dorsal/arxiv
View SchemaTime Optimal Control in Spin Systems
| Authors | Navin Khaneja, Roger Brockett, Steffen Glaser |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0006114 |
| URL | https://arxiv.org/abs/quant-ph/0006114 |
| DOI | 10.1103/PhysRevA.63.032308 |
Abstract
In this paper, we study the design of pulse sequences for NMR spectroscopy as a problem of time optimal control of the unitary propagator. Radio frequency pulses are used in coherent spectroscopy to implement a unitary transfer of state. Pulse sequences that accomplish a desired transfer should be as short as possible in order to minimize the effects of relaxation and to optimize the sensitivity of the experiments. Here, we give an analytical characterization of such time optimal pulse sequences applicable to coherence transfer experiments in multiple-spin systems. We have adopted a general mathematical formulation, and present many of our results in this setting, mindful of the fact that new structures in optimal pulse design are constantly arising. Moreover, the general proofs are no more difficult than the specific problems of current interest. From a general control theory perspective, the problems we want to study have the following character. Suppose we are given a controllable right invariant system on a compact Lie group, what is the minimum time required to steer the system from some initial point to a specified final point? In NMR spectroscopy and quantum computing, this translates to, what is the minimum time required to produce a unitary propagator? We also give an analytical characterization of maximum achievable transfer in a given time for the two spin system.
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"abstract": "In this paper, we study the design of pulse sequences for NMR spectroscopy as\na problem of time optimal control of the unitary propagator. Radio frequency\npulses are used in coherent spectroscopy to implement a unitary transfer of\nstate. Pulse sequences that accomplish a desired transfer should be as short as\npossible in order to minimize the effects of relaxation and to optimize the\nsensitivity of the experiments. Here, we give an analytical characterization of\nsuch time optimal pulse sequences applicable to coherence transfer experiments\nin multiple-spin systems. We have adopted a general mathematical formulation,\nand present many of our results in this setting, mindful of the fact that new\nstructures in optimal pulse design are constantly arising. Moreover, the\ngeneral proofs are no more difficult than the specific problems of current\ninterest. From a general control theory perspective, the problems we want to\nstudy have the following character. Suppose we are given a controllable right\ninvariant system on a compact Lie group, what is the minimum time required to\nsteer the system from some initial point to a specified final point? In NMR\nspectroscopy and quantum computing, this translates to, what is the minimum\ntime required to produce a unitary propagator? We also give an analytical\ncharacterization of maximum achievable transfer in a given time for the two\nspin system.",
"arxiv_id": "quant-ph/0006114",
"authors": [
"Navin Khaneja",
"Roger Brockett",
"Steffen Glaser"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.63.032308",
"title": "Time Optimal Control in Spin Systems",
"url": "https://arxiv.org/abs/quant-ph/0006114"
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