Phases of interacting bosons in a hybrid Harper-Hofstadter system with a synthetic dimension of harmonic trap states

This paper numerically investigates the impact of inhomogeneous, long-range interactions within a hybrid Harper-Hofstadter system utilizing a synthetic dimension of harmonic trap states, revealing familiar vortex and Meissner phases in ladder geometries while uncovering novel ground states like the "Meissner stripe" in two-dimensional models.

The Gist

Imagine you are trying to build a complex city for tiny, invisible particles called bosons (a type of atom that loves to act in unison). Usually, to study how these particles interact with magnetic fields, physicists build a physical grid of traps, like a chessboard made of light. But building a huge, 2D chessboard is hard and expensive.

This paper introduces a clever trick: The Synthetic Dimension.

The Magic Trick: Turning "States" into "Places"

Instead of building a physical grid in two directions (left-right and up-down), the scientists build a grid in only one direction (left-right) and use a "magic ladder" for the second direction.

• The Real World: The atoms sit in a long, cigar-shaped trap. They can move left and right along a real track.
• The Synthetic World: The atoms also have different "energy levels" (like rungs on a ladder). By shaking the trap at just the right rhythm, the scientists make the atoms jump between these energy levels as if they were moving up and down a physical staircase.

So, an atom moving "up" the energy ladder is treated by the math as if it's moving "up" a physical street. This creates a hybrid city: one real street and one "energy street."

The Problem: The Neighborhood is Weird

In a normal city, if two neighbors talk, they only talk to the person standing right next to them. If you are far away, you can't hear them.

But in this Synthetic City, the rules are bizarre:

1. Long-Range Chatter: Because the atoms are physically sitting in the same spot (just in different energy states), they can "talk" to each other even if they are on opposite ends of the energy ladder. It's like you being able to have a loud conversation with your neighbor who lives on the 50th floor, even though you are both standing in the same elevator shaft.
2. Changing Identities: When these atoms interact, they don't just bump into each other; they might swap energy levels. It's like two people meeting, and suddenly one turns into a giant and the other into a mouse, or they swap clothes.

The Experiment: What Happens When They Interact?

The researchers wanted to see what happens when you put these chatty, shape-shifting atoms into a magnetic field (which makes them want to swirl in circles). They studied two setups:

1. The Two-Lane Highway (The Ladder)

Imagine a highway with two lanes.

• Normal Physics: Usually, you get two main traffic patterns:
  • The Meissner Phase: Cars drive fast in opposite directions in each lane, creating a smooth flow that blocks the magnetic field (like a shield).
  • The Vortex Phase: Cars get stuck in little whirlpools (vortices) in the middle of the road.
• The Discovery: When the scientists added the "weird synthetic interactions," the traffic patterns stayed mostly the same. The cars still formed shields or whirlpools.
• The Twist: The only big change was that the cars started piling up at the very ends of the highway. The "long-range chatter" made them crowd toward the edges, but the basic flow patterns remained familiar.

2. The Full City Grid (The 2D Model)

Now, imagine a full city grid with many streets and avenues.

• Normal Physics: You expect the cars to form a neat, repeating pattern of whirlpools (like a honeycomb of traffic jams).
• The Discovery (The Big Surprise): When they turned on the synthetic interactions, the city didn't form a honeycomb. Instead, it formed something completely new called the "Meissner Stripe."

What is a Meissner Stripe?
Imagine a city where the traffic flows in straight, parallel lines (like a river), but the density of the cars changes in a striped pattern.

• Some "streets" (energy levels) are packed with cars.
• The next "street" is almost empty.
• The next is packed again.
• The cars in the packed streets flow in opposite directions, creating a strong, rhythmic current.

It's like a zebra crossing that never ends, where the cars are constantly rushing in opposite directions on alternating lanes. This state is very different from the usual "whirlpool" city and only appears because of the unique, long-range way the atoms talk to each other in this synthetic setup.

Why Does This Matter?

This paper is like finding a new flavor of ice cream that you didn't know existed.

• For Scientists: It shows that using "energy levels" as a dimension isn't just a math trick; it creates entirely new physical laws where long-range interactions create strange, striped states that you can't get in normal materials.
• For the Future: This opens the door to simulating exotic quantum states (like those found in the most extreme corners of the universe) using relatively simple lab equipment. It suggests that if we can control these "long-range conversations" between atoms, we might be able to build new types of quantum computers or sensors.

In a nutshell: The scientists built a fake 2D world using energy levels. They found that while a simple 2-lane road looked mostly normal, a full city grid turned into a strange, striped traffic jam that had never been seen before, proving that the "rules of the neighborhood" in this synthetic world are delightfully weird.

Technical Summary

1. Problem Statement

The paper addresses the challenge of understanding many-body physics in hybrid quantum systems that combine real spatial dimensions with synthetic dimensions. While synthetic dimensions (constructed from internal atomic states or, in this case, harmonic trap states) have successfully simulated topological lattice models like the Harper-Hofstadter (HH) model, the nature of inter-particle interactions in these systems remains poorly explored.

Specifically, the authors focus on a synthetic dimension formed by harmonic trap states coupled via periodic shaking of the trapping potential. Unlike conventional systems or other synthetic dimension schemes (e.g., hyperfine states or momentum states), interactions in this setup are:

• Inhomogeneous: Interaction strength varies with position along the synthetic dimension.
• Long-ranged: Particles physically close in real space but far apart in the synthetic dimension still interact strongly.
• Non-state-preserving: Collisions can change the synthetic dimension index (state-changing collisions), leading to complex terms like correlated pair tunneling and dipole-like interactions.

The central question is: How do these exotic, inhomogeneous, and long-range interactions alter the ground-state phases of bosons in Harper-Hofstadter ladders and 2D lattices?

2. Methodology

The authors employ a mean-field theoretical approach (weakly interacting limit) using the Gross-Pitaevskii equation.

• System Setup:
  • Real Space: A 1D optical lattice along the $x$-direction.
  • Synthetic Dimension: A set of harmonic trap states along the $y$-direction, indexed by $\lambda$.
  • Coupling: A time-periodic driving potential $V(x,y,t)$ is applied to couple neighboring trap states, inducing an effective hopping amplitude $J_\lambda$ with a position-dependent phase $e^{-i\phi_n}$, mimicking a magnetic flux $\Phi$.
• Hamiltonian:
  • The single-particle part is modeled as a Harper-Hofstadter Hamiltonian with a synthetic dimension.
  • The interaction term ($H_{int}$) is derived by projecting real-space contact interactions onto the harmonic oscillator basis. This results in a complex interaction Hamiltonian containing:
    1. Onsite interactions: Decay algebraically along the synthetic dimension.
    2. Correlated pair tunneling: Terms where two atoms scatter into/out of the same state (e.g., $\lambda \to \lambda \pm m$).
    3. Dipole-like interactions: Terms where atoms exchange states or scatter to symmetric positions.
• Numerical Simulation:
  • Ground states are found by evolving the wavefunction in imaginary time using the fourth-order Runge-Kutta method until convergence.
  • The study covers two geometries:
    1. Two-legged HH Ladder: One leg is real space ($x$), the other is synthetic ($\lambda$).
    2. 2D HH Model: One direction is real space, the other is synthetic.
  • Observables: Local density ($|\psi|^2$), local currents ($J$), chiral current ($C$), and ladder density bias (LDB).

3. Key Contributions

• Characterization of Synthetic Interactions: The paper provides a detailed breakdown of how real-space contact interactions map to exotic processes in the synthetic dimension of harmonic trap states, specifically highlighting the roles of correlated pair tunneling and dipole-like terms.
• Discovery of the "Meissner Stripe" Phase: The authors identify a novel ground state unique to the 2D hybrid system with synthetic interactions, distinct from standard vortex lattices or Meissner phases.
• System-Size Dependence: The study demonstrates that the ground state in these hybrid systems is highly sensitive to the length of the synthetic dimension, a feature not typically prominent in standard short-range interaction models.

4. Results

A. Two-Legged HH Ladder

• Standard vs. Synthetic Interactions:
  • In the single-particle and standard contact interaction limits, the system exhibits known Meissner (uniform chiral currents, no density bias) and Vortex-Lattice (circulating currents, modulated density) phases.
  • Standard contact interactions also produce a Biased Ladder Phase (density imbalance between legs).
• Effect of Synthetic Interactions:
  • Preservation of Topology: The Meissner and Vortex-Lattice phases remain qualitatively similar. The inhomogeneity of interactions causes a global density modulation (particles accumulate at the edges of the synthetic dimension) but does not destroy the phase structure.
  • Suppression of Bias: A key finding is the disappearance of the Biased Ladder Phase. The specific structure of synthetic interactions (particularly dipole-like and correlated pair tunneling) prevents the formation of the density imbalance seen in standard contact models.

B. 2D Harper-Hofstadter Model

• Standard Interactions: Reproduce known results: density modulations forming $q \times q$ unit cells (where $\alpha = p/q$) with circulating currents (Abrikosov-like vortex lattices).
• Synthetic Interactions (The "Meissner Stripe" Phase):
  • For small synthetic dimension lengths ($N_\lambda$), the ground state is a "Meissner stripe" state.
  • Characteristics:
    • Currents: Strong counter-propagating currents flow along the real spatial dimension (similar to the Meissner effect), rather than circulating around plaquettes.
    • Density: Strong density modulations occur along the synthetic dimension with a period matching the magnetic flux (e.g., period of 3 for $\alpha=1/3$).
    • Structure: Alternating rows of high-density (negligible current) and low-density (strong opposing currents).
  • Mechanism: Analysis of interaction subsets reveals that dipole-like interactions are the primary driver stabilizing the Meissner stripe phase.
  • Transition: As the synthetic dimension length ($N_\lambda$) increases, the system transitions from the Meissner stripe state to states resembling vortex lattices with alternating circulating currents. This indicates a competition between different interaction subsets that is sensitive to system size.
  • Robustness: The Meissner stripe phase persists even with non-uniform hopping amplitudes (realistic shaking potentials) and alternative driving schemes.

5. Significance and Outlook

• New Platform for Correlated Physics: This work establishes harmonic trap states as a viable platform for exploring the interplay between long-range interactions and magnetic flux effects. The inhomogeneous nature of these interactions offers a unique testing ground for many-body physics that differs fundamentally from condensed matter models with short-range interactions.
• Experimental Relevance: The predicted phases, particularly the Meissner stripe, are accessible in current cold-atom experiments (referencing recent work by Oliver et al., 2023). The paper suggests that these states should be observable via real-space density measurements, which map to synthetic dimension occupation.
• Future Directions:
  • Strong Interactions: The current study is mean-field. The authors propose extending this to the strong-interaction regime to search for fractional quantum Hall-like states, which may be fundamentally different from those in standard lattices due to the unique interaction terms.
  • Thermodynamic Limit: Investigating whether the Meissner stripe phase survives in the thermodynamic limit or if it is a finite-size effect driven by boundary conditions and interaction inhomogeneity.

In summary, the paper demonstrates that while synthetic dimensions of harmonic trap states preserve the topological features of the Harper-Hofstadter model, they introduce unique interaction-driven phenomena, most notably the suppression of biased phases in ladders and the emergence of exotic "Meissner stripe" states in 2D systems.