Calculation of first- and second-order symmetries in Jaynes-Cummings model
Description
We provide three files created in the symbolic computation software Maple (licensed to Universidad de Guadalajara), which were used to calculate the higher symmetries of the system of differential equations associated with the Jaynes-Cummings model (see ArXiV https://arxiv.org/abs/2606.15538). The file Jaynes_CL_2025_2026_NEW_LINEAR_v6_c1_-1.mw contains a semi-automated process for finding the generating functions for first- and second-order symmetries (see Steps to reproduce for details). File verification_Heun_polynomial_final.mw verifies the fulfillment of the sufficient condition for the confluent Heun function to be a polynomial, given the parameters specified in the article. File Luis_graficas_Heun_2025_2026.mw handles the graphics of Heun polynomials and radial components for both invariant solutions; it is also used to generate 3D plots of the characteristic functions at the initial time.
Files
Steps to reproduce
File Jaynes_CL_2025_2026_NEW_LINEAR_v6_c1_-1.mw begins by defining the total derivative operators with respect to the variables $t$, $\phi$, and $R$ (section "operadores de derivada total"). The components of the function $\vec{u}$ are denoted by u1,..., and the derivatives as follows u[1,t] = $\partial u^1/\partial t$. The initial system of differential equations is given in (1)–(4). Subsequently, its differential consequences—necessary for transitioning to the system's manifold—are formulated (section "forming differential sequences"). Then, equations describing the action of the operator $l_F$ on the components of the generating function $\vec{\phi}$ are formulated. They are denoted as e1, ..., e4. The equations involving a transition to the system manifold are designated as EEE1, ..., EEE4. The lists listU1, ..., listU4 contain certain initial defining equations obtained by splitting with respect to higher-order derivatives. Next, the governing equations are solved in a semi-automatic mode. This refines the components of the generating function and leads to the formation of new equations (listEXTRA, listEXTRA2, listEXTRA3). Finally, the general form of the symmetry generating function is given in equations (70)–(73), and its components are denoted by Phi1, ..., Phi4. The "simplification of PSI" section contains the simplification of the obtained generating functions based on the system. In file verification_Heun_polynomial_final.mw, the necessary condition for the confluent Heun function to be a polynomial is designated as NC in (4). Equations (11) through (1.15) verify that the first six determinants vanish (a sufficient condition for polynomiality; see P.Fiziev, “Novel relations and new properties of confluent Heun’s functions and their derivatives of arbitrary order,” J.Phys.A: Math. 43, 035203 (2009)). Equations (13)–(26) contain special cases of the quantity $S_p$. Equation (28) is the general formula for $S_p$. Relations (32)–(34) show that the formula holds for $S_{p+1}$ as well. Expressions (2.1.1)–(2.3.9) for $n= 0,1,2$ show that the components $u^{2,3}$ of the characteristic function for the second invariant solution do not have a singularity at $r = 2a$, since they contain a factor of $r - 2a$ in the numerator. Relations (35)–(39) and (3.1)–(3.15) provide the basis for proving that the radial part of the numerator of the components $u_n^{2,3}$ vanishes at $r = 2a$ for any $n$. Equations (4.1) and those following compare the Heun polynomials—obtained using the derived general formula for the coefficients $v_k$—with the series expansion coefficients of the HeunC function for $n=0,1,2$. At the beginning of the file Luis_graficas_Heun_2025_2026.mw, plots of Heun polynomials for $n= 0,1,2,3$ and the functions $Y_6(r)$ are presented, as well as 3D plots of the characteristic functions $w_f$ at $t=0$ for both invariant solutions. Animated images of these functions, illustrating their dynamics, have been generated for $t = 0..100$.
Institutions
- Universidad de GuadalajaraJalisco, Guadalajara