2. Physical Background

2.1. Angular Momentum

The possibility to use optical radiation for exciting and detecting spin polarization can be traced back to the angular momentum of the photon. Photons as the carriers of the electromagnetic interaction carry one unit (h) of angular momentum, which is oriented either parallel or antiparallel to the direction of propagation of the light. Since angular momentum is a conserved quantity, the total angular momentum of the system (radiation and matter) cannot be changed during absorption and emission of radiation. When a photon is absorbed by an atom or molecule, its angular momentum must therefore be transferred to the atom. The resulting angular momentum of the atom is then equal to the vector sum of its initial angular momentum plus the angular momentum of the absorbed photon.

Figure 2.1.

Conservation of angular momentum during absorption of a photon

Magnetic resonance spectroscopy requires a spin polarization inside the medium. In conventional magnetic resonance experiments, this polarization is established by thermal contact of the spins with the lattice. This process is relatively slow, especially at low temperatures, where relaxation times can be many hours, and it leads to polarizations that are limited by the Boltzmann factor, which is typically less than 10^-5. Photon angular momentum, in contrast, can be created in arbitrary quantities with a polarization that can be arbitrarily close to unity. If it is possible to transfer this polarization to nuclear or electronic spins, their polarization can reach the same values.

2.2. Optical Pumping

This possibility was first suggested by Alfred Kastler [Kastler, 1950, Kastler, 1967]. Figure 2.2 illustrates the principle of operation.

Figure 2.2.

Principle of optical pumping illustrated for a simple atomic system.

The model atomic system consists of two electronic states, labelled |g> for ground state and |e> for excited state. Both are assumed to have an angular momentum J = 1/2 and therefore two states corresponding to Jz = +1/2 and Jz = -1/2. If the system is irradiated by circularly polarized light, the photons have a spin quantum number ms = +1. Since the absorption of a photon is possible only if both, the energy and the angular momentum of the system are conserved, only those ground state atoms that are initially in the Jz = -1/2 state can absorb photons. If an atom that is initially in the state Jz = +1/2, the resulting excited state would have to be a Jz = +3/2 state, which does not exist in the atom of our model system.

An atom that has absorbed a photon will in general reemit one after the excited state lifetime. Spontaneous emission can occur in an arbitrary direction in space and is therefore not limited by the same selection rules as the excitation process with a laser beam of definite direction of propagation. The spontaneously emitted photons carry away angular momentum with different orientations and the atom can therefore end up in either of the two ground states. If it ends up in the original state, it can absorb another photon and repeat the cycle; if it ends up in the other state, it is no longer coupled to the laser field and remains therefore in this state indefinitely. The net effect of the absorption and emission processes is therefore a transfer of population from one spin state to the other and thereby a polarization of the atomic system. Optical pumping represents therefore a very efficient method for polarizing a spin system.

2.3. Dynamics

Like in conventional magnetic resonance, the spin polarization remains in general not static, but undergoes a Larmor precession if an external magnetic field is applied. If we use light to drive the spin system, it has an additional effect on the spin dynamics: If the laser couples to a particular transition, it appears to shift the energy of the levels to which it couples [Barrat and Cohen-Tannoudji, 1961, Kastler, 1963]. Shifts of energy levels always affect the dynamics of the corresponding system. In this case, the energy level shift is exactly the same effect as a magnetic field parallel to the direction of the laser beam would have. The light shift effects are therefore often analysed in terms of virtual magnetic fields. The strength of this virtual magnetic field depends on the detuning of the laser from the electronic transition frequency. In addition to these level shifts, the laser light also causes a damping of the spin polarization. In contrast to the light shift effect, which has a dispersive dependence on the laser detuning, the damping effect has an absorptive behaviour, i.e. it its maximum occurs when the laser frequency is tuned exactly to the optical transition frequency. Light shift and damping are the main contributers to laser-induced dynamics in atomic spin systems [Suter and Mlynek, 1991].

2.4. Observation

The last requirement for optically enhanced magnetic resonance is a method for observing the spin polarization. An early suggestion that magnetic resonance transitions should be observable in optical experiments is due to Bitter [Bitter, 1949]. The physical process used in such experiments may be considered as the complement of optical pumping: the spin angular momentum is transferred to the photons and a polarization selective detection measures the photon angular momentum.

Figure 2.3.

Principle of observation of spin polarization by optical radiation.

Figure 2.3. illustrates this for the same model system that we considered for optical pumping. Light with a given circular polarization interacts only with one of the ground state sublevels. Since the absorption of the medium is directly proportional to the number of atoms that interact with the light, a comparison of the absorption of the medium for the two opposite circular polarizations yields directly the population difference between the two spin states. This population difference is directly proportional to the component of the magnetisation that is parallel to the laser beam. This analysis of the transmitted light allows the observation of ground state spin polarization. Another possibility is the analysis of the fluorescence light: angular momentum conservation imposes correlations between the direction and polarization of the spontaneously emitted photons which depend on the angular momentum state of the excited atom.

Figure 2.4.

Polarization of the fluorescence depends on the spin state of the excited atoms.

As shown schematically in figure 2.4., the angular momentum Jn of the spontaneously emitted photon is determined by the difference between the atomic angular momentum of the excited and ground states. This condition determines the polarization of the emitted radiation for a given direction. In the first experiments on optical pumping, observation of the fluorescence was used not only for measuring the excited state polarization, but also to infer also the ground state polarization [Brossel, Kastler, 1952].

2.5. Angular Momentum Reservoirs

Atoms and molecules may contain different types of angular momentum. The most important reservoirs include rotational motion of molecules, orbital angular momentum of electrons, and spin angular momentum of electrons and nuclei. Not all these types of angular momentum couple directly to the radiation field: in free atoms, only the orbital angular momentum of the electrons is directly coupled to the optical transitions. However, the different types of angular momentum are in general coupled to each other by various interactions which allow the polarization to flow from the photon spin reservoir through the electron orbital to all the other reservoirs, as shown schematically in figure 2.5.

Figure 2.5.

Summary of the most important reservoirs of angular momentum

This is of course of spatial interest for nuclear magnetic resonance, since there is no direct transfer to nuclear spins. However, the coupling between electronic and nuclear angular momentum is usually strong enough to provide an efficient transfer mechanism. This allows even the polarization of nuclear spin systems in diamagnetic ground states.

Figure 2.6.

Polarization of nuclear spin reservoirs in diamagnetic ground states.

In the example of figure 2.6, the electronic ground state is diamagnetic and has a nuclear spin I = 1/2. Since the absorption of light is not affected by the nuclear spin, both spin states interact with a circularly polarized laser beam. Angular momentum conservation requires that only the states with an electronic angular momentum of mJ = 1 are populated by the optical excitation. If electronic and nuclear spin are parallel in the excited state (mj = 1, mI = 1/2, dashed line in figure 2.6), the resulting state does not evolve until it reemits a photon and decays into the state from which it was excited. If, however, the nuclear spin is oriented antiparallel to the electronic angular momentum (mj = 1, mI = -1/2), the hyperfine interaction can induce simultaneous spin flips that conserve the total angular momentum and transfer the atom into the (mj = 0, mI = 1/2) state. Spontaneous decay from this state again leaves the nuclear spin unchanged and thus brings the atom into the mI = +1/2 ground state. The net effect of the absorption - hyperfine - emission cycle is therefore the transfer of an atom from the -1/2 to the +1/2 nuclear spin state. A sequence of such cycles polarises the nuclear spin system in complete analogy to the case of electronic spin polarization.

Spin polarization can be transferred between different reservoirs not only within one atomic species, but also between different particles. This was first demonstrated by Dehmelt who used transfer to free electrons to polarize them [Dehmelt, 1958]. Another frequently used transfer process uses optical pumping of alkali atoms, in particular Rb and Cs and transfer of their spin polarization to noble gas atoms like Xe. This method was pioneered by Happer [Happer, Miron, 1984] and applied to the study of surfaces in systems with high surface to volume ratios like graphitized carbon [Raftery, Long, 1991] or to the construction of NMR gyroscopes [Mehring, Appelt, 1988, Grover, Kanegsberg, ]. The transfer from alkali to noble gas atoms is relatively efficient because they form van der Waals complexes. During the lifetime of this quasi-molecule, the two spins couple, mainly by dipole-dipole interaction. This coupling allows simultaneous spin flips of the two species which transfer polarization from the Rb atoms to the Xe nuclear spin. Typical cross-polarization times are on the order of minutes, but the long lifetime of the Xe polarization permits to reach polarizations close to unity. The spin polarization survives freezing [Cates, Benton, 1990] and can be transferred to other spins by thermal mixing [Bowers, Long, 1993].

At low magnetic fields, the coupling between the reservoirs can exceed the Zeeman interaction between the individual spins and the magnetic field. This implies that the spins do not evolve independently, but as a collective entity that may include electronic as well as nuclear spins. Under these conditions, the traditional distinction between ESR and NMR loses its meaning: in a single experiment, one usually flips electronic and nuclear spins. Nevertheless, it may be possible to extract the physical parameters for the various interactions.

Figure 2.7.

Example of an optically detected magnetic resonance spectrum in a strongly coupled system.

Figure 2.7. shows an example: in Na atoms, the electron (S=1/2) and nuclear spin (I=3/2) are coupled by the hyperfine interaction with a coupling constant of 1.8 GHz. In fields of less than 0.1 T, the hyperfine interaction is therefore significantly stronger than the Zeeman interaction. For the spectrum shown above, the atoms were placed in a field of 0.7 mT. At these field strengths, the two spins remain strongly coupled, but as shown in the spectrum, the electron Zeeman interaction can be determined as 5 MHz and the nuclear Zeeman interaction as 19 kHz.

2.6. Laser Magnetic Resonance

A method for optical detection of magnetic resonance transitions that does not directly rely on the conservation of angular momentum is laser magnetic resonance. It uses transitions between states that differ both in their electronic or vibrational and angular momentum quantum numbers. Transitions between such states depend on magnetic interactions but fall into the optical frequency range. The population difference between the two states is thus close to unity and the detection of the radiation can proceed with a high efficiency.

Figure 2.8.

Principle of laser magnetic resonance.

Figure 2.8. illustrates the principle of the method: a magnetic field lifts the degeneracy of both the ground and excited states. For the figure, only a single spin ë = 1/2 was assumed. If the laser induces transitions that change both the vibrational and the spin quantum number, like the transitions indicated with the arrows in figure 2.8., the resonance frequency depends clearly on the magnetic field strength. The resulting spectra contain thus the information about the magnetic interactions in both the excited and the ground state. Experiments of this type were performed in molecular gases [Sears, Bunker, 1982] as well as in semiconductor materials, where the process is referred to as spin-flip Raman scattering [Slusher, Patel, 1967], [Brueck and Mooradian, 1973].

While this method allows high sensitivity, its resolution is lower than with direct detection. The resolution of the magnetic resonance transitions is limited by the laser linewidth. It can in principle be overcome by optical-rf double resonance methods [Hofmann, Pascher, 1992] or by a modification of the basic Raman experiment which is known as coherent Raman scattering.

2.7. Coherent Raman Processes

Raman processes can be considered as an interaction between two optical photons and a material excitation, as shown schematically in figure 2.9.

Figure 2.9.

Raman processes couple two electromagnetic fields (w1, w2) with a material excitation.

The arrows labelled w1, w2 represent two optical fields that couple to two allowed optical transitions which share the energy level |3>. If two laser fields with these frequencies are incident on the three level system, they excite coherences in all three transitions of the three level system, in particular also the coherence labelled w12 in the transition that is not directly coupled to the laser fields. If, conversely, the coherence in transition |1> « |2> is already present in the material, and a single laser field at frequency w1 is incident on the system, it excites a Raman field at frequency w2 = w1 + w12. This Raman field propagates with the incident laser field and the frequency w12 can be measured as the difference between the two optical frequencies. If the laser frequency drifts, the frequency of the incident field as well as that of the Raman field are shifted by the same amount. As a result, the difference frequency is not affected and the resolution of the measurement is not affected by laser frequency jitter or broad optical resonance lines [Bai and Kachru, 1991]. Coherent Raman processes provide therefore a combination of high resolution with high sensitivity.

In order to implement the coherent Raman scattering, the coherent excitation of the material has to be excited. This can be achieved either with optical fields [Shelby, Tropper, 1983]; Shelby, 1984 #1973; Blasberg, 1994 #2477] or with radio frequency irradiation [Mlynek, Wong, 1983].

2.8. Sensitivity

The increase in sensitivity that can be gained with optical methods in magnetic resonance spectroscopy can be traced to several reasons: The first is the spin polarization that can be achieved. If thermal relaxation establishes the spin polarization, it cannot exceed the Boltzmann factor which is at most of the order of 10^-5. Optically, it is possible to polarize the spins completely and thereby gain some 5 orders of magnitude [Suter, 1992]. In addition, the optical detection process occurs at much higher energies: optical photons have energies some seven order of magnitude higher than that of rf photons. Detecting a small number of optical photons is therefore significantly easier than detection of rf photons. At the same time, thermal noise is almost negligible at optical frequencies. A third reason for the increased sensitivity is the fact that the polarization of the spins can proceed much faster if optical irradiation is used: Depending primarily on the laser intensity, the spin system can be completely polarized in less than 1 µsec [Suter, Rosatzin, 1990].

Since optical detection measures directly the magnetisation, in contrast to pick-up coils which measure its time-derivative, the detection sensitivity is independent of the resonance frequency. It is therefore possible to perform experiments at low or vanishing fields with the same detection efficiency as at high fields. This is of particular interest in cases where small effects like rotational velocities are to be investigated, which cannot be seen in high fields [HŠrle, WŠckerle, 1993].

2.9. Information Content

Apart from the sensitivity advantage, optically enhanced magnetic resonance is sometimes capable of providing information that cannot be found with conventional methods. We illustrate this with the measurement of the sign of the nuclear quadrupole interaction.

The nonspherical part of the charge distribution of atomic nuclei with spin I > 1/2 is a sensitive probe of the electric field at the site of the nucleus. Measurements of the interaction between the nuclear quadrupole moment and the electric field gradient (EFG) tensor can provide information about the electronic and structural environment of the nuclei, as well as about motional processes. Many experiments in magnetic resonance are therefore performed to measure quadrupole couplings [Cohen and Reif, 1957]. However, as is well known [Abragam, 1961], conventional magnetic resonance experiments cannot provide the sign of the coupling.

In the simplest case of axial symmetry, the Hamiltonian HQ of the nuclear quadrupole interaction is given by a coupling constant D times the square of the nuclear spin operator Iz, HQ = D I\s(2,z). The coupling constant D is determined by the size of the nuclear quadrupole moment and the electric field gradient. It can be measured either in the absence of a magnetic field, which corresponds to the case of pure quadrupole coupling, or in a high magnetic field, which corresponds to the case of high-field NMR. In both cases, the spectra are identical for positive and negative sign of the coupling constant D.

Figure 2.10. Principle of the measurement of nuclear quadrupole interaction by laser spectroscopy.

Figure 2.10. shows schematically how laser spectroscopy can be used to measure the nuclear quadrupole interaction with the sign information. In the model system considered here, the spin is 5/2 and a measurement is performed in zero magnetic field. In the electronic ground state, the system has then three sets of doubly degenerate states. If we can neglect the quadrupole splitting in the excited state, as assumed in figure 2.10., the splittings between the levels can be measured directly in the absorption spectrum. Reversal of the sign of the quadrupole splitting leads to a reversal of the line positions in the spectrum.

Figure 2.11.

Comparison of the experimental spectrum (top) with theoretical stick spectra for negative and positive quadrupole coupling.

In actual systems, the inhomogeneous broadening of the optical resonance lines complicates the procedure. Closely analogous measurements are nevertheless possible and have been used to measure quadrupole coupling constant of Pr in the host material YAlO3 [Blasberg and Suter, 1993].

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