Deterministic and stochastic sampling of two coupled Kerr parametric oscillators Gabriel Margiani1Javier del Pino2Toni L. Heugel2Nicholas E. Bousse3Sebasti an Guerrero1Thomas W. Kenny4Oded Zilberberg5Deividas Sabonis1and Alexander Eichler1_2

2025-05-06 0 0 794.73KB 9 页 10玖币

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Deterministic and stochastic sampling of two coupled Kerr parametric oscillators

Gabriel Margiani,1Javier del Pino,2Toni L. Heugel,2Nicholas E. Bousse,3Sebasti´an

Guerrero,1Thomas W. Kenny,4Oded Zilberberg,5Deividas Sabonis,1and Alexander Eichler1

1Laboratory for Solid State Physics, ETH Z¨urich, CH-8093 Z¨urich, Switzerland.

2Institute for Theoretical Physics, ETH Z¨urich, CH-8093 Z¨urich, Switzerland.

3Departments of Mechanical Engineering, Stanford University, Stanford, California 94305, USA

4Departments of Mechanical and Electrical Engineering,

Stanford University, Stanford, California 94305, USA

5Department of Physics, University of Konstanz, D-78457 Konstanz, Germany.

(Dated: March 6, 2023)

The vision of building computational hardware for problem optimization has spurred large eﬀorts

in the physics community. In particular, networks of Kerr parametric oscillators (KPOs) are envi-

sioned as simulators for ﬁnding the ground states of Ising Hamiltonians. It was shown, however, that

KPO networks can feature large numbers of unexpected solutions that are diﬃcult to sample with

the existing deterministic (i.e., adiabatic) protocols. In this work, we experimentally realize a system

of two classical coupled KPOs, and we ﬁnd good agreement with the predicted mapping to Ising

states. We then introduce a protocol based on stochastic sampling of the system, and we show how

the resulting probability distribution can be used to identify the ground state of the corresponding

Ising Hamiltonian. This method is akin to a Monte Carlo sampling of multiple out-of-equilibrium

stationary states and is less prone to become trapped in local minima than deterministic protocols.

The Kerr parametric oscillator (KPO) is a nonlinear

system whose potential energy is modulated at a fre-

quency fpclose to twice its resonance frequency, fp≈

2f0[1–16]. When the modulation depth λexceeds a

threshold λth, the system features two stationary oscil-

lation solutions. The solutions have a frequency fp/2,

an amplitude Xdetermined by λrelative to the Kerr

nonlinearity, and phases that diﬀer by π. These ‘phase

states’ can be mapped to the two states σ∈ {−1,1}of

an Ising spin. Building on that analogy, it was proposed

that networks of KPOs can be utilized to simulate the

ground state of coupled spin ensembles, as captured by

the Ising model Hamiltonian [17]:

HIsing =−X

i,j

Ki,j σiσj,

where Ki,j is the coupling coeﬃcient between two spins

with states σi,j . Interestingly, ﬁnding this ground state

is equivalent to many computational problems that are

nearly intractable with conventional computers [18], such

as the number partitioning problem [19], the MAX-CUT

problem [20,21], and the famous traveling salesman

problem [22].

Various physical implementations have been proposed

or realized as “Ising solvers” [14,23–29]. A well known

example is the Coherent Ising Machine, a network of de-

generate optical parametric oscillators (DOPOs) that are

coupled through a programmable electronic feedback el-

ement [20,25,28,30–32] (note that DOPOs diﬀer from

KPOs in that their amplitude is not determined by their

Kerr nonlinearity but rather by two-photon loss). The

feedback breaks the energy conservation of the network

and imparts dissipative coupling (mutual damping) be-

tween the oscillators. As a consequence, diﬀerent network

conﬁgurations corresponding to diﬀerent Ising solutions

become stable at diﬀerent driving thresholds. The opti-

mal solution is assumed to possess the lowest threshold

and therefore to appear as the solution when driving the

network.

A diﬀerent line of investigation focuses on energy-

conserving, bilinear coupling between KPOs [19,27,33–

37]. In this works, the coupled oscillator solutions can

be approximated as decoupled normal modes with split

resonance frequencies. In contrast to the case of dis-

sipative coupling, the lowest threshold that is encoun-

tered when ramping up the driving strength depends

here on the detuning ∆ between the external drive and

the resonance frequency. This control parameter opens

up the possibility for speciﬁc protocols to ﬁnd diﬀer-

ent solutions [19,27,33]. It was experimentally demon-

strated, however, that the combination of nonlinearities

and strong bilinear coupling can give rise to a rich so-

lution space beyond that of a simple Ising model [37].

Careful validation and testing of small systems is there-

fore important before larger networks can be understood

and operated correctly.

In this paper, we experimentally test the validity of the

Ising analogy for a system of two classical coupled KPOs.

In a ﬁrst step, we apply an adiabatic ramping protocol

to ﬁnd one particular solution for each selected combina-

tion of ∆ and λin a deterministic way. In a second step,

we use strong force noise to explore the solution space of

the system: this method is based on transitions between

diﬀerent stationary KPO solutions [38–42], and it allows

for the visualization of a probability distribution for all

accessible states. Such “stochastic sampling” presents a

way of quantifying the occupation probability of each so-

lution, and thus selecting the optimal state. Surprisingly,

we ﬁnd that the oscillation state corresponding to the

expected Ising ground state has not the highest but the

lowest occupation probability over the entire parameter

space. We reconcile this result with theory and predict

arXiv:2210.14731v3 [cond-mat.mes-hall] 3 Mar 2023

X (µV)

(c)

−40 −20 0 20 40

∆(Hz)

−2.5

0.0

2.5

ϕ(rad)

(d)

(a)

Ud,1

Ubias,1

Uout,1

Ud,2

Ubias,2

Uout,2

Ud,1

Ubias,1

Uout,1

Ud,1Ud,2

z1z2

Ub,1Ub,2

−100 0 100

∆(Hz)

Ud(V)

(b)

X (µV)

FIG. 1. Device and basic characterization. (a) Simpli-

ﬁed schematic of our setup: Two mechanical resonators are

coupled with a strength Jvia their common substrate. The

devices are charged (and tuned) with bias voltages Ub,i and

driven by Ud,i. Their displacement zicapacitively translates

into a voltage xithat is read out. (b) The Arnold tongue of

a single KPO indicates the region in a space spanned by ∆

and Ud∝λwhere the zero-amplitude state becomes unstable

and the resonators respond with a ﬁnite amplitude X. The

solid gray line indicates the theoretical threshold Uth (see Ap-

pendix A). (c) Amplitude and (d) phase of the two resonators

in a frequency sweep from low to high frequencies (direction of

arrows) with Ud= 3 V; cf. the green dashed line in (b). Close

to ∆ = 25 Hz, one of the KPOs (dark green) ﬂips its phase

while the other (bright green) remains in the same phase state.

The system’s state changes from antisymmetric to symmetric

at this point. Outside of the Arnold tongue, the phases are

undeﬁned. Note that for the opposite sweep direction, the

state change does not occur. Instead, the system follows the

symmetric state as long as it remains stable.

how the method can be used for larger networks.

Our system comprises two microelectromechanical res-

onators (MEMS) made from highly-doped single-crystal

silicon. Both resonators are fabricated on the same chip

in a wafer-scale encapsulation process [43], and they have

the shape of double-ended tuning forks with branches

200 µm long and 6 µm thick; see Fig. 1(a) and Ap-

pendix A. They have resonance frequencies of roughly

f0≈1.124 MHz and quality factors of Q≈13500.

Bias voltages Ubcan be used to ﬁne-tune the resonator

frequencies by a few kHz and induce negative Kerr-

nonlinear coeﬃcients of β≈ −63.2×1017 V−2s−2due

to the nonlinear electrostatic forces between the biased

tuning fork and the electrodes next to it [44]. Those

electrodes capacitively transduce the motion into elec-

trical signals that are measured with a Zurich Instru-

ments HF2LI lock-in ampliﬁer. The capacitive driving

and measurement allows us to write eﬀective equations

of motion [10] as

¨xi+ω2

0[1 −λcos (2πfpt)] xi+βx3

i+γ˙xi−Jxj=Uξ,i ,

where ω0/2π=f0,γ=ω0

Q,xiis the measured volt-

age signal of resonator i,Jquantiﬁes the coupling to

resonator j, and Uξ,i indicate uncorrelated white noise

sources. The electrical tuning eﬀect allows us to para-

metrically modulate (drive) the resonator potentials at

frequency fp[42]. The required oscillating driving volt-

age is Ud=λU0

thQ/2, where U0

th ≈2.4 V is the mea-

sured parametric threshold voltage on resonance. As a

function of detuning ∆ = fp/2−f0, the driving thresh-

old for parametric oscillation, Uth, is described by a so-

called ‘Arnold tongue’, see Fig. 1(b). Outside the Arnold

tongue, a resonator is stable at zero amplitude, while in-

side the tongue the zero-amplitude solution becomes un-

stable and the resonator oscillates at fp/2 with a ﬁnite

eﬀective amplitude Xin one out of two possible phase

states.

The resonators are mechanically coupled via their

common substrate [45]. We calibrate the coupling

strength from the normal-mode splitting and ﬁnd ∆f=

(−2.6±0.3) Hz, corresponding to a coupling coeﬃcient

J= 4π2∆ff0=−(113 ±13) ×106Hz2between the two

resonators; see Appendix Afor details. Even though the

coupling is weak, |∆f|  f0/Q, we can use a normal-

mode basis of antisymmetric and symmetric oscillations

to describe our system in the following.

When both resonators are operated as KPOs with a

parametric drive voltage Ud> Uth, each of them selects

one of its two phase states. The resonators can respond

either in the same (symmetric) or in opposite (antisym-

metric) phases. Which of those two solutions is preferred

depends on the signs of Jand ∆. In Figs. 1(c) and 1(d),

we show the experimental results of sweeping the driv-

ing frequency from negative to positive ∆ at a ﬁxed Ud.

The system ﬁrst rings up into the antisymmetric state at

∆ = −35 Hz before it ﬂips to a symmetric conﬁguration

close to ∆ = 25 Hz.

The ordering observed in Figs. 1(c) and 1(d) reﬂects

the fact that for J < 0, the antisymmetric normal mode

has a lower eigenfrequency than the symmetric mode.

We can therefore expect to ﬁnd separate normal-mode

Arnold tongues for symmetric and antisymmetric oscil-

lations with a splitting in frequency; see Fig. 2(a) [37,46].

This splitting has important consequences for the driven

system: when the drive is suddenly activated at a speciﬁc

detuning ∆ above threshold, the KPOs should preferen-

tially select the normal-mode oscillation state with the

lowest threshold. We experimentally test this prediction

in Fig. 2(b) by measuring the chosen state once for each

pixel individually, and we ﬁnd very good agreement with

the schematic in Fig. 2(a). Note the narrow regions on

the left and right boundaries where only one state can be

activated. These regions are a direct conﬁrmation of the

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DeterministicandstochasticsamplingoftwocoupledKerrparametricoscillatorsGabrielMargiani,1JavierdelPino,2ToniL.Heugel,2NicholasE.Bousse,3SebastianGuerrero,1ThomasW.Kenny,4OdedZilberberg,5DeividasSabonis,1andAlexanderEichler11LaboratoryforSolidStatePhysics,ETHZurich,CH-8093Zurich,Switzerland.2Instit...

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Deterministic and stochastic sampling of two coupled Kerr parametric oscillators Gabriel Margiani1Javier del Pino2Toni L. Heugel2Nicholas E. Bousse3Sebasti an Guerrero1Thomas W. Kenny4Oded Zilberberg5Deividas Sabonis1and Alexander Eichler1_2.pdf

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Deterministic and stochastic sampling of two coupled Kerr parametric oscillators Gabriel Margiani1Javier del Pino2Toni L. Heugel2Nicholas E. Bousse3Sebasti an Guerrero1Thomas W. Kenny4Oded Zilberberg5Deividas Sabonis1and Alexander Eichler1_2

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