Multistability and condensation of exciton-polaritons below threshold

Jiun-Yi Lien, Yueh-Nan Chen, Natsuko Ishida, Hong-Bin Chen, Chi-Chuan Hwang, and Franco Nori

Phys. Rev. B 91, 024511 – Published 20 January 2015

Abstract

Exciton-polaritons can condense to a macroscopic quantum state through a nonequilibrium process of pumping and decay. In recent experiments, polariton condensates are used to observe, for a short time, nonlinear Josephson phenomena by coupling two condensates. However, it is still not clear how these phenomena are affected by the pumping and decay at long times and how the coupling alters the polariton condensation. Here, we consider a polariton Josephson junction pumped on one side and study its dynamics within the mean-field theory. The Josephson current is found to give rise to multistability of the stationary states, which are sensitive to the initial conditions and incoherent noises. These states can be attributed to either the self-trapping effect or the parity-time (PT) symmetry of the system. These results can be used to explain the emission spectra and the $π$ -phase locking observed in recent experiments. We further predict that the multistability can reduce to the self-trapped state if the PT symmetry is broken. Moreover, the polaritons can condense even below the threshold, exhibiting hysteresis.

Received 16 July 2014
Revised 8 January 2015

DOI:https://doi.org/10.1103/PhysRevB.91.024511

Authors & Affiliations

Jiun-Yi Lien^1,2,*, Yueh-Nan Chen^1,†, Natsuko Ishida³, Hong-Bin Chen¹, Chi-Chuan Hwang², and Franco Nori^3,4

¹Department of Physics and Center for Theoretical Sciences, National Cheng Kung University, Tainan 701, Taiwan
²Department of Engineering Science, National Cheng Kung University, Tainan 701, Taiwan
³CEMS, RIKEN, Wako-shi, Saitama 351-0198, Japan
⁴Physics Department, University of Michigan, Ann Arbor, Michigan 48104-4313, USA

^*jiunyi.lien@gmail.com
^†yuehnan@mail.ncku.edu.tw

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Issue

Vol. 91, Iss. 2 — 1 January 2015

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Figure 1
(a) A polariton Josephson junction made of semiconducting micropillars pumped on one side. (b) Model of exciton-polariton condensation of the junction. A nonresonant continuous-wave laser is used to excite high-energy excitons as a pumping to the polariton reservoir, and a resonant pulsed laser can be used to control the initial condition.
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Figure 2
(a) Imaginary part of the excitation spectrum $ω^{(0)}$ without condensation. (b) Threshold pumping $P_{t h}$ as a function of the Josephson coupling strength $J$ with and without the detuning $Δ ɛ_{12}$ , where $P_{t h}$ is defined as the first point of $I m [ω_{\pm}^{(0)}] = 0$ in (a). The decay rates are $γ_{1} = γ_{2} = 0.1$ meV, which are close to those in the micropillar experiment, $γ_{R 1} = 0.5$ meV and $R_{1}^{'} = 0.01$ meV.
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Figure 3
Signatures of four stationary condensate states with respect to the pumping strength $P_{1}$ : (a) energy spectrum corresponding to the real eigenvalues $Ω_{+}$ or $Ω_{-}$ ; (b) reservoir-polariton number $N_{R 1}$ in response to the condensates; (c) total condensate-polariton number $N_{c T}$ ; (d) population imbalance $ζ$ ; (e) stability determined by the maximum imaginary part of the fluctuation spectrum; and (f) phase difference $Δ φ$ . These states are not necessarily orthogonal with each other due to the nonlinearity, and they can be classified into two groups. Thin solid (blue) and dash-dotted (red) curves represent the asymmetric (self-trapped and untrapped) states, and both of them are antibonding states ( $Ω_{+}$ ) with a phase difference close to $π$ . Thick solid (yellow) and long-dashed (green) curves correspond to the PT-symmetric states with $Ω_{+}$ and $Ω_{-}$ , respectively, and they coincide in (b)–(d) except for having different energy and phase. Above the threshold, two (for large pumping) or three (for small pumping) states are stable. Below the threshold there exists a hysteresis behavior between the self-trapped state and the noncondensed state, shown by the vertical short-dashed (blue) lines in (c). The threshold pumping $P_{t h} = 10$ is determined by the noncondensed state, which is shown by the black lines [not shown in (d) and (f)] and becomes unstable above the threshold. Parameters used are $ε_{1} = ε_{2} = 0$ , $U_{1} = U_{2} = 0.01$ meV, $γ_{1} = γ_{2} = 0.1$ meV, $γ_{R 1} = 0.5$ meV, $R_{1}^{'} = 0.01$ meV, and $J = 0.1$ meV.
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Figure 4
Time dependence of the imbalance $ζ (t)$ , the condensate-polariton number $N_{c T} (t)$ , and the phase difference $Δ φ (t)$ of the self-trapped state with zero initial phase difference for (a) $P_{1} = 11$ and (b) $P_{1} = 0$ (b). Insets: $π$ -phase locking for $Δ φ (t \to \infty)$ .
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Figure 5
Detuning effects of the stationary states: (a) number $N_{c 1}$ of condensate polaritons at the pumped side for stationary solutions with $R_{1} \geq γ_{1} = 0.1$ meV and (b) corresponding pumping $P_{1}$ , both as functions of the stimulated scattering $R_{1}$ for three detunings, $Δ ɛ_{12} \equiv ε_{12} + V_{12}^{R}$ . Dark gray (blue) and light gray (red) curves represent antibonding ( $Ω_{+}$ ) and bonding ( $Ω_{-}$ ) asymmetric states, respectively. The vertical solid (green) line corresponds to both the PT-symmetric states ( $Ω_{\pm}$ ) for zero detuning. These solutions make sense only for $N_{c 1} \geq 0$ and $P_{1} \geq 0$ , i.e., the upper half-plane in (a) and the area above the thin dashed line (corresponding to $P_{1} = γ_{R 1} N_{R 1}$ ) in (b). The detuning introduces hybridization and anticrossing between the PT-symmetric and the asymmetric states and changes the dynamics for small pumping. However, increasing the pumping strength would restore the zero-detuning behavior, i.e., the bistability of the self-trapped state and the PT-symmetric bonding state.
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Figure 6
Simulation of the spectrally resolved emission distributions derived from stable states (a) close to the threshold (with $P_{1} = 11$ ) and (b) at high pumping (with $P_{1} = 50$ and an additional potential difference $Δ ɛ_{12} = 1$ meV induced by the incoherent polaritons and high-energy excitons). Micropillars are centered at $x_{1 / 2} = \mp 5 μ m$ , with radii of 5 $μ m$ . The emission strengths and distributions of the $n th$ stable state $| Ψ_{n} {(x, Ω) |}^{2} = {| Ψ_{1 n} (x - x_{1}, Ω - Ω_{n}) + Ψ_{2 n} (x - x_{2}, Ω - Ω_{n}) |}^{2}$ are determined by its energy $Ω_{n}$ , particle numbers, population imbalance, and phase difference, where the spectral and spatial distributions on each pillar $Ψ_{j n} (x - x_{j}, Ω - Ω_{n})$ are defined by Gaussian functions. The three states in (a), from bottom to top, are the PT-symmetric $Ω_{-}$ state, the PT-symmetric $Ω_{+}$ state, and the self-trapped state. In (b), the PT-symmetric $Ω_{+}$ state becomes unstable and only two states are left.
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