Parity measurement in the strong dispersive regime of circuit quantum acoustodynamics

Authors: Uwe von Lüpke, Yu Yang, Marius Bild, Laurent Michaud, Matteo Fadel, and Yiwen Chu

First Author’s Primary Affiliation: Department of Physics, ETH Zurich, Zurich, Switzerland

Manuscript: Published in Nature Physics

Introduction

Superconducting qubits are a promising candidate for functional quantum computation as well as investigating fundamental physics of composite quantum systems where superconducting qubits are coupled to other quantum degrees of freedom. The most common example of this is circuit quantum electrodynamics (cQED), where a superconducting qubit is coupled to an electromagnetic resonator, and the resonator can be used to control and read out the quantum state of the qubit. In an analog to cQED, it is possible to replace this electromagnetic resonator with a mechanical resonator – this now allows for the study the quantum limits of mechanical excitations in a field commonly known as circuit quantum acoustodynamics (cQAD). By coupling a superconducting qubit to a mechanical resonator in this fashion, physicists are able to draw upon the rich and developed field of cQED to study not only further applications in quantum information science using cQAD as a building block, but also the fundamental physics of mechanical resonators in their quantum limit. In addition to the ability to study new physics, acoustic resonators are much more compact due to the slow speed of sound (relative to the speed of light which would be used in an electromagnetic cavity!) leading to much smaller wavelengths at high frequencies. In cQED/cQAD the interaction between the qubit and the resonator is often described by the Jaynes-Cummings Hamiltonian:

$\hat{H}/ \hbar = \omega_c \hat{a}^{\dagger} \hat{a} + \frac{\omega_q}{2} \hat{\sigma}_{z} + g\left( \hat{a}\hat{\sigma}_{+} + \hat{a}^{\dagger} \hat{\sigma}_{-}\right)$

Here the first term in the Hamiltonian describes the resonator as a harmonic oscillator with a transition frequency $\omega_c$ , and the second term describes the qubit as a two level system with transition frequency $\omega_q$ . The interesting physics described by this Hamiltonian is contained in the third term, which contains the interaction between the qubit and the resonator. Because the terms $\hat{a} \hat{\sigma}_{+}$ and $\hat{a}^{\dagger} \hat{\sigma}_{-}$ conserve total excitation number, we can think of this interaction term as the qubit and the resonator “trading” excitations with a rate $g$ !

In this recent paper published in Nature Physics, the authors demonstrate strong coupling between a superconducting qubit and an HBAR (high bulk overtone acoustic resonator)[1]. HBAR devices launch mechanical excitations (called phonons) by using the piezoelectric effect. This means that the polarization and the mechanical strain in the material are not independent – by applying an electric field to a piezoelectric material it is possible to create mechanical excitations! The device in this experiment uses a thin film of piezoelectric aluminum nitride (AlN) patterned onto a small sapphire chip. This substrate is then sandwiched together with another chip containing a superconducting qubit which acts as an anharmonic oscillator. By carefully aligning the two chips relative to each other, the authors are able to couple the electric field of the qubit to the piezoelectric material on the chip containing the HBAR and thus couple the degrees of freedom of the qubit to the phonon modes in the HBAR (see Fig. 1 for a description of the device). The joint quantum acoustics system is then loaded into an electromagnetic cavity, which will also couple to the qubit and allow for the control and measurement of the device.

**Figure 1**
(a) View of the hybrid flip-chip device. Both large substrates are sapphire, which is transparent and thus makes it much easier to align the two chips to a high precision. The substrate that contains the superconducting qubit is on the bottom and slightly larger than the top chip which contains the HBAR resonator. (b) Optical microscopy image of the small disc of piezoelectric AlN which is responsible for the creation of phonons.

Measurement of Phonon Lifetime

By applying strong microwave signals into the system, the qubit frequency is able to be moved around by a small amount such that the qubit’s resonant frequency can be equal to the resonant frequency of the phonon mode. In this case, the qubit and mechanical system will transfer exctations to each other in the time $\pi/2g$ . This can be used as a tool to measure how long phonons will remain in the HBAR device by first promoting the qubit to its excited state and then shifting the qubit’s frequency so that it’s resonant frequency is the same as that of the mechanical mode for a time $\pi/2g$ . This is often called a “swap” operation. Once the excitation has been fully transferred to the mechanical mode, the qubit’s resonant frequency is then quickly moved far away in frequency so that the two systems stop exchanging energy. After a variable amount of time the qubit is then brought back to the mechanical resonator and another swap operation is performed. Then, by measuring the probability of the qubit being in its ground or excited state, experimentalists are able to measure whether or not the phonon was lost to the environment during the time the qubit was not resonant with the HBAR device. Another similar measurement is preformed to measure the phase coherence of the phonon mode, this is done by preparing a superposition state in the qubit and measuring the evolution of its phase (see Fig. 2).

Measurement of Phonon Coherent States

By applying a strong tone to the system which is resonant with the HBAR device, the HBAR device will be placed into a coherent state which can be written down as a sum of Fock states:

$|\alpha\rangle = \sum_{n = 0}^\infty \frac{\alpha^n}{\sqrt{n!}}|n\rangle$

In order to determine how this will impact the spectral features of the qubit, it can be helpful to look at the probability of having $m$ phonons given a certain coherent state $|\alpha\rangle$ , which is found to be $|\langle m | \alpha\rangle|^2 = e^{-\overline{n}}\frac{\overline{n}^m}{m!}$ , where the mean phonon number $\overline{n} = |\alpha|^2$ has been introduced. This is simply a Poisson distribution in phonon number, and interestingly by measuring the mean phonon number it’s possible to learn about the quantum mechanical fluctuations in the phonon resonator!

The Hamiltonian which describes the interaction between the qubit and mechanical modes in the regime where the detuning ( $\Delta = \omega_q - \omega_m$ is the difference in resonant frequency between the qubit and mechanical mode) is much larger than the coupling rate, $g \ll |\Delta|$ can be approximated as:

$\hat{H}_{dispersive}/\hbar = \omega_m \hat{a}^{\dagger}\hat{a} + \frac{1}{2}\left(\omega_q + \chi \hat{a}^{\dagger}\hat{a}\right)\hat{\sigma}_z$

Where here the dispersive shift $\chi \simeq 2g^2 / \Delta$ has been introduced. Writing the system Hamiltonian down in this from is typically called the dispersive regime, and this allows us to see that the effective qubit frequency $\omega_q' = \omega_q +\chi\hat{a}^{\dagger}\hat{a}$ is now shifted by the number of excitations in the mechanical resonator! Oftentimes, in order to investigate the dispersive interaction between a qubit and a resonator, the authors will measure the qubit’s absorption spectrum, which is the frequency at which the qubit absorbs energy and is driven from its ground to excited state. This is also often called the qubit spectrum. If the qubit and resonator both have extremely low loss (both loss rates must be much less than $\chi$ ), the system is said to be in the strong dispersive regime, and the qubit spectrum is split into many peaks where the transition energy between the ground and excited states is shifted by $\chi$ for each phonon.

By changing the amplitude of the signal, the authors are able to vary the mean phonon number injected. This is measured by observing the qubit spectra split into multiple peaks each representing different phonon numbers, with each peak. Then, by comparing the relative height of each peak, the authors are able to determine the corresponding phononic coherent state. See Fig. 3 for the resulting measurement. Additionally, the authors see that there is a linear relationship between the mean phonon number and the strength of the signal generating the phononic coherent state, as expected.

**Figure 3**
(a) Measurement of the qubit absorption spectrum as a function of drive amplitude for the signal displacing the HBAR’s coherent state. At low drive amplitudes, the qubit spectra has a single peak, while at high drive powers, the qubit spectra splits into many peaks, each one indicating a different phononic Fock state. (b) Extraction of the probability of each Fock state at different drive amplitudes. The extracted probability distribution in Poissonian in the phonon number, as would be expected for a coherent state. The authors also find that the mean phonon number in the HBAR is linear in drive amplitude (inset).

Parity Measurement of Phonon Number

After investigating the qubit’s response to phonon states in the frequency domain, the authors look to the qubit response in time to learn about how the presence of phonons impacts the qubit. By repeatedly preparing the qubit into its excited state and preforming multiple swap operations between the qubit and the HBAR device, it is possible to prepare higher number Fock states (by quickly adding many excitations into the HBAR device one at a time). This is done by first exciting the qubit, swapping the excitation into the mechanical resonator, and repeating to add more excitations to the HBAR. After preparing the mechanical resonator’s state, the authors put the qubit into a superposition state: $|\psi_q\rangle = \frac{1}{\sqrt{2}}\left(|g\rangle + |e\rangle\right)$ . As a function of time, the qubit will accumulate a phase on the component of its wavefunction corresponding to the excited state of: $\phi = -n \chi t$ , where $n$ is the number of excitations in the HBAR device. It’s important to note here that because the HBAR is in a Fock state, there is not a distribution of phonon numbers now as there would be for a coherent state, but rather one single Fock state describes the quantum state of the HBAR! After allowing the qubit state to accumulate phase for some amount of time, the qubit’s state is then rotated with the same phase as the pulse that prepared the original superposition state. This means that if the qubit accumulated no extra phase, it would be repositioned to the excited state (assuming that there are no losses). In reality the probability of measuring the qubit in its excited state will always decay in time, but the presence of phonons in the HBAR device can be measured from the frequency of oscillation from this measurement. The frequency of oscillation can be calculated to be equal to $M|\chi_{Ramsey}|/ (2 \pi)$ , where $M$ is the phonon number in the HBAR resonator. Fig. 4 details this measurement as a function of time for several different swap operations. At a time of approximately $t = 7\mu s$ , which corresponds to the time $\pi/\chi$ , the authors are able to tell whether or not the resonator has an odd or even number of phonons based on whether or not the Ramsey decay is at a maximum or minimum! At this time, if there are an even number of phonons in the HBAR, the qubit phase has accumulated an even integer multiple of $\pi$ so that qubit superposition states are re projected to the excited state prior to measurement. Similarly, an odd number of phonons in the HBAR results in an odd multiple of $\pi$ phase accumulation so that the qubit is re projected to its ground state prior to measurement. This measurement allows the authors to quickly measure the parity of the phonon resonator in a single shot, rather than measuring the entire qubit spectra, which takes much more time.

**Figure 4**
(a) Measurement of qubit absorption spectra when the HBAR is prepared in different Fock states. Rather than seeing the spectra split into several different peaks here (as in Fig. 3a), the authors find that the qubit spectra remains largely constant, only the center frequency of the qubit is shifted proportional to the number of phonons in the HBAR resonator. (b) Ramsey measurements of the qubit. Here, the frequency of oscillation of the Ramsey decay allows the authors to extract the phonon number in the HBAR resonator. Importantly, at $t\simeq 7 \mu s$ , the authors are able to determine the parity (even-ness or odd-ness) of the phonon number in the HBAR resonator. (c) Comparison of measured parity based on two measurement schemes described by spectroscopic measurements (panel (a)), as well as measured parity by Ramsey measurements (panel (b)), the authors find very good agreement in measured parity between the two measurement schemes!

**Figure 4**
(a) Measurement of qubit absorption spectra when the HBAR is prepared in different Fock states. Rather than seeing the spectra split into several different peaks here (as in Fig. 3a), the authors find that the qubit spectra remains largely constant, only the center frequency of the qubit is shifted proportional to the number of phonons in the HBAR resonator. (b) Ramsey measurements of the qubit. Here, the frequency of oscillation of the Ramsey decay allows the authors to extract the phonon number in the HBAR resonator. Importantly, at $t\simeq 7 \mu s$ , the authors are able to determine the parity (even-ness or odd-ness) of the phonon number in the HBAR resonator. (c) Comparison of measured parity based on two measurement schemes described by spectroscopic measurements (panel (a)), as well as measured parity by Ramsey measurements (panel (b)), the authors find very good agreement in measured parity between the two measurement schemes!

Conclusion

In this experiment, the authors demonstrate a hybrid quantum acoustics experiment which operates in the strong dispersive regime, where the dispersive interaction between a superconducting qubit is much stronger than either the loss of the qubit or the loss of the HBAR resonator. By entering this special regime of circuit quantum acoustodynamics (cQAD), the authors are able to perform experiments which allow them to probe the quantum properties of high frequency sound. By using special experimental techniques, the authors are able to create non-classical phonon states in the HBAR resonator (Fock states) and determine phonon parity based on two separate measurement schemes.

References:

[1] U. von Lupke, Y. Yang, M. Bild, L. Michaud, M. Fadel, and Y. Chu, Parity measurement in the strong dispersive regime of circuit quantum acoustodynamics, Nature Physics 10.1038/s41567-022-01591-2 (2022)

Many thanks to Akash Dixit for his many helpful comments and suggestions in the writing of this summary!

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