Bilkent University

Senior Thesis 2 - Final ReportNumerical Solution of the non-Local Gross-Pitaevskii Equation

Author:S. Furkan Ozturk

Supervisor:Assoc. Prof. M. Ozgur Oktel

5 February - 14 May 2018

Contents

1 Introduction 2

2 Theory 22.1 Dipolar Gross-Pitaevskii Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Dipolar Energy Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Fourier Transform of the Dipolar Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Variational Energy Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 Real Space Hartree Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Numerical Solution of the Dipolar GP Equation 113.1 Imaginary Time Evolution for Contact Interaction in 3D . . . . . . . . . . . . . . . . . . . . . . . 113.2 Dipolar Thomas-Fermi Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Imaginary Time Evolution for Dipolar Interaction in 3D . . . . . . . . . . . . . . . . . . . . . . . 13

4 Conclusion 15

5 Future Scope of the Project 16

1

Abstract

The effect of of interactions on the physical dynamics of many-body systems is crucial. Dipole-dipoleinteraction is one that creates many new interesting emergent properties in cold atoms due to its long rangeand anisotropic character. Recent observations [1-4] have revealed two new kinds of self bound droplets thatare stabled by repulsive and attractive forces. Yet their experimental observation cannot be justified fromthe conventional Gross-Pitaevskii theory. This senior thesis aims to generate a computational scheme thatis based on the Gross-Pitaevskii equation including the higher order fluctuations to justify the existence ofself-bound quantum droplets. For that purpose, this project involves the development of an imaginary timeevolution code to calculate the ground state density profile of a Bose gas with dipolar interactions, to formthe theoretical basis of the recent observations.

1 Introduction

In this second part of the senior thesis, numerical analysis of the Gross-Pitaevskii equation is advanced to dipolarBose gases. In the first part, various methods are developed to tackle the nonlinear Gross-Pitaevskii equationin the presence of contact interactions modeled by the delta function pseudo-potential. The nature of theinteractions among atoms are responsible for various emergent properties of the quantum gases. Therefore, theproperties of the interactions are studied extensively. In order obtain even more interesting and unprecedentedproperties, gases with a different interaction, the dipolar interaction, is studied in this thesis. Having acquiredvarious skills to solve the GP equation with contact interaction, the numerical solution of the dipolar quantumgas is going to be carried out and the ground state density profile is going to be sought. In this progress report,the theoretical analysis of the Gross-Pitaevskii equation with dipole-dipole interaction is made. The propertiesof the dipolar interaction is analyzed and the dipolar energy functional in Fourier space is studied. In order tounderstand the effect of physical parameters on the system and to experience numerical complexities, variationalenergy calculation is made. The effect of dipolar interaction strength and the coefficient of trap anisotropy onthe density profile is analyzed. Moreover, the dipolar term in the Hamiltonian, the so-called Hartree potentialis generated in real space. Imaginary time evolution code for contact interactions, which was developed for 1and 2D in the last semester, is extended to 3D. Also, the Thomas-Fermi limit is studied for the dipolar gases tocompare the numerically found density profile. After all, the imaginary time evolution code to find the groundstate density profile of a 3D dipolar Bose gas is written. For various parameter regimes the numerically founddensity profile has been tested. Having reached the aim of the senior thesis, which is to create the computationalframework to tackle the dipolar GP equation, we are going to try to account for the existence of self-bounddroplets.

2 Theory

2.1 Dipolar Gross-Pitaevskii Equation

Dipolar interaction leads to rich emergent properties and unlike the short range contact interaction it has aslightly complicated expression.

For two particles with dipole moments along the unit vectors e1 and e2, and whose are separated by r, thepotential energy due to the dipole–dipole interaction reads:

Udd(r) =Cdd4π

(e1 · e2)− 3(e1 · r)(e2 · r)r3

(1)

Where Cdd is the dipolar coupling constant. When all the dipoles are aligned in the z direction, as shownin the figure above, the dipolar interaction simplifies to the following expression:

Udd(r) =Cdd4π

1− 3 cos2 θ

r3(2)

2

The dipolar interaction has two significant properties that result in many emergent physical phenomena.One is its long range, ∼ 1/r3, and the other is its anisotropic nature. Unlike short range contact interactionwhich is pronounced when the particles are close to each other compared to atomic scale, dipolar interaction islong-range in its nature. This property leads to the fact that the energy per particle of a dipolar gas does notonly depend on the particle density, but also on the number of particles. The anisotropic nature on the otherhand is due to the 1− 3 cos2 θ term and as θ varies between 0 and π/2, this term varies between −2 and 1. Asa result the dipolar interaction becomes repulsive, for θ = π/2, for dipoles standing side by side, however forθ = 0, for dipole sitting up and down, it is attractive.

Having included the contact interaction and dipolar interaction as the interaction potential energy, theGross-Pitaevskii equation becomes:[

−h2

2m∇2 + Vtrap (r) + Uint (r)

]Ψ (r) = µΨ (r) (3)

where Uint is,

Uint(r) =

∫d3r′Ψ∗(r′)U(r − r′)Ψ(r′)

= g |Ψ|2 +

∫d3r′ |Ψ(r′)|2

[Cdd4π

1− 3 cos2 θ′

|r − r′|3

] (4)

For a polarized sample where all the dipoles are along the z direction.

2.2 Dipolar Energy Functional

The time dependent Gross-Pitaevskii equation within the mean-field approximation is given as below.

ih∂ψ

∂t= − h2

2m∇2ψ +

(Vtrap + g |ψ|2 + Φdd

)ψ (5)

The non-linear term proportional to g is due to the contact interaction in the mean-field limit. The externalpotential V is usually the trap potential. Moreover, dipolar effects is included as an extra interaction term tothe mean-field interaction potential g|ψ|2 and denoted by Φdd.

Φdd(r, t) =

∫|ψ(r′, t)|2 Udd(r − r′)d3r′ (6)

From the time-dependent GP equation the below given energy functional is obtained from which the time-independent form can be derived via minimization.

E [ψ] =

∫ {− h2

2m|∇ψ|2 + Vtrap |∇ψ|2 +

g

2|∇ψ|4 +

[1

2|ψ|2

∫Udd(r − r′) |ψ(r′)|2 d3r′

]}d3r (7)

The dipolar interaction term is under special interest and it represents the dipolar interaction energy betweenthe dipole density at r and at r’ as shown in the diagram above.

The dipolar term per se is going to be analyzed. The dipolar variational energy is given as below:

3

Edd [ψ] =1

2

∫ ∫d3rd3r′n(r)n(r′)

Cdd4π

1− 3cos2θr,r′

|r − r′|3(8)

where n(r) and n(r′) are the local particle densities at r and r′ respectively and the pre-factor 1/2 is attachedto avoid overcounting. Minimizing the energy the functional given above and finding out the correspondingdensity profile is a convenient approach that enables to understand the effect of parameters on the physics ofthe system. Also, by this way the numerical complexities are going to be faced from the beginning and thesolution strategies are going to be applied before the tackling the equation numerically. However, due to severalnumerical complexities that are going to be discussed the variational minimization is done not in the real spacebut in the Fourier space. This transform is scrutinized in the next section.

2.3 Fourier Transform of the Dipolar Interaction

The dipolar energy functional (8) is hard to tackle numerically in real space. Due to the 1/r3 nature of the dipolarinteraction the energy blows up at small distances and it changes its sign due to aforementioned anisotropydrastically based on the choice of increment ∆r. This drastically change based on the space increment changesthe value of the integral significantly especially at small distances and results in a significant numerical error.However, in Fourier space the nature of the interaction becomes much simpler and numerical implementationbecomes easier.

In real space, as shown in (8) the dipolar part of the energy functional is given as:

Edd =1

2

∫ ∫d3rd3r′n(r)n(r′)Udd(r − r′)

In order to rewrite this equation in Fourier space following transformations are introduced.

n(r) =1

(2π)3

∫n(k1)eik1·rd3k1 (9)

n(r′) =1

(2π)3

∫n(k2)eik2·r

′d3k2 (10)

Udd(r − r′) =1

(2π)3

∫U(k3)eik3·(r−r

′)d3k3 (11)

Now using transformed variables given in (8),(9), and, (10) the above given Edd expression is written as:

Edd =1

2

(1

(2π)3

)3 ∫ ∫d3rd3r′

∫ ∫ ∫d3k1d

3k2d3k3n(k1)n(k1)Udd(k3)ei(k1+k3)·rei(k2−k3)·r (12)

Each space integral creates a delta function and uses a pre-factor 1/(2π)3 and simplifies (12) to the belowexpression.

Edd =1

2

1

(2π)3

∫ ∫ ∫d3k1d

3k2d3k3n(k1)n(k1)Udd(k3)δ(k1 + k3)δ(k2 − k3) (13)

Obviously each delta function drops out an integral in k space and at the end only one integral remainswritten in the Fourier space.

Edd =1

2

1

(2π)3

∫d3k |n(k)|2 Udd(k) (14)

Here n(k) is the density in Fourier space which can be found numerically and Udd(k) is the dipolar interactionenergy in Fourier space which can be found analytically.

In order find Udd(k) below shown coordinate choice is made and following variables are introduced.

4

In this coordinate choice, the first dipole is confined to the xz plane and aligned parallel to the second dipole.Also the Fourier space vector k is chosen in the z direction.

Udd(r) =Cdd4π

(e1 · e2)− 3(e1 · r)(e2 · r)

r3=Cdd4π

1− 3 cos2(γ)

r3(15)

Accordingly, the Fourier transform of the interaction potential is given as:

Udd =

∫d3r

Cdd4π

1− cos2 γ

r3e−ik·r (16)

In order to solve this integral, usage of the below given geometrical expressions is crucial.

e1 = x sin(α) + z cos(α)

r = x sin(θ) cos(φ) + y sin(θ) sin(φ) + z cos(θ)

cos(γ) = e1 · r = sin(α) sin(θ) cos(φ) + cos(α) cos(θ)

k · r = cos(θ)

Having put these into [16], the integral can be evaluated step by step. dφ integral is trivial. After that,having made the variable changes cosθ = u and kr = β the integral becomes:

Udd =Cdd2

∫ 1

−1

∫ ∞β0

e−iβu

β

[1− 3

2sin2 α+ 3u2

(1− 1

2sin2 α

)]dβdu

Where β0 = kr0 is a small cut-off distance to avoid divergences when dipoles are very close. In order to find∫ 1

−1u2e−iβudu second derivative of

∫ 1

−1e−iβudu with respect to β is considered. Having evaluated the u integral

one gets,

Udd = Cdd(1− 3 cos2 α)

∫ ∞β0

(sinβ

β2+

3 cosβ

β3− 3 sinβ

β4

)dβ

The part in the integral is just a number.∫ ∞β0

(sinβ

β2+

3 cosβ

β3− 3 sinβ

β4

)dβ =

cosβ0

β20

− sinβ0

β30

As β0 is a small number introduced to as a cut-off, its limit to 0 can be considered. In this limit the aboveexpression becomes −1/3. Thus the Fourier transform of the dipolar interaction is:

5

Udd = Cdd(cos2 α− 1/3) (17)

where α is the angle between k and the polarization direction of the dipole. In 3D the Fourier transformof the dipole-dipole interaction is independent of the norm of the wave-vector k and this nature simplifies thenumerical implementation in k-space.

2.4 Variational Energy Calculation

Variational calculation gives a quantitative picture of the dipolar Bose gas and helps us to understand parameterregimes and critical values of them. Also, in preparation for tackling the GP equation numerically to find theground state properties, it is informative to study the variational approach.

For this purpose, as the variational condensate wavefunction, a Gaussian with a radial size of σρ and anaxial size of σz, normalized to the particle number N is considered.

Ψ(x, y, z) =

√N

π3/2σ2ρσz

exp

[−1

2

(x2 + y2

σ2ρ

+z2

σ2z

)](18)

Edd =1

2

∫ ∫d3rd3r′n(r)n(r′)Udd(r − r′)

=1

2

∫ ∫ (N

π3/2σ2ρσz

)2

exp

(−x

2 + y2

σ2ρ

− z2

σ2z

)exp

(−x′2 + y′2

σ2ρ

− z′2

σ2z

)Cdd4π

1− 3 cos2 θ

|r − r′|3d3rd3r′

(19)

By introducing the scaling r = r/σρ and r′ = r′/σρ equation (19) is rewritten.

Edd =1

2σ3ρ

∫ ∫N2κ2

π3exp

(−x2 − y2 − κ2z2

)exp

(−x′2 − y′2 − κ2z′2

) Cdd4π

1− 3 cos2 θr,r′

|r − r′|3d3rd3r′ (20)

where κ = σρ/σz. Now in order write this equation in Fourier domain, following Fourier transform pairs areused:

Nκ

π3/2exp

(−x2 − y2 − κ2z2

)↔ Nπ3/2

κexp

[1

4

(−k2

x − k2y − k2

z/κ2)]

(21)

Cdd4π

1− 3 cos2 θr,r′

|r − r′|3↔ Cdd

(cos2 α− 1/3

)(22)

Using Fourier domain transformation pairs (21) and (22) as in equation (14) we get:

Edd =1

2σ3ρ

CddN2π3

κ2

1

(2π)3

∫exp

(−1/2(k2

x + k2y + k2

z/κ2)) (

cos2 α− 1)d3k

=1

2

N2Cdd8σ3

ρκ2I(κ)

(23)

where I(κ) is the integral that depends only on the aspect ratio of the cloud κ which is going to be evaluatednumerically.

While variational calculation of the energy is made, the real and Fourier space density profiles are plotted.Also the integrand of the calculation n(k)2U(k) is plotted in order to see if its geometry is compatible with itsanalytical expression or not. When κ < 1 the real space density is cigar shaped and the Fourier space densityis pancake shaped. Also the profile of the integrand is shaped by the cos2α− 1. They are presented below.

6

When κ = 1 the real and Fourier space densities are spherically symmetric. Also the dipolar energy densityintegrated over all k-space is 0 due to symmetry.

When κ > 1 the real space density is pancake shaped and the Fourier space density is cigar shaped. Alsothe profile of the integrand is shaped by the cos2α− 1. They are presented below.

7

I(κ) integral is analytically calculated to find the analytic equation of the variational energy and found as:

I(κ) = − (2π)3/2

3f(κ)κ (24)

where f(κ) is a function of the aspect ratio of the cloud.

f(κ) =1 + 2κ2

1− κ2− 3κ2 arctanh

√1− κ2

(1− κ2)3/2

Therefore the expected analytic relation between variational energy and the trap aspect ratio κ is found as:

Eexpdd = −1

2

√2π3/2

12

N2Cddσ3ρ

f(κ)

κ(25)

Even though the integral I(κ) seems easy to tackle numerically, sampling of the Fourier space should becarefully done. When κ > 1 for example the cloud is elongated in the z direction and thus the k-space in thez-direction must be sampled to include that elongated cigar shaped cloud. Thus kx, ky and kz directions mustbe extended differently, proportional to κ. Also the step sizes in real and Fourier space play a crucial role forcorrect numerical calculation. The real space (similarly the Fourier space) sampling for a step size of dx andfor a Gaussian to be sampled with a standard deviation of σx is done as shown below:

8

For the sampling to be accurate, σx � dx relation must be satisfied. As Nxdx = Lx, the inequality ofLx � σx must be satisfied where Nx is the number of points. Also, for the Fourier space, σkx � dkx mustbe true. We know that, dkx = 2π/dx and σkx = 2π/σx. Then, 2π = Nxdxdkx is the constraint for choosingdx and dkx. We need both space steps to be maximum for a given number of points. We cannot increase thenumber of points as much as we desire due to its computational cost. Thus the following choice is the best forreal and Fourier space:

dx = dkx =

√2π

Nx(26)

For our case, the parameters are scaled by σρ = σx = σy. Therefore σx = σy = 1 and σz = 1/κ. For that,the implementation of the real and k-space sampling is shown below:

1 Nx = 32; Nx=Ny; Nz=Nx; %Spatial Resolution2 Kappa = 2; %Cloud Aspect Ratio3

4 %Real Space5 dx = sqrt((2*pi)/Nx); %Space step in x6 dy = dx; %Space step in y7 dz = (1/Kappa)*dx; %Space step in z8

9 ax = dx*Nx; %Size in x10 xx = linspace(-ax/2,ax/2-dx,Nx);11 ay = dy*Ny; %Size in y12 yy = linspace(-ay/2,ay/2-dy,Ny);13 az = dz*Nz; %Size in z14 zz = linspace(-az/2,az/2-dz,Nz);15

16 %Fourier Space17 kx=linspace(-Nx/2,Nx/2-1,Nx); %Fourier space kx18 kkx=(2*pi/ax)*kx;19 dkx=kkx(2)-kkx(1); %Fourier space step in kx20 ky=linspace(-Ny/2,Ny/2-1,Ny);21 kky=(2*pi/ay)*ky;22 kz=linspace(-Nz/2,Nz/2-1,Nz);23 kkz=(2*pi/az)*kz;

2.5 Real Space Hartree Potential

In order to apply the imaginary time evolution scheme for dipolar gases, we need to numerically create thedipolar potential energy operator in the Hamiltonian, known as the Hartree potential. As discussed earlier, theimplementation in the Fourier domain is easier due to the functional nature of the dipolar interaction. Sinceit critically changes sign depending on the angle and diverges at small separation distances, its real space formcauses numerical difficulties which are not present in its Fourier domain representation. Also, we learned lotsof numerical techniques and solved several issues regarding real and Fourier space sampling. Therefore, theHartree potential is better calculated in Fourier space and then transformed back into the real space. Letsrecall the representation of the Hartree potential:

Φ(x, y, z) =

∫n(r′)Udd(r − r′)d3r′ (27)

In order to transform the Hartree potential into the Fourier space, following Fourier domain representationsare used.

n(r′) =1

(2π)3

∫n(k1)eik1·r

′d3k1 (28)

Udd(r − r′) =1

(2π)3

∫Vdd(k2)eik2·(r−r

′)d3k2 (29)

Inserting (28) and (29) into (27) and evaluating the Delta function, we get the real space form of the Hartreepotential in terms of the Fourier domain representations of the density and dipolar interaction potential insidea single integral.

Φ(x, y, z) =

∫d3k

(2π)3n(k)Udd(k)eik·r =

∫d3k

(2π)3n(k)Cdd

(cos2(α)− 1/3

)eik·r (30)

9

Which is simply the inverse Fourier transform of the Fourier domain Hartree potential, which is easy tocalculate, yet the implementation is slightly tricky. Here, regarding the implementation of the Fourier transformin MatLab one remark must be made. In MatLab (fft), both forward and inverse Fourier transforms considersthe first data of the array as the reference and calculates the rest accordingly. So if the data point correspondingto x = 0 is not the first data point of the array a discrepancy occurs, the Fourier transform becomes alternatingbetween positive and negative values. This was the case for us, as we created the real space from negative L/2to positive L/2 and had the x = 0 in the middle. To deal with this, before transforming the function, it isshifted by fftshift and the x = 0 point became the first data point of the function. Then this shifted functionis Fourier transformed and then shifted back. This solved the issue.

For a Gaussian real space density with various κ values, Hartree potential is calculated and plotted. Theyare presented below:

-20

-15

50

-10

5

-5

0

0

0

-50-5

-100 -10

-1.5

-1

5

-0.5

0

50

0.5

0

-5-5

-10 -10

-0.6

-0.4

4

-0.2

0

2

0.2

0.4

5

0.6

0

0.8

0-2

-5-4

-10

0

0.2

0.5

0.4

5

0.6

0

0.8

0

-0.5-5

-1 -10

Moreover, the Hartree potential is calculated for the Thomas-Fermi density of a dipolar gas, which is aninverted parabola to be introduced in the next section. The result is compared to the Hartree potential in [6].The plot of the calculated Hartree potential for the Thomas-Fermi density in x− z plane is given together withthe one in [6]:

-0.5

-6

-4

-2

5

0

0

02

4-5

6

0.5

10

3 Numerical Solution of the Dipolar GP Equation

So far, the theory of dipolar gas is widely studied. The nature of the dipolar interaction is explained and thedipolar GP equation is introduced. The ways to tackle the dipolar term is discussed and a variational analysisis made to understand the effect of parameters. Now, the real ground state of the 3D dipolar Bose gas is goingto be sought. For that purpose, imaginary time evolution scheme, which was extensively studied in the previousthesis, has been used.

3.1 Imaginary Time Evolution for Contact Interaction in 3D

Imaginary time evolution is an evolution in time defined as τ = it. When a wave function is evolved in tau, onlythe ground state contribution is left in the wave function as eigenfuncitons corresponding to higher energiesdecay faster than the ground state eigenfunction. Thus, when an arbitrary initial wavefunction is evolved inimaginary time and normalized at each time step, the real ground state is reached.

Imaginary time evolution scheme with time splitting spectral methods was the main numerical algorithmto find the ground state density profile of a Bose gas in the first part of the senior thesis. However, it wasdeveloped in 1 and 2 dimensions. Now, before attempting to find the ground state density profile of a dipolarBose gas, to understand the indexing and numerical details of MatLab in 3D, an imaginary time evolution codefor the short range GP equation is written in 3D. By tackling with contact interaction before including thedipolar one, complexities are surpassed step by step and code has developed with a greater confidence.

Imaginary time evolution for of a gas with short range interactions in an anisotropic trap is made. Groundstate density profile is found in 3D and presented as 3D contour plots using isosurface of MatLab. For a clearerview, cuts in each direction are taken and presented as 1D plots. Below, the ground state density profile for agas, in a trap with a coefficient of asymmetry γ = ωz/ωx = 2 is presented together with Thomas-Fermi density.Each plot corresponds to a separate cut in one spatial direction. Cuts are taken from the middle of the trap ineach direction.

-6 -4 -2 0 2 4 6

0

1

2

3

4

5

6

7

8

Numerical Solution

Thomas-Fermi

-6 -4 -2 0 2 4 6

0

1

2

3

4

5

6

7

8

Numerical Solution

Thomas-Fermi

-6 -4 -2 0 2 4 6

0

1

2

3

4

5

6

7

8Numerical Solution

Thomas-Fermi

As one can infer, the gas squeezed in the z direction two times more than x and y directions due to trapanisotropy. The overlap between Thomas-Fermi and the numerical solution is great as the particle number isN = 500. 3D contour plots are also drawn and they are instructive to understand the shape of the cloud. Forγ = 1 the cloud is spherical, for γ < 1 the cloud is pancake shaped and for γ > 1 the cloud is cigar shaped.This can be seen in the plots below. They correspond to a cloud with γ = 4, 0.25 respectively:

11

3.2 Dipolar Thomas-Fermi Density

Thomas-Fermi limit of a Bose gas is widely reached in the cold atom experiments when the kinetic energycontribution is negligible compared to interaction and trap potential energies. When the particle number ishigher, the Thomas-Fermi limit is more accurate. As shown in the previous thesis by a variational estimation, thekinetic energy contribution becomes 2 orders of magnitude smaller than the potential energy contribution evenfor N = 1000 particles. Therefore, when the kinetic energy term in the Hamiltonian is neglected beforehand,an accurate limit of the trapped BEC is reached. Furthermore, as shown by O’Dell et al in [7] an exact solutionof the dipolar GP equation with dipole-dipole and contact interactions in a harmonic trap, within the Thomas-Fermi limit is obtainable. In [7] the condensate density within the Thomas-Fermi limit is given as an invertedparabola:

n(r) = n0

[1− ρ2

R2x

− z2

R2z

](31)

for n(r) > 0 where n0 = 15N/(8πR2xRz).

Rx = Ry =

[15gNκ

4πmω2x

{1 + εdd

(3κ2f(κ)

2(1− κ2)− 1

)}]1/5

(32)

where εdd = Cdd/3g which is a dimensionless measure of the strength of the dipole-dipole interaction Cddcompared to the short range scattering energy g. To be able to use the above density in numerical calculationswe need to non-dimensional it. For our case, the characteristic length of the harmonic oscillator is a convenientscaling parameter. Having introduced a0 =

√h/mωx to non-dimensionalize the equation, we get:

Rx = Rx/a0 =

[15gNκ

4π

{1 + εdd

(3κ2f(κ)

2(1− κ2)− 1

)}]1/5

(33)

where g = g/(a30hωx). O’Dell et al solves the GP equation by resembling it to the Poisson’s equation within

the Thomas-Fermi limit and finds out that the aspect ratio of the gas, κ, is calculated by solving the followingtranscendental equation of εdd and γ.

3κ2εdd

[(γ2

2+ 1

)f(κ)

1− κ2− 1

]+ (εdd − 1)(κ2 − γ2) = 0 (34)

This equation needs to be solved for every εdd and γ which are the experimental parameters of the dipolarinteraction strength and trap anisotropy. The output κ both becomes a numerical parameter to scale our realand Fourier space as explained 2.4 and also is the aspect ratio of the Thomas-Fermi density which is a limitthat we use to check our numerical results. Eq. (34) is solved numerically. First a vector of κ is created from0 to γ, as γ is the highest value that κ takes when εdd = 0. Then for every εdd and γ the zeros of the equationare found and stored for that values. The results are plotted in the κ− εdd plane with the contours of γ. Belowthe plot is given together with the one in O’Dell et al [7] as a comparison.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Having solved the equation, all of the parameters needed to create the Thomas-Fermi density are obtained.For εdd = 0.5 and γ = 2 the density profile is plotted. Again, for clarity cuts in each direction are taken. Noticethe inverted parabolic nature and the effect of trap anisotropy on the density.

12

-4 -2 0 2 4

0

0.5

1

1.5

2

2.5

3

3.5

4

-4 -2 0 2 4

0

0.5

1

1.5

2

2.5

3

3.5

4

-2 0 2

0

0.5

1

1.5

2

2.5

3

3.5

4

3D figures for various parameters are given below which show the shape of the cloud in 3D. Also, above thefigures all of the three parameters,εdd, γ, κ are given and their effect on the shape of the cloud are visible.

3.3 Imaginary Time Evolution for Dipolar Interaction in 3D

The real ground-state of the dipolar Gross-Pitaevskii equation has been sought for and lots analytically andnumerical techniques are developed for that purpose. The main scheme is the imaginary time evolution withtime splitting spectral methods which can be summarized with the equation below:

Ψ(τ + dτ) = e−dτ2 (V+Us+Φdd)e−dτKe−

dτ2 (V+Us+Φdd)Ψ(τ) +O

(dτ3)

(35)

There, for each (imaginary) time step the wave function is evolved with a time splitted propagator. Theintegral time step evolved kinetic energy part is sandwiched between half time step evolved potential energyparts. Also, the kinetic energy part is evolved in the Fourier domain as it is computationally easier. Thereforethe evolution is better written as:

Ψ(τ + dτ) = e−dτ2 (V+Us+Φdd)

{F−1e−dτk

2/2F}e−

dτ2 (V+Us+Φdd)Ψ(τ) +O

(dτ3)

(36)

The kinetic and potential energy parts, being linear, are written once and then used to propagate the wavefunction. Their numerical implementation however is slight non-trivial and the indexing is crucial.

1 n run=1:(Nx*Ny);2 nx0=1:Nx;3 nx0=repmat(nx0,[1,Ny*Nz]);4 ny0=ceil(n run/(Nx));5 ny0=repmat(ny0,[1,Nz]);6

7 for n k=1:(Nx*Ny*Nz)8 nn=ceil(n k/Nx);9 nx=nx0(n k); ny=ny0(n k); nz=ceil(n k/(Nx*Nz));

10 ind=(nn-1)*Nx+1;11

12 %Kinetic Energy in dt as vector13 K(ind:(ind-1)+Nx) = exp(-1*(dt)*0.5*(KX(nx,:,nx).ˆ2+KY(ny,:,ny).ˆ2+KZ(nz,:,nz).ˆ2));14 %External potential in dt/2 as vector15 V(ind:(ind-1)+Nx) = ...

exp(-1*(dt/2).*(0.5*(XX(nx,:,nx)'.ˆ2+YY(ny,:,ny)'.ˆ2+gammaˆ2*ZZ(nz,:,nz)'.ˆ2)));16 end

On the other hand, the non-linear parts of potential energy Us = g|Ψ|2 and Φdd are always re-created (orevolved) when the wave function is being evolved. This is done under a while loop and iteration is done whenthe wave function is not changing compared to its previous value. While being evolved, the wave function is

13

multiplied by every term, one by one, in the propagator given in Eq. (36). The dipolar part of the propagatorhas the Hartree potential and takes the wave function as an input which is implemented in the code as separatefunction. Having taken the Ψ input, the evolution is done by propagating the wave-function by multiplyingthat term with the wave-function. Of course at the end of each step the wave-function is re-normalized as theimaginary time propagation does not conserve the normalization. Having normalized the wave function, thedifference between the previously calculated density and the newly found one is taken. When that difference issmaller than a particular value, which is the convergence parameter, the code stops running. The implementationof this process is given in the listing below.

1 while max dif > 0.0012 Psi0=Psi; n0=abs(Psi0).ˆ2; %Previous wave-function3 %H=V+U, without kinetic energy4 H = V + exp(-1*(dt/2)*(g*abs(Psi).ˆ2));5 %H(dt/2)6 Psi = H.*Psi;7 %Udd(dt/2)8 Psi=reshape(Psi,[Nx,Ny,Nz]);9 Psi=exp(-1*(dt/2)*Hartree v3(Psi,Kappa,Cdd,N,Nx)).*Psi;

10 %K(dt)11 HatPsi=fftshift(fftn(Psi,[Nx,Ny,Nz]));12 HatPsi=reshape(HatPsi,[Nx*Ny*Nz,1]);13 EvolvedHatPsi=K.*HatPsi;14 EvolvedHatPsi=reshape(EvolvedHatPsi,[Nx,Ny,Nz]);15 Psi=ifftn(ifftshift(EvolvedHatPsi),[Nx,Ny,Nz]);16 %Udd(dt/2)17 Psi=exp(-1*(dt/2)*Hartree v3(Psi,Kappa,Cdd,N,Nx)).*Psi;18 %H(dt/2)19 Psi=reshape(Psi,[Nx*Ny*Nz,1]);20 Psi=H.*Psi;21 %Normalization after imaginary time evolution22 norm=sum(abs(Psi).ˆ2)*(dx*dy*dz);23 Psi=sqrt(N)*(Psi/sqrt(norm));24 %Convergence Parameter25 max dif=max(n0-abs(Psi.ˆ2))26 end

The behavior of a dipolar Bose can be illustrated in an experiment with large number of atoms, in a harmonictrap with aspect ratio γ where the value of dipolar interaction strength relative to short range interactionstrength εdd is slowly increased from zero with an external magnetic field. The system more or less follows oneof the contour curves given in (3.2) within the Thomas-Fermi limit up to εdd = 1. When there is no magneticfield, εdd is zero and the condensate aspect ratio κ becomes exactly same with the trap aspect ratio γ. Onthe other hand, the condensate behavior for εdd > 1 is more complicated and beyond the Thomas-Fermi limit.After that value, the condensate becomes meta-stable or unstable and a more careful analysis is needed.

The code is run for various parameters and the resulting ground state density profile is compared to Thomas-Fermi density. The overlap is with Thomas-Fermi is higher for more number of particles and for 0 < εdd < 1where the gas is stable. For different values of εdd and γ the ground state density profile of a dipolar Bose gas isgiven below, together with the Thomas-Fermi density for that parameter settings, as cuts in 3 spatial directions.

For N = 10, 000 particles, and for εdd = 0.1, g = 0.1, and γ = 0.5 the density of the dipolar gas becomes:

-15 -10 -5 0 5 10 15

0

10

20

30

40

50

60

70 Numerical Solution

Thomas-Fermi

-15 -10 -5 0 5 10 15

0

10

20

30

40

50

60

70 Numerical Solution

Thomas-Fermi

-15 -10 -5 0 5 10 15

0

10

20

30

40

50

60

70 Numerical Solution

Thomas-Fermi

And in 3D, the numerical density and Thomas-Fermi density are given as:

14

The overlap with the Thomas-Fermi solution is great as there are N = 10, 000 particles and the calculationsare done with parameters corresponding to the Thomas-Fermi regime shown in figure in (3.2). For a differentset of parameters: εdd = 0.9, g = 0.1, and γ = 2 the shape of the cloud is different and more pancake shape,unlike above given cigar shaped cloud.

-8 -6 -4 -2 0 2 4 6 8

0

10

20

30

40

50

60

70

80

90

100Numerical Solution

Thomas-Fermi

-8 -6 -4 -2 0 2 4 6 8

0

10

20

30

40

50

60

70

80

90

100Numerical Solution

Thomas-Fermi

-8 -6 -4 -2 0 2 4 6 8

0

10

20

30

40

50

60

70

80

90

100Numerical Solution

Thomas-Fermi

And in 3D, the numerical density and Thomas-Fermi density are given as:

4 Conclusion

In this senior thesis, the Gross-Pitaevskii equation with dipole-dipole interaction potential is theoretically an-alyzed and variational energy calculation is made. It is found out that tackling the equation in Fourier spaceis useful due to numerical convenience hence the Fourier transform of the dipolar interaction is taken. Thisanalysis of the dipolar interaction enabled us to numerically implement the dipolar potential to the imaginarytime evolution scheme. To do this, first the imaginary time evolution scheme using time splitting spectralmethods is extended to 3D which was written for 1 and 2D. Then the Hartree potential, the part that accountsfor the dipolar interaction, is embedded in the potential energy part of the code as a separate unit. This unittakes the real space density as an input, then Fourier transforms it and calculates the Hartree potential in theFourier domain. Afterwards, it transforms the Fourier domain potential back into the real space. As a result,

15

the ground state density profile of a dipolar gas is found and a numerical framework to tackle the non-local GPequation is created.

5 Future Scope of the Project

Having found the density profile of a dipolar Bose gas and created a numerical framework to tackle the dipolarGross-Pitaevskii equation, we are going to try to account for the existence of self bound droplets. Thesedroplets are self bound to each other, like a fluid droplet balanced by the surface tension, without the need ofa trapping potential. The particles in a dipolar gas interact through short range repulsion g = 4πh2as/M and

long range dipolar potential, Vint(x) = g[δ(x) + 3εdd

4π|x|3

(1− 3 z2

|x|2

)], where εdd = Cdd/3g is the dimensionless

dipolar interaction strength expressed in terms of s-wave scattering length as. However the self-bound natureof the droplets cannot be explained by the Gross-Pitaevskii equation as the GP approximation becomes invalidwhen many-body correlations become significant, when the mean field theory collapses. Hence a non-orthodoxway of rendering the dipolar droplets is needed. Aybar and Oktel [8] extends the theory of dipolar droplets tonon-zero temperatures using Hartree–Fock–Bogoluibov theory, and propose the following Hamiltonian includingthe many-body correlations.

[− h

2∇2

2M+ Utr(x)− µ+

∫d3x′Vint(x− x′)

(|Ψ(x′)|2+n(x′)

)+

32

3g

√a3s

πQ5(εdd)|Ψ(x)|3

]Ψ(x) = 0 (37)

where Ql(εdd) =∫ 1

0du[1 + εdd(3u

2 − 1)]l/2 and n(x) = 83

√a3sπ Q3(εdd)|Ψ(x)|3 which is usually neglected in

droplet experiments.Therefore, the future scope of my work is to use the Hamiltonian put forward by Aybar and Oktel in the

imaginary time evolution scheme, without a trapping potential, and try to find out if the resulting densityprofile matches with the profile of the observed self-bound quantum droplets.

References

[1] arXiv:1708.07806v1

[2] M. Schmitt, M. Wenzel, B. Bottcher, I. Ferrier-Barbut, and T. Pfau, Self-bound droplets of a dilute magneticquantum liquid. Nature 539, 259 (2016).

[3] D. Baillie, R. M. Wilson, R. N. Bisset, and P. B. Blakie, Self-bound dipolar droplet: A localized matterwave in free space. Phys. Rev. A 94, 021602(R) (2016).

[4] I. Ferrier-Barbut, H. Kadau, M. Schmitt, M.Wenzel, and T. Pfau, Observation of quantum droplets in astrongly dipolar Bose gas. Phys. Rev. Lett. 116, 215301 (2016).

[5] Lahaye, T., Menotti, C., Santos, L., Lewenstein, M. Pfau, T. The physics of dipolar bosonic quantum gases.Rep. Prog. Phys. 72, 126401 (2009).

[6] Stuhler J, Griesmaier A, Koch T, Fattori M, Pfau T, Giovanazzi S, Pedri P and Santos L. Observation ofDipole-Dipole Interaction in a Degenerate Quantum Gas. Phys. Rev. Lett. 95 150406 (2005).

[7] C. Eberlein, S. Giovanazzi, and D. H. J. O’Dell, Exact solution of the Thomas-Fermi equation for a trappedBose-Einstein condensate with dipole-dipole interactions. Phys. Rev. A 71, 033618 (2005).

[8] Aybar, E., Oktel, M. O. Temperature Dependent Density Profiles of Dipolar Droplets. Manuscript in prepa-ration. (2018)

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