Revealing Hidden Correlations in a Fermi-Hubbard system via Interaction Ramps

This paper demonstrates that rapidly increasing interaction strength in an attractive Hubbard model enhances charge-density-wave correlations by converting nonlocal pairs into doublons, thereby providing a method to distinguish between unpaired Fermi liquids and pseudogap phases of preformed pairs in cold-atom systems.

The Gist

Imagine a crowded dance floor where pairs of dancers (atoms) are holding hands. In a very calm, slow-moving crowd, these pairs are easy to spot; they are standing right next to each other. But in a highly energetic, chaotic crowd, the pairs start to stretch out. One dancer might be on one spot, while their partner is a few steps away. They are still a pair, but they are "nonlocal"—spread out across the room.

This spreading out makes it very hard to see the patterns they are making. It's like trying to see a checkerboard pattern on a floor when the people standing on the squares are constantly stretching their arms to hold hands with people three squares away. The pattern gets blurry and invisible.

The Problem
Scientists have been studying a specific type of "dance floor" made of ultracold atoms (called the attractive Hubbard model). They know that in certain conditions, these atoms should form a beautiful, ordered pattern called a "charge-density wave" (like a checkerboard of pairs). However, when the atoms are strongly interacting, the pairs stretch out so much that the cameras can't see the checkerboard pattern. It's hidden in plain sight.

The Solution: The "Snap" Trick
The researchers in this paper came up with a clever trick to reveal this hidden pattern. They call it an "interaction ramp," but you can think of it as a magnetic "snap."

1. The Setup: They start with the atoms in their natural, stretched-out state.
2. The Snap: Just before they take a photo, they rapidly change the magnetic field. This acts like a sudden, strong magnet that pulls the stretched-out pairs tight.
3. The Result: The pairs that were spread out across the room instantly snap together into tight, local bundles (called "doublons").

What They Found
Once they took this "snap" photo, the hidden checkerboard pattern suddenly became crystal clear.

• Before the snap: The photo looked messy. The pairs were too spread out to show the pattern.
• After the snap: The photo showed a strong, clear checkerboard pattern.

This proved that the pattern was there all along; it was just hidden because the pairs were too stretched out to be seen. The "snap" didn't create the pattern; it just revealed it by pulling the pairs back together.

Why It Matters
The researchers found that this trick works best in the "Goldilocks" zone—not too weak, not too strong, but just right. In this zone, the pairs are naturally very stretched out, making the pattern hardest to see without the trick.

They also used this method to distinguish between two different states of matter:

1. The "Fermi Liquid": A state where atoms aren't really paired up at all (like solo dancers).
2. The "Pseudogap": A state where pairs exist but are stretched out and dancing in a weird, pre-formed way.

By using the "snap," they could instantly tell the difference. If the atoms were truly paired, the snap pulled them into tight bundles, and the photo showed the pattern. If they weren't paired, the snap did nothing special.

The Big Picture
This technique is like a new pair of glasses for scientists. It allows them to see "exotic" forms of order in atoms that were previously invisible. The authors suggest this could help them find even stranger patterns in the future, such as specific types of superconductivity or "stripe" patterns, by simply taking a picture after giving the atoms a quick magnetic "snap."

Technical Summary

Technical Summary: Revealing Hidden Correlations in a Fermi-Hubbard System via Interaction Ramps

Problem Statement
In strongly correlated electron systems, such as high-temperature cuprates and bismuthates, the interplay between fermion pairing and charge order (e.g., charge-density waves, CDW) is fundamental. The attractive Hubbard model serves as a paradigm for this physics, where superfluid and CDW states are degenerate at half-filling. While ultracold atoms in optical lattices have successfully realized this model and observed pairing and CDW correlations, a significant detection barrier exists in the strongly interacting regime ($U/t \sim$ bandwidth). In this regime, the pair size becomes comparable to the interparticle spacing, resulting in "nonlocal" pairs where the two fermions occupy neighboring sites rather than the same site. This delocalization obscures the direct detection of the pairs and their spatial order, as a pair spread across two sites does not contribute to the alternating doublon-hole pattern characteristic of CDW correlations.

Methodology
The authors utilize a two-dimensional attractive Fermi-Hubbard gas realized with $^{40}$K atoms in a deep sinusoidal square lattice ($4.3(2)E_r$). The system is prepared in an equilibrium state with a density per site of $n \sim 0.8$ and a temperature of $T/t \sim 0.5$.

To probe the hidden order, the authors introduce an interaction ramp technique (or quench):

1. Rapid Interaction Boost: Immediately prior to imaging, the attraction strength $U$ is rapidly increased via a Feshbach resonance. The ramp is a linear magnetic field sweep from the initial value to the resonance over $1$ ms ($\approx 0.3 \hbar/t$).
2. Freezing and Imaging: The lattice depth is increased to $\sim 70 E_r$ within $0.1$ ms to freeze the atom distribution. The cloud is then imaged with single-site and spin resolution.
3. Mechanism: The rapid ramp projects extended, nonlocal pairs onto local doublons (doubly occupied sites). The doublon density subsequently serves as a local proxy for the presence of pairs, effectively "collapsing" the nonlocal wavefunction to reveal the underlying spatial arrangement.

The ramp duration is calibrated to be slow enough for pair constituents to merge into doublons (timescale $\hbar/t$) but fast enough to preserve the global pair arrangement and prevent the creation of new pairs (timescale $\hbar/J$, where $J$ is the pair-hopping scale).

Key Results

• Enhancement of CDW Visibility: The interaction ramp significantly enhances the visibility of charge-density-wave correlations. This enhancement is most pronounced in the crossover regime ($4 \lesssim U/t \lesssim 8$), where pairs are initially nonlocal.
• Quantitative Evidence: Density-density correlators and the density structure factor $\chi_n(\vec{k})$ show a dramatic increase in CDW correlations after the ramp. Specifically, the CDW strength $\chi_n(\pi, \pi)$ measured after the ramp at $U/t \approx 6$ exceeds the maximum attainable without the ramp at any interaction strength. This confirms that the order existed prior to the ramp but was hidden by pair delocalization.
• Regime Distinction:
  • Strong Attraction ($U/t \gtrsim 12$): Pairs are already local; the ramp has negligible effect.
  • Crossover Regime ($4 \lesssim U/t \lesssim 8$): The ramp produces the largest enhancement, coinciding with the region of predicted highest superfluid critical temperatures.
  • Weak Attraction ($U/t \lesssim 4$): The system behaves as a Fermi liquid governed by Pauli repulsion; no CDW structure is observed with or without the ramp.
• Pairing Characterization: The technique distinguishes between the unpaired Fermi liquid and the pseudogap phase of preformed pairs. By analyzing atom-resolved spin-charge correlations, the authors show that the ramp converts nonlocal pairs into local doublons, allowing a single image to reveal the presence of a pairing gap (vanishing spin susceptibility) without the need for statistical analysis of hundreds of repetitions required by previous magnetization fluctuation methods.
• Spin Correlations: The ramp reveals a transition at $U/t \sim 4$ from spin-dependent Fermi liquid behavior to spin-independent nonlocal correlations expected for the paired pseudogap regime. It effectively eliminates intra-pair contributions, leaving correlations that purely capture inter-pair repulsion.

Significance and Claims
The paper claims to establish a robust technique for probing spatial order in fermion pairing that is otherwise obscured by nonlocality. The primary significance lies in:

1. Revealing Hidden Order: The ability to detect CDW order in the strongly correlated regime where direct observation fails due to pair delocalization.
2. Simplified Detection: The method allows for the characterization of the crossover from a Fermi liquid to a paired regime using single-shot images, contrasting with previous methods requiring extensive statistical averaging.
3. Future Applications: The authors suggest this technique may facilitate the observation of exotic forms of pair order in spin-imbalanced systems (where spin fluctuations of the majority component might otherwise obscure order) and the elusive Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state. Additionally, they propose the technique could serve as a local thermometer, as the singlon fraction after the ramp is directly sensitive to entropy and temperature. Finally, they note the potential applicability to the dual case of the doped repulsive Hubbard model to observe stripe order.