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<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"/> </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="285" 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="109" left="342" top="12" width="69" height="87.75" align-x="342" align-y="12" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <picture> <png item-idref="2" display-width="67.5" display-height="86.25"/> </picture> <rendering item-idref="3"/> </region> <region region-id="67" left="24" top="12.75" width="276" height="14.25" align-x="24" align-y="24" 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"> <f size="12"> Time development of a coupled two level system </f> </text> </region> <region region-id="84" left="246" top="36" width="60.75" height="45" align-x="246" align-y="36" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <picture> <png item-idref="4" display-width="59.25" display-height="43.5"/> </picture> <rendering item-idref="5"/> </region> <region region-id="132" left="48" top="56.25" width="139.5" height="12" align-x="55.5" align-y="66" 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"> The Hamiltonian for this case is: </text> </region> <region region-id="131" left="48" top="104.25" width="438.75" height="12" align-x="48" align-y="114" 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"> Given that you start in state |a>, what is the time-dependent probability of ending up in state |b>? </text> </region> <region region-id="130" left="48" top="134.25" width="85.5" height="12" align-x="69" 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="true"> Frequency splitting: </text> </region> <region region-id="103" left="144" top="135" width="27.75" height="12.75" align-x="159" align-y="144" 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">Δ</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="138" left="48" top="158.25" width="67.5" height="12" align-x="48" align-y="168" 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"> Rabi frequency: </text> </region> <region region-id="139" left="138" top="151.5" width="78" height="20.25" align-x="165.75" align-y="168" 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">V</ml:id> </ml:boundVars> </ml:function> <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:id xml:space="preserve">Δ</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="140" left="282" top="158.25" width="159.75" height="24" align-x="282" align-y="168" 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"> Note here I have expressed energy as inverse time. E/hbar </text> </region> <region region-id="141" left="138" top="188.25" width="148.5" height="39.75" align-x="171" align-y="210" 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">V</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">V</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:parens> <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:id xml:space="preserve">Δ</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:parens> <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">V</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="8"/> </region> <region region-id="150" left="48" top="242.25" width="78.75" height="12" align-x="60.75" align-y="252" 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"> Define a time grid: </text> </region> <region region-id="151" left="138" top="243" width="42.75" height="12.75" align-x="147" align-y="252" 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="9"/> </region> <region region-id="152" left="204" top="236.25" width="35.25" height="27.75" align-x="219" align-y="252" 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="10"/> </region> <region region-id="126" left="48" top="272.25" width="331.5" height="12" align-x="66" align-y="282" 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"> Calculate probabilities for different values of the coupling: V/Δ = 0.1, 0.5, 1, 5 </text> </region> <region region-id="125" left="48" top="303" width="57" height="18" align-x="66.75" align-y="312" 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:real>5</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="11"/> </region> <region region-id="123" left="138" top="303" width="57" height="18" align-x="156.75" align-y="312" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math 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</region> <region region-id="131" left="48" top="104.25" width="438.75" height="12" align-x="48" align-y="114" 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"> Given that you start in state |a>, what is the time-dependent probability of ending up in state |b>? </text> </region> <region region-id="130" left="48" top="134.25" width="85.5" height="12" align-x="69" 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="true"> Frequency splitting: </text> </region> <region region-id="103" left="144" top="135" width="27.75" height="12.75" align-x="159" align-y="144" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define 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<ml:apply> <ml:pow/> <ml:id xml:space="preserve">V</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">Δ</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="140" left="282" top="158.25" width="159.75" height="24" align-x="282" align-y="168" 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"> Note here I have expressed energy as inverse time. E/hbar </text> </region> <region region-id="141" left="138" top="188.25" width="148.5" height="39.75" align-x="171" align-y="210" 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">V</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:div/> <ml:apply> <ml:pow/> <ml:id xml:space="preserve">V</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:parens> <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:id xml:space="preserve">Δ</ml:id> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:parens> </ml:apply> </ml:parens> <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">V</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="8"/> </region> <region region-id="150" left="48" top="242.25" width="78.75" height="12" align-x="60.75" align-y="252" 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"> Define a time grid: </text> </region> <region region-id="151" left="138" top="243" width="42.75" height="12.75" align-x="147" align-y="252" 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="9"/> </region> <region region-id="152" left="204" top="236.25" width="35.25" height="27.75" align-x="219" align-y="252" 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="10"/> </region> <region region-id="126" left="48" top="272.25" width="331.5" height="12" align-x="66" align-y="282" 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"> Calculate probabilities for different values of the coupling: V/Δ = 0.1, 0.5, 1, 5 </text> </region> <region region-id="125" left="48" top="303" width="57" height="18" align-x="66.75" align-y="312" 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:real>5</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="11"/> </region> <region region-id="123" left="138" top="303" width="57" height="18" align-x="156.75" align-y="312" 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">P2</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:real>1</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="122" left="240" top="303" width="63.75" height="18" align-x="258.75" align-y="312" 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">P3</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:real>0.5</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="121" left="348" top="303" width="63.75" height="18" align-x="366.75" align-y="312" 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">P4</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:real>0.1</ml:real> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="137" left="54" top="336" width="312" height="174.75" 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