Quantitative analysis of the transient response of the refractive index to conditions of electromagnetically induced transparency F. Meinert, C. Basler, A. Lambrecht, S. Welte, and H. Helm Department of Molecular and Optical Physics, University of Freiburg, Stefan-Meier-Str. 19, 79104 Freiburg, Germany. (Dated: December 29, 2011) At the example of the D1 line of 87 Rb we analyze the experimental parameters which control the transient response of electromagnetically-induced transparency. Quantum coherent free induction decay is observed on timescales exceeding several milliseconds in a buffer gas vapor cell. Numerical solutions of the quantum master equation and approximate analytical solutions are tested and absolute comparisons of the transient timescales, the power broadening, the resonance contrast and frequency shifts are made. Two actively phase-locked lasers are employed. Effects of not fully correlated laser redphase noise are studied. PACS numbers: 33.80.Rv 33.60+q 33.20.Xx I. INTRODUCTION The optical properties of atomic media under conditions of electromagnetically-induced transparency (EIT), have attracted wide interest, both from fundamental aspects of an open quantum system in which a tailored laser field shields the system from its environment, as well as for spe- cific applications [1–4]. The coherent preparation of an atomic sample, which allows near dissipation-free propa- gation of light-fields, has applications in the enhancement of nonlinear optical processes [5, 6], in laser cooling [7– 9], in quantum nondemolition measurements [10, 11], for ultraslow group velocities [12, 13] as well as for all-optical miniature atomic clocks [14–18]. While the study of transient excitation of three-level sys- tems is a mature field, first laid out in the theoretical paper of Berman and Salomaa [19], quantitative experimental studies on the dynamic response on how fast an incoherent sample of atoms enters the conditions of EIT are rare. Pri- mary attention has been paid to the free-induction decay which appears when a coherent superposition of atomic hyperfine levels exits from the dark state when one [20] or both of the lasers are suddenly detuned from resonance [21] or when one laser is entirely turned off [22]. The transient response of transparency is highly sen- sitive to environmental parameters such as collisional, magneto-optic and temperature effects. To an even greater extent it is sensitive to noise in the cross correlation of the two light fields which drive EIT. Thus transient EIT- signatures serve as a useful laboratory tool for daily con- trol of such parameters. The transient response of trans- parency is also a critical feature when recording spectral scans of EIT-resonances but hardly ever referred to. This dependence arises from the finite response time required for building up coherent superpositions. It may lead to distortion of the line shape and of the frequency position of an EIT resonance. Here we attempt to characterize parameters which are relevant for the transient response in both experiment and theory. Using two independently tunable, but phase-locked lasers we present a quantitative study of EIT transients at the example of the D 1 line of 87 Rb, specifically the atomic clock transitions in the presence of weak magnetic fields. We interpret our findings first with simple three- state model predictions and provide a toolbox of analytic solutions of the quantum master equation for analyzing and predicting transient EIT. We also quantify shortcom- ings of three-level models as they appear even for spec- trally isolated dark resonances. II. EXPERIMENT Our experimental setup is shown schematically in Fig. 1. It employs two redexternal-cavity diode lasers [23]. A master laser is frequency locked by Rb Doppler-free FM- spectroscopy and drives the transition 2 S 1/2 (F ′ = 2) ↔ 2 P 1/2 (F = 1). This laser is set to resonance as cali- brated in an unbuffered Rb spectroscopy cell. A second control-level signal Master Slave OPLL feedback controller PD shielded Rb buffer gas cell to FM spectroscopy GHz PD 6.8 Ghz beatnode FET and piezo modulation LabView Control Mixer GPS controlled timebase opt. fiber tunable reference ref ref HF Gen. MHz HF Gen. GHz Figure 1: Schematic of experiment and optical phase-locked loop (OPLL) between master and slave laser, PD: photodiodes, GPS: global positioning system, HF Gen: high-frequency generators. laser is stabilized at a variable detuning from the master by employing a digital phase-lock [24, 25]. The differ- ence frequency, near the ground-state hyperfine splitting, ν hfs = 6 834 682 611 Hz [26] can continuously be tuned while the slave laser is actively stabilized against the mas- ter laser with an RMS-phase error of better 0.5 rad over periods of hours. The difference frequency is controlled by two frequency generators which are actively stabilized. In lack of an atomic frequency standard we linked all fre- quency references to a GPS disciplined reference oscillator [27]. This proved highly important in searching for sources of instability and for reliably locating absolute frequency positions of narrow dark resonances. The frequency gen- erators serve as bases for the digital multiplication scheme of the phase lock and allow quasi-continuous or sudden changes of the difference frequency under computer con- trol. The absolute difference is controlled by the frequency