Skip to main content
Top
Published in: Quantum Information Processing 7/2015

01-07-2015

Approximate controllability of the two trapped ions system

Authors: Esteban Paduro, Mario Sigalotti

Published in: Quantum Information Processing | Issue 7/2015

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

We prove the approximate controllability of a bilinear Schrödinger equation modelling a two trapped ions system. A new spectral decoupling technique is introduced, which allows to analyze the controllability of the infinite-dimensional system through finite-dimensional considerations.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Appendix
Available only for authorised users
Literature
1.
go back to reference Adami, R., Boscain, U.: Controllability of the Schrödinger Equation via Intersection of Eigenvalues. In: Proceedings of the 44th IEEE Conference on Decision and Control, IEEE, December 2005, pp. 1080–1085 Adami, R., Boscain, U.: Controllability of the Schrödinger Equation via Intersection of Eigenvalues. In: Proceedings of the 44th IEEE Conference on Decision and Control, IEEE, December 2005, pp. 1080–1085
2.
go back to reference Albertini, F., D’Alessandro, D.: Notions of controllability for bilinear multilevel quantum systems. IEEE Trans. Automat. Control 48(8), 1399–1403 (2003). MR MR2004373 (2004g:93018)MathSciNetCrossRef Albertini, F., D’Alessandro, D.: Notions of controllability for bilinear multilevel quantum systems. IEEE Trans. Automat. Control 48(8), 1399–1403 (2003). MR MR2004373 (2004g:93018)MathSciNetCrossRef
3.
go back to reference Bliss, R.S., Burgarth, D.: Quantum control of infinite-dimensional many-body systems. Phys. Rev. A 89, 032309 (2014)ADSCrossRef Bliss, R.S., Burgarth, D.: Quantum control of infinite-dimensional many-body systems. Phys. Rev. A 89, 032309 (2014)ADSCrossRef
4.
go back to reference Bloch, A.M., Brockett, R.W., Rangan, C.: Finite Controllability of Infinite-Dimensional Quantum Systems. IEEE Trans. Autom. Control 55(8), 1797–1805 (2010)MathSciNetCrossRef Bloch, A.M., Brockett, R.W., Rangan, C.: Finite Controllability of Infinite-Dimensional Quantum Systems. IEEE Trans. Autom. Control 55(8), 1797–1805 (2010)MathSciNetCrossRef
5.
go back to reference Boscain, U., Caponigro, M., Chambrion, T., Sigalotti, M.: A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule. Commun. Math. Phys. 311(2), 423–455 (2012)MATHMathSciNetADSCrossRef Boscain, U., Caponigro, M., Chambrion, T., Sigalotti, M.: A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule. Commun. Math. Phys. 311(2), 423–455 (2012)MATHMathSciNetADSCrossRef
6.
go back to reference Boscain, U., Caponigro, M., Sigalotti, M.: Multi-input Schrödinger equation: controllability, tracking, and application to the quantum angular momentum. J. Differ. Equa. 256(11), 3524–3551 (2014)MATHMathSciNetADSCrossRef Boscain, U., Caponigro, M., Sigalotti, M.: Multi-input Schrödinger equation: controllability, tracking, and application to the quantum angular momentum. J. Differ. Equa. 256(11), 3524–3551 (2014)MATHMathSciNetADSCrossRef
7.
go back to reference Boscain, U.V., Mason, P., Panati, G., Sigalotti, M: On the control of spin-boson systems. In: Proceedings of the 12th European Control Conference, Zurich, Switzerland, 2013, pp. 2110–2115 Boscain, U.V., Mason, P., Panati, G., Sigalotti, M: On the control of spin-boson systems. In: Proceedings of the 12th European Control Conference, Zurich, Switzerland, 2013, pp. 2110–2115
9.
go back to reference Chambrion, T., Mason, P., Sigalotti, M., Boscain, U.: Controllability of the discrete-spectrum Schrödinger equation driven by an external field. Annales de l’Institut Henri Poincare (C) Non Linear Analysis 26(1), 329–349 (2009)MATHMathSciNetADSCrossRef Chambrion, T., Mason, P., Sigalotti, M., Boscain, U.: Controllability of the discrete-spectrum Schrödinger equation driven by an external field. Annales de l’Institut Henri Poincare (C) Non Linear Analysis 26(1), 329–349 (2009)MATHMathSciNetADSCrossRef
10.
go back to reference Childs, A., Chuang, I.: Universal quantum computation with two-level trapped ions. Phys. Rev. A 63, 012306 (2000)ADSCrossRef Childs, A., Chuang, I.: Universal quantum computation with two-level trapped ions. Phys. Rev. A 63, 012306 (2000)ADSCrossRef
11.
go back to reference Cirac, J., Zoller, P.: Quantum computations with cold trapped ions. Phys. Rev. Lett. 74(20), 4091–4094 (1995)ADSCrossRef Cirac, J., Zoller, P.: Quantum computations with cold trapped ions. Phys. Rev. Lett. 74(20), 4091–4094 (1995)ADSCrossRef
12.
go back to reference Ervedoza, S., Puel, J.-P.: Approximate controllability for a system of Schrödinger equations modeling a single trapped ion. Annales de l’Institut Henri Poincare (C) Non Linear Analysis 26(6), 2111–2136 (2009)MATHMathSciNetADSCrossRef Ervedoza, S., Puel, J.-P.: Approximate controllability for a system of Schrödinger equations modeling a single trapped ion. Annales de l’Institut Henri Poincare (C) Non Linear Analysis 26(6), 2111–2136 (2009)MATHMathSciNetADSCrossRef
13.
go back to reference Gulde, S., Riebe, M., Lancaster, G., Becher, C., Eschner, J., Häffner, H., Schmidt-Kaler, F., Chuang, I.L., Blatt, R.: Implementation of the Deutsch-Jozsa algorithm on an ion-trap quantum computer. Nature 421(6918), 48–50 (2003)ADSCrossRef Gulde, S., Riebe, M., Lancaster, G., Becher, C., Eschner, J., Häffner, H., Schmidt-Kaler, F., Chuang, I.L., Blatt, R.: Implementation of the Deutsch-Jozsa algorithm on an ion-trap quantum computer. Nature 421(6918), 48–50 (2003)ADSCrossRef
14.
go back to reference Jonathan, D., Plenio, M., Knight, P.: Fast quantum gates for cold trapped ions. Phys. Rev. A 62(4), 042307 (2000)ADSCrossRef Jonathan, D., Plenio, M., Knight, P.: Fast quantum gates for cold trapped ions. Phys. Rev. A 62(4), 042307 (2000)ADSCrossRef
15.
go back to reference Jurdjevic, V.: Geometric Control Theory. Cambridge University Press, Cambridge (1996)CrossRef Jurdjevic, V.: Geometric Control Theory. Cambridge University Press, Cambridge (1996)CrossRef
16.
go back to reference Katok, A., Hasselblatt, B.: Introduction to the modern theory of dynamical systems. In: Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, (1995) With a supplementary chapter by Katok and Leonardo Mendoza Katok, A., Hasselblatt, B.: Introduction to the modern theory of dynamical systems. In: Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, (1995) With a supplementary chapter by Katok and Leonardo Mendoza
17.
go back to reference Keyl, M., Zeier, R., Schulte-Herbrueggen, T.: Controlling several atoms in a cavity. New J. Phys. 16, 065010 (2014)ADSCrossRef Keyl, M., Zeier, R., Schulte-Herbrueggen, T.: Controlling several atoms in a cavity. New J. Phys. 16, 065010 (2014)ADSCrossRef
18.
go back to reference Kneer, B., Law, C.: Preparation of arbitrary entangled quantum states of a trapped ion. Phys. Rev. A 57(3), 2096–2104 (1998)ADSCrossRef Kneer, B., Law, C.: Preparation of arbitrary entangled quantum states of a trapped ion. Phys. Rev. A 57(3), 2096–2104 (1998)ADSCrossRef
19.
go back to reference Law, C., Eberly, J.: Arbitrary control of a quantum electromagnetic field. Phys. Rev. Lett. 76(7), 1055–1058 (1996)ADSCrossRef Law, C., Eberly, J.: Arbitrary control of a quantum electromagnetic field. Phys. Rev. Lett. 76(7), 1055–1058 (1996)ADSCrossRef
20.
go back to reference Paduro, E.: Approximate Controllability for the Two Trapped Ions System, Master thesis. Universidad Tecnica Federico Santa Maria, 2013, pp. 1–123 Paduro, E.: Approximate Controllability for the Two Trapped Ions System, Master thesis. Universidad Tecnica Federico Santa Maria, 2013, pp. 1–123
21.
go back to reference Rangan, C., Bloch, A., Monroe, C., Bucksbaum, P.: Control of trapped-ion quantum states with optical pulses. Phys. Rev. Lett. 92(11), 10 (2004)CrossRef Rangan, C., Bloch, A., Monroe, C., Bucksbaum, P.: Control of trapped-ion quantum states with optical pulses. Phys. Rev. Lett. 92(11), 10 (2004)CrossRef
22.
go back to reference Rouchon, P.: Quantum systems and control. Int. Conf. Honor Claude Lobry 9, 325–357 (2008)MathSciNet Rouchon, P.: Quantum systems and control. Int. Conf. Honor Claude Lobry 9, 325–357 (2008)MathSciNet
23.
go back to reference Yuan, H., Lloyd, S.: Controllability of the coupled spin-\(\frac{1}{2}\) harmonic oscillator system. Phys. Rev. A 75, 052331 (2007)MathSciNetADSCrossRef Yuan, H., Lloyd, S.: Controllability of the coupled spin-\(\frac{1}{2}\) harmonic oscillator system. Phys. Rev. A 75, 052331 (2007)MathSciNetADSCrossRef
Metadata
Title
Approximate controllability of the two trapped ions system
Authors
Esteban Paduro
Mario Sigalotti
Publication date
01-07-2015
Publisher
Springer US
Published in
Quantum Information Processing / Issue 7/2015
Print ISSN: 1570-0755
Electronic ISSN: 1573-1332
DOI
https://doi.org/10.1007/s11128-015-0991-3

Other articles of this Issue 7/2015

Quantum Information Processing 7/2015 Go to the issue