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Metastability is a quintessential feature of first order quantum phase transitions, which is lost either by dynamical instability or by nucleating bubbles of a true vacuum through quantum tunneling. By considering a drive across the first order quantum phase transition in the quantum Ising chain in the presence of both transverse and longitudinal fields, we reveal multiple regions in the parameter space where the initial metastable state loses its metastability in successive stages. The mechanism responsible is found to be semi-degenerate resonant tunnelings to states with specific bubble sizes. We show that such dynamics of quantized bubble nucleations can be understood in terms of Landau-Zener transitions, which provide quantitative predictions of nucleation probabilities for different bubble sizes., Comment: 7+4 pages, 5+3 figures. Close to published version

Driving a homogeneous system across a quantum phase transition in a quench-time $\tau_Q$ generates excitations on wavelengths longer than the Kibble-Zurek (KZ) length $\hat\xi\propto\tau_Q^{\nu/(1+z\nu)}$ within the KZ time window $\hat t\propto\tau_Q^{z\nu/(1+z\nu)}$, where $z$ and $\nu$ are the critical exponents. Quenches designed with local time-dependent inhomogeneity can introduce a gap in the spectrum. For a variety of setups with short-range interactions, they have been shown to suppress excitations if the spatial velocity of the inhomogenous front is below the characteristic KZ velocity $\hat v \propto \hat\xi/\hat t$. Ising-like models with long-range interactions can have no sonic horizon, spreading information instantaneously across the system. Usually, this should imply that inhomogenous transitions will render the dynamics adiabatic regardless of the front velocity. However, we show that we get an adiabatic transition with no defects only when the inhomogeneous front moves slower than a characteristic crossover velocity $\tilde v \propto \theta^{(z-1)\nu/(1+\nu)}$, where $\theta$ is the slope of the inhomogeneous front at the critical point. The existence of this crossover velocity and adiabaticity of the model is a consequence of the energy gap in the quasiparticle spectrum that is opened by the inhomogeneity. This effect can be employed for efficient adiabatic quantum state preparation in systems with long-range interactions., Comment: 12 pages, 10 figures (closed to published version)

The Aubry-Andr\'e 1D lattice model describes a particle hopping in a pseudo-random potential. Depending on its strength $\lambda$, all eigenstates are either localized ($\lambda>1$) or delocalized ($\lambda<1$). Near the transition, the localization length diverges like $\xi\sim(\lambda-1)^{-\nu}$ with $\nu=1$. We show that when the particle is initially prepared in a localized ground state and the potential strength is slowly ramped down across the transition, then -- in analogy with the Kibble-Zurek mechanism -- it enters the delocalized phase having finite localization length $\hat\xi\sim\tau_Q^{\nu/(1+z\nu)}$. Here $\tau_Q$ is ramp/quench time and $z$ is a dynamical exponent. At $\lambda=1$ we determine $z\simeq2.37$ from the power law scaling of energy gap with lattice size $L$. Even though for infinite $L$ the model is gapless, we show that the gap relevant for excitation during the ramp remains finite. Close to the critical point it scales like $\xi^{-z}$ with the value of $z$ determined by the finite size scaling. It is the gap between the ground state and the lowest of those excited states that overlap with the ground state enough to be accessible for excitation. We propose an experiment with a non-interacting BEC to test our prediction. Our hypothesis is further supported by considering a generalized version of Aubry-Andr\'{e} model possessing an energy-dependent mobility edge., Comment: 7 pages, 9 figures; close to the published version; new section on generalized Aubry-Andr\'e model