The first step to obtaining stationary, controllable atoms to be used as qubits is to slow them down by cooling and contain them in a trap. To cool the atoms, a highly successful and powerful technique is laser cooling [49]. This technique relies on the Doppler effect, so it is also known as Doppler cooling. The principle behind the technique is illustrated in Figure 3.2. At room temperature, atoms
have a large velocity on the order of magnitude of 100 m s−1in a random direction. A laser providing resonant electromagnetic radiation can excite the atom into a higher energy level. The absorption of a photon causes the atom to recoil and gain momentum equal to the photon momentum, ~k where k = 2π/λ is the wavenumber, in the propagation direction of the photon. When the atom then spontaneously emits a photon and decays back to the lower energy level, the photon will be emitted in a random direction, leading to a loss of momentum for the atom. For continuous incident radiation, the photon scattering rate of the atom is given by equation3.23. After many cycles of this process, the momentum change due to spontaneous emission averages to zero due to the isotropy of the emission, whereas the momentum gained by absorption increasingly builds in one direction. This is the mechanism behind radiation pressure, in which a beam of photons can apply an effective force on an atom as a result of the overall momentum change in the direction of incidence. This scattering force is given by the photon momentum ~k multiplied by the scattering rate,
Fscatt= ~kRscatt= ~kΓ
2
I/Isat
1 + I/Isat+ 4∆2/Γ2
. (3.26)
Due to the Doppler effect, which is the change in frequency experienced by an atom due to the motion of the atom relative to the frequency source, an atom moving in the same direction as the laser experiences a lower frequency than the actual laser frequency and an atom moving opposite to the laser direction experiences a higher frequency. The detuning of the cooling laser can therefore be tuned to slow only those atoms which are moving opposite to the photon propagation direction. If the laser frequency is red-detuned, then by the Doppler
mv ~k
mv − ~k
1)
2)
3)
Figure 3.2: Doppler cooling with red-detuned lasers. In step (1), a photon with momentum ~k is travelling in the opposite direction to an atom with momentum mv. In step (2), the atom absorbs the photon as it is Doppler shifted into resonance by the atomic motion, exciting the atom and decreasing the momentum of the atom to mv − ~k. Finally, in step (3), the atom decays back to the ground state, spontaneously emitting a photon in a random direction. Over many cycles of this process, the momentum loss caused by the cooling laser builds up, whereas the momentum change from the photon emission averages to zero.
effect the frequency ‘seen’ by the atom is higher than the actual frequency, and is shifted towards the resonant frequency of the cooling transition. This allows the cooling cycle described previously to occur, and the atom is affected by radiation pressure in the opposite direction to its motion. The radiation pressure acts as a force causing the atom to slow down, until the laser frequency observed by the atom is no longer within its naturally-broadened absorption linewidth.
Optical molasses
Using a pair of counter-propagating red-detuned lasers will push atoms travelling either way in the opposite direction to their motion. This is because atoms travelling towards one of the lasers will experience a frequency which is Doppler-shifted into resonance with the cooling transition and be pushed in the opposite direction by that laser, whereas they remain unaffected by the
σ− σ+ σ− σ+ σ− σ+ B = 0 B B z (a) σ− σ+ B = 0 E J = 0 mJ= 0 J = 1 mJ= 0 mJ = +1 mJ = −1 ∆ B B z (b)
Figure 3.3: The magneto-optical trap (MOT). In (a), a diagram of the MOT is shown. Three orthogonal pairs of counter propagating, oppositely circularly-polarised red-detuned lasers are passed through the trap centre. Atoms in the beam path are cooled by the Doppler cooling technique. A non-uniform magnetic field is produced by external anti-Helmholtz coils, causing the atomic energy levels to be Zeeman shifted into resonance with the beams, red-detuned by ∆, in such a way as to produce a trapping force towards the trap centre. In (b), the Zeeman shift induced in the excited state (J = 1) sublevels in one dimension is shown.
counter-propagating laser as it is out of resonance with the cooling transition. This provides a frictional force acting against the motion of the atoms in one dimension, known as optical molasses. Extending this setup to three pairs of counter-propagating beams orthogonal to each other as demonstrated in Figure 3.3 applies a viscous force in all three dimensions, which pushes atoms towards the point at which the beams cross. This technique is capable of cooling atoms down to the Doppler temperature limit, which arises from the non-zero minimum momentum gained from absorbing photons. The Doppler limit for rubidium is 146 µK [42,43].
A large assumption in the laser cooling method is that the atoms can be described as a two-level system. This is not true in practice for rubidium atoms, which have many energy levels above the ground state of the valence electron including fine and hyperfine splitting, as shown in Figure 2.4. In our experiment, a cooling transition is selected for which selection rules confine the system to be almost two-level. This transition is in the D2 line, 52S1/2 F = 2 → 52P3/2 F = 3, which
is excited by the resonant laser wavelength 780.24 nm [43]. The selection rule ∆F = 0, ±1 ensures that the upper level can only spontaneously decay to the 52S
1/2 F = 2 state. There is a chance that the ground state will be excited into
the 52P
3/2 F = 2 state, at which point it could decay into the 52S1/2 F = 1 state,
causing atoms to escape from the cooling cycle. In order to restore the atoms back into the cooling process a second laser is required, called the ‘repump’ laser, which drives the 52S1/2 F = 1 → 52P3/2 F = 2 transition with resonant light of
780.24 nm. The energy level structure with the cooling and repump transitions highlighted is shown in Figure 4.1.