Slowing and stopping light with an optomechanical crystal
array
D.E. Chang
∗
, A.H. Safavi-Naeini
†
, M. Hafezi
∗∗
and O. Painter
†
∗
Institute for Quantum Information, California Institute of Technology, Pasadena, CA 91125
†
Thomas J. Watson, Sr., Laboratory of Applied Physics, California Institute of Technology, Pasadena, CA 91125
∗∗
Joint Quantum Institute and Department of Physics, University of Maryland, College Park, MD 20742
Abstract.
The ability to coherently store and retrieve optical information in a rapidly tunable manner is an important
ingredient for all-optical information processing. In the classical domain, this optical buffering is necessary to manage
information flow in complex networks. In quantum information processing, such a system can also serve as a long-term
memory capable of storing the full quantum information contained in an optical pulse. Here we suggest a novel approach
to light storage involving an optical waveguide coupled to an optomechanical crystal array, where light in the waveguide
can be dynamically and reversibly mapped into long-lived mechanical vibrations in the array. This technique enables large
bandwidths and long storage and delay times in a compact, on-chip platform.
Keywords:
optomechanics, nanomechanics, slow light
PACS:
07.10.Cm,42.50.Wk,42.50.Ex
INTRODUCTION
A number of schemes to coherently delay and store
optical information are being actively explored. These
range from tunable coupled resonator optical waveg-
uide (CROW) structures [1, 2], where the propagation
of light is dynamically altered by modulating the refrac-
tive index of the system, to electromagnetically induced
transparency (EIT) in atomic media [3, 4], where the
optical pulse is reversibly mapped into internal atomic
degrees of freedom. While these schemes have been
demonstrated in a number of remarkable experiments [5,
6, 7, 8], they remain difficult to implement in a practical
setting.
Here, we present a novel approach to store or stop an
optical pulse propagating through a waveguide, wherein
coupling between the waveguide and a nearby nano-
mechanical resonator array enables one to map the op-
tical field into long-lived mechanical excitations. Our
scheme combines many of the best attributes of pre-
viously proposed approaches, in that it simultaneously
allows for large bandwidths of operation, on-chip inte-
gration, relatively long delay/storage times, and ease of
external control. The possibility of observing quantum
behavior in nano- and opto-mechanical systems has at-
tracted considerable interest in recent years [9]. Beyond
fundamental interest, our present work opens up the pos-
sibility of a novel major application for such systems –
quantum or classical all-optical information processing.
DESCRIPTION OF SYSTEM: AN
OPTOMECHANICAL CRYSTAL ARRAY
An optomechanical crystal [10] is a periodic structure
that constitutes both a photonic [11] and a phononic [12]
crystal. The ability to engineer optical and mechanical
properties in the same structure should enable unprece-
dented control over light-matter interactions. Planar two-
dimensional (2D) photonic crystals, formed from pat-
terned thin dielectric films on the surface of a microchip,
have been succesfully employed as nanoscale optical cir-
cuits capable of efficiently routing, diffracting, and trap-
ping light. Fabrication techniques for such 2D photonic
crystals have matured significantly over the last decade,
with experiments on a Si chip [13] demonstrating excel-
lent optical transmission through long (
N
>
100
) linear
arrays of coupled photonic crystal cavities. In a similar
Si chip platform it has recently been shown that suit-
ably designed photonic crystal cavities also contain lo-
calized acoustic resonances which are strongly coupled
to the optical field via radiation pressure [10]. These pla-
nar optomechanical crystals (OMCs) are thus a natural
candidate for implementation of our proposed slow-light
scheme, although the phenomenon is quite general to any
array of optomechanical systems.
In the following we consider an optomechanical crys-
tal containing a periodic array of such defect cavities (see
Fig. 1a). Each element of the array contains two opti-
cal cavity modes (denoted
1
,
2
) and a co-localized me-
chanical resonance (denoted
m
). The modes
2
are classi-
cally pumped, and the frequencies are chosen such that
ω
1
=
ω
2
+
ω
m
. The pump photons facilitate Raman scat-
13
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FIGURE 1.
a) Illustration of a double optical cavity system forming the unit cell of the optomechanical array. A two-way optical
waveguide is coupled to a pair of optical cavity modes
a
1
and
a
2
, whose resonance frequencies differ by the frequency of the
mechanical mode
b
. Both optical modes leak energy into the waveguide at a rate
κ
ex
and have an inherent decay rate
κ
in
. The
mechanical resonator optomechanically couples the two optical resonances with a cross coupling rate of
h
. b) The band structure
of the system, for a range of driving strengths between
Ω
m
=
0
and
Ω
m
=
κ
. The blue shaded regions indicate band gaps, while
the color of the bands elucidates the fractional occupation (red for energy in the optical waveguide, green for the optical cavity, and
blue for mechanical excitations). The dynamic compression of the bandwidth is clearly visible as
Ω
m
→
0
. c) Band structure for
the case
Ω
m
=
κ
/
10
is shown in greater detail. d) The fractional occupation for each band in c) is plotted separately. It can be seen
that the polaritonic slow-light band is mostly mechanical in nature, with a small mixing with the waveguide modes and negligible
mixing with the optical cavity mode. Zoom-ins of figures c) and d) are shown in e) and f).
tering events where a photon in mode
1
is destroyed and
a phonon is created, or vice versa. The Rabi frequency
Ω
m
(
t
) =
h
α
2
(
t
)
of this process depends on the intra-
cavity field ampiltude of mode
2
(which is readily tun-
able) and the per-phonon cross-coupling strength
h
be-
tween modes
1
and
2
. In addition, the cavity modes
1
are coupled to a common two-way waveguide. The tun-
able coupling between phonons and photons is the key
mechanism to reversibly map propagating fields in the
waveguide (through mode
1
) to mechanical excitations
along the array. This Raman scattering process is a well-
known feature of optomechanical systems. For instance,
this mechanism has been suggested as a way to opti-
cally cool the mechanical motion to its quantum ground
state [14, 15, 16, 17, 18, 9]. Here, we show that the op-
tomechanical interaction not only facilitates cooling, but
also yields a rapidly varying susceptibility for an incom-
ing optical field in the waveguide, which in turn enables
a tunable slow group velocity.
The design considerations for the optomechani-
cal crystal are described in detail below. For now,
we take as realistic parameters
ω
1
/
2
π
=
200
THz,
ω
m
/
2
π
=
10
GHz,
h
/
2
π
=
0
.
35
MHz, and me-
chanical and (unloaded) optical quality factors of
Q
m
≡
ω
m
/
γ
m
∼
10
3
(room temperature)-
10
5
(low tem-
perature) and
Q
1
≡
ω
1
/
κ
1
,
in
=
3
×
10
6
, where
γ
m
is the
14
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mechanical decay rate and
κ
1
,
in
is the intrinsic optical
cavity decay rate. In practice, one can also over-couple
cavity mode
1
to the waveguide, with a waveguide-
induced optical decay rate
κ
ex
that is much larger than
κ
in
.
SLOWING AND STOPPING LIGHT
The propagation characteristics of light in the waveguide
and interacting with the optomechanical crystal is char-
acterized by an optical band structure, which is shown in
Figs. 1b-f. The color coding of the dispersion curves (red
for waveguide, green for optical cavity, blue for mechan-
ical resonance) indicates the distribution of energy or
fractional occupation in the various degrees of freedom
of the system in steady-state. Far away from the cav-
ity resonance, the dispersion relation is nearly linear and
simply reflects the character of the input optical waveg-
uide, while the propagation is strongly modified near res-
onance (
ω
=
ω
1
=
ω
2
+
ω
m
). In the absence of optome-
chanical coupling (
Ω
m
=
0
), a transmission band gap of
width
∼
κ
forms around the optical cavity resonance (re-
flections from the bare optical cavity elements construc-
tively interfere). In the presence of optomechanical driv-
ing, the band gap splits in two (blue shaded regions) and
a new propagation band centered around the cavity res-
onance appears in the middle of the band gap. For weak
driving (
Ω
m
<
κ
) the width of this band is
∼
4
Ω
2
m
/
κ
,
while for strong driving (
Ω
m
>
κ
) one recovers the “nor-
mal mode splitting” of width
∼
2
Ω
m
[19]. This relatively
flat polaritonic band yields the slow-light propagation of
interest. Indeed, for small
Ω
m
the steady-state energy in
this band is almost completely mechanical in character,
indicating the strong mixing and conversion of energy in
the waveguide to mechanical excitations along the array.
The Bloch wavevector near resonance is given by
k
eff
≈
k
0
+
κ
ex
δ
k
2
d
Ω
2
m
+
i
κ
ex
κ
in
δ
2
k
4
d
Ω
4
m
+
(
2
κ
3
ex
+
12
κ
ex
Ω
2
m
)
δ
3
k
24
d
Ω
6
m
.
(1)
Here
k
0
is the resonant wavevector,
d
is the distance
between elements, and
δ
k
is the frequency detuning
from resonance. The group velocity through the coupled
waveguide
v
g
= (
dk
eff
/
d
δ
k
)
−
1
=
2
d
Ω
2
m
/
κ
ex
can be dra-
matically slowed by an amount that is tunable through
the optomechanical coupling strength
Ω
m
. The quadratic
and cubic terms in
k
eff
characterize pulse absorption and
group velocity dispersion, respectively. For a system of
N
elements, these processes yield a static bandwidth-delay
product of
∆
ω τ
delay
∼
min
(
√
2
N
κ
ex
/
κ
in
,
(
6
π
N
2
)
1
/
3
)
(2)
for constant
Ω
m
and negligible mechanical losses. The
first term on the right is the limit set by absorption, while
the second term is the limit set by pulse distortion. When
intrinsic optical cavity losses are negligible, and if one
is not concerned with pulse distortion, one can propa-
gate light over the full bandwidth
∼
4
Ω
2
m
/
κ
of the slow-
light polariton band and the bandwidth-delay product in-
creases to
∆
ω τ
delay
∼
N
.
In the static regime, the scaling of bandwidth-delay
product obtained here is identical to CROW systems [2].
In the case of EIT, a static bandwidth-delay product of
∆
ω τ
delay
∼
√
OD results, where OD is the optical depth
of the atomic medium. This product is limited by photon
absorption and re-scattering into other directions, and is
analogous to our result
∆
ω τ
delay
∼
√
N
κ
ex
/
κ
in
in the case
of large intrinsic cavity linewidth. On the other hand,
when
κ
in
is negligible, photons are never lost and reflec-
tions can be suppressed by interference. This yields an
improved scaling
∆
ω τ
delay
∼
N
2
/
3
or
∼
N
, depending on
whether one is concerned with group velocity dispersion.
In atomic media, the weak atom-photon coupling makes
achieving OD
>
100
very challenging. In contrast, in our
system as few as
N
∼
10
elements would be equivalently
dense.
While we have thus far shown the static dispersion
curves, we now argue that the group velocity
v
g
(
t
) =
2
d
Ω
2
m
(
t
)
/
κ
ex
can in fact be adiabatically changed once a
pulse is completely localized inside the system, leading
to distortion-less propagation at a dynamically tunable
speed. In particular, by tuning
v
g
(
t
)
→
0
, the pulse can be
completely stopped and stored.
This phenomenon can be understood in terms of a “dy-
namic compression” of the pulse bandwidth. The same
physics applies for CROW structures [1, 20], and the ar-
gument is re-summarized here. First, under constant
Ω
m
,
an optical pulse completely enters the medium within the
bandwidth of the polariton band. Once the pulse is inside,
we consider the effect of a gradual reduction in
Ω
m
(
t
)
.
Decomposing the pulse into Bloch wavevector compo-
nents, it is clear that each Bloch wavevector is conserved
under arbitrary changes of
Ω
m
(as it is fixed by the sys-
tem periodicity). Furthermore, transitions to other bands
are negligible provided that the energy levels are varied
adiabatically compared to the size of the gap. In this case,
the bandwidth of the pulse is dynamically compressed,
and the reduction in slope of the polariton band (Fig. 1)
causes the pulse to propagate at an instantaneous group
velocity
v
g
(
t
) =
2
d
Ω
2
m
(
t
)
/
κ
ex
without any distortion. In
the limit that
Ω
m
→
0
, the polaritonic band becomes flat
and completely mechanical in character, indicating that
the pulse has been reversibly and coherently mapped
onto stationary mechanical excitations within the array.
15
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LIMITATIONS TO STORAGE
Given that the polaritonic band is mostly mechanical
for realistic systems, the maximum storage time is set
by the mechanical decay rate,
∼
1
/
γ
m
. For parameters
ω
m
/
2
π
=
10
GHz and
Q
m
=
10
5
, this yields a storage
time of
∼
10
μ
s, which is much longer than that of re-
alistic CROW structures. The key feature of our system
is that we effectively “down-convert” the high-frequency
optical fields to low-frequency mechanical excitations,
which naturally decay over much longer time scales.
While storage times of
∼
10
ms are possible using atomic
media [21], their bandwidths so far have been limited to
<
1
MHz [20]. In our system, the width of the polari-
ton band depends on the circulating intensity in the “tun-
ing” cavities, but as an example, a width of
∼
1
GHz is
possible using an intra-cavity average photon number of
|
α
2
2
|∼
10
7
.
One of the major sources of error in our device will
be mechanical noise, which through the optomechanical
coupling can be mapped into noise power in the optical
waveguide output. This is essentially the price that one
pays for down-converting optical excitations to mechan-
ical to yield longer storage times – in turn, any mechani-
cal noise gets “up-converted” to optical energy (whereas,
for example, the probability of having a thermal optical
photon is negligible). Defining an effective optically in-
duced cooling rate
Γ
opt
=
4
Ω
2
m
/
κ
, it can be shown that
when
Γ
opt
À
γ
m
the optical noise power emerging from
one end of the waveguide is given by
P
noise
≈
N
̄
h
ω
1
2
κ
ex
κ
(
γ
m
̄
n
th
+
Γ
opt
(
κ
4
ω
m
)
2
)
.
(3)
The first term on the right corresponds to the upcon-
version of thermal mechanical noise (
̄
n
th
= (
e
̄
h
ω
m
/
k
B
T
b
−
1
)
−
1
is the Bose occupation number at the mechanical
frequency and
T
b
is the bath temperature), while the sec-
ond term corresponds to optically-induced Stokes scat-
tering (where a pump photon in cavity 2 creates both a
photon in cavity 1 and a phonon). Generally, the ther-
mal noise can be suppressed by working at lower tem-
peratures, while Stokes scattering can be suppressed with
good sideband resolution,
κ
¿
ω
m
.
At room temperature,
̄
n
th
≈
k
B
T
b
/
̄
h
ω
m
is large and ther-
mal noise will dominate, yielding a noise power of
∼
0
.
4
nW per element for previously given system pa-
rameters and
κ
ex
/
κ
≈
1
. We note that the thermal noise
scales inversely with mechanical frequency, and the use
of high-frequency mechanical oscillators ensures that
such noise remains easily tolerable even at room temper-
ature. At cryogenic temperatures, these high-frequency
oscillators can be thermally cooled to the ground state,
which enables our device to operate as a quantum mem-
ory for photons. A more detailed analysis shows that up
to
∼
100
single-photon pulses may be stored in our sys-
tem under realistic conditions.
OPTOMECHANICAL CRYSTAL DESIGN
A schematic showing a few periods of our proposed
2D OMC slow-light structure is given in Fig. 2. The
structure is built around a “snowflake” crystal pattern
of etched holes into a Silicon slab [22]. This pattern,
when implemented with a physical lattice constant of
a
=
400
nm, snowflake radius
r
=
168
nm, and snowflake
width
w
=
60
nm (see Fig. 2a), provides a simultane-
ous phononic bandgap from
8
.
6
to
12
.
6
GHz and a pho-
tonic pseudo-bandgap from
180
to
230
THz. A single
point defect, formed by removing two adjacent holes (a
so-called “L2” defect), yields the co-localized phononic
and photonic resonances shown in Figs. 2b and c, re-
spectively. The optomechanical coupling between the
two resonances can be quantified by a coupling rate,
g
,
which corresponds to the frequency shift in the optical
resonance line introduced by a single phonon displace-
ment. Numerical finite-element-method (FEM) simula-
tions of the L2 defect indicate the mechanical reso-
nance occurs at
ω
m
/
2
π
=
11
.
2
GHz, with a coupling
rate of
g
/
2
π
=
489
kHz to the optical mode at frequency
ω
o
/
2
π
=
199
THz.
In order to form the double-cavity system described
in the slow-light scheme above, a pair of L2 cavities
are placed in the near-field of each other as shown in
the dashed box region of Fig. 2a. Modes of the two de-
generate L2 cavities mix, forming supermodes of the
double-cavity system which are split in frequency. The
frequency splitting between modes can be tuned via the
number of snowflake periods between the cavities. The
optomechanical cross-coupling of the odd (
E
−
) and even
(
E
+
) optical supermodes mediated by the motion of the
odd parity mechanical supermode (
Q
−
) of the double-
cavity drives the slow-light behavior of the system. We
let cavity modes
1
,
2
in our system correspond to
E
−
and
E
+
, respectively. Since
Q
−
is a displacement field that is
antisymmetric about the two cavities, there is no optome-
chanical self-coupling between the optical supermodes
and this mechanical mode. On the other hand, the cross-
coupling between the two different parity optical super-
modes is large and given by
h
=
g
/
√
2
=
2
π
(
346
kHz
)
.
The different spatial symmetries of the optical cavity
supermodes allow them to be addressed independently.
To achieve this we create a pair of linear defects in the
snowflake lattice as shown in Fig. 2a, each acting as a
single-mode optical waveguide at the desired frequency
of roughly
200
THz. Sending light down both waveg-
uides, with the individual waveguide modes either in or
out of phase with each other, will then excite the even
or odd supermode of the double cavity, respectively. The
16
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FIGURE 2.
a) Top view of our proposed implementation of
the optomechanical crystal array, with the superlattice unit-cell
of length
d
highlighted in the center. The displacement field
amplitude
|
Q
(
r
)
|
of the mechanical mode and in-plane electric
field amplitude
|
E
(
r
)
|
of the optical mode are shown in b) and
c), respectively, for a single L2 defect cavity.
waveguide width and proximity to the L2 cavities can be
used to tune the cavity loading, which for the structure in
Fig. 2a results in the desired
κ
ex
/
2
π
=
2
.
4
GHz. It should
be noted that these line-defect waveguides do not guide
phonons at the frequency of
Q
−
, and thus no additional
phonon leakage is induced in the localized mechanical
resonance.
OUTLOOK
The possibility of using optomechanical systems to fa-
cilitate major tasks in classical optical networks has
been suggested in several recent proposals [23, 24]. This
present work not only extends these prospects, but pro-
poses a fundamentally new direction where optome-
chanical systems can be used to control and manipulate
light at a quantum mechanical level. Such efforts would
closely mirror proposals to perform similar tasks using
EIT and atomic ensembles [4]. At the same time, the op-
tomechanical array has a number of novel features com-
pared to atoms, in that each element can be deterministi-
cally positioned, addressed, and manipulated, and a sin-
gle element is already optically dense. Taken together,
this raises the possibility that mechanical systems can
provide a novel, highly configurable on-chip platform for
realizing quantum optics and “atomic” physics.
ACKNOWLEDGMENTS
This work was supported by the DARPA/MTO ORCHID
program through a grant from AFOSR. DC acknowl-
edges support from the Gordon and Betty More Foun-
dation through Caltech’s Center for the Physics of Infor-
mation. ASN acknowledges support from NSERC.
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