Figure 1

Schema of the systems of interest. The main object of study—two coupled qubits—is sketched in panel (c). It is compared throughout with other coupled systems: (a), (b), (d), and (e).Reuse & Permissions

Figure 2

Energy levels for the two qubits described by Hamiltonian (1). Weak coupling (a) and strong coupling (b) in the absence of pump are well defined in terms of the bare and dressed states $|\pm \rangle$, respectively. As shown in Sec. 5b, the SC regime gives rise to new regions when pump is taken into account: (c) FSC where dressed states remain $|\pm \rangle$, and (d) SSC and MC, where dressed states form new sets $|I_{\pm }\rangle$, $|O_{\pm }\rangle$. Only SSC, with splitting of all the new dressed states, is shown. MC corresponds to the case where $|I_{\pm }\rangle$ have closed (both collapsing on ${\omega }_1$). The thickness of the levels represents the uncertainty in energy due to (a,b) the decay and (c,d) both the pump and the decay. The arrows linking the levels due to pump and decay are blue for the lower ($l_i$) and red for the upper ($u_i$) transitions. Transitions labeled $A$, $C$ (involving $|+\rangle$ or $|{\mathrm{I}}_{\pm }\rangle$, in green) occur at frequencies determined by $z_1$, while $B$ and $D$ (involving $|-\rangle$ or $|{\mathrm{O}}_{\pm }\rangle$, in orange) are determined by $z_2$. The same color code is used in the rest of the figures to plot the decomposition of the spectra.Reuse & Permissions

Figure 3

Phase space of the steady-state strong and weak coupling regimes as a function of pump for ${\gamma }_1=g$ and ${\gamma }_2=g/2$. In strong coupling (SC, blue), one can distinguish two regions: first-order (FSC, light blue) and second-order (SSC, dark blue) strong coupling. Weak coupling (WC) is in purple and mixed coupling (MC) in green. The dashed blue lines enclose the two regions where two peaks can be resolved in the power spectrum of the first qubit, $S_1(\omega )$. One falls in SC and the other in WC.Reuse & Permissions

Figure 4

Factors contributing to $D^{\mathrm{s}}$. (a) $\sqrt{{\Gamma }_1{\Gamma }_2}/[({\Gamma }_1+{\Gamma }_2)/2]$ as a function of ${\Gamma }_1/g,{\Gamma }_2/g$ and (b) $\sqrt{r_1+r_2-2r_1{r}_2}$ as a function of $r_1,r_2$. The values corresponding to the contour lines are marked on the plots. Both functions take values from 0 (dark blue) to 1 (light gray).Reuse & Permissions

Figure 5

Phase space of FSC, SSC, MC, and WC as functions of $P_1/g={\gamma }_2/g$ and $P_2/g={\gamma }_1/g$. The color code is that of Fig. 3. (Inset) Possible line shapes of $S_1(\omega )$: doublet (red), distorted doublet (green), distorted singlet (blue), and singlet (white).Reuse & Permissions

Figure 6

(a) Positions of the peaks (inner ${\omega }_A$ in thick green, outer ${\omega }_B$ in thick orange) and populations ($n_1$ increasing in dashed blue, $n_2$ decreasing in dashed purple, $\left|n_{\mathrm{corr}}\right|$ in dotted brown) as function of $P_1/g={\gamma }_2/g=\gamma /g$ for $P_2={\gamma }_1=0$. The system goes from FSC (at 0) to SSC, to MC, and to WC. The most interesting line shapes, which can only appear in SSC and MC, are the distorted doublet (I) and the singlet (II). The total spectra (in black) is decomposed into inner (green) and outer (orange) peaks coming from the transitions sketched in Fig. 2d.Reuse & Permissions

Figure 7

Comparison between the results of Fig. 6 (solid lines) and those obtained for a 4LS (dashed black lines). The positions of the peaks composing the spectra, are marked with vertical lines in (I) and (II). Different types of splitting lead to dramatic differences in the spectral shapes. In the 4LS, only plain doublets (or singlets, not shown) arise.Reuse & Permissions

Figure 8

(a) Positions of the peaks (inner ${\omega }_{A,C}$ in thick green, outer ${\omega }_{B,D}$ in thick orange) as a function of detuning for six cases picked from Fig. 6 with the specified parameter $\gamma$. The positions for a 4LS appear in dashed black and the bare modes in thin black, for comparison. The first two plots are those labeled (I) and (II) in previous figures.Reuse & Permissions

Figure 9

Spectra for varying positive detuning (anticrossings) in SSC (I) and in MC (II) for the corresponding cases in Fig. 6. The inset in (II) shows again the positions of the three or four peaks composing the spectra as a function of detuning. Close to resonance, the inner peaks converge to the average transition energy ${\omega }_{11}/2=\Delta /2$. At large detunings, the first dot (which is pumped) emits at ${\omega }_1=0$ and dominates over the second dot (which decays) with emission at ${\omega }_2=-\Delta$.Reuse & Permissions

Figure 10

(a) Contributions from state $|1,0\rangle$ to the dressed states appearing in an optimally pumped case: $P_1={\gamma }_2=\gamma$, $P_2={\gamma }_1=0$. The $x$ axis is $\gamma /g$, as in Figs. 6 and 7 describing the same case. Thin dashed black lines show the contribution to the 4LS dressed states $|\pm \rangle$. Thick lines show the qubit dressed states: orange for $|O_{\pm }\rangle$ and green for $|I_{\pm }\rangle$. The splitting of each pair of lines $\pm$ from equal to different contributions, marks their transition into WC. All the “$-$” states tend toward $|1,0\rangle$ in the very weakly coupled regime.Reuse & Permissions

Figure 11

Circulation of coherence among the transitions between bare states for two qubits in the case where ${\gamma }_1=P_2=0$. A loop is formed that involves all the transitions, thanks to the pumping and decay terms that participate in the flow of coherence in a constructive way. The direct coupling goes both in tune with the loop or against it. When ${\gamma }_2=P_1=2g$ the transfer is optimal.Reuse & Permissions

Figure 12

Chain of correlators—indexed by $\{\eta \}=(m,n,\mu ,\nu )$—linked by the Hamiltonian dynamics with pump and decay for two coupled qubits. On the left (right) is the set ${\bigcup }_k{\mathcal{N}}_k$ (${\bigcup }_k{\tilde{\mathcal{N}}}_k$) involved in the equations of the two-time (single-time) correlators. The first manifolds ${\mathcal{N}}_1$ and ${\tilde{\mathcal{N}}}_1$ that correspond to the linear model [14] are shown in green; the second manifold ${\mathcal{N}}_2$ and ${\tilde{\mathcal{N}}}_2$ in blue. The equation of motion $\langle {\sigma }_1^{†}(t)C_{\{\eta \}}(t+\tau )\rangle$ with $\eta \in {\mathcal{N}}_k$ requires for its initial value the correlator $\langle C_{\{\mathrm{\eta ̃}\}}\rangle$ with $\{\mathrm{\eta ̃}\}\in {\tilde{\mathcal{N}}}_k$ defined from $\{\eta \}=(m,n,\mu ,\nu )$ by $\{\mathrm{\eta ̃}\}=(m+1,n,\mu ,\nu )$, as seen in the diagram. The red arrows indicate which elements are linked by the coherent (SC) dynamics, through the coupling strength $g$, while the green and blue arrows show the connections due to the incoherent quantum-dot pumpings. The sense of the arrows indicates which element is “calling” which in its equations. The self-coupling of each node to itself is not shown. This is where ${\omega }_{1,2}$ and ${\Gamma }_{1,2}$ enter. These links are obtained from the rules in Eqs. (B2), which result in the matrices ${\mathbf{M}}_1$ and ${\mathbf{M}}_0$. The number of correlators needed to compute the spectrum is self-truncated (with four elements) thanks to the saturation of the qubit.Reuse & Permissions

Figure 13

Steady-state values of $g^{(2)}(\tau )$ (solid blue) $g_{12}^{(2)}(\tau )$ (dashed purple) at resonance. (a) The system is in FSC: ${\gamma }_1=g$, ${\gamma }_2=g/2$, $P_1=g/2$, and $P_2=g/10$. (b) In MC with optimum coupling $G=\sqrt{2}g$: ${\gamma }_1=P_2=0$ and $P_1={\gamma }_2=2g$ (see Fig. 6).Reuse & Permissions