Optical absorptions activated by an ultrashort halfcycle pulse in metallic and superconducting states of the Hubbard model

The development of strong ultrashort pulse lasers has enabled the observation of non-equilibrium electron dynamics on femtosecond and even attosecond timescales Krausz2009 ; Gallmann2012 . A novel technique called subcycle-pulse engineering has been proposed, involving the application of a subcycle pulse with less than one cycle oscillation within a pulse envelope Tsuji2012 ; Lenarcic2014 ; Shinjo2022 ; Shinjo2023 ; Tohyama2023 ; Shinjo2024 . This approach aims to generate electronic states that cannot be accessed using conventional multicycle pulses. It has been suggested that applying ultrashort subcycle pulses to metallic or superconducting states can induce current with inversion-symmetry breaking, which is detectable by second-harmonic generations Kawakami2020 ; Shinjo2023 .

In a current-induced state with a broken inversion symmetry, exotic phenomena are anticipated, including the Doppler shifts of quasi-particle excitations known as the Volovik effect Volovik1993 . Particularly in absorption spectra, supercurrents in a dirty superconductor with intermediate strength of disorder can activate absorptions at a Higgs-mode energy $\omega=2\Delta_{\text{s}}$ between quasiparticle excitations with a gap $\Delta_{\text{s}}$ Moor2017 . This behavior has been observed in a superconductor NbN with a current generated by a DC bias Nakamura2019 ; Shimano2020 . While disorder effects are often essential in understanding absorptions in superconductors Mattis1958 ; Laplae1983 ; Zimmermann1991 , it has been pointed out that inversion-symmetry breaking due to supercurrent can change a selection rule, leading to a significant absorption at $\omega=2\Delta_{\text{s}}$ even in clean superconductors Seibold2020 ; Papaj2022 ; Crowley2022 .

In this paper, we investigate how ultrashort halfcycle pulses, which can induce a current with an inversion-symmetry breaking, contribute to transient absorption spectra of strongly correlated electron systems. Using the time-dependent density-matrix renormalization group (tDMRG) method, we demonstrate that the halfcycle pump pulse leads to the enhancement of optical absorptions in metallic and superconducting states of the Hubbard model.

In the one-dimensional (1D) and two-dimensional (2D) Hubbard models with attractive $U<0$ on-site interactions, we find the enhancement of absorptions at energies corresponding to amplitude modes associated with superconducting and charge-density-wave (CDW) orders. This behavior is the manifestation of a current-induced activation of absorptions at amplitude modes in a clean superconducting state. In a 2D system, we find that the amplitude-mode absorptions emerge in both parallel and perpendicular directions to the pump pulse for $U=-4$ (in the unit of the electron hopping), but only in parallel for $U=-8$. This $U$ dependence comes from the difference in the localized nature of superconducting pairing at weak and strong couplings.

In a metallic state of the 2D Hubbard model at nearly half filling with repulsive $U>0$ on-site interactions, we find another type of absorption enhancement broadly distributed in the Mott gap. The mid-gap absorptions appear in both directions parallel and perpendicular to the pump pulse, whose nonlocal nature can be attributed to the effect of magnetic excitations.

The rest of this paper is organized as follows. We introduce the Hubbard model with a time-dependent electric field in Sec. II. We also show a method to obtain time-dependent optical conductivity by the tDMRG method. In Sec. III, we show transient absorption spectra of the Hubbard models with attractive interactions $U<0$. We demonstrate that a halfcycle pulse activates absorptions at superconducting and CDW amplitude modes. In Sec. IV, we show transient absorption spectra of the Hubbard models with repulsive interactions $U>0$. We show the enhancement of absorptions at energies corresponding to spin-exchange interactions. Finally in Sec. V, we give a summary of this paper. Additional results for the $t$-$J$ model are provided in Appendix A.

To investigate nonequilibrium properties of strongly correlated electron systems, we consider the Hubbard model on a ladder-shaped cluster of $L=L_{x}\times L_{y}$ sites with a time ($t$) dependent vector potential $\bm{A}(t)=[A_{x}(t),A_{y}(t)]=A_{x}(t)\bm{e}_{x}+A_{y}(t)\bm{e}_{y}$, with $\bm{e}_{x}$ ($\bm{e}_{y}$) being the unit vector along the $x$-axis ($y$-axis) direction, described by the following Hamiltonian:

where $B_{ij,\sigma}=e^{-i\bm{A}(t)\cdot\bm{R}_{ij}}c_{i,\sigma}^{{\dagger}}c_{j,\sigma}+\text{H.c.}$, $c_{i,\sigma}^{\dagger}$ is the electron creation operator at site $i$, located at $\bm{r}_{i}$ in the cluster, with spin $\sigma\,(=\uparrow,\downarrow)$, and $n_{i,\sigma}=c_{i,\sigma}^{\dagger}c_{i,\sigma}$. $t_{\text{h}}$ is nearest-neighbor hopping and $U$ is on-site Coulomb interaction. Hereafter, we set $t_{\text{h}}=1$ as the energy unit. The sum indicated by $\langle i,j\rangle$ runs over all pair of nearest-neighbor sites $i$ and $j$, and $\bm{R}_{ij}=\bm{r}_{i}-\bm{r}_{j}$. We impose open boundary conditions along the $x$- and $y$-axis directions. A spatially homogeneous electric field $\bm{E}(t)=-\partial_{t}\bm{A}(t)$ is incorporated via the Peierls substitution in the hopping terms Peierls1933 . Note that we set the light velocity $c$, the elementary charge $e$, the Dirac constant $\hbar$, and the lattice constant to 1.

The effect of a strong ultrashort pump pulse applied to the Hubbard model can be investigated with a vector potential $\bm{A}_{\text{pump}}(t)=A_{\text{pump}}(t)\bm{e}_{x}$ introducing a phase shift Lenarcic2014 at $t=0$ in the $x$-axis direction with

where we set the width of the pump pulse to $t_{\text{d}}=0.02$. For $t_{\text{d}}\ll 2\pi/t_{\text{h}}$, $\bm{E}_{\text{pump}}(t)=-\partial_{t}\bm{A}_{\text{pump}}(t)$ leads to a halfcycle pulse that can quench the flux of $A_{0}$ and twists the phase of many-body wavefunctions Shinjo2023 . As a result, the pump pulse can induce an inversion-symmetry breaking associated with a current in metallic and superconducting states.

To obtain transient absorptions of the Hubbard model parallel (perpendicular) to the pump pulse, we subsequently apply a probe pulse $\bm{A}_{\text{probe}}(t,\tau)=A_{\text{probe}}(t,\tau)\bm{e}_{x}$ [$A_{\text{probe}}(t,\tau)\bm{e}_{y}$] at time $t=\tau$ polarized in the $x$-axis ($y$-axis) direction with

We set $A_{0}^{\text{pr}}=0.001$, $\Omega^{\text{pr}}=10$, and $t_{\text{d}}^{\text{pr}}=0.02$. Note that $\tau$ indicates the delay time between pump and probe pulses. We obtain a current

with a time-dependent many-body wave function $|\psi(t)\rangle$ evaluated by the tDMRG method implemented with the Legendre polynomial Shinjo2021 ; Shinjo2021b ; Shinjo2022 . Here, $\langle\cdots\rangle$ implies $\langle\psi(t)|\cdots|\psi(t)\rangle$ and the initial state of $|\psi(t)\rangle$ is the ground state of the Hubbard model $\mathcal{H}$ with $\bm{A}(t)=0$ in Eq. (1). With the present methodology, we obtain both singular and regular parts of the optical conductivity in nonequilibrium Shao2016 ; Shinjo2018 ; Shinjo2022 by directly calculating

where $\bm{A}_{\text{probe}}(\omega,\tau)$ and $\bm{j}_{\text{probe}}(\omega,\tau)$ are the Fourier transform of $\bm{A}_{\text{probe}}(t,\tau)$ and $\bm{j}_{\text{probe}}(t,\tau)=\bm{j}_{\text{total}}(t,\tau)-\bm{j}_{\text{pump}}(t)$, respectively. Here, $\bm{j}_{\text{total}}(t,\tau)$ and $\bm{j}_{\text{pump}}(t)$ are obtained by taking $\bm{A}(t)=\bm{A}_{\text{pump}}(t)+\bm{A}_{\text{probe}}(t,\tau)$ and $\bm{A}(t)=\bm{A}_{\text{pump}}(t)$, respectively, in Eq. (4). For the Fourier transformation, we evaluate $\bm{j}_{\text{probe}}(t,\tau)$ for $t$ up to $t_{\text{max}}=20$. We take a broadening factor $\gamma=0.4>2\pi/t_{\text{max}}$ in this paper. In a finite system with open boundary conditions, we obtain the singular part of $\sigma^{xx(yy)}(\omega,\tau)$ at $\omega\propto 1/L>0$, where $\sigma^{xx}(\omega,\tau)$ and $\sigma^{yy}(\omega,\tau)$ are the diagonal components of $\bm{\sigma}(\omega,\tau)$ along the $x$- and $y$-axis directions, respectively.

In the 2D Hubbard models with $U=10$ noteu , we find another type of absorption induced by a halfcycle pulse. The results of Re$[\sigma^{xx}(\omega,\tau<0)]$ for $L=16\times 2$ are shown in Fig. 3(a), while those of Re$[\sigma^{xx}(\omega,\tau<0)]$ and Re$[\sigma^{yy}(\omega,\tau<0)]$ for $L=6\times 6$ are shown in Fig. 3(b). In the tDMRG calculations, the bond dimensions are set to $\chi=4500$ and $7000$ for the $L=16\times 2$ and $6\times 6$ systems, respectively. Since we consider carrier densities of $\delta=1/8$ for the $L=16\times 2$ system and $\delta=1/9$ for the $L=6\times 6$ system, both of which are near half filling, there are absorptions above the remnant Mott gap at $\omega\gtrsim 6$, as shown by the black lines in Fig. 3. At low energies, in addition to the Drude weight at $\omega=0$, we find incoherent spectral weights in the ranges $2<\omega<4$ for $L=16\times 2$ and $2<\omega<6$ for $L=6\times 6$. These incoherent weights originate from string-type antiferromagnetic excitations present in 2D Mott insulators Dagotto1992 ; Stephan1990 ; Inoue1990 ; Jaklic2000 ; Tohyama2005 ; Shinjo2021 , and have been experimentally observed in La_2-xSr_xCuO₄ Uchida1991 and Sr_14-xCa_xCu₂₄O₄₁ Osafune1997 in the mid-infrared (MIR) region. The incoherent spectral weights in the MIR band extend to higher energies, up to $\omega\lesssim 6$ in 2D systems Tohyama2005 , compared to $\omega\lesssim 4$ in two-leg ladder systems. Furthermore, the Drude weight is larger in the $L=16\times 2$ system than in the $L=6\times 6$ system, since the 1D nature is retained in the two-leg ladder Osafune1997 .

After pumping with a halfcycle pulse, we find a redshift of the remnant Mott gap in the two-leg ladder system with $L=16\times 2$, as shown in Fig. 3(a), along with the suppression of the Drude weight. We attribute this redshift to the Stark shift, as a halfcycle pulse induces polarization Shinjo2023 . A similar redshift behavior of the remnant Mott gap is also observed in 1D systems (not shown). However, in contrast to quasi-1D systems, no redshift of the remnant Mott gap is observed in a square lattice, as shown in Fig. 3(b). This is because the induced polarization after pumping is smaller in a square lattice compared to quasi-1D systems (not shown).

In the mid-gap energy range $2<\omega<3.5$, we find a slight enhancement in absorption. The mid-gap absorptions are attributed to magnetic excitations activated by a halfcycle pulse, with peak positions characterized by spin-exchange interactions (see Appendix A). This behavior suggests that antiferromagnetic order partially remains in the nearly half-filled Mott insulator, leading to spin excitations caused by charge motion. A similar mid-gap structure has been observed in pump-probe spectroscopy of Sr_14-xCa_xCu₂₄O₄₁ using a near-infrared laser Fukaya2015 ; Hashimoto2016 ; Shao2019 .

The mid-gap absorptions are more pronounced in the 2D system with $L=6\times 6$, as shown by colored lines in Fig. 3(b), where they are broadly distributed in the range $1\lesssim\omega\lesssim 5$. A similar large and broad enhancement of mid-gap absorptions has also been observed in a carrier-doped Bi-based cuprate Peli2017 , excited by a near-infrared pulse. If we assume that the enhanced mid-gap absorption observed in this experiment shares the same origin as our results, we expect that the structure found in Fig. 3(b) can be reproduced not only by pumping with a halfcycle pulse but also with a near-infrared multicycle pulse, as investigated in Ref. Shinjo2017 .

Finally, we compare Re$[\sigma^{xx}(\omega,\tau=2)]$ parallel to the pump pulse [the red line in Fig. 3(b)] with Re$[\sigma^{yy}(\omega,\tau=2)]$ perpendicular to the pump pulse [the green line in Fig. 3(b)]. We find that the effect of the pump pulse extends to the perpendicular direction, causing changes in absorption both parallel and perpendicular to the pump pulse. This behavior can be attributed to spin-charge coupling in the square lattice. Indeed, it has been reported that charge dynamics in the direction parallel to the electric field is coupled with that in the perpendicular direction via magnetic excitations Tsutsui2021 .

Using the tDMRG method, we investigated the transient absorption spectra of superconducting and metallic states in the Hubbard models excited by an ultrashort halfcycle pulse. In the superconducting state of the Hubbard model with attractive $U<0$ interactions, we found an enhancement of absorption induced by the halfcycle pump pulse. The energies of these induced absorptions correspond to the amplitude modes of superconducting and CDW orders, indicating the current-induced activation of amplitude-mode absorptions in a clean superconducting state. In the square lattice, we found that the behavior of amplitude-mode absorptions depends on the strength of $U$. At $U=-8$ in the strong coupling region, where pairing is formed locally, the amplitude-mode absorptions are activated only in the direction parallel to the pump pulse. In contrast, at $U=-4$, where pairing is formed nonlocally, absorptions are activated both in the parallel and perpendicular directions.

In the metallic state of the Hubbard model on a square lattice near half filling with repulsive $U=10$ interactions, we found a different behavior in the transient absorption spectra. A halfcycle pump pulse induces absorptions that are broadly distributed in the mid-gap region. These mid-gap absorptions appear in both directions parallel and perpendicular to the pump pulse, and their nonlocal nature can be attributed to the effect of magnetic excitations.