Towards qubit state manipulation of magnetically trapped 87Rb atoms Rydberg excitation and Raman transitions at individual lattice sites

University

of

Am

sterd

am

Quantum

Gases

&

Quantum

Inf

orma

tion

Towards qubit state manipulation of

magnetically trapped

87

Rb atoms

Rydberg excitation and Raman transitions at individual lattice sites

A thesis submitted for the degree:

Master of Science in Chemistry

Author:

_{Jannie Vos}

September 2014

Daily Supervisor:

Julian Naber

Supervisor:

Dr. Robert Spreeuw

Abstract

In the Quantum Gases & Quantum Information group at the University of Amsterdam a quantum information platform is being created in the form of a permanent magnetised atom chip with trapped rubidium-87 (87Rb) atoms. This creates an unique scalable platform where quantum information is stored in qubit states encoded in the hyperfine levels of the ground state. Operations on qubit states are created by quantum logic gates involving single-qubit transitions and often a control mechanism evoked by Rydberg excitation.

Extension of single-qubit to two-qubit quantum gates require controlled interactions, which will be created via Rydberg excitation and the concomitant Rydberg blockade mechanism will serve as a control mechanism in quantum logic gates. Electromagnetically Induced Transparency (EIT) spectroscopy is used to calibrate the wavelength of the coupling laser used during Rydberg excitation. The experimental set-up allows for spectroscopy on Rydberg states n = 19− 28 in a room temperature vapour cell. The Rydberg level hyperfine splittings in these states is studied and the influence of magnetic fields on excitation pathways is investigated experimentally and theoretically.

Additionally, these quantum gates call for single-site addressing which will be realised via the use of a Spatial Light Modulator (SLM). The pixelated structure behind the liquid-crystal allows for phase modulation by altering the optical axis through applying a varying voltage across the pixels, thereby creating arbitrary shaped light fields in the image plane. For optimal operation the phase response as a function of applied voltage is linearised. Secondly, the backplane curvature is compensated by applying a corrective phase pattern obtained through the use of Phase Shifting Interferometry (PSI). Combined these two topics are an important progression towards the realisation quantum information on an atom chip.

Samenvatting

De Quantum Gases & Quantum Information onderzoeksgroep aan de Universiteit van Amsterdam werkt aan de ontwikkeling van een quantum informatie platform. Dit platform wordt gerealiseerd in de vorm van een per-manent gemagnetiseerde atoom chip, waarop koude rubidium-87 (87_{Rb) atomen zijn gevangen. Op deze manier} is een uniek schaalbaar platform ontwikkeld. Quantum informatie wordt opgeslagen in de zogenaamde qubit toestanden gecodeerd in de hyperfijnstructure van de grondtoestand. Manipulatie van deze qubit toestanden wordt vervolgens ge¨ıntroduceerd door quantum logic gates bestaande uit transities tussen qubit toestanden en vaak een controlemechanisme opgewekt via Rydberg excitatie.

Voor de overgang tussen verschillende qubit toestanden zijn de interacties tussen de atomen onmisbaar. Inter-actie tussen naastgelegen atoomvallen worden gecre¨eerd door middel van Rydberg excitatie en het bijkomende Rydberg blockade effect, waardoor gecontroleerde quantum gates uitvoerbaar zijn. Electromagnetically Induced Transparency (EIT) spectroscopie wordt hier gebruikt om de golflengte van het coupling licht nodig tijdens Ry-dberg excitatie te kalibreren. De experimentele methode laat onderzoek naar RyRy-dberg toestanden in87_{Rb toe.} De afstand tussen de hyperfijn spectraal lijnen is onderzocht voor Rydberg toestanden met n = 19_{− 28, en de} invloed van een magnetisch veld op de voorkeur voor een excitatie route is bestudeerd zowel experimenteel als theoretisch.

Hiernaast is het belang om gedurende de manipulatie van qubit toestanden een enkele specifieke atoomval te adresseren van groot belang. Dit zal ge¨ımplementeerd worden met behulp van een Spatial Light Modulator (SLM). Deze modulator maakt gebruik van een pixel structuur bedekt met een vloeibaar kristal, waarbij elke pixel kan worden onderworpen aan een bepaald voltage dat de optische as van het kristal en daarmee de fase van het inkomende licht veranderd. Om het apparaat optimaal te gebruiken is de faserespons als functie van voltage gelineariseerd. Daarnaast is de kromming in de achterzijde, ontstaan als gevolg van het fabricage process, gecorrigeerd. Door gebruik te maken van een techniek genaamd Phase Shifting Interferometry (PSI) is een corrigerend fase-patroon berekend. Al deze voorgaande stappen in het onderzoek uitgevoerd in de groep leidt tot een belangrijke vooruitgang richting quantum informatie op een atoomchip.

Contents

Abstract

i

Samenvatting

i

1

Introduction

1

2

Rydberg states in

87

Rb

4

Rydberg Atoms & Properties . . . .

4

2.1

Electromagnetically Induced Transparency . . . .

5

2.1.1

Theory . . . .

5

Dressed states

. . . .

5

Probability amplitude . . . .

6

2.1.2

EIT spectra . . . .

7

2.1.3

Our system . . . .

7

2.2

Experimental setup . . . .

9

2.2.1

Previous experimental work . . . .

9

2.2.2

Further investigation . . . .

9

2.2.3

Experimental setup . . . .

10

Two-photon Rydberg excitation

. . . .

11

Laser stabilisation via PDH technique . . . .

12

Sideband locking . . . .

13

Linewidth . . . .

14

2.2.4

Spectroscopy . . . .

14

Electromagnetically Induced Transparency

. . . .

15

Magnetic field control . . . .

16

2.3

Results . . . .

17

2.3.1

Rydberg states and hyperfine splitting . . . .

17

2.3.2

Rydberg states in magnetic fields . . . .

18

σ

+

σ

+

polarisation . . . .

18

σ

−

σ

+

polarisation . . . .

19

σ

−

σ

−

polarisation . . . .

20

2.3.3

Theoretical model . . . .

21

2.3.4

Conclusion . . . .

25

3

Spatial Light Modulation

28

SLM prerequisites . . . .

28

3.1

Phase modulation

. . . .

29

3.1.1

Theory . . . .

29

3.1.2

Our SLM . . . .

30

3.2

Imaging . . . .

30

3.2.1

Application software . . . .

31

3.2.2

Experimental setup . . . .

32

3.2.3

Efficiency characterisation . . . .

32

3.3

Phase response linearisation . . . .

36

3.4

Nonlinear phase response . . . .

36

3.4.1

Experimental setup . . . .

36

Calibration software . . . .

38

Calibration procedure . . . .

39

Default HoloEye gamma curve . . . .

39

New gamma curves . . . .

39

3.4.2

Linear phase response . . . .

43

3.5

Backplane curvature correction . . . .

44

3.5.1

Backplane Curvature . . . .

44

3.5.2

Experimental setup . . . .

45

Coordinate transformation function . . . .

45

Phase Shifting Interferometry . . . .

46

3.5.3

Corrective phase pattern . . . .

47

3.6

Conclusion

. . . .

48

Appendix A

56

Chapter 1

Introduction

Predictions of a physical system’s behaviour can be made in two fashions, either by describing the system mathematically or by studying a simpler, similar system simulating its evolution [1]. This is true for any classical physical system. With the help of a classical computer the behaviour of a classical system can be analysed mathematically. Meantime, when studying a quantum mechanical system with a classical device the complexity of these simulations grows rapidly. The number of quantum states describing the system increases exponentially with the number of particles comprising the system [2]. The computational power needed to describe a quantum mechanical system grows fast, resulting in a less accurate mathematical analysis of the system. We are limited to study quantum many-body systems composed of up to only a few tens of particles. Clearly, another method for simulating body quantum systems is required in order to overcome the many-body problem. This can be done by using a quantum mechanical system. Such a quantum simulator would consist of particles that mimic more complex physical systems.

Another method to overcome the scaling problem is by making use of quantum mechanical effects to perform computations or store information. Since the proposal of Richard Feynman to make use of quantum mechan-ical effects to perform computations or store information the discipline of quantum computing and quantum information has received vast attention [3]. The first papers in which quantum models for computation were proposed appeared in the 1980s [3, 4]. At that time research was mostly driven to test the limits of computation, contrary to todays motivation which is mostly driven by the modern demand for even smaller microprocessors that perform complex calculations. Devices are made smaller and smaller; approaching feature sizes as small as a number of atoms. In this field of expertise, atoms will be used to store and process information in the unit of quantum bits, or simply qubits. As opposed to their classical bit counterparts, which are used to store information in the classical bit states 0 and 1, qubits can be used to store information in a linear combination of quantum bit states_{|0i and |1i. That is, a system can occur in a superposition of both states |ψi = α|0i + β|1i,} where the coefficients α and β are normalized. This gives the important distinction between classical bits, which can only appear in two classical states. In addition, the qubit can appear as a continuous superposition. With this, problems can be solved more efficiently than with classical computation, thereby overcoming the many-body problem.

Many physical systems, such as trapped ions [5, 6], solid state devices [7], cavity QED [8] and superconducting devices have been proposed as good candidates for quantum computation [9]. Although the requirements for suitable candidates are clear, the experimental implementation remains challenging [10]. A difficult task to achieve is coherent manipulation which is a necessity for quantum computation [11]. In the Quantum Gases & Quantum Information group at the University of Amsterdam this challenge will be overcome by trapping neutral atoms in a regular spaced lattice on an atom chip, whereafter they can be manipulated by optical fields. Rather than using an optical lattice to trap atoms, a permanent magnetised atom chip is used. Atom chips are considered as unique solutions in the development of quantum information science as they can be used to trap neutral atoms which exhibit weak interactions with the environment [12, 13]. The atom chip is patterned with a magnetised FePt layer to create a lattice of potential wells wherein neutral rubidium-87 (87_{Rb) atoms} are trapped. A beneficial feature of this platform for quantum information and quantum simulation is the scalability of the lattice while the structure can be shaped arbitrarily. With this chip design the platform can be used for different research directions. The lattice period can be varied to create a spacing of_{∼ 100 − 500}

2 Introduction

nm between microtraps, herewith devising a platform for a direct analog quantum simulator. At larger lattice periods of 4_{− 10 µm the atom chip can be used as a platform for quantum information.}

Currently, a regular lattice of approximately hundreds of tightly confining potential wells has been created with an inter-trap distance of 10 µm. Here,87_{Rb atoms are trapped in these microtraps after sufficient cooling} (to a few µK) via radio-frequency forced evaporative cooling. Each microtrap contains up to ∼ 300 atoms per trapping site. Not only can these alkali atoms be cooled easily, they also show weak couplings to the environment hence they possess long coherence times, thereby making them ideal candidates for applications in the field of quantum information. The objective of the Quantum Gases & Quantum Information group is to store quantum information in the hyperfine ground state levels of 87_{Rb atoms. The magnetic trappable states} |0i ≡ |F = 1, mF =−1i and |1i ≡ |F = 2, mF = +1i of87Rb are the ideal choice for logical qubit states [14]. More information on the magnetic atom chip, cooling and trapping of atoms can be found in [12, 13].

With the chip design currently in use, each trapping site will serve as a mesoscopic ensemble qubit [12]. Precise control and manipulation of these ensemble qubits is a precondition for the implementation in the quantum information. The logical qubit states are defined by the whole ensemble, resulting in states _{|0i ≡ |0i}⊗N _and |1i ≡ |0i⊗(N−1)_{|1i, where N defines the size of the atomic ensemble [12]. This single collective hyperfine ground} state excitation can be actualised by making use of Rydberg atoms [11, 15]. As a consequence of this excitation towards high lying states, that is Rydberg states, long-range interactions between atoms are created. This dipole-dipole interaction between Rydberg atoms induces an energy shift in the atomic energy levels depending on the interatomic distance, meaning only one atom within the interaction region can be excited towards the Rydberg state [11, 16–18]. Excitation of neighbouring atoms can be controlled. Prohibition of a second excitation towards the same Rydberg state is called the Rydberg blockade mechanism. It is proposed to use this blockade mechanism in the design of quantum logic gates [19].

These quantum logic gates can in turn be used to perform simple Boolean functions to produce an logical output from logical input(s) in the same fashion as their classical counterparts. Now, the Rydberg blockade mechanism offers a way to control manipulations of qubit states. In order to comprise a full set of quantum gates, single-qubit gates and two-qubit gates need to be implemented. For example, single-qubit gates rotate the qubit states from_{|0i → |1i, or vice versa. Driving coherent transitions between these logical states will be} possible with two-photon Raman pulses [9, 13].

Extension of single-qubit to two-qubit quantum gates have thus been enabled by the Rydberg atom mediated blockade mechanism. Two-qubit operations based on neutral atoms require long-range atom-atom interactions which can be created by exciting rubidium atoms towards high lying Rydberg states [13]. The concomitant Rydberg blockade mechanism will be exploited to create controllable interactions which will form the building blocks for two-qubit or n-qubit quantum gates [19–21]. With 100 nm confinements and trap separations of 10µm, all atoms are within the intra- and inter-trap radius of the Rydberg blockade mechanism for sufficiently high-lying Rydberg states. Consequently, state dependent Rydberg excitations can be used as a control mechanism in the manipulation of the target qubit [22].

In the field of quantum information these controlling techniques for atomic interactions are essential, although these techniques are proposed theoretically, experimental implementation is limited. Most schemes for neutral atom gates rely on the Rydberg mediated dipole-dipole interactions. With the ability to address individual lattice sites and coherently manipulate the atoms in these sites, the foundation of a logic quantum gate on an atom chip is created [12]. The current chip design allows for resolvable single-site addressing of microtraps and interactions between atoms within and between microtraps can be created via Rydberg excitation at these separations [17], while at the same time providing optical resolvable trapping sites during detection.

3

In this thesis we will focus on the opportunity to create a platform for quantum information using qubit states in trapped 87_{Rb atoms and the manipulation of these states to structure quantum gates. Hence the title;} Towards qubit state manipulation of magnetically trapped 87Rb atoms. This thesis is divided in two subjects, namely Rydberg atoms in magnetic fields and Spatial Light Modulator (SLM) wavefront shaping for single-site addressing.

For Rydberg excitation on the atom chip, precise knowledge of a stable wavelength of the two light sources is essential. In this thesis a experimental setup for the calibration and stabilisation for both wavelengths is described. This system can be used as a direct method for the study of Rydberg87_{Rb atoms using excited state} Electromagnetically Induced Transparency (EIT) in a room temperature vapour cell. The hyperfine splitting of Rydberg states with n = 19_{− 28 is investigated. Additionally, the influence of magnetic fields on preferred} excitation pathways is examined both experimentally and theoretically. The experiment itself will be discussed in detail in chapter 2.

Secondly, gate operations call for single-site addressing of qubit states on an atom chip. Shaping the light field at high resolution is thus of great importance, especially with an inter-trap distance of 10µm. Arbitrary light patterns can be created by modulating the phase of the wavefront. A Liquid Crystal SLM is used to shape these light landscapes. Important first steps in optimising the performance of this modulator will be explained in the third part of this thesis.

Combined, the study of hyperfine splitting of Rydberg state atomic levels, the behaviour of Rydberg state excitation pathways in the presence of magnetic fields and the first steps towards shaping light fields for single-site addressing of a mesoscopic ensemble qubit on an atom chip are important progressions towards quantum information on an atom chip.

Chapter 2

Rydberg states in

87

Rb

Quantum gate sequences require excitation of atoms towards excited states with high principle quantum number n, i.e. it calls for excitation towards Rydberg states. In this Rydberg state the atom acquires exaggerated large transition dipole moments _{∝ n}2 _{and and electric polarisabilities}

∝ n7_{. This results in long-range interactions} and increased atom-atom interactions, respectively. This property will be used to evoke controllable interac-tions between particles separated by several micrometer. Rydberg atoms can be used in quantum information processing to control interactions between neighbouring trapping sites on an atom chip, hence they prove useful candidates in the development of fast quantum gates. As these Rydberg atoms are indispensable in the field of quantum information we explain the useful properties in more detail.

The work presented in this part involves the study of Rydberg states in rubidium atoms in a room temperature vapour cell. As a consequence of the interaction between the nuclear spin I and the total angular momentum J (resulting from the spin-orbit coupling) the fine structure levels split in hyperfine structure levels defined by F = J_{±I. The hyperfine structure of Rydberg s-states will be studied by making use of EIT. Excitation to high} lying Rydberg states can be reached using a two step excitation scheme. This ladder type excitation scheme allows for the use of the optical phenomenon EIT.

The experiments shown in this chapter are performed in the view of creating Rydberg excitation in rubidium atoms on an atom chip. Excellent knowledge of Rydberg state hyperfine splitting in the presence of electric and magnetic fields is required for nearing atom chip experiments. Here, we will focus on the influence of the magnetic field present in the experiment and the coherent Zeeman splitting. As a consequence of the Zeeman effect the spectral lines of the Rb spectrum will split. Knowledge of the splitting and shifting of these spectral lines for Rydberg states is crucial for Rydberg excitation on the atom chip. In section 2.3 we will present the study of87_{Rb Rydberg atoms in the presence of magnetic fields, and especially focus on the hyperfine splitting} in the presence of these fields.

Both experiments are an extension of previous performed experiments, these will be described in section 2.2.1. With the experimental setup described the hyperfine structure in 87_{Rb can be resolved for Rydberg s-states} 19_{≤ n ≤ 28. These measurements were performed in a shielded, coil-wrapped vapour cell. With this method} the magnetic field can be controlled with high precision.

Rydberg Atoms & Properties

The alkali Rb atom, with a single valence electron, excited towards a high lying electronic state with large principal quantum number n i.e. a Rydberg state, can be thought of as Hydrogen-like. It can be considered as a positively charged ionic core orbited by a single negatively charged electron. For Hydrogen the eigenstates and eigenenergies can be found by solving the time-independent Schr¨odinger equation, yielding the solution for the energy

E = I₋ 1

2n2 = I− Ry

n2 (2.1)

where the ionization energy is given by I and the Rydberg unit of energy Ry is given by hcR∞[23].

2.1.1 Theory 5

This description can be used to express the energy for Rydberg states with high n and high angular momentum l. For low l-states, such as the s-states studied, the valence electron penetrates the ionic core where it is no longer shielded from the nuclear charge by the core electrons. The energy states of these low l-state Rydberg atoms are shifted. Taking this into account a correction to the energy states returns

E = I−_nRy_∗2 (2.2)

where the principle quantum number n is corrected with the angular dependent quantum defect δlto give the effective quantum number n∗= n− δl[23].

Similar to the fine structure interval, the hyperfine splitting in Rydberg atoms depends on the presence of the valence electron at the nucleus. It scales with the probability of finding the electron at the positively charged ionic core. When the angular momentum of the Rydberg states is low compared to n this scales with n−3 [23]. The relation between the hyperfine splitting in Rydberg s-states and the principle quantum number n has previous been studied in [24].

2.1

Electromagnetically Induced Transparency

As Electromagnetically Induced Transparency will be used as a direct observation technique for Rydberg state characteristics, and future Rydberg state mediated dipole blockade mechanism on the atom chip, it will be explained in this section. Simply, EIT can be described as the cancellation of the linear response by destructive interference in a laser dressed medium [25]. The destructive interference in a laser dressed system leads to the elimination of absorption at resonant frequencies, while increasing the nonlinear susceptibility [25]. The absorption of a medium is thus diminished, creating a transparency window for the probe laser field [25–27].

2.1.1

Theory

The theory behind the phenomenon of EIT can be described in either bare atomic states or in dressed state picture. Both descriptions will explained below. In both cases the destructive interference between excitation pathways lead to a change in the optical response in the form of eliminated absorption.

The principle of EIT can be explained by means of three different three-levels schemes displayed in figure 2.1. In this figure three possible EIT schemes are depicted. Each can be described by the states _{|1i, |2i and |3i.} Transitions between|1i ↔ |3i and |3i ↔ |2i are allowed as they are dipole coupled. The transition between states |1i and |2i is dipole-forbidden, hence not allowed. The strong coupling laser with Rabi frequency Ωc= µ23E2/~ at frequency ω2and dipole transition moment µ23is used to couple the|3i → |2i states, so that the probe beam passes through the medium with eliminated absorption [27–29].

Dressed states

The underlying physics of the absorption elimination in EIT is closely related to dark state preparation and coherent population trapping. In the presence of the driving fields Ωpand Ωcthe Hamiltonian of the bare atom has to be modified to include the interaction with the driving field, so that it becomes H = H0+ Hint. Here H0 describes the bare atom and Hintthe interaction with the driving fields. The response of the system in the

6 Electromagnetically Induced Transparency

Chapter 1

Electromagnetically Induced Transparency

As Electromagnetically Induced Transparency will be used as a direct observation technique for Rydberg state characteristics, and future Rydberg state mediated dipole blockade mechanism on the atom chip, it will be explained in this section. Simply, EIT can be described as the cancellation of the linear response by destructive interference in a laser dressed medium [25]. The destructive interference in a laser dressed system leads to the elimination of absorption at resonant frequencies, while increasing the nonlinear susceptibility [25]. The absorption of a medium is thus diminished, creating a transparency window for the probe laser field [25–27].

1.1 Theory

The theory behind the phenomenon of EIT can be described in either bare atomic states or in dressed state picture. Both descriptions will explained below. In both cases the destructive interference between excitation pathways lead to a change in the optical response in the form of eliminated absorption.

The principle of EIT can be explained by means of three di↵erent three-levels schemes displayed in figure1.1. In this figure three possible EIT schemes are depicted. Each can be described by the states|1i, |2i and |3i. Transitions between|1i $ |3i and |3i $ |2i are allowed as they are dipole coupled. The transition between states |1i and |2i is dipole-forbidden, hence not allowed. The strong coupling laser with Rabi frequency ⌦c= µ23E2/~

at frequency !2and dipole transition moment µ23is used to couple the|3i ! |2i states, so that the probe beam

passes through the medium with eliminated absorption [28].

|1i |3i |2i |1i |3i |2i |3i |1i |2i

Figure 1.1: EIT three-level schemes

the states|1i, |2i and |3i are described by the energies given by E1, E2and E3. Classification of the

schemes is based on the relative energies, so that the ladder-type scheme (a) is described by E1< E3< E2, the lambda-type scheme is described by E1< E2< E3and the v-type scheme is

described by E3< E2& E3< E1[28].

6

Figure 2.1: EIT three-level schemes

the states _{|1i, |2i and |3i are described by the energies given by E}1, E2and E3. Classification of the schemes is based on the relative energies, so that the ladder-type scheme (a) is described by E1< E3< E2, the lambda-type scheme is described by E1< E2< E3and the v-type scheme is

described by E3< E2& E3< E1[28].

presence of a weak probe field such that Ωp Ωc and Ωp  Γ3, where Γ3gives the decay rate of state 3, can be expressed by the Hint eigenstates [25]:

|a+i = q Ωp 2(Ω2 p+ Ω2c) |1i + q Ωc 2(Ω2 p+ Ω2c) |2i + √1 2|3i (2.3) |a0i = q Ωc (Ω2 p+ Ω2c) |1i −q Ωp (Ω2 p+ Ω2c) |2i (2.4) |a−i = q Ωp 2(Ω2 p+ Ω2c) |1i + q Ωc 2(Ω2 p+ Ω2c) |2i − √1 2|3i (2.5)

where_|a0i remains at zero energy. It becomes a dark state as it has no contribution of the bare atomic state |3i, meaning if the atom is formed in this state there is no possible excitation and successive spontaneous emission for_{|3i. The states |a}+

i and |a−_{i are shifted up and down in energy by an amount:} ~ω±₌ ~

2(∆± q

∆2_{+ Ω}2

p+ Ω2c) (2.6)

After initially being on resonance with the bare state at zero energy, the probe laser is off-resonance with the new dressed states. The detuning is equal for both states, but opposite of sign. As Ωp  Ωc, this system can now be regarded as two-level sub-system of states_{|3i and |2i. The new dressed states can be described as}

|+i = √1

2(|2i + |3i) (2.7)

|−i = √1

2(|2i − |3i) (2.8)

Both excitation paths lead to the same end state and will thus interfere [27]. As they are of opposite sign they will interfere destructively resulting in a cancelation of the medium response much similar to Fano-interference between auto-ionization decay channels [25]. The medium becomes transparent for the probe beam.

Probability amplitudes

The concept of EIT can also be explained using transition probability amplitudes for the transitions. The transition between state|1i and |3i can take place via the direct route, this transition has a large probability

2.1.3 Our system 7

Figure 2.2: EIT spectrum

In the absence of the coupling field a typical absorption profile is shown in red. The probe field is scanned across a frequency range given by detuning -15− +15 MHz, normalized to the resonance frequency of the transition. In the presence of the coupling field, curve shown in blue, the absorption

is cancelled as a consequence of destructive interference. Figure adapted from [26].

amplitude. Likewise, the transition can follow the indirect path defined by _{|1i → |3i → |2i → |3i. This} indirect transition towards state _{|3i can only arise in the presence of a strong coupling field. The probability} amplitude for this transition is very small [28]. This path will have a negative sign relative to the direct path, resulting in an overall cancelation of the absorption of the probe beam. Again, this situation is much similar to Fano-interference between auto-ionisation decay channels [25, 28].

2.1.2

EIT spectra

A typical EIT spectrum of a ladder type transition is used to illustrate our system, which will be described more elaborate in the next section 2.1.3. In the absence of the coupling field, the probe field is scanned across a certain frequency range (from -15 MHz to +15 MHz, with 0 MHz given by the resonance frequency) to give a typical absorption profile. In the presence of the coupling field, the system is subjected to the EIT conditions and a diminished absorption is found. The spectra obtained in the presence and absence of the coupling field are shown in figure 2.2.

2.1.3

Our system

Throughout this work EIT will be created in a ladder-type system. This ladder type system is shown in figure 2.3. Here, the_|5s1/2i state of87Rb will be coupled to|5p3/2i by a weak probe field Ωp at a wavelength of 780 nm. The strong coupling field will be at a wavelength of 480 nm, to couple the |5p3/2i to a |ns1/2i Rydberg state with Ωc.

When both beams are on resonance the|ns1/2i dressed Rydberg state interferes with the dressed intermediate |5p3/2i state to cancel the absorption of the probe beam. That is the interference renders the medium transparent for the probe beam observable as an increased transmission. The width of the transparency window is determined

8 Electromagnetically Induced Transparency

1.3 Our system 9 |5s1/2i |5p3/2i |nsi | i |+i Dressed States Fano Interference ⌦c F0_{= 3} ⌦p F0_{= 2} F0= 1 F0= 0 F = 2 F = 1 F00= 2 87_Rb 780 nm 480 nm F00_{= 1}

Figure 1.3: Ladder-type EIT system

The ladder-type system used to invoke EIT is a system where the ground state 5s1/2(F = 2) is

weakly coupled to the excited state 5p3/2(F0= 2) by the probe laser field at 780 nm. The excited

state and Rydberg state ns1/2(F00= 1, 2) is coupled by a strong coupling field at 480 nm. The

involved energy states are depicted on the left, with hyperfine splitting depicted in the middle. On the right side the dressed state picture due to the presence of the strong coupling laser is illustrated.

Figure adapted from [26]

the linear response of the medium is described by the first-order susceptibility (!) = 0_{(!) i} 00_{(!). The}

real part 0_{(!) is related to the index of refraction while the imaginary part} 00_{(!) is related to the absorption}

or the dissipation of the field. The macroscopic polarization can be related to the microscopic polarization P = nµ13⇢13. Therewith, giving the following relation to describe the response of the medium

00_{/ ⇢} 13= i p i p+ ⌦ 2 c/4 c 1( p+ c) (1.7)

This is valid in the limit of low probe intensity and with negligible population in the intermediate and Rydberg state. p and c are the detuning of the probe and coupling field respectively, and p = 1/2 5p3/2 and

c= 1/2 ns1/2. With this the transmission spectrum as a function coupling laser frequency can be calculated. Figure 2.3: Ladder-type EIT system

The ladder-type system used to invoke EIT is a system where the ground state 5s1/2(F = 2) is weakly coupled to the excited state 5p3/2(F0 = 2) by the probe laser field at 780 nm. The excited

state and Rydberg state ns1/2(F00= 1, 2) is coupled by a strong coupling field at 480 nm. The involved energy states are depicted on the left, with hyperfine splitting depicted in the middle. On the right side the dressed state picture due to the presence of the strong coupling laser is illustrated.

Figure adapted from [26].

by the strength of the coupling field. By setting the probe beam to an optical resonance and sweeping the coupling beam across resonances in the Rydberg state (F00 _{= 1, 2), we can observe an increased transmission} of the probe beam when the coupling beam is on resonance with |5p3/2i → |ns1/2i. Opposed to the spectrum shown in figure 2.2, we will scan the coupling field instead of scanning the probe field. Only when both beams are on resonance an increased transmission of the probe beam is observed. Thus, by scanning the coupling field across_|ns1/2(F00= 1)i and |ns1/2(F00= 2)i, two transmission peaks in the probe beam spectrum are expected. Alike all atomic optical properties EIT can be described by interaction between medium and the light electric field described by the macroscopic polarisation of the medium P13= χ(1)E + χ(2)EE + χ(3)EEE + etc where the linear response of the medium is described by the first-order susceptibility χ(ω) = χ0_{(ω)− iχ}00_{(ω). The} real part χ0_{(ω) is related to the index of refraction while the imaginary part χ}00_{(ω) is related to the absorption} or the dissipation of the field. The macroscopic polarisation can be related to the microscopic polarisation P13= nµ13ρ13. Therewith, giving the following relation to describe the response of the medium

χ00∝ ρ13= i γp− i∆p+ Ω 2 c/4 γc−1(∆p+∆c) (2.9)

The optical response of the system can be described by the coherence ρ13, which can in turn be calculated by solving the Optical Bloch Equations [26]. Relation 2.9 is valid in the limit of low probe intensity and with negligible population in the intermediate and Rydberg state. ∆p and ∆c are the detuning of the probe and coupling field respectively, and γp = 1/2Γ5p3/2 and γc = 1/2Γns1/2. With this the transmission spectrum as a

2.2.2 Further investigation 9

However, in room-temperature vapour cells the Doppler shift will show a significant broadening in the trans-mission spectra. Although, the counter-propagating arrangement reduces the Doppler shift, it is not completely cancelled due to a difference in wavelength for the probe and coupling beams. The detuning ∆p and ∆c must be replaced for an effective detuning in which the Doppler shift is taken into account.

∆p,eff = ∆p− kpv ∆c,eff = ∆c+ kcv

Different velocity classes of the Maxwell-Boltzmann distribution contribute to the signal. Assuming the weak probe limit and taking the effective detuning and Maxwell-Boltzmann distribution into account the spectra can be fit to the following equation:

χ00∝ ∞ Z −∞ i γp− i∆p,eff+ Ω 2 c/4

γc−i(∆p,eff+∆c,eff)

· N(v)dv (2.10)

Here, N (v) is given by the Maxwell-Boltzmann distributionpm/2πkBT N0e−mv

2_/2k

BT _{with k}

B the Boltzmann constant, m the mass of87_{Rb. With this equation the hyperfine splitting for the Rydberg states can be extracted.}

2.2

Experimental setup

2.2.1

Previous experimental work

Before studying Rydberg atoms and the associated processes on an atom chip, Rydberg atom properties are investigated in an independent setup. It will be shown that the Rydberg state hyperfine splittings can be investigated by making use of a room temperature vapour cell containing rubidium atoms, where the ratio of the two isotopes is given by their natural abundances (85_{Rb(72%) and}87_{Rb(28%)) . The setup illustrated} throughout this chapter can be used to study the Rydberg state (n = 19_{− 28) hyperfine splitting in}87_{Rb with} high frequency resolution. Therewith, providing an optical setup to examine the influence of magnetic fields on the Rydberg state hyperfine splitting.

Previously, this EIT scheme has been used to study the hyperfine splitting in 87_{Rb Rydberg states. The} frequency of the coupling laser was scanned across the resonances F00 _{= 1, 2 in the Rydberg s-state. An} increased transmission of the probe beam was observed when the coupling field was on resonance with the |5p3/2(F0 = 2)i → |ns1/2(F = 1, 2)i transition. When the laser linewidth of the coupling field is sufficiently narrow both Rydberg hyperfine levels in the s-states can be observed in the transmission spectrum of the probe beam. Previously, this direct measurement of the hyperfine splitting of Rydberg states in 87_{Rb using} EIT spectroscopy in a room-temperature vapour cell was conducted. With this method Rydberg s-states with n = 20_{− 24 were investigated [24]. The spectra were measured with an accuracy of ∼ 100 kHz. The hyperfine} splittings were extracted and fitted to the scaling law given by Υn∗−3 _[24].

2.2.2

Further investigation

By locking both lasers for Rydberg excitation to a high finesse reference cavity the laser linewidths can be reduced to_{∼ 5 kHz. With this method the hyperfine splitting in Rydberg s-states up to n = 28 can be resolved.} The results, plotted in figure 2.4 show deviating behaviour with respect to theory. On the left side the hyperfine

10 Experimental setup

Figure 2.4: Hyperfine splitting for 87 _{Rb Rydberg s-states with n = 20} − 28. On the left the effective quantum number hyperfine splitting between F00_{= 1 and F}00_{= 2 is given in} MHz as a function of effective quantum number n∗_{. On the right these results are rescaled to give the}

hyperfine splitting hf s/(n∗₎3_{as a function of effective quantum number. Figure adapted from [31].}

splitting of the Rydberg s-states are plotted as a function of effective quantum number n∗_{= n−δ, with δ=3.131} [30]. These results are rescaled by (hfs· (n − δ)3_{) to give Υ, which is predicted to give a single value for different} n-states. A horizontal line is thus predicted. The new data is not in agreement with predicted theory.

The cause for this error can be the presence of an external fields, most probably the earth magnetic field. The influence of magnetic fields on the EIT spectra was further investigated [31]. When subjected to a magnetic field the normally degenerate magnetic sublevels split according to E = gFµBBmF. The ladder type EIT system can no longer be treated as a simple system, all 18 magnetic sublevels have to be taken into account.

Initially, the influence of a magnetic field was measured and the transition from Zeeman to Paschen-Back splitting of the spectral lines was observed for the 20 s state (depicted in figure 2.5). The spectrum was measured for various well-defined polarisation states of the coupling and probe beams for the 20s state. One of the resulting spectrum is shown in 2.5. At low magnetic fields a broadening for the spectral lines can be observed. Further increasing the field results in Zeeman splitting whereafter at hνhfs/µB = 5.5G the Paschen-Back regime is entered.

Both results required further investigation. Deviations from the scaling law are ascribed to a systematic error in the presence of external fields. Secondly, The influence of magnetic fields on the appearance of the EIT spectra is investigated. To obtain further insight in previous obtained results the room-temperature vapour cell is placed in a coil surrounded by Mu-metal therewith cancelling earth’s magnetic field. In this section the experimental setup will be explained in more detail.

2.2.3

Experimental setup

The experimental setup used for the investigation of Rydberg state hyperfine splitting consists of three parts. The spectroscopy part for EIT, the cavity locking for linewidth reduction and the part for linewidth measure-ments. With these three sections it is possible to obtain a high degree of control over the wavelengths of both lasers, as well as achieving narrow linewidths for measurements with high resolution. In the following sections the lasers for the two-photon Rydberg transition in87_{Rb will be discussed (section 2.2.3), followed by the} dis-cussion of the Pound-Drever-Hall method for linewidth reduction (section 2.2.3). The narrow linewidth coupling laser interrogates the frequency of the atomic spectral lines by scanning the laser over a frequency range. The

2.2.3 Experimental setup 11

Figure 20: Spectrum of the 20s Rydberg state as a function of magnetic field for linearly polarized light in both laser beams. The relevant field scale describing the transition from Zeeman to Paschen-Back splitting should be:(hνhf s)/µB≈ 5.55

Gauss.

27

Figure 2.5: 20s Rydberg state in a magnetic field

The EIT spectra for various magnetic fields are measured with probe and coupling fields in linear polarisation states. The relevant field scale describing the transition from Zeeman to Paschen-back

splitting should be (hνhf s)/µB≈ 5.55 Gauss [31]. Figure adapted from [31]

method for linewidth reduction while at the same time preserving the freedom to sweep the frequency of the coupling field will be illustrated. This is followed by a detailed explanation of the spectroscopy setup, where saturated absorption spectroscopy is used as a frequency reference of the probe beam and EIT is used to find the resonance in87_{Rb (section 2.2.4). The frequency of the coupling field is stabilized by making use of the EIT} signal.

Two-photon Rydberg excitation

87_{Rb atoms in Rydberg s-states will be created by making use of two-photon excitation. The use of this} two-photon excitation scheme accommodates the use of EIT as a probing mechanism, further this offers a practical simplicity for the required wavelengths [16]. Rydberg excitation will be performed with both light sources detuned from atomic resonances, whereas during experiments where EIT will be probed as a function of coupling field frequency both fields will be tuned to be on resonance with the atomic transitions (indicated in figure 2.3) The ladder type EIT configuration is used, in which the absorption from 5s_{− 5p is probed with the} 5p strongly coupled to a high lying Rydberg s-state of87_{Rb. The ground state|5s}

1/2(F = 2)i is coupled to the |5p3/2(F0 = 2)i excited state by the probe laser Ωp at 780 nm. The coupling laser Ωc, connecting the excited state|5p3/2(F0= 2)i to the Rydberg state |ns1/2(F00= 1, 2)i with n = 19 − 28, is at a wavelength of ∼ 480 nm. Both wavelength are obtained from commercial available light sources. A Toptica DL Pro is used as a source for 780 nm wavelength and a Toptica TA-SHG Pro1 _{is used as a source for 480 nm. The Toptica TA-SHG}

1_{Initially an other light source was present. After replacing this source with the source mentioned in the text an} additional telescope arrangement was introduced to match the beam waist of the new and previous light source. The maximum output power approximates 400 mW, and is set to ∼ 200 − 250 mW for each measurement. With this source the 19s1/2(F

00

12 Experimental setup

Figure 2.6: Laser source for 480 nm coupling light

960 nm laser diode in Littrow configuration, where the grating micrometer screw allows for coarse frequency tuning. The light is guided through optical isolators (OI) to prevent feedback to the diode, after which it is amplified (Tapered Amplifier TA). Subsequently the light is coupled into the bow-tie

cavity. Here, the second harmonic is created in the bow-tie arranged doubling stage.

comprises a 960 nm laser head, a high power semi-conductor tapered amplifier (TA) and a second harmonic generation cavity containing a KNbO3-crystal, see figure 2.6.

Both beams are in the Littrow-configuration, so that the first-order reflection stemming from the grating is reflected back into the diode thereupon giving optical feedback. The frequency is set by turning the grating. Then, the frequency can be controlled over a broad range by the pi¨ezo, whereas the current is used to modify small frequency changes.

Laser stabilisation via PDH technique

Both lasers are frequency stabilised by a frequency lock based on the Pound-Drever-Hall (PDH) technique. With this technique an error signal is created which can be used to make the laser resistant to frequency changes via PID feedback. The technique will be discussed briefly, a more elaborate explanation can be found in [31]. The crucial component for a stable frequency lock is the high finesse cavity placed under vacuum and which is temperature stabilised. The finesse of this cavity is determined to be_{∼ 29000 for 780 nm and ∼ 25000 for 960} nm, and the free spectral range equals∼ 1500 MHz. The setup for locking the laser to this reference cavity is shown in figure 2.7. Light stemming from both sources is passed through an EOM used for phase modulation, hereafter the beams are overlapped and mode-matched before entering the cavity. The light is modulated with a signal of 2 MHz, which is significantly larger than the approximate 50_{− 60 kHz linewidth of the cavity. The} modulating signal creates sidebands on the carrier frequency. When the carrier frequency is resonant with the

2.2.3 Experimental setup 13

Figure 2.7: Cavity locking

The electronics and optics for a stable frequency loch to one of the cavity modes is depicted. Light stemming from either the 780 or 960 nm source is phase modulated by an electro-optic modulator (EOM) to create sidebands (originating from the 2 MHz and Evaluation Boards (EB)) on the carrier

frequency. Light reflected off the cavity will interfere with these sidebands, whereafter the signal recorded by the photodiode (PD) is used to create an error signal by mixing this beating signal with

the 2 MHz source. This signal is led to the proportional-integral-derivative (PID) feedback for frequency stabilisation.

free spectral range of the cavity, the sidebands are off-resonance. The sideband will be reflected off the cavity and interfere with the incoming carrier frequency. The resulting beat signal is measured on a photodiode. In order to obtain an error signal for frequency locking the lasers via PID feedback the beat signal must be modified. The modulation signal is split, so that one part is led to the EOM for phase modulation whereas the other part is led to a mixer. This mixer receives the modulation signal and the beat signal. The error signal is subsequently created by subtracting the beat signal. This signal shows a steep linear slope around zero, which is ideally be used for PID feedback. More detailed information on laser stabilisation via the PDH technique can be found in [31].

Sideband locking

In the interest of maintaining freedom for frequency scanning while conserving the lock an additional sideband is imposed on the carrier frequency. The additional sideband can be created by frequency synthesisers in a range of 40_{− 4400 MHz by evaluation boards ADF4351 supplied Analog Devices [32]. The added sideband is,} together with the 2 MHz modulated signal, led to the reference cavity. Now, the reflected 2 MHz modulated signal will interfere with the incoming sideband frequency. The beating signal will produce an error signal in the same fashion as the error signal stemming from the 2 MHz sideband and carrier frequency. This error signal can in turn be fed back to the light source via PID feedback.

The sidebands generated by the evaluation boards can be varied in frequency, therewith varying the frequency at which the error signal occurs. The frequency spacing between carrier frequency and the sideband is well defined and can be varied. Locking the laser to this sideband offers advantage tuning of the carrier frequency.

14 Experimental setup

Figure 2.8: Self-heterodyne linewidth measurement

The linewidth of the 780 or 960 nm sources are measured by superimposing a delayed version of the output spectrum upon a frequency shifted (shifted by an acousto-optic modulator (AOM)) output

spectrum. Herewith a self convolution of the output spectrum is created for sufficient long delay times. The resulting beat note is detected on the fast photodiode.

By locking to the sideband and scanning the frequency of the modulating signal the carrier frequency is affected. This allows for the possibility of scanning the coupling light field at the same time reducing the laser linewidth through PID feedback. Unfortunately, by modulating the sideband frequency the error signal scans over a certain frequency where it can encounter other modes. This can lead to unlocking or locking to another mode.

Linewidth

Subsequently, it is important to investigate the laser linewidth as an indication for the locking performance. The linewidth is determined by employing a self heterodyne laser linewidth measurement setup, displayed in figure 2.8. The laser beam, this can either be 780 nm or 960 nm, is split in two paths. One is led through a 12.4 km long telecom fiber, by that introducing a delay of τ0∼ 60.7µs. The other path is guided through an AOM in double-pass configuration, which introduces a frequency shift of 160 MHz. The first order diffraction is selected and overlapped with the delayed beam at the beam splitter cube, so that the output spectrum becomes a self convolution of the laser. The resulting beat signal is examined by a fast photo diode and spectrum analyser. From the previous linewidth measurements and comparison to these spectra it can be concluded that linewidths are in the same range as previously determined. The coherence length of both sources exceeds the delay length of the telecommunication fiber. This is observed in the beat spectrum as a single sharp peak with fringes. From this we can conclude that the linewidth of both sources is < 1/τ0 = 16.5 kHz, where τ0 is the delay time introduced by the telecom fiber. More thorough investigation of the linewidths results in an approximate linewidth of 4.7 kHz for 480 nm source (this is two times the measured linewidth of 2.4 kHz at the wavelength of 960 nm) and 6.0 kHz for the 780 nm source [31].

2.2.4

Spectroscopy

In order to measure the EIT spectra, where the probe interrogates the atomic resonance lines in the 87_Rb Rydberg s-states a high degree of frequency control for both the probe and coupling field is required. The probe frequency is tuned to match the |5s1/2(F = 2)i → |5p3/2(F0 = 2)i transition. This is accomplished with the

2.2.4 Spectroscopy 15

Figure 2.9: EIT spectroscopy

EIT spectra are measured in room temperature vapour cell containing 87_{Rb atoms. The probe and} coupling field are arranged in a counter-propagating fashion to reduce Doppler shifts. The signal is detected by a photodiode, from where the signal is send to a lock-in amplifier. The reference signal

for lock-in amplification is given by chopper wheel modulated coupling field.

help of saturated absorption spectroscopy on87_{Rb in a vapour cell. Subsequently, the coupling field is tuned to} match the_|5p3/2(F0= 2)i → |ns1/2(F00= 1, 2)i transitions, where n = 19 − 28, the resonance frequency will be probed by the presence of an EIT signal.

Electromagnetically Induced Transparency

The EIT spectra of the hyperfine levels (F00_{= 1, 2) in}87_{Rb Rydberg atoms are measured in a room-temperature} vapour cell, depicted in figure 2.9. The frequency of the probe beam is set to the|5s1/2(F = 2)i → |5p/3/2(F0= 2)_{i transition by saturated absorption spectroscopy. For this, a separate vapour cell containing rubidium is used} where the transitions are saturated by an intense beam traversing the cell. After passing through the vapour cell, the beam is reflected traversing the cell a second time. Only atoms which fulfil to resonance condition for both beams, that is only stationary atoms (moving in the plane perpendicular to laser beam propagation), show a saturated absorption for the reflected beam. The Lamb dips caused by spectral hole burning visible in the spectrum are used as a frequency reference for the probe beam.

The frequency is set to the desired transition at approximately 780.24 nm, whereafter a sideband is created on the carrier frequency (at 255.5 MHz) with an ADF4351 evaluation board. This sideband generates the error signal at the desired frequency appropriate for frequency locking, in this way offering PID feedback for frequency stabilisation.

The 780 nm probe beam is, after passing the λ/4 and λ/2 waveplates, guided through the EIT rubidium vapour cell. After traversing the cell the probe intensity is collected on a photodiode. This is sent to a lock-in amplifier

16 Results

as the signal, whereas the coupling beam is collected as a reference signal. Lock-in amplifiers are frequently used as a method to increase signal-to-noise ratio. Before entering the vapour cell the coupling beam is passed through a chopper wheel and the two waveplates, namely λ/4 and λ/2 making it suitable as a reference for lock-in detection. The waveplates are used to set a defined polarisation state for both beams.

The probe beam is counter-propagated in the vapour cell by the coupling beam in order to reduce Doppler-broadening. The coupling field wavelength is adjusted to match the frequency difference between the interme-diate level and the Rydberg state. Coarse tuning of the wavelength is performed by using the micrometer screw and fine tuning is completed by adjusting the pi¨ezo and/or current. The wavelength of the 960 nm beam is monitored while the wavelength of the frequency doubled 480 nm beam is set by scanning the spectrum for the EIT signal. A sideband is created at the EIT signal frequency in a similar fashion as sideband generation for the probe beam. After a sideband is created on the carrier frequency at the desired EIT resonance frequency the coupling laser is locked for frequency stabilisation.

The control software can subsequently be used to control the scan, the scan range, step size and number of repeats for the coupling laser and be controlled. Initially the frequency spacing between the carrier and sideband frequency is set, whereafter scanning of the sideband frequency will result in a displacement of the carrier frequency. Whenever the coupling laser is resonant with a transition, either_|5p3/2(F0= 2)i → |ns1/2(F00= 1)i or_|5p3/2(F0= 2)i → |ns1/2(F00= 2)i an increase in probe transmission will be observed.

The beams are in a counter-propagation arrangement to reduce Doppler shifts present in the room-temperature vapour. When the coupling beam Ωc  ∆vDopp splitting of the energy levels in the intermediate state is maintained and interference will result in EIT. At room-temperature the atoms within the vapour will move in and out the radius of the overlapping beams at a rate of_{∼ 580 kHz causing transit broadening. This is valid} when the beam waist of the probe beam has a waist of 0.5 mm and the coupling laser waist (1.0 mm) is larger than the probe beam. In this way, all atoms interacting with the probe beam are subjected to dressing via the coupling beam. For each measurement the relation Ωc Ωp is satisfied.

Magnetic field control

One of the possible systematic errors present in previously conducted EIT measurements on the hyperfine splitting in 87_{Rb is the presence of external fields (as described in section 2.2.1). The earth magnetic field} present in the vapour cell is one of the possible candidates for this systematic error. By placing the vapour cell in a housing of high magnetic permeability Mu-metal external magnetic fields will be shielded.

Without this shielding the magnetic field along the axis of the vapour cell (z-axis) approximates 150 mG (measured ‘ outside’ the optical table, on the table the field is given by 20 mG). Shielding the vapour cell with a casing of Mu-metal (length = 175 mm and diameter = 100 mm) results in magnetic field in the centre of the vapour cell (length = 100 mm) of 2.5 mG.

Next, well controlled magnetic fields are created by placing a coil around the vapour cell. The current of the coil can be controlled precisely. A coil of N = 80 windings and length l = 11 cm the field dependency_{∼ 10} Gauss/Amp´ere. During measurements the coil will be subjected to -80− +80 mA, resulting in magnetic fields in the range of -800_{− +800 mG. Each subsequent measurement is varied by 200 mG.}

2.3.1 Rydberg states and hyperfine splitting 17

2.3

Results

2.3.1

Rydberg states and hyperfine splitting

With the setup described in the previous section Rydberg s-states with varying principal quantum number were investigated. The coupling laser is tuned to be resonant with n = 19− 28. The probe transmission spectra are measured and the hyperfine splitting between_|ns1/2(F00 = 1)i and |ns1/2(F00= 2)i is extracted, as previously described in section 2.2.1. From earlier results, it appears that the rescaled splitting between the hyperfine levels in various Rydberg state does not yield a single value [31]. Two possible explanations for this deviating behaviour can be found in the polarisation state used and the presence of magnetic fields during the experiments. The probe transmission spectra are re-measured, now with well-defined polarisation states for both beams. Three different polarisation configurations are defined. Two polarisations states are characterised as identical polarisation for both beams, this is either σ+_σ+ _{or σ}−_σ−_{. Likewise, the third polarisation state is defined} as opposite polarisations, namely σ−_σ+_{. Rydberg states in} 87_{Rb can be measured up to the 28s state with} the current setup. Excitation towards higher states results in unresolvable hyperfine splitting, as can be seen in both figures, shown in figures 2.12 and 2.13. Restricted accessibility for the lower s-states is giving by the tuning range of the coupling laser. For each measurement the magnetic field is controlled, so that the following spectrum for each state can be obtained.

- 0.6 - 0.4 - 0.2 0.2 0.4 0.6 Magnetic Fieldf(Gauss) 7.5 7.6 7.7 7.8 Hyperfine- Splitting(MHz) σ+ σ+ 20 s1/2 (a)

Figure 2.10: Hyperfine splitting as a function of applied field

The hyperfine splitting between_|20s1/2(F00= 1)i and |20s1/2(F002)i is measured for a well defined polarization state given by σ+_{, σ}+_{. The applied magnetic field is varied between -800}

− 800 mG. Figure obtained from [33]

In this picture a clear influence of the magnetic field on the hyperfine splitting is visible. A similar curve is measured for the other Rydberg states. It appears that the hyperfine splitting can not be described by the simple analytical model previously used [24], see section 2.3. Most probably, the system is not described by the relative simple three-level system (|gi : |5s1/2(F002)i, |ei : |5p3/2(F0 = 2)i and |ri : |ns1/2(F00 = 1 or 2)i). Instead all 18 sub-magnetic sub levels have to be taken into consideration, this leaves a system as shown in figure 2.11.

It is expected that all the magnetic sub levels play a role in a correct description of the system, where transitions are determined by the polarization state and the magnetic field applied to the system. The probe transmission

18 Results

(a)

Figure 2.11: 18 level system

In this figure all magnetic sub levels belonging to the levels indicated previously in figure 2.3 are indicated. Figure obtained from [33]

spectra are re-measured for all n-states with a well-defined polarization state. In figure 2.12 the transmission spectrum for the probe beam is shown for the principle quantum numbers n = 19− 28, in these figures the polarisation state is given by σ−_σ+ _{(other polarisation states, σ}+_σ+ _{and σ}−_σ−_{, are shown in figure 2.13}2_). Subsequently, the applied magnetic field is varied to study its influence on the EIT spectra.

2.3.2

Rydberg states in magnetic fields

All the previous spectra were measured in the shielded vapour cell with zero magnetic field, contrary to the previous results obtained by [31] as described in section 2.2.1. Varying the magnetic field between_{−800 − +800} mG in steps of 200 mG results in the spectra shown in Appendix A. As all spectra show similar results, only the spectra for the 19s, 22s and 26s states are shown in figures 2.14 - 2.16.

In these spectra a clear influence of the magnetic field can be seen. In the spectra for the higher Rydberg states with n = 27, 28 the splitting between F00_{= 1, 2 is almost unresolved, see Appendix A. For the Rydberg states} 19s− 26s general trends can be seen. In the next sections these general trends for each well defined polarisation state will be illustrated in more detail.

σ

+

σ

+

polarisation

All the spectra measured for the configuration indicated as σ+_σ+ _{shows two resolved hyperfine states for the} Rydberg states up to 26s. The hyperfine splitting for the states 27s and 28s are unresolved. For the lower lying s-states the signal at the lower frequency side of the spectrum is dominant for the states 19s-24s at any applied magnetic field. Although the left peak remains dominant in the spectrum, the transmission is clearly influenced by the magnetic field. Applying a magnetic field from -800_{− +800 mG results in a increased transmission for} the left (lower frequency) peak.

2_{The EIT spectrum for the n = 25 state shows a zero signal outside the frequency range of −5 − +5 MHz as the} frequency lock for the coupling laser field is lost. The error signal used for frequency locking the coupling field shows a low amplitude, thereby the lock is easily lost when coming across higher order cavity modes of comparable amplitude.

2.3.2 Rydberg states in magnetic fields 19

3.2 Rydberg states in magnetic fields 19

0

-20

-10

0

10

20

E

IT

si

gn

al

in

ar

b

.

u

n

it

s

Frequency in MHz

Rydberg state hyperfine splitting for

,

+

19s

20s

21s

22s

(a)

0

-20

-10

0

10

20

E

IT

si

gn

al

in

ar

b

.

u

n

it

s

Frequency in MHz

Rydberg state hyperfine splitting for

,

+

23s

24s

25s

(b)

0

-20

-10

0

10

20

E

IT

si

gn

al

in

ar

b

.

u

n

it

s

Frequency in MHz

Rydberg state hyperfine splitting for

,

+

26s

27s

28s

(c)

Figure 3.1: Hyperfine splitting of 87Rb Rydberg atoms

The EIT spectra showing the hyperfine splitting between ns1/2(F00= 1) and ns1/2(F00 = 2) is

measured for Rydberg s-states with principle quantum number (a) n = 19 22, (b) n = 23 25 and (c) n = 26 28. The polarisation states of the probe and coupling lasers are defined as +_.

+ + +

⌫

hf s

(MHz)

95% confidence

⌫

hf s

(MHz)

95% confidence

⌫

hf s

(MHz)

95% confidence

19s

20s

21s

22s

23s

24s

25s

26s

27s

28s

Table 3.1:

3.2

Rydberg states in magnetic fields

All the previous spectra were measured in the shielded vapour cell with zero magnetic field, in contrary to the previous results obtained by [29] as described in section2.1. Varying the magnetic field between 800 +800

(a)

3.2 Rydberg states in magnetic fields 19

0

-20

-10

0

10

20

E

IT

si

gn

al

in

ar

b

.

u

n

it

s

Frequency in MHz

Rydberg state hyperfine splitting for

,

+

19s

20s

21s

22s

(a)

0

-20

-10

0

10

20

E

IT

si

gn

al

in

ar

b

.

u

n

it

s

Frequency in MHz

Rydberg state hyperfine splitting for

,

+

23s

24s

25s

(b)

0

-20

-10

0

10

20

E

IT

si

gn

al

in

ar

b

.

u

n

it

s

Frequency in MHz

Rydberg state hyperfine splitting for

,

+

26s

27s

28s

(c)

Figure 3.1: Hyperfine splitting of 87_{Rb Rydberg atoms}

The EIT spectra showing the hyperfine splitting between ns1/2(F00 = 1) and ns1/2(F00 = 2) is

measured for Rydberg s-states with principle quantum number (a) n = 19 22, (b) n = 23 25 and (c) n = 26 28. The polarisation states of the probe and coupling lasers are defined as +_.

+ + +

⌫

hf s

(MHz)

95% confidence

⌫

hf s

(MHz)

95% confidence

⌫

hf s

(MHz)

95% confidence

19s

20s

21s

22s

23s

24s

25s

26s

27s

28s

Table 3.1:

3.2

Rydberg states in magnetic fields

All the previous spectra were measured in the shielded vapour cell with zero magnetic field, in contrary to the previous results obtained by [29] as described in section2.1. Varying the magnetic field between 800 +800

(b)

3.2 Rydberg states in magnetic fields 19

0

-20

-10

0

10

20

E

IT

si

gn

al

in

ar

b

.

u

n

it

s

Frequency in MHz

Rydberg state hyperfine splitting for

,

+

19s

20s

21s

22s

(a)

0

-20

-10

0

10

20

E

IT

si

gn

al

in

ar

b

.

u

n

it

s

Frequency in MHz

Rydberg state hyperfine splitting for

,

+

23s

24s

25s

(b)

0

-20

-10

0

10

20

E

IT

si

gn

al

in

ar

b

.

u

n

it

s

Frequency in MHz

Rydberg state hyperfine splitting for

,

+

26s

27s

28s

(c)

Figure 3.1: Hyperfine splitting of87Rb Rydberg atoms

The EIT spectra showing the hyperfine splitting between ns1/2(F00= 1) and ns1/2(F00= 2) is

measured for Rydberg s-states with principle quantum number (a) n = 19 22, (b) n = 23 25 and (c) n = 26 28. The polarisation states of the probe and coupling lasers are defined as +_.

+ + +

⌫

hf s

(MHz)

95% confidence

⌫

hf s

(MHz)

95% confidence

⌫

hf s

(MHz)

95% confidence

19s

20s

21s

22s

23s

24s

25s

26s

27s

28s

Table 3.1:

3.2

Rydberg states in magnetic fields

All the previous spectra were measured in the shielded vapour cell with zero magnetic field, in contrary to the previous results obtained by [29] as described in section2.1. Varying the magnetic field between 800 +800

(c)

Figure 2.12: Hyperfine splitting of 87_{Rb Rydberg atoms}

The EIT spectra showing the hyperfine splitting between ns1/2(F00= 1) and ns1/2(F00= 2) is measured for Rydberg s-states with principle quantum number (a) n = 19− 22, (b) n = 23 − 25 and

(c) n = 26_{− 28. The polarisation states of the probe and coupling lasers are defined as σ}−_σ+_{. The} magnetic field in the centre of the vapour cell is defined as 2.5 mG.

σ

−

σ

+

polarisation

Comparable spectra for the polarisation state σ+_σ+_{and σ}−_σ+_{are found for the Rydberg states with n = 19} −25. For spectra 21s_{− 25s measured with σ}+_σ+ _{polarisation there is a clear imbalance in signal strength for both} peaks. Instead, the transmission spectra for the 21s-25s Rydberg states measured with σ−_σ+_{polarisation show} approximate equal signals at zero magnetic field. Applying a negative field shows increased transmission for the peak at the higher frequency side of the spectrum. Application of a positive magnetic field results in opposite behaviour, the lower frequency side of the spectrum shows an increased signal strength. Already for the 23s Rydberg state this behaviour results in different dominant transmission peaks at alternating sign of applied magnetic field. Again, much similar to the transmission spectrum with σ+σ+ polarisation the Rydberg state with n = 26 shows an almost unresolved hyperfine splitting, see figure 2.16.