Flutuations around the mean eld - (1)ultraold atoms in an optial resonator Ph.D

in the dispersive interation limit. As it was alulated for a probe eld propagating

freelythrough the ondensate [117℄,the depletion sales asthe absorptionwhihwillbe

suppressed by hoosing verylarge detuning. Therefore the avity is essentialin reating

aspeiouplingofthe many-atomsystem totheradiation.Beauseof thefast

round-trips of photons, the avity eld experienes the olletive behavior of atoms. This kind

of oupling gives rise to the noise ampliation mehanism analogous to the so alled

exess noise in laser physis [118,119, 120, 121℄.

We omit the atom-atom s-wave ollision, in order to distil the eets due to the

eld-mediatedouplingbetween atoms.Notethatthetwotypesofinterationinvolvedistint

lasses of transitions between the Bloh states of the

λ/2

^periodi^potential

U (x)

^. ^The

atom-photon interation term is

λ/2

^periodi ^and ^it ^does ^not ^ouple ^Bloh ^states ^with

dierent quasimomenta

q

^. ^W^e ^an ^thus^onsider wavefuntionsbeingalsoperiodiwith

λ/2

^, ^the

q = 0

^Bloh ^state representing the entire band. By ontrast, ollisions indue transitionsmostly within aband. The twotypes ofinterationsale dierentlywith the

density and an be treated separately. We onsider low densities, where ollisions are

suppressed and the depletion is due to avity-indued `band-to-band' transitions. The

small amount of additionaldepletion into higher

q

^states ^due ^to ^s-wave ^sattering ^an

be independently estimated by the usual formula of the Bogoliubov-theory [106℄. The

validity of our approah is restrited to parameters where the gas loaded in the avity

lattie eld isin the superuidphase, far fromthe Motttransition[115, 116℄.

Following the desriptionpresented inSe. 2.2.2, the eldoperatorsare deomposed

intomean and utuating parts aording to

ˆ

a(t) = α(t) + δˆ a(t),

^(4.2a)

Ψ(x, t) = ˆ √

Nψ(x, t) + δ Ψ(x, t), ˆ

^(4.2b)

with

α(t)

^being^the^oherent^omponent^of^the^radiation^eld,^and^similarly

ψ(x, t)

^stands

forthe ondensatewavefuntion normalizedtounity, while

N

^is ^the^number^of^atoms ⁱⁿ

the ondensate. Substitutingthese into Eqs. (4.1) and negleting those terms whih are

notpure -numbers leadstoaoupledpairofnonlinear equationssimilarinspirittothe

Gross-Pitaevskiiequations:

i ∂

∂t α(t) =

− ∆ C + N h U i − iκ

α(t) + iη,

^(4.3a)

i ℏ ∂

∂t ψ(x, t) =

− ℏ ² ∆

2m + ℏ | α(t) | ² U (x)

ψ(x, t).

^(4.3b)

The ondensate wavefuntion

ψ(x)

^ouples ^into ^the ^evolution ^of ^the ^eld ^amplitude

through theaverage

h U i = R

ψ ^∗ (x, t)U (x)ψ(x, t)dx

^.^The ^solution^of^Eqs. ^(4.3)^yields^the

possible steady-states of the BEC-avity system.

The Heisenberg-Langevin equations (4.1) an be linearized in the utuations

δˆ a(t)

and

δ Ψ(x, t) ˆ

^around^the^steady ^state ^of^Eqs.^(4.3). ^The^proedure^is^similar ^to^that^used

in Ref. [108℄ and, for a dierent geometry in Ref. [92℄, however, here we will onsider

utuationswith ouplingstotheir hermitianadjoints.Fora ompat notationthe

u-tuations are arranged in a vetor

R ˆ ≡ [δˆ a, δˆ a ^† , δ Ψ(x), δ ˆ Ψ ˆ ^† (x)]

^for ^whih ^the ^linearized

equation of motion,

i ∂

∂t R ˆ = M R ˆ + i ξ , ˆ

^(4.4)

where

ξ ˆ = [ˆ ξ, ξ ˆ ^† , 0, 0]

^, ^and ^the ^matrix

M

^is

M =







A 0 α X ˆ ^∗ α X ˆ 0 − A ^∗ − α ^∗ X ˆ ^∗ − α ^∗ X ˆ α ^∗ Y αY H ˆ 0 0

− α ^∗ Y ^∗ − αY ^∗ 0 − H ˆ 0







,

^(4.5)

with

A ≡ − ∆ _C + N h U i − iκ

Y ≡ √

N ψ(x)U (x)

Xf(x) ˆ ≡ R

ψ(x)U (x)f(x)dx

^and

H ˆ 0 ≡ − ℏ ∆/(2m) + | α | ² U (x)

^. ^The ^key ^point ^with ^respet ^to ^the ^depletion ^of ^a ^BEC

is that the matter eld utuation

δ Ψ(x) ˆ

^ouples ^to ^both

δˆ a

^and

δˆ a ^†

^, ^this ^latter ^being

driven by the `noise reation' operator

ξ ^†

The matrix

M

^has ^an ^important ^symmetry ^property, ^namely

ΓMΓ = − M ^∗

origi-natingfromthefatthat

R ˆ ^† = Γ R ˆ

^,^with

Γ

^being^the permutationmatrixthatexhanges the rst row with the seond and the third row with the fourth one. This property

de-termines the struture of the spetrum. The eigenvalues ome in pairs, namely if

ω k

^is

aneigenvalue, then

− ω _k ^∗

^will^also^be ^aneigenvalue. The other onsequene of the above symmetryisthatthematrix

M

^isnon-normal,i.e.,itdoesn'tommutewithitshermitian adjoint. Therefore, its eigenvetors are not orthogonal to eah other, hene the analogy

to exess noise in lasers and to the Petermann-fator arises [122, 123, 124, 125℄. The

stabilityanbeproperlyharaterizedintermsof pseudo-spetra by usingthe theoryof

non-normal operators [126, 127℄. However, we will restrit this work to the alulation

of the depletion of the BEC.

Theeigenvalues

ω k

^and^theorrespondingleftand righteigenvetors,

l ^(k)

^and

r ^(k)

re-spetively,of

M

^are^alulated ^numerially^.^First^the^oupledGross-Pitaevskiiequations (4.3) are solved on a

200

^point ^grid ^with ^imaginary ^time propagation. From the wave-funtions,thematrix

M

^an^be^formed^and^thendiagonalizednumerially.Thespetrum of utuationsisexhibited inFig.4.1, inunits of the reoilfrequeny

ω R = ~ k ² /m

^.^The

setting

κ = 100 ω R

^and ^the ^eetive ^number ^of ^atoms ⁱⁿ ^the ^mode

N = 1000

^are ⁱⁿ

aordane with Ref. [86℄, the other parameters,

∆ C

η

^, ^and

U 0

^are ^rather ^tunable ⁱⁿ

pratie. At

U 0 = 0

^, ^the ^real ^part ^(left ^panel) ^renders ^the ^spetrum ^of ^a homogeneous ideal gas. In a large region

0 < | U ₀ | . 0.8 ω _R

^the ^solid ^lines orresponding to motional

| U 0 | [in units of ω R ] R e ω [i n u n it s of ω R ]

1.2 1 0.8 0.6 0.4 0.2 0 120 100 80 60 40 20 0

| U 0 | [in units of ω R ]

− Im ω [i n u n it s of ω R ]

1.2 1 0.8 0.6 0.4 0.2 0 10 ² 10 ⁰ 10 ⁻ ² 10 ⁻ ⁴ 10 ⁻ ⁶ 10 ⁻ ⁸

Figure4.1:The real andimaginarypartsof theeigenvalues ofthe linearstabilitymatrix

M

^vs. ^the ^one ^atom ^light ^shift

U 0

^. ^Only ^those eigenvalues are plotted whih have a

nonnegative real part. Every eigenvalue is degenerate at

U 0 = 0

^, ^and ^the ^lift ^of ^the

degeneray an be resolved only lose to

U 0 ≈ 1

^for ^the ^low-lying ^levels, ^where ^one ^of

theeigenvalue inthequasi-degenerate pairisplottedbythin, dashedline.Theseare not

oupled tothe radiation mode and have vanishing imaginary part. The highest plotted

eigenvalue at

U 0 = 0

^(thik ^dashed ^line) ^is ^divided ^by ^a ^fator ^of

10

^and ^orresponds ^to

a dominantlyradiation eld exitation;notie that its imaginary part is at

− κ

^. ^As

| U 0 |

grows this mode rosses the other exitation energies and reahes zero at

NU 0 ≈ ∆ C

The parameters are

∆ C = − 1000 ω R

κ = 100 ω R

η = 1000ω R

N = 1000

to the homogeneous one. There is a ritial point where the ondensate pulls the

fre-queny of the avity mode intoresonane withthe pump laser frequeny (

N h U i = ∆ C

^),

marked by that the real part of the eigenvalue of the dominantlyavity-type exitation

(thik,dashedline) rosseszero.Atthis point,thesteady-state wavefuntionisloalized

inthe dipolepotential(itsdepth is about 100

ω R

^for ^the ^parameters ^of ^the ^gure).^Due

to the nite spread of the wavefuntion around the antinode, the resonane ours for

U 0

^slightly^shifted ^from

∆ C /N

Coupling ofthe motionalmodestotheradiationmode issigniedbythe appearane

ofnon-vanishingimaginaryparts(Fig.4.1,rightpanel).Abovethe ritialpointpositive

imaginary parts appear indiating a dynamial instability of the steady-state solution.

Thisisinaordanewiththesemilassialtheoryofthemehanialfores onatomsina

avity [63℄,whihstatesthat theavity eldheatsup the motionofatomsinthe regime

∆ _C − N h U i > 0

^. ^W^e ^will^disregard ^this ^regime^and ^onsider ^only^utuations ^around

parts reets the semilassialresult that the avity ooling mehanismismost eient

athalf alinewidth belowresonane [67℄.

In document (1)ultraold atoms in an optial resonator Ph.D (Pldal 60-64)

Flutuations around the mean eld

λ/2

U (x)

λ/2

q

λ/2

q = 0

q

ˆ

a(t) = α(t) + δˆ a(t),

Ψ(x, t) = ˆ √

Nψ(x, t) + δ Ψ(x, t), ˆ

α(t)

ψ(x, t)

N

i ∂

∂t α(t) =

− ∆ C + N h U i − iκ

α(t) + iη,

i ℏ ∂

∂t ψ(x, t) =

− ℏ 2 ∆

2m + ℏ | α(t) | 2 U (x)

ψ(x, t).

ψ(x)

h U i = R

ψ ∗ (x, t)U (x)ψ(x, t)dx

δˆ a(t)

δ Ψ(x, t) ˆ

R ˆ ≡ [δˆ a, δˆ a † , δ Ψ(x), δ ˆ Ψ ˆ † (x)]

i ∂

∂t R ˆ = M R ˆ + i ξ , ˆ

ξ ˆ = [ˆ ξ, ξ ˆ † , 0, 0]

M

M =













A 0 α X ˆ ∗ α X ˆ 0 − A ∗ − α ∗ X ˆ ∗ − α ∗ X ˆ α ∗ Y αY H ˆ 0 0

− α ∗ Y ∗ − αY ∗ 0 − H ˆ 0













,

A ≡ − ∆ C + N h U i − iκ

Y ≡ √

N ψ(x)U (x)

Xf(x) ˆ ≡ R

ψ(x)U (x)f(x)dx

H ˆ 0 ≡ − ℏ ∆/(2m) + | α | 2 U (x)

δ Ψ(x) ˆ

δˆ a

δˆ a †

ξ †

M

ΓMΓ = − M ∗

R ˆ † = Γ R ˆ

Γ

ω k

− ω k ∗

M

ω k

l (k)

r (k)

M

200

M

ω R = ~ k 2 /m

κ = 100 ω R

N = 1000

∆ C

η

U 0

U 0 = 0

− ℏ ² ∆

2m + ℏ | α(t) | ² U (x)

ψ ^∗ (x, t)U (x)ψ(x, t)dx

R ˆ ≡ [δˆ a, δˆ a ^† , δ Ψ(x), δ ˆ Ψ ˆ ^† (x)]

ξ ˆ = [ˆ ξ, ξ ˆ ^† , 0, 0]

A 0 α X ˆ ^∗ α X ˆ 0 − A ^∗ − α ^∗ X ˆ ^∗ − α ^∗ X ˆ α ^∗ Y αY H ˆ 0 0

− α ^∗ Y ^∗ − αY ^∗ 0 − H ˆ 0

A ≡ − ∆ _C + N h U i − iκ

H ˆ 0 ≡ − ℏ ∆/(2m) + | α | ² U (x)

δˆ a ^†

ξ ^†

ΓMΓ = − M ^∗

R ˆ ^† = Γ R ˆ

− ω _k ^∗

l ^(k)

r ^(k)

ω R = ~ k ² /m

0 < | U ₀ | . 0.8 ω _R

1.2 1 0.8 0.6 0.4 0.2 0 10 ² 10 ⁰ 10 ⁻ ² 10 ⁻ ⁴ 10 ⁻ ⁶ 10 ⁻ ⁸

∆ _C − N h U i > 0