Static and dynamic probes of strongly interacting low-dimensional atomic systems.

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Static and dynamic probes of strongly interacting
low-dimensional atomic systems.
Anatoli Polkovnikov, Boston University
Collaboration
Ehud Altman - The Weizmann Institute of
Science Antonio Castro Neto - Boston
University Eugene Demler - Harvard
University Vladimir Gritsev - Harvard
University Corinna Kollath - University of
Geneva Ludwig Mattey - Harvard University
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Why low dimensions?

1. Existence of many low-dimensional correlated
   phases unconventional superconductivity,
   fractionalized and topological phases, QHE,
   Luttinger liquids, TG gas, etc.
2. Excellent laboratory for studying dynamics and
   thermalization in nonintegrable and integrable
   systems.
3. Realization of new accurate interference probes,
   which are not available in 3D systems.

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This talk

• Interference between two 1D systems of
  interacting bosons
• shot noise
• noise due to phase fluctuations
• full distribution function
• Quench experiments in 1D and 2D systems
• excitations in coupled 1D condensates
• dynamics after the quench
• quenching coupled 2D systems Kibble Zurek
  mechanism of topological defect formation.

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Interference between two condensates.

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What do we observe?
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Define an observable (interference amplitude
squared )
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Fluctuating Condensates.
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Scaling with L two limiting cases
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Formal derivation
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Intermediate case (quasi long-range order).
1D condensates (Luttinger liquids)
z
Repulsive bosons with short range interactions
Finite temperature
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Angular Dependence.
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Higher Moments.
is an observable quantum operator
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Sketch of the derivation
Action
With periodic boundary conditions we find
These integrals can be evaluated using Jack
polynomials (Fendley, Lesage, Saleur, J. Stat.
Phys. 79799 (1995))
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Two simple limits
Strongly interacting Tonks-Girardeau regime
Central limit theorem! Also at finite T.
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Connection to the impurity in a Luttinger liquid
problem.
Boundary Sine-Gordon theory
P. Fendley, F. Lesage, H. Saleur (1995).
Same integrals as in the expressions for
(we rely on Euclidean invariance).
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Experimental simulation of the quantum impurity
problem

1. Do a series of experiments and determine the
   distribution function.

1. Read the result.

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can be found using Bethe ansatz methods for half
integer K.
In principle we can find W
Difficulties need to do analytic continuation.
The problem becomes increasingly harder as K
increases.
Use a different approach based on spectral
determinant
Dorey, Tateo, J.Phys. A. Math. Gen. 32L419
(1999) Bazhanov, Lukyanov, Zamolodchikov, J.
Stat. Phys. 102567 (2001)
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Evolution of the distribution function.
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Universal Gumbel distribution at large K
(?-1)/??
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Generalized extreme value distribution
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Quench experiments in 1D and 2D systems
T. Schumm . et. al., Nature Physics 1, 57 - 62
(01 Oct 2005)
Study dephasing as a function of time. What sort
of information can we get?
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Excitations solitons and breathers.
solitons
Can create solitons only in pairs. Expect damped
oscillations
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Analogy with a Josephson junction.
En
soliton pairs (only with q?0)
breathers
f
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Numerical simulations
Hubbard model, 2 chains, 6 sites each
b02
b24
2s01
b46
b04
2b02
b26
Fourier analysis of the oscillations is a way to
perform spectroscopy.
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Quench in 2D condensates
Expect a very sharp change in TKT as a function
of the layer separation.
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RG calculation for various values of vortex
fugacity. Neglect dependence ?c(T).
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Conclusions.

• Analysis of interference between independent
  condensates reveals a wealth of information about
  their internal structure.
• Scaling of interference amplitudes with L or ?
  correlation function exponents.
• Probability distribution of amplitudes
  information about higher order correlation
  functions.
• Interference of two Luttinger liquids partition
  function of 1D quantum impurity problem (also
  related to variety of other problems).
• Quench experiments in 1D and 2D systems are a
  possible new way of performing spectroscopy in
  cold atom systems
• Detecting solitons and breathers in 1D coupled
  condensates
• Kibble-Zurec mechanism in 2D condensates
• Such experiments are much simpler and more
  robust than e.g. parametric resonance.

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Angular (momentum) Dependence.