Michele Correggi- Vortices in Rotating Bose-Einstein Condensates: A Review of (Recent) Mathematical...

8/3/2019 Michele Correggi- Vortices in Rotating Bose-Einstein Condensates: A Review of (Recent) Mathematical Results

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Vortices in Rotating Bose-Einstein CondensatesA Review of (Recent) Mathematical Results

Michele Correggi

CIRMFBK, Trento

15/09/2009Mathematical Models of Quantum FluidsUniversita di Verona

M. Correggi (CIRM) Vortices in Rotating BE Condensates Verona 15/09/2009 1 / 35


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Outline

1 General Setting and Background: the Gross-Pitaevskii (GP) Theory

for Rotating Bose-Einstein (BE) Condensates [A].

2 Vortices and Rotational Symmetry Breaking.3 The Thomas-Fermi (TF) Limit of the GP Theory:

Harmonic trapping potentials [IM].

(Strongly) Anharmonic trapping potentials [CY].

Main References

[A] A. Aftalion, Vortices in Bose-Einstein Condensates, 2006.

[CY] M.C., J. Yngvason, J. Phys. A 41 (2008), 445002.

[IM] R. Ignat, V. Millot, J. Funct. Anal. 233 (2006), 260306.

Physics: A.L. Fetter, Rev. Mod. Phys. 81 (2009), 647691.

Numerics: W. Bao, in Dynamics in Models of Coarsening,Coagulation, Condensation and Quantization, 2007.



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General Setting and Background

The GP Theory of a Rotating BE Condensate in a Trap

2d BE condensate rotating along the zaxis with angular velocity .External trap given by a potential V(r) (V(r) as r ).The stationary ground state properties of a rotating BE condensatecan be described through minimizers of the GP energy functional (inthe non-inertial rotating frame).

L = i(xy yx) is the zcomponent of the angular momentum.2 is the coupling parameter ( scattering length).The TF limit is 0 (large coupling and/or fast rotation).

The GP Energy Functional

EGP [] =R2

dr

||2 L + V(r) ||2 + ||

4

2

.



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Minimization of the GP Functional

EGP

[] =R2

dr iA

2

+

V(r) 2r2

4||2 + |

|4

2

The vector potential is A = ez r/2.GP ground state energy EGP = inf2=1 EGP[].GP stands for any corresponding minimizer.

Boundedness from Below of EGPAssume that V L2loc(R3) and either V(r) r2, < 2, or V(r) rs,

s > 2, as r = EGP

is bounded from below. Usually one considersthe harmonic potential V(r) = r2 (upper bound on !),

(strongly) anharmonic (homogeneous) potentials V(r) rs, s > 2(no upper bound on !): the simplest example is a confinement tothe unitary disc

B1 with Neumann boundary conditions (s =

).


G l S i d B k d


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Existence of a Minimizer GP

Under the same hypothesis on V, (at least) one minimizer GP.

GP

solves the GP time-independent equation

GP LGP + VGP + 22GP2 GP = GPGP.

The chemical potential GP is fixed by the L2 normalization:

GP

= EGP

+ 2

GP

44.

If V is smooth, the same is GP.

Uniqueness of GP

The minimizer is not necessarily unique.Non-uniqueness is due to the presence of the angular momentumterm. If = 0, EGP is strictly convex in = ||2.Non-uniqueness is strictly related to the occurence of isolated vortices

rotational symmetry breaking.


G l S tti d B k d


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Ginzburg-Landau vs. Gross-Pitaevskii

[GL] Minimize EGL

w.r.t. u andA with h = curl(

A), hex = (uniform magnetic field),

EGL[u, A] =Ddr

iA u2 + |h hex|2 + 2(1 |u|2)2

.

[GP] MinimizeEGP w.r.t. L2 normalized (r) = (r)u(r),

EGP [] = ETF[] +Sdr

iA

u2 + 2 1 |u|2

.

Main Differences:

In GL there is an additional minimization w.r.t. A, whereas in GP A is

given = major differences in the minimizers but the ground stateenergies can be close in certain regimes.No L2 normalization in GL = no chemical potential in GL = nodensity in GL. Not only a technicality since there is some physicsbehind it!


Vortices and Rotational Symmetry Breaking


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EGP

[] =R2

dr iA

2

+

V(r) 2r2

4||2 + ||

4

2

Rotational Symmetry Breaking

The functionalEGP is invariant under rotations around the z-axis.

If is fixed and large, the GP minimizer is not an eigenfunction ofthe angular momentum (rotational symmetry breaking), due to theoccurrence of isolated vortices.

VorticesGP has a vortex at r0 (0 = x0 + iy0) with winding number d, if

GP(r0) = 0, (locally)GP

|GP

|

0

|

0

|

d.




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Vortices in a Rotating BE Condensate [Dalibard et al 05]

Formation of quantized vortices in a rotating Rb BE condensate.

Vortices in a fast rotating Rb BE condensate in a quartic+quadratic trap.




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Why Are Vortices Energetically Favorable?

EGP

[] =B1

dr||2 L + 2 ||4

Energy Compensation

For small rotational velocities the condensate is at rest in the inertial

frame (superfluidity) = |GP

| = const.At higher angular velocities vortices start to occur: For small , avortex of degree d at the origin has the form f(r)exp{id}, with fapprox. constant outside B (= nonlinear term).

Kinetic energyB1\Bdr ||2 d2

B1\B dr

1

r2 Cd2| log |.

Angular momentum B1\B

dr L d.

If

C

|log

|, a vortex can be energetically favorable.




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Why/When More Vortices?

EGP

[] =B1

dr||

2

L +

2

||4

Energy Optimization

So far we have not justified the breaking of the rotational symmetry,

since a vortex at the origin is an eigenfunction of L!GP can contain more than one vortex (= nonlinearity). fixes the total winding number d (angular momentum) of GP

and, if is sufficiently large, d > 1.

Suppose d = 2: The kinetic energy is d2

= 2 vortices of windingnumber 1 have a smaller kinetic energy (1 + 1 = 2) than 1 vortex ofwinding number 2 (22 = 4), but almost the same angular momentum.

If 1 the vortex cores are small ( ) and the interaction energycan be neglected =

many vortices can be energetically favorable.




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y y g

Rotational Symmetry Breaking

Theorem (Symmetry Breaking [MC,Rindler-Daller,Yngvason 07])As 0, no minimizer ofEGP[] is an eigenfunction of the angularmomentum, if

6| log | + 3 < C

for any constant C R+.

Symmetry breaking is due to occurrence of isolated vortices outside ofthe origin.

The GP minimizer in no longer unique ( degeneracy).The estimate of the symmetry breaking threshold is not optimal (it isexpected to be 2| log | [Aftalion,Du 01]).The rotational symmetry is expected to be broken also for 1.


The Thomas-Fermi Limit of the Gross-Pitaevskii Theory


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y

Nucleation of Vortices (Harmonic Traps) [Ignat,Millot 06]

EGP [] = R2

dr||2 L + 2V(x, y)||2 + 2 ||4

Harmonic trapping potential (rescaled): V(x, y) = (x2 + 2y2).0 <

1 measures the asymmetry of the potential.

The coefficient 2 of V is chosen so that the first critical velocity isO(| log |) (it is equivalent to rescale all lengths):

1 =(2+1)

2| log |.

The vortex free profile is the (unique L2 normalized) minimizer of

EGP [] =R2

dr||2 + 2V||2 + 2 ||4

.

is real and positive. If = 1, it is also radial.

As

0, 2

TF(x, y) = 12 [

V(x, y)]+ in L

(

D).




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y

Theorem (Absence of Vortices below 1 [Ignat,Millot 06])

For any > 0, if 1 log | log |, then

GP 0 TF

(x, y) in Lloc(R

2

\ D),EGP = EGP eiS + o(1), where S(x, y) = 212+1 xy,Up to a subsequence (and a global phase factor , || = 1)

GP

0

TF(x, y)eiS,

in H1loc(D) = no vortices, i.e., for any R0 < and sufficientlysmall, GP does not vanish inside the region where x2 + 2y2 < R20 .

Theorem (Occurence of Vortices above 1 [Ignat,Millot 06])

For any > 0, if 1 + log | log | and is sufficiently small, thenGP has at least one vortex atr such thatdist(r, D) C.If in addition 1 + O(log | log |), then the vortex remains closeto the origin, i.e.,

|r

|= o(1).




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Critical Velocities [Ignat,Millot 06]

d

=(1+2)

2(|

log |

+ (d

1)log|

log |) , d

N

Theorem (Number and Distribution of Vortices [Ignat,Millot 06])

For any 0 < 1, if d + log | log | d+1 log | log |, thenFor any R0 < and sufficiently small, GP has exactly d vorticesof winding number 1 atri,, i = 1, . . . , d inside x

2 + 2y2 < R20 ,

Vortices remain close to the origin and close one another, i.e.,

|ri| C1/2, |ri rj| C1/2.Setting i = ri, the configuration (1, . . . , d) minimizes therenormalized energy

W

1, . . . , d

=

i

=j

logi j

+

1 + 2

di=1

x2i +

2y2i

.




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Distribution of Vortices [Gueron,Shafrir 00]

The distribution of vortices is determined by the minimization of W:

If = 1 and d 6, regular polygons and stars (d 1 side regularpolygon plus the origin) centered at the origin are (local) minimizingconfigurations for the renormalized energy W.

If d 11 neither regular polygons nor stars are local minimizers ofW. As d increases the minimizers approach a triangular lattice.

Larger Angular Velocities [Baldo,Jerrard,Orlandi,Soner 08]

If d for any d but = O(| log |), the number of vortices isnot uniformly bounded in .

The vortex distribution minimizes a (rescaled) free boundary problem.

Landau Regime [Aftalion et al 06]

When

1, the occupation of Landau levels become relevant...




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Rotating BE Condensates in Anharmonic Traps[MC,Rindler-Daller,Yngvason 07]

EGP [] =B1dr

iA

2 2r2 ||2

4+

||42

Motivations

There is no upper bound on the angular velocity , i.e., thecondensate is confined for any = one can explore regimes of veryfast rotation.

The unitary disc is the strongest anharmonic trap one can think of

since it can be formally obtained as the limit s of ahomogeneous potential V(r) = rs.Any homoheneous potential V(r) = crs, s > 2 can be mapped to theabove model by means of a suitable rescaling of all lenghts[MC,Rindler-Daller,Yngvason 07].




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Extraction of the TF Density

EGP [] = B1dr iA2

2r2

|

|2

4 + |

|4

2

If 2, the kinetic energy gives a smaller order correction.In anharmonic traps the second part of

EGP depends on .

TF Energy Functional

ETF[] =B1

dr

2

2

2r2

4

,

with ground state energy ETF = inf1=1 ETF [] and (L1 normalized)minimizer

TF(r) = 12 TF +

22r2

4 +.




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Asymptotics of the TF Functional

ETF[] =

B1

dr 2

2

2r2

4

Slow Rotation ( 1)ETF = 12 + O(2) and TF(r) = 1 + O(22).

Rapid Rotation ( 1)ETF = O(2) and TF(r) [C1 + C2r2]+.If > 4/(

), TF(r) = 0 for any r Rin

1 4

(hole).

Ultrarapid Rotation ( 1)ETF = 2/4 + O() and TF(r) = 228

r2 R2in

+

.

Rin = 1

O(11) =

TF approaches (1

r).




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Experimental Observations [Engels et al 03]

Condensate Density

Giant Vortex (Hole) for-mation in a rotating 87RbBE condensate (induced bya laser beam).




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Slow Rotation ( 1)

Theorem (GP Asymptotics [MC,Rindler-Daller,Yngvason 07])If 0 as 0,

EGP =1

2+ O(2),

|GP|2 1

L1(B1)

= O ().

1 is the GS density of the GP functional without rotation (withenergy 12) = the rotation has no leading order effect on theGS asymptotics.

If | log |, the GP minimizer is unique and strictly positive. Inthis case GP 1 as 0 in H1(B1) (no vortex).If | log |, vortices start to occur in GP = rotational symmetrybreaking.




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Occurrence and Distribution of Vortices ( 1)1

2

|log

|The GP minimizer is a unique and a strictly positive radial function.2 2| log | + O(log | log |)

Uniformly bounded (in ) number of vortices at ri, i = 1, . . . , n.n is fixed by the remainder in the angular velocity asymptotics(coefficient of log

|log

|).

Vortices are very close to the origin: |ri| | log |1/2 and|ri rj| | log |1/2. The vortex core is a ball of radius .Vortices arrange in regular polygonscentered at the origin to minimizethe interaction energy.

3

C

|log

|, C > 2

The number of vortices is no longer uniformly bounded.Vortices are confined to a subset ofB1 (free boundary problem).

4 2| log | 1The number of vortices is /2.Vortices with winding number 1 are uniformlydistributed over

B1.




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Vortex Energy Contribution ( 1)

Theorem (Improved Energy Asymptotics [MC,Yngvason 08])

For any | log | 1,

EGP = ETF +| log(2)|

2(1 + o(1)).

Since 1, ETF = 12(1 + o(1)).Vortices of winding number 1 are uniformly distributed over B1, theirnumber is /2 and their core is .Each vortex gives a kinetic contribution of order

|log(2)

|:

2

1/2

dr r1 | log(2)|.No proof of the existence of isolated vortices but only of a uniformdistribution of vorticity satisfying the above properties.




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Rapid Rotation ( 1): Numerical Simulations[Kasamatsu et al 02]

Condensate Density

Hole

The GP minimizer is exponen-tially small in a disc centeredat the origin (hole) and vortices

cover the whole trap.




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Rapid Rotation ( 1)Theorem (GP Asymptotics [MC,Yngvason 08])

For any 1,

EGP = ETF +| log |

2(1 + o(1)),

|GP|2 TFL1(B1) = O | log |.

GP contains a number /2 of vortices with winding number 1uniformly distributed over a regular lattice with spacing

.

The vortex core is a ball of radius .Each vortex gives an kinetic contribution of order | log |:

2

dr r1 | log |.




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Vortex Energy Contribution ( 1)

GP

TF(r)ei , where the phase minimizes the kinetic

energy and contains the vortices:

ei = i

| i| , =

arctan

y yix xi

.

Kinetic energy A

2

2

= Ar

2

2

, where we have

used the rotation =

log | i|, A Ar.

Electrostatic Analogy

Vortex at ri Unitary Point Charge + 1 at ri(vortex core of size ) (smeared over a ball of size )Vector Potential A Uniform Charge Density

2Vortex Kinetic Energy Energy of the Electric Field




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An Electrostatic Problem ( 1) [MC,Yngvason 08]Finding the optimal distribution of vortices is equivalent to minimize

the electrostatic energy of a charge distribution given by positivepoint charges and a negative uniform background.

As long as TF varies on a scale of order 1, the optimal distribution isuniform: Vortices on a regular lattice with fundamental cell Q andspacing

|Q|covering

B1 (triangular, square or hexagonal

lattice).

The cell volume is chosen so that the cell is neutral, i.e., |Q| = 2/.The dipole associated with any cell Qi vanishes because of the cellsymmetry = the electric field Ei generated by Qi decays very fast(at least r

3

) outside Qi = the leading order contribution is

given by the self-energy inside Qi (minimized by the triangularlattice).

Self-energy inside Qi:

Qi

\Bi

dr A

2= | log | + O(1).



1


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Slow and Rapid Rotation ( 1): Vorticity

Theorem (Uniform Distribution of Vorticity [MC,Yngvason 08])

Let > 0 be sufficiently small and | log | 1. Then there exists afinite family of disjoint balls

Bi supp(TF) such that1 the radius of any ball is at most of order 1/

,

2 the sum of all the radii is at most of order

,

3

GP C| log(2)|1 on Bi, for some C > 0,and, denoting byri, the center of each ballBi and by di, the windingnumber of |GP|1GP on Bi,

2

di, (rri,) w

0TF(r) dr.

If one could prove that GP has only isolated vortices, then theTheorem would yield their number and uniform distribution.



( 1)


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Ultrarapid Rotation ( 1)

The rotational energy dominates: ETF =

2

41 + O

(11)and TF tends to a distribution supported on B1.

Theorem (Energy and Density Asymptotics [MC,Yngvason 08])

For any 1

1

2| log |,

EGP = ETF +| log |

2(1 + o(1)).

The density

|GP

|2 is exponentially small in almost everywhere, except

for a thin layer of width 11 around the boundary B1.

If 2 the occupation of Landau levels becomes relevant (Landauregime). Why the upper bound 1

2

|log

|?



( 1) f G


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Ultrarapid Rotation ( 1): Emergence of the GiantVortex

As long as 12| log | there are vortices in the support of

TF

even though it is very thin.

If 1

2

|log

|

one can concentrate the whole vorticity in the center

and lower the energy (giant vortex).

Heuristic Comparison

For 1 the number of cells inside supp(TF) is 1 = themutual interaction is

# of pairs

2.

Vortices (one inside each cell) neutralize the mutual interaction buthave an energy cost | log |.The vortex energy is of the same order of the mutual energy if 2

|log

|1 =

a transition takes place at that threshold.



N i l Si l i [K l 02]


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Numerical Simulations [Kasamatsu et al 02]

Condensate Density

Giant Vortex

The GP minimizer is concen-trated in a thin annulus nearB1 (giant vortex) and expo-nentially small everywhere elsebut the essential support of

GP contains no vortices.



Ul id R i ( 1) Th Gi V


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Ultrarapid Rotation ( 1): The Giant Vortex

Rigorous Comparison [MC,Rougerie,Yngvason 09]

The upper bound EGP ETF + | log |2

(1 + o(1)) holds for any

| log | 2.A trial function of the form giant(r)

TF(r)eiN, with winding

number N /2 yieldsEGP[giant] = ETF + O(2) + O(22| log |)

but 2 | log | in this regime = giant lowers the energy.If > c =

O(2

|log

|1), vortices are expelled from the essential

support of GP.Inside the hole the GP minimizer is exponentially small in andcontains a very large number of vortices, i.e., GP giant onlyinside its essential support


References

R f (Ph i )


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References (Physics)

A. Aftalion, Q. Du, Phys. Rev A 64 (2001), 063603.

W. Bao, in Dynamics in Models of Coarsening, Coagulation,Condensation and Quantization, World Scientific, 2007.

V. Bretin, S. Stock, Y. Seurin, J. Dalibard, Phys. Rev.Lett. 92 (2004), 050403.

Y. Castin, R. Dum, Eur. Phys. J. D 7 (1999), 399-412.F. Dalfovo, S. Giorgini, L.P. Pitaevskii, S. Stringari,Rev. Mod. Phys. 71 (1999), 463532.

A.L. Fetter, Rev. Mod. Phys. 81 (2009), 647691.

U.R. Fischer, G. Baym, Phys. Rev. Lett. 90 (2003), 140402.

K.W. Madison, F. Chevy, W. Wohlleben, J. Dalibard,Phys. Rev. Lett. 84 (2000), 806.

S. Stock, B. Battelier, V. Bretin, Z. Hadzibabic, J.

Dalibard, Laser Phys. Lett. 2 (2005), 275284.


References

R f (M th)


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References (Math)

A. Aftalion, Vortices in Bose-Einstein Condensates, Progress inNonlinear Differential Equations and their Applications 67,Birkhauser, 2006.

A. Aftalion, X. Blanc, Vortex Lattices in Rotating Bose EinsteinCondensates, SIAM J. Math. Anal. 38 (2006), 874893.

A. Aftalion, X. Blanc, Reduced Energy Functionals for aThree-dimensional Fast Rotating Bose Einstein Condensates, Ann. I.H. Poincare Anal. Nonlineaire25 (2008), 339355.

A. Aftalion, X. Blanc, F. Nier, Lowest Landau Level

Functional and Bargmann Spaces for Bose Einstein Condensates, J.Funct. Anal. 241 (2006), 661702.

S. Baldo, R.L. Jerrard, G. Orlandi, H.M. Soner, inpreparation.


References

R f s (M th)


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References (Math)

M.C., T. Rindler-Daller, J. Yngvason, Rapidly RotatingBose-Einstein Condensates in Strongly Anharmonic Traps, J. Math.Phys. 48 (2007), 042104.

M.C., T. Rindler-Daller, J. Yngvason, Rapidly RotatingBose-Einstein Condensates in Homogeneous Traps, J. Math. Phys.48 (2007), 102103.

M.C., N. Rougerie, J. Yngvason, in preparation.

M.C., J. Yngvason, Energy and Vorticity in Fast RotatingBose-Einstein Condensates, J. Phys. A: Math. Theor. 41 (2008)

445002.

S. Gueron, I. Shafrir, On a Discrete Variational ProblemInvolving Interacting Particles, SIAM J. Appl. Math. 60 (2000), 117.


References

References (Math)


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References (Math)

R. Ignat, V. Millot, The Critical Velocity for Vortex Existence ina Two-dimensional Rotating Bose-Einstein Condensate, J. Funct.Anal. 233 (2006), 260306.

R. Ignat, V. Millot, Energy Expansion and Vortex Location for aTwo Dimensional Rotating Bose-Einstein Condensate, Rev. Math.

Phys. 18 (2006), 119162.N. Rougerie, Transition to the Giant Vortex State for aBose-Einstein Condensate in a Rotating Anharmonic Trap, preprintarXiv:0809.1818v2 [math-ph] (2008).

R. Seiringer, Gross-Pitaevskii Theory of the Rotating Bose Gas,Commun. Math. Phys. 229 (2002), 491509.

R. Seiringer, Vortices and Spontaneous Symmetry Breaking inRotating Bose Gases, preprint arXiv:0801.0427v1 [math-ph] (2008).


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