. The Linewidth of Ramsey Laser with Bad Cavity Yang Li, Wei Zhuang, Jinbiao Chen, ∗ and Hong Guo † CREAM Group, State Key Laboratory of Advanced Optical Communication Systems and Networks (Peking University) and Institute of Quantum Electronics, School of Electronics Engineering and Computer Science, and Center for Computational Science and Engineering (CCSE), Peking University, Beijing 100871, P. R. China (Dated: October 29, 2018) We investigate a new laser scheme by using Ramsey separated-field technique with bad cavity. By studying the linewidth of the stimulated-emission spectrum of this kind of laser inside the cavity, we find its linewidth is more than two orders of magnitude narrower than atomic natural linewidth, and it is far superior to that of conventional optical Ramsey method and any other available subnatural linewidth spectroscopy at present. Since any cavity related noise is reduced to cavity-pulling e ff ect in bad cavity laser, this Ramsey laser provides the possibility of precision subnatural linewidth spectroscopy, which is critical for the next generation of optical clock and atom interferometers. PACS numbers: 42.55.Ah, 42.50.Ar, 42.60.Da, 32.30.-r Introduction: Since the invention of the separated-field technique [1], it has played an important role in the field of precision spectroscopy due to its linewidth narrowing e ff ect via multiple coherent interaction. Atomic clocks based on this technique have greatly extended our ability for frequency measurement, further, almost all the atom interferometers are based on this technique [2]. Though, the natural linewidth of quantum transition was regarded as the ultimate limit to high-resolution laser spec- troscopy [4], several methods of subnatural linewidth spec- troscopy have been proposed to gain subnatural linewidth [3– 10]. However, in all these e ff orts, including optical Ramsey spectroscopy, subnatural line is realized at the expense of a quick reduction in signal-to-noise (SNR) ratio due to the ex- ponential decaying of signal, thus all these schemes can only get the linewidth several times narrower than the atomic nat- ural linewidth. In the past three decades, this situation does not change in the field of the precision laser spectroscopy. On the other hand, the thermal noise of the cavity mirrors is the main obstacle for further linewidth reduction of a laser [11, 12], and it is a challenge to substantially reduce this noise further[13]. Recently, a new scheme, called active optical clock [14–18], was proposed to substantially reduce the las er linewidth. With lattice trapped atoms, it is possible to reach mHz linewidth laser based on the mechanism of active optical clock [14, 15, 19]. The principal mechanism of active optical clock is to directly extract light emitted from the ultranarrow atomic transition with a cavity mode linewidth much wider than that of lasing. This bad cavity ensures that any frequency shift due to cavity noise reduces to cavity-pulling e ff ect [15– 17], then the thermal noise is not the major obstacle again for reducing the linewidth. This means the bad cavity can play an indispensable role in new subnatural linewidth spectroscopy. In this Letter, we propose a new scheme called Ramsey laser with bad cavity. Distinct from any previous applications of conventional Ramsey separated oscillating fields method [1], which focuses on the absorption spectrum, we here fo- where ˆ a , ˆ a † are the annihilation and creation operators of the field mode inside the cavity, with the frequency ω , σ j a = $^{(}$| $^{a}$〉 〈$^{a}$| ) j and σ j b = $^{(}$| $^{b}$〉 〈$^{b}$| ) j are the projection operators for the jth atom corresponding to the upper and lower lasing levels, Theoretical framework: We consider the case of a two-level atomic beam interacting with a single-mode Ramsey cavity of separated-oscillating-field resonators with the cavitymode linewidth is much wider than the atomic gain linewidth. Thus we call it bad-cavity Ramsey laser. All atoms are pumped onto the upper lasing state a before entering the first cavity of seperated field, and the lower lasing state is b . We assume all the atoms have the same velocities υ , that means what we consider here is a homogeneous laser system. And for the sake of simplicity, we consider the two-standing waves linear optical Ramsey configuration with a grid as spatial selector [20, 21]. Our treatment can be extended to other configura- tions as in [22–24]. The length of each oscillating part is l , and the length of the free drift region is L . The corresponding Hamiltonian is cus on the stimulated emission spectrum via multiple coher- ent interactions inside the cavity. We find this Ramsey laser can provide a stimulated-emission spectrum with a linewidth much narrower than that of any conventional optical Ramsey seperated-field spectroscopy, which is commonly applied in optical atomic clock. Our results also show that a subnatural linewidth spectroscopy, superior to any other available subnat- ural spectroscopy technique at present [3–10], can be reach ed by this kind of laser, if a suitable atomic level structure is cho- sen. Thus, this method can provide an e ff ective subnatural spectroscopy, and the possibilities for the new optical clock scheme [15] and atom interferometers [2]. arXiv:1001.2670v1 [quant-ph] 15 Jan 2010 H = ℏ ω ˆ a † a + ℏ ∑ j [ ω j $_{a}$(t ) σ j a + ω j $_{b}$(t ) σ j $_{b}$] + ℏ g ∑ j Γ$_{j}$ ( t )(ˆ a † ˆ σ j − e − i ⃗ $^{k}$· ⃗ r$_{j}$ + ˆ σ j $_{+}$ˆ ae i ⃗ $^{k}$· ⃗ $^{r$_{j}$}$) , (1)