On the observation of nonclassical excitations in Bose–Einstein condensates

Abstract

In the recent experimental and theoretical literature well-established nonclassicality criteria from the field of quantum optics have been directly applied to the case of excitations in matter-waves. Among these are violations of Cauchy-Schwarz inequalities, Glauber-Sudarshan P-nonclassicality, sub-Poissonian number-difference squeezing (also known as the two-mode variance) and the criterion of nonseparability. We review the strong connection of these criteria and their meaning in quantum optics, and point out differences in the interpretation between light and matter waves. We then calculate observables for a homogeneous Bose-Einstein condensate undergoing an arbitrary modulation in the interaction parameter at finite initial temperature, within both the quantum theory as well as a classical reference. We conclude that to date in experiments relevant for this scenario nonclassical effects have not conclusively been observed and conjecture that additional, noncommuting, observables have to be measured to this end. Moreover this has important implications for proposed analog gravity models where the observation of nonclassical effects is a major goal.Bose-Einstein condensation (BEC) is a macroscopic quantum phenomenon where a large fraction of the bosons occupy the same lowest energy quantum state, and thus form a coherent matter wave. Many properties of the condensate can be captured by a Schrödinger-type equation for a complex field with a nonlinear potential referred to as the Gross-Pitaevskii equation (GPE) [1-3]. To cite one example, the GPE correctly predicts discrete values for the circulation of the velocity field in a BEC [4]. Nevertheless, within this description the evolution of the condensate can be considered a classical process [5]. On the other hand it is possible to excite small fluctuations of the condensate, for example by means of rapid changes in the condensate parameters [6][7][8][9][10][11], which may activate their nonclassical behavior. An ongoing line of research is to investigate experimentally the manipulation and detection of fluctuations in BECs, e.g. exploring their quantum nature [12][13][14] and correlations [11,15]. This is partially motivated by various analogue gravity studies [16][17][18][19], where BECs are utilized as quantum simulators [5] of quantum field theories in curved spacetime (QFTCS). In QFTCS one is interested in how quantum fields propagate on a classical curved spacetime geometry acting as a non-trivial background configuration. Within analogue gravity studies the ultimate goal is to mimic and capture genuine quantum effects predicted from QFTCS. The field of experimental analogue gravity in BECs started only a few years ago, and with recent advances, e.g.mimicking black hole evaporation [9,10,[20][21][22] and cosmological particle production in our Universe [13,[23][24][25][26], and the debate on the nonclassicality of the observed effects is as timely as ever. Hence, it is essential to find suitable observables that can distinguish between classical and nonclas...

Assessing degrees of entanglement of phonon states in atomic Bose gases through the measurement of commuting observables

“…vac (k), giving rise to inequality (17). This condition is represented by the dashed vertical line in Fig.…”

“…It should be noticed that it crosses the outermost limit of physical states with n 1 = n 2 ≡ n for a finite value of the mean occupation number n. This can be understood from the fact that the realisation of inequality (17) requires sufficiently large oscillations of the G (2) (k, t) with respect to G (2) vac (k). It is interesting to notice that it is the symmetric condition (A.12) that is used in the recent work [46], where the threshold is correctly pointed out.…”

Momentum correlations as signature of sonic Hawking radiation in Bose-Einstein condensates

“…The study of g 2 is of interest because the occurrence of entanglement and the quantum nature of the Hawking process can be tested through the violation of the Cauchy-Schwarz inequality, as recently studied in a similar context in Refs. [32,33,[40][41][42][43][44][45][46]. For instance, the Cauchy-Schwarz inequality is violated along the characteristic Hawking quanta-partner correlation lines u|out-d2|out of Figs.…”

Towards a Gordon form of the Kerr spacetime

“…Note that not transforming the coordinate φ as done in equation (35) would have resulted in obtaining the somewhat more usual diagonal form of oblate spheroidal coordinates (without the dr dφ cross term).…”