Scaling and full counting statistics of interference between independent fluctuating condensates

1
Scaling and full counting statistics of
interference between independent fluctuating
condensates
Anatoli Polkovnikov, Boston University
Collaboration
Ehud Altman - Weizmann Eugene Demler -
Harvard Vladimir Gritsev - Harvard
2
Interference between two condensates.

3
What do we observe?
c) Initial number state. No phases?
4
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).
5
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.
6
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.
• Direct simulator (solver) for interacting
  problems. Quantum impurity in a 1D system of
  interacting fermions (an example).
• Potential applications to many other systems.

7
(No Transcript)
8
Scaling with L two limiting cases
9
Formal derivation
10
Intermediate case (quasi long-range order).
1D condensates (Luttinger liquids)
z
11
Angular Dependence.
12
Angular (momentum) Dependence.
13
Two-dimensional condensates at finite temperature
(picture by Z. Hadzibabic)
14
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
15
Observing the Kosterlitz-Thouless transition
Above KT transition
Lx??Ly
16
Zoran Hadzibabic, Peter Kruger, Marc Cheneau,
Baptiste Battelier, Sabine Stock, and Jean
Dalibard (2006).
17
Z. Hadzibabic et. al.
universal jump in the superfluid density
c.f. Bishop and Reppy
18
Higher Moments.
is an observable quantum operator
19
1D condensates at zero temperature
Low energy action
Then
Similarly
Easy to generalize to all orders.
20
Changing open boundary conditions to periodic find
These integrals can be evaluated using Jack
polynomials (Fendley, Lesage, Saleur, J. Stat.
Phys. 79799 (1995))
Explicit expressions are cumbersome (slowly
converging series of products).
21
Two simple limits
Strongly interacting Tonks-Girardeau regime
(also in thermal case)
Weakly interacting BEC like regime.
22
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).
23
Experimental simulation of the quantum impurity
problem

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

1. Read the result.

24
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.
25
Evolution of the distribution function.
26
Universal asymmetric distribution at large K
(?-1)/??
27
Further extensions
is the Baxter Q-operator, related to the transfer
matrix of conformal field theories with negative
charge
2D quantum gravity, non-intersecting loops on 2D
lattice
Yang-Lee singularity
28
Spinless Fermions.
However for KK-1 ? 3 there is a universal cusp
at nonzero momentum as well as at 2kf
Higher dimensions nesting of Fermi surfaces,
CDW, Not a low energy probe!
29
Fermions in optical lattices.
Possible efficient probes of superconductivity
(in particular, d-wave vs. s-wave).
Not yet, but coming!
30
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. Working example
  detecting KT phase transition.
• Probability distribution of amplitudes ( full
  counting statistics of atoms) 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 like 2D
  quantum gravity).
• Vast potential applications to many other
  systems, e.g.
• Fermionic systems superconductivity, CDW orders,
  etc..
• Rotating condensates instantaneous measurement
  of the correlation functions in the rotating
  frame.
• Correlation functions near continuous phase
  transitions.
• Systems away from equilibrium.

31
Universal adiabatic dynamics across a quantum
critical point
Consider slow tuning of a system through a
critical point.
? ? ? t, ? ? 0
Gap vanishes at the transition. No true adiabatic
limit!
How does the number of excitations scale with ? ?
This question is valid for isolated systems with
stable excitations conserved quantities,
topological excitations, integrable models.
32
Use a general many-body perturbation theory.
Expand the wave-function in many-body basis.
33
Uniform system can characterize excitations by
momentum
34
Caveats

1. Need to check convergence of integrals (no cutoff
   dependence)

Scaling fails in high dimensions.

1. Implicit assumption in derivation small density
   of excitations does not change much the matrix
   element to create other excitations.

1. The probabilities of isolated excitations

should be smaller than one. Otherwise need to
solve Landau-Zeener problem. The scaling argument
gives that they are of the order of one. Thus the
scaling is not affected.
35
Simple derivation of scaling (similar to
Kibble-Zurek mechanism)
In a non-uniform system we find in a similar
manner
36
Example transverse field Ising model.
There is a phase transition at g1.
This problem can be exactly solved using
Jordan-Wigner transformation
37
Spectrum
Critical exponents z?1 ? d?/(z? 1)1/2.
Correct result (J. Dziarmaga 2005)
Other possible applications quantum phase
transitions in cold atoms, adiabatic quantum
computations, etc.