Coherence decay

It follows that there are two kinds of processes required for an arbitrary initial state to relax to an equilibrium state the diagonal elements must redistribute to a Boltzmaim distribution and the off-diagonal elements must decay to zero. The first of these processes is called population decay in two-level systems this time scale is called Ty The second of these processes is called dephasmg, or coherence decay in two-level systems there is a single time scale for this process called T. There is a well-known relationship in two level systems, valid for weak system-bath coupling, that... 

We now make two coimections with topics discussed earlier. First, at the begiiming of this section we defined 1/Jj as the rate constant for population decay and 1/J2 as the rate constant for coherence decay. Equation (A1.6.63) shows that for spontaneous emission MT = y, while 1/J2 = y/2 comparing with equation (A1.6.60) we see that for spontaneous emission, 1/J2 = 0- Second, note that y is the rate constant for population transfer due to spontaneous emission it is identical to the Einstein A coefficient which we defined in equation (Al.6.3). 

Others which formed a coherent decay scheme down to Em ... 

As far as the controls are concerned, we here consider time-continuous modulation of the system Hamiltonian, which allows for vastly more freedom compared to control that is restricted to stroboscopic pulses as in DD [42, 55, 91]. We do not rely on rapidly changing control fields that are required to approximate stroboscopic a -pulses. These features allow efficient optimization under energy constraint. On the other hand, the generation of a sequence of well-defined pulses may be preferable experimentally. We may choose the pulse timings and/or areas as continuous control parameters and optimize them with respect to a given bath spectrum. Hence, our approach encompasses both pulsed and continuous modulation as special cases. The same approach can also be applied to map out the bath spectrum by measuring the coherence decay rate for a narrow-band modulation centered at different frequencies [117]. 

Figure 3. Field-matter interactions for a pair of electronic states. The zero and first excited vibrational levels are shown for each state (A). The fields are resonant with the electronic transitions. A horizontal bar represents an eigenstate, and a solid (dashed) vertical arrow represents a single field-matter interaction on a ket (bra) state. (See Refs. 1 and 54 for more details.) A single field-matter interaction creates an electronic superposition (coherence) state (B) that decays by electronic dephasing. Two interactions with positive and negative frequencies create electronic populations (C) or vibrational coherences either in the excited (D) or in the ground ( ) electronic states. In the latter cases (D and E) the evolution of coherence is decoupled from electronic dephasing, and the coherences decay by the vibrational dephasing process.

The mixed quantum classical description of EET can be achieved in using Eq. (49) together with the electronic ground-state classical path version of Eq. (50). As already indicated this approach is valid for any ratio between the excitonic coupling and the exciton vibrational interaction. If an ensemble average has been taken appropriately we may also expect the manifestation of electronic excitation energy dissipation and coherence decay, however, always in the limit of an infinite temperature approach. 

Thus we cannot escape the depressing reality that 7 2 will get shorter and linewidth will get bigger as we increase the size of the protein studied. The reduced T2 is not only a problem for linewidth, but also causes loss of sensitivity as coherence decays during the defocusing and refocusing delays (1/(2J)) required for INEPT transfer in our 2D experiments. The only ray of hope comes in the form of a new technique called TROSY (transverse relaxation optimized spectroscopy), which takes advantage of the cancellation of dipole-dipole relaxation by CSA relaxation to get an effectively much longer 7 2 value we will briefly discuss TROSY at the end of this chapter. 

The majority of studies of vibrational dephasing have looked at the width or shape of the isotropic Raman line. The Raman line shape is the Fourier transform of the coherence decay function that characterizes dephasing (20,21). The coherence decay can also be measured directly in... 

For the moment, assume that the VE picture is correct and inertial solvent motion causes negligible dephasing. Diffusive motion must be the primary cause of coherence decay. In the VE theory, the diffusive motion is the relaxation of stress fluctuations in the solvent by viscous flow. The VE theory calculates both the magnitude Am and lifetime z0J of the resulting vibrational frequency perturbations. A Kubo-like treatment then predicts the coherence decay as a function of the viscosity of the solvent. Figure 19 shows results for typical solvent parameters. At low viscosity, the modulation is in the fast limit, so the decay is slow and nearly exponential. Under these conditions, the dephasing time is inversely proportional to the viscosity, as in previous theories [Equation (19)]. As the viscosity increases, the modulation rate slows. The decay becomes faster and approaches a... 

Figure 19 The VE prediction of the diffusive component of the vibrational coherence decay CpiD as a function of the solvent viscosity (ij = 1, 2, 4, 8, 16, 32 and oo cP) for typical parameters. At low viscosity, the decay is exponential, its rate is inversely proportional to the viscosity, and the corresponding Raman line is homogeneously broadened. At high viscosity, the decay becomes Gaussian, its decay time reaches a limiting value, and the Raman line is inhomogeneously broadened. (From Ref. 8.)...

Within the Mtuliiov approximation, the intermolecular dephasing constant associated with the coherence decay between i<->j and i ->j vibrational Raman transitions is expressed in terms of the population decay constant from each state and intermolecular pure dephasing constant as ... 

Figure 9. Measured coherent decay rates (2/T2 ps" ) for V4, polariton in NH4CI as a function of polariton frequency in cm ( ) at crystal temperature of 78 K. Also shown ( ) is decay rate at 78 K of nonpropagating longitudinal component v, at 1418 cm" .

Figure 6. Examples on the breakdown of the positivity. Top off-diagonal memory is too short compared to the diagonal memory and the coherences decay too slowly. Middle coherences oscillate too much. Bottom diagonal and off-diagonal oscillations have different frequencies and the zeros do not occur simultaneously.

If at time zero, after the field has been switched off, the system is found in a state with nonvanishing coherences, oy (z 7), Eqs (18.43b,c) tell us that these coherences decay with the dephasing rate constant k. k was shown in turn to consist of two parts (cf. Eq. (10.176) The lifetime contribution to the decay rate of Oy is the sum of half the population relaxation rates out of states z and j, in the present case for 012 and <721 this is (l/2)( 2 i + 1 2)- Another contribution that we called pure dephasing is of the form (again from (10.176)) Zi 2C(0)(K — 22) - system operator that couples to the thermal bath so that — V22... 

In the case of vibrational responses the population relaxation times may be dominating the coherence decays. In addition, it can be essential to incorporate the multilevel nature of molecular vibrators into the response. The rate of repopulation of the ground state is seldom equal to the decay of the fundamental V = 1 state, so there can be bottlenecks in the ground state recovery. Following... 

Figure 46. Simulated fluorescence decays showing the effects of purely rotational coherence. Decays were calculated using the results of Refs. 47 and 49 and parameters given in the text. The top two decays correspond to infinite temporal resolution and infinite fluorescence lifetime. "Parallel and perpendicular refer to the orientation of with respect to 4,. The bottom two decays are just convolutions of the top two decays with a finite detection response function (80 psec FWHM), assuming, in addition, a 2.6 nsec fluorescence lifetime.

To become observable by NMR, the polarizations next have to find their way into diamagnetic products. This depends on the chemistry of the system and on the concentrations, but very often occurs on a timescale of several tens of microseconds. If an rf pulse is applied at some point of time during that period, it converts the polarizations present at that very moment into coherences and isolates them from the further development of the polarizations, which are longitudinal magnetizations only hence, these coherences decay in the same way as any normal NMR signal. 

In molecules, as noted by de Vries and Wiersma, the application of free induction decay to study optical dephasing may be frustrated by the presence of an intermediate triplet state. The level scheme, which is representative for most molecules with an even number of electrons, is shown in Fig. 26. For an applied laser field E =EQCOs t-k ), that is resonant with the (2 <- 1) transition, we may write, in the RWA approximation, the following steady-state density matrix equations, which describe the coherent decay after laser frequency switching ... 

A second important conclusion to be made is that (41) show that coherence feeding and decay is due to distinctly different processes. In fact, the coherence decay itself is also due to two different types of effects. 

The nonadiabatic (nonsecular) contributions T, and T34 to the coherence decay are caused by inelastic 7 ,-type processes. Equation (41b) shows that these inelastic scattering processes are induced by anharmonic-ity (k ) in the ground state and a combination of anharmonicity and electron-phonon coupling (Vg ) in the excited state. Here describes the decay (creation) of the pseudolocalized phonon into (from) two band phonons. The relevant part of is in (39) the last term, which describes in the excited state the exchange of a pseudolocalized phonon with a band phonon. At low temperature (/c7 phonon scattering processes in the ground and excited state. 

Although the general roots of Eq. (55) are easily obtained, there are two limiting cases in which particularly simple and useful results are found. The first is for the symmetric case, A = 0. In this limit, the population and coherence equations decouple and the population relaxation rate in the site representation is precisely the coherence decay rate... 

Note, that the line width of a single-frequency laser is also strongly related to temporal coherence a narrow fine width means high temporal coherence. The line width can be used to estimate the coherence time, but the conversion depends on the spectral shape, and the relationship between optical bandwidth and temporal coherence is not always simple. In the case of exponential coherence decay, e.g. as encountered for a laser whose performance is limited by noise, the width of the frequency distribution function (ftdl width at half maximum (FWHM)) is... 

Besides various detection mechanisms (e.g. stimulated emission or ionization), there exist moreover numerous possible detection schemes. For example, we may either directly detect the emitted polarization (oc PP, so-called homodyne detection), thus measuring the decay of the electronic coherence via the photon-echo effect, or we may employ a heterodyne detection scheme (oc EP ), thus monitoring the time evolution of the electronic populations In the ground and excited electronic states via resonance Raman and stimulated emission processes. Furthermore, one may use polarization-sensitive detection techniques (transient birefringence and dichroism spectroscopy ), employ frequency-integrated (see, e.g. Ref. 53) or dispersed (see, e.g. Ref. 54) detection of the emission, and use laser fields with definite phase relation. On top of that, there are modern coherent multi-pulse techniques, which combine several of the above mentioned options. For example, phase-locked heterodyne-detected four-pulse photon-echo experiments make it possible to monitor all three time evolutions inherent to the third-order polarization, namely, the electronic coherence decay induced by the pump field, the djmamics of the system occurring after the preparation by the pump, and the electronic coherence decay induced by the probe field. For a theoretical survey of the various spectroscopic detection schemes, see Ref. 10. 

---