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
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.
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  and can play a role for quantum computation technologies [9, 10]
and superconducting units  as they can be expanded and stabilized in su-
perconducting metals . 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 . 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 . 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  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  deﬁned the Hamiltonian operator for anyons as
H=|~/i∇ − (e/c)~
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)~
where the Hamiltonian is simply the normal square of the covariant deriva-
tive. Supersymmetry rules  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
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 T∗are 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:
which commute by the relation:
[T T ∗]−[T∗T] = 0 (8)
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,
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  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
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
 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:
∂y −y∂ ψ
∂x )−β|ψ|2ψ+ ( ~
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, Ω, 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/2−1/10 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/2−1/10, 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 , 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= 3√2.
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= 3√2 .
Figure 3.The amplitude of the wave function at diﬀerent instants of time cal-
culated for B0=√2 - two upper lines, and for B0= 2√2 - 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= 4√2, 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/2−0.1 - 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/2−0.1
- 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/2−1/10, 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 . 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.
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.
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