Interference between fluctuating condensates

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Interference between fluctuating condensates
Anatoli Polkovnikov, Boston University
Collaboration
Ehud Altman - Weizmann Eugene Demler -
Harvard Vladimir Gritsev - Harvard
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What do we observe interfering ideal condensates?
TOF
Measure (interference part)
d

Andrews et. al. 1997
a) Correlated phases (?? 0) ? ? I (x)? ? N
cos(Qx)
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c) Initial number state. No phases?
Work with original bosonic fields
The same answer as in the case b) with random but
well defined phases!
First theoretical explanation I. Casten and J.
Dalibard (1997) showed that the measurement
induces random phases in a thought experiment.
Experimental observation of interference between
30 condensates in a strong 1D optical lattice
Hadzibabic et.al. (2004).
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Z. Hadzibabic et. al., Phys. Rev. Lett. 93,
180401 (2004).
Polar plots of the fringe amplitudes and phases
for 200 images obtained for the interference of
about 30 condensates. (a) Phase-uncorrelated
condensates. (b) Phase correlated condensates.
Insets Axial density profiles averaged over the
200 images.
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What if the condensates are fluctuating?
This talk

• Access to correlation functions.
• Scaling of ? AQ2 ? with L and ? power-law
  exponents. Luttinger liquid physics in 1D,
  Kosterlitz-Thouless phase transition in 2D.
• Probability distribution W(AQ2) all order
  correlation functions.
• Fermions cusp singularities in ? AQ2 (? )?
  corresponding to kf.
• Direct simulator (solver) for interacting
  problems. Quantum impurity in a 1D system of
  interacting fermions (an example).
• Potential applications to many other systems.

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What are the advantages compared to the
conventional TOF imaging?

• TOF relies on free atom expansion. Often not true
  in strongly correlated regimes. Interference
  method does not have this problem.
• It is often preferable to have a direct access to
  the spatial correlations. TOF images give access
  either to the momentum distribution or the
  momentum correlation functions.
• Free expansion in low dimensional systems occurs
  predominantly in the transverse directions. This
  renders bad signal to noise. In the interference
  method this is advantage longitudinal
  correlations remain intact.

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One dimensional systems.

• Algebraic correlations at zero temperature
  (Luttinger liquids). Exponential decay of
  correlations at finite temperature.
• Fermionization of bosons, bosonization of
  fermions. (There is not much distinction between
  fermions and bosons in 1D).
• 1D systems are well understood. So they can be a
  good laboratory for testing various ideas.

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Scaling with L two limiting cases
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The phase distribution of an elongated 2D Bose
gas. (courtesy of Zoran Hadzibabic)
Matter wave interferometry
very low temperature straight fringes which
reveal a uniform phase in each plane
from time to time dislocation which reveals the
presence of a free vortex
higher temperature bended fringes
S. Stock, Z. Hadzibabic, B. Battelier, M.
Cheneau, and J. Dalibard Phys. Rev. Lett. 95,
190403 (2005)
atom lasers
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Formal derivation.
Independent condensates
for identical homogeneous systems
Long range order
short range correlations
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Intermediate case (quasi long-range order).
L
1D condensates (Luttinger liquids)
z
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Angular Dependence.
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Angular (momentum) Dependence.
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Higher moments (need exactly two condensates).
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Evolution of the distribution function.
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Connection to the impurity in a Luttinger liquid
problem.
Boundary Sine-Gordon theory
P. Fendley, F. Lesage, H.Saleur (1995).
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Sounds complicated? Not really.

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

• Read the result. Direct experimental simulation
  of a quantum impurity problem.

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Spinless Fermions.
However for KK-1 ? 3 there is a universal cusp
at nonzero momentum as well as at 2kf
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(No Transcript)
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Two dimensional condensates at finite temperature.
Similar setup to 1D.
S. Stock et.al Phys. Rev. Lett. 95, 190403
(2005)
Can also study size or angular (momentum)
dependence.
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Observing the Kosterlitz-Thouless transition
Above KT transition
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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 the
  system size or the probing beam angle gives the
  correlation functions exponents.
• Probability distribution ( full counting
  statistics in TOF) of amplitudes for two
  condensates contains information about higher
  order correlation functions.
• Interference of two Luttinger liquids directly
  realizes the statistical partition function of a
  one-dimensional quantum impurity problem.
• Vast potential applications to many other
  systems, e.g.
• Spin-charge separation in spin ½ 1D fermionic
  systems.
• Rotating condensates (instantaneous measurement
  of the correlation functions in the rotating
  frame).
• Correlation functions near continuous phase
  transitions.
• Systems away from equilibrium.