Abstract Ultracold interacting atoms are an excellent tool to study correlation functions of many-body systems that are generally eluding detection and manipulation. Herein, we investigate the ground state of bosons in a tilted triple-well potential and characterize the many-body state by the eigenvalues of its reduced one-body density matrix and Glauber correlation functions. We unveil how the interplay between the interaction strength and the tilt can be used to control the number of correlated wells as well as the fragmentation, i.e. the number of macroscopic eigenvalues of the reduced one-body density matrix.
We investigate the appearance of quantum chaos in a single many-body wave function by analyzing the statistical properties of the eigenvalues of its reduced density matrix ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{A}$ of a spatial subsystem
A. We find that (i): the spectrum of the density matrix is described by so-called Wishart random matrix theory, which (ii): exhibits besides level repulsion, spectral rigidity, and universal spectral correlations between eigenvalues separated by distances ranging from one up to many mean level spacings, which we investigate. We use these universal spectral characteristics of the reduced density matrix as a definition of chaos in the wave function. A simple and precise characterization of such universal correlations in a spectrum is a segment of strictly linear growth at sufficiently long times, recently called the ``ramp,'' of the spectral form factor which is the Fourier transform of the correlation function between a pair of eigenvalues. It turns out that Wishart and standard random matrix theory have the same universal ``ramp.'' Specifically, here numerical results for the spectral form factor of the density matrix of generic nonintegrable many-body systems, such as one-dimensional quantum Ising and Floquet spin models, are found to exhibit a universal ``ramp'' identical to that appearing for a ``random pure state'' (``Page state,'' or ``Haar state''). The density matrix of the latter is precisely the Wishart random matrix, the reduced density matrix of a completely random wave function. In addition, we study the development of chaos in the wave function by letting an initial direct product state evolve under the unitary time evolution. We find that the universal spectral correlations as manifested by the ``ramp'' set in as soon as the entanglement entropy begins to grow, and first develop for the eigenvalues at the top of the spectrum of the density matrix ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{A}$, subsequently spreading over the entire spectrum at later times. Finally, we study a prethermalized regime described by a generalized Gibbs ensemble, which develops in a rapidly driven Floquet model at intermediate times. We find that the prethermalized regime exhibits no chaos, as evidenced by the absence of a ``ramp'' in the spectral form factor of the density matrix, while the universal spectral correlations start to develop when the prethermalized regime finally relaxes at late times to the fully thermalized (infinite temperature) chaotic regime.
The internal one-particle density matrix is discussed for Bose-Einstein condensates with a finite number of particles in a harmonic trap. It is found that the outcome of the diagonalization of the single-particle density matrix depends on the choice of the internal coordinates: The Pethick-Pitaevskii (PP) type of internal density matrix, whose analytical eigenvalues and eigenfunctions are evaluated, yields a fragmented condensate, while the Jacobi type of internal density matrix leads to an ideal condensate. We give a criterion for the choice of the internal coordinates: In the macroscopic limit the internal density matrix should have the same eigenvalues and eigenfunctions as those of the corresponding ideal Bose-Einstein condensate in the laboratory frame, this being a very physical condition. One choice satisfying this boundary condition is given by the internal Jacobi coordinates, while the internal coordinates with respect to the center of mass of the PP density matrix do not satisfy this condition. Based on our criterion, a general definition of the internal one-particle density matrix is presented in a self-bound system consisting of interacting bosons.