Tobias DonnerETH Zürich, SwitzerlandMarch 2016

Bose-EinsteinCondensationandCriticalBehavior

Outline

• Introduction toBose-Einsteincondensation• Long-rangeorder• Criticalphenomena• MeasuringacriticalexponentoftheBEC-normaltransition

• Long-rangeinteractionsinquantumgases• CavityQEDintro• Realizingthesuperfluid – supersolidphasetransition• Studying criticalpropertiesofthephasetransition• (RealizinganextendedHubbardmodel: superfluid –

Mottinsulator– supersolid – chargedensitywave

L1

L2

Agasat room temperatureT>>Tc

AnultracoldgasT>Tc λdb=h/mvα T-1/2

CondensationT<Tc λdb= d

PurecondensateT<<Tc

Bose-Einsteinstatistics(grand-canonical ensemble)

�

�Bose-Einsteindistribution function: Chemicalpotential:

orsmaller

Bose-EinsteinstatisticsTotalnumber ofparticles

Goodapproximation forlargedensityofstates

KeepingTfixedaddparticles:

Densityofstatesin3D:

increase however,

Bose-Einsteinstatistics

Bose-Einsteinstatistics

Herewemadeamistake:

Ourwayoutistosingleouttheground state:

Nowanyadditionalparticleisaccommodatedinthegroundstate!

Atthecriticalpoint:

T<<Tc

100nK

Getting cold:Liquidhelium?

Roomtemperature

LiquidHe:4K

Getting cold:Dilutionfridge??

Roomtemperature

LiquidHe:4KDilutionfridge:10mK

T<<Tc

300K…

100uK

LaserCooling

0.1mK

300K

LithiumMOT

T<<Tc

100uK…

100nK

magnetic quadrupole trap

V = µ B

x

B

Magnetic Trapping

Evaporativecooling

Gettingcold:Evaporativecooling

Lasercooling:0.1mK

Roomtemperature

Evaporativecooling:100nK

Howcoldis100nK? 23 2mm/s2 Bmv k T v= → ≈

AbsorptionImaging

CCD

Atoms

UHVglasscell

lens

Expansionofacoldgas

500nK

0.6mm

Condensation

430nK

Condensation

300nK

Condensation

200nK

PureCondensate

100nK

CharacteristicsofBEC?

(MIT1997)

( ((

Off-diagonallongrangeorder

Single-particledensitymatrixhowevercapturesmostrelevantphysics:

with

Many-bodycorrelationscangetcomplicated.

e.g.densityofgasatposition r:

alsocapturesmomentumdistribution(that’swhereBECtakesplace!):

annihilatesparticleatr

Off-diagonallongrangeorderConsiderhomogeneous gasinthethermodynamic limit:

translationalinvariantsystem

BECtakesplaceinmomentumspace,thus

Correlationfunction

Distinguishcoherencelengthfromcorrelation length!

Penrose+OnsagercriterionforBEC:

Firstordercorrelationfunction:

Fieldoperator:

Fieldoperator

BEC excitations

Bogoliubov approximation:

with

BECwavefunction:

density(“diagonaldensity”)

Phase(“off-diagonalelements”)

condensate fluctuations

Correlationfunction:

Optics:YoungsDoubleSlit Experiment

Measuring the Coherence of aBEC

VOLUME 82, NUMBER 15 P HY S I CA L REV I EW LE T T ER S 12 APRIL 1999

Atom Laser with a cw Output Coupler

Immanuel Bloch, Theodor W. Hänsch, and Tilman EsslingerSektion Physik, Ludwig-Maximilians-Universität, Schellingstrasse 4/III, D-80799 Munich, Germany

and Max-Planck-Institut für Quantenoptik, D-85748 Garching, Germany(Received 3 December 1998)

We demonstrate a continuous output coupler for magnetically trapped atoms. Over a period of up to100 ms, a collimated and monoenergetic beam of atoms is continuously extracted from a Bose-Einsteincondensate. The intensity and kinetic energy of the output beam of this atom laser are controlled bya weak rf field that induces spin flips between trapped and untrapped states. Furthermore, the outputcoupler is used to perform a spectroscopic measurement of the condensate, which reveals the spatialdistribution of the magnetically trapped condensate and allows manipulation of the condensate on amicrometer scale. [S0031-9007(99)08914-0]

PACS numbers: 03.75.Fi, 05.30.Jp, 32.80.Pj, 42.55.– f

Four decades ago the first optical lasers [1] were demon-strated [2,3], marking a scientific breakthrough: Ultimatecontrol over frequency, intensity, and direction of opticalwaves had been achieved. Since then, lasers have foundinnumerable applications, for both scientific and generaluse. It may now be possible to control matter wavesin a similar way, as Bose-Einstein condensation (BEC)has been attained in a dilute gas of trapped atoms [4–6].In a Bose-Einstein condensate a macroscopic number ofbosonic atoms occupy the ground state of the system,which can be described by a single wave function. Apulsed output coupler which coherently extracts atomsfrom a condensate was demonstrated recently [7,8]. Be-cause of its properties, this source is often referred to as anatom laser.In this Letter we report on the successful demonstration

of an atom laser with a continuous output. The durationof the output is limited only by the number of atoms inthe condensate. The Bose-Einstein condensate is producedin a novel magnetic trap [9] which provides an extremelystable trapping potential. A weak rf field is used to ex-tract the atoms from the condensate over a period of up to100 ms, thereby forming an atomic beam of unprecedentedbrightness. The output coupling mechanism can be visual-ized as a small, spatially localized leak in the trapping po-tential. The condensate wave function passes through theleak and forms a collimated atomic beam. This continu-ous output coupler allows the condensate wave function tobe studied and manipulated with high spatial resolution.Let us consider a simple model for a cw output coupler

[10–12] and assume that atoms which are in a magneticallytrapped state are transferred into an untrapped state by amonochromatic resonant rf field. In the untrapped state theatoms experience the repulsive mean-field potential of thecondensate, which can be approximated by the parabolicform of a Thomas-Fermi distribution. When the atomsleave the trap the corresponding potential energy of theatoms is transformed into kinetic energy. The velocity ofthe atoms leaving the trap can be adjusted by the frequency

of the rf field, because it determines the spatial regionwhere the atoms are transferred into the untrapped state.So far, only pulsed output couplers have been demon-

strated [7,13,14]. If a short and, hence, broadband rf pulseis applied, the output pulse has a correspondingly largeenergy spread. Each portion of the condensate that leavesthe trap has a kinetic energy distribution with a widthcomparable to the mean-field energy of the condensate.For this case the output coupling process is no longer spa-tially selective and therefore largely insensitive to fluctua-tions in the magnetic field.Continuous wave output coupling, as done in this paper,

can be achieved only if the magnetic field fluctuations thatthe trapped atoms experience are minimized. The level offluctuations in the magnetic field has to be much less thanthe change of the magnetic trapping field over the spatialsize of the condensate.In the experiment, we use a novel magnetic trap [9],

the quadrupole and the Ioffe configuration (QUIC) trap,which is particularly compact and operates at a currentof just 25 A. The compactness of the trap allowedus to place it inside a m-metal box, which reduces themagnetic field of the environment and its fluctuations bya factor of approximately 100. In combination with anextremely stable current supply sDIyI , 10

24d we areable to reduce the residual fluctuations in the magneticfield to a level below 0.1 mG. The rf field for the outputcoupler is produced by a synthesizer (HP-33120A) andis radiated from the same coil as is used for evaporativecooling. The coil has 10 windings, a diameter of 25 mm,and is mounted 30 mm away from the trap center. Themagnetic field vector of the rf is oriented in the horizontalplane, perpendicular to the magnetic bias field of the trap.To obtain Bose-Einstein condensates we use the same

setup and experimental procedure as in our previous paper[9]. Typically, 10

9 rubidium atoms are trapped and cooledin a magneto-optical trap. Then the atoms are transferredinto the QUIC trap where they are further cooled byrf-induced evaporation. The QUIC trap is operated with

3008 0031-9007y99y82(15)y3008(4)$15.00 © 1999 The American Physical Society

87Rb condensate

2×106 atoms

|mF=-1〉

|mF=0〉

Radio-frequencyoutputcoupling

Radio-frequency output coupling

B = const.

RF

50µm

ProbingFirstOrderCoherence(Experiment)

Thermal gas T >Tc Partly condensed T<Tc Fully condensed T <<Tc

V(Δz,T)

SpatialCorrelationFunctionofaTrappedBoseGas

Constant correlationfunction indicates thepresence of long-range phasecoherence !

I.Bloch,T.W.Hänsch,andT.Esslinger,Nature403,166-170(2000)

Butwhathappensrightatthecriticalpointofthephasetransition?

?

ChangeTandmeasure?

L Influenceofoutput coupling30%ofcloudL Signal-to-noiseofvisibility

Critical behavior at a phase transition

Historyofphasetransitions• 1869:AndrewsobservescriticalopalescenceinCO2

wikipedia

Phasediagramofcarbondioxide

Criticalpointdryer(MEMS,spices,vaccines,SEMsamples,decaffination)

wikipedia

Phasediagramofcarbondioxide

Passingthecriticalpoint (Andrews1869)

Historyofphasetransitions• 1869:AndrewsobservescriticalopalescenceinCO2

• 1873:vanderWaalsformulatesmicroscopictheory

• 1937:Landautheory• Globalorderparameter• Expandfreeenergy inpotentialsof• ->predictscriticalexponentsdescribing

divergencesat2nd orderphasetransitions

• 1944:Onsagersolves2DIsing modelexactly• CriticalexponentdiffersfromLandautheory• Experimentsalsoshowdifferentexponents

• 1950:Landau-Ginzburg theory• ExtendsLandautheorybyspace- andtime-varying

orderparameter• Nowfluctuationscanbecaptured!• Closeto the critical point, fluctuations dominate,

i.e.Landau-theory breaks down• Ginzburg criterionpredictssizeofcriticalregion

www.math.duke.edu

T=Tc

T >Tc T<T

c

©

a b

Historyofphasetransitions• 1950:Stillhardtocalculatephysicsinthecriticalregime

• 1971:Wilsonrenormalizationgroup theory• SeelecturesofZinn-Justin• Theorypredictscriticalexponentsmatchingto

experiments

K1

K2

isorderparameterofthenormal-to-superfluid phasetransition:

OrderparameterSystemundergoing a2ndorderphasetransitionfromT>TctoT<Tcreducesitssymmetry.

e.g.Ising magnet:directionofmagnetizatione.g.BEC:phaseofwavefunction

->extraparameterneeded todescribestateofsystem:orderparameter

for

forChoiceoforderparameterisnotalwaysobvious,hastobedone foreverysystemafresh.Theorderparameterhastoreflectthesymmetriesofthesystem.

Magnetization isorderparameterofIsing model

BECwavefunction

for for

Landautheory T=Tc

T >Tc T<T

c

©

a b

Expandfreeenergy inpowersoftheorderparameter:

Externalfieldh(=0forBEC).

Findminimum infreeenergywithrespecttoorderparameter:

Oneglobalminimum for

Twominima for

Criticalexponentoforderparameter:

Parameterize

T=Tc

T >Tc T<T

c

©

a b

1Dorderparameter 2Dorderparameter

Landautheory:Mexicanhat

Landau-Ginzburg theory

Takefluctuations intoaccount:

“Phi-4model”makesprettyaccuratepredictions. IncreasinglyhigherordershavetobeincludedwhenapproachingTc

Ginzburg criterionforsizeofcriticalregion:

Comparemagnitudeoffluctuationsoforderparameterwithmeanvalueoforderparameter.

ForaninteractingBEC:

Prediction fordecayofcorrelation function

or

(frommicroscopicarguments)

CriticalexponentsAtcriticalpoints, somepropertiescandiverge:

• Correlationlengthoffluctuations (criticalopalescence)

• Heatcapacityofliquidheliumdivergesatthecriticalpoint

Mostprecisemeasurementsofheatcapacityexponentdone inaspacemission(Lipa etal.PRL76,944(1996))

LG:

Divergenceofcorrelationlength

Scalinghypothesis (Griffiths):

Atthecriticalpoint, theonlyrelevantlengthscalediverges.Thusthesystemisinvariantunderscaletransformations!

correlationfunction forlarger:

For

correlation length:typical range in which fluctuations are correlated

DivergingCorrelationLengthT<Tc T=Tc T>Tc

Criticalexponentsarenotindependent, butarerelatedviascalingrelations:

FarawayfromTc:

Inthecriticalregion:

Scaleinvariancealsomeansself-similarity.

Scalinghypothesis

UniversalityclassesAtthecriticalpoint, thecorrelationlengthdiverges:

• theonlylengthscaleinthesystemiscorrelationlength

• microscopicpropertiesofthesystembecomeirrelevant

Systemscanbesorted intouniversalityclassesgivenbysymmetryanddimensionality!

• Eachuniversalityclassisdescribedbythesamesetofcriticalexponents.

uniaxial magnet

critical opalescence

Ising-model: XY-model

superconductor superfluid He

BEC

CriticalslowdownandfinitesizeFinitesize:

• Thermodynamic limitisoutofexperimentalreach• Finitesizewillroundoffdivergences(thinkofcorrelationlength)• Uncleariftrappedsystembelongs todifferentuniversalityclass

Criticalslowdown:

• Correlationsoverinfinitedistancesalsoneedinfinite time!Alsotherelaxationtimediverges:“criticalslowdown”

• Criticalslowdowncanbeexperiencedbothexperimentallyandnumerically.

• Depending onquenchspeed,correlationswill“freezeout”atacertainpointandremainasexcitationinthesystem:Kibble-Zurek mechanism

Butwhathappensrightatthecriticalpointofthephasetransition?

?

0 1 2 3 4 50

distance r (¸dB

)

1

N0

N

NV

N0

V

T<Tc

T2>T

cT1>T

2

T=0

0

½(1)(r) g

(1)(r)

¸dB

Correlationfunction

0.2

0.4

0.6

0.8

1

00 0.5 1 1.5 2 2.5

distance ¢z (µm)

g(1)(¢z)

Correlationfunctionincriticalregion

Butwhathappensrightatthecriticalpointofthephasetransition?

?

ChangeTandmeasure?

L Influenceofoutput coupling20%ofcloud

L Signal-to-noiseofvisibility

L “Sizeofcriticalregion isproblemofexperimentalists”(Zinn Justin)

RFoutput coupling

Single atom detector

RF

Outputcoupleonlyfewatoms(<1%)

Detectwithsingle-atomefficiency

Havingluck:criticalregion

or

L Influenceofoutput coupling20%ofcloud

L Signal-to-noiseofvisibility

L “Sizeofcriticalregion isproblemofexperimentalists”(Zinn Justin)

Detecting single atoms

cavity

Singleatom

laserSinglephotoncounter

Cavity Transmission:

0 200 400 600 8000

20

40

60

80

100

time(μs)

photonsper4

0μs

firstobservationbyH.Mabuchi etal.,Opt.Lett.21,1393(1996)

OptischeHoch-Finesse-Cavity

length =178μmModewaist =25μmFinesse ≈300.000

Atom light coupling g0= 2π 10 MHzAtomic decay rate γ = 2π 6.0 MHzCavity decay rate κ = 2π 1.4 MHz

Strongcoupling regime of cQED

κγ ,0 >g ü

γκ

g0

Apparatus

transport

transport

magnetic

magnetic

MOTpump &dispensers

vibrationisolationsystem

cavity

outputcoupledatoms

BEC

10-9 mbar

10-11 mbar

to pumps

bottom flange

top flange

Apparatus

transfer coil

Ioffe coil

quadrupole coil

gradient coil

gradient coil

cavity coils (5)

transport bracket

cooling circuit

M8 thread bar

towards cooling liquid feedthrough

MOT chamber

cavity

BEC

offset coil

sapphire spacers

Ioffe frame

vibration isolationsystem

gasket & screen

window(teflon sealed)

differentialpumping tube

MOT coil

extra windings

tight fit bushing

ceramic spacer

Apparatus

Apparatus

15stabilized lasers

>1000optical elements

>200moptical fibers

Detecting single atoms

g

position

Detectingsingleatoms

Detection efficiency >23%

Detectingsingleatoms

dete

cted

ato

ms

(in 0

.5s)

time (s)0 2 4 6 8 10

0

500

1000

1500

2000

2500

0

1

2

3

4123 nK

128 nK

134 nK

output coupling position (µm)

ato

m c

ount

rate

(ms-1

)

20 400-20-40absorption

images

dete

cted

ato

ms

(in 0

.5s)

time (s)0 2 4 6 8 10

0

500

1000

1500

2000

2500

“Ramping”thetemperature

->changetemperaturebywaiting:heatingrate4nK/s

Detectinginterferenceonthesingleatomlevel

ACloseLookatthePhaseTransition

temperature resolution: 300 picokelvinTc=146 nK

(Tc+0.6 nK)(Tc+2.8 nK)(Tc+8.4 nK)

DeterminingtheCriticalExponent

T. Donner , S. Ritter, T. Bourdel, A. Öttl, M. Köhl, T. Esslinger, Science 315, 1556 (2007)

Result

Liquid Helium: Lipa et al., Phys. Rev. B 68, 174518 (2003)Theory: Campostrini et al., Phys. Rev. B 74, 144506 (2006)

Non-interacting system:homogeneous ν =0.5trapped ν =1

Landau theory homogeneous system ν =0.5

Renormalization group theory ν =0.6717(1)

Liquid Helium spaceborne experiment ν =0.67056(6)(scaling relation α = 2 - 3ν)ten orders of

magnitudedifferencein density !

InfluenceoftrappingpotentialLengthscales:• ThermaldeBrogliewavelength 0.4um• Criticalregion inspace(vialocalchemicalpotential): 50um

• Experimentprobesbetween0.4umand2.2um– expecttoobservehomogeneous results(?!)

• Experimentalprobesymmetricwithrespecttotrapcenter!

PRA79,033611(2009)

Smallersystem:

Rounding offduetofinitesizeeffects