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014

Hyperparametric effects in a whispering-gallery mode rutile dielectric resonator at

liquid helium temperatures

Nitin R. Nand*, Maxim Goryachev, Jean-Michel Le Floch, Daniel L. Creedon, and Michael E. TobarARC Centre for Engineered Quantum Systems, School of Physics,

The University of Western Australia, 35 Stirling Highway, Crawley 6009, Western Australia,*now at the ARC Centre for Engineered Quantum Systems,Department of Physics and Astronomy, Macquarie University,

1 Balaclava Road, North Ryde 2109, NSW, Australia(Dated: November 5, 2014)

We report the first observation of low power drive level sensitivity, hyperparametric amplification,and single-mode hyperparametric oscillations in a dielectric rutile whispering-gallery mode resonatorat 4.2 K. The latter gives rise to a comb of sidebands at 19.756 GHz. Whereas most frequency combsin the literature have been observed in optical systems using an ensemble of equally spaced modesin microresonators or fibers, the present work represents generation of a frequency comb usingonly a single-mode. The experimental observations are explained by an additional 1/2 degree-of-freedom originating from an intrinsic material nonlinearity at optical frequencies, which affects themicrowave properties due to the extremely low loss of rutile. Using a model based on lumped circuits,we demonstrate that the resonance between the photonic and material 1/2 degree-of-freedom, isresponsible for the hyperparametric energy transfer in the system.

INTRODUCTION

Rutile (TiO2) is a centrosymmetric tetragonal crys-tal which is a useful and widely studied material in sci-ence and engineering. The crystal is a wide band gapsemiconductor, and demonstrates favorable optical prop-erties, such as high refractive index, birefringence, andtransparency to wavelengths ≥ 400 nm. These proper-ties make rutile suitable for nonlinear applications, suchas ultra fast switching, logic, and wavelength conver-sion [1]. Rutile is well known to exhibit a host of non-linear phenomena including polaritonic [2, 3] and pola-ronic effects [4], and near ‘ferroelectric catastrophe’ be-havior [5]. Third harmonic generation in particular, hasbeen investigated in numerous publications [6–11]. Manyof these works [2, 3, 6] characterize an effective third or-der susceptibility χ(3) of the rutile crystal which leads tofour wave mixing, but unlike the aforementioned effects,is always observed at optical frequencies.

At microwave frequencies, the linear dielectric prop-erties of rutile have been well studied, however ratherlittle work has been published on nonlinear effects in thisfrequency regime. In much of the previous work the ru-tile crystal has been used as a whispering-gallery (WG)mode resonator [12–14] below 5 GHz. Rutile is a particu-larly interesting low loss material due to its high uniaxialanisotropic permittivity, an order of magnitude higherthan that of sapphire, and its negative temperature co-efficient of permittivity. Thus, WG mode resonators ofmuch smaller size and higher frequency can be made fromrutile compared to sapphire. By virtue of its low mi-crowave losses, very high quality factor (Q) WG modescan be supported in such resonators at cryogenic temper-atures, where electromagnetic energy is strongly confinedaround the inner surface of the resonator at the interface

of the crystal medium and external environment due tototal internal reflection.

Over the past few years, WG mode resonators havefeatured in numerous publications describing the genera-tion of frequency combs [15]. Many applications includ-ing optical spectroscopy, precision frequency metrology,the development of optical clocks, and precise calibra-tion and measurement in observational astronomy all relyon such combs. Optical hyperparametric oscillations inCaF2 resonators have been reported [16] and theoreti-cally analyzed [17, 18], and comb generation has beenseen in a range of crystals, such as CaF2 [19, 20], and sil-ica toroidal microcavities [21]. More recently, frequencycomb generation via four wavemixing in MgF2 resonatorswas reported [22, 23], and comb generation has begun toextend into the mid-infrared regime [15].

In the present work, we describe new experimental re-sults and show that a nonlinear phenomenological modelpossessing 1 1

2 degrees-of-freedom explain the results. Thesingle degree-of-freedom represents the WG mode reso-nance at the frequency of interest, while the additionalhalf degree-of-freedom represents a resonance highly de-tuned from the optical domain, which defines the lossesat microwave frequencies and possesses nonlinear losses.The model predicts three new types of nonlinear behav-ior that generally do not exist for nonlinear systems, suchas Duffing or Van der Pol oscillators. Single-mode hyper-parametric oscillations are observed, which gives rise toa comb of sidebands and low power overdamping. Allof these effects are simultaneously observed in the WGmode resonator as predicted by the model, and hencewe rule out the causes due to temperature, paramagneticions, semiconductor, and even electron spin resonance ef-fects.

Most of the existing literature on nonlinear resonant

2

systems, such as WG mode resonators employs simpleone-dimensional models, usually related in some way tothe well studied Duffing or Van der Pol oscillators. Theformer model is a damped oscillator with a nonlinear an-gular frequency, whereas the latter is a single degree-of-freedom oscillator with a nonlinear loss term. In the caseof optical WG resonators, nonlinear effects are modeledas nonlinear correction terms for the refractive index orsusceptibility, proportional to the square of the intensityof the electric field [24, 25], or by direct inclusion of non-linear interaction terms into a Hamiltonian of interactingmodes and the corresponding Langevin equations [17].Nonlinear behavior in such systems is well studied and isobserved usually in the form of a pitchfork bifurcation,excess losses at high powers, or hyperparametric oscilla-tions in many-mode systems. Nevertheless, the numberof nonlinear phenomena in resonant systems can be en-riched simply by increasing the system degree-of-freedomby one half. In this section, we analytically show thatsuch a system can lead to three new phenomena; lowpower overdamping; single-mode hyperparametric oscil-lations; and hyperparametric amplification.

MODEL

The model consists of two branches; a resonant pho-tonic branch representing a WG mode, and a dissipa-tive nonlinear branch representing an intrinsic materialdegree-of-freedom, the optical phonon mode. The latteris taken to be highly detuned, so that its overall responsecould be represented only by a half degree-of-freedom (ora single reactance). Thus, in terms of an equivalent cir-cuit (see Fig. 1), the system is modeled by a resonantseries RLC circuit (the photonic branch) in parallel tothe material branch consisting of a nonlinear capacitancewith dissipation. The effective inductance can be ignoreddue to the high detuning of the material resonance fromoptical frequencies. In mathematical language, this rep-resentation is a degenerate case of a class of models rep-resented by two coupled nonlinear oscillators. For sim-plicity, we include in the description just the nonlinearitydue to the material behavior, which means only the halfdegree-of-freedom is assumed to be nonlinear.A Hamiltonian in terms of the variables of the equiva-

lent circuit model may be written as

H =1

2Lx

φ2x +

1

2Cx

q2x +1

2Lp

φ2p +

1

2Cp

q2p +Lpη

4ηq4p,

≈1

2Lx

φ2x +

1

2Cx

q2x +ηLp

4q4p,

(1)where qx is a charge in the resonant photonic branch,qp is a charge in the nonlinear loss branch, Vinp is thedriving signal (the incident traveling wave in the trans-mission line at the point of the resonator), γx and ωx

are the damping coefficient and angular frequency of the

Input line Output line

Photonic

resonant

branch

Dissipative

branch

L

C

R

x

p

x

x

C

R

p

C CL Li i o o

up

Lp

FIG. 1. An equivalent lumped circuit model representing anonlinear 1 1

2degree-of-freedom system, with a photonic WG

mode (Cx, Rx, Lx) and the material dissipative half degree-of-freedom (Cp, Rp, Lp).

photonic branch, γc is a coupling coefficient, and η is thecoefficient of the nonlinear term derived from the voltage-charge relation of the nonlinear capacitance up = ηq3p−φ.The approximation only holds for strong nonlinearity.The Cx − Lx pair represents the photonic mode reso-nance, and Cp − Lp refers to the optical phonon mode.Additional dissipative elements in Fig. 1 are due to thedissipative bath

∑

ωi. The equations of motion for theequivalent system are written in terms of the charge intwo capacitances, such that

qx + 2γxqx + ω2xqx + 2γcqp = Vinp

2γpqp + ηq3p + 2γcqx = Vinp(2)

Note that the Hamiltonian corresponding to the equa-tions of motion (Eq. 2) could be written using only threeoperators; flux φx and charge qx for the photonic mode,and only charge for the material mode qp.It can be shown analytically that in the linear limit

(ηq3p → 0), the total frequency response of the system(qx(ω) + qp(ω)) can be made totally independent of de-tuning frequency (ωx − ω) in the limit γp → γc. Thisis equivalent to the effective nonexistence of the mode atlow powers (see Figs. 5 and 6). Although when nonlinear-ity exists in the second branch, this leads to a situationwhen external pump photons could be ‘damped’ by thedissipative branch, or stored and re-emitted from the res-onant photon branch depending on their number. Onlywhen the dissipative branch is saturated can the photonsbe transmitted through the mode resonance.We attribute the notion of a hyperparametric process

to the process of energy transfer in a nonlinear system be-tween frequencies whose difference is much less than thefrequencies themselves. This notion was first introducedin the domain of optical comb generation [15, 19, 20].In the case of parametric oscillators, the energy of thepump will sustain oscillations when it is an integer mul-tiple of the oscillation frequency, i.e. pump frequency ofnω, where n is an integer and ω the oscillation frequency.

3

Under hyperparametric oscillation, energy is transferredfrom the pump ω into its sidebands ω ± n∆ω, where n

is an integer. Although some of the literature on opti-cal comb generation refers to these effects as paramet-ric, we use the term ‘hyperparametric’ to emphasize theunique situation in which the energy transfer is madebetween signals with a frequency spacing on the order ofone mode bandwidth. The hyperparametric problem canbe reduced to the usual parametric case simply by trans-forming the problem from the domain of actual signalsinto a domain of its quadratures by applying the methodof slowly varying amplitudes relative to some referencefrequency (see Fig. 2(a)).The 1 1

2 degrees-of-freedom system exhibits two typesof resonant behavior, the first being a series resonance(a pure resonance of the photonic degree-of-freedom)between the inductive element Lx and capacitive ele-ment Cx. The second resonant behavior is antiresonance(mixed resonance of the photonic half-degree-of-freedomand the material half-degree-of-freedom), which is a par-allel resonance between Cx and the inductance Lp in thenonlinear branch. The notion of antiresonance is com-monly used in the acoustic device community where aseries mechanical resonance is always accompanied by anantiresonance - parallel resonance with capacitive elec-trodes. In the present work, it is the antiresonancebehavior that is responsible for hyperparametric energytransfer. To demonstrate this, a corresponding equationof motion can be written in terms of flux, such that

φ2

3

(

Φ− φ)

2

3

Φ + 2δaΦ + ω2aΦ = kvVinp + kpqx, (3)

where 2δa = 3Ziη1

3φ−2

3 , ω2a = 3

Lpη

1

3φ−2

3 , kv = 3η1

3φ−2

3 ,

kp = 3η1

3Rx

Lpφ−

2

3 , Φ = up, and qx is a charge from the

pure photonic mode. The flux calculated using this equa-tion of motion is fed back into the resonant part creatinga closed-loop feedback system (see Fig. 2(b)).The nonlinear subsystem described by Eq. 3 exhibits

two types of external forcing: Vinp due to the externalpump signal, and qx which is a signal coming out ofthe pure photonic subsystem. To demonstrate the en-ergy transfer from the former to the latter, we apply themethod of slowly varying amplitudes [32, 33] (a rotatingwave approximation) assuming forcing terms of the form

Vinp = V0 sinωpt,

qx = A sinωxt+B cosωxt = qx0 cos(

ωxt+ ϕ)

,

(4)where ωp and ωx are angular frequencies of the pumpand signal in the absolute frequency scale. The solutionis then represented in the form

Φ = U cosωRt+ V sinωRt, (5)

where ωR is some reference frequency. It is chosen in

such a way that

n∆x = n(ωx − ωR) = ωp − ωR = ∆p, (6)

where n ∈ Z, ∆p, and ∆x are angular frequencies of thepump and signal relative to the reference frequency ωR.Thus, in the new relative frequency scale, the pump fre-quency is an integer (n) multiple of the signal frequency.The relationships between reference frequencies and ab-solute frequencies are shown in Fig. 2. Thus, Eq. 5 trans-forms the initial hyperparametric problem in terms offlux Φ and charge qx into an nth order parametric prob-lem in terms of quadratures U and V .

Flux, chargePump

0

Reference

Frame

Absolute Frequency

Scale

Relative Frequency

Scale

Signal

Quadratures

Resonator (Photonic

degree-of-freedom)

(Antiresonance:

½ photonic + ½ material)

Hyperparametric amplifier

(Antiresonance: ½ photonic

+ ½ material)

Pump

(Vinp

)

qx

Gs

Hx

Equivalent feedback system

Resonator

(Photonic degree-of-

freedom)

(a)

(b)

Hyperparametric Amplifier

FIG. 2. (a) Relationship between the frequencies of the hyper-parametric system: the introduction of a reference frequencytransforms the problem into a pure parametric one in therelative frequency scale, where pump frequency is an inte-ger multiple of signal. (b) Equivalent block diagram of thesystem representing resonant and antiresonant behavior in aclosed-loop feedback system.

The nth order parametric problem corresponding tothe hyperparametric amplifier under analysis can be writ-

4

ten as

V = −

[

Ωa − µ(

3V 2− U2

)

]

U − δa

[

1− χ(

V 2 + U2)

]

V+

+kv

2ωR

V0 sin∆p +kpωx

2ωR

qx0 sin(

∆x + ϕ)

,

U = −δa

[

1− χ(

V 2 + U2)

]

U +[

Ωa − µ(

3U2− V 2

)

]

V−

−kv

2ωR

V0 cos∆p −kpωx

2ωR

qx0 cos(

∆x + ϕ)

,

(7)

where Ωa =ω2

a−ω2

R

2ωR, χ = 15

9·8ω2φ−2, and µ = χ

ω2

a

ω. It

is assumed that V ≪ UωR and U ≪ V ωR, i.e. thereference frequency is much larger than ∆x and ∆p. Alsothe system is considered to be low loss, i.e. ωR ≫ δa.This approximation allows us to neglect terms appearingin Eq. 7 associated with multiplication of kvVinp and

kpqx terms by(

Φ− φ)

2

3

terms in Eq. 3. Hence, we

consider the case

δa ∼ χΥ2, ωR ≫ χΥ2 (8)

where Υ is both U or V . This is a very important ad-vantage of the hyperparametric amplifier in comparisonwith a parametric system. In the former case, the non-linearity resulting in a parametric process must be of theorder of mode losses (δa ∼ χΥ2), whereas in the lattercase it must be comparable with the mode frequency it-self ωR ∼ χΥ2.The small signal gain of the system described by Eq.

7 for the internal resonator signal qx is calculated nu-merically using the harmonic balance method, up to the5th harmonic. The result is shown in Fig. 3. The figuredemonstrates amplification of the signal from the res-onator part of the system. Since both parts are inter-connected into a feedback system, the entire closed-loopsystem becomes unstable when the loop amplification ex-ceeds unity and phase matching conditions are met. Inthis situation, the parametric amplifier part of the sys-tem can compensate the losses in the resonant part. Suchregimes are predicted by the numerical simulations. Thissituation is a regime of sustainable parametric oscilla-tions in the quadrature space of relative frequency thatcorresponds to hyperparametric sideband generation inthe original space.In the degenerate regime (n = 0), the system described

by Eq. 7 is able to transfer energy from the strong pumpto a signal degenerate with it in frequency, but associatedwith different input source, e.g. noise generated by Rx.In this case, the system behaves as a hyperparametric de-generate amplifier, so that the noise signal of a frequencycoming into the resonator would be amplified, and thenread out upon transmission.In summary, the method of slowly varying amplitudes

was applied to the original problem producing hyperpara-metric oscillations in the domain of actual real physicalquantities, with the signal represented by a set of cou-pled nonlinear differential equations. The order of the

Ma

gn

itu

de

re

sp

on

se

Gs

Relative Frequency

0 = V01

< V02

<V03

V01

V02

V03

FIG. 3. Small signal magnitude response Gs(∆) =

√

u2+v2

qx0

of the hyperparametric amplifier in the quadrature space fordifferent values of the pump power, where u and v are quadra-tures of the system response to small signal qx0.

problem is one and a half due to large detuning of thematerial degree-of-freedom. The applied technique trans-forms the original system from this domain into a domainof oscillating quadratures in the relative frequency frame.The system of nonlinear differential equations which wederived can be considered itself as a nonlinear problemrelating quantities in the quadrature domain. The sys-tems output signal in the original domain is a comb, a setof equally spaced (∆x) spectral lines near the referencefrequency (ωR). The energy is transferred through a hy-perparametric process from the pump into the sidebands.In the quadrature domain, there are several signals ofthe frequencies k∆x, where k is an integer. The energyis transferred from the pump that corresponds to one ofthem through the usual parametric process. Thus, hy-perparametric problems in the domain of actual physicalquantities may be approximately reduced to a parametricproblem in the domain of quadratures.

EXPERIMENTAL OBSERVATIONS

Whispering-gallery modes in the frequency range 9 to22 GHz with Q-factors of a few million were excited in acylindrical rutile crystal of diameter 14.08 mm and height5.57 mm. The crystal was supported inside a copper cav-ity (internal diameter 21.06 mm and height 9.01 mm) us-ing two teflon supports via a thin capillary bored throughthe center of the rutile specimen. The cavity was cooledinside a cryocooler with the temperature regulated near4.2K. Two loop probes were used to couple to the res-onator, one to excite WGE modes (coupling to the Er

field component) and the other, a detection probe placedat 180 to the excitation probe. The resonator was char-acterized in transmission using a vector network analyzer(VNA), allowing mode characterization over a wide rangeof frequencies and incident powers. The coupling coeffi-

5

Pro

be

sig

na

l tr

an

sm

issio

n

Absolute frequency of the probe signal

0=V01

<V02

<V03

V01

V02

V03

Pump FrequencyResonance Frequency

5 dB

FIG. 4. Numerical calculation of the transmission coefficientof the small probe signal through the system in the presenceof the slightly detuned pump.

cient of the probes was set low so that the measuredmodes were undercoupled. In this way, the measuredproperties (such as Q) are close to their intrinsic values,and effects such as loading of the Q-factor due to themeasurement apparatus are minimized.By varying the input signal power, three distinct types

of nonlinear behavior have been observed in the rutilecrystal; a) low power overdamping; b) single-mode hyper-parametric oscillations; and b) hyperparametric amplifi-cation which are explained in the following paragraphs.These nonlinear features are common for every mode.

Firstly, a drive level sensitivity or excess loss is ob-served at low driving power. Below a certain lower powerthreshold, or in what is supposed to be a linear limit,mode profiles are unmeasurable as if the mode has van-ished. This effect is clearly seen in Figs. 5 and 6, wheremodes do not exist below −12 and −25 dBm correspond-ingly. The model presented in the previous section ex-plains this behavior, which cannot be demonstrated fora model with one degree-of-freedom without introducinga singularity.Secondly, the mode vanishes at high drive power lev-

els which defines an upper threshold. This effect is alsoobserved in Fig. 5 at approximately 7 dBm. Such ob-servations can be made for a Duffing-like model whenthe nonlinearity dominates over linear frequency selec-tive behavior. In other words, when the nonlinear termin the equation of motion becomes larger than the lineardynamic term.Thirdly, we demonstrated that when a weak probe sig-

nal is tuned on resonance, it exhibits amplification in thepresence of a strong detuned pump. The effect existsonly within the mode bandwidth and significantly weak-ens when the probe is being detuned from the resonance.This amplification has been simulated numerically usingthe model in its original charge representation by theHarmonic balance method (see Fig. 4).Finally, as a manifestation of the previous effect, the

excess noise near the discontinuity is explained by hyper-

-12

10

0

2

8

6

4

-2

-4

-6

-8

-10

Inp

ut P

ow

er

(dB

m)

-100 0-20-80 -60 -40 20 40 60 80 100

Offset (kHz) from 17.187 GHz

-20

-25

-30

-35

Ptr

an

sm

itte

d P

incid

en

t (d

B)

FIG. 5. Intensity plot of the ratio of transmitted signal toincident signal for a typical WG mode around the center fre-quency of 17.2 GHz for different power levels of the inputsignal. The mode demonstrates low drive level sensitivity(disappears below −12 dBm), and strong nonlinear effects athigh powers.

parametric amplification of the noise (see Fig. 7). This ef-fect is also explained by the 1 1

2 degrees-of-freedommodel.It corresponds to a degenerate regime (n = 0) for small∆x when the detected signal contains an amount of sys-tem noise hyperparametrically amplified by an incidentpump. This effect has been numerically confirmed bycalculating a small signal transfer function from the res-onator internal noise source to the output.

Offset (kHz) from 9.500,110 GHz

Inp

ut P

ow

er

(dB

m)

-30

0

-25

-20

-5

-10

-15

-50 0 50 100

2

1

0

-1

-2

-3

-4

-5

-6

-7

-8

Ptr

an

sm

itte

d P

incid

en

t (d

B)

FIG. 6. Intensity plot of the ratio of a transmitted signalto incident signal for a typical WG mode around the centralfrequency of 9.5 GHz for different power levels of the inputsignal. The mode demonstrates low drive level sensitivity.

6

MICROWAVE HYPERPARAMETRIC

OSCILLATIONS

Another manifestation of nonlinear phenomena in therutile WG mode resonator is hyperparametric single-mode microwave oscillation. This type of oscillation wasobserved as sideband frequencies generated around thetransmitted signal when the input pump frequency wastuned to near that of a WG mode at 19.756 GHz. Theexperimental setup consisted of a microwave synthesizer,a cryocooled rutile resonator, and a spectrum analyzer.A typical spectrum of the transmitted signal is shown

in Fig. 8. The oscillations could only be excited for acertain range of input powers and detuning frequencies(several tens of kilohertz blue-detuned from the WG res-onance frequency). None of the additional tones coin-cided with any other WGM. The closest WGM is sev-eral orders of magnitude away compared to the frequencyrange analyzed. The system demonstrated an upper andlower threshold power (16 mW and 8 mW respectively)at which the sidebands totally disappeared.The intensity of the sidebands strongly depended on

the power and detuning of the input signal (see Fig. 9).The intensity decreased with increasing pump power.This effect has been predicted by numerical simulation.As is predicted by the theory, the comb spacing ∆x is thepump detuning from the resonance, such that

∆x =ωp − ωx

n− 1. (9)

The power dependence of the comb spacing ∆x is due tothe power dependence of the ωx component in Eq. 9 thatappears due to Kerr nonlinearity of the photonic mode,and nonlinear effects of higher order. Variation in thecomb spacing over such a wide range implies that theeffect does not depend on some characteristically well-

-30

-29

-28

-27

-26

-30 -20 -10 0 10 20 30

0 dBm-10 dBm

Tra

nsm

issio

n S

21

(d

B)

Frequency Offset (kHz) from 19.755,964 GHz

Pump

0

V (

qu

ad

ratu

re)

Noise

Antiresonance

Resonance

(A)

(B)

FIG. 7. (a) The WG 19.756 GHz mode showing a power effectfor input powers of 0 dBm and -10 dBm. (b) The energytransfer diagram for the degenerate (n = 0) internal noiseamplification process.

-100

-80

-60

-40

-20

-120 -80 -40 0 40 80 120

Frequency Offset (kHz) from 19.755,964 GHz

Am

plit

ud

e (

dB

m)

ω4-

ω3-

ω2-ω1-

ω5-

ω1+

ω2+

ω3+

ω4+ ω5+

ω0

Ppump = 10.70 dBm

FIG. 8. Spectrum analyzer trace of the sidebands generatedin rutile with a WGE mode at 19.756 GHz. The comb spacingis 20 kHz at the given power and proportional to the pumpfrequency detuning.

-85

-80

-75

-70

-65

-60

-55

10.5 11 11.5 12 12.5

Pump

ω3-

ω2-

ω1-

ω1+

ω2+

ω3+

-24

-25

5 dBm

Incident (pump) power

Ou

tpu

t P

ow

er,

dB

m

Second (pump) harmonic

First harmonic

Third harmonic

(B)

(A)

Ou

tpu

t P

ow

er,

dB

m

Incident (pump) power, dBm

FIG. 9. (a) Experimental observation of power dependanceof sidebands as labeled in Fig. 8. (b) Numerical calculationwith the proposed model showing the decrease of generatedsignal power with increasing pump power.

defined frequency, such as those expected for mechanicalvibrations.

The model predicts that for the same values of sys-tem parameters including ωp and ωx, there could bemore than one solution corresponding to different val-ues of n (thus different reference frequencies ωR). Thecase of multiple co-existing solutions is possible when theNyquist criterion of stability is not fulfilled for more thanone value of n. In this case, an effect of comb crowding

7

(the appearance of intermediate tones) is observed. Theprocess is shown in Fig. 10. The figure shows separatesolutions for n = 2 (a) and n = 3 (b), and for theirco-existent pumping in (c) and (d). Due to much lowermagnitude instability margins of the n = 3 solution, itis considerably less stable, thus it is observed experimen-tally that for the same values of the pump it may or maynot appear. In the ideal situation, solutions with dif-ferent n are almost independent since their interactionterms are orders of magnitude smaller than the solutionsthemselves. The experimental observations shown in Fig.10(d) demonstrates plots for different values of the pumppower, where there is a small difference between two re-alizations of the n = 2 solution. This discrepancy isattributed to the power dependance of ωR due to a selfKerr effect of the photonic mode that has not been takeninto account.The effects observed cannot be ascribed to tempera-

ture, as temperature effects ordinarily occur over muchlonger time scales, especially for such a macroscopic ob-ject. In addition, no correlation between cavity tempera-ture and sideband mid-term and long-term stability hasbeen measured. In case of temperature-related effects,it would be natural to expect that temperature fluctua-tions produce long and mid-term evolution of the non-linear system that can be detected by sideband stabilitymeasurements.

DISCUSSION AND CONCLUSION

In this work, we related the hyperparametric phe-nomenon to other nonlinear observations in a modelbased on lumped circuits. It must be noted that theimportant difference with optical hyperparametric sys-tems [16, 17] is that optical hyperparametric oscillation(HPO) requires several near-equidistant modes with afree-spectral range (FSR) on the order of a gigahertz.Thus, they are based on nonlinear coupling between pho-tons of different resonator modes. In our case, a comb isgenerated with only a single resonator mode. The presentwork should therefore be compared to the well-knownparametric oscillators, for example, those based on highquality quartz resonators [34]. Such oscillators are phys-ical implementations of the well studied Mathieu oscilla-tors, however instead of transferring the energy from apump which is a harmonic of the oscillating signal, thepower is transferred from a pump which is detuned fromthe resonance frequency.Any prior work on hyperparametric oscillations in the

optical domain does not discuss the physical origin ofthe nonlinearity in the bulk resonator [24]. The originsof the Kerr nonlinearity have been studied previously inmeasuring contributions from the Raman process [35]. Ithas also been demonstrated that instability phenomenain optical WG resonators are related to the effects of fast

thermal relaxations in localized areas of the resonator [36,37]. However, we rule out the possibility of temperatureeffects of this sort for a number of reasons.

First, the thermal model and the fast Kerr effect [37]do not predict low power drive level sensitivity, where amode disappears at low drive level. Second, the temper-ature coefficient measured for the comb generating mode(β = −3.9 × 10−7 K−1) is the lowest amongst all themodes characterized. Additionally, it is at least two or-ders of magnitude lower than that in case of optical WGmode resonators [37]. Third, the present experimentsare made at liquid helium temperatures, where temper-ature relaxation times are much shorter, including thatof the resonator itself and the cavity. Moreover, the highquality of our temperature control is able to stabilize theresonator to within 5 mK, with no temperature increaseobserved. Finally, microwave WG mode resonators aresignificantly larger than optical microspheres, and thusthe field intensity is much lower for the same amount ofstored energy. Significant bulk temperature effects havebeen observed only down to 200 mK where the coolingpower is at minimum and no temperature control is in-volved. However, no parametric instability effects as de-scribed in [37] are seen.The observed oscillatory effects cannot be explained

by the interaction of a few WG modes. Predictions us-ing method of lines of the resonator and experimentalobservations confirm that no other WG modes lie withinthe range of hyperparametric oscillations. Also, no spu-rious signal is observed outside this range. Additionally,the comb tunability and comb co-existence cannot be ex-plained by the interference between modes. Similar os-cillatory instabilities found in the literature on opticalWG mode resonators have been ascribed to free carrierdynamics [38]. However, such polaronic effects are onlyphysically observable in the optical range of frequenciesdue to the wide band-gap of rutile, which exceeds the en-ergy of microwave frequency photons by orders of mag-nitude.

By applying an external magnetic field at low temper-atures, the effects of paramagnetic ion frequency tran-sitions were measured and spectroscopy was performed,allowing measurement of some ion g-factors. However,the properties of the resonances under study in this workwere determined to be independent of magnetic field andwere much stronger than the paramagnetic ion effects.This observation rules out the phenomena being param-agnetic.

We conclude that the nonlinear phenomena are mostlyrelated to dielectric properties of the material. In themodel proposed here, the additional dissipative branchwith 1

2 a degree-of-freedom is no more than a degeneratecase of a coupled, highly detuned oscillator representing amaterial degree-of-freedom. The microwave properties oflow loss materials are determined mainly from the opticaldomain. For example, a similar (but linear) relation be-

8

V (

qu

ad

ratu

re)

Pump

0

Second harmonic pumping (n=2)

(a) (b)

Simultaneous pumping (n=2 and n=3)V

(q

ua

dra

ture

)

Pump

0

Third harmonic pumping (n=3)

V (

qu

ad

ratu

re) Pump

(c)

-120

-100

-80

-60

-40

10.16 dBm

Am

plit

ud

e (

dB

)

Frequency Offset (kHz) from 19.975,590 GHz

-100

-80

-60

-40

10.36 dBm

05-05- 0

(d)

FIG. 10. Effect of co-existence of different solutions: (a) Second harmonic pumping (n = 2), (b) Third harmonic pumping(n = 3), (c) Simultaneous pumping of n = 2 and n = 3 , and (d) Experimental observations. Note that in both cases (a) and(b), the frequency of the oscillating signal ωx and the pump ωp are the same, only the reference frequency ωR (and thus n) aredifferent. Small differences in frequencies for n = 2 in (d) are due to power dependence of ωx.

tween the microwave and optical domains is known to bepresent in sapphire resonators and other dielectrics wherethe permittivity and absorption coefficient for microwavemodes is determined by losses in an optical phonon reso-nance in the infrared frequency range [28], where the firstinfrared phonon mode is even higher in frequency than inrutile. This property creates a velocity damping effect,which produces a Q× f = constant law in such systems.

ACKNOWLEDGEMENTS

This work was funded by the Australian ResearchCouncil under grants CE110001013 and FL0992016.

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