Second excited state of ${}^4\mathrm{He}$ tetramer

Using a momentum-space transition operator framework with realistic interatomic potentials, this study rigorously calculates the atom-trimer scattering of cold ${}^4\mathrm{He}$ atoms to confirm the existence of a second excited tetramer resonance below the excited trimer threshold, determining its position and width while accounting for significant finite-range effects and nonresonant partial wave contributions.

The Gist

The Big Picture: A Family of Floating Balloons

Imagine you have a room full of invisible, super-cold helium balloons (these are Helium-4 atoms). Because they are so cold, they move very slowly, and they have a strange, magical property: they love to stick together in groups, but only in very specific ways.

Physicists have long known that if you have two of these balloons, they can form a weak pair (a dimer). If you have three, they can form a trio (a trimer). The big question this paper answers is: What happens when you try to get four of them to dance together?

The "Ghost" Dance Partner

In the world of quantum physics, there's a famous rule called Efimov Physics. It predicts that if you have a group of three particles that are just barely holding hands, there should be a "cousin" group of four particles that tries to join the dance.

Usually, these groups of four are stable. They sit quietly in a corner. But this paper is about a very special, very unstable group of four.

Think of it like this:

• The Stable Tetramer: A family of four holding hands in a circle. They are solid and real.
• The "Second Excited" Tetramer (The Star of this paper): Imagine a fourth balloon trying to join a trio that is already dancing. The trio is spinning so fast that the fourth balloon can't quite grab on. Instead of sticking, it bumps into them, spins around wildly for a split second, and then flies away.

In physics terms, this isn't a "bound state" (a stable family). It's a resonance. It's a fleeting moment where the four atoms briefly synchronize before falling apart. It's like a "ghost" family that exists for a tiny fraction of a second.

The Challenge: The Sticky and the Bouncy

Why was this so hard to calculate?

1. The "Ghost" Problem: Most computer simulations are great at finding things that stay still (like a stable family). They are terrible at finding things that are fleeting and unstable (like the ghost family). It's like trying to photograph a hummingbird's wingbeat with a camera that only takes pictures of statues.
2. The "Velcro vs. Magnet" Problem: Helium atoms have a weird personality. When they are far apart, they are slightly attracted to each other (like weak Velcro). But if they get too close, they repel each other with the force of a giant spring (like a super-strong magnet pushing away). This makes the math incredibly difficult because the atoms are constantly trying to snap together and then bounce apart.

The Solution: The "Softening" Trick

The author, A. Deltuva, used a clever mathematical trick to solve this. Imagine trying to solve a puzzle where the pieces are made of glass and keep shattering.

• The Trick: He first solved the puzzle using pieces made of soft rubber (a "softened" potential). This was easy.
• The Extrapolation: Once he knew how the rubber pieces fit, he mathematically "stretched" the solution back to the hard glass pieces (the real helium atoms).

This allowed him to calculate exactly how the four atoms behave when they collide, even though they don't form a stable group.

The Discovery: A Sharp "Bump" in the Data

When the author ran the numbers, he found something exciting:

• The Resonance: At a very specific energy level, the four atoms line up perfectly. The "scattering" (how they bounce off each other) spikes dramatically.
• The Analogy: Imagine you are pushing a child on a swing. If you push at the exact right rhythm, the swing goes super high. If you push at the wrong time, nothing happens. This paper found the exact "rhythm" where the four helium atoms swing together in a massive, synchronized bump before falling apart.

However, there's a twist:
The paper found that while this "four-atom swing" is the main event, other "dancers" (atoms spinning in different directions, called P and D waves) are also in the room. They aren't dancing in sync, but they are moving enough that they make the whole scene a bit messy.

• The Result: The "bump" in the data is still there and very noticeable, but it's not as perfectly sharp as the theory predicted for a perfect, ideal world. The "messiness" of the other dancers blurs the signal slightly.

Why Does This Matter?

1. It Confirms a Prediction: It proves that the "Efimov physics" predictions about these ghostly four-atom states are real, even for real helium atoms (not just theoretical models).
2. It Shows the "Real World" Effect: The paper shows that in the real world, atoms have a "size" (finite range). In a perfect, theoretical world, the resonance would be super sharp. In the real world, because the atoms take up space and have that "springy" repulsion, the resonance is a bit wider and fuzzier.
3. Future Experiments: This gives experimentalists a roadmap. If they cool helium atoms down and smash them together, they now know exactly what energy level to look for to see this fleeting "ghost family" appear.

Summary in One Sentence

This paper uses advanced math to prove that four helium atoms can briefly form a "ghostly" dancing group that resonates like a musical note, confirming that even in the messy real world, the beautiful predictions of quantum physics hold true.

Technical Summary

1. Problem Statement

The paper addresses a gap in the understanding of few-body quantum systems, specifically the four-boson universality predicted by Efimov physics. While the existence of ground and first excited states for helium-4 ($^4$He) trimers and tetramers is well-established, the second excited tetramer state has remained uncalculated for realistic systems.

The primary challenges in calculating this state are:

• Nature of the State: Unlike the ground and first excited states, the second excited tetramer is not a bound state. It exists as an unstable bound state (UBS) or resonance located in the continuum, specifically in the atom-trimer scattering channel, well above the two-cluster threshold.
• Interaction Complexity: Realistic $^4$He interatomic potentials feature a weakly attractive van der Waals tail and a very strong short-range repulsion. This makes numerical solutions for spatially extended scattering states highly sensitive and computationally difficult.
• Previous Limitations: Most existing four-body calculations focus on bound states or scattering near the atom-trimer threshold, lacking the capability to resolve resonances deep in the continuum.

2. Methodology

The author employs a rigorous momentum-space transition operator framework to solve the four-body scattering problem.

• Equations: The study utilizes the Alt, Grassberger, and Sandhas (AGS) equations, which are the integral version of the Faddeev-Yakubovsky equations. These equations relate four-particle transition operators ($U_{\beta\alpha}$) to two-particle and three-particle subsystem transition operators.
• Potentials: Two realistic interatomic potentials for $^4$He are used to test model dependence:
  1. LM2M2: A widely used parametrization by Aziz and Slaman.
  2. PCJS: A more recent and sophisticated potential by Przybytek et al.
• Numerical Technique: To handle the strong short-range repulsion, the author applies the "softening and extrapolation" method (previously established in Ref. [17]). This involves gradually reducing the strength of the repulsive core to enable stable numerical solutions, then extrapolating the results back to the original potential strength.
• Partial Wave Expansion: The equations are solved in the momentum-space partial-wave representation using Jacobi momenta. Calculations include partial waves up to orbital angular momenta $l_x, l_y, l_z \leq 6$.
• Observables: The study calculates the on-shell matrix elements of the transition operator to derive the S-wave phase shifts ($\delta_J$) and scattering cross sections ($\sigma_J$) for the atom-trimer system.

3. Key Contributions

• First Calculation of the Second Excited Tetramer: This work provides the first rigorous calculation of the second excited state of the $^4$He tetramer in a realistic system.
• Resonance Characterization: It successfully identifies and characterizes the state as a resonance in atom-trimer scattering, determining its precise position (energy) and width.
• Quantification of Finite-Range Effects: The paper explicitly compares realistic potential results with the strictly universal (zero-range) Efimov scenario, quantifying how finite-range effects alter the resonance properties.
• Analysis of Non-Resonant Background: It highlights the significant contribution of non-resonant partial waves (P and D waves) to the total cross section, a factor often overlooked in idealized Efimov scenarios.

4. Key Results

• Resonant Behavior: A clear resonant behavior is observed in the $J=0$ (S-wave) channel at an energy of approximately $E \approx -1.8 B^*_3$ (where $B^*_3$ is the binding energy of the excited trimer).
  • The phase shift rises sharply, and the cross section increases by a factor of 100 compared to the background.
• Resonance Parameters:
  • Using the LM2M2 potential, the resonance position is $B^{**}_4 / B^*_3 = 1.82(2)$ with a width $\Gamma / B^*_3 = 0.010(1)$.
  • Using the PCJS potential, the position is $1.79(2)$ with a width $\Gamma / B^*_3 = 0.009(1)$.
  • In absolute energy units, the resonance is located at roughly 4.1–4.7 mK with a width of 0.023–0.024 mK.
• Finite-Range Effects:
  • Comparison with universal zero-range predictions shows sizable finite-range effects.
  • Specifically, the resonance width in the realistic system is twice as large as the universal prediction ($\Gamma_{real} \approx 2 \times \Gamma_{universal}$).
  • The model dependence between the two potentials is relatively small compared to the deviation from the universal limit.
• Role of Higher Partial Waves:
  • While the resonance is in the $J=0$ channel, higher angular momentum states ($J=1, 2$) contribute significantly to the non-resonant background.
  • At the resonance energy, the $J=1$ contribution to the cross section actually exceeds the resonant $J=0$ contribution.
  • Consequently, the total atom-trimer cross section at the peak increases by only about 60% (rather than the factor of 100 seen in the $J=0$ component alone) due to the masking effect of these non-resonant waves.

5. Significance

• Observability: The study concludes that the second excited tetramer state is observable in cold atom experiments as a sharp resonance in atom-trimer collisions, despite being masked by non-resonant background contributions.
• Validation of Universality: The work confirms the existence of the second excited tetramer predicted by Efimov physics but demonstrates that finite-range effects are critical for accurate quantitative predictions in realistic systems like $^4$He.
• Methodological Advancement: It validates the "softening and extrapolation" method combined with momentum-space AGS equations as a powerful tool for solving complex four-body scattering problems involving strong short-range repulsion.
• Experimental Guidance: The calculated resonance energy and width provide specific targets for experimentalists working with ultracold $^4$He gases to detect this elusive state.