Strong optical coupling through superfluid Brillouin lasing

TL;DR

This paper demonstrates ultralow threshold Brillouin lasing and strong phonon-mediated optical coupling in superfluid helium, enabling advanced applications in sensing, quantum information, and fluid dynamics exploration.

Contribution

It introduces a novel optically-induced surface deformation mechanism in liquids that surpasses electrostrictive interactions, enabling strong optical coupling and lasing in superfluid helium.

Findings

Achieved ultralow threshold Brillouin lasing in superfluid helium.

Demonstrated strong phonon-mediated optical coupling.

Proposed superfluid Brillouin gyroscope for precise superfluid circulation measurement.

Abstract

Brillouin scattering has applications ranging from signal processing, sensing and microscopy, to quantum information and fundamental science. Most of these applications rely on the electrostrictive interaction between light and phonons. Here we show that in liquids optically-induced surface deformations can provide an alternative and far stronger interaction. This allows the demonstration of ultralow threshold Brillouin lasing and strong phonon-mediated optical coupling for the first time. This form of strong coupling is a key capability for Brillouin-reconfigurable optical switches and circuits, for photonic quantum interfaces, and to generate synthetic electromagnetic fields. While applicable to liquids quite generally, our demonstration uses superfluid helium. Configured as a Brillouin gyroscope this provides the prospect of measuring superfluid circulation with unprecedented precision, and to explore the rich physics of quantum fluid dynamics, from quantized vorticity to quantum turbulence.

Equations

$${\Gamma }_{\mathrm{eff}}=\Gamma -\frac{4g_{0,\mathrm{tot}}{g}_{0,\mathrm{rp}}}{\kappa }\zeta n_{\mathrm{cav}},$$

$${\omega }_{\pm }=\bar{\omega }\pm \sqrt{{\Omega }_B^2/4+g_{\mathrm{opt}}^2},$$

Full text

Strong optical coupling through superfluid Brillouin lasing

Xin He∗

ARC Centre of Excellence for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, St Lucia, Australia.

∗These authors contributed equally. †To whom correspondence should be addressed, c.baker3@uq.edu.au

  

Glen I. Harris∗

ARC Centre of Excellence for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, St Lucia, Australia.

∗These authors contributed equally. †To whom correspondence should be addressed, c.baker3@uq.edu.au

  

Christopher G. Baker*∗,†*

ARC Centre of Excellence for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, St Lucia, Australia.

∗These authors contributed equally. †To whom correspondence should be addressed, c.baker3@uq.edu.au

  

Andreas Sawadsky

ARC Centre of Excellence for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, St Lucia, Australia.

∗These authors contributed equally. †To whom correspondence should be addressed, c.baker3@uq.edu.au

  

Yasmine L. Sfendla

ARC Centre of Excellence for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, St Lucia, Australia.

∗These authors contributed equally. †To whom correspondence should be addressed, c.baker3@uq.edu.au

  

Yauhen P. Sachkou

ARC Centre of Excellence for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, St Lucia, Australia.

∗These authors contributed equally. †To whom correspondence should be addressed, c.baker3@uq.edu.au

  

Stefan Forstner

ARC Centre of Excellence for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, St Lucia, Australia.

∗These authors contributed equally. †To whom correspondence should be addressed, c.baker3@uq.edu.au

  

Warwick P. Bowen

ARC Centre of Excellence for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, St Lucia, Australia.

∗These authors contributed equally. †To whom correspondence should be addressed, c.baker3@uq.edu.au

Brillouin, Superfluid

Brillouin scattering has applications ranging from signal processing Kittlaus_NatComm_17 ; MicrowaveSynthesizer , sensing Eggleton_Review and microscopy scarcelli2008confocal , to quantum information Renninger_NatPhys_18 and fundamental science HighFrequency ; kashkanova2017superfluid . Most of these applications rely on the electrostrictive interaction between light and phonons Eggleton_Review ; kashkanova2017superfluid ; giorgini_stimulated_2018 . Here we show that in liquids optically-induced surface deformations can provide an alternative and far stronger interaction. This allows the demonstration of ultralow threshold Brillouin lasing and strong phonon-mediated optical coupling for the first time. This form of strong coupling is a key capability for Brillouin-reconfigurable optical switches and circuits Ruesink_NatComms_18 ; Shen_NatComms_18 , for photonic quantum interfaces SafaviNaeini_NJP_11 , and to generate synthetic electromagnetic fields Fang_NatPhys_17 ; Schmidt_Optica_15 . While applicable to liquids quite generally, our demonstration uses superfluid helium. Configured as a Brillouin gyroscope BrillouinGyroscope this provides the prospect of measuring superfluid circulation with unprecedented precision, and to explore the rich physics of quantum fluid dynamics, from quantized vorticity to quantum turbulence sachkou_coherent_2019 ; Gauthier1264 .

Brillouin scattering is an optomechanical process that couples two optical waves via their interaction with travelling acoustic phonons. In the electrostrictive interaction usually employed, the optical electric field induces strain in a bulk medium, and the generated phonons scatter light between the two optical waves via refractive index changes caused by the medium’s photoelasticity. However, the inherent weakness of this interaction presents a significant challenge Eggleton_Review , necessitating the use of high optical powers and prohibiting some applications. This can be alleviated by resonant enhancement in an optical cavity, which has allowed recent demonstrations of ultralow linewidth lasers ObservationOfBrillouin ; Gundavarapu_IntegratedBrillouin , Brillouin lasing in liquid droplets giorgini_stimulated_2018 , non-reciprocal optical transport Dong_BIT_reciprocity , Brillouin gyroscopes BrillouinGyroscope and low-noise microwave oscillators MicrowaveSynthesizer . Alternatively, optically-induced deformations of the boundary of the medium can be leveraged to provide a Brillouin interaction, with scattering induced by the effective refractive index-modulation caused by the deformation. In purpose-engineered solid structures these surface interactions can be made comparable to, or even exceed, the native electrostriction Shin_NatCom_13 ; VanLaer_NatPhot_2015 ; wolff_stimulated_2015 .

Here, we transfer the concept of deformation-induced Brillouin scattering to liquid media, specifically a few-nanometer-thick superfluid helium film that coats the surface of a silica microdisk cavity and couples to its whispering gallery modes (see Fig. 1a,b) via perturbation of their evanescent field. Similar to other liquids kaminski_ripplon_2016 , the superfluid film has an exceedingly weak restoring force, affording a compliant dielectric interface that easily deforms in the presence of optical forces harris_laser_2016 ; baker_theoretical_2016 , as illustrated in Fig. 1c. This offers the potential for very large surface deformations and consequently extreme interaction strengths. We show that it allows radiation-pressure interactions with acoustic phonons that are over two orders of magnitude stronger than have been achieved using electrostriction in similar-sized microcavities ObservationOfSpontaneous ; Bahl_NatComm_2013 . These interactions are enhanced by a further order of magnitude by the superfluid fountain pressure, where optical absorption-induced entropy gradients induce superfluid flow mcauslan_microphotonic_2016 .

In addition to confining light, the microdisk used in our experiments provides confinement for travelling whispering gallery-like surface acoustic waves in the superfluid film. These acoustic waves are co-located with the optical whispering gallery modes close to the perimeter of the disk (see Fig. 1a) and are a class of third-sound ThirdSound , which manifests as film thickness fluctuations with a restoring force provided by the van der Waals interaction. They generate an effective refractive index grating that scatters photons between forward and backward propagating optical whispering gallery modes, forming a Brillouin optomechanical system. The thin wedge-shaped perimeter of the microdisk (see Fig. 1b) is engineered to maximize the optical evanescent field at the location of the acoustic wave baker_theoretical_2016 . Combined with the co-location of light and acoustic waves, this enhances the achievable radiation pressure coupling by several orders of magnitude compared to conventional optomechanics experiments previously performed with thin superfluid films harris_laser_2016 ; mcauslan_microphotonic_2016 ; sachkou_coherent_2019 .

In the backward Brillouin scattering process demonstrated here, a pump photon of frequency $\omega_{p}$ is scattered into a lower frequency counter-propagating Stokes photon at $\omega_{s}$, and a forward-travelling acoustic phonon of frequency $\Omega_{B}$ and wavelength $\lambda_{B}$ in the superfluid, where energy and momentum conservation dictate that $\omega_{p}=\omega_{s}+\Omega_{B}$ and that the optical wavelength $\lambda_{\mathrm{light}}=2\lambda_{B}$ (see Fig. 1a). In solids, the deformation due to electrostriction is determined by the Young’s modulus, $E$. By contrast, in our experiments, regions of high light intensity inside the resonator continuously deform a fluid interface by drawing in more superfluid by means of an optical gradient force and the superfluid fountain effect ashkin1986observation ; mcauslan_microphotonic_2016 . Here, the equivalent of the Young’s modulus is the van der Waals pressure on the surface of the fluid, $P_{v}=3\alpha_{v}\rho/d^{3}$, where $\alpha_{v}$ is the van der Waals coefficient, $\rho$ is the fluid density and $d$ is its thickness. While both electrostriction and surface deformation result in a periodically modulated refractive index grating which scatters pump light (illustrated for our case in Fig. 1(c)), for typical few nanometer-thick films, the van der Waals pressure is over six orders of magnitude lower than the Young’s modulus of typical Brillouin active solids ($\sim$ kPa versus GPa). As a result, the modulation achieved per photon can be much larger, enabling the ultra-low threshold lasing reported here, and providing the future prospect to reach quantum regimes. The reduced stiffness greatly reduces the speed of sound $c$, from kilometers per second for solids to meters per second for superfluid films. This results in a Brillouin shift $\Omega_{B}/2\pi=2c/\lambda_{\mathrm{light}}$ in the megahertz range, versus gigahertz range in many solids at telecom wavelengths.

These unique features offer two advantages. First, because of the small frequency shift, the Stokes beam is naturally resonant with the degenerate whispering gallery mode that counter-propagates with the pump mode, alleviating the need for potentially complex spectral engineering of the resonator BrillouinGyroscope ; ObservationOfBrillouin and allowing resonant enhancement that is independent of the device size, down to mode volumes in the range of the optical wavelength cubed. Quite generally, such miniaturization is desirable since it enhances the strength of the Brillouin interaction, but so far this has been hampered by the inability to meet momentum and energy conservation criteria in micron-scale devices with sparse optical spectra. Second, and a key result of this work, the strong Brillouin scattering enables the emergence of mechanically-mediated strong optical coupling, hybridising the initially degenerate forward and backward propagating fields.

The optical pumping of phonons into the forward traveling acoustic wave in backward Brillouin scattering amplifies the acoustic wave, modifying its linewidth to

[TABLE]

where $\Gamma$ is the intrinsic linewidth, $n_{\mathrm{cav}}$ is the intra-cavity photon number, $\kappa$ is the loaded optical linewidth, and $g_{0,\mathrm{tot}}=g_{0,\mathrm{rp}}+g_{0,\mathrm{fp}}$ is the single photon optomechanical coupling rate which includes radiation pressure ($g_{0,\mathrm{rp}}$) and fountain pressure ($g_{0,\mathrm{fp}}$) components (see Supplemental Material). The parameter $\zeta<1$ quantifies a reduction in Brillouin amplification from the presence of optical backscattering, caused by geometric imperfections of the microdisk. This reduction is due both to the backscattering-induced depletion of the pump field, and to active cooling of the acoustic mode by the backscattered photons, which directly competes with the amplification from the pump. Once the effective linewidth crosses zero, the acoustic damping is offset by the Brillouin amplification. In this regime, spontaneously occurring thermal excitations in the superfluid film are exponentially amplified to a large coherent amplitude, i.e. the system exhibits phonon lasing.

The apparatus used to experimentally probe superfluid thin-film Brillouin lasing is shown in Fig. 2, with details given in the Methods. Figure 3(a) shows the observed Stokes signal peak power as a function of input laser power, while the inset shows the Stokes linewidth versus power. As can be seen, the Stokes wave is amplified with increasing input power in good agreement with theory, and crosses the threshold to lasing at a power of 1.8 $\mu$W. Fitting the linewidth data to Eq. (1) yields an intrinsic acoustic linewidth of $\Gamma/2\pi=85\pm 6$ kHz. We extract a total single photon coupling rate of $g_{0,\mathrm{tot}}/2\pi=133$ kHz, independently of $\zeta$, through numerical simulations that include all dynamic couplings between optical and acoustic modes induced by backscattering. The radiation pressure component is estimated to be $g_{0,\mathrm{rp}}/2\pi=11$ kHz from finite element modeling (see Supplemental Material), leaving a fountain pressure contribution of $g_{0,\mathrm{fp}}/2\pi=122$ kHz. The total single photon coupling rate is more than three orders of magnitude larger than what is achieved using the electrostrictive interaction in similar-sized silica microspheres ObservationOfSpontaneous . Furthermore, it represents a several order-of-magnitude increase compared to previous superfluid optomechanics experiments with microtoroidal cavities harris_laser_2016 ; mcauslan_microphotonic_2016 ; sachkou_coherent_2019 .

The fountain pressure is introduced by optical heating through the temperature dependence of the entropy of the film. Gradients in entropy induce a flow of the superfluid component in an analogous manner to optical gradient forces. Despite the observed strength of this interaction, we verify that it comes in concert with only minimal levels of bulk heating, and with feasible modifications could allow ground state cooling Metzger_Nat_04 ; Jourdan_PRL_08 (see Supplemental Material).

Above the lasing threshold the pump light is depleted by scattering off the strong superfluid travelling refractive index grating, such that the linearized theory is no longer valid. Furthermore, the presence of optical backscattering introduces a dynamic coupling between optical modes. Therefore, the above-threshold behavior is fitted by numerically solving the full equations of motion of the optical and acoustic fields (see Supplemental Material). These simulations are in good agreement with both the measured power scaling (dashed red curve in Fig. 3(a)) and the optical power spectra (Fig. 3(b)), providing a confirmation of the saturation behavior through pump depletion.

In the Brillouin interaction, the lasing of the co-propagating acoustic wave creates an effective refractive index grating travelling at the speed of sound. If the backscattering generated from this grating is sufficiently strong, it is possible to enter a regime of mechanically-mediated photon-photon strong coupling, where photons cycle between the pump and Stokes resonances at a rate faster than the decay rate of the optical fields. This should be distinguished from the phonon-photon strong coupling previously demonstrated in both conventional Verhagen_Nat_2012 and Brillouin ObservationOfBrillouin optomechanics. The signature of this strong coupling is that the pump and Stokes resonances hybridize into a new pair of frequency-shifted optical eigenmodes, with frequencies given by:

[TABLE]

where $\bar{\omega}=(\omega_{p}+\omega_{s})/2$ is the average of the two bare optical modes that support the pump and Stokes photons, and $g_{\mathrm{opt}}=g_{\mathrm{0,rp}}\,\beta$ is the phonon-boosted optomechanical coupling rate with $\beta$ the amplitude of the co-propagating acoustic wave (see Supplemental Material).

Our superfluid thin-film optomechanical Brillouin laser is naturally suited to observe the mode hybridisation induced by strong coupling due to its combination of high radiation pressure coupling rate $g_{0,\mathrm{rp}}$ and low Brillouin frequency $\Omega_{B}$. This allows access to the regime where $g_{\rm opt}\gg\Omega_{B}$ in which the optical modes are shifted by frequencies of $\pm\left[(\Omega_{B}^{2}/4+g_{\mathrm{opt}}^{2})^{1/2}-\Omega_{B}/2\right]\sim\pm g_{\mathrm{opt}}$. By comparison, in the usual regime where the Brillouin frequency is high enough that $\Omega_{B}\gg g_{\rm opt}$, the optical mode shifts are suppressed by a factor of $\Omega_{B}/g_{\rm opt}$.

To spectroscopically explore the regime of strong coupling we implement a pump-probe measurement scheme, shown in Fig. 2 and described in the Methods. Figure 4 shows the optical resonance measured with this setup. With the pump laser off (Fig. 4(a)), no splitting is evident between the clockwise (pump) and counter-clockwise (Stokes) optical modes. Injecting 5 $\mu$W of pump light initiates spontaneous Brillouin lasing, as verified through the reflected light power spectrum. Due to the high contrast refractive index grating generated by the Brillouin lasing process, two distinct non-degenerate cavity resonances can now be resolved (Fig. 4(b)): a lower frequency resonance (red) sensing the peaks of the superfluid Brillouin wave, and a higher frequency resonance (blue) — spatially shifted by $\lambda_{B}/2$ — sensing the troughs, see inset of Fig. 4(b) and Supplemental Material. The magnitude of the splitting provides the coupling rate $g_{\mathrm{opt}}=2\pi\times 187$ MHz, which is larger than the half-width at half maximum of the optical mode ($\kappa/2=2\pi\times 142$ MHz), indicating mechanically mediated strong coupling between the two optical modes Verhagen_Nat_2012 .

In summary, we have shown that optically induced surface deformations in liquids can facilitate greatly enhanced Brillouin interactions. We use this concept to demonstrate ultralow threshold Brillouin lasing in few-nanometer-thick superfluid helium films and, for the first time, phonon-mediated strong coupling between optical cavity modes. While the microwatt-range lasing threshold presented here is already lower than all previously reported values grudinin_brillouin_2009 , even modest improvements in acoustic dissipation and optical linewidth – achieved for example by reducing sidewall roughness on our existing devices Chemically – could lower the threshold power into the picowatt range, significantly below what can be achieved with solid-state systems.

The system presented here represents a substantial departure from traditional travelling-wave Brillouin systems, which typically exhibit weak single photon coupling due to large mode volumes and high Young’s moduli. While this can be partially mitigated via resonant enhancement, doing so typically requires millimeter-scale devices to satisfy energy and momentum matching requirements BrillouinGyroscope ; MicrowaveSynthesizer . In contrast, superfluid helium, with its ultra-compliant fluid interface, affords a low-frequency Brillouin shift that is naturally resonant with the co-propagating whispering gallery mode, regardless of the device size. This paves the way towards resonantly enhanced backward Brillouin scattering on devices with mode volumes as small as the optical wavelength cubed and with optomechanical coupling rates in excess of $g_{\mathrm{0,rp}}/2\pi\sim 300$ kHz baker_theoretical_2016 .

Phonon lasing in superfluid helium opens up a new approach to explore quantum turbulence and quantum fluid dynamics in a strongly-interacting system sachkou_coherent_2019 . In this application, the combination of phonon-laser-induced mechanical line narrowing feng_self-sustaining_2008 with operation as a precision Brillouin gyroscope BrillouinGyroscope would provide unprecedented sensitivity in measurements of the quantized circulation of the superfluid. Applied to conventional fluids, the same Brillouin process could be used to optically mix microdroplets and to probe their material properties, important tools for picolitre-scale chemistry, biophysics and the biosciences destgeer_recent_2015 ; palombo_brillouin_2019 . Moreover, strong phonon-mediated optical coupling may allow the generation of synthetic electromagnetic fields Fang_NatPhys_17 ; Schmidt_Optica_15 , and superfluid based optical switches and reconfigurable optical circuits Ruesink_NatComms_18 ; Shen_NatComms_18 .

Methods

Experimental apparatus used to probe superfluid Brillouin lasing. To experimentally probe superfluid thin-film Brillouin lasing, we employ the experimental apparatus shown in Fig. 2. The microdisk is coupled to a tapered fiber and cooled to $\sim$20 mK in a sealed sample chamber within a dilution refrigerator. Helium-4 gas injected into the chamber forms a nanometer-thick self-assembling superfluid film coating the microresonator harris_laser_2016 (see Supplemental Material). A telecom wavelength pump laser drives the Brillouin lasing process, and the backwards emitted Stokes and anti-Stokes light is observed using optical heterodyne detection and characterized on a spectrum analyser. We select a microdisk optical mode at $\lambda_{\rm light}=1555$ nm with a loaded linewidth of $\kappa/2\pi=284$ MHz, corresponding to an optical quality factor of $Q\simeq 7\times 10^{5}$. This optical mode is also found to have a backscattering rate of $\kappa/2\pi=75$ MHz which suppresses the Brillouin induced linewidth narrowing by a factor of $\zeta=0.22$ (see Eq. 1 and Supplemental Material).

Pump-probe scheme used to characterize phonon-mediated strong coupling. To spectroscopically characterize phonon-mediated strong coupling a weak tunable diode probe laser is added to the heterodyne setup shown in Fig. 2. While the pump laser is set on resonance with the optical mode, the probe laser is repeatedly swept at 100 Hz across the optical resonance. The transmitted optical photodetector signal is low-pass filtered (filter bandwidth = 20 kHz) and averaged 256 times on an oscilloscope in order obtain the optical cavity spectrum, free from any potential modulation due to the Brillouin lasing process.

Acknowledgements

This work was funded by the U.S. Army Research Office through grant number W911NF17-1-0310 and the Australian Research Council Centre of Excellence for Engineered Quantum Systems (EQUS, Project No. CE170100009). W.P.B. and C.G.B respectively acknowledge Australian Research Council Fellowships FT140100650 and DE190100318. This work was performed in part at the Queensland node of the Australian National Fabrication Facility, a company established under the National Collaborative Research Infrastructure Strategy to provide nano and micro-fabrication facilities for Australia’s researchers.

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