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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.

www.fjordforsk.no

1

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

2

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)~

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-

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)~

A].(3)

which sign is changed to form a superpair of the factorized components:

HSU SY = [~/i∇ − (e/c)~

A]×[−~/i∇ − (e/c)~

A].(4)

3

In other words, the SUSY counter-part of the Hamiltonian from (3) becomes:

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 and one anoth-

ers Hilbert-adjoint operator.

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)

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)

4

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 [9] 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.

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:

e−iα ∂ψ

∂t =1

2∇2ψ+iΩ(x∂ψ

∂y −y∂ ψ

∂x )−β|ψ|2ψ+ ( ~

A)2ψ(12)

5

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= [ ~

B~r]/2.

Consequently we have

(~

A)2=B2

0z2

4+B2

0

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

6

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

7

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 [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.

8

.

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.

.

9

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

10

.

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.

.

11

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.

12

.

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-

13

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.

14

.

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.

15

.

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-

16

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.

17

References

[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

18

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.

19

[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,

2014.

[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,

2019.

[25] Fred Cooper, Avinash Khare, and Uday Sukhatme. Supersymmetry and

quantum mechanics. Physics Reports, 251(5-6):267–385, 1995.

20

[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.

21