Spontaneous emission rate

Einstein derived the relationship between spontaneous emission rate and the absorption intensity or stimulated emission rate in 1917 using a thennodynamic argument [13]. Both absorption intensity and emission rate depend on the transition moment integral of equation (B 1.1.1). so that gives us a way to relate them. The symbol A is often used for the rate constant for emission it is sometimes called the Einstein A coefficient. For emission in the gas phase from a state to a lower state j we can write... 

Note that negative Acoj (red detuning) produces a force attracting the atom to the intensity maximum while positive (blue detuning) repels the atom away from the intensity maximum. The spontaneous force or cooling force can also be written in tenns of the saturation parameter and the spontaneous emission rate. 

However, it is possible to concentrate most of the radiation onto a few modes in such a way that the number of photons in those modes becomes large and the stimulated emission in those modes will dominate (although the total spontaneous emission rate into all modes may still be larger than the induced rate in these few modes). Such selection of few modes is realized in a laser by using an appropriate resonator, which should exhibit a strong feedback for those modes. The resonator will allow an intense radiation field to be built in the modes with low losses, and will prevent oscillation from being reached in the modes with high losses. 

This effective dye relaxation time rp is the spontaneous fluorescence decay time shortened by stimulated emission which is more severe the higher the excitation and therefore the higher the population density w j. The dependence of fluorescence decay time on excitation intensity was shown in 34 35>. Thus, fluorescence decay times measured with high intensity laser excitation 3e>37> are often not the true molecular constants of the spontaneous emission rate which can only be measured under low excitation conditions. At the short time scale of modelocking the reorientation of the solvent cage after absorption has occurred plays a certain role 8 > as well as the rotational reorientation of the dye molecules 3M°)... 

It is possible to subdivide the radiative terms further into individual spontaneous emission rates, corresponding to transitions to a group of lower-energy states. That is,... 

Just as we previously summed the spontaneous emission rates over the possible final states to obtain the total spontaneous decay rate l/rnt, we can sum the black... 

Fig. 5.3 Spontaneous emission rates of the 18s state to lower lying n p states, An,P 18s, ( ) and the 300 K black body transition rates of the 18s state to higher and lower lying n p states, Kn p 18s, ( ) as a function of n (from ref. 5).

Spontaneous emission and radiative lifetime of lanthanide excited state in condensed phases is determined by the electromagnetic field and the index of refraction as shown in eq. (3). In nanocrystals, spontaneous emission of photons is influenced by two mechanisms (1) the non-solid medium surrounding the nanoparticles that changes the effective index of refraction thus influences the radiative lifetime (Meltzer et al., 1999 Schniepp and Sandoghdar, 2002), (2) size-dependent spontaneous emission rate due to interferences (Schniepp and Sandoghdar, 2002). 

Figure 1. Four-level molecular model. QiS is the collisional-transfer rate constant from level i to level j, TV is the sum of the electronic quenching and spontaneous emission rate constants, W,t is the absorption rate constant, and Wlt is the stimulated emission rate constant. WIt and WtI are proportional to the laser power PL. The dashed vertical line separates levels le and 2e, which are treated as an isolated system, from those levels not affected directly by the laser radiation.

This important relation allows us to replace the matrix element and the combined density of states in the spontaneous emission rate by the absorption coefficient a(fkj) and makes this treatment applicable to real materials once the absorption coefficient is known ... 

The above treatment is valid quite generally, even if f2 — fi > huj. when the denominator in (4.17) is negative. Under the same conditions, the absorption coefficient in (4.15) is also negative, and the spontaneous emission rate in (4.17) remains positive. When the absorption coefficient is negative, stimulated emission overcompensates the rate of upward transitions and the 2-level system amplifies the incident light exactly as in a laser. The condition f2 — fi > tko is also known as the lasing condition and is called inversion. 

Radiative Recombination. The rate of radiative recombination follows from the relations (4.17) and (4.11) for a 2-level system. Since we have expressed the spontaneous emission rate in terms of the absorption coefficient, integration over all transitions involving identical photon energies is already taken into account by using the absorption coefficient for the 2-band system. Integration over all photon energies occurring in transitions between the conduction band and the valence band yields the rate of radiative recombination ... 

Amos, R. M. and Baines, W. L. (1997). Modification of the spontaneous emission rate of Eu3+ ions close to a thin metal mirror Physical Review B 55 7249-7254. 

Forbidden pure rotational transitions of H3, following the selection rules Ak = +3, occur in the wide region from millimetre wave to mid-infrared.These transitions are caused by centrifugal distortions of the symmetric structure. No laboratory observation of them has been reported so far. These transitions are much weaker than the usual dipole-allowed rotational transitions in polar molecules, and their spontaneous emission rates range from ca. 10" s" to ca. 10" s". Nevertheless, such weak transitions may be observable in low-density regions just like the Hj quadrupole transitions. Also, the spontaneous emission lifetimes are short compared with the collisional time in low-density areas, making the forbidden rotational transitions important processes for cooling the rotational temperature of Hj. ... 

The Purcell s original idea [10] on modification of photon spontaneous emission rate is extended to modification of photon spontaneous scattering rate. Simultaneous account for local incident field and local density of photon states enhancements in close proximity to a silver nanoparticle is found to provide up to lO -fold Raman scattering cross-section rise up. Thus, single molecule Raman detection is found to be explained by consistent quantum electrodynamic description without chemical mechanisms involved. 

For the x-ray emission process, the transition probability( is also calculated from the dipole matrix similar to the case of the x-ray absorption, but the molecular state f in eq.(lO) is of occupied in this case. The transition probability corresponds to the spontaneous emission rate, then is given by Einstein formula as... 

As an application to the results obtained above we consider the spontaneous emission rate from our molecule after it is prepared in the excited state 2). In terms of zero-order states of the Hamiltonian Hq = + Hr the initial state is 2, 0 )... 

Surface enhanced fluorescence (SEE) takes place in the proximity of metal structures. The effect of fluorescence enhancement has been intensively studied by several groups [74]. In the proximity of metals, the fluorophore radiative properties are modified and an increase in the spontaneous emission rate is observed. 

Bjork, G. (1991). Modification of spontaneous emission rate in planar dielectric microcavity structures. Physical Review A, Vol. 44, No. 1, pp. 669—681. 

The first terms on the right-hand side of Eq. (109) determine the spontaneous emission rates from the symmetric and antisymmetric states, while the second terms determine the coherent interaction between these two states. Note that the... 

It is seen that the ratio (120) depends crucially on the spontaneous emission rate Teg, which depopulates the state e). Maximum inversion, with P, = 1 and Pg = 0, is obtained for I = 0, when the population is said to be shelved (trapped) in the state e) from which it cannot decay to the ground state. Thus, in the case of maximum inversion one could expect maximum amplification of a probe beam on the e) —> g) transition. However, the absorption rate W((Qp) of a probe beam of amplitude E ) and frequency cop monitoring the e) —> g) transition, as defined by Mollow [38], is... 

Consider the Menon-Agarwal approach to the Autler-Townes spectrum of a V-type three-level atom. The atom is composed of two excited states, 1) and 3), and the ground state 2) coupled by transition dipole moments with matrix elements p12 and p32, but with no dipole coupling between the excited states. The excited states are separated in frequency by A. The spontaneous emission rates from 1) and 3) to the ground state 2) are Tj and T2, respectively. The atom is driven by a strong laser field of the Rabi frequency il, coupled solely to the 1) —> 2) transition. This is a crucial assumption, which would be difficult to realize in practice since quantum interference requires almost parallel dipole moments. However, the difficulty can be overcome in atomic systems with specific selection rules for the transition dipole moments, or by applying fields with specific polarization properties [26]. 

In Fig. 10, we plot the absorption rate Wn as a function of 8 for p 0.95 and different ft. When ft / 2A the absorption rate exhibits the familar Mollow absorption spectrum [38] with small dispersive structures at 8 = fl The absorption rate changes dramatically when ft = 2A. Here, the dominant features of the rate are emissive and absorptive components at 8 = ft, indicating that at 8 = —ft the weaker field is absorbed, whereas at 8 = ft is amplified at the expense of the strong field. The weaker field is always absorbed (amplified) at 8 = —ft (8 = ft) independent of the ratio r = Ti/I between the spontaneous emission rates 14 and 14. We illustrate this in Fig. 11, where we plot the absorptive rate for different values of r. The absorptive (emissive) peak remains absorptive (emissive) independent of the ratio r. 

With the dressed states of the driven system available, we can easily predict transition frequencies and calculate transition dipole moments and spontaneous emission rates between the dressed states of the system. It is easily verified that nonzero dipole moments occur only between dressed states within neighboring manifolds. Using Eqs. (131) and (132), we find that the transition dipole moments between N, i) and /V — 1, j) are... 

The subject of correlated or collective spontaneous emission by a system of a large number of atoms was first proposed by Dicke [1], who introduced the concept of superradiance that the influence on each atomic dipole of the electromagnetic field produced by the other atomic dipoles could, in certain circumstances, cause each atom to decay with an enhanced spontaneous emission rate. The shortening of the atomic lifetime resulting from the interaction between N atoms could involve an enhancement of the intensity of radiation up to N2. 

An example of entangled states in a two-atom system are the symmetric and antisymmetric states, which correspond to the symmetric and antisymmetric combinations of the atomic dipole moments, respectively [1,7,21]. These states are created by the dipole-dipole interaction between the atoms and are characterized by different spontaneous decay rates that the symmetric state decays with an enhanced, whereas the antisymmetric state decays with a reduced spontaneous emission rate [7]. For the case of two atoms confined into the region much smaller than the optical wavelength, the antisymmetric state does not decay at all, and therefore can be regarded as a decoherence-free state. 

Substituting Eq. (26) into Eqs. (22) and (24), we obtain the following explicit expressions for the collective spontaneous emission rate... 

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