Nonequilibrium dynamics of ultracold atoms in optical lattices

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Nonequilibrium dynamics of ultracold atoms in
optical lattices
David Pekker, Rajdeep Sensarma, Takuya
Kitagawa, Susanne Pielawa, Vladmir Gritsev,
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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Nonequilibrium quantum dynamics of many-body
systems
Big Bang and Inflation. Structure of the
universe. From formation of galaxies to
fluctuations in the CMB radiation.
Jet production in particle decay. Heavy Ion
collisions.
Solid state devices
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Nonequilibrium quantum dynamics in artificial
many-body systems
Photons in strongly nonlinear medium
Example photon crystallization in nonlinear 1d
waveguides Chang et al (2008)
Strongly correlated systems of ultracold atoms
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Outline
Fermions in optical lattice. Decay of
repulsively bound pairs
Ramsey interferometry and many-body decoherence
Lattice modulation experiments
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Fermions in optical lattice.Decay of repulsively
bound pairs
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Fermions in optical lattice.Decay of repulsively
bound pairs
Experimets 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
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G
other spin combinations
Crossed diagram are not important
gk cos kx cos ky cos kz
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Self-consistent diagrammatics Neglect
fermion-fermion scattering
Calculate doublon lifetime from Im S
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Self-consistent diagrammatics Including
fermion-fermion scattering
For fermions it is easy to include non-crossing
diagrams
Diagrams not included
Diagrams included
No vertex functions to justify neglecting crossed
diagrams
Undercounting decay channels for doublons
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Self-consistent diagrammatics Including
fermion-fermion scattering
Correcting for missing diagrams
type present
type missing
Each diagram allows additional particle-hole pair
production. Decay rate is determined by the
number of particle-hole pairs. Correct the number
of decay channels by counting the number of
diagrams
e0 characteristic energy of
particle-hole pairs
Np number of diagrams included N total number
of diagrams
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Self-consistent diagrammatics Including
fermion-fermion scattering
Correcting for missing diagrams
Particle-hole self-energy
Doublon life-time
Typical energy transfer around 8 t
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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 et al., 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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Can we use quantum noise as a probe of dynamics?
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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, 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?
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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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• Fermions in optical lattice.
• Lattice modulation experiments as a probe of
  the Mott state

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Signatures of incompressible Mott state of
fermions in optical lattice
Suppression of double occupancies T. Esslinger
et al. arXiv0804.4009
Compressibility measurements I. Bloch et al.
arXiv0809.1464
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Lattice modulation experiments with fermions in
optical lattice.
Probing the Mott state of fermions
Related theory work Kollath et al., PRA
74416049R (2006)
Huber, Ruegg, arXiv08082350
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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., arXiv0804.4009
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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

• Low energy peak due to sharp 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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Summary
Fermions in optical lattice. Decay of
repulsively bound pairs
Ramsey inter- ferometry in 1d. Luttinger
liquid approach to many-body decoherence
T gtgt TN
T ltlt TN
Lattice modulation experiments as a probe of AF
order
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Thanks to
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