Interference between fluctuating condensates Anatoli Polkovnikov, Boston University Collaboration: Ehud Altman-Weizmann Eugene Demler - Harvard Vladimir.

Interference between fluctuating condensatesInterference between fluctuating condensates

Anatoli Polkovnikov,Anatoli Polkovnikov,Boston UniversityBoston University

Collaboration:

Ehud AltmanEhud Altman -- WeizmannWeizmannEugene Demler Eugene Demler - - HarvardHarvardVladimir Gritsev Vladimir Gritsev - - HarvardHarvard

What do we observe interfering ideal condensates?What do we observe interfering ideal condensates?

a) Correlated phases a) Correlated phases (( = 0) = 0) I (x)I (x) N N cos(cos(QxQx))

Measure (interference part):Measure (interference part):

dx

TOFTOF

1,2

† †1 2

1,2

( ) , ,

( ) cos

iQx iQxQ Q Q

i

mdI x A e A e A a a Q

t

a Ne I x N Qx

Andrews Andrews et. al. 1997et. al. 1997

b) Uncorrelated, but well defined phases b) Uncorrelated, but well defined phases I(x) I(x) 0 0

2

2

( ) ( ) cos cos

~ cos ( ) 0

I x I y N Qx Qy

N Q x y

Hanbury Brown-Hanbury Brown-Twiss EffectTwiss Effect

c) Initial number state. No phases?c) Initial number state. No phases?

† †1 1 2 2

2 2

( ) ( ) cos ( )

cos( ( )) cos ( )Q

I x I y a a a a Q x y

A Q x y N Q x y

Work with original bosonic fields:Work with original bosonic fields:

The same answer as in the case b) with random but well The same answer as in the case b) with random but well defined phases! defined phases!

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!

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.

c)c) Fermions: cusp singularities in Fermions: cusp singularities in AAQQ22 ( ( ) ) corresponding to kcorresponding to kff..

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?

What are the advantages compared to the What are the advantages compared to the conventional TOF imaging?conventional TOF imaging?

1.1. TOF relies on free atom expansion. Often not true in strongly TOF relies on free atom expansion. Often not true in strongly correlated regimes. Interference method does not have this correlated regimes. Interference method does not have this problem.problem.

2.2. It is often preferable to have a direct access to the spatial It is often preferable to have a direct access to the spatial correlations. TOF images give access either to the momentum correlations. TOF images give access either to the momentum distribution or the momentum correlation functions.distribution or the momentum correlation functions.

3.3. Free expansion in low dimensional systems occurs Free expansion in low dimensional systems occurs predominantly in the transverse directions. This renders bad predominantly in the transverse directions. This renders bad signal to noise. In the interference method this is advantage: signal to noise. In the interference method this is advantage: longitudinal correlations remain intact. longitudinal correlations remain intact.

1.1. Algebraic correlations at zero temperature (Luttinger Algebraic correlations at zero temperature (Luttinger liquids). Exponential decay of correlations at finite liquids). Exponential decay of correlations at finite temperature.temperature.

2.2. Fermionization of bosons, bosonization of fermions. Fermionization of bosons, bosonization of fermions. (There is not much distinction between fermions and (There is not much distinction between fermions and bosons in 1D).bosons in 1D).

3.3. 1D systems are well understood. So they can be a good 1D systems are well understood. So they can be a good laboratory for testing various ideas.laboratory for testing various ideas.

One dimensional systems.One dimensional systems.

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

cos zzI N Qx

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..

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

Formal derivation.Formal derivation.L

z

†1 20

2 † †1 1 2 1 2 2 1 2 1 20 0

( ) ( )

( ) ( ) ( ) ( )

L

Q

L L

Q

A a z a z dz

A a z a z a z a z dz dz

Independent condensates:Independent condensates:

2 † †1 1 1 2 2 1 2 2 1 20 0

2†1 10

( ) ( ) ( ) ( )

( ) (0)

L L

Q

L

A a z a z a z a z dz dz

L a z a dz

for identical homogeneous systems for identical homogeneous systems

Long range order:Long range order:† 2 21 1( ) (0) const Qa z a A L

short range correlations:short range correlations: † / 21 1( ) (0) e z

Qa z a A L

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

2 †1 10( ) (0)

L

QA L a z a dzL

z

1D condensates (Luttinger liquids):1D condensates (Luttinger liquids):

1/ 2†1 1( ) (0) /

K

ha z a z

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 )1 20

†1 20

( ) ( ) ,

( ) ( ) , tan

L iQ x z

L iqzQ

I a z a z e dz

A a z a z e dz q Q

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

L L iq z zQA a z a z a z a z e 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 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 K

Aq

Aq

has a cusp singularity for K<1, relevant for fermions.2QA

Higher moments (need exactly two condensates).Higher moments (need exactly two condensates).

Relative width of the distribution (no dependence on L at large L):Relative width of the distribution (no dependence on L at large L):

24 2 2

1, 1~

6 , 1

Q Q QA A A

K

K K

Wide Poissonian distribution in the Wide Poissonian distribution in the fermionized regime (and at finite fermionized regime (and at finite temperatures).temperatures).

Narrow distribution in the weakly Narrow distribution in the weakly interacting regime. Absence of interacting regime. Absence of amplitude fluctuations for true amplitude fluctuations for true condensates.condensates.

22 † †

1 1 10 0( ) ( ) ( ) ( )

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

0 1 2 3 4

Pro

babi

lity

P(x

)

x

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

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

2 2Q Qx A A

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

2nZ

21/ 2

22 , 2 ,!

nK

nn

xZ Z x g

n

Same integrals as in the expressions for Same integrals as in the expressions for 2nQA

2 20 00

( ) ( ) (2 / ) ,Z x P A I Ax A dA

1 1/ 20

KA L

P. Fendley, F. Lesage, H.~Saleur (1995).

Sounds complicated? Not really.Sounds complicated? Not really.

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 amplitudes (and phases) from two 1D condensates.Distribution of interference amplitudes (and phases) from two 1D condensates.

2.2. Evaluate the integral.Evaluate the integral. 2 20 00

( ) ( ) (2 / ) ,Z x P A I Ax A dA

3.3. Read the result. Direct experimental simulation of a quantum Read the result. Direct experimental simulation of a quantum

impurity problem.impurity problem.

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

-6 -4 -2 0 2 4 6

-0.4

-0.2

0.0

0.2

0.4

0.6

Inte

rfer

en

ce c

ontr

ast

A

2 Q

Incidence angle (relative momentum)

Interacting Fermions, K=3/2

Two dimensional condensates at finite temperature.Two dimensional condensates at finite temperature.

Similar setup to 1D.Similar setup to 1D.

S. Stock et.al : Phys. Rev. Lett. 95, 190403 (2005)

Ly

Lx

22 †

0 0

( , ) (0,0)yx

LL

Q x yA L L dx dy a x y a

Can also study size or Can also study size or angular (momentum) angular (momentum) dependence.dependence.

Observing the Kosterlitz-Thouless transition

Above KT transition Ly

Lx

2 2Q x yA L L

Below KT transition

22Q x yA L L

Universal jump in at TKT

KTT T 1/ 4

KTT T 1

Expect a similar jump in the Expect a similar jump in the distribution function.distribution function.

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 the system size or the Scaling of interference amplitudes with the system size or the probing beam angle gives the correlation functions exponents.probing beam angle gives the correlation functions exponents.

b)b) Probability distribution ( = full counting statistics in TOF) of Probability distribution ( = full counting statistics in TOF) of amplitudes for two condensates contains information about higher amplitudes for two condensates contains information about higher order correlation functions.order correlation functions.

c)c) Interference of two Luttinger liquids directly realizes the statistical Interference of two Luttinger liquids directly realizes the statistical partition function of a one-dimensional quantum impurity problem.partition function of a one-dimensional quantum impurity problem.

2.2. Vast potential applications to many other systems, e.g.:Vast potential applications to many other systems, e.g.:

a)a) Spin-charge separation in spin ½ 1D fermionic systems.Spin-charge separation in spin ½ 1D fermionic systems.

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.