Condensed Matter models for

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Condensed Matter models for many-body systems of
ultracold atoms
Eugene Demler Harvard
University
Collaborators Ehud Altman, Robert Cherng, Adilet
Imambekov, Vladimir Gritsev, Takuya Kitagawa,
Susanne Pielawa, David Pekker, Rajdeep
Sensarma Experiments Bloch et al., Esslinger
et al., Schmiedmayer et al., Stamper-Kurn et al.
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New Tricks for Old Dogs,Old Tricks for New Dogs
Condensed Matter models for many-body systems of
ultracold atoms
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Summary
Dipolar interactions. Magnetoroton softening,
spin textures, supersolid. New issues averaging
over Larmor precession, coupling of spin textures
and vortices R. Cherng, V. Gritsev. In
collaboration with D. Stamper-Kurn
Luttinger liquid. Ramsey interferometry and
many-body decoherence in 1d. New issues
nonequilibrium dynamics, analysis of quantum
noise. V. Gritsev, T. Kitagawa, S. Pielawa. In
collaboration with expt. groups of I. Bloch and
J. Schmiedmayer
Hubbard model. Fermions in optical lattice.
Decay of repulsively bound pairs. New issues
nonequilibrium dynamics in strongly interacting
regime. D. Pekker, R. Sensarma, E. Altman. In
collaboration with expt. group of T. Esslinger
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Dipolar interactions in spinor
condensates. Magnetoroton softening and spin
textures R. Cherng, V. Gritsev. In collaboration
with D. Stamper-Kurn
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Roton minimum in 4He
Glyde, J. Low. Temp. Phys. 93 861
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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)
D. Kirzhnits, Y. Nepomnyashchii (1970), T.
Schneider and C.P. Enz (1971). Formation of the
supersolid phase due to softening of roton
excitations
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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
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Cold atoms magnetic dipolar interactions
Short-ranged contact interactions
Long-ranged, anisotropic dipolar interactions
r
r
r
Contact vs. dipolar interactions
For 87Rb mmB and e0.007 For 52Cr m6mB and
e0.16
Menotti et al., arxiv0711.3422
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Magnetic dipolar interactions in spinor
condensates
Interaction of F1 atoms
Ferromagnetic Interactions for 87Rb
A. Widera, I. Bloch et al., New J. Phys. 8152
(2006)
a2-a0 -1.07 aB
Spin-depenent part of the interaction is small.
Dipolar interaction may be important (D.
Stamper-Kurn)
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Spinor condensates at Berkeley
M. Vengalattore et al., arXiv0901.3800
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Spinor condensates at Berkeley
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Competing energy scales
High energy scales
Precession (115 kHz)

• Spin independent
• S-wave scattering (gsn215 Hz)

Quasi-2D geometry
Low energy scales
Quadratic Zeeman (0-20 Hz)
Spin dependent S-wave scattering (gsn8 Hz)
Dipolar interaction (gdn0.8 Hz)
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Dipolar interactions after averaging over Larmor
precession
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Dipolar interactions
Static interaction
z
Averaging over Larmor precession
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Instabilities qualitative picture
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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
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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
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Instabilities technical details
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Hamiltonian
Quad Zeeman
Precession
Spin dep.
Spin indep.
Dipolar
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Effective dipolar interactionSpatial and time
averaging
Larmor precession comoving frame
Gaussian profile
Field Ansatz
Time-averaged Quasi-2D Effective dipolar
interaction
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Effective dipolar interaction
Time-averaged Quasi-2D Effective dipolar
interaction
dF
dF
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Collective Modes
Mean Field
Equations of Motion
Collective Fluctuations (Spin, Charge)
Spin Mode dfB longitudinal magnetization df
transverse orientation Charge Mode dn 2D
density d? global phase
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Instabilities of collective modes
Q measures the strength of quadratic Zeeman
effect
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Collective mode phase diagram
C-
DBC-
CB
D-CB
R0
D-
R
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Berkeley Experiments checkerboard phase
M. Vengalattore, et. al, PRL 100170403 (2008)
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Spin texture length scales
Spin modulation 10 µm

• Most unstable mode
• k2 cost in kinetic energy
• k gain in dipolar energy
• l 30 µm

Spin axis modulation 30 µm
M. Vengalattore et al., arXiv0901.3800
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Finding a stable ground state
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Non-linear sigma modelSpin textures cause phase
twists
Spinor vector potential
Energetic Constraints
Equations of motion for ?
Effective kinetic energy
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Non-linear sigma model
Topological charge (net vorticity)
Spin gradient
Vortex interaction
Dipolar interaction
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Spin Textures
Unit Cell
Qgt0
Max KE
Top. Charge Q
Kinetic Energy
Qlt0
Min KE
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Spin Textures Skyrmion Stripes
Unit Cell
Unit Cell
Top. Charge Q
Kinetic Energy
Top. Charge Q
Kinetic Energy
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Spin Textures Skyrmion Lattice
Unit Cell
Unit Cell
Top. Charge Q
Kinetic Energy
Top. Charge Q
Kinetic Energy
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Quantum noise as a probe of non-equilibrium
dynamics Ramsey interferometry and many-body
decoherence
T. Kitagawa, A. Imambekov, S. Pielawa, J.
Schmeidmayers group. Continues earlier work
with V. Gritsev, M. Lukin, I. Blochs
group. Phys. Rev. Lett. 100140401 (2008)
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Ramsey interference
Atomic clocks and Ramsey interference
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Interaction induced collapse of Ramsey fringes
Two component BEC. Single mode approximation
Ramsey fringe visibility
time
Experiments in 1d tubes A. Widera et al. PRL
100140401 (2008)
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Spin echo. Time reversal experiments
Single mode approximation
The Hamiltonian can be reversed by changing a12
Predicts perfect spin echo
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Spin echo. Time reversal experiments
Expts A. Widera et al., Phys. Rev. Lett. (2008)
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?
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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
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Time-dependent harmonic oscillator
See e.g. Lewis, Riesengeld (1969)
Malkin, Manko (1970)
Explicit quantum mechanical wavefunction can be
found
From the solution of classical problem
We solve this problem for each momentum component
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Interaction induced collapse of Ramsey fringesin
one dimensional systems
Only q0 mode shows complete spin echo Finite q
modes continue decay The net visibility is a
result of competition between q0 and other modes
Decoherence due to many-body dynamics of low
dimensional systems
Fundamental limit on Ramsey interferometry
How to distinquish decoherence due to many-body
dynamics?
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Interaction induced collapse of Ramsey fringes
Single mode analysis Kitagawa, Ueda, PRA 475138
(1993)
Multimode analysis evolution of spin distribution
functions
T. Kitagawa, S. Pielawa, A. Imambekov, et al.
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Noise measurements using BEC on a
chipIntereference of independent condensates
Hofferberth et al., Nature Physics 2008
Average contrast
Distribution function of fringe contrast
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Distribution function of interference fringe
contrast
Hofferberth et al., Nature Physics 4489 (2008)
Quantum fluctuations dominate asymetric Gumbel
distribution (low temp. T or short length L)
Thermal fluctuations dominate broad Poissonian
distribution (high temp. T or long length L)
Intermediate regime double peak structure
Comparison of theory and experiments no free
parameters Higher order correlation functions can
be obtained
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Fermions in optical lattice.Decay of repulsively
bound pairs
Experiment ETH Zurich, Esslinger et al., Theory
Sensarma, Pekker, Altman, Demler
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Fermions in optical lattice.Decay of repulsively
bound pairs
Experiments T. Esslinger et. al.
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Relaxation of repulsively bound pairs in the
Fermionic Hubbard model
U gtgt t
For a repulsive bound pair to decay, energy U
needs to be absorbed by other degrees of freedom
in the system
Relaxation timescale is important for quantum
simulations, adiabatic preparation
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Doublon decay in a compressible state
Perturbation theory to order nU/t Decay
probability
To calculate the rate consider processes which
maximize the number of particle-hole excitations
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Doublon decay in a compressible state
Doublon decay with generation of particle-hole
pairs
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Outline
Dipolar interactions. Magnetoroton softening
and spin textures in spinor condensates.
Luttinger liquid. Ramsey interferometry and
many-body decoherence in 1d.
Hubbard model. Fermions in optical lattice.
Decay of repulsively bound pairs.
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Thanks to