Download Ultra-cold atoms - University of St Andrews

Ultra-cold atoms
•New direction in atomic physics,
started about 20 years ago
•A gas can be cooled from room temperature
down to nK !
•Laser cooling, evaporative cooling
NIST
Why are cold atoms interesting?
Wave-particle duality:
v
m
λdeBroglie = h/mv
Low T ( = low speed)
wave-like behaviour becomes important
Atom optics = optics for matter waves
The motion of the atoms can be controlled using devices analogous
to those in optics, e.g. mirrors, lenses, diffraction gratings,
interferometers…..
Applications
Atom interferometry
acceleration sensors
Quantum computing
Bose-Einstein condensation (BEC), first observed in 1995
Our plan
How to achieve BEC in an atomic gas:
• Laser cooling
• Evaporative cooling
Nobel prizes were awarded for laser cooling and BEC
nλ3dB ≈ 1
N number
n=
V density
Based on this intuitive picture we can derive an expression for the
critical temperature Tc :
Equipartition theorem
Substituting p =
h
λdB
1 h2
3
= k BT
2
2m λdB
2
Substituting this in
N (3mk BTC ) 3 / 2
≈
V
h3
p2 3
= k BT
2m 2
(m=atomic mass)
we obtain:
λdB
h
=
3mk BT
“thermal de-Broglie
wavelength”
1
N
≈ 3 we obtain:
V λdB
2
N
1 h
Tc ≈
3 mk B V
23
A more rigorous calculation based on Bose-Einstein statistics leads to
a result that only differs by a numerical factor:
2
h
N
Tc = 0.0839
mk B V
23
Below TC, we have a macroscopic occupation of the ground state
given by:
N BEC (T )
T
N BEC (T ) = N 1 −
Tc
(see for instance Mandl,
Statistical Physics)
3/ 2
N
0
TC
T
This macroscopic occupation of the ground state of a bosonic system
is known as Bose-Einstein condensation.
E.g.for an ultracold gas of rubidium atoms:
N
≈ 10 20 m −3
V
Tc = 390nK
The first step towards achieving such a low temperature
is laser cooling.
Laser cooling (1) - Radiation pressure
Transfer of momentum from a resonant laser beam to an atom
A photon carries a momentum p photon = h / λ = k
(k = 2π / λ )
Due to the law of conservation of momentum, when the photon
is absorbed the atom receives a recoil momentum mvR = k
(e.g. 23 Na : λ = 589nm vR = 3cm/s)
To determine the force exerted by the laser beam, we need to know
the rate at which the atom absorbs photons….
An estimate of this rate is the time the atom takes to decay back
to the ground state, i.e. the lifetime τ of the excited state due to
spontaneous emission:
hυatom
absorption of a photon
atom acquires a recoil
R in laser propagation direction
de-excitation: spontaneous
emission is isotropic and
imparted momentum averages
to 0 over many events
result: net momentum transfer in
laser propagation direction
Can absorb many photons
R
big change in atom velocity
Radiation pressure - force on atom:
∆p = mv R =
h
λ
change of atomic momentum
for an absorption/spontaneous
emission event
∆p ∆p
h
F=
=
≈
∆t
τ
λτ
lifetime τ of excited state
due to spontaneous emission
Laser cooling (2)
How can radiation pressure be used for cooling?
h υ atom
v
hυ L
detuning
δ = υ L − υ atom
δ <0
An atom will be slowed down if it absorbs photons from a laser beam
directed opposite to its velocity. For this to happen, the laser frequency
υ L has to be below the atomic transition, i.e. the laser has to be
“red detuned”. The condition for absorption of photons is that their
Doppler shifted frequency in the atom’s frame is close to υ atom :
υ
observed
L
≈ υ atom where
υ
observed
L
v
v
= υ L 1 + = (υ atom + δ ) 1 +
c
c
Doppler effect
υ atom
v
≈ (υ atom + δ ) 1 +
c
We can solve this for δ:
δ≈
υ atom
v
1+
c
− υ atom
v
− υ atom
v
c
≈ − υ atom
=
v
c
1+
c can neglect if v << 1
c
δ
υ atom
v
=−
c
Let’s now add a second counterpropagating laser beam:
atomic velocity v
LASER
υlaser < υ atom
LASER
υlaser < υ atom
force F
If an atom is moving to the right (as in the diagram) for the Doppler
effect it will be closer to resonance with the laser on the right and
further from resonance with laser on the left. If an atom is moving to
the left, the opposite is true. Either way, the atom receives “kicks”
opposite to its velocity, or in other words it experiences a damping force:
F = − kv
If v=0, F=0 because the absorption is the same from both beams.
This scheme is known as Doppler cooling, as it relies on the Doppler
effect. It can be extended to the 3D case simply by adding pairs of
counterpropagating laser beams along orthogonal directions (“optical
molasses”):
The minimum temperature that can be achieved with Doppler cooling
is determined by the random recoils imparted to the atoms during
spontaneous emission. Although their average is 0 over many events,
their standard deviation is not and this leads to diffusion in momentum
space, analogous to Brownian motion. It is possible to show that:
Tmin ≈
k Bτ
E.g. for the sodium doublet τ=16ns, Tmin=400µK
Other laser cooling mechanisms exist that can decrease the
temperature even further, however they don’t achieve the critical
temperature for BEC
need evaporative cooling
How to get BEC:
• Trap a gas of atoms using laser cooling techniques
• Turn off the cooling laser
• Further cool the cloud with evaporative cooling until BEC is reached
Magnetic trapping
To implement evaporative cooling, we need to hold the cloud in
a trap. Very often this is a magnetic trap. Such a trap relies on the
(well known by now!) magnetic interaction in presence of a
inhomogeneous magnetic field:
B0 ( z )
from a set
of coils
z
Vmag ( z ) = − µ j ⋅ B0 ( z ) = µ B m j g j B0 ( z )
dB0 ( z )
Force = − µ B m j g j
dz
like in the Stern-Gerlach experiment
Case m j > 0 :
B0
mj
Vmag (z )
j
µj
B0 and µ j are anti - parallel
z
minimun
stable trap
atoms oscillate around
the minimum of B-field
Atoms in states with m j > 0 are low-field seekers.
Case m j < 0 :
B0
µj
Vmag (z )
z
j
mj
B0 and µ j are parallel
Atoms are repelled from centre
and move towards regions of
high B-field
Atoms in states with m j < 0 are high-field seekers.
We can only trap atoms that are in the right magnetic levels!
Evaporative cooling:
• The velocity distribution in the trap is Maxwell-Boltzmann
(as long as we are far in temperature from BEC).
• Reduce trap depth
faster atoms escape
• If done slowly, the gas is always close to thermal equilibrium,
and its temperature T is reduced
trap
depth
T
T’<T
Achieving Bose-Einstein condensation
Here evaporative cooling was stopped at three different temperatures.
The sharp peak appearing at 230 nK is the onset of BEC:
Rb-87 BEC
NATOM ~ 10 5- 106
530 nK
Tc = 230 nK
100 nK
Oxford, 2001
These pictures represent the density profile of the atomic cloud…
Detecting the atom cloud
probe laser beam
CCD
array
atom cloud
shadow is imaged onto CCD
The probe beam is in resonance with an atomic transition
the cloud absorbs photons and produces a shadow in the probe beam.
This shadow is detected by a CCD camera.
We switch off the magnetic trap a short time before taking the image.
The cloud has therefore expanded (hence the name “time of flight”
image) due to the initial velocity distribution of the atoms.
The condensed fraction is slower than the thermal atoms, hence it
appears as a sharp peak in the image.
Onset of BEC
JILA, 1995
The atoms in a condensate share the same wavefunction, which is
the ground state of the magnetic trap (essentially a simple harmonic
oscillator potential).
So it is a quantum object we are looking at with a simple camera!
Interference of matter waves (MIT)