You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'Level repulsion – the opening of a gap between two degenerate modes due to coupling – is ubiquitous anywhere from solid state theory to quantum chemistry. In contrast, if one mode has negative energy, the mode frequencies attract instead. They converge and develop imaginary components, leading to an instability; an exceptional point marks the transition. This, however, only occurs if the dissipation rates of the two modes are comparable. Here we expose a theoretical framework for the general phenomenon and realize it experimentally through engineered dissipation in a multimode superconducting microwave optomechanical circuit. Level attraction is observed for a mechanical oscillator and a superconducting microwave cavity, while an auxiliary cavity is used for sideband cooling. Two exceptional points are demonstrated that could be exploited for their topological properties.'
author:
- 'N. R. Bernier'
- 'L. D. Tóth'
- 'A. K. Feofanov'
- 'T. J. Kippenberg'
title: Level attraction in a microwave optomechanical circuit
---

Level repulsion of two coupled modes with an energy crossing has applications ranging from solid state theory [@ashcroft_solid_1976] to quantum chemistry [@atkins_atkins_2009]. While deceptively simple, it spawns a wealth of physics. With the introduction of dissipation or gain, an exceptional point [@heiss_physics_2012] appears that is topologically non-trivial [@dembowski_experimental_2001; @uzdin_observability_2011; @milburn_general_2015]. The special case of two modes with equal dissipation and gain rates is an example of parity-time symmetry [@bender_introduction_2005; @bender_observation_2013]. The spontaneous breaking of that symmetry is marked by the exceptional point. In recent years, exceptional points gathered significant interest and they were demonstrated in a variety of systems including active microwave circuits [@stehmann_observation_2004; @schindler_experimental_2011; @assawaworrarit_robust_2017], lasers [@brandstetter_reversing_2014; @peng_loss-induced_2014] and optical microresonators [@bender_twofold_2013; @peng_parity-time-symmetric_2014; @chen_exceptional_2017]. In particular, the topological transfer of energy between states by circling an exceptional point has been demonstrated with a microwave cavity [@dembowski_experimental_2001], a microwave waveguide [@doppler_dynamically_2016], as well as an optomechanical system [@xu_topological_2016; @jing_PT-symmetric_2014].

Strikingly, if one mode has negative energy, the energy levels of two interacting modes do not repel, but attract instead [@bernier_unstable_2014; @eleuch_exceptional_2014; @seyranian_coupling_2005]. The Hamiltonian leads to hybridized modes of complex eigenfrequencies, one of which is unstable. As in level repulsion, an exceptional point marks the transition between the regimes of real and complex frequencies. In the process, the real components of the frequencies become identical in a way that is reminiscent of the synchronization of driven oscillators [@pikovsky_synchronization:_2003].

![ Level repulsion and attraction. Two modes, whose bare frequencies depend on a parameter $\lambda$, have a level crossing (dotted lines). A coherent coupling will in general lift the degeneracy. (a) In the more usual case, level repulsion, the coupling opens a gap between the frequencies of the hybridized eigenmodes ${\tilde{\omega}}$ (blue solid lines) and the eigenfrequencies bend away from each other. (b) In contrast, if one of the modes has negative energy, level attraction occurs. The real components of the eigenfrequencies ${\tilde{\omega}}$ (blue solid lines) bend towards each other and converge. They meet at two exceptional points, where the curves have kinks. A gap opens in the imaginary components of the frequencies (orange dashed lines). The mode with a negative imaginary component to the frequency is unstable and grows exponentially. []{data-label="fig:LAvsLR"}](figure_LAvsLR.pdf)

Negative-energy modes (equivalent to harmonic oscillators with negative mass) have been studied in schemes to evade quantum measurement backaction [@tsang_evading_2012; @zhang_back-action-free_2013; @polzik_trajectories_2015]. Such a scheme was recently demonstrated with an atomic spin ensemble, prepared in its maximal-energy spin state in a magnetic field [@moller_quantum_2017]. Spin flips decrease the energy and correspond to excitations of a harmonic oscillator with a negative mass. Alternatively, the negative-energy mode can be effectively realized in a frame rotating faster than the mode itself [@woolley_two-mode_2013; @ockeloen-korppi_quantum_2016].

In cavity optomechanics [@aspelmeyer2014], a blue-detuned pump tone induces time-dependent interactions between the electromagnetic mode and the mechanical oscillator. In a frame rotating at the pump frequency, the Hamiltonian is time-independent, and the electromagnetic mode appears to have negative energy. While level repulsion was demonstrated in the strong coupling regime of cavity optomechanics [@teufel_circuit_2011; @verhagen_quantum-coherent_2012], level attraction has so far not been observed.

Here we construct a general theoretical framework to understand the phenomenon and, as an illustration, demonstrate level attraction in a microwave optomechanical circuit using engineered dissipation. In a first part, it is shown how a coherent coupling between modes of positive and negative energy gives rise to level attraction. The role of dissipation is discussed and explains the difficulty in observing level attraction in such systems, as the dissipation rates of the two modes must be similar. An intuitive way to classify different types of exceptional points in two-modes system is developed that allows to clearly distinguish the cases of level repulsion and attraction. In a second part, both level attraction and repulsion are demonstrated experimentally in the same microwave optomechanical circuit, where the mechanical dissipation rate can be engineered to match that of the microwave cavity.

We start with a general theoretical model of a positive-energy mode coherently coupled to a negative-energy one. The two modes, of annihilation operators ${\hat a}$ and ${\hat b}$ and coherently coupled with strength $g$, are described by the Hamiltonian $$\hat H =
  - \hbar \omega_1 (\lambda) {\hat a}^\dagger {\hat a}+
  \hbar \omega_2 (\lambda) {\hat b}^\dagger {\hat b}+
  \hbar g \left( {\hat a}{\hat b}+ {\hat a}^\dagger {\hat b}^\dagger \right)
  \label{eq:hamiltLA}$$ where the two positive frequencies $\omega_1$ and $\omega_2$ vary with respect to an external parameter $\lambda$. The linear coupling chosen here is quite general: if we assume the modes close in frequency, other linear terms ${\hat a}^\dagger {\hat b}$, ${\hat a}{\hat b}^\dagger$ can be neglected in the rotating wave approximation (valid only if the frequencies $\omega_{1,2}$ dominate over the dissipation rates for an open system). The coupling rate $g$ is chosen to be real, as any complex phase can be absorbed in a redefinition of ${\hat a}$ or ${\hat b}$.

In the Heisenberg picture, this leads to the equations of motion $$\frac{d}{dt}
  \begin{pmatrix}
    {\hat a}\\ {\hat b}^\dagger
  \end{pmatrix}
  =
  i
  \begin{pmatrix}
    \omega_1 & -g \\
    g & \omega_2
  \end{pmatrix}
  \begin{pmatrix}
    {\hat a}\\ {\hat b}^\dagger
  \end{pmatrix}
  \label{eq:EOM}$$ where we drop the explicit $\lambda$ dependence. We note that the uncoupled, bare modes evolve as ${\hat a}(t) = e^{i\omega_1 t} {\hat a}(0)$ and ${\hat b}^\dagger(t) = e^{i\omega_2 t} {\hat b}^\dagger(0)$ with a positive phase. The hybridized eigenmodes of the system are found by diagonalizing the matrix in [eq. (\[eq:EOM\])]{}, and have eigenfrequencies $${\tilde{\omega}}_{1,2}
  =
  \frac{\omega_1 + \omega_2}{2}
  \pm
  \sqrt{
  \left( \frac{\omega_1 - \omega_2}{2} \right)^2
  - g^2
  }.
  \label{eq:eigenfreqs}$$ The negative sign in front of $g^2$ is the only difference with the eigenfrequencies for the case of level repulsion (when ${\hat a}$ has positive energy) but dramatically impacts on the physics.

In [Fig. \[fig:LAvsLR\]]{}, level attraction is compared to level repulsion, with two striking features. First, instead of avoiding each other, the eigenfrequencies pull towards each other. Second, when they meet at $4g^2 = (\omega_1 - \omega_2)^2$, the frequencies acquire positive and negative imaginary parts, causing exponential decay and growth. The hybridized mode with a negative imaginary component grows exponentially and is therefore unstable.

![ The effect of dissipation on level attraction. Two modes, ${\hat a}$ and ${\hat b}$ cross in frequency (the mode ${\hat a}$ having negative energy), with respective dissipation rates $\kappa$ and $\Gamma$. The real component of the frequency is the solid blue line and the imaginary component the dashed orange line. (a) While a finite average dissipation rate simply translates the imaginary components of the frequency, a difference in the two rates ($\kappa\neq\Gamma$) affects qualitatively level attraction. The kinks and the exceptional points disappear and the picture is overall smoothed out. (b) When the dissipation rates differ significantly, no trace of level attraction is visible anymore. In both cases, one of the hybridized mode becomes unstable if the imaginary component of its frequency turns negative. []{data-label="fig:dissipation"}](figure_dissipation.pdf)

The transition between the regimes of real and complex eigenfrequencies is marked by exceptional points, which can be understood by studying the matrix of [eq. (\[eq:EOM\])]{}. Decomposed in terms of Pauli matrices and omitting the term proportional to the identity, it can be expressed as $
\frac{1}{2} (\omega_1 - \omega_2) \sigma_z
- ig \sigma_y
$. In contrast with level repulsion for which the interaction term would be $g\sigma_x$, here the Hermitian Pauli matrix is multiplied by an *imaginary* coefficient. The transition between the two regimes corresponds to a competition of the two terms. When the two Pauli matrices have coefficients of the same amplitude, the matrix is proportional to $\sigma_z - i\sigma_y$. At this point, the two eigenvectors coalesce and a single eigenvector with a single eigenvalue subsists: it is an exceptional point [@heiss_physics_2012]. More generally for all two-mode systems, any point when the dynamics is determined by a matrix proportional to $\sigma_\alpha + i \sigma_\beta$, with $\alpha\neq \beta$, is an exceptional point. In the supplementary information, we use this decomposition to construct an intuitive classification of the various realizations of exceptional points.

Level attraction arises whenever the coupling term consists of a Pauli matrix with an imaginary coefficient. In fact, coupled oscillators of positive and negative energy are only one way to achieve this. An alternative relies on dissipative interaction between two modes through one or multiple intermediary modes [@metelmann_quantum-limited_2014]. The mode hybridization observed between positive-energy oscillators with dissipative interactions [@xu_topological_2016; @gloppe_bidimensional_2014; @khanbekyan_unconventional_2015] can be interpreted as level attraction.

While level attraction of two linearly coupled modes displays intriguing similarities with the synchronization of driven oscillators, important differences exist. As in synchronization, the real components of the frequencies “lock” over a frequency range that increases with the coupling rate $g$, and form the equivalent of an Arnold tongue [@pikovsky_synchronization:_2003]. The physical process however differ. In synchronization, one starts with two oscillators that are driven nonlinearly to their limit-cycles, then a coupling is introduced that locks their frequencies and their phases [@weiss_noise-induced_2016]. In level attraction by contrast, the frequencies of the two modes attract through *linear* dynamics until they become identical. The state of the two hybridized modes remain independent and their phases can be set arbitrarily.

To understand why level attraction is in practice less common than level repulsion, the role of dissipation should be studied. We open the system and include in our treatment the energy dissipation rates $\kappa$ and $\Gamma$ respectively for the modes ${\hat a}$ and ${\hat b}$. They can be introduced as positive imaginary components of the bare frequencies in the equations of motion. The results of [eq. (\[eq:EOM\])]{} and (\[eq:eigenfreqs\]) can be extended by replacing $\omega_1$ with $\omega_1 + i \frac{\kappa}{2}$ and $\omega_2$ with $\omega_2 + i \frac{\Gamma}{2}$. In [Fig. \[fig:dissipation\]]{}, we compare the resulting eigenfrequencies. If the dissipation rates are equal ($\kappa=\Gamma$), the level structure of [Fig. \[fig:LAvsLR\]]{}b is reproduced with the imaginary components translated to a finite average. However, in the case of even slightly mismatched dissipation rates $\kappa\gtrsim\Gamma$ ([Fig. \[fig:dissipation\]]{}a), the exceptional points and the kinks in the frequencies all disappear. For increasingly dissimilar rates $\kappa \gg \Gamma$ ([Fig. \[fig:dissipation\]]{}b), the level-attraction picture progressively disappears until the modes seem to cross without interacting. Therefore, only in a system where dissipation rates can be tuned to closely match each other is level attraction observable.

Cavity optomechanics provides an ideal setting to study level attraction and compare it to level repulsion. We now take ${\hat a}$ to represent an electromagnetic mode and ${\hat b}$ a mechanical oscillator, coupled through the optomechanical interaction $\hbar g_0 {\hat a}^\dagger {\hat a}({\hat b}+ {\hat b}^\dagger)$, where $g_0$ is the vacuum optomechanical coupling [@aspelmeyer2014]. With a blue-detuned pump tone applied to the system, the three-wave-mixing coupling is linearized and the Hamiltonian reduces to the form of [eq. (\[eq:hamiltLA\])]{} $$\hat H = 
  - \hbar \Delta {\hat a}^\dagger {\hat a}+
  \hbar \Omega_{\mathrm{m}}{\hat b}^\dagger {\hat b}+ \hbar g
  \left( {\hat a}{\hat b}+ {\hat a}^\dagger {\hat b}^\dagger \right),
  \label{eq:OMHamilt}$$ where $\Delta$ is the detuning of the pump tone, $\Omega_{\mathrm{m}}$ the mechanical mode frequency and $g = g_0 \sqrt{n_{\mathrm{c}}}$ the linear coupling enhanced by the mean cavity photon number $n_{\mathrm{c}}$ due to the pump tone. As above, we neglect counter-rotating terms and assume the detuning $\Delta$ to be close to $\Omega_{\mathrm{m}}$. Critically, the Hamiltonian is expressed in a frame rotating at the pump frequency in order to be time-independent. Hence, for a blue detuning $\Delta>0$, the cavity mode effectively has a *negative* energy, since the photons have a negative relative frequency with respect to the pump. In this context, the well-known parametric instability of optomechanics [@aspelmeyer2014] can be interpreted as resulting from the physics of level attraction. The instability stems from the negative imaginary component that develops in the eigenfrequencies of the equations of motion, above the critical coupling $g_\mathrm{crit} = \sqrt{\kappa \Gamma}/2$. For level attraction to be observable, the magnitudes of $\kappa$ and $\Gamma$ should be close. For usual experimental parameters, however, the electromagnetic decay rate $\kappa$ is much larger than the mechanical rate $\Gamma$, and no attraction can be observed in practice for the mechanical and electromagnetic modes.

![ Engineering dissipation in a multimode optomechanical circuit. In order to observe level attraction, the dissipation rate $\Gamma$ of the mechanical mode ${\hat b}$ must be increased to match $\kappa$, the much larger dissipation rate of the primary electromagnetic mode ${\hat a}$. To that end, an auxiliary mode ${\hat a}_{\mathrm{aux}}$ is used for sideband cooling. (a) Schematic of the microwave optomechanical circuit, coupled inductively to a microwave feedline and measured in reflection. The two hybridized modes of the circuit ${\hat a}$ and ${\hat a}_{\mathrm{aux}}$ interact with the motion of the top membrane of a shared capacitor, acting as the mechanical oscillator ${\hat b}$ (in green). (b) Photograph of the circuit and scanning-electron micrograph of the vacuum-gap capacitor. (c) Diagram of the three interacting modes. (d) Frequency domain representation of the level-attraction experiment. A microwave pump tone (vertical red line), red-detuned by the mechanical frequency $\Omega_{\mathrm{m}}$ with respect to the auxiliary mode resonance $\omega_{\mathrm{aux}}$ (grey peak) is used for sideband cooling. Level attraction of the modes ${\hat a}$ and ${\hat b}$ is achieved by sweeping the detuning $\Delta$ of a pump tone (vertical blue line) near the blue sideband of the primary mode resonance $\omega_{\mathrm{c}}$ (blue peak). For level repulsion, the pump tone is instead swept near the red sideband. []{data-label="fig:scheme"}](figure_scheme.pdf)

![image](figure_data.pdf)

In our experiment, the effective mechanical energy decay rate $\Gamma_{\mathrm{eff}}$ is artificially increased to match $\kappa$ using sideband cooling with an auxiliary mode. We use a superconducting electromechanical circuit [@teufel_circuit_2011] containing two microwave LC modes interacting with the vibrational mode of a vacuum-gap capacitor (represented schematically in [Fig. \[fig:scheme\]]{}a and shown in [Fig. \[fig:scheme\]]{}b). The design, which was demonstrated in previous work [@toth_dissipative_2017], uses two hybridized electromagnetic modes of the circuit to ensure that one has a much larger external coupling rate to the microwave feedline than the other. The more dissipative, auxiliary mode ${\hat a}_{\mathrm{aux}}$ is used to perform sideband cooling of the mechanical oscillator with a red-detuned pump tone. This damps the oscillator and increases its effective dissipation rate to $\Gamma_{\mathrm{eff}}\approx\kappa$. Meanwhile, the less dissipative, primary mode ${\hat a}$ undergoes level attraction with the damped mechanical oscillator.

In the experiment, the device is placed inside a dilution refrigerator and cooled to the base temperature below 50 mK, at which the circuit is superconducting and therefore its internal $Q$-factor is enhanced. The two microwave modes ${\hat a}$ and ${\hat a}_{\mathrm{aux}}$ have respective resonance frequencies $\omega_{\mathrm{c}}\approx 2\pi\times 4.11$ GHz and $\omega_{\mathrm{aux}}\approx 2\pi\times 5.22$ GHz, and dissipation rates $\kappa \approx 2\pi\times 110$ kHz and $\kappa_{\mathrm{aux}}\approx 2\pi \times 1.8$ MHz. They interact with the fundamental mode of the top plate of the vacuum-gap capacitor that has a frequency $\Omega_{\mathrm{m}}\approx 2\pi \times6.3$ MHz. By placing a pump tone red-detuned by $\Omega_{\mathrm{m}}$ from the auxiliary mode resonance (see [Fig. \[fig:scheme\]]{}b), the mechanical oscillator is damped. The mechanical dissipation rate $\Gamma$, originally below $ 2\pi \times 100$ Hz, is tuned to an effective dissipation rate $\Gamma_{\mathrm{eff}}\approx \kappa \approx 2\pi \times 110$ kHz.

Level repulsion and attraction of the primary microwave mode and the damped mechanical oscillator are both measured. As a pump tone is tuned to the blue (or red) sideband of the primary microwave mode ([Fig. \[fig:scheme\]]{}b), the weak probe tone of a vector network analyser is applied to obtain its linear response. Due to the hybridization of the modes, the response carries information about both microwave and mechanical modes. In both cases, the same pump power is set to obtain a coupling strength $g \approx 2\pi \times 200$ kHz corresponding to a mean cavity photon number $n_{\mathrm{c}}\approx 4\times 10^6$. The known case of level repulsion is obtained with a red-detuned tone ([Fig. \[fig:LAresult\]]{}a). As the bare effective mechanical mode frequency comes near the microwave resonance, the two modes hybridize; their eigenfrequencies bend away from each other and a gap of $2g$ opens. If a blue-detuned tone is used instead, level attraction occurs, shown in [Fig. \[fig:LAresult\]]{}b which displays the characteristic level structure of [Fig. \[fig:LAvsLR\]]{}b. The resonance frequencies of the modes attract and converge at the points where the bare frequencies of the modes differ only by $\pm2g$. Data is omitted for clarity in the unstable region where the real component of the frequencies are identical. In order for the level attraction to be clearly visible, a large coupling rate $g$ is chosen that exceeds the dissipation rates $\kappa$ and $\Gamma_{\mathrm{eff}}$. It therefore exceeds the critical coupling $g_\mathrm{crit}$ as well and parametric instability occurs: in the unstable region, one of the modes grows exponentially until the conditions of the validity of [eq. (\[eq:OMHamilt\])]{} are no longer fulfilled. Namely, the fluctuating field is no more negligible compared to the mean cavity photon number $n_{\mathrm{c}}$. The original nonlinear optomechanical interaction constrain the system to a limit-cycle with a modified cavity resonance frequency, the description of which lies beyond the scope of this article [@marquardt_dynamical_2006].

In summary, level attraction was experimentally demonstrated using a dual-mode electromechanical circuit. Although related to the well-studied parametric instability of optomechanics, the vastly different dissipation rates for the mechanical and electromagnetic modes prevented its observation until now. Level attraction, similarly to level repulsion in open systems, gives rise to exceptional points. In both cases, the real part of the frequencies converge and a gap opens in the imaginary part (or vice versa) precisely at the exceptional point. In future work, the exceptional points of level attraction could be harnessed to demonstrate topological phenomena by circling such a point in a two-dimensional parameter space [@dembowski_experimental_2001; @xu_topological_2016; @doppler_dynamically_2016]. Since the exceptional point only exists when the dissipation rates of the two modes match exactly, the tunable mechanical damping rate $\Gamma_{\mathrm{eff}}$ can be used as one parameter in such an experiment, with the tunable coupling rate $g$ as the second.

We thank A. Nunnenkamp and D. Malz for useful discussions and comments. This work was supported by the SNF, the NCCR Quantum Science and Technology (QSIT), and the EU Horizon 2020 research and innovation programme under grant agreement No 732894 (FET Proactive HOT). TJK acknowledges financial support from an ERC AdG (QuREM). All samples were fabricated in the Center of MicroNanoTechnology (CMi) at EPFL.