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Register a new userGeneralised Kronig-Penney model for ultracold atomic quantum systems
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We study the properties of a quantum particle interacting with a one dimensional structure of equidistant scattering centres. We derive an analytical expression for the dispersion relation and for the Bloch functions in the presence of both even and odd scattering waves within the pseudopotential approximation. This generalises the well-known solid-state physics text-book result known as the Kronig-Penney model. Our generalised model can be used to describe systems such as degenerate Fermi gases interacting with ions or with another neutral atomic species confined in an optical lattice, thus enabling the investigation of polaron or Kondo physics within a simple formalism. We focus our attention on the specific atom-ion system and compare our findings with quantum defect theory. Excellent agreement is obtained within the regime of validity of the pseudopotential approximation. This enables us to derive a Bose-Hubbard Hamiltonian for a degenerate quantum Bose gas in a linear chain of ions.
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We generalize the textbook Kronig-Penney model to realistic conditions for a quantum-particle moving in the quasi-one-dimensional (quasi-1D) waveguide, where motion in the transverse direction is confined by a harmonic trapping potential. Along the waveguide, the particle scatters on an infinite array of regularized delta potentials. Our starting point is the Lippmann-Schwinger equation, which for quasi-1D geometry can be solved exactly, based on the analytical formula for the quasi-1D Greens function. We study the properties of eigen-energies as a function of particle quasi-momentum, which form band structure, as in standard Kronig-Penney model. We test our model by comparing it to the numerical calculations for an atom scattering on an infinite chain of ions in quasi-1D geometry. The agreement is fairly good and can be further improved by introducing energy-dependent scattering length in the regularized delta potential. The energy spectrum exhibits the presence of multiple overlapping bands resulting from excitations in the transverse direction. At large lattice constants, our model reduces to standard Kronig-Penney result with one-dimensional coupling constant for quasi-1D scattering, exhibiting confinement-induced resonances. In the opposite limit, when lattice constant becomes comparable to harmonic oscillator length of the transverse potential, we calculate the correction to the quasi-1D coupling constant due to the quantum interference between scatterers. Finally, we calculate the effective mass for the lowest band and show that it becomes negative for large and positive scattering lengths.
Exact solutions of the time-dependent Schrodinger equation for a quantum oscillator subject to periodical frequency delta-kicks are obtained. We show that the oscillator occurs in the squeezed state and calculate the corresponding squeezing coefficients and the energy increase rate in terms of Chebyshev polynomials.
Quantum technologies will ultimately require manipulating many-body quantum systems with high precision. Cold atom experiments represent a stepping stone in that direction: a high degree of control has been achieved on systems of increasing complexity, however, this control is still sub-optimal. Optimal control theory is the ideal candidate to bridge the gap between early stage and optimal experimental protocols, particularly since it was extended to encompass many-body quantum dynamics. Here, we experimentally demonstrate optimal control applied to two dynamical processes subject to interactions: the coherent manipulation of motional states of an atomic Bose-Einstein condensate and the crossing of a quantum phase transition in small systems of cold atoms in optical lattices. We show theoretically that these transformations can be made fast and robust with respect to perturbations, including temperature and atom number fluctuations, resulting in a good agreement between theoretical predictions and experimental results.
In this chapter we will present the one-dimensional (1d) quantum degenerate Bose gas (1d superfluid) as a testbed to experimentally illustrate some of the key aspects of quantum thermodynamics. Hard-core bosons in one-dimension are described by the integrable Lieb-Lininger model. Realistic systems, as they can be implemented, are only approximately integrable, and let us investigate the cross over to thermalisation. They show such fundamental properties as pre-thermalisation, general Gibbs ensembles and light-cone like spreading of de-coherence. On the other hand they are complex enough to illustrate that our limited ability to measure only (local) few-body observables determines the relevant description of the many-body system and its physics. One consequence is the observation of quantum recurrences in systems with thousand of interacting particles. The relaxation observed in 1D superfluids is universal for a large class of many-body systems, those where the relevant physics can be described by a set of long lived collective modes. The time window where the close to integrable dynamics can be observed is given by the lifetime of the quasi-particles associated with the collective modes. Based on these observations one can view (in a quantum field theory sense) a many-body quantum system at T=0 as vacuum and its excitations as the system to experiment with. This viewpoint leads to a new way to build thermal machines from the quasi-particles in 1D superfluids. We will give examples of how to realise these systems and point to a few interesting questions that might be addressed.
Quantum simulators allow to explore static and dynamical properties of otherwise intractable quantum many-body systems. In many instances, however, it is the read-out that limits such quantum simulations. In this work, we introduce a new paradigm of experimental read-out exploiting coherent non-interacting dynamics in order to extract otherwise inaccessible observables. Specifically, we present a novel tomographic recovery method allowing to indirectly measure second moments of relative density fluctuations in one-dimensional superfluids which until now eluded direct measurements. We achieve this by relating second moments of relative phase fluctuations which are measured at different evolution times through known dynamical equations arising from unitary non-interacting multi-mode dynamics. Applying methods from signal processing we reconstruct the full matrix of second moments, including the relative density fluctuations. We employ the method to investigate equilibrium states, the dynamics of phonon occupation numbers and even to predict recurrences. The method opens a new window for quantum simulations with one-dimensional superfluids, enabling a deeper analysis of their equilibration and thermalization dynamics.
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