Selective Addressing of High-Rank Atomic Polarization Moments

Abstract

We describe a method of selective generation and study of polarization moments of up to the highest rank κ = 2F possible for a quantum state with total angular momentum F . The technique is based on nonlinear magneto-optical rotation with frequency-modulated light. Various polarization moments are distinguished by the periodicity of light-polarization rotation induced by the atoms during Larmor precession and exhibit distinct light-intensity and frequency dependences. We apply the method to study polarization moments of 87 Rb atoms contained in a vapor cell with antirelaxation coating. Distinct ultra-narrow (1-Hz wide) resonances, corresponding to different multipoles, appear in the magnetic-field dependence of the optical rotation. The use of the highest-multipole resonances has important applications in quantum and nonlinear optics and in magnetometry.PACS numbers: PACS 32.80. Bx,95.75.Hi High-rank polarization moments (PM) and associated high-order coherences have recently drawn attention (see [1,2,3,4,5,6,7,8] While signatures of high-order PM were detected in several experiments [3,4,5,6,8], the methods used in these investigations are not sufficiently selective and/or do not allow real-time manipulation of particular multipoles. Here we describe a method, based on nonlinear optical rotation with frequency-modulated light (FM NMOR) [16], by which one can selectively induce, control, and study any possible multipole moment. Applying the method to 87 Rb atoms in a paraffin-coated cell [17,18], we have verified the expected power and spectral dependences of the resonant signals and obtained a quantitative comparison of relaxation rates for the even-rank moments.The density matrix in the M, M ′ representation for a state with total angular momentum F can be decomposed into PM of rank κ = 0 . . . 2F , uncoupled under rotations, with components q = −κ . . . κ: with surfaces for which the distance to the origin in a given direction is proportional to the probability of finding the projection M = F along this direction. (a): "pure" quadrupole κ = 2, q = 0; (b): κ = 4, q = 0 hexadecapole; (c): same as in (b), but rotated by π/2 around the x-axis; (d): the average of (b) and (c), which has a 4-fold symmetry with respect to rotations aroundx. In all cases, the minimum necessary amount of ρ (0) 0 was added to ensure that all sublevel populations are non-negative [20]. Probability surfaces (a) and (d) rotating aroundx-directed magnetic field with the Larmor frequency correspond to the polarization states produced in this experiment.is uniquely associated with the highest PM for a given state, e.g., the quadrupole moment (κ = 2) for F = 1, or the hexadecapole (κ = 4) for F = 2. The method introduced here exploits the different axial symmetries of the PM (2-fold and 4-fold for the quadrupole and hexadecapole, respectively; Fig. 1) to selectively create and detect them (see also [3,4,5,6]).While multipole moments of rank κ ≤ 2 can be easily generated and detected with weak light (since a photon has spin one), higher-ra...

Optical magnetometry

“…Several approaches have been proposed to alleviate the adverse effects of the nonlinear Zeeman effect in all-optical magnetometers, including synchronous optical pumping at the quantum revival frequency given by the quadratic correction to the Zeeman energies (Ref. [42] and references therein), and selective excitation and detection of coherences (that correspond to high-order polarization multipoles) between "stretched" Zeeman sublevels unaffected by nonlinear Zeeman effect [43,44,45].…”

Nonlinear magneto-optical rotation with frequency-modulated light in the geophysical field range

“…Consequently, the optical properties of the medium are modulated at the quantum-beat frequency, leading to modulation of the angle of the light polarization. If the time-dependent optical rotation is measured at the first harmonic of the modulation frequency Ω m , ultra-narrow resonances are observed at near-zero magnetic fields, and at fields where the modulation frequency Ω m is a subharmonic of one of the quantum-beat frequencies of the system [18,23,31]. It should be noted that this technique yields a scalar magnetometer: the precession frequency is dependent only on the magnitude of the magnetic field and not its direction.…”

“…Such signals are dominated by the precession of the quadrupole moment (atomic alignment) associated with ∆M F = 2 coherences, see Refs. [24,31,32]. Higher order multipole moments were studied in Refs.…”

“…Higher order multipole moments were studied in Refs. [31] and [33]. Figure 2 shows a schematic of the experimental apparatus.…”

Light-induced polarization effects in atoms with partially resolved hyperfine structure and applications to absorption, fluorescence, and nonlinear magneto-optical rotation

“…An atomic state with total angular momentum F can support multipole moments with rank up to κ = 2F [14,15]; multi-photon interactions and multipole transitions higher than dipole allow the higher-order moments to be created and detected. Magneto-optical techniques can be used to selectively address individual high-rank multipoles [17,18]. Recently, the possibility of using the κ = 4 hexadecapole moment to improve the characteristics of atomic magnetometers was studied (see Ref.…”