Technical ReportPDF Available

Quantum vorticity in a rotating magnetic field.

Abstract and Figures

Vortexes in superfluids are a critical part of quantum optics, quantum dynamics studies and quantum physics in general. The behavior of vortexes is modeled by models such as the Ginzburg-Landau formula, and other systems, where symmetry-breaking is an inevitable event upon vortex formation. In this paper, we present and study a supersymmetric Hamiltonian which allows formation of vortexes in a quantum hall system to occur from the boundaries, as in natural phenomena, without breaking the symmetry. We study the numerical solutions of the Hamiltonian under rotating magnetic field.
Content may be subject to copyright.
Quantum vorticity in a rotating magnetic field.
Sergio Manzetti 1,2and Alexander Trounev 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.
March 6, 2019
Citing information 1
1 Abstract
Vortexes in superfluids are a critical part of quantum optics, quantum dynamics
studies and quantum physics in general. The behavior of vortexes is modeled
by models such as the Ginzburg-Landau formula, and other systems, where
symmetry-breaking is an inevitable event upon vortex formation. In this paper,
we present and study a supersymmetric Hamiltonian which allows formation of
vortexes in a quantum hall system to occur from the boundaries, as in natural
phenomena, without breaking the symmetry. We study the numerical solutions
of the Hamiltonian under rotating magnetic field.
1Please cite as: Sergio Manzetti and Alexander Trounev. (2019) ”Quantum vorticity in a
rotating magnetic field.” In:Modeling of quantum vorticity and turbulence in two-dimensional
systems. Report no. 152020. Copyright Fjordforsk A/S Publications. Vangsnes, Norway.
2 Introduction
Vorticity in quantum fluids is a phenomenon that has attracted a wide interest
in recent years [1, 2, 3, 4, 5], and describes the quantized vortex states forming
in a superfluid subjected to a thermal cloud or an increase in energy. Vortexes
are important for superconducting technologies [6, 7], in quantum fluids and
optics [8] and can play a role for quantum computation technologies [9, 10]
and superconducting units [11] as they can be expanded and stabilized in su-
perconducting metals [12]. The study of vortexes is however deeply dependent
on the development of Hamiltonians, which accurate reflect and represent the
physical state of a vortex, and yield an exact description of the eigenvalues and
eigenfunctions of the vortexes. Vortexes are modelled by the Ginzburg Lan-
dua equation for Bose-Einstein condensates [13, 14, 15], the Abrikosov model
[16, 17] for lattice vortices, non-Hermitian Liueville Hamiltonians [18, 19], the
Hamiltonian developed by Murthy and Shanka recently [20, 21] and also the
well-known Hamiltonian developed by Laughlin [22]. Recently, we presented a
model for studying vortexes based on using supersymmetry to generate a com-
muting version of the Laughlin Hamiltonian for solid-state fermions [23, 24].
Compared to existing models, our model describes the origin and evolution of
Abrikosov vortices in a quantum fluid and follows specific natural patterns of
vortex formation where vortexes arise from the boundaries of the system, and
no symmetry breaking is observed [24]. The model we described, and show
below, is an extension of the well-known Ginzburg-Landau model to the case
of the dependence of the wave function on time. However, the supersymmetric
Hamiltonian [23, 24] pertains a principal difference from the Ginzburg-Landau
model as well, where it not only avoids symmetry breaking, but describes a state
of the wavefunction which leads to the arrangement of vortices in a quantum
hall system of specific symmetries at higher and higher levels of energy, avoiding
quantum chaos up to very high values of the momentum [24] under the influence
of an electromagnetic field. Our numerical analysis in the previous studies of this
supersymmetric operator show also that a higher number of vortices is formed
at higher values of the angular momentum, and that the population of vortexes
restricts to specific intervortex spaces, avoiding collision [23, 24] within specific
regions of the quantum hall system. Furthermore, our model showed also how
several small vortexes condense to form a larger vortex when they approach the
center of the quantum hall system [?, 24, 23]. These results indicate that the su-
persymmetric Hamiltonian describes the behaviour of Abrikosov-type vortexes
and can be attributed to the study of superconducting vortexes and quantum
fluids. We present here the form and derivation of the supersymmetric operator.
2.1 Supersymmetric representation of the Laughlin state
Laughlin [22] defined the Hamiltonian operator for anyons as
H=|~/i∇ − (e/c)~
where ~
2H0[xˆyyˆx], 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)~
where the Hamiltonian is simply the normal square of the covariant deriva-
tive. Supersymmetry rules [25] are here adapted to form a supersymmetric
model for the states of the quantum hall confined particles. We do this by look-
ing at the Hamiltonian in eqn. (2), which is composed of the two first-order
differential operators:
H= [~/i∇ − (e/c)~
A]×[~/i∇ − (e/c)~
which sign is changed to form a superpair of the factorized components:
HSU SY = [~/i∇ − (e/c)~
A]×[~/i∇ − (e/c)~
In other words, the SUSY counter-part of the Hamiltonian from (3) becomes:
HSU SY =T T ,(5)
T= [~/i∇ − (e/c)~
T= [~/i∇ − (e/c)~
where Tand Tare the superpair in the SUSY Hamiltonian and one anoth-
ers Hilbert-adjoint operator.
We consider then Tand T, with γ= (e/c)~
A, in one dimension:
dx γ·˜
dx γ·˜
which commute by the relation:
[T T ][TT] = 0 (8)
making Tand Ttwo commutating complex operators in Hilbert space H.
Both Tand Tare unbounded operators given the condition:
||T x|| c||x||,(9)
Tand Tare also non-linear given the γconstant, being a constant trans-
lation from the origin. Tand Tare non-self-adjoint in Has the following
condition is not satisfied:
hT φ, ψi=hφ, Tψi,(10)
From this, it follows that:
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 [9] it follows that HΨ =
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
2.2 Derivation of the unitary master equation with vari-
able coefficients from the SUSY Hamiltonian.
In Ref. [24, ?], 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,
which is a generalization of the well-known Hamiltonian proposed by Laughlin
[22] to describe the quantum fractional Hall effect. In the present paper, we
investigate the case of the nonlinear quantum system in a rotating magnetic
field. The corresponding equation in dimensionless variables has the form:
∂t =1
∂y y ψ
∂x )β|ψ|2ψ+ ( ~
Here ~
Ais a dimensionless vector potential, α, β, Ω - parameters of the model
describing the intermittency, number of particles and the angular momentum,
respectively. Note that nonlinear Schr¨odinger equation follows from (12) at
α=π/2, and the equation derived in our papers [24, ?] follows from (12)
for α= 0. Eq (12) describes evolution of the wave function from some initial
state ψ(x, y, 0) = ψ0(x, y) and up to the state describing a certain number of
vortices, which depends mainly on the angular velocity Ω. In the case of a
rotating magnetic field with ~
B=B0(cos(ωt),sin(ωt),0), we set ~
A= [ ~
Consequently we have
4(xsin(ωt) + ycos(ωt))2(13)
For 2D system we put z= 0.
For the equation (12) we consider the problem of the decay of the initial
state, which we set in the form
ψ(x, y, 0) = exp[(x2+y2)/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.
3 Results and discussion
The amplitude of the wave function is shown in Fig 1-10, at different states
depending on parameters α, β, , B0, ω, and at a fixed linear size of the region
L= 8. Fig. 1 shows the simulation data of the amplitude of the wave function
at different times in a rotating magnetic field with a frequency of ω= 4π, with
β= 100 and for various Ω = 0.5,1. In this case, vorticity is not formed, and the
evolution of the wave function is due to a rotating magnetic field. Comparing
the data with Ω = 0.5 and Ω = 1, we find that the distribution of the wave
function does not practically differ in two cases, although the angular velocity
differs by a factor of 2. In Fig. 2 we show the data of the amplitude of the
wave function at different times in a rotating magnetic field with a frequency
of ω= 4π, with β= 100 and for various Ω = 1.5,1.8. In this case vorticity is
formed, and the evolution of the wave function depends not only on the rotating
magnetic field, but also on the process of formation of the vortices. Comparing
the data with omega = 1.5 and 1.8, we find their essential difference, although
the angular velocity in these two cases differs only by 20 %. Changing the
frequency and amplitude of the magnetic field can effectively control the dis-
tribution of vorticity - Fig. 2-4. For a small amplitude of the magnetic field,
the vorticity distribution in the final state is practically static (Fig. 3), then
at a large amplitude there is a change in the distribution of the vorticity at
each cycle (Fig. 4). We note that the picture of the distribution of the vorticity
varies with a frequency decrease by a factor of two (Fig. 4). Along with isolated
vortices, there are horseshoe-shaped vortices that begin and end at the bound-
ary of the calculated region. These horseshoe-shaped vortices are apparently
associated with a rotating magnetic field, although this effect is observed in a
less pronounced form even in a static magnetic field. These vortices are further
described in the proceeding subsection. We also investigate the effect of the
parameter α, which is shown in Fig. 5. This figure shows that as we get closer
to the value α=π/2 (when equation (12) reduces to the Schrodinger equation),
wave turbulence arises and the quantum vorticity is suppressed. Conversely,
for α= 0, quantum vorticity develops and the wave turbulence is suppressed.
Therefore, we believe that the αparameter is associated with intermittency,
where the phase of the periodic wavefunction develops into quantum chaos.
The effect of the intermittency parameter can also be seen in Fig. 6, where
the amplitude of the wave function is represented in 3D. Intermittent behaviour
is regularly occurring in fluid flows which develop a turbulent dynamical be-
haviour, or are near the transition to turbulence. However, in highly turbulent
flows, intermittency is seen in the irregular dissipation of kinetic energy. In
Fig. 7-8 shows the 3D distribution of the amplitude of the wave function on an
enlarged scale for α=π/3, π/21/10 respectively and for combinations of pa-
rameters B0= 4, β = 100,Ω=1.8, ω = 2πat t= 2. There are nonlinear waves
and holes in the distribution that correspond to quantum vorticity. From these
data it is clearly seen that for α=π/3 the quantum vorticity prevails, whereas
for α=π/21/10, nonlinear waves dominate. The PDE (eqn. 12) developed
from the supersymmetric Hamiltonian in eqn. (11) can therefore describe both
vorticity and turbulence by modulating the factor α. This is of particular appeal
and can be beneficially used to analyze systems of quantum fluids at various
levels of the energy perturbing the system. These data shown in Figure 1-5 can
also answer part of the phenomena observed in the numerical studies published
recently by Dagnino et al [26], who observed the formation of turbulence at
a particular frequency on rotating Bose-Einstein condensates. In their model,
they described a quantum jump on the angular momentum occurring at Ω=0.75
which induced symmetry breaking. In our model however, we do not observe
any symmetry breaking, and derive the behaviour of turbulence by modulating
the factor αin eqn. 12.
It also follows from these data that the horseshoe vortices are formed against
the background of nonlinear waves that predominate at the boundary of the
region. Finally, these data can be compared with the case of a = 0, calculated
for the same values of the parameters - Fig. 9. From the data shown in Fig. 9 it
can be seen that the horseshoe-shaped vortices are similar to solitons, therefore,
they can possibly be calculated analytically.
Figure 1.The amplitude of the wave function at different instants of time cal-
culated for Ω = 0.5 - two upper lines, and for Ω = 1 - two lower lines with
α= 0, β = 100, ω = 4π, B0= 32.
Figure 2.The amplitude of the wave function at different instants of time cal-
culated for Ω = 1.5 - two upper lines, and for Ω = 1.8 - two lower lines with
α= 0, β = 100, ω = 4π, B0= 32 .
Figure 3.The amplitude of the wave function at different instants of time cal-
culated for B0=2 - two upper lines, and for B0= 22 - two lower lines with
α= 0, β = 100, ω = 4π, Ω=1.8
Figure 4.The amplitude of the wave function at different instants of time cal-
culated for B0= 42, ω = 4π- two upper lines, and for B0= 8, ω = 2π- two
lower lines with α= 0, β = 100,Ω=1.8.
Figure 5.The amplitude of the wave function at different instants of time cal-
culated for α=π/3 - two upper lines, and for α=π/20.1 - two lower lines
with B0= 4, β = 100,Ω=1.8, ω = 2π.
Figure 6.3D distributions of the amplitude of the wave function at different
instants of time computed for α=π/3 - two upper lines, and for α=π/20.1
- two lower lines with B0= 4, β = 100,Ω=1.8, ω = 2π.
Figure 7.3D distributions of the amplitude of the wave function computed at
t= 2 for combination of parameters α=π/3, B0= 4, β = 100,Ω=1.8, ω = 2π.
Figure 8.3D distributions of the amplitude of the wave function computed at
t= 2 for combination of parameters α=π/21/10, B0= 4, β = 100,Ω =
1.8, ω = 2π.
Figure 9.3D distributions of the amplitude of the wave function computed at
t= 2 for combination of parameters α= 0, B0= 4, β = 100,Ω=1.8, ω = 2π.
3.1 Effects from Boundary conditions
Let us consider the influence of boundary conditions. From the physical point
of view, the boundary conditions used by us mean that there is a sample of
finite size in which the particles do not penetrate the boundary (zero bound-
ary conditions) or penetrate with small probability (what we really put). If
the sample is a crystalline structure, it is appropriate to put periodic boundary
conditions. By following Fig. 10 we see that the calculations of the amplitude
of the wave function with periodic boundary conditions (and with initial data
as in Fig. 5-7) show that an essential change in the geometry of the vortex dis-
tribution occurs. This can also be seen by the fact that with periodic boundary
conditions the horseshoe vortices completely disappear (Fig. 11, 12 ). Conse-
quently, the formation of the horseshoe vortices are associated not only with the
rotating magnetic field, but also with the boundary conditions. The question of
horseshoe eddies requires further study, which will be published in our subse-
quent articles, as this particular formation of condensates has strong similarities
to the imaged vortices published by Embon et al [7]. The results by Embon
et al suggest that the horseshoe patterns we observe in our model of conden-
sates subjected to an electromagnetic field with zero boundary conditions (Fig
12, second row, t=2) are vortex flows which undergo bifurcation forming the
horseshoe patterns. The vortex flows, which are also clearly visible in Fig. 7,
induce superconductivity and superfluidity (zero viscosity) in respectively met-
als and superfluids, such as Niobium and supercooled Helium gas. Modelling
of superconductivity can therefore emerge from the successful representation of
vortex flows in a model system. It appears also from the results that wherever
the wavefunction decays towards zero at the boundaries, the bifurcations in the
vortex lines disappear. This suggests that the model of the SUSY Hamiltonian
describes vorticity wherever there is a contained physical system, however, the
same finite systems are not necessarily by the SUSY Hamiltonian for flows of
vortices, as there is no direction to where the vortexes can flow to. This tells us
therefore that even when adding a perturbation to the finite system, we cannot
expect vortices to flow to any direction, and infinity or some phase-transition
of the superfluid / superconductor is required to generate vortex flow.
Figure 10.3D and 2D distributions of the amplitude of the wave function
at different instants of time computed with periodic boundary conditions for
α=π/3, B0= 4, β = 100,Ω=1.8, ω = 2π.
Figure 11.The amplitude of the wave function at t= 2 computed with zero
boundary conditions (on the left) and with periodic boundary conditions (on
the right) for α=π/3, B0= 4, β = 100,Ω=1.8, ω = 2π.
Figure 12.The amplitude of the wave function computed with zero boundary
conditions (two upper rows) and with periodic boundary conditions (two lower
rows) for α= 0, B0= 4, β = 100,Ω=1.8, ω = 2π.
4 Conclusions
We have in this study analysed a SUSY Hamiltonian for Bose-Einstein con-
densates and how it describes vorticity under a rotating magnetic field. The
results show that the frequency and amplitude of the magnetic field can effec-
tively control the distribution of vorticity. The results show furthermore that
the SUSY Hamiltonian can describe both vorticity and turbulence by modulat-
ing the factor αin the derived master equation. We find this particularly useful
for numerical studies of vortex and turbulence. We also found that the formed
amplitudes in the quantum hall system which assume horseshoe-shaped vortices
are similar to solitons, which suggests that analytical solutions to these vortices
can be derived. Ultimately, our model suggests that the formation of soliton-
like amplitudes depends not only on the rotating magnetic field, but also by
the effect from the boundary conditions, when boundary conditions are absent,
soliton-like amplitudes do not form. Future work emphasize analytical solutions
to the SUSY Hamiltonian model and an investigation of its applicability into
the regime of turbulence and quantum chaos.
[1] Angela C White, Brian P Anderson, and Vanderlei S Bagnato. Vortices
and turbulence in trapped atomic condensates. Proceedings of the National
Academy of Sciences, page 201312737, 2014.
[2] Angela C White, Carlo F Barenghi, and Nick P Proukakis. Creation
and characterization of vortex clusters in atomic bose-einstein condensates.
Physical Review A, 86(1):013635, 2012.
[3] Lih-King Lim, C Morais Smith, and Andreas Hemmerich. Staggered-vortex
superfluid of ultracold bosons in an optical lattice. Physical review letters,
100(13):130402, 2008.
[4] Martin W Zwierlein, Jamil R Abo-Shaeer, Andre Schirotzek, Christian H
Schunck, and Wolfgang Ketterle. Vortices and superfluidity in a strongly
interacting fermi gas. Nature, 435(7045):1047, 2005.
[5] VMH Ruutu, VB Eltsov, AJ Gill, TWB Kibble, M Krusius, Yu G Makhlin,
B Placais, GE Volovik, and Wen Xu. Vortex formation in neutron-
irradiated superfluid 3he as an analogue of cosmological defect formation.
Nature, 382(6589):334, 1996.
[6] Daniel Ariad and Eytan Grosfeld. Signatures of the topological spin of
josephson vortices in topological superconductors. Physical Review B,
95(16):161401, 2017.
[7] Lior Embon, Yonathan Anahory, Zeljko L Jelic, Ella O Lachman, Yuri
Myasoedov, Martin E Huber, Grigori P Mikitik, Alejandro V Silhanek,
Milorad V Milosevic, Alexander Gurevich, et al. Imaging of super-fast
dynamics and flow instabilities of superconducting vortices. Nature com-
munications, 8(1):85, 2017.
[8] Gael Nardin, Gabriele Grosso, Yoan Leger, Barbara Pietka, Francois
Morier-Genoud, and Benoit Deveaud-Pledran. Hydrodynamic nucleation
of quantized vortex pairs in a polariton quantum fluid. Nature Physics,
7(8):635, 2011.
[9] Sumanta Tewari, S Das Sarma, Chetan Nayak, Chuanwei Zhang, and Peter
Zoller. Quantum computation using vortices and majorana zero modes of
a p x+ i p y superfluid of fermionic cold atoms. Physical review letters,
98(1):010506, 2007.
[10] Sergio Manzetti. Applied quantum physics for novel quantum computa-
tion approaches: an update. Computational Mathematics and Modeling,
29(2):244–251, 2018.
[11] Lucas Thiel, Dominik Rohner, Marc Ganzhorn, Patrick Appel, Elke Neu,
B Muller, R Kleiner, D Koelle, and P Maletinsky. Quantitative nanoscale
vortex imaging using a cryogenic quantum magnetometer. Nature nan-
otechnology, 11(8):677, 2016.
[12] Vasily S Stolyarov, Tristan Cren, Christophe Brun, Igor A Golovchanskiy,
Olga V Skryabina, Daniil I Kasatonov, Mikhail M Khapaev, Mikhail Yu
Kupriyanov, Alexander A Golubov, and Dimitri Roditchev. Expansion of a
superconducting vortex core into a diffusive metal. Nature communications,
9(1):2277, 2018.
[13] Lev Pitaevskii and Sandro Stringari. Bose-Einstein condensation and su-
perfluidity, volume 164. Oxford University Press, 2016.
[14] Vladimir Skarka, Najdan Aleksic, Wieslaw Krolikowski, Demetrios
Christodoulides, Branislav Aleksic, and Milivoj Belic. Linear modula-
tional stability analysis of ginzburg–landau dissipative vortices. Optical
and Quantum Electronics, 48(4):240, 2016.
[15] JP Eisenstein. Exciton condensation in bilayer quantum hall systems.
Annu. Rev. Condens. Matter Phys., 5(1):159–181, 2014.
[16] Aleksei Alekseevich Abrikosov. Fundamentals of the Theory of Metals.
Courier Dover Publications, 2017.
[17] V Kozhevnikov, A-M Valente-Feliciano, PJ Curran, G Richter, A Volodin,
A Suter, S Bending, and C Van Haesendonck. Ideal gas of vortices in
type-ii superconductors: Experiment and theoretical model. arXiv preprint
arXiv:1603.04105, 2016.
[18] David Krejcirk, Petr Siegl, and Jakub Zelezny. On the similarity of sturm–
liouville operators with non-hermitian boundary conditions to self-adjoint
and normal operators. Complex analysis and operator theory, 8(1):255–281,
[19] Naomichi Hatano and David R. Nelson. Vortex pinning and non-hermitian
quantum mechanics. Phys. Rev. B, 56:8651–8673, Oct 1997.
[20] Ganpathy Murthy and R. Shankar. Hamiltonian theories of the fractional
quantum hall effect. Rev. Mod. Phys., 75:1101–1158, Oct 2003.
[21] Kang Yang, Mark Oliver Goerbig, and Benot Doucot. Hamiltonian theory
for quantum hall systems in a tilted magnetic field: robustness of activation
gaps. arXiv preprint arXiv:1807.09677, 2018.
[22] Robert B Laughlin. Anomalous quantum hall effect: an incompressible
quantum fluid with fractionally charged excitations. Physical Review Let-
ters, 50(18):1395, 1983.
[23] Sergio Manzetti and Alexander Trounev. Derivation of a hamiltonian for
formation of particles in a rotating system subjected to a homogeneous
magnetic field. Technical Reports, 14(2020):1–20, 2019.
[24] Sergio Manzetti and Alexander Trounev. Supersymmetric hamiltonian and
vortex formation model in a quantum nonlinear system in an inhomoge-
neous electromagnetic field. Advanced Theory and Simulations, In press,
[25] Fred Cooper, Avinash Khare, and Uday Sukhatme. Supersymmetry and
quantum mechanics. Physics Reports, 251(5-6):267–385, 1995.
[26] D Dagnino, N Barber´an, M Lewenstein, and J Dalibard. Vortex nucleation
as a case study of symmetry breaking in quantum systems. Nature Physics,
5(6):nphys1277, 2009.
... Such a study can further increase the understanding of the planet Earth, its climate change and the interaction between the magnetic field and the atmosphere. In order to develop this numerical treatise, we us the supersymmetric wave equation we have developed in the previous work [17,18,19,20]: ...
... This equation (1) previously reported [18,19,20] for 2D and 3D systems. The vorticity of this operator appears from the boundaries and moves towards the center as in natural phenomena [18,19,20], as the potential is repulsive. ...
... This equation (1) previously reported [18,19,20] for 2D and 3D systems. The vorticity of this operator appears from the boundaries and moves towards the center as in natural phenomena [18,19,20], as the potential is repulsive. As atmospheric vortices and electromagnetic vortices behave in the same fashion, rotating about a moving center in either directions, we postulate that the spinning direction for atmospheric vortices is perturbed not only by temperatures and humidity, but may be triggered by the phase of the interaction between the charged particles in the atmosphere and the electromagnetic vortices which result from the Earth's electromagnetic field, in a similar fashion to the mechanism observed by [16]. ...
Technical Report
Full-text available
Planetary magnetohydrodynamics is a central part of planetary sciences, where the magnetic field of the planet is studied in interaction with the Sun, solar storms and lunar cycles. In this paper, we study numerically the effects on the dynamics of electromagnetic vortices in the Northern hemisphere by the displacement of the magnetic North pole. Using a supersymmetric wave-equation, we consider the electronic gas of the whole planet as an autonomous system which produces electric current which interacts with the geomagnetic field. Our result suggest that electromagnetic vorticity lies at the core of the planet and affects the properties of the electromagnetic field in the atmosphere. The shift of the North Pole affects the distribution of electromagnetic vortices on the Northern hemisphere by two modes. The first mode is by displacement of the inclination of the central vortex of the planet, which funnels from the core of the planet. The second mode is by altering the distribution of the electric charge (vorticity) in the Northern hemisphere.
... By Fig 9-15, we focus in more detail on the behaviour of Helium acting as a superfluid, at various levels of resolution. The streamlines that we generate here are related to vorticity, which we have observed in previous calculations [12,13,14,15,16,11,17], and may therefore explain that superfluid Helium crosses barriers via forming vortices, and that these may be at the basis for superfluidity. Further studies into this theory are required. ...
Technical Report
Full-text available
Superfluid Helium4 has been known since the late 1960's to condense on the outside of its container when it reaches temperatures close to the absolute zero. Several models have been developed over the years to describe the dynamics of Helium, both thermodynamiacal models as well as non-equlibrium models. Here we introduce a new model of the behaviour of superfluid Helium and study it numerically. The model is derived from the Gross-Pitaevskii model for Bose-Einstein condensates, and describes a behaviour of superfluidity. From the results we see that an axially symmetric solution gives condensation on the outside of the container by complex wave-dynamics which flow upwards in the container supporting the existence of the Rollin effect. When we evaluate the streamlines of the wavefunction we observe however that the superfluid appears also to leak through the barrier. We hypothesize on the possibility of quantum tunneling of superfluid vortices and conclude the study with an outlook.
... The results show furthermore that vorticity is fueled by the electromagnetic field of the planet, and hence counterbalanced by the gravitational field. We 8 modelled the electromagnetic field by (13), which leads to that vorticity arises entirely from the nucleus of the planet and is projected towards the atmosphere ( Figure 5-8). In order to conceive in detail these results we look deeper at the analysis behind this model. ...
Technical Report
Full-text available
Macrophysical phenomena, such as turbulence, vorticity are normally modelled using fluid-mechanical differential equations and geodesics systems. Other methods may also be of importance to the meteorology community, such as quantum physical operators. In this study we use a novel Hamiltonian and study the vortex formation in the atmosphere of planet Earth under the effects of the gravity density stratification and the electromagnetic field of the planet. The results propose that vorticity in the atmosphere (high and low pressure systems) is driven in major part by the interplay between the earths magnetic field and gravity density. The results show that the quantized behaviour of atmospheric vortices lies in their dominant occurrence on the northern and southern hemispheres. The use of quantum mechanical operators in modelling planetary vorticity reveals also that these vortices arise from the core of the planet and manifest in a most pronounced manner on the surface of the earth where gravity density is experiencing an abrupt phase change. Further research is made on combining this model with earths atmospheric parameters, such as ocean temperatures and circulation, terrestrial oscillation and the sun's magnetic field. The results are important for future developments of climate and weather prediction models.
Full-text available
Vortices in quantum condensates exist owing to a macroscopic phase coherence. Here we show, both experimentally and theoretically, that a quantum vortex with a well-defined core can exist in a rather thick normal metal, proximized with a superconductor. Using scanning tunneling spectroscopy we reveal a proximity vortex lattice at the surface of 50 nm-thick Cu-layer deposited on Nb. We demonstrate that these vortices have regular round cores in the centers of which the proximity minigap vanishes. The cores are found to be significantly larger than the Abrikosov vortex cores in Nb, which is related to the effective coherence length in the proximity region. We develop a theoretical approach that provides a fully self-consistent picture of the evolution of the vortex with the distance from Cu/Nb interface, the interface impedance, applied magnetic field, and temperature. Our work opens a way for the accurate tuning of the superconducting properties of quantum hybrids.
Full-text available
Quantum information processing is a critical part of the development of future computers, quantum computers, and quantum algorithms, where elementary particles such as photons and electrons can be applied in optomagnetic or optoelectronic devices. The computational physics behind these emerging approaches is also experiencing dramatic developments. In this paper I report on the most recent mathematical basics for quantum algorithms and quantum computing approaches. Some of these described approaches show intriguing methods for determining the states and wavefunction properties for anyons, bosons, and fermions in quantum wells that have been developed in the last years. The study also shows approaches based on N-quantum states and the reduced 1- and 2-fermion picture, which can be used for developing models for anyons and multi-fermionic states in quantum algorithms. Also, antisymmetry and generalized Pauli constraints have been given particular emphasis and include establishing a basis for pinning and quasipinning for exploring the symmetric states of many-fermionic subsets, as a foundation for quantum information processing, and are here briefly revisited. This study summarizes the developments in recent years of an advanced and important field for future computational techniques.
Full-text available
Quantized magnetic vortices driven by electric current determine key electromagnetic properties of superconductors. While the dynamic behavior of slow vortices has been thoroughly investigated, the physics of ultrafast vortices under strong currents remains largely unexplored. Here we use a nanoscale scanning superconducting quantum interference device to image vortices penetrating into a superconducting Pb film at rates of tens of GHz and moving with velocities up to tens of km/s, which are not only much larger than the speed of sound but also exceed the pair-breaking speed limit of superconducting condensate. These experiments reveal formation of mesoscopic vortex channels which undergo cascades of bifurcations as the current and magnetic field increase. Our numerical simulations predict metamorphosis of fast Abrikosov vortices into mixed Abrikosov-Josephson vortices at even higher velocities. This work offers an insight into the fundamental physics of dynamic vortex states of superconductors at high current densities, crucial for many applications.
Full-text available
We consider a modified setup for measuring the Aharonov-Casher phase which consists of a Josephson vortex trapped in an annular topological superconducting junction. The junction encloses both electric charge and magnetic flux. We discover a deviation from the Aharonov-Casher prediction whose origin we identify in an additive universal topological phase that remarkably depends only on the parity of the number of vortices enclosed by the junction. We show that this phase is ±2π times the topological spin of the Josephson vortex and is proportional to the Chern number. The presence of this phase can be measured through its effect on the junction's voltage characteristics, thus revealing the topological properties of the Josephson vortex and the superconducting state.
Full-text available
Equilibrium magnetic properties of the mixed state in type-II superconductors were measured with high purity bulk and film niobium samples in parallel and perpendicular magnetic fields using dc magnetometry and scanning Hall-probe microscopy. Equilibrium magnetization data for the perpendicular geometry were obtained for the first time. It was found that none of the existing theories is consistent with these new data. To address this problem, a theoretical model is developed and experimentally validated. The new model describes the mixed state in an averaged limit, i.e. %without detailing the samples' magnetic structure and therefore ignoring interactions between vortices. It is quantitatively consistent with the data obtained in a perpendicular field and provides new insights on properties of vortices. % and the entire mixed state. At low values of the Ginzburg-Landau parameter, the model converts to that of Peierls and London for the intermediate state in type-I superconductors. It is shown that description of the vortex matter in superconductors in terms of a 2D gas is more appropriate than the frequently used crystal- and glass-like scenarios.
Full-text available
Two-dimensional dissipative solitons are described by the complex Ginzburg–Landau equation, with cubic-quintic nonlinearity compensating for diffraction, while linear and nonlinear losses are simultaneously balanced by the gain. Vortices with zero electric field in the center, corresponding to a topological singularity, are particularly sensitive to the azimuthal modulational instability that causes filamentation for some values of dissipative parameters. We perform linear stability analysis, in order to determine for which values of parameters the dissipative vortex either splits into filaments or becomes stable dissipative vortex soliton. The growth rates of different modulational instability modes is established. In the domain of dissipative parameters corresponding to the zero maximal growth rate, steady state solutions are stable. Analytical results are confirmed by numerical simulations of the full complex radially asymmetric cubic-quintic Ginzburg–Landau equation.
Full-text available
Microscopic studies of superconductors and their vortices play a pivotal role in our understanding of the mechanisms underlying superconductivity. Local measurements of penetration depths or magnetic stray-fields enable access to fundamental aspects of superconductors such as nanoscale variations of superfluid densities or the symmetry of their order parameter. However, experimental tools, which offer quantitative, nanoscale magnetometry and operate over the large range of temperature and magnetic fields relevant to address many outstanding questions in superconductivity, are still missing. Here, we demonstrate quantitative, nanoscale magnetic imaging of Pearl vortices in the cuprate superconductor YBCO, using a scanning quantum sensor in form of a single Nitrogen-Vacancy (NV) electronic spin in diamond. The sensor-to-sample distance of ~10nm we achieve allows us to observe striking deviations from the prevalent monopole approximation in our vortex stray-field images, while we find excellent quantitative agreement with Pearl's analytic model. Our experiments yield a non-invasive and unambiguous determination of the system's local London penetration depth, and are readily extended to higher temperatures and magnetic fields. These results demonstrate the potential of quantitative quantum sensors in benchmarking microscopic models of complex electronic systems and open the door for further exploration of strongly correlated electron physics using scanning NV magnetometry.
Vortices and quasiparticles in 2D systems are studied for the last decades in relation to the development of technologies in quantum electromagnetics, optics, and quantum computation. Hamiltonians are a fundamental part for studying quasiparticles and vortices, and provide models to calculate the eigenvalues and eigenfunctions that describe a real physical state. By devising supersymmetry, a Hamiltonian for the description of vortices is developed forming in an 2D electron gas and study the numerical solutions under the effects of an alternating electromagnetic field. The numerical analysis shows that vorticity is formed spontaneously without symmetry‐breaking and vortices arise from the boundaries and converge toward the center of the system in a similar fashion to natural hydrodynamic phenomena in water or plasma. Additionally, the equation under damping conditions is studied, a homogenous magnetic field and under the absence of an electromagnetic field. The results and the study of the parameters indicate that the supersymmetic wave equation (SWE) may be a good model equation to describe vorticity for quantum electromagnetics, hydrodynamics, and other physical phenomena in the realm of physical and quantum physical sciences. A supersymmetric wave‐equation (SWE) is developed for the analysis of vortex dynamics in 2D quantum systems. The supersymmetry‐derived wave‐equation describes vortex formation without symmetry breaking. Numerical analysis of the vortex formation is carried out under damped conditions and under a variating electromagnetic field and an electric field. Results show that the SWE describes quantized vortex systems, and can be applied to large‐ and small‐ scale vorticity phenomena.
We use the Hamiltonian theory developed by Shankar and Murthy to study a quantum Hall system in a tilted magnetic field. With a finite width of the system in the z direction, the parallel component of the magnetic field introduces anisotropy into the effective two-dimensional interactions. The effects of such anisotropy can be effectively captured by the recently proposed generalized pseudopotentials. We find that the off-diagonal components of the pseudopotentials lead to mixing of composite fermions Landau levels, which is a perturbation to the picture of p filled Landau levels in composite-fermion theory. By changing the internal geometry of the composite fermions, such a perturbation can be minimized and one can find the corresponding activation gaps for different tilting angles, and we calculate the associated optimal metric. Our results show that the activation gap is remarkably robust against the in-plane magnetic field in the lowest and first Landau levels.