Tmp Lecture 23

Rabi frequencies, the dark state is predominantly the state with the weaker coupling. For e.g., $\omega _{1}=0$, the dark state is trivially $|D>=|g>$.

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

clarification on coherence and dipole moment

When considering coherence in the atom (after coherent excitation with a short pulse), or in the light, we have to be careful about the basis used. In the atom with states $|g>,|e>$, clearly the coherence is maximum for $C_{e}=C_{g}={\frac {1}{\sqrt {2}}}$, i.e. for ? ${\frac {\pi }{2}}$ pulse. Since emission maps atomic states onto photon number states $a,|g>\rightarrow |o>,|e>\rightarrow |1?$ (considering only a single em mode, i.e. an atom strongly coupled to a cavity), a ${\frac {\pi }{2}}$ pulse also maximizes the coherence <underline> <attributes> </attributes> between photon number states |o>,|1>. </underline>. On the other hand, for continuous excitation we know that the <underline> <attributes> </attributes> light is coherent </underline> in the Rayleigh scattering limit (i.e. then the frequency spectrum of the scattered light is a $\delta$-function at the incident frequency, for an infinitely heavy atom), while saturation of the atom leads to emission of increasingly incoherent light (Mollow triplet). Monochromatic, coherent light is represented qm-ly by a coherent state$|\alpha >$ that has a Poissonian distribution of photon numbers. Thus is <underline> <attributes> </attributes> the full basis of photon number states </underline> light is coherent for the state $\alpha >=e^{-{\frac {1}{z}}}|+|^{2}\sigma _{h=0}^{\varpropto }{\frac {\alpha ^{n}}{\sqrt {n+1}}}|n>$. For $|\varpropto |\ll |$, the population of the states with $n>1$ is negligible, and the atom prepared in a state $|g>+\epsilon |e>$ with $|\epsilon |\ll 1$ emits a coherent state of light(albeit with very small electric-field amplitude), in agreement with what is expected for small saturation.

Back to dark state in a $\lambda$-system.

First observation of coherent population trapping CPT

Multimode laser

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with regular frequency spacing Gas in a cylindrical volume with gradient of magnetic field applied, observe fluorescence

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dark region where Zeeman shift between magnetic sublevels equals frequency difference between laser modes.

Absorption calculation by interference, goin without inversion

(Steve Harris, PRL <underline> <attributes> </attributes> 62 </underline>, 1033 (1989))

It is commonly believed that we need $Ne>N_{g}$ for optical gain. But: Consider a V system with two unstable states that decay by coupling to the <underline> <attributes> </attributes> same </underline> continuum.

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i.e. if an atom is placed in $|e_{1}>$ or 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 |e_2>} and decays to the continuum, it is impossible to tell whether it came from 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 |e_1>} or $|e_{2}>$.

(This is a fairly special situation, e.g. different m-levels do not qualify, since they emit photons of different polarizations, thus the continue 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 |k_1>} or 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 |k_2>} are distinguishable.) Then the two-photon scattering process $|g>\rightarrow |continuum>$ can proceed via two pathways that are fundamentally indistinguishable

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and we must add the amplitudes. The second order matrix element in perturbation theory

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 ?=\stackrel{?}{i}\frac{<continuum|V'|e_ i><ei|V|g>}{\Delta _ i-i\frac{r_:}{2}}}

vanishes (almost exactly) for a certain frequency $\omega _{o}$ that corresponds to an energy between the two levels. 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 _ o} depends on the two matrix elements and we assume $(\Delta ,1,|\Delta _{2}||ggr_{1},r_{2}$. Then tat frequency is not absorbed by atoms in 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 |g>} , although it would be absorbed if there was only a single excited level. Now assume that with some mechanism we populate, say, 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 |e_2>} with a small number of atoms $N_{2}<N_{g}$.

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These atoms have maximum stimulated emission probability on resonance, 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 |e_2>\rightarrow |g>} , but there is also even larger absorption, since 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_ g>N_2} . However, because of the finite linewidth $r_{2}$ of level 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 |e_2>} , there is also stimulated emission gain at the "magical" (absorption-free) 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 _ o} . Since the $N_{g}$ atoms do not absorb here, there is net gain at this frequency in spite of 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_2<N_ g} , which can lead to "lasing without inversion." Note: this only works if the two excited states decay to the same continuum, such that the paths are indistinguishable. How can a system for lasing without inversion be realized?

Possibility 1: hydrogen and dc electric field

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Possibility 2: use ac electric field to mix non-degenerate s state with p state.

Electromagnetically induced transparency

"Is it possible to send a laser beam through a brick wall?"

Radio Yerevan: "In principle yes, but you need another very powerful laser..."

Steve Harris thought initially of special, ionizing excited states. However it is possible to realize the requirement of identical decay paths in 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 lambda} -system with a a(strong) coupling laser. The phenomenon is closely related to coherent population trapping.

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For resonant fields $\omega _{1}=\omega _{ge},\omega _{2}=\omega _{fe}$, we have

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As we turn up the power of the coupling laser the transmission improves and then broadens (in the realistic case of a finite decoherence rate 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 ?_{gf}=0} , an infinitesimally small coupling Rabi frequency, but the frequency window over which transmission occurs is very narrow and given by 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 \omega =\Omega c} .

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