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Physical Review A

Revealing quantum-control mechanisms through Hamiltonian encoding in different representations

Abhra Mitra, Ignacio R. Solá, HERSCHEL A. RABITZ

Published Apr 28, 2003• 36 citations
Publisher

Abstract

The Hamiltonian encoding is a means for revealing the mechanism of controlled quantum dynamics. In this context, the mechanism is defined by the dominant quantum pathways starting from the initial state and proceeding through a set of intermediate states to end at the final state. The nature and interpretation of the mechanism depends on the choice of the states to represent the dynamics. Alternative representations may provide distinct insights into the system mechanism, and representations producing fewer pathways are especially interesting. In addition, a suitable choice of representation may highlight the role of certain couplings in a system that would normally be masked by other, higher magnitude couplings. A simple three-level system is chosen for illustration, where different values for the Rabi frequencies lead to mechanistic analyses that are best described in terms of particular representations. As an examlple, the role of the nonadiabatic terms in stimulated Raman adiabatic passage dynamics is analyzed through the Hamiltonian encoding.

DOI
10.1103/physreva.67.043409

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The Hamiltonian encoding is a means for revealing the mechanism of controlled quantum dynamics. In this context, the mechanism is defined by the dominant quantum pathways starting from the initial state and proceeding through a set of intermediate states to end at the final state. The nature and interpretation of the mechanism depends on the choice of the states to represent the dynamics. Alternative representations may provide distinct insights into the system mechanism, and representations producing fewer pathways are especially interesting. In addition, a suitable choice of representation may highlight the role of certain couplings in a system that would normally be masked by other, higher magnitude couplings. A simple three-level system is chosen for illustration, where different values for the Rabi frequencies lead to mechanistic analyses that are best described in terms of particular representations. As an examlple, the role of the nonadiabatic terms in stimulated Raman adiabatic passage dynamics is analyzed through the Hamiltonian encoding.
Hamiltonian Encoding (HE) has been proposed as a technique for analyzing the mechanism of controlled quantum dynamics, where mechanism is understood in terms of the set of amplitudes of the dominant pathways connecting the initial and final states of the system. The choice of representation for the system wave function is often motivated by seeking simplicity for the structure of the Hamiltonian and not necessarily for the generated dynamics. However, the mechanism revealed by HE is strongly dependent on the basis in which the wave function is represented. The degree of mechanistic complexity is indicated by the relevant orders of the Dyson series contributing to the dynamics. An appropriate choice of representation can yield a simpler view of the dynamical mechanism by shifting some of the complexity into the representation itself. In this work the choice of representation is set up as the solution to a variational optimization problem. For unconstrained basis transformations, the optimization of the representation is formally equivalent to solving the time-dependent Schr\"odinger equation; different constrained basis transformations provide distinct dynamical perspectives. Specific constrained variational Ans\"atze are compared and analyzed by performing HE on several simple Hamiltonians with an observation of the extent to which the mechanism assessment varies with representation. The general variational formulation for determining representation can flexibly admit other Ans\"atze with the ultimate aim of balancing the ease of determining and understanding the representation with the reduction in mechanistic complexity.
Optimal control of quantum dynamics in the laboratory is proving to be increasingly successful. The control fields can be complex, and the mechanisms by which they operate have often remained obscure. Hamiltonian encoding (HE) has been proposed as a method for understanding mechanisms in quantum dynamics. In this context mechanism is defined in terms of the dominant quantum pathways leading to the final state of the controlled system. HE operates by encoding a special modulation into the Hamiltonian and decoding its signature in the dynamics to determine the dominant pathway amplitudes. Earlier work encoded the modulation directly into the Hamiltonian operators. This present work introduces the alternative scheme of field based HE, where the modulation is encoded into the control field and not directly into the Hamiltonian operators. This distinct form of modulation yields a new perspective on mechanism and is computationally faster than the earlier approach. Field based encoding is also an important step towards a laboratory based algorithm for HE as it is the only form of encoding that may be experimentally executed. HE is also extended to cover systems with noise and uncertainty and finally, a hierarchical algorithm is introduced to reveal mechanism in a stepwise fashion of ever increasing detail as desired. This new hierarchical algorithm is an improvement over earlier approaches to HE where the entire mechanism was determined in one stroke. The improvement comes from the use of less complex modulation schemes, which leads to fewer evaluations of Schrodinger's equation. A number of simulations are presented on simple systems to illustrate the new field based encoding technique for mechanism assessment.
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 procedures 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.
There are increasing numbers of successful numerical simulations as well as laboratory tests of controlled quantum phenomena. However, little is understood about the control mechanisms, especially in strongly driven systems. In this paper, mechanism is defined in terms of the significant quantum pathways induced by the control field, connecting the initial state to the final target state. The present work focuses on understanding mechanism in the coordinate representation, which is natural for many dynamical systems. The quantum amplitudes defining the mechanistic pathways are determined by a special procedure called coding Hamiltonians to access mechanistic pathways (CHAMP). Numerical procedures are presented for performing CHAMP calculations in a stable fashion. Mechanism determination is illustrated for excitation of a model linear triatomic molecule. Mechanisms are revealed for optimal control fields as well as for suboptimally driven systems including snapshots in time of the evolving mechanism and the effect of field noise on the mechanism. The high order multiphoton processes involved in the example produce many interfering pathways to create the final state, yet the dynamics and the mechanism are robust to reasonable levels of noise.
Hamiltonian encoding was introduced as a technique for revealing the mechanism of controlled quantum systems. It does so by decomposing the evolution into pathways between the computational basis states, where each pathway has an associated complex amplitude. The magnitude of a pathway amplitude determines its significance and many pathways constructively and/or destructively interfere to produce the final evolution of the system. In this paper, we apply Hamiltonian encoding to reveal the mechanism behind creating qubit gates implemented via optimal control pulses. An X gate, two CNOT gates, and a SWAP gate are examined to determine the degree of interference involved and to demonstrate that different optimal controls produce distinct mechanisms. Although the detailed mechanism for creating any gate depends on the nature of the control field, the mechanism analysis tools are generic. The presented gates and their mechanisms in this paper are thus illustrative and a researcher may apply these same tools to any gate with a suitable optimal control field.
Hamiltonian encoding is a methodology for revealing the mechanism behind the dynamics governing controlled quantum systems. In this paper, following Mitra and Rabitz \cite{abhra_1}, we define mechanism via pathways of eigenstates that describe the evolution of the system, where each pathway is associated with a complex-valued amplitude corresponding to a term in the Dyson series. The evolution of the system is determined by the constructive and destructive interference of these pathway amplitudes. Pathways with similar attributes can be grouped together into pathway classes. The amplitudes of pathway classes are computed by modulating the Hamiltonian matrix elements and decoding the subsequent evolution of the system rather than by direct computation of the individual terms in the Dyson series. The original implementation of Hamiltonian encoding was computationally intensive and became prohibitively expensive in large quantum systems. This paper presents two new encoding algorithms that calculate the amplitudes of pathway classes by using techniques from graph theory and algebraic topology to exploit patterns in the set of allowed transitions, greatly reducing the number of matrix elements that need to be modulated. These new algorithms provide an exponential decrease in both computation time and memory utilization with respect to the Hilbert space dimension of the system. To demonstrate the use of these techniques, they are applied to two illustrative state-to-state transition problems.