CN110008611B - Spontaneous radiation calculation method of three-energy-level atoms in multilayer topological insulator structure - Google Patents

Spontaneous radiation calculation method of three-energy-level atoms in multilayer topological insulator structure Download PDF

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CN110008611B
CN110008611B CN201910295167.8A CN201910295167A CN110008611B CN 110008611 B CN110008611 B CN 110008611B CN 201910295167 A CN201910295167 A CN 201910295167A CN 110008611 B CN110008611 B CN 110008611B
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曾然
仲佼佼
张猛
陈芳芳
胡淼
李浩珍
毕美华
李齐良
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Hangzhou Dianzi University
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Abstract

The invention provides a spontaneous radiation calculation method of three-energy-level atoms in a multilayer topological insulator structure, which comprises the following steps: the first step is as follows: establishing a model of a multilayer topological insulator structure; the second step is that: determining electromagnetic properties of the topological insulator; the third step: determining boundary conditions of the electromagnetic waves on the interface; the fourth step: obtaining a transmission matrix of the multilayer topological insulator by using boundary conditions; the fifth step: obtaining the reflection coefficient of the multilayer topological insulator structure; and a sixth step: and solving a spontaneous emissivity expression of the three-energy-level atoms in the multilayer topological insulator structure. The method can accurately analyze the spontaneous radiance and the quantum interference intensity of the three-energy-level atoms near the multilayer topological insulator structure and in the cavity formed by the multilayer topological insulator structure, and accurately position the influence and the root cause of various parameters and multilayer topological insulator materials on the spontaneous radiance and the quantum interference, so as to regulate and control the atomic spontaneous radiance and the quantum interference.

Description

Spontaneous radiation calculation method of three-energy-level atoms in multilayer topological insulator structure
Technical Field
The invention belongs to the field of quantum optics, and particularly relates to a spontaneous radiation calculation method of three-energy-level atoms in a multilayer topological insulator structure.
Background
The spontaneous emission of atoms is the content of important research in the quantum optics category, and has important significance for manufacturing devices such as light emitting diodes, lasers, solar cells and the like. The phase and direction of photons released by spontaneous emission are uncertain, and the randomness often destroys the stability of a system, so that the spontaneous emission influences the research of the field of quantum information, and the application of a laser and the quantum information is limited, so that the control of the spontaneous emission becomes a main research work. In recent years, many special materials and materials with non-trivial electromagnetic properties have been discovered in succession and rapidly applied to regulate the spontaneous emission of atoms. Such as topological insulator, left-handed material, plasma waveguide, hyperbolic specific material, etc., and broadens the research direction of spontaneous radiation regulation.
The Topological Insulator (TI) has an energy gap inside and exhibits an insulating state as a conventional insulator, but the surface is a conducting state without an energy gap and has time reversal symmetry, and the surface state represents the Topological property of the Topological insulator, relates to hilbert space, and is a brand new description in quantum mechanics. The origin of the "topology" in the topological insulator is that the hubert space contained therein has a topological structure because of the electronic states corresponding to the wave functions describing the topological insulator. Due to the unique properties of multilayer topological insulator materials, spontaneous emission and quantum interference of nearby atoms can be significantly different from those of other materials. Through retrieval, the research on the spontaneous radiation and quantum interference characteristics of three-level atoms near the multilayer topological insulator material and in the cavity of the multilayer topological insulator material is not related in the prior art. Therefore, the method can be used for obtaining the spontaneous radiation and the interference intensity of the three-energy-level atomic system in the multilayer topological insulator structure.
Disclosure of Invention
The invention provides a method for calculating spontaneous emission of a three-energy-level atomic system in a cavity formed by the vicinity of a multilayer topological insulator through a transmission matrix. The theoretical model of the multilayer topological insulator material used in the invention is closer to the actual multilayer topological insulator material, and has application value as a test material.
The invention adopts the following technical scheme:
a spontaneous radiation calculation method of three-energy-level atoms in a multilayer topological insulator structure is carried out according to the following steps:
the first step is as follows: establishing a model of a multilayer topological insulator structure;
the multilayer topological insulator structure consists of topological insulators and a conventional insulator periodic arrangement, wherein the dielectric constant and the magnetic permeability of the conventional insulator are epsilon 1 And mu 1 Dielectric constant and permeability of three-dimensional topological insulator are epsilon 2 And mu 2 。
The second step is that: determining electromagnetic properties of the topological insulator;
according to the topological field theory, the general electromagnetic response term in the topological insulator is as follows:
S 0 ＝∫dx 3 dt(ε 2 E 2 -B 2 /μ 2 ) (1)
the electromagnetic response terms related to the topological magnetoelectric response are as follows:
S Θ ＝(αΘ/4π 2 )∫dx 3 dtE·B (2)
due to existence of topological magnetoelectric coupling effect, a topological contribution term is added in the constitutive relation of the three-dimensional topological insulator, and the expression is as follows:
Figure BDA0002026246220000021
deducing the relation between components of the electric field and the magnetic field in the TI:
Figure BDA0002026246220000031
the third step: boundary conditions of the electromagnetic wave on the interface are determined.
The boundary conditions at the interface of the topological insulator multilayer structure are:
z×E 1 ＝z×E 2 (5)
z×H 1 ＝z×H 2 (6)
the fourth step: and solving the transmission matrix of the multilayer topological insulator by using the boundary condition.
According to the boundary condition of the electromagnetic field, the system of equations among the incident electric field, the reflected electric field and the transmitted electric field at the interface of the conventional insulator (medium 1) and the topological insulator (medium 2) is obtained as follows:
Figure BDA0002026246220000032
wherein
Figure BDA0002026246220000033
Wave numbers, k, in media 1 and 2, respectively || Is the wavenumber component parallel to the interface.
According to boundary conditions on the interface, the obtained reflection matrix is as follows:
Figure BDA0002026246220000034
Figure BDA0002026246220000035
is a transmission matrix of electromagnetic waves from medium 1 to medium 2, is combined>
Figure BDA0002026246220000036
According to the transmission characteristics and the propagation theory of the electromagnetic waves in the medium, the transmission matrix of the electromagnetic waves in the j-th layer of medium is obtained as follows:
Figure BDA0002026246220000041
k js represents the wave number, d, of the electromagnetic wave in the vertical direction j The thickness of the j-th layer of the medium is shown.
The fifth step: and obtaining the reflection coefficient of the multilayer topological insulator structure.
For a periodic structure with N layers of media, the transmission matrix of the whole system is the multiplication of a transfer matrix and a propagation matrix in sequence of a multilayer structure, and the total transmission matrix is as follows:
Figure BDA0002026246220000042
Figure BDA0002026246220000043
indicating that the final exit medium is a vacuum. The reflection coefficient matrix is defined as:
Figure BDA0002026246220000044
where the superscripts i and r denote incidence and reflection, respectively. According to a transmission matrix M N An expression for the reflection coefficient can be obtained as
Figure BDA0002026246220000045
When Θ =0, the deflection reflection coefficient r sp And r ps Turning to 0, the reflection coefficient matrix of the topological insulator reduces to the fresnel reflection coefficient matrix of the conventional insulator.
And a sixth step: and solving a spontaneous emissivity expression of the three-energy-level atoms in the multilayer topological insulator structure. According to the Fermi gold rule, the expression of the atomic spontaneous decay rate is as follows:
Figure BDA0002026246220000051
deducing a parallel Green function expression of a parallel dipole and a vertical dipole when an atom is positioned in a cavity as
Figure BDA0002026246220000052
/>
Figure BDA0002026246220000053
K 0 Is the wave number in vacuum, K || And K 0z Respectively, parallel and perpendicular to the interface.
Combining the dyadic Green function of the parallel dipole and the vertical dipole when the atoms are positioned in the cavity formed by the multilayer topological insulator, calculating the spontaneous emissivity of the parallel dipole and the vertical dipole as follows:
Figure BDA0002026246220000054
Figure BDA0002026246220000055
wherein
Figure BDA0002026246220000056
The index u (l) indicates the reflection coefficient of the upper layer (lower layer) of the cavity. When an atom is located near a single multilayer topological insulator structure, r l ss ＝r l pp ＝0，D p0 ＝D s0 ＝1。
According to the reflection coefficient calculated in the fifth step and the expression of atomic spontaneous emission and quantum interference in the sixth step, the influence of the multilayer topological insulator structure and the cavity structure formed by the multilayer topological insulator structure on atomic spontaneous emission and quantum interference can be analyzed.
The invention has the following characteristics:
1. according to the invention, the spontaneous radiation and quantum interference of the three-level atoms in the multilayer topological insulator structure and the cavity thereof are calculated according to the transmission matrix, and the spontaneous radiation and quantum interference characteristics of the three-level atoms in the multilayer topological insulator structure and the cavity thereof can be accurately analyzed.
2. The invention can accurately reflect the influence of the dissipation of the material, the dipole direction of the atoms and the position of the atoms on the spontaneous radiation and quantum interference of the atoms.
Drawings
FIG. 1 is an analytical flow chart of the present invention.
FIG. 2 is a schematic diagram of a model of a multilayer topological insulator and its cavity in accordance with the present invention: topological Insulator (TI) surfaces are covered with magnetic films, with atoms located near one of the multilayer TI structures or in the cavities formed by the multilayer TI.
FIG. 3 is a schematic diagram of system input and output.
FIG. 4 (a) is a graph showing the change of spontaneous emissivity of parallel dipoles with atom positions when the TI surface is magnetized in parallel with dissipation;
FIG. 4 (b) is a graph showing the variation of spontaneous emissivity of a perpendicular dipole with atomic position when the TI surface is magnetized in parallel with dissipation;
FIG. 5 (a) is a plot of spontaneous emissivity of a five-layer TI atom-parallel dipole as a function of atom position without dissipation;
FIG. 5 (b) is a plot of spontaneous emissivity of a five-layer TI atomic vertical dipole as a function of atomic position without dissipation;
FIG. 5 (c) is a plot of quantum interference intensity of five-layered TI atoms as a function of atomic position without dissipation;
FIG. 6 is a curve of the three-layer TI atomic quantum interference intensity as a function of the cavity length.
Detailed Description
The following describes embodiments of the present invention in detail with reference to the accompanying drawings.
The method for calculating the spontaneous radiation of the three-energy-level atoms in the multilayer topological insulator structure, as shown in fig. 1, comprises the following six steps:
the first step is as follows: establishing a model of the multilayer topological insulator structure, as shown in fig. 2;
the multilayer topological insulator structure consists of topological insulators and a conventional insulator periodic arrangement, wherein the dielectric constant and the magnetic permeability of the conventional insulator are epsilon 1 And mu 1 The dielectric constant and the magnetic permeability of the three-dimensional topological insulator are epsilon 2 And mu 2 。
The second step is that: determining electromagnetic properties of the topological insulator;
according to the topological field theory, the general electromagnetic response terms in the topological insulator are:
S 0 ＝∫dx 3 dt(ε 2 E 2 -B 2 /μ 2 ) (1)
the electromagnetic response terms related to the topological magnetoelectric response are as follows:
S Θ ＝(αΘ/4π 2 )∫dx 3 dtE·B (2)
due to existence of topological magnetoelectric coupling effect, a topological contribution term is added in the constitutive relation of the three-dimensional topological insulator, and the expression is as follows:
Figure BDA0002026246220000071
deducing the relation between components of the electric field and the magnetic field in the TI:
Figure BDA0002026246220000072
the third step: boundary conditions of the electromagnetic wave on the interface are determined.
The boundary conditions at the interface of the topological insulator multilayer structure are:
z×E 1 ＝z×E 2 (5)
z×H 1 ＝z×H 2 (6)
the fourth step: and solving the transmission matrix of the multilayer topological insulator by using the boundary condition.
According to the boundary condition of the electromagnetic field, the system of equations among the incident electric field, the reflected electric field and the transmitted electric field at the interface of the conventional insulator (medium 1) and the topological insulator (medium 2) is obtained as follows:
Figure BDA0002026246220000081
wherein
Figure BDA0002026246220000082
Wave numbers, k, in media 1 and 2, respectively || Is the wavenumber component parallel to the interface.
According to boundary conditions on the interface, the obtained reflection matrix is as follows:
Figure BDA0002026246220000083
Figure BDA0002026246220000084
is a transmission matrix of electromagnetic waves from medium 1 to medium 2, is combined>
Figure BDA0002026246220000085
According to the transmission characteristics and the propagation theory of the electromagnetic waves in the medium, the transmission matrix of the electromagnetic waves in the j-th layer of medium is obtained as follows:
Figure BDA0002026246220000091
k js represents the wave number of the electromagnetic wave in the vertical direction, d j The thickness of the j-th layer of the medium is shown.
The fifth step: and (5) obtaining the reflection coefficient of the multilayer topological insulator structure.
For a periodic structure with N layers of media, the transmission matrix of the whole system is the multiplication of a transfer matrix and a propagation matrix in sequence of a multilayer structure, and the total transmission matrix is as follows:
Figure BDA0002026246220000092
Figure BDA0002026246220000093
indicating that the final exit medium is a vacuum. The reflection coefficient matrix is defined as:
Figure BDA0002026246220000094
where the superscripts i and r denote incidence and reflection, respectively. According to a transmission matrix M N An expression for the reflection coefficient can be obtained as
Figure BDA0002026246220000095
When Θ =0, the deflection reflection coefficient r sp And r ps Turning to 0, the reflection coefficient matrix of the topological insulator reduces to the fresnel reflection coefficient matrix of the conventional insulator.
And a sixth step: and solving a spontaneous emissivity expression of the three-energy-level atoms in the multilayer topological insulator structure. According to the Fermi gold rule, the expression of the atomic spontaneous decay rate is as follows:
Figure BDA0002026246220000096
deducing a parallel Green function expression of a parallel dipole and a vertical dipole when an atom is positioned in a cavity as
Figure BDA0002026246220000101
Figure BDA0002026246220000102
K 0 Is the wave number in vacuum, K || And K 0z Respectively, parallel and perpendicular to the interface.
Combining the dyadic Green function of the parallel dipole and the vertical dipole when atoms are positioned in a cavity formed by the multilayer topological insulator, calculating the spontaneous emittance of the parallel dipole and the vertical dipole as follows:
Figure BDA0002026246220000103
Figure BDA0002026246220000104
wherein
Figure BDA0002026246220000105
The index u (l) indicates the reflection coefficient of the upper layer (lower layer) of the cavity. When the atoms are in a single multilayer topological insulator structure, r l ss ＝r l pp ＝0，D p0 ＝D s0 ＝1。
According to the reflection coefficient calculated in the fifth step and the expressions of atomic spontaneous emission and quantum interference in the sixth step, the influence of the multilayer topological insulator structure and the cavity structure formed by the multilayer topological insulator structure on atomic spontaneous emission and quantum interference can be analyzed.
In the present embodiment, as shown in fig. 3, spontaneous radiation and quantum interference of atoms with three energy levels are discussed in terms of magnetization direction of the surface of the topological insulator, dipole moment direction of the atoms, atom position, material dissipation, cavity length, number of TI layers, and difference between TI and the conventional insulator (NI, Θ = 0), respectively. Selecting a nonmagnetic material, wherein the magnetic permeability mu =1, the topological magnetic susceptibility is theta = | pi |, epsilon TI =36+ δ, δ being the dissipation of the material.
The relevant parameters of the atoms, such as the dipole moment and transition frequency of the three-energy-level atoms, are input into the A port. And inputting parameters of the multilayer topological insulator material at the port B, wherein the parameters comprise dielectric constant, surface magnetization direction, dissipation, layer number and the like. Parameters of the input cavity at the C-port, such as cavity length, are not input to represent a single TI multilayer. And the spontaneous radiance of the model is output at the D port, and the quantum interference intensity is output at the E port.
In this embodiment, the related parameter of the atom, the transition frequency ω of the atom, is input at the A port α =30THz, corresponding wavelength λ 0 The input magnetic permeability mu =1 of the port B, the topological magnetic susceptibility theta = | pi |, and epsilon TI With a number of TI layers of 3, input material dissipation δ =0.01, 0.02, 0.04, no input cavity length, representing a single TI multilayer, fig. 4 (a) -4 (b) are spontaneous emissivity curves for atoms located near the multilayer TI structure, taking the case of TI surface parallel magnetization as an example. The B port dissipation is set to (δ = 0), the number of TI layers is changed to 5, and the curves of the atomic spontaneous emission and quantum interference with the atomic position are shown in fig. 5 (a) -5 (c). The number of the B port input TI layers is 3, and the length d of the C port input cavity 0 ＜λ 0 The variation of the atomic quantum interference intensity with cavity length is shown in FIG. 6.
According to the method, the three-level atomic spontaneous radiation and quantum interference in the multilayer topological insulator structure and the cavity of the multilayer topological insulator structure are obtained based on the transmission matrix, the influences of the dipole direction of the three-level atoms, the dissipation of materials and the cavity length of the model on the atomic spontaneous radiation and the quantum interference can be analyzed, and the influences of various parameters on the spontaneous radiation and the quantum interference can be accurately positioned. Because the material cost is higher in practice, the theoretical model used in the invention is closer to the actual multilayer topological insulator material, and the theoretical model has application value when being used as a test model. The system can calculate and test to obtain the required atomic spontaneous emissivity and quantum interference intensity, can provide a new application for multilayer topological insulator materials, and also provides a new way for controlling the spontaneous emission of atoms.
While the preferred embodiments and principles of this invention have been described in detail, it will be apparent to those skilled in the art that variations may be made in the embodiments based on the teachings of the invention and such variations are considered to be within the scope of the invention.

Claims (1)

1. The spontaneous radiation calculation method of the three-energy-level atoms in the multilayer topological insulator structure is characterized by comprising the following steps of: the method comprises the following steps:
the first step is as follows: establishing a model of a multilayer topological insulator structure;
the second step is that: determining electromagnetic properties of the topological insulator;
the third step: determining boundary conditions of the electromagnetic waves on the interface;
the fourth step: obtaining a transmission matrix of the multilayer topological insulator by using boundary conditions;
the fifth step: solving the reflection coefficient of the multilayer topological insulator structure;
and a sixth step: solving a spontaneous radiation expression of the three-energy-level atoms in the multilayer topological insulator structure and a cavity formed by the multilayer topological insulator structure;
in the first step: the multilayer topological insulator structure is formed by a topological insulator and a conventional topological insulatorMulti-layer structure of periodic arrangement of insulators, in which conventional insulators have dielectric constants and magnetic permeabilities of epsilon 1 And mu 1 The dielectric constant and permeability of the topological insulator are epsilon 2 And mu 2 ；
The second step is as follows:
according to the topological field theory, the electromagnetic response term in the topological insulator is as follows:
S 0 ＝∫dx 3 dt(ε 2 E 2 -B 2 /μ 2 ) (1)
the electromagnetic response terms related to the topological magnetoelectric response are as follows:
S Θ ＝(αΘ/4π 2 )∫dx 3 dtE·B (2)
theta is the topological magnetic susceptibility;
adding a topological contribution term in a constitutive relation of the three-dimensional topological insulator, wherein the expression is as follows:
Figure FDA0003971907240000011
relationship between electric field and magnetic field components in topological insulator:
Figure FDA0003971907240000021
the third step is as follows:
the boundary conditions at the interface of the topological insulator multilayer structure are:
z×E 1 ＝z×E 2 (5)
z×H 1 ＝z×H 2 (6)
the fourth step is as follows:
according to the boundary condition of the electromagnetic field, obtaining an equation system among an incident electric field, a reflection electric field and a transmission electric field at the interface of the conventional insulator and the topological insulator as follows:
Figure FDA0003971907240000022
wherein
Figure FDA0003971907240000023
Figure FDA0003971907240000024
Wave numbers, k, in media 1 and 2, respectively || Is the wavenumber component parallel to the interface;
the resulting reflection matrix is:
Figure FDA0003971907240000025
Figure FDA0003971907240000026
is the transmission matrix of the electromagnetic wave from the medium 1 to the medium 2, is based on>
Figure FDA0003971907240000027
According to the transmission characteristics and the propagation theory of the electromagnetic waves in the medium, the transmission matrix of the electromagnetic waves in the j-th layer of medium is obtained as follows:
Figure FDA0003971907240000031
k js represents the wave number of the electromagnetic wave in the vertical direction, d j Represents the thickness of the j-th layer medium;
the fifth step is as follows:
for a periodic structure with N layers of media, the transmission matrix of the whole system is the multiplication of a transfer matrix and a propagation matrix in sequence of a multilayer structure, and the total transmission matrix is as follows:
Figure FDA0003971907240000032
Figure FDA0003971907240000033
indicating that the final exit medium is vacuum;
the reflection coefficient matrix is defined as:
Figure FDA0003971907240000034
where i and r denote incident and reflected, respectively;
according to a transmission matrix M N Obtaining an expression of the reflection coefficient of
Figure FDA0003971907240000035
The sixth step is as follows:
according to the Fermi gold rule, the expression of the atomic spontaneous decay rate is as follows:
Figure FDA0003971907240000036
dipole parallel and perpendicular to boundary when bound atoms are located in cavity formed by multilayer topological insulator
And (3) calculating the spontaneous radiance of the parallel dipole and the vertical dipole by the parallel Green function in the time-plane as follows:
Figure FDA0003971907240000041
K 0 is the wave number in vacuum, K || And K 0z Components parallel and perpendicular to the interface, respectively;
Figure FDA0003971907240000042
wherein
Figure FDA0003971907240000043
The reflection coefficient indicated by the subscript u is the reflection coefficient of the superstructure of the cavity; the index l indicates the reflection coefficient of the underlying structure of the cavity;
when the atoms are located in a single multi-layer topological insulator structure,
r l ss ＝r l pp ＝0，D p0 ＝D s0 ＝1；
and analyzing the influence of the multilayer topological insulator structure and the cavity structure formed by the multilayer topological insulator structure on atomic spontaneous emission and quantum interference according to the reflection coefficient calculated in the fifth step and the expressions of the atomic spontaneous emission rate and the quantum interference intensity in the sixth step.
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