Learning about order from noise Quantum noise studies of ultracold atoms Eugene Demler Harvard University Funded by NSF, Harvard-MIT CUA, AFOSR, DARPA, MURI Collaborators: Ehud Altman, Robert Cherng, Adilet Imambekov, Vladimir Gritsev, Mikhail Lukin, Anatoli Polkovnikov
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Learning about order from noise
Quantum noise studies of ultracold atoms
Eugene Demler Harvard University
Funded by NSF, Harvard-MIT CUA, AFOSR, DARPA, MURI
Collaborators:Ehud Altman, Robert Cherng, Adilet Imambekov,
Vladimir Gritsev, Mikhail Lukin, Anatoli Polkovnikov
Introduction. Historical review
Quantum noise analysis of time-of-flightexperiments with atoms in optical lattices:HBT experiments and beyond
Quantum noise in interference experiments with independent condensates
Outline
Adiabaticity of creating many-body fermionic states in optical lattices
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 θ‘s and the state
Experimentally measured value B=2.697. Phys. Rev. Let. 49:92 (1982)
+
-
+
-1 2θ1 θ2
S
Hanburry-Brown-Twiss experimentsClassical theory of the second order coherence
Measurements of the angular diameter of Sirius
Proc. Roy. Soc. (London), A, 248, pp. 222-237
Hanbury Brown and Twiss,
Proc. Roy. Soc. (London),
A, 242, pp. 300-324
Quantum theory of HBT experiments
For bosons
For fermions
Glauber,Quantum Optics and Electronics (1965)
HBT experiments with matter
Experiments with 4He, 3He
Westbrook et al., Nature (2007)
Experiments with neutrons
Ianuzzi et al., Phys Rev Lett (2006)
Experiments with electrons
Kiesel et al., Nature (2002)
Experiments with ultracold atoms
Bloch et al., Nature (2005,2006)
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 time-of-flight
experiments with atoms in optical lattices:
Hanburry-Brown-Twiss experiments
and beyond
Theory: Altman, Demler, Lukin, PRA 70:13603 (2004)
Experiment: Folling et al., Nature 434:481 (2005); Spielman et al., PRL 98:80404 (2007);
Tom et al. Nature 444:733 (2006)
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
4
Bose Hubbard model
1+n
Uµ
02
0
M.P.A. Fisher et al.,
PRB40:546 (1989)
MottN=1
N=2
N=3
Superfluid
Superfluid phase
Mott insulator phase
Weak interactions
Strong interactions
Mott
Mott
Superfluid to insulator transition in an optical lattice
M. Greiner et al., Nature 415 (2002)
U
µ
1−n
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
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)
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 is more than HBT interference
Second order interference from the BCS superfluid
)'()()',( rrrr nnn −≡∆
n(r)
n(r’)
n(k)
k
0),( =Ψ−∆ BCSn rr
BCS
BEC
kF
Theory: Altman et al., PRA 70:13603 (2004)
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
Interference experiments
with cold atoms
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 ∆φ 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 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
Distribution function of interference fringe contrastExperiments: Hofferberth et al., arXiv0710.1575
Theory: 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
Interference of two dimensional condensates
Ly
Lx
Lx
Experiments: Hadzibabic et al. Nature (2006)
Probe beam parallel to the plane of the condensates
Gati et al., PRL (2006)
Interference of two dimensional condensates.Quasi long range order and the KT transition
Ly
Lx
Below KT transitionAbove KT transition
x
z
Time of
flight
low temperature higher temperature
Typical interference patterns
Experiments with 2D Bose gasHadzibabic, Dalibard et al., Nature 441:1118 (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 441:1118 (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 441:1118 (2006)
Experiments with 2D Bose gas. Proliferation of
thermal vortices Hadzibabic et al., Nature 441:1118 (2006)
The onset of proliferation
coincides with α shifting to 0.5!
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
Adiabaticity of creating many-body
fermionic states in optical lattices
Formation of molecules with increasing interaction strength
Strohmaier et al., arXiv:0707.314
Saturation in the number ofmolecules created is related
to the finite rate of changing
interaction strength U(t)
Formation of molecules with increasing interaction strength
U
As U is increased, the excess energy of two unpaired atoms should be converted to the kinetic energy of bound pairs.
The kinetic energy of a single molecule is set by .When U>>t many particles will have to be involved in the relaxation process.
During adiabatic evolution with increasing attractive U, all single atomsshould be converted to pairs. Entropy is put into the kinetic energy of bound pairs.
Hubbard model with repulsion:
dynamics of breaking up pairs
Energy of on-site repulsion Energy of spin domain walls
U
E
Hubbard model with repulsion:
dynamics of breaking up pairs
Energy of on-site repulsion UEnergy of spin domain wall
Stringent requirements on the rate of change of the interaction strength to maintain adiabaticity at the level crossing
Hubbard model with repulsion:
dynamics of breaking up pairs
Hubbard model with repulsion:
dynamics of breaking up pairs
Dynamics of recombination: a moving pair pulls out a spin domain wall
High order perturbation theory
Hubbard model with repulsion:
dynamics of breaking up pairs
N itself is a function of U/t :
U
E
Hubbard model with repulsion:
dynamics of breaking up pairs
Probability of nonadiabatic transition
ω12 – Rabi frequency at crossing point
τd – crossing time
Extra geometrical factor to account for different configurations of domain walls
Formation of molecules with increasing interaction strength
U
Value of U/t for which one finds saturation in the production of molecules
V0/ER=10, 7.5, 5.0, 2.5
Rey, Sensarma, Demler
Summary
Experiments with ultracold atoms provide a new
perspective on the physics of strongly correlated
many-body systems. Quantum noise is a powerful
tool for analyzing many body states of ultracold atoms