1. Dynamics of spinor condensates: dipolar interactions and more
Eugene Demler Harvard University
Collaborators:
Ehud Altman, Robert Cherng, Vladimir Gritsev, Mikhail Lukin,
Anatoli Polkovnikov, Ana Maria Rey
Experimental collaborators:
Immanuel Bloch’s group and Dan Stamper-Kurn’s group
Funded by NSF, DARPA, MURI, AFOSR, Harvard-MIT CUA

2. Outline Dipolar interactions in spinor condensates
Larmor precession and dipolar interactions. Roton instabilities. Following experiments of D. Stamper-Kurn
Superexchange interaction in double well systems
Towards quantum magnetism of ultracold atoms.
Collaboration with I. Bloch’s group.
Many-body decoherence and Ramsey interferometry
Luttinger liquids and non-equilibrium dynamics.

3. Dipolar interactions in spinor condensates.
Introduction

7. Possible supersolid phase in 4He
Phase diagram of 4He
A.F. Andreev and I.M. Lifshits (1969):
Melting of vacancies in a crystal due
to strong quantum fluctuations.
Also
G. Chester (1970); A.J. Leggett (1970)
T. Schneider and C.P. Enz (1971).
Formation of the supersolid phase due to
softening of roton excitations

8. Resonant period as a function of T

9. Interlayer coherence in bilayer quantum Hall systems at n=1
Hartree-Fock predicts roton softening
and transition into a state with both
interlayer coherence and stripe order.
Transport experiments suggest first
order transition into a compressible
state.
Eisenstein, Boebinger et al. (1994)
L. Brey and H. Fertig (2000)

10. Roton spectrum in pancake polar condensates
Santos, Shlyapnikov, Lewenstein (2000)
Fischer (2006)
Origin of roton softening
Repulsion at long distances
Attraction at short distances
Stability of the supersolid phase is a subject of debate

11. Magnetic dipolar interactions in spinor condensatesq
Comparison of contact and dipolar interactions.
Typical value a=100aB
For 87Rb m=mB and e= For 52Cr m=6mB and e=0.16
Bose condensation
of 52Cr.
T. Pfau et al. (2005)
Review:
Menotti et al.,
arXiv
take a=100a_0

12. Magnetic dipolar interactions in spinor condensates
Interaction of F=1 atoms
Ferromagnetic Interactions for 87Rb
A. Widera, I. Bloch et al.,
New J. Phys. 8:152 (2006)
a2-a0= aB
Spin-depenent part of the interaction is small.
Dipolar interaction may be important (D. Stamper-Kurn)

13. Spontaneously modulated textures in spinor condensates
Vengalattore et al.
PRL (2008)
Fourier spectrum of the
fragmented condensate

14. This talk: Instabilities of F=1 spinor condensates due to dipolar interactions. New phenomena due to averaging over Larmor precession
Theory: unstable modes in the regime
corresponding to Berkeley experiments
Results of Berkeley experiments
Wide range of instabilities
tuned by quadratic
Zeeman, AC Stark shift,
initial spiral spin winding

15. Instabilities of F=1 spinor condensates due to
dipolar interactions and roton softening
Earlier theoretical work on dipolar interactions in spinor condensates:
Meystre et al. (2002), Ueda et. al. (2006), Lamacraft (2007).
New phenomena: interplay of finite transverse size and dipolar interaction
in the presence of fast Larmor precession

16. Dipolar interactions after averaging over Larmor precession

17. Energy scales Magnetic Field B F S-wave Scattering Dipolar Interaction
Larmor Precession (100 kHz)
Quadratic Zeeman (0-20 Hz) B F
S-wave Scattering
Spin independent (g0n = kHz)
Spin dependent (gsn = 10 Hz)
Dipolar Interaction
Anisotropic (gdn=10 Hz)
Long-ranged
Reduced Dimensionality
Quasi-2D geometry

18. Dipolar interactions Static interaction
parallel to is preferred
“Head to tail” component dominates z
Averaging over Larmor precession
perpendicular to is preferred. “Head to tail”
component is averaged with the “side by side”

19. Instabilities: qualitative picture

20. Stability of systems with static dipolar interactions
Ferromagnetic configuration is robust against small
perturbations. Any rotation of the spins conflicts with
the “head to tail” arrangement
Large fluctuation required to reach a lower energy configuration

21. Dipolar interaction averaged after precession
“Head to tail” order of the transverse spin components is violated
by precession. Only need to check whether spins are parallel
XY components of the
spins can lower the energy
using modulation along z. X X
Z components of the
spins can lower the energy
using modulation along x
Strong instabilities of systems with dipolar interactions
after averaging over precession

22. Instabilities: technical details

23. From Spinless to Spinor Condensates
Charge mode:
n is density and h is the overall phase
I would like to begin this talk by discussing what distinguishes a spinor condensate from a spinless one. Spinless particles have a rather simple structure characterized by two conjugate variables: phase and particle number. The hallmark of Bose-Einstein condensation is development of coherence in the phase variable. In turn, the experimental signature of this condensate is the macroscopic occupation of the ground state.
What makes spinor condensates more interesting is that they have a richer and more complex internal structure. In addition to developing phase coherence, spinor condensates still have additional degrees of freedom describing the orientation of magnetization for example. Experimentally, this means that this rather featureless condensate peak describing spinless condensates can develop complex patterns and structure in spinor condensates.
However understanding and manipulating this complex internal structure poses a number of difficult challenges for experiment (which have been the focus of the Stamper-Kurn group at Berkeley) and theory (some aspects of which I will be discussing today).
Spin mode:
f determines spin orientation in the XY plane
c determines longitudinal magnetization (Z-component)

24. Hamiltonian Magnetic Field Quasi-2D Dipolar Interaction
S-wave Scattering
The geometry I want to consider is the following:
n is the normal to the 2D plane of the condensate and B points along the magnetic field at an angle alpha wrt n.
k is the momentum vector in the plane and kappa is the wavevector for the spiral which are at angles theta and beta wrt to e1, which is the projection of the magnetic field vector in the plane. The thickness of the plane is dn.
In order to study these competing effects, we consider the following Hamiltonian. Here psi is the spin 1 field describing the condensate and there is the usual kinetic energy and chemical potential terms.
Harmonic confinement along n gives rise to the 2D geometry. The magnetic field enters through the linear and quadratic Zeeman terms.
There are also interaction terms coupling the total density and spin densities with strength guv. S-wave scattering terms are contact interactions while the dipolar interaction is long ranged.

25. Precessional and Quasi-2D Averaging
Rotating Frame
Gaussian Profile
It turns out the bare 3D dipolar interaction is strongly modified due to the precessional dynamics of the spins and the quasi-2D nature of the system.
To take into account the precessional dynamics, we use the fact that the time scale for precession 115 Hz is much faster than the time scale for dipolar interactions 10 Hz. By transforming to a rotating frame comoving with the spins, we can then obtain the effective dipolar interaction by time averaging.
To take into account the reduced dimensionality, we assume the spins are frozen along the n direction and take a gaussian trial wavefunction along this direction.
The final, quasi-2d time averaged dipolar interaction relevant for the spin dynamics is heavily modified by these two effects.
Quasi-2D Time Averaged Dipolar Interaction

26. Quasi-2D Time Averaged Dipolar Interaction

27. Collective Modes Equations of Motion δφ δfB δη δn Ψ0 Spin Mode
Mean Field
Equations of Motion
Collective Fluctuations
(Spin, Charge) δφ
δfB δη δn Ψ0
Spin Mode
δfB – longitudinal magnetization
δφ – transverse orientation
Charge Mode
δn – 2D density
δη – global phase

28. Instabilities of collective modes
Q measures
the strength
of quadratic
Zeeman effect

29. Instabilities of collective modes

30. Berkeley Experiments: checkerboard phase
To original goal of this talk was to try to explain some of the observations of the Berkeley group on spinor condensates. The left hand side shows the measured modulated spin texture in real space and momentum space with dipolar interactions at full strength and strength reduced by sequences of random pi/2 pulses. In real space there is a distinct checkerboard pattern with a roughly length scale of 10 microns in either direction. In momentum space, the spin texture has a distinctive shape with two lobes connected to the origin and two lobes disconnected from the origin.
The right hand side shows the magnitude of the imaginary part of the excitation energies in momentum space. There are two lobes connected to the origin along the direction perpendicular to the axis of the field while the two lobes along the field are disconnected from the origin. In addition the two lobes are of roughly equal magnitude and are peaked near a length scale of 10 microns.
M. Vengalattore, et. al, arXiv:

31. Dipolar interaction averaged after precession
XY components of the
spins can lower the energy
using modulation along z. X X
Z components of the
spins can lower the energy
using modulation along x

32. Instabilities of collective modes

33. Instabilities of collective modes. Spiral configurations
Spiral wavelength
Spiral spin winding
introduces a separate
branch of unstable
modes

34. Superexchange interaction in experiments with double wells
Refs:
Theory: A.M. Rey et al., Phys. Rev. Lett. 99: (2007)
Experiment: S. Trotzky et al., Science 319:295 (2008)

35. Two component Bose mixture in optical lattice
Example: Mandel et al., Nature 425:937 (2003) t t
Two component Bose Hubbard model

36. Quantum magnetism of bosons in optical lattices
Duan, Demler, Lukin, PRL 91:94514 (2003)
Altman et al., NJP 5:113 (2003)
Ferromagnetic
Antiferromagnetic

37. Observation of superexchange in a double well potential
Theory: A.M. Rey et al., PRL (2007) J J
Use magnetic field gradient to prepare a state
Observe oscillations between and states
Experiment:
Trotzky et al.,
Science (2008)

38. Preparation and detection of Mott states
of atoms in a double well potential

39. Comparison to the Hubbard model
Experiments: I. Bloch et al.

40. Beyond the basic Hubbard model
Basic Hubbard model includes
only local interaction
Extended Hubbard model
takes into account non-local
interaction

41. Beyond the basic Hubbard model

42. Observation of superexchange in a double well potential.
Reversing the sign of exchange interactions

43. Many-body decoherence and Ramsey interferometry
Collaboration with A. Widera, S. Trotzky, P. Cheinet,
S. Fölling, F. Gerbier, I. Bloch, V. Gritsev, M. Lukin
Phys. Rev. Lett. (2008)

44. Ramsey interference Working with N atoms improves the precision by .1
Working with N atoms improves
the precision by
Need spin squeezed states to
improve frequency spectroscopy

45. Squeezed spin states for spectroscopy
Motivation: improved spectroscopy, e.g. Wineland et. al. PRA 50:67 (1994)
Generation of spin squeezing using interactions.
Two component BEC. Single mode approximation
Kitagawa, Ueda, PRA 47:5138 (1993)
In the single mode approximation we can neglect kinetic energy terms

46. Interaction induced collapse of Ramsey fringes
Ramsey fringe visibility
- volume of the system
time
Experiments in 1d tubes:
A. Widera, I. Bloch et al.

47. Spin echo. Time reversal experiments
Single mode approximation
The Hamiltonian can be reversed by changing a12
Predicts perfect spin echo

48. Spin echo. Time reversal experiments
Expts: A. Widera, I. Bloch et al.
Experiments done in array of tubes.
Strong fluctuations in 1d systems.
Single mode approximation does not apply.
Need to analyze the full model
No revival?

49. Interaction induced collapse of Ramsey fringes. Multimode analysis
Low energy effective theory: Luttinger liquid approach
Luttinger model
Changing the sign of the interaction reverses the interaction part
of the Hamiltonian but not the kinetic energy
Time dependent harmonic oscillators
can be analyzed exactly

50. Time-dependent harmonic oscillator
See e.g. Lewis, Riesengeld (1969)
Malkin, Man’ko (1970)
Explicit quantum mechanical wavefunction can be found
From the solution of classical problem
We solve this problem for each
momentum component

51. Fundamental limit on Ramsey interferometry
Interaction induced collapse of Ramsey fringes in one dimensional systems
Only q=0 mode shows complete spin echo
Finite q modes continue decay
The net visibility is a result of competition
between q=0 and other modes
Conceptually similar to experiments with
dynamics of split condensates.
T. Schumm’s talk
Fundamental limit on Ramsey interferometry

52. Summary Dipolar interactions in spinor condensates
Larmor precession and dipolar interactions. Roton instabilities. Following experiments of D. Stamper-Kurn
Superexchange interaction in double well systems
Towards quantum magnetism of ultracold atoms.
Collaboration with I. Bloch’s group.
Many-body decoherence and Ramsey interferometry
Luttinger liquids and non-equilibrium dynamics.