Interacting dynamic Wannier-Stark ladder driven by a periodic pulse train

The electronic structures of the Floquet states of the dynamic Wannier-Stark ladder (DWSL) are examined, where the DWSL is formed by driving the biased superlattices (SLs) by the periodic pulse train (PPT) with the electric field $F\left(t\right)$—with time $t$—and the temporal period $2\pi ∕\omega$. For a strong $F\left(t\right)$, interminiband interactions, namely, the ac-Zener tunneling (ac-ZT), are predominantly caused in the DWSL. Such a system is termed the interacting DWSL. In order to understand the details of the Floquet states and the modulation patterns by alteration of a couple of the PPT laser parameters, the linear absorption spectra, ${\alpha }_{\mathit{abs}}({\omega }_p;\omega )$, of optical interband transitions invoked by the monochromatic probe laser $f_p\left(t\right)$ with the frequency ${\omega }_p$ are calculated, where the spectra are not only linear in $f_p\left(t\right)$ but also nonlinear in $F\left(t\right)$. The exciton effect is not included for the sake of simplicity. For the PPT driving with unit-pulse shapes largely deviated from the square and saw-toothed profiles, the spectra show unexpected dent structures, differing a great deal from the corresponding ac-ZT-free spectra basically similar to those of the original SLs just showing the ascending steplike structure. To deepen the understanding of this anomaly, the spectra of ${\alpha }_{\mathit{abs}}^0({\omega }_p;\omega )\propto \partial {\alpha }_{\mathit{abs}}({\omega }_p;\omega )∕\partial {\omega }_p$ are also calculated, whereby the dent structures become spectral dips showing the negative absorption. It is found that such anomalous behavior is attributed to the ac-ZT between different minibands that accompanies emission/absorption of the nonzero net number of photons with $J\omega$ (with $J$ a nonzero integer). This anomaly also shows the unusual time dependence in the dual-time optical susceptibility associated with ${\alpha }_{\mathit{abs}}^0({\omega }_p;\omega )$. Moreover, the possibility of existence of the negative absorption in the more realistic excitonic spectra is speculated.

Figure 1

The illustration of the wave form $F\left(t\right)$ as a function of the elapsed time $t$ for the PPT with $x=0.8$, $\mu =0.7$, and $\delta =\pi ∕6$. For $\tau$ and $T$, consult the text.

Figure 2

The wave forms of (a) the PPT-1 $(\mu =1,\delta =\pi ∕2)$ and (b) the PPT-2 $(\mu =0.5,\delta =\pi )$ for $x=0.1$, 0.5, and 0.9.

Figure 3

The joint-miniband quasienergies $E_{\nu }\left(k\right)$ as a function of $Z=F_{\mathit{dc}}d∕\Omega$ for the PPT-1 with $x=0.1$, 0.5, and 0.9 and $N_{\omega }=3$. The quasienergies depicted in panels (a)–(c) are obtained with the ac-ZT, while those in panels (d)–(f) are obtained without the ac-ZT. The arrowed portions in panels (c) and (f) at $Z=10$ are used to aid the discussion made in Sec. 3C. For more details, consult the text.

Figure 4

The same as Fig. 3 but for the PPT-2.

Figure 5

(Color online) The effective absorption coefficients ${\alpha }_{\mathit{abs}}({\omega }_p;\omega )$ as a function of the probe light energy ${\omega }_p$ for the PPT-1 with $x=0.1–0.9$ and $N_{\omega }=3$. The spectra with the ac-ZT are depicted by the red line, while those without it are denoted by the blue line.

Figure 6

(Color online) The same as Fig. 5 but for the PPT-2.

Figure 7

(Color online) The effective absorption coefficients ${\alpha }_{\mathit{abs}}^0({\omega }_p;\omega )$ as a function of the probe light energy ${\omega }_p$ for the PPT-1 with $x=0.1–0.9$ and $N_{\omega }=3$. The spectra with the ac-ZT are depicted by the red line, while those without it are denoted by the blue line. The arrowed portions in the spectra for $x=0.9$ are used to aid the discussion later made in Sec. 3C. For more details, consult the text.

Figure 8

(Color online) The same as Fig. 7 but for the PPT-2.

Figure 9

The Feynman diagrams indicating the perturbation expansion of ${\chi }_J^0({\omega }_p;\omega )$ with respect to the ac-ZT interaction ${\mathcal{V}}_{\left[BJ\right],\left[B^{\prime }{J}^{\prime }\right]}$. Here, time evolves upward. In the first equality, the diagram shows the time evolution based on Eq. (66), where the large filled circles represent ${\mathcal{C}}_{-J\nu }\left(k\right)$ and ${\mathcal{C}}_{0\nu }^{*}\left(k\right)$, the thick vertical line represents the propagator for the Floquet state $\nu$ given by $1∕[E_{\nu }\left(k\right)-{\omega }_p-i\gamma ]$, the small open circles represent the vertices $d_0^{\left(\mathit{vc}\right)}$ and $d_0^{\left(\mathit{vc}\right)*}$ due to the probe photon emission and absorption, respectively, and the wiggly line represents $J$-photon emission due to the ac ZT. In the second equality, the cross represents the vertex of ${\mathcal{V}}_{\left[BJ\right],\left[B^{\prime }{J}^{\prime }\right]}$, and the thick and thin vertical lines represent the propagators $G_{\left[B_0{J}_0\right]}\left({\omega }_p\right)$ and $G_{\left[BJ\right],\left[B_0{J}_0\right]}^{\left(ZT\right)}\left({\omega }_p\right)$, respectively. The labels put on the propagators, such as $[B^{\prime },J^{\prime }]$, show the mediating ZT-free Floquet states. For more details, consult the text.

Figure 10

The $N_{\omega }$ dependence on the effective absorption coefficient ${\alpha }_{\mathit{abs}}^0({\omega }_p;\omega )$ for the PPT-1 with $N_{\omega }=1$, 3, and 10, where panels (a) and (b) are for $x=0.3$ and 0.9, respectively.

Figure 11

The $Z$ dependence on the effective absorption coefficient ${\alpha }_{\mathit{abs}}^0({\omega }_p;\omega )$ for the PPT-1 with $Z=5$ and 10, where $N_{\omega }=1$ and $x=0.4$.

Figure 12

The lowest three miniband energies of the electrons $(b=1–3)$ as a function of the Bloch momentum $k$ in the $c$ band [panel (a)] and the $v$ band [panel (b)] for the SLs of 35 ML GaAs/11 ML ${\mathrm{Ga}}_{0.75}{\mathrm{Al}}_{0.25}\mathrm{As}$. The solid curves represent the results obtained by the fully numerical calculations, while the dashed curves show the results given by the NNTB model of Eq. (32). For more details, consult the text.

Figure 13

The DWSL quasienergy of the $c$-band electron as a function of $F_{\mathit{ac}}$ in the system of the SLs of 35 ML GaAs/11 ML ${\mathrm{Ga}}_{0.75}{\mathrm{Al}}_{0.25}\mathrm{As}$ exposed to the monochromatic ac field and the dc field with $F_{\mathit{dc}}^{\prime }=F_{\mathit{dc}}=24\mathrm{kV}∕\mathrm{cm}$. The results in panels (a), (b), and (c) are obtained by the IPBQW model, the fully numerical calculations with both dc- and ac-ZTs, and the fully numerical calculations with just the ac-ZT, respectively. The results of (b) and (c) are cited from Ref. 21. For more details, consult the text.

Figure 14

(Color online) The $\Delta$ dependence on the effective absorption coefficient ${\alpha }_{\mathit{abs}}^0({\omega }_p;\omega )$ for the PPT-1 with $x=0.6$ and $N_{\omega }=3$. In the red and green traces, the $\Delta$ dependence is included and excluded, respectively. For more details, consult the text.