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Bose-Einstein condensation

Lecturer: Yong-il Shin (SNU)

2010 KIAS-SNU Physics camp

Outline

1. What is Bose-Einstein condensation?

2. BEC in ultracold atomic gases

3. Phase coherence of BEC

4. Superfluidity and BEC

5. BEC in an optical lattice

Outline

1. What is Bose-Einstein condensation?

2. BEC in ultracold atomic gases

3. Phase coherence of BEC

4. Superfluidity and BEC

5. BEC in an optical lattice

Three boarders on slopes

A

B C

Here, we can’t distinguish them.They all look same.

Classical counting Quantum counting

Total number of cases: 27Probability of having all of them

in the same slope: 1/9Total number of cases: 10Probability of having all of them

in the same slope: 3/10

Indistinguishability makes them more likely to be together.

Saturation of occupation

Example) N-particles in a two-level system

Classical counting Quantum countingdue to the indistinguishability of particles

2N N+1

1

2E

ppNN+

=12 1

1

2 1)1(

1 +

+

−+−

−= N

N

ppN

ppN

TkE Bep /−=

In a thermodynamic limit N→∞,the occupation number of the excited state is saturated

Bose statistics

Occupation number of a state with energy

Total number of particles

iε

( )CN T≤

Density of states

When , the remaining particles are put into the ground state with .

CN N>0µ ε=

ε

( )n ε

Bose-Einstein condensate: Macroscopic occupation of a single quantum state

Criterion of Bose-Einstein condensation

: density matrix of a given many-body state of N-bosons

: single-particle density matrix

: corresponding eigenfunctions and values

Macroscopic occupation of a single quantum state

: wavefunction of a condensate

Penrose & Onsager (1956)

Yang (1962): off-diagonal long-range order (ODLRO)

Birth of the BEC idea (1920’s)

S. N. Bose

A. Einstein

E. Schrodinger

de Broglie

• Bose derived Plank distribution of Black-body radiation with a new photon counting way, but failed to publish his results.

• Einstein immediately agreed with Bose, and they described the indistinguishabilityof photons and Bose-Einstein statistics.

• Einstein extended this idea to include systems with a conserved particle number, adopting de Broglie’s new idea of matter waves.

• Einstein pointed a peculiar feature of the distribution: at low temperature it saturates.

• Schrodinger first heard about de Broglie’s idea from reading Einstein’s paper and later he developed his wave equation.

Matter wave picture of BEC

TmkBdB

22 πλ =

612.23 ≅dBnλ

de Broglie’s wavelength

Critical condition of BEC

BEC systems Superfluid Helium Lasers and Masers (macroscopic occupation in the same state) Superconductors Ultracold atomic gases

Outline

1. What is Bose-Einstein condensation?

2. BEC in ultracold atomic gases

3. Phase coherence of BEC

4. Superfluidity and BEC

5. BEC in an optical lattice

Ultracold Atom Cloud

Typical sample size

Atom number ~ 106

Spatial size ~ 100 umn=1011~1015 /cm3

T= 100 nK (~ 1 Hz)

Bose-Einstein condensate (BEC)

A Bose-Einstein condensate is the macroscopic occupation of the ground state of a system.

T > TBEC T = 0

BECTBEC ~ 100nK

How to cool down atoms ?

Laser cooling

ground state

excited state

• photon has momentum• atom absorbs and emits photons

• Doppler effect : Optical molasses

kv+ω

kv−ω

Laser cooling(1997 Nobel prize)

v

T ~ 100 µK

How to cool down atoms ?

Evaporative cooling

Removing the tail of thermal distribution leads to lower average energy, i.e., cooling the sample.

Method 1: transitions to untrapable states

ωRF

Method 2: Reducing the trap depth

How to cool down atoms ?

Evaporative cooling

Removing the tail of thermal distribution leads to lower average energy, i.e., cooling the sample.

T ~ 100 nK

Time-of-flight Imaging

The expanded cloud reveals the momentum distribution of the sample.

Bose-Einstein condensation in a dilute gas

BEC @ JILA, 1995(2001 Nobel prize)

MOT

Many-body Hamiltonian in cold atom gases

∑∑<=

−+

+=

jiji

N

iiext

i rrVrVm

pH )()(21

2

∑∑<=

−+

+=

jiji

N

iiext

i rrUrVm

pH )()(2 0

1

2

δ

Model system

Scattering problem

)(rVFor a given Interparticle potential

refe

ikrikz )(θψ +=

)(θf : Scattering amplitude

Partial wave description

)(cos)()12()(0

θθ ll

l Pkflf ∑∞

=

+=

ikkikekf

l

i

l

l

−=

−=

δ

δ

cot1

21)(

2

∑∞

=

+=0

22 sin)12(4

lltot l

kδπσ

Phase shift

Cold atom collisions

2

2

2)1()()(

mrllhrVrVeff+

+=For non-zero l,

van der Waals attractionrc ac

V(r)

r

4/1

262

=

Cma rcCharacteristic length

Centrifugal barrier mK 12

22

≈≈cr

c amlE

For gases in the sub-milikelvin regime, only s-wave collisions are relevant.

s-wave scattering length

Physical meaning of scattering length

rc acV(r)

r

krkrei

out)sin( 0

0 δψδ +

≅

At r >>ac

a<0

a<<0

a>0

Sign of scattering length and energy shift

Corresponding energy shift?

Positive a : repulsive Negative a : attractive

L

a

2 22 2 2 201

3

2( )2 2 2

kk a naEm m m L m

πδ = −

0kLπ

=

1kL aπ

=−

Effective potential

In the regime of ultracold collision, kac<<1The two-body collision is completely specified by a single parameter, a

Effective pseudopotential

∑∑<=

−+

+=

jiji

N

iiext

i rrm

arVm

pH )(4)(2

2

1

2

δπ

Realizing the toy-model Hamiltonian,

Mean-field description of a dilute Bose gas

),(),()(2

),( 20

22

trtrUrVm

trt

i ext Φ

Φ++∇−=Φ

∂∂

Gross-Pitaevskii (GP) equation

A simplest approximation for many-body states a product of a single-particle wavefunction:

2

01

( ) ( )2

Ni

ext i i ji i j

pH V r U r rm

δ= <

= + + −

∑ ∑

Wave function of condensate

Outline

1. What is Bose-Einstein condensation?

2. Ultracold atomic gases

3. Phase coherence of BEC

4. Superfluidity and BEC

5. BEC in an optical lattice

Laser lightOrdinary light

diffraction limited (directional)coherentone big wavesingle mode (monochromatic)

divergentincoherentmany small wavesmany modes

Interference @ MIT, 1997(2001 Nobel Prize)

Interference of two BECs

Hanbury Brown – Twiss Effect

Hanbury Brown & Twiss, Nature 177 (1956)

Photon bunching in light emitted by a chaotic source Highlight the importance of two-photon correlations Modern quantum optics

Quantum theory of optical coherenceGlauber, PRL 10 (1963)

)()();( 2,2)(

1,1)(

2,21,1)1( trEtrEtrtrG +−=

First-order coherence function

Laser light

Chaotic light

a a+

How to describe the state of light

Correlations in many-body systems First-order coherence function

)(ˆ)(ˆ),( 2121)1( xxxxG ψψ += one-particle density matrix

0),(lim 021)1(

21

≠=∞→−

nxxGxx

: condensate fraction

1),(),(

),(),(22

)1(11

)1(21

)1(

21)1( ≤=

xxGxxGxxGxxg Normalized first-order coherence function

For a translational invariant system

−=

Tcl

rrgλπ 2

)1( exp)(

Tmkp

kBen 2

2

~−

For a classical gas

nnrgr

/)(lim 0)1( =

∞→With a BEC

At T~Tc

−

)(exp~)()1(

Trrg

ξ

r

Spatial coherence of a trapped Bose gas

Bloch et al., Nature 403 (2000)

T<<Tc T~Tc T>>Tc

Two-slit experiment to measure spatial coherenceUsing two rf waves, outcouple two atomic beams in different positionsVisibility of the interference pattern indicates spatial coherence

Spatial coherence of a trapped Bose gasBloch et al., Nature 403 (2000)

1),(),(),(

),(21

)1(

22)1(

11)1(

21)1(

≤== xxgxxGxxG

xxGV

Quantum theory of optical coherence (2)

)()()()();( 1,1)(

2,2)(

2,2)(

1,1)(

2,21,1)2( trEtrEtrEtrEtrtrG ++−−=

Second-order coherence function

Fluorescence From a single atom

Laser light

Chaotic light

How to describe the state of light Glauber, PRL 10 (1963)

Correlations in many-body systems (2) Second-order coherence function

)(ˆ)(ˆ)(ˆ)(ˆ),( 122121)2( xxxxxxG ψψψψ ++=

),()()()()()(ˆ)(ˆ 21)2(

2121121 xxgxnxnxxxnxnxn

+−= δ

)()(),(),(

21

21)2(

21)2(

xnxnxxGxxg =

Normalized second-order coherence function

Density-density correlation function

Prob. To have another particle in a shell [r, r+dr]drrgnr )(4 )2(22π

Higher order phase coherenceOttl et al., PRL 95 (2005)

Higher order phase coherenceOttl et al., PRL 95 (2005)

Coherent outcoupling Incoherent outcoupling

Bunching and anti-bunching

Using 3He* (fermion) and 4He* (boson)

Schellekens et al., Science 310 (2005) / Jeltes et al., Nature 445 (2007)

Bunching and anti-bunching

Using 3He* (fermion) and 4He* (boson)

Schellekens et al., Science 310 (2005) / Jeltes et al., Nature 445 (2007)

Higher order phase coherence

6!3)0()3( ==thg

1)0()3( =BECg

Three-body decay rate is six-times smaller for condensates.

Burt et al., PRL 79 (1997)

Outline

1. What is Bose-Einstein condensation?

2. Ultracold atomic gases

3. Phase coherence of BEC

4. Superfluidity and BEC

5. BEC in an optical lattice

Superfluid

Superfluid, having a phenomenological definition, can flow without dissipation.

Q) Can this particle excite this fluid, or give its kinetic energy to this fluid?

Landau Criterion of Superfluidity

L.D. Landau, J. Phys. (USSR) 5, 71 (1941).

min

)(

=

pp

cευ

Critical velocity

2 21 1 ( )2 2

( )

mv m v v mv v

p mv m v v m v

ε δ δ

δ δ

= − − ≈

= − − =

If the particle excite the fluid, ( )pvp

ε=

Excitation energy for momentum p

Excitation spectrum of Superfluid Helium

Phonon

Roton

Maxon

∆pC1=ε

r

ppµ

ε2

)( 20−

+∆=

Excitation spectrum of Superconductor

Normal

Super

Excitation spectrum of a non-interacting Bose gas

2( )p pε ∝

0Cv =p

( )pε

2

1 2

N

i

pHm=

=∑

Excitation spectrum of a non-interacting Bose gas

p

( )pε

2

01

( )2

N

i ji i j

pH U r rm

δ= <

= + −∑ ∑

Microscopic theory of a Bose gas at T=0

20000 )(2 qqq Un εεε +=

Bogoliubov approximation: replacing a0 with c-number N01/2

Diagonalize with canonical transformation:

Elementary excitation of an interacting Bose gas

20000 )(2 qqq Un εεε +=

2

01

( )2

N

i ji i j

pH U r rm

δ= <

= + −∑ ∑

mUns

qsUn qq

/

2

00

000

=

=≈ εε

000 Unqq +≈ εεPhonon regime

Free particle regime

Many-body ground state

Dominant scattering processes at T~0

)()()0()0( pp −++⇔+Two atoms in condensate collide into +p and –p atoms.

02 ≠== +pppp vaan

Non-condensed atom number

Quantum depletion

Outline

1. What is Bose-Einstein condensation?

2. Ultracold atomic gases

3. Phase coherence of BEC

4. Superfluidity and BEC

5. BEC in an optical lattice

Optical dipole trap

R. Grimm, et al, Adv. At., Mol., Opt. Phys. 42, 95 (2000)

Complex polarizability

Far detuning limit (∆ << Γ)

Optical lattice

When two laser beams overlap, they interfere, leading to a periodic pattern of the intensity, i.e. a periodic potential for atoms.

Standing potential

Moving lattice potential21

21

ωωωω

≠=

Lattice period is controlled by the angle between the two beams.

Optical lattice2D optical lattice / quantum wire, 1D physics

3D optical lattice

Superlattice potentials

Optical lattice

Atoms moving in an optical lattice have the same basic physics as electrons in a crystal lattice in solids.

Lattice constant

Solid crystal ~10-10mOptical lattice ~10-7m

Lattice barrier height

Solid crystal ~105 KOptical lattice ~10-5 K

Optical lattice: Magnifying laboratory for condensed matter physics.

Band structure

The presence of an optical lattice modifies the single-particle energy spectrum to a band structure.

Energy band structure

Bloch wave function for nth band with momentum q-distributed over all lattice sites

Bloch wavefunction

)()()()(

xudxuxuex ikx

=+=ψ

Time-of-flight image of a BEC in an optical lattice

BEC from a harmonic trap

BEC from a lattice

Sudden release from a trap Revealing the in-trap momentum distribution. Diffraction from an optical grating

Adiabatic mapping of quasimomentumUnder adiabatic transformation of the lattice depth the quasimomentum q is preserved during slow turn-off process

PRL 87, 160405 (2001).

Brillouin zones

BEC in a double-well potential

BEC1 BEC2

Relative phase of two condensates

Tunneling of particles between the wells

Time evolution of the phase and the atom number: Josephson dynamics

Simple description

++−= ++

=

++∑ )(21

21122,1

aaaaJaaaaUHi

iiii

Interaction term Tunneling term

Ground state for the non-interacting case U=0

−+−+

++ +−= aJaaJaH 2/)(

2/)(

21

21

aaa

aaa

−=

+=

−

+Symmetric state

Anti-symmetric state

Symmetric ground state If we start with a BEC in one well, it will oscillate at hJ /2

Two-mode approximation

Coherent state and number state

Non-interacting case U=0

0)( 21Ni

coh aea ++ +∝ φψ : Coherent state with a well-defined relative phase

Strongly-interacting case U>>J

0)()( 22

21

NN

num aa ++∝ψ : Number (Fock) statewith well-defined atom numbers

++−= ++

=

++∑ )(21

21122,1

aaaaJaaaaUHi

iiii

Particle number uncertainty ~ N

Bose-Hubbard model

Kinetic energyHopping to nearest neighbors

On-site interactions

Both the thermal and mean interaction energies at a single site are much smaller than the separation to the first excited band. Only the lowest band is involved.

Wannier functions decay essentially within a singel lattice constant. Only the hopping to nearest neighbors are counted.

In the limit of a sufficient deep optical lattice.

Superfluid-Mott-insulator transition

The many-body ground state is determined via the competition between the kinetic energy and the interaction energy.

Superfluid phasenn

NNn L

=∆

= /

Mott-insulating phase0

1=∆=n

n

can be controlled by the lattice intensity.

Quantum phase transition from SF to Mott-Insulating phase

Superfluid-Mott-insulator transitionNature 415, 39 (2001).

V=0 Er 3 7 10

13 14 16 20

Interference peaks disappear Loosing superfluidity

Phase coherence in SF-to-MI transition

Existence of BECSuperfluid phase

Perfect Mott regime, J=0

vanishes exponentially beyond R=0.

The momentum distribution is a structureless Guassian.

With non-zero J, a coherent admixture of particle-hole pairs

Short range coherence

Summary

Ultracold atom gases

: Model system for many-body physics

BEC has laser-like properties

BEC with interactions is a superfluid

Optical lattice systems simulate solid state physics.