Reconstruction method and reconstruction system of non-hermitian system eigenstate

Abstract

The application relates to a reconstruction method and a reconstruction system of a non-hermitian system eigenstate, and the reconstruction method comprises the following steps: constructing an n-order non-hermitian system, wherein n is an integer greater than or equal to 2; coupling a driving source to the n-order non-hermitian system, and transforming an equivalent Hamiltonian of the n-order non-hermitian system with external excitation into an anti-PT symmetric form by using a suitable unitary transformation; calculating eigenvalues of the n-order non-hermitian system based on the equivalent Hamiltonian of the anti-PT symmetric form, finding a critical condition corresponding to a chirality inversion between two types of EPs in the n-order non-hermitian system; and adjusting the size of the excitation coupling coefficient based on the critical condition, so as to realize the chirality inversion between the two types of EPs. The application can actively manipulate the eigenstate of the EP without changing the physical structure of the non-hermitian system.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of wireless energy transmission, and particularly relates to a reconstruction method and a reconstruction system of eigenstates of a non-Hermitian system. BACKGROUND

[0002] Quantum mechanics is generally used to describe a closed physical system, and the system is represented by a Hermitian operator, which ensures that the eigenenergy is a real number. The eigenvalue of a Hermitian Hamiltonian is usually used to describe it, which ensures that the system satisfies the law of conservation of energy. The Hamiltonian of an open system is non-Hermitian, and its eigenvalue is usually a complex number. Hermiticity is not a necessary condition for an operator to have real eigenvalues, so it is of great significance to find real eigenvalues of non-Hermitian systems. As early as 1998, Bender and Boettche proposed that a non-Hermitian system with parity-time (PT) symmetry would have a pure real energy spectrum before the PT phase transition. This phase transition singularity is called an exceptional point (EP). It is worth noting that in a non-Hermitian system, there are many interesting and counterintuitive phenomena in EPs that are simultaneously degenerate with two or more eigenvalues and corresponding eigenstates, for example: n-order EPs usually satisfy the nth root dependence relationship in the eigenvalues, that is, E∝δ 1 / n , where E represents energy, and δ represents the perturbation of the EP in the parameter space. Therefore, EPs of non-Hermitian systems can be used to realize various high-sensitivity sensors. On the other hand, EPs are singularities of self-intersecting Riemann surfaces in non-Hermitian parameter space. When the parameters of a non-Hermitian system evolve adiabatically around EPs, the initial eigenstate will transfer to other eigenstates and obtain a geometric phase. The topological properties of eigenstates near EPs have also been widely studied in physical systems such as coupled cavities, waveguides, and metasurfaces. In particular, another interesting property of EPs in a non-Hermitian system formed by two coupled resonant cavities is the self-orthogonal property of their eigenstates. Recently, researchers have proven the existence of "absorbing-type" EPs based on the clockwise and counterclockwise propagation modes in a whispering gallery resonant microcavity, which breaks the traditional "resonant-type" EPs in non-Hermitian systems. In particular, due to the special "missing dimension", EPs have inherent chirality, which has important application value in the design of chiral light sources.

[0003] Although it has been found that adjusting the sign (i.e., positive and negative) of the coupling coefficient of a non-Hermitian system can achieve the inversion of the chirality of the non-Hermitian system, it is usually necessary to construct different physical systems to achieve positive coupling and negative coupling. In practical applications, there is a lack of a simple and effective method for controlling the chirality of EPs in a non-Hermitian system. SUMMARY

[0004] In order to solve the above problems, the application provides a reconstruction method and a reconstruction system for eigenstates of a non-Hermitian system, which can actively manipulate the eigenstates of the EP without changing the physical structure of the non-Hermitian system.

[0005] The application aims to provide a reconstruction method for eigenstates of a non-Hermitian system, which comprises the following steps:

[0006] Constructing a non-Hermitian system of order n, wherein n is an integer greater than or equal to 2;

[0007] Coupling an excitation source to the non-Hermitian system of order n, and using a suitable unitary transformation to transform the equivalent Hamiltonian of the non-Hermitian system of order n under external excitation into an anti-PT symmetric form;

[0008] Calculating the eigenvalues of the non-Hermitian system of order n based on the equivalent Hamiltonian in the anti-PT symmetric form, and finding the critical condition corresponding to the inversion of chirality between two types of EPs in the non-Hermitian system of order n;

[0009] Adjusting the size of the excitation coupling coefficient based on the critical condition to realize the inversion of chirality between the two types of EPs.

[0010] The further improvement of the reconstruction method for eigenstates of a non-Hermitian system is that the constructed non-Hermitian system of order n is a non-Hermitian system of order 2; and the method for using a suitable unitary transformation to transform the equivalent Hamiltonian of the non-Hermitian system of order 2 under external excitation into an anti-PT symmetric form is:

[0011] Let the excitation coupling coefficient g = γ A , wherein γ A represents the radiation loss of a resonant atom A coupled to the excitation source in the non-Hermitian system of order 2, and the dynamic equation of the non-Hermitian system of order 2 under external excitation is:

[0012]

[0013] , wherein, and respectively represent harmonic modes in two resonant atoms A and B in the non-Hermitian system of order 2; ω0 and κ respectively represent the resonance frequency and the internal coupling coefficient between the two resonant atoms A and B in the non-Hermitian system of order 2; γ B represents the radiation loss of the resonant atom B in the non-Hermitian system of order 2; Γ A and Γ B respectively represent the dissipation loss of the two resonant atoms A and B in the non-Hermitian system of order 2; is an input signal;

[0014] Let γ A = γ B , and the zero reflection condition is The equivalent Hamiltonian H of the second-order non-Hermitian system with external excitation is obtained as follows:

[0015]

[0016] The basis vector of the expression (2) is changed by using the unitary transformation formula (3),

[0017]

[0018] Wherein, A and B are the original basis vectors, and a and b are the changed basis vectors, and the equivalent Hamiltonian H is transformed into the anti-PT symmetric form H':

[0019]

[0020] Wherein, g0=g / 2, ΔΓ=(Γ A -Γ B ) / 2 and Γ0=(Γ A +Γ B ) / 2.

[0021] The further improvement of the reconstruction method of the eigenstate of the non-Hermitian system is that the eigenvalue of the second-order non-Hermitian system calculated based on the equivalent Hamiltonian of the anti-PT symmetric form is:

[0022]

[0023] The further improvement of the reconstruction method of the eigenstate of the non-Hermitian system is that the step of finding the critical condition corresponding to the inversion of chirality between the two types of EPs in the second-order non-Hermitian system comprises:

[0024] The corresponding eigenstate is calculated based on the formula (4):

[0025]

[0026] Suppose g0-ΔΓ=±κ, and the eigenstate of the EPs is And the critical condition g0=ΔΓ is obtained.

[0027] The further improvement of the reconstruction method of the eigenstate of the non-Hermitian system is that while adjusting the size of the excitation coupling coefficient based on the critical condition, the size of the internal coupling coefficient of the n-order non-Hermitian system is also adjusted, the inversion of the resonance-type EP and the absorption-type EP is realized by adjusting the excitation coupling coefficient, and the counterclockwise rotation under the resonance-type EP or the clockwise rotation under the absorption-type EP is realized by adjusting the internal coupling coefficient.

[0028] Another object of the present application is to provide a reconstruction system of the eigenstate of a non-Hermitian system, which is used to realize the reconstruction method of the eigenstate of a non-Hermitian system as claimed in any one of the above, and the reconstruction system comprises:

[0029] An n-order non-hermitian system composed of n resonant coils, wherein n is an integer greater than or equal to 2;

[0030] A source coil coupled to the n-order non-hermitian system, a coupling coefficient between a resonant coil A adjacent to the source coil in the n-order non-hermitian system and the source coil is the excitation coupling coefficient;

[0031] A load coil coupled to one end of the n-order non-hermitian system away from the source coil.

[0032] The further improvement of the reconstruction system of the eigenstate of the non-hermitian system is that the source coil, the load coil and the n resonant coils are movably connected to the track, and the adjustment of the excitation coupling coefficient can be realized by adjusting the distance between the source coil and the resonant coil A, and the adjustment of the internal coupling coefficient between the adjacent resonant coils can be realized by adjusting the distance between the adjacent resonant coils.

[0033] The further improvement of the reconstruction system of the eigenstate of the non-hermitian system is that a scale is arranged on the track in the axial direction.

[0034] The further improvement of the reconstruction system of the eigenstate of the non-hermitian system is that n=2, and the resonant coil B adjacent to the load coil is placed vertically to the resonant coil A.

[0035] The present application includes but is not limited to the following beneficial effects:

[0036] 1. By introducing external excitation, the establishment of the eigenstate in the non-hermitian system can be participated, and further, the inversion of the EP chirality can be actively controlled by adjusting the size of the external excitation, so that the eigenstate of the non-hermitian system can be manipulated without changing the physical structure of the non-hermitian system.

[0037] 2. A non-hermitian anti-PT symmetric system composed of simple resonant coils is provided to realize the inversion of the EP chirality, and the anti-PT symmetric system is directly constructed by the input of external excitation, without "virtual coupling".

[0038] 3. A new way is opened for the non-hermitian physical research in an open system, and a support platform is provided for the development of chiral antennas, polarization converters and wireless communication related devices, which can be popularized and applied to optical systems and acoustic systems. BRIEF DESCRIPTION OF DRAWINGS

[0039] Figure 1 A schematic diagram for the control of the eigenstate in a second-order non-hermitian system without external excitation is shown.

[0040] Figure 2The figure shows the schematic diagram of the control of eigenstate in a two-order non-hermitian system with external excitation.

[0041] Figure 3 The figure shows the schematic diagram of the reconstruction system of the present application aiming at a two-order non-hermitian system.

[0042] Figure 4 The figure shows the inherent chirality of "resonance type" EP in a two-order non-hermitian system under weak excitation.

[0043] Figure 5 The figure shows the inherent chirality of "absorption type" EP in a two-order non-hermitian system under strong excitation.

[0044] Figure 6 The figure shows the phase diagram of the Riemann surface and the chirality control of EP in a non-hermitian system under different external excitations. DETAILED DESCRIPTION

[0045] The principle of the control of eigenstate in a non-hermitian system is described in the following: Figure 1 as shown in Figure 1 The figure (a) shows a two-order non-hermitian system composed of two resonance atoms A and B, and the effective Hamiltonian of the system can be written as:

[0046]

[0047] where ω0 and κ represent the resonance frequency and internal coupling coefficient between the two resonance atoms A and B in the two-order non-hermitian system, respectively; γ A and γ B represent the radiation loss of the two resonance atoms A and B in the two-order non-hermitian system, respectively; Γ A and Γ B represent the dissipation loss of the two resonance atoms A and B in the two-order non-hermitian system, respectively.

[0048] The eigenvalue E ± of H is represented as:

[0049]

[0050] where γ1 = γ A + Γ A and γ2 = γ B + Γ B represent the total loss of the resonance atoms A and B, respectively. It can be found from the formula that the eigenvalue of the system is usually two complex numbers.

[0051] The phase difference of the two resonance atoms A and B can be determined by the corresponding eigenstate X ± :

[0052]

[0053] Based on the actual parameters f0= ω0 / 2π = 567 kHz, γ1= 0.4 kHz, γ2= 2.9 kHz, the evolution of eigenvalues under different coupling coefficients κ can be obtained, as shown in (b) of FIG. 1. Figure 1 When the condition κ = ± (γ1- γ2) / 2 is satisfied, the two eigenvalues will merge at the "resonance type" EP, so the eigenstate of the EP can be obtained as In particular, since the loss of the resonant atom B is greater than that of the resonant atom A (i.e. γ1< γ2), the EPs appearing when κ > 0 and κ < 0 are (-i, 1) and (i, 1) respectively, which are marked with light gray and dark gray respectively. The special chirality of the EP with different signs of the coupling coefficient is determined by the "missing dimension" of the EP. Although the sign (i.e. positive and negative) of the coupling coefficient can change the eigenstate of the non-Hermitian system, it is usually necessary to construct different physical systems to realize positive and negative coupling, and in practical applications, there is a lack of simple and effective method for realizing chirality control of EP in non-Hermitian systems.

[0054] In view of the above, the present application provides a method and system for reconstructing eigenstates of a non-Hermitian system, which can actively manipulate the eigenstates of the EP without changing the physical structure of the non-Hermitian system. The method and system for reconstructing eigenstates of a non-Hermitian system will be further described below with reference to specific embodiments and accompanying drawings.

[0055] Although it has been believed that the eigenstate of a system is intrinsic, for a determined physical system, external excitation is only used to observe the eigenstate and will not have a significant impact on it. However, the present application has found through a large number of experiments that the establishment of the eigenstate in a non-Hermitian system can be involved by introducing external excitation. Therefore, a method for reconstructing eigenstates of a non-Hermitian system is provided, comprising the steps of:

[0056] Step 1, constructing an n-order non-Hermitian system, wherein n is an integer greater than or equal to 2.

[0057] Step 2, coupling a source of excitation to the n-order non-Hermitian system, and using a suitable unitary transformation to transform the equivalent Hamiltonian of the n-order non-Hermitian system with external excitation into an anti-PT symmetric form.

[0058] Step 3, calculating the eigenvalues of the n-order non-Hermitian system based on the equivalent Hamiltonian of the anti-PT symmetric form, and finding the critical condition corresponding to the inversion of chirality between two types of EPs in the n-order non-Hermitian system.

[0059] Step 4, adjusting the size of the excitation coupling coefficient based on the critical condition to realize the inversion of chirality between the two types of EPs.

[0060] Specifically, taking a two-order (i.e. n = 2) non-Hermitian system as an example, referring to FIG. 1, Figure 2shown, Figure 2 Figure (c) in the figure shows a second-order non-hermitian system composed of two resonant atoms A and B, and the excitation source is coupled to the resonant atom A, and the corresponding excitation coupling coefficient is g. The method of transforming the equivalent Hamiltonian of the second-order non-hermitian system with external excitation into the anti-PT symmetric form by using a suitable unitary transformation is:

[0061] Let the excitation coupling coefficient g = γ A , where γ A represents the radiation loss of the resonant atom A in the second-order non-hermitian system coupled to the excitation source, and the dynamic equation of the second-order non-hermitian system with external excitation is:

[0062]

[0063] where, and represent the harmonic modes in the two resonant atoms A and B in the second-order non-hermitian system; ω0 and κ represent the resonance frequency and internal coupling coefficient between the two resonant atoms A and B in the second-order non-hermitian system; γ B represents the radiation loss of the resonant atom B in the second-order non-hermitian system; Γ A and Γ B represent the dissipation loss of the two resonant atoms A and B in the second-order non-hermitian system; is the input signal;

[0064] Let γ A = γ B , and under the zero reflection condition , the equivalent Hamiltonian H of the second-order non-hermitian system with external excitation is obtained:

[0065]

[0066] Using the unitary transformation formula (3) to change the basis vector of expression (2),

[0067]

[0068] where A and B are the original basis vectors, and a and b are the changed basis vectors, and the equivalent Hamiltonian H is transformed into the anti-PT symmetric form H':

[0069]

[0070] where g0 = g / 2, ΔΓ = (Γ A - Γ B ) / 2, and Γ0 = (Γ A + Γ B ) / 2.

[0071] It can be seen from equation (4) that the system is anti-PT symmetric about ω0, i.e. satisfies the following equation:

[0072] (PT)H′(PT) -1 = P(H′) * P = -H′

[0073] It is worth noting that the imaginary coupling coefficient of the equivalent Hamiltonian H′ in the anti-PT symmetric form is always realized by introducing indirect coupling of the external resonant atom. Therefore, it can be considered that the system in the anti-PT symmetric form is directly constructed by the input of external excitation. No additional imaginary coupling is needed. It is emphasized that the above transformation is for more intuitive comparison of g0and κ. In fact, equations (2) and (4) are equivalent, and in this case, the second-order non-Hermitian system eigenvalues calculated based on the equivalent Hamiltonian H′ in the anti-PT symmetric form are:

[0074]

[0075] The corresponding eigenstates are calculated based on equation (4):

[0076]

[0077] The transition between the anti-PT symmetric phase and the symmetry breaking phase occurs at the "absorbing type" EP, i.e. g0-ΔΓ=±κ, and the eigenstates of the EPs are Unlike the "resonant type" EP (g0<ΔΓ), the "absorbing type" EP mainly refers to the EP of the non-Hermitian system under the strong influence of external excitation (g0>ΔΓ). The eigenvalues and the corresponding eigenstates of the system under different coupling coefficients κ (κ>0) are shown in Fig. (d) of Figure 2 .

[0078] In conclusion, it can be concluded that the chirality of the EPs of the second-order non-Hermitian system directly depends on the external excitation g0, and in particular, the chirality inversion in the second-order non-Hermitian system occurs at g0>g c , where g c =ΔΓ=1.25 kHz corresponds to the critical condition. In Figure 2 Fig. (d), the evolution of the eigenvalues under the positive coupling coefficient κ when a stronger external excitation g0=3 kHz (i.e. g0>g c ) is introduced is represented by a dark gray dashed line, and the evolution of the eigenvalues under the positive coupling coefficient κ when a weaker external excitation g0=0.02 kHz (i.e. g0<g c ) is introduced is represented by a light gray solid line, and the angle of a pair of arrows represents the phase difference between the two resonant atoms A and B. It can be known from the comparison of the two cases that the EP chirality of the positive coupling coefficient is in the clockwise (counterclockwise) direction corresponding to the non-Hermitian with a stronger (weaker) external excitation.

[0079] The above only determined the critical conditions for second-order non-Hermitian systems. For other orders of non-Hermitian systems, the corresponding critical conditions can be determined based on the same principle (all of which are relationships with excitation coupling coefficients). Chirality reversal in non-Hermitian systems can be achieved by adjusting the excitation coupling coefficients based on the critical conditions, without changing the sign of the internal coupling coefficients of the non-Hermitian system, that is, without changing the physical structure of the non-Hermitian system.

[0080] To experimentally verify the above-mentioned method for reconstructing the eigenstates of a non-Hermitian system, this invention provides a reconstruction system for the eigenstates of a non-Hermitian system. The reconstruction system includes: an n-order non-Hermitian system composed of n resonant coils, where n is an integer greater than or equal to 2; a source coil coupled to the n-order non-Hermitian system, wherein the coupling coefficient between the resonant coil A adjacent to the source coil in the n-order non-Hermitian system and the source coil is the excitation coupling coefficient; and a load coil coupled to one end of the n-order non-Hermitian system away from the source coil.

[0081] See Figure 3 As shown, Figure 3 A reconstruction system for a second-order non-Hermitian system is shown. This system consists of two resonant coils, A and B, placed perpendicularly (in practice, they can also be placed coaxially and parallel as needed). Each coil has a diameter of 30 cm, 15 turns, and a lumped capacitor of 657 pF (with a withstand voltage greater than 2000 V) connected in parallel. The resonant frequencies of both coils A and B are 567 kHz. The internal coupling coefficient κ between coils A and B is a function of the distance d between them. Adjusting the value of d allows adjustment of the internal coupling coefficient κ. An active coil and a load coil are connected to this second-order non-Hermitian system. The active coil is coupled to the resonant coil A, and the excitation coupling coefficient g between them is a function of the distance D between the active coil and resonant coil A. Adjusting the value of D allows adjustment of the excitation coupling coefficient g. The active coil has a diameter of 30 cm and 3 turns. The load coil (non-resonant) is coupled to the resonant coil B, with a coupling coefficient denoted by l. It should be noted that the magnitude of this coupling coefficient l does not affect the entire system; the load coil is only used to detect the output signal. The dissipation loss Γ of the resonant coils A and B is... A and Γ B In the illustrated case, the value is constant, and the load coil has a diameter of 7 cm and is wound with 9 turns. All the coils described above are made of Litz wire (0.078 mm × 300 strands) tightly bonded to a hollow polymethyl methacrylate (PMMA) cylinder.

[0082] To facilitate the adjustment of distances d and D to adjust the excitation coupling coefficient g and the internal coupling coefficient κ, in a preferred embodiment, the reconstruction system further includes a track on which the source coil, the load coil, and all resonant coils are movably connected. The distances d and D are adjusted by regulating the position of each coil on the track. Preferably, a scale can also be provided along the axial direction on the track for precise adjustment.

[0083] This invention experimentally verifies the above-described reconstruction method for second-order non-Hermitian systems using the aforementioned reconstruction system:

[0084] Provide the above-mentioned reconstruction system, and connect the source coil and load coil to the input port and output port of the vector network analyzer (Keysight E5071C) respectively, with the source impedance of the input signal being 50Ω.

[0085] On the one hand, a weak excitation of g = 0.02 kHz is given, and the "resonance type" EP of the non-Hermitian system is characterized from the transmission spectrum. The chirality of the "resonance type" EP under the weak excitation condition is determined by adjusting the distance d between the two resonant coils A and B. Based on the dynamic equation of formula (1), the transmittance of the second-order non-Hermitian system can be obtained as:

[0086]

[0087] Where: the output signal is and γ B These represent the resonant mode and radiation loss of the resonant atom B, respectively.

[0088] See Figure 4 As shown, Figure 4 Figure (a) shows the relationship between the normalized transmittance and frequency f of a second-order non-Hermitian system under the condition that the weak excitation can be neglected. The system satisfies Figure 2 In the (d) diagram, g0 <g c Conditions. Here, the internal coupling coefficient κ is adjusted from 3.7 kHz to 1.3 kHz by changing the distance d from 16 cm to 22 cm. Experimental and theoretical results are represented by solid and dashed lines, respectively. When the distance d between the two resonant coils A and B is small (d = 16 cm), two resonance peaks appear in the transmission spectrum. As the distance d gradually increases, the two peaks gradually approach each other. When the distance d reaches a large value (d = 22 cm), the splitting disappears, and only one resonance peak exists. At this point, the transmission spectrum of the system approximates the transmission spectrum of a single resonant coil. This degeneracy point (f0 = 567 kHz) corresponds to a "resonant" EP in a second-order non-Hermitian system with d = 22 cm. The real eigenfrequency f and phase difference of the two resonant coils A and B are also considered. Each as Figure 4The phase difference is shown in Figures (b) and (c). in, and These represent the phases of resonant coil A and resonant coil B, respectively. It should be noted that due to the very weak coupling strength κ, the detection error is relatively large. The experimental (symbolized) and calculated (solid line) overall spectral shapes and the position of the "resonant" EP match very well. It can be seen that due to the phase difference... The chirality of the measured "resonance" EP is counterclockwise; therefore, in this weakly excited non-Hermitian system, the chirality of the "resonance" EP is counterclockwise.

[0089] On the other hand, by adjusting the distance D between the source coil and the resonant coil A, a strong excitation of g = 14.65 kHz is provided, making the second-order non-Hermitian system satisfy... Figure 2 In the (d) diagram, g0>g c Conditions. The relationship between the normalized transmittance and frequency f of an equivalent anti-PT symmetric second-order non-Hermitian system is as follows: Figure 5 As shown in Figure (a), the internal coupling coefficient κ was varied from 13.6 kHz to 24.2 kHz by changing the distance d from 3 cm to 7 cm. (Comparison) Figure 4 and Figure 5 As can be seen in Figure (a), Figure 5 The splitting of the transmission peak is more pronounced in figure (a). Furthermore, under strong excitation, a larger coupling strength (i.e., a larger internal coupling coefficient) leads to mode degeneracy, forming an "absorption-type" EP. The real eigenfrequency f and phase difference of the corresponding two resonant coils A and B are shown. Each as Figure 5 Figures (b) and (c) are shown in the table. The results indicate that the chirality of the measured "absorption-type" EP is... This indicates that the chirality of the "absorbent" EP is clockwise.

[0090] By comparison Figure 4 and Figure 5 It can be clearly seen that external stimuli participate in the establishment of eigenstates of non-Hermitian systems, and the chirality of the corresponding EP is also reversed.

[0091] To more clearly explain the eigenstates manipulated by external stimuli, see [reference needed]. Figure 6 As shown, Figure 6 Figure (a) shows the Riemann surface and EP lines of the system, which are functions of parameters on the (g, κ) surface. The white and black dashed lines represent two EPs with different chirality. For a fixed internal coupling coefficient κ = 0.75 kHz, the calculated eigenfrequency f is related to the external excitation parameter g as follows: Figure 5As shown in (b) of FIG. 1, with the increase of the excitation coupling coefficient g, it can be clearly seen that the phase difference is constantly changing. At f = 567 kHz, the counterclockwise and clockwise chirality of EPs are represented by light gray and dark gray arrows, respectively. Especially at EP1 and EP2, the measured phase difference is -91° and +90°, respectively, which is in good agreement with the calculated results. It is worth noting that although the equivalent anti-PT symmetric system is constructed by using the resonant coil and the EP chirality inversion controlled by external excitation is observed, the related results can be directly extended to other optical systems and acoustic systems. Based on this method, size-independent chiral devices such as antennas, topological excitations and near-field routing controlled by external excitation can also be constructed. In addition to providing a new way to control chirality, the EPs controlled by external excitation also provide a new scheme for the rich non-hermitian physical research.

[0092] The above detailed description of the embodiments of the present application is combined with the drawings, and those of ordinary skill in the art can make various changes to the present application according to the above description. Therefore, some details in the embodiments should not constitute a limitation on the present application, and the scope of protection of the present application will be defined by the appended claims.

Claims

1. A method for reconstructing the eigenstates of a non-Hermitian system, characterized in that, Including the following steps: Construct an n-order non-Hermitian system, where n is an integer greater than or equal to 2; A source of excitation is coupled to an nth-order non-Hermitian system, and an equivalent Hamiltonian of the nth-order non-Hermitian system under external excitation is transformed into an anti-PT symmetric form by using a suitable unitary transformation. Based on the equivalent Hamiltonian of the anti-PT symmetric form, the eigenvalues ​​of the nth-order non-Hermitian system are calculated, and the critical condition corresponding to the chiral inversion between two types of EPs in the nth-order non-Hermitian system is found. By adjusting the excitation coupling coefficient based on the aforementioned critical condition, the chirality inversion between the two types of EPs can be achieved; wherein... The constructed n-order non-Hermitian system is a second-order non-Hermitian system. The method for transforming the equivalent Hamiltonian of the second-order non-Hermitian system under external excitation into an inverse PT-symmetric form using a suitable unitary transformation is as follows: Let the excitation coupling coefficient be g = γ A , where γ A The kinetic equations for a second-order non-Hermitian system with external excitation represent the radiation loss of a resonant atom A coupled to the excitation source: in, and ω0 and κB represent the harmonic modes of two resonant atoms A and B in a second-order non-Hermitian system, respectively; ω0 and κB represent the resonant frequency and internal coupling coefficient between the two resonant atoms A and B in the second-order non-Hermitian system, respectively; γ0 B Γ represents the radiation loss of the resonant atom B in a second-order non-Hermitian system; A and Γ B These represent the dissipation losses of two resonant atoms A and B in a second-order non-Hermitian system, respectively. For input signals; Let γ A =γ B And under zero reflection conditions Below, we obtain the equivalent Hamiltonian H of a second-order non-Hermitian system with external excitation: By using the unitary transformation formula (3) to change the basis vectors of expression (2), Where A and B are the original basis vectors, and a and b are the modified basis vectors, the equivalent Hamiltonian H is transformed into the anti-PT symmetric form H': Where, g0=g / 2, ΔΓ=(Γ A -C B ) / 2 and Γ0=(Γ A +C B ) / 2。 2. The method for reconstructing the eigenstates of a non-Hermitian system as described in claim 1, characterized in that, The eigenvalues ​​of the second-order non-Hermitian system calculated based on the equivalent Hamiltonian of the anti-PT symmetric form are:

3. The method for reconstructing the eigenstates of a non-Hermitian system as described in claim 2, characterized in that, The steps to find the critical condition for chiral inversion between two classes of EPs in a second-order non-Hermitian system include: The corresponding eigenstates are calculated based on formula (4): Let g0-ΔΓ=±κ, then the eigenstates of EPs are: Therefore, the critical condition g0 = ΔΓ is obtained.

4. The method for reconstructing the eigenstates of a non-Hermitian system as described in any one of claims 1 to 3, characterized in that, While adjusting the magnitude of the excitation coupling coefficient based on the critical condition, the magnitude of the internal coupling coefficient of the nth-order non-Hermitian system is also adjusted. The reversal of the resonant EP and the absorptive EP is achieved by adjusting the excitation coupling coefficient, and the counterclockwise rotation under the resonant EP or the clockwise rotation under the absorptive EP is achieved by adjusting the internal coupling coefficient.

5. A system for reconstructing the eigenstates of a non-Hermitian system, characterized in that, A method for reconstructing the eigenstates of a non-Hermitian system as described in any one of claims 1 to 4, the reconstruction system comprising: An nth-order non-Hermitian system consisting of n resonant coils, where n is an integer greater than or equal to 2; The source coil is coupled to the nth-order non-Hermitian system, and the coupling coefficient between the resonant coil A adjacent to the source coil in the nth-order non-Hermitian system and the source coil is the excitation coupling coefficient. A load coil coupled to the end of the nth-order non-Hermitian system furthest from the source coil.

6. The system for reconstructing the eigenstates of a non-Hermitian system as described in claim 5, characterized in that, It also includes a track, on which the source coil, the load coil, and n resonant coils are movably connected. The excitation coupling coefficient can be adjusted by adjusting the distance between the source coil and the resonant coil A, and the internal coupling coefficient between adjacent resonant coils can be adjusted by adjusting the distance between adjacent resonant coils.

7. The system for reconstructing the eigenstates of a non-Hermitian system as described in claim 6, characterized in that, The track has graduations along its axial direction.

8. The system for reconstructing the eigenstates of a non-Hermitian system as described in claim 6, characterized in that, n=2, and the resonant coil B adjacent to the load coil is placed perpendicular to the resonant coil A.

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