Non-Linear Quantum Dynamics in Coupled Double-Quantum- Dot-Cavity Systems

Abstract

The steady-state quantum dynamics of a compound sample consisting of a semiconductor double-quantum-dot (DQD) system, non-linearly coupled with a leaking superconducting transmission line resonator, is theoretically investigated. Particularly, the transition frequency of the DQD is taken to be equal to the doubled resonator frequency, whereas the inter-dot Coulomb interaction is considered weak. As a consequence, the steady-state quantum dynamics of this complex non-linear system exhibit sudden changes in its features, occurring at a critical DQD-cavity coupling strength, suggesting perspectives for designing on-chip microwave quantum switches. Furthermore, we show that, above the threshold, the electrical current through the double-quantum dot follows the mean photon number into the microwave mode inside the resonator. This might not be the case any more below that critical coupling strength. Lastly, the photon quantum correlations vary from super-Poissonian to Poissonian photon statistics, i.e., towards single-qubit lasing phenomena at microwave frequencies.

1. Introduction

The various examples of hybrid circuit quantum electrodynamic (QED) structures exhibit a variety of quantum effects, previously achieved mainly in quantum optical setups, while presenting unquestionable advantages in the form of low-size on-chip implementation [1,2,3,4,5]. An emblematic example is the optical cavity coupling natural atoms and photons, where lasing phenomena are also achieved by additionally optically pumping the atom. A successful QED alternative system playing the role of an active medium that may induce light amplification in the resonant cavity includes artificial atoms, i.e., semiconducting double-quantum dots (DQDs) attached to normal metal leads, ensuring pumping via the applied biased voltage [6,7,8,9]. The circuit considered is driven by single-electron tunneling, leading to a population inversion in the DQD energy levels, such that, in the resonance condition with the cavity, electron transport takes place via the emission of energy quantum into the cavity [10,11,12]. In consequence, the excited microwave oscillator in the cavity may also facilitate current flow in the DQD [13,14].

The advantage of studying DQD microwave resonator systems is twofold. First, semiconducting DQDs are highly suitable candidates for qubits due to their higher degree of tunability and their experimentally attainable strong-coupling regime, as seen with the electromagnetic cavity demonstrated in Refs. [15,16,17,18]. Second, DQDs are highly efficient detectors of single-itinerant-microwave photons, which is a desirable trait of a quantum system for the realisation of quantum information processing tasks [19,20]. Thus, an electronic interface for the manipulation of photon statistics represents a promising environment for the implementation of various applications in nano-device technology [4,10,21].

DQDs coupled to a leaking resonator and immersed in various environments still show unexplored physics, fueling increasing theoretical interest and further investigations [6,14,20,22,23,24,25,26]. Also, different practical configurations may include relatively weak Coulomb interactions between charges [27,28,29,30,31], Kerr nonlinearity and a signal-driven cavity or DQD [14,32,33,34,35]. Thus, extensive theoretical inquiry is required to demonstrate the control of photon emission via transport properties in the DQD, as well as to investigate the variety of lasing resonances and their signatures in the transport current [9,27,36,37,38,39]. In addition, the Markovian master equation approach is applied in this study; this equation incorporates the dissipation and decoherence processes deteriorating the performance of a device due to its interaction with the degrees of freedom of the electronic and bosonic baths [40,41]. Further research on the creation of hybrid quantum systems and their features, involving circuit quantum electrodynamics, based on mechanical oscillators, quantum acoustodynamics via surface acoustic waves, quantum magnonics or coupling between superconducting circuits and with ensembles or single spins, can be found, for example, in Refs. [42,43,44,45].

In this study, we analyse the quantum dynamics of a complex system formed of a leaking superconducting transmission line resonator mode coupled with a semiconductor DQD qubit, while focusing on the two-photon resonance between the cavity mode and the energy separation levels of the DQD. Additionally, the interdot Coulomb interaction is considered weak enough, allowing for the simultaneous presence of two electrons in the DQD, but only one in each dot. Compared to the strong Coulomb interaction, under the two photon resonance conditions covered recently [46,47], in the current study, an additional incoherent electronic tunneling channel is established, contributing to the population of the DQD and facilitating stronger lasing effects in the resonator as a result. In the framework of open quantum systems, we employ the master equation for this system configuration and deduce the photon number characteristics and the average electronic current in the asymptotic limit of the steady state. We find a critical value for qubit-cavity coupling strengths, such that the mean number of the cavity photons is substantially enhanced or suppressed, as is the electric current through the DQD sample, demonstrating microwave quantum switching. This suggests that the electric current has a convenient mechanism of conversion into a microwave photon flux, or vice versa. Furthermore, above the threshold, the electrical current through the double-quantum dot follows the behaviours of mean photon numbers into the microwave mode inside the resonator, while this might not be the case below that critical point. Finally, the photon correlations vary from super-Poissonian to Poissonian photon statistics, that is, towards single-qubit lasing effects at microwave frequencies.

This paper is organized as follows. In Section 2, we describe the analytical approach and the system of interest, while in Section 3, we present the equations of motion characterising the considered system. Section 4 presents and analyses the obtained results. The article concludes with a summary given in Section 5.

2. Analytical Approach

We consider a hybrid setup made of a semiconducting DQD qubit which is coupled to an electromagnetic microwave resonator such as a superconducting transmission line. Furthermore, the DQD system is pumped incoherently via electron tunneling through the dots, by means of the bias voltage that is assumed to be relatively high. The Coulomb interaction is considered relatively weak, signifying that the DQD is described by the following possible states (see, e.g., Ref. [27]): the null-electron subspace or the empty-dot state, i.e., $|o\rangle$, along with the single-electron subspaces, where the electron is localized either on the left, $|L\rangle$, or the right, $|R\rangle$, dots as well as the mixed state, $|LR\rangle$, denoting the situation when two electrons are simultaneously localised separately in each of the two quantum dots, with $|o\rangle \langle o|+|L\rangle \langle L|+|R\rangle \langle R|+|LR\rangle \langle RL|=1$.

Assuming that the corresponding resonator and the DQD’s interactions with the photon or electronic environmental reservoirs are quite weak, in the Born–Markov approximations and with a low-temperature limit [6,27,46,47,48,49], one obtains the master equation

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}\rho \left(t\right)+{\displaystyle \frac{i}{\hslash }}[H,\rho ]={\mathrm{\Lambda }}_L\rho +{\mathrm{\Lambda }}_R\rho +{\mathrm{\Lambda }}_c\rho ,\end{array}$$

(1)

(with t denoting the time and ℏ the reduced Planck constant) for the reduced density matrix $\rho$ describing the qubit plus leaking resonator mode subsystem only, where the complete Hamiltonian, describing the investigated system, is given by

$$\begin{array}{c}\hfill H=H_q+H_r,\end{array}$$

(2)

where (see, e.g., [6,27])

$$\begin{array}{c}\hfill H_q={\displaystyle \frac{\hslash ϵ}{2}}{\sigma }_z+\hslash \tau ({\sigma }^{+}+{\sigma }^{-})+\hslash u|m\rangle \langle m|,\end{array}$$

(3)

is the Hamiltonian of two quantum dots, forming the DQD qubit, with $ϵ$ being their energy separation, while $\tau$ is the inter-dot tunnelling amplitude, whereas u denotes the inter-dot Coulomb interaction and $|m\rangle \equiv |LR\rangle$. The Hamiltonian

$$\begin{array}{c}\hfill H_r=\hslash {\omega }_r{a}^{†}a+\hslash g{\sigma }_z(a^{†}+a),\end{array}$$

(4)

describes the microwave resonator free energy and qubit–cavity interaction, with ${\omega }_r$ and g being the corresponding resonator frequency and coupling strength, respectively. The cavity photon creation (annihilation) operators, $a^{†}\left(a\right)$, obey the standard commutation relations: $[a,a^{†}]=1$, $[a,a]=[a^{†},a^{†}]=0$. On the other hand, the qubit operators are defined as follows: ${\sigma }_z=|L\rangle \langle L|-|R\rangle \langle R|$, ${\sigma }^{+}=|L\rangle \langle R|$ and ${\sigma }^{-}=|R\rangle \langle L|$, which obey the commutation relations for su(2) algebra, that is, $[{\sigma }^{+},{\sigma }^{-}]={\sigma }_z$ and $[{\sigma }_z,{\sigma }^{\pm }]=\pm 2{\sigma }^{\pm }$. The damping terms, entering in the right-hand side part of the master Equation (1) read

$$\begin{array}{ccc}\hfill {\mathrm{\Lambda }}_L\rho & =& -{\displaystyle \frac{{\mathrm{\Gamma }}_L}{2}}[s_L,s_L^{+}\rho ]+\mathrm{h}.\mathrm{c}.,\hfill \\ \hfill {\mathrm{\Lambda }}_R\rho & =& -{\displaystyle \frac{{\mathrm{\Gamma }}_R}{2}}[s_R^{+},s_R\rho ]+\mathrm{h}.\mathrm{c}.,\hfill \\ \hfill {\mathrm{\Lambda }}_c\rho & =& -{\displaystyle \frac{\kappa }{2}}[a^{†},a\rho ]+\mathrm{h}.\mathrm{c}.,\hfill \end{array}$$

(5)

where “h.c.” stands for Hermitian conjugate, ${\mathrm{\Gamma }}_L$ and ${\mathrm{\Gamma }}_R$ are the corresponding electron tunneling rates and $\kappa$ is the cavity decay rate, while [27] $s_L=|o\rangle \langle L|+|R\rangle \langle m|$ and $s_R=|o\rangle \langle R|-|L\rangle \langle m|$. Note that the master equation’s terms given by the expressionsin the system (5) are represented in the second order in the corresponding small coupling strengths of the DQD and the cavity mode to the corresponding environmental electronic and photonic reservoirs.

Diagonalizing the first two terms of the Hamiltonian (3), using the transformation

$$\begin{array}{c}\hfill |L\rangle =cos(\theta /2)|e\rangle -sin(\theta /2)|g\rangle ,\\ \hfill |R\rangle =sin(\theta /2)|e\rangle +cos(\theta /2)|g\rangle ,\end{array}$$

(6)

with $cos\theta =ϵ/\mathrm{\Omega }$, $sin\theta =2\tau /\mathrm{\Omega }$ and the generalized frequency $\mathrm{\Omega }=\sqrt{{ϵ}^2+{\left(2\tau \right)}^2}$, one arrives at a Hamiltonian H represented via new qubit quasispin operators, namely $R_{\alpha \beta }$ = $|\alpha \rangle \langle \beta |$, $\{\alpha ,\beta \in o,e,g,m\}$, satisfying the commutation relations: $[R_{\alpha \beta },R_{{\beta }^{\prime }{\alpha }^{\prime }}]$ = ${\delta }_{\beta {\beta }^{\prime }}{R}_{\alpha {\alpha }^{\prime }}-{\delta }_{{\alpha }^{\prime }\alpha }{R}_{{\beta }^{\prime }\beta }$ (with ${\delta }_{\alpha \beta }$ the Kronecker delta), that is,

$$\begin{array}{ccc}\hfill H& =& \hslash \mathrm{\Omega }{R}_z/2+\hslash {\omega }_r{a}^{†}a+\hslash u|m\rangle \langle m|+\hslash g(cos\theta R_z\hfill \\ & -& sin\theta (R_{eg}+R_{ge}))\left(a+a^{†}\right),\hfill \end{array}$$

(7)

with $R_z=|e\rangle \langle e|-|g\rangle \langle g|$ being the inversion operator.

In what follows, it is assumed that the semiconductor DQD exchanges energy with the resonator mode when its generalized frequency $\mathrm{\Omega }$ is equal approximately to the doubled value of the cavity frequency ${\omega }_r$. Then, in the interaction picture, given by the unitary operator $U\left(t\right)=exp(i{H}_{0}t/\hslash )$ with $H_0=\hslash \mathrm{\Omega }{R}_z/2+\hslash {\omega }_r{a}^{†}a$, one arrives at a time-dependent Hamiltonian, $H_f\left(t\right)$, oscillating considerably quickly in time. Its contribution to the established quantum dynamics can be obtained via the following relation:

$$\begin{array}{c}\hfill \overline{H}=-{\displaystyle \frac{i}{\hslash }}{H}_f\left(t\right)\int dt{H}_f\left(t\right),\end{array}$$

(8)

when $g\ll {\omega }_r$; see [50,51], for example. As a result, the master Equation (1) transforms into the following:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d}{dt}}\rho \left(t\right)& +& {\displaystyle \frac{i}{\hslash }}[\overline{H},\rho ]=-{\mathrm{\Gamma }}_{Lc}[R_{oe}+R_{gm},(R_{eo}+R_{mg})\rho ]\hfill \\ & -& {\mathrm{\Gamma }}_{Ls}[R_{og}-R_{em},(R_{go}-R_{me})\rho ]\hfill \\ & -& {\mathrm{\Gamma }}_{Rs}[R_{eo}+R_{mg},(R_{oe}+R_{gm})\rho ]\hfill \\ & -& {\mathrm{\Gamma }}_{Rc}[R_{go}-R_{me},(R_{og}-R_{em})\rho ]\hfill \\ & -& {\displaystyle \frac{\kappa }{2}}[a^{†},a\rho ]+\mathrm{h}.\mathrm{c}.,\hfill \end{array}$$

(9)

where

$$\begin{array}{ccc}\hfill \overline{H}& =& \hslash {\omega }_e{R}_{ee}-\hslash {\omega }_g{R}_{gg}+\hslash \delta a^{†}a+\hslash u|m\rangle \langle m|+\hslash \beta R_z{a}^{†}a\hfill \\ & +& \hslash \overline{g}\left(a^{†2}{R}_{ge}+R_{eg}{a}^2\right).\hfill \end{array}$$

(10)

Thus, due to the presence of rapidly oscillating terms in the complete Hamiltonian, arrive at an effective novel DQD-resonator Hamiltonian, $\overline{H}$, characterizing non-linear coherent two-photon effects. Here, ${\omega }_e=\beta /2-{\omega }_s$, ${\omega }_g=\beta /2+{\omega }_s$, $\beta =2g^2\mathrm{\Omega }{sin}^2\theta /({\mathrm{\Omega }}^2-{\omega }_r^2)$, ${\omega }_s=g^2({\mathrm{\Omega }}^2{cos}^2\theta -{\omega }_r^2)/\left[{\omega }_r({\mathrm{\Omega }}^2-{\omega }_r^2)\right]$, $\delta ={\omega }_r-\mathrm{\Omega }/2$ and $\overline{g}=g^2\mathrm{\Omega }sin2\theta /\left[2{\omega }_r(\mathrm{\Omega }-{\omega }_r)\right]$. The corresponding rates, entering in Equation (9), read

$$\begin{array}{ccc}\hfill {\mathrm{\Gamma }}_{Lc}& =& {\mathrm{\Gamma }}_L{cos}^2(\theta /2)/2,{\mathrm{\Gamma }}_{Ls}={\mathrm{\Gamma }}_L{sin}^2(\theta /2)/2,\hfill \\ \hfill {\mathrm{\Gamma }}_{Rc}& =& {\mathrm{\Gamma }}_R{cos}^2(\theta /2)/2,{\mathrm{\Gamma }}_{Rs}={\mathrm{\Gamma }}_R{sin}^2(\theta /2)/2.\hfill \end{array}$$

Notice that the master Equation (9) is obtained under the secular approximation, i.e., we considered that $\mathrm{\Omega }\pm u\gg {\mathrm{\Gamma }}_{L/R}$; we also considered lower-temperature environments. As a result we involve parameters satisfying the adopted approximations which are also experimentally achievable [8], namely the frequency of the leaking resonator is taken of the order of ${\omega }_r\approx 1$ GHz and the cavity decay rate is taken of the order of $\kappa \approx 1$ MHz, respectively, whereas DQD–cavity coupling strengths may reach values up to $g\approx 100$ MHz.

In Section 3 just below, we make use of Equation (9) in order to obtain the corresponding equations of motion describing the quantum dynamics of the combined resonator–DQD subsystem.

3. The Equations of Motion

Using the master Equation (9), one obtains the exact system of equations of motion:

$$\begin{array}{ccc}\hfill {\dot{P}}_n^{\left(0\right)}& =& -{\mathrm{\Gamma }}_L{P}_n^{\left(0\right)}+2{\mathrm{\Gamma }}_{Rs}{P}_n^{\left(3\right)}+2{\mathrm{\Gamma }}_{Rc}{P}_n^{\left(1\right)}-\kappa (n{P}_n^{\left(0\right)}\hfill \\ & -& (n+1)P_{n+1}^{\left(0\right)}),\hfill \\ \hfill {\dot{P}}_n^{\left(1\right)}& =& -i\overline{g}{P}_n^{\left(6\right)}-2({\mathrm{\Gamma }}_{Lc}+{\mathrm{\Gamma }}_{Rc})P_n^{\left(1\right)}+2{\mathrm{\Gamma }}_{Ls}{P}_n^{\left(0\right)}\hfill \\ & +& 2{\mathrm{\Gamma }}_{Rs}{P}_n^{\left(2\right)}-\kappa \left(n{P}_n^{\left(1\right)}-(n+1)P_{n+1}^{\left(1\right)}\right),\hfill \\ \hfill {\dot{P}}_n^{\left(2\right)}& =& -{\mathrm{\Gamma }}_R{P}_n^{\left(2\right)}+2{\mathrm{\Gamma }}_{Lc}{P}_n^{\left(1\right)}+2{\mathrm{\Gamma }}_{Ls}{P}_n^{\left(3\right)}-\kappa (n{P}_n^{\left(2\right)}\hfill \\ & -& (n+1)P_{n+1}^{\left(2\right)}),\hfill \\ \hfill {\dot{P}}_n^{\left(3\right)}& =& -i\overline{g}{P}_n^{\left(4\right)}-2({\mathrm{\Gamma }}_{Ls}+{\mathrm{\Gamma }}_{Rs})P_n^{\left(3\right)}+2{\mathrm{\Gamma }}_{Lc}{P}_n^{\left(0\right)}\hfill \\ & +& 2{\mathrm{\Gamma }}_{Rc}{P}_n^{\left(2\right)}-\kappa \left(n{P}_n^{\left(3\right)}-(n+1)P_{n+1}^{\left(3\right)}\right),\hfill \\ \hfill {\dot{P}}_n^{\left(4\right)}& =& i\left((3+2n)\beta -2\delta \right)P_n^{\left(5\right)}-2i\overline{g}(n+1)(n+2)\hfill \\ & \times & \left(P_n^{\left(3\right)}-P_{n+2}^{\left(1\right)}\right)-(\mathrm{\Gamma }+\kappa )P_n^{\left(4\right)}-\kappa (n{P}_n^{\left(4\right)}\hfill \\ & -& (n+1)P_{n+1}^{\left(4\right)}),\hfill \\ \hfill {\dot{P}}_n^{\left(5\right)}& =& i\left((3+2n)\beta -2\delta \right)P_n^{\left(4\right)}-(\mathrm{\Gamma }+\kappa )P_n^{\left(5\right)}-\kappa (n{P}_n^{\left(5\right)}\hfill \\ & -& (n+1)P_{n+1}^{\left(5\right)}),\hfill \\ \hfill {\dot{P}}_n^{\left(6\right)}& =& i\left((1-2n)\beta +2\delta \right)P_n^{\left(7\right)}-2i\overline{g}n(n-1)\left(P_n^{\left(1\right)}-P_{n-2}^{\left(3\right)}\right)\hfill \\ & -& (\mathrm{\Gamma }-\kappa )P_n^{\left(6\right)}-\kappa \left(n{P}_n^{\left(6\right)}-(n+1)P_{n+1}^{\left(6\right)}+2P_n^{\left(8\right)}\right),\hfill \\ \hfill {\dot{P}}_n^{\left(7\right)}& =& i\left((1-2n)\beta +2\delta \right)P_n^{\left(6\right)}-(\mathrm{\Gamma }-\kappa )P_n^{\left(7\right)}-\kappa (n{P}_n^{\left(7\right)}\hfill \\ & -& (n+1)P_{n+1}^{\left(7\right)}+2P_n^{\left(9\right)}),\hfill \\ \hfill {\dot{P}}_n^{\left(8\right)}& =& -i\left((1+2n)\beta -2\delta \right)P_n^{\left(9\right)}-2i\overline{g}n(n+1)\left(P_{n+1}^{\left(1\right)}-P_{n-1}^{\left(3\right)}\right)\hfill \\ & -& \mathrm{\Gamma }{P}_n^{\left(8\right)}-\kappa \left(n{P}_n^{\left(8\right)}-(n+1)P_{n+1}^{\left(8\right)}-P_n^{\left(4\right)}\right),\hfill \\ \hfill {\dot{P}}_n^{\left(9\right)}& =& -i\left((1+2n)\beta -2\delta \right)P_n^{\left(8\right)}-\mathrm{\Gamma }{P}_n^{\left(9\right)}-\kappa (n{P}_n^{\left(9\right)}\hfill \\ & -& (n+1)P_{n+1}^{\left(9\right)}+P_n^{\left(5\right)}),\hfill \end{array}$$

(11)

describing the entire sample incorporating a semiconductor DQD qubit coupled in two-photon resonance to the cavity boson mode.

The system (11) can be straghtforwardly obtained using the master Equation (9), if first one obtains the equations of motion for ${\rho }_{\alpha \beta }$ = $\langle \alpha \left|\rho \right|\beta \rangle$, $\alpha ,\beta \in \{o,e,g,m\}$ (see Refs. [52,53]), and then writing the corresponding equations for the following variables: ${\rho }^{\left(0\right)}={\rho }_{oo}$, ${\rho }^{\left(1\right)}={\rho }_{gg}$, ${\rho }^{\left(2\right)}={\rho }_{mm}$, ${\rho }^{\left(3\right)}={\rho }_{ee}$, ${\rho }^{\left(4\right)}=a^2{\rho }_{ge}-{\rho }_{eg}{a}^{†2}$, ${\rho }^{\left(5\right)}=a^2{\rho }_{ge}+{\rho }_{eg}{a}^{†2}$, ${\rho }^{\left(6\right)}=a^{†2}{\rho }_{eg}-{\rho }_{ge}{a}^2$, ${\rho }^{\left(7\right)}=a^{†2}{\rho }_{eg}+{\rho }_{ge}{a}^2$, ${\rho }^{\left(8\right)}=a^{†}{\rho }_{eg}{a}^{†}-a{\rho }_{ge}a$ and ${\rho }^{\left(9\right)}=a^{†}{\rho }_{eg}{a}^{†}+a{\rho }_{ge}a$. The projection on the Fock states, $|n\rangle$, i.e., $P_n^{\left(j\right)}=\langle n|{\rho }^{\left(j\right)}|n\rangle$, with $j\in \{0,\dots ,9\}$ and $n\in \{0,\infty \}$, leads to the system (11), where $\mathrm{\Gamma }=({\mathrm{\Gamma }}_L+{\mathrm{\Gamma }}_R)/2$. Notice that the involved analytical approach allows us to avoid decoupling the qubit–cavity correlators, when obtaining the system (11).

In general, to solve the infinite system (11), one may truncate it at a certain maximum value $n=n_{\max }$ so that a further increase in its value, i.e., $n_{\max }$, does not modify the obtained results. As a consequence, the steady-state cavity-mode photon mean number, i.e., $\langle n\rangle =\langle a^{†}a\rangle$, is expressed as follows:

$$\begin{array}{c}\hfill \langle n\rangle =\sum _{n=0}^{n_{\max }}n\left(P_n^{\left(0\right)}+P_n^{\left(1\right)}+P_n^{\left(2\right)}+P_n^{\left(3\right)}\right),\end{array}$$

(12)

with

$$\begin{array}{c}\hfill \sum _{n=0}^{n_{\max }}\left(P_n^{\left(0\right)}+P_n^{\left(1\right)}+P_n^{\left(2\right)}+P_n^{\left(3\right)}\right)=1.\end{array}$$

The current $I_q$ through the DQD qubit is proportional to the population in the state $|R\rangle$ and $|m\rangle$, that is,

$$\begin{array}{ccc}\hfill I_q& =& \left|e\right|{\mathrm{\Gamma }}_R\left(|R\rangle \langle R|+|m\rangle \langle m|\right)\hfill \\ & =& \left|e\right|{\mathrm{\Gamma }}_R\sum _{n=0}^{n_{\max }}\left({cos}^2(\theta /2)P_n^{\left(1\right)}+P_n^{\left(2\right)}+{sin}^2(\theta /2)P_n^{\left(3\right)}\right),\hfill \end{array}$$

(13)

where e is the electron charge. In Equation (13), we apply the transformation (6) and the secular approximation.

The corresponding steady-state second-order cavity photon correlation function is defined commonly as [54]

$$\begin{array}{ccc}\hfill g^{\left(2\right)}\left(0\right)& =& {\displaystyle \frac{\langle a^{†2}{a}^2\rangle }{{\langle n\rangle }^2}}\hfill \\ & =& {\displaystyle \frac{1}{{\langle n\rangle }^2}}\sum _{n=0}^{n_{\max }}n(n-1)\left(P_n^{\left(0\right)}+P_n^{\left(1\right)}+P_n^{\left(2\right)}+P_n^{\left(3\right)}\right).\hfill \end{array}$$

(14)

Note that $g^{\left(2\right)}\left(0\right)<1$ indicates sub-Poissonian, $g^{\left(2\right)}\left(0\right)>1$ indicates super-Poissonian, and $g^{\left(2\right)}\left(0\right)=1$ indicates Poissonian photon statistics.

In Section 4 just below, we describe the photon resonator quantum dynamics, as well as the steady-state behaviours of the electrical current, based on results obtained from Equations (11)–(14).

4. Results and Discussion

Figure 1a shows the steady-state dependence of the mean cavity photon number as a function of the DQD-resonator coupling strength, near two-photon resonance, i.e., $\mathrm{\Omega }\approx 2{\omega }_r$, extracted from the system of Equation (11). Particularly, the decay rates were selected within the secular approximation, that is, they were smaller than qubit frequency $\mathrm{\Omega }$, in concordance with the valability of the master Equation (9). As a result, one observes a threshold transition when the mean photon number abruptly changes from lower to higher values, or, conversely, when varying to some degree the qubit–resonator coupling strength. Actually, this behaviour is typical for an externally coherently pumped and leaking cavity mode, containing a Kerr-like non-linearity [32,33], for instance. Here, however, this behaviour is due to the unilateral electronic pumping of the DQD sample, i.e., from the left to the right.

On the other hand, Figure 1b depicts the steady-state current through the DQD qubit, which is proportional to the populations in the state $|R\rangle$ and $|m\rangle$. The current follows the same behaviour as the mean photon number: compare the behaviours by contrasting Figure 1a,b. Therefore, measuring the current through the DQD, one can estimate the photon flux leaking out from the cavity mode, or vice versa. In practice, the described process can be a practical tool with which to design on-chip microwave quantum switches, based on electric current conversion into microwave photons in hybrid DQD–resonator systems. Furthermore, the cavity photons obey the super-Poissonian statistics below the threshold, i.e., $g^{\left(2\right)}\left(0\right)\ge 2$ (see Figure 2), which changes to Poissonian photon statistics above the threshold, i.e., to microwave photon lasing effects, where $g^{\left(2\right)}\left(0\right)=1$. Around the threshold, the photon correlation evidently enhances. Actually, $g^{\left(2\right)}\left(0\right)$ does not converge for $g/{\omega }_r=0$, since $\langle n\rangle =0$, so one should proceed with non-zero values for DQD–cavity coupling strengths in this particular case.

We proceed further by describing the corresponding quantum dynamics versus the experimentally controllable DQD energy separation $ϵ$, while all other parameters are being fixed. The resonator–DQD coupling strength g selected is below or above the threshold:see Figure 1a or Figure 1b, respectively. Thus, Figure 3a depicts the mean cavity photon number in these situations. Particularly, in the near-resonance condition, i.e., when $\mathrm{\Omega }\approx 2{\omega }_r$, one observes a plateau where the mean cavity photon number does not highly vary, while the scaled energy separation $ϵ/{\omega }_r$ is scanned around the resonance. Then, it drops to zero out of the resonance. Above the threshold, i.e., when $g/{\omega }_r=0.06$, the current, again, follows the behaviour of the resonator mean photon number; see Figure 3b. However, this is no longer the case below the threshold, that is, when $g/{\omega }_r=0.04$; compare the insets in Figure 3a,b. Furthermore, within the plateau region, the photon statistics are Poissonian one since $g^{\left(2\right)}\left(0\right)\approx 1$; see Figure 4 above the threshold. On the other hand, below the threshold, the photon statistics are super-Poissonian; see the inset in Figure 4. Thus, one may observe stable single-qubit lasing effects above the critical value of the cavity–DQD coupling strength, that is, above the threshold.

5. Summary

In summary, we have investigated a semiconducting DQD qubit coupled in two-photon resonance with a leaking microwave resonator, where the pumping of the qubit is achieved via the environmental electronic reservoirs. Particularly, the inter-dot Coulomb interaction is assumed to be weak, while the focus is on calculating the mean photon cavity number, under the secular approximation, with a corresponding accent on an obtained critical value for the qubit–resonator coupling strength, where the steady-state photon number changes abruptly. This can be practical for engineering quantum microwave switches, for instance. Moreover, the corresponding photon statistics is obtanied to vary from super-Poissonian to Poissonian photon statistics. Also, the electric current through the DQD system follows the behaviour of the cavity-mode steady-state photon number, particularly above the threshold. This way, one may achieve lasing effects in microwave-frequency domains, while having the ability to convert electric current in a photon flux. Thus, one may estimate the intensity of the photon flux, generated in cavity mode, via measuring the electric current, or vice versa.