Entropic comparison of Landau–Zener and Demkov interactions in the phase space of a quadrupole billiard

At first, let us briefly recapitulate the feature of the Demkov interaction, comparing with the well-know Landau–Zener interaction. The formula associated with Landau–Zener interaction is a solution governing the transition dynamics of a two-level quantum system JE29 . Thus, the eigenvalue trajectories of the Landau–Zener interaction are just solutions of two by two Hamiltonian ($\textsf{H}_{LZ}$). For the simplification, the difference between two eigenvalues are linearly proportional to the time: $\Delta E=E_{1}-E_{2}=\xi t$, and under this condition, the probability of a diabatic transition is given by $T_{LZ}=\exp^{-2\pi\eta}$ with $\eta=\frac{v^{2}}{\hbar|\xi|}$ JE29 . Here, $v$ is an off-diagonal term of $\textsf{H}_{LZ}$. It should be noticed that the two-level quantum system in a real physical system can be justified only when the magnitude of the level repulsion is so small compared with the distance to the other levels.

On the other hand, the Demkov interaction can take places in the opposite case to the Landau–Zener interaction, i.e., the magnitude of the level repulsion is much larger compared with the distance to the other levels so that the overall shapes of interacting two modes look so broad YV68 . Moreover, this condition indicates that the Demkov interaction is generally not restricted to two-level systems, yielding non-isolated avoided crossings FF98 ; JJ17 because the ending part of one of the interacting two level can interact with other levels in general. The perturbed term of Hamiltonian for Demkov interaction ($\textsf{H}_{D}$) is given by $\beta t\left|\varphi\right>\left<\varphi\right|$ with $\beta>0$ and $t$ as a parameter. Here, $\left|\varphi\right>\left<\varphi\right|$ is a projection operator onto a zero state (initial state) $\left|\varphi\right>$. Hence, the eigenvalue equation of $\textsf{H}_{D}$ is given by $[(\textsf{H}_{0}-E)+\beta t\left|\varphi\right>\left<\varphi\right|]\left|\psi\right>=0$ with a non-perturbed Hamiltonian $\textsf{H}_{0}$. Accordingly, the eigenvalues are poles of the function $\left<\varphi\right|G(E)\left|\varphi\right>$ on a contour integration, where the $G(E)=(\textsf{H}_{0}-E)^{-1}$ is a resolvent operator. See the details in the reference YV68 . Furthermore, the transition probability from the zero state $\left|\varphi\right>$ to another state $\left|n\right>$ with an energy larger than $E$ is given by $T_{D}=\exp(-2\pi\beta^{-1}\sum_{E_{n}<E}\textsf{H}_{n}^{2})$ with $\textsf{H}_{n}=\left<n\right|\textsf{H}_{D}\left|0\right>$ YV68 . Thus, $T_{D}$ goes to $1$ as $\beta$ goes to infinity.

Implementing Demkov interaction in a quantum billiard system is not addressed well mathematically or quantitatively up to now. Instead of that, indirectly, Fermi resonance YH15 ; JJ17 and RAT KY15 ; FR19 have been studied in a quantum billiard under the Demkov interaction. Thus, we exploit the Fermi resonance and RAT to investigate the features of the Demkov interaction in this paper.

The avoided crossings associated with classical resonance in the phase space can be formulated in terms of Fermi resonance. The Hamiltonians of two eigenstates can be described by $H(I_{1},I_{2})$ and $H(I^{\prime}_{1},I^{\prime}_{2})$ where the $I_{i}$ and $I^{\prime}_{i}$ are action variables for the two eigenstates. Hence, the crossing of a pair of eigenstates indicates semiclassically $H(I_{1},I_{2})=H(I^{\prime}_{1},I^{\prime}_{2})$. When $|I_{i}-I^{\prime}_{i}|$ is small, we can expand $H(I^{\prime}_{1},I^{\prime}_{2})$ around $I^{\prime}_{i}=I_{i}$ to yield: $H(I^{\prime}_{1},I^{\prime}_{2})\simeq H(I_{1},I_{2})+(I_{1}-I^{\prime}_{1})\omega_{1}+(I_{2}-I^{\prime}_{2})\omega_{2}+$…, where $\omega_{i}=\partial H/\partial I_{i}$ are the frequencies associated with the actions $I_{i}$. Substituting the semiclassical quantization condition $H_{i}=(n_{i}+\frac{\alpha_{i}}{4})$ with Maslov index $\alpha_{i}$ result in the relation: $(n_{1}-n^{\prime}_{1})\omega=(n_{2}-n^{\prime}_{2})$ YH15 ; JJ17 . If the winding number $\omega_{1}/\omega_{2}$ is rational, the corresponding orbit is periodic, and it gives rise to the stable orbit or unstable orbit (i.e., scar).

The RAT induces an enhanced interaction of unperturbed modes (WGMs) that live along nearby invariant tori, owing to nonlinear resonance chains located between unperturbed modes when the quantum number of unperturbed modes differ in a multiple of the order of the resonance chain (selection rule) OP01 ; OP02 ; KY15 , and the selection rules are associated with Fermi resonance FF98 ; CY17 . Hence, RAT also implies a leading to the wavefunction localization on the periodic orbits associated with the nonlinear resonance chain DM11 ; D14 .

It should be stressed that because both of the Fermi resonance and the RAT need well-defined classical resonance chains or invariant tori, and also quantum numbers, the Demkov type interactions take place in a regular or mixed phase space rather than a fully chaotic phase space.

Let us briefly introduce a dynamical billiard in a classical and quantum system, respectively, before addressing the interactions in the quantum billiard in detail. A dynamical billiard is a dynamical system in which particles move through without loss of energy as a straight line. Moreover, when the particles hit the boundary, they reflect from it without loss of energy and all reflections are specular, i.e., the incident angle just before the collision is equal to the reflected angle just after the collision. The classical Hamiltonian $H$ for describing a particle of mass $m$ moving freely without friction in billiard system ($\Omega$) is $H(q,p)=\frac{p^{2}}{2m}+v(q)$ where $v(q)$ is zero inside the billiards and infinity otherwise: $v(q)=0$ when $q\in\Omega$ and $v(q)=\infty$ when $q\notin\Omega$. This infinity potential ensures a specular reflection on the boundary of the billiard system.

For a dynamical billiard in quantum systems, the Hamiltonian equations is replaced by the time independent Schrödinger equation: $-\frac{\hbar^{2}}{2m}\nabla^{2}\psi_{n}(q)=E_{n}\psi_{n}(q)$. The potential $v(q)$ given above lead to the Dirichlet boundary conditions: $\psi_{n}(q)=0$ for all $q\notin\Omega$ and also to the Helmholtz equation $(\nabla^{2}+k^{2})\psi=0$ with a wave number $k=\frac{1}{\hbar}\sqrt{2mE_{n}}$.

If the classical billiards belong to integrable systems such as a circular and rectangular system, then the corresponding quantum billiards are completely solvable and the spectral properties coincide with those of Poissonian random numbers. On the other hand, when the classical billiard is chaotic, then the corresponding quantum billiards are generally not exactly solvable and the spectral properties coincide with those of random matrices from the Gaussian ensembles F10 .

In this study, we considered a closed quadrupole cavity as a model for a chaotic quantum billiard in a Hermitian system, because this quadrupole billiard has been extensively exploited for studying the physical phenomena in the phase space PN03 ; WLL+21 and for the candidate of deformed micro cavity lasers ZYX+09 ; CW15 . Its geometrical boundary shape is described by $\rho(\theta)=(1+\epsilon\cos(2\theta))$. Here, $\theta$ is an angle in polar coordinates and $\epsilon$ is the deformation parameter. We numerically calculated the eigenvalues and their eigenfunctions using the boundary element method J03 by solving the Helmholtz equation $\nabla^{2}\psi+n^{2}k^{2}\psi=0$ for transverse-magnetic modes that satisfy the Dirichlet boundary conditions. Here, $\psi$ is the vertical component of the electric field, $k$ denotes the vacuum wavenumber, and $n=3.3$ is the refractive index of the cavity. The eigenvalue trajectories in the region $k\in[9.1,10.1]$ are plotted as a function of the deformation parameter $\epsilon$ in Fig. 1.

The red solid curve I and blue solid curve II are a pair of eigenvalue trajectories for the Demkov interactions. We cannot easily identify avoided crossing between these two curves. However, the black solid curve in the lower inset panel in Fig. 1 resolves this problem. It is the relative difference between I and II, and is minimized at $\epsilon\simeq 0.055$. This indicates that the coupling strength of the Demkov interactions is maximized at $\epsilon\simeq 0.055$ JJ17 . The green solid curve III and orange solid curve IV in the upper inset panel are a pair of eigenvalue trajectories for the Landau–Zener interactions. We observe that the Landau–Zener interaction occurs in a highly narrow parameter region compared with the Demkov interaction. Note that the center of the avoided crossing is located at $\epsilon\simeq 0.0603$.

The figures on the left panels in Figs. 3 and 4 are the intensities of a few of the representative eigenfunctions as indicated in Fig. 1. In the case of the Landau–Zener interactions (Fig. 3), the eigenfunctions are mixed around the center of the avoided crossing and exchanged across the avoided crossing KS18 . In contrast, the panels on the left in Fig. 4 for the Demkov interaction show a highly disparate aspect. The whispering galley-type modes (WGMs) with ($l=2,m=22$) and ($l=1,m=26$) at $\epsilon=0$ (i.e., Fig. 4(g) and Fig. 4(j)) are deformed to period-$4$ orbits, diamond (i) and rectangular (l)-shaped orbits, respectively. This is the result of the Fermi resonance YH15 . That is, the quantum number difference equals the period of the orbits: $|\Delta N_{l}|,|\Delta N_{m}|=(1,4)$. Here, $N_{l}$ and $N_{m}$ are the radial and angular quantum numbers, respectively.

The classical billiard dynamic is completely described by using the bounce map defined by the Poincaré-Birkhoff coordinates $(s,p)$. Here, $s$ is a boundary length of the cavity and its conjugate momentum $p=\textrm{sin}(\chi)$ and $\chi$ is the incident angle with respect to the normal to the boundary wall MM76 ; M89 . The Fig. 2 shows an example of Poincaré-surface of section (classical phase space) defined in terms of Poincaré-Birkhoff coordinates at $\varepsilon=0.05$. In addition, the $(s,p)$ fulfill the Poisson bracket relation: $\{s,p\}=1$. Thus, to study quantum billiard dynamics, it is natural to employ the Husimi distributions K40 ; BG93 in the Poincaré-Birkhoff coordinate defined below:

with the angular momentum-dependent weighting factor $F=\sqrt{n\cos(\chi)}$ and $h^{\prime}(s,p)=\int ds^{\prime}\partial\psi(s^{\prime})\xi(s^{\prime};s,p)$ AS04 . Here, $\partial\psi(s^{\prime})$ is the normal derivative of the boundary wavefunction, and $\xi(s^{\prime};s,p)$ is the minimal-uncertainty wave packet (coherent state) centered at a position $(s,p)$ in the phase space. The minimum-uncertainty wave packet is given by

Here, $L$ and $k$ are the total length of the boundary and wave number, respectively. We set the aspect ratio factor as $\sigma=\frac{1}{\sqrt{(\sqrt{2}/k)}}$ on this coherent state.

The right panels in Fig. 3, denoted as (a), (b), (c), (d), (e), and (f), are the Husimi distributions corresponding to each eigenfunction in the left panels for the Landau–Zerner interactions. The Husimi distributions displayed in Fig. 3(a) and (d) are exchanged with each other across the center of the avoided crossing. Meanwhile, those displayed in Fig. 3(b) and 2(e) at the center of the avoided crossing are more intricate than the others (Figs. 3(a), 2(d), 2(c), and 2(f)).

On the other hand, the right panels (g), (h), (i), (j), (k), and (l) in Fig. 4 show a highly distinct tendency. We can easily observe that the WGM modes ((g), (j)) are transformed to the period-$4$ orbits ((i), (l)). The bright 4 spots of Husimi distributions in Fig. 4(i) and (l) support this observation. Moreover, we also observe that (h) and (k) at the center of the avoided crossing are transient modes that settle down into a diamond-shaped stable period-$4$ orbit as well as a rectangle-shaped unstable period-$4$ orbit corresponding to a scar mode, respectively. We quantitatively analyze these features within the framework of information theory in Sec. IV.

The green dashed line at the bottom of Fig. 5(a) is the overlap of eigenfunctions $O_{L}^{\psi}(n,m)$, whereas the orange solid line in the Fig. 5(a) is the overlap of the Husimi distributions $O_{L}^{\tilde{H}}(n,m)$. Both of these are for the Landau–Zener interaction in Fig. 3. We observe that the value of the green dashed line at the bottom is zero throughout the interaction. This verifies the orthogonality in the Hermitian system. In contrast, the overlap of the Husimi distributions shows a distinct behavior. That is, it is maximized at the center of the avoided crossing ($\epsilon=0.0603$). This indicates that the degree of occupation in the same region in a phase space is maximized at the center of the interaction.

In Fig. 5(b), the blue dashed line at the bottom is the overlap of wavefunctions $O_{L}^{\psi}(n,m)$, whereas the red solid line is that of the Husimi distributions $O_{L}^{\tilde{H}}(n,m)$. We can easily observe that the orthogonality of eigenfunctions is still valid on the Demkov interaction because the value of the blue dashed line at the bottom of Fig. 5(b) is also zero throughout the interaction. However, the overlap of the Husimi distributions $O_{L}^{\tilde{H}}(n,m)$ exhibits a behavior different from that of the orange solid line in Fig. 5(a). That is, the center of the avoided crossing is located at $\epsilon\simeq 0.055$, whereas the maximum value of $O_{L}^{\tilde{H}}(n,m)$ is located at $\epsilon\simeq 0.09$. These two values are significantly different from each other. We analyze these features by exploring Shannon entropy in the next section.

The Shannon entropies of the two eigenfunctions in a phase space for the Landau–Zener interaction are plotted in Fig. 6(a). These are maximized around the center of the avoided crossing ($\epsilon\simeq 0.0603$) and exchanged across the avoided crossing. Note that these two Shannon entropies have similar maximum values, i.e., $\emph{{S}}_{\textnormal{max}}\simeq 8.57$ for the green dashed line and $\emph{{S}}_{\textnormal{max}}\simeq 8.56$ for the yellow solid line. These features are in good agreement with the results of the Shannon entropies in a configuration space KS18 . Moreover, our previous study verified that the Shannon entropy as well as inverse participation ratio and image contrast can measure the localization properties of eigenfunctions on configuration space KJ21 . Thus, the maximum values of Shannon entropies around the center of the avoided crossing can directly indicate that the Husimi distributions are maximally delocalized in the phase space around the center of the avoided crossing.

Therefore, we can explain why we obtain the maximum value of the overlap of the Husimi distributions $O_{L}^{\tilde{H}}(n,m)$ at the center of avoided crossing. The increased delocalization or diffusion of the two Husimi distributions causes these two distributions to occupy a larger space of the same region in a phase space. An examination of the patterns inside the two boxes in Figs. 3(b) and 3(e) reveals this fact. In this manner, the overlap of Husimi distributions can capture the effects of the avoided crossing, even though the overlap of the eigenfunction cannot do so.

Figure 6(b) displays the Shannon entropies of the two eigenfunctions in a phase space for Demkov interaction. The behaviors of these Shannon entropies differ significantly from those in Fig. 5(a). They decrease (rather than be maximized) as they go through the center of avoided crossing with increased deformation. The Husimi distributions of the eigenfunctions at $\epsilon=0.0$ (circles) have rotational symmetries, so that these distributions spread along the boundary. However, once the deformation increases, the Husimi distributions start to move toward the periodic orbits owing to the Fermi resonance YH15 . This process continues until they settle down into periodic orbits. This can be revealed by the local minimum values of Shannon entropies at $\epsilon\simeq 0.1$ because the Shannon entropy can also measure the degree of (de)localization KJ21 . Here, we can quantitatively state the main difference between the Landau–Zener interaction and Demkov interaction. The former induces delocalization in a phase space, whereas the latter induces localization in a phase space.

Moreover, the values of the blue solid curve are larger than those of the red dotted curve throughout the process, not being exchanged with the red dotted-curve. In other words, the values of the Shannon entropy of the stable period $4$ are larger than those of the unstable period $4$ (scar). This is because the stable period-$4$ orbit settles down on the island structures having finite areas, whereas the unstable period-$4$ orbit corresponding to scar settles down on the points of measure zero. Therefore, we can presume that the Shannon entropy in a phase space can be a new indicator of scar formation.

Finally, these behaviors explain why the maximum values of $O_{L}(\tilde{H}_{(n,m)})$ are obtained in the transient regime. Two Husimi distributions of eigenfunctions (WGM) in the circle ($\epsilon=0$) are placed at highly distinct values of the $p=\textrm{sin}(\chi)$ (center of $p$: $p_{c}\simeq 0.67$ and $p_{c}\simeq 0.82$, respectively) as shown in Figs. 4(g) and (j) such that these two WGMs occupy distinct striped regions in the phase space. However, when the Demkov interaction occurs, these two Husimi distributions shift toward the period-$4$ orbits. These shifts lead to the occupation of a larger space of the same region in a phase space. It should be noted that the maximum value of $O_{L}(\tilde{H}_{(n,m)})$ is attained in the transient regime at $\epsilon=0.09$, rather than at $\epsilon=0.1$, where the two Husimi distributions are maximally localized on the stable period-$4$ orbit and unstable period-$4$ orbit, respectively. In this manner, the phase space overlap and Shannon entropy in a phase space can be related to each other under the influences of avoided crossing.

In addition, it should be noticed that RAT partly can explain the results above. The value of $O_{L}\simeq 0.0$ at $\epsilon=0.0$ comes from the condition that one unperturbed mode (with $p_{c}\simeq 0.82$) must be placed above the $t=4:r=1$ resonance chain with $P_{{t=4}:{r=1}}=\cos(\pi r/t)=1/\sqrt{2}\simeq 0.707$ while the other one (with $p_{c}\simeq 0.67$) must be placed below the $P_{{t=4}:{r=1}}$ so that RAT can occur. Here, the $P_{{t=4}:{r=1}}$ corresponds to the adiabatically invariant curve interpolating through the nonlinear resonance FR19 . The increased value of the $O_{L}$ can be due to the increased coupling strength by increased area of separatrix of the resonance chain. However, the center of avoided crossing is located at $\epsilon\simeq 0.055$ far from $\epsilon=0.09$ (maximum of value of the $O_{L}$) or $\epsilon=0.1$ (minimum of value of the S). It is also worth mentioning that the Demkov interaction does not always induce a scar formation. For example, regarding an excited-state quantum phase transition, the Husimi distribution is more localized at the center of avoided crossing with a transition between two regular states, i.e., tori of rotational motion and that of the libration motion, without giving rise to any scar state IE21 . Those problem should be addressed in the future works.

Lastly, the absolute value of the overlap of Husimi distributions $O_{L}^{\tilde{H}}(n,m)$ for the Demkov interaction is ten times smaller than that of $O_{L}^{\tilde{H}}(n,m)$ for the Landau–Zener interaction. This is because there is negligible resemblance in the Demkov cases, whereas we can observe resemblance in the Landau–Zener case shown in the two boxes of Fig. 3(b) and 3(e).

Figure 7(a) shows the Fisher information $I(\epsilon$) as a function of the deformation parameter $\epsilon$ in the case of the Landau–Zener interactions. $I(\epsilon$) has relatively high marginal values (of the order of $10^{0}$) well away from the avoided crossings ($\epsilon\simeq 0.059,\epsilon\simeq 0.061$). However, it increases dramatically as the center of interaction ($\epsilon\simeq 0.0603$) is approached and attains values of the order of $10^{5}$. This directly indicates that the Landau–Zener interaction results in an extensive variation in the eigenfunctions over a highly narrow parameter range. Consequently, we can quantify the degree of the effects of the avoided crossing on the eigenfunctions depending on the system parameter. Furthermore, note that $I(\epsilon$) of the two interacting modes shows no significant exchange across the avoided crossing, unlike the Shannon entropies in Fig. 6. This implies that the degree of variations in eigenfunctions under an interaction on the system parameter is similar, whereas the degree of eigenfunctions localization under an interaction on the system parameter is not.

Contrary to the Landau–Zener interaction, the Fisher information of the Demkov interactions exhibits various features as shown in Fig. 7(b). First, the absolute values of $I(\epsilon$) on the Demkov interactions (which are of the order of $10^{1}\sim 10^{2}$) are extremely small compared to that on the Landau–Zener interactions. Such small values quantitatively support that the Demkov interaction occurs over a broad region, whereas the Landau–Zener interactions occur over a highly narrow region.

What is more noteworthy is that the Fisher information $I(\epsilon$) on Demkov interaction gradually (almost linearly) increases across the avoided crossing rather than being maximized at the center of the avoided crossing ($\epsilon=0.055$). This implies that the rate of variations in the eigenfunctions increase across the center of the avoided crossing. Subsequently, the Fisher informations are barely changed in the range $0.09\leq\epsilon\leq 0.11$. This behavior can be attributed to their settling down into the periodic orbits. This is consistent with the results shown in Fig. 6(b), i.e., the Shannon entropies on the Demkov interactions are locally minimized around the $\epsilon\simeq 0.1$. The substantial increase in $I(\epsilon$) in the region $\epsilon\geq 0.12$ is owing to leakage from the periodic orbits shown in the insets of Fig. 7(b). We can check that there are considerable amount of probability below the $P_{4:1}\simeq 0.707$ at $\epsilon=0.125$.