Reaching a target quantum state from an initial state within a finite temporal window is a challenging problem due to non-adiabaticity. We study the optimal protocol for swithcing on interactions to reach the ground state of a weakly interacting Luttinger liquid within a finite time $τ$, starting from the non-interacting ground state. The protocol is optimized by minimizing the excess energy at the end of the quench, or by maximizing the overlap with the interacting ground state. We find that the optimal protocol is symmetric with respect to $τ/2$, and can be expressed as a functional of the occupation numbers of the bosonic modes in the final state. For short quench durations, the optimal protocol exhibits fast oscillation and excites high energy modes. In the limit of large $τ$, minimizing energy requires a smooth protocol while maximizing overlap requires a linear quench protocol. In this limit, the minimal energy and maximal overlap are both universal functions of the system size and the duration of the protocol.
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 non universal power-law behavior of a zero-temperature non-quenched 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 s…
We consider finite-time quantum quenches in the interacting Tomonaga--Luttinger model, for example time-dependent changes of the nearest-neighbour interactions for spinless fermions. We use the exact solutions for specific protocols including the linear and cosine ramps (or, more generally, periodic pumping). We study the dynamics of the total and kinetic energy as well as the Green functions during as well as after the quench. For the latter we find that the light-cone picture remains applicable, however, the propagating front is delayed as compared to the sudden quench. We extract the univer…
We study the escort probability distribution function of work done during an interaction quantum quench of Luttinger liquids. It crosses over from the thermodynamic to the small system limit with increasing $a$, the order of the escort distribution, and depends on the universal combination $(|K_i-K_f|/(K_i+K_F))^a$ with $K_i$, $K_f$ the initial and final Luttinger liquid parameters, respectively. From its characteristic function, the diagonal Rényi entropies and the many body inverse participation ratio (IPR) are determined to evaluate the information content of the time evolved wavefunction i…
We analyze the probability distribution function (PDF) of work done on a Luttinger liquid for an arbitrary finite duration interaction quench and show that it can be described in terms a generalized Gibbs ensemble. We construct the corresponding density matrix with explicit intermode correlations, and determine the duration and interaction dependence of the probability of an adiabatic transition and the PDF of non-adiabatic processes. In the thermodynamic limit, the PDF of work exhibits a non-Gaussian maximum around the excess heat, carrying almost all spectral weight. In contrast, in the smal…
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" perturbatio…
We study the quantum quench in two coupled Tomonaga-Luttinger Liquids (TLLs), from the off-critical to the critical regime, relying on the conformal field theory approach and the known solutions for single TLLs. We consider a squeezed form of the initial state, whose low energy limit is fixed in a way to describe a massive and a massless mode, and we encode the non-equilibrium dynamics in a proper rescaling of the time. In this way, we compute several correlation functions, which at leading order factorize into multipoint functions evaluated at different times for the two modes. Depending on t…
The study of the non-equilibrium dynamics of many-body systems after a quantum quench received a considerable boost and a deep theoretical understanding from the path integral formulation in imaginary time. However, the celebrated problem of a quench in the Luttinger parameter of a one dimensional quantum critical system (massless quench) has so far only been solved in the real-time Heisenberg picture. In order to bridge this theoretical gap and to understand on the same ground massive and massless quenches, we study the problem of a gaussian field characterized by a coupling parameter K withi…
We study the time evolution of the two-point correlation functions in the XXZ Heisenberg chain after a finite-time quantum quench in the anisotropy. We compare results from numerical simulations to ones obtained in the Luttinger model and find good agreement. We analyse the spreading of the correlations and the associated light-cone features. We observe a delay in the appearance of the light cone as compared to the sudden-quench setup, and link this delay to the properties of the quench protocol.
We study variable-rate linear quenches in the anisotropic Heisenberg (XXZ) chain, starting at the XX point. This is equivalent to switching on a nearest neighbour interaction for hard-core bosons or an interaction quench for free fermions. The physical observables we investigate are: the energy pumped into the system during the quench, the spin-flip correlation function, and the bipartite fluctuations of the z component of the spin in a box. We find excellent agreement between exact numerics (infinite system time-evolving block decimation, iTEBD) and analytical results from bosonization, as a …
Motivated by recent experiments on interacting cold atoms, we analyze interaction quenches in Luttinger liquids (LL), 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 regimes, and a LL regime with 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 r…
Using a Luttinger liquid theory we investigate the time evolution of the particle density of a one-dimensional fermionic system with open boundaries and subject to a finite duration quench of the inter-particle interaction. We provide analytical and asymptotic solutions to the unitary time evolution of the system, showing that both switching on and switching off the quench ramp create light-cone perturbations in the density. The post-quench dynamics is strongly affected by the interference between these two perturbations. In particular, we find that the discrepancy between the time-dependent d…
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 form are investigated all leading to universal Luttinger liquid physics in equilibrium. The steady-state fermionic momentum distribution shows universal behavior in the sense of the Luttinger liquid phenomenology. For generic regular potentials the large time decay of the momentum distribution function towards the steady-state value is characterized by a pow…
We develop a detailed theory for spin transport in a one-dimensional quantum wire described by Luttinger liquid theory. A hydrodynamic description for the quantum wire is supplemented by boundary conditions taking into account the exchange coupling between the magnetization of ferromagnetic reservoirs and the boundary magnetization in the wire. Spin-charge separation is shown to imply drastic and qualitative consequences for spin-dependent transport. In particular, the spin accumulation effect is quenched except for fine-tuned parameter regimes. We propose several feasible setups involving an …
We study the time evolution in the transverse-field Ising chain subject to quantum quenches of finite duration, ie, a continuous change in the transverse magnetic field over a finite time. Specifically, we consider the dynamics of the total energy, one- and two-point correlation functions and Loschmidt echo during and after the quench as well as their stationary behaviour at late times. We investigate how different quench protocols affect the dynamics and identify universal properties of the relaxation.
The dynamics of the Luttinger model and the sine-Gordon model (at the Luther-Emery point and in the semiclassical approximation) after a quantum quench is studied. We compute in detail one and two-point correlation functions for different types of quenches: from a non-interacting to an interacting Luttinger model and vice-versa, and from the gapped to the gapless phase of the sine-Gordon model and vice-versa. A progressive destruction of the Fermi gas features in the momentum distribution is found in the case of a quench into an interacting state in the Luttinger model. The critical exponents …
Results are presented for a two-point correlation function of a spin-chain after a quantum quench for an intermediate time regime where inelastic effects are weak. A Callan-Symanzik like equation for the correlation function is explicitly constructed which is used to show the appearance of three distinct scaling regimes. One is for spatial separations within a light-cone, the second is for spatial separations on the light-cone, and the third is for spatial separations outside the light-cone. In these three regimes, the correlation function is found to decay with power-laws with nonequilibrium …
The dynamics of quantum phase transitions are inevitably accompanied by the formation of defects when crossing a quantum critical point. For a generic class of quantum critical systems, we solve the problem of minimizing the production of defects through the use of a gradient-based deterministic optimal control algorithm. By considering a finite size quantum Ising model with a tunable global transverse field, we show that an optimal power law quench of the transverse field across the Ising critical point works well at minimizing the number of defects, in spite of being drawn from a subset of q…
