Decoherence control

While the formalism of DD is quite different from the formalism presented here, it can be easily incorporated into the general framework of universal dynamical decoherence control by introducing impulsive PM. Let the phase of the modulation function periodically jump by an amount 4> at times r, 2t,. .. Such modulation can be achieved by a train of identical, equidistant, narrow pulses of nonres-onant radiation, which produce pulsed AC-Stark shifts of co. When (/> = tt, this modulation corresponds to DD pulses. 

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... 

As shown in Refs [19,28, 30, 112], each QIP application has its own figure of merit that assesses the effectiveness of the dynamical decoherence control scheme. In general, we have shown that this effectiveness is related to the decoherence matrix of the form given by Eq. (4.203). Yet, if some of its elements cannot be drastically reduced, one can still have a dramatic effect and increase the appropriate figure of merit. 

In Section 4.5, we have introduced the Euler-Lagrange optimization approach to decoherence control that is optimally tailored to the bath spectrum in question. This approach has then been applied in Section 4.6 to optimized state transfer in hybrid system, from noisy to quiet qubits or through noisy spin channels. 

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. 

The final section deals with known examples of molecular spin qubits based on lanthanide SIMs. Distinction is made between single-qubit molecules and molecules which embody more than one qubit. This section includes some comments about decoherence in these molecular systems and strategies to control it. 

The underlying issue is broader Coherent control was originally conceived for closed systems, and it is a priori unclear to what extent it is applicable to open quantum systems, that is, systems embedded in their ubiquitous environment and subject to omnipresent decoherence effects. These may have different physical origins, such as the coupling of the system to an external environment (bath), noise in the classical fields controlling the system, or population leakage out of a relevant system subspace. Their consequence is always a deviation of the quantum-state evolution (error) with respect to the unitary evolution expected... 

We first survey in Section 4.2 the Kurizki-Shapiro-Brumer scheme [5-7] of phase-coherent photocurrent control and focus on its robustness to decoherence, relaxation, and quantum (Langevin) noise induced by the environment. We then... 

Remarkably, when our general ME is applied to either AN or PN in Section 4.4, the resulting dynamically controlled relaxation or decoherence rates obey analogous formulae provided the corresponding density matrix (generalized Bloch) equations are written in the appropriate basis. This underscores the universality of our treatment. It allows us to present a PN treatment that does not describe noise phenomenologically, but rather dynamically, starting from the ubiquitous spin-boson Hamiltonian. 

Figure 4.4 Frequency-domain representation of the dynamically controlled decoherence rate in various limits (Section 4.4). (a) Golden-Rule limit, (b) Anti-Zeno effect (AZE) limit (c) Quantum Zeno effect (QZE) limit. Here, F,( ) and G(w) are the modulation and bath spectra, respectively and F are the interval of change and width of G( ), respectively and is the interruption rate.

For either AN or PN we may control the decoherence by either off-resonant or near-resonant modulations, respectively. The modulation spectrum has the same form for both (see Section 4.4.4) ... 

The DD sequences [42, 50, 55, 80-82] have not shown optimality with respect to the decoherence suppression for any given coupling spectrum. Here we apply variational principles to our universal-formula dynamical control of decoherence in order to find the optimal modulation for any given decoherence process. We first derive an equation for the optimal, energy-constrained control by modulation that minimizes decoherence for any given bath-coupling spectrum and then numerically solve this equation, and compare the optimal modulation to energy-constrained DD pulses. 

We use this general approach to optimize a reliable transfer of a quantum state from a fragile (noisy) qubit to a robust (quiet) qubit. We choose to focus on the case of two resonant qubits with temporally controlled coupling strength. The free Hamiltonian without decoherence is then... 

In order to affect the system-bath coupling and control, or modulate, the decoherence due to this coupling, one must dynamically modulate the system faster than the correlation time. Slower modulation will have no effect on the loss of coherence and will thus not be able to control it. Modulating the system faster than the correlation time can effectively reset the clock. Applying a modulation sequence repeatedly can thus drastically change the decoherence and impose a continued coherent evolution of the system-bath coupling [46, 94]. 

In this review, we have expounded our universal approach to the dynamical control of qubits subject to noise or decoherence. It is based on a general non-Markovian ME valid for weak System-bath coupling and arbitrary modulations, since it does not invoke the RWA. The resulting universal convolution formula provide intuitive clues as to the optimal tailoring of modulation and noise spectra. 

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 ... 

In this chapter, we have expounded our comprehensive approach to the dynamical control of decay and decoherence. Our analysis of dynamically modified coupling between a qubit and a bath has resulted in the universal formula (4.49) for the dynamically modified decay rate into a zero-temperature bath, as well as its counterparts (4.114) for excited- and ground-state dynamical decay into finite-temperature baths. This ground-state dynamically induced decay results from RWA violation by ultrafast modulation. 

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