Figure 1

(a) Origin of the peaks in the Mollow triplet: The three different frequencies ($p=0,\pm$) are found in the four possible transitions between two Jaynes-Cummings rungs at high intensities. Below, different Mollow triplets when varying detuning [(b), (d)] or dephasing (c), with ${\Omega }_{\mathrm{L}}=1.5{\gamma }_{\sigma }$. The symmetry of the triplet is broken only under the combined action of detuning and dephasing.Reuse & Permissions

Figure 2

Degree of asymmetry $\mathcal{V}$ of the Mollow triplet under coherent excitation, as a function of detuning $\Delta$ and pure dephasing ${\gamma }_{\phi }$ for ${\Omega }_{\mathrm{L}}=1.5{\gamma }_{\sigma }$. It is 0 (dark blue) when the side peaks have the same intensity and 1 (bright yellow) when one of them disappears completely.Reuse & Permissions

Figure 3

Exact results computed numerically for (a) ${\gamma }_a{n}_a/g$, (b) $n_{\sigma }$, and (c) $g^{\left(2\right)}$, as a function of pumping for various systems, with ${\gamma }_a/g\in \{0.1,0.22,0.46,1,2.15,4.64,10\}$ from lighter to darker shades [top to bottom in (a)]. Other parameters are ${\gamma }_{\sigma }=0.00334g$, from Ref. [55], and $P_a={\gamma }_{\phi }=\Delta =0$. The different regimes of operation are designated in (a).Reuse & Permissions

Figure 4

Comparison between the numerical results for $n_a$ [from Fig. 3a], in thick yellowish lines, with different approximated solutions. In each panel, ${\gamma }_a$ decreases exponentially from bottom to top curves, ranking from $10g$ (darkest curve) to $0.1g$ (lightest curve). The limiting case ${\gamma }_a=0$ is also shown on the left upper corner, where the divergence is a feature of the Jaynes-Cummings model. In (a), we superimpose the linear model (dotted red) and Jaynes-Cummings truncated at one excitation (dashed blue), given by Eq. (24a). In (b), the approximated semiclassical solution, Eq. (32), which provides an accurate description in the lasing regime given by Eq. (61). In (c), the thermal approximation, given by Eq. (39) that converges to the models in (a) at low and high pumps. In (d), the cothermal approximation, given by the numerical solution of Eqs. (44a). In this case, we extend solutions to values of ${\gamma }_a$ down to $0.01g$, out of reach numerically. A vertical guideline marks the value of ${\gamma }_{\sigma }=0.00334g$. Other parameters are $P_a=\Delta ={\gamma }_{\phi }=0$.Reuse & Permissions

Figure 5

Comparison between the numerical $g^{\left(2\right)}$ (in thick yellow lines) and the cothermal approximation (dashed black lines), from Eq. (43) with $n_a$ of Fig. 4. It interpolates neatly between the lasing and the thermal regions ($1\to 2$) but also provides a very good agreement in the linear regime. Here as well, we extend solutions to values of ${\gamma }_a$ down to $0.01g$, out of reach numerically.Reuse & Permissions

Figure 6

(a) $n_a$, (b) $g^{\left(2\right)}$, and (c) $Q$-Mandel parameter, as a function of the pumping rate, extracted with the cothermal approximation. Detuning is fixed in each column to the value while pure dephasing increases from 0 (thick curves) to $12g$ (dashed curves) in steps of $2g$. Detuning brings a threshold for lasing and, while dephasing can help to reduce it, it also makes the system less robust to quenching. Parameters are ${\gamma }_a=0.1g$, $P_a=0={\gamma }_{\sigma }=0$.Reuse & Permissions

Figure 7

Cavity (left blue) and emitter (right pink) spectra of emission computed numerically. As a function of pumping, a transition from (a) the quantum linear regime to (f) lasing can be followed passing through (b) the quantum nonlinear regime that (c) melts into (d) and (e) structures of much reduced complexity, namely, triplets. A Mollow triplet is neatly visible in the emitter spectrum, whereas the cavity gives rise to a single narrowing line. Note that in the cuts (a)–(f), spectra are displayed in log scale. Parameters are ${\gamma }_a=0.1g$, ${\gamma }_{\sigma }=0.00334g$, $P_a={\gamma }_{\phi }=\Delta =0$.Reuse & Permissions

Figure 8

Lasing as a condensation of dressed states: Transition energies between the dressed states when solving numerically the dynamics exactly are shown. The color code (online) has blue shades corresponding to a positive weight of the transition in the cavity emission and red shades corresponding to a negative weight. In (a) all regimes are shown over all energies, in (b) a close-up of (a) is given for frequencies $|{\omega }_p|<g$ lying between the vacuum Rabi splitting, and in (c) a close-up of (b) is given in the lasing regime, showing an extremely complex structure of the exact solution obtained in the full-quantization picture, although the final result washes out completely most of this underlying information, to provide only the single narrow line of a lasing system.Reuse & Permissions

Figure 9

Approximated spectra of emission, Eq. (59), as pumping is increased and the system undergoes the quantum-to-classical transition. Parameters: ${\gamma }_a=0.1g$, $P_a={\gamma }_{\sigma }={\gamma }_{\phi }=\Delta =0$. In inset, the comparison between the approximated-analytical (thin blue) and exact-numerical (thick black) spectra for the lowest pump $P_{\sigma }=0.1g$. Although the approximation breaks down in the quantum regime, it shows how the Mollow triplet is formed.Reuse & Permissions

Figure 10

Transition energies (positions of the spectral peaks) as a function of decoherence $\left|\Gamma \right|/\left(4g\right)$ from rungs $n=1,2,3,4$ (from dark to light lines) for two cases: spontaneous emission [$\Gamma ={\gamma }_a-{\gamma }_{\sigma }$, Eq. (62), thin dashed lines] and in the presence of pumping [$\Gamma ={\Gamma }_{\sigma }-{\gamma }_{\phi }$, Eq. (60a), thick lines]. The line starting at $\omega =g$ corresponds to the linear regime or first-rung-to-vacuum transition [$\mathrm{Re}\left(R_0\right)$, thick black line] and is common to both cases. The curves starting above (resp. below) 1 correspond to the outer (inner) transitions.Reuse & Permissions

Figure 11

Comparison between the position of the side peaks [$\mathrm{Re}\left(R_{\mathrm{O}}\right)$, in solid blue] and those observed in the emitter spectrum (in solid purple, above the filling), as a function of pumping. In dashed is shown the neck of the side peak so that the filled area delimits approximately its half-width. Parameters: ${\gamma }_a=0.1g$, ${\gamma }_{\sigma }=0$.Reuse & Permissions

Figure 12

Mollow triplet in the direct PL spectra from the emitter, $S_{\sigma }\left(\omega \right)$ for ${\gamma }_a=0.1g$, $P_a=0$, ${\gamma }_{\sigma }=0.00334g$, and $P_{\sigma }=7g$ ($n_a\approx 11.6$ and $n_{\sigma }\approx 0.53$), for various configurations of detuning and dephasing. The exact numerical results are plotted with a dark-blue dashed line, overlapping almost exactly our analytical formula, in light-blue solid. The coherent scattering peak is featured in the analytical solution in (a) while only the incoherent spectrum is shown in (b) and (c). The dotted line in (a) represents the Mollow triplet obtained under coherent excitation, with a laser intensity equivalent to the cavity occupation in the incoherent case, that is, for ${\Omega }_{\mathrm{L}}\to \sqrt{n_a}g$ and also an equivalent emitter broadening, ${\gamma }_{\sigma }\to {\gamma }_{\sigma }+P_{\sigma }$. The two types of Mollow line shapes are clearly different even though their peak positions and broadening are equal. Also, as compared to the coherent excitation case, the Mollow triplet under incoherent pumping is a resonant structure that becomes strongly asymmetric with detuning.Reuse & Permissions

Figure 13

Contributions to the Mollow triplet under (a) incoherent and (b) coherent excitation for the three types of peaks: the central Lorentzian peak (constant at $1/2$, solid line), the sidebands (dashed), and the elastic scattering peak (dotted). The parameters are chosen so that the two Mollow triplets can be compared on equal grounds, as explained in the text, for ${\gamma }_a=0.1g$, $P_a=0$, ${\gamma }_{\sigma }={\gamma }_{\phi }=0$, and $\Delta =0$.Reuse & Permissions

Figure 14

Counterpart of Fig. 1 for the Mollow triplet under incoherent excitation. In contrast to the coherent excitation case, detuning alone breaks the Mollow triplet. Parameters are ${\gamma }_a=0.1g$, ${\gamma }_{\sigma }=0.01g$, $P_a=0$, and $P_{\sigma }=7g$.Reuse & Permissions