Suppressed fluctuations in Fermi gases

Suppose you have a large volume of an ordinary gas and you examined a small subvolume within it that contains a mean number of particles N. You would expect that the variance of that number, δN2, resulting from repeated measurements, is also equal to N. By “ordinary,” I mean two things. First, the gas is ideal, i.e., composed of particles with negligible interactions, and second, the gas is “classical,” i.e., composed of distinguishable particles. We often refer to this behavior of the variance as shot noise. A common example of this noise is what we hear when rain drops hit a tin roof, although in that example, N is the number of hits within a time interval, rather than particles within a sub-volume. But if the particles are quantum mechanically identical (that is, are subject to quantum mechanical exchange symmetry), the above expectation is incorrect. Ensembles of identical particles can be either bosons or fermions and their fluctuation properties are very different. Most statistical physics textbooks show that δN2 = N ± N2/Z, where Z is the number of elementary phase-space cells occupied by the small volume Z =∆x∆y∆z∆px∆py∆pz/h, where h is Planck’s constant, and the plus and minus signs refer to bosons and fermions, respectively [1]. This formula, at least for the case of bosons, was already demonstrated by Einstein in 1925 in one of his famous papers on the theory of the ideal quantum gas [2]. Thus the number of particles in a small subvolume of a gas will exhibit fluctuations either above or below shot noise, depending on whether it is composed of bosons or fermions (Fig. 1). Now, two groups, one at the Eidgenössische Technische Hochschule (ETH) in Zurich, Switzerland [3], and the other at the Massachusetts Institute of Technology [4] FIG. 1: A schematic illustration of the relationship between density, density fluctuations, and temperature in a onedimensional Fermi gas. The three grids represent a 1D phase space, with axes x and px. Each box represents a phase-space cell (with volume = h). At most, one particle is permitted per box. The density corresponds to the number of atoms per column. The temperature is related to how the number of atoms per row decreases with px. A higher temperature means more population in high momentum states. (a) A cold dense gas. (b) A cold but less dense gas. (c) A dense but hotter gas. The density fluctuations (that is, the variance in the number of particles per column) are lowest in (a). If the absolute density is known, a measurement of the density fluctuations gives information about the absolute temperature. This relationship is embodied in the fluctuation and dissipation theorem.