capbtabboxtable[][]

# An effective density matrix approach for intersubband plasmons coupled to a
cavity field: electrical extraction/injection of intersubband polaritons

M. Lagrée<EMAIL_ADDRESS>III-V Lab, Campus Polytechnique,
1, Avenue Augustin Fresnel, RD 128, 91767 Palaiseau cedex, France  M. Jeannin
Centre de Nanosciences et de Nanotechnologies (C2N), CNRS UMR 9001, Université
Paris-Saclay, 91120 Palaiseau, France  G. Quinchard  III-V Lab, Campus
Polytechnique, 1, Avenue Augustin Fresnel, RD 128, 91767 Palaiseau cedex,
France  S. Pes  III-V Lab, Campus Polytechnique, 1, Avenue Augustin Fresnel,
RD 128, 91767 Palaiseau cedex, France  A. Evirgen  III-V Lab, Campus
Polytechnique, 1, Avenue Augustin Fresnel, RD 128, 91767 Palaiseau cedex,
France  A. Delga  III-V Lab, Campus Polytechnique, 1, Avenue Augustin Fresnel,
RD 128, 91767 Palaiseau cedex, France  V. Trinité<EMAIL_ADDRESS>III-V Lab, Campus Polytechnique, 1, Avenue Augustin Fresnel, RD 128, 91767
Palaiseau cedex, France  R. Colombelli<EMAIL_ADDRESS>Centre de Nanosciences et de Nanotechnologies (C2N), CNRS UMR 9001, Université
Paris-Saclay, 91120 Palaiseau, France

###### Abstract

The main technological obstacle hampering the dissemination of modern
optoelectronic devices operating with large light-matter coupling strength
$\Omega$ is an in-depth comprehension of the carrier current extraction and
injection from and into strongly coupled light-matter states, the so-called
polaritonic states. The main challenge lies in modeling the interaction
between excitations of different nature, namely bosonic excitations (the
plasmonic ISB excitations) with fermionic excitations (the electrons within
the extraction or injection subband). In this work, we introduce a
comprehensive quantum framework that encompasses both the ISB plasmonic mode
and the extractor/injector mode, with a specific emphasis on accurately
describing the coherent nature of transport. This reveals inherent selection
rules dictating the interaction between the ISB plasmon and the
extraction/injection subband. To incorporate the dynamics of the system, this
framework is combined to a density matrix model and a quantum master equation
which have the key property to distinguish intra and intersubband mechanisms.
These theoretical developments are confronted to experimental photocurrent
measurements from midinfrared quantum cascade detectors ($\lambda=10$ µm)
embedded in metal-semiconductor-metal microcavities, operating at the onset of
the strong light-matter coupling regime ($2\Omega=9.3$ meV). We are able to
reproduce quantitatively the different features of the photocurrent spectra,
notably the relative amplitude evolution of the polaritonic peaks with respect
to the voltage bias applied to the structure. These results on extraction
allow us to elucidate the possibility to effectively inject electronic
excitations into ISB plasmonic states, and thus polaritonic states.

##  I Introduction

The use of electromagnetic resonators like antennas or cavities is an
established tool to tailor and improve the properties of optoelectronic
devices, whether by increasing the sensitivity, reducing the electronic noise,
improving the wall-plug efficiency. In general, the strategy is to engineer,
and typically increase, the interaction strength between light and an
electronic transition in matter. However, the interaction strength in
practical devices is always limited to a small fraction of the photon or
electronic transition lifetimes, which places the device in the so-called weak
coupling regime. On the contrary, when the light-matter interaction strength
overcomes the losses in the system, the latter enters the strong coupling
regime. The new constituents of this system are mixed light-matter states
called polaritons, which can be formed by hybridizing any polarization-
carrying matter excitation and a photon field.

Polariton physics thus emerged as a transverse research field studying the
fundamental properties of strongly coupled systems. It revealed a plethora of
phenomena, the most recognized being the out-of-equilibrium Bose-Einstein
condensation of exciton-polaritons [1, 2, 3]. However, most experiments on
polaritons are performed by optical means, whereas practical devices require
electrical injection or extraction of charge carriers. Recent experiments
sparked new interest in electrical transport in systems under strong light-
matter coupling conditions, with the report of increased conductivity in
organic molecules [4], or the breakdown of topological protection in quantum
Hall systems [5, 6]. Intense research effort is thus currently devoted to
provide an accurate description of transport in systems strongly coupled to a
cavity field.

In this context, intersubband (ISB) polaritons, that originate from the
coupling between an intersubband transition in doped semiconductor quantum
wells (QW) and a cavity mode, are of particular interest. They were first
reported in 2003 [7] with absorption experiments, and that same year
electronic detection of the signature of strong coupling was also reported
[8]. However, proposals for electrical injection and electroluminescence of
ISB polariton devices [9, 10], that were quickly followed by experimental work
[11, 12], faced the problem of inefficient electrical injection in a
polaritonic state. That issue proved insurmountable in the following years
[12, 13, 14, 15]. To circumvent the problem, the study of the ”reverse”
process (photo-detection) was proposed to elucidate transport mechanisms in
polaritonic ISB electronic devices, with experiments on quantum well infrared
photodetectors (QWIP) operating in the strong light-matter coupling regime
[16].

In this context, we have recently presented a semi-empirical model to describe
the electronic photoresponse of quantum cascade detectors (QCD) operating in
the strong light-matter coupling regime [17]. Based solely on classical
oscillators, it allowed us to shine new light on the polariton-to-electron
process, and in particular to conjecture that a direct polariton-to-electron
tunnel mechanism may play a major role in such devices. This result was
obtained at the expense of great simplifications. In particular, because the
model is based on classical theory, it cannot include any consideration on the
coherence of the involved processes.

Nevertheless, coherence is of paramount importance when dealing with systems
operating in the strong-coupling regime, and even more so for ISB polaritons,
that originate from the coupling between a cavity mode and a collective
excitation. ISB transitions, that are more rigorously defined as ISB
plasmons[18, 19, 20], are collective matter excitations originating from the
electronic plasma inside a semiconductor quantum well, subject to its own
Coulomb interaction. This is in stark contrast to, for instance, exciton-
polaritons that result from an ensemble of single-particle transitions. The
main consequence is the presence of dark states, that do not couple to the
electromagnetic field, but do participate in electronic transport. This has
important consequences on the behavior of ISB polariton systems under
electrical injection.

Figure 1: (center) Schematic representation of the system in momentum space.
Bright and dark states are represented for both $\alpha$ (0$\rightarrow$1) and
$\beta$ (0$\rightarrow$2) transitions. Note that the $\beta$-bright state is
degenerated with the $\beta$-dark states. The different important operators
and their effect on the transport of the excitations are represented. The blue
path represents a detection process, whereas the red path represents an
injection process. (Top) Typical bandstructure for a quantum cascade detector.
The main extraction pathway is represented in blue. (Bottom) Typical
bandstructure for a quantum cascade emitter. The main injection pathway is
represente in red. The cavity electric field $E_{z}$ is also schematically
superimposed on the figure.

In this paper, we propose a quantum description of QCDs based on a density
matrix formalism, that we compare to a complete set of experimental data.
Crucially, this approach allows us to describe (de)coherence and dissipation
in the system. Our goal is to develop a theoretical description that permits
to explain the electronic extraction process (photo-detection), and that - at
the same time - provides a more suitable vantage point to elucidate the more
complex electronic injection process leading to light emission.  We note that
a very recent work reports experimental results and proposes an alternative
transport model for similar QCD structures operating in the strong coupling
regime [21]. It works explicitly with the Fermionic approach, without
performing the bozonisation steps. While similar conclusions are drawn in the
photo-detection case, the work we present raises fundamental open questions
and presents ways forward to the case of electrically pumped polaritonic light
emitters.

In the first part, we develop the model and derive the main observable
quantities, notably the photocurrent generated by an exciting external photon
field.
In the second part, we validate the theoretical results by studying the
photoresponse of quantum cascade detectors operating in the strong coupling
regime as a function of the applied bias. We compare the values obtained in
our model with a in-house code based on Ref. [22] that models the electronic
transport in a more rigorous way, but does not incorporate the cavity effects
[23, 24].
In the last part, we discuss the implications of the main assumption at the
basis of our new model, and extend them to the case of electrical injection.

The system under study is sketched in the central part of Fig. 1. It consists
of two electronic subbands confined inside a QW, here represented in momentum
space. The second subband is tunnel-coupled to the fundamental state of an
adjacent QW, and the whole system is embedded inside a cavity. The system can
operate as a detector, acting as a QCD (top sketch), when it is excited by a
photon that generates a photocurrent. This path is represented by blue arrows.
It is also possible to inject electrons in the system (red arrows and bottom
sketch), when an electric bias is applied, that can eventually lead to photon
emission. In this case the device behaves as a polaritonic LED.

##  II An effective density matrix approach for electronic transport in
cavity-coupled QCDs

###  II.1 Bosonization of the active optical transition

We start by defining the annihilation and creation operators
$c_{\lambda\mathbf{k}}$ and $c_{\lambda\mathbf{k}}^{\dagger}$, the fermionic
operators related to the creation and annihilation of electrons in subbands
$\lambda=\\{0,1,2\\}$ (see Fig. 1). We impose $T=0$K and we assume that all
$N$ electrons are contained inside the $0$-subband without external
excitation. The one-particle quantum state $|1,\mathbf{k}\rangle$ of
electronic wave vector $\mathbf{k}$, representing a state where one electron
is in subband $\lambda=1$ is:

$\ket{1,\bf{k}}=c_{1\bf{k}}^{\dagger}c_{0\bf{k}}|F\rangle$ (1)

where $|F\rangle$ denotes the fundamental Fermi state (equilibrium state,
where all the electrons are contained in subband $\lambda=0$). For now, we
restrain the problem to the $\lambda=0,1$ subbands, that form the intersubband
optical transition. This transition will be denoted as $\alpha$. Following the
developments of Ref. [25], to describe the photo-excitation of an electron in
the $\alpha$-transition, it is relevant to switch from the fermionic basis
formed by the $\ket{1,\bf{k}}$ states to a new basis of states
$\\{|B_{i}^{\alpha}\rangle\\}_{i=[1:N]}$.

We have:

$\ket{B_{i}^{\alpha}}=\sum_{|\mathbf{k}|<\mathbf{k}_{F}}w_{i\mathbf{k}}^{\alpha}\ket{1,\mathbf{k}}$
(2)

Since the system is considered at $T=0$ K, only $|\mathbf{k}|<\mathbf{k}_{F}$
states are occupied, $\mathbf{k}_{F}$ the module of the $k$-wavevector
corresponding to the Fermi level of 0-subband. The
$\\{|B_{i}^{\alpha}\rangle\\}_{i=[1:N]}$ basis only covers the single-
excitation subspace (only one photo-excited electron per-subband), which is
sufficient in the case of a weak excitation regime. The coefficients
$w_{i\mathbf{k}}^{\alpha}$ are defined as:

$\displaystyle w_{1\mathbf{k}}^{\alpha}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N}}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \forall\mathbf{k}$ (3)
$\displaystyle\sum_{\mathbf{k}}w_{i\mathbf{k}}^{\alpha}$ $\displaystyle=$
$\displaystyle 0\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\forall i\neq 1$ (4)

The $|B_{1}^{\alpha}\rangle$ state, of eigenenergy equal to the ISB transition
energy $\omega_{\alpha}=\omega_{1}-\omega_{0}$ (assuming parabolic
dispersion), has the remarkable property of holding the entire oscillator
strength of the $\alpha$ transition:

$\langle F|\hat{d}|B_{1}^{\alpha}\rangle=z_{\alpha}\sqrt{N}$ (5)

where $\hat{d}$ denotes the dipole operator and $z_{\alpha}$ the dipole
strength of one electronic transition. The $|B_{1}^{\alpha}\rangle$ state is
called the bright state: it is formed by the coherent superposition of the
one-particle fermionic states $|1,\mathbf{k}\rangle$ of the
$\alpha$-transition and it holds the entire capacity of light-matter
interaction. The $\\{|B_{i}^{\alpha}\rangle\\}_{i=[2:N]}$ are called the dark
states since they can not interact with the light:

$\langle F|\hat{d}|B_{i}^{\alpha}\rangle=0\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ i\neq 1$ (6)

From these developments, one can define bright state destruction and creation
operators $b_{\alpha}$ and $b_{\alpha}^{\dagger}$ which describe the
collective excitation of the $\alpha$-transition:

$b_{\alpha}^{\dagger}=\frac{1}{\sqrt{N}}\sum_{\mathbf{k}}c_{1\mathbf{k}}^{\dagger}c_{0\mathbf{k}}$
(7)

In a weak excitation regime and for a large number of electrons $N$,
$b_{\alpha}$ can be approximated as a bosonic operator. $b_{\alpha}$ and
$b_{\alpha}^{\dagger}$ respectively demote and promote excitations inside the
bright state $|B_{1}^{\alpha}\rangle$.

The final step in this development is to include the plasmonic effect
$\omega_{P}$ of the electronic polarizations. The diagonalization of the
plasmonic Hamiltonian leads to the emergence of new operators of eigen-energy
$\tilde{\omega}_{\alpha}=\sqrt{\omega_{\alpha}^{2}+\omega_{P}^{2}}$ and a
plasmonic bright state that is still orthogonal to the dark states [25].
Mathematically, this new state is essentially the same as the previous bright
state, except that it is no longer degenerated with the dark states: for
simplicity, we will keep the notation $|B_{1}^{\alpha}\rangle$ and
$b_{\alpha}$ for respectively the bright state and the corresponding creation
operator. Note: at this stage we did not introduce strong light-matter
coupling yet. This derivation is therefore valid in any coupling regime.

###  II.2 Bosonization of the extractor: the tunnel-coupling Hamiltonian

We now turn to the insertion of the extraction subband in the formalism. As
outlined in Refs. [10, 26], the mixing of bosonic (the plasmonic ISB
excitations) and fermionic (the electrons in the extraction subband) degrees
of freedom is necessary to correctly model the transport mechanisms that take
place in an optically excited ISB system. The focus of our paper is on ISB
systems strongly coupled to a photonic mode, but we stress that the above
consideration is valid also in the weak-coupling regime. When a photon is
absorbed by an ISB transition, it generates a bosonic excitation: an ISB
plasmon. But the measured current, in a detector, is of course of fermionic
nature.

In the case where the extraction subband is explicitly included in the system
dynamics (and not only in the form of an external bath), it becomes an
extremely tedious task to keep track of all these degrees of freedom.
Effectively, one correct way to describe the interaction between these
excitations of different nature is to use a full fermionic Hamiltonian of
extremely large dimension. It is a significant mathematical challenge that
demands considerable effort, and the nature of transport cannot be
straightforwardly interpreted due to this complexity.

In this work, we overcome this strong limitation with a key modification: we
propose to depict the subband $\lambda=2$ with a bosonic operator in the
context of an extraction process. This approach has several advantages, and -
as we will discuss later on - it might also permit to address the scenario
involving an injection process. To explicitly incorporate subband $\lambda=2$
into our formalism, we introduce the one-particle fermionic states
$|2,\mathbf{k}\rangle$ of the $\beta$-transition:

$\displaystyle|2,\mathbf{k}\rangle=c_{2\mathbf{k}}^{\dagger}c_{0\mathbf{k}}|F\rangle$
(8)

Analogous to the $\alpha$-transition, we will not use this fermionic state
basis and instead employ a new ortho-normal basis
$\\{|B_{i}^{\beta}\rangle\\}_{i=[1:N]}$ defined as:

$\ket{B_{i}^{\beta}}=\sum_{|\mathbf{k}|<\mathbf{k}_{f}}w_{i\mathbf{k}}^{\beta}\ket{2,\mathbf{k}}$
(9)

where the coefficients $w_{i\mathbf{k}}^{\beta}$ are chosen such that:

$\displaystyle w_{1\mathbf{k}}^{\beta}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N}}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \forall k$ (10)
$\displaystyle\sum_{k}w_{i\mathbf{k}}^{\beta}$ $\displaystyle=$ $\displaystyle
0\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \forall i\neq
1$ (11)

The construction of this basis follows a similar approach as that of the
$\\{|B_{i}^{\alpha}\rangle\\}_{i=[1:N]}$ basis. Specifically, the first state
$|B_{1}^{\beta}\rangle$ is the bright state of the $\beta$-transition, while
the remaining states $\\{|B_{i}^{\beta}\rangle\\}_{i=[2:N]}$ are the dark
states of this same transition. However, this time, the oscillator strength of
a diagonal transition being very small, we have $z_{\beta}\ll z_{\alpha}$ and
thus the bright and dark states of the extractor are degenerated. Note that
the one excitation subspace describing subband $1$ and $2$, of dimension $2N$,
is spanned by the concatenation of the
$\\{|B_{i}^{\alpha}\rangle\\}_{i=[1:N]}$ and
$\\{|B_{i}^{\beta}\rangle\\}_{i=[1:N]}$ basis.

The introduction of this new basis is valuable to evaluate the tunnel coupling
between subbands 1 and 2 within the regime of strong light-matter coupling.
The tunnel coupling operator $\hat{T}$ can be defined as:

$\displaystyle\hat{T}=\Omega_{T}\sum_{\mathbf{k}}(c_{2\mathbf{k}}c_{1\mathbf{k}}^{\dagger}+c_{2\mathbf{k}}^{\dagger}c_{1\mathbf{k}})$
(12)

where $\Omega_{T}$ is the tunnel coupling strength. Using equations (3), (4),
(10) and (11), we compute the tunnel interaction between subbands $1$ and $2$:

$\displaystyle\braket{B_{1}^{\alpha}}{\hat{T}}{B_{1}^{\beta}}=$
$\displaystyle\Omega_{T}$ (13)
$\displaystyle\braket{B_{1}^{\alpha}}{\hat{T}}{B_{j}^{\beta}}=$ $\displaystyle
0$ $\displaystyle j\neq 1$ (14)
$\displaystyle\braket{B_{i}^{\alpha}}{\hat{T}}{B_{1}^{\beta}}=$ $\displaystyle
0$ $\displaystyle i\neq 1$ (15)
$\displaystyle\braket{B_{i}^{\alpha}}{\hat{T}}{B_{j}^{\beta}}=$
$\displaystyle\Omega_{T}\sum_{\mathbf{k}}w_{i\mathbf{k}}^{\alpha\ast}w_{j\mathbf{k}}^{\beta}$
$\displaystyle i\neq 1,j\neq 1$ (16)

The above relations, that are de facto selection rules, are one of the key
results of this work: through tunnel interaction, it is not possible to
transition from a dark state to a bright state (Eq. (14)) or vice versa (Eq.
(15)). Obviously, dark states can interact with each other through tunnel
coupling (Eq. (16)), and the same applies to bright states as well (Eq. (13)).

These results have crucial implications on the nature of electronic transport
in a QCD. For a detection process, where light promotes excitations into the
$|B_{1}^{\alpha}\rangle$ bright state, the previous results suggest that an
optical excitation can generate an electronic current in only two ways:

  1. 1.

Direct tunnelling into the extractor bright state $|B_{1}^{\beta}\rangle$,
preserving the coherent nature of the excitation, and subsequent decay - with
loss of coherence - into an extractor dark state $|B_{i\neq 1}^{\beta}\rangle$

or

  2. 2.

First decay - with loss of coherence - into an ISB dark state $|B_{i\neq
1}^{\alpha}\rangle$ in the active region, and subsequent tunneling into an
extractor dark state $|B_{i\neq 1}^{\beta}\rangle$

Other channels involving bright-to-dark tunneling should not be considered, as
they are prohibited by selection rules (14)(15). Once in the extractor dark
states, the electronic excitation will simply decay in the remaining cascade,
generating photocurrent. We stress that the construction of the new $\beta$
basis merely extended the procedure applied to the $\alpha$ transition
(detailed in reference [25]) to the $\beta$ transition, without additional
hypothesis. By implementing this basis transformation, the comprehension of
the transport process is streamlined, leading to the natural emergence of the
selection rules presented in Equation (13) to (16). In the following section,
we will assess the need to actually incorporate the dark states from both the
$\alpha$ and $\beta$-transitions to replicate the experimental photocurrent
measurements from a QCD operating in the strong light-matter coupling regime.
The implications of this section for an electronic injection process into
polaritonic states will be discussed in section IV.

###  II.3 Introducing dissipation and decoherence in the model

In the following, we develop an effective density matrix model of the
photocurrent extraction. We apply a drastic choice in the description of the
system: we limit the extraction model to the transport induced by the bright
states $|B_{1}^{\alpha}\rangle$ and $|B_{1}^{\beta}\rangle$. The dark states
from both the $\alpha$ and $\beta$-transitions are omitted. Both subbands $1$
and $2$ will thus be described only using bosonic operators. This is
equivalent to choose scenario (1) among the two described at the end of the
previous section: direct tunnelling into the extractor bright state
$|B_{1}^{\beta}\rangle$ (preserving the coherent nature of the excitation),
and subsequent decay - with loss of coherence - into an extractor dark state
$|B_{i\neq 1}^{\eta}\rangle$. This choice was already implicit in the approach
that we have employed in our previous work based on a classical description of
the electronic transport, using coupled mode theory [17].

We now go beyond this classical model using a quantum master equation. The key
addition is the introduction of decoherence in the system, that is distinct
from dissipation.

In terms of spectral effects, decoherence impacts the broadening of the
photocurrent peaks, while dissipation primarily affects their amplitude. In
the experimental study we will report in Sec. III, bias will be varied, and -
as a result - the amplitude of the peaks will be affected more than their
broadening. It will be essential to differentiate between the effects of
decoherence and dissipation, a distinction that was previously impossible to
achieve with the classical model.

We define the operator $b_{\beta}$ using our new basis from equations (9) and
(10):

$\displaystyle b^{\dagger}_{\beta}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N}}\sum_{\mathbf{k}}c_{2\mathbf{k}}^{\dagger}c_{0\mathbf{k}}$
(17) $\displaystyle b^{\dagger}_{\beta}|F\rangle$ $\displaystyle=$
$\displaystyle|B_{1}^{\beta}\rangle$ (18)

Using the fermionic commutation rules and a weak excitation regime, we have:

$\displaystyle[b_{\beta},b_{\beta}^{\dagger}]=\frac{\hat{N}_{0}-\hat{N}_{2}}{N}\approx\hat{\mathcal{I}_{d}}$
(19)

where $\hat{N}_{i}$ is the population operator of subband $i$ and
$\hat{\mathcal{I}_{d}}$ the identity operator. $b_{\beta}$ can thus be
approximated as a bosonic operator: $b_{\beta}$ and $b_{\beta}^{\dagger}$
describe the destruction and creation of electronic excitations inside the
extraction mode, of eigen-frequency $\omega_{\beta}=\omega_{2}-\omega_{0}$.
The related Hamiltonian is:

$\hat{\mathcal{H}}_{\beta}=\omega_{\beta}b_{\beta}^{\dagger}b_{\beta}$ (20)

We restrict the tunnel interaction to the interaction between the plasmonic
bright mode and this new extraction mode. This drastically simplifies the
tunnel interaction Hamiltonian described in Eq. (13). The restricted
Hamiltonian $\hat{T}_{\text{bright}}$ is:

$\hat{T}_{\text{bright}}=\Omega_{T}(b_{\alpha}^{\dagger}b_{\beta}+b_{\alpha}b_{\beta}^{\dagger})$
(21)

The TM01 electromagnetic mode confined in the patch antennas will be modeled
as a standard optical resonator of frequency $\omega_{c}$, using $a_{c}$ and
$a_{c}^{\dagger}$ bosonic destruction and creation operators. Using the
rotating wave approximation to describe the light-matter interaction, the time
dependent Hamiltonian $\mathcal{H}(t)$ of the whole system reads:

$\displaystyle\hat{\mathcal{H}}(t)$ $\displaystyle=$
$\displaystyle\omega_{c}a_{c}^{\dagger}a_{c}+\tilde{\omega}_{\alpha}b_{\alpha}^{\dagger}b_{\alpha}+\omega_{\beta}b_{\beta}^{\dagger}b_{\beta}$
$\displaystyle+\Omega\left(a_{c}^{\dagger}b_{\alpha}+a_{c}b_{\alpha}^{\dagger}\right)+\Omega_{T}\left(b_{\alpha}^{\dagger}b_{\beta}+b_{\alpha}b^{\dagger}_{\beta}\right)$
$\displaystyle+\kappa_{c}s_{+}\left(a_{c}e^{i\omega t}+a_{c}e^{-i\omega
t}\right)$

where $s_{+}$ is the amplitude of the incoming light excitation, $\omega$ its
frequency, and $\kappa_{c}$ is the coupling constant between this external
field and the confined optical mode inside the cavity.

We map this system on an equivalent open quantum system described by the
reduced density matrix $\rho$. Under standard Born-Markov approximations, the
time evolution of the density matrix $\rho$ obey the following quantum master
equation [27] ($\hbar=1$ for clarity):

$\displaystyle\frac{\text{d}\rho(t)}{\text{d}t}$ $\displaystyle=$
$\displaystyle-i\big{[}\mathcal{H}(t),\rho\big{]}$
$\displaystyle+\gamma_{\alpha}\mathcal{L}\left[b_{\alpha},\rho\right]+\gamma_{\beta}\mathcal{L}\left[b_{\beta},\rho\right]+(\gamma_{c}+\Gamma_{c})\mathcal{L}\left[a_{c},\rho\right]$
$\displaystyle+\gamma_{\alpha}^{\text{intra}}\mathcal{L}\left[b_{\alpha}^{\dagger}b_{\alpha},\rho\right]+\gamma_{\beta}^{\text{intra}}\mathcal{L}\left[b_{\beta}^{\dagger}b_{\beta},\rho\right]$

where the $\mathcal{L}$ are Lindblad super-operators modeling the dissipative
and decoherent interactions of the environment with the system. For any
operator $\hat{A}$, a super-operator $\mathcal{L}$ reads:

$\displaystyle\mathcal{L}[\hat{A},\rho]=2\hat{A}\rho\hat{A}^{\dagger}-(\hat{A}^{\dagger}\hat{A}\rho+\rho\hat{A}^{\dagger}\hat{A})$
(24)

The plasmonic ISB excitations are mainly dissipated through their interaction
with interface roughness, at a non-radiative rate $\gamma_{\alpha}$.
Similarly, the extractor dissipates electrons into the next period at a non-
radiative rate $\gamma_{\beta}$, and is responsible for the generation of
electrical current inside the structure. $\gamma_{\beta}$ represents an
effective dissipation rate that takes into consideration the remaining
electronic cascade. The cavity also dissipates photons (mainly through
undesired free-carriers absorption) at a rate $\gamma_{c}$, but also through a
spontaneous emission channel, at a radiative rate $\Gamma_{c}$. Note that the
radiative coupling $\kappa_{c}$ is related to the radiative damping through
$\kappa_{c}=\sqrt{2\Gamma_{c}}$ [28].

The main difference with our previous work [17] lies in the ability to
explicitly introduce the intra-subband scattering through the pure decoherence
terms
$\gamma_{\alpha}^{\text{intra}}\mathcal{L}\left[b_{\alpha}^{\dagger}b_{\alpha},\rho\right]$
(resp.
$\gamma_{\beta}^{\text{intra}}\mathcal{L}[b_{\beta}^{\dagger}b_{\beta},\rho]$)
[29]. These terms model pure decoherence damping without excitation
dissipation (the intra-subband scattering thermalize excitations inside a
subband without dissipating them into an other subband). By using the density
matrix formalism, it thus becomes possible to differentiate between the
effects of inter-subband (dissipation) and intra-subband (pure decoherence)
processes on the evolution of the system (and ultimately on the shape of the
calculated photoresponse spectra). More details on the necessity to
distinguish intra and intersubband scatterings can be found in Appendix A.

###  II.4 Deriving observable quantities for comparison with experiments

Equation II.3 can be solved numerically in steady state. The solution is a
stationnary reduced density matrix $\rho_{S}$, and any observable $\hat{O}$
can then be computed using:

$\displaystyle\langle\hat{O}\rangle=\text{Tr}(\hat{O}\rho_{s})$ (25)

where Tr represents the trace function. We can then compute the different
interesting quantities of the system. The system total absorption is the sum
of the power dissipated into the different decay channels, normalized by the
incoming power $|s_{+}|^{2}$:

$\displaystyle\mathcal{A}_{\text{tot}}$ $\displaystyle=$
$\displaystyle\mathcal{A}_{c}+\mathcal{A}_{\alpha}+\mathcal{A}_{\beta}$ (26)
$\displaystyle=$ $\displaystyle
2\gamma_{c}\frac{\braket{a_{c}^{\dagger}a_{c}}}{\left|s^{+}\right|^{2}}+2\gamma_{\alpha}\frac{\braket{b_{\alpha}^{\dagger}b_{\alpha}}}{\left|s^{+}\right|^{2}}+2\gamma_{\beta}\frac{\braket{b_{\beta}^{\dagger}b_{\beta}}}{\left|s^{+}\right|^{2}}$
(27)

where $\mathcal{A}_{c}$, $\mathcal{A}_{\alpha}$ and $\mathcal{A}_{\beta}$
represent respectively the cavity, ISB and extraction absorptions.

The net photocurrent $\mathcal{J}_{\beta}$ is defined as the current under
illumination. $\mathcal{J}_{\beta}$ is proportional to the power dissipated
from a period to the next adjacent period. This is exactly the power
dissipated by the extraction mode $\beta$:

$\displaystyle\mathcal{J}_{\beta}=2\gamma_{\beta}\braket{b_{\beta}^{\dagger}b_{\beta}}$
(28)

Note: this is a phenomenological interpretation of the photocurrent. It is in
fact expected that an excitation inside the bright extractor state
$|B_{1}^{\beta}\rangle$ should first decay in the dark states $|B_{i\neq
1}^{\beta}\rangle$ before being extracted in the electronic cascade and
contribute to the photocurrent. We choose to neglect these dark extractor
states such that the power is directly dissipated from the bright extractor
state. This also applies on the ISB dissipation, where the $|B_{i\neq
1}^{\alpha}\rangle$ dark states are neglected when considering the non-
radiative dissipation $\gamma_{\alpha}$.

##  III Experimental validation in photo-detection: the polariton-to-current
process

###  III.1 Experimental details

The samples investigated in this study are the same as those already studied
in Ref. [17]. They are processed into 8 $\times$ 8 (approximately equal to 50
$\times$ 50 µm2) patch antenna arrays, with the patches connected through
250-nm thin metallic wires (see Fig. 6 in Appendix B). Details of the
processing can be found in [30]. The samples are cooled down to T = 78K in a
cryostat, and they are illuminated by light from a globar source at normal
incidence. The photocurrent spectra are acquired in rapid scan mode, after
amplification using a low-noise transimpedance amplifier.

We extend the data presented in [17], and now present measurements with
voltage bias applied to the samples. The applied electric field ranges from
$F=-25\leavevmode\nobreak\ $kV.cm-1 to $F=8\leavevmode\nobreak\ $kV.cm-1. We
have fabricated several array designs ($p$, $s$), with $p$ the inter-patch
period of the array, and $s$ the lateral dimensions of the patches. However,
to allow for a quantitative comparison, we present measurements under an
applied electric field for two samples only, with same $p=$7 µm, and $s$= 1.5
µm and $s$=1.55 µm, respectively, as reported in Fig. 2 (continuous lines).
Additional measurements can be found in Appendix C. While the relative
amplitude of the spectra when varying the bias contains meaningful information
of the electronic transport, one should exercise caution when comparing the
amplitudes of different pairs ($p$, $s$) as the experimental protocol does not
ensure a consistent illumination between each measurement of the device.

Figure 2: Normalized photocurrent measurements (continuous lines) and quantum
master equation global fit (dashed lines), for two cavity geometries [a]
$s=1.50$ µm, $p=7$ µm and [b] $s=1.55$ µm, $p=7$ µm. Offsets are added for
clarity. Filled areas represent the errors of the fit parameters propagated
onto the spectra. The extractor frequency $\omega_{\beta}(F)$, dependant of
the electric field $F$, and the plasma-shifted ISB transition
$\tilde{\omega}_{\alpha}$ are both superimposed on the spectra. Additional
results can be found in Appendix C.

Two photocurrent peaks are clearly visible in Fig. 2, signature of the strong
light-matter coupling regime. Note: the peaks under consideration cannot be
confused with the two peaks arising from coupled subbands (tunnel coupling),
since the peak positions would change with the applied bias in the latter
case. Here, the energy splitting (for a given pair $p$, $s$) is constant
regardless of the applied field. For all ($p$, $s$) couples studied, the
global amplitude of the photocurrent spectra evolves with the applied electric
field $F$. A maximum amplitude is observed around $F=-10$ kV.cm-1. The noise
level increases strongly when the absolute amplitude of the field $|F|$
increases. The noise level is the direct consequence of the increase of the
parasitic dark current with the electric field and - as is well known [31, 32]
\- it affects the range of exploitable field $F$ for device applications.

The relative amplitude of these peaks inverts with respect to the applied
field $F$, with the equal amplitude condition of the two polaritonic photo-
detection peak found for a negative field $F\approx-5$ kV.cm-1. Below this
threshold, the low energy peak dominates. Inversely, for $F>-5$kV.cm-1, it is
the high energy peak that dominates. This phenomenon can be attributed to the
realignment of the subbands under the influence of the applied bias. When a
highly negative voltage is applied, the subbands follow a clear staircase
structure (see Fig. 7 in Appendix B for the QCD bandstructure), which
facilitates the extraction process. Conversely, at positive voltages, the
subband cascade becomes less organized, hindering the extraction process.

###  III.2 System parameters and constraints

Before applying the theoretical developments of section II to the experimental
data, let us detail the system parameters and the constraints applied to them.

The photonic degrees of freedom are the cavity parameters $\omega_{c}$,
$\gamma_{c}$ and $\Gamma_{c}$, that are independent of the applied electric
field $F$. They only depend on the geometrical parameters ($p$, $s$) of the
cavities [33, 34, 35]:

$\displaystyle\omega_{c}(s)$ $\displaystyle=$ $\displaystyle\frac{\pi
c_{0}}{n_{\text{eff}}s}$ (29) $\displaystyle\Gamma_{c}(p)$ $\displaystyle=$
$\displaystyle\frac{\alpha_{c}}{p^{2}}$ (30)

where $c_{0}$ is the light velocity, $n_{\text{eff}}$ is the effective index
of the cavity, that represents the effective medium composed of the
semiconductor contacts and of the undoped periodic structure embedded between
the gold layers forming that cavity, and $\alpha_{c}$ is the cavity dispersion
loss factor. We choose to constrain $n_{\text{eff}}$, $\alpha_{c}$ and
$\gamma_{c}$ to the values obtained from our prior investigation of the same
samples [17], where the photocurrent of several samples with different
($s$,$p$) couples have been studied for $F=0$ kV.cm-1:

$\displaystyle n_{\text{eff}}$ $\displaystyle=$ $\displaystyle 3.22$ (31)
$\displaystyle\alpha_{c}$ $\displaystyle=$ $\displaystyle
29.1\leavevmode\nobreak\ \leavevmode\nobreak\ \text{meV.\textmu m${}^{2}$}$
(32) $\displaystyle\gamma_{c}$ $\displaystyle=$ $\displaystyle
3.4\leavevmode\nobreak\ \leavevmode\nobreak\ \text{meV}$ (33)

The cavity parameters are thus excluded from the fitting process.

Several electronic degree of freedom can also be fixed or constrained
independently of our density matrix model. The parameters of the ISB
transition in the active QW ($\alpha$) are assumed independent of the applied
electric field $F$: the transition is vertical in a single quantum well and
therefore is very marginally affected by the applied bias. The ISB frequency
$\omega_{\alpha}$ and the plasma frequency $\omega_{P}$ could be computed from
our sequential transport software [22]. However, it is common to observe
disparities between expected and measured doping levels (up to 15%).
Experimental discrepancies also affect the ISB frequency (up to 5%), usually
caused by the quality of the quantum wells interfaces during the epitaxial
process. To account for these disparities, and since both $\omega_{\alpha}$
and $\omega_{P}$ are crucial parameters to reproduce the strong coupling
measurements, we chose to let these parameters free during the fitting
process:

$\displaystyle\tilde{\omega}_{\alpha}$ $\displaystyle=$
$\displaystyle\sqrt{\omega_{\alpha}^{2}+\omega_{P}^{2}}$ (34)

Note: the light-matter coupling constant $\Omega$ is parametrized using
$\omega_{P}$:

$\displaystyle\Omega=\frac{\omega_{P}}{2}\sqrt{f_{w}}$ (35)

with $f_{w}$ ($\approx$ 0.17), the computed overlap factor between the cavity
field and the doped active quantum wells.
Two additional $\alpha$ parameters can be computed using our sequential
transport software: the non-radiative dissipation rate $\gamma_{\alpha}$ of
the $\alpha$ plasmon from the excited subband to the fundamental subband, and
the tunnel coupling $\Omega_{T}$. We compute $\gamma_{\alpha}=0.66$ meV and
$\Omega_{T}=4.2$ meV, respectively. The new parameter of our transport model
in the strong coupling regime, the intra-subband rate
$\gamma_{\alpha}^{\text{intra}}$, will instead be fitted.

The parameters related to the extractor $\beta$ are instead dependent on the
electric field $F$: the extractor energy shifts with respect to the upper
excited state of the ISB transition when a bias is applied to the structure.
The misalignment is approximated as linear:

$\displaystyle\omega_{\beta}(F)=\alpha_{F}F+\omega_{\beta}^{0}$ (36)

where $\alpha_{F}$ is the linear coefficient and $\omega_{\beta}^{0}$ is the
extractor energy for $F=0$. This dispersion can be computed using our
sequential transport software and is injected into the model:

$\displaystyle\alpha_{F}$ $\displaystyle=$ $\displaystyle
1.12\leavevmode\nobreak\ \leavevmode\nobreak\ \text{meV/(kV.cm${}^{-1}$)}$
(37) $\displaystyle\omega_{\beta}^{0}$ $\displaystyle=$ $\displaystyle
124\leavevmode\nobreak\ \leavevmode\nobreak\ \text{meV}$ (38)

Similarly to $\gamma_{\alpha}^{\text{intra}}$, $\gamma_{\beta}^{\text{intra}}$
will be a fitting parameter common to the whole data set.
Finally, we expect the misalignment of the cascade with the electric field to
modify the value of the effective extraction rate $\gamma_{\beta}(F)$.
$\gamma_{\beta}$ is one of the most important parameters of the fitting
process, as it controls the relative amplitude of the spectra. Although we
suspect that it might closely match with the actual extraction rate calculated
from our sequential transport model, we decided to keep it as a free
parameter: for each measured electric field value $F_{i}$, we fit one
extraction rate $\gamma_{\beta}(F_{i})$. Note: $\gamma_{\beta}(F_{i})$ is
independent of the geometrical parameters $p$ and $s$. In summary,
$\omega_{\alpha}$, $\omega_{P}$ and $\gamma_{\alpha}^{\text{intra}}$ and
$\gamma_{\beta}^{\text{intra}}$ are fitting parameters common to the whole
data set, and their initial values for the fit will be based on the ones
derived by our software.

###  III.3 Discussion on the validity of the fit

In this section, we perform a global fit on the whole experimental
photocurrent dataset (Fig. 2), using the parameters constraints exposed in the
previous section. We solve Eq.(II.3) in the stationary regime (using the QuTiP
python library [36]) to evaluate the theoretical photocurrent $J_{\beta}$, as
per Eq.(28). The parameters resulting from the fit are presented in Table 1.

Fit parameters |  Fit results
---|---
$\omega_{\alpha}\text{ (meV)}$ |  116.9 $\pm$ 0.1 meV
$\gamma_{\alpha}^{\text{intra}}\text{ (meV)}$ |  2.4$\pm$ 0.1 meV
$\gamma_{\beta}^{\text{intra}}\text{ (meV)}$ |  9.3 $\pm$ 0.1 meV
$\omega_{P}\text{ (meV)}$ |  23.4$\pm$ 0.1 meV
Table 1: Parameters returned by the global fit using a quantum master equation
model.

The returned values are consistent with the previous fits performed with the
coupled mode theory in [17]. In particular, the extraction rate
$\gamma_{\beta}$ as a function of the applied electric field is plotted in
Fig. 3 and compared with the values computed through our sequential transport
model. The right order of magnitude is obtained ($\gamma_{\beta}<1$ meV) and
the evolution trends are relatively well reproduced ($\gamma_{\beta}$
decreasing for $F>0$, slope break around $F=-4$ kV.cm-1). These results on
$\gamma_{\beta}$ are also consistent with the evolution of the integrated
amplitude of the spectra (Fig. 3, right-side scale): when the electric field
is below $F=-4$ kV.cm-1, the electronic cascade is efficiently aligned, and
the effective extraction rate $\gamma_{\beta}$ is high. This leads to a
significant photocurrent signal.

The spectrally resolved photocurrent calculated using the parameters returned
by the global fit procedure is compared to the experimental data in Fig. 3,
with a quantitative agreement obtained on the set of triplets ($p$, $s$, $F$).
Two important trends are reproduced as a function of the bias, i.e. as a
function of the $\omega_{\alpha}-\omega_{\beta}$ alignment: (i) the overall
amplitude of the spectra, and (ii) the relative amplitude inversion between
the peaks of the two polaritonic branches.

Figure 3: Left-side scale: extraction rate $\gamma_{\beta}$ as a function of
the applied electric field $F$. Red cross: predicted values computed using a
standard sequential transport model. Blue plus sign: values returned by the
global fit using a quantum master equation model. Right-side scale:
experimental photocurrent integrated amplitude, for two different ($s$, $p$)
couples of cavity parameters.

This study quantitatively confirms that the extractor (the electronic cascade
of the QCD) and its relative alignment with respect to the ISB transition
controls the overall amplitude of the spectra, and also the relative amplitude
of the peaks of the polaritonic branches. Applying an electric field to the
structure enables the selective extraction of excitations from a polaritonic
state towards the electronic cascade, while also providing control over the
efficiency of this extraction. This selective extraction capacity is enabled
by the sharp transfer function and the $2\Omega$ spacing (the Rabi splitting)
between the polaritonic peaks: a finer transfer function and a stronger
coupling would allow for better selectivity of $\omega_{\pm}$ polaritons. More
details on a QCD transfer function in the strong coupling regime can be found
in Appendix A.

The good agreement between the experimental data and the theoretical model
provides strong evidence that the dark states for both transitions $\alpha$
and $\beta$ do not need to be included in the model to depict an extraction
process. The bright tunnel interaction $\hat{T}_{\text{bright}}$ and the
phenomenological dissipation rate $\gamma_{\beta}$ from the extractor bright
state are sufficient to quantitatively reproduce the experimental
measurements. As previously postulated in [17], this result confirms that the
polaritonic nature of the excitation is carried on during the extraction
process through the coherent tunnel coupling. The extraction is a coherent
process, mainly involving the bright states from both $\alpha$ and $\beta$
transitions.

This model permits however a step forward in the comprehension of the
polariton-to-electron process. Chronologically, the early attempts were
limited to the observation of a polariton splitting in photo-detection [8,
37]. A phenomenological transfer function was then introduced in the study of
QWIPs operating in strong coupling [16]. Recently, the Coupled Mode Theory
(CMT) permitted a more rigorous modeling of the transfer function, and an
initial indication of direct tunneling into the extractor bright state, with
no role for the polaritonic dark states [17]. The model presented in this
paper gets rid of the transfer function - a phenomenological concept - and
replaces it with a rigorous tunnel coupling Hamiltonian between the $\alpha$
and $\beta$ transitions, with a complete description of bright and dark
states. The latter do not play a major role for the polariton extraction
process, but they have a crucial role for polariton injection. Our model
integrates them, and might constitute a valid vantage point to study
electrically injected polariton emitters. More information on the transfer
function and the difference between the CMT and the effective density matrix
approach can be found in Appendix A.

##  IV Implications of the model for electrically pumped polariton emitters:
the electron-to-polariton process

The validity of the density matrix approach to describe electrical extraction
from optically excited polaritons, motivates to study the implications of
these findings on the electrical injection and subsequent photon emission,
represented by the red arrows in Fig. 1. As discussed in Ref. [26], the main
difficulty describing an intersubband emitter operating in the strong light-
matter coupling regime lies in the simultaneous description of both optical
(bosonic) and electronic (fermionic) excitations. The injection process fills
subband $2$ with fermionic excitations in the form of electrons, while the
plasmonic excitations that occupy the $\alpha$ bright state are bosonic.
Working with the full Fermionic Hamiltonian is an arduous task [26], that
could hinder the development of an intuitive understanding of the transport,
although very recently a Fermionic approach was successfully used to model
QCDs operating in the strong coupling regime [21].

The previous section II.2 suggests that the bosonization procedure of the
extractor, that we employed to describe the extraction process, is a novel and
readily interpretable approach for examining the injection process. In
particular the selection rules for the tunnel Hamiltonian, eqs. (13)-(16)
might prove a powerful tool. Due to the impossibility of conducting an
experimental study resembling the one carried out for QCDs for a detection
process, the following discussion will be supported by the quantitative
arguments previously presented in section II.3. Note: the $\beta$ extractor
states are now referred to as injector states.

An injection process is inherently incoherent because it introduces electrical
excitations into an intersubband system through an incoherent external bath of
electrons.

The relevant coherence here is the one of the ISB plasmon[18, 19, 20], that is
a collective - and coherent - matter excitation originating from the
electronic plasma inside a semiconductor quantum well (QW). In this respect,
an intuitive picture suggests that for an ISB polariton system, the electrical
injection process is not the reverse of the electrical extraction. In the
latter, coherence (induced by light) is destroyed to generate an electrical
current, while in the former it appears that coherence must be created.

More formally, in the framework of a bosonized injector, we expect most of the
electronic population to be located in the dark states $|B_{i}^{\beta}\rangle$
($i\neq 1$) upon electrical injection. Furthermore, to emit light, excitations
must be transferred to the plasmonic bright state $|B_{1}^{\alpha}\rangle$,
which holds the entire oscillator strength of the system. However, the
selection rules (14) and (15) are clear: it is impossible for a dark state
from the injector to interact with the plasmonic bright state through a tunnel
interaction. In other words, the primary injection pathway, which would
involve direct transfer from the injector states to the bright plasmonic
state, can not be taken. The bosonized injector formalism confirms that
polaritonic emitters do not operate as reversed polaritonic detectors.

In QCDs, the coherence is established through the photonic mode and maintained
up to the extractor using both light-matter coupling $\Omega$ and tunnel
coupling $\Omega_{T}$. Coherence can also be lost through the irreversible
intrasubband scatterings $\gamma_{\alpha}^{\text{intra}}$ in the plasmonic
mode, although we have demonstrated that it is not the main extraction scheme.
However, the extraction process can still take place, since the usual dark-to-
dark tunnel interactions are possible (Eq. (16)). On the contrary, in a LED
the injection mechanism is incoherent, and coherence cannot emerge
spontaneously during the transport. Additionally, we showed that incoherent
(dark) states cannot interact with a coherent (bright) via the tunneling
Hamiltonian (Eq. (15)) and (Eq. (14)). As a result, it seems unfeasible to
efficiently transfer excitations to the optically active bright state
$\alpha$, and thus to the polaritonic states, in the absence of an additional
mechanism to generate coherence.
In the case where the electrical injection would be uniform among the $N$
states of $|B_{i}^{\beta}\rangle$, light could be emitted since the system
would start with excitations in $|B_{1}^{\beta}\rangle$, but the expected
efficiency would be at most $1/N$, without considering intrasubband
decoherence.

There are however two points that need to be discussed further. First, light
emission from another kind of polariton states under electrical injection is
well documented, namely in exciton-polariton devices [2, 38, 39], with
additional reports of polariton lasing under electrical injection [40, 41].
The key difference is that exciton-polaritons states do not result from a
collective matter excitation, but rather from an ensemble of single-particle
transitions. As a consequence, non-resonant pumping schemes can apply to
exciton polaritons, as demonstrated in optical experiments.

Second, several reports of electroluminescence from electrically-injected
polariton LEDs exist in the literature. Some of them clearly determine that
thermally assisted emission processes have a major role [14, 42], but in many
other ones simple thermal models cannot explain the data [11, 13, 12, 15]. We
can only conjecture possible ways forward to elucidate electrical injection of
polaritonic LEDs. On one hand, one might wonder if the application of the
generalized, local Kirchoff [43] law to ISB polariton LEDs can shine new light
on the electrical injection process, and possibly explain all the existing
experimental data in the literature. On the other, the problem of electrical
excitation of coherent electronic motion - which is essentially the mechanism
at play in electrically pumped polariton emitters - is well known from the
field of surface plasmon polaritons (SPPs) [44, 45, 46, 47, 48, 49, 50]. The
extremely low efficiency of the electron-to-plasmon and electron-to-photon
processes is well known, although recent theoretical works, supported by one
experimental finding, have demonstrated that the efficiency could be
drastically increased by tailoring the electronic landscape to favor inelastic
over elastic tunneling, as long as the electronic coherence is preserved in
the process [51, 52].

###### Acknowledgements.

We thank S. De Liberato, J-M Manceau, I. Carusotto, A. Bousseksou for helpful
discussions. We acknowledge financial support from the European Union Future
and Emerging Technologies (FET) Grant No. 737017 (MIR-BOSE), and by the French
National Research Agency: project SOLID (No. ANR-19-CE24-0003), HISPANID
(ANR-17-ASTR-0008-01), and EVEREST (ANR-21-CE24-0021).

##  Appendix A Quantum master equation model for a QCD operating in the strong
light-matter coupling regime: parametric study of the impact of the light-
matter coupling strength on the transfer function

Figure 4: Light-matter coupling $\Omega$ parametric study for the different
quantities of interest: [a] Total absorption of the system (ISB absorption,
extractor dissipation and cavity absorption) [b] Internal QCD absorption (ISB
absorption and extractor dissipation) [c] Photocurrent (extractor dissipation)
[d] Transfer function between the extractor dissipation
($\mathcal{A}_{\beta}:=\mathcal{J}_{\beta}/|s_{+}|^{2}$) and the internal QCD
absorption ($\mathcal{A}_{\alpha}+\mathcal{A}_{\beta}$). The blue dashed line
represents an equivalent situation weak coupling situation where the cavity is
not included in the model, and the ISB $\alpha$ transition is directly pumped
(every other parameters are exactly the same as the strong coupling
situations).

The transfer function between the photocurrent and the total power dissipated
inside the QCD
($\mathcal{A}_{\text{QCD}}=\mathcal{A}_{\alpha}+\mathcal{A}_{\beta}$) is
defined as $\mathcal{T}$:

$\displaystyle\mathcal{T}(\omega)=\frac{\mathcal{A}_{\beta}}{\mathcal{A}_{\alpha}+\mathcal{A}_{\beta}}$
(39)

$\mathcal{T}$ is dependent on the light frequency $\omega$.

###  A.1 Parametric study

Fig. 4 plots the different quantities $\mathcal{A}_{\text{tot}}$
($\mathcal{A}_{\text{tot}}=\mathcal{A}_{\text{QCD}}+\mathcal{A}_{c}$),
$\mathcal{A}_{\text{QCD}}$, $\mathcal{J}_{\beta}$ and $\mathcal{T}$ computed
from the solution of equation (II.3). We impose a realistic situation between
the inter and intra-subband dynamics within the QCD such that 90% of the total
broadening is due to the intrasubband scattering:

$\displaystyle\gamma_{\alpha}^{\text{intra}}+\gamma_{\beta}^{\text{intra}}=0.9\cdot\gamma_{\alpha\beta}$
(40)

where
$\gamma_{\alpha\beta}=\gamma_{\alpha}^{\text{intra}}+\gamma_{\beta}^{\text{intra}}+\gamma_{\alpha}+\gamma_{\beta}$
represents the total contribution to the broadening from the $\alpha$ and
$\beta$ transitions, including intersubband and intrasubband scatterings.
This assumption is equivalent to set $T_{1}\approx 10\cdot T_{2}$, where
$T_{2}$ ($T_{1}$) are the dephasing (upper state) lifetime, respectively. For
a typical mid-IR ISB transition this is verified, as we have $T_{1}$ of the
order of the ps, and $T_{2}$ of the order of a few hundreds fs. The cavity
resonance $\omega_{c}$ and the extractor resonance are also voluntarily
mismatched with the ISB transition:

$\displaystyle\omega_{c}=1.05\omega_{\alpha},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\omega_{\beta}=0.95\omega_{\alpha}$ (42)

$\mathcal{A}_{\text{tot}}$, $\mathcal{A}_{\text{QCD}}$, $\mathcal{J}_{\beta}$
and $\mathcal{T}$ are computed for different light-matter coupling amplitudes
$\Omega$, up to 10% of the ISB transition $\omega_{\alpha}$.

When the light-matter coupling ratio $\Omega/\omega_{\alpha}$ increases, the
system progressively moves from a weak coupling regime to a strong coupling
regime: around the spectral resolution criteria
$2\Omega>\gamma_{\alpha\beta}$, we compute the characteristic splitting of the
polaritonic peaks, for each spectra $\mathcal{A}_{\text{tot}}$ (A),
$\mathcal{A}_{\text{QCD}}$ (B) and $\mathcal{J}_{T}$ (C). The model is able to
reproduce the smaller splitting of the QCD absorption (B) compared to the
splitting of the total absorption (A) for a same coupling situation
$\Omega/\omega_{\alpha}$, something previously observed in [17]. The important
novelties that brings the model are found in the transfer function
$\mathcal{T}$. In weak coupling (small ratios $\Omega/\omega_{\alpha}$), the
transfer function is almost scalar: it coincides with the transfer function
computed in the framework of a QCD that is not inside a cavity. As the ratio
$\Omega/\omega_{\alpha}$ increases, the baseline of the transfer function
gradually falls, and the amplitude of its peak increases: increasing $\Omega$
enables the transfer function to reach a Lorentzian shape.

Therefore, in a model where the intra-subband dynamics is explicitly
described, the progressive increase of the light-matter coupling allows us to
move continuously from a sequential transport in QCDs (flat, quasi-scalar
transfer function $\mathcal{T}(\omega)$) to a delocalized description of the
transport (sharp, Lorentzian transfer function). Again, when the strong light-
matter coupling $\Omega$ is sufficiently intense, the coherent nature of the
transport is maintained during the extraction process.

The previous discussion explains the satisfactory description of the
photocurrent experimental data produced by the semi-classical CMT obtained in
our previous work [17], despite the impossibility in this previous model to
describe the intrasubband dynamic. By default, the CMT predicts a sharp
Lorentzian transfer function $\mathcal{T}$. While this description is not
suited for a weak coupling scenario, where the sequential transport should be
described with a scalar transfer function, Fig. 4-[D] illustrates that it is
on the other hand quite adapted to a strong coupling scenario and a
delocalized transport scheme. However, being a semi-classical model, the CMT
also lacked the ability to distinguish between the inter and intrasubband
dynamic which would prevent the distanglement between the spectra broadening
and their amplitude.

###  A.2 Tunneling current

Another quantity of interest is the tunneling current $\mathcal{J}_{T}$
between the plasmonic mode $\alpha$ and the electronic extraction mode
$\beta$. It is defined as:

$\displaystyle\mathcal{J}_{T}=\Omega_{T}(\braket{b_{\alpha}^{\dagger}b_{\beta}}-\braket{b_{\alpha}b_{\beta}^{\dagger}})$
(43)

Using Eq. (II.3) in the low excitation regime, and developing the expressions
of the coherences, $\mathcal{J}_{T}$ can be approximated as:

$\displaystyle\mathcal{J}_{T}=$
$\displaystyle\frac{\displaystyle{2\Omega_{T}^{2}\gamma_{\alpha\beta}}}{\displaystyle{(\tilde{\omega}_{\alpha}-\omega_{\beta})^{2}+(\gamma_{\alpha\beta})^{2}}}\left(\braket{b_{\alpha}^{\dagger}b_{\alpha}}-\braket{b_{\beta}^{\dagger}b_{\beta}}\right)$
(44) $\displaystyle+\Re\left[\frac{\displaystyle
2i\Omega_{T}\Omega}{\displaystyle(\tilde{\omega}_{\alpha}-\omega_{\beta})^{2}+(\gamma_{\alpha\beta})^{2}}\braket{a_{c}b_{\beta}^{\dagger}}\right]$

where
$\gamma_{\alpha\beta}=\gamma_{\alpha}+\gamma_{\alpha}^{\text{intra}}+\gamma_{\beta}+\gamma_{\beta}^{\text{intra}}$
is the sum of the different contributions to the coherences damping inside the
QCD. The obtained expression of $\mathcal{J}_{T}$ in Eq.(44) is decomposed in
two contributions. The fist term is the standard sequential tunnel current
[53, 54] (in its first order expression) which is broadely used for the
electronic transport in QCD operating in the weak coupling regime [22]. It is
a semi-classical expression of the current, in the sense that it directly
involves the population difference between the modes involved in the tunneling
process
$\braket{b_{\alpha}^{\dagger}b_{\alpha}}-\braket{b_{\beta}^{\dagger}b_{\beta}}$.
The second term is a new addition to the tunnel current. It involves the
coherences $\braket{a_{c}b_{\beta}^{\dagger}}$ between the cavity and the
extractor modes, which could be qualified as long-range coherences (the two
modes are only coupled through their mutual coupling to the ISB mode). It thus
expresses the system capacity to transport current between modes that are not
directly coupled. We will refer to this current as delocalized current.

The amplitude of the delocalized current of Eq.(44) is controlled by a
Lorentzian function and involves the cross product of the couplings
$\Omega_{T}$ (tunnel coupling) and $\Omega$ (light-matter coupling). In the
case of a weakly coupled QCD, it is thus expected that the delocalized current
is null. Note that it can be numerically checked that the current expression
is independent of the considered interface where its computed, thus
$\mathcal{J}_{T}=\mathcal{J}_{\beta}$.

###  A.3 Validity domains of the different models

Figure 5: Sharpness of the transfer function $\mathcal{T}$ as a function of
the light-matter coupling strength ratio $\Omega/\omega_{\alpha}$ and the
intrasubband scattering predominance $\gamma^{\text{intra}}$ in both $\alpha$
and $\beta$ transitions with respect to the total scattering
$\gamma_{\alpha\beta}$. This map gives intuitive validity domains for all the
different physical models considered up to now, but the limits between the
domains are arbitrary: a flat transfer function ($r<0.3$) suggests a
sequential transport model, whereas a sharp transfer function ($r>0.7$)
suggests a delocalized transport model, described by the Coupled Mode Theory.
The quantum master equation model presented in this article covers the whole
domain.

To explore the validity domains of the different models introduced here and in
our previous work [17] (sequential model, CMT model, quantum master equation
model), we define a criterion based on the spectral shape of the transfer
function $\mathcal{T}(\omega)$, the sharpness $r$:

$\displaystyle
r(\Omega,\gamma_{\alpha}^{\text{intra}}+\gamma_{\beta}^{\text{intra}})=\frac{\text{Max}\\{\mathcal{T}(\omega)\\}-\text{Min}\\{\mathcal{T}(\omega)\\}}{\text{Max}\\{\mathcal{T}(\omega)\\}}$
(45)

$r=1$ thus indicates that the transfer function $\mathcal{T}(\omega)$ is a
sharp Lorentzian function, $r=0$ indicates that $\mathcal{T}(\omega)$ is a
flat scalar function. Fig. 5 summarizes the results of the parametric
exploration on both the total intrasubband scattering and the light-matter
coupling strength. We differentiate three domains D1, D2 and D3:

  1. 1.

Domain D1: sequential transport model, flat scalar transfer function
($\mathcal{T}(\omega)\approx p_{E}$). This domain is correctly described by
the standard thermalized subband model for QCD.  It corresponds for instance
to QCDs operating in the weak-coupling regime

  2. 2.

Domain D3: delocalized transport model, sharp Lorentzian transfer function.
This domain is correctly described by the CMT.

  3. 3.

Domain D2: intermediate domain, where transport combines contributions from
different sources. D1, D2 and D3 are correctly described by the density matrix
formalism of equation (II.3) and the capability to distinguish intra and
inter-subband dynamic.

##  Appendix B Experimental system and QCD bandstructure

In this section, we present additional information about the samples used in
this work. Fig. 6 presents a scanning electron microscope (SEM) image of a
patch cavity array, and Fig. 7 presents the bandstructure of the QCD embedded
inside the patches. The bandstructure is computed using our sequential
transport software [22].

Figure 6:  SEM image of a patch cavity-embedded QCD detector. $s$ is the patch
lateral size and $p$ is the array period. Patches are electrically connected
using gold wires deposited on a dielectric bridge layer. The active layers,
the QCDs, are embedded between gold layers (Au).

Figure 7: Bandstructure of the QCD used for the photocurrent measurements
studied in Fig. 2. Here the electric field applied on the structure is $F=-10$
kV.cm-1. The extraction probability computed for this field and $T=78$ K is
$p_{E}=0.43$ [55]. The potential of the quantum wells is calculated by
considering the gradual variation in the composition profile.

##  Appendix C Additionnal photocurrent measurements and computational results

In this section, we present additional photocurrent measurements and
computational results to supplement the results of Fig. 2.

Figure 8: Normalized photocurrent measurements (continuous lines) and quantum
master equation global fit (dashed lines), for two cavity geometries [a]
$s=1.50$ µm, $p=5$ µm and [b] $s=1.6$ µm, $p=7$ µm. Offsets are added for
clarity. Filled areas represent the errors of the fit parameters propagated on
the spectra. The extractor frequency $\omega_{\beta}(F)$, dependent of the
electric field $F$, and the plasma-shifted ISB transition
$\tilde{\omega}_{\alpha}$ are both superimposed on the spectra.

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