Bath systems

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... 

Anderson S M, Zink J I and Neuhauser D 1998 A simple and accurate approximation for a coupled system-bath locally propagating Gaussians Chem. Phys. Lett. 291 387... 

This modulation may pertain to any intervention in the system-bath dynamics (i) measurements that effectively interrupt and completely dephase the evolution. 

This limit is that of the quantum Zeno effect (QZE), namely, the suppression of relaxation as the interval between interruptions decreases [64-66]. In this limit, the system-bath exchange is reversible and the system coherence is fully maintained (Figure 4.4c). Namely, the essence of the QZE is that sufficiently rapid interventions prevent the excitation escape to the continuum, by reversing the exchange with the bath. 

Despite the differences in the long-time behavior (due to the lower cutoff in the 1 // case), these two examples allow us to generalize to any dephasing spectrum with a monotonically decreasing system-bath coupling strength as a function of frequency. The optimal modulation for such spectra will be an energy-constrained chirped modulation, with modifications due to other spectral characteristics, for example, cutoffs. 

Equation (4.150) expresses the score P as an overlap between the gradient P and the change of system state Ap. In order to find expressions for P in terms of physically insightful quantities, we decompose the total Hamiltonian into system, bath, and interaction parts. 

Aperiodic DD sequences such as Uhrig dynamical decoupling (UDD) [55] suppress low-frequency components (to the left of the main peak) in the system spectrum, which retain the system-bath coupling even if the main peak of the system spectrum has been shifted beyond the bath cutoff frequency (Figure 4.11). The plots indicate that this suppression of low-frequency components is achieved at the price of a smaller shift of the main peak, that is, shifting the main peak beyond a given cutoff requires more pulses in UDD than in FDD. Note that optimized DD sequences with improved asymptotics exist [91], which we will not consider here. 

To give an example of the opposite case, where the goal is to maximize the system-bath coupling, we apply our approach of constrained optimization to the linear entropy Si = 2[1 - Tr(p )] of a qubit. (Note that here Si has been normalized to 1 by setting the coefficient k= dl d- 1) = 2, cf. Section 4.5.4.) We assume an initial mixture... 

Figure 4.15 Change of linear entropy in units of the system-bath coupling strength obtained by minimization (attempted cooling) and maximization (attempted heating) of A5 under different constraints E = 0,1,100 as a function of the initial p for a bath as shown in Figure 4.14.

A possibility to achieve negative (positive) A5 by its minimization (maximization) even for weak modulation (i.e., for small E) is to adapt the temperature of the bath such that for an undriven system Hamiltonian Hq, no change is observed, A5 = 0, which is a necessary condition for a system-bath equilibrium. This would require nonunitary system modulation, for example, the effect of repeated measurements [92-94],... 

To this end, we resort to a novel general approach to the control of arbitrary multidimensional quantum operations in open systems described by the reduced density matrix p(t) if the desired operation is disturbed by linear couplings to a bath, via operators S B (where S is the traceless system operator and B is the bath operator), one can choose controls to maximize the operation fidelity according to the following recipe, which holds to second order in the system-bath coupling (i) The control (modulation) transforms the system-bath coupling operators to the time-dependent form S t) (S) B(t) in the interaction picture, via the rotation matrix e,(t) a set of time-dependent coefficients in the operator basis, (Pauli matrices in the case of a qubit), such that ... 

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]. 

The goal of any practical modulation scheme is to reduce, or, if possible, eliminate, all the elements of the decoherence matrix in Eq. (4.203) (see Ref. [20] for an alternative solution). However, in order to obtain the optimal modulation [29], one must first know the system-bath coupling spectra of the qubits in question. This information is usually not available a priori and thus most experimentalists have resorted to the suboptimal DD (or bang-bang) modulation, which does not require this knowledge. 

Use known, prescribed modulations to reveal the unknown decoherence parameters, that is, the system-bath coupling spectra. 

After one obtains the system-bath coupling spectra by applying specific, parameterized modulation schemes, one can finally tailor the specific modulation that would optimally reduce or eliminate decoherence. However, two aspects should be... 

However, these schemes can only be applied after obtaining an approximate form of the system-bath coupling spectra, without which one cannot tailor the local modulations that equate or eliminate specific elements of the decoherence matrix. 

To summarize, after obtaining the general shape of the system-bath coupling spectra matrix of the multipartite system by applying parameterized, known modulations, one can optimize the appropriate QIP-dependent figure of merit while obeying the constraints of the available modulations. 

A novel interplay between entanglement as a QIP resource and entanglement as the source of decoherence was detailed [116]. Two entangled qubits were analyzed, each coupled to a bath via common modes. The non-Markovian timescale was considered, as well as dynamical modulations. It was shown how the entanglement of the qubits could vanish after a finite time (entanglement sudden death, ESD), but later restored by non-Markovian modulation-induced oscillations of the system-bath coherence. 

Thus, an exchange between two-qubit entanglement and system-bath entanglement may take place and be dynamically controlled by modulations. If the systems couple to two baths that have common modes, one can observe the transfer of coherence and buildup of entanglement between the two systems via these modes. If, on the other hand, the baths were completely separate, the coherence transfer between each system and its bath can modulate the amount of system-system entanglement, first lost and then regained. 

We have shown that immediately after the measurement, the system and bath always heat-up, that is, get excited. Remarkably, for certain system-bath coupling spectra one can also observe a system that has lower excited-state population than the equilibrium state, that is, a purer system. This occurs despite the fact that the system has effectively recoupled with the bath and has become entangled with it. 

This interplay between system-bath entanglement and system-probe entanglement and its effect on system cooling is still under active investigation by our group. 

Entanglement in open systems is a fascinating concept that is not yet fully understood. It is essential for QIP, yet it is also the origin of decoherence. The interplay between system-system entanglement, system-probe entanglement and system-bath entanglement on non-Markovian timescales is an intriguing field that deserves more attention. 

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. 

Open issues An open issue of the approach is the inclusion of higher orders in the system-bath coupling, which becomes important for strong or resonant system-bath coupling, so that a perturbative expansion cannot be applied. This may be the case especially when this coupling is to be enhanced in order to achieve... 

Another concern regards the initial conditions. Here we have assumed a factorized initial state of the system and bath. This prevents us from taking into account system-bath interactions that may have occurred prior to that time. In particular, if the system is in equilibrium with the bath, their states are entangled or correlated [24, 94]. 

Quantum-dynamical modeling of ultrafast processes in complex molecular systems multiconfigurational system-bath dynamics using Gaussian wavepackets... 

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