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Fast and Scalable Entangling Gatein Trapped Ions via RydbergInteraction Chi Zhang

Chi Zhang Fast and Scalable En

tanglin

g Gate in

Trapped Ions via Rydberg In

teraction

Doctoral Thesis in Physics at Stockholm University, Sweden 2020

Department of Physics

ISBN 978-91-7911-238-7

Chi Zhang

Fast and Scalable Entangling Gate in TrappedIons via Rydberg InteractionChi Zhang

Academic dissertation for the Degree of Doctor of Philosophy in Physics at StockholmUniversity to be publicly defended on Monday 21 September 2020 at 10.00 in sal FB42,AlbaNova universitetscentrum, Roslagstullsbacken 21.

AbstractTrapped Rydberg ions are a novel platform for quantum information processing. This approach combines the advancedquantum control of trapped ions and the strong dipolar interaction of Rydberg atoms. In this thesis, a strong dipole-dipole interaction has been demonstrated and a sub-microsecond entangling gate has been implemented in a cold two-ioncrystal. After minimizing the polarizability of the Rydberg state by microwave dressing and understanding the effect ofthe quadrupole radio-frequency trap,the entangling gate has been applied in a warm 12-ion crystal.

Keywords: trapped ions, Rydberg interaction, quantum entanglement.

Stockholm 2020http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-183740

ISBN 978-91-7911-238-7ISBN 978-91-7911-239-4

Department of Physics

Stockholm University, 106 91 Stockholm

FAST AND SCALABLE ENTANGLING GATE IN TRAPPED IONS VIARYDBERG INTERACTION

Chi Zhang

Fast and Scalable EntanglingGate in Trapped Ions viaRydberg Interaction

Chi Zhang

©Chi Zhang, Stockholm University 2020 ISBN print 978-91-7911-238-7ISBN PDF 978-91-7911-239-4 Printed in Sweden by Universitetsservice US-AB, Stockholm 2020

Publications

• Sub-microsecond entangling gate between trapped ions via

Rydberg interactionChi Zhang, Fabian Pokorny, Weibin Li, Gerard Higgins,Andreas Pöschl, Igor Lesanovsky, and Markus HennrichNature 580, 345 (2020)

• Tracking the dynamics of an ideal quantum measurementFabian Pokorny, Chi Zhang, Gerard Higgins, Adán Cabello,Matthias Kleinmann, and Markus HennrichPhys. Rev. Lett., 124, 080401 (2020)

• Highly-polarizable ion in a Paul trapGerard Higgins, Fabian Pokorny, Chi Zhang, and MarkusHennrichPhys. Rev. Lett., 123, 153602 (2019)

• Coherent Control of a Single Trapped Rydberg IonGerard Higgins, Fabian Pokorny, Chi Zhang, Quentin Bodart,and Markus HennrichPhys. Rev. Lett., 119, 220501 (2017)

• Single Strontium Rydberg Ion Confined in a Paul TrapGerard Higgins, Weibin Li, Fabian Pokorny, Chi Zhang, FlorianKress, Christine Maier, Johannes Haag, Quentin Bodart, IgorLesanovsky, and Markus HennrichPhys. Rev. X, 7, 021038 (2017)

• Fast two-qubit gate in a 12-ion crystalChi Zhang, Fabian Pokorny, Gerard Higgins, and MarkusHennrichIn preparation (2020)

• Magic trapping of a Rydberg ion with a diminished staticpolarizabilityFabian Pokorny, Chi Zhang, Gerard Higgins, and MarkusHennricharXiv: 2005.12422 (2020)

• Long-range multi-body interactions and three-body anti-

blockade in a trapped Rydberg ion chainFilippo Gambetta, Chi Zhang, Markus Hennrich, IgorLesanovsky, and Weibin LiarXiv: 2005.05726 (2020)

• Observation of effects due to an atom's electric quadrupolepolarizabilityGerard Higgins, Chi Zhang, Fabian Pokorny, Harry Parke, ErikJansson, Shalina Salim, and Markus HennricharXiv: 2005.01957 (2020)

Acknowledgement

First of all I thank my supervisor Markus Hennrich, for lettingme work with Rydberg ions, and for his detailed and helpfulinstructions on understanding physics, working in the lab,presenting in conferences and writing papers and thesis. I thank the pioneers in this project, Gerard Higgins and FabianPokorny, for working together in the lab from the beginning tothe end of my PhD. I learnt how to operate the experimentalsystem from Gerard and improved my physics knowledge a lotfrom our discussions. Working with Fabian I didn't need toworry about computer issues since he's good at classicalcomputers and always very helpful, I even didn't need toremember many passwords in our group systems. I thank Harry, Shalina, Erik, Anders for working together, andour previous members Andreas and Quentin for setting up theablation loading system that makes my life easier. I thank theorists Weibin Li, Igor Lesanovsky and FilippoGambetta for our collaborations and discussions. I thank Sten and Per-Erik for answering numerous questionsabout the thesis and defense, and Isa for helping me with theresidence permit. I thank all friends from KOMKO for having lunch together. Lastly I thank my wife for skiing, hiking and studying physicstogether.

Stockholm University

Fast and Scalable Entangling Gatein Trapped Ions via Rydberg Interaction

A thesissubmitted to the Department of Physicsin partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY (PHYSICS)

carried out at the Department of Physicsunder the guidance of Assoc. Prof. Dr. Markus Thomas Hennrich,

presented by

CHI ZHANG

August 2020

iii

SammanfattningFangade Rydberg-joner ar en ny plattform for kvantinformation. Detta tillvagagangssattkombinerar den avancerade kvantstyrningen av fangade joner och den starka dipolaravaxelverkan mellan Rydberg-atomer. I denna avhandling har en stark dipol-dipolinteraktiondemonstrerats och en sub-mikrosekund sammanflatningsgrind har implementerats i enkall tvajonskristall. Efter att ha minimerat polariserbarheten i Rydberg-tillstandet genommikrovagsbekladning och forstatt effekten av quadrupol-radiofrekvensfallan har sam-manflatningsgrinden applicerats i en varm 12-jonskristall.

AbstractTrapped Rydberg ions are a novel platform for quantum information processing. Thisapproach combines the advanced quantum control of trapped ions and the strong dipo-lar interaction of Rydberg atoms. In this thesis, a strong dipole-dipole interaction hasbeen demonstrated and a sub-microsecond entangling gate has been implemented in acold two-ion crystal. After minimizing the polarizability of the Rydberg state by mi-crowave dressing and understanding the effect of the quadrupole radio-frequency trap,the entangling gate has been applied in a warm 12-ion crystal.

Contents

1 Introduction 11.1 Publications included in this thesis and my contributions . . . . . . . . 3

1.1.1 Publication I . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Publication II . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.3 Publication III . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.4 Publication IV . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.5 Publication V . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Experimental background 62.1 Ion trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Trap effects on Rydberg states . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Modification of trapping frequencies . . . . . . . . . . . . . . . 82.2.2 Quadrupole couplings . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Rydberg interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Lifetime of Rydberg states . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Rydberg gate schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6 88Sr+ level scheme and laser system . . . . . . . . . . . . . . . . . . . 112.7 Experimental sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Sub-microsecond entangling gate in a two-ion crystal 16

4 Microwave-dressed Rydberg state with vanishing polarizability 17

5 Quadrupole effects in high Rydberg states 18

6 Sub-microsecond entangling gate in a 12-ion crystal 19

7 Proposal for quantum simulation with Rydberg ions 20

8 Summary and outlook 21

Bibliography 25

iv

Chapter 1

Introduction

Quantum theory has been tested stringently and can precisely describe simple systemsranging from sub-nuclear particles to condensations of millions of atoms. However,when a quantum system is strongly correlated, the computational resource required tounderstand the system increases exponentially with the system’s size. Therefore, manyinteresting and important problems are extremely difficult or impossible to be solvedby conventional computers, for example, the quark–gluon plasma [1], high tempera-ture superconductor [2] and large bio-molecules [3]. Quantum computers [4] are wellcontrolled quantum systems and can be used to study other quantum systems and solvecertain problems more efficiently than any conventional computers. The most basic el-ements of a quantum computer are the quantum bits (qubits) and qubit gate operations[5].

Trapped atomic ions are one of the most promising systems to realize universalquantum computers [6, 7]. Ions can be trapped by their charges in radio-frequencytraps for months. Their internal electronic states are ideal qubits, they are well isolatedfrom the environment and coherence time has exceeded 10 min [8]. They can be con-trolled precisely by stable laser fields and fidelities for single-qubit operations higherthan 99.99% have been achieved [9].

When two ions are confined in the same trap, their motion is coupled via the Coulombinteraction, though their electronic states do not interact directly. Electronic states andcollective motional states can be coupled by laser fields via motional sideband transi-tions [9–11] or spin-dependent forces [12, 13]. Entanglement operation between twoions can be realized by these indirect interactions with fidelity higher than 99.9% [9],which is sufficiently high for fault-tolerant quantum error correction schemes [14–16].Two-ion entanglement gates together with single-ion rotations form a complete set ofoperations for quantum computation [5]. However, so far quantum computation andsimulation have only been demonstrated in few-ion systems [17–19]. To exceed thecapability of classical supercomputers and realize a useful quantum computer, a largenumber of qubits and a large number of entangling gate operations (within the coherencetime) will be required.

The most established gate schemes operated via a common motional mode of theion crystal are rather slow, typically between 40µs and 100µs in a two-ion crystal.They slow down further in large ion crystals, since the motional mode spectral density

1

2 Chapter 1. Introduction

increases as N3 with N the number of ions, therefore a longer gate time is needed toaddress a single mode. Much faster gate schemes have been proposed [20–23] andimplemented [24, 25], by driving multiple motional modes simultaneously. Gate timesof 1.6µs with 99.8% fidelity and 480 ns with 60% fidelity have been achieved in two-ion crystals [25]. The current research focus in this field is to realize fast entanglinggates that do not slow down significantly in large systems. This usually requires morecomplicated pulse sequences [25–27], or more sophisticated high power pulsed lasers[28, 29], since all motional modes used to mediate the interaction need to be decoupledfrom qubits at the end while the number of modes increases with 3N . Alternatively, ifqubits can interact directly without using motional modes, this can also provide a wayto scale up trapped ion systems.

In other systems, such as Rydberg atoms [30, 31] and polar molecules [32], becauseof the large polarizabilities, the electronic (qubit) states interact directly via dipole-dipole interactions [33, 34]. Therefore, the interaction is strong and does not rely on(but still may couple to) the collective motion of the entire system. Fast and scal-able entangling gates are possible [35, 36], as demonstrated by recent works [37–39]in Rydberg atoms with entanglement fidelities up to 99.5% [39]. Nevertheless, in mostexperiments the traps have to be switched off during Rydberg excitation, since the red-detuned optical dipole potential is usually anti-trapping for Rydberg states. This canresult in unwanted coupling between qubits and motional environment and cause de-phasing [31, 40].

Using blue-detuned optical tweezers at a magic wavelength (the polarizabilities fordifferent states are the same) [41] or alkaline earth elements with two valence electrons[39, 42, 43] (when one electron is excited to Rydberg state the other one can be usedfor trapping), one may realize confinement of Rydberg states. As another option, if theatom is charged (then it is an ion), it can be confined in the ion trap, which is not onlyhighly state-independent, but also strong so that Doppler effects from laser excitationcan be mitigated.

Rydberg ions [44] are a promising approach for scalable quantum computation.They combine the state-independent interactions of trapped ions and state-dependentinteractions of Rydberg atoms. State-independent Coulomb interaction can be used fortrapping and sympathetic cooling without unwanted entanglement with the environment(motional states or coolant ions). State-dependent Rydberg interaction, without usingthe motional modes, allows fast and qubit number-independent entangling operations.

Before the start of this thesis, trapped ions had been excited to Rydberg states fortwo different species, 40Ca+ in Mainz [45] using single photon excitation [46] and 88Sr+

in our group in Stockholm [47] using two-photon excitation. During the period of thisthesis, in our group, coherent control of Rydberg state has been realized in the PhDthesis of Gerard Higgins [48, 49], microwave dressing and magic trapping of Rydbergstates have been realized in the PhD project of Fabian Pokorny.

The topic of this thesis follows the path started by the previous works, it focuseson the realization of interaction between Rydberg ions and the implementation of a fastentangling gate in two- and longer-ion crystals. The structure of the thesis is as follows:The experimental background, including the properties of the 88Sr+ Rydberg ion, thetrap, the laser system and the basic experimental sequence, are introduced in Chapter

Chapter 1. Introduction 3

2. The interaction between two microwave-dressed Rydberg ions as well as a sub-microsecond entangling gate are described in Chapter 3 (Publication I [50]). In orderto extend the entangling gate to a large ion crystal, a vanishing-polarizability Rydbergstate is required, and this is presented in Chapter 4 (Publication II [51]). When usingRydberg states with principal quantum number higher than 50, the interaction with theradio-frequency quadrupole field used for trapping the ions becomes significant andneeds to be considered and understood, and this is presented in Chapter 5 (PublicationIII [52]). Subsequently, the potential of scalability of the Rydberg interaction gate hasbeen demonstrated by performing the two-ion entangling gate in a 12-ion crystal, andthis is reported in Chapter 6 (Publication IV [53]). Chapter 7 is a theory proposal ofquantum simulation with trapped Rydberg ions by coupling Rydberg excitations to ionmotion for multi-body interactions (Publication V [54]). Chapter 8 concludes the thesiswith a summary and outlook.

1.1 Publications included in this thesis and my contri-butions

1.1.1 Publication ISubmicrosecond entangling gate between trapped ions via Rydberg interaction. ChiZhang, Fabian Pokorny, Weibin Li, Gerard Higgins, Andreas Poschl, Igor Lesanovskyand Markus Hennrich. Nature 580, 345 (2020).

My contributions to this work are:

• I measured the tunable dipole-dipole interaction between two microwave-dressedRydberg ions.

• I studied the two-ion stimulated Raman adiabatic passage (STIRAP) process the-oretically and proposed a scheme to combine microwave-dressing with STIRAPfor the interaction gate (similar to the theoretical work [55] where STIRAP isproposed for exciting atoms to interacting Rydberg states). I realized STIRAP fortwo interacting ions experimentally.

• Our microwave horn antenna does not maintain polarization when emitting themicrowave field to the ions. This caused coupling between all Zeeman sublevelsof the Rydberg states and reduced the gate fidelity. I recognized this problemfrom the spectroscopy signal, and implemented a microwave polarizer to purifythe polarization.

• I stabilized the Rydberg excitation lasers on the new cavity with higher finesseand stability.

• I implemented a 674 nm laser beam to address one of the two ions for the two-ioncorrelation measurement and wrote a script to analyze two-ion data taken by thecamera (based on the camera software developed by Andreas Poschl [56]).

4 Chapter 1. Introduction

• I carried out the gate measurement and analyzed the result.

• I did numerical simulations for all measurements, and analyzed the gate error.

• I wrote the manuscript with the help and discussion of all other authors.

1.1.2 Publication IIMagic trapping of a Rydberg ion with a diminished static polarizability. Fabian Poko-rny, Chi Zhang, Gerard Higgins and Markus Hennrich. arXiv:2005.12422 (2020). Sub-mitted to Physical Review Letters.


• At the time of the experiment for part of the data, the Rydberg excitation laserswere locked on a cavity with substantial resonance frequency drifts over time.This caused around ∼ 1 MHz/min drifts of the Rydberg excitation laser frequen-cies. I implemented the method to track these drifts and keep the lasers always onresonance, this made it possible to do the spectroscopy measurement continuouslyfor a week.

• Fabian and I carried out all measurements.

• I simulated the Rabi oscillation measurements for states with different polariz-abilities and different temperatures.

• I contributed to discussion and writing of the manuscript.

1.1.3 Publication IIIObservation of effects due to an atom’s electric quadrupole polarizability. Gerard Hig-gins, Chi Zhang, Fabian Pokorny, Harry Parke, Erik Jansson, Shalina Salim and MarkusHennrich. arXiv:2005.01957 (2020). Submitted to Physical Review Letters.


• I recognized that the energy shifts and sidebands in the spectroscopy signal werecaused by the electric quadrupole coupling. I did a qualitative calculation andexplained the spectroscopy.

• I proposed a coherent spectroscopy method to measure the Rydberg P states moreprecisely with much less double-ionization probability compare to the method weused before. This new method is explained in the supplemental material sectionV of this paper.

• I did the full diagonalization calculation of the Rydberg P1/2 states in quadrupolefield using the open source python package Alkali Rydberg Calculator (modifiedby Gerard Higgins for Rydberg ions).


Chapter 1. Introduction 5

1.1.4 Publication IVFast two-qubit gate in a 12-ion crystal. Chi Zhang, Fabian Pokorny, Gerard Higginsand Markus Hennrich. In preparation.


• Ions may be double ionized when they are excited to Rydberg states. Thesedoubly-charged ions cannot be used as qubits and affect the ion crystal config-uration and need to be replaced by normal ions to continue the experiment. Iproposed and tested a method to remove the doubly-charged ion deterministicallywithout affecting other normal (singly-charged) ions. This is important for work-ing with large ion crystal because double-ionization is likely and loading an entirecrystal is slow.

• Gerard, Fabian and I set up polarization gradient cooling to cool the large ioncrystal.

• I carried out the measurement and analyzed the results.

• I simulated the experiment and analyzed the gate error.

• I wrote the manuscript with the help and discussion of all authors.

1.1.5 Publication VLong-range multi-body interactions and three-body anti-blockade in a trapped Ryd-berg ion chain. Filippo Gambetta, Chi Zhang, Markus Hennrich, Igor Lesanovsky andWeibin Li. arXiv:2005.05726 (2020). Submitted to Physical Review Letters.


• An important requirement for this theory proposal is that the interaction potentialneeds to have a strong gradient (force) but a weak interaction energy shift be-tween the neighboring ions. I proposed and calculated the scheme to shape theinteraction potential between Rydberg ions using microwave dressing.


Chapter 2

Experimental background

This chapter introduces the experimental system, including the linear Paul trap to con-fine the ions, the effect of the trapping field on Rydberg states, the interactions betweenRydberg states, the level scheme of 88Sr+ ions and the lasers for cooling, state manip-ulation and detection, and the basic experimental sequence for Rydberg excitation andentangling gate. More details of the experimental setup can be found in the Masterthesis of Fabian Pokorny [57] and the PhD thesis of Gerard Higgins [49].

2.1 Ion trapA linear Paul trap [58, 59] is used in our experiment to confine the ions in ultra-highvacuum (pressure less than 10−10 mbar). In this section, the basic properties of thetrap are discussed following reference [60]. A more detailed derivation of the trappingpotential can be found in references [61–63], The assembly of our trap is described inthe Master thesis of Fabian Pokorny [57].

The linear Paul trap applies a combination of static and time-dependent electricfields to confine charged particles in three dimensions. A typical configuration of thelinear Paul trap is shown in Fig. 2.1. Four rod electrodes generate a potential for con-finement in the xy-plane, each electrode is connected to the diagonally opposite one anda radio-frequency (rf) voltage V = V0 cos(Ωt) is applied between the two pairs. Theconfinement in the z-direction is achieved by a d.c. voltage U0 applied on the endcapelectrodes. In the xy-plane, near the trap center, the time-dependent potential is givenby

Φ(x, y, t) =x2 − y2

2r20V0 cos Ωt+

x2 + y2

2z20U0, x, y r0 (2.1)

where r0 is the distance from the trap center to the electrodes and z0 is the distancebetween two endcaps. In this section we assume the charged particle in this potentialis not polarizable, though the large polarizability of the Rydberg state will introduce anadditional trapping potential due to the a.c. Stark effect. Additionally, the quadrupolefield of the trap couples Rydberg states via quadrupole transitions. These effects will bediscussed in section 2.2 and more details can be found in references [64, 52].

The motion of the ion in the xy-plane can be decomposed into micromotion and

6

Chapter 2. Experimental background 7

d.c. endcaps

Vrf rf blades

a b

Figure 2.1: Schematics of a linear Paul trap. (a) the view along the trapping axis and theconnection of the radio frequency to the trap blade electrodes. (b) the endcap electrodes and thetrapping region. Figure adapted from [61]

secular motion. The secular motion can be described as the motion in an effectiveharmonic trapping potential

Ψ =M

2(ω2

xx2 + ω2

yy2), (2.2)

where the trapping frequencies ωx, ωy are determined by the rf voltage and frequencyand the trap geometry. The micromotion is a small driven amplitude oscillation of theharmonic oscillator wavefunction of the ion at the rf frequency Ω. Offset electric fieldscan shift the equilibrium position of the ion away from the rf center and cause excessmicromotion. This excess micromotion is usually minimized by applying voltages oncompensation electrodes. This is particularly important for Rydberg ions which arehighly sensitive to this unwanted electric field [64]. The degeneracy of x and y motionalmodes can be lifted by a small d.c. offset voltage (normally 1.5 V in our experiment) onthe rf voltage. It is required to address both radial motional modes using a single laserbeam.

The axial confinement by the voltage applied to the endcap electrodes is parabolicnear the trap center, whose depth can be approximated by setting Mω2

zz20/2 = keU0,

where ωz is the axial trapping frequency, and k is a geometric factor (k ≈ 0.042 for ourtrap [57]) that account for the complicated geometry of the trap. The three-dimensionalharmonic pseudo-potential near the trap center is given by

Ψ =M

2(ω2

xx2 + ω2

yy2 + ω2

zz2), (2.3)

In our experiment, the typical trapping frequencies for a single ion are

ωx ∼ 2π × 1.7 MHz, ωy ∼ 2π × 1.7 MHz, ωz ∼ 2π × 800 kHz. (2.4)

Usually an ion can be confined in the trap for weeks.

8 Chapter 2. Experimental background

2.2 Trap effects on Rydberg statesRydberg states are highly polarizable and extremely sensitive to external fields. Toexcited ions to Rydberg states the effects caused by the rf trap need to be understood,especially for high Rydberg states since most properties of Rydberg ions scale with theprincipal quantum number. Two effects are discussed in this section, and more detailscan be found in references [64, 51] and Chapters 4 and 5.

2.2.1 Modification of trapping frequenciesAn ion in a Rydberg state with polarizability α experiences an additional energy shift∆U = −1

2α〈ε2〉 [64], where ε = −OΦ is the electric field and 〈〉 denotes time average.

Taking Φ from Eq. 2.1 we have ∆U ≈ −αA2(x2 + y2) where A = V02r20

is the oscillationelectric field gradient. Therefore, the Rydberg state experiences an altered trappingpotential U ′ = U + ∆U with altered trapping frequencies [64]

ω′x,y ≈√ω2x,y −

2αA2

M, ω′z ≈ ωz. (2.5)

For our typical experimental settings and Rydberg states with principal quantumnumber∼ 50, the trapping frequencies are altered by 2 or 3%. This causes the transitionfrequency between low-lying and Rydberg states to depend on motional phonon number,typically around 50 kHz per phonon. For a thermal motional state with a broad phonondistribution, this causes decoherence in Rydberg excitation.

To mitigate this effect, a microwave field can be used to dress Rydberg S and Pstates. Since S and P states have polarizabilities of opposite signs, the polarizability ofthe dressed state can be tuned to zero. This is discussed in Chapter 4 and reference [51].

2.2.2 Quadrupole couplingsThe electric potential (Eq. 2.1) of the linear Paul trap introduces a coupling Hamiltonian[52]

H(r, t) = eΦ = er2

[−2

√π

5BY 0

2 (θ, φ)−√

8π

15A cos ΩtY 2

2 (θ, φ) + h.c.

], (2.6)

whereB = U0

2z20is the static electric field gradient and Y m

l (θ, φ) are spherical harmonics.Normally the oscillating electric field is much stronger than the static one, as a result

A B and the time-dependent part of the Hamiltonian has much stronger effects thanthe time-independent part. The time-independent part shifts the energy levels of J > 1states by ∼ MHz for typical experimental parameters. And the time-dependent partcouples states with ∆L = 0,±2 and ∆m = ±2 by up to ∼ GHz coupling strengthsoscillating at the trap drive frequency Ω. This strongly affects states with J > 1 since theZeeman sublevels (level spacing ∼ 10MHz) are coupled. Although J = 1/2 electronicstates (S1/2 and P1/2) do not have a quadrupole moment and are not directly shifted inenergy by Hamiltonian (Eq. 2.6), their energy levels are strongly shifted, especially in


high Rydberg states (n ∼ 50), because of higher order effects of the couplings betweenstates S1/2 ↔ D3/2, D5/2 and P1/2 ↔ P3/2, with ∼ GHz coupling strengths and ∼100GHz level spacing. It shifts the energy levels by ∼ 100MHz oscillating at 2Ω ≈36MHz, and thus generates sidebands on the spectrum [52]. All these sidebands need tobe considered when using higher Rydberg states. More details are presented in Chapter5 and reference [52].

2.3 Rydberg interactionsThe main motivation of exciting trapped ions to Rydberg states is that the strong Ry-dberg interaction can be used to realize an entangling gate between ions. This sectionestimates the interaction between Rydberg states of atoms and ions and the scaling withprincipal quantum number and net core charge (it is one for atoms and two for ions).

For two ions trapped in their equilibrium positions or two neutral atoms, when theseparation is much larger than the size of their electronic wavefunctions, the dominantinteraction is the dipole-dipole interaction:

Vdd =1

4πε0

(µ1 · µ2 − 3(µ1 · n)(µ2 · n)

|R|3)

(2.7)

where µi is the electric dipole moment operator of ion i (i = 1, 2), R = R2 − R1 isthe relative ion position and n = R/|R|.

Ions in atomic eigenstates (for example |a〉 and |b〉) do not have dipole moments andVdd has no first-order effect. Vdd couples pair state to other pair states with couplingstrength C = 〈ab|Vdd|cd〉 ∝ 〈a|µ|c〉〈b|µ|d〉R−3. The coupling by µ between two statesis the transition dipole moment 〈a|µ|c〉 ∝ n2 ·Z−1 with n the principal quantum numberand Z the net core charge. Therefore the coupling strength between pair states C ∝n4 ·R−3 · Z−2.

This coupling causes the two ions to polarize each other, i.e., one ion in state |a〉is polarized with |c〉 by the transition dipole moment of the other ion 〈b|µ|d〉, and viceversa. Usually the two pair states are not degenerate, the oscillating frequencies of thetransition dipole moments 〈a|µ|c〉 and 〈b|µ|d〉 are off-resonance by the energy difference∆E ∝ n−3·Z2 between |ab〉 and |cd〉. The two ions can only slightly polarize each otherand this off-resonance coupling C causes a second-order energy shift ∼ C2/(4∆E) ∝n11 ·R−6 ·Z−6. This is the most common interaction in Rydberg atom systems and it iscalled van der Waals interaction. The Z−6 scaling makes it ∼ 64 times weaker for ionsthan for atoms.

When the two pair states are close to degeneracy, or the coupling is strong C ∆E, the two ions can fully polarize each other and the total energy of the pair stateis shifted by ∼ C/2. This is called Forster resonance and the interaction strength is∼ n4 ·R−3 ·Z−2. The Forster resonance interaction is much stronger than van der Waalsinteraction, sometimes the atomic level can be shifted by an external electric field or amicrowave field to tune the pair state close to a Forster resonance for strong interactions.

The Rydberg ions can also be polarized directly by an external field and acquiredipole moments to interact with each other, the interaction strength is proportional to


µ2. Full polarization can be achieved with an external field that couples much strongerthan the energy separation between the two neighboring levels, or with an oscillatingfield that matches the energy difference between the levels. When the ions are fullypolarized, the eigenstates in the external field is |±〉 ≡ (|a〉 ± |b〉)/

√2, the upper limit

of µ is realized and 〈±|µ|±〉 ∝ n2 · Z−1. As a result, the dipole-dipole interactioncauses an energy shift 〈+ + |Vdd|+ +〉 ∝ n4 ·R−3 ·Z−2. This is the scheme used in ourexperiments [50, 53].

For typical experimental settings of ion distance ≈ 5µm and principal quantumnumber ∼ 50, the maximum interaction strength is ≈ 2MHz by using a maximallydressed state (|S〉 ± |P 〉)/

√2 where |S〉 and |P 〉 are Zeeman sublevels of Rydberg

S1/2 and P1/2 states respectively. The trap quadrupole coupling discussed in section2.3.2 may slightly affect the interaction strength since populations from |S〉 and |P 〉are hybridized slightly with other states via the quadrupole coupling. This effect alsomodulates the interaction strength by ∼ 1% at a frequency of 2Ω. This is discussed inChapter 6 and reference [53].

2.4 Lifetime of Rydberg states

Rydberg states have finite lifetimes because of decay to other states, which places anintrinsic limit for the gate fidelity. This section discusses the lifetimes of Rydberg statesand the scaling with principal quantum number and the net core charge.

The decay rate Γ from a Rydberg state |r〉 to another state |f〉 by emitting a pho-ton is proportional to the energy density of the photon mode and the transition dipolemoment squared |〈r|µ|f〉|2. For a vacuum state of the light field, the energy density isproportional to ω3 where ω is the frequency of the field. At a high temperature whenkBT & ~ω, the energy density of black-body radiation is proportional to ω2. The atomin Rydberg state |r〉 can decay to a different Rydberg state |r′〉 or a low-lying state |g〉.When decaying to |r′〉, the level spacing ωrr′ ∝ n−3 · Z2 and the transition dipole mo-ment 〈r|µ|r′〉 ∝ n2 · Z−1, the decay rate Γrr′ ∝ n−5 · Z4. When decaying to |g〉, thelevel spacing ωrg ∝ n0 · Z2 and the transition dipole moment 〈r|µ|g〉 ∝ n−3/2 · Z−1,the decay rate Γrr′ ∝ n−3 ·Z4. In cryogenic environment, the decays to low-lying statesdominate and the Rydberg state lifetime τ ∝ n3 ·Z−4. At a high temperature, the decaysto other Rydberg states dominate and τ ∝ n2 · Z−2.

In our experiment at room temperature, the Rydberg state lifetimes are limited byblack-body induced decay to other Rydberg states. For n ∼ 50, τ ∼ 5µs.

2.5 Rydberg gate schemes

The Rydberg interaction can be used to implement an entangling gate between twoatoms. A very common gate scheme, the blockade gate [30], uses the Rydberg block-ade effect. When two atoms are close to each other, only one of them can be excited toRydberg state because the two-atom Rydberg-excited state is shifted out of laser reso-nance by the Rydberg interaction [33]. The lifetime limited gate error and the optimized


gate time are εgate = 3(7π)2/3

81

(τV )2/3and Tgate = 1

(7π)1/3τ1/3

V 2/3 where V is the interactionstrength [30].

Another way to entangle two atoms, which is used in our experiment [50], is theinteraction gate [30, 55]. Both atoms are excited to Rydberg states and and their inter-action introduces a dynamical phase. The lifetime limited gate error and the optimizedgate time are εgate = 1

τVand Tgate = 1

V. The interaction gate is faster and the in-

trinsic error is lower than the blockade gate. However, it requires coupling laser Rabifrequency Ω V , and it couples Rydberg excitation to motion because of the force(potential gradient) between two atoms in Rydberg states.

The interaction gate scheme suits our system better, because the interaction betweenour Rydberg ions is not strong enough for the blockade gate, but weak enough for bothions to be excited to Rydberg states. In addition, the motional effect caused by the forceis negligible (∼ 10−4 gate error [50]) in trapped ions because of the strong confinement.The realization of the interaction gate is explained in Chapter 3. When applying theinteraction gate in a longer ion crystal with a higher Rydberg state, the effects of therf trap (section 2.2) need to be considered, more details are discussed in Chapter 6 andreference [53].

2.6 88Sr+ level scheme and laser systemThis section explains the level structure of 88Sr+ ion and the lasers used to create andmanipulate the ion.

To load ions in the trap, strontium atoms are produced by ablation of a target with515 nm nanosecond laser pulses1, and then ionized inside the trap by a two-photon ion-ization process by a 461 nm diode laser2 and a 405 nm diode laser3 and confined by thetrapping potential. More details about ablation loading can be found in the Master thesisof Andreas Poschl [56].

The relevant electronic states and transitions of 88Sr+ ion are presented in the levelscheme in Fig. 2.2. A magnetic field of 3.6 G created by permanent magnets (moredetails in the Master thesis of Anders Lindberg [65]) is applied to split the levels intoZeeman sublevels.

The ground state configuration is 5S1/2. A 422 nm laser field driving the transitionbetween ground state and excited state 5P1/2 is used for Doppler cooling and state de-tection. Since the population from 5P1/2 may decay to a metastable state 4D3/2 witha probability of 6% [66], a 1092 nm repump laser field is used to close the cycle. Forquantum information processing, one of the ground state sublevel (5S1/2,mJ = −1/2)and one of the metastable state sublevel (4D5/2,mJ = −5/2) are used as qubit states|0〉 and |1〉, respectively. The electric quadrupole transition between the two qubit statesis driven by a 674 nm laser field. A 1033 nm laser field is used to initialize the qubit to|1〉 by optical pumping via state 5P3/2. In our experiment, interaction between differentqubits is realized by the dipole-dipole interaction of microwave-dressed Rydberg states.

1Coherent Flare-NX2Toptica DL pro3Thorlabs L405-SF10


Rydberg excitation from |0〉 is driven by two ultraviolet lasers, the first ultraviolet lasernear 243 nm couples |0〉 and the intermediate state 6P3/2, the second tunable ultravioletlaser (tunable between 305 nm and 309 nm) couples the intermediate state and the Ry-dberg S1/2 states. Rydberg S1/2 and P1/2 states in a large range of principal quantumnumbers (43 ≤ n ≤ 60) can be coupled by a tunable microwave field (50− 75 GHz and90−140 GHz) to form dressed superposition states with strong rotating dipole moments[50].

Rydberg

304 nm –

309 nm

4D3/2

(0.37 Hz)

5P1/2

(22 MHz)

5P3/2

(24 MHz)

nS1/2

6P3/2

(4.9 MHz)

n’P3/2

n’P1/2

Rydberg

243 nm

quench

1033 nmrepump

1092 nm

qubit

674 nm

uorescence

detection &

Doppler

cooling

422 nm

4D5/2

(0.41 Hz) - |0

microwave

- 5S1/2

|1

Figure 2.2: 88Sr+ ion level scheme with natural decay rate Γ/2π (taken from references [49, 67–69]). Zeeman sublevels are not shown. (Figure adapted from [49])

All these lasers for the transitions of the ion are frequency stabilized on optical cavi-ties by the Pound-Drever-Hall technique [70]. The 422 nm, 1092 nm and 1033 nm lasers


are diode lasers4 and are stabilized on cavities with finesses ∼ 1000 and the stabilizedlaser linewidths are∼ 2π×100 kHz. The 674 nm laser is a diode-laser-pumped tapered-amplifier system5 and is stabilized on a commercial optical cavity6 with a high finesse∼ 100 000 and the stabilized laser linewidth is ∼ 2π × 600 Hz [71]. The first Rydbergexcitation laser at 243 nm is generated in a commercial system7 by upconverting 970 nmwith two stages of second harmonic generation (SHG). The 970 nm fundamental laseris locked on a cavity8 with a finesse ∼ 20 000, the stabilized 243 nm laser linewidthis ∼ 2π × 10 kHz. The tunable 304 nm to 309 nm laser for the second Rydberg exci-tation step is generated in the following way: 1550 nm laser light from a fibre laser9

amplifier10 system and 1000 nm to 1030 nm tunable laser from a diode-laser-pumpedtapered-amplifier system11 are mixed in a Periodically-Poled Lithium Niobate (PPLN)crystal to generate 608 nm to 618 nm laser light by sum-frequency generation [72]. Thislaser is upconverted to 304 nm to 309 nm by SHG in a commercial system12. The 608 nmto 618 nm laser light is locked on a cavity13 with a finesse ∼ 20 000 and the linewidthof 304 nm to 309 nm laser is stabilized to ∼ 2π × 10 kHz. The microwave is generatedby multiplication and amplification of a fundamental frequency at 2 − 20 GHz from asignal generator. The output of the microwave (50 − 75 GHz and 90 − 140 GHz) isemitted by a horn antenna. More details about the microwave setup can be found in theLicentiate thesis of Fabian Pokorny [73].

All lasers except for the pulsed laser used for ablation loading of the ion are contin-uous wave (CW) lasers. The typical length of the pulses sent to the ion is between 10µsand 10 ms. These pulses are controlled by driving acousto-optic modulators (AOM)with phase coherent radio-frequency (RF) signals at around 200 MHz produced by directdigital synthesizers (DDS) from the bus system or the pulse sequencer box described inreferences [74–76]. These systems are controlled by the computer via the Trapped IonControl Software (TrICS) developed by the group of Rainer Blatt at the University ofInnsbruck, Austria.

The beams are sent to the ion in the configuration shown in Fig. 2.3.

4Toptica DL pro5Toptica TA pro6Stable Laser System7Toptica TA-FHG pro8Stable Laser System9NKT Koheras BASIK E15 DFB10Manlight EYFA-CW-SLM-P-TKS11Toptica TA pro12Toptica SHG pro13Stable Laser System


304 nm –

309 nm

Rydberg

step 2

magnetic eld B

quantisation axis

243nm

Rydberg

step 1

z

y

x

qubit 674 nm

(45° to trap axis)

cooling 422 nm

repump 1092 & 1033 nm

photomultiplier

tube

a

qubit 674 nm

(radial beam)

to EMCCD camera

z

y

xb

up

Figure 2.3: Experimental system, (a) side view, (b) along the trap axis. An ion is trapped insidethe linear Paul trap, the quantization axis is along the trap axis defined by a magnetic field.Lasers for Doppler cooling and repumping are sent from an angled direction 45 to the trap axisthrough the trap. Two laser beams for qubit manipulation are sent from the angled direction andthe radial direction (x + y)/

√2. Two counter propagating ultraviolet lasers are sent through

the holes on the endcap electrodes. The microwave field is sent in from the opposite radialdirection. Fluorescence photons are collected in the radial direction by an electron multiplyingCCD (EMCCD) camera 16and in the up direction (x − y)/

√2 by a photomultiplier tube 17.

(Figure adapted from [49])

17Andor iXon3 89717Hamamatsu Photonics H10682-210


2.7 Experimental sequenceThe typical experimental sequence is briefly described in this section. More details canbe found in [49]. A typical experimental cycle consists of the following steps:

1. Doppler cooling and state initialization: As shown in Fig. 2.3, the 422 nm, 1092 nmand 1033 nm laser beams are sent from 45 to the trap axis to cool all three mo-tional degree of freedoms and pump the ion to the ground state 5S1/2.

2. Optical pumping: After Doppler cooling, the ion is in a mixed state of groundstate sublevels 5S1/2,mJ = −1/2(|1〉) and 5S1/2,mJ = 1/2. The population in5S1/2,mJ = 1/2 is transferred to one of the 4D5/2 sublevels by the 674 nm laserand then pumped to the ground state by the 1033 nm laser. After typically 10 to20 pumping cycles, the ion will end up in |1〉 with very high probability. Thesetransitions normally do not heat the ion in the Lamb-Dicke regime.

3. Sideband cooling (optional): Doppler cooling cannot cool the ion to motionalground state. If we need the ion in motional ground state, red sideband transitionsfrom |1〉 to one of the 4D5/2 sublevels is driven to remove one phonon, and 4D5/2

is pumped to |1〉 via the 5P3/2 state by the 1033 nm laser. By repeating this processfor ∼ 1 ms, the ion is prepared with around 99% probability in the motionalground state.

4. Initial state preparation: Depending on the experiment we want to carry out,sometimes we start from the qubit state |0〉, sometimes we start from a super-position of |0〉 and |1〉. The initial state is prepared by a 674 nm laser pulse with acontrolled pulse area and phase.

5. Rydberg excitation: Two counter propagating ultraviolet laser beams are used forRydberg excitation. For Rydberg excitation with the stimulated Raman adiabaticpassage (STIRAP) pulse sequence, the sinusoidal pulse shapes of the two ultra-violet beams are generated using amplitude modulation of the radio-frequencysignals from the pulse sequencer box, the amplitudes are modulated by pulsesfrom external arbitrary waveform generators via a radio-frequency mixer (moredetails in [49]).

6. Measurement: States |0〉 and |1〉 can be distinguished by the electron shelvingtechnique [47]. 422 nm and 1092 nm laser beams are sent to the ion, if the ion isin state |1〉, 422 nm laser light is scattered and detected by the photomultiplier tube(PMT), whereas if it is in |0〉, it does not scatter 422 nm laser light and the PMTcount rate is low. Electron shelving is a projective measurement in |0〉 and |1〉basis, when measuring in an arbitrary set of basis, electron shelving is combinedwith a rotation in the qubit space by a 674 nm laser pulse with a certain pulse areaand phase.

The cycle above is repeated 20 to 50 times with the same experimental parameters,the probabilities of the final state being in the measurement basis is measured. Then inthe measurement the parameters, such as laser frequencies, phases, pulse lengths, canbe scanned in different sets of experimental cycles.

Chapter 3

Sub-microsecond entangling gate in atwo-ion crystal

In most trapped ion quantum computation systems, the qubits are encoded in the elec-tronic states of the ions. The qubits can interact strongly with lasers but not with eachother, therefore the qubit interactions are normally mediated by the common vibrationalmodes. In large systems with a lot of motional modes, the gate becomes slow and theperformance becomes sensitive to small changes in the laser parameters. By excitingthe ions to microwave-dressed Rydberg states, we realized direct dipolar Rydberg inter-action between the ion qubits. This interaction is strong and does not rely on ion motionand thus provides a new promising way to scale up trapped ion systems. In addition, wedemonstrated a 700-ns entangling gate between two ions. This is one of the fastest en-tangling gates in a trapped ion system. By numerical simulations we identified the gateerror sources and predicted that the total gate error can be reduced to less than 0.2%with reasonable technical improvements. These are the fundamental steps towards atrapped Rydberg ion quantum computer or simulator. The main result is published inNature 580, 345 (2020) [50].

16

Chapter 4

Microwave-dressed Rydberg state withvanishing polarizability

In the previous Chapter, we realized a fast entangling gate between two ions using Ry-dberg interaction. The gate does not rely on ion motion, but because the Rydberg statesare extremely polarizable, they experience a different trapping potential compared tolow-lying electronic states and Rydberg excitation becomes correlated to ion motion[64]. As a result, decoherence will be caused by ion motion. To avoid this problem,in the two-ion system of the previous Chapter, we applied resolved sideband cooling toprepare the ions in motional ground state. To implement the gate in large ion crystalswhere motional ground state cooling of all motional modes is not feasible, a Rydbergstate with vanishing polarizability is required to achieve state-independent confinement,also called magic trapping. The Rydberg state polarizability can be tuned by microwavedressing of two Rydberg states with polarizabilities of opposite sign. We showed thatby reducing the polarizability, coherent Rydberg excitation is highly independent of ionmotion. The result is described by the included preprint [51], which is central part ofthe PhD thesis by Fabian Pokorny and listed here for completeness. My contributionsto this work is explained in section 1.1.2 of Chapter 1.

17

Chapter 5

Quadrupole effects in high Rydbergstates

In the previous Chapter we realized magic trapping of Rydberg ions by using the van-ishing polarizability Rydberg state and showed that coherent Rydberg excitation can beindependent of ion motion. However, the vanishing polarzability Rydberg state is notmaximally dressed by the microwave field and has weaker oscillating dipole moment[50]. Therefore the interaction strength is reduced by∼ 41% compare to the maximallydressed state. The reduced interaction strength can be compensated by using higher Ry-dberg states, but higher Rydberg states are also more sensitive to the electric quadrupolefield used for trapping. The quadrupole field does not cause a change in trapping poten-tial, but it causes the energy levels of the Rydberg states to oscillate. The quadrupoleeffects is discussed in the included preprint [52].

18

Chapter 6

Sub-microsecond entangling gate in a12-ion crystal

The fast two-ion entangling gate has been introduced in Chapter 3, but a powerful quan-tum computer requires a large number of qubits. In Chapters 4 and 5, the main dif-ficulties have been overcome. State-independent trapping has been realized with thevanishing-polarizability Rydberg state, Rydberg excitation is decoupled from ion mo-tion. The reduced interaction strength of the vanishing-polarizability state is compen-sated by using Rydberg states with higher principal quantum number, and the effects ofthe quadrupole field have been studied. In this Chapter, we discuss the realization of thefast Rydberg entangling gate in a 12-ion crystal, the same gate time and a slightly lowerfidelity have been realized. The additional error is not intrisic, and numerical simulationshows that it can be reduced to less than 0.2% by reasonable technical improvements.This indicates that the Rydberg interaction gate is largely independent of ion numberand temperature of the ion crystal, thus it is suitable for large systems and promising forscalable quantum computation. The result is described in the following preprint [53].

19

Chapter 7

Proposal for quantum simulation withRydberg ions

In the previous Chapters we have shown that trapped Rydberg ions are a promisingplatform for scalable quantum computation, the main advantage is that this system com-bines the state-independent interaction of ions and state-dependent interaction of Ryd-berg states. This combination also makes trapped Rydberg ions promising for quantumsimulation. For example, in large molecules, the dynamics of the motional structuresare often strongly correlated with the electronic interactions. As a result these systemsare difficult to model and understand. Trapped Rydberg ions are suitable for simu-lating these systems since both electronic and motional states of the ions can be wellcontrolled. To enhance the coupling between motion and electronic excitation, the in-teraction potential between Rydberg ions can be shaped via microwave dressing. Thegradient of the interaction potential, which is the force between ions, couples pair ex-citation to vibrational modes. The interplay between the Rydberg interaction and vi-brational modes of an ion chain also enables studying complex quantum phenomena,like the long-range multi-body interactions or frustrated quantum systems. A theoryproposal for engineering multi-body interaction is described in the included preprint[54].

20

Chapter 8

Summary and outlook

Rydberg ions combine the advantages of both trapped ion and neutral atom systems.In this thesis, the interaction between two microwave-dressed Rydberg ions has beenobserved and a sub-microsecond entangling gate has been realized by exciting the ionsto Rydberg states. This gate is one of the fastest in two-ion systems and, unlike anyother trapped ion gates, it does not rely on ion motion. Subsequently, after studying theeffect of the trapping field, the same gate has been implemented in a warm 12-ion crystalusing the vanishing-polarizability microwave-dressed state and a bit lower fidelity hasbeen achieved. The speed of the Rydberg interaction gate is largely independent ofthe number of ions and the temperature of the ion crystal. Thus the gate is fast andsuitable in large systems. The possibility of quantum simulation using the interactionbetween electronic excitation and collective motion of the system in trapped Rydbergion systems has also been discussed. These are fundamental steps towards a scalabletrapped Rydberg ion quantum computer or quantum simulator.

However, the gate fidelity still needs to be significantly improved for fault-tolerantquantum computation. Technical improvements that can be done in the near future toimprove the gate fidelity include:

• Send the UV beams from a radial direction with a high trapping frequency. Sincein a stronger trap the motional wavefunction spread is smaller and lower temper-ature can be achieved, the gate error related to the Doppler effect [53] can be re-duced. This can also improve the UV beam focusing on the ions and improve theadiabaticity during STIRAP process. To implement this in our current system, weneed new optics and setups for single ion addressing, for example acousto-opticdeflectors for the two UV beams.

• Improve the polarization gradient cooling. Polarization gradient cooling is effi-cient in large systems. By reducing the ion temperature, the sizes of the motionalwavefunctions in large crystals can be reduced and gate fidelity can be improved[53]. Better cooling can be achieved by higher 422 nm laser power at a largerdetuning. This can be done by using new acousto-optic modulators with higherefficiency at 200 MHz.

• Stabilize the microwave power. Microwave power fluctuation is the main sourceof gate error [50]. Because the energy level of the microwave-dressed Rydberg

21

22 Chapter 8. Summary and outlook

state depends on the microwave Rabi frequency, fluctuating microwave powercauses fluctuating Rydberg state energies and therefore dephasing between Ry-dberg and low-lying electronic states. The stabilization of the microwave powercan be done by measuring the microwave by the ion or a detector and sending thefeedback to the microwave source.

• Compensate the quadrupole shifts. Rydberg S1/2 and P1/2 state energies areshifted by the second order coupling of the oscillating quadrupole field of theion trap [52]. Their energy levels oscillate at twice the trap rf frequency. Thiscan be compensated by modulating the UV and microwave frequencies at twicethe trap rf frequency. This way, even though the atomic levels are oscillating, thecoupling fields remain resonant to the atomic transition.

• Currently the main gate error is from Rydberg excitation. Once high efficiency hasbeen achieved for Rydberg excitation, the qubit operation fidelity and coherencetime need to be improved. As the first step, magnetic field shielding and fibernoise cancellation need to be implemented. Magnetic field shielding is ready tobe set up, fiber noise cancellation has been tested in the Master thesis of AndersLindberg [65].

• Test isotope selective loading. In our system isotope selective loading has notbeen realized. This may cause problems for quantum computation in a large ioncrystal, because other isotopes cannot be manipulated or detected by the lasers for88Sr+ and therefore cannot be used for quantum computation. Isotope selectiveloading may be possible by ionizing or heating up other isotope, similar to themethods proposed for 133Ba+ ion [77, 78].

In the long term, other improvements may be implemented:

• Implement a cryogenic ion trap system. Ions may become double ionized whenthey are excited to Rydberg states. These doubly-charged ions cannot be used asqubits and affect the ion crystal configuration, thus they need to be replaced bynormal (singly-charged) ions to continue the experiment. Black-body radiation atroom temperature in the ion trap may account for the observed double-ionizationrate [50], this can be reduced by cooling the system to cryogenic temperature. Ad-ditionally, in cryogenic systems lower pressures can be achieved and backgroundgas collisions can be reduced [79]. This is important for long experimental se-quences in large ion crystals.

• Implement a rotating quadrupole trap [80]. The quadrupole field strength ina rotating trap does not oscillate, thus the energy shifts on the Rydberg stateswill not oscillate and the spectral sidebands can be removed [52]. Althoughthe quadrupole field dressed Rydberg states wavefunction still rotate with thequadrupole field, the J = 1/2 components are symmetric about the rotationalaxis. If the excitation from low lying states only couple to the J = 1/2 statesand the dipole-dipole interaction only relies on J = 1/2 states, the gate processis invariant under trap rotation.

Chapter 8. Summary and outlook 23

• Implement a microwave cavity to realize full connectivity and distance indepen-dent interaction for large Rydberg ion systems. If we consider two ions that areinside a microwave cavity and two Rydberg states |s〉 and |p〉 are strongly coupledvia a near resonant cavity mode with vacuum coupling strength g. In the emptycavity limit, we only consider states that couple to the vacuum mode |0〉. For oneion, the cavity couples |p0〉 ↔ |s1〉 and the dressed states (|s1〉 ± |p0〉)/

√2 have

eigenenergies±g/2. For two ions, the cavity couples |sp0〉 ↔ |ss1〉 ↔ |ps0〉, theeigenstates (|sp0〉 ±

√2|ss1〉+ |ps0〉)/2 and (|sp0〉 − |ps0〉))/

√2 have eigenen-

ergies ±√

2g and 0, only the ±√

2g dressed states can be excited from low-lyingstate. The effective interaction energy mediated by the cavity mode is the dif-ference between the two-ion eigenenergy and the sum of two independent ioneigenenergies V = (

√2− 1)g. To implement an entangling gate using this inter-

action, the minimum required gate time is T ≈ 1/V = (√

2 + 1)/g. The cavitydressed two-ion Rydberg state (|sp0〉±

√2|ss1〉+ |ps0〉)/2 can decay with a rate

of Γ = (Γs2/3 + Γp1/3 + κ1/2), where Γs, Γp and κ are the decay rates of theRydberg states |s〉 and |p〉 and the cavity. The gate error caused by spontaneousdecay will be ε ≈ Γ · T/2 ∼ (Γs + Γp + κ)/g. In a superconducting one-dimensional transmission-line microwave cavity [81], the geometry is similar toa coaxial cable and the mode volume V = πr2λ/2 can be much less than a cubicwavelength, where λ is the wavelength and r is the distance between the wires.This results in strong coupling between the cavity mode and the Rydberg states.For instance, for Rydberg states with principal quantum number n ≈ 60, thetransition wavelength is around 6 mm, and the cavity wires can be placed around100µm away from the ion string to avoid laser beams on the wires. This resultsin a strong coupling g ≈ 2π × 20 MHz. The cavity microwave photon lifetimeis about tens of microseconds [81], similar to the Rydberg state lifetimes. As aresult, the gate error caused by spontaneous decays of the Rydberg states and thecavity photon will be in the order of 10−3. This intrinsic error is sufficiently lowfor quantum error correction codes, and other errors can be diminished by techni-cal improvements. This interaction is long range and its strength is independent ofthe distance between two ions since they are in the same microwave cavity mode(λ/2, it is much larger than the size of typical trapped ion crystals). Because ofthe distance-independence, there is no force between the two ions and the gate islargely independent of ion motion. The same gate scheme can be implementedbetween arbitrary pairs of ions, entangling gates between more than two ions oreven over the entire ion crystal may also be possible.

• Implement sympathetic cooling. Electric field noise or background gas collisioncan cause heating of the ions and thus reduction of the gate fidelity. Sympatheticcooling can take the energy of the crystal away by cooling the non-qubit ions. Forexample, a different ion species can be trapped in the same crystal for sympatheticcooling of the qubit ions. The sympathetic cooling lasers does not interact withthe qubit ions.

In addition to these technical improvements, the following experiments could be

24 Chapter 8. Summary and outlook

performed in the existing system:

• Rydberg blockade gate. The interaction gate requires high laser power to achievehigh fidelity (see supplemental information of [50]). A Rydberg blockade gaterequires stronger interaction but lower excitation laser power compared with theinteraction gate. For higher Rydberg states, for which the interaction is stronger,high fidelity blockade gate may be realized with the current Rydberg excitationlaser powers.

• Microwave dressing can be used to tune some Rydberg states close to a Forsterresonance. A high fidelity entangling gate proposal using the dark state of two-ionFoster resonance [82] may be realized.

• Operate the microwave in pulsed mode and test sweeping the microwave fre-quency and power using a microwave upconverter or mixer, this will enable adia-batic control of Rydberg states. The dipole-dipole interaction can move the pointsof degeneracy in the parameter space and thus change the topological structure.A robust geometric phase gate may be implemented.

• Enhance the interaction between Rydberg excitation and ion motion, oppositeto the requirements for a gate operation, by shaping the interaction potential orsetting the parameters close to a linear-zigzag structure phase transition [83]. Thisenables quantum simulations of many-body systems and large molecules, wheremotion and electronic states are strongly coupled.

In summary, in trapped Rydberg ion systems the Coulomb interaction allows state-independent trapping and state-independent sympathetic cooling. The strong dipolarRydberg interaction provides fast, direct and motional-structure-independent qubit cou-pling. The combination of these two types of interactions may enable a large numberof gate operations in large systems. The interplay between the charge and dipolar in-teractions also offers a new platform for quantum simulation. Thus Rydberg ions are apromising quantum system for scalable quantum computation and quantum simulation.

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