Abstract
Soft-clamped silicon nitride membrane resonators are capable of coherence times τ exceeding 100 ms at millikelvin bath temperatures. However, harnessing strong optomechanical coupling in dry dilution refrigerators remains a challenge due to vibration issues and heating by optical absorption. Here, we address these issues with an actuator-free optical cavity and mechanical resonator design, with the cavity mounted on a simple vibration-isolation platform. We observe dynamical backaction when the cavity is driven with a free-space optical beam stabilized close to the red sideband using a two-beam locking scheme. Finally, we characterize the effect of absorption heating on coherence time, finding it scales with the intracavity power P as τ ∝ P−(0.34±0.04).
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Cavity optomechanics has emerged as a dynamic field over the past few decades, fuelled by great progress in the fabrication of integrated, low-loss mechanical resonators [1]. Coupling light to mechanical motion through radiation pressure effects in optomechanical systems has led to advances both in fundamental physics and technological applications. Membrane-in-the-middle (MIM) systems, which utilize a partially reflective membrane resonator inside an optical cavity, have been of particular significance [2]. They have been used in a wide variety of experiments, ranging from investigations of the tenets of continuous quantum measurement [3–7], to topological [8] and parametric [9] energy transfer, and for applications in quantum information processing [10], gravitational wave detection [11], and force sensing [12,13].
When operating MIM systems in the regime where motion of the mechanical oscillator is dominated by quantum uncertainties, it is necessary for the optomechanical coupling rates to exceed the thermal decoherence rate of the system. Towards this goal, a large focus has been placed on developing mechanical resonators that demonstrate ultralow decoherence. Soft-clamped membrane resonators [14], which comprise a phononic crystal pattern with an isolated defect, were used to reach the quantum regime at moderate ($T\sim 10$K) cryogenic temperatures [5], as well as approach the quantum regime at room temperature [15]. The operation of such membranes in a dry dilution refrigerator is critical for applications such as electro-optic transduction [10,16] and realizing long-lived quantum memories [17]. However, this introduces several challenges associated with (i) maintaining stability within the high-finesse cavity, and (ii) optical absorption heating of the membrane resonator at high intracavity fields.
Challenge (i) involves aligning to and locking the high-finesse cavity. Various approaches have been explored, including misaligning fiber-coupled cavities at room temperature to achieve high coupling efficiencies at low temperatures [18]. To realize optical lock, actuators in fiber-coupled cavities [19] and free-space coupling to optomechanical cavities [10] have been used. However, dry dilution refrigerators tend to possess significant vibrations due to the use of a pulse tube system to maintain the Helium 4 at sufficiently low temperatures [20]. These vibrations often result in large excursions that complicate locking to high-finesse cavities.
Challenge (ii) arises from the optical absorption induced heating of membrane resonators at high intracavity fields. This phenomenon, which has been observed in numerous optomechanical and electro-optic experiments [21,22], is problematic as it can lead to higher mechanical bath temperatures. Studies have shown that patterned SiN membranes demonstrate low thermal conductivity [23] and absorption heating effects can become significant at millikelvin temperatures, especially within the near-infrared regime [11]. Plain membranes with sub-millimeter dimensions have been employed in (wet) dilution refrigerators before, in which no significant heating was observed when driven with laser radiation at $\sim 1$-$\mathrm{\mu}$m wavelenghth. ( [24] made this observation by exposing a 100 nm thick membrane to 7.4 mW intracavity power, while [10] reports such a finding while shining 200 mW of intracavity power on a 40 nm thick membrane.) Here, we investigate soft-clamped membranes resonators, which while offering higher quality factors ($Q>10^9$) and coherence times $\tau > 140 \text { ms}$ [17], are expected to suffer from increased heating due to their patterned structure and larger dimensions ($\sim$cm).
In response to these issues, here we present a design for a sideband-resolved optomechanical assembly that offers a method for effective coupling and locking a laser to the cavity within a dilution refrigerator. Figure 1 illustrates the soft-clamped membrane design with a mechanical resonance of interest at 1.32 MHz. The chip design incorporates additional coupling to a microwave cavity, although the details of this feature are outside the scope of this manuscript. We also investigate the absorption heating effect of the membrane from $805 \text { nm}$ wavelength laser light.
2. Mechanical design
The cavity design shown in Fig. 2(a) employs a plano-convex, over-coupled Fabry-Pérot cavity with highly reflective mirrors. The wavelength-dependent reflectivities of these mirrors allows tuning both the finesse of the cavity and the over-coupling ratio by adjusting the wavelength. These features enable us to achieve a cavity finesse over 30,000, a cavity linewidth below 300 kHz, and substantial over-coupling greater than 95%. The cavity assembly is made of oxygen-free high-conductivity copper. The individual parts are clamped together tightly using short stainless steel screws to reduce fluctuations within the cavity due to differences in thermal contraction. Within the assembly, the membrane frame lies parallel to the flat cavity mirror, separated by a 500 $\mathrm{\mu}$m silicon spacer. With an equally thick membrane chip, the membrane is located at approximately 1 mm from the flat mirror’s surface [25]. The total cavity length is $\sim 24 \text { mm}$. With the convex mirror’s radius of curvature around $25 \text { mm}$, the waist of the optical mode at the position of the membrane is $\sim 43\;\mathrm{\mu}$m.
The light is coupled to the cavity by aiming a free-space laser beam through windows in the cryostat, onto the more transmissive cavity mirror. Movements of the cavity assembly in directions orthogonal to the longitudinal cavity axis can induce rather dramatic fluctuations in the intracavity field. Likewise, movements of the cavity in the axial direction can lead to a motion of the mechanical resonator not limited by thermal noise. To address those two issues, the cavity assembly is affixed to the simple home-built vibration isolation platform shown in Fig. 2(b) and (c). It consists of a heavy ($1.9 \text { kg}$) rectangular (264 mm $\times$ 130 mm $\times$ 6.25 mm) copper plate that is suspended from the mixing chamber plate of a dry dilution refrigerator (LD250 by Bluefors) via thin copper sheets (24 cm$\times$5.1 cm $\times$ 0.6 mm), anchored at four points forming an inner rectangle of 170 mm $\times$ 104 mm. This construction is ’soft’ for oscillations along the cavity axis, allowing the platform to swing at a low eigenfrequency of 2.3 Hz, with the aim of mitigating MHz mechanical noise due to non-thermal, external vibrations. For oscillations orthogonal to the cavity axis, it is much stiffer, with the first eigenfrequency appearing at 145 Hz in the horizontal direction. Here, the goal is to avoid large amplitude, low frequency excursions that lead to cavity axis pointing noise. This platform allowed us to lock the laser to cavities with finesses up to $\sim 31,000$, despite an active pulse tube disturbing the system (albeit without an intracavity membrane).
3. Resonator design
This work utilizes a variant of Lotus-class soft-clamped membranes, which have demonstrated quality factors exceeding 1 billion in electromechanical experiments [17]. Our experiment specifically employs a phononic dimer membrane containing two coupled defects leading to a pair of hybridized mechanical modes, one symmetric and the other antisymmetric [13]. The released membrane has a rectangular extent of $5.4 \text { mm} \times 4.8 \text { mm}$. In our case, only one mechanical mode, at a frequency of $\Omega _m/2\pi \sim 1.32$ MHz will be considered. Figure 3(a) shows a zoom-in of the mechanical defect of a ’Lotus’ membrane described in Fig. 1.
If the cavity mode diameter at the membrane position exceeds the defect size of a patterned membrane, additional optical losses occur. These ‘clipping losses’ originate from the phase difference between the light field that travels through the material and that outside the defect. This results in altering the cavity wavefront and leads to coupling into higher-order modes. Minimizing these additional cavity losses is crucial to attain sideband resolution, a prerequisite for sideband cooling of a mechanical resonators to the quantum regime. To gain insights into this cavity loss effect, we carried out a series of tests using optomechanical cavity assemblies with varied defect sizes. Figure 3(b) shows the hexagonal hole pattern etched into 200 nm thick SiN membranes, which we used in these tests to simulate the defect of a soft-clamped membrane.
In the assembly procedure for these tests, we first clamp the membrane and flat cavity mirror together. Then, we scan a laser beam across the membrane, and measure the back-reflected power. These scans typically show interference between the light reflected directly from the membrane and off the flat cavity mirror behind the membrane (Fig. 3(c)). With 2D scans of the membrane plane, we can calculate the tilt between the membrane and cavity mirror from the interference fringe. In the example shown in Fig. 3(c), this leads to an angle of approx. $1 \text { mrad}$ for one interference fringe across 0.8 mm with a 830 nm laser beam. For all test assemblies, the angle is kept below 1 mrad to ensure tilt is not a dominant source of cavity loss.
The same scan method is then used to find the center of the membrane defect with the laser beam. In the example shown in Fig. 3(d), it occurs at a position around 7.93 mm. Once we have found the center position in the membrane plane, we position and fix the curved mirror in the assembly, forming the membrane-in-the-middle cavity.
The longitudinal position of the membrane with respect to the standing wave of the optical cavity determines the optomechanical coupling, among other factors. To identify positions of high coupling, we measure a series of subsequent fundamental cavity resonance frequencies. The position-dependent perturbation of the intracavity field through the dielectric SiN material causes a deviation of the cavity spectrum from equidistant modes separated by a constant free spectral range (FSR). This deviation $\Delta \omega _\text {FSR}$ can be converted into a coupling point using the model provided in [26]. It shows a periodicity of $2kz = 2\pi N z / L$, where $N$ is the mode number, $z$ the distance to the closest cavity mirror (here 1 mm), and $L$ is the cavity length (here 24 mm). We therefore record the resonance frequency of 24 subsequent cavity modes, in order to observe all coupling points and obtain the graph in Fig. 3(e). Then we perform cavity loss measurements via cavity ringdowns at various coupling points, yielding the data presented in Fig. 3(f). The findings demonstrate that the defect size does impact the cavity loss. We find that sufficiently small defects lead to significant enough cavity losses that sideband resolution becomes unattainable. Based on these results, we use Lotus-class defects with an innermost diameter of 230 $\mathrm{\mu}$m for relevant optical cavities.
4. Noise thermometry
For the optomechanical measurements at millikelvin temperatures, we clamp the cavity assembly to the vibration isolation platform described in Fig. 2(b). We shine laser light towards the cavity, through the cryostat’s windows, from fiber couplers mounted on a breadboard, which is itself clamped to the outer shield of the dilution refrigerator. Figure 4 shows a simplified version of the optical setup, which uses two orthogonally polarized beams derived from the same laser via two acousto-optic modulators (AOMs) [10]. In our experiments we use a widely tuneable, low-noise Ti:sapphire laser SolsTiS from MSquared at a wavelength around 800 nm. The first, weak “lock” beam is parked at resonance with the cavity and its reflection is used to derive a Pound-Drever-Hall (PDH) error signal to lock the laser frequency to the cavity. The second, stronger, red-detuned, “science” beam probes (and cools) the mechanical resonance. Its detuning from the cavity resonance is set by the frequencies at which the two free-space AOMs from CSRayzer are driven. We record mechanical noise spectra by direct detection of the science beam using avalanche photodiodes of type APD410A2 from Thorlabs.
We initially measure the thermal noise spectrum at room temperature, revealing the fundamental mode of the soft-clamped membrane at approximately 1.32 MHz (see Fig. 1(b)). As the setup cools, thermal contraction of the silicon frame triggers an approx. 2% down-shift in mechanical frequency. At millikelvin temperatures we observe the same resonance close to 1.30 MHz. As we increase the science beam power, we observe dynamical backaction damping as a broadening mechanical linewidth [1], as shown in Fig. 5(a).
For a systematic laser cooling series, we maintain the dilution refrigerator temperature at 20 mK and perform measurements at the science beam detunings [−1.0, −1.5, −2.0] MHz. The science beam’s input power is swept up to $10\;\mathrm{\mu}$W, at an estimated mode-matching efficiency of $0{.}8$. A fit to the PDH error signal gives a cavity linewidth $\kappa / 2 \pi = \left ( 2.0 \pm 0.2 \right ) \text { MHz}$, corresponding to a finesse of $\left ( 3.1 \pm 0.3 \right ) \times 10^3$.
Figures 5(b-d) show the results of Lorentzian fits to these spectra. We note resonance frequency shifts of different signs for distinct detunings, attributed to the anticipated optical spring effect. Moreover, the linewidth increase directly relates to the dynamical backaction effect, from which we infer a vacuum optomechanical coupling rate [1] of $g_0 / 2 \pi \approx 1.2 \text { Hz}$. For comparison, the maximum achievable coupling rate in a membrane-in-the-middle configuration can be approximated as $g_0 ^\text {max} \approx 2 (\omega _\text {c}/{L}) |r| x_\text {zpf} \xi$, where $\omega _\text {c}/2\pi$ is the cavity resonance frequency, $L$ is the cavity length, $r$ the optical field reflectivity of the membrane, $x_\text {zpf}$ is the zero-point fluctuation, and $\xi$ is the mode overlap between membrane displacement and optical field [15]. With the $50 \text { nm}$ thick membrane and $24 \text { mm}$ long optical cavity, we anticipate $g_0 ^\text {max} / 2 \pi \approx 8 \text { Hz}$ at perfect mode-overlap. We attribute the discrepancy with the measured $g_0$ to the unoptimized positioning of the membrane along the cavity axis and a potentially imperfect transverse overlap ($\xi <1$).
Finally, in Fig. 5(e) we plot the area $A$ of the mechanical peak over the intracavity power $P$ squared. In a direct-detection measurement, as in this setup, $A/P^2$ is proportional to the steady-state phonon occupation number $\bar {n}_\text {f}$. In our experiment’s regime, quantum backaction is negligible and the occupation is approximately given by
By combining Eqs. (2) and (3), a crude approximation would then suggest a scaling:
that is $\alpha _\text {lit.} \approx 0.55$, which has to be compared to our findings of $\alpha \approx 0.33$. We attribute the small discrepancy between those values to the questionable extrapolation we use to combine Eqs. (2) and (3), which have been measured in very different experimental situations.5. Conclusion
We have presented a detailed experimental procedure to construct and analyze an optomechanical setup with a Lotus-class soft-clamped membrane within an over-coupled Fabry-Pérot cavity. The setup operates in a dry dilution refrigeration thanks to a stable mechanical design and a vibration isolation platform. We have explored how the finite size of the membrane defect can introduce additional cavity losses. We have developed an efficient procedure of assembling MIM cavities and derived design restrictions on soft-clamped membranes, here being a defect size $>200\;\mathrm{\mu}$m. The thermal noise spectra observed during the noise thermometry experiments shows clear dynamical backaction cooling. From the noise thermometry measurement, we derived the power-law scaling of the decoherence rate with the optical power with an exponent of $\alpha = 0.33 \pm 0.04$, in reasonable agreement with existing literature. This suggests that soft-clamped membranes are subject to optical absorption heating and mechanical linewidth broadening.
Our findings provide insights for researchers in quantum optomechanics to optimize their experimental procedures. The techniques and approaches developed here could facilitate the realization of new optomechanical systems that operate in the quantum regime. In the future, coupling the second defect of the phononic dimer to a microwave resonator could enable quantum transduction experiments, with long-term intermediate storage of the quantum state, opening new possibilities in quantum communication and computation. Other applications include compact setups for manipulating optical quantum noise [11,29], as well as the search for unconventinoal decoherence [30] and dark matter [31].
Funding
Horizon 2020 Framework Programme (101107341, 801199); Danmarks Grundforskningsfond (Hy-Q); Novo Nordisk Fonden (NNF20OC0061866); European Research Council (101002179).
Acknowledgements
This work was supported by the European Research Council project PHOQS (grant no. 101002179), the Novo Nordisk Foundation (grant no. NNF20OC0061866) and the Danish National Research Foundation (Center of Excellence "Hy-Q"). This project has furthermore received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreements No. 801199 and 101107341.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available in [32].
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