*6.2. Results*

To date we have used the Markovian version of the master equation to investigate properties of the steady state, and how the system with electrons being transported through a photon cavity reaches it. We assume GaAs parameters with effective mass *m* = 0.067*me*, effective relative dielectric constant  *r* = 12.3, and effective Landé *g*-factor *g* = −0.44. The characteristic energy of the parabolic confinement of the semi-infinite leads and the central system in the *y*-direction is *h*¯ Ω0 = 2.0 meV. The length of the short quantum wire is *Lx*, and the overall coupling coefficient for the leads to the system is *gLRa*3/2 *w* = 0.124 meV.

We start with a central system made of a finite parabolic quantum wire without any embedded quantum dots. Figure 10 demonstrates that the approach to build and solve the Markovian master Equations (62)–(63) works for an interacting system with 120 many-body states participating in the transport [108].

**Figure 10.** (**upper left**) For the closed system as functions of the number of the eigenstate *μ*, the many-body energy (squares), the mean photon (*γ*) and electron content (*e*), and the mean spin *z*-component (*Sz*). The horizontal yellow lines represent the chemical potentials of the left (*μL*) and right leads (*μR*) when the system will be coupled to them. (**upper right**) The mean electron (solid) and photon number (dashed) in the central system as a function of time. The mean occupation of the many-body eigenstates of the system for *g*EM = 1 × 10−<sup>6</sup> meV (**lower left**), and *g*EM = 0.05 meV (**lower right**). *Vg* = −1.6 mV, *h*¯ *ω* = 0.8 meV, *x*-polarization, *κ* = 1 × 10−<sup>5</sup> meV, *Lx* = 150 nm, and *B* = 0.1 T. No quantum dots in the short wire.

The upper right panel displays the properties of the lowest 32 many-body states at the plunger gate voltage *Vg* = −1.6 mV. With *μL* = 1.4 meV and *μL* = 1.1 meV there are 8 states below the bias window and five states within it. In the bias window is one spin singlet two-electron state (the two-electron ground state) and two spin components of two one-electron states with a non-integer mean photon content indicating a Rabi splitting. The upper left panel of Figure 10 show the mean electron and photon numbers in the central system when it is initially empty. With a very low coupling, *g*EM = 1 × 10−<sup>6</sup> meV, between the electrons and photons, the charging is very slow with the probability approaching unity around *t* ≈ 10<sup>8</sup> s. With increasing *g*EM the charging becomes faster, and during the phase the mean photon number in the system rises. The lower panels of Figure 10 reveal what is happening. With the low photon coupling (lower left panel) electrons tunnel non-resonantly into the two spin components of the ground state, |1) and |2) as the vacuum state |3) looses occupation, and to a small fraction the two-electron state |9) gets occupied. When the coupling of the electrons and the photons is not vanishingly small (lower right panel) the charging of the system takes a different rout. The finite *g*EM allows the incoming electron to enter the Rabi-split one-electron states in the bias window as these are a linear combination of electron states with a different photon number. This explains the growing mean number of photons in the system for intermediate times. These states are eigenstates of the central system, but not of the open system, so at a later time they decay into the the one- and two-electron ground states as before bringing the system into the same Coulomb blocked steady state as before. We thus observe electromagnetically active transitions in the system in an intermediate time regime [108].

The on-set of the steady state regime is difficult to judge only from the shape of the charge being accumulated in the system or the current through or into it as a function of time [85]. For a system of two parallel quantum dots embedded in a short quantum wire (*Lx* = 150) nm the charging and the current as functions of time look the same (see Figures 4 and 5 in ref. [85]), but when the occupation of the eigenstates of the closed system is analyzed, see Figure 11, a clear difference is seen for the approach to the steady state depending on whether the initial state contains only one or no photon [85]. In the case of neither photon nor an electron in the cavity initially an electron tunnels into the system into the two spin components of the one-electron ground state, which happens to be in the bias window for *Vg* = −2.0 mV. Thus, the steady state is a combination of the empty state and these two one-electron states. In the case of one photon and no electron initially in the system an electron tunnels non-resonantly into the 1-electron states |8) and |9) with energy slightly below 2 meV, and thus well above the bias window. The mean photon content of these states is close to unity and at a later time the electron ends up in the two spin components of the one-electron ground state via a radiative transition [85]).

**Figure 11.** The mean occupation of the many-body eigenstates of the system when the initial state is the ground state |1) (**left**), or the first photon replica of the ground state |2) (**right**). *g*EM = 0.05 meV. *Vg* = −2.0 mV, *h*¯ *ω* = 0.8 meV, *x*-polarization, *Lx* = 150 nm, and *B* = 0.1 T. Two parallel quantum dots embedded in the short wire, but no photon reservoir.

Note that the "irregularly" looking structure around *t* ≈ 2000 ps will be addressed below. Please note that the numbering of interacting many-body state depends on the structure of the system, and the plunger gate voltage *Vg*.

In the steady state all the mean values of the open system have reached a constant value. In order to query about the active underlying processes it is necessary to calculate the spectral densities of the photon or current correlations. We present these for the central system consisting of a short quantum wire (*Lx* = 150) nm with two embedded quantum dots in Figure 12 (see refs. [107,109]).

**Figure 12.** The spectral density *S*(*E*) of the emitted cavity radiation for the central system in a steady state (**left**), and the spectral densities for the current–current correlations *Dll*(*E*) (**right**). *g*EM = 0.1 meV, *Vg* = −2.0 mV, *h*¯ *ω* = 0.72 meV, *κ* = 1 × 10−<sup>3</sup> meV, and *Lx* = 150 nm. Two parallel quantum dots embedded in the short wire.

Importantly we show in Ref. [50] that both the paramagnetic and the diamagnetic electron-photon interactions can lead to a Rabi resonance. The resonance for the diamagnetic interactions is much smaller, but the symmetry of the two parallel quantum dots leads to selection rules where for *x*-polarized cavity photon field the paramagnetic interaction is blocked, but both are present for the *y*-polarized field. Here, the active states are the one-electron ground state and the first excited one-electron state, with which the first photon replica of the ground state interacts for *h*¯ *ω* = 0.72 meV. The spectral density of the photon-photon two-time correlation function, *S*(*E*) seen in the left panel of Figure 12 shows one peak at the energy of the cavity mode *h*¯ *ω* = 0.72 meV, and two side peaks for the *y*-polarization. The central peak is the ground state state electroluminescence and the side peaks are caused by the Rabi-split states [34,109,110]. Here, we observe the ground state electroluminescence even though the electron-photon coupling is not in the ultra strong regime, as we diagonalize the Hamiltonian in a large many-body Fock space instead of applying conventional perturbative calculations.

For the *x*-polarized cavity field we find a much weaker ground state electroluminescence caused by the diamagnetic electron-photon interaction [109]. In addition, we identify these effects for the fully interacting two-electron ground state, where they are partially masked by many concurrently active transitions. The spectral density for the current–current correlation functions *Dll*(*E*) displayed in the right panel of Figure 12 show only peaks at the Rabi-satellites, as could be expected [107]. An inspection of *Dll*(*E*) over a larger range of energy reveals more transitions active in maintaining the steady state, both radiative transitions and non radiative [107]. Moreover, we notice that when the steady state is not in a Coulomb blocking regime the spectral density of the current-current correlations always shows a background to the peaks with a structure reminiscent of a 1/ *f* behavior, that is known in multiscale systems.

An "irregularly" looking structure in the mean occupation, the current current, and the mean number of electrons and photons. This is a general structure seen in all types of central system we have investigated in the continuous model. In Figure 13 we analyze it in a short parabolically confined quantum wire of length *Lx* = 180 nm with two asymmetrically embedded quantum dots [111].

**Figure 13.** The mean electron (*e*), photon (*γ*), *z*-component of the spin (*Sz*), trace of the reduced density matrix, and the Réniy-2 entropy (*S*) as functions of time. *h*¯*ω* = 0.373 meV, *x*-polarization, *κ* = 1 × 10−<sup>5</sup> meV, *g*EM = 0.05 meV, and *Lx* = 180 nm. Two asymmetrically embedded quantum dots in the short wire.

An increased number of time points on the logarithmic scale shows regular oscillations. A careful analysis reveals two independent oscillations: A spatial charge oscillation between the quantum dots with the Rabi frequency in the system, and a still slower nonequilibrium oscillation of the spin populations residing as the system is brought to a steady state [111].

The steady-state Markovian formalism has been used to investigate oscillations in the transport current as the photon energy or the electron–photon coupling strenght are varied with or without flow of photons from the external reservoir [112,113]. Moreover, the formalism has been used to establish the signs of the Purcell effect [114] in the transport current [98].

In light of the experimental interest of using a two-dimensional electron gas in a GaAs heterostructure [84] we have calculated the exact matrix elements for the electron-photon interaction taking into account the spatial variation of the vector field **A** of the electronic system. This is a small correction in most cases but may be important when studying high order transitions or nonperturbational effects caused by the photon field. This has led us to discover a very slow high order transition between the ground states of two slightly dissimilar quantum dots [115].

The fist steps have been taken to investigate thermoelectric effects in the central system coupled to cavity photons, in the steady state. In a short quantum wire with one embedded quantum dot in the resonant regime, an inversion of thermoelectric current is found caused by the Rabi-splitting. The photon field can change both the magnitude and the sign of the thermoelectric current induced by the temperature gradient in the absence of a voltage bias between the leads [116].
