Excitation with quantum light. I. Exciting a harmonic oscillator

We present a two-part study of the excitation of an optical target by quantum light. In this first part, we introduce the problematic and address the first case of interest, that of exciting the quantum harmonic oscillator, corresponding to, e.g., a single-mode passive cavity or a noninteracting bosonic field. We introduce a mapping of the Hilbert space that allows to chart usefully the accessible regions. We then consider the quantum excitation from single-photon sources in the form of a two-level system under various regimes of (classical) pumping: incoherent, coherent, and in the Mollow triplet regime. We close this first part with an overview of the material to be covered in the subsequent work.

Figure 1

Upper part: the two main types of classical excitation. The harmonic oscillator can be implemented by a single-mode cavity, sketched here as two distributed Bragg reflectors facing each other. The SPS can be implemented by a quantum dot, sketched here as a little pyramid, as it would be grown by self-assembly [44]. (a) Incoherent excitation, typically corresponding to thermal light. (b) Coherent excitation, typically corresponding to driving the system with a laser. Lower part: upgrading the classical sources of the upper panel with SPS. (c) is the counterpart of (a), to be referred to as the “incoherent SPS.” The coherent case yields two very different types of quantum light at low and high pumping: (d) the coherent SPS and (e) the Mollow Photon Source (Mollow PS) which can emit more than one photon at a time. Other types of quantum sources and other types of targets constitute the topic of our future work.

Figure 2

Charting the Hilbert space. The shaded area shows the region where a physical quantum state exists with the corresponding joint population and antibunching. It is delimited by the boundary (11) of “Fock duos,” defined by Eq. (12). Higher-order correlators (thin dashed) set other boundaries, also delimited by Fock duos.

Figure 3

Trajectories in the Hilbert space charted by $n_a$ and $g^{\left(2\right)}$ for the states excited by an incoherent SPS, for various values of ${\gamma }_a/{\gamma }_{\sigma }$. The implicit parameter is pumping. Highlighted are the cases ${\gamma }_a/{\gamma }_{\sigma }={10}^{-1}$ (blue), 1 (green), 10 (yellow), and ${10}^2$ (red). The dark green thick envelope shows the closest one can get in this configuration to the ideal antibunching, which is zero. All the states, and only these states, above this line and below $g^{\left(2\right)}=2$, are accessible with an incoherent SPS.

Figure 4

Upper row: populations (isolines) and $g^{\left(2\right)}$ (color) of the target excited by an incoherent SPS, through (a) the cascaded coupling or (b), (c) the Hamiltonian coupling with (b) and without (c) decay of the 2LS. (d) Minimum $g^{\left(2\right)}$ that can be reached at a given population for all these cases. The color circles mark the parameters $P_{\sigma }/{\gamma }_{\sigma }$ and ${\gamma }_a/{\gamma }_{\sigma }$ at which the population isoline meets the minimum $g^{\left(2\right)}$. Lower row: same but for a coherent SPS, with (e) the cascaded coupling and (f), (g) the Hamiltonian coupling with (f) and without (g) decay of the 2LS. (h) Minimum $g^{\left(2\right)}$ that can be reached at a given population for all these cases. The color circles mark the parameters ${\mathrm{\Omega }}_{\sigma }/{\gamma }_{\sigma }$ and ${\gamma }_a/{\gamma }_{\sigma }$ at which the population isoline meets the minimum $g^{\left(2\right)}$.

Figure 5

Same as Fig. 3 but for the excitation from a coherent SPS. In (a-c) the implicit parameter is pumping and the green line shows the closest one can get from the Fock-duos limit from the case at resonance. In (d) the implicit parameter is the detuning between the source and its target. (a) Excitation at resonance. The optimum antibunching is the envelope in this configuration of the family of curves varying with ${\gamma }_a/{\gamma }_{\sigma }$. Highlighted are the cases ${\gamma }_a/{\gamma }_{\sigma }={10}^{-2}$ (blue), ${10}^{-1}$ (green), 1 (chartreuse), 10 (yellow), and ${10}^2$ (red). (b)–(d) Excitation in other configurations, namely (b) with the leapfrog processes, (c) with the satellite, and (d) as a function of frequency. Highlighted are the cases ${\gamma }_a/{\gamma }_{\sigma }={10}^{-1}$ (blue), 0.5 (chartreuse), 1 (green), 5 (orange), and 10 (red). The dashed-dotted line in panel (b) and the dashed line in panel (c) show the limit of large intensities and are merely a visual aid rather than a physical boundary. They are also shown in panel (d), along with the black solid line for the envelope of the curves at ${\omega }_a={\omega }_{\sigma }$.

Figure 6

Wigner function (a)–(c) and density matrix (d)–(f) for the states marked by black circles along the green line in Fig. 4. These states have populations $n_a=1$, 1.5, and 3 from top to bottom and all feature antibunching.

Figure 7

States accessible in the $(g^{\left(2\right)},g^{\left(3\right)})$ space by coherent SPS excitation, at resonance with the central peak (orange line) or with the leapfrog emission (red line). The dashed blue line corresponds to the “Fock duos” [Eq. (A1)]. The same result is shown in both (a) log-log and (b) linear scales.

Figure 8

Variations in the model. (a), (b) Adding a dissipative channel to the source in the cascaded coupling spoils the antibunching. (c), (d) Increasing the coupling strength $g$ in the Hamiltonian coupling (with decay in the source) enhances the antibunching until it reaches the one obtained for the Hamiltonian coupling without decay of the source (when $\tilde{g}\approx 10g$). However, the enhancement is never enough to top the antibunching obtained using the coherently driven 2LS and the cascaded coupling.

Figure 9

Summary of our results: the Hilbert space of a harmonic oscillator charted by $n_a$ (population) and $g^{\left(2\right)}$ (second-order correlation function) is accessible for any point in the blue region I (the white region is forbidden). Of this region, the wider span is achieved by exciting with a coherent SPS, covering the green region. This region breaks down into II, the region achieved at resonance between the source and the target, III, that achieved with the excitation from a satellite of the Mollow triplet and IV, that achieved with the leapfrog emission. The incoherent SPS covers a smaller region, delimited by the red dotted curve. The Hamiltonian coupling, between the two dotted-dashed yellow curves, also covers a large span of the Hilbert space and is neither included by, nor fully excluded of, the SPS area. In particular, the excitation by a SPS allows to provide a stronger antibunching at small populations where the antibunching is the largest, as compared to a Hamiltonian coupling, although it still remains some distance away from the theoretical limit.

Figure 10

Excitation with quantum light: the basic configurations we propose to study in this and future works. Paper I is the current work, exciting a harmonic oscillator (single-cavity mode) with a SPS. Paper II [53] addresses the excitation of a 2LS with a SPS. In the future we will study excited coupled light-matter systems, or polaritons, still with a SPS. Following the comprehensive treatments of these several targets under the cw excitation from a SPS, we then trade this simple quantum source for a $N$-photon emitter (bundler) and a pulsed quantum source, impinging on all the targets previously considered.