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arXiv:1407.7167 · Physical Review X

Glimmers of a Quantum KAM Theorem: Insights from Quantum Quenches in One-Dimensional Bose Gases

Giuseppe Piero Brandino, Jean-Sébastien Caux, Robert Konik

Published Dec 16, 2015• 115 citations
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Abstract

Theorists demonstrate a quantum variant of the Kolmogorov-Arnold-Moser theorem, a theorem in classical mechanics that concerns the crossover between integrability and chaos. Integrability breaking in the Lieb-Liniger model, a model describing one-dimensional Bose gases, leads to a deformation, not destruction, of the model's conserved quantities.

DOI
10.1103/physrevx.5.041043

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Theorists demonstrate a quantum variant of the Kolmogorov-Arnold-Moser theorem, a theorem in classical mechanics that concerns the crossover between integrability and chaos. Integrability breaking in the Lieb-Liniger model, a model describing one-dimensional Bose gases, leads to a deformation, not destruction, of the model's conserved quantities.
Using a numerical renormalization group based on exploiting an underlying exactly solvable non-relativistic theory, we study the out-of-equilibrium dynamics of a 1D Bose gas (as described by the Lieb-Liniger model) released from a parabolic trap into a weak cosine potential. The presence of the cosine potential breaks integrability and leads the formerly conserved charges of Lieb-Liniger to be time dependent. We however argue that from these charges we are able to construct approximately conserved quantities despite the presence of the cosine term. How good the time invariance of these quasi-conserved quantities is can be related to the strength of the post-quench cosine potential and the width of this perturbation in Fourier space. This gives then an analog in a quantum example to the classical KAM theorem.
We show that for any finite-dimensional quantum systems the conserved quantities can be characterized by their robustness to small perturbations: for fragile symmetries, small perturbations can lead to large deviations over long times, while for robust symmetries, their expectation values remain close to their initial values for all times. This is in analogy with the celebrated Kolmogorov-Arnold-Moser theorem in classical mechanics. To prove this result, we introduce a resummation of a perturbation series, which generalizes the Hamiltonian of the quantum Zeno dynamics.
In this work we report preliminary results on the relaxational dynamics of one dimensional Bose gases, as described by the Lieb-Liniger model, upon release from a parabolic trap. We explore the effects of integrability and integrability breaking upon these dynamics by placing the gas post-release in an integrability breaking one-body cosine potential of variable amplitude. By studying the post-quench evolution of the conserved charges that would exist in the purely integrable limit, we begin to quantify the effects of the weak breaking of integrability on the long time thermalization of the gas.
The quasienergy spectrum of a particle in an infinite square well perturbed by a monochromatic external field is analyzed in terms of single- and double-resonance Hamiltonians. At small-enough perturbations the quasienergies are accurately given by those obtained from a single-resonance, integrable Hamiltonian. This indicates the existence of a local constant of motion and is a quantum manifestation of the Kolmogorov-Arnold-Moser theorem. At larger perturbations, the quasienergy spacing distributions are Poissonian for the single-resonance Hamiltonian. But for the double-resonance Hamiltonian, the distributions undergo a transition from Poissonian toward Wigner-like behavior when the perturbation is increased from a regime without resonance overlap to a regime with resonance overlap. This indicates the destruction of the local constant of motion through quantum-resonance overlap and its associated quantum number. It is a quantum manifestation of classical-resonance overlap.
The one-dimensional δ-function interacting Bose gas (the Lieb-Liniger model) is an integrable system, which can model experiments with ultra-cold atoms in one-dimensional traps. Even though the model is integrable, integrability breaking effects are always present in the real-world experiments. In this work we consider the integrability breaking due to atomic loss, which is the most relevant effect in the experiments. We set up a framework for the exact computation of the losses of the canonical charges of the model, and compute an exact result for the energy loss due to the local K-body processes, valid for arbitrary K. Our result takes the form of multiple integrals, which are explicitly factorized in the experimentally relevant cases of K=1,2,3.
We study the decay of Bogoliubov quasiparticles in one-dimensional Bose gases. Starting from the hydrodynamic Hamiltonian, we develop a microscopic theory that enables one to systematically study both the excitations and their decay. At zero temperature, the leading mechanism of decay of a quasiparticle is disintegration into three others. We find that low-energy quasiparticles (phonons) decay with the rate that scales with the seventh power of momentum, whereas the rate of decay of the high-energy quasiparticles does not depend on momentum. In addition, our approach allows us to study analytically the quasiparticle decay in the whole crossover region between the two limiting cases. When applied to integrable models, including the Lieb-Liniger model of bosons with contact repulsion, our theory confirms the absence of the decay of quasiparticle excitations. We account for two types of integrability-breaking perturbations that enable finite decay: three-body interaction between the bosons and two-body interaction of finite range.