Theory of bound-state coherences generated and probed by optical attosecond pulses

We investigate the excitation and probing of electronic coherences in atoms by a sequence of optical attosecond pulses. Wave packets representing the coherent superposition of bound states in atoms are generated by a strong optical attosecond pulse. Amplitudes and phases of induced coherences can be retrieved from quantum beats in the radiative emission signal induced by a time-delayed weaker optical attosecond probe pulse. Such an attosecond-pump attosecond-probe scenario promises access to the excitation amplitudes and the off-diagonal elements of the density matrix generated by strong-field multiphoton processes. We illustrate this attosecond quantum beat spectroscopy with simulations for atomic hydrogen.

Figure 1

(a) Temporal and (b) spectral profile of a typical optical attosecond (OAS) pulse (see Ref. [29]). (A Gaussian window function with width of 10 fs has been applied to remove pre- and postpulse tails).

Figure 2

(a) Quantum-beat spectroscopy and interfering pathways in hydrogen, schematically. The three-photon spectrum of the pump pulse $F^{\left(3\right)}\left(\omega \right)=\int d{\omega }_1\int d{\omega }_{2}F\left({\omega }_1\right)F\left({\omega }_2\right)F(\omega -{\omega }_1-{\omega }_2)$ is shown on the left. At $t=0$, impulsive excitation forms a coherent superposition of excited states $|k\rangle$ ($k=e,i,j,...$). A second OAS pulse at $t=\tau$ transfers the coherent population to the state $|e\rangle$. The time-resolved VUV signal from the radiative decay of $|e\rangle$ to the ground state $|0\rangle$, $S(t,\tau )$ [see Eq. (14)], features in addition to the exponential decay $\propto e^{-\mathrm{\Gamma }t}$ quantum beats due to interfering pathways reaching the state $|e\rangle$, thereby providing information on impulsively induced coherences. The dynamical quantum beat phase is determined by the green shaded area, $|E_i-E_j|\tau$, in the energy-time diagram enclosed by different pathways. (b) The beat signal of the photon emission at energy of $|E_e-E_0|$ (left) and its frequency spectrum (right).

Figure 3

Photon emission induced by the pump OAS pulse with peak intensity $7\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$: (a) spontaneous emission $S_s\left(\omega \right)$, (b) strong-field-induced signal $S_i\left(\omega \right)$ (note the different scales).

Figure 4

The population of several excited states as a function of the intensity of the pump OAS pulse for (a) $n=2$, (b) $n=3$.

Figure 5

The population of several excited states after the conclusion of the probe pulse as functions of the time delay $\tau$. The peak intensity of the pump pulse is $I_{\text{p}}=7\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$, and the peak intensity of the probe pulse is $I_{\text{pr}}=2\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$; (a) $n=2$, (b) $n=3$.

Figure 6

The VUV emission spectrum normalized to the maximum as function of time delay $\tau$ between the OAS pump pulse ($I_{\text{p}}=7\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$) and OAS probe pulse ($I_{\text{p}}=2\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$). The emission lines back to the ground state are labeled by the emitting states from which the transition occurs. The spectrum is completely dominated by spontaneous emission. The induced component would appear only when a logarithmic rather than a linear color scale would be used.

Figure 7

Fourier quantum beat map normalized to the maximum as a function of the beat frequency in units of energy (eV) calculated by a Fourier transform of the time-delay resolved VUV emission spectrum (Fig. 6). Similar to Fig. 6, the horizontal emission lines back to the ground state are labeled by the emitting states ($2p,3p$) from which the transition occurs. Intersections of tilted lines A ($\omega ={\omega }_b$), B ($\omega ={\omega }_b+E_{n=2}-E_{1s}$), and C ($\omega ={\omega }_b+E_{n=3}-E_{1s}$) with the horizonal emission lines mark the contribution of different interfering pairs of paths (Fig. 2): pairs involving the emitting state and the ground state lie on line A, those with the emitting state and an $n=2$ state lie on line B, those with the emitting state and an $n=3$ state line on line C. Note that intermediate $nl$ states can have any angular momentum. The vertical line D (${\omega }_b=E_{n=2}-E_{1s}$) marks the position of the interference contribution from the fixed path pair ($1s,2p$) of intermediate states to different emission lines with which the line intersects.

Figure 8

(a) Quantum beat spectrum of the $2p$ emission line (10.20 eV) for three different intensities of the pump OAS pulse ($I_{\text{p}}=4\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$, $5\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$, and $7\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$) and fixed probe intensity $I_{\text{pr}}=2\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ with a resolution $\sim 1/\tau$ of $\mathrm{\Delta }E=0.1\mathrm{eV}$, and (b) for different probe intensities ($I_{\text{pr}}=2\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ and $2\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$) at a fixed pump intensity $I_{\text{p}}=5\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$. The quantum numbers of the coherently excited state pairs $n\text{−}{n}^{\prime }$ contributing to the most prominent peaks are labeled.

Figure 9

The same as Fig. 8 but for the higher beat frequencies. The green labels denote the coherently excited state pairs (labeled by quantum numbers $n\text{−}{n}^{\prime }$) contributing to the peaks.

Figure 10

(a) Same as Fig. 8 and (b) same as Fig. 9 but including decoherence due to intensity average over the focal volume or due to the phase fluctuations. The peak intensities of the pump pulse and the probe pulse are $I_{\text{p}}=7\times {10}^{13}$ and $I_{\text{pr}}=2\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$, respectively. The results without ensemble average (Figs. 8 and 9) are shown as reference. Phase fluctuations are simulated by assuming uniformly distributed random time delays between pump and probe corresponding to phase shifts $[-\pi /10,\pi /10]$ or $[-\pi ,\pi ]$. Intensity focal averaging over the pump field is performed for seven intensities from $I_{\text{p}}=1\times {10}^{13}$ to $I_{\text{p}}=7\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ with the weights given in Refs. [44, 45, 46]. The green labels denote the coherently excited state pairs (labeled by $n$ quantum numbers) contributing to the peaks.

Figure 11

The beat phase $\theta$ [Eq. (24)] as a function of the intensity of the pump pulse $I_{\text{p}}$ for beat oscillations in the emission signal at beating frequencies corresponding to excitation of the path pairs (a) $\left[2p\text{−}3\right(s,d\left)\right]$ and (b) $\left[2p\text{−}4\right(s,p,d\left)\right]$. Results for two intensities of the probe pulse $I_{\text{pr}}=2\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ and $2\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$ are shown. The perturbative lowest-order one- and two-photon transition phases are indicated by green circles near $I_{\text{p}}=0$ in each panel.