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arXiv:1604.07373 · Physical review. B./Physical review. B

Decay of Bogoliubov excitations in one-dimensional Bose gases

Zoran Ristivojević, К. А. Матвеев

Published Jul 11, 2016• 22 citations
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Abstract

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.

DOI
10.1103/physrevb.94.024506

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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.
We study the relaxation of excitations in a system of one-dimensional weakly interacting bosons. Due to residual weak interactions, Bogoliubov quasiparticles in this system have finite lifetimes. As a result of the conservation laws in one dimension, at zero temperature the leading mechanism of decay of a quasiparticle is disintegration into three others. We focus on phonon quasiparticles and find that their decay rate is proportional to the seventh power of momentum. In the integrable case of contact interaction between the bosons, the decay rate vanishes.
In a one-dimensional weakly interacting Bose-Fermi mixture, one branch of elementary excitations is well described by the Bogoliubov spectrum. Here we use the microscopic theory to study the decay of such quasiparticle excitations. The main scattering process which leads to their decay is the backscattering of a Bogoliubov quasiparticle off the Fermi sea, where a particle-hole pair is excited. For a low-momentum quasiparticle (phonon) of momentum qqq, we find that the decay rate scales as q3{q}^{3}q3 provided qqq is smaller than the Fermi momentum kF{k}_{F}kF​, while in the opposite case the decay behaves as q2{q}^{2}q2. If the ratio of the masses of fermions and bosons is equal to the ratio of the boson-fermion and the boson-boson interaction strengths, the decay rate changes dramatically. It scales as q7{q}^{7}q7 for q<kFq<{k}_{F}q<kF​, while we find q6{q}^{6}q6 scaling at q>kFq>{k}_{F}q>kF​. For a high-momentum Bogoliubov quasiparticle, we find a constant decay rate for q<kFq<{k}_{F}q<kF​, while it scales as 1/q1/q1/q for q>kFq>{k}_{F}q>kF​. We also find an analytic expression for the decay rate in the crossover region between low and high momenta. The decay rate is a continuous, but nonanalytic function of the momentum at q=kFq={k}_{F}q=kF​. In the special case when the parameters of our system correspond to the integrable model, we observe that the decay rate vanishes.
We study the integrable model of one-dimensional bosons with contact repulsion. In the limit of weak interaction, we use the microscopic hydrodynamic theory to obtain the excitation spectrum. The statistics of quasiparticles changes with the increase of momentum. At lowest momenta good quasiparticles are fermions, while at higher momenta they are Bogoliubov bosons, in accordance with recent studies. In the limit of strong interaction, we analyze the exact solution and find exact results for the spectrum in terms of the asymptotic series. Those results undoubtedly suggest that fermionic quasiparticle excitations actually exist at all momenta for moderate and strong interaction and also at lowest momenta for arbitrary interaction. Moreover, at strong interaction we find highly accurate analytical results for several relevant quantities of the Lieb-Liniger model.
In the presence of dipolar interactions, the excitation spectrum of a Bose gas can acquire a local minimum.The corresponding quasiparticles are known as rotons.They are gapped and do not decay at zero temperature.Here we study the decay of rotons in one-dimensional Bose gases at low temperatures.It predominantly occurs due to the backscattering of thermal phonons on rotons.The resulting rate scales with the third power of temperature and is inversely proportional to the sixth power of the roton gap near the solidification phase transition.The hydrodynamic approach used here enables us to find the decay rate for quasiparticles at practically any momenta, with minimal assumptions on the exact form of the interparticle interactions.Our results are an essential prerequisite for the description of all the dissipative phenomena in dipolar gases and have direct experimental relevance.
Describing properties of a strongly interacting quantum many-body system poses a serious challenge both for theory and experiment. In this work, we study excitations of one-dimensional repulsive Bose gas for arbitrary interaction strength using a hydrodynamic approach. We use linearization to study particle (type-I) excitations and numerical minimization to study hole (type-II) excitations. We observe a good agreement between our approach and exact solutions of the Lieb-Liniger model for the particle modes and discrepancies for the hole modes. Therefore, the hydrodynamical equations find to be useful for long-wave structures like phonons and of a limited range of applicability for short-wave ones like narrow solitons. We discuss potential further applications of the method.
In a one-dimensional Bose gas, there is no nontrivial scattering channel involving three Bogoliubov quasiparticles that conserves both energy and momentum. Nevertheless, we show that such three-wave mixing processes (Beliaev and Landau damping) account for their decay via interactions with thermal fluctuations. Within an appropriate time window where the Fermi golden rule is expected to apply, the occupation number of the initially occupied mode decays exponentially and the rate takes a simple analytic form. The result is shown to compare favorably with simulations based on the truncated Wigner approximation. It is also shown that the same processes slow down the exponential growth of phonons induced by a parametric oscillation.