Trapped

Ion

Quantum

Information

ETH Zürich, Department of Physics

SEMESTER THESIS

Numerical Simulation of

Electromagnetically-Induced-

Transparency-Based Laser Cooling of

Motional States in40Ca

+

Supervisor

Prof. Dr. Jonathan Home

Hsiang-Yu Lo

Student

Peter Clemens Strassmann

2013

Acknowledgements

I wish to thank all who supported me during my semester project in our group.First of all professor Jonathan Home for giving me the opportunity of participatingat an active research project in our group, Hsiang-Yu Lo who supported me withhis knowledge and patience, our research group for the help that I received andthe good time that I had during my time as a member of the TIQI group. It wasan honorful and great experience for me to work with them as I received greatinsight into ion-trap physics with all its diversity.

i

Abstract

This thesis presents numerical calculations of the optical Bloch equations for a four-level atomic system, including the eects of AC Stark shift and nite scatteringinduced by an additional far-detuned laser beam. We simulate the laser coolingof motional states for 40Ca+ ions using electromagnetically induced transparency.The simulation results show that it is possible to cool the ion's motional state ton = 0.005 with reasonable experimental parameters.

ii

Contents

Acknowledgements i

Abstract ii

1 Introduction 1

2 Optical Bloch Equations 3

2.1 The master equation . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.1 Dissipative processes and Lindblad operators . . . . . . . . . 42.1.2 Optical Bloch equations for a four-level system . . . . . . . . 5

2.2 Simulation results and discussion . . . . . . . . . . . . . . . . . . . 6

3 EIT Cooling 11

3.1 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Conclusion 16

A Electronic structure of the experimental setup 17

B Derivation of the Hamiltonian 19

B.1 Hamiltonian for OBE . . . . . . . . . . . . . . . . . . . . . . . . . . 21B.1.1 Four Level System . . . . . . . . . . . . . . . . . . . . . . . 22

B.2 Calculation of the Clebsch Gordan coecients . . . . . . . . . . . . 22B.2.1 Calculation of the expectation values of the dipole operator . 23

Bibliography 28

iii

Chapter 1

Introduction

The eld of quantum information processing (QIP) has grown very rapidly in recentyears and became one of the most interesting subjects in quantum physics. Dif-ferent schemes for realizing a quantum information processor are developed in thelast decade using e.g. trapped ions and atoms, nuclear magnetic resonance, cavityquantum electrodynamics, superconducting electronic circuit, quantum dots, andhybrid systems [4]. The requirements for QIP are commanded by the DiVincenzocriteria [2]:

• the scalable quantum system with a well characterized qubits

• the feasibility to initialize the state of the qubits

• qubit states with long coherence time, compared to the gate operation time

• a universal set of quantum logic gates

• a qubit-specic measurement capacity

Trapped ions are one of the most promising technologies for realizing quantuminformation processing and quantum simulation. The ions are well isolated fromthe environment, so quantum states can be stored robustly in the internal andmotional states. In our group, we use beryllium (9Be+) and calcium (40Ca+)ions to implement quantum simulations of open quantum systems and quantumstate engineering. The advantages of using two species of ion have shown in manydierent aspects [3]. Pre-requisites for this work are the ability to perform Dopplerlaser cooling and to read-out the nal state by observing uorescence. In orderto do this, we need to have a good understanding of the internal state dynamicsunder interaction with the laser light.

For the 9Be+ ions, we use two hyperne ground states to store and manipulatequantum states. One of the principal problems in the laboratory is uctuations

1

of the magnetic eld. This has motivated us to work with two qubit states whosefrequency separation has zero rst-order dependence on the magnetic eld. Atransition of this form is found in the internal ground state of beryllium at amagnetic eld of 119.64G. Coherence time of the eld-insensitive qubit stateshave been observed up to 15 seconds by Ion Storage Group at NIST [5]. On theother hand, calcium has a more complex level structure than beryllium [6, 12].Therefore, the dynamics of the uorescence generation is more complex due to thelarge energy level splitting at the magnetic eld of B ≈ 119.64G (these energylevel splittings are much larger than the natural linewidth of the transition). Inorder to nd optimal settings for our lasers, we have carried out the simulationsof the S1/2, P1/2, and D3/2 levels for the Doppler cooling by solving the opticalBloch equations. Using these simulations, we have devised a set of frequencies andpolarizations which should allow us to overcome the aforementioned diculties.

To further cool down the ion's motion to near ground state, the Doppler coolingis not sucient such that a second stage of cooling is necessary. There are threedierent methods to date which have been used to cool the ion to the motionalground state [7]. One of them is using the eect of electromagnetically inducedtransparency (EIT). The EIT cooling has been studied and demonstrated to bemore simpler, ecient and faster [1, 8, 9, 11]. In this work, we simulate the eectsof EIT cooling for 40Ca+ ions starting from solving the optical Bloch equationsfor a four-level atomic system (here we only consider S1/2 and P1/2 states of the40Ca+ ion). The inuence of the AC Stark shift and the scattering terms on theexited-state population and EIT cooling limit induced by the a far detuned laserbeam [13] are discussed in this thesis. These eects not only can be revealed fromthe master equation for a multilevel system but also have an inuence on theachieved nal temperature and the cooling rate.

The thesis is organized as follows: Chapters 2 and 3 are dedicated to thesimulations of optical Bloch equations and EIT cooling, respectively. The eectsof AC Stark shift and scattering term induced by a far detuned laser beam arealso discussed in Chapter 2. Both eects that we mentioned above are included inthe simulation of EIT cooling and the results are discussed in Chapter 3.

2

Chapter 2

Optical Bloch Equations

|1〉-1/2

|2〉+1/2

|3〉-1/2 |4〉

+1/2

42S1/2

42P1/2

Ω13 π

Ω14 σ+

Ω23 σ−

Ω24 π

∆14

∆24

Figure 2.1: The 4 energy levels with the detunedlasers.

In this chapter, we study theinteraction of a 40Ca+ ion withlaser light at the magnetic eldof 119.64 G. Since the nucleusof 40Ca+ has no spin (I=0),there is no hyperne splittingof the atomic energy levels.However, the energy levels splitup into several Zeeman compo-nents for a given applied mag-netic eld. The Fig. 2.1 showsthe energy levels of the 40Ca+

ion that we are interested in.Here, only the atomic transi-tion at 397 nm, driven betweenS1/2 and P1/2 states, is consid-ered. We numerically solve theoptical Bloch equations for thefour-level system.1 Our aim isto obtain higher scattering ratefrom the excited states of theion. The chapter starts withtheoretical deviations of opti-cal Bloch equations (OBEs) forthe four-level system. Next,the eects of AC Stark shift

1 This part of the thesis follows mostly the procedure of the thesis "Resonance uorescenceof single Barium ions" by Hilmar Oberst [10].

3

and the scattering term are taken into account in the model.

2.1 The master equation

For studying the evolution of a quantum system coupled to an environment, themaster equation is an essential tool. In the Schrödinger picture, it takes the form

dρ

dt= Lρ,

where ρ is the density operator. The Liouvillian operator, L, is dened as

Lρ :=1

i~[H, ρ] +Kρ,

where the rst term describes the coherent evolution of the quantum system andthe second one is the dissipation of the system, K is called the Lindblad operator.

2.1.1 Dissipative processes and Lindblad operators

The Lindblad operator K characterizes the coupling of the ion to an environmentand has the following general form

Kρ =1

2

∑m

C†mCmρ+ ρC†mCm − 2CmρC

†m

, (2.1)

where Cm describes dierent dissipative processes [10]:Finite laser linewidth ΛPS for the 397 nm light

C1 =È

2ΛPS(|1〉 〈1|+ |2〉 〈2|). (2.2)

The spontaneous decay rates via the σ+ transition at 397 nm are represented as

C2 =

Ê2

3ΓPS |1〉 〈4| ,

via the σ− transition at 397 nm

C3 =

Ê2

3ΓPS |2〉 〈3| ,

via the π transitions at 397 nm

C4 =

Ê1

3ΓPS |1〉 〈3| , (2.3)

C5 =

Ê1

3ΓPS |2〉 〈4| , (2.4)

where ΓPS is the spontaneous decay rate of the P1/2 state and the coecients arethe branching ratios for dierent transitions.

4

2.1.2 Optical Bloch equations for a four-level system

This section provides the explicit form of the Hamiltonian of the laser light andion which is derived in appendix B.

Hamiltonian for a single 40Ca+ ion neglecting the D states

Looking only at 4 levels, terms like AC Stark shift or the inuence of the transitionrate of the far detuned laser |2〉 ↔ |3〉 can be discussed more comprehensive thanfor 8 levels. For the present treatment of a single 40Ca+ ion, the Hamiltonian inthe rotating frame reduces to

HRot.Frame,4 = ~

∆14

Ω13

2√

3Ω14√

6

0 ∆24 −Ω23√6e−2i(−2u−∆14+∆24)t − Ω24

2√

3Ω13

2√

3−Ω23√

6e2i(−2u−∆14+∆24)t 4u

3+ ∆14 −∆24 0

Ω14√6

− Ω24

2√

30 0

(2.5)

with frequency detunings

∆14 = ωσ − (ω4 − ω1),

∆24 = ωπ − (ω4 − ω2),

Rabi frequencies Ω13, Ω14, Ω23, and Ω24, and magnetic eld strength u = µB~ |B|.

The dashed laser frequency in Fig. 2.1 is assumed to be far detuned fromthe transition |2〉 ↔ |7〉 which means that the rotating wave approximation isapplicable and can be implemented by setting Ω23 = 0 what is in reality notcorrect. Additionally the Rabi frequencies of the π transitions are assumed to bethe same Ω13 = Ω24. Hence the Hamiltonian gets the simpler form

HRot.Frame,4 = ~

∆14

Ω13

2√

3Ω14√

6

0 ∆24 0 − Ω24

2√

3Ω13

2√

30 4u

3+ ∆14 −∆24 0

Ω14√6− Ω24

2√

30 0

. (2.6)

The physical parameters relevant in the experiment are

• the detunings of the laser beams with respect to the electronic transition,

• the laser powers which relate to the Rabi frequencies by P ∝ Ω2, and

• the magnetic eld B, linewidth ΓPS.

5

AC stark shifted energy levels The far detuned laser of the σ transition stillshifts the energy levels |2〉 and |7〉 which is described by the term Ω2

4∆. Including

these terms, the Hamiltonian extends to

HRot.Frame,4 = ~

∆14Ω13

2√

3Ω14√

6

0 ∆24 +Ω2

14

4(∆14+ 8u3 )

0 − Ω24

2√

3

Ω13

2√

30 4u

3+ ∆14 −∆24 − Ω2

14

4(∆14+ 8u3 )

0

Ω14√6

− Ω24

2√

30 0

(2.7)

by using that in reality the Rabi frequencies of the σ polarized light are equal.

Induced transition rate The other eect of the far detuned laser in Fig. 2.1 isthe induced transition rate [13]

C6 =Ω23

2

ÌΓPS

∆14 + 8u3

2+

Γ2PS

4

|3〉 〈2| (2.8)

as a dissipative process. The Ω23 dropped out of the Hamiltonian in the rotatingframe approximation as it is far detuned. Hence we are dealing only with one Rabifrequency Ω14 which is equal to Ω23 for the Eq. (2.8).

2.2 Simulation results and discussion

The steady-state solution of the OBE gives a statement of the relative population ofthe states in equilibrium. Depending on given experimental values, the populationand therefore also the uorescence can be predicted where uorescence stands forthe incoherent spontaneous emission of light.

The parameters chosen in the simulations are

• the decay rate ΓPS = 21.58MHz,

• the Rabi frequencies Ω13 = Ω24 = 0.1ΓPS of the π transition andΩ14 = Ω23 = 1ΓPS of the σ transition,

• the laser linewidth ΛPS = 0MHz,

• the π transition detuning ∆24 = 80MHz,

• the magnetic eld B = (0, 0, 119.64G).

6

−100 −80 −60 −40 −20 0 20 40 60 80 1000

1

2

3

4

5

6

7

8

9x 10

−3

Detuning of 397 nm (σ) laser (MHz)

Rel

ativ

e po

pula

tion

Figure 2.2: The curve depicts the population dependent on the detuning of thelaser σ transition. This is the bare behavior in contrast to the following diagrams.

The detuning of the σ transition ∆14 is the variable parameter of the simulations.At rst we have a look at the bare behavior of the population of the excited P

states |3〉 and |4〉 which is shown in Fig. 2.2. The broader excitation of populationin the P states has its maximum of 6.33 · 10−3 at a detuning of −959 kHz of theσ transition. The full width half maximum of this main maximum is 26.8MHzwhich is approximately a factor of 1.24 broader than the decay rate ΓPS of theexcited P states into the S states. The dark state where there is no excitedpopulation is at equal detuning ∆14 = ∆24 = 80MHz as there is no eect takeninto account. The bright state close to equal detuning has a relative population of3.71 · 10−3 at a detuning of ∆14 = 81.0MHz.

The Fig. 2.3 shows the eect of the AC Stark shift on the population of theexcited states in comparison to the bare population. The eect induces simply ared shift of the peaks about 607(2) kHz whereas the shape stays to high accuracysame.

In Fig. 2.4, the result of scattering is clearly visible. On the tails far fromresonance, there is much more population. The dierence of the population atthe tails seems to converge to 8 · 10−4. The maximal population is increased onlyto 6.71 · 10−3 at a detuning of −959 kHz and to 4.05 · 10−3 at 81.0MHz which

7

−100 −80 −60 −40 −20 0 20 40 60 80 1000

1

2

3

4

5

6

7

8

9x 10

−3

Detuning of 397 nm (σ) laser (MHz)

Rel

ativ

e po

pula

tion

with AC Stark shiftwithout any effect

Figure 2.3: The red curve in the graph depicts the AC Stark shifted populationwhereas the blue dashed curve depicts the population without additional term.

is an absolute increase of 3.76 · 10−4 and 3.45 · 10−4 respectively. The minimumat equal detuning ∆14 = ∆24 = 80MHz increased for nite scattering to a valueof 8.06 · 10−4. At resonance there is more population too because the scatteringsuppresses other damping terms. The full width half maximum is broadened to30.6MHz which is almost a factor 1.42 broader than the decay rate into the Sstates.

The Fig. 2.5 shows that the AC Stark shift and the scattering term have relevantimpact on the population of the excited state. The AC Stark shift is compared tothe eect of the scattering as the general behavior stays the same.

8

−100 −80 −60 −40 −20 0 20 40 60 80 1000

1

2

3

4

5

6

7

8

9x 10

−3

Detuning of 397 nm (σ) laser (MHz)

Rel

ativ

e po

pula

tion

with scattering termwithout any effect

Figure 2.4: The red curve depicts the population including the scattering processesof the far detuned laser and the blue dashed curve depicts the population withoutadditional term.

9

−100 −80 −60 −40 −20 0 20 40 60 80 1000

1

2

3

4

5

6

7

8

9x 10

−3

Detuning of 397 nm (σ) laser (MHz)

Rel

ativ

e po

pula

tion

with both effectswithout any effect

Figure 2.5: The red curve in the graph depicts the population under considerationof both eects the AC Stark shift and the scattering term. The blue dashedcurve depicts the population without additional term.

10

Chapter 3

EIT Cooling

For any type of laser cooling in a trap, the system achieves a balance betweenthe cooling and heating mechanisms. The ion or atom in principle can be cooledfurther when the events that dissipate kinetic energy are more than the events thatinduce heat. Electromagnetically induced transparency (EIT) is one of the coolingtechniques that we can use to cool an ion to its motional ground state. The EITcooling scheme utilizes a two-photon dark resonance to suppress the scattering inthe carrier transition and has higher scattering rate on the red sideband comparedto that on the blue sideband, which means the cooling events are dominant duringthe whole process. The detailed theoretical works can be found in [1, 7, 9]. Thismethod has been demonstrated in Innsbruck Group with a single 40Ca+ ion [11].Recently NIST Group demonstrated sympathetic cooling of 9Be+ ions by EITcooling of 24Mg+ ions in a linear ion chain [8]. In this chapter, we will study theeects of EIT cooling on the 40Ca+ ions and try to nd the optimal laser settingsfor future experiments.

3.1 Simulation results and discussion

The principle of the EIT-cooling method is the following. In Fig. 2.2, we candene δ as the frequency dierence between the absorption null and the relativelynarrow peak on the right. If δ approximates the frequency of the motional modes,the carrier transition would be completely suppressed (at the absorption null), thered sideband is resonant within the sharp slope of the right-hand peak, and theblue-sideband transition is on the wing of the broad bright resonance. Therefore,due to a higher scattering rate for the red sideband, the ion has higher probabilityto scatter a photon and simultaneously loses a quanta of motion. Additionally, theother advantage of EIT cooling is that the width of the right-hand peak can betuned such that it is broad enough for cooling multiple modes at the same time [8].

11

In our simulation, the goal is to nd conditions where the scattering rate for thered sideband is as large as possible, while the scattering rate for the blue sidebandis a comparatively small.

0 20 40 60 80 100 120 140 160 180 20010

−3

10−2

10−1

100

101

102

103

Detuning of 397 nm laser (MHz)

<n>

Ω14

= 2ΓPS

Ω14

= 1ΓPS

Figure 3.1: The green dashed and magenta solid line in the graph depict theachievable minimal temperature dependent on the detuning without any eectbut with dierent Rabi frequencies Ω14 = 2ΓPS and Ω14 = 1ΓPS, respectively.

The theoretical model we use is based on the treatment in Ref. [7]. The eectsof AC Stark shift and the scattering terms that we have described in Chapter2 are also included in the theoretical model. The parameters chosen for thesesimulations are

• Spontaneous decay rate: ΓPS = 2π × 21.58MHz

• Magnetic eld: B = 119.64G

• Trap frequency: ωt = 2π × 3MHz

• Laser linewidth: ΛPS = 0MHz

The detunings (∆14 and ∆24) and the laser intensities (the Rabi frequencies) arethe variables in the simulation.

The Fig. 3.1 shows the minimum motional state n as a function of laser detun-ings for dierent laser-intensity settings. The magenta one with a small ratio of

12

Ω14 to Ω24 has a minimum n = 0.05 at ∆14 = ∆24 = 2π×28.2MHz. With a largerΩ14 and a constant Ω24, it is able to achieve a smaller n = 0.005 due to the moreecient EIT eect at larger laser detuning of ∆14 = ∆24 = 2π × 104MHz.

The AC Stark shift leads to an achievable minimal cooling limit of n = 0.0032at a detuning of 79.3MHz.

0 20 40 60 80 100 120 140 160 180 20010

−3

10−2

10−1

100

101

102

103

Detuning of 397 nm laser (MHz)

<n>

without any effectwith AC Stark shift

Figure 3.2: The green dashed and magenta solid line in the graph depict theachievable minimal temperature dependent on the detuning without any eectand with AC Stark shift, respectively.

The scattering term has negligible eect to the achievable cooling limit as shownin Fig. 3.3

Considering both eects in Fig. 3.4, the achievable minimal temperature is thesame as with AC Stark shift because the AC Stark shift only has relevant inuenceon the cooling limit.

13

0 20 40 60 80 100 120 140 160 180 20010

−3

10−2

10−1

100

101

102

103

Detuning of 397 nm laser (MHz)

<n>

without any effectwith scattering term

Figure 3.3: The green dashed and magenta solid line in the graph depict theachievable minimal temperature dependent on the detuning without any eectand with scattering process, respectively.

14

0 20 40 60 80 100 120 140 160 180 20010

−3

10−2

10−1

100

101

102

103

Detuning of 397 nm laser (MHz)

<n>

without any effectwith both effects

Figure 3.4: The green dashed and magenta solid line in the graph depict theachievable minimal temperature dependent on the detuning without any eectand with both eects, respectively.

15

Chapter 4

Conclusion

For the OBEs, the AC Stark shift induces just the predicted shift of the levels,whereas the scattering term induces an additional population of about 8 ·10−4 andan additional broadening of the central peak.

For the electromagnetically induced transparency (EIT) cooling, the additionalterms introduce an improvement of the cooling minimum to an average phononnumber of n = 0.34 which is much below the original value of n = 0.60.

Outlook One possible issue of further projects would be to take into account thefour 3D3/2 states and look nd parameters at which EIT cooling is still possible.An other issue is to verify the provided simulations in experiments.

16

Appendix A

Electronic structure of 40Ca+ in the

experimental setup

This section gives an introductory description of the energy levels of the singlyionized 40Ca+ ion with the experimental conditions which are considered in thesimulations.

,

|1〉−1/2

|2〉+1/2

|3〉−1/2 |4〉

+1/2

|5〉−3/2

|6〉−1/2

|7〉+1/2

|8〉+3/2

42S1/2

32D3/2

42P1/2

Figure A.1: The relevant energy levels of the 40Ca+ ion.

In the present thesis, the electronic structure is restricted to the three lowestenergy levels with dipole-allowed transitions 42S 1

2, 32D 3

2, and 42P 1

2. For under-

17

standing the additional eects, the 32D 32states are neglected (electrons decay with

signicantly lower probability into these states). In the 42S 12conguration are two

degenerate states

|1〉 =∣∣∣n = 4, l = 0, s = 1

2, j = 1

2,mj = −1

2

¶,

|2〉 =∣∣∣n = 4, l = 0, s = 1

2, j = 1

2,mj = +1

2

¶,

in the 32D 32conguration are four degenerate states

|5〉 =∣∣∣n = 3, l = 2, s = 1

2, j = 3

2,mj = −3

2

¶,

|6〉 =∣∣∣n = 3, l = 2, s = 1

2, j = 3

2,mj = −1

2

¶,

|7〉 =∣∣∣n = 3, l = 2, s = 1

2, j = 3

2,mj = +1

2

¶,

|8〉 =∣∣∣n = 3, l = 2, s = 1

2, j = 3

2,mj = +3

2

¶,

and in the 42P 12conguration are again two degenerate states

|3〉 =∣∣∣n = 4, l = 1, s = 1

2, j = 1

2,mj = −1

2

¶,

|4〉 =∣∣∣n = 4, l = 1, s = 1

2, j = 1

2,mj = +1

2

¶,

with the principal quantum number n, the azimuthal quantum number l, the spinquantum number s, the total angular momentum quantum number j, and theprojection of the total angular momentum along the axis of the magnetic eld mj.

The degeneracy is split up by applying a magnetic eld which leads to a Zeemansplitting. In quantum optics, energies often are expressed in frequencies by therelation E = ~ω. The Zeeman splittings between the energy levels at a magneticeld of 119.64G are

∆ωj = 2∆mjgju, (A.1)

∆νS = 2u

2π= 334.90MHz,

∆νD = 22

5

u

2π= 133.96MHz,

∆νP = 21

3

u

2π= 111.63MHz,

where gj are the Landé factors and u = µB~ |B|. The ion 40Ca+ has no hyperne

splitting due to its absence of nuclear spin.

18

Appendix B

Derivation of the Hamiltonian

In this chapter the derivation of the Hamiltonian is presented in more details.1 Atthe beginning, the general Hamiltonian is derived from the basic equations. Thereis a explicit discussion of the Hamiltonian for OBEs of the 4 level system in aseparate section.

The total Hamiltonian in the Schrödinger picture

Htot,S = Hatom,S + Held,S + Hmotion,S + Hint,S (B.1)

is composed into the Hamiltonian for the internal electronic states of the atomHatom,S, the eld Hamiltonian Held,S which describes the electromagnetic eld,

the motional Hamiltonian Hmotion,S describing the ions center-of-mass motion withmomentum p inside the trap potential connecting all three terms together and thecommon interaction Hamiltonian Hint,S. The atomic Hamiltonian is

Hatom,S := ~N∑a=1

(ωa − ωRef) |a〉 〈a| , (B.2)

with ωa = ωa + ∆ωa where the Zeeman shift ∆ωa = ∆megau splits the degenerateenergy ωa ∈ ωS, ωP , ωD of the levels to the amount of Eq. (A.1) into eigenstates|a〉. As the amount of photons is classically high, the system can be described interms of the coherent states |α〉 [ [10], p.8]. For the current thesis the quantiza-tion of the motional states is not necessary as the splitting will not be discussed.

Therefore, eld and motion are treated classically and the terms Held,S + e2A2(r)2m

and Hmotion,S + p2

2mcan be assumed to be constant for the moment. Nevertheless,

1The derivation in this appendix applies to all simple ionized alkaline earth metals withoutnuclear spin and can be generalized to higher order transitions like quadrupole transition.

19

the interaction with the internal electronic states is discussed in details.

Hint,S :=

p + eA(r)

22m

+ eφ(r)−eA(r)

22m

− p2

2m

:= − e

mp · A(r) + eφ(r). (B.3)

because the commutator of [p, A] = 0.In the Coulomb gauge, where ∇r ·A(r) = 0, φ(r) = 0, the Hamiltonian reduces

to

Hint,S = − e

mp · A(r)

= − e

mp

N∑f=1

Afefe

i(kr−ωf t)af + e∗fe−i(kr−ωf t)a†f

(B.4)

for the vector potential

A(r) =N∑f=1

Afefe

i(kr−ωf t)af + e∗fe−i(kr−ωf t)a†f

where Af =

√~

2ε0V ωf. A number of N distinct energy levels is assumed.

The next step is to change into the rotating frame and applying approximations.The total Hamiltonian is transformed into a rotating frame2 and gets

HRot.Frame = UHtotU† − i~U dU†

dt(B.5)

by applying the corresponding uniform transformation3 . In this rotating framethe rotating frame approximation can be applied to simplify the interaction Hamil-tonian to

Hint = − e

m

N∑f,a,a′=1

|a〉 〈a′|Af 〈a| pefe

ikraf + e∗fe−ikra†f

|a′〉

· δ(∆a,a′ , ωf − (ωa′ − ωa))

whereas the rest of the Hamiltonian stays the same. In this step, the identity1=

∑fa=1 |a〉 〈a| was inserted from left and right.

2This corresponds to the Interaction picture Hamiltonian if there was whether Zeeman split-ting nor detuning between laser frequency and transition frequency.

3The unitary transform for the four level system is

U = e−iωσt |1〉 〈1|+ e−iωπt |2〉 〈2|+ e−i(ωσ−ωπ)t |3〉 〈3|+ |4〉 〈4|

20

B.1 Hamiltonian for OBE

For the OBEs, the next simplication is the dipole approximation which meansthat 〈a′| pe±ikr |a〉 ≈ 〈a′| p |a〉. Hence the interaction Hamiltonian is

Hint = −ieN∑

f,a,a′=1

Af |a〉 〈a′| (ωa′ − ωa)Da′,a

ef af + e∗f a

†f

δ(∆a,a′ , ωf − (ωa′ − ωa))

= eN∑

f,a,a′=1

Ef |a〉 〈a′|Da′,a

ef af + e∗f a

†f

δ(∆a,a′ , ωf − (ωa′ − ωa))

where the expectation value of the momentum operator can be rewritten with theequation of motion p = m∂tr = m i

~ [Hatom, r]4 using Eq. (B.2) as

〈a′| p |a〉 = mi

~〈a′| [Hatom, r] |a〉 = im(ωa′ − ωa) 〈a′| r |a〉 . (B.7)

for the internal electronic states |a〉 where the dipole matrix element is dened asDa′,a = 〈a′| r |a〉, which is symmetric. The amplitude Af of the vector potentialcan be rewritten in terms of the amplitude of the electric eld

Ef = 2ωfAf |αf | ≈ (ωa − ωa′)Af |αf | = (ωa − ωa′)Afαfi

due to the relation E = −∂tA and by choosing the phase of the coherent stateαf = |αf |ei

π2 .

The normalized Rabi frequency is dened as

Ωa′,a,f :=2

~(ef · da′,a/

√3)Ef (B.8)

with eld amplitude Ef = 2ωfAf |α| and the normalized expectation value for the

dipole transition da′,a =Da′,aCa′,a

. Hence the interaction Hamiltonian in the rotating

frame is

Hint = ~∑f,a,a′

a6=a′

Ca,a′Ωa,a′,f

2δ(∆a,a′ , ωf − (ωa′ − ωa)) (B.9)

4In the interaction picture, any operator OI can be written as

OI = ei~ HatomtOSe

− i~ Hatomt, (B.6)

in terms of the operator OS in the Schrödinger picture and therefore the time derivative of r is

∂trI =i

~[Hatom, rI(t)] + e

i~ Hatomt(∂trS)e

− i~ Hatomt =

i

~[Hatom, rI(t)].

21

where Ca,a′ is the corresponding Clebsch Gordan coecient. More details aboutthe explicit coecients is provided in section B.2

Hence the total Hamiltonian from Eq. (B.5) gets the more explicite form

HRot.Frame = ~

N∑a=1

(ωa − ωRef ) |a〉 〈a|+∑f,a,a′

a6=a′

Ca,a′Ωa,a′,f

2δ(∆a,a′ , ωf − (ωa′ − ωa))− iU

dU†dt

(B.10)

as the atomic Hamiltonian commutes with the unitary.

B.1.1 Four Level System

For the System with the states |1〉 , . . . , |4〉, the total Hamiltonian in terms ofdetunings

∆14 = ωσ − (ω4 − ω1),

∆24 = ωπ − (ω4 − ω2).

writes

HRot.Frame,4 = ~

∆14

Ω13

2√

3Ω14√

6

0 ∆24 −Ω23√6e−2i(−2u−∆14+∆24)t − Ω24

2√

3Ω13

2√

3−Ω23√

6e2i(−2u−∆14+∆24)t 4u

3+ ∆14 −∆24 0

Ω14√6

− Ω24

2√

30 0

and by applying the rotating frame approximation again, it further reduces to

HRot.Frame,4 = ~

∆14

Ω13

2√

3Ω14√

6

0 ∆24 0 − Ω24

2√

3Ω13

2√

30 4u

3+ ∆14 −∆24 0

Ω14√6− Ω24

2√

30 0

. (B.11)

B.2 Calculation of the Clebsch Gordan

coecients

The Rabi frequency is dened by

Ωa,b :=2

~(ef · da,b/

√3)Ef (B.12)

such that the Rabi frequency is normalized correctly and Cab = Dab

dab= 〈a|er|b〉

dabis

the expectation value of the dipole operator and ef is the polarization vector.

22

B.2.1 Calculation of the expectation values of the dipole

operator Dab

The dipole operator of the relevant transitions is calculated using the form

Dab = 〈a| er |b〉 = e 〈a|

xyz

|b〉 = e 〈a|

1√2(T

(1)−1 − T

(1)1 )

i√2(T

(1)−1 − T

(1)1 )

T(1)0

|b〉 (B.13)

by transforming to an irreducible tensor operator representation of rank 1 T (1)q

x =1√2

(T(1)−1 − T

(1)1 ), (B.14)

y =i√2

(T(1)−1 + T

(1)1 ), (B.15)

z = T(1)0 . (B.16)

Remember that T(1)1 , T

(1)0 , and T

(1)−1 corresponds to σ+, π

0, and σ− polarized light,respectively. On the other hand, the back-transformation is

T(1)1 = − 1√

2(x+ iy), (B.17)

T(1)0 = z, (B.18)

T(1)−1 =

1√2

(x− iy). (B.19)

The matrix elements of the electric multipole operators can be written as

〈a|T (1)q |b〉 = 〈(na, la, sa), ja,mja|T (1)

q |(nb, lb, sb), jb,mjb〉

= 〈ja,mja|

∑mla ,msa

|la,mla , sa,msa〉 〈la,mla , sa,msa |

T (1)q ∑

mlb ,msb

|lb,mlb , sb,msb〉 〈lb,mlb , sb,msb|

|jb,mjb〉

=∑

mla ,msa

∑mlb ,msb

〈ja,mja | la,mla , sa,msa〉

〈la,mla , sa,msa |T (1)q |lb,mlb , sb,msb〉

〈lb,mlb , sb,msb | jb,mjb〉=

∑mla ,msa

∑mlb ,msb

〈la,mla , sa,msa | ja,mja〉 〈lb,mlb , sb,msb | jb,mjb〉

〈la,mla , sa,msa |T (1)q |lb,mlb , sb,msb〉 , (B.20)

23

remembering that only the terms with mli +msi = mji contribute to the sums, i.e.

〈ji,mji | li,mli , si,msi〉 = 〈li,mli , si,msi | ji,mji〉= 〈li,mli , si,msi | ji,mli +msi〉 δmji ,mli+msi , (B.21)

where the symmetry of the Clebsch Gordan coecients is applied and i ∈ 1, . . . , 8 .The electric multipole operators act only on the angular momentum and not onspin. Therefore, the matrix elements reduce to

〈a|T (1)q |b〉 =

∑mla ,msa

∑mlb ,msb

〈la,mla , sa,msa | ja,mja〉 〈lb,mlb , sb,msb | jb,mjb〉

〈la,mla |T (1)q |lb,mlb〉 〈sa,msa | sb,msb〉︸ ︷︷ ︸

=δsa,sbδmsa,msb

=∑

mla ,msa

∑mlb ,msb

〈la,mla , sa,msa | ja,mja〉 〈lb,mlb , sb,msb | jb,mjb〉

〈la,mla |T (1)q |lb,mlb〉 δmsa ,msb , (B.22)

using the fact that electrons are spin-12particles such that δsa,sb = 1.

Denition B.2.1. Let k ∈ N0. An irreducible tensor operator of rank k is a setof 2k + 1 operators

¦T (k)q

©q∈[−k,k]∩Z that satisfy

[Jz, T(1)q ] = ~qT (1)

q ,

[J+, T(1)q ] = ~

Èk(k + 1)− q(q + 1)T

(1)q+1,

[J−, T(1)q ] = ~

Èk(k + 1)− q(q − 1)T

(1)q−1.

and J is the total angular momentum of the physical system.

Theorem B.2.2 (Wigner-Eckart theorem). The Matrix elements of a tensor op-

erator T (1)q can be rewritten as

〈n, j,m|T (1)q |n′, j′,m′〉 =

1√2j + 1

〈n, j|T (1) |n′, j′〉 〈j′,m′, 1, q | j,m〉 . (B.23)

Applying the Wigner-Eckart theorem to the dipole operators irreducible tensoroperator elements results in

〈a|T (1)q |b〉 =

∑mla ,msa

∑mlb ,msb

〈la,mla , sa,msa | ja,mja〉 〈lb,mlb , sb,msb | jb,mjb〉

1√2la + 1

〈na, la|T (1) |nb, lb〉 〈lb, 1;mlb , q | la,mla〉 δmsa ,msb .

(B.24)

24

The resulting expectation value da,b := 〈na, la|T (1) |nb, lb〉 is constant for each tran-sition. The conditions

|la| ≤ lb + 1 (B.25)

mlb + q = mla (B.26)

follow immediately from the contributing Clebsch-Gordan coecient from Wigner-Eckart theorem (similar to Eq. (B.21)). Additionally the parity transformation ofthe tensor operator

〈na, la|T (1) |nb, lb〉 = −〈na, la| PT (1)P |nb, lb〉= −(−1)la+lb 〈na, la|T (1) |nb, lb〉 . (B.27)

leads to the resulting conditions

la − lb = ±1 (B.28)

mlb + q = mla . (B.29)

Their reason is the allowed polarization of the corresponding transition, i.e. thecontributing terms of forbidden transitions vanish.

〈a|T (1)q |b〉 =

da,b√2la + 1

∑mla ,msa

∑mlb ,msb

〈lb,mlb , 1, q | la,mla〉 δmsa ,msb

〈la,mla , sa,msa | ja,mja〉 〈lb,mlb , sb,msb | jb,mjb〉

(B.30)

This formula is applied directly to the transitions of interest in table ??.Hence the Clebsch Gordan coecients areC13 = 1√

3, C23 =

È23, C53 = 1√

2, C63 = 1√

3, C73 = 1√

6, C83 = 0,

C14 =È

23, C24 = − 1√

3, C54 = 0, C64 = 1√

6, C74 = − 1√

3, C84 = 1√

2.

25

〈3|T (1)1 |1〉 = 0 〈3|T (1)

0 |1〉 = d3,13

〈3|T (1)−1 |1〉 = 0

〈4|T (1)1 |1〉 =

√2d4,13

〈4|T (1)0 |1〉 = 0 〈4|T (1)

−1 |1〉 = 0

〈3|T (1)1 |2〉 = 0 〈3|T (1)

0 |2〉 = 0 〈3|T (1)−1 |2〉 = −

√2d3,23

〈4|T (1)1 |2〉 = 0 〈4|T (1)

0 |2〉 = −d4,23〈4|T (1)

−1 |2〉 = 0

〈3|T (1)1 |5〉 = d3,5√

6〈3|T (1)

0 |5〉 = 0 〈3|T (1)−1 |5〉 = 0

〈4|T (1)1 |5〉 = 0 〈4|T (1)

0 |5〉 = 0 〈4|T (1)−1 |5〉 = 0

〈3|T (1)1 |6〉 = 0 〈3|T (1)

0 |6〉 = −d3,63〈3|T (1)

−1 |6〉 = 0

〈4|T (1)1 |6〉 = d4,6

3√

2〈4|T (1)

0 |6〉 = 0 〈4|T (1)−1 |6〉 = 0

〈3|T (1)1 |7〉 = 0 〈3|T (1)

0 |7〉 = 0 〈3|T (1)−1 |7〉 = d3,7

3√

2

〈4|T (1)1 |7〉 = 0 〈4|T (1)

0 |7〉 = −d4,73〈4|T (1)

−1 |7〉 = 0

〈3|T (1)1 |8〉 = 0 〈3|T (1)

0 |8〉 = 0 〈3|T (1)−1 |8〉 = 0

〈4|T (1)1 |8〉 = 0 〈4|T (1)

0 |8〉 = 0 〈4|T (1)−1 |8〉 = d4,8√

6

Table B.1: The matrix elements of the transitions between the lowest energy levelsof 40Ca+.

26

Acronyms

EIT electromagnetically induced transparency

OBE optical Bloch equation

27

Bibliography

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[2] David P. DiVincenzo. The physical implementation of quantum computation.Fortschritte der Physik, 48(9-11):771783, 2000.

[3] Jonathan P. Home. Quantum science and metrology with mixed-species ionchains. pre-print, 2013.

[4] Thaddeus D. Ladd, Fedor Jelezko, Raymond Laamme, Yasunobu Nakamura,Christopher Monroe, and Jeremy L. O'Brien. Quantum computers. Nature,464(7285):4553, 2010.

[5] C. Langer, R. Ozeri, J. D. Jost, J. Chiaverini, B. DeMarco, A. Ben-Kish, R. B.Blakestad, J. Britton, D. B. Hume, W. M. Itano, D. Leibfried, R. Reichle,T. Rosenband, T. Schaetz, P. O. Schmidt, and D. J. Wineland. Long-livedqubit memory using atomic ions. Phys. Rev. Lett., 95(6):060502, Aug 2005.

[6] Christopher E. Langer. High Fidelity Quantum Information Processing with

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[8] Y. Lin, J. P. Gaebler, T. R. Tan, R. Bowler, J. D. Jost, D. Leibfried, andD. J. Wineland. Sympathetic electromagnetically-induced-transparency lasercooling of motional modes in an ion chain. Phys. Rev. Lett., 110(15):153002,Apr 2013.

[9] Giovanna Morigi, Jürgen Eschner, and Christoph H. Keitel. Ground statelaser cooling using electromagnetically induced transparency. Phys. Rev. Lett.,85(21):44584461, Nov 2000.

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28

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[12] Christian Felix Roos. Controlling the quantum state of trapped ions. PhDthesis, Leopold-Franzens-Universität Innsbruck, 2000.

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29

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Numerical Simulation of Electromagnetically-Induced-Transparency-Based Laser Cooling of Motional States in 40Ca+

Last nameStrassmann

First namePeter Clemens

Last nameHome

First nameJonathan

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