Observation of an exceptional point in a two-dimensional ultrasonic cavity of concentric circular shells

We report observation of an exceptional point in circular shell ultrasonic cavities in both theory and experiment. In our theoretical analysis we first observe two interacting mode groups, fluid- and solid-based modes, in the acoustic cavities and then show the existence of an EP of these mode groups exhibiting a branch-point topological structure of eigenfrequencies around the EP. We then confirm the mode patterns as well as eigenfrequency structure around the EP in experiments employing the schlieren method, thereby demonstrating utility of ultrasound cavities as experimental platform for investigating non-Hermitian physics.

. Structure of our 2D shell cavity. It consists of three sub-regions: inner fluid, a solid shell, and outer fluid. Ultrasound fields are described in terms of pressure P in and P out inside the inner and outer fluid and displacement u inside the solid. We investigated the mode interactions based on the decomposition of shell modes into FBM's and SBM's. If we vary the outer radius R b with the inner radius R a fixed, the resonance frequencies of FBM's are almost invariant. It is because the modes localized in the internal fluid are hardly affected by the changes of the outer shell boundary. On the other hand, the frequencies of SBM's are inversely proportional to the outer radius R b of the shell because the size parameter (k f R b for SBM) is a constant for a mode regardless of the system size. Accordingly, FBM's and SBM's can move closer to or move away from each other with varying R b , allowing interactions between two groups of modes across the inner boundary.
This behavior is shown in Fig. 3. As mentioned above, FBM's are not affected by the change of R b /R a with R a fixed. When the two mode groups are far apart, Re[k f R a ] values of the FBM's more or less follow a constant horizontal line, which is often called the diabatic transition line. Similarly, Re[k f R a ] values of SBM's follow another diabatic transition line with the inverse dependence on R b . As the ratio R b /R a is varied for different angular quantum numbers m's, FBM's and SBM's repel (avoided crossing, AC) or cross (mode crossing, MC) each other near the crossing point of the diabatic lines. Here, the angular quantum number m equals a half of the number of anti-nodes of the wavefunction in the direction of the azimuthal angle. It is also proportional to the angular momentum χ = = L m kR sin   of a fictitious particle associated with the wave solution in the semi-classical limit with χ the incident angle of the particle on the circular boundary of radius R. Note that the angular quantum number m was used as an internal system parameter in the previous studies 9, 28 . In the case of MC, FBM's and SBM's just follow their diabatic lines. In the case of AC, however, FBM's and SBM's do not follow their diabatic lines but follow the paths of instantaneous solutions accompanying a mode gap which is approximately proportional to the strength of the interaction between two groups. By following these paths, spatial mode patterns change from FBM to SBM or vice versa. Such mode pattern exchange has been experimentally observed in other systems such as in microwave billiards 7 and in exciton-polariton billiards 13 .
Exceptional point. Exceptional point (EP) is a singular point in parametric space where two interacting modes coalesce into one mode. EP condition is satisfied when the coupling equals their differential loss. Occurrence of an EP can be easily understood in a simple 2 × 2 matrix model. Let us consider a non-Hermitian Hamiltonian given by where unperturbed modes have real energy E 1 , E 2 and decay rates γ 1 , γ 2 (γ 1 > γ 2 ). The coupling C between the modes is assumed to be real. After diagonalization of the Hamiltonian, we get eigenvalues , where E ± = (E 1 ± E 2 )/2 and γ ± = (γ 1 ± γ 2 )/2. When E 1 = E 2 (i.e., E − = 0),    ∆ = − + − depends on the coupling C and the differential decay rate γ − . If γ − > C, then the energy difference is given by  γ ∆ = − − i C 2 2 2 . Therefore, when we vary the detuning E − across zero, the real parts of the energy cross but the imaginary parts repel each other. If γ − < C, on the other hand, we get , which means avoided crossing in real parts and crossing in imaginary parts as the detuning E − is varied. Lastly, if γ − = C, the real and imaginary parts of two modes have the same values. Moreover, two eigenfunctions become the same in this case, differently from the usual energy degeneracy. This coalesced mode is called an EP mode.
In Fig. 4(a) and (b), the resonance modes of the shell cavity are plotted as R b /R a is varied. Solid (open) symbols represent the modes followed from FBM's (SBM's) in the lower (R b /R a ) range. In Fig. 4(a), real eigenvalues Re[k f R a ] are plotted whereas in Fig. 4(b) the imaginary parts are plotted. In these plots, we observe a transition between MC and AC. When m = 17, 18, FBM's and SBM's are undergoing MC (AC) in real (imaginary) parts. For the smaller m values, the modes are undergoing AC (MC) in real (imaginary) parts. In Fig. 5(a-d), the trajectories of the complex eigenvalues are plotted as R b /R a value is increased. Blue (red) dots are followed from the FBM's (SBM's) in the lower R b /R a range.
It is evident that an EP exists somewhere between m = 16 and 17 when .
in the parameter space. Note the internal parameter m controlling E − (detuning) is an integer and thus discrete. For this reason it is difficult to hit the exact position of an EP in the (m, R b /R a ) parameter space. However, it is in principle possible to reach the EP by changing a continuous system parameter such as density of fluid, instead of m, which is accessible by mixing two different types of fluids. For example, in Fig. 6, complex eigenvalues at crossing points of the diabatic lines for Re[k f R a ] are displayed. Figure 6(a) is the results for the parameters in Table 1. As we mention above, it is impossible to reach an EP with only varying the discrete parameter m. If we slightly change the sound velocity in the fluid -by changing the Lamé's parameters -as in Fig. 6(b), however, we can hit the EP accurately. In this case, m = 17 modes become an EP mode. Another way to reach an EP is to include additional loss in the fluid, which can be simulated by introducing a complex sound velocity.

Experiment.
We now present our experimental results to verify our theoretical predictions. Frequencies and mode patterns of resonance modes obtained with the schlieren method are shown in Fig. 7, where experimental data are marked by black dots. Blue and red lines are the theoretical paths of instantaneous solutions, followed from FBM and SBM in the lower R b /R a region, respectively. We observe a good agreement between theory and experiment. Mode patterns visualized by the schlieren method are displayed below the mode spectrum. As already shown in the theoretical analysis or in Fig. 3(a), we observe AC in the spectrum as well as the mode pattern exchange in Fig. 7(a). Note that the intensity of the mode pattern in the fluid is gradually reduced if we follow the path (iii) → (ii) → (i) or (iv) → (v) → (vi). This is because unperturbed SBM's do not have any spatial distributions in the fluid. In Fig. 7(b), however, we observe FBM's with a constant Re[k f R a ]. In this MC case, there is neither mode splitting nor noticeable spatial mode pattern mixing. As a result, mode patterns of the SBM's could not be visualized because they have negligible spatial distribution in the inner fluid. In addition, the mode patterns of the FBM's are hardly affected by the change of R b /R a as expected in the theoretical analysis or in Fig. 3(b).
In   experiment. This small imaginary component corresponds to a medium-loss quality factor  Q 3400 loss , consistent with the loss-broadened linewidths of otherwise high-Q modes in the experiment. It is seen that AC (MC) occurs for m ≤ 16 while MC (AC) occurs for m ≥ 17 in the real (imaginary) parts of resonance frequencies.
Although we can measure only modes with spatial distribution in the fluid by the schlieren method, the transition from AC to MC can be clearly seen in Fig. 8 as m is increased. This observation implies the existence of an EP with 16 < m < 17 and

Discussion
In both theory and experiment, we have observed the transition from AC to MC by increasing angular quantum number m. This transition is due to the reduced C compared to γ − . The transition can be analyzed in more details as follows.
If m is increased with the radial quantum number l fixed, the size parameter Re[k f R a ] of both FBM and SBM increases since the size parameter is approximately equal to the number of wavelengths fitting the inner circumference of the shell. Moreover, the distributions of FBM and SBM are shifted to the internal and external boundaries, respectively, corresponding to an increased incident angle of waves on the boundaries (recall m = kRsinχ). As a result, the loss of FBM is reduced whereas that of SBM is increased. The coupling decreases much more than the loss of FBM. The reason is as follows. As m is increased, the distribution of FBM in the solid region is reduced because of the decreased loss of FBM, and at the same time the distribution of SBM further shifts to the external boundary. Therefore, the wavefunction overlap between FBM and SBM is greatly reduced, resulting in the coupling much more decreased than the loss of FBM. Therefore, we can induce a transition from AC to MC by increasing the angular quantum number m.
It is apparent that the schlieren method cannot visualize the mode patterns inside the opaque shell (aluminium). As shown for m ≥ 17 in Fig. 8, this limitation is pronounced in the weak-coupling regime. However, with smaller m values, for which the coupling is strong, it was possible to measure the SBM-like modes partially even quite away from the R b /R a point where the diabatic lines cross. It is because the SMB-like modes still have some distribution in the internal fluid due to the mode mixing arising from the intermode interaction between SBM . As shown in the theoretical analysis, one can reach an EP by adjusting the sound velocity or the medium loss continuously.
Scientific RepoRts | 6:38826 | DOI: 10.1038/srep38826 and FBM. As a result, we could observe AC in real eigenvalues despite the limitation of the schlieren method. Note it is still impossible to visualize unperturbed SBM's in the opaque solid with the schlieren method since little mode-mixing exists with FBM's. This limitation, however, can be easily overcome by adopting transparent solid such as glass or acrylic. Ultrasound cavities made of fluid enclosed in transparent solid would thus be a promising platform for studying intermode interactions in non-Hermitian systems. In particular, there are many interesting phenomena expected to occur near EP's such as adiabaticity breaking when an EP is dynamically encircled 4,5,6,29,30 , chirality of EP modes 31 and mode evolution near a triple EP 32 . We expect these phenomena can be effectively investigated without disturbing the system by using our approach in terms of both eigenvalues and eigenfunctions.

Methods
Solving wave equations numerically. The shell cavity has three sub-regions: inner fluid, a solid shell, and outer fluid (see Fig. 1). In the frequency domain, the harmonic ultrasound fields are described by the Helmholtz equation in the fluid and by Cauchy-Navier equation in the solid: . Obviously ψ has only z component in a 2D system described in x and y coordinates. By substituting the potential form of u in Eq. (3) and after rearranging terms according to their polarization, we obtain two Helmholtz equations for ϕ and y as m m s m m s where N m is the Neumann function of order m. Outside the shell, the pressure field is also found from Eq. (2), but in order to satisfy the outgoing wave condition we take the first kind Hankel function instead of the Bessel function: Our goal now is to find the resonant frequencies of the normal modes. To do this, we need six boundary conditions for the six unknowns {A m , B m , … , F m } for a given m. The boundary conditions are as follows. The first is the continuity of normal components of the stress, which is just the equilibrium of surface normal forces to maintain the interface. Next is the continuity of the displacement, i.e., the solid and the fluid should contact each other all the time. The last is that the tangential stress at the inner (r = R a ) and outer (r = R b ) interfaces should vanish because there cannot be shear stress in the fluid. These conditions are explicitly given by (1) where σ ij is the stress tensor within the shell defined as The superscripts f and s in the displacement u refer to fluid and solid. Indices i, j in the stress tensor σ denote orthogonal coordinates r and φ.
After substituting the expressions for u and P into the boundary conditions, one finds six linear equations for six unknowns which depend on the complex frequency ω. Accordingly, those equations can be written in the 6 × 6 matrix form M(ω)b = 0 for a given m, where b consists of the field coefficients {A m , … , F m }.
In the cylindrical coordinates, the surface-normal displacements and the components of the stress tensor are easily found to be as follows.
By substituting P, ϕ, ψ for the boundary conditions given in the main text and after some algebra, we get six homogeneous linear equations for the coefficients {A m , … , F m } of the field.     The equations are summarized to a simple matrix form M(ω)b = 0, where b is a column vector consisting of the field coefficients {A m , … , F m }. As mentioned in the main text, to find nontrivial solutions one need to search complex ω's such that det(M(w)) = 0. These ω's can be found by using the Newton-Raphson method in complex space, as in an optical microcavity 33 . Because we take the convention that the fields have the form of e i(k·r−wt) , ω is obviously expressed by ω = ω r + iω i = ω r − i|ω i | (ω i is negative), where ω r mainly determines the spatial distribution of the field and (− ω i ) gives the decay rate of the resonant mode. Then the quality factor Q of a mode is given by Q = − ω r /2ω i .

Experimental setup.
We fabricated aluminium shells with R a = 5 mm and R b ranging from 2.65R a to 3.0R a in total of 11 steps. The surface roughness is about 10μm, which is negligible compared to the sound wavelength of interest (order of 1 mm). The cavity is immersed in distilled water. The water is first heated to the boiling temperature to remove dissolved air. It is then rapidly cooled down to the room temperature by a immersion chiller in order to avoid re-dissolving of air. In addition, we cover the surface of water with polyethylene spheres for the same reason. With this procedure, small air bubbles which act as scatterers of the sound waves are mostly eliminated, allowing high-Q modes with Q ~ 10 4 .
The cavity modes are excited by an immersion ultrasonic transducer which is driven by a function generator with an RF amplifier (Fig. 9). The driving sine wave frequency is scanned in the range of 800 kHz-1.3 MHz. Spatial intensity patterns are measured by using the schlieren method, which is widely used to visualize the refractive index modulations in transparent media. It is well established that the schlieren image represents the sonic pressure intensity |P| 2 at low pressure 34 . When the driving frequency is on resonance with a FBM, one can observe a bright image of the pressure field in the internal fluid. In addition, the spectrum of FBM's can be obtained by integrating the pressure field distribution seen in the schlieren image as a function of the excitation frequency. Therefore, with our setup, we are able to measure the mode patterns as well as the mode spectrum simultaneously. Spatial mode patterns around an EP have been observed in microwave billiards before by scanning a perturbative Scientific RepoRts | 6:38826 | DOI: 10.1038/srep38826 probe 35 . Our setup does not need such a physical probe, which is known to introduce unwanted perturbation to the system 36 .
Inclusion of medium loss. In actual experiments, scattering and absorption loss inevitably occurs in media, mostly in the fluid in our experiment. The loss in the fluid can be included in our theoretical calculation by introducing an imaginary component v i (< 0) in the longitudinal sound velocity v f in the fluid. Note k f in Eqs (14), (16), (17) and (19) are replaced with The new matrix equation M(ω)b = 0 is solved for complex frequency ω, which is now given by ω ω ω ω ω where k i , v i < 0 is assumed. This equation indicates that the total loss 1/Q is composed of 1/Q k = − 2k i /k r accounting for the wave-tunneling loss and 1/Q v = − 2v i /v r for absorption and scattering loss in the medium. The quality factor Q v corresponding to the medium loss has been estimated to be approximately 3400 from the observed linewidth of otherwise high-Q mode (Q k ~ 10 5 ) in our experiment. The estimated medium loss is found to be consistent with our choice of v i = − 0.22 m/s.