# Non-mesonic decay of the $\eta$-mesic ${}^{3}\mbox{He}$ via
$pd\rightarrow(^{3}\mbox{He}$-$\eta)_{bound}\rightarrow$
${}^{3}\mbox{He}2\gamma(6\gamma$) reaction

M. Skurzok<EMAIL_ADDRESS>S. Hirenzaki  S. Kinutani  H. Konishi
P. Moskal  H. Nagahiro  O. Rundel  Institute of Physics, Jagiellonian
University, prof. Stanisława Łojasiewicza str. 11, 30-348 Kraków, Poland
INFN, Laboratori Nazionali di Frascati, 00044 Frascati, Italy  Department of
Physics, Nara Women’s University, Nara 630-8506, Japan  Research Center for
Nuclear Physics (RCNP), Osaka University, Ibaraki 567-0047, Japan

###### Abstract

In this article a theoretical model for the $\eta$-mesic ${}^{3}\mbox{He}$
non-mesonic decay channels is presented. We present the resultant relative
momentum distribution of bound ${}^{3}\mbox{He}$-$\eta$ as well as in-medium
branching ratios of $\eta\rightarrow 2\gamma$ and $\eta\rightarrow 3\pi^{0}$,
which are crucial for the Monte Carlo simulations of measured processes and
thus for the experimental data interpretation. As an example we also apply the
model for the estimation of the detection efficiency of the WASA-at-COSY
detector.

###### keywords:

mesic nuclei, non-mesonic decay, optical potential, Monte Carlo Simulations

††journal: Journal of LaTeX Templates

##  1 Introduction

In the surrounding universe, aside from usual matter like atomic nuclei, a
variety of uncommon exotic objects can be found. Although many of them, such
as hypernuclei [1], tetraquarks [2], pentaquarks [3] or dibaryons [4, 5, 6]
have already been discovered and investigated, still many exotic systems are
theoretically predicted but never experimentally confirmed. Mesic nuclei,
consisting of a nucleus bound via the strong interaction with a neutral meson
are one example. They are currently one of the hottest topics in nuclear and
hadronic physics, both from experimental [7, 8, 9, 10, 11, 12] and theoretical
[13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,
32, 33, 34] standpoints. Some of the most promising candidates for such bound
states are $\eta(\eta^{\prime})$-mesic nuclei since the $\eta$-nucleon
interaction was found to be attractive [35, 36] while the imaginary
$\eta^{\prime}$-nucleus potential for near threshold is significantly smaller
than the modulus of the real part [37].

Recently performed studies of hadron- and photo-production of the $\eta$ meson
result in a wide range of values of the $\eta N$ scattering length indicating
that the $\eta$ meson-nucleon interaction is attractive and strong enough to
create even light $\eta$-nucleus bound systems [15, 16, 17, 18, 19, 38, 39,
40]. However, none of the experiments performed till now have found a clear
signature confirming their existence. They only provided signals which might
be interpreted as indications of the $\eta$-mesic nuclei as well as allow the
determination of the upper limits of the total cross section for the bound
state formation [7, 8, 9, 11, 12, 23, 25, 41, 42, 43, 44, 45, 46, 47, 48, 49,
50, 51, 54, 52, 53].

One of the most recent and promising experiments related to $\eta$-mesic
Helium nuclei have been performed by the WASA-at-COSY Collaboration [7, 8, 9,
55]. The measurements were carried out with high statistics with the WASA
detection setup in deuteron-deuteron (${}^{4}\mbox{He}$-$\eta$) [7, 8, 9] and
proton-deuteron (${}^{3}\mbox{He}$-$\eta$) [56, 57] fusion reactions using the
ramped beam technique.

The analysis dedicated to search for ${}^{4}\mbox{He}$-$\eta$ mesic nuclei in
$dd\rightarrow$ ${}^{3}\mbox{He}n\pi{}^{0}$ and $dd\rightarrow$
${}^{3}\mbox{He}p\pi^{-}$ processes resulted in the upper limits of the total
cross section at a 90% confidence level equal to roughly 3 nb and 6 nb,
respectively [8]. The determined excitation functions were compared with the
predictions of the model proposed in Ref. [14], this allowed to put a
constraint on the $\eta$-4He optical potential parameters [7].

The analyses and interpretations of all experiments up till now have been
performed assuming a mechanism according to which (after the $\eta$-mesic
nucleus creation) the $\eta$ meson is absorbed on one of the nucleons inside
helium and may propagate in the nucleus via consecutive excitations of
nucleons to the $N^{*}$(1535) state, until the resonance decays into the
nucleon-pion pair. Thus far, Monte Carlo simulations, used for the estimation
of the detector systems registration efficiency, have been carried out
assuming that the $N^{*}$ momentum distribution is the same as the
distribution of nucleons [58]. Only recently the first model describing the
$N^{*}$ momentum in the $N^{*}$-${}^{3}\mbox{He}$ bound state was proposed in
references [22, 59]. Another theoretical model predicting non-$N\pi$ decays of
the mesic-helium formed via two-nucleon absorption process was pointed in Refs
[33, 60].

Recently, a new mechanism of the hypothetical $\eta$-mesic helium decay was
considered, namely via $\eta$ meson decay while it is still ”orbiting” around
a nucleus. In order to avoid complications due to final state interactions
neutral decay channels such as $\eta\to 2\gamma$ and $\eta\to 3\pi^{0}\to
6\gamma$ constituting more than 70% of $\eta$ meson decays [61] are best
suited for these studies. The dedicated measurement was performed for the
first time with the WASA-at-COSY facility to search for $\eta$-mesic
${}^{3}\mbox{He}$ in $pd\rightarrow$ ${}^{3}\mbox{He}2\gamma$ and
$pd\rightarrow$ ${}^{3}\mbox{He}6\gamma$ reactions.

In this article we present a theoretical model for the $\eta$-mesic
${}^{3}\mbox{He}$ non-mesonic decay channels. We present the resulting
relative momentum distribution of bound ${}^{3}\mbox{He}$-$\eta$ as well as
in-medium branching ratios of $\eta\rightarrow 2\gamma$ and $\eta\rightarrow
3\pi^{0}$, which are crucial for the Monte Carlo simulations of measured
processes and hence for the experimental data interpretation. As an example we
apply the model for the estimation of the detection efficiency of the WASA-at-
COSY detector.

Our goal is to know the $\eta$-nucleus interaction and hence the
(non)existence of the quasi-stable bound states. The criterion of the
existence of quasi-stable bound states with long enough lifetime can be
verified by the measurement of binding energy and the width of the state and
by experimental determination of the in-medium branching ratios of $\eta$
decay into $2\gamma$ and $3\pi^{0}$.

##  2 Theoretical model

A theoretical model has been developed in order to describe the kinematics of
the decay of $\eta$-mesic ${}^{3}\mbox{He}$ nucleus in $pd\rightarrow$
${}^{3}\mbox{He}2\gamma$ and $pd\rightarrow$ ${}^{3}\mbox{He}6\gamma$
processes. The proposed mechanism proceeds according to the scheme shown in
Fig. 1.

Figure 1: Scheme of the ${}^{3}\mbox{He}$-$\eta$ mesic nucleus production and
decay in $pd\rightarrow$ ${}^{3}\mbox{He}2\gamma$ reaction.

The proton-deuteron collision leads to the creation of ${}^{3}\mbox{He}$
nucleus bound with the $\eta$ meson via strong interaction. The bound $\eta$
meson is orbiting inside the nucleus until it decays into $2\gamma$ or
$3\pi^{0}$, wheres $3\pi^{0}$ decay almost immediately to $6\gamma$. The
${}^{3}\mbox{He}$ nucleus plays the role of a spectator in the decay
processes. It is assumed, that the $\eta$ meson is moving with Fermi momentum
inside mesic nucleus before decaying into $2\gamma$ or $3\pi^{0}$. The Fermi
momentum distribution is evaluated based on the calculated $\eta$ wavefunction
described in detail in the next subsection.

###  2.1 Structure of the $\eta$ bound state and in-medium branching ratios of
$\eta\rightarrow 2\gamma$ and $\eta\rightarrow 3\pi^{0}$

The structure of the hypothetical $\eta$ bound state produced in
$pd\rightarrow$ ${}^{3}\mbox{He}\eta$ reaction can be described as the
solution of the Klein-Gordon equation:

$~{}\left[-\vec{\bigtriangledown}^{2}+\mu^{2}+2\mu U_{\rm
opt}(r)\right]\psi(\vec{r})=E^{2}_{\rm KG}\psi(\vec{r}),$ (1)

with $E_{\rm KG}$ and $\mu$ denoting the Klein-Gordon energy and 3He-$\eta$
reduced mass, respectively. The $U_{\rm opt}(r)$ is the optical potential
describing the interaction between 3He and $\eta$, and is assumed to have the
functional form:

$U_{\rm opt}(r)=(V_{0}+iW_{0})\frac{\rho(r)}{\rho_{0}}\ ,$ (2)

where $\rho(r)$ is the density distribution of 3He and $\rho_{0}$ is the
normal nuclear density $\rho_{0}=0.17$ fm-3. The 3He density distribution
$\rho(r)$ is obtained by the theoretical calculation described in Refs. [62,
63, 64].

The equation is solved numerically for several sets of real $(V_{0})$ and
imaginary $(W_{0})$ optical potential parameters to obtain the Klein-Gordon
energy $E_{\rm KG}$ and the wavefunction $\psi(\vec{r})$. The binding energy
$B_{\rm s}$ and the nuclear absorption width $\Gamma_{\rm abs}$ of the $\eta$
bound state are defined via the Klein-Gordon energy $E_{\rm KG}$ as $B_{\rm
s}={\rm Re}(\mu-E_{\rm KG})$ and $\Gamma_{\rm abs}=-2{\rm Im}(E_{\rm KG})$,
respectively. We show the assumed strength of the optical potential parameters
($V_{0}$ and $W_{0}$), the obtained binding energy $B_{\rm s}$, and nuclear
absorption width $\Gamma_{\rm abs}$ in Table 1. In addition to the strongly
absorptive potential with $W_{0}=-20$ MeV, we also assumed the weakly
absorptive potential with $W_{0}=-1$ MeV as indicated in Ref. [7] for the
4He-$\eta$ system.

$(V_{0},W_{0})$ [MeV] | ($B_{\rm s}$, $\Gamma_{\rm abs}$) [MeV] | BR${}^{*}_{\eta\rightarrow 2\gamma}$ | BR${}^{*}_{\eta\rightarrow 3\pi^{0}}$
---|---|---|---
$-$(75,20) | (4.06, 15.66) | $3.30\times 10^{-5}$ | $2.73\times 10^{-5}$
$-$(90,20) | (11.16, 20.65) | $2.50\times 10^{-5}$ | $2.07\times 10^{-5}$
$-$(75,1) | (5.96, 0.76) | $6.78\times 10^{-4}$ | $5.62\times 10^{-4}$
$-$(90,1) | (12.67, 1.02) | $5.06\times 10^{-4}$ | $4.20\times 10^{-4}$
Table 1: The binding energies $B_{\rm s}$ and nuclear absorption widths
$\Gamma_{\rm abs}$ values for the 3He-$\eta$ ground $(0s)$ states obtained by
solving Eq. (1) are listed with the optical potential parameters
$(V_{0},W_{0})$ assumed in the present calculation. Evaluated in-medium
branching ratios BR∗ are also shown (See details in text).

By transforming the coordinate space wavefunction $\psi(\vec{r})$ obtained by
solving Eq. (1), we can derive the momentum space wavefunction in the form
$\phi(\vec{p})=R(p)Y_{\ell m}(\hat{p})$ using:

$\phi(\vec{p})=\frac{1}{(2\pi)^{3/2}}\int
e^{i\vec{p}\cdot\vec{r}}\psi(\vec{r})d\vec{r}\ ,$ (3)

and we can evaluate the relative 3He-$\eta$ momentum distribution using
$|R(p)|^{2}$, where $R(p)$ satisfies the normalization condition
$\displaystyle\int|R(p)|^{2}p^{2}dp=1$. The momentum distributions
$|R(p)|^{2}p^{2}$ for four sets of potential parameter values are presented in
Fig. 2. We also derive the momentum space wavefunction of the nucleon
$\phi_{N}(\vec{p})=R_{N}(p)Y_{00}(\hat{p})$ in the same manner using the
coordinate space $s$-wave nucleon wavefunction $\Psi_{N}(\vec{r})$, which is
assumed to be related to the nucleon density distribution $\rho(r)$ via
$\rho(r)=A|\Psi_{N}(\vec{r})|^{2}$ with the nuclear mass number $A$.

Figure 2: Fermi momentum distribution of the $\eta$ meson in
${}^{3}\mbox{He}$-$\eta$ bound system estimated for ($V_{0},W_{0}$)=$-$(75,20)
MeV (thick solid line), ($V_{0},W_{0}$)=$-$(75,1) MeV (thick dotted line),
($V_{0},W_{0}$)=$-$(90,20) MeV (thin solid line), and
($V_{0},W_{0}$)=$-$(90,1) MeV (thin dotted line). The distributions are
normalized to be $1$ in the whole momentum range.

We show in Fig. 3 the calculated momentum distribution of the nucleon
$|R_{N}(p)|^{2}p^{2}$ based on the 3He and 4He density distributions [62, 63,
64].

Figure 3: Fermi momentum distribution of a nucleon in 3He (solid line) and 4He
(dotted line) nuclei evaluated by the theoretical nuclear density
distributions (see details in text).

One can see that in general the momentum distributions for a nucleon inside
3He and in 4He are broader then the distributions of the relative 3He-$\eta$
momentum, though the distributions for a nucleon inside 3He are comparable
with the one for 3He-$\eta$ for $V_{0}$=-75 MeV.

We can also evaluate the in-medium branching ratios BR∗ for $\eta\rightarrow
2\gamma$ and $\eta\rightarrow 3\pi^{0}$ decay channels below the $\eta$
threshold using the calculated nuclear absorption width $\Gamma_{\rm abs}$ as,

${\rm BR}^{*}_{\eta\rightarrow 2\gamma/\eta\rightarrow
3\pi^{0}}=\frac{\Gamma_{\eta\rightarrow 2\gamma/\eta\rightarrow
3\pi^{0}}}{(\Gamma^{\rm tot}_{\eta}+\Gamma_{\rm abs})}\ ,$ (4)

where $\Gamma_{\eta\rightarrow 2\gamma/\eta\rightarrow 3\pi^{0}}$ is the width
of the in-vacuum $\eta$ decay to $2\gamma$ or $3\pi^{0}$, $\Gamma^{\rm
tot}_{\eta}$ is the total width of $\eta$ meson in vacuum (1.31 keV) [61] and
$\Gamma_{\rm abs}$ is the nuclear absorption width obtained from Klein-Gordon
equation. From the in-vacuum branching ratios BR${}_{\eta\rightarrow
2\gamma/\eta\rightarrow 3\pi^{0}}$ reported in [61], $\Gamma_{\eta\rightarrow
2\gamma}$ and $\Gamma_{\eta\rightarrow 3\pi^{0}}$ can be calculated as

$\displaystyle\Gamma_{\eta\rightarrow 2\gamma}=0.3941\times 1.31\ {\rm keV}\
=0.516\ {\rm keV},$ (5) $\displaystyle\Gamma_{\eta\rightarrow
3\pi^{0}}=0.3268\times 1.31\ {\rm keV}\ =0.428\ {\rm keV}\ .$ (6)

The estimated branching ratios in medium BR∗ are listed in the 3rd and 4th
column in Table 1. They vary from about 2 $\times$ 10-5 to 7 $\times$ 10-4
depending on the optical potential parameters.

The calculated BRs has to be corrected due to the Final State Interaction
(FSI) effects of pions with the 3He nucleus. We found that the FSI reduces the
strength of the signal to half and the effect should be taken into account for
the planning and/or analyzing the experiments.

###  2.2 Monte Carlo simulation

The theoretical model described in previous subsection will be applied in the
realistic Monte Carlo simulations of the $\eta$-mesic production in the
$pd\rightarrow$ ${}^{3}\mbox{He}2\gamma$ and $pd\rightarrow$
${}^{3}\mbox{He}6\gamma$ reactions. In the first step, the geometrical
acceptance of the WASA detector [65, 66] as a function of the excess energy
$Q$ near the kinematical threshold for $\eta$ meson production was evaluated.
In the simulation, the ${}^{3}\mbox{He}$ nucleus is assumed to be a spectator
and the bound $\eta$ has the energy determined by its mass and binding energy.
The decay distribution is assumed to be isotropic in the $\eta$ meson rest
frame and the sum of the momenta of the emitted particles are the same as the
$\eta$ Fermi momentum in the center of mass frame. In the final state, the
${}^{3}\mbox{He}$ nucleus has the recoil momentum and energy.

The acceptance was determined for the simultaneous registration of
${}^{3}\mbox{He}$ in the Forward Detector (covering polar angles from 3∘ to
18∘) and $\gamma$ quanta in the Central Detector (covering polar angles from
20∘ to 169∘) and was found to be about 60% and 40% for $pd\rightarrow$
$(^{3}\mbox{He}$-$\eta)_{bound}$ $\rightarrow$ ${}^{3}\mbox{He}2\gamma$ and
$pd\rightarrow$ $(^{3}\mbox{He}$-$\eta)_{bound}$ $\rightarrow$
${}^{3}\mbox{He}6\gamma$ reactions, respectively.

The realistic simulations including the detector responses and all analysis
conditions, will be crucial for the interpretation of the experimental data,
in particular data collected by WASA-at-COSY Collaboration. The simulation
results will be compared with experimental data to choose the most optimal
analysis selection conditions as well as to allow the estimation of the
overall detection and reconstruction efficiency for the considered processes
including all analysis criteria.

##  3 Conclusions

In May 2014, WASA-at-COSY Collaboration performed a search for $\eta$-mesic
Helium in proton-deuteron collisions. For the first time the hypothetical
${}^{3}\mbox{He}$-$\eta$ bound state was searched for in non-mesonic
$pd\rightarrow$ ${}^{3}\mbox{He}2\gamma$ and $pd\rightarrow$
${}^{3}\mbox{He}6\gamma$ decays. For the purpose of the experimental data
interpretation, a new theoretical model for the $\eta$-mesic helium was
developed, according to which the mesic nucleus decays without $\eta$ meson
absorption. The Fermi momentum distribution was determined for a bound $\eta$
meson orbiting around the ${}^{3}\mbox{He}$ nucleus for different combinations
of the ${}^{3}\mbox{He}$-$\eta$ optical potential parameters. The performed
calculations allowed, for the first time, the estimation of the branching
ratio for $\eta\rightarrow 2\gamma$ and $\eta\rightarrow 3\pi^{0}$ decay
channels in the nuclear medium.

The obtained Fermi momentum distribution will allow the determination of the
efficiency for $\eta$-mesic ${}^{3}\mbox{He}$ production processes and its
non-mesonic decays, namely $pd\rightarrow$ $(^{3}\mbox{He}$-$\eta)_{bound}$
$\rightarrow$ ${}^{3}\mbox{He}2\gamma$ and $pd\rightarrow$
$(^{3}\mbox{He}$-$\eta)_{bound}$ $\rightarrow$ ${}^{3}\mbox{He}6\gamma$.

##  4 Acknowledgements

We acknowledge the support from the Polish National Science Center through
grant No. 2016/23/B/ST2/00784. This work was partly supported by JSPS KAKENHI
Grant Numbers JP16K05355 (S.H.) and JP17K05443 (H.N.) in Japan.

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# Thermal, quantum anti-bunching and lasing thresholds from single emitters to
macroscopic devices

Mark Anthony Carroll  Department of Physics, University of Strathclyde, 107
Rottenrow, Glasgow G4 0NG, UK.  Giampaolo D’Alessandro  School of Mathematics,
University of Southampton, Southampton SO17 1BJ, United Kingdom  Gian Luca
Lippi  Université Côte d’Azur, Institut de Physique de Nice, UMR 7710 CNRS,
1361 Route des Lucioles, 06560 Valbonne, France  Gian-Luca Oppo  Department of
Physics, University of Strathclyde, 107 Rottenrow, Glasgow G4 0NG, UK.
Francesco Papoff<EMAIL_ADDRESS>Department of Physics, University of
Strathclyde, 107 Rottenrow, Glasgow G4 0NG, UK.

###### Abstract

Starting from a fully quantized Hamiltonian for an ensemble of identical
emitters coupled to the modes of an optical cavity, we determine analytically
regimes of thermal, collective anti-bunching and laser emission that depend
explicitly on the number of emitters. The lasing regime is reached for a
number of emitters above a critical number – which depends on the light-matter
coupling, detuning and the dissipation rates – via a universal transition from
thermal emission to collective anti-bunching to lasing as the pump increases.
Cases where the second order intensity correlation fails to predict laser
action are also presented.

††preprint: APS/123-QED

Optical cavities containing emitters with discrete energy levels such as
atoms, ions or quantum dots, have proved extremely effective both as a tool to
investigate fundamental properties of light-matter interaction and as a way to
produce light with engineered statistical properties. Historically, the
earliest optical cavities contained very large numbers of emitters to overcome
losses and occupied macroscopic volumes. Significant improvements in cavity
quality have led in recent years to micro and nano cavities [1] that provide
detectable fields even with very few emitters and promise commercial
applications with uses ranging from components on integrated circuits to
medicine [2]. Their small size comes with the benefit of lower energy
consumption and increased energy efficiency, making them attractive for
extreme miniaturization. Although the basic quantum interaction process
between the cavity modes and the emitters is the same for all cavities,
different approximations of the expectation values of light-matter interaction
and photon number (correspondiong to the classical intensity) lead to two
classes of models: one for micro and nano systems, the other for macroscopic
systems. In the latter, e.g. the Maxwell-Bloch semi-classical models for
macroscopic systems [3], the only expectation values considered are those of
the emitters’ raising and lowering operators, representing excitation and de-
excitation of an electron, and of the cavity mode creation and destruction
operators, representing emission and absorption of a photon. The expectation
values of the destruction and creation operators correspond to the complex
amplitude of the classical coherent field and its complex conjugate.
Similarly, the expectation values of raising and lowering operators correspond
to the amplitude of the medium polarization and its complex conjugate.
Macroscopic models neglect correlations among these operators, both in the
intensity and in the photon-matter coupling, also called the photon-assisted
polarization. This approximation allows one to predict the threshold for laser
emission, but is not suitable for the analysis of non-lasing emission because
photon-assisted polarization is essential to model correctly spontaneous
emission [4]. Quantum models for micro and nano lasers take the exact opposite
approach: they consider only the correlations and neglect the expectation
values [5, 6, 7, 4] corresponding to the amplitudes of the classical coherent
field and polarization. With this approximation non-lasing emission can be
modelled, but it is not possible to identify the onset of lasing. The same
problem also affects rate equation models of micro- and nanolasers that add
average spontaneous emission to coherent emission [8, 9, 10]. A key parameter
related to the size of the system is the spontaneous emission factor, $\beta$,
defined as the ratio of the spontaneous emission rate into the lasing mode to
the total spontaneous emission. $\beta^{-1}$, proportional to the number of
electromagnetic modes in the cavity volume, thus characterises the system
size. In macroscopic systems $\beta\ll 1$, while $\beta=1$ is the nanoscale
limit, in which only the lasing mode remains accessible to spontaneous and
stimulated emission. For this value of $\beta$ the output power linearly
follows the input power, and for this reason this laser is considered
“thresholdless” [8, 11]. This regime poses questions on how to define the
laser threshold [9] and identify coherent emission [12], which we address
unambiguously in this paper.

Starting from a fully quantized Jaynes-Cummings Hamiltonian in the Heisenberg
picture, see Supplementary Material (SM) Eq.(1), we derive a model for the
emission of any number of identical two-level emitters coupled to one mode of
the cavity [13, 14]. However, we do not approximate the expectation values of
intensity and light-matter interaction, including the variables of both
macroscopic and nano lasers, and apply dynamical system methods defined for
systems of any dimension [15] to identify the laser threshold. Analytical
solutions seamlessly connect single emitter devices with devices containing
millions of emitters, predicting where the thermal, quantum and coherent
emission regimes lie with respect to one another. While the extent and
existence of these regions in the space of parameters depend on the number of
emitters, we find two universal features that are common to all lasing
devices. The first is that lasing, when possible, is reached via a universal
sequence of transitions as the pump is increased. The emission of the non-
lasing state evolves continuously from thermal (with second order intensity
correlation $1<g^{(2)}(0)\leq 2$) to anti-bunching (with $0\leq g^{(2)}(0)<1$)
until the laser threshold is crossed and the non-lasing state becomes
unstable. A coherent laser field, due to a lasing collective state, appears at
this threshold and its amplitude increases as a function of the pump. This is
the same instability predicted for macroscopic laser by Maxwell-Bloch models.
However, in these models the total field before the instability is zero, while
in our theory before the threshold only the coherent field is zero, while the
incoherent field is non zero. The second universal feature of lasers is that
the emerging coherent field has a well defined frequency: our model shows that
neither the number of emitters nor the effective cavity volume ($\beta$
factor) influence the frequency value, which remains a general feature of the
cavity-emitter interaction. We give examples where the measurement of
$g^{(2)}(0)$ cannot identify laser emission.

We consider identical emitters with two energy levels and one electron, and
assume that all transitions conserve the electron spin. This model applies to
atoms and ions with suitable energy level structure, and also to shallow
quantum dots (possessing two localized levels) at temperatures low enough to
neglect Coulomb and phonon interactions. Assuming that detuning and coupling
coefficients with the mode are identical [7, 16] is justified by numerical
simulations for emitters with $10\%$ random variations of detuning and
coupling coefficients. In these simulations all emitters’ variables – started
from random initial conditions – after a transient converge to common values
that match extremely well those obtained using the same parameters and
expectation values for all emitters, see SM Fig. 2. Since current technology
enables very accurate positioning [17], the range of fluctuations that we are
considering is quite realistic and covers estimates which can be expected from
QD positions inside typical cavity realizations and other experimental
fluctuations.

In the following $c^{\dagger}$, $v^{\dagger}$, $b^{\dagger}$ ($c$, $v$, $b$)
are operators that create (annihilate), respectively, an electron in the upper
energy level (conduction level in quantum dot terminology), in the lower
energy level (valence level), and a photon in the laser mode. We take the
expectation values of the Heisenberg equations for each operator in the
Hamiltonian, truncating the resulting infinite hierarchy of coupled
differential equations by keeping only correlation functions that appear in
the cluster expansion of the Hamiltonian [13, 4]. The variables of the model
are the expectation values of carrier population, $\langle
c^{\dagger}c\rangle$, mode destruction operator, $\langle b\rangle$, and
emitter lowering operator, $\langle v^{\dagger}c\rangle$, and the intensity
and photon-polarization correlations, $\delta\langle b^{\dagger}b\rangle$ and
$\delta\langle bc^{\dagger}v\rangle$. These correlations appear through the
cluster expansions $\langle bc^{\dagger}v\rangle=\delta\langle
bc^{\dagger}v\rangle+\langle b\rangle\langle c^{\dagger}v\rangle$ and $\langle
b^{\dagger}b\rangle=\delta\langle b^{\dagger}b\rangle+\langle
b^{\dagger}\rangle\langle b\rangle$ of terms in the Hamiltonian. Note that the
equations for $\langle b^{\dagger}\rangle$, $\langle c^{\dagger}v\rangle$,
$\langle b^{\dagger}v^{\dagger}c\rangle$ are the complex conjugated of the
equation for $\langle b\rangle$, $\langle v^{\dagger}c\rangle$, $\langle
bc^{\dagger}v\rangle$, so they are not explicitly included in the model. The
model equations are

$\displaystyle d_{t}\langle c^{\dagger}c\rangle=$
$\displaystyle-(\gamma_{nl}+\gamma_{nr})\langle
c^{\dagger}c\rangle+r(1-\langle c^{\dagger}c\rangle)$
$\displaystyle-2\mathfrak{Re}[g(\delta\langle bc^{\dagger}v\rangle+\langle
b\rangle\langle c^{\dagger}v\rangle)],$ (1) $\displaystyle d_{t}\delta\langle
bc^{\dagger}v\rangle=$
$\displaystyle-(\gamma_{c}+\gamma+i\Delta\nu)\delta\langle
bc^{\dagger}v\rangle+g^{*}[\langle c^{\dagger}c\rangle$
$\displaystyle+\delta\langle b^{\dagger}b\rangle(2\langle
c^{\dagger}c\rangle-1)-|\langle v^{\dagger}c\rangle|^{2}],$ (2) $\displaystyle
d_{t}\delta\langle b^{\dagger}b\rangle=$
$\displaystyle-2\gamma_{c}\delta\langle
b^{\dagger}b\rangle+2N\mathfrak{Re}(g\delta\langle bc^{\dagger}v\rangle),$ (3)
$\displaystyle d_{t}\langle b\rangle=$ $\displaystyle-(\gamma_{c}+i\nu)\langle
b\rangle+g^{*}N\langle v^{\dagger}c\rangle,$ (4) $\displaystyle d_{t}\langle
v^{\dagger}c\rangle=$ $\displaystyle-(\gamma+i\nu_{\varepsilon})\langle
v^{\dagger}c\rangle+g[\langle b\rangle(2\langle c^{\dagger}c\rangle-1)],$ (5)

where $r$ is the pump rate per emitter (the one for the whole system is $Nr$),
with non resonant pump photons absorption and carrier transport effects
approximated via an injection rate [7]. $\nu$ is the frequency of the resonant
mode and $\nu_{\varepsilon}$ is the resonant frequency of emitters,
$\Delta\nu=\nu_{\varepsilon}-\nu$ is the detuning between the field and
emitter, $g$ the light-matter coupling strength and $N$ the number of
emitters. The decay rates for the laser mode, $\gamma_{c}$, the population,
$\gamma_{nr}$, and polarisation, $\gamma$, introduce dissipation. The
dissipative part of these equations can be obtained by considering Lindblad
terms describing the coupling to a Markovian bath [18, 19]. $\gamma_{nr}$ is
due to non-radiative transitions, while the additional decay rate for the
population, $\gamma_{nl}$, results from the adiabatic elimination of rapidly
decaying non lasing modes [20]. The spontaneous emission factor, $\beta$, is
related to the cavity losses by
$\beta=\frac{\gamma_{l}}{\gamma_{nl}+\gamma_{l}}$, where $\gamma_{l}$ is the
rate of spontaneous emission into the lasing mode. The expectation value of
lower level population, $\langle v^{\dagger}v\rangle$ has been eliminated
using $\langle c^{\dagger}c\rangle+\langle v^{\dagger}v\rangle=1$. The
variables $\langle b\rangle$ and $\langle v^{\dagger}c\rangle$ correspond to
the amplitudes of the coherent field and the medium polarization of semi-
classical models of macroscopic lasers and were previously neglected in
nanolasers. The imaginary coefficients in their the equations show that they
have fast oscillations with frequency of the order of that of the cavity mode.
Hence, we call them “fast” and the remaining variables “slow”.

Figure 1:  I/O curves: Intensity, $\langle b^{\dagger}b\rangle=\delta\langle
b^{\dagger}b\rangle+\langle b^{\dagger}\rangle\langle b\rangle$, and coherent
field amplitude for different $N$ above and below $N_{c}$ versus pump for
$\beta=1$ ($\gamma_{nl}=0$), (a) and (b); and $\beta=7\times 10^{-4}$
($\gamma_{nl}=1.4\times 10^{12}s^{-1}$ and $\gamma_{l}=9.68\times
10^{8}s^{-1}$), (c) and (d). Laser and anti-bunching thresholds are marked by
red stars and crosses, respectively. The purple triangles mark the anti-
bunching threshold where we include higher order correlations (this shows our
approximation to be valid). Note that a laser threshold is found even for a
“thresholdless” device (green line in (a)). Parameters:
$\gamma=10^{13}s^{-1}$, $\gamma_{c}=10^{10}s^{-1}$,
$\gamma_{nr}=10^{9}s^{-1}$, and $g=7\times 10^{10}s^{-1}$. Losses are kept
constant in all figures.

The non-lasing stationary state is found by setting to zero the fast variables
and the time derivatives in Eqs.(1-3). This state exists for all values of the
control parameters and its emission evolves continuously from thermal to anti-
bunching. The boundary between these two regimes is identified by the curve
$g^{(2)}(0)=1$. We find a very good analytical approximation of the
$g^{(2)}(0)=1$ curve assuming that the correlations used here are independent
of higher order correlations [6], so that $g^{(2)}(0)\sim 2+A/B$, see SM
Eqs.(12-14). We determine the stability of the non-lasing state by deriving
the equations that govern the evolution of small perturbations in the linear
regime. We find that the perturbations of the fast variables are decoupled
from the perturbations of the slow variables allowing us to determine
analytically the stability of the non-lasing state (see SM Eqs.(22-24)). The
non-lasing state is stable for $\langle c^{\dagger}c\rangle<\langle
c^{\dagger}c\rangle_{th}$ and unstable for $\langle
c^{\dagger}c\rangle>\langle c^{\dagger}c\rangle_{th}$ where

$\langle
c^{\dagger}c\rangle_{th}=\frac{1}{2}+\frac{\gamma_{c}\gamma}{2N|g|^{2}}\left[1+\left(\frac{\Delta\nu}{\gamma_{c}+\gamma}\right)^{2}\right]$
(6)

is the laser threshold. Because $\langle c^{\dagger}c\rangle<1$, lasing can
only happen when the number of emitters satisfies the condition

$\frac{\gamma_{c}\gamma}{|g|^{2}}\left[1+\left(\frac{\Delta\nu}{\gamma_{c}+\gamma}\right)^{2}\right]\leq
N.$ (7)

The laser frequency is determined using trial solutions $\langle
b\rangle,\langle v^{\dagger}c\rangle\propto e^{-i\Omega t}$ in Eqs.(4,5) and
is

$\Omega=\nu+\frac{\gamma_{c}\Delta\nu}{\gamma_{c}+\gamma}.$ (8)

A few points are worth highlighting. First, the threshold can be calculated
for any values of the decay rate $\gamma_{nl}$, including the so called
“thresholdless” case $\gamma_{nl}=0$. Second, neither $\langle
c^{\dagger}c\rangle_{th}$ nor the critical number of emitters necessary to
lase depend on $\beta$. However, the value of the pump per emitter required to
reach $\langle c^{\dagger}c\rangle_{th}$ depends on $\beta$ and decreases as
$\beta$ increases, see SM Eq.(20) for the analytic expression for $r$ at laser
threshold. Third, the laser frequency $\Omega$ is independent of the number of
emitters, $N$.

Figure 2: Phase Diagrams: thermal, anti-bunching and lasing regimes correspond
to the white, yellow and blue regions, respectively. Where $\beta=1$ and
$g=10^{11}s^{-1}$ in (a); $\beta=7\times 10^{-4}$ and $g=10^{11}s^{-1}$ in
(b); $\beta=1$ and $g=7\times 10^{10}s^{-1}$ in (c); $\beta=7\times 10^{-4}$
and $g=7\times 10^{10}s^{-1}$ in (d). For large $g$ the thermal region that
extends to large pump values vanishes, and for a lasing a device with
$\beta=1$ the anti-bunching regime exists over greater values of the pump.

In Fig. 1a and Fig. 1b the intensity and coherent field versus pump for a
$\beta=1$ device are shown. The different number of emitters correspond to
values above (blue and olive curves) and below (black curves) the critical
number, $N_{c}$, required for a laser using Eq.(7). Below $N_{c}$, the minimum
integer that satisfies Eq.(7), the coherent field amplitude is always zero and
the intensity saturates at high pump. Above $N_{c}$, $\langle b\rangle\neq 0$
emerges through a pitchfork bifurcation. The emergence of a pitchfork
bifurcation also coincides with the growth of the intensity and a clear
qualitative difference is apparent between the curves corresponding to lasing
and non lasing devices. The I/O curve for 40 quantum dots (Fig. 1a)
illustrates the impossible task of determining the laser threshold for a
$\beta=1$ device, identifiable only through the fast variables. Comparison of
these thresholds with standard estimates is given in SM Fig. 4. Notice that
all graphs plot pump for a single quantum dot. Thus, comparison between
devices with different $N$ requires multiplication of each horizontal scale by
$N$. Variations in the I/O similar to those shown in Fig. 1a for $N=21$ and
$N=40$ can be obtained with the same value of N and changing the detuning.
This is observed in experiments where detuning decreases the effective number
of quantum dots interacting with the field. See SM Fig. 6 for the effect of
detuning. Experiments have obtained this kind of I/O response through cavity
or thermal tuning [21, 22, 23]. While so far explained only through ad-hoc
calculations, here the continuous transformation with change in characteristic
I/O shape emerges thanks to self-consistent modeling.

Fig. 1c and Fig. 1d show the intensity and coherent field for a device where
$\beta\ll 1$. One clear difference is that the intensity profile is no longer
linear and there is the characteristic s-shaped I/O curve. Notable is the
appearance of the bifurcation at the knee of the upper branch of the I/O
curve, rather than at the inflection point (as from [9]). This points to a
substantial contribution from the incoherent emission to the intensity growth
before the bifurcation and highlights the difficulty intrinsic in determining
threshold through the I/O curve.

Figure 3: $g^{(2)}(0)$ versus pump for different $N$ and $\beta$ where the
coloured regions correspond to the same in Fig. 2. In (a) and (c)
$\beta=7\times 10^{-4}$; and in (b) and (d) $\beta=1$. Note the large range of
pump values in (a) and (b) where $g^{(2)}(0)$ is smaller, but very close to
$1$, making $g^{(2)}(0)$ difficult to use as an experimental indicator of
lasing. All curves in (a) converge to the same value for larger pump values
not shown. All curves are obtained for $g=7\times 10^{10}s^{-1}$ thus lasing
occurs for $N=21$.

Fig. 2 shows the analytic solutions of $g^{(2)}(0)=1$ and the laser threshold
as functions of the pump and $N$ for different values of $g$ and $\beta$. For
a device capable of lasing there is a well defined path of emission as the
pump increases (vertical cut in the graph): from thermal to anti-bunching;
then from anti-bunching to lasing. This is independent of the value of
$\beta$. Fig. 2a and Fig. 2c show an anti-bunching regime that exists for a
large region of the pump and extends from nano- to macroscopic lasers with
$\beta=1$. Anti-bunching has been observed in an experiment with a high cavity
Q micropillar containing $N\approx 15$ pumped quantum dots [24] and in
numerical simulations of quantum dot nanocavities [6],[25]. The observation is
consistent with our prediction (Fig. 3), which shows stronger anti-bunching
close to $N_{c}$; an increase in $N$ reduces the range and amplitude, thus
making an experimental observation much more difficult (no anti-bunching was
in fact observed upon doubling of the quantum dot number [24]). Fig. 3a shows
the three regimes for lasers with $\beta=7\times 10^{-4}$. Although anti-
bunching is present in macroscopic lasers, $g^{(2)}(0)$ is so close to unity
that it becomes very difficult to distinguish between a laser and a device
emitting anti-bunched light, thus stressing that $g^{(2)}(0)$ is not a
sufficient indicator of laser action. We show in Fig. 3 of the SM a comparison
of Eqs. (1-5) with the master equation given in [9] and a cQED model [7].

In conclusion, we have derived a model that includes coherent and incoherent
emission and derived analytically thresholds that separate thermal, anti-
bunching and lasing emission regimes from single emitter to macroscopic
devices. We predict the qualitative differences between the I/O curves of the
photon number above and below threshold and find analytically the number of
intracavity emitters necessary for laser action to occur. We identify a
universal route leading from thermal to coherent emission, through a
collective anti-bunching regime that always precedes lasing. The coherent
laser field always emerges from a bifurcation with a well defined threshold
and has a frequency independent from the number of emitters. Interferometric
measurements of the coherence time [26, 27] ($g^{(1)}(\tau)$) – equivalent to
detecting the existence of a well-defined frequency at the bifurcation – can
in principle be used to unambiguously identify lasing in all devices. This
also holds for $\beta=1$ (or close to this boundary), where the I/O
characteristics cannot carry useful information and measurements of
$g^{(2)}(0)$ are poor indicators of lasing, leading to wrong conclusions on
the existence or position of the laser threshold. The accurate
characterization of the coherence properties of nanolasers holds promise for
quantitative predictions to be used in a variety of applications, ranging from
telecommunications and optical chips [28], to the spectroscopic use of
nanosources [29], nanosensing [30] and biophysical applications [31]. These
applications can greatly benefit from a detailed understanding and predictive
power of designs where control in the number of quantum dots (now
technologically feasible [17, 32]) coupled to effective coupling allow for
tailored emission properties. In addition, at the micro and nano scale this
new theory can be used to investigate the interaction of emitters with
nanostructures and arrays of particles [33, 34, 35, 36, 37] and obtain
predictions for new effects emerging from quantum interactions.

The Strathclyde and UCA groups are grateful to the the CNRS and its
Laboratoire International Associé (LIA) “Solace” for support. The PhD of Mark
Anthony Carroll is supported by the University of Strathclyde. GD and FP wish
to thank the Mathematisches Forschungsinstitut Oberwolfach for support. GLL is
grateful to L. Chusseau for discussions.

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