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Universita degli Studi di MilanoFacolta di Scienze Matematiche, Fisiche e Naturali
Laurea Triennale in Fisica
Expansion of an Interacting
Boson Condensate: the Role of
Dimensionality
RELATORE: Prof. Nicola Manini
CORRELATORE: Dott. Luca Salasnich
Michael KorbmanMatricola n◦ 666896
A.A. 2005/2006
Codice PACS: 03.75.-b
2
Expansion of an Interacting Boson
Condensate: the Role of Dimensionality
Michael Korbman
Dipartimento di Fisica, Universita di Milano,
Via Celoria 16, 20133 Milano, Italia
2
Contents
1 Introduction 5
1.1 The Gross-Pitaevskii Equation . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 The role of dimensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Exact results 9
2.1 Free Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 1D Solitonic Solution of an Attractive Boson Cloud . . . . . . . . . . . . . 10
3 The Hydrodynamic Limit 13
3.1 A local analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.1 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.2 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4 Numerical Analysis 17
4.1 Notes about the Programs and Scaling . . . . . . . . . . . . . . . . . . . . 174.2 Pure 1D with repulsive particles . . . . . . . . . . . . . . . . . . . . . . . . 184.3 Cylindrically-confined wavefunction . . . . . . . . . . . . . . . . . . . . . . 194.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Bibliography 25
4 CONTENTS
Chapter 1
Introduction
When a sample of bosons (i.e. atoms) is isolated, usually by eletromagnetic traps, and thetemperature of the gas is lowered the most particles partecipate to the condensate: thisis what is referred to as a Bose-Einstein condensation. The Bose-Einstein condensates(BEC) are very important for the role they play in the study of quantum mechanics: theopportunity of realising a mesoscopic (∼ 104 particles) sample which move coherently ina single quantum state is a very powerful tool in quantum research. In this documentwe investigate the expansion of an interacting-bosons cloud, after the trap is removed inone or more direction, both with analytical and computational tools: we will focus ourattention on the role of space dimensionality in these processes, and compare in particularexpansion in a single direction (1D) to the regular three-dimensional isotropic case.
1.1 The Gross-Pitaevskii Equation
The dynamics of N interacting bosons is a formidable problem, which would requiresophisticate numerical many-body techniques to address exactly. However a fairly satis-factory approximate dynamics is obtained by means of a mean-field share, the so-calledGross-Pitaevskii Equation (GPE) [1]. The stationary GPE is a variational approximationbased on the best non-interacting wavefunction.
Our sample is formed by N identical bosons moving in an external potential Vext. Fora diluted gas it is a good approximation to only consider pairwise interaction Vint, so thatthe total Hamiltonian is
H =N∑
i=1
[
− h2
2m∇2i + Vext (ri)
]
+1
2
N∑
i6=jVint (ri − rj) , (1.1)
where rk is the position of the kth boson. For a diluted gas only s-wave scattering betweenpairs of bosons remains significant. It is characterized by scattering lenght as, so that theinteraction Vint is approximated by a pseudopotential
Vint(ri − rj) = gδ3(ri − rj) (1.2)
6 Introduction
where g = 4πh2
mas is the parameter which measure the strenght of the interaction. Gener-
ally, the ground state of (1.1) cannot be determined exactly. In the absence of interactionshowever, it is a product state: all the bosons are in the ground state Φ(r) of the singleparticle Hamiltonian:
Ψ (r1, . . . , rN) = Φ(r1) · · ·Φ(rN ) (1.3)
This is a trusting N-particle wavefunction and be taken as a variational ansatz. Startingfrom (1.3) we can carry on the variational calculation, minimizing the energy (of single-particle)
E[Ψ] = 〈Ψ|H|Ψ〉 (1.4)
where the expectation value of H is
〈Ψ|H|Ψ〉 =∫
d3r1 · · · d3rNΨ∗ (r1, . . . , rN)HΨ (r1, . . . , rN)
=∫
d3r1 · · · d3rNΦ∗ (r1) · · ·Φ∗ (rN)HΦ (r1) · · ·Φ (rN) (1.5)
and substituting the Hamiltonian (1.1) with the interaction potential (1.2):
〈Ψ|H|Ψ〉 =N∑
i=1
∫
d3r1 · · ·d3rNΦ∗ (r1) · · ·Φ∗ (rN)
[
− h2
2m∇2i + Vext (ri)
]
Φ (r1) · · ·Φ (rN) +
+1
2
N∑
i6=j
∫
d3r1 · · · d3rNΦ∗ (r1) · · ·Φ∗ (rN) gδ3(ri − rj)Φ (r1) · · ·Φ (rN) (1.6)
The first integral is simple as integrations over j, with j different from i yields unit dueto normalization of Φ(r). This yields the sum of N identical terms:
〈Ψ|T + Vext|Ψ〉 =N∑
i=1
∫
d3riΦ∗ (ri)
[
− h2
2m∇2i + Vext (ri)
]
Φ (ri) =
N
∫
d3r′Φ∗(r)
[
− h2
2m∇2 + Vext (r)
]
Φ(r) . (1.7)
The interaction term are factorized, as the potential depends only on the inter-distancesbetween particles
1
2
N∑
i6=j
∫
d3r1 · · · d3rNΦ∗ (r1) · · ·Φ∗ (rN) gδ3(ri − rj)Φ (r1) · · ·Φ (rN) =
1
2
N∑
i6=j
∫
d3rid3rjΦ
∗ (ri)Φ∗ (rj) gδ3(ri − rj)Φ (ri) Φ (rj) =
N(N − 1)
2g
∫
d3r′d3rΦ∗ (r)Φ∗ (r′) δ3(r − r′)Φ (r) Φ (r′) . (1.8)
1.1 The Gross-Pitaevskii Equation 7
The r′ integration is easily carried out:
N(N − 1)
2g
∫
d3rd3r′δ3(r − r′)|Φ (r)Φ (r′) |2 =N(N − 1)
2g
∫
d3r′ · |Φ(r)|4 . (1.9)
The term N(N − 1) behave like N2 for large enough N ; we can finally collect the piecescomposing the total energy (1.4):
E[Φ] = 〈Ψ|H|Ψ〉 = N
∫
d3r
[
Φ∗(r)
(
− h2
2m∇2 + V (r)
)
Φ(r) + gN
2|Φ(r)|4
]
. (1.10)
Our goal is to minimize this functional over the function space L2(ℜ3) under thenormalization costraint
∫
d3r|Φ|2 = 1 (1.11)
(which implies that also Ψ is normalized) to obtain the equation for the single-particleground state. Costrained minimization is obtained through the method of the Lagrangemultipliers: introduce the multiplier µ which will take count of the normalization costraintand differentiate E[Φ] − µN w.r.t. Φ∗(r), obtaining the estremal condition for Φ(r):
δN
δΦ∗(r)
∫
d3r′Φ∗(r′)
(
− h2
2m∇′2 + V (r′) +
g
2N |Φ(r′)|2 − µ
)
Φ(r′) =
N
∫
d3r
(
− h2
2m∇2 + V (r) +
g
2N |Φ(r)|2 − µ
)
Φ(r) = 0. (1.12)
This condition is equivalent to the nullifying of the integrand term because (1.12) mustbe true for each variation. We then obtain the stationary GPE:
(
− h2
2m∇2 + V (r) + gN |Φ(r)|2
)
Φ(r) = µΦ(r). (1.13)
The GPE equation is a sort of nonlinear Schrodinger equation, because of the inter-action term: any particle is assumed to feel a mean-field potential created by the otherparticles. The GPE (1.13) is formally a stationary Schroedinger equation with Hamilto-nian
H = − h2
2m∇2 + V (r) + gN |Φ(r)|2 (1.14)
In analogy to the Schrodinger time evolution one can postulate a time-dependent GPE:
(
− h2
2m∇2 + V (r) + gN |Φ(r, t)|2
)
Φ(r) = ih∂Φ(r, t)
∂t. (1.15)
The self-interaction term affect the time-evolution of nonhomogeneus in a highlynon trivialway.
8 Introduction
In the following we will study the solution of this equation with different initial states,interaction strenghts g, and shapes of the confining potential. Numerical methods havebeen used to observe these distinct situations changing the environment around the sam-ple: some differences have been pointed out letting the sample evolving in computationalsimulations. In particular we investigate a clear qualitative feature, namely the depletionof the density at the cloud center, which is found in the 3D isotropic expansion startingfrom a Gaussian profile.
1.2 The role of dimensionality
We investigate if this phenomenon is also present in 1D, what differences arise in differentdimensions. We will mainly use numerical methods.
We consider two different kinds of 1D expansion: an ideal one, the pure 1D dynamicswhere the particles exist actually in one dimension only and can move only there, andthe realistic one, with a 3D condensate which is left free to expand in one directiononly, while the other two remain confined. The ”pure” 1-D equation represent the limitof extreme confinement on a line. The repulsive case g > 0 is solved numerically inChapter 4; instead in the attractive case g < 0, the reducted number of dimensionsallows for an exact resolution of (1.13). The attraction between the particles balancesthe natural kinetic spreadment, leading to a self-bound solitonic state. A more realistic1D setup involves 3D sample allowed to expand freely in one direction only. We simulatenumerically the 3D GPE (1.15) in cylindrically symmetry, in the presence of a confiningharmonic potential in two dimensions. The harmonic potential is actually the one whichis used in experiment, mainly because in absence of the interaction we can solve exactlythe Schroedinger equation with this potential: the non-interactive bound eigenstates ofthe harmonic potential are Gaussians. Then the wavefunction ψ(z, r, t) is initially takenas a Gaussian of finite width σr in the transversal (radial) direction where it experencesthe confining potential V (r), and a Gaussian of width σz in the z (axial) direction wherethe motion is free. This condensate evolves to a ”cigar” like shape [2]. We use someof the pre-existing results in our simulation, mainly to adjust the starting wavefunction.To investigate the expansion we analyze the condensate density along the cylinder axisr = 0 and the radial |ψ(z = 0, r)|2. Anyway what can be directly accessed by laboratorymeasurements is the ”shadow” of the cloud which means the integrated density in oneparticular direction The integration along the axial direction of the cylinder yields afunction of the radial distance from the cylinder center:
ρ(r) =∫ ∞
−∞|ψ(r, z)|2dz. (1.16)
In addition a direct comparison with the 1D model can be obtained by considering
ρ(z) = 2π∫ ∞
−∞|ψ(r, z)|2rdr. (1.17)
Chapter 2
Exact results
We begin our investigation reporting some exact analytic results related to different kindsof expansions. These exact result provide insight in the less trivial situations. We willanlyze the free evolution of a wavepacket in the simple case of non-interacting particles,a necessary knowledge to approach the non-linear analysis. We also report exact analyticsolution in the 1D attractive case, a bound soliton.
2.1 Free Expansion
In order to study the non-linear evolution of a system of interacting bosons it is usefulto analyze the simpler situation with non-interacting particles, which allows for an exactanalytic description [3]. This calculation will give us a rough idea of what can happen inthe interacting regime and will also provide some useful relations for subsequent problems.Consider a Gaussian wavepacket
Ψt=0(x) = Z · exp
(
−1
2
(
x
σ
)2)
, (2.1)
where Z = (πσ2)− 1
4 is the normalization costant. The probability distribution is
|Ψt=0(x)|2 =exp
(
−(
xσ
)2)
√πσ
. (2.2)
If we consider the free expansion, the Hamiltonian is H = p2
2m. Therefore, it is wise to
use the momentum rapresentation where H is diagonal and acts by multiplication on thebase’s states:
Φt=0(p) =1√2πh
∫ +∞
−∞e
−ipxh Ψt=0(x)dx (2.3)
Φt=0(p) =1√2πh
· 14√πσ2
∫ +∞
−∞e
−ipxh exp
(
− x2
2σ2
)
dx (2.4)
10 Exact results
=(
4h2π3σ2)− 1
4
∫ +∞
−∞exp
[
−(
x2
2σ2+ipx
h
)]
dx (2.5)
Using the standard trick of the square completion, the initial wavefunction in momentumspace turns out
Φt=0(p) =4
√
σ2
h2π· exp
(
−p2σ2
2h2
)
. (2.6)
The time-evolved wavefunction is
Φ(p) =4
√
σ2
h2πexp
[
− p2
2h
(
it
m+σ2
h
)]
. (2.7)
Returning to the position rapresentation
Ψ(x) =4
√
σ2
4π3h2
∫ +∞
−∞exp
[
−(
it
2hm+
σ2
2h2
)
p2 +ipx
h
]
dp
= 4
√
√
√
√
(
σ2h2m2
π
)
√
(
1
iht+mσ2
)
exp
[
−x2 · m2σ2 − ihmt
2m2σ4 + 2h2t2
]
(2.8)
where we have applied the same formula as in eq. (2.6).
The time evolution of the wavepacket is characterized therefore by a progressive broad-ening of its shape; we can write down the variance s of the probability distribution|Ψ(x, t)|2, i.e. the square modulus of the wavefunction, as a function of time t.
s(t)2 =m2σ4 + h2t2
m2σ2= σ2 +
h2t2
m2σ2. (2.9)
Obviously for t = 0 s reduces to σ, the initial variance. Moreover for large t the evolutionof s becomes proportional to t, so the wavepacket expands linearly with a slope (speed)dsdt
≈ hmσ
, and s is inversely proportional to the initial σ, so the more the initial distributionis peaked, the faster it will expands, in accord with Heisenberg uncertainty principle.For repulsive particles then we expect the same effect though accelerated. On the contrary,for attractive particles, the natural broadening can balance the attraction, giving birthto an equilibrium point. Owing to isotropy and the simmetry of the system, the samecalculation remains valid for a 2D or 3D expansion: this implies that in the non-interactingcase we observe no difference among the 1D and the 3D expansion starting off with aninitial packet which is Gaussian distributed in all direction of space: the evolution isexactly the same except fot the substitution of x with ~r.
2.2 1D Solitonic Solution of an Attractive Boson Cloud
For negative g it is possible to find explicitly the shape of the soliton wavefunction (1.15)in 1D. The differential stationary equation is
Ψ′′(x) − 2m
h2 gNΨ3(x) = −2m
h2 EΨ(x) , (2.10)
2.2 1D Solitonic Solution of an Attractive Boson Cloud 11
where we assume that Ψ is real and Ψ′′(x) = d2Ψdx2 . By making the substitutions
α = −2m
h2 gN > 0 and β = −2mE
h2 > 0 (2.11)
the equation becomesΨ′′(x) + αΨ3(x) − βΨ(x) = 0 . (2.12)
Now we can multiply both sides for Ψ′ obtaining
Ψ′Ψ′′ + αΨ′Ψ3 − βΨ′Ψ = 0 . (2.13)
We notice that
Ψ′Ψ′′ =1
2
d
dx
[
Ψ′2(x)]
(2.14)
Ψ′Ψ3 =1
4
d
dx
[
Ψ4(x)]
(2.15)
Ψ′Ψ =1
2
d
dx
[
Ψ2(x)]
. (2.16)
(2.17)
We can now write the differential equation (2.13) as
d
dx
[
Ψ′2
2− β
2Ψ2(x) +
α
4Ψ4(x)
]
= 0 (2.18)
This indicates that the contents of the square bracket is x-independent. We assume itequals zero, so that we obtain
Ψ′ = Ψ
√
−α2
Ψ2 + β, (2.19)
which can be integrated explicitly.
∫
dΨ
Ψ√
−α2Ψ + β
=1√β
∫
dΨ
Ψ√
1 − α2β
Ψ2=
1√β
∫
dz
z√
1 − z2, (2.20)
where we have renamed z =√
α2β
Ψ. The integration is straightforward once the substitu-
tion z = sech(t) is made. We find
∫
dΨ
Ψ√
−α2Ψ + β
=1√β
arcsech
(√
α
2βΨ
)
. (2.21)
The stationary wavefunction is therefore
Ψ(x) =
√
2β
αsech
[
√
β(x− x0)]
. (2.22)
12 Exact results
This wavefunction belongs to L2(−∞,+∞) thus represents a bound state of the system.Now we need to impose the normalization condition to the wavefunction: ‖Ψ‖2 = 1. Thenorm of the wavefunction in L2 is
‖Ψ‖2 =2β
α
∫ +∞
−∞sech2
(
√
β x dx
)
=
[
2√β
αtanh
(
√
βz
)
]+∞
−∞=
4√β
α, (2.23)
which requires that4√β
α= 1. By substituting back α and β we find some information
about this bound state:4h
gN
√
−E2m
= 1 . (2.24)
This relation yields the energy of this bound state
E = −mg2N2
8h2 (2.25)
which is negative as it should. This calculation is carried out also in [4] where it describesthe evolution of an electromagnetic wave.
Chapter 3
The Hydrodynamic Limit
The hydrodynamic approach is a useful tool to study the interesting quntum dynamicsprovidede by the GPE. Here we show that the time-dependent Schroedinger equation isequivalent to the Euler equations for the fluid dynamic. We identify the square modulus ofthe wavefunction as the density ρ of the fluid and the spatial derivative of the wavefunctionphase as the velocity v:
ψ(x, t) =√
ρ(x, t)eiθ(x,t), v =∂
∂xθ(x, t). (3.1)
The GPE equation for this fluid is immediatly written in terms of ρ and θ:
ih∂
∂t
(√ρeiθ
)
= − h2
2m∇2
(√ρeiθ
)
+ gρ(√
ρeiθ)
(3.2)
For brevity we use the notation f ′(x) = ∇f and f = ∂∂tf :
ihρ
2√ρeiθ − h
√ρeiθθ = (3.3)
= − h2
2m
[
−ρ′2eiθ
4ρ3
2
+ρ′′eiθ
2√ρ
+iρ′eiθ
2√ρθ′ +
ieiθ
2√ρρ′θ′ −√
ρeiθθ′2 + i√ρeiθθ′′
]
+ gρ(√
ρeiθ)
.
(3.4)Multiplying both sides by e−iθ2
√ρ we obtain:
ihρ− 2hρθ = − h2
2m
[
−ρ′2
2ρ+ ρ′′ + 2iρ′θ′ − 2ρθ′2 + 2iρθ′′
]
+ 2gρ2. (3.5)
Now we split the real and the imaginary part of the equation, leading to a system of twodifferential equations:
hρ = − h2
2m[2ρ′v + 2ρv′]
2hρθ = h2
2m
[
ρ′2
2ρ+ ρ′′ − 2ρv2
]
− 2gρ2,
(3.6)
14 The Hydrodynamic Limit
Where we have operated the substitution v = ∇θ.
ρ+ hm∇ (ρv) = 0
hθ + gρ+ h2
m
[
v2 − 12√ρ∇2√ρ
]
= 0,(3.7)
ρ+ hm∇ (ρv) = 0
h ddtv + g∇(ρ) + h2
m∇[
v2 − 12√ρ∇2√ρ
]
= 0,(3.8)
Equation (3.8) only involves the scalar field ρ(x, t) and the irrotational vector field v(x, t).In this hydrodynamical form the GPE is apparently more complicated, but it has theadvantage of involving real fields, of transparent physical significance. Simple approxima-tions can be applied [5] .
3.1 A local analysis
Our purpose is to use the hydrodynamic approach to study locally the boson cloud ina neighborhood of the center. We have a particular interest in the point x = 0 becausenumerical simulation indicate that 1D and 3D differ in particular in keeping/loosing thedensity maximum at the center, for strong interaction. The ”depletion” of the cloudis signaled by a sign change of the second spatial derivative of the density d2
dx2ρ. Thehydrodynamical equations (3.8) for large interaction g should approximately produce thesame evolution if the quantum pressure term,in square brackets in Eq. (3.8) is neglected.The equations then take the Euler form
ρ+ ∇ (ρv) = 0
v + g∇(ρ) = 0.(3.9)
We analyze a free expansion, in both 1 and 3D, starting from a real Gaussian wavepacket:by symmetry we can safely assume that the center of the expanding cloud remains sta-tionary at all times: at x = 0, ρ′(0, t) = 0 ∀t.
3.1.1 1D
We start our analysis by noting that ρ is even and v is odd for exchanging x → −x.Similar properties apply to the derivatives of ρ and v. We apply a third-order Taylorexpansion of the first equation of the first equation (3.9) around x = 0:
ρ+ ρ′v + v′ρ ≈ ρ(t) + v′(t)ρ(t) +x2
2[ρ′′(t) + 3ρ′′(t)v′(t) + v′′′(t)ρ(t)] + o(x4) = 0, (3.10)
v(t) + gρ′(t) ≈ x [v′(t) + gρ′′(t)] + o(x3) = 0, (3.11)
3.1 A local analysis 15
where we have omitted the x = 0 indication for brevity.
We have now three differential equations which describe the evolution of the densityand of its derivative in a neighborhood of the origin, two of which come from Eq. (3.10)order by order
ρ(t) + v′(t)ρ(t) = 0 (3.12)
f(t) + 3f(t)v′(t) + v′′′(t)ρ(t) = 0 (3.13)
v′(t) + gf(t) = 0. (3.14)
Here we have renamed f = ρ′′, as the behaviour of the sign of this function is the centralpoint. Looking at the equations we can have a rough idea about the possibilities of itsevolution. Since the particles repel each other, at the beginning of the expansion v′ ispositive, i.e. the particles dash away from the origin. Moreover we expect it keeping itssign even if the depletion occurs: then the term 3f(t)v′(t) pulls f on the x axes frombelow. The effect of the term v′′′(t)ρ(t) is harder to determin and could have differenteffects on the sing of f .
3.1.2 3D
Carrying on the same analysis in the 3D isotropic expansion allow us to verify the sensitiv-ity of the method. In the symmetric 3D expansion, the density is a function of the distanceonly, ρ = ρ(r, t), and the velocity vector ~v has only the radial component ~v ≡ vrur . Thusthe ∇ appearing in the equations 3.9 takes the place of the differential spatial operator d
dx:
ρ+ vr∂ρ∂r
+ ρ · 1r2
∂∂r
(r2vr) = 0
vr + g ∂∂rρ = 0
(3.15)
In spherical symmetry we can use the notation f ′(r) = ∂f(r)∂r
and v = vr without causingambiguity.
ρ+ vρ′ + v′ρ+ 2ρvr
= 0
v + gρ′ = 0(3.16)
Comparing the Eq. (3.16) with the 1D version (3.8) we notice that the only difference isthe presence of the term 2ρv
r. Following the same procedure as above we use the Taylor
expansion on the numerator of the fraction:
2ρv = 2 ·(
ρ(t) +r2
2ρ′′(t) +O(r4)
)(
rv′(t) +r3
6v′′′(t) +O(r5)
)
(3.17)
= 2rρ(t)v′(t) +r3
3ρ(t)v′′′(t) + r3ρ′′v′ +O(r4) . (3.18)
16 The Hydrodynamic Limit
Now we can insert this term in Eq. (3.16), using the expansion (3.10) with r in place ofx:
ρ+ vρ′ + v′ρ+ 2ρv
r= ρ+ 3ρv′ +
r2
2
[
ρ′′ + 5v′ρ′′ +5
3ρv′′′
]
= 0 . (3.19)
The comparison of this last equation with Eq. (3.10) shows no qualitative difference, butfor the numeric coefficients. Owing to this we conclude that this method does not permitto demonstrate analitically the difference among the 1D and the 3D expansion which isno especially surprising for a completely local analysis.
Chapter 4
Numerical Analysis
The bulk of our work has consist of numerical simulation of the BEC expansion. Asmentioned above, to trace the wavefunction at later times we use two programs one forthe pure 1D and one for the 3D cylindrically expansion.In this chapter we are going to display the results and to compare them. From here on,we will refer to the square moudulus of the wavefunction |ψ|2 as ρ, the density of the gas.
4.1 Notes about the Programs and Scaling
The numerical programs we use to integrate the GPE are both based on the Cranck-Nicholson algorithm [6] . The wavefunction is represented in a finite vector, and also thetime is discretizedin finite steps.We have carefully tested the convergence of the resultsagainst the discretization steps. As is often the case, here we use scaled variables: threephysical natural scales are selected so that all variables are expressed in combination ofthose quantities. As it was explained before the fundamental equation of our system isthe GPE (1.15). Even if our system is not spherical symmetric during its evolution it isin the initial state: the scaling will be carried on the isotropic GPE. We scale the spatialcoordinate r and the time t substituting them with the new variables
r = br (4.1)
t = ct. (4.2)
With this substitution, for the condition of normalization (1.11) we get ψ(r, ·) = ψ(r,·)b3/2 ,
thus the GPE with the new variables and with an armonic potential is:
ih∂
c∂tψ(r, t) = − h2
2m
1
b2r
∂
∂r
(
r∂ψ(r, t)
∂r
)
+mω2b2r2
2ψ(r, t)+
4πasmb3
N |ψ(r, t)|2ψ(r, t). (4.3)
A great semplification is obtained if the conditions
hc
mb2= 1 (4.4)
18 Numerical Analysis
00.10.20.30.40.5
|Ψ(z
)|2 t=0t=0.7t=2
00.10.20.30.40.5
|Ψ(z
)|2 t=0t=0.7t=2
-10 -5 0 5 10z
00.10.20.30.40.5
|Ψ(z
)|2 t=0t=0.7t=2
g=0
g=10
g=100
Figure 4.1: Expansion for different values of g : 0, 10, 100 from top to bottom. Here wehave drawn the earlier wavefunction (straight line) and the ones at time 0.7 (point-dashed)and at time 2 (dashed).
mω2
h(cb2) = 1 (4.5)
are satisfied: to obtain this we choose
c =1
ω(4.6)
b =
√
h
mω(4.7)
From here on we will use r and t in the place of r and t, but will omit the tildes: thesystem will be ruled by equation (1.15) with all numerical constants set to unity.
4.2 Pure 1D with repulsive particles
In this section we restrict ourselves to the repulsive case, g > 0. We take a Gaussian initialstate and we have evolve it in time, the particular choice of the initial state allows us tocompare this numerical calculation with the non-interacting one which has been solvedanalytically. In figure 4.1 we can observe directly the effect of the internal repulsion.The largerer is g the faster the broadening of the wavefunction is; moreover the Gaussianprofile which is conserved in the non-interacting cloud (Eq. 2.8) here is modified rapidlylost due to the interaction. A part from these expected consequences no particular effectsare shown in the fig. 4.1; even with raise the value of g (as large as 2000) the only changeis the growing in the velocity of the evolution with no sign of depletion.
To clarify this difference we attempt a different kind of expansion as explained in Sec-tion 1.2, a middle way between the pure 1D and the isotropic 3D. We take a 3D cloud ofbosons and let it expanding only in free direction, say z, confining the others, say x and y
4.3 Cylindrically-confined wavefunction 19
0 1 2 3 4 5 6r
0
0.005
0.01
0.015
0.02
0.025
|Ψ|2
t=0.4g=100g=250g=500g=1000
Figure 4.2: 3D isotropic expansion at fixed time t = 0.4 for different values of g :100, 250, 500, 1000.
with an armonic potential: here the expansion is actually one-dimensional but the othertwo dimension are not ignored.
4.3 Cylindrically-confined wavefunction
This particular cigar-shaped kind of evolution allow more choices then the 1D or 3Disotropic ones for the initial wavefunction: we have used two different initial states. Thefirst is the isotropic one:
ψ(z, r) = f(z) · 1√πe−r
2/2, (4.8)
where f(z) =1
4√πe−z
2/2. (4.9)
The second initial state takes the form
ψ = exp[− (x2 + y2)]
2σ2(z)· f(z), (4.10)
where σ2(z) is given (in natural units) by
σ2(z) =√
1 + g|f(z)|2 = σz. (4.11)
20 Numerical Analysis
0
0.25
0.5
ρ(z)
t=0t=0.5t=1t=1.7
0
0.25
0.5
ρ(z)
t=0t=0.5t=1t=1.7
-12 -6 0 6 12z
0
0.25
0.5
ρ(z)
t=0t=0.5t=1t=1.7
1D
Cylind.
Cylind. σz
Figure 4.3: Different expansions, changing the dimension and the initial state; all thesimulation have been made with g = 100.
The choice in equations (4.10) (4.11) is motivated by the non-linear Schrodinger equationapproach of Refs [7] [2]. Figure 4.3 compares the 1D evolution and the 3D cylindricallyconfined one, using both discussed inital states (4.8) (4.10). The 3D plottes represent theradially integrated function (1.17).
The simulated wavefunctions show no important qualitative differences: the behavioris always, a rapid broadening of the wavepacket; this is what we could expect, and meansthat the 1D evolution is not a bad description of the full solution in 3D. Some differencesemerge mainly in the speed of the broadening. The 1D is the fastest: this is due to aneffective ’compression’ we have made, squeezing all the bosons in one dimension only andmaking g effectivly larger there.
It might be difficult experimentaly to actually access the internal density of cloud,though we can take some advantage of our theoretical tool to explore the internal dynamicsof the gas expansion The figures 4.4 and 4.5 show the density along the cylinder axis,|ψ(r = 0, z)|2 in successive times of the expansion. These figures show that even a 1Dexpansion actually shows some degree of depletion (this is illustrated more clearly in Fig4.6 which is however washed out by the lateral integration.
Figure 4.5 reports the second part of the evolution, where another interesting fact isdetectable. Around t = 1.5 the wavefunction along the axis begins to increase globally:this effect is caused by the radial component which tends to return to the axis repelledback by the harmonic confining potential. The radial evolution at z = 0, |Ψ(r, z = 0)|2,shown in Fig 4.7 is similar to that of |Ψ(r = 0, z)|2, also here we observe depletionsand a final growing (fig 4.7). However in this case the explanation is very different: the
4.3 Cylindrically-confined wavefunction 21
0
0.25
0.5
ρ(z)
t=0t=0.3t=0.5t=0.7
-4 -2 0 2 4z
0
0.06
0.12
0.18|Ψ
(r=
0,z)
|2 t=0t=0.3t=0.5t=0.7
Figure 4.4: The initial part of the evolution of the interacting (g = 100) condensate withisotropic initial wavefunction. The two panels compare the integrated density ρ(z), (Eq.(1.16)), and the density along the cylindrical axis |ψ(0, z)|2 .
0
0.05
0.1
0.15
0.2
ρ(z)
t=1.2t=1.6t=2
-6 0 6z
0
0.01
|Ψ(r
=0,
z)|2
t=1.2t=1.6t=2
Figure 4.5: Same as fig 4.4, at later times.
22 Numerical Analysis
-2 0 2z
0.01
0.02
0.03
0.04
|Ψ(r
=0,
z)|2
Figure 4.6: Zoom on the depletion in fig (4.4). Same times: t = 0, 0.3, 0.5, 0.7
depletion occurs despite the confining potential, indicating that the initial wavefunctionwas too compressed and had a strong tendency to expand (see eq (2.9), the speed of thebroadening is inversely proportional to the initial narrowing). After this first expansionhowever the potential pull back the bosons which return to the origin, giving birth to thegrowth observed in Fig 4.5.
For strong enough interaction, the Fig. 4.7 depletion shown for the radial cut atz = 0, in Fig. 4.8 is also visible in the z-integrated density ρ(r) and should therefore beobservable experimentally. The initial depletion of the radial wavefunction, even the oneintegrated in the axial direction, is a hint that we have taken an initial wavefunction toonarrow: when a large repulsion is considered, the bosons form a radial shockwave too.
This problem is conditioned by the choice of initial state: now we consider the modu-lated gaussian (4.10), we study a strongly interacting case, g = 500. The evolution of ther-integrated ρ(z) doesn’t show special differences with the previous behaviors (Fig 4.9).On the contrary, ρ(r) (fig 4.10) initially shrinks, indicating that the initial radial widthis too large: at later times it keeps growing but forming the depletion in the center, andit ends filling the central hole. We can suppose that the sigma modified has enlargedtoo much the initial wavefunction, costrained by the confining potential to return to thecentre.
In analogy with Figs 4.4 and 4.5 we plot in Fig. 4.11 the comparison between theintegrated density and the square-modulus of the wavefunction in r = 0 for g = 100.|ψ(0, z)|2 shows a peculiar evolution mainly due to the drop of the density caused by therapid expansion: after an initial broadening it starts expiriensing the return of the densityto the cylinder axis, thus growing back.
4.3 Cylindrically-confined wavefunction 23
0 2 4r
0
0.05
0.1
0.15
|Ψ(z
=0,
r)|2
t=0t=0.1t=0.3t=0.5
0 2 4 6r
0
0.01
|Ψ(r
,z=
0)|2
t=1.4t=1.8t=2
Figure 4.7: Evolution of the radial square-modulus of the wavefunction for g = 500 inz = 0.
0 5 10r
0
0.05
ρ(r)
t=0.3t=1.4t=1.7t=2.0
Figure 4.8: Several frames of the evolution of the radial density ρ(r) Eq (1.16) for g = 500.
24 Numerical Analysis
-10 0 10z
0
0.1
0.2
0.3
0.4
0.5
0.6
ρ(z)
t=0t=0.5t=1.2t=2
Figure 4.9: Evolution of the radial density with the σ of Eq.(4.11).
0 2 4r
0
0.06
0.12
ρ(r)
t=00.50.9t=1.3
Figure 4.10: Evolution of the radial density with the radial σ of Eq.(4.11).
4.4 Discussion 25
0
0.25
0.5
ρ(z)
t=0t=0.7t=1.6t=2.2
-10 0 10z
0
0.05
|Ψ(r
=0,
z)|2 t=0
t=0.7t=1.6t=2.2
Figure 4.11: Evolution of the axial wavefunction with the radial σ of Eq.(4.11) .
4.4 Discussion
The present calculations show that, even though 1D expansion is different to the 3D one innot showing depletion at the center, in fact the actual density does deplete when it is ana-lyzed in a realistic 3D context, but the laterally-integrated does not. This effect might beobservable in experiment by looking along the axial direction, rather then perpendicularlyto it.
26 Numerical Analysis
Bibliography
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[2] L. Salasnich, A. Parola and L. Reatto, Phys. Rev. A 65 043614 (2002).
[3] J. J. Sakurai, Modern Quantum Mechanics, Addison-Wesley, (1994).
[4] A.C. Newell, J.V. Moloney, Nonlinear Optics, Addison Wesley, (1992).
[5] G. Diana, N. Manini, and L. Salasnich, Phys. Rev. A 73, 065601 (2006).
[6] A. Brown and W. J. Meath, Phys. Rev. A 65, 060702 (2002).
[7] L. Salasnich, A. Parola and L. Reatto, Jour. of Phys. B 39, 2839 (2006)
