Quantum magnetism of ultracold atoms
Eugene Demler (Harvard)
Theory collaborators:
Robert Cherng, Adilet Imambekov,
Vladimir Gritsev, Takuya Kitagawa, Mikhail Lukin,
Susanne Pielawa, Joerg Schmiedmayer
Experiments:
Bloch et al., Schmiedmayer et al., Stamper-Kurn et al.
Harvard-MIT
$$ NSF, AFOSR MURI, DARPA
Magnetism in condensed matter systems
Ferromagnetism
in itinerant systems
Stoner instability.
Double exchange
Antiferromagnetism
Frustrated magnetic
systems
?
Quantum magnetism of ultracold atoms
Familiar models, New questions
Spin dynamics in 1d systems
Luttinger model and nonequilibrium dynamics.
New characterization: full distribution functions
Ferromagnetic F=1 spinor condensates
Quantum Hall ferromagnets in disguise.
Skyrmion crystal phases
Spin dynamics in 1d systems:
Ramsey interference experiments
arXiv:0912.4643
T. Kitagawa, S. Pielawa, A. Imambekov, J.Schmiedmayer,
V. Gritsev, E. Demler
Ramsey interference
1
0
Atomic clocks and Ramsey interference:
Working with N atoms improves
the precision by
.
t
Ramsey Interference with BEC
Single mode
approximation
Amplitude of
Ramsey fringes
Interactions should
lead to collapse and
revival of Ramsey fringes
time
Ramsey Interference with 1d BEC
1d systems in optical
lattices
Ramsey interference in 1d tubes:
A.Widera et al.,
B. PRL 100:140401 (2008)
1d systems in microchips
Two component BEC
in microchip
Treutlein et.al, PRL 2004,
also Schmiedmayer, Van Druten
Ramsey interference in 1d condensates
A. Widera, et al, PRL 2008
Collapse but no revivals
Ramsey interference in 1d condensates
Spin echo experiments
A. Widera, et al, PRL 2008
Expect full revival of fringes
Only partial revival
after spin echo!
Spin echo experiments in 1d tubes
Single mode approximation does not apply.
Need to analyze the full model
Ramsey interference in 1d
Time evolution
Luttinger liquid provides good agreement with experiments.
A. Widera et al., PRL 2008. Theory: V. Gritsev
Technical noise could also
lead to the absence of echo
Need “smoking gun” signatures
of many-body decoherece
Distribution
Probing spin dynamics using
distribution functions
Distribution contains information
about all the moments
→ It can probe the system
Hamiltonian
Joint distribution function can
also be obtained!
Distribution function of fringe contrast
as a probe of many-body dynamics
Short segments
Radius =
Amplitude
Angle =
Phase
Long segments
Distribution function of fringe contrast
as a probe of many-body dynamics
Splitting one
condensate
into two.
Preliminary results
by J. Schmiedmayer’s group
Short segments
Long segments
l =20 mm
l =110 mm
Expt
Theory
Data: Schmiedmayer et al.,
unpublished
Skyrmion crystals in
ferromagnetic F=1
spinor condensates
R. Cherng, Ph.D. Thesis
Spinor condensates. F=1
Three component order parameter: mF=-1,0,+1
Contact interaction depends on relative spin orientation
When g2>0 interaction is antiferromagnetic. Example 23Na
Favors condensation into mF=0 state (or its rotation)
When g2<0 interaction is ferromagnetic. Example 87Rb
Favors condensation into mF=1 state (or its rotation)
Spin textures in ferromagnetic Rb condensates
Imbalanced
(non-equilibrium)
Initial populations
mF=-1
mF=0
mF=+1
Equal
(equilibrium)
Initial populations
Vengalattore et al., PRL (2008)
mF=-1
mF=0
mF=+1
Spin textures: checkerboard pattern
Equal populations
Vengalattore et al.,
PRL (2008)
Transverse
Spectrum in
Momentum
Space
Longitudinal
Magnetic dipolar interactions in spinor condensates
q
Comparison of contact and dipolar interactions.
Typical value a=100aB
For 87Rb m=mB and e=0.007
Interaction of F=1 atoms
Spin dependent interactions in 87Rb are small
a2-a0= -1.07 aB
A. Widera, I. Bloch et al.,
New J. Phys. 8:152 (2006)
Energy scales
High energy scales
Spin independent
S-wave scattering
(gsn=215 Hz)
Precession
(115 kHz)
Quasi-2D geometry
d  spin
B
F
Low energy scales
Spin dependent
S-wave scattering
(gsn=8 Hz)
Quadratic Zeeman
(1 Hz)


E ~  B F 




2
Dipolar interaction
(gdn = 1 Hz)
~r
3
Dipolar interactions
Fast Larmor precession strongly modifies effective dipolar interactions
Fourier components of effective interaction (in-plane field)

ˆ
B F  Bˆ

ˆ
B F // Bˆ
Instabilities of ferromagnetic F=1 Rb
condensate due to dipolar interactions
Theory: unstable modes in the regime
corresponding to Berkeley experiments.
Cherng, Demler, PRL (2009)
Experiments.
Vengalattore et al. PRL (2008)
From microscopic Hamiltonian to
effective low energy theory
Dipolar and
quadratic Zeeman
A. Lamacraft, PRA (2008)
Fixed density
Maximally polarized
Low energy manifold
Magnetization
Condensate phase
Superfluid velocity
Mermin-Ho relation
Magnetization
Superfluid velocity
Divergence flow
Mermin-Ho
Skyrmion density
Skyrmion density
Superfluid velocity
Non-linear sigma model
Low-energy Lagrangian
Superfluid flow related
to skyrmion density
Superfluid kinetic energy
Spin Stiffness
Skyrmion interaction (Log)
Magnetic crystals in spinor
condensates
Effective Hamiltonian
Spin dependent interactions
Skyrmion interaction
Interaction strengths
Minimal energy spin texture
Find all spin groups consistent with
constraints
Intrinsic constraints
a) Zero net skyrmion
charge
b) Maximally polarized
magnetization
c) Explicit symmetry
breaking via external
field
D2 point group
SG = p2mm, p2mg, p2gg
Phenomenological constraints
d) Rectangular lattice
e) No easy-axis or easy plane
f) Zero net magnetization
Minimal energy spin texture
Understanding spin textures
Skyrmions in ferromagnets
Ordinary ferromagnets. Equations of motion
Single skyrmion
Spin space Real space
solution
Radial
coordinate
Azimuthal
coordinate
Spinor ferromagnets. Equations of motion
~
Exact solutions for spinor condensates
Spin space
Stereographic
coordinates
Real space
nˆ
nˆ
Separation of variables
static solution ansatz
Holomorphic
coordinates
Single skyrmion solutions
Ordinary ferromagnet
Spinor condensate ferromagnet
Lattice of skyrmions
Ordinary ferromagnet
Spinor condensate ferromagnet
Spin textures: skyrmion lattice
Equal populations
Skyrmion lattice solution
without dipolar interactions
Transverse
Longitudinal
Spin textures
Equal populations
Skyrmion lattice solution
with dipolar interactions
Transverse
Longitudinal
Quantum magnetism of ultracold atoms
New questions, interesting physics
Spin dynamics in 1d systems
Luttinger model and nonequilibrium dynamics.
New characterization: full distribution functions
Ferromagnetic F=1 spinor condensates
Quantum Hall ferromagnets in disguise.
Skyrmion crystal phases
Harvard-MIT