Expected values for SUSY hierarchies of Jaynes-Cummings Hamiltonian

TL;DR

The paper investigates how supersymmetric (SUSY) partner hierarchies of the Jaynes-Cummings Hamiltonian affect dynamical observables. It constructs k-SUSY JC partners via intertwining operators, derives the almost-isospectral spectra, and computes the time evolution of atomic inversion, field operators a^±, and quadratures for both Fock and coherent initial states, including classical and revival time analyses. It finds that atomic inversion exhibits SUSY-dependent classical and revival times, while field-operator dynamics remain largely JC-like over several revivals, with no universal SUSY fingerprint across the observables. The work sheds light on how spectral structure from SUSY manifests (or not) in measurable quantum-optical dynamics and informs potential experimental probes of SUSY in JC-like systems.

Abstract

The aim of this letter is to compute the evolution of some expected values, such as the field operators $a^{\pm}$, quadratures and atomic inversion, under SUSY partner Hamiltonians associated to the Jaynes-Cummings Hamiltonian of quantum optics. This kind of SUSY partners are characterized by having spectra which differ in a finite number of energy levels. We wish to elucidate if the partner connection has any influence on these expected values. In particular, we want also to know in which way the classical and revival times are affected by such SUSY partners.

Figures (11)

• Figure 1: Two distinct ways to represent the lower energy levels for the JC Hamiltonian (with $\delta=3$, $\lambda=1$). (a) Represents the pairs of eigenvalues $\varepsilon_n^\pm$ depending on $n$, and (b) represents two columns, the left corresponding to $\varepsilon_n^+$, the right to $\varepsilon_n^-$ eigenvalues, ordered by their values.
• Figure 2: (a) Plot of the first energy levels for both partner Hamiltonians: $H_{\rm JC}$ (in black) and $\tilde{H}_{\rm JC}^{(2)}$ (in red) with $\delta=3$, $\lambda=1$. (b) The common points of their spectrum are linked by dashing lines. Their spectra differ in 4 points, two of them from the 'left spectrum' ($\varepsilon^+_1$ and $\varepsilon^+_2$) and two from the 'right spectrum' ($\varepsilon^-_0$ and $\varepsilon^-_1$) of $H_{\rm JC}$, each at the botton of its respective column. They do not have corresponding isospectral points in the partner Hamiltonian $\tilde{H}_{\rm JC}^{(2)}$.
• Figure 3: (a) Classical times for $\tilde{H}_{\rm JC}^{(k)}$ for $\delta=3,\lambda=1, n=4$, $k=-13,\dots,10$. (b) The same for revival times. (c) Differences of revival times $\Delta t_r$. (d) Superposition of classical and differences of revival times.
• Figure 4: Evolution of atomic inversion for a JC Hamiltonian with $\delta=3, \lambda=1, \alpha=2$ (black) and two next SUSY partners (for $k=1$ in red and for $k=2$ in green).
• Figure 5: Details of Fig. \ref{['fig4']} showing (a) the revival times and (b) the classical times for the three cases.