Magneto-optical spectroscopy with polarization-modulated light

Transcription

1 Published in " " which should be cited to refer to this work. Magneto-optical spectroscopy with polarization-odulated light E. Breschi, * Z. D. Gruijć, P. Knowles, and A. Weis Departent of Physics, University of Fribourg, 7 Fribourg, Switzerland We present a cobined theoretical and experiental study of agnetic resonance transitions induced by polarization-odulated light in cesiu vapor exposed to a transverse agnetic field. Signals are obtained by phase-sensitive analysis of the light power traversing the vapor cell at six haronics of the polarization odulation freuency. Resonances appear whenever the Laror freuency atches an integer ultiple of the odulation freuency. We have further investigated the odifications of the spectra when varying the odulation duty cycle. The resonance aplitudes of both in-phase and uadrature coponents are well described in ters of the Fourier coefficients of the odulation function. The background-free signals generated by the polarization odulation schee have a high application potential in atoic agnetoetry. I. INTRODUCTION Magneto-optical spectroscopy of atoic edia is a powerful tool that uses resonant optical interactions for detecting agnetic resonance in atos (a coprehensive review is given by Alexandrov et al. []). Magnetic resonance transitions are conventionally excited by a agnetic field oscillating at the atos Laror freuency, but it has been known since the seinal work by Bell and Bloo in 9 [] that an aplitude-odulated (AM) resonant light bea can also induce agnetic resonance transitions. More recently, the Bell-Bloo ethod, in cobination with phase-sensitive detection, has found renewed interest [,], in particular for its application in high-sensitivity atoic agnetoetry []. Alternatively, odulation of either the light s freuency or polarization at the Laror freuency induces agnetic resonance. While freuency-odulation spectroscopy has becoe a well-established ethod for high-sensitivity agnetoetry [,7], little work has been devoted to polarization odulation [8 ]. Recently, resonant polarization odulation has been used to solve specific technical issues. The authors of Ref. [] have deonstrated that polarization odulation between circular and linear polarization states eliinates both dead zones and heading errors in an alkali-etal-ato agnetoeter. Polarization odulation between circular polarization states at the ground-state hyperfine transition freuency was shown to increase the contrast of the clock resonance as described in Ref. [], and references therein. A theoretical odel deriving algebraic expressions for the rich resonance structure of agnetic resonance transitions driven by aplitude, freuency, and polarization odulation has recently been presented []. So far the odel predictions were shown to give an excellent description of experiental results obtained with aplitude odulation using phasesensitive detection [] and for polarization odulation with low-pass-filtered detection []. Here, we address polarization odulation in cobination with phase-sensitive detection. We characterize, both experientally and theoretically, the agneto-optical resonance spectra observed in a transverse agnetic field, when the helicity of the exciting laser bea * evelina.breschi@unifr.ch is periodically reversed with different duty cycles. We first introduce our experiental and theoretical ethods and then discuss the full agneto-optical resonance spectra as a function of the odulation duty cycle. We show that resonances appear when the Laror freuency is a haronic of the odulation freuency and that the relative resonance aplitudes are well described in ters of the Fourier coefficients of the polarization odulation function. II. METHOD Experients are carried out in a conventional agnetooptical spectroscopy apparatus [,]. Light fro a 89-n distributed feedback diode laser passes an electro-optical odulator, which flips the light polarization between σ + and σ polarizations at a constant freuency ω od = π 7 Hz. The degrees of circular polarization were deterined to be 99.7% and 99.8%, for σ + and σ, respectively. The laser freuency can be tuned to any hyperfine coponent of the cesiu D line (S / P / ); however, the experients reported here were carried out on the F g = F e = transition, which provides the highest agneto-optical resonance contrast []. The odulated bea passes an evacuated paraffin-wall-coated spherical cell containing cesiu vapor at roo teperature ( C) saturated vapor pressure. The laser bea diaeter is about ties saller than the cell diaeter (d = ) so that lensing or birefringence effects fro the cell itself do not critically affect the polarization uality. This is testable in that linear polarization coponents will create resonances at half of the Laror freuency due to the syetry and evolution of any created spin alignent. We did not see resonances in the experiental spectra arising fro spin alignent. The final excellent agreeent between odel and experiental results (cf. Sec. III) strongly iplies that the cell has negligible effect on the light polarization. The cell is isolated in a agnetically controlled environent [], with noinal residual field coponents below a few nt. The aplitude of a static agnetic field applied orthogonally to the laser propagation direction, and hence the Laror freuency ω L, is scanned in the range ± ω od.the experiental signal is fored by detecting the light power P (t) transitted through the cell: The photocurrent signal is converted into voltage, aplified, and then deodulated via a

2 lock-in aplifier, which allows the siultaneous recording of the in-phase and uadrature signals at six haronics ω od of the odulation freuency. A. Model For signal interpretation we apply the odel developed in Ref. [] for a spin-oriented atoic saple under periodically odulated excitation and detection. The key results of that odel are suarized here. The power transitted through the cell is given by the Labert-Beer law, which, for an optically thin atoic vapor of length L, is well approxiated by P = P P κl, where κ is the optical absorption coefficient and L is the saple length. For a laser bea resonant with a F g F e hyperfine transition, incident on atos with a longitudinal spin orientation S z, the absorption coefficient becoes κ = κ ( αξs z ). Here, κ is the peak absorption coefficient of unpolarized atos, ξ is the light helicity, and α α Fg,F e is the orientation analyzing power. Dropping tie-independent ters, the transitted laser power is given by P (t) = (ακ LP ) S z (t)ξ(t), () where S z (t) is the tie-dependent longitudinal spin orientation resulting fro the tie-dependent solutions of the Bloch euations under polarization odulation. A suare-wave odulation function ξ(t) is chosen to ake a fast switch between the two circular polarization states without spending significant tie in linearly polarized interediate states. We allow for odulation with arbitrary duty cycle ( <<) chosen to be syetric with respect to t = and constrained to have ξ(t) =, whose Fourier expansion is given by ξ(t) = + j= g j () cos (jωt), () g () = and g j () = sin (πj). () π j In-phase I and uadrature Q parts of P (t) extracted by deodulation at ω od can be written as I () = a, ()A, Q () = d, ()D, () where A and D are, respectively, absorptive and dispersive Lorentzian resonances given by γ A = (ω od ω L ) + γ, () D = γ (ω od ω L ) (ω od ω L ) + γ, () with aplitudes a, () = ακ L P P s g ()[g () + g + ()], (7) d, () = ακ L P g ()[g () g + ()], (8) P s where P s is the optical puping saturation power. We note that the odel was developed for the low power liit (a,,d, P ), and that the experients were carried out in that range. III. ANALYSIS OF THE RESONANCE SPECTRUM We have studied the deodulated signals as a function of the detection haronic and odulation duty cycle. Since in the Fourier expansion of a syetric = suare wave the even coefficients vanish, one sees fro Es. (7) and (8) that lock-in resonances appear only at odd and even. To copare experiental and theoretical results we noralize the in-phase and uadrature signals to the highest aplitude absorptive and dispersive signals, respectively, found when =, =, and =. No other scaling factors are needed. The noralized odel signals thus read and I () a, = Q () d, = a, A, (9) a, d, D. () d, Theory also predicts d, /a, = /. Experiental aplitudes are obtained by fitting the recorded data with the absorptive and dispersive Lorentzians of Es. () and (), yielding fit aplitudes that represent the odel aplitudes (7) and (8), respectively. We then noralize all fitted in-phase (uadrature) aplitudes to the in-phase (uadrature) aplitude of the ( =, =, = ) resonance. The experiental d, /a, ratio was found to be 9(), in accordance with the theory prediction. In this way, we eliinate all secondary experiental paraeters affecting the signal, e.g., detector uantu efficiency, current-voltage converter gains, etc. The noralization of the odel predictions, of course, reoves unknown theory paraeters (e.g., P s, α Fg,F e, κ, etc). In Figs. and we copare experiental and theoretical spectra for different values of when = and =.. No scaling has been applied beyond the noralization procedure above. As predicted, resonances are observed when the Laror freuency is a ultiple of the odulation freuency, i.e., ω L = ω od. An iportant point to note is that in contrast to experients with freuency- or aplitude-odulated light, polarization odulation yields background-free in-phase and uadrature signals. The aplitude of the zero-field levelcrossing (Hanle) resonance is proportional to g, E. (), and thus vanishes for % duty cycle. Figure presents the signals obtained for =, where one can see the largest signals at = and = (the absorptive and dispersive reference resonances for noralization). This resonance is the ost interesting for atoic agnetoetry, and work is ongoing to fully characterize the signal and study the final sensitivity liit of the ethod. The = duty cycle puping, i.e., excitation with a syetric suare wave, essentially corresponds to the so-called push-pull optical puping [7], whose effect is the synchronization of the puping light s polarization odulation with the haronic evolution of the atoic uantu state in the external field. Notable in the = case is that all = odd resonances are predicted to be zero by the odel (as discussed at the start of the section), and the data (only one trace for = is plotted in the figure) show no signal at the detection liit in those cases.

3 Quadrature = In Phase FIG.. (Color online) Magnetic field dependence at % duty cycle ( = ) of noralized in-phase and uadrature lock-in signals, deodulated at even haronics of ω od ; the odd haronic signal is predicted to be zero, and the data (here = is shown) support the prediction. The ratio between the Laror freuency and the odulation freuency is, with = ω L /ω od, where ω od = π 7 Hz. Black lines are easureents, and red (gray) lines are the odel prediction. Figure presents the signals obtained for =., where = odd resonances are not null. The agnetic resonance linewidths show no dependence on and, consistent with the fact that the linewidth is deterined by relaxation echaniss and power broadening coon to all resonances. In the antirelaxation wall-coated cell, the priary relaxation echaniss arise fro spinexchange collisions and loss of the atos to the alkali reservoir via the entrance channel, followed in iportance by collisions between alkali atos and the antirelaxation coating, and agnetic field inhoogeneities, none of which has an expected dependence on or. We have investigated the dependence of the spectra on the duty cycle by varying between. and.9 in steps of. For each value of we easured the aplitudes of the resonances occurring at ω L = ω od,ω od, and ω od (i.e., for =,,) in each of the six deodulation channels = (,...,) ω od. Figure shows the experiental aplitudes (after noralization) as a function of and, together with the odel predictions that are in excellent agreeent with experient. We note the alternating syetric-antisyetric dependence with respect to = for odd and even, respectively. The agnetic resonance linewidth depends on because the effective power driving either the σ + or σ transition and hence the power broadening of the line depends on the duty Quadrature =. In Phase FIG.. (Color online) Magnetic field dependence of noralized in-phase and uadrature lock-in signals, deodulated at haronic of ω od, for laser light resonant with the F g = F e = transition. Black lines are easureents, and red (gray) lines are the odel prediction. The duty cycle was % ( =.). cycle. For this reason the balanced σ + /σ excitation with = is expected to produce the narrowest resonances, as observed in the experient. IV. SUMMARY AND CONCLUSIONS We have studied agneto-optical resonances that occur at ultiples of the Laror freuency when the polarization of a resonant laser bea traversing an alkali ato vapor is switched between left and right circular polarization. The lock-in deodulated signals have a rich spectral structure that is well reproduced by algebraic odel predictions. The linear zero crossings of the uadrature resonances can be used as discriinator signals for agnetic field easureents. The optial resonance for agnetoetry applications is the one with =, =, and =, since it has the largest aplitude and siultaneously the narrowest linewidth. The application of the polarization-odulation approach reported

4 Noralized in phase aplitude Noralized uadrature aplitude FIG.. (Color online) Noralized aplitude of the in-phase (left) and uadrature (right) signal for =,, (fro top to botto). The aplitudes are function of the detection haronic ( =,...,) and the duty cycle ( <<). here to atoic agnetoetry has a distinctive advantage copared to siilar approaches using aplitude or freuency odulation, viz., the absence of a dc background on the inphase signal. In a feedback-locked agnetoeter, an iperfect phase setting transfers a part of the dc background onto the uadrature signal, thereby introducing a shift in the agnetic field easure and increasing power noise. Finally polarization odulation with variable duty cycle can also find applications in the preparation and anipulation of specific atoic states, i.e., in etrology [8] and in uantu inforation processing [9]. ACKNOWLEDGMENTS This work is supported by SNF-Abizione Grant No. PZP_9. We thank the echanical workshop and the electronics pool of the Physics Departent for expert technical support.

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