Time evolution formalism in the complex scaling method: Application to the E1 response of $^6$He

This paper establishes a novel time-evolution formalism within the complex scaling method using the extended completeness relation to successfully model the real-time dynamics of $^6$He's E1 excitation, revealing how its initial correlated configuration evolves into continuum states via both sequential decay and direct breakup.

The Gist

Imagine you are watching a slow-motion video of a fragile glass vase shattering. You want to understand not just that it broke, but exactly how the pieces flew apart, which ones stayed together for a moment, and how the energy spread out.

This paper is about building a new, high-speed camera for the quantum world, specifically for atoms that are barely holding together.

Here is the breakdown of what the scientists did, using simple analogies:

1. The Problem: The "Frozen Photo" vs. The "Movie"

In nuclear physics, scientists often study unstable atoms (like Helium-6, which has an alpha particle core and two extra neutrons). These atoms are like a wobbly tower of blocks; they are on the edge of falling apart.

• The Old Way (The Photo Album): Scientists have a powerful tool called the Complex Scaling Method (CSM). Think of this as a super-advanced camera that takes a perfect, frozen photo of the atom. It can tell you exactly what the atom looks like when it's stable, or what it looks like when it's a "resonance" (a short-lived vibration before it breaks). It's great for cataloging the states of the atom.
• The Missing Piece: But a photo doesn't show movement. It doesn't show the movie of the atom breaking apart. Until now, scientists couldn't use this powerful camera to watch the actual time-evolution of the atom falling apart. They had to switch to a different, less precise method to see the motion.

2. The Solution: A New "Time-Machine" Lens

The authors (Kikuchi, Katō, and Myo) figured out how to upgrade the camera. They created a Time-Evolution Formalism.

• The Analogy: Imagine you have a map of a city (the energy states). Usually, you just look at the map to see where the buildings are. The authors figured out how to use that same map to simulate a car driving through the city in real-time.
• How it works: They used a mathematical trick called the "Extended Completeness Relation." Think of this as a universal translator. It allows them to take all the different "snapshots" of the atom (bound states, resonances, and scattered pieces) and stitch them together into a continuous, flowing movie.

3. The Test Drive: The Simple Model

Before applying this to a real atom, they tested it on a simple, made-up system (a single particle moving in a potential well).

• The Result: They compared their new "movie" against a standard, direct simulation (like a physics engine in a video game). The two matched perfectly. This proved their new camera works and can accurately predict how a quantum wave packet moves and spreads out over time.

4. The Main Event: Watching Helium-6 Break Apart

Now, they applied this to Helium-6 ($^6$He). This is a "halo nucleus," meaning it has a tight core (an alpha particle) and two "valence" neutrons that are loosely orbiting it, like two kids holding onto a parent's hand while running in a circle.

They gave the atom a little "kick" (an Electric Dipole or E1 excitation) to start the breakup. Here is what they saw in their new movie:

• The Starting Pose: At the very beginning, the two neutrons are huddled close together, acting like a "dineutron" (a pair of best friends holding hands).
• The Breakup: As time passes, the movie shows two different ways the atom falls apart happening at the same time:
  1. The Relay Race (Sequential Decay): One neutron lets go and runs away, leaving the other neutron still attached to the core for a moment. It's like a relay race where the baton is passed. The atom briefly becomes a "Helium-5" (Core + 1 neutron) before the second neutron also flies off.
  2. The Explosion (Direct Breakup): Both neutrons let go of the core and each other simultaneously, flying off in different directions like an explosion.

5. Why This Matters

The beauty of this paper is that it unifies two worlds.

• Before: Scientists used one tool to study the structure (what the atom looks like) and a different tool to study the dynamics (how it moves).
• Now: They have one unified framework. They can look at the same atom and see its structure, its resonances, and watch its entire breakup process in real-time, all within the same mathematical language.

In Summary:
The authors built a new mathematical lens that turns a static photo of a fragile atom into a high-definition movie. By watching this movie, they discovered that when Helium-6 breaks apart, it doesn't just fall apart in one way; it does a complex dance involving both a "relay race" style breakup and a simultaneous explosion, all happening at once. This helps us understand the fundamental rules of how matter holds together and falls apart in the quantum world.

Technical Summary

1. Problem Statement

The Complex Scaling Method (CSM) is a well-established non-Hermitian quantum mechanical framework used to describe resonant states and continuum spectra in nuclear few-body systems. By rotating spatial coordinates into the complex plane ($r \to re^{i\theta}$), resonant states appear as isolated eigenvalues of the complex-scaled Hamiltonian, separable from the continuum.

However, a significant gap existed in the CSM framework: while it successfully handled spectral properties (energy eigenvalues, strength functions, scattering observables), it lacked a general formulation for explicit time evolution in the continuum. Existing time-dependent approaches (e.g., Crank-Nicolson methods) operate in coordinate space but struggle with the consistent treatment of resonances and continuum couplings that CSM provides. The authors aimed to bridge this gap by formulating a time-evolution formalism within CSM to study real-time dynamics, specifically how initial correlations in weakly bound nuclei evolve into continuum states.

2. Methodology

A. Theoretical Framework

The authors developed a time-evolution formalism based on the Extended Completeness Relation (ECR).

• Complex Scaling: The Hamiltonian $\hat{H}$ is transformed via a similarity operator $\hat{U}(\theta)$ into a complex-scaled Hamiltonian $\hat{H}_\theta$.
• Spectral Representation: Using the ECR, the identity operator is expanded over the complete set of eigenstates (bound, resonant, and discretized continuum) of $\hat{H}_\theta$.
• Time-Evolution Operator: The standard time-evolution operator $e^{-i\hat{H}t/\hbar}$ is rewritten using the complex-scaled basis:
  $$|\Phi(t)\rangle = \hat{U}^{-1}(\theta) \sum_{\nu} |\Psi^\theta_\nu\rangle e^{-iE^\theta_\nu t/\hbar} \langle \tilde{\Psi}^\theta_\nu | \hat{U}(\theta) | \Phi(0) \rangle$$
  Here, $|\Psi^\theta_\nu\rangle$ and $\langle \tilde{\Psi}^\theta_\nu |$ are the right and left eigenstates of $\hat{H}_\theta$, and $E^\theta_\nu$ are the complex eigenvalues.
• Physical Interpretation:
  • Resonances: Exhibit exponential decay ($e^{-\Gamma t/2\hbar}$) naturally.
  • Continuum: Discretized continuum components acquire exponential damping factors due to the complex rotation. This acts mathematically as an absorbing boundary condition, preventing the reflection of outgoing flux, similar to Complex Absorbing Potentials (CAPs) but derived rigorously from the spectral decomposition.

B. Numerical Implementation

1. Test Calculation (Two-Body Model):

   • A simple radial model with a Gaussian potential and $l=1$ (p-wave) was used to generate a broad resonance.
   • The time evolution of a Gaussian wave packet was calculated using the new CSM formalism.
   • Validation: Results were compared against a direct numerical solution of the time-dependent Schrödinger equation using the Crank-Nicolson method.

2. Application to $^6$He (Three-Body Model):

   • System: $^6$He modeled as an $\alpha + n + n$ three-body system.
   • Hamiltonian: Included $n-\alpha$ (KKNN potential), $n-n$ (Minnesota force), a phenomenological three-body force, and the Orthogonality Condition Model (OCM) to handle Pauli exclusion.
   • Excitation: The system was excited from the ground state ($0^+$) to the continuum via an Electric Dipole (E1) operator.
   • Basis: Gaussian Expansion Method (GEM) with complex scaling ($\theta = 10^\circ$).
   • Analysis: The time evolution of the density distribution was analyzed in two different Jacobi coordinate systems:
     • $(r_{n-n}, R_{\alpha-nn})$: Relative motion of two neutrons vs. $\alpha$ core.
     • $(r_{\alpha-n}, R_{n-\alpha n})$: Relative motion of one neutron vs. $\alpha$ core vs. the remaining neutron.

3. Key Contributions

• Formal Extension of CSM: Established the first general framework for time evolution within CSM, unifying the description of structure (eigenvalues), scattering (observables), and dynamics (time evolution) under the Extended Completeness Relation.
• Unified Dynamics: Demonstrated that the CSM can naturally describe the decay of resonances and the propagation of continuum wave packets without ad-hoc boundary conditions, as the complex scaling inherently absorbs outgoing flux.
• Visualization of Decay Modes: Provided a method to visualize the emergence of different decay channels (sequential vs. direct) in real-time within a single theoretical framework.

4. Results

A. Validation (Two-Body Test)

• The CSM time-evolution results for the radial density distribution $\rho(r,t)$ showed excellent agreement with the direct Crank-Nicolson solution at various time steps ($ct = 20, 40, 80$ fm).
• The integrated norm $N(t)$ decreased over time, confirming the absorbing nature of the complex scaling (damping of outgoing flux). This behavior was found to be stable and independent of the scaling angle $\theta$, proving it is a physical feature of the method rather than a numerical artifact.

B. Application to $^6$He E1 Excitation

• Initial State: Immediately after E1 excitation, the wave packet exhibited a dineutron-like configuration (strong correlation between the two neutrons), concentrated at small inter-neutron distances ($r_{n-n}$).
• Time Evolution:
  • As time progressed ($ct = 50 \to 300$ fm), the wave packet expanded spatially.
  • The initial dineutron correlation weakened, and the system evolved into spatially extended continuum states.
• Decay Mode Identification:
  • Sequential Decay: In the $(r_{\alpha-n}, R_{n-\alpha n})$ coordinate system, a distinct vertical structure appeared at small $r_{\alpha-n}$ (indicating a bound $^5$He core-neutron subsystem) while the third neutron propagated outward. This component was visible even at early times ($ct=50$ fm).
  • Direct Breakup: A simultaneous extension in both coordinates ($r_{\alpha-n}$ and $R_{n-\alpha n}$) indicated a direct breakup where all three constituents separate without forming an intermediate resonance.
• Coexistence: The analysis revealed that both sequential decay and direct breakup coexist in the E1 excitation of $^6$He, with the sequential component forming early in the evolution.

5. Significance

This work fundamentally broadens the scope of the Complex Scaling Method. Previously limited to energy-domain spectroscopy and scattering observables, CSM is now capable of describing real-time continuum dynamics.

The significance lies in:

1. Unified Perspective: It connects initial-state correlations, continuum structure, and decay dynamics within a single, rigorous non-Hermitian framework.
2. Mechanism Insight: It provides a clear, visualizable mechanism for how weakly bound nuclei (like halo nuclei) decay, distinguishing between sequential and direct breakup processes dynamically.
3. Methodological Robustness: It validates that complex scaling acts as a natural absorbing boundary condition for time-dependent problems, offering a powerful alternative to traditional grid-based time-evolution methods for systems with complex resonance structures.

This formalism opens new avenues for studying time-dependent phenomena in nuclear physics, such as multi-particle emission and the real-time response of exotic nuclei to external fields.