Theory of time-resolved Raman scattering in correlated systems: Ultrafast engineering of spin dynamics and detection of thermalization

Ultrafast characterization and control of many-body interactions and elementary excitations are critical to understanding and manipulating emergent phenomena in strongly correlated systems. In particular, spin interaction plays an important role in unconventional superconductivity, but efficient tools for probing spin dynamics, especially out of equilibrium, are still lacking. To address this question, we develop a theory for nonresonant time-resolved Raman scattering, which can be a generic and powerful tool for nonequilibrium studies. We also use exact diagonalization to simulate the pump-probe dynamics of correlated electrons in the square-lattice single-band Hubbard model. Different ultrafast processes are shown to exist in the time-resolved Raman spectra and dominate under different pump conditions. For high-frequency and off-resonance pumps, we show that the Floquet theory works well in capturing the softening of bimagnon excitation. By comparing the Stokes and anti-Stokes spectra, we also show that effective heating dominates at small pump fluences, while a coherent many-body effect starts to take over at larger pump amplitudes and frequencies on resonance to the Mott gap. Time-resolved Raman scattering thereby provides the platform to explore different ultrafast processes and design material properties out of equilibrium.

Figure 1

Different ultrafast processes induced by a pump field: effective heating or thermalization, transient (Floquet) band renormalization, and nonthermal many-body excitation.

Figure 2

The Betts 12A cluster in (a) real space and (b) momentum space. The gray arrows denote basis vectors. The solid and dotted lines in (a) represent respectively the intra- and intercluster hopping terms.

Figure 3

Time-resolved $B_{1g}$ Raman spectra with pump polarization ${\mathbf{e}}_{\mathrm{pol}}={\mathbf{e}}_{\mathrm{x}}$, frequency $\mathrm{\Omega }=4t_h$, and amplitude $A_0=0.6$. The bimagnon and Stokes/anti-Stokes charge excitations are marked on the right to guide the eye. Energy zero (defined as the equilibrium ground-state energy) is denoted by the dashed white line. The oscillatory Gaussian pump is drawn as a solid white curve.

Figure 4

Time-resolved $B_{1g}$ Raman spectra with ${\mathbf{e}}_{\mathrm{pol}}={\mathbf{e}}_{\mathrm{x}},\mathrm{\Omega }=6t_h$, and $A_0=0.3$ for (a) $U=8t_h$ and (b) $U=6t_h$. The plots are drawn in the same manner as in Fig. 3. The shaded curves to the left of each panel show the corresponding equilibrium spectra; the bimagnon peak is denoted by a darker color.

Figure 5

The $B_{1g}$ Raman spectra with pump amplitude $A_0$ varying from 0 (bottom) to 1.5 (top) at pump frequency (a) $\mathrm{\Omega }=2t_h$ and (b) $\mathrm{\Omega }=8t_h$. The time is fixed at $t=0$ (the center of the pump). (c) Bimagnon energies as a function of $A_0$ under high-frequency pumps $\left(\mathrm{\Omega }>U\right)$. The solid curves are predicted by the Floquet theory in the nonresonant limit. (d) Schematic cartoon showing specific spin exchange bonds altered by a pump pulse field.

Figure 6

Effective temperatures $T_{\mathrm{eff}}$ extracted from the postpump $\left(t=10t_h^{-1}\right)$ ratio of Stokes/anti-Stokes responses averaged over the energy range $4t_h<\omega <12t_h$ for the $B_{1g}$ Raman spectra. The colors denote different pump frequencies $\mathrm{\Omega }$. The inset shows mean values and standard deviations of $T_{\mathrm{eff}}$ for $\mathrm{\Omega }=2t_h,4t_h$, and $6t_h$.

Figure 7

(a) Time-resolved $A_{1g}$ Raman spectra with ${\mathbf{e}}_{\mathrm{pol}}={\mathbf{e}}_{\mathrm{x}},\mathrm{\Omega }=4t_h$, and $A_0=0.6$. The pump profile is identical to that in Fig. 3. The plot is drawn in the same manner as in Fig. 3. (b) and (c) Comparison of effective temperatures $T_{\mathrm{eff}}$ extracted from $A_{1\mathrm{g}}$ and $B_{1\mathrm{g}}$ Raman spectra at a postpump time $t=10t_h^{-1}$ under different pump frequencies, using the ratio of Stokes/anti-Stokes responses averaged over the energy range $4t_h<\omega <12t_h$.

Figure 8

Time-resolved $B_{1g}$ Raman spectra with $\mathrm{\Omega }=4t_h$ and $A_0=0.6$ for the tilted polarization corresponding to the horizontal momentum-space direction. The plot is drawn in the same manner as in Fig. 3.

Figure 9

Time-resolved $B_{1g}$ Raman spectra with probe pulse width (a) ${\sigma }_{\mathrm{pr}}=4t_h$ and (b) ${\sigma }_{\mathrm{pr}}=6t_h$. The pump profile is identical to that in Fig. 3. The plots are drawn in the same manner as in Fig. 3.