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Scaling and full counting statistics of interference
between independent fluctuating condensates
Anatoli Polkovnikov,
Boston University
Collaboration:
Ehud Altman
Eugene Demler Vladimir Gritsev -
Weizmann
Harvard
Harvard
Interference between two condensates.
TOF
 ( x, t )   a1† ( x, t )  a2† ( x, t )  a1 ( x, t )  a2 ( x, t ) 
  ( x, t )  int ( x, t )
d
int ( x, t )  a1† ( x, t )a2 ( x, t )  a2† ( x, t )a1 ( x, t )
x
Free expansion: t  
m( x  d / 2)
a1 ( x, t ) ~ a1 exp  iQ1 x  , Q1 

t
mv2 m( x  d / 2)
a2 ( x, t ) ~ a2 exp  iQ2 x  , Q2 

t
mv1
int ( x, t ) a a exp(iQx)  a a exp(iQx),
†
1 2
a1,2
Ne
i1,2
†
2 1
md
Q
t
 int ( x)  N cos  Qx   
Andrews et. al. 1997
What do we observe?
TOF
x
int ( x)  N cos Qx   
a) Correlated phases ( = 0)
 int ( x)  N cos  Qx 
b) Uncorrelated, but well defined phases  int(x)=0
int ( x) int ( y ) ~ N 2 cos  Qx    cos  Qy    ~ N 2 cos  Q( x  y)   0
Hanbury Brown-Twiss Effect
c) Initial number state. No phases?
Work with original bosonic fields:
int ( x) ~ a1† a2 exp(iQx)  a2† a1 exp(iQx) =0
int ( x) int ( y ) ~ a1† a1 a2† a2 cos  Q( x  y )  ~ N 2 cos  Q( x  y ) 
int ( x) int ( y ) ~ a1† a1 a2†a2 cos  Q( x  y )   AQ2 cos  Q( x  y ) 
AQ2  a1†a1 a2†a2
Easy to check
that at large N:
Interference amplitude squared. Observable!
2 2
Q
 A  A
4
Q
AQ2  0
The interference amplitude does not fluctuate!
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).
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.
Imaging beam
What if the condensates are fluctuating?

L
This talk:
1. Access to correlation functions.
a) Scaling of  AQ2  with L and : power-law exponents. Luttinger
liquid physics in 1D, Kosterlitz-Thouless phase transition in 2D.
b) Probability distribution W(AQ2): all order correlation functions.
2. Direct simulator (solver) for interacting problems.
Quantum impurity in a 1D system of interacting fermions (an example).
3. Potential applications to many other systems.
L
int ( x) exp(iQx) a1† ( z)a2 ( z)dz  c.c. 
0
int ( x) int ( y )  AQ2 cos Q( x  y )
z1
A
z2
A
AQ
L
0
0
a1† ( z1 )a2 ( z1 )a2† ( z2 )a1 ( z2 )dz1dz2
L
L
0
0

2
Q
z
L
 
2
Q
a1† ( z1 )a1 ( z2 ) a2 ( z1 )a2† ( z2 ) dz1dz2
Identical homogeneous condensates:
x
2
Q
A
L
L
0
†
1
2
a ( z )a1 (0) dz
Interference amplitude contains information about fluctuations
within each condensate.
Scaling with L: two limiting cases
int ( x)   z a1† ( z)a2 ( z)exp(iQx)  c.c.   z N z exp(iQx  iz )  c.c.
Dephased condensates:
z
AQ  L
Interference contrast
does not depend on L.
L
x
L
x
Ideal condensates:
z
AQ  L
Contrast scales
as L-1/2.
Formal derivation:
AQ2
L
L
0
2
a1† ( z )a1 (0) dz
Ideal condensate:
L
a1† ( z )a1 (0)  c
AQ2
c L2
Thermal gas:
L
a1† ( z )a1 (0) ~  exp( z /  )
AQ2
 L
Intermediate case (quasi long-range order).
L
L
AQ2
L
0
2
a1† ( z )a1 (0) dz
1D condensates (Luttinger liquids):
a ( z)a1 (0)   h / z 
z
2
Q
A
21/ K
L

1/ K
h
1/ 2 K
†
1
, Interference contrast  h / L 
1/ 2 K
Repulsive bosons with short range interactions:
Weak interactions K
1 
AQ2
L2
1
Strong interactions (Fermionized regime) K
Finite temperature:
A
11/ K

1
L h  

 m h T 
2
2
Q
AQ2
2
L
x(z1)
x(z2)
Angular Dependence.
z
(for the imaging beam
orthogonal to the
page,  is the angle of
the integration axis
with respect to z.)


int ( x)

L
0
a1† ( z )a2 ( z )eiQ ( x  z tan  ) dz  c.c.
L
 exp(iQx)  a1† ( z )a2 ( z )e iqz dz +c.c., q  Q tan 
0
2
Q
A (q)
L
L
0
0

a1† ( z1 )a1 ( z2 ) a2 ( z1 )a2† ( z2 )  cos  q( z2  z1 )  dz1dz2
q is equivalent to the relative momentum of the two condensates
(always present e.g. if there are dipolar oscillations).
Angular (momentum) Dependence.
L
2
Q
A ( q)
qL
L
0
†
a ( z )a(0)
2
cos(qz ) dz
1
A (q )    q  ,
2
Q
ideal condensates ( K
1);
1
A (q) 
, finite T (short range correlations);
2 2
1 q 
2
Q
A (q) 
2
Q
2
Q
A (q)
1
11/ K
q
,
quasi-condensates finite K.
has a cusp singularity for K<1, relevant for fermions.
Two-dimensional condensates at finite temperature
CCD z
camera
z
Time of
flight
y
x
x
imaging
laser
(picture by Z. Hadzibabic)
Elongated condensates: Lx>>Ly .
2
Q
A (X )
X
X Ly  dx 
0
Ly
0
†
a ( x ', y)a(0,0)
2
The phase distribution of an elongated 2D Bose gas.
(courtesy of Zoran Hadzibabic)
Matter wave
interferometry
0
p
very low temperature:
straight fringes which
reveal a uniform phase
in each plane
“atom lasers”
higher temperature:
bended fringes
from time to time:
dislocation which
reveals the presence
of a free vortex
S. Stock, Z. Hadzibabic, B. Battelier, M. Cheneau,
and J. Dalibard: Phys. Rev. Lett. 95, 190403 (2005)
Observing the Kosterlitz-Thouless transition
Above KT transition
Ly
LxLy
Lx
A (X )   X
2
Q
Below KT transition
2
Q
A
X
2  2
AQ2 ( X )  X 22
Universal jump of  at TKT
T  TKT
     1/ 2
T

TKT
   1/ 4
Always algebraic scaling, easy to detect.
Zoran Hadzibabic, Peter Kruger, Marc Cheneau, Baptiste Battelier, Sabine
Stock, and Jean Dalibard (2006).
Interference contrast:
x
C2(X ) ~
AQ2 ( X )
X2
1
0  g1 (0, x) dx ~  X 
X
1
~
X
2
2
z
Contrast after integration
0.4
integration
low T
over x axis
z
middle T
0.2
integration
high T
over x axis
X
z
0
0
10
20
30
integration distance X
(pixels)
Exponent 
Z. Hadzibabic et. al.
“universal jump in
the superfluid density”
0.5
c.f. Bishop and Reppy
0.4
0.3
high T
0
low T
0.1
0.2
0.3
central contrast
1.0
0
1.0
1.1
Vortex proliferation
Fraction of images showing at least one dislocation:
30%
20%
10%
low T
high T
0
0
0.1
0.2
0.3
0.4
central contrast
T (K)
1.2
Higher Moments.
2
Q
A
L
L
0
0

a ( z1 )a1 ( z2 ) a2 ( z1 )a ( z2 )  dz1dz2
†
1
†
2
is an observable
quantum operator
Identical condensates. Mean:
AQ2

L
0
L
dz1  dz2 a1† ( z1 )a1 ( z2 )
2
0
Similarly higher moments
2n
Q
A
L
L
0
0
 ...
†
1
†
1
dz1...dzn dz1...dzn a ( z1 )...a ( zn )a1 ( z1 )...a1 ( zn )
Probe of the higher order correlation functions.
Distribution function (= full counting statistics):
W ( AQ2 ) :
AQ2 n   AQ2 n W ( AQ2 ) dAQ2
Non-interacting non-condensed
W ( AQ2 )  C exp(CAQ2 )
regime (Wick’s theorem):
Nontrivial statistics if the Wick’s theorem is not fulfilled!
2
1D condensates at zero temperature:
Low energy action:
1/ 2 K
Then
a( y )
c e
ip ( y )
,
†
a ( y )a ( y )
 h 
c 

 y y 
Similarly
†
†
a ( y1 )a ( y2 )a( y1 )a( y2 )

 y1  y2 y1  y2
 
 y1  y2 y1  y2 y1  y1 y2  y2
2
c
Easy to generalize to all orders.
2
h
1/ 2 K



2n
Q
A
 C  

1/ 2 K 11/ 2 K n
c h
L
Z 2n
Changing open boundary conditions to periodic find
These integrals can be evaluated using Jack polynomials
(Fendley,
Lesage, Saleur, J. Stat. Phys. 79:799 (1995))
Explicit expressions are cumbersome (slowly converging series
of products).
 2 (1  1/ 2 K )
1
(1  1/ K )
Z2  2
 2

2
 (1/ 2 K ) 01  (1  1 )
 (1  1/ 2 K )
   1  1/ K    2  1/ 2 K  
4
Z4  4



 (1/ 2 K ) 02 1    1  1/ 2 K  1   2  1 
2
Two simple limits:
Strongly interacting Tonks-Girardeau regime
K  1: Z2n  n!, W ( AQ2 )  Cexp(CAQ2 )
z1
(also in
thermal case)
z2
z
Weakly interacting BEC like regime.
K   : Z 2 n  1, W ( A )    A  A  ,
2
Q
 AQ2
AQ2
Z 4  Z 22

Z2
p
6K
2
Q
2
0
AQ
x
Connection to the impurity in a Luttinger liquid problem.
Boundary Sine-Gordon theory:
Z   D exp  S ,
S
pK




2
Z ( x)  
n
dx  d
0
x2n
 n!
P. Fendley, F. Lesage, H. Saleur (1995).
           2g 
2

2
x
0
Z 2 n , x  g   2p  
1/ 2 K
2

d cos 2p (0, )
,
Same integrals as in the expressions for AQ2n (we rely on Euclidean
invariance).
Z ( x)  

0
1/ 2 K 11/ 2 K
A

C


L
W ( A ) I 0 (2 Ax / A0 )dA , 0
c h
2
W (A )  2
A0
2
2


0
2
Z (ix) J 0 (2 Ax / A0 ) xdx,
Experimental simulation of the quantum impurity problem
1. Do a series of experiments and determine the distribution function.
Distribution of interference phases (and amplitudes) from two 1D condensates.
T. Schumm, et.
al., Nature Phys.
1, 57 (2005).

2. Evaluate the integral.
3. Read the result.
Z ( x)   W ( A2 ) I 0 (2 Ax / A0 )dA2 ,
0
Z ( x) can be found using Bethe ansatz methods for half integer K.
In principle we can find W:
2
W (A )  2
A0
2


0
Z (ix) J 0 (2 Ax / A0 ) xdx,
Difficulties: need to do analytic continuation.
The problem becomes increasingly harder as K increases.
Use a different approach based on spectral determinant:
  2 
Z (ix)   1 
   x sin p / 2K
En 
n 0 
p

Dorey, Tateo, J.Phys. A. Math. Gen. 32:L419 (1999);
Bazhanov, Lukyanov, Zamolodchikov, J. Stat. Phys. 102:567 (2001)
Evolution of the distribution function.
Probability W()
K=1
K=1.5
K=3
K=5
0
1

2
A
2
Q
3
2
Q
A
4
Universal asymmetric distribution at large K
(-1)/
Further extensions:
Z (ix)  Qvac ( )
Q ( )
x
sin p / 2K
p
is the Baxter Q-operator, related to the transfer matrix of
conformal field theories with negative charge:
2 K  1

c  1 3
2
K
2D quantum gravity,
non-intersecting loops on 2D lattice
Yang-Lee singularity
Spinless Fermions.
2
Q
A
L
L
0
†
a ( z )a (0)
2
a ( z )a (0) 
†
dz ,
sin(k f z )
z

1 2 K  K 1

Note that K+K-1  2, so AQ  L and the distribution function
is always Poissonian.
However for K+K-1  3 there is a universal cusp at nonzero
momentum as well as at 2kf:
2
2
Q
A ( q)
L
L
0
†
a ( z)a(0)
A (q)  A (0)  q
2
Q
2
Q
2
cos  qz  dz,
K  K 1 1
q  Q tan 
. There is a similar cusp at 2kf
Higher dimensions: nesting of Fermi surfaces, CDW, …
Not a low energy probe!
Fermions in optical lattices.
Possible efficient probes of superconductivity (in
particular, d-wave vs. s-wave). Not yet, but coming!
Rapidly rotating two dimensional condensates
Time of flight experiments with rotating
condensates correspond to density
measurements
Interference experiments measure single particle
correlation functions in the rotating frame
Conclusions.
1. Analysis of interference between independent condensates reveals
a wealth of information about their internal structure.
a) Scaling of interference amplitudes with L or  : correlation function
exponents. Working example: detecting KT phase transition.
b) Probability distribution of amplitudes (= full counting statistics of
atoms): information about higher order correlation functions.
c) Interference of two Luttinger liquids: partition function of 1D
quantum impurity problem (also related to variety of other
problems like 2D quantum gravity).
2. Vast potential applications to many other systems, e.g.:
a) Fermionic systems: superconductivity, CDW orders, etc..
b) Rotating condensates: instantaneous measurement of the correlation
functions in the rotating frame.
c) Correlation functions near continuous phase transitions.
d) Systems away from equilibrium.
Universal adiabatic dynamics across a
quantum critical point
gap 
Consider slow tuning of a system through a critical point.
   t,   0
tuning parameter 
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.
Use a general many-body perturbation theory.
Expand the wave-function in many-body basis.

i
 H
t
Substitute into Schrödinger equation.
Uniform system: can characterize excitations by momentum:
Use scaling relations:
Find:
Caveats:
1. Need to check convergence of integrals (no cutoff dependence)
Scaling fails in high dimensions.
2. Implicit assumption in derivation: small density of
excitations does not change much the matrix element to
create other excitations.
3. The probabilities of isolated excitations:
should be smaller than one. Otherwise need to solve LandauZeener problem. The scaling argument gives that they are of
the order of one. Thus the scaling is not affected.
Simple derivation of scaling (similar to Kibble-Zurek mechanism):
Breakdown of adiabaticity:
From
   t and
we get

In a non-uniform system we find in a similar manner:
Example: transverse field Ising model.
g 0

 iz zj  1
g 

 ix  1
There is a phase transition at g=1.
This problem can be exactly solved using Jordan-Wigner
transformation:
 ix  2ci†ci  1,  iz   (c†j c j  1) (c j  c†j )
j i 1
Spectrum:
 k  2 1  g  2 g cos k  2 1  g   k 2
2
2
Critical exponents: z==1  d/(z +1)=1/2.
 nex  0.18 
Correct result (J. Dziarmaga 2005):
nex  0.11 
Other possible applications: quantum phase transitions in cold
atoms, adiabatic quantum computations, etc.