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arXiv:quant-ph/0312212 · Physical Review A

Efficient extraction of quantum Hamiltonians from optimal laboratory data

J. M. Geremia, HERSCHEL A. RABITZ

Published Aug 13, 2004• 8 citations
arXivHTMLPDF

Abstract

Optimal identification (OI) is a recently developed procedure for extracting information about quantum Hamiltonians from experimental data. It employs techniques from coherent learning control to drive the quantum system such that dynamical measurements provide maximal information about its Hamiltonian. OI is an optimal procedure as initially presented; however, the data inversion component is computationally expensive. Here, we demonstrate that highly efficient global, nonlinear, map-facilitated inversion procedures can be combined with the OI concept to make it more suitable for laboratory implementation. A simulation of map-facilitated OI illustrates how the input-output maps can greatly accelerate the data inversion process.

DOI
10.1103/physreva.70.023804

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Optimal identification (OI) is a recently developed procedure for extracting information about quantum Hamiltonians from experimental data. It employs techniques from coherent learning control to drive the quantum system such that dynamical measurements provide maximal information about its Hamiltonian. OI is an optimal procedure as initially presented; however, the data inversion component is computationally expensive. Here, we demonstrate that highly efficient global, nonlinear, map-facilitated inversion procedures can be combined with the OI concept to make it more suitable for laboratory implementation. A simulation of map-facilitated OI illustrates how the input-output maps can greatly accelerate the data inversion process.
We introduce optimal identification (OI), a collaborative laboratory/computational algorithm for extracting quantum Hamiltonians from experimental data specifically sought to minimize the inversion error. OI incorporates the components of quantum control and inversion by combining ultrafast pulse shaping technology and high throughput experiments with global inversion techniques to actively identify quantum Hamiltonians from tailored observations. The OI concept rests on the general notion that optimal data can be measured under the influence of suitable controls to minimize uncertainty in the extracted Hamiltonian despite data limitations such as finite resolution and noise. As an illustration of the operating principles of OI, the transition dipole moments of a multilevel quantum Hamiltonian are extracted from simulated population transfer experiments. The OI algorithm revealed a simple optimal experiment that determined the Hamiltonian matrix elements to an accuracy two orders of magnitude better than obtained from inverting 500 random data sets. The optimal and nonlinear nature of the algorithm were shown to be capable of reliably identifying the Hamiltonian even when there were more variables than observations. Furthermore, the optimal experiment acted as a tailored filter to prevent the laboratory noise from significantly propagating into the extracted Hamiltonian.
This paper presents a general self-guided algorithm for direct laboratory learning of controls to manipulate quantum-mechanical systems. The primary focus is on an algorithm based on the learning of a linear laboratory input–output map from a sequence of controls, and their observed impact on the quantum-mechanical system. This map is then employed in an iterative fashion, to sequentially home in on the desired objective. The objective may be a target state at a final time, or a continuously weighted observational trajectory. The self-guided aspects of the algorithm are based on implementing a cost functional that only contains laboratory-accessible information. Through choice of the weights in this functional, the algorithm can automatically stay within the bounds of each local linear map and indicate when a new map is necessary for additional iterative improvement. Finally, these concepts can be generalized to include the possibility of employing nonlinear maps, as well as just the laboratory control instrument settings, rather than observation of the control itself. An illustrative simulation of the concepts is presented for the control of a four-level quantum system.
A closed loop learning control concept is introduced for teaching lasers to manipulate quantum systems for the purpose of optimally identifying Hamiltonian information. The closed loop optimal identification algorithm operates by revealing the distribution of Hamiltonians consistent with the data. The control laser is guided to perform additional experiments, based on minimizing the dispersion of the distribution. Operation of such an apparatus is simulated for two model finite dimensional quantum systems.
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.
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.
Understanding and controlling engineered quantum systems is key to developing practical quantum technology. However, given the current technological limitations, such as fabrication imperfections and environmental noise, this is not always possible. To address these issues, a great deal of theoretical and numerical methods for quantum system identification and control have been developed. These methods range from traditional curve fittings, which are limited by the accuracy of the model that describes the system, to machine learning methods, which provide efficient control solutions but no control beyond the output of the model, nor insights into the underlying physical process. Here we experimentally demonstrate a "graybox" approach to construct a physical model of a quantum system and use it to design optimal control. We report superior performance over model fitting, while generating unitaries and Hamiltonians, which are quantities not available from the structure of standard supervised machine learning models. Our approach combines physics principles with high-accuracy machine learning and is effective with any problem where the required controlled quantities cannot be directly measured in experiments. This method naturally extends to time-dependent and open quantum systems, with applications in quantum noise spectroscopy and cancellation.