Synchronization and clustering in a quantum dot laser Evgeny Viktorov Paul Mandel Université Libre de Bruxelles Yann Tanguy John Houlihan Guillaume Huyet National University of Ireland, Cork, Ireland Andrei Vladimirov Weierstrass Institute, Berlin, Germany

Outline Self-assembled quantum dot lasers: some properties of a different laser Multimode lasing: clustering Correlation measurements: antiphase dynamics- from disordered to « regular » switching Modeling: physical background Modeling: (non)degenerate Hopf, normal forms

Quantum dots: nanocrystalline gain medium - nanoscale islands -form spontaneously during the epitaxial growth process on a semiconductor substrate -atomlike properties - "artificial atoms" -discrete energy spectrum - 1011 dots per cm2 Applications: lasers optical amplifiers information storage quantum computing quantum cryptography

A summary of laser performance: Low threshold current < 30 A/cm2 (Huang 2000; Park 2000) Modulation characteristics: 10 Gb/s (Hatori 2004, Kuntz 2005) CW operation up to 80°C Small a-factor < 1 or as low as 0.7 (Martinez, 2005) ?? Prospects: Lowest threshold current High temperature operation Tunability High quality beam Low sensitivity to feedback Reliable lasing without filamentation and parasitic instabilities for ultrahigh-speed applications «  dream » laser

Quantum dot laser is an ensemble of independent nanolasers ??? Seminal picture Quantum dot laser is an ensemble of independent nanolasers ???

Multimode lasing We know: We measure/calculate: Quantum dots have different shapes and sizes: - strong inhomogeneous broadening and multimode lasing up to 50 modes We measure/calculate: modal oscillation frequencies modal timetraces modal correlations Hilbert phases

We measure with: Important limitation: - a high bandwidth (4 GHz) detector. an electronic spectrum analyzer a low bandwidth amplified InGaAs detector (50 MHz, Thorlabs PDA255). Important limitation: Only TWO modes can be measured simultaneously

Control parameter (pumping current) leads to increasing : number of lasing modes asymmetry in the gain profile a-factor - a global measure of the phase-amplitude coupling.

Experimental timetraces Antiphase fluctuations : strongly chaotic 40 % of the amplitude low frequency range : up to 50 MHz 50 MHz << 5GHz (relaxation oscillation frequency – timescale of field-matter interaction): Mode-to-Mode coupling???

Experimental timetraces "Chaos must shimmer through the veil of order“, Novalis Observations: total output remains nearly constant antiphase fluctuations : perfect antisynchronization, correlation???

Experimental power spectra: different modes can have different averaged frequencies of fluctuations clustering in averaged frequencies the spread of frequencies narrows with increasing current from 11 MHz to 3 MHz.

Experimental Hilbert phases: We define: two modes belong to the same cluster if the difference between two Hilbert phases is bounded

Detection noise influence two detectors, the same mode, phase difference.

Equally separated clusters?

Correlation dimension vs clustering We measure the correlation dimension of the modal signals. the modes from the central region of the spectrum have lower values of the correlation dimension, ≈2.8, and the modes at the edges have higher values, ≈3.3. the trend is similar (?) to the distribution of frequencies across the spectrum suggesting that the modes from different clusters can (?) exhibit different levels of complexity. we link the difference to the stochastic processes which govern the appearance of new lasing modes at the edges of optical spectrum (stronger influence of noise).

Cross-correlation measurements: We measure: normalized cross-correlation function of all modes with respect to a reference mode. a maximum value of 1 is expected when the two modes are identical in amplitude and phase and, therefore, perfectly correlated. the time corresponding to maximum correlations between the modes as a function of modal frequency difference between the two recorded modes Results: this time changes randomly for low currents when the spread of the frequencies is large becomes linear for the higher current when the frequency spread of the fluctuations among the modes was smaller linear dependence indicates the propagation of perturbations through the spectrum “from blue to red” - oscillations are equally phase-shifted

Cross-correlation measurements: from disorder to regularity

MODE-TO-MODE COUPLING Main Results clustering in averaged frequencies the spread of frequencies narrows oscillations can be equally phase-shifted Switching « from blue to red » MODE-TO-MODE COUPLING

Quantum Well Laser: experiment nearly constant total output similar frequency range periodic modal switching « from blue to red» originate from the Hopf bifurcation (statistical analysis) Experiments: Institut Non-lineaire de Nice, France, 2004

Quantum Well Laser: more advanced modeling dominant mechanism – four-wave mixing  large a-factor (asymmetry in phase-amplitude coupling) defines the unique sequence of switching from »blue to red» Hopf? Heteroclinic? Simplified equations, four-wave mixing and global coupling:

Two types of semiconductor lasers Quantum Well Laser a-factor 4-5 homogemeous material strong carrier diffusion Quantum Dot Laser a-factor <1, increasing with the current inhomogemeous material small carrier diffusion total output remains nearly constant antiphase fluctuations low frequency range periodic, 100 % of the amplitude the same frequency of oscillations for all modes total output remains nearly constant antiphase fluctuations low frequency range chaotic, 40 % of the amplitude different frequences of oscillations, clustering

Physical model Equations: The modal gains and the cross-coupling coefficient typically depend on four-wave-mixing processes and inhomogeneous broadening, but physical mechanisms are complex and not fully understood yet

Challenge Quantum dot laser is an ensemble of independent nanolasers ??? carrier capture and recombination in individual quantum dots are random processes so each quantum dot couples to its own excited carrier Conclusion: UNCORRELATED OUTPUT FROM THE DIFFERNET QUANTUM DOTS We assume: Modes are globally coupled Hopf bifurcation Inhomogeneous broadening (different shapes/sizes) results in different frequencies of oscillations Two main effects to describe: -frequency clustering -antiphase state

Degenerate Hopf Equations: First “good” approximation: frequency dependent parameters are equal Degenerate Hopf, normal form equations:

Hopf: nondegeneracy - the modes have different average oscillation frequencies. - we relate this non-degeneracy to the high degree of inhomogeneous broadening. weak perturbation of the linear part Phase approximation: Kuramoto, Hansel Global linear coupling do not exhibit phase clustering behavior right after Hopf bifurcation (Okuda,1993) Nonlinear coupling: frequency clustering? antiphase state?

Normal forms, N=5

Normal forms, N=5: clustering

conclusion Modal oscillations in quantum dot laser result from the global coupling and exhibit clustering and antiphase state.

Thank you!