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<Input><Text-field style="Text" layout="Normal" alignment="centred"><Font size="24" bold="true">Optical Control of Chemical Systems</Font></Text-field>
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<Input><Text-field style="Text" layout="Normal" alignment="centred">Ryan Choi 2023</Text-field>
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<Section collapsed="false" isCollapsible="true" drawButton="true" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title><Text-field style="Heading 1" layout="Heading 1">Abstract</Text-field></Title><Text-field style="Text" layout="Normal">Chemical reactions can be controlled in many ways, such as reagent selection and temperature. These macroscopic methods of regulation are good for controlling and investigating reactions in bulk; however, even more precise control is possible with electromagnetic radiation.  Light is frequently used in chemistry through spectroscopy, with which we can make observations about our system of interest, but we are not limited to only making measurements. With the advancement of laser technology, we can also control chemical processes using ultrashort laser pulses. </Text-field>
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<Section collapsed="false" isCollapsible="true" drawButton="true" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title><Text-field style="Heading 1" layout="Heading 1">Introduction: Tannor-Rice Control Method</Text-field></Title><Text-field style="Text" layout="Normal"></Text-field><Text-field style="Text" layout="Normal">The Tannor-Rice method is an early method to control simple chemical reactions. Though it has been succeed by more sophisticated schemes such as STIRAP control, it illustrates fundamental principles of optical control. This introduction will provide a conceptual overview of the method. </Text-field><Text-field style="Text" layout="Normal"></Text-field><Text-field style="Text" layout="Normal">The goal behind the Tannor-Rice method is to steer a chemical reaction into producing a specified product. In terms of a potential energy &quot;landscape,&quot; the reactants could end up in a variety of product stable states separated by potential energy barriers. This method involves electronic (and vibronic) excitation of the reactant molecule with a short laser pulse, on the order of femtoseconds, which produces a localized wave packet on the excited state surface. As time passes, the wave packet  propagates on the excited state surface, and by careful timing of a second pulse, one can select the products when the wave packet is aligned with the desired state.  </Text-field><Text-field style="Text" layout="Normal"></Text-field><Text-field style="Text" layout="Normal">Optical control of chemical reactions was not feasible until the invention of ultrashort laser pulses. Pulses on the order of femtoseconds is necessary for a few reasons. First, the propagation of the wave packet on the excited state surface occurs on such a timescale, so naturally, temporally precise optics are needed.<Equation executable="false" style="Text" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2I1EhRictRiM2JUYuLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjhRJ25vcm1hbEYn">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2I1EhRictRiM2JUYuLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjhRJ25vcm1hbEYn</Equation> Second, energy dissipates from the excited state quickly. Lastly, a short pulse localized in space will be spectrally broad due to the uncertainty relation. This, combined with the Franck-Condon relation, helps produce a localized wavepacket when the pump pulse is delivered. </Text-field><Text-field style="Text" layout="Normal"></Text-field><Text-field style="Text" layout="Normal"></Text-field>
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opaque="false" subscript="false" underline="false" superscript="false" bold="true" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman" spaceabove="0" linebreak="space" rightmargin="0" linespacing="0.0" initial="0" firstindent="0" leftmargin="0" bulletsuffix="" alignment="centred" bullet="none" spacebelow="0" pagebreak-before="false"><Font opaque="false" subscript="false" underline="false" superscript="false" bold="true" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman">Figure 1. </Font><Font opaque="false" subscript="false" underline="false" superscript="false" bold="false" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman">Illustration of highly localized but spectrally broad wavepacket. (Licensed under Creative Commons, [3])</Font></Text-field>
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<Group labelreference="L14752" drawlabel="true" applyint="true" applyrational="true" applyexponent="false"><Text-field opaque="false" subscript="false" underline="false" superscript="false" bold="true" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman" spaceabove="0" linebreak="space" rightmargin="0" linespacing="0.0" initial="0" firstindent="0" leftmargin="0" bulletsuffix="" alignment="left" bullet="none" spacebelow="0" pagebreak-before="false"><Font opaque="false" subscript="false" underline="false" superscript="false" bold="false" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman">The Frank-Condon relation describes the probability of a vibrational transition from an initial state i to a final state f when an electronic transition occurs.</Font></Text-field><Text-field opaque="false" subscript="false" underline="false" superscript="false" bold="true" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman" spaceabove="0" linebreak="space" rightmargin="0" linespacing="0.0" initial="0" firstindent="0" leftmargin="0" bulletsuffix="" alignment="left" bullet="none" spacebelow="0" pagebreak-before="false"></Text-field><Text-field opaque="false" subscript="false" underline="false" superscript="false" bold="true" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman" spaceabove="0" linebreak="space" rightmargin="0" linespacing="0.0" initial="0" firstindent="0" leftmargin="0" bulletsuffix="" alignment="centred" bullet="none" spacebelow="0" pagebreak-before="false"><Equation executable="false" input-equation="" 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<Group labelreference="L14751" drawlabel="true" applyint="true" applyrational="true" applyexponent="false"><Text-field opaque="false" subscript="false" underline="false" superscript="false" bold="false" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman" spaceabove="0" linebreak="space" rightmargin="0" linespacing="0.0" initial="0" firstindent="0" leftmargin="0" bulletsuffix="" alignment="centred" bullet="none" spacebelow="0" pagebreak-before="false"></Text-field><Text-field opaque="false" subscript="false" underline="false" superscript="false" bold="true" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman" spaceabove="0" linebreak="space" rightmargin="0" linespacing="0.0" initial="0" firstindent="0" leftmargin="0" bulletsuffix="" alignment="centred" bullet="none" spacebelow="0" pagebreak-before="false"><Font opaque="false" subscript="false" underline="false" superscript="false" bold="true" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman">Figure 2. </Font><Font opaque="false" subscript="false" underline="false" superscript="false" bold="false" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman">Illustration of Franck-Condon principle. (Licensed under Creative Commons, [4])</Font></Text-field><Text-field opaque="false" subscript="false" underline="false" superscript="false" bold="true" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman" spaceabove="0" linebreak="space" rightmargin="0" linespacing="0.0" initial="0" firstindent="0" leftmargin="0" bulletsuffix="" alignment="centred" bullet="none" spacebelow="0" pagebreak-before="false"></Text-field><Text-field opaque="false" subscript="false" underline="false" superscript="false" bold="true" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman" spaceabove="0" linebreak="space" rightmargin="0" linespacing="0.0" initial="0" firstindent="0" leftmargin="0" bulletsuffix="" alignment="left" bullet="none" spacebelow="0" pagebreak-before="false"><Font opaque="false" subscript="false" underline="false" superscript="false" bold="false" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman">With a ultrashort laser pulse, we can coherently excite multiple vibrational states at once, creating a wavepacket. This then propagates on the excited state surface. </Font></Text-field><Text-field opaque="false" subscript="false" underline="false" superscript="false" bold="true" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman" spaceabove="0" linebreak="space" rightmargin="0" linespacing="0.0" initial="0" firstindent="0" leftmargin="0" bulletsuffix="" alignment="left" bullet="none" spacebelow="0" pagebreak-before="false"></Text-field>
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</Group><Text-field style="Text" layout="Normal"></Text-field><Text-field style="Text" layout="Normal">To illustrate how the Tannor-Rice control method works, we can take a look at an example. 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</Group><Text-field style="Text" layout="Normal">To see how this reaction is controlled, we can look at the potential energy surfaces of several electronic excited states of Na<Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>Refer to Figure 3, on page 53, in reference [5] for the following description.</Text-field>
<Group labelreference="L14720" drawlabel="true" applyint="true" applyrational="true" applyexponent="false"><Text-field opaque="false" subscript="false" underline="false" superscript="false" bold="false" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman" spaceabove="0" linebreak="space" rightmargin="0" linespacing="0.0" initial="0" firstindent="0" leftmargin="0" bulletsuffix="" alignment="left" bullet="none" spacebelow="0" pagebreak-before="false"><Font opaque="false" subscript="false" underline="false" superscript="false" bold="false" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman">Starting at the ground state, we can use laser pulses to excite the molecule to the 2 </Font><Equation executable="false" input-equation="" display="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">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</Equation><Font opaque="false" subscript="false" underline="false" superscript="false" bold="false" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman">state, which is where we have the opportunity to influence the reaction. With a spectrally broad laser pulse, one can coherently excite multiple vibrational states, forming a wave packet. Upon excitation to the 2 </Font><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkubW11bHRpc2NyaXB0c0dGJDYoLUYjNiktSSNtaUdGJDYmUScmIzkyODtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJStleGVjdXRhYmxlR0Y2LyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy9GNVEldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRK1swLDE2MCw4MF1GJ0Y3LyUscGxhY2Vob2xkZXJHRj0vJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRj0vRjpRJ2l0YWxpY0YnLUYjNiYtRjE2JlEiZ0YnRjxGN0ZFRjxGN0ZFLUYxNiNRIUYnLUksbXByZXNjcmlwdHNHRiRGJ0ZMLUYjNiYtSSNtbkdGJDYlUSIxRidGN0Y5RjxGN0ZFLUkjbW9HRiQ2LlEifkYnRjdGOS8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmVvRjdGOQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkubW11bHRpc2NyaXB0c0dGJDYoLUYjNiktSSNtaUdGJDYmUScmIzkyODtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJStleGVjdXRhYmxlR0Y2LyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy9GNVEldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRK1swLDE2MCw4MF1GJ0Y3LyUscGxhY2Vob2xkZXJHRj0vJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRj0vRjpRJ2l0YWxpY0YnLUYjNiYtRjE2JlEiZ0YnRjxGN0ZFRjxGN0ZFLUYxNiNRIUYnLUksbXByZXNjcmlwdHNHRiRGJ0ZMLUYjNiYtSSNtbkdGJDYlUSIxRidGN0Y5RjxGN0ZFLUkjbW9HRiQ2LlEifkYnRjdGOS8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmVvRjdGOQ==</Equation><Font opaque="false" subscript="false" underline="false" superscript="false" bold="false" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman">state, the wave packet will be localized to the left turning point of the curve (the bottom of arrow 1). Since the wavefunction for the ground state Hamiltonian is not a stationary state of an excited state Hamiltonian, the wave packet will time evolve and move towards the right turning point of the curve (the bottom of arrow 2). </Font></Text-field><Text-field opaque="false" subscript="false" underline="false" superscript="false" bold="false" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman" spaceabove="0" linebreak="space" rightmargin="0" linespacing="0.0" initial="0" firstindent="0" leftmargin="0" bulletsuffix="" alignment="left" bullet="none" spacebelow="0" pagebreak-before="false"></Text-field><Text-field opaque="false" subscript="false" underline="false" superscript="false" bold="false" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman" spaceabove="0" linebreak="space" rightmargin="0" linespacing="0.0" initial="0" firstindent="0" leftmargin="0" bulletsuffix="" alignment="left" bullet="none" spacebelow="0" pagebreak-before="false"><Font opaque="false" subscript="false" underline="false" superscript="false" bold="false" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman">Eventually, the wave packet will decohere, but before it does so, we can deliver a final laser pulse to select the desired product. Exciting the molecule when the wave packet is under arrow 1 will yield the diatomic ion, since the wave packet can &quot;settle&quot; into the well of the </Font><Equation executable="false" input-equation="" display="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">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</Equation><Font opaque="false" subscript="false" underline="false" superscript="false" bold="false" foreground="[0,0,0]" italic="false" executable="false" selection-placeholder="false" size="12" readonly="false" background="[255,255,255]" placeholder="false" family="Times New Roman">state. Exciting the molecule when the wave packet is under arrow 2 will yield the dissociated ionized product, since the wave packet will travel outwards towards increasing bond length. Arrow 3 shows that one can excite the diatomic ion to produce the dissociated product. In literature, the first laser pulse(s) are referred to as the &quot;pump,&quot; and the final laser pulse that determines the product is called the &quot;dump&quot; or the &quot;probe.&quot; Figure 4 from reference [5] shows how the delay between the pump and dump pulses selects for one product, with the periodic nature corresponding to the wave packet oscillating back and forth between the two turning points. </Font></Text-field>
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</Section>
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<Section collapsed="false" isCollapsible="true" drawButton="true" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title><Text-field style="Heading 1" layout="Heading 1">Theory</Text-field></Title><Text-field style="Text" layout="Normal"><Font encoding="UTF-8">To mathematically represent the Tannor-Rice method, we will start with a coupled system of differential equations representing the time-dependent Schr\303\266dinger equations of a ground state and an excited state. The Hamiltonian of the entire system will contain the Hamiltonians of the ground and excited states, as well as the interaction of the transition dipole moment with an electromagnetic field, \316\265(t). It is convenient to write the Hamiltonian as a matrix. </Font></Text-field><Text-field style="Text" layout="Normal"></Text-field><Text-field style="Text" layout="Normal" alignment="centred"><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field><Text-field style="Text" layout="Normal" alignment="right">(1)</Text-field><Text-field style="Text" layout="Normal" alignment="centred"></Text-field><Text-field style="Text" layout="Normal">The coupled differential equations are written with this matrix. </Text-field><Text-field style="Text" layout="Normal"></Text-field><Text-field style="Text" layout="Normal" alignment="centred"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=">JSFH</Equation></Text-field><Text-field style="Text" layout="Normal" alignment="centred"><Equation executable="true" style="2D Math" input-equation="" display="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">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</Equation></Text-field><Text-field style="Text" layout="Normal" alignment="right">(2)</Text-field><Text-field style="Text" layout="Normal" alignment="centred"></Text-field><Text-field style="Text" layout="Normal">The system will start in the ground state, so we will set the wavefunction to be <Equation executable="true" style="2D Math" input-equation="" display="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">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</Equation>when t = 0. We can then write equation (2) in integral form. </Text-field><Text-field style="Text" layout="Normal"></Text-field><Text-field style="Text" layout="Normal" alignment="centred"><Equation executable="true" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUklbXN1YkdGJDYmLUYsNiVRJiZwc2k7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GLDYlUSJnRicvRjhRJXRydWVGJy9GO1EnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRitGOkYrRjo=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUklbXN1YkdGJDYmLUYsNiVRJiZwc2k7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GLDYlUSJnRicvRjhRJXRydWVGJy9GO1EnaXRhbGljRicvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRitGOkYrRjo=</Equation>(t) = <Equation executable="false" style="Text" input-equation="" display="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">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</Equation></Text-field><Text-field style="Text" layout="Normal" alignment="right">(3)</Text-field><Text-field style="Text" layout="Normal" alignment="centred"></Text-field><Text-field style="Text" layout="Normal" alignment="centred"><Equation executable="true" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUklbXN1YkdGJDYmLUYsNiVRJiZwc2k7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUYsNiVRImVGJy9GOFEldHJ1ZUYnL0Y7USdpdGFsaWNGJ0ZCRkQvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRitGOkYrRjo=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUklbXN1YkdGJDYmLUYsNiVRJiZwc2k7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUYsNiVRImVGJy9GOFEldHJ1ZUYnL0Y7USdpdGFsaWNGJ0ZCRkQvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRitGOkYrRjo=</Equation>(t) = <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field><Text-field style="Text" layout="Normal" alignment="right">(4)</Text-field><Text-field style="Text" layout="Normal" alignment="centred"></Text-field><Text-field style="Text" layout="Normal">We can use second-order time-dependent pertubation theory to derive an expression for the motion of the wavefunction. First, equation (4) is set to the following.  </Text-field><Text-field style="Text" layout="Normal"></Text-field><Text-field style="Text" layout="Normal" alignment="centred"><Equation executable="true" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUklbXN1YkdGJDYmLUYsNiVRJiZwc2k7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUYsNiVRImVGJy9GOFEldHJ1ZUYnL0Y7USdpdGFsaWNGJ0ZCRkQvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRitGOkYrRjo=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUklbXN1YkdGJDYmLUYsNiVRJiZwc2k7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYlLUYsNiVRImVGJy9GOFEldHJ1ZUYnL0Y7USdpdGFsaWNGJ0ZCRkQvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9JK21zZW1hbnRpY3NHRiRRJ2F0b21pY0YnRitGOkYrRjo=</Equation>(<Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNictSSNtbkdGJDYlUSIyRidGNS9GOVEnbm9ybWFsRidGMkY1LyUwZm9udF9zdHlsZV9uYW1lR1ElVGV4dEYnRjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1RkNGQQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNictSSNtbkdGJDYlUSIyRidGNS9GOVEnbm9ybWFsRidGMkY1LyUwZm9udF9zdHlsZV9uYW1lR1ElVGV4dEYnRjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1RkNGQQ==</Equation>) = <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field><Text-field style="Text" layout="Normal" alignment="right">(5)</Text-field><Text-field style="Text" layout="Normal" alignment="centred"></Text-field><Text-field style="Text" layout="Normal">Then, within pertubation theory, the ground state wavefunction is:</Text-field><Text-field style="Text" layout="Normal"></Text-field><Text-field style="Text" layout="Normal" alignment="centred"><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=</Equation><Equation executable="true" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEmJnBzaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiUtRi82JVEiZ0YnL0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnRj1GPy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRictSSVtc3VwR0YkNiUtRi82I1EhRictRiM2JS1JKG1mZW5jZWRHRiQ2JC1GIzYlLUkjbW5HRiQ2JFEiMkYnRjVGPUY/RjVGPUY/LyUxc3VwZXJzY3JpcHRzaGlmdEdGQ0Y1">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JVEmJnBzaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiUtRi82JVEiZ0YnL0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnRj1GPy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRictSSVtc3VwR0YkNiUtRi82I1EhRictRiM2JS1JKG1mZW5jZWRHRiQ2JC1GIzYlLUkjbW5HRiQ2JFEiMkYnRjVGPUY/RjVGPUY/LyUxc3VwZXJzY3JpcHRzaGlmdEdGQ0Y1</Equation>(t) = <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field><Text-field style="Text" layout="Normal" alignment="right">(6)</Text-field><Text-field style="Text" layout="Normal" alignment="centred"></Text-field><Text-field style="Text" layout="Normal">This equation represents the ground state wavefunction evolving on the ground state potential energy surface from <Equation executable="true" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1JI21uR0YkNiRRIjBGJ0Y5RjUvJStleGVjdXRhYmxlR0Y9Rjk=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1JI21uR0YkNiRRIjBGJ0Y5RjUvJStleGVjdXRhYmxlR0Y9Rjk=</Equation>to <Equation executable="true" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlMtRiw2I1EhRidGOQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlMtRiw2I1EhRidGOQ==</Equation><Equation executable="true" style="2D Math" input-equation="" display="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">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</Equation>then transitioning instantaneously to the excited state surface where it evolves from <Equation executable="true" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlMtRiw2I1EhRidGOQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlMtRiw2I1EhRidGOQ==</Equation><Equation executable="true" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSJ+RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSS8lKXN0cmV0Y2h5R0ZJLyUqc3ltbWV0cmljR0ZJLyUobGFyZ2VvcEdGSS8lLm1vdmFibGVsaW1pdHNHRkkvJSdhY2NlbnRHRkkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZYRkMvJStleGVjdXRhYmxlR0ZJRj4=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSJ+RidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSS8lKXN0cmV0Y2h5R0ZJLyUqc3ltbWV0cmljR0ZJLyUobGFyZ2VvcEdGSS8lLm1vdmFibGVsaW1pdHNHRkkvJSdhY2NlbnRHRkkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZYRkMvJStleGVjdXRhYmxlR0ZJRj4=</Equation>to <Equation executable="true" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlMtRiw2I1EhRidGOQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlMtRiw2I1EhRidGOQ==</Equation><Equation executable="true" style="2D Math" input-equation="" display="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">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</Equation>Lastly, the wavefunction relaxes back to the ground state where it evolves from <Equation executable="true" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlMtRiw2I1EhRidGOQ==">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSI9RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlMtRiw2I1EhRidGOQ==</Equation><Equation executable="true" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRi82I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJ0Y+">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRi82I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJ0Y+</Equation> onwards (until time <Equation executable="true" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=</Equation>). As <Equation executable="true" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY1Rjk=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTEY1Rjk=</Equation>increase beyond the times when the transitions occured, the system will enter one of the product states, and a projection operator onto that state can be used to find the probability of forming that product. </Text-field><Text-field style="Text" layout="Normal"></Text-field><Text-field style="Text" layout="Normal">Because the dump pulse determines product selection, an important question is what the optimal dump pulse shape should be given a certain pump pulse. <Font encoding="UTF-8">If we have two discrete inital and final states, \317\207</Font><Equation executable="false" style="Text" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLzYlUSJpRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjdGOi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEifkYnL0Y7USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZXRkQ=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLzYlUSJpRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjdGOi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEifkYnL0Y7USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZXRkQ=</Equation>and <Font encoding="UTF-8">\317\207</Font><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field><Text-field style="Text" layout="Normal">with <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation><Font encoding="UTF-8">\317\207</Font><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation>and <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation><Font encoding="UTF-8">\317\207</Font><Equation executable="false" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUklbXN1YkdGJDYlLUkjbWlHRiQ2I1EhRictRiM2Jy1GLzYmUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRidGN0Y6LyUwZm9udF9zdHlsZV9uYW1lR1ElVGV4dEYnRj0vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi5RIn5GJ0Y6L0Y+USdub3JtYWxGJy8lJmZlbmNlR0Y8LyUqc2VwYXJhdG9yR0Y8LyUpc3RyZXRjaHlHRjwvJSpzeW1tZXRyaWNHRjwvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGPC8lJ2FjY2VudEdGPC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmZuLUZHNi5RIixGJ0Y6RkpGTC9GT0Y5RlBGUkZURlZGWEZaL0ZoblEsMC4zMzMzMzMzZW1GJ0ZGRjpGQEZK">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</Equation>then the probability of a transition between the inital and final states is determined by:</Text-field><Text-field style="Text" layout="Normal"><Equation executable="true" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic=">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=</Equation></Text-field><Text-field style="Text" layout="Normal" alignment="centred"><Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field><Text-field style="Text" layout="Normal" alignment="right">(7)</Text-field><Text-field style="Text" layout="Normal" alignment="centred"></Text-field><Text-field style="Text" layout="Normal">A normalization constraint is also enforced, since there is a finite number of photons in the pulse. </Text-field><Text-field style="Text" layout="Normal" alignment="centred"></Text-field><Text-field style="Text" layout="Normal" alignment="centred"><Equation executable="true" style="2D Math" input-equation="" display="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">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkobXN1YnN1cEdGJDYnLUkjbW9HRiQ2L1ErJkludGVncmFsO0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmdW5zZXRGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYjNiUtRi82L1EqJnVtaW51czA7RidGMkY1RjgvRjxGNC9GP0Y0L0ZBRjQvRkRGNC9GRkY0L0ZIRjQvRkpGNC9GTFEsMC4yMjIyMjIyZW1GJy9GT0Znbi1GLzYvUSgmaW5maW47RidGMkY1RjhGVUZWRldGWEZZRlpGZW5GS0ZORjgtRiM2JEZpbkY4LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy8lL3N1YnNjcmlwdHNoaWZ0R0Zgby1JJW1zdXBHRiQ2JS1JKG1mZW5jZWRHRiQ2Jy1GIzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEnJiM5NDk7RicvJSdpdGFsaWNHRjRGNUY4LUYjNiQtSSNtbkdGJDYmUSIyRidGMkY1RjhGOEZhby1GZ282JS1GIzYkLUZfcDYmUSJ0RicvRmNwRkJGNS9GOVEnaXRhbGljRidGOEY1RjhGOEY1RjgvJSVvcGVuR1EifGdyRicvJSZjbG9zZUdGZnEtRiM2JS1GZ3A2JEZpcEY4RmFxRmJxRl5vLUZfcDYlUSNkdEYnRmFxRmJxLUYvNi1RIn5GJ0Y4RlVGVkZXRlhGWUZaRmVuRktGTi1GLzYtUSI9RidGOEZVRlZGV0ZYRllGWkZlbi9GTFEsMC4yNzc3Nzc4ZW1GJy9GT0ZnckZgci1GZ3A2JFEiMUYnRjhGOA==</Equation></Text-field><Text-field style="Text" layout="Normal" alignment="right">(8)</Text-field><Text-field style="Text" layout="Normal">At this point, Tannor and Rice's derivations use variational analysis to determine the optimal pulse shape. The following functional, using equations (7) and (8), is maximized. </Text-field><Text-field style="Text" layout="Normal" alignment="centred"></Text-field><Text-field style="Text" layout="Normal" alignment="centred">I = <Equation executable="true" style="2D Math" input-equation="" display="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">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</Equation></Text-field><Text-field style="Text" layout="Normal" alignment="right">(9)</Text-field><Text-field style="Text" layout="Normal" alignment="centred"></Text-field><Text-field style="Text" layout="Normal">The resulting equation is:</Text-field><Text-field style="Text" layout="Normal" alignment="centred"></Text-field><Text-field style="Text" layout="Normal" alignment="centred"><Equation executable="true" style="2D Math" input-equation="" display="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">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</Equation> = <Equation executable="false" style="2D Math" input-equation="" display="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">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</Equation></Text-field><Text-field style="Text" layout="Normal" alignment="right">(10)</Text-field><Text-field style="Text" layout="Normal">where <Equation executable="true" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiUtRiM2JC1GLDYmUSJ0RidGL0YyRjUvRjZRJ25vcm1hbEYnRjJGQC1JI21vR0YkNi5RIn5GJ0YyRkAvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkgvJSlzdHJldGNoeUdGSC8lKnN5bW1ldHJpY0dGSC8lKGxhcmdlb3BHRkgvJS5tb3ZhYmxlbGltaXRzR0ZILyUnYWNjZW50R0ZILyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGV0ZA">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiUtRiM2JC1GLDYmUSJ0RidGL0YyRjUvRjZRJ25vcm1hbEYnRjJGQC1JI21vR0YkNi5RIn5GJ0YyRkAvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkgvJSlzdHJldGNoeUdGSC8lKnN5bW1ldHJpY0dGSC8lKGxhcmdlb3BHRkgvJS5tb3ZhYmxlbGltaXRzR0ZILyUnYWNjZW50R0ZILyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGV0ZA</Equation>is defined as follows. </Text-field><Text-field style="Text" layout="Normal" alignment="centred"></Text-field><Text-field style="Text" layout="Normal" alignment="centred"><Equation executable="true" style="2D Math" input-equation="" display="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">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</Equation></Text-field><Text-field style="Text" layout="Normal" alignment="right">(11)</Text-field><Text-field style="Text" layout="Normal" alignment="centred"></Text-field><Text-field style="Text" layout="Normal"></Text-field><Text-field style="Text" layout="Normal"><Equation executable="true" style="2D Math" input-equation="" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiUtRiM2JC1GLDYmUSJ0RidGL0YyRjUvRjZRJ25vcm1hbEYnRjJGQEZA">LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiUtRiM2JC1GLDYmUSJ0RidGL0YyRjUvRjZRJ25vcm1hbEYnRjJGQEZA</Equation> can be interpreted as a convolution of the pump pulse and the time-evolving wavepacket, and one is trying to maximize the response of the system via the dump pulse. </Text-field>
</Section><Text-field style="Text" layout="Normal" alignment="centred"></Text-field>
<Section collapsed="false" isCollapsible="true" drawButton="true" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title><Text-field style="Heading 1" layout="Heading 1">Applications</Text-field></Title><Text-field style="Text" layout="Normal"></Text-field><Text-field style="Text" layout="Normal">Optical manipulation of quantum systems on nanosecond or femtosecond scales has opened the door for useful technologies. One notable example is pump-probe spectroscopy, in which one is not necessarily attempting to control a chemical system, but investigating its properties. The transmission of the probe pulse in response to the pump pulse can reveal the identity of a sample, characterize nanostructures, and track reaction dyanamics. This technique has advantages such as high chemical selectivity and the ability to detect weakly fluorescent molecules. It also has promising applications in biological imaging, since it is a nonlinear optical technique and thus less affected by the thickness of the medium, namely tissue. </Text-field><Text-field style="Text" layout="Normal"></Text-field><Text-field style="Text" layout="Normal">Stimulated Raman adiabatic passage (STIRAP) is another optical control scheme that has applications in trapped, ultracold molecular systems, such as Bose-Einstein condensates, which have interesting properties. These ultracold systems are usually formed from atoms using laser-cooling, but the same techniques cannot be directly applied to diatomic molecules because the energy from photons can be absorbed into rotational and vibrational channels. With STIRAP, one can prepare diatomics in their rotational-vibrational ground state, overcoming this difficulty. Optics are also used to control ultracold qubits in quantum computers. </Text-field>
</Section><Text-field style="Text" layout="Normal" alignment="centred"></Text-field>
<Section collapsed="false" isCollapsible="true" drawButton="true" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title><Text-field style="Heading 1" layout="Heading 1">Conclusion</Text-field></Title><Text-field style="Text" layout="Normal">The Tannor-Rice method is a foundational method that introduces how one can control chemical reactions with optics. By delivering an appropriately timed dump pulse after a pump pulse, the reaction can be directed into a specific product. Beyond the Tannor-Rice method, many other optical methods are used for precise control and measurement of quantum systems. </Text-field>
</Section><Text-field style="Text" layout="Normal"></Text-field>
<Section collapsed="false" isCollapsible="true" drawButton="true" MultipleChoiceAnswerIndex="-1" MultipleChoiceRandomizeChoices="false" TrueFalseAnswerIndex="-1" EssayAnswerRows="5" EssayAnswerColumns="60"><Title><Text-field style="Heading 1" layout="Heading 1">References</Text-field></Title><Text-field style="Text" layout="Normal">[1] Stuart Alan Rice; Meishan Zhao. Optical Control of Molecular Dynamics; John Wiley: New York, 2000.</Text-field><Text-field style="Text" layout="Normal"></Text-field><Text-field style="Text" layout="Normal"><Font encoding="UTF-8">[2] Levis, R. J. Principles of the Quantum Control of Molecular Processes By Moshe Shapiro (Weizmann Institute of Science) and Paul Brumer (University of Toronto). John Wiley &amp; Sons, Inc.:\342\200\211 Hoboken. 2003. Xiv + 354 Pp. $79.95. ISBN 0-471-24184-9. J. Am. Chem. Soc. 2003, 125 (45), 13906\342\200\22313906. https://doi.org/10.1021/ja033533r.</Font></Text-field><Text-field style="Text" layout="Normal"></Text-field><Text-field style="Text" layout="Normal">[3] 7.3: The Heisenberg Uncertainty Principle. Physics LibreTexts. https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/University_Physics_III_-_Optics_and_Modern_Physics_(OpenStax)/07%3A_Quantum_Mechanics/7.03%3A_The_Heisenberg_Uncertainty_Principle (accessed 2023-02-23).
</Text-field><Text-field style="Text" layout="Normal"><Font encoding="UTF-8">[4] Franck\342\200\223Condon Principle. Wikipedia; 2022.</Font></Text-field><Text-field style="Text" layout="Normal"></Text-field><Text-field style="Text" layout="Normal"><Font encoding="UTF-8">[5] Baumert, T.; Helbing, J.; Gerber, G. Coherent Control With Femtosecond Laser Pulses. In Advances in Chemical Physics; John Wiley &amp; Sons, Ltd, 1997; pp 47\342\200\22382. https://doi.org/10.1002/9780470141601.ch2.</Font></Text-field><Text-field style="Text" layout="Normal"></Text-field><Text-field style="Text" layout="Normal"><Font encoding="UTF-8">[6] Dong, P.-T.; Cheng, J.-X. Pump\342\200\223Probe Microscopy: Theory, Instrumentation, and Applications. Spectroscopy 2017, 32 (4), 24\342\200\22336.</Font></Text-field><Text-field style="Text" layout="Normal"></Text-field><Text-field style="Text" layout="Normal"><Font encoding="UTF-8">[7] Bergmann, K.; N\303\244gerl, H.-C.; Panda, C.; Gabrielse, G.; Miloglyadov, E.; Quack, M.; Seyfang, G.; Wichmann, G.; Ospelkaus, S.; Kuhn, A.; Longhi, S.; Szameit, A.; Pirro, P.; Hillebrands, B.; Zhu, X.-F.; Zhu, J.; Drewsen, M.; Hensinger, W. K.; Weidt, S.; Halfmann, T.; Wang, H.-L.; Paraoanu, G. S.; Vitanov, N. V.; Mompart, J.; Busch, T.; Barnum, T. J.; Grimes, D. D.; Field, R. W.; Raizen, M. G.; Narevicius, E.; Auzinsh, M.; Budker, D.; P\303\241lffy, A.; Keitel, C. H. Roadmap on STIRAP Applications. J. Phys. B: At. Mol. Opt. Phys. 2019, 52 (20), 202001. https://doi.org/10.1088/1361-6455/ab3995.
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