Nonexponential fidelity decay in isolated interacting quantum systems

E. J. Torres-Herrera and Lea F. Santos

Phys. Rev. A 90, 033623 – Published 22 September 2014

Abstract

We study isolated finite interacting quantum systems after an instantaneous perturbation and show three scenarios in which the probability for finding the initial state later in time (fidelity) decays nonexponentially, often all the way to saturation. The decays analyzed involve Gaussian, Bessel of the first kind, and cosine squared functions. The Gaussian behavior emerges in systems with two-body interactions in the limit of strong perturbation. The Bessel function, associated with the evolution under full random matrices, is obtained with surprisingly sparse random matrices. The cosine squared behavior, established by the energy-time uncertainty relation, is approached after a local perturbation in space.

Received 17 July 2014

DOI:https://doi.org/10.1103/PhysRevA.90.033623

Authors & Affiliations

E. J. Torres-Herrera and Lea F. Santos

Department of Physics, Yeshiva University, New York, New York 10016, USA

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 90, Iss. 3 — September 2014

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Local density of states (a, b) and fidelity decay (c–f). Quench from the XX to the XXZ Hamiltonian with $Δ_{F} = 1.5$ , $E_{ini} = - 0.87$ , $σ_{ini} = 1.47$ , and $\bar{F} = 1.19 \times 10^{- 3}$ (a, c). Quench from the XXZ to the NNN model with $λ_{F} = 1$ , $Δ_{I} = Δ_{F} = 0.5$ , $E_{ini} = - 0.82$ , $σ_{ini} = 1.17$ , and $\bar{F} = 1.326 \times 10^{- 3}$ (b, d). In panels (e) and (f), the initial state is an eigenstate of a full random matrix from a GOE projected onto the XXZ Hamiltonian, $E_{ini} = - 0.39$ , $σ_{ini} = 2.01$ , and $\bar{F} = 2.32 \times 10^{- 4}$ (e), and onto the NNN model, $E_{ini} = - 0.24$ , $σ_{ini} = 2.07$ , and $\bar{F} = 2.35 \times 10^{- 4}$ (f). The solid lines give the analytical Gaussian expressions [Eq. (10)], and the shaded area (top panels) and circles (middle and bottom panels) are numerical results. The saturation value of the fidelity is indicated with the dashed horizontal line; $L = 16$ , ${\hat{S}}^{z} = 0$ , and $D = 12 870$ ; and $J = 1$ sets the energy scale.
Reuse & Permissions
Figure 2
Quench from the XXZ to the NNN model, $Δ_{I} = Δ_{F} = 0.5$ . Local density of states (a) and fidelity decay (b) for $λ_{F} = 0.45$ . Parameter $η$ of the pseudo-Voigt distribution vs $λ_{F}$ (c); critical time and $t_{R}$ vs $λ_{F}$ (d). The initial state for all panels has $E_{5445} = - 0.37$ . In panels (a) and (b), $E_{ini} = - 0.57$ , $σ_{ini} = 0.53$ , and $\bar{F} = 2.32 \times 10^{- 3}$ . The fittings lead to $Γ_{ini} = 0.64$ , $σ_{pV} = 0.37$ , and $η = 0.41$ . Numerical results, shaded area and symbols; Gaussian, black solid lines; Lorentzian, blue dashed lines; pseudo-Voigt, green dot-dashed lines; saturation point, horizontal line; and value of $t_{c}$ , vertical line. (c) Fitting for the numerical data (dashed line). (d) $t_{c}$ (triangles), $t_{R}$ (squares), and $t_{R}$ for $λ_{F} = 1$ (horizontal dashed line). $L = 16$ , ${\hat{S}}^{z} = 0$ , and $D = 12 870$ ; $J = 1$ sets the energy scale.
Reuse & Permissions
Figure 3
Quench from the XXZ to the NNN model, $Δ_{I} = Δ_{F} = 0.5$ and $λ_{F} = 1$ . Local density of states (a) and fidelity decay (b). Skewness $γ_{1}$ (c) and excess kurtosis $γ_{2}$ (d) vs $E_{ini}$ . In panels (a) and (b), $E_{ini} = - 2.96$ , $σ_{ini} = 1.05$ , and $\bar{F} = 2.40 \times 10^{- 3}$ . Numerical results, shaded area and circles; Gaussian, black solid lines; skewed Gaussian, blue dashed lines; and saturation point, horizontal line. In panels (c) and (d), the symbols are numerical results and the solid lines are guides for the eye. $L = 18$ , ${\hat{S}}^{z} = - 3$ , and $D = 18 564$ ; $J = 1$ sets the energy scale.
Reuse & Permissions
Figure 4
Density of states $ρ$ for the XXZ model with $Δ_{F} = 0.5$ and added random flip-flop terms between sites $i$ and $j$ , where $j - 1 \geq 2$ (a), and with those elements replaced with uncorrelated random numbers (b). Average of the absolute value of the off-diagonal elements vs the distance $m$ from the diagonal (c) for the Hamiltonian from system (a) (decaying curve) and from system (b) (flat curve). Fidelity decay (d) for system (a) (Gaussian analytical expression is the black solid line; numerical data are the circles; saturation point is the highest horizontal line) and for system (b) (analytical expression is the green dashed line; numerical data are the squares; saturation point is the lowest horizontal line). $Δ_{I} = 0.5$ , $E_{ini} \sim 0$ , $σ_{ini} = 4.42$ (a), $σ_{ini} = 4.03$ (b), $L = 16$ , ${\hat{S}}^{z} = 0$ , and $D = 12 870$ ; $J = 1$ sets the energy scale.
Reuse & Permissions
Figure 5
Local density of states (a, c) and corresponding fidelity decay (b, d) for a quench from the XXZ model to the impurity model with $d_{F} = 1.2$ (a, b) and $d_{F} = 8.0$ (c, d). The initial state is in the middle of the spectrum of ${\hat{H}}_{I}$ , $E_{D / 2}$ . (a, c) Numerical data, shaded red area; two Lorentzians with $E_{1} = - 0.44$ , $E_{2} = 0.19$ , and $Γ_{1} = Γ_{2} = 0.39$ , blue dashed line (a); two Gaussians with $E_{1} = - 3.98$ , $E_{2} = 3.90$ , $σ_{1} = 0.48$ , and $σ_{2} = 0.54$ , black solid line (c). (b, d) Numerical results, circles; Eq. (18), blue dashed line; and Fourier transform of the two Gaussians from panel (c), black solid lines. The saturation points are the horizontal lines. $Δ_{I} = Δ_{F} = 0.48$ , $L = 16$ , ${\hat{S}}^{z} = 0$ , and $D = 12 870$ ; $J = 1$ sets the energy scale.
Reuse & Permissions

Physical Review A

covering atomic, molecular, and optical physics and quantum information