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arXiv:1212.2279 · Communications in Theoretical Physics

Control Protocol of Finite Dimensional Quantum Systems

Jianju Tang, Hong‐Chen Fu

Published Dec 1, 2013• 0 citations
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Abstract

An analytic control protocol of two types of finite dimensional quantum systems is proposed. The system can be driven to an arbitrary target state using cosine classical fields in finite cycles. The control parameters which are time periods of interaction between systems and control fields in each cycle are connected with the probability amplitudes of target states via trigonometrical functions and can be determined analytically.

DOI
10.1088/0253-6102/60/6/17

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An analytic control protocol of two types of finite dimensional quantum systems is proposed. The system can be driven to an arbitrary target state using cosine classical fields in finite cycles. The control parameters which are time periods of interaction between systems and control fields in each cycle are connected with the probability amplitudes of target states via trigonometrical functions and can be determined analytically.
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In this paper, we consider the problem of the controllability of a finite-dimensional quantum system in both the Schrödinger and interaction pictures. Introducing a Quantum Transfer Graph, we elucidate the role of Lie algebra rank conditions and the complex nature of the control matrices. We analyze the example of a sequentially coupled N-level system: a spin-12 particle coupled to a finite quantum harmonic oscillator. This models an important physical paradigm of quantum computers—the trapped ion. We describe the control of the finite model obtained, under the right conditions, from the original infinite-dimensional system.
In this work, we derive exact solutions of a dynamical equation, which can represent all two-level Hermitian systems driven by periodic NNN-step driving fields. For different physical parameters, this dynamical equation displays various phenomena for periodic NNN-step driven systems. The time-dependent transition probability can be expressed by a general formula that consists of cosine functions with discrete frequencies, and, remarkably, this formula is suitable for arbitrary parameter regimes. Moreover, only a few cosine functions (i.e., one to three main frequencies) are sufficient to describe the actual dynamics of the periodic NNN-step driven system. Furthermore, we find that a beating in the transition probability emerges when two (or three) main frequencies are similar. Some applications are also demonstrated in quantum state manipulations by periodic NNN-step driving fields.
A quantum system subject to external fields is said to be controllable if these fields can be adjusted to guide the state vector to a desired destination in the state space of the system. Fundamental results on controllability are reviewed against the background of recent ideas and advances in two seemingly disparate endeavours: (i) laser control of chemical reactions and (ii) quantum computation. Using Lie-algebraic methods, sufficient conditions have been derived for global controllability on a finite-dimensional manifold of an infinite-dimensional Hilbert space, in the case that the Hamiltonian and control operators, possibly unbounded, possess a common dense domain of analytic vectors. Some simple examples are presented. A synergism between quantum control and quantum computation is creating a host of exciting new opportunities for both activities. The impact of these developments on computational many-body theory could be profound.