Suppression of inhomogeneous broadening using the ac Stark shift

Inhomogeneous broadening is ubiquitous in light-matter interactions [1]. When a laser beam interacts with a gas, the dominant broadening is typically due to the Doppler effect, which causes atoms to experience different laser frequencies depending on their velocities. For atoms embedded in crystals, the inhomogeneous broadening is a result of the shifts in the energy levels due to local variations in the crystal field [2]. Inhomogeneous broadening is not necessarily an undesired effect; many physical processes require an inhomogeneously broadened atomic ensemble. For example, by using an appropriate sequence of excitation pulses, photon echoes utilize the rephasing of different oscillation frequencies in an inhomogeneously broadened ensemble. Optical memories that use spectral hole burning rely on choosing a particular subset of a broadened ensemble using various frequency-selective spectroscopic techniques. The ratio of the inhomogeneous linewidth to the homogeneous linewidth determines the capacity of such memory, and this ratio is typically referred to as the figure of merit for such devices [3]. There are also many physical effects where inhomogeneous broadening is quite detrimental. For example, the magnitudes of the linear and nonlinear susceptibilities of an atomic medium typically scale as the inverse of the total linewidth of a given transition. Thus, if the inhomogeneous linewidth is larger, then the ensemble-averaged optical response is weaker. In the simplest case of a laser beam interacting with a two-level atomic system, the inhomogeneous broadening reduces the optical depth and the maximum refractive index that can be achieved. In experiments that rely on quantum interference, such as electromagnetically induced transparency (EIT), inhomogeneous broadening puts stringent constraints on the collinearity of the interacting lasers, severely limiting the nonlinearities that can be obtained at single-photon energies [4,5]. In quantum computing, Doppler broadening is a key limitation on the speed and fidelity of singleand two-qubit gates [6,7]. About three decades ago, Cohen-Tannoudji and colleagues discussed and experimentally demonstrated how Doppler broadening can be suppressed by the appropriate use of the light (Stark) shift, in the emission spectrum of an excited level [8–10]. The key idea is to compensate the frequency shifts due to Doppler effect by using an intense laser to provide an equal and opposite Stark shift. In this paper, we extend this idea to the excitation processes from the ground level. We investigate the feasibility of this approach by performing numerical simulations of the density matrix under realistic experimental conditions. Our results show almost complete suppression of Doppler broadening and, as a result, this approach may provide a powerful tool in a wide range of experiments ranging from nonlinear optics to quantum computing. Before proceeding, we cite other related prior work. Agarwal and colleagues discussed sub-Doppler line shapes in inhomogeneously broadened media under the conditions of EIT [11]. Popov et al. investigated suppression of inhomogeneous broadening using coherent fields for enhanced fourwave mixing [12]. Kaplan and co-workers demonstrated the suppression of inhomogeneous broadening in rf spectroscopy of optically trapped atoms using a compensating laser beam [13]. We also would like to clearly differentiate our approach from sublinewidth spectroscopic techniques such as spectral hole burning and saturated absorption spectroscopy. These techniques rely on selecting a particular class of atoms in an inhomogeneously broadened ensemble. For example, in saturated absorption spectroscopy, an intense saturating beam is used to select a particular velocity class of atoms and saturate the transition of this smaller group. Only the atoms in the specific velocity class contribute to the optical response. In contrast, the approach that we discuss below eliminates inhomogeneous broadening and forces all the atoms in the ensemble to respond in a similar way. This is achieved with the use of an appropriately tuned Stark-shift laser so that Doppler shift is canceled by the Stark shift. Because the Doppler shift is effectively absent, all the atoms in the ensemble respond to the probe laser as if they are at zero velocity. Thus, in terms of their response to the probe laser beam, the atoms in a hot vapor cell act as though they were an ultracold ensemble. For concreteness, we will focus on the specific example of Doppler broadening, although this idea can be extended to other inhomogeneous processes. Following Ref. [8] and as shown in Fig. 1, we consider the interaction of a four-level atomic system with two laser beams, a weak probe laser (EP ) and an intense beam that is used to Stark shift the ground level (ES). The probe laser is tuned close to the |1〉 → |2〉 transition. Due to atomic motion, this transition is Doppler broadened, and we will be interested in the limit where the Doppler width is much larger than the homogeneous linewidth of the transition. To suppress this broadening, we will utilize the Stark-shift laser, which couples the ground level to two other excited levels, |3〉 and |4〉 [14]. We define the Rabi frequencies of the Stark-shift beam for the |1〉 → |3〉 and |1〉 → |4〉 transitions to be 13 ≡ ESμ13/ and 14 ≡ ESμ14/ , respectively. Here the quantitiesμ13 andμ14 are the dipole matrix elements of the respective transitions. With these definitions, the Stark shift of