Phase-space microscopes for quantum gases: Measuring conjugate variables and momentum-weighted densities

This paper proposes concrete protocols for extending quantum gas microscopes to measure conjugate variables and momentum-weighted densities in phase space by mapping momentum onto auxiliary degrees of freedom, distinguishing between a Husimi-Q mode that jointly measures position and momentum with fundamental noise in position, and an averaged mode that retrieves spatially resolved momentum density averages.

The Gist

Imagine you are trying to take a perfect photograph of a swarm of bees in a garden.

The Old Way (Standard Quantum Gas Microscopes):
Currently, scientists have a super-powerful camera (the Quantum Gas Microscope) that can freeze time and take a picture of exactly where every single bee is sitting on a flower. This is amazing! They can count them, see how they cluster, and map out the garden.

The Problem:
But there's a catch. In the quantum world, bees are also flying. If you try to take a picture of where they are, you lose information about how fast and in what direction they are flying. It's like the famous rule of quantum mechanics: you can know the location OR the speed, but not both perfectly at the same time. If you try to measure both, the camera lens gets blurry, and the picture gets fuzzy with "quantum noise."

The New Idea (Phase-Space Microscopes):
This paper proposes a new, clever trick to build a "Phase-Space Microscope." This new device doesn't just take a flat photo; it creates a 3D movie that shows you both where the bees are and how fast they are moving, all at the same time.

Here is how they do it, using two different "modes" or tricks:

Trick 1: The "Ghost Shadow" Method (Husimi-Q Mode)

Imagine you have a bee flying in the air. You want to know its speed, but you can't just look at it.

1. The Magic Lens: First, the scientists use a special "lens" (a harmonic trap) that turns the bee's speed into a position. It's like a magic mirror where a fast bee appears far to the left, and a slow bee appears to the right.
2. The Shadow Tag: Now, they give the bee a tiny "tag" in a hidden dimension (let's call it the "Z-axis," a direction we can't normally see). They push the bee sideways in this hidden dimension. The harder the bee was flying originally, the harder it gets pushed in this hidden dimension.
3. The Photo: Finally, they take a picture of the bee in the real world (where it is) and the hidden world (how hard it was pushed).

The Result: You get a picture where every dot represents a bee with a specific location and a specific speed.

• The Catch: Because of quantum rules, the picture isn't perfectly sharp. It's a bit fuzzy, like a watercolor painting. You can't pinpoint the exact speed and exact location simultaneously with infinite precision, but you get a beautiful, fuzzy map that tells you the probability of finding a bee there with that speed. This is called the Husimi-Q measurement.

Trick 2: The "Spin-Tag" Method (Averaged Mode)

This is a different approach for when you don't need a fuzzy picture of every single bee, but rather want to know the average speed of bees in a specific spot.

Imagine the bees have tiny internal compasses (spins).

1. The Spin Rotation: The scientists use a magnetic field to spin the bees' compasses. The faster a bee is flying, the more its compass spins. A slow bee barely moves its compass; a fast bee spins it wildly.
2. The Measurement: They don't try to see the speed directly. Instead, they look at the compasses. If they see a compass pointing North, they know the bees in that area were moving slowly. If they see a compass pointing South, they know the bees were moving fast.

The Result: This method gives you a super-sharp map of the average energy or average speed at every single point in the garden. Because they aren't trying to measure the exact speed of one specific bee, but rather the "average spin" of the group, they don't get the fuzzy quantum noise. They can see the speed map with incredible, microscopic detail.

Why is this a Big Deal? (Real-World Applications)

The authors show how this new microscope can solve puzzles that were previously impossible:

• Seeing Invisible Edges: Imagine a wall in the garden that is so sharp it's invisible to a normal camera. A normal camera just sees a blur. But this new microscope sees the "high-speed bees" that bounce off the sharp edge. It can detect the edge even if it's smaller than the camera's blur limit.
• Mapping Vortex Whirlpools: In super-cold gases, particles sometimes spin around in tiny tornadoes (vortices). This microscope can see the "wind speed" of the tornado, showing exactly where the center is and how fast the air is spinning around it.
• Local Thermometers: Just like you can tell how hot a room is by how fast the air molecules are moving, this microscope can measure the temperature of a tiny, specific spot in the gas cloud. It can tell you if one corner of the cloud is hotter than the other, which is crucial for understanding how these quantum systems work.
• Detecting Hidden Interactions: It can measure how much the particles are "bumping" into each other, revealing secrets about the forces holding the quantum world together.

The Bottom Line

This paper is a blueprint for upgrading our "quantum cameras." By using clever tricks to map speed onto hidden dimensions or internal compasses, scientists can finally see the full picture of quantum gases: where the particles are and how they are moving, simultaneously. It turns a blurry snapshot into a high-definition, 3D movie of the quantum world.

Technical Summary

1. Problem Statement

Quantum Gas Microscopes (QGMs) have revolutionized the study of ultracold atomic gases by enabling spatially resolved detection of individual particles, allowing for the measurement of density-density correlators and string order. However, a fundamental limitation exists: standard QGMs measure atomic positions (position basis) or, with specific lattice manipulations, local currents and phases. They cannot directly measure conjugate variables (position $x$ and momentum $p$) simultaneously with high spatial resolution.

While joint measurements of non-commuting observables are theoretically permitted via Positive Operator-Valued Measures (POVMs), they introduce fundamental quantum noise (Heisenberg uncertainty). Existing techniques struggle to map momentum information onto spatially resolved images without losing the ability to resolve local features in the position basis. The paper addresses the need for a protocol to extend QGMs into phase-space microscopes that can access atomic occupations and correlations jointly in both position and momentum.

2. Methodology

The authors propose two distinct operational modes for phase-space microscopes, both relying on mapping momentum (or functions of momentum) onto auxiliary degrees of freedom (an auxiliary spatial dimension or internal spin states) using matter-wave Fourier transformations.

Core Mechanism: The Fourier Plane Mapping

The protocols utilize a "matter-wave microscope" approach:

1. Fourier Transformation: A quarter-period ($T/4$) evolution in a harmonic trap transforms the system from position space to momentum space (the Fourier plane), where the spatial coordinate encodes the initial momentum.
2. Coupling: In this Fourier plane, a spatially dependent potential or field couples the momentum-dependent variable to an auxiliary degree of freedom.
3. Inverse Transformation: A second $T/4$ evolution maps the system back to real space, converting the auxiliary information into a measurable spatial displacement or spin state.

Mode A: Husimi-Q Phase-Space Microscope

• Goal: Jointly measure position ($x$) and momentum ($p$) for a single particle (or weakly interacting gas).
• Protocol:
  • An auxiliary dimension ($z$) is used, initially prepared in a ground state $\phi_0(z)$ with width $\sigma$.
  • A unitary "kick" operator $\hat{U}_{kick} = e^{i\alpha \hat{p}_x \hat{z}/\hbar}$ is applied. This is realized by transforming to the Fourier plane, applying a saddle potential $V_{kick}(x,z) \propto -xz$ (which imparts a $z$-momentum impulse proportional to $x$-position/momentum), and transforming back.
  • The final state is measured in both $x$ and $z$ (momentum).
• Result: The probability distribution $P(x_m, p_m)$ corresponds to the Husimi-Q representation of the quantum state. This is a positive-definite measure obtained by convolving the Wigner function with a Gaussian.
• Noise Trade-off: The protocol introduces quantum noise. The uncertainty in momentum ($\Delta p_m$) is inversely proportional to the auxiliary width $\sigma$, while the uncertainty in position ($\Delta x_m$) is proportional to $\sigma$. The product satisfies the uncertainty principle.

Mode B: Averaged-Mode Phase-Space Microscope

• Goal: Measure spatially resolved averages of momentum density and its moments (e.g., kinetic energy, quartic momentum) without the Heisenberg noise penalty of joint $x$-$p$ measurement.
• Protocol:
  • The auxiliary degree of freedom is an internal spin-$S$ system (or a finite set of harmonic trap states).
  • The system is prepared in a polarized spin state $|S, S\rangle$.
  • In the Fourier plane, a momentum-dependent spin rotation $\hat{U}_{rot} = \exp[-i\vec{\theta}(\hat{p}) \cdot \hat{\vec{S}}]$ is applied.
  • Two specific examples:
    1. Spin-rotation-(i): A circulating Rabi field maps 2D momentum to the Bloch sphere. Projective measurement of $\hat{S}_z$ yields the kinetic energy density ($\propto |\nabla \psi|^2$).
    2. Spin-rotation-(ii): A quadratically varying Zeeman shift combined with a uniform Rabi field maps $|p|^2$ to the spin state. Projective measurement yields the quartic momentum density ($\propto |\nabla^2 \psi|^2$).
• Result: These measurements provide spatial maps of momentum moments. Crucially, because they measure specific observables rather than joint conjugate variables, they are not limited by Heisenberg noise regarding spatial resolution.

3. Key Contributions

• New Measurement Protocols: The paper introduces concrete experimental protocols to realize phase-space microscopes using current state-of-the-art QGM technology (harmonic traps, optical lattices, and spin manipulation).
• Husimi-Q Realization: It demonstrates how to implement the Arthurs-Kelly proposal for joint measurement of non-commuting variables in cold gases, providing a direct experimental realization of the Husimi-Q function.
• Momentum-Weighted Densities: It defines and proposes methods to measure new observables: local kinetic energy density and higher-order momentum moments, which are difficult to access with standard QGMs.
• Resolution Analysis: The authors clarify that while joint measurements (Husimi-Q) are subject to noise trade-offs, the "averaged-mode" measurements can achieve arbitrary spatial resolution limited only by atomic shot noise, not quantum uncertainty.

4. Results and Applications

The paper illustrates the utility of these microscopes in several physical scenarios:

• Sharp Potential Steps: A Husimi-Q microscope can detect the high-momentum tails generated by sharp potential edges (length scale $\delta$) even when $\delta$ is smaller than the optical resolution limit ($w$), whereas a standard density measurement would blur the edge.
• Quantum Vortices: Averaged-mode microscopes can image the kinetic energy density (peaked at the vortex core) and quartic density (a ring around the core), allowing for detailed characterization of vortex cores and potential 3D vortex line tracking via spin-flips.
• Local Thermometry: By measuring local kinetic energy density, one can determine local temperature in strongly interacting systems or partially condensed clouds, improving the fit of bimodal profiles and enabling the study of temperature gradients and second sound.
• Local Tan Contact: The quartic momentum density measurement allows for the extraction of the Tan contact ($C$) in spatially inhomogeneous systems (e.g., inside vortex cores), providing insight into interaction strength variations.
• Correlations and Entanglement: The ability to measure conjugate variables with single-particle resolution opens pathways to study spatial correlations in momentum space, hidden order in exotic phases, and entanglement structures (e.g., squeezing).

5. Significance

This work significantly expands the toolbox for cold-atom experiments. By bridging the gap between real-space imaging and momentum-space analysis, phase-space microscopes allow researchers to:

1. Access Hidden Information: Reveal local momentum distributions and correlations that are invisible to standard density imaging.
2. Overcome Diffraction Limits: Use high-momentum tails to resolve features smaller than the optical point-spread function.
3. Probe Non-Equilibrium and Interacting Systems: Provide direct access to local thermodynamic quantities (temperature, contact) and kinetic energy in complex, inhomogeneous many-body states.

The proposed techniques are directly implementable with existing quantum gas microscope setups, suggesting an immediate pathway for experimental realization and new discoveries in quantum many-body physics.