1D Bose Gases

1
1D Bose Gases Single Atom Imaging
Xiao Li Karl Nelson Jean-Felix Riou Neal
Meyer Kunyan Zhu Aaron Reinhard Phillip
Schafer David Weiss
Vladimir Yurovsky Tel Aviv U.
Penn State
recent group members Trevor Wenger Toshiya
Kinoshita (Kyoto U.) Fang Fang (LANL)
Supported by NSF, ARO, NIST and DARPA
2
Optical Lattices
Calculable, versatile atom traps
UAC ? Intensity
Far from resonance, no light scattering
1D Bose gases
electron electric dipole moment search
1D
2D
3D
quantum computing
3
3D lattice Experimental Goal
Use individual neutral atoms as qubits in a
quantum computer.
Approach Make a large array of single neutral
atoms. Image them individually. Initialize and
measure qubit states. Cool atoms to their trap
ground states. Demonstrate site-resolved quantum
gates. Perform a series of gates. Incorporate
error correction.
4
3D Optical Lattice with Large Spacing
? 10
Blue-detuned atoms trapped at intensity minima
5
Loading the Lattice
Turn on the lattice around MOT atoms. Very good
optical access remains.
6
Cooling Single Atoms in a 3D Optical Lattice
Start with an average of 6 atoms per site in the
lattice
Before cooling
After laser cooling
A random half of the lattice sites are occupied
by a single atom
DePue, McCormick, Winoto, Oliver, DSW, PRL 82
2262 (1999)
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3D Array of Single Atoms
Image with the cooling light
8
3 Planes Imaged
250 atoms in central region
Highly reliable occupancy identification
Negligible site hopping during detection
Nelson, Li DSW, Nature Physics 3, 556 (2007).
9
3D Raman Sideband Cooling
??-1
??1
??0
We are pursuing cooling variants ?
Goal a well-initialized many atom array.
10
Lattice compacting
Focused beam microwaves gives site-selective
state change
Rotating polarizations gives state-selective
translations
In 3D, compact N atoms in lt4N1/3 steps.
0.5 s for 4000 atoms
Can check for and fix errors
DSW,Vala,Thapliyal,Myrgren,Vazirani, Whaley, PRA
70, 040302 (2004)
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Addressing and Entangling Atoms
single qubit gate
two (or more) qubit gate using Rydberg transitions
(Much progress at Institut dOptique and
Wisconsin)
Each atom has 26 pretty near neighbors 10s of
seconds decoherence times, 200 ns 30 µs gates.
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1D Bose Gases
Theory of 1D Bose gas with d-interactions Experi
ments with equilibrium d-interacting 1D Bose
gases Experiments with non-equilibrium
d-interacting 1D Bose gases Experiments with
not quite d-interacting, not quite 1D Bose
gases
Motivations
1D Bose gases integrable many-body systems
exact theoretical results weak or strong
coupling control over integrability
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1D Bose gases with infinite hard core interactions
Lewi Tonks, 1936 Eq. of state of a 1D classical
gas of hard spheres
Marvin Girardeau, 1960 1D Bose gases with
infinite hard core repulsion
It costs 8 energy for two particles to be in the
same place.
?
fermionization
Each particles wavefunction varies smoothly, but
no two overlap a highly correlated many body
wavefunction.
single particles
two particle correlations
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1D Bose gases with variable point-like
interactions
Elliot Lieb and Werner Liniger, 1963 Exact
solutions for 1D Bose gases with arbitrary ?(z)
interactions
Solutions parameterized by
?gtgt1 Tonks-Girardeau gas
large g1D low density
kinetic energy dominates
?ltlt1 mean field theory
(Thomas-Fermi gas)
mean field energy dominates
small g1D high density
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1D Bose atomic gases
Maxim Olshanii, 1998 Adaptation to real atoms
a3D 3D scattering length a? transverse
oscillator length
The 3D scattering problems maps onto the 1D
problem when (p/?)a?ltlt1.
?? when a 3D?, n1D? or a??
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Bundles of 1D Systems
For 1Dall energies ltlt h?? negligible tunneling
Blue-detuned lattice minimizes spontaneous
emission and barely effects axial trapping
Recall ?? when a?? or n1D?
So ?? when the lattice power ? or the dipole trap
power ?
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Expansion in the 1D tubes
50-250 atoms/tube
1000-5000 tubes
? up
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1D energy parameterized by ?
Exact theory
Kinoshita, Wenger, DSW, Science 305, 1125 (2004)
Expt
Lieb Liniger, PR 130, 1605 (1963) Dunjko,
Lorent, Olshanii, PRL 86, 5413 (2001)
2
no free parameters
1.5
Tonks-Girardeau gas
1
0.5
1D quasi-BEC
0
0.4
0.7
1
4
weak coupling
strong coupling
g
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Normalized Local Pair Correlations
By photo-association
Theory Gangardt Shlyapnikov, PRL 90 010401
(2003)
Expt Kinoshita, Wenger, DSW, PRL 95 190406 (2005)
g(2) of the 3D BEC is 1.
0.8
Other 1D expts Aspect Bloch Esslinger Hulet Ingus
cio Ketterle Phillips/Porto Rempe/Dürr
Schmiedmayer Sengstock Van Druten
0.7
0.6
Pauli exclusion for Bosons
0.5
0.4
0.3
0.2
0.1
0
.3
10
1
3
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Collisions in 1D
For identical particles, reflection looks just
like transmission !
?
Two-body collisions between distinct bosons
cannot change their momentum distribution.
But, the momentum distribution of a freely
expanding 1D Bose gas does change, for all ?.
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Does a Real 1D Gas Thermalize?
1D Bose gases with d-fn interactions are
integrable systems ? they do not ergodically
sample phase space become chaotic
thermalize
Thermalization in a real 1D Bose gas has been a
somewhat open question.
Do imperfectly d-fn interactions lift
integrability enough to allow the atoms to
thermalize? Do longitudinal potentials matter?
Procedure take the 1D gas out of equilibrium and
see how it evolves.
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Creating Non-Equlibrium Distributions
Optical thickness
1 standing wave pulse
Position (µm)
2 standing wave pulses
Optical thickness
Wang, et al., PRL 94, 090405 (2005)
Position (µm)
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Harmonic Trap Motion
x
A classical Newtons cradle
v
We make thousands of parallel quantum Newtons
cradles, each with 50-300 oscillating atoms.
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1D Evolution in a Harmonic Trap
Kinoshita, Wenger, DSW Nature 440, 900 (2006)
Position (µm)
ms
-500
0
500
40 µm
0
1st cycle average
15?
195 ms
5
30?
390 ms
10
Each 1D gas is 1 mm ? 40 nm ! 25,0001
aspect ratio
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Dephased Momentum Distributions
1st cycle average 15 ? distribution 40 ?
distribution
evolution without grating pulses
?18 ?3.2 ? 1.4
Optical thickness (normalized)
Position (µm)
Position (µm)
Project the evolution
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Negligible Thermalization
Projected curves and actual curves at 30 ? or
40 ?
?18 ?3.2 ? 1.4
After dephasing, the 1D gases reach a steady
state that is not thermal equilibrium
Each atom continues to oscillate with its
original amplitude
Optical thickness (normalized)
Lower limit thousands of collisions without
thermalization
(near) integrability ? no thermalization
Position (µm)
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What happens in 3D?
Thermalization is known to occur in 3 collisions.
0 ?
2 ?
4 ?
9 ?
These collisions occur above the Landau critical
velocity for the 3D BEC.
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Is there a non-integrability threshold for
thermalization?
The classical KAM theorem shows that if a
non-integrable system is sufficiently close to
integrable, it will not ergodically sample phase
space.
Is there a quantum mechanical analog?
Procedure controllably lift
integrability and measure thermalization.
Ways to lift integrability Allow tunneling among
tubes (1D ? 2D and 3D behavior) Finite range 1D
interactions Add axial potentials
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Making 1D gases thermalize
Lower the 2D lattice depth
e. g. Ux UY 21 Erec
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Thermalization in a 2D array of tubes
Lattice Depth (Erec)
Fraction of energy in 1D
0.02
1
Thermalization Rate (per collision)
0.8
0.015
0.6
0.01
no tails
0.4
0.005
0.2
2-body collisions are well below threshold for
transverse excitation.
0
equipartition
0
2
4
6
8
10
12
Lattice Depth (uK)
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Is there a threshold?
0.005
instrumental resolution
0
Tunneling Amplitude (nK)

The experiment says maybe.
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Thermalization scenario (Yurovsky)
A combined effect of tunneling and 3-body
collisions.
when 3-body collisions are spread out over time
and multiple tubes.
2nd order perturbation theory to generate kinetic
equations
J, the tunneling energy
Ua(J) is the atom-atom interaction energy
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Preliminary theory-expt comparison
More experimental checks and tests will be done.
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Summary

• We have imaged and cooled single atoms in a 3D
  array. Its a step toward a neutral atom quantum
  computer.
• Experiments with equilibrium 1D Bose gases
  across coupling regimes total energy, cloud
  lengths, momentum distributions, local pair
  correlations agree with the exact 1D Bose gas
  theory.
• Non-equilibrium 1D Bose gases quantum Newtons
  cradle. Independent d-int. 1D Bose gases do not
  thermalize!
• Relaxed conditions allow 1D Bose gases do
  thermalize. We have a theory to test.
• We can also lift integrability in other ways. Is
  there universal behavior?