Generalized nonreciprocity in an optomechanical circuit via synthetic magnetism and
reservoir engineering
Kejie Fang,
1, 2
Jie Luo,
1, 2
Anja Metelmann,
3, 4
Matthew H. Matheny,
1, 5
Florian Marquardt,
6, 7
Aashish A. Clerk,
3
and Oskar Painter
1, 2
1
Kavli Nanoscience Institute, California Institute of Technology, Pasadena, California 91125, USA
2
Institute for Quantum Information and Matter and Thomas J. Watson, Sr.,
Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
3
Department of Physics, McGill University, 3600 rue University, Montr ́eal, Quebec H3A 2T8, Canada
4
Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
5
Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
6
Max Planck Institute for the Science of Light, G ̈unther-Scharowsky-Straße 1/Bau 24, 91058 Erlangen, Germany
7
Institute for Theoretical Physics, Department of Physics, Universit ̈at Erlangen-N ̈urnberg, 91058 Erlangen
Synthetic magnetism has been used to control charge neutral excitations for applications ranging
from classical beam steering to quantum simulation. In optomechanics, radiation-pressure-induced
parametric coupling between optical (photon) and mechanical (phonon) excitations may be used to
break time-reversal symmetry, providing the prerequisite for synthetic magnetism. Here we design
and fabricate a silicon optomechanical circuit with both optical and mechanical connectivity between
two optomechanical cavities. Driving the two cavities with phase-correlated laser light results in
a synthetic magnetic flux, which in combination with dissipative coupling to the mechanical bath,
leads to nonreciprocal transport of photons with 35 dB of isolation. Additionally, optical pumping
with blue-detuned light manifests as a particle non-conserving interaction between photons and
phonons, resulting in directional optical amplification of 12 dB in the isolator through direction.
These results indicate the feasibility of utilizing optomechanical circuits to create a more general
class of nonreciprocal optical devices, and further, to enable novel topological phases for both light
and sound on a microchip.
Synthetic magnetism involving charge neutral elements
such as atoms [2], polaritons [3–5], and photons [6–
10] is an area of active theoretical and experimental
research, driven by the potential to simulate quantum
many-body phenomena [11], reveal new topological wave
effects [12, 13], and create defect-immune devices for in-
formation communication [7, 10]. Optomechanical sys-
tems [14], involving the coupling of light intensity to
mechanical motion via radiation pressure, are a partic-
ularly promising venue for studying synthetic fields, as
they can be used to create the requisite large optical
nonlinearities [15]. By applying external optical driving
fields time-reversal symmetry may be explicitly broken
in these systems. It was predicted that this could enable
optically tunable nonreciprocal propagation in few-port
devices [16–19], or in the case of a lattice of optomechan-
ical cavities, topological phases of light and sound [1, 20].
Here we demonstrate a generalized form of optical non-
reciprocity in a silicon optomechanical crystal circuit [21]
that goes beyond simple directional propagation; this is
achieved using a combination of synthetic magnetism,
reservoir engineering, and parametric squeezing.
Distinct from recent demonstrations of optomechan-
ical nonreciprocity in degenerate whispering-gallery res-
onators with inherent nontrivial topology [22–24], we em-
ploy a scheme similar to that proposed in Refs. [1, 18] in
which a synthetic magnetic field is generated via optical
pumping of the effective lattice formed by coupled op-
tomechanical cavities. In such a scenario, the resulting
synthetic field amplitude is set by the spatial variation
of the pump field phase and the field lines thread op-
tomechanical plaquettes between the photon and phonon
lattices (see Fig. 1). To achieve nonreciprocal transmis-
sion of intensity in the two-port device of this work –
i.e., bonafide phonon or photon transport effects, not just
nonreciprocal transmission phase – one can combine this
synthetic field with dissipation to implement the general
reservoir engineering strategy outlined in Ref. [25]. This
approach requires one to balance coherent and dissipa-
tive couplings between optical cavities. In our system the
combination of the optical drives and mechanical dissipa-
tion provide the “engineered reservoir” which is needed
to mediate the required dissipative coupling.
To highlight the flexibility of our approach, we use it
to implement a novel kind of nonreciprocal device ex-
hibiting gain [26, 27]. By using an optical pump which is
tuned to the upper motional sideband of the optical cav-
ities, we realize a two-mode squeezing interaction which
creates and destroys photon and phonon excitations in
pairs. These particle non-conserving interactions can be
used to break time-reversal symmetry in a manner that is
distinct from a standard synthetic gauge field. In a lattice
system, this can enable unusual topological phases and
surprising behavior such as protected chiral edge states
involving inelastic scattering [28] and amplification [29].
Here, we use these interactions along with our reservoir-
engineering approach to create a cavity-based optical di-
rectional amplifier: backward propagating signals and
noise are extinguished by 35 dB relative to forward prop-
agating waves which are amplified with an internal gain
of 12 dB (1 dB port-to-port).
The optomechanical system considered in this work is
arXiv:1608.03620v1 [physics.optics] 11 Aug 2016
2
a
J
V
O
L
O
R
M
R
M
L
b
Forward
Backward
i
i
O
L
O
R
M
R
M
L
e
i
R
e
i
L
O
L
O
R
M
R
M
L
e
i
L
e
i
R
B
L
e
i
L
R
e
i
R
Energy
FIG. 1.
Synthetic magnetic field in an optomechani-
cal cavity system. a
, In this scheme consisting of only two
optomechanical cavities, a two-dimensional plaquette can be
formed from the synthetic dimension [1] created by radia-
tion pressure coupling from the optical modes to the mechan-
ical modes. Photon hopping at rate
J
and phonon hopping
at rate
V
occurs between the optical and mechanical cavi-
ties, respectively, with
J
and
V
real for appropriate choice
of gauge. Pumping of the optomechancial cavities with phase
correlated laser light (
|
α
L
|
e
iφ
L
for the left cavity and (
|
α
R
|
e
iφ
R
for the right cavity) results in a synthetic flux Φ
B
=
φ
L
−
φ
R
threading the 4-mode plaquette.
b
, Scheme for detecting the
synthetic flux through nonreciprocal power transmission of
an optical probe laser field. For forward (L
→
R) prop-
agation, constructive interference set by the flux-dependent
phase Φ
B
≈
π/
2 of the dissipative phonon coupling path re-
sults in efficient optical power transmission. The accumulated
phase in the phonon coupling path is reversed for the back-
ward (R
→
L) propagation direction resulting in destructive
interference and reduced optical power transmission in the
left output waveguide. The power in this case is sunk into
the mechanical baths.
shown schematically in Fig. 1a and consists of two in-
teracting optomechanical cavities, labeled
L
(left) and
R
(right), with each cavity supporting one optical mode
O
L(R)
and one mechanical mode
M
L(R)
. Both the optical
and mechanical modes of each cavity are coupled together
via a photon-phonon waveguide, resulting in optical and
mechanical inter-cavity hopping rates of
J
and
V
, re-
spectively (here we choose a local definition of the cavity
amplitudes so both are real). The radiation pressure in-
teraction between the co-localized optical and mechanical
modes of a single cavity can be described by a Hamilto-
nian
ˆ
H
=
~
g
0
ˆ
a
†
ˆ
a
(
ˆ
b
+
ˆ
b
†
), where ˆ
a
(
ˆ
b
) is the annihilation
operator of the optical (mechanical) mode and
g
0
is the
vacuum optomechanical coupling rate [14] (here we have
omitted the cavity labeling).
To enhance the effective photon-phonon interaction
strength each cavity is driven by an optical pump field
with frequency relatively detuned from the optical cavity
resonance by the mechanical frequency (∆
≡
ω
p
−
ω
c
≈
±
ω
m
), with a resulting intra-cavity optical field ampli-
tude
|
α
|
e
iφ
. In the good-cavity limit, where
ω
m
κ
(
κ
being the optical cavity linewidth), spectral filter-
ing by the optical cavity preferentially selects resonant
photon-phonon scattering, leading to a linearized Hamil-
tonian with either a two-mode squeezing form
ˆ
H
ent
=
~
G
(
e
iφ
ˆ
d
†
ˆ
b
†
+
e
−
iφ
ˆ
d
ˆ
b
) (blue detuned pumping) or a beam-
splitter form
ˆ
H
ex
=
~
G
(
e
iφ
ˆ
d
†
ˆ
b
+
e
−
iφ
ˆ
d
ˆ
b
†
) (red detuned
pumping). Here
G
=
g
0
|
α
|
is the parametrically en-
hanced optomechanical coupling rate and
ˆ
d
= ˆ
a
−
α
con-
tains the small signal sidebands of the pump. For both
cases the phase of the resulting coupling coefficient is
nonreciprocal in terms of the generation and annihilation
of photon-phonon excitations. As has been pointed out
before, such a nonreciprocal phase resembles the Peierls
phase that a charged particle accumulates in a mag-
netic vector potential [30]. Crucially, the relative phase
Φ
B
=
φ
L
−
φ
R
is gauge independent (i.e. independent of
local redefinitions of the ˆ
a
and
ˆ
b
cavity amplitudes), im-
plying it should have an observable effect. In the simple
case of ∆ =
−
ω
m
, Φ
B
is formally equivalent to having a
synthetic magnetic flux threading the plaquette formed
by the four coupled optomechanical modes (two optical
and two mechanical)[1, 8, 18]. For ∆ = +
ω
m
, a non-zero
Φ
B
still results in the breaking of time-reversal symmetry,
though the lack of particle number conservation means
that it is not simply equivalent to a synthetic gauge field.
Nonetheless, we will refer to it as a flux in what follows
for simplicity.
To detect the presence of the effective flux Φ
B
, con-
sider the transmission of an optical probe signal, on res-
onance with the optical cavity resonances and coupled in
from either the left or the right side via external optical
coupling waveguides as depicted in Fig. 1b. The probe
light can propagate via two different paths simultane-
ously: (i) direct photon hopping between cavities via the
connecting optical waveguide, and (ii) photon-phonon
conversion in conjunction with intervening phonon hop-
ping via the mechanical waveguide between the cavities.
As in the Aharonov-Bohm effect for electrons [32], the
synthetic magnetic flux set up by the phase-correlated
optical pump beams in the two cavities causes a flux-
dependent interference between the two paths. We de-
fine the forward (backward) transmission amplitude as
T
R
→
L(L
→
R)
≡
d
out,L(R)
/d
in,R(L)
, where
d
out(in)
is the am-
plitude of the outgoing (incoming) electromagnetic signal
field in the corresponding coupling waveguide in units of
square root of photon flux. The optical transmission am-
plitude in the forward direction has the general form
T
L
→
R
[
ω
; ∆ =
±
ω
m
] =
A
±
[
ω
]
(
J
−
Γ
±
[
ω
]
e
−
i
Φ
B
)
,
(1)
3
2.5 μm
Wavelength (nm)
1534.4
1534.45
1534.55
1534.6
Normalized reection
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1534.5
Frequency (GHz)
5.75
5.77
5.79
5.81
5.83
PSD (dBm/Hz)
-132
-131
-130
-129
-128
-127
e
f
cavity
L
cavity
R
M
W
M
+
M
10 μm
0
1
-1
0
1
O
L
M
L
optical coupler
L
photon+phonon waveguide
optical coupler
R
waveguide modes
optical ber-to-chip couplers
L
R
a
b
c
d
FIG. 2.
Silicon optomechanical crystal circuit. a
, Scanning electron microscopy (SEM) image of the optomechanical
crystal circuit studied in this work. The circuit is fabricated from a silicon-on-insulator microchip (see App. A).
b
, SEM of the
main part of the circuit, which consists of a left and a right nanobeam optomechanical crystal cavity with a central unpatterned
nanobeam waveguide connecting the two cavities. A left and right optical coupler, which are each fed by an adiabatic fiber-to-
chip coupler [31], are used to evanescently couple light into either of the two optical cavities.
c
, FEM simulated electrical field
E
y
and magnitude of the displacement field for the localized optical and mechanical cavity modes, respectively, of the nanobeam.
d
, FEM simulated section of the corresponding optical and mechanical modes of the connecting waveguide.
e,
Optical reflection
spectrum of the left (blue) and right (orange) optical cavities.
f
, Optically transduced mechanical power spectral density (PSD)
measured from the left (blue) and right (orange) optical cavities.
M
±
are the two hybridized mechanical cavity modes with
frequency
ω
M
+(
−
)
/
2
π
= 5788
.
4 (5779
.
1) MHz and
M
W
is a mechanical waveguide mode with frequency
ω
M
W
/
2
π
= 5818
.
3 MHz.
where
ω
≡
ω
s
−
ω
p
and
ω
s
is the frequency of the probe
light. Γ
±
is the amplitude of the effective mechanically-
mediated coupling between the two optical cavities, and
is given by
Γ
±
[
ω
] =
V G
L
G
R
(
−
i
(
ω
±
ω
mL
) +
γ
iL
2
)(
−
i
(
ω
±
ω
mR
) +
γ
iR
2
) +
V
2
.
(2)
The prefactor
A
±
[
ω
] in Eq. (1) accounts for reflection
and loss at the optical cavity couplers, as well as the
mechanically-induced back-action on the optical cavities.
This prefactor is independent of the transmission direc-
tion, and for the reverse transmission amplitude
T
R
→
L
,
only the sign in front of Φ
B
changes.
The directional nature of the optical probe transmis-
sion may be studied via the frequency-dependent ratio
(
T
L
→
R
T
R
→
L
)
[
ω
; ∆ =
±
ω
m
] =
J
−
Γ
±
[
ω
]
e
−
i
Φ
B
J
−
Γ
±
[
ω
]
e
+
i
Φ
B
.
(3)
Although the presence of the synthetic flux breaks time-
reversal symmetry, it does not in and of itself result
in nonreciprocal photon transmission magnitudes upon
swapping input and output ports [25, 33]. In our sys-
tem, if one takes the limit of zero intrinsic mechanical
damping (i.e.
γ
ik
= 0), the mechanically-mediated cou-
pling amplitude Γ
±
[
ω
] is real at all frequencies. This
implies
|
T
L
→
R
|
=
|
T
R
→
L
|
, irrespective of the value of Φ
B
.
We thus find that non-zero mechanical dissipation will be
crucial in achieving any non-reciprocity in the magnitude
of the optical transmission amplitudes.
The general reservoir-engineering approach to nonre-
ciprocity introduced in Ref. [25] provides a framework
for both understanding and exploiting the above obser-
vation. It demonstrates that nonreciprocity is generically
achieved by balancing a direct (Hamiltonian) coupling
between two cavities against a dissipative coupling of the
cavities; such a dissipative coupling can arise when both
cavities couple to the same dissipative reservoir. The
balancing requires both a tuning of the magnitude of the
coupling to the bath, as well as a relative phase which
plays a role akin to the flux Φ
B
. In our case, the damped
mechanical modes can play the role of the needed reser-
voir, with the optical drives controlling how the optical
cavities couple to this effective reservoir. One finds that
at any given frequency
ω
, the mechanical modes induce
both an additional coherent coupling between the two
cavities (equivalent to an additional coupling term in the
Hamiltonian) as well as a dissipative coupling (which is
not describable by a Hamiltonian). As is shown explic-
itly in App, B, in the present setting these correspond
directly to the real and imaginary parts of Γ
±
[
ω
]. Hence,
the requirement of having Im Γ[
ω
]
6
= 0 is equivalent to
requiring a non-zero mechanically-mediated dissipative
coupling between the cavities.
Achieving directionality requires working at a fre-
quency where the dissipative coupling has the correct
magnitude to balance the coherent coupling
J
, and a tun-
4
ing of the flux Φ
B
. For
|
Γ
±
[
ω
]
|
=
J
and arg(Γ
±
) =
−
Φ
B
(
6
= 0
,π
), one obtains purely uni-directional transport
where the right optical cavity is driven by the left optical
cavity but not vice versa. One finds from Eq. (3) that
the mechanically-mediated dissipative coupling between
the cavities is maximized at frequencies near the mechan-
ical normal mode frequencies
ω
≈ −
ω
m
±
V
; to achieve
the correct magnitude of coupling, the optical pumping
needs to realize a many-photon optomechanical coupling
G
k
≈
(
Jγ
ik
)
1
/
2
(see App. B for details). Note that our
discussion applies to both the choices of red-detuned and
blue-detuned pumping. While the basic recipe for direc-
tionality is the same, in the blue-detuned pump case the
effective reservoir seen by the cavity modes can give rise
to negative damping, with the result that the forward
transmission magnitude can be larger than one. We ex-
plore this more in what follows.
In order to realize the optomechanical circuit depicted
in Fig. 1 we employ the device architecture of optome-
chanical crystals [34–36], which allows for the realization
of integrated cavity-optomechanical circuits with versa-
tile connectivity and cavity coupling rates [21, 37]. Fig-
ure 2a shows the optomechanical crystal circuit fabri-
cated on a silicon-on-insulator microchip. The main sec-
tion of the circuit, shown zoomed-in in Fig. 2b, con-
tains two optomechanical crystal nanobeam cavities,
each of which has an optical resonance of wavelength
λ
≈
1530 nm and a mechanical resonance of frequency
ω
m
/
2
π
≈
6 GHz. The two optical cavities can be ex-
cited through two separate optical coupling paths, one
for coupling to the left cavity and one for the right cav-
ity. Both the left and right optical coupling paths con-
sist of an adiabatic fiber-to-chip coupler which couples
light from an optical fiber to a silicon waveguide, and
a near-field waveguide-to-cavity reflective coupler. This
allows separate optical pumping of each cavity and opti-
cal transmission measurements to be carried out in either
direction. The two nanobeam cavities are physically con-
nected together via a central silicon beam section which
is designed to act as both an optical waveguide and an
acoustic waveguide. The central beam thus mediates
both photon hopping and phonon hopping between the
two cavities even though the cavities are separated by a
distance much larger than the cavity mode size [21, 38].
The numerically simulated mode profiles for the local-
ized cavities and the connecting waveguide are shown in
Fig. 2c and 2d, respectively. The hopping rate for pho-
tons and phonons can be engineered by adjusting the
number and shape of the holes in the mirror section of
the optomechanical crystal cavity along with the free-
spectral range of the connecting waveguide section [21].
Here we aim for a design with
J/
2
π
≈
100 MHz and
V/
2
π
≈
3 MHz so that nonreciprocity can be realized at
low optical pump power, yet still with high transmission
efficiency.
As will be presented elsewhere [39], the optical and me-
chanical frequencies of the optomechanical cavities are
independently trimmed into alignment post-fabrication
using an atomic force microscope to oxidize nanoscale
regions of the cavity.
After nano-oxidation tuning,
the left (right) cavity has optical resonance wavelength
λ
L(R)
= 1534
.
502 (1534
.
499) nm, total loaded damping
rate
κ
L(R)
/
2
π
= 1
.
03 (0
.
75) GHz, and intrinsic cavity
damping rate
κ
iL(R)
/
2
π
= 0
.
29 (0
.
31) GHz (c.f. Fig. 2e).
Note that hybridization of the optical cavity resonances
is too weak to be observable in the measured left and
right cavity spectra due to the fact that the optical cav-
ity linewidths are much larger than the designed cavity
coupling
J
. The thermal mechanical spectra, as mea-
sured from the two cavities using a blue-detuned pump
laser (see App. A), are shown in Fig. 2f where one can
see hybridized resonances
M
±
which are mixtures of
the localized mechanical cavity modes
M
L
and
M
R
. A
nearby phonon waveguide mode (
M
W
) is also observable
in both left and right cavity spectra. The optomechanical
coupling rate and mechanical dissipation rate of
M
L(R)
were measured before nano-oxidation tuning, yielding
g
0
,
L(R)
/
2
π
= 0
.
76 (0
.
84) MHz and
γ
iL(R)
/
2
π
= 4
.
3
(5
.
9) MHz.
The experimental apparatus used to drive and probe
the optomechanical circuit is shown schematically in
Fig. 3a. As indicated, an optical pump field for the
left and right cavities is generated from a common diode
laser. The phase difference of the pump fields at the in-
put to the cavities, and thus the synthetic magnetic flux,
is tuned by a stretchable fiber phase shifter and stabilized
by locking the interference intensity of the reflected pump
signals from the cavities. To highlight the unique kinds of
nonreciprocal transport possible in our setup, we present
results for an experiment performed with blue-detuned
pump fields with frequency
ω
p
≈
ω
c
+
ω
m
; as discussed,
this will enable non-reciprocal transport with gain. An
input optical probe signal is generated from either of the
left or right cavity pump beams by sending them through
an electro-optic modulator (EOM). A vector network an-
alyzer (VNA) is used to drive the EOMs at modulation
frequency
ω
mod
and detect the photocurrent generated by
the beating of the transmitted probe and reflected pump
signals, thus providing amplitude and phase information
of the transmitted probe signal. Owing to the spectral
filtering of the cavities, only the generated lower sideband
of the blue-detuned pump at
ω
=
−
ω
mod
is transmitted
through the circuit as a probe signal.
Figure 3b shows the ratio of the forward and back-
ward optical power transmission coefficients of the probe
light (
|
T
L
→
R
/T
R
→
L
|
2
) for several magnetic flux values be-
tween Φ
B
= 0 and
π
. For these measurements the pump
powers at the input to the left and right cavity were set
to
P
pL
=
−
14
.
2 dBm and
P
pR
=
−
10
.
8 dBm, respec-
tively, corresponding to intra-cavity photon numbers of
n
cL
= 1000 and
n
cR
= 1420. So as to remove differences
in the forward and reverse transmission paths external
to the optomechanical circuit, here the
|
T
L
→
R
/T
R
→
L
|
2
ratio is normalized to 0 dB for a modulation frequency
ω
mod
/
2
π
≈
5
.
74 GHz, detuned far from mechanical res-
onance in a frequency range where reciprocal transmis-
5
λ-meter
EDFA
VOA
EOM
VNA
EOM
PD
FPC
FPC
FPC
FPC
φ
-shifter
PD
5.77
5.78
5.79
5.8
-20
-10
0
10
20
30
40
5.77
5.78
5.79
5.8
5.77
5.78
5.79
5.8
-20
-10
0
10
20
30
40
5.77
5.78
5.79
5.8
5.77
5.78
5.79
5.8
5.77
5.78
5.79
5.8
Frequency (GHz)
Transmission ratio (dB)
a
b
d
Frequency (GHz)
c
0
5
10
-10
0
-20
-26
-24
-22
-20
-18
-16
-14
P
p
(dBm)
Amplication (dB)
5.775
5.78
5.785
5.79
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
dB
Frequency (GHz)
e
-30
-20
-10
0
10
20
30
p
p
L
R
cL
cR
|
L
|
e
i
L
|
R
|
e
i
R
B
=
0.18
B
=
0.26
B
=
0.34
B
=
1.18
B
=
1.26
B
=
1.34
B
/
Transmission ratio (dB)
p
p
Attenuation (dB)
FIG. 3.
Measurement of optical nonreciprocity. a
, Experiment set-up. Red (blue) lines are optical (electronic) wiring.
Blue-detuned pump light from a tunable diode laser is split into two paths and fed into the two cavities (red arrows). Part of
the reflected pump laser light from the cavities (purple arrows) is collected by a photodetector (PD) and fed into a stretchable
fiber phase shifter (
φ
-shifter) to tune and lock the phase difference of the optical pumps. Each optical path can be modulated
by an electro-optic modulator (EOM) to generate an optical sideband which we use as the optical probe signal. The microwave
modulation signal with frequency
ω
mod
is generated by port 1 of a vector network analyzer (VNA). After optical amplification
and photodetection, the transmitted optical probe signal through the optomechanical circuit is sent back to port 2 of the VNA
to measure the phase and amplitude of the optical probe transmission coefficient. EDFA: Erbium doped fiber amplifier, FPC:
fiber polarization controller,
λ
-meter: wavelength meter.
b
, The ratio of optical power transmission coefficients for right- and
left-propagation versus modulation frequency (
ω
mod
=
−
ω
=
ω
p
−
ω
s
), for three different synthetic flux values Φ
B
/π
= 0
.
18,
0
.
26, and 0
.
34. The blue curves correspond to the fit of the theoretical model (c.f. Eq. 3) to the measured spectra.
c
, The
power transmission coefficient ratio for Φ
B
with an additional
π
flux relative to those in
b
.
d
, Theoretical calculation of the
power transmission coefficient ratio for 0
≤
Φ
B
≤
2
π
, where the six grey lines correspond to the six measured Φ
B
values in
b
and
c
.
e
, Peak forward signal amplification above background level (blue squares) and corresponding signal attenuation in the
reverse direction (red circles) versus average optical pump power (
̄
P
p
=
√
P
pL
P
pR
) for fixed flux value of Φ
B
= 0
.
28
π
. The
solid curves are theoretical calculations based upon the theoretical model (c.f. Eq. 3 and SI) fit to the data in
b
and
c
.
sion is expected. Closer to mechanical resonance, strong
nonreciprocity in the optically transmitted power is ob-
served, with a peak and a dip in
|
T
L
→
R
/T
R
→
L
|
2
oc-
curring roughly at the resonance frequencies of the hy-
bridized mechanical modes
M
+
and
M
−
, respectively
(c.f. Fig. 2c). The maximum contrast ratio between
forward and backward probe transmission – the isola-
tion level – is measured to be 35 dB for Φ
B
= 0
.
34
π
near the
M
+
resonance. The forward transmission is
also amplified in this configuration (blue-detuned pump,
∆ = +
ω
m
), with a measured peak probe signal amplifi-
cation of 12 dB above the background level set by photon
hopping alone (
J/
|
Γ
±
|
1). The corresponding port-to-
port net gain is only 1 dB due to impedance mismatching
(
J
6
=
κ/
2) and intrinsic optical cavity losses (see SI for
details).
From a two-parameter fit to the measured optical
power transmission ratio spectra using Eq. 3 (see blue
curves in Figs. 3b and 3c), we obtain a waveguide-
mediated optical and mechanical hopping rate of
J/
2
π
=
110 MHz and
V/
2
π
= 2
.
8 MHz, respectively, consistent
with our design parameters. Figure 3d shows the theo-
retical calculation of
|
T
L
→
R
/T
R
→
L
|
2
for a full 2
π
range of
Φ
B
with the measured and fit optomechanical circuit pa-
rameters. The pattern is seen to be odd symmetric with
respect to Φ
B
=
π
. Inserting an additional magnetic flux
π
into the measurements performed in Fig. 3b yields the
spectra shown in Fig. 3c which displays a switch in the
isolation direction as predicted by the model. The pump
power dependence of the peak (in frequency) forward sig-
nal amplification and the corresponding backward signal
attenuation relative to the background level far from me-
chanical resonance are shown in Fig. 3e for a fixed mag-
netic flux of Φ
B
= 0
.
28
π
. Good correspondence with the
theoretical power dependence (solid curves) is observed,
with nonreciprocal amplification vanishing at low pump
power.
One can also obtain nonreciprocal optical power trans-
mission utilizing an even simpler system involving a sin-
gle mechanical cavity. This is the situation we have for
the Fabry-Perot-like mechanical resonances that exist in
the central coupling waveguide (see
M
W
resonance of
6
5.81
5.82
-5
-2.5
0
2.5
5
5.81
5.82
5.81
5.82
5.81
5.82
-5
-2.5
0
2.5
5
5.81
5.82
5.81
5.82
Frequency (GHz)
Frequency (GHz)
dB
dB
a
J
b
c
G
wR
e
i
R
G
wL
e
i
L
O
L
O
R
M
W
i
B
B
=
0.06
B
=
0.31
B
=
0.56
B
=
1.06
B
=
1.31
B
=
1.56
FIG. 4.
Synthetic magnetic field with a single me-
chanical cavity. a
, Physical configuration for generation of
a synthetic magnetic field and optical nonreciprocity with two
optical modes parametrically coupled with a common dissi-
pative mechanical waveguide mode.
b,c
The ratio of optical
power transmission coefficients for right and left propagation
versus modulation frequency
ω
mod
around the frequency of
the waveguide mode
M
W
for various Φ
B
. The blue curves
correspond to a fit of the theoretical model (see App. B) to
the measured data.
Fig. 2c). As depicted in Fig. 4a, the mode configura-
tion in this case consists of two optical cavity modes (
O
L
and
O
R
) coupled together via the optical waveguide, one
mechanical waveguide mode
M
W
which is parametrically
coupled to each of the optical cavity modes, and the syn-
thetic magnetic flux Φ
B
=
φ
L
−
φ
R
due to the relative
phases of the optical pump fields threading the triangular
mode space. In Fig. 4b and 4c we show the measurement
of
|
T
L
→
R
/T
R
→
L
|
2
for a series of different flux values Φ
B
with blue-detuned pumping (∆
≈
+
ω
M
W
) at levels of
n
cL
= 770 and
n
cR
= 1090. In this single mechanical
mode case the direction of the signal propagation is de-
termined by the magnitude of the flux; Φ
B
≤
π
leads
to backward propagation and Φ
B
≥
π
to forward prop-
agation. The lower contrast ratio observed is a result of
the weaker coupling between the localized optical cavity
modes and the external waveguide mode, which for the
modest pump power levels used here (
.
100
μ
W) does
not allow us to reach the parametric coupling required
for strong directional transmission.
While our focus has been on the propagation of in-
jected coherent signals through the optomechanical cir-
cuit, it is also interesting to consider the flow of noise.
As might be expected, the induced directionality of our
system also applies to noise photons generated by the up-
conversion of both thermal and quantum fluctuations of
the mechanics; see App. C for detailed calculations. One
finds that for the system of Fig. 2, the spectrally-resolved
photon noise flux shows high directionality, but that the
sign of this directionality changes as a function of fre-
quency (analogous to what happens in the transmission
amplitudes). In contrast, in the single-mechanical mode
setup of Fig. 4 the sign of the directionality is constant
with frequency, and thus the total (frequency-integrated)
noise photon flux is directional depending upon the flux
magnitude. The laser pump fields can thus effectively
act as a heat pump, creating a temperature difference
between the left and right waveguide output fields. The
corresponding directional flow of quantum noise is espe-
cially useful for quantum information applications, as it
can suppress noise-induced damage of a delicate signal
source like a qubit [25, 27].
The device studied in this work highlights the poten-
tial for optomechanics to realize synthetic gauge fields
and novel forms of nonreciprocity enabled by harness-
ing mechanical dissipation. Using just a few modes, it
was possible to go beyond simply mimicking the physics
of an isolator and realize a directional optical amplifier.
By adding modes, an even greater variety of behaviours
could be achieved. For example, the simple addition of
a third optical cavity mode, tunnel-coupled to the first
two cavities but with no mechanical coupling, would real-
ize a photon circulator similar to the phonon circulators
considered in Ref. [18]. Scaling the synthetic gauge field
mechanism realized here to a full lattice of optomechani-
cal cavities would allow the study of topological phenom-
ena in the propgation of both light and sound. Predicted
effects include the formation of back-scattering immune
photonic [1] and phononic [20] chiral edge states, topo-
logically nontrivial phases of hybrid photon-phonon exci-
tations [20], dynamical gauge fields [40], and, in the case
of non-particle-conserving interactions enabled by blue-
detuned optical pumping, topologically protected inelas-
tic scattering of photons [28] and even protected ampli-
fying edge states [29].
ACKNOWLEDGMENTS
The authors would like to thank Michael Roukes for
the use of his atomic force microscope in the nanoox-
idation tuning of the cavities.
This work was sup-
ported by the AFOSR-MURI Quantum Photonic Mat-
ter, the ARO-MURI Quantum Opto-Mechanics with
Atoms and Nanostructured Diamond (grant N00014-15-
1-2761), the University of Chicago Quantum Engineer-
ing Program (AAC,AM), the ERC Starting Grant OP-
TOMECH (FM), the Institute for Quantum Information
and Matter, an NSF Physics Frontiers Center with sup-
port of the Gordon and Betty Moore Foundation, and
the Kavli Nanoscience Institute at Caltech.
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Appendix A: Device Fabrication and Methods
1. Device fabrication and atomic force microscope nano-oxidation tuning
The devices were fabricated from a silicon-on-insulator wafer with a silicon device layer thickness of 220 nm and
buried-oxide layer thickness of 2
μ
m. The device geometry was defined by electron-beam lithography followed by
inductively coupled plasma reactive ion etching to transfer the pattern through the 220 nm silicon device layer. The
devices were then undercut using an HF:H
2
O solution to remove the buried oxide layer and cleaned using a piranha
etch.
After device fabrication, we used an atomic force microscope to draw nanoscale oxide patterns on the silicon device
surface. This process modifies the optical and mechanical cavity frequencies in a controllable and independent way
with the appropriate choice of oxide pattern. The nano-oxidation process was carried out using an Asylum MFP-3D
atomic force microscope and conductive diamond tips (NaDiaProbes) in an environment with relative humidity of
48%. The tip was biased at a voltage of
−
11
.
5 V, scanned with a velocity of 100 nm/s, and run in tapping mode with
an amplitude of 10 nm. The unpassivated silicon device surface was grounded.
2. Optical transmission coefficient measurement
Frequency (GHz
)
5.74
5.76
5.78
5.8
5.82
Transmission coefficient (dB)
-80
-75
-70
-65
-60
-55
-50
-45
-40
-35
Frequency (GHz
)
5.74
5.76
5.78
5.8
5.82
-80
-75
-70
-65
-60
-55
-50
-45
-40
-35
b
a
Transmission coefficient (dB)
B
=
0.34
B
=
1.34
FIG. 5.
a
Microwave signal power transmission through the optomechanical circuit for forward (right-propagation; blue) and
backward (left-propagation; blue curve) directions, with flux set to Φ
B
= 0
.
34
π
and cavity photon number
n
cL
= 1000 and
n
cR
= 1420.
b
Same as
a
but with Φ
B
= 1
.
34
π
.
To measure the optical power transmission through the optomechanical circuit we used a vector network analyzer
(VNA). The VNA outputs a microwave tone from port 1 with frequency
ω
mod
to an electro-optic modulator which
modulates the optical pump to generate an optical sideband corresponding to the optical probe. In the case of a
blue-detuned pump from the optical cavity resonance, the probe field corresponds to the lower sideband (selected by
the filtering properties of the cavity itself). Both the optical probe and pump are launched into one optomechanical
cavity in the circuit. At the other cavity, the transmitted optical probe combines with a second pump and the beating
of the two is detected by a high-speed photodetector (both the first and second pump beams are from the same
laser source, and thus phase coherent). The photocurrent signal from the photodetector is sent into port 2 of the
VNA to measure the microwave signal transmission coefficient
T
μ
. Fig. 5 shows
|
T
μ
|
2
for forward (right-propagating;
blue curve) and backward (left-propagating; red curve) directions through the optomechanical circuit as a function
of the modulation frequency
ω
mod
. In Fig. 5a the synthetic flux value is locked to Φ
B
= 0
.
34
π
whereas in Fig. 5b
Φ
B
= 1
.
34
π
. In both flux settings the optical pumping levels were such that the left and right cavity photon numbers