Coherent dark states

Different from these normal dark states are the coherent dark states. Here two lasers are used (Fig. 7.27c). If laser 1 excites the atoms from 11) into 3) they can be transferred by laser 2 into level 2). From here they can be pumped again by laser 2 into level 3) and further by laser 1 into 11). For sufficiently large laser intensities the stimulated processes are fast compared with the spontaneous emission and the Raby oscillations dominate which periodically alter the populations of 1) and 2). 

Such a dark state is created by optical pumping. After a few excitation cycles the action of stimulated and spontaneous processes reaches the correct mixture of the levels 1) and 2). Then a steady state condition is reached where the level populations do not change any longer, because now no more transitions to level 3> occur. The population ratio iVi/N2 depends on the ratio of the transition probabilities ( i,3/ 2,3) - This coherent dark state fl ) is called a trapping state. 

Up to now the hyperfine transition in the ground state of the Cs-atom at 9.192 GHz represents the accepted frequency standard. An alternative to the cesium atomic fountain is the dark resonance of Cs atoms in a cell when a coherent dark state of the hyperfine levels is realized where the optical transition is excited by a frequency modulated laser with a modulation frequency which matches the hyperfine splitting in the Cs ground state. This modulation frequency can be used for the stabilization of the microwave which modulates the laser output. Since the dark resonance is very narrow, the uncertainty of the stable frequency is small. 

A. Nagel et al.. Experimental retilization of a coherent dark state magnetometer. Europhys. Lett. 44, 31-36 (1998) ... 

We have shown in Ref. [19] that if the systems in question have three levels, one can completely eliminate decoherence and disentanglement by imposing a special symmetry using the appropriate modulation. Thus, even if drastic reduction of all the decoherence matrix elements is not possible, then by using local modulations, one may equate the intraparticle elements, eliminate the interparficle elements, and code the QI in the ground and antisymmetric dark state of the two excited levels, and consequently completely preserve coherence and entanglement. 

EIT is based on the phenomenon of coherent population trapping [Harris 1997 Scully 1997 Liu 2001], in which the application of two laser fields to a three-level A system creates the so-called "dark state", which is stable against absorption of both fields. Dark states are also found in several other... 

A number of interesting conclusions follow from Eq. (81). In the first place, we note that the superposition states decay at different rates, the symmetric state decays with an enhanced rate (F I T ), while the antisymmetric state decays at a reduced rate (r — 1)2). For F12 = T, the antisymmetric state does not decay at all. In this case the antisymmetric state can be regarded as a dark state in the sense that the state is decoupled from the environment. Second, we note from Eq. (81) that the state a) is coupled to the state j) through the splitting A, which plays a role here similar to the Rabi frequency of the coherent interaction between the symmetric and antisymmetric states. Consequently, an initial population in the state a) can be coherently transferred to the state j), which rapidly decays to the ground state. When A = 0, that is, the excited states are degenerate, the coherent interaction does not take place and then any initial population in a) will stay in this state for all times. In this case we can say that the population is trapped in the state u). 

Fig, 7.27 (a) and (b) Normal incoherent dark states (c) coherent dark... 

The transition amplitude for the dipole transition from the dark state ui)coherent to the upper state 3> is... 

The numerical examples illustrates that the usual techniques of optical cooling are not sufficient to reach BEC at realistic atomic densities unless one uses the experimentally difficult Raman-cooling or the coherent generation of dark states (see Sect. 7.10). 

The basic idea of STIRAP to transfer population between two quantum states relies on a particularly clever implementation of a coherent two-photon Raman transition, involving a dark state during the transfer. For a detailed description of its principles and an overview of early applications, the reader may refer to Ref. [86]. 

An essential property of such a coherently coupled three-level system is the existence of a dark state D) as an eigenstate of the system. This state generally occurs if both laser fields have the same resonance detuning with respect to the corresponding transition, that is, if the two-photon demning is zero. The state is dark in the sense that it is decoupled from the excited state e) and thus not influenced by its radiative decay. The dark state can be understood as a coherent superposition of state a) and state b),... 

Double-dark resonances have been demonstrated in a variety of four-level systems [39-50], where the probe absorption spectrum is characterized by two HIT windows, separated by a sharp absorption peak [51]. The appearance of the central narrow structure is due to the coherent interaction between the two dark states [39], which greatly enhances the Kerr nonlinear susceptibility [55]. In this section, we present our atom localization schemes based on double-dark resonance effects in two different four-level atomic systems. 

Emission processes always lead to a photon-related recoil velocity for an atom. Thus, in the quest for stiU lower temperatures it is necessaury to ascertain that an atom, which for some reason is brought to a standstill, can be exempt from further interaction with the fight. This is possible if the atom is placed in a so-called dark state [9.440]. If the atom is in a coherent superposition of two ground-state sublevels, from which the transition amplitudes exhibit a total destructive interference, a dark state is achieved (See also Sect. 9.5.3). It can be shown that for counter-propagating beams with circular polariza-... 

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