Transcript Slide 1

Measuring correlation functions in
interacting systems of cold atoms
Detection and characterization of
many-body quantum phases
Anatoli Polkovnikov
Ehud Altman
Vladimir Gritsev
Mikhail Lukin
Vito Scarola
Sankar Das Sarma
Eugene Demler
Harvard/Boston University
Harvard/Weizmann
Harvard
Harvard
University of Maryland
University of Maryland
Harvard
Outline
Measuring correlation functions in intereference experiments
1. Interference of independent condensates
2. Interference of interacting 1D systems
3. Interference of 2D systems
4. Full counting statistics of intereference experiments.
Connection to quantum impurity problem
Quantum noise interferometry in time of flight experiments
References:
1. Polkovnikov, Altman, Demler, cond-mat/0511675
Gritsev, Altman, Demler, Polkovnikov, cond-mat/0602475
2. Altman, Demler, Lukin, PRA 70:13603 (2004)
Scarola, Das Sarma, Demler, cond-mat/0602319
Measuring correlation functions
in intereference experiments
Interference of two independent condensates
Andrews et al., Science 275:637 (1997)
Interference of two independent condensates
r’
r
1
r+d
d
2
Clouds 1 and 2 do not have a well defined phase difference.
However each individual measurement shows an interference pattern
Interference of one dimensional condensates
Experiments: Schmiedmayer et al., Nature Physics 1 (05)
d
Amplitude of interference fringes,
,
contains information about phase fluctuations
within individual condensates
x1
x2
x
y
Interference amplitude and correlations
Polkovnikov, Altman, Demler, cond-mat/0511675
L
For identical condensates
Instantaneous correlation function
Interference between Luttinger liquids
Luttinger liquid at T=0
K – Luttinger parameter
L
For non-interacting bosons
For impenetrable bosons
and
and
Luttinger liquid at finite temperature
Analysis of
can be used for thermometry
Interference between two-dimensional
BECs at finite temperature.
Kosteritz-Thouless transition
Interference of two dimensional condensates
Experiments: Stock, Hadzibabic, Dalibard, et al., cond-mat/0506559
Gati, Oberthaler, et al., cond-mat/0601392
Ly
Lx
Lx
Probe beam parallel to the plane of the condensates
Interference of two dimensional condensates.
Quasi long range order and the KT transition
Ly
Lx
Above KT transition
Theory: Polkovnikov, Altman, Demler,
cond-mat/0511675
Below KT transition
Experiments with 2D Bose gas
z
Hadzibabic, Stock, Dalibard, et al.
Time of
flight
x
Typical interference patterns
low temperature
higher temperature
Experiments with 2D Bose gas
Hadzibabic, Stock, Dalibard, et al.
x
integration
over x axis z
z
Contrast after
integration
0.4
low T
integration
middle T
0.2
over x axis
z
high T
integration
over x axis
Dx
0
z
0
10
20
30
integration distance Dx
(pixels)
Experiments with 2D Bose gas
Integrated contrast
Hadzibabic, Stock, Dalibard, et al.
0.4
fit by:
C2 ~
low T
1
Dx
 1 

 Dx 
Dx
2


g
(
0
,
x
)
dx ~ 
1

middle T
0.2
Exponent a
high T
0
0
10
20
30
integration distance Dx
if g1(r) decays exponentially
with
:
0.5
0.4
0.3
high T
0
if g1(r) decays algebraically or
exponentially with a large
:
0.1
low T
0.2
0.3
central contrast
“Sudden” jump!?
2a
Experiments with 2D Bose gas
Hadzibabic, Stock, Dalibard, et al.
Exponent a
c.f. Bishop and Reppy
0.4
1.0
0
0.5
1.0
1.1
T (K)
1.2
0.3
high T
0
0.1
low T
0.2
0.3
central contrast
He experiments:
universal jump in
the superfluid density
Ultracold atoms experiments:
jump in the correlation function.
KT theory predicts a=1/4
just below the transition
Experiments with 2D Bose gas. Proliferation of
thermal vortices
Hadzibabic, Stock, Dalibard, et al.
30%
Fraction of images showing
at least one dislocation
Exponent a
20%
0.5
10%
0.4
low T
high T
0
0
0.1
0.2
0.3
central contrast
The onset of proliferation
coincides with a shifting to 0.5!
Z. Hadzibabic et al., in preparation
0.4
0.3
0
0.1
0.2
central contrast
0.3
Full counting statistics
of interference between
two interacting one
dimensional Bose liquids
Gritsev, Altman, Demler, Polkovnikov, cond-mat/0602475
Higher moments of interference amplitude
is a quantum operator. The measured value of
will fluctuate from shot to shot.
Can we predict the distribution function of
?
L
Higher moments
Changing to periodic boundary conditions (long condensates)
Explicit expressions for
are available but cumbersome
Fendley, Lesage, Saleur, J. Stat. Phys. 79:799 (1995)
Impurity in a Luttinger liquid
Expansion of the partition function in powers of g
Partition function of the impurity contains correlation functions
taken at the same point and at different times. Moments
of interference experiments come from correlations functions
taken at the same time but in different points. Euclidean invariance
ensures that the two are the same
Relation between quantum impurity problem
and interference of fluctuating condensates
Normalized amplitude
of interference fringes
Distribution function
of fringe amplitudes
Relation to the impurity partition function
Distribution function can be reconstructed from
using completeness relations for the Bessel functions
Bethe ansatz solution for a quantum impurity
can be obtained from the Bethe ansatz following
Zamolodchikov, Phys. Lett. B 253:391 (91); Fendley, et al., J. Stat. Phys. 79:799 (95)
Making analytic continuation is possible but cumbersome
Interference amplitude and spectral determinant
is related to a Schroedinger equation
Dorey, Tateo, J.Phys. A. Math. Gen. 32:L419 (1999)
Bazhanov, Lukyanov, Zamolodchikov, J. Stat. Phys. 102:567 (2001)
Spectral determinant
Evolution of the distribution function
Probability P(x)
K=1
K=1.5
K=3
K=5
Narrow distribution
for
.
Distribution width
approaches
Wide Poissonian
distribution for
0
1
x
2
3
4
From interference amplitudes to conformal field theories
correspond to vacuum eigenvalues of Q operators of CFT
Bazhanov, Lukyanov, Zamolodchikov, Comm. Math. Phys.1996, 1997, 1999
When K>1,
is related to Q operators of
CFT with c<0. This includes 2D quantum gravity, nonintersecting loop model on 2D lattice, growth of random
fractal stochastic interface, high energy limit of multicolor
QCD, …
2D quantum gravity,
non-intersecting loops on 2D lattice
Yang-Lee singularity
Outlook
Full counting statistics of interference between fluctuating
2D condensates
One and two dimensional systems
with tunneling. Competition of single
particle tunneling, quantum, and
thermal fluctuations. Full counting
statistics of the phase
and the amplitude
Expts: Shmiedmayer et al. (1d),
Oberthaler et al. (2d)
Time dependent evolution of distribution functions
Extensions, e.g. spin counting in Mott states of multicomponent systems
Ultimate goal: Creation, characterization, and manipulation
of quantum many-body states of atoms
Quantum noise interferometry
in time of flight experiments
Atoms in an optical lattice.
Superfluid to Insulator transition
Greiner et al., Nature 415:39 (2002)

U
Mott insulator
Superfluid
n 1
t/U
Time of flight experiments
Quantum noise interferometry of atoms in an optical lattice
Second order coherence
Second order coherence in the insulating state of bosons.
Hanburry-Brown-Twiss experiment
Theory: Altman et al., PRA 70:13603 (2004)
Experiment: Folling et al., Nature 434:481 (2005)
Hanburry-Brown-Twiss stellar interferometer
Second order coherence in the insulating state of bosons
Bosons at quasimomentum
expand as plane waves
with wavevectors
First order coherence:
Oscillations in density disappear after summing over
Second order coherence:
Correlation function acquires oscillations at reciprocal lattice vectors
Second order coherence in the insulating state of bosons.
Hanburry-Brown-Twiss experiment
Theory: Altman et al., PRA 70:13603 (2004)
Experiment: Folling et al., Nature 434:481 (2005)
Effect of parabolic potential on the second order coherence
Experiment: Spielman, Porto, et al.,
Theory: Scarola, Das Sarma, Demler, cond-mat/0602319
Width of the correlation peak changes across the
transition, reflecting the evolution of Mott domains
Potential applications of quantum noise intereferometry
Altman et al., PRA 70:13603 (2004)
Detection of magnetically ordered Mott states
Detection of paired states of fermions
Experiment: Greiner et al. PRL
Conclusions
Interference of extended condensates is a powerful
tool for analyzing correlation functions in one and
two dimensional systems
Noise interferometry can be used to probe
quantum many-body states in optical lattices