Coherent coupling completing an unambiguous optomechanical classification framework

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Xiang Li,1 Mikhail Korobko,2 Yiqiu Ma,1 Roman Schnabel,2 and Yanbei Chen1 1

Theoretical Astrophysics 350-17, California Institute of Technology, Pasadena, California 91125, USA

2_{Institut für Laserphysik und Zentrum für Optische Quantentechnologien,}

Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany

In most optomechanical systems a movable mirror is a part of an optical cavity, and its oscillation modulates either the resonance frequency of the cavity, or its coupling to the environment. There exists the third option – which we call a "coherent coupling" – when the mechanical oscillation couples several non-degenerate optical modes supported by the cavity. Identifying the nature of the coupling can be an important step in designing the setup for a specific application. In order to unambiguously distinguish between different optomechanical couplings, we develop a general framework based on the Hamiltonian of the system. Using this framework we give examples of different couplings, and discuss in details one particular case of a purely coherent coupling in a ring cavity with a movable mirror inside. We demonstrate that in certain cases coherent coupling can be beneficial for cooling the motion of the mechanical oscillator. Our general framework allows to approach the design of optomechanical experiments in a methodological way, for precise exploitation of the strengths of particular optomechanical couplings.

I. INTRODUCTION

Cavity optomechanics [1–4] studies the interaction be-tween light and mechanical systems embedded into opti-cal resonators. The precision at which modern optome-chanical experiments operate allows to study the quan-tum properties of light and matter, including the cool-ing of macroscopic oscillators to their quantum ground state [5–7], optomechanical squeezing of quantum fluc-tuations in light [8–11], quantum entanglement between optical and mechanical degrees of freedom [12–15] as well as between space-like separated mechanical oscil-lators [13, 16–19] and non-classical states of mechani-cal oscillators [12, 19–22]. Optomechanics has become an experimental platform for testing quantum mechan-ics in the macroscopic world [13, 16, 23, 24] and look-ing for potential paths towards quantum gravity [19, 25– 27]. Optomechanics has also been established as a tool-box for computational and metrological tasks, such as: frequency-converting microwaves to optical light [28–31], on-chip signal modulation and processing [32], nanoscale torque detection [33] and the detection of gravitational waves [34–36] with kilometre-scale detectors (Advanced LIGO [37, 38], Advanced Virgo [39, 40], GEO600 [41, 42], KAGRA [43, 44]).

While optomechanical systems vary in scale, frequency and complexity, their theoretical description on the fun-damental level can be reduced to simple Hamiltonians. Conventionally the coupling between the optical and me-chanical degrees of freedom is classified based on intuitive physical picture of the setup. Most common are the sys-tems withdispersive coupling, where the mechanical os-cillation modulates the cavity’s resonance frequency [1]. The simplest case of such systems is a Fabry-Pérot cavity with a movable end mirror [45]. Another type of coupling

isdissipative[46]: the oscillation modulates the coupling

between the system and the environment. Any system with a movable mirror that simultaneously couples with the cavity modes and the pumping field can be viewed

as having dissipative coupling, and the simplest example is a Fabry-Pérot cavity with a movable front mirror [47]. There exists another type of interaction where the me-chanical oscillation modulates the coupling between two or more cavity modes [48]. In practice, a complex op-tomechanical system might not fit into one single type of interaction presented above, or might be misclassified. Therefore, we want amutually exclusive and collectively

exhaustive way of classification.

To illustrate the necessity for such a classification, we show how the coupling could be identified ambiguously when the description of one optomechanical system has different forms depending on the choice of cavity basis modes. We consider two optical modes ˆa1,ˆa2 with fre-quenciesω1, ω2coupled via the mechanical oscillationx. Such system is described by the following intuitive Hamil-tonian:

ˆ

Hcav= ¯hω1â†1a1ˆ + ¯hω2â†2a2ˆ + ¯hg12x(â†1â2+ h.c.). (1)

If the system is classified simply based on this Hamil-tonian, it could fall into the category of "optical modes coupled by mechanical oscillation". However, this Hamil-tonian would have two different forms based on the pa-rameters of the system. The first case is when the fre-quencies of the modesˆa1,ˆa2 are equal (ω1 =ω2). Then Eq. (1) can be presented in another form:

ˆ

Hcav0 = ¯h(ω1−g12x)â01†â10 + ¯h(ω1+g12x)â02†â02, (2)

with the following choice of basis modes:

ˆ a01= ˆ a1−â2 √ 2 , â 0 2= ˆ a1+ â2 √ 2 . (3)

Such Hamiltonian is a dispersive one: the resonant fre-quencies of the modes are modulated by the mechanical oscillation. The second case is when the eigenfrequencies of the modes ˆa1,2 are separated by ∆ω _≡ (ω2₋ω1)/2. Then we can define a new basis of x-dependent modes

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ˆ a001,2(x): ˆ a001(x) = â1− g12 2∆ωxâ2, â 00 2(x) = â2+ g12 2∆ωxa1,ˆ (4)

where we assume mechanical oscillation to be small (g12x_|∆ω_|) and keep only the terms linear inx. Un-der this basis the Hamiltonian in Eq. (1) takes another form:

ˆ

Hcav00 = ¯hω1â001†(x)â001(x) + ¯hω2â002†(x)â002(x), (5)

where the modes themselves havex-dependence. Such a form of the optomechanical coupling is distinct from ei-ther the dispersive or the dissipative coupling. We call it

coherent coupling and will define rigorously in the next

section. These three different forms of the Hamiltonian illustrate the ambiguity: Eq. (1) describes the coupling between the two modes via the mechanical oscillation, but in different regimes depending on∆ω, it could also be either classified as dispersive coupling in Eq (2), or have some new form in Eq. (5). However, one system should have a unique classification, which is determined by the physical properties, not by the choice of basis. Identifying the coupling correctly and uniquely is impor-tant for optimizing the design of the experiment. Thus, an unambiguous classification framework is necessary.

In this paper, we establish a general framework for the unambiguous classification of the optomechanical sys-tems. The paper is organized as follows: Sec. II provides a step-by-step strategy for expressing the Hamiltonian in a canonical form and discriminating between different

x-dependence. We make emphasis in this section on the coherent coupling, which has not been widely recognized as a separate type of optomechanical coupling. Sec. III gives some examples from the literature, including possi-ble ambiguities that could arise in identifying couplings and how our approach helps to resolve them. We provide a further focus on the pure coherent coupling in Sec. IV, where we investigate an optomechanical ring cavity sys-tem [49–51]. We provide an application example where in certain cases this coupling is beneficial for laser cooling of the mechanical oscillator to its ground state. Sec. V includes a summary of the paper and further discussion.

II. CLASSIFICATION OF OPTOMECHANICAL

COUPLINGS

In this section we provide a step-by-step strategy that will lead to a unique classification for each cavity op-tomechanical system and help to avoid potential amigu-ity. We start by expressing the total optical Hamiltonian in a canonical form that can describe any optomechani-cal system with multiple optioptomechani-cal and mechanioptomechani-cal degrees of freedom:

ˆ

H(x) = ¯hâ†(x)Ω(x)â(x) +ih¯â†(x)Γ(x)ˆb₋h.c., (6) wherex=_{x1, x2, ..._} are the displacements of mechan-ical oscillators from their equilibrium positions; aˆ(x) =

(ˆa1(x),ˆa2(x), ...)T _{are the cavity eigenmodes, such that}

Ω(x) = diag(ω1(x), ω2(x), ...) is a diagonal matrix with the corresponding eigenfrequencies; bˆ = (ˆb1,ˆb2, ...)T

are the external electromagnetic modes, which cou-ple to cavity eigenmodes with coupling rates Γ(x) = diag(p2γ1(x),p2γ2(x), ...) and the optical linewidths

are γ1,2,...(x). Note that x can be treated as

quasi-stationary parameters here because the time scale for optical relaxation is much smaller than the mechani-cal one. For practimechani-cal mechani-calculation, thex-dependence in

ˆ

a(x),Ω(x),Γ(x)can be expanded in series andxcan be upgraded to dynamical variables and quantum operators

ˆ

xfollowing the canonical formulation [52–54]. We further consider a conventional linear regime, where the mechan-ical oscillationx is much smaller than the optical wave-lengthλ, allowing the Hamiltonian to remain only linear

xterms for a good approximation. While this approxi-mation is not necessarily applicable to all optomechanical systems [55–58], it covers most of the popular ones.

One system can be described by different Hamiltonians under different choices of basis, as shown from Eq. (1) to Eq. (5), but the canonical form in Eq. (6) is always

unique. This serves as the starting point for

establish-ing an unambiguous classification. Position-dependence inΩ(x)andΓ(x)are intuitive and can be directly under-stood as dispersive and dissipative couplings separately. We construct a mutually exclusive and collectively ex-haustive way of classification by considering the last the last possiblex-dependence: ˆa(x), which we call coherent coupling. Suchx-dependent modesaˆ(x)can be presented as linear combinations of unperturbed modesaˆ(0) cou-pled via the mechanical oscillationx, as follows from the linearity of the optical system:

ˆ

ai(x) =X

l

fil(x)ˆal(0), (7)

wherefil(x)are the coupling coefficients and fii(x)_≡1

(no summation for the repetitivei).

In order to classify an optomechanical system without any ambiguity, we formulate the following steps:

1. Write the total Hamiltonian of the optomechanical system including all the optical and mechanical de-grees of freedom and the coupling among them in any convenient basis.

2. Transform the Hamiltonian to the canonical form shown by Eq. (6), whereΩ(x)andΓ(x)are diagonal andˆa(x)is the set of cavity eigenmodes. Environ-mental modesbˆ can be chosen correspondingly. 3. Classify the type of optomechanical interaction by

thex-dependence feature inaˆ(x), Ω(x)andΓ(x). We can follow these steps to illustrate the classification of the optomechanical couplings into three types mentioned in the Introduction. We consider a specific example with two cavity modes ˆa1,a2ˆ and one mechanical degree of

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freedom x, and expand the x-dependence up to a lin-ear order in x, where g1,2 and gγ1,γ2 are the expansion coefficients of diagonal terms inΩ(x)andΓ(x)matrices:

1.Dispersive coupling, where the eigenfrequencies

de-pend on the mechanical oscillation: Ω(x), the ex-ample reads: Ω(x) = ω1₋g1x 0 0 ω2−g2x . (8)

2.Dissipative coupling, where the rates of coupling

to the external modes depend on the mechanical oscillation: Γ(x), the example reads:

Γ(x) =

√₂_γ1₊_gγ1x ₀

0 √2γ2+gγ2x

. (9)

3.Coherent coupling, where the eigenmodes depend

on the mechanical oscillation: aˆ(x), and the deriva-tive of any optical modeaiˆ with respect to any me-chanical displacement xj includes only the other optical modes (see Eq. (7)):

∂ˆai(x) ∂xj x=0 =X l6=i ∂fil(x) ∂xj x=0 ˆ al(0). (10)

The simple example can be taken from Eq. (5) where the original modes ˆa1,2 become mixed byx

(see Eq. (4)) and the corresponding eigenfrequency matrix: Ω00(x) = ω1 0 0 ω2 , (11) doesn’t depend onx: dΩ00(x)/dx= 0.

Following our classification strategy, one can clearly dis-tinguish among the different types of interactions, even in cases where several couplings coexist.

In addition to the definition above, a physical picture of coherent coupling also helps to understand this new concept. The difference between dispersive coupling and coherent coupling can sometimes be not obvious: both of them can be expressed as coupling of optical modes by mechanical motion under some specific cavity basis, as showed the example in Eq. (1). The distinction be-tween them is illustrated in FIG. 1: when mechanical motion couples different optical modes, these modes are either frequency-degenerate or have different frequencies. When the unperturbed modes are frequency-degenerate, the coupling via the mechanical motion breaks the de-generacy and leads to newx-dependent eigenfrequencies, which are the sign of dispersive coupling. On the other hand, when the unperturbed modes have different fre-quencies, the mechanical displacement leads to a coherent energy transfer between these modes, and such coupling is coherent. Expressed in the canonical Hamiltonian, up to linear order in x, the eigenfrequencies remains un-changed but the eigenmodes are the original ones mixed in ax-dependent way.

Coherent

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Dispersive

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!2 ↵2x

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!3+↵3x

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↵12x

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...

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0 B

B

@

1 C

C

A

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Dispersive

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!16=!26=!3

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FIG. 1. The comparison between coherent coupling and

dis-persive coupling: the Ω(x) matrix under the basis of ˆa(0).

Each color block represents a frequency degenerate subspace, which can have one or more modes. Different blocks have

different frequencies,e.g. ω16=ω26=ω3. The influence of

me-chanical oscillation shows up as x-dependent perturbation,

which will either directly add on to diagonal terms as

dis-persive coupling,e.g. the α3xterm, or will show up in

off-diagonal terms to couple different modes. The off-off-diagonal

coupling within one color block,e.g. theα2xterm, will open

the degeneracy and also cause dispersive coupling. While the

coupling between blocks, e.g. the α12xterm, won’t change

the eigenfrequencies ω1,2 and will cause coherent coupling.

See main text for detailed discussion.

III. EXAMPLES OF DIFFERENT COUPLINGS

In this section, we provide some detailed examples of optomechanical coupling of the above three categories. We also cover cases with coexisting couplings.

A. Dispersive coupling

Dispersive coupling is the most well-studied type of optomechanical interactions [1]. The physical origin of dispersive coupling is the dependence of cavity resonant frequencies on the mechanical oscillationx. The Hamil-tonian of a single cavity, shown in FIG. 2, reads:

ˆ

Hcav= ¯h(ωa−gωx)ˆa†ˆa, (12)

where ωa is the resonant frequency not affected by the mechanical oscillation,gω =ωa/Lis the dispersive cou-pling strength, x is the end mirror displacement from its equilibrium position (see detailed derivation in Ap-pendix A).

In this section, we discuss a metrological system that features the dispersive coupling: the Laser Interferometer Gravitational-Wave Observatory (LIGO) [38, 48]. This detector takes advantage of two Fabry-Pérot cavities in the arms of the Michelson interferometer (arm cavities), which sense the gravitational-wave-induced displacement of the test masses. The two arm cavity modes are rep-resented by ˆa,ˆb and their resonance frequencies are by

ω0. These two modes have the same dispersive coupling strengthg, but they couple to two different displacements

x1, x2. The cavity Hamiltonian can be expressed as:

ˆ

(4)

x

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FIG. 2. Single cavity with a movable end mirror. L is the

original cavity length,xis the end mirror displacement from

its equilibrium position,ˆais the cavity optical mode.

Defining the common and differential mechanical and op-tical modes asx+ = (x1+x2)/√2, x₋ = (x1₋x2)/√2;

ˆ

c+ = (ˆa+ˆb)/√2,cˆ− = (ˆa−ˆb)/√2, the transformed

Hamil-tonian takes the form:

ˆ

Hcav = ¯h(ω0₋gx+)ˆc†cˆ+ ¯hgx−ˆc†σxˆ ˆc, (14)

whereˆc= (ˆc+,ˆc−)Tandσxis thex-component of Pauli

matrix. Only the differential motionx−carries the

gravi-tational wave strain signal, so we don’t consider the com-mon motionx+. After this operation the transformation from c+,ˆ cˆ− to a,ˆ ˆb is equivalent to the transformation

fromˆa1,ˆa2toaˆ0

1,ˆa02in Eq. (3). Even though the

Hamilto-nian can be expressed in different forms in Eqs. (13)(14), in our classification strategy, the coupling will always

be classified as dispersive with eigenmodes ˆa,ˆb and x−

-dependent eigenfrequencies: ω_±(x₋) =ω0_±gx₋.

B. Dissipative coupling

Dissipative coupling happens when the coupling of cav-ity modes to external modes depends onx. For example, for a single cavity modeˆa:

ˆ

Hγ =i¯hp2γ+gγx aˆ†ˆb₋h.c., (15)

whereˆb is the external mode and√2γ+gγxis the cou-pling rate, which gives rise to the finite cavity linewidth. The gγx term describes the dependence of the dissipa-tion rate on the mechanical oscilladissipa-tion x. The form of dissipative coupling strength gγ depends on the specific physical realization.

One recent example is the on-chip dissipative op-tomechanical resonator [32]. As schematically shown in FIG. 3, this system consists of a racetrack optical cavity, which is also a mechanical resonator with out-of-plane vibrations, and a curved input waveguide. Except for the material refractive indices, the optical coupling rate between the racetrack cavity and the input waveguide is determined by the distance between them. The racetrack cavity supports optical modeˆaand the out-of-plane oscil-lation expressed by x, while the input waveguide carries optical mode ˆb. The mechanical oscillation x changes the distance between the racetrack cavity and the input waveguide and thus changes the optical coupling rate be-tween modes ˆa and ˆb. The Hamiltonian describing the

Mechanical Resonator<latexit sha1_base64="94bcyjzbO8XHlbVnvs0EBeebV/c=">AAACAnicbZDLSgMxFIYz9VbrrepK3ASL4KrMiKDLohs3QhV7gc5QMumZNjSTDElGKENx46u4caGIW5/CnW9j2s5CW38IfPznHE7OHyacaeO6305haXllda24XtrY3NreKe/uNbVMFYUGlVyqdkg0cCagYZjh0E4UkDjk0AqHV5N66wGUZlLcm1ECQUz6gkWMEmOtbvnAVzG+ATogwnrcx3egpSBGqm654lbdqfAieDlUUK56t/zl9yRNYxCGcqJ1x3MTE2REGUY5jEt+qiEhdEj60LEoSAw6yKYnjPGxdXo4kso+YfDU/T2RkVjrURzazpiYgZ6vTcz/ap3URBdBxkSSGhB0tihKOTYST/LAPaaAGj6yQKhi9q/YhqEINTa1kg3Bmz95EZqnVc/y7VmldpnHUUSH6AidIA+doxq6RnXUQBQ9omf0it6cJ+fFeXc+Zq0FJ5/ZR3/kfP4A6w6XHg==</latexit><latexit sha1_base64="94bcyjzbO8XHlbVnvs0EBeebV/c=">AAACAnicbZDLSgMxFIYz9VbrrepK3ASL4KrMiKDLohs3QhV7gc5QMumZNjSTDElGKENx46u4caGIW5/CnW9j2s5CW38IfPznHE7OHyacaeO6305haXllda24XtrY3NreKe/uNbVMFYUGlVyqdkg0cCagYZjh0E4UkDjk0AqHV5N66wGUZlLcm1ECQUz6gkWMEmOtbvnAVzG+ATogwnrcx3egpSBGqm654lbdqfAieDlUUK56t/zl9yRNYxCGcqJ1x3MTE2REGUY5jEt+qiEhdEj60LEoSAw6yKYnjPGxdXo4kso+YfDU/T2RkVjrURzazpiYgZ6vTcz/ap3URBdBxkSSGhB0tihKOTYST/LAPaaAGj6yQKhi9q/YhqEINTa1kg3Bmz95EZqnVc/y7VmldpnHUUSH6AidIA+doxq6RnXUQBQ9omf0it6cJ+fFeXc+Zq0FJ5/ZR3/kfP4A6w6XHg==</latexit><latexit sha1_base64="94bcyjzbO8XHlbVnvs0EBeebV/c=">AAACAnicbZDLSgMxFIYz9VbrrepK3ASL4KrMiKDLohs3QhV7gc5QMumZNjSTDElGKENx46u4caGIW5/CnW9j2s5CW38IfPznHE7OHyacaeO6305haXllda24XtrY3NreKe/uNbVMFYUGlVyqdkg0cCagYZjh0E4UkDjk0AqHV5N66wGUZlLcm1ECQUz6gkWMEmOtbvnAVzG+ATogwnrcx3egpSBGqm654lbdqfAieDlUUK56t/zl9yRNYxCGcqJ1x3MTE2REGUY5jEt+qiEhdEj60LEoSAw6yKYnjPGxdXo4kso+YfDU/T2RkVjrURzazpiYgZ6vTcz/ap3URBdBxkSSGhB0tihKOTYST/LAPaaAGj6yQKhi9q/YhqEINTa1kg3Bmz95EZqnVc/y7VmldpnHUUSH6AidIA+doxq6RnXUQBQ9omf0it6cJ+fFeXc+Zq0FJ5/ZR3/kfP4A6w6XHg==</latexit><latexit sha1_base64="94bcyjzbO8XHlbVnvs0EBeebV/c=">AAACAnicbZDLSgMxFIYz9VbrrepK3ASL4KrMiKDLohs3QhV7gc5QMumZNjSTDElGKENx46u4caGIW5/CnW9j2s5CW38IfPznHE7OHyacaeO6305haXllda24XtrY3NreKe/uNbVMFYUGlVyqdkg0cCagYZjh0E4UkDjk0AqHV5N66wGUZlLcm1ECQUz6gkWMEmOtbvnAVzG+ATogwnrcx3egpSBGqm654lbdqfAieDlUUK56t/zl9yRNYxCGcqJ1x3MTE2REGUY5jEt+qiEhdEj60LEoSAw6yKYnjPGxdXo4kso+YfDU/T2RkVjrURzazpiYgZ6vTcz/ap3URBdBxkSSGhB0tihKOTYST/LAPaaAGj6yQKhi9q/YhqEINTa1kg3Bmz95EZqnVc/y7VmldpnHUUSH6AidIA+doxq6RnXUQBQ9omf0it6cJ+fFeXc+Zq0FJ5/ZR3/kfP4A6w6XHg==</latexit>

Input Waveguide

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Optical Racetrack Cavity

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x

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Optical

Coupli_ng

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FIG. 3. On-chip Optomechanical coupling between the curved input waveguide and the optical racetrack cavity, adapted from FIG. 1 in Ref. [32]. The upper right is the top view of the chip, where blue line represents the input waveguide and the red one represents that optical racetrack cavity which is also a mechanical resonator that can have out-of-plane vibrations. The lower left is the schematic of the cross-section inside the

dashed area of the upper right. See the main context for

discussion.

whole system reads:

ˆ

H= ¯hωaaˆ†ˆa+i¯hp2γ+gγx ˆa†ˆb₋h.c., (16)

where neither the cavity mode ˆa nor its resonance fre-quencyωa depends on x. There exists only one cavity eigenmode and it already satisfies the canonical form of Eq. (6). Thus, the x-dependence in ˆa,ˆb coupling rate shows the feature of dissipative coupling.

C. Coherent coupling

The last interaction category to be discussed is co-herent coupling where the x-dependence appears in the eigenmodes themselves rather than the eigenfrequencies of the optical modes.

One notable example of coherent coupling is the three-modes optoacoustic interaction [59]. It can give rise to important non-linear optomechanical effects such as parametric instability [60, 61], which complicates the op-eration of gravitational-wave detectors. In a simplified model [59], there are two orthogonal transverse optical-cavity modesaˆandˆb with different resonant frequencies

ω1 and ω2. The acoustic mode has a torsional mode profile and x is its generalized coordinate. The cavity Hamiltonian in this case has the form (see Appendix B for detailed derivation):

ˆ

Hcav = ¯hω1ˆa†aˆ+ ¯hω2ˆb†ˆb+ ¯hG0xaˆ†ˆb+h.c.. (17)

Note that Eq. (17) has the same structure as Eq. (1) and thus follows the same transformation process as in