measuring correlation functions in interacting systems of cold atoms
DESCRIPTION
Measuring correlation functions in interacting systems of cold atoms. Anatoli Polkovnikov Harvard/Boston University Ehud Altman Harvard/Weizmann Vladimir Gritsev Harvard Mikhail Lukin Harvard Eugene Demler Harvard. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/1.jpg)
Measuring correlation functions in interacting systems of cold atoms
Thanks to: M. Greiner , Z. Hadzibabic, M. Oberthaler, J. Schmiedmayer, V. Vuletic
Anatoli Polkovnikov Harvard/Boston UniversityEhud Altman Harvard/WeizmannVladimir Gritsev HarvardMikhail Lukin HarvardEugene Demler Harvard
![Page 2: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/2.jpg)
Correlation functions in condensed matter physics
Most experiments in condensed matter physics measure correlation functions
Example: neutron scattering measures spin and density correlation functions
Neutron diffraction patterns for MnO
Shull et al., Phys. Rev. 83:333 (1951)
![Page 3: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/3.jpg)
Outline
Lecture I: Measuring correlation functions in intereference experiments
Lecture II: Quantum noise interferometry in time of flight experiments
Emphasis of these lectures: detection and characterization of many-body quantum states
![Page 4: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/4.jpg)
Lecture I
Measuring correlation functions in intereference experiments 1. Interference of independent condensates 2. Interference of interacting 1D systems 3. Interference of 2D systems 4. Full distribution function of the fringe amplitudes in intereference experiments. 5. Studying coherent dynamics of strongly interacting systems in interference experiments
![Page 5: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/5.jpg)
Lecture II
Quantum noise interferometry in time of flight experiments
1. Detection of spin order in Mott states of atomic mixtures 2. Detection of fermion pairing
![Page 6: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/6.jpg)
Analysis of high order correlation functions in low dimensional systems
Polkovnikov, Altman, Demler, PNAS (2006)
Measuring correlation functions in intereference experiments
![Page 7: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/7.jpg)
Interference of two independent condensates
Andrews et al., Science 275:637 (1997)
![Page 8: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/8.jpg)
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
![Page 9: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/9.jpg)
Amplitude of interference fringes, , contains information about phase fluctuations within individual condensates
y
x
Interference of one dimensional condensates
x1
d Experiments: Schmiedmayer et al., Nature Physics (2005)
x2
![Page 10: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/10.jpg)
Interference amplitude and correlations
For identical condensates
Instantaneous correlation function
L
![Page 11: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/11.jpg)
Interacting bosons in 1d at T=0
K – Luttinger parameter
Low energy excitations and long distance correlation functions can be described by the Luttinger Hamiltonian.
Small K corresponds to strong quantum fluctuations
Connection to original bosonic particles
![Page 12: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/12.jpg)
Luttinger liquids in 1d
For non-interacting bosons
For impenetrable bosons
Correlation function decays rapidly for small K. This decay comes from strong quantum fluctuations
![Page 13: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/13.jpg)
For impenetrable bosons and
Interference between 1d interacting bosonsLuttinger liquid at T=0
K – Luttinger parameter
Luttinger liquid at finite temperature
For non-interacting bosons and
Analysis of can be used for thermometry
L
![Page 14: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/14.jpg)
Luttinger parameter K may beextracted from the angular dependence of
Rotated probe beam experiment
For large imaging angle, ,
![Page 15: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/15.jpg)
Interference between two-dimensional BECs at finite temperature.Kosteritz-Thouless transition
![Page 16: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/16.jpg)
Interference of two dimensional condensates
Ly
Lx
Lx
Experiments: Stock, Hadzibabic, Dalibard, et al., cond-mat/0506559
Probe beam parallel to the plane of the condensates
Gati, Oberthaler, et al., cond-mat/0601392
![Page 17: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/17.jpg)
Interference of two dimensional condensates.Quasi long range order and the KT transition
Ly
Lx
Below KT transitionAbove KT transition
Below Kosterlitz-Thouless transition:Vortices confined. Quasi long range order
Above Kosterlitz-Thouless transition:Vortices proliferate. Short range order
![Page 18: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/18.jpg)
x
z
Time of
flight
low temperature higher temperature
Typical interference patterns
Experiments with 2D Bose gasHadzibabic et al., Nature (2006)
![Page 19: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/19.jpg)
integration
over x axis
Dx
z
z
integration
over x axisz
x
integration distance Dx
(pixels)
Contrast afterintegration
0.4
0.2
00 10 20 30
middle Tlow T
high T
integration
over x axis z
Experiments with 2D Bose gasHadzibabic et al., Nature (2006)
![Page 20: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/20.jpg)
fit by:
integration distance Dx
Inte
grat
ed c
ontr
ast 0.4
0.2
00 10 20 30
low Tmiddle T
high T
if g1(r) decays exponentially with :
if g1(r) decays algebraically or exponentially with a large :
Exponent
central contrast
0.5
0 0.1 0.2 0.3
0.4
0.3high T low T
2
21
2 1~),0(
1~
x
D
x Ddxxg
DC
x
“Sudden” jump!?
Experiments with 2D Bose gasHadzibabic et al., Nature (2006)
![Page 21: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/21.jpg)
Exponent
central contrast
0.5
0 0.1 0.2 0.3
0.4
0.3 high T low T
He experiments:universal jump in
the superfluid density
T (K)1.0 1.1 1.2
1.0
0
c.f. Bishop and Reppy
Experiments with 2D Bose gasHadzibabic et al., Nature (2006)
Ultracold atoms experiments:jump in the correlation function.
KT theory predicts =1/4 just below the transition
![Page 22: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/22.jpg)
Experiments with 2D Bose gas. Proliferation of thermal vortices Haddzibabic et al., Nature (2006)
Fraction of images showing at least one dislocation
0
10%
20%
30%
central contrast
0 0.1 0.2 0.3 0.4
high T low T
![Page 23: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/23.jpg)
Rapidly rotating two dimensional condensates
Time of flight experiments with rotating condensates correspond to density measurements
Interference experiments measure singleparticle correlation functions in the rotatingframe
![Page 24: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/24.jpg)
Interference between two interacting one dimensional Bose liquids
Full distribution function of the amplitude of interference fringes
Gritsev, Altman, Demler, Polkovnikov, cond-mat/0602475
![Page 25: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/25.jpg)
Higher moments of interference amplitude
L Higher moments
Changing to periodic boundary conditions (long condensates)
Explicit expressions for are available but cumbersome Fendley, Lesage, Saleur, J. Stat. Phys. 79:799 (1995)
is a quantum operator. The measured value of will fluctuate from shot to shot.Can we predict the distribution function of ?
![Page 26: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/26.jpg)
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
![Page 27: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/27.jpg)
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
![Page 28: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/28.jpg)
is related to the single particle 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
![Page 29: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/29.jpg)
![Page 30: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/30.jpg)
0 1 2 3 4
P
roba
bilit
y P
(x)
x
K=1 K=1.5 K=3 K=5
Evolution of the distribution function
Narrow distributionfor .Approaches Gumbledistribution. Width
Wide Poissoniandistribution for
![Page 31: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/31.jpg)
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
![Page 32: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/32.jpg)
Studying coherent dynamics of strongly interacting systemsin interference experiments
![Page 33: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/33.jpg)
Coupled 1d systems
J
Interactions lead to phase fluctuations within individual condensates
Tunneling favors aligning of the two phases
Interference experiments measure only the relative phase
Motivated by experiments of Schmiedmayer et al.
![Page 34: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/34.jpg)
Coupled 1d systems
J
Relative phase Particle number imbalance
Conjugate variables
Small K corresponds to strong quantum flcutuations
![Page 35: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/35.jpg)
Quantum Sine-Gordon model
Quantum Sine-Gordon model is exactly integrable
Excitations of the quantum Sine-Gordon model
Hamiltonian
Imaginary time action
soliton antisoliton breather
![Page 36: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/36.jpg)
Coherent dynamics of quantum Sine-Gordon model Motivated by experiments of Schmiedmayer et al.
J
Prepare a system at t=0 Take to the regime of finitetunneling and let evolve for some time
Measure amplitude of interference pattern
![Page 37: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/37.jpg)
time
Amplitude ofinterference fringes
Oscillations or decay?
Coherent dynamics of quantum Sine-Gordon model
![Page 38: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/38.jpg)
From integrability to coherent dynamics
At t=0 we have a state with for all
This state can be written as a “squeezed” state
Time evolution can be easily written
Matrix can be constructed using connection to boundary SG modelCalabrese, Cardy (2006); Ghoshal, Zamolodchikov (1994)
Interference amplitude can be calculated using form factor approachSmirnov (1992), Lukyanov (1997)
![Page 39: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/39.jpg)
Coherent dynamics of quantum Sine-Gordon model
J
Prepare a system at t=0 Take to the regime of finitetunneling and let evolve for some time
Measure amplitude of interference pattern
![Page 40: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/40.jpg)
time
Amplitude ofinterference fringes
Amplitude of interference fringes showsoscillations at frequencies that correspond to energies of breater
Coherent dynamics of quantum Sine-Gordon model
![Page 41: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/41.jpg)
Conclusions for part I
Interference of fluctuating condensates can be usedto probe correlation functions in one and two dimensional systems. Interference experiments canalso be used to study coherent dynamics of interactingsystems
![Page 42: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/42.jpg)
Measuring correlation functions in interacting systems of cold atoms
Lecture II Quantum noise interferometry in time of flight experiments
1. Time of flight experiments. Second order coherence in Mott states of spinless bosons 2. Detection of spin order in Mott states of atomic mixtures 3. Detection of fermion pairing
Emphasis of these lectures: detection and characterization of many-body quantum states
![Page 43: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/43.jpg)
Bose-Einstein condensation
Cornell et al., Science 269, 198 (1995)
Ultralow density condensed matter system
Interactions are weak and can be described theoretically from first principles
![Page 44: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/44.jpg)
Superfluid to Insulator transitionGreiner et al., Nature 415:39 (2002)
U
1n
t/U
SuperfluidMott insulator
![Page 45: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/45.jpg)
Time of flight experiments
Quantum noise interferometry of atoms in an optical lattice
Second order coherence
![Page 46: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/46.jpg)
Second order coherence in the insulating state of bosons.Hanburry-Brown-Twiss experiment
Theory: Altman et al., PRA 70:13603 (2004)
Experiment: Folling et al., Nature 434:481 (2005)
![Page 47: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/47.jpg)
Hanburry-Brown-Twiss stellar interferometer
![Page 48: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/48.jpg)
Hanburry-Brown-Twiss interferometer
![Page 49: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/49.jpg)
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
![Page 50: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/50.jpg)
Second order coherence in the insulating state of bosons.Hanburry-Brown-Twiss experiment
Theory: Altman et al., PRA 70:13603 (2004)
Experiment: Folling et al., Nature 434:481 (2005)
![Page 51: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/51.jpg)
Effect of parabolic potential on the second order coherence
Experiment: Spielman, Porto, et al.,Theory: Scarola, Das Sarma, Demler, PRA (2006)
Width of the correlation peak changes across the transition, reflecting the evolution of Mott domains
![Page 52: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/52.jpg)
Width of the noise peaks
![Page 53: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/53.jpg)
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
![Page 54: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/54.jpg)
Applications of quantum noise interferometry in time of flight experiments
Detection of spin order in Mott states of boson boson mixtures
![Page 55: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/55.jpg)
Engineering magnetic systems using cold atoms in an optical lattice
See also lectures by A. Georges and I. Cirac in this school
![Page 56: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/56.jpg)
Spin interactions using controlled collisions
Experiment: Mandel et al., Nature 425:937(2003)
Theory: Jaksch et al., PRL 82:1975 (1999)
![Page 57: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/57.jpg)
t
t
Two component Bose mixture in optical latticeExample: . Mandel et al., Nature 425:937 (2003)
Two component Bose Hubbard model
![Page 58: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/58.jpg)
Quantum magnetism of bosons in optical lattices
Duan et al., PRL (2003)
• Ferromagnetic• Antiferromagnetic
Kuklov and Svistunov, PRL (2003)
![Page 59: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/59.jpg)
Exchange Interactions in Solids
antibonding
bonding
Kinetic energy dominates: antiferromagnetic state
Coulomb energy dominates: ferromagnetic state
![Page 60: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/60.jpg)
Two component Bose mixture in optical lattice.Mean field theory + Quantum fluctuations
2 nd order line
Hysteresis
1st order
Altman et al., NJP 5:113 (2003)
![Page 61: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/61.jpg)
Probing spin order of bosons
Correlation Function Measurements
![Page 62: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/62.jpg)
Engineering exotic phases
• Optical lattice in 2 or 3 dimensions: polarizations & frequenciesof standing waves can be different for different directions
ZZ
YY
• Example: exactly solvable modelKitaev (2002), honeycomb lattice with
H Jx
i, jx
ix j
x Jy
i, jy
iy j
y Jz
i, jz
iz j
z
• Can be created with 3 sets of standing wave light beams !• Non-trivial topological order, “spin liquid” + non-abelian anyons …those has not been seen in controlled experiments
![Page 63: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/63.jpg)
Applications of quantum noise interferometry in time of flight experiments
Detection of fermion pairing
![Page 64: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/64.jpg)
Fermionic atoms in optical lattices
Pairing in systems with repulsive interactions. Unconventional pairing. High Tc mechanism
![Page 65: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/65.jpg)
Fermionic atoms in a three dimensional optical lattice
Kohl et al., PRL 94:80403 (2005)
See also lectures of T. Esslinger and W. Ketterle in this school
![Page 66: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/66.jpg)
Fermions with repulsive interactions
t
U
tPossible d-wave pairing of fermions
![Page 67: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/67.jpg)
Picture courtesy of UBC Superconductivity group
High temperature superconductors
Superconducting Tc 93 K
Hubbard model – minimal model for cuprate superconductors
P.W. Anderson, cond-mat/0201429
After many years of work we still do not understand the fermionic Hubbard model
![Page 68: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/68.jpg)
Positive U Hubbard model
Possible phase diagram. Scalapino, Phys. Rep. 250:329 (1995)
Antiferromagnetic insulator
D-wave superconductor
![Page 69: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/69.jpg)
Second order correlations in the BCS superfluid
)'()()',( rrrr nnn
n(r)
n(r’)
n(k)
k
0),( BCSn rr
BCS
BEC
kF
Expansion of atoms in TOF maps k into r
![Page 70: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/70.jpg)
Momentum correlations in paired fermionsGreiner et al., PRL 94:110401 (2005)
![Page 71: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/71.jpg)
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
![Page 72: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/72.jpg)
Simulation of condensed matter systems: Hubbard Model and high Tc superconductivity
t
U
t
Using cold atoms to go beyond “plain vanilla” Hubbard model a) Boson-Fermion mixtures: Hubbard model + phonons b) Inhomogeneous systems, role of disorder
Personal opinion:
The fermionic Hubbard model contains 90% of the physics of cuprates. The remaining 10% may be crucial for getting high Tcsuperconductivity. Understanding Hubbard model means findingwhat these missing 10% are. Electron-phonon interaction?Mesoscopic structures (stripes)?
![Page 73: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/73.jpg)
Boson Fermion mixtures
Fermions interacting with phonons
![Page 74: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/74.jpg)
Boson Fermion mixtures
BEC
See lectures by T. Esslinger and G. Modugno in this school
Bosons provide cooling for fermionsand mediate interactions. They createnon-local attraction between fermions
Charge Density Wave Phase
Periodic arrangement of atoms
Non-local Fermion Pairing
P-wave, D-wave, …
![Page 75: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/75.jpg)
Boson Fermion mixtures
“Phonons” :Bogoliubov (phase) mode
Effective fermion-”phonon” interaction
Fermion-”phonon” vertex Similar to electron-phonon systems
![Page 76: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/76.jpg)
Boson Fermion mixtures in 1d optical latticesCazalila et al., PRL (2003); Mathey et al., PRL (2004)
Spinless fermions Spin ½ fermions
![Page 77: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/77.jpg)
Boson Fermion mixtures in 2d optical lattices
40K -- 87Rb 40K -- 23Na
=1060 nm(a) =1060nm
(b) =765.5nm
Wang et al., PRA (2005)
![Page 78: Measuring correlation functions in interacting systems of cold atoms](https://reader036.vdocuments.site/reader036/viewer/2022062720/56813342550346895d9a37a6/html5/thumbnails/78.jpg)
Conclusions
Interference of extended condensates is a powerful tool for analyzing correlation functions in one and two dimensional systems
Noise interferometry can be used to probequantum many-body states in optical lattices