Summary

InthishapterIstudiedtheinterationofaBose-Einsteinondensateandasingle-mode

optialavity in afull quantum model. In the weak ouplinglimit,the avity mode

ap-points a single ondensate exitationmode that dominantly ouples to the avity eld.

Keepingonlythissingleexitationmode,theBEC-avitysystembeomesformally

anal-ogoustoalass ofopto-mehanialsystems.Byeliminatingthe photoneld,I derived a

quantum master equationfor the BECexitation mode playing the role of the

mehan-ial osillator. Adiabati eliminationof the eld led to an eetive optial potential in

the master equation that has been showed to produe optial bistability.Dissipation of

the avity eld istaken intoaount by diusion and frition terms, with frition being

determinedasthe rst orderorretion totheadiabatieliminationinthe small

param-eter

ω R /κ

^.Linearization ofthe master equationestablishes a onnetionwith linearized

models desribing vibrating mirrors in the literature, and it enabledus toalulate

an-alytiallyan equilibriumexitation number in the extra weak ouplinglimit.Using the

eetivemasterequation,Isimulatedthenonlinear quantum dynamisofthe BEC

exi-tationinthebistabilityregime.Igenerallyshowedthatquantumtunnelingishinderedby

the dephasing proess aompaniedby photonloss. Nevertheless, oherent matterwave

osillationsarepossible,wherediusionislessdominantforlargeramplitudeosillations.

Dipole-dipole instability of atom louds

in a far-detuned optial dipole trap

Inthis hapter,I onsider theradiativeinteration between oldatoms trapped inside a

far-o-resonanedipoletrap(FORT).Althoughthisworkissomewhatlateraltothemain

bodyoftheThesis,itprovidesaomplementtothemodelsusedintheprevioushapters,

sine it allows for treating an eet whih is usually omittedfrom the desription. The

light-indued dipole-dipole interation between the atoms in a laser eld has indeed

a small eet at typial atom densities and laser powers, however, it is instrutive to

alulatethe limitabove whih this type of atomatom interation beomes signiant.

In dense samples of old atoms, two-atomproesses strongly inuene the trapping.

For example, ollisional proesses (photo-assoiation, hyperne ground state hanging

ollisions)are known toresultintraplosses andthey an limitthe maximumahievable

density. Collisions usually depend heavily on the internal eletroni struture of the

speies.Forinteratomidistanes intherange ofthe optialwavelength, the atom-atom

interation is dominatedby the radiativeeletromagneti oupling. In this ase one an

distinguish two limits: the photon sattering is dominated (i)by spontaneous emission,

e.g. in magneto-optial traps (MOT), and (ii) by stimulated emission, whih ours in

far-o-resonant dipoletraps (FORT). Sine the MOT enabled us rst toapture dense

atom louds from vapour, it was rst analyzed with respet to radiative many-body

eets [23, 140℄. In a MOT, the ooling laser is quasi-resonant with the atoms and the

sampleformsanoptiallythik medium.The depletionof thelaser beamstogether with

the multiple spontaneous sattering of resonant photons within the sample an lead to

instability[23℄andextraheating[141,142℄.Athighdensities,thereabsorptionofphotons

ina highlynonlinear olletivedynamis of the atom loud ina MOT [145, 146℄.

TheFORToperatesatanextremelylowspontaneoussatteringrate,thustheeets

of multiple spontaneous photon sattering are strongly suppressed. The mehanism of

trappingreliesontheproessofabsorptionandstimulatedemissionoflaserphotons.This

proesspolarizesthepartiles,henethedipoleforeisaompaniedbythedipole-dipole

ouplingbetweenatoms.Inpreviousworksaphenomenologialtermproportionaltothe

square of the atom density was introdued to desribe the atom-atom interation, e.g.,

the eet of ollisions on the loading of a FORT [147, 148℄. This dominates in strongly

loalized traps,ollisionalblokade an prevent us fromonning even twoatoms in an

extremely tiny FORT[149℄.

High atom densities an be an issue for many kinds of experiments with FORT's.

One example is the attempt to ahieve Bose-Einsteinondensation with alkali gases in

optial rather than magneti traps [150, 151℄. The stability of an atom loud against

dipole-dipole oupling is an issue also in the ultraold temperature regime where the

atoms forma degenerate Bose-ondensate. The orresponding anisotropi potentialan

beinludedintheGross-Pitaevkiequation[152℄.Thereisastablesolutionfortheatomi

mean-eld wavefuntion depending on the trap aspet ratio [153℄, or on the sattering

length [154, 155℄. The dipole-dipoleinteration gives rise to density modulations[156℄,

solitons [157℄. In these works the atoms possess a permanent dipolemoment. Reently,

the eets of the magneti dipole-dipole oupling has been observed in a hromium

ondensate[158℄.Thelaser indueddipole-dipoleouplingatlowtemperatureshas been

treatedin [159℄ and was shown tolead to nonlinear atom optial eets. An interesting

question arises that whether the dipole-dipole interation exludes the possibility of

forming aBose-Einstein ondensatein anoptial dipoletrap.

6.1 Dipole-dipole interation

The theory of radiative atom-atom interation in the presene of a driving laser eld

is desribed in several papers, e.g. Refs. [144, 159, 160, 161℄, we will mostly use the

approahpresented in [161℄.

Weonsider anumber

N

^{of atoms interating with a single Gaussian}standing-wave laser eld mode along the

z ˆ

^{diretionwhihhas amode funtion}

f (r) = ǫ cos(k _L z) exp

− 2π ² (x ² + y ² )/w ²

,

^(6.1)

where

ǫ

^{is the eld} polarization, and the mode is paraxial,

k L ≫ 2π/w

^{. The atomi}

transition frequeny is

ω A

^{, that of the laser eld mode is}

ω L

^{, and the detuning is}

de-ned by

∆ _A = ω _L − ω _A

^{. The atom-mode interation strength is desribed by the Rabi}

frequeny

Ω

^{inthe positionofmaximum oupling.Weassumean}

S ↔ P

^{transitionwith}

a degenerate manifold of exited states and keep the three-dimensional polarizability

of the atoms. Atually the xed eld polarization selets two levels taking part in the

dynamis. The atomi internal degree of freedom is desribed by the vetorial lowering

operator

σ = P

q ǫ _q σ ^q

^with

q = ± 1

^and

q = 0

orresponding to the irular and linear polarizations, respetively. The quantizationaxis willbedened inaordane with the

hoie of the eld polarization

ǫ

^.

The equation of motionfor the density operator inthe Markov approximationreads

˙ ρ = 1

i ~ [H, ρ] + L ρ .

^(6.2)

In aframe rotatingatthe laser frequeny

ω L

^,the Hamiltonianis

H =

N

X

n=1

"

p ²

n

2m − ~ ∆ A σ ^† _n σ _n − i ~ Ω f (r _n ) σ ^† _n − σ _n

#

− ~ γ

N

X

n,m=1 n 6 =m

σ ^† _m β(R _mn )σ _n ,

^(6.3)

where

r _n

^,

p

n

^and

σ _n

^{are the position, the momentum and the} polarization of the

n

^th

atom (

n = 1, . . . , N

^{).Next to the single atomterms, i.e.kineti energy, internal energy,}

and atom-eld oupling, the last term ontains the indued dipole-dipole interation

energyof theatoms. Notethatthe naturallinewidth

γ

haraterizesthe strength ofthis interation. The tensor

β

^{depends on the oordinate dierene}

R _mn ≡ r _m − r _n

^{of the}

interating pairs of atoms.

Thedipole-dipoleinteration ismediatedby thebroadbandvauum,and istherefore

aompanied by inoherent evolution. This is represented by additional terms in the

Liouvilleoperatorresponsible for the dissipation:

L ρ = − γ

N

X

n=1

{ σ ^† _n σ _n , ρ } − 2 X

q

Z

d ² u N q (u)σ ^q _n e ^{− ik A ur n} ρe ^{ik A ur n} σ _n ^{q †}

− γ

N

X

n,m=1 n 6 =m

{ σ ^† _m α(R _mn )σ _n , ρ } − 2 Z

d ² u σ _n N(u)e ^{− ik A ur n} ρe ^{ik A ur m} σ ^† _m

,

^(6.4)

where

{ , }

^{denotes the}antiommutator.Thesingleatomtermsinludethespontaneous deay aompanied by momentum reoil. The tensor

N(u) = ^3γ _8π (1 − u ◦ u)

^{, and its}

diagonal elements

N _q (u) = ǫ _q ^† N(u)ǫ _q

^{are the angular momentum} distribution of the spontaneousemissionfromthe

q

^{-state inthe exited manifold.Here}

1

^{is thetwo-by-two}

unitmatrix, and

◦

^{denotes dyadi produt.The doublesumdesribesthelosseet due}

tothe dipole-dipoleoupling.

In free spae the tensors

α

^and

β

^{assume the followingform :}

α(R _mn ) = 3 2

(

(1 − R ˆ mn ◦ R ˆ mn ) sin kR mn

kR _mn

+ (1 − 3 ˆ R mn ◦ R ˆ mn ) cos kR mn

(kR mn ) ² − sin kR mn

(kR mn ) ³

!)

,

^(6.5)

β(R _mn ) = 3 2

(

(1 − R ˆ mn ◦ R ˆ mn ) cos kR mn

kR mn

− (1 − 3 ˆ R mn ◦ R ˆ mn ) sin kR mn

(kR _mn ) ² + cos kR mn

(kR _mn ) ³

!)

,

^(6.6)

where

k = | k | ≈ k L

^,

R mn = | R _mn |

^,and

R ˆ mn

^{is a unit vetor along the diretion of}

R _mn

^.

The xed eld polarizationselets the exited state and the atom redues toa

two-level system with

σ _n = σ n ǫ

⁽

n = 1, . . . , N

^{). The tensors}

α

^and

β

^{have to be projeted}

ontothis partiular polarization,

σ ^† _m β(R _mn )σ _n = β(R _mn )σ _m ^† σ n ,

^(6.7)

where

β(R _mn ) = ǫ ^† β(R _mn )ǫ

^{. We now evaluate this projetion in two ases: for linear}

polarizationalong

x ˆ

^{,and for irular one inthe}

x ˆ

y ˆ

^plane.

6.1.1 Linear polarization

When the polarization of the beam is linear,

ǫ = ˆ x

^{, the atomi} quantization axis is taken inthis diretion. Theatomi polarizationisdesribed by theoperator

σ 0

^{,and the}

projetion given inEq. (6.7)results in

β mn = 3

2 (3 cos ² φ mn − 1)

"

sin kR mn

(kR mn ) ² + cos kR mn

(kR mn ) ³

# + 3

2 (1 − cos ² φ mn ) cos kR mn

kR mn

,

^(6.8)

where

φ mn = ∡(R _mn , x) ˆ

^{is the angle between the distane vetor of the two atoms and}

theaxisof thepolarization,

x ˆ

^{.Asthis funtionissingularatthe origin,}

kR mn → 0

^,later

we willuse rather itsFourier transform [162℄,

β(k) = lim ˜

η → 0

√ 3 2πk _L ³

(k _z ² + k ² _y ) (k ² − k ² _L + η ² )

(k ² − k ² _L ) ² + 2η ² (k ² + k _L ² ) .

^(6.9)

6.1.2 Cirular polarization

Ifthe polarizationisirular,

ǫ = − ^{√ 1} ₂ (ˆ x + i y) ˆ

^{, the atomi}quantizationaxis is the eld propagation diretion

z ˆ

^{. The atomi} polarizationis desribed by the operator

σ ⁺¹

^{, and}

the projetion Eq. (6.7) gives

β mn = 3

4 (1 − 3 cos ² θ mn )

"

sin kR mn

(kR mn ) ² + cos kR mn

(kR mn ) ³

# + 3

4 (cos ² θ mn + 1) cos kR mn

kR mn

.

^(6.10)

The angle

θ mn = ∡(R _mn , z) ˆ

^{is the angle between the distane vetor of the two atoms}

and the axis

z ˆ

^{. The Fouriertransform is}

β(k) = lim ˜

η → 0

3 2 √

2πk ³ _L

(k ² _z + k ² ) (k ² − k _L ² + η ² )

(k ² − k _L ² ) ² + 2η ² (k ² + k ² _L ) .

^(6.11)

In document (1)ultraold atoms in an optial resonator Ph.D (Pldal 86-92)