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Q U E N C H E S A C R O S SS T R U C T U R A L T R A N S I T I O N SI N I O N C O U L O M B C RY S TA L S

Dynamics of Ion Coulomb Crystals in State-Dependent Potentialsand their Characterization by Ramsey Interferometry

D I S S E RTAT I O N

zur Erlangung des Grades des Doktors der Naturwissenschaften

der Naturwissenschaftlich-Technischen Fakulttder U N I V E R S I TT D E S S A A R L A N D E S

von

jens domagoj baltrusch

Saarbrcken

2016

Tag des Kolloquiums: 14.07.2016

Dekan: Prof. Dr. Guido Kickelbick

Mitglieder desPrfungsausschusses: Prof. Dr. Christoph Becher

Prof. Dr. Giovanna MorigiProf. Dr. Heiko Rieger

Dr. Reza Shaebani

Q U E N C H E S A C R O S SS T R U C T U R A L T R A N S I T I O N SI N I O N C O U L O M B C RY S TA L S

Dynamics of Ion Coulomb Crystals in State-Dependent Potentialsand their Characterization by Ramsey Interferometry

Jens D. Baltrusch: Quenches Across Structural Transitions in Ion Coulomb Crys-tals: Dynamics of Ion Coulomb Crystals in State-Dependent Potentials and theirCharacterization by Ramsey Interferometry.

Dissertation zur Erlangung des Grades des Doktors der Naturwissenschaftender Naturwissenschaftlich-Technischen Fakultt der Universitt des Saarlandes.

Betreuerin: Prof. Dr. Giovanna Morigi.

Plichtexemplar

February 2016 by Jens D. Baltrusch exceptFigures 5.1, 6.7 and 6.8 by American Physical Society.

This work except the parts that are by the American Physical Society is licensed under theCreative Commons Attribution-NonCommercial-NoDerivatives 4.0 International Licence. Toview a copy of this license, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.

http://creativecommons.org/licenses/by-nc-nd/4.0/

E I D E S S TAT T L I C H E V E R S I C H E R U N G

Hiermit versichere ich an Eides statt, dass ich die vorliegende Ar-beit selbststndig und ohne Benutzung anderer als der angegebenenHilfsmittel angefertigt habe. Die aus anderen Quellen oder indirektbernommenen Daten und Konzepte sind unter Angabe der Quelle ge-kennzeichnet. Die Arbeit wurde bisher weder im In- noch im Auslandin gleicher oder hnlicher Form in einem Verfahren zur Erlangungeines akademischen Grades vorgelegt.

Ulm, 27. Juni 2017

Jens Domagoj Baltrusch

A B S T R A C T

This thesis theoretically discusses the dynamics of small Coulombcrystals of ions confined in state-dependent potentials following asudden quench of the mechanical forces on a single ion embedded inthe crystal. This dynamics is analysed using the principle of Ramseyinterferometry, for which purpose the electronic state of the ion is putinto a superposition, thereby entangling the ions internal degrees offreedom with the crystal wavefunction due to the state-dependentdynamics. Measuring the electronic state after a time of free evolutionand determining the interferometric visibility enables us to deduceinformation about the motional state of the crystal. We analyse thetemporal variation of this visibility in dependence on the trap par-ameters, the crystal size, and the temperature close to a structuraltransition, which allows us to infer the equilibrium properties ofthe crystal close to criticality as well as the crystals features as anon-Markovian bath.

Z U S A M M E N FA S S U N G

Diese theoretische Arbeit behandelt die Dynamik von kleinen Ionen-Coulomb-Kristallen in zustandsabhngigen Potentialen nach einerrasch erfolgten nderung des Fallenpotentials fr ein einzelnes Ion.Diese Dynamik analysieren wir mittels des Konzepts der Ramsey-Interferometrie indem wir den elektronischen Zustand in eine berla-gerung bringen, sodass sich dieser aufgrund der zustandsabhngigenDynamik mit der Wellenfunktion des gesamten Kristalls verschrnkt.Die Messung des elektronischen Zustands des Ions nach einer freienZeitentwicklung und die Bestimmung des interferometrischen Kon-trasts dieser Messung ermglichen es uns, Rckschlsse auf den Bewe-gungszustand des gesamten Kristalls zu ziehen. Wir analysieren diezeitliche Vernderung des Kontrasts fr verschiedene Fallenparameter,Kristallgren und Temperaturen nahe eines strukturellen bergan-ges, worber sich die Gleichgewichtseigenschaften des Kristalls nahedes kritischen berganges sowie die Charakteristik des Kristalls alsnicht-markovsches Bad ableiten lassen.

In memory of

Frederick William Ayer(1941 2010)

C O N T E N T S

introduction 1

I ion coulomb crystals in state-dependent poten-tials 5

1 trapped ions and ion coulomb crystals 71.1 Trapping Ions . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Physics of Trapped Ions Plasmas and Crystals . . . . 171.3 Crystalline Structures and Structural Transitions . . . . 24

2 state-dependent structures of ion coulomb crys-tals 312.1 Small Ion Coulomb Crystals in Harmonic Potentials . . 312.2 Structural Superposition States . . . . . . . . . . . . . . 442.3 State-Dependent Crystalline Structures . . . . . . . . . 55

3 dynamics of state-dependent ion coulomb crystals 613.1 Dynamics of State-Dependent Harmonic Oscillators . . 613.2 Transformations between Dynamical Variables . . . . . 633.3 Transformations between Quantum States . . . . . . . . 70

II ramsey interferometry 874 ramsey interferometry with ion coulomb crystals 89

4.1 The Principle of Ramsey Interferometry . . . . . . . . . 894.2 Ramsey Interferometry as a Probe . . . . . . . . . . . . 964.3 Implementation of Ramsey Interferometry with Trapped

Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.4 Ramsey Interferometry at Finite Temperatures . . . . . 113

5 quantum quenches at the linearzigzag transition1215.1 Quantum Quenches in Ion Coulomb Crystals . . . . . 1215.2 Analytical Formula for the Visibility . . . . . . . . . . . 1305.3 Analysis of Quenches out of the Ground State . . . . . 135

6 quantum quenches of thermally excited ion cou-lomb crystals 1576.1 Evaluation of the Visibility for Thermal States . . . . . 1576.2 Analysis of Quenches Including the Photon Recoil . . . 1676.3 Analysis of Quenches for Thermal States . . . . . . . . 168

discussion and conclusions 181

Appendix 189a calculation of the normal modes 191

iii

iv contents

b equilibrium configurations for three ions 195c the disentangling theorem 201d gaussian integrals 207e derivation of the visibility for thermal states 217

publications 225references 227acronyms 249

L I S T O F F I G U R E S

Figure 1.1 Sketch of a Penning trap . . . . . . . . . . . . . 11Figure 1.2 Potential and fields of a Penning trap . . . . . 11Figure 1.3 Sketch of a linear Paul trap . . . . . . . . . . . 12Figure 1.4 Electric potential of a Paul trap . . . . . . . . . 12Figure 1.5 Schematic drawings of different crystal structures 25Figure 2.1 Equilibrium positions for 3 ions in a homogen-

eous potential . . . . . . . . . . . . . . . . . . . 40Figure 2.2 Normal modes linear chain for 3 ions in a ho-

mogeneous potential . . . . . . . . . . . . . . . 41Figure 2.3 Normal modes zigzag chain for 3 ions in a

homogeneous potential . . . . . . . . . . . . . . 42Figure 2.4 Normal mode angular frequencies for homo-

geneously trapped three-ion crystal . . . . . . 43Figure 2.5 Equilibrium positions for crystals with up to 15

ions . . . . . . . . . . . . . . . . . . . . . . . . . 45Figure 2.6 Dressed states of the atom-light interaction . . 50Figure 2.7 Spatial variation of the dressed states in an

inhomogeneous laser beam . . . . . . . . . . . 51Figure 2.8 Linewidths of the dressed states . . . . . . . . 52Figure 2.9 Level scheme and laser setup for dipole potential 53Figure 2.10 Approximation of a Gaussian beam . . . . . . 54Figure 2.11 The linearzigzag transition for three ions where

the central ion is subjected to an additional po-tential. . . . . . . . . . . . . . . . . . . . . . . . 58

Figure 2.12 Normal mode frequencies for a three-ion crys-tal in state-dependent potential . . . . . . . . . 58

Figure 3.1 Expansion of the position vector around differ-ent equilibrium positions . . . . . . . . . . . . . 65

Figure 3.2 Transformation between normal modes of dif-ferent configurations . . . . . . . . . . . . . . . 66

Figure 3.3 Graphical representation of index contraction . 79Figure 3.4 Graphs of the linked cluster expansion in first

order . . . . . . . . . . . . . . . . . . . . . . . . 80Figure 3.5 Graphs of the linked cluster expansion in second

order . . . . . . . . . . . . . . . . . . . . . . . . 81Figure 3.6 Counting of the possibilities drawing an l-cluster 83Figure 4.1 Principle of magnetic resonance . . . . . . . . . 91

v

vi List of Figures

Figure 4.2 Sequence of Ramseys method of separated os-cillatory fields . . . . . . . . . . . . . . . . . . . 91

Figure 4.3 Effect of the second Ramsey pulse dependenton the phase between spin and field . . . . . . 92

Figure 4.4 Ramsey fringes . . . . . . . . . . . . . . . . . . 92Figure 4.5 Visibility of the Ramsey fringes . . . . . . . . . 103Figure 4.6 Analogy to the Mach-Zender interferometer . 103Figure 4.7 Ramsey interferometry as a probe . . . . . . . 104Figure 5.1 Quantum quench for an ion Coulomb crystal 127Figure 5.2 Parameter space for a quench of an ion Cou-

lomb crystal . . . . . . . . . . . . . . . . . . . . 127Figure 5.3 Validity region for the quench . . . . . . . . . 128Figure 5.4 Sweep scheme for the quantum quench . . . . 128Figure 5.5 Visibility in the linear regime for varied g . . . 137Figure 5.6 Visibility in the linear regime for varied . . . 137Figure 5.7 Visibility in the zigzag regime for varied g . . 138Figure 5.8 Visibility in the zigzag regime for varied

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