Phase Dislocations in Bose-Einstein Condensates

Phase Dislocations in Bose-Einstein

Condensates

- Literature Review -

Shekhar Suresh Chandra∗

May 18, 2005

1 Bose-Einstein Condensates (BECs)

A Bose-Einstein Condensate is a macroscopic quantum object consisting of integral intrinsic angu-

lar momentum or spin particles called Bosons. The condensate arises when the bosons undergo a

phase transition after being cooled so that their de Broglie wavelength becomes comparable to their

separation with other bosons, with all eventually occupying the ground state energy level[25][7]. In

this phase transition the bosons lose their individuality to become this single macroscopic quantum

object - the condensate.

1.1 A Prediction of Statistical Mechanics

In 1924, an Indian Physicist by the name of Satyendra Nath Bose provided an alternative derivation

of the Planck Blackbody Radiation Law, in which he applied statistical mechanics and treated

photons as indistinguishable particles. He did this with a function which he introduced later named

the Bose-Einstein function[13][34]. The quantum distribution for bosons, which is now called the

Bose-Einstein Distribution, has the form

fBE(E) ≈ 1eE/kbT − 1

(1)

where E is the energy and kb is the Boltzmann constant. The most important feature pretenant

to this review is the behaviour of this distribution as E → 0. The distribution in this case, tends∗Monash University. Email: [email protected]

1

mailto:[email protected]

1 BOSE-EINSTEIN CONDENSATES (BECS) 2

to infinity (i.e. the occupancy of energy states tends to infinity and so the number of bosons

occupying the ground state can be infinite). Therefore, the distribution predicts the existence of a

condensate, i.e. a phase consisting of the ground state energy level to which all bosons eventually

occupy at some critical temperature.

1.2 The Experimental Verification

It was in 1995, after improvements in magnetic traps (see section 1.3.1 for details), that the first

Bose-Einstein Condensate was observed independently by three competing groups[26]. The first

group was from Boulder[3] and they had their breakthrough in June. The next group was from

MIT[17], who had their observation in September. Lastly, the Rice group obtained evidence of

quantum degenerate regime in July[14].

1.3 Forming an Atomic Bose-Einstein Condensate

In order to form a particular Bose-Einstein condensate, one needs to cool the relevant bosons until

the interbosonic spacing is of the order of the de Broglie wavelength given by λdB = 2πh2/mkbT .

This cooling must continue until the phase transition occurs, which normally depends on the

peak bosonic density as nλ3dB ≈ 2.612. Sophisticated techniques are required to cool and contain

the condensate, as temperatures involved as of the order of µK and so is very sensitive to the

environment[26][7].

1.3.1 Laser pre-cooling, Magnetic Trapping & Forced Evaporative cooling

Will use [8][30].

1.3.2 Micro-Traps

Will use [32][23].

1.3.3 All Optical Cooling

Will use [9].

1.4 The Gross-Pitaevskii-Bogoliubov Theory

Will use [21][33][12][4].

2 VORTICES IN BOSE-EINSTEIN CONDENSATES 3

1.5 Many-Body Interactions

Will use [15][40].

1.6 Imaging of Bose-Einstein Condensates

2 Vortices in Bose-Einstein Condensates

(Under Construction)

2.1 Theoretical Prediction

Will use [37].

2.2 Forming Vortices

Will use [31][29][1][22][24][20][28][35][16][5][18][27].

2.3 Imaging of Vortices

Will use [1][20].

3 Conducted Theoretical Simulations of Vorticity in BEC’s


Will use [19][39][16][5][6]

4 Wavefunction dislocations

4.1 Dislocations in the Aharonov-Bohm effect

Will use [10][11].

4.2 The respective water wave analogue

Will use [11].

5 WAVEFUNCTION RECONSTRUCTION TECHNIQUES 4

4.3 Dislocations in Bose-Einstein condensate vortices

5 Wavefunction Reconstruction Techniques


5.1 Tan, Paganin, Yu & Morgan Method (Working Title)

Will use [36].

6 Preliminary Results

A quick review of scientific libraries available for C++ was conducted (including libraries such as

the GNU Scientific Library (GSL) and Class Library for Numbers (CLN)) revealing that Blitz++

Scientific Library1 was the most appropriate and computationally fastest. Analytically, the Gross-

Pitaevskii Equation was variable separated to determine the time dependent and independents

parts2. The time independent part was then discretized with second order finite differences and

the time dependent part with the Semi-Implicit Method (which utilizes the Crank-Nicolson Implicit

method with fractional stepping and the Method of Approximate Factorization3. Computationally,

the steady state solution was evaluated using the fixed point method and used as the inital state,

then evolved using the time dependent part of the Gross-Pitaevskii Equation. Plots were made of

the time independent solution and the evolved solution4.

Appendix

A Blitz++ Scientific Library

Blitz++ is a library that offers the same features that Fortran 77/90 offers but in C++. It is

as fast and in some instances faster than Fortran. The library is designed with Object Oriented

Design and also has special support for dynamic (run-time variable) arrays, tensor notation and

differential operators through stencils5.1refer to Appendix A for more info2refer to Appendix B.2 for the exact prescription3refer to Appendix B.3 for details4refer to Appendix C5see Blitz++ Homepage

http://www.oonumerics.org/blitz/

B THE GROSS-PITAEVSKII EQUATION 5

B The Gross-Pitaevskii Equation

B.1 Construction

B.2 Separation of Variables

The Gross-Pitaevskii Equation of a BEC, in a trap, with applied angular momentum, in one time

and 2-spatial dimensions is given by

ih∂Ψ∂t

=[− h2

2m∇2⊥ +

12mω2|q|2 − LzΩ + g|Ψ|2

]Ψ (2)

where q is a generalised coordinate and

∇2⊥ =

∂2

∂x2+

∂2

∂y2(3)

=∂2

∂r2+

1r

∂

∂r+

1r2

∂2

∂φ2(4)

=1r

∂

∂r

(r∂

∂r

)+

1r2

∂2

∂φ2. (5)

Lz = −ih(x∂

∂y− y

∂

∂x

)(6)

= −ih ∂

∂φ. (7)

Let us consider that the above is Variable Separable of the form

Ψ(x, y, t) = Q(q)T (t) (8)

where the T (t) is the time part and Q(q) the spatial part in generalised coordinates. Inserting this

into equation 2, we get

ih∂

∂tQ(q)T (t) = − h2

2m∇2⊥Q(q)T (t) +

12mω2|q|2Q(q)T (t)− LzΩQ(q)T (t) + g|Q(q)T (t)|2Q(q)T (t)

The partial derivatives now become full derivatives and hence

ihQ(q) ˙T (t) = − h2

2mQ′′(q)T (t) +

12mω2|q|2Q(q)T (t)− ΩQ′(q)T (t) + g|Q(q)T (t)|2Q(q)T (t)

Dividing through by Q(q)T (t) and noting that |T (t)|2 = 1 because of the unitary requirement of

time evolution, one gets

ih1

T (t)˙T (t) = − h2

2m1

Q(q)Q′′(q) +

12mω2|q|2 − Ω

1Q(q)

Q′(q) + g|Q(q)|2 (9)

Note that the left hand side is only a function of time and time derivatives and that the right hand

side is only a function of space and spatial derivatives. In order for the time derivatives to equal


spatial derivatives, both must equal a constant, the constant of separation, which we shall call µ.

This constant allows us to separate the time and spatial parts of the above into

ih1

T (t)dT (t)dt

= µ (10)

− h2

2m1

Q(q)Q′′(q) +

12mω2|q|2 − Ω

1Q(q)

Q′(q) + g|Q(q)|2 = µ (11)

Let us proceed to determine the steady state or time independent equation of the system by writing

the spatial equation above in Cylinderical Co-ordinates (i.e. Q(q) = R(r)φ(φ))

− h2

2m∇2⊥R(r)φ(φ)+

12mω2r2R(r)φ(φ)−LzΩR(r)φ(φ)+g|R(r)φ(φ)|2R(r)φ(φ) = µR(r)φ(φ) (12)

We use equations 4 & 7 to write above as

− h2

2m

[∂2R(r)∂r2

+1r

∂R(r)∂r

+1r2∂2φ(φ)∂φ2

]+

12mω2r2R(r)φ(φ)

+ ihΩR(r)∂φ(φ)∂φ

+ g|Ψ|2R(r)φ(φ) = µR(r)φ(φ) (13)

Now for our case, we wish to setup a vortex, so we now take our angular solution as

φ(φ) = e−inφ (14)∂φ(φ)∂φ

= −inφ(φ)

∂2φ(φ)∂φ2

= −n2φ(φ)

where n is the winding number of the vortex6 and φ is the phase angle. Substituting this into

equation 13 and multiplying through by φ(φ), our stead state ψ(r) of the system becomes

− h2

2m

(d2ψ(r)dr2

+1r

dψ(r)dr

− n2

r2

)+

12mω2r2ψ(r) + nhΩψ(r) + g|ψ(r)|2ψ(r) = µψ(r) (15)

B.3 Discretization

In order to numerical simulate using differential equations, one must discretise the equations.

Firstly we discretize the time independent or steady state form of the Gross-Pitaevskii Equation,

namely equation 15. This is known as a boundary value problem and is treated accordingly in

section B.4. Secondly, we discretize the time evolution of the inital state (that of equation 15),

which is an initial value problem and described in section B.5.6This is due to the quantization of vorticity


B.4 The Boundary Value Problem

Equation 15 needs to be discretized using a stable and reasonably accurate numerical method. For

accuracy, we shall keep it to the order of the derivatives involved, in this case, it is of second order.

The second order accurate of the second derivative is given as

d2ψ

dr2=(ψj+1 − 2ψj + ψj−1

∆r2

)(16)

where now the j represents the grid position. The above is a centered difference method. We shall

discretize the first order also with a second order accurate scheme

dψ

dr=(ψj+1 − ψj−1

2∆r

)(17)

Therefore, in using the above, our finite difference form of the steady state equation is

− h2

2m

[(ψj+1 − 2ψj + ψj−1

∆r2

)+

1r

(ψj+1 − ψj−1

2∆r

)− n2

r2

]+

12mω2r2ψj + nhΩψj + g|ψj |2ψj = µψr (18)

This can be solved using the Fixed Point Method to acquire the steady state solution.

B.5 The Initial Value Problem

In order to evolve the system in time, one needs to solve equation 10 and write the full wavefunction,

which is then

Ψ(q, t) = Ψ(q)e−iµt/h (19)

Here the evolution of the system is governed by the Schrodinger Time Evolution Equation

|Ψ(t)> = e−iHt/h|Ψ(0)> (20)

Because the evolution must be unitary for the conservation of probability and it involves complex

values, the method of integration must be unitary as well as stable in the complex domain and

reasonably accurate. Such a method is the Crank-Nicolson Method. We begin with the Cartesian

form of the Gross-Pitaevskii Equation

ih∂Ψ∂t

= − h2

2m∂2

∂x2Ψ− h2

2m∂2

∂y2Ψ +

12mω2x2Ψ +

12mω2y2Ψ + ihΩ

(x∂Ψ∂y

− y∂Ψ∂x

)+ g|Ψ|2Ψ (21)

Using n to denote the nth timestep k & ` for x & y spatial points respectively, we use forward

difference approximation of the left-hand side of above

ih∂Ψ∂t

= ih

(Ψn+1

k,` −Ψnk,`

∆t

)(22)


Explicit centred difference approximation for the x part of the spatial derivatives

− h2

2m∂2

∂x2Ψ = − h2

2m

(Ψnk+1,` − 2Ψn

k,` + Ψnk−1,`

∆x2

)(23)

and the y part of the spatial derivatives

− h2

2m∂2

∂y2Ψ = − h2

2m

(Ψnk,`+1 − 2Ψn

k,` + Ψnk,`−1

∆y2

)(24)

Lastly, forward difference approximation to the angular momentum derivatives

x∂Ψ∂y

= x

(Ψnk,`+1 −Ψn

k,`

∆y

)(25)

y∂Ψ∂x

= y

(Ψnk+1,` −Ψn

k,`

∆x

)(26)

The implicit versions of the spatial differentials above are when n = n+1. For the Crank-Nicolson

method, we use the average of the explicit and implicit forms. This gives us

− h2

2m∂2

∂x2Ψ = −1

2h2

2m∆x2

[(Ψn

k+1,` − 2Ψnk,` + Ψn

k−1,`) + (Ψn+1k+1,` − 2Ψn+1

k,` + Ψn+1k−1,`)

](27)

− h2

2m∂2

∂y2Ψ = −1

2h2

2m∆y2

[(Ψn

k,`+1 − 2Ψnk,` + Ψn

k,`−1) + (Ψn+1k,`+1 − 2Ψn+1

k,` + Ψn+1k,`−1)

](28)

x∂Ψ∂y

=x

2∆y

[Ψn

k,`+1 −Ψnk,` + Ψn+1

k,`+1 −Ψn+1k,`

](29)

y∂Ψ∂x

=y

2∆x

[Ψn

k+1,` −Ψnk,` + Ψn+1

k+1,` −Ψn+1k,`

](30)

Likewise we take the average of any other functions of Ψn

12mω2(x2 + y2)Ψ =

12mω2(x2 + y2)

(Ψn

k,` + Ψn+1k,`

)(31)

g|Ψ|2Ψ =12g(|Ψn

k,`|2Ψnk,` + |Ψn+1

k,` |2Ψn+1

k,`

)(32)

Combining equations 22, 27, 28, 29, 30, 31 & 32, we get the Gross-Pitaevskii Equation in implicit

finite difference form

ih

(Ψn+1

k,` −Ψnk,`

∆t

)= −1

2h2

2m∆x2

[(Ψn

k+1,` − 2Ψnk,` + Ψn

k−1,`) + (Ψn+1k+1,` − 2Ψn+1

k,` + Ψn+1k−1,`)

]− 1

2h2

2m∆y2

[(Ψn

k,`+1 − 2Ψnk,` + Ψn

k,`−1) + (Ψn+1k,`+1 − 2Ψn+1

k,` + Ψn+1k,`−1)

]+

xihΩ2∆y

[Ψn

k,`+1 −Ψnk,` + Ψn+1

k,`+1 −Ψn+1k,`

]− yihΩ

2∆x

[Ψn

k+1,` −Ψnk,` + Ψn+1

k+1,` −Ψn+1k,`

]+

12mω2(x2 + y2)

(Ψn

k,` + Ψn+1k,`

)+

12g(|Ψn

k,`|2Ψnk,` + |Ψn+1

k,` |2Ψn+1

k,`

)(33)


We need to separate the known values (those at timestep n) and those that are unknown. We do

this first by breaking up the time finite difference

Ψn+1k,` = Ψn

k,` +ih∆t

4m∆x2

[(Ψn

k+1,` − 2Ψnk,` + Ψn

k−1,`) + (Ψn+1k+1,` − 2Ψn+1

k,` + Ψn+1k−1,`)

]+

ih∆t4m∆y2

[(Ψn

k,`+1 − 2Ψnk,` + Ψn

k,`−1) + (Ψn+1k,`+1 − 2Ψn+1

k,` + Ψn+1k,`−1)

]+

xΩ2∆y

[Ψn

k,`+1 −Ψnk,` + Ψn+1

k,`+1 −Ψn+1k,`

]− yΩ

2∆x

[Ψn

k+1,` −Ψnk,` + Ψn+1

k+1,` −Ψn+1k,`

]+

∆t2ih

mω2(x2 + y2)(Ψn

k,` + Ψn+1k,`

)+

∆t2ih

g(|Ψn

k,`|2Ψnk,` + |Ψn+1

k,` |2Ψn+1

k,`

)(34)

For convenience, we can write

δ2xΨn+1 =(Ψn+1

k+1,` − 2Ψn+1k,` + Ψn+1

k−1,`

)(35)

δ2xΨn =(Ψn

k+1,` − 2Ψnk,` + Ψn

k−1,`

)(36)

δxΨn+1 =(Ψn+1

k+1,` −Ψn+1k,`

)(37)

δxΨn =(Ψn

k+1,` −Ψnk,`

)(38)

The same will be used for the y component. Then equation 34 can be more conveniently rewriten

as

Ψn+1k,` = Ψn

k,` +ih∆t

4m∆x2

[δ2xΨn + δ2xΨn+1

]+

ih∆t4m∆y2

[δ2yΨn + δ2yΨn+1

]+

xΩ2∆y

[δyΨn + δyΨn+1

]− yΩ

2∆x[δxΨn + δxΨn+1

]+

∆t2ih

mω2(x2 + y2)(Ψn

k,` + Ψn+1k,`

)+

∆t2ih

g(|Ψn

k,`|2Ψnk,` + |Ψn+1

k,` |2Ψn+1

k,`

)(39)

We now take separate implicit from the explicit by taking the implicits terms to the left-hand side

of the above equation. This left-hand side is then

Ψn+1k,` − ih∆t

4m

[δ2xΨn+1

∆x2+δ2yΨn+1

∆y2

]− Ω

2

[xδyΨn+1

∆y− y

δxΨn+1

∆x

]−∆t

2ihmω2(x2 + y2)Ψn+1

k,` − ∆t2ih

g|Ψn+1k,` |

2Ψn+1k,` (40)

and the right-hand side is now

Ψnk,` +

ih∆t4m

[δ2xΨn

∆x2+δ2yΨn

∆y2

]+

Ω2

[xδyΨn

∆y− y

δxΨn

∆x

]+

∆t2ih

mω2(x2 + y2)Ψnk,` +

∆t2ih

g|Ψnk,`|2Ψn

k,` (41)


And the right side of the equation can be simplified to a single value d, since all values are known

and can be evaluated explicitly. Therefore, the Gross-Pitaevskii Equation in semi-implicit form

can be written simply as

Ψn+1k,` − ih∆t

4m

[δ2xΨn+1

∆x2+δ2yΨn+1

∆y2

]− Ω

2

[xδyΨn+1

∆y− y

δxΨn+1

∆x

]−∆t

2ihmω2(x2 + y2)Ψn+1

k,` − ∆t2ih

g|Ψn+1k,` |

2Ψn+1k,` = d (42)

The above equation has to be put into tridiagonal form for each spatial dimension to be solved

conveniently. There are two approaches which can be taken.

B.5.1 2D Alternating Direction Implicit (ADI) Approach

Evaluating one spatial dimension per timestep is not unconditionally stable. However, by dividing

each timestep into smaller fractions according to the number of dimensions being evaluated, the

conditionally stable fractions combine to form a unconditionally stable method if the dimensions

are even. So we break equation 42 into fractional timesteps as

Ψn+1/2k,` − ih∆t

8m

[δ2xΨn+1/2

∆x2+δ2yΨn

∆y2

]− Ω

2

[xδyΨn

∆y− y

δxΨn+1/2

∆x

]−∆t

4ihmω2(x2 + y2)Ψn+1/2

k,` − ∆t4ih

g|Ψn+1/2k,` |2Ψn+1/2

k,` = dn (43)

Ψn+1k,` − ih∆t

8m

[δ2xΨn+1/2

∆x2+δ2yΨn+1

∆y2

]− Ω

2

[xδyΨn+1

∆y− y

δxΨn+1/2

∆x

]= dn+1/2 (44)

where the superscripts denote the timesteps to evaluate the term at. This is second order in both

space and time. However, each of the equations on their own is unstable, but together they make

the method stable. The method is unstable for odd dimensions due to the lack of symmetry in

the stability ratio as pairs alternate directions and stablize. The ADI can be improved for any

number of dimensions[2], but is not needed here. To solve the above as a tridiagonal, we rearrange

the above equations. The first fractional timestep then becomes

axΨn+1/2k+1,` + bxΨn+1/2

k,` + cxΨn+1/2k−1,` = dn

x + fn+1/2 (45)

Here the tridiagonal elements are

ax = −αx − βx (46)

bx = (1− 2αx − βx) (47)

cx = −αx (48)

dnx = dn +

ih∆t8m

δ2yΨn

∆y2+xΩ2δyΨn

∆y(49)


where

αx =ih∆t

8m∆x2βx =

yΩ2∆x

and

fn+1/2 =∆t4ih

mω2(x2 + y2)Ψn+1/2k,` +

∆t4ih

g|Ψn+1/2k,` |2Ψn+1/2

k,`

The second fractional timestep then becomes

ayΨn+1k,`+1 + byΨn+1

k,` + cyΨn+1k,`−1 = dn+1/2

y (50)

Here the tridiagonal elements are

ay = −αy − βy (51)

by = (1− 2αy − βy) (52)

cy = −αy (53)

dn+1/2y = dn+1/2 +

ih∆t8m

δ2xΨn+1/2

∆x2+

Ω2yδxΨn+1/2

∆x(54)

where

αy =ih∆t

8m∆y2βy =

xΩ2∆y

Both equations 45 & 50 have the right-hand side in semi-implicit form and so have to be solved with

multiple iterations for each timestep. The semi implicit nature is required due to the non-linear

terms involved in the Gross-Pitaevskii Equation.

B.5.2 Method of Approximate Factorization

Using the Gross-Pitaevskii Equation in the form[1− ih∆t

4mδ2x

∆x2− ih∆t

4mδ2y

∆y2

]Ψn+1 =

[1 +

ih∆t4m

δ2x∆x2

+ih∆t4m

δ2y∆y2

]Ψn +

∆t2[fn+1 + fn

](55)

where

fn+1 =∆t2ih

mω2(x2 + y2)Ψn+1k,` +

∆t2ih

g|Ψn+1k,` |

2Ψn+1k,` +

Ω2

[xδyΨn+1

∆y− y

δxΨn+1

∆x

](56)

We approximately factorise the terms containing δ2 in the following way[1− ih∆t

4m∆x2δ2x

] [1− ih∆t

4m∆y2δ2y

]Ψn+1 =

[1 +

ih∆t4m∆x2

δ2x

] [1 +

ih∆t4m∆y2

δ2y

]Ψn

+∆t2[fn+1 + fn

](57)


Notice how the left-hand is semi-implicit, this helps to stablize the method. Letting

αx =ih∆t

4m∆x2(58)

αy =ih∆t

4m∆y2(59)

Each of the factors on the left is of the form of a tridiagonal system to solve, i.e.

Ax = Ψn+1k,` − αx

(Ψn+1

k+1,` − 2Ψn+1k,` + Ψn+1

k−1,`

)(60)

= −αxΨn+1k+1,` + (1 + 2αx) Ψn+1

k,` − αxΨn+1k−1,` (61)

where the tridiagonal system is of the form

Ai ·Ψ = d (62)bi ci 0 . . .

ai bi ci . . .

0 ai bi . . ....

......

. . .

Ψ0

Ψ1

...

ΨN

=

d0

d1

...

dN

(63)

In the case of equation 61, the elements for Ax

ax = −αx (64)

bx = (1 + 2αx) (65)

cx = −αx (66)

Therefore equation 57 can be rewriten as

AxAyΨn+1 = BxByΨn +∆t2[fn+1 + fn

](67)

where each of the A’s is a tridiagonal to be solved. To complete the method, the tridiagonals must

be solved using fractional stepping, i.e.

AxΨn+1/2 = BxByΨn +∆t2[fn+1 + fn

](68)

AyΨn+1 = Ψn+1/2 (69)

on a similar reasoning to that of the ADI method in the previous section. This method can be

applied to multiple dimensions and is not restricted to just even number of dimensions as with

the simple ADI case. In order to evaluate the fn+1 term, a number of iterations has to be done

per timestep because of the non-linear terms in the equation. At the first iteration, the value

fn+1 = fn can be used and updated values used in further iterations. Winiecki & Adams[38]

showed that it converges when following this proceedure and that it has an optimum number of

iterations of three per timestep.

C RESULTS 13

C Results

C.1 Graphs

Figure 1 shows the solution of equation 18. Figures 2 & 3 shows the initial topological phase and

probability density in the system. Finally, figure 4 shows the solution of the numerical integration

after 125 timesteps.

C.2 Discussion

The solution to the time independent Gross-Pitaevskii equation was as expected, with a singularity

at r = 0. Although this singularity was introduced by our boundary conditions, rest of the radial

solution reflects the expected probability density for a vortex. The topological phase induces the

rotation, which is maintained by the application of angular momentum in the Gross-Pitaevskii

equation. The intial state was evolved 125 timesteps and due to no external forces or disturbances,

the vortex remained stationary and undisturbed as expected.

C RESULTS 14

Figure 1: The Radial Solution of the BEC with a Vortex

C RESULTS 15

Figure 2: The Initial Topological Phases of the BEC with a Vortex

C RESULTS 16

Figure 3: The Initial Probability Density of the BEC with a Vortex

C RESULTS 17

Figure 4: The Probability Density 125 timesteps later of the BEC with a Vortex

REFERENCES 18

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Phase Dislocations in Bose-Einstein Condensates

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