non-reciprocity in microring resonators
Mohammad Hafezi1∗, Peter Rabl2
1Joint Quantum Institute, NIST/University of Maryland, College Park 20742
2Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences,
6020 Innsbruck, Austria
We describe a new approach for on-chip optical non-reciprocity which
makes use of strong optomechanical interaction in microring resonators. By
optically pumping the ring resonator in one direction, the optomechanical
coupling is only enhanced in that direction, and consequently, the system
exhibits a non-reciprocal response. For different configurations, this system
can function either as an optical isolator or a coherent non-reciprocal
phase shifter. We show that the operation of such a device on the level of
single-photon could be achieved with existing technology.
© 2011 Optical Society of America
OCIS codes: (120.4880,230.3240,270.1670)
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The development of integrated photonic circuits is a rapidly progressing field which aims at the
realization of micron scale photonic elements and the integration of these elements into a single
chip-based device. Apart from conventional optical signal processing and telecommunication
applications , this technology might eventually also provide the basis for applications on a
more fundamental level such as optical quantum computation [2, 3, 4, 5] or photonic quantum
simulation schemes [6, 7, 8]. A remaining challenge in integrated photonic circuits is on-chip
optical isolation, that is, filtering of photons propagating in different directions in the circuit, or
more generally, the implementation of non-reciprocal optical elements on a micrometer scale.
Standard approaches for optical isolation make use of magneto-optical properties (e.g. Fara-
day rotation), which however require large magnetic fields , and thus make it difficult for
integration [10, 11, 12] on a small scale. To overcome this problem, other non-magnetic ap-
proaches have been proposed which, for example, rely on a dynamical modulation of the index
of refraction , stimulated inter-polarization scattering based on opto-acoustic effects ,
modulated dielectric constant  or on optical non-linearities that lead to an intensity depen-
dent isolation [16, 17, 18].
The suitability of different optical isolation schemes will depend very much on the specific
task. While for many commercial applications high bandwidth and robust fabrication tech-
niques are key requirements for optical isolators, this can be different for on-chip quantum
computing and quantum simulation schemes, where low losses, the operation on a single pho-
ton level and also the implementation of coherent non-reciprocal phase shifters are the most
important aspects. An intriguing new direction in this context is the study of quantum Hall
physics with photons, which has recently attracted a lot of interest in the microwave as well as
in the optical domain [19, 20, 21, 22, 23, 24]. Here, apart from new possibilities to simulate
quantum many body systems with light, the appearance of edge states in quantum Hall sys-
tem could also be exploited for a robust transfer of photons and optical delay lines. However,
previous proposals (except Ref. ) can not be easily integrated on chip, while the scheme
in Ref.  does not break the time reversal symmetry and therefore, it is not suitable for
non-reciprocal robust waveguides , or the emulation of real magnetic fields for light.
In this work, we propose a new approach for on-chip optical non-reciprocity which makes
use of the recent advances in the fabrication of on-chip and micron sized optomechanical (OM)
devices [25, 26, 27, 28]. In our scheme, the non-linear coupling between light and a mechanical
mode inside a ring resonator leads to a non-reciprocal response of the OM system, which is
induced and fully controlled by an external driving field. We characterize the input-output rela-
tions of such a device and show that by choosing different configurations the same mechanism
can be employed for optical isolation as well as non-reciprocal phase shifting and routing ap-
plications. We describe under which conditions non-reciprocity is optimized and in particular,
we find that even in the presence of a finite intrinsic mode coupling inside the ring resonator,
non-reciprocal effects remain large for a sufficiently strong OM coupling. In contrast to optical
isolation based on a non-linear response of the OM system , our schemes can in princi-
ple be applied on a single photon level, limited by the up-conversion of thermal phonons only
Fig. 1. Non-reciprocal optomechanical device. (a) A strong pump field enhances the op-
tomechanical coupling between an isolated vibrational mode and the right-circulating opti-
cal mode inside a ring resonator. This results in different transmission properties for right-
and left-moving fields in the waveguide. (b) Optical isolation. (c) Non-reciprocal phase
[29, 30, 31]. Our analysis shows that a noise level below a single photon can be achieved
with present technology, which makes this device a suitable building block for various non-
reciprocal applications in the classical as well as the quantum regime.
2.Optomechanically induced non-reciprocity: A toy model
Before starting with a more general treatment below, we first outline in this section the essence
of the OM induced non-reciprocity for an idealized and slightly simplified setting. Specifically,
to a waveguide as shown in Fig. 1 (a). This configuration is commonly referred to as an all-
pass filter (APF). The ring resonator supports two degenerate right- and left-circulating optical
modes with frequency ωcand bosonic operators aRand aLrespectively. Radial vibrations of the
resonator lead to a modulation of ωcwhich can be modeled by the standard OM Hamiltonian
[32, 33, 34, 35] (¯ h = 1),
Here b is the bosonic operator for the mechanical mode of frequency ωmand g0is the OM
coupling, which corresponds to the optical frequency shift per quantum of motion. Note that
the mechanical mode is extended and varies slowly over the scale of the optical wavelength
. Therefore, the optomechanical coupling does not induce a mixing between the right- and
left-circulating optical modes. In typical experiments g0is very weak and to enhance OM inter-
actions we now assume that the right-circulating resonator mode is excited by an external laser
field of frequency ωL= ωc+∆. In the limit |αR| ? 1, where αRis the classical field amplitude
of the driven mode, we can make a unitary transformation aR→ aR+αRand linearize the OM
coupling around αR. As a result, we obtain an effective Hamiltonian which in the frame rotating
with ωLis given by [32, 33, 34, 35]
where an additional OM frequency shift has been reabsorbed into the definition of ∆ (see Sec. 3
for a more detailed derivation). In Eq. 2, we have introduced the enhanced OM coupling GR=
g0αR, and in view of |GR| ? g0, neglected residual OM interactions ∼ O(g0). We see that the
external driving field creates an asymmetry between left- and right-circulating modes, which
we can exploit for generating non-reciprocal effects.
In order to study the transport properties of light through the OM APF, we use the input-
output formalism [36, 32]. For both propagation directions, we define in- and outgoing fields
which are related by
fR/L,out(t) = fR/L,in(t)+√2κaR/L(t),
where 2κ is the resonator decay rate into the waveguide. Due to the linearity of the effective
OM Hamiltonian (Eq. 2), we can solve the resulting equations of motion in frequency space.
In the following, we are primarily interested in the case where the resonator is driven at or
close to the mechanical red sideband (∆ ≈ −ωm), and only a†b+ab†terms in Eq. 2 will be
resonant. Therefore, in the sense of a rotating wave approximation, we can ignore other off-
resonant contributions in Eq. 2. In the appendix we show that this approximation is justified
and it allows us to describe the transport properties of the systems in terms of a simple 2×2
where δ = ω +∆ is the detuning of the incoming field from the optical resonator resonance.
For our idealized model, there is no scattering between left- and right-moving modes and rR=
rL= 0. In turn, the transmission coefficients are given by
where 2κindenotes the intrinsic photon loss rate of the optical resonator and γmthe mechanical
damping rate. We see that the transmission spectrum of the left-going field is simply that of a
bare APF, while the transmission of the right-going mode is modified by the presence of the
mechanical oscillator. In particular, the expression for tRresembles that of electromagnetically
induced transparency (EIT) , if we assume GRto be the “control field”. As in atomic EIT,
the coupling of the light field to the long-lived mechanical mode (γm? κ) leads to a dip in
the weak coupling limit and to a splitting of the transmission resonance in the strong coupling
regime. This feature has already been suggested in previous works for slowing and stopping
of light using OM systems [38, 30, 39, 40]. The same effect can be used for achieving non-
reciprocity in the following ways:
Optical isolation. Let us first consider a critically coupled ring resonator where κ = κin. In
this case, we see that for frequencies around the ring optical resonance (δ ≈ 0),
Therefore, this configuration realizes an optical diode, where light passes unaltered in one di-
rection, but is completely absorbed in the other direction, as schematically shown in Fig. 1(b).
The frequency window over which this isolation is efficient is approximately G2
coupling limit (GR?κ) and κ in the strong coupling limit, where the width of the EIT window
is 2GRand exceeds the resonator linewidth. A typical non-reciprocal transmission spectrum for
the strong coupling regime is shown in Fig. 2(a), which is that of an optical diode for frequen-
cies around δ ≈ 0. Note that in contrast to conventional optical isolation, no magnetic field is
applied and instead the optical pump breaks the left-right symmetry.
Non-reciprocal phase shifter. Let us now consider the so-called over-coupled regime where
the intrinsic loss is much smaller than the resonator-waveguide coupling (κin?κ). In this case,
the transmittance is close to unity in both directions. However, the left- and right-going fields
R/κ in the weak
Fig. 2. (a) Transmission |tR/L|2of the OM system when operated as an optical isola-
tor (κin= κ). Within the resonator bandwidth, the left-moving field is attenuated while
the right-moving field is almost completely transmitted. For this plot GR= 5κ. (b) Non-
reciprocal phase shifter (κin= 0.01κ). Both the left and the right input field are almost
completely transmitted (> 98%), but acquire different phases, ∆θ = θR−θL. For these
plots γm= 0.
experience a different dispersion and
In general, the phases θRand θLwill be different, and therefore, in this configuration, our
devices acts as a non-reciprocal phase shifter, as schematically shown in Fig. 1(c). Again, in
contrast to conventional magnetic field induced non-reciprocal phases, e.g. Faraday rotation,
our scheme does not require large magnetic fields. As shown in Fig. 2(b), the OM induced
phase difference |θR−θL| can easily be controlled by changing the pump intensity and can
be tuned from zero to about π over a large range of frequencies. Therefore, a maximal non-
reciprocal phase shift can already be achieved for light passing through a single device.
3. General formalism
In this section we present the general formalism for investigating OM induced non-reciprocity.
In particular, we now include the effect of energy non-conserving terms as well as a finite
coupling between the left- and right-circulating resonator modes which have been neglected
in our simplified discussion above. For completeness, we will also extend our discussion to a
slightly more general configuration shown in Fig. 3, where the ring resonator is side-coupled
to two optical waveguides with rates κ and κ?. For κ?= 0, this setting reduces to the APF
configuration from above. Moreover, in the so-called add-drop configuration (κ?= κ, κin≈ 0),
this device can be used for non-reciprocal routing of light.
To account for a more realistic situation we now include the presence of intrinsic defects
inside the ring resonator and model the system by the total Hamiltonian
H = Hom+βa†
Here, in addition to the OM interaction Homgiven in Eq. 2, the second and third terms in this
Hamiltonian represent a coherent scatting of strength β between the two degenerate optical
modes, which is associated with bulk or surface imperfections [41, 42]. The system dynamics
Fig. 3. General add-drop configuration, which can be employed for non-reciprocal photon
routing between the upper and lower waveguide. It reduces to the APF configuration, if the
coupling to the lower waveguide is absent (κ?= 0).
is fully described by the set of quantum Langevin equations (i = L,R)
i,out(t) = f1
i,in(t)+√2κai(t) and f2
together with the relations f1
tween the in- and out-fields. In these equations, κt=κ+κ?+κinis the total ring resonator field
decay rate and the operators f1,2
and lower waveguide (see Fig. 3) and fi,0(t) is a vacuum noise operator associated with the
intrinsic photon loss. Finally, γm= ωm/Qmis the mechanical damping rate for a quality factor
Qmand ξ(t) is the corresponding noise operator. In contrast to the optical fields, the mechan-
ical mode is coupled to a reservoir of finite temperature T such that [ξ(t),ξ†(t?)] = δ(t −t?)
and ?ξ(t)ξ†(t?)? = (Nth+1)δ(t −t?) where Nthis the thermal equilibrium occupation number
of the mechanical mode. Note that the Langevin equation for the mechanical mode, Eq. 10, is
only valid for γm? ωm. For typical mechanical quality factors Qm∼ 104−105this condition
is well satisfied and for most of the results discussed below we will consider the limit γm→ 0,
while keeping a finite thermal heating rate γmNth? kBT/(¯ hQm).
As before, we assume that the clockwise mode of the resonator is driven by a strong classical
field of frequency ωL= ωc+∆0and amplitude E. We make the transformation fR,in(t) →
fR,in(t)+√2κE and write the average field expectation values in the frame rotating with ωL,
In the steady-state, we find that ?b? = −g0(|?aR?|2+|?aL?|2)/ωm. By redefining the detuning
to absorb the OM shift, ∆ = ∆0+2g2
equations in the steady state as
i,out(t) = f2
i,in(t) are δ-correlated field operators for the in-fields in the upper
0(|?aR?|2+|?aL?|2)/ωm, we can rewrite the optical field
In the absence of mode coupling (β = 0), the counter clockwise mode remains empty (?aL? =
0), and we obtain ?aR? = 2κE/(i∆−κt). However, in the presence of mode coupling, we have
and in general both optical modes are excited. As above, we proceed by making the unitary
transformations ai→ ai+?ai? and b → b+?b? and after neglecting terms of O(g0), we arrive
at the linearized OM Hamiltonian
where due to the mode coupling, both circulating modes exhibit an enhanced coupling (Gi=
g0αi) to the mechanical mode. We are primarily interested in the case where the resonator is
driven near the mechanical red sideband (∆=−ωm), where the terms of the form a†
dominant. However, small corrections due to the off-resonant couplings a†
in our general formalism.
We group the OM field operators into a vector v(t) = (b(t),aR(t),aL(t),b†(t),a†
and write the equations of motion in the form
∂tv(t) = −Mv(t)−√2κI1(t)−
Here the coupling matrix M is given by
and the input field vectors are defined as Ii(t) = (0, fi
i = 1,2 and Im(t) = (ξ(t),0,0,ξ†(t),0,0)T. Note that in Eq. 13, we have already omitted con-
tributions from the intrinsic noise operators fi,0(t) which act on the vacuum and therefore do
not contribute to the results discussed below. We solve the equations of motion for the OM
degrees of freedom (a’s and b’s) in the Fourier domain and obtain
˜ v(ω) = (−M+iωI)−1(√2κ˜I1(ω)+
By defining the output field vector
ib†+aib are included
M = i
˜O1(ω) = (0,˜f1
we can rewrite the input-output relation as
and a similar expression can be derived for the out-fields in the second waveguide ˜O2(ω).
Combining Eqs. (15-17), the output fields can be evaluated as a function of the input field
for arbitrary system parameters. Note that due to the presence of non-resonant OM interactions
this means that different quadratures of the input fields have different transmission properties,
an effect which is related to OM squeezing [32, 43, 44]. However, in the appendix we show
that for the relevant parameter regimes this effect is negligible in our device and for a more
transparent discussion we will evaluate below only the relevant phase independent part of the
ib†+aib), the scattering matrix mixes the fi,inwith the conjugate fields f†
i,in. In other words
4.Results and discussion
In the four port device shown in Fig. 3, we can study various different non-reciprocal effects
and apart from the optical diode and phase-shifter settings outlined above the add-drop config-
uration (κ = κ?, κin= 0) could be used to realize a non-reciprocal optical router between the
two waveguides where, e.g., f1
equivalent to the optical diode by interchanging the role of κ?and κinand therefore we can
restrict the following discussion to the transmission amplitudes tR,L(ω) as defined in the two
port scattering matrix in Eq. 4.
Compared to the ideal situation described in Sec. 2, we are now in particular interested in
OM non-reciprocity in the presence of a finite intrinsic mode coupling, β ?=0, where photons in
the left- and right-circulating modes of both the probe and pump field can no longer propagate
independently. Such a coupling is found in many experiments with high-Q micro-resonators
and often attributed to bulk or surface imperfections [41, 42]. As already mentioned above, a
first consequence of this mode mixing is that the pump field is scattered into the left-circulating
mode and we obtain enhanced OM couplings GR,L∼ αR,Lfor both propagation directions (see
Eq. 11). More specifically, for a purely right-going pump field, the intra-resonator fields are
L,out. However, this situation is formally
and these expressions are plotted in Fig. 4 as a function of the pump detuning ∆ and for the
case of large mode coupling (β ? κt). We see that in principle an asymmetric pumping can
be achieved either for ∆ = 0 or |∆| ? β. However, to achieve a resonant OM coupling, we
should choose ∆ ? −ωm. Therefore, |GL|/|GR| ∼ β/?ω2
out that a complete cancellation of GLcould be achieved by adding a second pump beam
in the left-circulating direction. In particular, if the strength of the left input pump is chosen
as E?= −iβ/(i∆−κt)E, then ?αR? = 2κE/(i∆−κt) and ?αL? = 0. In the following, we will
simply assume that |GL| is suppressed either by a large detuning or by adding a reverse pumping
field to cancel the coupling exactly.
In addition to pump backscattering, the probe photons are also mixed by the coupling term
∼ β in Hamiltonian (12) and even for |GL| → 0 a degradation of the non-reciprocal response
of the device will occur. Let us first consider the case of weak mode mixing, β ? κ, and
assume that the system is pumped in the right-circulating mode at the OM red sideband (∆ =
−ωm,ωm? β), as indicated in Fig. 4(b). In this regime, the rate of backscattering of photons
inside the resonator is smaller than the decay rate, and therefore, the non-reciprocal response
of the device is qualitatively the same as in the ideal case. This is shown in Fig. 5(a) where the
mode coupling only slightly reduces the operational bandwidth, i.e., κ → κ(1−β2/G2
In contrast, when the mode coupling is strong (β ? κ), the backscattering strongly redis-
tributes the probe field in between right- and left-circulating modes, and as shown in Fig. 5(b),
the EIT width and the associated non-reciprocal effects can be significantly reduced. In Fig. 6,
we have plotted the bandwidth of an optical diode as a function of the mode mixing and the
strength of the OM coupling |GR|. While the bandwidth decrease with increasing β, we observe
that this effect can be compensated for by using a stronger pump to achieve GR> β. There-
fore, we conclude that the presence of a finite intrinsic mode mixing does not fundamentally
limit the operation of our device, and even if this coupling exceeds the ring resonator linewidth,
non-reciprocal effects can persist, provided that the OM coupling is sufficiently strong.
To put our results in relation with existing experimental parameters, we consider the system
presented in Ref. , where an optical whispering gallery mode inside a toroidal microres-
m+κ2which means that the parasitic
coupling can be suppressed by choosing high frequency mechanical modes. Further, we point
mode coupling β and as a function of the pump detuning ∆=ωL−ωc. For this plot we have
assumed that the pump field only drives the right-circulating mode and that the resonator is
coupled to a single waveguide (κ?=0). The other parameters are (β,κin)/κ =(4,1). At the
normal mode frequencies ω ? ±β, the left- and right-circulating modes are almost equally
populated, while everywhere else, there is an intensity imbalance between left- and right-
circulating modes. (b) The diagram shows the relation between the relevant frequencies in
bare resonator frequency ωcand the resonator is pumped at the mechanical red sideband.
20 100 10 20
20100 10 20
Fig. 5. Transmittance for light propagating in a waveguide coupled to a resonator (AFP),
in the presence of (a) weak (β = 2κ) and (b) strong (β = 8κ) mode mixing. For these plots
we have assumed (ωm,GR,κin,γm)/κ = (20,5,1,0) and ∆ = −ωm.
onator is coupled to a mechanical mode of frequency ωm/(2π) = 78 MHz. In this system the
single-photon OM coupling is g0/(2π) = 3.4 kHz and the directional enhanced coupling can
reach G/(2π) = 11.4 MHz. The resonator decay rate is κt/(2π) = 7.1 MHz. Therefore, this
system can be operated in the strong coupling regime |G| > κt, and assuming that intrinsic de-
fects can be reduced to a level |β| < |G| ∼ 10 MHz, this device can be used for implementing
the different non-reciprocal effects described in this work. In particular, if κ ? κin, then the op-
tical isolation can be observed within the resonator bandwidth. Note that recents experiments
have demonstrated OM systems supporting optical whispering gallery modes with mechani-
cal frequencies ωm∼ GHz [27, 28]. A further optimization of such devices could be used to
achieve non-reciprocal OM effects at a much higher frequencies and to push the operational
bandwidth of such devices into the 100 MHz regime.
Fig. 6. Operational bandwidth of an optical diode in the presence of a finite mode coupling
β and different values of the enhanced OM coupling GR. For this plot we have assumed
GL=0 and (ωm,κin,γm)/κ =(20,1,0), ∆=−ωm. In the absence of the mode coupling the
bandwidth is 4κ, which for a finite β can be recovered by using a strong pump to enhance
5.Thermal noise and the single photon limit
So far we have only considered the scattering relations between the optical in- and out-fields,
which due to the linearity of the equations of motion are the same for large classical fields
as well as single photons. In practice additional noise sources will limit the operation of the
device to a minimal power level, or equivalently to a minimal number of photons in the probe
beams. A fundamental noise source in our system stems from the thermal Langevin force ξ(t)
which excites the mechanical resonator. The OM coupling up-converts mechanical excitations
into optical photons which then appear as noise in the output fields [29, 30, 31]. To estimate
the effect of this noise, we investigate the contribution of thermal phonons in the noise power
of the right moving out-field
where B denotes frequency band of interest centered around the optical resonance. We can
use Eq. 17 to express˜f1
conditions and ∆ = −ωm, we obtain the approximate result
As described above, non-reciprocal effects are most effective in a small band around the me-
chanical frequency and we can set B = [ωm−∆B,ωm+∆B] where ∆B ? ωmis the operation
for strong OM coupling we obtain – up to a numerical factor O(1) – the general relation
Pnoise= ¯ hωc×
R,out(ω) in terms of the noise operator ξ(ω) and under the relevant
Pnoise? ¯ hωc
Pnoise≈ ¯ hωc×γmNth×κ∆B
For weak coupling and a maximal bandwidth ∆B = G2
γmNth?kBT/(¯ hQm) at which phonons in the mechanical resonator are excited. This means, that
R/κ, the noise power is given by the rate
if we send a signal pulse of length ∆B−1through the device a number Nnoise≈ γmNth/(G2
noise photons is generated during this time. Therefore, in this case the condition for achieving
non-reciprocal effects on a single photon level, i.e. Nnoise< 1, is equivalent to OM ground state
cooling [33, 34], which is achievable in a cryogenic environment [25, 26]. Eq. 21 also shows
that the thermal noise level can be further reduced in the strong coupling regime. In this case
the maximal operation bandwidth is ∆B=κ and the noise power is suppressed by an additional
factor (κ/GR)2?1. This is due to the fact that thermal noise is mainly produced at the two split
mode frequencies ωm±GR, while the non-reciprocal effects rely on the transparency window
between those modes. Note that while OM cooling saturates at GR≈ κ, the noise suppression
in our device can always be improved with increasing ratio GR/κ, eventually limited by the
onset of the OM instability at GR= ωm/2.
6.Conclusions and outlook
In summary, we have shown that optomechanics can induce non-reciprocity in the optical do-
main. In particular, an optomechanical ring resonator coupled to a waveguide induces a non-
reciprocal phase in the under-coupled regime (κin? κ) and forms an optical isolator in the
critically coupled regime (κ ? κin).
From an application perspective, this system provides an optical isolator that can be inte-
grated on-chip. The bandwidth of such a device will be limited by the amount of pump power
that the system can tolerate, before nonlinear effects become significant. In current experimen-
tal settings this amounts to bandwidths in the few MHz regime, which however could be further
improved in optimized designs.
From a fundamental point of view, the relevant features of our technique are the possibility
to implement coherent non-reciprocal phase shifts, to operate on the single photon level and the
ability to dynamically control non-reciprocal effects by tuning the power of the pump beam.
For example, one can consider a 2D array of optical resonators connected to each other via such
non-reciprocal phase shifters. If the phase-shifts are chosen appropriately (e.g. according to the
Landau gauge), then a tight-binding model of photons with an effective magnetic field can be
simulated . In other words, one can simulate quantum Hall physics with photons where the
time-reversal symmetry is broken. In future experiments, it might be possible to combine these
techniques with single photon non-linearities which could be either induced by the intrinsic
non-linearity of the OM interaction itself  or by interfacing the OM system with other
atomic  or solid state qubits . Combined with such strong interaction between photons,
the implementation of magnetic Hamiltonians using micron-sized OM elements could pave the
way for the exploration of fractional quantum Hall physics [47, 48] and various other exotic
states of light.
The authors gratefully thank A. Safavi-Naini, K. Srinvasan, J. Taylor, K. Stannigel and M.
Lukin for fruitful discussions. This research was supported by the U.S. Army Research Office
MURI award W911NF0910406, the EU Network AQUTE and by the Austrian Science Fund
(FWF): Y 591-N16.
We study the effect of off-resonant OM interactions ∼ (a†
sensitive transmission or equivalently, a partial squeezing of light in the output field. For sim-
plicity, we consider the APF configuration without mode coupling (β = 0), assume GR= G
and GL= 0, and set the pump field to the red sideband (∆ = −ωm). In this situation, the incom-
Phase sensitive transmission effects
ib†+aib), which lead to a phase
ing right (left)-going field will exit the system as only right (left)-going field, respectively. The Download full-text
out-field in the right-going channel is then given by
α(ω) =4|G|2ωm(ωm+iκ)−?ω2−ωm2??(ω +iκin)2−(ωm+iκ)2?
(4|G|2ωm2+(ω2−ωm2) ((κ +κin−iω)2+ωm2))
The diagonal elements are the phase insensitive transmissions amplitudes, which we have dis-
cussed in the main part of the paper and which are related to the resonant OM coupling terms.
In general the presence of non-zero off-diagonal terms, η(ω) ?= 0, mixes the finand f†
ponents. This implies a different transmission for different quadratures of the probe light, an
effect which is exploited for OM squeezing, but is unwanted in the present context. However,
as shown in Fig.7, these effects are strongly suppressed in the parameter regime of interest.
In particular squeezing effects are negligible within the transparency window |δ| < G, where
non-reciprocal effects are most pronounced.
Fig. 7. Ratio between phase sensitive squeezing terms (η) and the phase-insensitive trans-
mission amplitudes (α). For this plot we have assumed ∆ = −ωmand (G,κin,γm)/κ =