Adiabatic preparation of a fractional quantum Hall fluid by coherently pumping atoms from a Bose-Einstein condensate

This paper proposes and numerically validates a protocol for adiabatically preparing a bosonic fractional quantum Hall fluid by coherently pumping atoms from a Bose-Einstein condensate using Laguerre-Gauss Raman beams and anharmonic confinement, thereby avoiding topological phase transitions and maintaining a sizable adiabatic gap for large particle numbers.

The Gist

Imagine you are trying to build a very delicate, intricate sandcastle. Usually, to get the sand into the perfect shape, you have to pour it in all at once or carefully sculpt it while it's already there. But in the world of quantum physics, building a "Fractional Quantum Hall" (FQH) fluid—a special state of matter where atoms dance in a highly coordinated, topological pattern—is incredibly hard. If you try to build it particle by particle using old methods, the structure tends to collapse as it gets bigger because the "energy gap" (the stability holding it together) shrinks to nothing.

This paper proposes a new, clever way to build this quantum sandcastle, not by forcing the atoms into place, but by coherently pumping them in, like filling a bucket with a steady, controlled stream of water.

Here is how the authors' proposal works, broken down into simple concepts:

1. The Setup: Two Buckets and a Magic Hose

Imagine you have two buckets of atoms:

• Bucket A (The Reservoir): This is a huge, calm pool of atoms (a Bose-Einstein Condensate) that are easy to handle and don't interact much with each other.
• Bucket B (The Target): This is an empty, tight, two-dimensional trap where the atoms are supposed to form the special FQH fluid. These atoms are "strongly interacting," meaning they are very sensitive and want to dance in a specific, complex pattern.

The authors propose connecting these two buckets with a "magic hose" made of laser beams (specifically, Raman beams with a special spiral shape called Laguerre-Gauss). This hose doesn't just move atoms; it spins them as it transfers them, giving each atom a specific amount of "twist" (angular momentum) as it moves from the calm pool to the empty trap.

2. The Problem with Old Methods: The Narrow Bridge

In previous experiments, scientists tried to build these states by starting with a fixed number of atoms and slowly changing the environment (like turning a dial) to force them into the FQH state.

• The Analogy: Imagine trying to cross a river by walking across a bridge that gets thinner and thinner the further you go. For a few steps (a few atoms), it's fine. But as you add more weight (more atoms), the bridge becomes so thin that you fall through. In physics terms, the "energy gap" that protects the state disappears as the system grows, making it impossible to build large, stable FQH fluids.

3. The New Solution: A Wide, Adjustable Path

The authors' new method avoids this "narrow bridge" problem entirely.

• The Analogy: Instead of walking across a thinning bridge, imagine you are in a large elevator shaft. You start at the bottom (an empty trap). You have a control panel that lets you adjust the "floor" (energy levels) and the "speed" of the elevator (the laser coupling).
• How it works:
  1. Start Empty: The trap is empty.
  2. The Pump: You turn on the laser hose. It starts pulling atoms from the reservoir one by one (or in small groups) into the trap.
  3. The Twist: Because the laser gives each atom a specific "twist," the atoms naturally fall into the correct dance pattern (the Laughlin state) as they arrive.
  4. The Safety Net: The most important part is that the "gap" (the stability of the state) is not determined by how many atoms are in the trap. Instead, it is controlled by the strength of the laser hose. The authors can keep the "bridge" wide and sturdy no matter how many atoms they add.

4. The "Tilted Lattice" Visualization

The paper uses a visual metaphor to explain the process:

• Imagine a row of stepping stones labeled 0, 1, 2, 3... (representing the number of atoms).
• Initially, the stone labeled "0" is the lowest and most comfortable.
• As the experiment runs, the scientists slowly tilt the row of stones so that the higher-numbered stones (more atoms) become lower and more comfortable.
• Simultaneously, they turn up the "hopping" power (the laser) so the atoms can easily jump from one stone to the next.
• By the end, the "lowest" stone is the one with the target number of atoms (e.g., 4 or 8), and the system settles there naturally. Because the laser keeps the stones connected, the atoms never get stuck or fall off the edge.

5. Why This Matters (According to the Paper)

• Scalability: The authors ran computer simulations showing this works well for up to 8 atoms (and potentially many more). This is a huge jump from previous experiments that were stuck at 3 atoms.
• Robustness: They found that adding a slight "anharmonic" shape to the trap (making the bowl slightly different from a perfect circle) actually helps. It acts like a guide rail, keeping the atoms in the right pattern and preventing them from getting confused or slowing down.
• Flexibility: This method isn't just for the basic "Laughlin" state (the ground state). They showed it can also create "quasi-hole" states (excited states with a missing piece in the middle), which are important for studying exotic quantum properties.

Summary

In short, the paper proposes a way to build complex quantum fluids by growing them from an empty state using a laser pump, rather than trying to reshape an existing group of atoms. This avoids the "collapsing bridge" problem of previous methods, allowing for the creation of much larger and more stable quantum states than ever before. The authors suggest this could be a key step toward using these fluids for future quantum technologies, though their current focus is strictly on the method of creation itself.

Technical Summary

Technical Summary: Adiabatic Preparation of a Fractional Quantum Hall Fluid by Coherently Pumping Atoms from a Bose-Einstein Condensate

Problem Statement
Realizing fractional quantum Hall (FQH) states in ultracold atomic systems remains a significant challenge, particularly for systems containing more than a few particles. While FQH physics has been observed in two-dimensional electron gases and recently in photonic and atomic platforms, current atomic experiments are limited to a maximum of three particles. The primary bottleneck in scaling these systems is the difficulty of directly cooling atoms into the FQH regime. Existing protocols typically involve preparing an easy-to-cool state and then adiabatically tuning system parameters to enter the FQH regime. However, this approach necessitates crossing a topological phase transition. As the particle number ($N$) increases, the energy gap protecting the FQH state decreases rapidly (often exponentially) due to finite-size effects, severely limiting the achievable system size and the fidelity of the preparation.

Methodology
The authors propose a novel protocol to adiabatically prepare bosonic Laughlin states by exploiting a time-dependent coherent coupling between a strongly interacting atomic state and a large, dilute Bose-Einstein condensate (BEC) acting as a particle reservoir. The scheme operates in a larger Hilbert space where the particle number is not fixed but varies via an external coherent pump.

Key elements of the methodology include:

• System Configuration: The system consists of atoms in two internal states: $|1\rangle$ (weakly interacting, populated by a BEC) and $|2\rangle$ (strongly interacting, initially empty and trapped in a 2D geometry).
• Synthetic Magnetic Field: The $|2\rangle$ state is subjected to a rotating frame with frequency $\Omega$, generating a synthetic gauge potential and a uniform synthetic magnetic field.
• Coherent Pumping: Atoms are transferred from $|1\rangle$ to $|2\rangle$ using a pair of Raman beams with a Laguerre-Gauss profile. This coupling is angular-momentum-selective, transferring an orbital angular momentum $l$ per particle.
• Adiabatic Path: The protocol starts with an empty $|2\rangle$ state. By adiabatically varying the pump strength $F(t)$ and the effective detuning $\Delta(t)$, the system evolves from the vacuum to a target Laughlin state with $N_t$ particles and total angular momentum $L = N_t(N_t - 1)$.
• Hamiltonian Engineering: The effective Hamiltonian in the Lowest Landau Level (LLL) includes a time-dependent term $\hbar F(t)(\hat{a}^\dagger_l + \hat{a}_l)$ representing the coherent pump. The detuning $\Delta(t)$ acts as a chemical potential, shifting the energy of different particle number sectors.

Key Contributions

1. Avoidance of Topological Phase Transitions: Unlike number-conserving schemes that require crossing a topological phase transition (where the gap closes), this protocol identifies an adiabatic path in Fock space that avoids such crossings. The system evolves through a sequence of states with increasing particle number without encountering a vanishing gap.
2. Externally Controlled Adiabatic Gap: The energy gap protecting the evolution is not determined by many-body physics or finite-size effects but is externally controlled by the amplitude of the coherent coupling ($F$). This allows the gap to remain sizable throughout the entire evolution, regardless of the particle number.
3. Single-Ramp Generation: The proposal generates the $N$-particle Laughlin state in a single adiabatic ramp, avoiding the sequential accumulation of infidelity found in multi-step protocols.
4. Role of Anharmonic Confinement: The authors highlight the crucial advantage of including an anharmonic confinement potential (e.g., a quartic term). This breaks the degeneracy of states in the $N < N_t$ sectors, further opening the adiabatic gap and making the scheme robust against deconfinement issues and imperfections in cylindrical symmetry.

Numerical Results
The authors performed numerical simulations for typical experimental parameters of $^7$Li atoms, assessing the protocol's efficiency for particle numbers exceeding current experimental capabilities (up to $N_t = 8$).

• Fidelity: For a target of $N_t = 4$, a fidelity of $F = 0.99$ is achievable with a ramp time of $T = 95 T_\Omega$ (where $T_\Omega$ is the trapping period). This represents a six-fold improvement over theoretical proposals based on ramping trapping frequency and ellipticity.
• Scalability: The infidelity shows an exponential decrease with ramp time for smooth ramps. Crucially, the required ramp time does not scale exponentially with $N_t$ as it does in fixed-particle-number schemes, because the gap is maintained by the pump strength.
• Quasi-hole States: The scheme is also applicable to excited states, such as single quasi-holes. By using a larger angular momentum pump ($l = N_t$) and a localized repulsive "plug" potential, the degeneracy of the quasi-hole state is lifted, resulting in a large adiabatic gap and high fidelity.
• Robustness: The inclusion of an anharmonic potential significantly improves the scaling of the ramp time required to reach a desired fidelity as $N_t$ increases.

Significance and Claims
The paper claims that this proposal offers a significant improvement in preparation times and scalability for FQH samples. By decoupling the adiabatic gap from finite-size many-body properties and making it tunable via external drives and confinement, the protocol promises the realization of FQH samples with particle numbers far exceeding the current experimental state-of-the-art (potentially up to a few tens of particles). The authors assert that this approach opens the way to investigating the properties of fractional quantum Hall fluids in synthetic quantum matter, including non-Abelian anyonic excitations, in a well-controlled and tunable platform. While the focus is on atomic systems, the authors note the proposal is directly applicable to photonic systems where tunable coherent pumps and synthetic magnetic fields are already realized.