1. Laser Cooling and Trapping
Quantum Optics – 2006/7
Shimon Machluf

2. Outline Temperatures Scheme Deceleration of Atomic Beam – 1D
Mathematical Review
The Forces
Experiments
Optical Molasses – 2D
Experiments and Results
Sub-Doppler Cooling
Linear Polarization
Circular Polarization
Recoil limit
MOT
BEC
Dipole Trap
Optical Lattice

3. Temperatures Scheme
Difference in definitions of temperature and entropy
We still define:
Capturing temp.
Doppler limit
Recoil limit
Thermodynamics' temperature is a state of the system in thermal equilibrium with its surrounding, needs heat exchange. Here the atom scatter light so they change the environment and light can’t exchange heat.
With entropy it’s the same – there is no equilibrium.

4. Some Definitions Optical Bloch Eqs: where: The steady state solution,
gives: where:

5. Mathematical Background
Ehrenfest theorem:
The interaction Hamiltonian:
(using the electric dipole approximation, in the last part, which allows to interchange the gradient and the expected value)
Using: we’ll get:
Defining:
and the general solution:
In the interaction Hamiltonian: epsilon dot r is the perturbed Hamiltonian, meaning the energy of the interaction of the atom with the field.

6. The Forces on the Atoms
Substituting the steady-state solution of OBE into the last force Eq, gives:
The first term is proportional to detuning and the second one to the decay rate
For zero detuning:
Radiation force:
This force caused by the momentum the atom get when absorbing a photon, but when scattering a photon the momentum averaged out
Radiation force applet
This force saturates at large intensity
The direction of the force is the direction of the light
Dipole Force:
This force direction depends on the detuning sign
For zero detuning: we get the momentum of photon times scattering rate.
Radiation force: explain that this force is actually because each photon give some momentum (hbar k) to the atom, and that the recoil velocity is v=hbar times k over m, about few cm/s

7. Deceleration of an Atomic Beam – 1D
Doppler Shift
Define Doppler dependent force:
From saturation of the intensity:
Lmin and tmin where calculated with basic calculations
After some cycles the slowed atoms are out of resonance. It is enough to reduce the velocity a few m/s.
Doppler force applet
In order to keep the deceleration, it is necessary to maintain:
aMax: if the intensity is too high then stimulated emission occurs, and on average the atom is cooled and them get hotter because of the stim.em. The factor 2 is because the atom is only half of the time in the ground state.
To maintain the detuning about zero: we need to change the Doppler freq., the atom freq. or the light freq.

8. Keeping the Deceleration
Laser frequency sweep
Sweeping the laser frequency at the same rate as the Doppler frequency changes (just to the other direction)
Easiest to use with diode lasers
The slow atoms arrive in pulses
Broadband light
Using light that is not spectrally narrow
High power laser is needed to keep the spectral density
Diffuse light
Because , atoms moving through diffused monochromatic light see different frequencies, depend of the angle
Changing the Magnetic Field

9. Real (or Multilevel) Atoms
Alkali-Metal atoms
The frequency from the ground to the first exited state: in visible light
Large vapor pressure at low temperature: easy to evaporate to vacuum
Have closed shell with one valance electron: only that electron contribute to the atom angular momentum
Fine structure
Hyper-Fine structure

10. Zeeman Splitting There are also Zeeman sublevels
The energy of the sublevels:
Where:

11. Changing the Magnetic Field
For uniform deceleration, , the appropriate field profile:
Where:

12. Measurements and Results
Na atoms leave the source trough 1mm opening at 3000C
The slowing beam (laser 2) is focused at the source
Atomic beam at 1000 m/s and spread of 0.01 radians, comes to 50 m/s and 0.2 radians

13. Measurements and Results
The TOF technique:
Since the pump and probe are perpendicular to the beam it excite all velocities.
The pump is wide enough, 0.5mm, to excite 98% of the atoms
To measure velocity, the pump is interrupted for
At that time atoms with certain velocity are not excited by the pump beam, so the probe will excite them
The resolution of the measurement:
typically less then 1 m/s, and for atoms at 80 m/s it’s better then the Doppler limit, 30 cm/s

14. Measurements and Results
Cooling the atoms
For cooling the atoms, the velocity distribution must be compressed, as the measured velocity shows compression in the velocity distribution
Another method is to turn off the cooling beam while the atoms are in the solenoid.
That give the 2D graphs of the density and position in the solenoid
The atoms are moving at constant , and they arrive to the probe beam after
The dashed line is the resonance velocity
Optical pumping
Between the source and the solenoid, where the beam is focused, there is movement (pumping) of atoms between HFS levels.
That appends because of power broadening and Doppler shift
After the dashed line there is drop in the intensity

15. Optical Molasses The Doppler force:
And the total (in low intensity regime):
The direction of the force is against the speed (regardless off the light) when the light is red detuned
In the usual realm of OM, , the force is linear with the velocity when:
Capture velocity:
Optical molasses applet

16. Doppler Limit The recoil energy:
The average energy the light loose (at rate for 2 beams), becomes kinetic energy (heat)
The competition between cooling ( ) and heating ( ) has steady-state at:
Since the steady-state energy depends on , and the minimum is when , Doppler temperature is defined as:
The Doppler temperature is about 1 mK which is about 30 cm/s
Another way to understand the Doppler limit is that the total momentum averaged to 0, but the rms doesn’t.

17. Squeezing Atomic beam - 2D

18. Atomic beam collimation - 1D
1500C atoms leaving an oven with aperture of , and another aperture of the same size after 24 cm

19. Atomic beam collimation - 2D

20. The Measuring System

21. Optical Molasses – 3D

22. Measuring the Temperature

23. Detuning Results

24. Light Shifts
Absorbing the diagonal elements of into , and assuming that , is the only nonzero element.
Then the Eq. becomes (for two levels atom):
Multiplying on the left by , and integrating over

25. Light Shifts … Transforming to a rotating frame
Inserting the new c’s into the last solution:
The new Hamiltonian is:
And the new energies (solutions) are:

26. Linear Polarization The Eq. of linear polarized light:
Two perpendicular counter propagating linear beams:
At the origin there is linear light at an angle of
Similarly,
But between,
Since the 2 components are perpendicular and out of phase, they represent circular polarization, in the negative sense
At , the light is circularly polarized but in the positive sense
There is strong polarization gradient

27. Sub-Doppler Cooling - Linear light
The light shift of multilevel atoms at low intensity limit:
Since depends on the magnetic quantum numbers and the polarization, the light shifts are different for different sublevels
The ground state light shift is negative for red detuned light
For light, the light shift for substate is 3 times larger then for substate
For it’s the opposite
Absorption of light produced transitions, and spontaneous emission produced , so on average
light will pump toward the
When atom at the origin in the M=+1/2 state move in the light field it moves up the potential, because the polarization of the light is changing and the +1/2 state becomes less coupled. After , the atom arrive to a position where the light is polarized, and the atom is pumped to M=-1/2. Now the atom is again at the bottom of a potential hill.

28. Sub-Doppler Cooling - Linear light

29. Circular Polarization
The Eq. of circular polarized light:
Two perpendicular counter propagating circular beams:
This represent a linear light with constant magnitude that rotate around z axis over one wavelength

30. Sub-Doppler Cooling - Circular light
When it come to circular polarized light, the explanation is totally different
The atom now are real atoms with 3 hyper-fine levels
When atom is moving toward one of the beams, assume , it is pumped toward the sublevel because of the Doppler shift
Now, when it’s in the sublevel, because of the difference in the Clebsch-Gordon coefficient, it scatter light 6 times more (for the other direction it is opposite)
This cooling can go below the Doppler limit because it is based of the atoms “want” to scatter light from the opposing beam not because of the Doppler effect but because of different population of the sublevels

31. Recoil Limit
Since the cooling happens because of momentum transfer from the light to the atoms, it can happen only in a discrete way
One quant of momentum is:

32. Magneto-Optical Traps
A MOT is created with 3 pair of counter propagating laser beams and an inhomogeneous magnetic field – quadrupole:
The combined force is:
When the force becomes:
For fast atoms the first term is the dominant term
For atoms far from the center of the trap the second term is dominant (which is relative to the distance and the magnetic field gradient)

33. MOT Apparatus

34. MOT Arrangement in 1D

35. Capturing Atoms in a MOT
Capture velocity
Repumper
The cooling is a closed transition
The excited HPS are close but the ground ones are not

36. Evaporative Cooling
Since the atoms have magnetic momentum, the fill a potential, , and want to be at the minimum – Magnetic Trap
Evaporative cooling applet

37. Bose-Einstein Condensation
The pathway to BEC
Capturing atoms from the room temperature in a MOT (only the low velocity tail)
The magnetic fields are turned off and an Optical Molasses stage begins – cooling the atoms to almost recoil temperature
The OM beams are switched off and the magnetic trap turned on
The atoms are trapped in a dark magnetic trap and the Evaporative Cooling stage starts
After the Evaporative Cooling end there are in the trap about 1% of the atoms in a BEC state
To go through the phase transition, the atoms need to be in a phase space density (density in both the place and momentum) of:

38. BEC
BEC at 400, nK

39. BEC - Interference
Interference patterns obtained after 14 ms potential-free time-of-flight expansion of the two BECs.
Matter-wave interferometry in a double well on an atom chip, Nature Physics 1 (2005)
T. Schumm, S. Hofferberth, L. M. Andersson, S. Wildermuth, S. Groth, I. Bar-Joseph, J. chmiedmayer and P. Krüger

40. Dipole Traps
When the light intensity is not homogeneous in space, the light shifts are not the same, so there is difference in the potential (hence, a force) and the atoms will want to go to lowest energy
When the light is red detuned, the lowest energy is where the maximum intensity, and when it’s blue detuned the maximum is at maximum intensity
When a Gaussian laser beam is focused, the intensity is:
Assuming , the force is:
And for the Gaussian beam:

41. Dipole Traps There is also low gradient in the longitudinal direction
There is no radiation force because the laser is far detuned
radiation force goes like , but dipole force goes like
When far detuned the atoms feel only the dipole force
The first dipole traps were made with 220 mW dye laser, about 650 GHz red detuned and focused to waist
The trap depth was 7 mK
Another far-of-resonance trap was with 0.5 or 1 W laser focused to , the laser was 795 nm instead of 780 nm
Two another cases used Nd:YAG laser with 1064 nm to trap Na atom with nearest transition of 596 nm. Cs atoms with resonance at 852 nm were trapped with 10,600 nm CO2 laser

42. Optical Lattices
The “egg-crate” potential, created by 2 (or 3) standing waves
The lattice sites size is
If the atoms are cold enough they will be trapped in the periodic potential
For a MOT the fill factor is a few percents because the density in a MOT is limited to a few 1011 cm-3 but the lattice site density is a few 1013 cm-3
Some laboratories use far-off-resonance CO2 laser to create the lattice
It is possible to fill the lattice with atoms in a BEC state – higher fill factor

43. Optical lattices – Other Considerations
If there are sub-wavelength movements of the mirrors the “standing wave” the relative phase is changed

44. Optical lattices – Other Considerations
The momentum of the trapped atoms is a “few” times , so their deBroglie wavelength is equal to the trap size divided by a “few”
They don’t oscillate as classical particles, but instead they occupy discrete levels

45. References
The presentation is based on the book: “Laser Cooling and Trapping”, by Harold J. Metcaf and Peter van der Straten
The references are from that book