Quantum optomechanics in tripartite systems Ryan O. Behunin Department of Applied Physics and Materials Science

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Quantum optomechanics in tripartite systems

Ryan O. Behunin

Department of Applied Physics and Materials Science,

Northern Arizona University, Flagstaﬀ, AZ 86011 and

Center for Materials Interfaces in Research and Applications (¡MIRA!),

Northern Arizona University, Flagstaﬀ, Arizona 86011

Peter T. Rakich

Department of Applied Physics, Yale University, New Haven, Connecticut 06520

(Dated: November 7, 2022)

Owing to their long-lifetimes at cryogenic temperatures, mechanical oscillators are recognized as

an attractive resource for quantum information science and as a testbed for fundamental physics.

Key to these applications is the ability to prepare, manipulate and measure quantum states of

mechanical motion. Through an exact formal solution to the Schrodinger equation, we show how

tripartite optomechanical interactions, involving the mutual coupling between two distinct optical

modes and an acoustic resonance enables quantum states of mechanical oscillators to be synthesized

and interrogated.

New methods to prepare and interrogate nonclassical

states of mechanical oscillators could enable novel quan-

tum technologies as well as the exploration of funda-

mental physics [1–12]. If the astounding lifetimes ex-

hibited by phonons at cryogenic temperatures [2, 13] can

be translated to quantum coherence times, phononic sys-

tems could form the basis for high-dimensional quantum

memories [14]. In addition, the ability to interrogate and

manipulate these acoustic modes using superconducting

qubits [15–18], electrical signals [2–4], or telecommunica-

tions wavelengths of light [7, 8, 13] make them compelling

candidates for quantum repeaters [14] and high-ﬁdelity

quantum state-transfer [11, 17, 19]. Whereas, mechani-

cal oscillators with large eﬀective mass may shed light on

the quantum-to-classical transition [20, 21], the nature

of dark matter [12, 22], and the impacts of gravity on

decoherence [1, 23, 24].

Generation, control, and measurement of quantum

states of mechanical oscillators has recently been ex-

plored in a variety electromechanical and optomechanical

systems [15–17, 25–31]. Within circuit systems, nonlin-

earity provided by a superconducting qubit has enabled

quantum state preparation and readout in the mechan-

ical domain [15–17, 25, 27, 30, 31]. Canonical cavity

optomechanical interactions, that utilize nonlinear cou-

pling between a single electromagnetic mode and a sin-

gle mechanical mode (i.e., bi-partite system), permit an

array of state preparation, control and readout function-

alities [26, 29, 32–34]. By detuning a strong coherent

drive from resonance, a linearized optomechanical cou-

pling can be realized, enabling coherent state swaps be-

tween the mechanical and optical domains, ground-state

cooling, entanglement generation, two-mode squeezing,

and when combined with photon number measurements,

the synethsis of single-phonon Fock states [26, 29, 32].

Looking beyond these demonstrations, it is challenging

to access more exotic quantum states using conventional

bipartite cavity optomechanical systems. While it is pos-

sible to create multi-component cat states, macroscopi-

cally distinguishable superpositions, and phonon-photon

entanglement if one can reach the ultrastrong coupling

regime, this regime requires coupling rates on par with

the phonon frequency [34]. Moreover, relatively weak op-

tomechanical nonlinearities make this regime diﬃcult to

access with GHz-frequency phonons, which oﬀer long co-

herence times at cryogenic temperatures. Alternatively,

high frequency phonons can be accessed using a tripartite

system consisting of a single phonon mode that mediates

coupling between two optical modes [7, 8]. Moreover, the

distinct structure of the tripartite system may oﬀer some

unique advantages as we consider new strategies to gen-

erate and detect exotic quantum states with mechanical

systems.

Here, we show that the nonlinear quantum dynamics of

tripartite optomechanical systems can enable the prepa-

ration of highly nonclassical phononic states. Consider-

ing a triply resonant system, we derive a formal solution

for the exact time-evolution of the total system wave-

function, enabling analytical and numerical calculations

for the quantum state dynamics. Our results show that

experimentally accessible initial states (e.g., prepared us-

ing a coherent classical drive) evolve into wavefunctions

exhibiting entanglement between optical and mechani-

cal degrees of freedom. Leveraging this entanglement,

we show that conditional measurements on the optical

modes of the system, such as homodyne detection and/or

photon counting, can project the mechanical oscillator

into highly nonclassical states that depend sensitively on

the initial system wavefunction. By simulating the sys-

tem evolution including the eﬀects of decoherence, we

identify regimes where quantum states can be robustly

synthesized. Moreover, in the presence of a classical co-

herent drive, we show that the phonon’s reduced den-

sity matrix exhibits nonclassicality even without state-

collapsing conditional measurements. We also illustrate

how π/2- and π-pulses can be used to entangle optical

and mechanical modes, or transfer quantum states be-

tween the optical and mechanical domains.

arXiv:2210.14967v2 [quant-ph] 3 Nov 2022

While closely related, this tripartite system has im-

portant features not present in the canonical cavity op-

tomechanical interaction that aﬀord unique quantum dy-

namics. First, access to two optical modes greatly ex-

pands the number and complexity of the phonon states

that can be heralded, where projective measurements

produce families of phonon states parameterized by two

sets of observables. Second, photon number measure-

ments can herald highly nonclassical phonon states, in

contrast with standard resonant cavity optomechanical

interactions, even in systems with weak coupling [34].

Third, for telecommunications wavelengths of light the

relevant phonon frequencies are of order ∼10 GHz, en-

abling ground state cooling with standard cryogenics.

Put together, these results reveal an unexplored regime of

nonlinear quantum dynamics in systems spanning from

chip-scale optomechanical devices [9, 10] to bulk crystals

[7, 8].

Quantum Dynamics: To illustrate how quantum state

generation can be accomplished using multi-mode op-

tomechanical coupling, we explore the dynamics of a sys-

tem described by the Hamiltonian H=H0+Hint,

H0=~ωpa†

pap+~ωSa†

SaS+~Ωb†b, (1)

Hint =~g(apa†

Sb†+a†

paSb).

Here, ap,aSand b, are the annihilation operators of

the pump, Stokes, and phonon modes, with angular fre-

quencies ωp,ωS, and Ω, respectively. This interaction

Hamiltonian, Hint, describes phonon-mediated coupling

between these two electromagnetic modes. Through-

out, we assume that our system satisﬁes the condition

ωp=ωS+ Ω, necessary for the phonon mode to medi-

ate resonant coupling between the photon modes (i.e.,

inter-modal scattering). Systems that are well described

by this Hamiltonian typically utilize a high-frequency

elastic wave to mediate resonant inter-modal scattering

(e.g., through Brillouin interactions [35]), with couplings

(g) that can be produced by electrostriction or radiation

pressure [36, 37]. In the analysis that follows, we con-

sider the dynamics of this system for times that are much

shorter than the decoherence time of our phonon mode

[15, 31], permitting us to neglect the eﬀects of phonon

decoherence.

Neglecting decoherence, application of the time

evolution operator to the initial wavefunction gives

the quantum dynamics of this system in terms of

the time-dependent wavefunction, given by the for-

mal solution to the Schrodinger equation |ψ(t)i=

exp{−iH0t/~}exp{−iHintt/~}|ψ(0)i.Because ωp=

ωS+ Ω, H0and Hint commute, permitting the time-

evolution operator to be factorized. While the opera-

tor exp{−iHintt/~}is an exponent of non-commuting

operators, a symmetry of the system provides a path

to a formal analytical solution: for a Fock state, the

total number of phonons and pump photons np+nb

is conserved, reducing the Hilbert space to a compact

(np+nb+ 1)-dimensional subspace. Within this com-

pact Hilbert space, Hint can be diagonalized, where the

Hamiltonian given by Eq. (1) is formally equivalent to

a Jaynes-Cummings model describing the interaction be-

tween a bosonic mode and a spin-(np+nb)/2 system (see

Supplementary Information A) [38].

For the initial state |n, m, 0i ≡ |nip⊗ |miS⊗ |0iph,

where the pump, Stokes and phonon modes respectively

have n,mand 0 quanta and using Eq. (1), the time-

dependent wavefunction |ψnm0(t)iin the interaction pic-

ture is generally represented by

|ψnm0(t)i=

k=0

An,m,k(t)|n−k, m +k, ki.(2)

Truncation of the sum over kat nis a consequence of the

compact nature of the Hilbert space for Fock state evolu-

tion. Inserting |ψnm0(t)iinto the Schrodinger equation

yields a linear matrix diﬀerential equation for the com-

plex probability amplitudes An,m,k(t) given by

An,m =−iMnm ·~

An,m.(3)

Here, ~

An,m is a column vector of the probability am-

plitudes ~

An,m = (An,m,n, An,m,n−1, . . . , An,m,0)T, and

Mnm is the symmetric matrix

Mnm =







0 Λnm

n0 0 . . .

Λnm

n0 Λnm

n−10

0 Λnm

n−10 0

0 0 ....

. 0 Λnm

. . . Λnm







(4)

with matrix elements given by Λnm

g√n−k+ 1√m+k√k. The solution to Eq. (3)

can be obtained by diagonalizing the matrix Mnm,

yielding

An,m(t) = Vnm ·e−iΩnm t·V†

nm ·~

An,m(0) (5)

where Vnm is a unitary matrix diagonalizing Mnm, and

the diagonal matrix of eigenvalues Ωnm =V†

nm ·Mnm ·

Vnm [39] (see Supplementary Information Sec. B).

Focusing on initial states that can be prepared in the

laboratory using classical light sources, photon squeez-

ing, or single photon emitters, we calculate the system

wavefunction. With the phonon cooled to the ground

state, we consider initial wavefunctions given by

|ψ(0)i=

∞

n=0

∞

m=0 PnSm|n, m, 0i(6)

where Pn(Sm) is the probability amplitude for the pump

(Stokes) mode to be found initially in the nth (mth) Fock

state. Using Eqs. (5) & (6), the time-dependent wave-

function is given by

|ψ(t)i=

∞

n,m=0

k=0 PnSmAn,m,k(t)|n−k, m +k, ki.(7)

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QuantumoptomechanicsintripartitesystemsRyanO.BehuninDepartmentofAppliedPhysicsandMaterialsScience,NorthernArizonaUniversity,Flagsta,AZ86011andCenterforMaterialsInterfacesinResearchandApplications(

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