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Register a new userFew-body techniques using momentum space for bound and continuum states
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Authors
M. T. Yamashita
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These notes were written for a set of three lectures given in a school at the Max Planck Institute for the Physics of Complex Systems in October/2017 before the workshop Critical Stability of Quantum Few-Body Systems. These lectures are primarily dedicated to the students and represent a very idiosyncratic vision of the author, mainly in the last part of the text related to applications. These notes are only a tentative to show a technique, among many others, to solve problems in a very rich area of the contemporary physics - the Few-Body Physics - many times unknown by a considerable part of the students.
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We present a distinct mechanism for the formation of bound states in the continuum (BICs). In chiral quantum systems there appear zero-energy states in which the wave function has finite amplitude only in one of the subsystems defined by the chiral symmetry. When the system is coupled to leads with a continuum energy band, part of these states remain bound. We derive some algebraic rules for the number of these states depending on the dimensionality and rank of the total Hamiltonian. We examine the transport properties of such systems including the appearance of Fano resonances in some limiting cases. Finally, we discuss experimental setups based on microwave dielectric resonators and atoms in optical lattices where these predictions can be tested.
The exotic phenomenon of time translation symmetry breaking under periodic driving - the time crystal - has been shown to occur in many-body systems even in clean setups where disorder is absent. In this work, we propose the realization of time-crystals in few-body systems, both in the context of trapped cold atoms with strong interactions and of a circuit of superconducting qubits. We show how these two models can be treated in a fairly similar way by adopting an effective spin chain description, to which we apply a simple driving protocol. We focus on the response of the magnetization in the presence of imperfect pulses and interactions, and show how the results can be interpreted, in the cold atomic case, in the context of experiments with trapped bosons and fermions. Furthermore, we provide a set of realistic parameters for the implementation of the superconducting circuit.
We present a technique to diagnose the condensate fraction in a one-dimensional optical lattice of weakly interacting bosons based on the dynamics of the trapped atoms under the influence of a momentum kick. It is shown using the Multi-Configuration Time Dependent Hartree method for Bosons (MCTDHB) that the two extreme cases of the superfluid and Mott insulator states exhibit different behaviors when the lattice is briefly tilted. The current induced by the momentum boost caused by the tilt which depends directly on the amount of phase coherence between the lattice sites is linearly proportional to the condensate fraction. The atom-atom interactions only change the slope of the linear relationship. We discuss the applications of this scheme in magnetic field gradiometery.
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We illustrate the existence of single-excitation bound states for propagating photons interacting with $N$ two-level atoms. These bound states can be calculated from an effective spin model, and their existence relies on dissipation in the system. The appearance of these bound states is in a one-to-one correspondence with zeros in the single-photon transmission and with divergent bunching in the second-order photon-photon correlation function. We also formulate a dissipative version of Levinsons theorem for this system by looking at the relation between the number of bound states and the winding number of the transmission phases. This theorem allows a direct experimental measurement of the number of bound states using the measured transmission phases.
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