From Symmetry Breaking via Charge Migration to Symmetry Restoration in Electronic Ground and Excited States: Quantum Control on the Attosecond Time Scale
<p>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. (<b>Bottom</b>) One-electron density of the oriented benzene molecule in the ground state labeled <math display="inline"><semantics> <msub> <mi>A</mi> <mrow> <mn>1</mn> <mi>g</mi> </mrow> </msub> </semantics></math> (symmetry <math display="inline"><semantics> <msub> <mi>D</mi> <mrow> <mn>6</mn> <mi>h</mi> </mrow> </msub> </semantics></math>, irreducible representation <math display="inline"><semantics> <mrow> <mi>I</mi> <mi>R</mi> <mi>R</mi> <mi>E</mi> <msub> <mi>P</mi> <mi>g</mi> </msub> <mo>=</mo> <msub> <mi>A</mi> <mrow> <mn>1</mn> <mi>g</mi> </mrow> </msub> </mrow> </semantics></math>). The first circularly-polarized laser pulse is centered at time <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mi>b</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>4.5</mn> <mi>T</mi> </mrow> </semantics></math>. It breaks symmetry by exciting the ground state to the superposition labeled “<math display="inline"><semantics> <mrow> <msub> <mi>A</mi> <mrow> <mn>1</mn> <mi>g</mi> </mrow> </msub> <mo>−</mo> <mi>i</mi> <msub> <mi>E</mi> <mrow> <mn>1</mn> <mi>u</mi> </mrow> </msub> </mrow> </semantics></math>” of the ground state and an excited state with <math display="inline"><semantics> <mrow> <mi>I</mi> <mi>R</mi> <mi>R</mi> <mi>E</mi> <msub> <mi>P</mi> <mi>e</mi> </msub> <mo>=</mo> <msub> <mi>E</mi> <mrow> <mn>1</mn> <mi>u</mi> </mrow> </msub> </mrow> </semantics></math>. This laser excitation is symbolized by the first red arrow. The superposition state has symmetry <math display="inline"><semantics> <msub> <mi>C</mi> <mi>s</mi> </msub> </semantics></math>. (<b>Middle</b>) Periodic charge migration from “<math display="inline"><semantics> <mrow> <msub> <mi>A</mi> <mrow> <mn>1</mn> <mi>g</mi> </mrow> </msub> <mo>−</mo> <mi>i</mi> <msub> <mi>E</mi> <mrow> <mn>1</mn> <mi>u</mi> </mrow> </msub> </mrow> </semantics></math>” via “<math display="inline"><semantics> <mrow> <msub> <mi>A</mi> <mrow> <mn>1</mn> <mi>g</mi> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>E</mi> <mrow> <mn>1</mn> <mi>u</mi> </mrow> </msub> </mrow> </semantics></math>” back to “<math display="inline"><semantics> <mrow> <msub> <mi>A</mi> <mrow> <mn>1</mn> <mi>g</mi> </mrow> </msub> <mo>−</mo> <mi>i</mi> <msub> <mi>E</mi> <mrow> <mn>1</mn> <mi>u</mi> </mrow> </msub> </mrow> </semantics></math>”, with period <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>504</mn> </mrow> </semantics></math> as. This is symbolized by the two curved arrows, with snapshots of the one-electron densities for state “<math display="inline"><semantics> <mrow> <msub> <mi>A</mi> <mrow> <mn>1</mn> <mi>g</mi> </mrow> </msub> <mo>−</mo> <mi>i</mi> <msub> <mi>E</mi> <mrow> <mn>1</mn> <mi>u</mi> </mrow> </msub> </mrow> </semantics></math>” (left) at central time <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <msub> <mi>t</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> (and also at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mi>T</mi> <mo>,</mo> <mn>2</mn> <mi>T</mi> </mrow> </semantics></math>, etc.) and for state “<math display="inline"><semantics> <mrow> <msub> <mi>A</mi> <mrow> <mn>1</mn> <mi>g</mi> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mi>E</mi> <mrow> <mn>1</mn> <mi>u</mi> </mrow> </msub> </mrow> </semantics></math>” (right) at time <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mi>T</mi> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math> (and also at <math display="inline"><semantics> <mrow> <mn>3</mn> <mi>T</mi> <mo>/</mo> <mn>2</mn> <mo>,</mo> <mn>5</mn> <mi>T</mi> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>, etc.) The second laser pulse centered at <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mi>r</mi> </msub> <mo>=</mo> <mn>4.5</mn> <mi>T</mi> </mrow> </semantics></math> restores <math display="inline"><semantics> <msub> <mi>D</mi> <mrow> <mn>6</mn> <mi>h</mi> </mrow> </msub> </semantics></math> symmetry by transferring the superposition state to the excited state with <math display="inline"><semantics> <mrow> <mi>I</mi> <mi>R</mi> <mi>R</mi> <mi>E</mi> <msub> <mi>P</mi> <mi>e</mi> </msub> <mo>=</mo> <msub> <mi>E</mi> <mrow> <mn>1</mn> <mi>u</mi> </mrow> </msub> </mrow> </semantics></math>. This laser excitation is symbolized by the second red arrow. (<b>Top</b>) One-electron density of the excited target state labeled <math display="inline"><semantics> <msub> <mi>E</mi> <mrow> <mn>1</mn> <mi>u</mi> </mrow> </msub> </semantics></math>. The time delay <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mi>d</mi> </msub> <mo>=</mo> <msub> <mi>t</mi> <mi>r</mi> </msub> <mo>−</mo> <msub> <mi>t</mi> <mi>b</mi> </msub> <mo>=</mo> <mi>N</mi> <mi>T</mi> </mrow> </semantics></math> between the centers of the laser pulses must be equal to an integer number <span class="html-italic">N</span> of periods <span class="html-italic">T</span> of charge migration, with precision of few attoseconds. Here, <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math>. 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 <span class="html-italic">x</span>- and <span class="html-italic">y</span>-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 [<a href="#B43-applsci-09-00953" class="html-bibr">43</a>,<a href="#B44-applsci-09-00953" class="html-bibr">44</a>,<a href="#B45-applsci-09-00953" class="html-bibr">45</a>] and plotted using Matplotlib [<a href="#B46-applsci-09-00953" class="html-bibr">46</a>].</p> "> Figure 2
<p>(<b>Left</b>) 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 <math display="inline"><semantics> <msub> <mi>A</mi> <mrow> <mn>1</mn> <mi>g</mi> </mrow> </msub> </semantics></math> to the excited target state <math display="inline"><semantics> <msub> <mi>E</mi> <mrow> <mn>1</mn> <mi>u</mi> <mo>+</mo> </mrow> </msub> </semantics></math>, 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. (<b>Right</b>) Spectral profile of the laser pulses of duration 0.47 fs (red line), including a potential two-photon contribution at <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>E</mi> <mo>=</mo> <mn>2</mn> <mo>ℏ</mo> <mi>ω</mi> <mo>=</mo> <mn>16.42</mn> </mrow> </semantics></math> eV (grey line).</p> "> Figure 3
<p>Symmetry breaking of the electronic ground state of benzene labeled <math display="inline"><semantics> <msub> <mi>A</mi> <mrow> <mn>1</mn> <mi>g</mi> </mrow> </msub> </semantics></math> (symmetry <math display="inline"><semantics> <msub> <mi>D</mi> <mrow> <mn>6</mn> <mi>h</mi> </mrow> </msub> </semantics></math>, irreducible representation <math display="inline"><semantics> <mrow> <mi>I</mi> <mi>R</mi> <mi>R</mi> <mi>E</mi> <msub> <mi>P</mi> <mi>g</mi> </msub> <mo>=</mo> <msub> <mi>A</mi> <mrow> <mn>1</mn> <mi>g</mi> </mrow> </msub> </mrow> </semantics></math>) and symmetry restoration in an electronic excited labeled <math display="inline"><semantics> <msub> <mi>E</mi> <mrow> <mn>1</mn> <mi>u</mi> </mrow> </msub> </semantics></math> (symmetry <math display="inline"><semantics> <msub> <mi>D</mi> <mrow> <mn>6</mn> <mi>h</mi> </mrow> </msub> </semantics></math>, irreducible representation <math display="inline"><semantics> <mrow> <mi>I</mi> <mi>R</mi> <mi>R</mi> <mi>E</mi> <msub> <mi>P</mi> <mi>e</mi> </msub> <mo>=</mo> <msub> <mi>E</mi> <mrow> <mn>1</mn> <mi>u</mi> </mrow> </msub> </mrow> </semantics></math>) by two laser pulses according to the new strategy. (<b>a</b>) Gaussian envelopes (dashed lines) and the <span class="html-italic">x</span>- and <span class="html-italic">y</span>-components (red and green continuous lines) of the circularly right (+) polarized laser pulses centered at <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mi>b</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>4.5</mn> <mi>T</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mi>r</mi> </msub> <mo>=</mo> <mo>+</mo> <mn>4.5</mn> <mi>T</mi> </mrow> </semantics></math> with period <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>504</mn> <mspace width="3.33333pt"/> <mi>as</mi> </mrow> </semantics></math> of charge migration. The parameter of the circularly right (+) polarized laser pulses (Equation (<a href="#FD40-applsci-09-00953" class="html-disp-formula">40</a>)) are <math display="inline"><semantics> <mrow> <msub> <mi>ϵ</mi> <mi>b</mi> </msub> <mo>=</mo> <mn>4.207</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>7</mn> </msup> <mspace width="3.33333pt"/> <mrow> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi>cm</mi> </mrow> <mo>,</mo> <msub> <mi>ϵ</mi> <mi>r</mi> </msub> <mo>=</mo> <mn>7.192</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>7</mn> </msup> <mspace width="3.33333pt"/> <mrow> <mi mathvariant="normal">V</mi> <mo>/</mo> <mi>cm</mi> </mrow> <mo>,</mo> <mi>ω</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mo>/</mo> <mi>T</mi> <mo>,</mo> <mi>T</mi> <mo>=</mo> <mn>504</mn> <mspace width="3.33333pt"/> <mi>as</mi> <mo>,</mo> <msub> <mi>τ</mi> <mi>b</mi> </msub> <mo>=</mo> <msub> <mi>τ</mi> <mi>r</mi> </msub> <mo>=</mo> <mn>0.47</mn> <mspace width="3.33333pt"/> <mi>fs</mi> <mo>,</mo> <msub> <mi>t</mi> <mi>b</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>4.5</mn> <mi>T</mi> <mo>,</mo> <msub> <mi>t</mi> <mi>r</mi> </msub> <mo>=</mo> <mo>+</mo> <mn>4.5</mn> <mi>T</mi> <mo>.</mo> </mrow> </semantics></math> (<b>b</b>) Time evolution of the population of the excited state due to the first and second laser pulses shown in (<b>a</b>) for the case <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mi>r</mi> </msub> <mo>=</mo> <mo>+</mo> <mn>4.5</mn> <mi>T</mi> <mo>=</mo> <mn>2.267</mn> <mspace width="3.33333pt"/> <mi>fs</mi> </mrow> </semantics></math>. The results for sixteen different times <math display="inline"><semantics> <mrow> <msubsup> <mi>t</mi> <mi>r</mi> <mo>′</mo> </msubsup> <mo>=</mo> <msub> <mi>t</mi> <mi>r</mi> </msub> <mo>+</mo> <msup> <mi>t</mi> <mo>′</mo> </msup> </mrow> </semantics></math> where <math display="inline"><semantics> <mrow> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>k</mi> <mi>T</mi> <mo>/</mo> <mn>16</mn> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>16</mn> </mrow> </semantics></math> are also shown. (<b>c</b>) Numerical results (continuous blue line) and analytical result (dotted red line, Equation (<a href="#FD62-applsci-09-00953" class="html-disp-formula">62</a>)) for the final populations <math display="inline"><semantics> <mrow> <msub> <mi>P</mi> <mi>e</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>t</mi> <mi>f</mi> <mo>′</mo> </msubsup> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> of the excited state at time <math display="inline"><semantics> <mrow> <msubsup> <mi>t</mi> <mi>f</mi> <mo>′</mo> </msubsup> <mo>=</mo> <msub> <mi>t</mi> <mi>f</mi> </msub> <mo>+</mo> <msup> <mi>t</mi> <mo>′</mo> </msup> </mrow> </semantics></math> versus delay time <math display="inline"><semantics> <mrow> <msubsup> <mi>t</mi> <mi>d</mi> <mo>′</mo> </msubsup> <mo>=</mo> <msubsup> <mi>t</mi> <mi>r</mi> <mo>′</mo> </msubsup> <mo>−</mo> <msub> <mi>t</mi> <mi>b</mi> </msub> </mrow> </semantics></math>, in units of the period <span class="html-italic">T</span> (top abscissa) or fs (bottom abscissa, as in (<b>d</b>)). The results coincide within graphical resolution. (<b>d</b>) Phase difference <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>η</mi> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>t</mi> <mi>c</mi> <mo>′</mo> </msubsup> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>η</mi> <mi>e</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>t</mi> <mi>c</mi> <mo>′</mo> </msubsup> <mo stretchy="false">)</mo> </mrow> <mo>−</mo> <msub> <mi>η</mi> <mi>g</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>t</mi> <mi>c</mi> <mo>′</mo> </msubsup> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> of the wave functions in electronic excited and ground states at the central time <math display="inline"><semantics> <mrow> <msubsup> <mi>t</mi> <mi>c</mi> <mo>′</mo> </msubsup> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>t</mi> <mi>r</mi> <mo>′</mo> </msubsup> <mo>+</mo> <msub> <mi>t</mi> <mi>b</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>. (<b>e</b>) One-electron density of the electronic ground state of benzene labeled <math display="inline"><semantics> <msub> <mi>A</mi> <mrow> <mn>1</mn> <mi>g</mi> </mrow> </msub> </semantics></math>. (<b>f</b>) Five snapshots of the one-electron density during periodic charge migration of the superposition of the ground state labeled <math display="inline"><semantics> <msub> <mi>A</mi> <mrow> <mn>1</mn> <mi>g</mi> </mrow> </msub> </semantics></math> and the excited state labeled <math display="inline"><semantics> <msub> <mi>E</mi> <mrow> <mn>1</mn> <mi>u</mi> </mrow> </msub> </semantics></math> during one period, from <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <msub> <mi>t</mi> <mi>c</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> to <span class="html-italic">T</span>. (<b>g</b>) One-electron density of the excited target state labeled <math display="inline"><semantics> <msub> <mi>E</mi> <mrow> <mn>1</mn> <mi>u</mi> </mrow> </msub> </semantics></math>. All densities were created using detCI@ORBKIT [<a href="#B43-applsci-09-00953" class="html-bibr">43</a>,<a href="#B44-applsci-09-00953" class="html-bibr">44</a>,<a href="#B45-applsci-09-00953" class="html-bibr">45</a>] and plotted using Matplotlib [<a href="#B46-applsci-09-00953" class="html-bibr">46</a>].</p> ">
Abstract
: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
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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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
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 StyleLiu, 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