Dephasing decoherence

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 

We next consider the effect of thermal relaxation on the absorption lineshape. We start with the Bloch equations in the form (10.184), 

We use the subscript ss to denote steady state. The rate of energy change in the system is — daz/dt). kyt az ss — cPz.eq) must therefore be the rate at which energy is being dissipated, while — W2i oAtdy,ss is the rate at which it is absorbed from the radiation field. To find an explicit expression we solve (18.46b,c) for steady state, that is, with the time derivatives put to zero, and find 

The resulting absorption rate is proportional to the population difference CTz.ss = Tii,ss — T22,ss, as may have been expected. We have obtained a Lorentzian lineshape whose width is determined by the total phase relaxation rate kA, Eq. (10.176). 

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Second, we should keep in mind that between the two extreme limits discussed above there exists a regime of intermediate behavior, where dephasing/decoherence and molecular response occur on comparable timescales. In this case the scattering process may exhibit partial coherence. Detailed description of such situations requires treatment of optical response within a formalism that explicitly includes thermal interactions between the system and its environment. In Section 18.5 we will address these issues using the Bloch-Redfield theory of Section 10.5.2. 

However, wider exploitation of multidimensional solid-state NMR methods is hindered by low Si sensitivity, which results from low natural abundance of spin-1/2 isotope (4.7%), small gyro-magnetic ratio, and unusually slow longitudinal relaxation. Fortunately, the presence of a long relaxation time in inorganic solids is often accompanied by a long transverse dephasing (decoherence) time defined here as the decay time of the echo train due to time-dependent interactions... 

Phonon-induced pure dephasing/decoherence time for pairs of electronic states. Adapted from Ref. 43 (Copyright 2008 American Chemical Society). 

The previous sections focused on the case of isolated atoms or molecules, where coherence is fully maintained on relevant time scales, corresponding to molecular beam experiments. Here we proceed to extend the discussion to dense environments, where both population decay and pure dephasing [77] arise from interaction of a subsystem with a dissipative environment. Our interest is in the information content of the channel phase. It is relevant to note, however, that whereas the controllability of isolated molecules is both remarkable [24, 25, 27] and well understood [26], much less is known about the controllability of systems where dissipation is significant [78]. Although this question is not the thrust of the present chapter, this section bears implications to the problem of coherent control in the presence of dissipation, inasmuch as the channel phase serves as a sensitive measure of the extent of decoherence. 

Thus in the zero dephasing case, 8s reduces to the Breit-Wigner phase of the intermediate state resonance, elaborated on in the previous sections. In the dissipative environment, it is sensitive also to decay and decoherence mechanisms, as illustrated later. 

Quantum-state decay to a continuum or changes in its population via coupling to a thermal bath is known as amplitude noise (AN). It characterizes decoherence processes in many quantum systems, for example, spontaneous emission of photons by excited atoms [35], vibrational and collisional relaxation of trapped ions [36] and the relaxation of current-biased Josephson junctions [37], Another source of decoherence in the same systems is proper dephasing or phase noise (PN) [38], which does not affect the populations of quantum states but randomizes their energies or phases. 

Dynamical decoupling DD is one of the best known approaches to combat decoherence, especially dephasing [39-52,55,56]. We present its essential aspects and how it can be incorporated into the general framework described above. 

Figure 4.7 Average modified final decoherence rate R(T), normalized with respect to the unmodulated rate as a function of energy constraint. DD-dash, cyan. Optimal modulation-solid, dark green. Insets optimal modulation Q(t) for different energy constraints, (a) Single-peak resonant dephasing spectrum (inset E = 20). (b) Single-peak off-resonant spectrum (inset E = 50). (c) 1 // spectrum (inset E = 30). (d) Multipeaked spectrum (inset E = 30). (See color plate section for the color representation of this figure.)...

The universal dynamical decoherence control formula was first developed by us for single qubit decay due to interaction with a zero-temperature bath [9, 13] and was later extended to decay and dephasing due to interaction with finite-temperature baths in Refs [11, 21], and finally to multipartite systems in... 

For each modulation parameters pair, [(p,r, one experimentally measure the average decoherence rate of the qubit, R(t) (either decay of an initially excited state, or dephasing of an initial superposition of ground and excited states). Then one runs a simple numerical algorithm that finds the coupling spectrum, G( ), that best fits the observed results. 

Our analysis of multiple, field-driven qubits that are coupled to partly correlated or independent baths or undergo locally varying random dephasing allows one to come up with an optimal choice between global and local control, based on the observation that the maximal suppression of decoherence is not necessarily the best one. Instead, we demand an optimal phase relation between different but synchronous local modulations of each particle. The merits of local versus global modulations have been shown to be essentially twofold ... 

This coupling means that there is no population decay to the bath and the decoherence is only via the phase. In other pioneering works [39-41], off-diagonal coupling to a Lorentzian bath was considered within the RWA. The Hamiltonians in all of these works should be contrasted with the more general Hamiltonians (4.18), which account for both dephasing and decay into arbitrary baths, without the RWA. 

For both time-dependent and steady-state models, coherent dephasing is taken into account by phenomenological terms in the off diagonal time derivatives of the DM elements that appear in the Liouville dynamics. In the time dependent model, increasing the donor or bridge decoherences decreases the yield asymmetry and... 

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