Tmp Lecture 25

<framebox> <attributes> <width>None</width> <pos>None</pos> </attributes> Lecture XXV </framebox>

Two-phase absorption, Fano profiles

Let us assume large one-photon detuning, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta \gg \Gamma} , weak probe $\omega _{1}$ and strong control field $\omega _{2}$ (we also define the two-photon detuning $\delta =\Delta _{1}-\Delta _{2}$).

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In this limit analytic expressions for the absorption cross section for beam $\omega _{1}$ and the refractive index seen by beam $\omega _{1}$ exist, e.g. [Muller et al., PRA 56, 2385 (1997)]

The refractive index is given by: $n(d)=1+{\frac {3}{8\pi }}{\tilde {n}}\lambda ^{3}f(\Delta )$

where ${\tilde {n}}$ is the atomic density, $f(\delta )={\frac {\Gamma }{2}}\delta {\frac {A}{A^{2}+B^{2}}},A=\omega _{2}^{2}-\delta \Delta _{2},B=\delta \Gamma$.

For zero ground-state linewidth (decoherence between the ground-states) $\sigma (\delta )=\sigma _{0}.g\left(\delta \right)$ where $\sigma _{0}={\frac {3}{2\pi }}\lambda ^{2}$ is the resonant cross-section, and $g(\Delta )={\frac {\Gamma }{2}}{\frac {\delta B}{A^{2}+B^{2}}}$.

The absorption cross section is for $\Delta _{2}>0$

This is like a ground state coupling to one narrow and one wide excited state, except that there is EIT in between because both states decay to the same continuum.

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At $\delta =\Delta _{2}$ , we have one-photon absorption, which is a two-photon scattering process:

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At $\delta =-{\frac {\omega _{2}^{2}}{\Delta _{2}}}$, two-photon absorption, which is (at least) four-photon scattering process:

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For the EIT condition $\delta =0$, there is no coupling to the excited state, and the refractive index is zero. In the vicinity of EIT, there is steep dispersion, resulting in a strong alteration of the group velocity of light$\rightarrow$ slowing and stopping light.

Slow light, adiabatic changes of velocity of light

The group velocity of light in the presence of linear dispersion Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dn}{d\omega }} is given by (Harris and Han, PRC <underline> <attributes> </attributes> 82 </underline>, 4611 (1999))

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_ g=\frac{c}{1+\omega \frac{dn}{d\omega }}} for light of frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega }

A strong linear dispersion with positive slope near EIT then corresponds to very slow light

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Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=1\rightarrow } electric field ???$\rightarrow$ power per area

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{P}{A}=\frac{1}{2}\epsilon _ oC|E|^2} unchanged

Pulse is compressed in medium Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rightarrow } energy density is increased, light is partly in the form of an atomic excitation : polarization

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For sufficiently small Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega _2} , velocity of light may be very small...(L.V. Hau, S.E. Harris, Z. Dutton, and C.H. Behrooti, Nature <underline> <attributes> </attributes> 397 </underline>, 594 (1999)), as observed in a BEC. What happens if experiment performed in room temperature vapor when ???? ? The same if setup is Doppler free (co-propagating probe and control fields Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega _1} and $\omega _{2}$). If we change control field adiabatically while the pulse is inside the medium, we can coherently stop light, i.e. convert it into an atomic excitation or spin wave. With the reverse process we can then convert the stored spin-wave back into the original light field. The adiabatic conversion is made possible by the finite splitting between bright and dark states. In principle, all coherence properties and other (qm) features of the light are maintained, and it is possible to store non-classical states of light by mapping photon properties one-to-one onto quantized spin waves. More about these quanta called "dark-state polaritons" once we have introduced Dicke states. Is it possible to make use of EIT for, e.g. atom detection without absorption? Answer: no improvement for such linear processes. However: improvement for non-linear processes is possible.

Superradiance

Assume that two identical atoms, one in its ground and the other in its excited state, are placed within a distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\gg \lambda _{eg}} of each other. What happens?

For a simgle atom we have for the emission rate R(t) at time t an emission probability Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(t)=\int _ o^ tR(t')dt'} to have emitted a photon by time t:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_1(t)=re^{-rt}}

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Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_1(t)=\int _ o^ tdt'R(t')=1-e^{-rt}} for a single atom

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what about two atoms?

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It turns out that the correct answer is

$R_{2}(t)=re^{-2pt}$

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_2(t)=\int ^ t_ oR_2(t')dt'=\frac{1}{2}(1-e^{-2rt})}

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The photon is emitted with the same initial rate, but has only probability Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2}} of being emitted at all! How can we understand this? The interaction Hamiltonian is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V=-\vec{d}_1\cdot \vec{E}(\vec\gamma ,t)-\vec{d}_2\cdot \vec{E}(\vec\gamma ,t)=-\vec{D}\cdot \vec{E}(\vec\gamma ,t)} class.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V=\hbar g(\sigma ^+_1+\sigma ^-_1)(a+a^+)g(\sigma _2^++\sigma _2^-)(a+a^+) QED} $=\hbar g((\sigma _{1}^{+}+\sigma _{2}^{+}+\sigma _{1}^{-}+\sigma _{2}^{-})(a+a^{+})$ Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\hbar g(\Sigma ^++\Sigma ^-)(a+a^+)}

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma ^{\pm }\dot\Sigma ^ z_{i=1}\sigma ^{\pm }_ i, \vec{D}=\Sigma ^ z_{i=1}\vec{d}_ i}