Synchronization and clustering in a quantum dot laser

Evgeny Viktorov

Paul MandelUniversité Libre de Bruxelles

Yann TanguyJohn Houlihan

Guillaume Huyet

National University of Ireland, Cork, Ireland

Andrei VladimirovWeierstrass 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:

lasersoptical amplifiers information storage quantum computingquantum 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 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

Seminal picture

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

Multimode lasing

We know:

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

wavelength, nm

1235 1240 1245 1250

Pow

er (

a.u)

Multimode Spectrum: equally separated optical frequencies

We measure with:

- 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 • -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

Observations:

• total output remains nearly constant

• antiphase fluctuations : perfect antisynchronization, correlation???

• "Chaos must shimmer through the veil of order“, Novalis

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:

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

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

Cross-correlation measurements: from disorder to regularity

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

I 1,Itotal

I 2I 3

time(ns)

0 200 400 600 800 1000

I 4• Simplified equations,

four-wave mixing and global coupling:

dEm

dt 1

21 i Gm 1Em

k,p

pm EkEpEk p m ,

dFm

dt P Fm 1

k

N

mk |Ek |2.

#

#

• 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?

Two types of semiconductor lasers

Quantum Well Laser

-factor 4-5

• homogemeous material

• strong carrier diffusion

Quantum Dot Laser

-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

dE j

dt 1

21 i G j nk,Ek;k 1 N 1E j

dnj

dt P1 nj nj

k

N

gjkn,E; 1 N |Ek |2

#

#

G j nk,Ek;k 1 N

gjk n,E; 1 N

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:

dE j

dt 1

21 i G j nk,Ek;k 1 N 1E j

dnj

dt P1 nj nj

k

N

gjkn,E; 1 N |Ek |2

#

#

tzj i zj azj k 1

N

|zk |2 bzj

k 1

N

zk2 czj|zj |

2 cN

k 1

N

zk |zk |2 O5

G j nk,Ek;k 1 N fnj, |Ek |2

gjk n,E; 1 N g and gkk n,E; 1 N 1

#

#

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?

tzj i jzj azj k 1

N

|zk |2 bzj

k 1

N

zk2 czj |zj |

2 cN

k 1

N

zk |zk |2 O5

Normal forms, N=5

5000 10000 15000 20000 25000 30000

50

100

150

200

29920 29930 29940 29950 29960

0.3

0.4

0.5

0.6

0.7

5000 10000 15000 20000 25000 30000

0.45

0.5

0.55

0.6

0.65

0.7

Normal forms, N=5: clustering

2000 4000 6000 8000 100000.35

0.45

0.5

0.55

0.6

0.652000 4000 6000 8000 10000

-2000

-1500

-1000

-500

9920 9940 9960 9980 10000

-2060

-2040

-2020

-2000

9920 9940 9960 9980 10000

-2005

-2000

-1995

-1990

9992 9994 9996 9998 10000

0.2

0.3

0.4

0.5

0.6

9992 9994 9996 9998 10000

0.45

0.5

0.55

0.6

0.65

conclusion

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

Thank you!