Open Access
Add to BibSonomy
Abstract
We theoretically investigate the formation of the high-order fractional alignment echo in OCS molecule and systematically study the dependence of echo intensity on the intensities and time delay of the two excitation pulses. Our simulations reveal an intricate dependence of the intensity of high-order fractional alignment echo on the laser conditions. Based on the analysis with rotational density matrix, this intricate dependence is further demonstrated to arise from the interference of multiple quantum pathways that involve multilevel rotational transitions. Our result provides a comprehensive multilevel picture of the quantum dynamics of high-order fractional alignment echo in molecular ensembles, which will facilitate the development of “rotational echo spectroscopy.”
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Echo is a common phenomenon that occurs in a nonlinear system excited by a pair of delayed perturbation. Since E. L. Hahn first reported “spin echo” in a nuclear spin system in 1950 [1], echoes in various domains of physics have been discovered, such as cyclotron echoes [2,3], photon echoes [4], plasma-wave echoes [5], cold atom echoes in optical traps [6,7], echoes in cavity QED [8] and hadron colliders [9,10]. The application of echo phenomenon has been realized in magnetic resonance imaging (MRI) [11] and a wide range of imaging and spectroscopic applications, including 2D electronic [12–14], vibrational [15–18], and rotational spectroscopy [19], and also the generation of high-order harmonics in free-electron lasers [20–22].
In 2015, a new type of echo phenomenon, i.e., “molecular alignment echo”, was reported [23]. Excited by two laser pulses delayed by $\Delta \tau$, molecular alignment will appear at the time delays of $\Delta t= N\Delta \tau$ ($N$ is a positive integer) after the second excitation pulse. This phenomenon is called “molecular alignment echo”. Karras et al. observed alignment echo in CO$_2$ gas by measuring the laser-induced birefringence signal [23]. Later on, the alignment echo in iodomethane (CH$_3$I) [24] and carbonyl sulfide (OCS) [25] gas have also been reported. Unlike the conventional molecular rotational revivals, the arising time of alignment echo depends on the time delay between the two excitation pulses. Such time can be much shorter than the period of molecular rotational revivals, making alignment echo a powerful tool for studying the ultrafast rotational relaxation dynamics in dense gas medium and, potentially, in liquids [24–27].
In addition to the alignment echoes that occur at $\Delta t= N\Delta \tau$, molecular alignment can also recur at times that are rational fractions of the time delay between two excitation pulses [23,28]. These alignment events are called “(high-order) fractional alignment echoes”. For example, the alignment echoes occurring at $\Delta t= \Delta \tau /2$ and $\Delta t= \Delta \tau /3$ are referred to as “1/2 alignment echoes” and “1/3 alignment echoes”, respectively. While those occurring at $\Delta t= N\Delta \tau$ are referred to as “full alignment echoes”. Compared to full alignment echo, high-order fractional alignment echoes (HFAEs) dephase more rapidly, which is beneficial for very short-time pump-probe measurements in molecular gases. However, observing the HFAEs requires the measurement of higher-order moments of the molecular angular distribution [23,28], e.g. by means of high-order harmonic generation (HHG), which is more difficult than the full alignment echo. Hence, previous studies on the alignment echoes mainly focus on the full alignment echo. Until very recently, Coulomb explosion imaging (CEI) [29–33] method was reported enabling direct access to the evolution of molecular rotational wavepacket of the HFAEs in time and space. Besides, high-order harmonic generation (HHG), a highly non-linear phenomenon that occurs in laser-driven recollision process [34–37], intrinsically convolutes the information of structure and dynamics of the target atoms [38–44], molecules [45–54] and solids [55–59], which also provides an efficient way for probing the HFAEs [60]. For a better application of HFAEs in the future, further exploration of the features of HFAEs is needed.
In this work, we theoretically investigate the formation of the HFAEs in OCS molecule and study the dependence of the HFAE intensity on the intensities and time delay of the two excitation pulses. We find that the HFAE intensity has an intricate dependence on the intensity of the second excitation pulse and the time delay between the two excitation pulses. For a given time delay, there is an optimal intensity of the second excitation pulse that maximizes the alignment echo intensity. Moreover, the maximal alignment echo intensity is proportional to the intensity of the first excitation pulse and independent of the time delay between the two excitation pulses. Further analysis based on the rotational density matrix (RDM) suggests that the formation of the HFAEs mainly arises from the interference of multiple quantum pathways in a multilevel rotational system.
2. Numerical theory
In this work, we perform the simulations with OCS molecule, since it has a long revival period (T$_{rev}\sim 82 ps$) without quarter revivals, which can provide a broad time window for the study of the echo phenomena. To simulate the responses of the alignment echoes, we perform the theoretical simulations by solving the time-dependent Schrödinger equation (TDSE) of the molecular rotational wave packet [61–67]. For a given initial rotational state $|{ J_0,M_0}\rangle$ (the eigenstate of the field-free rigid rotor described by the spherical harmonic), the TDSE reads
Here, J is the angular momentum operator, B is the rotational constant, $\alpha _{\parallel }$ and $\alpha _{\perp }$ are the polarizabilities in parallel and perpendicular directions with respect to the molecular axis. For the OCS molecule, B$=$ 0.203 cm$^{-1}$, $\alpha _{\parallel }=4.058$ Å$^{3}$, $\alpha _{\perp }$=0.490 Å$^{3}$. $E(t)=E_0exp(-\frac {2ln2}{\sigma ^{2}}t^{2})$ is the envelope of the laser field with the duration $\sigma =$ 50 fs. The two excitation pulses are both linearly polarized along the Z-axis and propagate along the X-axis. $\theta$ is the polar angle between the molecular axis and Z-axis, $\phi$ is the azimuth angle of the molecular axis with respect to the X-axis. To model the echo responses, we calculate the time-dependent mean value of $cos (2n\theta )$, which is given by
Figures 1(a)-(c) show the results of $\langle cos(2\theta )\rangle (t)$, $\langle cos(4\theta )\rangle (t)$, and $\langle cos(6\theta )\rangle (t)$ calculated with the OCS molecule excited by two pump pulses (referred as P$_1$ and P$_2$) delayed by 10 ps. As shown in Fig. 1(a), one can see a clear modulation at $\Delta t=10$ ps after pump P$_2$ in the curve of $\langle cos(2\theta )\rangle (t)$, which is a signature of the full alignment echo response. While for HFAEs, it requires measuring higher order observables, i.e., $\langle cos (2n\theta )\rangle (t)$ with n$>$1. As shown in Figs. 1(b) and 1(c), similar modulations are found at $\Delta t=\Delta \tau /2$ and $\Delta \tau /3$ in the curves of $\langle cos(4\theta )\rangle (t)$ and $\langle cos(6\theta )\rangle (t)$, which just correspond to the 1/2 and 1/3 alignment echoes, respectively. In what follows, we use the peak-to-peak difference of the modulations (modulation depth) in the observables $\langle cos (2n\theta )\rangle (t)$ to quantify the intensities of the corresponding echo signals.
Fig. 1. The time-dependent mean value of (a) $\langle cos(2\theta )\rangle (t)$, (b) $\langle cos(4\theta )\rangle (t)$ and (c) $\langle cos(6\theta )\rangle (t)$ calculated with the OCS molecule excited by two pump pulses. Here, $\Delta t$ is the time elapsed after the second pulse, the rotational temperature is 100 K, the time delay between two pump pulses is 10 ps, the intensities of P$_1$ and P$_2$ are 4$\times$10$^{13}$W/cm$^{2}$ and 0.8$\times$10$^{13}$W/cm$^{2}$, respectively.
3. Results and discussion
We first investigate the dependence of the HFAE intensities on the intensity (I$_2$) of the second pump pulse P$_2$. Figure 2 depicts simulated results of the full (blue circle), 1/2 (green diamond) and 1/3 (red cross) alignment echoes for three different time delays of $\Delta \tau =0.05$ T$_{rev}$ [Fig. 2(a)], $0.075$ T$_{rev}$ [Fig. 2(b)], $0.125$ T$_{rev}$ [Fig. 2(c)]. Here, the intensity (I$_1$) of the first pump pulse P$_1$ is $4\times 10^{13}$W/cm$^{2}$, the rotational temperature is 100 K. As shown in Fig. 2(a), the intensity of the full alignment echo oscillates as $I_2$ increases. The oscillatory signal is proportional to sin$^{2}$(a$\cdot$I$_2$), where a is free parameter. This result is consistent with previous works [25]. Similar oscillatory dependence was also found for the HFAEs, e.g., 1/2 and 1/3 alignment echoes, but the oscillation behaves as sin$^{3}$(b$\cdot$I$_2$) and sin$^{4}$(c$\cdot$I$_2$) (b and c are free parameters), respectively. Hereafter, we will demonstrate that the exponential factor $n$ of the sin$^{n}$-oscillation is related to the number of Raman excitations with pump P$_2$ during the formation of alignment echoes. Besides, one can see that, for each echo, there is an optimal I$_2$ that maximizes the alignment echo intensity. The optimal I$_2$ varies with time delay $\Delta \tau$, and is higher for higher-order fractional alignment echo. More important, the maximal echo intensity $S_{max}$ is independent of the delay $\Delta \tau$ between the two pump pulses. This feature makes the HFAEs a potential tool for investigating the ultrafast collisional dynamics of molecules [26]. Note that, in Fig. 2, we have restricted our simulation by low intensity of P$_2$, further enhancement of the P$_2$ will result in the phase-inverted echo responses [25], which is beyond the scope of this work. In Fig. 3, we have also simulated the echo intensities as a function of the time delay $\Delta \tau$. The intensities of the full alignment echo and HFAEs are found to depend parabolically on $\Delta \tau$ and have a peak at $\Delta \tau =$T$_{rev}/8$. Note that the perturbation around $\Delta \tau =13.67$ ps in Fig. 3(a) is attributed to the temporal overlap of the full alignment echo and the first imaginary echo [26,30]. The imaginary echoes are the alignment events occurring at $\Delta \tau$ before the quantum revivals of P$_1$. For the OCS molecule, the first imaginary echo occurs at $t=T_{rev}/2-\Delta \tau$ after P$_1$.
Fig. 2. I$_2$-dependence of the alignment echoes for (a) $\Delta \tau$=0.05 T$_{rev}$, (b) 0.075 T$_{rev}$ and (c) 0.125 T$_{rev}$. Here, the rotational temperature is 100 K, the intensity of the first pump pulse is $4\times 10^{13}$W/cm$^{2}$.
Fig. 3. (a) TDSE simulations (blue circles) of $\Delta \tau$-dependence of full alignment echo intensity with I$_2$=0.6$\times$10$^{13}$W/cm$^{2}$. The red curve is the echo intensity calculated with the transition pathways described in Fig. 5(a). (b), (c) Same as (a), but for the 1/2 alignment echoes with I$_2$=0.8$\times$10$^{13}$W/cm$^{2}$ and 1/3 alignment echoes with I$_2$=1.0$\times$10$^{13}$W/cm$^{2}$, respectively. Here, the rotational temperature is 100 K, the intensity of the first pump pulse is $4\times 10^{13}$W/cm$^{2}$.
The dependence of the HFAE intensities on I$_2$ and $\Delta \tau$ is difficult to understand by commonly-used two-level model where the echo response is independent of $\Delta \tau$ [25]. In the following, we demonstrate that the intricate dependence of the HFAE intensities on I$_2$ and $\Delta \tau$ mainly arises from the interference among multiple quantum pathways that involve multilevel transitions. Considering a given initial state $|J_0,M_0\rangle$, the contribution to the observables is given by $\langle cos(2n\theta )\rangle _{J_0,M_0}(t)=\langle \Psi _{J_0,M_0}(\theta ,\phi ,t) | cos (2n\theta ) | \Psi _{J_0,M_0}(\theta ,\phi ,t) \rangle$. Expanding the time-dependent wave function $|\Psi _{J_0,M_0}(\theta ,\phi ,t)\rangle$ in the basis of the eigenstates of the field-free rigid rotor, i.e., $| \Psi _{J_0,M_0}(\theta ,\phi ,t) \rangle =\sum _{J,M}C_{J_0,M_0,J,M}|J,M\rangle e^{-iE_Jt/ \hbar }$, where $C_{J_0,M_0,J,M}$ is a complex coefficient and $E_J=BJ(J+1)$ is the eigenenergy of the rotational eigenstate $|J,M\rangle$, the contributed observable $\langle cos(2n\theta )\rangle _{J_0,M_0}(t)$ then reads
In our simulations, the two excitation pulses are both linearly polarized and parallel to each other, thus the magnetic quantum number M is conserved in the interactions. For convenience, we omit $M_0$ in Eq. (3) and restrict the discussion to the angular momentum quantum number J in the following.
According to Eq. (3), the observable $\langle cos(2n\theta )\rangle _{J_0}(t)$ is a summation of the terms with non-zero matrix elements $\Delta J=0$ and $\Delta J$ up to $2q$ $(0< q\leqslant n, q\in N)$. In Fig. 4, we have calculated the contributions of the terms in Eq. (3) to the observable $\langle cos(2n\theta )\rangle _{J_0}(t)$. For convenience, we use the abbreviations $|J\rangle \langle J|$ and $|J\rangle \langle J+2q|$ to represent the terms $C_{J_0,J}^{2}\langle J | cos (2n\theta ) | J \rangle$ and $C_{J_0,J+2q}^{\ast } C_{J_0,J}\langle J+2q | cos(2n\theta ) | J \rangle e^{-i(E_J-E_{J+2q})t/ \hbar }+c.c.$. For the full alignment echo, the observable $\langle cos(2\theta )\rangle _{J_0}(t)$ involves the terms with $\Delta J=0$ and $\Delta J= 2$. The transient alignment echo signal is only related to the terms with $\Delta J= 2$ [see Fig. 4(a)]. While for the HFAEs, e.g., 1/2 [$\langle cos(4\theta )\rangle _{J_0}(t)$] and 1/3 [$\langle cos(6\theta )\rangle _{J_0}(t)$] alignment echoes, they involve much higher rotational transitions with $\Delta J$ up to 4 and 6. Moreover, the terms with $\Delta J= 4$ and $\Delta J= 6$ play the leading roles in the formation of 1/2 [Fig. 4(b)] and 1/3 [Fig. 4(c)] alignment echoes, respectively.
Fig. 4. The contributions of the terms in Eq. (3) to the formation of the (a) full, (b) 1/2 and (c) 1/3 alignment echoes. Here, the rotational temperature is 100 K, $\Delta \tau$=T$_{rev}$/8, the intensities of P$_1$ and P$_2$ are 4$\times$10$^{13}$W/cm$^{2}$ and 0.8$\times$10$^{13}$W/cm$^{2}$, respectively.
For a deeper insight into the formation of HFAEs, we next analyze the dynamics of HFAEs with the RDM introduced in [25]. Before the first pump pulse P$_1$ is applied, the molecular system is described by a set of rotational eigenstates $|J\rangle$. The observable $\langle cos(2n\theta )\rangle (t)$ is a summation of terms $|J\rangle \langle J|$ (i.e., the dots along the diagonal of the RDMs in Fig. 5). In the following, we consider the rotational transitions starting from the same initial rotational eigenstate $|J\rangle$. For the full alignment echo [Fig. 5(a)], the initial population term $|J\rangle \langle J|$ can be excited to the final rephasing term $|J\rangle \langle J+2|$ through two pathways (blue dashed and green solid arrows) by one interaction with P$_1$ and two interactions with P$_2$. For the pathway$-1$ (blue dashed arrows), the initial state $|J\rangle$ is partly excited to $|J+2\rangle$ via one Raman excitation. Before P$_2$ is applied, the excited $|J+2\rangle$ and the remaining $|J\rangle$ evolve freely and accumulate the phase factors $e^{-iE_{J+2} \Delta \tau /\hbar }$ and $e^{-iE_{J} \Delta \tau /\hbar }$, respectively. In the interaction with P$_2$, the state $|J+2\rangle$ can be excited to $|J\rangle$ by one Raman excitation, meanwhile the remaining $|J\rangle$ can be excited to $|J+2\rangle$ by one Raman excitation. The new created $|J\rangle$ and $|J+2\rangle$ lead to the final rephasing term $|J\rangle \langle J+2|$ for the observable $\langle cos(2\theta )\rangle _J(t)$. The contribution of this pathway to the observable $\langle cos(2\theta )\rangle _J(t)$ can be written as
Fig. 5. Pictorial RDM representation of formations of (a) full, (b) 1/2 and (c) 1/3 alignment echoes. Here, we consider the transition pathways starting from the same initial rotational eigenstate $|J\rangle$ and interfere at the final rephasing term (red dot). The blue dashed arrows and green solid arrows represent pathway-1 and pathway-2, respectively.
In the pathway$-2$ (green solid arrows), P$_1$ excites the initial state $|J\rangle$ to $|J-2\rangle$ by one Raman excitation. Then the excited $|J-2\rangle$ and the remaining $|J\rangle$ accumulate the phase factors $e^{-iE_{J-2} \Delta \tau /\hbar }$ and $e^{-iE_J \Delta \tau /\hbar }$ before P$_2$ is applied. When P$_2$ is applied, the state $|J-2\rangle$ can be excited to $|J+2\rangle$ by two Raman excitations. The remaining $|J\rangle$ and the excited $|J+2\rangle$ lead to the final rephasing term $|J\rangle \langle J+2|$. The corresponding contribution to the observable $\langle cos(2\theta )\rangle _J(t)$ is given by
In our above analysis, only transitions with $\Delta$M= 0 and $\Delta$J=$\pm$2 are allowed. The transition possibilities of the pathway$-$1 and pathway$-$2 can be defined as $|C_{J,J+2}^{(P_1)}C_{J+2,J}^{(P_2)}C^{\ast (P_1)}_{J,J}C^{\ast (P_2)}_{J,J+2}|$ and $|C_{J,J}^{(P_1)}C_{J,J}^{(P_2)}C^{\ast (P_1)}_{J,J-2}C^{\ast (P_2)}_{J-2,J+2}|$, respectively. In Fig. 3(a), we have calculated the intensity of the full alignment echo with these two pathways [i.e., $\langle cos(2\theta )\rangle _{path-1}(t)$+$\langle cos(2\theta )\rangle _{path-2}(t)$]. The intensity of the alignment echo depends on the phase difference of these two pathways, which is given by $\Delta \phi =\frac {(E_{J+2}-E_{J})-(E_{J}-E_{J-2})}{\hbar }\Delta \tau =8\pi \Delta \tau /$T$_{rev}$. Thus, for $\Delta \tau$=T$_{rev}/8$, these two transition pathways interfere constructively, resulting in the maximal rephasing coherence and the largest alignment echo intensity. As $\Delta \tau$ deviates from T$_{rev}/8$, the vanishing constructive interference will lead to the decrease of the alignment echo intensity. These results (red curve) agree well with the TDSE simulations (blue circles). For the HFAEs, e.g., 1/2 and 1/3 alignment echoes in Figs. 5(b) and 5(c), P$_1$ can also excite the initial state $|J\rangle$ to $|J+2\rangle$ or $|J-2\rangle$ via one Raman excitation. While the interactions with P$_2$ lead to the final rephasing terms $|J\rangle \langle J+4|$ and $|J\rangle \langle J+6|$ for the observables $\langle cos(4\theta )\rangle _J(t)$ and $\langle cos(6\theta )\rangle _J (t)$ via three and four Raman excitations, respectively. Although the final rephasing terms of the transition pathways are different for the HFAEs, the phase difference of the related pathways is also $\Delta \phi =\frac {(E_{J+2}-E_{J})-(E_{J}-E_{J-2})}{\hbar }\Delta \tau =8\pi \Delta \tau /$T$_{rev}$. Thus, the intensities of the HFAEs are also peaked at $\Delta \tau =$T$_{rev}/8$ due to the constructive interference of the transition pathways. In Figs. 3(b) and 3(c), we have also calculated the intensities of 1/2 and 1/3 alignment echoes with the above mentioned transition pathways as a function of $\Delta \tau$. The results (red curves) also agree well with the TDSE simulations (blue circles). It’s worth mentioning that, in our above analysis we only consider the rotational transitions starting from the same initial rotational eigenstate with one interaction with P$_1$ and lowest number of interactions with P$_2$. In fact, multiple interactions with P$_1$, and a single interaction with P$_2$ can also contribute to the alignment echoes. But the contribution is smaller than the former. Besides, the formation of the alignment echoes also involves transition pathways starting from the adjacent initial rotational eigenstates, which is not considered in our above discussion but is included in the TDSE calculations.
To understand the I$_2$-dependence of the HFAE intensities, we have calculated the transition probabilities of the transition pathways in Fig. 5. As shown in Fig. 6, the transition probabilities of the pathway$-$1 and pathway$-$2 in Fig. 5(a) are both sin$^{2}$-dependent on I$_2$, which agree with the TDSE simulations in Fig. 2. Figures 6(c)-(d) and 6(e)-(f) show the results for 1/2 and 1/3 alignment echoes. The transition probabilities of the related transition pathways exhibit sin$^{3}$- and sin$^{4}$-dependence on I$_2$, respectively, which are also consistent with the TDSE simulations in Fig. 2. From the results in Fig. 6, it can be concluded that the exponential factor $n$ of sin$^{n}$-dependence of the HFAE intensities on I$_2$ is associated with the number of the Raman excitations excited by P$_2$. Since the formation of the HFAEs involves higher order Raman excitations, it requires higher pump intensity to achieve the maximal transition probability and then the maximal echo intensity. In Fig. 7, we have calculated the alignment echo intensities (for both the full alignment echo and HFAEs) as a function of I$_2$ with the transition pathways in Fig. 5. The results (red curves) agree well with the TDSE simulations (blue circles).
Fig. 6. (a)-(b) The transition probabilities of the transition pathways contributing to the full alignment echo described in Fig. 5(a). (c)-(d), (e)-(f) Same as (a)-(b), but for the 1/2 and 1/3 alignment echoes, respectively. Here, the rotational temperature is 100 K, I$_1=4\times$10$^{13}$W/cm$^{2}$, $\Delta \tau =$ T$_{rev}/8$.
Fig. 7. (a) The intensity of the full alignment echo calculated with the transition pathways (red curve). The blue circles are the echo intensities calculated by solving the TDSE. (b), (c) Same as (a), but for the 1/2 and 1/3 alignment echoes, respectively. Here, the rotational temperature is 100 K, I$_1$=4$\times$10$^{13}$W/cm$^{2}$, $\Delta \tau$=T$_{rev}/8$.
Finally, we investigate the I$_1$-dependence of the HFAEs. Figure 8(a) depicts the simulated full alignment echo intensity as a function of I$_2$ for I$_1$= 1$\times$10$^{13}$W/cm$^{2}$ (red dotted curve), 3$\times$10$^{13}$W/cm$^{2}$ (blue dashed curve), 5$\times$10$^{13}$W/cm$^{2}$ (green solid curve), respectively. Here $\Delta \tau$ is fixed at T$_{rev}/8$. It can be seen that, increasing I$_1$ results in overall increase in the full alignment echo intensity. In Fig. 8(b), we simulated the maximal intensity of the full alignment echo as a function of I$_1$ (blue circle), the fitting results (blue line) show that the maximal echo intensity is proportional to I$_1$. Similar I$_1$-dependence is also found for the HFAEs, e.g., 1/2 and 1/3 alignment echoes in Figs. 8(c)-(d) and 8(e)-(f), respectively. The result will provide a guidance for improving the HFAE intensity in experiment.
Fig. 8. (a) The simulated intensity of the full alignment echo as a function of I$_2$ for I$_1$=1$\times$10$^{13}$W/cm$^{2}$, 3$\times 10^{13}$W/cm$^{2}$, 5$\times$10$^{13}$W/cm$^{2}$. Here, the rotational temperature is 100 K, $\Delta \tau =$T$_{rev}/8$. (b) The dependence of the maximal intensity of the full alignment echo on I$_1$. (c)-(d), (e)-(f) Same as (a)-(b), but for the 1/2 and 1/3 alignment echoes.
4. Conclusion
In summary, we have investigated the formation of the HFAEs in OCS molecule and studied the dependence of HFAE intensity on the intensities and delay between the two excitation pulses. The results show that the HFAE intensity has an intricate dependence on the intensity of the second excitation pulse and the time delay between the two excitation pulses. For a given time delay, there is an optimal intensity of the second excitation pulse that maximizes the echo intensity. The optimal intensity of the second excitation pulse is higher for higher-order fractional alignment echo. Moreover, the maximal echo intensity is demonstrated to increase linearly with the intensity of the first excitation pulse and to be independent of the time delay between the two excitation pulses. This feature makes the HFAEs a potential tool for investigating the ultrafast collisional dynamics of molecules. For example, in very dense gases, the full alignment echoes can be completely suppressed, while HFAEs can be still observed as they emerge earlier (e.g. at $\Delta t=\Delta \tau /2$ instead of $\Delta t=\Delta \tau$). Based on the analysis with RDM, we further demonstrated that the formation of HFAEs mainly arises from the interference of multiple quantum pathways in the multilevel rotational system. Our result provides a comprehensive multilevel quantum picture of the formation of the HFAEs in molecular ensembles, which will stimulate new applications of “rotational echo spectroscopy”.
Funding
National Natural Science Foundation of China (11627809, 11704137, 11774109, 11874165, 12021004, 12074136); National Key Research and Development Program of China (2017YFE0116600, 2019YFA0308300); Fundamental Research Funds for the Central Universities (2017KFXKJC002); the Program for HUST Academic Frontier Youth Team.
Disclosures
The authors declare no conflicts of interest.
References
1. E. L. Hahn, “Spin Echoes,” Phys. Rev. 80(4), 580–594 (1950). [CrossRef]
2. R. M. Hill and D. E. Kaplan, “Cyclotron Resonance Echo,” Phys. Rev. Lett. 14(26), 1062–1063 (1965). [CrossRef]
3. R. Gould, “Echo phenomena,” Phys. Lett. 19(6), 477–478 (1965). [CrossRef]
4. N. A. Kurnit, I. D. Abella, and S. R. Hartmann, “Observation of a Photon Echo,” Phys. Rev. Lett. 13(19), 567–568 (1964). [CrossRef]
5. R. W. Gould, T. M. O’Neil, and J. H. Malmberg, “Plasma Wave Echo,” Phys. Rev. Lett. 19(5), 219–222 (1967). [CrossRef]
6. A. Bulatov, A. Kuklov, B. E. Vugmeister, and H. Rabitz, “Echo in optical lattices: Stimulated revival of breathing oscillations,” Phys. Rev. A 57(5), 3788–3792 (1998). [CrossRef]
7. M. Herrera, T. M. Antonsen, E. Ott, and S. Fishman, “Echoes and revival echoes in systems of anharmonically confined atoms,” Phys. Rev. A 86(2), 023613 (2012). [CrossRef]
8. T. Meunier, S. Gleyzes, P. Maioli, A. Auffeves, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Rabi Oscillations Revival Induced by Time Reversal: A Test of Mesoscopic Quantum Coherence,” Phys. Rev. Lett. 94(1), 010401 (2005). [CrossRef]
9. G. V. Stupakov, “Echo effect in hadron colliders,” SSC Report No. SSCL-579 (1992), https://doi.org/10.2172/7237216.
10. L. K. Spentzouris, J.-F. Ostiguy, and P. L. Colestock, “Direct Measurement of Diffusion Rates in High Energy Synchrotrons Using Longitudinal Beam Echoes,” Phys. Rev. Lett. 76(4), 620–623 (1996). [CrossRef]
11. R. Damadian, “Tumor Detection by Nuclear Magnetic Resonance,” Science 171(3976), 1151–1153 (1971). [CrossRef]
12. R. Inaba, K. Tominaga, M. Tasumi, K. A. Nelson, and K. Yoshihara, “Observation of homogeneous vibrational dephasing in benzonitrile by ultrafast Raman echoes,” Chem. Phys. Lett. 211(2-3), 183–188 (1993). [CrossRef]
13. J. D. Hybl, A. W. Albrecht, S. M. Gallagher Faeder, and D. M. Jonas, “Two-dimensional electronic spectroscopy,” Chem. Phys. Lett. 297(3-4), 307–313 (1998). [CrossRef]
14. V. I. Prokhorenko, A. Halpin, and R. J. D. Miller, “Coherently-controlled two-dimensional photon echo electronic spectroscopy,” Opt. Express 17(12), 9764–9779 (2009). [CrossRef]
15. K. A. Merchant, D. E. Thompson, and M. D. Fayer, “Two-Dimensional Time-Frequency Ultrafast Infrared Vibrational Echo Spectroscopy,” Phys. Rev. Lett. 86(17), 3899–3902 (2001). [CrossRef]
16. S. Park, K. Kwak, and M. D. Fayer, “Ultrafast 2D-IR vibrational echo spectroscopy: a probe of molecular dynamics,” Laser Phys. Lett. 4(10), 704–718 (2007). [CrossRef]
17. A. Shalit, S. Ahmed, J. Savolainen, and P. Hamm, “Terahertz echoes reveal the inhomogeneity of aqueous salt solutions,” Nat. Chem. 9(3), 273–278 (2017). [CrossRef]
18. J. Qiang, I. Tutunnikov, P. Lu, K. Lin, W. Zhang, F. Sun, Y. Silberberg, Y. Prior, I. S. Averbukh, and J. Wu, “Echo in a single vibrationally excited molecule,” Nat. Phys. 16(3), 328–333 (2020). [CrossRef]
19. J. Lu, Y. Zhang, H. Hwang, B. Ofori-Okai, S. Fleischer, and K. Nelson, “Nonlinear two-dimensional terahertz photon echo and rotational spectroscopy in the gas phase,” Proc. Natl. Acad. Sci. 113(42), 11800–11805 (2016). [CrossRef]
20. G. Stupakov, “Using the Beam-Echo Effect for Generation of Short-Wavelength Radiation,” Phys. Rev. Lett. 102(7), 074801 (2009). [CrossRef]
21. D. Xiang, E. Colby, M. Dunning, S. Gilevich, C. Hast, K. Jobe, D. McCormick, J. Nelson, T. O. Raubenheimer, K. Soong, G. Stupakov, Z. Szalata, D. Walz, S. Weathersby, M. Woodley, and P.-L. Pernet, “Demonstration of the Echo-Enabled Harmonic Generation Technique for Short-Wavelength Seeded Free Electron Lasers,” Phys. Rev. Lett. 105(11), 114801 (2010). [CrossRef]
22. Z. T. Zhao, D. Wang, J. H. Chen, Z. H. Chen, H. X. Deng, J. G. Ding, C. Feng, Q. Gu, M. M. Huang, T. H. Lan, Y. B. Leng, D. G. Li, G. Q. Lin, B. Liu, E. Prat, X. T. Wang, Z. S. Wang, K. R. Ye, L. Y. Yu, H. O. Zhang, J. Q. Zhang, M. Zhang, M. Zhang, T. Zhang, S. P. Zhong, and Q. G. Zhou, “First lasing of an echo-enabled harmonic generation free-electron laser,” Nat. Photonics 6(6), 360–363 (2012). [CrossRef]
23. G. Karras, E. Hertz, F. Billard, B. Lavorel, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. S. Averbukh, “Orientation and Alignment Echoes,” Phys. Rev. Lett. 114(15), 153601 (2015). [CrossRef]
24. D. Rosenberg, R. Damari, S. Kallush, and S. Fleischer, “Rotational Echoes: Rephasing of Centrifugal Distortion in Laser-Induced Molecular Alignment,” J. Phys. Chem. Lett. 8(20), 5128–5135 (2017). [CrossRef]
25. D. Rosenberg, R. Damari, and S. Fleischer, “Echo Spectroscopy in Multilevel Quantum-Mechanical Rotors,” Phys. Rev. Lett. 121(23), 234101 (2018). [CrossRef]
26. H. Zhang, B. Lavorel, F. Billard, J.-M. Hartmann, E. Hertz, O. Faucher, J. Ma, J. Wu, E. Gershnabel, Y. Prior, and I. S. Averbukh, “Rotational Echoes as a Tool for Investigating Ultrafast Collisional Dynamics of Molecules,” Phys. Rev. Lett. 122(19), 193401 (2019). [CrossRef]
27. J. Ma, H. Zhang, B. Lavorel, F. Billard, E. Hertz, J. Wu, C. Boulet, J.-M. Hartmann, and O. Faucher, “Observing collisions beyond the secular approximation limit,” Nat. Commun. 10(1), 5780 (2019). [CrossRef]
28. G. Karras, E. Hertz, F. Billard, B. Lavorel, G. Siour, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. S. Averbukh, “Experimental observation of fractional echoes,” Phys. Rev. A 94(3), 033404 (2016). [CrossRef]
29. K. Lin, Q. Song, X. Gong, Q. Ji, H. Pan, J. Ding, H. Zeng, and J. Wu, “Visualizing molecular unidirectional rotation,” Phys. Rev. A 92(1), 013410 (2015). [CrossRef]
30. K. Lin, P. Lu, J. Ma, X. Gong, Q. Song, Q. Ji, W. Zhang, H. Zeng, J. Wu, G. Karras, G. Siour, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. S. Averbukh, “Echoes in Space and Time,” Phys. Rev. X 6(4), 041056 (2016). [CrossRef]
31. K. Lin, J. Ma, X. Gong, Q. Song, Q. Ji, W. Zhang, H. Li, P. Lu, H. Li, H. Zeng, J. Wu, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. S. Averbukh, “Rotated echoes of molecular alignment: fractional, high order and imaginary,” Opt. Express 25(21), 24917–24926 (2017). [CrossRef]
32. L. Xu, I. Tutunnikov, L. Zhou, K. Lin, J. Qiang, P. Lu, Y. Prior, I. S. Averbukh, and J. Wu, “Echoes in Unidirectionally Rotating Molecules,” arXiv:2008.06776 (2020).
33. K. Lin, I. Tutunnikov, J. Ma, J. Qiang, L. Zhou, O. Faucher, Y. Prior, I. S. Averbukh, and J. Wu, “Spatiotemporal rotational dynamics of laser-driven molecules,” Adv. Photonics 2(2), 024002 (2020). [CrossRef]
34. P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993). [CrossRef]
35. M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49(3), 2117–2132 (1994). [CrossRef]
36. A.-T. Le, R. R. Lucchese, S. Tonzani, T. Morishita, and C. D. Lin, “Quantitative rescattering theory for high-order harmonic generation from molecules,” Phys. Rev. A 80(1), 013401 (2009). [CrossRef]
37. C. D. Lin, A.-T. Le, Z. Chen, T. Morishita, and R. Lucchese, “Strong-field rescattering physics—self-imaging of a molecule by its own electrons,” J. Phys. B: At., Mol. Opt. Phys. 43(12), 122001 (2010). [CrossRef]
38. R. Kienberger, M. Hentschel, M. Uiberacker, C. Spielmann, M. Kitzler, A. Scrinzi, M. Wieland, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Steering Attosecond Electron Wave Packets with Light,” Science 297(5584), 1144–1148 (2002). [CrossRef]
39. Y. Pertot, C. Schmidt, M. Matthews, A. Chauvet, M. Huppert, V. Svoboda, A. von Conta, A. Tehlar, D. Baykusheva, J.-P. Wolf, and H. J. Wörner, “Time-resolved x-ray absorption spectroscopy with a water window high-harmonic source,” Science 355(6322), 264–267 (2017). [CrossRef]
40. R. Shao, C. Zhai, Y. Zhang, N. Sun, W. Cao, P. Lan, and P. Lu, “Generation of isolated circularly polarized attosecond pulses by three-color laser field mixing,” Opt. Express 28(11), 15874–15884 (2020). [CrossRef]
41. K. Liu, M. Li, W. Xie, K. Guo, S. Luo, J. Yan, Y. Zhou, and P. Lu, “Revealing the effect of atomic orbitals on the phase distribution of an ionizing electron wave packet with circularly polarized two-color laser fields,” Opt. Express 28(8), 12439–12449 (2020). [CrossRef]
42. C. Zhai, R. Shao, P. Lan, B. Wang, Y. Zhang, H. Yuan, S. M. Njoroge, L. He, and P. Lu, “Ellipticity control of high-order harmonic generation with nearly orthogonal two-color laser fields,” Phys. Rev. A 101(5), 053407 (2020). [CrossRef]
43. R. Wang, Q. Zhang, C. Ran, W. Cao, and P. Lu, “Proposal for detecting ring current via electron vortices,” Opt. Lett. 45(6), 1383–1386 (2020). [CrossRef]
44. Z. Yang, W. Cao, Y. Mo, H. Xu, K. Mi, P. Lan, Q. Zhang, and P. Lu, “All-optical attosecond time domain interferometry,” National Science Review (2020).
45. J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pépin, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature 432(7019), 867–871 (2004). [CrossRef]
46. S. Baker, J. S. Robinson, C. A. Haworth, H. Teng, R. A. Smith, C. C. Chirilă, M. Lein, J. W. G. Tisch, and J. P. Marangos, “Probing Proton Dynamics in Molecules on an Attosecond Time Scale,” Science 312(5772), 424–427 (2006). [CrossRef]
47. O. Smirnova, Y. Mairesse, S. Patchkovskii, N. Dudovich, D. Villeneuve, P. Corkum, and M. Y. Ivanov, “High harmonic interferometry of multi-electron dynamics in molecules,” Nature 460(7258), 972–977 (2009). [CrossRef]
48. P. M. Kraus, B. Mignolet, D. Baykusheva, A. Rupenyan, L. Horný, E. F. Penka, G. Grassi, O. I. Tolstikhin, J. Schneider, F. Jensen, L. B. Madsen, A. D. Bandrauk, F. Remacle, and H. J. Wörner, “Measurement and laser control of attosecond charge migration in ionized iodoacetylene,” Science 350(6262), 790–795 (2015). [CrossRef]
49. L. He, P. Lan, A.-T. Le, B. Wang, B. Wang, X. Zhu, P. Lu, and C. D. Lin, “Real-Time Observation of Molecular Spinning with Angular High-Harmonic Spectroscopy,” Phys. Rev. Lett. 121(16), 163201 (2018). [CrossRef]
50. H. Yuan, L. He, S. M. Njoroge, D. Wang, R. Shao, P. Lan, and P. Lu, “Generation of Near-Circularly Polarized Attosecond Pulse with Tunable Helicity by Unidirectionally Rotating Laser Field,” Ann. Phys. 532(5), 1900570 (2020). [CrossRef]
51. N. Sun, X. Zhu, B. Wang, D. Wang, R. Shao, P. Lan, and P. Lu, “Near-circularly-polarized attosecond pulse generation from carbon monoxide molecules with a combination of linearly and circularly polarized fields,” Phys. Rev. A 101(5), 053437 (2020). [CrossRef]
52. Y. He, L. He, P. Wang, B. Wang, S. Sun, R. Liu, B. Wang, P. Lan, and P. Lu, “Measuring the rotational temperature and pump intensity in molecular alignment experiments via high harmonic generation,” Opt. Express 28(14), 21182–21191 (2020). [CrossRef]
53. B. Wang, Y. He, X. Zhao, L. He, P. Lan, P. Lu, and C. D. Lin, “Retrieval of full angular- and energy-dependent complex transition dipoles in the molecular frame from laser-induced high-order harmonic signals with aligned molecules,” Phys. Rev. A 101(6), 063417 (2020). [CrossRef]
54. J. Yan, W. Xie, M. Li, K. Liu, S. Luo, C. Cao, K. Guo, W. Cao, P. Lan, Q. Zhang, Y. Zhou, and P. Lu, “Photoelectron ionization time of aligned molecules clocked by attosecond angular streaking,” Phys. Rev. A 102(1), 013117 (2020). [CrossRef]
55. S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011). [CrossRef]
56. B. Zaks, R. B. Liu, and M. S. Sherwin, “Experimental observation of electron-hole recollisions,” Nature 483(7391), 580–583 (2012). [CrossRef]
57. L. Li, P. Lan, L. He, W. Cao, Q. Zhang, and P. Lu, “Determination of Electron Band Structure using Temporal Interferometry,” Phys. Rev. Lett. 124(15), 157403 (2020). [CrossRef]
58. H. Lakhotia, H. Y. Kim, M. Zhan, S. Hu, S. Meng, and E. Goulielmakis, “Laser picoscopy of valence electrons in solids,” Nature 583(7814), 55–59 (2020). [CrossRef]
59. Q. Liao, W. Cao, Q. Zhang, K. Liu, F. Wang, P. Lu, and U. Thumm, “Distinction of Electron Dispersion in Time-Resolved Photoemission Spectroscopy,” Phys. Rev. Lett. 125(4), 043201 (2020). [CrossRef]
60. B. Wang, L. He, Y. He, Y. Zhang, R. Shao, P. Lan, and P. Lu, “All-optical measurement of high-order fractional molecular echoes by high-order harmonic generation,” Opt. Express 27(21), 30172 (2019). [CrossRef]
61. S. Fleischer, Y. Khodorkovsky, Y. Prior, and I. S. Averbukh, “Controlling the sense of molecular rotation,” New J. Phys. 11(10), 105039 (2009). [CrossRef]
62. E. Gershnabel and I. S. Averbukh, “Deflection of Field-Free Aligned Molecules,” Phys. Rev. Lett. 104(15), 153001 (2010). [CrossRef]
63. E. Gershnabel and I. S. Averbukh, “Controlling molecular scattering by laser-induced field-free alignment,” Phys. Rev. A 82(3), 033401 (2010). [CrossRef]
64. S. Zhdanovich, A. A. Milner, C. Bloomquist, J. Floß, I. S. Averbukh, J. W. Hepburn, and V. Milner, “Control of Molecular Rotation with a Chiral Train of Ultrashort Pulses,” Phys. Rev. Lett. 107(24), 243004 (2011). [CrossRef]
65. J. Floß and I. S. Averbukh, “Molecular spinning by a chiral train of short laser pulses,” Phys. Rev. A 86(6), 063414 (2012). [CrossRef]
66. O. Korech, U. Steinitz, R. J. Gordon, I. S. Averbukh, and Y. Prior, “Observing molecular spinning via the rotational Doppler effect,” Nat. Photonics 7(9), 711–714 (2013). [CrossRef]
67. U. Steinitz, Y. Prior, and I. S. Averbukh, “Optics of a Gas of Coherently Spinning Molecules,” Phys. Rev. Lett. 112(1), 013004 (2014). [CrossRef]
