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Physica Scripta

Numerical simulation of Raman resonance due to the Ramsey interference induced by thermal motion of atoms

Z. D. Grujić, D. Arsenović, Milan Radonjić, Marina M. Mijailović, Branislav M. Jelenkovic

Published Jul 1, 2009• 0 citations
Publisher

Abstract

We use the optical Bloch equations to calculate spontaneous emission and Zeeman coherences of the probe laser beam in coaxial pump–probe configuration in rubidium (Rb) vapor. To properly model the open Fg=2→Fe=1 transition, the solution of the time-dependent optical Bloch equations was necessary. Also, due to the nature of a room temperature Rb vapor, we performed averaging of the total population of the excited Fe=1 state over different atomic velocities according to the Maxwell–Boltzmann distribution and over different directions of propagation of atoms through pump and probe laser beams. Time-dependent diagonal and nondiagonal density matrix elements for atoms are presented and discussed.

DOI
10.1088/0031-8949/2009/t135/014026

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Acta Physica Polonica A•Oct 1, 2009•0 citations
We use the optical Bloch equations to calculate spontaneous emission and Zeeman coherences of the probe laser beam in coaxial pump–probe configuration in rubidium (Rb) vapor. To properly model the open Fg=2→Fe=1 transition, the solution of the time-dependent optical Bloch equations was necessary. Also, due to the nature of a room temperature Rb vapor, we performed averaging of the total population of the excited Fe=1 state over different atomic velocities according to the Maxwell–Boltzmann distribution and over different directions of propagation of atoms through pump and probe laser beams. Time-dependent diagonal and nondiagonal density matrix elements for atoms are presented and discussed.
We investigate the evolution of the atomic coherence between Zeeman sublevels with $\ensuremath{\Delta}{m}_{F}=2$ in the region between pump and probe laser beams, in a Rb vapor cell using the Hanle configuration. The transmission of the probe beam, placed inside the hollow pump beam, shows the narrowing of the Hanle resonance (at external magnetic field B=0B=0B=0) by Ramsey fringes whose separation from the central peak depends on the distance between the beams. The temporal evolution of the pump induced alignment is evident from the changes of the probe transmission spectra with increasing the angle between linear polarizations of each beam. Experimental results are in agreement with the results of the model which calculates probe fluorescence by solving time-dependent optical Bloch equations, from the time when atom enters the pump beam to the time when it leaves the probe beam. The results for total excited state populations were averaged over atomic velocities distribution.
Counterpropagating, spatially separated hollow pump and coaxial probe laser beams generate narrow Zeeman electromagnetically induced transparency (EIT) resonances in the vacuum Rb cell. The lasers are locked to D2 line transition Fg=2→Fe=1 of Rb87. For the probe laser beam intensity between 0.1 and 3.0 mW/cm2 this Ramsey-type configuration yields dual-structured resonances having a narrow peak on top of a broader pedestal. Linewidths of the narrow peak are nearly independent of the probe laser beam intensity and of the probe diameter (for diameters 0.8 and 2.7 mm), provided that the dark region between the pump and the probe beams is fixed. At the probe laser beam intensities below 0.1 mW/cm2 Zeeman EIT is a single narrow resonance. With this geometry of laser beams, and at low probe intensity, the presence of the pump enables the probe EIT, i.e., the probe transmission becomes enhanced in a narrow spectral window. Accompanying theoretical model showed good quantitative agreement with the measurements.
Experimental and theoretical investigations show the strong effect of the pump beam, spatially separated from the probe beam, on the probe's electromagnetically induced absorption (EIA) and nonlinear magneto-optical rotation (NMOR). Linearly polarized pump and probe laser beams are locked to the Fg = 2 → Fe = 3 transition of the 87Rb D2 line and pass a vacuum Rb gas cell coaxially. We show that the observed narrowing of EIA and NMOR resonances is due to the Ramsey effect. Linewidths of the resonances decrease when the size of the dark region between pump and probe lasers increases. Variation of the angle between pump and probe linear polarizations strongly influences the phases of atomic coherences generated by the pump beam and consequently the line-shapes of the probe EIA and NMOR resonances. Complete change of the resonance sign is possible if the phases of the ground state coherences, Δmg = 2, are altered by π. The central EIA fringe becomes less pronounced if the probe intensity increases, due to the larger probe contribution to atomic evolution. Ramsey-like interference is a manifestation of the evolution of ground state Zeeman coherences, required for EIA, in the dark region in the presence of a small magnetic field.
We calculated line-shapes of the coherent population trapping (CPT) for the open transition in 87Rb by solving time-dependent optical Bloch equations and averaging results for the total excited state populations over atomic trajectories and velocities, while taking into account incident angles to the laser beam. Comparison with experiment produces good agreement for the laser intensity dependence of the CPT line-widths when the laser transverse beam profile is Gaussian. This theoretical model was used to calculate the probe laser CPT in the case of spatially separated pump beam. The probe beam is placed inside the cylindrically-shaped pump beam. The calculations as well as measurements show narrowing of the Hanle resonance due to Ramsey effect.
We follow theoretically the motion of the sodium atoms in vapor state under the influence of a laser mode in (1 + 1) D, which is achieved by introducing different optical filters. In the Dirac interaction representation, the equations of motion are represented via the Bloch form together with the Pauli operators to find the elements of the density matrix of the system. The emergence of the principle of coherence in varying the angles of the laser mode permits the evaluation of the average force affecting the atoms' acceleration or deceleration, and hence the corresponding velocities and temperatures are investigated. The atomic vapor is introduced in a region occupied by a heat bath presented by the laser field, such that the state of the atomic vapor is unstable inside the system due to the loss or gain of its kinetic energy to or from the laser field. This instability is studied by finding the eigenvalues of the system's entropy. Resorting to the assumption of Botin, Kazantsev, and Pusep, who issued in the presence of the weak and strong spontaneous emission, a coupling between the mean numbers of photons in terms of time, which allows the evaluation of the rate of entropy production of the system under study. No singularities are found throughout the process of equations solving and other calculations. Resorting to symbolic software, a set of figures illustrating the nonlinear behavior in the dynamics of the problem is present. In this paper, we introduce a theoretical study of the effect of two-counter propagation traveling plane waves on the motion of the sodium atoms in the vapor state by varying the coherence angles to investigate the atomic behavior. Good agreements are found with previous studies.
We study phenomenon of electromagnetically induced absorption in the Hanle configuration by solving time-dependent optical Bloch equations for the case of the closed multilevel F g = 1 → F e = 2 transition.Our model gives optical Bloch equations as a non-homogeneous system of ordinary linear differential equations.For weak laser fields (Ω Γ i.e.Rabi frequency small compared to spontaneous emission rate), a perturbative method to solve linear differential equations can be applied.Perturbative method is realized by solving (in the time-domain) higher-order corrections to the density matrix which in the sum converge to the exact solution of optical Bloch equations.By its form, each successive correction is also system of ordinary linear differential equations which depends on the solution of previous ones.Corrections are partitioned such that odd give corrections to optical coherences, while even give corrections to populations and Zeeman coherences.We present numerical results for the behavior of density matrix elements with successive corrections, and compare them with exact solution of optical Bloch equations.Electromagnetically induced absorption is observed as a 4th and higher (even) correction to populations, when behavior in respect to both time and magnetic field is viewed.Since in our method each correction depends on the solution of previous ones, we can analyze how (through mechanism of transfer of coherences and transfer of populations between Zeeman sublevels) electromagnetically induced absorption is formed.We also discuss qualitative differences in the behavior (with respect to time) of certain density matrix elements for magnetic fields "inside" and "outside" electromagnetically induced absorption resonance.