quantum coherence and interactions in many body...
TRANSCRIPT
Quantum coherence and
interactions in many body systems
Collaborators:Ehud Altman, Anton Burkov, Derrick Chang,
Adilet Imambekov, Vladimir Gritsev , Mikhail Lukin,
Giovanna Morigi, Anatoli Polkonikov
Eugene Demler Harvard University
Funded by NSF, AFOSR, Harvard-MIT CUA
Condensed matter
physics
Atomic
physics
Quantum optics
Quantum
coherence
Quantuminformation
Quantum Optics with atoms and
Condensed Matter Physics with photons
Interference of fluctuating condensatesFrom reduced contrast of fringes to correlation functionsDistribution function of fringe contrastNon-equilibrium dynamics probed in interference experiments
Luttinger liquid of photonsCan we get “fermionization” of photons?Non-equilibrium coherent dynamics of strongly interacting photons
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
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
Nature 4877:255 (1963)
Interference of one dimensional condensatesExperiments: Schmiedmayer et al., Nature Physics (2005,2006)
Transverse imaging
long. imaging
trans.imaging
Longitudialimaging
Figures courtesy of
J. Schmiedmayer
x1
d
Amplitude of interference fringes,
Interference of one dimensional 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)
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)
Fundamental noise in
interference experiments
Amplitude of interference fringes is a quantum operator. The measured value of the amplitude will fluctuate from shot to shot. We want to characterize not only the averagebut the fluctuations as well.
Shot noise in interference experiments
Interference with a finite number of atoms. How well can one measure the amplitude of interference fringes in a single shot?
One atom: NoVery many atoms: ExactlyFinite number of atoms: ?
Consider higher moments of the interference fringe amplitude
, , and so on
Obtain the entire distribution function of
Shot noise in interference experiments
Interference of two condensates with 100 atoms in each cloud
Coherent states
Number states
Polkovnikov, Europhys. Lett. 78:10006 (1997)
Imambekov, Gritsev, Demler, 2006 Varenna lecture notes
Distribution function of fringe amplitudes
for interference of fluctuating condensates
Lis 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 (2006)
Imambekov, Gritsev, Demler, cond-mat/0612011
We need the full distribution function of
0 1 2 3 4
Pro
ba
bili
ty 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
Interference of 1d condensates at finite temperature.
Distribution function of the fringe contrast
Luttinger parameter K=5
Experiments: Schmiedmayer et al.
Interference of 2d condensates at finite temperature.
Distribution function of the fringe contrast
T=TKT
T=2/3 TKT
T=2/5 TKT
From visibility of interference fringes
to other problems in physics
Quantum impurity problem: interacting one dimensional
electrons scattered on an impurity
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, …
Interference between interacting 1d Bose liquids.
Distribution function of the interference amplitude
is a quantum operator. The measured value of will fluctuate from shot to shot.
How to predict the distribution function of
Yang-Lee singularity
2D quantum gravity,
non-intersecting loops
Fringe visibility and statistics of random surfaces
)(ϕh
Proof of the Gumbel distribution of interfernece fringe amplitude for 1d weakly interacting bosons relied on the known relation between 1/f Noise and Extreme Value StatisticsT.Antal et al. Phys.Rev.Lett. 87, 240601(2001)
Fringe visibility
Roughness ϕϕ dh2
)(∫=
Non-equilibrium coherent
dynamics of low dimensional Bose
gases probed in interference
experiments
Studying dynamics using interference experiments.
Thermal decoherence
Prepare a system by
splitting one condensate
Take to the regime of
zero tunnelingMeasure time evolution
of fringe amplitudes
Relative phase dynamics
Quantum regime
1D systems
2D systems
Classical regime
1D systems
2D systems
Burkov, Lukin, Demler, cond-mat/0701058
Experiments:
Schmiedmayer et al.
Different from the earlier theoretical work based on a single
mode approximation, e.g. Gardiner and Zoller, Leggett
Quantum dynamics of coupled condensates. Studying
Sine-Gordon model in interference experiments
J
Prepare a system by
splitting one condensate
Take to the regime of finite
tunneling. System
described by the quantum
Sine-Gordon modelMeasure time evolution
of fringe amplitudes
Dynamics of quantum sine-Gordon model
Power spectrum
Gritsev, Demler, Lukin, Polkovnikov, cond-mat/0702343
A combination of
broad features
and sharp peaks.
Sharp peaks due
to collective many-body
excitations: breathers
Condensed matter physics with photons
Luttinger liquid of photons
Tonks gas of photons:photon “fermionization”
Chang, Demler, Gritsev, Lukin, Morigi, unpublished
Self-interaction effects for one-dimensional optical waves
kk0
ω
ω0
Nonlinear polarization for isotropic medium
Envelope function
Self-interaction effects for one-dimensional optical waves
Frame of reference moving with the group velocity
Gross-Pitaevskii type equation for light propagation
Competition of dispersion and non-linearity
Nonlinear Optics, Mills
Self-interaction effects for one-dimensional optical waves
BEFORE: two level systems and
insufficient mode confinement
Interaction corresponds to attraction.
Physics of solitons (e.g. Drummond)
Weak non-linearity due to insufficient
mode confining
Limit on non-linearity due to
photon decay
NOW: EIT and tight
mode confinement
Sign of the interaction can be tuned
Tight confinement of the
electromagnetic mode
enhances nonlinearity
Strong non-linearity without losses
can be achieved using EIT
Controlling self-interaction effects for photons
Ωcω
ω
∆ ω
Imamoglu et al.,
PRL 79:1467 (1997)
describes photons. We need to normalize to polaritons
Tonks gas of photons
Photon fermionization
Crystal of photons
Is it realistic?
Experimental signatures
Tonks gas of atoms
Small γ – weakly interacting Bose gas
Large γ – Tonks gas. Fermionized bosons
Additional effects for for photons:
Photons are moving with the group velocity
Limit on the cross section of photon interacting with one atom
Tonks gas of photons
Limit on strongly interacting 1d photon liquid due to finite group velocity
Concrete example: atoms in a hollow fiber
Experiments:
Cornell et al. PRL 75:3253 (1995);
Lukin, Vuletic, Zibrov et.al.
Theory: photonic crystal and non-linear medium
Deutsch et al., PRA 52:1394 (2005);
Pritchard et al., cond-mat/0607277
Atoms in a hollow fiber
k
ω
Without using
the “slow light” points
λ=1µm
A=10µm2
Ltot=1cm
A – cross section of e-m mode
Typical numbers
Experimental detection of the Luttinger liquid of photons
Control beam off.
Coherent pulse of
non-interacting photons
enters the fiber.
Control beam switched on adiabatically.
Converts the pulse into a Luttinger liquid
of photons.
“Fermionization” of photons detected by observing oscillations in g2
c
K – Luttinger parameter
Non-equilibrium dynamics of strongly
correlated many-body systems
g2 for expanding Tonks gas with
adiabatic switching of interactions
100 photons after expansion
Outlook
Atomic physics and quantum optics traditionally study non-equilibrium
coherent quantum dynamics of relatively simple systems.
Condensed matter physics analyzes complicated electron system
but focuses on the ground state and small excitations around it.
We will need the expertise of both fields
Next challenge in studying quantum coherence:understand non-equilibrium coherent quantum dynamics
of strongly correlated many-body systems
“…The primary objective of the JQI is to develop a world class research
institute that will explore coherent quantum phenomena and thereby
lay the foundation for engineering and controlling complex quantum systems…”
From the JQI web page http://jqi.umd.edu/