# From Symmetry Breaking via Charge Migration to Symmetry Restoration in Electronic Ground and Excited States: Quantum Control on the Attosecond Time Scale

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## Abstract

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## 1. Introduction

## 2. Model, Concept, Theory and Methods

#### 2.1. Model and Basic Theory

#### 2.2. Conceptual Background for the New Symmetry Restoration Strategy

#### 2.3. Extended Theory for the New Strategy

## 3. Results and Discussions

#### 3.1. The Proof-of-Principle for Quantum Control of Symmetry Breaking and Restoration of Molecules in Electronic Ground and Excited States with Different $IRRE{P}_{s}$

#### 3.2. The Requirement of Attosecond Precision for the Proper Time Delay Between the Laser Pulses for Electronic Structure Symmetry Breaking and Restoration

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The concept for symmetry breaking of the electronic ground state and symmetry restoration in an electronic excited state by two laser pulses according to the new strategy. (

**Bottom**) One-electron density of the oriented benzene molecule in the ground state labeled ${A}_{1g}$ (symmetry ${D}_{6h}$, irreducible representation $IRRE{P}_{g}={A}_{1g}$). The first circularly-polarized laser pulse is centered at time ${t}_{b}=-4.5T$. It breaks symmetry by exciting the ground state to the superposition labeled “${A}_{1g}-i{E}_{1u}$” of the ground state and an excited state with $IRRE{P}_{e}={E}_{1u}$. This laser excitation is symbolized by the first red arrow. The superposition state has symmetry ${C}_{s}$. (

**Middle**) Periodic charge migration from “${A}_{1g}-i{E}_{1u}$” via “${A}_{1g}+i{E}_{1u}$” back to “${A}_{1g}-i{E}_{1u}$”, with period $T=504$ as. This is symbolized by the two curved arrows, with snapshots of the one-electron densities for state “${A}_{1g}-i{E}_{1u}$” (left) at central time $t={t}_{c}=0$ (and also at $t=T,2T$, etc.) and for state “${A}_{1g}+i{E}_{1u}$” (right) at time $t=T/2$ (and also at $3T/2,5T/2$, etc.) The second laser pulse centered at ${t}_{r}=4.5T$ restores ${D}_{6h}$ symmetry by transferring the superposition state to the excited state with $IRRE{P}_{e}={E}_{1u}$. This laser excitation is symbolized by the second red arrow. (

**Top**) One-electron density of the excited target state labeled ${E}_{1u}$. The time delay ${t}_{d}={t}_{r}-{t}_{b}=NT$ between the centers of the laser pulses must be equal to an integer number N of periods T of charge migration, with precision of few attoseconds. Here, $N=9$. Any attempts to restore electronic structure symmetry at delay times that correspond to incomplete cycles of charge migration are useless—this is indicated by the crossed-out arrows. The Gaussian shape functions (dashed lines) and the x- and y-components of the electric field (red and green continuous lines) of the circularly-polarized laser pulses are also sketched. All densities were created using detCI@ORBKIT [43,44,45] and plotted using Matplotlib [46].

**Figure 2.**(

**Left**) Electronic energy levels of the lowest states of benzene, with assignment of the IRREPs. The present circularly polarized laser pulses yield exclusive population transfer from the electronic ground state ${A}_{1g}$ to the excited target state ${E}_{1u+}$, illustrated by the vertical arrow. All other transitions to excited states that are within the spectral width of the laser pulses with different IRREPs are dipole forbidden, symbolized by vertical arrows that are crossed out. The two-photon process at 16.42 eV is also found to be off-resonance. (

**Right**) Spectral profile of the laser pulses of duration 0.47 fs (red line), including a potential two-photon contribution at $\Delta E=2\hslash \omega =16.42$ eV (grey line).

**Figure 3.**Symmetry breaking of the electronic ground state of benzene labeled ${A}_{1g}$ (symmetry ${D}_{6h}$, irreducible representation $IRRE{P}_{g}={A}_{1g}$) and symmetry restoration in an electronic excited labeled ${E}_{1u}$ (symmetry ${D}_{6h}$, irreducible representation $IRRE{P}_{e}={E}_{1u}$) by two laser pulses according to the new strategy. (

**a**) Gaussian envelopes (dashed lines) and the x- and y-components (red and green continuous lines) of the circularly right (+) polarized laser pulses centered at ${t}_{b}=-4.5T$ and ${t}_{r}=+4.5T$ with period $T=504\phantom{\rule{3.33333pt}{0ex}}\mathrm{as}$ of charge migration. The parameter of the circularly right (+) polarized laser pulses (Equation (40)) are ${\u03f5}_{b}=4.207\times {10}^{7}\phantom{\rule{3.33333pt}{0ex}}\mathrm{V}/\mathrm{cm},{\u03f5}_{r}=7.192\times {10}^{7}\phantom{\rule{3.33333pt}{0ex}}\mathrm{V}/\mathrm{cm},\omega =2\pi /T,T=504\phantom{\rule{3.33333pt}{0ex}}\mathrm{as},{\tau}_{b}={\tau}_{r}=0.47\phantom{\rule{3.33333pt}{0ex}}\mathrm{fs},{t}_{b}=-4.5T,{t}_{r}=+4.5T.$ (

**b**) Time evolution of the population of the excited state due to the first and second laser pulses shown in (

**a**) for the case ${t}_{r}=+4.5T=2.267\phantom{\rule{3.33333pt}{0ex}}\mathrm{fs}$. The results for sixteen different times ${t}_{r}^{\prime}={t}_{r}+{t}^{\prime}$ where ${t}^{\prime}=kT/16,k=1,2,\dots ,16$ are also shown. (

**c**) Numerical results (continuous blue line) and analytical result (dotted red line, Equation (62)) for the final populations ${P}_{e}({t}_{f}^{\prime})$ of the excited state at time ${t}_{f}^{\prime}={t}_{f}+{t}^{\prime}$ versus delay time ${t}_{d}^{\prime}={t}_{r}^{\prime}-{t}_{b}$, in units of the period T (top abscissa) or fs (bottom abscissa, as in (

**d**)). The results coincide within graphical resolution. (

**d**) Phase difference $\Delta \eta ({t}_{c}^{\prime})={\eta}_{e}({t}_{c}^{\prime})-{\eta}_{g}({t}_{c}^{\prime})$ of the wave functions in electronic excited and ground states at the central time ${t}_{c}^{\prime}=({t}_{r}^{\prime}+{t}_{b})/2$. (

**e**) One-electron density of the electronic ground state of benzene labeled ${A}_{1g}$. (

**f**) Five snapshots of the one-electron density during periodic charge migration of the superposition of the ground state labeled ${A}_{1g}$ and the excited state labeled ${E}_{1u}$ during one period, from $t={t}_{c}=0$ to T. (

**g**) One-electron density of the excited target state labeled ${E}_{1u}$. All densities were created using detCI@ORBKIT [43,44,45] and plotted using Matplotlib [46].

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**MDPI and ACS Style**

Liu, C.; Manz, J.; Tremblay, J.C.
From Symmetry Breaking via Charge Migration to Symmetry Restoration in Electronic Ground and Excited States: Quantum Control on the Attosecond Time Scale. *Appl. Sci.* **2019**, *9*, 953.
https://doi.org/10.3390/app9050953

**AMA Style**

Liu C, Manz J, Tremblay JC.
From Symmetry Breaking via Charge Migration to Symmetry Restoration in Electronic Ground and Excited States: Quantum Control on the Attosecond Time Scale. *Applied Sciences*. 2019; 9(5):953.
https://doi.org/10.3390/app9050953

**Chicago/Turabian Style**

Liu, ChunMei, Jörn Manz, and Jean Christophe Tremblay.
2019. "From Symmetry Breaking via Charge Migration to Symmetry Restoration in Electronic Ground and Excited States: Quantum Control on the Attosecond Time Scale" *Applied Sciences* 9, no. 5: 953.
https://doi.org/10.3390/app9050953