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Physical Review A

Understanding the role of respresentation in controlled quantum-dynamical mechanism analysis

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

Published Apr 22, 2008• 10 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.77.043415

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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.
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.
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.
While closed-loop control of quantum dynamics in the laboratory is proving to be broadly successful, the control mechanisms induced by the fields are often left obscure. Hamiltonian encoding (HE) was originally introduced as a method for understanding mechanisms in quantum dynamics in the context of computational simulations, based on access to the system wavefunction. As a step towards laboratory implementation of HE, this paper addresses the issues raised by the use of observables rather than the wavefunction in HE. The goal of laboratory based HE is to obtain an understanding of control mechanism through a sequence of systematic control experiments, whose collective information can identify the underlying control mechanism defined as the set of significant amplitudes connecting the initial and final states. Mechanism is determined by means of observing the dynamics of special sequences of system Hamiltonians encoded through the control field. The proposed algorithm can handle complex systems, operates with no recourse to dynamical simulations, and functions with limited understanding of the system Hamiltonian. As with the closed-loop control experiments, the HE control mechanism identification algorithm performs a new experiment each time the dynamical outcome from an encoded Hamiltonian is called for. This paper presents the basic HE algorithm in the context of physical systems described by a finite dimensional Hilbert space. The method is simulated with simple models, and the extension to more complex systems is discussed.
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.
We report on the laboratory implementation of quantum-control-mechanism identification through Hamiltonian encoding and observable decoding (HE-OD). Over a sequence of experiments, HE-OD introduces a special encoded signature into the components of a previously determined control field expressed in a chosen representation. The outcome appears as a modulated signal in the controlled system observable. Decoding the modulated signal identifies the hierarchy of correlations between components of the control field in a particular representation. In cases where the initial quantum state and observable operator are fully known, then HE-OD can also identify the transition amplitudes of the various Dyson expansion orders contributing to the controlled dynamics. The basic principles of HE-OD are illustrated for second harmonic generation when the components of the field representation are simply taken as the pixels in the pulse shaper. The outcome of HE-OD agrees well with simulations, verifying the concept.
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.