Magnetic field measurements in Rb vapor by splitting Hanle resonances under the presence of a perpendicular scanning magnetic field

Eur. Phys. J. D Magnetic field measurements in Rb vapor by splitting Hanle resonances under the presence of a perpendicular scanning magnetic field Raghwinder Singh Grewal 0 1 Murari Pattabiraman 0 0 Department of Physics, Indian Institute of Technology Madras , Chennai 600036 , India 1 Current address: Institute of Physics, Jagiellonian University , Krakow , Poland We experimentally and computationally study the effect of an additional transverse magnetic field (TMF) on the Hanle resonance for a Fg = 2 → Fe = 3 transition of 87Rb D2 line for magnetic field scans perpendicular to the propagation direction of the optical field. It is shown that with a π-polarized light, no resonance signal is observed in absence of the TMF. When the TMF is applied, two peaks are observed on either side of zero scanning magnetic field in the transmission spectrum. The separation between the two peaks is linearly proportional to magnitude of the TMF, which can be used for magnetometry. We applied this technique to measure magnetic field from 0.1 to 0.5 G with a pure Rb vapor cell and from 10 to 30 mG with a Rb cell containing buffer gas. These observations are attributed to the population redistribution among the ground state sublevels due to the TMF. 1 Introduction function of the scanning magnetic field. As a result, no coherence develops among the ground state sublevels. In the above configuration, when an additional magnetic field is applied perpendicular to the scanning magnetic field, it redistributes the population and creates the coherence among the ground state sublevels. As a result, two dips are observed in absorption spectrum on either side of the zero scanning magnetic field. The separation between the two dips in the Hanle absorption spectrum is proportional to the magnitude of the additional transverse magnetic field (TMF), and can be used to measure the magnitude of the magnetic field. Fig. 1. (a) Scan configuration showing the directions of scanning magnetic field (Bscan), light wavevector (ky), light polarization (Ez) and transverse magnetic field (BTMF) and (b) atomic level configuration for the transition Fg = 2 → Fe = 3 of 87Rb D2 line. The transition probabilities are shown next to the solid arrows [17]. where Bx and By is the magnitude of the TMF along x and y axis, respectively. The total Hamiltonian H of the system is given by the sum of the unperturbed Hamiltonian H0, the atom-light interaction Hamiltonian HI , and the atom-magnetic field interaction Hamiltonian HB , with H0 = HI = where E is the electric field vector, dej gk = Fe, me| er |Fg, mg is the dipole matrix element that couples the sublevels of ground state |Fg, mg and excited state |Fe, me , and H.c. is the Hermitian conjugate of the first term in equation (3). Using the Wigner-Eckart theorem [18,19], the dipole matrix element dej gk can be written as: dej gk = Fe er Fg = Fe er Fg (−1)Fe+1−mg −me q mg q = 0, ±1, where the angled brackets denote the reduced matrix element and the term in parenthesis is a 3-j symbol. The total magnetic field-atom interaction Hamiltonian HB is sum of the scanning magnetic field-atom interaction Hamiltonian HBscan , and the additional TMF-atom interaction Hamiltonian HBTMF , HB = HBscan + HBTMF , HBTMF = where μB is the Bohr magneton, and g the gyromagnetic ± ratio. The Clebsch Gordan coefficient (CFama ), with a = (e, g), is given by: (Fa ∓ ma)(Fa ± ma + 1). The scanning magnetic field Bscan splits the Zeeman sublevels of Fg = 2 and Fe = 3 (see Fig. 1b) and this splitting is proportional to Bscan (see Eq. (6)). Since the TMF is perpendicular to the axis of quantization, it couples the Zeeman sublevels of ΔmF = ± 1 (see Eq. (7)) and causes population redistribution in the ground state (Fg = 2) and excited state (Fe = 3) sublevels. This is indicated by the curved arrows in Figure 1b. The time evolution of the density matrix ρ is given by the Liouville equation [20]: where H¯ is the total Hamiltonian of the system after making rotating wave approximation. On the right side of equation (8), the first and second terms represent the commutation and anticommutation operations, respectively. R is the diagonal matrix which represents the spontaneous decay rate, Γ of the excited state and the ground-state collisional decay rate, γ. The matrices Λγ and ΛΓ represent the repopulation of the ground state due to the relaxation terms γ and Γ , respectively. The Bloch equations were obtained after numerically solving equation (8) under steady state conditions [21,22]. The Hanle absorption is given by: where ωFe→Fg is the frequency difference between the ground and the excited states in the absence of the magnetic field and N is the density of atoms. 3 Computational results The calculated absorption spectra of the pump-probe beam for a closed Fg = 2 → Fe = 3 transition of 87Rb D2 line in the absence and presence of the TMF is shown in Figure 2. ) s sau 0.5 G ( )p0.4 e s ( 0.3 n o i t ra 0.2 a p e S 0.1 BTMF= (...truncated)