Stability analysis

We determine the solution ofEq. (6.16) by the iterative methoddisussed inSe. 2.2.1.

Initiallywetaketheanonialdistributionofanoninteratinggasinthedipoletrap.The

dipole-dipoleinterationtermgivenbytheonvolutionintegralinEq.(6.15)isalulated

thenfrom this distribution.It issuitable togointoFourier-spae wherethe onvolution

is a simple produt. The Fourier transform of the term

β(r)

^{is known, f. Eqs. (6.9)}

and (6.11),while theterm

p(r)f (r)

^is transformedby numerialFastFourierTransform (FFT). Due to the nite support of the distribution

p(k) ˜

^{and the mode funtion}

f(k) ˜

^,

thesingularityof

β(r)

^{,whihappearsasa}non-deayingFouriertransforminmomentum spae, is automatially regularized. On transforming the produt bak to real spae by

FFTweget thedipole-dipoleterm.Addingittothe dipoletraptermofEq.(6.14)yields

thetotalmeaneldpotentialwhihfurnishesanewatomidistributionviatheanonial

forminEq.(6.16).Theresulting

p(x)

^{an beused asthestarting}distributioninthenext stepof theiteration.Continuing thesteps ofiterationuntilonvergene, oneobtainsthe

self-onsistentsolution of Eq. (6.16).

Theiterationdoesnotneessarilyonverge:instabilityanbeinduedbylargeenough

atom number orintensity. The iteration method suggests that the instability ours as

aollapseof the atomilouddue toself-ontrationinthe enter ofthe trap. However,

in the lak of a preise modeling of the ollisional proesses, the ollapse itself annot

be aounted for by our approah. We must limitourselves todetermining the range of

onvergene.

The mean eld dipole-dipoleenergy Eq. (6.15) depends on the shape of the atomi

loud. It has a ylindrial symmetry and the laser-indued polarization is radial, the

optialeldbeingtransverse.Forthispolarization,itisthepanake-shaped trapwhere

thedipole-dipoleontributiontothe MFpotentialisnegative,deepeningthe trapdepth

N

I

10 ^{− 1} 10 ^{− 2}

10 ^{− 3} 10 ^{− 4}

10 ^{− 5} 10 ^{− 6}

100

10

Figure 6.1: The boundary of the stability range of the atomi loud is shown in

loga-rithmi phase diagram of the saled intensity

I

^{and the saled atom number}

N

^{. The}

eld polarizationis irularinthe

x ˆ − y ˆ

^{plane,and the waistis}

w/λ

^{=1.33 (irle),2.66}

(triangle), 5.33 (diamond), and 10.66 (pentagon). Straight lines represent a t on the

numerialdata.Asshown withemptyirles,below

I = 10

^{thephaseboundarydeviates}

fromthe orresponding ttedline(

w/λ = 1.33

^{),whihindiatesthe eet ofuntrapped}

atoms.

in the enter for a spherial atom distribution (the renement of this statement an be

foundin[163℄).Foraigar-shapedloudthe MFdipole-dipoleenergy wouldbepositive,

repellingatomsfromtheenter,andthustheinstabilitywedisussinthefollowingould

not our. Note that inusual onsiderationsof dipolarquantum gases, the stati dipole

moment is taken, oppositely, in the axial diretion and then the panake-shaped loud

is stableagainst dipole-dipoleattration [153℄.

InFig.6.1thestabilityrangeofthe iterationsispresentedforirularpolarizationin

the

x ˆ

y ˆ

^{plane, andfor beam waists}

w = 1.33

^{,2.66, 5.33,and 10.66 inunits of}

λ

^{.On the}

two-dimensional plot for the saled atom number and saled intensity variables (phase

diagram), border points of the stability region are shown and the onvergent iterative

solution of Eq. (6.16) exists in the region below the points. The border points an be

welltted by apowerlawdependene ofthe ritial saledintensity onthe saledatom

number,

I ∝ N ^{− c}

^.Thetisrepresentedbylines,theexponentis

c = 0.40( ± 0.01)

^.Shown

only for the

w/λ = 1.33

^{data with empty irles, the boundary bends away from the}

tted straight lineat the right-most end. This happens below a ertainsaled intensity

(

I < 10

^{),when a signiantportionof untrapped atoms appear.}

I

V d d /k B T p (0 ) λ 3 γ / ∆ A

5

1 160 80

40 20

10 1

0.1

0.01

Figure6.2:Leftsale:Maximumdensityoftheloud,

p(0)

^inunitsof

1/λ ³ × ∆ A /γ

^,plotted

againstthesaledintensity

I

^for

w/λ

^{=1.33(irle),2.66}(triangle),5.33(diamond).The straightlineisatonthe

w/λ = 5.33

^{data.The saledatomnumber}

N

^{isset suhthat}

the system remains slightly below the ritial point. Right sale: stars with a onstant

t represent the MF dipole-dipolepotential

V dd /k B T

^{alulated lose to the boundary}

of the range of stability. The beam waist is

w/λ = 5.33

^.

izationalong

x ˆ

^{.Wealsoheked thattheadditionofanarbitraryonstanttotheFourier}

transform

β(k) ˜

^{does not appreiablyshift the phase boundary. This justies the neglet}

of any type of Dira-

δ(r)

^{potential in the} Hamiltonian,e.g., the ontat potential[162℄

or

s

^{-wavesattering. Finally,themutualeet oflouds in}neighboring trappingsites of the optiallattiepotentialreated by the eldinEq. (6.1)proved tobe negligiblewith

respet to the phase diagram: the instability arises from the short-range part (

1/r ³

^{) of}

the dipole-dipoleoupling and isdetermined by the loud ata single trappingsite.

The dipole-dipole oupling enhanes the trap depth in the enter and inreases the

atom density there. The resulting self-ontration of the loud is ounterated by the

random motion of the atoms. We expet that instability ours when the energy shift

due to the dipole-dipoleoupling exeeds the thermal energy of the atoms. The

numer-ial approah has allowed us to onrm this expetation. In Fig. 6.2 the ratio of the

dipole-dipole interation potential and the thermal energy is plotted (with stars) on a

logarithmisale atthe edgeof the stableregion (where stillstablesolutionexists), and

one nds

V dd ≈ k B T

^{losely onstant.}

Aepting the instability ondition

V _dd ≈ k _B T

^{, the ritial exponent}

c = 0.4

^{an be}

obtainedbysimplearguments.ItfollowsfromEq.(6.15)that

V _dd ⁽⁰⁾ (0)/k B T

^isproportional

I V d d /V tr a p

140 120

100 80

60 40

20 0.1

0.01

0.001

1e-04

Figure 6.3: (Color online) The ratio of the MF dipolepotential and the total potential

at the origin as a funtion of the saled intensity

I

^for

w/λ = 5.33

^{, and at xed values}

of

N = 1

^,

2

^,

4 × 10 ^{− 5}

^.

to

IN

^{times the onvolution integral. Using the dominantterm of the potentialfor the}

distribution, i.e., the trap potentialin harmoni approximation, the resulting Gaussian

distribution has a normalization fator proportionalto

I ^3/2

^{. The remaining part is the}

onvolution involving the funtion

β

^{whih is singular at the origin. Thus the main}

ontribution must ome from this small domain, the loud size is irrelevant and the

integralmust bedetermined,atleast toleadingorder,bythe aspetratio ofthetrapped

loud. Altogether

V _dd ⁽⁰⁾ (0)/k B T ∝ I ^5/2 N

^{fromwhih the saling}

I ∝ N ^{− 0.4}

^follows.

Foronsisteny,letushekthevalidityoftheeetiveHamiltoniangiveninEq.(6.13)

whih keeps only the leading order term of the dipole-dipoleinteration. Figure6.3

de-pits the ratio of the MF dipole-dipole potential to the trap potential at the enter of

theloud,whihisthe sameratioasthatofthehigherordertermsofthepolarizationto

the leadingorder one [f.Eq. (6.12)℄. This ratio grows as afuntion of the saled

inten-sity untilreahing the ritial point whih islearly manifested inthis semi-logarithmi

plot. The main thing to observe is that the ritiality is reahed at fairly low ratio of

the dipole-dipole potential to the trapping one, at a value well below

0.1

^{. Therefore,}

in the onsidered parameter regime, the leading order desription is enoughto nd the

In document (1)ultraold atoms in an optial resonator Ph.D (Pldal 94-98)