Static and dynamic probes of Static and dynamic probes of strongly interacting low-dimensional strongly interacting low-dimensional atomic systems. atomic systems. Anatoli Polkovnikov, Anatoli Polkovnikov, Boston University Boston University Collaboration: Altman Altman - - The Weizmann Institute of Scie The Weizmann Institute of Scie nio Castro Neto nio Castro Neto - - Boston University Boston University ne Demler ne Demler - - Harvard University Harvard University mir Gritsev mir Gritsev - - Harvard University Harvard University na Kollath na Kollath - - University of Geneva University of Geneva ig Mattey ig Mattey - - Harvard University Harvard University
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Static and dynamic probes of strongly interacting low-dimensional atomic systems. Anatoli Polkovnikov, Boston University Collaboration: Ehud Altman- The.
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Static and dynamic probes of strongly interacting Static and dynamic probes of strongly interacting low-dimensional atomic systems.low-dimensional atomic systems.
Anatoli Polkovnikov,Anatoli Polkovnikov,Boston UniversityBoston University
Collaboration:
Ehud AltmanEhud Altman - - The Weizmann Institute of ScienceThe Weizmann Institute of ScienceAntonio Castro NetoAntonio Castro Neto -- Boston UniversityBoston UniversityEugene Demler Eugene Demler - - Harvard UniversityHarvard UniversityVladimir Gritsev Vladimir Gritsev - - Harvard UniversityHarvard UniversityCorinna KollathCorinna Kollath -- University of GenevaUniversity of GenevaLudwig MatteyLudwig Mattey -- Harvard UniversityHarvard University
Why low dimensions?Why low dimensions?
1.1. Existence of many low-dimensional correlated Existence of many low-dimensional correlated phases: unconventional superconductivity, phases: unconventional superconductivity, fractionalized and topological phases, QHE, fractionalized and topological phases, QHE, Luttinger liquids, TG gas, etc.Luttinger liquids, TG gas, etc.
2.2. Excellent laboratory for studying dynamics and Excellent laboratory for studying dynamics and thermalization in nonintegrable and integrable thermalization in nonintegrable and integrable systems.systems.
3.3. Realization of new accurate interference probes, Realization of new accurate interference probes, which are not available in 3D systems.which are not available in 3D systems.
This talk:This talk:
1.1. Interference between two 1D systems of interacting Interference between two 1D systems of interacting bosons: bosons: • shot noiseshot noise• noise due to phase fluctuationsnoise due to phase fluctuations• full distribution functionfull distribution function
2.2. Quench experiments in 1D and 2D systems:Quench experiments in 1D and 2D systems:• excitations in coupled 1D condensatesexcitations in coupled 1D condensates• dynamics after the quench dynamics after the quench • quenching coupled 2D systems: Kibble Zurek quenching coupled 2D systems: Kibble Zurek
mechanism of topological defect formation.mechanism of topological defect formation.
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
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 ( )
( ) ( ) ( ) ( )L L
x y A Q x y
A a z a z a z a z dz dz
Fluctuating Condensates.Fluctuating Condensates.
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
A 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:
A L
Contrast scalesContrast scales as L as L-1/2-1/2..
Formal derivation:Formal derivation:
2†
2 1 10( ) (0)
LA L a z a dz
Ideal condensate: Ideal condensate: †1 1( ) (0) ca z a
22 cA L
L
Thermal gas:Thermal gas:
†1 1( ) (0) ~ exp( / )a z a z
2A L
L
Intermediate case (quasi long-range order).Intermediate case (quasi long-range order).2
Repulsive bosons with short range interactions: Repulsive bosons with short range interactions: 2
2
2
Weak interactions 1
Strong interactions (Fermionized regime) 1
K A L
K A L
Finite temperature:Finite temperature:
1 1/22
2
1K
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
A 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.)
Higher Moments.Higher Moments.
2 † †1 1 1 2 2 1 2 2 1 20 0( ) ( ) ( ) ( )
L LA a z a z a z a z dz dz is an observable is an observable
Shot noise is subdominant for K>1 at T=0.Shot noise is subdominant for K>1 at T=0.
Sketch of the derivationSketch of the derivation
Action:Action:
With periodic boundary conditions we find:With periodic boundary conditions we find:
1/ 2 1 1/ 22 2~
ni K Kc n c h na e A 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))
Same integrals as in the expressions for (we rely on Euclidean Same integrals as in the expressions for (we rely on Euclidean invariance).invariance).
2nA
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
bxZ ix
E
2 1/ 2 sin / 2
4 (1 1/ 2 )K K
b K K
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.
22A A
Universal Gumbel distribution at large K Universal Gumbel distribution at large K
((-1)/-1)/
exp( / )e
/e
1
( 1)( 1) exp[ ( 1)]
exp( ) 1( ) exp( )
e 1K K
K x K x
x KW x x x
K
Generalized extreme value distribution:Generalized extreme value distribution:
Emergence of extreme value statistics on other instances: Emergence of extreme value statistics on other instances:
1max { }n n nz z z E. Bretin, Phys. Rev. Lett. E. Bretin, Phys. Rev. Lett. 9595, 170601 (2005) , 170601 (2005)
From independent From independent random variables to random variables to correlated intervalscorrelated intervals
Also Also 1/f1/f noise noise2
[0, ]( ) ( )
t Tw T h t h
Other examples of extreme value statistics.Other examples of extreme value statistics.
Quench experiments in 1D and 2D systems:Quench experiments in 1D and 2D systems:
T. Schumm . et. al., Nature Physics 1, 57 - 62 (01 Oct 2005)
Study dephasing as a function of time. What sort of Study dephasing as a function of time. What sort of information can we get?information can we get?
Analyze dynamics of phase coherence:Analyze dynamics of phase coherence:
1 2( ( ) ( )) ( )( ) e ei t t i tf t
Idea: extract energies of excited states Idea: extract energies of excited states and thus go beyond static probes.and thus go beyond static probes.
Relevant model:Relevant model:
Excitations: solitons and breathers.Excitations: solitons and breathers.
Can create solitons Can create solitons only in pairs. Expect only in pairs. Expect damped oscillations :damped oscillations :
2( ) sin
ei t sm t
t
solitonssolitons
breathers (bound solitons) breathers (bound solitons) Can create isolated Can create isolated breathers. Expect breathers. Expect undamped oscillations:undamped oscillations:
( )e sini tbm t
Analogy with a Josephson junction.Analogy with a Josephson junction.
2
2
22 cosH
K
2
0 1/ 4 22
1 1exp ,
42 h
EEnn
breathersbreatherssoliton pairs soliton pairs (only with q(only with q0)0)
Sudden change in T/TSudden change in T/TKTKT can result in the Kibble- can result in the Kibble-
Zurek mechanism of the topological defect formation.Zurek mechanism of the topological defect formation.
Start at T>TStart at T>TKTKT quench to T<T quench to T<TKTKT..
If quench is fast If quench is fast we expect that vortices do not thermalize. we expect that vortices do not thermalize. Have nonequilibrium vortex population.Have nonequilibrium vortex population.
RG calculation for various RG calculation for various values of vortex fugacity. values of vortex fugacity.
Neglect dependence Neglect dependence cc(T).(T).
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.
b)b) Probability distribution of amplitudes: information about higher Probability distribution of amplitudes: information about higher order correlation functions.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).problems).
2.2. Quench experiments in 1D and 2D systems are a possible new Quench experiments in 1D and 2D systems are a possible new way of performing spectroscopy in cold atom systems:way of performing spectroscopy in cold atom systems:
a)a) Detecting solitons and breathers in 1D coupled condensatesDetecting solitons and breathers in 1D coupled condensates
b)b) Kibble-Zurec mechanism in 2D condensatesKibble-Zurec mechanism in 2D condensates
Such experiments are much simpler and more robust than e.g. Such experiments are much simpler and more robust than e.g. parametric resonance.parametric resonance.