1. 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

2. 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

3. 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
dots per cm2
Applications:
lasers
optical amplifiers
information storage
quantum computing
quantum cryptography

4. 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

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

6. 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

7. 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

8. 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.

9. 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???

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

11. 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.

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

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

14. Equally separated clusters?

15. 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).

16. 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

17. Cross-correlation measurements: from disorder to regularity

18. 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

19. 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

20. 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:

21. 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

22. 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

23. 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

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

25. 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?

26. Normal forms, N=5

27. Normal forms, N=5: clustering

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

29. Thank you!