New Microscope Maps Quantum Gas In Phase Space

Researchers at the University of Cambridge and TU Dortmund University have detailed protocols for a new class of quantum gas microscope capable of mapping momentum in phase space. The work introduces two distinct operational modes, the Husimi-Q and averaged-mode phase space microscopes, each offering a different balance between position and momentum measurement precision.

The averaged-mode microscope can retrieve the spatial dependence of averages of momentum density with arbitrary spatial resolution, a capability exceeding that of standard quantum gas microscopy. These techniques extend existing methods by mapping momentum onto auxiliary degrees of freedom and using positive operator-valued measures.

Quantum Gas Microscopes and Phase-Space Mapping N.R. Cooper of the TCM Group and C. Weitenberg at TU Dortmund University co-authored the work detailing these techniques.

The core of this development lies in the ability to simultaneously access atomic occupations and correlations in both position and momentum, despite these being conjugate variables that do not typically commute. This mapping is accomplished through manipulation in matter-wave Fourier space, allowing for the creation of phase-space microscopes capable of revealing previously hidden features of quantum many-body systems.

Two distinct operational modes define the versatility of these new microscopes. The Husimi-Q phase space microscope jointly measures position and momentum, introducing quantum noise distributed between the two variables. “By exposing momentum-space features with spatial resolution, they open up many new ways to probe quantum many-body systems,” the authors write, highlighting the potential for deeper insights into complex quantum phenomena.

The protocols detailed in the work include methods for mapping momentum space to either an auxiliary dimension or an auxiliary spin, using resonant Rabi pulses to project momentum onto a Bloch sphere. The researchers note that these techniques approximate momentum densities accurately when momentum values are significantly less than a high-momentum cutoff, denoted as p0.

Illustrative examples demonstrate the utility of these microscopes in diverse physical settings, including the imaging of quantum vortices. The authors explain these advancements are particularly relevant for weakly trapped gases where the continuum momentum is the dominant factor.

By offering a new avenue for probing quantum many-body systems, these phase-space microscopes can facilitate a deeper understanding of the detailed behavior of matter at the quantum level and unlock new possibilities for exploring the fundamental laws of physics. Mapping Momentum Using Matter-Wave Fourier Space Extending the capabilities of quantum gas microscopy, researchers have devised protocols to measure momentum by mapping it onto auxiliary degrees of freedom, enabling phase-space imaging of quantum gases.

These new techniques use positive operator-valued measures ( POVMs ) to probe the behavior of cold atomic gases in a way previously inaccessible, opening avenues for studying complex quantum systems. The work details how momentum can be mapped to either an additional spatial dimension or an auxiliary spin state through precise manipulation in matter-wave Fourier space.

This approach allows for the construction of two distinct phase-space microscopes, each offering a unique way to analyze quantum gases. Conversely, the averaged-mode microscope focuses on extracting the spatial distribution of average momentum density and its related moments, achieving a level of spatial resolution not previously attainable with standard quantum gas microscopy.

The averaged-mode microscope’s ability to retrieve these averages with arbitrary spatial resolution stems from the use of specific protocols for mapping momentum. These protocols, demonstrated through sketches of experimental setups, involve applying tailored pulses in the Fourier plane to transform momentum information into measurable spin states.

For example, the application of quarter-period pulses to both x and y dimensions creates a configuration where the momentum can be analyzed. This allows for the measurement of mean momentum densities using projective measurements of spin components, providing insights into the kinetic energy density of the gas.

The researchers demonstrate that even without complete POVMs on the spin, the mapping of momentum to spin can be used to construct an averaged-mode phase space microscope. They illustrate this with examples for Dysprosium (S=8) and Holmium (S=11), showing how expectation values of spin components reveal mean momentum densities.

The team found that the spin coherent state aligned along a vector n≃(p_x/p_0,p_y/p_0,1-|p|^2/(2p_0^2)) allows for the extraction of these momentum-related properties. The general expression developed allows for analysis of the consequences of different “kicks” in the Fourier plane, providing a versatile tool for manipulating and measuring momentum distributions.

By using the fact that oscillator states take the same functional form in both position and momentum space, the researchers were able to refine the mapping process and correct for spatial inversion and momentum scaling. This precise control over momentum mapping is important for achieving the arbitrary spatial resolution offered by the averaged-mode phase space microscope, enhancing the ability to probe the complex behavior of quantum many-body systems.

The work illustrates the utility of these phase-space microscopes in a variety of physical settings. Husimi-Q Microscope: Joint Position-Momentum Measurement The Husimi-Q phase space microscope achieves a capability beyond conventional quantum gas microscopy by revealing momentum details even when position is sharply resolved.

While increasing positional accuracy typically obscures momentum information, this new approach maintains access to momentum distributions, allowing for the detection of subtle features like high-momentum tails. Quantum gas microscopes perform the spatially resolved detection of the individual particles in a quantum many-body system of ultracold atoms and allow for microscopic insights into quantum phases.

They were first realised in lattice systems and later in the continuum. Beyond characterizing potential steps, these microscopes also offer new ways to study more complex quantum phenomena. This is achieved by creating a coherent state in two-dimensional phase space, centered on the vortex’s position and momentum.

The measurement functions as a positive operator-valued measure (POVM) across the entire 2D phase space, realizing a two-dimensional Husimi-Q microscope. Positional uncertainty is governed by a parameter, while momentum uncertainty is controlled by the atomic spin S. The precision of momentum resolution in this averaged-mode microscope is directly linked to the size of the atomic spin.

For a fixed characteristic momentum p0, increasing the spin manifold allows for increasingly accurate momentum measurements. The researchers suggest that atomic species with large spin manifolds, such as Dysprosium with 17 states or Holmium with 23 states, would be particularly well-suited for this application.

This reliance on atomic spin provides a pathway to enhance the resolution of momentum measurements, offering a degree of control not previously available. The combination of these techniques provides a powerful toolkit for probing quantum many-body systems, offering insights into their complex behavior.

Averaged-Mode Microscopy of Momentum Density Averaged-mode phase space microscopy retrieves spatial information about the average momentum density with a level of detail previously unattainable, offering a new tool for examining quantum gases. This latter capability is particularly noteworthy, as these averages can be retrieved with arbitrary spatial resolution.

Researchers demonstrate this by outlining protocols to measure different moments of the momentum distribution, applying them in the Fourier plane where position is labeled by momentum p. One protocol, spin-rotation-(i), transforms an initial spin state using a resonant Rabi pulse, mapping two-dimensional momentum space onto the Bloch sphere.

Projective measurements of spin components then reveal information about the momentum density itself. Specifically, measurements of S z , when expanded for small |p|/p 0 , yield insights into both kinetic energy and orbital magnetization densities. Repeating measurements with Δ L = ± 1 allows for the extraction of these densities, while measurements of S x and S y provide the local momentum density.

The technique also allows for the measurement of quantities beyond simple momentum density. For Dysprosium (S = 8) this yields a resolution of approximately 1/8, and for Holmium (S = 11) approximately 1/11. The method functions effectively even without complete positive operator-valued measures (POVMs) on the spin.

They demonstrate that the expectation values of S x and S y provide information on the mean momentum densities, while the expectation value of S z reveals the mean kinetic energy density. Realizing Phase-Space Microscopes with Harmonic Traps Extending the capabilities of quantum gas microscopes, N.R.

Cooper, Y. Yang, and C. Weitenberg introduce new approaches that map momentum onto auxiliary degrees of freedom, sidestepping limitations inherent in directly measuring conjugate variables like position and momentum simultaneously.

This is achieved by using finite auxiliary spaces, either a finite set of harmonic trap states or a spin of finite length, to represent momentum information. This contrasts with the Husimi-Q microscope, which measures position and momentum jointly but introduces quantum noise due to the measurement apparatus.

The averaged-mode approach measures quantities like the sum of squared momenta weighted by a delta function, effectively probing momentum distributions without directly resolving individual momentum values. “Averages in phase space are readily obtained when one makes the auxiliary space finite,” the paper explains, highlighting the technique’s robustness.

These microscopes expand the range of measurable quantities beyond simple particle density. The work illustrates the utility of these techniques in diverse physical settings. 👉 More information 🗞 Phase-Space Microscopes for Quantum Gases: Imaging Conjugate Variables and Momentum-Weighted Densities ✍️ N.

R. Cooper, Y. Yang and C.

Weitenberg 🧠 DOI: http://link.aps.org/doi/10.1103/12rn-rlg8 More like this Stay current See today’s quantum computing news on Quantum Zeitgeist for the latest breakthroughs in qubits, hardware, algorithms, and industry deals.