We present a theoretical study of double-resonance alignment magnetometers using linearly polarized light, in which the effect of atomic high-order multipole moments is considered. Starting from the effective master equation of our system obtained by eliminating the excited state adiabatically, we derive the full evolution equations of the atomic multipole moments. The analytic solutions of resonance signals involving the four-order multipole moments effect are obtained by using the perturbation approach. We present that the four-order multipole moments effect is negligible in the weak laser field, and the results reduce to that obtained by a three-step approach. However, the role of four-order multipole moments coupled by two-order tensor moments is more significant with the increasing Rabi frequency of light, which cannot be ignored. Meanwhile, the analytic expressions of relaxation processes are also studied, which are a linear combination of the laser-induced equivalent relaxation rate Γ L and the spin-exchange collision rate Γ g . The expected domain of validity of the three-step approach on light power is roughly given by Γ L < 1 2 Γ g . In addition, the steady-state results of resonance signals are presented in a strong radio-frequency magnetic field; in that case, the physical mechanism of the splitting of resonance signals is discussed. These results are valid for arbitrary light power and for an arbitrarily oriented static magnetic field.
We present a theoretical study of double-resonance alignment magnetometers using linearly polarized light, in which the effect of atomic high-order multipole moments is considered. Starting from the effective master equation of our system obtained by eliminating the excited state adiabatically, we derive the full evolution equations of the atomic multipole moments. The analytic solutions of resonance signals involving the four-order multipole moments effect are obtained by using the perturbation approach. We present that the four-order multipole moments effect is negligible in the weak laser field, and the results reduce to that obtained by a three-step approach. However, the role of four-order multipole moments coupled by two-order tensor moments is more significant with the increasing Rabi frequency of light, which cannot be ignored. Meanwhile, the analytic expressions of relaxation processes are also studied, which are a linear combination of the laser-induced equivalent relaxation rate Γ L and the spin-exchange collision rate Γ g . The expected domain of validity of the three-step approach on light power is roughly given by Γ L < 1 2 Γ g . In addition, the steady-state results of resonance signals are presented in a strong radio-frequency magnetic field; in that case, the physical mechanism of the splitting of resonance signals is discussed. These results are valid for arbitrary light power and for an arbitrarily oriented static magnetic field.
Abstract We present a theoretical study of the optical-radio-frequency (rf) double resonance on hyperfine ground states of alkali-metal atoms in the geophysical magnetic-field range, in which resonance circularly polarized laser light is used in the optical pumping and detection processes. The analytical expressions of the rf resonance signals corresponding to two ground states are obtained based on atomic multipole moments, where the alignment effect is considered and that is responsible for the resonance shape and linewidth especially when the optical power is relatively strong. In addition, we also obtain the analytical expressions of the linewidths of two separated resonance peaks and one presenting the competition process of the optical pumping, rf field and spin-exchange collision. Two different types of light narrowing phenomena are investigated by comparing the longitudinal relaxation rates of atomic multipole moments for two ground states. Applied to rubidium and cesium atoms, we show a particularly close agreement of our analytical results with more elaborate calculations using density-matrix theory. Our theoretical model is relevant for optimizing the sensitivity of magnetometers.
We theoretically investigate the magneto-optical rotation (MOR) effect in cesium atoms in the Voigt geometry, in which an off-resonance linearly polarized laser beam serves as both pump and probe. By calculating the detailed evolution of atomic multipole moments truncated to second-rank, alignment-to-orientation conversion (AOC) effects are observed in two hyperfine ground states. The mechanisms responsible for this effect are demonstrated. The tensor AC-Stark shift produced by the optical pumping generates a nonlinear effect, resulting in atomic alignment directly coupled to orientation, which enables spin orientation to be obtained. Simultaneously, spin-exchange collisions lead to atomic alignment and orientation transfer between two ground-state manifolds. Additionally, we present the analytical expression of atomic spin polarization described by atomic multipole moments, and the contributions of the AOC effect to the optical-rotation signals are discussed in different light power regimes. Our results can be helpful for guiding MOR experiments by refining and optimizing the parameters.
As is well known a linearly polarized resonant laser will cause atoms to generate a magnetic tensor moment (MTM) by polarizing them. When there exists an external magnetic field, it is possible that the moment will precess around the field. In the presence of a radio frequency (RF) exciting source, we investigate theoretically the dependence of time-independent (direct current, DC), the first and second harmonic signal of the MTM precession on magnetic vector field, and obtain its analytical solution by solving the Liouville equation. The results show that the interference of both harmonic components will result in the precession spectrum evidently varying. A detailed explanation is described in the following. For the DC signal, Rabi frequency Ω of 1/(2√2) is a spectral splitting threshold. When it is greater than the threshold, the interference will cause single resonant absorption dip characterized usually to split into two dips, which has not been reported before to the best of our knowledge, and the separation between both the dips may be expressed as √3√Ω2+Ω4 -Ω2-1. For the first harmonic signal including symmetric and antisymmetric component, an interference fringe will appear near the center of antisymmetric part when Ω >1/(2√2), simultaneously its symmetric part behaves like the above dc component, such as splitting threshold and separation between both dips. With regard to the second harmonic signal, it is found that the interference can also lead to the width of the second harmonic decreasing to 38% compared with the case of the first harmonic signal. At the optimum RF Rabi frequency, on the assumption that noise spectral density is constant, it is theoretically shown that the most sensitive magnetometer, realized by the DC component or the first or second harmonic signal of the precession, depends only on the angle between the light polarization and the measured magnetic field.In fact, we are able to obtain the modules of the measured magnetic vector by RF resonant frequency. The angle between the magnetic field and the laser polarization is determined just by the ratio of the intensity of the DC component to the intensity of the second harmonic signal and the ratio between the intensities of the symmetric parts of two harmonic signals in resonance, and another orientation angle between the measured field projection at the plane perpendicular to the light polarization and the direction of RF source depends on the phase difference between the antisymmetric components of both harmonic signals. Consequently, we demonstrate a vectorial atomic magnetometer that is realized by using the RF source and the linearly polarized resonant laser without rotating laser polarization. This kind of atomic magnetometer with simple sensor structure is easy to integrate vector magnetometer array which will be suitable for solving the inverse problem and geomagnetic navigation.
In recent years diode laser sources have become widespread and reliable tools in magneto-optical spectroscopy. In particular, laser-driven atomic magnetometers have found a wide range of practical applications. More recently, so-called magnetically silent variants of atomic magnetometers have been developed. While in conventional magnetometers the magnetic resonance transitions between atomic sublevels are phase-coherently driven by a weak oscillating magnetic field, silent magnetometers use schemes in which either the frequency or the amplitude of the light beam is modulated. Here we present a theoretical model that yields algebraic expressions for the parameters of the multiple resonances that occur when either amplitude-, frequency-, or polarization-modulated light of circular polarization is used to drive the magnetic resonance transition in a transverse magnetic field. The relative magnitudes of the resonances that are observed in the transmitted light intensity at harmonic m of the Larmor frequency ${\ensuremath{\omega}}_{L}$ (either by DC or phase sensitive detection at harmonics q of the modulation frequency ${\ensuremath{\omega}}_{\mathrm{mod}}$) of the transmitted light are expressed in terms of the Fourier coefficients of the modulation function. Our approach is based on an atomic multipole moment representation that is valid for spin-oriented atomic states with arbitrary angular momentum F in the low light power limit. We find excellent quantitative agreement with an experimental case study using (square-wave) amplitude-modulated light.
We present an experimental study of the intrinsic magnetometric sensitivity of an optical or rf-frequency double-resonance magnetometer in which linearly polarized laser light is used in the optical pumping and detection processes. We show that a semiempirical model of the magnetometer can be used to describe the magnetic resonance spectra. Then, we present an efficient method to predict the optimum operating point of the magnetometer, i.e., the light power and rf Rabi frequency providing maximum magnetometric sensitivity. Finally, we apply the method to investigate the evolution of the optimum operating point with temperature. The method is very efficient to determine relaxation rates and thus allowed us to determine the three collisional disalignment cross sections for the components of the alignment tensor. Both first and second harmonic signals from the magnetometer are considered and compared.
Alkali-metal-vapor magnetometers, using coherent precession of polarized atomic spins for magnetic-field measurement, have become one of the most sensitive magnetic-field detectors. Their application areas range from practical uses such as NMR signal detection to fundamental physics research such as searches for permanent electric dipole moments. One of the main noise sources of atomic magnetometers comes from the light shift that depends on the frequency of the pump laser. In this work, we theoretically study the light shift, taking into account the relaxation due to the optical pumping and the collision between alkali-metal atoms and between alkali-metal atoms and the buffer gas. Starting from a full master equation containing both the ground and excited states, we adiabatically eliminate the excited states and obtain an effective master equation in the ground-state subspace that shows an intuitive picture and dramatically accelerates the numerical simulation. Solving this effective master equation, we find that in the light-narrowing regime, where the linewidth is reduced while the coherent precession signal is enhanced, the frequency dependence of the light shift is largely reduced, which agrees with experimental observations in cesium magnetometers. Since this effective master equation is general and is easily solved, it can be applied to an extensive parameter regime, and also to study other physical problems in alkali-metal-vapor magnetometers, such as heading errors.