Photon echo from lensing of fractional excitations in Tomonaga-Luttinger spin liquid

Progress in condensed matter physics is intimately connected to the development of new spectroscopic techniques. Among the many emerging spectroscopies, two-dimensional coherent spectroscopy (2DCS) stands out as a promising tool for investigating strongly correlated systems. The 2DCS use multiple coherent electromagnetic waves to probe the nonlinear optical properties of a sample, thereby producing a two-dimensional spectrum that visualizes the sample’s nonlinear response as a function of the frequencies of probing electromagnetic waves [1, 2].

Comparing to the more familiar one-dimensional spectroscopy that probes linear optical properties, the 2DCS reveals not only the optical excitations in a sample but also their relationship. In the infrared and higher frequency range, its ability to diagnose the interplay between optical excitations has been widely leveraged by chemists to unravel the structure of complex molecules and map out the kinetic pathways of chemical reactions [1, 2, 3]. The advent of terahertz (THz) 2DCS now puts this technique in the right energy window to study many-body phenomena. The THz 2DCS has offered new experimental insights into quantum wells [4], antiferromagnets [5], and electronic glasses [6]. On the theory front, it has recently been suggested that the THz 2DCS can resolve the spectral continua formed by optical excitations in several clean and disordered many-body system and characterize their dynamical properties, which would be challenging to accomplish with linear spectroscopy, if at all [7, 8, 9, 10, 11].

A main strength of the 2DCS lies in the photon echo [12] signal from the third-order nonlinear optical response $\chi^{(3)}$. The photon echo is an optical analogue of the spin echo in nuclear magnetic resonance (NMR) [13]. Schematically, one may observe the photon echo by exciting the system with three successive short optical pulses and detecting the resulted $\chi^{(3)}$ response (Fig. 1(a)). Let $\tau$ be the time delay between the second and first pulses, $t_{\mathrm{w}}$ (the waiting time) be the delay between the third and second, and $t$ be the delay between the time of detection and the last pulse. The photon echo signal appears as a surge of the nonlinear response at $t\approx\tau$ in close analogy with the nuclear magnetic resonance (NMR) spin echo (Fig. 1(b)).

Similar to its NMR cousin, the photon echo signal can diagnose dissipation in a few-body system by effecting a time-reversal operation — the system’s evolution during the time delay $t$ reverses the evolution occurred during the time delay $\tau$ [1, 2]. Had the dynamics been unitary, the quantum mechanical phase accumulated in $\tau$ would be completely removed when $t=\tau$, resulting in a perfect rephasing. This perfect rephasing process would produce a photon echo signal at $t=\tau$ regardless the value of $\tau$. Deviation from the perfect rephasing is thus a direct measure of dissipation in the few-body system. Specifically, the decay of echo signal with increasing $\tau$ is a manifestation of the decoherence time ($T_{2}$ time), whereas the decay of the signal as a function of $t_{\mathrm{w}}$ probes the population time ($T_{1}$ time).

This unique ability of diagnosing dissipation makes one naturally wonder if the photon echo could also find its success in strongly correlated many-body systems. Although the latest theoretical inquiries have suggested interesting applications of photon echo in gapped systems [7, 8, 10] and disordered systems [9], less attention is paid to clean, gapless many-body systems. Adapting this technique to a gapless many-body setting poses new theoretical challenges. The standard framework for analyzing the photon echo uses the language of energy levels [1, 2], which is best suited for few-body systems with discrete energy spectra. While it is possible to analyze the zero temperature optical response of a gapped system by truncating the Fock space to a subspace containing finite number of excitations [14] and thereby making use of the established framework, such truncation is not permitted in general for gapless systems. Important questions such as the existence of photon echo in gapless strongly-correlated systems, its underlying mechanism, and its features require theoretical investigation.

In this work, we address these questions by studying the nonlinear optical response of a prototypical gapless strongly-correlated system, namely the Tomonaga-Luttinger spin liquid (henceforth “Luttinger spin liquid” for short) [15, 16]. For concreteness, we consider the Luttinger spin liquid hosted by the XXZ spin chain, which possesses a global $U(1)$ spin rotational symmetry with respect to the spin $z$ axis. We consider exclusively the nonlinear magnetic response $\perp z$ and decompose the electromagnetic wave polarization in the left-handed ($+$) and right-handed ($-$) basis.

Using the bosonization Hamiltonian at the renormalization group (RG) fixed point, we find that, among the six symmetry-allowed third-order magnetic susceptibilities, $\chi^{(3)}_{+--+}$ and its complex conjugate $\chi^{(3)}_{-++-}$ show photon echo, which appears as a peak on the $t$ axis at $t\approx\tau$. The echo signal possesses a universal asymptotic form, which we obtain analytically and verify numerically. Crucially, the photon echo is “perfect” in the sense that the signal is a function of $t-\tau$ rather than $t$ and $\tau$ both, resembling the perfectly rephasing photon echo in a few-body system. This implies that the signal measured at a given value of $t-\tau$ is independent of $\tau$. Moreover, the echo signal depends weakly on the waiting time $t_{\mathrm{w}}$, and saturates when $t_{\mathrm{w}}\to\infty$.

This perfect photon echo, although resembles the one due to the prefect rephasing process in few-body systems, comes as a pleasant surprise — The rephasing process is understood as a result of the optical transitions between discrete energy levels. Here, the rephasing picture does not directly apply as the present system has a continuous energy spectrum. Instead, we trace its origin back to a unique “lensing” phenomenon of fractional excitations in Luttinger spin liquids (Fig. 6c): The first THz pulse creates two wave packets of fractional excitations with opposite chirality [17]. The second pulse converts the left-moving wave packet into a right-moving one, whereas the third pulse converts the right-moving wave packet into a left-moving one. These two wave packets then meet each other later at $t=\tau$, thereby producing an echo. For the fixed-point Hamiltonian, the phonon modes are exact eigenstates of the Hamiltonian, and their dispersion relation is linear. Consequently, the wave packets of fractional excitations can propagate through the system indefinitely without decay or dispersion. This naturally explains the perfect photon echo, namely the echo signal does not decay as either $\tau$ or $t_{\mathrm{w}}$ increases.

Our lensing picture immediately suggests that the photon echo is a sensitive diagnostic to the RG-irrelevant perturbations to the fixed point Hamiltonian. These RG-irrelevant corrections give only minor corrections to the most physical quantities at low temperature, and consequently they are often elusive to experimental probes. In the 2DCS, these corrections manifest themselves in the decay of the photon echo thanks to lensing.

In the XXZ spin chain, the RG-irrelevant corrections include the umklapp terms and higher order gradient terms [15, 16]. The umklapp terms give rise to the dissipation of phonon modes and therefore the decay of the wave packets at finite temperature. As a result, the lensing becomes unattainable when the pulse delay $\tau$ or $t_{\mathrm{w}}$ exceeds the lifetime of the wave packets. This is manifest as the decay of the echo signal as a function of $\tau$ or $t_{\mathrm{w}}$, which is analogous to the dissipation-induced decay of the phonon echo in few-body systems mentioned above.

The higher order gradient terms, on the other hand, may result in the decay of the photon echo through a different mechanism. By adding these terms, one may introduce a small curvature to the dispersion relation of the phonon modes while keeping them as the exact eigenstates. The curvature results in the dispersion of the wave packets. We expect the lensing to be ineffective beyond a time scale $\tau_{\mathrm{disp}}$, at which point the width of the wave packet is comparable with the correlation length.

This dispersion-induced decay of the photon echo is distinct from the dissipation-induced decay and finds no immediate analogue in few-body systems. We study this decay mechanism on a toy model, namely the harmonic chain, which is a lattice discretization of the fixed point Hamiltonian. The photon echo, when measured at $t=\tau$, decays as a stretched exponential $\sim\exp(-C(\tau/\tau_{\mathrm{disp}})^{1/2})$, where $C$ is a numerical constant. Meanwhile, the echo signal shows weak dependence on the waiting time $t_{\mathrm{w}}$ and saturates when $t_{\mathrm{w}}\to\infty$. We attribute the lack of $t_{\mathrm{w}}$ dependence to the absence of thermalization in the toy model — The decay of the photon echo as a function of $t_{\mathrm{w}}$ reflects the population time. Since the population of the phonon modes cannot relax, its population time is effectively infinity.

To summarize, our analysis shows that the $\chi^{(3)}$ photon echo from the Luttinger spin liquid is a sensitive diagnostic of the RG-irrelevant perturbations to the fixed point Hamiltonian, which are difficult to detect with linear optical spectroscopy. It also uncovers a dispersion-induced photon echo decay mechanism unique to many-body systems. Conceptually, the lensing of fractional excitations is a convenient picture for understanding the photon echo in the Luttinger spin liquid. The lensing picture extends the phase interference picture, commonly invoked for the photon echo in few-body systems [1, 2], from the time domain to the spacetime domain.

The rest of this work is organized as follows: In Sec. II, we describe the problem setup. We present the bosonization analysis in Sec.. III and the lensing picture in Sec. IV. We investigate the dispersion-induced photon echo decay in Sec. V. In Sec. VI, we point out a few interesting open problems.

We consider the $S=1/2$ XXZ spin chain:

$\displaystyle H=\sum_{j}\frac{J_{\perp}}{2}(S^{+}_{j}S^{-}_{j+1}+h.c.)+J_{z}S^{z}_{j}S^{z}_{j+1}-BS^{z}_{j}.$

(1)

$j$ labels the lattice sites. $S^{\pm}_{j},S^{z}$ are the $S=1/2$ spin operators. $J_{\perp}$ is the exchange constant in the spin $xy$ plane. We shall consider both “ferromagnetic” ($J_{\perp}<0$) and “antiferromagnetic” ($J_{\perp}>0$) chains. $J_{z}$ is the exchange constant in the spin $z$ axis. We include the Zeeman term due to an external field $B\parallel z$ (Fig. 1). Throughout this work, we use the natural units with $\hbar=k_{B}=\mu=1$ where $\mu$ is the magnetic moment carried by the spin. We may extend Eq. (1) by including additional terms so long as they preserve the symmetries. Our analysis is applicable to the Luttinger spin liquid phase of Eq. (1) and its extensions.

The 2DCS measures a sample’s nonlinear optical response [1, 2]. The electromagnetic wave interacts with an insulating spin system such as Eq. (1) primarily by the Zeeman coupling. Therefore, the nonlinear optical response of Eq. (1) is chiefly due to its leading-order nonlinear magnetic susceptibility. Note the photon echo that arises from this nonlinear magnetic response closely resembles the NMR spin echo in that both are induced by a sequence of magnetic field pulses, and the former may be justifiably called spin echo as well [5]. However, different from the canonical NMR spin echo set up [13], 2DCS does not require precise control of field pulse area. Here, we adhere to the term “photon echo” to emphasize this difference from the NMR spin echo.

The Hamiltonian Eq. (1) possesses a $U(1)$ spin rotational symmetry with respect to the $z$ axis. Since the total magnetization in $z$ commutes with $H$ and does not evolve in the Heisenberg picture, it is natural to consider the magnetic response in the $xy$ plane. Experimentally, this corresponds to the Faraday geometry where the propagation direction of the electromagnetic wave is parallel/anti-parallel to the external field (Fig. 1a) [18].

The $U(1)$ symmetry of the Hamiltonian Eq. (1) forbids second-order in-plane nonlinear magnetic susceptibilities. The same symmetry allows six third-order in-plane nonlinear magnetic susceptibilities, out of which three are independent: $\chi^{(3)}_{+-+-}$, $\chi^{(3)}_{+--+}$, and $\chi^{(3)}_{++--}$, where $+(-)$ refers to the left (right)-handed circular polarization of the electromagnetic wave. The other three susceptibilities, namely $\chi^{(3)}_{-+-+}$, $\chi^{(3)}_{-++-}$, and $\chi^{(3)}_{--++}$, are related to the former three by complex conjugation and thus offer no new information. Throughout this work, we focus on $\chi^{(3)}_{+--+}$, which exhibits the photon echo, and defer the discussion on the other two susceptibilities to Sec. IV.

The THz 2DCS typically measures the nonlinear response in the time domain [4]. Following the discussion in Sec. I, we consider a three-pulse set up (Fig. 1): three circularly polarized electromagnetic pulses arrive at the sample successively at time $0$, $\tau$, $\tau+t_{\mathrm{w}}$, where $\tau,t_{\mathrm{w}}>0$ are the delay between successive pulses. The signal from the sample detected at a later time $t>0$ after the last pulse contains contributions from both linear and nonlinear responses. Rerunning the experiment with individual pulses, one may subtract off the linear response and thereby isolate the nonlinear response.

The nonlinear signal is a convolution of the pulse profile and the third-order magnetic susceptibility:

$$\chi^{(3)}_{+--+}(t,t_{\mathrm{w}},\tau)=\iiint^{\infty}_{-\infty}dx_{1}dx_{2}dx_{3}\\ \widetilde{\chi}^{(3)}_{+--+}(t,x_{1};\,t+t_{\mathrm{w}},x_{2};\,t+t_{\mathrm{w}}+\tau,x_{3}).$$

(2)

Here, $\chi^{(3)}_{+--+}$ is the optical susceptibility that depends only on time, while $\widetilde{\chi}^{(3)}_{+--+}$ is the spacetime-dependent susceptibility. Note the time parametrization of the latter quantity follows the standard convention for nonlinear susceptibility [19], whereas the former does not. We use the symbol with and without the tilde to emphasize these differences.

We visualize $\chi^{(3)}_{+--+}(t,t_{\mathrm{w}},\tau)$ by holding $t_{\mathrm{w}}$ constant and scanning $\tau$ and $t$. We obtain the two-dimensional spectra $\chi^{(3)}(\omega_{t},t_{\mathrm{w}},\omega_{\tau})$ by performing a two-dimensional one-sided Fourier transform of $\chi^{(3)}(t,t_{\mathrm{w}},\tau)$ over the domain $t>0$ and $\tau>0$. Note alternative protocols for visualizing the nonlinear response exist [4, 6]. Ours is closely related to that of Refs. 5, 7.

In this section, we compute the nonlinear response of Eq. (1) by using bosonization. We show that the nonlinear susceptibility $\chi^{(3)}_{+--+}$ exhibits photon echo and characterize its features.

In the previous section, we have shown that the nonlinear magnetic susceptibility $\chi^{(3)}_{+--+}$ of the Luttinger spin liquid exhibits photon echo that resembles the perfectly rephasing echo in a few-body system such as a single spin. In this section, we provide an intuitive picture that clarifies its origin in the many-body system under consideration, namely Luttinger spin liquids. We illustrate our picture on the ferromagnetic spin chain and then generalize it to the antiferromagnetic chain, and discuss its various features and implications.

To set the stage, we review the effect of the bosonized spin raising operator $\exp(-i\theta)$. Let us consider the time evolution after the operator $\exp(-i\theta(0,0))$ at $t=x=0$ on the initial state $|\Psi(0)\rangle$ at $t=0$ which is assumed to be an energy eigenstate. At time $t$, the state is given by

	$\displaystyle|\Psi(t)\rangle$	$\displaystyle=e^{-i\mathcal{H}t}e^{-i\theta(0,0)}|\Psi(0)\rangle$
		$\displaystyle=e^{-i\mathcal{H}t}e^{-i\theta(-t,0)}e^{+i\mathcal{H}t}e^{-i\mathcal{H}t}|\Psi(0)\rangle$
		$\displaystyle\sim e^{-i\theta(-t,0)}|\Psi(0)\rangle,$		(26)

where we used the assumption that $\Psi(0)\rangle$ is an energy eigenstate and dropped the overall phase factor. We can decompose the field $\theta$ to chiral components as

$\displaystyle\theta(t,x)=$

$\displaystyle\theta_{L}(x^{+})+\theta_{R}(x^{-}),$

(27)

where

$\displaystyle\theta_{L}=\frac{1}{2}\left(\theta+\frac{\phi}{K}\right);\quad\theta_{R}=\frac{1}{2}\left(\theta-\frac{\phi}{K}\right).$

(28)

The equation of motion implies that each chiral components depends only on the corresponding light cone coordinate, as in Eq. (27). Using this, we can translate the time dependence into the position dependence, so that

$\displaystyle|\Psi(t)\rangle\sim$

$\displaystyle e^{-i\theta_{L}(-ut)}e^{-i\theta_{R}(ut)}|\Psi(0)\rangle,$

(29)

where $\theta_{L,R}(\mp ut)$ is now understood as a static operator at locations $\mp ut$, up to the overall phase factor which includes the one coming from the commutation relation between $\theta_{R}$ and $\theta_{L}$.

Now, recall the equal-time commutation relation:

$\displaystyle\phi(x)e^{-i\theta_{L,R}(y)}=e^{-i\theta_{L,R}(y)}\left(\phi(x)-\frac{\pi}{2}\Theta{(x-y)}\right).$

(30)

Namely, $\exp(-i\theta_{L,R}(y))$ creates a kink of step $\pi/2$ in $\phi$-field at the location $y$. Since this step corresponds to the magnetization density $(1/2)\,\delta(x-y)$, this kink can be interpreted as a spinon [17]. Eq. (29) then implies that the application of the operator $\exp(-i\theta)$ at $t=0$ is equivalent to creation of a right-moving spinon at $x=+ut$ and a left-moving spinon at $x=-ut$ at time $t$, when the other operations are applied after the time $t$. The chiral vertex operators $\exp(-i\theta_{L,R})$ can be viewed as spinon creation operators of the corresponding chirality. This justifies the following simple visual picture: Applying $\exp(-i\theta(0,0))$ creates a pair of the right-moving and left-moving spinons at $t=x=0$. Then these spinons propagate with the constant velocity $\pm u$.

Real-time correlation functions of the vertex operators can be qualitatively understood in terms of this visual picture. As the simplest example, consider the two-point correlation function (Fig. 6a)

	$\displaystyle\langle S^{-}(0)S^{+}(-1)\rangle\sim\langle e^{i\theta(0,0)}e^{-i\theta(-t_{1},-x_{1})}\rangle$
	$\displaystyle\quad\sim\langle e^{i\theta_{L}(0)}e^{i\theta_{R}(0)}e^{-i\theta_{L}(-x_{1}-ut_{1})}e^{-i\theta_{R}(-x_{1}+ut_{1})}\rangle$
	$\displaystyle\quad\sim\left|\frac{\sinh(\pi Tx^{+}_{1})}{\pi T\epsilon}\right|^{-\frac{1}{4K}}\left|\frac{\sinh(\pi Tx^{-}_{1})}{\pi T\epsilon}\right|^{-\frac{1}{4K}}.$		(31)

Here, we set $t_{1}>0$ and omit an overall phase factor. This correlation function corresponds to the creation of a pair of spinons at $(-t_{1},-x_{1})$, and the annihilation of a pair of spinons (or equivalently the creation of a pair of anti-spinons) at the origin. From the equation of motion (the second line), this correlation function can be interpreted as an expectation value of spinon creation operators at the split locations $-x_{1}\pm ut_{1}$ and two spinon annihilation operators at $0$. For a generic choice of $t_{1},x_{1}$, these three locations are different. As a result, the expectation value, although does not vanish completely, is exponentially suppressed with respect to the correlation length $u/T$ at finite temperature $T$.

If we wish to maximize the correlation function, we should try to “catch” the spinons created at $(-t_{1},-x_{1})$ at the later time and annihilate them. If we can manage to annihilate all the spinons created earlier, the correlation function would be reduced to the equal-time correlation function of the identity operator, which does not decay as a function of time $t$ at all! Unfortunately, it is impossible to annihilate both spinons created at $(-t_{1},-x_{1})$ by the single operator $\exp(i\theta(0,0))$, since spinons split and are located on different positions at later times.

We note it is possible to annihilate one of them, for example the left-moving spinon, by choosing the location of the second operator on the left side of the light cone ($x_{1}=-ut_{1}$, or $x^{+}_{1}=0$). Setting $x^{+}_{1}=0$ in Eq. (31) maximizes the first factor, which is formally divergent in the scaling limit (the cutoff $\epsilon\to 0^{+}$) but bounded to be a finite constant due to the finite $\epsilon$ in a realistic system. However, $x^{+}_{1}=0$ implies $x^{-}_{1}=2ut_{1}$, leading to the exponential suppression of the other factor in Eq. (31). Thus the two-point correlation function is exponentially suppressed for large $t_{1}$, for any choice of the relative spatial location $x_{1}$.

The situation is quite different for four-point correlation functions and corresponding response functions. It is convenient to rewrite the response function Eq. (15) as a sum over various four-point correlation functions by expanding the nested commutators:

	$\displaystyle\widetilde{\chi}^{(3)}_{+--+}(1,2,3)$	$\displaystyle\sim R_{1}+R_{2}+R_{3}+R_{4}$
		$\displaystyle-R^{\prime}_{1}-R^{\prime}_{2}-R^{\prime}_{3}-R^{\prime}_{4},$		(32a)
where
	$\displaystyle R_{1}$	$\displaystyle=\langle e^{-i\theta(0)}e^{i\theta(-1)}e^{i\theta(-2)}e^{-i\theta(-3)}\rangle;$		(32b)
	$\displaystyle R_{2}$	$\displaystyle=\langle e^{-i\theta(-3)}e^{i\theta(-1)}e^{-i\theta(0)}e^{i\theta(-2)}\rangle;$		(32c)
	$\displaystyle R_{3}$	$\displaystyle=\langle e^{-i\theta(-3)}e^{i\theta(-2)}e^{-i\theta(0)}e^{i\theta(-1)}\rangle;$		(32d)
	$\displaystyle R_{4}$	$\displaystyle=\langle e^{i\theta(-2)}e^{i\theta(-1)}e^{-i\theta(0)}e^{-i\theta(-3)}\rangle.$		(32e)

$R^{\prime}_{i}$ is related to $R_{i}$ by reversing the order of operators in the product. Following Ref. 1, 2, each contribution can be identified as a Liouville pathway.

We find that each four-point correlation function can be made free from exponential suppression for an appropriate choice of locations of the operators at given set of times. As exchanging the order of operators in $R_{i}$ only results in a phase factor, it is sufficient for our purpose to consider the correlation function $R_{1}$ (Fig. 6b). The spinon pairs are created at $(-t_{3},-x_{3})$ and $(0,0)$, and annihilated at $(-t_{2},-x_{2})$ and $(-t_{1},-x_{1})$. Unlike the case with the two-point correlation function (Eq. 31), here, we can catch and annihilate all the spinons created earlier by choosing (Fig. 6c):

$\displaystyle x_{1}=u\tau;\quad x_{2}=-u(\tau+t_{\mathrm{w}});\quad x_{3}=-ut_{\mathrm{w}}.$

(33)

For this configuration, the operator at $(-t_{2},-x_{2})$ annihilates the left-moving spinon created at $(-t_{3},-x_{3})$ and creates a right-moving antispinon. The operator at $(-t_{1},-x_{1})$ annihilates the right-moving spinon created at $(-t_{3},-x_{3})$ and creates a left-moving antispinon. The two antispinons moving to the opposite directions finally meet at the origin at time $0$, and are annihilated by the operator $\exp(i\theta)$! As a consequence, the magnitude of the four-point correlation function reaches its maximum and does not decay even in the long-time limit as long as the spatial coordinates are chosen according to Eq. (33). We call this phenomenon, which was absent in the two-point correlation function, the spinon lensing, that is, it is possible to focus the two (anti)spinons to the same point at time $0$, by placing the operators judiciously at earlier times. We also note that the spatial mirror reflection of Eq. (33) is equally valid for lensing.

The above reasoning can be made precise by applying the equation motion to reduce the $4$-point correlation function to an equal-time correlation function at time $0$ of multiple operators — spinon creation operators at locations $-x_{3}\pm ut_{3}$ and two at $0$, and antispinon creation operators at locations $-x_{2}\pm ut_{2}$ and $-x_{1}\pm ut_{1}$ (Fig. 6b). Under the condition Eq. (33), the locations of the spinon creation operators match with those of the spinon annihilation operators, and consequently the correlation function $R_{1}$ reaches the maximum.

The same phenomena occur in the other four-point correlation functions. Summing them up, the nonlinear response $\widetilde{\chi}^{(3)}_{+--+}$ as a function of spacetime coordinates $(t_{1},x_{1})$, $(t_{2},x_{2})$, and $(t_{3},x_{3})$, reaches a maximum at the lensing configuration Eq. (33). Fig. 6d shows the behavior of $\chi^{(3)}_{+--+}$ as a function of the detection position in a representative lensing configuration. Here we choose $\pi Tt=\pi T\tau=10$ and $\pi Tt_{\mathrm{w}}=10$. It shows that the response is maximum at the “focal point” $(0,0)$. As we have used a finite short distance cutoff $\epsilon$ to regularize the algebraic singularity at the light cone, we have slightly shifted $x_{1}$ and $x_{2}$ away from the light cone by a small distance $\epsilon$ to suppress the effect of the regularizer.

Now, the optical nonlinear response $\chi^{(3)}_{+--+}$ is related to $\widetilde{\chi}^{(3)}_{+--+}$ by spatial integration. When the time delays $t=\tau$, the spatial integration is dominated by the neighborhood of the lensing configuration Eq. (33) and its mirror reflection, and, as a result, remains non-vanishing even in the long-time limit $t=\tau\to\infty,t_{\mathrm{w}}\to\infty$. This explains the origin of the prefect photon echo observed in the $\chi^{(3)}_{+--+}$ of the ferromagnetic chain.

Turning to the antiferromagnetic chain, $\widetilde{\chi}^{(3)}_{+--+}$ is now a linear combination of various four-point response functions (Eq. (21)). Among these, $G^{R}_{a^{\dagger}aaa^{\dagger}}$ and $G^{R}_{b^{\dagger}bbb^{\dagger}}$ support the lensing phenonenon similar to Fig. 6 with the only difference being the type of fractional excitations created/annihilated by the spin operators. In the antiferromagnetic chain, $S^{+}$ creates a pair of spinons plus a pair of Laughlin quasiparticle and quasihole [17]. Note the created spinon and the Laughlin quasiparticle (and similarly the created antispinon and the Laughlin quasihole) are superimposed on each other, and therefore the world lines shown in Fig. 6 should be interpreted as the world line of the composite object. Our numerical calculations show that, indeed, only these two response functions produce photon echo whereas the others response functions do not.

In Sec. II, we have pointed out that the nonlinear susceptibilities $\chi^{(3)}_{+-+-}$ and $\chi^{(3)}_{++--}$ do not exhibit photon echo. This can now be easily understood in terms of lensing. For these two susceptibilities, it is impossible to arrange the spin operators in such a way that all the created fractional excitations get annihilated at a later time. Our calculations indeed confirm this.

Our discussion so far is based on the ideal Luttinger spin liquid at the RG fixed point. In general, there are RG-irrelevant perturbations to the fixed point Hamiltonian Eq. (3), but they only give sub-leading corrections to most of the physical quantities at low temperatures. This also makes these RG-irrelevant corrections difficult to detect in experiments. Nevertheless, in the 2DCS in Luttinger spin liquid, the RG-irrelevant perturbations can produce pronounced effects because the lensing of fractional excitations relies on two features of the fixed point Hamiltonian: First, the phonon modes are the exact eigenstates of the Hamiltonian. Second, the phonon dispersion relation is exactly linear. These features ensure that a wave packet of fractional excitation can propagate indefinitely through the system without dissipation or dispersion. RG-irrelevant corrections to the fixed point Hamiltonian would prevent the indefinite propagation of the wave packet, and, therefore, could be sensitively detected by the suppression of lensing.

In the XXZ spin chain, the RG-irrelevant perturbations include the umklapp terms such as $\cos(4\phi)$ [15, 16], which result in the damping of phonon modes at finite temperature. Consequently, the wave packets of fractional excitations acquire finite life time. The lensing phenomenon shown in Fig. 6 is suppressed when $\tau$ or $t_{\mathrm{w}}$ exceeds the life time of these excitations. This, in turn, will be manifest as the decay of the photon echo signal as $\tau$ or $t_{\mathrm{w}}$ increases, in close analogy with the dissipation-induced photon echo decay in few-body systems [1, 2]. Straightforward dimensional analysis suggests the decay rate vanishes as $T^{1+\eta}$ as $T\to 0$ where $\eta$ is the dimension of the umklapp term.