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David Pekker (Harvard/Caltech) Mehrtash Babadi (Harvard) Lode Pollet (Harvard/ETHZ) Rajdeep Sensarma (Harvard/JQI Maryland) Eric Vernier (Harvard/ENS) Nikolaj Zinner (Harvard/ Niels Bohr Institute) Antoine Georges ( Ecole Polytechnique ) Martin Zwierlein (MIT) - PowerPoint PPT Presentation
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Pairing and magnetism nearFeshbach resonance
$$ NSF, AFOSR MURI, DARPA, AROHarvard-MIT
David Pekker (Harvard/Caltech)Mehrtash Babadi (Harvard) Lode Pollet (Harvard/ETHZ)Rajdeep Sensarma (Harvard/JQI Maryland)Eric Vernier (Harvard/ENS) Nikolaj Zinner (Harvard/Niels Bohr Institute) Antoine Georges (Ecole Polytechnique)Martin Zwierlein (MIT)Eugene Demler (Harvard)
Thanks to W. Ketterle and other members of the MIT group
Outline
Competition of pairing and magnetism near Feshbach resonanceMotivated by experiments by Jo et al., Science (2009)
Shiba states in paired fermionic superfluidsMotivated by experiments in M. Zwierlein’s lab
Competition of pairing and magnetism near Feshbach resonance
Pekker, Babadi, Sensarma, Pollet, Zinner, Zwierlein, Demler arXiv:1005.2366
Stoner model of ferromagnetismSpontaneous spin polarizationdecreases interaction energybut increases kinetic energy ofelectrons
Mean-field criterion
U N(0) = 1
U – interaction strengthN(0) – density of states at Fermi level
Theoretical proposals for observing Stoner instability with cold gases: Salasnich et. al. (2000); Sogo, Yabu (2002); Duine, MacDonald (2005); Conduit, Simons (2009); LeBlanck et al. (2009); …
Recent work on hard sphere potentials: Pilati et al. (2010); Chang et al. (2010)
Experiments weredone dynamically.What are implicationsof dynamics?Why spin domains could not be observed?
Also earlier work by C. Salomon, D. Jin, others
Is it sufficient to consider effective model with repulsive interactions when analyzing experiments?
Feshbach physics beyond effective repulsive interaction
Feshbach resonanceInteractions between atoms are intrinsically attractiveEffective repulsion appears due to low energy bound states
Example:
scattering lengthV(x)
V0 tunable by the magnetic fieldCan tune through bound state
Feshbach resonanceTwo particle bound state formed in vacuum
BCS instabilityStoner instability
Molecule formationand condensation
This talk: Prepare Fermi state of weakly interacting atoms. Quench to the BEC side of Feshbach resonance. System unstable to both molecule formation and Stoner ferromagnetism. Which instability dominates ?
Pair formation
Two-particle scattering in vacuum
k -kp
-p
Lippman-Schwinger equation
For positive scattering length bound state atappears as a pole in the T-matrix
kk
-k
k
-k -p’-p -p
pp k pp’
-p
On-shell T-matrix. Universal low energy expression
CooperonTwo particle scattering in the presence of a Fermi sea
kp
-k
-p Cooperon equation
kk
-k
k
-k -p’-p -p
pp k pp’
-p
Cooper channel response functionLinear response theory
Induced pairing field
Response function
Poles of the Cooper channel response function are given by
Cooper channel response function
Linear response theory
Time dependent dynamics
When the mode frequency has imaginary part,the system is unstable to formation of paired state
Poles of the response functiondescribe collective modes
Pairing instability regularized
BCS side
Instability rate coincides with the equilibrium gap(Abrikosov, Gorkov, Dzyaloshinski)
Instability to pairing even on the BEC side
Related work: Lamacraft, Marchetti, 2008
Pairing instabilityIntuition: two body collisions do not lead to molecule formation on the BEC side of Feshbach resonance.Energy and momentum conservation laws can notbe satisfied.
This argument applies in vacuum. Fermi sea preventsformation of real Feshbach molecules by Pauli blocking.
Molecule Fermi sea
Pairing instability
From wide to narrow resonances
Pairing instabilityat different temperatures
Three body recombination as in Shlyapnikov et al., 1996; Petrov, 2003; Esry 2005
Magnetic instability
Stoner instability. Naïve theory
Spin response functionRelates induced spin polarizationto external Zeeman field
Spin collective modes are given by the poles of response function
Imaginary frequencies correspond to magnetic instability
Quench dynamics across Stoner instability
For U>Uc unstable collective modes
Magnetic Stoner instability
Unphysical divergence at unitarity
Stoner instability
Divergence in the scattering amplitude arises from bound state formation. Bound state is strongly affected by the Fermi sea.
Stoner instability is determined by two particlescattering amplitude
= + + + …= + + + …
Stoner instabilityRPA spin susceptibility
Interaction = Cooperon
Stoner instability
Pairing dominates over magnetic instability
If ferromagnetic domains form, they form at large q
Relation to experiments
Pairing instability vs experiments
Summary of part I
Competition of pairing and ferromagnetism near Feshbach resonance
Dynamics of competing orders is important for understanding experiments
Simple model with repulsive interactionsmay not be sufficient
Strong suppression of Stoner instability by Feshbach resonance physics + Pauli blocking
Alternative interpretation of experiments based on pair formation
Shiba states in fermionic superfluids
E. Vernier, D. Pekker, M. Zwierlein, E. Demler
Simplest example of interplay of magnetism and pairingMotivated by experiments in M. Zwierlein’s lab
Shiba states in superconductors
Classical spin impurity in SCquasiparitcle can make a localized in-gap state byaligning its spin with impurity
N(w)
w-D D
Local density of states
Gd on the surface of NbSTM measurements of LDOSYazdani et al., Science 1997
Ultracold atoms: Shiba states in paired Fermi superfluids
a aScattering lengths needto be computed includingimpurity confinement
Example
Shiba states may exist even when there are no Feshbach bound states
RF spectroscopy as a probe of Shiba states
Other possible probes: modulation type experiments on impurity confinement,interaction with fermions
Interesting open questions:• From Shiba states to midgap band• Gapless superconductivity• Suppression of pairing• Simplest example of interplay of magnetism and pairing
Why study Shiba states in paired Fermi superfluids?
Shiba states as a local probe(M. Zwierlein)• Local probe of pairing for FFLO, pseudogap, etc.• Probe of unconventional pairing
Zn impuritiesIn high HTSCDavis et al., 2003
SummaryCompetition of pairing and ferromagnetism near Feshbach resonance
Dynamics of competing orders is important for understanding experiments. Simple model with repulsive interactionsmay not be sufficient. Strong suppression of Stoner instability by Feshbach resonance physics + Pauli blocking. Alternative interpretation of experiments based on pair formation
State dependent interaction with localized magneticimpurities should allow the study of Shiba states.
Simplest example of the interplay of pairing and magnetismLocal probe of pairing
Harvard-MIT