BEC: many weakly interacting particles Gross
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Transcript BEC: many weakly interacting particles Gross
A review of atomic-gas Bose-Einstein
condensation experiments for
Workshop: “Hawking Radiation in
condensed-matter systems”
Eric Cornell
JILA
Boulder, Colorado
The basic loop.
Cooling. Minimum temperature. Stray heating.
Confinement. Magnetic. Optical. Reduced dimensions. Arrays
Observables. Images. Shot noise. Atom counting.
Interactions. The G-P equation. Speed of sound.
Time-varying interactions.
feshbach resonance. Reduced dimensions.
Thermal fluctuations.
A range of numbers.
The basic loop.(once every minute)
Cooling. Minimum temperature. Stray heating.
Confinement. Magnetic. Optical. Reduced dimensions. Arrays
Observables. Images. Shot noise. Atom counting.
Interactions. The G-P equation. Speed of sound.
Time-varying interactions.
feshbach resonance. Reduced dimensions.
Thermal fluctuations.
A range of numbers.
Laser cooling
2.5 cm
The basic loop.
Cooling. Minimum temperature.
Stray heating.
Confinement. Magnetic. Optical. Reduced dimensions. Arrays
Observables. Images. Shot noise. Atom counting.
Interactions. The G-P equation. Speed of sound.
Time-varying interactions.
feshbach resonance. Reduced dimensions.
Thermal fluctuations.
A range of numbers.
V(x)
m
x
Self-interacting condensate
expands to fill confining potential
to height m
n(x)
V(x)
m
x
Self-interacting condensate
expands to fill confining potential
to height m
n(x)
V(x)
kT
x
Cloud of thermal excitations
made up of atoms on trajectories
that go roughly to where the
confining potential reaches kT
n(x)
V(x)
m
kT
x
When kT < m then there are very few
thermal excitations extending outside of
condensate. Thus evaporation cooling power is
small.
BEC experiments must be completed
within limited time.
Bang!
Three-body molecular-formation process
causes condensate to decay, and heat!
Lifetime longer at lower density, but physics
goes more slowly at lower density!
Dominant source of heat:
Bang!
Decay products from three-body recombination
can collide “as they depart”, leaving behind
excess energy in still-trapped atoms.
Bose-Einstein Condensation in a Dilute Gas: Measurement of Energy and Ground-State
Occupation, J. R. Ensher, D. S. Jin, M. R. Matthews, C. E. Wieman, and E. A. Cornell,
Phys. Rev. Lett. 77, 4984 (1996)
The basic loop.
Cooling. Minimum temperature. Stray heating.
Confinement. Magnetic. Optical.
Reduced dimensions. Arrays
Observables. Images. Shot noise. Atom counting.
Interactions. The G-P equation. Speed of sound.
Time-varying interactions.
feshbach resonance. Reduced dimensions.
Thermal fluctuations.
A range of numbers.
m=2
Energy
F=2
m=-2
m=-1
m=0
F=1
m=1
Typical single-atom energy level diagram
in the presence of a magnetic field.
Note Zeeman splitting.
U(B)
m=-1
m=0
B
m=+1
B(x)
x
U(B)
m=-1
m=0
B
m=+1
B(x)
x
U(x)
m=-1
m=0
m=+1
x
Gravity (real gravity!) often is important force in experiments
U(z)
Why
do atoms
rest at equilibrium
here?
m=-1
m=0
m=+1
z
Gravity (real gravity!) often is important force in experiments
U(z)
Why
do atoms
rest at equilibrium
here?
UB(z)+mgz
m=-1
m=0
z
m=+1
m=-1
m=0
m=+1
z
Shape of magnetic potential.
U= a x2 + b y2 + g z2
Why?
R
L
D
electromagnet coils
typically much larger,
farther apart,
than size of atom
cloud
D, L >> R
so, order x3 terms are small.
Magnetic confining potential typically quadratic only,
except…
Cross-section of tiny
wire patterned on chip
Atom chip substrate
If electromagnets are based on “Atom chip”
design, one can have sharper, more structured
magnetic potentials.
If electromagnets are based on “Atom chip”
design, one can have sharper, more structured
magnetic potentials.
But there is a way to escape entirely
from the boring rules of magnetostatics….
Interaction of
light and atoms.
Laser beam
nlaser
Laser too red
+atom diffracts light
+light provides
conservative,
attractive potential
Laser quasi-resonant
+atom absorbs light
+light can provide
dissipative (heating,
cooling) forces
Laser too blue
+atom diffracts light
+light provides
conservative,
repulsive potential
One pair of beams: standing wave in intensity
Can provide tight confinement in 1-D with almost
free motion in 2-d. (pancakes)
This is a one-D array of quasi-2d condensates
Two pairs of beams:
Can provide tight confinement in 2-D, almost-free motion
in 1-d (tubes)
Can have a two-D array of quasi-1d condensates
Why K-T on a lattice?
1.Ease of quantitative comparison.
2. Makes it easier to get a quasi-2d system
3.This is the Frontier in Lattices Session!!!
(come on!)
Our aspect ratio, (2.8:1), is modest, but...
addition of 2-d lattice makes phase fluctuations much “cheaper” in 2 of
3 dimensions.
z, the
“thin
dimension”
x, y, the two “large dimemsions”.
Three pairs of beams:
Can provide tight confinement in 3-D, (dots)
Can have a three-D array of quasi-0d condensates
Single laser beam brought to a focus.
Rayleigh
range
Can provide a single potential (not an array of potentials)
with extreme, quasi-1D aspect ratio.
Additional, weaker beams can change free-particle dispersion
relation.
The basic loop.
Cooling. Minimum temperature. Stray heating.
Confinement. Magnetic. Optical. Reduced dimensions. Arrays
Observables. Images. Shot noise.
Atom counting.
Interactions. The G-P equation. Speed of sound.
Time-varying interactions.
feshbach resonance. Reduced dimensions.
Thermal fluctuations.
A range of numbers.
Observables: It’s all about images.
Temperature, pressure, viscosity,
all these quantities are inferred from
images of the atomic density.
Off-resonant imaging:
index of refraction mapping.
Signal-noise typically less good,
but (mostly) nondestructive..
Near-resonant imaging:
absorption. signal-noise
typically good, but sample
is destroyed.
Watching a shock coming into existence
110 ms
using a bigger beam
Absorption imaging: one usually
interprets images as of a continuous
density distribution, but
density distribution actually os made up
discrete quanta of mass known,
(in my subdiscipline of physics) as
“atoms”.. Hard to see an individual atom,
but can see effects of the discrete nature.
Absorption depth observed
in a given box of area A is
OD= (N0 +/- N01/2) s /A
The N01/2 term is the”atom
shot noise”. and it can dominate
technical noise in the image.
Absorption depth observed
in a given box of area A is
OD= (N0 +/- N01/2) s /A
The N01/2 term is the”atom
shot noise”. and it can dominate
technical noise in the image.
Atom shot noise will likely be an important background
effect which can obscure Hawking radiation unless
experiment is designed well.
Atom shot noise limited imaging
Data from lab of Debbie Jin.
N.B: imaging atoms with light is not the only way to
detect them:
I think we will hear from Chris Westbrook about
detecting individual metastable atoms.
The basic loop.
Cooling. Minimum temperature. Stray heating.
Confinement. Magnetic. Optical. Reduced dimensions. Arrays
Observables. Images. Shot noise. Atom counting.
Interactions. The G-P equation.
Speed of sound.
Time-varying interactions.
feshbach resonance. Reduced dimensions.
Thermal fluctuations.
A range of numbers.
Why are BECs so interesting?
QM: Particle described by Schrödinger equation
2
Vextern r Er
2m
BEC: many weakly interacting particles
Gross-Pitaevskii equation
2
4 2a 2
Vextern
r r mr
m
2m
The condensate is
self-interacting (usually self-repulsive)
BEC: many weakly interacting particles
Gross-Pitaevskii equation
4 a 2
r
Vextern
r r i
m
t
2m
2
2
Can be solved in various approximations.
The Thomas-Fermi approximation:
ignore KE term, look for stationary states
BEC: many weakly interacting particles
Gross-Pitaevskii equation
m
4 a 2
r
Vextern
r r i
m
t
2m
2
2
Can be solved in various approximations.
The Thomas-Fermi approximation:
ignore KE term, look for stationary states
4 2 a 2
r m Vextern
m
The Thomas-Fermi approximation:
ignore KE term, look for stationary states
4 2 a 2
r m Vextern
m
n(x)
V(x)
m
x
Self-interacting condensate
expands to fill confining potential
to height m
BEC: many weakly interacting particles
Gross-Pitaevskii equation
4 a 2
r
Vextern
r r i
m
t
2m
2
2
Can be solved in various approximations.
Ignore external potential, look for plane-wave
excitations
BEC: many weakly interacting particles
Gross-Pitaevskii equation
4 a 2
r
Vextern
r r i
m
t
2m
2
2
Can be solved in various approximations.
Ignore external potential, look for plane-wave
excitations
speed of sound:
c = (m/m)1/2
Healing length:
x = (hbar2/m)1/2
Data from Nir Davidson
Chemical potential:
m = 4 hbar2 a n /m
n(x)
f(x)
Long wavelength
excitations
(k << 1/x)
relatively little
density fluctuation,
large phase fluctuation
(which we can’t directly
image).
n(x)
f(x)
But, if we turn off
interactions suddenly
(m goes to zero), and wait a
little bit:
n(x)
f(x)
n(x)
f(x)
But, if we turn off
interactions suddenly
(m goes to zero), and wait a
little bit:
m goes to zero, x gets large
now (k > 1/x)
the same excitation now
evolves much larger density
fluctuations.
Can you DO that?
The basic loop.
Cooling. Minimum temperature. Stray heating.
Confinement. Magnetic. Optical. Reduced dimensions. Arrays
Observables. Images. Shot noise. Atom counting.
Interactions. The G-P equation. Speed of sound.
Time-varying interactions.
feshbach resonance.
Reduced dimensions.
Thermal fluctuations.
A range of numbers.
Magnetic-field Feshbach resonance
repulsive
free atoms
→←
attractive
molecules
> B
Single laser beam brought to a focus.
Suddenly turn off laser beam….
Data example from e.g. Ertmer.
The basic loop.
Cooling. Minimum temperature. Stray heating.
Confinement. Magnetic. Optical. Reduced dimensions. Arrays
Observables. Images. Shot noise. Atom counting.
Interactions. The G-P equation. Speed of sound.
Time-varying interactions.
feshbach resonance. Reduced dimensions.
Thermal fluctuations.(a serious problem)
A range of numbers.
The basic loop.
Cooling. Minimum temperature. Stray heating.
Confinement. Magnetic. Optical. Reduced dimensions. Arrays
Observables. Images. Shot noise. Atom counting.
Interactions. The G-P equation. Speed of sound.
Time-varying interactions.
feshbach resonance. Reduced dimensions.
Thermal fluctuations.
A range of numbers.(coming later)