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

Consider the coherence of the atom after coherent excitation with a short pulse (shorter than emission rate). Let the state of the atom be $|\psi >=C_{g}|g>+C_{e}|e>$. Then, the coherence between $|e>$ and $|g>$ is maximum for $C_{e}=C_{g}={\frac {1}{\sqrt {2}}}$, i.e. with ${\frac {\pi }{2}}$ pulse. (Coherence is $\rho _{ab}$ in the density matrix. For a pure state, it is $C_{g}C_{e}$).

Now, for simplicity of discussion, let us consider the case where there is only a single EM mode (ex, an atom strongly coupled to a cavity). Then, emission couples atomic states with photon number states: $|g,1>$ and $|e,0>$. Thus, a ${\frac {\pi }{2}}$ pulse also maximizes the coherence $|g,1>$ and $|e,0>$.

Since emission couples atomic states with 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 $|e_{2}>$ and decays to the continuum, it is impossible to tell whether it came from $|e_{1}>$ 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 $|k_{1}>$ or $|k_{2}>$ 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

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

vanishes (almost exactly) for a certain frequency $\omega _{o}$ that corresponds to an energy between the two levels. $\omega _{o}$ 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 $|g>$, although it would be absorbed if there was only a single excited level. Now assume that with some mechanism we populate, say, $|e_{2}>$ with a small number of atoms $N_{2}<N_{g}$.

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These atoms have maximum stimulated emission probability on resonance, $|e_{2}>\rightarrow |g>$, but there is also even larger absorption, since $N_{g}>N_{2}$. However, because of the finite linewidth $r_{2}$ of level $|e_{2}>$, there is also stimulated emission gain at the "magical" (absorption-free) frequency $\omega _{o}$. Since the $N_{g}$ atoms do not absorb here, there is net gain at this frequency in spite of $N_{2}<N_{g}$, 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 $lambda$-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 $?_{gf}=0$, an infinitesimally small coupling Rabi frequency, but the frequency window over which transmission occurs is very narrow and given by $\Delta \omega =\Omega c$.

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