Expected values for SUSY hierarchies of Jaynes-Cummings Hamiltonian

This paper investigates how supersymmetric partner Hamiltonians of the Jaynes-Cummings model, which differ by a finite number of energy levels, influence the time evolution of key quantum observables such as field operators, quadratures, and atomic inversion, as well as their associated classical and revival times.

The Gist

Imagine you are watching a complex dance between a tiny atom and a beam of light. In the world of quantum physics, this dance is described by something called the Jaynes-Cummings (JC) Hamiltonian. Think of this Hamiltonian not as a scary math formula, but as the "choreography rules" that dictate how the atom and light move together.

This paper asks a very specific question: If we slightly tweak the choreography rules to create a "partner" dance, does the actual performance look different to an observer?

Here is the breakdown of their findings using everyday analogies:

1. The "Almost Identical" Twins (SUSY Partners)

The authors are studying a special family of these dance rules called Supersymmetric (SUSY) partners.

• The Analogy: Imagine you have a song. Now, imagine you create a "partner song" that is identical to the original, except you removed the first two notes and shifted everything else up by one beat. To a listener, the songs sound almost exactly the same, but they aren't perfectly identical.
• The Science: These "partner" Hamiltonians have energy levels (the notes of the song) that are almost the same as the original, differing only in a few specific spots. The authors wanted to see if this tiny difference in the "rules" changed the "dance moves" we can actually measure.

2. The Atomic Inversion: The "Heartbeat" of the System

The first thing they measured was the atomic inversion.

• The Analogy: Think of the atom as a pendulum swinging back and forth between two states (like a light switch being flipped on and off). The "inversion" is just a measure of how much time the atom spends in the "on" state versus the "off" state. It's like measuring the atom's heartbeat.
• The Finding: When they compared the original dance to the SUSY partner dance, the heartbeats were shockingly different.
  • If you watched the original dance, the atom would pause and restart its rhythm at specific times (called "revivals").
  • If you watched the SUSY partner dance, those pauses happened at slightly different moments.
  • The Key Insight: The paper found that if you line up the original dance with its SUSY partners, their "heartbeats" line up perfectly in a predictable pattern. However, if you compare the original to a random dance (a non-partner), the rhythms are completely out of sync.
  • Conclusion: The atomic heartbeat is a perfect "fingerprint." If you see this specific rhythmic pattern, you know for sure you are watching a SUSY partner.

3. The Light Field: The "Foggy Mirror"

Next, they looked at the field operators (the light itself) and quadratures (which are like the position and momentum of the light waves).

• The Analogy: Imagine looking at a reflection in a mirror. The atomic inversion was like looking at a clear, sharp reflection where you could easily tell if the mirror was the "partner" version. But looking at the light field is like looking at a reflection in a foggy, rippling pond. The water is moving so fast and in so many directions that it's hard to tell what's going on.
• The Finding: When they watched the light itself, the difference between the original dance and the SUSY partner dance disappeared.
  • The light waves moved in a complex, chaotic way involving many different frequencies.
  • Whether the dance was the "original" or a "SUSY partner," the light looked the same. Even a "non-partner" dance looked just as similar.
  • Conclusion: The light field is too noisy and complex to act as a fingerprint. You cannot tell if you are watching a SUSY partner just by looking at the light; the subtle differences get washed out by the chaos of the waves.

4. The Big Picture

The authors' main takeaway is a lesson in what to measure:

• If you want to know if two quantum systems are "SUSY partners" (those special, almost-identical twins), look at the atom. Its rhythm will tell you the truth.
• If you look at the light, you will be confused because the signal is too messy to distinguish the subtle differences.

In summary: The paper proves that while these special "partner" systems share a deep mathematical connection, that connection only shows up clearly when you watch the atom's internal rhythm. The light they emit, however, hides that secret connection in a sea of complexity.

Technical Summary

Technical Summary: Expected Values for SUSY Hierarchies of Jaynes-Cummings Hamiltonians

Problem Statement
This work investigates the dynamical behavior of supersymmetric (SUSY) partner Hamiltonians associated with the Jaynes-Cummings (JC) model in quantum optics. While previous research established that a sequence of JC Hamiltonians with specific detuning parameters forms a SUSY hierarchy (sharing spectra up to a finite number of energy levels), the physical implications of this mathematical relationship on observable quantities remained unexplored. The central problem addressed is whether the "partner connection" between these Hamiltonians manifests in the time evolution of expected values, specifically regarding atomic inversion and field operators. The authors aim to determine if these expected values exhibit unique "fingerprints" that distinguish SUSY partners from non-partners, particularly concerning classical periods and quantum revival times.

Methodology
The study builds upon the formalism of SUSY quantum mechanics applied to the JC Hamiltonian under the rotating wave approximation (RWA).

1. Hamiltonian Construction: The authors utilize a hierarchy of JC Hamiltonians, $\tilde{H}^{(k)}_{JC}$, generated by iterative SUSY transformations. These partners are characterized by a modified detuning parameter $\delta_k = \sqrt{\delta^2 + k\lambda^2}$ and an energy shift, while the coupling constant $\lambda$ remains invariant. The spectrum of a $k$-partner is almost isospectral to the original, differing only in the lowest few energy levels.
2. Initial States: The evolution is analyzed using two types of initial states:
   • Fock states (number states) $|n\rangle$ coupled to atomic states.
   • Coherent states $|\alpha\rangle$ (Glauber states) coupled to excited atomic states, which are considered more appropriate for studying collapse and revival phenomena.
3. Time Evolution: The time-dependent spinors are derived by expanding the initial states in the eigenbasis of the specific SUSY partner Hamiltonian. The time evolution operator $U(t) = e^{-iHt}$ is applied to these expansions.
4. Observable Computation: The authors compute the time-dependent expectation values for:
   • Atomic Inversion ($\sigma_3$): Calculating $W(t) = \langle \Psi(t) | \sigma_3 | \Psi(t) \rangle$.
   • Field Operators ($a_\pm$): Computing $\langle \Psi(t) | a_\pm | \Psi(t) \rangle$, which are complex-valued.
   • Quadrature Operators ($x_1, x_2$): Derived from the field operators to represent real-valued observables analogous to position and momentum.
5. Characteristic Times: The study defines and compares the classical time ($t_c$) and revival time ($t_r$) for the evolution of these states across different members of the SUSY hierarchy and non-SUSY counterparts.

Key Contributions and Results

• Atomic Inversion as a SUSY Fingerprint:
  The evolution of the atomic inversion operator $\sigma_3$ is found to be highly sensitive to the SUSY partnership.

  • Revival Structure: For consecutive SUSY partners, the difference in revival times ($\Delta t_r$) is approximately equal to the classical oscillation period ($t_c$). This results in the maxima of the atomic inversion for consecutive partners being "in phase" near the revival times.
  • Differentiation: Non-SUSY partners (Hamiltonians with detuning parameters not linked by the SUSY hierarchy formula) exhibit a separation in revival times that does not match the classical period, leading to out-of-phase oscillations.
  • Conclusion: The atomic inversion serves as a robust indicator of SUSY hierarchy membership, clearly distinguishing partners from non-partners.

• Field Operators and Quadratures:
  In contrast to atomic inversion, the expectation values of the field creation/annihilation operators ($a_\pm$) and their quadratures ($x_1, x_2$) do not effectively differentiate SUSY partners.

  • Complex Dynamics: The evolution of $\langle a_\pm \rangle$ involves multiple frequencies, leading to complex parametric trajectories in the complex plane. While the modulus exhibits collapse and revival behavior similar to coherent states, the phase and frequency evolution become intricate over long times.
  • Lack of Distinction: Comparisons between SUSY partners and non-SUSY partners (with different detuning parameters) reveal no significant qualitative differences in the evolution of these field observables. The authors attribute this to the involvement of multiple fundamental frequencies in the field operator evolution, which washes out the specific signatures of the SUSY partnership.
  • Long-Time Behavior: At sufficiently long times, the intensity of revivals for field operators decreases, and interference effects dominate, a behavior consistent with previous studies on the JC model but not specific to the SUSY hierarchy.

Significance and Claims
The paper claims to elucidate the physical relevance of SUSY hierarchies in quantum optics by testing them against measurable dynamical quantities.

• Selective Sensitivity: The primary significance is the demonstration that the "partner connection" is not universally visible in all observables. It is clearly imprinted on the atomic inversion (a two-level system observable) but is obscured in field observables due to the complexity of the radiation field's frequency spectrum.
• Methodological Continuity: This work serves as a direct continuation of previous theoretical work establishing the existence of JC SUSY hierarchies. It moves from spectral analysis to dynamical analysis, confirming that while the spectra are almost isospectral, the dynamical signatures depend heavily on the specific operator being measured.
• Limitations: The authors modestly note that while SUSY methods are well-developed for Schrödinger and Dirac Hamiltonians (e.g., in graphene), their application to matter-radiation interaction is a developing area. They explicitly state that the results for field operators suggest that not all features of the SUSY hierarchy are "specific" to the partnership, as broader origins may explain the observed similarities.

The study concludes that while SUSY partners share spectral properties, their dynamical "fingerprints" are operator-dependent, with atomic inversion providing a clear signature of the hierarchy, whereas field operators do not.