Measuring correlation functions in interacting systems of cold atoms

Anatoli Polkovnikov Harvard/Boston UniversityEhud Altman Harvard/WeizmannVladimir Gritsev HarvardMikhail Lukin HarvardVito Scarola University of MarylandSankar Das Sarma University of MarylandEugene Demler Harvard

Detection and characterization of many-body quantum phases

Outline Measuring correlation functions in intereference experiments 1. Interference of independent condensates 2. Interference of interacting 1D systems 3. Interference of 2D systems 4. Full counting statistics of intereference experiments. Connection to quantum impurity problem Quantum noise interferometry in time of flight experiments

References:

1. Polkovnikov, Altman, Demler, cond-mat/0511675 Gritsev, Altman, Demler, Polkovnikov, cond-mat/06024752. Altman, Demler, Lukin, PRA 70:13603 (2004) Scarola, Das Sarma, Demler, cond-mat/0602319

Measuring correlation functions in intereference experiments

Interference of two independent condensates

Andrews et al., Science 275:637 (1997)

Interference of two independent condensates

1

2

r

r+d

d

r’

Clouds 1 and 2 do not have a well defined phase difference.However each individual measurement shows an interference pattern

Amplitude of interference fringes, , contains information about phase fluctuations within individual condensates

y

x

Interference of one dimensional condensates

x1

d Experiments: Schmiedmayer et al., Nature Physics 1 (05)

x2

Interference amplitude and correlations

For identical condensates

Instantaneous correlation function

L

Polkovnikov, Altman, Demler, cond-mat/0511675

For impenetrable bosons and

Interference between Luttinger liquidsLuttinger liquid at T=0

K – Luttinger parameter

Luttinger liquid at finite temperature

For non-interacting bosons and

Analysis of can be used for thermometry

L

Interference between two-dimensional BECs at finite temperature.Kosteritz-Thouless transition

Interference of two dimensional condensates

Ly

Lx

Lx

Experiments: Stock, Hadzibabic, Dalibard, et al., cond-mat/0506559

Probe beam parallel to the plane of the condensates

Gati, Oberthaler, et al., cond-mat/0601392

Interference of two dimensional condensates.Quasi long range order and the KT transition

Ly

Lx

Below KT transitionAbove KT transition

Theory: Polkovnikov, Altman, Demler, cond-mat/0511675

x

z

Time of

flight

low temperature higher temperature

Typical interference patterns

Experiments with 2D Bose gasHadzibabic, Stock, Dalibard, et al.

integration

over x axis

Dx

z

z

integration

over x axisz

x

integration distance Dx

(pixels)

Contrast afterintegration

0.4

0.2

00 10 20 30

middle Tlow T

high T

integration

over x axis z

Experiments with 2D Bose gasHadzibabic, Stock, Dalibard, et al.

fit by:

integration distance Dx

Inte

grat

ed c

ontr

ast 0.4

0.2

00 10 20 30

low Tmiddle T

high T

if g1(r) decays exponentially with :

if g1(r) decays algebraically or exponentially with a large :

Exponent

central contrast

0.5

0 0.1 0.2 0.3

0.4

0.3high T low T

2

21

2 1~),0(

1~

x

D

x Ddxxg

DC

x

“Sudden” jump!?

Experiments with 2D Bose gasHadzibabic, Stock, Dalibard, et al.

Exponent

central contrast

0.5

0 0.1 0.2 0.3

0.4

0.3 high T low T

He experiments:universal jump in

the superfluid density

T (K)1.0 1.1 1.2

1.0

0

c.f. Bishop and Reppy

Experiments with 2D Bose gasHadzibabic, Stock, Dalibard, et al.

Ultracold atoms experiments:jump in the correlation function.

KT theory predicts =1/4 just below the transition

Experiments with 2D Bose gas. Proliferation of thermal vortices Hadzibabic, Stock, Dalibard, et al.

Fraction of images showing at least one dislocation

Exponent

0.5

0 0.1 0.2 0.3

0.4

0.3

central contrast

The onset of proliferation coincides with shifting to 0.5!

0

10%

20%

30%

central contrast

0 0.1 0.2 0.3 0.4

high T low T

Z. Hadzibabic et al., in preparation

Full counting statistics of interference between two interacting one dimensional Bose liquids

Gritsev, Altman, Demler, Polkovnikov, cond-mat/0602475

Higher moments of interference amplitude

L Higher moments

Changing to periodic boundary conditions (long condensates)

Explicit expressions for are available but cumbersome Fendley, Lesage, Saleur, J. Stat. Phys. 79:799 (1995)

is a quantum operator. The measured value of will fluctuate from shot to shot.Can we predict the distribution function of ?

Impurity in a Luttinger liquid

Expansion of the partition function in powers of g

Partition function of the impurity contains correlation functions taken at the same point and at different times. Momentsof interference experiments come from correlations functionstaken at the same time but in different points. Euclidean invarianceensures that the two are the same

Relation between quantum impurity problemand interference of fluctuating condensates

Distribution function of fringe amplitudes

Distribution function can be reconstructed fromusing completeness relations for the Bessel functions

Normalized amplitude of interference fringes

Relation to the impurity partition function

is related to a Schroedinger equation Dorey, Tateo, J.Phys. A. Math. Gen. 32:L419 (1999) Bazhanov, Lukyanov, Zamolodchikov, J. Stat. Phys. 102:567 (2001)

Spectral determinant

Bethe ansatz solution for a quantum impurity can be obtained from the Bethe ansatz followingZamolodchikov, Phys. Lett. B 253:391 (91); Fendley, et al., J. Stat. Phys. 79:799 (95)Making analytic continuation is possible but cumbersome

Interference amplitude and spectral determinant

0 1 2 3 4

P

roba

bilit

y P

(x)

x

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

Evolution of the distribution function

Narrow distributionfor .Distribution widthapproaches

Wide Poissoniandistribution for

When K>1, is related to Q operators of CFT with c<0. This includes 2D quantum gravity, non-intersecting loop model on 2D lattice, growth of randomfractal stochastic interface, high energy limit of multicolor QCD, …

Yang-Lee singularity

2D quantum gravity,non-intersecting loops on 2D lattice

correspond to vacuum eigenvalues of Q operators of CFT Bazhanov, Lukyanov, Zamolodchikov, Comm. Math. Phys.1996, 1997, 1999

From interference amplitudes to conformal field theories

Outlook

One and two dimensional systems with tunneling. Competition of single particle tunneling, quantum, and thermal fluctuations. Full counting statistics of the phase and the amplitude

Full counting statistics of interference between fluctuating 2D condensates

Time dependent evolution of distribution functions

Expts: Shmiedmayer et al. (1d), Oberthaler et al. (2d)

Extensions, e.g. spin counting in Mott states of multicomponent systems

Ultimate goal: Creation, characterization, and manipulation of quantum many-body states of atoms

Quantum noise interferometry in time of flight experiments

Atoms in an optical lattice.Superfluid to Insulator transition

Greiner et al., Nature 415:39 (2002)

U

1n

t/U

SuperfluidMott insulator

Time of flight experiments

Quantum noise interferometry of atoms in an optical lattice

Second order coherence

Second order coherence in the insulating state of bosons.Hanburry-Brown-Twiss experiment

Theory: Altman et al., PRA 70:13603 (2004)

Experiment: Folling et al., Nature 434:481 (2005)

Hanburry-Brown-Twiss stellar interferometer

Second order coherence in the insulating state of bosons

Bosons at quasimomentum expand as plane waves

with wavevectors

First order coherence:

Oscillations in density disappear after summing over

Second order coherence:

Correlation function acquires oscillations at reciprocal lattice vectors

Second order coherence in the insulating state of bosons.Hanburry-Brown-Twiss experiment

Theory: Altman et al., PRA 70:13603 (2004)

Experiment: Folling et al., Nature 434:481 (2005)

Effect of parabolic potential on the second order coherence

Experiment: Spielman, Porto, et al.,Theory: Scarola, Das Sarma, Demler, cond-mat/0602319

Width of the correlation peak changes across the transition, reflecting the evolution of Mott domains

Potential applications of quantum noise intereferometry

Detection of magnetically ordered Mott states

Detection of paired states of fermions

Altman et al., PRA 70:13603 (2004)

Experiment: Greiner et al. PRL

Conclusions

Interference of extended condensates is a powerful tool for analyzing correlation functions in one and two dimensional systems

Noise interferometry can be used to probequantum many-body states in optical lattices