coherence and correlations in an atomic mott insulator
TRANSCRIPT
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 pictureDries van OostenUlrich SchneiderHerwig Ott
Outline
Optical lattices and the superfluid to Mott insulatortransition
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 theground 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)@ λL=850 nm
• Lattice spacing :
alat=λL/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
Loading a BEC in the lattice
• Produce a 87Rb Bose-Einstein condensatein a purely magnetic QUIC trap
• Expand the condensate to reduce its density (and avoid losses)
• Ramp up slowlylattice beams intensity
• Switch off the trap, expand and take an absorption image
0
10
Latti
cede
oth
(ER
)
160 msHold time
Time of flightinterference pattern
• Interference between all waves coherentlyemitted from each lattice site
Tim
e of
flig
ht
Wannierenveloppe
Grating-likeinterference
20 ms
Periodicity of thereciprocal lattice
Reversible lossof coherence
in deep lattices
0 Erecoil 22 Erecoil12 Erecoil 16Erecoil
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
Interactions matter:Bose-Hubbard model
Describes interacting Bose gas in a lattice, in the tight-binding limit
Compétition between tunneling and on-site interactions :
M.P.A. Fisher et al., PRB 40, 546 (1989)D. Jaksch et al., PRL 81, 3108 (1998)
Lattice depth
Ground state in thezero tunneling limit
The system try to form an atom distribution as regularas 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
Gapped excitations: incompressibleNo off-diagonal long range order orsuperfluid currents
Gapless excitations: compressibleLong-range phase coherenceSuperfluid 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 theinterference pattern
SF to MI transition
minmax
minmax
nnnnV
+−
=
Excitations in thezero 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 stateby an interaction gap U
n0: filling factorHere n0=1
Deviations from theperfect Mott Insulator
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
JU
+ + +KJU
Predictions for the visibility
Perfect MI
MI withparticle/hole pairs
0V =
( )04 13
zJV nU
≈ +
Perturbation approach predicts a finite visibility, scaling as (U/J)-1
Comparison with experiments
Average slope measured to be -0.97(7)
A more careful theory
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.
More details in : D. van Oosten et al. PRA 63, 053601 (2001) and following papersD. 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)
Comparisonwith 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 Ern=3 Mott shell 15.9 Er
Reproduced in munerical calculationsby the GSI Darmstadt group (R. Roth et al,
unpublished)
Spatial correlations in expandingatom 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
ExperimentTheory
Seminal experimentby Hanbury-Brown and Twissin 1952
Joint detection probabilitytwice as large for superimposed detectors
Second order coherence function :g(2) = 1 : uncorrelated particlesg(2) > 1 : bunching, typical for Bose statistics
τ
g(2)
τ2 6
1
Intensity interferometryvia the Hanbury Brown and Twiss effect
Bunching as consequence of Bose statistics
(Quantum-statistical) Noise analysis as a sensitive probe of thesource 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 physicsMasuda & 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 intime of fight images
Bunching effectfor relative distancesequal to a reciprocallattice vector
Correlation function(normalized)
dx
dy
Hanbury Brown-TwissEffect for Atoms (1)
Detector 1 Detector 2
Hanbury Brown-TwissEffect for Atoms (2)
There‘s another way ...
Detector 1 Detector 2
Hanbury Brown-TwissEffect for Atoms (3)
Cannot fundamentally distinguish between both paths...
Two Particle
Detection probability
Detector 1 Detector 2
ie φ±2
Relative phase accumulatedWhen propagating from sourceto detector
Hanbury Brown-TwissEffect for Atoms (4)
Interference in Two-Particle Detection Probability
ie φ±2
alat
Detector 1 Detector 2
depends on source separation alat
Multiple Wave Hanbury Brown-Twiss Effect
Interference in Two-Particle Detection Probability
Calculation for Ns=6 sites
Detector 1 Detector 2
alat
Detection system
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)
Atom density is in fact integrated over a column parallel to the probe.
In each bin, Nbin>>1 atoms are counted.
Bin geometry :
σ : 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 >> σ > Lcoh
σ
w
Scaling of correlations
Comparison of the results to a more sophisticated model, taking shell structureinto account :Scaling of the correlation amplitude with 1/N and t2 approximately verifiedHowever correlation amplitude is too small by 40 %
Noise floor
2
Applications to thedetection of magnetic phases
E. Altman, E. Demler & M. Lukin, PRA 70, 013603 (2004).Antiferromagnet (Bose/Fermi)
Spin waves (Bose/Fermi)
Charge density wave (predicted in Bosons/Fermions mixtures)
Conclusion and perspectives
• Fundamental deviations from a perfectMott state can be observed in the visibilitySignature for particle/hole pairsEvidence for n>1 shell formation ?
Implications for the fidelity of entanglementschemes in a lattice
• Spatial correlations of density fluctuationsIn expanding cloudssignature of lattice orderingApplications to the study of magnetic systems; also works for fermions
Other directions
• Visibility in a 2D latticeD. Gangardt et al., cond-mat (2004) :possible signature of correlationsin each tube (Tonks-Girardeau)
• Dynamical studies
• Resolve shell structure (microwave or rfspectroscopy)
• Detection of magnetic ordering
Adiabatic or diabatic loadingHow fast can we go to stay close to the ground state ?
Loading a BEC in the lattice
• Produce a 87Rb Bose-Einstein condensatein a purely magnetic QUIC trap
• Expand the condensate to reduce its density (and avoid losses)
• Ramp up slowlylattice beams intensity
• Switch off the trap, expand and take an absorption image
0
10
Latti
cede
oth
(ER
)
160 msHold time
Adiabatic loading in the lattice ?10
Latti
cede
pth
(ER)
Smooth profile to ramp upthe intensity of the lattice beams
Typically ramp time = 160 ms
0
Hold timeRamp timeAdiabaticity wrt the band structure : easy to fulfill (µs 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)
160 ms
Lattice depth V0 = 10 ER
Visi
bilit
y
0.5
0
Time constant ~ 100 msMuch longer than microscopic time scales
N=2.2 105
N=3.6 105N=4.3 105N=5.9 105
Adiabaticity in the MI state
Compare calculated to measuredvisibility in the deep MI state
Breakdown around V0~25 ER, whereTunneling time ~ 200 ms
• 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 timeSingle particle redistribution