Mathcad Professional 14.0 <description/> <author>Andrei Tokmakoff</author> <company/> <keywords/> <revisedBy>Andrei Tokmakoff</revisedBy> </userData> <identityInfo> <revision>2</revision> <documentID>8B791799-6769-4B81-9793-111E2C2B8310</documentID> <versionID>1500E17D-C7D0-4A33-954D-E6D59CE567B2</versionID> <parentVersionID>00000000-0000-0000-0000-000000000000</parentVersionID> <branchID>00000000-0000-0000-0000-000000000000</branchID> </identityInfo> </metadata> <settings> <presentation> <textRendering> <textStyles> <textStyle name="Normal"> <blockAttr margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"/> <inlineAttr font-family="Arial" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> <textStyle name="Heading 1"> <blockAttr margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"/> 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<inlineAttr font-family="Times New Roman" font-charset="0" font-size="24" font-weight="bold" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> <textStyle name="Subtitle" base-style="Title"> <blockAttr margin-left="0" margin-right="0" text-indent="0" text-align="center" list-style-type="none" tabs="inherit"/> <inlineAttr font-family="Times New Roman" font-charset="0" font-size="18" font-weight="normal" font-style="normal" underline="false" line-through="false" vertical-align="baseline"/> </textStyle> </textStyles> </textRendering> <mathRendering equation-color="#000"> <operators multiplication="narrow-dot" derivative="derivative" literal-subscript="small" definition="colon-equal" global-definition="triple-equal" local-definition="left-arrow" equality="bold-equal" symbolic-evaluation="right-arrow"/> <mathStyles> <mathStyle name="Variables" font-family="Times New Roman" font-charset="0" font-size="10" font-weight="normal" 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</mathRendering> <pageModel show-page-frame="false" show-header-frame="false" show-footer-frame="false" header-footer-start-page="1" paper-code="1" orientation="portrait" print-single-page-width="false" page-width="612" page-height="792"> <margins left="50.4" right="50.4" top="50.4" bottom="50.4"/> <header use-full-page-width="true"/> <footer use-full-page-width="true"> <center>{n}</center> </footer> </pageModel> <colorModel background-color="#fff" default-highlight-color="#ffff80"/> <language math="en" UI="en"/> </presentation> <calculation> <builtInVariables array-origin="0" convergence-tolerance="0.001" constraint-tolerance="0.001" random-seed="1" prn-precision="4" prn-col-width="8"/> <calculationBehavior automatic-recalculation="true" matrix-strict-singularity-check="true" optimize-expressions="false" exact-boolean="false" strings-use-origin="false" zero-over-zero="0"> <compatibility multiple-assignment="MC11" local-assignment="MC11"/> </calculationBehavior> <units> <currentUnitSystem name="si" customized="false"/> </units> </calculation> <editor view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="in"/> <plotTemplate> <xy item-idref="1"/> </plotTemplate> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="true" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="292" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="179" left="24" top="18.75" width="210.75" height="14.25" align-x="24" align-y="30" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit"> <f size="12"> <b> <u>Resonant driving of two-level system</u> </b> </f> </p> </text> </region> <region region-id="184" left="42" top="44.25" width="147.75" height="12" align-x="42" align-y="54" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Input energy splitting and coupling</p> </text> </region> <region region-id="185" left="246" top="45" width="32.25" height="15" align-x="265.5" align-y="54" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="kl">ω</ml:id> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="2"/> </region> <region region-id="186" left="300" top="45" width="25.5" height="12.75" align-x="312.75" align-y="54" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">V</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="3"/> </region> <region region-id="187" left="42" top="74.25" width="67.5" height="12" align-x="42" align-y="84" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Rabi frequency:</p> </text> </region> <region region-id="193" left="42" top="91.5" width="127.5" height="28.5" align-x="70.5" align-y="108" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define warning="WarnRedefinedBIUnit" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">Ω</ml:id> <ml:boundVars> <ml:id xml:space="preserve">ω</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:parens> <ml:apply> <ml:sqrt/> <ml:apply> <ml:plus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">V</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve" subscript="kl">ω</ml:id> <ml:id xml:space="preserve">ω</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="190" left="42" top="134.25" width="243" height="12" align-x="63" align-y="144" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Probability of transfer from intial state (l) to final state (k)</p> </text> </region> <region region-id="191" left="42" top="164.25" width="165.75" height="42" align-x="75.75" align-y="186" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">P</ml:id> <ml:boundVars> <ml:id xml:space="preserve">t</ml:id> <ml:id xml:space="preserve">ω</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">V</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">V</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve" subscript="kl">ω</ml:id> <ml:id xml:space="preserve">ω</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:pow/> <ml:apply> <ml:id xml:space="preserve">sin</ml:id> <ml:apply> <ml:mult/> <ml:apply> <ml:id xml:space="preserve">Ω</ml:id> <ml:id xml:space="preserve">ω</ml:id> </ml:apply> <ml:id xml:space="preserve">t</ml:id> </ml:apply> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="197" left="42" top="230.25" width="45" height="12" align-x="51" align-y="240" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Time axis:</p> </text> </region> <region region-id="194" left="126" top="231" width="42.75" height="12.75" align-x="135" align-y="240" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">i</ml:id> <ml:range> <ml:real>0</ml:real> <ml:real>500</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="195" left="186" top="224.25" width="35.25" height="27.75" align-x="201" align-y="240" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">τ</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:id xml:space="preserve">i</ml:id> <ml:real>50</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="201" left="42" top="260.25" width="273" height="15" align-x="60" align-y="270" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Evalulate time-dependence for four values of the detuning ω<sub>kl</sub>-ω.</p> </text> </region> <region region-id="251" left="48" top="285" width="36.75" height="12.75" align-x="60" align-y="294" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">n</ml:id> <ml:range> <ml:real>0</ml:real> <ml:real>3</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="8"/> </region> <region region-id="274" left="372" top="290.25" width="120.75" height="36" align-x="381" align-y="300" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Here the units of detuning should be measured with respect to the coupling:</p> </text> </region> <region region-id="252" left="216" top="288" width="24" height="84.75" align-x="221.25" align-y="306" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">ω</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:sequence> <ml:real>0</ml:real> <ml:real>0.5</ml:real> <ml:real>1</ml:real> <ml:real>3</ml:real> </ml:sequence> </ml:define> </math> <rendering item-idref="9"/> </region> <region region-id="272" left="48" top="326.25" width="44.25" height="12" align-x="66" align-y="336" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Detuning :</p> </text> </region> <region region-id="273" left="114" top="327" width="63.75" height="18" align-x="133.5" align-y="336" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">Δ</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:apply> <ml:minus/> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">ω</ml:id> <ml:id xml:space="preserve">n</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="kl">ω</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="255" left="288" top="308.25" width="51" height="62.25" align-x="301.5" align-y="342" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">Δ</ml:id> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="4" cols="1"> <ml:real>0</ml:real> <ml:real>0.5</ml:real> <ml:real>1</ml:real> <ml:real>3</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="11"/> </region> <region region-id="275" left="402" top="326.25" width="54" height="62.25" align-x="418.5" align-y="360" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:div/> <ml:id xml:space="preserve">Δ</ml:id> <ml:id xml:space="preserve">V</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="4" cols="1"> <ml:real>0</ml:real> <ml:real>0.5</ml:real> <ml:real>1</ml:real> <ml:real>3</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="12"/> </region> <region region-id="258" left="48" top="423" width="63.75" height="18" align-x="66.75" align-y="432" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">P1</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">P</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">τ</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">ω</ml:id> <ml:real>0</ml:real> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="13"/> </region> <region 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