The QD system is hard-wall confined in the

x-direction and parabolically confined in the

y-direction. In two-dimensional systems, the Hamiltonian can be described by [

29,

30,

31]

The electron field operator is

$\widehat{\psi}$, the magnetic vector potential is

${\mathbf{A}}_{\mathrm{B}}(\mathbf{r})=-By\widehat{x}$ introduced in the Landau gauge, and

${\widehat{\mathbf{A}}}_{\gamma}$ is the vector potential of the photon cavity written as

Herein,

A is the amplitude of the photon cavity field determined by the strength of electron–photon coupling

${g}_{\gamma}=eA{a}_{w}{\mathsf{\Omega}}_{w}/c$,

$\mathbf{e}={\mathbf{e}}_{x}$ (

${\mathbf{e}}_{y}$) when the photon field is parallel (perpendicular) to the direction of electron motion, and

${\mathsf{\Omega}}_{w}$ refers to the effective confinement frequency determined by the external static magnetic field

B and the bare confinement frequency

${\mathsf{\Omega}}_{0}$, via

${\mathsf{\Omega}}_{w}=\sqrt{{\mathsf{\Omega}}_{0}^{2}+{\omega}_{c}^{2}}$, where

${\omega}_{c}=eB/{m}^{*}$. The photon creation and annihilation operators are represented by

${\widehat{a}}^{\u2020}$ and

$\widehat{a}$, respectively.

To investigate the transport characteristics of the total system in the steady-state, we use a projection formalism built on the density operator [

20,

21]. Before coupling the QD system to the leads, the density operator of the total system is given by the tensor product of the individual density operators

$\widehat{\rho}(t<{t}_{0})={\widehat{\rho}}_{\mathrm{L}}{\widehat{\rho}}_{\mathrm{R}}{\widehat{\rho}}_{\mathrm{S}}(t<{t}_{0})$, where

${\widehat{\rho}}_{\mathrm{L}}$ and

${\widehat{\rho}}_{\mathrm{R}}$ indicate the density operator of the left (L) and the right (R) leads, respectively. After coupling the QD system to the leads, one can find the reduced density operator

${\widehat{\rho}}_{\mathrm{S}}$ that introduces the state of the electrons in the QD system under the effect of the leads as

where the trace is over the Fock space of the leads. We derive the equation of motion for the reduced density operator as a non-Markovian integrodifferential equation with a kernel evaluated up to the second order in the system–lead coupling [

33]. As we are interested in the long-time evolution and the steady state of the system, we further transform the equation into a corresponding Markovian equation for the reduced density operator of the QD system [

22]:

Herein,

${\mathsf{\Lambda}}^{L}$ and

${\mathsf{\Lambda}}^{R}$ represent the “dissipation” processes caused by both leads,

$\overline{\kappa}=1.0\times {10}^{-5}$ meV is the photon decay constant, and

${\overline{n}}_{\mathrm{R}}$ indicates the mean photon number of the reservoir. The second and the third lines in Equation (

5) display the photon dissipation of the cavity.

${\alpha}^{\u2020}$ (

$\alpha $) stands for the original operator in the non-interacting photon number basis,

${a}^{\u2020}$ (

a), transformed to the interacting electron photon basis using the rotating wave approximation [

34], where care has been taken in constructing a non-white noise spectrum appropriate for strong electron–photon coupling [

35,

36,

37].

#### Results

The total system, the QD system, and the leads are considered to be in a GaAs heterostructure where the relative dielectric constant is

$\kappa =12.4$ and the effective mass is

${m}^{*}=0.067{\mathrm{m}}_{\mathrm{e}}$ [

38,

39]. The electron confinement energy in both the QD system and the leads is considered to be

$\hslash {\mathsf{\Omega}}_{0}=\hslash {\mathsf{\Omega}}_{\mathrm{L},\mathrm{R}}=2.0$ meV, and the cyclotron energy is

$\hslash {\omega}_{c}=0.172$ meV at the weak magnetic field

$B=0.1$ T applied to the total system, leading to

${a}_{w}=23.8$ nm.

Figure 2 shows the Many-Body (MB) energy spectrum as a function of the plunger-gate voltage

${V}_{\mathrm{p}}$ for the QD system coupled to the cavity. The golden horizontal line indicates the chemical potential of the leads

${\mu}_{\mathrm{L}}={\mu}_{\mathrm{R}}=1.2$ meV. It is clearly seen that the ground-state (GS) at

${V}_{\mathrm{p}}=1.95$ mV and the first-excited state (FES) at

${V}_{\mathrm{p}}=0.271$ mV are touching (reaching) the chemical potential of the leads. Therefore, one can expect that these two states are responsible for the electron transport in the selected range of the gate voltage in the case of no photon cavity. The photon energy is assumed to be

$\hslash {\omega}_{\gamma}=1.31$ meV, which is smaller than the energy spacing between the GS and the FES at

${g}_{\gamma}=0.05$ meV. Under these conditions, the QD system is not in resonance with the photon field. In addition to the two major states, the GS and the FES, there appear photon replica states. For instance, the lowest photon replica of the ground-state (

$\gamma $GS) appearing in the energy spectrum is in resonance with the chemical potential of the leads at

${V}_{\mathrm{p}}=0.65$ meV. We note that the energy spectrum for the

x- and the

y-polarized photon fields is almost the same here.

To understand the properties of the thermoelectric current due to the temperature gradient, we start by considering the case of no photon cavity. In this case, the relevant states contributing to the transport are the original pure electron states, such as the GS and the FES. The left thermoelectric current

${I}_{\mathrm{L}}$ into the QD system and the right thermoelectric current

${I}_{\mathrm{R}}$ out of it for these two states as a function of the gate voltage are presented in

Figure 3a. The left and the right thermoelectric currents are equal but with opposite signs, indicating the onset of a steady-state regime already at time just before

$t=1\times {10}^{8}$ ps, even though we follow the evolution to

$t=1\times {10}^{11}$ ps.

The thermoelectric current emerges due to the occupation or the difference between the two Fermi functions of the leads or the electron reservoirs. The thermoelectric current is observed when the Fermi functions of the leads have different widths but the same chemical potential. It can be described as follows. The thermoelectric current is zero in two situations: First, when the two Fermi functions of the leads or their occupations (see

Figure 3b) are equal to

$0.5$ (half filling); second, when both Fermi functions or occupations are 0 or 1 (integer filling) [

40,

41]. As a result, the thermoelectric current is approximately zero at

${V}_{\mathrm{p}}=0.271$ and

$1.95$ mV, corresponding to half filling of the FES and the GS, respectively [

23]. The thermoelectric current is approximately zero at

${V}_{\mathrm{p}}=1.8$ and

$2.4$ mV for an integer filling or occupation of 0 and 1 around the GS, respectively.

We note that the electron or charge occupation of the system is large when the GS or the FES are in or close to resonance with the the chemical potential of the leads. Without cavity photons, the charge almost exclusively resides in the corresponding resonant states and is vanishingly small for

${\mathrm{V}}_{\mathrm{p}}$ in the range between 1.0 and 1.7 mV. This can be understood keeping in mind that the temperatures

${\mathrm{T}}_{\mathrm{L}}$ and

${\mathrm{T}}_{\mathrm{R}}$ are very low, the GS localized in the quantum dot is very weakly coupled to the leads, and the electron density of states of the quasi-1D leads has a peak at the lowest sub-band bottom at 1.0 meV, while the GS is well below this value for this range of the

${\mathrm{V}}_{\mathrm{p}}$. The coupling to the leads depends on the spatial extension of the corresponding wave functions into the contact areas of width

${a}_{w}$ at the ends of each subsystem. In addition, the coupling depends on the electron affinity defined by

$exp(-|{E}_{a}-\u03f5(q)|/{\mathsf{\Delta}}_{E})$, where

${E}_{a}$ stands for the states of the original single-electron basis for the central system,

$\u03f5(q)$ is the energy spectrum of a lead, and

${\mathsf{\Delta}}_{E}=0.5$ meV here [

42]. The electron occupation or charge cumulation in the central system will be strongly affected by the cavity photon field, as will be reported below.

Let us now assume the situation where a photon field is applied to the QD system. In the off-resonant regime, the photon energy is considered to be $\hslash {\omega}_{\gamma}=1.31$ meV, which is smaller than the energy spacing between the two lowest states of the QD system (${\mathrm{E}}_{\mathrm{FES}}-{\mathrm{E}}_{\mathrm{GS}}=1.682$ meV) for ${g}_{\gamma}=0.05$ meV and an x-polarized photon field.

Figure 4 demonstrates the left thermoelectric current (

${I}_{\mathrm{L}}$) for the off-resonant regime when the mean photon number is

${\overline{n}}_{\mathrm{R}}=0$ (a) and 1 (b). In addition, its occupation versus the gate voltage is shown in

Figure 4c. The occupation is almost the same for both cases of

${\overline{n}}_{\mathrm{R}}=0$ and 1. Compared to the case of no photon field (blue color), extra current oscillation, from negative to positive, around the

$\gamma $GS at

${V}_{\mathrm{p}}=0.65$ mV is observed in the presence of the photon field for both

${\overline{n}}_{\mathrm{R}}=0$ and 1. The additional current oscillation arises due to a photon-assisted tunneling (PAT) [

43]. An additional “peak” in the occupation around

${V}_{\mathrm{p}}=0.65$ mV, shown in

Figure 4c, is found corresponding to the extra current oscillation. The photon-assisted thermal transport has also been calculated for a simple two-level system using a Green function formalism [

40,

44]. We have not seen the extra thermal current peak in the transient regime [

23,

24]; however, the photon-assisted charge current peak can be clearly seen in the transient regime [

32,

45].

We should mention that the thermoelectric current is almost unchanged when

${\overline{n}}_{\mathrm{R}}=0$, and a suppression of thermoelectric current around the GS and FES for

${\overline{n}}_{\mathrm{R}}=1$ is recorded due to the contribution of their photon replica states to the transport. The processes of current transport in the presence of the photon field is totally different here. For example, the contributed ratio of the GS to the transport is approximately

$90\%$ in the range

${V}_{\mathrm{p}}=[1.8-2.2]$ mV where there is no photon field. But the GS is no longer the most active state that is responsible for the transport in the presence of the photon field.

$\gamma $GS together with the GS contribute to the transport in this range (

${V}_{\mathrm{p}}=[1.8-2.2]$ mV), and the mechanism of thermal transport is totally different for these two states. Thermoelectric current flows from the left lead to the right lead through the GS. Surprisingly, the direction of current through the

$\gamma $GS is contrary, going from the right lead to the left lead, as is shown in

Figure 5, irrespective of the direction of the thermal gradient. Therefore, the thermoelectric current is reduced. The reversed transport via the

$\gamma $GS can be related to the location of the chemical potential of the leads. For instance, if the chemical potential is located between the GS and the

$\gamma $GS, the GS (

$\gamma $GS) is located below (above) the Fermi function of the leads. In this case, the current must flow from the left lead to the right lead via the GS because it is below the Fermi function, and the opposite direction of flow may occur for the

$\gamma $GS as it is above the Fermi function.

The same explanation can be applied to the transport mechanism through the FES for the range

${V}_{\mathrm{p}}=[0\u20130.5]$ mV, but instead of

$\gamma $GS, the one-photon replica of the first-excited state,

$\gamma $FES, contributes to the transport here.

Figure 6 demonstrates the thermoelectric current versus the gate voltage for both

x- (red color) and

y-polarized photon fields (green color). It seems that the photon polarization for the off-resonance regime does not play an important role in the transport. The reason is that the location of the photon replica states in the MB energy spectrum is not sufficiently changed by tuning the photon polarization from the

x- to the

y-direction. Therefore, the contribution of the

$\gamma $GS to the transport is almost the same for both polarizations.

The total electron occupation in the central system, as displayed in

Figure 4c, is similar to the results in

Figure 3c except for the contribution around the

$\gamma $GS peak, but the partial occupation shows strong influences of the photon field. Around

${\mathrm{V}}_{\mathrm{p}}=1.8$ mV, the GS is occupied as before, but now around 1/4 of the charge resides in the FES. A total change takes place for

${\mathrm{V}}_{\mathrm{p}}$ in the 0.0–1.0 mV range. There, now 60–70% of the charge is in the GS and the rest is in the FES and the

$\gamma $GS. If the time evolution is analyzed, all of the charge enters the central system through the FES and the

$\gamma $GS, but the GS mainly gets occupied through slower radiative processes made possible by the photon field.

To recognize and further see the effects of the photon field on thermal transport, we display the thermoelectric current for a different electron–photon coupling strength

${g}_{\gamma}$ in

Figure 7, where

${\overline{n}}_{\mathrm{R}}=1$, and the photon field is polarized in the

x-direction. By increasing the electron–photon coupling strength, the thermoelectric current is suppressed and a nearly zero current is recorded at

${g}_{\gamma}=0.15$ meV over the same interval of voltage as before. This happens because the contributing ratio of the GS and the

$\gamma $GS to transport are almost equal at a higher electron–photon coupling strength. As a result, the current is vanishing, and a plateau of nearly zero values is obtained.

We now investigate the resonant regime when the energy spacing between the GS and the FES of the QD system is approximately equal to the photon energy,

$\hslash {\omega}_{\gamma}\simeq {E}_{\mathrm{FES}}-{E}_{\mathrm{GS}}$. The photon energy is considered to be

$\hslash {\omega}_{\gamma}=1.68$ meV, the electron–photon coupling strength is

${g}_{\gamma}=0.05$ meV, and the mean photon number is

${\overline{n}}_{\mathrm{R}}=1$. The MB energy spectrum is plotted against the gate voltage in

Figure 8 for the

x- (a) and

y-polarization (b) of the photon field. The Rabi-splitting between the

$\gamma $GS and FES emerges and is larger for the

x-polarized photon field. To confirm this, we display the MB energy spectrum of these two states as a function of the photon energy for

x- (

Figure 8c) and

y-polarization (

Figure 8d). The anti-crossings at the photon energy

$\hslash {\omega}_{\gamma}=1.68$ meV indicates a Rabi-splitting, and it is quite small for the

y-polarized photon field. The Rabi-splitting is larger for the x-polarization because the quantum dot system is an anisotropic system and the geometry of the QD system makes the charge densities of the states a bit more polarizable in that direction.

The thermoelectric current for the on-resonant regime is shown in

Figure 9. We find that the thermoelectric current through the GS is almost unchanged for both polarizations, but the characteristics of the thermoelectric current of the FES, which is in resonance with the

$\gamma $GS, is drastically modified. The effect of the resonant photon field is to invert the thermoelectric current from “positive” to “negative” values, or vice versa, around the FES at

${V}_{\mathrm{p}}=0.271$ mV. The more

$\gamma $GS-like state at

${V}_{\mathrm{p}}\simeq 0.279$ mV participates in the transport of the electrons, with the more FES-like state leading to the current flip from “positive” to “negative” values. Furthermore, the first photon replica of the first-excited state (

$\gamma $FES) becomes active in the transport here. It should be noted that the current inversion is larger for the smaller Rabi-splitting in the

y-polarized photon field. It indicates that the photon replica states have a major contribution in the transport; the resonance condition activates higher-lying states in the spectrum in the transport.