Collaboration: Ehud Ehud AltmanAltman - - The Weizmann ...€¦ · Ehud AltmanAltman - - The Weizmann . Weizmann Institute of Science. Eugene Demler . Demler - - Harvard University.

Interference between independent cold atom systemsInterference between independent cold atom systems

Anatoli Polkovnikov,Anatoli Polkovnikov,Boston UniversityBoston University

Collaboration:

EhudEhud AltmanAltman -- The The WeizmannWeizmann Institute of ScienceInstitute of ScienceEugene Eugene DemlerDemler -- Harvard UniversityHarvard UniversityVladimir Vladimir GritsevGritsev -- Harvard UniversityHarvard University

AFOSRAFOSR

Workshop on Quantum Noise in Strongly Workshop on Quantum Noise in Strongly Correlated Systems. Correlated Systems.

WeizmannWeizmann Institute, Jan. 2008Institute, Jan. 2008

Interference between independent sources.Interference between independent sources.((HanburyHanbury--Brown Brown TwissTwiss effect)effect)

Interference between independent sources.Interference between independent sources.((HanburyHanbury--Brown Brown TwissTwiss effect)effect)

xx11

xx22

Origin of interference Origin of interference –– superposition principle.superposition principle.

1 1 1 2 1

2 1 2 2 2

( ) ( ) ( )( ) ( ) ( )

E x E x E xE x E x E x

= += +

2 * *1 1 1 1 2 1 1 1 2 1 2 1 1 1( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )I x E x I x I x E x E x E x E x∝ = + + +

Interference term drops out for uncorrelated sourcesInterference term drops out for uncorrelated sources* *

1 2 1 1 1 2 2 2 2 1 1 2( ) ( ) ... ( ) ( ) ( ) ( )I x I x E x E x E x E x x x∝ + + ↔

IntensityIntensity--intensity intensity correlatorcorrelator survives!survives!

HBT effect is the classical wave phenomenon!HBT effect is the classical wave phenomenon!

From classical waves to quantum particlesFrom classical waves to quantum particles

xx11

xx22

Origin of interference Origin of interference –– superposition principle.superposition principle.

1 1 1 2 1

2 1 2 2 2

ˆ ˆ ˆ( ) ( ) ( )ˆ ˆ ˆ( ) ( ) ( )a x a x a xa x a x a x

= += +

† † †1 1 1 1 1 2 1 1 1 2 1 2 1 1 1( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )n x a x a x n x n x a x a x a x a x∝ = + + +

Interference term drops out for uncorrelated sourcesInterference term drops out for uncorrelated sources† †

1 2 1 1 1 2 2 2 2 1 1 2( ) ( ) ... ( ) ( ) ( ) ( )n x n x a x a x a x a x x x∝ ± + ↔

DensityDensity--Density Density correlatorcorrelator survives!survives!

Note: at this level it is not important whether sources have ranNote: at this level it is not important whether sources have random dom phases or have fixed number of particles. phases or have fixed number of particles. ((CastinCastin DalibardDalibard 1997)1997)

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

ρ

ρ ρ

= + +

= +

† †int 1 2 2 1( , ) ( , ) ( , ) ( , ) ( , )x t a x t a x t a x t a x tρ = +

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 Qt

mv m x da x t a iQ x Qt

−→∞ = =

+= =

h h

h h

† †int 1 2 2 1( , ) exp( ) exp( ), mdx t a a iQx a a iQx Q

tρ + − =

h

( )1,21,2 int ( ) cosia Ne x N Qxϕ ρ δϕ⇒ ∝ +

What do we observe?What do we observe?

b) Uncorrelated, but well defined phases b) Uncorrelated, but well defined phases ⇒⇒

<<ρρintint (x(x))>>=0=0

( ) ( ) ( )2 2int int( ) ( ) ~ cos cos ~ cos ( ) 0x y N Qx Qy N Q x yρ ρ δϕ δϕ+ + − ≠

HanburyHanbury BrownBrown--TwissTwiss EffectEffect

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 yρ ρ − −

Work with original Work with original bosonicbosonic fields:fields:† †

int 1 2 2 1( ) ~ exp( ) exp( ) =0x a a iQx a a iQxρ + −

( )int ( ) cosx N Qxρ δϕ∝ +

a)a) Correlated phases Correlated phases ((δϕδϕ = 0= 0) ) ⇒⇒ ( )int ( ) cosx N Qxρ ∝

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 tρ ρ ρ= − −∫∫ ∫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≡ = + + +

Shot (counting) noise distinguishes quantum particles from classShot (counting) noise distinguishes quantum particles from classical ical waves (only affects fluctuations of the interference amplitude):waves (only affects fluctuations of the interference amplitude):

24 2

22

2 11 two coherent states

2 1 11 two Fock states2

A A NNA

N N N

⎧+ −⎪− ⎪= ⎨

⎪ + − −⎪⎩

The interference amplitude The interference amplitude does notdoes not fluctuate at large N!fluctuate at large N! CastinCastin, , DalibardDalibard 1997.1997.

A.P. 2006A.P. 2006ImambekovImambekov, , GritsevGritsev, , DemlerDemler, 2007, 2007

Interference from multiple sourcesInterference from multiple sources

dx

Having many particles in each Having many particles in each condensate does not reduce noise in condensate does not reduce noise in the interference amplitude.the interference amplitude.

Redefine the observableRedefine the observable

( )2 ( , ) ( , ) exp ( ) ( , )nn

A dxdy x t y t iQ x y dx x tρ ρ ρ= − −∑∫∫ ∫

nmdQ nQ n

t= =

hUtilize higher momentum harmonics.Utilize higher momentum harmonics.

24 2

2

( ) , ( ) 1.6, (1) 4.2A A C Nw C C

A M−

= ≅ ∞ ≅ ≅

Noiseless HBT signal for large number of sources, true both for Noiseless HBT signal for large number of sources, true both for bosonsbosons and for and for fermionsfermions. We can have classical . We can have classical GrassmanGrassman waves for fermions!waves for fermions!

x

z

z1

z2

AQ

†int 1 20

( ) exp( ) ( ) ( ) c.c.L

x iQx a z a z dzρ + ⇒∫

† †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

ρ ρ = −

∫ ∫

Extended Condensates.Extended Condensates.

Scaling with L: two limiting casesScaling with L: two limiting cases†

int 1 2( ) ( ) ( ) exp( ) . . exp( ) . .z zz zx a z a z iQx c c N iQx i c cρ δϕ∝ + ∝ + +∑ ∑

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

DephasedDephased condensates:condensates:

A L∝

Contrast scalesContrast scales as Las L--1/21/2..

Formal derivation:Formal derivation:2

†2 1 10

( ) (0)L

A L a z a dz∫Ideal condensate: Ideal condensate:

†1 1( ) (0) ca z a ρ→

22 cA Lρ

L

Thermal gas:Thermal gas:

†1 1( ) (0) ~ exp( / )a z a zρ ξ−

2A Lρ ξ

L

Intermediate case (quasi longIntermediate case (quasi long--range order).range order).2

†2 1 10

( ) (0)L

A L a z a dz∫

z

1D condensates (1D condensates (LuttingerLuttinger liquids):liquids):

( )1/ 2†1 1( ) (0) / K

ha z a zρ ξ≅

L

( )1/ 22 1/ 1/2 , Interference contrast / KK K

h hA L Lξ ξ− ∝

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 LK A L

→→

Finite temperature:Finite temperature:1 1/2

22

1K

hh

A Lm T

ξ ρξ

−⎛ ⎞⎜ ⎟⎝ ⎠

h

Observing the Kosterlitz-Thouless transition

Above KT transition Ly

Lx

2 ( )QA X Xξ∝

Below KT transition

2 2 2QA X α−∝

LLxx >>>>LLyy

Universal jump of γ

at TKT

KTT T 1/ 4γ γ − =

KTT T> 1/ 2γ γ += =

2 2 2( )QA X X γ−∝

Always algebraic scaling, easy to detect.Always algebraic scaling, easy to detect.

Higher Moments Higher Moments (More in talk by V. (More in talk by V. GritsevGritsev))

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 Kn 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: δδAA2n 2n / A/ A2n2n ~ 1~ 1 / L/ L11--1/K1/K

Shot noise is subdominant for K>1 at T=0.Shot noise is subdominant for K>1 at T=0.

Two simple limits:Two simple limits:

2 21: ( ) exp( )K W A C CA→ → −

Central limit theorem! Also at finite T.

x

z

z1

z2

A

Strongly interacting Strongly interacting TonksTonks--Girardeau regimeGirardeau regime

( )2 2 20

224 2

22

: ( ) ,

6

K W A A A

Z ZAZA K

δ

δ π

→∞ → −

−=

Weakly interacting BEC like regime.Weakly interacting BEC like regime.

Connection to the impurity in a Connection to the impurity in a LuttingerLuttinger liquid problem.liquid problem.

2 20 00

( ) ( ) (2 / ) ,Z W A I A A dAλ λ∞

= ∫ 1/ 2 1 1/ 20

K Kc hA C Lρ ξ −=

20 02 0

0

2( ) ( ) (2 / ) ,W A Z i J A A dA

λ λ λ λ∞

= ∫

Experimental simulation of the quantum impurity problemExperimental simulation of the quantum impurity problem1.1. Do a series of experiments and determine the distribution Do a series of experiments and determine the distribution

function of the interference amplitude.function of the interference amplitude.

2.2. Evaluate the integral.Evaluate the integral.

3.3. Read the result. Read the result.

(more in V. (more in V. GritsevGritsev talk on Thursday).talk on Thursday).

0 1 2 3 4

Pro

babi

lity

W(α

)

α

K=1 K=1.5 K=3 K=5

Evolution of the distribution function.Evolution of the distribution function.

22A Aα =

Universal Universal GumbelGumbel distribution at large K distribution at large K

((αα--1)/1)/δαδα

exp( / )e α δα−

/eα δα

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. E. BretinBretin, Phys. Rev. , Phys. Rev. LettLett. . 9595, 170601 (2005) , 170601 (2005)

From independent From independent random variables to random variables to correlated intervalscorrelated intervals

Also Also 1/f1/f noisenoise 2

[0, ]( ) ( )

t Tw T h t h

∈= −

Other examples of extreme value statistics.Other examples of extreme value statistics.

Detecting Detecting fermionicfermionic superfluiditysuperfluidity (D(D--wave…)wave…)

† †int 1 2 2 1( , ) exp( ) exp( ),x t a a iQx a a iQx

mdQt

ρ + −

=hd

TOFTOF

( )2 ( , ) ( , ) exp ( )A dxdy x t y t iQ x yρ ρ= −∫∫BeforeBefore

2 † † † †2 1 1 2 2 1 1 2 2 A A a a a a a a a a≡ = = Good for shot noise but not Good for shot noise but not

for for superfluiditysuperfluidity..

( )2 ( , ) ( , ) exp ( )B dxdy x t y t iQ x yρ ρ= +∫∫NowNow

2 † † *2 1 21 1 2 2 B B a a a a↑ ↓ ↓ ↑≡ = = Δ Δ Problem with gauge invariance Problem with gauge invariance

(undefined relative phase)(undefined relative phase)

Can go to 4Can go to 4thth order correlation functions. Shot noise will order correlation functions. Shot noise will kill us. kill us.

Two ways aroundTwo ways around

Introduce weak tunneling coupling such that Introduce weak tunneling coupling such that macroscopic phases are locked but macroscopic phases are locked but correlation functions are not yet affected.2 correlation functions are not yet affected.2

2*2 1 2 0B = Δ Δ ≅ Δ ≥

1.1.

2.2. * * *2 2 1 1 2 2LR L R L R R LB B B= Δ Δ Δ Δ

GuageGuage invariance is restored invariance is restored because of long range coherence. because of long range coherence. Still measure local gap!Still measure local gap!

Fermions on a lattice.Fermions on a lattice.

Different orientations of Different orientations of imaging beams give strong imaging beams give strong angular dependence of the angular dependence of the signal in Dsignal in D--wave case.wave case.

Can detect even the sign of Can detect even the sign of the pairing gap using two the pairing gap using two different imaging beams different imaging beams for Bfor BLL and Band BRR and having and having one Done D--condensate and one condensate and one SS--condensate. condensate.

* * *2 2 1 1 2 2LR L R L R R LB B B= Δ Δ Δ Δ

Conclusions.Conclusions.

Two sources of noise in the interference:Two sources of noise in the interference:a) thermal or quantum fluctuationsa) thermal or quantum fluctuations b) shot noiseb) shot noise

Mean amplitude of interference contains information on Mean amplitude of interference contains information on twotwo--particle correlation functions. Higher moments particle correlation functions. Higher moments contain additional information.contain additional information.

Interference is a powerful tool for studying correlated Interference is a powerful tool for studying correlated cold atom systems.cold atom systems.