the experimental ratios (4.4.44) correspond to just one of the possible
solutions of the GP equations characterized by a m = 2 potential family.
Proceeding further, one can determine the resonances belonging to the experimentally found
ratios (4.4.44) to be [cf. with equations (4.2.16), (4.2.17)]
j1 = 2 + m = 4,
j2 = 1 − m = −1.
(4.4.48)
This result means that no further work is needed, the underlying potential falls into the category
defined by the compatibility condition for j = 4; a possible representation of such potentials is
given by (4.3.40a-b). Indeed, the quadratic trap potential used in the experiments suits well to the
general form of potentials obtained from the analysis of the resonance at j = 4.
4.5
Summary
In this chapter the first step towards verification of integrability of the coupled GP equations by
means of the P-analysis has been carried out (see also [1]). It has been shown that the GP equations
pass the P-test provided a special relation among the system parameters (masses, interaction
strengths) is satisfied [cf. with (4.2.18)]. One of the recent experiments has been taken as an
example. In this experiment [65, 66, 165] and a subsequent numerical study [179], the vortex
stability of a two component BEC has been investigated. It is found that the system parameters
at which stability occurs are just in the proportion which fits our relation (4.2.18) with m = 2, a
condition necessary for the GP equation to pass the P-test.
As the GP-equations play a great role in describing BEC, a particular attention has been paid
to establish the admissible forms of the confining trap potentials of experimental interest. It has
been found that, in addition to the prescribed form resulted by the P-analysis of a single GP
equation, there is a possibility of introducing an extra potential term of arbitrary shape into the
external potentials [cf. with equations (4.3.40a-b)]. Also, some discussion of the results has been
added which includes the comparison of the earlier results obtained for the one component GP
74
CHAPTER 4. PAINLEVÉ TEST OF CGP
4.5. SUMMARY
equation, the role of the source (imaginary) terms β(t) and δ(t) in the potentials, and the sign of
the additional potential terms.
Finally, we add a remark to the fulfillment of equation (4.2.18) with integer m’s. In the light
of experimental errors the above agreement m = 2 may seem to be accidental. We note however
that soliton-type structures (e.g. vortices in 3D) possess an outstanding stability sometimes called
’robustness’ which enables these particle-like formations to survive for a long time or even to arise
in circumstances that do not fit the exact constraint of mathematics. Therefore equation (4.2.18)
may prove also useful in exploring other regions of parameters where such stable structures are to
be observed in binary condensates.
75
Chapter 5
Relation between optical and
atomic solitons
In this chapter a relation is established [2] between the solutions of the coupled non-linear Schrödinger
equations with and without external potentials. The external confining potential is assumed to
modify both amplitudes and phases together with the spatial and temporal variables appearing in
the transformation relation. On examples of trap potentials depending linearly and quadratically
on spatial coordinate it is demonstrated how the transformation ansatz is working in practice for
obtaining atomic solitons of Bose-Einstein condensates consisting of sodium and rubidium atoms
when one starts from known optical soliton solution of the non-linear Schrödinger equation without
potential.
5.1
Introduction
Analytical solutions of nonlinear partial differential equations in terms of spatial solitons represents a fruitful field of theoretical and experimental physics. In particular, solutions of different
versions of nonlinear Schrödinger equations (NLSEs) describing optical pulse propagation in nonlinear medium have lead to the rapidly evolving telecommunication industry to date [42, 180–182].
Utilizing various mechanisms (such as e.g., stimulated Raman process) of the medium as well as
soliton solutions of the governing NLSEs, Hasegawa has proposed [183] a ”repeaterless” communication which has evolved nowadays to distances much larger than trans-oceanic ones. Generally,
soliton solutions of the NLSEs are called optical solitons.
Another kind of NLSE called Gross-Pitaevskii equation (GPE) [155, 156] is used to describe
a dilute gas of weakly interacting atomic particles. In this case the NLSE is supplemented by
an external trap potential which confines the gas particles. The GPE has become an important
theoretical tool in recent Bose-Einstein condensate (BEC) experiments. For example, it enables
prediction of the outcome of experiments and helps in design and calibrating the apparatus [61,62].
Also, several soliton-like phenomena have recently been observed such as vortices [165, 184], dark
solitons [6], or their decay into vortex rings [185]. Soliton solutions of GPEs are also known and,
in analogy with the nonlinear optics, we may call them atomic solitons.
In the course of obtaining soliton solutions to nonlinear partial differential equations a great
role is played by the singular value (Painlevé) analysis introduced by Refs. [186]. A Painlevé
77
CHAPTER 5. RELATION BETWEEN . . .
5.1. INTRODUCTION
(P) analysis of the single NLSE has been carried out in Ref. [174], that of two coupled NLSEs
(CNLSEs) in Ref. [176, 187]. Bright and dark soliton solutions to the CNLSEs have been given
in the work [175, 188]. The single GPE has been P analyzed by Ref. [135], the coupled GPEs
(CGPEs) has been P tested by Ref. [1].
In this chapter we show that there exists a simple transformation between the solutions of the
CGPEs and the CNLSEs provided the external potentials of the CGPEs equation have a special
form. The transformation ansatz is based on the similarity between the CNLS and CGP equations
and on the general assumption that the external potential modifies both amplitudes and phases of
the solutions of the CNLSEs together with the spatial and temporal variables. Switching on/off
the external potentials does not change the functional forms of the solutions of the CNLSEs which
are therefore supposed to be given a priori. Equipped with such a general transformation one is
able to employ the numerous special analytical and numerical solutions obtained so far in optics
directly to the CGPEs describing multi-component BECs. Interestingly enough, the special form
of the external potentials enabling working with the above transformation ansatz is very similar
though more restricted compared to our earlier result [1] obtained by applying the P test to the
CGPEs. Recall that the P test has nothing to do with the present transformation because it aims
simply to establish integrability conditions, i.e., conditions of solvability of the nonlinear differential
equations by means of the inverse scattering transform [186].
It has been found in Ref. [1] that the P test in the j = 4 resonance analysis of the (1+1)
dimensional CGPEs requires that the external potentials depend linearly or quadratically on the
spatial coordinate x. (Performing the P test for j > 4 one may expect more general potentials.)
In the quadratical (x2 ) case the potential must have also an imaginary term β(t). Although such
potentials cannot model a simple harmonic field which is usual in forming BECs, absorption/feeding
corresponding to negative/positive β(t) is always present in the experiments. In the linear (x)
case the potentials are real, and may have an arbitrary time dependence. Such potentials can
be produced in recent miniaturized BEC apparatus [119] to perform experiments with different
samples [189]
Although our transformation can be used for any solution of CNLSE (e.g., vector-solitons [190],
bimodal solitons [191], Josephson-type oscillation [67]), we take the simple bright-dark (bd) static
soliton solutions of Shalaby and Barthelemy [192]. The parametrization corresponds to coupled
BEC species consisting of N1 = 106 sodium atoms as the bright component and N2 = 106 rubidium
atoms as the dark one. Note that each specimen alone is found to form a stable BEC state in
a number of experiments [see e.g., Ref. [193] for 23 Na and Ref. [194] for 87 Rb], but as a twocomponent specimen they have not yet been realized. Both types of atoms have a repulsive
two-body interaction which means that the (intra) scattering lengths are positive; we adopt the
values a11 = 2.75 nm [195] for the sodium component and a22 = 5.5 nm [65] for the rubidium
one. Since the scattering length between 23 Na and 87 Rb is not yet available we choose it to be
a12 = a21 = 4.9 nm, a value which lies within the allowed range of stability (determined by the
reality of the amplitudes) and meets the j = 4 resonance condition of the singular values (P test)
analysis [1].
The free bd solitons will be placed numerically into different confining potentials which are
assumed to be linear or quadratic in the spatial variable. As it will be shown the potentials
determine the transformation amplitudes and phases, so that we can study how various trapping
potentials modify the motion of the different (bd) solitons. In particular, we show how a potential
which is linear in the spatial variable and behaves like a sine function in the temporal coordinate,
acts on the bd solitons, when it is switched on. Also the effect of a harmonic potential with an
imaginary part is demonstrated.
78
CHAPTER 5. RELATION BETWEEN . . .
5.2. SOME RESULTS OF P ANALYSIS . . .
The organization of the chapter is as follows. In Sec. 5.2 some results of the P analysis of
the CGPEs will be recalled from Ref. [1]. In Sec. 5.3 the transformation between solutions of
CNLS and CGP equations will be carried out. Section 5.4 contains the necessary transformation
functions which are determined solely by the given trap potential. In Sec. 5.5 it is demonstrated
how the proposed transformation works on the example of bright and dark solitons and Sec. 5.6
is left for a short summary.
5.2
Some results of P analysis of CGP equations
In (1 + 1) dimension the coupled Gross-Pitaevskii (CGP) equations are usually written as
∂
~2 2
2
2
i~ ψ1 (x, t) =
−
∇ + U1 (x, t) + U11 |ψ1 (x, t)| + U12 |ψ2 (x, t)| ψ1 (x, t) (5.2.1a)
∂t
2m1
∂
~2 2
2
2
i~ ψ2 (x, t) =
−
∇ + U2 (x, t) + U21 |ψ1 (x, t)| + U22 |ψ2 (x, t)| ψ2 (x, t) (5.2.1b)
∂t
2m2
where mi denotes the mass of the atomic species i (i = 1, 2) of the two-component BEC gas
and Uij is related with the interactions between the atoms i and j (i, j = 1, 2) via the relation
Uij = 2π~2 aij Nj /Aµij with Nj meaning the number of atoms in the jth component of the BEC,
aij being the 3D scattering length characterizing the interaction between atoms i and j confined
to the spatial dimension x with a general transverse crossing area A, and µij = mi mj /(mi + mj ) is
the reduced mass. In the case of real trap potentials
U2 the normalization of wave functions
R ∞ U1 and
2
reads, after subtracting the background, as 1 = −∞|ψj | dx.
By introducing the new parameters: λ = ~/2m1 , ϑ = ~/2m2 , Tij = −Uij /~, and also, by
changing the notations: Vi = Ui /~ for the external potentials and u = ψ1 , w = ψ2 for the envelope
wave functions, one arrives at another form of the CGP equations
2
2
(5.2.2a)
2
2
(5.2.2b)
iut + λuxx + T11 |u| u + T12 |w| u = V1 u
iwt + ϑwxx + T21 |u| w + T22 |w| w = V2 w
which is a usual starting point for the P analysis.
In order that the CGP equations (5.2.2) pass the P test (which is a requirement for the equations
to be integrable by the inverse scattering transform [186]), it has been shown in Ref. [1] that the
family of the admissible potentials may have a real and an imaginary part. The imaginary parts
may depend only on time, the real parts may have both spatial and temporal dependencies. The
external potentials thus can be written in the form
V1 (x, t)
V2 (x, t)
= α(x, t) + iβ(t),
= γ(x, t) + iδ(t)
(5.2.3a)
(5.2.3b)
where α, γ and β, δ are real functions, not completely independent of each other. A further
result of Ref. [1] concerns the real parts of the potentials which should satisfy a power-like (x n )
relationship up to the second order in the spatial variable as follows
T12 w0 z0 γ + T21 u0 v0 α = C1 (t)x2 + C2 (t)x + C3 (t)
(5.2.4)
where both C2 (t) and C3 (t) are arbitrary time dependent real functions but C1 (t), the coefficient
of the quadratic term, is related to the arbitrary functions β(t), δ(t) of the imaginary parts of the
79
CHAPTER 5. RELATION BETWEEN . . .
5.2. SOME RESULTS OF P ANALYSIS . . .
external potentials V1 and V2 by the formula
T21
1 dβ
T12
1 dδ
2
2
C1 (t) =
u0 v 0
−β +
w0 z0
−δ
.
λ
2 dt
ϑ
2 dt
(5.2.5)
In the above equations the quantities u0 v0 , w0 z0 denote real constants which can be calculated
by the expression
2
T22 −T12
λ
u0 v 0
(5.2.6)
=
T11
ϑ
w0 z0
∆ −T21
with ∆ = T11 T22 −T12 T21 . If accidentally ∆ = 0 happens then we may use the relation u0 v0 /w0 z0 =
const instead of (5.2.6), which case needs a special consideration.
As it has been shown in Ref. [1] the above result does not pose a severe restriction for the
individual form of the real part of the external potentials V1 and V2 since one may always choose
a completely arbitrary real potential Ṽ (x, t) which plays the role of the common potential term in
both V1 and V2 .
A possible form of the external potentials can thus be written as [1]
1 (2) 2
V x +
λ 1
1 (2) 2
V x +
=
ϑ 2
V1 =
V2
(1)
(1)
(0)
1 (1)
V (t)x +
λ 1
1 (1)
V (t)x +
ϑ 2
1 (0)
V (t) + iβ(t)
λ 1
1 (0)
V (t) + iδ(t)
ϑ 2
(5.2.7a)
(5.2.7b)
(0)
where V1 , V2 , V1 , V2 are arbitrary real functions of time t and the coefficients of the
harmonic (x2 ) terms are related to the imaginary parts β and δ by
1 dβ(t)
1 dδ(t)
(2)
(2)
2
2
V1 =
− β (t) ,
V2 =
− δ (t) .
(5.2.8)
2 dt
2 dt
This form (5.2.7) of the external potentials is considered as a direct generalization of the results
obtained for the one-channel case by [135]. We note that also for the one channel case an exact
soliton solution has been found quite recently [196] assuming harmonic trap potentials with a
constant imaginary part, i.e., V = −β 2 x2 + iβ.
Another interesting result provided by the P test [1] interrelates the interaction strengths
T11 , T12 , T21 , T22 or the respective scattering lengths a11 , a12 = a21 , a22 characterizing the BEC
samples:
2 T11 T22 − (ϑ/λ)T11 T12 − (λ/ϑ)T21 T22
=
(5.2.9)
T11 T22 − T12 T21
2(a11 /a21 )(a22 /a12 ) − (a11 /a21 )(m2 /m1 ) − (a22 /a12 )(m1 /m2 )
1
=
=
(2m + 1)2 + 7 ,
(a11 /a21 )(a22 /a12 ) − 1
16
where m should be a non-negative integer number. Incidentally m = 2 happens for the parameter
ratios a11 : a12 : a22 = 1.03 : 1 : 0.97 which characterize a two-component BEC consisting of
rubidium atoms in the phase-sensitive population transfer experiments [65, 66].
Although in view of experimental errors which are always present, the above agreement with
m = 2 may seem to be accidental, we note that soliton-like structures possess an outstanding
stability which enables these particle-like formations to survive or to arise in cases, in which the
constraint (5.2.9) of the P test is only approximately satisfied. Moreover, since the magnitude and
the sign of the s-wave scattering length can be controlled via the magnetic field-induced Feshbach
resonances [117], equation (5.2.9) may prove also useful in exploring other regions of parameters
where stable structures are to be created in (1D) binary condensates.
80
CHAPTER 5. RELATION BETWEEN . . .
5.3
5.3. TRANSFORMATION BETWEEN . . .
Transformation between solutions of
CNLS and CGP equations
In the absence of outer potentials (Vi = 0), the CGPEs become the CNLSEs
2
2
iQ1,τ + λQ1,ξξ + Ω11 |Q1 | + Ω12 |Q2 | Q1 = 0
iQ2,τ + ϑQ2,ξξ + Ω21 |Q1 |2 + Ω22 |Q2 |2 Q2 = 0
(5.3.10a)
(5.3.10b)
which plays a major role in optical transmission industry so that their various properties and
solutions are thoroughly studied and available in optics literature [42, 180, 181].
The similarity between the structures of the CGP and CNLS equations (5.2.2) and (5.3.10)
suggests that there may exist general transformations that relate solutions of the CNLSEs and
the CGPEs to each other. It is also clear that the difference between the solutions comes entirely
from the external potentials Vi , i = 1, 2. Quite generally we may assume that the modification
due to the external potentials may be expressed as independent factors changing the amplitudes
and the phases of the known solutions Qi of the CNLS equations. Thus, using the arbitrary real
coupling constants Ωij as well as the general temporal τ (x, t) and spatial ξ(x, t) variables, valid in
the CNLS equations (5.3.10), we write the solutions of the CGP equations (5.2.2) as
u(x, t) = A1 (x, t)eiΘ1 (x,t) Q1 (ξ(x, t), τ (x, t)) ,
(5.3.11a)
iΘ2 (x,t)
(5.3.11b)
w(x, t) = A2 (x, t)e
Q2 (ξ(x, t), τ (x, t)) .
Here A1 , A2 , Θ1 , Θ2 represent differentiable, real functions of the temporal (t) and spatial (x)
coordinates, valid in the CGP system, and Q1 and Q2 are solutions taken from the CNLS equations
(5.3.10). By substituting the above ansatz (5.3.11) into the CGP equations (5.2.2) and collecting
terms containing Qi ’s and their derivatives together, the following equations can be obtained (γ is
a real constant)
τt = A21 ,
(ξx )2 = A21 ,
τx = 0,
A2 = γA1 ,
(5.3.12a)
Ω22 = γ 2 T22 ,
(5.3.12b)
−A1 Θ1,t + iA1,t + 2iλA1,x Θ1,x − λA1 (Θ1,x ) + λA1,xx + iλA1 Θ1,xx − V1 A1 = 0
(5.3.12c)
Ω11 = T11 ,
Ω12 = γ 2 T12 ,
Ω21 = T21 ,
2
iA1 ξt + 2iλA1 Θ1,x ξx + 2λA1,x ξx + λA1 ξxx = 0 (5.3.12d)
2
−A2 Θ2,t + iA2,t + 2iϑA2,x Θ2,x − ϑA2 (Θ2,x ) + ϑA2,xx + iϑA2 Θ2,xx − V2 A2 = 0
iA2 ξt + 2iϑA2 Θ2,x ξx + 2ϑA2,x ξx + ϑA2 ξxx = 0.
(5.3.12e)
(5.3.12f)
One may integrate these equations to determine the unknown functions τ , ξ, Θ1 to be
τ (x, t)
ξ(x, t)
Θ1 (x, t)
(5.3.12a)
=
(5.3.12a)
=
(5.3.12d)
=
τ (t)
(5.3.12a)
=
Zt
2
[A1 (s)] ds
κA1 (t)x + A0 (t)
1
κ
dA0 (t)
1
− β̃(t)x2 −
x − B1 (t)
2λ
2λA1 (t)
dt
λ
(5.3.13a)
(5.3.13b)
(5.3.13c)
with κ = ±1, and A0 (t) and B1 (t) being arbitrary functions of time resulted from the spatial
integrations, and the function β̃(t) is related to the amplitude function A1 by
β̃(t) =
1 d
ln A1 (t).
2 dt
81
(5.3.13d)
CHAPTER 5. RELATION BETWEEN . . .
5.3. TRANSFORMATION BETWEEN . . .
We thus learn that the transformation amplitudes may depend only on time t: Ai = Ai (t), i = 1, 2
as does the function τ = τ (t). However the spatial variable ξ(x, t) of the CNLS system may possess,
as yet, an arbitrary time dependence via the function A0 (t).
Analogously, one finds from Eq. (5.3.12e) for the transformation phase of index 2 the relation
1
κ
dA0 (t)
1
Θ2 (x, t) = − β̃(t)x2 −
x − B2 (t)
(5.3.13e)
2ϑ
2ϑA1 (t)
dt
ϑ
with B2 (t) being an arbitrary function.
Equation (5.3.12f) remains to check for the consistency of all the above derivations.
We may conclude that, apart from the mass factors λ and ϑ, the phases Θ1 and Θ2 have the
same dependence on the spatial coordinate x. On the other hand, their temporal dependence on
t may depart from each other via the arbitrary functions B1 (t) and B2 (t).
Finally, one may determine the possible shapes of the potentials V1 (x, t) and V2 (x, t) that enable
employment of the simple transformation ansatz (5.3.11) between the solutions of CNLS and CGP
equations. Inserting the above functions β̃(t), A0 (t), A1 (t), B1 (t), B2 (t) into Eqs. (5.3.12c) and
(5.3.12e) and performing the derivations, one obtains
"
#
2
κ
d A0 (t)
dA0 (t)
1 1 dβ̃(t)
2
2
V1 (x, t) =
− β̃ (t) x +
− 4β̃(t)
x
λ 2 dt
2λA1 (t)
dt2
dt
"
2 #
1 dB1 (t)
1
dA0 (t)
+
−
+ iβ̃(t), (5.3.14a)
λ
dt
4A21 (t)
dt
"
#
2
1 1 dβ̃(t)
κ
d A0 (t)
dA0 (t)
2
2
V2 (x, t) =
− β̃ (t) x +
− 4β̃(t)
x
ϑ 2 dt
2ϑA1 (t)
dt2
dt
"
2 #
1 dB2 (t)
1
dA0 (t)
+
−
+ iβ̃(t). (5.3.14b)
ϑ
dt
4A21 (t)
dt
This result is in harmony with the findings obtained from the Painlevé test by Ref. [1] for the
CGP equations, also presented in the preceding section [see Eq. (5.2.7)]. Comparing Eqs. (5.2.7)
and (5.3.14) we obtain further specifications (restrictions and inter-relations) among the various
time dependent parts of the potentials. For example, a comparison of the imaginary parts of Eqs.
(5.2.7) and (5.3.14) yields the relation
β̃(t) = β(t) = δ(t)
(5.3.15)
from which it follows, by using Eq. (5.2.8), that the coefficients of the harmonic (x2 ) terms in Eqs.
(5.2.7) and (5.3.14) must agree:
(2)
(2)
V (2) (t) := V1 (t) = V2 (t) =
1 dβ(t)
− β 2 (t).
2 dt
(5.3.16)
We have introduced here the common denotation V (2) (t). From the equality of the linear (x) terms
in (5.2.7) and (5.3.14) we obtain the restriction
2
κ
d A0 (t)
dA0 (t)
(1)
(1)
−
4β(t)
(5.3.17)
V (1) (t) := V1 (t) = V2 (t) =
2A1 (t)
dt2
dt
82
CHAPTER 5. RELATION BETWEEN . . .
5.4. DERIVATION OF TRANSFORMATION . . .
where another collective denotation V (1) (t) has been introduced. These last two equalities (5.3.16)
and (5.3.17) should be fulfilled in order that the transformation ansatz (5.3.11) be applicable
between the solutions of CNLS and CGP equations.
Checking the x-independent real terms in Eqs. (5.2.7) and (5.3.14) yields the relations
(0)
V1 (t)
=
(0)
V2 (t) =
(0)
2
dB1 (t)
1
dA0 (t)
−
dt
4A21 (t)
dt
2
dB2 (t)
1
dA0 (t)
−
dt
4A21 (t)
dt
(5.3.18a)
(5.3.18b)
(0)
which indicates that V1 (t) and V2 (t) remain, as yet, independent of each other because of the
arbitrariness of the functions B1 (t) and B2 (t).
5.4
Derivation of transformation functions
from given trap potentials
We now determine the transformation functions τ , ξ, Θi and Ai , i = 1, 2, explicitly appearing in the
ansatz (5.3.11), from the given trap potentials V1 and V2 . These functions help us to employ any
solutions of the CNLS equations found so far either numerically or analytically to the description
of BECs of confined atomic species.
It is clear from Eqs. (5.3.13) that only the amplitude function A1 (t) should be known together
with the functions β(t), A0 (t) and Bi (t), i = 1, 2 in order that the transformation functions τ, ξ, θi
as defined by Eqs.(5.3.13) be completely specified by the potentials.
In agreement with the P test [1] and the preceding analysis we suppose that the external trap
potentials are given in the form
V1 (x, t) =
1
1 (0)
1 (2)
V (t)x2 + V (1) (t)x + V1 (t) + iβ(t)
λ
λ
λ
(5.4.19a)
V2 (x, t) =
1 (2)
1
1 (0)
V (t)x2 + V (1) (t)x + V2 (t) + iβ(t)
ϑ
ϑ
ϑ
(5.4.19b)
and
(0)
where the known coefficient functions V (2) , V (1) , Vi , i = 1, 2 are x-independent, but they may
depend on the temporal variable t. Moreover the relation (5.3.16) holds between the functions
V (2) (t) and β(t).
In order to obtain the transformation functions A1 (t), τ , ξ, Θi from the various potential coef(0)
(0)
ficients V (2) (t), V (1) (t), V1 (t), V2 (t) we simply solve the equations (5.3.16), (5.3.17), (5.3.18a),
and (5.3.18b). But equation (5.3.16) for the unknown function β(t) is a nonlinear ordinary differential equation to which there exist no general analytical solution methods. Therefore we always
suppose in the following that β(t) is a priori given. This assumption is reasonable since β(t) represents the imaginary part of the potentials too, which should be given and may be accounted
for the loss or the gain of the specimen. For the remaining three differential equations (5.3.17),
(5.3.18a), and (5.3.18b) one may apply the method of variation of parameters to obtain, after some
83
CHAPTER 5. RELATION BETWEEN . . .
5.5. EXAMPLES
calculation, the following results
A1 (t)
A0 (t)
B1 (t)
B2 (t)
(5.3.13d)
=
(5.3.17)
=
(5.3.18a)
=
(5.3.18b)
=
Zt
A10 exp 2 β(s)ds
Zt
Zs
P0 (s) exp 4 β(s0 )ds0 ds
2κ


Zt
Zs
(0)
V1 (s) + P02 (s) exp 4 β(s0 )ds0  ds
Zt

V2(0) (s)

Zs
+ P02 (s) exp 4 β(s0 )ds0  ds
(5.4.20a)
(5.4.20b)
(5.4.20c)
(5.4.20d)
with A10 being a real constant of integration and for the sake of transparency we have introduced
the abbreviation


Zt
Zs
P0 (t) =
V (1) (s) exp −2 β(s0 )ds0  ds.
(5.4.20e)
Then, inserting the above result into Eqs. (5.3.13) and (5.3.13e) one obtains the transformation
functions τ , ξ, Θi in terms of the given potential functions.
In summary, the given external potentials V1 (x, t) and V2 (x, t) determine the functions V (2) (t),
(0)
(0)
(1)
V (t), V1 (t), V2 (t) and β(t) from which the transformation functions A1 (t), τ (t), ξ(t) Θi (x, t)
i = 1, 2 can be calculated by making use of Eqs. (5.4.20) inserted into Eqs. (5.3.13c-e). Then
using the transformation ansatz (5.3.11) one is in a position to calculate the solutions of the CGP
equations, if the corresponding solutions (analytical or numerical ones) of the CNLS equations
are at hand. In this way all solutions developed so far in the field of CNLSEs can be utilized
straightforwardly in the CGP theories.
5.5
Examples
In this section we apply our results to several external potentials corresponding to various experimental arrangements. For creating BECs, there are two main techniques such as the magnetic
and optical confinement. Both techniques lead to sophisticated trap potentials which may depend
quadratically and linearly on the spatial coordinate so that we shall study the effect of these types
of potentials on the free spatial solitons which we define first. Because these free solitons are solutions of the CNLS equations (5.3.10) they can be called as optical solitons, although they consist
of atomic particles.
In the literature there are indications that a great experimental effort is made on realizing twocomponent systems from different atomic species such as 41 K-87 Rb [120], 7 Li-133 Cs [76]. Also there
are some theoretical work on the stability of ground state configurations especially for the 23 Na87
Rb binary system [71,197,198]. We shall consider here a simple dynamics of the sodium-rubidium
system.
84
CHAPTER 5. RELATION BETWEEN . . .
5.5.1
5.5. EXAMPLES
Bright-dark optical solitons
We start from the simplest solutions of the one component NLS equation namely from the static
bright and the static dark solitons as given by the solutions sech(ξ) and tanh(ξ). Adding to these
solutions some amplitudes and phases one obtains a coupled solution of bright-dark pair of the
CNLS equations (5.3.10) as [192]
Q1 (ξ, τ ) = q1 sech (kξ)eiω1 τ
Q2 (ξ, τ ) = q2 tanh (kξ)eiω2 τ .
(5.5.21)
The real amplitudes q1 , q2 and the real phases ω1 , ω2 are determined by insertion of (5.5.21) into
(5.3.10) to be
2k 2
(λΩ22 − ϑΩ12 ) ,
D
i
2h
k
ω1 =
Ω11 (λΩ22 − ϑΩ12 ) − Ω12 (ϑΩ11 − λΩ21 ) ,
D
|q1 |2 =
|q2 |2 =
ω2 =
2k 2
(λΩ21 − ϑΩ11 ) (5.5.22a)
D
2k 2
Ω22 (λΩ21 − ϑΩ11 ) (5.5.22b)
D
10
10
5
5
ξ [µm]
ξ [µm]
with D = Ω11 Ω22 − Ω12 Ω21 = γ 2 ∆, and k being an arbitrary real parameter characterizing the
extension of the solitons.
Figure 5.1 exhibits these solutions in the coordinate system (ξ, τ ) with the parametrization
τ [ms] 4
τ [ms] 4
-2
0
2
6
-2
0
2
6
0
PSfrag replacements
0
PSfrag replacements
–5
–5
–10
–10
Figure 5.1: The figures show the squares |Q1 |2 and |Q2 |2 of the static bright-dark solution of the coupled nonlinear
Schrödinger equations (5.3.10) as a function of τ and ξ with the parameters characteristic of a BEC composed
from (a) N1 = 106 sodium (23 Na) atoms and (b) N2 = 106 rubidium (87 Rb) atoms: k = 106 m−1 , γ = 1, λ =
1.37 × 10−9 m2 s−1 , ϑ = 0.36 × 10−9 m2 s−1 , Ω11 = −31.55 ms−1 , Ω12 = Ω21 = −35.54 ms−1 , Ω22 = −16.68 ms−1 .
Darker regions are those with less densities.
k = 106 m−1 , γ = 1, λ = 1.37 × 10−9 m2 s−1 , ϑ = 0.36 × 10−9 m2 s−1 , Ω11 = −31.55 ms−1,
Ω12 = Ω21 = −35.54 ms−1, Ω22 = −16.68 ms−1, which corresponds to a BEC consisting of N1 = 106
sodium 23 Na atoms as the bright component with scattering length a11 = 2.75 nm [195] and
of N1 = 106 rubidium 87 Rb atoms as the dark component with a22 = 5.5 nm [65]. Since the
inter-specimen scattering length between sodium and rubidium is as yet unknown we fix it to be
a12 = a21 = 4.9 nm. This value lies within the allowed range of 3.1 nm ≤ a12 ≤ 8.7 nm determined
by the condition of reality of the amplitudes of q1 and q2 and also corresponds to the value m = 2
in Eq. (5.2.9) being required by the j = 4 resonance analysis of the P test [1]. (We note that if the
scattering length between sodium and rubidium atoms turns out to be close to this selected value
of 4.9 nm then the system 23 Na-87 Rb would be highly suitable for performing two-component BEC
85
CHAPTER 5. RELATION BETWEEN . . .
5.5. EXAMPLES
2
2
experiments.) In Fig. 5.1 the absolute squares |Q1 | and |Q2 | of the bd soliton solution (5.5.21)
of the CNLSEs (5.3.10) can be seen as a function of the variables ξ and τ . In Fig. 5.1 darker
regions correspond to less density and the static behavior of the solitons is evidently observed.
5.5.2
Linear potentials
To demonstrate the efficiency of our results we next calculate the transformation functions (A i , τ , ξ,
Θi , i = 1, 2) for the case of a real external potential which depends linearly on the spatial coordinate
x. Similar potentials may arise in BEC experiments when earth gravity, or a time-dependent
potential (e.g., low frequency laser) are taken into account. Moreover, recent miniaturized magnetic
trap arrangements [119] provide possibilities to study BEC transport in such a linear potential [189].
We have seen in the preceding sections that the transformation ansatz (5.3.11) for the solutions
(1)
of the CNLS and CGP equations demands that the temporal coefficients Vi (t), i = 1, 2 of the
linear terms of the trap potentials V1 (x, t) and V2 (x, t) be the same. Thus we shall take our linear
potential example in the form
0
t<0
V1 (x, t) =
,
(5.5.23a)
1
V
sin(ωt
+
ϕ)(x
−
x
)
t≥0
0
0
λ
0
t<0
,
(5.5.23b)
V2 (x, t) =
1
V
sin(ωt
+
ϕ)(x
−
x
)
t≥0
0
ϑ 0
which define the potential coefficient functions to be
V (2) (t) = 0,
β(t) = 0,
(5.5.23c)
and
V
(0)
(1)
(t)
(0)
V1 (t) = V2 (t)
=
=
0
t<0
,
V0 sin(ωt + ϕ) t ≥ 0
0
−V0 x0 sin(ωt + ϕ)
t<0
.
t≥0
(5.5.23d)
(5.5.23e)
Because of β(t) = 0, we obtain the transformation functions A1 (t), τ (t) and ξ(x, t) for t ≥ 0
immediately to be
A1 (t) = A2 /γ = A10 e2
τ (t)
ξ(x, t)
A210 t
Rt
β(s)ds
= A10 = const
=
= κA10 x + A0 (t)
(5.5.24a)
(5.5.24b)
(5.5.24c)
where use of Eqs. (5.4.20a), (5.3.13a), (5.3.13b) has been made. To determine the function A 0 (t)
from Eqs. (5.4.20b) we first calculate the abbreviating function P0 (t) (see Eq. (5.4.20e)) to be
(t ≥ 0)
V0
P0 (t) =
[cos (ϕ) − cos (ωt + ϕ)]
(5.5.25)
A10 ω
from which
A0 (t) =
2κV0 A10
[ω cos (ϕ)t − sin(ωt + ϕ) + sin (ϕ)] .
ω2
86
(5.5.26)
CHAPTER 5. RELATION BETWEEN . . .
5.5. EXAMPLES
To determine the transformation phases Θ1 (x, t) and Θ2 (x, t), we next calculate the functions
B1 (t), B2 (t) by making use of Eqs. (5.4.20c) and (5.4.20d) to be (t ≥ 0):
B1 (t) = B2 (t) =
V0 h 2
4ω x0 (cos (ωt + ϕ) − cos (ϕ)) + V0 (sin (2[ωt + ϕ]) + 3 sin (2ϕ))
4ω 3
i
+2V0 ωt (2 + cos (2ϕ)) − 4V0 (sin (ωt + 2ϕ) + sin (ωt)) .
(5.5.27)
Employing these results in Eqs. (5.3.13c) and (5.3.13e) we obtain the wanted phases Θ 1 (x, t) and
Θ2 (x, t) to be (t ≥ 0)
λΘ1 (x, t) = ϑΘ2 (x, t) =
V0
[cos(ωt + ϕ) − cos (ϕ)] x − B1 (t).
ω
(5.5.28)
Thus, we obtain a result that, in the case of a real potential depending linearly on the variable x, the
transformation phases Θ1 (x, t) and Θ2 (x, t) also have the same temporal and spatial dependence
(apart from the trivial mass factors λ and ϑ).
2
2
Figure 5.2 exhibits the absolute squares |u| and |w| of solutions of the CGP equations (5.2.2) as
PSfrag replacements
0
2
t [ms] 4
6
-2
6
4
4
x [µm]
x [µm]
-2
6
2
0
PSfrag replacements
–2
0
2
t [ms] 4
6
2
0
–2
Figure 5.2: The figures show the squares |u|2 and |w|2 of the bright-dark solution of the coupled Gross-Pitaevskii
equations (5.2.2) with the linear potentials (5.5.23) as a function of t and x. The parameters chosen are characteristic
of a BEC composed from N1 = 106 sodium (23 Na) atoms and N2 = 106 rubidium (87 Rb) atoms: k = 106 m−1 , γ = 1,
λ = 1.37×10−9 m2 s−1 , ϑ = 0.36×10−9 m2 s−1 , T11 = −31.55 ms−1 , T12 = T21 = −35.54 ms−1 , T22 = −16.68 ms−1 ,
V0 = 1 ms−2 , ω = (ω1 + ω2 )/2, ϕ = 1 rad, x0 = 1 µm. For explanation see text. Darker regions are those with less
densities.
a function of x and t obtained from the optical solitons of Eq. (5.5.21) by using the transformation
ansatz (5.3.11) with the aid of transformation functions (5.5.24), (5.5.26), and (5.5.28). The set of
parameters used is supplemented by the additional values: γ = 1, V0 = 1 ms−2 , ω = (ω1 + ω2 )/2,
ϕ = 1 rad, x0 = 1 µm.
In Fig. 5.2 one can clearly see the effect of the potential switched on at t = 0 s, thereby causing
an initial velocity v0 = 2κV0 A10 cos (ϕ)/ω of about 55 [µm/ms]. The solitons begin to accelerate
and decelerate periodically with a period T = 2π/ω ≈ 3.2 [ms], according to the periodical change
of sign of the spatially linear potential. Dark and bright components behave similarly as a result
of the common potentials confining them. The figures have been generated by the Maple [178]
program package and the fulfillment of CGPEs has been always tested and proved to be better
than one part in a million.
87
CHAPTER 5. RELATION BETWEEN . . .
5.5.3
5.5. EXAMPLES
Quadratic potentials with imaginary parts
From the P analysis [1] of the CGP equations we know that the external potentials, displayed by
Eqs.(5.2.7) and (5.2.8), may possess a quadratic (x2 ) spatial behavior and the coefficient of this
term should be in a strict relationship with the imaginary part. (Note however that this result of
the P test follows from the resonance analysis at j = 4 while at higher j 0 s more generic potentials
are possible.) On the other hand we have found in Sect. 5.3 that the results of the P test are
in harmony with our transformation ansatz (5.3.11) in that it results also in quadratic potentials
given by Eqs. (5.3.14). Moreover we have found additional restrictions among the various parts of
the potential terms which are related to the transformation functions Ai , Θi , τ , ξ . Therefore, as
a last example, we shall take a (shifted) quadratic potential for t ≥ 0 in the form
V1 (x, t)
=
V2 (x, t)
=
1 2
C (x − x0 )2 + iβ(t)
λ
1 2
C (x − x0 )2 + iβ(t)
ϑ
(5.5.29a)
(5.5.29b)
which defines the necessary coefficient functions to be
V (2) (x, t) = C 2 ,
V (1) (x, t) = −2C 2 x0 ,
V (0) (t) = −C 2 x20 .
(5.5.30)
The (constant) coefficient V (2) of the harmonic potential terms is related to the imaginary part by
the relation
1 dβ(t)
V (2) =
− β 2 (t) = C 2 .
(5.5.31)
2
dt
Solving the above differential equation for β(t) we obtain
β(t) = C tan(2Ct) .
(5.5.32)
Using this result and the general formulas (5.3.13) and (5.4.20), the necessary transformation
functions and variables read as follows (t > 0)
A1 (t) = γA2 (t)
=
τ (t)
=
ξ(x, t)
=
2λΘ1 (x, t) = 2ϑΘ2 (x, t)
κA10
|cos(2C t)|
A210
[tan(2C t) − 2Ct]
2C
κA10 (x − x0 )
+ κA10 x0
|cos(2C t)|
= −C tan (2C t)(x − x0 )2
(5.5.33a)
(5.5.33b)
(5.5.33c)
(5.5.33d)
Figure 5.3 shows the absolute squares |u|2 and |w|2 of solutions of the CGP equations (5.2.2) as
a function of x and t obtained from the optical solitons of Eq. (5.5.21) by using the transformation
ansatz (5.3.11) with the aid of the transformation functions (5.5.33). The additional parameters
are: γ = 1, C = −100 s−1, x0 = −5 µm, A10 = 1.
In Fig. 5.3 we see the effect of the shifted quadratic potentials (5.5.29) on the bd solitons.
The solitons perform a periodic motion in the spatial region 0 ≤ x ≤ x0 as one can also infer
this behavior from Eq. (5.5.33c). The time period of the motion is determined by the strength of
the potential to be T = π/C. We note that in this example the amplitudes Ai (t) of the solitons
become infinite with the same period T because of Eqs. (5.5.32) and (5.5.33a).
88
CHAPTER 5. RELATION BETWEEN . . .
5.6. SUMMARY
t [ms]
1
2
t [ms]
3
4
0
4
2
2
x [µm]
x [µm]
0
4
0
-2
2
3
4
0
-2
-4
PSfrag replacements
1
-4
PSfrag replacements
-6
-6
Figure 5.3: The figures show the squares |u|2 and |w|2 of the bright-dark solution of the coupled Gross-Pitaevskii
equations (5.2.2) with the quadratic potentials (5.5.29) as a function of t and x. The parameters chosen are
characteristic of a BEC composed from N1 = 106 sodium (23 Na) atoms and N2 = 106 rubidium (87 Rb) atoms:
k = 106 m−1 , γ = 1, λ = 1.37×10−9 m2 s−1 , ϑ = 0.36×10−9 m2 s−1 , T11 = −31.55 ms−1 , T12 = T21 = −35.54 ms−1 ,
T22 = −16.68 ms−1 , C = −100 s−1 , x0 = −5 µm, A10 = 1. For explanation see text. Darker regions are those with
less densities.
5.6
Summary
A formal transformation has been worked out between solutions of the CNLSEs and the CGPEs
using the assumption that the external potentials modify both amplitudes and phases of the solutions and affect also the temporal and spatial variables. Solutions of the CNLSEs are usually
optical solitons enabling powerful communication to date. Solutions of CGPEs describe, among
others, groups of weakly interacting identical atomic particles, specifically those being able to undergo Bose-Einstein condensation. Thus, the transformation established in this paper enables a
direct use of all the previous work on optical soliton theories of optics community in the growing
field of (multi-component) BECs theories and experiments.
In order that our transformation ansatz [see Eq. (5.3.11)] be applicable, the external trap potentials should have a power-like behavior in the spatial variable upto the second order. Also, they
may possess an imaginary part (responsible for loss/gain of particles from/into the trap) depending
only on the temporal coordinate. Both conditions for the external potentials are in a harmony
with a recent analysis of P type of the CGPEs [1], although in a restricted sense. Nevertheless, analytical formulae can be given which determine the transformation functions (amplitudes, phases,
and variables) from the given external potentials.
On the example of potentials depending linearly and quadratically on the spatial coordinate
it has been demonstrated how the potentials modify the motion of a two-component BEC system
consisting of 106 sodium atoms composing the bright soliton and of 106 rubidium atoms forming
the dark soliton. The scattering lengths characterizing the interaction between sodium atoms and
rubidium atoms are taken to be a11 = 2.75 nm [195] and a22 = 5.5 nm [65], respectively. To
describe the inter-specimen interaction between sodium and rubidium atoms we have adopted the
value a12 = a21 = 4.9 nm which meets a stability requirement originating from the P test analysis
[see also Eq. (5.2.9)] and the existence requirement of the bright-dark soliton [see Eqs. (5.5.21)
and (5.5.22)].
Finally we remark that the method developed in this work can be extended into various di89
CHAPTER 5. RELATION BETWEEN . . .
5.6. SUMMARY
rections, e.g., to generalize the potentials by performing a P analysis for j > 4, or to establish a
more sophisticated ansatz suitable to other special soliton solutions, or to develop a transformation between more than two-component coupled solutions of CNLSEs and CGPEs. The analytical
formulas as described here may serve also as a useful test of direct numerical solutions of CGPEs.
90
Chapter 6
Stability of static solitonic
excitations of two-component
Bose-Einstein condensates in
finite range of interspecies
scattering length a12
In this chapter a simple method is presented [3] to analyze the stability of static solitonic excitations
of two-component Bose-Einstein condensates described within the Gross-Pitaevskii approximation.
First, a quick assessment can be made for the values of interspecies scattering length a 12 allowing
existence of static solitons independent of the particle numbers N1 and N2 . If the true value of a12
falls into that acceptable interval, then a fine tuning is performed to get the ratio N2 /N1 at which
the static solitons may be created. The technique is illustrated for four two-component systems,
lithium-rubidium, sodium-rubidium, lithium-sodium and potassium-rubidium condensates.
6.1
Introduction
Over the past few years an increasing interest is observed in the field of atomic Bose-Einstein condensates (BECs). The great interest can be explained by the fact that the study of BECs extends
our knowledge in different branches of theoretical and experimental physics, such as in optics,
statistical physics, thermodynamics, atomic collision theory, quantum properties of mesoscopic
systems, etc. (For an overview and references see, e.g., Leggett [61], and Dalfovo [62].)
Mostly one-component BECs have been studied so far (see references [98,115,116,121,194,199,
200], respectively, for the elements 1 H, 7 Li, 23 Na, 41 K, 87 Rb, 85 Rb, 133 Cs), but BECs involving
two hyperfine states, or two isotopes of the same element have also been investigated exhaustively
[63–70]. The fast development of experimental techniques makes it possible in near future to
engineer also genuine two-component condensates that are composed of two different atoms. The
two-component systems Na-Rb [71–73] and K-Rb [74] have been analyzed theoretically and the
mixture Cs-Li [75, 76] has been investigated experimentally without reaching the BEC phase.
91
CHAPTER 6. STABILITY OF STATIC SOLITONIC EXCITATIONS . . .
6.1. INTRODUCTION
A main question of the two-species BECs is how to identify the stability properties of the
mixtures. In engineering BECs the scattering lengths aij are the most important quantities characterizing the strength of interactions between the atoms of types i and j forming the condensate.
The magnitude and sign of these parameters sensitively influence the existence and stability of
the states of this ultracold boson system. For one-species system, therefore, the sign of the atomatom interaction can be treated as a stability signature. As a rule, repulsive interactions (positive
scattering length) maintain stable condensates whereas attractive interactions (negative scattering
length) may lead to implosion of the BEC if the number of atoms involved exceeds a critical number. However, in the case of two-component condensates the sign of the interactions alone does
not determine the stability of the system as it has been shown [201] numerically. Instability may
occur even if all interactions (intraspecies and interspecies) are repulsive.
Of course, the realization of two-component BECs depends on three scattering lengths, namely
the intraspecies scattering lengths a11 , a22 and the interspecies ones a12 = a21 . The accurate
knowledge of these three quantities is of great importance in design and implementation of twocomponent BECs. However, the problem is that the scattering lengths between many like alkali
atoms are known, whereas those between unlike alkalis have not yet been measured or calculated,
except for very few cases.
In this chapter we give a simple treatment of the stability of the two-species condensate mixture.
We shall consider the existence of static solitonic excitations as a stability signature of the twocomponent BECs composed of two different types of alkali atoms. The motivation behind this
proposition relies upon the obvious fact that background equilibrium BEC-density should exist if
solitonic excitations can be triggered by some mechanism within the condensate. Solitons have
already been created, for example, by a phase imprinting technique in one-component BECs [185],
or by tuning the interspecies scattering length [5, 6, 202] via Feshbach resonances. Also, soliton
stabilities and interactions within one- and two-component BEC media have been extensively
studied in recent years in numerous works [203–218]. These investigations reveal the well known
”robustness” of soliton solutions against system perturbations. One therefore intuitively expects
that those regions of a12 supporting soliton solutions of the coupled GP equations, may encompass
the true value itself and, therefore, guide experimentalists when engineering two-component BECs.
For the sake of simplicity we restrict the treatment of the dynamics to one dimension but, by using
a standard technique, we shall employ three-dimensional scattering lengths. Note that ”cigarlike” one-dimensional BECs have already been produced by micro-trap devices developed recently
[119, 177, 219, 220].
To carry out the analysis we start from the static coupled ground state solutions Φ i (x), i = 1, 2
calculated in the standard Thomas-Fermi approximation. Then we treat the static excited states of
bright (B) or dark (D) solitonic type of the mixture in the form ψ̃i (x, t) = Φi (x)φi (x) exp(−iẼi t/~)
and derive the corresponding Gross-Pitaevskii (GP) equations for the soliton wave functions φ i (x)
with appropriate boundary conditions. In order to solve the coupled solitonic equations we first
insert the static one-soliton solutions of B and D type with generalized amplitudes and range
parameters into the coupled GP equations with no trap potentials. We then obtain relations among
the system parameters (masses, interaction strengths, etc.) and the excitation amplitudes, whose
trivial property, i.e., that the modulus of the amplitudes should be non-negative, can be satisfied
by simple inequalities, not involving the particle numbers N1 and N2 . By solving these inequalities
with given intraspecies scattering lengths a11 and a22 , we get ranges for the interspecies scattering
length parameter a12 enabling the existence of stable BD or DB solitons of the two-component
BECs described within the GP approximation. (Note that similar calculation can be carried out
for the DD solitonic excitations.) If the actual value of a12 falls within this range then, by solving
92
CHAPTER 6. STABILITY OF STATIC SOLITONIC EXCITATIONS . . .
6.2. STABILITY ANALYSIS
a system of coupled nonlinear equations expressing particle number conservation, one can finally
determine the ratio N2 /N1 at which static solitons may be created.
Stability analysis of two-component BECs has also been performed without involving solitons
in a number of important papers including references [64,73,99,167,171,197,201,221–226]. In these
investigations predominantly the excitation spectrum of the condensate has been calculated using
the well-known Bogoliubov-Hartree theory and the parameter regions where complex frequencies
arise have been identified with the instability domains. These instability regions are usually exhibited as a function of a12 so that a comparison between our findings and theirs can be made easily,
despite the slightly different concept of the approaches.
The results of the analysis presented in this chapter indicate that almost in all cases of the twocomponent BECs composed from alkali atoms there is a finite range of the interspecies scattering
length a12 which supports static BD or DB soliton solutions of the coupled GP equations. Similar
conclusion might be obtained also from the DD analysis. This means the quite trivial results
found also in all other works dealing with stability problem of two-component BECs that strong
interspecies scattering length leads to instability of the mixture. Moreover, one observes that
there are several pairs of alkali atoms which support BD solitonic excitations but do not favor
the opposite DB formations (or vice versa). We shall also show that both the amplitude and the
spatial extension of the components of the condensates are sensitive functions of the interspecies
scattering length parameter a12 lying within the stability domain. This knowledge may prove
useful in producing genuine two-component BECs.
6.2
Stability analysis
We use the zero temperature mean-field theory for two interacting dilute Bose condensates, so that
the collisions between the condensed atoms and the thermal cloud are neglected. The macroscopic
dynamics of this physical system is governed accurately by two coupled time-dependent cubic
nonlinear Schrödinger equations (NLSs). In the condensed matter context this equation is also
known as the Gross-Pitaevskii (GP) equation [155, 156], where the interaction is well described
by self- and cross-interaction energies depending on the densities of the species and the s-wave
scattering lengths. Restricting ourselves to (1+1) dimension the coupled GP equations can be
written as follows:
i~ψ1,t
i~ψ2,t
~2
2
2
= −
∂xx + Ω11 |ψ1 | + Ω12 |ψ2 | + V1 ψ1
2m1
~2
2
2
= −
∂xx + Ω21 |ψ1 | + Ω22 |ψ2 | + V2 ψ2
2m2
(6.2.1a)
(6.2.1b)
where mi denotes the individual mass of the ith atomic species, Ωij = 2π~2 aij /Aµij with aij
being the 3D scattering length, A represents a general transverse crossing area of the cigar-shape
BEC, µij = mi mj /(mi + mj ) is the reduced mass and Vi , (i = 1, 2), means the external trapping
potentials.
R ∞ In2 the case of real trap potentials the normalization of the wave functions reads as
Ni = −∞|ψi | dx, (i = 1, 2) with Ni denoting the number of atoms in the ith component of the
BEC. Here we do not allow for the species to transform into each other, so the numbers of particles
are conserved quantities for both alkalies.
93
CHAPTER 6. STABILITY OF STATIC SOLITONIC EXCITATIONS . . .
6.2.1
6.2. STABILITY ANALYSIS
Ground state profiles
We look for stationary solutions of the coupled GP equations (6.2.1) in the form
ψi (x, t) = Φi (x) exp(−iEi t/~)
(6.2.2)
where Ei means the one-particle energy of the i−th component particle (i = 1, 2). By inserting
the above ansatz into equations (6.2.1) and neglecting the kinetic terms, we get the solutions in
the Thomas-Fermi (TF) approximation as
Ω22 (E1 − V1 (x)) − Ω12 (E2 − V2 (x))
∆
Ω11 (E2 − V2 (x)) − Ω21 (E1 − V1 (x))
2
|Φ2 (x)| =
∆
2
|Φ1 (x)| =
(6.2.3a)
(6.2.3b)
with ∆ = Ω11 Ω22 − Ω12 Ω21 . By specifying the external potential as usual to be a harmonic one,
Vi (x) =
1
mi ωi2 x2 ,
2
i = 1, 2,
(6.2.4)
the TF ground state solutions can be written in the following more explicit form (i = 1, 2)
2
|Φi (x)| =
(
Ai 2
xi − x 2
if |x| ≤ xi
∆
0
if |x| > xi
(6.2.5)
with the turning points defined by xi = (3∆Ni /4Ai )1/3 and the potential constants given by
A1 = (Ω22 m1 ω12 − Ω12 m2 ω22 )/2 and A2 = (Ω11 m2 ω22 − Ω21 m1 ω12 )/2.
With this notation the TF energies are obtained as
1
E1 = ∆
Ω11 A1 x21 + Ω12 A2 x22 ,
(6.2.6a)
1
E2 = ∆
Ω21 A1 x21 + Ω22 A2 x22 .
(6.2.6b)
It is observed from the formulae for xi that the quantity Ai /∆ must be positive in order that
the TF solution be exist.R As can be checked easily the above ground state solutions fulfill the
x
normalization conditions −xi i |Φi (x)|2 dx = Ni (i = 1, 2).
Figures 6.1(a) and 6.1(b) show the density profiles |Φi |2 derived in the TF approximation
given by equation (6.2.5) as a function of the distance x, for two types of BEC, namely, for the
mixture 7 Li−87 Rb with parameters a11 = −1.4 nm, a22 = 5.5 nm, a12 = 4.5 nm, N1 = 2 × 103 ,
N2 = 104 , and for the system 23 Na−87 Rb with the data a11 = 2.7 nm, a22 = 4.7 nm, a12 = 4.8 nm,
N1 = N2 = 2×104. Confining harmonic potentials are employed with frequencies 2π×530 Hz (7 Li),
2π × 310 Hz (23 Na), and 2π × 150 Hz (87 Rb). Furthermore, figures 6.2 present the Thomas-Fermi
ground state for the case of 7 Li-23 Na and of 41 K-87 Rb systems (for parameters see the captions).
All these parameters are known from earlier works, except for the interspecies scattering lengths
a12 which are chosen to be in the middle of the region given later in tabulated form by using triplet
and singlet intraspecies scattering lengths aii (i = 1, 2) as input. We remark that the profiles shown
in figure 6.1(b) compare well to the results of Pu and Bigelow who performed a 3D analysis of the
23
Na−87 Rb system (cf. figure 1b of reference [72]).
In order to get a more thorough insight into the behavior of the ground state profiles whose
actual measure xi have been indicated by arrows in the upper part of figures 6.1 and 6.2, we also
94
CHAPTER 6. STABILITY OF STATIC SOLITONIC EXCITATIONS . . .
|Φ i |2 [ µm-1]
(a)
x
x
1
4
6.2. STABILITY ANALYSIS
|Φ i |2 [ µm-1]
(b)
2
x
x
1
7
2
6
3
87Rb
4
2
PSfrag replacements
87Rb
2
23Na
[µm]
[µm−1 ]
|Φi (x)|
3
PSfrag replacements
1
7Li
[µm]
5
2 –2
0
–1
x 2 –3
|Φi2(x)|
1
–2
6
5
5
4
4
(d)
[mm]
3
x2
2
PSfrag replacements
x1
[mm]
3x
2
1
x
2
3
x1
2
PSfrag replacements
1
[µm]
[µm−1 ]
0
–1
7
[mm]
6
(c)
2
|Φi (x)| –10
1
−1 ]
[mm]
[µm
–6
–2
0
1
[µm]
2
a12 [nm]
[µm−1 ]
2
6
|Φ10
(x)| –10
i
–6
–2
0
2
a12 [nm]
6
10
Figure 6.1: (a) Absolute squares of the TF wave functions |Φi |2 given by equation (6.2.5) as a function of the
distance x for a two-component BEC consisting of 7 Li atoms (N1 = 2 × 103 ) and 87 Rb atoms (N2 = 104 ) and
confined by oscillator potentials with frequencies ω1 = 2π × 530 Hz and ω2 = 2π × 150 Hz. The corresponding
triplet intraspecies scattering lengths are a11 = −1.4 nm (7 Li) and a22 = 5.5 nm (87 Rb) and the chosen interspecies
scattering length is a12 = 4.5 nm. (b) The same as in (a) for the two-component BEC composed of 23 Na atoms
(N1 = 2 × 104 ) and 87 Rb atoms (N2 = 2 × 104 ) confined by harmonic potentials with frequencies ω1 = 2π × 310 Hz
and ω2 = 2π × 150 Hz. The corresponding singlet intraspecies scattering lengths are a 11 = 2.7 nm (23 Na) and
a22 = 4.7 nm (87 Rb) and the chosen interspecies scattering length is a12 = 4.8 nm. (c) Extension parameters x1 , x2
as a function of the interspecies scattering length a12 for the system 7 Li-87 Rb. Except for a12 , the parameters are
the same as given in the upper left part (a) of figure 6.1. (d) The same as in (c) for the system 23 Na−87 Rb with
parameters given in the upper right part (b) of figure 6.1.
show the extension parameter functions xi (a12 ) in the bottom part (c)-(d) of figures 6.1 and 6.2.
Here we have employed a broader range of the interspecies scattering length a 12 . The functions
xi (a12 ) are obtained by using the definition given after equation (6.2.5) and have been calculated
with the same system parameters, except for a12 , as given above. One observes a very strong
dependence of the xi values on the parameter a12 which may result even in the interchange of the
spatial extension of the components. Also, several characteristic values of the functions x 1 (a12 ) and
x2 (a12 ) can be noticed in figures 6.1(d) and 6.2(d). For example, the extension parameter functions
become zero where ∆ vanishes. This happens if a11 a22 > 0 and the interspecies scattering length
takes the values a012 = ±2(a11 a22 µ12 /(m1 + m2 ))1/2 . Note that these values of a12 separate the
critical domains of BEC beyond which a phase separation occurs as a result of a 3D investigation
performed by Esry and Greene [227]. The extension parameter functions become positive infinite,
xi (a∞
ij ) = ∞, i 6= j = 1, 2, if Ai ’s vanish. This happens at values of interspecies scattering length
95
CHAPTER 6. STABILITY OF STATIC SOLITONIC EXCITATIONS . . .
2
|Φ i |
(a)
[
7
µm-1
6.2. STABILITY ANALYSIS
-
|Φ i |2 [ µm 1]
(b)
16
]
x1
x1
14
6
5
7Li
12
41
K
x2
10
4
8
3
6
2
Sfrag replacements
23Na
[µm]
[µm−1 ]
|Φi (x)|
2 –6
–4
87Rb
PSfrag replacements
1
0
–2
4
2
10
(c)
x[µm]
[mm]
[µm−1 ]
6
|Φi (x)|
x2
4
x
[mm]
2
–1
–2
2
1
0
(d)
2
7
[mm]
[mm]
6
8
5
x1
6
4
x2
3
4
x2
2
Sfrag replacements
[µm]
[µm−1 ]
|Φi (x)|
2 –10
1
[µm]
–6
–2
x1
PSfrag replacements
2
0
2
6
a12 [nm]
[µm−1 ]
10
|Φi (x)|
2 –10
a12 [nm]
–6
–2
0
2
6
10
Figure 6.2: (a) Absolute squares of the TF wave functions |Φi |2 given by equation (6.2.5) as a function of the
distance x for a two-component BEC consisting of 7 Li atoms (N1 = 1.8 × 104 ) and 23 Na atoms (N2 = 104 ) and
confined by oscillator potentials with frequencies ω1 = 2π × 530 Hz and ω2 = 2π × 292 Hz. The corresponding triplet
intraspecies scattering lengths are a11 = −1.4 nm (7 Li) and a22 = 4.0 nm (23 Na) and the chosen interspecies
scattering length is a12 = 3.42 nm. (b) The same as in (a) for the two-component BEC composed of 41 K atoms
(N1 = 4.9 × 104 ) and 87 Rb atoms (N2 = 104 ) confined by harmonic potentials with frequencies ω1 = 2π × 220 Hz
and ω2 = 2π × 150 Hz. The corresponding triplet intraspecies scattering lengths are a 11 = 3.4 nm (41 K) and
a22 = 5.5 nm (87 Rb) and the chosen interspecies scattering length is a12 = 4.85 nm. (c) Extension parameters x1 ,
x2 as a function of the interspecies scattering length a12 for the system 7 Li-23 Na. Except for a12 , the parameters
are the same as given in the upper left part (a) of figure 6.2. (d) The same as in (c) for the system 41 K−87 Rb with
parameters given in the upper right part (b) of figure 6.2.
2
2
a∞
ij = 2ajj (mi ωi ) /(mi + mj )mj ωj . An interesting relation holds between these characteristic
0
∞ ∞ 1/2
values of a12 , namely a12 = ±(a12 a21 ) , which tells us that one of the zero places of xi lies at
the geometric mean of the two divergence points [see figures 6.1(d) and 6.2(d)]. The sensitive
dependence of the spatial extension of the two-component BEC on the interspecies scattering
length a12 has been observed recently also by Ho and Shenoy [71], and Pu and Bigelow [72, 73].
96
CHAPTER 6. STABILITY OF STATIC SOLITONIC EXCITATIONS . . .
6.2.2
6.2. STABILITY ANALYSIS
Static solitonic excitation
Now one looks for the excited static solutions in the form ψ̃i (x, t) = Φi (x)φi (x) exp(−iẼi t/~)
with φi (x) being an excess or defect of the i−th component of the background density, triggered
by an excitation mechanism within a spatial range of Li < xi . Inserting the above ansatz into
the GP equations, equations (6.2.1), and assuming that the excitation mechanism restricts the
change of density only to a small interval x ∈ (−Li , Li ) around x = 0 where the TF solutions
can be approximated by Φi (0), one obtains the following two coupled equations for the perturbing
functions
~2
2
2
˜
E 1 φ1 = −
∂xx + Ω̃11 |φ1 | + Ω̃12 |φ2 | φ1
(6.2.7a)
2m1
~2
2
2
E˜2 φ2 = −
∂xx + Ω̃21 |φ1 | + Ω̃22 |φ2 | φ2
(6.2.7b)
2m2
where the notation Ω̃ij = Ωij Aj x2j /∆ (i, j = 1, 2) has been introduced and the very small potential
terms have been neglected. (Apart from the fact that the neglect of trap potential is an accurate
approximations in this case [208], it does not mean at all a restriction because there exists a
general transformation between coupled solutions of the GP equations with and without trap
potentials [2].)
The coupled equations above determine the perturbing functions φi (x) within the range |x| ≤
Li < xi provided the boundary conditions are specified. If there is no interspecies interaction
(a12 = 0) then the equations formally reduce to the single NLS or GP equations which admit the
well known static soliton solution of B or D type [228]. Since the B and D solitons can be produced
in BECs [6, 185] by using phase imprinting optical technique and magnetic-field-tuned Feshbach
resonance method, we shall try to use the static uncoupled soliton solutions
φB1 (x)
= q1 sech (k1 x),
φD2 (x)
= q2 tanh (k2 x)
(6.2.8a)
(6.2.8b)
L−1
i
with generic complex amplitudes qi and real range parameters ki ∼
in the description of the
excitation of the two-component BECs. In accordance with the soliton characters we impose the
appropriate boundary conditions φB1 (x → ±∞) = 0, and φD2 (x → ±∞) = ±q2 .
The insertion of the above solitonic ansatz into equations (6.2.7) yields the condition
k1 = k 2 ≡ k
(6.2.9)
for the range parameters, and the relations
|q1 |
|q2 |
2
2
=
~2 k 2
∆
Ω̃12
Ω̃22
−
m2
m1
!
=
~2 k 2
∆
Ω̃11
Ω̃21
−
m2
m1
!
,
(6.2.10a)
(6.2.10b)
for the modulus of the amplitudes. The requirement that the modulus of the two amplitudes q1
and q2 be positive real numbers gives the stability conditions (Aij = aij (1 + mi /mj ))
fB1 (a12 ) =
fD2 (a12 ) =
A12 − A22
≥ 0,
det(A)
A11 − A21
≥0
det(A)
97
(6.2.11a)
(6.2.11b)
CHAPTER 6. STABILITY OF STATIC SOLITONIC EXCITATIONS . . .
6.2. STABILITY ANALYSIS
for the existence of BD solitonic excitation within the two-component BEC. (Note that for DD
solitonic excitation the stability condition is obtained when fB1 (a12 ) is replaced by −fD1 (a12 ).)
Although the above conditions do not involve the particle numbers Ni , the latter introduces a
constraint by the particle number conservation because the normalization for the bright and dark
solitonic excitation reads as
N1
N2
A1
n(kx1 ),
∆ 2 A2 2
= 2 |q2 |
(kx2 )3 − n(kx2 ) ,
∆ 3
= 2 |q1 |2
(6.2.12a)
(6.2.12b)
with
n(w) = −w2 + 2w ln(1 + exp (2w)) + dilog(1 + exp (2w)) + π 2 /12.
(6.2.12c)
Table 6.1: Intervals of interspecies scattering lengths a12 enabling static solitonic excitations of two component
BECs composed of alkalis indicated. The column at gives the triplet intraspecies scattering length aii taken from
the references: 1 H [229] , 7 Li [230–234], 23 Na [115,235,236], 39 K, 41 K [120,199], 83 Rb, 85 Rb, 87 Rb [237,238], 133 Cs,
135 Cs [239–241]. All values are in unit of nm.
Element
Dark-component
1
H
0.1
Li
−1.4
Na
4.0
7
23
at
39
K
41
−0.9
1H
Bright-component
41 K
83 Rb
85 Rb
7 Li
23 Na
39 K
-2.4;0.0
0.0;0.2
-1.8;0.0
0.0;0.1
-37.6;0.0 0.0;0.2
2.8;3.5
1.7;3.4
-30;1.7
-1.5;-0.8
0.2;7.7
-0.6;6.1
-1.1;3.0
-0.4;-0.1
K
3.4
0.1;6.6
-0.4;5.8
3.5;4.4
-0.9;3.5
4.2
0.1;8.3
-0.2;7.7
3.4;6.6
-0.6;5.7
85
Rb
−19.0
-2.7;-0.2
-3.8;-0.6
87
Rb
5.5
0.2;10.9
-0.2;10.2 3.8;8.7
-0.6;7.6
4.1;7.5
0.0;4.8
-0.1;4.6
-0.4;3.7
2.4;3.7
0.1;14.3
-0.1;13.7 3.8;12.3
(2.4)
135
Cs
7.2
135 Cs
0.0;0.1
1.7;3.8
1.2;2.2
1.2;3.8
-26;-3.8
Rb
Cs
133 Cs
0.0;0.1
-35.1;-2.7
83
133
87 Rb
0.0;0.2
2.2;4.1
2.3;3.6
-25.4;2.2 2.2;4.0
1.6;2.4
1.6;4.2
-19.2;4.1 4.1;4.8
3.1;3.2
3.2;5.3
4.8;5.6
-18.8;5.6
3.6;4.4
4.3;6.1
3.0;3.1
-14.8;2.9 2.9;3.6
5.3;8.9
-14.7;8.9 6.1;8.8
3.6;5.6
-0.4;11.2 4.2;11.1
2.4;4.2
4.2;7.3
Assuming now that the system parameters m1 , m2 , a11 and a22 are given, and not regarding the
particle numbers N1 and N2 , one can determine the broadest range of the interspecies scattering
length a12 for which the existence of BD or DB solitons in the two-component BEC might be
expected. This task is accomplished by solving the inequalities (6.2.11). Moreover, if the actual
value of a12 determined by some experimental or theoretical method turns out to fall within the
range just determined, then one may solve the coupled equations (6.2.12) to fine tune the particle
number ratio N2 /N1 (and the size parameter k) in order to get more information about the twocomponent system which may exhibit static solitonic features.
In the following we shall determine those common regions of values of a12 for which the modulus
of both amplitudes qi (i = 1, 2) takes positive values. In the next section the regions of a12 obtained
for simultaneous fulfillment of the stability conditions 6.2.11. will be given and analyzed further
in tabulated form, together with two selected examples for different alkali systems of bosonic type,
for which also the fine tuning of the ratio N2 /N1 will be performed.
98
CHAPTER 6. STABILITY OF STATIC SOLITONIC EXCITATIONS . . .
6.2. STABILITY ANALYSIS
Table 6.2: Intervals of interspecies scattering lengths a12 enabling static solitonic excitations of two component
BECs composed of alkali pairs indicated. The column as gives the singlet scattering length aii of atoms taken from
the references: 7 Li [230,233,234,242], 23 Na [235,236], 39 K, 41 K [199], 83 Rb, 85 Rb, 87 Rb [238], 133 Cs, 135 Cs [239,240].
(For 1 H the triplet value has been used [229]). All values are in unit of nm.
Element
Dark-component
1
1H
Bright-component
41 K
83 Rb
85 Rb
7 Li
23 Na
39 K
-0.0;0.2
0.0;0.2
0.0;0.2
0.0;0.2
0.0;0.1
0.8;1.8
0.5;2.5
0.5;1.6
0.3;1.3
2.0;4.3
1.9;3.3
5.7;7.0
H
(0.1)
Li
1.7
0.2;3.0
Na
2.7
0.2;5.3
1.8;4.2
7
23
as
87 Rb
133 Cs
135 Cs
0.0;0.6
0.0;0.1
0.0;0.1
0.0;0.2
0.3;7.7
0.3;1.5
0.2;0.9
0.2;2.9
1.2;2.5
1.2;15.1
1.1;2.9
0.8;1.8
0.8;6.0
4.6;4.7
4.6;28.0
4.5;5.4
3.3;3.5
3.3;11.5
3.0;3.7
2.9;22.1
2.9;4.3
2.1;2.8
2.0;9.0
3.4;20.7
3.3;4.0
2.6;2.8
2.6;9.2
24.2;123 16.9;97
55.5;96
39
K
7.3
0.2;14.2
2.5;12.4
4.3;9.2
41
K
4.4
0.2;8.6
1.2;5.4
3.3;5.7
4.6;5.7
83
Rb
3.4
0.1;6.8
1.3;6.3
2.5;5.4
4.6;4.7
85
Rb
124.8
0.6;246
7.7;230
15.1;196 28.0;171 22.1;167 20.7;126
87
Rb
4.7
0.1;9.2
1.5;8.7
2.9;7.4
5.4;6.5
4.3;6.3
4.0;4.8
4.7;24.2
2.8;3.7
2.8;3.0
3.0;16.9
3.7;4.6
133
Cs
(2.4)
0.1;4.8
0.9;4.6
1.8;4.0
3.5;3.7
135
Cs
26.0
0.2;51.6
2.9;49.5
6.0;44.4
11.5;40.3 12.1;39.9 9.1;32.3
6.2.3
3.3;3.8
2.9;3.3
3.7;10.8
2.7;7.9
32.0;55.5 10.8;31.6 7.9;26.2
Static solitonic excitation profiles
In order to illustrate the sensitivity of the solitonic excitations to the system parameters we show
in figure 6.3 the density profiles |ψ̃i (x)|2 of the two BEC systems composed of lithium-rubidium
and sodium-rubidium atoms. The first is chosen to be in the triplet scattering state and the
second example is taken to form a singlet spin state. The amplitudes |qi |2 are calculated now from
equations (6.2.12) with the same parameters treated in connection with the ground state BECs
shown in figure 6.1, except for the particle numbers for which only the ratio N2 /N1 can be specified
as a function of a12 . We have kept N2 fixed, and therefore the dark soliton particle number is
N2 = 104 for the 7 Li-87 Rb triplet system and N2 = 2 × 104 for the 23 Na-87 Rb singlet case.
Each of the three possible excited states shown in figure 6.3 differs in the values of the interspecies scattering length parameter a12 (chosen from the allowable range determined by equations
(6.2.11)) and of the ratio N2 /N1 . Specifically, the values a12 = 3.6, 4.5, 5.0 nm and N2 /N1 = 1, 5,
11 have been used for the 7 Li-87 Rb triplet case in figures 6.3(a1)-(a3), whereas the values a12 = 4.3,
4.8, 5.5 nm and N2 /N1 = 0.6, 1, 3.6 have been adopted for the 23 Na-87 Rb singlet configuration
in figures 6.3(b1)-(b3). The excitation of the BEC is manifested by the appearance of the B and
D solitons on the background density. As a rule we can observe a strong dependence of both the
extensions xi and magnitudes |qi |2 (i = 1, 2) in all the six examples shown. Thus, the knowledge of the physical value of a12 is of utmost importance from point of view of production of
two-component BECs with stable BD or DB excitations.
Figures 6.3(a4) and (b4) contain more information about the dependence of the corresponding
ratio N2 /N1 on the values of a12 . We observe here a rather smooth behavior in the allowable intervals a12 = −0.2; 10.2 for the 7 Li-87 Rb system and a12 = 2.9; 7.4 for the 23 Na-87 Rb configuration
(as determined by equations (6.2.11)). In figure 6.4 the same analysis can be seen for the systems
7
Li-23 Na and 41 K-87 Rb with parameters given in the captions.
99
CHAPTER 6. STABILITY OF STATIC SOLITONIC EXCITATIONS . . .
(a1)
6.2. STABILITY ANALYSIS
(b1)
6
14
5
12
4
10
3
8
6
2
4
1
2
[mm]
–2
–1
0
x
2
1
–3
–2
0
–1
2
1
3
[mm]
x
(b2)
(a2)
6
5
5
4
4
3
3
2
2
1
–2
1
0
–1
1
[mm]
2 x
–2
0
–1
2
1
[mm]
x
(b3)
(a3)
5
4
4
3
3
2
2
1
1
–2
0
–1
1
8
[mm]
2 x
(a4)
N
ln __2
N1
6
4
0
–1
2
1
8
[mm]
x
(b4)
N2
ln ___
N1
6
4
2
0
–2
2
2
4
6
a12 [nm]
8
0
–2
–2
–4
–4
4
5
6
a12 [nm]
8
7
Figure 6.3: Absolute squares of wave functions, |ψ˜1 |2 and |ψ˜2 |2 in unit [µm−1 ], of the BD solution of the coupled
Gross-Pitaevskii equations (6.2.1) as a function of the position x for different BEC systems: (a) triplet 7 Li-87 Rb
system with N2 = 104 , (b) singlet 23 Na-87 Rb components with N2 = 2 × 104 . The difference in magnitude and
extension of the corresponding profiles resides in the interspecies scattering lengths and particle number N 1 used
as follows: (a1) a12 = 3.6 nm, N1 ≈ 104 , (a2) a12 = 4.5 nm, N1 ≈ 2 × 103 , (a3) a12 = 5.0 nm, N1 ≈ 103 , (b1)
a12 = 4.3 nm, N1 ≈ 3.3 × 104 , (b2) a12 = 4.8 nm, N1 ≈ 2 × 104 , (b3) a12 = 5.5 nm, N1 ≈ 5.5 × 103 . Other essential
data are the same as given in figure 6.1. Figs. (a4) and (b4) show the dependence of the logarithmic ratio ln (N 2 /N1 )
on the interspecies scattering length a12 for these multicomponent BECs.
100
CHAPTER 6. STABILITY OF STATIC SOLITONIC EXCITATIONS . . .
6.2. STABILITY ANALYSIS
(b1)
(a1)
20
14
12
41K
7
Li
10
8
15
10
6
4
–6
23
Na
2
–2
0
–4
87
Rb
5
[mm]
[mm]
2
–2
6x
4
(a2)
–1
0
x
2
1
(b2)
10
5
8
4
6
3
4
2
2
1
[mm]
–3
–2
0
–1
3 x
2
1
(a3)
[mm]
–2
0
–1
(b3)
2
1
x
6
5
5
4
4
3
3
2
2
1
1
[mm]
–2
0
–1
2
1
8
6
x
(a4)
N
ln __2
N1
[mm]
–2
6
4
2
2
a12 [nm]
2
3
4
5
6
2 x
1
8
4
0 1
0
–1
0
–2
–2
–4
–4
(b2)
N
ln __2
N1
a12 [nm]
3
4
5
6
7
8
Figure 6.4: Absolute squares of wave functions, |ψ˜1 |2 and |ψ˜2 |2 in unit [µm−1 ], of the BD solution of the coupled
Gross-Pitaevskii equations (6.2.1-b) as a function of the position x for different BEC systems: (a) triplet 7 Li-23 Na
system with N2 = 104 , (b) triplet 41 K-87 Rb components with N2 = 104 . The difference in magnitude and extension
of the corresponding profiles resides in the interspecies scattering lengths and particle number N 1 used as follows:
(a1) a12 = 2.90 nm, N1 ≈ 4.7 × 104 , (a2) a12 = 3.74 nm, N1 ≈ 9.8 × 103 , (a3) a12 = 4.21 nm, N1 ≈ 3.8 × 103 , (b1)
a12 = 4.85 nm, N1 ≈ 4.9 × 104 , (b2) a12 = 5.69 nm, N1 ≈ 1.6 × 104 , (b3) a12 = 6.54 nm, N1 ≈ 1.6 × 103 . Other
essential data are the same as given in figure 6.2. Figs. (a4) and (b4) show the dependence of the logarithmic ratio
ln(N2 /N1 ) on the interspecies scattering length a12 for these multicomponent BECs.
101
CHAPTER 6. STABILITY OF STATIC SOLITONIC EXCITATIONS . . .
6.3
6.3.1
6.3. ILLUSTRATIVE RESULTS
Illustrative Results
a12 values allowing stability and solitonic excitations of BECs
Tables 6.1 and 6.2 give, separated from each other by semicolons, the limiting point values of the
intervals of the interspecies scattering length a12 satisfying our stability conditions encoded into
inequalities (6.2.11-b). These conditions express the possibility of developing solitonic excitations
of static BD or DB type from the ground state of the two-component BECs, irrespective of the
particle numbers N1 and N2 of the mixture. The results shown in Tabs. 6.1 and 6.2 have been
calculated, respectively, with triplet and singlet intraspecies scattering lengths a ii (i = 1, 2) listed
133
in the columns denoted by the labels
Cs atom where only the
p at and as . The exception is the
average scattering length (aii = + σexp /8π) has been available (therefore the parenthesis).
In general we observe in Tabs. 6.1 and 6.2 that one of the limiting points of the intervals
of the corresponding DB and BD cases coincides; a fact, easily derivable by using the relation
0 = fB1 (a12 ) = fD2 (a12 ). The diagonal elements of Tabs. 6.1 and 6.2 represent one-component
BECs which is out of scope of the present interest so that the data are missing there. However the
rows corresponding to 7 Li, 39 K, and 85 Rb of Tab. 6.1 do not contain any results as well, if their
partner owns a positive triplet intraspecies scattering length. The lack of information means here
that we do not find any region of triplet interspecies scattering length a12 which gives rise to a
stable DB type excitation, with those atoms being the D component. On the other hand all items
of the column headed by 7 Li, 39 K, and 85 Rb are filled by data which means that there are definite
ranges of triplet a12 which support BECs with solitonic excitations of BD type in which the 7 Li,
39
K, and 85 Rb atoms play the role of the B component. This result can easily be understood, for
those atoms possess a negative intraspecies scattering length a11 , corresponding to an attractive
interaction which favors the formation of B-soliton component. (The above argumentation can be
confined by using stability conditions (6.2.11-b) with a11 < 0 and a22 > 0.)
In the case when both a11 and a22 are positive we observe the existence of BECs with both DB
or BD excitations within a finite interval of positive interspecies scattering length a 12 . This can
be understood physically in the following way. In the context of the mean field theory, an effective
potential is created by the i-th atoms acting repulsively among each other (positive scattering
lengths) and thus forming the cavity of the D component. Inside this cavity there is a possibility
for the j-th atoms to form a bunch representing the B component of the BECs.
6.3.2
a12 from earlier studies
There are very few interspecies scattering length a12 data in the literature. The cases studied so
far include the pairs composed of different isotopes of rubidium atoms [238], of those of potassium
atoms [199], of potassium and rubidium atoms [74], of cesium and rubidium atoms [243], and of
lithium and cesium atoms [75]. In the latter case the sign of a12 remains undetermined since we
can derive, by using the relation σel = 4πa212 , only the absolute value of the interspecies scattering
length from the measured unpolarized elastic cross-section data σel (7 Li −133 Cs) ≈ 5 · 10−12 cm2
to be a12 (7 Li −133 Cs) = ±6.3 nm. This value of a12 is to be compared to our ranges of a12
characterized by the limiting points −0.1; 4.6 and 0.9; 4.6 nm obtained, respectively, with triplet
and singlet intraspecies parameters a11 = −1.4 and 1.7 nm for the lithium atoms, and a22 = 2.4
nm for the cesium atom (where the latter value also represents unpolarized data [239] with known
positive sign). From the comparison we predict the plus sign for the unpolarized interspecies
scattering length, i.e., a12 (7 Li −133 Cs) = +6.3 nm, provided the stability can be attained.
102
CHAPTER 6. STABILITY OF STATIC SOLITONIC EXCITATIONS . . .
6.3. ILLUSTRATIVE RESULTS
For the potassium-rubidium system we compare our ranges a12 (39 K−87 Rb) = −0.6; 7.6 nm and
a12 (41 K −87 Rb) = 4.1; 7.5 nm obtained using triplet intraspecies data (i.e. with a11 (39 K) = −0.9
nm, a11 (41 K) = 3.4 nm, and a22 (87 Rb) = 5.5 nm) to the triplet interspecies values a12 (39 K −87
Rb) = 1.6 nm and a12 (41 K −87 Rb) = 8.5 nm given in reference [74]. We see here that our findings
are comparable with the calculated values of a12 , though the upper limiting point in the 41 K−87 Rb
case is ten percent smaller than that given in the literature.
For the cesium-rubidium system a recent work [243] lists various a12 values calculated using
six different ab-initio potential for both triplet and singlet collisions. The calculated data lying
between a12 = 2; 19 nm for the singlet and between a12 = −8.5; 3.2 nm for the triplet 133 Cs-85 Rb
system compare well to ours given by the BD ranges of a12 (133 Cs −85 Rb) = 16.9; 97 nm and
a12 (85 Rb −133 Cs) = 3.0; 16.9 nm for the singlet and a12 (85 Rb −133 Cs) = −14.8; 2.9 nm for the
triplet arrangement, respectively. In the case of the 87 Rb-133 Cs DB system the singlet scattering
lengths are comparable; for example, the calculated singlet value of a12 = 3.0 nm of the ab-initio
result VI [243] compares well to our range of a12 (87 Rb −133 Cs) = 2.9; 3.3 nm.
Finally, we mention the singlet interspecies scattering length for different isotopes of the potassium atoms, namely the value a12 (39 K −41 Kb) = 5.9 nm [199] to be compared to our range
a12 (39 K −41 K) = 5.7; 7.0 nm of the DB case. Again, a clear compatibility is observed.
We may thus conclude that the solitonic excitation method proposed in section 6.2.3 is capable
to assess approximate intervals of interspecies scattering length a12 which may encompass the
actual values themselves.
103
CHAPTER 6. STABILITY OF STATIC SOLITONIC EXCITATIONS . . .
6.3.3
6.3. ILLUSTRATIVE RESULTS
Comparison to results derived from Painlevé test
In reference [1] the Painlevé (P) test of the coupled Gross-Pitaevskii equations has been carried out.
This test provides an efficient method for analyzing the singularity structure of a nonlinear partial
differential equation (see e.g., reference [149]). It has been shown in reference [1] that the GP
equations pass the P test provided the following special condition among the system parameters
(masses, interaction strengths) is satisfied
M≡
i 2Ω Ω − Ω Ω m /m − Ω Ω m /m
1 h
11 22
11 12 1
2
21 22 2
1
2
(2m + 1) + 7 =
,
16
∆
(6.3.13)
where m can be treated as an arbitrary positive integer number (m = 0, 1, 2, . . .). By recalling
the definitions Ωij = 2π~2 aij /Aµij and ∆ = det(Ω), it is clear that this expression depends only
on the ratios m1 /m2 , a11 /a12 , and a12 /a22 involving the characteristic parameters of the GP
equations. Furthermore, in reference [1] it was shown that the P test of the GP equations depends
on the external potential as well. It is sufficient to mention here only the fact, that the different
potential families belong to the different values of m, so that m acts as a classification number.
Experimentally preferred spatially harmonic (∼ x2 ) potential-families fall into the category m = 2.
Using the notation Aij = aij (1 + mi /mj ) and the functions fB1 and fD2 introduced in connection with inequalities (6.2.11), the conditions given by equation (6.3.13) can be written in the
following remarkable form
M = A22 fD2 (a12 ) − A11 fB1 (a12 ).
(6.3.14)
Formula (6.3.14) connects the stability conditions (6.2.11) with the results of the P test encoded
into equation (6.3.13).
7
(b)
(a)
0
7
(c)
6
5
5
–5
7
6
6
5 a12 1
5
m=4
m=4
m=4
m=3
m=3
m=3
m=2
m=2
m=2
m=1
m=1
m=1
[nm]
–2
15
0
2 a12 4
6
[nm]
8
–3
–2
–1 a12 0
1
[nm]
2
Figure 6.5: Classification number m as a function of interspecies scattering length a 12 [see equation (6.3.13)].
Different potential families belong to every positive integer m indicated by the horizontal lines. We plot the three
typical shapes of the function m(a12 ), for (a) 23 Na– 87 Rb system where both of the (singlet) interspecies scattering
lengths are positive, (b) 23 Na– 7 Li system where the corresponding (singlet) aii ’s have opposite signs, (c) 7 Li– 39 K
system where both of the (triplet) aii ’s are negative.
In figure 6.5 we exhibit the classification number m as a function of a12 for three cases discriminated by the signs of a11 and a22 . The horizontal lines labelled by m = integer indicate the different
trap potential families in accordance with the P test of the GP equations. We see in figure 6.5
that the function m(a12 ) has two branches when a11 a22 > 0, and both lines go to infinity at values
√
a12 = ±2 m1 m2 a11 a22 /(m1 + m2 ). These values of a12 agree exactly with those a012 at which
104
CHAPTER 6. STABILITY OF STATIC SOLITONIC EXCITATIONS . . .
6.3. ILLUSTRATIVE RESULTS
the spatial extension of BEC vanishes, i.e. xi (a012 ) = 0, and we remarked that these same critical
values of a12 was found also in reference [227] by performing a 3D stability analysis of the twocomponent BEC systems. On the other hand if the intraspecies scattering lengths have different
signs, a11 a22 < 0, the function m(a12 ) remains finite. This is important, because it means that the
P-test of the coupled Gross-Pitaevskii equations passes only for some restricted potential families.
In figure 6.5(b) we show this case for the 23 Na– 7 Li system with singlet intraspecies parameters.
Intersections can be found only for m = 2, which is the aforementioned (∼ x2 ) potential-family
(m = 1 gives no restriction on the potential).
Table 6.3: Values of interspecies scattering length a12 solving the P test condition, equation (6.3.13), at classification
numbers m = 2 preferring harmonic oscillator trapping potentials. Upper diagonal contains results obtained with
singlet intraspecies scattering lengths as listed in the second column, lower diagonal exhibits those with triplet aii ’s
given in the third column as at . All values are in unit of nm.
Element
1
at
1H
7 Li
H
(0.1)
0.1
Li
1.7
−1.4
Na
2.75
4.0
3.8
2.9
7.28
−0.9
-0.9
-0.9
7
23
as
39
K
41
-1.3
-1.2
23 Na
39 K
41 K
83 Rb
85 Rb
135 Cs
2.6
7.1
4.3
3.4
123.3
4.6
2.4
25.8
6.4
2.9
3.3
115.6
4.5
2.3
24.8
3.8
3.3
196.7
4.3
2.5
22.6
5.8
4.7
87.9
5.5
3.5
21.8
5.6
0.9
K
3.12
3.4
3.3
2.8
3.6
1.3
3.43
4.2
4.2
3.8
4.1
2.6
3.9
85
Rb
124.8
−19.0
-18.8
-17.6
-14.1
-13.3
-11.6
-7.5
87
Rb
4.68
5.5
5.4
5.0
5.2
3.5
4.8
4.8
-6.6
3.8
85.0
4.6
2.6
20.6
64.8
4.1
2.8
17.4
64.0
50.3
64.3
3.3
17.6
133
Cs
(2.4)
(2.4)
2.4
2.2
2.6
1.6
2.6
3.1
-6.0
3.6
135
Cs
26.0
7.2
0.7
6.8
6.7
5.4
6.3
6.1
-2.9
6.5
6.3.4
133 Cs
2.5
Rb
83
87 Rb
4.8
4.8
Summary of the static solitonic excitation method
The assessment of the interspecies interaction parameter a12 at which two-component atomic BEC
exhibits stable configuration is an important task, and can be carried out with the present method
in the following way.
In principle, all the ground state parameters including a12 can be extracted from the TF equilibrium density profiles. This method is rather ambiguous and, therefore, restricting the number of
uncertain parameters or their ranges may prove useful in design of two-component BECs. Moreover,
in this fitting procedure the particle numbers are not coupled, they can be chosen independently
of each other. The coupling is represented mostly by the inter- and intraspecies scattering length
parameters aij (i, j = 1, 2) being strongly correlated with each other.
If one is interested in producing excitations of BEC in a form of static solitons then one first
determines a region of interspecies scattering length a12 which may support the soliton formation.
This task can be performed by simultaneous solution of the inequalities (6.2.11-b) for a 12 , provided
the interaction parameters a11 and a22 are given. This step is again free of the particle number
parameters N1 and N2 and gives the broadest region of a12 for which DB or BD static solitons
ever can be created inside the stable two-component BEC. For the two examples treated explicitly,
table 6.1 contains such intervals between a12 = −0.6 : 6.1 nm for the 7 Li-23 Na triplet system, and
a12 = 4.1; 7.5 nm in the case of the 41 K-87 Rb triplet scattering.
If the actual value of intraspecies scattering length a12 which must be known from other source,
is within the interval determined above, then one may perform a fine tuning for the particle number
105
CHAPTER 6. STABILITY OF STATIC SOLITONIC EXCITATIONS . . .
6.4. CONCLUSIONS
ratio N2 /N1 at which the solitonic excitation may be implemented. This fine tuning is accomplished
by solving the coupled nonlinear equations (11) of Reference [3] for the ratio N 2 /N1 and the size
parameter k of the solitons. For the two explicit examples chosen, figures 6.3 and 6.4 contain the
results with N2 kept fixed.
Bright-dark soliton combinations in one- and two-component condensates have already been
considered in a variety of papers, including [206, 208, 214, 244]. Experimental creation of both
bright [5] and dark [6] solitons have been observed in case of one-component condensates. We hope
that fine tuning the system parameters (scattering lengths, particle numbers, trapping frequencies)
as outlined above may also help in realizing coupled solitonic excitations in two-component BoseEinstein condensates.
6.4
Conclusions
In this chapter a simple procedure has been suggested to investigate static solitonic excitations
for use in engineering two-component BECs. This technique provides results representing an alternative to those given by other stability investigations. The method is based on stable solitonic
excitation of BD or DB type of the condensate. The analysis can be extended into various directions, e.g. to include DD static solitons or other types of dynamical solitons but in this latter case
the treatment may loose its simplicity.
We have found that the interspecies scattering length values a12 is a sensitive parameter of the
existence of two component BEC. Its value heavily influences the extensions x i and the amplitudes
|qi |2 of the excited formation. We have recovered some earlier stability limits as simple existence
condition for the TF solutions [Ai /∆ ≥ 0, see after equation (6.2.5)]. On the other hand, by
imposing existence condition of the solitonic excitation [see inequalities (6.2.11)], we have further
restricted the possible range of interspecies scattering length domain. To get condition for the
solitonic formation at a specific values of a12 belonging to this range, one has to solve the system
of coupled nonlinear equations (6.2.12) which specifies the ratio N2 /N1 of the, as yet unrestricted,
particle numbers.
Moreover, a relation between the Painlevé-test for integrability and the stability of solitonic
excitation has been derived [see equation (6.3.14)] which joins a general procedure known from
the theory of singular value analysis of nonlinear partial differential equations with the well known
formalism of the description of two-component BEC using the coupled GP equations. Especially
the values of a12 obtained at m = 2 corresponding to a harmonic trap potential well fit well into the
ranges of a12 obtained by the present stability analysis. In this context the interrelation between
the case m → ∞ and the instability points a012 deserves a further study. Also, the a12 values
derivable for the case m = 1 can be used as an assessment for practical orientation.
The present day experimental technique enables us to vary the interactions among the atoms
within the trap in such a way to adjust the environment favorable for the condensation. Therefore
we believe that the intervals calculable by using equations (6.2.11) and (6.2.12) for singlet and
triplet interspecies scattering lengths may be reached by experiments in order to attain stable
two-component Bose-Einstein condensates which may exhibit static solitonic features.
106
Appendices
107
Appendix A
Theory of
Bose-Einstein condensation
Consider a system of N non-interacting spinless, free, non-relativistic bosons1 with mass m in a
three-dimensional box of volume V . The energy of the system can be labelled by a set of occupation
numbers {nk }
~2 X 2
k nk
E ({nk }) =
2m
k
where nk takes the values of non-negative integers (0, 1, . . . ). It is clear from this formula that E is
totally independent of the occupation number n0 . This is the decisive difference to fermions where
n0 may be zero or one, and nothing else. In the case of bosons n0 may be a macroscopic number,
so that the system can lower its energy if more and more particles condense into this state, because
this is thermodynamically favourable. Therefore, in extreme case n0 is equal to the total particle
number N , subsequently E = 0 from which we cannot derive any thermodynamic functions. This
result forces us to treat the ground-state, and occupation of the ground-state detached from the
summand. For a mathematically strict and simple derivation of thermodynamical properties we
suggest reading [96].
We are interested in the thermodynamic properties of the system at temperature T . The mean
value of occupation number of a given quantum state is
hn(k)i =
1
eβ(k −µ)
−1
where k is the energy of the level identified with set of quantum numbers k, β = 1/kB T , kB is
the Boltzmann constant and µ is the chemical potential.
Studying permanent bosons we must take into account the constraint on the particle number:
X
N=
hn(k)i
k
In the case of bosons, µ cannot be greater than the lowest single-particle energy level 0 which is
taken hereafter equal to zero for the sake of simplicity (this choice does not influence the results).
1 The expression ”spinless boson” might seem to be a contradiction, but it means here that Bose-Einstein statistics
will only be applied in the description. In other area of formulation this degree of freedom of particles will not be
taken into account.
109
APPENDIX A. THEORY OF
BOSE-EINSTEIN CONDENSATION
A.1. HOMOGENOUS INFINITE IDEAL BOSE-GAS
On the other hand, this condition for µ is sufficient in order that the mean value of the occupation
number, hn(k)i be positive.
A.1
Homogenous infinite ideal Bose-gas
The energy-levels of single-particle states in a sufficiently large quantum mechanical system are
very close to each other which implies that the particle-density, n, can be introduced and the
summation over the states can be approximated by an integral. The average value of number of
particles fallen into the range and + d can be expressed as
dn = hn(k)i
dΓ
h3
(A.1.1)
where dΓ and h3 represent the infinitesimal volume of phase-space and phase-cell, respectively.
Z
Z
1
dΓ
(A.1.2)
N = dn = 3
β(
−µ) − 1
k
h
e
Let us consider the case of free ideal Bose-gas in three dimension. The single-particle energy
( = p2 /2m) is independent of space coordinates (q), so that the integration with respect to q can
be easily performed.
N=
V
h3
Z
d3 p
eβ((p)−µ)
(A.1.3)
−1
Let us change the coordinate parametrization to polar coordinate system in momentum space, so
that d3p = p2 sin(ϑ) dϕdϑdr.
Z
4πV
p2 dp
N= 3
(A.1.4)
h
eβ(p2 /2m−µ) − 1
In order to simplify the integrand let us transform the independent variable from p to the energy 2πV (2m)3/2
N=
h3
Z∞
0
√
d
eβ(−µ) − 1
(A.1.5)
In the low-temperature limit the parameter β = (kB T )−1 diverges. Taken into account this fact
1/β may be used as a small parameter in the Taylor-series expansion
1
eβ(k −µ) − 1
=
∞
∞
k=0
k=1
X
X
e−β(−µ)
x
=
xk =
xk
=x
−β(−µ)
1−x
1−e
(if |x| < 1)
(A.1.6)
where the denotation x is used instead of e−β(−µ) . Thus,
∞
X
e−β eβµ
=
e−nβ σ n
1 − e−β eβµ n=1
where σ = eβµ is the so called fugacity or ”degeneracy parameter”.
110
(A.1.7)
APPENDIX A. THEORY OF
BOSE-EINSTEIN CONDENSATION
2π(2m)3/2 V
N=
h3
Z∞
0
Z∞
1/2
A.1. HOMOGENOUS INFINITE IDEAL BOSE-GAS
∞
X
e
n=1
1/2 e−nβ d =
0
N=
∞
Z
∞
2π(2m)3/2 V X n
σ d =
σ
1/2 e−nβ d
h3
n=1
−nβ n
∞
2π(2m)3/2 V X n
σ
h3
n=1
N=
(A.1.8)
0
1
nβ
3/2 Z∞
v 1/2 e−v dv =
0
kB T
n
√ 3/2
π
1
2
nβ
3/2 √
∞
π
(2πmkB T )3/2 V X σ n
=
2
h3
n3/2
n=1
∞
(2πmkB T )3/2 V X nµ/kB T −3/2
e
n
h3
n=1
(A.1.9)
(A.1.10)
(A.1.11)
It can be seen clearly that if the temperature is decreased then the value of summation should grow
in order that the total number of particles remains constant. But there was also a constraint on the
chemical potential µ when Bose-Einstein quantum statistic was derived, namely µ must be smaller
than 0 , the energy of the lowest lying eigenstate of the system. Therefore the summation can not
increase infinitely and there should exist an upper limit which we can easily get with substitution
µ = 0 to be
∞
∞
X
X
3
nµ/kB T −3/2
−3/2
e
n
≤
n
=ζ
≈ 2.612375
(A.1.12)
2
n=1
n=1
P∞
where ζ is the famous Riemann-zeta function defined as ζ(z) = k=1 1/k z . This implies also that
formulae obtained above are only valid if temperature is greater than a critical temperature T c .
This value is fixed if we reorder the equation (A.1.11) and insert µ = 0 into it:
h2
TC =
2πmkB
ζ
N
3
2
V
!2/3
(A.1.13)
What will happen if we would be able to cool the particles in the gas below TC ? It is self-evident
that we must modify something on the description used above. This alteration had been made
firstly by Einstein in 1924, when he pointed out that a finite fraction of particles, N −Nmax , secedes
from the gaseous phase and occupy the state with the lowest eigenenergy, = 0. The particles
possess either zero energy or zero momentum. Similar process happened when water in liquid
phase condensed out from saturated steam. This is why we call this phenomena Bose-Einstein
condensation, but here the separation between the two state takes place only in momentum space.
The gas atoms and condensed particles are mixed together henceforward in space. We have to
keep in our mind this essential feature.
Let us detach the occupation number of lowest energy state
N
∞
(2πmkB T )3/2 V X nµ/kB T −3/2
e
n
e−βµ − 1
h3
n=1
3/2
∞
T
1 X n −3/2
= N0 + N
σ n
TC
ζ 23 n=1
=
1
+
111
(A.1.14)
APPENDIX A. THEORY OF
BOSE-EINSTEIN CONDENSATION
A.2. CONDENSATION IN PRESENCE OF CONFINING POTENTIAL
If T < TC then µ = 0− and for the particle number in the ground state we obtain the following
expression
"
3/2 #
T
N0 = N 1 −
(A.1.15)
TC
The macroscopic occupation of the lowest energy level is the phenomenon of Bose-Einstein condensation. This prediction for the transition temperature of Bose-Einstein condensation and the
condensate fraction have been explored in a number of observations. In the initial study of BEC
at JILA [93] and MIT [115, 245], the transition temperature, TC , and number of particles in the
condensed part were in rough agreement with predictions (A.1.13) and (A.1.15). In a later experiment at JILA [246] this agreement was proven within few percents which is just the experimental
error. The effects originating from the finite size of the system and from the presence of particle
interaction have been estimated by W. Ketterle and N. J. van Druten [94] and announced that the
difference of the transition temperature between the case of a finite size condensate (T CN ) and the
case of thermodynamic limit (TC∞ ) can be measured in experimental set-ups [93, 98, 115].
A.2
Condensation in presence of confining potential
In the calculations presented below we follow the main steps of [247].
In the semiclassical limit, the D-dimensional system is described by a continuum of states and
instead of k , one uses the classical single-particle phase-space energy (q; p). In this way, the
single-particle phase-space distribution is
n(q; p) =
1
eβ((q;p)−µ) − 1
(A.2.16)
The accuracy of the semiclassical approximation is expected to be 5% if the number of particles is
large and the energy level spacing is smaller than kB T [91].
The number of particles in D-dimensional space can be written as:
1
N= 3
h
Z
D
D
n(q; p) d q d p =
Z∞
0
ρ() d
eβ(−µ) − 1
where ρ() is the density of states. It can be obtained from the semiclassical formula
Z
1
ρ() = D
δ − (q; p) dDq dDp
h
(A.2.17)
(A.2.18)
where δ(x) is the Dirac-delta distribution. In the case of bosons, below the BEC transition temperature TC the single-particle phase-space distribution (A.2.16) describes only the non-condensed
thermal cloud.
Let us consider the ideal Bose-gas in a confining external potential U (q) and in a D-dimensional
space. The single-particle energy is defined as
(q; p) =
1 2
p + U (q)
2m
Then the semiclassical density (A.2.18) of states can be written as:
Z 1
1 2
δ −
p + U (q) dDq dDp
ρ() = D
h
2m
112
(A.2.19)
(A.2.20)
APPENDIX A. THEORY OF
BOSE-EINSTEIN CONDENSATION
A.2. CONDENSATION IN PRESENCE OF CONFINING POTENTIAL
Integration with respect to the momentum can be calculated. We have just to know some basic
fact about Dirac-delta distribution, namely, we shall use that
1 γ
δ(αz + γ) =
δ z+
(A.2.21)
|α|
α
δ(f (z)) =
N
X
k=1
1
δ(z − zk )
|f 0 (zk )|
(A.2.22)
where α and γ are arbitrary constants and the points, zk , are the zeros of f (z), and we assume that
the derivative of f (z) does not vanish at z = zk . Using these important relations we immediately
get
1 2
δ − U (q) −
p = 2mδ p2 − 2m [ − U (q)]
(A.2.23)
2m
Due to the spherical symmetry in the momentum space, since only p2 appears in the formulae, we
can change to spherical coordinates either under the integrations or in the argument of Dirac-delta
distribution.
s
" #
q
q
1 2
1
2m
δ − U (q) −
p =
δ p − 2m( − U (q)) + δ p + 2m( − U (q))
2m
2 − U (q)
(A.2.24)
This expression can be substituted into the D-dimensional integral with respect to momentum.
The applied spherical coordinate system enables us to reduce the D-dimensional integral to a
one-dimensional one, hence the angle variables do not occur in any expression. Differentiate the
expression of volume of the D-dimensional sphere with radius p we get
dD p = 2π D/2
1
Γ
D
2
pD−1 dp
(A.2.25)
where p ∈ [0, ∞). Substituting back we get:
s
Z D/2
π
1 2
2m
p
dDp =
×
(A.2.26)
δ − U (q) −
D
2m
− U (q)
Γ 2
" #
Z∞
q
q
pD−1 δ p − 2m[ − U (q)] + δ p + 2m[ − U (q)] dp
0
The second Dirac-delta term in the brackets can be neglected, since it would only contribute to
the result if p had been negative. Furthermore, from the first term one receives
s
q
D−1
Z D/2
1 2
π
2m
D
δ − U (q) −
p
d p =
2m[ − U (q)]
2m
− U (q)
Γ D
2
i D−2
(2πm)D/2 h
2
(A.2.27)
=
−
U
(q)
Γ D
2
Now, the final result of the calculation of density of states as a function of energy reads as
m D/2 1 Z h
i D−2
2
ρ() =
−
U
(q)
dDq
(A.2.28)
D
2π~2
Γ 2
113
APPENDIX A. THEORY OF
BOSE-EINSTEIN CONDENSATION
A.2. CONDENSATION IN PRESENCE OF CONFINING POTENTIAL
Let’s specialize the type of potential-family as a power-like potential, U (q) = Aq n , which is quite
similar used in experiments. This is also a spherical symmetric potential in the D-dimensional
space which suggests to use the suitable parametrization of coordinate-system, as we have done
above.
m D/2 1 Z h
i D−2
2
r
ρ() =
dDq
(A.2.29)
−
Aq
D
2
2π~
Γ 2
Calculate just the integral:
Z h
− Aq
n
i D−2
2
2π D/2
d q=
Γ D
2
D
qZmax
h
0
− Aq n
i D−2
2
q D−1 dq
(A.2.30)
where qmax is determined from the condition ( − Aqn ) = 0, accordingly qmax = (/A)1/n . Try to
transform this integrand into a dimensionless form
Z h
− Aq
n
i D−2
2
2π D/2 D−2
2
d q=
Γ D
2
D
qZmax
0
Aq n
1−
D−2
2
q D−1 dq
(A.2.31)
Introducing the dimensionless independent variable, y = Aq n /, we arrive at the simple expression
below
Z h
Dn 1 Z1
i D−2
D−2
D−n
2π D/2 D−2
2
n
D
2
− Aq
(1 − y) 2 y n dy
(A.2.32)
d q=
D
A
n
Γ 2
0
Using integral-tables we find two important formulae for our calculation
Z
xβ+1
(1 − x)α xβ dx =
F (−α ; β + 1 ; 2 + β ; x)
β+1
and
F (α ; β ; γ ; 1) =
Γ(γ)Γ(γ − α − β)
Γ(γ − α)Γ(γ − β)
where F denotes the hypergeometric function. In our case α =
of integration are 0 and 1. Using these terms we get
Z1
0
(1 − y)
D−2
2
y
D−n
n
dy
=
n
F
D
(A.2.34)
D−2
2 ,
2−D D D+n
; ;
;1
2
n
n
(A.2.33)
β=
D−n
n
and the limit points
D
nΓ D
n + 1 Γ 2
=
(A.2.35)
D
D Γ D
n + 2
Substituting all of these will result in the phase-space density:
m D/2 1 D/n Γ D + 1 Γ D + 1 D D
n
2
ρ() =
2 + n −1
D
2~2
A
Γ D
n + 2
(A.2.36)
Let’s calculate now the critical temperature, where the phase-transition can occur. First we
calculate the density in the momentum space
Z
1
n(p) = 3
n(q ; p) dDq
(A.2.37)
h
114
APPENDIX A. THEORY OF
BOSE-EINSTEIN CONDENSATION
A.2. CONDENSATION IN PRESENCE OF CONFINING POTENTIAL
For the sake of more transparent formulae we introduce the so-called Bose-function with the definition and its representation as an infinite sum
1
gn (z) =
Γ(n)
Z∞
0
n(p) =
=
∞
X zk
ze−y y n−1
dy =
−y
1 − ze
kn
Z
1 2π D/2
q D−1 dq
h 2
i
D
3
β(−µ)
p
h Γ 2
e
−1
exp β 2m
+ Aq n − µ
h
i
Z D−1 exp −β p2 − µ exp (−βAq n )
2m
1 2π D/2 q
h 2
i
dq
p
h3 Γ D
1 − exp −β 2m
− µ exp (−βAq n )
2
1
h3
Z
1
(A.2.38)
k=1
dD q =
(A.2.39)
h 2
i
p
If we make a variable transformation as y = βAq n , and use the denotation z = exp −β 2m
−µ
we finally obtain the formula
n(p) =
=
n(p) =
D/n Z
y D/n−1 e−y
dy
1 − ze−y
D/n 2π D/2 1
1
D
Γ
gD/n (z)
n
βA
n
hD Γ D
2
D/n D/n
Γ D
2
1
1
1
n + 1
√
gD/n e−β [p /2m−µ]
D
D
(2 π~) Γ 2 + 1
βA
βA
2π D/2 1
n
hD Γ D
2
1
βA
z
(A.2.40)
(A.2.41)
The total number of particles can be expressed explicitly using the power series representation
of Bose-function and assuming that the integration and summation are interchangeable
N
=
=
D/n Z
Γ D
1
1
−β [p2 /2m−µ]
n + 1
n(p) d p = √
g
e
dDp
D/n
(2 π~)D Γ D
βA
+
1
2
D/n X
Z∞
∞
D
Γ n +1
βk 2
2
1
1
βµ
e
pD−1 e− 2m p dp
D
D/n
(2~)D Γ D
βA
k
+
1
Γ
2
2
Z
D
k=1
0
p
Changing the independent variable p into y = βk/2m p we can calculate the integral and get a
close formula
D/n X
D2
∞
Γ D
1
2
1
1
1
D
βµ 2m
n +
N =
e
Γ
D
D
D
D/n
(2~) Γ 2 + 1 Γ 2
βA
βk
2
2
k
k=1
D ∞
D/n
1 Γ D
1
2m 2 X
1
n + 1
=
D
D
D/n+D/2
(2~) Γ 2 + 1
βA
β
k
k=1
D
D/n
1 Γ D
1
2m 2
D D
n + 1
=
ζ
+
(A.2.42)
(2~)D Γ D
βA
β
2
n
2 +1
115
APPENDIX A. THEORY OF
BOSE-EINSTEIN CONDENSATION
A.2. CONDENSATION IN PRESENCE OF CONFINING POTENTIAL
where we used the Riemann-zeta function, ζ, and substituted µ = 0. Finally we get
D
m D/2
D
D
D D
−D/nΓ n + 1
N=
A
ζ
+
(kB T ) 2 + n
D
2
2~
2
n
Γ 2 +1
(A.2.43)
Inverting this formula one can express the critical phase-transition temperature in a D-dimensional
power-like external potential, Aq n , trapping and holding together the Bose-gas.
1
Tc =
kB
"
Γ
Γ
D
2
D
n
D/2
#−( D2 + Dn )
+1
2~2
D
D/n −1 D
A
ζ
+
N
m
2
n
+1
(A.2.44)
If someone analyzes this expression carefully then he can conclude that Tc remains finite iff the
argument of zeta-function is greater than one, since lim x→1+ (ζ(x)) = ∞. This corollary has been
already obtained in Salasnich’s cited paper [247]
Salasnich’s lemma
Let us consider an ideal Bose gas in a power-law isotropic potential U (r) = Ar n with r = |r|. BEC
is possible if and only if the following condition is satisfied:
D D
+
>1
2
n
where D is the space dimension and n is the exponent of the confining power-law potential. This inequality justifies that Bose-Einstein condensation does not occur in homogeneous gas
and lower space dimension than 3, but can be achieved in both D = 1 and D = 2 cases if
trapping potential is applied in experiments. In spite of Hohenberg’s theorem [248] which prohibits
condensation in translational invariant one- or two dimensional boson-systems, in real experiments
this translational symmetry is broken by the spatially varying external confinement. This is a
very important theoretical verification for experimentalist whose endeavour is to fabricate nearly
one-dimensional trap [177].
Bagnato et al presented [91] a valuable theoretical result for critical temperature of BoseEinstein condensation trapped by a general power-law potential. Studying the table below one
immediately finds how sensitively the critical temperature, condensed fraction, and heat capacity
is influenced by intensity of confining potential. We direct the reader’s attention to the different
exponents in the formulae of Tc and of N0 /N , and to the disappearance of discontinuity of the
heat capacity.
116
APPENDIX A. THEORY OF
BOSE-EINSTEIN CONDENSATION
A.2. CONDENSATION IN PRESENCE OF CONFINING POTENTIAL
Table A.1: Critical temperature, ground-state population (if T < Tc ) and discontinuity in C(T ) for several cases
of three-dimensional 3D confinement. (V and S represents volume and area, respectively). In the first two cases
where the potential is one dimensional, rigid walls are assumed in other direction. For the harmonic oscillator, the
result agrees with the previous calculation.
Potential
U (z) =
(
U (z) = 3
3 (z/a)
∞
Tc
z>0
z<0
h3 N
5/2
1.4SkB (2πM )3/2
3h3 N
√
2 π 4 M 3/2
2 SkB
z 2
a
3D-box
U (r) = 1
2/5
1/2
h3 N
3/2
2.612kB (2πM )3/2 V
r 2
a
U (z, ρ) = 1
z 2
a
h3 N
3 π 3 (2M )3/2
1.202kB
+ 2
ρ 2
b
N0 /N
h3 N
3 π 3 (2M )3/2
1.202kB
3
a2
3 2/5
a
1/4
2/3
1/3
1
a2
1/3
1
a2
117
1/2
1/6
2
b2
1/3
∆C(Tc )
N kB
1−
T
TC
5/2
1−
T
TC
2
0
1−
T
TC
3/2
0
1−
T
TC
3
6.57
1−
T
TC
3
6.57
3.35
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