Nonequilibrium dynamics of ultracold atoms in optical lattices. Lattice modulation experiments and more

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Nonequilibrium dynamics of ultracold atoms in
optical lattices.Lattice modulation experiments
and more
David Pekker, Rajdeep Sensarma, Ehud
Altman, Takuya Kitagawa, Susanne Pielawa, Adilet
Imambekov, Mikhail Lukin, Eugene Demler
Harvard University
Collaboration with experimental groups of I.
Bloch, T. Esslinger, J. Schmiedmayer
NSF, AFOSR, MURI, DARPA,
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Outline
Lattice modulation experiments Probe of
antiferromagnetic order
Fermions in optical lattice. Decay of
repulsively bound pairs. Nonequilibrium dynamics
of Hubbard model
Ramsey interferometry and many-body
decoherence. Quantum noise as a probe of dynamics
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Lattice modulation experiments with fermions in
optical lattice.
Probing the Mott state of fermions
Sensarma, Pekker, Lukin, Demler, arXiv0902.2586
Related theory work Kollath et al., PRA
74416049R (2006)
Huber, Ruegg, arXiv08082350
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Atoms in optical lattice
Antiferromagnetism and pairing at sub-micro
Kelvin temperatures
Same microscopic model
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Probing many-body states
Beyond analogies with condensed matter
systems Far from equilibrium quantum many-body
dynamics
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Signatures of incompressible Mott state of
fermions in optical lattice
Suppression of double occupancies Jordens et al.,
Nature 455204 (2008)
Compressibility measurements Schneider et al.,
Science 51520 (2008)
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Lattice modulation experiments Probing
dynamics of the Hubbard model
Measure number of doubly occupied sites
Main effect of shaking modulation of tunneling
Doubly occupied sites created when frequency w
matches Hubbard U
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Lattice modulation experiments Probing
dynamics of the Hubbard model
R. Joerdens et al., Nature 455204 (2008)
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Mott state
Regime of strong interactions Ugtgtt.
High temperature regime
All spin configurations are equally likely. Can
neglect spin dynamics.
Spins are antiferromagnetically ordered or have
strong correlations
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Schwinger bosons and Slave Fermions
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Schwinger bosons and slave fermions
Fermion hopping
Propagation of holes and doublons is coupled to
spin excitations. Neglect spontaneous doublon
production and relaxation.
Doublon production due to lattice modulation
perturbation
Second order perturbation theory. Number of
doublons
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Schwinger bosons Bose condensed
Propagation of holes and doublons strongly
affected by interaction with spin waves
Assume independent propagation of hole and
doublon (neglect vertex corrections)
Self-consistent Born approximation Schmitt-Rink
et al (1988), Kane et al. (1989)
Spectral function for hole or doublon
Sharp coherent part dispersion set by J, weight
by J/t
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Propogation of doublons and holes
Spectral function Oscillations reflect
shake-off processes of spin waves
Comparison of Born approximation and exact
diagonalization Dagotto et al.
Hopping creates string of altered spins bound
states
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Rate of doublon production

• Sharp absorption edge due to coherent
  quasiparticles

• Broad continuum due to incoherent part

• Spin wave shake-off peaks

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High Temperature
Atomic limit. Neglect spin dynamics. All spin
configurations are equally likely.
Aij (t) replaced by probability of having a
singlet
Assume independent propagation of doublons and
holes. Rate of doublon production
Ad(h) is the spectral function of a single
doublon (holon)
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Propogation of doublons and holes
Hopping creates string of altered spins
Retraceable Path Approximation Brinkmann Rice,
1970
Consider the paths with no closed loops
Spectral Fn. of single hole
Doublon Production Rate
Experiments
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Lattice modulation experiments. Sum rule
Ad(h) is the spectral function of a single
doublon (holon)
Sum Rule
Experiments
Most likely reason for sum rule
violation nonlinearity
The total weight does not scale quadratically
with t
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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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Relaxation of doublon hole pairs in the Mott state
Energy U needs to be absorbed by spin
excitations

• Relaxation requires
• creation of U2/t2
• spin excitations

• Energy carried by
• spin excitations
• J 4t2/U

Relaxation rate
Very slow Relaxation
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Doublon decay in a compressible state
Excess energy U is converted to kinetic energy of
single atoms
Compressible state Fermi liquid description
Doublon can decay into a pair of quasiparticles
with many particle-hole pairs
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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
Single fermion hopping
Doublon decay
Doublon-fermion scattering
Fermion-fermion scattering due to projected
hopping
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Fermis golden rule Neglect
fermion-fermion scattering
2
G
other spin combinations
Particle-hole emission is incoherent Crossed
diagrams unimportant
gk cos kx cos ky cos kz
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Self-consistent diagrammatics Calculate
doublon lifetime from Im S Neglect
fermion-fermion scattering

Emission of particle-hole pairs is
incoherent Crossed diagrams are not important
Suppressed by vertex functions
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Self-consistent diagrammatics Neglect
fermion-fermion scattering
Comparison of Fermis Golden rule and
self-consistent diagrams
Need to include fermion-fermion scattering
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Self-consistent diagrammatics Including
fermion-fermion scattering
Treat emission of particle-hole pairs as
incoherent include only non-crossing diagrams
Analyzing particle-hole emission as coherent
process requires adding decay amplitudes and then
calculating net decay rate. Additional diagrams
in self-energy need to be included
No vertex functions to justify neglecting
crossed diagrams
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Including fermion-fermion scattering
Correcting for missing diagrams
Assume all amplitudes for particle-hole pair
production are the same. Assume constructive
interference between all decay amplitudes
For a given energy diagrams of a certain order
dominate. Lower order diagrams do not have enough
p-h pairs to absorb energy Higher order diagrams
suppressed by additional powers of (t/U)2
For each energy count number of missing crossed
diagrams
Rn0(w) is renormalization of the number of
diagrams
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Doublon decay in a compressible state
Doublon decay with generation of particle-hole
pairs
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Ramsey interferometry and many-body
decoherence Quantum noise as a probe of
non-equilibrium dynamics
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Interference between fluctuating condensates
high T
BKT
Time of flight
low T
2d BKT transition Hadzibabic et al, Claude et al
1d Luttinger liquid, Hofferberth et al., 2008
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Distribution function of interference fringe
contrast
Hofferberth, Gritsev, 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

• First demonstration of quantum fluctuations for
  1d ultracold atoms
• First measurements of distribution function of
  quantum
• noise in interacting many-body system

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Can we use quantum noise as a probe of
dynamics? Example Interaction induced collapse
of Ramsey fringes
A. Widera, V. Gritsev, et al. PRL 100140401
(2008), T. Kitagawa, S. Pielawa, A. Imambekov,
J. Schmiedmayer, E. Demler
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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.
Kitagawa, Ueda, PRA 475138 (1993)
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 analysis
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., PRL (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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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, J.
Schmiedmayer, E. Demler
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Summary
Lattice modulation experiments as a probe of AF
order
Fermions in optical lattice. Decay of
repulsively bound pairs
Ramsey interferometry in 1d. Luttinger
liquid approach to many-body decoherence
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Thanks to
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