# Coherence build up and laser thresholds from nanolasers to macroscopic
lasers

Mark Anthony Carroll  Department of Physics, University of Strathclyde, 107
Rottenrow, Glasgow G4 0NG, UK.  Giampaolo D’Alessandro  School of Mathematical
Sciences, 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

We detail the derivation of nanolaser models that include coherent and
incoherent variables and predict the existence of a laser threshold,
irrespective of cavity size and emitter number, for both single- and multi-
electron systems. The growth in photon number in the lasing mode is driven by
an increase in correlation between absorption and emission processes, leading
to the onset of self-sustained stimulated emission (laser threshold),
followed, in turn, by a correlation decrease and ending with the dominance of
coherent emission. The first-order coherence $g^{(1)}$ steadily increases, as
the pump grows towards the laser threshold value, and reaches unity at or
beyond threshold. The transition toward coherent emission becomes increasingly
sharp as the number of emitters and of the coupled electromagnetic cavity
modes increase, continuously connecting, in the thermodynamic limit, the
physics of nano- and macroscopic lasers at threshoold. Our predictions are in
remarkable agreement with experiments whose first-order coherence measurements
have so far been explained only phenomenologically. A consistent evaluation of
different threshold indicators provides a tool for a correct interpretation of
experimental measurements at the onset of laser action.

##  I Introduction

The rapid advancement in the design and manufacturing of laser resonators over
the past few decades has allowed the construction of lasing devices with mode
volume $V\propto\lambda^{3}$, where $\lambda$ is the emission wavelength [1].
Such small devices are far more compact and less energy hungry compared to
standard lasers, as lower input power is required to achieve coherent
emission. In addition to nanolasers, microlasers (e.g., $V=a\lambda^{3}$,
$2\lesssim a\lesssim 40$) hold promise for a number of uses spanning multiple
research disciplines and industrial applications, such as integrated optical
interconnects, sensing and biological probes, to name a few [2]. Photon number
squeezing is also expected to naturally emerge before the transition to
coherent emission, leading to cw photon fluxes for non-classical applications
[3]. The complexity of the transition between incoherent and coherent emission
in micro- and nanolasers is at the origin of interpretative problems, and
gives rise to new opportunities. The difficulties in threshold identification
in nanolasers [4] come from the intrinsic physical properties of the
transition in small systems rather than from technical measurement
limitations.

As the mode volume of a device decreases, so does the number of
electromagnetic cavity modes available for a spontaneous transfer of the
energy stored in the medium. In the Cavity Quantum Electrodynamics (CQED)
regime this number is significantly reduced. This “number” is characterized by
the spontaneous emission factor, $\beta$, which quantifies the ratio between
the spontaneous emission rate into the lasing mode and the total spontaneous
emission rate. For macroscopic devices $\beta\lesssim 10^{-6}$, while the
ideal nanolaser limit corresponds to the asymptote $\beta=1$. In other words,
$\beta$ is inversely proportional to the systems size.

The laser threshold is typically identified in macroscopic lasers by
inspecting the output power as a function of the input power. The input-output
(I-O) curves display a characteristic S-shape on a log-log plot with a steep
growth. The laser threshold is located at the inflection point of these curves
[5]. As the cavity volume decreases, the steep growth is progressively
smoothed, leading, in the nanolaser limit of $\beta=1$, to a straight line.
The extrapolation of this linear dependence down to zero pump power has
ushered the questionable concept of thresholdless lasers [6, 5].

In spite of the equivalence of the intracavity light-matter interaction in
macroscopic and microscopic lasers, two different approaches have emerged,
each with its own limitations. For macroscopic systems, the well established
semi-classical Maxwell-Bloch equations [7] describe coherent emission above
the laser threshold by considering the expectation values of the classical
coherent field amplitude and the standard medium polarization. The application
of classical factorization schemes to expectation values, which describe the
light-matter interaction, neglects quantum correlations [8], thus limiting the
theoretical description to above-threshold coherent emission, with no access
to the incoherent regimes below it. Quantum models for nanolasers, on the
other hand, neglect the classical variables associated with coherent emission
[9, 10, 11, 12] and apply factorization techniques, such as the cluster
expansion [13], keeping only the slowly varying quantum correlations.

In recent papers [14, 3] we combined these two approaches by including the
slowly-varying quantum correlations as well as the coherent variables into a
single Coherent-Incoherent (CI) model, whilst neglecting quantum correlations
between electromagnetic field and those medium operators which oscillate on a
fast timescale, as done in semi-classical theories [7]. We then used the
Linear Stability Analysis (LSA) of the CI model’s incoherent solution to
calculate analytically the laser threshold for all two-level emitter
nanolasers, including the so-called thresholdless lasers ($\beta=1$).

In this paper we examine in detail the derivation and predictions of the CI
model and extend it to multi-electron systems. We discover that – for both
single and multi-electron systems – the critical point identified by the
bifurcation analysis is the threshold beyond which stimulated emission becomes
self-sustained. This analysis is corroborated by experimental measurements
[15]. It is important to stress that our approach further contributes to
solving the confusion, first identified in [5], reigning around the concept of
a thresholdless laser by further dispelling the concept that an ideal CQED
laser would be a truly thresholdless device. In a future paper we will
consider models that retain fast quantum correlations between field and medium
operators, including more quantum aspects than those considered here at the
price of a significantly larger number of equations. These models confirm the
existence of lasing solutions in all nanolasers [16, 17] and predict laser
thresholds associated to the establishment of self-sustained stimulated
emission that become increasingly close to those calculated in this paper as
the number of emitters increases.

This paper is structured as follows. In Section II we outline the structure of
the system Hamiltonian. Section III covers the cluster expansion technique
needed to close the model equations and presents the derivation of the CI
model for single and multi-electron systems. Section IV presents the Linear
Stability Analysis, including a discussion of the conditions required for the
existence of instabilities in the system. Sections V and VI detail the effects
of emitter number $N$ and cavity “size” $\beta$. Section VII introduces the
characterization of coherence and conclusively interprets existing
experimental results in the framework of the models here introduced, while
Section VIII offers a brief overview of the work and conclusions.

##  II  The structure of the system Hamiltonian

Our investigation starts with writing the fully quantized Jaynes-Cummings
Hamiltonian [18] generalized to describe light-matter interaction between two
interacting levels with lasing and non-lasing modes,

$H=H_{free}+H_{int},$ (1)

where $H_{free}$ is the non-interacting part and $H_{int}$ the interacting
part of the Hamiltonian, respectively. The non-interacting part of the
Hamiltonian is itself made up of contributions from the free electromagnetic
field, $H_{E}$, and the free electrons in the quantum dot, $H_{QD}$,

$H_{free}=H_{E}+H_{QD}.$ (2)

The photon operators of the system Hamiltonian obey the Bosonic commutation
relations, and the carrier operators obey the Fermi anti-commutation
relations. The Bosonic operators $b$ and $b^{\dagger}$ correspond to single-
particle operators. It can be shown that 2$N$ Fermi operators are formally
equivalent to $N$ Bosonic operators under the requirement that the compound
Fermi operators contain equal numbers of creation and annihilation operators
[8]. Examples are the population of the excited state, $c^{\dagger}c$ ,and the
standard material polarization, $v^{\dagger}c$. Therefore, we refer to the
coherent field operator $b$ and standard polarization $v^{\dagger}c$ as single
particle operators and to the photon number operator $b^{\dagger}b$ and photon
assisted polarization $bc^{\dagger}v$ as two particle operators.

The first term in Eq. (2) reads

$H_{\mathrm{E}}=\hbar\sum_{q}\nu_{q}\left(b_{q}^{\dagger}b_{q}+\frac{1}{2}\right),$
(3)

where $\nu_{q}$ is the frequency of a photon in the $q$-th mode and the
quantum mechanical operators $b_{q},b_{q}^{\dagger}$ annihilate and create a
photon in the $q$-th mode, respectively. The sum over $q$ includes both lasing
and non-lasing modes.

The free electron part of the Hamiltonian describes charge carriers in the
conduction and valence band states of the $n$-th quantum dot with respective
energies $\epsilon_{c,n}$ and $\epsilon_{v,n}$

$H_{QD}=\sum_{n}\left(\varepsilon_{c,n}c_{n}^{\dagger}c_{n}+\varepsilon_{v,n}v_{n}^{\dagger}v_{n}\right),$
(4)

where $c_{n},c^{\dagger}_{n}$ and $v_{n},v^{\dagger}_{n}$ are the annihilation
and creation operators, respectively, for conduction and valence electrons of
the $n$-th quantum dot.

The two-particle light-matter interaction is described by

$\begin{split}H_{int}=-i\hbar\sum_{n,q}&\left[g_{nq}\left(b_{q}c_{n}^{\dagger}v_{n}+b_{q}v_{n}^{\dagger}c_{n}\right)\right.\\\
&\left.\,-g_{nq}^{*}\left(b_{q}^{\dagger}v_{n}^{\dagger}c_{n}+b_{q}^{\dagger}c_{n}^{\dagger}v_{n}\right)\right],\end{split}$
(5)

where $g_{nq}$ is the light-matter coupling strength between a photon in the
$q$-th mode and the $n$-th quantum dot.

In writing the quantum Hamiltonian we have made the standard assumption of
neglecting contributions from phonon and Coulomb interactions between charge
carriers. This is equivalent to assuming that the quantum dots are operating
at cryogenic temperatures ($\approx 4\mathrm{K}$) [10].

One final remark on the structure of the system Hamiltonian concerns the two-
particle operators coming from the interaction part of the Hamiltonian:
$b_{q}^{\dagger}c_{n}^{\dagger}v_{n}$ and $b_{q}v_{n}^{\dagger}c_{n}$. Quantum
mechanically, operators $b_{q}^{\dagger}c_{n}^{\dagger}v_{n}$ describe a
process where a photon in mode $q$ is created in conjunction with the
excitation of an electron from the valence to the conduction band of the
$n$-th quantum dot. Operators $b_{q}v_{n}^{\dagger}c_{n}$ describe its
symmetric counterpart, where a photon in mode $q$ is absorbed in conjunction
with the de-excitation of an electron from the conduction to the valence band
of the $n$-th quantum dot. These two processes do not individually conserve
energy, even if their sum is conservative, and oscillate at a frequency
approximately double that of the laser, thus in the following we eliminate
them from the interaction Hamiltonian (Rotating Wave Approximation).

##  III Cluster Expansion and Nonlinear QED Models

To derive the model equations we work in the interaction picture to obtain
Heisenberg’s equations of motion for the operators appearing in the system
Hamiltonian. The variables that appear in the CI models are the quantum
operator expectation values and correlations. The dynamics of an $M$ particle
expectation value is directly coupled to an $M+1$ expectation value through
equations of the form

$i\hbar d_{t}\langle M\rangle=Lo[\langle 1\rangle,\cdots,\langle
M\rangle]+Hi[\langle M+1\rangle],$ (6)

where $d_{t}$ is the first order derivative with respect to time, $\langle
1\rangle,\cdots,\langle M+1\rangle$ indicate the sets of the $1,\cdots,M+1$
particle operators and $Lo$ and $Hi$ are matrices that describe coupling to
terms of order $1,\cdots,M$ and $M+1$, respectively. As a result, there is an
infinite hierarchy in which each order $M$ depends on the higher order $M+1$.
We must therefore find a way of systematically breaking the infinite hierarchy
to obtain a closed set of solvable equations at any order $M$. This is
achieved through expressing the expectation values in terms of all possible
combinations of products of correlations of lower order operators and
introducing approximation schemes for the correlations to truncate the
infinite hierarchy [8].

The expectation values of photon number and photon-assisted polarization,
central to this work, are

$\displaystyle\langle b^{\dagger}b\rangle=\delta\langle
b^{\dagger}b\rangle+\langle b^{\dagger}\rangle\langle b\rangle,$ (7)
$\displaystyle\langle bc^{\dagger}v\rangle=\delta\langle
bc^{\dagger}v\rangle+\langle b\rangle\langle c^{\dagger}v\rangle,$ (8)

where $\delta\langle b^{\dagger}b\rangle$ is the two particle correlation
between emission and absorption and $\delta\langle bc^{\dagger}v\rangle$ is
the two particle correlation between photon absorption and electron jump from
lower to upper energy level.

We find a closed set of equations by including in the model the expectation
values $\langle b\rangle$ and $\langle c^{\dagger}v\rangle$. These correspond
to the complex amplitudes of the coherent field and of the medium polarization
and have been neglected in previous microscopic models [12, 19, 10]. They
display fast oscillations with frequencies of the order of that of the laser
mode. On the contrary, the two-particle quantum correlations Eqs. (7-8) that
appear in the cluster expansion of the Hamiltonian oscillate slowly. We call
the former variables “coherent” and the latter “incoherent”. Coherent quantum
correlations such as the correlation between population and field, $\langle
bc^{\dagger}c\rangle$, are neglected in the same way as they are in standard
semi-classical models. Models that include all possible two particle
correlations have also been shown to display laser thresholds [16] and will be
studied in detail in a future paper. We consider all quantum dots identical.
This is not a restrictive hypothesis: we have verified numerically that
variations in detuning and light-matter coupling strength up to $10\%$ have
negligible effects on the system. Thus, we drop the subscript for the Fermi
operators and replace the sum over $n$ with $N$. The resulting system of
equations is

$\displaystyle d_{t}\langle b\rangle=$ $\displaystyle-(\gamma_{c}+i\nu)\langle
b\rangle+Ng^{*}\langle v^{\dagger}c\rangle$ (9a) $\displaystyle d_{t}\langle
c^{\dagger}v\rangle=$ $\displaystyle-(\gamma-i\nu_{\epsilon})\langle
c^{\dagger}v\rangle+g^{*}\langle b^{\dagger}\rangle(2\langle
c^{\dagger}c\rangle-1)$ (9b) $\displaystyle d_{t}\langle c^{\dagger}c\rangle=$
$\displaystyle r(1-\langle
c^{\dagger}c\rangle)-(\gamma_{nr}+\gamma_{nl})\langle c^{\dagger}c\rangle$
(9c) $\displaystyle-2\Re\\{g(\delta\langle b_{q}c^{\dagger}v\rangle+\langle
b_{q}\rangle\langle v^{\dagger}c\rangle)\\}$ $\displaystyle d_{t}\delta\langle
bc^{\dagger}v\rangle=$
$\displaystyle-(\gamma_{c}+\gamma-i\Delta\nu)\delta\langle
bc^{\dagger}v\rangle$ (9d) $\displaystyle+g^{*}\left[\langle
c^{\dagger}c\rangle+\delta\langle b^{\dagger}b\rangle\left(2\langle
c^{\dagger}c\rangle-1\right)-|\langle c^{\dagger}v\rangle|^{2}\right]$
$\displaystyle d_{t}\delta\langle b^{\dagger}b\rangle=$
$\displaystyle-2\gamma_{c}\delta\langle
b^{\dagger}b\rangle+2N\Re\left(g\delta\langle bc^{\dagger}v\rangle\right),$
(9e)

where $\nu_{\epsilon}$ is the frequency of the inter-band energy,
$h\nu_{\epsilon}=\epsilon_{c}-\epsilon_{v}$,
$\Delta\nu\equiv\nu_{\epsilon}-\nu$ is the detuning, $\Re(\cdot)$ stands for
real part of its argument and a superscript ∗ indicates the complex conjugate.
The equations for $\langle b^{\dagger}\rangle$, $\langle v^{\dagger}c\rangle$
and $\langle b^{\dagger}v^{\dagger}c\rangle$ can be obtained from Eqs. (9)
through complex conjugation. The expectation value of the lower-level
population has been eliminated using $\langle c^{\dagger}c\rangle+\langle
v^{\dagger}v\rangle=1$. The dissipative part of the equations is obtained by
considering Lindblad terms describing the coupling to a Markovian heat bath
under the constraint that random excitations into the excited state are
neglected [20, 21], a condition which is fulfilled under the assumed cryogenic
temperatures. The cavity decay rate, $\gamma_{c}$, the polarization dephasing
rate, $\gamma$, and the non-radiative losses, $\gamma_{nr}$, describe the
dissipative channels. The coherent field amplitude, $\langle b\rangle$, and
standard polarization, $\langle v^{\dagger}c\rangle$, are analogous to their
amplitudes in semi-classical theories. They describe coherent inter-band
processes and therefore need to be externally driven to be sustained. In terms
of operators the population density of the excited state, $\langle
c^{\dagger}c\rangle$, describes an intra-band process and does not require any
externally driven source to exist; it is the probability of an electron being
in the excited state. The photon-assisted polarization describes a correlated
event between the annihilation of a photon with an inter-band transition or
the opposite scenario for its hermitian conjugate. Finally, the intensity
correlation describes the correlation between photon absorption and emission.

The model derived above assumes identical emitters, each with two discrete
energy levels and a single electron. We want to show that the existence of
coherent laser solutions is not specific to the single-electron nature of this
model, thus we include the coherent variables $\langle b\rangle$ and $\langle
v^{\dagger}c\rangle$ in the model given in Ref. [19], where the authors relax
the single-electron assumption. To ensure that radiative decays can take place
only if the upper level is occupied and the lower level empty, the radiative
decay terms are now proportional to the product of the probability that an
electron is in the excited level with the probability that the lower level is
empty. This gives rise to nonlinear terms in the equation for the photon-
assisted polarization and the population density, Eqs. (9c,9d), which in the
multi-electron model read

$\displaystyle d_{t}\langle c^{\dagger}c\rangle=$ $\displaystyle r(1-\langle
c^{\dagger}c\rangle)-(\gamma_{nr}+\gamma_{nl}\langle
c^{\dagger}c\rangle)\langle c^{\dagger}c\rangle$
$\displaystyle-2\Re\\{g(\delta\langle b_{q}c^{\dagger}v\rangle+\langle
b_{q}\rangle\langle v^{\dagger}c\rangle)\\}$ (10c) $\displaystyle
d_{t}\delta\langle bc^{\dagger}v\rangle=$
$\displaystyle-(\gamma_{c}+\gamma-i\Delta\nu)\delta\langle
bc^{\dagger}v\rangle+g^{*}\left[\langle c^{\dagger}c\rangle^{2}\right.$
$\displaystyle\left.+\delta\langle b^{\dagger}b\rangle(2\langle
c^{\dagger}c\rangle-1)-|\langle c^{\dagger}v\rangle|^{2}\right],$ (10d)

while all the other equations remain the same. Note that the population of the
lower level, $\langle v^{\dagger}v\rangle$, can be eliminated because $\langle
c^{\dagger}c\rangle+\langle v^{\dagger}v\rangle\rightarrow 1$ exponentially in
time (see difference between electron and holes densities in Ref. [19, Eq.
(3)].

The two sets of equations, (9) and (10), constitute the single-electron and
the multi-electron CI models respectively. They both contain coherent and
incoherent variables and differ only in the number of electrons in each
emitter. With a quantum theory containing variables that can describe both
coherent and incoherent processes, we can now investigate how coherence
emerges in nanolasers for single- and multi-electron systems.

##  IV Linear Stability Analysis

In analogy with the semi-classical theory of macroscopic lasers [7], we
identify the laser threshold as the instability threshold of a non lasing
solution where the incoherent variables are different from zero but the
amplitudes of the coherent field are $\langle b\rangle=\langle
v^{\dagger}c\rangle=0$. If perturbations of $\langle b\rangle,\langle
v^{\dagger}c\rangle$ grow, the non-lasing solution is unstable. If instead
they decay asymptotically to zero, then the solution is stable. The
bifurcation point, where the solution with zero values for coherent variables
becomes unstable, is the laser threshold. To study its existence as a function
of the parameter values, we perform a linear stability analysis of Eqs. (9)
and (10). We collect coherent and incoherent variables into two groups,
$\boldsymbol{c}=\\{\langle b\rangle,\langle b^{\dagger}\rangle,\langle
v^{\dagger}c\rangle,\langle c^{\dagger}v\rangle\\}$ and
$\boldsymbol{i}=\\{\langle c^{\dagger}c\rangle,\delta\langle
b^{\dagger}b\rangle,\delta\langle bc^{\dagger}v\rangle,\delta\langle
b^{\dagger}v^{\dagger}c\rangle\\}$, respectively, and write the two CI models
in a more compact form

$\displaystyle d_{t}\boldsymbol{i}$
$\displaystyle=\boldsymbol{F}(\boldsymbol{i},\boldsymbol{c}),$ (11)
$\displaystyle d_{t}\boldsymbol{c}$
$\displaystyle=\boldsymbol{G}(\boldsymbol{i},\boldsymbol{c}),$ (12)

where $\boldsymbol{G}(\boldsymbol{i},\boldsymbol{c})$ and
$\boldsymbol{F}(\boldsymbol{i},\boldsymbol{c})$ are non-linear vector
functions of $\boldsymbol{i}$ and $\boldsymbol{c}$ whose components are the
right-hand side of the two models. With this notation $G_{\langle
b\rangle}({\boldsymbol{i}},{\boldsymbol{c}})$ and $G_{\langle
v^{\dagger}c\rangle}(\boldsymbol{\boldsymbol{i}},{\boldsymbol{c}})$ are, for
example, the right-hand side of Eq. (9a) and of the complex conjugate of Eq.
(9b), respectively, evaluated at ${\boldsymbol{i}},{\boldsymbol{c}}$. The
linearized dynamics of small perturbations
$(\boldsymbol{\eta}_{\boldsymbol{i}},\boldsymbol{\eta}_{\boldsymbol{c}})$ of a
fixed point solution $(\boldsymbol{i},\boldsymbol{c})$ is given by

$d_{t}\left[\begin{array}[]{c}\boldsymbol{\eta}_{i}\\\
\boldsymbol{\eta}_{c}\end{array}\right]=\left[\begin{array}[]{cc}\boldsymbol{\nabla}_{\boldsymbol{i}}\otimes\boldsymbol{F}({\boldsymbol{i}},{\boldsymbol{c}})&\boldsymbol{\nabla}_{\boldsymbol{c}}\otimes\boldsymbol{F}({\boldsymbol{i}},{\boldsymbol{c}})\\\
\boldsymbol{\nabla}_{\boldsymbol{i}}\otimes\boldsymbol{G}({\boldsymbol{i}},{\boldsymbol{c}})&\boldsymbol{\nabla}_{\boldsymbol{c}}\otimes\boldsymbol{G}({\boldsymbol{i}},{\boldsymbol{c}})\end{array}\right]\left[\begin{array}[]{c}\boldsymbol{\eta}_{\boldsymbol{i}}\\\
\boldsymbol{\eta}_{\boldsymbol{c}}\end{array}\right].$ (13)

Each block in the matrix on the right-hand side of Eq. (13) is of dimension
$4\times 4$ (both sets ${\boldsymbol{i}}$ and ${\boldsymbol{c}}$ contain four
variables) and corresponds to the Jacobian with respect to the
$\boldsymbol{i}$ and $\boldsymbol{c}$ variables. $\otimes$ denotes the outer
product. For any solution with ${\boldsymbol{c}}=\boldsymbol{0}$ one has
$\boldsymbol{\nabla}_{\boldsymbol{i}}\otimes\boldsymbol{G}({\boldsymbol{i}},\boldsymbol{0})=\boldsymbol{0}$
and
$\boldsymbol{\nabla}_{\boldsymbol{c}}\otimes\boldsymbol{F}({\boldsymbol{i}},\boldsymbol{0})=\boldsymbol{0}$,
so that coherent and incoherent perturbations of the
$({\boldsymbol{i}},\boldsymbol{0})$ solutions decouple.

This is a general feature of _all_ models derived under the rotating wave
approximation independently of the order of the quantum correlations
considered. Its origin is the separation of time scale between the coherent
and the incoherent variables. The (fast) coherent variables oscillate at the
lasing frequency $\nu$, i.e., proportionally to $\sim e^{-i\nu t}$. They can
therefore only appear in complex conjugate quadratic pairs in the equations
for the (slow) incoherent variables. As the derivative of a quadratic term at
zero is zero, we have that
$\boldsymbol{\nabla}_{\boldsymbol{c}}\otimes\boldsymbol{F}({\boldsymbol{i}},\boldsymbol{0})=\boldsymbol{0}$.
Conversely, the (slow) incoherent variables can only appear in the equations
for the (fast) coherent variables if they are multiplied by a coherent
variable. Therefore,
$\boldsymbol{\nabla}_{\boldsymbol{i}}\otimes\boldsymbol{G}({\boldsymbol{i}},\boldsymbol{0})=\boldsymbol{0}$.

While these results are generic, in the specific case of the CI models the
incoherent perturbations of the incoherent solution always decay to zero.
Therefore, the existence of a laser threshold is determined solely by the
dynamics of the coherent perturbations, given by

$d_{t}\boldsymbol{\eta}_{\boldsymbol{c}}=\boldsymbol{\nabla}_{\boldsymbol{c}}\otimes\boldsymbol{G}({\boldsymbol{i}},\boldsymbol{0})\boldsymbol{\eta}_{\boldsymbol{c}}=\left[\begin{array}[]{cc}J&0\\\
0&J^{*}\end{array}\right]\boldsymbol{\eta}_{\boldsymbol{c}},$ (14)

where

$\displaystyle J$ $\displaystyle=\left[\begin{array}[]{cc}\partial_{\langle
b\rangle}G_{\langle
b\rangle}({\boldsymbol{i}},\boldsymbol{0})&\partial_{\langle
v^{\dagger}c\rangle}G_{\langle b\rangle}({\boldsymbol{i}},\boldsymbol{0})\\\
\partial_{\langle b\rangle}G_{\langle
v^{\dagger}c\rangle}({\boldsymbol{i}},\boldsymbol{0})&\partial_{\langle
v^{\dagger}c\rangle}G_{\langle
v^{\dagger}c\rangle}({\boldsymbol{i}},\boldsymbol{0})\end{array}\right]$ (17)
$\displaystyle=\left[\begin{array}[]{cc}-\gamma_{c}&g^{*}N\\\ g(2\langle
c^{\dagger}c\rangle-1)&-(\gamma+i\Delta\nu)\end{array}\right],$ (20)

and $J^{*}$ is the complex conjugate of $J$. For ease of notation and without
loss of generality, we have written $J$ in a frame rotating with $\langle
b\rangle$. This matrix depends on the system parameters and on the population
of the excited state. It is important to note that the structure of the
stability matrix is the same for both single- and multi-electron CI models.
However, since $J$ depends on the excited state population the eigenvalues of
these two models differ.

The lasing threshold condition is that there is at least one eigenvalue
$\lambda$ of $J$ such that $\Re(\lambda)>0$. Since $0\leq\langle
c^{\dagger}c\rangle\leq 1$, this can be satisfied only if

$N>\frac{\gamma\gamma_{c}}{|g|^{2}}\left[1+\left(\frac{\Delta\nu}{\gamma+\gamma_{c}}\right)^{2}\right],$
(21)

i.e., if the number of quantum dots is greater than a critical number given by
the right hand side of Eq. (21). This applies to both the single- and multi-
electron models, is independent of $\beta$, and increases with losses and
detuning. We conclude this section with two observations. The first is that
the CI models have been derived assuming weak light-matter coupling. Therefore
Eq. (21) does not apply to the strong coupling regime. The second, is that the
instability condition on the number of emitters in Eq. (21) is only a
necessary one: a sufficiently large pump rate is also necessary to cross the
laser threshold, as discussed in the following.

Figure 1: (a) The modulus of the coherent field amplitude $|\langle
b\rangle|$, (b) the correlation between photon absorption and emission,
$\delta\langle b^{\dagger}b\rangle$, and (c) the expectation value of the
photon number, $\langle b^{\dagger}b\rangle$ as a function of the pump for the
single- (solid line) and the multi-electron (dashed with circles) CI models,
for $N=\\{20,21,40\\}$ (blue, red and green lines respectively). In this and
all other figures time and decay and coupling parameters are scaled with
$\gamma_{nr}$, which is equivalent to setting $\gamma_{nr}=1$ in Eqs. (9) or
(10). The other parameter values are $g=70$, $\Delta\nu=0$, $\gamma=10^{4}$,
$\gamma_{c}=10$ and $\gamma_{nl}=1400$, equivalent to $\beta=7\times 10^{-4}$.

##  V Laser threshold: dependence on N

We now investigate photon emission processes in these single- and multi-
electron lasers below and above the instability threshold. Since $\langle
b\rangle=0$ when the device is not lasing, we see from the cluster expansion
in Eq. (7) that the photon number is given exclusively by the correlation
term, which dominates the spontaneous emission regime. Fig. 1 a) illustrates
the effect of including the fast variables for the single-electron (solid
lines) and the multi-electron (dashed lines with circles) CI models, Eqs. (9)
and (10). We compare three devices containing $N=\\{20,21,40\\}$ emitters
(blue, red and green lines respectively). For the parameter values of this
illustration (see caption of Fig. 1), an instability exists if $N>20$. Below
the critical number of quantum dots (blue lines), as the pump increases the
photon number saturates and the coherent field amplitude remains zero,
confirming the absence of laser emission. For a number of quantum dots just
above the minimum number required for lasing, i.e., $N=21$ (red line), there
is a clear jump in the photon number accompanied by an emerging non-zero
coherent field amplitude via a pitchfork bifurcation (Fig. 1a). We can see
from the graph of $\delta\langle b^{\dagger}b\rangle$, Fig. 1b, that the
initial growth in photon number is due to spontaneous emission, positively and
increasingly correlated to absorption, while the coherent part of the field is
zero. Indeed, the growth of $\delta\langle b^{\dagger}b\rangle$ in Fig. 1b
precedes the bifurcation (Fig. 1a) and occurs at pump rates toward the end of
the steeper growth in the photon number that is visible in Fig. 1c. This is a
characteristic feature which distinguishes small from macroscopic lasers. For
the latter, it is known that the inflection point of the steeper photon number
growth corresponds to the threshold [5] and, by extension, this points has
been taken as a reference also for small lasers with the help of clever
techniques [4]. Finite-size effects, instead, profoundly modify not only the
nature of threshold (as explained below), but also the pump value for which it
occurs. The consequences are important since the identification of coherent
emission becomes problematic. The difficulty is pragmatically circumvented, in
commercial microdevices, by manufacturers 111See, e.g., the Manufacturer’s
specification on the Thorlabs site for VCSEL980 (microVCSEL for data
transmission). whose laser characteristic sheets give a threshold current
which is placed well beyond the actual threshold, identified here through
linear stability analysis. A discussion of the various “kinds” of threshold
experimentally used is offered in [23] (Supplementary Material available in
[24]).

From threshold onward, the increase of the coherent field intensity $|\langle
b^{\dagger}\rangle|^{2}$ coincides with a sharp decrease in the correlation
between absorption and emission, $\delta\langle b^{\dagger}b\rangle$, as
expected in the presence of stimulated emission. When the correlation
$\delta\langle b^{\dagger}b\rangle$ becomes negative, stimulated emission
dominates and $\langle b^{\dagger}b\rangle=|\langle
b^{\dagger}\rangle|^{2}+\delta\langle b^{\dagger}b\rangle<|\langle
b^{\dagger}\rangle|^{2}$. In summary, Fig. 1b clearly shows two features:
($i$) during the steep parts of the emission growth the light is entirely
incoherent; ($ii$) immediately above threshold, the emitted field consists of
a mixture of coherent and incoherent photons and complete dominance of the
coherent component takes place only (well) beyond threshold. These features
are typical of nano- and microlasers.

The smoothness of the lasing transition imposed by the finite size of the
(small) devices paves the way towards new applications [25, 26]. In contrast,
the sharpness of the transition in macrolasers (e.g., Fig. 2) squeezes the
pump interval over which the evolution from entirely incoherent to dominantly
coherent emission takes place, explaining why, in macroscopic lasers, the
threshold can be considered as an on-off effect that corresponds to a single
well defined pump value. It is a strength of the CI models that they provide a
description of the continuous transformation in the laser emission features as
its size increases.

A device with twice the minimum number of quantum dots (e.g., $N=40$, dashed
lines in Fig. 1) crosses the laser threshold at a pump rate lower than that
for $N=21$ and with a sharper transition, see Fig. 1c.

As $N$ increases the differences between the single- and multi-electron model
become apparent (compare the solid and dashed curves in Fig. 1). The multi-
electron model reaches threshold for lower values of the pump rate and, hence,
the fraction of incoherent emission contributing to the initial growth in
photon number is reduced. This is due the lower losses of the upper level
population, $\langle c^{\dagger}c\rangle$, due to the term $\gamma_{nl}\langle
c^{\dagger}c\rangle^{2}$ in Eq. (10c) compared to the losses due to the term
$\gamma_{nl}\langle c^{\dagger}c\rangle$ in Eq. (9c). Both models have the
same critical number of emitters necessary for the instability to exist
($N=21$ for the chosen parameters). Only the pump power at the laser threshold
changes.

These results highlight the contributions of the fast variables, and the
necessity of their presence in the models to obtain a consistent description
of the emission processes in a laser. The position of the laser bifurcation in
the I-O curve shows that a simple visual inspection of the output
characteristics leads to an incorrect identification of the laser threshold
and fails to identify the true nature of the emission process, e.g., the
incoherent nature of the photon number in small lasers in the phase of steep
growth.

Figure 2:  (a) Numerical estimate of the pump threshold for the single- (S)
and multi-electron (M) CI models as a function of the spontaneous emission
factor, $\beta$, for different numbers of emitters. (b): Photon number
$\langle b^{\dagger}b\rangle$ as a function of the pump $r$ for the single-
(S) and multi-electron (M) CI models for different values of $\beta$ and
$N=40$ quantum dots . The black crosses identify the numerically established
laser thresholds for the two models. All other parameters as in Fig. 1.

##  VI Laser threshold: dependence on $\beta$

We now turn to the dependence of threshold on system size, $\beta$ [14]. Fig.
2a displays the value of pump at threshold as a function of $\beta$ for
devices with different $N$. This has been computed numerically by finding the
pump value for which the correlation $\delta\langle b^{\dagger}b\rangle$ is
maximum (see Fig. 1b). While the threshold pump rate decreases monotonically
as $N$ increases (for all values of $\beta$), the dependence on cavity size
shows the existence of two regimes: a rapid threshold decrease within the
realm of macroscopic lasers, and the onset of near-saturation (in double
logarithmic scale) for $\beta\gtrapprox 10^{-3}$ (i.e., micro- and
nanolasers). This latter feature would appear to contradict the common
knowledge according to which the threshold linearly decreases with slope
$\frac{1}{2}$ in double logarithmic scale [5, Eq. (20)]) as $\beta$ increases;
this property is, however, based exclusively on the (incorrect) assumption
that threshold is always placed at the inflection point of the I-O curve.
Instead, the saturation which emerges from the CI models results from the
identification of the true laser threshold (self-sustained stimulated
emission, Section IV) which progressively and substantially moves away from
the macroscopic definition as the laser size is decreased. The loss in
threshold reduction is, however, well compensated by the emergence of a
broader and richer transition region between incoherent and coherent emission,
whose features promise new applications (Sections IV and VII).

A clear visual illustration of the threshold displacement is provided in Fig.
2b, showing the I-O curves in double logarithmic scale for laser devices with
$N=40$ emitters and three different values of $\beta$. The straight,
superposed I-O curves correspond, as expected, to $\beta=1$, while those with
a gentle curvature to a microlaser: the respective thresholds are marked by
black crosses and appear well on the upper branch. It is only with a
macroscopic laser that the threshold appears at the inflection point of the
steeply growing photon number, matching the well-known properties of
macroscopic lasers [5].

##  VII First order correlation function $g^{(1)}(\tau)$

Surprisingly not included in the recommendations to identify laser threshold
in experiments [27], the autocorrelation functions remain the most sensitive
and most reliable way of obtaining pertinent threshold information, as long as
a meaningful model can be used for comparison. In this section, we tackle
precisely this aspect and examine the first order, time-delayed correlation
function. Here we study its properties and successfully compare them to the
experimental measurements in Ref. [15]. Due to its more complex experimental
implementation, it is more seldom used than its second order counterpart, but
it has the advantage of providing direct information on the coherence of the
emitted radiation [15]. Once the relationship between the two kinds of
correlations is clarified, comparison between the two indicators will
facilitate their individual use in the interpretation of experimental results.

In order to calculate the first-order correlation function

$g^{(1)}(\tau)=\frac{\langle b^{\dagger}(t)b(t+\tau)\rangle}{\langle
b^{\dagger}(t)b(t)\rangle},$ (22)

where $\tau$ is a delay time, we write the differential equation

$d_{\tau}g^{(1)}=\frac{1}{\langle b^{\dagger}(t)b(t)\rangle}d_{\tau}\langle
b^{\dagger}(t)b(t+\tau)\rangle$ (23)

which we solve with initial condition $g^{(1)}(0)=1$. To form a close set of
equations, we use the quantum regression formula, see Eqs. (1.105-1.107) of
Ref. [28]. In the Heisenberg picture, this reads $d_{\tau}\langle
A(t)B(t+\tau)\rangle=\langle A(t)d_{\tau}B(t+\tau)\rangle$ where $A$ and $B$
are operators and $d_{\tau}B$ is calculated applying the Hamiltonian and
Lindblad formalism at time $t+\tau$. We expand the $\tau$ derivative on the
right hand side of Eq. (23) and make use of Eqs. (9a) and (9d) to obtain

$\displaystyle
d_{\tau}\langle\tilde{b}^{\dagger}b\rangle=-(\gamma_{c}+i\nu)\langle\tilde{b}^{\dagger}b\rangle+Ng^{*}\langle\tilde{b}^{\dagger}v^{\dagger}c\rangle,$
(24a) $\displaystyle
d_{\tau}\langle\tilde{b}^{\dagger}v^{\dagger}c\rangle=-(\gamma+i\nu_{\epsilon})\langle\tilde{b}^{\dagger}v^{\dagger}c\rangle+g\left(2\langle\tilde{b}^{\dagger}bc^{\dagger}c\rangle-\langle\tilde{b}^{\dagger}b\rangle\right),$
(24b)

where $\tilde{b}^{\dagger}\equiv b^{\dagger}(t)$, and all other operators are
at time $t+\tau$. $\langle\tilde{b}^{\dagger}bc^{\dagger}c\rangle$ is the
expectation value of a $3$-particle operator. To find a closed set of
equations at two-particle level we use Eq. (7) and the cluster expansion

$\begin{split}\langle\tilde{b}^{\dagger}bc^{\dagger}c\rangle=&\delta\langle\tilde{b}^{\dagger}bc^{\dagger}c\rangle+\langle\tilde{b}^{\dagger}\rangle\delta\langle
bc^{\dagger}c\rangle\\\ &+\langle
b\rangle\delta\langle\tilde{b}^{\dagger}c^{\dagger}c\rangle+\langle
c^{\dagger}c\rangle\delta\langle\tilde{b}^{\dagger}b\rangle+\langle\tilde{b}^{\dagger}\rangle\langle
b\rangle\langle c^{\dagger}c\rangle\end{split}$ (25)

together with the semi-classical approximation used to derive the CI models,
which for these equations reduces to
$\delta\langle\tilde{b}^{\dagger}c^{\dagger}c\rangle\sim\delta\langle
bc^{\dagger}c\rangle\sim 0$. With these approximations Eqs. (24) become

$\displaystyle
d_{\tau}\langle\tilde{b}^{\dagger}b\rangle=-(\gamma_{c}+i\nu)\langle\tilde{b}^{\dagger}b\rangle+Ng^{*}\langle\tilde{b}^{\dagger}v^{\dagger}c\rangle,$
(26a) $\displaystyle
d_{\tau}\langle\tilde{b}^{\dagger}v^{\dagger}c\rangle=-(\gamma+i\nu_{\epsilon})\langle\tilde{b}^{\dagger}v^{\dagger}c\rangle+g\langle\tilde{b}^{\dagger}b\rangle(2\langle
c^{\dagger}c\rangle-1).$ (26b)

Eqs. (26) are formally identical for the single- and multi-electron models,
the only difference in $g^{(1)}(\tau)$ coming from the different values of the
term $\langle c^{\dagger}c\rangle$. This is due to the fact that the
Heisenberg equations and the dissipative Lindblad terms for the operators at
time $t+\tau$ do not depend on the losses of $\langle c^{\dagger}c\rangle$
that are proportional to $\gamma_{nl}$.

Figure 3: The main graph plots the coherence decay time $\tau_{c}$ as a
function of the spontaneous emission factor $\beta$ for the single- (S) and
multi-electron (M) CI models for a pump value equal to 15% of the threshold
for $N=40$ quantum dots. $\tau_{c}$ has been obtained by fitting with a
straight line $\log(|g^{(1)}(\tau)|)$ as a function of $\tau$. The inset is a
log plot $|g^{(1)}(\tau)|$ as a function of the delay time $\tau$ for a sample
of the values of $\beta$ used in the main plot. This confirms the exponential
decay of the correlation, Eq. (27). All other parameters as in Fig. 1.

With the help of these expressions, we can now plot the first order
autocorrelation as a function of the model parameters. We expect that below
threshold the correlation function decays exponentially with the delay time,

$g^{(1)}(\tau)\propto e^{-t/\tau_{c}},$ (27)

with $\tau_{c}$ the correlation decay time. This behavior is confirmed by the
log plot of $g^{(1)}(\tau)$ in the inset of Fig. 3, where we have set the pump
at 15% of the single-electron threshold value [29, Eq. (20)] for $N=40$
quantum dots. We have computed $\tau_{c}$ as a function of $\beta$ by fitting
these curves with a straight line. The decay rate has a sigmoidal behavior: it
is an increasing function of $\beta$ that jumps by two orders of magnitude as
$\beta$ changes from $10^{-4}$ to $10^{-2}$ and is approximately constant
outside this interval. This clearly illustrates a fundamental feature of small
lasers, whose coherence grows gradually as threshold is approached, in
agreement with the smooth response of their I-O curve. For macroscopic lasers,
on the other hand, we obtain results which are consistent with the standard
picture of a nearly incoherent output up until threshold, with a sudden
conversion to full coherence. The single- and multi-electron CI models have
similar behavior, with the multi-electron model having larger decay time. This
is an effect of the lower effective losses of the multi-electron with respect
to the single-electron model: at equal pump values the former is closer to
threshold than the latter (cf. the shift of threshold positions between the
two models in Fig. 2).

Figure 4:  $\left|g^{(1)}\right|$ for delay $\tau_{M}=60/\gamma_{nr}$ as a
function of the pump for $\beta=\\{1,7\times 10^{-4},3.4\times 10^{-6}\\}$
(blue, red and green lines respectively) for the single- (solid) and multi-
(dashed with circle) CI models. The number of quantum dots is
$N=\\{40,1000\\}$ in panels (a) and (b) respectively. The points where
$g^{(1)}(\tau_{M})$ reaches unity, and where the slopes of the curves suddenly
change, are the laser thresholds. All other parameters as in Fig. 1.

The evolution of coherence with pump power is examined in Fig. 4 for the
single- and multi-electron models at fixed number of emitters,
$N=\\{40,1000\\}$, and cavity volumes, $\beta=\\{1,7\times 10^{-4},3.4\times
10^{-6}\\}$. In order to clearly highlight the pump influence, a delay time
$\tau=60/\gamma_{nr}$ is fixed. Experimental information can be gathered, as
in [15], by fixing the difference in the Michelson interferometer arm lengths
and measuring the fringe visibility as a function of pump. The laser threshold
corresponds to the smallest pump value for which $g^{(1)}(\tau_{M})=1$; at
this point the curve slope is discontinuous. While in an experiments
unavoidable fluctuations of the control parameters, not included in the model,
will limit the coherence time even beyond the threshold, the behavior of the
coherence time as a a function of the pump will provide a clear indication of
the threshold value. Irrespective of laser size, there is a continuous growth
of coherence, driven by the increase in correlation between absorption and
emission properties (as in Fig. 2); however, while in smaller systems
coherence evolves steadily over a broad pump range below threshold, in
macroscopic lasers the change occurs over a narrow interval of pump values. In
other words, as $\beta$ decreases moving toward the macroscopic limit, it
becomes more and more difficult to obtain partially coherent emission. This
result does not depend on the choice of $\tau_{M}$, as long as
$\tau_{M}\gg\lambda_{0}/v$, with $\lambda_{0}$ and $v$ the light wavelength
and velocity in the interferometer, respectively. Changing $\tau_{M}$ only
changes the shape of the curves of Fig. 4. It is worth stressing again that
the deformation of the coherence curves progresses continuously from the nano-
to the macroscale.

Increasing the number of emitters from $N=40$, Fig. 4a, to $N=10^{3}$, Fig.
4b, while keeping the other parameters constant reduces the threshold values
and the range of pump values over which the transition toward
$g^{(1)}(\tau_{M})=1$ occurs for all values of $\beta$. While nanolasers are
typically built with tens of emitters, in macroscopic lasers their number will
easily be largely in excess of what we are showing here, thus further
enhancing the differences between the two categories of devices.

Figure 5: The main graph plots the coherence decay time $\tau_{c}$ as a
function of the pump power in units of the single electron CI threshold for
$\beta=7\times 10^{-4}$ and $N=40$ quantum dots. $\tau_{c}$ has been obtained
by fitting with a straight line $\log(|g^{(1)}(\tau)|)$ as a function of
$\tau$. The inset is a log plot $|g^{(1)}(\tau)|$ as a function of the delay
time $\tau$ for the pump values indicated by square symbols in the main plot.
All other parameters as in Fig. 1. This figure is the analogous of figures 2b
and 2c of Ref. [15]. For ease of comparison time units are expressed in
picoseconds. The dimensional time scale has been fixed by setting
$\gamma_{nr}=1$ ns.

We conclude this section by highlighting that these analytical and numerical
results are supported by independent experimental measurement of
$g^{(1)}(\tau)$ [15]. The first order coherence was experimentally obtained in
Ref. [15] by measuring the visibility of interference fringes resulting from
Michelson interferometry and plotted as a function of the pump power [15, Fig.
2b]. From these data the authors also computed the coherence decay time as a
function of the pump power. It is not possible from the experimental data
available in Ref. [15] to obtain unique values for the CI model parameters.
However, the parameter values used in the figures in this paper are reasonable
estimates. We plot in Fig. 5 the correlation decay time as a function of the
pump power, measured in units of the analytical threshold for the single
electron CI model [29, Eq. (20)]. The similarities between this figure and its
inset and figures 2c and 2b respectively of Ref. [15] are uncanny, keeping in
mind the uncertainty in the mapping of the experimental parameters. We can
therefore conclude that the CI model is capable of clearly and unequivocally
identifying the onset of coherence, matching it to the crossing of laser
threshold (self-sustained growth of stimulated emission), and to explain
experimental observations for which no model, derived from first principles,
had been available up until now.

##  VIII Conclusions

We have presented the details of a model for (semi-conducting) quantum
emitters (with single- or multiple-electrons) coupled to an electromagnetic
cavity of arbitrary size to describe the transition from thermal to coherent
emission. The joining of a fully quantum treatment, based on the explicit
description of incoherent fields and the correspondingly induced dipole
moments, and of a coherent field with its accompanying polarization, together
with an analysis based on nonlinear dynamical properties permits the clear and
unequivocal identification of a threshold for the emergence of a self-
sustained stimulated emission, i.e., the lasing onset. The Coherent-Incoherent
model marks an entirely new approach in the depiction of laser action, due to
the traditional attention brought to macroscopic devices and to the resulting
attempts at adjusting the latter to cover small devices through simple
modifications. This treatment shows that simple adjustments are not sufficient
and that a consistent treatment can be obtained only through fundamentally
revisiting the physics to explicitly introduce the two categories of
incoherent and coherent variables.

The main result is a proper definition of lasing threshold irrespective of
laser size, accompanied by a continuous description of the evolution of the
degree of coherence from the macro- to the nano- scale; we further find that
the number of coupled emitters contribute to sharpening or softening the more
extreme aspects of the system size.

In addition to the definition of threshold based on nonlinear physics
concepts, the quantum mechanical approach permits the direct evaluation of the
coherence properties of the electromagnetic field through the first order
coherence function. Its use shows that full coherence is attained at the
bifurcation point (laser threshold), which – at variance with scaling laws
established at the macroscale – is placed closer and closer to the upper
emission branch (or directly on it) as the finite system size contribution
increases through the reduced number of electromagnetic cavity modes.
Simultaneously, the quantum-mechanical analysis shows that the rapid growth in
photon number originates from an increase in correlation between absorption
and emission processes in the absence of self-sustained stimulated emission,
which account for the entirety of the transition to the upper emission branch
in the smallest devices. In macroscopic lasers, instead, this contribution is
limited to the lower portion of the (nearly) vertical growth in photon number.

A remarkable aspect of the CI model rests on its ability to predict features
experimentally observed in measurements of fringe visibility [15]; a good
qualitative agreement is obtained without any free parameters between
observations and the predictions shown in this paper. The topic is of great
interest since it allows for an unequivocal quantification – and for general
model-based predictions – of the amount of coherence, potentially paving the
way to numerous applications ranging from novel uses for micro- and
nanolasers, but also permitting better assessment of their performance as
sources for data treatment (e.g., interconnects in data centers with ultra-low
dissipation and small footprint [30, 31, 32, 33, 34, 35]).

The availability of a complete description of the threshold physics at all
scales permits the comparison with other experimental choices. For instance,
one can envisage computing the output of a mixing interferometer [36] to
interpret its results on the basis of a first-principle model, rather than
superposing ad hoc radiation packets with preset features.

The simultaneous availability of first-order and second-order
autocorrelations, in addition to the threshold information gathered through
the LSA, also permits a careful evaluation of the individual properties of
these indicators. This way, second-order autocorrelation measurements, easier
to perform and routinely used not only in Quantum-Dot-based devices [37, 38,
39, 19], but also with Quantum-Well emitters [40, 41, 42] and metallic
nanolasers [43, 44], can acquire a higher degree of reliability in the
determination of the nature of the emitted radiation. This can contribute to
reaching an agreement on a definite measurement technique for the
determination of laser threshold [27] thus sorting the different practical
definitions used over the past decades, which give concordant results only at
the macroscopic scale [23].

The CI models conclusively show that the transition from incoherent to
coherent radiation occurs in a negligibly small pump interval for macroscopic
devices. However, they also prove that the physics of laser threshold remains
the same even for large lasers, thus implying that the only obstacle in
obtaining information from an experiment is of practical nature. This
interpretation is consistent with the results of pioneering work of the 1960’s
and 70’s [45, 46], where statistical ensemble measurements gave evidence for a
gradual evolution in the nature of the emitted radiation at threshold
crossing. More information could become now available through the realization
of a novel system constituted by a broadband semiconducting amplifier, where
feedback is provided by a fiber loop (also containing adjustable filters)
which permit stroboscopic measurements of the light amplification as a
function of round trip [47]. The degree of spatio-temporal resolution gained
from this realization, thanks to the long delay time of the fibered cavity,
enables the measurement of the radiation properties at each round trip. This
scheme could garner detailed information to refine our understanding and
mathematical description of laser threshold.

## References

  * Hill and Gather [2014] M. T. Hill and M. C. Gather, Nature Photonics 8, 908 (2014).
  * Ma and Oulton [2019] R.-M. Ma and R. F. Oulton, Nature nanotechnology 14, 12 (2019).
  * Carroll  _et al._ [2021a] M. Carroll, G. D’Alessandro, G. L. Lippi, G.-L. Oppo, and F. Papoff, Appl. Phys. Lett. 119, 101102 (2021a).
  * Ning [2013] C.-Z. Ning, IEEE Journal of Selected Topics in Quantum Electronics 19, 1503604 (2013).
  * Rice and Carmichael [1994] P. R. Rice and H. Carmichael, Phys. Rev. A 50, 4318 (1994).
  * Yokoyama and Brorson [1989] H. Yokoyama and S. Brorson, J. Appl. Phys. 66, 4801 (1989).
  * Abraham and Narducci [1988] N. B. Abraham and L. M. Narducci,  _Laser physics and laser instabilities_ (World Scientific, 1988).
  * Kira and Koch [2011] M. Kira and S. W. Koch,  _Semiconductor quantum optics_ (Cambridge University Press, 2011).
  * Chow and Jahnke [2013] W. W. Chow and F. Jahnke, Progress in Quantum Electronics 37, 109 (2013).
  * Chow  _et al._ [2014] W. W. Chow, F. Jahnke, and C. Gies, Light: Science and Applications 3, 1 (2014).
  * Kreinberg  _et al._ [2017a] S. Kreinberg, W. W. Chow, J. Wolters, C. Schneider, C. Gies, F. Jahnke, S. Höfling, M. Kamp, and S. Reitzenstein, Light: Science and Applications 6, 1 (2017a), arXiv:1610.04129 .
  * Gies  _et al._ [2007] C. Gies, J. Wiersig, M. Lorke, and F. Jahnke, Physical Review A - Atomic, Molecular, and Optical Physics 75, 1 (2007).
  * Fricke [1996] J. Fricke, Annals of Physics 252, 479 (1996), arXiv:9610128 [cond-mat] .
  * Carroll  _et al._ [2021b] M. Carroll, G. D’Alessandro, G. L. Lippi, G.-L. Oppo, and F. Papoff, Phys. Rev. Lett. 126, 063902 (2021b).
  * Tempel  _et al._ [2011] J.-S. Tempel, I. Akimov, M. Aßmann, C. Schneider, S. Höfling, C. Kistner, S. Reitzenstein, L. Worschech, A. Forchel, and M. Bayer, J. Opt. Soc. Am. B 28, 1404 (2011).
  * Carroll  _et al._ [2022a] M. Carroll, G. D’Alessandro, G. L. Lippi, G.-L. Oppo, and F. Papoff, Phys. Rev. Lett. 128, 029402 (2022a).
  * Carroll  _et al._ [2022b] M. Carroll, G. D’Alessandro, G. L. Lippi, G.-L. Oppo, and F. Papoff, arXiv:2201.02468v1 [optics] (2022b).
  * Jaynes and Cummings [1963] E. T. Jaynes and F. W. Cummings, Proceedings of the IEEE 51, 89 (1963).
  * Kreinberg  _et al._ [2017b] S. Kreinberg, W. W. Chow, J. Wolters, C. Schneider, C. Gies, F. Jahnke, S. Höfling, M. Kamp, and S. Reitzenstein, Light: Science and Applications 6, e17030(1 (2017b), arXiv:1610.04129 .
  * Florian  _et al._ [2013] M. Florian, C. Gies, F. Jahnke, H. A. M. Leymann, and J. Wiersig, Phys. Rev. B 87, 165306 (2013).
  * Leymann  _et al._ [2014] H. Leymann, A. Foerster, and J. Wiersig, Phys. Rev. B 89, 085308 (2014).
  * Note [1] See, e.g., the Manufacturer’s specification on the Thorlabs site for VCSEL980 (microVCSEL for data transmission).
  * Lippi  _et al._ [2022a] G. L. Lippi, T. Wang, and G. P. Puccioni, Chaos, Solitons & Fractals 157, 111850 (2022a).
  * Lippi  _et al._ [2022b] G. L. Lippi, T. Wang, and G. P. Puccioni, arXiv:2202.06026 (2022b).
  * Wang  _et al._ [2021a] T. Wang, C. Jiang, J. Zou, H. Zhou, X. Lin, H. Chen, G. P. Puccioni, G. Wang, and G. L. Lippi, IEEE Sensors 18, 19948 (2021a).
  * Wang  _et al._ [2021b] T. Wang, C. Jiang, J. Zou, J. Yang, K. Xu, C. Jin, G. Wang, G. P. Puccioni, and G. L. Lippi, Nanomaterials 11, 3325 (2021b).
  * Editorial [2017] Editorial, Nat. Photonics 11, 139 (2017).
  * Carmichael [2002] H. Carmichael,  _Statistical Methods in Quantum Optics I_ (Springer-Verlag, Heidelberg, 2002).
  * Carroll  _et al._ [2021c] M. Carroll, G. D’Alessandro, G. L. Lippi, G.-L. Oppo, and F. Papoff, Phys. Rev. Lett. 126, 063902 (Supp. Mat.) (2021c).
  * Miller [2009] D. A. B. Miller, Proc. IEEE 97, 1166 (2009).
  * Service [2010] R. F. Service, Science 328, 810 (2010).
  * Smit  _et al._ [2015] M. Smit, J. van der Tol, and M. Hill, Lasers & Photonics Rev. 6, 1 (2015).
  * Notomi  _et al._ [2014] M. Notomi, K. Nozak, A. Shinya, S. Matsuo, and E. Kuramochi, Opt. Commun. 314, 3 (2014).
  * Soref [2018] R. Soref, APL Photonics 3, 021101 (2018).
  * Ning [2019] C. Z. Ning, Advanced Photonics 1, 014002 (2019).
  * Lebreton  _et al._ [2013] A. Lebreton, I. Abram, R. Braive, I. Sagnes, I. Robert-Philip, and A. Beveratos, Phys. Rev. Lett. 110, 163603 (2013).
  * Ulrich  _et al._ [2007] S. M. Ulrich, C. Gies, S. Ates, J. Wiersig, S. Reitzenstein, C. Hofmann, A. Löffler, A. Forchel, F. Jahnke, and P. Michler, Phys. Rev. Lett. 98, 043906 (2007).
  * Wiersig  _et al._ [2009] J. Wiersig, C. Gies, F. Jahnke, M. Aßmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, Nature 460, 245 (2009).
  * Ota  _et al._ [2017] Y. Ota, M. Kakuda, K. Watanabe, and S. I. Y. Arakawa, Opt. Express 25, 19981 (2017).
  * Wang  _et al._ [2015] T. Wang, G. P. Puccioni, and G. L. Lippi, Sci. Rep. 5, 15858 (2015).
  * Takiguchi  _et al._ [2016] M. Takiguchi, H. Taniyama, H. Sumikura, M. D. Birowosuto, E. Kuramochi, A. Shinya, T. Sato, K. Takeda, S. Matsuo, and M. Notomi, Opt. Express 24, 3441 (2016).
  * Wang  _et al._ [2020] T. Wang, D. Aktas, O. Alibart, É. Picholle, G. P. Puccioni, S. Tanzilli, and G. L. Lippi, Phys. Rev. A 101, 063835 (2020).
  * Hayenga  _et al._ [2016] W. E. Hayenga, H. García-Gracia, H. Hodaei, C. Reimer, R. Morandotti, P. Likamwa, and M. Khajavikhan, Optica 3, 1187 (2016).
  * Pan  _et al._ [2016] S. Pan, Q. Gu, A. E. Amili, F. Vallini, and Y. Fainman, Optica 3, 1260 (2016).
  * Arecchi  _et al._ [1966] F. Arecchi, A. Berné, A. Sona, and P. Burlamacchi, IEEE J. Quantum Electron. QE-2, 341 (1966).
  * Arecchi and Degiorgio [1971] F. Arecchi and V. Degiorgio, Phys. Rev. A 3, 1108 (1971).
  * Roche  _et al._ [2022] A. Roche, S. Slepneva, A. Kovalev, A. Pimenov, A. Vladimirov, M. Marconi, M. Giudici, and G. Huyet, arXiv:2212.11074 (2022).


# Enhancing Protocol Privacy with Blind Calibration of Quantum Devices

Ankit Khandelwal, Stephen DiAdamo © 2023 IEEE. Personal use of this material
is permitted. Permission from IEEE must be obtained for all other uses, in any
current or future media, including reprinting/republishing this material for
advertising or promotional purposes, creating new collective works, for resale
or redistribution to servers or lists, or reuse of any copyrighted component
of this work in other works.

###### Abstract

To mitigate the noise in quantum channels, calibration is used to tune the
devices to minimize error. Generally, calibration is performed by transmitting
pre-agreed-upon calibration states and determining an error cost so the two
parties can tune their devices accordingly. The calibration states can be the
same ones used for the desired protocol, and so an untrusted party could
potentially learn which protocol is being performed by gathering knowledge of
the calibration states and cost function. Here, we assume privacy of the
protocol is the goal and therefore the receiver should not be allowed to
determine the protocol states. We limit the information that is revealed to
the receiver, and in this regard, we propose a simple protocol that hides the
calibration states and cost function from the receiver, but still allows for
calibration to be performed efficiently, thereby increasing the privacy of the
protocol. We show various numerical results demonstrating the ability of the
protocol under various channel noise parameters and communication scenarios.

##  I Introduction

Quantum network technologies are rapidly developing and the potential
applications are becoming more realistic for commercial deployment. A key
benefit quantum networks bring is an unreachable level of privacy and security
that classical networks cannot currently achieve, with protocols such as
quantum key distribution [1], secret sharing [2], and blind quantum computing
[3]. A similarity that each of these protocols share is that, before the
protocols begins, and periodically during a continuous execution, the
communicating parties perform device calibration to ensure their devices are
aligned, minimizing any noise-induced errors caused by the quantum channel.
Optical components are affected by environmental changes like temperature
fluctuations [4], and so it is necessary to regularly calibrate the devices to
mitigate these effects.

By performing calibration, it opens the network up to potential attack. For
example, in the security proofs of some QKD protocols, it is assumed that the
devices begin in a calibrated state, but it has been shown that there are
attacks that can be performed during the calibration protocol [5, 6], thereby
diminishing the security. Further, in QKD it is critical that an accurate
estimate for the Quantum Bit Error Rate (QBER) is known so that information
reconciliation can be accurately and efficiently performed [7], and so
calibration is essential. It is therefore important for the calibration stage
to be as secure as the protocols themselves, not only for QKD, but in all
quantum secure communication scenarios.

An example approach to device calibration for quantum communication is to
align the transmitter and receiver using polarized states as described in [8].
Here the sender and receiver tune a polarization controller to calibrate,
where the sender and receiver both know the calibration states. The process
repeats until the two parties minimize the error rate. In this work, we
propose a calibration strategy that uses the same quantum signals that will be
used in the protocol to be executed, allowing calibration, but done in a way
such that the receiver cannot determine the states, thereby protecting the
secrecy of the protocol.

In this work, we propose a simple approach to overcoming the need for the
sender to reveal the calibration states and the cost function used for
calibration. We call this approach “blind calibration” because from the
perspective of the receiver, they are blind to which quantum states they are
receiving and why the tuning of their device is improving transmission cost.
As far as we know, this is the first proposal for blind calibration used for
quantum devices. In sensor networks [9] blind calibration was proposed as a
calibration method without accurate ground-truth, and so a different meaning
of “blind”. The goal here is to hide information intentionally from the
receiver. The approach here is to randomize the transmission order of the
calibration states, thereby preventing the receiver from performing quantum
state tomography to determine the state set. Moreover, the sender hides the
cost function by computing it at the their side of the channel, sending only
a—potentially obfuscated—value back to the receiver. The sender’s goal is
therefore to complete the calibration as efficiently as possible, while hiding
as much information as possible. In this work, we propose a communication
protocol that achieves this goal and we analyze various transmission scenarios
for proof-of-concept. With simulation, we test how well our approach works
over fiber channel models with various noise parameters. To gain advantage
using our protocol, although, calibration should be done infrequently enough
to outweigh its cost since our protocol is potentially more resource
intensive.

Throughout this work we use the following mathematical notation: the set of
quantum states on a finite dimensional, complex Hilbert space $\mathcal{H}$ of
dimension $n$ is denoted $\mathcal{S}(\mathcal{H})$; a quantum channel is a
Completely Positive Trace Preserving (CPTP) map over the state space
$\mathcal{S}(\mathcal{H})$; random variables are usually denoted as $X_{i}$
and vectors of random variables with a bold-face notation $\mathbf{X}_{i}$.

##  II Blind Calibration over Quantum Channels

In this section, we introduce our protocol for blind calibration. In order to
specify the protocol, we need to firstly state the assumptions we make.
Similar to the main assumptions for many quantum cryptographic protocols, we
assume the following: 1) Both parties act to achieve the goal of the protocol
and do not act to prevent it from being fulfilled; 2) The sender has a trusted
and private random number generator; 3) The sender operates in a secure
location and particularly their quantum source has no side channels leaking
information. Further, we rely on the fact that quantum states cannot be
perfectly copied, that is, the no-cloning theorem.

With this, we state our protocol for blind calibration. In Protocol 1, we list
the the protocol instructions for the sender and receiver. The high level idea
of the protocol is that the sender obfuscates the transmission statistics to
prevent the receiver from performing tomography [10], but at the same time,
using efficient optimization algorithms to find the tuning parameters that
minimize the desired cost function. To do this, for each iteration of the
protocol, the sender uses the uniform distribution $p$ over a set of
calibration states $S$ to determine the order to transmit $N$ quantum states.
From there, $N$ transmissions are made to the receiver after being processed
by an encoder, and the receiver then performs a decoding operation and
measures in a particular basis. The measurement results are sent back to the
sender with the basis information, and a cost of transmission is determined by
the sender. The cost is used to update the encoding strategy and is further
sent to the receiver to update the decoding strategy. An updating process
tunes the transmission and detection devices to perform calibration. This
process repeats until convergence or a maximum number of iterations $I_{\max}$
is reached. In this sense, any eavesdropper intercepting the quantum states in
transmission cannot perform a tomography attack to determine which quantum
states are being transmitted.

We further justify why tomography at the receiver alone cannot be performed on
the transmitted states to uncover the calibration state set $S$. The original
algorithm for tomography is as follows. For a state $\rho\in\mathcal{S(H)}$
from a Hilbert space $\mathcal{H}$ of $n$ dimensions, we have the following
equation,

$\rho=\frac{1}{n}\sum_{\vec{v}}\Tr(\sigma_{v_{1}}\otimes\dots\otimes\sigma_{v_{n}}\rho)\sigma_{v_{1}}\otimes\dots\otimes\sigma_{v_{n}}$
(1)

Here, the vectors $\vec{v}=(v_{1},\dots,v_{n})$ with entries $v_{i}$ are taken
from the set $\\{I,X,Y,Z\\}$ of Pauli matrices. In order to reconstruct a
density matrix with tomography, one needs to make many measurements in the
different basis such that the state statistics converge accordingly. Usually,
it is assumed that the state on which tomography is being performed is the
same for each measurement. In our protocol, we ensure that the receiver is
unable to determine when, from a set of calibration states, a specific quantum
state was transmitted because each state is transmitted uniformly randomly.
The best the receiver can do is compute an average state matrix, specifically
a maximally mixed state, which cannot reveal state information of the parts.
We prove the claim formally.

Protocol 1 Blind Calibration

Parameters

  * •

$S\subseteq\mathcal{S}(\mathcal{H})$, a set of calibration states, $|S|>0$

  * •

$N\in\mathbbm{N}$, the number of transmissions per iteration

  * •

$Cost:\mathbbm{R}^{N}\times\mathbbm{N}^{N}\rightarrow\mathbbm{R}$, a cost
function

  * •

$\epsilon_{\text{th}}\geq 0$, a calibration convergence threshold

  * •

$I_{\text{max}}$, the maximum number of iterations

  * •

Parameterized Encoder
$E_{\theta}:\mathcal{S}(\mathcal{H}_{A})\rightarrow\mathcal{S}(\mathcal{H}^{\prime}_{A})$

  * •

Quantum Channel
$C:\mathcal{S}(\mathcal{H}^{\prime}_{A})\rightarrow\mathcal{S}(\mathcal{H}^{\prime}_{B})$

  * •

Parameterized Decoder
$D_{\phi}:\mathcal{S}(\mathcal{H}^{\prime}_{B})\rightarrow\mathcal{S}(\mathcal{H}_{B})$

  * •

$Update:\mathbbm{R}^{M}\times\mathbbm{R}\rightarrow\mathbbm{R}^{M}$, a
parameter update function

Sender

1:$i\leftarrow 0$; $\theta^{i}\leftarrow$ Initialize

2:while $i<I_{\text{max}}$ and threshold $\epsilon_{\text{th}}$ not reached do

3: $statesTransmitted\leftarrow[\hskip 2.84526pt]$

4: for $t\in\\{0,...,N\\}$ do

5: choose State $\rho\in S$ with uniform probability

6: append Index of $\rho$ to $statesTransmitted$

7: transmit $E_{\theta^{i}}(\rho)$ over channel $C$

8: $M\leftarrow$ await Measurement results

9: $cost\leftarrow Cost(M,statesTransmitted)$

10: transmit $cost$ classically to Receiver

11: $\theta^{i+1}\leftarrow Update(\theta^{i},cost)$; $i\leftarrow i+1$

12:Terminate protocol

Receiver

1:$i\leftarrow 0$; $\phi^{i}\leftarrow$ Initialize

2:while protocol not terminated do

3: $M\leftarrow[\hskip 2.84526pt]$

4: for $t\in\\{0,...,N\\}$ do

5: $\tilde{\rho}\leftarrow$ await Quantum channel input

6: $m\leftarrow$ measure $D_{\phi^{i}}(\tilde{\rho})$

7: append $m$ to $M$ with basis information

8: transmit $M$ classically to sender

9: $cost\leftarrow$ await Cost value decreased

10: $\phi^{i+1}\leftarrow Update(\phi^{i},cost)$; $i\leftarrow i+1$

######  Theorem 1.

Given a state set
$S_{p}=\\{\rho_{1},...,\rho_{i}\\}\subseteq\mathcal{S}(\mathcal{H})$, where
$\dim(\mathcal{H})=n$, and an $\epsilon>0$, there exists calibration set
$S\supseteq S_{p}$, and $N$, the number of transmissions, such that, using
Protocol 1, a receiver following the protocol using quantum state tomography
can only recover a state set $\tilde{S}_{p}$ such that for each
$\tilde{\rho}\in\tilde{S}_{p}$, $\norm{I/n-\tilde{\rho}}\leq\epsilon$.

###### Proof.

Let $S_{p}=\\{\rho_{1},...,\rho_{k}\\}\subseteq\mathcal{S(H)}$ be a state set
and $\epsilon>0$ be given. We can construct $S_{p}$ in the following way. For
each state $\rho_{j}\in S_{p}$, $S$ will contain both $\rho_{j}$ and
$\overline{\rho_{j}}=U(\vec{\pi})\rho_{j}U^{\dagger}(\vec{\pi})$, that is, the
state from the state set and a state with the exact opposite behavior, where
$U$ is a rotation operator in $n$ dimensions. Let $p$ be the uniform
distribution over $S$. For each transmission, the sender sends a state
$\rho_{t}\in S$ with probability $p(\rho_{t})=1/|S|$. For state tomography,
the values $\Tr\left(\bigotimes_{i=1}^{n}\sigma_{v_{i}}\rho\right)$ are
estimated via,

$\displaystyle\frac{1}{N}\sum_{i=1}^{N}m_{i}=\frac{1}{N}\sum_{i=1}^{n}n_{i}m_{i}\overset{\makebox[0.0pt]{\mbox{\footnotesize$\epsilon/n$}}}{\approx}\frac{1}{n}\sum_{i=1}^{n}m_{i},$
(2)

where $n_{i}$ represents the number of times measurement outcome $m_{i}$ was
observed. Since either a state or its opposite is transmitted with equal
probability, in each basis, measuring one value is as probable as measuring
another, and so with the use of the law of large numbers, each $n_{i}$
approaches $N/n$ for large values of $N$, where such an $N$ can always be
found to be $\epsilon/n$ away from the true distribution. What results is
therefore a near equal mixture of all measurement outcomes. If we assume the
measurement outcomes $m_{i}$ are normalized to a $[-1,1]$ line, the sum (1)
becomes $\Tr(\rho)I/n\pm\epsilon/nI=I/n\pm\epsilon/nI$. ∎

##  III System Model and Simulation Configuration

In this section, we present a model for a communication system that performs
Protocol 1. The components of the system are a quantum source, an encoding
device, the noisy quantum channel, a decoding device, and a measurement
device. To model these components, we use the following. For simplicity of
analysis, we assume that the quantum source emits quantum states with perfect
fidelity, and therefore there is no noise model associated with it. For the
encoder and decoder, we assume that they are quantum channels parameterized by
a vector of variables
$\theta^{i}=(\theta^{i}_{0},\theta^{i}_{1},...,\theta^{i}_{n})$ for the
encoder and $\phi^{i}=(\phi_{0}^{i},\phi_{1}^{i},...,\phi_{m}^{i})$ for the
decoder, where the superscript $i$ represents the current iteration of the
protocol. For the measurement device, we again assume for simplicity that
measurement outcomes are accurate.

For the quantum channel, we consider noise models for coherent noise. Firstly
we model random noise by applying small, random, unitary rotations, and
secondly we consider length-dependent bit- and phase-flip errors. The noise
models are applied via quantum channels such that each qubit in the system has
an independent chance of experiencing the noise.

For the rotational noise model, we generate a random rotation vector
$\mathbf{\theta}_{R}$ with elements from $-\pi$ to $\pi$. Then, depending on
the length $L$ kilometers of the channel, we determine the percentage of the
rotation applied via, $p=1-10^{-\mu L/10}$, where $\mu$ is a parameter of the
channel. For the length-dependent noise, for a channel of length $L$ and
channel parameters $\mu_{1},\mu_{2}\geq 0$, the probability that a qubit
arrives with a bit-flip error or a phase-flip error is
$p_{b}=1-10^{-\mu_{1}L/10}$ and $p_{p}=1-10^{-\mu_{2}L/10}$ respectively.

For the protocol, we need to select a cost function. Here any valid cost
function can work, but in our simulations, we use the quantum fidelity or
error rates. Fidelity between states is defined as follows. Assuming the input
state is $\sigma\in\mathcal{S}(\mathcal{H})$ and the output state is
$\rho\in\mathcal{S}(\mathcal{H})$, the state fidelity between the two is given
by,
$F(\rho,\sigma)\coloneqq\Tr\left(\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\right)^{2}.$
To use fidelity as a cost, we define infidelity to be $1-F$. To determine the
fidelity in simulation, we implemented quantum state tomography [10] as
explained in the previous section. To compute the fidelity nonlocally in our
case, the receiver needs to perform the measurements and the sender needs to
perform the state reconstruction to compute the state fidelities of the
calibration states. This process is much the same as the calibration method in
[8], it differs only in that the measurement outcomes are sent back to the
sender for state reconstruction instead being computed locally.

The error rates as a cost function are calculated by comparing the classical
input to the measured output. For an input vector $in$ of classical messages
and output $out$, the error rate is computed as, $err(in,out)\coloneqq
1-\frac{1}{n}\sum_{i=1}^{n}\delta_{i}(in,out)$, where $\delta_{i}(a,b)=1$ if
$a_{i}=b_{i}$ and $0$ otherwise and $n=|in|=|out|$.

To numerically analyze Protocol 1, we use a collection of simulated
experiments developed using the quantum computing framework PennyLane [11]. We
configure our simulations as follows. We begin by generating a parameterized
quantum circuit in PennyLane that performs the encoding step, the quantum
channel, the decoding step, and the measurements as in Fig. 4. For updating
the encoder and decoder parameters, after each iteration, updated values are
passed to the circuit. For the update function, we use an optimizer to
minimize the cost function. The encoder and decoder parameters are then
updated during protocol execution according to the optimizer outputs, and in
the next iteration, noise compensation values are used. For noise parameters,
depending on the length of the channel, we provide the circuit a set of
parameters for the rotational and probabilistic bit- or phase-flip noise
values. The parameters of the protocol and the set of calibration states are
chosen and then simulation can run.

##  IV Protocol Performance Benchmarks

We benchmark the performance of Protocol 1 for various communication scenarios
under various noise parameters. To simplify the simulation model, we assume
that the sender simply prepares quantum sources but does not modify their
encoding during the protocol. In our notation, this implies $E_{\theta^{i}}$
is the identity map for all iterations $i$. Indeed, with tuning ability, a
sender can pre-calibrate their device locally, ensuring the states they intend
to transmit are calibrated to their standard, and so generality is not lost
with this assumption.

###  IV-A Random Single-Qubit States

As an initial test of the protocol, we analyze the case where the sender
randomly selects $n=5$ single-qubit states for transmission. During
transmission, the receiver measures the incoming states in one of the three
Pauli bases such that quantum state tomography can be performed at the
sender’s side. For a cost function, the sender computes an average state
fidelity using the measurements from the receiver to compare against the
originally chosen states.

For this test of the protocol, we fix the length of the fiber to 50 km and
determine how many iterations of the protocol are needed for approximate
convergence to determine a good parameter regime for a more in-depth study of
the protocol that we conduct in the later sections. The number of states
transmitted per iteration here is $N=15,000$. For this analysis, we use a
simple rotational noise model which rotates qubits as defined above with
$\mu=0.05$ dB/km. Bit- and phase-flip noise is also set to
$\mu_{1}=\mu_{2}=0.05$ dB/km. We then deterministically apply a Bloch sphere
rotation $p\cdot\mathbf{\theta}_{R}$ to each qubit that travels through the
channel. For flip errors, they are applied randomly with their respective
probability. For decoding, we apply a counter rotational correction before a
measurement is made. In Fig. 1, we can see that with approximately 100
iterations of the protocol, the protocol has learned how to overcome the
rotational noise effects in the channel. Based on these results, for the
remainder of this work, we use $I_{\max}=250$ iterations for our simulations
unless otherwise stated.

Figure 1: The average infidelity vs. number of protocol iterations for
calibration using $n=5$ randomly selected states over 50 km fiber under
deterministic rotational noise.

###  IV-B BB84 States

The BB84 protocol uses states $\\{\ket{0},\ket{1},\ket{+},\ket{-}\\}$ and
transmits them with equal probability. A naive calibration process is to tell
the receiver the basis in which the state is prepared and repeatedly transmit
the state. The receiver can then tune the decoder according to a quantum bit
error rate and this can be repeated for all states. The receiver then knows
exactly which states are the protocol states and what the cost function is,
and with this information, the receiver (or an eavesdropper imitating the
receiver), can potentially predict the protocol the sender intends to perform.
Using Protocol 1 instead, the sender need not disclose the calibration states
nor the cost function, thereby masking the protocol parameters from the
receiver, preventing the receiver from learning which protocol the sender
intends to perform.

The QKD results after performing Protocol 1 for a channel with only rotational
noise are plotted in Fig. 2 (a). We can see that as the fiber distance
increases, the QBER will tend to 0.6 in the pre-calibrated stage (blue). Using
Protocol 1 to calibrate the system, the trained decoder reduces the BB84 QBER
in the system within 250 iterations of training with 1,000 (lossless)
transmissions per iteration. In Fig. 2 (b) is the performance over a channel
that has both rotational noise and probabilistic bit- and phase-flip error.
What we can see is that until roughly 60 km, calibration makes no difference
to accommodate the noise in the channel, since the probability of the error
applying is smaller than 0.5. Once the flipping probability exceeds 0.5, the
QBER can be minimized by applying the inverse rotations deterministically,
which is observed in the bifurcation of the trends at around 60 km.

Figure 2: Calibration result for mitigating (a) rotational noise, and (b)
rotation, bit- and phase-flip noise optimized with
$\epsilon_{\text{th}}=10^{-7}$. Plotted is the average Quantum Bit Error Rate
(QBER) for 350 trials of performing the BB84 protocol with 1,000 qubits sent
per trial.

We test how many transmissions are required to minimize the QBER within 20
iterations. The number of transmissions build up accuracy in the statistics
regarding the noise properties of the channel. The results appear in Fig. 3.
In the case of a channel with both rotation noise and bit- and phase-flip
error with a fixed channel length of $L=120$ km, at approximately $N=2,000$,
further transmissions do not improve the calibration significantly. Here, we
do not consider loss in the system, so in this case we can consider $N$ to be
the number of single qubit detections. In a channel with attenuation
$\eta\in(0,1]$, we can expect the number of transmissions to increase by a
factor of $\eta^{-1}$.

Figure 3: The calibration accuracy using a fixed $I_{\max}=20$ iterations over
a noisy channel with both bit- and phase-flip and rotational error.

###  IV-C Entanglement-Swapping

In quantum networks, an important protocol for extending the range of quantum
state transmission is entanglement swapping [12]. To perform entanglement
swapping, three (or more) parties, a sender, a receiver, and a midpoint
receiver need to coordinate to perform the protocol. One approach to this
protocol is that the sender and receiver each locally generate an EPR pair and
send half of the EPR pair to the midpoint node. The midpoint node performs a
Bell State Measurement (BSM) and at the end of the protocol, the sender and
receiver share an EPR pair. This configuration is also seen in QKD protocols
such as Measurement Device Independent QKD (MDI-QKD) [13], making this use-
case a further practical scenario.

Figure 4: Entanglement swap circuit. The $BS$ gate prepares one of four Bell
states. $N(p)$ gates apply noise.

To implement the protocol, we use the following approach based on the circuit
in Fig. 4. We assume that the both the sender and receiver transmit half of a
Bell pair to a central node, and the sender selects a Bell state at random.
The central node performs a correction to overcome channel noise before
measuring, but is unaware of which Bell state is being distributed. The
central node performs a Bell state measurement on the incoming qubits. To
teleport the quantum state, two classical bits used for correction are sent to
one party. After this, the sender and receiver parties measure their remaining
half and can coordinate with each other to generate a cost function, using the
error cost defined previously for example. where the input is Bell states that
were transmitted, and the output is the resultant measured Bell pair.

We perform Protocol 1 over two error models. In Fig. 5 (a), we analyze the
effect of arbitrary rotation. In the Fig. 5 (b), we include bit- and phase-
flip error. With deterministic errors, as are the arbitrary rotations, it is
possible to correct for error. For probabilistic error, we see that until the
probability is high enough for error, no correction using rotational
correction can be effectively applied. Once it is the case, we see the
calibrated error rates diverge from the uncorrected error rates, as described
in the QKD example. Here again we do not simulate loss, and a scaling factor
should be applied to accommodate loss as explained in Section IV-B for each
channel and for detector flaws.

Figure 5: Calibration performance for entanglement swapping under (a)
rotational noise and (b) rotational and bit-flip noise optimized with a cost-
change tolerance of $\epsilon_{\text{th}}=10^{-7}$. Plotted is the average
error rate for transmitting performing Bell state transmissions.

###  IV-D Multipartite Entanglement Distribution

In this section, we simulate a communication setting for transmitting
multipartite entanglement, specifically GHZ and W states, over channels with
rotational noise. GHZ states and W states are defined as,
$\ket{GHZ}\coloneqq\frac{1}{\sqrt{2}}(\ket{00...0}+\ket{11...1})$ and
$\ket{W}\coloneqq\frac{1}{\sqrt{n}}(\ket{100...0}+\ket{010...0}+...+\ket{00...01})$.
GHZ states and W states are important for protocols like quantum secret
sharing [2] and others. Here we analyze a setting where a sender prepares $n$
qubit GHZ or W states and transmits them over a noisy channel.

In Fig. 6, we show the calibration effects for correcting the channel noise
for both a varying number of qubits in the entangled quantum systems and
length of the channel. With a growing number of qubits in the system, there is
a higher rate of infidelity, but as with the previous results, we see that the
protocol can robustly correct deterministic errors in the channel. Here we
have used the state infidelity as the cost, approximating the output state
density matrix using multi-qubit state tomography in Eq. (1). For better
performance of the simulation, we use $\epsilon_{\text{th}}=10^{-5}$ and
20,000 channel transmissions per protocol iteration. Here we do not simulate
probabilistic noise, as simulation performance rapidly deteriorates with
larger entangled states, but given the noise models are the same as the
previous cases, we suspect a similar trend would emerge, where until roughly
$L=60$ km no corrections can be made.

Figure 6: Calibration performance for transmitting $n=2,3,4,5$ qubit GHZ and W
states over a channel with rotational noise. Here we optimize with a cost-
change tolerance of $\epsilon_{\text{th}}=10^{-5}$, and use 20,000
transmissions per iteration. The calibrated trend is for all $n$ qubit cases.

##  V Conclusion

In conclusion, we show that masking the calibration states and cost while
still allowing the calibration to take place is possible. To do this, our
Blind Calibration protocol is used, preventing any other party from performing
state tomography to learn the calibration state set. By performing the error
cost calculation at the sender’s side, there is further masking of the
calibration protocol, minimizing any leakage of the calibration protocol. We
show that, under various communication settings, the protocol works well even
under noisy channel conditions. In our analysis, we used simple decoding
strategies to overcome noise, but more complex strategies can directly fit
with the protocol framework. The protocol we propose is general and
accommodates any quantum communication setting where calibration is required
making it practical and timely during the growing interest in quantum
communication and quantum networks.

## References

  * [1] S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani _et al._ , “Advances in quantum cryptography,” _Advances in optics and photonics_ , vol. 12, no. 4, pp. 1012–1236, 2020.
  * [2] M. Hillery, V. Bužek, and A. Berthiaume, “Quantum secret sharing,” _Physical Review A_ , vol. 59, no. 3, p. 1829, 1999.
  * [3] J. F. Fitzsimons, “Private quantum computation: an introduction to blind quantum computing and related protocols,” _npj Quantum Information_ , vol. 3, no. 1, pp. 1–11, 2017.
  * [4] W. F. Yeung and A. R. Johnston, “Effect of temperature on optical fiber transmission,” _Applied optics_ , vol. 17, no. 23, pp. 3703–3705, 1978.
  * [5] N. Jain, C. Wittmann, L. Lydersen, C. Wiechers, D. Elser, C. Marquardt, V. Makarov, and G. Leuchs, “Device calibration impacts security of quantum key distribution,” _Physical Review Letters_ , vol. 107, no. 11, p. 110501, 2011.
  * [6] Y.-Y. Fei, X.-D. Meng, M. Gao, H. Wang, and Z. Ma, “Quantum man-in-the-middle attack on the calibration process of quantum key distribution,” _Scientific reports_ , vol. 8, no. 1, pp. 1–10, 2018.
  * [7] D. Elkouss, J. Martinez-Mateo, and V. Martin, “Information reconciliation for quantum key distribution,” _arXiv preprint arXiv:1007.1616_ , 2010.
  * [8] G. Xavier, N. Walenta, G. V. de Faria, G. Temporão, N. Gisin, H. Zbinden, and J.-P. Von Der Weid, “Experimental polarization encoded quantum key distribution over optical fibres with real-time continuous birefringence compensation,” _New Journal of Physics_ , vol. 11, no. 4, p. 045015, 2009\.
  * [9] L. Balzano and R. Nowak, “Blind calibration of sensor networks,” in _Proceedings of the 6th international conference on Information processing in sensor networks_ , 2007, pp. 79–88.
  * [10] M. A. Nielsen and I. L. Chuang, _Quantum Computation and Quantum Information: 10th Anniversary Edition_. Cambridge University Press, 2010.
  * [11] V. Bergholm, J. Izaac, M. Schuld, C. Gogolin, M. S. Alam, S. Ahmed, J. M. Arrazola, C. Blank, A. Delgado, S. Jahangiri _et al._ , “Pennylane: Automatic differentiation of hybrid quantum-classical computations,” _arXiv preprint arXiv:1811.04968_ , 2018.
  * [12] M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, “Event-ready-detectors" bell experiment via entanglement swapping,” _Physical Review Letters_ , vol. 71, no. 26, 1993.
  * [13] H.-K. Lo, M. Curty, and B. Qi, “Measurement-device-independent quantum key distribution,” _Physical review letters_ , vol. 108, no. 13, p. 130503, 2012\.