Coherence and correlations in an atomic Mott insulator

Quantum Optics VI, Krynicka, Poland, 13-18 June 2005

Fabrice GerbierArtur WideraSimon FöllingOlaf Mandel

Tatjana GerickeImmanuel Bloch

Johannes Gutenberg Universität Mainz

www.physik.uni-mainz.de/quantum

Back rowImmanuel BlochSimon FöllingThomas BergTim Rom

Middle rowFGTatjana GerickeThorsten Best

Front rowArtur WideraOlaf MandelSusanne Kreim

Not on picture Dries van OostenUlrich SchneiderHerwig Ott

Outline

Optical lattices and the superfluid to Mott insulator transition

Reviews : I. Bloch, J. Phys. B 38, S629 (2005)D. Jaksch and P. Zoller, Annals of Physics 315, 52 (2005).W. Zwerger, J. Opt. B 5, 89 (2003)

Phase coherence of a Mott insulatorwhat does the interference pattern tell us about the nature of the ground state ?

Spatial correlations in expanding cloudstwo-particle correlations to probe phase-uncoherent

samples

1D optical lattice

Potential:

Natural scales :

• Recoil energy :

ER/h ~ 3.2 kHz (150 nK)@ L=850 nm

• Lattice spacing :

alat=L/2 = 425nm

3D Optical Lattices

• Three pairs of counter-propagating laser beams produce a simple cubic lattice

• Typically 20-60 sites occupied in each direction

• Mean atom number per site (filling factor) between 1 and 3

• Spontaneous emission rate ~ 1 Hz

• Produce a 87Rb Bose-Einstein condensate in a purely magnetic QUIC trap

• Expand the condensate to reduce its density (and avoid losses)

• Ramp up slowly lattice beams intensity

• Switch off the trap, expand and take an absorption image

Loading a BEC in the lattice

0

10

Latt

ice d

eoth

(E

R)

160 msHold time

Time of flight interference pattern

• Interference between all waves coherently

emitted from each lattice site

Tim

e o

f flig

ht

Periodicity of the reciprocal lattice

20 ms

Wannierenveloppe

Grating-likeinterference

Reversible loss of coherence

in deep lattices

0 Erecoil 22 Erecoil12 Erecoil16Erecoil

Phase coherence disappears with increasing lattice depth. This is reversible:

Generalization to a general matter wave:

Correlation functiondetermines the visibility

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

see also :C. Orzel et al., Science 291, 2386 (2001) Z. Hadzibabic et al., PRL 93, 180403 (2004)

before 0.1 ms 1 ms 4ms 14 ms

ramping

down

Describes interacting Bose gas in a lattice, in the tight-binding limit

Compétition between tunneling and on-site interactions :

Interactions matter:Bose-Hubbard model

M.P.A. Fisher et al., PRB 40, 546 (1989)D. Jaksch et al., PRL 81, 3108 (1998)

Lattice depth

Ground state in the zero tunneling limit

The system try to form an atom distribution as regular as possible to minimize locally the interaction

energy

Mott insulator ground state

Integer number of atoms per siteZero fluctuations

Survives at finite temperature << U

Intermediate regime

Superfluid ground state, J << U

Mott insulator ground stateU>> J

Gapless excitations: compressibleLong-range phase coherence Superfluid currents

Gapped excitations: incompressibleNo off-diagonal long range order or superfluid currents

Phase coherence of a Mott insulator

Does a Mott insulator produce an interference pattern ?

F. Gerbier et al., cond-mat/0503452, accepted in PRL.

Theory : V. N. Kashurnikov et al., PRA 66, 031601 (2002).R. Roth & K. Burnett, PRA 67, 031602 (2003).

Visibility of the interference pattern

minmax

minmax

nn

nnV

SF to MI transition

Excitations in the zero tunneling limit

Perfect Mott insulator ground state

• Low energy excitations :

• Particle/hole pairs couples to the ground state :

Energy E0

Energy E0+U, separated from the ground state by an interaction gap U

n0: filling factorHere n0=1

Ground state for t 0 :

``perfect´´ Mott insulator

Ground state for finite t<<U :

treat the hopping term Hhop in 1st order perturbation

=

Coherent admixture of particle/holes at finite t/U

Deviations from the perfect Mott Insulator

J

U J

U

Predictions for the visibility

Perfect MI

MI with

particle/hole pairs

0V

0

41

3

zJV n

U

Perturbation approach predicts a finite visibility, scaling as (U/J)-1

Comparison with experiments

Average slope measured to be -0.97(7)

Many-body calculation for the homogeneous case

• 1st order calculation : admixture of particle/hole pairs to the MI bound to neighboring lattice sites

• Higher order in J/U : particle/holes excitations become mobile

Dispersion relation of the excitations is still characterized by an interaction gap.

One can obtain analytically the interference pattern (momentum distribution) for a given n0.

A more careful theory

More details in : D. van Oosten et al. PRA 63, 053601 (2001) and following papers

D. Gangardt et al., cond-mat/0408437 (2004)

K. Sengupta and N. Dupuis, PRA 71, 033629 (2005)

Shell structure of a trapped MI

D. Jaksch et al. PRL 81, 3108 (1998)

Smooth ``external´´ potential present on top of the lattice potential (combination of magnetic trap +optical potential due to Gaussian profile)

Consequence: alternating MI/superfluid shells present at the same time

Figures courtesy of M. Niemeyer and H. Monien (Bonn)

Comparison with experiments

• Simplify shell structure :

ignore superfluid rings

No adjustable parameters

Extends to trapped system using the Local Density Approximation

F. Gerbier et al., in preparation

18 ER

Kinks in the visibility curve : evidence for n>1 Mott shell formation ?

Experiment:

Kink #1 14.1 (8) ErKink #2 16.6 (9) Er

Theory:

n=2 Mott shell 14.7 Er

n=3 Mott shell 15.9 Er

Reproduced in munerical

calculations by the GSI Darmstadt

group (R. Roth et al, unpublished)

Spatial correlations in expanding atom clouds

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

Theory : E. Altman, E. Demler & M. Lukin, PRA 70, 013603 (2004).

Related work : Z. Hadzibabic et al., PRL 93, 180403 (2004) . M. Greiner et al. , Phys. Rev. Lett. 94, 110401 (2005).

J. Grondalski et al., Opt. Exp. 5, 249 (1999)A. Kolovski, EPL 68, 330 (2005).

R. Bach and K. Rzazewski, PRA 70 (2005).

Hanbury Brown Twiss experiment

0,00

1,00

2,00

3,00

4,00

5,00

6,00

7,00

8,00

detectorssuperimposed

detectorsseparated

Experiment

Theory

Seminal experiment by Hanbury-Brown and Twiss in 1952

Joint detection probability twice as large for superimposed detectors

Second order coherence function :

g(2) = 1 : uncorrelated particles

g(2) > 1 : bunching, typical for Bose statistics

g(2)

2 6

1

Intensity interferometry

via the Hanbury Brown and Twiss effect

Bunching as consequence of Bose statistics

(Quantum-statistical) Noise analysis as a sensitive probe of the source properties, with a wide range of applications :

• Quantum optics• Nuclear and particle physics (angular correlations)• Condensed matter physics (electron antibunching, mesoscopics, …)

Emerging field in cold atom physics Masuda & Shimizu, PRL (1996); Orsay (2005) also pursued in optical cavity: Münich, Heidelberg, Zürich, Berkeley,

…

Hidden information in expanding atom clouds

For a cloud deep in the Mott state (here V0=50 ER), the interference pattern is unobservable.

Can we still extract information from such a picture ?

The answer is yes, if we use noise analysis.

Correlated fluctuations in time of fight images

Correlation function(normalized)

Bunching effectfor relative distances equal to a reciprocallattice vector

dx

dy

Hanbury Brown-Twiss Effect for Atoms (1)

Detector 1 Detector 2

Hanbury Brown-Twiss Effect for Atoms (2)

Detector 1 Detector 2

There‘s another way ...

Hanbury Brown-Twiss Effect for Atoms (3)

Detector 1 Detector 2

Cannot fundamentally distinguish between both paths...

Two Particle

Detection probability

ie 2

Relative phase accumulated When propagating from sourceto detector

Hanbury Brown-Twiss Effect for Atoms (4)

Interference in Two-Particle Detection Probability

Detector 1 Detector 2

ie 2

depends on source separation alat

alat

Multiple Wave Hanbury Brown-Twiss Effect

Interference in Two-Particle Detection Probability

Detector 1 Detector 2

alat

Calculation for Ns=6 sites

Detection system

Atom density is in fact integrated over a column parallel to the probe.

In each bin, Nbin>>1 atoms are counted.

Bin geometry :

HBT theory predicts a factor of 2 enhancement of fluctuations, or

In the experiment, the enhancement varies between 10-4 and 10-3 !(note the noise floor ~ 10-4)

: imaging resolution

w : cloud size

Coherence length : also ideal peak width

Great spatial resolution : fringe spacing l >> Lcoh >> res.

Poor spatial resolution : res. >> fringe spacing l >> Lcoh

Intermediate spatial resolution : fringe spacing l >> res. >> Lcoh

How large are the correlations ?

Probe direction : w >> l

Imaging plane:l >> > Lcoh

w

Scaling of correlations

Comparison of the results to a more sophisticated model, taking shell structure into account :

Scaling of the correlation amplitude with 1/N and t2 approximately verified However correlation amplitude is too small by 40 %

Noise floor

2

Applications to the detection of magnetic phases

Antiferromagnet (Bose/Fermi)

Spin waves (Bose/Fermi)

Charge density wave (predicted in Bosons/Fermions mixtures)

E. Altman, E. Demler & M. Lukin, PRA 70, 013603 (2004).

Conclusion and perspectives

• Fundamental deviations from a perfect

Mott state can be observed in the visibility

Signature for particle/hole pairs

Evidence for n>1 shell formation ?

Implications for the fidelity of entanglement

schemes in a lattice

• Spatial correlations of density fluctuations

In expanding clouds

signature of lattice ordering

Applications to the study of magnetic systems;

also works for fermions

Other directions

• Visibility in a 2D lattice D. Gangardt et al., cond-mat (2004) :possible signature of correlations

in each tube (Tonks-Girardeau)

• Dynamical studies

• Resolve shell structure (microwave or rf spectroscopy)

• Detection of magnetic ordering

Adiabatic or diabatic loading

How fast can we go to stay close to the ground state ?

• Produce a 87Rb Bose-Einstein condensate in a purely magnetic QUIC trap

• Expand the condensate to reduce its density (and avoid losses)

• Ramp up slowly lattice beams intensity

• Switch off the trap, expand and take an absorption image

Loading a BEC in the lattice

0

10

Latt

ice d

eoth

(E

R)

160 msHold time

Adiabatic loading in the lattice ?

0

10

Latt

ice d

epth

(E

R)Ramp time Hold time

Smooth profile to ramp upthe intensity of the lattice beams

Typically ramp time = 160 ms

Adiabaticity wrt the band structure : easy to fulfill (s time scale)

J. Hecker-Denschlag et al., J. Phys. B 35, 3095 (2002)

Adiabaticity wrt many body dynamics ?

S. Sklarz et al., PRA 66, 053620 (2002).S. Clark and D. Jaksch, PRA 70, 043612 (2004).J. Zakrzewski, PRA 71, 043601 (2005).

Influence of ramp time(SF regime)

Fix :

• Lattice depth V0 = 10 ER

• Hold time thold = 300 ms

Vary ramp time

1

20 100 200Ramp time (ms)

0

0.5

Vis

ibili

ty160 ms

Time constant ~ 100 msMuch longer than microscopic time scales

Lattice depth V0 = 10 ER

N=2.2 105

N=3.6 105N=4.3 105N=5.9 105

Adiabaticity in the MI state

• Superfluid regime :Time constant ~ 100 ms, much longer than tunneling time, trap frequencies, …

Long-lived collective excitations involved

• MI regime :Breakdown of adiabaticity for lattice depth such that the tunneling time is

comparable to ramp time Single particle redistribution

Compare calculated to measured visibility in the deep MI state

Breakdown around V0~25 ER, whereTunneling time ~ 200 ms