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Theoretical framework for thin film superfluid optomechanics: towards the quantum regime

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2016 New J. Phys. 18 123025

(http://iopscience.iop.org/1367-2630/18/12/123025)

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New J. Phys. 18 (2016) 123025 doi:10.1088/1367-2630/aa520d

PAPER

Theoretical framework for thin film superfluid optomechanics:towards the quantum regime

ChristopherGBaker, Glen IHarris, David LMcAuslan, Yauhen Sachkou, XinHe andWarwick PBowenQueenslandQuantumOptics Laboratory, University ofQueensland, Brisbane, Queensland 4072, Australia

E-mail: c.baker3@uq.edu.au

Keywords: cavity optomechanics, superfluidity, superfluid helium films, optical resonators, third sound, effectivemass, liquids

AbstractExcitations in superfluid helium represent attractivemechanical degrees of freedom for cavityoptomechanics schemes.Here we numerically and analytically investigate the properties ofoptomechanical resonators formed by thinfilms of superfluid 4He coveringmicrometer-scalewhispering gallerymode cavities.We predict that through proper optimization of the interactionbetweenfilm and opticalfield, large optomechanical coupling rates p> ´g 2 100 kHz0 and singlephoton cooperativities >C 100 are achievable. Our analyticalmodel reveals the unconventionalbehaviour of these thinfilms, such as thicker and heavier films exhibiting smaller effectivemass andlarger zero pointmotion. The optomechanical systemoutlined here provides access to unusualregimes such as > Wg M0 and opens the prospect of laser cooling a liquid into its quantum groundstate.

1. Introduction

Thefield of cavity optomechanics [1] focuses on the interaction between confined light and amechanical degreeof freedom.Optomechanical techniques enable an exquisite degree of control over themotion ofmicromechanical resonators, with successful examples including ground-state cooling [2] and squeezing of themechanicalmotion of a resonator [3]. Recently, in a push to extend the realmof applications to biologicalsystems, there has been a growing interest in the study of resonators immersed or interacting with liquids [4–6],or the use of liquids as resonators [7]. In parallel, a special type of quantum liquid, namely superfluid helium, hasalso garnered significant attention for optomechanical applications, but for a different reason: themotivationhere being the ultra-low optical andmechanical dissipation it can provide due to its reduced optical scattering[8] and absence of viscosity [9, 10]; with both properties being particularly desirable for quantumoperations.

To date, themajority of superfluid optomechanics schemes have relied on bulk helium, withimplementations taking for example the formof a gram-scale resonator coupled to a superconductingmicrowave resonator [9], a capacitively detected superfluidHelmholtz resonator [11] or a helium-filledfibercavity [12].

In contrast, our group recently demonstrated an approach to superfluid optomechanics based onfemtogram thin films of superfluid 4He condensed on the surface of amicrotoroidal whispering galleryresonator [13, 14]. Leveraging the techniques of cavity optomechanics, we demonstrated real-time observationof the superfluid Brownianmotion, laser cooling of the superfluid excitations [13], as well as the possibility toapply large optical forces at themicroscale arising from the atomic recoil of superfluid heliumflow [14].However these devices, while exhibiting strong photothermal coupling, suffered from reduced radiationpressure coupling due to a poor overlap between the optical field and the superfluidmechanical excitations(known as third sound [15], see section 3).

In this work, we theoretically design a superfluid thinfilm resonator from the ground up, carefullyoptimizing the interaction between superfluid film and optical field (section 2), in order tomaximize thedispersive radiation-pressure optomechanical coupling.We investigate three different resonator geometries(microdisk, annularmicrodisk andmicrosphere), and provide useful analytical expressions for the effective

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18November 2016

ACCEPTED FOR PUBLICATION

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22December 2016

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mass and zero-pointmotion of superfluid films (section 3), as well as the scaling of optomechanical figures ofmerit with experimental parameters such as resonator dimensions and superfluidfilm thickness (section 4).Based upon this analysis, we predict large optomechanical coupling rates p> ´g 2 100 kHz0 and singlephoton cooperativitiesC0 greater than 10 are achievable with experimentally accessible designs, as well asunconventional regimes such as g0 greater than themechanical resonance frequency WM .

Superfluid thinfilms present a number of desirable properties: they are naturally self-assembling on thesurface of any resonator due to a combination of ultra-low viscosity and attractive van derWaals forces [10], theycan be of extremelyminute volume, with third sound detected infilms only twomonolayers thick [16] and theyoffer a large degree of tunability as their thickness and the frequency of the excitations they sustain can be sweptin situ over a large range simply by changing the heliumpressure in the sample chamber. In addition, thirdsound can exhibit strongDuffing nonlinearities, be strongly coupled to quantized vortices [17–20] and interactwith electrons floating on thefilm [21]. All together, these properties underline the potential of superfluid thinfilms as a promising platform for cavity optomechanics.

2.Opticalfield optimization

Figure 1 shows a schematic illustration of the optomechanical coupling scheme investigated in this paper. Acircular whispering gallerymode (WGM) resonator [22] is uniformly coatedwith a thinfilm of superfluidhelium [13]. Acoustic waves in this thinfilm known as third sound [15, 23, 24]manifest as thickness variationswhich dispersively couple to the confinedWGMvia perturbations to its evanescent field. As shown infigure 1(b), the fluctuating thickness of the superfluid in the vicinity of theWGM induced by a third soundwavemodulates the amount of higher refractive indexmaterial in theWGM’s near field, thereby changing the opticalpath length of the resonator. The frequency shift wD experienced by aWGMof resonance frequency w0 due tothe presence of the superfluid thinfilm is given by a perturbation theory approach [25]:

ò

òw

w

e

eD

= --( ) ∣ ( )∣

( ) ∣ ( )∣( )

E r r

r E r r

1

2

1 d

d, 1

0

filmsf

2 3

allr

2 3

where Eis the unperturbedWGMelectric field calculated in the absence of superfluid, e ( )rr

is the relative

permittivity and e = 1.058sf is the relative permittivity of superfluid helium [26]. The numerator integral istaken over the volume of thefilmwhile the denominator integral is taken over all space; equation (1) thereforeessentially relates the electromagnetic (EM) energy ‘sensing’ the perturbing element (thefilm) to the total EMenergy in themode. Figure 2(a) plots a FEM simulation of ∣ ∣E 2

for a transverse electric (TE) [27]WGMof

wavelength l m= 1.5 m confined in a typical silica disk of radius m=R 40 m and 2 μmthickness [28].Overlayed in black is a plot of the vertical dependence of ∣ ∣E 2

along the dashed black line going through the

center of theWGM.Thefield ismostly confinedwithin the silica resonator, and has low intensity on the top andbottom interface where it is in contact with the superfluid film (red dashes), resulting inweak detectionsensitivity/optomechanical coupling through equation (1). This limitation can be overcome by a proper choiceof disk thickness andWGMpolarization so as tomaximize the electric field at the interface, as discussed in thefollowing. Figure 2(b) shows the same verticalmode profile for a transversemagnetic (TM) polarizedWGM in a400 nm thick silica disk of identical radius, and for a TEWGMconfined in a 200 nm thick disk. Reducing thethickness of the disk pushes the field out of the resonator, increasing ∣ ∣E 2

over the superfluid region, such that

anyfluctuation in the film thickness results in amuch larger frequency shift of theWGM. Figure 2(c)provides amore systematic investigation of thismechanism.We employ the effective indexmethod (EIM) [27] to calculate

Figure 1. (a)Artistic rendering of a disk-shaped optical resonator sustaining an opticalWGMresonance (red and blue) covered in asuperfluid helium thinfilm. (b)Radial cross-section showing a finite elementmethod (FEM) simulation of theWGMfield intensity.The oscillating superfluidwave on the top surface of the resonator (solid and dashedwhite lines) dispersively couples to theWGMviaequation (1). (c)The case of an annularmicrodisk (top) and amicrosphere resonator (bottom) are discussed separately in appendicesA andB.

2

New J. Phys. 18 (2016) 123025 CGBaker et al

themode profile for a TE andTMguidedwave inside a slabwaveguide of varying thickness (normalized suchthat ò e =

-¥

¥E zd 1r

2 ), recording for each thickness the value ofE2 at the interface. For each polarization there isan optimal thickness whichmaximizes E2 at the interface: too thick and the field ismostly confinedwithin theresonator, too thin and the field becomes very delocalized along z and its value at the interface drops again. Fromthis analysis, we obtain the optimal silica disk thickness for TM (TE)WGMs as approximately 400 nm (200 nm).Next, we repeat the same analysis for different values of the refractive index of the slab, while tracking the optimalthickness for TE andTMWGMs. These results are summarized infigure 2(d) and show for instance that for ahigh refractive indexmaterial such as silicon or gallium arsenide, a disk thickness of∼200 nm is optimal for TMpolarizedWGMs. Since these results are based on the EIM, they start to lose accuracy for strongly confininggeometries (R on the order of a fewλ); nevertheless they provide a useful starting point for designing optimizedstructures. The TMpolarizedWGMs,with dominant field componentEznormal to the upper and lowerinterfaces, provide a step increase in the field outside the disk (due to the continuity of e Ezr ), as shown infigure 2(b), and are thereforemore sensitive to the superfluid [29].

Since the evanescent field decays along zwith characteristic length on the order of hundreds of nanometers(figure 2(b)), the change in ∣ ∣E 2

over the typicalD =z 1 to 30 nm thickness of thefilm can safely be neglected.

Equation (1) can therefore be rearranged to give the optomechanical optical resonance frequency shift per unitdisplacement w= D DG z0 [1, 30, 31]:

ò

òw w e

e=

DD

= --( ) ∣ ( )∣

( ) ∣ ( )∣( )G

z

E r r

r E r r2

1 d

d, 20 0 interface

sf2 2

allr

2 3

where the numerator integral is now a surface integral over the resonator top interface1.We employ the one-

dimensional version of equation (2) òe e= - -w( )( ) ( ) ( ) ( )G E z E z z1 0 d2 sf

2r

20 to provide pG 2 as a function

of resonator thickness in the right axis offigure 2(c). (This 1Dmodel is valid in the same regime as the EIM). Asfor double-disk optomechanical resonators [32],G is independent of resonator radius and does also not dependon superfluid thickness. Proper choice of resonator thickness is important, resulting for instance in a 20-foldimprovement inG for TMmodeswhen going from a 2 μmto a 0.4 μmthick silica disk. Through this

Figure 2. (a) FEMsimulation showing the radial cross-section of a TEWGMofwavelength 1.5 μmconfined in a silica disk of 40 μmradius and 2 μmthickness. Overlayed in black is a plot of the ∣ ∣ ( )E z2

mode profile along the dashed black line going through the

center of theWGM.The red region indicates the superfluidfilm on top and bottom. (b)Verticalmode profiles ∣ ∣ ( )E z2

for a 1.5 μmwavelength TM (resp. TE)WGMconfined in a 400 nm (200 nm) thick, 40 μmradius disk. The dashed red linesmark the disk upperand lower boundaries. Inset: FEM radial cross-section of ∣ ∣E 2

for eachWGM. (c)Value ofE2 at the interface calculated using the EIM,

for TM (blue) andTE (orange) polarizedmodes. Right axis: optical frequency shift per nmof superfluid pG 2 . (d) Indicative optimaldisk thickness formaximalfield at the interface and superfluid detection, as a function of diskmaterial refractive index.

1For simplicity wewill in the following consider excitations on the top surface of the resonator, although the treatment follows the same

approach for the bottom surface, but with different boundary conditions for the third soundwave due to the presence of the pedestal.

3

New J. Phys. 18 (2016) 123025 CGBaker et al

optimization, it is possible to reach large values ofG upwards of 6 GHz nm–1, despite the challenge posed bysuperfluid helium’s optical properties being very close to those of vacuum (e = 1.058r , n=1.029) [26]. This isachieved in part thanks to the perfect spatial overlap provided by the self-assembling nature of the superfluidfilm: these predicted coupling rates are for instance nearly three orders ofmagnitude larger than those obtainedin experiments inwhich a silicon nitride stringwas approached in amicrotoroid’s near field [25]. Noteadditionaly that unlike the previouslymentioned scheme inwhich a perturbing element of constant volume isapproached in the near field of a resonator [25, 33–35], this detection approach does not rely on an electric fieldgradient along the z direction, as we are instead detecting a perturbing element of changing volume.

It is interesting to examinewhether large coupling rates can also be achieved for superfluidmodes confinedto the vertical sidewall of theWGM. For a circularWGMcavity, the sensitivity to a change in radius is given by

wGRradial

0 [36]. From equation (2), but integrated along the vertical, rather than top boundary, it can be shownthatfluctuations in the thickness of the superfluid film on the vertical boundary would translate to

p pw ee

--( )G 2 2

Rvertical1

10 sf

SiO2

; 0.25 GHz nm–1 with the above parameters.

Here we have discussed the sensitivity to thicknessfluctuations affecting either the top, bottomor verticalboundaries of the resonator. Naturally, a variation in themean thickness of the filmwould produce a frequencyshift of + >G G2 10 GHzvertical nm–1. Thismeans a change infilm thickness of 10 pm (i.e. 1

36th of a helium

monolayer [37])would be sufficient to shift aWGMresonancewith = ´Q 2 106 by one linewidth, therebyproviding an ultra-precise independentmeans to optically characterize the superfluidfilm thickness, asignificant improvement over capacitive detection schemes commonly used in the superfluidcommunity [24, 38].

3. Superfluid third soundmodes

Third soundwaves [13, 15, 16, 23, 24, 39] are a type of excitation unique to superfluid thin filmswhichmanifestas thickness fluctuationswith a restoring force provided by the van derWaals interaction; they are somewhatanalogous towaterwaves (where the restoring force is gravity) [40], see figure 3(a). Third sound propagates at aspeed c3 given by [10]:

rra

= ( )cd

3 , 33s vdw

3

with r rs the ratio of superfluid to totalfluid density [10], avdw the van derWaals coefficient characterizing thestrength of the attractive force between the helium atoms and the substrate, and d the superfluidfilmmeanthickness. The van derWaals coefficients for various resonatormaterials are provided in table 1. The d1 3

dependency in c3 neglects the retardation effects in the van derWaals potential and is a reasonable first orderapproximation forfilms 0–30 nm thick [37, 41]whichwewill consider here. The disk geometry, in addition toproviding optical confinement to theWGMs, also confines third sound excitations localized on the top surface,giving rise to third sound resonances. The shape of these resonantmodes is dictated by the confining geometryand, for circular resonators, these take the formof Besselmodes as shown infigure 3(a) [18, 24, 39]. The thirdsoundmode profile hm n, describes the out-of-plane deformation of the superfluid surface for the (m; n)mode asa function of time t and polar coordinates r and θ:

h q z q= W( ) ( ) ( ) ( )⎜ ⎟⎛⎝

⎞⎠r t A J

r

Rm t, , cos sin , 4m n m n m m n M, , ,

wherem and n are respectively the azimuthal and radialmode numbers,A themode amplitude, Jm the Besselfunction of thefirst kind of orderm, zW = ( )c RM m n, 3 themode frequency and zm n, a frequency parameterdepending on themode order and the boundary conditions, see table 2. In the followingwe only focus on therotationally invariantmodes (m=0), as these are the oneswith largest optomechanical coupling [36].

3.1. Boundary conditionsThemode profiles for thefirst three rotationally invariant third soundmodeswith fixed (h =( )R 0) and free( h¶ =( )R 0r ) boundary conditions [43] are shown infigure 3(b). The free boundary condition is also known asthe ‘noflow’ boundary condition, as it requires the radial velocity of the superfluid flow to be 0 at r=R [43], andis therefore volume conserving. On the contrary, thefixed boundary condition does not conserve volume(particularly visible for the (m=0; n=1)mode), and therefore requires significantflow across the confiningboundary, in the incompressible limit. In circular 3He third sound resonators a crossover from free tofixedboundary conditions has been observed for films thicker than∼200 nm [43, 44]. For our resonator design, withthinfilms and near atomically sharp ‘knife-edge like’ [45]microfabricated boundaries, we expectminimal thirdsound drivenflow to occur between the disk’s upper and lower surfaces. This implies free boundary conditions

4

New J. Phys. 18 (2016) 123025 CGBaker et al

for the third soundwave, consistent with those observed in circular 4He third sound resonatorsmost similar toour design [18, 39].

3.2. Third soundmode effectivemassCalculating the optomechanical single photon coupling strength g0 and single photon cooperativityC0—usefulfigures ofmerit of the optomechanical system [1] (see equation (5))—requires the third soundmode effectivemass meff

k= =

W=

G( )

g G x Gm

Cg

2

4. 5

M M0 zpf

eff0

02

Figure 3. Superfluid third sound. (a) Left: schematic illustration of a superfluid third soundwavewith profile h ( )ron a film ofmean

thickness d (dashed orange line). The normalfluid component [46] is viscously clamped to the surface, while the superfluidcomponent rs oscillatesmostly parallel to the substrate (blue arrows). Right: plot of the surface profile for the (m=0; n=3)Besselmodewith free boundary conditions. (b)Radial profile h ( )r along the dashed red line in (a) for the (m=0; n=1) –blue–, (m=0;n=2) –green– and (m=0; n=3) –red–Besselmodes withfixed (left) and free (right) boundary conditions. (c)Comparisonbetween the trajectory described by a ‘particle’ in a fluid (left) and a solidmembrane (right).

Table 1.VanderWaals coefficients for a fewresonatormaterials.

Material avdw Source

Silica ´ -2.6 10 24 m5 s−2 [16, 42]CaF2 ´ -2.2 10 24 m5 s−2 [42]Silicon ´ -3.5 10 24 m5 s−2 [42]MgO ´ -2.8 10 24 m5 s−2 [37]

Table 2. First eight values of the frequency parameter z n0, forfixed and free boundary conditions. In the case offixed (free) boundaryconditions, these correspond to the zeroes of J0 ( ¢J0).

n=1 n=2 n=3 n=4 n=5 n=6 n=7 n=8 n=9 n=10

Free 3.832 7.016 10.174 13.324 16.471 19.616 22.760 25.904 29.047 32.19

Fixed 2.405 5.520 8.654 11.792 14.931 18.071 21.212 24.353 27.49 30.63

5

New J. Phys. 18 (2016) 123025 CGBaker et al

In continuummechanics, the effectivemass at reduction point Ais obtained by reducing the system to a

pointmass meff movingwith velocity ( )v Apossessing the same kinetic energyEk as the original system, that is

=m E

veff2 k

A2 , or:

ò r=

( ) ( )

( )( )m

v r r

v A

d. 6V

eff

2 3

2

For rotationally invariantmodes of a thin solid circular resonator of thickness d, this leads to thewell knownexpression for the effectivemass of a point on the resonator boundary [47]:

òpr

h

h=

( )

( )( )m d

r r r

R2

d. 7

R

eff solid0

2

2

Going from equations (6) to (7) assumes the velocity ( )v r(and density) do not depend on the z coordinate,

which is valid for both in- and out-of-planemechanicalmodes.Here, in order to circumvent the question of thedistribution of superfluid velocity [15] and density [16] below thefilm surface, we use the equipartition theoremto replace the kinetic energy term in equation (6)with the van derWaals potential energy stored in thedeformation of the film surface. In analogy to gravity waves [40], this energy is given by:

ò ò ò ò ò ò

ò ò ò

r q r q

r q

= -

=

q

p h q

q

p

p h q

= = =

+

= = =

+

( ) ( )

( ) ( )

( )

( )

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

E U z z r r U z z r r

U z z r r

d d d d d d

d d d , 8

r

R

z

d r

r

R

z

d

R

d

d r

pot0

2

0 0

,

0

2

0 0

0

2

0

,

where ( )U z is the energy per unitmass of the film due to the van derWaals potential [10]:a

= -( ) ( )U zz

. 9vdw3

In the limit of small amplitude surface oscillation h d , we obtain:

òa h q a h q

= - +h q+

( ) ( ) ( ) ( )( )

U z zr

d

r

dd

, 3 ,

2. 10

d

d r,vdw

3vdw

2

4

Therefore equation (8) becomes for rotationally invariantmodes:

òpra h a h

= - +( ) ( ) ( )

⎛⎝⎜

⎞⎠⎟E r r

r

d

r

d2 d

3

2. 11

R

pot0

vdw3

vdw2

4

Asmentioned previously, free (‘noflow’) boundary conditions are volume conserving (ò h =( )r r rd 0R

0),

therefore we obtain for the effectivemass of a point on the film surface at r=R:

òp r a h

h= =

W

-

( )

( )

( )( )m

E

v R

d r r r

R

2 6 d. 12

R

Meff

pot

2

vdw4

02

2 2

Finally, using W = zM

c

R3 and equation (3)wefind:

òrr z

prh

h= ´

( )

( )( )⎜ ⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝

⎞⎠m

R

dd

r r r

R

12

d. 13

R

effs

2

20

2

2

Herewe recognize the effectivemass of the solid case (equation (7)), multiplied by a prefactor proportional to( )R d 2. Thismeans thatwhile for a solid such as a circularmembrane meff scales as expected as R d2 (like the realmass), for a third soundwave on a superfluid film meff scales as R d4 —with thicker and heavierfilms thereforemaking for lighter resonators possessing larger zero pointmotion. This radically different scaling can beunderstood by considering themicroscopicmotion of a ‘particle’ of the resonator in both cases, as illustrated infigure 3(c). Indeed, while the surface deformation in each case is governed by the samemathematical equation(equation (4)), a particle in the solid describes an essentially verticalmotionwhile that in afluid an extremelyflattened near horizontal trajectory. As the useful displacement for optomechanical coupling is in the z direction,the horizontal particle excursion (µR) and horizontal kinetic energy are ‘wasted’ and appear as a penalty term inthe effectivemass.

TheR4 dependence of meff underscores the dramatic gains achieved by going towards smallermicrofabricated third sound resonators. Indeed, going froma centimeter-scale third sound resonator [18] to a40micron radius resonator such as outlined here and demonstrated in [13] affords—with otherwise identicalparameters—a 5×108 reduction in effectivemass and identical boost in cooperativity (equation (5)).

6

New J. Phys. 18 (2016) 123025 CGBaker et al

Note interestingly that we recover exactly the same R d4 effectivemass scaling if we consider a gravitationalwave in a normal liquid (by substituting the gravitational potential g z in equation (9)), and consider the shallowwater limit (l d )where the speed of sound =c g d only depends on liquid height [40].

4.Optomechanical coupling

In this sectionwe evaluate the performance of superfluid thin films as optomechanical resonators andsuccessively address how this performance is influenced by resonator dimensions, film thickness andmechanicalmode order.

4.1. Influence of resonator radiusFigure 4(a) plots the dependence of third sound frequency pW 2M and pg 20 on resonator radius for thefundamental (m=0; n=1) third soundmode on a 30 nm thick superfluid film. The solid orange linecorresponds to the value of g0 given by equation (5), employing the previously determined value of meff

(equation (13)) and assuming a constant p =G 2 6.6 GHz nm–1 (see section 2). This is a good assumption fordisk radii above m~20 m, as themicron sized radial extension of theWGM is small compared toR, and thesuperfluid displacement is therefore essentially constant over the opticalmode (see inset). For smaller radii (andhigher ordermechanicalmodes—see section 4.3) themode overlap between the optical field and the third sounddisplacementfield needs to be taken into account when calculating g0:

ò

òw e

e= -

-( )∣ ( )∣

( )∣ ( )∣( )g

q E r r

r E r r2

1 d

d. 14

r0

0 interfacesf

2 2

all2 3

Here = hh

( )( )

q xr

R zpf is the superfluid displacement profile normalized to xzpf atR. The orange dots correspond to

the results of individual FEM simulations using equation (14)withTMWGMson a 380 nm thick silica disk. Theanalytical expression is in good agreementwith the FEM simulation down to m~R 20 m (<10 % error), below

Figure 4.Optomechanical parameters.(a) Single photon optomechanical coupling strength pg 20 andmechanical frequency pW 2M

for the fundamental (m=0; n=1) third soundmode as a function of resonator radius, for a d=30 nm thick superfluidfilm. Solidline: analytical formula, points: individual FEM simulations. Inset: FEM simulation displaying theWGMoverlayedwith the thirdsoundmode displacement profile (colored line). (b) pW 2M (blue) and pg 20 (orange) as a function offilm thickness for a

m=R 20 m disk. (c)Predicted single photon optomechanical cooperativityC0 as a function offilm thickness d, for a m=R 20 mdisk. (d) Influence of third sound radialmode order n on pg 20 , for m=R 20 m and d=30 nm. Inset:mechanical surfacedeformation profiles and FEMsimulation showing theWGMmode overlayedwith the displacement profile of the (m=0; n=14)third soundmode. All results are for free boundary conditions.

7

New J. Phys. 18 (2016) 123025 CGBaker et al

which it overestimates g0. Since themechanical frequency scales as -R 1 and the zero pointmotion as -R 3 2, g0increases faster than WM asR is reduced, and for mR 20 m the system enters the uncommon optomechanicalregime of > Wg M0 . Table 3 summarizes the relevant scaling parameters for a disk resonator based onequations (3)–(5) and (13).

4.2. Influence of superfluidfilm thicknessFigure 4(b) plots the dependence of pW 2M and pg 20 on superfluid film thickness, for a m=R 20 m

resonator. Since WM scales as -d 3 2 and g0 as d5 4 (see table 3), both parameters start with extremely dissimilarvalues for thin films and evolve towards one another as d increases. In this particular case,

pW ´g 2 13.5 kHzM0 for d=34 nm.We take into account theWGM ∣ ∣E 2

field decay as the film getsthicker, but this is only aminor correction for the thin filmswe consider here. Next we plot infigure 4(c) thedependence of the single photon cooperativityC0 (equation (5)) on d. For this estimationwe consider an opticalloss rate k p =2 20 MHz corresponding to an opticalQ of 107 as demonstrated in thin silica disks of identicalradius2 [35], and a conservative estimate for themechanical = W GQM M M of 4000, as demonstrated in ourpreviousworkwithmicrotoroid resonators [13]. (Note that third sound dissipation rates several orders ofmagnitude below these values have already been demonstrated [39]).The predicted cooperativity displays astrong dependence onfilm thickness, reaching large values above unity, with =C 60 for d=30 nm. This value—on parwith the state-of-the-art in optomechanical systems [2, 48]—would represent a significant increase inperformance compared to existing superfluid optomechanics systems; it corresponds for instance to an overfour orders ofmagnitude increase over recently demonstrated superfluid heliumfilled fiber cavities [12]. Highcooperativities and large optomechanical coupling rates are essential for such applications as ultra-highprecision quantum-limitedmeasurement [1] of the superfluidmotion,mechanical ground state cooling [2] andaccessing the strong coupling regime [49, 50] between light and third sound excitations.

4.3. Influence of third soundmode radial order nThe effectivemass of a third soundmode is inversely proportional to z2, as shown in equation (13). Thisrelationship arises because as ζ increases, the ratio of vertical to horizontal superfluidmotion becomesmorefavorable, as the distance between the peaks and the troughs of the third soundwave is reduced (see figure 3).Higher order radial third soundmodes (with higher ζ; see table 2) therefore exibit lower meff and larger xzpf (seetable 3). Note that this is the opposite behaviour to that for a solidmembrane, inwhich xzpf decreases withincreasingmode order. Figure 4(d) plots results of FEM simulations of g0 versus third soundmode radial ordern, obtained through equation (14) for a m=R 20 m diskwith d=30 nm. It reveals two competing trends. First,and initially dominating, is the increase in g0 due to the increase in xzpf . Second, for higher n, the third sounddisplacement becomes oscillatory over theWGM (see inset) leading to a dramatically reduced overlap integral,see equation (14). In this particular case g0 reaches itsmaximal value of p ´2 16 kHz for n=5 and

pW =2 71 kHzM . The optimal radial order will naturally depend on device dimensions, with largerR leadingto a larger optimal n. These higher ordermodes provide a twofold benefit: beyond the higher g0, they have ahigher frequency and therefore exhibit a lower thermal phonon occupancy n̄ for a given bath temperature.Starting from a base temperature of 50mK, on the order of = ´n 5 10c

3 intracavity photonswould besufficient to feedback cool [14, 51] thismode into its quantumground state [2], with the required nc increasinglinearly with base temperature.

Table 3. Scaling of experimental parameterswith resonator radiusR,film thickness d andmode order ζ. Table should be read horizontally.Symbols – and † respectively denote nodependence and a non-monotonousdependence.

R d ζ

m ∝ R2 d −meff ∝ R4 -d 1 z-2

WM ∝ -R 1 -d 3 2 ζ

xZPF ∝ -R 3 2 d5 4 z1 2

g0 ∝ † d5 4 †

2The presence of the superfluid heliumfilm around the resonator does not adversely affect the opticalQ due to superfluid helium’s ultralow

optical absorption in the infrared [12, 13].

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4.4.Miniaturized third sound resonatorsFinally we briefly address the potential ofmicrometer-sizedWGMresonatorsmade of high refractive indexsemiconductors such as silicon [29, 52] or gallium arsenide [5, 53] for thin-film superfluid optomechanics. Theoptimal resonator thicknesses given infigure 2(d) are chosen tomaximize theWGMdeconfinement, resultingin lowWGMeffective indices, and therefore do not lend themselves towavelength-sized radii without incurringsignificant bending losses [54]. For this reasonwe consider thickermore confining disks for this application,such as 200 nm thickness for TEmodes [53]. This trade-off between sensitivity and opticalQ results in a lower

pG 2 on the order of 1 GHz nm–1. The pW =2 330 kHzM (m=0; n=1) third soundmode of a 30 nm thickfilm confined on top of aR=1 μmdisk has a zero pointmotion = ´ -x 1.6 10zpf

13 mat the periphery and alarge optomechanical coupling rate reaching up to p= ´g 2 136 kHz0 . Because of the lower opticalQ of theseresonators however, the expected cooperativityC0 is on parwith that expected for larger silica resonators.

5. Conclusion

Wehave investigated the potential of thin films of superfluid 4He coveringmicrometer-scale whispering gallerymode cavities as optomechanical resonators. Our analysis predicts large optomechanical coupling andcooperativities are achievable, and provides useful tools for the design of third-sound optomechanicalresonators. Furthermore, beyond their solemerits for ‘conventional’ optomechanics, the ability to engineerinteractions between third sound phonons and quantized vortices [17–20] aswell as electrons [21], combinedwith the ability to control superfluidflowon-chip [14] and generate long-lived persistent flows [55], makes this apromising platformwith varied applications such as ground-state cooling of a liquid, on-chip inertial sensing[56], single-photon optomechanics [57, 58] and the study of the dynamics of strongly interacting quantumfluids.

Acknowledgments

Thisworkwas funded by theAustralian ResearchCouncil through theCentre of Excellence for EngineeredQuantumSystems (EQuS, CE110001013);WPB acknowledges theAustralian ResearchCouncil FutureFellowship FT140100650.

AppendixA. Annularmicrodisk

Herewe consider the case of a third soundwave confined on the surface of an annular disk resonator, asillustrated infigure A1(a). In this case, the surface deformation profile h q( )r,m n, is given by [59]:

h q q= +( ) ( ( ) ( )) ( ) ( )r A J k r B Y k r m, cos , A.1m n m n m m n m n m m n, , , , ,

wherem and n are respectively the azimuthal and radialmode numbers, Am n, and Bm n, mode amplitudecoefficients and Jm andYm respectively the Bessel functions of thefirst and second kind of orderm. For the free-free boundary condition in =r Rin and =r Rout, thewavenumber km n, is defined as the nth root of theequation:

Figure A1.Annular disk case. (a) Schematic of an annular disk resonator, of inner radiusRin and outer radiusRout. (b)Calculatedsingle photon optomechanical coupling strength pg 20 andmechanical frequency pW 2M for the (m=0; n=1) third soundmodewith free–free boundary conditions (see inset) as a function ofRin, for a d=30 nm thick superfluidfilm. The value of -R Rout in iskept constant at 4microns.We obtain p =G 2 4.6 GHz nm–1 (accounting for themode overlap between third sound profile andWGM) fromFEMsimulations (see inset). (c)Predicted single photon optomechanical cooperativityC0 and effectivemass meff as afunction of inner radius, for the same parameters as (b).

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¢ ¢ - ¢ ¢ =( ) ( ) ( ) ( ) ( )J k R Y k R Y k R J k R 0 A.2m m m mout in out in

and the coefficientsA andB are found by imposing the free boundary conditions inRin andRout (i.e.h h¶ = ¶ =( ) ( )R R 0r in r out ). Following the same approach outlined in themain text, we numerically calculate

the values of pW 2M and pg 20 as a function of resonator dimensions for the annular disk case. This is shown infigure A1(b), whereRin is swept from10−5m to 1 mm,while the annulus width ( -R Rout in) is kept constant at 4microns. As the third sound frequency only depends on the annular width, WM remains constant over the entirerange (blue line). For afixed annular width, increasingRin results in the effectivemass increasing nearly linearly(simply as the resonator surface area), resulting in amuch less dramatic decrease in g0 with resonator radiuswhen compared to the disk case (seefigure 4), with for example p> ´g 2 5 kHz0 onmillimeter-sized annularresonators. Figure A1(c) plots the dependence ofC0 and meff over the same parameter range. The predictedvalue ofC0 assumes a constantQM=4000 and =Q 10opt

7, as in the disk case (see section 4.2), and reaches 22for the smallest resonators.

Appendix B.Microsphere resonator

Herewe briefly address the optomechanical coupling between third sound and light respectively confined on thesurface of and inside amicrosphere resonator [22, 60], see figure B1(a). In analogy to the disk case, wemodel thethird soundmode profile hl m, on a sphere as:

h q j q j=( ) ( ) ( )A Y, , B.1l m l m lm

, ,

with Al m, themode amplitude andYlm the Laplace spherical harmonic of degree l and orderm, solution to

Laplace’s equation on a sphere. Following the same treatment outlined in section 3.2, we derive the effectivemass of a point situated on the sphere’s equator (j p= 2):

ò ò òr q j j

h q j p=

= Wj

p

q

p h q j

= =

+( )( ) ( )

( )( )

( )

mU z z R2 d d sin d

, 2. B.2

d

d

Meff, sphere

0 0

2 , 2

2 2

Which, substituting W = +( )c l l R1M 3 , for amode rotationally invariant along θ simplifies to:

òrr

pr h j j j

h j p=

+ =j

p

=

( )

( ) ( )

( )( )

⎛⎝⎜

⎞⎠⎟m

R

d l l

2

1

sin d

2. B.3eff, sphere

s

40

2

2

Herewe recognize the same R d4 effectivemass scaling as in the disk case (see equation (13)). Next, as discussed

in section 2, we set = w ee

--( )G

Rsphere1

10 sf

SiO2

and use this approximation to calculate the optomechanical coupling

rate g0. This is shown infigure B1(b) for the (l=2;m=0) third soundmode. Thismode corresponds to thesuperfluid sloshing back and forth between the equator and the poles, as illustrated in the inset, and naturallyexhibits good optomechanical coupling to aWGM localized on the sphere’s equator. As in the case of a diskresonator, WM scales as 1/R and xZPF scales as -R ;3 2 however the dependence of g0 onR is even steeper than inthe disk case, since for a sphereG is also inversely proportional toR. For a sphere of radiusR=20microns and a30 nm thick film, p= ´g 2 410 Hz0 , approximately 28 times less than for the (m=0; n=1)mode of a disk ofidentical radius (see figure 4(b)). This smaller value has two distinct origins: a 2.2 times smaller xZPF for the thirdsoundmode on the sphere, because of its larger effectivemass, and a 12.5 times smaller G compared to the diskcase because of theweaker interaction between the superfluid film and theWGM in the sphere, as discussed insection 2. Finallyfigure B1(c) plots the dependence ofC0 and meff on sphere radius, for a 30 nm thick superfluid

Figure B1.Microsphere case. (a) Schematic of amicrosphere on a pedestal, with spherical coordinatesR, θ andj. (b) Single photonoptomechanical coupling strength pg 20 andmechanical frequency pW 2M for the (l=2;m=0) third soundmode (see insetrepresenting the extremes of third sound displacement, with fluid pooled at the poles (left) and equator (right)) as a function of sphereradius, for a d=30 nm thick superfluidfilm. Solid lines: analytical formula. (c)Predicted single photon optomechanical cooperativityC0 and effectivemass meff as a function of sphere radius, for the same third soundmode and film thickness as (b).

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film. The predicted value ofC0 assumes a constantQM=4000 and =Q 10opt9 [60]. This plot underscores the

strong dependence of meff with sphere radius: asR goes from10microns to 2 mm, meff spans over 9 orders ofmagnitude. ForR=2 mm, meff reaches 130 g, i.e. over ´5 108 times the actualmass of the superfluid film.

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