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Dynamical excitation of the Bogoliubov vacuum in a two-component Bose-Einstein condensates

Salvatore Butera, Iacopo Carusotto

Published May 19, 2021• 0 citations
OpenAlex

Abstract

We investigate the parametric amplification of the zero-point fluctuations in the spin modes of a two-component Bose-Einstein condensate, that is triggered by the dynamical evolution of the background density. By working in the Thomas-Fermi limit, we first develop a theoretical model for the free dynamics of the Bogoliubov excitations, in a system that is trapped by a time-dependent harmonic potential. Furthermore, we perform an ab-initio numerical study of the two-body (density-density) correlations in the more general inhomogeneous configuration. We consider both the cases of the expanding and oscillating systems. In the former case, we find that the signal due to the particle creation in the spin modes, driven by the dynamical evolution of the overall density of the system, is weak and does not show any particular feature. In the oscillating case instead, because of the onset of resonance between the density oscillations and certain spin modes, the density correlations show a strong signature of the parametric amplification, displaying a clear imprint of the structure of the resonant mode.

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We investigate the parametric amplification of the zero-point fluctuations in the spin modes of a two-component Bose-Einstein condensate, that is triggered by the dynamical evolution of the background density. By working in the Thomas-Fermi limit, we first develop a theoretical model for the free dynamics of the Bogoliubov excitations, in a system that is trapped by a time-dependent harmonic potential. Furthermore, we perform an ab-initio numerical study of the two-body (density-density) correlations in the more general inhomogeneous configuration. We consider both the cases of the expanding and oscillating systems. In the former case, we find that the signal due to the particle creation in the spin modes, driven by the dynamical evolution of the overall density of the system, is weak and does not show any particular feature. In the oscillating case instead, because of the onset of resonance between the density oscillations and certain spin modes, the density correlations show a strong signature of the parametric amplification, displaying a clear imprint of the structure of the resonant mode.
We investigate the parametric amplification of the zero-point fluctuations in the spin modes of a two-component Bose-Einstein condensate, triggered by the dynamical evolution of the condensate density. We first make use of a Thomas-Fermi approximation to develop a tractable theoretical model of the quantum dynamics of the Bogoliubov excitations in a harmonically trapped condensate with a time-dependent trapping frequency. The predictions of this model are then compared to an ab initio numerical study of the correlation functions of density and spin fluctuations for general spatially inhomogeneous configurations. Results are shown for the two cases of expanding and oscillating condensates: while the quantum excitation of spin modes remains weak and relatively featureless in the case of an expanding condensate, clear and experimentally promising signatures of particle creation are anticipated for the oscillating case under suitable resonance conditions between the density and the spin modes.
We study the decay rate $\mathrm{\ensuremath{\Gamma}}(k)$ of density excitations of two-component Bose-Einstein condensates at zero temperature. Those excitations, where the two components oscillate in phase, include the Goldstone mode resulting from condensation. While within Bogoliubov approximation the density sector and the spin (out-of-phase) sector are independent, they couple at the three-phonon level. For a Bose-Bose mixture we find that the Belyaev decay is slightly modified due to the coupling with the gapless spin mode. At the phase separation point the decay rate changes instead from the standard k5{k}^{5}k5 to a k5/2{k}^{5/2}k5/2 behavior due to the parabolic nature of the spin mode. If instead a coherent coupling between the two components is present, the spin sector is gapped and, away from the ferromagnetic-like phase transition point, the decay of the density mode is not affected. On the other hand, at the transition point, when the spin fluctuations become critical, the Goldstone mode is not well defined anymore since $\mathrm{\ensuremath{\Gamma}}(k)\ensuremath{\propto}k$. As a consequence, we show that the friction induced by a moving impurity is enhanced---a feature which could be experimentally tested. Our results apply to every nonlinear 2-component quantum hydrodynamic Hamiltonian which is time-reversal invariant and possesses an $U(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry.
Ultra-cold spinor Bose gases present rich physics due to the special combination of superfluidity and magnetism in a quantum system. This system was first realized in 1998 by confining a Bose-Einstein condensate in an all-optical trap.\nIn this thesis, we consider the fluctuations of observable quantities in a spin-1 Bose gas. Understanding how fluctuations arise due to the spin excitations of the condensate is important because it offers an insight into how measurement noise reveals the many-body physics of the system.\nTo begin, we present the mean-field and Bogoliubov theory of a uniform spin-1 Bose gas subject to a constant magnetic field, describing the condensate and its low-energy collective excitations.\nWe then develop a formalism to describe the fluctuations in a general density-like observable. We start from the two-point correlation function and cast it in the form of a generalised static structure factor determined by the three Bogoliubov quasiparticle excitation branches. We derive analytic results for the fluctuation amplitudes and the temperature-dependent static structure factors for observables of total density and the three spin densities.\nFor all four magnetic phases, we analyse the spinor order parameter and quasiparticle spectra while numerically mapping out the fluctuation amplitudes and static structure factors for the total and spin density operators.\nWe describe the fluctuations in experimental measurements made within finite cells, which is an important step to making meaningful predictions for experiments. We consider cylindrical cells and gaussian cells as two limiting cases. We apply this analysis to an experimentally realisable system of a quasi-2D spinor gas in a harmonic trap, comparing extensive numeric results with analytic limits.
We theoretically investigate quadrupolar collective excitations in two-component Bose-Einstein condensates and their nonlinear dynamics associated with harmonic generation and mode coupling. Under the Thomas-Fermi approximation and the quadratic polynomial ansatz for density fluctuations, the linear analysis of the superfluid hydrodynamic equations predicts excitation frequencies of three normal modes constituted from monopole and quadrupole oscillations, and those of three scissors modes. These six modes are bifurcated into in-phase and out-of-phase modes by the intercomponent interaction, yielding the nonlinear dynamics that are absent in a single-component condensate. We obtain analytically the resonance conditions for the second-harmonic generation in terms of the trap aspect ratio and the strength of intercomponent interaction. The numerical simulation of the coupled Gross-Pitaevskii equations vindicates the validity of the analytical results and reveals the dynamics of the second-harmonic generation and nonlinear mode coupling that lead to nonlinear oscillations of the condensate with damping and recurrence reminiscent of the Fermi-Pasta-Ulam problem.
Bose-Einstein condensates of atoms with non-zero spin are known to constitute an ideal system to investigate fundamental properties of magnetic superfluids. More recently it was realized that they also provide the fascinating opportunity to investigate the macroscopic amplification of quantum and classical fluctuations. This is strikingly manifested in a sample initially prepared in the m F = 0 state, where spin-changing collisions triggered by quantum fluctuations may lead to the creation of correlated pairs in m F = ±1. We show that the pair creation efficiency is strongly influenced by the interplay between the external trapping potential and the Zeeman effect. It thus reflects the confinement-induced magnetic field dependence of elementary spin excitations of the condensate. Remarkably, pair production in our experiments is therefore characterized by a multi-resonant dependence on the magnetic field. Pair creation at these resonances acts as strong parametric matter-wave amplifier. Depending on the resonance condition, this amplification can be extremely sensitive or insensitive to the presence of seed atoms. We show that pair creation at a resonance which is insensitive to the presence of seed atoms is triggered purely by quantum fluctuations and thus the system acts as a matter-wave amplifier for the vacuum state.
Bose-Einstein condensates of atoms with non-zero spin makes up an ideal system for the study of macroscopic amplification of quantum and classical fluctuations. This study shows that the efficiency of pair creation is influenced by the interplay between the external trapping potential and the Zeeman effect. This reflects the confinement-induced magnetic-field dependence of elementary spin excitations of the trapped condensate. Pair production at these resonances serves as a strong parametric mater-wave amplifier. Results show that pair creation at resonance is triggered by quantum fluctuations. Therefore, in the vacuum state, this system acts as a matter-wave amplifier.