Abstract. Analytical expressions are given for the static structure factor S(k) and the pair correlation function g(r) for uniform ideal Bose-Einstein and Fermi-Dirac gases for all temperatures. In the vicinity of Bose Einstein condensation (BEC) temperature, g(r) becomes long ranged and remains so in the condensed phase. In the dilute gas limit, g(r) of bosons & fermions do not coincide with Maxwell-Boltzmann gas but exhibit bunching & anti-bunching effect respectively. The width of these functions depends on the temperature and is scaled as √ (inverse atomic mass). Our numerical results provide the precise quantitative values of suppression/increase (antibunching and bunching) of the density fluctuations at small distances in ideal quantum gases in qualitative agreement with the experimental observation for almost non-trapped dilute gases.
Analytical expressions are given for the static structure factor S(k) and the pair correlation function g(r) for uniform ideal Bose‐Einstein and Fermi‐Dirac gases for all temperatures. In the vicinity of Bose Einstein condensation (BEC) temperature, g(r) becomes long ranged and remains so in the condensed phase. In the dilute gas limit, g(r) of bosons & fermions do not coincide with Maxwell‐Boltzmann gas but exhibit bunching & anti‐bunching effect respectively. The width of these functions depends on the temperature and is scaled as √ (inverse atomic mass). Our numerical results provide the precise quantitative values of suppression/increase (antibunching and bunching) of the density fluctuations at small distances in ideal quantum gases in qualitative agreement with the experimental observation for almost non‐trapped dilute gases.
We derive an expression for the reduced density matrices of ideal Bose and Fermi gases in the canonical ensemble, which corresponds to the Bloch--De Dominicis (or Wick's) theorem in the grand canonical ensemble for normal-ordered products of operators. Using this expression, we study one- and two-body correlations of homogeneous ideal gases with
N particles. The pair distribution function
g(2)(r) of fermions clearly exhibits antibunching with
g(2)(0)=0 due to the Pauli exclusion principle at all temperatures, whereas that of normal bosons shows bunching with
g(2)(0)≈2, corresponding to the Hanbury Brown--Twiss effect. For bosons below the Bose--Einstein condensation temperature
T0, an off-diagonal long-range order develops in the one-particle density matrix to reach
g(1)(r)=1 at
T=0, and the pair correlation starts to decrease towards
g(2)(r)≈1 at
T=0. The results for
N→∞ are seen to converge to those of the grand canonical ensemble obtained by assuming the average
⟨ψ^(r)⟩ of the field operator
ψ^(r) below
T0. This fact justifies the introduction of the "anomalous" average
⟨ψ^(r)⟩=0 below
T0 in the grand canonical ensemble as a mathematical means of removing unphysical particle-number fluctuations to reproduce the canonical results in the thermodynamic limit.
This thesis presents two experiments on ultracold gases of metastable Helium that constitute the extension, to matter waves, of experiments that have been fundamental for the development of quantum optics. Both experiments have been realized by using a single atom detector, that allows one to reconstruct the position of the atoms in three dimensions. In the first experiment we directly compared the two-body correlation function for a cold cloud of identical fermions and for a cold cloud of identical bosons at the same temperature issued from the same experimental apparatus. Since bosons and fermions obey different quantum statistics, the two-body correlation functions are different: identical bosons tend to be detected bunched in pairs, while identical fermions tend to antibunch because of the Pauli exclusion principle. In the second experiment we studied the correlation between pairs of bosonic atoms generated in the collision of two Bose-Einstein condensates. The measurement of the correlation function allows us to demonstrate that atoms flying back to back in the centre of mass frame are correlated in pairs. Furthermore the observation of a Hanbury Brown Twiss correlation between atoms with collinear velocity demonstrates that our system cannot be interpreted in terms of classical mechanics, but that the bosonic statistics plays a crucial role. A new version of the experiment will allow us to measure the angular dependence of the population of the scattering modes and the number difference squeezing between two correlated modes.
Thermometry is essential for studying many-body physics with ultracold atoms. Accurately measuring low temperatures in these systems, however, remains a significant challenge due to the absence of a universal thermometer. Most widely applicable methods, such as fitting of in situ density profiles, are limited by the requirement of global thermal equilibrium and inapplicability to homogeneous systems. In this Letter, we introduce a novel in situ thermometry for quantum gases, leveraging single-atom-resolved measurements via quantum gas microscopy, and demonstrate it on an ideal Fermi gas. By analyzing number fluctuations in probe volumes containing as little as one atom on average, we extract both global and local temperatures over a broad dynamic range. Our method does not rely on the fluctuation-dissipation theorem and is based instead on the exact relationship between number fluctuations and density-density correlations. In the low-temperature regime, it allows us to observe significant deviations from fluctuation-dissipation predictions, uncovering subextensive fluctuations. Our method is applicable to systems with arbitrary trapping potentials, requiring neither precise trap calibration nor global thermal equilibrium. This thermometer for quantum gases overcomes key limitations of existing techniques, paving the way for more accurate and versatile temperature measurements in ultracold quantum systems.
The behavior of the spatial two-particle correlation function is surveyed in detail for a uniform one-dimensional Bose gas with repulsive contact interactions at finite temperatures. Long-, medium-, and short-range effects are investigated. The results span the entire range of physical regimes from ideal gas to strongly interacting and from zero temperature to high temperature (Gross-Pitaevskii) and strongly interacting (Tonks-Girardeau) gases. We present perturbative analytic methods, available at strong and weak couplings, and first-principles numerical results using imaginary time simulations with the gauge-
P representation in regimes where perturbative methods are invalid. Nontrivial effects are observed from the interplay of thermally induced bunching behavior versus interaction induced antibunching.