1. Introduction

Interactions of atoms and electromagnetic modes of an optical cavity forms the basic theme of cavity quantum electrodynamics (cavity QED) and are important for a variety of applications in quantum physics and quantum electronics [1]. The most fundamental cavity QED system consists of a single two-level atom coupled to a single cavity mode with a coupling coefficient g [2]. The composite atom-cavity system exhibits a double-peaked transmission spectrum representing the two lowest excited quantum states, the normal modes [3–4]. When the system is resonantly coupled, the two normal modes are separated in energy by 2g, commonly referred to as the vacuum Rabi splitting [3–4]. Observation of the two normal modes in the optical wavelength range requires a cavity-atom system with a g value greater or comparable with the decay rates of the cavity and the atomic system, which requires a high finesse cavity with a small mode volume [4]. On the other hand, if N two-level atoms collectively interact with the cavity mode, the coupling coefficient becomes G = √N g and the vacuum Rabi splitting of the normal modes for the collectively coupled atom-cavity system becomes 2G and may then be observed in a cavity with a moderate mode volume and finesse [5–9].

Studies of interactions between a cavity mode and multi-level atoms extend the regime of cavity QED and are important for practical applications. There are many earlier studies of optical bistability in two or three-level atoms confined in an optical cavity [10–11]. Recent studies of atom-cavity interactions are extended to a composite system of an optical cavity and coherently prepared multi-level atoms, in which the atomic coherence and interference may be used to manipulate the quantum interactions of the cavity and atomic system [12]. Experimental studies of coherently prepared atoms confined in a cavity have lead to the observations of the linewidth narrowing [13–15], coherent control of optical multistability [16], single photon generation [17], and cavity assisted slow light propagation [18].

Here we report an experimental study of the collectively coupled multiple atoms confined in a moderate-size cavity and interacting with a free-space control laser. We show that with the interaction of the control laser and the cavity mode with the atoms forming a three-level Lamda-type system, a three-peaked transmission spectrum is observed in the coherently coupled system: a central peak representing the dark state commonly observed in a coherently coupled three-level Lamda-type system and two sidebands representing the two normal modes observed in the cavity-coupled two-level system but their energy separation is modified by the frees-pace control laser and is manifested by the collective coupling of the cavity field with N atoms and the individual coupling of the control laser with single atoms. Such manifestation may open new ways to manipulate the cavity QED system.

2. Theoretical analysis

Consider a composite atom-cavity system that consists of N atoms confined in the cavity mode. The atoms are identical and have a Λ-type level structure shown in Fig. 1. The cavity is formed by two mirrors with reflectivity R and separated by a distance L. The schematic diagram of the coupled atom-cavity system is depicted in Fig. 1. The cavity mode couples the atomic transition |a>-|e> with the collective coupling coefficient G = √N g ($g=\mu \sqrt{{\omega }_a/2\mathit{ħ}{\epsilon }_{0}V},$ here μ is the atomic dipole moment, ω_a is the frequency of the cavity mode, V is the mode volume, ħ is the Plank constant, and ε₀ is the vacuum permittivity). A free-space control laser drives the atomic transition |b>-|e> with Rabi frequency 2Ω. Δ = v_con - v_eb is the control frequency detuning, Δ_c = v_c - v_ea is the cavity mode-atom detuning for the empty cavity (no atoms). When the atoms are confined inside the cavity mode, the atomic dispersion causes frequency pulling or frequency pushing, and then the resonant frequency of the cavity-atom system is different from the resonant frequency of the empty cavity. Since the control laser modifies the atomic dispersion, it can be used to manipulate the resonant frequency of the coupled atom-cavity system. To characterize the frequency response of the coupled cavity-atom system, a weak probe laser is coupled into the cavity mode and its transmission versus the probe frequency detuning Δ_p = v_p - v_ea is calculated. We treat the probe field coupled into the cavity mode classically. The susceptibility of the three-level Λ-type atomic medium at the frequency of the intra-cavity probe laser, χ(υ_p) = χ'(υ_p) + iχ"(υ_p), is derived by solving Schrodinger equations of the coupled atomic system under the condition ρ_aa ≈ 1 (the probe field is much weaker than the coupling field so the atomic population is concentrated in the ground state |a>) [18–19] and is given by

 

Fig. 1. Three-level atomic system coupled to a cavity mode (with a collective coupling coefficient g√N) and a free-space control laser (with a Rabi frequency Ω).

Download Full Size | PDF

Here K = n |μ_ea|² /ħε₀ (n is the atomic density), Γ is the decay rate of the excited state |e>, and γ_ab is the decoherence rate between the ground states |a> and |b>. The amplitude of the cavity-transmitted probe field is

here ${\phi }_t={\tan }^{-1}\left\{\frac{1+R}{1-R}\tan \left(k\left(L-\ell +\chi \text{'}\ell \right)\right)\right\}$ is the phase shift, E_in(v_p) is the input probe field, and ℓ is the length of the atomic medium, and k is the wave vector. Eq. (2) shows that the round-pass phase shift of the probe field inside the cavity consists of two parts: the empty cavity phase shift $2k\left(L-\ell \right)=2m\pi +{\phi }_1({\phi }_1=\frac{2\pi \left({\Delta }_p-{\Delta }_c\right)}{c/2L}$ is the effective phase shift) and the phase shift through the coherently coupled atomic medium, φ ₂ = 2kχ 'ℓ. Combination of the two phase shifts determines the transmission peaks of the probe field coupled into the cavity mode, or the resonant frequency of the coupled atom-cavity system. Fig. 2(a) plots the phase shifts φ ₁ (red curves), φ ₂ black curves), and φ ₁ + φ ₂ (blue curves) versus the probe frequency detuning normalized to the decay rate Γ for a cavity of length L=5 cm. The corresponding transmission spectrum of the probe light intensity through the cavity (E_t(v_p) E_t ^*(v_p)/(E_in(v_p)E_in ^* (v_p))) versus the normalized probe detuning is plotted in Fig. 2(b). The phase shift φ ₂ from the atomic medium is independent of the cavity length L, but the phase shift φ ₁ from the empty cavity depends on L. Due to the frequency pulling and pushing of the atomic dispersion, the resonant frequency of the cavity with the atomic medium occurs at the frequency where the combined phase shift φ ₁ + φ ₂ = 2jπ (j is an integer and j=0 in Fig.2). If the empty cavity is tuned to the atomic resonance, Δ_c = v_c - v_ea = 0 and the control laser is also tuned to the atomic resonance Δ = v_con - v_eb=0, then φ ₁ = and φ ₂ = 0 at Δ_p = v_p - v_ea = 0 for the probe field. Therefore, there is a cavity resonance at Δ_p =0 (for an arbitrary cavity length L). As the control laser creates electromagnetically induced transparency (EIT) in the three-level atomic system, the probe absorption at Δ_p = 0 is suppressed [19–20]. A transmission peak of the probe field with a narrow linewidth is observed at Δ_p = 0 as shown in Fig. 2(b), which corresponds to the cavity EIT reported in earlier studies [13–14]. The linewidth of the cavity EIT peak is given by $\Delta v=\frac{1-R_{\kappa }}{\sqrt{\kappa }\left(1-R\right)}\frac{{\Gamma }_c}{1+\eta }$ [12]. Here ${\Gamma }_c=\frac{c\left(1-R\right)}{2\pi L\sqrt{R}}$ is the empty cavity linewidth, and κ = exp(-kχ''ℓ) is the single pass medium absorption at Δ_p = 0, and $\eta =\frac{v_p\ell }{2L}\frac{\partial \chi \text{'}}{\partial v_p}$ is a coefficient characterizing the dispersion change at Δ_p = 0. With the steep slope of the dispersion around Δ_p = 0 for an EIT medium, η >> 1 can be readily obtained, which results in the linewidth of the cavity transmission peak much smaller than the empty cavity linewidth. The linewidth is ultimately limited by the ground state decoherence rate γ_ab that can be many orders of magnitude smaller than the atomic natural linewidth Γ [12]. The decoherence rate γ_ab also contributes to the residual probe absorption in the EIT medium and limits the transmitted light intensity. According to the Kramers-Kronig relation, the narrow central peak of the cavity EIT in Fig. 2 indicates the concomitant steep normal dispersion experienced by the intra-cavity probe field, which implies an ultra-slow group velocity for a pulsed probe field. This is consistent with the consideration of the probe light pulse propagating with a slow group velocity in an EIT medium and being reflected back and forth inside the cavity before the probe pulse leaks out of the cavity. Combination of the reduced group velocity in the EIT medium and the cavity feedback leads to a greater propagation time delay for the slow light pulse propagating through the cavity EIT system than through the same EIT medium without the cavity [18].

 

Fig. 2. (a) Calculated phase shifts φ ₁ (red curves), φ ₂ (black curves), and φ ₁ + φ ₂ (blue curves) versus the probe frequency detuning. (b) Calculated cavity transmission of the probe laser field versus the probe frequency detuning. The optical density of the atomic medium nσ_eaℓ=5. Other parameters are: Ω=Γ, γ_ab=0.002 Γ, δ = 0, Δ_c = 0, L=5 cm, and R=0.98.

Download Full Size | PDF

The calculations also show that there are two nonzero Δ_p values at which φ ₁ and φ ₂ have equal amplitudes but opposite signs, leading to ${\phi }_1+{\phi }_2=\frac{2\pi \left({\Delta }_p-{\Delta }_c\right)}{c/2L}+2k\chi \text{'}\ell =0,$ which results in two additional transmission peaks, or the resonant frequencies of the coupled atom cavity system. Solving the above equation with Δ_c=0, we derive

Here N=nAℓ is the total number of atoms in the cavity (A is the cross section of the intracavity probe beam and the cavity mode volume is taken as V=AL). In the cavity QED system where a cavity mode is coupled to N two-level atoms, the two normal modes of the coupled atom-cavity system are separated in frequency by the multi-atom vacuum Rabi frequency 2G = 2√N g, corresponding to two transmission peaks located at Δ_p = ±g √N and the peak linewdith (FWHM) is given by (Γ + Γ_c) / 2 [5–6]. The vacuum Rabi splitting derived in our classical analysis thus matches exactly the quantum mechanical results for Ω=0 (no control laser) [19]. With the control laser, the vacuum Rabi splitting is equally weighted by Ω characterizing the coupling of the control laser with individual atoms and g√N characterizing the collective coupling of the cavity and N atoms. For later discussions, $2\Omega \text{'}=2\sqrt{g^{2}N+{\mid \Omega \mid }^2}$ is referred to as the modified vacuum Rabi frequency. In the semiclassical dressed-state picture, the three transmission peaks represents the eigen-states of the coupled cavity and three-level atom system [21].

3. Experimental results

The experiment is done with cold ⁸⁵Rb atoms confined in a magneto-optical trap (MOT) produced at the center of a 10-ports stainless-steel vacuum chamber. A tapered-amplifier diode laser (TA-100, Toptica) with output power ~300 mW is used as the cooling laser and supplies three perpendicular retro-reflected beams. An extended-cavity diode laser with an output power of ~15 mW is used as the repump laser. The laser beam diameter is ~ 1 cm. The trapped⁸⁵Rb atom cloud is ~ 1 mm in diameter. The cavity-coupled Rb atomic system is shown in Fig. 3(a) and a schematic diagram of the cavity apparatus is depicted in Fig. 3(b). The standing-wave cavity consists of two mirrors of 5 cm curvature with a mirror separation of ~ 5 cm and is mounted on an Invar holder enclosed in the vacuum chamber. The empty cavity finesse is measured to be ~ 150. Movable anti-Helmholtz coils are used so the MOT position can be finely adjusted to coincide with the cavity center. A third extended-cavity diode laser with a beam diameter ~ 5 mm and output power ~ 20 mW is used as the control laser that propagates along the x direction and directed to overlap with the MOT through a vacuum viewport. A fourth extended-cavity diode laser is used as the probe laser and is propagating in the y direction. The probe laser is attenuated and then coupled into the cavity. The transmitted cavity light passes through an iris and is coupled into a single mode fiber, the output of which is collected by a photodiode detector.

 

Fig. 3. (a) ⁸⁵Rb atoms interacting with a control field and a cavity field, which forms a threelevel Λ-type system. The spontaneous decay rate of the excited state |e> is Γ =2πx5.4x10⁶ s^-1. (b) Schematic drawing of the cavity apparatus. The control laser is circularly polarized and the probe laser is linearly polarized along the x direction.

Download Full Size | PDF

The experiment is run in a sequential mode with a repetition rate of 10 Hz. All lasers are turned on or off by acousto-optic modulators (AOM) according to the time sequence described below. For each period of 100 ms, ~98 ms is used for cooling and trapping of the ⁸⁵Rb atoms, during which the trapping laser and the repump laser are turned on by two AOMs while the coupling laser and the probe laser are off. The time for the data collection lasts ~ 2 ms, during which the repump laser and the current to the anti-Helmholtz coils of the MOT are turned off first, and after a delay of ~0.2 ms, the trapping laser is turned off, and the coupling laser and the probe laser are turned on. After the coupling laser and probe laser are turned on by the AOMs for ~0.1 ms, the probe laser frequency is scanned across the ⁸⁵Rb D₁ F=2→F=3 transitions and the cavity transmission of the probe laser is then recorded versus the probe frequency detuning.

Figure 4 plots the measured cavity transmission of the probe laser versus the probe frequency detuning Δ_p for a series of the optical densities. During the experiment, the empty cavity frequency is tuned to v_c = v_ea and the control laser is on resonance (Δ=0). The measured Rabi frequency of the control laser is Ω≈19 MHz. By changing the cooling laser frequency and the vapor pressure of the Rb source, the optical density of the trapped Rb cloud, nσ₁₃ℓ, can be varied from ~ 0.4 to 5. The optical densities used in the calculations of Fig. 4(a) to Fig. 4(e) are 0.5, 0.8, 1.4, 2.8, and 4.0 respectively, which agree with the measured values of the optical density within a percentage error of ±15%. The other parameters used in the calculations are R=0.96 and γ_ab=0.025Γ. The measurements show that as the optical density increases, the central peak becomes narrower and the separation of the two sidebands gets larger.

 

Fig. 4. Cavity transmission versus the probe detuning Δ_p. Blue dotted lines are experimental data and red lines are calculations. From Fig. 4(a) to Fig. 4(e), the optical densities are 0.5, 0.8, 1.4, 2.8, and 4.0 respectively.

Download Full Size | PDF

Figure 5(a) plots the measured linewidth (FWHM) of the central peak (dots) at Δ_p=0 versus the optical density. The calculated linewidth is plotted as the red line. The maximum optical density obtainable in our experiment is nσ₁₃ℓ ≈5, at which the measured linewidth of the central peak is ~ 1.9 ±0.5 MHz and is limited by the laser linewidth of ~ 1 MHz, which corresponds to a reduction of the cavity linewidth by a factor of η ~21 times from the empty cavity linewidth. Figure 5(b) plots Ω’ (half of the frequency separation of the two sidebands) versus the optical density. The calculated Ω’ from Eq. (3) versus the optical density is plotted as the red line. For comparison, the calculated Rabi frequency G for the two-level atoms (no control laser) is plotted as the black line. The experimental results agree with the theoretical calculation and demonstrate clearly the effect of the coherent combination of the collective coupling between the cavity and t e multiple atoms with the individual coupling of the freespace control laser and the individual atoms.

 

Fig. 5. (a) The linewidth of the central peak versus the optical density nσ₁₃ℓ. (b) The modified vacuum Rabi frequency Ω' versus the optical density. The dots are the experimental data and the red lines are the calculations. The black line in (b) is the calculated vacuum Rabi splitting G for a two-level system. The parameters are Δ_c=0, Δ=0, Ω=19 MHz, R=0.96 and γ_ab=0.025Γ.

Download Full Size | PDF

4. Conclusion

In conclusion, we have analyzed a coherently coupled cavity-atom system that consists of multiple atoms in a Λ-type three-level configuration confined in an optical resonator and simultaneously driven by a free-space control laser. The coherent interactions of the atoms with both the cavity mode and the control laser lead to a three-peaked transmission spectrum for a weak probe laser coupled into the cavity mode. The narrow central peak represents the intra-cavity electromagnetically induced transparency and the peak linewidth is significantly reduced due to the collective enhancement of the cavity feedback. The two sidebands in the observed transmission spectrum represent two normal modes of the atom-cavity system that is modified by the control laser in comparison with the two-level atom-cavity system. The combined atom-cavity system demonstrates the coherent manifestation of the collective coupling between the cavity mode and the multiple atoms, and the coherent coupling between the control laser and the atoms. It shows that the cavity QED technique can be used to enhance atomic coherence and interference (such as the cavity EIT). On the other hand, the technique developed in the studies of the atomic coherence and interference can also be used to manipulate and modify the cavity QED. The late example can be found in a recent demonstration of the quantum interference in the excitation of the cavity normal modes [21].