Resolved Sideband Measurements of a Microwave Optomechanical Resonator

March 25, 2024 by Avishek Chowdhury

Introduction

Optomechanics [1] deals with interaction between mechanical and electromagentic fields. On one hand, such interactions could lead to ultra-precision sensors not limited by the so-called Heisenberg limit. While on the other hand, they allow study of fundamental concepts in quantum mechanics through a "true" quantum-mechanical oscillator. In most of the optomechanical systems, the mechanical resonator is placed inside an electromagentic cavity, which enhances the so-called optomecahnical interaction between the electromagnetic field and the mechanical resonator to many orders of magnitude. This interaction generates the so-called Stokes and anti-Stokes scattering tones around the resonance frequency of the cavity. The electromagnetic fields can have frequencies ranging from THz in the optical domain to GHz regime in the microwave domain. Recently, we had the opportunity to visit the group of Dr. Hans Hübl in the Walter-Meißner-Institute at the Technical University of Munich, where they are performing such optomechanics experiments by interfacing nanomechanical resonators with in-plane superconducting microwave cavities. There we were able to perform sideband measurements with a single high frequency lock-in amplifier going up to 8.5 GHz, the SHFLI Lock-in Amplifier.

Experimental Set-up

The experimental set-up includes a flux-controlled superconducting \(\lambda/4\) coplanar waveguide (CPW) strip short-circuited to ground via a dc-SQUID at one end [2]. The dc-SQUID has a flux-dependent inductance that allows full resonance control of the microwave resonator through an external magnetic field. Optomechanical interaction is enabled by using two Aluminum nanomechanical resonators as part of the SQUID loop (see Fig. 1) [2]. The time-varying displacement of the nanomechanical resonator modulates the resonance frequency of the microwave resonator by modulating the flux-threaded area of the SQUID-loop. This modulation creates two sidebands around the microwave resonance frequency, \(f_c \pm f_m\). Here, \(f_c\) is the cavity resonance frequency and \(f_m\) is the eigen-frequency of the mechanical oscillator. The sidebands are referred as the Stokes and anti-Stokes lines respectively. On one hand, observing these sidebands leads to highly sensitive sensing applications. While on the other hand, observing the mechanical motions in low temperatures lead to fundamental studies of mechanical objects embedded deep in the quantum regime. Usually, this is performed by looking at the both Stokes and anti-Stokes line simultaneously as the ratio of their scattering rates give direct estimation of the thermal occupation [3].

Microwave optomechanical resonator coupled to a cavity and driven by SHFLI.

Figure 1: Scanning Electron Micrograph of the waveguide SQUID system with the nanomechanical resonator [2] and the connection to the SHFLI. SEM image courtesy of Korbinian Rubenbauer and Thomas Luschmann from the Hübl group at the Technical University of Munich.

Characterization of the Microwave Resonator

The system operates in the so-called dispersive regime and the sidebands can be observed by performing transmission or reflection measurements of the microwave resonator. We start by characterizing the resonance of the microwave resonator. Using the built-in LabOne Sweeper tool, the amplitude and the phase of the reflected signal can be measured. To find the resonance frequency, we first perform a sweep with a large frequency span, and then zoom in to get a detailed measurement of the response. In the auto bandwidth mode, the sweeper module automatically selects a suitable filter bandwidth for each sweep point based on the desired \(\omega\) suppression. The power of the drive was set to -12 dBm or 162 mV peak-to-peak at the output of the SHFLI. The drive signal then goes through many attenuation stages to finally reach the resonator inside a dilution refrigerator. A total attenuation of 110 dB guarantees a microwave signal up to a single photon level at the sample. The measured signal is then again amplified by 72 dB before it is fed to the SHFLI. The amplitude and phase response are shown in Fig. 2 and the extracted resonance frequency of the cavity is \(f_c\) = 6.462 GHz. At resonance, stabilization of the microwave cavity can be implemented to (1) reduce drifts of resonance which effectively misplaces the drive or probe tones and, (2) to reduce unwanted optomechanical effects on the mechanical resonator. The stabilization can be achieved directly using the SHFLI-PID Quad PID/PLL option by choosing the resonant phase as the set-point and flux-bias as the control. Further discussion on the stabilization is scheme is out of scope of this blog post.

Characterization of a microwave resonator with the SHFLI lock-in amplifier.

Figure 2: Plot of amplitude (in µV) and phase (degrees) response of the microwave cavity as a function of drive frequency (GHz). The cavity is measured in reflection which can be identified by a dip in amplitude and a phase transition at resonance frequency of around 6.462 GHz.

Sideband Measurements

The optomechanical system is in the so-called resolved sideband regime. Therefore, to increase the visibility of the mechanical motion, the cavity is driven with a drive, such that the drive frequency \(f_0 = f_c +f_m+\Delta f\) [see Fig. 3(a)]. Ideally, with precise knowledge of the cavity and mechanical resonance, one ends up with \(\Delta f\)=0. However, since \(f_c\) and \(f_m\) are not precisely known, we consider a combined error as \(\Delta f\). This results in scattered sidebands at \(f_0 \pm f_m\) (Fig. 3(a)) and the Stokes sideband is then read very close to the cavity resonance, which leads to an increased visibility. This sideband is then measured by setting the demodulator of the lock-in at \(f_c\) and choosing the filter bandwidth such that \(f_{BW}\geq \Delta f\). Thanks to the SHFLI-MF Multi-Frequency option of the SHFLI, it would be possible to drive the blue-detuned and the red-detuned sideband simultaneously to obtain the Stokes and anti-Stokes sidebands, which would allow to extract of the quantum feature of the mechanical resonator. In this blog post, however, we only show the so-called Stokes sideband in a heterodyne configuration. Another obvious way to extract the mechanical motion is by driving and demodulating the resonator at the same frequency i.e. \(f_c\) (homodyne detection). However, often this is not a desired configuration as the exact quantum picture of the mechanical system is lost. Additionally, the 11 MHz filter bandwidth of the SHFLI around the demodulation frequency limits the total analysis bandwidth for homodyne detection.

The measurement schematic is shown in Fig. 3(a). The drive frequency \(f_0\) is chosen such that \(f_0 -f_c\) = 5.5 MHz. The filter bandwidth (grey shaded area) is chosen to be \(f_{BW}\) = 0.5 MHz with a filter order 4. A data transfer rate of 1 MSamples/sec allows us to observe the Stokes sideband directly using the spectrum tool in LabOne. The spectrum tool allows up to \(N_{samples}\) = 8 Million samples and the frequency resolution is defined by the following relation, \(\delta f=(Samples/sec)/N_{samples}\). Additionally, the spectrum tool allows to directly perform a filter bandwidth correction in the LabOne tool and it is then possible to directly save the data to the PC. The exported data is saved and plotted externally as shown in Fig. 3(b). The Stokes sideband is observed around \(\Delta f\) = -301 kHzaway from the demodulation frequency \(f_c\). Knowing all the frequency components in play, we could then easily determine the eigen-frequency of the mechanical oscillator, \(f_m = f_0 - f_c\) + 0.301 MHz = 5.801 MHz. This is consistent with previously measured values with the help of an alternate set-up including VNA, Spectrum analyzers, IQ down-conversion circuit and many other electronic components.

Sideband analysis of Optomechanical Resonator with SHFLI

Figure 3: (a) Schematic of the drive tone \(f_0\) relative to the cavity resonance at \(f_c\) and the sidebands at \(f_0 \pm f_m\). The sidebands are shown by blue and red color respectively. The shaded blue color indicates the anti-Stokes sideband which is filtered out by the cavity. The solid red color indicates the Stokes sideband which is residing inside the cavity resonance (shaded green) and is hence transmitted through the cavity. Detection of the sideband while filtering out every other tone is achieved by choosing  \(f_{BW} \geq \Delta f\), where, \(\Delta f=f_c - (f_0 +f_m)\). (b) The measured frequency response of the Stokes sideband with the spectrum tool.

Conclusion

In this blog post we discuss the measurement of sidebands in a microwave cavity optomechanical system. The multi-frequency capability of the SHFLI allows independent observation of the sidebands by shifting the measurement frequencies to a different numerical oscillator. While we focused on measuring the Stokes sideband in this blog post, it is possible to configure a second demodulator with another oscillator to measure the anti-Stokes sideband simultaneously. Since the local oscillator frequencies can be independently chosen, it allows a wide possibility of choosing the filter bandwidth and the data transfer rate independently as well. Modules inherent to LabOne such as the Sweeper, Plotter, and Spectrum tool allow full characterization of the microwave resonator and the mechanical oscillator with reduced complexity and time. Furthermore, it is also possible to use to PID/PLL toolbox of the SHFLI to stabilize the microwave resonator directly through the flux bias. More information about various stabilization techniques can be found in this blog post.

Acknowledgements

We thank Korbinian Rubenbauer, Thomas Luschmann and Hans Hübl for the possibility to perform the measurements in their lab and for sharing the data.

The measurements were done together with my colleague Heidi Potts.

References

  1. "Cavity optomechanics". Aspelmeyer, Kippenberg, and Marquardt. Rev. Mod. Phys. 86, 1391 (2014).  
  2. "Mechanical frequency control in inductively coupled electromechanical systems". Luschmann et al. Scientific Reports  12, 1608 (2022).
  3. "Sideband cooling of micromechanical motion to the quantum ground state". Teufel et al. Nature 475, 359–363 (2011).