QUANTUM MANY‐BODY SYSTEMS OF ULTRACOLD ATOMSSYSTEMS OF ULTRACOLD ATOMSEugene Demler Harvard University
Grad students: A. Imambekov (->Rice), Takuya KitagawaPostdocs: E. Altman (->Weizmann), A. Polkovnikov (->U. Boston)( ), ( )A.M. Rey (->U. Colorado), V. Gritsev (-> U. Fribourg), D. Pekker (-> Caltech), R. Sensarma (-> JQI Maryland)
Collaborations with experimental groups of I. Bloch (MPQ), T. Esslinger (ETH), J.Schmiedmayer (Vienna)
Supported by NSF, DARPA OLE, AFOSR MURI, ARO MURI
How cold are ultracold atoms?How cold are ultracold atoms?
keV MeV GeV TeVfeV peV µeV meV eVneV
pK nK µK mK K
He Nfirst BEC
roomtemperature
LHCcurrent experiments10-11 - 10-10 K
of alkali atoms
Bose-Einstein condensation of kl i t ti tweakly interacting atoms
Density 1013 cm-1
Typical distance between atoms 300 nmTypical scattering length 10 nm
Scattering length is much smaller than characteristic interparticle distances. Interactions are weak
yp g g
New Era in Cold Atoms ResearchFocus on Systems with Strong Interactions
• Feshbach resonances
• Rotating systems
At i ti l l tti
• Low dimensional systems
• Atoms in optical lattices
• Systems with long range dipolar interactions• Systems with long range dipolar interactions
Feshbach resonanceGreiner et al., Nature (2003); Ketterle et al., (2003), ( ); , ( )
Ketterle et al.,Nature 435, 1047-1051 (2005)
One dimensional systems
One dimensional systems in microtraps.Thywissen et al Eur J Phys D (99);
1D confinement in optical potentialWeiss et al., Science (05);Bloch et al Thywissen et al., Eur. J. Phys. D. (99);
Hansel et al., Nature (01);Folman et al., Adv. At. Mol. Opt. Phys. (02)
Bloch et al., Esslinger et al.,
Strongly interactingStrongly interacting regime can be reached for low densities
Atoms in optical lattices
Th J k h t l PRL (1998)Theory: Jaksch et al. PRL (1998)
Experiment: Kasevich et al., Science (2001);Greiner et al., Nature (2001);Phillips et al., J. Physics B (2002) Esslinger et al., PRL (2004);and many more …
Quantum simulations with ultracold atoms
Atoms in optical lattice
Antiferromagnetic and superconducting Tc
Atoms in optical lattice
Antiferromagnetism and pairing at nano Kelvin p g
of the order of 100 K temperatures
Same microscopic model
Strongly correated systemsStrongly correated systemsAtoms in optical latticesElectrons in Solids
Simple metalsPerturbation theory in Coulomb interaction applies. B d t t th d kBand structure methods work
Strongly Correlated Electron SystemsBand structure methods fail.
Novel phenomena in strongly correlated electron systems:Quantum magnetism, phase separation, unconventional superconductivity,Q g , p p , p y,high temperature superconductivity, fractionalization of electrons …
By studying strongly interacting systems of cold atoms we expect to get insights into the mysterious properties ofexpect to get insights into the mysterious properties of novel quantum materials: Quantum Simulators
BUTStrongly interacting systems of ultracold atoms :Strongly interacting systems of ultracold atoms :
are NOT direct analogues of condensed matter systemsThese are independent physical systems with their own “ liti ” h i l ti d th ti l h ll
Strongly correlated systems of ultracold atoms should
“personalities”, physical properties, and theoretical challenges
g y yalso be useful for applications in quantum information, high precision spectroscopy, metrology
First lecture: experiments with ultracold bosons
Cold atoms in optical lattices
Bose Hubbard model. Superfluid to Mott transitionLooking for Higgs particle in the Bose Hubbard modelQuantum magnetism with ultracold atoms in optical lattices
L di i l d tLow dimensional condensates
Observing quasi-long range order in interference experimentsObserving quasi long range order in interference experimentsObservation of prethermolization
Second lecture: Ultracold fermions
Fermions in optical lattices. Fermi Hubbard model.C t t t f i tCurrent state of experiments
Lattice modulation experimentsLattice modulation experiments
Doublon lifetimes
Strongly interacting fermions in continuum. Stoner instability
Ultracold Bose atoms in optical lattices
Bose Hubbard model
Bose Hubbard model
UU
t
l f b hb lltunneling of atoms between neighboring wells
repulsion of atoms sitting in the same well
In the presence of confining potential we also need to include
Typically
Bose Hubbard model. Phase diagramU
n=3 Mott M.P.A. Fisher et al.,PRB (1989)2
1n
1
n=2
n=3
Superfluid
Mott
Mott
1
0
Mottn=1
Weak lattice Superfluid phase
Strong lattice Mott insulator phase
Bose Hubbard model
l kHamiltonian eigenstates are Fock statesSet .
U
0 1
U
Away from level crossings Mott states have a gap. Hence they should be stable to small tunneling.
Bose Hubbard Model. Phase diagram
1n
U
n=3 Mott
2
1n=2 SuperfluidMott
0
Mottn=1
Particle‐hole
Mott insulator phase
Particle hole excitation
Tips of the Mott lobesTips of the Mott lobes
z‐ number of nearest neighbors, n – filling factor
Gutzwiller variational wavefunction
Normalization
Kinetic energy
z – number of nearest neighbors
Interaction energy favors a fixed number of atoms per well.Kinetic energy favors a superposition of the number states.
Bose Hubbard Model. Phase diagram
U
n=3 Mott
21n
n=2
n=3
Superfluid
Mott
Mott
1
Mottn=1
n 2 Mott
0
Note that the Mott state only exists for integer filling factors.For even when atoms are localized,
make a superfluid state. p
Nature 415:39 (2002)
Optical lattice and parabolic potentialParabolic potential acts as a “cut” through the phase diagram. Hence in a parabolic potential we find a “wedding cake” structureU potential we find a wedding cake structure.
21n
n=3 Mott
2
1
n=2 SuperfluidMott
0
Mottn=1
Jaksch et al.,Jaksch et al., PRL 81:3108 (1998)
Quantum gas microscopeQuantum gas microscopeBakr et al., Science 2010
y
y
density
x
Nature 2010
The Higgs (amplitude) mode in aThe Higgs (amplitude) mode in a trapped 2D superfluid on a lattice
Cold Atoms (Munich)Elementary Particles (CMS @ LHC)
Sherson et. al. Nature 2010
Theory: David Pekker, Eugene DemlerExperiments: Manuel Endres Takeshi Fukuhara Marc Cheneau Peter
Sherson et. al. Nature 2010
Experiments: Manuel Endres, Takeshi Fukuhara, Marc Cheneau, Peter Schauss, Christian Gross, Immanuel Bloch, Stefan Kuhr
Collective modes of strongly interactingsuperfluid bosons
Order parameter Breaks U(1) symmetry
superfluid bosons
Figure from Bissbort et al. (2010)
Phase (Goldstone) mode = gapless Bogoliubov mode
Gapped amplitude mode (Higgs mode)
Excitations of the Bose Hubbard modelExcitations of the Bose Hubbard model
U
22
1nn=3
Superfluid
Mott
2
1
n=2
p
Mott
Mott Superfluid
0
Mottn=1
Softening of the amplitude mode is the defining characteristicSoftening of the amplitude mode is the defining characteristicof the second order Quantum Phase Transition
Is there a Higgs mode in 2D ?Is there a Higgs mode in 2D ?neutron scattering
• Danger from scattering on phase modes
I 2D i f d di
Higgs
Higgs
• In 2D: infrared divergence
• Different susceptibility has no divergencel tti d l ti
S. Sachdev, Phys. Rev. B 59, 14054 (1999)
lattice modulation spectroscopy
W. Zwerger, Phys. Rev. Lett. 92, 027203 (2004) N. Lindner and A. Auerbach, Phys. Rev. B 81, 54512 (2010) Podolsky, Auerbach, Arovas, Phys. Rev. B 84, 174522 (2011)
Why it is difficult to observe the amplitude mode
Bissbort et al., PRL(2010)
Stoferle et al., PRL(2004)
Peak at U dominates and does not change as the system goes through the SF/Mott transition
Exciting the amplitude mode
Absorbed energy
Exciting the amplitude modeManuel Endres, Immanuel Bloch and MPQ team
Mottn=1 Mottn=1 Mottn=1
Experiments: full spectrump pManuel Endres, Immanuel Bloch and MPQ team
Time dependent mean‐field: Gutzwiller
Si il t L d Lif hit ti i tiSimilar to Landau-Lifshitz equations in magnetism
Keep twoKeep twostates per siteonly
Threshold for absorption i t d llis captured very well
Plaquette Mean Field “B G ill ”“Better Gutzwiller”
• Variational wave functions better captures local physics– better describes interactions between quasi‐particles
• Equivalent to MFT on plaquettes
Time dependent cluster mean‐fieldh h ( )Lattice height 9.5 Er: (1x1 vs 2x2)
breathing mode
single amplitude mode excited multiple modes
excited?single amplitude mode excited
2x2 captures width of spectral feature
e c ted
breathing mode
mode excited
Comparison of experiments and Gutzwiller theories
Experiment 2x2 ClustersKey experimental facts:
• “gap” disappears at QCP• gap disappears at QCP• wide band• band spreads out deep in SF
Single site Gutzwiller Plaquette Gutzwiller
Captures gapDoes not capture width
Captures gapCaptures most of the width
Beyond Gutzwiller: Scaling at low frequenciessignature of Higgs/Goldstone mode couplingsignature of Higgs/Goldstone mode coupling
Higgs
2 Goldstonesw
External drive couples vacuum to HiggsHiggs can be excited only virtually
vacuum
Higgs can be excited only virtuallyHiggs decays into a pair of Goldstone modes with conservation of energy Matrix element w2/w=wDensity of states wyFermi’s golden rule: w2xw = w3
Open question: observing discreet modes
disappearing amplitude mode
B thi dBreathing mode
details at the QCP
spectrum remains gapped due to trap
Higgs Drum Modes1x1 calculation, 20 oscillationsEabs rescaled so peak heights coincide
Quantum magnetism with ultracoldQuantum magnetism with ultracoldatoms in optical lattices
Two component Bose mixture in optical latticeExample: . Mandel et al., Nature (2003)
tt
Two component Bose Hubbard model
We consider two component Bose mixture in the n=1 Mott state with equal number of and atoms. We need to find spin arrangement in the ground state.
Quantum magnetism of bosons in optical latticesDuan et al., PRL (2003), ( )
• FerromagneticA tif ti• Antiferromagnetic
Two component Bose Hubbard modelIn the regime of deep optical lattice we can treat tunnelingas perturbation. We consider processes of the second order in t
We can combine these processes into anisotropic Heisenberg model
Two component Bose mixture in optical lattice.Mean field theory + Quantum fluctuations
HysteresisAltman et al., NJP (2003)
1st order
Two component Bose Hubbard model
+ infinitely large Uaa and Ubb
N f tNew feature:coexistence ofcheckerboard phasepand superfluidity
Exchange Interactions in SolidsExchange Interactions in Solidsantibonding
b dibonding
Kinetic energy dominates: antiferromagnetic state
Coulomb energy dominates: ferromagnetic stateCoulomb energy dominates: ferromagnetic state
Realization of spin liquid using cold atoms in an optical latticeusing cold atoms in an optical lattice
Theory: Duan, Demler, Lukin PRL (03)
Kitaev model Annals of Physics (2006)
H = - Jx ix j
x - Jy iy j
y - Jz iz j
z
y ( )
Questions:Detection of topological orderCreation and manipulation of spin liquid statesDetection of fractionalization, Abelian and non-Abelian anyonsMelting spin liquids. Nature of the superfluid state
Superexchange interaction in experiments with double wellsp
Theory: A.M. Rey et al., PRL 2008Experiments: S. Trotzky et al., Science 2008
Observation of superexchange in a double well potentialTh A M R t l PRL 2008
JTheory: A.M. Rey et al., PRL 2008
J
Use magnetic field gradient to prepare a state g g p p
Observe oscillations between and states
Experiments:S. Trotzky et al.Science 2008
Preparation and detection of Mott statesof atoms in a double well potentialof atoms in a double well potential
Reversing the sign of exchange interaction
Comparison to the Hubbard model
Beyond the basic Hubbard model
Basic Hubbard model includes
y
only local interaction
Extended Hubbard modeltakes into account non-localinteractioninteraction
Beyond the basic Hubbard model
Probing low dimensional d t ith i t fcondensates with interference
experimentsp
Quasi long range orderQuasi long range order
Prethermalization
Interference of independent condensates
Experiments: Andrews et al., Science 275:637 (1997)
Theory: Javanainen, Yoo, PRL 76:161 (1996)Ci Z ll t l PRA 54 R3714 (1996)Cirac, Zoller, et al. PRA 54:R3714 (1996)Castin, Dalibard, PRA 55:4330 (1997)and many more
zExperiments with 2D Bose gas
Hadzibabic, Dalibard et al., Nature 2006
Time of
fli h
, ,
xflight
Experiments with 1D Bose gas Hofferberth et al. Nat. Physics 2008
Interference of two independent condensates
rr’
1 r+dAssuming ballistic expansion
2d
Phase difference bet een clo ds 1 and 22 Phase difference between clouds 1 and 2is not well defined
Individual measurements show interference patternsIndividual measurements show interference patternsThey disappear after averaging over many shots
Interference of fluctuating condensatesPolkovnikov et al PNAS (2006); Gritsev et al Nature Physics (2006)
dAmplitude of interference fringes,
Polkovnikov et al., PNAS (2006); Gritsev et al., Nature Physics (2006)
x1For independent condensates Afr is finite frbut is random
x2
For identical condensates
Instantaneous correlation function
FDF of phase and contrast• Matter-wave interferometry
FDF of phase and contrast
phase, contrast
FDF of phase and contrast• Matter-wave interferometry
FDF of phase and contrast
phase, contrast
• Plot as circular statisticscontrast
phasephase
FDF of phase and contrast• Matter-wave interferometry: repeat
FDF of phase and contrast
many timesi>100phase, contrast
contrasti accumulate statistics
• Plot phase
Calculate average contrast
Fluctuations in 1d BECThermal fluctuations
Thermally energy of the superflow velocity
Quantum fluctuations
Interference between Luttinger liquids
Luttinger liquid at T=0
K Luttinger parameterK – Luttinger parameter
For non-interacting bosons and
For impenetrable bosons and
Finite temperature
Experiments: Hofferberth,Schumm SchmiedmayerSchumm, Schmiedmayer
Distribution function of fringe amplitudes for interference of fluctuating condensates
Gritsev, Altman, Demler, Polkovnikov, Nature Physics 2006
is a quantum operator. The measured value of
, , , , yImambekov, Gritsev, Demler, PRA (2007)
L
will fluctuate from shot to shot.
Higher moments reflect higher order correlation functions
We need the full distribution function ofWe need the full distribution function of
Distribution function of interference fringe contrastHofferberth et al Nature Physics 2009Hofferberth et al., Nature Physics 2009
Quantum fluctuations dominate:Quantum fluctuations dominate:asymetric Gumbel distribution(low temp. T or short length L)
Thermal fluctuations dominate:broad Poissonian distribution(high temp. T or long length L)
Intermediate regime:double peak structure
Comparison of theory and experiments: no free parametersComparison of theory and experiments: no free parametersHigher order correlation functions can be obtained
Interference between interacting 1d Bose liquids.Distribution function of the interference amplitudep
Distribution function of
Quantum impurity problem: interacting one dimensionalelectrons scattered on an impurity
Conformal field theories with negative central charges: 2D quantum gravity, non-intersecting loop model growth of
2D quantum gravity,non-intersecting loopsnon intersecting loop model, growth of
random fractal stochastic interface, high energy limit of multicolor QCD, …
Yang-Lee singularity
Fringe visibility and statistics of random surfaces
Distribution function of
Mapping between fringe pp g gvisibility and the problem of surface roughness for fluctuating random
)(h surfaces. Relation to 1/f Noise and Extreme Value Statistics
)(h
Roughness dh2
)(
Interference of two dimensional condensatesExperiments: Hadzibabic et al. Nature (2006)pe e ts ad bab c et a atu e ( 006)
Gati et al., PRL (2006)
Ly
LLxLx
Probe beam parallel to the plane of the condensates
Interference of two dimensional condensates.Quasi long range order and the BKT transitionQuasi long range order and the BKT transition
Ly
Lx
Below BKT transitionAbove BKT transition
zExperiments with 2D Bose gas
Time of
Hadzibabic, Dalibard et al., Nature 441:1118 (2006)
xflight
low temperature higher temperature
Typical interference patterns
Experiments with 2D Bose gas
xfi t ti
Hadzibabic et al., Nature 441:1118 (2006)
z
Contrast afterintegration
0.4
integration
over x axis z
integration
i 0 2middle T
low T
integration
over x axisz
0.2
high Tg
over x axisz integration distance Dx
00 10 20 30
Dx(pixels)
Experiments with 2D Bose gasH d ib bi t l N t 441 1118 (2006)
fit by:
trast 0.4
l T
22
12 1~),0(1~
D
Ddxxg
DC
x
Hadzibabic et al., Nature 441:1118 (2006)
rate
d co
nt
0.2low T
middle T
Exponent
xx DD
integration distance D
Inte
gr
00 10 20 30
high T
0.5integration distance Dx
if g1(r) decays exponentially ith
0.4
0.3 high T low Twith :
if g1(r) decays algebraically or central contrast
0 0.1 0.2 0.3
g ow
if g1(r) decays algebraically or exponentially with a large :
central contrast
“Sudden” jump!?
Experiments with 2D Bose gas. Proliferation of thermal vortices Hadzibabic et al Nature (2006)thermal vortices Hadzibabic et al., Nature (2006)
i f i h i30% Fraction of images showing at least one dislocation
Exponent
0.520%
30%
0.4
0 3
10%hi h T low T
0 0.1 0.2 0.3
0.3
l
00 0.1 0.2 0.3 0.4
high T low T
central contrast
The onset of proliferation
central contrast
pcoincides with shifting to 0.5!
Quantum dynamics of splitQuantum dynamics of split one dimensional condensatesP th li tiPrethermalization
Theory: Takuya Kitagawa et al., PRL (2010)New J. Phys. (2011)
Experiments: D. Smith, J. Schmiedmayer, et al.
arXiv:1112 0013arXiv:1112.0013
Relaxation to equilibriumThermalization: an isolated interacting systems approaches thermalThermalization: an isolated interacting systems approaches thermal equilibrium at long times (typically at microscopic timescales). All memory about the initial conditions except energy is lost.
Bolzmann equation
U S h id t lU. Schneider et al., arXiv:1005.3545
Prethermalization
Heavy ions collisionsHeavy ions collisionsQCD
We observe irreversibility and approximate thermalization. At large time the system approaches stationary solution in the vicinity of, but not identical to thermal equilibrium The ensemble therefore retainsnot identical to, thermal equilibrium. The ensemble therefore retains some memory beyond the conserved total energy…This holds for interacting systems and in the large volume limit.
Prethermalization in ultracold atoms, theory: Eckstein et al. (2009); Moeckel et al. (2010), L. Mathey et al. (2010), R. Barnett et al.(2010)
Measurements of dynamics of split condensate
Theoretical analysis of dephasingLuttinger liquid modelLuttinger liquid model
Luttinger liquid model of phase dynamics
Luttinger liquid model of phase dynamics
For each k-mode we have simple harmonic oscillators
Phase diffusion vs Contrast Decay
Segment size is smaller than the fluctuation lengthscale
y
Segment size is smaller than the fluctuation lengthscale
Segment si e is longer than the fl ct ation lengthscaleSegment size is longer than the fluctuation lengthscale
At long times the difference between the two regime occurs for
Length dependent phase dynamics
“Short segments” = phase diffusion
µm
10µm
15 ms 15.5 16 16.5 17 19 21 24 27 32 37 47 62 77 107 137 167 197
µm
30
µm
20µ
m
61µ
m
41µ
“L t ” t t
110
µm
“Long segments” = contrast decay
Energy distributionAt t=0 system is in a squeezed state with large number fluctuationsAt t=0 system is in a squeezed state with large number fluctuations
Energy stored in each mode initially
Equipartition of energy For 2d also pointed out by Mathey, Polkovnikov in PRA (2010) o d a so po ted out by at ey, o o o ( 0 0)
The system should look thermal like after different modes dephase.Effective temperature is not related to the physical temperature
Comparison of experiments and LL analysis
Do we have thermal-like distributions at longer times
Prethermalization
Interference contrast is described by thermal distributions but at temperature much lowerdistributions but at temperature much lower than the initial temperature
Testing PrethermalizationTesting Prethermali ation
First lecture: experiments with ultracold bosons
Cold atoms in optical lattices
Bose Hubbard model. Superfluid to Mott transitionLooking for Higgs particle in the Bose Hubbard modelQuantum magnetism with ultracold atoms in optical lattices
L di i l d tLow dimensional condensates
Observing quasi-long range order in interference experimentsObserving quasi long range order in interference experimentsObservation of prethermolization
Beyond Gutzwiller: Scaling at low frequencies
signature of Higgs/Goldstone mode coupling
Excite virtual Higgs excitationVirtual Higgs decays into a pair of Goldstone excitations Matrix element of Higgs to Goldstone coupling scales as w2
Phase space scales as 1/Phase space scales as 1/wFermi’s golden rule: (w2)2x(1/w) = w3
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