BCSBCS--BEC Crossover at Finite BEC Crossover at Finite Temperature in Cold Gases and Temperature in Cold Gases and

Condensed MatterCondensed Matter

KITPKITPMay 2007

Cold Atom Collaborators:Cold Atom Collaborators:

QijinQijin Chen Chen

J. Stajic (U Chicago; LANL)Yan He (U. Chicago)ChihChun Chien (U. Chicago)

J.E. Thomas, J. Kinast, A. Turlapov(Duke)

D.S. Jin, C. Regal and M. Greiner (JILA)

M. Holland, M. Chiofalo, J. Mil t i (JILA)

Outline of TalkOutline of Talk

Philosophy, motivation, requirements on theory..

FirstFirst generation experiments: The generation experiments: The discovery phase in cold gases.discovery phase in cold gases.

SecondSecond generation experiments: Evidence generation experiments: Evidence for a for a pseudogappseudogap in cold gases.in cold gases.

Theoretical Formalism with and without Theoretical Formalism with and without population imbalance.population imbalance.

Third Third generation experiments: Polarized generation experiments: Polarized gases.gases.

FourthFourth generation experiments: Lattices generation experiments: Lattices (optical as well as high (optical as well as high TcTc).).

Philosophy Philosophy and Motivationand Motivation

MotivationMotivation

High High TcTc ---- A. Leggett: A. Leggett: ““The small size of the The small size of the cupratecuprate pairs pairs puts us in the intermediate regime of the soputs us in the intermediate regime of the so--called BCScalled BCS--BEC BEC crossovercrossover”” (2006).(2006).

To understand cold Fermi gas To understand cold Fermi gas exptsexpts; opportunity to arrive at ; opportunity to arrive at counterpart to Gross counterpart to Gross PitaevskiiPitaevskii theory.theory.

Opportunity to generalize the paradigm of all condensed Opportunity to generalize the paradigm of all condensed matter theories: BCS theory.matter theories: BCS theory.

Novel form of Novel form of fermionicfermionic superfluiditysuperfluidity: pairing without : pairing without condensation.condensation.

cuprates

BECBCS

PhilosophyPhilosophy

Use BCSUse BCS--Leggett Leggett wavefunctionwavefunction for T=0for T=0

Why? Why? WavefunctionWavefunction is basis foris basis forBogoliubovBogoliubov de de GennesGennes theory (T=0).theory (T=0).

T =0 Gross T =0 Gross PitaevskiiPitaevskii Theory in BEC regime.Theory in BEC regime.

Unequal population theories.Unequal population theories.

Has simplicity and physical accessibility.Has simplicity and physical accessibility.

Essential Criteria for Successful Essential Criteria for Successful Finite T Crossover TheoryFinite T Crossover Theory

Must include Must include pseudogappseudogap effectseffects

SuperfluidSuperfluid density must be well density must be well behaved for all T from zero to behaved for all T from zero to TcTc..

BCS Unitary BECCharacter of Excitations

Comparison with Other Finite T Comparison with Other Finite T Theoretical ApproachesTheoretical Approaches

NozieresNozieres, Schmitt, Schmitt--Rink (NSR) includes pairing Rink (NSR) includes pairing fluctuations only in number equation. No inclusion fluctuations only in number equation. No inclusion of of pseudogappseudogap in gap equation. They takein gap equation. They take

Finite T NSR not designed to yield Leggett ground Finite T NSR not designed to yield Leggett ground state.state.

No other theory finds proper No other theory finds proper superfluidsuperfluid density density over entire range of T. Report first order over entire range of T. Report first order transitions (transitions (ZwergerZwerger--HaussmannHaussmann), or double valued ), or double valued functions (Griffin) or breakdown of theory below functions (Griffin) or breakdown of theory below TcTc ((StrinatiStrinati).).

First Generation ExperimentsFirst Generation Experiments

Evidence for Evidence for superfluditysuperfludity at at UnitarityUnitarity

JILA Unitary Phase DiagramJILA Unitary Phase Diagraminvolves two sweeps ( 2004)involves two sweeps ( 2004)

..

molecules

→ ←ΔB>

BCSBEC Unitary limit

Adiabatic sweep (sets temperature)

Quick sweep(Measure

s condensate)

Theory of Adiabatic Sweep ThermometryTheory of Adiabatic Sweep Thermometry

Adiabatic cooling leads to lower effective temperature.Adiabatic cooling leads to lower effective temperature.

TcTc curve changes shape when projected onto curve changes shape when projected onto temperatures measured in the temperatures measured in the noninteractingnoninteracting limit.limit.

Comparison of Theoretical and Comparison of Theoretical and Experimental Phase DiagramExperimental Phase Diagram

Our collaboration with JILA group.

Equilibrium phase diagram

TheoryJin et al

PRA 73, 041601(R) (2006)

ThermodynamicalThermodynamical Evidence for Evidence for Phase Transition near Phase Transition near unitarityunitarityProfiles used to Calibrate T.Profiles used to Calibrate T...

Our collaboration with Duke Group: John Thomas, Joe Kinast, Andrey Turlapov--- Feb 2005

Science 307, 1296Science 307, 1296

Second Generation ExperimentsSecond Generation Experiments

Evidence for a Evidence for a pseudogappseudogap

(i.e., pairing without (i.e., pairing without condensation)condensation)

BackdgroundBackdground: what goes on : what goes on inside a trap?inside a trap?

Particle density peaked at trap Particle density peaked at trap centercenter..

Gap decreases from center to edge: Gap decreases from center to edge: bosonicbosonic excitations in middle, excitations in middle, fermionicfermionic excitations at edgeexcitations at edge..Based on

LDA

Density Profiles and Density Profiles and PseudogapPseudogap Effects Effects as Condensate (Pair) Excitationsas Condensate (Pair) Excitations

Condensate Noncondensed pairs Fermions

At unitarity, pair excitations smooth out profiles—making it hard to tell if system is normal or superfluid.

PRL 95, 260405 (2005)PRL 95, 260405 (2005)

Thermodynamics and Thermodynamics and PseudogapPseudogapeffectseffects

Theory and Theory and exptexpt..

In data, T* appears as temperature where 2 curves meet

Duke data

Energy vs T

Science 307, 1296 (2005)Science 307, 1296 (2005)

RF Spectroscopy and RF Spectroscopy and PseudogapPseudogapEffectsEffects

0,0

0,4

0,0

0,4

-20 0 20 400,0

0,4

0,0

0,4

(d)

(c)

(a)

fract

iona

l los

s in

|2>

RF frequency offset (kHz)

(b)

C. Chin et al, Science 305, 1128 (2004).

-20 0 20 40

0

0.1

0.2

0.3

0.4

0.5

RF detuning (kHz)

-20 0 20 40

0

0.1

0.2

0.3

0.4

RF detuning (kHz)

Intermediate T

Low T

Temperatures inferred from R. Grimm updates

Theoretical FormalismTheoretical Formalism

BCSBCS--BEC Crossover at Finite BEC Crossover at Finite TemperaturesTemperatures

General approach to address finiteGeneral approach to address finitetemperature in BCStemperature in BCS--BEC Crossover: BEC Crossover:

TT--matrix schemematrix scheme

Treat pair propagators (Treat pair propagators (tt) and particles () and particles (GG) ) selfself--consistently. No higher order correlations.consistently. No higher order correlations.

ImportantImportant: this means that inter: this means that inter--boson boson interactions only treated at mean field level.interactions only treated at mean field level.

Solve coupled equations for two propagators: Solve coupled equations for two propagators: GG and and tt. .

Three Possible TThree Possible T--matrix approachesmatrix approaches

NSR pair susceptibility: NSR pair susceptibility:

Present work:Present work:

HaussmannHaussmann takestakes

Associated with BCS-Leggett ground state

DiagrammaticDiagrammatic Formalism Based on Formalism Based on BCSBCS--Leggett Ground StateLeggett Ground State

TT--matrixmatrix

FermionFermion selfself--energyenergy::

Pg = non condensed pairs

Δ2 = Δpg2 + Δsc2

propagator for non-condensed pairs (pg)

Note:

SelfSelf--consistent Equations Below consistent Equations Below TcTc

Gap equationGap equation: BEC condition: BEC condition

PseudogapPseudogap equationequation: Pair density (Boson number): Pair density (Boson number)

Number Number equation(equation(ss))Δ2 = Δpg2 + Δsc2

SummarySummary

Pair chemical potentialPair chemical potential::

Total ``numberTotal ``number”” of pairsof pairs

NoncondensedNoncondensed pairspairs::

……

……

……

……

……

……

….

Composite bosons Ideal Point bosons

Leads to BCS gap equation for

SuperfluidSuperfluid density vanishes at same density vanishes at same TcTc as found from as found from ““gap equationsgap equations””

Electromagnetic Vertex function

Because of Ward Identity, pg effects do not lead to Meissner effect.

MT= Maki-Thompson

AL= Aslamazov-Larkin

Critical TemperaturesCritical Temperatures

Homogeneous case:

Trap Case within LDA:

Third Generation ExperimentsThird Generation Experiments

Polarized GasesPolarized Gases

Three Ways to accommodate Three Ways to accommodate polarizationpolarization

Phase SeparationPhase Separation

Breached Pair (Breached Pair (SarmaSarma) State) State

FFLO (in principle)FFLO (in principle)Rice data

Real space phase separation:

Superfluid core followed by polarized normal fluid

Breached Pair or Sarma State

Gapless excitation spectrum Gapless excitation spectrum k-space “phase separation”

SarmaSarma--TcTc in homogeneous system: in homogeneous system: UnpolarizedUnpolarized and Polarized Case and Polarized Case

..

BCSBEC

BCS

P=0.05

BEC

P=0.16BCS

P=0.3

BCS

BEC

PRL 97, 090402

Away from BEC, Sarma only stable at finite T.

Unitary: BEC:

Solid lines separate different phases.BCS case: Phase separation and Sarma states retreat from each otBCS case: Phase separation and Sarma states retreat from each other her leaving possibility of new intervening state. FFLO, or normal,leaving possibility of new intervening state. FFLO, or normal,…… ??

BCS:

Polarized Superfluid

Trap Effects: Population imbalance Trap Effects: Population imbalance phase diagramsphase diagrams

Comparing Theory and Experiment in Comparing Theory and Experiment in Polarized GasesPolarized Gases

Unitary phase diagram:Unitary phase diagram:

Sweeps in temperature and polarizationSweeps in temperature and polarization

0 0.1 0.2 0.3 0.4 0.5T/TF

0

0.2

0.4

(n↑−

n ↓)/

n↑(

T=

0)

0

1

2

3

n ↑ /

n ↓

0 0.2 0.4 0.6 0.8 1p0

2

4

n ↑ /

n ↓ Tc

(a)

(b)

(c)

Transition line

Transition line

Tc

T*

MIT- sweeps in T

Rice- sweeps in p

Theory

Fourth Generation Experiments: Fourth Generation Experiments: Optical Lattice Effects and Optical Lattice Effects and

High High TcTc

Predictions for Optical Lattices: Attractive Hubbard Predictions for Optical Lattices: Attractive Hubbard Model Model (BCS(BCS--Leggett wave function with well behaved Leggett wave function with well behaved superfluidsuperfluid

density)density)

Behavior of Behavior of TcTc for variable for variable fermionfermion density:density:

Phase Diagrams:Phase Diagrams:

Mott-Like Effect: Pairs localize for moderate but non-integer n. Due to Pauli repulsion which inhibits pair hopping.

Shows T=0 (density induced) pseudogap

phase.Pair density wave order beyond Tc =0?

Applying Crossover Theory to dApplying Crossover Theory to d--wave Lattice casewave Lattice case

..

Theory Fit T*(x)

Pairs localize and Tc vanishes well before BEC.

Where is Mott Physics?Where is Mott Physics?

Attractive interaction derives from Attractive interaction derives from ““Mott physicsMott physics”” since pairing since pairing interaction gets stronger with interaction gets stronger with underdopingunderdoping, as seen by T*., as seen by T*.

We can now understand why We can now understand why TcTc behaves behaves oppositelyoppositely..

Trying To Understand the Trying To Understand the SuperfluidSuperfluidDensity in Density in CupratesCuprates

Paradox: Paradox: superfluidsuperfluid density does not directly reflect the density does not directly reflect the ((x,Tx,T) behavior of the ) behavior of the fermionicfermionic gap.gap.

Then what excitations drive Then what excitations drive superfluidsuperfluid density to zero?density to zero?

0 40 80T(K)

−30

−15

0ρ s

(T)−ρ s

(0)(

meV

)

Tc=90 K

Tc=90 K

Tc=88 K

Tc=65 K

Tc=55 K

Tc=50 K

Tc=48 K

0 1T / Tc

0

1

Δ(T

) / Δ

(0)

Tc = 48 K

Tc = 90 K

a) b)

Doping dependence of the fermionic gap.

Lemberger

Temp dependence of the gap.

PossiblePossible Evidence for pair Evidence for pair excitations in excitations in underdopedunderdoped

cupratescuprates..

Crossover Theory

Understanding universality in Penetration depth Understanding universality in Penetration depth vsvs temperature.temperature.

UBC experiments

ConclusionsConclusions

Cold gases present opportunity to explore biggerCold gases present opportunity to explore bigger--thanthan--BCS theory (BCS theory (ieie., rewrite the texts). ., rewrite the texts).

Possibly relevant to high Possibly relevant to high TcTc. .

Future: More Future: More ““exoticexotic”” phenomena in cold gases:phenomena in cold gases:

1. With optical lattices can test 1. With optical lattices can test attractive/repulsive Hubbard models.attractive/repulsive Hubbard models.

2. 2. Mott physicsMott physics vsvs ““small pair physicssmall pair physics””in high in high TcTc needs to be sorted out.needs to be sorted out.

Review ReferencesReview References

Phys. Reports 412, 1 (2005)

And

Varenna Summer School Cond-mat/ 0605039

Also (Former Soviet J) Low Temp. Phys. 32, 406 (2006)

Rationale for Applying BCSRationale for Applying BCS--BEC Crossover to high BEC Crossover to high TcTc

Pairs are small.Pairs are small.

““PseudogapPseudogap”” (normal state gap) very (normal state gap) very prominentprominent..

TcTc is high.is high.

Quasi 2 dimensional. Quasi 2 dimensional.

BCS-BEC Crossover at Finite Temperature in Cold Gases and Condensed MatterCold Atom Collaborators:�Qijin Chen ����Outline of TalkPhilosophy �and MotivationMotivationPhilosophyEssential Criteria for Successful Finite T Crossover TheoryComparison with Other Finite T Theoretical ApproachesFirst Generation Experiments����Evidence for superfludity at UnitarityJILA Unitary Phase Diagram�involves two sweeps ( 2004) Theory of Adiabatic Sweep ThermometryComparison of Theoretical and Experimental Phase Diagram �Thermodynamical Evidence for Phase Transition near unitarity�Profiles used to Calibrate T.Second Generation Experiments����Evidence for a pseudogap��(i.e., pairing without condensation)Backdground: what goes on inside a trap?Density Profiles and Pseudogap Effects as Condensate (Pair) ExcitationsThermodynamics and Pseudogap effects RF Spectroscopy and Pseudogap Effects Theoretical Formalism���BCS-BEC Crossover at Finite TemperaturesGeneral approach to address finite�temperature in BCS-BEC Crossover: �T-matrix scheme Three Possible T-matrix approachesDiagrammatic Formalism Based on BCS-Leggett Ground StateSelf-consistent Equations Below Tc SummarySuperfluid density vanishes at same Tc as found from “gap equations”Critical TemperaturesThird Generation Experiments��Polarized GasesThree Ways to accommodate polarizationSarma-Tc in homogeneous system: Unpolarized and Polarized Case Trap Effects: Population imbalance phase diagramsComparing Theory and Experiment in Polarized GasesFourth Generation Experiments: Optical Lattice Effects and High TcPredictions for Optical Lattices: Attractive Hubbard Model (BCS-Leggett wave function with well behaved superfluid density)�Applying Crossover Theory to d-wave Lattice caseWhere is Mott Physics?Trying To Understand the Superfluid Density in Cuprates Possible Evidence for pair excitations in underdoped cupratesConclusionsReview References Rationale for Applying BCS-BEC Crossover to high Tc