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Superfluid Bose and Fermi gases

Wolfgang KetterleMassachusetts Institute of Technology

MIT-Harvard Center for Ultracold Atoms

3/11/2013

Universal Themes of Bose-Einstein

Condensation

Leiden

Superfluidity and Superconductivity –

Macroscopic Quantum Phenomena

2

Single particle quantum mechanics

Single particle quantum mechanicsMany particle quantum mechanics

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Single particle quantum mechanicsMany particle quantum mechanics

Macroscopic quantum phenomena

Single particle quantum mechanicsMany particle quantum mechanics

Macroscopic quantum phenomena

4

This phenomenon, called Bose-Einstein condensation,

is at the heart of superfluidity and superconductivity

* 1925

5

Why did it take 70 years to

realize BEC in a gas?

Thermal de Broglie wavelength (∝T-1/2)

equals

distance between atoms (= n-1/3)

ncrit ∝T3/2

Criterion for BEC

nwater/109: T= 100 nK - 1 µK“Low” density:

seconds to minutes lifetime of the atomic gas

⇒ BEC ☺☺☺☺

nwater: T = 1 K“High” density:

BUT: molecule/cluster formation, solidification

⇒ no BEC ����

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Hydrogen

Sodium

advantage:

Three body recombination

rate coefficient is ten

orders of magnitude

smaller

but: elastic cross section

much smaller

BEC window for alkalis

is larger than for

hydrogen (and at lower

density)

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BEC @ JILA, June ‘95

(Rubidium)

BEC @ MIT, Sept. ‘95 (Sodium)

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Rotating superfluids

Superfluid described by macroscopic matter wave

A superfluid is irrotational

Velocity field:

unless

When going around a closed loop,

φ can only change by

multiples of 2π

Vorticity can enter the superfluid only in singularities,

the vortices

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non-rotating rotating (160 vortices)

Rotating condensates

J. Abo-Shaeer, C. Raman, J.M. Vogels, W.Ketterle,

Science, 4/20/2001

Two-component

vortex

Boulder, 1999

Single-component

vortices

Paris, 1999

Boulder, 2000

MIT 2001

Oxford 2001

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Discovery of superfluidity and superconductivity

1908 Liquefaction of helium (Kamerlingh-Onnes):

Superfluid helium created, but not recognized

1911 Discovery of superconductivity in mercury

(Kamerlingh-Onnes)

1937 Discovery of superfluidity in helium (Kapitza, Allen, Misener)

>100 years

1911:

First fermionic superfluids (superconducting mercury) were

“sympathetically” cooled by ultracold bosons (liquid helium)

Recently:

Fermi gases (e.g. 6Li) are cooled by sympathetic cooling

(evaporative cooling of bosonic gases, e.g. Na)

1911/1938:

Transition temperatures of He-II (2.2 K) and mercury (4.2 K),

tin (3.8 K), lead (6 K) similar:

purely technical reasons

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Degeneracy temperature

In cold gases: typically 200 nK – 2 µK

Footnote:

in extreme cases 0.5 – 5 nK

To avoid inelastic collisions (85Rb, JILA)

Low density for atom interferometry (87Rb, Virginia)

To achieve normal-incidence quantum reflection (Na, MIT)

same for bosons and fermions (at the same density and mass)

Superfluidity in fermions:

Usually requires much lower temperatures

than degeneracy temperature

But: Exponential factor is unity for a → ∞

Kamerlingh-Onnes: exponential (“pairing”) factor

was equal to Tfermi(electrons)/Tdegeneracy(4He)

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1911:

First fermionic superfluids (superconducting mercury) were

“sympathetically” cooled by ultracold bosons (liquid helium)

Recently:

Fermi gases (e.g. 6Li) are cooled by sympathetic cooling

(evaporative cooling of bosonic gases, e.g. Na)

1911/1938:

Transition temperatures of He-II (2.2 K) and mercury (4.2 K),

tin (3.8 K), lead (6 K) similar:

purely technical reasons

Recently:

Transition temperatures of Bose and Fermi gases similar

(Fermi gas with unitarity limited interactions):

“fundamental (?)” unitarity limit

10-5 I 10-4 normal superconductors

10-3 superfluid 3He

10-2 high Tc superconductors

0.15 high Tc superfluid

Transition temperature

Fermi temperature ∝(density)2/3

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How to vary a?

How to get a → ∞?

Particle A Particle B

Pair A-B

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Particle A Particle B

Pair A-B

Resonant interactions

have infinite strength

E

Feshbach resonance

Magnetic field

Free atoms

Molecule

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E

Feshbach resonance

Magnetic field

Free atoms

Molecule

Disclaimer: Drawing is schematic

and does not distinguish nuclear

and electron spin.

E

Feshbach resonance

Magnetic field

Molecule

I form a stable molecule

Free atoms

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E

Feshbach resonance

Magnetic field

Molecule

I form an unstable molecule

Free atoms

E

Feshbach resonance

Magnetic field

Molecule

Atoms attract each other

Free atoms

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E

Feshbach resonance

Magnetic field

Molecule

Atoms attract each otherAtoms repel each other

Free atoms

Fo

rce

be

twe

en

ato

ms

Sca

tte

rin

g le

ngth

Feshbach resonance

Magnetic field

Atoms attract each otherAtoms repel each other

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Molecules

Atoms

Energy

Magnetic field

Feshbach Resonance

Fo

rce

be

twe

en

ato

ms

Sca

tte

rin

g le

ngth

Feshbach resonance

Magnetic field

Atoms attract each otherAtoms repel each other

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Molecules

Atoms

Energy

Magnetic field

Molecules are unstableAtoms form stable molecules

Atoms repel each other

a>0

Atoms attract each other

a<0

BEC of Molecules:

Condensation of

tightly bound fermion pairs

BCS-limit:

Condensation of

long-range Cooper pairs

Feshbach Resonance

Bose Einstein condensate

of moleculesBCS Superconductor

Atom pairs Electron pairs

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Molecular BEC BCS superfluid

Molecular BEC BCS superfluid

Magnetic field

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Molecular BEC BCS superfluidCrossover superfluid

Preparation of an interacting Fermi system in 6Li

Optical trapping:

9 W @ 1064 nm

ω = 2π × (16,16, 0.19) kHz

Etrap = 800 µK

Setup:

States |1> and |2> correspond to

|↑> and |↓>

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Evidence for phase transition

Bose-Einstein condensationPeaks in the momentum distribution

(visible in spatial distribution after ballistic expansion)

SuperfluidityVortex lattices for rotating gas

Cold atomic gases: Realization of an

s-wave fermionic superfluid in the

strong coupling limit of BCS theory

JILA, Nature 426, 537

(2003).

Innsbruck, PRL 92,

120401 (2004).

ENS, PRL 93, 050401 (2004).

MIT, PRL 91, 250401 (2003)

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Vortex lattices in the BEC-BCS crossover

M.W. Zwierlein, J.R. Abo-Shaeer, A. Schirotzek, C.H. Schunck, W. Ketterle,

Nature 435, 1047-1051 (2005)

Superfluidity of fermions

requires pairing of fermions

Microscopic study of

the pairs by RF

spectroscopy

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RF spectroscopy

|�>

|3>

|�>

hf0

|�>

|3>

|�>

hf0+∆

Dissociation spectrum measures the Fourier transform

of the pair wavefunction

Width ∝ (1/pair size)2

Threshold ∝ (1/pair size)2

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Standard superconductors

ξ>> 1/kF

High Tc superconductors

ξ ≈ 6 I10 (1/kF)

Superfluid at unitarity

ξ = 2.6 (1/kF)

C. H. Schunck, Y. Shin,

A. Schirotzek, W. Ketterle, Nature 454, 739 (2008).

Rf spectra in the crossover

Confirms correlation between high Tc and small pairs

Interparticle spacing ~ 3.1 (1/kF)

“Molecular character”

of fermion pairs

Benchmarking the Fermi gas at unitarity

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Equation of State: Measuring density

2.5

2.0

1.5

1.0

0.5

0.0

De

nsity [

10

11/c

m3]

1.20.80.40.0

V [µK]

Experimental n(V)

from single profile

Exploiting cylindrical symmetry and careful characterization of trapping potential:

1.5

1.0

0.5

0.0

Specific

He

at C

V/N

1.51.00.50.0

T/TF

Heat capacity

Unitary Fermi Gas

( )0

0,

d / 3...

d 2

BV F

B N V

E NkC T P

Nk T T P

κκ

= = = −

For a resonant gas:

V

EP

3

2=

kB

Mark J. H. Ku, Ariel T. Sommer, Lawrence W. Cheuk, Martin W. Zwierlein

Science 335, 563-567 (2012)

Direct observation of the

superfluid transition

at TC/TF = 0.167(13)

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Experimental realization of the BCS-BEC crossover

Theory:

Late ’60s: Popov, Keldysh, Eagles

‘80’s Leggett, Nozières, Schmitt-Rink

Demonstrates that BEC and BCS are

two limiting cases of one theory

Su

pe

rflu

id tra

nsitio

n te

mp

era

ture

Strength of interactionsBCS BEC

Highest fermionic

transition temperature

BCS-BEC Crossover

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The BEC-BCS Crossover

( )NBEC

b vac+Φ =

+↓−

+↑

+ ∑=k

kkk ccb ϕ

vacccvukk

k

kkBCS)(

+↓−

+↑∏ +=Φ

The BCS wave function

can be written as a “BEC wave function” of pairs

However, the pair creation and annihilation operators fulfill

bosonic commutation relationships only in the “BEC limit” of

small pair size

This was known already soon after BCS theory was formulated.

F. Dyson (1957, cited by Bardeeen)

Now generally accepted:

Superconductivity is “kind of a” Bose-Einstein condensate

of electron pairs.

( )NBEC

b vac+Φ =

However:

Overlapping electron pairs are modified by Pauli

exclusion principle

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Novelty of ultracold atomic gases:

Many body physics is realized in an ultra-dilute gas, at

densities a billion times less than solids and liquids

“Superfluidity in a gas”

Realization of systems with truly short-range

interactions

What are the simplest interactions?

Short range

shorter than any other length scale

interatomic distance, de Broglie wavelength

characterized by only one parameter (strength)

approximated by delta functions

momentum space

scattering length

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Ultracold collisions

R ~50 a0 ~2 nm

Collisions parametrized by one single quantity:

scattering length a

At ultralow temperatures, only s-wave (“head-on”) collisions remain

de Broglie wavelength >> range of interatomic potential

Ultracold collisions

R λdB ~ µm

r

Collisions parametrized by one single quantity:

scattering length a

de Broglie wavelength >> range of interatomic potential

At ultralow temperatures, only s-wave (“head-on”) collisions remain

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Quantum simulators

New materials harnessing strong correlations in many-electron systems: Nanotubes, quantum magnets, superconductors, 6

Condensed matter models: Simple models which capture the relevant mechanism

?Approximations,

Impurities,

no exact solutions

Quantum simulators: Controlled, “simple” systemstesting models and verifying concepts

Cold atomic gases provide the building blocks of quantum simulators

Quantum “engineering” of interesting Hamiltonians

Ultracold Bose gases: superfluidity (like 4He)

Ultracold Fermi gases (with strong interactions near the unitartiy limit): pairing and superfluidity(BCS, like superconductors)

Now: strongly correlated systems

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New frontiers:

Interactions at the unitarity limit

Synthetic gauge field

Rapidly rotating gases

Quantum Hall effect

Spin-orbit coupling

Disorder – Anderson localization

Few-body correlations, Effimov states

Long-range interactions (Rydberg, dipolar)

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BEC II

Ultracold

fermions 6Li:

Lattice

density fluct.Ed Su

Wujie Huang

Junru Li

(Aviv Keshet)

(C. Sanner)

(J. Gillen)

BEC III

Na-Li molecules

Repulsive fermionsTout Wang

Timur Rvachov

Chenchen Luo

Myoung-Sun Heo

(Dylan Cotta)

(Ye-Ryoung Lee)

BEC IV

Rb BEC in

optical latticesHiro Miyake

Georgios Siviloglou

Colin Kennedy

Cody Burton

(David Weld (UCSB))

D.E. Pritchard

$$NSF

ONR

ARO

MURI-AFOSR

MURI-ARO

DARPA

BEC V

New exp: 7Li in

optical latticesJesse Amato-Grill

Ivana Dimitrova

Niklas Jepsen

(David Weld (UCSB))

(Graciana Puentes)