Symplectic tomography of ultracold gases in tight waveguides

Abstract

The phase space is the natural ground to smoothly extrapolate between local and non-local correlation functions. With this objective, we introduce the symplectic tomography of many-body quantum gases in tight-waveguides, and concentrate on the reduced single-particle symplectic tomogram (RSPST) whose marginals are the density profile and momentum distribution. We present an operational approach to measure the RSPST from the time evolution of the density profile after shutting off the interactions in a variety of relevant situations: free expansion, fall under gravity, and oscillations in a harmonic trap. From the RSPST, the one-body density matrix of the trapped state can be reconstructed. At low densities, ultracold bosonic gases exhibits universality. The interatomic interactions are then well described by the Fermi-Huang pseudo-potential, parametrized by a the 3D s-wave scattering length a s . If such gases are further confined in tight-waveguides, whenever the transverse excitation quantumhω ⊥ is larger than the longitudinal zero point and thermal energies, the system effectively becomes one-dimensional [1]. The interparticle pseudo-potential is then a simple delta function, so the system is well approximated by the LiebLiniger model [2]. Moreover, the strength of the interaction as a function of a s exhibits a confinement-induced 1D Feshbach resonance (CIR) [1,3], allowing to tune the 1D coupling constant g 1D from −∞ to +∞ and to reach both weak and strongly interacting regimes [4]. As a consequence, paradigmatic examples of the Bose-Fermi duality have been explored such as the Tonks-Girardeau gas [5], in which the strongly repulsive interactions between bosons leads to an effective Pauli exclusion principle [6]. In this regime, the system undergoes fermionization, all local correlation functions being identical to those of the spin-polarized ideal Fermi gas. Actually, the Fermi-Bose duality comes into play even with finite interactions [7]. However, quantum statistics invariably imposes an underlying signature manifested when looking at non-local correlations such as the momentum distribution or the one-body density matrix [8,9,10,11,12,13]. A natural ground to smoothly extrapolate between local and non-local correlations is the phase space. In this paper, we shall undertake the description of ultracold gases in tight-waveguides by means of the quantum tomographic technique [14]. We show that after shutting off the interactions in the system, the time evolution of the density profile governed by a quadratic Hamiltonian is tanta-

Quantum transients

“…Nevertheless, the sudden quench of interactions using Feschbach resonances [51] combined with a switch of the confining potential stands for its applications in tomography of trapped ultracold gases [55].…”

Frictionless quantum quenches in ultracold gases: A quantum-dynamical microscope

“…This mapping between local and non-local correlations is expected as the density decreases during expansion at a finite rate ḃ. Indeed, within the scheme of symplectic tomography, the evolution under quadratic Hamiltonians leads to a dynamical covering of correlations in phase-space [23]. In an isolated system, a sudden quench of the trapping frequency between two given finite values also induces undamped breathing of the density profile.…”

Observing Power-Law Dynamics of Position-Velocity Correlation in Anomalous Diffusion

“…C xv can reveal the fingerprint of the underlying model and in particular is essential for understanding concepts and techniques such as adiabatic cooling in lattices [24], stochastic cooling [25], point source atom interferometry [26,27] and enhanced velocity resolution [28,29], alongside elementary notions in quantum mechanics [30]. These correlations have been surprisingly overlooked in both theory and experiment, perhaps due to the lack of a direct method for imaging the phasespace of atomic clouds, which does not require cumbersome mathematical tools or a specific potential [31][32][33][34][35]. No analysis of the dynamics of the correlations has been reported to the best of our knowledge.…”