Variance of a multiple-species bosonic mixture

At the limit of an infinite number of particles, when the product of the number of bosons times the interaction strengths is kept fixed (i.e., at the mean-field or Gross–Pitaevskii limit), trapped Bose–Einstein condensates and mixtures thereof are known to be 100% condensed [1,2,3]. Furthermore, the many-body energy per particle and intraspecies and interspecies densities per particle exactly coincide with the mean-field results. Nonetheless, there can be massive correlations at this limit, which are readily exposed by variances of observables [4]. Such correlations emerge from the occupation of even a single particle in an excited natural orbital of the reduced one-particle density matrix that interacts with the large number of bosons in the condensed mode, and originate from the fundamental difference between the many-body and mean-field wavefunctions at this limit [5]. The differences between the many-body and mean-field variances can be used to characterize and explore such correlations [6,7]. In this work, we address some uncharted possibilities and consequences for the variance of observables in a multiple-species bosonic mixture by employing a generic solvable multiple-species model [8,9]. We analyze the variance of the position, momentum, and angular-momentum many-particle operators in a multiple-species bosonic mixture and identify how various correlations express themselves in fluctuations of observables. We investigate intraspecies and, in particular, interspecies observables in the mixture and their uncertainty products. A multiple-species uncertainty product is found to be structureless at the mean-field level of theory, whereas, at the many-body level of theory, it exhibits intricate properties. Generalizations, e.g., to a time-dependent scenario [10], are envisioned.

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