Excitation with quantum light. II. Exciting a two-level system

We study the excitation of a two-level system (2LS) by quantum light, thereby bringing our previous studies [see Paper I of this series, Phys. Rev. A 94, 063825 (2016)] to a target that is quantum itself. While there is no gain for the quantum state of the target as compared to driving it with classical light, its dynamical features, such as antibunching, can be improved. We propose a chain of two-level systems, i.e., setting the emission of each 2LS as the driving source of the following one, as an arrangement to provide better single-photon sources. At a fundamental level, we discuss the notion of strong coupling between quantum light from a source and its target, and the several versions of the Mollow triplet that follow from various types of driving light. We discuss the Heitler effect of antibunched photons from the scattered light off a laser.

Figure 1

(a) A 2LS is driven by the emission of a quantum source. In this particular case, the quantum source is made of a 2LS driven by a classical source. (b)–(f) Accessible states of the 2LS under different driving configurations, as seen by the accessible volume in the Bloch sphere. The states of the equator of the sphere are $\left|{\psi }_{\pm }\right.〉=\left(\left|0\right.〉\pm \left|1\right.〉\right)/\sqrt{2}$ and $\left|{\varphi }_{\pm }\right.〉=\left(\left|0\right.〉\pm i\left|1\right.〉\right)/\sqrt{2}$. (b) Incoherent excitation: this kind of excitation does not generate coherence in the 2LS, so all the accessible states lie along the $z$ axis of the Bloch sphere. (c) Coherent excitation: this kind of excitation does not allow the qubit to have a population larger than $\frac{1}{2}$, so all the accessible states lie in the south hemisphere of the Bloch sphere. The source 2LS is allowed to go only in one half of the ellipsoid, while the target 2LS can cover the ellipsoid completely. (d) Mixed excitation: under this driving the source 2LS can access a symmetrical region in the north hemisphere of the Bloch sphere. However, the region available to the target 2LS does not increase. (e) Rotating the direction of the excitation of the source 2LS allows it to access the regions otherwise out of reach, thus filling the entire ellipsoid. (f) Exciting with a source that generates the transition $\left|+\right.〉\left.〈0\right|$ allows the source 2LS to access states with mean population $\frac{1}{2}$ but not completely mixed.

Figure 2

Excitation by a coherent source. (a) Energy spectrum of a coherently driven 2LS. The energies that contribute to the total emission spectrum with positive Lorentzians are shown in blue, whereas those contributing as negative Lorentzians are shown in red. (b)–(d) Emission spectra of the target 2LS (dashed, black lines) for the values $\mathrm{\Omega }/{\gamma }_{\sigma }$ marked by the vertical dashed lines in (a). The total emission spectrum is made of the sum of positive (blue lines) and negative (red) Lorentzians, and dispersive functions (dotted purple).

Figure 3

Excitation by a single-photon source. (a)–(i) Energy spectra of the target 2LS. The energies that contribute to the total emission spectrum with positive Lorentzians are shown in blue, whereas those that contribute with negative Lorentzians are shown in red. (j)–(l) Emission spectra of the target 2LS (dashed, black lines) for the values $P_{\sigma }/{\gamma }_{\sigma }$ marked by the vertical dashed lines in (f). The total emission spectrum is made of the sum of positive (blue lines) and negative (red lines) Lorentzians. To compute the figures, we set ${\gamma }_{\sigma }$ as the unit, and set ${\gamma }_{\xi }/{\gamma }_{\sigma }=0.1$ in (a)–(c), ${\gamma }_{\xi }/{\gamma }_{\sigma }=1$ in (d)–(f), and ${\gamma }_{\xi }/{\gamma }_{\sigma }=10$ in (g)–(i).

Figure 4

Energy structure of a 2LS driven by a single-photon source. (a) Energy change due to the allowed transitions in the 2LS. (b) Linewidth of the emission of the 2LS for the allowed transitions. The transitions that involved the dressed states are shown by the dashed red and dashed blue lines. (a) Shows a splitting in the transition energies, while (b) shows that those transitions have the same linewidth. (c) The energy structure of the 2LS driven by a single-photon source is reconstructed from the allowed transitions in (a).

Figure 5

Excitation by a Mollow triplet. (a)–(i) Energy spectra of the target 2LS. The energies that contribute to the total emission spectrum with positive Lorentzians are shown in blue, whereas those that contribute with negative Lorentzians are shown in red. (j)–(l) Emission spectra of the target 2LS (dashed, black lines) for the values $\mathrm{\Omega }/{\gamma }_{\sigma }$ marked by the vertical dashed lines in (c). The total emission spectrum is made of the sum of positive (blue lines) and negative (red lines) Lorentzians. To compute the figures, we set ${\gamma }_{\sigma }$ as the unit, and set ${\gamma }_{\xi }/{\gamma }_{\sigma }=0.1$ in (a)–(c), ${\gamma }_{\xi }/{\gamma }_{\sigma }=1$ in (d)–(f), and ${\gamma }_{\xi }/{\gamma }_{\sigma }=10$ in (g)–(i).

Figure 6

Excitation by an incoherently driven cavity. (a)–(f) Energy spectra of the target 2LS. The energies that contribute to the total emission spectrum with positive Lorentzians are shown in blue, whereas those that contribute with negative Lorentzians are shown in red. (g), (h) Emission spectra of the target 2LS (dashed, black lines) for the values $P_a/{\gamma }_{\sigma }$ marked by the vertical dashed lines in (c). The total emission spectrum is made of the sum of positive (blue lines) and negative (red lines) Lorentzians. To compute the figures, we set ${\gamma }_a$ as the unit, and set ${\gamma }_{\xi }/{\gamma }_a=0.1$ in (a)–(c), and ${\gamma }_{\xi }/{\gamma }_a=1$ in (d)–(f).

Figure 7

Excitation by a coherently driven cavity. (a)–(f) Energy spectra of the target 2LS. The energies that contribute to the total emission spectrum with positive Lorentzians are shown in blue, whereas those that contribute with negative Lorentzians are shown in red. (g), (h) Emission spectra of the target 2LS (dashed, black lines) for the values $\mathrm{\Omega }/{\gamma }_{\sigma }$ marked by the vertical dashed lines in (c). The total emission spectrum is made of the sum of positive (blue lines) and negative (red lines) Lorentzians. To compute the figures, we set ${\gamma }_a$ as the unit, and set ${\gamma }_{\xi }/{\gamma }_a=0.1$ in (a)–(c), and ${\gamma }_{\xi }/{\gamma }_a=1$ in (d)–(f).

Figure 8

Dressing of the target 2LS with a one-atom laser. We compare the normalized emission spectra of a 2LS driven by the emission of a one-atom laser (solid color lines) with the normalized emission spectra of a 2LS driven coherently by a classical laser (dashed black lines). The incoherent driving of the one-atom laser increases towards the upper right corner of the figure, and makes the emission spectra of the target 2LS to change from a filter of the emission of the cavity, as is clear from the lateral peaks due to the strong coupling between the cavity and the atom inside, to the splitting of the emission line until it reaches a triplet shape equivalent to that of a coherently driven 2LS. To compute the figures, we set the atom, the cavity, and the target 2LS in resonance, and the rest of the parameters were as follows: ${\gamma }_{\xi }$ was set as the unit, ${\gamma }_{\sigma }/{\gamma }_{\xi }={10}^{-2}, {\gamma }_a/{\gamma }_{\xi }=1, g/{\gamma }_{\xi }=10$. The incoherent driving rate $P_{\sigma }$ is different for each spectrum; for the spectrum at the bottom we took the limit $P_{\sigma }/{\gamma }_{\xi }\to 0$. For the next seven lines we used the $P_{\sigma }$ such that $\mathrm{\Omega }/{\gamma }_{\xi }=\sqrt{{\gamma }_a{\gamma }_{\xi }{n}_a}/{\gamma }_{\xi }=0.5$, 1.0, 1.5, 2.0, 2.5, 3.0, and 3.5. The last spectrum, at the top right of the figure, was obtained for a cavity with ${\gamma }_a/{\gamma }_{\xi }=0.1$ and $P_{\sigma }/{\gamma }_{\xi }=20.115$, so that the population in the cavity was 100 photons. The inset shows a zoom on this last spectrum very close to the resonance of the target 2LS, to show the Rayleigh peak due to the driving by the laser.

Figure 9

Photoluminescence spectra of the two-level system driven by all the sources studied in the text. (a) A classical laser resulting in the conventional Mollow triplet. (b) An incoherent 2LS providing a single line that merely broadens. (c) A coherent 2LS providing a Mollow splitting but of weak intensity and that vanishes with increasing driving intensity. (d) A thermal cavity providing a single line whose broadening is reminiscent of the Mollow structure. (e) A one-atom laser showing the transition from the nonlasing regime of the driving source to the exact recovering of the conventional Mollow triplet. At low driving, the target echoes the structure from a Jaynes-Cummings–type coupling with the source.

Figure 10

Scheme of the cascaded single-photon sources. (a) A series of SPS is connected through a cascaded architecture. In each step of the cascade, a SPS is driven by the emission of the previous SPS. (b)–(d) Emission spectra of the initial SPS (red, dotted), the first (green, dashed), and second (blue, solid) cascade steps. In each step of the cascade, the emission line becomes narrower and more peaked around the central frequency. Panels (b) [linear scale] and (c) [log scale] were made with ${\gamma }_{\xi }/{\gamma }_{\sigma }=P_{\sigma }/{\gamma }_{\sigma }=1$. In Panel (d), the coherent driving excites all the SPS in the cascade while in Panel (e), it drives only the first SPS. In both cases, each SPS drives the next one with its fluorescent output.

Figure 11

Improvement in the $g_{\mathrm{\Gamma }}^{\left(2\right)}$ due to the cascaded scheme. Once the linewidth of the first SPS is fixed, the degree of freedom of the second SPS spans a new accessible region shown by the color shading. The antibunching provided by a incoherent SPS is enhanced by the cascaded scheme, but for a wide range of parameters it is overcome by the antibunching provided by the coherent SPS. However, by allowing all the SPS in the cascade to be driven by a coherent field, a destructive interference sets limits on the enhancement of the $g_{\mathrm{\Gamma }}^{\left(2\right)}$ at the first step. The enhancement on the second cascade is obtained by driving the cascade with only the emission of the coherent SPS. In the axis, ${\gamma }_{\sigma }$ has the dimension of inverse time and its numerical value sets the unit.

Figure 12

Getting closer to the limit: adding a cascade step to excite a harmonic oscillator pushed the border of the accessible states from the green (first cascade) to the orange (second cascade) region. Adding more steps to the cascade could take the accessible region closer to the Fock Duos limit.