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QUANTUM MANY‐BODY SYSTEMS OF ULTRACOLD ATOMS
Eugene 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
keV MeV GeV TeVfeV peV µeV meV eV
pK nK µK mK K
neV
roomtemperature
LHC
He N
current experiments10-11 - 10-10 K
How cold are ultracold atoms?
first BECof alkali atoms
Bose-Einstein condensation of weakly 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
New Era in Cold Atoms ResearchFocus on Systems with Strong Interactions
• Atoms in optical lattices
• Feshbach resonances
• Low dimensional systems
• Systems with long range dipolar interactions
• Rotating systems
Feshbach resonanceGreiner et al., Nature (2003); Ketterle et al., (2003)
Ketterle et al.,Nature 435, 1047-1051 (2005)
One dimensional systems
1D confinement in optical potentialWeiss et al., Science (05);Bloch et al., Esslinger et al.,
One dimensional systems in microtraps.Thywissen et al., Eur. J. Phys. D. (99);Hansel et al., Nature (01);Folman et al., Adv. At. Mol. Opt. Phys. (02)
Strongly interacting regime can be reached for low densities
Atoms in optical lattices
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
Antiferromagnetic and superconducting Tc of the order of 100 K
Atoms in optical lattice
Antiferromagnetism and pairing at nano Kelvin temperatures
Same microscopic model
Strongly correated systemsAtoms in optical latticesElectrons in Solids
Simple metalsPerturbation theory in Coulomb interaction applies. Band structure methods work
Strongly Correlated Electron SystemsBand structure methods fail.
Novel phenomena in strongly correlated electron systems:Quantum magnetism, phase separation, unconventional superconductivity,high temperature superconductivity, fractionalization of electrons …
By studying strongly interacting systems of cold atoms we expect to get insights into the mysterious properties of novel quantum materials: Quantum Simulators
BUTStrongly interacting systems of ultracold atoms :
are NOT direct analogues of condensed matter systemsThese are independent physical systems with their own “personalities”, physical properties, and theoretical challenges
Strongly correlated systems of ultracold atoms shouldalso 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
Low dimensional condensates
Observing quasi-long range order in interference experimentsObservation of prethermolization
Second lecture: Ultracold fermions
Fermions in optical lattices. Fermi Hubbard model.Current state of experiments
Lattice modulation experiments
Doublon lifetimes
Strongly interacting fermions in continuum. Stoner instability
Ultracold Bose atoms in optical lattices
Bose Hubbard model
Bose Hubbard model
tunneling of atoms between neighboring wells
repulsion of atoms sitting in the same well
U
t
In the presence of confining potential we also need to include
Typically
Bose Hubbard model. Phase diagramM.P.A. Fisher et al.,PRB (1989)2
1+n
Uµ
1
0
Mottn=1
n=2
n=3
Superfluid
Mott
Mott
Weak lattice Superfluid phase
Strong lattice Mott insulator phase
Bose Hubbard model
Hamiltonian eigenstates are Fock states
Uµ
0 1
Set .
Away from level crossings Mott states have a gap. Hence they should be stable to small tunneling.
Bose Hubbard Model. Phase diagram
21+n
Uµ
1
0
Mottn=1
n=2
n=3
Superfluid
Mott
Mott
Mott insulator phase
Particle‐hole excitation
Tips 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
21+n
Uµ
1
0
Mottn=1
n=2
n=3
Superfluid
Mott
Mott
Note that the Mott state only exists for integer filling factors.For even when atoms are localized,
make a superfluid state.
Nature 415:39 (2002)
Optical lattice and parabolic potential
Jaksch et al., PRL 81:3108 (1998)
Parabolic potential acts as a “cut” through the phase diagram. Hence in a parabolic potential we find a “wedding cake” structure.
21+n
Uµ
1
0
Mottn=1
n=2
n=3
Superfluid
Mott
Mott
Quantum gas microscopeBakr et al., Science 2010
xy
density
Nature 2010
The Higgs (amplitude) mode in a trapped 2D superfluid on a lattice
Sherson et. al. Nature 2010
Cold Atoms (Munich)Elementary Particles (CMS @ LHC)
Theory: David Pekker, Eugene DemlerExperiments: 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
Figure from Bissbort et al. (2010)
Phase (Goldstone) mode = gapless Bogoliubov mode
Gapped amplitude mode (Higgs mode)
Excitations of the Bose Hubbard model
2
Mott Superfluid
21+n
Uµ
1
0
Mottn=1
n=2
n=3
Superfluid
Mott
Mott
Softening of the amplitude mode is the defining characteristicof the second order Quantum Phase Transition
Is there a Higgs mode in 2D ?neutron scattering
lattice modulation spectroscopy
• Danger from scattering on phase modes
• In 2D: infrared divergence
• Different susceptibility has no divergence
φHiggs
φHiggs
S. Sachdev, Phys. Rev. B 59, 14054 (1999) 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
Stoferle et al., PRL(2004)
Bissbort et al., PRL(2010)
Peak at U dominates and does not change as the system goes through the SF/Mott transition
Exciting the amplitude mode
Absorbed energy
Mottn=1 Mottn=1 Mottn=1
Exciting the amplitude modeManuel Endres, Immanuel Bloch and MPQ team
Experiments: full spectrumManuel Endres, Immanuel Bloch and MPQ team
Time dependent mean‐field: Gutzwiller
Similar to Landau-Lifshitz equations in magnetism
Keep twostates per siteonly
Threshold for absorption is captured very well
Plaquette Mean Field “Better Gutzwiller”
• Variational wave functions better captures local physics– better describes interactions between quasi‐particles
• Equivalent to MFT on plaquettes
Time dependent cluster mean‐fieldLattice height 9.5 Er: (1x1 vs 2x2)
2x2 captures width of spectral feature
breathing mode
single amplitude mode excited multiple modes
excited?single amplitude mode excited
breathing mode
Comparison of experiments and Gutzwiller theories
Experiment 2x2 ClustersKey experimental facts:
• “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 coupling
2 Goldstones
Higgs
wvacuum
External drive couples vacuum to HiggsHiggs can be excited only virtuallyHiggs decays into a pair of Goldstone modes with conservation of energy Matrix element w2/w=wDensity of states wFermi’s golden rule: w2xw = w3
Open question: observing discreet modes
disappearing amplitude mode
Breathing 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 ultracold atoms in optical lattices
tt
Two component Bose mixture in optical latticeExample: . Mandel et al., Nature (2003)
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)
• Ferromagnetic• 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
2 ndorder line
Hysteresis
1st order
Altman et al., NJP (2003)
Two component Bose Hubbard model
+ infinitely large Uaa and Ubb
New feature:coexistence ofcheckerboard phaseand superfluidity
Exchange Interactions in Solidsantibonding
bonding
Kinetic energy dominates: antiferromagnetic state
Coulomb energy dominates: ferromagnetic state
Realization of spin liquid using cold atoms in an optical lattice
Theory: Duan, Demler, Lukin PRL (03)
Kitaev model Annals of Physics (2006)
H = - Jx Σ σix σj
x - Jy Σ σiy σj
y - Jz Σ σiz σj
z
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 wells
Theory: A.M. Rey et al., PRL 2008Experiments: S. Trotzky et al., Science 2008
Observation of superexchange in a double well potentialTheory: A.M. Rey et al., PRL 2008
J
J
Use magnetic field gradient to prepare a state
Observe oscillations between and states
Experiments:S. Trotzky et al.Science 2008
Preparation and detection of Mott statesof 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 includesonly local interaction
Extended Hubbard modeltakes into account non-localinteraction
Beyond the basic Hubbard model
Probing low dimensional condensates with interference
experiments
Quasi long range order
Prethermalization
Interference of independent condensates
Experiments: Andrews et al., Science 275:637 (1997)
Theory: Javanainen, Yoo, PRL 76:161 (1996)Cirac, Zoller, et al. PRA 54:R3714 (1996)Castin, Dalibard, PRA 55:4330 (1997)and many more
x
z
Time of
flight
Experiments with 2D Bose gasHadzibabic, Dalibard et al., Nature 2006
Experiments with 1D Bose gas Hofferberth et al. Nat. Physics 2008
Interference of two independent condensates
r’r
Assuming ballistic expansion1 r+d
d
2 Phase difference between clouds 1 and 2is not well defined
Individual measurements show interference patternsThey disappear after averaging over many shots
Interference of fluctuating condensates
x1
dAmplitude of interference fringes,
For independent condensates Afr is finite but ∆φ is random
x2
Polkovnikov et al., PNAS (2006); Gritsev et al., Nature Physics (2006)
For identical condensates
Instantaneous correlation function
FDF of phase and contrast• Matter-wave interferometry
phase, contrast
FDF of phase and contrast• Matter-wave interferometry
contrast
phase, contrast
phase
• Plot as circular statistics
• Matter-wave interferometry: repeat many times
• Plot
i>100
contrasti
phase
phase, contrast
FDF of phase and contrast
accumulate statistics
Calculate average contrast
Fluctuations in 1d BECThermal fluctuations
Thermally energy of the superflow velocity
Quantum fluctuations
For impenetrable bosons and
Interference between Luttinger liquids
Luttinger liquid at T=0
K – Luttinger parameter
Finite temperature
Experiments: Hofferberth,Schumm, Schmiedmayer
For non-interacting bosons and
Distribution function of fringe amplitudes for interference of fluctuating condensates
L
is a quantum operator. The measured value of will fluctuate from shot to shot.
Higher moments reflect higher order correlation functions
Gritsev, Altman, Demler, Polkovnikov, Nature Physics 2006Imambekov, Gritsev, Demler, PRA (2007)
We need the full distribution function of
Distribution function of interference fringe contrastHofferberth et al., Nature Physics 2009
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 parametersHigher order correlation functions can be obtained
Interference between interacting 1d Bose liquids.Distribution function of the interference amplitude
Distribution function of
Quantum impurity problem: interacting one dimensionalelectrons scattered on an impurity
Yang-Lee singularity
2D quantum gravity,non-intersecting loops
Conformal field theories with negative central charges: 2D quantum gravity, non-intersecting loop model, growth of random fractal stochastic interface, high energy limit of multicolor QCD, …
Fringe visibility and statistics of random surfaces
Distribution function of
)(ϕh
Roughness ϕϕ dh2
)(∫=
Mapping between fringe visibility and the problem of surface roughness for fluctuating random surfaces. Relation to 1/f Noise and Extreme Value Statistics
Interference of two dimensional condensatesExperiments: Hadzibabic et al. Nature (2006)
Gati et al., PRL (2006)
Ly
LxLx
Probe beam parallel to the plane of the condensates
Interference of two dimensional condensates.Quasi long range order and the BKT transition
Ly
Lx
Above BKT transition Below BKT transition
x
z
Time of
flight
Experiments with 2D Bose gasHadzibabic, Dalibard et al., Nature 441:1118 (2006)
Typical interference patterns
low temperature higher temperature
Experiments with 2D Bose gasHadzibabic et al., Nature 441:1118 (2006)
integration
over x axis
Dx
z
z
integration
over x axisz
xintegration
over x axis z
integration distance Dx (pixels)
Contrast afterintegration
0.4
0.2
00 10 20 30
middle Tlow T
high T
Experiments with 2D Bose gasHadzibabic et al., Nature 441:1118 (2006)
[ ]α2
21
2 1~),0(1~ ⎟⎟⎠
⎞⎜⎜⎝
⎛∫
x
D
x Ddxxg
DC
x0.4
0.2
00 10 20 30
low Tmiddle T
high T
fit by:
Inte
grat
ed c
ontra
st
Exponent α
central contrast
0.5
0 0.1 0.2 0.3
0.4
0.3 high T low T
“Sudden” jump!?
integration distance Dx
if g1(r) decays exponentially with :
if g1(r) decays algebraically or exponentially with a large :
Experiments with 2D Bose gas. Proliferation of thermal vortices Hadzibabic et al., Nature (2006)
Fraction of images showing at least one dislocation
Exponent α
0.5
0 0.1 0.2 0.3
0.4
0.3
central contrast
The onset of proliferation coincides with α shifting to 0.5!
0
10%
20%
30%
central contrast0 0.1 0.2 0.3 0.4
high T low T
Quantum dynamics of split one dimensional condensatesPrethermalization
Theory: Takuya Kitagawa et al., PRL (2010)New J. Phys. (2011)
Experiments: D. Smith, J. Schmiedmayer, et al.
arXiv:1112.0013
Relaxation to equilibriumThermalization: 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. Schneider et al., arXiv:1005.3545
Prethermalization
Heavy 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 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 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
Segment 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
110
µm
61µ
m
41µ
m
30
µm
20µm
10
µm
15 ms 15.5 16 16.5 17 19 21 24 27 32 37 47 62 77 107 137 167 197
“Long segments” = contrast decay
Energy distributionAt 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)
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 lower than the initial temperature
Testing Prethermalization
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
Low dimensional condensates
Observing 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/wFermi’s golden rule: (w2)2x(1/w) = w3