Coherent optical wavelength conversion via cavity-optomechanics
Jeff T. Hill,
1,
∗
Amir H. Safavi-Naeini,
1,
∗
Jasper Chan,
1
and Oskar Painter
1, †
1
Thomas J. Watson, Sr., Laboratory of Applied Physics,
California Institute of Technology, Pasadena, CA 91125
(Dated: June 5, 2012)
We theoretically propose and experimentally demonstrate coherent wavelength conversion of optical photons
using photon-phonon translation in a cavity-optomechanical system. For an engineered silicon optomechanical
crystal nanocavity supporting a 4 GHz localized phonon mode, optical signals in a 1
.
5 MHz bandwidth are
coherently converted over a 11
.
2 THz frequency span between one cavity mode at wavelength 1460 nm and a
second cavity mode at 1545 nm with a 93% internal (2% external) peak efficiency. The thermal and quantum
limiting noise involved in the conversion process is also analyzed, and in terms of an equivalent photon number
signal level are found to correspond to an internal noise level of only 6 and 4
×
10
−
3
quanta, respectively.
The interaction of light with acoustic and molecular me-
chanical vibrations enables a great many optical functions used
in communication systems today, such as amplification, modu-
lation, wavelength conversion, and switching [1]. Convention-
ally, these functions are realized in long (centimeter to many
meter) waveguide based devices, relying on inherent materi-
als’ properties and requiring intense optical pump beams. With
the technological advancements in the fields of nanomechan-
ics and nanophotonics, it is now possible to engineer inter-
actions of light and mechanics. Progress in this area has in-
cluded enhanced nonlinear optical interactions in structured
silica fibers [2], near quantum-limited detection of nanome-
chanical motion [3], and the radiation pressure cooling of a
mesoscopic mechanical resonator to its quantum ground state
of motion [3, 4]. Coupling of electromagnetic and mechani-
cal degrees of freedom, in which the coherent interaction rate
is larger than the thermal decoherence rate of the system, as
realized in the ground-state cooling experiments, opens up an
array of new applications in classical and quantum optics. This
realization, along with the inherently broadband nature of radi-
ation pressure, has spawned a variety of proposals for convert-
ing between photons of disparate frequencies [5–8] through
interaction with a mechanical degree of freedom - proposals
which are but one of many possible expressions of hybrid op-
tomechanical systems [9].
The ability to coherently convert photons between disparate
wavelengths has broad technological implications not only for
classical communication systems, but also future quantum net-
works [10–12]. For example, hybrid quantum networks re-
quire a low loss interface capable of maintaining quantum
coherence while connecting spatially separate systems oper-
ating at incompatible frequencies [9]. For this reason, pho-
tons operating in the low loss telecommunications band are
often proposed as a conduit for connecting different physical
quantum systems [13]. It has also been realized that a wide
variety of quantum systems lend themselves to coupling with
mechanical elements. A coherent interface between mechan-
ics and optics, then, could provide the required quantum links
∗
These authors contributed equally to this work
†
Electronic address:
opainter@caltech.edu;
URL:
http://copilot.
caltech.edu
of a hybrid quantum network [6]. Until now, most experi-
ments demonstrating either classical or quantum wavelength
conversion have used intrinsic optical nonlinearities of mate-
rials [1, 14, 15]. In this Letter we show that by patterning an
optomechanical crystal (OMC) nanobeam from a thin film of
silicon, engineered GHz mechanical resonances can be used to
convert photons within a 1
.
5 MHz bandwidth coherently over
an optical frequency span of 11.2 THz, at internal efficiencies
exceeding 90%, and with a thermal(quantum)-limited noise of
only 6 (4
×
10
−
3
) quanta. Such cavity optomechanical systems
are not just limited to the optical frequency domain, but may
also find application to the interconversion of microwave and
optical photons[7, 16], enabling a quantum-optical interface to
superconducting quantum circuits[17].
Conceptually, the proposed system for wavelength conver-
sion can be understood from Fig. 1a. Two optical cavity modes
ˆ
a
k
(
k
=
1
,
2) are coupled to a common mechanical mode
ˆ
b
via
an interaction Hamiltonian
H
=
∑
k
~
g
k
ˆ
a
†
k
ˆ
a
k
(
ˆ
b
+
ˆ
b
†
)
, where ˆ
a
k
,
ˆ
b
are the annihilation operators and
g
k
is the optomechani-
cal coupling rate between the mechanical mode and the
k
th
cavity mode. Physically,
g
k
represents the frequency shift of
cavity mode
k
due to the zero-point motion of the mechani-
cal resonator. Wavelength conversion is driven by two con-
trol laser beams (
α
k
in Fig. 1b), of frequency
ω
l
,
k
and nom-
inal detuning
δ
k
≡
ω
k
−
ω
l
,
k
=
ω
m
to the red of cavity reso-
nance at frequency
ω
k
. In the resolved sideband regime, where
ω
m
κ
k
(
κ
k
the bandwidth of the
k
th cavity mode), the spec-
tral filtering of each cavity preferentially enhances photon-
phonon exchange. The resulting beam-splitter-like Hamilto-
nian is
H
=
∑
k
~
G
k
(
ˆ
a
†
k
ˆ
b
+
ˆ
a
k
ˆ
b
†
)
[7, 18], where
G
k
≡
g
k
√
n
c
,
k
is
the parametrically enhanced optomechanical coupling rate due
to the
α
k
control beam (
n
c
,
k
is the control-beam induced intra-
cavity photon number). In the weak-coupling limit,
G
k
κ
k
,
this interaction effectively leads to an additional mechanical
damping rate,
γ
OM
,
k
=
4
G
2
k
/
κ
k
. The degree to which this
optomechanical loss rate dominates the intrinsic mechanical
loss is called the cooperativity,
C
k
=
γ
OM
,
k
/
γ
i
. At large co-
operativities, the optomechanical damping has been used as a
nearly noiseless loss channel to cool the mechanical mode to
its ground state [3, 4]. In the case of a single optical cavity sys-
tem, it has also been used as a coherent channel allowing inter-
conversion of photons and phonons leading to the observation
of electromagnetically-induced transparency (EIT) [19, 20].
arXiv:1206.0704v1 [physics.optics] 4 Jun 2012
2
1
0
d
1
0
e
f
c
1 μm
b
a
γ
i
G
1
G
2
a
1
a
2
b
11.2 THz
a
out
a
in
!
l;2
!
l;1
!
m
!
m
(=4GHz)
!
l;1
+
¢
1
!
l;2
+
¢
1
a
in
a
out
®
2
®
1
·
e;1
·
e;2
·
i;1
·
i;2
FIG. 1:
System model and physical realization. a
, Diagram of the wavelength conversion process as realized via two separate Fabry-Perot
cavities. The two optical cavity modes,
a
1
and
a
2
, are coupled to the same mechanical mode,
b
, with coupling strengths
G
1
and
G
2
, respectively.
The optical cavity modes are each coupled to an external waveguide (with coupling strengths
κ
e
,
1
,
κ
e
,
2
), through which optical input and output
signals are sent. The optical cavities also have parasitic (intrinsic) loss channels, labeled
κ
i
,
1
and
κ
i
,
2
, whereas the mechanical mode is coupled
to its thermal bath at rate
γ
i
.
b
, Schematic indicating the relevant optical frequencies involved in the wavelength conversion process. The cavity
control laser beams, labeled
α
1
and
α
2
, are tuned a mechanical frequency red of the corresponding optical cavity resonances. An input signal
(
a
in
) is sent into the input cavity at frequency
ω
l
,
1
+
∆
1
. The input signal is converted into an output signal (
a
out
) at frequency
ω
l
,
2
+
∆
1
via
the optomechanical interaction.
c
, Scanning electron micrograph (SEM) of the fabricated silicon nanobeam optomechanical cavity.
d
, Finite
element method (FEM) simulation of the electromagnetic energy density of the first and,
e
, second order optical cavity modes of the silicon
nanobeam.
f
, FEM simulation of the displacement field of the co-localized mechanical mode.
In the double optical cavity system presented here, coherent
wavelength conversion of photons results.
As shown in Fig. 1a, each optical cavity is coupled not
just to a common mechanical mode, but also to an optical
bath at rate
κ
i
,
k
and to an external photonic waveguide at rate
κ
e
,
k
(the total cavity linewidth is
κ
k
=
κ
i
,
k
+
κ
e
,
k
). The ex-
ternal waveguide coupling provides an optical interface to the
wavelength converter, and in this work consists of a single-
transverse-mode waveguide, bi-directionally coupled to each
cavity mode. The efficiency of the input/output coupling is
defined as
η
k
=
κ
e
,
k
/
2
κ
k
, half that of the total bi-directional
rate. Although the wavelength converter operates symmetri-
cally, here we will designate the higher frequency cavity mode
(
k
=
1) as the input cavity and the lower frequency cavity
(
k
=
2) as the output cavity. As shown in Fig. 1b, photons sent
into the wavelength converter with detuning
∆
1
∼
ω
m
from the
control laser
α
1
, are converted to photons
∆
1
detuned from the
control laser
α
2
, an 11
.
2 THz frequency span for the device
studied here.
The details of the conversion process can be understood by
solving the Heisenberg-Langevin equations (see Appendix).
We linearize the system and work in the frequency domain,
obtaining through some algebra the scattering matrix element
s
21
(
ω
)
, which is the complex, frequency-dependent conver-
sion coefficient between the input field at cavity ˆ
a
1
, and the
output field at cavity ˆ
a
2
. This coefficient is given by the ex-
pression
s
21
(
ω
) =
√
η
2
η
1
√
γ
OM
,
2
γ
OM
,
1
i
(
ω
m
−
ω
)+
γ
/
2
,
(1)
where
γ
=
γ
i
+
γ
OM
,
2
+
γ
OM
,
1
is the total mechanical damp-
ing rate and equal to the bandwidth of the conversion process.
From this expression, the spectral density of a converted signal
S
out
,
2
(
ω
)
, given the input signal spectral density
S
in
,
1
(
ω
)
, may
be found and is given by
S
out
,
2
(
ω
) =
η
2
η
1
γ
OM
,
2
γ
OM
,
1
(
ω
+
ω
m
)
2
+(
γ
/
2
)
2
(
n
added
+
S
in
,
1
(
ω
))
.
(2)
These spectral densities have units of photons/Hz
·
s and are
proportional to optical power. The added noise,
n
added
, arises
from thermal fluctuations of the mechanical system and the
quantum back-action noise of light present in each optical
mode. From here, we see that in a system with ideal cavity-
waveguide coupling, (
η
1
,
η
2
=
1), the peak internal photon
conversion efficiency is given by
η
max,int
=
4
C
1
C
2
(
1
+
C
1
+
C
2
)
2
.
(3)
This efficiency only depends on the internal coupling of the
optomechanical system, and for both
C
1
=
C
2
and
C
1
,
C
2
1,
approaches unity. The latter condition can be understood from
requiring the coupling between the optical and mechanical
modes to overtake the intrinsic mechanical loss rate, while the