1. Introduction

The coherent interaction between atomic systems and optical fields is a major focus of scientific interest owing to multiple applications ranging from the present-day frontier of metrology to new developments in quantum information science. Among a broad variety of phenomena, Electromagnetically Induced Transparency (EIT) and the closely related Coherent Population Trapping (CPT) play an important role [1–3]. In the simplest configuration, two optical fields excite an ensemble of three-level atoms in Λ configuration and, as a result, the medium may become highly transparent owing to optical pumping into a dark state [1]. EIT is widely employed, for example, for mapping quantum information into and out of atomic memories, a crucial task for the construction of quantum networks with atomic ensembles [4]. On a different front, the narrow features of the CPT spectrum are also the basis for a new generation of miniature atomic clocks [5, 6].

Two central aspects of EIT are responsible for these different applications. On the one hand, we have the spectrally-narrow transparency window itself, which allows both the efficient extraction of information from dense atomic media [2, 4] and the spectral resolution of CPT clocks [5]. On the other hand, the steep normal dispersion around EIT resonances may lead to very small group velocities for wave packets of light propagating in the sample, a key ingredient for efficiently writing information into atomic ensembles [2, 3].

During the last decade, a series of new features of EIT emerged in connection with the spectra of correlation measurements between the optical fields participating in the process. Both correlation and anti-correlation were initially observed in the system in different conditions [7, 8], with correlation prevailing at the center of the EIT resonance. Later the transition between correlation and anti-correlation even for the center of the EIT line was reported and carefully investigated [9], being controlled by the intensity of the exciting fields, with anti-correlation dominating for high powers. Another observed feature, particularly useful for spectroscopic or clock applications, was that the linewdith of the correlation peak in the vicinity of the EIT resonance may become free of power broadening [8,10], resulting in the unusual situation of the narrowest spectral structures being observed simultaneously with the strongest signals (see also Ref. [11] for a different set of power-broadening free measurements). All these new features of EIT were observed in warm vapor cells containing alkali atoms. Such atomic systems, however, present typically large contributions to the signal from different velocity classes or adjacent hyperfine levels [9, 12], eluding the direct comparison of experiments to simpler theories that could lead to a more intuitive understanding of the phenomenon.

We seek to clarify the physical mechanism behind these observations. Our approach is twofold. From the experimental side, we employ an ensemble of cold atoms, with a narrow velocity distribution. From the theoretical side, we introduce a simple analytical theory describing the observed results. Such theory directly relates all the observed features to the interplay between the two central aspects of EIT discussed above: the transparency itself and the steep normal dispersion around EIT resonances. These two different aspects of the same phenomenon compete in the correlation spectrum by the mapping, induced by laser phase noise, of the dispersive part of the medium’s susceptibility into the fluctuation of its absorptive part.

2. Experiment

In our experimental setup two laser fields, labeled 1 and 2, forming a small angle θ ≈ 2°, excite a cold ensemble of cesium atoms. The fields have powers P₁ ≈ P₂/2 and come from the same extended-cavity diode laser, with frequencies ω₁, ω₂ tuned close to the 6S_1/2(F = 3) → 6P_3/2(F′ = 2) transition of cesium (frequency ω₀) by independent acousto-optic modulators. They have orthogonal circular polarizations, thereby exciting different sets of transitions connected to one another in Λ configuration. Such a scheme is well known to exhibit EIT-like behavior [13, 14], which can be modeled by an effective three-level system [Fig. 1(a)].

 

Fig. 1 (a) Effective three-level system describing our experiment. Fields 1,2 have frequencies ω₁, ω₂.ω₀ is the frequency for the transition between either ground states 1,2 and the excited state 0. The corresponding detunings are δ_i = ω_i − ω₀. (b) and (c) present plots for real-time signals at the detectors for fields 1,2 (gray and black curves) after interaction with the sample, and for (b) P₂ ≈ 30μW and (c) P₂ ≈ 500μW, with δ₂ = 0 and δ₁ = 0.73MHz in both cases.

Download Full Size | PDF

The cold atoms are obtained from a magneto-optical trap (MOT). The MOT is off during 2 ms, enough to allow the magnetic field to relax to zero and to optically pump the atoms to the 6S_1/2(F = 3) manifold of the ground state. Each time the MOT is off, the EIT fields are turned on for 1.5 ms before we record 8 μs of the signals for fields 1,2 in a digital oscilloscope (400 MHz bandwidth). The detectors (ET-2030A, Electro-Optics Technology) are used to measure the fields after propagation in the sample, with optical depth ≈ 3. They are AC coupled with bandwidth from 30 kHz to 1.2 GHz, recording then only the fluctuations δI₁, δI₂ of the field intensities I₁, I₂. Since we observe that the ensemble contributes to field fluctuations up to about 15 MHz, we further pass the recorded signal through a low-pass 15 MHz filter (Butterworth, third order) to avoid high-frequency electronic noises.

In Figs. 1(b) and 1(c), we observe examples of the recorded time signals in situations corresponding to correlation and anti-correlation, respectively, between fields 1,2. The detunings are set to δ₂ = ω₂ − ω₀ = 0 and δ₁ = ω₁ − ω₀ = 0.73 MHz. In this way, the relative detuning δ₁₂ = δ₁ − δ₂ is such that the fields are on the side of the EIT-resonance peak, occuring at δ₁ = 0.37 MHz. The parameter that is varied between the two panels is the power of the exciting fields, going from P₂ ≈ 30μW for Fig. 1(b) to 500μW for Fig. 1(c). This relative detuning at the side of the two-photon resonance maximizes the contrast between the correlation and anti-correlation signals for the various laser powers employed in the experiment.

A more systematic way to quantify such real-time correlations is to compute the g₂(0) function, defined by

 

Fig. 2 Measurement of g₂(0) as a function of δ₁, for δ₂ = 0 and P₂ ≈ 30 μW (squares), 100 μW (circles), 300 μW (triangles), and 500 μW (diamonds), respectively. The lines are just guides to the eyes.

Download Full Size | PDF

The observed shift of the correlation peak from δ₁ = 0 is consistent with the level of residual DC magnetic fields we expect in our setup. In order to cancel stray DC magnetic fields, we employ a set of three pairs of independent bias coils in Helmholtz configuration. The nulling of the magnetic field is obtained by optimizing the memory time in a light storage measurement [15]. This technique, however, is more sensitive to fields perpendicular to the propagation direction, which generate strong collapses and revivals in the efficiency to read the atomic memory [16]. DC Magnetic fields parallel to the light propagation direction shift the position of the two-photon resonance (as in Fig. 2), but do not affect strongly the readout efficiency or storage time of the atomic memory.

Such behavior is consistent with all previous observations in vapor cells. As in Ref. [9], we observe strong anti-correlations developing as the exciting power increases. We also observe the characteristic correlation peak on two-photon resonance with the wings of the peak reaching the anti-correlation regime, as in Refs. [8] and [10]. Finally, the correlation peak becomes more and more pronounced, but still maintaining the same width, as power increases, as reported in Ref. [10].

3. Theory

In order to model the behavior described above we consider the simplest case of an optically thin sample of three-level atoms in the Λ configuration of Fig. 1(a). In this situation, g₂(0) may be written as [9, 17]

The fluctuations of the imaginary parts of p_i (i = 1, 2) are given by δImp_i = Imp_i − Im〈p_i〉, with 〈p_i〉 denoting the average, steady-state value of p_i. For a three-level atom in Λ configuration it is straightforward to calculate 〈p_i〉, which is proportional to the coherence of the respective optical transition. Since we are modelling a cold atomic ensemble, we neglect any inhomogeneous Doppler broadening, considering only the radiative broadening of the excited state (linewidth Γ) and an effective decoherence rate γ between the two ground states to take into account decoherence induced by spurious, inhomogeneous magnetic fields (see the set of Eqs. 6 of Ref. [15]). In this situation, with the two exciting fields coming from the same laser, it is then straightforward to obtain

In order to calculate g₂(0) we have to compute the average, steady-state values of the quantities 〈δImp_iδImp_j〉, with i, j = 1 or 2. As an alternative to the approaches of Refs. [8–10] and [17], we developed the following analysis to obtain a simple analytical expression for g₂(0) that captures the essential physical elements behind the observed experimental features. Consider a random phase variation ϕ over the steady-state of p_i. From Eqs. (3), such phase variation modifies Ω_i in a way that 〈p_i〉 becomes p_i = 〈p_i〉e^iϕ. This fluctuation then leads to a mixture of the real and imaginary parts 〈p_i〉 into the imaginary part of the fluctuating p_i:

A plot of g₂(0) as a function of δ₁ according to Eq. (8) is given in Fig. 3, for four Rabi frequencies varying in proportion to the experimental variations of intensities for the curves in Fig. 2. The other parameters are δ₂ = 0, Γ/2π = 5.2 MHz, and γ/2π = 150 kHz. The maximum Ω₂ = 0.37 Γ and the value of γ were chosen to obtain a best approximation between theoretical and experimental curves. The relation Ω₁ = 0.75Ω₂ was chosen in accordance to the experimental relation between I₁, I₂.

 

Fig. 3 Theoretical g₂(0) as a function of δ₁, for δ₂ = 0. The Rabi frequencies for each curve are Ω₁ = 0.75Ω₂ and (solid curve) Ω₂ = 0.09 Γ, (dashed) 0.17Γ, (dotted) 0.29Γ, and (dash-dotted) 0.37Γ, respectively. The other parameters are Γ/2π = 5.2 MHz and γ/2π = 150 kHz.

Download Full Size | PDF

From Fig. 3, we observe that the theoretical g₂(0) exhibits all the crucial features of the analogous experimental curves as the Rabi frequency increases. We observe a similar transition from a regime of completely dominant positive correlations, at low powers, to a regime with strong anti-correlations around a resonant correlation peak, at high powers. We also observe that the correlation-peak width at high powers is independent of power broadening. Shifts of the correlation peak from δ₁ = 0 are obtained by introducing a spurious detuning, as would come from any residual DC magnetic field. On the other hand, this simple model does not account for the observed asymmetries of the experimental curves in Fig. 2.

Since the theory presented above is able to provide good qualitative agreement with the variations of the correlation spectra as intensity increases, it can be used to elucidate the physical mechanisms behind the observed features. First of all, from Eq. (7), we have that the sign of the observed correlation depends basically on the interplay between the real and imaginary parts of the atomic coherences. On the other hand, from Eqs. (3) and (4) we have that the imaginary, absorptive, parts of p₁ and p₂ always have the same sign. In this way, they always lead to correlation between fields 1,2.

The anti-correlation originates from the real parts of p₁, p₂. In Fig. 4(a) we plot Re〈p₁〉,Re〈p₂〉 for two different intensities, corresponding to Ω₂ = 0.09 Γ and 0.29Γ. For Ω₂ = 0.09 Γ, the solid (dashed) curve corresponds to Re〈p₁〉 (Re〈p₂〉). For Re〈p₁〉, we observe then the steep normal dispersion around the two-photon resonance that is characteristic of EIT. On the other hand, around the two-photon resonance, Re〈p₂〉 has an inverted sign with respect to Re〈p₁〉, since the positive two-photon detuning for one field is necessarily negative for the other. The multiplication Re〈p₁〉Re〈p₂〉 results then negative around the EIT resonance, being responsible for the anti-correlation in that region. Note that any absorption profile has dispersion peaks on both sides that lead to an enhanced transfer of phase to amplitude noise [18, 19]. Figure 4(a), however, highlights that in a Λ two-photon transition the two fields approach resonance from opposite detunings. Since the fields originate from the same laser, they fluctuate in phase together, leading to simultaneous amplitude fluctuations in opposite directions as well. As can be seen from the Re〈p₁〉 (dotted) and Re〈p₂〉 (dash-dotted) curves for Ω₂ = 0.29 Γ, the increase in intensity enlarges the anti-correlation region around the EIT resonance.

 

Fig. 4 (Color online) (a) Real part of the coherences as a function of δ₁ for two Rabi frequencies. The solid and dashed (dotted and dash-dotted) curves are the results for Re〈p₁〉 and Re〈p₂〉, respectively, for Ω₂ = 0.09 Γ (Ω₂ = 0.29 Γ). (b) Imaginary part of 〈p₁〉 as a function of δ₁ for each of the four curves in Fig. 3, following the same convention for line types as in that figure.

Download Full Size | PDF

In recent work, the same anti-correlation mechanism was applied as the basis of a new method to accurately measure the position of the center of an EIT resonance [20]. A laser was subject to FM modulation, leading to strong, periodic, amplitude oscillations in the fields after interaction with the atomic ensemble. Small deviations from the symmetrical, two-photon-resonance condition would directly result in oscillations with opposite phases, owing to the opposite signs of the respective dispersive curves. In this way, the resonance center could be determined free from the limitation of the ground state coherence lifetime [20].

In our case, the real part of the coherences is zero exactly on the two-photon resonance, and the intrinsic random fluctuations of the laser fields result in a small dispersion around it. This region is then dominated by the nonzero imaginary part of the coherences, leading to the correlation peak. This correlation is originated then by the transparency process itself, which maps amplitude fluctuations in one field in an increase or decrease of transparency to the other. A plot of Im〈p₁〉, for various Rabi frequencies, is shown in Fig. 4(b). Im〈p₂〉 follows closely Im〈p₁〉 and is not shown. If the intensity is low, the real parts of the coherences would have opposite signs only in a small region around zero detuning. Since that region is dominated by Im〈p_i〉, positive correlations are observed for all detunings. As power increases, however, the region of inverted signs in Re〈p_i〉 enlarges and eventually reaches detunings for which the real parts of the coherences dominate, resulting in anti-correlation.

Finally, in Fig. 4(b), for |Ω₁|, |Ω₂|, Γ >> γ the saturation of the transparency as intensity increases leads to a flattening around δ₁ = δ₁₂ = 0, with Im 〈p_i〉 = A_iγ (i = 1, 2) and A_i being constants that depend on all parameters of the problem. On the other hand, in the same limit, we have a near linear dependence in Fig. 4(a), with Re〈p₁〉 = −A₁δ₁ and Re 〈p₂〉 = A₂δ₁. In this way, the transition between the regions dominated by the imaginary or real parts of the coherence is determined simply by δ₁ = ±γ, which gives the crossing points between correlated and anti-correlated behavior and allows for an experimental determination of γ free of power broadening. Note that it is not unusual to have such close relation between imaginary and real parts of the polarization, since these quantities are connected by Kramers-Kronig relations. In this situation, however, it is striking to observe their direct competition in the same experimental signal.

4. Conclusion

We presented an investigation of the physical mechanisms leading to the correlation spectra of electromagnetically-induced-transparency resonances, comparing data obtained from cold atoms with a simple theory. The analysis reveals the role of the laser phase noise in the mapping of the dispersive part of the medium’s susceptibility into the fluctuations of its absorptive part, resulting in marked transitions between strongly correlated and anti-correlated behavior. We are then able to relate the observed features of the correlation spectrum with the two most characteristic features of the susceptibility in these conditions: the transparency peak itself and the sharp normal dispersion in its vicinity. Particularly, we explained how these different contributions come together to provide a power-broadening-independent measure of the ground-state coherence time. This analysis may lead to the formulation of new spectroscopic methods to determine linewidths with clear advantages over the regular excitation or absorption spectra. Beyond its spectroscopic applications, the clarification of such correlation mechanisms may also lead to new methods to manipulate the correlation properties of light fields interacting with atomic ensembles, with possible applications in the field of quantum information science.