aurel bulgac university of washington, seattlefaculty.washington.edu/bulgac/media/bad_honnef.pdfno...
TRANSCRIPT
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What have we learned so far about dilute Fermi gases?What have we learned so far about dilute Fermi gases?
AurelAurel BulgacBulgacUniversity of Washington, SeattleUniversity of Washington, Seattle
These slides will be posted shortly athttp://www.phys.washington.edu/~bulgac/
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What is the What is the Holy GrailHoly Grail of this field?of this field?
FermionicFermionic superfluiditysuperfluidity!!
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• Dilute atomic Fermi gases Dilute atomic Fermi gases TTcc ≈≈ 1010--1212 –– 1010-9 eVeV
Liquid Liquid 33He He TTcc ≈ 1010--77 eVeV
Metals, composite materials Metals, composite materials TTcc ≈ 1010--3 3 –– 1010--22 eVeV
Nuclei, neutron stars Nuclei, neutron stars TTcc ≈ 101055 –– 101066 eVeV
•• QCD color superconductivity QCD color superconductivity TTcc ≈ 10107 7 –– 10108 8 eVeV
Superconductivity and Superconductivity and superfluiditysuperfluidity in Fermi systemsin Fermi systems
units (1 eV ≈ 104 K)
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• 1913 1913 KamerlinghKamerlingh OnnesOnnes
•• 1972 1972 BardeenBardeen, Cooper and , Cooper and SchriefferSchrieffer
•• 1973 1973 EsakiEsaki, , GiaeverGiaever and and JosephsonJosephson
•• 1987 1987 BednorzBednorz and Mullerand Muller
•• 1996 Lee, 1996 Lee, OsheroffOsheroff and Richardsonand Richardson
•• 2003 2003 AbrikosovAbrikosov, , GinzburgGinzburg and Leggettand Leggett
Memorable years in the history of Memorable years in the history of superfluiditysuperfluidity and and superconductivity of Fermi systemssuperconductivity of Fermi systems
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Robert B. Laughlin, Nobel Lecture, December 8, 1998Robert B. Laughlin, Nobel Lecture, December 8, 1998
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Topics to be covered Topics to be covered (not necessarily in this order):
• A single atom in magnetic fieldA single atom in magnetic field•• Two atoms in magnetic fieldTwo atoms in magnetic field•• Atomic trapsAtomic traps•• Basic parameters of Basic parameters of fermionicfermionic dilute atomic clouds dilute atomic clouds •• A review of a number of key experimental resultsA review of a number of key experimental results•• What theory tells us so far? What theory tells us so far?
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1995 BEC was observed.2000 vortices in BEC were createdthus BEC confirmed un-ambiguously.
In 1999 DeMarco and Jin created a degenerate atomic Fermi gas.2002 O’Hara, Hammer, Gehm, Granada and Thomas observed expansion of a Fermi cloud compatible with the existence of a superfluid fermionic phase.2003 Jin’s, Grimm’s, Ketterle’s groups and othersultracold molecules, mBEC from Fermi gas2004 Jin’s group (and bit later Ketterle’s group too) announces the observation of the resonance condensation of fermionic atomic pairs ?Grimm’s group reports measurements of the gapThomas’ group reports measurements of the specific heat
HISTORY
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One One fermionicfermionic atom in magnetic fieldatom in magnetic field
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Regal and Jin Phys. Rev. Lett. 90, 230404 (2003)
Feshbach resonance
BSSVSSa
V
VPrVPrVVVpH
nzn
eze
Znehfhf
dZi
i
hfi
r
)(,
)()()(2
2
1100
2
1
2
γγ
µ
−=⋅=
+++++= ∑=
Tiesinga, Verhaar, StoofPhys. Rev. A47, 4114 (1993)
Atomicseperation
Energy
Eres
Eth
Channel coupling
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Atomicseperation
Energy
Eres
Eth
Köhler, Gasenzer, Jullienne and BurnettPRL 91, 230401 (2003), inspired byBraaten, Hammer and Kusunokicond-mat/0301489
NB The size of the “Feshbach molecule”(closed channel state) is largely B-independenand smaller than the interparticle separation.
0
0 0
( ) 1 ( )
P r r aP r r r
>∝ >>
<
Halo Halo dimerdimer
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11 12 2
21 22
211
222 0 0 0
0 0 0 00 0
012 1 1
21
2 20
u uv v
u u
2( ) sin , u ( ) sin( ), v ( ) ( )
( ) ( ), 2
max( , )exp( ) sin( )u( ) sin( ) ,
H Vk
V H
H k
H
rr r kr r rr r
rV r g r r r
r r rg ikr krr krrk
φ κ φ
πφ φ
δ
κ>> <
<
⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦⎣ ⎦⎧ =⎪⎨
=⎪⎩
= = ∝
= − =
== +
−
0
1
2 20
21
1
2 22
02 2 2 21 1
R2 2
00
min( , )
1
u( ) exp( ) sin ( ),
1 d v( ) = = O( ) ( 1)
d u( ) d sin ( ) O( ) O( ) R
r
r r
ka g r
r i k r a r r
k ar r rk a g r g r
r r r kr R r
κ
δ
δ
⎧⎨ =⎩
⎧ −− =⎪
⎨⎪ = − >⎩⎧
≤⎪ +⎪⎨⎪ ≈ + =⎪⎩
∫
∫ ∫ Open channel wf (triplet)
Closed channel wf (singlet)
Closed channelOpen channel
0
1/ 46
0 2
30 0 0
2 2 00 2
0
ln u( ) 1
1, , O( ) 12O( ) O( )
2
r r
F
B
F
d rdr a
C mr
nr a r k rB mg
r R
µκ
λ
=
= −
⎛ ⎞≈ ⎜ ⎟⎝ ⎠
= =
≈
Halo dimersParticles in a pair spend most of the time outside the interaction zone, in the triplet state.
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Optical trap for evaporative coolingborrowed from R. Grimm
special feature #1precise controlof laser power
10 W → few 100µW
special feature #1precise controlof laser power
10 W → few 100µW
special feature #2: axial magnetic confinement- spatial compression at very weak optical traps- perfectly harmonic !!- precisely known trap frequency for weak optical trap
special feature #2: axial magnetic confinement- spatial compression at very weak optical traps- perfectly harmonic !!- precisely known trap frequency for weak optical trap
νz = 24.5 Hz @ 1kG
Umagn
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What is in a trap?What is in a trap?• Typically about 10Typically about 1055--10106 6 atoms divided 50atoms divided 50--50 among 50 among the lowest two hyperfine statesthe lowest two hyperfine states
•• Due to the high diluteness atoms in the same hyperfineDue to the high diluteness atoms in the same hyperfinestate do not interact with one anotherstate do not interact with one another
•• Atoms in different hyperfine states experience interactions Atoms in different hyperfine states experience interactions only in sonly in s--wave.wave. The strength of this interaction is fully tunable!The strength of this interaction is fully tunable!
Who does experiments?Who does experiments?
•• Jin’s group at Boulder Jin’s group at Boulder •• Grimm’s group in InnsbruckGrimm’s group in Innsbruck•• Thomas’ group at DukeThomas’ group at Duke•• Ketterle’sKetterle’s group at MIT group at MIT •• Salomon’s group in ParisSalomon’s group in Paris•• Hulet’sHulet’s group at Ricegroup at Rice
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F
z
r
1 3
h y p e r f in e s p li t t in g 8 0 M H z F e rm i e n e rg y 2 0 5 0 k H z T e m p e ra tu re T 0 2
2 5 H z t r a p f r e q u e n c ie s c lo u d h a s c ig a r s h a p e7 5 0 H z
n u m b e r d e n s i ty n 1 0 a t
Fε ενν⎧⎪⎨⎪⎩
≈
3
0
5 6
0 030
o m s /c m ty p ic a l a to m s e p a ra t io n 5 0 0 0
N u m b e r o f a to m s N 1 0 1 0
r a d iu s o f in te r a c t io n r 1 0 0 d i lu te n e s s n r 1
a
a
Typical parametersTypical parameters
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In dilute Fermi systems only very few characteristics are relevant.
• These systems are typically very cold
• A dilute Fermi system is degenerate and the fastest particle has a momentum of the order of the Fermi momentum
• The wave functions are basically constant over the interactionvolume of two particles and thus they cannot “see” any details,except the scattering length typically.
2 1 / 3 2 1 / 3(6 ) (6 )
2
Fp n nnn n
π π↑ ↓
↑ ↓
= =
= =
m
p εk
TT FFB
FF 2
,2
==
-
1/a
T
a0shallow 2-body bound state
halo halo dimersdimers
BCS BCS SuperfluidSuperfluid
High T, normal atomic (plus a few molecules) phase
?Molecular BEC and/orMolecular BEC and/orAtomic + Molecular Atomic + Molecular SuperfluidsSuperfluids
Expected phases of a two species dilute Fermi system BCS-BEC crossover
weak interactionweak interaction weak interactionsweak interactions
Strong interactionStrong interaction
F
akπ
≤
-
BCS → BEC crossover
If a>∆
=
-
BertschBertsch ManyMany--Body X challenge, Seattle, 1999Body X challenge, Seattle, 1999
What are the ground state properties of the manyWhat are the ground state properties of the many--body system composed of body system composed of spin ½ fermions interacting via a zerospin ½ fermions interacting via a zero--range, infinite scatteringrange, infinite scattering--length contactlength contactinteraction. interaction.
In 1999 it was not yet clear, either theoretically or experimentally, whether such fermion matter is stable or not.
- systems of bosons are unstable (Efimov effect)- systems of three or more fermion species are unstable (Efimov effect)
• Baker (winner of the MBX challenge) concluded that the system is stable.See also Heiselberg (entry to the same competition)
• Carlson et al (2003) Fixed-Node Green Function Monte Carloand Astrakharchik et al. (2004) FN-DMC provided best the theoretical estimates for the ground state energy of such systems.
• Thomas’ Duke group (2002) demonstrated experimentally that such systemsare (meta)stable.
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Experimental signatures/predictions of the large size pairs/ halo dimer modelversus fermion-boson model:
1) Near the Feshbach resonance the pair is in triplet statein agreement with Grimm’s group experiments (fermion-boson model predicts a singlet state)
2) Particle loss is consistent with large spatial size pairsin agreement with Grimm’s and Jin’s groups experiments(fermion-boson model is consistent with small spatial size pairs)
3) Spatial size of the cloud in agreement with Grimm’s group experiment(fermion-boson model disagrees with experiment)
4) Frequencies and frequency shift of the frequencies of the collective oscillations(if at the Feshbach resonance the system would be made of small size pairs the frequency would be higher and the frequency shift would be much smaller than observed in experiments, see Pitaevskii and Stringari, PRL 81, 4541 (1998), Braaten and Pearson, PRL 82, 255 (1999) )
Theory is now in such a state that it can Theory is now in such a state that it can make verifiable or falsifiable predictions.make verifiable or falsifiable predictions.
30
63 (0) 1128
M
M
n rδω πω
≈
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Ohashi, Levico 2004
Closed (singlet) channel
Open (triplet) channel
Atomicseperation
Energy
Eres
Eth
-
Chang et al. physics/0404115
Astrakharchik et al, cond-mat/0406113
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Even though two atoms can bind, there is no binding among dimers!
Fixed node GFMC results, J. Carlson et al. (2003)
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⎟⎟⎠
⎞⎜⎜⎝
⎛=∆
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛=∆
akmk
e
akmk
e
F
FBCS
F
FGorkov
2exp
28
2exp
22
22
2
223/7
π
π
Fixed node GFMC results, J. Carlson et al. (2003)
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S. Jochim et al. PRL 91, 240402 (2003)
1)2)
Electron spin along B
Electron spin opposite B
see also Petrov et al. cond-mat/0309010Regal et al. PRL 92, 083201 (2004)
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Ohashi and Griffin, Phys. Rev. Lett. 89, 130402 (2002) Bruun, cond-mat/0401497
The fermion-boson model predict that at resonance an atomic Fermi cloud consists predominantly of molecules in the closed channel (singlet), which thus have an almost vanishing magnetic moment.
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ħω=0.568 x 10-12eV, a = -12.63nm (when finite)
40K (Fermi) atoms in a spherical harmonic trap
Effect of interaction, with and without weak and strong pairing correlations with fixed particle number, N = 5200.
0 2 4 6 8 10 12−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
r [µm]
U(r
) or
∆(r
) [1
0−10
eV
]
Mean−field and Pairing field
U∞
Ua
U0∞
U0 a
∆∞(r)
∆a(r)
0 2 4 6 8 10 120
1
2
3
4
5
6
7
8
9
r [µm]n(
r) [1
012
cm−3
]
Particle number density
no interaction mean−field + pairing mean−field + no pairing
Unpublished, fully self-consistent SLDA (Kohn-Sham generalized to pairing)calculation performed by Yongle Yu in July 2003.
3)
-
Ohashi, Levico 2004 Grimm, Levico 2004
The fermion-boson model predict that at resonance the size of the cloud is significantly smaller than the observed one.
3)
-
First soundSpherical
CollisionalregimeCompressionalCompressionalmodemode
Landau’s zero soundNormal Fermi fluidcollisionlessIncompressionalIncompressionalmodemode
Anderson-BogoliubovsoundSpherical
SuperfluidSuperfluidcollisionlesscollisionlessCompressionalCompressionalmodemode
Soundvelocity
Local shapeof Fermi surface
Elongated along propagation direction
s Fv v1
ss=
>
vskω =Sound in infinite Sound in infinite fermionicfermionic mattermatter
vv3F
s ≈
vv3F
s ≈
-
( ) ( )
( )
3
2 2
2
2 2 2 2 20
2
2
3 5 1( ) O5 2 3
0.44, 1, 1
21(0)
F
F F F
F
knm k a k a k a
m x y zU
Kk a
ς ιε ξ
ξ ς ι
ω λ
δω ςω ξ
⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟= − − +⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦≈ ≈ ≈
+ +=
=Perturbation theory result usingGFMC equation of state in a trap
Adiabatic regimeSpherical Fermi surface
Frequency shifts inthese modes mightcarry information about possible atom-halo dimer mixture
-
Hu et al. cond-mat/0404012, a semi-quantitative analysis (gap and chemical potential inaccurate) assuming a polytropic equation of state
For a more careful analysis, using GFMC equation of state in a trap see Bulgac and Bertsch, cond-mat/0404687
Grimm’s group experiment
Thomas’ group experiment
4)
Fermion-boson model would predict here
-
a0
0
0
0
( ) 1
174
7.8 0.6 G 202.10 0.07 G
bg
bg
wa B aB B
a a
wB
⎛ ⎞= −⎜ ⎟−⎝ ⎠
=
= ±= ±
-
Falco and Stoof, Phys. Rev. Lett. 92, 130401 (2004)
Theory declared full victory here!Regal, Greiner and Jin,Phys. Rev. Lett. 92, 040403 (2004)
Zwierlein et al.Phys. Rev. Lett. 92, 120403 (2004)
3
112 ( )m F
nnk a
⎡ ⎤−⎢ ⎥
⎢ ⎥⎣ ⎦
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Fermion superfluidity, more specificaly superflowsuperflow,, has not yet been demonstratedunambiguously experimentally. There is lots of circumstantial evidence and facts in agreement with theoretical models assuming its existence.
Theory is able to make very precise predictions in this regime and the agreementwith experiment can be check quantitatively.
-
radio-frequency spectroscopy
high B-field
1
0
mI=-1
meas. of mol. bind. energy in 40KRegal et al., Nature 424, 47 (2003)
rf spectroscopy of 6Li: Gupta et al., Science 300, 1723 (2003)
rf
~80MHz
breaking moleculescosts energy
→molecular signal
up-shifted
breaking moleculescosts energy
→molecular signal
up-shifted
breaking pairscosts energy
→pair signalup-shifted
breaking pairspairscosts energy
→pairpair signalup-shifted
borrowed from R. Grimm
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rf spectra in crossover regime
rf offsetborrowed from R. Grimm
-
temperature dependence of pairing
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)
T < 0.1 TF
≈ 0.2 TF
≈ 0.4 TF
≈ 0.5 TF
borrowed from R. Grimm
-
J. Thomas’ group at Duke
Consistent with E≈aT5/2
Consistent with E≈bT2
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[ ] [ ]
[ ] [ ]
22
0 0
*k
0
2 23 5/3 *
2
1/3
( ) 2 v ( ) , ( ) 2 v ( )
( ) v ( )u ( )
E ( ) ( ) ( ) ( ) ( )2
( ) 1( ) ( ), ( )( ) ( )
( ) (
k c k c
k c
k kE E E E
kE E
eff
F
n r r r r
r r r
d r r x r n r x r rm m
x rx r r x r
m n r k r a
T U r
τ
ν
τ β ν
γν
µ
< < < <
< <
= = ∇
=
⎧ ⎫= + −∆⎨ ⎬
⎩ ⎭
∆ = − =
+ − ∆
∑ ∑
∑
∫
*
u ( ) u ( ))v ( ) v ( )( ) ( ( ) )
k ki
k k
r rrE
r rr T U r µ⎛ ⎞ ⎛ ⎞⎛ ⎞
=⎜ ⎟ ⎜ ⎟⎜ ⎟∆ − + −⎝ ⎠⎝ ⎠ ⎝ ⎠
Superfluid LDA (SLDA)
number and kinetic densities
anomalous density
Bogoliubov-de Gennes like equations.Correlations are however included by default!
Cutoff and position running coupling constant!
Divergent!
-
Vortex in fermion matter
kn
kn
u ( ) u ( )exp[ ( 1/ 2) ], n - half-integer
v ( ) v ( )exp[ ( 1/ 2) ]
( ) ( )exp( ), ( , , ) [cyllindrical coordinates]
r i n ikzr i n ikz
r i r z
α α
α α
ρ φρ φ
ρ φ ρ φ
+ −⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠
∆ = ∆ =
( ) ( )v v2
Oz - vortex symmetry axis
Ideal vortex, Onsager's quantization (one per Cooper pair)
1V ( , ,0) V2 2 2C
r y x r drm mρ π
= − ⇐ ⋅ =∫
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How can one put in evidence a vortexin a Fermi superfluid?
Hard to see, since density changes are not expected, unlike the case of a Bose superfluid.
However, if the gap is not small, one can expect a noticeable density depletion along the vortex core, and the bigger the gap the bigger the depletion, due to an extremely fast extremely fast vorticalvortical motionmotion.
s c
F F
v Tv 2 TFε
∆< ∝
NB Tc unknown in the strong coupling limit!
-
Number density and pairing field profiles
The depletion along the vortex coreis reminiscent of the correspondingdensity depletion in the case of a vortex in a Bose superfluid, when the density vanishes exactly along the axis for 100% BEC.
Extremely fast quantum vortical motion!
Local vortical speed as fraction of Fermi speed
Fermions with 1/kFermions with 1/kFFa = 0.3, 0.1, 0, a = 0.3, 0.1, 0, --0.1, 0.1, --0.50.5
Bosons with naBosons with na33 = 10= 10--33 and 10and 10--55
From Ketterle’sgroup
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1/a
T
a0shallow 2-body bound state
halo halo dimersdimers
Phases of a two species dilute Fermi system BCS-BEC crossover
weak interactionweak interaction weak interactionsweak interactionsStrong interactionStrong interaction
“Before”- Life
Atoms form rather unstablecouples, spread over largedistances, with many othercouples occupying thesame space
After – Life
Atoms form very strong couples, mostly inert and at rest, halo dimers, widely separated from one another and weakly interacting
LIFE!LIFE!
Atoms form strong couples, living as in an apartment building, in close proximity of one another, and obviouslyanoying each other quite a bit
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Conclusions:Conclusions:
The field of dilute atomic systems is going to be for many years to comeone of the most exciting fields in physics, with lots surprises at every corner.