Quantum electron transport controlled by cavity vacuum fields

We explore theoretically how the coupling to cavity vacuum fields affects the electron transport in quantum conductors due to the counter-rotating-wave terms of light-matter interaction. We determine the quantum conductance in terms of the transmission coefficients predicted by an effective electron Hamiltonian. The coupling between bare electronic states is mediated by virtual processes involving intermediate states with one electron (or one hole) on top of the Fermi sea and one virtual cavity photon. We study the behavior of the quantum conductance in the presence of artificial or disordered single-particle potentials, as well as a spatially varying cavity mode. As illustrative examples, we apply our theory to one-dimensional conductors and to disordered two-dimensional quantum Hall systems. We show how the cavity vacuum fields can lead to both large enhancement or suppression of electron conductance in the ballistic regime, as well as modification of the conductance quantization and fluctuations.

Figure 1

Top panel: Sketch of the considered electronic system, described by a single-band tight-binding lattice model with hopping $J$ between nearest neighbors. The goal is to study how the linear quantum electron transport is affected by the cavity vacuum fields (no illumination). Bottom panel: Sketch of the process creating an effective coupling between single-particle states $|{\varphi }_{\lambda }\rangle$ and $|{\varphi }_{{\lambda }^{\prime }}\rangle$ via two types of intermediate states consisting of one virtual cavity photon and one electron (or hole) in another state $|{\varphi }_{\mu }\rangle$.

Figure 2

Example of conductance enhancement by cavity vacuum fields in a 1D system. The cavity mode is spatially homogeneous, linearly polarized along the $x$ direction, with frequency ${\omega }_{\mathrm{cav}}=2\pi \times 1$ THz and vacuum field $A_{\mathrm{vac}}$ that corresponds to ${\epsilon }_r\eta ={10}^{-9}$ (see text). Other parameters: $m=0.067m_0$ and $N=180$. (a) Single-particle potential $U\left(x\right)$ (thick black line). The shapes of the wave functions for the energy eigenstates are also shown, vertically offset by their corresponding energies. (b) Conductance versus $E_{\mathrm{F}}$ for the bare case (gray line) and for the cavity case (thick red line).

Figure 3

Example of conductance suppression in a 1D system. Same plots and parameters as in Fig. 2 (except $N=200$), but with a symmetric double well. In panel (b), the red line corresponds to a flat cavity mode (as in Fig. 2). The thicker blue line corresponds to a vacuum field $A_{\mathrm{vac}}\left(x\right)$ varying linearly from $0.75A_0$ to $1.25A_0$, where $A_0$ corresponds to the same field as in Fig. 2.

Figure 4

Same plots and parameters as in Fig. 2 (except $N=800$), but with a smooth random potential $U\left(x\right)$ with a finite correlation length (thick black line) and with a spatially flat cavity mode.

Figure 5

Two-terminal conductance for a 2D system ($N=60\times 40$) with a perpendicular magnetic field $B$ for the bare case (gray line) and the cavity case (blue line). The current flows along the $x$ direction. Parameters: $B=1$ T, $m=0.067m_0, L_x=0.5 \mu \mathrm{m}, L_y/L_x=2/3, E_{\mathrm{Zeeman}}=0.2\hslash {\omega }_{\mathrm{cyc}}$, with ${\omega }_{\mathrm{cyc}}=eB/m$ being the cyclotron frequency, and ${\omega }_{\mathrm{cav}}=2\pi \times 1$ THz. The vacuum field is polarized along the $y$ direction and its amplitude $A_{\mathrm{vac}}\left(y\right)$ varies linearly from $0.5\overline{A}$ to $1.5\overline{A}$, where the average $\overline{A}$ corresponds to ${\epsilon }_r\eta =4\times {10}^{-10}$. The considered smooth 2D random potential $U\left(\mathbf{r}\right)$ with a finite correlation length is shown in the inset.

Figure 6

This figure reports the conductance for a 1D symmetric double-well potential as in Fig. 3. For the sake of comparison, panel (b) contains the same plot as in Fig. 3. (a) Same, but with a width of the middle potential barrier $25\%$ thinner than in Fig. 3. (c) Same, but with a width of the middle potential barrier $25\%$ thicker than in Fig. 3. All other parameters are the same as those in Fig. 3.

Figure 7

The effect of a cavity field gradient for the asymmetric double-well potential case shown in Fig. 2. The blue line shows the conductance for a field with a gradient varying linearly from 0.75 to 1.25 times the average value, while the red line shows the conductance for the case of a flat mode. All parameters are the same as those in Fig. 2. As shown here, in the asymmetric case, the effect of a nonhomogeneous cavity field is negligible.

Figure 8

Results showing the effect of asymmetry of the electronic potential. Here we show the conductance for a flat mode and potential $U_s\left(x\right)$ with: $s=0$ (left panels), $s=0.1$ (middle panels) $s=0.5$ (right panels), where $s$ quantifies the symmetry of the potential (see text). All parameters are the same as those in Fig. 2.

Figure 9

Results for different spatial profiles of the cavity mode. Here we consider cosine (red), sine (blue), and asymmetric (green) cavity modes, as shown in the figure insets. Panel (a) shows the conductance for the asymmetric potential and the same parameters of Fig. 2. Instead, panel (b) shows the conductance for the symmetric potential and the same parameters of Fig. 3.