# Theoretical proposals to measure resonator-induced modifications of the
electronic ground-state in doped quantum wells

Yuan Wang  Simone De Liberato<EMAIL_ADDRESS>School of Physics and
Astronomy, University of Southampton, Southampton, SO17 1BJ, United Kingdom

###### Abstract

Recent interest in the physics of non-perturbative light-matter coupling led
to the development of solid-state cavity quantum electrodynamics setups in
which the interaction energies are comparable with the bare ones. In such a
regime the ground state of the coupled system becomes interaction-dependent
and is predicted to contain a population of virtual excitations which,
notwithstanding having been object of many investigations, remain still
unobserved. In this paper we investigate how virtual electronic excitations in
quantum wells modify the ground-state charge distribution, and propose two
methods to measure such a cavity-induced perturbation. The first approach we
consider is based on spectroscopic mapping of the electronic population at a
specific location in the quantum well using localised defect states. The
second approach exploits instead the photonic equivalent of a Kelvin probe to
measure the average change distribution across the quantum well. We find both
effects observable with present-day or near-future technology. Our results
thus provide a route toward a demonstration of cavity-induced modulation of
ground-state electronic properties.

##  I Introduction

The many advances in the fabrication of nanophotonic resonators and
nanostructured materials have made solid-state cavity quantum electrodynamics
(CQED) an interdisciplinary research domain, with applications ranging from
chemistry [1] to machine learning [2]. One of the figures of merit of CQED
setups which has seen sustained improvements is the coupling strength between
light and matter. When their mutual interaction energy becomes comparable with
the excitation energy, higher order effects become observable, a regime called
ultrastrong coupling [3, 4, 5]. In 2009 the impact of these higher-order
perturbative effects was observed for the first time in the anomalous shift of
intersubband polariton resonances: transitions between conduction subbands in
doped semiconductor quantum wells (QWs) strongly coupled to photonic
resonators [6]. Large couplings with the photonic vacuum is expected to modify
not only the system’s optical response, but also the underlying matter degrees
of freedom. Interest in the impact of CQED effects on electronic and molecular
degrees of freedom dates back to Khurgin’s idea of very strong coupling [7, 8,
9, 10], but only in more recent years a broader interest followed [11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21, 22].

Crucially for this work, the ground state of an ultrastrongly coupled CQED
system is predicted to host a cloud of virtual excitations, both photonic and
matter ones [23]. These excitations are predicted to be stable also in
realistic dissipative environments [24] but, notwithstanding many theoretical
works proposing different approaches to observe them, a direct measurement is
still missing. Except a proposal to observe the virtual excitations via
electro-optical sampling [25], whose efficacy has been questioned [26], and
one to use the Lamb shift of an ancilla qubit [27], all the other proposals we
are aware of deal with variants of one basic idea: to non-adiabatically
modulate the system in order to make some of these excitations real [28, 29,
30, 31, 32, 33, 34]. One problem with this idea is that we need to consider a
modulated, time-dependent system, in analogy with the dynamical Casimir effect
[35, 36, 37], and similarly sensitive to the density of dressed states, not to
the presence of vacuum excitations. A second problem is that, in order to
achieve non-vanishing emission, perturbation frequencies of the order of the
bare optical frequency are necessary [38, 39], a requirement which has until
now thwarted any attempt to observe vacuum excitations.

This paper explores a novel approach to these problems, by noticing that
cavity-induced virtual electronic excitations do modify the ground-state
charge distribution, and they can thus be measured exploiting already well-
tested methods used to map charge distributions in nanoscopic systems without
requiring any non-adiabatic modulation. Our results thus provide both a route
to a first direct observation of virtual excitations in the ultrastrong-
coupling vacuum, and a test bench for theoretical and numerical approaches
studying CQED-induced modifications of the electronic ground state.

##  II System description

Figure 1: Example of an asymmetric QW potential $V(z)$ with its envelope
wavefunctions (a) and in-plane dispersions (b) of the first two conduction
subbands. The vertical dotted lines represent the average charge position in
each subband. The green arrow shows the localised defect wavefunction and the
horizontal dotted lines the envelope wavefunction at the defect position in
the two subbands. The definition of all the marked quantities can be found in
the main text.

We consider a stack of $n_{QW}$ QWs, doped with a two-dimensional electron gas
of density $\sigma_{e}$. The electrons in each QW occupy parallel parabolic
subbands with dispersion $\omega_{jk}$, function of the subband $j$ and in-
plane wavevector $\mathbf{k}$. The Fermi wave vector $k_{F}$ is chosen such
that the Fermi energy $E_{F}$ lies between the first and the second subbands.
Electronic wavefunctions, dispersions, and other quantities introduced in this
Section are shown in Fig. 1.

The wavefunctions in th $n$th QW can be written in the envelope function
approximation

$\displaystyle\phi_{jn\mathbf{k}}(\mathbf{r})$
$\displaystyle=\psi_{j}(z-z_{n})\frac{e^{i\boldsymbol{k}\cdot\boldsymbol{\rho}}}{\sqrt{S}},$
(1)

where $\mathbf{r}=(z,\boldsymbol{\rho})$ is the position vector decomposed in
cylindrical coordinates, $S$ is the sample surface, and $z_{n}$ the $n$th QW
position with the origin fixed at $z_{1}=0$. We introduce the corresponding
annihilation operators $\hat{c}_{jn\mathbf{k}}$, obeying Fermionic
anticommutation rules

$\displaystyle\left\\{\hat{c}_{jn\mathbf{k}},\hat{c}^{\dagger}_{j^{\prime}n^{\prime}\mathbf{k}^{\prime}}\right\\}=\delta_{jj^{\prime}}\delta_{nn^{\prime}}\delta(\mathbf{k}-\mathbf{k}^{\prime}).$
(2)

Here and in the following the electron spin is not explicitly marked and the
spin multiplicity is implicitly summed over. The ground state of the uncoupled
system, describing the electron gas in the electromagnetic vacuum is then

$\displaystyle\mathinner{|{F}\rangle}$
$\displaystyle=\prod_{n=1}^{n_{QW}}\prod_{k<k_{F}}\hat{c}^{\dagger}_{1n\mathbf{k}}\mathinner{|{0}\rangle},$
(3)

where $\mathinner{|{0}\rangle}$ is the vacuum state
$\hat{c}_{jn\mathbf{k}}\mathinner{|{0}\rangle}=0$. The unperturbed charge
distribution corresponding to the uncoupled state $\mathinner{|{F}\rangle}$
can be written as

$\displaystyle\rho_{F}(z)$
$\displaystyle=\sum_{n}\lvert\psi_{1}(z-z_{n})\rvert^{2}\sigma_{e}.$ (4)

We use the formalism initially developed for the multi-subband case [40], in
order to be able to treat the most general cases, but for the sake of clarity
in this paper we will specialise it to consider only the first two electronic
subbands, an approximation which, in the proper gauge, can be justified when
higher-lying subbands are either detuned or weakly coupled [41].

We consider the intersubband transitions to be almost vertical in momentum
space. This implies that the frequency of the resonant transition with in-
plane momentum $\mathbf{q}$ is given by

$\displaystyle\omega_{2\mathbf{k+q}}-\omega_{1\mathbf{k}}\approx\omega_{12},$
(5)

thus neglecting terms of the order of $qv_{F}$, with $v_{F}$ the Fermi
velocity. The QWs are embedded in a double metal nanopatch resonator of length
$L$ of which we retain only transverse-magnetic (TM) modes due to selection
rules of intersubband transitions. The normalised TMm mode profile can be
written as

$\displaystyle f_{m\mathbf{q}}(\mathbf{r})$
$\displaystyle=\sqrt{\frac{2}{(1+\delta_{m0})LS}}\cos(\frac{\pi
mz}{L})e^{i\mathbf{k}\cdot\boldsymbol{\rho}},$ (6)

with in-plane wavevector $\mathbf{q}$, out-of-plane one $\frac{\pi m}{L}$, and
$m\in\mathbb{N}_{0}$. The $\delta_{mn}$ is the Kronecker symbol. Photons in
these modes will be described by the boson annihilation operators
$\hat{a}_{m\mathbf{q}}$, whose bare frequencies $\omega_{mq}$ obey the
dispersion relation

$\displaystyle\epsilon_{r}\frac{\omega_{mq}^{2}}{c^{2}}$
$\displaystyle=q^{2}+\frac{\pi^{2}m^{2}}{L^{2}},$ (7)

with $\epsilon_{r}$ the background dielectric constant and $c$ the speed of
light.

We are interested to determine the modification in the charge distribution
along the $z$-axis induced in the electron gas by the coupling to the photonic
resonator. As such we can employ a theory whose degrees of freedom are not the
electron themselves, but their transitions and in particular those coupled
with the photonic field. We thus introduce the collective, bright intersubband
transition in QW $n$, with in-plane wave vector $\mathbf{q}$

$\displaystyle\hat{b}_{n\mathbf{q}}^{\dagger}$
$\displaystyle=\frac{1}{\sqrt{S\sigma_{e}}}\sum_{\mathbf{k}}\hat{c}^{\dagger}_{2n\mathbf{k+q}}\hat{c}_{1n\mathbf{k}}.$
(8)

These excitations obey quasi-bosonic commutation relations

$\displaystyle\left[\hat{b}_{n\mathbf{q}},\hat{b}_{n^{\prime}\mathbf{q}^{\prime}}^{\dagger}\right]$
$\displaystyle=\delta_{nn^{\prime}}\delta(\mathbf{q}-\mathbf{q}^{\prime})+\delta_{nn^{\prime}}O(\frac{\sigma_{n}}{\sigma_{e}}),$
(9)

and they can thus be approximated as bosons in the sector of the Hilbert space
where the excitation density in the $n$th QW $\sigma_{n}$ is much smaller than
the electron density [42]

$\displaystyle\sigma_{n}\ll\sigma_{e}.$ (10)

The coupling of intersubband transitions to the electromagnetic field can be
described using the Power-Zienau-Woolley (PZW) Hamiltonian including self-
interaction terms leading to the depolarization shift [43]. As shown in
Appendix A, such an Hamiltonian can be diagonalised in the bosonic regime
using a Hopfield-Bogoliubov rotation. The coupled theory is then described by
free bosonic polaritonic operators, linear superpositions of the bare ones

$\displaystyle\hat{p}_{s\mathbf{q}}$
$\displaystyle=\sum_{m}\left(x_{smq}\hat{a}_{m\mathbf{q}}+z_{smq}\hat{a}^{\dagger}_{m\mathbf{-q}}\right)$
(11)
$\displaystyle+\sum_{n}\left(y_{snq}\hat{b}_{n\mathbf{q}}+w_{snq}\hat{b}_{n\mathbf{-q}}^{\dagger}\right).$

The linear transformation in Eq. (11) can then be inverted to yield

$\displaystyle\hat{b}_{n\mathbf{q}}$
$\displaystyle=\sum_{s}\left(\bar{y}_{snq}\hat{p}_{s\mathbf{q}}-\bar{w}_{snq}\hat{p}^{\dagger}_{s-\mathbf{q}}\right).$
(12)

From Eq. (12) we can calculate the total density of matter excitations in the
$n$th QW in the coupled ground state defined by
$\hat{p}_{s\mathbf{q}}\mathinner{|{G}\rangle}=0$

$\displaystyle\sigma_{n}$
$\displaystyle=\frac{1}{S}\sum_{\mathbf{q}}\mathinner{\langle{G}|}\hat{b}_{n\mathbf{q}}^{\dagger}\hat{b}_{n\mathbf{q}}\mathinner{|{G}\rangle}=\frac{1}{S}\sum_{s\mathbf{q}}\lvert
w_{snq}\rvert^{2}.$ (13)

Intersubband transitions modify the electronic density because, as clearly
shown in the example in Fig. 1, the electronic charge density in each subband
is different. In Appendix B we extend the approach from Ref. [44] to express
the the ground state electronic wavefunction in term of bosonised excitations.
We are then able to put the ground state electronic distribution in the form

$\displaystyle\rho_{G}(z)$
$\displaystyle=\rho_{F}(z)+\sum_{n}\Xi(z-z_{n})\sigma_{n},$ (14)

with

$\displaystyle\Xi(z)$
$\displaystyle=\left[\lvert\psi_{2}(z)\rvert^{2}-\lvert\psi_{1}(z)\rvert^{2}\right].$
(15)

In the following we will discuss two different experimental procedures to
measure the quantity in Eq. (13), exploiting two different observables linked
to the modifications it induces on the charge distribution. On the one hand,
electronic excitations modify the charge distribution at a specific position
$z$, as shown in Eq. (14). On the other hand, if the QW potential is
asymmetric, electrons in different subbands have a different average positions
along the sample growth axis $z$, leading to an average charge displacement
when an electron jumps between the first two subbands. In the two-subband
approximation we are employing the level of asymmetry can be quantified by a
single parameter: the average electron displacement induced by an intersubband
transition

$\displaystyle\Delta z$
$\displaystyle=\mathinner{\langle{\psi_{2}}|}\hat{z}\mathinner{|{\psi_{2}}\rangle}-\mathinner{\langle{\psi_{1}}|}\hat{z}\mathinner{|{\psi_{1}}\rangle}.$
(16)

A graphical illustration of both $\Xi(z)$ and $\Delta z$ can be found in Fig.
1.

An estimate of the expected excitation density, to the leading order in the
coupling, can be obtained by performing a lowest order expansion of
$\sigma_{n}$ as a function of the plasma frequency of the electron gas in the
QW $\omega_{P}$, which quantifies the strength of the collective light-matter
coupling. To the first perturbative order the coupled ground state can be
written as

$\displaystyle\mathinner{|{G}\rangle}$
$\displaystyle\approx\mathinner{|{F}\rangle}-\sum_{mn\mathbf{q}}\frac{\omega_{P}K_{mnq}}{\omega_{12}+\omega_{mq}}\hat{a}^{\dagger}_{m-\mathbf{q}}\hat{b}^{\dagger}_{n\mathbf{q}}\mathinner{|{F}\rangle},$
(17)

where the dimensionless coefficient $K_{mnq}$, defined in Appendix A, embeds
all the other microscopic details of the CQED setup. From Eq. (13), to the
lowest non-trivial order, we thus obtain the following expression for the
electronic excitation density in the $n$th QW

$\displaystyle{\sigma}_{n}$
$\displaystyle\approx\frac{1}{S}\sum_{m\mathbf{q}}\frac{\omega_{P}^{2}K_{mnq}^{2}}{(\omega_{12}+\omega_{mq})^{2}}.$
(18)

The sum in Eq. (18) is mathematically divergent, also due to the dipolar
approximation used in our theory which breaks down for photonic wavelengths of
the order of the QW width. Still, for $\omega_{mq}$ larger than a much smaller
cut-off of the order of the plasma frequency of the metallic mirrors
$\omega_{C}$, the mirrors become transparent and their position can not
efficiently affect the electromagnetic confinement. As we will see in the
following Sections, the two measurement schemes we will propose aim to measure
the impact of the cavity length on the excitation density, while filtering out
the potentially much larger signal independent from the cavity length. It will
thus be useful to partition the sum in Eq. (18) into two components

$\displaystyle{\sigma}_{n}\approx{\sigma}_{n}^{<}+\sigma_{n}^{>}.$ (19)

The first component of Eq. (19) takes into account the sum of the modes well
confined by the photonic cavity and it thus depends upon the cavity length $L$

$\displaystyle{\sigma}_{n}^{<}$
$\displaystyle=\frac{1}{S}\sum_{m\mathbf{q}}\frac{\omega_{P}^{2}K_{mnq}^{2}}{(\omega_{12}+\omega_{mq})^{2}}\Theta(\omega_{C}-\omega_{mq}),$
(20)

where $\Theta$ is the Heaviside function. The second component of Eq. (19)
includes instead all unconfined modes, and it can be considered independent of
$L$

$\displaystyle\partial_{L}\sigma_{n}^{>}$ $\displaystyle=0.$ (21)

We will develop all our theory assuming zero temperature, which is a good
approximation even at room temperature for intersubband energies
$\hbar\omega_{12}\gg 25$meV. Spurious thermal excitations will be present at
any non-zero temperature, but their density will be independent of $L$ and we
will thus operationally consider them as included in the $\sigma_{n}^{>}$
component.

For the sake of simplicity and generality in the following we will retain only
the contribution of the first cavity mode $m=0$ to $\sigma_{n}^{<}$. We can
see from Eq. (18) that the total population of virtual excitations is a sum of
positive terms corresponding to each value of $m$. Considering only the $m=0$
mode provides thus a lower bound on the intensity of the signal to be
measured, allowing us to fix the minimal experimental requirements to observe
it while avoiding complications due to time-variation in the number of
involved photonic modes and other secondary details as the exact position of
each QW in the cavity. Inserting the coefficient $K_{mnq}$ defined in Appendix
A into Eq. (18) we thus obtain, to the leading order in $\omega_{C}$

$\displaystyle\sigma^{<}(L)$
$\displaystyle\approx\frac{\omega_{C}z_{12}^{2}e^{2}\sigma_{e}}{4\pi\epsilon_{0}c^{2}\hbar
L},$ (22)

where we marked explicitly the $L$-dependence. Given that in such an
approximation the electronic excitation density is the same in each QW, in Eq.
(22) and in the following we will not mark the QW index $n$ explicitly, but it
remains intended that all the electronic densities are expressed per QW.

##  III Spectroscopic measurement

Figure 2: Sketch of the setup (a) and measurement scheme (b) described in Sec.
III. The pump laser induces absorption from the two subbands into the excited
defect state at position $z_{d}$. When the position of the top mirror is
modulated in time with amplitude $aL_{0}$ the relative population of the two
subbands, and thus the electronic density at $z_{d}$, is also modified,
leading to a time-dependet absorption $\Gamma(t)$.

In this Section we discuss how to spectroscopically measure $\sigma^{<}$ by
using defect modes grown in each QW to map the perturbation of the electronic
density. A conceptually similar procedure has already been used to measure the
real-space charge density in GaAs QWs using indium and aluminum defects [45].
The apparatus and measurement scheme we will consider are schematised in Fig.
2. The top mirror of the photonic resonator is mechanically connected to the
tip of an atomic force microscope (AFM). As the tip is driven at its resonant
mechanical frequency $\omega_{m}$ the cavity length $L$ is modulated around
its equilibrium value $L_{0}$ with the normalised amplitude $a<1$

$\displaystyle L$ $\displaystyle=L_{0}\left[1+a\cos(\omega_{m}t)\right].$ (23)

The oscillation changes the photonic cavity length, thus modifying the number
of ground-state virtual excitations $\sigma^{<}$. The mechanical resonance of
the cantilever $\omega_{m}$ is normally in the kHz range, much smaller than
all the other relevant frequency scales of the system. We can thus consider
the driven evolution adiabatic, and the system at equilibrium in its ground
state.

We consider that a planar layer of defect electronic states with energy
$\omega_{d}$ above the first subband $\omega_{1\mathbf{k}}$ and Lorentzian
lineshape with linewidth $\gamma_{d}$ is introduced in each QW. The defect
layer is placed at the position $z_{d}$ corresponding to a non-vanishing value
of $\Xi(z)$ in Eq. (15) as shown in Fig. 1(a). Note that in such a Figure we
represent an explicitly asymmetric QW potential, although asymmetry does not
play a role in this scheme. The asymmetry of the two subbands’ wavefunctions
will instead be crucial to the scheme described in the next Section, relying
on a non-vanishing value of $\Delta z$.

A laser pump at frequency $\omega_{d}$ and power $P$ is then shone on the
system, leading to electrons being excited from the ground state
$\mathinner{|{G}\rangle}$ to the defect state. The underlying idea is that the
absorption from each QW will be proportional to the charge density
$\rho_{G}(z_{d})$. The spectral component of the absorption at the cantilever
frequency $\omega_{m}$ will then be proportional to $\Xi(z_{d})$ and thus to
$\sigma^{<}$, providing a measure of the CQED-induced ground-state excitation
density. Note that the light-matter coupling in the QW is a collective
phenomenon depending on the total plasma frequency of the QW $\omega_{P}$.
This implies that while Eq. (10) is verified we can safely neglect the change
in the ground state energy due to transitions of $\sigma^{<}$ electrons to
defect states [46].

Using a simple Fermi golden rule approach we can write the total absorption
rate as

$\displaystyle\Gamma(t)$
$\displaystyle=\frac{4Pl_{d}n_{QW}\mu^{2}}{\hbar^{2}\gamma_{d}c\sqrt{\epsilon_{r}}\epsilon_{0}}\rho_{G}(z_{d},t),$
(24)

where $l_{d}\ll L_{QW}$ is the defect layer thickness, $\mu$ the defect dipole
moment, and we explicitly marked the time-dependency of $\rho_{G}$ through the
cavity length. Using Eq. (10), Eq. (22), and Eq. (23), and assuming that the
cantilever oscillations are small $a\ll 1$, Eq. (25) takes the form

$\displaystyle\rho_{G}(z_{d},t)$
$\displaystyle\approx\lvert\psi_{1}(z_{d})\rvert^{2}\sigma_{e}-\Xi(z_{d})\sigma^{<}(L_{0})a\cos(\omega_{m}t).$
(25)

The mirror oscillation thus translates in an oscillation of the charge coupled
to the defect mode and, through Eq. (24), in an harmonic modulation of the
laser absorption. Defining $\beta$ to be the quantum efficiency of the
detector, the total photocurrent can be written as

$\displaystyle I_{pc}(t)$ $\displaystyle=\beta
e\left[\frac{P}{\hbar\omega_{d}}-\Gamma(t)\right],$ (26)

and the variation in the photocurrent due to the QWs

$\displaystyle I_{s}(t)$ $\displaystyle=\beta e\Gamma(t).$ (27)

This leads to the time-independent shot-noise current

$\displaystyle i_{sh}$ $\displaystyle=\sqrt{2eI_{pc}\Delta\nu},$ (28)

where $\Delta\nu$ is the detection bandwidth. Assuming the shot-noise to be
the dominant noise term, that only a small fraction of the light is absorbed
by the QW $I_{s}\ll I_{pc}$, and after having applied a pass-band filter
around the frequency $\omega_{m}$ to remove the time-independent part of the
signal, we can use Eq. (24) to Eq. (28) to write the signal-to-noise ratio
[47]

$\displaystyle SNR$
$\displaystyle=\frac{I_{s}}{i_{sh}}=\sqrt{\frac{2\hbar\omega_{d}P\beta}{\Delta\nu}}\frac{l_{d}n_{QW}\mu^{2}\Xi(z_{d})a\sigma^{<}(L_{0})}{\hbar^{2}\gamma_{d}c\sqrt{\epsilon_{r}}\epsilon_{0}}.$
(29)

It is worth to notice that the scheme presented above is solid against both
fluctuations in the doping density $\sigma_{e}$, which gives only a continuum
signal easily filtered out, and against spurious mirror-induced forces. This
last point is quantitatively discussed in Appendix C, but it can be otherwise
inferred by the fact that appreciable modifications of the electronic QW
wavefunctions by metallic mirrors in close proximity would have otherwise
became apparent in precise measurements of intersubband polariton systems with
and without a metallic mirror [6].

##  IV Electrostatic measurement

Figure 3: Sketch of the setup described in Sec. IV (a) and its schematic
circuit representation (b). Each of the QWs, by creating a built-in potential
due to the photon-induced charge displacement, generates a potential
difference. When the position of the top mirror is modulated in time a current
flows through the circuit. The definition of all the marked quantities can be
found in the main text. In (c) we plot the discernability from Eq. (41) (solid
red) and them norm of the signal component from Eq. (38) (dashed blue) as a
function of the normalised oscillation amplitude $a$.

In the previous Section we investigated the ground-state excited electronic
population through a spectroscopic measurement of different charge
distribution between the two subbands at a specific position $z_{d}$,
quantified by the quantity $\Xi(z_{d})$. Here instead we will discuss how to
electrostatically measure the average charge displacement $\Delta z$ from Eq.
(16). While the former procedure is arguably simpler to implement, being more
robust against spurious effects, the latter one has the conceptual advantage
to measure a macroscopic real-space charge displacement, thus providing a more
direct proof of the CQED-induced ground-state modification.

The scheme here proposed is based on the fact that resonator-induced charge
perturbation described by the second term in Eq. (14) creates a built-in
electrostatic potential in each QW. Considering a sample surface $S\gg L^{2}$
large enough to neglect border effects, such a potential can be determined
solving the Poisson equation

$\displaystyle\partial^{2}_{z}V(z)$
$\displaystyle=-\frac{e\sigma\Xi(z)}{\epsilon_{0}\epsilon_{r}}.$ (30)

Writing a formal solution of Eq. (30)

$\displaystyle
V(z)=-\int_{-\infty}^{z}dz^{\prime}\int_{-\infty}^{z^{\prime}}dz^{\prime\prime}\;\frac{e\sigma\Xi(z^{\prime\prime})}{\epsilon_{0}\epsilon_{r}},$
(31)

integrating by part the derivative of $V(z)$, and exploiting the charge
conservation in each QW

$\displaystyle\int_{\mathbb{R}}\Xi(z)dz=0,$ (32)

we can calculate the potential drop across the QW

$\displaystyle\Delta V$
$\displaystyle=\frac{e\sigma}{\epsilon_{0}\epsilon_{r}}\int dzz\Xi(z).$ (33)

Using Eq. (14) and Eq. (16) we can put Eq. (33) in the form

$\displaystyle\Delta V$ $\displaystyle=\frac{e\sigma\Delta
z}{\epsilon_{0}\epsilon_{r}}.$ (34)

From Eq. (34) we see that the voltage induced across each QW by the coupling
with the transverse field of the resonator is the same created by two planes
of electron density $\sigma$ at a distance $\Delta z$. Given that $\Delta z$
can be calculated or independently measured, a measure of $\Delta V$ will thus
constitute a direct measure of the virtual excitation density in the ground
state.

The induced potential difference can not be measured with a standard voltmeter
because it is not a difference in the Fermi level but an in-built equilibrium
potential. In can nevertheless be measured using a modified Kelvin probe [48,
49]. Kelvin probes are usually employed to measure the difference in work
function between a known reference material and a sample, by using an AFM tip
as the top plate of a capacitor whose lower plate is the sample. When the
distance between the two is modulated by the tip oscillation, the capacitance
also varies. Being the potential difference fixed by the work functions, a
change in the capacitance leads to a change in the charge on the capacitor’s
plates, and thus to a measurable current. The apparatus we will consider,
schematised in Fig. 3(a), is thus similar to the one considered in the
previous Section, with the top mirror now doubling as capacitor top plate.
Moreover, the top and bottom mirrors are now electrically connected and the
time-dependent current flowing between them is measured as the tip oscillates.

Applying the Kirchoff’s law to the circuit equivalent of the apparatus shown
in Fig. 3(b) and exploiting Eq. (34) we can find the relation between the
photon-induced electron density $\sigma$ and the electronic density on the
mirrors’ surfaces $\sigma_{M}$

$\displaystyle\Delta zn_{QW}\sigma+D$ $\displaystyle=L\sigma_{M},$ (35)

where we added a fixed dipole density $D$, modelling the in-built potentials
independent of the mirror position which would be measured by a standard
Kelvin probe. Deriving over time the electron density on the mirror and using
Eq. (19) and Eq. (21) we can then calculate the electric current generated in
the Kelvin probe by the tip oscillation

$\displaystyle I_{K}$
$\displaystyle=eS\dot{\sigma}_{M}=eS\dot{L}\partial_{L}\frac{\Delta
zn_{QW}\sigma+D}{L}$ (36) $\displaystyle=eS\dot{L}\left[\Delta
zn_{QW}\partial_{L}\frac{\sigma^{<}}{L}-\frac{\tilde{D}}{L^{2}}\right].$

The first term in the square bracket in Eq. (36) represents the current
induced by the virtual excitations caused by photonic modes confined by the
cavity, and it is thus dependent on its length $L$, while in the second term
we have grouped all the $L$-independent background polarizations by defining
$\tilde{D}=\Delta zn_{QW}\sigma^{>}+D$.

Note that in Eq. (25) only the term proportional to $\sigma^{<}$ is time-
dependent, allowing for a simple suppression of the potentially much larger
time-independent term proportional to $\sigma_{e}$. In Eq. (36) instead, both
$\sigma^{<}$ and the background polarization $\tilde{D}$ lead to time-
dependent signals. This is to be expected given that a standard Kelvin probe
is meant to measure the static in-built potentials $\tilde{D}$, but it also
implies extra care has to be paid in order to extract $\sigma^{<}$ from the
measured signal. To this aim we can notice, plugging Eq. (22) into Eq. (36)
that the term due to CQED-induced charge modulation proportional to
$\sigma^{<}$ scales as $L^{-3}$, while the background polarization as
$L^{-2}$. Given that we know the time evolution of the cavity length $L(t)$
from Eq. (23), it is a priori possible to distinguish the two contributions,
but only if the tip explores a non-vanishing range of $L$, substantially
changing the cavity length. This time we will thus not be able to limit
ourselves to the perturbative regime $a\ll 1$.

In order to filter out the background polarization we can introduce the two
functions $g_{p}$ with $p=\\{2,3\\}$, periodic on the interval
$\left[0,2\pi\right]$

$\displaystyle g_{p}(x)$
$\displaystyle=\frac{\sin(x)}{\left[1+a\cos(x)\right]^{p}},$ (37)

their $l^{2}$-norm

$\displaystyle l_{p}$ $\displaystyle=\sqrt{\int_{0}^{2\pi}g_{p}(x)^{2}dx},$
(38)

and their normalised versions $\tilde{g}_{p}(x)=g_{p}(x)/l_{p}$. Substituting
Eq. (22) and Eq. (23) into Eq. (36) we can write the current as

$\displaystyle I_{K}(t)$
$\displaystyle=I^{<}\tilde{g}_{3}(\omega_{m}t)+\frac{eSa\omega_{m}\tilde{D}l_{2}}{L_{0}}\tilde{g}_{2}(\omega_{m}t),$
(39)

with

$\displaystyle I^{<}$
$\displaystyle=\frac{Sa\omega_{C}\omega_{m}z_{12}^{2}\Delta
ze^{3}n_{QW}\sigma_{e}l_{3}}{2\pi\epsilon_{0}c^{2}\hbar L_{0}^{2}}.$ (40)

By projecting Eq. (39) on the non-orthogonal normalised basis $\tilde{f}_{p}$
using the $l^{2}$ inner product we can then obtain a system of two equations
in two unknowns which can be solved determining the coefficients, assuming
their discernability

$\displaystyle\eta$
$\displaystyle=1-\lvert\int_{0}^{2\pi}dx\tilde{g}_{2}(x)\tilde{g}_{3}(x)\rvert^{2},$
(41)

is not vanishing. In Fig. 3(c) we plot $\eta$ showing that for large
oscillations ($a\approx 0.9$) we can reach a distinguishability of order
$10\%$, while having a norm $l_{3}$ of order 100. This procedure should allow
us to measure the intensity of the current due to the cavity-induced ground-
state modifications $I^{<}$ while filtering out static in-built potentials and
all the other contributions not dependent on the position of the metallic
mirror. The discrimination procedure could be further improved by performing
control experiments either removing the bottom metallic mirror, or using doped
semiconductors with lower plasma frequencies as oscillating mirrors, in order
to obtain independent measurements of $\tilde{D}$. Moreover, methods to
analyse time-resolved data using non-orthogonal decompositions as the one we
just discussed but integrating both known and unknown components also exist
and could be exploited in this context although their analysis is beyond the
scope of this paper [50].

Note that, beyond the doped QW system considered in this work, other
asymmetric CQED platforms have been considered in the literature, both
dielectric [51, 52] and superconducting [53]. It remains to be seen whether
the electrostatic measurement scheme we introduced could be extended to these
systems, further broadening our capability to measure and interact with the
light-matter coupled ground state.

##  V Numerical results

In this Section we will estimate the magnitude of the SNR expected from Eq.
(29) and of the current from Eq. (40), with the objective to ascertain whether
the two schemes we proposed are realisically implementable with present-day or
near-future technology.

In order to determine a test QW structure, which for simplicity we will
consider to be the same for the two schemes, we notice that strongly
asymmetric potentials increase the average charge separation and thus $\Delta
z$, but segregating the wavefunctions they also reduce the overlap between
$\psi_{1}(z)$ and $\psi_{2}(z)$ and thus the dipole moment $z_{12}$. This same
problem has been studied in the context of dipolar emission in driven
asymmetric quantum wells, where the dipole $e\Delta z$ oscillates and causes
emission at the vacuum [54] or pump [55] Rabi-frequency. In Ref. [56] the
parameter space of a simple asymmetric GaAs-based QW geometry has been
explored in order to identify structures with both non-negligible $\Delta z$
and $z_{12}$. In the following we will use the parameters of a structure
highlighted in such a publication, which are the ones we used in Fig. 1(a), to
have comparatively large values of both figures of merit while being solid
against fabrication tolerances.

We thus consider $n_{QW}=10$ GaAs-based QWs of length $L_{QW}=11.6$nm, with
$\hbar\omega_{12}=125$meV, $z_{12}=0.18L_{QW}$, $\Delta z=0.11L_{QW}$, and the
total QW stack height, including barriers, is taken to be $160$nm. Considering
a typical AFM scanning frequency $\omega_{m}=2\pi\times 100$kHz and an
equilibrium cavity length $L_{0}=2\mu$m, this makes it possible to choose
values of the normalised oscillation amplitude $a$ up to $0.9$ before the tip
touches the QW stack. The QWs are each doped at a surface density
$\sigma_{e}=10^{12}$cm-2, corresponding to a Fermi energy $E_{F}\approx 35$meV
from the bottom of the first subband, and we consider metallic mirrors with
$\hbar\omega_{C}=10$eV. The corresponding cut-off wave vector
$q_{C}=\frac{c}{\sqrt{\epsilon_{r}}\omega_{C}}$ is smaller than $0.5v_{F}$,
assuring that Eq. (5) remains at least qualitatively correct even close to the
cut-off.

We can thus numerically calculate the induced electronic excitation density
per QW from Eq. (22), obtaining $\sigma^{<}\approx 0.8\times
10^{6}$cm${}^{-2}\ll\sigma_{e}$, justifying a posteriori our bosonic approach
and our choice not to consider the backaction of the photon-induced charge
displacement upon the electronic wavefunctions, which would have obliged us to
solve a Schrödinger-Poisson equation instead of Eq. (30).

For the sample to evaluate Eq. (29) we place the defect layer at
$z_{d}=0.5L_{QW}$, corresponding to $\Xi(z_{d})\approx\frac{2}{L_{QW}}$, and
use a defect layer width of the order of the GaAs lattice parameter
$l_{d}=0.5$nm, with a dipole moment $\mu=4$D, and a linewidth
$\gamma_{d}=0.1$meV [57]. We consider a $100$ps plused laser with pulse energy
of $1$mJ and repetition rate of $100$kHz, synced with the AFM tip having a
small oscillation amplitude $a=0.1$. With a quantum efficiency $\beta=0.5$ and
an integration time of ten minutes $\Delta\nu=\frac{\omega_{m}}{1200\pi}$ Eq.
(29) leads to $SNR\approx 1$. Given the realistic parameters used, and the
fact the virtual electronic population in Eq. (22) represents a lower estimate
of the real value of $\sigma^{<}$, makes us confident that the spectroscopic
measurement scheme we proposed is within technological reach.

In order to evaluate Eq. (40), we choose instead $a=0.7$ and consider a device
surface $S=50\mu$m2, leading to a lower bound of the expected current of the
order of $I_{K}\approx 0.5$fA, small but still detectable with low-noise
electronics. In this scheme the larger value of $a$ could a priori bring the
mirror close enough to the QWs for the Casimir effect to exert a non-
negligible force on the electron gas. While such a force remains very small,
leading to a shift in the electronic position much smaller than $\Delta z$, it
would affect all the electrons, potentially leading to an effect which could
overshadow the larger shift of the few virtually-excited electrons. While such
contribution scales with higher powers of $L$, and it could thus potentially
be filtered out extending the approach described in Sec. IV, in Appendix C we
perform a rough overestimate of such an effect, showing that for the
parameters chosen it should not pose a critical hurdle to the detection of
CQED-induced effects.

Until now we completely neglected the effect of losses in our formalism. While
the coupling with the environment will modify the population of virtual
excitations in the ground state, in Ref. [24] one of us demonstrated that such
changes are limited even for over-damped systems.

##  VI Conclusions

In this paper we have explored the possibility of detecting resonator-induced
effects on the electronic ground state of a CQED system. In particular we
discussed two schemes which can achieve such an objective by providing a first
direct measurement of virtual electronic excitations in the ground state of a
CQED system. Instead of focussing, as most proposals do, on the detection of
photonic excitations through non-adiabatic modulation, we aim at detecting
material excitations using an adiabatic modulation. Our estimates show that
detection of electronic CQED-induced modifications of the electronic ground
state could be within technological reach. Although highly not trivial, the
experiments we propose would not incur in the many hurdles linked with sub-
cycle optical modulation and detection, which have until now thwarted multiple
efforts to observe ground-state virtual photons through the non-adiabatic
route.

We conclude by pointing out that the two schemes we proposed are not intended
to represent an exhaustive list. The general theory developed in Sec. II shows
how the ground-state charge distribution in doped quantum wells is modified by
the coupling with the electromagentic field. The results of many experiments
which can be performed on QWs depend on different details of the ground-state
charge distribution and we thus expect many different experiments could be
used to highlight its CQED-induced modifications. Other possible examples
which are not investigated in this paper include nonlinear emission in
asymmetric QWs [58] or lifting of selection rules is symmetric ones.

We hope this work will stimulate novel interest toward the physics of virtual
excitations and more broadly on the study of CQED effects on electronic
degrees of freedom.

## Acknowledgements

The authors acknowledge R. Colombelli and A. Tredicucci for useful
discussions. S.D.L. acknowledges support from a Royal Society Research
Fellowship and from the Philip Leverhulme Prize of the Leverhulme Trust.
Y.W.’s studentship was financed by the Royal Society RGF\EA\180062 grant.

##  Appendix A Diagonalization of the PZW Hamiltonian

Following Ref. [40], the PZW Hamiltonian describing the system can be written
in the bosonic approximation

$\displaystyle\hat{H}=\sum_{\mathbf{q}}\hbar\left(\hat{W}_{0\mathbf{q}}+\hat{W}_{I\mathbf{q}}+\hat{W}_{D\mathbf{q}}\right),$
(42)

with the first part describing the bare fields

$\displaystyle\hat{W}_{0\mathbf{q}}=$
$\displaystyle\sum_{n}\omega_{12}\hat{b}^{\dagger}_{n\mathbf{q}}\hat{b}_{n\mathbf{q}}+\sum_{m}\omega_{mq}\hat{a}^{\dagger}_{m\mathbf{q}}\hat{a}_{m\mathbf{q}},$
(43)

the second their interaction

$\displaystyle\hat{W}_{I\mathbf{q}}=$
$\displaystyle\sum_{mn}\omega_{P}K_{mnq}(\hat{a}^{\dagger}_{m\mathbf{-q}}+\hat{a}_{m\mathbf{q}})(\hat{b}^{\dagger}_{n\mathbf{q}}+\hat{b}_{n\mathbf{-q}}),$
(44)

and the third the self-interaction of the electronic polarization

$\displaystyle\hat{W}_{D\mathbf{q}}=$
$\displaystyle\sum_{n}\frac{\omega_{P}^{2}}{4\omega_{12}}(\hat{b}^{\dagger}_{n\mathbf{q}}+\hat{b}_{n\mathbf{-q}})(\hat{b}_{n\mathbf{-q}}^{\dagger}+\hat{b}_{n\mathbf{q}}).$
(45)

In Eq. (44) we have the plasma frequency for a single QW

$\displaystyle\omega_{P}^{2}$ $\displaystyle=$ $\displaystyle\frac{I_{12}\hbar
e^{2}\sigma_{e}}{2m_{e}^{*2}\epsilon_{0}\epsilon_{r}\omega_{12}},$ (46)

with $m_{e}^{*}$ the effective mass of conduction electrons and the
dimensionless coupling coefficient

$\displaystyle
K_{mnq}^{2}=\omega_{mq}\omega_{12}\frac{z_{12}^{2}m_{e}^{*2}}{\hbar^{2}LI_{12}}\frac{2\cos^{2}\theta_{mq}}{(1+\delta_{m0})}\cos^{2}(\frac{\pi
mz_{n}}{L}),$ (47)

parametrising the coupling between the bright matter mode in the $n$th QW and
the TMm photonic mode, where we defined $\theta_{mq}$ the angle of propagation
relative to the $z$-axis

$\displaystyle\cos\theta_{mq}=\frac{cq}{\sqrt{\epsilon_{r}}\omega_{mq}}.$ (48)

Note that the term in Eq. (45), quadratic in the plasma frequency
$\omega_{P}$, is responsible for the phenomenon of polarization shift [59,
60]. This can be seen by considering only the terms of the full Hamiltonian
describing the bare field and the electron-electron interaction

$\displaystyle\hat{W}_{0\mathbf{q}}^{\prime}=\hat{W}_{0\mathbf{q}}+\hat{W}_{D\mathbf{q}},$
(49)

and performing a Bogoliubov rotation over the $\hat{b}_{n\mathbf{q}}$
operators, leading to a new Hamiltonian describing modified intersubband
transitions operators $\hat{d}_{n\mathbf{q}}$ whose frequency is renormalised
by the interaction

$\displaystyle\hat{W}_{0\mathbf{q}}^{\prime}=$
$\displaystyle\sum_{n}\sqrt{\omega_{12}^{2}+\omega_{P}^{2}}\,\hat{d}^{\dagger}_{n\mathbf{q}}\hat{d}_{n\mathbf{q}}+\sum_{m}\omega_{mq}\hat{a}^{\dagger}_{m\mathbf{q}}\hat{a}_{m\mathbf{q}}.$
(50)

The electronic wavefunctions in the two subbands enter in play through the
definition of the intersubband dipole

$\displaystyle z_{12}$ $\displaystyle=$
$\displaystyle\frac{\hbar}{2m_{e}^{*}\omega_{12}}\int\left[\bar{\psi}_{1}(z)\partial_{z}\psi_{2}(z)-\psi_{2}(z)\partial_{z}\bar{\psi}_{1}(z)\right]dz,$

and of the normalization factor

$\displaystyle I_{12}$ $\displaystyle=$
$\displaystyle\int\left[\bar{\psi}_{1}(z)\partial_{z}\psi_{2}(z)-\psi_{2}(z)\partial_{z}\bar{\psi}_{1}(z)\right]^{2}dz.$
(52)

In the bosonic regime we can then use the Hopfield-Bogoliubov approach to
diagonalise $\hat{H}$ in Eq. (42) in terms of polaritonic modes

$\displaystyle\hat{p}_{s\mathbf{q}}$
$\displaystyle=\sum_{m}\left(x_{smq}\hat{a}_{m\mathbf{q}}+z_{smq}\hat{a}^{\dagger}_{m\mathbf{-q}}\right)$
$\displaystyle+\sum_{n}\left(y_{snq}\hat{b}_{n\mathbf{q}}+w_{snq}\hat{b}_{n\mathbf{-q}}^{\dagger}\right),$

whose coupled ground state $\mathinner{|{G}\rangle}$ is defined by
$\hat{p}_{s\mathbf{q}}\mathinner{|{G}\rangle}=0$. The linear transformation in
Eq. (A) can then be inverted to obtain

$\displaystyle\hat{b}_{n\mathbf{q}}$
$\displaystyle=\sum_{s}\bar{y}_{snq}\hat{p}_{s\mathbf{q}}-\bar{w}_{snq}\hat{p}^{\dagger}_{s\mathbf{q}}.$
(54)

##  Appendix B Derivation of the bosonic expression for the electron density

Here we will provide a derivation of the equation

$\displaystyle\rho_{G}(z)-\rho_{F}(z)=$
$\displaystyle\frac{1}{S}\sum_{n\mathbf{q}}\left[\lvert\psi_{2}(z-z_{n})\rvert^{2}-\lvert\psi_{1}(z-z_{n})\rvert^{2}\right]$
(55)
$\displaystyle\times\mathinner{\langle{G}|}\hat{b}^{\dagger}_{n\mathbf{q}}\hat{b}_{n\mathbf{q}}\mathinner{|{G}\rangle},$

describing the change of the electronic density in the ground state created by
the vacuum excitations. While physically intuitive its derivation requires
some care as it links fermionic to bosonised quantities. We will thus expand
the approach originally developed in Ref. [44] to calculate the electronic
wavefunctions of photon-bound excitons.

We start by introducing the electron field operator projected on the first two
subbands

$\displaystyle\hat{\Psi}(\mathbf{r})=\sum_{n\mathbf{k}}\left[\phi_{1n\mathbf{k}}(\mathbf{r})\hat{c}_{1n\mathbf{k}}+\phi_{2n\mathbf{k}}(\mathbf{r})\hat{c}_{2n\mathbf{k}}\right].$
(56)

Using as reference the electronic distribution in the absence of the
resonator, we can then write the induced electron density as

$\displaystyle\rho_{G}(z)-\rho_{F}(z)$
$\displaystyle=\mathinner{\langle{G}|}\hat{\Psi}^{\dagger}(\mathbf{r})\hat{\Psi}(\mathbf{r})\mathinner{|{G}\rangle}-\mathinner{\langle{F}|}\hat{\Psi}^{\dagger}(\mathbf{r})\hat{\Psi}(\mathbf{r})\mathinner{|{F}\rangle}.$
(57)

The expression in Eq. (57) can be simplified by noticing that both the free
and coupled Hamiltonians, with ground states $\mathinner{|{F}\rangle}$ and
$\mathinner{|{G}\rangle}$ respectively, commute with the parity of the total
excitation number operator

$\displaystyle\hat{T}$
$\displaystyle=\sum_{m\mathbf{q}}\hat{a}^{\dagger}_{m\mathbf{q}}\hat{a}_{m\mathbf{q}}+\frac{1}{2}\sum_{n\mathbf{k}}\left(\hat{c}^{\dagger}_{2n\mathbf{k}}\hat{c}_{2n\mathbf{k}}-\hat{c}^{\dagger}_{1n\mathbf{k}}\hat{c}_{1n\mathbf{k}}\right).$
(58)

Both the coupled and free ground states have thus a well defined excitation
number parity. This implies that all the ground states expectation values
involving terms which increase $\hat{T}$ by one, like
$\hat{c}^{\dagger}_{2n\mathbf{k}}\hat{c}_{1n^{\prime}\mathbf{k}}$, have to
vanish. Exploiting the fact that the number of electrons in each QW is fixed

$\displaystyle\sum_{\mathbf{k}}\left(\hat{c}^{\dagger}_{1n\mathbf{k}}\hat{c}_{1n\mathbf{k}}+\hat{c}^{\dagger}_{2n\mathbf{k}}\hat{c}_{2n\mathbf{k}}\right)$
$\displaystyle=\sigma_{e}S,$ (59)

we can then put Eq. (57) in the form

$\displaystyle\rho_{G}(z)-\rho_{F}(z)=$
$\displaystyle\frac{1}{S}\sum_{n\mathbf{k}}\left[\lvert\psi_{2}(z-z_{n})\rvert^{2}-\lvert\psi_{1}(z-z_{n})\rvert^{2}\right]$
$\displaystyle\times\mathinner{\langle{G}|}\hat{c}^{\dagger}_{2n\mathbf{k}}\hat{c}_{2n\mathbf{k}}\mathinner{|{G}\rangle}.$
(60)

In order to prove Eq. (55) we have thus to demonstrate that

$\displaystyle\mathinner{\langle{G}|}\hat{N}_{Fn}\mathinner{|{G}\rangle}$
$\displaystyle=\mathinner{\langle{G}|}\hat{N}_{Bn}\mathinner{|{G}\rangle},$
(61)

that is that the total number of electrons in the second subband of any QW

$\displaystyle\hat{N}_{Fn}=\sum_{\mathbf{k}}\hat{c}^{\dagger}_{2n\mathbf{k}}\hat{c}_{2n\mathbf{k}},$
(62)

and the total number of matter excitations in the same QW

$\displaystyle\hat{N}_{Bn}=\sum_{\mathbf{q}}\hat{b}^{\dagger}_{n\mathbf{q}}\hat{b}_{n\mathbf{q}},$
(63)

have the same expectation value in the coupled ground state. Note that by
construction

$\displaystyle\hat{N}_{Fn}\mathinner{|{F}\rangle}$
$\displaystyle=\hat{N}_{Bn}\mathinner{|{F}\rangle}=0,$ (64)

and they thus also trivially coincide in the uncoupled ground state. Using the
definition of the intersubband transition operator in terms of electron
operators

$\displaystyle\hat{b}_{n\mathbf{q}}^{\dagger}$
$\displaystyle=\frac{1}{\sqrt{S\sigma_{e}}}\sum_{\mathbf{k}}\hat{c}^{\dagger}_{2n\mathbf{k+q}}\hat{c}_{1n\mathbf{k}},$
(65)

we can verify that the fermionic commutator of $\hat{N}_{Fn}$ with the right-
hand-side of Eq. (65)

$\displaystyle\left[\hat{N}_{Fn},\hat{b}_{n\mathbf{q}}^{\dagger}\right]$
$\displaystyle=\hat{b}_{n\mathbf{q}}^{\dagger},$ (66)

generates the same algebra as the commutator of $\hat{N}_{Bn}$ with the left-
hand-side of the same equation, when considering the $\hat{b}_{n\mathbf{q}}$
as perfect bosons,

$\displaystyle\left[\hat{N}_{Bn},\hat{b}_{n\mathbf{q}}^{\dagger}\right]$
$\displaystyle=\hat{b}_{n\mathbf{q}}^{\dagger}.$ (67)

Using the same bosonic assumption the coupled ground state
$\mathinner{|{G}\rangle}$ can be easily written in a perturbative expansion as
a sum over orthogonal vectors
$\mathinner{|{G}\rangle}=\sum_{\zeta}\mathinner{|{\zeta}\rangle}$, with

$\displaystyle\mathinner{|{\zeta}\rangle}$
$\displaystyle=\chi_{\zeta}\prod_{j=1}^{j_{\zeta}}\hat{b}^{\dagger}_{\mathbf{q}_{\zeta
j}}\prod_{h=1}^{h_{\zeta}}\hat{a}^{\dagger}_{m_{\zeta h}\mathbf{q}_{\zeta
h}}\mathinner{|{F}\rangle}.$ (68)

Exploiting Eq. (64) and the fact that $N_{Fn}$ and $N_{Bn}$ have the same
commutation relations with all the operators appearing in the
$\mathinner{|{\zeta}\rangle}$ states, their expectation value in the ground
state is necessarily the same, proving Eq. (61) and thus concluding our proof
of Eq. (55).

##  Appendix C Casimir-induced polarization

In the main body of the paper we considered two sources of induced
polarization: the virtual excitations due to the transverse electromagnetic
mode confinement, leading to $\sigma_{n}\propto L^{-1}$, and in-built
potentials independent from $L$. At small scales there are nevertheless other
interactions between the mirror and the electron gas which can lead to induced
polarization and, should those forces be comparable or larger than the ones
due to the transverse field, they would need to be taken into account when
performing data analysis. In this Appendix we will derive the general form for
the polarization induced on the $n$th QW from an arbitrary mirror-induced
pressure $P(d_{n})$ depending upon the distance between the mirror and the
quantum well $d_{n}=L-z_{n}$. We will also provide an upper estimate of such
an effect for the Casimir force, showing that, although potentially
measurable, such a force is not large enough to cover the electrostatic signal
whose measurement we describe in Sec. IV.

A pressure $P(d_{n})$ translates in a force per single electron
$F(d_{n})=P(d_{n})/\sigma_{e}$ and its impact on one electron in the nth QW
can thus be described by the Hamiltonian

$\displaystyle H_{n}$ $\displaystyle=F(d_{n})\hat{z},$ (69)

with $\hat{z}$ the electron coordinate. To the lowest order the modified
wavefunction in the first subband can thus be written as

$\displaystyle\mathinner{|{\psi_{1}^{\prime}}\rangle}$
$\displaystyle=\mathinner{|{\psi_{1}}\rangle}-\frac{F(d_{n})}{\hbar\omega_{12}}\mathinner{\langle{\psi_{2}}|}\hat{z}\mathinner{|{\psi_{1}}\rangle}\mathinner{|{\psi_{2}}\rangle},$
(70)

and the corresponding shift in electron position is thus, to the lowest order

$\displaystyle\Delta z^{\prime}$
$\displaystyle=\mathinner{\langle{\psi_{1}^{\prime}}|}\hat{z}\mathinner{|{\psi_{1}^{\prime}}\rangle}-\mathinner{\langle{\psi_{1}}|}\hat{z}\mathinner{|{\psi_{1}}\rangle}\approx\frac{2F(d_{n})z_{12}^{2}}{\hbar\omega_{12}}.$
(71)

The total dipole density induced in the QW is thus

$\displaystyle P^{\prime}_{n}=\sigma_{e}\Delta z^{\prime},$ (72)

which needs to be added to Eq. (35).

While precise calculations of Casimir force in planar systems are possible
[61] they are beyond the scope of the present paper. We can instead obtain an
upper bound of such an effect by assuming that the electrons gas in a single
QW is dense enough to act as a perfect mirror. Note that this would only be
true for unphysical densities leading to coupling frequencies larger than
$\omega_{12}$ as shown in Ref. [62]. In the sample we are considering the
effect of a single QW is instead largely perturbative and the result we will
obtain only represents a very loose overestimate of the actual effect. We can
then consider each QW as a perfect mirror and use the analytical expression
for the Casimir pressure between two planar metallic mirrors

$\displaystyle P(d_{n})$ $\displaystyle=-\frac{\hbar
c\pi^{2}}{240\sqrt{\epsilon_{r}}d_{n}^{4}}.$ (73)

Plugging Eq. (73) into Eq. (71) we obtain from Eq. (72) the induced dipole,
and using the parameters from Sec. V we conclude that even at the point of the
oscillation in which the mirror is the closest to the QW stack, and
considering a QW on the very top of the stack, the induced dipole is only one-
half of the CQED-induced one, dropping to $10^{-4}$ times smaller for a QW on
the bottom of the stack when the mirror is at its furthest point. These
estimates, as rough as they are, demonstrate that the Casimir effect, if not
always completely negligible, does not represent a critical obstacle to the
measurement scheme described in Sec. IV and a fortiori for the one from Sec.
III.

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