Direct observation of Landau-Zener tunneling in a curved optical waveguide coupler
F. Dreisow,
1,
*
A. Szameit,
1
M. Heinrich,
1
S. Nolte,
1,2
and A. Tünnermann
1,2
1
Institute of Applied Physics, Friedrich-Schiller-University Jena, Max-Wien-Platz 1, 07743 Jena, Germany
2
Fraunhofer Institute for Applied Optics and Precision Engineering, Albert-Einstein-Strasse 7, 07745 Jena, Germany
M. Ornigotti and S. Longhi
Dipartimento di Fisica and Istituto di Fotonica e Nanotecnologie del Consiglio Nazionale delle Ricerche,
Politecnico di Milano, Piazza L da Vinci 32, I-20133 Milan, Italy
Received 9 February 2009; published 5 May 2009
An electromagnetic realization of Landau-Zener LZ tunneling is experimentally demonstrated in
femtosecond-laser written waveguide couplers with a cubically bent axis. Quantitative measurements of light
evolution inside the coupler, based on fluorescence imaging, enable to trace the detailed dynamics of the LZ
process. The experimental results are in good agreement with the theoretical LZ model for linear crossing of
energy levels with constant coupling of finite duration.
DOI: 10.1103/PhysRevA.79.055802 PACS numbers: 42.82.Et, 03.65.Xp, 33.80.Be
First introduced by Landau in the context of atomic scat-
tering processes 1 and by Zener in the study of the elec-
tronic properties of a biatomic molecule 2, the Landau-
Zener LZ transition represents a fundamental dynamical
process which occurs at the intersection of two energy levels
that repel each other. Owing to its general character, the LZ
model is encountered in different physical fields and sys-
tems. Among others, LZ transitions have been investigated
for Rydberg atoms 3, molecular nanomagnets 4,5, cold
atoms and Bose-Einstein condensates in accelerated optical
lattices 6,7, field-driven superlattices 8, current-driven Jo-
sephson junctions 9, and Cooper-pair box qubits 10. Clas-
sical analogs of LZ transitions have been also investigated,
including polarization rotation in an optical cavity 11 and
LZ tunneling of light waves in coupled waveguides 12,13.
Many extensions of the original LZ model have been studied
in the past few years as well, including the effect of different
temporal interaction profiles 14, nonlinearities 15, finite-
coupling duration effects 16, multistate dynamics 17,18,
decoherence, noise and dissipation 19,20, to name a few. In
spite of the vast literature on LZ models, direct observations
of time-resolved evolution of level occupancy during LZ
transition are very few, and mainly reported for light polar-
ization dynamics in classical optical cavities 11. Coupled
optical waveguides, on the other hand, have been recently
shown to provide an accessible laboratory system to mimic,
at a classical level, the coherent control of quantum mechani-
cal tunneling 21,22. In such structures, the fast temporal
evolution of the quantum mechanical wave function is re-
placed by spatial light propagation along the waveguides,
and occupancy probabilities in the two wells can be mea-
sured by tracing the flow of light using fluorescence imaging
or scanning optical microscopy techniques.
In this Brief Report we present spatially resolved mea-
surements of LZ tunneling dynamics for light waves in cu-
bically curved coupled waveguides fabricated with the
femtosecond-laser writing technique, which provides an op-
tical realization of the LZ model with linear energy level
crossing and with constant coupling of finite duration 13.
We consider the propagation of a monochromatic wave at
the wavelength =2 / k in an optical directional coupler of
length L made of two identical single-mode waveguides
separated by the distance d in the transverse x direction. The
propagation axis of the coupler is assumed to be weakly
curved along the paraxial propagation direction z Fig. 1a.
To mimic LZ tunneling with linear crossing of energy levels,
a cubically bent profile x
0
z for the waveguide axis is as-
sumed according to 13
*
f.dreisow@uni-jena.de
FIG. 1. Color onlinea Schematic of a cubically curved di-
rectional coupler for the observation of LZ dynamics. b Fluores-
cence measurement of curved directional coupler with spacing d
=17 m and cubic profile A = 300 m, L
1
= 31.25 mm, the arrow
indicates the propagation direction.
PHYSICAL REVIEW A 79, 055802 2009
1050-2947/2009/795/0558024 ©2009 The American Physical Society 055802-1