Quantum ampliﬁcation of spin currents in cavity magnonics by a parametric drive induced long-lived mode Debsuvra Mukhopadhyay1Jayakrishnan M. P. Nair1yand G. S. Agarwal1 2z

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Quantum ampliﬁcation of spin currents in cavity magnonics by a parametric drive induced

long-lived mode

Debsuvra Mukhopadhyay,1, ∗Jayakrishnan M. P. Nair,1, †and G. S. Agarwal1, 2, ‡

1Institute for Quantum Science and Engineering, Department of Physics and Astronomy,

Texas A&M University, College Station, TX 77843, USA

2Department of Biological and Agricultural Engineering,

Texas A&M University, College Station, TX 77843, USA

(Dated: October 13, 2022)

Cavity-mediated magnon-magnon coupling can lead to a transfer of spin-wave excitations between two spa-

tially separated magnetic samples. We enunciate how the application of a two-photon parametric drive to the

cavity can lead to stark ampliﬁcation in this transfer eﬃciency. The recurrent multiphoton absorption by the cav-

ity opens up an inﬁnite ladder of accessible energy levels, which can induce higher-order transitions within the

magnon Fock space. This is reﬂected in a heightened spin-current response from one of the magnetic samples

when the neighboring sample is coherently pumped. The enhancement induced by the parametric drive can be

considerably high within the stable dynamical region. Speciﬁcally, near the periphery of the stability boundary,

the spin current is ampliﬁed by several orders of magnitude. Such striking enhancement factors are attributed to

the emergence of parametrically induced strong coherences precipitated by a long-lived mode. While contextu-

alized in magnonics, the generality of the principle would allow applications to energy transfer between systems

contained in parametric cavities.

Photon mediated interactions are a quintessential resource

in various branches of sciences. A prime example is the well

known dipole-dipole interaction (DDI) , which, according to

quantum electrodynamics, arises from the exchange of a pho-

ton [1] between two atoms. The DDI consists, in general,

of a dissipative component as well which can vanish at cer-

tain separation of the dipoles. This DDI determines the en-

ergy/excitation transfer [2, 3] between, say, the donor and the

acceptor molecules, and is of paramount importance in the

generation of quantum gates [4, 5]. It is therefore desirable to

have a mechanism which can control and potentially improve

such an interaction. One routine technique is to use high-

quality cavities and employ large coupling between the cavity

photons [6–8]. A natural question then arises - can one fur-

ther improve the cavity-mediated energy transfer which would

instrumental to the realization of quantum-enhanced funda-

mental interactions and development of sophisticated quan-

tum machines and networks?

In this letter, we provide a deﬁnite answer to this question

and demonstrate the possibility of enhancing photon-mediated

transfer of excitations. Guided by the developments in quan-

tum metrology using squeezed states [9–13], we propose to

use parametric interactions, which give rise to squeezed states

of matter and light, to enhance the cavity-mediated DDI.

We would speciﬁcally apply the idea in the context of cav-

ity magnonics [14–19] and show signiﬁcant ampliﬁcation of

the photon-mediated transfer of spin currents between mag-

netic samples [20] by using parametric interactions in cavi-

ties. We note that many other applications of parametric in-

teractions have appeared in literature: enhanced cooling [21];

exponentially enhanced spin-cavity photon coupling [22, 23];

enhanced phonon-mediated spin-spin coupling in a system of

spins coupled to a cantilever [24]; possibility of ﬁrst order su-

perradiant phase transitions [25]; enhancement in the genera-

tion of entangling gates [26]; ampliﬁcation of small displace-

ments of trapped ions [27].

Here, we focus on cavity magnonics involving the coupling

of high-quality microcavities and YIG spheres. These sys-

tems are attracting increasing attention [28–30] as favorable

candidates to observe various semiclassical [31–39] and quan-

tum phenomena [40–43] at the macroscopic level. Some of

the key developments include the coupling of magnons to a

superconducting qubit [40] and phonons [41], microwave-to-

optical interconversion [31, 32], exceptional points [33], en-

tanglement [42, 43] and many more. One of the remarkable

signatures of such a coupling was the observation of a cavity-

mediated transfer of spin excitations [20]. Two YIG samples

were placed at the opposite ends of a microwave cavity and

by manipulating the cooperativity of one of them, researchers

could detect the modiﬁcations in magnon population, namely,

the spin current of the other. Here we demonstrate how the

photon-mediated transfer of spin-wave excitations can be sig-

niﬁcantly boosted by a two-photon parametric drive applied

to the cavity. The parametric interaction could be produced

either from a χ(2)-type or χ(3)-type nonlinearity. By coher-

ently pumping one of the samples, we probe modiﬁcations to

the steady-state magnon occupancy in the neighboring sam-

ple as a function of the parametric drive strength. Our anal-

ysis showcases the emergence of parametrically induced co-

herences characterized by a long-lived mode leading to pre-

cipitous enhancement in the spin current response. The en-

hancement can be extraordinarily large around the two-photon

resonance condition ωd=ωp/2, where ωdand ωpare the fre-

quencies of the magnon drive and the parametric pump ﬁeld

respectively. This phenomenon is brought to bear by the cu-

mulative eﬀect of two-photon-excitation events. The cascaded

absorption of photons by the concerned magnon mode opens

up higher-order transition pathways within the magnon Fock

space. The fact that magniﬁcent enhancement factors can be

achieved even within the permissible stable regime of the non-

arXiv:2210.05898v1 [quant-ph] 12 Oct 2022

linear dynamics underscores the utility of this scheme. The

analysis presented in this letter is generic, pertinent to a wide

class of systems with a parametrically driven component. This

is because the DDI is ubiquitous. The investigated paradigm

would apply, for example, to nonlinear Kerr boson systems

driven far from equilibrium [44].

To set the stage, we shortly recapitulate the problem of two

spatially separated macroscopic ferrite samples of YIG cou-

pled to the microcavity ﬁeld. Owing to an eﬀective cavity-

mediated coupling between the two spheres, an external driv-

ing ﬁeld applied to the ﬁrst YIG sample would elicit a spin

current from the second one. The system Hamiltonian, in the

reference frame of the driving ﬁeld, assumes the form [17, 45]

H0/~= ∆ca†a+

j=1

∆jm†

jmj+

j=1

gj[a†mj+am†]

+iΩ(m†

1−m1),(1)

where ais the annihilation operator representing the cavity,

∆c=ωc−ωdis the cavity detuning, m1,m2are the two Kittel

modes representing magnonic excitations in the two samples,

and ∆1=ω1−ωd,∆2=ω2−ωdthe respective detunings.

For each j, the parameter gj=(√5/2)γepNjBvac denotes the

coherent magnon-photon interaction strength, with γebeing

the gyromagnetic ratio, Bvac =rµ0~ωc

2Vc

the magnetic ﬁeld

of vacuum, and Njthe total number of spins in the sample.

Plus, Ω = γe

2r5µ0ρ1d1Dp

3cis the Rabi frequency of the ap-

plied drive, where ρ1and d1are the respective spin density

and diameter of the ﬁrst sample, while Dpis incident power

of the applied drive. To understand this transfer of spin-wave

excitations at the level of transitions among energy levels, we

note that in the absence of the extrinsic magnon drive, the

Hamiltonian is excitation-preserving, with only an oscillatory

energy transfer between the magnon and the cavity modes.

Now let us label the eigenstates of the noninteracting sys-

tem as |na,n1,n2i, where na,n1and n2indicate populations

of the cavity and the two magnon modes respectively. If a

weak coherent drive (at the single-photon level) on m1excites

the system into the state |0,1,0i, energy would be exchanged

back and forth among the energy levels |0,1,0i,|1,0,0i, and

|0,0,1i, provided the dissipation is negligible. The transfer

happens via the pathway |0,1,0i→|1,0,0i→|0,0,1i, and

this simple scheme can be easily extended to the case of co-

herent drives.

Ampliﬁed spin current: We now demonstrate the impact

of a parametric drive applied to the cavity on the associated

transfer eﬃciency in the system considered above. Precisely,

we would be leveraging the potential of parametrically en-

hanced spin-photon interactions to amplify the spin currents

from magnetic samples loaded into a cavity resonator. The

new schematic is portrayed in Fig. 1, where the cavity ﬁeld

is now parametrically driven. The preceding Hamiltonian has

to be supplemented by an additional contribution of the form

FIG. 1: Schematic of two ferrimagnetic samples of YIG, co-

herently coupled to a single-mode cavity, which is driven ex-

ternally by a two-photon parametric drive. A uniform bias

magnetic ﬁeld B0applied to either of the YIG spheres gener-

ates the corresponding Kittel mode and the YIG1 is driven

externally by a coherent drive of low photon occupancy.

Hp/~=(G/2)(a2+a†2), so that the Hamiltonian for the para-

metric system [22, 23, 25] would be given by H=H0+Hp.

To quell the time-dependence of the Hamiltonian, the fre-

quency of the applied magnon drive ωdhas been set equal

to ωp/2. Our objective is to investigate the steady-state spin-

current response from the second YIG. Under the semiclas-

sical approximation, the dynamical equations for the mode

amplitudes at the level of mean ﬁelds can be obtained from

the master equation of the system. These equations can be

condensed in the form of a 6 ×6-matrix-diﬀerential equation

X=−iHeﬀX+ ΩFin,(2)

where X=(a m1m2a†m†

1m†

2)T,Heﬀ= H(0)

eﬀJ

−J−H∗(0)

eﬀ!

is a 6 ×6 coupling matrix, and Fin =(0 1 0 0 1 0)T.

The expectation-value notations h.ihave been dropped for

brevity. The constituent block elements of Heﬀare given by

H(0)

eﬀ=





∆c−iκg1g2

g1∆1−iγ10

g20∆2−iγ2







and J=





G0 0

0 0 0







, where

2κ, 2γ1,2γ2are the respective relaxation rates of the cavity

mode and the two magnon modes. Thus H(0)

eﬀdenotes the cou-

pling matrix in the absence of the parametric drive, i.e., with

Gset equal to 0. Eq. (4) would permit a steady-state so-

lution insofar as the eigenmodes of Heﬀhave decaying char-

acter. Subject to this assumption, the steady-state spin-current

response from the second magnetic sample could be expressed

as M=|m2|2, wherein m2=Pj=2,5−iH−1

eﬀ2j

Ω.

To keep the analysis straightforward, we henceforth work

with the assumption that ∆1= ∆2= ∆c= ∆,κ=γ1=γ2=γ,

and g1=g2=g. As was just stated, the stability of the steady

state hinges on the imaginary parts of the eigenvalues, which

we label as λi’s. This is formally equivalent to the Routh-

Hurwitz criterion for the stability of nonlinear dynamics. The

stable and the unstable regimes for this parametric model are

numerically plotted in Fig. 2(a), for the ratio g/κ =2, with

the demarcating partition between the two indentiﬁed by the

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Quantumamplicationofspincurrentsincavitymagnonicsbyaparametricdriveinducedlong-livedmodeDebsuvraMukhopadhyay,1,JayakrishnanM.P.Nair,1,yandG.S.Agarwal1,2,z1InstituteforQuantumScienceandEngineering,DepartmentofPhysicsandAstronomy,TexasA&MUniversity,CollegeStation,TX77843,USA2DepartmentofBiologicalan...

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Quantum ampliﬁcation of spin currents in cavity magnonics by a parametric drive induced long-lived mode Debsuvra Mukhopadhyay1Jayakrishnan M. P. Nair1yand G. S. Agarwal1 2z.pdf

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Quantum ampliﬁcation of spin currents in cavity magnonics by a parametric drive induced long-lived mode Debsuvra Mukhopadhyay1Jayakrishnan M. P. Nair1yand G. S. Agarwal1 2z

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