Abstract
Transport properties of a quantum dot coupled to a photon cavity are investigated using a quantum master equation in the steadystate regime. In the offresonance regime, when the photon energy is smaller than the energy spacing between the lowest electron states of the quantum dot, we calculate the current that is generated by photon replica states as the electronic system is pumped with photons. Tuning the electronphoton coupling strength, the photocurrent can be enhanced by the influences of the photon polarization, and the cavityphoton coupling strength of the environment. We show that the current generated through the photon replicas is very sensitive to the photon polarization, but it is not strongly dependent on the average number of photons in the environment.
Introduction
The lightmatter interaction in nanoscale systems is one of the most fundamental and interesting topic of modern nanodevices^{1,2}, especially if the light consists of few photons. In this case the light must be treated as fully quantized^{3,4}. Few photons interacting with a quantized electronic system, both in weak and strong coupling regimes, are very attractive for fundamental research and applications of nanotechnology^{5,6}. For instance, an unconventional photon blockade is observed in a quantum dot (QD) weakly coupled to a quantum electrodynamics (QED) cavity, in which a photon blockade effect can control the efficiency of single photon sources^{7}. In addition, in the strong coupling limit a quantum cavity coupled to a double quantum dot (DQD) system offers the capability of a coherent spectroscopy of a DQD qubit in the dispersive regime^{8}.
The coupling strength between an electronic system and a photon field in a cavity, g_{γ}, can be compared to the coupling strength of the cavity to the environment, κ^{9}. The system is said to be in the strong coupling regime if g_{γ} > κ. Note though that in addition to this condition the strength of the interaction could also be compared to a characteristic energy spacing of the electron system^{10}. In the strong coupling regime, several interesting phenomena have been observed such as photoninduced tunneling in a vacuum Rabi split two level quantum dot system^{11} and photon blockade in the presence of effective photonphoton interactions in a qubit system^{12}. Recently, it has been proposed that the entanglement properties of the cavityphotons with electrons in a strong coupling regime can be used to mediate nonCoulombic entanglement between two distant electrons^{13}. In the weak coupling regime, when g_{γ} < κ, the photon losses in the cavity overcome the electronphoton coupling element. As a result, photons may leave the cavity faster than being absorbed by the electronic system. In the weak coupling regime a highperformance singlephoton source achieves very high efficiency and it can be used as a multiphoton interferometric solidstate device^{14}.
The strong coupling regime opens the way towards a deterministic building of qubitphoton entanglement^{15}, single photon states^{16}, and longrange coupling of semiconductor qubits^{17,18}. In addition, the strong coupling regime is most desirable for optoelectronic nanodevices and it is an emerging technology in the microelectronics industry in the form of solid statebased quantum processors^{19}.
In previous publications, we demonstrated the effects of a photon cavity on the timedependent electron transport in a strong electronphoton coupling regime for both charge and thermoelectric transport through a QD^{20,21}, a DQD^{22}, quantum wires^{23,24,25} and quantum rings^{26}. In theses publications, we have shown that the photon field in the cavity can be used to control the transport properties of the systems in the early transient regime were nonMarkovian effects may be important. In the present work, we assume a quantum dot system coupled to a photon cavity with a single photon mode in the steadystate regime. We consider a strong coupling between the QD system and the photon cavity, (g_{γ} > κ). We use a Markovian quantum master equation to investigate the current through the QD system that is generated due to photon replica states under the influences of the photon polarization, the cavityenvironment coupling strength and the average photon number in the environment.
Results
The system under investigation is a QD embedded in a short twodimensional GaAs quantum wire with hardwall confinement in the xdirection and parabolic confinement in the ydirection with characteristic confinement energy \(\hslash \)Ω_{0} = 2.0 meV. The QD system and the external leads are assumed to be formed in a AlGaAsGaAs heterstructure. The system is exposed to a weak external perpendicular magnetic field, B = 0.1 T, in the zdirection leading to a cyclotron energy \(\hslash \)ω_{c} = 0.172 meV. Therefore, the effective confinement energy is given by \(\hslash \)Ω_{w} = \(\hslash \)[Ω_{0}^{2} + ω_{c}^{2}]^{1/2}. With the weak external magnetic field we avoid the effects of the Lorentz force on electron transport in the QD system. The role of this magnetic field is to lift the spin degeneracy by a small Zeeman splitting.
The QD system is connected to two leads with μ_{L} = 1.25 meV the chemical potential of the left lead and for the right lead μ_{R} = 1.15 meV. The gate voltage is V_{g} = 0.651 meV which is set to place the first photon replica of the oneelectron ground state in the bias window. The temperature of the leads is fixed at T = 0.5 K. The weak external perpendicular magnetic field is also applied to both leads.
The wireQD system is coupled to a photon field with energy \(\hslash \)ω_{γ} = 1.31 meV. The photon energy is smaller than the energy distance between the oneelectron ground state and the first excitation thereof. Under this condition the QD system is offresonant with the cavity, but the anisotropic polarizability of the charge in the system is different for the two linear photon polarizations along the x and the ydirection, as will be seen below. We intentionally choose the offresonant regime, with a photon replica in the bias window to study the current yield generated by such states.
The first photon replica of the oneelectron ground state can be understood as a state with the same electron part, but one photon added in the case of a vanishing electronphoton interaction. As the interaction is increased the replica state acquires a more complex photon component consisting of a small amount of a 0, 2 and higher number of photons. A strong interaction makes the mean number of photons in the replica state to deviate from an integer number. As the electron states of the system have a shape reflecting the geometry of it, we can similarly interpret the slight changes in the electron component of the replica as a polarization of the electron charge caused by the cavity field. We should be talking about cavityphoton dressed electron states. If the energy of cavity mode brings the first photon replica close to the first excitation of the ground state we obtain a Rabi resonance resulting in states with a mean photon number far from an integer. But, even out of resonance, the electronphoton interaction does vary the photon content of the replica state. A replica state in the bias window does thus offer an incoming electron increased probability of tunneling as it presents it with an increasing 0photon component as the interaction strength increases. At the same time we have to have in mind that the polarization of the charge changes the coupling of the state to the external leads, as the coupling depends on the shape of the charge distribution.
A schematic figure of the QD system (black circle) connected to two leads (blue) and coupled to a photon field (red) in a 3Dcavity with a single photon mode is shown in Fig. 1.
Figure 2 shows the manybody energy spectrum of the wireQD system as a function of the electronphoton coupling strength, g_{γ}, for the x (a) and ypolarized (b) photon field. 0 indicates the oneelectron groundstate and 1^{st} refers to the firstexcited oneelectron state. In addition, 1γ0 and 2γ0 are the first and the second photon replicas of the oneelectron groundstate, respectively.
At the given gate voltage, V_{g} = 0.651 meV, the first photon replica of the groundstate, 1γ0, is found to be in the bias window. Tuning the electronphoton coupling strength, the energy of the states is shifted up or down and can form anticrossings, especially in the xpolarized photon field. For instance, 1γ0 is shifted down with increasing g_{γ} and leaves the bias window at high electronphoton coupling strength in the xpolarized photon field, while it remains in the bias window for the ypolarization. In addition, the 1^{st} state is approaching 2γ0 starting to form an anticrossing at high electronphoton coupling strength in the xpolarization while the same phenomenon can not be seen for the ypolarization. The states are effectively stronger coupled to the xpolarized photon field compared to the ypolarization as the anisotropy of the system makes the charge more polarizable in the xdirection.
The mean photon number or photon content of the aforementioned four oneelectron states is shown in Fig. 3 for the xpolarized (a) and ypolarized photon field (b). Increasing the electronphoton coupling strength, photonexchange between the 1^{st} and 2γ0 states for the xpolarization is observed. The photon content of 1^{st} is enhanced to \(\simeq 0.4\) and the photon content of 2γ0 decreases to \(\simeq 1.2\). In addition, the photon content of 1γ0 is suppressed to \(\simeq 0.65\) in the xpolarization. The characteristics of the photon content here together with the energy spectra shown in Fig. 2 indicate that the states 1^{st} and 2γ0 are approaching a Rabiresonance in the case of an xpolarized photon field^{27}.
We now present the properties of the electron transport displayed by the current through the system. The current carried by the electrons depends on the width and location of the bias window. Here we assume the chemical potential of the left(right) lead to be 1.25(1.15 meV), respectively. The current from the left lead into the QD system is displayed in Fig. 4 for x (purple rectangles) and ypolarized (green rectangles) photon field. The current is enhanced with the electronphoton coupling strength reaching a maximum at g_{γ} = 0.1 meV for the xpolarized photon field while the current is very small and remains almost constant with increasing g_{γ} in the ypolarization. The enhancement of current is related to the charge and the electronphoton dressed states which are more polarizable in the xdirection. In addition, the transport through the photon replica state such as 1γ0 located in the bias window is enhanced with g_{γ}. As expected the current after g_{γ} = 0.1 meV, decreases as the photon replica state, 1γ0, leaves the bias window. We stress that the characteristics of current would not be the same as is shown in Fig. 4 if there would only be slightly photon dressed states of the QD system or additional electronphoton dressed states together with the photon replica states in the bias window^{28}. We note that the left and the right currents here, i.e. the current through the QD, are equal in magnitude because the system is in the steadystate regime.
To understand the detailed characteristics of the electron transport we present Fig. 5 which shows the partial currents, i.e. the currents going through individual states, for the x (a) and ypolarized (b) photon field. For the given chemical potentials the first photon replica, 1γ0, is located in the bias window. In xpolarized photon field, 1γ0 is the most active state in the transport as is shown in Fig. 5(a) in which the partial current for the four lowest states of the QD system coupled to the cavity is plotted. The current through 1γ0− with spin down (green hollow circle) and 1γ0+ for spin up (green filled circle) is enhanced by increased electronphoton coupling strength up to g_{γ} = 0.15 meV. The current is suppressed at higher electronphoton coupling strength in the case of xpolarization. The reason for the current suppression through 1γ0 after g_{γ} = 0.15 meV is that 1γ0 is moving out of the bias window as the coupling increases. In addition to the first photon replica state, the second photon replica state 2γ0 participates in the electron motion and a small current through 2γ0 is observed for both spin down (orange hollow circles) and spin up (orange filled circles). Again, the current through 2γ0 for both spin components is approaching zero but at the same time the firstexcited state gains charge leading to generation of a small current via 1^{st} with both spin components (blue) at high electronphoton coupling strength, g_{γ} > 0.15 meV. The discharging of 2γ0 and charging of 1^{st} is related to the photonexchange between these two states shown in Fig. 3(a) leading to an intraband transition that occurs between them. The characteristics of total current shown in Fig. 4 follow the partial currents of the photon replica states 1γ0 and 2γ0 shown in Fig. 5(a). It indicates that more than 95% of the total current is generated due to the contribution of the photon replica state, 1γ0, to the electron transport. Therefore, we can call the total current the photogenerated, or the photocurrent of the QD system.
We should mention that the current through the aforementioned states for the ypolarized photon field is much smaller, 100 times smaller, than that for the xpolarization as is displayed in Fig. 5(b). This confirms that the ypolarized photon field does not influence much the electron transport and no important effects of the photons can be seen in the energy spectrum and the photon content. This is related to the anisotropy of the QD system in which charge is more polarizable in the xdirection^{22}.
We now further study the properties of the current in the case of an xpolarized photon field and neglect the transport properties for the ypolarization because the current is too small. We now tune the cavityenvironment coupling κ, but keep in mind that the electronphoton coupling strength is still greater than the cavityenvironment coupling, g_{γ} > κ. As n_{R}, the mean value of photons in the reservoir, is not zero the parameter κ both influences the rate of flow of photons out and into the cavity. The current versus the electronphoton coupling strength for different values of κ is shown in Fig. 6. Clearly, the current is enhanced with increasing κ. To explain, we refer to the partial occupation and current of the QD system. Figure 7 demonstrates the partial occupation of the first photon replica of the oneelectron ground state, 1γ0 (a), and the electronic state, 1^{st} (b), versus the electronphoton coupling strength. We should remember that only 1γ0 is located in the bias window and 1^{st} is above the bias window (see Fig. 2a). The occupation of 1γ0 decreases with increasing photonreservoir coupling rate for all electronphoton coupling strength (see Fig. 7a) while the occupation of 1^{st} is enhanced (see Fig. 7b). This behavior indicates that the participation of photon replica states to the transport becomes weak at high photonreservoir coupling while the pure electronic states are populated at the same cavityreservoir coupling. In addition, for low photonreservoir coupling the most active state is 1γ0 which is due to a photon accumulation in the QD system leading to intraband transitions between photon replicas. The occupation of 1^{st} at high photonreservoir coupling enhances the current through it (until it moves outside the bias window) as is shown in Fig. 8 for κ = 10^{−4} meV (a) and κ = 10^{−3} meV (b). The current through 1^{st} seems to be blocked by the photon cavity for low values of κ. As a result the total current through the QD system is enhanced with increasing κ.
Another feature of our system is the effect of mean photon number \({\bar{n}}_{R}\) in the reservoir on the transport properties. We assume the cavityreservoir coupling, κ = 10^{−5} meV, and the chemical potential of the leads to be fixed as the above calculations. We keep in mind that only 1γ0 is located in the bias window for the given values of the chemical potentials and the gate voltage. Figure 9(a) shows the current as a function of the electronphoton coupling strength for different mean photon numbers in the reservoir. The current is almost zero when the mean number of photons is zero, \({\bar{n}}_{R}\) = 0, which is expected because the photon replica states are not active in the transport in the present situation. For \({\bar{n}}_{R}\) = 0 the system enters a Coulomb blocking regime in the steady state. In this case, the occupation of 1γ0 is almost zero and in turn the current vanishes. Clearly, the most occupied state here is the ground state which does not contribute to the transferred current through the QD system because it is far below the bias window (see Fig. 2). If we assume \({\bar{n}}_{R}\) = 2 (blue diamonds), the contribution of 1γ0 and especially 2γ0 is slightly enhanced which can be seen from the occupation of these two states (now shown). Therefore, the current is slightly enhanced for the case of two photons. This happens as the photon replicas are not pure simple perturbational states with an integer number of photons, but instead contain states with 0, 1, and 2 photons at least to some amount. The total mean photon number N_{γ} displayed in Fig. 9(b) is invariably lower than mean photon number in the reservoir \({\bar{n}}_{R}\), as the flow of electrons through the system is maintained by the “consumption” of photons, i.e. a photocurrent is maintained in the system. In preparation of Fig. 9 the coupling g_{γ} is never put lower than 0.001 meV, but the coupling of the cavity to the environment κ is kept constant. For the lowest g_{γ} the system still achieves a steadystate, but now in a very long time. (We do not set g_{γ} exactly equal to zero as in that unphysical limit it is technically difficult to account for the approach to the steady state properly within the numerical accuracy set by the time scale needed).
The partial current through the four lowest states is displayed in Fig. 10 for \({\bar{n}}_{R}\) = 2. It can clearly be seen that the current through 1γ0 and 2γ0 is slightly increased in the case of two photons leading to a slight enhancement of the total current.
Discussion
We have calculated the transport properties through a quantum dot system connected to leads and coupled to a photon cavity with a single photon mode. We focus on the transport properties of the photon replica states that are formed in the presence of the photon field coupled to the QD system. These photon replica states can be confined in the bias voltage by setting the chemical potential of the leads. In this way, one can see 95% of the current in the system can be obtained due to the photon replica states. We can thus show the influences of photon polarization, mean photon number in the reservoir, and photonreservoir coupling rate on the transport properties in the system. We find that the photon polarization plays an important role and can be used to control the photocurrent generated in the system. In addition, the total current is enhanced with increasing the photonreservoir coupling rate because the partial current carried by both the almost pure electronic states and the photon replica states is increased.
It is important to have in mind that we have not considered very strong electron photon coupling if we compare the photon coupling strength with the energy difference between the oneelectron ground state and the first electronic excitation thereof. We have also not selected the photon energy to be very close to this energy difference, but as we account for geometrical effects in our anisotropic system it is clear that the effective electronphoton coupling becomes rather strong, and the only way to approach this regime sincerely is by using a step wise exact numerical diagonalization scheme for all interactions in the central system. Even though, we do consider the variation of the photon fields to be small on the size scale of the electronic system, we are not using a traditional dipole approximation and the higher order interaction effects are important in delivering a more appropriate cavity photon dressed electron states to describe transport of electron through our system. Radiative transitions in our system with a FIR photon mode take time, and the variation of the cavityphoton reservoir coupling strength, κ, can be used to activate them or reduce their effects.
Methods
The Hamiltonian describing the QDsystem coupled to a photon cavity in the manybody (MB) basis is given as^{20,29,30,31}
with H_{e} the Hamiltonian of the QDwire system
including the electronelectron Coulomb interaction. Herein, π: = p + (e/c)A with p being canonical momentum, \({\bf{A}}=\,By\hat{{\bf{x}}}\) is the magnetic vector potential with \({\bf{B}}=B\hat{{\bf{z}}}\), and \(\hat{\psi }({\bf{r}})={\sum }_{i}\,{\psi }_{i}(r){d}_{i}\) and \({\hat{\psi }}^{\dagger }({\bf{r}})={\sum }_{i}\,{\psi }_{i}^{\ast }(r){d}_{i}^{\dagger }\) are the electron field operators with d_{i}(\({d}_{i}^{\dagger }\)) the annihilation(creation) operators for the singleelectron state i corresponding to ψ_{i}. The electron confinement frequency due to the lateral parabolic potential is defined by Ω_{0} in the short quantum wire and the potential of the QD is described by
with V_{0} its strength, and γ_{x}(γ_{y}) are constants that define the diameter of the QD, respectively. The gate voltage, V_{g}, moves the energy states of the QDwire system with respect to the chemical potential of the leads, and it is assumed to be constant in our calculations. The Zeeman Hamiltonian referring to the interaction between the magnetic moment of an electron and the external magnetic field (B), is given by H_{Z} = ±g^{*}μ_{B}B/2 with μ_{B} the Bohr magneton and g^{*} = −0.44 the effective gfactor for GaAs. In addition, the electronelectron interaction is shown in the second line of Eq. 2 with V_{c} being the Coulomb interaction potential^{20}. The Coulomb interaction in the leads is neglected.
The second term of Eq. 1 is the Hamiltonian of the free photon field defined via
with \(\hslash \)ω_{γ} the energy of the photons in the cavity, and a(\({a}^{\dagger }\)) the photon annihilation(creation) operators, respectively.
The last part of the Eq. 1 stands for the interaction between the electron in the QD system and the photons in the cavity
where the first part of Eq. 5 is the paramagnetic and the second part is the diamagnetic electronphoton interaction. The charge density is given by \(\rho =\,e{\psi }^{\dagger }\psi \) and the charge current density can be introduced by
with ψ the field operator of the QD system. In addition, the phoon vector potential, A_{γ}, in the Coulomb gauge is
Herein, A is the amplitude of the photon field introduced by the electronphoton coupling constant g_{γ} = eAa_{w}Ω_{w}/c. The photon polarization can be determined by e, in which e = e_{x} in the xdirection and e = e_{y} in the ydirection. We assume the wavelength of the FIR cavity photons to be much larger than the size of the short quantum wire and the quantum dot. As a step wise exact numerical diagonalization technique is used to treat the electronphoton and the Coulomb interactions interaction in appropriately truncated Fockspaces, the electronphoton interaction is treated well beyond a traditional dipole approximation^{32}.
A quantum master equation is utilized to study the transport properties of the system in the steadystate regime in which a projection formalism based on the density operator is used^{33,34}. The leads and the central system are assumed to be weakly coupled leading to terms of higher than second order in terms of the coupling to be neglected in the dissipation kernel of the resulting integrodifferential equation. We assume that the QD system and leads are uncorrelated before the coupling
where \({\hat{\rho }}_{{\rm{L}}}\) and \({\hat{\rho }}_{{\rm{R}}}\) are the density operators of the left (L) and the right (R) leads, respectively.
As we are interested in the state of the wireQD system after the coupling, we can obtain the reduced density operator of the QD system from a partial trace over the combined QD system and leads
The nonMarkovian generalized master equation is
where the coupling of the single cavity photon mode is assumed Markovian, and a rotating wave approximation has been used only for this coupling. The second term of the first line of Eq. 10 describes the electron “dissipation” processes caused by both leads, and the second and the third lines of Eq. 10 represents the photon reservoir where κ is the photoncavity coupling constant to the environment (seen as a photon reservoir), \({\bar{n}}_{R}\) is the mean photon number in the reservoir. The photon operators in the cavity, a and \({a}^{\dagger }\), are replaced by α and \({\alpha }^{\dagger }\), that lead to the correct steady state by removing all high frequency creation terms from the annihilation operator, and high frequency annihilation terms from the creation operator^{35,36,37,38}. Subsequently, a Markovian approximation is applied to the master Eq. (10), and a vectorization together with a Kronecker tensor product transforms it from the manybody Fock space of photon dressed electron states into a Liouville space of transitions to facilitate numerical and analytical solutions^{39}.
We assume the chemical potential of the left lead is higher than that of the right lead producing a bias voltage that generates current through the QDsystem coupled to the leads. The charge current from the left lead into the QDsystem, I_{L}^{c}, and the current from it into the right lead, I_{R}^{c}, can be introduced as
The charge operator of the QDsystem is \(Q=\,e{\sum }_{i}\,{d}_{i}^{\dagger }{d}_{i}\) with \({\hat{d}}^{\dagger }(\hat{d})\) the electron creation (annihilation) operator of the central system, respectively. Λ^{L,R} stand for the “dissipation” processes caused by both electron leads^{39,40}. The average total number of photons in the cavity is evaluated as
and each term in the trace operation performed in the basis of interacting electrons and photons can be regarded as the photon content of the corresponding dressed electron state.
References
 1.
Delbecq, M. R. et al. Photonmediated interaction between distant quantum dot circuits. Nat. Commun. 4, 1400 EP – Article (2013).
 2.
Sánchez, R., Platero, G. & Brandes, T. Resonance fluorescence in driven quantum dots: Electron and photon correlations. Phys. Rev. B 78, 125308, https://doi.org/10.1103/PhysRevB.78.125308 (2008).
 3.
Giannelli, L. et al. Optimal storage of a single photon by a single intracavity atom. New J. Phys. 20, 105009, https://doi.org/10.1088/13672630/aae725 (2018).
 4.
Kreinberg, S. et al. Quantumoptical spectroscopy of a twolevel system using an electrically driven micropillar laser as a resonant excitation source. Light. Sci. & Appl. 7, 41, https://doi.org/10.1038/s4137701800456 (2018).
 5.
Hümmer, T., GarcíaVidal, F. J., MartínMoreno, L. & Zueco, D. Weak and strong coupling regimes in plasmonic qed. Phys. Rev. B 87, 115419, https://doi.org/10.1103/PhysRevB.87.115419 (2013).
 6.
Cottet, A., Kontos, T. & Douçot, B. Electronphoton coupling in mesoscopic quantum electrodynamics. Phys. Rev. B 91, 205417, https://doi.org/10.1103/PhysRevB.91.205417 (2015).
 7.
Snijders, H. J. et al. Observation of the unconventional photon blockade. Phys. Rev. Lett. 121, 043601, https://doi.org/10.1103/PhysRevLett.121.043601 (2018).
 8.
Stockklauser, A. et al. Strong coupling cavity qed with gatedefined double quantum dots enabled by a high impedance resonator. Phys. Rev. X 7, 011030, https://doi.org/10.1103/PhysRevX.7.011030 (2017).
 9.
Dewhurst, S. J. et al. Slowlightenhanced single quantum dot emission in a unidirectional photonic crystal waveguide. Appl. Phys. Lett. 96, 031109, https://doi.org/10.1063/1.3294298 (2010).
 10.
Frisk Kockum, A., Miranowicz, A., De Liberato, S., Savasta, S. & Nori, F. Ultrastrong coupling between light and matter. Nat. Rev. Phys. 1, 19–40, https://doi.org/10.1038/s4225401800062 (2019).
 11.
Faraon, A. et al. Coherent generation of nonclassical light on a chip via photoninduced tunnelling and blockade. Nat. Phys. 4, 859 EP (2008).
 12.
Lang, C. et al. Observation of resonant photon blockade at microwave frequencies using correlation function measurements. Phys. Rev. Lett. 106, 243601, https://doi.org/10.1103/PhysRevLett.106.243601 (2011).
 13.
Kfir, O. Phys. Rev. Lett. 123, 103602 (2019).
 14.
Ding, X. et al. Ondemand single photons with high extraction efficiency and nearunity indistinguishability from a resonantly driven quantum dot in a micropillar. Phys. Rev. Lett. 116, 020401, https://doi.org/10.1103/PhysRevLett.116.020401 (2016).
 15.
Eichler, C. et al. Observation of entanglement between itinerant microwave photons and a superconducting qubit. Phys. Rev. Lett. 109, 240501, https://doi.org/10.1103/PhysRevLett.109.240501 (2012).
 16.
Houck, A. A. et al. Generating single microwave photons in a circuit. Nature 449, 328 EP (2007).
 17.
Majer, J. et al. Coupling superconducting qubits via a cavity bus. Nature 449, 443 EP (2007).
 18.
Sillanpää, M. A., Park, J. I. & Simmonds, R. W. Coherent quantum state storage and transfer between two phase qubits via a resonant cavity. Nature 449, 438 EP (2007).
 19.
Wu, Y., Wang, Y., Qin, X., Rong, X. & Du, J. A programmable twoqubit solidstate quantum processor under ambient conditions. npj Quantum Inf. 5, 9, https://doi.org/10.1038/s415340190129z (2019).
 20.
Abdullah, N. R., Tang, C. S., Manolescu, A. & Gudmundsson, V. Electron transport through a quantum dot assisted by cavity photons. J. Physics:Condensed Matter 25, 465302 (2013).
 21.
Abdullah, N. R., Marif, R. B. & Rashid, H. O. Photonmediated thermoelectric and heat currents through a resonant quantum wirecavity system. Energies 12, https://doi.org/10.3390/en12061082 (2019).
 22.
Abdullah, N. R., Tang, C. S., Manolescu, A. & Gudmundsson, V. Delocalization of electrons by cavity photons in transport through a quantum dot molecule. Phys. E 64, 254–262 (2014).
 23.
Abdullah, N. R., Tang, C.S., Manolescu, A. & Gudmundsson, V. ACS Photonics 3, 249–254, https://doi.org/10.1021/acsphotonics.5b00532 (2016).
 24.
Abdullah, N. R., Tang, C. S., Manolescu, A. & Gudmundsson, V. Coherent transient transport of interacting electrons through a quantum waveguide switch. J. physics: Condens. matter 27, 015301 (2015).
 25.
Gudmundsson, V., Jonasson, O., Tang, C.S., Goan, H.S. & Manolescu, A. Timedependent transport of electrons through a photon cavity. Phys. Rev. B 85, 075306, https://doi.org/10.1103/PhysRevB.85.075306 (2012).
 26.
Abdullah, N. R., Tang, C.S., Manolescu, A. & Gudmundsson, V. Spindependent heat and thermoelectric currents in a Rashba ring coupled to a photon cavity. Phys. E: Lowdimensional Syst. Nanostructures https://doi.org/10.1016/j.physe.2017.09.011 (2017).
 27.
Abdullah, N. R., Tang, C.S., Manolescu, A. & Gudmundsson, V. arXiv:1903.03655 (2019).
 28.
Gudmundsson, V. et al. Electroluminescence caused by the transport of interacting electrons through parallel quantum dots in a photon cavity. Annalen der Physik 530, 1700334, https://doi.org/10.1002/andp.201700334.
 29.
Arnold, T., Tang, C.S., Manolescu, A. & Gudmundsson, V. Excitation spectra of a quantum ring embedded in a photon cavity. J. Opt. 17, 015201 (2015).
 30.
Jonasson, O., Tang, C.S., Goan, H.S., Manolescu, A. & Gudmundsson, V. Nonperturbative approach to circuit quantum electrodynamics. Phys. Rev. E 86, 046701, https://doi.org/10.1103/PhysRevE.86.046701 (2012).
 31.
Abdullah, N. R., Tang, C.S., Manolescu, A. & Gudmundsson, V. Competition of static magnetic and dynamic photon forces in electronic transport through a quantum dot. J. Physics: Condens. Matter 28, 375301 (2016).
 32.
Gudmundsson, V. et al. Stepwise introduction of model complexity in a general master equation approach to timedependent transport. Fortschr. Phys. 61, 305 (2013).
 33.
Zwanzig, R. Ensemble Method in the Theory of Irreversibility. The J. Chem. Phys. 33, 1338–1341, https://doi.org/10.1063/1.1731409 (1960).
 34.
Nakajima, S. On quantum theory of transport phenomena steady diffusion. Prog. Theor. Phys. 20, 948 (1958).
 35.
Gudmundsson, V. et al. Current correlations for the transport of interacting electrons through parallel quantum dots in a photon cavity. Phys. Lett. A 382, 1672–1678, https://doi.org/10.1016/j.physleta.2018.04.017 (2018).
 36.
Lax, M. Formal Theory of Quantum Fluctuations from a Driven State. Phys. Rev. 129, 2342–2348, https://doi.org/10.1103/PhysRev.129.2342 (1963).
 37.
Gardiner, C. W. & Collett, M. J. Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation. Phys. Rev. A 31, 3761–3774, https://doi.org/10.1103/PhysRevA.31.3761 (1985).
 38.
Beaudoin, F., Gambetta, J. M. & Blais, A. Dissipation and ultrastrong coupling in circuit QED. Phys. Rev. A 84, 043832, https://doi.org/10.1103/PhysRevA.84.043832 (2011).
 39.
Jonsson, T. H. et al. Efficient determination of the Markovian timeevolution towards a steadystate of a complex open quantum system. Comput. Phys. Commun. 220, 81–90, https://doi.org/10.1016/j.cpc.2017.06.018 (2017).
 40.
Gudmundsson, V. et al. Coexisting spin and Rabi oscillations at intermediate time regimes in electron transport through a photon cavity. Beilstein J. Nanotechnol. 10, 606–616, https://doi.org/10.3762/bjnano.10.61 (2019).
Acknowledgements
This work was financially supported by the Research Fund of the University of Iceland, the Icelandic Research Fund, grant no. 163082051, and the Icelandic Infrastructure Fund. The computations were performed on resources provided by the Icelandic High Performance Computing Center at the University of Iceland. NRA acknowledges support from University of Sulaimani and Komar University of Science and Technology. CST acknowledges support from Ministry of Science and Technology of Taiwan under grant No. 1062112M239001MY3.
Author information
Affiliations
Contributions
N.R.A. conceived the idea of the paper, N.R.A. and V.G. performed the calculations, N.R.A. wrote the manuscript, N.R.A., C.T., A.M., and V.G. analysed the results. All authors reviewed the manuscript.
Corresponding authors
Ethics declarations
Competing Interests
The authors declare no competing interests.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Abdullah, N.R., Tang, CS., Manolescu, A. et al. The photocurrent generated by photon replica states of an offresonantly coupled dotcavity system. Sci Rep 9, 14703 (2019). https://doi.org/10.1038/s41598019513208
Received:
Accepted:
Published:
Further reading

Role of interlayer spacing on electronic, thermal and optical properties of BNcodoped bilayer graphene: Influence of the interlayer and the induced dipoledipole interactions
Journal of Physics and Chemistry of Solids (2021)

Properties of BSi6N monolayers derived by firstprinciple computation
Physica E: Lowdimensional Systems and Nanostructures (2021)

The interplay of electron–photon and cavityenvironment coupling on the electron transport through a quantum dot system
Physica E: Lowdimensional Systems and Nanostructures (2020)

Optical absorption microscopy of localized atoms at microwave domain: twodimensional localization based on the projection of threedimensional localization
Scientific Reports (2020)

Electronic, thermal, and optical properties of graphene like SiC structures: Significant effects of Si atom configurations
Physics Letters A (2020)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.