Stability of Floquet sidebands and quantum coherence in 1D strongly interacting spinless fermions

TL;DR

The paper investigates how periodic driving forms and stabilizes Floquet-Bloch sidebands (FBs) in a 1D, strongly interacting spinless fermion system. By computing time-dependent spectral functions $A^{\rm ret}_k(t,\\omega)$, real-space propagators, and momentum distributions using ED and time-dependent MPS, it reveals a distinct frequency dependence: high-frequency driving ($\\Omega \gg W$) yields long-lived FBs with a renormalized bandwidth $t_h^{\rm eff}=t_h J_0(A_0)$, while low-frequency driving ($\\Omega < W$) leads to strong heating and suppression of FBs due to replica overlap. Noise in the driving also degrades FB coherence, even at high frequency, indicating finite-time stability windows. Real-space correlators and momentum-space QFI relations corroborate the spectral findings, offering experimentally accessible markers. Overall, the results illuminate the conditions under which Floquet engineering can be realized in correlated materials and highlight heating and overlap as key factors limiting FB lifetimes.

Abstract

For strongly correlated quantum systems, fundamental questions about the formation and stability of Floquet-Bloch sidebands (FBs) upon periodic driving remain unresolved. Here, we investigate the impact of electron-electron interactions and perturbations in the coherence of the driving on the lifetime of FBs by directly computing time-dependent single-particle spectral functions using exact diagonalization (ED) and matrix product states (MPS). We study interacting metallic and correlated insulating phases in a chain of correlated spinless fermions. At high-frequency driving we obtain clearly separated, long-lived FBs of the full many-body excitation continuum. However, if there is significant overlap of the features, which is more probable in the low-frequency regime, the interactions lead to strong heating, which results in a significant loss of quantum coherence and of the FBs. Similar suppression of FBs is obtained in the presence of noise. The emerging picture is further elucidated by the behavior of real-space single-particle propagators, of the energy gain, and of the momentum distribution function, which is related to a quantum Fisher information that is directly accessible by spectroscopic measurements.

Figures (15)

• Figure 1: Summary of the two extreme cases of our findings. (a) Sketch of a typical equilibrium result for the retarded spectral function $A^{\rm ret}_k(\omega)$ in the charge density wave (CDW) insulating phase at half filling for model \ref{['eq:Peierls_Hamiltonian']}. We denote by $W$ the spectral width (see Sec. \ref{['sec:results']}), in which the spectral function has populated continua. (b) and (c) illustrate the general expectation for FBs in such a strongly correlated system: In both sketches, the magenta region illustrates the zeroth Floquet sector, which is located at the position of the original equilibrium spectral function. Note that, due to the driving, in comparison to the equilibrium result the shape can be modified Dunlap1986, and additional effects (e.g., in-gap bands Osterkorn2023) can come into appearance. This spectral function in the zeroth Floquet sector then is replicated at energies, which are multiples of the driving frequency $\pm n\Omega$. Here, we only sketch the 1st replicas at $\pm \Omega$ (light blue). In (b) we sketch the situation, when the driving frequency $\Omega < W$. This leads to substantial overlap of the different Floquet sectors. In contrast to this, in (c) we sketch the situation if $\Omega > W$, so that there is no overlap between the Floquet sectors. (d) and (e) illustrate the effect of the driving on correlations in real space. They display the time evolution of the single particle propagator Eq. \ref{['eq:propagator']} in real space in the CDW phase for driving frequencies $\Omega = 3.0$ and $\Omega = 20.0$, respectively, at waiting time $t=5$, indicating the stability of quantum coherence in both regimes. (f) and (g) show examples for the resulting time dependent spectral function $A^{\rm ret}_k(t,\omega)$ [Eq. \ref{['eq:Spectral_function']}] in the CDW state at low and high frequency driving, $\Omega = 3.0$ (no FBs visible) and $\Omega=20$ (FBs visible at $\pm n\Omega$ indicated by the arrows), respectively, at waiting time $t=5$. All results are at $V/t_h=5$.
• Figure 2: Retarded single-particle spectral function $A^{\rm ret}_k(t,\omega)$ [Eq. \ref{['eq:Spectral_function']}] for the $tV$-chain \ref{['eq:Peierls_Hamiltonian']}. (a)-(c): Results in the CDW phase for $V/t_h=5$ for $L=32$ and OBC at half filling obtained with MPS. (a) Results at equilibrium. $W_{\rm CDW} \approx 18$ estimates the spectral width, in which $A^{\rm ret}_k(\omega) > \epsilon$, where $\epsilon$ is $2\%$ of the peak value of $A^{\rm ret}_k(t,\omega)$ (see main text). (b) Driven case with $\Omega = 3$ and $A_0=1.0$ at waiting time $t=0$. (c) The same driven case at waiting time $t=20$. The inset in (c) shows the result at fixed $k=\pi/2$ for $t=20$. (d)-(f): Results in the LL phase $V/t_h=1.5$ for $L=18$ and PBC at half filling obtained with ED. (d) Results at equilibrium, $W_{\rm LL} \approx 10$ denotes the spectral width, as before. (e) and (f) nonequilibrium results for the same driving parameters as in (b) and (c). In (e) the first two side bands are indicated with arrows. The inset in (f) shows the results at fixed $k=\pi/2$ for $t=20$, vertical lines indicate the position of the side bands. Dashed lines in (e) and (f) are guides to the eye for locating the side bands.
• Figure 3: The same as in Fig. \ref{['fig:Low_freq_Eq_and_driven']} for $V/t_h=5$, but at high-frequency driving $\Omega=20$ and driving amplitude $A_0=1.6$. (a) shows MPS results for $L=32$ and OBC, while the long-time results in (b) are obtained using Lanczos time evolution with $L=18$ sites and PBC. The inset in (a) shows the results at fixed $k=\pi/2$; the green highlighted region indicates the spectral function in the 0th Floquet sector, the blue shaded regions indicate the 1st FBs, and the magenta shaded regions indicate the 2nd FBs.
• Figure 4: The same as in Fig. \ref{['fig:High_freq_drive']}, but in the presence of time-dependent noise in the driving, where $\Omega = 20 \pm \Omega_{\rm noise}$ and $\Omega_{\rm noise} \in [0,0.1]$. The results are obtained using Lanczos time evolution with $L=18$ and PBC. Panels (a)-(c) show $A_k^{\rm ret}(t,\omega)$ at waiting times $t=0$, $t=20$, and $t=100$, respectively. The inset in (c) shows results at fixed $k=\pi/2$ at $t=100$. All results show an average over 8 realizations of the random noise.
• Figure 5: Results at $V/t_h=5$: (a)-(c) show the imaginary part of $G_{r,L/2}(t,\tau)$ [Eq. \ref{['eq:propagator']}] (MPS, $L=32$, OBC) for the equilibrium case, and for the driving parameters of Figs. \ref{['fig:Low_freq_Eq_and_driven']} and \ref{['fig:High_freq_drive']} at waiting time $t=0$. (d) shows $\langle n_k\rangle(t)$ at $\Omega=3$ and at $\Omega=20$, with and without noise for $L=18$ and PBC. (e) shows typical time evolution of the energy gain compared to the initial state (ground state energy $E_{\rm GS}$), $\Delta E(t) = \langle H(t) \rangle - E_{\rm GS}$ for different driving frequencies for $L=14$ and PBC.