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Visualization of two-photon Rabi oscillations in evanescently coupled optical waveguides
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2008 J. Phys. B: At. Mol. Opt. Phys. 41 085402
(http://iopscience.iop.org/0953-4075/41/8/085402)
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IOP PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS
J. Phys. B: At. Mol. Opt. Phys. 41 (2008) 085402 (6pp) doi:10.1088/0953-4075/41/8/085402
Visualization of two-photon Rabi
oscillations in evanescently coupled
optical waveguides
M Ornigotti1, G Della Valle1, T Toney Fernandez1, A Coppa2, V Foglietti2,
P Laporta1and S Longhi1
1Dipartimento di Fisica and Istituto di Fotonica e Nanotecnologie del CNR, Politecnico di Milano,
Piazza L. da Vinci 32, I-20133 Milano, Italy
2Istituto di Fotonica e Nanotecnologie del CNR, sezione di Roma, Via Cineto Romano 42, 00156 Roma,
Italy
E-mail: longhi@fisi.polimi.it
Received 10 December 2007, in final form 12 March 2008
Published 4 April 2008
Online at stacks.iop.org/JPhysB/41/085402
Abstract
An optical analogue of two-photon Rabi oscillations, occurring in a three-level atomic or
molecular system coherently driven by two detuned laser fields, is theoretically proposed and
experimentally demonstrated using three evanescently coupled optical waveguides realized on
an active glass substrate. The optical analogue stems from the formal analogy between spatial
propagation of light waves in the three-waveguide structure and the coherent temporal
evolution of populations in a three-level atomic medium driven by two laser fields under
two-photon resonance. In our optical experiment, two-photon Rabi oscillations are thus
visualized as a slow spatial oscillatory exchange of light power between the two outer
waveguides of the structure with a small excitation of the central waveguide.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
The coherent evolution of a two-level system subjected to a
harmonic perturbation, such as the coherent interaction of a
two-level atom with a resonant optical laser radiation or the
dynamics of driven superconducting qubit systems, is a basic
process in quantum physics. A rather universal feature of
driven two-level systems is an oscillatory behaviour in the
populations of the two levels, a phenomenon which is usually
referred to as a Rabi oscillation (see, for instance, [1]). Simple
Rabi oscillations in level populations can be observed even
in coherent excitation of multilevel systems via multiphoton
processes, and a rather vast literature exists on this subject (see,
for instance, [2–14] and references therein). The simplest case
is that of a three-level system |1,|2and |3with forbidden
single-photon transition between levels |1and |3.Ifthe
system is driven by two fields nearly resonant with the single-
photon allowed transitions |1↔|2and |2↔|3,itis
possible to excite two-photon transitions between levels |1
and |3using the off-resonance level |2as the intermediate
state. The detuning of the field frequencies with single
photon transitions, involving the intermediate state |2, should
be sufficiently small to enhance the transition probability
significantly, but large enough to avoid populating the
intermediate state |2. If the two-photon resonance condition is
satisfied, the three-level system behaves approximately like a
two-level system involving only levels |1and |3, with a slow
periodic oscillation of the populations of these two levels (two-
photon Rabi oscillations) while the occupation probability
of the intermediate level oscillates rapidly with very small
amplitude (see, for instance, [6,10,11]). Two-photon Rabi
oscillations have been experimentally observed, among others,
in three-level nuclear magnetic resonance experiments [3,6],
in microwave transitions between Rydberg states of calcium
[13] and between rotovibrational levels of SF6[14].
In a different physical context, the exploitation of suitable
analogies between quantum mechanics and wave optics
(see [15–19] and references therein) has allowed, on many
occasions, the observation at a macroscopic level of the optical
analogues of quantum dynamical phenomena encountered in
0953-4075/08/085402+06$30.00 1© 2008 IOP Publishing Ltd Printed in the UK
J. Phys. B: At. Mol. Opt. Phys. 41 (2008) 085402 M Ornigotti et al
solid-state or atomic and molecular physical systems. For
instance, the optical analogues of Bloch oscillations [20–25],
Zener tunnelling [24,26], dynamic localization [27,28],
coherent enhancement and destruction of tunnelling [29,30],
adiabatic stabilization of atoms in strong fields [31]and
Anderson localization [32] have been recently demonstrated
using engineered photonic structures. Optical tunnelling
among two or three evanescently coupled optical waveguides
has also provided a very appealing laboratory tool for a
direct visualization of some basic dynamical effects typically
encountered in the coherent interaction of laser fields with two-
level or three-level atomic or molecular systems, including
the optical analogues of stimulated Raman adiabatic passage
and electromagnetically induced transparency [33–37], level
crossing and Landau–Zener dynamics [38], Raman chirped
adiabatic passage [39], and multiphoton transitions in strongly
driven two-level systems with permanent dipole moments [40].
In this work we report on the experimental observation
of two-photon Rabi oscillations for light waves in three
evanescently coupled optical waveguides which provide the
optical analogue of two-photon Rabi oscillations in a three-
level atomic or molecular system. In our optical system the
visualization of the flow of light along the three-waveguide
structure using fluorescence imaging allows us to mimic
two-photon Rabi oscillations as a slow oscillatory transfer
of light power between the two outer waveguides of the
structure with small excitation of the central waveguide. The
paper is organized as follows. In section 2the basic model
of light transport in a three-waveguide optical structure is
presented and its analogy with two-photon Rabi oscillations
is highlighted. Section 3presents the experimental results
on light dynamics in a few waveguide structures realized on
an active glass substrate, and the occurrence of the optical
analogue of two-photon Rabi oscillations is demonstrated.
Finally, in section 4the main conclusions are outlined.
2. Optical analogue of two-photon Rabi oscillations:
theory
2.1. Light transfer dynamics in three evanescently coupled
optical waveguides: basic model
The starting point of our analysis is provided by a rather
standard model describing propagation of monochromatic
light waves in a waveguide structure composed by three
straight and parallel channel waveguides of length L, placed at
a distance d0each other as shown schematically in figure 1(a).
We assume that the fundamental modes of the outer
waveguides |1and |3have the same propagation constant,
whereas the fundamental mode of the central waveguide |2
has a propagation constant shift σ. In practice, as shown
in figure 1(a) such a condition can be realized by designing
the outer waveguides with the same channel width w1=
w3, whereas the central waveguide has a slightly larger (or
narrower) channel width w2. Indicating by c1(z), c2(z) and
c3(z) the amplitudes of the fundamental modes in each of the
three waveguides at plane z, in the weak coupling limit the
power transfer of light waves among the three waveguides is
d0d0
0
1>
w1w2w3
2> 3>
z
0
L
x
E
E2
E3
E11>
2>
3>
a
b
(a) (b)
0
Figure 1. (a) Schematic of the three-waveguide structure designed
for the experimental demonstration of the optical analogue of
two-photon Rabi oscillations, and (b) the corresponding three-level
atomic system driven by two laser fields. a,b are the Rabi
frequencies of the driving fields nearly resonant with the
electric-dipole-allowed transitions |1↔|2and |2↔|3;σis the
one-photon detuning. The two-photon resonance condition
ω21 −ωa=ω23 −ωb=σis assumed.
described by the following coupled-mode equations (see, for
instance, [41]),
id
dz⎛
⎝
c1
c2
c3
⎞
⎠=⎛
⎝
000
0σ
0
000
⎞
⎠⎛
⎝
c1
c2
c3
⎞
⎠,(1)
where 0is the coupling rate between the adjacent waveguides
|1,|2and |2,|3. In their present form, the optical
coupled-mode equations (1) are analogous to the Schr¨
odinger
equation of a three-level atomic or molecular medium,
projected onto the bare states |1,|2and |3of the atom or
molecule, interacting with two continuous-wave laser fields at
frequencies ωaand ωbnearly resonant with the two electric-
dipole-allowed transitions |1–|2and |2–|3, respectively,
with equal Rabi frequencies a=b=20and under the
two-photon resonance condition ωa−ω21 =ωb−ω23,where
ω21 =(E2−E1)/¯hand ω23 =(E2−E3)/¯hare the atomic
transition frequencies (see figure 1(b)). In the corresponding
atomic problem, the detuning σentering in equation (1)
corresponds to the one-photon detuning σ=ω21 −ωa=
ω23 −ωb, whereas the spatial propagation distance zalong
the waveguides plays the same role as time. Owing to such a
formal analogy, the dynamics of light power transfer among
the three waveguides exactly mimics the population evolution
in the three-level atomic system driven by the two continuous-
wave laser fields in the two-photon resonance regime.
2.2. Two-photon Rabi oscillations
In the atomic physics context, it is known that for a three-level
atomic or molecular system driven by two laser fields it is
possible to excite two-photon transitions between levels |1
and |3using the off-resonance level |2as the intermediate
state. The detuning σof the field frequencies with single
photon transitions, involving the intermediate state |2, should
be sufficiently small to enhance the transition probability
significantly, but large enough to avoid populating the
intermediate state |2. If the two-photon resonance condition
is satisfied, the three-level system behaves approximately like
a two-level system involving only levels |1and |3, with
2
J. Phys. B: At. Mol. Opt. Phys. 41 (2008) 085402 M Ornigotti et al
a slow periodic oscillation of the populations of these two
levels (two-photon Rabi oscillations) while the occupation
probability of the intermediate level oscillates rapidly with
very small amplitude (see, for instance, [6,10,11]). This
can be readily shown by considering the exact solution for
the populations |c1(z)|2,|c2(z)|2and |c3(z)|2in the three
levels as derived from equation (1) with the initial condition
c1(0)=1,c
2(0)=c3(0)=0,
|c2(z)|2=2
0
M2sin2(Mz) (2)
|c3(z)|2=1
41+cos
2(Mz) −2cosσz
2cos(Mz)
−σ
Msin σz
2sin(Mz) +σ2
4M2sin2(Mz)(3)
|c1(z)|2=1−|c2(z)|2−|c3(z)|2,(4)
wherewehaveset
M=22
0+σ2
4.(5)
In the large detuning limit |σ|0, the solution given
by equations (2)–(4) can be expanded in series of the
small parameter =0/σ , yielding at leading order the
approximate relations
|c1(z)|2=cos2(Rz) +O(2)(6)
|c2(z)|2=2
0
M2sin2(Mz) (7)
|c3(z)|2=sin2(Rz) +O(2), (8)
wherewehaveset
R=−
σ
4+1
2σ2
4+22
0≈2
0
σ.(9)
Note that in this approximation the populations of levels |1
and |3periodically oscillate between ∼0and∼1 with an
effective two-level Rabi frequency Rabi =2R(Ris given
by equation (9)), whereas the population in the intermediate
level |2remains small (of the order ∼2). The optical
analogue of such two-photon Rabi oscillations corresponds
to a sinusoidal exchange of light power between the two
outer waveguides, with small excitation of the intermediate
waveguide. Note that, owing to the finite length Lof the
waveguides, an observation of such a light transfer dynamics
requires a careful design of the coupling strength 0and
detuning σso as to satisfy the large one-photon detuning limit
|σ|0but yet keeping the spatial Rabi period LR=π/R
to a value smaller than the waveguide length L.
3. Optical analogue of two-photon Rabi oscillations:
experimental results
3.1. Waveguide fabrication and experimental setup
In our experiment, the waveguides were fabricated using
the Ag–Na ion exchange technique in a phosphate glass
Sample
System
Microscope
Translational
Stage
z
x
980-nm
Figure 2. Experimental setup for the visualization of light transfer
dynamics in a triplet of evanescently coupled waveguides.
substrate (Schott IOG1) doped with erbium and ytterbium and
containing about 20% in weight of Na2O to allow waveguide
fabrication (see, for instance, [42] and references therein). A
set of three parallel and straight channel waveguides of length
L=30 mm was fabricated using a single step ion diffusion
process in which a titanium mask with three linear apertures,
of width w1=3.5µm, w2=4.5µmandw3=w1=3.5µm,
was deposited on the surface of the rectangular IOG-1 glass
sample. The sample was then dipped for 3 min into a molten
salts bath (8% AgNO3, 5.25% NaNO3, 86.75% KNO3)at
330 ◦C inside a temperature controlled resistive furnace.
During this time Ag+ions diffuse inside the glass throughout
the apertures of the mask, replacing Na+ions, and producing
a localized refractive index increase. With the fabrication
parameters reported above, we obtained an absolute refractive
index change of n 0.0124 at the surface. Three different
sets of waveguides were manufactured corresponding to three
different values of waveguide spacing d0(see figure 1(a)),
namely d0=9.6µm,d
0=10.4µmandd0=11.4µm. Note
that, as the channel width unbalance |w2−w1,3|determines
the propagation constant shift σentering in equation (1), a
change in waveguide spacing d0basically corresponds to a
change of the tunnelling rate 0between adjacent waveguides.
In order to accurately estimate the values of σand of 0
for the three sets of triple waveguides, we also fabricated
on the same sample three asymmetric couples of straight
waveguides (i.e. three asymmetric directional couplers) with
the same parameters as the triplets (in particular with the same
channel widths w1,w
2and three different values of waveguide
separation d0). The estimated values of σand 0, retrieved
from fluorescence measurements as discussed e.g. in [30], turn
out to be σ15.2mm
−1and 02mm
−1,
01.5mm
−1
and 00.98 mm−1for the three distances d0=9.6µm,
d0=10.4µmandd0=11.4µm, respectively.
The experimental setup for our measurements is
schematically shown in figure 2. A fibre-coupled probing
laser radiation at 980 nm emitted by a semiconductor laser is
delivered into the sample and focused on its input polished
facet to excite the fundamental mode of one of the lateral
waveguides. The probing light is partially absorbed by the
3
J. Phys. B: At. Mol. Opt. Phys. 41 (2008) 085402 M Ornigotti et al
(b)
0.5
1
0
z(mm)
510 15 20 25 30
0
0.5
1
00
z(mm)
510 15 20 25 30
0510 15 20 25 30
x)m(
-50
0
50
25
-25
z(mm)
(a)
||c1
2
||c2
2
||c3
2||c1
2||c3
2
||c2
2
(c)
Figure 3. (a) Measured fluorescence image for the structure with d0=9.6µm. (b) Fractional light power trapped in the three waveguides
versus propagation distance. (c) Theoretical behaviour of normalized modal powers |c1(z)|2,|c2(z)|2and |c3(z)|2as given by
equations (2)–(4).
0
z(mm)
510 15 20 25 30
x)m(
-50
0
50
25
-25
0
z(mm)
510 15 20 25 30
0
0.5
1
(a)
(b)
0
z(mm)
510 15 20 25 30
0
0.5
1
||c1
2
||c2
2
||c3
2||c1
2
||c2
2
||c3
2
(c)
Figure 4. Same as figure 3, but for waveguide separation d0=10.4µm.
Yb3+ ion (absorption length of ∼6 mm) and generates a
green up-conversion fluorescence arising from the radiative
decay of high-energy levels of Er3+ ions, excited by resonant
energy transfer via Yb3+. The fluorescence pattern is recorded
by using a CCD camera connected with a microscope and
moved along the propagation direction zby a computer-
controlled micro-positioning system. This setup provides a
high resolution over a huge scanning area in the (x, z) plane,
thus allowing an accurate mapping of the flow of light along
the structure.
3.2. Experimental results
The experimental results of light propagation in the three
sets of triplet waveguides, showing the measured fluorescence
patterns for the three different values of waveguide spacing
d0,areshowninfigures3(a), 4(a) and 5(a). A digital analysis
of the fluorescence images was also performed to retrieve
quantitative data about the fractional light power trapped in
each waveguide versus propagation distance. The results
of the analysis are depicted in figures 3(b), 4(b) and 5(b),
4
J. Phys. B: At. Mol. Opt. Phys. 41 (2008) 085402 M Ornigotti et al
x)m(
0
z(mm)
510 15 20 25 30
-50
0
50
25
-25
(a)
0
z(mm)
510 15 20 25 30
0.5
1
0
(b)
||c1
2
||c2
2
||c3
2
0.5
1
0
0
z(mm)
510 15 20 25 30
||c1
2
||c2
2
||c3
2
(c)
Figure 5. Same as figure 3, but for waveguide separation d0=11.6µm.
together with the theoretical behaviour of |c1(z)|2,|c2(z)|2and
|c3(z)|2predicted by solutions (2)–(4) of the coupled-mode
equations (see figures 3(c), 4(c) and 5(c)). The experimental
measurements clearly show that an almost complete periodic
exchange of light between the outer waveguides, with a small
excitation of the central waveguide, occurs. Such oscillatory
behaviour of light power along the propagation distance
represents the spatial analogue of temporal two-photon Rabi
oscillations in the atomic system of figure 1(b), as discussed
in the previous section. As compared to the theoretical
predictions, the oscillations in the experimental measurements
turn out to be imperfect and slightly damped, a result which
may be ascribed to radiation losses and local propagation
constant mismatch induced by imperfections and roughness of
waveguide channels. Note that, as the waveguide spacing d0
increases and waveguide coupling 0decreases (from figure 3
to figure 5), the spatial period of Rabi oscillations LRincreases
and, correspondingly, excitation of the intermediate waveguide
decreases according to equations (7)and(9). In particular,
the measured spatial period Lmeas
Rof the oscillations versus
coupling strength 0turns out to be in good agreement with
the theoretical value Lth
R=π/R≈πσ/2
0: for the case of
figure 3one has Lmeas
R≈12.5mmandLth
R≈12.4 mm; for the
case of figure 4one has Lmeas
R≈22.4mmandLth
R≈22.5 mm;
finally, for the case of figure 5one can estimate Lmeas
R≈
46 mm and Lth
R≈48 mm.
4. Conclusions
In conclusion, an experimental demonstration of an optical
analogue of two-photon Rabi oscillations in an engineered
triplet waveguide structure has been reported. In our optical
system, the temporal Rabi oscillations in the populations of
a three-level atomic system driven by two laser fields are
mimicked by a spatial oscillatory exchange of light power
in the waveguides which is visualized in our experiment using
a fluorescence imaging technique. By varying the channel
width of the central waveguide and the separation distance of
the outer waveguides, we were able to control the spatial period
LRof the Rabi oscillations and experimentally confirmed the
dependence of LRon the coupling strength 0.Weenvisage
that our results, besides providing an additional demonstration
of the use of wave optics for studies of the classical analogues
of quantum effects, may suggest new ideas for the control of
the flow of light in engineered photonic structures borrowing
concepts and methods from quantum physics.
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