Mechanical On-Chip Microwave Circulator
S. Barzanjeh,
1,
∗
M. Wulf,
1
M. Peruzzo,
1
M. Kalaee,
2, 3
P. B. Dieterle,
2, 3
O. Painter,
2, 3
and J. M. Fink
1,
†
1
Institute of Science and Technology Austria, 3400 Klosterneuburg, Austria
2
Kavli Nanoscience Institute and Thomas J. Watson, Sr., Laboratory of Applied Physics,
California Institute of Technology, Pasadena, CA 91125, USA
3
Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, CA 91125, USA
(Dated: June 7, 2017)
Nonreciprocal circuit elements form an inte-
gral part of modern measurement and commu-
nication systems. Mathematically they require
breaking of time-reversal symmetry, typically
achieved using magnetic materials [1] and more
recently using the quantum Hall effect [2], para-
metric permittivity modulation [3] or Josephson
nonlinearities [4, 5]. Here, we demonstrate an
on-chip magnetic-free circulator based on reser-
voir engineered optomechanical interactions [6,
7]. Directional circulation is achieved with con-
trolled phase-sensitive interference of six distinct
electro-mechanical signal conversion paths. The
presented circulator is compact, its silicon-on-
insulator platform is compatible with both su-
perconducting qubits [8] and silicon photonics,
and its noise performance is close to the quan-
tum limit. With a high dynamic range, a tun-
able bandwidth of up to 30 MHz and an in-situ
reconfigurability as beam splitter or wavelength
converter [9, 10], it could pave the way for su-
perconducting qubit processors with multiplexed
on-chip signal processing and readout.
Nonreciprocal devices are quintessential tools to sup-
press spurious modes, interferences and unwanted signal
paths. More generally, circulators can be used to realize
chiral networks [11] in systems where directional matter-
light coupling is not easily accessible. In circuit quantum
electrodynamics circulators are used for single port cou-
pling or as isolators to protect the vulnerable cavity and
qubit states from electromagnetic noise. State of the art
passive microwave circulators are based on magneto-optic
effects which require sizable magnetic fields incompati-
ble with ultra-low loss superconducting circuits forming
a major roadblock towards a fully integrated quantum
processor based on superconducting qubits.
Many recent theoretical and experimental efforts have
been devoted to overcome these limitations both in the
optical [12–14] and microwave regimes [2–5, 15–18]. In
parallel, the rapidly growing field of optomechanical and
electromechanical systems has shown promising potential
for applications in quantum information processing and
communication, in particular for microwave to optical
∗
shabir.barzanjeh@ist.ac.at
†
jfink@ist.ac.at
conversion [19, 20] and amplification [21]. Very recently,
several theoretical proposals [6, 22, 23] have pointed out
that optomechanical systems can lead to nonreciprocity
and first isolators have just been demonstrated in the
optical domain [24–26]. Here, we present an on-chip mi-
crowave circulator using a new and tunable silicon elec-
tromechanical system.
The main elements of the microchip circulator device
are shown in Fig. 1 a-b. The circuit is comprised of
three high-impedance spiral inductors (
L
i
) capacitively
coupled to the in-plane vibrational modes of a dielectric
nanostring mechanical resonator. The nanostring oscilla-
tor consists of two thin silicon beams that are connected
by two symmetric tethers and fabricated from a high re-
sistivity silicon-on-insulator device layer [27]. Four alu-
minum electrodes are aligned and evaporated on top of
the two nanostrings, forming one half of the vacuum gap
capacitors that are coupled to three microwave resonators
and one DC voltage bias line as shown schematically in
Fig. 1c (see App. A for details).
The voltage bias line can be used to generate an attrac-
tive force which pulls the nanobeam and tunes the oper-
ating point frequencies of the device [9]. Fig. 1 d shows
the measured resonance frequency change as a function
of the applied bias voltage
V
dc
. As expected, resonators
1 and 3 are tuned to higher frequency due to an increased
vacuum gap while resonator 2 is tuned to lower frequency.
A large tunable bandwidth of up to 30 MHz as obtained
for resonator 2, the ability to excite the motion directly
and to modulate the electromechanical coupling in-situ
represents an important step towards new optomechani-
cal experiments and more practicable on-chip reciprocal
and nonreciprocal devices.
As a first step we carefully calibrate and character-
ize the individual electromechanical couplings and noise
properties. We then measure the bidirectional frequency
conversion between two microwave resonator modes as
mediated by one mechanical mode [10]. The incoming
signal photons can also be distributed to two ports with
varying probability as a function of the parametric drive
strength and in direct analogy to a tunable beam split-
ter. We present the experimental results, the relevant
sample parameters and the theoretical analysis of this
bidirectional frequency conversion process in App. B.
Directionality is achieved by engaging the second me-
chanical mode, a method which was developed in paral-
lel to this work [28, 29] for demonstrating nonreciproc-
arXiv:1706.00376v2 [quant-ph] 6 Jun 2017
2
Resonator 1
Nanostring
Port 1
Port 2
V
dc
(V)
- 4
- 2
0
2
4
11.38
11.37
9.58
9.59
9.60
9.77
9.78
9.79
9.80
Frequency (GHz)
C
s,1
C
m,1
C
m,3
C
m,2
C
m
L
1
C
s,3
L
3
C
s,2
L
2
a
1,in
a
2,out
port 1
a
3,in
a
3,out
port 2
a
2,in
a
2,out
V
dc
mode 2
mode 1
a
d
c
b
V
dc
10
¹
m
100
¹
m
Resonator 3
Resonator 2
Resonator 1
Resonator 2
Resonator 3
FIG. 1.
Microchip circulator and tunability. a
, Scan-
ning electron micrograph of the electromechanical device in-
cluding three microwave resonators, two physical ports, one
voltage bias input (
V
dc
) and an inset of the spiral inductor
cross-overs (green dashed boxed area).
b
, Enlarged view of
the silicon nanostring mechanical oscillator with four vacuum-
gap capacitors coupled to the three inductors and one voltage
bias. Insets show details of the nanobeam as indicated by the
dashed and dotted rectangles.
c
, Electrode design and elec-
trical circuit diagram of the device. The input modes
a
i,in
couple inductively to the microwave resonators with induc-
tances
L
i
, coil capacitances
C
i
, additional stray capacitances
C
s,i
, and the motional capacitances
C
m,i
. The reflected tones
a
i,out
pass through a separate chain of amplifiers each, and are
measured at room temperature using a phase locked spectrum
analyzer (not shown). The simulated displacement of the low-
est frequency in-plane flexural modes of the nanostring are
shown in the two insets.
d
, Resonator reflection measurement
of the three microwave resonators of an identical device, as a
function of the applied bias voltage and a fit (dashed lines)
to ∆
ω
=
α
1
V
2
+
α
2
V
4
with the tunabilties
α
1
/
2
π
= 0
.
53
MHz/V
2
and
α
2
/
2
π
= 0
.
05 MHz/V
4
with a total tunable
bandwidth of 30 MHz for resonator 2 at 9.8 GHz.
ity in single-port electromechanical systems. We begin
with the theoretical model describing two microwave cav-
ities with resonance frequencies
ω
i
and total linewidths
κ
i
with
i
= 1
,
2 parametrically coupled to two distinct
modes of a mechanical resonator with resonance frequen-
cies
ω
m,j
and damping rates
γ
m,j
with
j
= 1
,
2. To es-
tablish the parametric coupling, we apply four microwave
tones, with frequencies detuned by
δ
j
from the lower mo-
tional sidebands of the resonances, as shown in Fig. 2a.
In a reference frame rotating at the frequencies
ω
i
and
ω
m,j
+
δ
j
, the linearized Hamiltonian in the resolved side-
band regime (
ω
m,j
κ
1
,κ
2
) is given by ( ̄
h
= 1)
H
=
−
∑
j
=1
,
2
δ
j
b
†
j
b
j
+
∑
i,j
=1
,
2
G
ij
(
e
iφ
ij
a
i
b
†
j
+
e
−
iφ
ij
a
†
i
b
j
)
+
H
off
,
(1)
where
a
i
(
b
j
) is the annihilation operator for the cavity
i
(mechanics
j
),
G
ij
=
g
0
,ij
√
n
ij
and
g
0
ij
are the ef-
fective and vacuum electromechanical coupling rates be-
tween the mechanical mode
j
and cavity
i
respectively,
while
n
ij
is the total number of photons inside the cavity
i
due to the drive with detuning ∆
ij
, and
φ
ij
is the rel-
ative phase set by drives. Here, ∆
11
= ∆
21
=
ω
m,
1
+
δ
1
and ∆
22
= ∆
12
=
ω
m,
2
+
δ
2
are the detunings of the
drive tones with respect to the cavities and
H
off
de-
scribes the time dependent coupling of the mechanical
modes to the cavity fields due to the off-resonant drive
tones.
These additional coupling terms create cross-
damping [30] and renormalize the mechanical modes, and
can only be neglected in the weak coupling regime for
G
ij
,κ
j
ω
j
,
|
ω
m,
2
−
ω
m,
1
|
.
To see how the nonreciprocity arises we use the quan-
tum Langevin equations of motion along with the input-
output theorem to express the scattering matrix
S
ij
of
the system described by the Hamiltonian (1), and relat-
ing the input photons
a
in
,i
(
ω
i
) at port
i
to the output
photons
a
out
,j
(
ω
j
) at port
j
via
a
out
,i
=
∑
j
=1
,
2
S
ij
a
in
,i
with
i
= 1
,
2. The dynamics of the four-mode sys-
tem described by Hamiltonian (1) is fully captured by
a set of linear equations of motion as verified in App. C.
Solving these equations in the frequency domain, us-
ing the input-output relations, and setting
φ
22
=
φ,
φ
11
=
φ
21
=
φ
12
= 0, the ratio of backward to forward
transmission reads
λ
:=
S
12
(
ω
)
S
21
(
ω
)
=
√
C
11
C
21
Σ
m,
2
(
ω
) +
√
C
12
C
22
Σ
m,
1
(
ω
)
e
iφ
√
C
11
C
21
Σ
m,
2
(
ω
) +
√
C
12
C
22
Σ
m,
1
(
ω
)
e
−
iφ
.
(2)
Here, Σ
m,j
= 1 + 2
i
[
(
−
1)
j
δ
−
ω
]
/γ
m,j
is the inverse of
the mechanical susceptibility divided by the mechanical
linewidth
γ
m,j
and
C
ij
= 4
G
2
ij
/
(
κ
i
γ
m,j
) is the optome-
chanical cooperativity. Note that, in Eq. (2) we assume
the device satisfies the impedance matching condition on
resonance i.e.
S
ii
(
ω
= 0) = 0 which can be achieved in
the high cooperativity limit (
C
ij
1).
Inspection of equation (2) reveals the crucial role of
the relative phase between the drive tones
φ
and the de-
tuning
δ
to obtain nonreciprocal transmission. When the
cooperativities for all four optomechanical couplings are
equal (
C
ij
=
C
) then perfect isolation, i.e.
λ
= 0, occurs
for
tan[
φ
(
ω
)] =
δ
(
γ
m,
1
+
γ
m,
2
) +
ω
(
γ
m,
2
−
γ
m,
1
)
γ
m,
1
γ
m,
2
/
2
−
2(
δ
2
−
ω
2
)
.
(3)
Equation 3 shows that on resonance (
ω
= 0) tan[
φ
]
∝
δ
,
highlighting the importance of the detuning
δ
to obtain
nonreciprocity. Tuning all four drives to the exact red
sideband frequencies (
δ
= 0) results in bidirectional be-
havior (
λ
= 1). At the optimum phase
φ
given by
Eq. (3),
ω
= 0, and for two mechanical modes with iden-
tical decay rates (
γ
m,
1
=
γ
m,
2
=
γ
) the transmission in
forward direction is given by
S
21
=
−
√
η
1
η
2
4
iδ
(1
−
2
iδ/γ
)
C
γ
(
1 +
1+4
δ
2
/γ
2
2
C
)
2
(4)
3
0
-20
-40
dB
Phase = -102.6
1.9 dB ca
vit y loss
2.4 dB inser
tion loss
C
2
C
1
1.9 dB ca
vit y loss
2.8 dB inser
tion loss
2
C
Phase = +102.6
C
2
C
1
C
Data
Data
Theory
Theory
C
1
M
2
M
1
C
2
a
b
c
G
11
G
12
a
1,in
a
1,out
a
2,in
a
2,out
G
21
G
22
PSD
±
1
+
!
m,1
!
d,11
±
2
+
!
m,2
!
d,12
±
1
+
!
m,1
!
d,21
±
2
+
!
m,2
!
d,22
!
S
12
≠
S
21
Transmission (dB)
Probe detuning (Hz)
Pump phase (deg)
-40
-700
-180
-90
0
90
180
-180
-90
0
90
180
-350
350
700
0
-700
-350
350
700
0
-30
-20
-10
0
j
S
21
j
2
j
S
21
j
2
j
S
12
j
2
j
S
12
j
2
j
S
21
j
2
j
S
12
j
2
j
S
21
j
2
j
S
12
j
2
FIG. 2.
Optomechanical isolator. a
, Mode coupling dia-
gram for optomechanically induced nonreciprocity. Two mi-
crowave cavities (
C
1
and
C
2
) are coupled to two mechanical
modes (
M
1
and
M
2
) with the optomechanical coupling rates
G
ij
(where
i,j
= 1
,
2), inducing two distinct signal conversion
paths. Power spectral density (PSD) of the two microwave
cavities and arrows indicating the frequency of the four mi-
crowave pump tones slightly detuned by
δ
i
from the lower
motional sidebands of the resonances. All four pumps are
phase-locked while the signal tone is applied. Only one of
the microwave source phases is varied to find the optimal in-
terference condition for directional transmission between port
1 and 2.
b
, Measured power transmission (dots) in forward
|
S
21
|
2
(cavity 1
→
cavity 2) and backward directions
|
S
12
|
2
(cavity 2
→
cavity 1) as a function of probe detuning for two
different phases
φ
=
±
102
.
6 degrees. The solid lines show the
results of the coupled-mode theory model discussed in the
text.
c
, Experimental data (top) and theoretical model (bot-
tom) of measured transmission coefficients
|
S
12
|
2
and
|
S
21
|
2
as
a function of signal detuning and pump phase
φ
. Dashed-lines
indicate the line plot locations of panel b.
where
η
1(2)
=
κ
ext
,
1(2)
/κ
1(2)
is the resonator coupling ra-
tio and
κ
i
=
κ
int
,i
+
κ
ext
,i
is the total damping rate.
Here
κ
int
,i
denotes the internal loss rate and
κ
ext
,i
the
loss rate due to the cavity to waveguide coupling. Equa-
tion (4) shows that the maximum of the transmission in
forward direction,
|
S
21
|
2
=
η
1
η
2
[1
−
(2
C
)
−
1
], occurs when
2
C
= 1 + 4
δ
2
/γ
2
and for large cooperativities
C
1.
These conditions, as implemented in our experiment, en-
able the observation of asymmetric frequency conversion
with strong isolation in the backward direction and small
insertion loss in forward direction.
Using the on-chip electromechanical microwave circuit
shown in Fig. 1 a, we experimentally realize directional
wavelength conversion between two superconducting coil
resonators at (
ω
1
,ω
2
)
/
2
π
= (9
.
55
,
9
.
82) GHz coupled to
two different physical waveguide ports and measurement
lines with (
η
1
,η
2
) = (0
.
74
,
0
.
86). Here, we use the two
lowest-frequency vibrational in-plane modes of the me-
chanical resonator at (
ω
m,
1
,ω
m,
2
)
/
2
π
= (4
.
34
,
5
.
64) MHz
with intrinsic damping rates (
γ
m,
1
,γ
m,
2
)
/
2
π
= (4
,
8) Hz.
The vacuum optomechanical coupling strengths for these
mode combinations are (
g
0
,
11
,g
0
,
12
,g
0
,
21
,g
0
,
22
)
/
2
π
=
(33
,
34
,
13
,
31) Hz.
Figure 2 b shows the measured transmission of the
wavelength conversion in the forward
|
S
21
|
2
and back-
ward directions
|
S
12
|
2
as a function of probe detuning
for two different phases as set by one out of the four
phase locked microwave drives. At
φ
=
−
102
.
6 degree
and over a frequency range of 1
.
5 kHz we measure high
transmission from cavity 1 to 2 with an insertion loss of
2
.
4 dB while in the backward direction the transmission
is suppressed by up to 40
.
4 dB. Likewise, at the positive
phase of
φ
= 102
.
6 degree the transmission from cavity 1
to 2 is suppressed while the transmission from cavity 2 to
1 is high. In both cases we observe excellent agreement
with theory (solid lines). Fig. 2 c shows the S parameters
for the whole range of phases
φ
, which are symmetric and
bidirectional around
φ
= 0. We find excellent agreement
with theory over the full range of measured phases with
less than 10% deviation to independently calibrated drive
photon numbers and without any other free parameters.
For bidirectional wavelength conversion, higher coop-
erativity enhances the bandwidth. In contrast, the band-
width of the nonreciprocal conversion is independent of
cooperativity and set only by the intrinsic mechanical
linewidths
γ
m,i
, which can be seen in Eq (2). This high-
lights the fact that the isolation appears when the en-
tire signal energy is dissipated in the mechanical environ-
ment, a lossy bath that can be engineered effectively [7].
In the present case it is the off-resonant coupling be-
tween the resonators and the mechanical oscillator which
modifies this bath. The applied drives create an effec-
tive interaction between the mechanical modes, where
one mode acts as a reservoir for the other and vice versa.
This changes both the damping rates and the eigenfre-
quencies of the mechanical modes. It therefore increases
the instantaneous bandwidth of the conversion and auto-
matically introduces the needed detuning, which is fully
taken into account in the theory.
The described two-port isolator can be extended to an
effective three-port device by parametrically coupling the
third microwave resonator capacitively to the dielectric
nanostring, as shown in Fig. 1 a. The third resonator at
a resonance frequency of
ω
3
/
2
π
= 11
.
30 GHz is coupled
to the waveguide with
η
3
= 0
.
52 and to the two in-plane
mechanical modes with (
g
0
,
31
,g
0
,
32
)
/
2
π
= (22
,
45) Hz.
Similar to the isolator, we establish a parametric cou-