Atom counts reveal temperature in ultracold quantum gases

Researchers at Laboratoire Kastler Brossel have developed a new way to measure temperature in ultracold quantum gases by analyzing number fluctuations within probe volumes containing, on average, only one atom. This novel in-situ thermometry, demonstrated on an ideal Fermi gas using quantum gas microscopy, moves beyond the limitations of established methods requiring global thermal equilibrium. Unlike traditional techniques, the work, which extracts both global and local temperatures, does not rely on the fluctuation-dissipation theorem, instead utilizing the relationship between number fluctuations and density-density correlations. This nearly universal thermometer.

Quantum Gas Microscopy Measures Atom Fluctuations for Thermometry

The approach allows for both global and local temperature determination across a broad dynamic range, a capability previously limited in these systems. Researchers found that in the low-temperature regime, their measurements revealed significant deviations from predictions based on the fluctuation-dissipation theorem, uncovering the presence of sub-extensive fluctuations, similar to those observed in lattice systems. These sub-extensive fluctuations scale slower than the number of atoms, becoming particularly important when probing small volumes for local thermometry.

This method’s applicability extends beyond specific system configurations; it functions without demanding global thermal equilibrium, a common constraint of traditional thermometry. The team demonstrated this capability using a two-dimensional, spin-polarized cloud of ⁶Li atoms, using a technique called continuum quantum gas microscopy, previously introduced in their work.

Analysis of the variance of atom number as a function of average atom number within each probe volume allowed for precise temperature extraction. The observed quantum fluctuations at zero temperature scale as $L^{d–1}$ log(L^d), where L represents the system size and d is the number of dimensions, reflecting a violation of the area law for entanglement entropy. “When the size of the probe volume is small, as is required for local thermometry, sub-extensive number fluctuations are an important contribution,” the paper states, highlighting the sensitivity of the technique.

This sensitivity allows for the study of strongly-correlated systems even within the complex environments created by trapping potentials. The researchers conclude that their method provides a nearly-universal local and global thermometer for quantum gases, overcoming limitations inherent in existing techniques and opening new avenues for investigating ultracold atomic systems.

Number Fluctuations Correlate to Temperature in Probe Volumes

The method’s sensitivity is demonstrated through the observation of deviations from fluctuation-dissipation predictions, revealing sub-extensive fluctuations that become increasingly important as probe volume shrinks. Researchers confirmed that these sub-extensive fluctuations contribute a substantial fraction of the overall variance at low temperatures, quantifying the temperature dependence by analyzing probe volumes of fixed size, 2.2 micrometers, and varying the reduced temperature. Results show that the ratio of quantum fluctuations to total number fluctuations decreases monotonically with increasing temperature, aligning with expected behavior.

To improve confidence in the measurements, the team analyzed data from probe volumes ranging from 2.6 to 3.7 micrometers, averaging the resulting confidence intervals. The technique also accounts for correlations between atoms in adjacent probe volumes, and these are important to consider for fermionic gases where atom numbers exhibit anti-correlation.

In classical gases, number fluctuations in non-overlapping regions are independent, justifying a simple division of the error estimate; however, this is not the case for fermions. By dividing the temperature uncertainty by the square root of four, representing the number of non-overlapping squares within the probe volume, researchers corrected for these anti-correlations, yielding more accurate temperature readings.

“For a homogeneous system, there are no fundamental constraints on the size L of the square probe volumes used to partition our images when estimating the temperature via the density fluctuations,” the study reports, highlighting the method’s adaptability to various system configurations. This flexibility, combined with the ability to extract both global and local temperatures, positions this new thermometry as a nearly-universal tool for studying quantum gases.

Beyond Fluctuation-Dissipation: Observing Sub-Extensive Fluctuations

By measuring atom number fluctuations within volumes containing on the order of one atom, researchers have moved beyond reliance on the fluctuation-dissipation theorem to establish a new method for determining temperature in ultracold quantum gases. This approach reveals sub-extensive fluctuations, deviations from predictions based on the standard theorem, particularly noticeable at low temperatures, and offers a path to thermometry applicable to systems with complex trapping potentials without requiring precise calibration of those potentials.

Analysis of these fluctuations revealed a correction term, designated ${\Delta }_Q$, which captures sub-extensive atom number fluctuations scaling slower than the total number of atoms within the measured volume. This finding, detailed in the paper, suggests that even relatively small systems exhibit behavior predicted by this correction, with number fluctuations converging toward the fluctuation-dissipation theorem only at higher temperatures. Similar behavior has been previously observed in studies of Fermi-Hubbard gases, strengthening the validity of this new approach.

Researchers derived a general expression to further understand these sub-extensive fluctuations, considering a total system composed of a larger volume and the probe volume itself. This relation indicates that the size of the larger volume must exceed the probe volume by several times the correlation length, confirming that ${\Delta }_Q$ is a non-local quantity dependent on the shape and size of the probe volume.

Importantly, the team found that both zero- and finite-temperature contributions to sub-extensive fluctuations share the same microscopic origin, stemming from the interplay between the boundaries of the probe volume and the functional form of the two-particle correlation function. “This shows that even for a small system, the number fluctuations converge to the result of the fluctuation-dissipation theorem at high temperature,” the paper states. This advancement complements existing fluctuation thermometry methods and extends temperature measurement capabilities to quantum gases existing far from thermal equilibrium.

The ability to accurately measure both global and local temperatures, coupled with the uncovering of sub-extensive fluctuations, highlights the importance of further theoretical studies focused on accurately computing spatial density-density correlations within many-body systems. The work, the researchers conclude, underscores the need for a deeper understanding of these correlations to refine temperature measurements in complex quantum systems.

Local and Global Temperature Extraction with Single-Atom Resolution

This technique, detailed in recent work, moves beyond conventional methods by directly linking number fluctuations to density-density correlations, bypassing reliance on the fluctuation-dissipation theorem. The method’s versatility extends to systems with arbitrary trapping potentials, eliminating the need for precise calibration of those potentials, a significant advantage over existing thermometry techniques. By scanning a smaller square box within the primary probe volume and extracting temperature measurements from various positions, the team established confidence intervals for local temperature readings.

Results demonstrate that the extracted local temperature can be reliably determined as a function of the ratio between the size of the smaller square and the overall probe volume, providing a spatially resolved thermal map. This capability is crucial for understanding non-equilibrium phenomena and complex quantum systems where temperature variations are expected.

Limitations of Traditional Density Profile and Fluctuation Thermometry

Fitting in-situ density profiles and standard fluctuation thermometry, while widely used, demand precise knowledge of trapping potentials and are inapplicable to spatially homogeneous systems, limitations this work addresses through a new approach. This new method diverges from established principles by dispensing with reliance on the fluctuation-dissipation theorem, a cornerstone of statistical mechanics used in conventional fluctuation thermometry.

The ability to bypass the need for precise trap calibration is particularly significant, as accurately characterizing trapping potentials can be a substantial experimental hurdle. The limitations of existing techniques become particularly acute when investigating out-of-equilibrium scenarios, such as the breakdown of thermalization, the response to a sudden quantum quench, or heat transport within the system.

Traditional methods struggle to provide accurate local temperature readings in these dynamic situations, hindering the study of non-equilibrium phenomena. “To date, temperature measurements have relied on a range of methods, each with its own strengths and limitations,” the paper states, highlighting the ongoing need for improved thermometry techniques. The demonstrated method, applicable to systems with arbitrary trapping potentials, offers a solution by providing both global and local temperature measurements without requiring the system to be in full thermal equilibrium.

The researchers point out that while the local density approximation can be used in the horizontal plane of the light sheet potential, the significant vertical trapping frequency necessitates a fully quantized treatment of motion in that direction. They detail how the occupation of motional levels is determined by the chemical potential, providing a theoretical basis for their analysis.

⁶Li Atom Imaging with Continuum Quantum Gas Microscopy

Researchers observed increasingly suppressed density fluctuations with decreasing temperature, a phenomenon also seen in three-dimensional bulk systems via absorption imaging and in two-dimensional lattices using quantum gas microscopy, but now with unprecedented precision for a bulk gas. Fitting data sets of atom number variance against average atom number using a modified equation allowed extraction of temperature as the sole fitting parameter, with details available in supplementary materials.

This advancement builds on prior work employing quantum gas microscopy to study Fermi-Hubbard gases, performing thermometry at scales not specified relative to the correlation length. However, the current method goes further, revealing the presence of sub-extensive number fluctuations and quantitatively studying their behavior as a function of temperature. The exact relation connecting number fluctuations to compressibility within a homogeneous probe volume is central to this observation; a correction term captures these sub-extensive fluctuations, becoming negligible only in the thermodynamic limit at finite temperature. (1),” the paper states, detailing the mathematical basis for this observation.

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