Proposal for an optomechanical traveling wave phonon–photon translator
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Proposal for an optomechanical traveling wave
phonon–photon translator
Amir H Safavi-Naeini
1
and Oskar Painter
1
Thomas J Watson, Sr., Laboratory of Applied Physics, California Institute of
Technology, Pasadena, CA 91125, USA
E-mail:
safavi@caltech.edu
and
opainter@caltech.edu
New Journal of Physics
13
(2011) 013017 (30pp)
Received 1 October 2010
Published 13 January 2011
Online at
http://www.njp.org/
doi:10.1088/1367-2630/13/1/013017
Abstract.
In this paper, we describe a general optomechanical system for
converting photons to phonons in an efficient and reversible manner. We analyze
classically and quantum mechanically the conversion process and proceed to a
more concrete description of a phonon–photon translator (PPT) formed from
coupled photonic and phononic crystal planar circuits. The application of the
PPT to RF-microwave photonics and circuit QED, including proposals utilizing
this system for optical wavelength conversion, long-lived quantum memory and
state transfer from optical to superconducting qubits, is considered.
1
Authors to whom any correspondence should be addressed.
New Journal of Physics
13
(2011) 013017
1367-2630/11/013017+30
$
33.00
© IOP Publishing Ltd and Deutsche Physikalische Gesellschaft
2
Contents
1. Introduction
2
2. Outline of the proposed system
3
3. Analysis
7
3.1. Simplified dynamics of the system
. . . . . . . . . . . . . . . . . . . . . . . .
7
3.2. Scattering matrix formulation of the phonon–photon translator (PPT)
. . . . . .
9
3.3. Effects of thermal and quantum noises
. . . . . . . . . . . . . . . . . . . . . .
14
4. Proposed on-chip implementation
16
4.1. Single and double cavity systems
. . . . . . . . . . . . . . . . . . . . . . . . .
16
4.2. Optomechanical coupling rates
. . . . . . . . . . . . . . . . . . . . . . . . . .
19
4.3. Implementation of waveguides
. . . . . . . . . . . . . . . . . . . . . . . . . .
19
4.4. Cavity–waveguide coupling
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
5. Applications
21
5.1. Delay lines
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
5.2. Wavelength conversion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
5.3. Quantum state transfer and networking between circuit QED and optics
. . . .
24
6. Summary
27
Acknowledgments
27
References
27
1. Introduction
Classical and quantum information processing network architectures utilize light (optical
photons) for the transmission of information over extended distances, ranging from hundreds
of meters to hundreds of kilometers [
1
,
2
]. The utility of optical photons stems from their weak
interaction with the environment, large bandwidth of transmission and resilience to thermal
noise due to their high frequency (
∼
200 THz). Acoustic excitations (phonons), although limited
in terms of bandwidth and their ability to transmit information farther than a few millimeters,
can be delayed and stored for significantly longer times and can interact resonantly with RF-
microwave electronic systems [
3
]. This complimentary nature of photons and phonons suggests
hybrid phononic–photonic systems as a fruitful avenue of research, where a new class of
optomechanical
circuitry could be made to perform a range of tasks out of the reach of purely
photonic and phononic systems. A building block of such a hybrid architecture would be
elements coherently interfacing optical and acoustic circuits. The optomechanical translator we
propose in this paper acts as a chip-scale
transparent, coherent interface
between phonons and
photons and fulfills a key requirement in such a program.
In the quantum realm, systems involving optical, superconducting, spin or charge
qubits coupled to mechanical degrees of freedom [
4
]–[
9
] have been explored. The recent
demonstration of coherent coupling between a superconducting qubit and a mechanical
resonance by O’Connell
et al
[
10
] has provided an experimental backing for this vision and
is the latest testament to the versatility of mechanics as a connecting element in hybrid quantum
systems. In the specific case of phonon–photon state transfer, systems involving trapped atoms,
ions, nanospheres [
11
]–[
15
] and mechanically compliant optical cavity structures [
16
] have
New Journal of Physics
13
(2011) 013017 (
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)
3
all been considered. In these previous studies, the state of an incoming light field is usually
mapped onto the motional state of an atom, ion or macroscopic mirror through an exact timing
of control pulses, turning the interaction on and off between the light and mechanical motion in
a precise way. The ability to simultaneously implement phononic and photonic waveguides
in optomechanical crystal (OMC) structures [
17
] opens up the opportunity to implement
a
traveling-wave
phonon–photon translator (PPT). Such a device, operating continuously,
connects acoustic and optical waves to each other in a symmetric manner and allows for on-the-
fly conversion between phonons and photons without having to precisely time the information
and control pulses. In effect, the problem of engineering control pulses is converted into a
problem of engineering coupling rates.
Our proposal for a PPT is motivated strongly by recent work [
18
,
19
] on radiation
pressure effects in micro- and nano-scale mechanical systems [
20
]–[
26
]. Furthermore, the
concrete realization of a PPT is aided by the considerable advances made in the last decade
in the theory, design and engineering of thin-film artificial quasi-2D (patterned membrane)
crystal structures containing photonic [
27
]–[
32
] and phononic [
33
]–[
37
] ‘bandgaps’. Such
systems promise unprecedented control over photons and phonons and have been separately
subjected to extensive investigation. Their unification, in the form of OMCs that possess a
simultaneous phononic and photonic bandgap [
17
], [
38
]–[
41
] and in which the interaction
between the photons and phonons can be controlled, promises to further expand the capabilities
of both photonic and phononic architectures and forms the basis for the proposed PPT
implementation.
An outline of this paper is as follows. In sections
2
and
3
, we introduce and study the PPT
system as an abstraction, at first classically and then quantum mechanically. After introducing
the basic system, its properties and its scattering matrix, we study the effects of quantum and
classical noises on device operation. In section
4
, we design and simulate a possible physical
implementation of the system, utilizing recent results for simultaneous phononic–photonic
bandgap materials [
39
]. Finally, in section
5
, we demonstrate a few possible applications of
the PPT. Focusing first on ‘classical’ applications, we evaluate the performance of the PPT
when used for the implementation of an optical delay line and wavelength converter. Finally,
we show how such a system could be used in theory to carry out high-fidelity quantum state
transfer between optical and superconducting qubits.
2. Outline of the proposed system
The proposed PPT system, shown in figure
1
, consists of a localized mechanical resonance (
b
)
which couples the two optical resonances (
a
1
,
a
2
) of an optomechanical cavity via radiation
pressure. External coupling to and from the mechanical resonance is provided by an acoustic
waveguide, while each of the optical resonances is coupled via separate optical waveguides.
Multi-optical-mode optomechanical systems have been proposed and experimentally studied
previously in the context of enhancing quantum back-action, reduced lasing threshold and
parametric instabilities [
42
]–[
48
]. Here we use a two-moded optical cavity because it allows
for the spatial filtering and separation of signal and pump optical beams while reducing the
required input power, as explained below.
A general description of the radiation–pressure coupling of the mechanical and optical
degrees of freedom in such a structure is as follows. For each of the two high-
Q
optical resonances of the cavity, we associate an annihilation operator
ˆ
a
k
and a frequency
New Journal of Physics
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(2011) 013017 (
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)
4
b
a
1
a
2
b
in
2,i
1,i
2,e
1,e
e
i
h
b
out
a
2,in
a
2,out
a
1,in
a
1,out
Figure 1.
Full system diagram. Circles represent resonant modes, while
rectangles represent waveguides. Blue is for photonics and beige is for
phononics, a color scheme followed in other parts of this paper. The coupling
h
between the two optical modes is modulated by the intervening phonon
resonance.
ω
k
(
k
=
1
,
2). Geometric deformation of the optomechanical cavity parameterized by
x
changes
the frequencies of the optical modes by
g
k
(
x
)
. The deformation, due to the localized mechanical
resonance with annihilation operator
ˆ
b
and frequency
, can be quantized and given by
ˆ
x
=
x
ZPF
(
ˆ
b
+
ˆ
b
†
)
. There is also a coupling between the two optical cavity modes given by
h
(
x
)
,
where for resonant intermodal mechanical coupling the cavity structure must be engineered
such that
=
ω
2
−
ω
1
. In a traveling-wave PPT device consisting of the two optical cavity
resonances and a single mechanical resonance, the lower frequency cavity mode (
a
1
in this case)
is used as a ‘pump’ cavity, which enables the inter-conversion of phonons in the mechanical
resonance (
b
) to photons in the second, higher frequency, optical cavity mode (
a
2
) through
a two-photon process in which pump photons are either absorbed or emitted as needed. The
‘signals’ representing the phonon and photon quanta to be exchanged will thus be contained in
b
and
a
1
, respectively.
As described, the Hamiltonian of this system is
ˆ
H
=
̄
h
ω
1
ˆ
a
†
1
ˆ
a
1
+
̄
h
ω
2
ˆ
a
†
2
ˆ
a
2
+
̄
h
ˆ
b
†
ˆ
b
+
̄
h
(
g
1
(
ˆ
x
)
ˆ
a
†
1
ˆ
a
1
+
g
2
(
ˆ
x
)
ˆ
a
†
2
ˆ
a
2
)
+
̄
hh
(
ˆ
x
)
(
ˆ
a
†
2
ˆ
a
1
+
ˆ
a
†
1
ˆ
a
2
)
+ i
√
2
κ
1
,
e
E
pump
(
e
−
i
ω
L
t
ˆ
a
†
1
+ e
i
ω
L
t
ˆ
a
1
)
,
(1)
where we have added a classical optical pumping term of electric field amplitude
E
pump
and
frequency
ω
L
. Optical pumping is performed through one of the optical waveguides with (field)
coupling rate to the
a
1
cavity resonance given by
κ
1
,
e
. In addition to the waveguide loading of
each optical resonance (
κ
k
,
e
), the total optical loss rate of each cavity mode includes an intrinsic
component (
κ
k
,
i
) of field decay due to radiation, scattering and absorption. Similarly, for the
mechanical resonance, we have a field decay rate given by
γ
=
γ
e
+
γ
i
, which is a combination
of waveguide loading and intrinsic losses. The constant parts of both
h
(
ˆ
x
)
and
g
k
(
ˆ
x
)
(
h
(
0
)
,
g
k
(
0
)
) can be eliminated by a change of basis and are thus taken to be zero. As discussed below,
it is advantageous to choose a cavity structure symmetry in which
g
k
(
ˆ
x
)
=
0 up to linear order
in
ˆ
x
. In fact, we can generally assume that the mechanical displacements are small enough
New Journal of Physics
13
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)
5
1
-
1
2
State Transfer
Processes
Noise
mode 1
mode 2
Phonon Emission
Phonon Absorption
1
+
2,-
2,+
2,-
2,+
1,0
1,0
Cavity Line
Photon Sideband
Phonon Absorption/Emission
Inter-sideband Photon Scattering
Figure 2.
Optical sidebands and processes for phonon–photon translation. The
optical pump is located on sideband
α
1
,
0
, on resonance with the first cavity
mode at frequency
ω
1
. There are three relevant optical sidebands to consider in
the translation process,
α
2
,
+
,
α
1
,
0
and
α
2
,
−
. The inter-sideband photon scattering
gives rise to phonon emission and absorption. The state transfer occurs through
scattering between sidebands
α
2
,
+
and
α
1
,
0
, whereas inter-sideband scattering
between
α
1
,
0
and
α
2
,
−
can be thought of as phonon noise. Note that for the
g
=
0
case considered here, there are no sidebands at
ω
1
±
for cavity mode
a
1
and
no sidebands at
ω
1
for mode
a
2
.
to make the linear order the only important term in the interaction. Assuming then a properly
chosen cavity symmetry,
g
k
(
ˆ
x
)
=
g
·
(
ˆ
b
+
ˆ
b
†
)
=
0 and
h
(
ˆ
x
)
=
h
·
(
ˆ
b
+
ˆ
b
†
),
which yields a simplified Hamiltonian,
ˆ
H
=
̄
h
ω
1
ˆ
a
†
1
ˆ
a
1
+
̄
h
ω
2
ˆ
a
†
2
ˆ
a
2
+
̄
h
ˆ
b
†
ˆ
b
+
̄
hh
(
ˆ
b
+
ˆ
b
†
)
(
ˆ
a
†
2
ˆ
a
1
+
ˆ
a
†
1
ˆ
a
2
)
+ i
√
2
κ
1
,
e
E
pump
(
e
−
i
ω
L
t
ˆ
a
†
1
+ e
i
ω
L
t
ˆ
a
1
)
.
(2)
Treating the system classically and approximately, we can write each intracavity photon
and phonon amplitude, and their inputs (see figure
2
) as a Fourier decomposition of a few
relevant sidebands,
a
1
(
t
)
=
α
1
,
0
e
−
i
ω
1
t
+
α
1
,
+
e
−
i
(ω
1
+
)
t
+
α
1
,
−
e
−
i
(ω
1
−
)
t
,
(3)
a
2
(
t
)
=
α
2
,
0
e
−
i
ω
1
t
+
α
2
,
+
e
−
i
(ω
1
+
)
t
+
α
2
,
−
e
−
i
(ω
1
−
)
t
,
(4)
New Journal of Physics
13
(2011) 013017 (
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)
6
b
(
t
)
=
β
0
+
β
+
e
−
i
t
,
(5)
b
in
(
t
)
=
β
in
,
+
e
−
i
t
,
(6)
a
2
,
in
(
t
)
=
α
in
,
+
e
−
i
(ω
1
+
)
t
,
(7)
b
out
(
t
)
=
β
out
,
+
e
−
i
t
,
(8)
a
2
,
out
(
t
)
=
α
out
,
0
e
−
i
ω
1
t
+
α
out
,
+
e
−
i
(ω
1
+
)
t
+
α
out
,
−
e
−
i
(ω
1
−
)
t
.
(9)
The equations of motion arrived at from the system Hamiltonian (presented generally in the
following section) then become algebraic relations between the
α
and
β
sideband amplitudes.
By ignoring the self-coupling term (
g
=
0), pumping on-resonance with cavity mode
a
1
(
ω
L
=
ω
1
), and engineering the optical cavity mode splitting for mechanical resonance (
ω
2
=
ω
1
+
),
we arrive at classical sideband amplitudes,
α
1
,
+
=
α
1
,
−
=
α
2
,
0
=
β
0
=
0
,
(10)
α
1
,
0
=
√
2
κ
e
κ
E
pump
+
O
(
h
α
k
,
±
β
+
),
(11)
α
2
,
+
=−
i
h
α
1
,
0
κ
β
+
−
√
2
κ
e
κ
α
in
,
+
,
(12)
α
2
,
−
=−
i
h
α
1
,
0
κ
+ 2i
β
∗
+
,
(13)
β
+
=−
i
h
α
∗
1
,
0
γ
α
2
,
+
−
√
2
γ
e
γ
β
in
,
+
−
i
h
α
1
,
0
γ
α
∗
2
,
−
.
(14)
From here we see that the central sideband amplitude of cavity mode
a
1
,
α
1
,
0
is proportional
to the sum of a term containing the pump field
E
pump
and terms containing products of the optical
and mechanical sideband amplitudes. By increasing
E
pump
, the effect of the other sidebands on
the pump resonance amplitude can be made negligible, and we assume here and elsewhere in
this paper that the pump sideband is generally left unaffected by the dynamics of the rest of the
system. As desired, the optical sideband that contains mechanical information is
α
2
,
+
because it
is the only sideband directly proportional to
β
+
. The constant of proportionality between these
two terms is seen to contain both
h
and
α
1
,
0
, demonstrating the role of the pump beam in the
conversion process. Since coherent information transfer between the optics and mechanics is
occurring between
β
+
and
α
2
,
+
, it is desirable to remove the effects of the lower-energy photonic
sideband,
α
2
,
−
. This sideband can be made significantly smaller in magnitude than
α
2
,
+
in the
sideband-resolved regime where
κ
. A convenient way to visualize all of the processes
in the system is shown in figure
2
, where the photonic sideband amplitudes
α
2
,
±
and
α
1
,
0
are
represented as ‘energy levels’, with transitions between them being due to the emission and
absorption of phonons.
From this approximate analysis, it is clearly suggested that in a sideband resolved
optomechanical system, a state-transfer process is possible between the phononic and photonic
resonances, and the process is controlled by a pump beam [
16
]. A more in-depth study of the
system dynamics required to understand how such processes may be used for traveling wave
phonon–photon conversion, and a full investigation of all relevant noise sources required to
understand the applicability of such a system to quantum information, is carried out in the
following section.
New Journal of Physics
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)
7
3. Analysis
A detailed treatment of the operation of a traveling PPT is carried out in this section. At first, the
dynamics of the system are simplified while still taking into account the noise processes related
to the sideband
α
2
,
−
. In this way, one is left with an effective ‘beam-splitter’ Hamiltonian, which
describes the coherent interaction between the optics and mechanics, while the aforementioned
noise processes are accounted for through an effective increase in the thermal bath temperature.
This is followed by a treatment of the traveling-wave problem through a scattering matrix
formulation, which provides an insight into the role of the intracavity pump photon number
(
|
α
1
,
0
|
2
) and optimizing the state-transfer efficiency.
3.1. Simplified dynamics of the system
Starting from the Hamiltonian in equation (
2
), a set of Heisenberg–Langevin equations can be
written down [
49
],
̇
ˆ
a
1
=−
(κ
1
+ i
ω
1
)
ˆ
a
1
−
i
h
(
ˆ
b
+
ˆ
b
†
)
ˆ
a
2
+
√
2
κ
1
,
e
E
pump
e
−
i
ω
L
t
−
√
2
κ
1
ˆ
a
′
1
,
in
,
(15)
̇
ˆ
a
2
=−
(κ
2
+ i
ω
2
)
ˆ
a
2
−
i
h
(
ˆ
b
+
ˆ
b
†
)
ˆ
a
1
−
√
2
κ
2
ˆ
a
′
2
,
in
,
(16)
̇
ˆ
b
=−
i
ˆ
b
−
i
h
(
ˆ
a
†
2
ˆ
a
1
+
ˆ
a
†
1
ˆ
a
2
)
−
γ
ˆ
b
+
γ
ˆ
b
†
−
√
2
γ
ˆ
b
′
in
.
(17)
The input coupling terms as written above include both external waveguide coupling and
intrinsic coupling due to lossy channels (see figure
1
). Separated, the intrinsic (with subscript i)
and extrinsic (with no subscript) components look as follows,
√
2
κ
k
ˆ
a
′
k
,
in
=
√
2
κ
k
,
e
ˆ
a
k
,
in
+
√
2
κ
k
,
i
ˆ
a
k
,
in
,
i
,
(18)
√
2
γ
ˆ
b
′
in
=
√
2
γ
ˆ
b
in
+
√
2
γ
i
ˆ
b
in
,
i
,
(19)
κ
k
=
κ
k
,
i
+
κ
k
,
e
,
(20)
γ
=
γ
i
+
γ
e
.
(21)
As the fluctuations in the fields are of primary interest, each Heisenberg operator can be
rewritten as a fluctuation term around a steady-state value,
ˆ
a
1
(
t
)
→
(α
1
+
ˆ
a
1
)
e
−
i
ω
1
t
,
(22)
ˆ
a
2
(
t
)
→
(α
2
+
ˆ
a
2
)
e
−
i
ω
2
t
,
(23)
ˆ
b
(
t
)
→
(β
+
ˆ
b
)
e
−
i
t
.
(24)
Assuming that the pump beam is driven resonantly with
a
1
(
ω
L
=
ω
1
), the
c
-number steady-state
values are equal to
(α
1
,α
2
,β)
=
((
√
2
κ
1
,
e
/κ)
E
pump
,
0
,
0
)
.
For the fluctuation dynamics, with
1
≡
(ω
2
−
ω
1
)
−
, the resulting equations are
̇
ˆ
a
1
=−
κ
1
ˆ
a
1
−
i
h
ˆ
a
2
(
ˆ
b
e
−
i2
t
+
ˆ
b
†
)
e
−
i
1
t
−
√
2
κ
ˆ
a
′
1
,
in
e
i
ω
1
t
,
(25)
̇
ˆ
a
2
=−
κ
2
ˆ
a
2
−
i
h
(α
1
+
ˆ
a
1
)(
ˆ
b
+
ˆ
b
†
e
+i2
t
)
e
+i
1
t
−
√
2
κ
ˆ
a
′
2
,
in
e
i
ω
2
t
,
(26)
̇
ˆ
b
=−
γ(
ˆ
b
−
ˆ
b
†
e
+i2
t
)
−
i
h
(α
1
+
ˆ
a
1
)
ˆ
a
†
2
e
+i
(1
+2
)
t
−
i
h
(α
1
+
ˆ
a
1
)
†
ˆ
a
2
e
−
i
1
t
−
√
2
γ
ˆ
b
′
in
e
i
t
.
(27)
New Journal of Physics
13
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8
Ignoring all of the mechanically anti-resonant terms for now and invoking the rotating wave
approximation (RWA) valid when
1
and
|
α
1
|
h
, we arrive at the simplified set of
fluctuation equations,
̇
ˆ
a
1
=−
κ
1
ˆ
a
1
−
i
h
ˆ
a
2
ˆ
b
†
e
−
i
1
t
−
√
2
κ
ˆ
a
′
1
,
in
e
i
ω
1
t
,
(28)
̇
ˆ
a
2
=−
κ
2
ˆ
a
2
−
i
h
(
̃
α
1
+
ˆ
a
1
)
ˆ
b
e
+i
1
t
−
√
2
κ
ˆ
a
′
2
,
in
e
i
ω
2
t
,
(29)
̇
ˆ
b
=−
γ
ˆ
b
−
i
h
(
̃
α
1
+
ˆ
a
1
)
†
ˆ
a
2
e
−
i
1
t
−
√
2
γ
ˆ
b
′
in
e
i
t
.
(30)
By ignoring all of the counter-rotating terms proportional to e
+i2
t
, we have also neglected
the noise processes alluded to previously due to the
α
2
,
−
sideband. Of the mechanically
anti-resonant terms that have been dropped, the terms proportional to
α
1
(
h
α
1
ˆ
b
†
e
+i2
t
in
equation (
26
) and
h
α
1
ˆ
a
†
2
e
+i2
t
in equation (
27
)) are the largest and most significant in terms
of error in RWA. These terms correspond to
ˆ
a
1
ˆ
a
†
2
ˆ
b
†
+ h.c. in the Hamiltonian and cause inter-
sideband photon scattering between the pump,
α
1
,
0
, and its lower frequency sideband,
α
2
,
−
, as
shown in figure
2
. This inter-sideband scattering process causes the emission and absorption of
phonons in the mechanical part of the PPT; thus, in principle, even when the extrinsic phonon
inputs are in the vacuum state, spontaneous scattering of photons from
α
1
,
0
to
α
2
,
−
may populate
the mechanical cavity with a phonon.
This effect was studied in [
50
,
51
] in the context of quantum limits to optomechanical
cooling. Similar to that work, a master equation for the phononic mode with the
ω
1
−
optical
sideband adiabatically eliminated leads to an additional
phononic
spontaneous emission term
given by
̇
ρ
b
,
spon
=
G
2
κ
1
(
2
/κ
)
2
+ 1
(
2
ˆ
b
†
ρ
ˆ
b
−
ˆ
b
ˆ
b
†
ρ
−
ρ
ˆ
b
ˆ
b
†
),
(31)
where
G
=
h
| ̃
α
ss
1
|
. The master equation for the mechanics, found by tracing over all bath
and optical variables, is of the form ̇
ρ
=
̇
ρ
b
,
i
+ ̇
ρ
b
,
spon
+ ̇
ρ
b
,
e
−
i
/
̄
h
[
H
b
,ρ
], where the terms on
the right-hand side of the equation are, respectively, the intrinsic phononic loss, the phonon
spontaneous emission, the phonon–waveguide coupling and the coherent evolution of the
system. The first two terms can be lumped together into an effective intrinsic loss, ̇
ρ
′
b
,
i
=
̇
ρ
b
,
i
+ ̇
ρ
b
,
spon
,
̇
ρ
′
b
,
i
=
γ
′
i
(
̄
n
′
+ 1
)(
2
ˆ
b
ρ
ˆ
b
†
−
ˆ
b
†
ˆ
b
ρ
−
ρ
ˆ
b
†
ˆ
b
)
+
γ
′
i
̄
n
′
(
2
ˆ
b
†
ρ
ˆ
b
−
ˆ
b
ˆ
b
†
ρ
−
ρ
ˆ
b
ˆ
b
†
),
(32)
γ
′
i
=
γ
i
(
1
−
n
spon
),
(33)
̄
n
′
=
̄
n
+
n
spon
1
−
n
spon
,
(34)
and
n
spon
≡
G
2
κγ
i
1
(
2
/κ
)
2
+ 1
.
(35)
Hence, by assuming that the intrinsic loss phonon bath is at a modified temperature with
occupation number
̄
n
′
and changing the intrinsic phonon loss rate to
γ
′
i
, the spontaneous
emission and intrinsic loss noises are lumped into one effective thermal noise Liouvillian for the
New Journal of Physics
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)
9
a
b
b
in
i
e
e
i
G
Simplified System
PPT
Mechanical
Optical
a
in
a
out
b
in
b
out
b
out
a
in
a
out
System Symbol
Figure 3.
The simplified PPT system diagram and symbol. The coupling rate
G
is proportional to
h
and
√
n
1
, where
n
1
is the number of photons in the pump
mode.
mechanics. Note that it is possible in this model to have
γ
′
i
negative when
n
spon
>
1; however,
the motional decoherence rate is always positive and given by
γ
′
i
̄
n
′
=
γ
i
(
̄
n
+
n
spon
)
.
In section
3.2
, we will see that the optimal state-transfer efficiency is given by
2
G
2
=
γκ
,
in which case
n
spon
≈
γ
γ
i
(
κ
2
)
2
(36)
in the sideband resolved limit. In the case where
g
6=
0 (recall that this is the self-coupling
radiation pressure term), photons scattering from the pump into the lower-frequency sideband
(
ω
1
−
) can scatter into the
a
1
cavity mode, which is only detuned by
. This is to be compared
with the
g
=
0 case considered so far in which the detuning is 2
for scattering into the
a
2
mode.
As such, for the
g
6=
0 case, there will be roughly four times the spontaneous emission noise,
with
n
g
6=
0
spon
≈
γ/γ
i
(κ/)
2
.
A final simplification can be made by neglecting the fluctuations in the strong optical pump
of cavity mode
a
1
. Considering that the fluctuations in the variables are all of the same order
and that
ˆ
a
1
always appears as
α
1
+
ˆ
a
1
in the equations of motion for
ˆ
b
and
ˆ
a
2
, we can ignore
the dynamics of the pump fluctuations in the case where
|
α
1
|〈ˆ
a
1
〉
,
〈ˆ
a
2
〉
and
〈
ˆ
b
〉
. This is the
undepleted pump approximation. Adiabatically removing the pump from the dynamics of the
system yields a pump-enhanced optomechanical coupling
G
=
h
|
α
1
|
between the optical cavity
mode
a
2
and the mechanical resonance
b
. Dropping the subscript from the cavity mode
a
2
and
moving to a rotating reference frame results in the new effective Hamiltonian [
16
],
ˆ
H
eff
=−
1
ˆ
b
†
ˆ
b
+
G
ˆ
a
†
ˆ
b
+
G
∗
ˆ
a
ˆ
b
†
.
(37)
The system diagram and symbol corresponding to this simplified model of the PPT are shown
in figure
3
.
3.2. Scattering matrix formulation of the phonon–photon translator
(
PPT
)
To understand the properties of the PPT as a waveguide adapter, we begin with a study
of its scattering matrix. Starting from the effective Hamiltonian given in equation (
37
), the
2
The
γ
in this case is in principle
γ
e
+
γ
′
i
as opposed to
γ
e
+
γ
i
, and the equations must therefore be solved
self-consistently. This is discussed in section
3.2.1
.
New Journal of Physics
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)
10
Heisenberg–Langevin equations of motion for the Hamiltonian (
37
) are written under a Markov
approximation in the frequency domain,
−
i
ω
̃
a
(ω)
=−
κ
̃
a
(ω)
−
i
G
̃
b
(ω)
−
√
2
κ
e
̃
a
in
(ω)
−
√
2
κ
i
̃
a
in
,
i
(ω),
(38)
−
i
ω
̃
b
(ω)
=−
(γ
−
i
1)
̃
b
(ω)
−
i
G
∗
̃
a
(ω)
−
√
2
γ
e
̃
b
in
(ω)
−
√
2
γ
i
̃
b
in
,
i
(ω).
(39)
The intrinsic noise terms
̃
a
in
,
i
(ω)
and
̃
b
in
,
i
(ω)
are the initial-state boson annihilation operators for
the baths, whereas the extrinsic terms
̃
a
in
(ω)
and
̃
b
in
(ω)
are annihilation operators for the optical
and mechanical guided modes for each respective waveguide. Since the effective Hamiltonian
(
37
) has been used to derive equations (
38
) and (
39
), thereby neglecting the counter-rotating
terms present in the full system dynamics, the effects of phonon spontaneous emission noise
is included separately. Following the discussion in section
3.1
, the effective Liouvillian (
32
)
corresponds to replacing
γ
i
with
γ
′
i
in (
39
) and using a Langevin force
̃
b
in
,
i
(ω)
satisfying the
relations
〈
̃
b
†
in
,
i
(ω)
̃
b
in
,
i
(ω
′
)
〉= ̄
n
′
δ(ω
−
ω
′
),
(40)
〈
̃
b
in
,
i
(ω
′
)
̃
b
†
in
,
i
(ω)
〉=
(
̄
n
′
+ 1
)δ(ω
−
ω
′
),
(41)
where
γ
′
i
and
̄
n
′
are given in equations (
33
) and (
34
). The intrinsic optical noise correlations are
only due to vacuum fluctuations and given by
〈 ̃
a
in
,
i
〉
(ω
′
)
̃
a
†
in
,
i
(ω)
〉=
δ(ω
−
ω
′
)
.
To ensure efficient translation, competing requirements of matching and strong coupling
between waveguide and resonator must be satisfied. This is similar to the problem of designing
integrated optical filters using resonators and waveguides [
52
,
53
]. From the above equations
and the input–output boundary condition [
49
,
54
], we arrive at the matrix equation
(
̃
a
rmout
(ω)
̃
b
rmout
(ω)
)
=
S
(
̃
a
in
(ω)
̃
b
in
(ω)
)
+
N
(
̃
a
in
,
i
(ω)
̃
b
in
,
i
(ω)
)
,
(42)
with scattering and noise matrices
S
=
(
s
11
(ω)
s
12
(ω)
s
21
(ω)
s
22
(ω)
)
and
N
=
(
n
11
(ω)
n
12
(ω)
n
21
(ω)
n
22
(ω)
)
.
(43)
The elements of the scattering matrix
S
are
s
11
(ω)
=
1
−
2
κ
e
(γ
−
i
(1
+
ω))
|
G
|
2
+
(γ
−
i
(ω
+
1))(κ
−
i
ω)
,
(44)
s
12
(ω)
=
2i
G
∗
√
γ
e
κ
e
|
G
|
2
+
(γ
−
i
(ω
+
1))(κ
−
i
ω)
,
(45)
s
21
(ω)
=
s
∗
12
(ω),
(46)
s
22
(ω)
=
1
−
2
γ
e
(κ
−
i
ω)
|
G
|
2
+
(γ
−
i
(ω
+
1))(κ
−
i
ω)
.
(47)
New Journal of Physics
13
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)
11
Similar expressions are also found for the noise scattering matrix elements
n
i j
(ω)
, with their
extrema reported below.
In order to obtain efficient conversion, the cavities must be over-coupled to their respective
waveguides, ensuring that the phonon (photon) has a higher chance of leaking into the
waveguide continuum modes than escaping into other loss channels. In this regime,
κ
≈
κ
e
and
γ
≈
γ
e
. In the weak-coupling regime,
G
<κ
, the response of the system exhibits a maximum
for
s
12
and
s
21
at
ω
=
0 and a minimum at the same point for
s
11
and
s
22
. In fact, with realistic
system parameters, only the weak-coupling regime leads to efficient translation. In the strong-
coupling regime (
G
>κ
and
κ
γ
), the photon is converted to a phonon at the rate
G
and
then back to a photon before it has a chance of leaving through the much slower phononic loss
channel at rate
γ
, causing there to be significant reflections and reduced conversion efficiency.
To find the optimal value of
G
, we consider the extrema given by
|
s
11
|
min
=
∣
∣
∣
∣
G
2
+
γκ
i
−
γκ
e
G
2
+
γκ
i
+
γκ
e
∣
∣
∣
∣
,
(48)
|
s
12
|
max
=
∣
∣
∣
∣
2
G
√
γ
e
κ
e
G
2
+
γκ
∣
∣
∣
∣
,
(49)
|
s
22
|
min
=
∣
∣
∣
∣
G
2
+
κγ
i
−
κγ
e
G
2
+
κγ
i
+
κγ
e
∣
∣
∣
∣
.
(50)
In the over-coupled approximation and in the case where
κ
i
=
γ
i
=
0, it is easy to see that the
full translation condition
|
s
12
|
max
=
1 is achievable by setting
G
equal to
G
o
=
√
γκ.
(51)
This result has a simple interpretation as a matching requirement. The photonic loss channel
viewed from the phononic mode has a loss rate of
G
2
/κ
. Matching this to the purely mechanical
loss rate of the same phononic mode,
γ
, one arrives at
G
o
=
√
γκ
. The same argument can be
used for the photonic mode, giving the same result. Under this matched condition, the linewidth
of the translation peak in
|
s
12
|
2
is simply
γ
transfer
=
4
|
G
o
|
2
κ
=
2
γ.
(52)
With intrinsic losses taken into account, either
|
s
11
|
or
|
s
22
|
(but not both) can be made
exactly 0 by setting
G
2
=
γ(κ
e
−
κ
i
)
or
G
2
=
(γ
e
−
γ
i
)κ
, respectively. The optimal state-transfer
condition, however, still occurs for
G
o
=
√
γκ
. The extremal values (
ω
=
0) of the scattering
matrix are in this case
|
s
11
|
optimal
ω
=
0
=
κ
i
κ
e
+
κ
i
,
(53)
|
s
12
|
optimal
ω
=
0
=
√
γ
e
κ
e
γκ
,
(54)
|
s
22
|
optimal
ω
=
0
=
γ
i
γ
e
+
γ
i
,
(55)
with corresponding noise matrix elements of
|
n
11
|
optimal
ω
=
0
=
√
κ
i
κ
e
κ
e
+
κ
i
,
(56)
New Journal of Physics
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)
12
-500
0
500
0
1
|s
11
|
amplitude
0
|s
12
|
0
1
|s
21
|
amplitude
1
|s
22
|
amplitude
2
2
2
2
(MHz)
-500
0
500
(MHz)
-4000
-3000
-2000
-1000
0
1000
2000
3000
4000
|s
11
|
2
(MHz)
(a)
(b)
(c)
(d)
(e)
Figure 4.
Phonon–photon scattering matrix amplitudes for
(γ
e
,γ
i
,κ
e
,κ
i
,
G
)
=
2
π
×
(
10
,
1
,
2000
,
200
,
155
.
6
)
MHz. (a) Plot of the frequency dependence of the
optical reflection
s
11
. The broad over-coupled optical line is visible, along with
the PPT feature near the center in the unshaded region. This unshaded region
is shown in more detail in plots (b)–(e), showing the frequency dependence of
the scattering matrix elements
s
11
,
s
12
,
s
21
and
s
22
, respectively. In each plot,
the curves (
−
), (– –) and (
·−
) represent detunings of
1
=
0
,
2
π
×
200 and
2
π
×
400 MHz, respectively.
|
n
12
|
optimal
ω
=
0
=
√
γ
i
κ
e
γκ
,
(57)
|
n
21
|
optimal
ω
=
0
=
√
γ
e
κ
i
γκ
,
(58)
|
n
22
|
optimal
ω
=
0
=
√
γ
i
γ
e
γ
e
+
γ
i
.
(59)
For a set of parameters typical of an OMC system, the magnitudes of the scattering matrix
elements versus frequency are plotted in figure
4
. In these plots, we have assumed resonant
optical pumping of the
a
1
cavity mode and considered several different detuning values
1
.
The normalized optical reflection spectrum (
|
s
11
|
2
) is shown in figure
4
(a), in which the broad
optical cavity resonance can be seen along with a deeper, narrowband resonance that tunes
with
1
. This narrowband resonance is highlighted in figure
4
(b), showing that the optical
reflection is nearly completely eliminated on resonance. Photons on resonance, instead of
New Journal of Physics
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)