Elimination of the photon eld

amplitude

c 2

^assoiated ^with ^the

cos 2kx

^exitation ^mode ^of ^the ^BEC. ^By ^introduing

the quadratures (real and imaginary parts) of

c ₂

X = 1

√ 2 (c ^† ₂ + c 2 ) ; Y = i

√ 2 (c ^† ₂ − c 2 ),

^(5.6)

the Hamiltoniantakes the form

H = − δ C a ^† a + iη (a ^† − a) + 2ω R (X ² + Y ² ) + u a ^† a X .

^(5.7)

Here

δ C = ∆ C − NU 0 /2

^is ^the ^shifted ^avity ^detuning, ^and

u = √

NU 0 /2

^, ^i.e., ^the

single-atom oupling strength

U ₀

^is ^magnied ^by ^the ^square ^root ^of ^the ^atom ^number,

whih an mean several orders of magnitude. Even if the dispersive oupling is very

weak due to large detuning,the bosoni enhanement an lead to asigniant oupling

betweenthetwomodes

a

^and

c 2

^.^TheHamiltonian(5.7)orrespondstothewidelystudied radiationpressureouplinginopto-mehanialsystems[128℄.There,intheordinary

opto-mehanial sheme, the avity photon number is oupled to the position operator of a

mirror. Here, in the BEC-avity system, the

X

^quadrature ^operator ^of ^the ^atom ^eld

exitation mode an be interpreted only as a titious `position'. Although the formal

analogy is omplete between the two systems, the typial parameters lead to dierent

regimes.Inpartiular,hereintheaseoftheBECthe`mehanial'vibrationalfrequeny

ω R

^is ^typially^muh ^less ^than ^other ^frequenies ^of ^the ^system, ^whih ^is ^not ^the ^ase ⁱⁿ

usualopto-mehanialsetups.Therefore,itislegitimatetomakeafurtherapproximation

for the BEC-avity system, whih onsists in using an expansion in terms of the small

parameter

ω R /κ

The avity noise is taken into aount by the noise operator

ξ

^. ^It ^has ^a ^vanishing

expetation value:

h ξ i = 0

^, ^meanwhile ^its ^seond ^order ^orrelation ^is ^assumed ^to ^be ^a

Dira-

δ

^funtion,^deribing ^white ^noise:

h ξ(t) ξ ^† (t ^′ ) i = 2κ δ(t − t ^′ ) ,

^(5.9)

allthe other orrelations vanish,

h ξ ^† ξ ^† i = h ξ ^† ξ i = h ξ ξ i = 0 .

5.2.1 Adiabati elimination of the eld

On integrating Eq.(5.8a), weget the timedependent solutionof the eld operator

a(t)

The time sale of the atomimotion whih weare interested in is muh longerthan

1/κ

^,^thus ^the homogeneoussolution with the initialvalue

a(0)

^vanishes. ^The ^partiular

solutionis

a(t) = η κ − i∆ +

Z t 0

e ^(i∆ ⁻ ^κ)(t ⁻ ^t ^′ ⁾ ξ(t ^′ )dt ^′ ,

^(5.10)

where

∆ ≡ ∆(X) = δ C − uX

^. ^The ^rst ^term ^is ^the steady-state oherent amplitude of the eld,

α(t) = η/(κ − i∆(X(t))) ,

^(5.11)

whihisnowanoperator-valuedfuntionoftheposition

X

^.^The^seond^termⁱⁿ^Eq.^(5.10)

representsan`aggregate' noise,denoted by

Σ(t)

^.^Its ^expetation ^value ^is^obviously^zero:

h Σ(t) i = 0

^.^The ^seond ^order^orrelation ^funtion^an ^be^readily ^derived ^from ^Eq.^(5.9),

h Σ(t 1 )Σ ^† (t 2 ) i = e ^iδ(t ¹ ⁻ ^t ² ⁾ e ⁻ ^κ ^| ^t ¹ ⁻ ^t ² ^| − e ⁻ ^κ(t ¹ ^+t ² ⁾

.

^(5.12)

The`aggregate'noise,havinganexponentiallydeayingorrelationfuntion,isaoloured

noisewithaspetralwidth

κ

^.^However,^on^a^time^sale^muh^longer^than

1/κ

^,^the^seond

term inthe braket vanishes,and the remainingpart an be approximated by aDira-

δ

funtion. Fulllingproper normalization,the approximate orrelationfuntion beomes

h Σ(t ₁ )Σ ^† (t ₂ ) i ∼ = 2 κ

∆ ² + κ ² δ(t ₁ − t ₂ ) .

^(5.13)

The dipole fore to be substituted into equation (5.8) is the sum of an adiabati

potentialand a noise term:

ua ^† a = u | α(t) | ² + Ξ(t) ,

^(5.14)

with the intensity noise dened as

Ξ(t) = u α(t)Σ ^† (t) + α ^† (t)Σ(t) + Σ ^† (t)Σ(t)

.

^(5.15)

The noise orrelation funtionis

h Ξ(t 1 )Ξ(t 2 ) i = u ² h α ^† (t 1 )Σ(t 1 )α(t 2 )Σ ^† (t 2 ) i = D(X) δ κ (t 1 − t 2 ) ,

^(5.16)

wherethe diusion operatoris

D(X) = u ² | α | ² 2 κ

∆ ² + κ ² = u ² 2κη ²

(∆ ² (X) + κ ² ) ² .

^(5.17)

Thisdiusionproessrepresentsthequantumnoisetransmittedfromthephotonnumber

utuationsintothe momentumviathe optomehanialoupling. Wenotethat herethe

diusion isan operator-valuedfuntion of the positionoperator

X

5.2.2 First-order orretion to the adiabati elimination

WhengettingtheadiabatieldamplitudeinEq.(5.10),theondensatedynamisis

as-sumedtobefrozen.Thisassumptionanberelaxedand theatomimotionanbetaken

intoaountbymeansofasystematiexpansionintermsofthe titiousmomentum

Y

Thesmallparameterinourmodelis

ω R /κ

^,^beause^the^reovery ^time^of^the^photon^eld

1/κ

^is ^small^ompared ^to ^the harateristi time sale of the atomi motion determined by the reoil frequeny

ω R

^. ^At ^the ^same ^time ^we^will^neglet^the ^eet ^of^motion ^on^the

utuating fore

Σ

^sine ^it ^would ^ontribute ^to ^the ^variation ^of ^momentum ^orrelation

funtions inhigher orders.

Inpriniple,theeldderivativeinEq.(5.8a)anbeintegratedandtheinstantaneous

eld an be obtained from the knowledge of the full trajetory

X(t)

^. ^The ^adiabati

approximation means that the eld depends only on the atual position. Close to the

adiabatiregime,thehistorywellinthe pastisirrelevant,andjusttheloalbehaviouris

important.Thetrajetoryloallyanbedesribedbythepositionanditsrstderivative,

i.e., the momentum.Thus welook forthe eld amplitude inthe form

α(X, t) ∼ = α 0 (X) + 1

2 { Y, α 1 (X) } .

^(5.18)

where

{ , }

^denotes antiommutation.Wemust takeare ofoperatorordering, beause

Y

does not ommute with

α 1 (X)

^.^Throughout ^the ^alulation ^we ^use ^symmetri ^ordering.

The time-derivativeof a funtion of anoperator, suh as

α(X, t)

^, ^is

d

dt α(X, t) = ∂

∂t α + i [H, α(X, t)] .

^(5.19)

In the Hamilton operator of Eq. (5.7), it is only the term

2ω R Y ²

^that ^gives ^nonzero

ommutator with

X

^. ^On ^using ^that

[Y, α(X)] = − i ^∂α(X,t) _∂X

^, ^one ^gets

d

dt α(X, t) = ∂

∂t α + 4ω _R 1 2

Y, ∂α(X, t)

∂X

,

^(5.20)

whihissymmetriallyordered.Usingthisderivativeontheleft-hand-side,andinserting

theansatzEq.(5.18)totheright-hand-sideofEq.(5.8a),wegetahierarhy ofequations

indierentpowersof theoperator

Y

^.^In^zeroth^order,^we^get^the ^adiabati^oherent^eld

amplitude

α 0 (X) = η

− i∆(X) + κ .

^(5.21)

In rst order one gets the orretion:

α 1 (X) = 4ω _R i∆ − κ

∂α ₀ (X)

∂X = i 4ω _R uη

(κ − i∆(X)) ³ .

^(5.22)

We express the photon number up to rst order in

Y

^with ^the ^ansatz ^Eq. ^(5.18) ^as

a ^† a = 1

2 (αα ^∗ + α ^∗ α) = | α 0 | ² + 1

2 { Y, α ^∗ ₀ α 1 + α 0 α ^∗ ₁ } .

^(5.23)

This expansion gives rise to a frition fore in the equation of motion of

Y

^Eq. ^(5.8),

that is inthe form

F f = − 1

2 { Y, Γ(X) } ,

^(5.24)

wherethe frition oeient isan operator-valuedfuntion of the positionoperator

X

Γ(X) = − 16ω R u ² ∆(X)κη ²

(∆ ² (X) + κ ² ) ³ .

^(5.25)

Note that damping of the atomi motion is possible via the photon loss hannel, if the

detuning is anegative-valued funtion,i.e.,

h ∆(X) i < 0

^, ^for^all ^states.

5.2.3 Eetive master equation

Puttingtheabovealulateddiusion(5.14)andfrition(5.24)intoaneetive

Heisenberg-Langevin equations,we get the replaementof the Eqs. (5.8b,), namely,

X ˙ = 4ω R Y ,

^(5.26a)

Y ˙ = − 4ω R X − u | α 0 (X) | ² − 1

2 { Y, Γ(X) } + Ξ ,

^(5.26b)

Given the eetiveHeisenberg-Langevin equations,inthe nextstep wean onstrut an

equivalentquantum masterequationforthe densitymatrix oftheBECexitationmode,

˙

ρ = − i [H eff , ρ] + L

^di

ρ + L

^fri

ρ .

^(5.27)

IntegratingthedipoleforeinEq.(5.26b)wegettheadiabatidipolepotentialwhih,

added tothe harmoni osillator energy, formsthe eetiveHamilton operator

H _eff = 4ω _R 1

2 (X ² + Y ² ) + η ²

κ arctan

uX − δ _C κ

.

^(5.28)

It has to be emphasized, that the adiabati dipole potential is not proportional to the

intensity as inthe ase of anexternal laser potential[102℄. The reasonlies in the

bak-ationof the atoms onthe avity photoneld.

The sum of the harmonipotentialand the arus tangentterm an lead totwo loal

energy minimafor ertain settings of the parameters. This is the bistabilityregime and

the eet is losely related to the dispersive optial bistability [83, 132℄. Here, the two

minima(

X c

^and

X h

⁾^are^at^opposite^sides^of^the^avity^resonane.^The^one^on^the^ooling

side,

∆(X _c ) < 0

^, ^is ^generally ^situated ^farther ^from ^the ^resonane ^than ^the ^one ^on ^the

heating side,

∆(X h ) > 0

^, ^i. ^e.,

| ∆(X c ) | > | ∆(X h ) |

^. ^Therefore ^low ^photon ^number ^and

shallowoptialpotentialorrespondtotheminimumontheoolingside,aordingly,the

BECis hardlymodulated. Whereasat theheatingside minimum,the photonnumberis

larger,hene the deeperoptial potentialleads toa moremodulatedondensate ground

state.

We generalize the standard diusion and frition terms that appear in the quantum

Brownian motion [133℄ for position dependent diusion and frition oeients. The

momentumdiusionduetothenoisesoure

Ξ

^an ^be^desribed^by^a^double^ommutator

with

d(X)

L

^di

ρ = − [d(X), [d(X), ρ]] ,

^(5.29)

provided the funtion

d(X)

^obeys

D(X) = 2

∂d(X)

∂X 2

.

^(5.30)

Thisdierentialequationisobtained bymathingthevariationofthemomentumsquare

mean,

h Y ² i

ⁱⁿ ^the Heisenberg-Langevin and in the master equation approahes. After integration,

d(X) = η

√ κ arctan

∆(X) κ

.

^(5.31)

Thediusion anbeinterpretedas ameasurement-induedbak-ation onthe quantum

state of a BEC. The diusion eet of the eld on the BEC has been inluded in a

masterequation for adierentsheme where the optialeld freely propagates through

the dispersive mediumof BEC [117℄. In our sheme the measurementan be assoiated

with the irreversible detetion of photons leaking out fromthe avity.

Finally,the fritionterm in Eq. (5.26b)an bereprodued by a termlike

L

^fri

ρ = − i

2 [g(X), { Y, ρ } ]

^(5.32)

inthe master equation, provided

Γ(X) = ∂

∂X g(X) ,

^(5.33)

whih follows frommathing the variationof

h Y i

ⁱⁿ ^the ^two^pitures. ^The ^result ^is

g(X) = − 4ω R u κη ²

(∆ ² (X) + κ ² ) ² .

^(5.34)

Withthis,wehaveentirelydenedtheeetivemasterequation(5.27)forthebosoni

exitationmode of the BEC. The equationrelies on that the ratio

ω R /κ

^is ^small.

5.2.4 Relation with linearized models

Quantum utuationsinopto-mehanialsystemsare usuallydesribed inmodelsbased

on linearization: the state of the system is assumed to remain in a small viinity of a

stable stationary solution [134, 135, 136℄. Although the master equation (5.27) derived

in the previous subsetion does not rely on any restrition onerning the range of the

variable

X

^, ^it^is ^instrutive ^to ^redue ^our ^more ^general^model^to^the ^speial ^ase ^where

thestateofthe

c 2

^mode^is^well^loalized^around^a^stationary^position

X 0

^andlinearization anbeinvoked. On theother hand,our modelisrestrited tosmallvalues of

ω R /κ

^,^thus

weanmakeadiretonnetiontothe resultsof,e.g.,the referene[134℄,providedthese

latter are expanded up toleading orderin

ω _R /κ

Linearization of the master equation an beperformedaround the energy minimum

oftheeetiveHamiltonianEq.(5.28).Weassumethatweareoutofthebistableregime,

thus, in position representation, the eetive potential has a well-dened minimum at

X 0

^.Considering utuations aroundthis stationary position,

X = X ₀ + δX ; Y = δY ,

^(5.35)

the linearized Hamiltoniantakes the form:

H _eff ^′ = 2ω R (2X 0 δX + δX ² + δY ² ) + uη ² δX

(uX 0 − δ C ) ² + κ ² .

^(5.36)

In the linearized regime,the optialpotentialterm must have the only eet of shifting

theenter oftheharmoni potentialinto

X 0

^.^As ^the^potential^has ^a^minimum^at

X 0

^,^the

terms linear in

δX

^should ^anel. ^Aordingly^, ^we ^get ^a self-onsistent equation for the equilibriumposition:

X 0 = − u 4ω _R

η ²

(uX ₀ − δ _C ) ² + κ ² .

^(5.37)

Outside the bistable regime, this third order equation has only one real solution. If

| uX 0 | ≪ κ, δ C

^, ^then ^anapproximate value of

X 0

^an ^be ^obtained ^by ^negleting ^the

X 0

-dependeneofthe denominatorontherighthandside.Linearizationofthediusionand

frition terms isreadily availablefrom Eq. (5.30)and Eq. (5.33),thus

d(X) = d(X 0 ) +

r D(X 0 )

2 δX,

^(5.38)

g(X) = g (X 0 ) + Γ(X 0 ) δX.

^(5.39)

The onstant terms drop from the ommutators, and we reover the famous

Caldeira-Legett master equation [137℄ with diusion and frition oeients depending on the

stationaryposition

X 0

˙

ρ = − i [H eff , ρ] − D(X 0 )

2 [δX, [δX, ρ]] − i

2 Γ(X 0 ) [δX, { δY, ρ } ] .

^(5.40)

Notethatafterthelinearization,the diusionand thefritionarenolongerdesribed by

operators,but by realnumbers. Hene,we aneasilyexpress the steady-state exitation

number of the osillator:

¯

n(X ₀ ) = D(X 0 )

Γ(X 0 ) = − ∆ ² (X 0 ) + κ ²

8ω R ∆(X 0 ) .

^(5.41)

Let us analyze the weak oupling limit, where

| uX 0 | ≪ κ, δ C

^, ^and ^one ^an ^neglet ^the

X 0

^-dependene ^of ^the ^avity ^detuning, ^thus

∆(X 0 ) ≈ δ C = ∆ C − NU 0 /2

^. ^In ^this ^ase,

the expression Eq. (5.41)simplies to

¯

n = (∆ _C − NU ₀ /2) ² + κ ²

8ω R ( − ∆ C + NU 0 /2) .

^(5.42)

ThisagreeswithEq.(6)ofRef.[134℄,byusingtheorrespondenebetweenthenotations,

ω m → 4ω R

4τ ² → 1/κ ²

∆ L → δ C

^. ^Let ^us ^also ^note, ^that ^for ^a ^BEC, ^the ^exitation

numberof the osillator orresponds tothe populationin the

c 2

^mode, ^that^is ^nothing

else but the ondensate depletion already alulated in Chapter 4. The steady-state

exitation number of the osillator Eq. (5.42), whih we obtained from the linearized

master equation, is equal to the ondensate depletion Eq. (4.12) derived in the weak

We an dedue the riterion of linearization, sine the mean exitation number

de-terminesthe magnitudeof the utuations

δX

^,^as

n ¯ = h δX ² i /2

^.^The ^small^parameter ⁱⁿ

the Taylor-seriesexpansion of the arus tangent funtion was

δX u κ

"

1 +

∆(X 0 ) κ

2 # ₋ 1

≪ 1 .

^(5.43)

Raising the expression to seond power, taking the average and replaing

h δX ² i

^by

Eq. (5.41),one gets arestrition to the strength of the atom-eldinteration:

u ² ≪ 8ω R | ∆(X 0 ) |

"

1 +

∆(X 0 ) κ

2 #

.

^(5.44)

The eetive avity detuning

∆(X 0 )

^needs ^to^beⁱⁿ ^the ^order ^of ^the ^avity ^deay ^rate

κ

for the avity to have onsiderableeets, henethe ondition of linearizationbeomes,

roughly,

u ² ≪ 8κω R

In document (1)ultraold atoms in an optial resonator Ph.D (Pldal 72-79)

c 2

cos 2kx

c 2

X = 1

√ 2 (c † 2 + c 2 ) ; Y = i

√ 2 (c † 2 − c 2 ),

H = − δ C a † a + iη (a † − a) + 2ω R (X 2 + Y 2 ) + u a † a X .

δ C = ∆ C − NU 0 /2

u = √

NU 0 /2

U 0

a

c 2

X

ω R

ω R /κ

ξ

h ξ i = 0

δ

h ξ(t) ξ † (t ′ ) i = 2κ δ(t − t ′ ) ,

h ξ † ξ † i = h ξ † ξ i = h ξ ξ i = 0 .

a(t)

1/κ

a(0)

a(t) = η κ − i∆ +

Z t 0

e (i∆ − κ)(t − t ′ ) ξ(t ′ )dt ′ ,

∆ ≡ ∆(X) = δ C − uX

α(t) = η/(κ − i∆(X(t))) ,

X

Σ(t)

h Σ(t) i = 0

h Σ(t 1 )Σ † (t 2 ) i = e iδ(t 1 − t 2 ) e − κ | t 1 − t 2 | − e − κ(t 1 +t 2 )

.

κ

1/κ

δ

h Σ(t 1 )Σ † (t 2 ) i ∼ = 2 κ

∆ 2 + κ 2 δ(t 1 − t 2 ) .

ua † a = u | α(t) | 2 + Ξ(t) ,

Ξ(t) = u α(t)Σ † (t) + α † (t)Σ(t) + Σ † (t)Σ(t)

.

h Ξ(t 1 )Ξ(t 2 ) i = u 2 h α † (t 1 )Σ(t 1 )α(t 2 )Σ † (t 2 ) i = D(X) δ κ (t 1 − t 2 ) ,

D(X) = u 2 | α | 2 2 κ

∆ 2 + κ 2 = u 2 2κη 2

(∆ 2 (X) + κ 2 ) 2 .

X

Y

ω R /κ

1/κ

ω R

Σ

X(t)

α(X, t) ∼ = α 0 (X) + 1

2 { Y, α 1 (X) } .

{ , }

Y

α 1 (X)

α(X, t)

d

dt α(X, t) = ∂

∂t α + i [H, α(X, t)] .

2ω R Y 2

X

[Y, α(X)] = − i ∂α(X,t) ∂X

d

dt α(X, t) = ∂

∂t α + 4ω R 1 2

Y, ∂α(X, t)

∂X

,

Y

α 0 (X) = η

− i∆(X) + κ .

α 1 (X) = 4ω R i∆ − κ

∂α 0 (X)

∂X = i 4ω R uη

(κ − i∆(X)) 3 .

Y

c ₂

√ 2 (c ^† ₂ + c 2 ) ; Y = i

√ 2 (c ^† ₂ − c 2 ),

H = − δ C a ^† a + iη (a ^† − a) + 2ω R (X ² + Y ² ) + u a ^† a X .

U ₀

h ξ(t) ξ ^† (t ^′ ) i = 2κ δ(t − t ^′ ) ,

h ξ ^† ξ ^† i = h ξ ^† ξ i = h ξ ξ i = 0 .

e ^(i∆ ⁻ ^κ)(t ⁻ ^t ^′ ⁾ ξ(t ^′ )dt ^′ ,

h Σ(t 1 )Σ ^† (t 2 ) i = e ^iδ(t ¹ ⁻ ^t ² ⁾ e ⁻ ^κ ^| ^t ¹ ⁻ ^t ² ^| − e ⁻ ^κ(t ¹ ^+t ² ⁾

h Σ(t ₁ )Σ ^† (t ₂ ) i ∼ = 2 κ

∆ ² + κ ² δ(t ₁ − t ₂ ) .

ua ^† a = u | α(t) | ² + Ξ(t) ,

Ξ(t) = u α(t)Σ ^† (t) + α ^† (t)Σ(t) + Σ ^† (t)Σ(t)

h Ξ(t 1 )Ξ(t 2 ) i = u ² h α ^† (t 1 )Σ(t 1 )α(t 2 )Σ ^† (t 2 ) i = D(X) δ κ (t 1 − t 2 ) ,

D(X) = u ² | α | ² 2 κ

∆ ² + κ ² = u ² 2κη ²

(∆ ² (X) + κ ² ) ² .

2ω R Y ²

[Y, α(X)] = − i ^∂α(X,t) _∂X

∂t α + 4ω _R 1 2

α 1 (X) = 4ω _R i∆ − κ

∂α ₀ (X)

∂X = i 4ω _R uη

(κ − i∆(X)) ³ .

a ^† a = 1

2 (αα ^∗ + α ^∗ α) = | α 0 | ² + 1

2 { Y, α ^∗ ₀ α 1 + α 0 α ^∗ ₁ } .

Γ(X) = − 16ω R u ² ∆(X)κη ²

(∆ ² (X) + κ ² ) ³ .

Y ˙ = − 4ω R X − u | α 0 (X) | ² − 1

H _eff = 4ω _R 1

2 (X ² + Y ² ) + η ²

uX − δ _C κ

∆(X _c ) < 0

h Y ² i

g(X) = − 4ω R u κη ²

(∆ ² (X) + κ ² ) ² .

ω _R /κ

X = X ₀ + δX ; Y = δY ,

H _eff ^′ = 2ω R (2X 0 δX + δX ² + δY ² ) + uη ² δX

(uX 0 − δ C ) ² + κ ² .

X 0 = − u 4ω _R

η ²

(uX ₀ − δ _C ) ² + κ ² .