Time-dependent transport of electrons through a photon cavity

We use a non-Markovian master equation to describe the transport of Coulomb-interacting electrons through an electromagnetic cavity with one quantized photon mode. The central system is a finite-parabolic quantum wire that is coupled weakly to external parabolic quasi-one-dimensional leads at $t=0$. With a stepwise introduction of complexity to the description of the system and a corresponding stepwise truncation of the ensuing many-body spaces, we are able to describe the time-dependent transport of Coulomb-interacting electrons through a geometrically complex central system. We take the full electromagnetic interaction of electrons and cavity photons without resorting to the rotating-wave approximation or reduction of the electron states to two levels into account. We observe that the number of initial cavity photons and their polarizations can have important effects on the transport properties of the system. The quasiparticles formed in the central system have lifetimes limited by the coupling to the leads and radiation processes active on a much longer time scale.

Figure 1

A schematic of the total system consisting of a finite-quantum wire (the central electronic system) in a photon cavity (bold), coupled to external L and R leads at different chemical potentials. An external homogeneous magnetic field $\mathbf{B}$ is perpendicular to the system and the leads. The lengths are not shown to scale as the length of the system is 300 nm, but the characteristic lengths of the cavity are on the millimeter scale. The GaAs leads and system contain a Q1D electron gas in one to four sub-bands of width 60–120 nm for the parameters to be introduced below. The cavity photon modes are standing waves in the $z$ direction with an electric component in the $x$ or $y$ direction. The applied bias between the L and the R leads is indicated by placing them at a different level with respect to the central system.

Figure 2

The many-body energy spectra for Coulomb-interacting electrons coupled to quantized cavity photon modes with the electric component polarized along the finite-quantum wire ($x$ polarization, left panel) and perpendicular to the wire ($y$ polarization, right panel) versus the electron-photon coupling strength $g^{\mathrm{EM}}={\mathcal{E}}_{\mathrm{c}}$. In the energy range shown, there are states with no electrons (horizontal solid green lines), one electron (solid red lines), and two electrons (dashed blue lines). $B=0.1\mathrm{T},\hslash {\Omega }_0=1.0,\hslash \omega =0.4\mathrm{meV},L_x=300\mathrm{nm},m^{*}=0.067m_e$, and $\kappa =12.4$, the dielectric constant of GaAs.

Figure 3

The number of electrons in the MBS $|\breve{\mu })$ (solid red lines), and the diagonal element of the photon-number operator $N_{\mu \mu }^{\mathrm{ph}}$ (dashed blue lines). $B=0.1\mathrm{T},g^{\mathrm{EM}}=0.1,\hslash {\Omega }_0=1.0,\hslash \omega =0.4\mathrm{meV},L_x=300\mathrm{nm},m^{*}=0.067m_e$, and $\kappa =12.4$.

Figure 4

The SE energy spectrum of the leads. $B=0.1\mathrm{T}$ and $\hslash {\Omega }_0^l=1.0\mathrm{meV}$.

Figure 5

The spectral density $J_i^l\left(E\right)$ (L) and the probability amplitude for the corresponding SE eigenstate (R). $B=0.1\mathrm{T},\hslash {\Omega }_0=1.0\mathrm{meV},L_x=300\mathrm{nm},{\Delta }_E^l=0.25,g_0^l{a}_w^{3/2}=53.14\mathrm{meV},{\delta }_{1,2}^l{a}_w^2=0.4916$.

Figure 6

The total mean number of electrons $\langle N_{\mathrm{e}}\left(t\right)\rangle$ as a function of time for $g^{\mathrm{EM}}=0.01\mathrm{meV}$ (upper panel) and $g^{\mathrm{EM}}=0.10\mathrm{meV}$ (lower panel). $B=0.1\mathrm{T},\hslash {\Omega }_0=1.0\mathrm{meV},L_x=300\mathrm{nm},\hslash \omega =0.4,{\mu }_{\mathrm{L}}=2.0,{\mu }_{\mathrm{R}}=1.4,{\Delta }_E^l=0.25,g_0^l{a}_w^{3/2}=53.14\mathrm{meV},{\delta }_{1,2}^l{a}_w^2=0.4916,m^{*}=0.067m_e$, and $\kappa =12.4$.

Figure 7

The mean number of electrons $\langle N_{\mathrm{e}}\left(t\right)\rangle$ in a MBS $|\breve{\mu })$ for $x$ polarization (L) and $y$ polarization (R) as a function of time. Initially, at $t=0$, there is no photon in the cavity (top panels), one photon (middle panels), or four photons (bottom panels). The initial number of electrons is zero in all cases. $B=0.1\mathrm{T},g^{\mathrm{EM}}=0.01,\hslash \omega =0.4,{\mu }_{\mathrm{L}}=2.0,{\mu }_{\mathrm{R}}=1.4,\hslash {\Omega }_0=1.0,{\Delta }_E^l=0.25,g_0^l{a}_w^{3/2}=53.14\mathrm{meV},{\delta }_{1,2}^l{a}_w^2=0.4916,L_x=300\mathrm{nm},m^{*}=0.067m_e$, and $\kappa =12.4$.

Figure 8

The mean number of electrons $\langle N_{\mathrm{e}}\left(t\right)\rangle$ in a MBS $|\breve{\mu })$ for $x$ polarization (L) and $y$ polarization (R) as a function of time. Initially, at $t=0$, there is no photon in the cavity (top panels), one photon (middle panels), or four photons (bottom panels). The initial number of electrons is zero in all cases. $B=0.1\mathrm{T},g^{\mathrm{EM}}=0.10,\hslash \omega =0.4,{\mu }_{\mathrm{L}}=2.0,{\mu }_{\mathrm{R}}=1.4,\hslash {\Omega }_0=1.0,{\Delta }_E^l=0.25,g_0^l{a}_w^{3/2}=53.14\mathrm{meV},{\delta }_{1,2}^l{a}_w^2=0.4916,L_x=300\mathrm{nm},m^{*}=0.067m_e$, and $\kappa =12.4$.

Figure 9

The total mean number of photons $\langle N_{\mathrm{ph}}\rangle$ as a function of time for $g^{\mathrm{EM}}=0.01\mathrm{meV}$ (upper panel) and $g^{\mathrm{EM}}=0.10\mathrm{meV}$ (lower panel). $x$ polarization (solid red lines) and $y$ polarization (dashed blue lines). $B=0.1\mathrm{T},\hslash {\Omega }_0=1.0,\hslash \omega =0.4,{\mu }_{\mathrm{L}}=2.0,{\mu }_{\mathrm{R}}=1.4\mathrm{meV},L_x=300\mathrm{nm},{\Delta }_E^l=0.25,g_0^l{a}_w^{3/2}=53.14\mathrm{meV},{\delta }_{1,2}^l{a}_w^2=0.4916,m^{*}=0.067m_e$, and $\kappa =12.4$.

Figure 10

The mean number of photons $\langle N_{\mathrm{ph}}\left(t\right)\rangle$ in a MBS $|\breve{\mu })$ for $x$ polarization (L) and $y$ polarization (R) as a function of time. Initially, at $t=0$, there are no photons in the cavity (top panels), one photon (middle panels), or four photons (bottom panels). $B=0.1\mathrm{T},g^{\mathrm{EM}}=0.01,\hslash \omega =0.4,{\mu }_{\mathrm{L}}=2.0,{\mu }_{\mathrm{R}}=1.4,\hslash {\Omega }_0=1.0,{\Delta }_E^l=0.25,g_0^l{a}_w^{3/2}=53.14\mathrm{meV},{\delta }_{1,2}^l{a}_w^2=0.4916,L_x=300\mathrm{nm},m^{*}=0.067m_e$, and $\kappa =12.4$.

Figure 11

The mean number of photons $\langle N_{\mathrm{ph}}\left(t\right)\rangle$ in a MBS $|\breve{\mu })$ for $x$ polarization (L) and $y$ polarization (R) as a function of time. Initially, at $t=0$, there are no photons in the cavity (top panels), one photon (middle panels), or four photons (bottom panels). $B=0.1\mathrm{T},g^{\mathrm{EM}}=0.10,\hslash \omega =0.4,{\mu }_{\mathrm{L}}=2.0,{\mu }_{\mathrm{R}}=1.4,\hslash {\Omega }_0=1.0,{\Delta }_E^l=0.25,g_0^l{a}_w^{3/2}=53.14\mathrm{meV},{\delta }_{1,2}^l{a}_w^2=0.4916,L_x=300\mathrm{nm},m^{*}=0.067m_e$, and $\kappa =12.4$.

Figure 12

The total current from the L lead and into the R lead as a function of time for $g^{\mathrm{EM}}=0.01\mathrm{meV}$. Initially, at $t=0$, there are no photons in the cavity (top panel) or four photons (bottom panel). $B=0.1\mathrm{T},\hslash {\Omega }_0=1.0\mathrm{meV},L_x=300\mathrm{nm},\hslash \omega =0.4,{\mu }_{\mathrm{L}}=2.0,{\mu }_{\mathrm{R}}=1.4,{\Delta }_E^l=0.25,g_0^l{a}_w^{3/2}=53.14\mathrm{meV},{\delta }_{1,2}^l{a}_w^2=0.4916,m^{*}=0.067m_e$, and $\kappa =12.4$.

Figure 13

The total current from the L lead and into the R lead as a function of time for $g^{\mathrm{EM}}=0.1\mathrm{meV}$. Initially, at $t=0$, there are no photons in the cavity (top panel) or four photons (bottom panel). $B=0.1\mathrm{T},\hslash {\Omega }_0=1.0\mathrm{meV},L_x=300\mathrm{nm},\hslash \omega =0.4,{\mu }_{\mathrm{L}}=2.0,{\mu }_{\mathrm{R}}=1.4,{\Delta }_E^l=0.25,g_0^l{a}_w^{3/2}=53.14\mathrm{meV},{\delta }_{1,2}^l{a}_w^2=0.4916,m^{*}=0.067m_e$, and $\kappa =12.4$.