How this works
Kinetic energy change from entering or leaving a magnetic field
The change in kinetic energy of a Cs atom in the M = ±4 state, initially at rest (moving very slowly), entering or leaving a magnetic field B is:
½ m (vf 2 - vi 2) = μB gF M B = ±μB B,
where vi and vf are the initial and final velocities respectively, m is the mass of the Cs atom μB is the Bohr magneton, and gF is the “g” factor. If vi is small, we have:
vf = 9.2 B½.
Thus a Cs atom in the M=-4 (F=4 ground) state leaving a 0.6 Tesla magnetic field will accelerate to 7 m/s, the velocity needed for a very tall, 2 m high fountain. To make this happen, the magnetic field is initially off and is turned on after the atom has entered the region of maximum magnetic field.
Magnetic field gradients
For our simulations and first experiments, we are using solenoids (Fig. 1) because they have axial symmetry, they can have a long uniform region in which to turn the field on or off, and their fields and gradients can be precisely calculated.
The acceleration (m/s2) of a Cs atom in a M = ± 4 state, due to a magnetic field gradient ∂B/∂z (T/m) in the axial (z) direction, is 42.2 ∂B/∂z , and similarly, 42.2 ∂B/∂r for a magnetic field gradient in the radial (r) direction.
Unless the atom is exactly on the solenoid axis, it will be subject to focusing/defocusing forces which change sign as the atom enters or leaves the solenoid. In a solenoid, used for focusing, and where the current is constant, atoms in a M <0 state will slow down as they approach and enter the solenoid, and speed up as they leave the solenoid. Because the forces are conservative, the magnitudes of the initial and final velocities are the same. Fig. 2 shows the calculated magnetic field gradient in the radial direction (focus/defocus) for a solenoid with a length to radius ratio of six.
Simulation and experiment design
For simulation results, scroll to the bottom of this page
A cloud of cold atoms has an initial spatial and velocity distribution (phase space), which must be managed to prevent beam loss. This is done by transverse focusing and bunching (longitudinal focusing) the atoms.
Determining the number, size, and placement of the focusing and acceleration elements, and their magnetic field strength, which can be changed as the cloud passes through, is done using a simulation code run by Ben Feinberg, written by Hiroshi Nishimura, and modeled on codes used in accelerator charged particle beam transport.
In Fig. 2, the radial forces are linear out to r/a ≈ 0.3. While it is not impossible to model with non linear forces, working in the linear region makes the simulation and beam tuning much simpler, while avoiding phase space growth and loss of atoms.
Our first design is for Cs atoms in the M=-4 state, released from a magneto optical trap (MOT) to fall 0.4 m into the bore of a 0.2 T solenoid. The solenoid is shown in Fig. 3 and the results of the simulation are shown in Fig. 4 below.
Atoms falling towards the solenoid will slow, stop, reverse direction, and rise back towards the MOT with the same magnitude velocity as they had entering the solenoid.
In the simulation we place a focus coil about 5 cm below the MOT. Actually this is just the lower MOT coil which we will turn back on after the atoms have been released from the MOT and is shown in red at the left in Fig. 4).
This coil focuses the atoms on their way down and on their way back up. The atoms at 130 μK have velocity spreads in excess tens of cm/s. Atoms with initial downward components of velocity will pass through the focusing coil faster than other atoms. We compensate by using a time-dependent current in the focus coil. For the 130 μK atoms, we change the focus coil current seven times during the first bounce but are able to leave it fixed during subsequent bounces.
The focus point of the falling atoms is the axis of the bounce coil at its entrance (Fig. 4). if the atoms are not close to the axis, transverse forces from the strong bounce coil fields over focuses the atoms on their ascent, causing losses.
After several hundred simulation runs we have found solutions, for which 85 - 90% of the 130 μK Cs atoms will survice two bounces and 70% of the Cs atoms will survive 15 bounces (assuming perfect vacuum). If we use ultra cold (2 μK) Cs atoms in our simulation, the 15 bounce survival rate increases to 98%. We are now modifying our MOT apparatus to test this simulation.
Our next simulation is to take the returning atoms and accelerate them so they travel a half meter above the MOT. This requires adding bunching coils, acceleration coils, and more focusing coils to the simulation. Preliminary results were presented at the 2017 DAMOP meeting.
Fig. 4 shows twenty four of the time steps
(T = 0 to T = 1.5 s) of
the simulation of bouncing 500 Cs atoms.
Scroll to see later times.
The atoms are released from a MOT at z = 0 and T=0. Only the right hand half of the radially symmetric cloud is shown. Atoms that go beyond r = 4.5 cm are lost. The first frame shows the focusing coil (red) and the bounce solenoid (blue). The number of surviving atoms is shown in small print near the top of each frame. For the first 900 ms, each frame is 50 ms later, then 100 ms later.
Fig. 4: Simulation of dropping a 130 μK cloud of 500 Cs atoms.