Optical Magnetic Resonance - Optical polarization/Optical detection (all-optical MR)

Optical pumping of a Rb (alkali-metal) atomic population (used in atomic/optical magnetometers)

"the fact that the electrons are being pumped into a state of maximal mF means that their magnetic dipole moments orient themselves parallel to the external magnetic field."

"rubidium atoms behave like a single electron in the 5s ground state."

Light consists of photons which carry energy E, momentum p and angular momentum j=1 (two degrees of freedom are defined i.e. mj=±1). The angular momentum eigenstates are called σ+ (mj=+1) and σ− (mj=−1). Photons of right circular polarization or positive helicity are σ+ and photons of left circular polarization or negative helicity are σ-. Linearly polarized light is a combination of two circular polarizations, right and left corresponding to σ+ and σ− respectively.

Alkali-metal atoms (Li, Na, K, Rb, or Cs) have a single electron in the outer shell. For example, rubidium (Z=37) has the configuration 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 5s with the 5s shell having one single electron.

"The spin-orbit interaction pushes the 5s shell lower than the 4d and 4f shells, so that the rubidium atoms behave like a single electron in the 5s ground state."

Orbital angular momentum is represented by the orbital quantum number l. Spin angular momentum or spin (s) is represented by the spin quantum number ms, which takes the values +1/2 for "spin up" and -1/2 for "spin down". The total angular momentum is represented by the total angular momentum number j which is equal to j=l+s and therefore for the electron it is j=l±1/2.

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Fine structure - Spin-orbit interaction

The 5 state with l=0 (s shell) is the ground state and has j=1/2. It is termed 5s(1/2).
The 5 state with l=1 (p shell) is split due to the spin-orbit interaction in two states (energetic levels), one with j=1+1/2=3/2 termed 5p(3/2) and the other with j=1-1/2=1/2 termed 5p(1/2).

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Hyperfine structure - Zeeman interaction

The total angular momentum of the atom F is given by F=J+K where J and K are the angular momenta of the electrons and the protons/neutrons respectively.

The 5s(1/2) state is split in two states, F1 and F2, with magnetic quantum numbers mj=+1/2 and mj=-1/2.
The 5p(1/2) state is split in two states, F1 and F2, with magnetic quantum numbers mj=+1/2 and mj=-1/2.
The 5p(3/2) state is split in four states, F0 to F3, with magnetic quantum numbers mj=+1/2, mj=-1/2, mj=+3/2, mj=-3/2.

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Energetic Transitions - Selection rules

The D1 transition of Rb: From 5s(1/2) to 5p(1/2) by absorption of photons of 794.7nm
The D2 transition of Rb: From 5s(1/2) to 5p(3/2) by absorption of photons of 780 nm

"When atoms absorb light, they must absorb both energy and angular momentum in the transition from the ground state to an excited state. The most basic, leading order process is called the electric dipole, or E1, transition and has the selection rules for the atom ∆l = ±1, ∆F = ±1 and ∆m = 0, ±1, depending on the polarization of the light. For circularly polarized photon absorption ∆m = mj. For linearly polarized photons ∆m = 0. These restrictions of ∆l, ∆F, etc. are called selection rules. Emission of light by F1 radiation has the same selection rules. Also if ∆m = ±1, the photon emitted is circularly polarized, and if ∆m = ±0, the emitted photon is linearly polarized, that is a superposition of σ+ and σ−."

"σ+ photons can affect 2S1/2 →2 P1/2 transitions according to the following dipole selection rules (these give the dominant transition processes):
∆L = ±1 ∆J = 0, ±1 ∆F = 0, ±1 ∆mF = +1

Note that σ+ photons can only cause ∆mF = +1 transitions. Likewise, photons of negative helicity - so-called σ− photons - can only cause ∆mF = −1 transitions, with otherwise identical selection rules. So when a σ+ photon hits a Rubidium 87 atom, it not only excites the electron to the 2P1/2 level, but also raises its mF by one unit. Of course, the excited state is unstable and spontaneously decays back to the ground state by emitting another photon. However, the spontaneously emitted photon has arbitrary polarization and the decay can have ∆mF = 0, ±1. Thus, on average we have ∆mF = +1 for the excitation by σ+ photons, but ∆mF = 0 for the spontaneous decay, giving a net average ∆mF = +1: on average, the electrons will migrate into states of higher and higher mF . This phenomenon is referred to as optical pumping."

"Physically, the fact that the electrons are being pumped into a state of maximal mF means that their magnetic dipole moments orient themselves parallel to the external magnetic field.

In a state of maximal mF , dipole selection rules prohibit the atoms from absorbing any more σ+ photons, and thus these photons pass through the Rubidium vapor bulb unaffected: in its fully pumped state the bulb is transparent.

Now imagine that there was - loosely speaking - a way to disorient the magnetic moments we just talked about. Then the Rubidium gas would not be in its fully pumped state anymore and could again absorb σ+ photons: it would temporarily become opaque.

By monitoring the depletion of the incident beam of σ+ photons using the optical detector behind the Rubidium vapor bulb, we can thus tell when the Rubidium vapor is in its fully pumped state and when it has become temporarily disoriented: temporary opacity shows up as a dip in the intensity of the incident photon beam measured by the optical detector.

This is the key idea which makes optical pumping a powerful experimental technique: we can use it to tell when the Rubidium vapor becomes temporarily disoriented."

Figure: On the left from Wikipedia (https://en.wikipedia.org/wiki/Hyperpolarization_(physics)_- on the right from second reference

## Atomic optical magnetometry with visible/near-infrared light

Spin Exchange Relaxation-Free (SERF) magnetometers

[1] "Alkali atoms couple in a resonant fashion to light, with frequency in the visible and near-infrared spectrum. The transitions between the ground state and excited state energy levels are often called optical transitions. The transitions that couple the ground state to the P1/2 and P3/2 excited states are often referred as the D1 and D2 optical transitions, respectively."

"In the absence of external fields, the hyperfine energy spectrum of alkali atoms is determined by the interaction between the nucleus, with angular momentum I, and the unpaired electron of the atom, with angular momentum J."

[2] "Magnetic resonance involves the coupling of energy from the applied magnetic field to the atomic spin system, which has a natural frequency due to the Zeeman splitting of spin up and spin down atoms in the presence of an applied, static magnetic field B0.

The energy levels for an atom with J = 1/2 and two hyperfine levels F = I ± 1/2 are given by the Breit–Rabi formula (...) .

"The magnetic resonance transitions correspond to ∆F = 0 and ∆m = ±1. For B0 of a few Gauss, x <<1, and the Breit Rabi formula can be used to show, for example that the
corresponding frequency is:
hν ≈ h γA B0
where γA is a gyromagnetic ratio for each isotope 85Rb with I85 = 5/2 and 87Rb with I87=3/2."

"The phenomenon of magnetic resonance is observed by changing the frequency and magnitude of the applied RF magnetic field. The mathematics is identical to that of a driven LRC oscillator, where the damping rate constant is Γ = R/2L."

Spin Exchange Relaxation

[2] "The dynamics of a spin coupled to a magnetic field Bo through the Zeeman interaction consists of the precession of the spin about the field. In the absence of a driving field, spin relaxation mechanisms restrict the synchronous precession of the spins about the magnetic field, which results in dephasing. In the presence of spin decoherence and optical pumping the response of the atomic spin ensemble to magnetic fields can be modeled phenomenologically through Bloch equations as described in chapter 2.”

When a pair of rubidium atoms move past each other "the magnetic dipole moment of one atoms causes a rapidly changing magnetic field in the rest frame of the other leading to spin flips and the phenomena of spin exchange. Spin exchange is an important effect because it is very rapid and affects the electron spins of atoms that may be in different hyperfine levels or even different isotopes."

Also "when a rubidium atom moves near the wall, the rapidly varying field in the atom’s rest frame can cause transitions between mj = +1/2 and mj = −1/2. This is called “wall relaxation.”"

[1][e-page 32] "Figure 2.2: (a) A single atom interacting with a magnetic field. All degrees of freedom are frozen, but its spin. The effect of the interaction is the precession of the spin about the magnetic field. In the real world the atom cannot be completely isolated from its environment. The spin will interact with this environment resulting in effects such as the sudden change in its precession phase.

(b) An ensemble of atoms interacting with the same magnetic field as in (a). Here all degrees of freedom of the atoms are unfrozen. Thus in addition to interacting with the magnetic field B, the atoms move within the cell, collide and interact with each other and with the cell walls, which in turn causes the dephasing of the precession in the spin ensemble."

[1] Thesis by R. Jimenez Martinez, University of Colorado at Boulder (in collaboration with NIST)

e-page 32] Figure 2.2: (a) A single atom interacting with a magnetic field. All degrees of freedom are frozen, but its spin. The effect of the interaction is the precession of the spin about the magnetic field. In the real world the atom cannot be completely isolated from its environment. The spin will interact with this environment resulting in effects such as the sudden change in its precession phase. (b) An ensemble of atoms interacting with the same magnetic field as in (a). Here all degrees of freedom of the atoms are unfrozen. Thus in addition to interacting with the magnetic field B, the atoms move within the cell, collide and interact with each other and with the cell walls, which in turn causes the dephasing of the precession in the spin ensemble.