$$ NSF, AFOSR MURI, DARPA

1. Magnetism in ultracold Fermi gasesandNew physics with ultracold ions: many-body systems with non-equilibrium noise 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) Harvard-MIT $$ NSF, AFOSR MURI, DARPA

2. Outline • Stoner instability in ultracold atoms Motivated 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 Competition of molecule formation and Stoner instability Ref: M. Babadi et al., arXiv:0909.3483 and unpublished • New physics with ultracold ions Quantum many-body systems in the presence of non-equilibrium noise Ref: Dalla Torre et al., arXiv:0908.0868

3. U N(0) = 1 Stoner model of ferromagnetism Spontaneous spin polarization decreases interaction energy but increases kinetic energy of electrons Mean-field criterion U – interaction strength N(0) – density of states at Fermi level Kanamori’s counter-argument: renormalization of U then Theoretical proposals for observing Stoner instability with ultracold Fermi gases: Salasnich et. al. (2000); Sogo, Yabu (2002); Duine, MacDonald (2005); Conduit, Simons (2009); LeBlanck et al. (2009); …

4. Magnetic domains could not be resolved. Why? T.L. Ho (2009)

5. Stoner Instability New feature of cold atoms systems: non-adiabatic crossing of Uc Two timescales in the system: screening and magnetic domain formation Screening of U (Kanamori) occurs on times 1/EF Magnetic domain formation takes place on much longer time scales: critical slowing down

6. Quench dynamics across Stoner instability For U>Uc unstable collective modes Unstable modes determine characteristic lengthscale of magnetic domains Find collective modes

7. slow growth domains freeze domains coarsen u* 0 u Dynamics of magnetic domain formation near Stoner transition M. Babadi et al. (2009) Quench dynamics in D=3 Moving across transition at a finite rate Domains freeze when Growth rate of magnetic domains Domain size at “freezing” point Domain size For MIT experiments domain sizes of the order of a fewlF

8. Is it sufficient to consider effective model with repulsive interactions when analyzing experiments? Feshbach physics beyond effective repulsive interaction

9. Feshbach resonance Two particle bound state formed in vacuum Review: Duine and Stoof, 2004 Chin et al., 2009 Stoner instability BCS instability Molecule formation and 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 ?

10. Many-body instabilities Imaginary frequencies of collective modes Magnetic Stoner instability Pairing instability

11. Pairing instability Change from bare interaction to the scattering length Instability to pairing even on the BEC side

12. Pairing instability Intuition: two body collisions do not lead to molecule formation on the BEC side of Feshbach resonance. Energy and momentum conservation laws can not be satisfied. This argument applies in vacuum. Fermi sea prevents formation of real Feshbach molecules by Pauli blocking. Molecule Fermi sea

13. Pairing instability Time dependent variational wavefunction Time dependence of uk(t) and vk(t) due to DBCS(t) For small DBCS(t):

14. Pairing instability From wide to narrow resonances

15. Stoner vs pairing Does Stoner instability really exceed molecule formation rate?

16. = Stoner instability Stoner instability is determined by two particle scattering amplitude Divergence in the scattering amplitude arises from bound state formation. Bound state is strongly affected by the Fermi sea.

17. Stoner instability Spin susceptibility

18. RPA with bare scattering length RPA with Cooperon Stoner instability Growth rate of magnetic Stoner instability Growth rate of pairing instability Stoner instability suppressed when using a Cooperon. Strong suppression due to Pauli blocking Changing from scattering length to Cooperon gives strong suppression of the Stoner instability ?

19. Stoner vs pairing G.B. Jo et al., Science (2009)

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

21. Conclusions for part I Competition of pairing and Stoner instabilities New features due to dynamical character of experiments Simple model with contact repulsive interactions may 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.

22. NEW PHYSICS WITH ULTRACOLD IONS QUANTUM MANY-BODY SYSTEMS IN THE PRESENSE OF NONEQUILIBRIUM NOISE

23. Question: Trapped ions Ultracold polar molecules E What happens to low dimensional quantum systems when they are subjected to external non-equilibrium noise? 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

24. A brief review:Universal long-wavelength theory of 1D systems Haldane (81) Displacement field: Long wavelength density fluctuations (phonons): Weak interactions: K >>1Hard core bosons: K = 1Strong long range interactions: K < 1

25. 1D review cont’d: Wigner crystal correlations Wigner crystal order parameter: No crystalline order ! Scale invariant critical state (Luttinger liquid)

26. 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)

27. + + + + + + + + + + - - - - - - - - - - New systems more prone to external disturbance Ultracold polar molecules E Trapped ions R. Blatt’s talk at this conference (from NIST group )

28. Linear ion trap Linear coupling to the noise:

29. Measured noise spectrum in ion trap From dependence of heating rate on trap frequency. f Monroe group, PRL (06), Chuang group, PRL (08) • Direct evidence that noise spectrum is 1/f • Short range spatial correlations (~ distance from electrodes)

30. + + + + + + + + + + - - - - - - - - - - Ultra cold polar molecules E Polarizing electric field: Molecule polarizability System is subject to electric field noise from the electrodes !

31. + + + + + + + + + + - - - - - - - - - - Long wavelength description of noisy low D systems

32. + + + + + + + + + + - - - - - - - - - - Component of noise at wavelengths near the inter-particle spacing Long wavelength component of noise Effective coupling to external noise >> The “backscattering” z can be neglected if the distance to the noisy electrode is much larger than the inter-particle spacing.

33. + + + + + + + + + + - - - - - - - - - - Effective harmonic theory of the noisy system (Quantum) Langevin dynamics: Thermal bath External noise Dissipative coupling to bath needed to ensure steady state (removes the energy pumped in by the external noise) Implementation of bath: continuous cooling

34. Wigner crystal correlations Case of local 1/f noise: • Decay of crystal correlations remains power-law. • Decay exponent tuned by the 1/f noise power. 1/f noise is a marginal perturbation ! Critical steady state

35. + + + + + + + + + + - - - - - - - - - - Kc Critical state 2 Localized F0 /h Effect of a weak commensurate lattice potential Without lattice: Scale invariant steady state. How does the lattice change under a scale transformation? Phase transition tuned by noise power (Supported also by a full RG analysis within the Keldysh formalism)

36. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Kc 2D superfluid 1D critical 1/4 F0 /h 1D-2D transition of coupled tubes Scaling of the inter-tube hopping:

37. Kc 2D superfluid 2 1D critical 2D crystal F0 /h Global phase diagram Inter-tube tunneling Inter-tube interactions Kc Kc Critical state 1 2D superfluid 1D critical 2D crystal 1/4 F0 /h F0 /h Both perturbations

38. Conclusions for part II New perspectives on many-body physics from chains of ions and polar molecules • Effects of external noise on quantum critical states • new critical state • new phases and phase transitions tuned by noise

39. Harvard-MIT Summary • Stoner instability in ultracold atoms Motivated 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 instability Ref: M. Babadi et al., arXiv:0909.3483 and unpublished • New physics with ultracold ions Quantum many-body systems in the presence of non-equilibrium noise Ref: Dalla Torre et al., arXiv:0908.0868