Magnetism in ultracold Fermi gasesand

New physics with ultracold ions: many-body systems with non-equilibrium noise

$$ NSF, AFOSR MURI, DARPAHarvard-MIT

David Pekker (Harvard),Rajdeep Sensarma (Harvard/JQI Maryland), Mehrtash Babadi (Harvard), Nikolaj Zinner (Harvard/Niels Bohr Institute), Antoine Georges (Ecole Polytechnique),Ehud Altman (Weizmann),Emanuele Dalla Torre (Weizmann),Thierry Giamarchi (Geneva),Eugene Demler (Harvard)

Outline

• Stoner instability in ultracold atomsMotivated by experiments of G.-B. Jo et al., Science (2009) Introduction to Stoner instability. Possible observation of Stoner instability in MIT experiments. Domain formation.Competition of molecule formation and Stoner instabilityRef: M. Babadi et al., arXiv:0909.3483 and unpublished

• New physics with ultracold ionsQuantum many-body systems in the presence of non-equilibrium noiseRef: Dalla Torre et al., arXiv:0908.0868

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 instabilitywith ultracold Fermi gases:Salasnich et. al. (2000); Sogo, Yabu (2002); Duine, MacDonald (2005); Conduit, Simons (2009); LeBlanck et al. (2009); …

Kanamori’s counter-argument: renormalization of U

then

Magnetic domainscould not be resolved.Why? T.L. Ho (2009)

Stoner InstabilityNew feature of cold atoms systems: non-adiabatic crossing of Uc

Screening of U (Kanamori) occurs on times 1/EF

Two timescales in the system: screening and magnetic domain formation

Magnetic domain formation takes place on much longer time scales: critical slowing down

Quench dynamics across Stoner instability

Find collective modes

Unstable modes determine characteristic lengthscale of magnetic domains

For U>Uc unstable collective modes

For MIT experiments domain

sizes of the order of a few F

Quench dynamics in D=3

Dynamics of magnetic domain formation near Stoner transition

Moving across transition at a finite rate

0 uu*

domains coarsen

slowgrowth

domainsfreeze

Growth rate of magnetic domains

Domain size

Domains freeze when

Domain size at “freezing” point

M. Babadi et al. (2009)

Is it sufficient to consider effective model with repulsive interactions when analyzing experiments?

Feshbach physics beyond effective repulsive interaction

Feshbach resonanceReview: Duine and Stoof, 2004 Chin et al., 2009

Two 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 ?

Many-body instabilitiesImaginary frequencies of collective modes

Magnetic Stoner instability

Pairing instability

Change from bare interaction to the scattering length

Instability to pairing even on the BEC side

Pairing instability

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 instabilityTime dependent variational wavefunction

Time dependence of uk(t) and vk(t) due to BCS(t)

For small BCS(t):

Pairing instabilityFrom wide to narrow resonances

Stoner vs pairing

Does Stoner instability really exceed molecule formation rate?

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 instabilitySpin susceptibility

Growth rate of pairing instability

Growth rate of magneticStoner instability

RPA with barescattering length

RPA with Cooperon

Changing from scattering length to Cooperon gives strong suppressionof the Stoner instability

Stoner instability suppressedwhen using a Cooperon. Strong suppression due to Pauli blocking

Stoner instability

?

Stoner vs pairing

G.B. Jo et al., Science (2009)

Stoner vs pairing

Increase in the kinetic energy:consistent with pairing.In the BCS state kineticenergy goes up and the interaction energy goes down

Conclusions for part I

Competition of pairing and Stoner instabilities

New features due to dynamical character of experiments

Simple model with contact repulsive interactionsmay not be sufficient to understand experiments

Strong suppression of Stoner instability by Fechbach resonance physics + Pauli blocking

Possible ways to recover Stoner instability:many-body correlations, e.g. effective mass renormalization.Interesting questions beyond linear instability analysis.

QUANTUM MANY-BODY SYSTEMS

IN THE PRESENSE OF

NONEQUILIBRIUM NOISE

NEW PHYSICS WITH ULTRACOLD IONS

Question:

What happens to low dimensional quantum systems when they are subjected to external non-equilibrium noise?

Ultracold polar moleculesTrapped ions

E

One dimensional Luttinger state can evolve into a new critical state. This new state has intriguing interplay of quantum critical and external noise driven fluctuations

A brief review:Universal long-wavelength theory of 1D systems

Displacement field:

Long wavelength density fluctuations (phonons):

Haldane (81)

Weak interactions: K >>1Hard core bosons: K = 1Strong long range interactions: K < 1

1D review cont’d: Wigner crystal correlations

No crystalline order !

Scale invariant critical state (Luttinger liquid)

Wigner crystal order parameter:

1D review cont’d:Effect of a weak commensurate lattice potential

How does the lattice potential change under rescaling ?

Quantum phase transition: K<2 – Pinning by the lattice (“Mott insulator”) K>2 – Critical phase (Luttinger liquid)

New systems more prone to external disturbance

+-

+-

+-

+-

+-

+-

+-

+-

+- E+

-

Ultracold polar molecules

Trapped ions R. Blatt’s talk at this conference

(from NIST group )

Linear ion trap

Linear coupling to the noise:

Measured noise spectrum in ion trap

f

From dependence of heating rate on trap frequency.

- Direct evidence that noise spectrum is 1/f

- Short range spatial correlations (~ distance from electrodes)

Monroe group, PRL (06), Chuang group, PRL (08)

Ultra cold polar molecules

+-

+-

+-

+-

+-

+-

+-

+-

+- E

Polarizing electric field:

+-

System is subject to electric field noise from the electrodes !

Molecule polarizability

Long wavelength description of noisy low D systems

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

Effective coupling to external noise

Long wavelength component of noise

Component of noise at wavelengths near the inter-particle spacing

The “backscattering” can be neglected if the distance to the noisy electrode is much larger than the inter-particle spacing.

>>

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

Effective harmonic theory of the noisy system

+- +- +- +-+-+- +-+- +-+-

Dissipative coupling to bath needed to ensure steady state (removes the energy pumped in by the external noise)

Implementation of bath: continuous cooling

(Quantum) Langevin dynamics:

Thermal bath

External noise

Wigner crystal correlations

1/f noise is a marginal perturbation ! Critical steady state

Case of local 1/f noise:

- Decay of crystal correlations remains power-law.

- Decay exponent tuned by the 1/f noise power.

Effect of a weak commensurate lattice potential

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

How does the lattice change under a scale transformation?

Without lattice: Scale invariant steady state.

Phase transition tuned by noise power

(Supported also by a full RG analysis within the Keldysh formalism)

Kc

F0 /

Localized

Critical state2

1D-2D transition of coupled tubes

+- +- +- +-+-+- +-+- +-+-+- +- +- +-+-+- +-+- +-+-

+- +- +- +-+-+- +-+- +-+-+- +- +- +-+-+- +-+- +-+-

+- +- +- +-+-+- +-+- +-+-

Scaling of the inter-tube hopping:Kc

F0 /1/4

2D superfluid

1D critical

Global phase diagram

Kc

F0 /

2D crystal

Critical state1

Kc

F0 /1/4

2D superfluid

1D critical

Inter-tube interactionsInter-tube tunneling

Both perturbations

2

Kc

F0 /

2D superfluid

2D crystal

1D critical

Conclusions for part II

New perspectives on many-body physics fromchains of ions and polar molecules

Effects of external noise on quantum critical states- new critical state- new phases and phase transitions tuned by noise

Summary

• Stoner instability in ultracold atomsMotivated by experiments of G.-B. Jo et al., Science (2009) Introduction to Stoner instability. Possible observation of Stoner instability in MIT experiments. Domain formation.Competition of molecule formation and Stoner instabilityRef: M. Babadi et al., arXiv:0909.3483 and unpublished

• New physics with ultracold ionsQuantum many-body systems in the presence of non-equilibrium noiseRef: Dalla Torre et al., arXiv:0908.0868

Harvard-MIT