Correlated steady states in continuously pumped and probed atomic ensembles

In this section, we recapitulate the two-level-system model of the superradiant laser and its most general features, which was treated in great detail in meiser_prospects_2009; kolobov_role_1993; bohnet_steady-state_2012. The general setup is shown schematically in Fig. 1a. We consider an ensemble of $N$ two-level atoms placed in a cavity with linewidth $\kappa$. The cavity frequency $\omega_{c}$ is set to be resonant with the atom transition frequency ${\omega_{\text{eg}}}$, i.e. $\omega_{c}={\omega_{\text{eg}}}$. The transition between the ground state $\left|g\right\rangle$ and the excited state $\left|e\right\rangle$ couples to the cavity mode $\hat{a}$ with a single photon Rabi frequency $\Omega/2$, cf. Fig. 1b. The system is described by the Lindblad master equation

	$\displaystyle\frac{d}{dt}\rho$	$\displaystyle=-i\frac{{\omega_{\text{eg}}}}{2}\left[J^{z}+\hat{a}^{\dagger}\hat{a},\,\rho\right]-i\frac{\Omega}{2}\left[J^{+}\hat{a}+J^{-}\hat{a}^{\dagger},\,\rho\right]$
		$\displaystyle\phantom{=}\,+w\sum_{i}{\mathcal{D}}\left[\sigma^{+}_{i}\right]{\rho}+\kappa{\mathcal{D}}\left[\hat{a}\right]{\rho},$		(1)

where ${\mathcal{D}}\left[\hat{A}\right]{\rho}=A\rho A^{\dagger}-1/2\left[A^{\dagger}A,\,\rho\right]_{+}$ is a Lindblad superoperator, $J^{z}=\sum_{i=1}^{N}\sigma_{i}^{z}$ and $J^{\pm}=\sum_{i=1}^{N}\sigma_{i}^{\pm}$ are collective spin operators, written in terms of the Pauli operator $\sigma^{z}_{i}$ and the ladder operators $\sigma^{+}_{i}=(\sigma^{-}_{i})^{\dagger}$ for the $i$-th atom. In addition to the cavity decay, the model takes in an incoherent, non-collective pumping process causing population inversion at an effective rate $w$ . This pumping process could correspond e.g. to an additional $\Lambda$-type two-photon process involving a laser assisted excitation followed by a spontaneous emission as shown in Fig. 1c.

The finite linewidth of the atomic transition could be reflected in an additional Lindblad term in Eq. (1). However, the physics of the superradiant laser relies in particular on the excited state being long lived on the scale of the cavity decay rate. In this limit spontaneous decay plays a minor role and we choose to suppress it here for the sake of clarity. Its role has been dicussed carefully in meiser_prospects_2009; meiser_steady-state_2010; meiser_intensity_2010 where long lived transitions in Alkaline earth atoms were considered. Another realization of narrow band transitions can be found in $\Lambda$-type Raman transitions as shown in Fig. 1d. This corresponds also to the way the first proof-of-principle realizations of superradiant (Raman) lasing have been achieved bohnet_steady-state_2012; bohnet_active_2013. We note that the following section will expand on this correspondence, and investigate more complicated two-photon transitions and lasing transitions in multi-level atoms.

In contrast to the conventional laser, the superradiant laser relies on collective effects in the atomic medium to store its coherence, instead of relying on the long coherence time of photons inside the cavity bohnet_steady-state_2012. Therefore, we consider the atomic ensemble coupling to the light field in an extreme bad-cavity regime. In this regime the cavity decay is much faster than all other processes, i.e. $\kappa\gg w,\Omega$, and can be adiabatically eliminated meiser_prospects_2009, resulting in the permutation invariant master equation, taken here in a frame rotating at the atomic transition frequency ${\omega_{\text{eg}}}$,

$\displaystyle\frac{d}{dt}\rho$

$\displaystyle=w\sum_{i}{\mathcal{D}}\left[\sigma^{+}_{i}\right]{\rho}+\gamma{\mathcal{D}}\left[J^{-}\right]{\rho}$

(2)

with the rate $\gamma=\Omega^{2}/\kappa$ of the collective decay term. From (2) the evolution of the expectation values of $\left\langle\sigma^{z}_{1}\right\rangle$ and $\left\langle\sigma^{+}_{1}\sigma^{-}_{2}\right\rangle$ follows as

	$\displaystyle\frac{d}{dt}\left\langle\sigma^{z}_{1}\right\rangle$	$\displaystyle=w\left(1-\left\langle\sigma^{z}_{1}\right\rangle\right)-\gamma\left(1+\left\langle\sigma^{z}_{1}\right\rangle\right)-2N\gamma\left\langle\sigma^{+}_{1}\sigma^{-}_{2}\right\rangle,$
	$\displaystyle\frac{d}{dt}\left\langle\sigma^{+}_{1}\hat{\sigma}^{-}_{2}\right\rangle$	$\displaystyle=\Big{\{}N\gamma\left\langle\sigma^{z}_{1}\right\rangle-\left(w+\gamma\right)\Big{\}}\left\langle\sigma^{+}_{1}\sigma^{-}_{2}\right\rangle+\frac{\gamma}{2}\left(\left\langle\sigma^{z}_{1}\right\rangle+1\right)\left\langle\sigma^{z}_{1}\right\rangle,$		(3)

where we used the cumulant expansion and factorized $\left\langle\sigma^{+}_{1}\sigma^{-}_{2}\sigma^{z}_{3}\right\rangle=\left\langle\sigma^{+}_{1}\sigma^{-}_{2}\right\rangle\left\langle\sigma^{z}_{1}\right\rangle$ and $\left\langle\sigma^{z}_{1}\right\rangle\left\langle\sigma^{z}_{2}\right\rangle=\left\langle\sigma^{z}_{1}\right\rangle^{2}$ due to negligible cumulants $\left\langle\sigma^{z}_{1}\sigma^{z}_{2}\right\rangle_{c}$ and $\left\langle\sigma^{+}_{1}\sigma^{-}_{2}\sigma^{z}_{3}\right\rangle_{c}$, cf xu_synchronization_2014.

The steady state expectation values can be obtained by setting the left hand sides of the equations (3) to zero and solving the resulting quadratic equation. Fig. 2 shows the characteristic linearly increasing polarization $\left\langle\sigma^{z}_{1}\right\rangle$ (blue solid line) and the inverted parabola of the correlations $\left\langle\sigma^{+}_{1}\sigma^{-}_{2}\right\rangle$ (red dashed line) over the single-atom pumping rate $w/N\gamma$. As one can see, the non-zero two-atom correlations for a large atomic ensemble ($N\gg 1$) corresponding to the superradiant laser regime exist only when the pumping $w$ fulfills the inequalities

$\displaystyle\gamma<w<N\gamma.$

(4)

At the lower threshold ($w=\gamma$), the pumping overcomes the atomic losses, and the population inversion is established. At the same time, the two-atom correlations build up signifying the onset of superradiance, i.e. atom decay rate $\gamma$ through the cavity is enhanced by a factor proportional to $N$ (cf. Fig. 1a). It is the minimum condition for lasing, which is in contrast to a conventional laser where the threshold is obtained when the pumping overcomes the cavity losses. At the upper threshold ($w=N\gamma$), the two-atom correlations vanish due to the noise imposed by the pumping. Thus, in this case the ensemble consists of random radiators producing thermal light.

The spectrum of light leaving the cavity is $S(\omega)=\mathcal{F}[\left\langle{\hat{a}}^{\dagger}(t)\hat{a}(0)\right\rangle](\omega)=\frac{\Omega^{2}}{\kappa^{2}}\mathcal{F}[\left\langle J^{+}(t)J^{-}(0)\right\rangle](\omega)$ where $\mathcal{F}$ denotes Fourier transform ¹¹1We use the convention $\mathcal{F}[f(t)](\omega)=\frac{1}{\sqrt{2\pi}}\int^{+\infty}_{-\infty}\;dte^{-i\omega t}f(t)$. The equation of motion for the two-time collective dipole correlation function

$\displaystyle\frac{d}{dt}\left\langle J^{+}(t)J^{-}(0)\right\rangle$

$\displaystyle=\left(i\delta-\frac{\Gamma}{2}\right)\left\langle J^{+}(t)J^{-}(0)\right\rangle$

(5)

with $\Gamma=w+\gamma-(N-1)\gamma\left\langle\sigma^{z}_{1}\right\rangle$ follows from the Quantum Regression Theorem carmichael_open_1993. As a result, the spectrum of the output light of the cavity is Lorentzian with a linewidth $\Gamma$, which is on the order of $\gamma$ meiser_prospects_2009.

At the pumping strength $w_{\mathrm{opt}}=N\gamma/2$ the atom-atom correlations $\left\langle\sigma^{+}_{1}\hat{\sigma}^{-}_{2}\right\rangle$ reach their maximum, meaning optimal synchronization of the dipole moment of individual atoms and a corresponding maximal collective atomic dipole moment. This results in the maximal intensity and the relatively narrow linewidth of the output laser light bohnet_steady-state_2012. Thus, the superradiant laser regime corresponds to a quite delicate balance between the collective and non-collective processes given in Eq. (2).

We now consider a generalization of the superradiant laser master equation where we allow for additional processes which can arise in more complicated level schemes such as shown in Fig. 3a. These processes correspond to counter-rotating terms in the picture of the effective two-level system, cf. Fig. 3b, which may still arise as resonant processes from suitable $\Lambda$-type transitions. Thus, we consider a Jaynes-Cummings like coupling of each atom to the cavity mode at (effective) single photon Rabi rate $\Omega_{+}$ and an anti-Jaynes-Cummings type interaction at rate $\Omega_{-}$. Moreover, we also account for individual pumping at rate $w_{+}$ and at individual depumping from the excited to the ground state at rate $w_{-}$. All of these processes may arise in double-$\Lambda$ like transitions as shown in Fig. 3 and in more complex level schemes as discussed in Sec. III.

After eliminating the excited states, the master equation that accounts for these additional processes in the effective two-level system corresponds to,

	$\displaystyle\frac{d}{dt}\rho$	$\displaystyle=-i\frac{{\omega_{\text{eg}}}}{2}\left[J^{z},\,\rho\right]-i\left[\left(\frac{\Omega_{+}}{2}J^{+}+\frac{\Omega_{-}}{2}J^{-}\right)\hat{a}+h.c.,\,\rho\right]$
		$\displaystyle\phantom{=}\,+w_{+}\sum_{i}{\mathcal{D}}\left[\sigma^{+}_{i}\right]{\rho}+w_{-}\sum_{i}{\mathcal{D}}\left[\sigma^{-}_{i}\right]{\rho}+\kappa{\mathcal{D}}\left[\hat{a}\right]{\rho}.$		(6)

Here, ${\omega_{\text{eg}}}$ accounts for a (possible) energy difference between the two states which physically corresponds to an energy splitting between the two ground states. Considering the cavity decay as the fastest timescale, i.e., $\kappa\gg\Omega_{\pm},{w_{\pm}}$, we perform its adiabatic elimination as before, resulting in a field that is slaved to the collective atomic dipole of the atomic ensemble, $\hat{a}\simeq-i(\Omega_{+}J_{+}+\Omega_{-}J_{-})/\kappa$. The master equation for atoms only becomes

$\displaystyle\frac{d}{dt}\rho$

$\displaystyle=w_{+}\sum_{i}{\mathcal{D}}\left[\hat{\sigma}^{+}_{i}\right]{\rho}+\gamma_{-}{\mathcal{D}}\left[\hat{J}^{-}\right]{\rho}+w_{-}\sum_{i}{\mathcal{D}}\left[\hat{\sigma}^{-}_{i}\right]{\rho}+\gamma_{+}{\mathcal{D}}\left[\hat{J}^{+}\right]{\rho}$

(7)

with rates $\gamma_{-}=\Omega_{-}^{2}/\kappa$ and $\gamma_{+}=\Omega_{+}^{2}/\kappa$ of the collective terms.

The first two terms are identical to the simplified model of the superradiant laser, which were considered in the previous section, while the third and fourth can be regarded as a superradiant laser with interchanged levels. The model considered here thus is unchanged by relabelling $+\leftrightarrow-$. We exploit this symmetry here and assume without loss of generality that the single atom pumping generates population inversion in $\left|e\right\rangle$, that is $w_{+}>w_{-}$. Furthermore, we are interested in the regime of $w_{+},w_{-}\gg\gamma_{-},\gamma_{+}$ where only collectively enhanced rates $N\gamma_{\pm}$ are comparable to $w_{\pm}$.

We proceed as in the previous section, and derive the evolution of expectation values from (7)

	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left\langle\sigma^{z}_{1}\right\rangle$	$\displaystyle=w_{+}\left(1-\left\langle\sigma^{z}_{1}\right\rangle\right)-w_{-}\left(1+\left\langle\sigma^{z}_{1}\right\rangle\right)$
		$\displaystyle\phantom{=}\,-2(N-1)\left(\gamma_{-}-\gamma_{+}\right)\left\langle\sigma^{+}_{1}\sigma^{-}_{2}\right\rangle$
	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left\langle\sigma^{+}_{1}\sigma^{-}_{2}\right\rangle$	$\displaystyle=\Big{\{}(N-2)\left(\gamma_{-}-\gamma_{+}\right)\left\langle\sigma^{z}_{1}\right\rangle$
		$\displaystyle\phantom{=}\,-\left(w_{+}+w_{-}+\gamma_{-}+\gamma_{+}\right)\Big{\}}\left\langle\sigma^{+}_{1}\sigma^{-}_{2}\right\rangle$
		$\displaystyle\phantom{=}\,+\frac{1}{2}\left((\gamma_{-}-\gamma_{+})+(\gamma_{-}+\gamma_{+})\left\langle\sigma^{z}_{1}\right\rangle]\right)\left\langle\sigma^{z}_{1}\right\rangle$		(8)

where we factorized $\left\langle\sigma^{z}_{1}\sigma^{z}_{2}\right\rangle\approx\left\langle\sigma^{z}_{1}\right\rangle^{2}$ and $\left\langle\sigma^{+}_{1}\sigma^{-}_{2}\sigma^{z}_{3}\right\rangle\approx\left\langle\sigma^{+}_{1}\sigma^{-}_{2}\right\rangle\left\langle\sigma^{z}_{1}\right\rangle$ as in xu_synchronization_2014 assuming negligible cumulants $\left\langle\sigma^{z}_{1}\sigma^{z}_{2}\right\rangle_{c}$, $\left\langle\sigma^{+}_{1}\sigma^{-}_{2}\sigma^{z}_{3}\right\rangle_{c}$.

The steady state expectation values can be obtained by setting the left hand sides of Eqs. (8) to zero and solving the resulting quadratic equation. The steady state solution of $\langle\sigma^{+}_{1}\sigma^{-}_{2}\rangle$ shows that atom-atom correlations, witnessing the regime of superradiant lasing, exist if and only if the single-atom pumping rate $w_{+}$ fulfills the inequality

$\displaystyle w_{+}<N\left(\gamma_{-}-\gamma_{+}\right)\frac{w_{+}-w_{-}}{w_{+}+w_{-}}-w_{-}.$

(9)

Here, we assume the limit of a large atom numbers $N\gg 1$ and restrict equations to leading order in $1/N$. Moreover, we assume strong pumping towards level inversion, $w_{+}\gg\gamma_{\pm}$. In comparison with (4), the threshold condition for the generalized superradiance laser has a nonlinear dependence on $w_{+}$ for the upper and for the lower bounds. We also recall that we took $w_{+}>w_{-}$ and conclude that a dominant collective emission rate $\gamma_{-}>\gamma_{+}$ is necessary for superradiance to occur.

Compared to the model of the superradiant laser, which depended only on the ratio $w/N\gamma$, there are now four independent parameters $N\gamma_{\pm}$ und $w_{\pm}$. It will be useful to discuss the steady state physics in terms of the ratios of single atom pumping to collective decay, $w_{+}/N\gamma_{-}$, collective excitation to collective decay, $\gamma_{+}/\gamma_{-}$, and single atom depumping to collective decay, $w_{-}/N\gamma_{-}$.

Fig. 4 illustrates the case where $\gamma_{+}/\gamma_{-}=0$, and shows atomic polarization and dipole correlations versus $w_{+}/N\gamma_{-}$, in analogy to what was shown for the superradiant laser in Fig. 2. The overall behaviour is similar, but in comparison, the superradiant regime is somewhat reduced for nonzero single atom depumping $w_{-}/N\gamma_{-}$, as is to be expected. Fig. 5 provides a more complete overview, and shows the steady state polarization $\langle\sigma^{z}_{1}\rangle$ and atom-atom correlation $\langle\sigma^{+}_{1}\sigma^{-}_{2}\rangle$ versus individual pumping $w_{+}/N\gamma_{-}$ and collective decay rate $N\gamma_{-}/\gamma_{-}$. The left (right) column of Fig. 5 refers to vanishing (nonzero) single atom depumping $w_{-}/N\gamma_{-}$. The Figure illustrates the superradiant domain and shows that it is excellently characterized by condition (9). Most importantly, Fig. 5 reveals a rich dependence of the steady state properties on the ratio of collective excitation and decay ratios $\gamma_{+}/\gamma_{-}$. Corresponding cuts along this axis are shown in Fig. (6). The behaviour of the system along these will be of importance for our discussion of multi-level atoms in the next section, where we will show that geometrical aspects of the light-matter interactions determine the ratio of the rates $\Omega_{\pm}$ and with it the ratio of $\gamma_{\pm}$. The maximal atom-atom correlations are

$\displaystyle\max_{w_{+}}\left\langle\sigma^{+}_{1}\sigma^{-}_{2}\right\rangle=\frac{1}{8}-\frac{w_{-}}{N\gamma_{+}}\frac{1}{\frac{\gamma_{-}}{\gamma_{+}}-1}$

in leading order in $1/N$, for $\gamma_{+}\geq\gamma_{-}$ at the optimal pumping strength $w_{+,\text{opt}}=N\left(\gamma_{-}-{\gamma_{+}}\right)/2-w_{-}$.

In the generalized superradiant laser the linewidth is still on the order of the atomic linewidth $\gamma_{-}$, even though the ensemble is incoherently pumped with a much stronger rate $w_{+}$. We will now show that we essentially have two superradiant lasing transitions, $\left|e\right\rangle\to\left|g\right\rangle$ and $\left|g\right\rangle\to\left|e\right\rangle$, radiating at the same time with identical linewidth of the order of $\gamma_{-}$, but different intensities.

As before, the spectrum of the output light $S(\omega)=\mathcal{F}[\left\langle{\hat{a}}^{\dagger}(t)\hat{a}(0)\right\rangle](\omega)=\frac{\gamma_{+}}{\kappa}\mathcal{F}[\left\langle J^{+}(t)J^{-}(0)\right\rangle](\omega)+\frac{\gamma_{-}}{\kappa}\mathcal{F}[\left\langle J^{-}(t)J^{+}(0)\right\rangle](\omega)$ is evaluated using the Quantum Regression Theorem gardiner_quantum_2004 based on the equation of motion

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\left\langle J^{-}(t)J^{+}(0)\right\rangle$

$\displaystyle=\left(-{\text{i}}\nu-\frac{\Gamma}{2}\right)\left\langle J^{-}(t)J^{+}(0)\right\rangle.$

with linewidth $\Gamma=\gamma_{-}+\gamma_{+}+w_{+}+w_{-}-(N-1)\left(\gamma_{-}-\gamma_{+}\right)\left\langle\sigma^{z}_{1}\right\rangle$. The corresponding spectrum is given by two Lorentz functions at $\pm\nu$ with heights $S_{\pm}$ and identical linewidth (Full-width at half maximum) $\Gamma$. The linewidth $\Gamma$ for $w_{+}>w_{-}$ to leading order $1/N$ is

$\displaystyle\frac{\Gamma}{\gamma_{-}}$

$\displaystyle\approx\frac{W_{+}+W_{+}W_{-}(W_{-}-W_{+}W_{-}-1)}{\left(W_{+}-1\right)\left(W_{-}-1\right)W_{-}}\left(1-\frac{\gamma_{+}}{\gamma_{-}}\right),$

where we defined the dimensionless variables

$\displaystyle W_{\pm}:=\frac{\left(w_{+}\pm w_{-}\right)\left(w_{+}+w_{-}\right)}{N\left(\gamma_{-}-\gamma_{+}\right)\left(w_{+}-w_{-}\right)}.$

The linewidth $\Gamma$ is shown in Figs. 5, panel c) and f). We see that the generalized superradiant laser preserves the remarkable feature of the superradiant laser – the linewidth on the order of the effective atomic decay rate $\gamma_{-}$ – even for non-vanishing $\gamma_{+}$. And even with additional single-atom depumping rate $w_{-}$ the linewidth increases only slightly (see Fig. 5f). The spectrum exhibits two asymmetric peaks at the sideband frequencies $\omega=\pm\nu$. The ratio of the sideband intensities $S_{\pm}$ is given by

$\displaystyle\frac{S_{+}}{S_{-}}$

$\displaystyle=\frac{\gamma_{+}}{\gamma_{-}}\frac{1}{1-\frac{\left\langle J^{z}\right\rangle}{\left\langle J^{+}J^{-}\right\rangle}}\approx\frac{\gamma_{+}}{\gamma_{-}}$

(10)

for $N\gg 1$ in the superradiant regime, which is simply the ratio of collective emission rates at the sidebands. We want to point out that equal Lorentz peak height is not possible, because the superradiant condition (9) can not be fulfilled for $\gamma_{+}=\gamma_{-}$.

The model of the generalized superradiant laser will be a helpful reference to understand the physics of continuously pumped and probed atomic ensembles, which will be treated in the following.

We consider $N$ Alkali atoms which are continuously probed by an off-resonant laser of wavelength $\lambda_{c}$ propagating in $z$ direction with linear polarization enclosing an angle $\theta$ relative to the $x$-axis, the axis of mean atomic polarization, cf. Fig. 7. The laser couples off-resonantly to one of the atomic $D$-lines with ground state spin $F$ and excited state spins $F^{\prime}$. The respective Zeeman states will be denoted by $\left|F,m_{F}\right\rangle$ and $\left|F^{\prime},m_{F^{\prime}}\right\rangle$. We assume a spatial distribution of atoms exhibiting a large optical depth $D=\frac{N\sigma_{0}}{A}$ along the axis of the probe field. Here, $\sigma_{0}=\frac{3\lambda^{2}}{2\pi}$ is the scattering cross section on resonance, and $A$ is the beam cross section. In this limit, the scattering of photons in the $z$ direction occurs in the same spatial mode, which we model here by a cavity mode with linewidth $\kappa$ to which all atoms couple equally loudon_quantum_2000. This (virtual) cavity mode is then adiabatically elimated in the limit $\kappa\rightarrow\infty$, which yields a master equation for the atoms that represents collective emissions in the $z$ direction in free space. Scattering in all other directions is non-collective, and will be covered by suitable Lindblad terms in the master equation. Regarding motion of atom, we will follow the approximations of hammerer_quantum_2010 suitable for treating an ensemble of thermal atoms in a cell. Through thermal averaging, the motion of the atoms is almost decoupled from their spin and the forward-scattered photons.

Our starting point is the master equation for $N$ atoms interacting with the electromagnetic field in dipole approximation. In the electromagnetic field we distinguish the forward scattering modes, which are modelled as a running wave cavity, and all other field modes,

$\displaystyle\dot{\rho}$

$\displaystyle=\frac{1}{{\text{i}}\hbar}\left[H_{\mathrm{at}}+H_{\mathrm{cav}}+H_{{\mathrm{field}}}+H_{{\mathrm{int}}},\,\rho\right]+\kappa{\mathcal{D}}\left[{\hat{a}}\right]{\rho}.$

(11)

The Hamiltonians are

	$\displaystyle H_{\mathrm{at}}$	$\displaystyle=\hbar\sum_{i=1}^{N}\Bigg{\{}\sum_{F^{\prime},m_{F^{\prime}}}\omega^{\prime}_{F^{\prime},m_{F^{\prime}}}\left|F^{\prime},m_{F^{\prime}}\right\rangle\left\langle F^{\prime},m_{F^{\prime}}\right|_{i}$
		$\displaystyle\phantom{=}\,+\sum_{\begin{subarray}{c}m_{F}\end{subarray}}\omega_{F,m_{F}}\left|F,m_{F}\right\rangle\left\langle F,m_{F}\right|_{i}\Bigg{\}}$
	$\displaystyle H_{\mathrm{cav}}$	$\displaystyle=\hbar\omega_{\text{c}}\hat{a}^{\dagger}{\hat{a}}$
	$\displaystyle H_{\mathrm{field}}$	$\displaystyle=\hbar\sum_{\lambda}\int\mathrm{d}\bm{k}\;\omega_{\bm{k}}\hat{a}^{\dagger}_{\bm{k},\lambda}{\hat{a}}_{\bm{k},\lambda}$
	$\displaystyle H_{\mathrm{int}}$	$\displaystyle=\sum_{i=1}^{N}\sum_{F^{\prime}}{\bm{{E}}}^{-}(\bm{r}_{i},t)\bm{d}^{-}_{i,FF^{\prime}}+\text{h.c.}.$

We expand the electric field into its coherent ($\mathds{C}$-number) component, the (quantized) cavity field orthogonally polarized to it, and all other field modes, which are, respectively,

	$\displaystyle{\bm{{E}}}^{-}(\bm{r},t)$	$\displaystyle={{\bm{\mathcal{E}}}}^{-}(z,t)+{\bm{{E}}}_{\mathrm{cav}}^{-}(z)+\bm{E}_{\mathrm{field}}^{-}(\bm{r}),$		(12)
	$\displaystyle{{\bm{\mathcal{E}}}}^{-}(z,t)$	$\displaystyle=\rho_{c}\sqrt{\Phi}e^{-{\text{i}}\left(k_{c}z-\omega_{\text{c}}t\right)}\bm{e}_{c},$		(13)
	$\displaystyle{\bm{{E}}}_{{\mathrm{cav}}}^{-}(z,t)$	$\displaystyle=\rho_{q}\hat{a}^{\dagger}e^{-{\text{i}}k_{c}z}\bm{e}_{q},$		(14)
	$\displaystyle\bm{E}_{\mathrm{field}}^{-}(\bm{r})$	$\displaystyle=\sum_{\lambda}\int\mathrm{d}\bm{k}\;\rho_{\omega}\hat{a}^{\dagger}_{\bm{k},\lambda}e^{-{\text{i}}\bm{k}\bm{r}}\bm{e}_{\bm{k},\lambda}.$		(15)

Here, $\rho_{c}=\sqrt{\frac{\hbar\omega_{\text{c}}}{2\epsilon_{0}cA}}$, $\rho_{q}=\sqrt{{\kappa}{}}\rho_{c}/2$, and $\rho_{\omega}=\sqrt{\frac{\hbar\omega}{2\epsilon_{0}(2\pi)^{3}}}$ is the electrical field per photon for classical, cavity and free field, respectively. $\omega_{\text{c}}$ is the laser frequency, $k_{c}$ its wave number, $\Phi$ the photonflux, and $\bm{e}_{c}=(\begin{matrix}\cos\theta&\sin\theta&0\end{matrix})^{T}$ the linear polarization of the laser field. Regarding the forward scattered quantum field (i.e. the cavity field), $\bm{e}_{q}=(\begin{matrix}-\sin\theta&\cos\theta&0\end{matrix})^{T}$ is the linear polarization vector orthogonal to $\bm{e}_{c}$, and $\kappa$ the cavity line width. Since we are eventually interested in the free space limit $\kappa\rightarrow\infty$, we take the cavity resonance frequency $\omega_{\text{c}}$ to be identical to the laser frequency. The dipole operator in $H_{\mathrm{int}}$ is expanded as $\bm{d}_{i}=\bm{d}^{+}_{i,F^{\prime}F}+\bm{d}^{-}_{i,F^{\prime}F}$ as $\bm{d}^{+}_{i,F^{\prime}F}=\pi_{i}^{F^{\prime}}\bm{d}_{i}\pi_{i}^{F}$, where $\pi_{i}^{F}=\sum_{m_{F}}\left|F,m_{F}\right\rangle\left\langle F,m_{F}\right|_{i}$ are projectors in the spin-$F$-subspace of atom $i$, and $\bm{d}^{-}_{i,F^{\prime}F}=(\bm{d}^{+}_{i,F^{\prime}F})^{\dagger}$.

In a first step, we consider the dispersive limit of light-matter interaction where the detuning $\Delta$ of the laser frequency $\omega_{\text{c}}$ from the closest atomic $F\leftrightarrow F^{\prime}$ transition is large, and only resonant two photon transitions can occur. In this limit, the excited states can be adiabatically eliminated reiter_effective_2012. In the same step, we eliminate the field modes in Born-Markov approximation breuer_theory_2007. This results in a master equation for the ground state spins $F$ and the cavity mode, covering forward scattering of photons,

	$\displaystyle\dot{\rho}$	$\displaystyle=\frac{1}{{\text{i}}\hbar}\left[H_{{\mathrm{at}},g}+H_{\mathrm{cav}}+H^{\text{eff}}_{\mathrm{int}},\,\rho\right]+\kappa{\mathcal{D}}\left[{\hat{a}}\right]{\rho}$
		$\displaystyle\quad+\sum_{i=1}^{N}\sum_{\mu=1}^{3}{\mathcal{D}}\left[L_{{\mathrm{at}},i,\mu}^{\text{eff}}\right]{\rho}.$		(16)

From the atomic Hamiltonian $H_{{\mathrm{at}}}$ only the ground state manifold remains,

$\displaystyle H_{{\mathrm{at}},g}$

$\displaystyle:=\hbar\sum_{i=1}^{N}\sum_{\begin{subarray}{c}m_{F}\end{subarray}}\omega_{F,m_{F}}\left|F,m_{F}\right\rangle\left\langle F,m_{F}\right|_{i}.$

Here the effective interaction Hamiltonian for the ground state spins with light is reiter_effective_2012

$\displaystyle H^{\text{eff}}_{\mathrm{int}}$

$\displaystyle\approx-\sum_{i=1}^{N}\left(\left({\bm{{E}}}_{\mathrm{cav}}^{-}(z_{i},t)\right)^{T}\mathchoice{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\displaystyle\alpha\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\textstyle\alpha\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptstyle\alpha\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptscriptstyle\alpha\hfil$\crcr}}}_{i}{{\bm{\mathcal{E}}}}^{+}(z_{i},t)+\text{h.c.}\right)$

where we use the polarizability tensor

$\displaystyle\mathchoice{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\displaystyle\alpha\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\textstyle\alpha\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptstyle\alpha\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptscriptstyle\alpha\hfil$\crcr}}}_{i}$

$\displaystyle:=\frac{\left|d\right|^{2}}{\hbar\Delta}\sum_{\begin{subarray}{c}k=0\end{subarray}}^{2}{s_{k}}\hat{T}_{i}^{(k)}$

(25)

with scalar, vector and tensor polarizability operators geremia_tensor_2006; brink_angular_1994

	$\displaystyle\hat{T}_{i}^{(0)}$	$\displaystyle=-\frac{1}{\sqrt{3}}\mathds{1}_{i},$
	$\displaystyle\hat{T}_{i}^{(1)}$	$\displaystyle=\frac{{\text{i}}}{\sqrt{2}}\bm{F}_{i}\bm{\times},$
	$\displaystyle\hat{T}_{i}^{(2)}$	$\displaystyle=\frac{1}{2}\left(2\bm{F}_{i}\otimes\,\bm{F}_{i}+{\text{i}}\bm{F}_{i}\bm{\times}.-\frac{2}{3}\left(\bm{F}_{i}\right)^{2}\mathds{1}_{i}\right).$

$d$ denotes the reduced dipole matrix element (in the convention of brink_angular_1994), and $s_{k}$ are dimensionless, real coefficients which depend on the detuning. For detunings much larger than the excited states’ hyperfine splitting, the tensor polarizability does not contribute, $s_{2}\rightarrow 0$ julsgaard_brian_2003. In the effective light matter interaction we keep terms linear in the coherent field, and drop terms which are quadratic in the coherent field (Stark shift of atomic levels) or in the quantum field (no mean field enhancement). The Stark shift is dropped here for simplicity, but could be easily taken into account in this framework. We note that in the Hamiltonian in Eq. (III.1) the atomic coordinates drop out, such that the atomic positions decouple from the dynamics. This is due to the fact that we consider forward scattering only.

The Lindblad terms in the second line of Eq. (16) account for individual spontaneous emission of each atom. The jump operators can be conveniently labelled in a Cartesian basis with index $\mu=1,2,3$,

$\displaystyle L_{\mathrm{at},i,\mu}^{\text{eff}}$

$\displaystyle\approx\sqrt{\gamma^{\prime}}\left(\mathchoice{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\displaystyle\alpha\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptstyle}\crcr\nointerlineskip\cr$\hfil\textstyle\alpha\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptstyle\alpha\hfil$\crcr}}}{\vbox{\halign{#\cr\leftrightarrowfill@{\scriptscriptstyle}\crcr\nointerlineskip\cr$\hfil\scriptscriptstyle\alpha\hfil$\crcr}}}_{i}{{\bm{\mathcal{E}}}}^{+}\right)_{\mu}$

where we define $\gamma^{\prime}=\frac{\omega_{\text{c}}^{3}}{6\pi\hbar\epsilon_{0}c^{3}}$. We note that, due to the structure of Lindblad terms, the second line of Eq. (16) is actually basis independent. A convenient choice will be to use $\left\{\bm{e}_{z},\bm{e}_{c},\bm{e}_{q}\right\}$.

In the next step we eliminate the cavity field in the free space limit based on the methods of breuer_theory_2007; gardiner_quantum_2004. The resulting master equation, written in a rotating frame with respect to $H_{{\mathrm{at}},g}$, in the limit $\kappa\rightarrow\infty$ is

$\displaystyle\dot{\rho}$

$\displaystyle=\sum_{\omega}\Bigg{\{}\gamma\,{\mathcal{D}}\left[V^{q}(\omega)\right]{\rho}+\gamma_{\mathrm{dec}}\sum_{i=1}^{N}\sum_{\begin{subarray}{c}\mu=c,q,z\end{subarray}}{\mathcal{D}}\left[V^{\mu}_{i}(\omega)\right]{\rho}\Bigg{\}}+w\sum_{i=1}^{N}{\mathcal{D}}\left[F_{i}^{+}\right]{\rho}$

We introduce here the dimensionless jump operators

	$\displaystyle V^{q}(\omega)$	$\displaystyle=\sum_{i=1}^{N}V^{q}_{i}(\omega),$		(26)
	$\displaystyle V^{\mu}_{i}(\omega)$	$\displaystyle=\sum_{\begin{subarray}{c}m_{F},m_{F}^{\prime}\\ \omega_{m_{F}}-\omega_{m_{F}^{\prime}}=\omega\end{subarray}}\left|m_{F}\right\rangle\left\langle m_{F}\right|V^{\mu}_{i}\left|m_{F}^{\prime}\right\rangle\left\langle m_{F}^{\prime}\right|$		(27)

for atom $i$ for $\mu\in\left\{c,q,z\right\}$. The sum is over all pairs $(m_{F},m^{\prime}_{F})$ with a given energy splitting $\hbar\omega=\hbar(\omega_{m_{F}}-\omega_{m^{\prime}_{F}})$, and

$\displaystyle V^{\mu}_{i}$

$\displaystyle:=\sum_{\begin{subarray}{c}k=0\end{subarray}}^{2}{s_{k}}\bm{e}_{\mu}^{T}\hat{T}_{i}^{(k)}\bm{e}_{c}.$

(28)

In Eq. (III.1) we introduced the decoherence rate due to spontaneous emission,

$\displaystyle\gamma_{\mathrm{dec}}=\frac{\Phi}{8}\frac{\sigma_{0}}{A}\left(\frac{\gamma_{0}}{\Delta}\right)^{2},$

and the rate of collective forward scattering,

$\displaystyle\gamma$

$\displaystyle=\frac{\Phi}{16}\left(\frac{\sigma_{0}}{A}\right)^{2}\left(\frac{\gamma_{0}}{\Delta}\right)^{2}.$

(29)

We use here the spontaneous emission rate $\gamma_{0}=\frac{\omega_{\text{c}}^{3}|d|^{2}}{3\pi\epsilon_{0}\hbar c^{3}}$. We note that due to the collective nature of the jump term associated with collective scattering, the effective rate of these processes is $N\gamma$. Therefore, the relative strength of collective scattering with respect to decoherence due to spontaneous emission,

$$\frac{N\gamma}{\gamma_{\mathrm{dec}}}=\frac{D}{2}$$

, becomes large for sufficiently large optical depth.

Furthermore, we add in the last line of Eq. (III.1) a Lindblad term accounting for optical pumping to the ground state with $m_{F}=F$. As explained earlier, we employ a phenomenological description for this process, as our main aim here is to provide a microscopic picture for the non-collective and collective effects of the continuous probe. The microscopic theory of optical pumping is of course well established, and can in principle be used to give a more realistic account than the minimal model used here. The master equation (III.1) is the main result of this section. For more details on its derivation we refer to roth_collective_2018.

It is instructive to consider in more detail the form of the collective jump operator in (26) which is a sum over single particle operators

$\displaystyle V^{q}_{i}$

$\displaystyle={{\text{i}}\frac{s_{1}}{2}\left(F_{i}^{-}-F_{i}^{+}\right)-s_{2}\left(\frac{{\text{i}}\cos(2\theta)}{\sqrt{2}}W_{1}+\frac{\sin(2\theta)}{4}W_{2}\right)},$

(30)

where we defined the operators

	$\displaystyle W_{1}$	$\displaystyle:={\left(F_{i}^{0}+\frac{1}{2}\right)F_{i}^{-}+\left(F_{i}^{0}-\frac{1}{2}\right)F_{i}^{+}},$
	$\displaystyle W_{2}$	$\displaystyle:={3\left(F_{i}^{0}\right)^{2}-\left(\bm{F}_{i}\right)^{2}+\left(F_{i}^{-}\right)^{2}+\left(F_{i}^{+}\right)^{2}}.$

The operator $W_{1}$ collects processes which change $m$ by $\pm 1$, and $W_{2}$ contains changes by $0$ or $\pm 2$. We emphasize that the $\theta$-dependence is an effect of the tensor component $\hat{T}^{(2)}$ in the polarizability tensor.

We will now evaluate the master equation in Eq. (III.1) for the case of spin $F=1$. In order to highlight the most important features more clearly, we deliberately omit the $W_{2}$ components in the jump operators in Eq. (30) for now. With this simplification, the master equation becomes

	$\displaystyle\dot{\rho}$	$\displaystyle=\frac{1}{i}\sum_{\begin{subarray}{c}i=1\end{subarray}}^{N}\left[\sum_{\begin{subarray}{c}m=-1\end{subarray}}^{1}\omega_{m}\left|m\right\rangle\left\langle m\right|_{i},\,\rho\right]$
		$\displaystyle\phantom{=}\,+\gamma{\mathcal{D}}\left[V^{+}(\theta)\right]{\rho}+\gamma{\mathcal{D}}\left[V^{-}(\theta)\right]{\rho}$
		$\displaystyle\phantom{=}\,+w_{+}\sum_{\begin{subarray}{c}i=1\end{subarray}}^{N}{\mathcal{D}}\left[F_{i}^{+}\right]{\rho}+w_{-}\sum_{\begin{subarray}{c}i=1\end{subarray}}^{N}{\mathcal{D}}\left[F_{i}^{-}\right]{\rho}.$		(31)

Here, the first term on the right hand side accounts for the splitting of the levels $\left|m\right\rangle$ in the external magnetic field with Zeemann energies $\omega_{m}$ where now $m=-1,0,1$. The terms in the second line represent the effect of collective scattering of photons in the $z$-direction. The collective jump operators depend on the angle $\theta$ between the polarizations of atoms and light, and are given by

$\displaystyle V^{\pm}(\theta)=\sum_{\begin{subarray}{c}i=1\end{subarray}}^{N}V^{\pm}_{i}(\theta)$

(32)

with single atom operators

$\displaystyle V^{\pm}_{i}(\theta)$

$\displaystyle=s_{1}\left(1+\epsilon\cos(2\theta)\left(\mp F_{i}^{0}+\mathds{1}/2\right)\right)F_{i}^{\pm}.$

(33)

We define $\epsilon=\sqrt{2}\left|s_{2}/s_{1}\right|$ measuring the relative weight of the ground states’ tensor to vector polarizability. In the limit of large detuning $\epsilon$ vanishes asymptotically. The terms in the last line describe individual optical pumping and depumping at rate $w_{\pm}$, respectively. As in the case of the generalized superradiant laser in Sec. II.2, we restrict the analysis to $w_{+}>w_{-}$. The collective jump operators are associated with transitions between Zeeman states $\left|n\right\rangle$ to $\left|m\right\rangle$ where $\Delta m=m-n=\pm 1$ for $V^{\pm}(\theta)$, respectively. It will be useful to define the single-atom transition rates for these transitions

	$\displaystyle\gamma_{m,n}$	$\displaystyle=\gamma\left|\left\langle m\right|V^{m-n}_{i}(\theta)\left|n\right\rangle\right|^{2}$		(34)
		$\displaystyle=\gamma s_{1}\left({1+\epsilon\cos(2\theta)\left(\mp m+1/2\right)}\right)^{2}$

It can be seen that the angle $\theta$ controls the balance between $\Delta m=\pm 1$ transitions. Fig. 8a illustrates how the relative weight of $\gamma_{0,\pm 1}$ and $\gamma_{\pm 1,0}$ shifts with $\theta$. From the discussion of the generalized superradiant laser model in Sec. II.2, it should be expected that the relative weight crucially determines the regimes of superradiance, as shown schematically in Fig. 8b.

As in the previous sections, the master equation (31) is solved for the steady state in a cumulant expansion. For this purpose, the master equation is expanded in an operator basis (with elements $A^{\alpha}_{i}$ for particle $i$), and $3$-particle correlators are approximated as $\left\langle A_{1}^{\alpha_{1}}A_{2}^{\alpha_{2}}A_{3}^{\alpha_{3}}\right\rangle\approx\left\langle A_{1}^{\alpha_{1}}A_{2}^{\alpha_{2}}\right\rangle\left\langle A_{3}^{\alpha_{3}}\right\rangle+\left\langle A_{1}^{\alpha_{1}}A_{3}^{\alpha_{3}}\right\rangle\left\langle A_{2}^{\alpha_{2}}\right\rangle+\left\langle A_{2}^{\alpha_{2}}A_{3}^{\alpha_{3}}\right\rangle\left\langle A_{1}^{\alpha_{1}}\right\rangle-2\left\langle A_{1}^{\alpha_{1}}\right\rangle\left\langle A_{2}^{\alpha_{2}}\right\rangle\left\langle A_{3}^{\alpha_{3}}\right\rangle$. From this approximate solution we can extract information on single particle observables such as level populations and mean polarization, as well as on the magnitude of two-particle correlations. The latter we quantify by the norm $||\tau_{2}||_{2}$ of $\tau_{2}=\rho_{2}-\rho_{1}\otimes\rho_{1}$, where $\rho_{n}$ denotes the $n$-body reduced density operator. The dependence of these quantities on the angle $\theta$ are shown in Fig. 9 and Fig. 10.

The mean polarization $\langle F_{i}^{0}\rangle/F$ along the $x$-direction strongly depends on the parameter $\theta$ as a result of the interplay between the optical pumping along $x$ and quantum jumps described by the collective jump operators $V^{\pm}_{i}(\theta)$. We can understand this behavior by considering each transition in Fig. 8b involving only two levels and comparing it with the condition for superradiance (9) of the generalized superradiant laser. For the upper transition $1\leftrightarrow 0$ and $\theta=0$ the collective emission with rate $\gamma_{0,1}$ is dominant, due to $\gamma_{0,1}/\gamma_{1,0}=\left((2+\epsilon)/(2-\epsilon)\right)^{2}>1$. This allows superradiance, meaning correlations between atoms build up and the atoms emit collectively such that the emitted intensity scales with $N^{2}$. For the upper transition and For $\theta>\pi/4$ the collective excitations are dominant, due to $\gamma_{0,1}/\gamma_{1,0}\leq 1$, meaning the superradiant condition (9) cannot be fulfilled. Tuning $\theta$ between $0$ and $\pi/4$ gives a polarization curve in Fig. 9 similar to Fig. 6. This similarity is somewhat surprising, as the change of $\theta$ in Fig.  9 entails a nonlinear change of the both rates $\gamma_{0,1}$, $\gamma_{1,0}$ (see Fig. 8), while in Fig. 6 only $\gamma_{+}$ is linearly changed. The lower transition $-1\leftrightarrow 0$ can fulfill the superradiant condition (9) only for $\theta>\pi/4$, with a maximum dominant collective down rate $\gamma_{-1,0}$ for $\theta=\pi/2$, resulting in a polarization similar to Fig. 6 with inverted $x$-Axis.

For both, transitions $-1\leftrightarrow 0$ and $0\leftrightarrow 1$, superradiance implies an enhanced collective jump rate proportional to $N$, necessarily decreasing the polarization $\langle F_{i}^{0}\rangle/F$. The superradiant transition in the red shaded region in Fig. 10 shifts much of the population from $\left|1\right\rangle$ to $\left|0\right\rangle$, as is shown in Fig. 10. The small change in population of $\left|-1\right\rangle$ is a result of the single-atom depumpings with rate $w_{-}$ shifting the population of $\left|0\right\rangle$ downwards. The superradiant transition in the blue shaded region in Fig. 10 shifts the population from $\left|0\right\rangle$ to $\left|-1\right\rangle$. The change in population of $\left|1\right\rangle$ is a result of the single-atom depumpings with rate $w_{-}$ shifting the population of $\left|1\right\rangle$ downwards.

The significant $\theta$-dependent redistribution of populations away from the fully polarized state is shown in Fig. 10. It is also clearly visible that the red shaded regime corresponding to superradiance of the $0\leftrightarrow 1$ transition involves a much larger population than the blue shaded regime corresponding to superradiance of the $-1\leftrightarrow 0$ transition. This will be visible also in terms of the intensity of collectively scattered photons.

The spectrum of light collectively scattered to the polarization orthogonal to the laser polarization, $S(\omega)\propto\sum_{i,j=1}^{N}\mathcal{F}\left[\left\langle V_{i}(\tau)V_{j}(0)\right\rangle\right]\left(\omega\right)$, follows from a Fourier transform of the atomic two-time correlation functions

	$\displaystyle\sum_{i,j=1}^{N}\left\langle V_{i}(t+\tau)V_{j}(t)\right\rangle$
	$\displaystyle=N(N-1)\left\langle V_{2}(t+\tau)V_{1}(t)\right\rangle+N\left\langle V_{1}(t+\tau)V_{1}(t)\right\rangle.$

Here we defined $V_{i}:=V_{i}^{+}(\theta)+V_{i}^{-}(\theta)$. In order to distinguish contributions from $0\leftrightarrow 1$ and $-1\leftrightarrow 0$ transitions in the spectrum we assume a nonlinear Zeeman splitting with, for concreteness, $\omega_{-1}=0$, $\omega_{0}=10\gamma$, and $\omega_{1}=30\gamma$. This particular level splitting is chosen here such that photons generated on the lower transition occur at a sideband frequency $\omega_{0}-\omega_{-1}=10\gamma$ and for the upper transition at $\omega_{1}-\omega_{0}=20\gamma$. The spectrum in Fig. 11 reveals clearly that for $0\leq\theta<\pi/4$ only the upper transition can be superradiant and for $\pi/4\leq\theta\leq\pi/2$ only the lower transition can be superradiant as expected from the superradiant condition (9) and indicated in figure 8 (b).

In addition, we extract the full-width at half maximum $\Gamma$ of the dominant Lorentz peak, as shown in Fig. 12. In the red and blue shaded superradiant regions, the linewidth $\Gamma$ is on the same order of magnitude as the collective jump rate $\gamma$. At $\theta=\pi/4$ the dynamics is well approximated by single atom dynamics for which the linewidth is given by $\Gamma=2\gamma+w_{-}+w_{+}$.

Finally, we consider as an example the case of the Caesium $D_{2}$-line with $F=4$ and $F^{\prime}=3,4,5$. Here, we consider the complete master equation (III.1) without any approximation. The steady state is determined as before in cumulant expansion assuming vanishing cumulants of three or more atoms, that is, keeping only two-atom correlations.

Because the full jump operators (30) generate transition rates $\gamma_{n\pm 1,n}$ with similar $\theta$ dependence (see Fig. 13), multiple transitions can fulfill the superradiant condition (9) and we expect multiple transitions contributing to the superradiance at the same time. An independent indication of which transitions are involved in the superradiance is the population distribution over the different hyperfine levels plotted in Fig. 15. For uncorrelated atoms around $\theta\approx\pi/4$ the single-atom pumpings dominate, due to the pumping rate $w$, giving an exponential population distribution. For $\theta=0$ Fig. 15 shows the approximately flat distribution for $m\geq 0$, indicating that all upper transitions have net collective emissions balancing the single-atom pumpings dominantly created by the pumping rate $w$. This implies that all transitions between levels $m\geq 0$ are radiating collectively enhanced, i.e., are superradiant. For $\theta=\pi/2$ one has an inverted behavior: The population of the upper levels is almost exponential, while the lower levels $m\leq 0$ show a flat distribution. In the lower levels the collective emissions are balancing the single-atom pumpings, meaning the transitions between the levels $m\leq 0$ are radiating superradiantly.

Fig. 14 shows the same qualitative behavior for the polarization and correlations as Fig. 9 and confirms that the choice of the simplified jump operators (33) captured the dominant effect of the full jump operators (30).