System-bath coupling

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... 

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. 

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.

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. 

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. 

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... 

The time-local approach is based on the Hashitsume-Shibata-Takahashi identity and is also denoted as time-convolutionless formalism [43], partial time ordering prescription (POP) [40-42], or Tokuyama-Mori approach [46]. This can be derived formally from a second-order cumulant expansion of the time-ordered exponential function and yields a resummation of the COP expression [40,42]. Sometimes the approach is also called the time-dependent Redfield theory [47]. As was shown by Gzyl [48] the time-convolutionless formulation of Shibata et al. [10,11] is equivalent to the antecedent version by Fulinski and Kramarczyk [49, 50]. Using the Hashitsume-Shibata-Takahashi identity whose derivation is reviewed in the appendix, one yields in second-order in the system-bath coupling [51]... 

Fig. 2 Population dynamics of the third excited state for TL (left) and TNL (right) truncation in different orders N of the system-bath coupling calculated using a Drude spectral density with r] = 0.544 and ujd = 0.5 a.u. (Reproduced from Ref. [28]. Copyright 2007, American Institute of Physics.)...

Fig. 1 Plots of B(t) = population difference = (az)(t) for the symmetric spin-boson (e = 0) as functions of time Exact quantum results from Ref. [41] (solid circles), TQCL algorithm (squares), ILDM propagation (open circles). Upper panel presents results for low temperature, weak system-bath coupling case (3 = 12.5, = 0.09 and 12 = 0.4. Lower panel presents exact quantum results from Ref. [42] (solid circles), TQCL algorithm (squares), and ILDM propagation (open circles) for intermediate temperature and strong system-bath coupling (3 = 3, = 0.5 and 12 = 0.333.

Here Hs is the Hamiltonian of the relevant system S, Z/8 is that of the heat bath B, while Z/S B denotes the system-bath coupling. Subscribe m describes the projection onto (pm r) states. The inter-state coupling V m (Q,Z) can be written as... 

Here we apply the LAND-map approach to compute of the time dependent average population difference, A t) = az t)), between the spin states of a spin-boson model. Here az = [ 1)(1 — 2)(2 ]. Within the limits of linear response theory, this model describes the dissipative dynamics of a two level system coupled to an environment [59,63-65]. The environment is represented by an infinite set of harmonic oscillators, linearly coupled to the quantum subsystem. The characteristics of the system-bath coupling are completely described by the spectral density J(w). In the following, we shall restrict ourselves to the case of an Ohmic spectral density... 

Q. Shi and E. Geva (2004) A semiclassical generalized quantum master equation for an arbitrary system-bath coupling. J. Chem. Phys. 120, p. 10647... 

Our previously elaborated approach, [Kofman 2000 Kofman 2001 (a)], to dynamical control of states coupled to an arbitrary zero-temperature "bath" or continuum has reaffirmed the intuitive anticipation that, in order to suppress their decay, we must modulate the system-bath coupling at a rate exceeding the spectral interval over which the coupling is significant. The spectra of baths (continua) corresponding to vibrational or collisional decay or decoherence typically allow dynamical suppression, using realistic rates of modulation, [Kofman 2000 Kofman 2001 (a)]. 

Here Hs(t) is the driven (and modulated) system Hamiltonian, S(t) is a system operator and B(t) is a bath operator, whose choice depends on the system-bath coupling (linear or quadratic, diagonal or off-diagonal). These operators vary with time due to the external fields. This general form of ///(/.), unlike common treatments, does not invoke the RWA, [Cohen-Tannoudji 1992], which may fail for ultrafast modulation. The combined state of the system and the bath is described by the density matrix ps+B(t). 

T+ being the time-ordering operator. In the derivation of Eq. (168) we assumed that B(t)) = 0. It needs to be stressed that Eq. (168) generalizes previously known master equations to arbitrary time-dependent hamiltonians, Hs t) for the system and Hi(t) for system-bath coupling, [Cohen-Tannoudji 1992], Henceforth, we explicitly consider a driven TLS undergoing decay, whose resonant frequency and dipolar coupling to the reservoir are dynamically modulated, so that... 

Such correlation functions are often encountered in treatments of systems coupled to their thennal environment, where the mode 1 for the system-bath interaction is taken as a product of A or B with a system variable. In such treatments the coefficients Cj reflect the distribution of the system-bath coupling among the different modes. In classical mechanics these functions can be easily evaluated explicitly from the definition (6.6) by using the general solution of the harmonic oscillator equations of motion... 

The spectral density (see also Sections (7-5.2) and (8-2.5)) plays a prominent role in models of thermal relaxation that use harmonic oscillators description of the thermal environment and where the system-bath coupling is taken linear in the bath coordinates and/or momenta. We will see (an explicit example is given in Section 8.2.5) that /(co) characterizes the dynamics of the thermal environment as seen by the relaxing system, and consequently determines the relaxation behavior of the system itself. Two simple models for this function are often used ... 

In Eqs (8,13) and (8.14), both y and C originate from the system-bath coupling, and should therefore be somehow related to each other. In order to obtain this relation it is sufficient to consider Eq. (8.13) for the case where V does not depend on position, whereupon... 

It shows the relevant (system) part of the density operator at time r (1) coupled to the bath (2), propagated in the bath subspace from time r to time t (3) and affecting again the system via the system-bath coupling (4). This is a mathematical expression of what we often refer to as a reaction field effect The system at some time r appears to act on itself at some later time t, and the origin of this action is the reaction of the system at time t to the effect made by the same system on the bath at some earlier time r. 

Practical solutions of dynamical problems are almost always perturbative. We are interested in the effect of the thennal environment on the dynamical behavior of a given system, so a natural viewpoint is to assume that the dynamics of the system alone is known and to take the system-bath coupling as the perturbation. We have seen (Section 2.7.3) that time dependent perturbation theory in Hilbert space is most easily discussed in the framework of the interaction representation. Following this route" we start from the Liouville equation in this representation (cf. Eq. (10.21))... 

As just stated, we henceforth use the term quantum master equation (QME) to denote the approximate time evolution equation for the system s density matrix d obtained in second order in the system-bath coupling V. To obtain this equation we start from Eq. (10.104) and use a simplified version of Eq. (10.110)... 

This fast phase oscillation, or its remaining signature in the presence of system-bath coupling, should not be taken out of the integral. We therefore use the interaction representation of d (see Eq. (10.65))... 

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