# Quantum Electrodynamical Metamaterials

Josephine Yu1∗, Jamison Sloan3∗, Nicholas Rivera2, and Marin Soljačić2 1
Department of Applied Physics, Stanford University, Stanford, CA 94305, USA
2 Department of Physics, MIT, Cambridge, MA 02139, USA  3 Research Laboratory
of Electronics, MIT, Cambridge, MA 02139, USA
∗These authors contributed equally.

###### Abstract

Recent experiments have revealed ultrastrong coupling between light and matter
as a promising avenue for modifying material properties, such as electrical
transport, chemical reaction rates, and even superconductivity. Here, we
explore (ultra)strong coupling as a means for manipulating the optical
response of metamaterials based on ensembles of constituent units individually
in the ultrastrong coupling regime. We develop a framework based on linear
response for quantum electrodynamical systems to study how light-matter
coupling affects the optical response. We begin by applying this framework to
find the optical response of a two-level emitter coupled to a single cavity
mode, which could be seen as a “meta-atom” of a metamaterial built from
repeated units of this system. We find optical behaviors ranging from that of
a simple two-level system (Lorentz-oscillator) to effectively transparent, as
the coupling goes from the weak to deep strong coupling regimes. We explore a
one-dimensional chain of these meta-atoms, demonstrating the tunability of its
optical behavior. Our scheme may ultimately provide a framework for designing
new metamaterials with low-loss, highly-confined modes, as well as tunable
(single-photon) nonlinearities.

The interactions between light and matter form the basis for describing, and
ultimately controlling, the optical response of materials. The optical
response of a material is governed by its microscopic dynamics at a quantum
mechanical level, often understood through the energy eigenstates of
electrons, phonons, or other excitations of the system. Typically,
descriptions of a material’s optical properties rely on a quantum mechanical
description of the constituent matter of the system, but only a classical
description of the electromagnetic field. This is usually sufficient, because
in most cases, the constituent particles of a material are only weakly coupled
to the surrounding electromagnetic field modes.

However, recent experiments in strong and ultrastrong coupling have shown that
the the quantized nature of the electromagnetic field can strongly influence
material properties. In fact, from its first experimental observation in 2009
Anappara et al. (2009), ultrastrong coupling has revealed itself as a
promising avenue for controlling various material properties Ashhab and Nori
(2010); Galego et al. (2015); Orgiu et al. (2015); Herrera and Spano (2016);
Kockum et al. (2017a), including chemical reaction rates Hutchison et al.
(2012), superconducting temperatures Thomas et al. (2019), and
magnetotransport in 2D electron gases Paravicini-Bagliani et al. (2019).
Moreover, ultrastrong coupling is a ubiquitous phenomenon, observed in a
variety of experimental platforms ranging from those involving collective
excitations of emitters Gambino et al. (2014); Todisco et al. (2018); Mazzeo
et al. (2014); Günter et al. (2009); Delteil et al. (2012); Auer and Burkard
(2012), to systems such as superconducting circuits Niemczyk et al. (2010);
Forn-Díaz et al. (2010) which demonstrate ultrastrong coupling between a
single two-level emitter (qubit) and light. Recent experiments using
superconducting circuits have also demonstrated ultrastrong coupling between
artificial atoms and a continuum of electromagnetic modes Forn-Díaz et al.
(2017); Puertas Martínez et al. (2019); Magazzù et al. (2018), further
widening the scope of possibilities for ultrastrong coupling.

Despite the promise of these recent developments, strong and ultrastrong
coupling have not yet been considered as a platform on which to design new
macroscopic-scale optical metamaterials. This is a particularly intriguing
prospect, since control over optical materials may present opportunities for
optical excitations with low loss and high confinement, tunable
nonlinearities, and at frequency ranges which are naturally difficult to
achieve. Here, we propose using ultrastrong coupling between light and matter
to create novel optical materials with tunable properties. We show that even a
two-level system coupled to a single electromagnetic mode, which we refer to
as a “USC meta-atom,” or just “meta-atom,” exhibits a rich variety of optical
behavior depending on the coupling strength between the emitter and cavity,
ranging from the well-known behavior of a two-level system (Lorentz
oscillator) to effectively transparent. We propose to design new metamaterials
based on meta-atoms and show that their macroscopic optical response can be
reshaped by tuning the coupling strength between the light and matter in the
constituent meta-atoms of the metamaterial; by extension, we also control the
new electromagnetic modes which propagate through such a metamaterial. We
discuss how the quantum electrodynamical optical materials we propose could be
realistically implemented via systems such as superconducting qubits Niemczyk
et al. (2010) or molecules in picocavities Chikkaraddy et al. (2016), as these
systems have achieved the requisite USC strengths and could be arranged into a
metamaterial with straightforward geometry, such as a one- or two-dimensional
array.

Figure 1: (a) Schematic of the proposed “strongly coupled metamaterial” and
interacting field described by $E(t)$ with driving frequency $\omega$. The
meta-atoms comprising the metamaterial can interact via dipole or cavity
coupling, or via a more general interaction involving both the dipole and
cavity. An energy level diagram for the meta-atom shows an example of the
allowed transitions at a particular coupling strength. (b) A plot displaying
how the first few energy levels evolve as a function of $g$. As the coupling
increases, the two lowest energy levels converge, bringing the lowest resonant
transition energy to zero. (c) Schematic of the proposed physical realization
of a 1D material composed of meta-atoms. Each meta-atom consists of a flux
qubit inductively coupled to an LC resonator with an effective coupling $g$.
Neighboring units are coupled capacitively through their charges, generating
an inter-unit coupling $g_{c}\sigma_{x}^{i}\sigma_{x}^{i+1}$ between the $i$th
and $i+1$th qubits.

We now elaborate on our proposed model for “strongly coupled metamaterials”
which are composed of individual strongly coupled “meta-atoms” as their
constituent components. Each meta-atom is comprised of a two-level system with
frequency $\omega_{a}$ coupled to a single electromagnetic mode of frequency
$\omega_{c}$, represented as a resonant cavity. This model of a meta-atom
serves as a versatile proxy for physical systems in which ultrastrong coupling
has been achieved. We also note that while we focus on the case of a two level
system coupled to a single mode, the concept of a meta-atom is quite general,
and encompasses possibilities such as multi-level emitters (molecules,
artificial atoms, spins, etc.) and multi-mode electromagnetic excitations
(fiber modes, photonic crystal modes, surface polaritons, etc.). Once the
platform has been chosen, a metamaterial can be conceived as a sufficiently
dense arrangement of many meta-atoms, shown in Fig. 1a.

To provide a thorough analysis of an example of the proposed material, we
consider one possible realization of the meta-atoms: flux qubits, functioning
as the two-level systems, coupled to LC resonators, functioning as the
cavities. We will later focus on a chain of such units and explore the modes
supported in such a one-dimensional material. In our example, the meta-atoms
interact through their dipoles (Fig. 1c), but interactions between meta-atoms
can in general involve both the cavity and two-level system.

We begin by describing the optical response of an individual meta-atom. We
assume that each meta-atom is a two-level system coupled to a single cavity
mode, and is thus described by the Rabi Hamiltonian

$H_{\text{Rabi}}=\frac{\hbar\omega_{a}}{2}\sigma_{z}+\hbar\omega_{c}\hat{a}_{c}^{\dagger}\hat{a}_{c}+\hbar
g\sigma_{x}\left(\hat{a}_{c}^{\dagger}+\hat{a}_{c}\right)+\frac{\hbar
g^{2}}{\omega_{c}},$ (1)

where $\omega_{a}$ is the transition frequency of the two-level system,
$\omega_{c}$ is the cavity frequency, $\hat{a}^{(\dagger)}$ is the
annihilation (creation) operator for the cavity mode, $g$ is the light-matter
coupling strength, and $\sigma_{x}$ and $\sigma_{z}$ are Pauli matrices. We
consider here the case that the emitter and cavity are on resonance at a
frequency $\omega_{0}$ such that $\omega_{a}=\omega_{c}=\omega_{0}$. As the
emitter-cavity coupling strength $g$ changes, so do the energy levels and
allowed transitions in the meta-atom, as highlighted in Fig. 1b. The nature of
the interactions between light and matter in a system is characterized by
comparing $g$ to the losses or transition energies of the system Frisk Kockum
et al. (2019). When the coupling strength is lower than the losses in the
system, the system is in the weak coupling (WC) regime. Here, interactions may
be treated perturbatively, resulting in relatively well-studied phenomena in
which the eigenstates of the emitter are not drastically impacted by the
coupling to electromagnetic modes, such as Purcell enhancement of spontaneous
emission Purcell (1995). In contrast, the strong-coupling (SC) regime is
defined by a coupling strength $g$ which exceeds the losses in the system.
Physically, the matter and photon form a single entity whose energy levels
differ substantially from those of the individual components, giving rise to
phenomena such as Rabi oscillations. As the coupling increases to the order of
the transition energies, the system enters the ultrastrong coupling (USC)
regime, in which even properties of the quantum-mechanical ground state are
strongly modified Ashhab and Nori (2010); Hines et al. (2004).

To compute the optical response of the meta-atom, we introduce a classical
probe-field of frequency $\omega$ through a time-dependent perturbation
$E(t)$. We assume that this probe-field couples to the dipole moment $d$ of
the two-level system in the meta-atom. While more general driving mechanisms
exist, the direct drive of the two-level system is readily realizable in the
flux qubit systems that we use as an example here. Then, taking the
perturbation to be sinusoidal, the full Hamiltonian takes the form

$\displaystyle H=H_{\text{Rabi}}+dE_{0}\sigma_{x}\left(e^{i\omega
t}+e^{-i\omega t}\right).$ (2)

For a system like the proposed flux qubit (two-level system) coupled to an LC
resonator (cavity), the natural observable quantities are those involving the
transitions between eigenstates of the Rabi Hamiltonian (Rabi basis states).
Since we aim to characterize the response of the meta-atom, rather than the
emitter alone, we should analyze the expectation values of the “effective”
dipole operators corresponding to each transition between Rabi basis states
$m$ and $n$: $\tau_{x}^{nm}\equiv\ket{n}\bra{m}+\ket{m}\bra{n}$.

We now consider the case that the driving frequency $\omega$ in Eq. 2 is close
to a transition frequency $\omega_{n0}$ of the meta-atom between the $n$-th
excited state of the Rabi Hamiltonian and the ground state. Then, it is
possible to relate the emitter’s dipole operator $\sigma_{x}$ and the
effective dipole operator for the meta-atom $\tau_{x}^{n0}$. By expanding the
emitter’s dipole operator $\sigma_{x}$ explicitly in the Rabi basis, we find
$\sigma_{x}=\sum_{m,n}\sigma_{x}^{mn}\ket{m}\bra{n}$. When
$\omega\approx\omega_{n0}$, the system is well-approximated by a two-level
system with levels $\ket{0}$ and $\ket{n}$. In this case, the terms
$\sigma_{x}^{n0}\ket{n}\bra{0}$ and $\sigma_{x}^{0n}\ket{0}\bra{n}$ contribute
most in the expansion of $\sigma$, so we can approximate the dipole operator
operator $\tau_{x}^{n0}$ in terms of the emitter’s effective dipole operator,

$\tau_{x}^{n0}=\frac{\sigma_{x}}{\sigma_{x}^{0n}},$ (3)

where $O^{mn}\equiv\innerproduct{m|O}{n}$ denotes matrix elements between
eigenstates of the coupled system. The full meta-atom is then approximately a
collection of two-level systems with transition frequencies which can be tuned
via the emitter-cavity coupling $g$.

Figure 2: (a) Atomic polarizability of the coupled atom-cavity system with
phenomenological loss $\gamma_{a}=0.01\omega_{0}$ (and omitting cavity loss
$\gamma_{c}$) as a function of driving frequency $\omega$ and coupling
strength $g$. Colored lines at $g=0$ $g=0.1\omega_{0}$, $g=\omega_{0}$, and
$g=1.5\omega_{0}$ highlight signature behaviors at various coupling strengths,
which correspond to the various regimes of light-matter interaction. (b)
Polarizability of a single meta-atom at various values of the coupling
strength $g$ highlighted in the line cuts of (a).

Though the Hamiltonian in Eq. 2 is valid for arbitrary drive strengths, we
focus on weak probe-fields so that we may treat the probe-field interaction
$E(t)$ as a small perturbation on the meta-atom Hamiltonian $H_{\text{Rabi}}$.
The optical response of the meta-atom is then well-captured by the first-order
correction of the effective dipole expectation value (Eq. 3) in response to
the applied external field. This correction is proportional to the atomic
polarizability $\alpha(\omega)$, a linear response function which relates the
applied electric field $E(\omega)$ to the change in dipole moment $\langle
d(\omega)\rangle=\alpha(\omega)E(\omega)$ in frequency space. Using time-
dependent perturbation theory to find the first-order correction to the dipole
expectation value, we find that the frequency-dependent polarizability
$\alpha(\omega)$ for a strongly coupled system initially in the ground state
can be approximated as the sum of responses from all excited levels:

$\displaystyle\begin{split}\alpha(\omega)&=\frac{d^{2}}{\hbar}\sum_{n}\alpha_{n0}(\omega),\\\
\alpha_{n0}(\omega)&=-I_{n0}\left[\frac{1}{\omega-\omega_{n0}-i\Gamma_{n}}-\frac{1}{\omega+\omega_{n0}-i\Gamma_{n}}\right].\end{split}$
(4)

Here, $\omega_{n0}$ is the transition frequency in the coupled system,
$I_{n0}$ is 0 if $\innerproduct{n|\sigma_{x}}{0}=0$ and 1 otherwise, and
$\Gamma_{n}$ is the loss associated with the transition from the $n$-th
excited state to the ground state of the coupled system. This result from
perturbation theory has been corroborated by a numerical approach via master
equation evolution.

In the weak coupling regime, the linewidth $\Gamma_{n}$ is given simply by the
atomic linewidth of the two-level system transition. However, the onset of the
ultrastrong coupling regime invalidates this assumption. Since the ground
state of an ultrastrongly coupled system contains virtual excitations, naively
relaxing the system according to the losses and transitions of the emitter
would result in unphysical photon generation in the cavity. In Eq. 4, we
account for the effects of (ultra)strong coupling and define the loss
associated with each transition according to Beaudoin et al. (2011) as
$\Gamma_{n}\equiv|\innerproduct{n|\sigma_{x}}{0}|^{2}\gamma_{a}+|\innerproduct{n|(\hat{a}+\hat{a}^{\dagger})}{0}|^{2}\gamma_{c}$,
where $\gamma_{a}$ and $\gamma_{c}$ are the relaxation rates of the emitter
and cavity in the absence of coupling, respectively. We assume for
concreteness that the atomic relaxation rate dominates the overall loss and
therefore neglect the cavity relaxation rate. Qualitatively, the inclusion of
the cavity loss would not change our results. Additionally, we assume that the
atomic relaxation rate is substantially smaller than the differences between
transition frequencies in the system, ensuring that the transition frequencies
of the coupled system remain distinct. This assumption ensures that the
relaxation due to losses in the emitter and the cavity may be treated
independently (see Beaudoin et al. (2011) for details). We note that for the
atomic relaxation rate of $\gamma_{a}=0.02\omega_{0}$ used throughout our
calculations, the previously stated assumptions are valid up until a coupling
strength of $g=1.5\omega_{0}$, at which the lowest transition frequency
approaches the value of the $\gamma_{a}$: $\omega_{10}\sim 0.01\omega_{0}$.

The form of Eq. 4 shows how the optical response of a single meta-atom –
namely, the location, height, and width of each peak – is strongly controlled
by the coupling. Both the eigenstates (which contribute to the matrix
elements) and the eigen-energies (which contribute to the transition
frequencies, as seen in Fig. 1b) evolve as a function of the coupling constant
$g$. In Fig. 2a, the changes in transition energies are reflected in the
evolving location of the peaks in the atomic polarizability, as well as in the
introduction of new peaks at sufficiently high coupling strengths. The line
cuts at various coupling strengths (Fig. 2b) highlight features in the meta-
atom system representative of the corresponding regimes of light-matter
interaction. In the WC regime ($g\ll\gamma_{a}$), the system unsurprisingly
behaves as a single Lorentz oscillator with resonant frequency $\omega_{0}$,
since the atomic energy structure is completely unaffected by the presence of
the vacuum electromagnetic field. As the coupling strength increases
($g=0.1\omega_{0}$), the polarizability exhibits a signature of the strong
coupling regime: the single peak splits into two, indicative of Rabi
splitting. Under strong coupling ($g\sim\gamma_{a}$), the system, which
previously had just one resonant transition energy, now has an expanded set of
energy levels, causing the previously degenerate excited states to split into
distinct upper and lower polariton states, as can be described by the Jaynes-
Cummings (JC) Hamiltonian Jaynes and Cummings (1963). Once the system enters
the ultra-strong coupling regime ($g=\omega_{0}$), the JC model breaks down,
and new resonant frequencies are introduced as a result of the inclusion of
counter-rotating terms in the Rabi Hamiltonian (Eq. 2). This results in the
appearance of multiple new peaks between $g=0.1\omega_{0}$ and $g=\omega_{0}$
in Fig. 2b. The introduction of new resonant frequencies originates physically
from the mixing between light and matter states, which expands the number of
possible transitions (and therefore transition frequencies) as the coupling
increases. The line cut at $g=\omega_{0}$ also highlights a region in which
the lowest-frequency peak in the response approaches $\omega=0$. This occurs
as the ground and first excited states of the coupled system converge (see
Fig. 1b), thus causing the corresponding transition energy to approach zero.

Beyond dictating the number and location of resonances (and therefore peaks)
in the system, the coupling strength $g$ also plays a key role in determining
the losses associated with each transition, manifested through the widths and
heights of the corresponding peaks. The evolving eigenstates dictate the
matrix elements of $\sigma_{x}$ by which the loss parameter $\Gamma_{n}$ is
scaled; as the coupling $g$ increases, these matrix elements approach zero in
all transitions except for the lowest, thus bringing the scaled loss
$\Gamma_{n}$ to zero for $n\neq 1$ and thereby narrowing the corresponding
peaks of the transitions from the $n-$th excited level to the ground state. In
Fig. 2b, the narrowing of peaks can be seen especially clearly in regions of
higher $g$ in the colorplots. Indeed, the line cut at $g=1.5\omega_{0}$ in
Fig. 2b highlights how the polarizability resembles that of a transparent
emitter which does not interact with the external probe at nonzero frequency
and sufficiently high coupling strengths.

So far, we have shown that even the simplest possible strongly coupled light-
matter system — a two-level system coupled to a single mode — exhibits a rich
variety of optical behavior depending on the coupling strength. The optical
response of a material is often dictated by the quantum mechanical properties
of the constituent particles, so the tunable and varied behaviors of the meta-
atom make it promising foundation for designing similarly behaved bulk
materials. We now focus on one example of the individual meta-atom properties
translating to a material. Here, we show that even a simple 1D chain of meta-
atoms inherits their strong- and ultrastrong-coupling behaviors to give rise
to novel modes.

To be concrete, we consider a 1D chain of the emitter-resonator units realized
for example through flux qubits inductively coupled to LC resonators (Fig.
1c). Units are capacitively coupled to only their nearest neighbors through
charge interactions, which may be expressed as
$g_{c}\sigma_{x}^{i}\sigma_{x}^{j}$ between the qubits at neighboring sites
$i$ and $j$ Krantz et al. (2019). Here, the coupling constant $g_{c}$ is
determined by the circuit elements (i.e., the capacitance within each qubit
and the coupling capacitance). Since the only coupling is facilitated through
macroscopic wires, it is only the charges across different meta-atom units
that interact with each other. We assume that the capacitances involved are
frequency-independent in the range of interest, and we set the coupling
constant to take the value $g_{c}=0.08\omega_{0}$ Ozfidan et al. (2020); Yan
et al. (2016); Krantz et al. (2019). Since the coupling between meta-atoms is
weak compared to all other energy scales, we expect that the effect of
coupling is perturbative and that the supported modes are close in frequency
to the resonant frequencies of the individual units. Near each resonant
frequency of the meta-atom, we may express the interaction in terms of the
transition operators in the Rabi basis (Eq. 3) as
$g_{c}|\sigma^{0n}|^{2}\tau_{x}^{i}\tau_{x}^{j}$. One can interpret this as a
dipole $\tau_{x}^{j}$ interacting with a field
$g_{c}|\sigma^{0n}|^{2}\tau_{x}^{i}$ generated by its neighbor.

Figure 3: (a)-(d) Dispersion relations for a one-dimensional chain of meta-
atoms for various coupling values $g$, illustrated through a colormap of the
spectral function $s(k,\omega)$. Overlaid is Re$[\omega(k)]$. (e) Quality
factor for the lowest-frequency mode as a function of $ka$, for various
coupling values $g$.

Each unit has an effective polarizability as calculated in Eq. 4, so we model
the geometry as a line of polarizable units and calculate the supported modes
by solving the equation of motion for the self-consistent local dipole moment
$\tau_{x}$ of each unit. In a spatially periodic system,
$\tau_{x}^{i}(t)=e^{ikx}\tilde{\tau}_{x}^{i}(t)$, where $\tilde{\tau}_{x}$ is
the envelope of the periodic function and has no spatial dependence. Without
any inter-unit coupling, the effective dipole moment oscillates at any of its
resonant frequencies, $\omega_{n0}$. The dipole moment is also affected by the
fields from its neighbors. In full, the equation of motion for the $i$-th
qubit’s dipole moment is:

$-\omega^{2}\tilde{\tau}_{x}^{i}=\omega_{n0}^{2}(\tilde{\tau}_{x}^{i}+\alpha_{n0}(\omega)g_{c}|\sigma^{0n}|^{2}(\tilde{\tau}_{x}^{i-1}e^{-ika}+\tilde{\tau}_{x}^{i+1}e^{ika})),$
(5)

where $a$ is the lattice spacing, and $k$ is the wavevector of the propagating
mode. Since $\tilde{\tau}_{x}^{i}$ has no spatial dependence, Eq. 5 may be
simplified and solved to find the supported modes of frequency

$\omega=\omega_{n0}\sqrt{1+\alpha_{n0}(\omega)|\sigma^{0n}|^{2}g_{c}\cos ka}.$
(6)

We now describe the properties of the modes described by Eq. 6. We solve for
the form of the dispersion relation $\omega(k)$ via both an analytical
approximation and numerical methods. Analytically, we approximate
$\alpha_{n0}(\omega)$ by the term $1/(\omega-\omega_{n0}-i\Gamma_{n})$, which
dominates near resonance. The resulting cubic equation can be solved
numerically, yielding only one solution which respects passivity (i.e., it
decays over time). We focus here on that solution.

Fig. 3b-e shows the exact dispersion relations (found numerically from Eq. 6)
for the allowed modes of the chain for various coupling strengths $g$ between
the emitter and resonator of each unit. Like in the case of a single unit, the
allowed modes at a given emitter-cavity coupling value $g$ are dictated by the
resonant frequencies of the system. As expected, the chain supports a mode
only near the transition frequency of the emitter if there is no emitter-
resonator coupling ($g=0$). Small values of $g$ result in Rabi splitting
($g=0.1\omega_{0}$) of the single mode into two, mirroring the behavior of the
energies of the individual meta-atoms. The weak coupling behavior of this
system is qualitatively similar to that which is observed in photonic crystals
Chong et al. (2007). Meanwhile, larger coupling values reveal evidence of
light-matter decoupling ($g=1.5\omega_{0})$, corresponding to the lowest
transition frequency of the individual units being driven to zero with
increasing coupling. The behaviors of the chain resemble those of the
constituent meta-atoms, inheriting the signatures of strong and ultrastrong
coupling behavior.

Near $|ka|=\pi/2$, the change in sign of $\cos ka$ creates a cusp in the real
part of the frequency. This is most evident from the cubic approximation used
to solve Eq. 6, which can be further approximated as a quadratic by neglecting
the unphysical solution near $-\omega_{n0}$. The resulting quadratic equation
is solved by

$\omega(k)_{\pm}=\omega_{n0}-\frac{i\Gamma_{n}}{2}\pm
i\sqrt{2\omega_{n0}g_{c}|\sigma^{n0}|^{2}\cos ka+\Gamma_{n}^{2}}.$ (7)

For a passive system such as this one, the only physical solutions are those
which decay in time, and in this case only $\omega_{-}(k)$ satisfies this
condition. Since the coefficient of $\cos ka$ is much larger than
$\Gamma_{n}^{2}$, the behavior of the mode on either side of $|ka|=\pi/2$ is
dictated by the sign of $\cos ka$. For $|ka|<\pi/2$,
Re$[\omega_{-}(k)]\approx\omega_{n0}$ while for $|ka|>\pi/2$,
Re$[\omega_{-}(k)]\approx\omega_{n0}+\sqrt{2\omega_{n0}g_{c}|\sigma^{n0}|^{2}\cos
ka}$. This sharp change in behavior is present across all coupling values and
is a consequence of the geometry, coupling mechanism, and the constituent
meta-atoms’ optical properties.

The modes supported by the chain structure may be further understood by
examining their associated losses. For this purpose, we define the spectral
function $s(k,\omega)$ which describes the relative modal loss,

$s(k,\omega)\equiv\frac{\omega^{\prime\prime}(k)/2}{\omega^{\prime\prime}(k)^{2}/4+(\omega-\omega^{\prime}(k))^{2}},$
(8)

where the frequency is decomposed into real and imaginary parts as
$\omega(k)=\omega^{\prime}(k)+i\omega^{\prime\prime}(k)$. A colormap of
$s(k,\omega)$ is shown in Fig. 3b-e. For all values of the coupling, the loss
associated with modes for which $|ka|>\pi/2$ is significantly less than the
loss for those with $|ka|<\pi/2$, as seen by the brighter halos around the
bands for $|ka|>\pi/2$. This can be understood by examining the solution
$\omega_{-}(k)$ in Eq. 7; for $|ka|>\pi/2$, there is a constant loss of
magnitude $\Gamma_{n}/2$ as opposed to a loss of
$\Gamma_{n}/2+\sqrt{2\omega_{n0}g_{c}|\sigma^{n0}|^{2}\cos ka}$ for
$|ka|<\pi/2$. For small values of $g$, $\omega_{n0}|\sigma^{n0}|^{2}$ is
largest (Fig, 3a), so the difference between the behaviors of the mode in
regions $|ka|>\pi/2$ and $|ka|<\pi/2$ is the clearest. As the emitter-cavity
coupling within each meta-atom increases, the loss associated with the mode in
the region $|ka|<\pi/2$ decreases. This is due to the lowest transition
frequency $\omega_{10}$ approaching zero, causing the loss in the region
$|ka|<\pi/2$ to be dramatically reduced (Fig. 3e).

The quality factor (Fig. 3f) is another metric by which to gauge the modal
losses as functions of both the coupling $g$ and the wavevector $k$. Examining
$Q\equiv\omega^{\prime}(k)/\omega^{\prime\prime}(k)$, we see that the quality
factor of the lowest-frequency mode is significantly higher in the region
$|ka|>\pi/2$ than in the region $|ka|<\pi/2$. This behavior is also seen in
Fig.3b-e, in which the losses for $|ka|>\pi/2$ are smaller than for
$|ka|<\pi/2$. As the coupling $g$ increases, the quality factor actually
decreases even though the $\omega^{\prime\prime}(k)$ is also decreasing; this
is due to $\omega^{\prime}(k)$ decreasing linearly with $\omega_{n0}$ to zero,
while $\omega^{\prime\prime}(k)$ decreases like $\sqrt{\omega_{n0}}$.
Additionally, the constant loss $\Gamma_{n}$ also increases, approaching
$\gamma_{a}$ as $g$ increases, since the matrix element
$|\sigma_{x}^{01}|^{2}$ approaches 1 with increasing $g$. The overall quality
factor then decreases with increasing coupling.

The properties of the modes sustained in this 1D chain of meta-atoms are
closely related to those of the constituent meta-atoms. By tuning the emitter-
cavity coupling within each meta-atom unit, one can manipulate the resonant
frequencies, losses, and band structures of the resulting material
excitations. As demonstrated through this example, the mode properties depend
strongly not only on the meta-atom responses but also on the geometry and
coupling mechanisms.

In summary, we have presented a framework to design novel optical materials
based on quantum electrodynamical strong coupling. This framework shows quite
clearly the important role played by the vacuum field in determining optical
properties outside of the weak coupling regime. We described the optical
response of the simplest possible light-matter system — a meta-atom composed
of a coupled cavity mode and two-level emitter — and found that it depends
strongly on the coupling between the emitter and cavity. Notably, we found
that by adjusting the coupling, we can change the optical behavior in a unit
of this type from that of a Lorentz oscillator at a given frequency to
effectively transparent. We were then able to create and control new
electromagnetic excitations in an example material built of meta-atoms which
inherits similar properties from its constituents. Specifically, we showed
that the modes propagating through a 1D geometry of the meta-atoms inherit
similar tunable properties from the consitutent meta-atoms, with modes ranging
from single-emitter type responses to effectively transparent. The interplay
between the geometry and the optical properties of the meta-atoms result in
unique behaviors for the modes supported by the 1D material, as seen in the
varied behavior of the modes across different wavenumbers $k$.

Though we focus here on a material built of single-emitters coupled to single
electromagnetic modes, our methods may also be extended to study the optical
behaviors of materials in which (ultra)strong coupling occurs between more
general collections of emitters and electromagnetic excitations. As a result,
our work may also be related to recent experimental observations such as
modified conductivity due to ultrastrong coupling Orgiu et al. (2015). Our
approach may also be used to study the optical properties of composite systems
in which USC has been proposed, such as a material formed by interfacing 2D
and 3D materials Kurman and Kaminer (2020).

Additionally, while our work focuses on linear optical response, our methods
may be extended to study nonlinear response in (ultra)strongly coupled
materials, which is recent topic of interest Kockum et al. (2017a); Garziano
et al. (2015); Kockum et al. (2017b). One potential approach is to consider
higher-order terms in perturbation theory in our calculation of the dipole
expectation, which would offer finer corrections to the calculations presented
here in the linear regime. This approach could be used to study effects such
as frequency conversion in strongly coupled materials. We can also generalize
our results to nonperturbative regimes by using stronger probe fields, thus
allowing the study of phenomena such as high harmonic generation in strongly
coupled materials. Moreover, while we have focused on classical light in
strongly coupled systems, one could also consider how these systems interact
with light at a single photon level, presenting opportunities for quantum
optics.

##  I Acknowledgements

JY was supported by the National Science Foundation Graduate Research
Fellowship under Grant No. DGE-1656518. This material is also based upon work
supported in part by the U. S. Army Research Office through the Institute for
Soldier Nanotechnologies at MIT, under Collaborative Agreement Number
W911NF-18-2-0048.

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# Active User Detection and Channel Estimation for Spatial-based Random Access
in Crowded Massive MIMO Systems via Blind Super-resolution

Abolghasem Afshar, Vahid Tabataba Vakili, Sajad Daei The authors are with the
school of Electrical Engineering, Iran University of Science and Technology

###### Abstract

This work presents a novel framework for random access (RA) in crowded
scenarios of massive multiple-input multiple-output (MIMO) systems. A huge
portion of the system resources is dedicated as orthogonal pilots for accurate
channel estimation which imposes a huge training overhead. This overhead can
be highly mitigated by exploiting intrinsic angular domain sparsity of massive
MIMO channels and the sporadic traffic of users, i.e., few number of users are
active to send or receive data in each coherence interval. Besides, the
continuous-valued angles of arrival (AoA) corresponding to each active user
are alongside each other forming a specific cluster. To exploit these features
in this work, we propose a blind clustering algorithm based on super-
resolution techniques that not only detects the spatial features of the active
users but also provides accurate channel estimation. Specifically, an off-grid
atomic norm minimization is proposed to obtain the AoAs and then a clustering-
based approach is employed to identify which AoAs correspond to which active
users. After active user detection, an alternating-based optimization approach
is performed to obtain the channels and transmitted data. Simulation results
demonstrate the effectiveness of our approach in AoA detection as well as data
recovery which indeed provides a high performance spatial-based RA in crowded
massive MIMO systems.

###### Index Terms:

Crowded massive MIMO, Random access, Super-resolution, Atomic norm, Semi-
definite programming, Convex optimization.

##  I Introduction

Nowadays, the number of wirelessly connected devices has been vastly increased
by the development of applications such as Internet of Things (IoT), social
networking and next generations of cellular communications including massive
machine-type communications (mMTC), enhanced mobile broadband communications
(eMBB) and ultra-reliable low-latency communications (URLLC). There are lots
of advantages with massive MIMO systems such as increasing the system
throughput and the energy efficiency [1, 2]. As such, massive multiple-input
multiple-output (MIMO) systems have received remarkable attention during the
past few years. However, all of these advantages are dependent on accurate
channel state information (CSI) in coherent transmission which is a highly
challenging task. Due to the reciprocity of channel estimation (CE), Time
Division Duplexing (TDD) mode is often preferred to Frequecny Division
Duplexing (FDD) in MIMO systems. However, it requires a portion of resources
as pilots or training signal in each Coherence Interval (CI) to estimate the
channels corresponding to active users. In this regard, Random Access (RA) to
pilots (RAP) is a promising solution for this pilot allocation which divides
into two categories: grant-based and grant-free. In grant-based schemes (see
[3, 4]), multiple active user equipment (UE)s first transmit dedicated
preambles selected from a pool of pilot sequences to access the Base Station
(BS). Contention resolution schemes are then required if multiple users select
the same pilots. Since large number of collisions occurs, the BS cannot
resolve all of the contentions and thus many users are not able to access the
BS. [4] proposes a strongest user collision resolution protocol to resolve
this issue in crowded scenarios (which neglects the users in the edge of the
cells) by using orthogonal pilots. [3] employs a deterministic Compressed
Sensing (CS)-based RA scheme. As the signaling overhead and access latency is
directly proportional to the number of users, conventional grant-based RA
fails to support massive connectivity. In contrast, in grant-free RAP
protocol, each active UE transmits pilots with embedded data without waiting
for permission by BS [1, 5]. By using orthogonal pilots in this case, the
number of active users that can access the BS is confined to the number of
orthogonal pilots which is severely limited due to the short channel coherence
time. Moreover, using non-orthogonal pilots in grant-free case complicates the
task of active user detection (AUD) due to the intra- and inter-cell
interference caused by correlated pilots [6, 7]. A large number of research
works consider coordinated grant-free schemes in which the BS knows the pilots
in advance (e.g. [6, 8, 9, 10]). As an example of this category, [6] proposes
a statistical Approximate Message Passing (AMP)-based algorithm for joint AUD
and CE. Their method exploits the sporadic traffic of active users and the
sparse nature of the channel but needs the full knowledge of the channels and
noise distributions and assumes the angle of arrivals (AoAs) to be on a
predefined domain of grids. Both of these assumptions are not generally
satisfied in practice. Another line of research works lying in the coordinated
grant-free category devotes to covariance-based activity identification which
formulates the problem as an maximum likelihood estimation (see e.g. [8, 10,
11, 12]). It also has been shown that covariance-based algorithms outperforms
AMP with the same length of the pilots [8]. Such kinds of schemes are based on
the accuracy of the sample covariance of the measurement matrix. In order to
have an accurate covariance, the number of required measurements has to be far
more than the degrees of freedom (the true number of unknowns) and imposes a
huge waste in bandwidth resources and cost. Overall, there are specific
disadvantages with the mentioned prior works: full knowledge of channel and
noise distribution, e.g. [6, 13]; the number of active users that can access
the BS is severely limited, e.g. [14, 6, 8]; A very large number of antennas
and measurements are required, e.g. [8, 12, 7]. However, there is a common
issue that the mentioned prior works are coordinated. This means that the BS
has to be first know some pilots in advance for AUD which increases access
latency, subsequently preventing to achieve a high spectral efficiency and
seems to be not practical since the BS does know which user lies in which
cell. It should be mentioned that [15] provides a statistical uncoordinated
data recovery and channel estimation (not necessarily designed for RA),
however, besides its high complexity and the issue regarding the discrete
nature of AoAs, it needs full knowledge of prior distributions of data and
channel in advance which seems to be impractical. To solve the mentioned
challenges, we propose an uncoordinated grant-free RAP deterministic scheme in
crowded scenarios of massive MIMO systems which leverages the intrinsic
features of massive MIMO channels (angular domain sparsity of massive MIMO
channels) as well as the sporadic traffic of users. The task of extracting
such features from a few number of measurements builds upon the well-known
framework of CS [16]. Precisely, CS suggests a framework for recovering
discrete-index parameters i.e. the unknown parameters are confined to be on a
predefined domain of grids. There is also a more recent framework called
continuous CS (or super-resolution) [17, 18, 19, 20] which assumes that the
unknown parameters can lie anywhere and are not confined to be on the
predefined grids. The aforementioned features are available in massive MIMO
systems with massive number of users. For example, the physical channel of
massive MIMO systems, employed in high frequencies (millimeter wave), have
continuous sparse structure i.e. signal is received out of few off-grid
(continuous) angles in BS antenna arrays and they can have any arbitrary
values [21]. This is due to the fact that signals with higher frequencies are
more likely to be blocked by obstructions and few multi-path components (MPCs)
contribute to the channel. Another feature is sporadic traffic of massive
users which means only few users want to send their data at the same time. It
has also been shown that the channels between users and BS exhibit a clustered
continuous sparsity pattern [6]. This implies that the AoAs corresponding to
each user are alongside each other, few number of clusters are active and the
AoAs have continuous-index values. For AUD and channel estimation, we design a
deterministic optimization framework to encourage the mentioned features which
does not need any distributions of channels and users’ data. Our framework has
three stages: first, we find the AoAs by solving an optimization problem, then
a clustering-based algorithm is proposed to detach the angles corresponding to
each active user and finally an alternative optimization algorithm is
developed to estimate complex amplitudes of the channels and data/primary
pilots transmitted by active users in a blind way. By our approach, the
limitation in the number of adopted simultaneous users in crowded scenarios of
massive MIMO systems would be resolved without any need for coordination
between BS and users in advance. The organization of the paper is as follows:
In Section II, the system model of massive MIMO is presented. Section III is
about our proposed blind super resolution method and provides an algorithm for
blind detection and CE. Lastly, Section IV provides some numerical experiments
to verify our proposed method. Lastly, the paper is concluded in Section V.

Notations: We use boldface lower-and upper-case letters for vectors and
matrices, respectively. The $i$-th element of a vector e.g. $\bm{x}$ and the
$(i,j)$ element of a matrix e.g. $\bm{X}$ are respectively shown by $x_{i}$
and $X_{(i,j)}$. For vector $\bm{x}\in\mathbb{C}^{n}$ and matrix
$\bm{X}\in\mathbb{C}^{n_{1}\times n_{2}}$, the $\ell_{2}$ norm and Frobenius
norm are defined respectively as
$\|\bm{x}\|_{2}:=({\sum_{i=1}^{n}|x(i)|^{2}})^{\tfrac{1}{2}}$
$\|\bm{X}\|_{F}:=\sqrt{\sum_{i=1}^{n_{1}}\sum_{j=1}^{n_{2}}|X(i,j)|^{2}}$.
$\bm{X}\succeq\bm{0}$ means that $\bm{X}$ is a positive semidefinite matrix.
For two arbitrary matrices $\bm{A},\bm{B}$, $\langle\bm{A},\bm{B}\rangle_{R}$
represents the trace of $\bm{B}^{H}\bm{A}$. $\mathcal{P}_{\Omega}(\cdot)$ is a
operator transforming an arbitrary matrix to a reduced matrix with rows
indexed by $\Omega$.

##  II System Model

We consider a typical uplink access scenario for MIMO-OFDM systems where there
are one BS equipped with an $N$-element uniform linear array (ULA) and $K$
single-antenna users along with OFDM modulation to combat inter-symbol
interference [6, Section II]. The sub-channel corresponding to each OFDM sub-
carrier between the $k$-th user and the BS is modeled as (see [6, Equ. 2] or
[15, Equ. 7]):

$\displaystyle\bm{h}_{k}=\sum_{l=1}^{L_{k}}\alpha_{l}^{k}\bm{a}(\theta_{l}^{k})=\bm{A}_{k}\bm{\alpha}^{k}\in\mathbb{C}^{N\times
1},$ (1)

in which $L_{k}$ is the number of physical paths between $k$-th user and BS,
$\theta_{l}^{k}$ is the Angle of Arrival (AoA) of the $l$-th path,
$\alpha_{l}^{k}$ is the complex gain of the $l$-th path,

$\displaystyle\bm{a}_{r}(\theta)=\tfrac{1}{\sqrt{N}}[1,{\rm
e}^{-j2\pi\Delta_{r}\cos(\theta)},...,{\rm
e}^{-j2\pi\Delta_{r}(N-1)\cos(\theta)}]^{T}$ (2)

is the receive steering vector,
$\bm{\alpha}^{k}:=[\alpha_{1}^{k},...,\alpha_{L_{k}}^{k}]^{T}$, and
$\bm{A}_{k}:=[\bm{a}_{r}(\theta_{1}^{k}),...,\bm{a}_{r}(\theta_{L_{k}}^{k})]\in\mathbb{C}^{N\times
L_{k}}$. Due the sparse characteristics of massive MIMO channels, it holds
that $L_{k}\ll N$. The sub-channel is considered to be block fading, i.e., it
is constant during several CIs where each is denoted by $T$. In each sub-
carrier, the received signal at the BS after $T$ time slots at the sensors
indexed by $\Omega\subseteq\\{1,...,N\\}$ (with length $|\Omega|:=M<N$)
becomes in the form of ([22], [6, Equ. 3] or [15, Equ. 1]):

$\displaystyle\bm{Y}_{\Omega}=\mathcal{P}_{\Omega}(\bm{Y})=\mathcal{P}_{\Omega}(\sum_{k=1}^{K}\underbrace{\bm{h}_{k}\bm{s}_{k}^{H})}_{:=\bm{X}_{k}}+\bm{W}\in\mathbb{C}^{M\times
T},$ (3)

where $\bm{s}_{k}\in\mathbb{C}^{T}$ is the transmitted signal from $k$-th UE,
$\bm{W}\in\mathbb{C}^{M\times T}$ is the additive noise matrix, each element
of which is distributed as $\mathcal{CN}(0,\sigma^{2})$ and
$\bm{X}_{k}:=\sum_{l=1}^{L_{k}}\alpha_{l}^{k}\bm{a}(\theta_{l}^{k})\bm{s}_{k}^{H}$.
Inspired by [23], and by defining
$c_{l}^{k}:=\alpha_{l}^{k}\|\bm{s}_{k}\|_{2}$ and
$\bm{\phi}_{k}:=\tfrac{\bm{s}_{k}}{\|\bm{s}_{k}\|_{2}}$, $\bm{X}_{k}$ can be
expressed as a sparse linear combinations of the matrix atoms in the atomic
set
${\mathcal{A}_{k}=\\{\bm{a}_{r}(\theta)\bm{\phi}_{k}^{H}:\|{\bm{\phi}}_{k}\|_{2}=1,\theta\in(0,\pi)\\},}$
which are regarded as building blocks of
$\bm{X}_{k}=\sum_{l=1}^{L_{k}}c_{l}^{k}\bm{a}(\theta_{l}^{k})\bm{\phi}_{k}^{H}$.
The aim is to extract the continuous parameters of $\bm{X}_{k}$ (i.e. the
angles $\bm{\theta}_{k}$) by observing $\bm{Y}_{\Omega}$. Note that for
inactive users, $\bm{\phi}_{k}$ and thus $\bm{X}_{k}$ are equal to zero.

##  III Proposed Blind Super-resolution method

In (3), we have an under-determined set of equations with $NT$ observations
and $KNT$ unknowns. While this problem has infinite number of solutions, it
could be transformed to a tractable problem by assuming that $L_{k}\ll N$
which is reasonable in massive MIMO systems. This strategy is built upon well-
known continuous CS approaches [17, 18, 24] and provides a unique optimal set
of solutions for matrices $\bm{X}_{k}$s in (3) leading to the least number of
atoms under the affine constraints of (3). Thus, we form the following
optimization problem to reflect the structure of $\bm{X}_{k}$s:

$\displaystyle\min_{\begin{subarray}{c}\bm{Z}_{k}\in\mathbb{C}^{N\times T}\\\
k=1,...,K\end{subarray}}\sum_{k=1}^{K}\|{\bm{Z}_{k}}\|_{\mathcal{A}_{k},0}~{}s.t.\|\bm{Y}_{\Omega}-\sum_{k=1}^{K}\mathcal{P}_{\Omega}(\bm{Z}_{k})\|_{F}\leq\eta$
(4)

where
$\|\bm{Z}_{k}\|_{\mathcal{A},0}:=\inf\big{\\{}L_{k}:\bm{Z}_{k}=\sum_{l=1}^{L_{k}}c_{l}^{k}\bm{a}({\theta}_{l}^{k})\bm{\phi}_{k}^{H},c_{l}^{k}>0,\bm{a}(\theta_{l}^{k})\bm{\phi}_{k}^{H}\in\mathcal{A}_{k}\big{\\}}$
is the atomic $\ell_{0}$ function which computes the least number of atoms to
describe $\bm{Z}_{k}$. As (4) is an NP-hard problem in general, we relax  (4)
into its closest convex optimization problem which is stated as:

$\displaystyle\min_{\begin{subarray}{c}\bm{Z}_{k}\in\mathbb{C}^{N\times T}\\\
k=1,...,K\end{subarray}}\sum_{k=1}^{K}\|{\bm{Z}_{k}}\|_{\mathcal{A}_{k}}~{}s.t.~{}\|\bm{Y}_{\Omega}-\sum_{k=1}^{K}\mathcal{P}_{\Omega}(\bm{Z}_{k})\|_{F}\leq\eta,$
(5)

where  the atomic norm $\|\cdot\|_{\mathcal{A}_{k}}$ is the best convex
surrogate for the number of atoms composing $\bm{Z}_{k}$ (i.e.
$\|\cdot\|_{\mathcal{A}_{k},0}$) and is defined as the minimum of the
$\ell_{1}$ norm of the coefficients forming $\bm{Z}_{k}$:

$\displaystyle\|\bm{Z}_{k}\|_{\mathcal{A}}:=\inf\\{t>0:\bm{Z}_{k}\in t{\rm
conv}(\mathcal{A}_{k})\\}=$
$\displaystyle\inf\\{\sum_{l=1}^{L_{k}}c_{l}^{k}:\bm{Z}_{k}=\sum_{l=1}^{L_{k}}c_{l}^{k}\bm{a}({\theta}_{l}^{k})\bm{\phi}_{k}^{H},c_{l}^{k}>0,\bm{a}(\theta_{l}^{k})\bm{\phi}_{k}^{H}\in\mathcal{A}_{k}\\}$
(6)

where ${\rm conv(\mathcal{A})}$ is the convex hull of $\mathcal{A}$. To
identify the AoAs (which we used in our simulations) is by leveraging the
solution of the dual problem of (5) which is provided below:

$\displaystyle\min_{\bm{V}\in\mathbb{C}^{N\times
T},\bm{Z}\in\mathbb{C}^{T\times T}}2{\rm
Re}\langle\bm{V}_{\Omega},\bm{Y}_{\Omega}\rangle_{F}+2\eta\|\bm{V}_{\Omega}\|_{F}~{}~{}s.t.$
$\displaystyle\begin{bmatrix}\bm{I}&\bm{V}^{H}\\\
\bm{V}&\bm{Z}_{i}\end{bmatrix}\succeq\bm{0},\mathcal{T}^{*}(\bm{Z}_{i})=\mathcal{T}^{*}(\bm{I}),i=1,...,K,~{}\mathcal{P}_{\Omega}(\bm{V})=\bm{0}$
(7)

where
$(\mathcal{T}^{*}(\bm{Z}))_{k}=\sum_{i=\max(1,k+1)}^{\min(k+T,T)}\bm{Z}_{i,i-k},k=-(T-1),...,(T-1)$
is the adjoint operator of $\mathcal{T}$. Then, we use the following lemma
(adapted from [20, Lemma 1] and [19, Theorem 1]) which guarantees the
uniqueness of the solution in the noiseless case:

######  Lemma 1.

Denote the set of AoAs from $i$-th users by
$\mathcal{S}_{a}^{i}=\\{\theta_{l}^{i}\\}_{l=1}^{L_{i}}$. The solutions of
$\bm{Z}_{k}$ obtained from (5) in the noiseless case ($\eta=0$) are unique if
there exist dual matrices $\bm{V}\in\mathbb{C}^{N\times T}$ such that the
vector-valued dual polynomials
$\bm{q}_{i}(\theta)=\bm{V}^{H}\bm{a}_{r}(\theta)$ satisfy the conditions

$\displaystyle\bm{q}_{i}(\theta)=\bm{\phi}_{i},\forall\theta\in\mathcal{S}^{i}_{a},~{}\|\bm{q}_{i}(\theta)\|_{2}<1\forall\theta\in[0,\pi)\setminus\mathcal{S}^{i}_{a},i=1,...,K.$

This lemma shows that the AoAs can be easily estimated by identifying
locations where $\|\bm{q}_{i}(\theta)\|_{2}$ achieves $1$. As stated in [25],
this provides a good insight about the procedure of finding AoAs in the noisy
case. Specifically, we find the AoAs by identifying the ones that
$\|\bm{q}_{i}\|_{2}=1,i=1,...,K$.

After obtaining the estimated AOAs, a clustering-based algorithm [26] is
employed to detect the AOAs of  each cluster (active user) denoted by
$\bm{\theta}^{k}=[\theta_{1}^{k},...,\theta^{k}_{\widehat{L}_{k}}]^{T},k\in\widehat{\mathcal{S}}_{u}$.
Here, $\widehat{\mathcal{S}}_{u}$ is the estimated set of indices
corresponding to clusters (estimated active users) with known length
$|\widehat{\mathcal{S}}_{u}|=K_{a}$. By knowing the AOAs corresponding to each
cluster (estimated active user), (3) turns into the following equation:

$\displaystyle\bm{Y}_{\Omega}=\sum_{k\in\widehat{\mathcal{S}}_{u}}\bm{A}^{k}_{\Omega}\bm{c}^{k}\bm{\phi}_{k}^{H}+\bm{W}_{M\times
T},$ (8)

where
$\bm{A}^{k}_{\Omega}=\mathcal{P}_{\Omega}([\bm{a}_{r}(\theta^{k}_{1}),...,\bm{a}({\theta}^{k}_{\widehat{L}_{k}})])\in\mathbb{C}^{M\times
K_{a}}$ is the steering matrix of $k$-th active user and
$\bm{c}^{k}=[c_{1}^{k},...,c^{k}_{\widehat{L}_{k}}]^{T}$. The task of
recovering the unknown matrices $\bm{c}^{k}$ and $\bm{\phi}_{k}$ from
$\bm{Y}_{\Omega}$ is a bi-linear inverse problem. For this task, we propose an
alternating optimization to jointly estimate complex channel coefficients and
transmitted data corresponding to active users. First, we begin with a random
$\widehat{\bm{\phi}}_{k}$ distributed on the unit sphere. By replacing
$\widehat{\bm{\phi}}_{k}$ in (8), we deal with the following least square
problem:

$\displaystyle[\widehat{\bm{c}}^{1},...,\widehat{\bm{c}}^{K_{a}}]=\mathop{\arg\min}_{\bm{c}^{k},k=1,...,K_{a}}\|\bm{Y}_{\Omega}-\sum_{k\in\mathcal{S}_{u}}\bm{A}^{k}_{\Omega}\bm{c}^{k}\widehat{\bm{\phi}}_{k}^{H}\|_{F}$
(9)

which can be easily solved by numerical optimization. By integrating the
latter expression into (8), we must solve the following least square
optimization: ${\widehat{\bm{\Phi}}=\mathop{\arg\min}_{\bm{\Phi}_{K_{a}\times
T}}\|\bm{Y}_{\Omega}-\bm{B}\bm{\Phi}\|_{F},}$ where
$\bm{B}:=[\bm{A}_{\Omega}^{1}\widehat{\bm{c}}^{1},...,\bm{A}_{\Omega}^{K_{a}}\widehat{\bm{c}}^{K_{a}}]\in\mathbb{C}^{M\times
K_{a}}$ and $\bm{\Phi}:=[\bm{\phi}_{1},...,\bm{\phi}_{K_{a}}]^{T}$. The latter
optimization has also the closed-form solution

$\displaystyle[\widehat{\bm{\phi}}_{1},...,\widehat{\bm{\phi}}_{K_{a}}]^{T}=\widehat{\bm{\Phi}}=\bm{B}^{\dagger}\bm{{Y}}_{\Omega}.$
(10)

Finally, the steps (9) and (10) are alternatively performed to yield the final
solution. The pseudo code of the proposed method which is indeed a summary of
the aforementioned steps is provided in Algorithm 1.  Algorithm 1

0: $\bm{Y}\in\mathbb{C}^{M\times T}$,$\eta$, $K_{a}$, maxiter  1: Select a
uniformly distributed random vector for transmitted data as
$\widehat{\bm{\phi}}_{k}=5rand(T,1)\forall~{}k=1~{}\text{to}~{}K_{a}$ 2:
$\bm{A}_{total}=\emptyset$ • Solve the dual problem (III) to obtain $\bm{V}$
as follows: • Obtain the dual polynomial
$\bm{q}_{i}(\theta)=\bm{V}^{H}\bm{a}_{r}(\theta),i=1,...,K$. • Localize the
estimated angle $\widehat{\theta}\in[0,1]$ by the following two methods: •
Discretize $\widehat{\theta}$ on a fine grid up to a desired accuracy and find
$\widehat{\theta}$ and by identifying locations where
$\|\bm{q}_{i}(\theta)\|_{2},i=1,...,K$ achieves to $1$ according to Lemma 1.
The total number of angles reaching $1$ specifies an estimate for the total
number of MPCs i.e. $\sum_{k=1}^{K_{a}}L_{k}$ 2: 3: Apply the k-means methods
to cluster the angles of channel UEs.  4: $[{\rm label}]={\rm
k-means}(\widehat{\theta},K_{a})$ 5: for $k=1$ to $K_{a}$ do 6: Identify the
corresponding indices with the $k$-th label.  7: Estimate the length of $k$-th
cluster i.e. $L_{k}$.  8: Obtain the angles corresponding to the $k$-th
cluster (UE) i.e. $\widehat{\theta}_{1}^{k},...,\widehat{\theta}_{L_{k}}^{k}$
9: Estimate the steering matrix as  10:
$\bm{A}_{total}\leftarrow[\bm{A}_{total},\bm{A}_{k}]$ 11: end for 12: for
$i=1$ to maxiter do 13: Recover
$[\widehat{\bm{c}}^{1},...,\widehat{\bm{c}}^{K_{a}}]$ according to (9).  14:
Recover $[\widehat{\bm{\phi}}_{1},...,\widehat{\bm{\phi}}_{K_{a}}]$ according
to (10).  15:
$\widehat{\bm{\phi}}_{k}\leftarrow\tfrac{\widehat{\bm{\phi}}_{k}}{\|\widehat{\bm{\phi}}_{k}\|_{2}}$.
16: end for

Return:
$\widehat{\bm{\theta}}^{k},\widehat{\bm{\alpha}}^{k},\widehat{\bm{s}}_{k},~{}\forall
k\in\mathcal{S}_{u}$.

Discussion: Our algorithm provides a blind spatial-based RA scheme which
simultaneously estimates data as well as channels. The only assumption on the
data to be unambiguously recovered is positivity i.e. $\bm{s}_{i}>0$ and
normalized power $\|\bm{s}_{i}\|_{2}=1$ for all active users $i=1,...,K_{a}$.
The proposed method has implications for data recovery in mMTC as well as RAP
and AOA detection in crowded mobile broadband communications (cMBB). For
example, CI in cMBB divides into 2 parts: RAP and data transmission blocks. By
utilizing this novel approach in the RAP block, AUD and AoA estimation are
performed by the BS [3, Section 5]. By our method, BS can identify many users
at the same time with their AoAs and with the lowest level of spending system
resources. In fact, the number of active users that can access the network
depends on the complexity that BS can bear. There is no need for orthogonality
of RA pilots needed for RAP process in [4] and prior distribution of pilots
and channels as is the case in AMP-based approaches [15, 6]. After RAP, in the
coherent transmission step, BS can easily allocate dedicated orthogonal pilots
to non-overlapped UEs and estimates their corresponding data via (10) by
knowing the exact AoAs of active users in the RAP stage.

##  IV Simulations

In this section, we perform some numerical experiments to evaluate the
performance of our proposed algorithm in blind channel and data
reconstruction. We use SDPT3 package of CVX [27] in MATLAB for solving problem
(III). The number of BS antennas is set to $N=64$. We assume the one-ring
model for the channel [28]. The AoAs are randomly chosen from $[0,\pi]$. The
path amplitudes are distributed as $\mathcal{CN}({0},1)$. The separation
between any receive antennas at BS are set to $\Delta_{r}=0.5$. The observed
sensors at BS ($\Omega$) are randomly chosen out of $\\{1,...,N\\}$. Also, the
maximum number of iterations in Algorithm 1 denoted by maxiter is fixed to
$5$. The upper bound of noise variance is chosen as $\eta=\|\bm{W}\|_{F}$. The
signal to noise ratio is defined by ${\rm
SNR}=10\log_{10}(\frac{\|\mathcal{P}_{\Omega}(\bm{X})\|_{F}^{2}}{MT\sigma^{2}})$.
First, in the top-left image of Figure 1, we show the successful procedure of
AUE and clustering with parameters $M=30$, $K=10$, $K_{a}=3$, $T=2$, ${\rm
SNR}=10~{}dB$. The maximum number of MPCs is fixed to $L_{\max}=3$. This image
shows the $\ell_{2}$ norm of the vector-valued dual polynomial at different
angles in terms of radian. The estimated angles are found by identifying
locations that $\|\bm{q}(\theta)_{i}\|_{2}=1,i=1,...,K$. The number of peaks
provides an estimate for $\sum_{k\in\mathcal{S}_{u}}L_{k}$. After finding the
angles, we apply k-means method to cluster the angles corresponding to $K_{a}$
active users. The number of elements inside each cluster provides an estimate
for $\widehat{L}_{k}$. Then, steps 12 to 16 of Algorithm 1 are employed to
obtain the pilots and channels. The performance of our algorithm in recovering
users’ data, channel amplitudes and AoAs is evaluated using normalized mean
square error (NMSE) respectively defined by ${\rm
NMSE}_{\bm{\phi}}:=\mathds{E}\sqrt{\frac{\sum_{k=1}^{K_{a}}\|\bm{\phi}_{k}-\widehat{\bm{\phi}}_{k}\|_{2}^{2}}{\sum_{k=1}^{K_{a}}\|\bm{\phi}_{k}\|_{2}^{2}}}$,
${\rm
NMSE}_{\bm{\alpha}}:=\mathds{E}\sqrt{\frac{\sum_{k=1}^{K_{a}}\|\bm{\alpha}_{k}-\widehat{\bm{\alpha}}_{k}\|_{2}^{2}}{\sum_{k=1}^{K_{a}}\|\bm{\alpha}_{k}\|_{2}^{2}}}$
and ${\rm
NMSE}_{\bm{\theta}}:=\mathds{E}\sqrt{\frac{\sum_{k=1}^{K_{a}}\|\bm{\theta}^{k}-\widehat{\bm{\theta}}^{k}\|_{2}^{2}}{\sum_{k=1}^{K_{a}}\|\bm{\theta}^{k}\|_{2}^{2}}}$.
The Monte-Carlo iterations to approximate the expectation is set to $50$. The
evaluation for the first experiment are as follows: ${\rm
NMSE}_{\bm{\Phi}}=10^{-6},{\rm NMSE}_{\bm{\alpha}}=10^{-5},{\rm
NMSE}_{\theta}=10^{-8}$. In the second experiment, we evaluate the performance
of Algorithm 1 in different noise values in a more practical scenario with
parameters $N=64,L_{\max}=3,K_{a}=12,K=40,T=10$. As it turns out from the
bottom image of bottom-right image of Figure 1, NMSEs tends to zero at high
SNRs which in turn implies that our proposed method performs well in
estimating users’ data, complex channel amplitudes and AoAs of active users.
In the experiment shown in the top-right image of Figure 1, we compare our
method with [3] for different number of antennas. For both methods, we obtain
NMSE of the channel matrix defined by ${\rm
NMSE}_{\bm{h}}:=\mathds{E}\sqrt{\frac{\sum_{k=1}^{K_{a}}\|\bm{h}_{k}-\widehat{\bm{h}}_{k}\|_{2}^{2}}{\sum_{k=1}^{K_{a}}\|\bm{h}_{k}\|_{2}^{2}}}$.
As it can be observed, our blind method performs better in CE than [3] which
assumes the users’ data known. In the last experiment, the performance of AUE
in our algorithm is compared with [12, Algorithm 1] by a detection rate
criterion defined as
$DR=\frac{|\mathcal{S}_{u}-\widehat{\mathcal{S}_{u}}|}{K_{a}}$ where the
numerator returns the number of differences between the true active users and
the estimates. As shown in the bottom-left image of Figure 1, the probability
of detection enhances by increasing the  number $N$ of BS antennas.

Figure 1: Top-left image: This image depicts $\ell_{2}$ norm of the dual
polynomial vector. One can find the angles of active users by identifying
angles with maximum amplitude. The angles of active user channels are
clustered using kmeans method. The used parameters are
$N=64,M=30,K_{a}=3,K=10,T=2,SNR=10~{}dB.$ Top-right image: This image compares
the performance of our algorithm in CE with [3] for different number of BS
antennas with settings $L_{\max}=4,T=10,K_{a}=3,K=10,{\rm SNR}=3dB$. Bottom
left image: The performance of AUE is compared with [12] versus the number of
observed arrays with parameters $N=60,M=60,K_{a}=5,K=50,L_{\max}=3,{\rm
SNR}=5~{}dB$. Bottom right image: This image shows the performance of our
algorithm in estimating angles, pilots and complex amplitudes with parameters
$N=64,L_{\max}=3,K_{a}=12,K=40,T=10$.

##  V Conclusion

In this work, we designed a novel blind spatial-based random access solution
which is applicable to crowded massive MIMO systems. Specifically, we showed
that the recovery of both pilots and AoAs are possible via observing a few
noisy measurements in blind manner. For this task, we used a clustering method
to demix the AoAs corresponding to active user and an alternating-based
approach is designed to recover the pilots and the complex amplitudes of the
channels.

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