Protocols for a many-body phase microscope: From coherences and d-wave superconductivity to Green's functions

This paper proposes a protocol for a many-body phase microscope using Fourier-space manipulation in quantum gas microscopes to directly measure off-diagonal correlators, including d-wave superconducting order parameters, spectral functions, and hidden orders in exotic quantum states.

The Gist

Imagine you are trying to understand a complex dance performance happening in a dark room. You have a special camera (a Quantum Gas Microscope) that can take incredibly sharp photos of the dancers' positions. You can see exactly where every dancer is standing and how they are spinning relative to their neighbors. This is what scientists have been able to do for years: they can map the "density" of the crowd.

But there's a catch. You can't see the rhythm, the flow, or the secret handshakes between dancers who are far apart. In the quantum world, this missing information is called phase and coherence. It's the invisible glue that holds exotic states of matter together, like superconductors (materials that conduct electricity with zero resistance) or "fractional" states where particles act like a single fluid.

This paper proposes a new set of "protocols" (recipes) to build a Many-Body Phase Microscope. It's like upgrading your camera from taking still photos to recording a high-speed, 3D movie that reveals the invisible dance moves.

Here is how they do it, using some creative analogies:

1. The Magic Trick: The "Time-Traveling" Lens

The core of their idea is a Matter-Wave Microscope. Imagine the atoms in the experiment are like marbles on a trampoline.

• The Problem: Usually, you can only see where the marbles are sitting.
• The Solution: The scientists use a "lens" (a specific magnetic pulse) that acts like a time-traveling camera.
  • Step 1: They let the marbles bounce for a tiny fraction of a second. This transforms the "position" of the marbles into their "momentum" (how fast and in what direction they are moving). It's like taking a photo of the speed of the dancers instead of their location.
  • Step 2: In this "momentum world," they use a laser (a Raman pulse) to give a specific group of dancers a gentle nudge. This is like whispering a secret instruction to a dancer in the middle of the crowd.
  • Step 3: They let the marbles bounce again, turning the "momentum" back into "position."
  • The Result: Because of the nudge, the dancers who received the instruction now land in a slightly different spot than the ones who didn't. When you take the final photo, the two groups overlap and create an interference pattern (like ripples in a pond meeting).

By looking at these ripples, the scientists can calculate the phase—the invisible timing and connection between atoms that were far apart.

2. What Can This New Microscope See?

The paper shows three specific "superpowers" this new method unlocks:

A. Finding the "Superconducting Dance" (D-Wave Superconductivity)

• The Mystery: Scientists want to know if certain materials can become superconductors at higher temperatures. In these materials, electrons pair up and dance in a specific pattern called d-wave (shaped like a four-leaf clover).
• The Old Way: It was like trying to guess the dance pattern by looking at individual dancers. Very hard.
• The New Way: This protocol allows them to look at pairs of dancers simultaneously. They can check if two dancers, far apart, are holding hands in that specific "clover" shape. If they are, they've found the superconducting order.

B. Listening to the "Echo" (Spectral Functions)

• The Mystery: In solid-state physics, scientists use a technique called ARPES to see the energy levels of electrons. In cold atom labs, this is very hard to do because the atoms are too crowded.
• The New Way: Imagine pulling one single dancer out of the crowd, letting them dance alone in a quiet room for a while, and then putting them back.
  • The protocol isolates one specific "momentum" (one specific dance move).
  • It lets the rest of the crowd dance around it for a moment.
  • Then, it brings the solo dancer back and sees how their rhythm has changed compared to when they left.
  • This "echo" tells scientists exactly what the energy levels of the system are, revealing the "spectrum" of the quantum material.

C. Seeing the "Hidden Order" (Fractional Quantum Hall)

• The Mystery: In some exotic states, particles act like they are carrying invisible "flux tubes" (like carrying a balloon). The order of the system isn't about where the particles are, but about how these invisible balloons are tangled. This is called hidden order.
• The New Way: The protocol uses a "spotlight" (a focused laser) to check the connection between two specific dancers, while simultaneously checking the positions of everyone else in the room.
  • It's like checking if two people are holding hands, but only if everyone else in the room is standing in a specific formation.
  • This reveals the "hidden" dance steps that define these topological states, which were previously impossible to measure directly.

Why Does This Matter?

Think of the current Quantum Gas Microscope as a black-and-white photo. It tells you where things are.
This new proposal turns it into a 3D, full-color, high-speed video.

It allows scientists to:

1. Design better materials: By understanding exactly how electrons pair up, we might design room-temperature superconductors (which would revolutionize power grids and electronics).
2. Solve complex puzzles: It helps verify theories about how quantum computers might work or how exotic states of matter behave.
3. See the invisible: It turns abstract mathematical concepts (like "off-diagonal correlations") into something you can actually measure and see in a lab.

In short, the authors have built a new set of glasses that let us see the invisible rhythm of the quantum world, not just the static positions of the particles.

Technical Summary

1. Problem Statement

Quantum gas microscopes have revolutionized the study of ultracold atoms in optical lattices by enabling projective measurements with single-site resolution. However, a fundamental limitation exists: these microscopes typically measure density in the occupation basis. Consequently, they cannot directly access phase information, off-diagonal long-range correlations, or complex coherences, which are essential for characterizing exotic quantum states like superconductors, topological phases, and non-equilibrium dynamics.

While recent experiments have measured local currents and phase fluctuations by mapping them to density imbalances, a general, scalable protocol to measure long-range off-diagonal correlators (such as Green's functions and pairing correlations) in momentum space or arbitrary real-space separations has been lacking.

2. Methodology: The Many-Body Interferometer

The authors propose a unified protocol utilizing a matter-wave microscope combined with Fourier-space manipulation to realize a "many-body phase microscope." The core mechanism is a many-body interferometer scheme involving the following steps:

1. Fourier Transformation: The system (initially in an optical lattice) is released, and a $T/4$ evolution in a harmonic trap (a matter-wave lens) transforms the system from real space to Fourier (momentum) space.
2. Raman Coupling in Momentum Space: A $\pi/2$ Raman pulse is applied in Fourier space. This pulse transfers atoms to an auxiliary spin state while imparting a controlled momentum kick ($q$). This kick corresponds to a specific displacement ($d$) in real space after the next lens.
3. Real-Space Displacement: A second $T/4$ matter-wave lens transforms the system back to real space. The momentum kick acquired in step 2 now manifests as a spatial displacement ($d$) of the auxiliary spin component relative to the original spin component.
4. Interference: A final Raman pulse (without momentum transfer) recombines the two spin components.
5. Measurement: Spin-resolved single-atom imaging is performed. The interference fringes observed in the density distribution as a function of the Raman phase ($\phi$) reveal the phase coherence between sites separated by distance $d$.

The measured density $\langle \hat{n}_{-x, \uparrow} \rangle$ contains an interference term proportional to the correlation function $g^{(1)}(d) = \langle \hat{a}^\dagger_{x+d} \hat{a}_x \rangle$. By varying the Raman phase, both the magnitude (contrast) and the complex phase of the correlator can be extracted.

3. Key Contributions and Specific Protocols

The paper demonstrates how this general interferometer framework can be adapted to measure three distinct classes of physically significant correlators:

A. Equal-Time Green's Functions and Off-Diagonal Order

• Goal: Measure the single-particle coherence $g^{(1)}(d)$ over arbitrary distances.
• Mechanism: The standard protocol described above.
• Significance: Allows direct mapping of phase coherence and detection of symmetry breaking (e.g., time-reversal symmetry breaking) via the phase offset of interference fringes.

B. d-Wave Superconducting Order

• Goal: Measure the four-point pairing correlation function $C_{\mu,\nu}(d) = \langle \hat{a}^\dagger_{i\uparrow} \hat{a}^\dagger_{i+e_\mu\downarrow} \hat{a}_{j\uparrow} \hat{a}_{j+e_\nu\downarrow} \rangle$ in repulsive Fermi-Hubbard models.
• Mechanism: A spinful extension of the protocol using two physical spins and two auxiliary spins. Independent Raman pulses apply different momentum kicks ($d_1, d_2$) to the up and down spin components.
• Differentiation: By choosing specific displacements ($d_1 = i-j$ and $d_2 = i-j + e_\mu - e_\nu$), the protocol can distinguish between s-wave (in-phase bonds) and d-wave (out-of-phase bonds) symmetries.
• Significance: Provides a direct experimental route to verify d-wave superconductivity in the repulsive Hubbard model, a long-standing open problem in condensed matter physics, without relying on the mapping to attractive models (which fails for models with diagonal hopping $t'$).

C. Non-Equal-Time Correlations and ARPES Spectra

• Goal: Measure the retarded Green's function $G(k_0, t)$ and the spectral function $A(k, \omega)$, which are central to understanding excitations (e.g., polarons, dopants).
• Mechanism:
  1. A focused Raman beam in Fourier space extracts a single particle in a specific momentum mode $k_0$ into an auxiliary spin state.
  2. This particle is spatially isolated from the many-body system.
  3. The remaining system (with a hole at $k_0$) evolves for time $t$ under the full interacting Hamiltonian.
  4. The isolated particle is recombined with the system via a second Raman pulse.
• Significance: Unlike linear-response spectroscopy (which requires weak perturbations), this real-time approach allows for high-resolution spectral measurements without the linear-response limitation. It directly accesses the time-evolution of the system's excitations.

D. Hidden Off-Diagonal Order in Fractional Chern Insulators

• Goal: Detect the hidden order of composite bosons in fractional quantum Hall (FQH) states.
• Mechanism: A variation of the protocol where the final Raman pulse is locally applied via a focused tweezer to a single site $j$, while the rest of the system is measured globally.
• Significance: This allows the measurement of the correlation function of composite bosons (particles attached to flux tubes), which exhibit algebraic decay (long-range order) rather than the exponential decay of bare particles. This bypasses the need for complex "folded" lattice geometries previously proposed to measure such non-local order.

4. Experimental Considerations and Feasibility

The authors provide a realistic roadmap for implementation:

• Atoms: Proposes using $^{133}$Cs (bosons) for FQH protocols and $^6$Li (fermions) for Hubbard/superconductivity protocols.
• Interaction Suppression: A critical requirement is switching off interactions during the matter-wave evolution to maintain single-particle coherence.
  • For $^{133}$Cs: Quenching the magnetic field to a zero-crossing point between Feshbach resonances.
  • For $^6$Li: Rapidly flipping spins to weakly interacting states or ramping the magnetic field to near zero.
• Band Mapping: To re-immerse the isolated momentum mode into the lattice for time evolution, the protocol uses inverse band mapping (adiabatic ramping) to ensure the particle returns to a specific quasimomentum eigenstate without heating or Umklapp scattering.
• Resolution: The system can achieve shifts of $\sim 10$ lattice sites with standard optical setups (NA=0.5 objectives) and Raman beam angles.

5. Results and Significance

• Theoretical Validation: The authors derive the Heisenberg picture evolution of the operators, proving that the density measurements in the final step directly yield the target correlators (e.g., Eq. 2, 4, 6, and 9).
• Signal Strength: Numerical estimates suggest that while universal long-range signals (e.g., in d-wave superconductors) are small ($\sim 10^{-4}$), the non-universal short-range signals are strong enough to verify pairing symmetry with $\sim 10^4$ snapshots.
• Broad Impact:
  • Superconductivity: Offers a definitive test for d-wave order in the repulsive Hubbard model.
  • Spectroscopy: Enables high-fidelity ARPES-like measurements in cold atoms, bridging the gap between cold atom simulations and solid-state spectroscopy.
  • Topology: Provides a method to detect topological order in fractional Chern insulators without geometric folding.
  • Future Directions: The framework is extensible to out-of-time-order correlators (OTOCs) for diagnosing chaos, Majorana zero modes, and entanglement entropy.

Conclusion

This paper establishes a powerful, versatile "many-body phase microscope" protocol. By leveraging matter-wave lenses and Raman interferometry, it overcomes the fundamental limitation of quantum gas microscopes regarding phase information. The proposed protocols provide a direct experimental pathway to characterize some of the most elusive and important states in quantum many-body physics, including high-temperature superconductors, topological phases, and non-equilibrium spectral functions.