Quantum dynamics in lattices in presence of bulk dephasing and a localized source

TL;DR

The paper investigates how a localized source injects particles into 1D lattices subject to bulk dephasing, across both non-interacting and interacting regimes and including long-range hopping. It combines GKSL-based analytic treatments (adiabatic approximation) for the short-range non-interacting case with TEBD simulations for the interacting case, and extends to power-law hopping with exponent $\alpha$. The main finding is a universal late-time diffusive growth $N(t)\sim t^{1/2}$ with diffusion constant $D = \sqrt{\frac{8 J^2}{\pi \Gamma_d}}$ in the non-interacting limit, and a $\Delta$- and $\Gamma_d$-dependent diffusion in the interacting case, along with an intermediate sub-diffusive regime; for long-range hopping, the system exhibits anomalous scaling $N(t)\sim t^{1/(2\alpha-1)}$ for $1<\alpha<3/2$. These results reveal robust universal scaling across regimes and offer insights for open quantum system dynamics and quantum simulation experiments.

Abstract

The aim of this work is to study the dynamics of quantum systems subjected to a localized fermionic source in the presence of bulk dephasing. We consider two classes of one-dimensional lattice systems: (i) a non-interacting lattice with nearest-neighbor and beyond, i.e., long-ranged (power-law) hopping, and (ii) a lattice that is interacting via short-range interactions modeled by a fermionic quartic Hamiltonian. We study the evolution of the local density profile $n_i(t)$ within the system and the growth of the total particle number $N(t)$ in it. For case (i), we provide analytical insights into the dynamics of the nearest-neighbor model using an adiabatic approximation, which relies on assuming faster relaxation of coherences of the single particle density matrix. For case (ii), we perform numerical computations using the time-evolving block decimation (TEBD) algorithm and analyze the density profile and the growth exponent in $N(t)$. Our detailed study reveals an interesting interplay between Hamiltonian dynamics and various environmentally induced mechanisms in open quantum systems, such as local source and bulk dephasing. It brings out rich dynamics, including universal dynamical scaling and anomalous behavior across various time scales and is of relevance to various quantum simulation platforms.

Figures (5)

• Figure 1: Schematic for nearest-neighbor lattice system of size $L$ with hopping strength $J$ subjected to onsite local dephasing with strength $\Gamma_d$ at each of its site. A local source is injecting quantum particles from the left end with rate $\Gamma_G$. The setup is modeled by Eq. \ref{['eq:lindblad']}.
• Figure 2: Quantum dynamics via exact numerics for total number of particles $N(t)$ with time $t$ (in units of $1/J$) when particles are injected from one end of the lattice and the non-interacting lattice given in Eq. \ref{['eq:system_hamiltonian_non_int']} is subjected to bulk dephasing. For (a) $\Gamma_d = 10 J$ and for (b) $\Gamma_d = 100 J$, with results shown for different values of $\Gamma_G$. Here, we take $L = 200$ and the hopping amplitude as $J=1$. The growth of $N(t)$ shows a crossover from linear growth ($N(t)\propto t$) to sub-diffusive growth ($N(t) \propto t^{\nu}$ with $\nu<1/2$) and finally diffusive ($N(t)\propto t^{1/2}$) before saturating to a system-size dependent value. In (a), note that when $\Gamma_G \gtrsim \Gamma_d$, the diffusive regime ($t^{1/2}$) appears at a later stage in time, as opposed to the opposite case, $\Gamma_G \lesssim \Gamma_d$. The local density profile $n_i(t)$ is plotted as a function of scaled lattice coordinate $i/\sqrt{t}$ at different time instances in the regime denoted by the dashed red line in (a) and (b), for (c) $\Gamma_d = 10 J$ and (d) $\Gamma_d = 100 J$, respectively, and $\Gamma_G = 10 J$. The plots show a perfect collapse indicating diffusive scaling in the long-time limit.
• Figure 3: The dynamics of total number of particles $N(t)$ in the adiabatic limit with time $t$ (in units of $1/J$) when particles are injected. For (a) $\Gamma_d = 10.0J$ and (b) $\Gamma_d = 100.0J$, for different values of $\Gamma_G$ and $J = 1$. $N(t)$ shows a crossover from ballistic ($N(t)\propto t$), to sub-diffusive ($N(t) \propto t^{\nu}$ with $\nu<1/2$) and finally to diffusive ($N(t)\propto t^{1/2}$). The density profile $n_i(t)$ is plotted as a function of $i/\sqrt{t}$ at different time instances in the inset of (a) for $\Gamma_d = 10.0J$ and $\Gamma_G = 2/\Gamma_d$, showing agreement with the analytical result obtained in Eq. \ref{['eq:n_xt_adiabatic_special_case']}. In the inset of (b) $N(t)$ is plotted using exact numerics [Eq. \ref{['eq:EOM-2pt_matrix']}] and adiabatic approximation [Eq. \ref{['eq:c_diag_adiabatic_solution']}] for $L = 200, \Gamma_d = 100.0J$, and $\Gamma_G = 10.0J$, showing good agreement between the two. (c) $N(t)$ is plotted as a function of scaled time axis $t/\Gamma_d$ for $\Gamma_G = 1.0J$, and different values of $\Gamma_d$. The system size is fixed to $L = 1000$ for all cases, unless otherwise specified.
• Figure 4: The dynamics of total number of particles $N(t)$ with time $t$ for the long range lattice system, defined in Eq. \ref{['eq:HLR']}, when the particles are injected from the left. For different long-range hopping exponent $\alpha$, the late time dynamics is different. For $1<\alpha<1.5$, the dynamics is superdiffusive with time dependence $t^{1/2\alpha-1}$ (black dashed line) and for $\alpha>1.5$ (black dotted line), the dynamics is diffusive with $t^{1/2}$ scaling.
• Figure 5: Plot for quantum dynamics of average total particle number $N(t)$ with time $t$ (in units of $1/J$) for the XXZ lattice given in Eq. \ref{['eq:XXZ_hamil']}. (a) Growth of $N(t)$ is plotted for different system sizes $L=20, 40$, and $80$. The other parameters are given as $\Gamma_d=1.0J$, $\Gamma_G=1.0J$, and $\Delta=1.0J$ (isotropic case). Initially, $N(t)$ grows linearly as $\Gamma_G t$ until the dephasing effect emerges. At late times, $N(t)$ grows diffusively. In the inset, the density profile $n_i$ is plotted with respect to the scaled lattice site $i/\sqrt{t}$ for different time instants and for system size $L=80$. A perfect scaling collapse of $n_i$ data is observed for different time instants, confirming the late-time diffusive dynamics. (b) Plot for $N(t)$ with time $t$ for different $\Gamma_G$ values with $\Gamma_d=10.0J$, $\Delta=1.5J$. We observe that, though the initial dynamics are sensitive to the value of $\Gamma_G$ because initially $N(t)\sim \Gamma_G t$ (dashed black line), the late-time dynamics are insensitive to the value of $\Gamma_G$. In (c), we have shown that the diffusion constant depends on $\Delta$. We fix the parameters to be $L = 80, \Gamma_d = 10.0J$, and $\Gamma_G = 1.0J$. We have fixed the bond dimension at $\chi = 200$ in all the figures.