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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
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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
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This phenomenon, called Bose-Einstein condensation,
is at the heart of superfluidity and superconductivity
* 1925
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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)