measuring correlation functions in interacting systems of cold...
TRANSCRIPT
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/Weizmann
Vladimir Gritsev Harvard
Mikhail Lukin HarvardEugene Demler Harvard
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)
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
Lecture I
Measuring correlation functions in intereference experiments
1. Interference of independent condensates2. Interference of interacting 1D systems
3. Interference of 2D systems4. Full distribution function of the fringe amplitudes
in intereference experiments. 5. Studying coherent dynamics of strongly interacting
systems in interference experiments
Lecture II
Quantum noise interferometryin time of flight experiments
1. Detection of spin order in Mott states of atomic mixtures2. Detection of fermion pairing
Analysis of high order correlation
functions in low dimensional systems
Polkovnikov, Altman, Demler, PNAS (2006)
Measuring correlation functions in intereference experiments
Interference of two independent condensates
Andrews et al., Science 275:637 (1997)
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
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
Interference amplitude and correlations
For identical condensates
Instantaneous correlation function
L
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
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
For impenetrable bosons and
Interference between 1d interacting bosons
Luttinger 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
Luttinger parameter K may beextracted from the angular
dependence of
Rotated probe beam experiment
θ
For large imaging angle, ,
Interference between two-dimensional
BECs at finite temperature.
Kosteritz-Thouless transition
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
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
x
z
Time of
flight
low temperature higher temperature
Typical interference patterns
Experiments with 2D Bose gasHadzibabic et al., Nature (2006)
integration
over x axis
Dx
z
z
integration
over x axisz
x
integration distance Dx
(pixels)
Contrast after
integration
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)
fit by:
integration distance Dx
Inte
gra
ted
con
tras
t 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 α
central contrast
0.5
0 0.1 0.2 0.3
0.4
0.3high T low T
[ ]α2
2
1
2 1~),0(
1~
∫
x
D
x Ddxxg
DC
x
“Sudden” jump!?
Experiments with 2D Bose gasHadzibabic et al., Nature (2006)
Exponent α
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 α=1/4
just below the transition
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
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
Interference between two interacting
one dimensional Bose liquids
Full distribution function
of the amplitude of
interference fringes
Gritsev, Altman, Demler, Polkovnikov, cond-mat/0602475
Higher moments of interference amplitude
L Higher moments
Changing to periodic boundary conditions (long condensates)
Explicit expressions for are available but cumbersomeFendley, 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 ?
Impurity in a Luttinger liquid
Expansion of the partition function in powers of g
Partition function of the impurity contains correlation functionstaken 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 problem
and 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 the single particle Schroedinger equationDorey, 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
P
rob
ab
ility
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
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 CFTBazhanov, Lukyanov, Zamolodchikov, Comm. Math. Phys.1996, 1997, 1999
From interference amplitudes to conformal field theories
Studying coherent dynamics
of strongly interacting systems
in interference experiments
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.
Coupled 1d systems
J
Relative phase Particle number imbalance
Conjugate variables
Small K corresponds to strong quantum flcutuations
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
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 finite
tunneling and let evolve
for some time
Measure amplitude
of interference pattern
time
Amplitude of
interference fringes
Oscillations or decay?
Coherent dynamics of quantum Sine-Gordon model
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 model
Calabrese, Cardy (2006); Ghoshal, Zamolodchikov (1994)
Interference amplitude can be calculated using form factor approach
Smirnov (1992), Lukyanov (1997)
Coherent dynamics of quantum Sine-Gordon model
J
Prepare a system at t=0 Take to the regime of finite
tunneling and let evolve
for some time
Measure amplitude
of interference pattern
time
Amplitude of
interference fringes
Amplitude of interference fringes shows
oscillations at frequencies that correspond
to energies of breater
Coherent dynamics of quantum Sine-Gordon model
Conclusions for part I
Interference of fluctuating condensates can be usedto probe correlation functions in one and two
dimensional systems. Interference experiments can
also be used to study coherent dynamics of interactingsystems
Measuring correlation functions in interacting systems of cold atoms
Lecture II
Quantum noise interferometryin time of flight experiments
1. Time of flight experiments.
Second order coherence in Mott states of spinless bosons2. 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
Bose-Einstein condensation
Cornell et al., Science 269, 198 (1995)
Ultralow density condensed matter systemInteractions are weak and can be described theoretically from first principles
Superfluid to Insulator transitionGreiner et al., Nature 415:39 (2002)
U
µ
1−n
t/U
SuperfluidMott insulator
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
Theory: Altman et al., PRA 70:13603 (2004)
Experiment: Folling et al., Nature 434:481 (2005)
Hanburry-Brown-Twiss stellar interferometer
Hanburry-Brown-Twiss interferometer
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
Theory: Altman et al., PRA 70:13603 (2004)
Experiment: Folling et al., Nature 434:481 (2005)
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
Width of the noise peaks
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
Applications of quantum noise interferometryin time of flight experiments
Detection of spin order in Mott states of boson boson mixtures
Engineering magnetic systems
using cold atoms in an optical lattice
See also lectures by A. Georges and I. Cirac in this school
Spin interactions using controlled collisions
Experiment: Mandel et al., Nature 425:937(2003)
Theory: Jaksch et al., PRL 82:1975 (1999)
t
t
Two component Bose mixture in optical latticeExample: . Mandel et al., Nature 425:937 (2003)
Two component Bose Hubbard model
Quantum magnetism of bosons in optical lattices
Duan et al., PRL (2003)
• Ferromagnetic
• Antiferromagnetic
Kuklov and Svistunov, PRL (2003)
Exchange Interactions in Solids
antibonding
bonding
Kinetic energy dominates: antiferromagnetic state
Coulomb energy dominates: ferromagnetic state
Two component Bose mixture in optical lattice.
Mean field theory + Quantum fluctuations
2 ndorder line
Hysteresis
1st order
Altman et al., NJP 5:113 (2003)
Probing spin order of bosons
Correlation Function Measurements
Engineering exotic phases
• Optical lattice in 2 or 3 dimensions: polarizations & frequencies
of standing waves can be different for different directions
ZZ
YY
• Example: exactly solvable model
Kitaev (2002), honeycomb lattice with
H = Jx
< i, j>∈x
∑ σ i
xσ j
x + Jy
< i, j>∈y
∑ σ i
yσ j
y + Jz
< i, j>∈z
∑ σ i
zσ 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
Applications of quantum noise interferometryin time of flight experiments
Detection of fermion pairing
Fermionic atoms in optical lattices
Pairing in systems with repulsive interactions. Unconventional pairing. High Tc mechanism
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
Fermions with repulsive interactions
t
U
t
Possible d-wave pairing of fermions
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
Positive U Hubbard model
Possible phase diagram. Scalapino, Phys. Rep. 250:329 (1995)
Antiferromagnetic insulator
D-wave superconductor
Second order correlations in the BCS superfluid
)'()()',( rrrr nnn −≡∆
n(r)
n(r’)
n(k)
k
0),( =Ψ−∆ BCSn rr
BCS
BEC
kF
Expansion of atoms in TOF maps k into r
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
Simulation of condensed matter systems:Hubbard Model and high Tc superconductivity
t
U
t
Using cold atoms to go beyond “plain vanilla” Hubbard modela) Boson-Fermion mixtures: Hubbard model + phononsb) 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 Tc
superconductivity. Understanding Hubbard model means finding
what these missing 10% are. Electron-phonon interaction?
Mesoscopic structures (stripes)?
Boson Fermion mixtures
Fermions interacting with phonons
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, …
Boson Fermion mixtures
q/kF
/EF
ω
4
3
2
1
0 2 1 0
“Phonons” :Bogoliubov (phase) mode
Effective fermion-”phonon” interaction
Fermion-”phonon” vertex Similar to electron-phonon systems
Boson Fermion mixtures in 1d optical lattices
Cazalila et al., PRL (2003); Mathey et al., PRL (2004)
Spinless fermions Spin ½ fermions
Boson Fermion mixtures in 2d optical lattices
40K -- 87Rb 40K -- 23Na
=1060 nm
(a) =1060nm
(b) =765.5nm
Wang et al., PRA (2005)
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