Question

The first term in Equation 9.25 comes from the ${\mathit{e}}^{\mathbf{i}\mathbf{\omega }\mathbf{t}}\mathbf{/}\mathbf{2}$, and the second from $e^{-i\omega t}/2.$. Thus dropping the first term is formally equivalent to writing $\hat{H}=(V/2)e^{-i\omega t}$, which is to say,

$\begin{array}{l}{c}_b\left(l\right)-\frac{i}{h}{v}_{ba}{\int }_0^{t}cos\left(\omega t^{\text{'}}\right)e^{i{\omega }_0{t}^{\text{'}}}d{t}^{\text{'}}=-\frac{i{V}_{ba}}{2\sout{h}}{\int }_0^t\left[e^{j({\omega }_0+\omega )t^{\text{'}}}+e^{j({\omega }_0-\omega )t^{\text{'}}}\right]d{t}^{\text{'}}\\ =--\frac{i{V}_{ba}}{2\sout{h}}\left[\frac{e^{j({\omega }_0+\omega )t^{\text{'}}}-1}{{\omega }_0+\omega }+\frac{e^{j({\omega }_0-\omega )t^{\text{'}}}-1}{{\omega }_0-\omega }\right]\mathbf{}\mathbf{(}\mathbf{9}\mathbf{.}\mathbf{25}\mathbf{)}\mathbf{.}\\ H_{ba}^{\text{'}}=\frac{V_{ba}}{2}{e}^{-i\omega t},H_{ab}^{\text{'}}=\frac{V_{ab}}{2}{e}^{i\omega t}\mathbf{}\mathbf{}\mathbf{(}\mathbf{9}\mathbf{.}\mathbf{29}\mathbf{)}\mathbf{.}\end{array}$

(The latter is required to make the Hamiltonian matrix hermitian—or, if you prefer, to pick out the dominant term in the formula analogous to Equation 9.25 for$c_a\left(t\right)$. ) Rabi noticed that if you make this so-called rotating wave approximation at the beginning of the calculation, Equation 9.13 can be solved exactly, with no need for perturbation theory, and no assumption about the strength of the field.

${\stackrel{.}{c}}_a=-\frac{i}{\sout{h}}{H}_{ab}^{\text{'}}{e}^{-i{\omega }_{0}t}{c}_b,{\stackrel{.}{c}}_b=-\frac{i}{\sout{h}}{H}_{ba}^{\text{'}}{e}^{-i{\omega }_{0}t}{c}_a,$

(a) Solve Equation 9.13 in the rotating wave approximation (Equation 9.29), for the usual initial conditions: $c_a\left(0\right)=1,c_b\left(0\right)=0$. Express your results $\left({\mathrm{c}}_{\mathrm{a}}\right(\mathrm{t})\mathbf{and}{\mathrm{c}}_{\mathrm{b}}(\mathrm{t}\left)\right)$in terms of the Rabi flopping frequency,

${\mathrm{\omega }}_{\mathrm{r}}=\frac{1}{2}\sqrt{{(\mathrm{\omega }-{\mathrm{\omega }}_0)}^2+{(\left|{\mathrm{V}}_{\mathrm{ab}}\right|/\sout{\mathrm{h}})}^2}$ (9.30).

(b) Determine the transition probability,$P_{a\to b}\left(t\right)$, and show that it never exceeds 1. Confirm that.

${\left|c_a\left(t\right)\right|}^2+{\left|c_b\left(t\right)\right|}^2=1.$

(c) Check that $P_{a\to b}\left(t\right)$reduces to the perturbation theory result (Equation 9.28) when the perturbation is “small,” and state precisely what small means in this context, as a constraint on V.

$P_{a\to b}\left(t\right)={\left|c_b\left(t\right)\right|}^2\approx \frac{{\left|V_{ab}\right|}^2}{\sout{h}}\frac{si{n}^2\left[\left({\omega }_0-\omega \right)t/2\right]}{{\left({\omega }_0-\omega \right)}^2}\mathbf{(}\mathbf{9}\mathbf{.}\mathbf{28}\mathbf{)}$

(d) At what time does the system first return to its initial state?

Short answer

$$\begin{gathered} \mathbf{\left(}\mathit{a}\mathbf{\right)}{c}_b\left(t\right)=-\frac{i}{2\sout{h{\omega }_r}}{V}_{ba}{e}^{i({\omega }_0-\omega )t/2}sin\left({\omega }_{r}t\right),=e^{i(\omega -{\omega }_0)1/2}\left[cos\left({\omega }_{r}t\right)+i\left(\frac{{\omega }_0-\omega }{2{\omega }_0}\right)sin\left({\omega }_{r}t\right)\right] \\ \mathbf{\left(}\mathit{b}\mathbf{\right)}{\left|c_a\right|}^2+{\left|c_b\right|}^2=co{s}^2\left({\omega }_{r}t\right)+{\left(\frac{{\omega }_0-\omega }{2{\omega }_0}\right)}^{2}si{n}^2\left({\omega }_{r}t\right)+{\left(\frac{\left|V_{ab}\right|}{2\sout{h}{\omega }_r}\right)}^{2}si{n}^2\left({\omega }_{r}t\right). \\ \mathbf{\left(}\mathit{c}\mathbf{\right)}{P}_{a\to b}\left(t\right)={\left|c_b\left(t\right)\right|}^2\approx \frac{{\left|V_{ab}\right|}^2}{{\sout{h}}^2}\frac{si{n}^2\left[({\omega }_0-\omega )t/2\right]}{{({\omega }_0-\omega )}^2} \\ \mathbf{\left(}\mathit{d}\mathbf{\right)}{\omega }_{r}t=\mathrm{\pi }\Rightarrow \mathrm{t}=\mathrm{\pi }/{\mathrm{\omega }}_{\mathrm{r}}. \end{gathered}$$

Step-by-step solution

(a) Solving the equation 9.13 in wave rotating approximation

$\stackrel{}{c_a=-\frac{i}{2\sout{h}}}{V}_{ab}{e}^{i\omega t}{e}^{-i{\omega }_{0}t}{c}_b;\stackrel{.}{c_b}=-\frac{i}{2\sout{h}}{V}_{ba}{e}^{-i\omega t}{e}^{-i{\omega }_{0}t}{c}_a$

Differentiate the latter, and substitute in the former:

$$\begin{gathered} C_b=-i\frac{V_{ba}}{2\sout{h}}\left[i({\omega }_0-\omega )e^{i({\omega }_0-\omega )t}{c}_a+e^{i({\omega }_0-\omega )t}\dot{c_a}\right] \\ =i({\omega }_0-\omega )\left[-i\frac{V_{ba}}{2\sout{h}}{e}^{i({\omega }_0-\omega )t}{c}_a\right]-i\frac{V_{ab}}{2\sout{h}}{e}^{i({\omega }_0-\omega )t}\left[-i\frac{V_{ab}}{2\sout{h}}{e}^{i({\omega }_0-\omega )t}{c}_b\right]=i({\omega }_0-\omega )\dot{c_b}-\frac{{\left|V_{ab}\right|}^2}{{\left(2\sout{h}\right)}^2}{c}_b. \\ \frac{d^2{c}_b}{d{t}^2}+i(\omega -{\omega }_0)\frac{d{c}_b}{dt}+\frac{{\left|V_{ab}\right|}^2}{{\left(2\sout{h}\right)}^2}{c}_b=0.\mathrm{Soluation}\mathrm{of}\mathrm{the}\mathrm{form}{\mathrm{c}}_{\mathrm{b}}={\mathrm{e}}^{\mathrm{\lambda t}}:{\mathrm{\lambda }}^2+\mathrm{i}({\mathrm{\omega }}_0-\mathrm{\omega })\mathrm{\lambda }+\frac{{\left|{\mathrm{V}}_{\mathrm{ab}}\right|}^2}{4{\sout{\mathrm{h}}}^2} \\ \mathrm{\lambda }=\frac{1}{2}\left[-\mathrm{i}(\mathrm{\omega }-{\mathrm{\omega }}_0)\pm \sqrt{-{(\mathrm{\omega }-{\mathrm{\omega }}_0)}^2-\frac{\left|\mathrm{V}\right|}{}}\right] \end{gathered}$$_ab²h²=i-(ω-ω0)2±ωr,withωr.Generalsoluation:cb(t)=Aei(ω-ω0)2+ωrt+Bei(ω-ω0)2+ωrt=e-i(ω-ω0)t/2Aeiωrt+Be-iωrt,or,moreconveniently:cb(t)=e-i(ω-ω0)t/2Ccos(ωrt)+Dsin(ωrt).Butcb(0)=0soC=0cb(t)=Dei(ω0-ω)t/2sin(ωrt).cb=Diω0-ω2ei(ω0-ω)t/2sin˙uncaught exception: Invalid chunk

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#0 /var/www/html/integration/lib/php/Boot.class.php(769): com_wiris_plugin_impl_HttpImpl_1(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Invalid chunk') #1 /var/www/html/integration/lib/haxe/Http.class.php(532): _hx_lambda->execute('Invalid chunk') #2 /var/www/html/integration/lib/php/Boot.class.php(769): haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Invalid chunk') #3 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(30): _hx_lambda->execute('Invalid chunk') #4 /var/www/html/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Invalid chunk') #5 /var/www/html/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), Object(sys_net_Socket), NULL) #6 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(43): haxe_Http->request(true) #7 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(268): com_wiris_plugin_impl_HttpImpl->request(true) #8 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(307): com_wiris_plugin_impl_RenderImpl->showImage('b21ce702a20675c...', NULL, Object(PhpParamsProvider)) #9 /var/www/html/integration/createimage.php(17): com_wiris_plugin_impl_RenderImpl->createImage('uncaught exception: Invalid chunk

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$$\begin{gathered} \mathrm{conveniently}:{\mathrm{c}}_{\mathrm{b}}\left(\mathrm{t}\right)={\mathrm{e}}^{-\mathrm{i}(\mathrm{\omega }-{\mathrm{\omega }}_0)\mathrm{t}/2}\left[\mathrm{C}\cos \left({\mathrm{\omega }}_{\mathrm{r}}\mathrm{t}\right)+\mathrm{D}\sin \left({\mathrm{\omega }}_{\mathrm{r}}\mathrm{t}\right)\right].\mathrm{But}{\mathrm{c}}_{\mathrm{b}}\left(0\right)=0\mathrm{so}\mathrm{C}=0 \\ {\mathrm{c}}_{\mathrm{b}}\left(\mathrm{t}\right)={\mathrm{De}}^{\mathrm{i}({\mathrm{\omega }}_0-\mathrm{\omega })\mathrm{t}/2}\sin \left({\mathrm{\omega }}_{\mathrm{r}}\mathrm{t}\right).\dot{{\mathrm{c}}_{\mathrm{b}}=\mathrm{D}\left[\mathrm{i}\left(\frac{{\mathrm{\omega }}_0-\mathrm{\omega }}{2}\right){\mathrm{e}}^{\mathrm{i}({\mathrm{\omega }}_0-\mathrm{\omega })\mathrm{t}/2}\sin \left({\mathrm{\omega }}_{\mathrm{r}}\mathrm{t}\right)+{\mathrm{\omega }}_{\mathrm{r}}{\mathrm{e}}^{\mathrm{i}(\mathrm{\omega }-{\mathrm{\omega }}_0)\mathrm{t}/2}\cos \left({\mathrm{\omega }}_{\mathrm{r}}\mathrm{t}\right)\right]} \\ {\mathrm{c}}_a\left(\mathrm{t}\right)=i\frac{2\sout{h}}{V_{ba}}{e}^{\mathrm{i}({\mathrm{\omega }}_0-\mathrm{\omega })\mathrm{t}}\dot{c_b=}i\frac{2\sout{h}}{V_{ba}}{e}^{\mathrm{i}({\mathrm{\omega }}_0-\mathrm{\omega })\mathrm{t}/2}D\left[\mathrm{i}\left(\frac{{\mathrm{\omega }}_0-\mathrm{\omega }}{2}\right)\sin \left({\mathrm{\omega }}_{\mathrm{r}}\mathrm{t}\right)+{\mathrm{\omega }}_{\mathrm{r}}\cos \left({\mathrm{\omega }}_{\mathrm{r}}\mathrm{t}\right)\right].\mathrm{But}{\mathrm{c}}_{\mathrm{a}} \\ 1=\mathrm{i}\frac{2\sout{\mathrm{h}}}{{\mathrm{V}}_{\mathrm{ba}}}{\mathrm{D\omega }}_{\mathrm{r}},\mathrm{or}\mathrm{D}=\frac{-{\mathrm{iV}}_{\mathrm{ba}}}{2\sout{\mathrm{h}}{\mathrm{\omega }}_{\mathrm{r}}} \\ {\mathrm{c}}_{\mathrm{b}}\left(\mathrm{t}\right)=-\frac{\mathrm{i}}{2\sout{\mathrm{h}}{\mathrm{\omega }}_{\mathrm{r}}}{\mathrm{V}}_{\mathrm{ba}}{\mathrm{e}}^{\mathrm{i}({\mathrm{\omega }}_0-\mathrm{\omega })\mathrm{t}/2}\sin \left({\mathrm{\omega }}_{\mathrm{r}}\mathrm{t}\right),{\mathrm{c}}_{\mathrm{a}}\left(\mathrm{t}\right)={\mathrm{e}}^{\mathrm{i}(\mathrm{\omega }-{\mathrm{\omega }}_0)\mathrm{t}/2}\left[\cos \left({\mathrm{\omega }}_{\mathrm{r}}\mathrm{t}\right)+\mathrm{i}\left(\frac{{\mathrm{\omega }}_0-\mathrm{\omega }}{2}\right)\sin \left({\mathrm{\omega }}_{\mathrm{r}}\mathrm{t}\right)\right]. \end{gathered}$$

(b) Determining the transition probability

$$\begin{gathered} C_b=-i\frac{V_{ba}}{2\sout{h}}\left[i({\omega }_0-\omega )e^{i({\omega }_0-\omega )t}{c}_a+e^{i({\omega }_0-\omega )t}\dot{c_a}\right] \\ =i({\omega }_0-\omega )\left[-i\frac{V_{ba}}{2\sout{h}}{e}^{i({\omega }_0-\omega )t}{c}_a\right]-i\frac{V_{ab}}{2\sout{h}}{e}^{i({\omega }_0-\omega )t}\left[-i\frac{V_{ab}}{2\sout{h}}{e}^{i({\omega }_0-\omega )t}{c}_b\right]=i({\omega }_0-\omega )\dot{c_b}-\frac{{\left|V_{ab}\right|}^2}{{\left(2\sout{h}\right)}^2}{c}_b. \\ \frac{d^2{c}_b}{d{t}^2}+i(\omega -{\omega }_0)\frac{d{c}_b}{dt}+\frac{{\left|V_{ab}\right|}^2}{{\left(2\sout{h}\right)}^2}{c}_b=0.\mathrm{Soluation}\mathrm{of}\mathrm{the}\mathrm{form}{\mathrm{c}}_{\mathrm{b}}={\mathrm{e}}^{\mathrm{\lambda t}}:{\mathrm{\lambda }}^2+\mathrm{i}({\mathrm{\omega }}_0-\mathrm{\omega })\mathrm{\lambda }+\frac{{\left|{\mathrm{V}}_{\mathrm{ab}}\right|}^2}{4{\sout{\mathrm{h}}}^2} \\ \mathrm{\lambda }=\frac{1}{2}\left[-\mathrm{i}(\mathrm{\omega }-{\mathrm{\omega }}_0)\pm \sqrt{-{(\mathrm{\omega }-{\mathrm{\omega }}_0)}^2-\frac{{\left|{\mathrm{V}}_{\mathrm{ab}}\right|}^2}{{\sout{\mathrm{h}}}^2}}\right]=\mathrm{i}\left[-\frac{(\mathrm{\omega }-{\mathrm{\omega }}_0)}{2}\pm {\mathrm{\omega }}_{\mathrm{r}}\right],\mathrm{with}{\mathrm{\omega }}_{\mathrm{r}}. \\ \mathrm{General}\mathrm{soluation}:{\mathrm{c}}_{\mathrm{b}}\left(\mathrm{t}\right)={\mathrm{Ae}}^{\mathrm{i}\left[\frac{(\mathrm{\omega }-{\mathrm{\omega }}_0)}{2}+{\mathrm{\omega }}_{\mathrm{r}}\right]\mathrm{t}}+{\mathrm{Be}}^{\mathrm{i}\left[\frac{(\mathrm{\omega }-{\mathrm{\omega }}_0)}{2}+{\mathrm{\omega }}_{\mathrm{r}}\right]\mathrm{t}}={\mathrm{e}}^{-\mathrm{i}(\mathrm{\omega }-{\mathrm{\omega }}_0)\mathrm{t}/2}\left[{\mathrm{Ae}}^{{\mathrm{i\omega }}_{\mathrm{r}}\mathrm{t}}+{\mathrm{Be}}^{-{\mathrm{i\omega }}_{\mathrm{r}}\mathrm{t}}\right], \\ \mathrm{or},\mathrm{more} \\ \mathrm{conveniently}:{\mathrm{c}}_{\mathrm{b}}\left(\mathrm{t}\right)={\mathrm{e}}^{-\mathrm{i}(\mathrm{\omega }-{\mathrm{\omega }}_0)\mathrm{t}/2}\left[\mathrm{C}\cos \left({\mathrm{\omega }}_{\mathrm{r}}\mathrm{t}\right)+\mathrm{D}\sin \left({\mathrm{\omega }}_{\mathrm{r}}\mathrm{t}\right)\right].\mathrm{But}{\mathrm{c}}_{\mathrm{b}}\left(0\right)=0\mathrm{so}\mathrm{C}=0 \\ {\mathrm{c}}_{\mathrm{b}}\left(\mathrm{t}\right)={\mathrm{De}}^{\mathrm{i}({\mathrm{\omega }}_0-\mathrm{\omega })\mathrm{t}/2}\sin \left({\mathrm{\omega }}_{\mathrm{r}}\mathrm{t}\right).\dot{{\mathrm{c}}_{\mathrm{b}}=\mathrm{D}\left[\mathrm{i}\left(\frac{{\mathrm{\omega }}_0-\mathrm{\omega }}{2}\right){\mathrm{e}}^{\mathrm{i}({\mathrm{\omega }}_0-\mathrm{\omega })\mathrm{t}/2}\sin \left({\mathrm{\omega }}_{\mathrm{r}}\mathrm{t}\right)+{\mathrm{\omega }}_{\mathrm{r}}{\mathrm{e}}^{\mathrm{i}(\mathrm{\omega }-{\mathrm{\omega }}_0)\mathrm{t}/2}\cos \left({\mathrm{\omega }}_{\mathrm{r}}\mathrm{t}\right)\right]} \end{gathered}$$uncaught exception: Invalid chunk

in file: /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php line 68
#0 /var/www/html/integration/lib/php/Boot.class.php(769): com_wiris_plugin_impl_HttpImpl_1(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Invalid chunk') #1 /var/www/html/integration/lib/haxe/Http.class.php(532): _hx_lambda->execute('Invalid chunk') #2 /var/www/html/integration/lib/php/Boot.class.php(769): haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Invalid chunk') #3 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(30): _hx_lambda->execute('Invalid chunk') #4 /var/www/html/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Invalid chunk') #5 /var/www/html/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), Object(sys_net_Socket), NULL) #6 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(43): haxe_Http->request(true) #7 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(268): com_wiris_plugin_impl_HttpImpl->request(true) #8 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(307): com_wiris_plugin_impl_RenderImpl->showImage('46fca5c470c333f...', NULL, Object(PhpParamsProvider)) #9 /var/www/html/integration/createimage.php(17): com_wiris_plugin_impl_RenderImpl->createImage('$$\begin{gathered} P_{a\to b}\left(t\right)={\left|c_b\left(t\right)\right|}^2={\left(\frac{\left|V_{ab}\right|}{2\sout{h}{\omega }_r}\right)}^2{\sin }^2\left({\omega }_{r}t\right). \\ Thelargestthisgets(whwnsi{n}^2=1whensin2=1)is\frac{{\left|V_{ab}\right|}^2/\sout{h^2}}{4{\omega }_r^2} \\ Andthedenominatorexceedsthenumerator,soP>1(and1onlyif\omega ={\omega }_0) \\ {\left|c_a\right|}^2+{\left|c_b\right|}^2={\cos }^2\left({\omega }_{r}t\right)+{\left(\frac{{\omega }_0-\omega }{2{\omega }_r}\right)}^2{\sin }^2\left({\omega }_{r}t\right)+{\left(\frac{\left|V_{ab}\right|}{2h{\omega }_r}\right)}^2{\sin }^2\left({\omega }_{r}t\right). \\ ={\cos }^2\left({\omega }_{r}t\right)+\frac{{(\omega ={\omega }_0)}^2+(\left|V\right|}{} \end{gathered}$$_ab/h)²4ωr2sin2(ωrt)=cos2(ωrt)+sin2(ωrt)=1

:(c) Checking Pa→b(t)reduces to perturbation theory

If

$$\begin{gathered} {\left|V_{ab}\right|}^2\times \sout{h^2}{(\omega -{\omega }_0)}^2,then{\omega }_r\approx \frac{1}{2}\left|\omega -{\omega }_0\right|, \\ and{P}_{a\to b}\approx \frac{{\left|V_{ab}\right|}^2}{{\sout{h}}^2}\frac{{\sin }^2\left({\displaystyle \frac{\omega -{\omega }_0}{2}}t\right)}{{(\omega -{\omega }_0)}^2} \\ {P}_{a\to b}\left(t\right)={\left|c_b\left(t\right)\right|}^2\approx \frac{{\left|V_{ab}\right|}^2}{{\sout{h}}^2}\frac{{\sin }^2\left[\left(\omega -{\omega }_0)t/2\right)\right]}{{(\omega -{\omega }_0)}^2} \end{gathered}$$

(d) At time the system first returns to its initial stage

${\omega }_{r}t=\pi \Rightarrow t=\pi /{\omega }_r.$