Pulse excitation to continuous-wave excitation in a low-dimensional interacting quantum system

Real-time dynamics in a one-dimensional transverse Ising model coupled with the time-dependent oscillating field is analyzed by using the infinite time-evolving block decimation algorithm and Floquet theory. In particular, the transient dynamics induced by the pulse field and their connections to the dynamics by a continuous-wave field are focused on. During the pulse-field irradiation, the order parameter shows a characteristic oscillation, in which the frequency shifts from the pulse-field frequency. This is considered as a kind of the Rabi oscillation, but the frequency strongly depends on the intersite Ising interaction. After turning off the pulse field, the oscillation remains with a frequency $\mathrm{\Omega }$ and a damping constant $\gamma$. In the case of low fluence, both $\mathrm{\Omega }$ and $\gamma$ are scaled by the pulse amplitude over a wide range of the parameter values of the model. In the case of high fluence, $\mathrm{\Omega }$ and $\gamma$ are arranged by a product of the pulse amplitude and the pulse width, which governs the population of the excited state when the pulse field is turned off. Experimental feasibility of the present results is briefly discussed.

Figure 1

(a)–(c) Time profiles of the $x$ component of the magnetization induced by a short pulse field. Amplitude of the external field is (a) $A/J=0.001$, (b) 0.003, and (c) 0.005. Time profiles of the $y$ and $z$ components of the magnetization in the case of $A/J=0.005$ are shown in panels (d) and (e), respectively. The time profile of energy is presented in panel (e). Other parameter values are chosen to be $h_z/J=1.5,t_{\mathrm{width}}=0,{\omega }_{\mathrm{p}}/J=1$, and $t_{\mathrm{p}}=20/J$. Shaded areas represents the time domain, when the pump pulses are irradiated. Bold dashed lines in panels (a), (b), and (c) represent fitting curves for a time interval $t=100/J–200/J$.

Figure 2

Contour maps of the short-time Fourier transformation of $m^x$ in the case of the short-pulse field. Amplitudes of the pulse external field are chosen to be (a) $A/J=0.002$ and (b) 0.01. Other parameter values are chosen to be $h_z/J=1.5,t_{\mathrm{width}}=0,{\omega }_{\mathrm{p}}/J=1$, and $t_{\mathrm{p}}=20/J$. Arrows represent the time domain $\left(-2t_{\mathrm{p}}<t<2t_{\mathrm{p}}\right)$ when the pump pulses are irradiated. In the short-time Fourier transformation, the Gaussian width is chosen to be $\tau =24/J$.

Figure 3

(a) A contour map of the short-time Fourier transformation of $m^x\left(t\right)$ obtained by the iTEBD algorithm in an infinite size system, and (b) a result obtained by the exact diagonalization method in a finite size cluster with $N=4$. Parameter values are chosen to be $A/J=0.02,h_z/J=1.5,t_{\mathrm{width}}=150/J,{\omega }_{\mathrm{p}}=\mathrm{\Delta }$, and $t_{\mathrm{p}}=4/J$. Arrows represent the time domains, when the pump pulses are applied. (c) Frequencies of the major oscillating components during the external-field irradiation $\left(0<t<t_{\mathrm{width}}\right)$ obtained from the data in panel (a), and (d) the results obtained from the data in panel (b). Numerical data in $M^x(\omega ,t)$ are fit by the three Gaussian functions. Bold lines in panel (c) are obtained by fitting, and dotted lines in panel (d) are the results obtained by Floquet theory presented in Sec. 3b.

Figure 4

(a) Oscillation frequency $\mathrm{\Omega }$ subtracted by $\mathrm{\Delta }$ and (b) damping constant $\gamma$ plotted as functions of $A^2/\mathrm{\Delta }$ after turning off the pulse field. Both $\mathrm{\Omega }$ and $\gamma$ are obtained by fitting the numerical data after turning off the pulse field. Parameter values are taken to be $h_z/J=1.1–1.9,t_{\mathrm{width}}=0,{\omega }_{\mathrm{p}}=\mathrm{\Delta }$, and $t_{\mathrm{p}}=20/J$. Solid lines represent ${(A^2/\mathrm{\Delta })}^{\alpha }$ with $\alpha =1.2$ in panel (a) and 1.0 in panel (b).

Figure 5

(a) Dominant oscillation frequencies in $M^x(\omega ,t)$ after turning off the pulse field $(t=t_{\mathrm{width}}+80/J)$ and those during the pulse irradiation $\left(t=t_{\mathrm{width}}/2\right)$ plotted by the bold line with circles and the bold line with squares, respectively. The energy increment $\delta E$ and the damping constant $\gamma$ are also plotted. The width of the pulse fields are taken to be $t_{\mathrm{width}}=150/J$. Oscillation frequencies are plotted as functions of $A/J$ in panel (b), and those as functions of $A{t}_{\mathrm{width}}$ in panel (c). Parameter values are chosen to be $h_z/J=1.5,{\omega }_{\mathrm{p}}=\mathrm{\Delta }=1.0J$, and $t_{\mathrm{p}}=4/J$.

Figure 6

(a) Intensity plot of $\left|W\right|$ defined in Eq. (18) in a $\delta \varepsilon \text{−}A$ plane. A finite size cluster with $N=2$ is adopted. Other parameter values are chosen to be $J/h_z=0.05$ and ${\omega }_{\mathrm{p}}=\mathrm{\Delta }$. (b) Slopes of the ${\varepsilon }_1-{\varepsilon }_2+{\omega }_{\mathrm{p}}$ versus $A$ curve in panel (a) at $A/h_z={10}^{-5}$ and 0.1. Bold line represents the proportional coefficient between ${\varepsilon }_1-{\varepsilon }_2$ and $A$ given in Eq. (21).