Quantum feedback

Keywords chaos, conditioned evolution, continuous measurement, density matrix, quantum backaction, quantum feedback. 

The ar tide is organized as follows. We will begin with a discussion of the various possibilities of dynamical description, clarify what is meant by nonlinear quantum dynamics , discuss its connection to nonlinear classical dynamics, and then study two experimentally relevant examples of quantum nonlinearity - (i) the existence of chaos in quantum dynamical systems far from the classical regime, and (ii) real-time quantum feedback control. 

To illustrate an application of nonlinear quantum dynamics, we now consider real-time control of quantum dynamical systems. Feedback control is essential for the operation of complex engineered systems, such as aircraft and industrial plants. As active manipulation and engineering of quantum systems becomes routine, quantum feedback control is expected to play a key role in applications such as precision measurement and quantum information processing. The primary difference between the quantum and classical situations, aside from dynamical differences, is the active nature of quantum measurements. As an example, in classical theory the more information one extracts from a system, the better one is potentially able to control it, but, due to backaction, this no longer holds true quantum mechanically. 

The notion of quantum feedback control naturally suggests a closed-loop process in the laboratory to stabilize or guide a system to a desired state. In addition, feedback is important in the design of molecular controls. These points will be made clear below, starting with considerations of design followed by a discussion of its role in the laboratory and finally leading to feedback concepts for the inversion of laboratory data. 

R.J. Nelson, Y. Weinstein, D. Cory, S. Lloyd, Experimental demonstration of fully coherent quantum feedback, Phys. Rev. Lett. 85 (2000) 3045. 

Experimental demonstration of fully coherent quantum feedback - This paper of 1999 by Nelson and co-workers describes a quantum circuit which implements a coherent quantum feedback by NMR [15]. The idea is to transmit a quantum state with its correlations from a quantum register to a target. A classical feedback can be used to control quantum states, but it involves a measurement step, which destroys quantum correlations. On the contrary, quantum feedback control is able to transmit full quantum states with correlations altogether. The experiment involves three spins (qubits) A, B, and C. The proposal is to use the quantum feedback control circuit to transmit the quantum correlations, initially between B and C, to A and C. Starting from the equilibrium state, a sequence of pulses applied over qubits B and C creates a density matrix proportional to the spin operators -1 + 21 if - Such a density matrix describes a thermal... 

Bardeen C J, Yakovlev V V, Wilson K R, Carpenter S D, Weber P M and Warren W S 1997 Feedback quantum control of molecular electronic population transfer Chem. Phys. Lett. 280 151... 

The approach described represents one more step toward the realization of a completely stand-alone single-electron junction based on nanoparticles and produced in organic matrix. Quantum dot synthesis directly on the tip of a metal stylus does not require the use of STM for localizing the particle position and requires only the use of atomically flat electrodes and a feedback system for maintaining a desirable double-barrier structure. 

I have reported this last example not for the sake of completeness in our discussion, but to underline a different point. Quantum chemistry, in the work of CTOup 1 and even more in the work of group II, put the emphasis on some properties which by tradition are not object of direct experimental determination. Electron charge distribution and MEP arejust two examples. The use of these quantities by theoreticians has spurred the elaboration of experimental methods able to measure them. This positive feedback between theory and experiment is an indication that quantum and experimental chemistry do not live in separate worlds. 

Infrared spectroscopy is the workhorse in this field, because it can quickly provide dynamical details, discriminate between different cluster sizes and phases [40], and sample a wide spectral range. It often yields valuable feedback for quantum chemical calculations. In contrast to some action spectroscopy techniques, IR absorption spectroscopy is not intrinsically size-selective. All cluster sizes generated in the expansion are observed together, and indirect methods of size assignment are needed. 

The s terms in Eq. (80) contribute only the term E,2 in Eq. (97). Thus, the term represents the quantum diffusional. v-terms in the Fokker-Planck equation. The other terms in Eqs. (93)-(100) originate in the drift terms of the Fokker-Planck equation. The terms B12 and C in Eqs. (93)-(94) play the role of feedback terms that pump quantum fluctuations into the classical Bloembergen equations. If the s terms in Eq. (80) do not appear (the classical case), the term in Eq. (97) does not appear, either. In this case the subset (95)—(100) with zero initial conditions has zero solutions and in consequence leads to the first truncation [171]. 

Fig. 6. A closed loop apparatus for optimally identifying quantum Hamiltonian information. The closed loop operations aim to reveal one or more control experiments that identify the best quality Hamiltonian information. Hamiltonian quality is used as the feedback signal for the learning algorithm guiding the laboratory experiments.

All of the analyses described above are used in a predictive mode. That is, given the molecular Hamiltonian, the sources of the external fields, the constraints, and the disturbances, the focus has been on designing an optimal control field for a particular quantum dynamical transformation. Given the imperfections in our knowledge and the unavoidable external disturbances, it is desirable to devise a control scheme that has feedback that can be used to correct the evolution of the system in real time. A schematic outline of the feedback scheme starts with a proposed control field, applies that field to the molecular system that is to be controlled, measures the success of the application, and then uses the difference between the achieved and desired final state to design a change that improves the control field. Two issues must be addressed. First, does a feedback mechanism of the type suggested exist Second, which features of the overall control process are most efficiently subject to feedback control ... 

This chapter has emphasized the special and central role that feedback plays in virtually all aspects of control over molecular quantum phenomena. In terms of applications, the manipulation of chemical reactions still stands as a prime historical objective. However, other rich applications abound. For example, the growing interest in the field of quantum computing is a potentially exciting area [14], and any practical realization of quantum computers will surely entail control over quantum phenomena. Other unforeseen applications may also lie ahead. 

M. Demiralp and H. Rabitz, Feedback Controlled Quantum Solitons, to be published. 

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