Summary

Nonlinear quantum dynamis of two

BEC modes dispersively oupled by an

optial avity

PhysisofaoupledBECavitysystemisinterestingintheontextoftherapidly

devel-opingeldofopto-mehanialoupling[128℄.Opto-mehanisisordinarilyonsidered in

the ontextof a avity system in whih the vibratingmirror isoupledto the lighteld

viaradiationpressure.Infat,densitymodulationsoftheatomloudamounttoeetive

avity lengthhanges as does the vibrating mirror.In a reent paper[87℄, Brenneke et

al. have pointed out that the BEC-avity system in a restrited spae of modes an be

desribed by the same, so-alled radiation pressure Hamiltonian. The interrelation of

the parameters in the Hamiltonian, however, are realized in a dierent regime with a

BEC than with a moving mirror. Namely, in the BEC-avity system the frequeny of

the mehanial motion, given by the atomi reoil frequeny, is well below the dipole

ouplingstrength. Byontrast,inthease ofavibratingmirror,itsosillationfrequeny

isonthesameorderofmagnitudeastheouplingstrength. Theotheressentialdierene

is that the initialstate of the BEC-avity system is readily the low-exitation quantum

regimewhih,onthe otherhand,isjusttheultimategoaltobereahed withmehanial

osillatorsvia various opto-mehanialooling shemes.

InthefollowingIshallstudythisquantumregimeoftheBEC-avityopto-mehanial

systeminthelowtemperaturelimit.Ishallestablishthequantumversionofthelassial

modelpresented in Ref. [87℄ in the form of a quantum master equation whih aounts

fornon-onservativeeetsdue totheirreversiblephotonleakagefromthe avity.Then,

possibility of observing generi quantum eets, suh as, for example, the tunneling in

aneetive double-wellpotential.

Our model will be based on the two-mode approximation in whih atoms from the

quasi-homogeneous ondensate an be exited only into one other mode, typially into

the one with

cos 2kx

^spatial ^variation ^seleted ^by ^the ^avity ^mode ^funtion. ^This

ap-proximation is supported by the analytial result of Chapter 4, whih shows that the

dominant ontribution to the depletion omes from the

cos 2kx

^BEC ^exitation ^mode.

Hene, the interation with the avity eld ouples a homogeneous ondensate mainly

tothis spei mode. The two-mode approah isused in several papers together with a

semilassial approximation [87, 129℄, in whih the exited mode is assumed to be in a

oherent state with omplex amplitude

c 2

^. ^Suh ^a ^simplisti^model^an ^explain, ^for

ex-ample,the oherent density wave osillationsof aBose-Einstein ondensateobserved in

reentexperiments[87,95℄.Westepforwardbykeepingtheexitedstatemodeamplitude

asan operator

ˆ c ₂

^, ^thereby ^inluding^quantum ^statistial ^eets ^of ^the ^atom ^loud.

5.1 The BEC-avity system and its analogy to

opto-mehanial systems

We onsider a zero-temperature Bose-Einstein ondensate inside a single-mode

high-Q optial avity. The avity mode of frequeny

ω C

^is ^driven ^by ^a ^oherent ^laser ^eld

of frequeny

ω

^. ^The ^pump ^laser ^frequeny ^is ^detuned ^far ^above ^the ^atomi ^resonane

frequeny

ω A

^, ^so ^that ^the ^atom-pump ^detuning

∆ A = ω − ω A

^far ^exeeds ^the ^rate

of spontaneous emission. One an adiabatialy eliminate the exited atomi level, and

an deriveadispersive atom-eldinteration withstrength

U ₀ = g ₀ ² /∆ _A

^,^where

g ₀

^is^the

single-photonRabifrequeny[63℄.Wedesribetheondensatedynamisinonedimension

alongside the avity axis

x ˆ

^, ^and ^we ^assume ^a ^single ^avity ^mode ^funtion

cos kx

^, ^with

k = ω/c

We begin with the master equation of the total density operator of the BEC-avity

system,

˙

ρ = − i [H, ρ] + L ρ .

^(5.1)

Themany-partileHamiltonoperatorintherotatingframeofthe pumpfrequenyreads

H = − ∆ C a ^† a + iη(a ^† − a) + Z

Ψ ^† (x)

− 1 2 ~ m

d ²

dx ² + U 0 a ^† a cos ² (kx)

Ψ(x)dx,

^(5.2)

where

Ψ(x)

^and

a

^are ^the annihilation operators of the atom and the avity elds,

respetively. The parameter

∆ C = ω − ω C

^is ^the ^avity ^detuning,

m

^is ^the ^atomi^mass.

The dissipation is taken into aount by the Liouvillian terms

L ρ = κ ( 2 aρa ^† − a ^† aρ − ρa ^† a ) ,

^(5.3)

with

2κ

^being ^the ^photon ^loss^rate.

As the oupling term in Eq. (5.2) depends on a osine funtion, it is onvenient to

perform the seond quantization of the atom eld in the disrete basis of normalized

harmoni funtions

{ 1

√

2 cos nkx

√

2 sin nkx } ^∞ n=1

^. ^Sine^the ^oupling^is

cos ² kx = (1 + cos 2kx)/2

^,^just^the^symmetri^modes

{ 1

√

2 cos 2nkx } ^∞ n=1

^are^involved ⁱⁿ^the ^dynamis.

Thekinetienergyofthesemodes grows quadratiallywiththe index

n

E n = (2n) ² ~ ω R

where

ω R = ~ k ² /(2m)

^is ^the ^reoil ^frequeny. ^The ^energy ^dierene ^between ^adjaent

modes inreases linearly,therefore there is a hierarhy in the respetive populations. In

the weak-oupling regime, where

U 0 h a ^† a i ≤ 10ω R

^, ^we ^an approximately restrit the dynamis to the rst two modes, i.e. to the onstant mode, initially marosopially

populated by the ondensate, and to the one that is populated by the interation with

the avity eld in rst order. Notethat weonsidered this latter mode in the analytial

alulationofthe exess noisedepletion (Eq.4.12),whih gavegoodagreementwith the

numerial results in the weak oupling limit. Aordingly, we write the eld operator

Ψ(x)

ⁱⁿ^the ^form:

Ψ(x) = c 0 + √

2 c 2 cos 2kx ,

^(5.4)

with

c 0

^and

c 2

^being ^the ^bosoni annihilation operators of the orresponding modes. In the following, we shall omit the hat from the operators. By inserting this ansatz into

Eq. (5.2), weget the following seond quantized Hamiltonoperator:

H =

− ∆ C + NU 0

2 a ^† a + iη (a ^† − a) + 4ω R c ^† ₂ c 2 +

√ 2U 0

4 a ^† a (c ^† ₀ c 2 + c ^† ₂ c 0 ).

^(5.5)

If there is a xed number of atoms in the two modes

c ^† ₀ c 0 + c ^† ₂ c 2 = N

^, ^we ^an ^onsider

the atomi system as a marosopi spin with

J = N/2

^. ^In ^this ^ase ^the Hamiltonian (5.5) is quitesimilar to the one of the famous Dike model [130, 131℄. However, instead

of the Dike-type ouplingterm

(a ^† + a) J x

^, ^here ^the ^interation ^term,

a ^† a J x

^represents

ahigher-order nonlinearity.

Having a ondensate of a large number of atoms in

c 0

^together ^with ^having ^a ^weak

interation, we assume that the ondensate ground state is undepleted, and formally

set

c ₀ ≡ √

N

^. ^Then ^we ^end ^up ^with ^the ^model ^that ^desribes ^the ^oupling ^between

two harmoni osillators, the driven and lossy avity mode with amplitude

a

^and ^the

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

In document (1)ultraold atoms in an optial resonator Ph.D (Pldal 68-72)

cos 2kx

cos 2kx

c 2

ˆ c 2

ω C

ω

ω A

∆ A = ω − ω A

U 0 = g 0 2 /∆ A

g 0

x ˆ

cos kx

k = ω/c

˙

ρ = − i [H, ρ] + L ρ .

H = − ∆ C a † a + iη(a † − a) + Z

Ψ † (x)

− 1 2 ~ m

d 2

dx 2 + U 0 a † a cos 2 (kx)

Ψ(x)dx,

Ψ(x)

a

∆ C = ω − ω C

m

L ρ = κ ( 2 aρa † − a † aρ − ρa † a ) ,

2κ

{ 1

√

2 cos nkx

√

2 sin nkx } ∞ n=1

cos 2 kx = (1 + cos 2kx)/2

{ 1

√

2 cos 2nkx } ∞ n=1

n

E n = (2n) 2 ~ ω R

ω R = ~ k 2 /(2m)

U 0 h a † a i ≤ 10ω R

Ψ(x)

Ψ(x) = c 0 + √

2 c 2 cos 2kx ,

c 0

c 2

H =

− ∆ C + NU 0

2

a † a + iη (a † − a) + 4ω R c † 2 c 2 +

√ 2U 0

4 a † a (c † 0 c 2 + c † 2 c 0 ).

c † 0 c 0 + c † 2 c 2 = N

J = N/2

(a † + a) J x

a † a J x

c 0

c 0 ≡ √

N

a

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

ˆ c ₂

U ₀ = g ₀ ² /∆ _A

g ₀

H = − ∆ C a ^† a + iη(a ^† − a) + Z

Ψ ^† (x)

d ²

dx ² + U 0 a ^† a cos ² (kx)

L ρ = κ ( 2 aρa ^† − a ^† aρ − ρa ^† a ) ,

2 sin nkx } ^∞ n=1

cos ² kx = (1 + cos 2kx)/2

2 cos 2nkx } ^∞ n=1

E n = (2n) ² ~ ω R

ω R = ~ k ² /(2m)

U 0 h a ^† a i ≤ 10ω R

a ^† a + iη (a ^† − a) + 4ω R c ^† ₂ c 2 +

4 a ^† a (c ^† ₀ c 2 + c ^† ₂ c 0 ).

c ^† ₀ c 0 + c ^† ₂ c 2 = N

(a ^† + a) J x

a ^† a J x

c ₀ ≡ √

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