Identifying mechanisms in the control of quantum dynamics through Hamiltonian encoding Abhra Mitra 1, * and Herschel Rabitz 2 1 Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544 2 Department of Chemistry, Princeton University, Princeton, New Jersey 08544 Received 1 July 2002; published 26 March 2003 A variety of means are now available to design control fields for manipulating the evolution of quantum systems. However, the underlying physical mechanisms often remain obscure, especially in the cases of strong fields and high quantum state congestion. This paper proposes a method to quantitatively determine the various pathways taken by a quantum system in going from the initial state to the final target. The mechanism is revealed by encoding a signal in the system Hamiltonian and decoding the resultant nonlinear distortion of the signal in the system time-evolution operator. The relevant interfering pathways determined by this analysis give insight into the physical mechanisms operative during the evolution of the quantum system. A hierarchy of mechanism identification algorithms with increasing ability to extract more detailed pathway information is presented. The mechanism identification concept is presented in the context of analyzing computer simulations of controlled dynamics. As illustrations of the concept, mechanisms are identified in the control of several simple, discrete-state quantum systems. The mechanism analysis tools reveal the roles of multiple interacting quantum pathways to maximally take advantage of constructive and destructive interference. Similar proce- dures may be applied directly in the laboratory to identify control mechanisms without resort to computer modeling, although this extension is not addressed in this paper. DOI: 10.1103/PhysRevA.67.033407 PACS numbers: 32.80.Qk I. INTRODUCTION Optimal control theory is an effective technique for de- signing electric fields to manipulate the evolution of quantum-mechanical systems 1–6. Closed-loop learning algorithms 2 combined with advances in laser pulse- shaping techniques have enabled the direct discovery of laboratory optimal controls, even for complex systems 7–14. However, the mechanisms by which the target state is reached often remain obscure, in both computer simula- tions and experiments. Under favorable conditions informa- tion about the control mechanism may be deduced from an analysis of the temporal, frequency, or time-frequency struc- ture of the control fields 15,16. However, under general circumstances caution is called for as the mechanism can depend in a nonlinear fashion on the control field. Thus, a more systematic technique is required, which addresses the nonlinearities of the mechanism identification problem. This paper presents the means to understand the control mecha- nism in the theoretical design of fields and their simulated dynamic response. The control mechanism is revealed by identifying the dominant quantum pathways contributing to the observable final state achieved by the control field. The pathways, and thus the system mechanism, can be resolved at various levels of detail. The notion of a quantum pathway is also subject to the definition associated with the choice of representation of the Hamiltonian, and this paper uses a natu- ral definition in the context of applications described by a discrete set of states. However, some systems might lend themselves to other definitions of mechanism, which may be similarly revealed. The mechanism identification concept The essence of the mechanism identification MI concept will be explained below with the remainder of the paper presenting the details of the procedure and its illustration on several simple problems. A quantum control pathway analy- sis can be used for post-field-design MI as well as during the design procedure, to actively steer the dynamics to favor certain pathways. Analogous MI pathway analyses could be performed directly in the laboratory 17. This paper concen- trates on introducing the MI concept in the context of analy- sis after computational control field design. The basic proce- dure for MI remains the same when working with laboratory data, but additional complexities must be dealt with as direct access to the wave function is not available. The quantum systems analyzed in this paper are described by Hamiltonians of the form H =H 0 +V ( t ), where H 0 is the field-free Hamiltonian and V ( t ) accounts for the external field. For many quantum control applications typically V ( t ) =- E( t ) where  is the dipole and E( t ) is the control electric field. Although the paper will assume this form for V ( t ), the general formulation of Hamiltonian encoding does not require the Hamiltonian to be linear in the control field. The time evolution of the system is prescribed by the equa- tion i  dU t  dt = H 0 - E t  U t  , U 0  =1. 1 The eigenvalues E i and eigenfunctions | n i  of H 0 satisfy H 0 | n i  =E i | n i  for i =1,2, . . . , d where d is the dimension of the state space of the quantum system. We define  ij =( E i -E j )/  , and the control field can be conveniently expressed as *Electronic address: abhra@princeton.edu PHYSICAL REVIEW A 67, 033407 2003 1050-2947/2003/673/03340716/$20.00 ©2003 The American Physical Society 67 033407-1