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Formation of quantum vortices in a rotating

sphere in an electromagnetic ﬁeld.

Sergio Manzetti 1,2and Alexander Trounev 1,3

1. Fjordforsk A/S, Bygdavegen 155, 6894 Vangsnes, Norway.

2. Uppsala University, BMC, Dept Mol. Cell Biol, Box 596,

SE-75124 Uppsala, Sweden.

3. A & E Trounev IT Consulting, Toronto, Canada.

December 17, 2018

1

1 Abstract

Vortexes in three dimensional systems are an emerging topic of study in the

realm of quantum physics, particle physics and magnetism. In this study we de-

scribe a new Hamiltonian, and solve it numerically for the description of vortices

in a spherical model subjected to an electromagnetic ﬁeld. The numerical anal-

ysis shows also that supersymmetric Hamiltonian describes quantized orbitals

for atomic systems with vorticity included, which is to our knowledge novel and

also describes electricity as composed of quasi-symmetric vortex bundles. Fur-

ther work on analytical solutions is under development. The 3D solutions of

1Please cite as: Sergio Manzetti and Alexander Trounev. (2019) ”Formation of quan-

tum vortices in a rotating sphere in an electromagnetic ﬁeld.” In: Modeling of quantum

vorticity and turbulence with applications to quantum computations and the quantum Hall

eﬀect. Report no. 142019. Copyright Fjordforsk A/S Publications. Vangsnes, Norway.

www.fjordforsk.no

1

the Hamiltonian are relevant for nuclear physics, in the physics of elementary

particles, in geophysics, and in the physics of stellar bodies.

2 Introduction

Three-dimensional vortex systems are used to model several phenomena in

physics, ranging from modeling sunspot-dynamics [1, 2, 3], cyclones and anti-

cyclones in planetary systems [2, 4, 5, 6], vortex formation in ﬂuids and vortex

formation in quantum ﬂuids and superﬂuids [7, 8]. Additionally, vortex mod-

els were the ﬁrst to be proposed to describe the forces which hold the nuclear

subparticles together [9] and were later published and accepted as a ”center

vortices” [10], nevertheless not empirically demonstrated. The vortices oc-

curring at a subatomic level are deﬁned as magnetic vortices or vortex ﬂuxes

which are modelled by a Wilson loop, also known as the Wilson loop average

in nonperturbative quantum-chromodynamics [11]. In exonucelar systems, the

Gross-Pitaevskii equation is widely used to model the magnetic vortex behaviour

in Fermi holes and in superﬂuids [7, 8, 12, 13], where some studies provide also

exact solutions to the superﬂuid vortices [14]. In one study in particular [8]

the authors model bose-eisntein condensates (BECs) with induced vorticity, by

applying a moving object with an elliptical paddle shape to create the vortices.

The authors use the Gross-Pitaevskii equation and generate a spherical sys-

tem with vortices formed on the surface boundary. The authors generate hence

long-lived vortices which are randomly distributed with turbulence [8]. In an-

other study, Bradley and Anderson [15] model the Gross-Pitaevskii equation

numerically and introduce the concept of a clustered fraction for vortices, which

describes that vortices of the same sign cluster as nearest neighbours, which

has signiﬁcant implications for vortex condensation and annihilation. We also

found of particular relevance the results of Chesler et al [16], who modelled

vortex dynamics in a 2D system by using holographic duality as a gravitational

description for the vortex-system where vortex condensation and annihilation

was described numerically. Chesler et al reported that the intervortex spacing

2

remains in the inertial range and vortices avoided condensation. In view of

these important results, we present a new model Hamiltonian which is solved

for numerical solutions in a spherical system, yielding a 3D description of the

formation of vortices on the surface of a sphere. This model diﬀers from pre-

vious models in that it allows for the formation of vortices without symmetry

breaking, and includes both vortex-condensation as well as ﬁxed intervortex

spacing in respect to speciﬁc energy levels [17, 18]. It is also, to our knowl-

edge, the ﬁrst model of vortex formation on a spherical system, which describes

the dynamics of vortex formation in the same fashion as for two-dimensional

systems, as previously published [17, 18], regulated by an internal dipole ﬁeld

located at the center of the rotating sphere. This model can be applied to any

level of physics, where vortices occur in a spherical regime, with a modulated

center-dipole ﬁeld.

2.1 Supersymmetric Hamiltonian for vortex formation

We present here the derivation of the supersymmetric operator for vortex sys-

tems in a sphere or in a 2D system. We develop the operator from the model

published by Laughlin [19], who deﬁned the Hamiltonian operator for anyons

as

H=|~/i∇ − (e/c)~

A|2,(1)

where ~

A=1

2H0[xˆy−yˆx], is the symmetric gauge vector potential, where

H0is the magnetic ﬁeld intensity. However, presenting this treatise in a ﬁrst

quantized formalism, we use the regular form:

H= [~/i∇ − (e/c)~

A]2,(2)

where the Hamiltonian is simply the normal square of the covariant deriva-

3

tive. At this point, looking at the factorized form of (2):

H= [~/i∇ − (e/c)~

A]×[~/i∇ − (e/c)~

A].(3)

we apply supersymmetry rules [20] on the two factorized components in eqn.

(3), and get the superpair of the Laughlin Hamiltonian:

HSU SY = [~/i∇ − (e/c)~

A]×[−~/i∇ − (e/c)~

A].(4)

The SUSY counter-part of the Hamiltonian from (2) becomes therefore:

HSU SY =T T ∗,(5)

with

T= [~/i∇ − (e/c)~

A],(6)

and

T∗= [−~/i∇ − (e/c)~

A],(7)

where Tand T∗are the superpair in the SUSY Hamiltonian in (4) and one

anothers complex conjugate.

We consider then Tand T∗, with γ= (e/c)~

A, in one dimension:

T=h

i

d

dx −γ·˜

I

T∗=−h

i

d

dx −γ·˜

I.

which commute by the relation:

[T T ∗]−[T∗T] = 0 (8)

4

making Tand T∗two commutating complex operators in Hilbert space H.

Both Tand T∗are unbounded operators given the condition:

||T x|| c||x||,(9)

Tand T∗are also non-linear given the γconstant, being a constant trans-

lation from the origin. Tand T∗are non-self-adjoint in Has the following

condition is not satisﬁed:

hT φ, ψi=hφ, T∗ψi,(10)

From this, it follows that:

HSU SY :D(HS U SY )−→ H,

and

D(HSU SY )⊂H,

where HSU SY is an unbounded linear and non-self-adjoint operator complex

on Hilbert space H=L2[−∞,+∞]. However, HSUS Y is self-adjoint in its

domain, D(HSU SY ) on L2[a, b].

From supersymmetry theory in quantum mechanics [20] it follows that

H†Ψ = HΨ†=EΨ = EΨ†, therefore we can assume that:

HSU SY Ψ = EΨ,(11)

where E is the energy of the system, which generates the boundaries of

D(HSU SY ) on L2[0, L], where the interval of quantization (L) is contained in

the zero-point energy term E= n2~2π2

2mL2.

5

2.2 Numerical study of vorticity dynamics in a nonlinear

quantum system in an electromagnetic ﬁeld

In Ref. [17, 18], we investigated the formation of vortices in a homogeneous

and alternating magnetic ﬁeld within the framework of the modiﬁed Gross-

Pitaevskii model. To derive the basic equation, we used the supersymmetric

Hamiltonian in (4), which is a generalization of the well-known Hamiltonian

proposed by Laughlin [19] to describe the quantum fractional Hall eﬀect. In

the present paper, we investigate the case of the 3D nonlinear quantum system in

an electromagnetic ﬁeld. The corresponding equation in dimensionless variables

has the form:

∂ψ

∂t =1

2∇2ψ+iΩ(x∂ψ

∂y −y∂ ψ

∂x )−β|ψ|2ψ+ ( ~

A)2ψ(12)

Here ~

Ais a dimensionless vector potential, β, Ω - parameters of the model de-

scribing the number of particles and the angular momentum, respectively. Eq

(12) describes evolution of the wave function from some initial state ψ(x, y, z, 0) =

ψ0(x, y, z) and up to the state describing a certain number of vortices, which

depends mainly on the angular velocity Ω. In the case of a homogeneous mag-

netic ﬁeld with ~

B=B0(sin θ, 0, cosθ), we set ~

A= [ ~

B~r]/2. In the case of a

magnetic dipole ~m =m0(sin θ, 0, cosθ) and the electric charge qcombined with

it, we assume scalar potential ϕ= 0, and the vector potential

~

A(x, y, z, t) = −[~m~r]

r3−qt~r

r3(13)

For the equation (12) we consider the problem of the decay of the initial

state, which we set in the form

ψ(x, y, z, 0) = exp[−(x2+y2+z2)/2] (14)

As boundary conditions, we will use the function of the initial state (14) given

on the boundaries of the computational domain. We note that for suﬃciently

large dimensions of the domain this is equivalent to zero boundary conditions.

The amplitude of the wave function is shown in Fig 1-10, at diﬀerent states

depending on parameters β, Ω, B0, m0, q, θ, and at a ﬁxed linear size of the region

6

x2+y2+z2≤16. Fig. 1 shows the simulation data of the amplitude of the wave

function in cross section z= 0 at diﬀerent times in a homogeneous magnetic

ﬁeld with B0= 2√2, θ = 0, and for β= 100,Ω = 2. In this case, vorticity is

formed, and in Fig. 1.1, one can see the vorticity distribution in diﬀerent cross

sections z= 0,0.5,1,1.5,2,2.5,3,3.5, and in Fig. 1.2 shows the distribution

of vorticity in the volume of the sphere. We note that the ﬁnal state is not

symmetric, which is due to the presence of a system of vortices interacting in

the volume of the sphere. Vortices begin and end on the surface of the sphere,

some of them have a horseshoe shape, others are stretched from one hemisphere

to another.

.

Figure 1.The amplitude of the wave function at diﬀerent instants of time cal-

culated for z= 0 with Ω = 2, β = 100, B0= 2√2, θ = 0.

7

.

Figure 1.1The amplitude of the wave function at diﬀerent cross sections z=

0,0.5,1,1.5,2,2.5,3,3.5 calculated for t= 1 with Ω = 2, β = 100, B0= 2√2, θ =

0.

8

.

Figure 1.2The vorticity distribution in the volume of the sphere computed for

t= 1 with Ω = 2, β = 100, B0= 2√2, θ = 0.

It should be noted that vorticity is formed even at A= 0 - Fig. 2. However,

even if A2>0, then the main parameter inﬂuencing the vorticity is Ω. Fig.

2 shows the vorticity distributions in the volume of the sphere computed for

various Ω = 1.8,2,3 at t= 1 with β= 100, B0= 0,2√2, θ = 0.In the case of

Ω=1.8,2, the vorticity is distributed throughout the volume, whereas at Ω = 3

the vortices are concentrated near the axis of rotation.

9

.

Figure 2 The vorticity distribution in the volume of the sphere computed for

t= 1 with Ω = 1.8,2,3; β= 100; B0= 0,2√2; θ= 0.

Part of the calculations were performed for the modiﬁed model (12), in which

the sign of the term A2ψchanges to the opposite. In this case we have the Gross-

Pitaevsky model, in which vorticity under these conditions is not formed - Fig.

3-4. In Fig. 3 shows the distribution of the amplitude of the wave function

in the modiﬁed model (above) and in the model (12) - below, calculated for

t= 1 with parameters Ω = 1.8, β = 100, B0= 2√2; θ= 0. Fig. 4 shows the

10

distribution of the amplitude of the wave function calculated for t= 1 for the

values of the parameters Ω = 1.8, β = 100, B0= 2√2; θ=π/2 (in this case the

magnetic ﬁeld is oriented perpendicular to the axis of rotation). From these

data it follows that in the modiﬁed model the wave function has the usual form

for quantum mechanics, while in the model (12) vortices are formed, distributed

over the volume of the sphere.

.

Figure 3.The amplitude of the wave function computed for t= 1 with Ω =

1.8, β = 100, B0= 2√2, θ = 0 in the modiﬁed model (above) and in the model

(12) - below.

11

.

Figure 4.The amplitude of the wave function computed for t= 1 with Ω =

1.8, β = 100, B0= 2√2, θ =π/2 in the modiﬁed model (above) and in the model

(12) - below.

In the case when the magnetic ﬁeld perpendicular to the axis of rotation,

vortices appear that have the form of a spindle - Fig. 4. Fig. 4.1 shows the

simulation data of the amplitude of the wave function in cross section z= 0

12

at diﬀerent times in a homogeneous magnetic ﬁeld with B0=2√2, θ =π/2,

and for β= 100,Ω = 2. In this case, vorticity is formed (while in the modiﬁed

model is not formed), and in Fig. 4.2, one can see the vorticity distribution in

diﬀerent cross sections.

.

Figure 4.1The amplitude of the wave function at diﬀerent instants of time cal-

culated for z= 0 with Ω = 1.8, β = 100, B0= 2√2, θ =π/2.

.

Figure 4.2The amplitude of the wave function at diﬀerent cross sections z=

0,0.5,1,1.5,2,2.5,3,3.5 calculated for t= 1 with Ω = 1.8, β = 100, B0=

2√2, θ =π/2.

13

The change in the orientation of the magnetic ﬁeld relative to the axis of

rotation aﬀects signiﬁcantly the vorticity distribution. In Fig. 5 shows the

amplitude of the wave function in the cross section z= 0 at diﬀerent times

computed for Ω = 2, β = 100, B0= 2√2, θ =π/4. In Fig. 5.1 and 5.2 are the

vorticity distributions at the time t= 1 in diﬀerent sections and in the volume

of the sphere.

.

Figure 5.The amplitude of the wave function at diﬀerent instants of time cal-

culated for z= 0 with Ω = 2, β = 100, B0= 2√2, θ =π/4.

.

14

Figure 5.1The amplitude of the wave function at diﬀerent cross sections z=

0,0.5,1,1.5,2,2.5,3,3.5 calculated for t= 1 with Ω = 2, β = 100, B0= 2√2, θ =

π/4.

.

Figure 5.2The vorticity distribution in the volume of the sphere computed for

t= 1 with Ω = 2, β = 100, B0= 2√2, θ =π/4.

2.3 Vorticity in a charged ﬁeld

The process of vorticity formation in the dipole and charge ﬁeld (Fig. 6-10)

diﬀers substantially from the analogous process in a homogeneous magnetic

15

ﬁeld (Fig. 1-5). In the magnetic dipole ﬁeld a dense core is formed around

which a ﬁnite number of vortices are distributed - Fig 6-7. In Figs. 6 and 6.1

show the distribution of vorticity around the central nucleus in diﬀerent cross

sections and in the volume of the sphere, calculated for t= 1 with parameters

Ω=2, β = 100, m0= 1, θ = 0, q = 0. In Fig. 6.2 shows the process of formation

of the central core and vortices in the cross section z= 0.

.

Figure 6 The amplitude of the wave function at diﬀerent cross sections z=

0,0.5,1,1.5,2,2.5,3,3.5 calculated for t= 1 with Ω = 2, β = 100, m0= 1, θ =

0, q = 0.

16

.

Figure 6.1The vorticity distribution in the volume of the sphere computed for

t= 1 with Ω = 2, β = 100, m0= 1, θ = 0, q = 0.

17

.

Figure 6.1The amplitude of the wave function at diﬀerent tin cross sections

z= 0 calculated with Ω = 2, β = 100, m0= 1, θ = 0, q = 0.

As the intensity of the magnetic dipole increases, the density of the central

nucleus increases (Fig. 7). Changing the orientation of the axis of the magnetic

dipole relative to the rotation axis also aﬀects the vorticity distribution. In Fig.

7 and 7.1 the vorticity distribution in diﬀerent sections and in the volume of

the sphere is shown, and in Fig. 7.2 shows the formation of a dense core and

vortices in the cross section z= 0 at Ω = 2, β = 100, m0= 12, θ =π/6, q = 0.

18

.

Figure 7 The amplitude of the wave function at diﬀerent cross sections z=

0,0.5,1,1.5,2,2.5,3,3.5 calculated for t= 1 with Ω = 2, β = 100, m0= 12, θ =

π/6, q = 0.

19

.

Figure 7.1The vorticity distribution in the volume of the sphere computed for

t= 1 with Ω = 2, β = 100, m0= 12, θ =π/6, q = 0.

20

.

Figure 7.2The amplitude of the wave function at diﬀerent tin cross sections

z= 0 calculated with Ω = 2, β = 100, m0= 12, θ =π/6, q = 0.

In the ﬁeld of the magnetic dipole and electric charge, a dense core is formed

on which vortices are closed (Fig. 8-9). In Fig. 8-9 shows the distribution of

vortices around a dense nucleus calculated at t= 1 with parameters Ω = 2, β =

100, m0= 7, θ = 0, q =±5. Vortices in this case have a pronounced funnel-

shaped form. In Fig. 8.1 and 9.1 shows the distribution of vortices in the cross

section z= 0, calculated at t= 1 with parameters Ω = 2, β = 100, m0= 7, θ =

0, q =±5. From the data in Fig. 8.2 and 9.2 it can be seen that the process of

forming a dense core and vortices occurs cyclically with a certain period.

21

.

Figure 8 The vorticity distribution around a dense nucleus computed for t= 1

with Ω = 2, β = 100, m0= 7, θ = 0, q =−5.

.

Figure 8.1The amplitude of the wave function at diﬀerent cross sections cal-

culated for t= 1 with Ω = 2, β = 100, m0= 7, θ = 0, q =−5.

22

.

Figure 8.2The amplitude of the wave function at diﬀerent tin cross sections

z= 0 calculated with Ω = 2, β = 100, m0= 7, θ = 0, q =−5.

23

.

Figure 9 The vorticity distribution around a dense nucleus computed for t= 1

with Ω = 2, β = 100, m0= 7, θ = 0, q = 5.

.

Figure 9.1The amplitude of the wave function at diﬀerent cross sections cal-

culated for t= 1 with Ω = 2, β = 100, m0= 7, θ = 0, q = 5.

24

.

Figure 9.2The amplitude of the wave function at diﬀerent tin cross sections

z= 0 calculated with Ω = 2, β = 100, m0= 7, θ = 0, q = 5.

The electric charge also forms a dense core (Figure 10), around which vortices

are distributed (Figure 10.1), and the process of formation of the nucleus and

vortices occurs cyclically (Figure 10.2). In Fig. 10 shows the distribution of

vortices around a dense nucleus in the ﬁeld of a point charge computed at t= 1

with parameters Ω = 2, β = 100, m0= 0, θ = 0, q = 12. In Fig. 10.1 and 10.2

show the amplitude of the wave function in diﬀerent sections and at various

times calculated with the parameters as in Fig. 10.

25

.

Figure 9 The vorticity distribution around a dense nucleus computed for t= 1

with Ω = 2, β = 100, m0= 0, θ = 0, q = 12.

.

Figure 10.1The amplitude of the wave function at diﬀerent cross sections cal-

culated for t= 1 with Ω = 2, β = 100, m0= 0, θ = 0, q = 12.

26

.

Figure 10.2The amplitude of the wave function at diﬀerent tin cross sections

z= 0 calculated with Ω = 2, β = 100, m0= 0, θ = 0, q = 12.

3 Discussion

By ﬁg 4 we see preserved structural similarities between the p-orbitals in quan-

tum mechanics for atomic systems, and the same orbitals with vorticity formed

around their nodular shapes. Our model describes hence the magnetic ﬁeld

formed around the orbitals of quantized systems, also accounting for the s-orbital

displayed in ﬁg 3. Here also, the ground state is reproduced with and without

vorticity, showing how the vortex forms in the center of the spherical structure

of the s-orbital. This shows a validity of our model in describing quantized

electronic systems, and also potential subatomic magnetic behaviours. From ﬁg

10.1 we observe another intriguing point of the SUSY Hamiltonian, where an

electric ﬁeld also generates vortices, as magnetic ﬁelds do. This is to our knowl-

27

edge new, and normally vorticity is attributed only to magnetic ﬁelds. The

similarities between the vortic behaviour of electric ﬁeld (Fig 10) and magnetic

ﬁeld (Fig 6, 7) are such that the electric ﬁeld vortexes are more structurally

order in quasi-symmetric ensembles than are magnetic ﬁeld vortices. The impli-

cations of the model are also signiﬁcant for describing subatomic forces, where

the evolution of the wavefunction displays a pulsating character, expanding and

contracting periodically.

4 Conclusions

From the results we see that the spherical solutions to the SUSY Hamiltonian

are good models for atomic and nuclear magnetic ﬁeld structures. In particular,

it is the evolution of the wavefunction and the vorticity patterns which arouse

interest. The evolution of the wavefunction of vortices suggests that magnetic

ﬁelds in spherical systems pulsate from the center to the boundaries of the

sphere in harmonical fashion. The results are also showing that the model

Hamiltonian can be used to describe the nature of the magnetic ﬁeld vortexes

around electronic orbitals, which is a signiﬁcant addition to orbital physics and

quantum physics.

28

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