Superfluid Optomechanics With Phononic Nanostructures

In quantum optomechanics, finding materials and strategies to limit losses has been crucial to the progress of the field. Recently, superfluid ${}^4\mathrm{He}$ was proposed as a promising mechanical element for quantum optomechanics. This quantum fluid shows highly desirable properties (e.g., extremely low acoustic loss) for a quantum optomechanical system. In current implementations, superfluid optomechanical systems suffer from external sources of loss, which spoils the quality factor of resonators. In this work, we propose an alternate implementation, exploiting nanofluidic confinement. Our approach, based on acoustic resonators formed within phononic nanostructures, aims at limiting radiation losses to preserve the intrinsic properties of superfluid ${}^4\mathrm{He}$. In this work, we estimate the optomechanical system parameters. Using recent theory, we derive the expected quality factors for acoustic resonators in different thermodynamic conditions. We calculate the sources of loss induced by the phononic nanostructures with numerical simulations. Our results indicate the feasibility of the proposed approach in a broad range of parameters, which opens prospects for more complex geometries.

Figure 1

2D sonic crystal composed of an array of cylindrical pillars patterned in a substrate (gray regions). Hollow volumes in-between pillars are filled with superfluid ⁴He (blue regions). The square unit cell is shown on the right-hand side with side length $a_1\sim 100\mu \mathrm{m}$ and pillar diameter $a_2\sim 80\mu \mathrm{m}$. A point defect in the lattice will host the acoustic mode of the superfluid optomechanical system. The nanofluidic geometry is enclosed by bonding another substrate on top (not shown here for clarity), confining acoustic propagation within a thin superfluid ⁴He slab ($d\sim 100$ nm).

Figure 2

Schematic of the superfluid ⁴He dispersion curve, showing the low-energy branch of the excitation spectrum. Thanks to liquid ⁴He’s isotropy, the dispersion curve does not depend on the orientation of the excitation wavevector $\mathbf{q}$.

Figure 3

Phonon contribution ${\rho }_n^{\mathrm{ph}}/{\rho }_n$ (blue curve) and roton contribution ${\rho }_n^{\mathrm{rot}}/{\rho }_n$ (yellow curve) to the normal fluid density fraction. These two contributions are equal at a temperature $T\simeq 0.6$ K.

Figure 4

Beliaev-Landau processes where acoustic phonons are represented by wavy lines and thermal phonons by double lines. (a) Landau process ($q+q_2\to q_1$) where an acoustic phonon $q$ interacts with a thermal phonon $q_2$ to give a thermal phonon $q_1$, (b) Beliaev process ($q\to q_2+q_3$) where an acoustic phonon $q$ decays into two thermal phonons $q_2$ and $q_3$.

Figure 5

Quality factor of an acoustic mode $\mathbf{q}$, computed using the expression of the damping rate given in Eq. (29) for three-phonon interaction, as a function of the mode frequency ${\mathrm{\Omega }}_q/2\pi$ at temperatures of 10 (blue line), 20 (orange line), 50 (green line), and 100 mK (red line). The black dashed line indicates the ${\mathrm{\Omega }}_q^{-4}$ frequency dependence at frequencies higher than $k_{B}T/\hslash$.

Figure 6

Quality factor of an acoustic mode $\mathbf{q}$, computed using the expression of the damping rate given in Eq. (29) for three-phonon interaction as a function of the mode frequency ${\mathrm{\Omega }}_q/2\pi$. The different lines show the quality factor at different temperatures: 10 (blue line), 20 (orange line), 50 (green line), and 100 mK (red line). The dashed lines show the asymptotic limits of the damping rate for the four-phonon interaction [Eq. (36)], taken at 50 mK, at low $\tilde{q}$ (purple dashed line) and at high $\tilde{q}$ (pink dashed line). The superimposed black line is a guide to the eye to show the trend of the quality factor. The exact frequency dependence has a more complex structure around $\tilde{q}\sim 1$, which could be obtained by computing the full $\tilde{q}$ dependence of the four-phonon damping rate.

Figure 7

Quality factor of an acoustic mode $\mathbf{q}$, computed using the expression of the damping rate given in Eq. (29) for three-phonon interaction, as a function of the mode frequency ${\mathrm{\Omega }}_q/2\pi$ at temperatures of 10 (blue line), 20 (orange line), 50 (green line), and 100 mK (red line). The quality factor is computed from Eq. (40) for phonon-³He interaction processes and for different ³He concentrations: $x_3={10}^{-6}$ (blue dashed line), $x_3={10}^{-9}$ (green dashed line), and $x_3={10}^{-12}$ (orange dashed line).

Figure 8

Schematic of the nanofabrication process steps of nanofluidic geometries. (a) Substrate (gray) is covered with a masking layer (pink) prior to the lithography step, (b) lithography followed by substrate etching, (c) substrate cleaning, and (d) direct wafer bonding.

Figure 9

(a) Schematic of a rectangular nanofluidic channel made in a solid substrate (gray) and filled with superfluid ⁴He (blue) (b) Cross section of the nanofluidic channel taken along the black dashed line rectangle drawn in (a) showing an acoustic wave propagating in the $y$ direction. In the range of parameters used in this proposal, such a nanofluidic channel acts like an acoustic waveguide.

Figure 10

(a) Calculated phononic band structure for a square lattice of cylindrical pillars in superfluid ⁴He. The lattice spacing is $a_1=100\mu \mathrm{m}$ and the pillar diameter is $a_2=90\mu \mathrm{m}$. Three bandgaps are visible (blue bands) in this structure. (b) The reciprocal lattice showing the irreducible Brillouin zone (orange triangle) and the characteristic points $\mathrm{\Gamma }$, $M$, and $X$.

Figure 11

Phononic band structure enlarged around the lowest frequency bandgap calculated for a lattice spacing $a_1=100\mu \mathrm{m}$, at different values of the pillar diameter $a_2$, from $90\mu \mathrm{m}$ (black line) to $70\mu \mathrm{m}$ in steps of 5 $\mu \mathrm{m}$ (gray lines). The mode shape of the pressure field is shown at different characteristic points (color plot).

Figure 12

Mode shape of a point defect acoustic mode (${\mathrm{\Omega }}_a/2\pi =1.34$ MHz) at the center of a 2D sonic crystal of parameters $a_1=100\mu \mathrm{m}$ and $a_2=80\mu \mathrm{m}$. (a) Acoustic pressure field mode shape; (b) velocity field mode shape indicating the velocity magnitude (color scale) and direction (black arrows).

Figure 13

Quality factor $Q_a^{\mathrm{SC}}$ of a point defect mode obtained by considering planar radiation loss in helium caused by the finite size $N$ of the sonic crystal for different values of $a_2$ from $90\mu \mathrm{m}$ (black squares) to 70 $\mu \mathrm{m}$ (gray squares) in steps of 5 $\mu \mathrm{m}$, as indicated in the figure. The lines are linear fits of the datapoint on this log scale calculated for a crystal of size $N>17$.

Figure 14

Calculated pressure field outside of the defect at the mode frequency (${\mathrm{\Omega }}_a/2\pi =1.34$ MHz) as a function of the distance to the center of the point defect for three sonic crystals with parameters $a_1=100\mu \mathrm{m}$, $a_2=75\mu \mathrm{m}$ (blue line), 80 $\mu \mathrm{m}$ (red line), and 85 $\mu \mathrm{m}$ (black line). The pressure field is normalized by the maximum of the field at the center of the point defect. The envelope of the normalized pressure field is fitted with an exponential function $p(r)=\exp (-r/l_c)$, where $l_c$ is the characteristic decay length extracted from the fit and given by $l_c=176\mu \mathrm{m}$ (dashed blue line), $l_c=143\mu \mathrm{m}$ (dashed red line), $l_c=115\mu \mathrm{m}$ (dashed black line).

Figure 15

3D numerical simulation showing both the acoustic field (blue to red color scale) in helium and the displacement field (white to purple color scale) in the substrate taken at the mode frequency (${\mathrm{\Omega }}_a=1.34$ MHz). The fields are normalized to their maximum value. Panel (a) shows a 3D view and panel (b) shows an enlarged view of the cross section of the rectangular channel.

Figure 16

3D numerical simulation showing the acoustic field in the vicinity of the sonic crystal defect, and its coupling to elastic deformation in the substrate taken at the mode frequency (${\mathrm{\Omega }}_a=1.34$ MHz). Both the acoustic field (blue to red color scale) in helium and the displacement field (white to purple color scale) in the substrate are normalized to their maximum value.

Figure 17

Acoustic pressure field mode shape of a point defect acoustic mode (${\mathrm{\Omega }}_a/2\pi =1.34$ MHz) at the center of a 2D sonic crystal of parameters $a_1=100\mu \mathrm{m}$ and $a_2=80\mu \mathrm{m}$, in the presence of imperfections.

Figure 18

(a) Schematic of a nanofluidic device (light red area) composed of a nanoscale capacitor (dark red area) terminated by two large antennas (black area), embedded in a 3D superconducting microwave cavity (blue area), which can be read out with a pair of pin couplers (green areas). (b) Numerical simulation of the electric field mode shape in this type of resonator showing the focusing of the electric field inside the nanoscale capacitor with an arbitrary unit scale (red is for high intensity and blue is for low intensity).

Figure 19

Circuit diagram showing the coupling between the nanoscale capacitor ($C_{\mathrm{nano}}$) and the 3D superconducting microwave cavity with effective inductance $L$ and capacitance $C$. The coupling capacitors ($C_c$) couple the in-cavity field to the nanoscale capacitor. The in and out coupling capacitors ($C_{\mathrm{in}}$ and $C_{\mathrm{out}}$) represent the pin coupling to the measurement transmission line.

Figure 20

Illustration of the overlap between the pressure field (normalized color scale) of the sonic crystal’s defect mode and the electric field confined in the nanoscale capacitor located at the center of defect where the pressure field is maximum, leading to a mode overlap function close to unity.