Photon Echo from Lensing of Fractional Excitations in Tomonaga-Luttinger Spin Liquid

We study theoretically the nonlinear optical response of Tomonaga-Luttinger spin liquid in the context of terahertz (THz) two-dimensional coherent spectroscopy (2DCS). Using the gapless phase of the $XXZ$-type spin chain as an example, we show that its third-order nonlinear magnetic susceptibilities ${\chi }_{+--+}^{(3)}$ and ${\chi }_{-++-}^{(3)}$ exhibit photon echo, where $\pm$ refers to the left- or right-hand circular polarization with respect to the $S^z$ axis. The photon echo arises from a “lensing” phenomenon in which the wave packets of fractional excitations move apart and then come back toward each other, amounting to a refocusing of the excitations’ world lines. Renormalization-group-irrelevant corrections to the fixed-point Hamiltonian result in dispersion and/or damping of the wave packets, which can be sensitively detected by lensing and consequently the photon echo. Our results thus unveil the strength of THz 2DCS in probing the dynamical properties of the collective excitations in a prototypical gapless many-body system.

Popular Summary

Physicists routinely diagnose the properties of a material by its linear responses to external stimuli. The recently developed nonlinear spectroscopy measures a material’s nonlinear optical responses to obtain more information. Yet, it is often unclear if there is a straightforward connection between such nonlinear responses and the material’s properties, especially when its constituent particles strongly interact with one another. We address this question by studying theoretically a prototypical strongly interacting system, the Tomonaga-Luttinger spin liquid. We find a direct link between a specific nonlinear response, known as photon echo, and the dynamical properties of the excitations in this system. From the photon echo, one could read off how fast these excitations decay or disperse, which is often hard to do with conventional experimental tools.

The photon echo is measured by three successive optical pulses. It appears as a surge in the response after the last pulse. In the Luttinger spin liquid, we trace the origin of the photon echo to a unique “lensing” phenomenon: The first pulse creates a pair of excitations moving in opposite directions. The second and third pulses change their directions of motion. These two excitations thus come back toward each other and reunite, thereby giving rise to the echo. Dissipation and dispersion effects, which suppress the lensing phenomenon, are sensitively picked up by the echo signal.

Our work thus uncovers one aspect of the many uses of the nonlinear spectroscopy and related dynamical phenomena. We think more interesting physics is yet to be explored in the nonlinear responses of strongly correlated systems.

Figure 1

(a) The Faraday configuration considered in this work. A magnetic field $B$ is applied in the $z$ axis. Three short electromagnetic pulses with circular polarizations pass through the $S=1/2$ spin chain. The propagation direction is parallel with the spin $z$ axis. The first pulse is right-handed, whereas the second and the third are left-handed. We denote the time delay between the first and the second pulse by $\tau$, the second and the third by $t_w$, and the third pulse and the time of detection by $t$. (b) The photon echo appears as the surge of nonlinear optical response at the time $t\approx \tau$.

Figure 2

(a) Nonlinear magnetic susceptibility ${\chi }_{+--+}^{(3)}$ of a ferromagnetic chain as a function of $\pi T(t-\tau )$ at $\pi T\tau =20$ (blue) and $\pi T\tau =40$ (cyan). The waiting time $\pi T{t}_w=1$. Luttinger parameter $K=0.7$. Data for different $\tau$ are scaled by the same factor such that the maximum of $\pi T\tau =20$ data is 1. Upper right inset: the $t>\tau$ part of the data multiplied by $\exp (\gamma |t-\tau |)$ with $\gamma =\pi T/(2K)$ plotted against $[\pi T(t-\tau ){]}^3$. Lower left inset: the $t<\tau$ part of the data multiplied by $\exp (\gamma |t-\tau |)$ plotted against $\pi T(t-\tau )$. (b) The same as (a) but for $K=1$. (c) The same as (a) but for $K=1.5$.

Figure 3

(a) Nonlinear magnetic susceptibility ${\chi }_{+--+}^{(3)}$ of a ferromagnetic chain as function of $\pi Tt$ and $\pi T\tau$. The waiting time $\pi T{t}_w=1$. The Luttinger parameter $K=1$. The data are real numbers and are rescaled such that the maximum is 1. (b),(c) The real and imaginary parts of the two-dimensional spectrum obtained by Fourier transforming the data of (a). Only the first and fourth quadrants are shown. The data in the other two quadrants are obtained by complex conjugation; i.e., the real part (b) is invariant under the transformation ${\omega }_{\tau }\to -{\omega }_{\tau },{\omega }_t\to -{\omega }_t$, whereas the imaginary part (c) changes sign.

Figure 4

(a) Nonlinear magnetic susceptibility ${\chi }_{+--+}^{(3)}$ of an antiferromagnetic chain as a function of $\pi T(t-\tau )$ for $\pi T\tau =50$ (blue) and $\pi T\tau =80$ (cyan). The waiting time $\pi T{t}_w=1$. Luttinger parameter $K=0.7$. The data for different $\tau$ are rescaled by the same factor such that the maximum of the complex modulus of the $\pi T\tau =50$ data is 1. Upper right inset shows the complex modulus of the $t>\tau$ part of the data multiplied by $\exp (\gamma |t-\tau |)$ with $\gamma =\pi (\mathrm{\Delta }-2)T$. Lower left inset shows the complex modulus of the $t<\tau$ part of the data multiplied by $\exp (\gamma |t-\tau |)$. (b) The same as (a) but for $K=1$. (c) The same as (a) but for $K=1.5$.

Figure 5

(a),(b) The real and imaginary parts of the nonlinear magnetic susceptibility ${\chi }_{+--+}^{(3)}$ of an antiferromagnetic chain as a function of $\pi Tt$ and $\pi T\tau$. The Luttinger parameter $K=1$. Magnetization density $2mu/T=1.15$. The waiting time $\pi T{t}_w=1$. The data are scaled such that the maximum of the absolute value of the data is 1. (c),(d) The real and imaginary parts of the two-dimensional spectrum.

Figure 6

(a) The two-point correlation function [Eq. (25)] can be understood in terms of a spinon-creation or -annihilation process. The spin-raising operator at $(-t_1,-x_1)$ creates a pair of spinons with left and right chirality, which then propagate with velocities $\pm u$ (blue solid lines). The spin-lowering operator tries to annihilate the spinon pair at (0,0). (b) The four-point correlation function Eq. (26b) corresponds to the spinon creation at $(-t_3,-x_3)$ and (0,0) (blue solid lines), and antispinon creation at $(-t_2,-x_2)$ and $(-t_3,-x_3)$ (blue dashed lines). (c) In the lensing configuration [Eq. (27)], the left-moving spinon created at $(-t_3,-x_3)$ is captured by the spin-lowering operator at $(-t_2,-x_2)$ and converted to a right-moving antispinon. Likewise, the right-moving spinon is captured by the spin-lowering operator at $(-t_1,-x_1)$ and converted to a left-moving antispinon. The antispinons then meet at (0,0) and are annihilated by the spin-raising operator. (d) Behavior of ${\tilde{\chi }}_{+--+}^{(3)}$ in the shaded region in (c), which shows its magnitude reaches the maximum near the “focal point” (0,0). Here, $\pi Tt=\pi T\tau =10$, and $\pi T{t}_w=10$.

Figure 7

(a) The nonlinear magnetic susceptibility ${\chi }_{+--+}^{(3)}$ as a function of $t$ and $\tau$ computed from the harmonic chain model Eq. (28). The waiting time $t_w=4a/u$. The temperature $T=1.2u/a$. The data are scaled such that the maximum is 1. (b),(c) The real and imaginary parts of the two-dimensional spectrum obtained by Fourier transforming the data in panel (a). Only the first and the fourth quadrants are shown. The other two quadrants are obtained by complex conjugation. (d) ${\chi }_{+--+}^{(3)}$ measured at $t=\tau$ as a function of $(a^{2}Tt/u^2{)}^{1/2}$ for various temperatures. $t_w=4a/u$. (e) ${\chi }_{+--+}^{(3)}$ measured at $t=\tau$ as a function of waiting time $t_w$ for two representative values of $\tau$. The temperature $T=0.8u/a$.

Figure 8

(a)–(c) illustrate three possible cases in which the spacetime points $(-t_1,-x_1)$ (red dot), $(-t_2,-x_2)$ (green dot), $(-t_3,-x_3)$ (blue dot) lie outside of the past light cone (dashed line) of the origin (black dot). In any of the three cases, the response function $G^R$ [Eq. (8)] vanishes. (d) Causality requires that all three points lie inside the past light cone of the origin to induce a nonzero response.