Optical Trapping Theory

Magnetic Field and Laser Cooling

The goal of the magneto-optical trap is to constrain the movement of sample atoms using a linearly spatially varying magnetic field, along with red-detuned lasers along three axes. The magnetic field takes on a quadrupolar form oriented in the ẑ direction, produced by anti-Helmholtz coils (Fig. 1). This produces Zeeman splitting of the sample atoms into magnetic substates. For an atom with a single ground state and excited state, denoted |J = 0⟩,|J = 1⟩, this gives possible values of mf = -1, 0, 1. The magnetic field varies linearly in space, increasing from 0 at the trap's center, and therefore produces increased splitting as atoms travel away from this point.

In order to cool the atoms, red-detuned lasers are employed along 3 orthogonal axes, as shown in Fig. 2. The process of red-detuning shifts the frequency of the lasers away from the resonant frequency ω0 of the atoms to be trapped. At the trap's center, where B = 0, the transition energy from |J=0⟩ to |J=1⟩ is greater than the energy of the incident photons, thus no absorption occurs. As atoms move away from the center, the increasing magnetic field and thus increased energy level splitting lowers the gap between the |J=0⟩ and |J=1, mf = -1 ⟩ states. This effectively lowers the frequency of the transition towards the red-detuned value ω = ω0 - δ, leading to greater photon absorption farther from the trap's center.

When an atom absorbs a photon with energy ℏω, conservation of momentum requires that it receives a momentum kick equal to ℏω/c in the photon's initial direction. In the present setup, only atoms moving away from the trap's center absorb photons, and only in the direction opposite their travel, resulting in an effect that pushes atoms back toward the center. Spontaneous emission may then occur, in which the atom produces a photon in a random direction. This process returns the atom to the ground state, decreases the magnitude of its momentum, and shifts the direction of its momentum toward the direction opposite the emission. Over the course of many absorptions and emissions, the result is to bring the atom back to the trap's center with little or momentum, hence trapping it.

Figure 1: Quadrupolar Magnetic Field. Because Maxwell's equations require that ∇.B = 0, the magnitude in the z direction will be twice that in the x and y directions [1]

Figure 2: The Magneto-Optical Trap. Two anti-Helmholtz coils produce a linear spatially varying magnetic field along the z axis. Three orthogonal beams intersect to trap and cool the Rubidium atoms in the center. Along each axis, red detuned σ+ and σ− counter-propagating beams are sent in, along with a repump beam. [6]

Additional Requirements: Photon Polarization and Repumping Light

In the MOT, the laser light must be correctly polarized. Photon absorption requires conservation of angular momentum between the photon and atom. Transitions from |J=0, mf = 0⟩ to |J=1, mf = -1 ⟩ are driven only by negatively circularly polarized σ- light, and the |J=0, mf = 0⟩ to |J=1, mf = 1⟩ transition is driven by circularly polarized σ+ light. For atoms traveling in the +ẑ direction, the energy gap of the |J=0, mf = 0⟩ to |J=1, mf = -1⟩ decreases, thus we wish to induce a Δmf = -1 change in momentum and only σ- light can be absorbed. For atoms with velocity in the -ẑ direction, where B < 0, the |J=0, mf = 0⟩ to |J=1, mf = 1⟩ transition reaches the red-detuned frequency, and σ+ light is required for absorption.

Atom trapping also requires a ``closed optical loop" of transition states, such that emission from its excited state returns it to the same ground state, allowing a single laser frequency to continue inducing transitions. In 85Rb, the 5S1/2 (f=3) to 5P3/2(f=4) transition forms a closed loop. Selection rules, as discussed in [2], ensure that de-excitations to 5S1/2(f=2) and 5P1/2 states cannot occur. The first would require Δmf = 2, where Δmf can only be $\pm 1$ or 0, and spin-orbit selection forbids 5P3/2 → 5P1/2. However, it is possible that some atoms may reach the $5P3/2 (f=3) state, resulting in emission to the inaccessible 5S1/2 (f=2) state, whose transition energy is too great for the red-detuned laser to induce.

To reintroduce these atoms to the cycle, a repumping laser is used with a frequency equal to the transition from this lower state to the excited state of the cycle. For 85Rb, this requires increasing the frequency by approximately 3 GHz [3].

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