Elsevier

Optics Communications

Volume 199, Issues 1–4, 15 November 2001, Pages 127-142
Optics Communications

Coherent control of magneto-optical rotation in inhomogeneously broadened medium

https://doi.org/10.1016/S0030-4018(01)01534-6Get rights and content

Abstract

We extend our earlier investigations [Opt. Commun. 179 (2000) 97] on the enhancement of magneto-optical rotation (MOR) to include inhomogeneous broadening. We introduce a control field that counter-propagates with respect to the probe field. We derive analytical results for the susceptibilities corresponding to the two circular polarization components of the probe field. From the analytical results we identify and numerically demonstrate the region of parameters where significantly large MOR can be obtained. From the numerical results we isolate the significance of the magnetic field and the control field in enhancement of MOR. The control field opens up many new regions of the frequencies of the probe where large MOR occurs. We also report that a large enhancement of MOR can be obtained by operating the probe and control field in two-photon resonance condition.

Introduction

A magnetic field, when applied to an initially isotropic medium containing gaseous atoms having m-degenerate sublevels, can cause birefringence in the medium. Because, the applied magnetic field creates asymmetry between the susceptibilities χ± of the medium corresponding to the two circularly polarized components σ of the probe field. That results in magneto-optical rotation (MOR), i.e. the plane of polarization of a weak probe field is rotated when it passes through the medium. For a small absorption the rotation angle θ is given byθ=πkpl(χ−χ+),where kp corresponds to propagation vector of the probe and l is length of the cell along kp. Further we note that, χ± depend on the atomic density and the oscillator strength of the atomic transition.

Traditionally MOR was used as a tool in polarization spectroscopy using continuum sources [1]. The interest in MOR was intensified in the atomic and molecular physics with the availability of intense light sources of definite polarization [2] and frequency [3]. Several reviews exist in the literature on this subject including several interesting applications (e.g., see Ref. [4]). Using saturating fields the non-linear MOR has also been studied at length [5], [6], [7], [8]. MOR with a transverse magnetic fields [9] (known as Voigt effect) and with inclined magnetic fields [10] have also been studied. Recently large MOR has been reported in dense and cold atomic cloud of rubidium [11]. On the other hand, laser field alone can also break the symmetry in the response of an atomic gas to different polarization components of a probe field. For example, let us consider j=0↔j=1 transitions of an atomic gas containing V systems. When a linearly polarized weak probe field passes through the medium, the σ± components of the probe field couple the |j=0,m=0〉 with the degenerate states |j=1,m=∓1〉. The susceptibilities χ± of the medium to these two components σ are same; i.e., response of the medium is symmetric to both the components. However when a σ polarized strong field is applied on the |j=0,m=0〉↔|j=1,m=1〉 transition, the susceptibility χ+ is modified by the control field parameters creating asymmetry between χ+ and χ. Thus the plane of polarization of the probe field rotates (due to Eq. (1)). Note that, this rotation is solely due to the laser field and is a function of control field parameters. Resonant birefringence due to optically induced level shift by a coherent source was observed [12]. The light-induced polarization rotation in optical pumping experiments with incoherent light has also been extensively studied [13]. Liao and Bjorklund [14] were the first to observe polarization rotation in a three-level system by resonant enhancement of two-photon dispersion in the 3s2 S1/2↔5s2 S1/2 of sodium vapor. Hänch and coworkers [15] have used this polarization rotation as a high resolution spectroscopic technique. Heller et al. [16] extended this idea to atomic systems involving the ionization continuum. Experimental and theoretical work has been reported by Ståhlberg et al. [17] on laser induced dispersion in a three-level cascade system of Ne discharge.

Recently, combining the ideas of enhancement of refractive index using atomic coherence [18] and the non-linear MOR, Scully and his coworkers have investigated a possible application to high-precision optical magnetometry [19], [20]. They have demonstrated this possibility both theoretically [19] and experimentally [20], considering the rotation of polarization of a strong linearly polarized probe caused by an optically thick cell containing 87Rb vapor. The maximum sensitivity reported in their experiment is ∼6×10−12 G/(Hz)1/2, which is superior to other existing high precision magnetometers. Budker et al. at Berkeley have also reported high sensitive optical magnetometry in a series of papers [21], based on the non-linear MOR involving ultra-narrow resonances (≃2π×1.3Hz) using special cell with high quality anti-relaxation paraffin coating that enables the atomic coherence to survive even after a large number of collisions with the wall. Using a similar configuration, Budker et al. have shown reduction of the group velocity of light to ≃8 m/s in a non-linear magneto-optical system [22]. Further, Pavone et al. [23] introduced the idea of coherent control to obtain significant atomic birefringence in presence of electromagnetically induced transparency (EIT) [24]. Wielandy and Gaeta [25] used quantum coherence to control the polarization state of a probe field. They reported a large birefringence and hence a large polarization rotation in a three-level cascade of 85Rb (see also Refs. [26], [27]). Using a similar configuration, Fortson and coworkers [28] have showed a possible utility of the polarization rotation at EIT to measure the atomic parity non-conservation signal with a better efficiency. A detailed discussion on the role of degenerate sublevels and effect of the polarized fields on EIT has been discussed in Ref. [29].

However, it is interesting to investigate the combined effects of the laser field and the magnetic field in the context of coherent control of the rotation of polarization. In our earlier work [26], we have reported laser field induced enhancement of MOR in cold atoms. In the present paper we generalize the above work [26] by including the thermal motions of the atoms inside the cell (see Fig. 1). Here a large broadening is introduced in the rotation signal. This could be desirable to get large rotations for a broad range of probe frequencies in presence of a control field. But on the other hand, broadening reduces the magnitude of rotation considerably. However, one can work with a denser medium when Doppler effect is included in the calculation. Moreover, we have included all spontaneous decay events involved in the j=0→j=1→j=0 transitions of the system (unlike in Ref. [26]). Further we discuss a special case when the weak probe field and strong control field are in two-photon resonance with |e〉↔|g〉 transition, that gives rise to large enhancement of MOR.

The organization of the paper is the following. In Section 2, we describe the model scheme and determine the susceptibility of a moving atom using density matrix formalism. In Section 3, we present the analytical results for the susceptibilities of the Doppler broadened medium. In Section 4, we give a measure of rotation of plane of polarization. In Section 5, we show how one identifies the regions of interest by suitably choosing the control field parameters. In Section 6, we present numerical results that substantiates the analytical results. We show that indeed large MOR could be obtained due to the control field. We analyze different probe frequency regions to understand the contributions of electric and magnetic field to the large polarization rotation. In Section 7, we discuss a special case where the counter-propagating probe and control field are in two-photon resonance with the |e〉↔|g〉 transition (see Fig. 2). We show both analytically and numerically that this configuration can be advantageous for enhancement of MOR. We conclude with a summary of the results in Section 8.

Section snippets

The model and the susceptibilities

The MOR consists of the propagation of linearly polarized light Ep tuned close to the transition jj in presence of a magnetic field B. The susceptibilities χ± for the two circularly polarized components of the probe beam would be different as B≠0. We can now consider coherent control of MOR in a configuration as depicted in Fig. 1 with a control field Ec which can be tuned close to another transition say jj′′. The atoms move randomly inside the cell with velocity v. The probe field Ep

Susceptibilities χ± of the Doppler broadened medium

Next we calculate the χ± of a Doppler broadened medium. Here, as mentioned in Section 2, one needs to average s± over the atomic velocity distribution σ(vz) inside the cell to obtain the response of the medium〈s±〉=∫−∞s±(vz)σ(vz)dvz.It is assumed that at thermal equilibrium, the atoms inside the cell follow Maxwell–Boltzmann velocity distributionσ(vz)=(2πKBT/M)−1/2exp(−Mvz2/2KBT),where mass of the moving atom is M, temperature of the cell T and KB is Boltzmann constant. For convenience,

Measure of rotation

Using the 〈s±〉 obtained above, the rotation of polarization θ of the probe can be determined from Eq. (1) which, however, is valid only if the absorption of the medium is very small. Since we consider the resonant or near-resonant MOR, one also needs to take into account the large absorption associated with the large dispersions near resonance. Absorption contributes to the polarization rotation via dichroism (rotation solely due to Im〈s±〉) but large absorption attenuates the MOR signal at the

Condition for enhancement of MOR

In this section we identify the regions of our interest. We determine the criteria to choose the control field parameters to efficiently control and hence enhance the MOR. From Eq. (34), one observes the following:

  • (i) When 〈s+〉≈〈s〉, Ty→0.

  • (ii) When Re〈s+〉≃Re〈s〉 but Im〈s+〉≠Im〈s〉, Ty reduces toTy14expαl2Im〈s+expαl2Im〈s2.


If both (αl/2)Im〈s±〉 are large, Ty→0. However if (αl/2)Im〈s+〉 is large but (αl/2)Im〈s〉 is small (or vice versa), we obtainTyexp(αlIm〈s〉)414which is the rotation due

Numerical results on coherent control of MOR

Based on the above observations, we present some interesting numerical results for different parameters to demonstrate the large enhancement of MOR. We define the MOR signal enhancement factorη=TyG1≠0TyG1=0.For a given δ, η represents the enhancement (ifη>1) or suppression (ifη<1) of MOR signal by a control field, when compared to the MOR without control field. We use the notation 〈s±0〉 to represent susceptibilities corresponding to σ components of the probe when control field is absent and 〈s+

MOR in two-photon resonance condition

In this section we consider the enhancement of MOR when the σ+ polarized control field and the probe field are always on two-photon resonance with |e〉↔|g〉 transition (Δ+δ=0). In the following discussion, we consider both the cases of stationary atom and homogeneously broadened atom with the above condition.

Summary

In summary, we have shown how a control field can be used to control birefringence and hence enhance MOR in a Doppler broadened medium. We have shown how control laser can modify the susceptibilities and hence result significantly large MOR in frequency regions, where MOR otherwise is small. The key to large enhancement of MOR consists of utilizing the large asymmetry in the susceptibilities caused by the Autler–Townes splitting. We have derived conditions to select frequency regions where one

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    Present address: Department of Applied Physics and Chemistry, University of Electro-Communications, Chofu, Tokyo 182-8585, Japan.

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