Level attraction in a microwave optomechanical circuit

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.

Level repulsion of two coupled modes with an energy crossing has applications ranging from solid state theory [1] to quantum chemistry [2]. While deceptively simple, it spawns a wealth of physics. With the introduction of dissipation or gain, an exceptional point [3] appears that is topologically non-trivial [4][5][6]. The special case of two modes with equal dissipation and gain rates is an example of parity-time symmetry [7,8]. 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 [9][10][11], lasers [12,13] and optical microresonators [14][15][16]. In particular, the topological transfer of energy between states by circling an exceptional point has been demonstrated with a microwave cavity [4], a microwave waveguide [17], as well as an optomechanical system [18,19].
Strikingly, if one mode has negative energy, the energy levels of two interacting modes do not repel, but attract instead [20][21][22]. 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 [23].
Negative-energy modes (equivalent to harmonic oscillators with negative mass) have been studied in schemes to evade quantum measurement backaction [24][25][26]. Such a scheme was recently demonstrated with an atomic spin ensemble, prepared in its maximal-energy spin state in a magnetic field [27]. 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 [28,29].
In cavity optomechanics [30], a blue-detuned pump tone induces time-dependent interactions between the * alexey.feofanov@epfl.ch † tobias.kippenberg@epfl.ch 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 [31,32], 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 positiveenergy mode coherently coupled to a negative-energy one. The two modes, of annihilation operatorsâ andb and coherently coupled with strength g, are described by the Hamiltonian where the two positive frequencies ω 1 and ω 2 vary with respect to an external parameter λ. The linear coupling chosen here is quite general: if we assume the modes close in frequency, other linear termsâ †b ,âb † can be neglected in the rotating wave approximation (valid only if the frequencies ω 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 ofâ orb.
In the Heisenberg picture, this leads to the equations of motion d dt where we drop the explicit λ dependence. We note that the uncoupled, bare modes evolve asâ(t) = e iω1tâ (0) and b † (t) = e iω2tb † (0) with a positive phase. The hybridized eigenmodes of the system are found by diagonalizing the matrix in eq. (2), and have eigenfrequencies The negative sign in front of g 2 is the only difference with the eigenfrequencies for the case of level repulsion (when a has positive energy) but dramatically impacts on the physics. In Fig. 1, 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 = (ω 1 − ω 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 transition between the regimes of real and complex eigenfrequencies is marked by exceptional points, which can be understood by studying the matrix of eq. (2). Decomposed in terms of Pauli matrices and omitting the term proportional to the identity, it can be expressed as 1 2 (ω 1 − ω 2 )σ z − igσ y . In contrast with level repulsion for which the interaction term would be gσ 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 σ z − iσ y . At this point, the two eigenvectors coalesce and a single eigenvector with a single eigenvalue subsists: it is an exceptional point [3]. More generally for all two-mode systems, any point when the dynamics is determined by a matrix proportional to σ α + iσ β , with α = β, 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 [33]. The mode hybridization observed between positive-energy oscillators with dissipative interactions [18,34,35] 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 [23]. 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 [36]. 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 κ and Γ respectively for the modesâ andb. They can be introduced as positive imaginary components of the bare frequencies in the equations of motion. The results of eq. (2) and (3) can be extended by replacing ω 1 with ω 1 + i κ 2 and ω 2 with ω 2 + i Γ 2 . In Fig. 2, we compare the resulting eigenfrequencies. If the dissipation rates are equal (κ = Γ), the level structure of Fig. 1b is reproduced with the imaginary components translated to a finite average. However, in the case of even slightly mismatched dissipation rates κ Γ (Fig. 2a), the exceptional points and the kinks in the frequencies all disappear. For increasingly dissimilar rates κ Γ (Fig. 2b), 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 takeâ to represent an electromagnetic mode and b a mechanical oscillator, coupled through the optomechanical interaction g 0â †â (b +b † ), where g 0 is the vacuum optomechanical coupling [30]. 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. (1) where ∆ is the detuning of the pump tone, Ω m the mechanical mode frequency and g = g 0 √ n c the linear coupling enhanced by the mean cavity photon number n c due to the pump tone. As above, we neglect counter-rotating terms and assume the detuning ∆ to be close to Ω 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 ∆ > 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 [30] 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 crit = √ κΓ/2. For level attraction to be observable, the magnitudes of κ and Γ should be close. For usual experimental parameters, however, the electromagnetic decay rate κ is much larger than the mechanical in out Engineering dissipation in a multimode optomechanical circuit. In order to observe level attraction, the dissipation rate Γ of the mechanical modeb must be increased to match κ, the much larger dissipation rate of the primary electromagnetic modeâ. To that end, an auxiliary modeâaux is used for sideband cooling. rate Γ, and no attraction can be observed in practice for the mechanical and electromagnetic modes.
In our experiment, the effective mechanical energy decay rate Γ eff is artificially increased to match κ using sideband cooling with an auxiliary mode. We use a superconducting electromechanical circuit [31] containing two microwave LC modes interacting with the vibrational mode of a vacuum-gap capacitor (represented schematically in Fig. 3a and shown in Fig. 3b). The design, which was demonstrated in previous work [37], 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 modeâ 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 Γ eff ≈ κ. Meanwhile, the less dissipative, primary modeâ 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 modesâ andâ aux have respective resonance frequencies ω c ≈ 2π ×4.11 GHz and ω aux ≈ 2π ×5.22 GHz, and dissipation rates κ ≈ 2π × 110 kHz and κ aux ≈ 2π × 1.8 MHz. They interact with the fundamental mode of the top plate of the vacuum-gap capacitor that has a frequency Ω m ≈ 2π×6.3 MHz. By placing a pump tone red-detuned by Ω m from the auxiliary mode resonance (see Fig. 3b), the mechanical oscillator is damped. The mechanical dissipation rate Γ, originally below 2π × 100 Hz, is tuned to an effective dissipation rate Γ eff ≈ κ ≈ 2π × 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. 3b), 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 ≈ 2π × 200 kHz corresponding to a mean cavity photon number n c ≈ 4 × 10 6 . The known case of level repulsion is obtained with a red-detuned tone (Fig. 4a). 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. 4b which displays the characteristic level structure of Fig. 1b. The resonance frequencies of the modes attract and converge at the points where the bare frequencies of the modes differ only by ±2g. 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 κ and Γ eff . It therefore exceeds the critical coupling g 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. (4) are no longer fulfilled. Namely, the fluctuating field is no more negligible compared to the mean cavity photon number n 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 [38].
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 [4,17,18]. Since the exceptional point only exists when the dissipation rates of the two modes match exactly, the tunable mechanical damping rate Γ eff can be used as one parameter in such an experiment, with the tunable coupling rate g as the second. ACKNOWLEDGMENTS 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.

I. SYMMETRY BETWEEN LEVEL REPULSION AND ATTRACTION
We explicitly derive here two minimal models, one for the usual level repulsion of two coherently coupled modes, the other for level attraction of a negative-energy mode coherently coupled to a positive-energy mode. We show that a symmetry relation links the two cases and all the physics of one is mirrored in the other if the frequency and dissipation rates are exchanged.
First, we review level repulsion. Two modes of positive energy (with annihilation operatorsâ andb, and respective frequencies ω 1 and ω 2 ) interact, as described by the Hamiltonian with g the linear coupling strength. The equation of motion in the Heisenberg picture is given by To solve the system in terms of eigenmodes, the matrix is diagonalized. The eigenfrequencies are given bỹ The model is extended to describe modes with dissipation by adding imaginary components to the bare frequencies, substituting ω 1 → ω 1 − iΓ 1 /2 and ω 2 → ω 2 − iΓ 2 /2. Note that a negative sign is required to obtain decaying exponentials. For simplicity, we define ω 0 = ω1+ω2 2 , Γ 0 = Γ1+Γ2 2 and ∆ω = ω 1 −ω 2 , ∆Γ = Γ 1 −Γ 2 . The eigenfrequencies can now be written asω The frequency of oscillation of the eigenmodes is given by the real component ofω LR 1,2 and the dissipation rate by half its imaginary component (with a minus sign).
We now consider the case of level attraction. One (and only one) mode has negative energy. The system is now described by the HamiltonianĤ The equations of motion are given by with eigenfrequenciesω The only difference with eq. (3) is the sign in front of g 2 , which can result in a complex eigenvalue meaning an instability for the system. It might appear at first disconcerting that the Hermitian Hamiltonian in eq. (5) leads to complex eigenfrequencies and unstable dynamics. In fact, for such an infinite Hilbert space, an operator must be compact as well as Hermitian to guarantee the existence of real eigenvalues [1], which is not the case here. We also note that only when the eigenfrequencies are real can the eigenoperators be interpreted as Bogolyubov modes [2]. When the eigenfrequencies are complex, the required commutation relations cannot be satisfied. In order to include dissipation, we now substitute ω 1 → ω 1 + iΓ 1 /2 and ω 2 → ω 2 + iΓ 2 /2. Note the opposite sign as above is required to obtain decaying exponentials. The eigenfrequencies including dissipation becomẽ There is a symmetry between eq. (4) and eq. (8). They are equivalent under the transformation ω = Γ/2 and Γ = 2ω, if eq. (8) is multiplied by a factor −i: We conclude that level attraction and repulsion are equivalent to each other through the exchange of frequency and dissipation rates (within a factor of 2). In Fig. S1, this symmetry is highlighted by contrasting equivalent situations for both level repulsion and level attraction Hamiltonians. In Fig. S1a-d, the curves for the real and imaginary parts of the eigenfrequencies are interchanged when going from the left column (level repulsion) to the right (level attraction). Moreover, Fig. S1e-f can be compared with Fig. 1 of the main text, where the characteristic shape of level attraction is here seen for the dissipation rates for the Hamiltonian of level repulsion and vice versa. In general, forĤ LR the frequencies repel, while the dissipation rates attract. The exact opposite is true forĤ LA : it is the dissipation rates that repel and the frequencies that attract.

II. CLASSIFICATION OF EXCEPTIONAL POINTS FOR SYSTEMS OF TWO COUPLED MODES
We describe here how to classify exceptional points of 2 × 2 matrices using Pauli matrices and proceed to sort out under which cases fall recent experimental demonstrations of exceptional points.
In general, the equations of motion for a 2-modes system can be written in the form in the form M = a 0 1 + a 1 σ x + a 2 σ y + a 3 σ z with 1 the identity matrix and a i complex numbers. The eigenvalues of M are now easily expressed as Note that the first term proportional to the identity in eq. (12) only shifts the eigenvalues by a constant and has no effect on the eigenvectors. Anytime that the matrix M can be written as a multiple of σ α + iσ β (α = β) (plus the identity), there is only one eigenvalue and this is an exceptional point [3]. We can use this decomposition to classify examples of exceptional points.
(I) The most common case is the level repulsion of two (positive-energy) modes of degenerate frequencies due to a coherent coupling. Most of the experimentally demonstrated exceptional points have realized such a system [4][5][6][7][8][9][10][11]. The two modes ared 1 =â andd 2 =b in our previous notation. The matrix M can be written as For low coupling rate g, M is diagonal with a gap in dissipation rates. For large g, the last term dominates such that the eigenmodes are eigenvectors of σ x : g opens a gap in frequency, while the dissipation rates are shared equally. The transition between the regimes is marked by an exceptional point at g = ∆Γ/4. (II) In this article, we consider the case of level attraction of modes of negative and positive energies and matching dissipation rates. Here the operators ared 1 =â andd 2 =b † in our notation. The matrix M can be written as At low coupling g, there is a gap in frequency, while at high coupling, the σ y term opens a gap in dissipation rates and the frequencies are identical. An exceptional point marks the transition at g = ∆ω/2. Note that the coherent coupling corresponds to a term with an imaginary coefficient for a system with one mode of negative energy.
(III) Level attraction of two modes can be realized in any system in which the coupling term has an imaginary component. In particular, dissipative interactions [12] represent an alternative way to the one presented in this article. The matrix M is there expressed as where for simplification the effective dissipation rate Γ eff of the two coupled modes is taken to be approximately equal. The effective interaction between the two modes has an imaginary coefficient and they have an increased effective dissipation rate Γ eff due to the dissipative interaction as well. Similarly to the previous case, by increasing the coupling g dis the original gap in frequency is closed and a difference in dissipation rates is created. In contrast however, Γ eff grows proportionally with g dis , such that the gap in dissipation rates does not result in an instability. The mode hybridization between modes of positive energy coupled with dissipative interactions can be interpreted as level attraction. This is the case for the experiments by Xu et al. [13], where two mechanical oscillators are effectively coupled by both interacting with the same optical cavity, by Gloppe et al. [14] where two modes of a nanowire interact through the non-conservative force of an optical field, and by Khanbeykyan et al. [15] where two modes of an optical resonator interact through multiple quantum dots.
(IV) Finally, yet another way to implement an exceptional point was realized in the experiment of Chen et al. [16]. The clockwise and counterclockwise modes of a whispering-gallery-mode resonator (of degenerate frequencies and dissipation rates) interact through two Rayleigh scatterers. This results in a combination of coherent and dissipative interaction that can be described by As the coupling g coh and g dis are varied, the relative phases of the bare modes change in the hybrid eigenmodes. When the two coupling strengths match (g coh = g dis ), the two eigenmodes coalesce and there is an exceptional point. Interestingly, this corresponds to a point of maximal nonreciprocity, with one bare mode coupled to the second but not the other way around [17].