Maximal Shannon entropy in the vicinity of an exceptional point in an open microcavity

The Shannon entropy as a measure of information contents is investigated around an exceptional point (EP) in an open elliptical microcavity as a non-Hermitian system. The Shannon entropy is maximized near the EP in the parameter space for two interacting modes, but the exact maximum position is slightly off the EP toward the weak interaction region while the slopes of the Shannon entropies diverge at the EP. The Shannon entropies also show discontinuity across a specific line in the parameter space, directly related to the exchange of the Shannon entropy as well as the mode patterns with that line as a boundary. This feature results in a nontrivial topological structure of the Shannon entropy surfaces.

Scientific RepoRtS | (2020) 10:12551 | https://doi.org/10.1038/s41598-020-69479-w www.nature.com/scientificreports/ In this paper, we take on this intriguing question by introducing the Shannon entropy for the probability density of eigenmodes and applying it around an EP in a dielectric microcavity. The Shannon entropy is originally defined as a measure of average information contents associated with random outcomes in data communication 33 and information theory 34 . It has been utilized in diverse research areas to quantify amount of information, including black holes 35 , confined hydrogen-like systems 36 and quantum entanglement 37 . The Shannon entropy has also been used as an indicator for avoided crossing in dielectric microcavities 38 and related to the quantum transition from order to chaos 39 . eigenmodes and eigenvalues in an elliptical dielectric microcavity. Before we evaluate Shannon entropy, let us first discuss the physical system that we consider. Assume a closed physical system described by a Hermitian Hamiltonian H S . Let us suppose it is allowed to interact with a bath and thus becomes an open system. The resulting open system can be described by a non-Hermitian Hamiltonian formulated by where G (out) B is an outgoing Green function in a bath, and V SB (V BS ) is the interaction from the bath (the closed system) to the closed system (the bath) 1 . It should be noted that the domain of H and thus that of its eigenvectors is restricted to the part of the system excluding the bath 1,40 . If we pay attention to two particular eigenstates interacting with each other, with the interaction with the other states negligible, the non-Hermitian Hamiltonian can modeled in a 2-by-2 matrix form as where ǫ i ∈ C(complex) and its eigenvalues are We denote the eigenstates corresponding to eigenvalues E ± as ψ ± , respectively. Typically, we assume the coupling g to be of a real value to simplify the consideration of strong and weak interactions. Under this condition, there is a repulsion in the real part of the energy eigenvalue with a crossing in the imaginary part for 2g > |Im(ǫ 1 ) − Im(ǫ 2 )| . On the other hand, there is a repulsion in the imaginary part with a crossing in the real part for 2g < |Im(ǫ 1 ) − Im(ǫ 2 )| . The former (latter) case corresponds to the strong (weak) interaction. Especially, when Z = 0 , the eigenvalues E ± are degenerate while the eigenfunctions ψ ± coalesce to |ψ EP � ∝ |ψ 1 � + i|ψ 2 � (with a choice of π/2 relative phase) 41 , corresponding to an EP, where |ψ 1,2 � are eigenfunctions with eigenvalues ǫ 1,2 when g = 0 . The EP is a singular point where the transition between the strong and the weak interactions takes place 27,42 .
In the present work, we consider an elliptical dielectric two-dimensional microcavity as our open system. The closed version of this system is integrable and can be associated with a Hermitian Hamiltonian. The off-diagonal elements in Eq. (2) comes then only from the external interaction ( V SB G (out) B V BS ) in Eq. (1) in our system. The domain of eigenmodes should then be restricted to the inside of the ellipse as discussed above.
In order to study EP by two interacting modes, we need two external parameters to vary. We choose n the refractive index of the cavity medium and χ the deformation parameter associated with the major axis a = R(1 + χ) and the minor axis b = R 1+χ . These two parameters can be independently varied and easily controllable in actual experiments. Since functional form of the matrix elements ǫ 1,2 and g on these parameters are not explicitly known, which is usually the case in many open physical systems, we rely on numerical methods to obtain the eigenvalues and their eigenmodes of electromagnetic wave confined in the dielectric cavity.
We obtained the eigenvalues and eigenmodes by solving the Helmholtz equation ∇ 2 ψ + n 2 k 2 ψ = 0 with the boundary element method 43 for TM electromagnetic modes in the elliptical dielectric two-dimensional cavity (in xy plane), where k is the wave number and ψ is the z-component of the electric field. Shannon entropies in dielectric microcavity. The Shannon entropy for a discrete probability distribution ρ i given at N number of different states is defined as with a normalized condition N i=1 ρ i = 1 . Here, we choose the mode intensity pattern inside our cavity as the probability distribution and the N-mesh points for the mode intensity pattern as the N spatial-coordinate states of a fictitious particle in the corresponding billiard or as our N different states. The probability distributions are discretized at the N-mesh points. Note that the Shannon entropy in our case is different from the von Neumann entropy: the former is defined in terms of the probability distributions corresponding to the discretized spatialcoordinate states whereas the latter is defined by the distribution of the eigenvalue spectrum.

Results
We pay attention to two particular modes, which form an EP in the parameter space s = (n, χ) . Their eigenvalues are depicted in Fig. 1 as a function of n and χ in the form of eigenvalue surfaces. In order to display the EP structure more clearly, we consider the eigenvalue offsets, (2020) 10:12551 | https://doi.org/10.1038/s41598-020-69479-w www.nature.com/scientificreports/ average values E AV instead of the eigenvalue themselves E ± . The eigenvalues are presented in kR with k the complex wave number. An EP is located at n EP ≃ 2.9772, χ EP ≃ 0.16657 . The line n = n EP in Fig. 1 separates the two regimes of interactions, i.e., the strong ( n > n EP ) and weak ( n < n EP ) interactions. The mode patterns |ψ ± (x)| 2 of two interacting modes at three representative points (A, B and C) in the parameter space are plotted in Fig. 1c. Note that the mode pattern at the EP (B) has more uniform and chaotic probability distribution than the others (A 1,2 , C 1,2 ). Fig. 2, the Shannon entropies S(ρ; n, χ) of probability density for the two interacting modes around the EP considered in Fig. 1 are plotted in the parameter space. The plots reveal two interesting features, an extreme value in the vicinity of the EP and a nontrivial topological structure around it, resembling that of the imaginary part of energy eigenvalues. For the extreme value, we first note that the Shannon entropy is maximized near the center of interaction at a fixed refractive index n in both weak and strong interaction regimes. It is because the coherent superposition of eigenfunctions in either weak or strong interaction regime makes the intensity distribution more uniform. The dotted black arrows in Fig. 2a suggest that the trace of these maximum points would reach a peak of S(ρ) ≃ 7.89 in the vicinity of the EP. We can understand this feature by recalling that the mode function at an EP is given by ψ EP ∝ ψ 1 + iψ 2 , where both ψ 1 and ψ 2 can be approximately described by real wavefunctions far from the EP. As a result, the nodes of the intensity distribution occur only where two mode functions vanish simultaneously, which is extremely rare, giving rise to a distribution with a weak contrast and increased uniformity, though chaotic in pattern, and hence an increased Shannon entropy. On the other hand, when an avoided crossing occurs, we have ψ ± ∝ ψ 1 ± ψ 2 , which still can give a clear node structure with a strong contrast or a smaller Shannon entropy. A similar consideration can be made for mode crossing.

Shannon entropies in the vicinity of an ep. In
A close examination reveals that the maximum Shannon entropy occurs slightly off the EP. Magnified views of the Shannon entropies in the vicinity of the EP, Fig. 3a,c, show that only the mean values (dashed gray lines) of the Shannon entropies of two interacting modes show a maximum at the EP. Individual Shannon entropies either exhibit a peak and a dip structure, respectively, in the weak interaction region ( n < n EP ) or show rapid transitions from one branch to another in the strong interaction region ( n > n EP ) in the vicinity of the EP. In Fig. 3a, the Shannon entropies S(χ) are plotted as a function of χ with the refractive index fixed at n − = n EP − δn (weak interaction regime) with δn = 1 × 10 −10 . In this case, mode crossing without exchange of mode patterns results in the repulsion of Shannon entropies with an extreme local minimum and a local maximum, respectively,  Fig. 2b resemble the imaginary part of the energy eigenvalue surfaces in Fig. 1b. Interestingly, the slopes Ṡ (χ) ≡ ∂S ∂χ | n=n ∓ of the Shannon entropies in Fig. 3b,d become divergently large as the EP is approached in both n − and n + cases. S(n) and Ṡ (n) at χ ± = χ EP ± δχ with δχ = 10 −10 behave in a similar way.

Discussion
Diverging slopes of Shannon entropies very near an ep. In order to understand the behaviors of the Shannon entropy and its divergent slope very near the EP, let us consider the Shannon entropy as a function of small perturbation ǫ in the vicinity of the EP by using the Newton-Puiseux series 44 . The eigenvalue equations in the vicinity of an EP can be written as (5) (H EP + ǫH 1 + ···) |ψ EP � + ǫ  where c = √ �φ EP |H 1 |ψ EP � with �φ EP | the adjoint of |ψ EP � and J EP (x) is the Jordan vector at the EP (see "Methods"). Note that the probability density is given by ρ(x) ≡ |ψ(x)| 2 for the electromagnetic eigenmode ψ(x) of a dielectric microcavity. Consequently, the Shannon entropy near the EP can be expanded to the lowest order of ǫ as and its derivative as and therefore dS ± (ǫ) dǫ → ∓∞ as ǫ → 0 . Equations (7) and (8) explain the branching of the Shannon entropies at the EP with divergent slopes as shown in Fig. 3. The fact that the slopes diverge at the EP indicates the Shannon entropy would change rapidly under perturbations in the vicinity of the EP. These property can be used to build a sensor operating at an EP by monitoring the eigenmode distribution.
It is noteworthy that the separation (repelling) of two Shannon entropies is larger for χ < χ EP (also for n < n EP although not shown) than for χ > χ EP (or n > n EP ) in Fig. 3a,c. This is due to the fact that the strong (weak) interactions between two modes induce the strong (weak) mixture of their eigenfunctions, resulting in the more (less) similar Shannon entropies. Consequently, the global maximum (S max ∼ = 7.89996) of the Shannon entropy occurs toward the doubly weak interaction region specified by χ < χ EP and n < n EP as shown in Fig. 4. The displacement of the global maximum of Shannon entropy from the EP is extremely small, in the order of �χ/χ EP ≃ 3 × 10 −4 and �n/n EP ≃ 7 × 10 −4 . The global maximum of the average of two Shannon entropies occurs exactly at the EP as already discussed in Fig. 3. www.nature.com/scientificreports/ These two observations support our conjecture that the mode distribution at the EP might contain the largest information contents or exhibit the most uniformly complex spatial patterns, roughly speaking. The statement is true for the average Shannon entropy and approximately true for the global maximum. Based on this confirmation, one may then legitimately ask the relation of the most uniformly complex mode pattern at the EP and the increased vacuum fluctuations occurring in the EP mode. The answers to such investigation, which is beyond the scope of the present work and is thus left for a future study, would then shed light on the current conflicting experimental observation of an increased linewidth 29 to the classical theory predicting no such enhancement 30 . topological structure of the Shannon entropy in the vicinity of an ep. For the nontrivial topological structure of the Shannon entropy seen in Fig. 2a,b, we note that the two cyclic variations are required for the Shannon entropy values to return to the original values on the Shannon entropy surfaces, just like the imaginary part of the complex energy surfaces shown in Fig. 1b. The surface discontinuity is exhibited along the line n ≃ n EP for χ > χ EP -let us call this line the interaction branch (IB)-for both modes in Fig. 5a,b. This feature can be quantified by δS(ρ) = S(ρ; n − , χ) − S(ρ; n + , χ) as shown in Fig. 5c, where δS(ρ) remains almost zero for χ < χ EP whereas it increases significantly for χ > χ EP . The discontinuity of the Shannon entropy surfaces across the IB is directly related to the exchange of the Shannon entropy as well as the mode pattern exchange.
This exchange property can be further quantified by introducing the relative entropy or the Kullback-Leibler (KL) divergence, which is a measure of the distance between two probability distributions of a random variable 45 . The KL divergence from Q to P, D KL P Q , is defined by The KL divergence for the two interacting modes along the n ± lines, respectively, is plotted in Fig. 5d. It is seen that the KL divergences are almost the same when χ < χ EP and they become zero at the EP. However, their difference D w,s KL becomes larger across the interaction branch when χ > χ EP . These results are consistent with the fact that the mode patterns as well as the Shannon entropies in the weak interaction regime are not exchanged whereas those in the strong interaction regime are exchanged.

Methods
Series expansion of Shannon entropy and its derivative near an ep. We performed the series expansion of the eigenvectors and eigenvalues near an EP following the method described in Ref. 44 . At an EP, two eigenvectors coalesce to one eigenvector |ψ EP � with an eigenvalue EP .

Its adjoint satisfies
For completeness at the EP, we can define the so-called Jordan vector |J EP � with the relation and its adjoint by   In order to obtain the expressions for the eigenvalues and eigenvectors near the EP, let us consider the Taylor expansion of the Hamiltonian in terms of a small perturbative parameter ǫ ≪ 1 near the EP.
Note that the eigenvalues and eigenvectors in the vicinity of EP can be expanded in fractional powers of ǫ (Newton-Puiseux series). Hence, the eigenvalue equations in the vicinity of EP can be written as Equating terms corresponding to different powers of ǫ , we obtain the following relations for the leading orders.
(H EP |ψ 2 � + H 1 |ψ EP �) =( EP |ψ 2 � + 1 |ψ 1 � + 2 |ψ EP �).   Now let us calculate the Shannon entropy and its derivative. Shannon entropy in our case is defined as for the probability density ρ(x) ≡ |ψ(x)| 2 of a classical eigenmode ψ(x) of a two-dimensional dielectric microcavity. Near an EP, the probability density ρ ± (x i , ǫ) = |ψ ± (x i , ǫ)| 2 are given by up to the lowest order of ǫ . Then, the Shannon entropy in the vicinity of the EP can be written as where S EP ≡ − N i=1 |ψ EP (x i )| 2 log |ψ EP (x i )| 2 and W 1 ≡ Re N i=1 cψ * EP (x i )J EP (x i ) log |ψ EP (x i )| 2 + 1 . Hence, the dominant term �S(ǫ) for the separation of the Shannon entropy at EP is given by The derivative of the Shannon entropy with respect to ǫ is given by Using as ǫ → 0 , we obtain As a result, dS ± (ǫ) dǫ → ∓∞ as ǫ → 0 . Figure 3 shows these features clearly.