Ultrafast Mapping of Coherent Dynamics and Density Matrix Reconstruction in a Terahertz-Assisted Laser Field

A time-resolved spectroscopic protocol exploiting terahertz-assisted photoionization is proposed to reconstruct transient density matrix. Population and coherence elements are effectively mapped onto spectrally separated peaks in photoionization spectra. The beatings of coherence dynamics can be temporally resolved beyond the pulse duration, and the relative phase between involved states is directly readable from the oscillatory spectral distribution. As demonstrated by a photoexcited multilevel open quantum system, the method shows potential applications for subfemtosecond time-resolved measurements of coherent dynamics with free electron lasers and tabletop laser fields.

Figure 1

Schematic of the THz streaking-assisted photoionization experiment that allows for the real-time density matrix imaging. (a) The probing pulse configuration in the measurement. To observe the unknown quantum dynamics of an open quantum system within the temporal window (gray box), a femtosecond XUV pulse (purple) with a well-synchronized THz streaking pulse (blue) as a probe beam scans over time delay $\tau$. The XUV pulse is locked at the zero crossing of the THz vector potential. (b) The energy levels and the time evolution of the density matrix elements ${\rho }_{ij}$ within the time window. (c) Photoelectron momentum distribution without THz field. Populations ${\rho }_{ii}$ are mapped into spectral peaks of different longitudinal momentum $p$, as indicated by colors of solid lines. The $\tau$-dependent spectral yields, labeled by red, yellow, green, and cyan lines for ${\rho }_{gg}$, ${\rho }_{e_1{e}_1}$, ${\rho }_{e_2{e}_2}$, and ${\rho }_{e_3{e}_3}$, respectively, reveal the evolution of populations. (d) Photoelectron momentum distribution with THz streaking. Spectral peaks are broadened and the emergent oscillating fringes, labeled along blue and purple dashed lines, can be shown to account for coherences ${\rho }_{e_1{e}_2}$ and ${\rho }_{e_2{e}_3}$, respectively. The fringe can be well resolved, though the duration of the probing XUV pulse is much longer than its period of oscillation. The effective separation of ${\rho }_{ij}$ along $p$ and the temporally resolved coherence dynamics allow one to reconstruct the time evolution of ${\rho }_{ij}$.

Figure 2

The temporally resolved quantum coherence with streaking-assisted photoionization. In a two-level system, we assume the amplitudes of the two states are constant with initial relative phase ${\phi }_{eg}(t_0)={\phi }_e-{\phi }_g=0$ at $t_0=0\mathrm{fs}$. Panels (a) and (b) show $w(p;\tau )$ without and with THz-streaking, respectively. Panel (c) shows both the coherence $\mathrm{Re}[{\rho }_{eg}(\tau )]$ (blue) and the temporal profile of the simulated spectral signal $w(p=p_{eg};\tau )$ (black dotted line), as extracted along the white dashed line in (b). Panel (d) presents $w(p;\tau =t_0)$ as extracted along the black dashed line in (b). The extracted result (yellow) and the sum over model-based Gaussian peaks (blue), including the contributions from $W_{\mathrm{pop}}$ in Eq. (1) (red) and $W_{\mathrm{coh}}$ in Eq. (2) (green), are compared. The relative phase ${\phi }_{eg}(t_0)$ can be reconstructed with a single-$\tau$ measurement at time $\tau =t_0$ simply by examining $w(p;t_0)$ along the $p$ axis. When ${\phi }_{eg}(t_0)=0$, the coherence (green) at $p=p_{eg}$ is maximized as $\cos [{\phi }_{eg}(t_0)]=1$. By comparison, assuming ${\phi }_{eg}(t_0)=\pi /2$, the distribution is shown in (e), where the coherence at $p=p_{eg}$ becomes 0.

Figure 3

Reconstruction of ${\rho }_{ij}(\tau )$ in a multilevel open system. The four-level system under the excitation of two-color fields is shown in (a) with energies $(E_g,E_{e_1},E_{e_2},E_{e_3})=(-0.5,-0.29,-0.18,-0.06)\mathrm{a}.\mathrm{u}.$. The initial coherences at time $t_0$ are generated by, e.g., a two-color pulse sequence, depicted by the gray and orange arrows. Relaxation processes are assumed to occur within the excited manifold, where populations in $|e_2⟩$ and $|e_3⟩$ may jump randomly to the lowest excited state $|e_1⟩$ with constant rates, as shown by the wiggly arrows. Panels (b1) and (b2) show momentum distributions without and with streaking-THz field, respectively. The peaks of characteristic momenta $p_{ij}$, along which the results are extracted for ${\rho }_{ij}$-reconstruction, are indicated by dashed lines of the same color code as used for energy levels in (a). In (c), the THz-streaking signals extracted from (b2) (dotted lines with the same color code) are compared with the true ${\rho }_{ij}$ (solid lines). Panel (d) shows the ${\rho }_{ij}(\tau )$ reconstructed with the protocol described in the main text.