Coherent Control of Diabolic Points of a Hermitian Hamiltonian in a Four-Level Atomic System Using Structured Light Fields

Abstract

A four-level atomic medium is used to manipulate the diabolic points of the Hermitian Hamiltonian using driving fields of structured light. The diabolic points of the fourth, third, and second orders are observed by the real and imaginary parts of the eigenvalues of the Hermitian Hamiltonian. The diabolic points and degeneracy regions are studied with variation in Rabi frequencies, detuning, and topological charges. The structured light has a key impact on diabolic points. By changing the topological charges, the number of diabolic points and the degeneracy regions are changing. The imaginary part of eigenvalues shows fourth-order diabolic points. At topological charge $\mathsf{\ell }=even$, the real part of eigenvalues does not show higher-order diabolic points. The obtained results of the diabolic point are helpful in the fields of deformation space, entanglement physics, optomechanical systems, and crystal optics.

1. Introduction

The diabolic point (DP) is the degeneracy in the eigenvalues of Hermitian Hamiltonian of a medium where two or more eigenvalues intersect. The DP is different from an exceptional point (EP), where both eigenvalues and eigenvectors intersect. This concept originates from quantum mechanics and classical wave physics, mainly in systems governed by Hermitian Hamiltonian. The concept of diabolic point was first introduced by Berry and Wilkinson [1]. Using structured light can engineer diabolic points in photonic systems. By adding an orbital angular momentum or topological charge term to a Hamiltonian in the form of Rabi frequency, can modify level crossings, leading to diabolic degeneracies [2,3].

The diabolic points are classified according to their order as first-, second-, third-, and higher-order diabolic points. The order refers to the degree of interaction that causes the degeneracy between energy levels. In a second-order diabolic point, only two eigenvalues intersect at one point. The third-order diabolic point refers to the intersection of three eigenvalues, and higher order denotes how many eigenvalues are coinciding at a single point. The field of Hermitian, i.e., diabolic, points received special interest for its wide range of applications in different fields and technologies. Systems close to diabolic points show extreme sensitivity to perturbations, valuable for precision measurements. Diabolic points ensure stable, mode-locked laser emission in microresonators. Diabolic points have applications in quantum exciton-polariton [4], entanglement physics [5], fully automated train control [6], periodic plasmonic nanostructures [7], deformation space [8], complex matrices [9], crystal optics [10], quantum evolution [11], magnetization in an atomic spin chain [12], magnetic molecules [13], conical intersections [14], magnetic spectrum [15], molecular magnets [16], and optomechanical systems [17]. Because of these numerous applications in the field of diabolic points, it is an interesting topic in this era. Many researchers have studied the diabolic point using different methods and models.

Optical gratings are broadly used to control the energy levels of atoms by generating periodic light patterns, known as optical lattices. Optical gratings allow researchers to control how atoms move, how their energy states split, and how they interact with light [18]. This method gives fine control over atomic states, level crossings, and coherence. In our study, we use structured light, which also has a spatially varying phase and intensity. This variation forms position-dependent Rabi frequencies, similar to how optical gratings create periodic potentials. In this manner, structured light can act like a dynamic optical lattice, allowing us to control the energy states and obtain different types of diabolic points. So, the physical idea is related; both approaches use spatial light structures to control atomic energy levels [19,20].

In the modern era, structured light fields have emerged as an important tool in controlling light–matter interactions due to their spatially varying phase and amplitude profiles [21,22]. Unlike simple Gaussian beams, structured light [23,24] can carry orbital angular momentum and exhibit nontrivial topological charge, enabling enhanced spatial control over atomic transitions. In this work, we consider a vortex-type structured light field whose Rabi frequency is defined in Equation (18), where r and $\theta$ are the radial and azimuthal coordinates, respectively, W is the beam waist, and $l_i$ is the topological charge. This expression describes a beam with a ring-shaped intensity distribution and a helical phase front. While this form shares structural similarity with Laguerre–Gaussian (LG) modes, it is not restricted to the orthogonal basis of LG solutions but rather represents a more general class of structured fields. The Laguerre–Gaussian beam [25], characterized by an azimuthal phase factor $e^{-i{tan}^{-1}(y/x)}$ and a spatial intensity profile dependent on the radial axis $\sqrt{x^2+y^2}$, scaled by the beam waist W. The field displays a Gaussian envelope $exp\left[-{\displaystyle \frac{x^2+y^2}{W^2}}\right]$, ensuring transverse confinement. This form corresponds to a vortex beam carrying orbital angular momentum and has been widely employed to manipulate light–matter interactions in atomic systems.

The coherent manipulation of diabolic points have extensively studied in quantum optics. For example, the diabolic points at the resonance of two coupled microdisks has been demonstrated by controlling backscattering directions through cavity size optimization [26,27,28]. The observed diabolic points match theoretical predictions, considering backscattering effects. These points arise in active optical structures, enabling geometric phase control. Verevkin et al. [29] worked on the dynamical meaning of the diabolic points. Sadovskii [30] and Wiersig [31] examined the separation between diabolic points. Arkhipov et al. [32] studied dynamically crossing diabolic points while encircling exceptional curves. Sadovskii et al. [33] worked on diabolic point in molecular energy spectrum manifestation. Berry [1] studied diabolical points in the spectra of triangles. Whitaker [34] studied shining light on diabolic points. Krivtsun et al. [35] examined diabolic points in spherical top molecules’ rovibrational energy spectra. Rowley et al. [36] worked on level interactions and diabolic points. Bid et al. [37] investigated higher-order Weyl semimetals with surface diabolic points.

Here, a four-level atomic medium is used to examine the diabolic points of Hermitian Hamiltonian using driving fields of structured light. The multiple diabolic points of the second, third, and fourth orders of the Hermitian Hamiltonian of a four-level atomic medium are controlled by the real and imaginary parts of eigenvalues. The modified work have significance for high-technological applications of optics, complex matrices, and magnetic spectra.

2. Four-Level Atomic System and Derivation of Eigenvalues

Consider a four-level Hermitian atomic system having one ground state and three excited states as shown in Figure 1. The ground state $|1\rangle$ and state $|2\rangle$ are coupled by control field $E_1$ of Rabi frequency $J_1$ with detuning ${\delta }_1$. The ground state $|1\rangle$ and state $|3\rangle$ are coupled by control field $E_2$ of Rabi frequency $J_2$ with detuning ${\delta }_2$. The ground state $|1\rangle$ and state $|4\rangle$ are coupled by probe field $E_p$ of Rabi frequency $J_p$ with detuning ${\delta }_p$. The excited state $|2\rangle$ and state $|3\rangle$ are coupled by controlled field $E_3$ of Rabi frequency $J_3$ with detuning ${\delta }_3$. The excited state $|2\rangle$ and state $|4\rangle$ are coupled by controlled field $E_4$ of Rabi frequency $J_4$ with detuning ${\delta }_4$. The excited state $|4\rangle$ and state $|5\rangle$ are coupled by controlled field $E_5$ of Rabi frequency $J_5$ with detuning ${\delta }_5$.

In this theoretical work, the proposed Hermitian four-level atomic system is implemented in a cavity-based cold atom platform. A suitable setup involves ultracold $R{b}^{87}$ atoms confined within a cavity, where long coherence times enable decay-free dynamics over the relevant timescales. The model levels can be mapped to specific hyperfine states, such as $|1\rangle \equiv 5S_{1/2},F=1$ and $|2\rangle ,|3\rangle ,|4\rangle \equiv 5P_{3/2}$ sublevels. The required probe and control fields can be introduced into the cavity using structured light beams. In such a model, the effective Hamiltonian parameters, specifically Rabi frequencies and detunings, can be precisely tuned by adjusting the intensity and frequency of the external fields. The cavity supports mode-resolved measurements and the eigenvalue spectrum, including energy crossings and diabolic point formation.

In quantum systems, degeneracies in the energy spectrum can be studied in two concepts, diabolic points and exceptional points. Diabolic points arise in Hermitian systems, where only the eigenvalues intersect, and the eigenstates remain distinct. These points reflect idealized, closed quantum systems without decoherence or decay. On the other side, exceptional points appear in non-Hermitian systems, where both the eigenvalues and the eigenstates join. These typically arise in systems with loss, gain, or decay processes. In this study, we focus on a Hermitian Hamiltonian where no decoherence is present, which is an essential condition for the observation of true diabolic points, including decoherence would render the non-Hermitian Hamiltonian and lead instead to exceptional points. This distinction is explored in recent work [38], where the authors studied exceptional surfaces in a three-level atomic system under Doppler and Zeeman-induced non-Hermitian Hamiltonian.

The Hamiltonian for four-level atomic system can be written as

$$\begin{array}{ccc}\hfill H& =& \hslash \left[{\delta }_1\right|2\rangle \langle 2|+({\delta }_2+{\delta }_3)|3\rangle \langle 3|+({\delta }_p+{\delta }_5+{\delta }_4-{\delta }_3-{\delta }_2)|4\rangle \langle 4|+J_1|1\rangle \langle 2|\hfill \\ & & +J_2|1\rangle \langle 3|+J_p|1\rangle \langle 4|+J_3|2\rangle \langle 3|+J_4|2\rangle \langle 4|+J_5|3\rangle \langle 4|+H.c.].\hfill \end{array}$$

(1)

The Hamiltonian matrix is given is

$$H=\hslash \left(\begin{array}{cccc}0& J_1& J_2& J_p\\ J_1^{\ast }& {\delta }_1& J_3& J_4\\ J_2^{\ast }& J_3^{\ast }& P& J_5\\ J_p^{\ast }& J_4^{\ast }& J_5^{\ast }& Q\end{array}\right)$$

(2)

The eigenvalues are derived from the given Hamiltonian matrix using the secular equation $|H-\lambda I|$, where I is the $4\times 4$ identity matrix. The determinant was computed symbolically by this secular equation, and the resulting quartic polynomial in $\lambda$ was solved to derive eigenvalues. The determinant and then the root equation are provided in the Appendix A.

To prove the Hermiticity of the Hamiltonian under structured light, we considered the matrix in Equation (2), where the complex Rabi frequencies are defined by $J_i$, and $J_i^{\ast }$ denotes their complex conjugates. To take the Hermitian conjugate $H^{†}={\left(H^T\right)}^{\ast }$, first take the transpose of the matrix as

$$H^T=\hslash \left(\begin{array}{cccc}0& J_1^{\ast }& J_2^{\ast }& J_p^{\ast }\\ J_1& {\delta }_1& J_3^{\ast }& J_4^{\ast }\\ J_2& J_3^{\ast }& P& J_5^{\ast }\\ J_p& J_4& J_5& Q\end{array}\right),$$

(3)

as taking the Hermitian conjugate $H^{†}={\left(H^T\right)}^{\ast }$, we obtain

$$H^{†}={\left(H^T\right)}^{\ast }=\hslash \left(\begin{array}{cccc}0& J_1& J_2& J_p\\ J_1^{\ast }& {\delta }_1& J_3& J_4\\ J_2^{\ast }& J_3^{\ast }& P& J_5\\ J_p^{\ast }& J_4^{\ast }& J_5^{\ast }& Q\end{array}\right),$$

(4)

which is the same as H. To confirm the Hermitian nature of the Hamiltonian under spatially structured light, we consider both the matrix structure and the physical properties of the applied fields. The structured Rabi frequencies used in our model contain spatially varying complex phases of the form $e^{i\mathsf{\ell }\theta }$. These complex terms appear in the off-diagonal elements of the Hamiltonian and are always accompanied by their complex conjugates in the corresponding symmetric positions, thereby satisfying the Hermiticity condition $H_{ij}=H_{ji}^{\ast }$. All diagonal elements remain strictly real, satisfying $H_{ii}=H_{ii}^{\ast }$, which is needed for the overall Hermiticity of the system. Along with matrix symmetry, we also verify the spatial behavior of the structured fields. The vortex beams used in our considered model are square integrable, meaning their intensities yield a finite value when integrated over the transverse plane. Specifically, the field profiles satisfy

$$\begin{array}{c}\hfill \int |{\Omega }_j{(r,\theta )|}^{2}rdrd\theta <\infty ,\end{array}$$

that vanish at spatial infinity,

$$\begin{array}{c}\hfill \underset{r\to \infty }{lim}{\Omega }_j(r,\theta )=0,\end{array}$$

ensuring that the fields are localized and physically normalizable. These properties confirm that the Hamiltonian remains Hermitian not only from a matrix algebra perspective but also under the complete spatial structure of the light–matter interaction. The inclusion of field normalization and boundary behavior ensures physical consistency of the model.

The computed eigenvalues from above matrix are given as

$$\begin{array}{cc}& {\lambda }_1=-{\displaystyle \frac{q}{4}}-{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{q^2}{4}}-{\displaystyle \frac{2r_1}{3}}+m+n}-{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{q^2}{2}}-{\displaystyle \frac{4r_1}{3}}-m-n+{\displaystyle \frac{q^3-4q{r}_1+8s}{4\sqrt{\frac{q^2}{4}-\frac{2r_1}{3}+m+n}}}},\end{array}$$

(5)

$$\begin{array}{cc}& {\lambda }_2=-{\displaystyle \frac{q}{4}}-{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{q^2}{4}}-{\displaystyle \frac{2r_1}{3}}+m+n}+{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{q^2}{2}}-{\displaystyle \frac{4r_1}{3}}-m-n+{\displaystyle \frac{q^3-4q{r}_1+8s}{4\sqrt{\frac{q^2}{4}-\frac{2r_1}{3}+m+n}}}},\end{array}$$

(6)

$$\begin{array}{cc}& {\lambda }_3=-{\displaystyle \frac{q}{4}}+{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{q^2}{4}}-{\displaystyle \frac{2r_1}{3}}+m+n}-{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{q^2}{2}}-{\displaystyle \frac{4r_1}{3}}-m-n-{\displaystyle \frac{q^3-4q{r}_1+8s}{2\sqrt{q^2-\frac{8r_1}{3}+4(m+n)}}}},\end{array}$$

(7)

$$\begin{array}{cc}& {\lambda }_4=-{\displaystyle \frac{q}{4}}+{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{q^2}{4}}-{\displaystyle \frac{2r_1}{3}}+m+n}+{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{q^2}{2}}-{\displaystyle \frac{4r_1}{3}}-m-n-{\displaystyle \frac{q^3-4q{r}_1+8s}{2\sqrt{q^2-\frac{8r_1}{3}+4(m+n)}}}}.\end{array}$$

(8)

The suppositions in the above Equations are

$$\begin{array}{c}P={\delta }_2+{\delta }_3,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \hfill \end{array}$$

(9)

$$\begin{array}{c}Q={\delta }_p+{\delta }_5+{\delta }_4-{\delta }_3-{\delta }_2,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hfill \end{array}$$

(10)

$$\begin{array}{c}q=-Q-P-{\delta }_1,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hfill \end{array}$$

(11)

$$\begin{array}{c}{r}_1=QP+Q{\delta }_1+P{\delta }_1-J_1^2-J_2^2-J_3^2-J_4^2-J_5^2-J_p^2,\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill & \hfill & s=-QP{\delta }_1+Q{J}_1^2+Q{J}_2^2+Q{J}_3^2+P{J}_1^2+{\delta }_1{J}_2^2-J_1{J}_3{J}_2^{\ast }-J_2{J}_1^{\ast }{J}_3^{\ast }+P{J}_4^2-J_4^{\ast }{J}_3{J}_5\hfill \\ \hfill & \hfill & -J_4^{\ast }{J}_p{J}_1^{\ast }+{\delta }_1{J}_5^2+J_p^{2}P+J_p^2{\delta }_1-J_p^{\ast }{J}_2{J}_5-J_5^{\ast }{J}_p{J}_2^{\ast }-J_5^{\ast }{J}_4{J}_3^{\ast }-J_p^{\ast }{J}_1{J}_4,\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill & \hfill & t=-QP{J}_1^2-Q{\delta }_1{J}_2^2+Q{J}_1{J}_3{J}_2^{\ast }+Q{J}_2{J}_1^{\ast }{J}_3^{\ast }-J_4^{\ast }{J}_2{J}_5{J}_1^{\ast }+J_4^{\ast }P{J}_p{J}_1^{\ast }+J_4^2{J}_2^2-J_4^{\ast }{J}_3{J}_p{J}_2^{\ast }\hfill \\ & \hfill & +J_5^2{J}_1^2-J_5^{\ast }{J}_1{J}_4{J}_2^{\ast }+J_5^{\ast }{\delta }_1{J}_p{J}_2^{\ast }-J_5^{\ast }{J}_p{J}_1^{\ast }{J}_3^{\ast }+J_p^{\ast }P{J}_1{J}_4+J_p^{\ast }{\delta }_1{J}_2{J}_5-J_p^{\ast }{J}_1{J}_3{J}_5-\hfill \\ & \hfill & J_p^{2}P{\delta }_1-J_p^{\ast }{J}_2{J}_4{J}_3^{\ast }+J_p^2{J}_3^2,\hfill \end{array}$$

(14)

$$\begin{array}{ccc}& & m={\displaystyle \frac{2^{1/3}(r_1^2-3qs+12t)}{3{\left(z+\sqrt{-4{(r_1^2-3qs+12t)}^3+{(2r_1^3-9q{r}_{1}s+27s^2+27q^{2}t-72r_{1}t)}^2}\right)}^{1/3}}},\hfill \end{array}$$

(15)

$$\begin{array}{ccc}& & n={32}^{-{\displaystyle \frac{1}{3}}}{\left(z+\sqrt{{(2r_1^3-9q{r}_{1}s+27s^2+27q^{2}t-72r_{1}t)}^2-4{(r_1^2-3qs+12t)}^3}\right)}^{{\displaystyle \frac{1}{3}}},\hfill \end{array}$$

(16)

$$\begin{array}{ccc}& & z=2r_1^3-9q{r}_{1}s+27s^2+27q^{2}t-72r_{1}t.\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hfill \end{array}$$

(17)

The Rabi frequency in structured light [23,24] form are described as

$$\begin{array}{ccc}\hfill J_i& =& J_i{\left[{\displaystyle \frac{r}{W}}\right]}^{l_i}exp(i{l}_i\theta -{\left[{\displaystyle \frac{r}{W}}\right]}^2),\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill J_i^{\ast }& =& J_i{\left[{\displaystyle \frac{r}{W}}\right]}^{l_i}exp(-i{l}_i\theta -{\left[{\displaystyle \frac{r}{W}}\right]}^2).\hfill \end{array}$$

(19)

Here, W is the beam width, $J_i$ is the coefficient of field strength, and $l_i$ is the winding number, where $i=1,2,3,4,5,p$, which can be positive or negative depending on the twisted direction of the beam. The values of r and $\theta$ are given as

$$\begin{array}{ccc}& & r^2=x^2+y^2,\hspace{1em}\hspace{1em}\hspace{1em}\hfill \end{array}$$

(20)

$$\begin{array}{ccc}& & \theta ={tan}^{-1}\left({\displaystyle \frac{y}{x+r}}\right).\hfill \end{array}$$

(21)

In the next section, the identification of second-, third-, and fourth-order diabolic points is primarily explained by graphical observation of the real and imaginary parts of the eigenvalues of the Hermitian Hamiltonian. However, to confirm it mathematically, we have now introduced both analytical and numerical eigenvalue coalescence metrics that offer a systematic method for classifying the order of DPs. A diabolic point is defined as a parameter value at which two or more eigenvalues of a Hermitian Hamiltonian coalesce while their corresponding eigenvectors remain linearly independent. This makes different diabolic points from exceptional points, where both eigenvalues and eigenvectors coalesce, typically occurring in non-Hermitian systems. Since the Hamiltonian used in this study is Hermitian, only diabolic points are considered. The eigenvalues ${\lambda }_i$$(i=1,2,3,4)$ are obtained by solving the characteristic equation:

$$\begin{array}{c}\hfill {\lambda }^4+q{\lambda }^3+r_1{\lambda }^2+s\lambda +t=0,\end{array}$$

(22)

where the coefficients q, $r_1$, s, and t are functions of the detuning $\left({\delta }_i\right)$ and Rabi frequencies $\left(J_i\right)$, as explicitly defined in Equations (11)–(14). To quantitatively classify the order of a DP, we compute the set of eigenvalue differences:

$$\begin{array}{c}\hfill {\Delta }_{ij}\left(x\right)=|{\lambda }_i\left(x\right)-{\lambda }_j\left(x\right)|,\mathrm{for}i<j,i,j\in \{1,2,3,4\}.\end{array}$$

When ${\Delta }_{ij}\left(x\right)$ gives 0, it indicates that the $i$th and $j$th eigenvalues are degenerate at position $x/\lambda$. The order of the DP is then defined as the number of eigenvalues that simultaneously coalesce.

• Second-order DP: A diabolic point is classified as second-order when exactly two eigenvalues become degenerate, i.e., ${\Delta }_{ij}\to 0$ for a single pair $(i,j)$.
• Third-order DP: A third-order diabolic point occurs when three eigenvalues coalesce simultaneously, satisfying ${\Delta }_{ij},{\Delta }_{ik},{\Delta }_{jk}\to 0$ for distinct $i,j,k$.
• Fourth-order DP: A fourth-order diabolic point is identified when all four eigenvalues become equal, such that ${\lambda }_1={\lambda }_2={\lambda }_3={\lambda }_4$, or equivalently, all pairwise differences ${\Delta }_{ij}\to 0$ for $i<j$.

On the analytical side, we also examine the discriminant $\mathcal{D}$ of the quartic characteristic equation. As the general formula for the discriminant of a quartic (fourth-degree) polynomial is given as

$$\begin{array}{ccc}\mathcal{D}\hfill & =\hfill & -27q^4{t}^2+18q^3{r}_{1}st-4q^3{s}^3-4q^2{r}_1^{3}t+q^2{r}_1^2{s}^2+144q^2{r}_1{t}^2-6q^2{s}^{2}t-80q{r}_1^{2}st\hfill \\ & \hfill & +18q{r}_1{s}^3-192qs{t}^2+16r_1^{4}t-4r_1^3{s}^2-128r_1^2{t}^2+144r_1{s}^{2}t-27s^4+256t^3.\hfill \end{array}$$

(23)

When the discriminant $\mathcal{D}=0$, it signals that the polynomial has multiple roots, thus confirming the presence of eigenvalue degeneracies, or we can say diabolic points. In our numerical analysis, we track the eigenvalues as a function of the spatial coordinate $x/\lambda$ and topological charge ${\mathsf{\ell }}_i$. The coalescence metrics ${\Delta }_{ij}$ are computed pointwise, and degeneracy orders are recorded accordingly. For example, the fourth-order DP observed at $x=0\lambda$ which is verified by the condition ${\Delta }_{12}={\Delta }_{13}={\Delta }_{14}={\Delta }_{23}={\Delta }_{24}={\Delta }_{34}\approx 0$. This framework enables the meaningful classification of diabolic points and offers a more rigorous alternative to purely visual inspection. It also ensures numerical accuracy, particularly in distinguishing between multiple second-order DPs and true higher-order degeneracies.

3. Results and Discussion

This study shows the diabolic points of the eigenvalues for a four-level medium with a Hermitian Hamiltonian, affected by structured control and probe light fields. The second-, third-, and fourth-order diabolic points are recorded. The atomic units are used in the manuscript. In this study, the parameter $\gamma =1$ GHz is introduced as an arbitrary [39,40] but consistent scaling factor to normalize all medium parameters, such as Rabi frequencies and detunings. Although the model is Hermitian and does not involve any decay or dissipation, using $\gamma$ allows all parameters to be expressed in dimensionless form, which makes the analysis simpler and enhances the clarity of the results.

The graphs in Figure 2 are displayed for the imaginary and real parts of eigenvalues ${\lambda }_i$, where $i=1,2,3,4$, against position $x/\lambda$. The diabolic points are studied at topological charges ${\mathsf{\ell }}_{1,3,p}=-1$, ${\mathsf{\ell }}_{2,4}=1$, and ${\mathsf{\ell }}_5=-2$. The diabolic points are controlled at Rabi frequency $j_{1,2}=0.2\gamma$, $j_3=0.8\gamma$, $j_{4,5}=0.5\gamma$, and $j_p=0.6\gamma$, and detuning ${\delta }_{1,2,3,4}=0.1\gamma$, ${\delta }_{5,p}=0.05\gamma$. The imaginary part of the eigenvalue of the Hermitian Hamiltonian denotes two fourth-order and two second-order diabolic points. The two eigenvalues ${\lambda }_1$ and ${\lambda }_2$ intersect at $\pm 1.5$ and $x=0\lambda$ and represent two second-order diabolic points. The four eigenvalues of ${\lambda }_i$ intersect at $\pm 0.6$ and $x=0\lambda$ and show two fourth-order diabolic points. The real part of eigenvalues coincides at six points and denotes six second-order diabolic points. The two eigenvalues ${\lambda }_2$ and ${\lambda }_4$ coincide at two points. $0.14$ and $x=\pm 0.45\lambda$ and show two second-order diabolic points. The other two eigenvalues, ${\lambda }_2$ and ${\lambda }_3$, intersect at two points, $0.08$ and $x=\pm 0.2\lambda$, and show two second-order diabolic points. The other two eigenvalues, ${\lambda }_3$ and ${\lambda }_4$, coincide at $0.1$ and $x=0\lambda$, which denotes a second-order diabolic point. The other two eigenvalues, ${\lambda }_1$ and ${\lambda }_2$, intersect at $0.05$ and $x=0\lambda$ and represent a second-order diabolic point.

Figure 3 displays the imaginary and real parts of eigenvalues ${\lambda }_i$, where $i=1,2,3,4$, against position $x/\lambda$. The diabolic points are studied at topological charges ${\mathsf{\ell }}_{1,3,p}=-1$, ${\mathsf{\ell }}_{2,4}=3$, and ${\mathsf{\ell }}_5=-2$. The diabolic points are controlled at Rabi frequency $j_{1,2}=0.2\gamma$, $j_3=0.8\gamma$, $j_{4,5}=0.5\gamma$, and $j_p=0.6\gamma$, and detuning ${\delta }_{1,2,3,4}=0.1\gamma$, ${\delta }_{5,p}=0.05\gamma$. The imaginary part of the eigenvalue of the Hermitian Hamiltonian denotes two fourth-order and two second-order diabolic points. The two eigenvalues ${\lambda }_1$ and ${\lambda }_2$ intersect at $\pm 1.5$ and $x=0\lambda$ and represent two second-order diabolic points. The four eigenvalues of ${\lambda }_i$ intersect at $\pm 0.6$ and $x=0\lambda$ and show two fourth-order diabolic points. The real part of eigenvalues coincides at one point at the origin and denotes one fourth-order diabolic point. The values of the diabolic point are 0 and $x=0\lambda$.

The graphs in Figure 4 are displayed for the imaginary and real parts of eigenvalues ${\lambda }_i$, where $i=1,2,3,4$, against position $x/\lambda$. The diabolic points are studied at topological charges ${\mathsf{\ell }}_{1,p}=5$, ${\mathsf{\ell }}_{2,4}=-4$, and ${\mathsf{\ell }}_{3,5}=2$. The diabolic points are controlled at Rabi frequency $j_{1,2,3,4,5}=5\gamma$, and $j_p=5\gamma$, and detuning ${\delta }_{1,2}=0.2\gamma$, ${\delta }_{3,4}=2\gamma$, ${\delta }_5=2.2\gamma$, and ${\delta }_{5,p}=0.1\gamma$. The imaginary part of the eigenvalue of the Hermitian Hamiltonian denotes one fourth-order and two second-order diabolic points. The two eigenvalues ${\lambda }_1$ and ${\lambda }_2$ intersect at $\pm 14$ and $x=0\lambda$ and represent two second-order diabolic points. The four eigenvalues of ${\lambda }_i$ intersect at 0 and $x=0\lambda$ and show one fourth-order diabolic point. The real part of eigenvalues coincides at one point at the origin and denotes one fourth-order diabolic point. The values of the diabolic point are 0 and $x=0\lambda$. The two eigenvalues ${\lambda }_3$ and ${\lambda }_4$ show degeneracy in the region $-0.5\lambda \le x\le 0.5$.

The graphs in Figure 5 are displayed for the imaginary and real parts of eigenvalues ${\lambda }_i$, where $i=1,2,3,4$, against position $x/\lambda$. The diabolic points are studied at topological charges ${\mathsf{\ell }}_{1,2,3,4,5,p}=1$. The diabolic points are controlled at Rabi frequency $j_1=15\gamma$, $j_2=1.5\gamma$, $j_{3,4,5}=0.5\gamma$, and $j_p=10\gamma$, and detuning ${\delta }_{1,2,4}=0.2\gamma$, ${\delta }_{3,p}=0.1\gamma$, and ${\delta }_5=0\gamma$. The imaginary part of the eigenvalue of the Hermitian Hamiltonian denotes one fourth-order and two second-order diabolic points. The two eigenvalues ${\lambda }_1$ and ${\lambda }_2$ intersect at $\pm 1.5$ and $x=0\lambda$ and represent two second-order diabolic points. The two eigenvalues ${\lambda }_1$ and ${\lambda }_2$ intersect at $\pm 0.0006$ and $x=0\lambda$ and show two second-order diabolic points. The four eigenvalues of ${\lambda }_i$ coincide at 0 and $x=0\lambda$ and show a fourth-order diabolic point. The real part of eigenvalues illustrates one third-order and two second-order diabolic points. The two eigenvalues ${\lambda }_2$ and ${\lambda }_4$ coincide at two points, $0.03$ and $x=\pm 0.35\lambda$, and show two second-order diabolic points. The four eigenvalues of ${\lambda }_i$ coincide at $0.005$ and $x=0\lambda$ and denote a fourth-order diabolic point.

The graphs in Figure 6 are displayed for the imaginary and real parts of eigenvalues ${\lambda }_i$, where $i=1,2,3,4$, against position $x/\lambda$. The diabolic points are studied at topological charges ${\mathsf{\ell }}_{1,2,3,4,5,p}=2$. The diabolic points are controlled at identical parameters of Figure 5. The imaginary part of the eigenvalue of the Hermitian Hamiltonian denotes one fourth-order diabolic point. The four eigenvalues of ${\lambda }_i$ coincide at 0 and $x=0\lambda$ and show a fourth-order diabolic point. The two eigenvalues ${\lambda }_2$ and ${\lambda }_3$ show degeneracy in the region $-0.5\lambda \le x\le 0.5$. The real part of eigenvalues illustrates one second-order diabolic point. The two eigenvalues ${\lambda }_2$ and ${\lambda }_3$ show degeneracy in the region $-0.5\lambda \le x\le 0.5$. At topological charges of ${\mathsf{\ell }}_i=2$, there are no third- and fourth-order diabolic points, and also the two eigenvalues ${\lambda }_1$ and ${\lambda }_4$ are not coinciding at all.

The graphs in Figure 7 are displayed for the imaginary and real parts of eigenvalues ${\lambda }_i$, where $i=1,2,3,4$, against position $x/\lambda$. The diabolic points are studied at topological charges ${\mathsf{\ell }}_{1,2,3,4,5,p}=3$. The diabolic points are controlled at Rabi frequency $j_1=15\gamma$, $j_2=1.5\gamma$, $j_{3,4,5}=0.5\gamma$, and $j_p=10\gamma$, and detuning ${\delta }_{1,2,4}=0.2\gamma$, ${\delta }_{3,p}=0.1\gamma$, and ${\delta }_5=0\gamma$. The imaginary part of the eigenvalue of the Hermitian Hamiltonian denotes one fourth-order and two second-order diabolic points. The two eigenvalues ${\lambda }_2$ and ${\lambda }_3$ intersect at $\pm 1.5$ and $x=0\lambda$ and represent two second-order diabolic points. The four eigenvalues of ${\lambda }_i$ coincide at 0 and $x=0\lambda$ and show a fourth-order diabolic point. The two eigenvalues ${\lambda }_1$ and ${\lambda }_2$ denote degeneracy in the region $-0.5\lambda \le x\le 0$, and the other two eigenvalues ${\lambda }_3$ and ${\lambda }_4$ denote degeneracy in the region $0\lambda \le x\le 0.5$. The real part of eigenvalues illustrates one fourth-order diabolic point. The four eigenvalues of ${\lambda }_i$ coincide at the origin at points 0 and $x=0\lambda$ and show a fourth-order diabolic point. The two eigenvalues ${\lambda }_1$ and ${\lambda }_2$ denote degeneracy in the region $-0.5\lambda \le x\le 0$, and the other two eigenvalues ${\lambda }_3$ and ${\lambda }_4$ denote degeneracy in the region $0\lambda \le x\le 0.5$.

The graphs in Figure 8 are displayed for the imaginary and real parts of eigenvalues ${\lambda }_i$, where $i=1,2,3,4$, against position $x/\lambda$. The diabolic points are studied at topological charges ${\mathsf{\ell }}_{1,2,3,4,5,p}=4$. The diabolic points are executed at the same parameters as Figure 5. The imaginary part of the eigenvalue of the Hermitian Hamiltonian denotes one fourth-order diabolic point. The four eigenvalues of ${\lambda }_i$ coincide at 0 and $x=0\lambda$ and show a fourth-order diabolic point. The two eigenvalues ${\lambda }_2$ and ${\lambda }_3$ show degeneracy in the region $-0.5\lambda \le x\le 0.5$. The real part of eigenvalues illustrates one second-order diabolic point. The two eigenvalues ${\lambda }_2$ and ${\lambda }_3$ show degeneracy in the region $-0.5\lambda \le x\le 0.5$. At topological charges of ${\mathsf{\ell }}_i=4$, there are no third- and fourth-order diabolic points, and also the two eigenvalues ${\lambda }_1$ and ${\lambda }_4$ are not coinciding at all. The behavior of Figure 8 is similar to that of Figure 6. It represents that at even values of topological charges ${\mathsf{\ell }}_i$, the real part of eigenvalues shows no higher-order diabolic points.

The graphs in Figure 9 are displayed for the imaginary and real parts of eigenvalues ${\lambda }_i$ against position $x/\lambda$. The diabolic points are studied at topological charges ${\mathsf{\ell }}_{1,2,3,4,5,p}=5$. The diabolic points are controlled at identical parameters of Figure 5. The imaginary part of the eigenvalue of the Hermitian Hamiltonian denotes one fourth-order diabolic point. The four eigenvalues of ${\lambda }_i$ coincide at 0 and $x=0\lambda$ and show a fourth-order diabolic point. The two eigenvalues ${\lambda }_3$ and ${\lambda }_4$ denote degeneracy in the region $-0.5\lambda \le x\le 0$, and the other two eigenvalues ${\lambda }_1$ and ${\lambda }_2$ denote degeneracy in the region $0\lambda \le x\le 0.5$. The real part of eigenvalues illustrates one fourth-order and two second-order diabolic points. The four eigenvalues of ${\lambda }_i$ coincide at the origin at points 0 and $x=0\lambda$ and show a fourth-order diabolic point. The two eigenvalues ${\lambda }_1$ and ${\lambda }_4$ intersect at points 40 and $x=0\lambda$ and represent a second-order diabolic point. The two eigenvalues ${\lambda }_1$ and ${\lambda }_4$ intersect at points 40 and $x=0\lambda$ and represent a second-order diabolic point. The other two eigenvalues, ${\lambda }_2$ and ${\lambda }_3$, intersect at points $-40$ and $x=0\lambda$ and represent second-order diabolic points. The two eigenvalues ${\lambda }_3$ and ${\lambda }_4$ denote degeneracy in the region $-0.5\lambda \le x\le 0$, and the other two eigenvalues ${\lambda }_1$ and ${\lambda }_2$ denote degeneracy in the region $0\lambda \le x\le 0.5$.

4. Conclusions

The diabolic points of the Hermitian Hamiltonian using driving fields of structured light in a four-level atomic medium are examined. The eigenvalues of a Hermitian Hamiltonian are calculated by using the secular equation $|H-\lambda I|=0$. Diabolic points are connected to the interception of real and imaginary parts of eigenvalues. Real and imaginary components of the Hermitian Hamiltonian’s eigenvalues report multiple diabolic points of the second, third, and fourth orders of the Hamiltonian. The diabolic points of the fourth, third, and second orders are observed by the real and imaginary parts of the eigenvalues of the Hermitian Hamiltonian. The diabolic points and degeneracy regions are studied with variations of Rabi frequencies, detuning, and topological charges. The structured light has a key impact on diabolic points. By changing the topological charges, the number of diabolic points and the degeneracy region are changing. At topological charge $\mathsf{\ell }=even$, the real part of eigenvalues only shows second order DP, and no higher order DP is reported. The obtained results of the diabolic point are helpful in the fields of deformation space, entanglement physics, optomechanical systems, and crystal optics.