Correlations and linewidth of the atomic beam continuous superradiant laser

We first thank the referee for his detailed and valuable comments. We appreciate his general very positive assessment of our work which aimed at accounting as simply as possible for the main properties of a beam-based superradiant laser.
About the referee main question:
First, it is important to clarify that an atom entering the cavity does not merely undergo Rabi oscillations in the cavity field. This picture can be used as a toy model to guess the main physical properties and/or regimes, but does not accurately picture the physics. Rather, the system is a many-body system, and the field is created by the atoms themselves such that the picture of atoms coupled to a pre-existing cavity field should be taken with a grain of salt. This is especially true because in the superradiant regime, the number of photons is smaller than the number of atoms, so that describing atoms as merely undergoing Rabi oscillation in a pre-existent field is necessarily insufficient. As a consequence, it does not appear to us that one can necessarily reach the regime where each atom will send exactly one atom in the cavity.
Let us however stick to this picture to try and answer the referee’s question on why we assume $<s^-_{j-N}>=< S^- >/N$. The Rabi frequency can be written as $\Omega = g b$ where $b$ is the cavity field. Using Eq. 12 and 23, we have $(\Omega/\Gamma_R)^2 \approx N g^2 / \kappa \Gamma_R$. Therefore, in the relevant regime for steady-state superradiance, $\Omega / \Gamma_R >>1$. That is to say: each atom has had the time to undergo many cycles of Rabi oscillation before leaving the cavity after a time $1/\Gamma_R = w_0/v$ (w_0 is the waist of the mode, and v the velocity of atoms). Our main answer to the referee’s question is a follow up on the following sentence from the paper : « the many atoms entering the cavity (in practice at random times and random velocities) follow different trajectories, such that, at the exit, the statistical mean for such random realizations is identical to the average inside the cavity (ergodicity argument). » Let us try and be more precise: if the beam has a spread in velocities $\delta_v$ (as it will always have), the uncertainty on the phase of out-coupled atom is given by $\delta ( \Omega / \Gamma_R) = \Omega / \Gamma_R \times \delta v/v$. Given that $\Omega / \Gamma_R >>1$, a relatively small $\delta v / v$ is sufficient to insure that the atoms leave the cavity at a random phase of their Rabi oscillation, which justifies the claim $<s^-_{j-N}>=< S^- >/N$. We have added a footnote to provide such a reasoning and better justify the approximation (footnote [24]). Note that in addition to this argument, if a 3D cylindrically symmetric cavity mode is assumed (as will be in reality), atoms crossing at different distances from the cavity axis will experience different interaction times, and exit the cavity at different phases of their Rabi cycle.
Note finally that our other guess for a possibility to justify $<s^-_{j-N}>=< S^- >/N$ was « This assumption requires (either) that each atom reaches a steady state shortly after entering the cavity » but did not imply that we had in mind an additional dephasing mechanism compared to those already included in the model. As we explain above, the system cannot be described by a simple Rabi oscillation, and we simply do not know whether the dynamics of a given atom within an ensemble of many atoms can lead to an apparent steady state shortly after entering the cavity. We simply meant to point this as a possibility.
About the referee’s other comments :
(a) Indeed Eq. (20) includes dissipative terms and therefore is not related to a \textit{local} hermitian Hamiltonian. However, our approach is to consider a larger system (of infinite dimension) where, a priori, all atoms that will ever cross the cavity are included, and which also a priori contain all possible electromagnetic modes. Our point is that the system can then formally be considered as Hamiltonian. Only when we focus on the local system at a time t, containing N atoms, do we recover Eq. (20), after some explicit approximations made in the manuscript.
(b) We have now properly defined the spin operators in terms of projectors (now page 3), as well as the variance and covariance (now page 21).
(c) Indeed, the outcome of the integral is in general a bit singular. The key point is that the model only considers propagation in 1D, and in vacuum, so that the density of states is constant and the frequency $\omega_k$ is strictly proportional to $k$. Note however, that things can be more complicated when the coupling $\Omega$ depends on $\omega_k$, which is not taken into account in our model, for simplicity. As the referee himself points out, however, the associated shift only renormalizes the cavity resonances, and therefore does not modify our description of the superradiant laser. In addition to the sentence in our paper “The coherent coupling $\Omega$ of the cavity mode to all these modes is assumed to be independent of k », we now have also written: “The first term is odd in $\omega_k$ so that the summation over $k$ is zero, and the associated frequency shift is therefore neglected.”
(d) We agree that $N g^2/\kappa$ can be small. We have modified the sentence accordingly: ''In other words, the laser spectrum is at best Fourier limited by the pulse envelope. For metrological applications, it can therefore be useful to reach a sustained or CW regime, in order to further reduce the linewidth.''
(e) We have added the word “qualitatively”.
(f) We do not agree with the referee that our equations are identical to the trapped case equations. In the trapped case, repumping only affects ground-state atoms, whereas our loss mechanism (atoms leaving the cavity) is independent of the atomic internal state. This translates into slightly different equations, and different behavior. As expected, the only difference arises from the repumping/loading term. In the trapped case, the repumping leads to $dS^-/dt = -w/2 S^-$ and $dS^z/dt = -w S^-$ whereas in the case of losses $dS^-/dt = -Gamma/N S^-$ and $dS^z/dt = -Gamma/N S^-$. There is a factor of two difference in the ratio of decoherence in the $z$ and $–$ components of the collective spin in the trapped case, which is not the case in our setting. Both cases are therefore different, which is for example highlighted in the paper by the sentence “Indeed, repumping atoms removes one atom in the ground state to create an atom in the excited state, while the re-loading approach corresponds to losing one atom (irrespective of its internal state) and gaining another atom in the excited state.” In addition, on a less formal level, there is also a key practical distinction: repumping $w$ in the trapped case qualitatively corresponds to the loading rate divided by $N$ in the beam case, $\Gamma/N$. Therefore, the atom number scales differently with the tunable experimental knobs; in practice for example, this is why the light intensity scales as $N$ in the beam architecture, and $N^2$ in the trapped case.