From the expression for the energy it becomes clear that the Hamiltonian
$\hat{H}^{\prime}$, which does not include $\hat{\varepsilon}_{dip}$, is
unbounded from below, because the parameter $a$ can be chosen arbitrarily
($F_{a}$ can be moved further and further away from the origin) and we can
therefore lower the energy of $\hat{H}^{{}^{\prime}}$ indefinitely. Thus, we
come to the conclusion that without the dipole self-energy the length gauge
Hamiltonian becomes unbounded from below and has no ground-state [48]. We
would like to note that this result is not so surprising as we subtracted a
harmonic potential from the Hamiltonian (3.29) and despite claims in
literature this term is dominant and cannot be discarded.

Comments on Assumptions.—Finally, we would like to comment on the choice of
considering the particles to be in full space while the photon field was
quantized in a box with periodic boundary conditions (see section 2.2), as
well as considering a purely negative external potential
$v_{ext}(\mathbf{r})$. First of all, we want to emphasize that allowing for
different lengths in the different directions of the quantization volume is
straightforward, and in this way we can model not only free space but also a
planar cavity. Further, it would be not a problem to use other boundary
conditions, like zero boundary conditions in the $z$-direction and periodic
ones in the $(x,y)$-plane and in the very end take the limit to infinity for
the open directions [57]. But all these considerations become superfluous in
the long-wavelength limit (or dipole approximation), where the spatial profile
of the electromagnetic modes is not taken into account. Further, the explicit
spatial form of the photon field does not change the harmonic nature of the
dipole self-energy, and for that reason we employed periodic boundary
conditions. However, for the investigation of the ground-state in the length
gauge Hamiltonian it makes a difference if we enclose the particles in a
finite volume. In this case it would not be possible to lower the energy of
$\hat{H}^{{}^{\prime}}$ indefinitely. But what would happen in this case is
that we would find a ground-state that is localized at the edge of the
quantization box. This was demonstrated exactly in a numeric fashion in [49].
Such a wavefunction however would not be a physical ground-state of an atom
because the electrons of the atom would not be localised around the nucleus.
Further, in this case we would also have a maximally allowed box length for a
given atomic or molecular system which would just cut the region in which the
dipole self-energy becomes dominant. Keeping the dipole self-energy in the
length-gauge Hamiltonian has the advantage of allowing for a treatment
independent of the box-size, which is definitely a physically desirable
property that yields physically acceptable localised ground-states. Lastly, we
would like to mention that in the case of a high-Q cavity, the cavity mirrors
can be modeled as barrier for the matter-particles by adding a very large
repulsive potential at the assumed positions of the mirrors. But even with the
inclusion of such a potential barrier would not invalidate our proof that the
length gauge Hamiltonian without the dipole self-energy is unbounded from
below, because the potentials we consider here are in the Banach space
$L^{2}(\mathbb{R}^{3})+L^{\infty}(\mathbb{R}^{3})$ and in the limit
$|\mathbf{r}|\rightarrow\infty$ only the bounded part of the potential
survives. Thus, by shifting $F_{a}$ arbitrarily, only a contribution
proportional to the limiting constant $v_{ext}(\mathbf{r})\rightarrow
v_{ext}^{\infty}$ contributes. As a consequence, such a positive potential in
$L^{2}(\mathbb{R}^{3})+L^{\infty}(\mathbb{R}^{3})$ cannot compensate the
linear decrease in energy due to the bilinear interaction $\hat{V}_{int}$
between the photons and the charged particles.

####  3.5 Maxwell’s Equations in Matter

Clearly, as we showed in the previous section, despite the claims in the
literature, the dipole self-energy is a very important term for the length-
gauge Hamiltonian and neglecting this term dramatically changes the properties
of the interacting light-matter system. Without the dipole self-energy a
ground-state becomes impossible, and even for arbitrarily small but finite
coupling to the photon field the light-matter system decays and no stable
ground-state exists. Therefore, the dipole self-energy is necessary for
describing the static properties of the combined matter-photon system.

Besides this extremely important effect, there is also another crucial point
for which the dipole self-energy cannot be neglected which has to do with the
fact that Maxwell’s equations for the electric field are not satisfied if
$\hat{\varepsilon}_{dip}$ is omitted from $\hat{H}_{L}$ [48]. So let us see
how this actually happens.

As we stated in the previous sections, a peculiarity of the length gauge is
that it mixes the electronic and the photonic degrees of freedom. This fact
can be understood also from the expression of the electric field in the
length-gauge picture. From the definition of the vector potential operator
(3.7) and performing the transformation on the photonic coordinates defined in
Eq. (3.12) we find that the $\mathbf{A}$-field in the length gauge is

$\hat{\mathbf{A}}=\sum_{\bm{\kappa},\lambda}\hat{\mathbf{A}}_{\bm{\kappa},\lambda},\qquad\textrm{where}\qquad\hat{\mathbf{A}}_{\bm{\kappa},\lambda}=-\textrm{i}\sqrt{\frac{\hbar}{\epsilon_{0}V\omega(\bm{\kappa})}}\bm{\varepsilon}_{\lambda}(\bm{\kappa})\frac{\partial}{\partial
p_{\bm{\kappa},\lambda}}.$ (3.43)

Then, from the Heisenberg equations of motion [84] we can obtain the
expression for the electric field

$\hat{\mathbf{E}}=-\frac{d\hat{\mathbf{A}}}{dt}=-\frac{\textrm{i}}{\hbar}[\hat{H}_{L},\hat{\mathbf{A}}]=\sum_{\bm{\kappa},\lambda}\hat{\mathbf{E}}_{\bm{\kappa},\lambda}\;\;\textrm{with}\;\;\hat{\mathbf{E}}_{\bm{\kappa},\lambda}=\sqrt{\frac{\hbar\omega(\bm{\kappa})}{\epsilon_{0}V}}\bm{\varepsilon}_{\lambda}(\bm{\kappa})\left(p_{\bm{\kappa},\lambda}-\frac{\bm{\varepsilon}_{\lambda}(\bm{\kappa})\cdot\mathbf{R}}{\sqrt{\epsilon_{0}V\hbar\omega(\bm{\kappa})}}\right).$
(3.44)

Further, by defining the polarization operator $\hat{\mathbf{P}}$ as

$\hat{\mathbf{P}}=\epsilon_{0}\sum_{\bm{\kappa},\lambda}\bm{\varepsilon}_{\lambda}(\bm{\kappa})\frac{e\bm{\varepsilon}_{\lambda}(\bm{\kappa})\cdot\mathbf{R}}{\epsilon_{0}V},$
(3.45)

we find that the photonic coordinates $p_{\bm{\kappa},\lambda}$ actually
correspond to the displacement field $\hat{\mathbf{D}}$

$\hat{\mathbf{D}}=\epsilon_{0}\sum_{\bm{\kappa},\lambda}\sqrt{\frac{\hbar\omega(\bm{\kappa})}{\epsilon_{0}V}}\bm{\varepsilon}_{\lambda}(\bm{\kappa})p_{\bm{\kappa},\lambda}$
(3.46)

of the Maxwell equations in matter, i.e.,
$\hat{\mathbf{D}}=\epsilon_{0}\hat{\mathbf{E}}+\hat{\mathbf{P}}$ [79]. We note
that this result would be true even if we discarded the dipole self-energy
from the length-gauge Hamiltonian. The consequences of omitting the dipole
self-energy term only show up in higher derivatives of the vector potential
and lead to a violation of the equations of motion.

To see how this happens we will compute the equations of motion for the
$\mathbf{A}$-field and the $\mathbf{E}$-field, with and without the dipole
self-energy. Let us start with the case where $\hat{\varepsilon}_{dip}$ is
present. Firstly, we compute the time-derivative for the electric field which
relates to the second time-derivative of the vector potential
$\dot{\mathbf{E}}=-\ddot{\mathbf{A}}$, and we obtain

$\frac{d^{2}}{dt^{2}}\hat{\mathbf{A}}+\sum_{\bm{\kappa},\lambda}\omega^{2}(\bm{\kappa})\hat{\mathbf{A}}_{\bm{\kappa},\lambda}=\frac{\textrm{i}\hbar
e}{\epsilon_{0}Vm_{\textrm{e}}}\sum_{\bm{\kappa},\lambda}\bm{\varepsilon}_{\lambda}(\bm{\kappa})\sum^{N}_{i=1}\bm{\varepsilon}_{\lambda}(\bm{\kappa})\cdot\nabla_{i}.$
(3.47)

The equation above is the mode resolved inhomogeneous Maxwell equation, with
the inhomogeneity coming from the presence of the paramagnetic current
operator on the right-hand side [79]. For the computation of the equation of
motion for the electric field we make the choice $v_{ext}(\mathbf{r})=0$ to
simplify the further analysis. We would like to emphasize that this choice
corresponds to the paradigmatic system of the homogeneous electron gas, also
known as the jellium model [63, 110]. Then, for the equation of motion for the
electric field we have [48]

$\frac{d^{2}}{dt^{2}}\hat{\mathbf{E}}=-\sum_{\bm{\kappa},\lambda}\omega^{2}(\bm{\kappa})\hat{\mathbf{E}}_{\bm{\kappa},\lambda}-\frac{Ne^{2}}{\epsilon_{0}Vm_{\textrm{e}}}\hat{\mathbf{E}}=-\sum_{\bm{\kappa},\lambda}\left(\omega^{2}(\bm{\kappa})+\omega^{2}_{p}\right)\hat{\mathbf{E}}_{\bm{\kappa},\lambda}.$
(3.48)

The equation of motion of the electric field is the well-known mode resolved
Maxwell equation. Moreover, we see that due to the interaction with matter
there is an additional term contributing to the oscillation frequencies of the
of the electric field. The bare photon frequencies $\omega(\bm{\kappa})$ get
dressed by the contribution $\omega_{p}$ coming from matter

$\omega_{p}=\sqrt{\frac{e^{2}N}{\epsilon_{0}Vm_{\textrm{e}}}}=\sqrt{\frac{e^{2}n_{\textrm{e}}}{\epsilon_{0}m_{\textrm{e}}}},$
(3.49)

where $n_{\textrm{e}}=N/V$ is the electron density. The total frequencies of
the electric field therefore are

$\widetilde{\omega}^{2}(\bm{\kappa})=\omega^{2}(\bm{\kappa})+\omega_{p}^{2}.$
(3.50)

This change in the frequency spectrum of the electric field is a diamagnetic
shift due to the interactions with matter and depends on the full electron
density $n_{\textrm{e}}$ via the plasma frequency $\omega_{p}$ [48, 58]. This
diamagnetic contribution has been observed experimentally in resonant matter-
photon systems in the ultra-strong coupling regime [119, 120, 121].

However, if we ignore the dipole self-energy and compute the equations of
motion for the electric field in the case where
$v_{\textrm{ext}}(\mathbf{r})=0$ we find that the equations of motion differ
significantly [48]

$\frac{d^{2}}{dt^{2}}\hat{\mathbf{E}}+\sum_{\bm{\kappa},\lambda}\omega^{2}(\bm{\kappa})\hat{\mathbf{E}}_{\bm{\kappa},\lambda}=-\omega^{2}_{p}\hat{\mathbf{D}}.$
(3.51)

As it is clear from the above result, neglecting the dipole self-energy leads
to a wrong description of the electromagnetic field, because on the right-hand
side of equation (3.51) we do not have the electric field $\hat{\mathbf{E}}$
but the displacement field $\hat{\mathbf{D}}$. This implies that if we neglect
the dipole self-energy the electric field does not satisfy the mode-resolved
Maxwell equation and we get a completely wrong description for the
electromagnetic field coupled to matter. Therefore, the dipole self-energy
cannot be ignored and must always be included in order to have a complete and
consistent physical description of an interacting photon-matter system [48].

##  Part II The Free Electron Gas in Cavity QED

### CHAPTER 4 The Free Electron Gas

The model of the free electron gas introduced by Sommerfeld in 1928 [62] is a
paradigmatic model for solid state and condensed matter physics. Originally it
was introduced for the description of thermal and conduction properties of
metals. Since then it has served as one of the fundamental models for
understanding and describing materials. Further, with the inclusion of the
electron-electron interactions and the positive ion charges in terms of a
background medium, the free electron gas was transformed into the homogeneous
electron gas, also known as the jellium model [63, 110], which with the advent
of density functional theory (DFT) and the local density approximation (LDA)
[122] has become one of the most useful and successful computational tools for
physics, chemistry and materials science [123]. Also within Landau’s Fermi
liquid theory [124], the free electron gas was used as the fundamental
building block [125]. In addition, the free electron gas in the presence of
strong homogeneous magnetic fields has also proven to be extremely important
for the description of the integer and the fractional quantum Hall effects
[70, 126, 75, 74]. Due to the importance of the free electron gas for
condensed matter physics and its wide applicability, in what follows we focus
on this system.

####  4.1 Free Electrons in a “Periodic” Box

The Sommerfeld model of the free electron gas consists of $N$ non-interacting
electrons confined in a cube whose sides are of length $L$ and volume
$V=L^{3}$. Because the electrons do not interact with one another the
Hamiltonian describing this system is the sum of the kinetic energy operators
of all the particles

$\displaystyle\hat{H}=\sum^{N}_{j=1}\hat{H}_{j}=\frac{-\hbar^{2}}{2m_{\textrm{e}}}\sum^{N}_{j=1}\nabla^{2}_{j}.$
(4.1)

This implies that we can find the full set of eigenstates of the system by
solving the single-particle Hamiltonian $\hat{H}_{j}$ and then construct the
many-body eigenstates from the single-particle ones. To describe the fact that
the electrons are confined within the cube of volume $V$ we need to impose
boundary conditions on the wavefunctions of the electrons. The natural choice
would be the wavefunctions to satisfy zero boundary conditions which make
clear that the electrons do not escape outside of the material. Zero boundary
conditions though lead to solutions which are standing waves, which are not
convenient to work with. Another possibility is to impose periodic boundary
conditions [63], which as it was proven by Lebowitz and Lieb in [127], do not
affect the bulk properties of a system. Thus, we choose to impose periodic
boundary conditions which are more convenient to work with from a mathematical
point of view. Such boundary conditions imply that the wavefunctions
describing our system need to satisfy

$\displaystyle\phi(x+L,y,z)=\phi(x,y+L,z)=\phi(x,y,z+L)=\phi(x,y,z).$ (4.2)

With such boundary conditions and because the Hamiltonian commutes with the
momentum operator, $[\hat{H},\nabla]=0$, the single-particle eigenfunctions
are plane waves of the form

$\displaystyle\phi_{\mathbf{k}}(\mathbf{r})=\frac{e^{\textrm{i}\mathbf{k}\cdot\mathbf{r}}}{\sqrt{V}}\;\;\;\textrm{with}\;\;\;\mathbf{k}=\frac{2\pi\mathbf{n}}{L}\;\;\;\textrm{and}\;\;\;\mathbf{n}\in\mathbb{Z}^{3}.$
(4.3)

The quantum number $\mathbf{k}$ multiplied by $\hbar$ is the momentum
$\mathbf{p}=\hbar\mathbf{k}$ of the free particle, which due to translational
invariance of the system, is a conserved quantity. The eigenenergy of the
eigenfunction $\phi_{\mathbf{k}}(\mathbf{r})$ is

$\displaystyle
E_{\mathbf{k}}=\frac{\hbar^{2}\mathbf{k}^{2}}{2m_{\textrm{e}}}.$ (4.4)

From the single-particle eigenfunctions we can construct the many-body
eigenstates of the system. Because electrons are fermions the many-body
eigenstates must be antisymmetric under the exchange of any two electrons. To
satisfy the fermionic statistics we use a Slater determinant [63].

$\displaystyle\Psi_{\mathbf{K}}(\mathbf{r}_{1}\sigma_{1},..,\mathbf{r}_{N}\sigma_{N})=\frac{1}{\sqrt{N!}}\;\begin{tabular}[]{|c
c c
c|}$\phi_{\mathbf{k}_{1}}(\mathbf{r}_{1}\sigma_{1})$&$\phi_{\mathbf{k}_{1}}(\mathbf{r}_{2}\sigma_{2})$&$\cdot\cdot\cdot$&$\phi_{\mathbf{k}_{1}}(\mathbf{r}_{N}\sigma_{N})$\\\
$\phi_{\mathbf{k}_{2}}(\mathbf{r}_{1}\sigma_{1})$&$\phi_{\mathbf{k}_{2}}(\mathbf{r}_{2}\sigma_{2})$&$\cdot\cdot\cdot$&$\phi_{\mathbf{k}_{2}}(\mathbf{r}_{N}\sigma_{N})$\\\
$\cdot$&$\cdot$&$\cdot\cdot\cdot$&$\cdot$\\\
$\cdot$&$\cdot$&$\cdot\cdot\cdot$&$\cdot$\\\
$\phi_{\mathbf{k}_{N}}(\mathbf{r}_{1}\sigma_{1})$&$\phi_{\mathbf{k}_{N}}(\mathbf{r}_{2}\sigma_{2})$&$\cdot\cdot\cdot$&$\phi_{\mathbf{k}_{N}}(\mathbf{r}_{N}\sigma_{N})$\end{tabular}$
(4.10)

We note that $\mathbf{K}=\sum_{j}\mathbf{k}_{j}$ is the collective momentum of
the electrons and we introduced it here in order to indicate the fact that
each many-body eigenstate is defined with respect to a particular distribution
of the electrons in $\mathbf{k}$-space and depends on the collective momentum
of the electron gas. Thus, the collective momentum $\mathbf{K}$ is used as a
label to distinguish different many-body states of the electron gas. The
energy $E_{\mathbf{K}}$ of the Slater determinant
$\Phi_{\mathbf{K}}(\mathbf{r}_{1},..,\mathbf{r}_{N})$ is then the sum of the
single-particle energies

$\displaystyle
E_{\mathbf{K}}=\sum^{N}_{j=1}E_{\mathbf{k}_{j}}=\sum^{N}_{j=1}\frac{\hbar^{2}\mathbf{k}^{2}_{j}}{2m_{\textrm{e}}}.$
(4.11)

####  4.2 Ground State Energy in the Thermodynamic Limit

Our aim now is to compute the ground-state energy of the free electron gas in
the so-called thermodynamic limit, in which the number of particles and the
volume of the system tend to infinity but in such a way that the electron
density $n_{\textrm{e}}=N/V$ stays fixed [63, 110]. To achieve this, first we
need to find what is actually the ground-state of the system in the
thermodynamic limit.

In the previous section we found that the full set of eigenstates of the
system is described by Slater determinants
$\Psi_{\mathbf{K}}(\mathbf{r}_{1},..,\mathbf{r}_{N})$ of Eq. (4.10). But as we
already mentioned these states are defined by a distribution in
$\mathbf{k}$-space. Thus, to find the ground state of the system we need to
find the optimal distribution for the electrons in $\mathbf{k}$-space.

The allowed values for $\mathbf{k}$ are discrete as we can see from Eq. (4.3).
This implies that the allowed $\mathbf{k}$ form a cubic grid in
$\mathbf{k}$-space. From this fact one might expect intuitively that the
ground state distribution for small number of particles should be a cube since
all three directions $k_{x},k_{y}$ and $k_{z}$ are equivalent. However, here
we are interested in very large amount of particles and very large
$\mathbf{k}$-space regions which eventually become so regular that the ground
of distribution in $\mathbf{k}$-space can be regarded to be a sphere. This
sphere is known as the Fermi sphere [63]. The fact that the ground state
distribution has to be a sphere can be understood from the fact that the
Hamiltonian of Eq. (4.1) is spherically symmetric.

The volume $\Omega_{\mathcal{S}}$ of the Fermi sphere is defined by its radius
$k_{\textrm{F}}$ which is the highest occupied wave vector in
$\mathbf{k}$-space and it is known as the Fermi wave vector. Then, the volume
$\Omega_{\mathcal{S}}$ is

$\displaystyle\Omega_{\mathcal{S}}=\frac{4\pi k^{3}_{\textrm{F}}}{3}.$ (4.12)

We note that using the Fermi wave vector we can also define the highest
occupied energy in the system as
$E_{\textrm{F}}=\hbar^{2}k^{2}_{\textrm{F}}/2m_{\textrm{e}}$ which is known as
the Fermi energy. Having found the volume of the distribution in
$\mathbf{k}$-space, in terms of $k_{\textrm{F}}$, we can also find the number
of allowed states in $\mathbf{k}$-space as the quotient between
$\Omega_{\mathcal{S}}$ and the volume occupied by a single state in
$\mathbf{k}$-space, which is $(2\pi/L)^{3}$. We therefore conclude that number
of allowed states contained in the Fermi sphere $\Omega_{\mathcal{S}}$ is

$\displaystyle\\#\textrm{states}=\frac{\Omega_{S}}{(2\pi/L)^{3}}=\frac{\Omega_{\mathcal{S}}V}{8\pi^{3}},$
(4.13)

where $V$ is the volume of the material in real space. Since each wave vector
$\mathbf{k}$ can be occupied by two electrons, due to the spin degeneracy, we
find that the number of electrons that can be accommodated in the Fermi sphere
is

$\displaystyle N=2\\#\textrm{states}=\frac{\Omega_{\mathcal{S}}V}{4\pi^{3}}.$
(4.14)

As a consequence the electron density in the system as a function of the Fermi
wave vector is

$\displaystyle
n_{\textrm{e}}=\frac{N}{V}=\frac{k^{3}_{\textrm{F}}}{3\pi^{2}}.$ (4.15)

Having defined the ground-state distribution in $\mathbf{k}$-space and the
electron density, in the thermodynamic limit, we can attempt to compute the
ground-state energy density of the system. The total energy of the system
inside the Fermi sphere for doubly occupied wave vectors is

$\displaystyle
E_{gs}=2\sum_{|\mathbf{k}|<k_{F}}\frac{\hbar^{2}}{2m_{\textrm{e}}}\mathbf{k}^{2}.$
(4.16)

The summation of a smooth function as the one appearing above can be done as
follows [63]: The volume of $\mathbf{k}$-space per allowed $\mathbf{k}$ value
is $\Delta\mathbf{k}=8\pi^{3}/V$ and consequently we can write the sum of the
previous equation as

$\displaystyle\sum_{|\mathbf{k}|<k_{F}}\mathbf{k}^{2}=\frac{V}{8\pi^{3}}\sum_{|\mathbf{k}|<k_{F}}\mathbf{k}^{2}\Delta\mathbf{k}.$
(4.17)

In the limit where the volume of the system approaches infinity
($V\rightarrow\infty$) the measure $\Delta\mathbf{k}$ goes to zero
($\Delta\mathbf{k}\rightarrow 0$) and the sum approaches an integral. Thus,
the energy per volume in the limit where $V\rightarrow\infty$ is

$\displaystyle\mathcal{E}_{gs}\equiv\lim_{V\to\infty}\frac{E_{gs}}{V}=\frac{\hbar^{2}}{8\pi^{3}m_{\textrm{e}}}\iiint\limits_{|\mathbf{k}|<k_{F}}d^{3}k\mathbf{k}^{2}.$
(4.18)

Going into spherical coordinates in $\mathbf{k}$-space and performing the
integral, we find the energy density as a function of the Fermi momentum

$\displaystyle\mathcal{E}_{gs}(k_{\textrm{F}})=\frac{\hbar^{2}k^{5}_{\textrm{F}}}{10\pi^{2}m_{\textrm{e}}}.$
(4.19)

Further, using the relation for the electron density $n_{\textrm{e}}$ as a
function of the Fermi momentum in Eq. (4.15) we can find the energy density
$\mathcal{E}_{gs}$ as a function of the electron density

$\displaystyle\mathcal{E}_{gs}(n)=\frac{3\hbar^{2}(3\pi)^{2/3}}{10m_{\textrm{e}}}n^{5/3}_{\textrm{e}}.$
(4.20)

Such expressions of the energy density in terms of the electron density are
very important for the construction of functional approximations in density
functional theory, like for example in the case of the LDA functional [122].
Moreover, we note that from the expression of the energy density one can
compute also other kinds of thermodynamic properties of interest like the
energy per particle, the pressure exerted by the electron gas or the
compressibility of the system [63].

####  4.3 Density of States

Another very important quantity for solid state systems is the density of
states. The density of states describes the number of states that are
available in a system in a particular energy neighborhood, and is essential
for determining the carrier concentrations and energy distributions of
carriers. Furthermore, the density of states is connected to the transport
properties of materials [63, 110]. Generally the density of states
$\mathcal{D}(E)$ of a system of volume $V$ and with $N$ countable energy
levels is defined as

$\displaystyle\mathcal{D}(E)=\frac{2}{V}\sum^{N}_{i=1}\delta(E-E_{i})$ (4.21)

where $E_{i}$ are the energy levels of the system and the prefactor $2$
appears due to the spin degeneracy. In our case we have a system which in the
thermodynamic limit has a continuous spectrum. Following the procedure we
described before, on how to perform summations in the thermodynamic limit, the
density of states in the limit $V\rightarrow\infty$ and for a continuous
spectrum becomes

$\displaystyle\mathcal{D}(E)=\lim_{V\to\infty}\frac{2}{V}\sum_{\mathbf{k}}\delta(E-E_{\mathbf{k}})=\frac{2}{(2\pi)^{3}}\iiint
d^{3}k\delta(E-E_{\mathbf{k}}).$ (4.22)

The energy spectrum of the free electrons is
$E_{\mathbf{k}}=\hbar^{2}\mathbf{k}^{2}/2m_{\textrm{e}}$. Substituting this
energy expression and going into spherical coordinates
$(k_{x},k_{y},k_{z})\longrightarrow(k_{r},k_{\theta},k_{\phi})$ we obtain

$\displaystyle\mathcal{D}(E)=\frac{2}{(2\pi)^{3}}\iint\sin
k_{\theta}dk_{\theta}dk_{\phi}\int
dk_{r}k^{2}_{r}\delta\left(E-\frac{\hbar^{2}k^{2}_{r}}{2m_{\textrm{e}}}\right).$
(4.23)

We integrate over the angle coordinates

$\displaystyle\mathcal{D}(E)=\frac{8\pi}{(2\pi)^{3}}\int
dk_{r}k^{2}_{r}\delta\left(E-\frac{\hbar^{2}k^{2}_{r}}{2m_{\textrm{e}}}\right).$
(4.24)

For the last integral over $k_{r}$ we introduce the variable $s$

$\displaystyle
s=\frac{\hbar^{2}k^{2}_{r}}{2m_{\textrm{e}}}\;\;\;\textrm{with}\;\;\;ds=\frac{\hbar^{2}k_{r}dk_{r}}{m_{\textrm{e}}}$
(4.25)

and the integral for $\mathcal{D}(E)$ transforms as

$\displaystyle\mathcal{D}(E)=\frac{\sqrt{2}}{\pi^{2}}\left(\frac{m_{\textrm{e}}}{\hbar^{2}}\right)^{3/2}\int
ds\sqrt{s}\delta(E-s),$ (4.26)

and after integrating over the variable $s$ we obtain the expression for the
density of states

$\displaystyle\mathcal{D}(E)=\frac{\sqrt{2}}{\pi^{2}}\left(\frac{m_{\textrm{e}}}{\hbar^{2}}\right)^{3/2}\sqrt{E}\;\;\;\textrm{for}\;\;\;E<E_{F}.$
(4.27)

We note that the expression above for the density of states holds for energies
smaller than the Fermi energy and that for energies above the Fermi energy the
density of states is zero [63].

### CHAPTER 5 The Free Electron Gas in Cavity QED

> I remember Fermi used to ask “Where is the hydrogen atom of this problem?”
> Where, in what domain, will we find a simple system with a relatively simple
> law for its description, which will be the forerunner or the test of a real
> theory?

Murray Gell-Mann
Particles and Principles [128]

Most of our understanding of light-matter interactions is based on finite-
system (or few-level) models coming from the field of quantum optics, like the
Rabi [102], Jaynes-Cummings [101] or the Dicke model [103]. These models have
been proven to be extremely successful for the description of coupled light-
matter systems and particularly for the field of cavity QED. However, during
the last decade these fundamental models are being challenged by developments
in the emerging field of cavity QED materials [7], in which also extended
solid state systems are coupled to the quantized cavity-field.

It is well-known that extended solid state or condensed matter systems behave
very much differently than finite-system models. This brings into question
whether the few-level models of quantum optics can be used for the description
of macroscopic systems, like materials, strongly coupled to the quantized
field of a cavity. Consequently, it becomes desirable to have a an example of
an exactly solvable extended system strongly coupled to the quantized
electromagnetic field of a cavity, analogously to the Rabi and the Dicke
model.

For that purpose, in this chapter our aim is to revisit the paradigmatic
Sommerfeld model [62] of the free electron gas in the framework of cavity QED.

####  5.1 Electron Gas in Cavity QED

Here we are interested in the two-dimensional free electron gas (2DEG)
confined inside a cavity, as depicted in Fig. 5.1. For that purpose we
consider the Pauli-Fierz Hamiltonian of Eq. (2.114) in the absence of any
external potential, $v_{\textrm{ext}}(\mathbf{r})=0$, and we neglect the
Coulomb interaction as in the original Sommerfeld model [62]. Further, we
neglect the Pauli (Stern-Gerlach) term
$\hat{\bm{\sigma}}\cdot\hat{\mathbf{B}}(\mathbf{r})$ and we consider the
vector potential of the electromagnetic field in the long-wavelength limit.
Under these assumptions the Hamiltonian for the 2DEG in the cavity is

$\displaystyle\hat{H}=\frac{1}{2m_{\textrm{e}}}\sum\limits^{N}_{j=1}\left(\textrm{i}\hbar\mathbf{\nabla}_{j}+e\hat{\mathbf{A}}\right)^{2}+\sum\limits_{\bm{\kappa},\lambda}\hbar\omega(\bm{\kappa})\left(\hat{a}^{\dagger}_{\bm{\kappa},\lambda}\hat{a}_{\bm{\kappa},\lambda}+\frac{1}{2}\right).$
(5.1)

We note that for our system the long-wavelength limit or dipole approximation
is respected and justified, because we are considering a 2D material
(effectively of zero thickness) confined in the cavity, as shown in Fig. 5.1.
Since we restrict our considerations to two dimensions, the momentum operator
has only two components $\nabla=(\partial_{x},\partial_{y})$. Moreover, the
2DEG in the $(x,y)$ plane is assumed to be macroscopic and consequently the
electrons can be described with the use of periodic boundary conditions, as in
the original Sommerfeld model [62].

In cavity QED it is very typical to employ the single-mode approximation in
which the cavity is considered to have a particular mode which is on resonance
with the matter system and the rest of the electromagnetic modes are
neglected. As a starting point we will also take the approximation where the
field consists of a single mode with frequency $\omega$ but we will keep both
polarizations of the field. As we will see later, the single-mode case will be
used for the construction of an effective quantum field theory in which the
full 2D continuum of electromagnetic modes interacts with the 2DEG. The
polarization vectors of the quantized field are chosen to be in the $(x,y)$
plane such that the mode interacts with the 2DEG. For the polarizations to be
orthogonal we choose $\bm{\varepsilon}_{1}=\mathbf{e}_{x}$ and
$\bm{\varepsilon}_{2}=\mathbf{e}_{y}$. We note that these two polarization
vectors with respect to the general definition for the polarization vectors in
Eq. (2.1.2) correspond to $\kappa_{x}=\kappa_{y}=0$. This implies that the
photon field in the $(x,y)$ plane carries no momentum, and thus translational
symmetry is preserved in the $(x,y)$ plane, in accordance with the dipole-
approximation. Also, it is important to note that the analytic solution for
the single-mode case that we will present in what follows, can also be
generalized for an arbitrary, finite amount of modes. This is presented in
detail in appendix B.

With these extra assumptions enforced, the Pauli-Fierz Hamiltonian of Eq.
(5.1), after expanding the covariant kinetic energy, reads

$\displaystyle\hat{H}=\sum\limits^{N}_{j=1}\left[-\frac{\hbar^{2}}{2m_{\textrm{e}}}\nabla^{2}_{j}+\frac{\textrm{i}e\hbar}{m_{\textrm{e}}}\hat{\mathbf{A}}\cdot\nabla_{j}\right]+\underbrace{\frac{Ne^{2}}{2m_{\textrm{e}}}\hat{\mathbf{A}}^{2}+\sum\limits^{2}_{\lambda=1}\hbar\omega\left(\hat{a}^{\dagger}_{\lambda}\hat{a}_{\lambda}+\frac{1}{2}\right)}_{\hat{H}_{p}}.$
(5.2)

and the quantized vector potential takes the simple form

$\displaystyle\hat{\mathbf{A}}=\left(\frac{\hbar}{\epsilon_{0}V}\right)^{\frac{1}{2}}\sum^{2}_{\lambda=1}\frac{\bm{\varepsilon}_{\lambda}}{\sqrt{2\omega}}\left(\hat{a}_{\lambda}+\hat{a}^{\dagger}_{\lambda}\right).$
(5.3)

Figure 5.1:  Schematic depiction of a 2D material confined inside a cavity.
The cavity mirrors are of length $L$ and area $S=L^{2}$. The area of the
material is also $S$, while the distance between the mirrors of the cavity is
$L_{z}$.

In Eq. (5.2) there exists a purely photonic part $\hat{H}_{p}$ which depends
only on the photonic operators
$\hat{a}^{\dagger}_{\lambda},\hat{a}_{\lambda}$. Inserting the expression for
$\hat{\mathbf{A}}$ given by Eq. (5.3) and introducing the diamagnetic shift
$\omega_{p}$

$\displaystyle\omega_{p}=\sqrt{\frac{e^{2}N}{m_{\textrm{e}}\epsilon_{0}V}}=\sqrt{\frac{e^{2}n_{\textrm{e}}}{m_{\textrm{e}}\epsilon_{0}}},$
(5.4)

the purely photonic part $\hat{H}_{p}$ reads

$\displaystyle\hat{H}_{p}=\sum^{2}_{\lambda=1}\left[\hbar\omega\left(\hat{a}^{\dagger}_{\lambda}\hat{a}_{\lambda}+\frac{1}{2}\right)+\frac{\hbar\omega^{2}_{p}}{4\omega}\left(\hat{a}_{\lambda}+\hat{a}^{\dagger}_{\lambda}\right)^{2}\right].$
(5.5)

The diamagnetic shift $\omega_{p}$ is a consequence of having the
$\mathbf{A}^{2}$ in the Pauli-Fierz Hamiltonian and shows up due to the
collective coupling of the full electron density $n_{\textrm{e}}=N/V$ to the
quantized cavity-field [48, 11, 119, 120, 92]. This implies that
$\omega_{p}=\sqrt{e^{2}n_{\textrm{e}}/m_{\textrm{e}}\epsilon_{0}}$ is the
plasma frequency inside the cavity. We would like to emphasize that the 3D
electron density $n_{\textrm{e}}=N/V$ is defined via the 2D electron density
of the material $n_{2\textrm{D}}=N/S$ and the distance between the cavity
mirrors $L_{z}$ as $n_{\textrm{e}}=n_{2\textrm{D}}/L_{z}$.

The photonic part $\hat{H}_{p}$ can be diagonalized by introducing a new set
of bosonic annihilation and creation operators
$\hat{b}^{\dagger}_{\lambda},\hat{b}_{\lambda}$

$\displaystyle\hat{b}_{\lambda}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{\omega\widetilde{\omega}}}\left[\hat{a}_{\lambda}\left(\widetilde{\omega}+\omega\right)+\hat{a}^{\dagger}_{\lambda}\left(\widetilde{\omega}-\omega\right)\right]$
(5.6) $\displaystyle\hat{b}^{\dagger}_{\lambda}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{\omega\widetilde{\omega}}}\left[\hat{a}_{\lambda}\left(\widetilde{\omega}-\omega\right)+\hat{a}^{\dagger}_{\lambda}\left(\widetilde{\omega}+\omega\right)\right].$

where the frequency $\widetilde{\omega}$ is defined as

$\displaystyle\widetilde{\omega}=\sqrt{\omega^{2}+\omega^{2}_{p}}.$ (5.7)

The above frequency is a dressed frequency which depends on the cavity
frequency $\omega$ and the diamagnetic shift (or plasma frequency)
$\omega_{p}$. Thus, $\widetilde{\omega}$ should be interpreted as a plasmon-
polariton frequency. As we will see later in section 6, the dressed frequency
corresponds to a plasmon-polariton excitation of the light-matter hybrid
system. The operators $\hat{b}_{\lambda},\hat{b}^{\dagger}_{\lambda}$ satisfy
bosonic commutation relations
$[\hat{b}_{\lambda},\hat{b}^{\dagger}_{\lambda^{\prime}}]=\delta_{\lambda,\lambda^{\prime}}$
for $\lambda,\lambda^{\prime}=1,2$. In terms of the operators
$\hat{b}_{\lambda},\hat{b}^{\dagger}_{\lambda}$ the photonic part
$\hat{H}_{p}$ is equal to the sum of two non-interacting harmonic oscillators

$\hat{H}_{p}=\sum^{2}_{\lambda=1}\hbar\widetilde{\omega}\left(\hat{b}^{\dagger}_{\lambda}\hat{b}_{\lambda}+\frac{1}{2}\right)$
(5.8)

and the vector potential $\hat{\mathbf{A}}$ is

$\displaystyle\hat{\mathbf{A}}=\left(\frac{\hbar}{\epsilon_{0}V}\right)^{\frac{1}{2}}\sum^{2}_{\lambda=1}\frac{\bm{\varepsilon}_{\lambda}}{\sqrt{2\widetilde{\omega}}}\left(\hat{b}_{\lambda}+\hat{b}^{\dagger}_{\lambda}\right).$
(5.9)

From the above expression we see that the quantized vector potential
$\hat{\mathbf{A}}$ got renormalized and now depends on the plasmon-polariton
$\widetilde{\omega}$ [11]. Substituting back into Eq. (5.2) the expressions
for $\hat{H}_{p}$ and $\hat{\mathbf{A}}$ given by Eqs. (5.8) and (5.9)
respectively, and introducing the parameter $g_{0}$

$\displaystyle
g_{0}=\frac{e\hbar}{m_{\textrm{e}}}\left(\frac{\hbar}{\epsilon_{0}V2\widetilde{\omega}}\right)^{\frac{1}{2}},$
(5.10)

we obtain the following expression for the the Hamiltonian of Eq. (5.2)

$\displaystyle\hat{H}=-\frac{\hbar^{2}}{2m_{\textrm{e}}}\sum^{N}_{j=1}\nabla^{2}_{j}+\sum^{2}_{\lambda=1}\hbar\widetilde{\omega}\left(\hat{b}^{\dagger}_{\lambda}\hat{b}_{\lambda}+\frac{1}{2}\right)+\textrm{i}g_{0}\sum^{2}_{\lambda=1}\left(\hat{b}_{\lambda}+\hat{b}^{\dagger}_{\lambda}\right)\bm{\varepsilon}_{\lambda}\cdot\sum^{N}_{j=1}\nabla_{j}.$
(5.11)

We note that $g_{0}$ in Eq. (5.10) can be interpreted as the single-particle
light-matter coupling constant. The Hamiltonian is invariant under
translations in the electronic space, because it includes only the electronic
momentum operators. Thus, $\hat{H}$ commutes with $\nabla$,
$[\hat{H},\nabla]$=0, and the two operators share eigenfunctions. For the
electronic wavefunctions we employ periodic boundary conditions. Thus, the
eigenfunctions of the momentum operator $\nabla$ and the Hamiltonian are plane
waves of the form [62, 63]

$\displaystyle\phi_{\mathbf{k}_{j}}(\mathbf{r}_{j})=\frac{e^{\textrm{i}\mathbf{k}_{j}\cdot\mathbf{r}_{j}}}{\sqrt{S}}\;\;\textrm{with}\;\;1\leq
j\leq N,$ (5.12)

where $\mathbf{k}_{j}=2\pi(n^{x}_{j}/L,n^{y}_{j}/L)$ are the momenta of the
electrons, with
$\mathbf{n}_{j}=\left(n^{x}_{j},n^{y}_{j}\right)\in\mathbb{Z}^{2}$, and
$S=L^{2}$ is the area of the 2D material inside the cavity. The plane waves in
Eq. (5.12) are the single-particle eigenfunctions. To construct the many-body
eigenfunctions which satisfy the proper fermionic statistics we will use a
Slater determinant built out of the single-particle eigenfunctions of Eq.
(5.12)

$\displaystyle\Phi_{\mathbf{K}}(\mathbf{r}_{1}\sigma_{1},..,\mathbf{r}_{N}\sigma_{N})=\frac{1}{\sqrt{N!}}\;\begin{tabular}[]{|c
c c
c|}$\phi_{\mathbf{k}_{1}}(\mathbf{r}_{1}\sigma_{1})$&$\phi_{\mathbf{k}_{1}}(\mathbf{r}_{2}\sigma_{2})$&$\cdot\cdot\cdot$&$\phi_{\mathbf{k}_{1}}(\mathbf{r}_{N}\sigma_{N})$\\\
$\phi_{\mathbf{k}_{2}}(\mathbf{r}_{1}\sigma_{1})$&$\phi_{\mathbf{k}_{2}}(\mathbf{r}_{2}\sigma_{2})$&$\cdot\cdot\cdot$&$\phi_{\mathbf{k}_{2}}(\mathbf{r}_{N}\sigma_{N})$\\\
$\cdot$&$\cdot$&$\cdot\cdot\cdot$&$\cdot$\\\
$\cdot$&$\cdot$&$\cdot\cdot\cdot$&$\cdot$\\\
$\phi_{\mathbf{k}_{N}}(\mathbf{r}_{1}\sigma_{1})$&$\phi_{\mathbf{k}_{N}}(\mathbf{r}_{2}\sigma_{2})$&$\cdot\cdot\cdot$&$\phi_{\mathbf{k}_{N}}(\mathbf{r}_{N}\sigma_{N})$\end{tabular}$
(5.18)

where $\mathbf{K}=\sum_{j}\mathbf{k}_{j}$ is the collective momentum of the
electrons and we introduced it to indicate the fact that the ground state and
the excited states of the system depend on the distribution of the electrons
in $\mathbf{k}$-space and particularly on the collective momentum of the
electron gas. As we did in the previous chapter we denote the Slater
determinant as
$\Phi_{\mathbf{K}}\equiv\Phi_{\mathbf{K}}(\mathbf{r}_{1}\sigma_{1},..,\mathbf{r}_{N}\sigma_{N})$.
We apply now the Hamiltonian $\hat{H}$ of Eq. (5.11) on the eigenfunction
$\Phi_{\mathbf{K}}$ and we have

$\displaystyle\hat{H}\Phi_{\mathbf{K}}=\Bigg{[}\sum^{2}_{\lambda=1}\left[\hbar\widetilde{\omega}\left(\hat{b}^{\dagger}_{\lambda}\hat{b}_{\lambda}+\frac{1}{2}\right)-g_{0}\left(\hat{b}_{\lambda}+\hat{b}^{\dagger}_{\lambda}\right)\bm{\varepsilon}_{\lambda}\cdot\mathbf{K}\right]+\sum^{N}_{j=1}\frac{\hbar^{2}\mathbf{k}^{2}_{j}}{2m_{\textrm{e}}}\Bigg{]}\Phi_{\mathbf{K}}\;\;\textrm{with}\;\;\mathbf{K}=\sum^{N}_{j=1}\mathbf{k}_{j}.$

As a next step we define another set of bosonic operators
$\hat{c}^{\dagger}_{\lambda},\hat{c}_{\lambda}$

$\displaystyle\hat{c}_{\lambda}=\hat{b}_{\lambda}-\frac{g_{0}\bm{\varepsilon}_{\lambda}\cdot\mathbf{K}}{\hbar\widetilde{\omega}}\;\;\;\textrm{and}\;\;\;\hat{c}^{\dagger}_{\lambda}=\hat{b}^{\dagger}_{\lambda}-\frac{g_{0}\bm{\varepsilon}_{\lambda}\cdot\mathbf{K}}{\hbar\widetilde{\omega}},$
(5.21)

and the operator $\hat{H}\Phi_{\mathbf{K}}$ of Eq. (5.1) takes the form

$\displaystyle\hat{H}\Phi_{\mathbf{K}}=\Bigg{[}\sum^{2}_{\lambda=1}\left[\hbar\widetilde{\omega}\left(\hat{c}^{\dagger}_{\lambda}\hat{c}_{\lambda}+\frac{1}{2}\right)-\frac{g^{2}_{0}}{\hbar\widetilde{\omega}}\left(\bm{\varepsilon}_{\lambda}\cdot\mathbf{K}\right)^{2}\right]+\frac{\hbar^{2}}{2m_{\textrm{e}}}\sum\limits^{N}_{j=1}\mathbf{k}^{2}_{j}\Bigg{]}\Phi_{\mathbf{K}}.$
(5.22)

We note that also the operators
$\hat{c}^{\dagger}_{\lambda},\hat{c}_{\lambda}$ satisfy bosonic cummutation
relations
$[\hat{c},\hat{c}^{\dagger}_{\lambda^{\prime}}]=\delta_{\lambda\lambda^{\prime}}$
for $\lambda,\lambda^{\prime}=1,2$. For the operator
$\hat{H}_{\lambda}=\hbar\widetilde{\omega}\left(\hat{c}^{\dagger}_{\lambda}\hat{c}_{\lambda}+1/2\right)$
which has the form of a harmonic oscillator it is well-known that the complete
set of eigenstates is [84]

$\displaystyle|n_{\lambda},\bm{\varepsilon}_{\lambda}\cdot\mathbf{K}\rangle_{\lambda}=\frac{(\hat{c}^{\dagger}_{\lambda})^{n_{\lambda}}}{\sqrt{n_{\lambda}!}}|0,\bm{\varepsilon}_{\lambda}\cdot\mathbf{K}\rangle_{\lambda}\;\;\textrm{with}\;\;n_{\lambda}\in\mathbb{Z},\lambda=1,2$

where $|0,\bm{\varepsilon}_{\lambda}\cdot\mathbf{K}\rangle_{\lambda}$ is the
ground-state of $\hat{H}_{\lambda}$, which gets annihilated by
$\hat{c}_{\lambda}$ [84]. Further, the eigenvalues of $\hat{H}_{\lambda}$ are
$\hbar\widetilde{\omega}(n_{\lambda}+1/2)$. The operator
$\hat{H}\Phi_{\mathbf{K}}$ in Eq. (5.22) in terms of the bosonic operators
$\hat{c}^{\dagger}_{\lambda},\hat{c}_{\lambda}$ contains only the sum over
$\hat{H}_{\lambda}$, as a consequence by applying $\hat{H}\Phi_{\mathbf{K}}$
on the states
$\prod_{\lambda}|n_{\lambda},\bm{\varepsilon}_{\lambda}\cdot\mathbf{K}\rangle_{\lambda}$
we find

$\displaystyle\hat{H}\Phi_{\mathbf{K}}\prod^{2}_{\lambda=1}|n_{\lambda},\bm{\varepsilon}_{\lambda}\cdot\mathbf{K}\rangle_{\lambda}=\left[\sum^{2}_{\lambda=1}\left[\hbar\widetilde{\omega}\left(n_{\lambda}+\frac{1}{2}\right)-\frac{g^{2}_{0}\left(\bm{\varepsilon}_{\lambda}\cdot\mathbf{K}\right)^{2}}{\hbar\widetilde{\omega}}\right]+\sum\limits^{N}_{j=1}\frac{\hbar^{2}\mathbf{k}^{2}_{j}}{2m_{\textrm{e}}}\right]\Phi_{\mathbf{K}}\prod^{2}_{\lambda=1}|n_{\lambda},\bm{\varepsilon}_{\lambda}\cdot\mathbf{K}\rangle_{\lambda}.$

From the equation above we conclude that the complete set of eigenstates of
the Hamiltonian in Eq. (5.2) describing the 2DEG coupled to the cavity field
is

$\displaystyle\Phi_{\mathbf{K}}\prod^{2}_{\lambda=1}|n_{\lambda},\bm{\varepsilon}_{\lambda}\cdot\mathbf{K}\rangle_{\lambda}\;\;\textrm{with}\;\;\lambda=1,2\;\;\textrm{and}\;\;\mathbf{K}=\sum^{N}_{j=1}\mathbf{k}_{j}$
(5.25)

and the full eigenspectrum reads as

$\displaystyle
E_{n_{\lambda},\mathbf{k}}=\sum^{2}_{\lambda=1}\left[\hbar\widetilde{\omega}\left(n_{\lambda}+\frac{1}{2}\right)-\frac{\gamma}{N}\frac{\left(\bm{\varepsilon}_{\lambda}\cdot\hbar\mathbf{K}\right)^{2}}{2m_{\textrm{e}}}\right]+\sum\limits^{N}_{j=1}\frac{\hbar^{2}\mathbf{k}^{2}_{j}}{2m_{\textrm{e}}}.$
(5.26)

In the above expression for the energy spectrum there exists also a negative
term proportional to the square of collective momentum of all the electrons in
the 2DEG,
$\sim\left(\bm{\varepsilon}_{\lambda}\cdot\hbar\mathbf{K}\right)^{2}$. This
particular term is an all-to-all interaction between the electrons, mediated
by the photon field. We call it all-to-all because the momentum of each
individual electron couples to the momenta of all the other electrons.

Moreover, we would like to mention that to obtain the expression of Eq. (5.26)
we substituted the single-particle coupling constant $g_{0}$ given by Eq.
(5.10) in Eq. (5.1) and introduced the parameter $\gamma$

$\displaystyle\gamma=\frac{2m_{\textrm{e}}N}{\hbar^{2}}\frac{g^{2}_{0}}{\hbar\widetilde{\omega}}=\frac{\omega^{2}_{p}}{\widetilde{\omega}^{2}}=\frac{\omega^{2}_{p}}{\omega^{2}+\omega^{2}_{p}}\leq
1.$ (5.27)

The parameter $\gamma$ must be interpreted as the collective coupling constant
of the 2DEG to the cavity field. The collective coupling $\gamma$ depends on
the cavity frequency and the electron density in the cavity $n_{\textit{e}}$
via the dressed frequency $\omega_{p}$ defined in Eq. (5.4). This means that
the more charges in the system, the stronger the light-matter coupling.
Further, we emphasize that the collective coupling $\gamma$ is dimensionless
and most importantly $\gamma$ has an upper bound and cannot exceed one. This
upper bound, as we will later see in section 5.3, guarantees the stability of
the electron-photon system.

####  5.2 Ground State in the Thermodynamic Limit

> In this limit, a large collection $N\rightarrow\infty$ of objects can behave
> completely differently—have different symmetry from anything that the
> separate objects can themselves exhibit.

Philip Anderson about the thermodynamic limit
More and Different [129]

Having diagonalized the Hamiltonian $\hat{H}$ of Eq. (5.2) for $N$ free
electrons coupled to a single mode, we want now to find the ground state of
this many-body system in the thermodynamic (or large $N$) limit. To do so, we
need to minimize the energy of the many-body spectrum given by Eq. (5.26) in
the limit where the number of electrons $N$ and the area of the materials
$S=L^{2}$ become arbitrarily large and approach the thermodynamic limit. This
procedure needs to be performed in such a way that the 2D electron density
$n_{2\textrm{D}}=N/S$ stays fixed.

To define properly the electron density we need to compute the number of
allowed states in a region of $\mathbf{k}$-space of volume
$\Omega_{\mathcal{D}}$. The volume $\Omega_{\mathcal{D}}$ is defined with
respect to a distribution $\mathcal{D}$ which has particular shape in
$\mathbf{k}$-space. The number of allowed states in the volume
$\Omega_{\mathcal{D}}$ is

$\displaystyle\\#\textrm{states}=\frac{\Omega_{\mathcal{D}}}{(2\pi)^{2}/L^{2}}=\frac{\Omega_{\mathcal{D}}S}{(2\pi)^{2}}.$
(5.28)

The volume $\Omega_{\mathcal{D}}$ with respect to the generic distribution
$\mathcal{D}(\mathbf{k}-\mathbf{q})$ whose origin $\mathbf{q}$ is an arbitrary
point in $\mathbf{k}$-space (see Fig. 5.2) is defined as

$\displaystyle\Omega_{\mathcal{D}}=\iint\limits_{-\infty}^{+\infty}\mathcal{D}(\mathbf{k}-\mathbf{q})d^{2}k=\iint\limits_{-\infty}^{+\infty}\mathcal{D}(\mathbf{u})d^{2}u,$
(5.29)

where we performed the shift $\mathbf{u}=\mathbf{k}-\mathbf{q}$. With the use
of $\Omega_{\mathcal{D}}$ we can now define the 2D electron density. The
number of electrons $N$ we can accommodate within the volume
$\Omega_{\mathcal{D}}$ is 2 times (due to the spin degeneracy) the number of
allowed states $N=2\\#\textrm{states}=2\Omega_{\mathcal{D}}S/(2\pi)^{2}.$
Thus, with find that the 2D electron density is

$\displaystyle
n_{2\textrm{D}}=\frac{N}{S}=\frac{2\Omega_{\mathcal{D}}}{(2\pi)^{2}}.$ (5.30)

As we already stated, our aim is to find the ground state of the system. From
Eq. (5.26) it is clear that the energy minimizes for $n_{\lambda}=0$ for both
$\lambda=1,2$. Thus, the photonic contribution to the ground state energy is
$E_{\textrm{p}}=\hbar\widetilde{\omega}$, which is constant and independent of
the momenta of the electrons. As a consequence, the photonic contribution can
be neglected for finding the ground-state distribution of the electrons in
$\mathbf{k}$-space. Thus, the ground-state energy of the system as a function
of the momenta $\mathbf{k}_{j}$ is

$\displaystyle
E_{\mathbf{k}}=\frac{\hbar^{2}}{2m_{\textrm{e}}}\left[\sum\limits^{N}_{j=1}\mathbf{k}^{2}_{j}-\frac{\gamma}{N}\sum^{2}_{\lambda=1}\left(\bm{\varepsilon}_{\lambda}\cdot\mathbf{K}\right)^{2}\right],$
(5.31)

In the expression above we have two contributions: (i) a positive one, which
is the sum over the individual kinetic energies of all the electrons and we
denote by $T$, and (ii) a negative one which is minus the square of the
collective momentum $\mathbf{K}=\sum_{j}\mathbf{k}_{j}$ of the electrons.

To find the ground-state distribution in $\mathbf{k}$-space we need to
minimize the energy density $E_{\mathbf{k}}/S$ with respect to the
distribution $\mathcal{D}(\mathbf{k}-\mathbf{q})$. In the thermodynamic limit,
the number of particles $N$ and the area $S$ of the 2D material tend to
infinity and the sums in the expression for the energy density
$E_{\mathbf{k}}/S$ turn into integrals [63]. Thus, for the kinetic energy per
area $T/S$ in the large $N,S$ limit with doubly occupied momenta we have [63].

$\displaystyle\frac{T}{S}=\frac{\hbar^{2}}{2m_{\textrm{e}}}\lim_{S\to\infty}\frac{2}{S}\sum_{\mathbf{k}}\mathbf{k}^{2}=\frac{\hbar^{2}}{2m_{\textrm{e}}}\frac{2}{(2\pi)^{2}}\iint\limits^{+\infty}_{-\infty}d^{2}k\mathcal{D}(\mathbf{k}-\mathbf{q})\mathbf{k}^{2}$

We perform the transformation $\mathbf{u}=\mathbf{k}-\mathbf{q}$ and we obtain

$\displaystyle\frac{T}{S}=\frac{\hbar^{2}}{2m_{\textrm{e}}}\left[\underbrace{\frac{2}{(2\pi)^{2}}\iint\limits^{+\infty}_{-\infty}d^{2}u\mathcal{D}(\mathbf{u})\mathbf{u}^{2}}_{t_{\mathcal{D}}}+2\mathbf{q}\underbrace{\frac{2}{(2\pi)^{2}}\iint\limits^{+\infty}_{-\infty}d^{2}u\mathcal{D}(\mathbf{u})\mathbf{u}}_{\mathbf{K}_{\mathcal{D}}}+\mathbf{q}^{2}\underbrace{\frac{2}{(2\pi)^{2}}\iint\limits^{+\infty}_{-\infty}d^{2}u\mathcal{D}(\mathbf{u})}_{n_{2\textrm{D}}}\right]$

Thus, the kinetic energy per area $\tau[\mathcal{D}]=T/S$ is

$\displaystyle\tau[\mathcal{D}]=\frac{\hbar^{2}}{2m_{\textrm{e}}}\left(t_{\mathcal{D}}+2\mathbf{q}\cdot\mathbf{K}_{\mathcal{D}}+\mathbf{q}^{2}n_{2\textrm{D}}\right).$
(5.34)

Before we continue, let us comment and give an interpretation of each term
appearing above. The term $t_{\mathcal{D}}$ is the standard kinetic energy of
the electron gas with respect to the distribution centered at zero
$\mathcal{D}(\mathbf{u})$ [63]. The term $\mathbf{K}_{\mathcal{D}}$ is the
collective momentum of the electrons with respect to
$\mathcal{D}(\mathbf{u})$, and $\mathbf{q}^{2}n_{2\textrm{D}}$ is the kinetic
energy due to the arbitrary origin of the distribution,
$\mathcal{D}(\mathbf{k}-\mathbf{q})$ (see Fig. 5.2). This last term depends on
the 2D density $n_{2\textrm{D}}$ and the origin of the distribution
$\mathbf{q}$, but it is independent of the shape of the distribution
$\mathcal{D}$.

Figure 5.2: Illustration of a generic distribution
$\mathcal{D}(\mathbf{k}-\mathbf{q})$ in $\mathbf{k}$-space. The shape
$\mathcal{D}$ as well as the origin $\mathbf{q}$ of the distribution are
arbitrary. To find the ground-state distribution in $\mathbf{k}$-space we have
to minimize the energy density of the system with respect to both the shape
$\mathcal{D}$ and the origin $\mathbf{q}$.

Let us continue by computing the second term appearing in Eq. (5.31),
$\bm{\varepsilon}_{\lambda}\cdot\mathbf{K}$. The collective momentum per area
$\mathbf{K}/S$, with doubly occupied momenta, in the thermodynamic limit is

$\displaystyle\frac{\mathbf{K}}{S}$ $\displaystyle=$
$\displaystyle\frac{2}{(2\pi)^{2}}\iint\limits^{+\infty}_{-\infty}d^{2}k\mathcal{D}(\mathbf{k}-\mathbf{q})\mathbf{k}.$
(5.35)

Performing again the shift $\mathbf{u}=\mathbf{k}-\mathbf{q}$ we find

$\displaystyle\frac{\mathbf{K}}{S}=\frac{2}{(2\pi)^{2}}\iint\limits^{+\infty}_{-\infty}d^{2}u\mathcal{D}(\mathbf{u})\mathbf{u}+\mathbf{q}\frac{2}{(2\pi)^{2}}\iint\limits^{+\infty}_{-\infty}d^{2}u\mathcal{D}(\mathbf{u})=\mathbf{K}_{\mathcal{D}}+\mathbf{q}n_{2\textrm{D}}.$
(5.36)

Taking now the inner product of the above quantity with the polarization
vectors $\bm{\varepsilon}_{\lambda}\cdot\mathbf{K}/S$, squaring it and then
multiplying by $S$, we find the following result for the second term in Eq.
(5.31)

$\displaystyle\frac{\left(\bm{\varepsilon}_{\lambda}\cdot\mathbf{K}\right)^{2}}{S}=S\left(\bm{\varepsilon}_{\lambda}\cdot\mathbf{K}_{\mathcal{D}}+\bm{\varepsilon}_{\lambda}\cdot\mathbf{q}n_{2\textrm{D}}\right)^{2}.$
(5.37)

Adding the two contributions which we found in Eqs. (5.34) and (5.37) we
obtain the expression for the energy density as a function of the shape of the
distribution $\mathcal{D}$ and the origin $\mathbf{q}$

$\displaystyle\mathcal{E}[\mathcal{D}]\equiv\frac{E_{\mathbf{k}}}{S}=\frac{\hbar^{2}}{2m_{\textrm{e}}}\left[t_{\mathcal{D}}+2\mathbf{q}\cdot\mathbf{K}_{\mathcal{D}}+\mathbf{q}^{2}n_{2\textrm{D}}-\frac{\gamma}{n_{2\textrm{D}}}\sum^{2}_{\lambda=1}\left(\bm{\varepsilon}_{\lambda}\cdot\mathbf{K}_{\mathcal{D}}+\bm{\varepsilon}_{\lambda}\cdot\mathbf{q}n_{2\textrm{D}}\right)^{2}\right].$
(5.38)

The energy density first has to be minimized with respect to the origin of the
distribution $\mathbf{q}=(q_{x},q_{y})$. For that we compute the derivative of
the energy density $\mathcal{E}[\mathcal{D}]$ with respect to both components
of the vector $\mathbf{q}=(q_{x},q_{y})$,

$\displaystyle\frac{\partial\mathcal{E}[\mathcal{D}]}{\partial
q_{x}}=\frac{\hbar^{2}}{2m_{\textrm{e}}}2(1-\gamma)\left(K^{x}_{\mathcal{D}}+q_{x}n_{2\textrm{D}}\right)=0,$
$\displaystyle\frac{\partial\mathcal{E}[\mathcal{D}]}{\partial
q_{y}}=\frac{\hbar^{2}}{2m_{\textrm{e}}}2(1-\gamma)\left(K^{y}_{\mathcal{D}}+q_{y}n_{2\textrm{D}}\right)=0.$
(5.39)

From the above set of equations we find the optimal vector $\mathbf{q}_{0}$
that minimizes the energy density

$\displaystyle\mathbf{q}_{0}=-\frac{\mathbf{K_{\mathcal{D}}}}{n_{2\textrm{D}}},$
(5.40)

where we see surprisingly that the optimal vector $\mathbf{q}_{0}$ is
independent of the coupling constant $\gamma$. Substituting the optimal vector
$\mathbf{q}_{0}$ into the Eq. (5.38) we find for the energy density

$\displaystyle\mathcal{E}[\mathcal{D}]|_{\mathbf{q_{0}}}=\frac{\hbar^{2}}{2m_{\textrm{e}}}\left[t_{\mathcal{D}}-\frac{\mathbf{K}^{2}_{\mathcal{D}}}{n_{2\textrm{D}}}\right].$
(5.41)

Having optimized the energy density with respect to the origin $\mathbf{q}$ of
the distribution $\mathcal{D}(\mathbf{k}-\mathbf{q})$ the remaining task is to
optimize the energy with respect to the shape $\mathcal{D}$ of the
distribution in momentum space. In general to perform such a minimization of
the energy functional $\mathcal{E}[\mathcal{D}]$ it is a cumbersome task.
Thus, in order to find the optimal shape of the distribution we will use some
physical intuition.

The energy density $\mathcal{E}[\mathcal{D}]$ and the optimal origin
$\mathbf{q}_{0}$, given by Eqs. (5.41) and (5.40) respectively, are both
independent of the coupling constant $\gamma$. This indicates that the ground-
state and the ground-state energy of the electrons in the thermodynamic limit
is independent of the coupling to the cavity mode. Guided by this observation
let us compare the energy density in Eq. (5.41) with the energy density of the
original Sommerfeld model [63] not coupled to a cavity mode.

#####  5.2.1 Comparison to Free Electron Gas Not Coupled to a Cavity

In the original free electron model introduced by Sommerfeld, the energy of
the system is the sum over the kinetic energies of all the electrons [62, 63]

$\displaystyle
E^{nc}_{\mathbf{k}}=\frac{\hbar^{2}}{2m_{\textrm{e}}}\sum^{N}_{j=1}\mathbf{k}_{j}.$
(5.42)

Due to rotational symmetry, the ground-state $\mathbf{k}$-space distribution
is the standard 2D Fermi sphere $\mathcal{S}(\mathbf{k})$ [63] as shown in
Fig. 5.3.

Figure 5.3: Schematic representation of the ground-state distribution of the
2D free electron model not coupled to a cavity. The ground state distribution
is the 2D Fermi sphere $\mathcal{S}(\mathbf{k})$ with radius
$|\mathbf{k}_{\textrm{F}}|$ (Fermi momentum). For the 2DEG coupled to the
cavity, the ground-state distribution in $\mathbf{k}$-space is also the 2D
Fermi sphere $\mathcal{S}(\mathbf{k})$ with radius
$|\mathbf{k}_{\textrm{F}}|$.

But let us forget for a moment the fact that we know that the ground state
distribution in $\mathbf{k}$-space is the 2D Fermi sphere, and consider again
a generic distribution in $\mathbf{k}$-space
$\mathcal{D}(\mathbf{k}-\mathbf{q})$ with an arbitrary origin $\mathbf{q}$ as
the one in Fig. 5.2. For such a generic distribution the ground state energy
density, which we already computed it in Eq. (5.34), is

$\displaystyle\mathcal{E}^{nc}[\mathcal{D}]=\frac{\hbar^{2}}{2m_{\textrm{e}}}\frac{2}{(2\pi)^{2}}\iint\limits^{+\infty}_{-\infty}d^{2}k\mathcal{D}(\mathbf{k}-\mathbf{q})\mathbf{k}^{2}=\frac{\hbar^{2}}{2m_{\textrm{e}}}\left(t_{\mathcal{D}}+2\mathbf{q}\cdot\mathbf{K}_{\mathcal{D}}+\mathbf{q}^{2}n_{2\textrm{D}}\right).$
(5.43)

Minimizing the above energy density with respect to the origin $\mathbf{q}$ of
the distribution we find the optimal origin $\mathbf{q}_{0}$

$\displaystyle\frac{\partial\mathcal{E}^{nc}[\mathcal{D}]}{\partial\mathbf{q}}=\frac{\hbar^{2}}{2m_{\textrm{e}}}2\left(\mathbf{K}_{\mathcal{D}}+\mathbf{q}n_{2\textrm{D}}\right)=0\Longrightarrow\mathbf{q}_{0}=-\frac{\mathbf{K}_{\mathcal{D}}}{n_{2\textrm{D}}}$
(5.44)

which is the same with the one we found in Eq. (5.40) for the electron gas
coupled to the cavity mode. Substituting $\mathbf{q}_{0}$ into the expression
for the energy density $\mathcal{E}^{nc}[\mathcal{D}]$ of the uncoupled
electron gas in Eq. (5.43) we obtain

$\displaystyle\mathcal{E}^{nc}[\mathcal{D}]|_{\mathbf{q_{0}}}=\frac{\hbar^{2}}{2m_{\textrm{e}}}\left[t_{\mathcal{D}}-\frac{\mathbf{K}^{2}_{\mathcal{D}}}{n_{2\textrm{D}}}\right].$
(5.45)

Comparing the energy density above, for the electron gas not coupled to the
cavity $\mathcal{E}^{nc}[\mathcal{D}]|_{\mathbf{q_{0}}}$, to the energy
density $\mathcal{E}[\mathcal{D}]|_{\mathbf{q_{0}}}$ in Eq. (5.41) of the
electron gas coupled to the cavity mode, we see that they are exactly the same

$\displaystyle\mathcal{E}[\mathcal{D}]|_{\mathbf{q_{0}}}=\mathcal{E}^{nc}[\mathcal{D}]|_{\mathbf{q_{0}}}.$
(5.46)

This means that both energy functionals, the coupled and the uncoupled, get
minimized exactly by the same distribution in $\mathbf{k}$-space. As we
already mentioned for the uncoupled free electrons gas, the shape of the
ground-state distribution in $\mathbf{k}$-space is the 2D Fermi sphere
$\mathcal{S}$. For a sphere the collective momentum is zero,
$\mathbf{K}_{\mathcal{S}}=0$, due to the parity symmetry of the Fermi sphere,
$\mathbf{k}\rightarrow-\mathbf{k}$. As consequence the optimal origin of the
sphere is also zero $\mathbf{q}_{0}=0$. Thus, we find that for the coupled
system the ground-state momentum distribution is also the 2D the Fermi sphere
$\mathcal{S}(\mathbf{k})$ centered at zero, as depicted in Fig. 5.3.

Since the collective momentum over the Fermi sphere is zero,
$\mathbf{K}=\sum_{j}\mathbf{k}_{j}=0$, the ground-state of the electron gas
coupled to the cavity is

$\displaystyle|\Psi_{gs}\rangle=|\Phi_{0}\rangle\otimes\prod^{2}_{\lambda=1}|0,0\rangle_{\lambda},$
(5.47)

where $|\Phi_{0}\rangle$ is the Slater determinant given by Eq. (5.18) with
zero collective momentum $\mathbf{K}=0$. The fact that the ground state
distribution of the electrons in $\mathbf{k}$-space is the Fermi sphere
implies that the 2DEG coupled to the cavity field is a Fermi liquid [130,
124].

Moreover, for the 2D Fermi sphere, the ground state energy density is equal to
the kinetic energy density $t_{\mathcal{D}}$, whose expression as a function
of the Fermi momentum $\mathbf{k}_{\textrm{F}}$ is

$\displaystyle\mathcal{E}[\mathcal{S}]=\frac{\hbar^{2}}{2m_{\textrm{e}}}\frac{2}{(2\pi)^{2}}\iint\limits^{+\infty}_{-\infty}d^{2}k\mathcal{S}(\mathbf{k})\mathbf{k}^{2}=\frac{\hbar^{2}k^{4}_{\textrm{F}}}{16\pi
m_{\textrm{e}}}.$ (5.48)

We note that a discrepancy shows up between the energy density of the
electrons and of the energy density of the photon field. The contribution of
the photon field to the ground-state energy density as we can deduce from Eq.
(5.26) is $E_{p}/S=\hbar\widetilde{\omega}/S$, and in the thermodynamic limit,
where the area of the material $S$ goes macroscopic,
$E_{p}/S=\hbar\widetilde{\omega}/S$ is miniscule and strictly speaking goes to
zero. On the other hand the energy density, of the electronic sector is
finite. This implies that only the electron gas contributes to the ground
state energy density of the interacting electron-photon system in the cavity.
This hints towards the fact that for the photons and electrons to contribute
equally on the ground state a continuum of modes for the photon field need to
be taken into account.

Lastly, from the fact that the electronic ground-state is the standard Fermi
sphere and that the energy density of the cavity field in the thermodynamic
limit is zero, one might conclude that the ground-state of the hybrid
electron-photon system is trivial and there are no quantum fluctuation
effects. However, this is not true. In order to classify completely the
electron-photon ground-state we need to look also at the photon occupation.

#####  5.2.2 Ground State Photon Occupation

The photon number operator is defined as

$\displaystyle\hat{N}_{\textrm{ph}}=\sum^{2}_{\lambda=1}\hat{a}^{\dagger}_{\lambda}\hat{a}_{\lambda}.$
(5.49)

To calculate the ground-state photon occupation we need to express the photon
number operator in terms of the bosonic operators
$\hat{c}^{\dagger}_{\lambda},\hat{c}_{\lambda}$ defined in Eq. (5.21). Using
Eqs. (5.6) and (5.21) we find for the photon number operator

$\displaystyle\hat{N}_{\textrm{ph}}$ $\displaystyle=$
$\displaystyle\sum^{2}_{\lambda=1}\frac{1}{4\omega\widetilde{\omega}}\Bigg{[}\left(\omega^{2}-\widetilde{\omega}^{2}\right)\left(\hat{c}_{\lambda}+\frac{g\bm{\varepsilon}_{\lambda}\cdot\mathbf{K}}{\hbar\widetilde{\omega}}\right)^{2}+\left(\omega-\widetilde{\omega}\right)^{2}\left(\hat{c}_{\lambda}+\frac{g\bm{\varepsilon}_{\lambda}\cdot\mathbf{K}}{\hbar\widetilde{\omega}}\right)\left(\hat{c}^{\dagger}_{\lambda}+\frac{g\bm{\varepsilon}_{\lambda}\cdot\mathbf{K}}{\hbar\widetilde{\omega}}\right)$
(5.50) $\displaystyle+$
$\displaystyle\left(\omega+\widetilde{\omega}\right)^{2}\left(\hat{c}_{\lambda}+\frac{g\bm{\varepsilon}_{\lambda}\cdot\mathbf{K}}{\hbar\widetilde{\omega}}\right)\left(\hat{c}^{\dagger}_{\lambda}+\frac{g\bm{\varepsilon}_{\lambda}\cdot\mathbf{K}}{\hbar\widetilde{\omega}}\right)+\left(\omega^{2}-\widetilde{\omega}^{2}\right)\left(\hat{c}^{\dagger}_{\lambda}+\frac{g\bm{\varepsilon}_{\lambda}\cdot\mathbf{K}}{\hbar\widetilde{\omega}}\right)^{2}\Bigg{]}.$

In the ground-state the collective momentum is zero, $\mathbf{K}=0$. Moreover,
from all the terms appearing above only the term that first creates and then
destroys a bosonic excitation $\hat{c}_{\lambda}\hat{c}^{\dagger}_{\lambda}$
gives a non-zero contribution. As a consequence we find that the ground-state
photon occupation is

$\displaystyle\langle\hat{N}_{\textrm{ph}}\rangle_{gs}\equiv\langle\Psi_{gs}|\hat{N}_{\textrm{ph}}|\Psi_{gs}\rangle=\frac{\left(\widetilde{\omega}-\omega\right)^{2}}{2\omega\widetilde{\omega}}.$
(5.51)

The result above shows that the photon occupation is non-zero. This implies
that there are virtual photons in the ground-state of the hybrid electron-
photon system. This phenomenon has also been reported for dissipative systems
[131]. From the fact that the ground-state of the 2DEG in the cavity contains
photons we conclude that there are quantum fluctuations of the photon field in
the ground-state due to the electron-photon coupling. Thus, our system is not
a trivial Fermi liquid, but rather it is a Fermi liquid dressed with photons.

Moreover, the ground-state photon occupation exhibits an interesting behavior
with respect to the electron density. For electron densities small enough for
the diamagnetic shift
$\omega_{p}=\sqrt{e^{2}n_{\textrm{e}}/m_{\textrm{e}}\epsilon_{0}}$ to be much
smaller than the cavity frequency, $\omega_{p}\ll\omega$, the dressed
frequency $\widetilde{\omega}=\sqrt{\omega^{2}_{p}+\omega^{2}}$ is
approximately equal to the cavity frequency,
$\widetilde{\omega}\approx\omega$. In this case the ground-state photon
occupation is zero, $\langle\hat{N}_{\textrm{ph}}\rangle_{gs}=0$. This
recovers nicely the correct decoupling limit.

However, for large electronic densities such that $\omega_{p}\gg\omega$, the
dressed frequency is $\widetilde{\omega}\approx\omega_{p}$ and the numerator
in the expression for $\langle\hat{N}_{\textrm{ph}}\rangle_{gs}$ is
approximately $\omega^{2}_{p}$. In this case, we find that the ground-state
photon occupation has a square root dependence on the electronic density

$\displaystyle\langle\hat{N}_{\textrm{ph}}\rangle_{gs}\sim\sqrt{n_{\textrm{e}}}.$
(5.52)

This shows that the number of photons in the ground-state increases by adding
more electrons to the system. This behavior of the photon occupation might be
related to the superradiant phase transition [50] and could potentially
provide some insights on how to achieve this correlated phase between light
and matter, which still remains elusive.

####  5.3 Critical Coupling, Instability & the Diamagnetic $\mathbf{A}^{2}$
term

Up to here we examined rigorously and in full generality the behavior of the
free electron gas coupled to the cavity, in the regime where the cavity mode
$\omega$ is non-zero and the collective coupling constant $\gamma$, defined in
Eq. (5.27), is smaller than one. But naturally, the following question arises:
what happens in the limit where the cavity frequency approaches zero,
$\omega\rightarrow 0$, and the collective coupling reaches its maximum value
$\gamma\rightarrow 1$?

We will refer to the maximum value of the coupling constant $\gamma$ as
critical coupling, $\gamma_{c}=1$. As we will see, an interesting transition
happens for the system at the critical coupling, from a stable phase to an
unstable phase, as it is summarized by the phase diagram in Fig. 5.6.

#####  5.3.1 Critical Coupling and Infinite Degeneracy

At the critical coupling $\gamma_{c}=1$ the energy density
$\mathcal{E}[\mathcal{D}]$ given by Eq. (5.38) becomes independent of the
origin $\mathbf{q}$

$\displaystyle\mathcal{E}[\mathcal{D}]|_{\gamma_{c}}=\frac{\hbar^{2}}{2m_{\textrm{e}}}\left[t_{\mathcal{D}}-\frac{\mathbf{K}^{2}_{\mathcal{D}}}{n_{2\textrm{D}}}\right].$
(5.53)

This means that the ground state of the system is not unique. Moreover, Eq.
(5.40) from which we determined the optimal value for the vector $\mathbf{q}$
is trivially satisfied and all $\mathbf{q}$ are possible.

In the previous section we showed that the energy density of Eq. (5.53)
minimizes for the 2D sphere $\mathcal{S}(\mathbf{k}-\mathbf{q})$. However, the
energy density $\mathcal{E}[\mathcal{D}]|_{\gamma_{c}}$ at the critical
coupling is independent of the origin $\mathbf{q}$ and the optimal
$\mathbf{q}$ cannot be determined from Eq. (5.40). This means that all spheres
of the form $\mathcal{S}(\mathbf{k}-\mathbf{q})$ are energetically degenerate
and have the same ground-state energy density. This is also depicted in Fig.
5.4. As a consequence, the ground-state $\mathbf{k}$-space distribution it is
not unique but rather at the critical coupling $\gamma_{c}=1$ it is infinitely
degenerate with respect to origin of the $\mathbf{k}$-space distribution of
the electrons.

We note that such an infinite degeneracy appears also for a 2D electron gas in
the presence of perpendicular, homogeneous magnetic field. For such systems we
have the Landau levels exhibiting exactly this behavior [71]. This infinite
degeneracy is directly connected to the quantum Hall effect and the
quantization of the macroscopic Hall conductance [70]. The link between QED
and the quantum Hall effect has also been investigated recently in the
framework of quantum electrodynamical Bloch theory [11].

We would also like to mention that the fact that all spheres
$\mathcal{S}(\mathbf{k}-\mathbf{q})$ of arbitrary origin $\mathbf{q}$ are
energetically degenerate means that the ground-state of our system is
invariant under translations in $\mathbf{k}$-space. This implies that it is
invariant under Galilean boosts.

Figure 5.4: Schematic depiction of the infinite degeneracy for $\gamma_{c}=1$.
All two-dimensional Fermi spheres of arbitrary center have exactly the same
ground-state energy.

#####  5.3.2 No Ground State Beyond the Critical Coupling

To complete our investigation, we will consider now also the case where the
collective coupling constant becomes larger than the critical coupling
$\gamma_{c}=1$. From its definition in Eq. (5.27) the collective coupling
$\gamma$ is not allowed to exceed the critical coupling, but exploring this
scenario will provide further physical insight why this should not happen.

Figure 5.5: For $\gamma>1$ by shifting the 2D Fermi sphere arbitrarily far in
$\mathbf{k}$-space the energy of the system can be lowered indefinitely. This
implies that the energy is unbounded from below and that the system is
unstable.

Without loss of generality, we simplify our consideration to the case where
the photon field has a single polarization vector
$\bm{\varepsilon}_{1}=\mathbf{e}_{x}$ and $\bm{\varepsilon}_{2}=0$. In this
case the energy density $\mathcal{E}[\mathcal{D}]$ given by Eq. (5.38) as a
function of the $x$-component of $\mathbf{q}=(q_{x},q_{y})$ is

$\displaystyle\mathcal{E}(q_{x})=\frac{\hbar^{2}}{2m_{\textrm{e}}}(1-\gamma)\left(2q_{x}K^{x}_{\mathcal{D}}+q^{2}_{x}n_{2\textrm{D}}\right).$
(5.54)

In the equation above we have neglected all terms in Eq. (5.38) independent of
$q_{x}$. For $\gamma>1$ the energy density above is unbounded from below and
has no minimum because $1-\gamma<0$ is negative and taking the limit for
$q_{x}$ to infinity the energy density goes to minus infinity

$\displaystyle\lim_{q_{x}\to\infty}\mathcal{E}(q_{x})=\frac{\hbar^{2}}{2m_{\textrm{e}}}\lim_{q_{x}\to\infty}(1-\gamma)\left(2q_{x}K^{x}_{\mathcal{D}}+q^{2}_{x}n_{\textrm{e}}\right)=-\infty.$

This shows that the 2DEG coupled to the cavity mode for $\gamma>1$ has no
ground state and the system in this case is unstable, because by shifting
further and further the origin $\mathbf{q}$ of the $\mathbf{k}$-space
distribution (see also Fig.5.5), the energy density can be lowered
indefinitely111We would like to highlight that this argument is similar to the
one for the lack of ground state in the length gauge when the dipole self-
energy is omitted. In the length gauge the energy can be lowered indefinitely
by shifting the electronic wavefunction further and further in real space
[48].. Thus, we come to the conclusion that the upper bound of the collective
coupling $\gamma$ in Eq. (5.27) guarantees the stability of the interacting
electron-photon system.

Figure 5.6:  Phase diagram for the 2DEG coupled to the cavity. The system is
stable and has a ground state for $\gamma<1$. At the critical coupling
$\gamma_{c}=1$ the ground state is infinitely degenerate. Beyond the critical
coupling $\gamma>1$ the system is unstable and has no ground-state.

#####  5.3.3 No-Go Theorem and the $\mathbf{A}^{2}$ Term

In this section now we would like to investigate the importance of the often
neglected [43] diamagnetic $\mathbf{A}^{2}$ term for our system. The influence
of the diamagnetic term has been studied theoretically in multiple
publications [49, 43, 47, 45] and its influence has also been experimentally
measured [14]. Moreover, the elimination of the diamagnetic $\mathbf{A}^{2}$
term is known to be responsible for the notorious superradiant phase
transition of the Dicke model [103]. The superradiant phase transition was
firstly predicted by Hepp and Lieb [50] for the Dicke model in the
thermodynamic limit and soon after derived in a different way by Wang and Hioe
[132]. However, the existence of the superradiant phase was challenged by a
no-go theorem [54] which demonstrated that the superradiant phase transition
in atomic systems appeared completely due to the elimination of the
$\mathbf{A}^{2}$ term. More recently, another demonstration of a superradiant
phase transition was predicted for circuit QED set-ups [51], which was
challenged again by another no-go theorem applicable also to circuit QED
systems [53]. The debate over the existence of the superradiant phase
transition is still ongoing, with new demonstrations emerging from the field
of cavity QED materials [108, 52] accompanied though by the respective no-go
theorems [56, 55].

To examine the importance of the diamagnetic $\mathbf{A}^{2}$ term for our
system, we will study the 2DEG coupled to the cavity field in the absence of
the $\mathbf{A}^{2}$ term. From the Hamiltonian $\hat{H}$ in Eq. (5.2) we can
obtain straightforwardly the Hamiltonian $\hat{H}^{\prime}$ for the 2DEG
coupled to the cavity mode when the $\mathbf{A}^{2}$ term is neglected
$\hat{H}^{\prime}=\hat{H}-Ne^{2}\hat{\mathbf{A}}^{2}/2m_{\textrm{e}}$

$\displaystyle\hat{H}^{\prime}$ $\displaystyle=$
$\displaystyle\sum\limits^{N}_{j=1}\left[-\frac{\hbar^{2}\nabla^{2}_{j}}{2m_{\textrm{e}}}+\frac{\textrm{i}e\hbar}{m_{\textrm{e}}}\hat{\mathbf{A}}\cdot\nabla_{j}\right]+\sum^{2}_{\lambda=1}\hbar\omega\left[\hat{a}^{\dagger}_{\lambda}\hat{a}_{\lambda}+\frac{1}{2}\right].$

As we explained in section 5.1, our system is translationally invariant in the
electronic space. Consequently, the many-body electronic eigenfunction is the
Slater determinant of plane waves given by Eq. (5.18). Introducing the
coupling parameter

$\displaystyle
g^{\prime}_{0}=\frac{e\hbar}{m_{\textrm{e}}}\left(\frac{\hbar}{2\epsilon_{0}\omega
V}\right)^{1/2},$ (5.57)

applying the Hamiltonian $\hat{H}^{\prime}$ on the Slater determinant
$\Phi_{\mathbf{K}}$ and substituting the definition for the quantized vector
potential $\hat{\mathbf{A}}$ given by Eq. (5.3) we find

$\displaystyle\hat{H}^{\prime}\Phi_{\mathbf{K}}=\Bigg{\\{}\sum^{2}_{\lambda=1}\left[\hbar\omega\left(\hat{a}^{\dagger}_{\lambda}\hat{a}_{\lambda}+\frac{1}{2}\right)-g^{\prime}_{0}\left(\hat{a}_{\lambda}+\hat{a}^{\dagger}_{\lambda}\right)\bm{\varepsilon}_{\lambda}\cdot\mathbf{K}\right]+\frac{\hbar^{2}}{2m_{\textrm{e}}}\sum^{N}_{j=1}\mathbf{k}^{2}_{j}\Bigg{\\}}\Phi_{\mathbf{K}}.$
(5.58)

The Hamiltonian $\hat{H}^{\prime}$ has precisely the same form with $\hat{H}$
in Eq. (5.1). Following the same procedure for diagonalizing $\hat{H}$, which
we demonstrated in detail in section 5.1, we can diagonalize the Hamiltonian
$\hat{H}^{\prime}$ as well, and we find that the full eigenspectrum is

$\displaystyle
E_{n_{\lambda},\mathbf{k}}=\sum^{2}_{\lambda=1}\left[\hbar\omega\left(n_{\lambda}+\frac{1}{2}\right)-\frac{\gamma^{\prime}}{N}\frac{\hbar^{2}\left(\bm{\varepsilon}_{\lambda}\cdot\mathbf{K}\right)^{2}}{2m_{\textrm{e}}}\right]+\sum\limits^{N}_{j=1}\frac{\hbar^{2}\mathbf{k}^{2}_{j}}{2m_{\textrm{e}}}$

where we substituted the parameter $g^{\prime}_{0}$ of Eq. (5.57) and we
introduced the coupling constant $\gamma^{\prime}$

$\displaystyle\gamma^{\prime}=\frac{2m_{\textrm{e}}N}{\hbar^{2}}\frac{(g^{\prime}_{0})^{2}}{\hbar\omega}=\frac{\omega^{2}_{p}}{\omega^{2}}$
(5.60)

in analogy to the collective coupling $\gamma$ given by Eq. (5.27). The
dressed frequency $\widetilde{\omega}$ does not appear anymore, neither in the
coupling $\gamma^{\prime}$ nor in the energy spectrum (5.3.3), because the
vector potential and the energy of the mode do not get renormalized by the
$\mathbf{A}^{2}$ term. Comparing the spectrum of Eq. (5.3.3) for the
Hamiltonian $\hat{H}^{\prime}$, to the spectrum in Eq. (5.26) derived for the
Hamiltonian of Eq. (5.2) which included the $\mathbf{A}^{2}$ term, we see that
they are the same, up to replacing $\widetilde{\omega}$ with $\omega$ and
$\gamma$ with $\gamma^{\prime}$. The last one is a crucial difference, because
the coupling $\gamma^{\prime}$ has no upper bound and can be arbitrarily
large, as $\omega_{p}$ can be larger than $\omega$. In section 5.3.2 we showed
that the spectrum in Eq. (5.3.3), has no ground-state if the coupling constant
becomes larger than one. Obviously, for large densities the diamagnetic shift
$\omega_{p}$ can be larger than the cavity frequency $\omega$ and
$\gamma^{\prime}$ can exceed the critical coupling one. As a consequence, the
Hamiltonian $\hat{H}^{\prime}$ will not have a ground-state.

This proves that eliminating the diamagnetic $\mathbf{A}^{2}$ term, is a no-go
situation for the 2DEG coupled to the cavity field. Thus, for the proper
description of such an extended solid-state system the diamagnetic
$\mathbf{A}^{2}$ term is absolutely necessary. For finite-system models, like
the Rabi, Jaynes-Cummings or the Dicke model, the diamagnetic term is of
course important, but these models have a stable ground-state even without the
$\mathbf{A}^{2}$ term. This is in stark contrast to the 2DEG coupled to the
cavity and demonstrates explicitly that finite-system models should be applied
to extended condensed matter systems with extra care. Our demonstration
strongly suggests that the diamagnetic term has to be included for the correct
description of extended systems, like 2D materials, coupled to a cavity. We
believe our proof contributes to the ongoing discussion about the proper
description of light-matter interactions [48, 49, 44, 47, 45, 43].

Lastly, we would like to highlight that our proof can be extended to the case
of interacting electrons. This is true because the Coulomb interaction
involves only the relative distances between the electrons and satisfies
translational invariance. Then, by going to the relative distances and center
of mass frame, the relative distances decouple from the quantized vector
potential $\mathbf{A}$ and from the center of mass. However, the center of
mass stays coupled to the quantized field $\mathbf{A}$. Then, one can go along
the lines of our proof and show that without the $\mathbf{A}^{2}$ term the
collective coupling constant has no upper bound and the center of mass can
obtain an arbitrarily large momentum which subsequently leads to an
arbitrarily negative energy. This of course implies that energy of the system
is unbounded from below and that the system consequently has no ground-state.

### CHAPTER 6 Cavity Modified Response Functions

> I was still technician, a problem-solver interested only in specific
> problems, not in buliding the general structure of the subject: I had even
> certain contempt for general formalism. Kubo saw that there were general
> possibilities in the correlation function formalism I had pioneered, and he
> pursued them rather than the specific answers I was after.

Philip Anderson about Ryogo Kubo
More and Different [129]

Linear response theory, also known as Kubo formalism [133], is the framework
which studies systems under the assumptions that they are originally at rest
and then perturbed by an external time-dependent perturbation whose strength
is considered to be small. If the latter is the case, then the response of the
system to the external perturbation can be expanded into a power series with
respect to the strength of the perturbation. To first order in this series,
the response of the system is a linear function of the external perturbation.
This is the regime of linear response theory and from a physical point of view
this is the definition of this particular framework.

Linear response theory finds many applications in the study of atoms,
molecules and solid-state systems, because in most cases experimental probes
can be regarded as small perturbations to the system. If the experimental
probes were not small, they would actually modify the system. This is, for
example, the case when materials are driven by strong lasers or when they are
studied in the presence of strong magnetic fields like in the quantum Hall
effect. Nevertheless, as long as the experimental probes do not modify the
system, linear response is applicable and we can describe such experiments
with linear response functions which are given by the properties of the
original unperturbed system.

####  6.1 Mathematical Formulation of Linear Response

We continue by presenting how linear response theory is usually formulated
mathematically. Suppose we have a system described by the Hamiltonian
$\hat{H}$ and we apply to the system a time-dependent external perturbation
described by the operator $\hat{H}_{\textrm{ext}}(t)$. The time-dependent
Schrödinger equation is

$\displaystyle\hat{H}(t)\Psi(t)=\left[\hat{H}+\hat{H}_{\textrm{ext}}(t)\right]\Psi(t)=\textrm{i}\hbar\partial_{t}\Psi(t).$
(6.1)

In the interaction picture a general state of the system is given by

$\displaystyle\Psi_{I}(t)=\hat{U}^{\dagger}(t)\Psi(t)=e^{\textrm{i}\hat{H}t/\hbar}\Psi(t)$
(6.2)

where $\Psi(t)$ is the wavefunction in the Schrödinger picture. In the
interaction picture any operator $\hat{\mathcal{O}}$ is defined as

$\displaystyle\hat{\mathcal{O}}_{I}(t)=\hat{U}^{\dagger}(t)\hat{\mathcal{O}}\hat{U}(t)=e^{\textrm{i}\hat{H}t/\hbar}\hat{\mathcal{O}}e^{-\textrm{i}\hat{H}t/\hbar}.$
(6.3)

Substituting the definition of the wavefunction $\Psi(t)$ (in the Schrödinger
picture) in terms of the wavefunction $\Psi_{I}(t)$ (in the interaction
picture) into the Schrödinger equation in Eq. (6.1) we find for $\Psi_{I}(t)$

$\displaystyle\hat{H}_{\textrm{ext}}(t)e^{-\textrm{i}\hat{H}t/\hbar}\Psi_{I}(t)=e^{-\textrm{i}\hat{H}t/\hbar}\;\textrm{i}\hbar\partial_{t}\Psi_{I}(t).$
(6.4)

We multiply now the equation above with the operator
$e^{\textrm{i}\hat{H}t/\hbar}$ and after introducing the operator
$\hat{H}_{\textrm{ext},I}(t)=e^{\textrm{i}\hat{H}t/\hbar}\hat{H}_{\textrm{ext}}(t)e^{-\textrm{i}\hat{H}t/\hbar}$
we obtain

$\displaystyle\hat{H}_{\textrm{ext},I}(t)\Psi_{I}(t)=\textrm{i}\hbar\partial_{t}\Psi_{I}(t).$
(6.5)

By integrating the equation above we find

$\displaystyle\Psi_{I}(t)=\Psi(t_{0})-\frac{\textrm{i}}{\hbar}\int^{t}_{t_{0}}dt^{\prime}\hat{H}_{\textrm{ext},I}(t^{\prime})\Psi_{I}(t^{\prime}).$
(6.6)

The equation above on a first glance might look rather simple, but it is
actually very complicated because to find the wavefunction $\Psi_{I}(t)$ at
time $t$ one already needs to know the wavefunction $\Psi_{I}(t^{\prime})$ at
all times $t^{\prime}$ prior to $t$. This means that Eq. (6.6) is not a closed
equation. There is a simple but also rather crude solution to this problem.
The solution is to approximate the wavefunction in the interaction picture at
time $t^{\prime}$ as $\Psi_{I}(t^{\prime})\simeq\Psi(t_{0})$. Under this
approximation and after multiplying Eq. (6.6) with the operator
$\hat{U}(t)=e^{-\textrm{i}\hat{H}t/\hbar}$ we obtain a closed equation for the
wavefunction $\Psi(t)$ in the Schrödinger picture

$\displaystyle\Psi(t)=\hat{U}(t)\Psi(t_{0})-\frac{\textrm{i}}{\hbar}\hat{U}(t)\int^{t}_{t_{0}}dt^{\prime}\hat{H}_{\textrm{ext},I}(t^{\prime})\Psi(t_{0}).$
(6.7)

In linear response theory we are not interested in the time evolution of the
wavefunction, but we are interested in observables. Namely, we are interested
in how an observable $\hat{\mathcal{O}}$ responds to the external
perturbation. The expectation value of the operator $\hat{\mathcal{O}}$ as a
function of time $\langle\hat{\mathcal{O}}(t)\rangle$ in the interaction
picture and the Schrödinger picture is the same

$\displaystyle\langle\hat{\mathcal{O}}(t)\rangle=\langle\Psi_{I}(t)|\hat{\mathcal{O}}_{I}(t)|\Psi_{I}(t)\rangle=\langle\Psi(t)|\hat{\mathcal{O}}|\Psi(t)\rangle.$
(6.8)

For simplicity we denote the ket $|\Psi(t_{0})\rangle$ at time $t_{0}$ as
$|\Psi_{0}\rangle$. It is important to highlight that at time $t_{0}$ the
system is unperturbed and consequently $|\Psi_{0}\rangle$ is the the ground
state of the system before the perturbation,
$|\Psi_{0}\rangle\equiv|\Psi_{gs}\rangle$. Substituting the expression we
derived for the wavefunction in the Schrödinger picture in Eq. (6.7) we find

$\displaystyle\langle\hat{\mathcal{O}}(t)\rangle$ $\displaystyle=$
$\displaystyle\langle\hat{\mathcal{O}}\rangle_{0}-\frac{\textrm{i}}{\hbar}\int^{t}_{t_{0}}dt^{\prime}\langle\Psi_{0}|\hat{U}^{\dagger}(t)\hat{\mathcal{O}}\hat{U}(t)\hat{H}_{\textrm{ext},I}(t^{\prime})|\Psi_{0}\rangle$
$\displaystyle+$
$\displaystyle\frac{\textrm{i}}{\hbar}\int^{t}_{t_{0}}dt^{\prime}\langle\Psi_{0}|\hat{H}_{\textrm{ext},I}(t^{\prime})\hat{U}^{\dagger}(t)\hat{\mathcal{O}}\hat{U}(t)|\Psi_{0}\rangle+(2_{\textrm{nd}}\;\textrm{order
terms in}\;\hat{H}_{\textrm{ext},I}),$

where
$\langle\hat{\mathcal{O}}\rangle_{0}=\langle\Psi_{0}|\hat{\mathcal{O}}|\Psi_{0}\rangle$.
Introducing now the definition of the operator $\hat{\mathcal{O}}$ in the
interaction picture
$\hat{\mathcal{O}}_{I}(t)=\hat{U}^{\dagger}(t)\hat{\mathcal{O}}\hat{U}(t)$, we
can write the expectation value $\langle\hat{\mathcal{O}}(t)\rangle$ of Eq.
(6.1) in terms of the commutator between $\hat{\mathcal{O}}_{I}(t)$ and
$\hat{H}_{\textrm{ext},I}(t^{\prime})$

$\displaystyle\langle\hat{\mathcal{O}}(t)\rangle$ $\displaystyle=$
$\displaystyle\langle\hat{\mathcal{O}}\rangle_{0}-\frac{\textrm{i}}{\hbar}\int^{t}_{t_{0}}dt^{\prime}\langle\Psi_{0}|[\hat{\mathcal{O}}_{I}(t),\hat{H}_{\textrm{ext},I}(t^{\prime})]|\Psi_{0}\rangle.$
(6.10)

To obtain the above result we neglected the higher order terms of the external
perturbation. The change $\delta\langle\hat{\mathcal{O}}(t)\rangle$ in the
expectation value of the observable $\hat{\mathcal{O}}$ is the difference
between the expectation value $\langle\hat{\mathcal{O}}(t)\rangle$ at time $t$
and the expectation value $\langle\hat{\mathcal{O}}\rangle_{0}$ at $t_{0}$ .
Thus, we find for $\delta\langle\hat{\mathcal{O}}(t)\rangle$

$\displaystyle\delta\langle\hat{\mathcal{O}}(t)\rangle=-\frac{\textrm{i}}{\hbar}\int^{t}_{t_{0}}dt^{\prime}\langle\Psi_{0}|[\hat{\mathcal{O}}_{I}(t),\hat{H}_{\textrm{ext},I}(t^{\prime})]|\Psi_{0}\rangle.$
(6.11)

The equation above gives the response of the observable $\mathcal{O}$,
represented by the operator $\hat{\mathcal{O}}$, due to the external
perturbation $\hat{H}_{\textrm{ext}}$ as a function of time. In most cases the
external perturbation is of the form
$\hat{H}_{\textrm{ext}}(t)=f_{\textrm{ext}}(t)\hat{\mathcal{P}}$ where
$f_{\textrm{ext}}(t)$ is some external classical force, field, current or
potential, which couples to some observable of the system represented by the
operator $\hat{\mathcal{P}}$. Substituting now this particular form of the
external perturbation into Eq. (6.11) for the response of $\hat{\mathcal{O}}$
we have

$\displaystyle\delta\langle\hat{\mathcal{O}}(t)\rangle=-\frac{\textrm{i}}{\hbar}\int^{t}_{t_{0}}dt^{\prime}\langle\Psi_{0}|[\hat{\mathcal{O}}_{I}(t),\hat{\mathcal{P}}_{I}(t^{\prime})]|\Psi_{0}\rangle
f_{\textrm{ext}}(t^{\prime}).$ (6.12)

By introducing the theta function $\Theta(t-t^{\prime})$ we can re-write the
response $\delta\langle\hat{\mathcal{O}}(t)\rangle$ with the help of a
function $\chi^{\mathcal{O}}_{\mathcal{P}}(t-t^{\prime})$ as

$\displaystyle\delta\langle\hat{\mathcal{O}}(t)\rangle=\int^{\infty}_{t_{0}}dt^{\prime}\chi^{\mathcal{O}}_{\mathcal{P}}(t-t^{\prime})f_{\textrm{ext}}(t^{\prime}),$
(6.13)

with the function $\chi^{\mathcal{O}}_{\mathcal{P}}(t-t^{\prime})$ defined as

$\displaystyle\chi^{\mathcal{O}}_{\mathcal{P}}(t-t^{\prime})=-\frac{\textrm{i}}{\hbar}\Theta(t-t^{\prime})\langle\Psi_{0}|[\hat{\mathcal{O}}_{I}(t),\hat{\mathcal{P}}_{I}(t^{\prime})]|\Psi_{0}\rangle.$
(6.14)

Functions of this form in linear response are known as response functions.
Such response functions are of great importance because they give us
information about how different observables of the system respond to a (small)
external perturbation. We note once more that the wavefunction $\Psi_{0}$ is
the ground state of the original system described by the Hamiltonian
$\hat{H}$, before the external perturbation $\hat{H}_{\textrm{ext}}(t)$ was
applied. Lastly, by performing a Laplace transform in Eq.(6.13), one can also
find how the observable $\mathcal{O}$ responds to the perturbation in the
frequency domain

$\displaystyle\delta\langle\hat{\mathcal{O}}(w)\rangle=\chi^{\mathcal{O}}_{\mathcal{P}}(w)f_{\textrm{ext}}(w),$
(6.15)

where $\chi^{\mathcal{O}}_{\mathcal{P}}(w)$ and $f_{\textrm{ext}}(w)$ are the
response function and the external perturbation respectively in the frequency
domain [133, 110, 134].

####  6.2 Radiation & Absorption in Linear Response

In the previous section we formulated the framework of linear response. Our
aim now is to apply the linear response formalism to the free electron gas
model coupled to the cavity in order to investigate the radiation and
absorption properties of the interacting electron-photon system. To do so we
will perform linear response on the photonic sector of the system by computing
response functions related to the electromagnetic field. From these response
functions we will obtain information about the absorption and the radiation
properties of the 2DEG coupled to the cavity field.

To probe these properties and responses of the electromagnetic field, we will
apply an external time dependent current $\mathbf{J}_{\textrm{ext}}(t)$ as
shown in Fig. 6.1. It is important to mention that in standard quantum
mechanics the possibility of perturbing a system with an external current does
not exist and only in QED this possibility arises.

To perturb our system by an external current we add the external time
dependent term
$\hat{H}_{\textrm{ext}}(t)=-\mathbf{J}_{\textrm{ext}}(t)\cdot\hat{\mathbf{A}}$
to the Hamiltonian $\hat{H}$ of Eq. (5.2), as it is done in QED [134, 57, 81].
Here for simplicity we choose the external current to flow only along the
$x$-direction
$\mathbf{J}_{\textrm{ext}}(t)=\mathbf{e}_{x}|\mathbf{J}_{\textrm{ext}}(t)|$.
With this term included, the full time-dependent Hamiltonian is

$\displaystyle\hat{H}(t)=\hat{H}-\mathbf{J}_{\textrm{ext}}(t)\cdot\hat{\mathbf{A}}.$
(6.16)

The external current influences the interacting system inside the cavity, and
generates electromagnetic fields, as shown in Fig. 6.1. The influence of the
external current on the observables of the photon field is what we are
interested in.

Figure 6.1:  2D material confined inside a cavity, perturbed by an external
time-dependent current $\mathbf{J}_{\textrm{ext}}(t)$. Due to the external
current, a time-dependent electric field is induced and the cavity radiates.
In an experiment the emitted radiation can be accessed through the openness of
the cavity.

Before we proceed by considering particular response functions we would like
first to derive a generic formula for the response of any observable
$\mathcal{O}$ due to the external time-dependent current. Following the linear
response formalism which we described in section 6.1, and using the definition
for the response of an observable given by Eq. (6.12), the response of any
observable $\hat{\mathcal{O}}$ due to
$\hat{H}_{\textrm{ext}}(t)=-\mathbf{J}_{\textrm{ext}}(t)\cdot\hat{\mathbf{A}}$
is

$\displaystyle\delta\langle\hat{\mathcal{O}}(t)\rangle=-\frac{\textrm{i}}{\hbar}\int^{t}_{0}\langle[\hat{\mathcal{O}}_{I}(t),\hat{\mathbf{A}}_{I}(t^{\prime})]\rangle\left(-\mathbf{J}_{\textrm{ext}}(t^{\prime})\right),$
(6.17)

where $\hat{\mathbf{A}}_{I}(t^{\prime})$ is the $\mathbf{A}$-field in the
interaction picture, and the correlator
$\langle[\hat{\mathcal{O}}_{I}(t),\hat{\mathbf{A}}_{I}(t^{\prime})]\rangle$ is
defined with respect to the ground state $|\Psi_{gs}\rangle$ of the
unperturbed Hamiltonian. Then, the respective response function is

$\displaystyle\chi^{\mathcal{O}}_{A}(t-t^{\prime})=-\frac{\textrm{i}\Theta(t-t^{\prime})}{\hbar}\langle[\hat{\mathcal{O}}_{I}(t),\hat{\mathbf{A}}_{I}(t^{\prime})]\rangle.$
(6.18)

In section 5.2 we found the ground-state $|\Psi_{gs}\rangle$ of the 2DEG
coupled to the cavity (before the current perturbation) in the thermodynamic
limit to be given by Eq. (5.47). Having $|\Psi_{gs}\rangle$ we can compute the
response function for any observable $\hat{\mathcal{O}}$. To find the response
function $\chi^{\mathcal{O}}_{A}(t-t^{\prime})$ of Eq. (6.18) it is necessary
to compute the commutator

$\displaystyle\langle[\hat{\mathcal{O}}_{I}(t),\hat{\mathbf{A}}_{I}(t^{\prime})]\rangle$
$\displaystyle=$
$\displaystyle\langle\hat{\mathcal{O}}_{I}(t)\hat{\mathbf{A}}_{I}(t^{\prime})\rangle-\langle\hat{\mathcal{O}}_{I}(t)\hat{\mathbf{A}}_{I}(t^{\prime})\rangle^{*}.$

In the equation above we used the hermiticity of the operator
$\hat{\mathcal{O}}_{I}(t)\hat{\mathbf{A}}_{I}(t^{\prime})$ which implies that
$\langle\hat{\mathbf{A}}_{I}(t^{\prime})\hat{\mathcal{O}}_{I}(t)\rangle=\langle\hat{\mathcal{O}}_{I}(t)\hat{\mathbf{A}}_{I}(t^{\prime})\rangle^{*}$.
As a consequence we have to compute only the correlator
$\langle\hat{\mathcal{O}}_{I}(t)\hat{\mathbf{A}}_{I}(t^{\prime})\rangle$. From
the fact that $|\Psi_{gs}\rangle$ is the ground state of the Hamiltonian
$\hat{H}$ we have

$\displaystyle
e^{-\textrm{i}\hat{H}t^{\prime}/\hbar}|\Psi_{gs}\rangle=e^{-\textrm{i}E_{0,\mathbf{k}}t^{\prime}/\hbar}|\Psi_{gs}\rangle.$
(6.20)

We use also the definition of the operators in the interaction picture
$\hat{\mathcal{O}}_{I}(t)=e^{\textrm{i}\hat{H}t/\hbar}\hat{\mathcal{O}}e^{-\textrm{i}\hat{H}t/\hbar}$
and we find for the expectation value in the commutator of Eq. (6.2)

$\displaystyle\langle\hat{\mathcal{O}}_{I}(t)\hat{\mathbf{A}}_{I}(t^{\prime})\rangle=e^{\frac{\textrm{i}E_{0,\mathbf{k}}(t-t^{\prime})}{\hbar}}\langle\hat{\mathcal{O}}e^{\frac{-\textrm{i}\hat{H}(t-t^{\prime})}{\hbar}}\hat{\mathbf{A}}\rangle,$
(6.21)

where $E_{0,\mathbf{k}}=E_{\mathbf{k}}+\hbar\widetilde{\omega}$ is the ground-
state energy given by Eq. (5.26), with $n_{\lambda}=0$ for both $\lambda=1,2$.
To proceed, we have to apply the quantized vector potential $\hat{\mathbf{A}}$
to the ground state $|\Psi_{gs}\rangle$. For that purpose we need the
expression of $\hat{\mathbf{A}}$ in terms of the bosonic operators
$\hat{c}_{\lambda},\hat{c}^{\dagger}_{\lambda}$. From Eqs. (5.9) and (5.21),
and for $\mathbf{K}=0$ (which is true in the ground state) we find

$\displaystyle\hat{\mathbf{A}}=\left(\frac{\hbar}{2\epsilon_{0}\tilde{\omega}V}\right)^{\frac{1}{2}}\left(\hat{c}_{1}+\hat{c}^{\dagger}_{1}\right)\mathbf{e}_{x}.$
(6.22)

We note that in the expression above for the vector potential, we have only
kept the polarization $\mathbf{e}_{x}$ which couples to the external current.
We apply the vector potential $\hat{\mathbf{A}}$ to the ground-state
$|\Psi_{gs}\rangle$

$\displaystyle\langle\hat{\mathcal{O}}_{I}(t)\hat{\mathbf{A}}_{I}(t^{\prime})\rangle=\left(\frac{\hbar}{2\epsilon_{0}\tilde{\omega}V}\right)^{\frac{1}{2}}e^{\frac{\textrm{i}E_{0,\mathbf{k}}(t-t^{\prime})}{\hbar}}\langle\Psi_{gs}|\hat{\mathcal{O}}e^{\frac{-\textrm{i}\hat{H}(t-t^{\prime})}{\hbar}}|\Phi_{0}\rangle\otimes|1,0\rangle_{1}|0,0\rangle_{2}.$
(6.23)

From the expression above it is clear that the quantized vector potential
$\hat{\mathbf{A}}$ gets the ground-state to the first excited state for
$n_{1}=1$. The state $|\Phi_{0}\rangle\otimes|1,0\rangle_{1}|0,0\rangle_{2}$
is the first excited state of $\hat{H}$ with eigenenergy
$E_{1,\mathbf{k}}=E_{\mathbf{k}}+2\hbar\widetilde{\omega}$ and we find

$\displaystyle\langle\hat{\mathcal{O}}_{I}(t)\hat{\mathbf{A}}_{I}(t^{\prime})\rangle=\left(\frac{\hbar}{2\epsilon_{0}\tilde{\omega}V}\right)^{\frac{1}{2}}e^{-\textrm{i}\widetilde{\omega}(t-t^{\prime})}\langle\Psi_{gs}|\hat{\mathcal{O}}|\Phi_{0}\rangle\otimes|1,0\rangle_{1}|0,0\rangle_{2}.$
(6.24)

Finally, using the above result we obtain the expression for the generic
commutator of Eq. (6.2)

$\displaystyle\langle[\hat{\mathcal{O}}_{I}(t),\hat{\mathbf{A}}_{I}(t^{\prime})]\rangle$
$\displaystyle=$
$\displaystyle\left(\frac{\hbar}{2\epsilon_{0}\tilde{\omega}V}\right)^{\frac{1}{2}}\Bigg{[}e^{-\textrm{i}\widetilde{\omega}(t-t^{\prime})}\langle\Psi_{gs}|\hat{\mathcal{O}}|\Phi_{0}\rangle\otimes|1,0\rangle_{1}|0,0\rangle_{2}$
$\displaystyle-$ $\displaystyle
e^{\textrm{i}\widetilde{\omega}(t-t^{\prime})}\left(\langle\Psi_{gs}|\hat{\mathcal{O}}|\Phi_{0}\rangle\otimes|1,0\rangle_{1}|0,0\rangle_{2}\right)^{*}\Bigg{]}.$

The formula above is very important because it applies to any observable
$\hat{\mathcal{O}}$ and in what follows we will make use of it for the
computation of several response functions.

#####  6.2.1 $\mathbf{A}$-Field Response & Absorption

The first response that we are interested in, is the response of the
$\mathbf{A}$-field. The response of the vector potential
$\delta\langle\hat{\mathbf{A}}(t)\rangle$ is defined via Eq. (6.11) and is
given by the $\mathbf{A}$-field response function
$\chi^{A}_{A}(t-t^{\prime})$.

$\displaystyle\chi^{\mathcal{A}}_{A}(t-t^{\prime})=-\frac{\textrm{i}\Theta(t-t^{\prime})}{\hbar}\langle[\hat{\mathbf{A}}_{I}(t),\hat{\mathbf{A}}_{I}(t^{\prime})]\rangle.$
(6.26)

From the generic formula that we derived in Eq. (6.2), it is clear that to
find the response function $\chi^{\mathcal{A}}_{A}(t-t^{\prime})$, all we have
to compute is
$\langle\Psi_{gs}|\hat{\mathbf{A}}|\Phi_{0}\rangle\otimes|1,0\rangle_{1}|0,0\rangle_{2}$.
For that we use Eq. (6.22) which gives the vector potential $\hat{\mathbf{A}}$
in terms of the operators $\hat{c}_{1},\hat{c}^{\dagger}_{1}$ and we have

$\displaystyle\langle\Psi_{gs}|\hat{\mathbf{A}}|\Phi_{0}\rangle\otimes|1,0\rangle_{1}|0,0\rangle_{2}=\left(\frac{\hbar}{2\epsilon_{0}\widetilde{\omega}V}\right)^{\frac{1}{2}}.$
(6.27)

Combining the above result together with Eqs. (6.2) and Eq. (6.18) we obtain
the expression for response function in time $\chi^{A}_{A}(t-t^{\prime})$

$\displaystyle\chi^{A}_{A}(t-t^{\prime})=-\frac{\Theta(t-t^{\prime})\sin(\widetilde{\omega}(t-t^{\prime}))}{\epsilon_{0}\widetilde{\omega}V}.$
(6.28)

The response function above is also known as the $\mathbf{A}$-field
propagator. Moreover, we use of the integral form of the $\Theta$-function

$\displaystyle\Theta(\tau)=-\frac{1}{2\pi\textrm{i}}\lim_{\eta\rightarrow
0^{+}}\int^{\infty}_{-\infty}\frac{e^{-\textrm{i}w\tau}}{w+\textrm{i}\eta}dw.$
(6.29)

We perform a Fourier transform for the response function
$\chi^{A}_{A}(t-t^{\prime})$ and we obtain the response of the quantized
vector potential $\mathbf{A}$ in the frequency domain $\chi^{A}_{A}(w)$

$\displaystyle\chi^{A}_{A}(w)=\frac{-1}{2\epsilon_{0}\widetilde{\omega}V}\lim_{\eta\to
0^{+}}\left[\frac{1}{w+\widetilde{\omega}+\textrm{i}\eta}-\frac{1}{w-\widetilde{\omega}+\textrm{i}\eta}\right].$
(6.30)

Then, from the expression above we can easily deduce the real
$\Re[\chi^{A}_{A}(w)]$ and the imaginary $\Im[\chi^{A}_{A}(w)]$ parts of
$\chi^{A}_{A}(w)$

$\displaystyle\Re[\chi^{A}_{A}(w)]$ $\displaystyle=$
$\displaystyle\frac{1}{2\epsilon_{0}\widetilde{\omega}V}\left[\frac{w-\widetilde{\omega}}{(w-\widetilde{\omega})^{2}+\eta^{2}}-\frac{w+\widetilde{\omega}}{(w+\widetilde{\omega})^{2}+\eta^{2}}\right],$
$\displaystyle\Im[\chi^{A}_{A}(w)]$ $\displaystyle=$
$\displaystyle\frac{\eta}{2\epsilon_{0}\widetilde{\omega}V}\left[\frac{1}{(w+\widetilde{\omega})^{2}+\eta^{2}}-\frac{1}{(w-\widetilde{\omega})^{2}+\eta^{2}}\right].$
(6.31)

From both, the real and the imaginary parts of the response function, we see
that the poles are at the frequency $w=\pm\widetilde{\omega}$. This is also
depicted in Fig. 6.2. The frequency $\widetilde{\omega}$ defined in Eq. (5.7)
depends on the cavity frequency $\omega$ and the plasma frequency $\omega_{p}$
in the cavity. This means that the 2DEG coupled to the cavity field exhibits a
plasmon-polariton resonance (or excitation).

Figure 6.2:  Real $\Re[\chi^{A}_{A}(w)]$ and imaginary $\Im[\chi^{A}_{A}(w)]$
parts of the $\mathbf{A}$-field response function $\chi^{A}_{A}(w)$ in the
frequency domain, plotted with a finite value for the artificial broadening
parameter $\eta$. The resonances for both parts appear at the plasmon-
polariton frequency $w=\pm\widetilde{\omega}$.

For a self-adjoint operator, like the $\mathbf{A}$-field, the real and the
imaginary part of any response function have to be even and odd respectively
[110]. As we see in Fig. 6.2 this is indeed true.

Further, before we proceed, we would like to briefly comment on how these
response functions should be interpreted. The real part $\Re[\chi^{A}_{A}(w)]$
is the component of the response function which is in-phase with the external
current and describes a polarization process in which the wavefunction is
modified periodically without any energy (on average) being absorbed or
released due to the external driving [110]. On the other side, the imaginary
part $\Im[\chi^{A}_{A}(w)]$ is the out-of-phase component of $\chi^{A}_{A}(w)$
and is responsible for the appearance of energy absorption in the system, with
the absorption rate $W$ defined as [110]

$\displaystyle W=-w\Im[\chi^{A}_{A}(w)]|\mathbf{J}_{\textrm{ext}}(w)|^{2}.$
(6.32)

#####  6.2.2 Electric Field Response & Current Induced Radiation

In addition to the response of the $\mathbf{A}$-field we would also like to
compute the response of the electric field $\mathbf{E}$ due to the external
time-dependent current $\mathbf{J}_{\textrm{ext}}(t)$. The electric field in
dipole approximation and polarized in the $x$-direction is [48]

$\hat{\mathbf{E}}=\textrm{i}\left(\frac{\hbar\omega}{2\epsilon_{0}V}\right)^{\frac{1}{2}}\left(\hat{a}_{1}-\hat{a}^{\dagger}_{1}\right)\mathbf{e}_{x}.$
(6.33)

Having the expression for the electric field we can compute the electric field
response function $\chi^{E}_{A}(t-t^{\prime})$

$\displaystyle\chi^{E}_{A}(t-t^{\prime})=-\frac{\textrm{i}\Theta(t-t^{\prime})}{\hbar}\langle[\hat{\mathbf{E}}_{I}(t),\hat{\mathbf{A}}_{I}(t^{\prime})]\rangle.$
(6.34)

By writing the electric field in terms of the bosonic operators
$\hat{c}_{1},\hat{c}^{\dagger}_{1}$

$\hat{\mathbf{E}}=\textrm{i}\left(\frac{\hbar\widetilde{\omega}}{2\epsilon_{0}V}\right)^{\frac{1}{2}}\left(\hat{c}_{1}-\hat{c}^{\dagger}_{1}\right)\mathbf{e}_{x}.$
(6.35)

we can use the generic formula that we derived for any observable in Eq. (6.2)
for the computation of the correlator
$\langle[\hat{\mathbf{E}}_{I}(t),\hat{\mathbf{A}}_{I}(t^{\prime})]\rangle$,
and we obtain the expression for the electric field response function

$\displaystyle\chi^{E}_{A}(t-t^{\prime})=\frac{\Theta(t-t^{\prime})\cos(\widetilde{\omega}(t-t^{\prime}))}{\epsilon_{0}V}.$
(6.36)

This response function describes the generation of a time-dependent electric
field due to the external current $\mathbf{J}_{\textrm{ext}}(t)$. This means
that the external current makes the interacting electron-photon system to
radiate. From Eq. (6.36) we see the radiation is at the plasmon-polariton
frequency $\widetilde{\omega}$, as the response function in time is a cosine
of $\widetilde{\omega}$.

Further, by performing a Fourier transform, we find the response function in
the frequency domain

$\displaystyle\chi^{E}_{A}(w)=\frac{\textrm{i}}{2\epsilon_{0}V}\lim_{\eta\to
0^{+}}\left[\frac{1}{w+\widetilde{\omega}+\textrm{i}\eta}+\frac{1}{w-\widetilde{\omega}+\textrm{i}\eta}\right],$
(6.37)

from which we deduce the real and the imaginary parts of $\chi^{E}_{A}(w)$

$\displaystyle\Re[\chi^{E}_{A}(w)]=\frac{\eta}{2\epsilon_{0}V}\left[\frac{1}{(w+\widetilde{\omega})^{2}+\eta^{2}}-\frac{1}{(w-\widetilde{\omega})^{2}+\eta^{2}}\right],$
$\displaystyle\Im[\chi^{E}_{A}(w)]=\frac{1}{2\epsilon_{0}V}\left[\frac{w+\widetilde{\omega}}{(w+\widetilde{\omega})^{2}+\eta^{2}}-\frac{w-\widetilde{\omega}}{(w-\widetilde{\omega})^{2}+\eta^{2}}\right].$
(6.38)

The fact that the radiation is emitted at the plasmon-polariton resonance
$w=\pm\widetilde{\omega}$ can be understood also from the pole structure of
the response function in the frequency domain. As depicted in Fig. 6.3 we see
that the poles are at $w=\pm\widetilde{\omega}$.

Figure 6.3:  Real $\Re[\chi^{E}_{A}(w)]$ and imaginary $\Im[\chi^{E}_{A}(w)]$
parts of the $\mathbf{E}$-field response function $\chi^{E}_{A}(w)$, plotted
with a finite $\eta$. The poles for both parts appear at the frequency
$w=\pm\widetilde{\omega}$ and signify the frequency at which a time-dependent
electric field is oscillating.

Lastly, we highlight that the response function of the electric field in time
$\chi^{E}_{A}(t-t^{\prime})$ given by Eq. (6.36), and the response function of
the $\mathbf{A}$-field $\chi^{A}_{A}(t-t^{\prime})$ given by Eq. (6.28)
satisfy Maxwell’s equation [79].

$\displaystyle\chi^{E}_{A}(t-t^{\prime})=-\partial_{t}\chi^{A}_{A}(t-t^{\prime})$
(6.39)

This is a very nice consistency check of our computations and of the linear
response formalism in QED [134], because it demonstrates that linear response
theory, even for interacting electron-photon systems, respects the classical
Maxwell equations.

####  6.3 Cavity Modified Conductivity

In this section we are interested in the conduction properties of the 2DEG
inside the cavity. More specifically we want to investigate whether the cavity
field modifies the conductive properties of the electron gas. This is a
question of current theoretical and experimental interest. Recently,
modifications of transport and conduction properties due to cavity confinement
have been observed for two-dimensional systems of Landau polaritons [17], as
well as modifications of the critical temperature of superconductors due to
strong coupling to the cavity field [20, 22].

To describe this process we will follow what is commonly done in condensed
matter physics, namely perturb the system with a uniform, external, time-
dependent electric field $\mathbf{E}_{\textrm{ext}}(t)$, as shown in Fig. 6.4,
and then compute how much current is generated and flows through the system
due to the external perturbation. Here we choose the electric field to be
polarized only along the $x$-direction
$\mathbf{E}_{\textrm{ext}}(t)=|\mathbf{E}_{\textrm{ext}}(t)|\mathbf{e}_{x}$.
The electric field can also be represented as the time derivative of a time-
dependent vector potential
$\mathbf{E}_{\textrm{ext}}(t)=-\partial_{t}\mathbf{A}_{\textrm{ext}}(t)$. We
would like to mention that for the external perturbation to be causal, the
electric field needs to be zero for all times prior to time $t_{0}$. This
implies that in the frequency domain the electric field and vector potential
are related via
$\mathbf{E}_{\textrm{ext}}(w)=\textrm{i}(w+\textrm{i}\eta)\mathbf{A}(w)$ with
$\eta\rightarrow 0^{+}$.

Figure 6.4:  An external time dependent electric field
$\mathbf{E}_{\textrm{ext}}(t)$ perturbs the combined light-matter system,
electrons start to flow, and a current is generated in the material.

In order to couple the external field we have to add the external vector
potential $\mathbf{A}_{\textrm{ext}}(t)$ in the covariant kinetic energy of
the Pauli-Fierz Hamiltonian of Eq. (5.1). With this, the covariant term takes
the form
$(\textrm{i}\hbar\mathbf{\nabla}_{j}+e\hat{\mathbf{A}}+e\mathbf{A}_{\textrm{ext}}(t))^{2}$
[71, 95, 57]. In linear response theory the current is computed to first order
in the external perturbation. Then, the conductivity is defined as the
function relating the induced current density to the external electric field
[133, 134, 110].

The Pauli-Fierz Hamiltonian with the electrons coupled to a single cavity
mode, in the dipole approximation, and to first order in the external vector
potential $\mathbf{A}_{\textrm{ext}}(t)$ reads as

$\displaystyle\hat{H}(t)=\hat{H}+\sum^{N}_{j=1}\left(\frac{\textrm{i}e\hbar}{m_{\textrm{e}}}\nabla_{j}+\frac{e^{2}}{m_{\textrm{e}}}\hat{\mathbf{A}}\right)\cdot\mathbf{A}_{\textrm{ext}}(t)=\hat{H}-\left(\hat{\mathbf{J}}_{\textrm{p}}+\hat{\mathbf{J}}_{\textrm{d}}\right)\cdot\mathbf{A}_{\textrm{ext}}(t)$
(6.40)

where $\hat{H}$ is the Hamiltonian of Eq. (5.2). The external vector potential
$\mathbf{A}_{\textrm{ext}}(t)$ couples to the internal parts of the current
operator, namely the paramagnetic part
$\hat{\mathbf{J}}_{\textrm{p}}=(-\textrm{i}e\hbar/m_{\textrm{e}})\sum_{j}\nabla_{j}$,
and the diamagnetic part
$\hat{\mathbf{J}}_{\textrm{d}}=-e^{2}N\hat{\mathbf{A}}/m_{\textrm{e}}$.
However, the full physical current includes also the contribution of the
external vector potential $\mathbf{A}_{\textrm{ext}}(t)$ [71, 95]

$\displaystyle\hat{\mathbf{J}}=\underbrace{-\frac{\textrm{i}e\hbar}{m_{\textrm{e}}}\sum^{N}_{j=1}\nabla_{j}}_{\hat{\mathbf{J}}_{\textrm{p}}}\underbrace{-\frac{e^{2}N}{m_{\textrm{e}}}\hat{\mathbf{A}}}_{\hat{\mathbf{J}}_{\textrm{d}}}-\frac{e^{2}N}{m_{\textrm{e}}}\mathbf{A}_{\textrm{ext}}(t).$
(6.41)

Following the linear response formalism, the expectation value for the full
physical current is [110, 133]

$\displaystyle\langle\hat{\mathbf{J}}(t)\rangle=\langle\hat{\mathbf{J}}\rangle+\delta\langle\hat{\mathbf{J}}(t)\rangle=\langle\hat{\mathbf{J}}\rangle-\int^{\infty}_{t_{0}}dt^{\prime}\chi^{J}_{J}(t-t^{\prime})\mathbf{A}_{\textrm{ext}}(t^{\prime})$
(6.42)

where $\delta\langle\hat{\mathbf{J}}(t)\rangle$ is the response of the current
in time. This can be computed from the current-current response function

$\displaystyle\chi^{J}_{J}(t-t^{\prime})=\frac{-\textrm{i}\Theta(t-t^{\prime})}{\hbar}\langle[\hat{\mathbf{J}}_{I}(t),\hat{\mathbf{J}}_{I}(t^{\prime})]\rangle.$
(6.43)

To keep the current response $\delta\langle\hat{\mathbf{J}}\rangle$ to first
order in the external field, we neglect all contributions, coming from
$\mathbf{A}_{\textrm{ext}}(t)$. Then, for the commutator of Eq. (6.43), we
find the following four terms

$\displaystyle[\hat{\mathbf{J}}_{I}(t),\hat{\mathbf{J}}_{I}(t^{\prime})]=[\hat{\mathbf{J}}_{\textrm{p},I}(t),\hat{\mathbf{J}}_{\textrm{p},I}(t^{\prime})]+[\hat{\mathbf{J}}_{\textrm{d},I}(t),\hat{\mathbf{J}}_{\textrm{p},I}(t^{\prime})]+[\hat{\mathbf{J}}_{\textrm{d},I}(t),\hat{\mathbf{J}}_{\textrm{p},I}(t^{\prime})]+[\hat{\mathbf{J}}_{\textrm{d},I}(t),\hat{\mathbf{J}}_{\textrm{d},I}(t^{\prime})].$

Using the self-adjointness of the paramagnetic current operator we find

$\displaystyle\langle[\hat{\mathbf{J}}_{\textrm{p},I}(t),\hat{\mathbf{J}}_{\textrm{p},I}(t^{\prime})]\rangle=\langle\hat{\mathbf{J}}_{\textrm{p},I}(t)\hat{\mathbf{J}}_{\textrm{p},I}(t^{\prime})\rangle-\langle\hat{\mathbf{J}}_{\textrm{p},I}(t)\hat{\mathbf{J}}_{\textrm{p},I}(t^{\prime})\rangle^{*}.$
(6.45)

In addition, we use the expression for the paramagnetic current operator in
the interaction picture and the fact that the expectation value is computed in
the ground-state which has energy $E_{0,\mathbf{k}}$, and we find

$\displaystyle\langle\hat{\mathbf{J}}_{\textrm{p},I}(t)\hat{\mathbf{J}}_{\textrm{p},I}(t^{\prime})\rangle=e^{\textrm{i}E_{0,\mathbf{k}}(t-t^{\prime})/\hbar}\langle\hat{\mathbf{J}}_{p}e^{\textrm{i}\hat{H}(t^{\prime}-t)/\hbar}\hat{\mathbf{J}}_{p}\rangle.$
(6.46)

Due to translational invariance, the momentum operator commutes with the
Hamiltonian $\hat{H}$, and the ground-state
$|\Psi_{gs}\rangle=|\Phi_{0}\rangle\otimes|0,0\rangle_{1}|0,0\rangle_{2}$ is
an eigenstate also of the paramagnetic current operator
$\hat{\mathbf{J}}_{\textrm{p}}\sim\sum_{j}\nabla_{j}$ as well. Acting with the
paramagnetic current operator on the ground-state we obtain the full
paramagnetic current
$\hat{\mathbf{J}}_{\textrm{p}}|\Psi_{gs}\rangle=\sum_{j}\mathbf{k}_{j}|\Psi_{gs}\rangle$.
In section 5.2 we showed that the ground state in the thermodynamic limit is
the Fermi sphere. As a consequence, the total paramagnetic current in the
ground state is zero and we have

$\displaystyle\hat{\mathbf{J}}_{\textrm{p}}|\Psi_{gs}\rangle=0.$ (6.47)

The above result implies that all expectation values and correlators which
involve the paramagnetic current $\hat{\mathbf{J}}_{\textrm{p}}$ will also be
zero. This argument applies of course to the mixed terms
$[\hat{\mathbf{J}}_{\textrm{d},I}(t),\hat{\mathbf{J}}_{\textrm{p},I}(t^{\prime})]$
and
$[\hat{\mathbf{J}}_{\textrm{p},I}(t),\hat{\mathbf{J}}_{\textrm{d},I}(t^{\prime})]$
as well. Thus, the response function $\chi^{J}_{J}(t-t^{\prime})$ in Eq.
(6.43) is given solely by the diamagnetic terms. Substituting the definition
for the diamagnetic current $\hat{\mathbf{J}}_{\textrm{d}}$ of Eq. (6.41) we
find the current-current response function $\chi^{J}_{J}(t-t^{\prime})$ to be
proportional to the $\mathbf{A}$-field response function
$\chi^{A}_{A}(t-t^{\prime})$

$\displaystyle\chi^{J}_{J}(t-t^{\prime})=\left(\frac{e^{2}N}{m_{\textrm{e}}}\right)^{2}\chi^{A}_{A}(t-t^{\prime}).$
(6.48)

with $\chi^{A}_{A}(t-t^{\prime})$ given by Eq. (6.28). Since
$\chi^{J}_{J}(t-t^{\prime})$ is proportional to $\chi^{A}_{A}(t-t^{\prime})$
the same will also hold in the frequency domain

$\displaystyle\chi^{J}_{J}(w)=\left(\frac{e^{2}N}{m_{\textrm{e}}}\right)^{2}\chi^{A}_{A}(w).$
(6.49)

Lastly, we have to compute the expectation value of the current
$\langle\hat{\mathbf{J}}\rangle$ which is

$\displaystyle\langle\hat{\mathbf{J}}\rangle=\langle\hat{\mathbf{J}}_{\textrm{p}}\rangle+\langle\hat{\mathbf{J}}_{\textrm{d}}\rangle-\frac{e^{2}N}{m_{\textrm{e}}}\langle\mathbf{A}_{\textrm{ext}}(t)\rangle.$
(6.50)

As we already explained, the contribution of the paramagnetic
$\hat{\mathbf{J}}_{\textrm{p}}$ current is zero in the ground state
$|\Psi_{gs}\rangle$. The diamagnetic current $\hat{\mathbf{J}}_{\textrm{d}}$
is proportional to the vector potential
$\hat{\mathbf{J}}_{\textrm{d}}\sim\hat{\mathbf{A}}$, which is the sum of an
annihilation and a creation operator. The expectation values of these
operators in the ground-state is zero. Consequently only the external field
contributes to the expectation value of the current

$\displaystyle\langle\hat{\mathbf{J}}\rangle=-\frac{e^{2}N}{m_{\textrm{e}}}\mathbf{A}_{\textrm{ext}}(t).$
(6.51)

This contribution comes from the the full background charge of the $N$
electrons in our system. Moreover, from Eq. (6.42) by performing a Fourier
transformation we can derive the relation between the current and the external
$\mathbf{A}_{\textrm{ext}}(w)$-field, in the frequency domain

$\displaystyle\langle\hat{\mathbf{J}}(w)\rangle=\left(-\frac{e^{2}N}{m_{\textrm{e}}}-\chi^{J}_{J}(w)\right)\mathbf{A}_{\textrm{ext}}(w).$
(6.52)

The electric field and vector potential in the frequency domain are related
via the relation
$\mathbf{A}_{\textrm{ext}}(w)=\mathbf{E}_{\textrm{ext}}(w)/\textrm{i}(w+\textrm{i}\eta)$.
Using the latter relation and dividing Eq. (6.52) by the volume $V$ in order
to introduce the current density
$\langle\hat{\mathbf{j}}(w)\rangle=\langle\hat{\mathbf{J}}(w)\rangle/V$ we can
define the frequency dependent conductivity $\sigma(w)$ as the ratio between
the external electric field $\mathbf{E}_{\textrm{ext}}(w)$ and the current
density $\langle\hat{\mathbf{j}}(w)\rangle$ [63, 110]

$\displaystyle\langle\hat{\mathbf{j}}(w)\rangle=\left(-\frac{e^{2}n_{\textrm{e}}}{m_{\textrm{e}}}-\frac{\chi^{J}_{J}(w)}{V}\right)\frac{\mathbf{E}_{\textrm{ext}}(w)}{\textrm{i}(w+\textrm{i}\eta)}=\sigma(w)\mathbf{E}_{\textrm{ext}}(w).$
(6.53)

The frequency dependent conductivity $\sigma(w)$ is also known as the optical
conductivity and the previous equation is known as the Kubo formula for the
electrical conductivity [133, 110]. Using the result for the current-current
response function $\chi^{J}_{J}(w)$ given by Eqs. (6.49) and (6.30), and
introducing $\omega_{p}^{2}=e^{2}n_{\textrm{e}}/m_{\textrm{e}}\epsilon_{0}$
which is the plasma frequency in the cavity, we obtain the expression for the
optical conductivity $\sigma(w)$

$\displaystyle\sigma(w)=\frac{\textrm{i}\epsilon_{0}\omega^{2}_{p}}{w+\textrm{i}\eta}-\frac{\textrm{i}\epsilon_{0}\omega^{4}_{p}}{(w+\textrm{i}\eta)2\widetilde{\omega}}\left[\frac{1}{w+\widetilde{\omega}+\textrm{i}\eta}-\frac{1}{w-\widetilde{\omega}+\textrm{i}\eta}\right]\;\;\textrm{with}\;\;\eta\rightarrow
0^{+}.$ (6.54)

The real $\Re[\sigma(w)]$ and imaginary $\Im[\sigma(w)]$ parts of the optical
conductivity are

$\displaystyle\Re[\sigma(w)]$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{0}\eta\omega^{2}_{p}}{w^{2}+\eta^{2}}-\frac{\eta\epsilon_{0}\omega^{4}_{p}}{2\widetilde{\omega}(w^{2}+\eta^{2})}\left[\frac{2w+\widetilde{\omega}}{(w+\widetilde{\omega})^{2}+\eta^{2}}-\frac{2w-\widetilde{\omega}}{(w-\widetilde{\omega})^{2}+\eta^{2}}\right],$
(6.55) $\displaystyle\Im[\sigma(w)]$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{0}w\omega^{2}_{p}}{w^{2}+\eta^{2}}-\frac{\epsilon_{0}\omega^{4}_{p}}{2\widetilde{\omega}(w^{2}+\eta^{2})}\left[\frac{w^{2}-\eta^{2}+w\widetilde{\omega}}{(w+\widetilde{\omega})^{2}+\eta^{2}}-\frac{w^{2}-\eta^{2}-w\widetilde{\omega}}{(w-\widetilde{\omega})^{2}+\eta^{2}}\right].$

Figure 6.5: Real part $\Re[\sigma(w)]$ of the conductivity $\sigma(w)$ of the
2DEG outside (blue dashed line) and inside (orange solid line) the cavity for
coupling $\gamma=0.2$. Inside the cavity the real part of the conductivity of
the 2DEG exhibits poles at the plasmon-polariton frequency
$w=\pm\widetilde{\omega}$. At frequency $w=0$ the Drude peak of the 2DEG gets
suppressed due to the cavity field.  Figure 6.6: Imaginary part
$\Im[\sigma(w)]$ of the conductivity $\sigma(w)$ of the 2DEG outside (blue
dashed line) and inside (orange solid line) the cavity for coupling
$\gamma=0.2$. Inside the cavity the imaginary part of the conductivity of the
2DEG has poles at the plasmon-polariton frequency $w=\pm\widetilde{\omega}$.
The peak at $w=0$ gets suppressed by the cavity field.

In the optical conductivity $\sigma(w)$ given by Eq. (6.54) there are two
contributions. The first one is due to the full electron density
$n_{\textrm{e}}$ and comes from the plasma frequency
$\omega^{2}_{p}=n_{\textrm{e}}e^{2}/m_{\textrm{e}}\epsilon_{0}$ and is of
second order in $\omega_{p}$. This is the standard Drude term of the free
electron gas [110]. The second contribution comes from the current-current
response function $\chi^{J}_{J}(w)$. This contribution is purely due to the
cavity field because $\chi^{J}_{J}(w)$ is proportional to the
$\mathbf{A}$-field response function $\chi^{A}_{A}(w)$. The current-current
response $\chi^{J}_{J}$ is of fourth order in $\omega_{p}$ and is a
diamagnetic modification to the standard free electron gas conductivity.

More specifically, both the real and the imaginary part of the optical
conductivity, depicted in Figs. 6.5 and 6.6 respectively, exhibit resonances
at the plasmon-polariton frequency $w=\pm\widetilde{\omega}$, which modify the
optical conductivity of the 2DEG. Most importantly, we see that in the real
part of the conductivity the Drude peak [135, 136] of the 2DEG at $w=0$ is
suppressed by the cavity field due to the higher-order diamagnetic
contributions. Since the Drude peak is of great importance for condensed
matter systems and materials let us have a closer look at it.

#####  6.3.1 Cavity Suppression of the Drude Peak

The Drude peak is defined as the $w\rightarrow 0$ limit of the real part of
the optical conductivity and gives the DC electrical conductivity of a
particular system $\sigma_{\textrm{dc}}=\lim_{w\rightarrow 0}\Re[\sigma(w)]$
[63, 135, 110]. In the standard free electron gas (without a cavity) the DC
conductivity is $\sigma^{0}_{\textrm{dc}}=\epsilon_{0}\omega^{2}_{p}/\eta$,
which is the first term in Eq. (6.55) for $w\rightarrow 0$.

However, in our case we consider the 2DEG coupled to the cavity and we have
the extra diamagnetic contributions and we find that the DC conductivity of
the electron gas in the cavity depends also on the the collective coupling
constant $\gamma$ (defined in Eq. (5.27))

$\displaystyle\sigma_{\textrm{dc}}(\gamma)=\sigma^{0}_{\textrm{dc}}\left(1-\frac{\gamma}{1+\eta^{2}/\widetilde{\omega}^{2}}\right)\;\;\textrm{with}\;\;\eta\rightarrow
0^{+}.$ (6.56)

Upon neglecting the infinitesimal parameter $\eta$ we find that the DC
conductivity in the cavity, i.e., the Drude peak, decreases linearly as
function of the coupling $\gamma$

$\displaystyle\sigma_{\textrm{dc}}(\gamma)=\sigma^{0}_{\textrm{dc}}\left(1-\gamma\right).$
(6.57)

This is an important result because it shows that by confining 2D materials
inside a cavity, the cavity field does not only modify the optical properties
of the material, like the optical conductivity, but the cavity can also
influence the static electrical conductivity of the material.

The fact that the cavity decreases the conduction of electrons means that the
cavity field acts like a viscous medium which slows down the motion of the
charged particles. Within this picture the suppression of the Drude peak can
also be interpreted as an increase in the effective mass of the electrons due
to the cavity photons. From Eq. (6.57) for the DC conductivity we find that
the effective (or renormalized) electron mass is

$\displaystyle m_{\textrm{e}}(\gamma)=\frac{m_{\textrm{e}}}{(1-\gamma)}.$
(6.58)

Such an increase of the effective electron mass we will encounter again in
section 7 when we will couple the 2DEG to the full continuum of
electromagnetic modes.

Finally, it is important to mention that due to the fact that the coupling
constant has an upper bound $\gamma<1$ (see Eq. 5.27) the Drude peak is always
positive and the 2DEG is a conductor. However, if $\gamma$ could reach the
critical value $1$ (which is forbidden) then the DC conductivity would be
zero. This would mean that the cavity can turn the 2DEG from a conductor to an
insulator. For $\gamma>1$ the Drude peak becomes negative which means that the
system is no longer stable. This explains from the linear-response point of
view why the collective coupling $\gamma$ must not exceed the value of 1.

####  6.4 Mixed Responses: Matter-Photon & Photon-Matter

What we have done so far is to perform linear response on the photonic and on
the electronic sectors of our system separately. In the photonic sector we
applied an external current and we computed response functions related to the
photon field. This is what is commonly done in the fields of quantum optics
and photonics [91, 79]. In the electronic sector we perturbed the system with
an external electric field and computed the current response of the system, as
it is done in condensed matter physics [110, 63].

Quantum electrodynamics combines both perspectives under a common unified
framework. In addition to the matter-matter and photon-photon responses QED
allows to access also cross-correlated response functions, like matter-photon
and photon-matter [7, 134]. The aim of this section is exactly to compute
these cross-correlated responses and to investigate how these responses relate
to the standard matter-matter and photon-photon response functions.

#####  6.4.1 Matter-Photon Response

First we would like to compute the response of the current
$\delta\langle\hat{\mathbf{J}}(t)\rangle$ due to the external time-dependent
current $\mathbf{J}_{\textrm{ext}}(t)$. The response of the current
$\delta\langle\hat{\mathbf{J}}(t)\rangle$ is defined via Eq. (6.11) and can be
computed from the mixed response function $\chi^{J}_{A}(t-t^{\prime})$

$\displaystyle\chi^{J}_{A}(t-t^{\prime})=\frac{-\textrm{i}\Theta(t-t^{\prime})}{\hbar}\langle[\hat{\mathbf{J}}_{I}(t),\hat{\mathbf{A}}_{I}(t^{\prime})]\rangle.$
(6.59)

As we explained in the previous section, the contributions of the paramagnetic
part of the current is zero. As a consequence only the diamagnetic component
contributes. Substituting the expression for the diamagnetic current
$\hat{\mathbf{J}}_{\textrm{d}}$ we find that the mixed response
$\chi^{J}_{A}(t-t^{\prime})$ is proportional to the $\mathbf{A}$-field
response function

$\displaystyle\chi^{J}_{A}(t-t^{\prime})=-\frac{e^{2}N}{m_{\textrm{e}}}\chi^{A}_{A}(t-t^{\prime})$
(6.60)

where $\chi^{A}_{A}(t-t^{\prime})$ is given by Eq. (6.28). The previous
relation between the two response functions will hold also in the frequency
domain

$\displaystyle\chi^{J}_{A}(w)=\left(\frac{-e^{2}N}{m_{\textrm{e}}}\right)\chi^{A}_{A}(w).$
(6.61)

Finally, we note that the mixed response function $\chi^{J}_{A}(w)$ is
dimensionless and describes the ratio between the induced current
$\delta\langle\hat{\mathbf{J}}(w)\rangle$ and the external current
$\mathbf{J}_{\textrm{ext}}(w)$

$\displaystyle\delta\langle\hat{\mathbf{J}}(w)\rangle=\chi^{J}_{A}(w)\mathbf{J}_{\textrm{ext}}(w).$
(6.62)

#####  6.4.2 Photon-Matter Response

Having computed the matter-photon response function $\chi^{J}_{A}$ we would
also like to compute the photon-matter response function $\chi^{A}_{J}$ which
corresponds to the inverse physical process. For this, we need to look into
the response of the vector potential
$\delta\langle\hat{\mathbf{A}}(t)\rangle$. The response of the vector
potential is given by the photon-matter response function
$\chi^{A}_{J}(t-t^{\prime})$

$\displaystyle\chi^{A}_{J}(t-t^{\prime})=\frac{-\textrm{i}\Theta(t-t^{\prime})}{\hbar}\langle[\hat{\mathbf{A}}_{I}(t),\hat{\mathbf{J}}_{I}(t^{\prime})]\rangle.$
(6.63)

To remain within linear response, we neglect the contribution of
$\mathbf{A}_{\textrm{ext}}(t)$ which would result into higher order
corrections. As we already stated the paramagnetic contribution is zero.
Substituting the definition for the diamagnetic current
$\hat{\mathbf{J}}_{\textrm{d}}$ we find that the mixed response function
$\chi^{A}_{J}(t-t^{\prime})$ is

$\displaystyle\chi^{A}_{J}(t-t^{\prime})=-\frac{e^{2}N}{m_{\textrm{e}}}\chi^{A}_{A}(t-t^{\prime})$
(6.64)

The above relation between the two responses will also hold in the frequency
domain

$\displaystyle\chi^{A}_{J}(w)=\left(\frac{-e^{2}N}{m_{\textrm{e}}}\right)\chi^{A}_{A}(w).$
(6.65)

From the equation above we see that the photon-matter response function
$\chi^{A}_{J}(w)$ is equal to the matter-photon response $\chi^{J}_{A}(w)$.
Finally, we would like to emphasize that the photon-matter response
$\chi^{A}_{J}(w)$ describes the dimensionless ratio between the induced
$\hat{\mathbf{A}}$-field and the external field $\mathbf{A}_{\textrm{ext}}$

$\displaystyle\delta\langle\hat{\mathbf{A}}(w)\rangle=\chi^{A}_{J}(w)\mathbf{A}_{\textrm{ext}}(w).$
(6.66)

The above relation implies that the external field gets screened by the
internal dynamical system, and the response function $\chi^{A}_{J}(w)$
describes precisely this screening process [134].

####  6.5 Equivalence Between the Electronic and the Photonic Sector

Let us now finally compare the four fundamental response sectors which we
introduced and computed throughout this chapter, the photon-photon, matter-
matter, photon-matter and matter-photon. From all the responses in the
different sectors we can can construct the following response matrix

$\displaystyle\left(\begin{tabular}[]{c}$\delta\langle\hat{\mathbf{J}}(w)\rangle$\\\
$\delta\langle\hat{\mathbf{A}}(w)\rangle$\end{tabular}\right)=\left(\begin{tabular}[]{
c c }$\chi^{J}_{J}(w)$&$\chi^{J}_{A}(w)$\\\
$\chi^{A}_{J}(w)$&$\chi^{A}_{A}(w)$\end{tabular}\right)\left(\begin{tabular}[]{c}$\mathbf{A}_{\textrm{ext}}(w)$\\\
$\mathbf{J}_{\textrm{ext}}(w)$\end{tabular}\right)$ (6.73)

which summarizes all the fundamental responses of the system. Using now the
Eqs. (6.49), (6.65) and (6.61) which give the response functions
$\chi^{J}_{J}(w)$, $\chi^{J}_{A}(w)$ and $\chi^{A}_{J}(w)$ respectively, we
see that all responses are proportional to the $\mathbf{A}$-field response
function $\chi^{A}_{A}(w)$. Thus, all elements of the response table can be
written in terms of $\chi^{A}_{A}(w)$

$\displaystyle\left(\begin{tabular}[]{c}$\delta\langle\hat{\mathbf{J}}(w)\rangle$\\\
$\delta\langle\hat{\mathbf{A}}(w)\rangle$\end{tabular}\right)=\chi^{A}_{A}(w)\left(\begin{tabular}[]{
c c }$\left(e^{2}N/m_{\textrm{e}}\right)^{2}$&\; $-e^{2}N/m_{\textrm{e}}$\\\
\\\
$-e^{2}N/m_{\textrm{e}}$&$1$\end{tabular}\right)\left(\begin{tabular}[]{c}$\mathbf{A}_{\textrm{ext}}(w)$\\\
$\mathbf{J}_{\textrm{ext}}(w)$\end{tabular}\right).$ (6.82)

The fact that all response functions are proportional to $\chi^{A}_{A}(w)$
means that all response functions have precisely the same pole structure. This
demonstrates a fundamental relation between the two sectors of the theory,
namely that the photonic and the electronic sectors have exactly the same
excitations and resonances. From an experimental point of view this means,
that in an experiment perturbing an interacting light-matter system with an
external time-dependent current (which couples to the photon field) and
perturbing with an external electric field (which couples to the current)
would result in exactly the same information about the excitations and the
resonances of the interacting system.

Moreover, from the response table in Eq. (6.82) we see that the current-
current response scales quadratically with the number of electrons
$\chi^{J}_{J}(w)\sim N^{2}\chi^{A}_{A}(w)$, while the mixed responses linearly
$\chi^{J}_{A}(w)=\chi^{A}_{J}(w)\sim N\chi^{A}_{A}(w)$. The photon-photon
response function $\chi^{A}_{A}(w)$ given by Eq. (6.2.1) depends also on the
area of the 2DEG as $1/S$. As a consequence in the large $N,S$ limit only the
responses involving matter ($\chi^{J}_{J},\chi^{J}_{A},\chi^{A}_{J}$) are
finite, due to the dependence on $N$, while $\chi^{A}_{A}$ strictly speaking
goes to zero. This is the same behavior that shows up also for the energy
densities of the two sectors as mentioned in section 5.2. Again, this hints
towards the fact that, to have a finite photon-photon response, we need to
include an infinite amount of modes by treating the photon field in the
continuum.

Lastly, we would like to emphasize that all response functions we computed
depend on the arbitrarily small auxiliary parameter $\eta$ which needs to be
introduced in linear response to have a well-defined Fourier transform [110,
134]. However, in the limit $\eta\rightarrow 0^{+}$ the response functions go
to zero (see for example Eq. (6.2.1)) except of the frequencies
$w=\pm\widetilde{\omega}$ where they diverge. This means that the broadening
parameter $\eta$ works like a regulator which spreads the resonance over a
finite range of frequencies and describes the coupling of the system to an
artificial environment and the dissipation of energy to this environment
[110]. To get rid of the artificial broadening parameter $\eta$, one has to
treat matter and photons on an equal footing and perform the continuum-limit
also for the electromagnetic field. This is a direction which we explore in
the next chapter, and as we will see allows for the description of absorption
and dissipation without the need of any artificial broadening.

### CHAPTER 7 Effective Quantum Field Theory in the Continuum

> In practice, quantum field theory is marvelously good for calculating
> answers to many physics questions. The answers involve approximations. These
> approximations seem to work very well: that is, they [produce] answers that
> match experiments. Unfortunately we do not fully understand, in a
> mathematically rigorous way, what these approximations are supposed to be