Multistability and high reflectance of a mono-layer of three-level quantum emitters with a doublet in the excited state

We study theoretically the nonlinear optical response of a mono-layer of three-level quantum emitters with a doublet in the excited state. It is shown that the layer’s response exhibits multistability. In a certain frequency range, the monolayer operates as a perfect bistable mirror. We conduct a theoretical study of the steady-state optical response of a monolayer of regularly spaced three-level quantum emitters (QEs) with a doublet in the excited state. The total (retarded) dipole-dipole interaction of QEs is taken into account. This interaction provides a positive feedback. The interplay of the latter and the immanent nonlinearity of QE’s gives rise to a multistability of the monolayer optical response. In a certain frequency range, the system operates as a nanometric bistable mirror. It is assumed that the monolayer undergoes an action of a CW external field of a Rabi amplitude Ω0 and frequency ω0, which is quasi-resonant with the QE’s allowed transitions. A constituent QE is modelled by a three-level V-type quantum system with the ground state |1⟩, and a doublet |2⟩ and |3⟩ in the excited states. The allowed optical transitions are |1⟩↔|2⟩ and |1⟩↔|3⟩. They are characterized by the transition dipole moments d21 and d31, transition frequencies ω21 and ω31, and spontaneous decay constants γ21 and γ31. The doublet is described by the splitting Δ32 and the relaxation constant γ32. The optical dynamics of a constituent QE is governed by the 3x3 density matrix ραβ (α,β = 1,2,3). The total field Ω acting on a given QE in the monolayer represents a sum of the external field Ω0 and the field produced by all others QEs in place of the given one. In this way, the total (retarded) QE-QE dipole-dipole interaction is taken into account. The nearzone (far-zone) part of the QE-QE interaction gives rise to a dynamic renormalization of the transition frequencies ω21 and ω32 (relaxation constants γ21 and γ31), depending on the population difference of corresponding transitions [1,2]. The effects are described by the constants ΔL (shift) and γR (relaxation). These parameters govern a positive feedback which is responsible for a sophisticated nonlinear optical properties of the monolayer. In Fig. 1, we present the results of the steady-state calculations performed for the case when the external field is tuned into the resonance with the transition |1⟩↔|3⟩ (Δ31 = ω31 – Corresponding author: d.bayramdurdiyev@gmail.com © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). EPJ Web of Conferences 220, 03004 (2019) https://doi.org/10.1051/epjconf/201922003004 IWQO-2019


Abstract.
We study theoretically the nonlinear optical response of a mono-layer of three-level quantum emitters with a doublet in the excited state. It is shown that the layer's response exhibits multistability. In a certain frequency range, the monolayer operates as a perfect bistable mirror.
We conduct a theoretical study of the steady-state optical response of a monolayer of regularly spaced three-level quantum emitters (QEs) with a doublet in the excited state. The total (retarded) dipole-dipole interaction of QEs is taken into account. This interaction provides a positive feedback. The interplay of the latter and the immanent nonlinearity of QE's gives rise to a multistability of the monolayer optical response. In a certain frequency range, the system operates as a nanometric bistable mirror.
It is assumed that the monolayer undergoes an action of a CW external field of a Rabi amplitude Ω0 and frequency ω0, which is quasi-resonant with the QE's allowed transitions. A constituent QE is modelled by a three-level V-type quantum system with the ground state |1⟩, and a doublet |2⟩ and |3⟩ in the excited states. The allowed optical transitions are |1⟩↔|2⟩ and |1⟩↔|3⟩. They are characterized by the transition dipole moments d21 and d31, transition frequencies ω21 and ω31, and spontaneous decay constants γ21 and γ31. The doublet is described by the splitting Δ32 and the relaxation constant γ32.
The optical dynamics of a constituent QE is governed by the 3x3 density matrix ραβ (α,β = 1,2,3). The total field Ω acting on a given QE in the monolayer represents a sum of the external field Ω0 and the field produced by all others QEs in place of the given one. In this way, the total (retarded) QE-QE dipole-dipole interaction is taken into account. The nearzone (far-zone) part of the QE-QE interaction gives rise to a dynamic renormalization of the transition frequencies ω21 and ω32 (relaxation constants γ21 and γ31), depending on the population difference of corresponding transitions [1,2]. The effects are described by the constants ΔL (shift) and γR (relaxation). These parameters govern a positive feedback which is responsible for a sophisticated nonlinear optical properties of the monolayer.
In Fig. 1, we present the results of the steady-state calculations performed for the case when the external field is tuned into the resonance with the transition |1⟩↔|3⟩ (Δ31 = ω31 -ω0 = 0). Panel (b) shows the acting field magnitude |Ω| as a function of the external field magnitude |Ω0| for the set of parameters typical for two-dimensional supercrystals built up of semiconductor quantum dots (SQD) [3]. As is seen from the plot, |Ω| appears to be a multi-valued function of |Ω0|, signaling multistability. The stability of different parts of the steady-state solution has been checked by analyzing the spectrum of Lyapunov exponents Λk (k = 1,2…8). The maximal real part of {Λk}, Max{Re[Λ]}, is plotted in panel (c).    2 shows the detuning and field dependence of the reflectance R = |Ωrefl/Ω0| 2 (left and right panels, respectively), Ωrefl = γR(ρ31 + ρ21) is the Rabi amplitude of the reflected field. As follows from the left plot, the linear reflectance (for a week |Ω0|, left panel) has a maximum at Δ31 = 2000γ31. Moreover, at this point R is approaches unity, i.e. the monolayer almost totally reflects the input field. The right panel in Fig. 2 demonstrates arising three solutions for R at a given |Ω0|, which means bistability of the reflectance.
Summarizing, we believe that a monolayer comprising V-type QEs may serve as a nanometric bistable mirror. These features might be of interest for nanophotonics. Supercrystals built up of SQDs with the degenerate valence band, e.g. CdSe, placed in magnetic field [4], can be considered as candidates for realization of such systems.