Energy-band echoes: Time-reversed light emission from optically driven quasiparticle wave packets

The at-will control of quantum states is a primary goal of quantum science and technology. The celebrated Hahn echo exemplifies such quantum-state control based on a time-reversal process in a few-level system. Here, we propose a different echo phenomenon associated with the energy-band structure in quantum many-body systems. We show that the dynamics of quasiparticle wave packets can be reversed by a driving electric-field pulse, yielding echoes with the time-reversed waveform of the optical excitation pulse when the quasiparticles recombine. The present echoes are observed not only in band insulators but also in correlated insulators, including a Mott insulator and a spontaneously-broken-symmetry charge-ordered insulator, in one- and higher-dimensional systems, irrespective of the integrability of the models. Analytical expressions reveal the conditions under which the echoes appear, and they also indicate that the frequency of the echo pulses reflects the dispersion relation for quasiparticles such as electron-hole pairs, doublon-holon pairs, and kink-antikink pairs. These findings provide a framework for all-optical momentum-resolved spectroscopy of the quasiparticles in quantum many-body systems.

Figure 1

(a) Time profile of the electric current in a one-dimensional band insulator. The dashed and dashed-dotted curves represent the electric fields of the excitation pulse $E_{\mathrm{e}}$ and the driving pulse $E_{\mathrm{d}}$, respectively. The parameter values are set to ${\omega }_{\mathrm{e}}=4\pi {\sigma }_{\mathrm{e}}^{-1}=5t_{\mathrm{h}}, {\omega }_{\mathrm{d}}={\sigma }_{\mathrm{d}}^{-1}=0.1t_{\mathrm{h}}, A_{\mathrm{e}}=0.002, A_{\mathrm{d}}=0.9$, and $N=1500$. (b) Sketch of the echo-generation process, which consists of the following three steps: (i) creation of a photocarrier by the excitation pulse, (ii) intraband acceleration by the driving pulse, and (iii) echo emission due to the recombination of the photocarrier wave packet. See the text for details. Note that the echoes also appear for optical pulses with $A_{\mathrm{f}}=0$ as shown in Sec. 3e.

Figure 2

Echoes in a one-dimensional band insulator. (a) Time profile of the electric current. The inset shows the Fourier spectrum of the electric current (solid curve) and the energy of the electron-hole pair ${\varepsilon }_{1\mathrm{bi}}(k_{\mathrm{e}}-A_{\mathrm{f}})$ calculated from Eq. (2) (vertical lines). The amplitude of the driving pulse is set to $A_{\mathrm{d}}=0.9$ (i.e., $A_{\mathrm{f}}/\pi \approx -0.44$). (b) Spectral map of the electric current. The red curves show $\omega ={\varepsilon }_{1\mathrm{bi}}(k_{\mathrm{e}}-A_{\mathrm{f}})$ as a function of $A_{\mathrm{f}}$. The inset shows a sketch of the intraband dynamics of two photoexcited pairs.

Figure 3

Echoes in a Mott insulator. (a) Time profile of the electric current calculated by the iTEBD method. The amplitude of the driving pulse is set to $A_{\mathrm{d}}=1.5$ (i.e., $A_{\mathrm{f}}/\pi \approx -0.73$) and the other parameters are ${\omega }_{\mathrm{e}}=U=8t_{\mathrm{h}}$ and $A_{\mathrm{e}}=0.004$. The inset shows the Fourier spectra for different bond dimensions $\chi$ and the energy of the doublon-holon pair. (b) Spectral map of the electric current. The bond dimension is set to $\chi =2000$. The red curves show $\omega ={\varepsilon }_{\mathrm{H}}(p^{-1}[p\left(k_{\mathrm{e}}\right)-A_{\mathrm{f}}])$ as a function of $A_{\mathrm{f}}$.

Figure 4

Echoes in a charge-ordered insulator. (a) Time profile of the electric current calculated by the iTEBD method. The amplitude of the driving pulse is set to $A_{\mathrm{d}}=1.5$ (i.e., $A_{\mathrm{f}}/\pi \approx -0.73$), and the other parameters are ${\omega }_{\mathrm{e}}=4.4t_{\mathrm{h}}$ and $V=1.6t_{\mathrm{h}}$. The inset shows the Fourier spectra for $\chi =100$ and 200; the energy of the kink-antikink pair is indicated by the vertical line. (b) Spectral map of the electric current. The red curve shows $\omega ={\varepsilon }_{\mathrm{TFI}}(k_{\mathrm{e}}-P_0{A}_{\mathrm{f}})$ as a function of $A_{\mathrm{f}}$ with $P_0\approx 0.94$ (see the text). The inset in (b) illustrates the motion of the kink-antikink pair in reciprocal space.

Figure 5

Echoes in a two-dimensional band insulator. (a) Energy-band structure ${\varepsilon }_{2\mathrm{bi}}\left(\mathit{k}\right)$. The contrast on the surface represents $|{\psi }_{\mathit{k}}^{\mathrm{cb}}{|}^2$ for an $x$-polarized excitation pulse. The double-headed arrow indicates the polarization of the excitation pulse. (b) Spectral map of the electric current with an $x$-polarized excitation pulse and an $x$-polarized driving pulse. The red and green curves represent $\omega ={\varepsilon }_{2\mathrm{bi}}(k_{\mathrm{e},x}^{\left(1\right)}-A_{\mathrm{f}},0)$ and ${\varepsilon }_{2\mathrm{bi}}(k_{\mathrm{e},x}^{\left(2\right)}-A_{\mathrm{f}},\pi )$, and $\omega ={\varepsilon }_{2\mathrm{bi}}(k_{\mathrm{e},x}-A_{\mathrm{f}},k_{\mathrm{e},y})$, respectively (see the text). (c) Fourier spectrum of the electric current for $A_{\mathrm{d}}=1.64$ (i.e., $A_{\mathrm{f}}/\pi \approx -0.79$). The vertical lines represent the calculated energies of the quasiparticles after the driving pulse. (d),(e) Sketches of the motions of the photoexcited electron and hole under an $x$-polarized driving pulse. (f) Spectral map of the electric current with an $x$-polarized excitation pulse and a $y$-polarized driving pulse. The parameters are set to ${\omega }_{\mathrm{d}}=0.1t_{\mathrm{h}}, {\omega }_{\mathrm{e}}=5t_{\mathrm{h}}, A_{\mathrm{e}}=0.002$, and $L=1500$ in (a)–(f).

Figure 6

Echoes in a nonintegrable Mott insulator. (a) Time profile of the electric current calculated by the iTEBD method. The amplitude of the driving pulse is set to $A_{\mathrm{d}}=1.4$ (i.e., $A_{\mathrm{f}}/\pi \approx -0.68$), and the other parameters are ${\omega }_{\mathrm{e}}=U=8t_{\mathrm{h}}, V=2t_{\mathrm{h}}$, and $A_{\mathrm{e}}=0.02$. The inset shows the Fourier spectra for $\chi =2000$ and 3000. (b) Spectral map of the electric current. The bond dimension is set to $\chi =3000$.

Figure 7

Echoes in a one-dimensional band insulator with an optical driving pulse. (a) Time profiles of the electric fields, vector potential, and electric current. The green curve shows the envelope of $J\left(\tau \right)$ calculated from the Hilbert transformation. The amplitude of the driving pulse is set to $A_{\mathrm{d}}=2.2$ [i.e., $|A_{\mathrm{d}}\left({\tau }_{\mathrm{e}}\right)|/\pi \approx 0.51$], and the other parameters are ${\omega }_{\mathrm{d}}=0.05t_{\mathrm{h}}, {\omega }_{\mathrm{e}}=5t_{\mathrm{h}}, A_{\mathrm{e}}=0.002, {\tau }_{\mathrm{e}}=0$, and $N=1500$. The inset shows the Fourier spectrum of $J\left(\tau \right)$. (b) Spectral map of the electric current. The green and red curves represent $\omega ={\varepsilon }_{1\mathrm{bi}}(k_{\mathrm{e}}-A_{\mathrm{d}}\left({\tau }_{\mathrm{r}}\right)+A_{\mathrm{d}}\left({\tau }_{\mathrm{e}}\right))$ as functions of $|A_{\mathrm{d}}\left({\tau }_{\mathrm{e}}\right)|$ for the first and second echoes, respectively, where $A_{\mathrm{d}}\left({\tau }_{\mathrm{r}}\right)-A_{\mathrm{d}}\left({\tau }_{\mathrm{e}}\right)$ is the vector-potential change to which the electron-hole pairs are subjected.

Figure 8

Light emission for a band-edge excitation pulse. (a) Sketch of band-edge excitation. Only a part of $\tilde{E}\left(\omega \right)$ (gray shading) can be transferred to the wave function of the electron-hole pair. (b) Spectral map of the electric current. The red, yellow, and green dashed curves represent $\omega ={\varepsilon }_{1\mathrm{bi}}(k_{\mathrm{e}}-A_{\mathrm{f}}), {\varepsilon }_{1\mathrm{bi}}(k_{\mathrm{e}}+\mathrm{\Delta }k-A_{\mathrm{f}})$, and ${\varepsilon }_{1\mathrm{bi}}(k_{\mathrm{e}}-\mathrm{\Delta }k-A_{\mathrm{f}})$, respectively, with $k_{\mathrm{e}}=\pi /2$ and $\mathrm{\Delta }k\approx 0.23$. The inset shows the time profile of the electric current for $A_{\mathrm{d}}=1.08$ (i.e., $A_{\mathrm{f}}/\pi \approx -0.52$). The parameters are set to ${\omega }_{\mathrm{d}}=0.1t_{\mathrm{h}}, A_{\mathrm{e}}=0.002, {\tau }_{\mathrm{e}}=-50t_{\mathrm{h}}^{-1}$, and ${\omega }_{\mathrm{e}}=3t_{\mathrm{h}}=E_{\mathrm{g}}$.

Figure 9

Level statistics for the one-dimensional extended Hubbard model. (a) Nearest-neighbor level-spacing distribution for $V=2t_{\mathrm{h}}$. The smoothing parameter and bin width are set to $\sigma =0.5$ and $\delta s=0.1$, respectively. (b) The parameter $q$ as a function of $V$, for $L=10–14$. The inset shows $q$ for $V=2t_{\mathrm{h}}$; the line is obtained by the weighted least-squares method.