Model

We onsider agas of

N

polarizablepartilesoupledtotwo degenerateoptialmodes of a ring resonator, desribed by the plane wave mode funtions

f ₁ (x) = e ^ikx

^and

f ₂ (x) = e ^{− ikx}

^{, with oherent amplitudes}

α 1

^and

α 2

^{. The sheme of the system is depited in}

Fig. 3.1. The partiles are driven by a pump laser oriented perpendiular to the avity

axis.Forsimpliity,weonsiderthe system inone dimension:the atomsaresupposed to

beonned nearthe resonatoraxisby e.g.astrongdipoletrap.Theinterationisinthe

dispersive regime,i.e. the pump laser is very far detuned with respet to the resonane

frequenies of the gas partiles: no real exitations need take plae. Thus, the model

Figure3.1: Sheme of a ring resonator with two ounterpropagating modes.

derived in Se. 2.1.4. For two avity modes, the eletri eld Eq. (2.13) is ompleted

to

E(x) = f 1 (x)a 1 + f 2 (x)a 2 + E pump

^{, where the last pumping eld term is assumed to}

be onstant along the resonator axis `

x

^{'. The atoms} redistribute photons by oherent sattering between the two modes and the pump eld. This proess feeds the avity

modes withaneetiveamplitude

η _t

^{.One obtainstheHamiltonoperatorofthe system,}

byinsertingtheneweletrield

E(x)

^{intotheeetive}Hamiltonian(2.17)andsumming up for atoms at positions

x j

^{with momentum}

p j

^{, where}

j = 1...N

^{. Assuming oherent}

eld in the avity modes with amplitudes

α 1

^and

α 2

^{, the} orresponding equations of motionread

d dt

α ₁ α ₂

!

= A α ₁ α ₂

!

− iη t N

X

j=1

f ₁ ^∗ (x _j ) f ₂ ^∗ (x _j )

!

,

^(3.8)

using aformalvetor notation, with the oupling matrix

A = i(∆ C − NU 0 ) − κ − iNU 0 σ

− iNU 0 σ ^∗ i(∆ C − NU 0 ) − κ

!

.

^(3.9)

The diagonalterms inludethe detuning between the pumplaser and the avity modes,

∆ C = ω − ω C

^{, the avity deay rate}

κ

^{, and the shift of the avity resonane with an}

amount of

U 0

^{per atom. This shift is due to forward sattering and is related to the}

atomiproperties by

U 0 = − ω C χ ^′ / V

^,where

χ ^′

^{isthe realpartof thelinear}polarizability and

V

^{isthemodevolume.Thetotalfrequeny shiftduetothe gas,expressedintermsof}

adimensionless olletiveouplingparameter

ζ = NU ₀ /κ

^{, willlater be used todesribe}

ouplingbetweenthe avity modesstemsfromthestimulatedbakreetiono thegas.

This proess is heavily dependent on the positions of the atoms, through the omplex

parameter

σ = _N ¹ P

j e ^{− 2ikx j}

^{desribingspatialorder.Similartothe}Debye-Wallerfator,

| σ |

^is

1

^{ifthegasformsaperfetlattiewithperiodofanintegermultipleof}

λ/2

^,andless

than 1for anon-perfet lattie;for anequidistributed gas, itis

| σ | ∝ 1/ √

N

^{.The phase}

of

σ

^gives

x 0

^{,thepositionof thelattie modulo}

λ/2

^{,withthe denition}

σ = | σ | e ^{− 2ikx 0}

^.

Notethatinontrasttotheorderparameterdened inEq.(3.3)forasinglemodeavity,

here we shall use

| σ |

^for haraterizing the spatial order in the system, sine

σ

^{is the}

primary quantity that appears inthe equations.

Interation between the gas and the avity eld lifts the degeneray of the avity

modes. It is instrutiveto alulatethe new eigenmodes of the avity whih are

h ₋ (x) = sin k(x − x 0 ) , h + (x) = cos k(x − x 0 ) ,

^(3.10)

havenodes, and antinodes at

x 0 + nλ/2

^, respetively. The orresponding amplitudes,

β ₋ = ie ^{ikx 0} α 1 − ie ^{− ikx 0} α 2 , β + = e ^{ikx 0} α 1 + e ^{− ikx 0} α 2 ,

^(3.11)

arenot oupled bythe interation withthe atomgas.The frequeniesofthe eigenmodes

h ₋

^and

h +

^{are shifted with respet tothe empty avity, givingthe eetivedetunings}

∆ ₋ = ∆ C − (1 − | σ | )NU 0 , ∆ + = ∆ C − (1 + | σ | )NU 0 .

^(3.12)

Withtheatomsatxed positions

x j

^{,the mode amplitudestakeonthestationaryvalues}

β ₋ = 2η t

∆ ₋ + iκ X

j

sin k(x j − x 0 ) , β + = 2η t

∆ ₊ + iκ X

j

cos k(x j − x 0 ) .

^(3.13)

Iftheatomsareuniformlydistributed,bothamplitudesvanishbydestrutiveinterferene

oftheomponentssatteredbydierentatoms(

kx 0

^{isanarbitraryphaseinthisase).If}

thegasroughlyformsalattiewithperiod

λ

⁽

x 0

^{givingthepositionofthelattiemodulo}

λ/2

^{), mode}

h ₋

^{will be weakly oupled, sine it has nodes at the lattie points. On the}

ontrary, mode

h ₊

^{has antinodes at the atomi positions and is driven by stimulated}

Bragg sattering o the gas [84, 77℄. Therefore, from the viewpoint of self-organization

the ring resonator an be simplied to an eetive one-mode avity that has a mode

funtion

cos k(x − x 0 )

^.Nevertheless,thefreepositionparameter

x 0

^{reetstheontinuous}

translation invariane of the ring avity, whihis not broken by the mirrors,but by the

atomidistribution.

The possibility of optial trapping of high eld seekers with negligiblespontaneous

emissionlossrequireslargereddetuning,wherebythe resonaneshift

U 0

^{isnegative.Itis}

knownfromavityoolingtheorythatthepumpeldhas tobedetunedbelowtheavity

frequenytoensuredampedmotionaldynamis[63℄.Forlargeenoughdensity

ζ

^,however,

if the gas forms a more and more periodi struture,

| σ |

^{inreases from 0 to 1, and the}

frequeny of mode

h +

^{may exeed the pump frequeny, i.e.}

∆ +

^{may beome positive.}

This yieldsheatingand instabilityof the gas. The eet is allthe more prominentsine

at

∆ + ≈ 0

^{the population of mode}

h +

^{is resonantly enhaned. We will disuss ways to}

avoidthisproblemand the eetof the detuning

∆ C

^{and thedensity}

ζ

^{onthe dynamis}

below.

Up to this point, the gas was onsidered as an inert bakground against whih the

dynamisof the avity eld takesplae. Nowwe relaxthis assumption,and inlude the

dynamisofthegaspartiles,whihisslowonthetimesaleofequilibrationoftheavity

eld,

κ ^{− 1}

^{.This separationof the timesalesallows ustotreatthe gas dynamis}

adiabat-ially, in a way reminisent of the Born-Oppenheimer approximation for moleules: the

motionofthe gas partilesisgovernedby the fores fromthelighteld thatdepends on

their positions.

In fat, we generalize the mean-eld model presented in Setion 2.2.1. For large

numberofatomsbut eahofthemweaklyoupledtotheavitymodes, onean takethe

thermodynami limit (where

N, V → ∞

^{, keeping the density}

N/ V = const

^{), and treat}

theatomiloudby ameaneldapproahusingaontinuouspositiondistribution

p(x)

^.

Furthermore, by assuming that the gas has a nite equilibrium temperature

T

^{, we an}

apply the anonial distribution, giving

p(x) = 1 Z exp

− V (x) k _B T

,

^(3.14)

with the partitionfuntion

Z

^ensuring normalization

R p(x)dx = 1

^{. Mirosopi}

alu-lations reveal that the temperatureis onthe order of

k B T = ~ κ

^{, when avity ooling is}

optimal.Themean-eldpotential

V (x)

^{isgivenbythelasttermoftheeetiveHamilton}

operator (2.17), when the steady-state oherent amplitudes

α 1

^and

α 2

^are substituted forthe operators

a 1

^and

a 2

^,respetively. Inturn,using theeigenmodes

h ₋ (x)

^and

h + (x)

with the orresponding oherent elds

β ₋

^and

β ₊

^{we have}

V (x) = 2 ~ η t

Reβ ₋ sin k(x − x 0 ) + Reβ + cos k(x − x 0 ) + + 1

2 ~ U 0

( | β + | ² − | β ₋ | ² ) cos 2k(x − x 0 ) + 2Re[β ₊ ^∗ β ₋ ] sin 2k(x − x 0 )

.

^(3.15)

The rst lineorresponds to sattering from the pump intothe avity: if the gas forms

a

λ

^{-periodigrating,this desribesapotentialwith attrativepointsatthe lattiesites,}

i.e.the gasdensity maxima.Theseondlineoriginatesfrommultiplesatteringbetween

the runningwavemodes.This latter has astrongdependene onthe densityand auses

deviations from a medium with linear refrative index: it attrats partiles to maxima

of the avity eld, and an reate defets inthe lattie, as explainedlater on.

Nowsine

V (x)

^{dependsonthemode amplitudes}

β ₋

^and

β +

^{,andtheselatterdepend}

onthe distribution

p(x)

^{ofthe satterersthrough Eqs.(3.13),Eq.(3.14)has tobesolved}

self-onsistently.

In document (1)ultraold atoms in an optial resonator Ph.D (Pldal 35-39)