Quasi-long-range order in trapped two-dimensional Bose gases

Abstract

We study the fate of algebraic decay of correlations in a harmonically trapped two-dimensional degenerate Bose gas. The analysis is inspired by recent experiments on ultracold atoms where power-law correlations have been observed despite the presence of the external potential. We generalize the spin wave description of phase fluctuations to the trapped case and obtain an analytical expression for the one-body density matrix within this approximation. We show that algebraic decay of the central correlation function persists to lengths of about 20% of the Thomas-Fermi radius. We establish that the trap-averaged correlation function decays algebraically with a strictly larger exponent weakly changing with trap size and find indications that the recently observed enhanced scaling exponents receive significant contributions from the normal component of the gas. We discuss radial and angular correlations and propose a local correlation approximation which captures the correlations very well. Our analysis goes beyond the usual local density approximation and the developed summation techniques constitute a powerful tool to investigate correlations in inhomogeneous systems.PACS numbers: 05.70. Jk, 64.60.an, With the achievement of quantum degeneracy in ultracold atom gases a new experimental platform for studying fundamental questions related to phase transitions and critical phenomena has appeared. In particular, correlation functions can be measured through interference and time-of-flight techniques [1][2][3][4][5][6][7]. A crucial aspect of the experiments, however, consists in the absence of translation invariance due to the presence of the external trapping potential. Consequently, correlations between points r and r are not uniquely determined by the value of r − r . This effect should be unimportant for short-range correlations, and, in fact, thermodynamic properties of ultracold gases are often very well-captured by a local density approximation. The situation changes drastically for a system at a critical point, where the correlation length diverges and thus competes with the inhomogeneity of the trapping potential.Whereas the experimental preparation of critical systems typically requires a highly fine-tuned setup, they are rather effortlessly realized in two-dimensional (2D) systems whose low-energy excitations can be mapped onto an XY model. Examples apart from 2D ultracold quantum gases [8][9][10][11][12] are given by thin Helium films [13], layered magnets [14,15], and 2D excitonpolariton condensates [16]. To quantify correlations in these systems we introduce the one-body density matrix ρ(r, r ) = Φ † (r)Φ(r ) , whereΦ † (r) is the creation operator for a particle at point r. For the spatially homogeneous case we then have ρ hom (r, r ) = f (|r − r |) with some function f (r) due to translation and rotation invariance. The Mermin-Wagner-Hohenberg theorem [17,18] forbids long-range order at any finite temperature such that lim r→∞ f (r) = 0. However, in the XY model an ordered low-temperature phase with inf...

Quantum scale anomaly and spatial coherence in a 2D Fermi superfluid

“…Experimentally, g 1 (r) is directly obtained from the n(k) through a Fourier transform. In our previous work [4], we observed the transition from exponential to algebraic decay in g 1 (r), in agreement with BKT theory and Quantum Monte Carlo computations [32]. Here, we use the same procedure described in [4] to extract g 1 (r) at the inner and outer turning points.…”

“…In addition, we extract the exponent η by fitting a power-law (f (r) ∼ r −η(t) ) to g 1 (r, t). Even though the exponents we measure are larger than the homogeneous BKT predictions, they have the same qualitative behavior [32], in particular a smaller exponent corresponds to a larger superfluid phase space density D s = ρ s λ 2 T , where ρ s is the superfluid density and λ T the thermal de Broglie wavelength.In the BEC regime, the two curves (g 1 (r, t i ) and g 1 (λr, t o )) collapse onto each other (see Fig. 3 A), whereas in the crossover regime, the correlation functions are substantially different with the inner g 1 (r, t i ) decaying slower than expected.…”

Observation of algebraic time order for two-dimensional dipolar excitons

“…Within our precision, the exponent η at criticality is compatible with η c 0.25 predicted for a BKT transition in the thermodynamic limit [13]. Averaging the contrast over a finite region with small density variations may result in significantly higher values for the apparent exponent than in strictly homogeneous systems [15,34]. Here, the interference contrast is extracted along one direction in the central density region, making our measurements less sensitive to small density variations, and thus closer to the homogeneous limit.…”

Fermi-to-Bose crossover in a trapped quasi-2D gas of fermionic atoms

“…Recent calculations of the algebraic-decay exponent have accounted for the trapped-gas inhomogeneity [108,109] and contribution of the normal fraction [109]. These calculations have shown η values close to the measured ones [109]. This agreement may be regarded as evidence against hydrodynamic expansion because such expansion would have changed the apparent η value, making it different from the actual in situ quantity.…”