TP02: On Bose–Einstein condensates with

spatially inhomogenous scattering length

Candidate Number: 468823 Supervisor: Dr Mason Porter

Wordcount: 6750

1

1Using TeXcount, http://app.uio.no/ifi/texcount/online.php

Abstract

I investigated a Bose–Einstein condensate confined harmonically to 1D, within the framework ofGross–Pitaevskii mean–field theory. I consider the 1D Gross–Pitaevskii equation with a combinedstep in its external potential and nonlinear terms. I generalise solitary wave solutions to the equationin its nonlinear limit, to take into account a constant potential and nonlinearity. I then generalise ane↵ective potential theory to take into account the combined step. The theory allows us to predictthe linear stability eigenvalues of the Bogoliubov–de Gennes equations. The theoretical work issupplemented by numerical investigations of the solitary wave solutions, mainly the e↵ect of varyingthe step width on the solitary wave solutions. It is found that dark solitary waves are unstable forall step widths: either with positive imaginary eigenvalues; or with a complex eigenvalue quartet.Bright solitary waves undergo a pitchfork bifurcation from stability to instability (or vice–versa,depending on the sign of the step) as the step width alters. The size of the relevant eigenvalues arepredicted well by the theory, becoming quantitatively less accurate as the step strength is increased,but maintaining their qualitative accuracy. Finally the time–development of an unstable solitarywave is shown. It is found that the solitary wave leaves the step region, emitting dispersive wavesas it crosses the step edge. It is hoped that this work will contribute to ongoing investigations intoBose–Einstein condensates with spatially inhomogenous scattering length.

1 INTRODUCTION

Predicted theoretically by Bose and Einstein in1924 [1, 2], a Bose–Einstein condensate (BEC) is aquantum phase entered by a gas of bosons below acritical temperature, T

c

. Below this temperature,a quantum state of the system is macroscopicallyoccupied. Following the theoretical suggestionsof Bose and Einstein, it was a long time beforeBEC would be achieved experimentally. The firsthints of the phenomena were in super–fluid 4He[3], where the transition temperature was reason-ably well predicted by an ideal Bose gas theory .However, the interactions in 4He are very strong(due to high particle density), and thus an idealBose gas theory is not applicable. Another BEC-like phenomena is superconductivity, which, in theBCS theory, relies on Cooper pairs of electrons(which are bosons) forming a macroscopic coher-ent state [4].The first true realisation of BEC was in ultra–

cold atoms, due to Anderson et al in 1995 [5], with87Rb atoms. Since then BEC has been achieved ingases of 23Na, 7Li, 41K, and 133Cs atoms, amongstothers [6, 7, 8, 9, 10, 11]. The atoms are firsttrapped in a magneto–optical trap before under-going laser cooling. These cold atoms are thenfurther cooled evaporatively (see [5] for furtherdetail). At the temperatures involved, typicallyµK and below, the equilibrium phase is the solidstate. To maintain the gaseous state a dilute gas

is used. In a dilute gas, three-body collisions arevery rare, the lifetime of the gaseous state is thuslong enough to perform experiments [12].

The large inter-particle distances in dilute al-kali gases lead to weak interactions. The theory ofweakly interacting Bose gases, in the low temper-ature limit, is built around the Gross–Pitaevskii(GP) equation. The GP equation arises from amean field treatment of the full many body the-ory, discussed in detail in section 2. Using a highlyanisotropic harmonic trap, it is possible to pro-duce a ‘cigar’ shaped, 1D condensate, which is de-scribed by the 1D GP equation:

i@

@t (z, t)=

✓�1

2

@2

@z2+V

ext

(z)+g(z)| |2◆ (z, t).(1)

The squared modulus of the order–parameter, (z, t), gives the atomic density. The second termon the right hand side is an external potentialterm, the third is a result of inter-atomic interac-tions. Both external potential and interactions areexperimentally controllable. The external poten-tial is implemented using magnetic or optical fields[18], the inter-atomic interactions are manipulatedvia Feshbach resonance management. Feshbachresonances can be exploited to tune the interac-tions to be attractive, repulsive, or zero, again, byapplication of a magnetic or optical field [20, 21].

In the case of Vext

= 0 and g = ±1 the GPequation is known as the non–linear Schrodinger

1

equation (NLS). Most weakly nonlinear, disper-sive, energy-preserving systems can be describedby the NLS in an appropriate limit [38]. The NLSis, therefore, important in many fields, apart fromthe study of BECs: Non–linear optics, plasmaphysics, fluid dynamics, and spin–waves are a fewexamples [22, 23, 24, 25]. The ease with whichthe potential and non–linearity can be varied isone of the main appeals of BECs in the study ofthis equation.

An important set of solutions to the NLS equa-tion are solitary waves, of the form (z, t) =�(z�vt)exp(i(kz�!t). There are two particularlyimportant families: bright solitary waves and darksolitary waves. Bright solitary waves occur in theattractive case of negative g, and take the form oftraveling, sech–shaped, density increases on zerobackground. Dark solitary waves occur in the re-pulsive case, and exist as traveling density depres-sions on a constant, non–zero, background [17]. Inthe literature, the term soliton is frequently usedinterchangeably with solitary wave in this context.Strictly, these solutions are not solitons, as theydo not behave elastically in a collision (a defin-ing property of a soliton) [26]. From this pointonwards we will use the term soliton to describethese solitary waves, to remain consistent with theliterature, and for brevity.

Recently, there has been an explosion of theo-retical interest in collisionally inhomogenous envi-ronments [27, 28, 29, 30, 31, 32, 33], where g variesin space. This follows the experimental demon-stration of such an environment in a Yb vapour[39]. The report to follow will focus on a 1D con-densate, confined in 2D as explained above, where,in the unconfined direction, we introduce a squarestep/well in both V

ext

and g (henceforth a com-

bined step). The steps are chosen in a mannersuch that they allow a constant density groundstate. The constant density constraint imposesa relationship between the step parameters; how-ever, there is still some freedom in the form of thestep. The report to follow will focus on the exis-tence and stability of soliton solutions in this set-ting, specifically as the width of the step is varied.In the literature there have been many studies ofspatially varying non–linearity (cited above), butfar fewer looking at both non–linear and potentialtogether, [34, §IV.B] provides a summary. Thecombined step setting we will investigate is inter-

esting, as it allows us to examine the competitionbetween the two e↵ects (and their e↵ects on thesolution of the GP equation), whilst still being arelatively simple setting.

There are techniques in the literature foranalysing the existence and stability of soliton so-lutions, summarised in [17]: these techniques areapplicable in settings with either spatially vary-ing non–linearities, external potentials, or both.I primarily used methods working from the non–linear limit, where we examine perturbations to anon–linear equation with a known set of solutions,in this case the NLS. The method I used involvesforming an e↵ective potential out of the perturba-tions to V

ext

and g. Solitons exist at the extremaof this e↵ective potential, their stability being de-termined by its curvature. The bright soliton casehas been treated previously in this manner, with adi↵erent form of variation in the non–linearity andpotential [36]. I took these results, and generalisedthe dark soliton methods in a similar manner. Therepulsive case is complicated slightly by the needto adjust for the non–zero background density.

The report to follow will begin with a quickoverview of the derivation of the GP equationand its validity. Following this, I will introducethe combined step setting more rigorously, beforelooking at what we might intuitively expect ourresults to be. Then, the theoretical tools neededto examine the e↵ects of the step on soliton solu-tions will be introduced (section 2). These are theBogoliubov-de Gennes (BdG) equations, whichgive the linear stability of the solutions; alongwith the e↵ective potential theory discussed in theprevious paragraph. We then move to numeri-cal analysis to examine the validity of our theory(section3): solving the 1D GP equations for sta-tionary solutions, and finding their stability eigen-values by numerically solving the BdG equations.We follow this with a conclusion and suggestionsfor future work.

2 THEORY

2.1 Gross–Pitaevskii mean field theory

In this section we will follow through the deriva-tion of the GP equation (more comprehensivetreatments of the topic being found in any of[12, 17, 15, 16]), introducing the relevant physi-cal quantities in more depth. We then derive thetime-independent GP equation which will be im-

2

portant later in the report.The derivation of the GP equation begins with

the second quantised Hamiltonian of the full manybody theory:

H =

Zdr †(r)

� ~2m

r2 + Vext

(r)

� (r) (2)

+1

2

Zdrdr0 †(r) †(r0)V (r� r0)0 (r0) (r),

where † and are the boson creation and an-nihilation operators at a point r, V (r � r0) is theinter–atomic interaction potential, and V

ext

is theexternal potential mentioned previously. The GPequation is a mean–field description of the abovetheory. Working now in the Heisenberg repre-sentation ( is function of space and time), wereplace the field operator in (2) by its expecta-tion value, plus a small perturbation , (r, t) = (r, t) + 0(r, t). The perturbation represents thenon–condensate part of the gas [17, under eqn.(1)].In a cold, dilute gas, we can replace the interac-tion potential, V (r � r0), by a delta–function po-tential of the form g(r)�(r� r0). The justificationfor this is that, in this regime, only low energy,two–body collisions are relevant, these are char-acterised by one physical parameter, the s–wavescattering length, a, independent of the detailsof V (r � r0) [12, 15, 16]. The s–wave scatteringlength is obtained by taking only the lowest orderterm in a partial–wave expansion (an expansionin terms of angular momentum eigenstates) of thetwo body scattering problem, and corresponds tolow–energy isotropic scattering [35]. The s–wavescattering length is related to g by g = 4⇡~2a/m:Positive g corresponds to repulsive interactions,negative g to attractive ones. Using the above ap-proximations for and V (r�r0), together with theHeisenberg evolution equation for the field opera-tor (r, t), gives the following mean–field equationfor the order parameter,

i~ @@t (r, t)=

✓�~2r2

2m+V

ext

(r)+g| |2◆ (r, t). (3)

The first term is a kinetic term, the second is theexternal potential term, and the third is the non–linear term.The validity of the dilute gas approximation

(which allows us to make the replacement forV (r� r0) above) is dependent upon the smallness

of the parameter hni|a|3 [12, 17], where hni is theaverage atomic density in the condensate. In typi-cal experiments this parameter is always less than10�3 [12]. The GP equation is strictly valid when 0 = 0 and all the atoms are in the condensate,even at T = 0 correlation e↵ects can lead to atomsleaving the condensate. Finite size e↵ects can alsocome into play, as the number of atoms is not trulymacroscopic. However, at low temperatures andwith large numbers of atoms (but such that hni|a|3is still small), the GP equation provides a gooddescription of experimentally observed behaviour.The GP equation can also be derived from the fullmany–body problem in a rigourous manner [14]

To reduce the GP equation to a 1D setting,as described in section 1, we use an harmonictrap to confine the condensate in two dimen-sions. The harmonic trap is described by, V

ext

=m/2((!

x

x)2 + (!y

y)2 + (!z

z)2), where !x,y,z

arethe trapping frequencies in each direction. Weconfine the condensate to the z axis, which be-comes the axial direction. We now define thetransverse oscillator frequency, !

r

⌘ !x

= !y

.By using a highly anisotropic harmonic trap asthe external potential, such that !

r

� !z

, it ispossible to reduce the full 3D theory to a quasi–1D one. The procedure for averaging over thetransverse dimensions is detailed in [37]. Theaveraging is performed using a multi–scale tech-nique which takes advantage of the disparate spa-tial scales in the two dimensions. The length scaleassociated with an harmonic trap is the oscilla-

tor length, ai

= (~/m!i

)1/2, which implies a ratioof length scales,a

z

/ar

= (!r

/!z

)1/2. A possibletrapping ratio would be 5.6⇥104 [12, Fig. 7], giv-ing a

z

/ar

= 236.6. Performing the averaging givesthe 1D GP equation. We use the rescaled variablest0 = t/!

r

, z0 = z/ar

, and | 0|2 = | |2/2|a|, to givethe equation in dimensionless form:

i@

@t (z, t)=

✓�1

2

@2

@z2+V

ext

(z)+g| |2◆ (z, t), (4)

where g is now rescaled to be ±1 for repulsive/at-tractive interactions1. For the rest of the report wewill neglect the weak trapping in the z–direction,for simplicity.

1The rescaled variables are taken from [17], but have

been altered for 1D, the paper uses a rescaled density ap-

propriate for a 3D equation.

3

The GP equation can be reduced to a time–independent form by assuming a solution of theform (z, t) = �(z)e�iµt, substituting gives,

✓�1

2

@2

@z2� µ+ V

ext

(z) + g|�|2◆�(z) = 0. (5)

The GP equation with a time–independent exter-nal potential has two conserved quantities asso-ciated with it: number of particles and energy[17, 12]. The number of particles, N , is given byN =

Rdz| (z, t)|2 which implies that � is nor-

malised to the number of particles. The param-eter µ is a dimensionless chemical potential, andis set to±1 for the rest of the report: the positivesign corresponds to the repulsive case, the nega-tive sign to the attractive one.

2.2 Non–linear and potential step withconstant density solution

The scattering length, and hence the non–linearity, can be altered by a process known asFeshbach resonance management. In a collisionprocess, the atoms involved will have spin struc-ture. Zeeman splitting in the energy levels, as a re-sult of the atomic spin, creates a number of chan-nels (possible routes from input to final states)which the process can occur in. These channelscan be divided into two types, open and closedchannels. The result of the scattering process canonly lead to atoms whose combined energy in theirfinal states is less than the energy available, this isan open channel process. Closed channel processesare those with final states above the energy avail-able, the scattering process cannot lead to atomsoutput into these final states. If the energy of abound state of a closed channel is su�ciently closeto the energy of the scattering state, this state canmix with the scattering state, leading to a modifi-cation of the scattering length. Varying the mag-netic field near one of these resonances allows usto alter the energy separation of the channels, andhence allows us to tune the scattering length. Wecan now replace g in (5) with g(z) [19].To create a constant density situation, both

within and external to the potential step, we workwithin the Thomas–Fermi (TF) approximation.The TF approximation consists of neglecting thesecond derivative term in the Hamiltonian, see[12], this is almost exact here due to the constantdensity nature of the ground state solution. The

result is the following equation for the density ineach region,

|�|2 = g�1(µ� Vext

), Vext

< µ. (6)

If we then apply this to both regions, and equatethe densities, we find that

g�1

1

(µ� V1

) = g�1

2

(µ� V2

), (7)

where g1,2

and V1,2

are the non–linearity and po-tential in the two regions. It is possible for g

2

todi↵er from g

1

, despite our rescaling, because thedensity is rescaled in terms of the scattering lengthoutside the step. We now define � ⌘ �g/�V =g1

/(V1

�µ), where �g = g2

� g1

, and similarly for�V . The step is determined fully by the parame-ter �, together with � ⌘ ��V (this choice of �gives �g with the same sign as �). Using � as aparameter is useful, as it makes it clear that therelative sizes of �g and �V are not independent.The step is now controlled by two parameters: thestep width, defined as w, and the step height, �.Unless stated otherwise, for the numerical sectionof the report (section 3) we will take g

1

and V1

to be given by their values in the non–linear limit(±1 and 0), giving � = �1. We will also imposea constraint on �, |�| < 1, this prevents g fromchanging sign or becoming zero within the stepregion (this situation will be discussed in furtherwork 4).

To avoid having non–smooth behaviour in ourstep, we will implement it as two tanh functions,

g = g1

+�g [tanh(z+

)� tanh(z�)] , (8)

V = V1

+�V [tanh(z+

)� tanh(z�)] , (9)

with z± = (z ± zw

)/s. This functional form givesa step with width w = 2z

w

, with an edge regionat z

w

, whose width is characterised by s. Theremainder of the report will primarily be an in-vestigation of how the step width, w, a↵ects thesoliton solutions at various ‘strengths’, �. This isa particularly interesting situation when the widthof the step becomes comparable with the width ofthe excitation. Using negative values for z

w

ineqns (8,9) leads to the same step, but with thesigns of �g and �V swapped. Henceforth, wewill keep � positive, using negative width to givenegative �g.

4

2.3 Soliton solutions to the GP equation

We begin to work towards our setting by firstconsidering the type of solutions we would expectfar from the step edges. If the solitons have veloc-ity 0, the functional form of the soliton solutionsare sech and tanh, for bright and dark solitons re-spectively [17]. The soliton solutions of the NLScan be generalised to include a constant V

ext

andg 6= ±1 (my derivations are in appendix 3):

�bs

(z � ⇠) =

r�2!

i

�gi

sech(p�2!

i

(z � ⇠)), (10)

�ds

(z � ⇠) =

r!i

gi

tanh(p!i

(z � ⇠)), (11)

with !i

= µ � Vi

. The parameter ⇠ is the soli-tons centre position. The constraint |�| < 1 en-sures that !

i

is the correct sign to give real rootsabove. The subscript, i 2 {1, 2}, refers to di↵erentvalues of the constant potential/non–linearity, asdefined in section 2.2. These soliton solutions arederived for an infinite domain, where we do nothave a change in V

ext

or g. The solution for i = 1corresponds to no step. The solution for i = 2corresponds to the limit of an infinitely wide step.

If we have a wide, finite, step in the equation,we expect the solution for i = 1 far outside thestep. The solution for i = 2 is what we wouldexpect near the centre of the step. We can seethis by considering a soliton located at the origin,and introducing a wide step: the edges of this stepwould be located in a region of constant or zerodensity (as they are far outside the soliton), andthus the kinetic term in (5) will be zero, by relation(7), we know that the regions either side of thestep allow the same density, thus, the solution isnot altered by the introduction of a step.

The solitons (10,11) reduce to those given bythe NLS for i = 1. When i = 2, the solitons dif-fer from those which solve the NLS: the parametergoverning the amplitude does not change (becauseof eqn. 7), however, the parameter governing thewidth does change. The soliton for i = 2 is widerthan the standard NLS soliton when � is the op-posite sign to µ, narrower when � has the samesign. When the step width becomes comparableto the width of the non–linear excitation, we canexpect that the solution inside the step may beperturbed away from either of these limits.

2.4 E↵ective potential landscapes andstability analysis

The methods elucidated in the following sec-tions are a powerful means of investigating thestability and persistence of soliton solutions to theGP equation when a perturbation is introduced.The combined step is introduced as a perturbationfrom the non–linear limit, characterised by a smallparameter, ✏. The procedure we will follow is sim-ilar to the analysis of equilibrium–points in ODEsystems: finding an equilibrium–point, examiningits stability by looking at small perturbations tothis, and seeing how the stability alters as a pa-rameter is varied. The PDE case is complicatedin that we are instead looking for stationary solu-tions, rather than points, and seeing how pertur-bations to these solutions grow or decay [43]. Weintroduce a small perturbation to the excitationwave function we have found,

(z, t) = e�iµt

�(z)

+X

j

(uj

(z)e�i!jt + v⇤j

(z)ei!jt)

�,(12)

where uj

(z), v⇤j

(z),!j

2 C are small, and describethe response to perturbations to � oscillating at±!

j

. Substituting (12) into the time–dependentGP equation, (4), and keeping only first orderterms in the perturbation, we get the Bogoliubov–de Gennes equations [12, 17]:

Auj

(z) +Bvj

(z) = !j

uj

(z),

Avj

(z) +B⇤uj

(z) = �!j

vj

(z), (13)

with

A = �1

2

@2

@z2+ V

ext

� µ+ 2g(z)|�|2,

B = g(z)�2(z). (14)

The BdG equations can be used to assess the lin-ear stability of any state �. If an eigenfrequency,!, solves the above, so also does �! and ±!⇤ . Apurely imaginary doublet of eigenfrequencies leadsto exponential growth of the relevant perturba-tion, and hence implies instability of that statealong one eigendirection. A complex eigenfre-quency quartet gives oscilllatory growth. Purelyreal eigenvalues imply the state is stable, the per-turbations neither grow nor decay.

5

The eigenfrequency spectrum of the NLS isknown for both bright and dark solitons. In thefirst case, there is a continuous spectrum of realeigenvalues above a certain minimum value, !

min

,below this value, there are four eigenvalues at theorigin [48, Fig. 1]. In the latter case the spec-trum fills the entire real axis, again, there is aneigenvalue with multiplicity 4 at the origin [49].The eigenvalues at the origin are particularly im-portant when considering perturbations from thenon–linear limit, as they are related to the sym-metries of the GP equation (4), namely phase in-variance and translational invariance, which leadto its associated conservation laws[41]. When webreak the translational invariance, by introducingthe step, we cause the two eigenvalues associatedwith this symmetry to leave the origin[41]. Whenthe eigenvalues leave the origin, they can moveeither along the imaginary axis, leading to insta-bility, or along the real axis. When eigenvaluesleave along the real axis they move away symmet-rically in opposite directions. If a collision with amember of the continuous spectrum occurs, it cancause the eigenvalues to leave the axis as a com-plex eigenfrequency quartet [41, §7.0.2]. Leavingalong the imaginary axis leads directly to instabil-ity.We rewrite the GP equation in an appropriate

form to consider perturbations to the potentialand non–linearity characterised by ✏,

i t

= �1

2 zz

+ (✏n1

(z) + (g1

+ ✏n2

(z))| |2) ,(15)

where we have used the subscripts to imply dif-ferentiation. The quantity ✏n

1

is the perturbationto V

1

, V � V1

. The quantity ✏n2

is the perturba-tion to g

1

, g� g1

. The energy associated with theunperturbed part is [17],

E0

[ ] =1

2

Z 1

�1(|

z

|2 + g1

| |4)dz. (16)

The first term is the kinetic energy term, the sec-ond is due to the non–linearity, and hence inter-actions. Note the form of the non–linear energyterm, dependent on the density squared.We attempt to simplify our initially complicated

problem by realising that we have identified a fam-ily of solutions to the NLS equation, characterisedby their centre position. When we introduce theperturbation, ✏, for certain centre positions there

will exist what is called a continuation from thesolution to the NLS equation, say �

0

(z� ⇠0

)e�iµt,to the solution to the GP equation, �

✏

(z�⇠✏

)e�iµt:For certain centre positions, soliton solutions willstill exist when the perturbation is introduced. Tofind the locations where these soliton solutionspersist, we will follow the procedure outlined in[40, 41]. In [40, 41] we find that, intuitively, thelocations where solitons persist are at the extremaof an energy functional for the perturbation. Thestability of these solitons is related to the cur-vature of this energy functional. We will followthrough the results for the dark soliton, these arecomplicated slightly by the presence of the back-ground, which must be taken into account whenwe define our energy functional. The bright soli-ton result will merely be quoted, but is achievedin a similar way.

The results of [40] are derived for a potentialwhich is bounded and decaying, these conditionsare satisfied by our step. We will generalise theresults, following [46, 41], to take into account aspatially varying nonlinear term, in addition to thepotential term. In the repulsive case, our solutionfor the NLS, (11), satisfies the conditions given in[40, Main results, i]. Combining the result of [40],for an external potential, and [46], for a spatiallyvarying nonlinearity, we find the following e↵ectivepotential,

M 0ds

(⇠0

) =

Z 1

�1

n01

(z)(⌘2ds

� �20

(z � ⇠0

))

+1

2n02

(z)(⌘4ds

� �40

(z � ⇠0

))

�dz = 0.(17)

The prime on n1,2

implies derivative with respectto their argument, that on the M is merely nota-tional. If 17 is fulfilled for some ⇠

0

, there exists aunique solution to the perturbed equation. Thissolution is �

✏

(z� ⇠✏

), for ⇠✏

✏–close to ⇠0

, and is ✏–close to �

0

(z� ⇠0

) in the L1–norm2. The proof isquite involved,see [40, §2]: essentially, persistenceof a solution upon introducing the perturbation isshown to be a result of the above condition (17),but for �

✏

rather than �0

. The condition on �✏

is less useful as the perturbed solutions are notknown. The technique of Lyapunov–Schmidt re-duction is used to prove that the above condition

2A measure of similarity between the functions. See []

for a definition

6

on �0

implies the condition on �✏

, and hence that(17) implies persistence of solutions in the per-turbed equation. Lyapunov–Schmidt reduction isa technique which allows us to reduce the prob-lem from an infinite dimensional one, to one witha small number of parameters [42]. The problemis reduced to that of dealing with a bifurcationequation in few dimensions, in our case these arethe parameters ✏ and ⇠. The bifurcation equationcan be solved and yields a correction to �

0

thatis unique, such that the corrected �

0

, �✏

, satisfiesthe conditions mentioned below (17).To analyse stability, it is necessary to look at,

M 00ds

(⇠0

) =

Z 1

�1

n001

(z)(⌘2ds

� �20

(z � ⇠0

))

+1

2n002

(z)(⌘4ds

� �40

(z � ⇠0

))

�dz 6= 0,(18)

if M 00ds

(⇠0

) < 0, the soliton is unstable with oneimaginary eigenvalue; if M 00

ds

(⇠0

) > 0, the soli-ton is unstable with a pair of complex eigenvalues.These unstable eigenvalues appear as part of theother spectral structure described below equations(13,14). Given M 00

ds

(⇠0

), it is possible to calculatean approximation to the eigenvalues for small ✏:

O(✏2) = �!2 +✏

4M 00

ds

(⇠0

)

✓1� i!

2

◆, (19)

The root picked is that with Real(i!) > 0 [40].The above result verifies the assertion in the pre-vious paragraph regarding stability, more impor-tantly, it allows us to get a quantitative handle onthe the eigenvalues.In the attractive case, one determines stabil-

ity in a similar manner, without having to worryabout the background. The resulting equationsfor bright solitons are

M 0bs

(⇠0

) =

Z 1

�1

n01

(z)�20

(z � ⇠0

)

+1

2n02

(z)�40

(z � ⇠0

)

�dz = 0, (20)

with M 00bs

being as above but with double primeswithin the integral. For the eigenvalues,

!2 =✏

2p�2µ

M 00bs

(⇠0

) +O(✏2). (21)

The simpler eigenvalue spectrum in the attractivecase arises because the NLS spectrum does not

include the origin, here the eigenvalues are imagi-nary, and hence unstable, for M

b

s00(⇠0

) < 0 whilstbeing real, and hence stable, for M

b

s00(⇠0

) > 0.If, whilst varying a parameter, there is a sudden

change in the qualitative nature of solutions (suchas the number of solutions), as well as their sta-bility, we say a bifurcation has occurred. We willfind in the numerical sections of the report, 3, thatour system exhibits a pitchfork bifurcation as thewidth of the step varies. Pitchfork bifurcations arecommon in physical systems [43] and are namedfor their shape, which resembles a pitchfork, theyhave a central branch which changes stability andtwo outer branches leaving the bifurcation point.

3 Numerical computations

We now build on our analytical results us-ing numerical computations. To solve the time-independent GP equation, a Newton–Raphson it-erative method is used. Given an initial guesswhich is su�ciently close to the initial solution,convergence is quick, around 5 iterations. Theanalytic solution to the NLS is a good place tostart, as the soliton solution to the GP equation,if it exists, is usually of a similar form. The BdGequations are solved numerically by using a combi-nation of MATLAB’s eig and eigs commands ona discretised version of the equations, further de-tails are in appendix B. Finally, to perform time–evolution simulations, we discretise the spatialpart of the GP equation and use a fourth–orderRunge–Kutta algorithmn to integrate in time. Fi-nite di↵erence methods are not the most appropri-ate for use with our equation, the NLS is bettersuited to split–step and fourier methods [44], theyare, however, quick to program and easily adaptedto a number of di↵erent situations: these are de-sirable traits given the length of time I had toapproach the investigation.

3.1 Bifurcation diagrams

The functions M 0(⇠) and M 00(⇠) are easy to cal-culate numerically. Finding the zeros, {⇠

i

}, ofM 0(⇠), and the sign of M 00(⇠

i

), allows us to findthe points at which soliton solutions to the GPequation exist, along with their associated stabil-ity. We will look first at bright solitons, as theyexhibit more interesting behaviour. We can sim-plify the equation for M 0 and M 00 by realising thepotential and non–linear step are just scaled ver-

7

sions of one another: if n01

(z) = �n0(z), thenn02

(z) = (�/�)n0(z) (see (8,9), n(z) is given bythe tanh functions). We can now replace n0

1,2

and�0

in (20) and simplify,

M 0bs

(⇠0

) =

Z 1

�1

n01

(z)�20

(z � ⇠0

)

+1

2n02

(z)�40

(z � ⇠0

)

�dz = 0, (22)

=

Z 1

�1n0(z)m(z � ⇠

0

)dz, (23)

m(x) ⌘ 2�µ

g1

(sech2 (p�2µx)

� sech4 (p

�2µx)). (24)

Equation (24) defining m(x) is interesting. Alter-ing any of our parameters only scales m(x) alongone of the axes, it does not alter the functionalform. The implication is that altering any of thoseparameters does not change the qualitative natureof the solutions, their number and stability for ex-ample, but only changes their centre positions bya scaling factor. The functional form of n0(z) in(22) is aproximately two delta–functions, of oppo-site signs, at ±z

w

: the gradient of the functionsdescribing the step, (8,9), is zero everywhere ex-cept at the step edges, where it is large. Substitut-ing the delta–function form for n0(z) in (22) givesM 0(⇠) = m(z

w

� ⇠) � m(�zw

� ⇠), this form isshown in figure 1 for four step widths. Looking atthe figure, we see that when the width becomescomparable to half of m(z � ⇠) the contributionsfrom both sides of the step begin to overlap, thisleads to more interesting structure arising.I calculated the zeroes, {⇠

i

}, of M 0(⇠) for stepwidths between -5 and 5. The set of points, {⇠

i

},correspond to the centres of solitons that have sta-bility given by the sign of M 00(⇠

i

). We can forma bifurcation diagram for the soliton centre posi-tions by plotting the ⇠

i

against width, colouringthe curves according to their stability. The resultis figure 2. At very narrow step widths there arethree zeros of M 0(⇠), these correspond to solitonscentred at 0 or a little outside the step. The cen-tral soliton is of opposite stability to the externalsolitons. As the step widens, the central soliton so-lution changes stability in a pitchfork bifurcation,the external solitons also move outwards, but moreslowly than the step widens, eventually meetingthe outer branches of the pitchfork at the step

−5 −4 −3 −2 −1 0 1 2 3 4 5−0.2

0

0.2

M’(ξ)

−5 −4 −3 −2 −1 0 1 2 3 4 5−0.2

0

0.2

M’(ξ)

−5 −4 −3 −2 −1 0 1 2 3 4 5−0.2

0

0.2

M’(ξ)

−5 −4 −3 −2 −1 0 1 2 3 4 5−0.2

0

0.2

z

M’(ξ)

Figure 1: A diagram showing M 0(⇠) and how thisevolves as the step (shown in red) narrows. Thegreen line is at zero and helps pick out whenM 0(⇠)touches/crosses the axis. The black arrows at-tempt to show the sequence of events describedin the text. The interesting behaviour in this caseis the formation of a further two fixed points whichtravel inwards to zero.

edge. When the step is at its widest, the branchat the origin widens into a region about zero, soli-tons can exist at any point within this region. Torboth signs of � (positive and negative width onthe diagram), the stability eigenvalues within thecentral region of solutions are vanishingly small:for unstable eigenvalues the instability develop-ment will be so prolonged that the solitons canbe considered to be stable.

The positions of the outer soliton solutions atvery narrow widths can be predicted by lettingn(z) tend to a delta–function. Substituting thisinto (23) and integrating by parts (to take care ofthe delta–function derivative) gives,

M 0(⇠) =�Z 1

�1�(z)m0(

p�2µ(z � ⇠

0

))dz,

m0(x) =2�µ

g1

(�(cosh (2x)� 3) tanh (x) sech4 (x)).(25)

The zeros of M 0(⇠) are at 0 and ±0.623. If weallow the newton raphson iterator to converge toa solution for very narrow widths (w = 0.1), wefind solitons just outside the step are centered at±0.634, the prediction from the theory is correctto within 1.7%.

The discussion of this section has focused on theattractive case, the reason being that the qual-itative nature of the solutions for dark solitonsare much simpler: only the centre solution existsfor dark solitons, and no bifurcations occur. The

8

width

z

−5 −4 −3 −2 −1 0 1 2 3 4−5

−4

−3

−2

−1

0

1

2

3

4

5

Figure 2: Bifurcation diagram for the bright soli-ton centre position, calculated for � = 0.1. Theorange regions do not admit soliton solutions, thelight curves indicate the centre positions of unsta-ble solitons, the dark curves stable ones. Outsidethe orange region, far from the step, the rest ofthe domain supports soliton solutions. The dot-ted lines indicate the step edge.

dark soliton branch at the origin is unstable fornegative widths, oscillatory–unstable for positivewidths: there is no change in stability as the stepwidens. Again, at very large widths the branchof solutions widens, becoming a region in whichsolutions can exist.

3.2 Soliton solutions and their stability

The previous section has confirmed the ear-lier intuitive arguments for wide steps (see sec-tion 2.3): soliton solutions exist, except withina narrow region about the step edge, in this re-gion the solution is strongly perturbed. We havealso revealed some interesting structure arisingfor narrow steps. To further investigate this be-haviour we will use our Newton–Raphson iterativemethod to find solutions to the time–independentGP equation (5). Given there is always a branchof solutions at zero, we can apply a continuationtype approach. Starting with a wide step, a soli-ton solution is allowed to converge to the correctsolution. This new solution is then passed as theinitial guess for the next width, in this manner, itis possible to get a set of solutions for all values ofthe width, more sophisticated methods are givenin [45]. The BdG equations (13,14) are solvednumerically to give the associated stability eigen-values for each solution, these are compared withthose given by the theory of section 2

−2 −1 0 1 20

0.5

1

1.5

2

z

φ(z)

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

z

φ(z)

Width = 4Width = 2Width = −2Width = −4No stepNegativePositive

Figure 3: Soliton solutions centred at zero for at-tractive (top) and repulsive (bottom) interactions.The legend holds for both sets of solutions. Thesolid line solution is barely visible in both cases,as it is overlapped by positive/negative.

−10 −8 −6 −4 −2 0 20

0.1

0.2

0.3

0.4

Width

Max(ω

i)

−1 0 1−0.2

0

0.2

ωr

ω i

Δ = 0.1Δ = 0.2Theory 0.1Theory 0.2

−1 0 1−0.2

0

0.2

ωr

ω i

b)

a)

Figure 4: a) Max imaginary part of eigenvaluesagainst width. b) Typical eigenvalue developmentin unstable (top) and stable (bottom) regions.

Figure 3 shows the soliton solutions for a varietyof widths, in both attractive and repulsive cases.The curves labelled positive and negative corre-spond to the solitons (10,11) with i = 2, wherepositive and negative refer to !

2

= µ±�. It canbe seen that these are the limiting forms of thesolitons centred at zero for wide steps. As thestep narrows, the solitons become di↵erent fromthese limiting forms, but do not tend to the solu-tion in the absence of the step. Instead, the soli-tons narrow and grow in amplitude, for positivewidth; widen and shrink, for negative. The oppo-site occurs for dark solitons, but with no changein amplitude.

9

−10 −5 00

0.05

0.1

0.15

0.2

0.25

Width

Max(ω

i)

−0.2 0 0.2−0.01

0

0.01

ωr

ω i

−0.2 0 0.2−0.2

0

0.2

ωr

ω i

Δ = 0.1Δ = 0.2Theory 0.1Theory 0.2

0 2 40

0.002

0.004

0.006

0.008

0.01

Width

Max(ω

i)

c)b)

a)

Figure 5: a) Max imaginary part of eigenvaluesagainst width. b) Typical eigenvalue developmentin unstable (top) and oscillatory–unstable (bot-tom) regions. c) How domain size a↵ects the nu-merical simulation: black being large domain, or-ange small.

Figure 4a) compares numerical and theoreti-cal (dashed) values for the imaginary part of theeigenvalues against width. The previously identi-fied bifurcations are clearly visible as sudden largeincreases in the magnitude of the stability eigen-value. The theory predicts the eigenvalues verywell, except very near the bifurcation. As theperturbation size is increased it can be seen thatthe numerical values are less well described quan-titatively, however qualitative agreement is stillgood. For both strengths the theoretical predic-tions are within ✏ of the numerical values, usuallywell within this region. With growing perturba-tion strength the size of the instability also grows.The insets show the translational eienvalues leav-ing the origin, along either the real or imaginaryaxis. When the eigenvalues move along the realaxis, they do not enter the continuous spectrumand hence cannot lead to oscillatory instabilitythrough collision (see section 2.4).

Figure 5a), shows the dark soliton case. No-table di↵erences are the immediate instability atzero width, the solutions being either unstable, oroscillatory–unstable, for all widths. As mentionedin section 2, this is a result of the eigenvalue spec-trum encompassing the origin. Figures 5b) showtypical eigenvalue development. Note the eigen-value quartet which results in oscillatory instabil-ity. Also noticable in the subfigures is the lack ofa gap about the origin, this is markedly di↵erent

t

z

20 40 60 80 100 120

−15

−10

−5

0

5

10

15

0

0.5

1

1.5

−50 0 500

0.51

x 10−3

t

|φ|2

t = 95t = 105t = 115

Figure 6: Main figure: unstable soliton leavingthe step. Inset: snapshots of the density at 3times. Both the counter propagating dispersivewave, and that infront of the soliton are clearlyvisible. Parameters: � = 0.5 with width of �2.

to the bright soliton case (see figure 4b)). Figure5c) shows an interesting numerical e↵ect: simulat-ing the infinite domain on a finite numerical one(which cannot be avoided) causes the continuouseigenvalue spectrum to become discrete, a resultof this is that occassionally the oscillatory eigen-values ‘drop’ back to the real axis when they en-counter a gap in the spectrum [], this leads to the‘spikes’ shown. The behaviour we see does not re-flect the actual physical situation, where the eigen-values should be a quartet with non–zero imagi-nary part throughout the region. The subfiguredemonstrates how the e↵ect lessens with increaseddomain size, where the gaps in the spectrum be-come smaller.

In order to study the development of instabil-ities, we use the Runge–Kutta integrator. Theinitial input for the integrator is a solution givenby the Newton–Raphson method. In both casesunstable solitons move o↵ from the origin after anamount of time dictated by the strength of the in-stability. As the soliton leaves the step it emitstwo dispersive waves: one travels ahead, one iscounter–propagating behind the soliton. Numberis still conserved, with the soliton decreasing inamplitude. These dispersive waves are mentionedin [46, 47], which both cover piecewise constant pe-riodic potentials, external and non–linear respec-tively, where the soliton dissipates constantly tozero. The emission of dispersive waves is a resultof the soliton moving over a spatially inhomoge-neous environment, and hence is concentrated atthe step edges. As a result, in our setting theemission of a wave occurs only once, when leav-

10

ing the step, and the soliton is persistent despiteits initial instability. The case shown is that of abright soliton, dark solitons behave similarly.

4 Conclusion

I investigated a combined nonlinear andexternal–potential step in the 1D GP equation,arranged to give a constant density ground state.Due to the simplicity of the setting, it was pos-sible to investigate how the two e↵ects competeor combine in a clear way, and the result on soli-ton solutions to the GP equation and their stabil-ity. I began by introducing the soliton solutionsto the GP equation, generalising them to take intoaccount the presence of a constant potential andnonlinearity. Doing so allowed us to consider theforms of the soliton solutions far from the edgesof the step. I picked the width as a step parame-ter to focus on, being interested in the behaviourof the solution when the step became of similarwidth to the nonlinear excitations. I then intro-duced various theoretical and numerical tools toinvestigate the kinds of solutions supported andtheir stability.A theoretical framework was introduced, which

aimed to explain the existence and linear stabil-ity of soliton solutions via an e↵ective potentiallandscape. Solitons persist at the extrema of thispotential landscape, with stability governed bywhether they exist at maxima, or minima, of saidpotential. I generalised previous results to ap-ply to a combined step setting. The linear sta-bility was given quantitatively by the BdG equa-tions for the stability eigenvalues, which describehow perturbations to solutions grow or decay. Itwas found that changing the parameters of thestep has little e↵ect on the qualitative nature ofthe solutions, only altering the magnitude of thestability eigenvalues, and shifting the centre po-sitions of solitons solutions. Bright solitons werefound to exhibit more interesting behaviour as thewidth was varied. A pitchfork bifurcation occursat narrow widths, as the width increases the outerbranches tend to towards two further branches ofsolutions at the edges of the step. Dark solitonsexhibit less interesting behaviour, having only onebranch of soliton solutions. The dark solitons areunstable for negative width, oscillatory–unstablefor postive width.We then moved to numerical studies to investi-

gate the nature of the soliton solutions supportedin more depth. For wide steps the solitons sup-ported are those given by the limit of an infinitelywide step, as the step narrows the solitons alter.Calculation of the stability eigenvalues, by numer-ically solving the BdG equations, leads to eigen-values in very good agreement with those givenby the e↵ective potential theory. In the limit ofvery wide steps, solitons centred at the origin, orin a region well within the step, are either stableor unstable with exponentially small eigenvalue.As the step narrows towards the point where thebifurcation occurs the eigenvalue grows in mag-nitude, before changing stability at the bifurca-tion point. The typical development of instabilitywas then shown, the soliton eventually becomesmobile, leaving the step and emitting dispersivewaves.

There are several avenues to pursue in fur-ther study, remaining with our setting in one–dimension, moving to higher dimensions or mov-ing towards a more experimentally feasible setup.One way forward, is to investigate the relationshipbetween narrow step states and defect–modes,which are solutions supported by a delta–functionpotential/non–linearity. At narrow step widths,the form (8,9) for the potential does not ade-quately represent the step. The limit of narrowwidth should be a delta–function, in my settingthe tanh functions overlap, leading to the stepeventually vanishing. It would be interesting tosimulate the very narrow step properly, and in-vestigate how the states supported by a delta–function relate to the solitons supported by narrowsteps. A second way forward would be to investi-gate the solutions in the case where the nonlinear-ity changes sign within the step, this may have aninteresting e↵ect on the edge states as presumablythe soliton centred at zero would no longer be sup-ported. It would also be interesting to look at thesetting either in higher dimensions, where othertypes of solutions such as vortices are supported,or to use a better one–dimensional reduction suchas [50].

5 Acknowledgments

I would like to thank my supervisor, MasonPorter, for his suggestions and help along the way.

11

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13

A Analytics

Here I derive the form of bright and dark solitons in the GP equation with constant external potential,V, and a constant nonlinearity, g. The derivation begins with the time–independent GP equation:

✓�1

2

@2

@z2� µ+ V + g|�(z)|2

◆�(z) = 0. (26)

The soliton solutions to the NLS are,

�bs

(z � ⇠) =p�2µ sech(

p�2µ(z � ⇠)), (27)

and,

�ds

(z � ⇠) =pµ tanh(

pµ(z � ⇠)), (28)

where µ is �1 for bright solitons, and +1 for dark solitons. The soliton solutions with constant V andg can be expected to be similar to those of the NLS, with a new chemical potential ! = µ � V , anddi↵erent constants within and without the sech function.

A.1 Bright soliton

We insert the ansatz,

�bs

(z) = a sech(bz), (29)

into (26) giving,

�1

4ab2(2 cosh2 (bz)� 4) sech3 (bz) = a(! � ga2 sech2 (bz)) sech (bz) (30)

�b2

2+ b2 sech2 (bz) = ! � ga2 sech2 (bz). (31)

Comparing the coe�cients in (31) gives b =p�2! and a = b/

p�g. The final result is (10).

A.2 Dark soliton

We insert the ansatz

�ds

(z) = a tanh(bz)), (32)

into (26) giving,

�1

2(�2ab2(1� tanh2 (bz)) tanh (bz) = a(�! + ga2 tanh2 bz) tanh (bz) (33)

b2 � b2 tanh2 (bz) = ! � ga2 tanh2 (bz). (34)

Comparing the coe�cients in (34) gives b =p! and a = b/

pg. The final result is (11).

14