Nature of Bosonic Excitations Revealed by High-Energy Charge Carriers

We address a long-standing problem concerning the origin of bosonic excitations that strongly interact with charge carriers. We show that the time-resolved pump-probe experiments are capable of distinguishing between regular bosonic degrees of freedom, e.g., phonons, and the hard-core bosons, e.g., magnons. The ability of phonon degrees of freedom to absorb essentially an unlimited amount of energy renders relaxation dynamics nearly independent of the absorbed energy or fluence. In contrast, the hard core effects pose limits on the density of energy stored in the bosonic subsystems resulting in a substantial dependence of the relaxation time on the fluence and/or excitation energy. Very similar effects can be observed also in a different setup when the system is driven by multiple pulses.

Figure 1

(a) $E_{\text{kin}}(t)$ vs time $t$ for different values of $k$. Dashed (full) lines represent results for HM (HCM). Other parameters of the Hamiltonian in Eq. (2) are $\omega =0.5$, $g=\sqrt{0.5}$ and the size of the system with periodic boundary conditions is $L=16$. Unless otherwise specified, identical parameters were used in all subsequent figures. The phase is set to $\varphi (t)=0$ in all figures except in Fig. 3. (b) Dependence of $\mathrm{\Delta }$ on the wave number $k$ of the initial free electron wave function. (c) Relaxation time ${\tau }_{\mathrm{HM}}$ (${\tau }_{\mathrm{HCM}}$) for HM (HCM) vs excitation energy $\mathrm{\Delta }$, extracted from data presented in (a) using the following analytical form $E_{\text{kin}}(t)=a\exp (-\sqrt{(t/\tau {)}^2+b^2}+b)+c$; for details of fitting see also Refs. [33, 34]. Error bars in (b) and (c) correspond to threefold standard deviation. Errors in specifying $\mathrm{\Delta }$ are due to small time fluctuation of $E_{\text{kin}}(t)$ in the long-time limit.

Figure 2

(a) Average number of bosonic excitations per site $n_{\text{bos}}$ vs $t$ for different values of $k$ corresponding to different $\mathrm{\Delta }$ as depicted in Fig. 2. Dashed (full) lines represent results for the HM (HCM). The horizontal line indicates the infinite-temperature value of $n_{\text{bos}}$ in the HCM. (b),(c) Spread of bosonic excitations $\gamma (j,t)$ for the HM and HCM, respectively, for $k=\pi$. Dashed white lines represent the Lieb Robinson bound as described in the text. To facilitate a straightforward comparison between models, identical color coding was used in both panels.

Figure 3

Kinetic energy $E_{\text{kin}}$, phonon energy $E_{\text{ph}}$ and hard core boson energy $E_{\mathrm{HCB}}$ vs time in (a). Dashed (full) lines represent results for the HM (HCM). $\omega =g=0.75$, and $L=16$ was used in this particular case. Note that the upper limit of $E_{\mathrm{HCB}}$, reached at infinite temperature, is $E_{\mathrm{HCB}}^{\max }=L\omega /2=6$. Multiple spikes in $E_{\text{kin}}$ are due to steplike jumps in the phase $\varphi (t)=\pi /2\sum _{l=0}^3\theta (t-l\mathrm{\Delta }t)$ where $\mathrm{\Delta }t=25$ and $\theta (x)$ is the Heaviside step function. In (b) we present relaxation time $\tau$ with squares (full circles) for the HM (HCM), extracted from the relaxation of $E_{\text{kin}}(t)$ in distinct time intervals between successive steplike changes of $\varphi (t)$.

Figure 4

Primary relaxation times ${\tau }_p$ of a generalized model containing both phonon as well as HC boson degrees of freedom with identical frequencies ${\omega }_{\text{phon}}={\omega }_{\text{HCbos}}=0.5$, respectively, but different coupling constants: ${\lambda }_{\text{HCbos}}=g_{\text{HCbos}}^2/2\omega t_0=0.5$ and ${\lambda }_{\text{phon}}=g_{\text{phon}}^2/2\omega t_0=0.1$ (full line) and ${\lambda }_{\text{HCbos}}=0.1$ and ${\lambda }_{\text{phon}}=0.5$ (dashed line). Smaller system with $L=12$ sites has been used in this case.