# Mirrorless lasing: a theoretical perspective

Aneesh Ramaswamy  Department of Physics, Stevens Institute of Technology,
Hoboken, NJ 07030, USA  Jabir Chathanathil  DEVCOM Army Research Laboratory,
Adelphi, MD 20783, USA  Dimitra Kanta  Helmholtz Institute Mainz, Johannes
Gutenberg University, 55099 Mainz, Germany  Emmanuel Klinger  Helmholtz
Institute Mainz, Johannes Gutenberg University, 55099 Mainz, Germany
Université de Franche-Comté, SupMicroTech-ENSMM, UMR 6174 CNRS, Institut
FEMTO-ST, 25000 Besançon, France  Aram Papoyan  Institute for Physical
Research, National Academy of Sciences of Armenia, 0203 Ashtarak-2, Armenia
Svetlana Shmavonyan  Institute for Physical Research, National Academy of
Sciences of Armenia, 0203 Ashtarak-2, Armenia  Aleksandr Khanbekyan  Institute
for Physical Research, National Academy of Sciences of Armenia, 0203
Ashtarak-2, Armenia  Arne Wickenbrock  Helmholtz Institute Mainz, Johannes
Gutenberg University, 55099 Mainz, Germany  Dmitry Budker  Helmholtz Institute
Mainz, Johannes Gutenberg University, 55099 Mainz, Germany  Department of
Physics, University of California, Berkeley, CA 94720, USA  Svetlana A.
Malinovskaya  Department of Physics, Stevens Institute of Technology, Hoboken,
NJ 07030, USA  Helmholtz Institute Mainz, Johannes Gutenberg University, 55099
Mainz, Germany

###### Abstract

Mirrorless lasing has been a topic of particular interest for about a decade
due to promising new horizons for quantum science and applications. In this
work, we review first-principles theory that describes this phenomenon, and
discuss degenerate mirrorless lasing in a vapor of Rb atoms, the mechanisms of
amplification of light generated in the medium with population inversion
between magnetic sublevels within the $D_{2}$ line, and challenges associated
with experimental realization.

Quantum Optics, Stimulated emission processes, Mirrorless lasing, Amplified
spontaneous emission, Alkali vapors

##  1 Introduction

Over sixty years of existence, lasers (Light Amplification by Stimulated
Emission of Radiation) have played a significant role in many areas of
scientific research, industry and defense [1], continuously growing as new
laser technologies are developed. There are three principal components usually
attributed to a laser: a gain medium, a pumping process and a feedback loop
[2]. Lasing usually requires pumping the medium to a state of population
inversion, although lasing without apparent inversion can occur in the case
where quantum coherence is induced between lower levels [3]. There is a debate
over whether lasing always requires a feedback loop. Lasing is often
distinguished from processes such as Amplified Spontaneous Emission (ASE),
Superradiance (SR) and Superflouorescence (SF) [4, 5]. Conventional lasers
usually incorporate an optical resonator setup where mirrors are used to have
light amplified over several round trips in the gain medium [2]. In mirrorless
lasing setups involving a feedback loop, the gain medium takes the role of the
resonator – usually through multiple scattering processes [4] – in systems
with a range of disorder, including random lasers [6] and distributed-feedback
systems [7]. Lasing in gain medium with a nonresonant feedback loop provided
by the same medium was considered by Letokhov [8] who gave a theoretical
treatment concerning light in the diffusive regime, see, for example, Ref.
[9]. Disordered random media provide coherent feedback loops that generate
lasing as seen in disordered ZnO nanoparticles in polycrystalline films [10]
and quantum-dot-doped liquid crystals [11]. In random lasers, feedback is
either resonant (phase sensitive, i.e. coherent) or non-resonant (frequency
and phase independent, i.e. incoherent) [10].

In this work, we adopt a broad definition of mirrorless lasing as directed
monochromatic emission from an ensemble of atoms or molecules excited with
pump laser light. The treatment of feedback loops arising from multiple
scattering is not considered here. There is great interest in the phenomenon
of mirrorless lasing in atomic vapors with theoretical [12, 3, 13, 14, 15, 16,
17, 18, 19], and experimental investigations [20, 21, 22, 23, 24, 25]
conducted in both cold atoms and hot vapors. The problem of mirrorless lasing
in atomic gases can be split into two: $(i)$ the problem of the gain mechanism
[14, 20] and $(ii)$ the problem of multiple-scattering feedback mechanism [16,
17, 4]. Mechanisms involving population inversion include Mollow gain (using a
near-resonant strong pumping field), Raman gain (by driving transitions using
off-resonance fields between two non-degenerate ground states, usually Zeeman
or hyperfine levels), and parametric gain (using degenerate four-wave mixing
(d-FWM) by way of counter-propagating pump fields) [14, 20].

Experiments in alkali metal vapor have shown the presence of gain through the
phenomenon of amplified spontaneous emission (ASE), see e.g. [26] and Refs.
therein. The combination of ASE and d-FWM at above-threshold scatterer
densities is suggested as a likely mechanism [22, 21].

The present work covers theoretical fundamentals of the lasing mechanisms of a
gas of atoms in free space, focuses on the phenomenon of amplification of
spontaneous emission and discusses degenerate mirrorless lasing from a gas of
alkali atoms with magnetically degenerate hyperfine states. We elucidate the
mechanisms for mirrorless lasing in both forward and backward direction with
respect to the pump laser beam and show that such process is possible even in
the degenerate case of the directed light being of the same frequency as the
pump. We reveal the role of population inversion among degenerate magnetic
sublevels of the hyperfine manifolds on the light amplification. The paper is
organized as follows. In Sec. 2 we present a first-principles theory
describing mirrorless lasing and mechanisms of light amplification in a gas of
atoms. In Sec. 3 we present a study of degenerate mirrorless lasing in
rubidium vapor using a semiclassical approach. Finally in Sec. 4 we review the
current status of experimental investigation of degenerate mirrorless lasing
followed by a summary.

##  2 First-principles formalism

We start with a microscopic approach for the problem of mirrorless lasing in a
gas of atoms. A system of N multilevel atoms located in free space and driven
by a classical pump field with envelope $E_{p}(t)$ and frequency $\omega_{p}$
is considered. Each atom $A_{i}$ has momentum $\bm{\mathrm{p}}_{i}$, center-
of-mass (COM) position $\bm{\mathrm{R}}_{i}$ and relative position
$\bm{\mathrm{r}}_{i}$. We define the central potentials $V_{i}(R_{i},r_{i})$
and the atom-atom interactions mainly composed of the potentials,
$W_{ij}(R_{i},R_{j},r_{i})$, introduced by the repulsion between the electrons
of atoms $A_{i}$ and $A_{j}$ respectively. The separation distance between any
pair of atoms is in general much greater than the transition wavelength
$\lambda_{p}$. Each atom $A_{i}$ is travelling with COM velocity
$\bm{\mathrm{v}}_{i}$, such that transition frequencies and dipole moments are
Doppler shifted. The dipole approximation is used such that the interaction
with the classical field driving transition $l$ for the $i$-th atom is given
by $-\mu_{il}E_{p}(t)\cos(\omega_{p}t)(\sigma_{il}^{+}+\sigma_{il}^{-})$. The
vacuum interaction is given by
$g_{\vec{k};il}^{(\mu)}a_{\vec{k}}^{(\mu)}\sigma_{il}^{+}+\mathrm{H.c}$. We
define the Hamiltonians:

$H_{A}=\sum_{i}\left(\dfrac{\bm{\mathrm{p}}_{i}^{2}}{2m}+V_{i}(\vec{R}_{i},\vec{r}_{i})+\sum_{j\neq
i}W_{ij}(\vec{R}_{i},\vec{R}_{j},\vec{r}_{i})\right),$ (1)
$H_{F}=\dfrac{1}{8\pi}\int
d^{3}r\text{}\left(\absolutevalue{\bm{\mathrm{E}}}^{2}+\absolutevalue{\bm{\mathrm{B}}}^{2}\right),$
(2)
$H_{AL}=-\sum_{il}\mu_{il}E_{p}(\vec{R}_{i},t)\cos(\omega_{p}t)\left[\sigma_{l}^{+}(\vec{R}_{i})+\sigma_{l}^{-}(\vec{R}_{i})\right],$
(3) $H_{AF}=-\sum_{il}\int d^{3}k\text{
}g_{\vec{k};il}^{(\mu)}(\vec{R}_{i})a_{\vec{k}}^{(\mu)}\sigma_{l}^{+}(\vec{R}_{i})+\mathrm{H.c.}$
(4)

From the above Hamiltonians, we determine the atomic density matrix equations
and the Heisenberg equation of motion for the photon operator
$\expectationvalue{a_{\vec{k}}^{(\mu){\dagger}}a_{\vec{k}}^{(\mu)}}$ [19]. We
ignore the contribution from the exchange potential terms and we use the
operator
$U_{L}(t,t_{1})=\exp{-i\int_{t_{1}}^{t}H_{AL,I}(t^{\prime})}\exp{-i\left(H_{A}+H_{F}\right)t}$
to transform the system Hamiltonian to the field interaction reference frame
where the atomic propagators are dressed with the pump field. This is to
include the case where the system is continuously driven, as opposed to where
the system is optically pumped and then left alone. The latter approach is
often used in solid-state lasing gain media, where the presence of non-
radiative emissions and quenching decrease the material radiative lifetimes
[27]. We note that there are separate, often competing [28], mechanisms for
gain. A seed pulse can be continuously amplified by the system or ASE can be
generated. We use the standard projection operator techniques to derive the
time non-local Liouville-von-Neumann equation [29]:

$\displaystyle\begin{split}\dfrac{d}{dt}\rho_{A,I^{\prime}}=&-i\left(\sum_{il}\text{
}g_{\vec{k}l}^{(\mu)}(\vec{R_{i}})\commutator{\sigma_{l,I^{\prime}}^{+}(\vec{R}_{i},t)}{\Omega^{0+}(\vec{R_{i}},t)}+\text{H.c}\right)\\\
&+\sum_{ilm}\int_{t_{0}}^{t}dt^{\prime}\left(\Gamma_{lm}(\vec{R_{i}},t,t^{\prime})\commutator{\hat{\sigma}^{+}_{l,I^{\prime}}(\vec{R_{i}},t)}{\hat{\sigma}^{-}_{m,I^{\prime}}(\vec{R_{i}},t^{\prime})\rho_{A,I^{\prime}}(t^{\prime})}\right.\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\
\left.-\Gamma_{lm}(\vec{R_{i}},t,t^{\prime})^{*}\commutator{\hat{\sigma}^{-}_{l,I^{\prime}}(\vec{R_{i}},t)}{\rho_{A,I^{\prime}}(t^{\prime})\hat{\sigma}^{+}_{m,I^{\prime}}(\vec{R_{i}},t^{\prime})}\right)\\\
&-\sum_{il}\sum_{jm}\dfrac{1}{\hbar^{2}}\int_{t_{0}}^{t}dt^{\prime}\biggl{(}\commutator{\hat{\sigma}^{+}_{l,I^{\prime}}(\vec{R_{i}},t)}{\hat{\sigma}^{-}_{m,I^{\prime}}(\vec{R_{j}},t^{\prime})\xi^{(\mu^{\prime}\mu)}_{m,l}(\vec{R_{j}},\vec{R_{i}},t^{\prime},t)}\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\
\left.-\commutator{\hat{\sigma}^{+}_{l,I^{\prime}}(\vec{R_{i}},t)}{\xi^{(\mu^{\prime}\mu)}_{m,l}(\vec{R_{j}},\vec{R_{i}},t^{\prime},t)^{*}\hat{\sigma}^{-}_{m,I^{\prime}}(\vec{R_{j}},t^{\prime})}\right.\,+\text{H.c.}\biggl{)},\end{split}$
(5)

where $\Omega^{0+}(\vec{R_{i}},t)$ is the vacuum Rabi field operator

$\Omega^{0+}(\vec{R_{i}},t)=\dfrac{1}{\hbar}\int
d^{3}k\expectationvalue{a_{\vec{k}}^{(\mu)}}_{F}(t_{0})e^{-i\omega_{k}t},$ (6)

$\Gamma_{lm}(\vec{R_{i}},t,t^{\prime})$ is the time-dependent correlator for
the decay

$\Gamma^{(\mu)}_{lm}(\vec{R_{i}},t,t^{\prime})=\int d^{3}k\text{
}g_{\vec{k};l}^{(\mu)}(\vec{R_{i}},t)g_{\vec{k};m}^{(\mu)*}(\vec{R_{i}},t^{\prime}),$
(7)

and $\xi^{(\mu^{\prime}\mu)}_{m,l}(\vec{R_{j}},\vec{R_{i}},t^{\prime},t)$
represents stimulated emission processes driven by the quantum light fields

$\xi^{(\mu^{\prime}\mu)}_{m,l}(\vec{R_{j}},\vec{R_{i}},t^{\prime},t)=\expectationvalue{\left\\{\vec{d}^{*}_{m}\cdot\vec{E}^{(\mu^{\prime})-}(\vec{R_{j}},t^{\prime})\right\\}\left\\{\vec{d}_{l}\cdot\vec{E}^{(\mu)+}(\vec{R_{i}},t)\right\\}}_{F}.$
(8)

The first term in Eq. (5) is the Langevin force that depends on vacuum
fluctuations. The second term is the spontaneous decay contribution and
$\hat{\sigma}^{\pm}_{l}(\vec{R_{i}},t)$ is the rotated dipole operator. Note
that the time and position dependent exponential is included into the coupling
rates. The final term is the stimulated absorption/emission term for a single
atom, ($i=j$), and two-atom events, ($i\not=j$). The former provides a
correction to spontaneous emission due to a non-zero quantum radiation field.
We note that there is a dependence on the expectation value
$\expectationvalue{a_{\vec{k}}^{(\mu){\dagger}*}a_{\vec{k}^{\prime}}^{(\mu^{\prime})}}_{F}(t^{\prime})$.
In the case where the density matrix is separable, and
$\vec{k}=\vec{k}^{\prime}$ and $\mu=\mu^{\prime}$, we get the product of the
usual photon number operator and the atomic density matrix. When any of the
last two conditions are not met, the final term represents a photonic
coherence between two modes. The Heisenberg equations of motion for the photon
number operator can be simply defined as

$\begin{split}\left[\dfrac{1}{c}\dfrac{d}{dt}+i(\omega_{k}-\omega_{k^{\prime}})\right]\expectationvalue{a_{\vec{k},I^{\prime}}^{(\mu){\dagger}}(t)a_{\vec{k}^{\prime},I^{\prime}}^{(\mu^{\prime})}(t)}(t)=S_{\text{s}}(t)+\mathcal{R}_{\text{s}}\left[\expectationvalue{a_{\vec{k}/\vec{q},I^{\prime}}^{(\mu){\dagger}}a_{\vec{q}/\vec{k}^{\prime},I^{\prime}}^{(\mu^{\prime})}}(t)\right],\end{split}$
(9)

where $S_{S}(t)$ is the source term due to atomic spontaneous decay and
$\mathcal{R}_{\text{s}}\left[\expectationvalue{a_{\vec{k}/\vec{q},I^{\prime}}^{(\mu){\dagger}}a_{\vec{q}/\vec{k}^{\prime},I^{\prime}}^{(\mu^{\prime})}}(t)\right]$
represents scattering contributions involving other modes. Equations (5) and
(9) are the microscopic equations required to model the time and space
dependent atomic density matrix and quantum field equations. For optically
thick systems with high disorder and/or non-ultracold temperatures,
statistical averaging of the equations, including averaging over the position
density and velocity distributions of atoms, and phase-space transforms are
required to obtain macroscopic equations of motion [30].

###  2.1 The probe field gain

The most common method for generating an intense monochromatic beam is to
amplify a weak probe pulse, $E_{pr}(t)$, using a medium which has undergone
population inversion, in the dressed state basis, with respect to a transition
having frequency close to the probe frequency [31]. Additional methods involve
using coherent fields to generate quantum interference between stimulated
emission and absorption pathways to create gain without population inversion
[32, 3, 20, 14]. The probe detuning, $\Delta_{pr}$, the magnetic sub-level
degeneracy, the presence of population inversion and the propagation direction
of the pump beam(s) determine which amplification mechanism dominates.

A Hamiltonian analogous to that in Eq. (3) is used to model the atom-probe
field interaction, which we call $H_{pr}$. We derive the rate equations for
the probe field $E_{pr}$. Using the steady state absorption rate of quanta for
a classical field [33], we write

$\dfrac{\partial}{\partial
t}E_{pr}(\vec{R},t)=n(\vec{R})\expectationvalue{-i\commutator{H_{pr;I}(t)}{\rho_{I}(t)}}\Delta(\vec{R},\vec{r}_{0})\,,$
(10)

where $n$ is the density of atoms in the medium, and $I$ denotes the field
interaction picture associated with transformation
$U_{0}(t,t_{1})=\exp{-i\left(H_{A}+H_{F}\right)t}$. The terms in $H_{AL;I}(t)$
are restricted to atoms within a volume $\Delta(\vec{R},\vec{r}_{0})$ of
radius $\vec{r}_{0}$ from the center $\vec{R}$.

To the first order in perturbation theory, the steady state of a driven atom
results in absorption spectrum $g_{A}(\omega)$ of a weak probe field of mode
$M$ and frequency $\omega$ [33]

$\displaystyle g_{A}(\omega)=\int_{0}^{\infty}d\tau\text{ }e^{-i\omega
t}\commutator{d_{I}^{-}(\tau)}{d_{I}^{+}(0)}\,,$ (11)

where $d_{I}^{+}(\tau)$ is the weighted sum of all dipole operators that
correspond to transitions involving radiation of mode $M$:

$\displaystyle
d_{I}^{+}(\tau)=\sum_{j}\left(\hat{\varepsilon}\cdot\vec{\mu}_{j}\right)\sigma_{j,I}^{+}(\tau).$
(12)

The averaged single-atom absorption rate, $\alpha(\omega,t)$, of quanta of
mode $M$ and energy $\hbar\omega$ from a weak probe field with slowly varying
amplitude $E_{pr}(t)$ and frequency $\omega$ according to [33] is

$\displaystyle\alpha(\omega,t)=\expectationvalue{g_{A}(\omega)}\left|\dfrac{\left(\hat{e}_{p}\cdot\hat{d_{I}}\right)E_{pr}(t)}{\sqrt{2}\hbar}\right|^{2}\,.$
(13)

Understanding of the scattering processes that contribute to stimulated
emission of the probe field in the presence of strong driving pump fields
requires a dressed state approach. In strong fields, the spectral state
composition of the atoms is understood in terms of multi-photon dressed states
$\ket{\lambda_{i},m}$ with energies $E_{i}=\lambda_{i}+m\omega_{p}$. In this
case, the unperturbed atomic dynamics at any time consists of ladder-type
transitions between different $m$-number states. Therefore, contributions to
the stimulated emission of probe field photons must take into account these
background ladder transitions [34].

The transition amplitudes for multi-photon processes can be found by looking
at the diagrammatic expansion of the self-energy $\Sigma(t,t_{0})$ [35]. This
follows from the below form of the Liouville-von-Neumann equation:

$\dot{\rho}_{I}(t)=\int_{t_{0}}^{t}dt^{\prime}{\rho}_{I}(t^{\prime})\Sigma(t^{\prime},t).$
(14)

For example, the sum of transition amplitudes for all three-photon processes
that involve absorption of two pump photons and emission of one single probe
photon is given by summing over three-vertex Keldysh diagrams with vertices
$(t_{1},\sigma_{j}^{+}E_{P}^{-})$, $(t_{2},\sigma_{l}^{-}E_{pr}^{+})$,
$(t,\sigma_{j}^{+}E_{P}^{-})$. The vertices can be on the forward or backwards
branch of the density matrix, and we sum over all permutations of the
vertices.

Figure 1: A diagram representing the three-photon scattering process.

For the three-photon scattering process involving absorption of two pump
photons and emission of one single probe photon, the contribution to the
evolution of the atomic density matrix is given by

$\begin{split}\delta^{(3)}\rho_{A,I}(t_{1})&=\dfrac{E_{P}^{2}E_{pr}}{\hbar^{3}}\int_{0}^{t}\int_{0}^{t_{2}}dt_{1}dt_{2}e^{-i\left((\Delta_{p})(t+t_{1})-\Delta_{pr}t_{2}\right)}\\\
&\leavevmode\nobreak\
\times\Big{[}d_{P}^{+}U(t,t_{2})d_{pr}^{-}U(t_{2},t_{1})d_{P}^{+}\rho_{A,I}(t_{1})U^{{\dagger}}(t,t_{1})\\\
&\leavevmode\nobreak\
+\,U(t,t_{1})\rho_{A,I}(t_{1})d_{P}^{+}(t_{2})U^{{\dagger}}(t_{2},t_{1})d_{pr}^{-}U^{{\dagger}}(t,t_{2})d_{P}^{+}(t)+...\Big{]},\end{split}$
(15)

where suspension points refer to all other possible time and branch (forward
or reverse of the density matrix) orderings of the three vertices, see Fig. 1.
The contribution in the steady state limit is determined by substituting the
steady state expression
$\bar{\rho}_{A,I;s}=\bar{\rho}_{A,I;0}+\sum_{j}\bar{\rho}_{A,I;j}e^{-i\nu_{j}t}$.
To look at the contribution for a specific transition, we compute the change
in energy of the probe field, $\expectationvalue{H_{p}\hat{X}}$, for the
particular transition described by the string of operators, $\hat{X}$.

In the following subsections, we described possible gain mechanisms using the
example of a two-level degenerate system modeling the $F=2\rightarrow F=3$
component of 87Rb $D_{2}$ transition.

###  2.2 Mollow gain

Figure 2: Mollow gain in a two-level system (TLS). The pump field
($\omega_{pr}$), with detuning $\Delta_{p}$, pumps atoms into the higher
energy dressed state. Depending on the sign of $\Delta_{p}$, the probe field
with frequency $\omega_{pr}$ and two-photon detuning $\delta$ is either
amplified or reduced due to a three-photon scattering process. The probe field
polarization is parallel to the polarization of the pump. Figure 3: Absorption
spectrum of light with parallel (to pump) polarization for the resonantly
driven 87Rb system with parameters $\Gamma=1.0$, $\Omega_{p}=4\,\Gamma$,
$\Delta_{p}=0$. Figure 4: Absorption spectrum of light with parallel (to pump)
polarization for the off-resonantly driven 87Rb system with parameters
$\Gamma=1.0$, $\Omega_{p}=4\Gamma$, $\Delta_{p}=2\Gamma$ Figure 5: Atom-photon
dressed state picture of the TLS in the case when the pump pulse is off
resonance; $\omega_{A}$ corresponds to an absorption feature that appears with
non-resonant driving; $\omega_{G}=2\omega_{pr}-\omega_{A}$ is the probe
frequency that corresponds to the three-photon scattering amplification side-
band.

In the case where the probe field is near resonance and a single pump field is
used, Mollow gain [33] dominates in which the gain mechanism is due to a
three-photon process involving the absorption of two pump photons and
stimulated emission of a probe photon [20], see Fig. 2. Population inversion
in the dressed state basis has occurred here and resulted in gain/loss in the
sidebands of the absorption spectrum about the resonance peak [36].

Figure 3 shows the absorption spectrum for the case of resonant driving field.
Note that, in this work, frequency parameters (such as $\Delta_{p,pr}$) are
expressed in units of $\Gamma$, the excited state decay rate, and time
parameters in the units of $1/\Gamma$. The feature seen at $\Delta_{p}=0$
corresponds to an interference between multiple two-photon processes. The
dominant resonant contribution is the two-photon Rayleigh scattering process
involving an absorption of the probe photon and spontaneous emission of a
photon having the same frequency [37]. Additionally there are processes
involving absorption of a pump photon and emission of a probe photon and vice
versa. The greatest gain components are observed at the sidebands centered at
the dressed state frequencies $\pm\Omega_{p}$ in the resonant driving case.
Increasing $\Delta_{p}$ from zero, results in an asymmetric spectrum where
amplification occurs only on one sideband while an absorption feature occurs
on the other, as shown in Fig. 4.

Figure 5 shows various spectral lines corresponding to features in the
absorption spectrum for the slightly detuned case as in Fig. 4. In the atom-
photon number dressed state basis, we see that the features correspond to
transitions between different dressed states involving absorption/emission of
a probe photon with energy $\hbar\omega_{pr}$ and absorption/emission of a
number of pump photons with energy $\hbar\omega_{p}$.

Figure 6: Raman gain in a detuned system with multiple magnetic sub-levels.
$\ket{g,m}$ is the ground state magnetic sub-level $m$. The pump field
($\omega_{p}$) and spontaneous decay create a population distribution among
the $\ket{g,m}$ states that depends on their Clebsch-Gordon coefficients. The
probe laser ($\omega_{pr}$) experiences gain/loss depending on the signs of
$\Delta_{p}$, $\delta$.

###  2.3 Raman gain using Zeeman sublevels

For the case of a two-level system with many magnetic sub-levels, a far-
detuned pump laser field can induce probe gain through population inversion
among the shifted sub-levels. Unlike Mollow gain, Raman gain on magnetic
sublevels requires a probe field with polarization orthogonal to the pump
filed polarization [14]. This is due to the necessary two-photon Raman
transition between different $m$ number states. The high one-photon detuning
creates level shifts of the dressed states corresponding to each magnetic sub-
level and introduces a dispersive structure for the absorption spectrum at
$\delta=0$ shown in Fig. 6. This results in gain on one side of the structure
and loss on the other, shown in Fig. 7.

Figure 7: Absorption spectrum of light with perpendicular polarization for the
off-resonantly driven 87Rb system with parameters $\Gamma=1.0$,
$\Omega_{p}=4\Gamma$, $\Delta_{p}=1.75\Gamma$.

###  2.4 Raman gain with coupling fields

Figure 8: Raman gain in a system with coupled hyperfine levels $g_{i}$. The
pump laser ($\omega_{pr}$) optically pumps the system to induce ground state
coherence. The coupling laser ($\omega_{C}$) is introduced to return atoms to
the ground states and create gain in the probe ($\omega_{p}$) due to the Raman
transition involving the coupling and probe.

Raman gain can be induced by sustaining a population inversion between two
different lower energy states, e.g. hyperfine ground states [20] through
optical pumping, and then producing gain with a stimulated two-photon
transition using a coupling field $E_{c}(t)$ [38, 20]. The probe polarization
in this case depends on the choice of level $\ket{e_{2}}$, see Fig. 8.

###  2.5 Four-wave mixing (FWM) gain

FWM is a phase-sensitive parametric nonlinear optical process that involves
the interactions of four fields with a nonlinear medium and transferring
energy between the fields [39]. When considering a degenerate FWM (d-FWM)
using counter-propagating pump fields and satisfying a phase matching
condition, gain is observed in the backward propagating reflection of the
probe field [20, 40].

###  2.6 Inversionless gain in coherently driven multilevel systems

Probe gain in coherently driven multilevel systems without inversion in any
meaningful basis has been described as resulting from coherence between
dressed states [32, 31]. For the case of the three-level V-system, each of the
coherent, linked dressed state contributes to the absorption spectrum with two
independent Lorentzians occurring at the dressed states resonances. The
presence of gain is suggested to be due to competition between n-photon
scattering processes, each of which is responsible for contributing to gain or
loss. In V and $\Lambda$ type systems, two-photon scattering is responsible
for inversionless gain, while one-photon processes cause gain in cascade type
systems [32].

###  2.7 Amplified Spontaneous Emission (ASE)

Spontaneous emission is a quantum process in which an atom in an excited state
spontaneously de-excites into a lower energy state by emitting a photon into
one of the modes of the quantum mechanical vacuum/reservoir field. It can be
semiclassically described as emission process stimulated by vacuum noise, (the
zero-point fluctuations of the electric field in the vacuum state) [41]. ASE
is a result of stimulated emission processes that amplify a spontaneous
radiation field propagating in a medium with gain and is often a pervasive
phenomenon in lasers and optical amplifiers [42]. We distinguish between ASE
and the cooperative phenomena of superfluorescence, the latter process
requiring negligible dephasing to form a cohrent macroscopic interaction [5].

The threshold for ASE in a particular gain medium is reached when the
contribution from optical gain processes is greater than that of absorption
and escape processes, (radiation scattering away from the medium). Unlike
lasing involving coherent amplification of a probe pulse, the temporal
coherence of ASE in media with multiple scatterers without coherent feedback
is often low, due to the large radiation bandwidth. In particular, the second-
order coherence $g^{(2)}(\omega,t)$ of ASE demonstrates super-Poissonian
statistics consistent with that of thermal light [43]. However the presence of
a coherent feedback loop realizes random lasing and the generation of more
coherent light with Poissonian statistics [44]. Coherent feedback loops can be
created by manipulating the scattering processes in the gain medium, as
multiple scattering can replace the requirement of mirrors for coherent
feedback. This leads to the formation of a new threshold for lasing with
coherent feedback that depends on the gain medium’s scattering properties
[45].

The spatial coherence of ASE can be very high even without a coherent feedback
loop. In atomic gas systems, the geometry of the lasing setup (the optically
active part of the system), as well as the internal degrees of freedom (atomic
populations, coherences and the spectral lineshape [46]), affect spatial
coherence and beam divergence. The treatment of spatial coherence becomes more
complicated when considering multiple scattering, especially in the strong
scattering regime where localization of light can occur [6].

The starting point for the microscopic treatment of ASE in a continuously
driven medium is the Heisenberg equations of motion for the photon number
operator Eq. (9). In general, it is difficult to solve for the photon number
operator, especially in the case of large systems. Instead, the operator
$\expectationvalue{E_{I^{\prime}}^{(\mu)-}(\vec{r})E_{I^{\prime}}^{(\mu^{\prime})+}(\vec{r})}(t)$
is considered. Before we look at the effects of feedback and multiple
scattering, we look into this operator in detail. The contribution of
spontaneous emission in
$\expectationvalue{E_{I^{\prime}}^{(\mu)-}(\vec{r},t)E_{I^{\prime}}^{(\mu^{\prime})+}(\vec{r},t)}$
is

$\displaystyle\begin{split}S_{\text{s}}(t)=\sum_{j,\alpha}\sum_{i,\beta;\nu=\pm}\int_{0}^{t}dt^{\prime}&\left(\Gamma^{(\mu\mu^{\prime})}_{ji}(\vec{R_{\alpha}},\vec{R_{\beta}},t,t^{\prime})\expectationvalue{{\sigma}_{i,I^{\prime}}^{+}(\vec{R}_{\alpha},t){\sigma}_{j,I^{\prime}}^{-}(\vec{R}_{\beta},t^{\prime})}\right.\\\
&\left.+\Gamma^{(\mu\mu^{\prime})}_{ij}(\vec{R_{\alpha}},\vec{R_{\beta}},t,t^{\prime})\expectationvalue{{\sigma}_{j,I^{\prime}}^{+}(\vec{R}_{\beta},t^{\prime}){\sigma}_{i,I^{\prime}}^{-}(\vec{R}_{\alpha},t)}\right),\end{split}$
(16)

where
$\Gamma^{(\mu\mu^{\prime})}_{ij}(\vec{R_{\alpha}},\vec{R_{\beta}},t,t^{\prime})$
is given by

$\int\int d^{3}k\text{ }d^{3}k^{\prime}\text{
}g_{\vec{k};i}^{(\mu)}(\vec{R_{\alpha}},t)g_{\vec{k^{\prime}};j}^{(\mu^{\prime})*}(\vec{R_{\beta}},t^{\prime}).$
(17)

We distinguish between the single-atom term ($\alpha=\beta$) and the two-atom
terms ($\alpha\not=\beta$). The latter contributes to collective emission, the
superfluorescence. The stimulated emission term is

$\displaystyle\begin{split}\mathcal{R}_{\text{s}}(t)=&\leavevmode\nobreak\
i\sum_{i,\alpha}\int\int d^{3}k\text{
}d^{3}k^{\prime}\left(g_{\vec{k}^{{}^{\prime}}i}^{\left(\mu^{\prime}\right)*}(\vec{R}_{\alpha})\Tr{a_{\vec{k},I^{\prime}}^{(\mu){\dagger}}(t){\sigma}_{i,I^{\prime}}^{-}(\vec{R}_{\alpha},t)\rho_{I^{\prime}}(t_{0})}\right.\\\
&\qquad\left.-g_{\vec{k}i}^{\left(\mu\right)}(\vec{R}_{\alpha})\Tr{a_{\vec{k}^{\prime},I^{\prime}}^{(\mu^{\prime})}(t){\sigma}_{i,I^{\prime}}^{+}(\vec{R}_{\alpha},t)\rho_{I^{\prime}}(t_{0})}\right)\\\
&-\sum_{ij,\alpha;\nu=\pm}\int_{0}^{t}dt^{\prime}\left(\expectationvalue{E_{I^{\prime}}^{(\mu)-}(\vec{r},t)\left[\vec{d}_{l}\cdot\vec{E}_{I^{\prime}}^{(\nu)+}(\vec{R}_{\alpha},t^{\prime})\right]\cdot\mathcal{C}^{(\mu^{\prime})}_{i,j}(\vec{r},\vec{R}_{\alpha},t,t^{\prime})}\right.\\\
&\qquad\left.+\expectationvalue{\left[\vec{d}^{*}_{l}\cdot\vec{E}_{I^{\prime}}^{(\nu)-}(\vec{R}_{\alpha},t^{\prime})\right]E_{I^{\prime}}^{(\mu^{\prime})+}(\vec{r},t)\cdot\mathcal{C}^{(\mu){\dagger}}_{i,j}(\vec{r},\vec{R}_{\alpha},t,t^{\prime})}\right).\end{split}$
(18)

The first term, the Langevin force, is non-zero only if there are initial
correlations at time $t_{0}$. The $\vec{d}_{l}$ is the unit transition dipole
vector for transition $l$. The term proportional to the single-atom dipole
correlation operator reads

$\displaystyle\mathcal{C}_{i,k}^{\mu^{\prime}}(\vec{r},\vec{R}_{\alpha},t,t^{\prime})$
$\displaystyle=\commutator{{\sigma}_{i,I^{\prime}}^{-}(\vec{R}_{\alpha},t)}{{\sigma}_{j,I^{\prime}}^{+}(\vec{R}_{\alpha},t^{\prime})}\times\int
d^{3}k^{\prime}\text{
}g_{\vec{k}^{{}^{\prime}}i}^{\left(\mu^{\prime}\right)*}(\vec{R}_{\alpha}-\vec{r})\varepsilon^{(\mu^{\prime})}_{\vec{k^{\prime}}}.$
(19)

The single-atom dipole-dipole functions are expanded in terms of projection
operators onto the populations and coherences to give us the contributions to
the spontaneous emission and gain/loss spectra. In the case where we have non-
degenerate levels and no driving fields, one transition is coupled to each
radiation mode, and single-atom dipole-dipole functions for transition $l$ are
equal to sums of $\sigma_{l}^{z}$ and $I_{l}$. Coupling multiple transitions
to a single mode introduces lower-state or upper-state coherence. The
contribution from each transition is weighted by their Clebsch-Gordan
coefficient and the transition frequency.

In the interaction picture where the atomic propagators are dressed with the
driving field, non-zero driving introduces the $\sigma_{l}^{\pm}$ operators in
the dipole-dipole functions, thus introducing dependence on the atomic
coherences. In this way, the dressed state structure of the degenerate system
appears with the presence of new Lorentzians and interference terms in the
spectral function. Since the atomic coupling to the vacuum is weak, it is
useful to switch to the more physically meaningful basis of the dressed states
which are the eigenstates of $U_{L}(t,t_{0})$.

We assume that driving fields are slowly varying, with small losses in the
intensities over large timescales. The dressed states,
$\ket{\Lambda_{k}(\vec{R}_{\alpha},t)}$, are defined along with their
corresponding energies $\lambda_{k}(t)$ such that

$\begin{split}&\sigma_{j;I^{\prime}}^{+}(\vec{R}_{\alpha},t)=\sum_{m^{\prime}m}C^{j}_{m^{\prime}m}(\vec{R}_{\alpha},t)e^{it(\omega_{j}+\lambda_{m^{\prime}}-\lambda_{m})}\ket{\Lambda_{m}(\vec{R}_{\alpha},t)}\bra{\Lambda_{m^{\prime}}(\vec{R}_{\alpha},t)}.\end{split}$
(20)

The dipole-dipole function consists of projection operators onto the dressed
state populations and coherences. This causes all fast time dependencies to be
stored in the exponentials and the density matrix. Due to small coupling
strength, and with the assumption that transition frequencies are in the
optical regime, the timescale for the evolution of the density matrix is
significantly larger than that for the decay of the correlation functions.
This justifies the use of the Markov approximation.

###  2.8 Paraxial approximation

A pencil-like geometry for the pump laser is assumed such that only a narrow
cone of wavevectors $\Sigma_{k}$, for forwards ($R$) and backwards ($L$)
components, contributes to the ASE modes. The electric field operator for a
single mode $\hat{E}^{(\mu)}(\vec{r},t)$ is therefore expanded as:

$\hat{E}^{(\mu)}(\vec{r},t)=\left[E^{(\mu)+}_{R}(\vec{r},t)e^{i\vec{k}_{0}\cdot\vec{r}}+E^{(\mu)+}_{L}(\vec{r},t)e^{-i\vec{k}_{0}\cdot\vec{r}}\right]e^{-i\omega_{0}t}-\text{H.c.}\,,$
(21)

where $\vec{k}_{0}$ is the forward propagating wavevector along the axis of
$\Sigma_{k}$ and $E^{(\mu)+}_{\hat{k}}(\vec{r},t)$ is a slowly varying
operator. This treatment is similar to the description of ASE for the case of
homogeneously broadened three-level atoms in a rod-like geometry [47]. We
assume that the atomic density is large enough that the dipole operator is now
a continuous function of position. Furthermore we assume in a small volume
$V_{r}$ with radius $r$ centered at any point $\vec{R}$, the
$\omega_{j}c^{-1}r\gg 1$, and the dipole correlation function and the atomic
density varies trivially. We introduce the volume-averaged dipole in a small
vertical slice of the cylindrical medium
$S_{j;I^{\prime}}^{+}(\vec{R}_{i},t)=n(\vec{R}_{i})\int_{V_{r}(\vec{R}_{i})}dV^{{}^{\prime}}\sigma_{j;I^{\prime}}^{+}(\vec{R}^{{}^{\prime}},t)$.
The interaction terms between pairs of atoms are assumed to contribute
negligently to the dynamics and are ignored. The equation for the density
matrix element in the dressed state basis reads

$\begin{split}\dfrac{d}{dt}\expectationvalue{\sigma_{\Lambda_{a}\Lambda_{b}}(\vec{R}_{a},t,t)}_{I^{{}^{\prime}}}&=\sum_{ml}\Gamma_{1}^{ab;ml}(\vec{R}_{\alpha},t)\expectationvalue{\sigma_{\Lambda_{m}\Lambda_{l}}(\vec{R}_{a},t,t)}_{I^{{}^{\prime}}}\\\
&\quad-\sum_{l}\Gamma_{2}^{ab;l}(\vec{R}_{\alpha},t)\expectationvalue{\sigma_{\Lambda_{a}\Lambda_{l}}(\vec{R}_{a},t,t)}_{I^{{}^{\prime}}}\\\
&\quad-\sum_{l}\Gamma_{3}^{ab;l}(\vec{R}_{\alpha},t)\expectationvalue{\sigma_{\Lambda_{l}\Lambda_{b}}(\vec{R}_{a},t,t)}_{I^{{}^{\prime}}}\\\
&\quad+\sum_{\hat{k}_{1},\hat{k}_{2}}I^{(\mu\mu^{\prime})}(\hat{k}_{1},\hat{k}_{2},\vec{R}_{\alpha},\vec{R}_{\alpha},t)\\\
&\qquad\times\left(\sum_{ml}B_{1}^{ab;ml}(\mu,\mu^{\prime},\vec{R}_{\alpha},t)\expectationvalue{\sigma_{\Lambda_{m}\Lambda_{l}}(\vec{R}_{a},t,t)}_{I^{{}^{\prime}}}\right.\\\
&\qquad\left.+\sum_{ml}B_{2}^{ab;l}(\mu,\mu^{\prime},\vec{R}_{\alpha},t)\expectationvalue{\sigma_{\Lambda_{a}\Lambda_{l}}(\vec{R}_{a},t,t)}_{I^{{}^{\prime}}}\right.\\\
&\qquad\left.+\sum_{ml}B_{3}^{ab;l}(\mu,\mu^{\prime},\vec{R}_{\alpha},t)\expectationvalue{\sigma_{\Lambda_{l}\Lambda_{b}}(\vec{R}_{a},t,t)}_{I^{{}^{\prime}}}\right).\end{split}$
(22)

We define the two-point field correlation function as
$I^{(\mu\mu^{\prime})}(\hat{k}_{1},\hat{k}_{2},\vec{R}_{1},\vec{R}_{2},t)=\expectationvalue{E_{\hat{k}_{1},I^{\prime}}^{(\mu)-}(\vec{R}_{1},t)\vec{E}_{\hat{k}_{2},I^{\prime}}^{(\mu^{\prime})+}(\vec{R}_{2},t^{\prime})}$.
The term $\vec{R}_{1,2}$ is the average polarization vector of the $\mp$ field
component and $\hat{k}_{1,2}\in\\{L,R\\}$. We use the paraxial approximation
formalism described in Ref. [47] to derive the radiative transport equation

$\begin{split}\left[\dfrac{\partial}{\partial
t}+c\left(\hat{k}_{1}\cdot\nabla_{\vec{R}_{1}}+\hat{k}_{2}\cdot\nabla_{\vec{R}_{2}}\right)\right.\qquad\quad&\\\
\left.-i\dfrac{c}{2k_{0}}\nabla_{\perp}^{2}\right]I^{(\mu\mu^{\prime})}(\hat{k}_{1},\hat{k}_{2},\vec{R}_{1},\vec{R}_{2},t)&=\sum_{\vec{x},ij}A^{ij;ml}_{\mu\mu^{\prime}}(\vec{x},t)\expectationvalue{S_{\Lambda_{m}\Lambda_{l}}(\vec{x},t,t)}_{I^{{}^{\prime}}}\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\
+\sum_{\hat{q}\in\\{L,R\\}}\sum_{\vec{x},ij}\delta_{\mu\mu^{\prime}}\expectationvalue{S_{\Lambda_{m}\Lambda_{l}}(\vec{x},t,t)}_{I^{{}^{\prime}}}\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\
\times\left[K^{ij;ml}(\hat{q}\rightarrow\hat{k}_{1},\hat{k}_{2},\vec{x},t)I^{(\mu\mu)}(\hat{q},\hat{k}_{2},\vec{R}_{1},\vec{R}_{2},t)\right.\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\
\left.+K^{ij;ml}(\hat{k}_{1},\hat{q}\rightarrow\hat{k}_{2},\vec{x},t)I^{(\mu\mu)}(\hat{k}_{1},\hat{q},\vec{R}_{1},\vec{R}_{2},t)\right],\end{split}$
(23)

where $\nabla_{\perp}$ is the gradient operator transverse to $\hat{k}_{1,2}$
and $A^{ij;ml}_{\mu\mu^{\prime}}(\vec{x},t)$ is given by

$\begin{split}A^{ij;ml}_{\mu\mu^{\prime}}(\vec{x},t)&=\int_{\Sigma_{k}}\int_{\Sigma_{k^{\prime}}}d^{3}k\text{
}d^{3}k^{\prime}\text{
}g_{\vec{k};i}^{(\mu)*}(\vec{x})g_{\vec{k^{\prime}};j}^{(\mu^{\prime})}(\vec{x})\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\
\times\sum_{l^{\prime}}\int_{0}^{\infty}d\tau\bigg{[}C_{l^{\prime}l}^{i*}(t)C_{ml^{\prime}}^{j}(t)e^{-i(\omega_{k}-\omega_{k^{\prime}}-\omega_{ij}+\lambda_{l^{\prime}l}-\lambda_{l^{\prime}m})t}\leavevmode\nobreak\
e^{i(\omega_{k^{\prime}}-\omega_{j}+\lambda_{l^{\prime}m})\tau}\\\
&\qquad\qquad\qquad\qquad+C_{ml^{\prime}}^{i}(t)C_{l^{\prime}l}^{j*}(t)e^{i(\omega_{k}-\omega_{k^{\prime}}-\omega_{ij}-\lambda_{l^{\prime}l}+\lambda_{l^{\prime}m})t}\leavevmode\nobreak\
e^{-i(\omega_{k^{\prime}}-\omega_{j}-\lambda_{l^{\prime}m})\tau}\bigg{]}.\end{split}$
(24)

The spontaneous emission contribution is integrated over the cone in momentum
space that contains wave-vectors that remain in the cylindrical medium and
contribute to ASE. The remaining wavevectors are considered to have escaped
from the medium and do not contribute to the evolution of the paraxial ASE
modes. $K^{ij;ml}(\hat{q}\rightarrow\hat{k}_{1},\hat{k}_{2},\vec{x},t)$ and
$K^{ij;ml}(\hat{k}_{1},\hat{q}\rightarrow\hat{k}_{2},\vec{x},t)$ are the
scattering probabilities for the mode with wavevector $\hat{q}$ to scatter
into the mode with wavevector $\hat{k}_{1,2}$ respectively through virtual
transitions between dressed states $\Lambda_{m}$ and $\Lambda_{l}$. This
corresponds to stimulated emission processes and is derived from Eq. (18)
using the dressed state expansion of the atomic operators Eq. (20).

The paraxial approximation equations for the dressed state atomic matrix and
the radiative equations for the field correlation effectively model the
dynamics of an ASE field in a cold random atomic gas, under the approximation
that interatomic scattering plays little to no effect. The gain condition that
results in ASE can be derived by calculating all the coefficients that depend
on the dressed state configuration and determining the choices of parameters,
including the pump field and detuning, that result in the right-hand side of
Eq. (23) being greater than zero in the steady state. The choice of parameters
can be informed by looking at the gain mechanisms, and the above section on
the probe field gain accounts for how dressed state inversions and coherences
contribute.

##  3 Degenerate Mirrorless Lasing

In this section, we study a prototypical system in which directional emission
was predicted and observed to occur at the same frequency and orthogonal
polarization as those of the excitation light.

Consider the eight-level system of the $D_{2}$ ($5^{2}S_{1/2}\rightarrow
5^{2}P_{3/2}$) line in 87Rb. The $5^{2}S_{1/2}$ and $5^{2}P_{3/2}$ states are
split into hyperfine structure components with total angular momentum
$F_{g}=2,1$ and $F_{e}=3,2,1,0$ respectively. We consider the
$F_{g}=1\rightarrow F_{e}=2$ transition. In the absence of magnetic field, the
ground state is three-fold and the excited state is five-fold degenerate.

Figure 9: The degenerate 8-level system interacting with a linearly polarized
pump field in $z$-direction (a) and linearly polarized probe field in the
orthogonal direction (b). The numbers given in parenthesis are the numbers
corresponding to the levels as they appear in the Hamiltonian.

Pump light linearly polarized in the $z$-direction induces the transition
between magnetic sublevels with the same quantum numbers: $m_{g}=m_{e}$. On
the other hand, circularly polarized light induces transition between energy
level with different magnetic quantum numbers depending on the direction of
polarization. This is due to the principle of angular momentum conservation,
as the sum of angular momenta of the light and the atom should remain
constant. If $E_{\sigma_{-}}$ and $E_{\sigma_{+}}$ represent the right and
left circular polarization components of the light, a superposition of the two
components chosen here as the probe gives linearly polarized field in the
$\hat{x}$ direction

$\displaystyle E_{x}=\frac{i}{\sqrt{2}}(E_{\sigma_{-}}+E_{\sigma_{+}})\,,$
(25)

where the overall phase factor is chosen for consistency with standard
definitions.

###  3.1 Population Dynamics

The interaction of the mutually orthogonal linearly polarized pump and probe
fields with atoms is depicted in Fig. 9. The pump field polarized along the
quantization axis $z$ couples the ground and excited states satisfying the
condition $\Delta m=m_{e}-m_{g}=0$, as shown in Fig. 9(a). The orthogonally
polarized probe field couples states satisfying $\Delta m=m_{e}-m_{g}=\pm 1$,
see 9(b).

The field-interaction Hamiltonian is given by:

$H=\frac{\hbar}{2}\begin{pmatrix}2\Delta_{pr}&-i\Omega_{21(pr)}&0&0&0&0&0&0\\\
i\Omega_{21(pr)}&0&\Omega_{23(p)}&0&i\Omega_{25(pr)}&0&0&0\\\
0&\Omega_{23(p)}&2(\Delta_{p}+\Delta_{pr})&-i\Omega_{43(pr)}&0&0&i\Omega_{47(pr)}&0\\\
0&-i\Omega_{25(pr)}&0&0&\Omega_{45(p)}&0&i\Omega_{47(pr)}&0\\\
0&-i\Omega_{25(pr)}&0&\Omega_{45(p)}&2(\Delta_{p}+\Delta_{pr})&-i\Omega_{65(pr)}&0&0\\\
0&0&0&0&i\Omega_{65}&0&\Omega_{67(p)}&i\Omega_{68(pr)}\\\
0&0&0&-i\Omega_{47}&0&\Omega_{67(p)}&2(\Delta_{p}+\Delta_{pr})&0\\\
0&0&0&0&0&-i\Omega_{68(pr)}&0&2\Delta_{pr}\\\ \end{pmatrix}\,,$ (26)

where $\Delta_{p}=\omega_{0}-\omega_{p}$ and
$\Delta_{pr}=\omega_{0}-\omega_{pr}$ are pump and probe detunings,
respectively, and $\Omega_{ij(p,pr)}$ are the Rabi frequencies for transitions
between $i$ and $j$ sublevels corresponding to either the pump or the probe
field. The Rabi frequencies are given by

$\displaystyle\Omega_{ij(p,pr)}$ $\displaystyle=$
$\displaystyle-\frac{E_{p,pr}\matrixelement{F_{g}\,m_{g}}{e\mathbf{r}}{F_{e}\,m_{e}}}{\hbar}\,,$
(27)

with the interaction strengths characterized by the dipole matrix elements
between a ground state $\ket{F_{g}\,m_{g}}$ and an excited state
$\ket{F_{e}\,m_{e}}$. To simplify the calculation of Rabi frequencies, the
matrix elements are expressed in terms of the 3-j sysmbol so that the angular
momentum dependence can be factored out:

$\displaystyle\matrixelement{F_{g}\,m_{g}}{e\mathbf{r}}{F_{e}\,m_{e}}=\bra{F_{g}}|e\mathbf{r}|\ket{F_{e}}(-1)^{F_{e}-1+m_{g}}\times\sqrt{2F_{g}+1}\begin{pmatrix}F_{e}&1&F_{g}\\\
m_{e}&q&-m_{g}\end{pmatrix}\,,$ (28)

where $\bra{F_{g}}|e\mathbf{r}|\ket{F_{e}}$ is the reduced matrix element,
independent of the magnetic quantum numbers and $q$ is the index of
$\mathbf{r}$ in the spherical basis. Note that the reduced matrix element will
be the same for all transitions. The pump (probe) reduced Rabi frequency
$\Omega_{p}$ ($\Omega_{pr}$) can now be defined in terms of the reduced matrix
element, such that

$\displaystyle\Omega_{p,pr}$ $\displaystyle=$
$\displaystyle-\frac{E_{p,pr}\bra{F_{g}}|e\mathbf{r}|\ket{F_{e}}}{\hbar}.$
(29)

Any population in the excited state decays to the ground sublevels satisfying
the selection rules, $\Delta m=0,\pm 1$. The decay is characterized by the
branching ratio. Considering that the total decay rate of the excited state is
$\Gamma$, the density matrix equation is given by

$\frac{d}{dt}\bm{\rho}=-\frac{i}{\hbar}[\mathbf{H},\bm{\rho}]-\frac{\Gamma}{2}\sum_{k=1}^{3}\left(\bm{\sigma}_{k}^{+}\bm{\sigma}_{k}^{-}\bm{\rho}+\bm{\rho}\bm{\sigma}_{k}^{+}\bm{\sigma}_{k}^{-}-2\bm{\sigma}_{k}^{-}\bm{\rho}\bm{\sigma}_{k}^{+}\right)\,,$
(30)

where $\bm{\sigma}^{+}_{k}$ and $\bm{\sigma}_{k}^{-}$ are the raising and
lowering operators for decay channels with $\Delta m=0$, $\Delta m=-1$ and
$\Delta m=+1$ denoted by $k=1,2,3$, respectively.

First, consider that the population is initially distributed equally in the
ground sublevels and a cw pump field, linearly polarized in the $z$-direction,
is applied. It induces population transfer with $\Delta m=0$. Consequently,
the states with $|m_{e}|=2$ do not obtain any population. The combined process
of strong pumping and decay moves populations to sublevels with lower magnetic
numbers and eventually brings the system into a steady-state. A useful
quantity called saturation parameter, denoted by S, can be defined in terms of
the reduced Rabi frequency, spontaneous decay rate and detuning

$S=\frac{\Omega_{p}^{2}}{\frac{\Gamma^{2}}{4}+\Delta_{p}^{2}}\,.$ (31)

As mentioned earlier, the frequency parameters are expressed in units of
$\Gamma$ and time parameters in the units of $1/\Gamma$. An example of the
time evolution of the populations for $S=36$ is shown in Fig. 10(a).
Initially, populations are equally distributed among the three ground
degenerate levels. As the system reaches the steady-state, a population
inversion between $|m_{e}|=0$ and $|m_{g}|=1$ levels has been achieved. The
one-photon resonance, $\Delta_{p}=0$, is favorable for achieving population
inversion at lower peak Rabi frequency.

Figure 10: The evolution of populations for $S=36$, (a), and the steady-state
populations vs saturation parameters (b) in the presence of linearly polarized
pump field. Under the combined effect of strong pumping and decay, the
populations tend to move towards the hyperfine levels with lower magnetic
numbers. While there is no inversion of population between the $m_{g}=0$ and
$m_{e}=0$ levels, there is a population inversion between $|m_{g}|=1$ and
$m_{e}=0$ levels. Since the pump is linearly polarized, the $|m_{e}|=2$ levels
are not populated at all. Here, $\Delta_{p}=0$ in both (a) and (b).

In Fig. 10(b), the steady-state populations vs. saturation parameter are
depicted. The population inversion is achieved for values of $S>4.0$, implying
a strong pumping is needed for inversion to occur. The pump field is taken
here to be in resonance with the transition in both figures 10(a) and 10(b).
Taichenachev et al. [48] have derived equations for the steady state
populations in a system with $j_{g}=j$ to $j_{e}=j+1$ energy levels, coupled
resonantly to a pump field. Those steady state equations were derived under
the condition that the time derivatives of the density matrix elements are
zero. They were used in Ref. [25] to calculate populations vs. saturation
parameter for the $F_{g}=2\rightarrow F_{e}=3$ and $F_{g}=3\rightarrow
F_{e}=4$ transitions in 87Rb and 85Rb respectively. Figure 10 shows the
populations obtained from the exact solution of the Liouville von Neuman
equation with relaxation; they are in a good agreement with the ones obtained
using steady state equations from [48] for the $F_{g}=1\rightarrow F_{e}=2$
case.

###  3.2 The probe field gain

Our objective is to find the conditions for positive gain for probe light
polarized orthogonally to the pump field. To achieve this, an equation for
amplification (or absorption) of the field is derived from the density matrix
equation. Every atom that is de-excited from the upper state emits a photon,
which means

$\displaystyle-\frac{d}{dt}n(\rho^{ex})=\frac{d}{dt}n_{p}\,,$ (32)

where $\rho^{ex}$ stands for the population of the excited state manifold
$\rho^{ex}=\sum_{i=1,3,5,7,8}\rho_{ii}$ ; $n$ and $n_{p}$ are the atom and
photon densities, respectively. Only the photons that are spontaneously
emitted within the solid angle $\Phi$ will contribute to the intensity in both
forward and backward directions. Considering this, the equation for the photon
density is written to account for photons emitted due to stimulated and
spontaneous emissions separately

$n_{p}=n_{p}^{st}+\frac{\Phi}{4\pi}n_{p}^{sp}\,.$ (33)

The light intensity is related to the photons density by $I=n_{p}c\hbar\omega$
where $c$ is the speed of light and $\hbar\omega$ is the energy of a photon.
From Eq. (30), one can identify that the first term is related to stimulated
photons while the second term is responsible for spontaneous emissions.
Considering this fact and making use of equality $dz=cdt$, the coupled
equations for the propagation of the pump and the probe fields are written as

$\begin{split}\frac{d}{dy}I_{z}&=-\alpha_{z}I_{z}+\frac{\Phi}{4\pi}n\hbar\omega\Gamma_{z},\\\
\frac{d}{dy}I_{x}&=-\alpha_{x}I_{x}+\frac{\Phi}{4\pi}n\hbar\omega\Gamma_{x},\end{split}$
(34)

where

$\begin{split}\alpha_{z}&=-\frac{n\omega}{2c\epsilon_{0}E_{z0}}\sum_{i=1,3,5,7,8}i[\bm{\mu}_{z},\bm{\rho}]_{ii}\,,\\\
\alpha_{x}&=-\frac{n\omega}{2c\epsilon_{0}E_{x0}}\sum_{i=1,3,5,7,8}i[\bm{\mu}_{x},\bm{\rho}]_{ii}\,,\\\
\Gamma_{z}&=\sum_{i=1,3,5,7,8}\frac{\Gamma}{2}(\bm{\sigma}_{1}^{+}\bm{\sigma}_{1}^{-}\bm{\rho}+\bm{\rho}\bm{\sigma}_{1}^{+}\bm{\sigma}_{1}^{-}-2\bm{\sigma}_{1}^{-}\bm{\rho}\bm{\sigma}_{1}^{+})_{ii}\,,\\\
\Gamma_{x}&=\sum_{i=1,3,5,7,8}\sum_{k=2,3}\frac{\Gamma}{2}(\bm{\sigma}_{k}^{+}\bm{\sigma}_{k}^{-}\bm{\rho}+\bm{\rho}\bm{\sigma}_{k}^{+}\bm{\sigma}_{k}^{-}-2\bm{\sigma}_{k}^{-}\bm{\rho}\bm{\sigma}_{k}^{+})_{ii}\,.\end{split}$
(35)

Here, $\mu_{z}$ and $\mu_{x}$ are the matrices with dipole elements
corresponding to the transitions of pump and probe fields respectively.

$\begin{split}\bm{\mu}_{z}&=\mu_{23}\outerproduct{2}{3}+\mu_{45}\outerproduct{4}{5}+\mu_{67}\outerproduct{6}{7}+\text{H.c.}\,,\\\
\bm{\mu}_{x}&=i\Big{(}\mu_{21}\outerproduct{2}{1}+\mu_{25}\outerproduct{2}{5}+\mu_{43}\outerproduct{4}{3}+\mu_{47}\outerproduct{4}{7}+\mu_{65}\outerproduct{6}{5}+\mu_{68}\outerproduct{6}{8}\Big{)}+\text{H.c.}\end{split}$
(36)

The spontaneous decay factors $\Gamma_{z}$ and $\Gamma_{x}$ are given by

$\begin{split}\Gamma_{z}&=\Gamma\Big{(}b_{32}\rho_{33}+b_{54}\rho_{55}+b_{76}\rho_{77}\Big{)}\,,\\\
\Gamma_{x}&=\Gamma\Big{(}b_{12}\rho_{11}+b_{34}\rho_{33}+(b_{52}+b_{56})\rho_{55}+b_{74}\rho_{77}+b_{86}\rho_{88}\Big{)}\,,\end{split}$
(37)

where $b_{ij}$ are the branching ratios corresponding to the transitions
between $i$ and $j$ states. They read

$\begin{split}b_{32}&=b_{76}=1/2\,,b_{54}=2/3\,,\\\ b_{12}&=b_{86}=1\,,\\\
b_{34}&=b_{74}=1/2\,,\\\ b_{52}&=b_{56}=1/6\,.\end{split}$ (38)

Figure 11: The geometry of the elongated medium responsible for amplified
spontaneous emission whose emission solid angle at both ends of the medium is
denoted as $\Phi$.

The second term of Eq. (34) gives the contribution to the change in intensity
due to spontaneous emission of photons of energy $\hbar\omega$ by $n$ atoms
per unit volume being in the excited state and decaying to respective ground
sub-states connected by $z$-polarized or $x$-polarized light. Since the
spontaneous decay can be assumed to be isotropic, the factor ${\Phi}/{4\pi}$
accounts for the fraction of spontaneously emitted photons into solid angle
$\Phi$. The solutions of Eqs. (34) are

$\begin{split}I_{z}(y)&=I_{z_{0}}e^{-\alpha_{z}y}-\frac{\Phi}{4\pi}n\hbar\omega\Gamma_{z}\left(\frac{e^{-\alpha_{z}y}-1}{\alpha_{z}}\right)\,,\\\
I_{x}(y)&=I_{x_{0}}e^{-\alpha_{x}y}-\frac{\Phi}{4\pi}n\hbar\omega\Gamma_{x}\left(\frac{e^{-\alpha_{x}y}-1}{\alpha_{x}}\right)\,,\end{split}$
(39)

where $I_{z_{0}}=I_{z}(0)$ and $I_{x_{0}}=I_{x}(0)$. In the absence of
contribution from spontaneous emission, the amplification (or absorption) of
the probe field is determined by the absorption coefficient $\alpha_{x}$. To
analyze this coefficient, it is useful to expand the commutator

$\begin{split}\sum_{i=1,3,5,7,8}i[\bm{\mu}_{x},\bm{\rho}]_{ii}&=2\Big{(}\mu_{21}\real[\rho_{12}]+\mu_{25}\real[\rho_{25}]+\mu_{43}\real[\rho_{34}]\\\
&\qquad+\mu_{47}\real[\rho_{47}]+\mu_{65}\real[\rho_{56}]+\mu_{68}\real[\rho_{68}]\Big{)}\end{split}$
(40)

Because of the symmetry in the system, $\rho_{21}=\rho_{68}$,
$\rho_{25}=\rho_{65}$ and $\rho_{43}=\rho_{47}$; the previous equation can
hence be reduced to

$\displaystyle\sum_{i=1,3,5,7,8}i[\bm{\mu}_{x},\bm{\rho}]_{ii}=4\Big{(}\mu_{21}\real[\rho_{12}]+\mu_{43}\real[\rho_{34}]+\mu_{65}\real[\rho_{56}]\Big{)}$
(41)

Figure 12: Coherence between states connected by the probe field. In (a) and
(c), $\Delta_{p}=0$, and in (b) and (d), $\Delta_{p}=0.1\Gamma$. When the pump
field is in resonance, imaginary parts of all the coherences are zero. In the
presence of detuning, non-zero imaginary parts contribute to a phase change of
the field. Figure 13: The probe field absorption as a function of the detuning
$\delta$ for different pump detunings considering a very weak probe field,
$\Omega_{pr}=10^{-3}\,\Omega_{p}$. The pump detunings are (a) $\Delta_{p}=0$,
(b) $\Delta_{p}=10\,\Gamma$ and (c) $\Delta_{p}=15\,\Gamma$. A part of the
plot in (b) is magnified and shown in the inset. While the absorption is
positive for all values in (a), it is negative around 0 in (b) and around
$-10\,\Gamma$ in (c), indicating that a high detuning offset is needed for the
amplification of the probe field due to stimulated emission.

The sign of the absorption coefficient $\alpha_{x}$ in Eq. (35) is opposite to
the sign of the quantity in Eq. (41). For the probe field to be amplified,
$\alpha_{x}$ needs to be negative. In Fig. 12, the real and imaginary parts of
the three coherences $\rho_{12},\rho_{34}$ and $\rho_{56}$ are plotted for
$\Omega_{p}=3.0\,\Gamma$ assuming a negligible probe intensity. Figures 12 (a)
and 12 (b) correspond to the resonant case ($\Delta_{p}=0$), and 12 (c) and 12
(d) correspond to the detuned case $\Delta_{p}=0.1\,\Gamma$. The sum of the
real parts is negative in both cases, leading to the coefficient $\alpha_{x}$
being positive. The positive value of the real part of coherence $\rho_{56}$
between inverted magnetic sublevels reduces the overall negative value of the
sum, thus reducing the degree of absorption. Presumably, there exists a system
in which the inverted state manifold makes dominant contribution, thus,
providing gain in the medium. The present case implies that there is no gain
in the medium. Note that all imaginary parts are zero in the resonant case. In
the detuned case, imaginary parts have non-zero values. However, they do not
contribute to the absorption coefficient; instead they contribute to the
change in phase of the electric field.

Figure 14: Graph of minimum absorption value over the entire spectrum of
radiation for the orthogonally polarized (to the pump) field (seeded by the
probe) versus the pump Rabi frequency. The 87Rb $F=2\rightarrow F=3$ system is
used with parameters $\Gamma=1.0$, $\Delta_{p}=0$. The probe field frequency
that corresponds to the minimum absorption feature varies with $\Delta_{p}$
and $\Omega_{p}$. Under the condition of the one-photon resonance there is no
gain for any value of $\Omega_{p}$. Figure 15: Graph of minimum absorption
value over the entire spectrum of radiation for the orthogonally polarized (to
the pump) field (seeded by the probe) versus the pump Rabi frequency. The 87Rb
$F=2\rightarrow F=3$ system is used with parameters $\Gamma=1.0$ and
$\Delta_{p}=0.75\,\Gamma$. Gain is observed when the one-photon detuning is
non-zero.

To analyze how detuning impacts the absorption coefficient $\alpha_{x}$, it is
useful to define a quantity called frequency offset given by
$\delta=\omega_{p}-\omega_{pr}$. The absorption coefficient, considering a
very weak probe field $\Omega_{pr}=10^{-3}\,\Omega_{p}$, is plotted against
$\delta$ for different pump detunings in Fig. 13. The pump detunings are
$\Delta_{p}=0,10\,\Gamma,15\,\Gamma$ in Figs. 13 (a), 13 (b) and 13 (c)
respectively. It is to be noted that there is no gain in the resonant case as
well as for small values of detuning offset $\delta$. For higher values of
pump detuning there is negative absorption for a certain range of $\delta$
which is a signature of gain. Such an outcome strongly depends on the value of
the Rabi frequency.

The minimum spectral response as a function of the Rabi frequency of the pump
field is shown in Fig. 14 for the resonant ($\Delta_{p}=0$) and Fig. 15 for
the detuned case ($\Delta_{p}=0.75\,\Gamma$). They are calculated using the
two-time correlation function in Eq. (11). The Qutip function library was used
to calculate the spectrum using fast Fourier transform of the two-time
correlation function. Under resonance, we find that there is no gain for any
value of the Rabi frequency. Meanwhile, non-zero detuning provides gain for
certain values of the Rabi frequency of the pump field.

We should note that the results presented here are strictly specific to the
degenerate manifolds of the $F_{g}=1\rightarrow F_{e}=2$ hyperfine transition.
For the system made of $F_{g}=2$ and $F_{e}=3$ states, the contribution from
the population inverted states into the amplification of spontaneous emission
may be more significant.

###  3.3 Propagation Results

In this section, we solve Eq. (34) to find the output intensity of the field
having linear polarization orthogonal to polarization of the incident pump
field. Once the system is pumped to a steady state, the spontaneous photons
generate light, which further gets amplified or absorbed depending on the
contribution from the first term of Eq. (34). The output field intensity for
varying values of the pump field intensity is given in Fig. 16. The output
intensity initially varies linearly and is followed by saturation. The
generated intensity is built from two contributions: one coming from the
directional, stimulated process due to induced polarization [first term in Eq.
(34)] and the second coming from the amplification of isotropic spontaneous
emission [second term in Eq. (34)]. Population inversion present in the steady
atomic state contributes to the stimulated emission/absorption mechanism yet
is not enough to overcome overall losses (absorption from non-inverted states)
and hence does not lead to gain but only reduced absorption. However, one
should expect the output measurement of the field having polarization
perpendicular to the pump field to be nonzero owing to the isotropic emission
of spontaneously emitted photons. The intensity of this field depends on the
Rabi frequency and the pump field detuning. As an example, assuming a decay
rate $\Gamma=2\pi\times 5.6$ MHz (87Rb $F=2\rightarrow F=3$), we propose using
a Rabi frequency $\Omega_{p}=0.4\Gamma$, and a one-photon detuning
$\Delta_{p}=0.75\Gamma$ for experimental realization.

Figure 16: The output intensity of the field linearly polarized in the $x$
direction as a function of the pump field linearly polarized in the
z-direction. The cell length is $0.1\,$m and the atomic density is
$n=1.16\times 10^{16}\,$m-3 [22].

##  4 Experimental situation

The idea of degenerate mirrorless lasing was born at the Institute for
Physical Research, in Ashtarak, Armenia [25, 23] and evolved into a fruitful
collaboration with the University of Mainz a few years later.

The principle idea, as discussed above, is that optical pumping redistributes
the population among the ground-state Zeeman sublevels. At sufficiently high
light powers of a linearly-polarized radiation, the population is partially
transferred to the excited state, which, in the case of an $F\rightarrow F+1$
transition may result in population inversions between certain sublevels of
the ground and excited states. Due to the pencil-shaped geometry of the light-
atom interaction volume, this inversion may, under right circumstances, result
in directional emission from the sample. In the case of degenerate mirrorless
lasing, amplified spontaneous emission is produced along or opposite to the
pump-beam direction, with polarization orthogonal to that of the pump.

Forward degenerate mirrorless lasing was detected both in Ashtarak and Mainz,
showing a typical threshold dependence of the outgoing beam on the incoming
beam intensity. The transitions for which we expected and observed the lasing
are $F_{e}>F_{g}$ cycling transitions in the $D_{2}$ line of rubidium,
specifically, 85Rb $F_{g}=3\rightarrow F_{e}=4$ and 87Rb $F_{g}=2\rightarrow
F_{e}=3$. The intensity of the emission polarized orthogonally with respect to
the incident light was studied as a function of the pump intensity and
magnetic field applied along the direction of the pump-light polarization. Up
to 1 % conversion efficiency was observed at zero-magnetic field, which
approached to zero with the applied magnetic field. The width of this feature
was about 15 – 100 mG (FWHM) depending on the pump intensity.

These findings [22] are a strong indication of amplified emission in the
forward direction, also known as forward mirrorless lasing. In parallel, we
detected degenerate four-wave mixing (FWM), which makes the signal more
complicated.

While forward mirrorless lasing is an interesting phenomenon, backward
mirrorless lasing is of greater interest, since it can be used for remote-
detection magnetometry, for example with laser guide stars [49]. Indeed, high
directivity of the backward beam will significantly facilitate recording of
the return flux, which will make the distant remote sensing much easier.

From the theoretical analysis presented in previous sections one can expect
that the mirrorless lasing should also take place along the pump beam in the
backward direction.

It would seem that mirrorless lasing in the backward direction should be
detected even more easily than in the forward direction, since there is no
strong background of pump radiation, and also no contribution from the
accompanying FWM. However, preliminary experiments have revealed complexities
that require further reflection and clarification.

It was possible to record a non-diverging backward emission in Ashtarak with
much lower conversion efficiency (almost 3 orders of magnitude less than for
the forward one) at specific experimental conditions. The recorded radiation
had an intensity threshold and was sensitive to the applied magnetic field,
similar to the forward case, but exhibited sharp sub-Doppler features, which
were not observed in the forward lasing. But this result could not be
reproduced in Mainz, even with the same vapor cell and the same
experimentalists.

However, these preliminary studies were useful in identifying the factors
important for realization of the backward mirrorless lasing. Among these
factors are: (i) careful avoiding of residual birefringence of the vapor cell
windows; (ii) proper cancellation and control of the magnetic field; (iii)
presence (or absence) of a weak seeding radiation; (iv) broad spectral
linewidth of the pump laser.

The much lower conversion efficiency, as well as the narrow spectral features
of the backward radiation, should be associated with the competition between
the emission and absorption in opposite directions by atoms with different
velocities for different pump detunings from Doppler-overlapped atomic
transitions.

Further theoretical analysis and experimental studies are needed for complete
understanding, as well as unambiguous and reproducible demonstration of
backward degenerate mirrorless lasing.

##  5 Conclusions and outlook

Degenerate mirrorless lasing is a beautiful phenomenon that has been realized
experimentally in the forward direction. However, in the case of backward
directed emission, more experimental work needs to be done. At the same time,
theoretical guidance is needed to guide and interpret the experimental work.

In this manuscript, we have reviewed the current status of the mirrorless
lasing and presented the work-in-progress developments in the general
theoretical analysis, hopefully, on the way to a comprehensive solution of the
problem.

## Conflicts of Interest

The authors declare no conflicts of interest.

We recognize the contribution of the tragically deceased Hrayr Azizbekyan.
S.M., J.Ch., and A.R. acknowledge support from the Office of Naval Research
under awards N00014-20-1-2086 and N00014-22-1-2374. S.M. acknowledges the
Helmholtz Institute Mainz Visitor Program and the Alexander von Humboldt
Foundation. J.Ch. was also partially supported by DEVCOM Army Research
Laboratory under Cooperative Agreement Number W911NF2320076. D.B. was
partially supported by DEVCOM Army Research Laboratory under Cooperative
Agreement Number W911NF2120180.

##  Appendix A Spontaneous emission modes

We use the formalism from [29]:

We define the field operator:

$E^{(\mu)+}_{H}(\vec{r},t)=\dfrac{i}{(2\pi)^{3}}\int d^{3}k\text{
}{a}_{\vec{k},H}^{(\mu)}(t)\sqrt{\dfrac{h\omega_{k}}{2\epsilon_{0}}}e^{i\vec{k}\cdot\vec{r}}\varepsilon^{(\mu)}_{\vec{k}}\,,$
(42)

and use the Heisenberg equations of $a_{\vec{k}}^{(\mu)}(t)$:

$\dot{a}_{\vec{k},H}^{(\mu)}(t)=-i\omega_{k}{a}_{\vec{k},H}^{(\mu)}(t)-i\sum_{\alpha,j}g_{\vec{k},j}^{(\mu)*}(\vec{R}_{\alpha})\sigma^{+}_{j,H}(\vec{R}_{\alpha},t)\,,$
(43)

to get the following equation for the Heisenberg operator:

$\begin{split}&\Box^{2}E^{(\mu)+}_{H}(\vec{r},t)=\dfrac{1}{(2\pi)^{3/2}c}\sum_{\alpha,j}\int
d^{3}k\text{
}\sqrt{\dfrac{h\omega_{k}}{2\epsilon_{0}}}g_{\vec{k},j}^{(\mu)*}(\vec{R}_{\alpha})\varepsilon^{(\mu)}_{\vec{k}}\\\
&\crossproduct(\omega_{k}-\omega_{j})\sigma^{+}_{j,H}\left(\vec{R}_{\alpha},t-\dfrac{\absolutevalue{\vec{r}-\vec{r}^{\prime}}}{c}\right)e^{i\vec{k}\cdot\vec{r}}.\end{split}$
(44)

$\Box^{2}$ is the D’Alembertian. The right hand side is the total current
density, summing over all atomic contributions. We use the Green’s function
for the D’Alembertian to get the integral:

$\begin{split}&E^{(\mu)+}_{j,H}(\vec{r},t)=\dfrac{1}{c}\int
d^{3}r^{\prime}\text{
}\dfrac{J_{j}^{+}\left(\vec{r}^{\prime},t-\dfrac{\absolutevalue{\vec{r}-\vec{r}^{\prime}}}{c}\right)}{\absolutevalue{\vec{r}-\vec{r}^{\prime}}}\,,\end{split}$
(45)

and obtain the general solution:

$\begin{split}&E^{(\mu)+}_{j,H}(\vec{r},t)=\sqrt{\dfrac{h}{2c^{2}\epsilon_{0}(2\pi)^{3}}}\sum_{\alpha,j}\int
d^{3}r^{\prime}\int dkd\Omega_{k}\\\ &\crossproduct
g_{\vec{k},j}^{(\mu)*}\left(\vec{R}_{\alpha}\right)k^{3}(\omega_{k}-\omega_{j})\dfrac{\hat{k}\crossproduct(\hat{k}\crossproduct\hat{d}_{j})}{\absolutevalue{\vec{r}-\vec{r}^{\prime}}}\\\
&\crossproduct\sigma^{+}_{j,H}\left(\vec{R}_{\alpha},t-\dfrac{\absolutevalue{\vec{r}-\vec{r}^{\prime}}}{c}\right)e^{i\vec{k}\cdot\vec{r}}.\end{split}$
(46)

In the far-field limit, the solution simplifies to:

$\begin{split}&E^{(\mu)+}_{j,H}(\vec{r},t)\propto\sum_{\alpha,j}\dfrac{\omega_{j}^{2}}{c^{2}}\dfrac{(\hat{r}_{\alpha}\cdot\hat{d}_{j})\hat{r}_{\alpha}-\hat{d}_{j}}{r_{\alpha}}\\\
&\crossproduct\sigma^{+}_{j,H}\left(\vec{R}_{\alpha},t-\dfrac{r}{c}\right)e^{-i\omega_{j}(\hat{r}\cdot\hat{r}_{j})}.\end{split}$
(47)

where $\vec{r}_{\alpha}=\vec{r}-\vec{R}_{\alpha}$.

In the far field, each atom acts as a dipole antenna. In the case where we
have many atoms and they’re all stationary, a very complicated spatially
constant interference pattern emerges. However if the atom has a statistical
velocity distribution and the number density is high enough for collisions to
be significant, the atomic medium behaves as a choatic light source.

##  Appendix B Paraxial approximation

The volume-averaged dipole operator
$S_{j;I^{\prime}}^{+}(\vec{R}_{i},t)=n(\vec{R}_{i})\int_{V_{r}(\vec{R}_{i})}dV^{{}^{\prime}}\sigma_{j;I^{\prime}}^{+}(\vec{R}^{{}^{\prime}},t)$
is introduced. The volume $V_{r}(\vec{R}_{i})$, center at $\vec{R}_{i})$ with
radius $r$ is chosen as our sampling region where the atomic density matrix
and the radiation fields vary trivially in $V_{r}(\vec{R}_{i})$. We use it to
simplify the below quantity:

$\begin{split}&\int_{V_{r}(\vec{R}_{i})}dV^{{}^{\prime}}\mathcal{C}^{(\mu^{\prime})}_{j,l}(\vec{r},\vec{R}^{{}^{\prime}},t,t^{\prime})e^{i(\vec{q}-\vec{k}^{\prime})\cdot\vec{R}}\\\
&=\mathcal{C}^{(\mu^{\prime})}_{j,l}(\vec{r},\vec{R}_{i},t,t^{\prime})\int_{V_{r}(\vec{R}_{i})}dV^{{}^{\prime}}e^{i(\vec{q}-\vec{k}^{\prime})\cdot\vec{R}}.\end{split}$
(48)

In the limit of high atomic number density, the condition
$\vec{q}=\vec{k}^{\prime}$ is enforced as other contributions average to 0 as
only contributions where the wavevectors are approximately the same survive
phase averaging over the volume. With the paraxial approximation, the dot
product
$\varepsilon^{(\mu)}_{\vec{k}}\cdot\varepsilon^{(\mu^{\prime})}_{\vec{k^{\prime}}}\approx\delta_{(\mu\mu^{\prime})}$.
We assume the averaged dipole moment vector
$\vec{D}_{l}(\vec{R}_{i})=\int_{V_{r}(\vec{R}_{i})}dV^{{}^{\prime}}\vec{d}_{l}(\vec{R}^{\prime})$
averages out all fluctuations such that the z-dipole moment,
$\vec{D}_{z}(\vec{R}_{i})$, is oriented along the pump polarization axis.
Ignoring the Langevin terms, Eq. (18) simplifies to:

$\displaystyle\begin{split}&\mathcal{R}_{\text{s}}(t)\\\
&=-\delta_{(\mu\mu^{\prime})}\sum_{jl,\alpha;\nu=\pm}\int_{0}^{t}dt^{\prime}\\\
&\left(\expectationvalue{(E_{I^{\prime}}^{(\mu)-}(\vec{r},t)\vec{E}_{I^{\prime}}^{(\mu)+}(\vec{r},t^{\prime})(\vec{d}_{l}\cdot\varepsilon^{(\nu)}_{\vec{x}})\mathcal{D}_{j,l}(\vec{R}_{i},t,t^{\prime})}\right.\\\
&\left.+\expectationvalue{\vec{E}_{I^{\prime}}^{(\mu)-}(\vec{r},t^{\prime})E_{I^{\prime}}^{(\mu)+}(\vec{r},t)(\vec{d}^{*}_{l}\cdot\varepsilon^{(\nu)*}_{\vec{x}})\mathcal{D}^{{\dagger}}_{j,l}(\vec{R}_{i},t,t^{\prime})}\right)/,,\end{split}$
(49)

Where $\mathcal{D}_{j,l}(\vec{R}_{i},t,t^{\prime})$ is the dipole correlation
function Eq. (19).

The expectation of single atom dipole products
$\expectationvalue{{\sigma}^{+}_{l,I^{\prime}}(\vec{R_{i}},t){\sigma}^{-}_{m,I^{\prime}}(\vec{R_{i}},t^{\prime})}$
is expanded in the dressed state basis in terms of operators, defined in the
picture $I^{\prime}$,
${\sigma}_{\Lambda_{r}\Lambda_{r^{\prime}}}(\vec{R_{i}},t,t^{\prime})=\ket{\Lambda_{r}(\vec{R_{i}},t)}\bra{\Lambda_{r^{\prime}}(\vec{R_{i}},t^{\prime})}$.
:

$\begin{split}&\sigma^{+}_{i,I^{\prime}}(\vec{R_{\alpha}},t){\sigma}^{-}_{j,I^{\prime}}(\vec{R_{\alpha}},t^{\prime})\\\
&=-{i}{\hbar}\sum_{i,l,l^{\prime}}\sum_{j,m}C^{i}_{ll^{\prime}}(\vec{R}_{\alpha},t)C^{j*}_{ml^{\prime}}(\vec{R}_{\alpha},t^{\prime})e^{it(\omega_{i}-\omega_{j}+\lambda_{l}-\lambda_{m})}\\\
&\crossproduct\sigma_{\Lambda_{l}\Lambda_{m}}(\vec{R}_{\alpha},t,t^{\prime})e^{i(\omega_{j}+\lambda_{m}-\lambda_{l^{\prime}})(t-t^{\prime})}/,,\end{split}$
(50)

Assuming the weak coupling approximation, the Markov approximation along with
the slow change of
$\sigma_{\Lambda_{l}\Lambda_{m}}(\vec{R}_{\alpha},t,t^{\prime})$, the
coefficients $C_{mm^{\prime}}(t^{\prime})$ and the operators are dependent
only on time $t$. The time evolution of the expectation of these operators is
given below (The contribution of the stimulated emission and pair-atom terms
are ignored):

$\begin{split}&\dfrac{d}{dt}\expectationvalue{\sigma_{\Lambda_{a}\Lambda_{b}}(\vec{R}_{\alpha},t,t)}\\\
&=-\dfrac{i}{\hbar}\int\int d^{3}k\text{ }d^{3}k^{\prime}\text{
}g_{\vec{k},i}^{(\mu)}(\vec{R}_{\alpha})g_{\vec{k}^{\prime},j}^{(\mu^{\prime})*}(\vec{R}_{\alpha})\\\
&\crossproduct\sum_{lm}C^{i}_{la}(\vec{R}_{\alpha},t)C^{j*}_{am}(\vec{R}_{\alpha},t)e^{it(\omega_{i}+\lambda_{l}-\omega_{k})}\\\
&\crossproduct\int_{t_{0}}^{t}dt^{\prime}\expectationvalue{\sigma_{\Lambda_{a}\Lambda_{m}}(\vec{R}_{\alpha},t,t)}e^{-it^{\prime}(\omega_{j}+\lambda_{m}-\omega_{k^{\prime}})}/,,\end{split}$
(51)

We now use the Markov approximation. From the standard methods of solving for
the spectral line shapes by solving the above equations for all dipole
operators in Laplace space, we can determine the power broadened spectral line
shapes, $f_{\Lambda_{a}\Lambda_{m}}(\omega^{\prime},t)$, corresponding to
$\expectationvalue{\sigma_{\Lambda_{a}\Lambda_{m}}(\vec{R}_{\alpha},t,t)}$ and
the decay rate $\Gamma_{\Lambda_{a}\Lambda_{m}}(t)$. The spontaneous decay
rates for the atomic density matrices in the dressed state basis are given
below:

$\begin{split}&\Gamma_{1}^{ab;ml}(t)=\int_{0}^{\infty}d\tau\int
d^{3}k\sum_{ij}\left(C_{bl}^{i*}(t)C_{ma}^{j}(t)g_{\vec{k},i}^{(\mu)*}g_{\vec{k},j}^{(\mu)}\right.\\\
&\left.e^{-i(\omega_{ij}+\lambda_{bl}+\lambda_{am})t}e^{i(\omega_{k}-\omega_{j}-\lambda_{am})\tau}\right.\\\
&+\left.C_{ma}^{i}(t)C_{bl}^{j*}(t)g_{\vec{k},i}^{(\mu)}g_{\vec{k},j}^{(\mu)*}e^{i(\omega_{ij}+\lambda_{bl}+\lambda_{am})t}e^{-i(\omega_{k}-\omega_{j}-\lambda_{bl})\tau}\right)/,,\end{split}$
(52)

$\begin{split}&\Gamma_{2}^{ab;l}(t)=\int_{0}^{\infty}d\tau\int
d^{3}k\sum_{ijl^{\prime}}\left(C_{l^{\prime}l}^{i*}(t)C_{l^{\prime}b}^{j}(t)g_{\vec{k},i}^{(\mu)*}g_{\vec{k},j}^{(\mu)}\right.\\\
&\left.e^{-i(\omega_{ij}+\lambda_{l^{\prime}l}+\lambda_{bl^{\prime}})t}e^{i(\omega_{k}-\omega_{j}-\lambda_{bl^{\prime}})\tau}\right)/,,\end{split}$
(53)

$\begin{split}&\Gamma_{3}^{ab;l}(t)=\int_{0}^{\infty}d\tau\int
d^{3}kd^{3}k^{\prime}\sum_{ijl^{\prime}}\left(C_{ll^{\prime}}^{i}(t)C_{l^{\prime}a}^{j*}(t)g_{\vec{k},i}^{(\mu)}g_{\vec{k},j}^{(\mu)*}\right.\\\
&\left.e^{i(\omega_{ij}+\lambda_{l^{\prime}a}+\lambda_{l^{\prime}l})t}e^{-i(\omega_{k}-\omega_{j}-\lambda_{l^{\prime}a})\tau}\right)/,,\end{split}$
(54)

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