Nonlinear Faraday Rotation and Superposition-State Detection in Cold Atoms Adam Wojciechowski, 1, 2 Eric Corsini, 3, 2 Jerzy Zachorowski, 1,2 and Wojciech Gawlik 1, 2 $^{1}$Institute of Physics, Jagiellonian University, Reymonta 4, PL-30-059 Kraków, Poland $^{2}$Joint Krakow-Berkeley Atomic Physics and Photonics Laboratory, Reymonta 4, PL-30-059 Kraków, Poland $^{3}$Department of Physics, University of California, Berkeley, CA 94720-7300, USA (Dated: October 29, 2018) We report on the first observation of nonlinear Faraday rotation with cold atoms at a temperature of ∼ 100 µ K. The observed nonlinear rotation of the light polarization plane is up to 0.1 rad over the 1 mm size atomic cloud in approximately 10 mG magnetic field. The nonlinearity of rotation results from long-lived coherence of ground-state Zeeman sublevels created by a near-resonant light. The method allows for creation, detection and control of atomic superposition states. It also allows applications for precision magnetometry with high spatial and temporal resolution. PACS numbers: 33.57.+c, 42.50.Dv, 42.50.Gy, 32.80.Xx The linear Faraday rotation (LFR) of the polarization plane of light propagating in the medium is a well known consequence of optical anisotropy caused by a longitudi- nal magnetic field. For thermal gases the Doppler effect broadens the range of the magnetic fields where the effect is visible and reduces the size of the maximum rotation relative to atoms at rest. The use of cold atoms with their Doppler width narrower than the natural linewidth dis- tinguishes this situation from experiments at room tem- perature. The experiments on LFR with cold atoms were performed in a magneto-optical trap (MOT) [1–3], and in an optical dipole trap [4]. Application of strong, near-resonant laser light may result in the creation of coherent superpositions of Zee- man sublevels of an atomic ground state. Such su- perpositions (Zeeman coherences) are known to be re- sponsible for a variety of coherent phenomena in light- matter interaction, like coherent population trapping [5], electromagnetically-induced transparency [6], nonlinear magneto-optical rotation or nonlinear Faraday rotation (NFR) [7] and their interplay [8]. Superposition states are also at the heart of quantum-state engineering (QSE). Most of QSE experiments require initial states of well de- fined atomic spin (or total angular momentum F ), usu- ally prepared in a stretched state, which is realized by putting most of (ideally all) atomic population into a Zee- man sublevel with extreme value of magnetic quantum number m [9]. Below we report how superpositions of specific Zeeman sublevels, or Zeeman coherences belong- ing to a given F are created in cold ( ∼ 100 µ K) atomic samples and observed with high sensitivity using non- linear Faraday rotation. In the experiment laser light both creates and detects the Zeeman coherences. The same detection technique can be applied to detect the presence of Zeeman coherences already introduced with other mechanisms. Furthermore, the time-dependent de- tection provides information on the temporal evolution of the superposition states. The described experiment shows the potential of NFR with cold atoms for precision magnetometry with The main difficulty in observation of NFR with cold For resonant excitation, rotation angle θ is a mea- sure of circular birefringence, θ ∝ ( n + − n − ) , where n ± are the refractive indices for σ ± polarized light and n ± − 1 ∝ E − 1 ∑$_{eg}$ Re ( d$_{eg}$ρ$_{eg}$ ) with E being the light electric field amplitude, d$_{eg}$ the dipole moment, and ρ$_{eg}$ the density matrix element. The summation goes over all ground- and excited-state sublevels g and e linked by the allowed transitions, as shown in Fig. 1b. In the stationary regime, ρ$_{eg}$ can be expressed as ρ$_{eg}$ = ∑$_{e}$ ′ g ′ (Ω$_{eg}$ ′ ρ$_{g}$ ′ g − ρ$_{ee}$ ′ Ω$_{e}$ ′ $_{g}$) / ( δ$_{eg}$ − i Γ / 2) , where δ$_{αβ}$ and Ω$_{αβ}$ denote respectively the light detuning and Rabi fre- quency for the α ↔ β transition, and Γ / 2 is the relax- ation rate of the optical coherence. This relation indi- cates that optical coherences, and consequently also the refractive indices and rotation angle, depend on the den- sity matrix elements ρ$_{g}$ ′ g and ρ$_{ee}$ ′ which represent pop- ulations of and coherences between Zeeman sublevels of the ground and excited states. For not-too-strong light, the excited-state coherences are negligible and all cou- plings shown in Fig. 1b form independent generic Λ - systems which involve coherences between ground-state sublevels with Δ m = ± 2 . prospective µ G sensitivity, large dynamic range (zero- field to several G), and sub-mm spatial resolution in magnetic field mapping. Magnetic field sensing with cold atoms utilizing Larmor precession of alkali atoms in a magnetic field has been discussed in: MOT [10], Bose-Einstein condensate [11, 12] and an optical dipole trap [13]. Our measurements apply a different principle: rather than measuring Larmor frequency (single atom quantity), we measure rotation of a polarization plane (a cumulative effect over the whole sample), which may offer higher accuracy in very low magnetic fields. In our ex- periment the rotation is mainly caused by the nonlinear medium’s birefringence resulting from the light-induced Zeeman coherences [14, 15], regarded as the diamagnetic effect. The rotation resulting from population imbalance (paramagnetic effect) was studied with cold atoms in re- cent experiments devoted to spin squeezing [16]. arXiv:1001.1629v1 [physics.atom-ph] 11 Jan 2010