Ultrastrong parametric coupling between a superconducting cavity and a mechanical resonator

TL;DR

This paper reports the development of an optomechanical device with ultrastrong parametric coupling between a superconducting microwave cavity and a mechanical resonator, enabling advanced quantum control applications.

Contribution

It demonstrates a novel architecture achieving ultrastrong coupling where the interaction surpasses dissipation and approaches the mechanical frequency.

Findings

Achieved 88% frequency splitting of hybrid modes

Coupling exceeds mechanical thermal decoherence rate

Enabled potential for ultrafast quantum state transfer

Abstract

We present a new optomechanical device where the motion of a micromechanical membrane couples to a microwave resonance of a three-dimensional superconducting cavity. With this architecture, we realize ultrastrong parametric coupling, where the coupling rate not only exceeds the dissipation rates in the system but also rivals the mechanical frequency itself. In this regime, the optomechanical interaction induces a frequency splitting between the hybridized normal modes that reaches 88% of the bare mechanical frequency, limited by the fundamental parametric instability. The coupling also exceeds the mechanical thermal decoherence rate, enabling new applications in ultrafast quantum state transfer and entanglement generation.

Equations

$${\Omega }_{\pm }\simeq \Omega \pm \sqrt{g^2-{\kappa }^2/16}.$$

$${\Omega }_{\pm }=\mathrm{Re}\sqrt{{\Omega }^2-\frac{{\kappa }^2}{16}\pm 2\Omega \sqrt{g^2-\frac{{\kappa }^2}{16}}}.$$

Full text

cavity optomechanics, superconducting circuits, ultrastrong coupling

The physics of coupled oscillators is used to understand a wide array of natural and man-made phenomena, from the microscopic vibrations of atoms and molecules to the interplay of planets and their moons. Coupling strengths can be grouped into different regimes that entail qualitatively different behavior. In particular, the regimes of weak and strong coupling are distinguished by whether or not the coupling between two oscillators exceeds their damping rates. In the strong coupling regime, the eigenfrequencies of the combined system split into normal modes where the energy swaps back and forth between the individual oscillators. For low-loss systems, this energy exchange can be fast compared to the lifetimes of the individual modes while still remaining slow compared to their periods. The strong coupling regime has become an essential tool in engineered quantum systems because it can allow the subsystems to exchange their quantum information before it is lost due to decoherence Mabuchi and Doherty (2002). As quantum devices continue to be engineered with increased coupling rates, a new regime known as ultrastrong coupling has become relevant. This regime occurs once the coupling rate becomes so large as to rival a bare resonance frequency, resulting in new quantum effects including multimode entanglement and virtually excited ground states Kockum et al. (2019); Forn-Díaz et al. (2019). Reaching ultrastrong coupling and studying its effect in quantum devices remains an active experimental challenge.

Cavity optomechanics is an area of engineered quantum systems in which a mechanical resonator and an electromagnetic mode form a coupled-oscillator system Aspelmeyer et al. (2014). Although the intrinsic coupling rate between single photons and phonons is typically small, cavity optomechanical systems allow an enhancement of the coupling proportional to the amplitude of a coherent cavity drive. This parametric enhancement has allowed demonstrations of strong coupling both at ambient temperatures Gröblacher et al. (2009); Enzian et al. (2019) and in cryogenic, quantum-coherent regimes Teufel et al. (2011a); Verhagen et al. (2012); Noguchi et al. (2016); Kharel et al. (2018). In practice, as the intensity of the coherent drive becomes large, it can induce other undesirable effects including heating and cavity nonlinearity, limiting the final parametric coupling. Operating deeply within the quantum-coherent ultrastrong coupling regime therefore requires a dramatic increase in either the single-photon optomechanical coupling rate or the cavity’s power handling capability. Specifically, in the microwave domain, one prominent optomechanical platform is a lumped element superconducting circuit formed from a mechanically compliant vacuum-gap capacitor shunted by a thin-film inductor. While this architecture has enabled strong coupling Teufel et al. (2011a), ground state cooling Teufel et al. (2011b), and entanglement Palomaki et al. (2013a), the coupling rate has remained well below the onset of ultrastrong coupling effects, limited by unwanted nonlinearity of the superconducting inductor Zmuidzinas (2012).

In this Letter, we introduce a new optomechanical architecture that mitigates the nonideality of previous designs, allowing us to reach ultrastrong coupling and approach the fundamental stability limit of the pure optomechanical interaction Aldana et al. (2013). Our device consists of a microfabricated vacuum-gap capacitor embedded in a three-dimensional superconducting microwave cavity, analogous to recent work in the field of circuit quantum electrodynamics Paik et al. (2011) and similar to other optomechanical demonstrations Tobar and Blair (1993); Yuan et al. (2015); Noguchi et al. (2016); Gunupudi et al. (2019). Our device takes advantage of the superior power handling of bulk cavity resonators compared to thin-film inductors. In general, the drawback of using a cavity resonator is a larger parasitic capacitance that would dilute the optomechanical coupling. Crucially, through careful microwave design and simulation, we maintain the relatively large single-photon coupling of lumped element vacuum-gap circuits Teufel et al. (2011a). As a result, we achieve ultrastrong parametric coupling by applying microwave drives with one hundred times larger power, ultimately limited by the instability inherent in the optomechanical Hamiltonian. Due to the low temperature operation, the quantum decoherence rates are kept sufficiently small to enable new regimes of entanglement Aldana et al. (2013); Hofer and Hammerer (2015), nonlinear quantum optomechanics Lemonde et al. (2013), and ultrafast quantum state transfer Palomaki et al. (2013b); Lecocq et al. (2015); Reed et al. (2017).

In a generic cavity optomechanical system, the natural frequency $\omega_{c}$ of an electromagnetic resonance depends on the position $\hat{x}$ of a mechanical harmonic oscillator Aspelmeyer et al. (2014). The interaction Hamiltonian is $\hat{H}_{\text{int}}=\hslash g_{0}\hat{n}\hat{x}/x_{zp}$, where $g_{0}$ is the single-photon coupling rate, $\hat{n}$ is the number operator for microwave photons, and $\hslash$ is the reduced Planck constant. $x_{zp}=\sqrt{\hslash/2m\Omega}$ is the mechanical zero-point fluctuation amplitude, where $m$ and $\Omega$ are the effective mass and the resonance frequency of the mechanical mode, respectively. Dissipation in the system is characterized by damping rates $\kappa$ for the electrical mode and $\Gamma$ for the mechanical mode, with $\Gamma\ll\kappa$. Even if $g_{0}$ is small, as is the case in most optomechanical systems, the coupling can be parametrically enhanced by driving the electrical mode to a coherent state of mean photon number $n_{d}$ at frequency $\omega_{d}$. Defining $\hat{a}$ as the annihilation operator for fluctuations around the driven state, we can approximate the interaction Hamiltonian as $\hat{H}_{\text{int}}=\hslash g(\hat{a}+\hat{a}^{\dagger})\hat{x}/x_{zp}$, where $g=g_{0}\sqrt{n_{d}}$ is the parametrically enhanced coupling rate, leading to a set of linear coupled equations of motion for $\hat{a}(t)$ and $\hat{x}(t)$. Just as two passively coupled oscillators interact most strongly when resonant with each other, for parametric coupling an effective resonance condition is reached when driving the system at the difference frequency ($\Delta\equiv\omega_{d}-\omega_{c}=-\Omega$) in the resolved sideband regime ($\kappa\ll\Omega$). These conditions optimize the coherent exchange of energy between the mechanical and electromagnetic modes.

As the driven coupling $g$ is increased from its small single-photon value, we encounter several distinct regimes of coupling. When the cooperativity $C=4g^{2}/\kappa\Gamma$ reaches $1$, the optical damping of the mechanical mode begins to dominate over its intrinsic dissipation. As $g$ increases further, the effective mechanical linewidth increases until it reaches the cavity dissipation rate when $g=\kappa/4$. Above this threshold, the system enters the strong-coupling regime where the cavity and mechanical mode hybridize, with the mechanical resonance frequency splitting into two solutions, which for $g\ll\Omega$ are given by

[TABLE]

This leads to the splitting frequency $\Omega_{s}=\Omega_{+}-\Omega_{-}\simeq 2g$ when the coupling overwhelms the cavity dissipation. In this regime of strong coupling, the two physical resonators exchange energy and information at a rate $\Omega_{s}$, faster than any dissipation in the system.

As the splitting frequency $\Omega_{s}$ approaches the bare mechanical frequency $\Omega$, however, counter-rotating terms of order $g/\Omega$ cannot be ignored, requiring the use of the exact eigenfrequency spectrum Sup

[TABLE]

The discrepancy between (1) and (2) is a measurable metric to distinguish strong and ultrastrong parametric coupling. In the resolved sideband ultrastrong coupling regime ($\kappa\ll 2g<\Omega$), Eq. (2) becomes $\Omega_{\pm}\simeq\Omega\sqrt{1\pm 2g/\Omega}$, showing that the splitting exceeds $2g$ until the system becomes parametrically unstable when $\Omega_{-}=0$ at $g=\Omega/2$, corresponding to a splitting frequency $\Omega_{s}=\sqrt{2}\Omega$.

The regimes discussed above are shown in Fig. 1 as a function of parametric coupling and sideband resolution $\Omega/\kappa$. The shaded region of ultrastrong coupling corresponds to $\Omega/5<\Omega_{s}$, roughly where terms of order $g/\Omega$ become relevant while still satisfying the condition for strong coupling. As $\Omega_{s}$ characterizes the rate at which the two physical resonators exchange energy, the instability sets a fundamental limit for both optomechanical coupling as well as coherent exchange of information between the resonators in the steady state. Reaching ultrastrong coupling therefore allows the exploration of the fundamental limitations of coupling in optomechanical systems.

For quantum applications, the coupling rate should be compared not only to dissipation but also to the decoherence rates in the system. In particular, the mechanical thermal decoherence rate $n_{\text{th}}\Gamma$ can be much larger than the intrinsic dissipation $\Gamma$, where $n_{\text{th}}$ is the mechanical occupancy when in equilibrium with its thermal environment. Ideally, the quantum cooperativity $C_{q}=4g^{2}/\kappa n_{\text{th}}\Gamma$ exceeds 1 before the onset of strong coupling, ensuring that the hybridized system is quantum coherent. In the following, we present an optomechanical device that achieves the hierarchy of rates desired for quantum coherent ultrastrong coupling: $n_{\text{th}}\Gamma\ll\kappa/2\ll 2g\lesssim\Omega\ll\omega_{c}$.

Our device is shown in Fig. 2. A microwave cavity resonator with inner dimensions 19\text{\,}\mathrm{mm}$\times$4\text{\,}\mathrm{mm}$\times$17\text{\,}\mathrm{mm} milled from bulk aluminum defines the electrical resonance of the system. We focus on the fundamental $\mathrm{TE}_{101}$ microwave mode, whose electric field is maximal at the center of the cavity, where we place a sapphire chip containing a microfabricated $20\text{\,}\mathrm{\SIUnitSymbolMicro m}$-diameter aluminum vacuum-gap capacitor Lecocq et al. (2015). The suspended top plate of the capacitor forms the mechanical resonator of the system. Reducing the parasitic capacitance of the cavity and thereby maximizing the optomechanical coupling rate requires a galvanic connection between the microfabricated capacitor and the cavity walls. To achieve this, we use aluminum bond wires to connect the cavity faces to lithographically patterned pads, which then connect to the capacitor through thin-film aluminum wires.

The vacuum-gap capacitor and sapphire substrate load the cavity resonance, pulling its frequency from around $12\text{\,}\mathrm{GHz}$ down to $\omega_{c}/2\pi\approx$6.506\text{,}\mathrm{GHz}. Two cavity ports with adjustable coupling pins allow signals to couple in and out to coaxial cables. We adjust the length of the pins at room temperature so the two ports contribute asymmetrically to the total dissipation $\kappa=\kappa_{1}+\kappa_{2}+\kappa_{i}$, where $\kappa_{1}/2\pi\approx$1.1\text{\,}\mathrm{MHz} and $\kappa_{2}/2\pi\approx$25\text{,}\mathrm{kHz} are the port coupling rates. The internal dissipation $\kappa_{i}$ of the cavity mode ranges from $\sim 30\text{\,}\mathrm{kHz}$ to $\sim 140\text{\,}\mathrm{kHz}$ depending on the circulating power Martinis *et al.* ([2005](#bib.bib25)). The cavity is therefore overcoupled with a total dissipation rate $\kappa/2\pi\approx$1.2\text{\,}\mathrm{MHz}. We place the device in a cryostat with a base temperature of $16\text{\,}\mathrm{mK}$ and probe and monitor the system with microwave signals applied near the cavity resonance frequency. Our setup allows us to measure all four elements of the scattering matrix for a broad range of parametric coupling parameters. With a weak cavity drive, we characterize the fundamental vibrational mode of the capacitor plate by its resonance frequency $\Omega/2\pi=$9.696\text{,}\mathrm{MHz} and its intrinsic damping rate $\Gamma/2\pi=$31\pm 1\text{\,}\mathrm{Hz}, measured spectroscopically, where the uncertainty represents the standard error of the mean. By varying the cryostat temperature and measuring mechanical thermal noise, we determine the single-photon optomechanical coupling rate $g_{0}/2\pi=$167\pm 2\text{,}\mathrm{Hz} Teufel *et al.* ([2011b](#bib.bib11)). We also find that at base temperature, the mechanical mode equilibrates to $35\pm 3\text{\,}\mathrm{mK}$, corresponding to a thermal phonon occupancy $n_{\text{th}}=$76\pm 6\text{\,}.

To probe the response of the coupled system, we measure the vector transmission of the microwave field through the cavity in the presence of a parametric drive Weis et al. (2010); Teufel et al. (2011a); Safavi-Naeini et al. (2011). In Fig. 3, we plot the magnitude of the transmission for a range of drive powers applied below the cavity resonance, with the drive frequency indicated as a red vertical line. We coherently cancel the drive after it leaves the cavity, allowing us to measure the transmission of a weak probe without saturating our microwave measurement Sup . We adjust the drive frequency as we increase drive power to maintain the condition $\Delta=-\Omega$ in the presence of two dominant nonlinear effects. Namely, we measure the pure optomechanical Kerr shift, $-2g_{0}^{2}/\Omega=$-5.8\pm 0.1\text{,}\mathrm{m}\mathrm{Hz}\mathrm{/}\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n}$$, and we attribute the remaining shift to the residual nonlinear kinetic inductance of the superconducting film, approximately $-4\text{\,}\mathrm{m}\mathrm{Hz}\mathrm{/}\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n}$ at our highest powers Zmuidzinas (2012).

At low drive power (Fig. 3a), we measure the bare cavity resonance, a single Lorentzian with linewidth $\kappa$. As the drive power increases (Fig. 3b), the optomechanical interaction appears as interference in the cavity response with a characteristic bandwidth of the damped mechanical linewidth $\sim 4g^{2}/\kappa$. At large enough power (Fig. 3c), the damped mechanical width reaches $\kappa/2$, after which the response splits into normal modes, marking the strong coupling regime. Additionally, we begin to resolve the next four vibrational modes of the membrane (Fig. 3d), which couple weakly to the cavity mode. At the highest power (Fig. 3e), the response acquires several features indicative of ultrastrong coupling. In addition to the splitting of the fundamental resonance becoming of order $\Omega$, the counter-rotating dynamics below the drive frequency become significant and easily observable. Lastly, the transmission at the drive frequency begins to increase, signifying a nonlinear relationship between input power and driven photon number in the cavity.

We fit the data in the complex plane to multimode optomechanical theory Sup , shown as a black line plotted over the data. The quantitative agreement between data and theory allows us to extract all the relevant system parameters as a function of drive power. These parameters are shown in Fig. 4a from very weak parametric coupling to the ultrastrong coupling regime. For very few drive photons (in this case $n_{d}\lesssim 10^{2}$), the bare mode properties are measured. As the drive power increases within the regime of weak coupling ($n_{d}\lesssim 10^{6}$), the mechanical mode is damped and cooled, passing through $C=1$ and entering the quantum-enabled regime $C_{q}>1$, where the occupancy is reduced below one quantum. Normal-mode splitting occurs at $n_{d}\approx 3\times 10^{6}$ where $4g=\kappa$, marking the beginning of strong coupling. Above $n_{d}\approx 2\times 10^{8}$, the splitting starts to deviate from $2g$ as the system enters the ultrastrong coupling regime. The threshold for parametric oscillation occurs at $n_{d}\approx 8.4\times 10^{8}$, above which no steady-state solution exists. We measure well into the regime where $\Omega_{-}<2g<\Omega_{s}$; that is, the energy swapping rate exceeds $2g$ as well as the lower eigenfrequency itself.

In Fig. 4b, the measured mechanical eigenfrequencies (blue circles) are plotted versus the optomechanical coupling rate. The black line shows the strong-coupling approximation (1), while the full solution (2) is shown as a blue line. The discrepancy between the two is a clear indication of ultrastrong coupling effects. At the highest power, the splitting between the two modes exceeds the frequency of the lower mode, reaching 88% of the bare mechanical frequency.

Our measurements of the driven cavity response allow us to reconstruct the mechanical susceptibility, defined as the ratio of induced motion to external force $\chi_{m}(\omega)=m\Omega^{2}x(\omega)/F(\omega)$, normalized by the intrinsic mechanical spring constant $m\Omega^{2}$. This is shown in Fig. 5. The gray line corresponds to the estimated susceptibility at zero coupling, which includes the bare resonance of the fundamental mode as well as the first four higher-order vibrational modes. The peak height on resonance is equal to the quality factor of the fundamental mode, $Q_{m}=\Omega/\Gamma\approx 3\times 10^{5}$, while the heights at the other resonances are reduced due to their lower vacuum coupling rates. The data plotted in blue (with a fit to theory in black) is the susceptibility inferred from our highest drive power in Fig. 3. The original peak of linewidth $\Gamma$ is split into two broad peaks with approximate widths $\kappa/2$, each representing a normal mode of the joint cavity-mechanics system. At this large coupling rate, the eigenfrequencies are no longer split symmetrically about the bare mechanical frequency, and the peak heights are unequal. As the lower eigenfrequency decreases, we also see an increase in the value of susceptibility at zero frequency, indicating an increased response to static forces. All of these effects agree with the full optomechanical theory that includes ultrastrong-coupling corrections. At our highest cooperativity of $C\approx 1.6\times 10^{6}$, we achieve a maximal splitting of $\Omega_{s}/2\pi\approx$8.5\text{,}\mathrm{MHz}. Here, the state of the mechanical mode is exchanged with the cavity mode in a characteristic time $\pi/\Omega_{s}\approx$60\text{\,}\mathrm{ns}, faster than the mechanical oscillation period $2\pi/\Omega\approx$100\text{,}\mathrm{ns}$$.

In conclusion, we have introduced a novel microwave optomechanical circuit architecture and experimentally characterized the eigenmodes of the system in the quantum regime. As the parametric coupling is increased in situ, we find quantitative agreement with the theoretical predictions from weak to ultrastrong coupling. Looking forward, ultrastrong coupling of bosonic modes is predicted to have interesting ground state properties Ciuti et al. (2005); Ciuti and Carusotto (2006); Fedortchenko et al. (2017); Marković et al. (2018). Quantum correlations induced by the counter-rotating terms in the ultrastrong coupling regime induce squeezing and entanglement between the mechanical and cavity modes Hofer and Hammerer (2015). Studying the noise properties in this regime would also allow us to observe these effects for the first time, as well as assess how these correlations could be exploited for quantum-enhanced sensing of forces, displacements, or gravitational waves Schnabel et al. (2010); Kampel et al. (2017); Mason et al. (2019). Furthermore, while the fundamental instability precludes traditional steady-state measurements when the swapping rate exceeds the mechanical frequency, pulsed measurements would allow the system to be characterized beyond this limit into the deep strong coupling regime Kockum et al. (2019); Forn-Díaz et al. (2019). These ultrafast pulsed measurements would allow new regimes of quantum state transfer Palomaki et al. (2013b); Reed et al. (2017), entanglement Palomaki et al. (2013a), topologically-protected operations Xu et al. (2016), and projective measurements Vanner et al. (2011). Finally, theoretical proposals have suggested using ultrastrong coupling to enhance the weak nonlinear terms of the optomechanical Hamiltonian Lemonde et al. (2013) allowing for nonlinear quantum optomechanics.