Mechanical Purcell Filters for Microwave Quantum Machines

TL;DR

This paper introduces a novel mechanical Purcell filter using nanomechanical resonators in lithium niobate to improve qubit measurement in microwave quantum systems by reducing decoherence.

Contribution

It proposes, fabricates, and characterizes a mechanical Purcell filter with a ladder topology, offering a compact and crosstalk-free alternative to electromagnetic filters.

Findings

The filter exhibits a bandpass response with steep edges.

It effectively isolates qubits from decoherence channels.

The design is suitable for integration in microwave quantum machines.

Abstract

In circuit quantum electrodynamics, measuring the state of a superconducting qubit introduces a loss channel which can enhance spontaneous emission through the Purcell effect, thus decreasing qubit lifetime. This decay can be mitigated by performing the measurement through a Purcell filter which forbids signal propagation at the qubit transition frequency. If the filter is also well-matched at the readout cavity frequency, it will protect the qubit from decoherence channels without sacrificing measurement speed. We propose and analyze design for a mechanical Purcell filter, which we also fabricate and characterize at room temperature. The filter is comprised of an array of nanomechanical resonators in thin-film lithium niobate, connected in a ladder topology, with series and parallel resonances arranged to produce a bandpass response. The modest footprint, steep band edges, and absence of cross-talk in these filters make them a novel and appealing alternative to analogous electromagnetic versions currently used in microwave quantum machines.

Equations

$$\hat{H}_{\text{JC}}=\hbar\omega_{\text{r}}\bigg{(}\hat{a}^{\dagger}\hat{a}+\frac{1}{2}\bigg{)}+\frac{\hbar\omega_{\text{q}}}{2}\hat{\sigma}_{z}+\hbar\,g\big{(}\hat{a}^{\dagger}\hat{\sigma}_{-}+\hat{a}\hat{\sigma}_{+}\big{)}$$

$$|+,1⟩\approx |e,0⟩+\frac{g}{\Delta }|g,1⟩.$$

$${\gamma }_{\text{q}}=\frac{g^2\kappa }{{\Delta }^2+(\kappa /2{)}^2}\underset{\Delta \gg \kappa ,g}{⟶}\frac{g^2}{{\Delta }^2}\kappa .$$

$$\mathbb{Y}+\left[\begin{array}[]{cc}\frac{1}{1/\text{i}\omega C_{1}+Z_{\text{q}}}&0\\ 0&\frac{1}{1/\text{i}\omega C_{2}+Z_{\text{ext}}}\end{array}\right]\approx\mathbb{\overline{Y}}+\left[\begin{array}[]{cc}\omega^{2}C_{1}^{2}Z_{\text{q}}&0\\ 0&\omega^{2}C_{1}^{2}Z_{\text{ext}}\end{array}\right]$$

$$\delta {\omega }_{\mathrm{r}}$$

$$\delta {\omega }_{\mathrm{q}}$$

$${\gamma }_{\text{q}}=\frac{g^2}{{\Delta }^2}\frac{\text{Re}{Z}_{\text{ext}}({\omega }_{\text{q}})}{\text{Re}{Z}_{\text{ext}}({\omega }_{\text{r}})}\kappa .$$

$$\text{Re}{Z}_{\text{ext}}=Z_0\frac{1-|S_{11}{|}^2}{|1-S_{11}{|}^2}.$$

Full text

††preprint: AIP/123-QED

Quantum information processing calls for systems which are well isolated from their environment, whose states can nonetheless be measured and manipulated with precision. DiVincenzo (2000) These fundamentally contradictory requirements can be satisfied by cleverly engineering devices and interactions between them. In the circuit QED platform,Devoret and Schoelkopf (2013); Koch et al. (2007); Neill et al. (2018) nonlinear superconducting circuits called qubits are used to store and process quantum information. Their internal states need to be read out rapidly and with low rates of error. An appealing approach for this is to couple the qubit circuit to an auxiliary, linear electromagnetic resonator (often called the readout cavity). Resonators have long been used to amplify emission from atoms, for instance, via Purcell enhancement. Purcell (1946); Goy et al. (1983); Girvin (2014) Conversely, qubit emission into the environment can be suppressed by tuning the qubit away from the resonator’s frequency by many times the linewidth and interaction energy. The qubit state is then measured “dispersively” by monitoring the resonator frequency for shifts induced by changes in the qubit state Koch et al. (2007) (although we note that alternative measurement strategies exist). Didier, Bourassa, and Blais (2015)

Dispersive shifts of the cavity can be measured and amplified to demonstrate extremely efficient single-shot measurements of qubits using this scheme. Vijay, Slichter, and Siddiqi (2011); Johnson et al. (2012) Nonetheless, the conflicting requirements of efficient readout and qubit isolation persist in the desired properties of the resonator. Fast, efficient readout requires strong coupling between the resonator and environment. This in turn increases the probability of qubit relaxation through the resonator in a process called Purcell decay.Purcell (1946); Houck et al. (2007); Gambetta, Houck, and Blais (2011); Houck et al. (2008) Purcell filters, often consisting of a second stage electromagnetic resonator, have been used effectively to mitigate this process.Reed et al. (2010); Jeffrey et al. (2014); Sete, Martinis, and Korotkov (2015); Bronn et al. (2015); Walter et al. (2017); Heinsoo et al. (2018) As qubit coherence times continue to improve, the basic limit imposed by Purcell decay will become more important. In principle, progressively higher order electromagnetic filters can be incorporated, requiring progressively larger components that take up valuable space on a chip.

We propose an alternative solution using ultra-compact, high-order microwave acoustic filters to isolate qubits from the environment and to curtail Purcell decay, using techniques adapted from the telecommunications industry. Low cross-talk and extremely small footprints make nanomechanical structures ideal for integration with emerging superconducting quantum machines.Arrangoiz-Arriola and Safavi-Naeini (2016); Satzinger et al. (2018); Chu et al. (2017); Pechal, Arrangoiz-Arriola, and Safavi-Naeini (2019); Arrangoiz-Arriola et al. (2018, 2019) Moreover, recent advances in fabrication and design of thin-film, high-$Q$, and strongly coupled lithium niobate (LN) devices make them well-suited for such applications.Arrangoiz-Arriola et al. (2018, 2019); Sarabalis et al. (2019) In this manuscript we outline the approach and present initial experimental results.

The cavity-qubit interaction is described by the Jaynes-Cummings Hamiltonian:

[TABLE]

where $\omega_{\text{r}}$ is the readout resonator frequency, $g$ is the qubit-resonator coupling strength, $\hat{a}$ annihilates photons in the resonator, and Pauli operators $\hat{\sigma}$ act on the qubit. This Hamiltonian can be diagonalized into a series of $n$-excitation subspaces, each spanned by $\ket{g,n}$ and $\ket{e,n-1}$. For $|g|\gg|\Delta|$, the “qubit-like” polariton in the first excited subspace is:

[TABLE]

The small but finite occupation of the resonator represented by $\ket{g,1}$ can decay to $\ket{g,0}$ through the resonator’s output channel, effectively causing the atom to relax to its ground state. This loss channel increases the decay rate of the qubit by $\gamma_{\text{q}}=\kappa|\langle g,0|\,\hat{a}\,|+,1\rangle|^{2}\approx{g^{2}}\kappa/{\Delta^{2}}$. More generally, we can use Fermi’s golden rule to calculate the decay of the qubit excited state $\ket{e,n}$ through the resonator:

[TABLE]

This allows us to choose $\Delta$ such that the maximum coherence time imposed by the Purcell effect $1/\gamma_{\text{q}}$ exceeds the qubit’s $T_{1}$ due to other sources. With improving qubit design, fabrication, and materials processing, larger detunings $\Delta$ will be required to avoid limitation by Purcell decay. However, increasing $\Delta$ also increases the amount of time required to make a measurement, which allows for more errors to be introduced. This inconvenience, as well as other practical concerns with operating at large detunings, has led the to design and implementation of Purcell filters. These filters, previously composed of electromagnetic resonators, protect the qubit from the Purcell decay channel while maintaining the ability to do fast measurements by operating with relatively small $\Delta$.

A Purcell filter can be considered to be a bandpass filter (Fig. 1), which performs an impedance transformation on the dissipative bath of the environment, through which the linear resonator can be probed. Placing the qubit frequency outside the passband where the filter presents an impedance mismatch isolates the qubit from decoherence channels of the vacuum; placing the resonator frequency within the passband allows a microwave tone to pass through unimpeded and probe the resonator frequency. The resonator can be strongly coupled to its feed line (large $\kappa$), allowing signal to pass through the cavity quickly for fast qubit state measurement, without risking a reduction in qubit lifetime.

In design and analysis, we treat the mechanical filter as a two port microwave system connected at one end to a regular transmission line and at the other end to the readout resonator (Fig. 1). The impedance $Z_{\text{ext}}(\omega)$ seen by the resonator captures all properties of the filter element relevant to qubit operation. In this section, we develop an understanding of how $Z_{\text{ext}}(\omega)$ affects qubit readout and Purcell decay.

A relevant figure of merit for a Purcell filter is the ratio of qubit lifetimes with and without the filter. In this ratio the capacitance $C_{2}$ is adjusted to keep the resonator linewidth fixed. We can derive this by analyzing the measurement circuitry depicted in Fig. 1c, in which the capacitances $C_{1}$ and $C_{2}$ are assumed to be small. The admittance matrix for the entire system, with respect to nodes 1 and 2, is given by:

[TABLE]

where we absorb the reactive part of the admittance matrix with $\mathbb{Y}$ into $\mathbb{\overline{Y}}$. We solve for the resonances by setting the determinant of the above expression to [math]. These solutions are small deviations from the uncoupled case: the resonator mode shifts to $\omega_{\text{r}}+\delta\omega_{\text{r}}$ and the qubit frequency that satisfies $Z_{\text{q}}(\omega_{\text{q}})\rightarrow\infty$ shifts to $\omega_{\text{q}}+\delta\omega_{\text{q}}$. Keeping lowest-order terms in the small capacitances we find:

[TABLE]

where we introduce \lambda=\frac{\text{d}}{\text{d}\omega}\det\mathbb{\overline{Y}}(\omega)\big{|}_{\omega=\omega_{\text{r}}}, $Y_{\text{q}}(\omega)=1/Z_{\text{q}}(\omega)$, and $Y^{\prime}=\text{d}Y/\text{d}\omega$. The coupling strength $g$ can be found by equating the shift of the cavity frequency to $2g^{2}/\Delta$ and approximating $|\Delta|\ll\omega_{\text{q,r}}$. This is given by $g^{2}\approx-{\omega_{\mathrm{q}}^{2}C_{1}^{2}\mathbb{\overline{Y}}_{22}(\omega_{\mathrm{q}})}/{Y^{\prime}_{\mathrm{q}}(\omega_{\mathrm{q}})\lambda}$. Dissipation from the external environment $Z_{\text{ext}}$ introduces a small imaginary component to the frequency shifts, from which we extract the resulting qubit and resonator linewidths $\gamma_{q},\kappa=2\,\text{Im}\,\delta\omega_{q,r}$ to find:

[TABLE]

Without a filter, when $Z_{\text{ext}}(\omega_{\text{r}})=Z_{\text{ext}}(\omega_{\text{q}})=Z_{0}$, this reduces to the familiar Purcell decay rate of Eq. (2). The filter adds an extra degree of protection from spontaneous emission, a “filter factor" which is the ratio of the resistances seen by the qubit and resonator at their respective frequencies.

The mechanical Purcell filter is based on a ladder network of piezoelectric oscillators, inspired by methods that are ubiquitous in classical RF and telecommunications technology. Morgan (2007); Hashimoto (2009) A ladder filter electrically connects series and shunt resonators with frequencies carefully chosen to produce a bandpass response (Fig. 2, 4a). The series resonators are identical to each other, as are the shunt (parallel) resonators.

Each single-resonator electrical response is described by its admittance $Y(\omega)=1/Z(\omega)$ where $Z(\omega)$ is the electrical impedance. This response is well-modeled by a Butterworth-van Dyke (BVD) equivalent circuit (Fig. 2). Morgan (2007); Hashimoto (2009); Pop et al. (2017); Lakin (1992) It is important that the antiresonance of the parallel resonators – the zero in the admittance $Y_{\text{p}}(\omega)$ – is placed at the series resonance, or the pole in $Y_{\text{s}}(\omega)$ (Fig. 4a). This frequency defines the center of the passband: here, the parallel resonators have maximal impedance, while the series resonators have minimal impedance, so a microwave signal passes easily through the filter. The spacing between each resonance $f_{\text{R}}$ and its antiresonance $f_{\text{A}}$ is given by $k^{2}=({\pi^{2}}/{8})({f_{\text{A}}^{2}-f_{\text{R}}^{2}})/{f_{\text{A}}^{2}}$.Pop et al. (2017) Thus $k^{2}$, which depends strongly on the material platform, determines filter bandwidth. The BVD circuit elements fully parameterize the frequency ($\omega_{\text{m}}={1}/{\sqrt{L\,C}}$), piezoelectric coupling ($k^{2}={\pi^{2}}C/{8}{C_{\text{g}}}$), and quality factor ($Q={\pi^{2}}/{8}{\,\omega_{\text{m}}\,C\,R\,}$).Pop et al. (2017)

Intriguingly, we find that while the quality factor of the resonators contributes to insertion loss in the passband as well as less sharply defined band edges, it is not a strong limiting factor in filter performance (Fig. 4b). Finally, we note that piezoelectric coupling efficiency can be tuned during fabrication by rotating the orientation of the IDTs with respect to the crystal axes (Fig. 5a). It can also be increased by patterning larger transducers, with longer electrodes (wider resonators) or a larger number of interdigitated pairs.

We fabricate our devices on X-cut LN with a process similar to that described in Ref. Arrangoiz-Arriola et al. (2018) We measure the scattering parameters of fabricated filters using a calibrated vector network analyzer (VNA) at room temperature and atmospheric pressure. Reflection and transmission are analyzed to calculate the filter enhancement factor on qubit lifetime.

[TABLE]

$Z_{\text{ext}}(\omega)$ and consequently the filter factor appearing in Eq. (3) can be deduced from calibrated microwave characterization. In particular, the environmental impedance can be extracted from a filter’s scattering parameters as:Pozar (2012)

[TABLE]

This expression is used to infer the reduction in Purcell decay rate $\gamma_{\text{q}}$ with the addition of the filter. We calculate unfiltered Purcell-limited as well as filter-enhanced coherence times for a qubit-resonator system with constant $g$ and $\kappa$. The bare Purcell rate is calculated by diagonalizing Eq. (1), using a single-mode resonator model, without assuming $g\ll\Delta$ in the mixing angle $\text{tan}\,2\theta_{n}=2g\sqrt{n}/\Delta$.Schuster (2007); Reed (2013) We see from Fig. 5b that the filter can be used to realize nearly two orders of magnitude enhancement of the qubit lifetime over the unfiltered system. The sharp dips in the filtered $T_{1}$ spectrum correspond to spurious mechanical resonances of the filter.

We have proposed and realized mechanical Purcell filters that use nanomechanical elements in a qubit-compatible platform. Using room temperature measurements, we quantify the expected enhancement of qubit coherence time for a range of qubit frequencies around the center of the passband. We demonstrate that the bandwidths of these filters can be tuned by design, reaching up to 220 MHz, which is broad enough to accommodate many strongly coupled resonators for fast, multiplexed qubit readout. Quantum acoustic systems have been proposed as means of realizing new regimes of quantum optics, quantum memory elements for processors, and quantum state converters for networking. Our work opens a new space in the field by suggesting an application for quantum acoustic systems that can impact development of quantum machines.

The authors thank P. Arrangoiz-Arriola and E. A. Wollack for useful discussions. This work was supported by the U.S. government through the Department of Energy Grant No. DE-SC0019174 and the National Science Foundation Grant No. ECCS-1808100. A. Y. C. was supported by a Stanford Graduate Fellowship. Part of this work was performed at the Stanford Nano Shared Facilities (SNSF), supported by the National Science Foundation under Grant No. ECCS-1542152, and the Stanford Nanofabrication Facility (SNF).