Webinar Q&A – Unlock the Full Potential of Your Optomechanical Set-up with Digital Lock-in Amplifiers

October 23, 2024 by Avishek Chowdhury

This blog post accompanies the webinar "Unlock the potential of your Optomechanical system with digital lock-in amplifiers". In this webinar, we learned how to use a digital lock-in amplifiers to control, measure and stabilize optomechanical experiments. We focused on certain applications emphasizing how to use our GHz lock-in amplifiers to perform

Injection locking on GHz optomechanical resonator
Sideband measurements of a mechanical resonator coupled to a GHz superconducting microwave cavity, and
How to implement Pound-Drever-Hall Stabilization on a GHz superconducting microwave resonator.

In addition we also discussed implementation of pulsed experiment on hybrid optomechanical systems.

Optomechanics:

Cavity optomechanics deals with interaction between optical or microwave photons and mechanical oscillations (or acoustic phonons) that are coupled to each other by the so-called optomechanical coupling. Typically an optical or microwave cavity traps an electromagnetic field which is modulated by the movement of the mechanical element (see Fig. 1). This could be in form of changing the cavity length or the dielectric constant. This coupling typically results in sidebands (\(f_c \pm f_m\)) around the photonic carrier (\(f_c\)) separated by one mechanical resonance frequency, \(f_m\) as shown in Fig. 1.

The crucial component of any optomechanical experiment is to measure these sidebands also referred to as Stokes and anti-Stokes sidebands. The ratio of Stokes and anti-Stokes sidebands is a crucial figure of merit in optomechanical experiments as it gives away the “quantumness” of the mechanical resonator.

Figure 1: A schematic of an optomechanical system. The mass spring system representing a mechanical oscillator forms the end mirror of an optical cavity thereby inducing the so-called optomechanical coupling. The vibrating mechanical resonator generates sidebands around the optical carrier knowns as Stokes and anti-Stokes sidebands.

Advanced applications:

We discussed mainly three advanced applications focusing on three distinctive applications. Here we discuss them briefly:

Injection Locking: The principle of injection locking is often used to realize stable optical reference clocks. In this example we discussed how a chip based reference clock is established on a Nano-Opto-Electro-Mechanical System (NOEMS). The discussion follows the technique to drive and measure such NOEMS platform at GHz frequency range with the SHFLI and how to track the “locked” regime directly using different toolsets in LabOne. For more information please follow the blog post here.
Resolved sideband measurement: As discussed previously, the challenge of any Optomechanical experiment is to measure the weak mechanical sidebands arising from the interaction between optical and mechanical elements in the experiment. In this example we discussed measuring sidebands in a microwave optomechanical system previously reported in this blog post.
Stabilization and PDH: In this example we discussed the blog post on stabilization of a microwave cavity resonator with Pound-Drever-Hall technique using the built-in frequency/phase modulation and PID locking unit of the SHFLI. Pound-Drever-Hall technique is often used in stabilization compared to Phase-Locked-Loops (PLL) due to its superior performance against phase noise.

Towards pulsed measurements:

Optomechanical systems are going towards GHz frequencies mainly to facilitate quantum operations and moreover coupling to other quantum systems. These are typically referred as hybrid quantum systems that require complicated measurements and circuit design. Such systems require traditional continuous lock-in operations to characterize, feedback control the resonators. Besides, it might also require additional pulse control for information transfer between one physical system to another. The higher operational frequency range requires shorter pulses which are more difficult to design, control and time.

An example of such operation is shown in Fig. 2. The challenge is to couple a nanomechanical oscillator to an optical and a microwave cavity simultaneously. Such a set-up can facilitate information transfer between microwave and optical domain with the mechanical oscillator acting as a switch. The applications include quantum memory, transducers enabling quantum state transfers and quantum sensing. This section of the webinar discusses how to implement such control and measurement protocols with a single instrument such as SHFLI. Due to the fact that all of our lock-in amplifiers operate from DC to the specified bandwidth (ex. for SHFLI it is 8.5 GHz); it is mow possible to perform continuous and pulsed measurement for all the physical elements involved in the set-up including mechanical, optical and microwave systems.

Figure 2: Example of a hybrid optomechanical system. A mechanical resonator is coupled simultaneously to an optical and microwave cavity. This opens the door of information transfer between optical and microwave regime. An example may include microwave pulses transduced by the mechanical element and finally imprinted on the optical network. The optomechanical couplings guarantee phase coherent information transfer.

Questions and Answers:

Does the mechanical resonance frequency depend on the mass of the system?

The mechanical resonance frequency depends on the mass of the system. Typically the square of the frequency scales as inverse of the mass of the system. Therefore, such systems are often approximated by a simple harmonic oscillator.

Where do the lock-ins typically work? In the microwave (GHz) or optical (THz) frequency ranges?

Lock-in amplifiers are available from DC up to microwave frequencies, the highest frequency being 8.5 GHz with the SHFLI Lock-in Amplifier. This means that microwave cavities up to 8.5 GHz can be driven with output channel of the lock-in amplifier directly. For optical cavities which are in the THz frequency range, a modulation scheme using an AOM or EOM is used. These modulations are typically in the MHz regime and can therefore also be controlled using the output of a lock-in amplifier.

Are UHFLI and SHFLI essentially the same instrument with different bandwidth? Or they have significant differences.

We currently have five different lock-in amplifiers, where the main difference is the operating frequency range, e.g. the UHFLI is from DC to 600 MHz, while the SHFLI is from DC to 8.5 GHz. The functionality of the different models is very similar, for example they all come with the same time- and frequency analysis toolset including an oscilloscope, a spectrum analyzer, a plotter and a parametric sweeper. Also, all models offer upgrade options, such as multi-frequency analysis and feedback with PID controllers. The differences between the different instruments are the number of channels, and the sampling rate of the input and output channels, and there are small variations of the functionality and the available options. If you are interested in understanding which would best meet your experimental needs, please don't hesitate to contact us.

Can you please comment on investigations about the coupling magnons and terahertz phonon waves created by absorption of ultrashort light pulses?

Yes this field is seeing significant progress in recent years. To my knowledge Helmholtz Zentrum in Dresden have published noteworthy works in recent years. I can recommend the following publication: Salikhov et al. Nature Physics 19, 529 - 535 (2023).

I am curious whether it is possible for the mechanical oscillator of a cavity optomechanical system to have resonance frequencies in hundreds of MHz, or even in GHz range? If yes, I would appreciate if you could provide a reference.

Achieving mechanical resonances in the GHz range is now indeed possible. These resonances are not the traditional drum modes but rather confined phononic modes operating at high frequencies. For further reference, you can explore the work of Oskar Painter's group at Caltech. You can look at the work by Meenehan et al. PRA, 90, 011803 (2014) more more insight. Additionally, we have a blog post discussing injection locking with these types of resonators.

Do your qubit controllers also have similar capabilities, such as a vector network analyzer (VNA), spectrum analyzer, and signal generator (SG), that can be used simultaneously?

The SHFQC+ Qubit Controller features an integrated analysis toolset that can be utilized concurrently, specifically designed to meet the demands of qubit control and readout in a streamlined manner. Among its capabilities is a "results logger with a spectroscopy mode," which facilitates the characterization of device response as a function of frequency. While our qubit controllers are optimized for qubit control and readout experiments, our lock-in amplifiers provide a versatile array of multi-purpose time and frequency analysis tools that can be configured to accommodate diverse experimental requirements. If you’re interested in discovering which functionalities best suit your experiments, please feel free to contact us. We would be delighted to discuss your needs and demonstrate the various possibilities.