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

Hanbury Brown-Twiss EffectHanbury Brown-Twiss Effect

x

TOFTOF

c) Initial number state. c) Initial number state.

† † 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

I. Casten and J. Dalibard (1997).I. Casten and J. Dalibard (1997).

Z. Hadzibabic et.al. (2004).Z. Hadzibabic et.al. (2004).

24 2

2

2

2 11 two coherent states

2 1 11 two Fock states

2

A A NNA

N N N

The interference amplitude The interference amplitude does notdoes not fluctuate at large N! fluctuate at large N!

Define an observable (Define an observable (interference amplitude squared interference amplitude squared ):):

2 ( , ) ( , ) exp ( ) ( , )A dxdy x t y t iQ x y dx x t 2 † † † †

2 1 1 2 2 1 1 2 2 A A a a a a a a a a depends only on Ndepends only on N

4 † 2 2 † 2 2 † 2 2 † † 2 2 † †4 1 1 2 2 1 1 2 2 1 1 1 1 2 24 4 2A A a a a a a a a a a a a a a a

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

A a z a z a z a z dz dz

Identical homogeneous condensates:Identical homogeneous condensates:

2†

2 1 10( ) (0)

LA 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 ( )

( ) ( ) ( ) ( )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

†2 1 10

( ) (0)L

A 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 1/ 1/2 , Interference contrast /

KK Kh 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

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

quantum operatorquantum operator

2† 2 1/

2 1 2 1 1 1 20 0( ) ( )

L L KA dz dz a z a z L

Identical condensates. Mean:Identical condensates. Mean:

Similarly higher momentsSimilarly higher moments2

† † (2 1/ )2 1 1 1 1 1 1 10 0

.. ... ( )... ( ) ( )... ( )L L n K

n n n nA dz dz a z a z a z a z L

Probe of the higher order correlation functions. Probe of the higher order correlation functions.

Universal (size independent) Universal (size independent) distribution function:distribution function:

2 2 2 2 2( ) : ( )n nW A A A W A dA

Shot noise contribution: Shot noise contribution: AA2n2n~ L~ Ln(2-1/K) n(2-1/K) / L/ L1-1/K1-1/K

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))

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( )nK Z n W A C CA

Central limit theorem! Also at finite T.

x

z

z1

z2

A

Strongly interacting Tonks-Girardeau regimeStrongly interacting Tonks-Girardeau regime

2 2 22 0

224 2

22

: 1, ( ) ,

6

nK Z W A A A

Z ZA

ZA K

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).

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)

Numerical simulationsNumerical simulations

0 10 20 30 40 500.80

0.85

0.90

0.95

1.00N=1, U=0.5, J=1

Pha

se C

oher

ence

Time

Hubbard model, Hubbard model, 2 chains, 6 sites each2 chains, 6 sites each

0.0 0.2 0.4 0.60.00

0.01

0.02

0.03

0.04

0.05

0.06N=1, U=0.5, J=1, J=0.1, M=6

Po

we

r S

pe

ctru

m

Frequency

bb0202

bb2424

bb4646 bb0404bb26262s2s0101 2b2b0202

Fourier analysis of the oscillations is a way to perform Fourier analysis of the oscillations is a way to perform spectroscopy.spectroscopy.

Quench in 2D condensatesQuench in 2D condensates

Expect a very sharp change in TExpect a very sharp change in TKTKT as as

a function of the layer separation. a function of the layer separation.

A simple entropic argument:A simple entropic argument:

rrEnergy ~ JEnergy ~ Jrr2 2 expect confinement expect confinement

KT argumentKT argument2 2

2 2 ln 02

c cKT KT

RT T

m m

JJ=0:=0:

2 2* *2 2 ln 0c c

KT KT

RT T

m m

JJ>0:>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.

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