*5.1. The Electron-Photon Coupling*

From the beginning our effort to model electron transport through a nano scale system placed in a photon cavity has been geared towards systems based on a two-dimensional electron gas in GaAs or similar heterostructures. We have emphasized intersubband transitions in the conduction band, active in the teraherz range, in anticipation of experiments in this promising system [84].

Here, subsystem *S*1 is a two-dimensional electronic nanostructure placed in a static (classical) external magnetic field. The leads are subjected to the same homogeneous external field. The electronic nanostructure, via split-gate configuration, is parabolically confined in the *y*-direction with a characteristic frequency Ω0. The ends of the nanostructure in the *x*-direction at *x* = ±*Lx*/2 are etched, forming a hard-wall confinement of length *Lx*. The external classical magnetic field is given by **B** = *B***zˆ** with a vector potential **A** = ( <sup>−</sup>*By*, 0, <sup>0</sup>). The single-particle Hamiltonian reads:

$$\begin{split} \hat{h}\_{S\_1}^{(0)} &= \frac{1}{2m} \left( \mathbf{p} + q \mathbf{A} \right)^2 + \frac{1}{2} m \Omega\_0^2 y^2 \\ &= \frac{1}{2m} p\_x^2 + \frac{1}{2m} p\_y^2 + \frac{1}{2} m \Omega\_w^2 y^2 + i \omega\_c \mathbf{y} p\_x \end{split} \tag{53}$$

where *m* is the effective mass of an electron, −*q* its charge, **p** the canonical momentum operator, *ωc* = *qB*/*m* is the cyclotron frequency and Ω*w* = / *ω*<sup>2</sup> *c* + Ω<sup>2</sup> 0 is the modified parabolic confinement. The spin degree of freedom is included with either a Zeeman term added to the Hamiltonian [85], or with Rashba and Dresselhaus spin orbit interactions, additionally [86].

*HS*2 is simply the free field photon term for one cavity mode and by ignoring the zero point energy can be written as *HS*2 = *h*¯ *<sup>ω</sup>pa*†*<sup>a</sup>* where *h*¯ *<sup>ω</sup>p* is the single photon energy and *a* (*a*†) is the bosonic annihilation (creation) operator. The electron-photon interaction term *<sup>V</sup>*el−p<sup>h</sup> can be split into two terms *<sup>V</sup>*el−p<sup>h</sup> = *V*(1) el−ph + *V*(2) el−ph where

$$V\_{\rm el-ph}^{(1)} \quad := \sum\_{ij} \sum\_{i,j} \left\langle \psi\_i \left| \frac{q}{2m} \left( \boldsymbol{\pi} \cdot \mathbf{A}\_{\rm EM} + \mathbf{A}\_{\rm EM} \cdot \boldsymbol{\pi} \right) \right| \psi\_j \right\rangle c\_i^\dagger c\_j \tag{54}$$

$$V\_{\rm el-ph}^{(2)} \quad := \sum\_{ij} \sum\_{i,j} \left\langle \psi\_i \left| \frac{q^2}{2m} \mathbf{A}\_{\rm EM}^2 \right| \psi\_j \right\rangle c\_i^\dagger c\_{j\nu} \tag{55}$$

with *π* ≡ **p** + *q***A** the mechanical momentum. The term in Equation (54) is the paramagnetic interaction, whereas the diamagnetic term is defined by Equation (55). By assuming that the photon wavelength is much larger than characteristic length scales of the system one can approximate the vector potential amplitude to be constant over the electronic system. Let us stress here that, in contrast to the usual dipole approximation, we will not omit the diamagnetic electron-photon interaction term. Then the vector potential is written as:

$$\mathbf{A}\_{\rm EM} \simeq \mathfrak{e} A\_{\rm EM} \left( a + a^{\dagger} \right) = \mathfrak{e} \frac{\mathcal{E}\_{\rm c}}{q \Omega\_{\rm w} a\_{\rm w}} \left( a + a^{\dagger} \right) \,, \tag{56}$$

where **eˆ** is the unit polarization vector and E*c* ≡ *qA*EMΩ*waw* is the electron-photon coupling strength. For a 3D rectangular Fabry Perot cavity we have *A*EM = /*h*¯ /(<sup>2</sup>*ωp<sup>V</sup>* <sup>0</sup>) where *V* is the cavity volume. Linear polarization in the *x*-direction is achieved for a **TE**011 mode, and in the *y*-direction with a **TE**101 mode.

Using the approximation in Equation (56), the expressions for the electron-photon interaction in Equations (54) and (55) are greatly simplified by pulling **A**EM in front of the integrals. For the paramagnetic term, we ge<sup>t</sup>

$$V\_{\rm el-ph}^{(1)} \simeq \mathcal{E}\_{\rm c} \left( a + a^{\dagger} \right) \sum\_{ij} \mathcal{g}\_{ij} c\_{i}^{\dagger} c\_{j} \,. \tag{57}$$

where we introduced the dimensionless coupling between the electrons and the cavity mode

$$\mathcal{g}\_{i\bar{j}} = \frac{a\_w}{2\hbar} \dot{\mathbf{e}} \cdot \int d\mathbf{r} \left[ \psi\_i^\*(\mathbf{r}) \left\{ \pi \pi \psi\_{\bar{j}}(\mathbf{r}) \right\} + \left\{ \pi \pi \psi\_i^\*(\mathbf{r}) \right\} \psi\_{\bar{j}}(\mathbf{r}) \right] \,. \tag{58}$$

As for the diamagnetic term, we ge<sup>t</sup>

$$\mathcal{N}\_{\rm el-ph}^{(2)} \simeq \frac{\mathcal{E}\_c^2}{\hbar \Omega\_w} \left[ \left( a^\dagger a + \frac{1}{2} \right) + \frac{1}{2} \left( a^\dagger a^\dagger + a a \right) \right] \mathcal{N}^\varepsilon \,, \tag{59}$$

where N *e* is the number operator in the electron Fock space. Note that *V*(2) el−ph does not depend on the photon polarization or geometry of the system in this approximation. We do not use the rotating wave approximation as in our multilevel systems even though a particular electron transition could be in resonance with the photon field we want to include the contribution form others not in resonance.

For the numerical diagonalization of *HS* we shall use the lowest *N*mesT *N*mes IMBS of *HS*1 and photon states containing up to *N*EM photons, resulting in a total of *N*mesT × (*<sup>N</sup>*EM + 1) states in the 'free' basis {|*<sup>ν</sup>*, *j*}.
