Slowing and stopping light using an optomechanical crystal array
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Slowing and stopping light using an optomechanical
crystal array
D E Chang
1
,
4
, A H Safavi-Naeini
2
,
4
, M Hafezi
3
and O Painter
2
,
5
1
Institute for Quantum Information and Center for the Physics of Information,
California Institute of Technology, Pasadena, CA 91125, USA
2
Thomas J Watson, Sr, Laboratory of Applied Physics, California Institute of
Technology, Pasadena, CA 91125, USA
3
Joint Quantum Institute and Department of Physics, University of Maryland,
College Park, MD 20742, USA
E-mail:
opainter@caltech.edu
New Journal of Physics
13
(2011) 023003 (26pp)
Received 24 November 2010
Published 1 February 2011
Online at
http://www.njp.org/
doi:10.1088/1367-2630/13/2/023003
Abstract.
One of the major advances needed to realize all-optical information
processing of light is the ability to delay or coherently store and retrieve optical
information in a rapidly tunable manner. In the classical domain, this optical
buffering is expected to be a key ingredient of managing the flow of information
over complex optical networks. Such a system also has profound implications for
quantum information processing, serving as a long-term memory that can store
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 coherently transferred into long-lived mechanical vibrations of the array.
Under realistic conditions, this system is capable of achieving large bandwidths
and storage/delay times in a compact, on-chip platform.
4
These authors contributed equally to this work.
5
Author to whom any correspondence should be addressed.
New Journal of Physics
13
(2011) 023003
1367-2630/11/023003+26
$
33.00
© IOP Publishing Ltd and Deutsche Physikalische Gesellschaft
2
Contents
1. Introduction
2
2. Description of the system: an optomechanical crystal (OMC) array
3
3. Slowing and stopping light
5
3.1. Static regime
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
3.2. Storage of optical pulse
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
3.3. Imperfections in storage
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
4. OMC design
10
5. Outlook
13
Acknowledgments
13
Appendix A. Equations of motion for an OMC array
13
Appendix B. Transfer matrix analysis of propagation
15
Appendix C. Optical noise power
16
Appendix D. The band structure analysis
17
Appendix E. Implementation in an OMC
20
References
24
1. Introduction
Light is a natural candidate for transmitting information across large networks owing to its high
speed and low propagation losses. A major obstacle to building more advanced optical networks
is the lack of an all-optically controlled device that can robustly delay or store optical wave
packets over a tunable amount of time. In the classical domain, such a device would enable all-
optical buffering and switching, bypassing the need to convert an optical pulse to an electronic
signal. In the quantum realm, such a device could serve as a memory to store the full quantum
information contained in a light pulse until it can be passed to a processing node at some
later time.
A number of schemes to coherently delay and store optical information are being
actively explored. These range from tunable coupled resonator optical waveguide (CROW)
structures [
1
,
2
], where the propagation of light is dynamically altered by modulating the
refractive 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
]–[
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 nanomechanical resonator array enables one to
map the optical field into long-lived mechanical excitations. This process is completely quantum
coherent and allows the delay and release of pulses to be rapidly and all-optically tuned. Our
scheme combines many of the best attributes of previously proposed approaches, in that it
simultaneously allows for large bandwidths of operation, on-chip integration, relatively long
delay/storage times and ease of external control. Beyond light storage, this work opens up the
intriguing possibility of a platform for quantum or classical all-optical information processing
using mechanical systems.
New Journal of Physics
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3
2. Description of the system: an optomechanical crystal (OMC) array
An optomechanical crystal (OMC) [
9
] is a periodic structure that comprises both a photonic [
10
]
and a phononic [
11
] crystal. The ability to engineer optical and mechanical properties in the
same structure should enable unprecedented control over light–matter interactions. Planar two-
dimensional (2D) photonic crystals, formed from patterned thin dielectric films on the surface of
a microchip, have been succesfully employed as nanoscale optical circuits capable of efficiently
routing, diffracting and trapping light. Fabrication techniques for such 2D photonic crystals have
matured significantly over the last decade, with experiments on an Si chip [
12
] demonstrating
excellent 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 suitably designed photonic
crystal cavities also contain localized acoustic resonances that are strongly coupled to the
optical field via radiation pressure [
9
]. These planar OMCs are thus a natural candidate for
the implementation of our proposed slow-light scheme.
In the following, we consider an OMC containing a periodic array of such defect cavities
(see figures
1
(a) and (b)). Each element of the array contains two optical cavity modes (denoted
1, 2) and a co-localized mechanical resonance. The Hamiltonian describing the dynamics of a
single element is of the form
̃
H
om
=
̄
h
ω
1
ˆ
a
†
1
ˆ
a
1
+
̄
h
ω
2
ˆ
a
†
2
ˆ
a
2
+
̄
h
ω
m
ˆ
b
†
ˆ
b
+
̄
hh
(
ˆ
b
+
ˆ
b
†
)(
ˆ
a
†
1
ˆ
a
2
+
ˆ
a
†
2
ˆ
a
1
).
(1)
Here,
ω
1
,
2
are the resonance frequencies of the two optical modes,
ω
m
is the mechanical
resonance frequency and
ˆ
a
1
,
ˆ
a
2
,
ˆ
b
are annihilation operators for these modes. The
optomechanical interaction cross-couples the cavity modes 1 and 2 with a strength characterized
by
h
and that depends linearly on the mechanical displacement
ˆ
x
∝
(
ˆ
b
+
ˆ
b
†
)
. While we formally
treat
ˆ
a
1
,
ˆ
a
2
,
ˆ
b
as quantum mechanical operators, for the most part it also suffices to treat these
terms as dimensionless classical quantities describing the positive-frequency components of the
optical fields and mechanical position. In addition to the optomechanical interaction described
by equation (
1
), cavity modes 1 are coupled to a common two-way waveguide (described
below). Each element is decoupled from the others except through the waveguide.
The design considerations necessary for achieving such a system are discussed in detail in
the section ‘Optomechanical crystal design’. For now, we take as typical parameters
ω
1
/
2
π
=
200 THz,
ω
m
/
2
π
=
10 GHz,
h
/
2
π
=
0
.
35 MHz and mechanical and (unloaded) optical quality
factors of
Q
m
≡
ω
m
/γ
m
∼
10
3
(room temperature)–10
5
(low temperature) and
Q
1
≡
ω
1
/κ
1
,
in
=
3
×
10
6
, where
γ
m
is the mechanical decay rate and
κ
1
,
in
is the intrinsic optical cavity decay
rate. Similar parameters have been experimentally observed in other OMC systems [
9
,
13
]. 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
.
For the purpose of slowing light, cavity modes 2 will be resonantly driven by an external
laser, so that to good approximation
ˆ
a
2
≈
α
2
(
t
)
e
−
i
ω
2
t
can be replaced by its mean-field value. We
furthermore consider the case where the frequencies are tuned such that
ω
1
=
ω
2
+
ω
m
. Keeping
only the resonant terms in the optomechanical interaction, we arrive at a simplified Hamiltonian
for a single array element (see figure
1
(b)),
H
om
=
̄
h
ω
1
ˆ
a
†
1
ˆ
a
1
+
̄
h
ω
m
ˆ
b
†
ˆ
b
+
̄
h
m
(
t
)(
ˆ
a
†
1
ˆ
b
e
−
i
(ω
1
−
ω
m
)
t
+ h
.
c
.).
(2)
Here, we have defined an effective optomechanical driving amplitude
m
(
t
)
=
h
α
2
(
t
)
and
assume that
α
2
(
t
)
is real. Mode 2 thus serves as a ‘tuning’ cavity that mediates population
New Journal of Physics
13
(2011) 023003 (
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)
4
b
a
2
a
1
ex
h
ex
in
in
a
R
(
z
)
optical
waveguide
mechanical
cavity
optical
cavities
a
L
(
z
)
a
R
(
z
+
d
)
a
L
(
z
+
d
)
a
ex
in
m
(t)
b
mechanical
cavity
optical
waveguide
a
R
(
z
)
a
L
(
z
)
a
R
(
z
+
d
)
a
L
(
z
+
d
)
(a)
(b)
|n , n
>
m
1
|n , n + 1
>
m1
|n + 1, n
>
m1
m
(t)
a
R
(
z
,
t
)
(d)
(c)
-3
-2
-1
0
1
2
3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
|t| ,
|r| ,
|t| ,
|r| ,
2
2
2
2
Ω
=
0
Ω
=
0
Ω
=
κ
/10
Ω
=
κ
/10
δ/κ
-0.1
0
0.1
0
0.5
1
“active”
optical
cavity
m
m
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) A simplified system diagram where the classically
driven cavity mode
a
2
is effectively eliminated to yield an optomechanical
driving amplitude
m
between the mechanical mode and the cavity mode
a
1
.
(c) Frequency-dependent reflectance (black curve) and transmittance (red) of
a single array element, in the case of no optomechanical driving amplitude
m
=
0 (dotted line) and an amplitude of
m
=
κ
ex
/
10 (solid line). The inherent
cavity decay is chosen to be
κ
in
=
0
.
1
κ
ex
. (Inset) The optomechanical coupling
creates a transparency window of width
∼
4
2
m
/κ
ex
for a single element and
enables perfect transmission on resonance,
δ
k
=
0. (d) Energy level structure
of the simplified system. The number of photons and phonons are denoted by
n
1
and
n
m
, respectively. The optomechanical driving amplitude
m
couples
states
|
n
m
+ 1
,
n
1
〉↔|
n
m
,
n
1
+ 1
〉
, while the light in the waveguide couples states
|
n
m
,
n
1
〉↔|
n
m
,
n
1
+ 1
〉
. The two couplings create a set of
3
-type transitions
analogous to that in EIT.
New Journal of Physics
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transfer (Rabi oscillations) between the ‘active’ cavity mode 1 and the mechanical resonator
at a controllable rate
(
t
)
, which is the key mechanism for our stopped-light protocol. In
the following analysis, we will focus exclusively on the active cavity mode and drop the ‘1’
subscript.
A Hamiltonian of the form (
2
) also describes an optomechanical system with a
single optical mode, when the cavity is driven off resonance at frequency
ω
1
−
ω
m
and
ˆ
a
1
corresponds to the sidebands generated at frequencies
±
ω
m
around the classical driving field.
For a single system, this Hamiltonian leads to efficient optical cooling of the mechanical
motion [
14
,
15
], a technique being used to cool nanomechanical systems toward their quantum
ground states [
16
]–[
19
]. While the majority of such work focuses on how optical fields
affect the mechanical dynamics, here we show that the optomechanical interaction strongly
modifies optical field propagation to yield the slow/stopped light phenomenon. Equation (
2
)
is quite general and thus this phenomenon could, in principle, be observed in any array of
optomechanical systems coupled to a waveguide. In practice, there are several considerations
that make the 2D OMC ‘ideal’. Firstly, our system exhibits an extremely large optomechanical
coupling
h
and contains a second optical tuning cavity that can be driven resonantly, which
enables large driving amplitudes
m
using reasonable input power [
20
]. Using two different
cavities also potentially allows for greater versatility and addressability of our system. For
instance, in our proposed design the photons in cavity 1 are spatially filtered from those in cavity
2 [
20
]. Secondly, the 2D OMC is an easily scalable and compact platform. Finally, as described
below, the high mechanical frequency of our device compared to typical optomechanical
systems allows for a good balance between long storage times and suppression of noise
processes.
3. Slowing and stopping light
3.1. Static regime
We first analyze propagation in the waveguide when
m
(
t
)
=
m
is static during the transit
interval of the signal pulse. As shown in the
appendix
, the evolution equations in a rotating
frame for a single element located at position
z
j
along the waveguide are given by
d
ˆ
a
d
t
=−
κ
2
ˆ
a
+ i
m
ˆ
b
+ i
√
c
κ
ex
2
(
ˆ
a
R
,
in
(
z
j
)
+
ˆ
a
L
,
in
(
z
j
))
+
√
c
κ
in
ˆ
a
N
(
z
j
),
(3)
d
ˆ
b
d
t
=−
γ
m
2
ˆ
b
+ i
m
ˆ
a
+
ˆ
F
N
(
t
).
(4)
Equation (
3
) is a standard input relation characterizing the coupling of right- (
ˆ
a
R
,
in
) and left-
propagating (
ˆ
a
L
,
in
) optical input fields in the waveguide with the cavity mode. Here
κ
=
κ
ex
+
κ
in
is the total optical cavity decay rate,
ˆ
a
N
(
z
)
is quantum noise associated with the inherent optical
cavity loss, and for simplicity we have assumed a linear dispersion relation
ω
k
=
c
|
k
|
in the
waveguide. Equation (
4
) describes the optically driven mechanical motion, which decays at a
rate
γ
m
and is subject to thermal noise
ˆ
F
N
(
t
)
. The cavity mode couples to the right-propagating
field through the equation
(
1
c
∂
∂
t
+
∂
∂
z
)
ˆ
a
R
(
z
,
t
)
=
i
√
κ
ex
2
c
δ(
z
−
z
j
)
ˆ
a
+ i
k
0
ˆ
a
R
,
(5)
New Journal of Physics
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6
where
k
0
=
ω
1
/
c
. We solve the above equations to find the reflection and transmission
coefficients
r
,
t
of a single element for a right-propagating incoming field of frequency
ω
k
(see
the
appendix
). In the limit where
γ
m
=
0, and defining
δ
k
≡
ω
k
−
ω
1
,
r
(δ
k
)
=−
δ
k
κ
ex
δ
k
(
−
2i
δ
k
+
κ)
+ 2i
2
m
,
(6)
while
t
=
1 +
r
. Example reflectance and transmittance curves are plotted in figure
1
(c). For
any non-zero
m
, a single element is perfectly transmitting on resonance, whereas for
m
=
0
resonant transmission past the cavity is blocked. When
m
6=
0, excitation of the cavity mode is
inhibited through destructive interference between the incoming field and the optomechanical
coupling. In EIT, a similar effect occurs via interference between two electronic transitions. This
analogy is further elucidated by considering the level structure of our optomechanical system
(figure
1
(d)), where the interference pathways and the ‘
3
’-type transition reminiscent of EIT
are clearly visible. The interference is accompanied by a steep phase variation in the transmitted
field around resonance, which can result in a slow group velocity. These steep features and
their similarity to EIT in a single optomechanical system have been theoretically [
21
,
22
] and
experimentally studied [
23
,
24
], while interference effects between a single cavity mode and
two mechanical modes have also been observed [
25
].
From
r
,
t
for a single element, the propagation characteristics through an infinite array
(figure
2
(a)) can be readily obtained via band structure calculations [
10
]. To maximize the
propagation bandwidth of the system, we choose the spacing
d
between elements such that
k
0
d
=
(
2
n
+ 1
)π/
2 where
n
is a non-negative integer. With this choice of phasing, the reflections
from multiple elements destructively interfere under optomechanical driving. Typical band
structures are illustrated in figures
2
(b)–(f). The color coding of the dispersion curves (red for
waveguide, green for optical cavity and blue for mechanical 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 cavity resonance, the dispersion relation is nearly linear and simply
reflects the character of the input optical waveguide, while the propagation is strongly modified
near resonance (
ω
=
ω
1
=
ω
2
+
ω
m
). In the absence of optomechanical coupling (
m
=
0), a
transmission band gap of width
∼
κ
forms around the optical cavity resonance (reflections from
the bare optical cavity elements constructively interfere). In the presence of optomechanical
driving, the band gap splits in two (blue shaded regions) and a new propagation band centered
on the cavity resonance appears in the middle of the band gap. For weak driving (
m
.
κ
), the
width of this band is
∼
4
2
m
/κ
, whereas for strong driving (
m
&
κ
), one recovers the ‘normal
mode splitting’ of width
∼
2
m
[
26
]. This relatively flat polaritonic band yields the slow-light
propagation of interest. Indeed, for small
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.
It can be shown that the Bloch wavevector near resonance is given by (see the
appendix
)
k
eff
≈
k
0
+
κ
ex
δ
k
2
d
2
m
+
i
κ
ex
κ
in
δ
2
k
4
d
4
m
+
(
2
κ
3
ex
−
3
κ
ex
κ
2
in
+ 12
κ
ex
2
m
)δ
3
k
24
d
6
m
.
(7)
The group velocity on resonance,
v
g
=
(
d
k
eff
/
d
δ
k
)
−
1
|
δ
k
=
0
=
2
d
2
m
/κ
ex
, can be dramatically
slowed by an amount that is tunable through the optomechanical coupling strength
m
. The
quadratic and cubic terms in
k
eff
characterize pulse absorption and group velocity dispersion,
New Journal of Physics
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(2011) 023003 (
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)