Free electron gas in cavity quantum electrodynamics

Cavity modification of material properties and phenomena is a novel research field largely motivated by the advances in strong light-matter interactions. Despite this progress, exact solutions for extended systems strongly coupled to the photon field are not available, and both theory and experiments rely mainly on finite-system models. Therefore, a paradigmatic example of an exactly solvable extended system in a cavity becomes highly desirable. To fill this gap we revisit Sommerfeld's theory of the free electron gas in cavity quantum electrodynamics. We solve this system analytically in the long-wavelength limit for an arbitrary number of noninteracting electrons, and we demonstrate that the electron-photon ground state is a Fermi liquid which contains virtual photons. In contrast to models of finite systems, no ground state exists if the diamagentic ${\mathbf{A}}^2$ term is omitted. Further, by performing linear response we show that the cavity field induces plasmon-polariton excitations and modifies the optical and the DC conductivity of the electron gas. Our exact solution allows us to consider the thermodynamic limit for both electrons and photons by constructing an effective quantum field theory. The continuum of modes leads to a many-body renormalization of the electron mass, which modifies the fermionic quasiparticle excitations of the Fermi liquid and the Wigner-Seitz radius of the interacting electron gas. Last, we show how the matter-modified photon field leads to a repulsive Casimir force and how the continuum of modes introduces dissipation into the light-matter system. Several of the presented findings should be experimentally accessible.

Figure 1

Schematic depiction of a 2D material inside a cavity with mirrors of length $L$ and area $S=L^2$. The area of the material is also $S$, and the distance between the mirrors is $L_z$. We note that the Pauli-Fierz Hamiltonian of Eq. (1) is in 3D, which is highlighted by the 3D Coulomb potential, while the electrons are restricted in the 2D plane. Further, we mention that in experimental setups, to increase the light-matter coupling, the space between the 2D material and the cavity is filled with a highly polarizable medium.

Figure 2

Schematic depiction of a generic distribution $\mathcal{D}(\mathbf{k}-\mathbf{q})$ in $\mathbf{k}$-space. The shape $\mathcal{D}$ of the distribution as well as its origin $\mathbf{q}$ are arbitrary. To find the ground-state distribution in $\mathbf{k}$-space one needs to minimize the energy density of the system with respect to both the shape $\mathcal{D}$ and the origin $\mathbf{q}$.

Figure 3

Graphic representation of the ground-state distribution of the 2DEG not coupled to a cavity. The ground-state distribution is the 2D Fermi sphere $\mathcal{S}\left(\mathbf{k}\right)$ (circle) with radius $|{\mathbf{k}}_{\text{F}}|$ (Fermi wave vector). For the 2DEG coupled to the cavity we find that the ground-state distribution in $\mathbf{k}$-space is also the 2D Fermi sphere $\mathcal{S}\left(\mathbf{k}\right)$ with radius $|{\mathbf{k}}_{\text{F}}|$.

Figure 4

Phase diagram for the free electron model in cavity QED. The system has a stable ground state for coupling constant $\gamma <1$. At the critical coupling ${\gamma }_c=1$ the ground state is infinitely degenerate. Beyond the critical coupling ${\gamma }_c=1$ the system is unstable and the system has no ground state.

Figure 5

Material confined inside a cavity, perturbed by external time dependent current ${\mathbf{J}}_{\text{ext}}\left(t\right)$. The external current perturbs the interacting light-matter system, a time-dependent electric field is induced, and the cavity radiates. We note that in an experiment the emitted radiation can be accessed through the openness of the cavity.

Figure 6

Real $\Re \left[{\chi }_A^A\left(w\right)\right]$ and imaginary $\Im \left[{\chi }_A^A\left(w\right)\right]$ parts of the $\mathbf{A}$-field response function ${\chi }_A^A\left(w\right)$ in the frequency domain, plotted with a finite $\eta$. The resonances for both parts appear at the plasmon-polariton frequency $w=\pm \tilde{\omega }$.

Figure 7

Real $\Re \left[{\chi }_A^E\left(w\right)\right]$ and imaginary $\Im \left[{\chi }_A^E\left(w\right)\right]$ parts of the $\mathbf{E}$-field response function ${\chi }_A^E\left(w\right)$ in the frequency domain with a finite broadening parameter $\eta$. The poles of $\Re \left[{\chi }_A^E\left(w\right)\right]$ and $\Im \left[{\chi }_A^E\left(w\right)\right]$ both appear at the frequency $w=\pm \tilde{\omega }$ and signify the frequency at which an time-dependent electric field is oscillating. Radiation should come out of the cavity at this frequency.

Figure 8

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

Figure 9

Real part $\Re \left[\sigma \right(w\left)\right]$ of the conductivity $\sigma \left(w\right)$ of the 2DEG outside (blue dashed) and inside (orange solid line) the cavity for collective coupling $\gamma =0.2$. Inside the cavity the real part exhibits poles at the plasmon-polariton resonance $w=\pm \tilde{\omega }$. At frequency $w=0$ the Drude peak of the 2DEG gets suppressed due to the cavity field.

Figure 10

Imaginary part $\Im \left[\sigma \right(w\left)\right]$ of the conductivity $\sigma \left(w\right)$ of the 2DEG outside (blue dashed) and inside (orange solid line) the cavity for collective coupling $\gamma =0.2$. Inside the cavity the imaginary part has poles at the plasmon-polariton resonance $w=\pm \tilde{\omega }$. The peak at $w=0$ gets suppressed by the cavity.

Figure 11

Representation of the frequency domain of the photon field in which the effective field theory is defined. The natural lower cutoff of the theory is ${\tilde{\omega }}^2\left({\kappa }_z\right)$, while the highest allowed mode is $\mathrm{\Lambda }$.

Figure 12

Relative percentage difference of the effective lower cutoff $\tilde{\omega }\left({\kappa }_z\right)$ and the lowest normal mode ${\mathrm{\Omega }}_l$ of the exact Hamiltonian (including the mode-mode interactions) plotted as function of the dimensionless ratio ${\omega }_p/\omega$. The computation is performed with different amount of modes. Above 100 modes the result is already converged. We assume all polarizations to be parallel to maximize the effect of the mode-mode interactions.

Figure 13

Exact many-mode coupling constant $g_{\text{ex}}^{\text{1D}}\left(M\right)$ normalized by ${({\omega }_p/\omega )}^2$ in 1D, as a function of the number of photon modes $M$. The exact coupling is plotted for three different values of the ratio ${\omega }_p/\omega$ corresponding to different regimes of light-matter interaction. In all cases we see that the exact coupling has arctangent dependence on the amount of photon modes. The exact coupling is normalized by ${({\omega }_p/\omega )}^2$ such that all graphs to be visible in one plot.

Figure 14

Schematic depiction of an electron in free space with mass $m_e$ and an electron coupled to the vacuum of the electromagnetic field. The virtual photons of the cavity vacuum “dress” the electron and provide a radiative correction to the electron mass $m_e\left(\mathrm{\Lambda }\right)$.

Figure 15

Renormalized mass $m_e\left(\mathrm{\Lambda }\right)$ as a function of the cutoff $\mathrm{\Lambda }$. For the cutoff $\mathrm{\Lambda }$ being equal to the natural lower cutoff ${\tilde{\omega }}^2\left({\kappa }_z\right)$ the renormalized mass is equal to the free-space electron mass $m_e\left[{\tilde{\omega }}^2\left({\kappa }_z\right)\right]=m_e$. As the cutoff increases the renormalized mass $m_e\left(\mathrm{\Lambda }\right)$ gets larger than the free-space mass $m_e$ and eventually goes to infinity at the Landau pole ${\mathrm{\Lambda }}_{\text{pole}}$.

Figure 16

Energy dispersion for electrons in a metal outside a cavity (black parabola) and for electrons in a metal coupled to a cavity (yellow parabola). From the curvature of the parabolas the effective mass of the electrons in the respective environment can be obtained. The dispersion for the electrons in the cavity is less steep, because the electron mass is larger in the cavity due to the “dressing” by the cavity photons.

Figure 17

Real $\Re \left[{\chi }_A^A\left(w\right)\right]$ and imaginary $\Im \left[{\chi }_A^A\left(w\right)\right]$ parts of the $\mathbf{A}$-field response function ${\chi }_A^A\left(w\right)$ in the effective field theory with $\eta =0$. The imaginary part has a finite value within the frequency window $\tilde{\omega }\left({\kappa }_z\right)<\left|w\right|<\sqrt{\mathrm{\Lambda }}$ which indicates that in this frequency range the system can continuously absorb energy. The real part though diverges at the natural lower cutoff $w=\pm \tilde{\omega }\left({\kappa }_z\right)$ and the upper cutoff $w=\pm \sqrt{\mathrm{\Lambda }}$ and shows that the system in the effective field theory has two scales.