Equilibrium and dynamical quantum phase transitions in dipolar atomic Josephson junctions

This study investigates how dipolar interactions in atomic Josephson junctions, modeled by an extended Bose-Hubbard Hamiltonian with nearest-neighbor pair tunneling, fundamentally alter both the equilibrium phase diagram—inducing parity modulations and shifting critical points—and the dynamical behavior, including macroscopic quantum self-trapping and dynamical quantum phase transitions.

The Gist

Imagine you have two buckets (let's call them Left and Right) and a bunch of tiny, magical marbles (atoms) that can jump between them. In the world of quantum physics, these marbles are special: they can exist in two places at once, and they can "tunnel" through the wall separating the buckets without actually climbing over it.

This setup is called a Josephson Junction. Usually, if you put marbles in one bucket, they just slosh back and forth like water in a swinging pendulum. This is the standard "Josephson effect."

However, this paper explores what happens when these marbles have a very specific, long-range "personality": they are dipolar. Think of them as tiny magnets or charged balloons. They don't just bump into each other when they touch; they feel each other's presence even when they are far apart.

Here is the simple breakdown of what the scientists discovered:

1. The New Rule: "The Buddy System" (Pair Tunneling)

In a normal quantum system, marbles jump one by one. But because these dipolar marbles are so sensitive to each other, they start doing something weird: they jump in pairs.

Imagine a dance floor where people usually dance alone. Suddenly, a rule is introduced where if one person moves, their partner must move with them, even if they are on opposite sides of the room. This is Pair Tunneling. The paper shows that this "buddy system" completely changes the rules of the game.

2. The Equilibrium: How the Marbles Settle Down

When the system is calm (equilibrium), the scientists looked at how the marbles distribute themselves between the two buckets.

• The Parity Puzzle: In the old world, the marbles could be in any number in a bucket. With the "buddy system," the marbles develop a strict preference for even or odd numbers. It's like a club where you can only enter if you are wearing a red or blue shirt, but never a mix. The paper shows that this creates a "striped" pattern in the probability of finding marbles, which wasn't there before.
• The Big Shift (Phase Transitions): Usually, if you make the marbles attract each other strongly enough, they all collapse into one bucket (a "NOON" state). The paper found that the "buddy system" changes how this happens.
  • Without buddies: It's a smooth slide into the single-bucket state.
  • With buddies: It becomes a sudden "snap" or a jump. It's the difference between slowly filling a glass with water versus a dam breaking. The "buddy system" also creates a new type of state where the marbles are split based on their timing (phase) rather than just their location.

3. The Dynamics: The Dance of the Marbles

Now, let's watch the marbles move over time.

• Self-Trapping: In the old model, if you push the system hard enough, the marbles get "stuck" in one bucket and refuse to go back. This is called Macroscopic Quantum Self-Trapping.
• The Twist: The "buddy system" changes the conditions for getting stuck. It's like adding a new gear to a bicycle. Sometimes, the marbles get stuck in a new way, oscillating in a weird rhythm that wasn't possible before. Sometimes, the "buddy system" is so strong it actually prevents them from getting stuck, forcing them to keep dancing back and forth.

4. The "Time Travel" Effect (Dynamical Phase Transitions)

This is the most mind-bending part. The scientists looked at how the system behaves over time, not just how it sits still.

They used a concept called the Loschmidt Echo. Imagine you take a photo of the marbles at the start, let them dance for a while, and then take another photo. The "Echo" measures how much the second photo looks like the first.

• The Cusp: Usually, this similarity fades smoothly. But at certain critical moments in time, the similarity drops sharply, like a cliff edge. This is a Dynamical Quantum Phase Transition (DQPT).
• The Analogy: Think of a spinning top. It wobbles smoothly. But at a specific moment, it suddenly jerks and changes its wobble pattern. That "jerky moment" is the DQPT.
• The Finding: The "buddy system" doesn't stop these jerky moments from happening, but it shifts the time they occur. It's like changing the tempo of a song; the beat is still there, but it hits at a different time.

Why Does This Matter?

Think of this research as learning how to tune a very complex musical instrument.

• The Instrument: The dipolar atoms in a double-well trap.
• The Strings: The interactions between the atoms.
• The Music: The quantum states and transitions.

By understanding how the "buddy system" (pair tunneling) changes the music, scientists can now design better quantum computers and sensors. They can engineer materials that switch states instantly or create new types of "super-fluids" (like the supersolids mentioned in the paper) that behave in ways we never thought possible.

In a nutshell:
The paper tells us that when quantum particles are "social" (dipolar) and insist on moving in pairs, the entire universe of their behavior changes. They organize themselves differently, jump into new states more abruptly, and dance to a different rhythm. This gives physicists a new "knob" to turn to control quantum matter.

Technical Summary

1. Problem Statement

The paper investigates the impact of dipolar interactions on the equilibrium and dynamical properties of an atomic Josephson junction realized with $N$ dipolar bosons in a double-well potential.

• Context: Standard atomic Josephson junctions are typically described by the Bose-Hubbard model, considering single-particle tunneling and on-site interactions. However, dipolar interactions introduce long-range forces that generate nearest-neighbor density-density interactions and, crucially, correlated pair tunneling.
• Gap: While pair tunneling is known to occur in strongly interacting systems, its specific role in modifying the quantum phase transitions (QPTs) and dynamical behaviors (such as Macroscopic Quantum Self-Trapping) in the context of dipolar atoms within a double-well geometry requires a systematic theoretical analysis.
• Objective: To determine how the inclusion of pair tunneling ($P$) alters the ground-state structure, shifts critical points, changes the order of phase transitions, and modifies the conditions for dynamical phenomena like self-trapping and Dynamical Quantum Phase Transitions (DQPTs).

2. Methodology

The authors employ a multi-faceted approach combining analytical mean-field theory with exact numerical diagonalization:

• Model Hamiltonian: They utilize an extended two-site Bose-Hubbard model. By integrating out the transverse degrees of freedom and applying a two-mode approximation, the many-body Hamiltonian is reduced to:
  $$\hat{H} = -J(\hat{a}^\dagger_L \hat{a}_R + h.c.) + \frac{U}{2}(\hat{n}_L^2 + \hat{n}_R^2) + P(\hat{a}^\dagger_L \hat{a}^\dagger_L \hat{a}_R \hat{a}_R + h.c.)$$
  Here, $J$ is the renormalized tunneling, $U$ is the effective on-site interaction (renormalized by nearest-neighbor interactions $V$), and $P$ is the pair tunneling amplitude induced by dipolar forces.
• Mean-Field Theory (MFT): They derive the semiclassical equations of motion using the principle of least action with Glauber coherent states. This yields a classical energy landscape $E(\phi, z)$ dependent on the relative phase $\phi$ and population imbalance $z$, allowing for the mapping of the zero-temperature phase diagram and the identification of fixed points (minima, saddles, maxima).
• Exact Diagonalization (ED): For finite particle numbers $N$, the Hamiltonian is solved exactly in the Fock basis $|i, N-i\rangle$. This allows for the calculation of:
  • Ground-state probabilities and entanglement/fragmentation entropies.
  • Energy gaps ($E_1 - E_0$) to identify QPTs.
  • Ground-state fidelity susceptibility ($\chi_F$) to locate critical points.
  • Real-time dynamics of atomic coherent states.
• Dynamical Analysis: The authors analyze Dynamical Quantum Phase Transitions (DQPTs) by computing the Loschmidt amplitude $Z(t) = \langle \Psi_0 | e^{-i\hat{H}t/\hbar} | \Psi_0 \rangle$. They use the method of Loschmidt cumulants to locate complex zeros of the amplitude and identify non-analyticities (cusps) in the return rate $\lambda(t)$.

3. Key Contributions and Results

A. Equilibrium Properties (Ground State)

1. Parity Modulations: The inclusion of pair tunneling induces a distinct parity modulation in the ground-state Fock probabilities ($p_i$). In the limit of vanishing tunneling ($J \to 0$), the Hamiltonian conserves parity, splitting the Hilbert space into even and odd sectors. Even with finite $J$, this symmetry breaking results in an oscillatory pattern in the probability distribution that depends on the parity of the occupation number.
2. Shift of Critical Points: Pair tunneling significantly shifts the critical interaction strength ($U_c$) required for the transition to a NOON state (a fragmented condensate $|N,0\rangle \pm |0,N\rangle$). Specifically, for attractive effective interactions, $P$ pushes the transition to more negative $U/J$ values.
3. Change in Transition Order:
   • Weak Pair Tunneling ($P < P_c$): The transition to the NOON state remains a continuous QPT (second-order), characterized by the softening of a collective mode and a closing energy gap.
   • Strong Pair Tunneling ($P > P_c$): The transition changes nature to a first-order QPT. This is driven by a level crossing between two macroscopically distinct ground states rather than mode softening. The energy gap exhibits a cusp rather than closing smoothly.
4. Emergence of Phase-NOON States: Strong pair tunneling induces a new continuous QPT toward a Phase-NOON state. This state corresponds to a fragmented condensate where the superposition involves states with different relative phases (e.g., $|\pi/2, 0\rangle \pm |-\pi/2, 0\rangle$) rather than just population imbalance.

B. Dynamical Properties

1. Modified Self-Trapping Conditions: The presence of pair tunneling alters the phase-space structure, introducing new fixed points (phase-imbalanced minima). This modifies the conditions for Macroscopic Quantum Self-Trapping (MQST).
   • For strong $P$, MQST can occur even for repulsive interactions where it would normally be forbidden, or the threshold for self-trapping is shifted.
   • New dynamical regimes emerge, such as phase-imbalanced Josephson oscillations where the time-averaged phase $\langle \phi(t) \rangle$ is neither $0$ nor $\pi$.
2. Quantum vs. Mean-Field Dynamics: While mean-field theory accurately predicts short-time dynamics, many-body effects (decoherence, wave packet spreading) become significant at longer times. Pair tunneling accelerates the departure from mean-field behavior in certain regimes but preserves the qualitative dynamical taxonomy (oscillations vs. self-trapping).
3. Dynamical Quantum Phase Transitions (DQPTs):
   • The authors demonstrate that DQPTs (non-analyticities in the return rate) occur in these systems.
   • Pair tunneling quantitatively modifies the DQPT structure: it shifts the critical times ($t_c$) and alters the periodicity between consecutive DQPTs.
   • Geometric Interpretation: The DQPTs are interpreted geometrically as a discontinuous switch in the "saddle point" of the phase-space path integral. As the wave packet evolves and distorts, the point minimizing the distance to the initial state jumps abruptly, causing the cusp in the return rate. Pair tunneling accelerates this distortion, causing the first DQPT to occur earlier ($t_c < T$, where $T$ is the classical period).

4. Significance

• Engineering Quantum Criticality: The study establishes dipolar interactions as a controllable mechanism to engineer higher-order tunneling processes. This allows experimentalists to tune the order of quantum phase transitions (from continuous to first-order) and access exotic states like Phase-NOON states.
• Beyond Standard Models: It highlights the insufficiency of the standard Bose-Hubbard model for dipolar systems, demonstrating that pair tunneling is not a minor correction but a fundamental driver of new physics in double-well potentials.
• Insight into Supersolids and Complex Systems: Given the current interest in supersolids and extended quantum systems, these results provide a theoretical framework for understanding how long-range interactions and correlated tunneling affect both static criticality and non-equilibrium dynamics.
• Experimental Relevance: The predicted parity modulations and shifts in critical points offer specific signatures that can be tested in current ultracold atom experiments using dipolar species (e.g., Dysprosium or Erbium).

In summary, the paper provides a comprehensive theoretical framework showing that dipolar-induced pair tunneling fundamentally reshapes the quantum landscape of Josephson junctions, creating new phases, altering the nature of phase transitions, and modifying the timing and geometry of dynamical critical phenomena.