Transcript Ketterle lecture notes July 13th - Quantum Optics and Spectroscopy

“Ultracold gases –
from the experimenters’
perspective (II)”
Wolfgang Ketterle
Massachusetts Institute of Technology
MIT-Harvard Center for Ultracold Atoms
7/13/06
Innsbruck ICAP Summer School
Bose-Einstein condensation
• Ideal Bose gas
• Weakly interacting homogenous Bose gas
• Inhomogeneous Bose gas
• Superfluid hydrodynamics
Ideal BEC
The shadow of a cloud of bosons
as the temperature is decreased
(Ballistic expansion for a fixed time-of-flight)
Temperature is linearly related to the rf frequency
which controls the evaporation
BEC @ JILA, June ‘95
(Rubidium)
BEC @ MIT, Sept. ‘95 (Sodium)
1-(T/Tc)3
Homogeneous BEC
Propagation of sound
Excitation of sound
Excitation of sound
Excitation of sound
Sound = propagating density perturbations
0.5
mm
Laser beam
1.3 ms per frame
(M. Andrews, D.M. Kurn, H.-J. Miesner, D.S. Durfee,
C.G. Townsend, S. Inouye, W.K., PRL 79, 549 (1997))
Quantum depletion
or
How to observe the transition from a
quantum gas to a quantum liquid
In 1D: Zürich
K. Xu, Y. Liu, D.E. Miller, J.K. Chin, W. Setiawan, W.K., PRL 96, 180405 (2006).
What is the wavefunction of a condensate?
Ideal gas:
 q0

N
H   U 0 (r )
Interacting gas:
† †

H  U0  a p aq ar as
  q0

N
H   U0a0 a0  a a
  q  0
† †
p p

N 2
q  p q   p  ...
Quantum depletion
Quantum depletion in 3-dimensional free space
 1.5 n
M
4
2
U0
Gaseous BEC: 0.2 %
He II: 90 %
Optical lattice: Increase n and Meff
Quantum Depletion
Free space
Lattice
: tunneling rate
: on-site interaction
2-D Mask Gaussian Fit
2-D Mask Gaussian Fit
Observed quantum depletion > 50 %
Dispersion relation
Laser light
Condensate
Absorption
image
Laser light
Condensate
Absorption
image
Laser light
Condensate
Absorption
image
Laser light
Condensate
+ excitation
Laser light
Condensate
Measure
momentum q
and frequency n
+ excitation
dynamic structure factor
S(q,n)
analogous to
neutron scattering
from 4He
Laser light
Condensate
dynamic structure factor
S(q,n)
Laser light
Condensate
Optical stimulation
dynamic structure factor
S(q,n)
Large and small momentum transfer to atoms
large momentum
(two single-photon recoil)
small momentum
Spectrum of small-angle Bragg scattering
low density
“free particles”
S(q)=1
high density
“phonons”
S(q)=q/2mc<1
frequency shift
large q
large q
small q
Inhomogeneous BEC
A live condensate in the magnetic trap
(seen by dark-ground imaging)
BEC peak
Thermal wings,
 Temperature
BEC peak
300 m
Thermal wings,
 Temperature
rms width of harmonic oscillator ground state 7 m
 (repulsive) interactions
 interesting many-body physics
Signatures of BEC: Anisotropic expansion
1 ms
5 ms
10 ms
20 ms
30 ms
45 ms
Length and energy scales in BEC
Size of the atom
Separation between
atoms
Matter wavelength
n
dB
200 nm
1 m
Healing length
2x
2m
Size of confinement
a
-1/3
<< n
kBTs-wave >> kBTc
Gas!
a
-1/3
3 nm
aosc
30m
 dB < 2x < aosc
 kBT
BEC
> Uint > 
=(h2/m)na
Vortices
Spinning a Bose-Einstein condensate
The rotating bucket experiment with a superfluid gas
100,000 thinner than air
Rotating
green laser beams
Two-component vortex
Boulder, 1999
Single-component vortices
Paris, 1999
Boulder, 2000
MIT 2001
Oxford 2001
Rotating condensates
non-rotating
rotating (160 vortices)
J. Abo-Shaeer, C. Raman, J.M. Vogels, W.Ketterle, Science, 4/20/2001
Sodium BEC in the magnetic trap
Resonant Drive:
Green beam Power
(arb. scale)
-21
-18
-15
-12
-9
-6
-3
0 dB
Immediately after
stirring
After 500 ms of
free evolution
Hydrodynamics
Collective excitations
(observed in ballistic expansion)
16 ms
23 ms
Absorption 0%
28 ms
41 ms
48 ms
100%
MIT, 1996
Shape oscillations
“Non-destructive” observation of a
time-dependent wave function
350
m
5 milliseconds per frame
m=0 quadrupole-type oscillation at 29 Hz
Low T
High T
Stamper-Kurn, Miesner, Inouye, Andrews, W.K, PRL 81, 500 (1998)
Tc
collisionless oscillation
osc/z= 1.569(4)
hydrodynamic oscillation
1.580 (prediction by Stringari)
thermal cloud
Temperature
dependence of
frequency condensate
“Beyond-mean field theory”
Onset of hydrodynamic
behavior
(Giorgini)
Landau damping
(Popov, Szefalusky, Condor,
Liu, Stringari, Pitaevskii,
Fedichev, Shlyapnikov, Burnett,
Edwards, Clark, Stoof, Olshanii)
Excitation of surface modes m=l
Radial cross section
of condensate
Focused IR beam
• Rapid switching between points (10 … 100 kHz)
• Slow variation of intensity or position
• Excitation of standing and travelling waves
Theory on surface modes: Stringari et al., Pethick et al.
Observation of m=2, l=2 collective excitation
Time of flight (20 msec), standing wave excitation
In-situ phase-contrast imaging (2 msec per frame)
rotating excitation
R. Onofrio, D.S. Durfee, C. Raman, M. Köhl, C.E. Kuklewicz, W.K.,
Phys. Rev. Lett. 84, 810 (2000)
Hexadecapole
Hexadecapole oscillation (
= 4)
“Ultracold gases –
from the experimenters’
perspective (III)”
Wolfgang Ketterle
Massachusetts Institute of Technology
MIT-Harvard Center for Ultracold Atoms
7/14/06
Innsbruck ICAP Summer School
The new frontier:
Strong interactions and
correlations
Strongly correlated bosons in optical lattices
The Superfluid to Mott Insulator Transition
BEC in 3D optical lattice
Courtesy Markus Greiner
The Superfluid-Mott Insulator transition
Shallow Lattices - Superfluid
Deep Lattices – Mott Insulator
Energy offset due to external
harmonic confinement. Not in
condensed matter systems.
tunneling term between
neighboring sites
on-site interaction
 4 2 a 
4
 w( x) d 3 x
U  
 m 
a = s-wave scattering length
Other exp: Mainz, Zurich, NIST Gaithersburg, Innsbruck, MPQ and others
The Superfluid-Mott Insulator transition
Shallow Lattices - Superfluid
5 Erec
9 Erec
The Superfluid-Mott Insulator transition
Deep Lattices – Mott Insulator
5 Erec
9 Erec
12 Erec
15 Erec
20 Erec
Diagnostics:
Loss of Coherence
Excitation Spectrum
Noise Correlations
Microwave Spectroscopy
As the lattice depth is increased, J decreases
exponentially, and U increases. For J/U<<1,
number fluctuations are suppressed, and the
atoms are localized
The Superfluid-Mott Insulator Transition in
Optical Lattices
MI phase
transition
Cold fermions
Lithium
Sodium
At absolute zero temperature …
Bosons
Particles with an even number of
protons, neutrons and electrons
Bose-Einstein condensation
 atoms as waves
 superfluidity
Fermions
Particles with an odd number of
protons, neutrons and electrons
Fermi sea:
 Atoms are not coherent
 No superfluidity
Pairs of fermions
Particles with an even number of
protons, neutrons and electrons
Two kinds of fermions
Fermi sea:
 Atoms are not coherent
 No superfluidity
At absolute zero temperature …
Pairs of fermions
Particles with an even number of
protons, neutrons and electrons
Bose-Einstein condensation
 atoms as waves
 superfluidity
Two kinds of fermions
Particles with an odd number of
protons, neutrons and electrons
Fermi sea:
 Atoms are not coherent
 No superfluidity
Weak attractive interactions
Cooper pairs
larger than interatomic distance
momentum correlations
 BCS superfluidity
Two kinds of fermions
Particles with an odd number of
protons, neutrons and electrons
Fermi sea:
 Atoms are not coherent
 No superfluidity
Atom pairs
Bose Einstein condensate
of molecules
Electron pairs
BCS Superconductor
BEC
BCS supe
BEC
Magnetic field
BCS supe
BEC
Crossover superfluid
BCS supe
Observation of Pair Condensates!
At 900 G (above dissociation limit of molecules)
Initial
T / TF = 0.2
temperature:
T / TF = 0.1
T / TF = 0.05
First observation: C. A. Regal et al., Phys. Rev. Lett. 92, 040403 (2004)
M.W. Zwierlein, C.A. Stan, C.H. Schunck, S.M.F. Raupach, A.J. Kerman, W.K.
Phys. Rev. Lett. 92, 120403 (2004).
„Phase diagram“ for pair condensation
kF|a| > 1