Macroscopic quantum self-trapping in bosonic Josephson junctions: an exact quantum treatment

This paper demonstrates that while mean-field theory predicts persistent macroscopic quantum self-trapping in symmetric Bose-Josephson junctions, exact quantum dynamics for any finite number of particles inevitably leads to the breakdown of this phenomenon after a finite time, with a quasi-self-trapping regime emerging only for large particle numbers due to specific spectral properties.

The Gist

Imagine you have a playground with two identical swings (let's call them Well A and Well B) connected by a small, bouncy bridge. You have a huge crowd of identical children (the bosons) who love to swing together.

In the world of quantum physics, these children can do something magical: they can tunnel through the bridge, moving back and forth between the two swings. This is the Josephson Junction.

The Two Ways They Move

Usually, if you push the children so that there are slightly more on one side than the other, they will swing back and forth in a perfect rhythm. The crowd moves from side to side, balancing out. This is called Josephson Oscillation. It's like a pendulum swinging left and right, always passing through the center.

But, there's a second, stranger behavior called Macroscopic Quantum Self-Trapping (MQST).
Imagine you push the children so hard that they pile up on one side. In the "classical" or "average" view of the world (called Mean-Field Theory), if the children interact strongly enough, they get "stuck" on that side. They oscillate a tiny bit, but they never fully cross over to the other side. They are self-trapped. It's like a crowd so stubbornly stuck on one side of a room that they refuse to leave, even though the door is open.

The Big Question

For a long time, physicists thought this "stuck" state was a permanent feature of the quantum world, just like it is in the classical average view. They asked: If we look at the exact, messy, individual behavior of every single child (the "exact quantum treatment"), will they still get stuck forever?

The Paper's Discovery: The "No-Go" Rule

The authors of this paper, A. Bardin, A. Minguzzi, and L. Salasnich, say: No. Not forever.

Here is the simple explanation of their findings:

1. The Finite Crowd Problem:
   If you have a finite number of children (even a huge number like 1,000 or 10,000), the "stuck" state is actually an illusion. Because the children are quantum particles, they are like waves. Even if they are mostly on the left side, there is a tiny, tiny wave of them on the right side. Over time, these waves interfere with each other.
   The authors proved mathematically that eventually, the waves will sync up in a way that forces the crowd to cross over to the other side. The "self-trapping" breaks down. The children will eventually visit the other swing. It just might take a very, very long time.

2. The "Branching" Analogy:
   Think of the energy levels of the children like a set of stairs.

   • Weak Interaction: If the children don't mind each other much, the stairs are spaced out evenly. The crowd swings back and forth quickly (Josephson Oscillation).
   • Strong Interaction: If the children are very grumpy and pushy (strong repulsion), the stairs change shape. Some steps get incredibly close together, almost merging.
   • The "Branching" Transition: The authors found a critical point where the stairs suddenly split into two distinct groups.
     • Below this point, the crowd swings normally.
     • Above this point, the crowd looks like it's stuck on one side for a very long time. This is the Quasi-MQST. It's not forever, but it lasts so long that for all practical purposes, it looks like they are trapped.

Why Does This Matter?

This paper bridges the gap between two worlds:

• The Classical World (Mean-Field): Where things are smooth, predictable, and "stuck" states are permanent.
• The Quantum World (Exact): Where everything is wiggly, probabilistic, and "stuck" states are only temporary.

The authors show that the "stuck" state we see in experiments is actually a long-lived illusion that emerges only when you have a massive number of particles. It's a transition from the messy, individual quantum world to the smooth, predictable classical world.

The Takeaway

Imagine you are watching a crowd of people in a room with two doors.

• Classical View: If they are grumpy enough, they will stay in one room forever.
• Quantum Reality: They will eventually wander into the other room, no matter how grumpy they are. But if the crowd is huge, it might take a million years for them to do it.

The paper tells us that true, permanent self-trapping is impossible in a finite quantum system. It's a beautiful reminder that at the deepest level of reality, nothing is ever truly "stuck"; everything is just waiting for the right moment to move.

Technical Summary

1. Problem Statement

The paper investigates the phenomenon of Macroscopic Quantum Self-Trapping (MQST) in Bose-Josephson junctions (BJJs) consisting of two weakly linked Bose-Einstein Condensates (BECs).

• Context: In the mean-field limit (large particle number $N$, weak interactions), BJJs exhibit two regimes: Josephson oscillations (small imbalance) and MQST (large initial imbalance where population remains trapped in one well).
• The Gap: Mean-field theory predicts persistent MQST. However, the validity of this prediction in the fully quantum regime for finite particle numbers is unclear. Previous studies suggest interactions with non-condensed particles or finite-size effects lead to decoherence, but a rigorous proof of MQST breakdown in finite quantum systems was lacking.
• Core Question: Does MQST truly exist for a finite number of particles in a perfectly symmetric double-well potential when treated with exact quantum mechanics, or is it an artifact of the mean-field approximation?

2. Methodology

The authors employ a rigorous exact quantum treatment rather than mean-field approximations.

• Model: The system is modeled using the two-mode Bose-Hubbard Hamiltonian for a symmetric double-well potential. The Hamiltonian is exactly solvable via Bethe ansatz methods.
  • The Hamiltonian is expressed in terms of the relative number operator $\hat{n}$ and relative phase operator $\hat{e}^{i\phi}$.
  • It is represented as a tridiagonal symmetric centrosymmetric matrix of size $(N+1) \times (N+1)$.
• Analytical Approach:
  • No-Go Theorem: The authors utilize the symmetry properties of the Hamiltonian matrix (centrosymmetry) to decompose the eigenbasis into symmetric and antisymmetric families. They analyze the time evolution of the population imbalance expectation value $\langle \hat{z}(t) \rangle$.
  • Spectral Analysis: They perform exact diagonalization to study the scaling of energy eigenvalue differences ($E_{ij}$) and the coefficients of the time-evolution expansion as a function of particle number $N$ and dimensionless interaction strength $\Lambda = UN/J$.
  • Comparison: Results are compared against mean-field predictions and atomic coherent state approximations.

3. Key Contributions and Results

A. The "No-Go" Theorem for Finite $N$

The most significant theoretical contribution is a mathematical proof that MQST cannot occur for any finite number of particles in a symmetric system, regardless of the initial state.

• Mechanism: The population imbalance $\langle \hat{z}(t) \rangle$ is expressed as a sum of cosines involving energy differences $E_{kk'} = E_k - E_{k'}$.
• Non-Degeneracy: For a finite $N$, the tridiagonal Hamiltonian matrix has non-zero off-diagonal elements, ensuring all eigenvalues are non-degenerate.
• Conclusion: Since no energy differences are exactly zero ($E_{kk'} \neq 0$), the oscillatory terms in $\langle \hat{z}(t) \rangle$ never cancel out to produce a constant non-zero offset. The imbalance is guaranteed to cross zero at finite times. Thus, true MQST (permanent trapping) is impossible in the exact quantum model for finite $N$.

B. Emergence of "Quasi-MQST" in the Large-$N$ Limit

While true MQST is forbidden for finite $N$, the authors identify a mechanism for the emergence of a quasi-MQST regime that mimics mean-field behavior as $N \to \infty$.

• Branching Transition: The authors identify a critical interaction strength $\Lambda \sim 1$ where the behavior of energy eigenvalue differences changes:
  • Weak Coupling ($\Lambda \lesssim 1$): Energy differences decay as a power law.
  • Strong Coupling ($\Lambda \gtrsim 1$): Energy differences decay exponentially with $N$.
• Critical Index $\sigma_c(\Lambda)$: A critical sum index $\sigma_c$ separates energy differences that remain finite from those that become infinitesimally small.
• Dominant Terms: For strong interactions ($\Lambda > \bar{\Lambda}$), the terms in the time-evolution sum that dominate correspond to infinitesimal energy differences. This results in oscillations with extremely long periods, creating a quasi-MQST regime where the population imbalance remains near a non-zero value for a very long (but finite) time.

C. Quantum Generalization of the Critical Point

The paper defines a quantum generalization of the mean-field critical interaction strength ($\Lambda_{c,MF}$).

• The transition from Josephson oscillations to quasi-MQST is marked by a jump discontinuity in the function $\sigma^*(\Lambda)$, which represents the index of the dominant energy difference contributing to the dynamics.
• This jump occurs at a critical value $\bar{\Lambda}$, which depends on the initial imbalance but remains finite even as $N \to \infty$.
• The authors propose alternative definitions for this critical point ($\Lambda_1, \Lambda_2$) based on the cumulative weight of contributions from the infinitesimal energy difference sector, all converging to the mean-field limit for large $N$.

4. Significance

• Bridging Quantum and Classical: The work provides a rigorous explanation of how classical nonlinear behavior (MQST) emerges from finite quantum many-body systems. It demonstrates that MQST is not a fundamental quantum state but an emergent phenomenon arising from the specific spectral structure (branching transition) in the large-$N$ limit.
• Experimental Relevance: The findings predict that state-of-the-art experiments with ultracold atoms can access the crossover regime between Josephson oscillations and quasi-MQST. By tuning the coupling strength $\Lambda$, experiments should observe strong deviations from mean-field predictions near the critical point, providing evidence of the underlying "branching transition."
• Fundamental Physics: The "No-Go" theorem clarifies the limitations of mean-field theory, emphasizing that finite-size quantum systems inherently possess dynamics (tunneling to the other well) that prevent permanent self-trapping, a feature often overlooked in approximate treatments.

Summary Conclusion

The paper mathematically proves that true macroscopic quantum self-trapping is impossible in finite quantum systems due to the non-degeneracy of the energy spectrum. However, it elucidates how a quasi-MQST regime emerges for large particle numbers through a branching transition in the energy spectrum, where specific energy differences become exponentially small. This transition acts as the quantum mechanism that reproduces mean-field MQST behavior, offering a precise framework for understanding the quantum-to-classical transition in Bose-Josephson junctions.