Non-integrability of the optimal control problem for n-level quantum systems
We study the problem of optimal laser-induced population transfer in n-level quantum systems. This problem can be represented as an energy-optimal control problem, related to a sub-Riemannian problem on SO(n), and it is known (Boscain, Chambrion, Charlot, Gauthier) that for n = 2 and n = 3 the H...
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Veröffentlicht in: | Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2017-04, Vol.50 (17), p.175202 |
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creator | Duval, Guillaume Maciejewski, Andrzej Respondek, Witold |
description | We study the problem of optimal laser-induced population transfer in n-level quantum systems. This problem can be represented as an energy-optimal control problem, related to a sub-Riemannian problem on SO(n), and it is known (Boscain, Chambrion, Charlot, Gauthier) that for n = 2 and n = 3 the Hamiltonian system associated with the Pontryagin Maximum Principle PMP is integrable. We will show that this changes completely for n 4. Specifically, for n = 4, we will prove that the adjoint equation of the PMP does not possess any meromorphic first integral independent of the Hamiltonian on the levels of the Casimir functions. This implies that the system is not B-integrable for any n 4. In proving our nonintegrability results we will use differential Galois theory. |
doi_str_mv | 10.1088/1751-8121/aa6203 |
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This problem can be represented as an energy-optimal control problem, related to a sub-Riemannian problem on SO(n), and it is known (Boscain, Chambrion, Charlot, Gauthier) that for n = 2 and n = 3 the Hamiltonian system associated with the Pontryagin Maximum Principle PMP is integrable. We will show that this changes completely for n 4. Specifically, for n = 4, we will prove that the adjoint equation of the PMP does not possess any meromorphic first integral independent of the Hamiltonian on the levels of the Casimir functions. This implies that the system is not B-integrable for any n 4. In proving our nonintegrability results we will use differential Galois theory.</description><identifier>ISSN: 1751-8113</identifier><identifier>EISSN: 1751-8121</identifier><identifier>DOI: 10.1088/1751-8121/aa6203</identifier><identifier>CODEN: JPHAC5</identifier><language>eng</language><publisher>IOP Publishing</publisher><subject>B-integrability ; control of quantum systems ; differential Galois theory ; Hamiltonian systems on Lie Groups ; Liouville integrability ; Mathematics ; optimal control ; Pontryagin Maximum Principle</subject><ispartof>Journal of physics. 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A, Mathematical and theoretical</title><addtitle>JPhysA</addtitle><addtitle>J. Phys. A: Math. Theor</addtitle><description>We study the problem of optimal laser-induced population transfer in n-level quantum systems. This problem can be represented as an energy-optimal control problem, related to a sub-Riemannian problem on SO(n), and it is known (Boscain, Chambrion, Charlot, Gauthier) that for n = 2 and n = 3 the Hamiltonian system associated with the Pontryagin Maximum Principle PMP is integrable. We will show that this changes completely for n 4. Specifically, for n = 4, we will prove that the adjoint equation of the PMP does not possess any meromorphic first integral independent of the Hamiltonian on the levels of the Casimir functions. This implies that the system is not B-integrable for any n 4. In proving our nonintegrability results we will use differential Galois theory.</description><subject>B-integrability</subject><subject>control of quantum systems</subject><subject>differential Galois theory</subject><subject>Hamiltonian systems on Lie Groups</subject><subject>Liouville integrability</subject><subject>Mathematics</subject><subject>optimal control</subject><subject>Pontryagin Maximum Principle</subject><issn>1751-8113</issn><issn>1751-8121</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2017</creationdate><recordtype>article</recordtype><recordid>eNp1kL1PwzAQxS0EEqWwM3pFIvQcx_kYqwooUgQLzJbjD5rKiYPtVup_T6Kgbix3p6f3Tnc_hO4JPBEoyxUpGElKkpKVEHkK9AItztLleSb0Gt2EsAdgGVTpAtXvrk_aPupvL5rWtvGEncFxp7EbYtsJi6Xro3cWD941VnfYOI_7xOqjtvjnIPp46HA4hai7cIuujLBB3_31Jfp6ef7cbJP64_Vts64TSUkWEwUNmIwZbUqVM2BKCUElyRsqy6LMCkXzVFRFUemxlEywRmslDZNVoxhISpfoYd67E5YPfjzTn7gTLd-uaz5pQKosA0KOZPTC7JXeheC1OQcI8Ikcn9DwCROfyY2RxznSuoHv3cH34zP_238BTwRvWA</recordid><startdate>20170428</startdate><enddate>20170428</enddate><creator>Duval, Guillaume</creator><creator>Maciejewski, Andrzej</creator><creator>Respondek, Witold</creator><general>IOP Publishing</general><scope>AAYXX</scope><scope>CITATION</scope><scope>1XC</scope><orcidid>https://orcid.org/0000-0002-3846-448X</orcidid></search><sort><creationdate>20170428</creationdate><title>Non-integrability of the optimal control problem for n-level quantum systems</title><author>Duval, Guillaume ; Maciejewski, Andrzej ; Respondek, Witold</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c314t-d0b0f45fef8d6505ddaa3c16b3c87847d362a9779e97785a5beedcf5c9bd50c33</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2017</creationdate><topic>B-integrability</topic><topic>control of quantum systems</topic><topic>differential Galois theory</topic><topic>Hamiltonian systems on Lie Groups</topic><topic>Liouville integrability</topic><topic>Mathematics</topic><topic>optimal control</topic><topic>Pontryagin Maximum Principle</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Duval, Guillaume</creatorcontrib><creatorcontrib>Maciejewski, Andrzej</creatorcontrib><creatorcontrib>Respondek, Witold</creatorcontrib><collection>CrossRef</collection><collection>Hyper Article en Ligne (HAL)</collection><jtitle>Journal of physics. A, Mathematical and theoretical</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Duval, Guillaume</au><au>Maciejewski, Andrzej</au><au>Respondek, Witold</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Non-integrability of the optimal control problem for n-level quantum systems</atitle><jtitle>Journal of physics. A, Mathematical and theoretical</jtitle><stitle>JPhysA</stitle><addtitle>J. Phys. A: Math. Theor</addtitle><date>2017-04-28</date><risdate>2017</risdate><volume>50</volume><issue>17</issue><spage>175202</spage><pages>175202-</pages><issn>1751-8113</issn><eissn>1751-8121</eissn><coden>JPHAC5</coden><abstract>We study the problem of optimal laser-induced population transfer in n-level quantum systems. This problem can be represented as an energy-optimal control problem, related to a sub-Riemannian problem on SO(n), and it is known (Boscain, Chambrion, Charlot, Gauthier) that for n = 2 and n = 3 the Hamiltonian system associated with the Pontryagin Maximum Principle PMP is integrable. We will show that this changes completely for n 4. Specifically, for n = 4, we will prove that the adjoint equation of the PMP does not possess any meromorphic first integral independent of the Hamiltonian on the levels of the Casimir functions. This implies that the system is not B-integrable for any n 4. In proving our nonintegrability results we will use differential Galois theory.</abstract><pub>IOP Publishing</pub><doi>10.1088/1751-8121/aa6203</doi><tpages>27</tpages><orcidid>https://orcid.org/0000-0002-3846-448X</orcidid></addata></record> |
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subjects | B-integrability control of quantum systems differential Galois theory Hamiltonian systems on Lie Groups Liouville integrability Mathematics optimal control Pontryagin Maximum Principle |
title | Non-integrability of the optimal control problem for n-level quantum systems |
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