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Scaling and full counting statistics of interference Scaling and full counting statistics of interference between independent fluctuating condensatesbetween independent fluctuating condensates
Anatoli Polkovnikov,Anatoli Polkovnikov,Boston UniversityBoston University
Collaboration:
Ehud AltmanEhud Altman -- WeizmannWeizmannEugene Demler Eugene Demler - - HarvardHarvardVladimir Gritsev Vladimir Gritsev - - HarvardHarvard
Interference between two condensates.Interference between two condensates.
dx
TOFTOF
† †1 2 1 2
int
( , ) ( , ) ( , ) ( , ) ( , )
( , ) ( , )
x t a x t a x t a x t a x t
x t x t
† †int 1 2 2 1( , ) ( , ) ( , ) ( , ) ( , )x t a x t a x t a x t a x t
Free expansion:Free expansion:
11 1 1 1
22 2 2 2
( / 2) ( , ) ~ exp ,
( / 2) ( , ) ~ exp ,
mv m x dt a x t a iQ x Q
tmv m x d
a x t a iQ x Qt
† †
int 1 2 2 1( , ) exp( ) exp( ), md
x t a a iQx a a iQx Qt
1,2
1,2 int ( ) cosia Ne x N Qx Andrews Andrews et. al. 1997et. al. 1997
What do we observe?What do we observe?
b) Uncorrelated, but well defined phases b) Uncorrelated, but well defined phases intint(x)(x)=0=0
2 2int int( ) ( ) ~ cos cos ~ cos ( ) 0x y N Qx Qy N Q x y
Hanbury Brown-Twiss EffectHanbury Brown-Twiss Effect
x
TOFTOF
c) Initial number state. No phases?c) Initial number state. No phases?
† † 2int int 1 1 2 2( ) ( ) ~ cos ( ) ~ cos ( )x y a a a a Q x y N Q x y
Work with original bosonic fields:Work with original bosonic fields:† †
int 1 2 2 1( ) ~ exp( ) exp( ) =0x a a iQx a a iQx
int ( ) cosx N Qx
a)a) Correlated phases Correlated phases (( = 0) = 0)
int ( ) cosx N Qx
First theoretical explanation: I. Casten and J. Dalibard (1997): showed that First theoretical explanation: I. Casten and J. Dalibard (1997): showed that the measurement induces random phases in a thought experiment.the measurement induces random phases in a thought experiment.
Experimental observation of interference between ~ 30 condensates Experimental observation of interference between ~ 30 condensates in a strong 1D optical lattice: Hadzibabic et.al. (2004).in a strong 1D optical lattice: Hadzibabic et.al. (2004).
24 2 2 0Q Q QA A A Easy to check Easy to check that at large N:that at large N:
The interference amplitude The interference amplitude does notdoes not fluctuate! fluctuate!
2 † †1 1 2 2 QA a a a a
† † 2int int 1 1 2 2( ) ( ) ~ cos ( ) cos ( )Qx y a a a a Q x y A Q x y
Interference amplitude squared. Observable!Interference amplitude squared. Observable!
Polar plots of the fringe amplitudes and phases for Polar plots of the fringe amplitudes and phases for 200 images obtained for the interference of about 200 images obtained for the interference of about 30 condensates. (a) Phase-uncorrelated 30 condensates. (a) Phase-uncorrelated condensates. (b) Phase correlated condensates. condensates. (b) Phase correlated condensates. Insets: Axial density profiles averaged over the 200 Insets: Axial density profiles averaged over the 200 images.images.
Z. Hadzibabic et. al., Phys. Rev. Lett. 93, 180401 (2004).
1.1. Access to correlation functions.Access to correlation functions.a)a) Scaling of Scaling of AAQQ
22 with L and with L and : power-law exponents. Luttinger : power-law exponents. Luttinger
liquid physics in 1D, Kosterlitz-Thouless phase transition in 2D.liquid physics in 1D, Kosterlitz-Thouless phase transition in 2D.b)b) Probability distribution W(Probability distribution W(AAQQ
22)): all order correlation functions.: all order correlation functions.
2.2. Direct simulator (solver) for interacting problems. Direct simulator (solver) for interacting problems. Quantum impurity in a 1D system of interacting fermions (an example).Quantum impurity in a 1D system of interacting fermions (an example).
3.3. Potential applications to many other systems.Potential applications to many other systems.
This talk:
Imag
ing
bea
m
L
What if the condensates are fluctuating?What if the condensates are fluctuating?
x
z
z1
z2
AQ
†int 1 20
( ) exp( ) ( ) ( ) c.c.L
x iQx a z a z dz
2 † †1 1 1 2 2 1 2 2 1 20 0( ) ( ) ( ) ( )
L L
QA a z a z a z a z dz dz
Identical homogeneous condensates:Identical homogeneous condensates:
22 †
1 10( ) (0)
L
QA L a z a dz
Interference amplitude contains information about fluctuations Interference amplitude contains information about fluctuations within each condensate.within each condensate.
2int int
2 † †1 1 2 1 2 2 1 2 1 20 0
( ) ( ) cos ( )
( ) ( ) ( ) ( )
Q
L L
Q
x y A Q x y
A a z a z a z a z dz dz
Scaling with L: two limiting casesScaling with L: two limiting cases
†int 1 2( ) ( ) ( ) exp( ) . . exp( ) . .z zz z
x a z a z iQx c c N iQx i c c
QA L
Ideal condensates:Ideal condensates:L x
z
Interference contrast Interference contrast does not depend on L.does not depend on L.
L x
z
Dephased condensates:Dephased condensates:
QA L
Contrast scalesContrast scales as L as L-1/2-1/2..
Formal derivation:Formal derivation:
22 †
1 10( ) (0)
L
QA L a z a dz
Ideal condensate: Ideal condensate: †1 1( ) (0) ca z a
2 2Q cA L
L
Thermal gas:Thermal gas:
†1 1( ) (0) ~ exp( / )a z a z
2QA L
L
Intermediate case (quasi long-range order).Intermediate case (quasi long-range order).2
2 †1 10( ) (0)
L
QA L a z a dz
z
1D condensates (Luttinger liquids):1D condensates (Luttinger liquids):
1/ 2†1 1( ) (0) /
K
ha z a z
L
1/ 22 2 1/ 1/ , Interference contrast /KK K
Q h hA L L
Repulsive bosons with short range interactions: Repulsive bosons with short range interactions: 2 2
2
Weak interactions 1
Strong interactions (Fermionized regime) 1
Q
Q
K A L
K A L
Finite temperature:Finite temperature:
1 1/22 2 1
K
Q hh
A Lm T
Angular Dependence.Angular Dependence.
† ( tan )int 1 20
†1 20
( ) ( ) ( ) c.c.
exp( ) ( ) ( ) +c.c., tan
L iQ x z
L iqz
x a z a z e dz
iQx a z a z e dz q Q
2 † †1 1 1 2 2 1 2 2 2 1 1 20 0
( ) ( ) ( ) ( ) ( ) cos ( )L L
QA q a z a z a z a z q z z dz dz
q is equivalent to the relative momentum of the two q is equivalent to the relative momentum of the two condensates (always present e.g. if there are dipolar condensates (always present e.g. if there are dipolar oscillations).oscillations).
z
x(z 1)
x(z 2)
(for the imaging beam (for the imaging beam orthogonal to the orthogonal to the page, page, is the angle of is the angle of the integration axis the integration axis with respect to z.)with respect to z.)
22 †
0( ) ( ) (0) cos( )
L
QA q L a z a qz dz
Angular (momentum) Dependence.Angular (momentum) Dependence.
1qL
2
22 2
21 1/
( ) , ideal condensates ( 1);
1( ) , finite T (short range correlations);
1
1( ) , quasi-condensates finite K.
Q
Q
Q K
A q q K
A qq
A qq
has a cusp singularity for K<1, relevant for fermions.2 ( )QA q
Two-dimensional condensates at finite temperatureTwo-dimensional condensates at finite temperatureCCD
camera
x
z
Time of
flight
z
xy
imaginglaser
(picture by Z. Hadzibabic)(picture by Z. Hadzibabic)
22 †
0 0( ) ( ', ) (0,0)
yX L
Q yA X X L dx a x y a
Elongated condensates: Elongated condensates: LLxx>>>>LLy y ..
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
“atom lasers”
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)
0
Observing the Kosterlitz-Thouless transition
Above KT transition Ly
Lx
2 ( )QA X X
Below KT transition
2 2 2QA X
LLxxLLyy
Universal jump of at TKT
KTT T 1/ 4
KTT T 1/ 2
2 2 2( )QA X X
Always algebraic scaling, easy to detect.Always algebraic scaling, easy to detect.
Zoran Hadzibabic, Zoran Hadzibabic, Peter Kruger, Marc Cheneau, Baptiste Battelier, Peter Kruger, Marc Cheneau, Baptiste Battelier, Sabine Sabine Stock,Stock, and Jean Dalibard (2006).and Jean Dalibard (2006).
integration
over x axis
X
z
z
integration
over x axisz
x
integration distance X
(pixels)
Contrast after integration0.4
0.2
00 10 20 30
middle Tlow T
high T
2 2
2212
0
( ) 1 1( ) ~ ~ (0, ) ~
XQA X
C X g x dxX X X
Interference contrast:Interference contrast:
Exp
onen
t
central contrast
0.5
0 0.1 0.2 0.3
0.4
0.3 high T low T
T (K)1.0 1.1 1.2
1.0
0
“universal jump in the superfluid density”
c.f. Bishop and Reppy
Z. Hadzibabic et. al.Z. Hadzibabic et. al.
Vortex proliferationVortex proliferation
Fraction of images showing at least one dislocation:
0
10%
20%
30%
central contrast0 0.1 0.2 0.3 0.4
high T low T
Higher Moments.Higher Moments.
2 † †1 1 1 2 2 1 2 2 1 20 0( ) ( ) ( ) ( )
L L
QA a z a z a z a z dz dz is an observable is an observable quantum operatorquantum operator
22 †
1 2 1 1 1 20 0( ) ( )
L L
QA dz dz a z a z
Identical condensates. Mean:Identical condensates. Mean:
Similarly higher momentsSimilarly higher moments2
2 † †1 1 1 1 1 1 1 10 0
... ... ... ( )... ( ) ( )... ( )L Ln
Q n n n nA dz dz dz dz a z a z a z a z
Probe of the higher order correlation functions. Probe of the higher order correlation functions.
Nontrivial statistics if the Wick’s theorem is not fulfilled!Nontrivial statistics if the Wick’s theorem is not fulfilled!
Distribution function (= full counting statistics):Distribution function (= full counting statistics):2 2 2 2 2( ) : ( )n nQ Q Q Q QW A A A W A dA
Non-interacting non-condensed Non-interacting non-condensed regime (Wick’s theorem):regime (Wick’s theorem):
2 2( ) exp( )Q QW A C CA
1D condensates at zero temperature:1D condensates at zero temperature:
Low energy action:Low energy action:
ThenThen
1/ 2
( ) †( ) , ( ) ( )
K
i y hc ca y e a y a y
y y
SimilarlySimilarly1/ 22
1 2 1 2† † 21 2 1 2
1 2 1 2 1 1 2 2
( ) ( ) ( ) ( )
K
hc
y y y ya y a y a y a y
y y y y y y y y
Easy to generalize to all orders.Easy to generalize to all orders.
Changing open boundary conditions to periodic findChanging open boundary conditions to periodic find
2 1/ 2 1 1/ 22
nn K KQ c h nA C L Z
These integrals can be evaluated using Jack polynomials These integrals can be evaluated using Jack polynomials ((Fendley, Lesage, Saleur, J. Stat. Phys. 79:799 (1995))
Explicit expressions are cumbersome (slowly converging series Explicit expressions are cumbersome (slowly converging series of products).of products).
1
21
2 2 2 20 1
( 1/ 2 )1 (1 1/ )
(1/ 2 ) (1 ) (1 1/ 2 )
K KZ
K K
2 1
2
1 24 4
0 1 2
1/ 1/ 24
(1/ 2 ) 1/ 2 1 1
K KZ
K K
Two simple limits:Two simple limits:
2 221: !, ( ) exp( )n Q QK Z n W A C CA
(also in thermal case)
x
z
z1
z2
AQ 2 2 22 0
2 24 2
22
: 1, ( ) ,
6
n Q Q
Q
Q
K Z W A A A
A Z Z
ZA K
Strongly interacting Tonks-Girardeau regimeStrongly interacting Tonks-Girardeau regime
Weakly interacting BEC like regime.Weakly interacting BEC like regime.
Connection to the impurity in a Luttinger liquid problem.Connection to the impurity in a Luttinger liquid problem.
Boundary Sine-Gordon theory:Boundary Sine-Gordon theory:
2 2
0 0
exp ,
2 cos 2 (0, )2 x
Z D S
KS dx d g d
21/ 2
22( ) , 2 ,!
nK
nn
xZ x Z x g
n
Same integrals as in the expressions for (we rely on Euclidean Same integrals as in the expressions for (we rely on Euclidean invariance).invariance).
2nQA
2 20 00
( ) ( ) (2 / ) ,Z x W A I Ax A dA
1/ 2 1 1/ 20
K Kc hA C L
P. Fendley, F. Lesage, H. Saleur (1995).
20 02 0
0
2( ) ( ) (2 / ) ,W A Z ix J Ax A xdx
A
Experimental simulation of the quantum impurity problemExperimental simulation of the quantum impurity problem
1.1. Do a series of experiments and determine the distribution function.Do a series of experiments and determine the distribution function.
T. Schumm, et. al., Nature Phys. 1, 57 (2005).
Distribution of interference phases (and amplitudes) from two 1D condensates.Distribution of interference phases (and amplitudes) from two 1D condensates.
2.2. Evaluate the integral.Evaluate the integral.2 2
0 00( ) ( ) (2 / ) ,Z x W A I Ax A dA
3.3. Read the result. Read the result.
( )Z x can be found using Bethe ansatz methods for half integer K.can be found using Bethe ansatz methods for half integer K.
In principle we can find In principle we can find WW::
20 02 0
0
2( ) ( ) (2 / ) ,W A Z ix J Ax A xdx
A
Difficulties: need to do analytic continuation. Difficulties: need to do analytic continuation. The problem becomes increasingly harder as The problem becomes increasingly harder as K K increases.increases.
Use a different approach based on spectral determinant:Use a different approach based on spectral determinant:
Dorey, Tateo, J.Phys. A. Math. Gen. 32:L419 (1999);
Bazhanov, Lukyanov, Zamolodchikov, J. Stat. Phys. 102:567 (2001)
2
0
( ) 1n n
Z ixE
sin / 2K
x
0 1 2 3 4
Pro
babi
lity
W(
)
K=1 K=1.5 K=3 K=5
Evolution of the distribution function.Evolution of the distribution function.
2 2Q QA A
Universal asymmetric distribution at large KUniversal asymmetric distribution at large K
((-1)/-1)/
Further extensions:Further extensions:
( ) ( )vacZ ix Q sin / 2K
x
( )Q is the Baxter is the Baxter QQ-operator, related to the transfer matrix of -operator, related to the transfer matrix of conformal field theories with negative charge:conformal field theories with negative charge:
Yang-Lee singularity
2D quantum gravity,non-intersecting loops on 2D lattice
22 1
1 3K
cK
Spinless Fermions.Spinless Fermions.
1
22 † †
1 20
sin( )( ) (0) , ( ) (0)
L fQ K K
k zA L a z a dz a z a
z
Note that K+K-1 2, so and the distribution function is always Poissonian.
2QA L
However for K+K-1 3 there is a universal cusp at nonzero momentum as well as at 2kf:
2
2 †
0( ) ( ) (0) cos , tan
L
QA q L a z a qz dz q Q
1 12 2( ) (0) .K K
Q QA q A q There is a similar cusp at There is a similar cusp at 2kf
Higher dimensions: nesting of Fermi surfaces, CDW, … Higher dimensions: nesting of Fermi surfaces, CDW, … Not a low energy probe!Not a low energy probe!
Fermions in optical lattices.Fermions in optical lattices.
Possible efficient probes of superconductivity (in Possible efficient probes of superconductivity (in particular, d-wave vs. s-wave). particular, d-wave vs. s-wave). Not yet, but coming!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.Conclusions.1.1. Analysis of interference between independent condensates Analysis of interference between independent condensates
reveals a wealth of information about their internal structure.reveals a wealth of information about their internal structure.
a)a) Scaling of interference amplitudes with Scaling of interference amplitudes with LL or or :: correlation function correlation function exponents. exponents. Working example: detecting KT phase transition.Working example: detecting KT phase transition.
b)b) Probability distribution of amplitudes (= full counting statistics of Probability distribution of amplitudes (= full counting statistics of atoms): information about higher order correlation functions.atoms): information about higher order correlation functions.
c)c) Interference of two Luttinger liquids: partition function of 1D Interference of two Luttinger liquids: partition function of 1D quantum impurity problem (also related to variety of other quantum impurity problem (also related to variety of other problems like 2D quantum gravity).problems like 2D quantum gravity).
2.2. Vast potential applications to many other systems, e.g.:Vast potential applications to many other systems, e.g.:
a)a) Fermionic systems: superconductivity, CDW orders, etc..Fermionic systems: superconductivity, CDW orders, etc..
b)b) Rotating condensates: instantaneous measurement of the correlation Rotating condensates: instantaneous measurement of the correlation functions in the rotating frame.functions in the rotating frame.
c)c) Correlation functions near continuous phase transitions.Correlation functions near continuous phase transitions.
d)d) Systems away from equilibrium.Systems away from equilibrium.
Consider slow tuning of a system through a critical point.Consider slow tuning of a system through a critical point.
tuning parameter tuning parameter
gap
gap
t, t, 0 0
Gap vanishes at the transition. Gap vanishes at the transition. No true adiabatic limit!No true adiabatic limit!
How does the number of excitations scale with How does the number of excitations scale with ? ?
This question is valid for isolated systems with stable excitations: This question is valid for isolated systems with stable excitations: conserved quantities, topological excitations, integrable models. conserved quantities, topological excitations, integrable models.
Universal adiabatic dynamics across a Universal adiabatic dynamics across a quantum critical pointquantum critical point
Use a general many-body perturbation theory.Use a general many-body perturbation theory.
Expand the wave-function in many-body basis.Expand the wave-function in many-body basis.
i Ht
Substitute into SchrSubstitute into Schrödinger equation.ödinger equation.
Uniform system: can characterize excitations by momentum:Uniform system: can characterize excitations by momentum:
Use scaling relations:Use scaling relations:
Find:Find:
Caveats:Caveats:
1.1. Need to check convergence of integrals (no cutoff dependence)Need to check convergence of integrals (no cutoff dependence)
Scaling fails in high dimensions.Scaling fails in high dimensions.
2.2. Implicit assumption in derivation: small density of Implicit assumption in derivation: small density of excitations does not change much the matrix element to excitations does not change much the matrix element to create other excitations.create other excitations.
3.3. The probabilities of isolated excitations:The probabilities of isolated excitations:
should be smaller than one. Otherwise need to solve Landau-should be smaller than one. Otherwise need to solve Landau-Zeener problem. The scaling argument gives that they are of Zeener problem. The scaling argument gives that they are of the order of one. Thus the scaling is not affected.the order of one. Thus the scaling is not affected.
Simple derivation of scaling (similar to Kibble-Zurek mechanism):Simple derivation of scaling (similar to Kibble-Zurek mechanism):
Breakdown of adiabaticity:Breakdown of adiabaticity:
From From t t and we getand we get
In a non-uniform system we find in a similar manner:In a non-uniform system we find in a similar manner:
Example: transverse field Ising model.Example: transverse field Ising model.
0 1z zi jg
1xig
There is a phase transition at There is a phase transition at g=g=11..
This problem can be exactly solved using Jordan-Wigner This problem can be exactly solved using Jordan-Wigner transformation:transformation:
† † †
1
2 1, ( 1) ( )x zi i i i j j j j
j i
c c c c c c
Spectrum:Spectrum:
22 22 1 2 cos 2 1k g g k g k
Critical exponents: Critical exponents: z=z===1 1 dd/(z/(z +1) +1)=1/2.=1/2.
Correct result (J. Dziarmaga 2005):Correct result (J. Dziarmaga 2005): 0.11exn
0.18exn
Other possible applications: quantum phase transitions in cold Other possible applications: quantum phase transitions in cold atoms, adiabatic quantum computations, etc.atoms, adiabatic quantum computations, etc.
