Emergence of microfrequency comb via limit cycles in dissipatively coupled condensates

Seonghoon Kim, Yuri G. Rubo, Timothy C. H. Liew, Sebastian Brodbeck, Christian Schneider, Sven Höfling, and Hui Deng

Phys. Rev. B 101, 085302 – Published 18 February 2020

Abstract

Self-sustained oscillations, limit cycles, are a fundamental phenomenon unique to nonlinear dynamic systems of high-dimensional phase space. They enable understanding of a wide range of cyclic processes in natural, social, and engineering systems. Here we show that limit cycles form in coupled polariton cavities following the breaking of Josephson coupling, leading to frequency-comb emission. The limit cycles and destruction of Josephson coupling both appear due to interplay between strong polariton-polariton interaction and a dissipative contribution to the cavity coupling. The resulting nonlinear dynamics of the condensates is characterized by asymmetric population distribution and nontrivial average phase difference between the two condensates, and by time-periodic modulation of their amplitudes and phases. The latter is manifested by coherent emission of new equidistant frequency components. The emission spectrum resembles that of a microfrequency comb, but originates from a fundamentally different mechanism than that of existing frequency combs. It allows nonresonant excitation with a power input much below the conventional semiconductor laser threshold. The comb line spacing is determined by the interaction and coupling strengths, and is adjustable up to multiterahertz frequency. The work establishes coupled polariton cavities as an experimental platform for rich nonlinear dynamic phenomena.

Received 26 February 2019
Revised 3 December 2019
Accepted 4 February 2020

DOI:https://doi.org/10.1103/PhysRevB.101.085302

Physics Subject Headings (PhySH)

Exciton polariton Frequency combs & self-phase locking Nonlinear optics Open quantum systems Quantum optics

Coupled oscillators Polariton condensate

Chaos & nonlinear dynamics

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & OpticalNonlinear Dynamics

Authors & Affiliations

Seonghoon Kim¹, Yuri G. Rubo ², Timothy C. H. Liew ³, Sebastian Brodbeck⁴, Christian Schneider⁴, Sven Höfling^4,5, and Hui Deng ^6,*

¹Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, Michigan 48109, USA
²Instituto de Energías Renovables, Universidad Nacional Autónoma de México, Temixco, Morelos 62580, Mexico
³Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore
⁴Technische Physik, Universität Würzburg, Am Hubland, Würzburg 97074, Germany
⁵SUPA, School of Physics and Astronomy, University of St Andrews, St Andrews KY16 9SS, United Kingdom
⁶Department of Physics, University of Michigan, 450 Church Street, Ann Arbor, Michigan 48109, USA

^*dengh@umich.edu

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Issue

Vol. 101, Iss. 8 — 15 February 2020

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Images

Figure 1
Bifurcation diagram and the sample properties at low excitation powers. (a),(b) Bifurcation diagrams of Eq. (1) in $P - γ$ parameter space for $J / Γ = 0.07$ , 2.5, respectively. For both diagrams, $α / Γ = 0.25, μ / Γ = 0.05$ . A standard lasing threshold in the absence of dissipative coupling is at $P = Γ$ . For certain values of $γ > 0$ , thresholds for stable and unstable fixed-point solutions emerge as $P$ increases, indicated by arrows for $γ / Γ = 0.1$ . (c) A schematic of the sample structure with a bent SWG mirror. Bending of the SWG is simulated by COMSOL and shown both by the color map in the schematic view and in the side view. The bending is less where there are open slots in the tethering pattern and vice versa. (d), The real-space photoluminescence (PL) spectrum of the coupled polariton system showing the discrete polariton states at low excitation powers. Color represents PL intensity. The lowest-energy single-particle state of LPs is the bonding state (B state), while the first excited single-particle state of LPs is the antibonding state (A state). The next excited single-particle state of LPs is labeled as the E state. The white dashed line illustrates the potential due to the bending of the SWG shown in (c). (e) The corresponding Fourier space spectrum showing the B state at $k = 0$ and the A state at $k = \pm π / a$ , where $a$ is the distance between the two coupled sites.
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Figure 2
Showing the time dependence of the site occupation numbers in the limit cycle regime, for $J / Γ = γ / Γ = 0.15, α / μ = 5$ , and pumpings (a) $p = 0.7 Γ$ , (b) $p = 0.76 Γ$ , (c) $p = 0.8 Γ$ . The positions and intensities of corresponding spectral lines are shown in the insets.
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Figure 3
Excitation power dependence of the intensity and real-space spectra of the polariton PL near the A and B states. (a) Bottom: Mean polariton number of the A and B states vs the excitation power for the L site (blue square), R site (red circle), and the sum of the L and R sites (black diamond). It shows clearly a threshold behavior and degenerate occupation at each site. Top: Relative fraction of the B state (blue plus) and A state (red cross) population vs the excitation power, showing switching of the dominant state upon transitions to stable weak lasing near A state followed by the limit cycles, and to stable lasing in the B state. (b)–(e) The real-space spectra at four different excitation powers as marked in (a), showing the transition from weak lasing, to limited cycles, toward B state lasing. The white dashed line marks the E state—the next lowest energy state above the B and A states. (f)–(i) Spectrum of each site obtained from (b) to (e), respectively. The solid lines are fits by equidistant Lorentzians.
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Figure 4
Experimental (a)–(c) and theoretical (d)–(f) $g^{(1)} (τ)$ for three different excitation powers, corresponding to Figs. 3–3, respectively. Red dots are shifted vertically by 0.1 for better visibility. The inset shows a zoom in of the revival peaks in (a). The lines are guides to the eye. The blue and red arrows indicate experimentally measured beating peaks and corresponding peaks in simulations for site L and R, respectively. The site with more frequency lines has more revival peaks and narrower $g^{(1)} (τ)$ linewidths at $τ = 0$ . The simulated $g^{(1)} (τ)$ is multiplied by an exponential decay with decay times of 20, 25, 30 ps respectively.
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Figure 5
Relative phase measurement between L and R sites at excitation powers of 2 mW (a),(d), 2.3 mW (b),(e), and 2.5 mW (c),(f). (a)–(c) Interference images from interfering both sites to a magnified single site. (d)–(f) Interference fringes for each site obtained from (a)–(c) along $x = \pm 3 μ m$ (dots). The solid lines are fits as described in the text. From the fit, we obtain the relative phase difference of $0.51 \pm 0.08 π, 0.21 \pm 0.06 π$ , and $0.15 \pm 0.04 π$ respectively.
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