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arXiv:1011.6655 · Physical Review Letters

Crossover from Adiabatic to Sudden Interaction Quench in a Luttinger Liquid

Balázs Dóra, Masudul Haque, Gergely Zaránd

Published Apr 14, 2011• 68 citations
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Abstract

Motivated by recent experiments on interacting cold atoms, we analyze interaction quenches in Luttinger liquids (LLs), where the interaction is ramped from zero to a finite value within a finite time. The fermionic single particle density matrix reveals several regions of spatial and temporal coordinates relative to the quench time, termed as Fermi liquid, sudden quench LL, adiabatic LL regime, and a LL regime with a time-dependent exponent. The various regimes can also be observed in the momentum distribution of the fermions, directly accessible through time of flight experiments. Most of our results apply to arbitrary quench protocols.

DOI
10.1103/physrevlett.106.156406

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Motivated by recent experiments on interacting cold atoms, we analyze interaction quenches in Luttinger liquids (LLs), where the interaction is ramped from zero to a finite value within a finite time. The fermionic single particle density matrix reveals several regions of spatial and temporal coordinates relative to the quench time, termed as Fermi liquid, sudden quench LL, adiabatic LL regime, and a LL regime with a time-dependent exponent. The various regimes can also be observed in the momentum distribution of the fermions, directly accessible through time of flight experiments. Most of our results apply to arbitrary quench protocols.
We study the relaxation dynamics of the one-dimensional Tomonaga–Luttinger model after an interaction quench, paying particular attention to the momentum dependence of the two-particle interaction. Several potentials of different analytical forms are investigated that all lead to universal Luttinger liquid (LL) physics in equilibrium. The steady-state fermionic momentum distribution shows universal behavior in the sense of the LL phenomenology. For generic regular potentials, the large time decay of the momentum distribution function toward the steady-state value is characterized by a power law with a universal exponent that depends only on the potential at zero momentum transfer. The commonly employed ad hoc procedure fails to give this exponent. In addition to quenches from zero to positive interactions, we also consider the abrupt changes in the interaction between two arbitrary values. Additionally, we discuss the appearance of a factor of two between the steady-state momentum distribution function and that obtained in equilibrium at equal two-particle interaction.
Using a Luttinger liquid theory we investigate the time evolution of the particle density of a one-dimensional spinful fermionic system with open boundaries and subject to a finite-duration quench of the inter-particle interaction. Taking into account also the turning on of an umklapp perturbation to the system Hamiltonian as a result of the quench, we study the possible formation of a Wigner molecule inside the system, focusing in particular on the sudden and adiabatic regimes. We show that the creation of this correlated state is essentially due to the propagation of "light-cone" perturbations through system which arise after both switching on and switching off the quenching protocol and that its behavior strongly depends on the quench duration.
We study the relaxation dynamics of the one-dimensional Tomonaga–Luttinger model after an interaction quench, paying particular attention to the momentum dependence of the two-particle interaction. Several potentials of different analytical forms are investigated that all lead to universal Luttinger liquid (LL) physics in equilibrium. The steady-state fermionic momentum distribution shows universal behavior in the sense of the LL phenomenology. For generic regular potentials, the large time decay of the momentum distribution function toward the steady-state value is characterized by a power law with a universal exponent that depends only on the potential at zero momentum transfer. The commonly employed ad hoc procedure fails to give this exponent. In addition to quenches from zero to positive interactions, we also consider the abrupt changes in the interaction between two arbitrary values. Additionally, we discuss the appearance of a factor of two between the steady-state momentum distribution function and that obtained in equilibrium at equal two-particle interaction.
We present a comprehensive analysis of the relaxation dynamics of a Luttinger liquid subject to a sequence of sudden interaction quenches. The critical exponent β governing the decay of the steady-state propagator is expressed as an explicit functional of the switching protocol. At long distances β depends only on the initial state while at short distances it is also history dependent. Continuous protocols of arbitrary complexity can be realized with infinitely long sequences. For quenches of finite duration we prove that there exists no protocol to bring the initial non-interacting system in the ground state of the Luttinger liquid, albeit thermalization occurs at short distances. The adiabatic theorem is then investigated with ramp switchings of increasing duration and several analytic results for both the propagator and the excitation energy are derived.
The transport dynamics of a quenched Luttinger liquid tunnel-coupled to a fermionic reservoir is investigated. In the transient dynamics, we show that for a sudden quench of the electron interaction universal power-law decay in time of the tunneling current occurs, ascribed to the presence of entangled compound excitations created by the quench. In sharp contrast to the usual nonuniversal power-law behavior of a zero-temperature nonquenched Luttinger liquid, the steady-state tunneling current is Ohmic and can be explained in terms of an effective quench-activated heating of the system. Our study unveils an unconventional dynamics for a quenched Luttinger liquid that could be identified in quenched cold Fermi gases.
The recent experimental implementation of condensed matter models in optical lattices has motivated research on their non-equilibrium behavior. Predictions about the dynamics of superconductors following a sudden quench of the pairing interaction have been made based on the effective Bardeen–Cooper–Schrieffer (BCS) Hamiltonian; however, their experimental verification requires the preparation of a suitable excited state of the Hubbard model along a twofold constraint: (i) a sufficiently non-adiabatic ramping scheme is essential to excite the non-equilibrium dynamics and (ii) overheating beyond the critical temperature of superconductivity must be avoided. For commonly discussed interaction ramps, there is no clear separation of the corresponding energy scales. Here we show that the matching of both conditions is simplified by the intrinsic relaxation behavior of ultracold fermionic systems: for the particular example of a linear ramp we examine the transient regime of prethermalization (Moeckel and Kehrein 2008 Phys. Rev. Lett. 100 175702) under the crossover from sudden to adiabatic switching using Keldysh perturbation theory. A real-time analysis of the momentum distribution exhibits a temporal separation of an early energy relaxation and its later thermalization by scattering events. For long but finite ramping times this separation can be large. In the prethermalization regime, the momentum distribution resembles a zero-temperature Fermi liquid as the energy inserted by the ramp remains located in high-energy modes. Thus ultracold fermions prove to be robust to heating, which simplifies the observation of non-equilibrium BCS dynamics in optical lattices.