Successive switching among four states in a gain-loss-assisted optical microcavity hosting exceptional points up to order four

Arnab Laha, Dinesh Beniwal, and Somnath Ghosh

Phys. Rev. A 103, 023526 – Published 22 February 2021

Abstract

The implementation of exceptional points (EPs), a special type of topological singularities, has emerged as a new paradigm for engineering the quantum-inspired or wave-based photonic systems. Even though there exists a range of investigations on EPs of order two and three (say, EP2s and EP3s, respectively), the hosting of fourth-order EPs (EP4s) in any real system and the exploration of associated topological features are lacking. Here we have designed a simple Fabry-Pérot type gain-loss-assisted open optical microcavity to host EPs up to order four. The scattering-matrix formalism has been used to analyze the microcavity numerically. With the appropriate modulation of the gain-loss profile in the same cavity geometry, we have encountered multiple different orders of EPs by investigating the simultaneous interactions among four coupled cavity states via level-repulsion phenomena. Besides affirming the second-order and third-order branch-point behaviors of the embedded EP2s and EP3s, the fourth-order branch-point functionality of an EP4 has been manifested by encircling three connecting EP2s simultaneously in the closed gain-loss parameter space. We have established a unique successive state-switching phenomenon among four coupled states by implementing such an EP4-encirclement scheme in the system's parameter space. The proposed scheme indeed offers potential applications in state-switching and control in quantum-inspired integrated photonic circuits, where the presence of an EP4 serves as a new light manipulation tool.

Received 5 November 2020
Accepted 5 February 2021

DOI:https://doi.org/10.1103/PhysRevA.103.023526

Physics Subject Headings (PhySH)

Classical optics

Optical microcavities

Atomic, Molecular & Optical

Authors & Affiliations

Arnab Laha ^1,2, Dinesh Beniwal ³, and Somnath Ghosh ^1,*

¹Unconventional Photonics Laboratory, Department of Physics, Indian Institute of Technology Jodhpur, Rajasthan-342037, India
²Institute of Radiophysics and Electronics, University of Calcutta, Kolkata-700009, India
³Department of Physics, National Institute of Science Education and Research Bhubaneswar, Odisha 752050, India

^*somiit@rediffmail.com

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Vol. 103, Iss. 2 — February 2021

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Images

Figure 1
(a) Schematic diagram of the proposed trilayer optical microcavity, occupying the region $0 \leq x \leq L$ with $n_{g} = 1.5, n_{c} = 2.65, L = 12 μ m, L_{1} = 0.5 μ m, L_{2} = 1.5 μ m, L_{3} = 10.5 μ m$ , and $L_{4} = 11.5 μ m$ . $\{Φ_{L}^{+}, Φ_{R}^{-}\}$ represents the set of incident waves from both sides, whereas $\{Φ_{L}^{-}, Φ_{R}^{+}\}$ represents the set scattered waves. (b) Complex refractive index profile [as given in Eq. (11)] along $x$ direction, where the dotted red line represents the $Re [n (x)]$ (associated with the left $y$ axis) and the solid blue line represents the $[Im n (x)]$ (associated with the right $y$ axis). (c) Initial positions of four chosen poles, say, $P_{1}, P_{2}, P_{3}$ , and $P_{4}$ within the chosen frequency ( $k$ ) range [3.35, 3.65] (in $μ m^{- 1}$ ).
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Figure 2
Trajectories of $Re (k)$ and $Im (k)$ associated with $P_{j} (j = 1, 2, 3, 4)$ for different $τ$ values, while varying $γ$ within [0, 0.5]. Here the dotted green, red, black, and blue curves represent the trajectories of $P_{1}, P_{2}, P_{3}$ , and $P_{4}$ , respectively. (a) $P_{2}$ and $P_{3}$ exhibit ARC (unaffecting $P_{1}$ and $P_{4}$ ) with (a.1) an anticrossing in $Re (k)$ and (a.2) a simultaneous crossing in $Im (k)$ for $τ = 0.622$ , and (a.3) a crossing in $Re (k)$ and (a.4) a simultaneous anticrossing in $Im (k)$ for $τ = 0.623$ ; (a.5) $P_{2}$ and $P_{3}$ coalesce at $γ \approx 0.083$ for $τ = 0.6288$ , referring to the encounter of $EP 2^{(2, 3)}$ . (b) $P_{2}$ and $P_{4}$ exhibit ARC (unaffecting $P_{1}$ and $P_{3}$ ) with (b.1) an anticrossing in $Re (k)$ and (b.2) a simultaneous crossing in $Im (k)$ for $τ = 0.8825$ , and (b.3) a crossing in $Re (k)$ and (b.4) a simultaneous anticrossing in $Im (k)$ for $τ = 0.883$ ; (b.5) $P_{2}$ and $P_{4}$ coalesce at $γ \approx 0.389$ for $τ = 0.8827$ , referring to the encounter of $EP 2^{(2, 4)}$ . (c) $P_{1}$ and $P_{2}$ exhibit ARC (unaffecting $P_{3}$ and $P_{4}$ ) with (c.1) an anticrossing in $Re (k)$ and (c.2) a simultaneous crossing in $Im (k)$ for $τ = 2.786$ , and (c.3) a crossing in $Re (k)$ and (c.4) a simultaneous anticrossing in $Im (k)$ for $τ = 2.787$ ; (c.5) $P_{1}$ and $P_{2}$ coalesce at $γ \approx 0.332$ for $τ = 2.7866$ , referring to the encounter of $EP 2^{(1, 2)}$ . In (a.5), (b.5), and (c.5), only the trajectories of the coalescing states have been shown for a clear visualization. (The unit of $k$ : $μ m^{- 1}$ .)
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Figure 3
(a) Three parametric loops in the $(γ, τ)$ plane [following Eq. (14)] to encircle the identified EP2s individually, where Loop-1, Loop-2, and Loop-3 individually enclose $EP 2^{(1, 2)}, EP 2^{(2, 3)}$ , and $EP 2^{(2, 4)}$ , respectively. Here we consider the clockwise encirclement direction. Three crosses indicate the location of three EP2s. (b) The trajectories of $P_{j} (j = 1, 2, 3, 4)$ following the parametric encirclement process along Loop-1, exhibiting an adiabatic state-flipping $P_{1} \to P_{2} \to P_{1}$ , unaffecting $P_{3} (\to P_{3})$ and $P_{4} (\to P_{4})$ . (c) Similar trajectories of $P_{j} (j = 1, 2, 3, 4)$ following the parametric encirclement process along Loop-2, exhibiting an adiabatic state-flipping $P_{2} \to P_{3} \to P_{2}$ , unaffecting $P_{1} (\to P_{1})$ and $P_{4} (\to P_{4})$ . (d) Similar trajectories following the parametric encirclement process along Loop-3, exhibiting an adiabatic state-flipping $P_{2} \to P_{4} \to P_{2}$ , unaffecting $P_{1} (\to P_{1})$ and $P_{3} (\to P_{3})$ . In (b) and (c) the circular markers of respective colors indicate the passive locations of $P_{j}$ , i.e., while initializing the encirclement from $γ = 0$ . Arrows of respective colors indicate their progression directions. (The unit of $k$ : $μ m^{- 1}$ .)
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Figure 4
Three parametric loops in the $(γ, τ)$ plane [following Eq. (14)], where each loop individually encloses at least two connecting EP2s among three identified EP2s. Here Loop-4, Loop-5, and Loop-6 individually enclose the pairs { $EP 2^{(1, 2)}, EP 2^{(2, 3)}$ }, { $EP 2^{(2, 3)}, EP 2^{(2, 4)}$ }, and { $EP 2^{(1, 2)}, EP 2^{(2, 4)}$ }, respectively. (b) The trajectories of $P_{j} (j = 1, 2, 3, 4)$ following the clockwise parametric encirclement process along Loop-4, exhibiting an adiabatic successive state-switching process $P_{1} \to P_{2} \to P_{3} \to P_{1}$ , unaffecting $P_{4} (\to P_{4})$ . (c) Similar trajectories of $P_{j} (j = 1, 2, 3, 4)$ following the parametric encirclement process along Loop-5, exhibiting an adiabatic successive state-switching process $P_{2} \to P_{3} \to P_{4} \to P_{2}$ , unaffecting $P_{1} (\to P_{1})$ . (d) Similar trajectories following the parametric encirclement process along Loop-6, exhibiting an adiabatic successive state-switching process $P_{1} \to P_{2} \to P_{4} \to P_{1}$ , unaffecting $P_{3} (\to P_{3})$ . The notations and colors bear the same meaning as we have already described in the caption of Fig. 3. (The unit of $k$ : $μ m^{- 1}$ .)
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Figure 5
(a) The trajectories of $P_{j} (j = 1, 2, 3, 4)$ in the complex $k$ plane, following a clockwise encirclement process along Loop-7 (as shown in the inset), which encloses all three identified EP2s (i.e., $EP 2^{(1, 2)}, EP 2^{(2, 3)}$ , and $EP 2^{(2, 4)})$ simultaneously. Here they execute a successive state-switching process following the manner $P_{1} \to P_{2} \to P_{3} \to P_{4} \to P_{1}$ with the completion of the encirclement process. (b) The formation of the Riemann surfaces corresponding to $P_{j}$ as a function of $γ$ and $τ$ with the overall variation of (b.1) $Re (k)$ and (b.2) $Im (k)$ within the interaction regime. The trajectories of $P_{j}$ following the parametric encirclement process along Loop-7 have been mapped on their respective Riemann sheets for a clear understanding of the successive state transfer mechanism among four coupled poles from their corresponding surfaces. The notations and colors bear the same meaning as we have already described in the caption of Fig. 3. (The unit of $k$ : $μ m^{- 1}$ .)
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