Strongly correlated many-body systems: from electronic materials to ultracold atoms to photons...
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Transcript of Strongly correlated many-body systems: from electronic materials to ultracold atoms to photons...
Strongly correlated many-body systems: from electronic materials
to ultracold atoms to photons
Eugene Demler Harvard University
Thanks to: E. Altman, I. Bloch, A. Burkov, H.P. Büchler, I. Cirac, D. Chang, R. Cherng, V. Gritsev, B. Halperin, W. Hofstetter, A. Imambekov, M. Lukin, G. Morigi, A. Polkovnikov, G. Pupillo, G. Refael, A.M. Rey, O. Romero-Isart , J. Schmiedmayer, R. Sensarma, D.W. Wang, P. Zoller, and many others
“Conventional” solid state materials
Bloch theorem for non-interacting electrons in a periodic potential
“Conventional” solid state materialsElectron-phonon and electron-electron interactions are irrelevant at low temperatures
kx
ky
kF
Landau Fermi liquid theory: when frequency and temperature are smaller than EF electron systems
are equivalent to systems of non-interacting fermions
Ag Ag
Ag
Strongly correlated electron systems
Quantum Hall systemskinetic energy suppressed by magnetic field
One dimensional electron systemsnon-perturbative effects of interactions in 1d
High temperature superconductors,Heavy fermion materials,Organic conductors and superconductorsmany puzzling non-Fermi liquid properties
Bose-Einstein condensation of weakly interacting atoms
Scattering length is much smaller than characteristic interparticle distances. Interactions are weak
Strongly correlated systems of cold atoms
• Low dimensional systems
• Optical lattices
• Feshbach resonances
Linear geometrical optics
Strong optical nonlinearities innanoscale surface plasmonsAkimov et al., Nature (2007)
Strongly interacting polaritons in coupled arrays of cavitiesM. Hartmann et al., Nature Physics (2006)
Crystallization (fermionization) of photonsin one dimensional optical waveguidesD. Chang et al., arXive:0712.1817
Strongly correlated systems of photons
We understand well: many-body systems of non-interactingor weakly interacting particles. For example, electron systems in semiconductors and simple metals,When the interaction energy is smaller than the kinetic energy,perturbation theory works well
We do not understand: many-body systems with stronginteractions and correlations. For example, electron systems in novel materials such as high temperature superconductors. When the interaction energy is comparable or larger than the kinetic energy, perturbation theory breaks down. Many surprising new phenomena occur, including unconventional superconductivity, magnetism, fractionalization of excitations
Ultracold atoms have energy scales of 10-6K, compared to 104 K for electron systems
We will also get new systems useful for applications in quantum information and communications, high precision spectroscopy, metrology
By engineering and studying strongly interacting systems of cold atoms we should get insights into the mysterious properties of novel quantum materials
Strongly interacting systems of ultracold atoms and photons: NOT the analogue simulatorsThese are independent physical systems with their own “personalities”, physical properties, and theoretical challenges
Focus of these lectures: challenges of new strongly correlated systems
Part IDetection and characterization of many body statesQuantum noise analysis and interference experiments
Part IINew challenges in quantum many-body theory: non-equilibrium coherent dynamics
Quantum noise studies of ultracold atoms
Part I
Introduction. Historical review
Quantum noise analysis of the timeof flight experiments with utlracold atoms (HBT correlations and beyond)
Quantum noise in interference experiments with independent condensates
Quantum noise analysis of spin systems
Outline of part I
Quantum noiseClassical measurement: collapse of the wavefunction into eigenstates of x
Histogram of measurements of x
Probabilistic nature of quantum mechanics
Bohr-Einstein debate on spooky action at a distance
Measuring spin of a particle in the left detectorinstantaneously determines its value in the right detector
Einstein-Podolsky-Rosen experiment
Aspect’s experiments:tests of Bell’s inequalities
SCorrelation function
Classical theories with hidden variable require
Quantum mechanics predicts B=2.7 for the appropriate choice of ‘s and the state
Experimentally measured value B=2.697. Phys. Rev. Let. 49:92 (1982)
+
-
+
-1 21 2
S
Hanburry-Brown-Twiss experimentsClassical theory of the second order coherence
Measurements of the angular diameter of SiriusProc. Roy. Soc. (London), A, 248, pp. 222-237
Hanbury Brown and Twiss, Proc. Roy. Soc. (London), A, 242, pp. 300-324
Shot noise in electron transport
e- e-
When shot noise dominates over thermal noise
Spectral density of the current noise
Proposed by Schottky to measure the electron charge in 1918
Related to variance of transmitted charge
Poisson process of independent transmission of electrons
Shot noise in electron transport
Current noise for tunneling across a Hall bar on the 1/3 plateau of FQE
Etien et al. PRL 79:2526 (1997)see also Heiblum et al. Nature (1997)
Quantum noise analysis of the time of flight experiments with utlracold atoms (HBT correlations and beyond)
Theory: Altman, Demler, Lukin, PRA 70:13603 (2004)
Experiments: Folling et al., Nature 434:481 (2005); Greiner et al., PRL 94:110401 (2005); Tom et al. Nature 444:733 (2006);
see also Hadzibabic et al., PRL 93:180403 (2004) Spielman et al., PRL 98:80404 (2007); Guarrera et al., preprint (2007)
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); Ketterle et al., PRL (2006)
Bose Hubbard model
tunneling of atoms between neighboring wells
repulsion of atoms sitting in the same well
U
t
Superfluid to insulator transition in an optical lattice
M. Greiner et al., Nature 415 (2002)
U
1n
t/U
SuperfluidMott insulator
Why study ultracold atoms in optical lattices?
t
U
t
Fermionic atoms in optical lattices
Experiments with fermions in optical lattice, Kohl et al., PRL 2005
Antiferromagnetic and superconducting Tc of the order of 100 K
Atoms in optical lattice
Antiferromagnetism and pairing at sub-micro Kelvin temperatures
Same microscopic model
Positive U Hubbard model
Possible phase diagram. Scalapino, Phys. Rep. 250:329 (1995)
Antiferromagnetic insulator
D-wave superconductor
Atoms in optical lattice
Same microscopic model
Quantum simulations of strongly correlated electron systems using ultracold atoms
Detection?
Quantum noise analysis as a probe of many-body states of ultracold
atoms
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
Experiment: Folling et al., Nature 434:481 (2005)
Hanburry-Brown-Twiss stellar interferometer
Hanburry-Brown-Twiss interferometer
Quantum theory of HBT experiments
For bosons
For fermions
Glauber,Quantum Optics and Electronics (1965)
HBT experiments with matter
Experiments with 4He, 3HeWestbrook et al., Nature (2007)
Experiments with neutronsIanuzzi et al., Phys Rev Lett (2006)
Experiments with electronsKiesel et al., Nature (2002)
Experiments with ultracold atomsBloch et al., Nature (2005,2006)
Second order coherence in the insulating state of bosons.Hanburry-Brown-Twiss experiment
Experiment: Folling et al., Nature 434:481 (2005)
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
Experiment: Folling et al., Nature 434:481 (2005)
0 200 400 600 800 1000 1200
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Interference of an array of independent condensates
Hadzibabic et al., PRL 93:180403 (2004)
Smooth structure is a result of finite experimental resolution (filtering)
0 200 400 600 800 1000 1200-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Quantum theory of HBT experiments
For bosons
For fermions
Second order coherence in the insulating state of fermions.Hanburry-Brown-Twiss experiment
Experiment: Tom et al. Nature 444:733 (2006)
How to detect antiferromagnetism
Probing spin order in optical lattices
Correlation Function Measurements
Extra Braggpeaks appearin the secondorder correlationfunction in theAF phase
How to detect fermion pairing
Quantum noise analysis of TOF images: beyond HBT interference
Second order interference from the BCS superfluid
)'()()',( rrrr nnn
n(r)
n(r’)
n(k)
k
0),( BCSn rr
BCS
BEC
kF
Momentum correlations in paired fermionsGreiner et al., PRL 94:110401 (2005)
Fermion pairing in an optical lattice
Second Order InterferenceIn the TOF images
Normal State
Superfluid State
measures the Cooper pair wavefunction
One can identify unconventional pairing
How to see a “cat” statein the collapse and revival experiments
Quantum noise analysis of TOF images: beyond HBT interference
Collapse and revival experimentswith bosons in an optical lattice
Coherent state in each well at t=0
Increase the heightof the optical latticeabruptly
Time evolution within each well
Initial superfluid state
Individual wells isolated
Collapse and revival experimentswith bosons in an optical lattice
At revival times tr =h/U, 2h/U, … all number states
are in phase again
The collapse occurs due to the loss of coherence betweendifferent number states
collapse
tc=h/NU
revival
tr=h/U
time
coherence
Collapse and revival experimentswith bosons in an optical lattice
Greiner et al., Nature 419:51 (2002)
Dynamical evolution of the interference pattern after jumping the optical lattice potential
Collapse and revival experimentswith bosons in an optical lattice
Greiner et al., Nature 419:51 (2002)
Quantum dynamics of a coherent states
Cat state at t=tr /2
How to see a “cat” statein collapse and revival experiments
pairing-like correlations
no first order coherence
Properties of the “cat” state:
In the time of flight experiments this should lead to correlations between and
Define correlation function(overlap original and flipped images)
Romero-Isart et al., unpublished
How to see a “cat” statein collapse and revival experiments
Exact diagonalization of the 1d lattice system. U/t=3, N=2
Collapse and revival experimentswith bosons in an optical lattice
Greiner et al., Nature 419:51 (2002)
Dynamical evolution of the interference pattern after jumping the optical lattice potential
Perfect correlations “hiding” in the image
Quantum noise in interference experiments with independent condensates
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
Nature 4877:255 (1963)
x
z
Time of
flight
Experiments with 2D Bose gasHadzibabic, Dalibard et al., Nature 441:1118 (2006)
Experiments with 1D Bose gas S. Hofferberth et al. arXiv0710.1575
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
x1
d
Amplitude of interference fringes,
Interference of fluctuating condensates
For identical condensates
Instantaneous correlation function
For independent condensates Afr is finite but is random
x2
Polkovnikov, Altman, Demler, PNAS 103:6125(2006)
Fluctuations in 1d BEC
Thermal fluctuations
Thermally energy of the superflow velocity
Quantum fluctuations
For a review see Shlyapnikov et al., J. Phys. IV France 116, 3-44 (2004)
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, cond-mat/0612011
We need the full distribution function of
Distribution function of interference fringe contrastExperiments: Hofferberth et al., arXiv0710.1575Theory: Imambekov et al. , cond-mat/0612011
Comparison of theory and experiments: no free parametersHigher order correlation functions can be obtained
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
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, cond-mat/0612011
We need the full distribution function of
L
Change to periodic boundary conditions (long condensates)
Explicit expressions for are available but cumbersome Fendley, Lesage, Saleur, J. Stat. Phys. 79:799 (1995)
Calculating distribution function of interference fringe amplitudes
Method I: mapping to quantum impurity problem
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
Pro
babi
lity
P(x
)
x
K=1 K=1.5 K=3 K=5
Interference of 1d condensates at T=0. Distribution function of the fringe contrast
Narrow distributionfor .Approaches Gumbeldistribution.
Width
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