ABSTRACT INVESTIGATION of ELECTROMAGNETICALLY INDUCED TRANSPARENCY and ABSORPTION in WARM Rb VAPOR by APPLICATION of a MAGNETIC

ABSTRACT

INVESTIGATION OF ELECTROMAGNETICALLY INDUCED TRANSPARENCY AND ABSORPTION IN WARM Rb VAPOR BY APPLICATION OF A MAGNETIC FIELD AND CO-PROPAGATING SINGLE LINEARLY POLARIZED LIGHT BEAM

by Jason Barkeloo

We have observed electromagnetically induced transparency (EIT) and electromagnetically induced absorption (EIA) in room temperature rubidium vapor, by coherent population trapping on the Zeeman substates formed by a magnetic ﬁeld co-linear with a laser beam passing through the vapor. We have observed with good signal-to-noise ratio EIT on the F = 2 → F 0 transitions in 85Rb and on the F = 1 → F 0 transitions in 87Rb. EIA has been observed on the F = 3 → F 0 transitions in 85Rb and on the F = 2 → F 0 transitions in 87Rb. INVESTIGATION OF ELECTROMAGNETICALLY INDUCED TRANSPARENCY AND ABSORPTION IN WARM Rb VAPOR BY APPLICATION OF A MAGNETIC FIELD AND CO-PROPAGATING SINGLE LINEARLY POLARIZED LIGHT BEAM

A Thesis

Submitted to the Faculty of Miami University in partial fulﬁllment of the requirements for the degree of Master of Science Department of Physics by Jason Barkeloo Miami University Oxford, Ohio 2012

Advisor: (Dr. Samir Bali)

Reader: (Dr. Perry Rice)

Reader: (Dr. James Clemens) Contents

1 Introduction1 1.1 Motivation and Background ...... 1 1.1.1 What are EIT and EIA? ...... 1 1.1.2 Why is EIT important? ...... 3 1.1.3 Methods of EIT and Their Applications ...... 3 1.2 Necessary Conditions for EIT/EIA ...... 4

2 Basic EIT Theory7 2.1 Theoretical Background ...... 7 2.1.1 The Zeeman Eﬀect in Our System ...... 8 2.1.2 The Dark State in a Three Level System ...... 9

3 Experimental Setup and Procedure 15 3.1 External Cavity Diode Laser System ...... 15 3.1.1 Construction of the ECDL ...... 15 3.1.2 Collimation ...... 21 3.1.3 Feedback ...... 22 3.1.4 Threshold ...... 23 3.1.5 Resonance ...... 24 3.2 Saturated Absorption Spectroscopy ...... 25 3.3 Optical Setup ...... 27 3.4 Magnetic Field Cancellation and Control ...... 30 3.4.1 Helmholtz Coils ...... 30 3.4.2 Magnetic Field Ramp Control ...... 34 3.5 The Atomic System ...... 35

ii 4 Results and Discussion 39 4.1 EIT on the 85Rb F = 2 → F 0 = 1, 2, 3 transition system ...... 40 4.2 EIA on the 85Rb F = 3 → F 0 = 2, 3, 4 transition system ...... 42 4.3 EIT on the 87Rb F = 1 → F 0 = 0, 1, 2 transition system ...... 45 4.4 EIA on the 87Rb F = 2 → F 0 = 1, 2, 3 transition system ...... 46

5 Conclusions and Future Outlook 49 5.1 Conclusion ...... 49 5.2 Future Outlook ...... 49

A Anamorphic Prism Pair Data 51

B Magnetic Field Uniformity 53

C Coil Driver Schematic 55

Bibliography 60

iii List of Tables

A.1 Anamorphic prism pair geometry setting values for a range of magniﬁcations.

Angles (α1 and α2) are in degrees and displacement lengths (d and e) are in mm...... 52

iv List of Figures

1.1 Basic 3-level atomic transition diagram for a) EIT and b) EIA...... 2

1.2 Traditional basic EIT level structure. The frequency of one laser beams ωL1

is held constand while a secondary copropagating beam with frequency ωL2 is frequency scanned through resonance. The EIT feature occurs when the secondary beam is on resonance with the atomic vapor...... 4 1.3 Zeeman EIT begins with a degenerate ground state. When a magnetic field is applied the magnetic substates split apart from each other causing the necessary level structure for EIT. A scaning magnetic field scans the resonant frequency for both degenerate ground states differently. This scan together with the fact there is only one laser beam means that EIT features should be seen only when the magnetic field is zero and the laser is able to resonantly excite both levels at the same time...... 5 1.4 Possible atomic transitions with σ+ or σ− laser light are shown. Bold lines represent transitions with F > F 0, solid lines with F = F 0, and dashed lines 0 with F < F . The probability distributions of the D2 hyperfine transitions that we use are shown in (a). The relative size of these lines is representative of the oscillator strength of that transition and the number next to the line is the F 0 state. EIT and EIA schemes are established with the highest probability hyperfine transitions with F > F 0 (b), F = F 0 (c), and F < F 0 (d) are shown. Shown in (e) is the F = 1 → F 0 = 0 transition system. Adapted from Dancheva et. al.[7]...... 6

2.1 Applying a magnetic ﬁeld creates Zeeman energy levels suitable for EIT using a single linearly polarized laser beam, which can be thought of as a superposition of σ+ and σ− light beams...... 8

v 3.1 The physical ECDL system with labeled components. The beam paths are shown in red and blue. Red correspond to the beam coming out out of the diode and the experimental beam. The light blue corresponds to the beam reflected from the diffraction grating that goes to the saturated absorption setup...... 16 3.2 The basic design of the ECDL system. Two orders of the diffraction grating reflection are shown and labeled by their order (0 and 1). The red beam is the beam incident upon the diffraction grating while the green beam is the -1 order copropagating beam...... 17 3.3 Shown is the position of the PZTs mounted behind the grating. The fine adjustment screws are also labeled. The third screw should not be adjusted. 19 3.4 A view of the laser system from behind the laser diode. Shown are the pertinant components (thermistor and TEC) for temperature control of the laser diode. The diffraction grating and mirror are shown as well...... 20 3.5 Shown is a sample saturated absorption setup used in all of our atomic physics experiments. The relevent beam lines are shown. The redlines are the output beam of laser and the strong beam going through the cell. The yellow lines are the weak beams reflected off of the thick glass plate and into photodiodes. Note that the strong red beam overlabs and is counter propagating to one of the weak yellow beams...... 25 3.6 Without saturated absorption spectroscopy the transmitted profile of the laser is Doppler broadened as shown in 3.6a. The transitions being shown are the F = 3 to F 0 in 85Rb and F = 2 to F 0 in 87Rb. Note that no hyperfine features are discernible due to this Doppler broadening that is caused by the motion of the atoms in the vapor cell. 3.6b shows the Doppler free ”holes”. These ”holes” are a result of the atomic vapor being saturated by a counter propagating strong beam and not interacting with the weak probe beam. 3.6c shows the Doppler free hyperfine spectrum that is the result of subtracting the two weak probe signals...... 26 3.7 Our setup to measure EIT and EIA. The beam is shaped and transmitted through the vapor cell and the transmittance is measured using a photodiode. 27 3.8 The geometry of an anamorphic prism pair as shown in the CVI Melles Griot catalog [12]...... 28

vi 3.9 To cancel the earth’s magnetic field we use the helmholtz coil arrangement shown here. Each coil has independent current control to create a static magnetic field that is the same magnitude and oppostie direction of earth’s field...... 31 3.10 Theoretical plots of magnetic field uniformity due to a Helmholtz coil. . . . . 32 3.11 Creating a longitudinal magnetic field: The rubidium vapor cell is wrapped with copper wire in the shape of a solenoid. This copper coil has 121 turns over the length of the vapor cell. This is located inside the Helmholtz coils and can be seen Fig. 3.9...... 34 3.12 Saturated absorption spectra shapes for each set of hyperfine transitions from

the upper ground states of the D2 lines. These result in EIT features. The horizontal (frequency) scales of the graphs are not the same...... 35 3.13 Saturated absorption spectra for each set of hyperﬁne transitions from the

lower ground states of the Rb D2 lines. These result in EIA features. The horizontal (frequency) scales of the graphs are not the same...... 36 85 3.14 Rb D2 line data. Level spacings are included for all transitions. EIT is seen on the transition from F = 2 to F 0 and EIA on F = 3 to F 0. Included in [14] 37 87 3.15 Rb D2 line data. Level spacings are included for all transitions. EIT is seen on the transition from F = 1 to F 0 and EIA on F = 2 to F 0. Included in [15] 38

4.1 The relative oscillator strengths for the hyperfine transition systems used in our experiment. Adapted from [7]. The lines follow the same rules as before in Fig. 1.4...... 40 4.2 Sample 85Rb EIT spectra taken at low intensities(0.5mW/cm2), low transverse magnetic field (0G), high intensities (1.5mW/cm2), and high transverse magnetic field (1G). A and δ are studied as a function of transverse magnetic field and intensity in Fig. 4.3...... 41 4.3 85Rb EIT feature height and width as functions of intensity and transverse magnetic field. The width is compared to the natural linewidth of 6MHz. Representative error bars are shown on one point...... 42 4.4 85Rb EIA spectra taken at low intensities( 0.5mW/cm2), low transverse magnetic field (0G), high intensities (1.5mW/cm2), and high transverse magnetic field (1G)...... 43

vii 4.5 85Rb EIA feature height and width as functions of intensity and transverse magnetic field. The width is compared to the natural linewidth of 6MHz. Representative error bars are shown on one point...... 44 4.6 87Rb EIT spectra taken at low intensities( 0.5mW/cm2), low transverse magnetic field (0G), high intensities (1.5mW/cm2), and high transverse magnetic field (1G)...... 45 4.7 87Rb EIT feature height and width as functions of intensity and transverse magnetic field. The width is compared to the natural linewidth of 6MHz. Representative error bars are shown on one point...... 46 4.8 87Rb EIA spectra taken at low intensities( 0.5mW/cm2), low transverse magnetic field (0G), high intensities (1.5mW/cm2), and high transverse magnetic field (1G)...... 47 4.9 85Rb EIT feature height and width as functions of intensity and transverse magnetic field. The width is compared to the natural linewidth of 6MHz. Representative error bars are shown on one point...... 48

5.1 Evidence for sign reversal using 85Rb. The probe beam intensity is 214µW/cm2. The coupling beam powers equal to a) 0µW/cm2, b) 224µW/cm2, c) 319µW/cm2, d) 553µW/cm2...... 50

A.1 Graphs containing information (as seen in Table A.1) on the seperation and angles necessary for given magniﬁcation...... 51

B.1 Magnetic field over a range of x positions blown up to show detail...... 53 B.2 Magnetic field over a range of y positions, blown up to show detail...... 54 B.3 Magnetic field over a range of z positions...... 54

viii Dedication

For Humphrey

ix Acknowledgments

I would gratefully acknowledge the invaluable assistance of Iris Zhang (MS 2007), Peter Har- nish (BS 2009), and Eric Williams (BS 2009) during the initial stages of this experiment.

In addition I would like to thank the contributions of John Camenisch, Amanda Day, and Brad Worth.

x Chapter 1

Introduction

1.1 Motivation and Background

Electromagnetically induced transparency(EIT) and electromagnetically induced absorption (EIA) are commonly known phenomena in the optical physics community. Traditionally these effects are seen in the combination of two co-propagating resonant laser beams interacting with an atomic vapor sample. The effect is characterized by the complete laser transmission through a medium for EIT or increased absorption for EIA, in both cases due to a quantum interference effect. EIT was first experimentally observed over two decades ago in 1991 by the Harris group at Stanford University [1]. EIA was first seen by the Akulshin group at the Instituto de F´ısica,Facultad de Ingenier´ıa[2].

1.1.1 What are EIT and EIA?

Interference experiments are routinely conducted to study a wide range of physical systems. These include the well known such as Young’s double slit experiment or Michelson interferometry, to the newer uses of interferometry in superconducting quantum interference devices or SQUIDs [3]. We use the concept of interferometry in the context of atomic transitions. EIT uses the concept of interference for two atomic transitions from two closely spaced ground states |g1i and |g2i. These states are excited by two resonant laser beams of frequency

1 ωL1 and ωL2 respectively to a common excited state |ei as seen in Fig. 1.1a.

Figure 1.1: Basic 3-level atomic transition diagram for a) EIT and b) EIA.

When the probability amplitude for the |g1i → |ei transition destructively interferes with the probability amplitude for the |g2i → |ei transition the atom is in a ‘dark state’. This dark state is a quantum superposition of both ground states which does not allow the atom to absorb a photon. This causes the atom to become completely transparent to both resonant laser beams L1 and L2. This method has been discussed in a recent review article [4]. These ideas can be applied to the phenomena of EIA as well. Consider the atomic level structure shown in Fig. 1.1b where we have two closely spaced excited states being coherently driven by L1 and L2 to decay to the same ground state. In this case the atom can be trapped in a superposition of the two excited states where it can no longer be stimulated by either L1 or L2 to emit a photon. However, spontaneous emission allows an alternative path for the atom to eventually decay to the ground state. The stimulated emission is redistributed as spontaneous emission, decreasing the intensity of the transmitted light. The ﬁrst experimental observation of EIA can be found in [2].

2 1.1.2 Why is EIT important?

Being able to coherently control the transmission through a sample allows for the pos- sibility of making an ultrafast (GHz) all-optical switch which could be used to modulate a material’s response to either ωL1 or ωL2 in Fig. 1.1a. Since EIT provides active control of the medium’s response to a beam it oﬀers a unique means for coherent control of photons. This control oﬀers great promise for many areas in information science including quantum computing as well as telecommunications [4], [5].

1.1.3 Methods of EIT and Their Applications

Traditional EIT

The traditional setup of EIT experiments involves using two lasers as denoted in Fig. 1.1 that correspond to two resonant frequencies of the corresponding ground states. To accomplish this we must 1) ﬁnd an atom with a suitable level structure, i.e., two close-lying states that have dipole-allowed transitions to a common excited state and 2) our laser beams must be phase locked and precisely collinear[6]. The experimental approach to see an EIT spectrum can be seen in Fig. 1.2. It should be noted that only one laser frequency is being scanned using this experimental setup for EIT.

Zeeman tuned method for EIT

We have implemented a different method to look at EIT features which offers a huge simplification in the experimental methodology over traditional EIT setups. We find the close lying ground states not by looking for an atom that naturally has this level structure but by creating Zeeman substates using an external magnetic field. Transitions between these Zeeman substates are excited by σ+ and σ− photons. We use the fact that we can change the detuning of these levels with the external magnetic field to take care of the first requirement mentioned above for EIT, as shown in Fig. 1.3.

3 Figure 1.2: Traditional basic EIT level structure. The frequency of one laser beams ωL1 is held constand while a secondary copropagating beam with frequency ωL2 is frequency scanned through resonance. The EIT feature occurs when the secondary beam is on resonance with the atomic vapor.

Furthermore, to satisfy the second requirement of being phase locked and collinear we use a single linearly polarized beam and align the external magnetic field along the direction of propagation of the laser beam. By doing this we can view this single linearly polarized laser beam as a superposition of right and left circularly polarized beams. We have now automatically satisfied the second condition for EIT since the beam is obviously both phase locked and collinear with itself. Zeeman EIT offers much experimental simplification over the traditional EIT method. However, since the magnetic field is being scanned both ground states are being tuned away from the zero field position. If we were to think of this as a system with two seperate laser beams interacting with the ground state an accurate theoretical model would have to include simultaneously scanning both laser beams. This makes creating a theoretical model of the system much more difficult and as we shall see the data becomes harder to interpret.

1.2 Necessary Conditions for EIT/EIA

The essential conditions needed for EIT are to have an energy level system with F → F 0 = F,F − 1. For EIA the necessary transition is F → F 0 = F + 1 and this transition

4 Figure 1.3: Zeeman EIT begins with a degenerate ground state. When a magnetic field is applied the magnetic substates split apart from each other causing the necessary level structure for EIT. A scaning magnetic field scans the resonant frequency for both degenerate ground states differently. This scan together with the fact there is only one laser beam means that EIT features should be seen only when the magnetic field is zero and the laser is able to resonantly excite both levels at the same time. must be closed, with no possible transitions to other ground states. Spontaneous emission destroys the coherence driving EIA more so if the transition is open. After we have Zeeman detuned our system the actual atomic level structure must be so that it mirrors the structure of Fig. 1.1a) for EIT and b) for EIA. All possible level schemes, based on probability of making transitions, are shown in Fig. 1.4. Any of these schemes can be simplified to the EIT or EIA like system as shown below. The bold and solid lines in Fig. 1.4 represent EIT like systems and dashed lines represent EIA like systems. When we are looking at EIT or EIA we set our frequency to one of the hyperfine groups of transitions as seen in Fig. 1.4. Our vapor cell system is at room temperature so any beam it sees will be Doppler broadened due to the moving atoms. This leads to all of the hyperfine transitions in that group being excited at the same time in the vapor cell. The spectral feature seen will be a combination of the three transitions. The strongest transition is the one that ultimately prevails. Thus, for example, in the hyperfine transitions system F = 3 → F 0 = 2, 3, 4 for 85Rb we expect a small amount of EIT on the weaker transitions

5 F = 3 → F 0 = 2, 3 and a strong EIA signal on the strongest transition F = 3 → F 0 = 4, yielding overall an EIA signal. Similar reasoning leads us to predict overall EIA signal on the F = 2 → F 0 = 1, 2, 3 hyperfine system for 87Rb, and an overall EIT signal on the F = 2 → F 0 = 1, 2, 3 hyperfine transitions of 85Rb. For the F = 1 → F 0 = 0, 1, 2 hyperfine transitions in 87Rb we see that the EIA effect on the F = 1 → F 0 = 2 transition cancels out the EIT on the F = 1 → F 0 = 1 transition, leading us to predict an overall EIT effect due to the leftover F = 1 → F 0 = 0 transition.

Figure 1.4: Possible atomic transitions with σ+ or σ− laser light are shown. Bold lines represent transitions with F > F 0, solid lines with F = F 0, and dashed lines with F < F 0. The probability distributions of the D2 hyperﬁne transitions that we use are shown in (a). The relative size of these lines is representative of the oscillator strength of that transition and the number next to the line is the F 0 state. EIT and EIA schemes are established with the highest probability hyperﬁne transitions with F > F 0 (b), F = F 0 (c), and F < F 0 (d) are shown. Shown in (e) is the F = 1 → F 0 = 0 transition system. Adapted from Dancheva et. al.[7].

6 Chapter 2

Basic EIT Theory

In this chapter I present the basic three level theory for EIT and EIA in the context of our experimental methodology.

2.1 Theoretical Background

The conditions for EIT mentioned in Sec.1.1.3 must be included in any model we wish to create to describe and derive this quantum dark state effect. We will start with a level structure similar to Fig. 1.1a. To fulfill the conditions of [6] we propose an atomic system with the minimum characteristics of Fig.2.1 with ground state total electron angular momentum J = 1 and excited state total angular momentum J = 0. This total electron angular momentum, J is the combination of both the orbital electron angular momentum,L, and electron spin, S, where J = L + S. The total angular momentum of the atom is F where F = J + I with I being the nuclear spin. For our theoretical model we will assume I = 0 for simplicity. Once these conditions are fulfilled and we apply a linearly polarized resonant light beam onto our atomic system we expect to see a sharp peak in the transmission spectrum. This peak should occur when our single beam is resonant with both ground state transitions. The only point at which the beam is on resonance with both transitions is when the splitting between the levels is zero which occurs at B = 0. This leads to an interesting application of this method of EIT as an extremely sensitive magnetometer[8].

7 Figure 2.1: Applying a magnetic ﬁeld creates Zeeman energy levels suitable for EIT using a single linearly polarized laser beam, which can be thought of as a superposition of σ+ and σ− light beams.

2.1.1 The Zeeman Eﬀect in Our System

Taking advantage of the Zeeman effect, the magnetic substates m = −1 and m = +1 split away from the m = 0 substate in energy by a known amount, U = −µ · B, with an applied external magnetic field where µ is the magnetic moment and B is the applied magnetic field. It should be noted that this energy level splitting only occurs along axis of quantization which is in the direction of the applied B-field. This obviously leads to a system with 3 ground state sublevels and a single excited state energy level. To further understand how we are using a three level model for our system we must look at the excitations possible with a known polarization of resonant light incident on the transitions. In Fig. 2.1 it should be noted that even though we are exciting these transitions with a linearly polarized beam the m = 0 substate is not excited. This is because the magnetic field creates our axis of quantization (alongz ˆ) and the magnetic field term of our Hamiltonian has a Jz term in it. This is not excited because our linearly polarized beam is polarized alongx ˆ which leads to a

Jx interaction with the light ﬁeld. Because we deﬁne our axis of quantization we are allowed to know Jz however we cannot also know Jx by basic quantum mechanical rules. This Jx

term then must consist of operators J+ and J− for the total electron angular momentum.

These operators (J+ and J−) are the operators that allow the atom to be driven to the states requiring a circularly polarized light fieldss. If our beam was polarized along the same direction of our magnetic field (ˆz) we would not see these EIT or EIA effects because both

8 of our interactions would include only Jz terms. The basic level structure for EIT in our scheme can be seen in Fig. 2.1. Due electric dipole transition rules light with σ+ polarization is only capable of taking the atom from the ground state |g1i, i.e. m = −1 state, to the excited state |ei. Also, σ− polarized light can only excite |g2i, i.e., m = +1 state to the excited state |ei. The third ground state, m = 0, can only be excited by π or linearly polarized light. These rules only apply to the absorption of photons however all 3 ground states are open for the atom to fall down to and emit the corresponding photon. These energy levels are the eigenstates of the atomic Hamiltonian and the magnetic ﬁeld. It is necessary to look at the Hamiltonian instead for the entire system of the atom and magnetic ﬁeld but also including the light atom interaction. The eigenstates for this Hamiltonian lead to the EIT and EIA phenomena.

2.1.2 The Dark State in a Three Level System

Starting with the total hamiltonian

ˆ H = HA + Hµ·B + H~ν + Hd·E, (2.1)

where HA is the bare atom Hamiltonian, Hµ·B is the atomic interaction with the magnetic

field, H~ν is the light shift caused by the laser field and Hd·E is the light atom interaction piece of the Hamiltonian. Hµ·B restults in the Zeeman level splitting mentioned before. The interesting physics comes out of the light atom interaction piece caused by the induced electric dipole moment. We know what happens to the eigenstates of the system with the first three pieces of Eqn. 2.1 so let us look at the induced dipole interaction. First we will define our circularly polarized light in the standard fashion:

E0 E0 σ+ ≡ √ (ˆx − iyˆ) and σ− ≡ √ (ˆx + iyˆ) 2 2

9 In addition for convenience we will move into a rotating coordinate system deﬁned in terms of Cartesian unit vectors with orthogonal unit vectors deﬁned as:

1 1 eˆ+ ≡ −√ (ˆx + iyˆ) ande ˆ− ≡ √ (ˆx − iyˆ) (2.2) 2 2

So if we continue to look at how our electric dipole is going to be changed when we move

to this new coordinate system we must see how the Cartesian −d·E = −dxEx −dyEy −dzEz changes. So ﬁrst let us write the dipole moment d in terms of these new unit vectors. Thez ˆ portion is unaﬀected as our rotating unit vectors only rotate in the xy-plane. If we add the two equations from Eqn. 2.2 we see:

2 eˆ+ +e ˆ− = −√ iyˆ 2

which leads to

i yˆ = √ (ˆe+ +e ˆ−) (2.3) 2

while subtraction leads to

1 xˆ = √ (ˆe− − eˆ+) (2.4) 2

Plugging Eqn. 2.3 and Eqn. 2.4 into the dipole moment equation and collecting terms we get:

d = dzzˆ + d−eˆ− + d+eˆ+

where d− and d+ are deﬁned as

1 1 d− ≡ √ (dx + idy) and d+ ≡ −√ (dx − idy) 2 2

10 Through a similar treatment we can deﬁne an electric ﬁeld in our new rotating circular coordinate system. Starting from E = Exxˆ + Eyyˆ + Ezzˆ we see

1 1 E+ = −√ (Ex − iEy) and E− = √ (Ex + iEy) 2 2 where the same thing holds for Ez as it did for dz. Now we are able to go back and look at the interaction Hamiltonian Hd·E to look at the −d · E combination in this rotating coordinate system.

−d · E = −(dzzˆ + d−eˆ− + d+eˆ+) · (Ezzˆ + E−eˆ− + E+eˆ+) (2.5)

However our beam will be propagating along thez ˆ direction which means it has no component of the electric ﬁeld along that direction such that Ez = 0 which means Eqn. 2.5 simpliﬁes to

Hd·E = −d−E− − d+E+ (2.6)

+ − The d− term corresponds to interaction with σ photons and d+ corresponds with σ interactions. From our basic assumptions about our level structure from Sec. 2.1 we start with a system in some superposition of magnetic substates with J = 1 such that

|ii = C−1 |J = 1, m = −1i + C0 |J = 1, m = 0i + C+1 |J = 1, m = +1i

|ii = C−1 |−1i + C0 |0i + C+1 |+1i

Our excited state that we are projecting onto is the |fi = |J 0 = 0, m0 = 0i = |0i state. What we are interested in looking at then is

hf| Hd·E |ii = h0| − d−E− − d+E+ |(C−1 |−1i + C0 |0i + C+1 |+1i)i (2.7)

11 The Case of X-Polarized Light

In the case of incident light polarizedalongn thex ˆ direction the electric ﬁeld

E = Exxˆ = E0 cos ωtxˆ (2.8)

with Ey = Ez = 0 and further plugging in these value of Ex and Ey to solve for E+ and

E we see that the Hamiltonian becomes H = √1 E (d − d ). Further we can calculate − d·E 2 x + − hf| Hd·E |ii for this polarization of light.

Ex hf| Hd·E |ii = 0 √ (d+ − d−) (C−1 |−1i + C0 |0i + C+1 |+1i) (2.9) 2 Ex hf| Hd·E |ii = √ (C−1 h0| d+ − d− |−1i + C0 h0| d+ − d− |0i + C+1 h0| d+ − d− |+1i) (2.10) 2

From Fig. 2.1 we know that any interaction with σ± light will only cause transitions from the m = ∓1 levels. This immediately makes the term h0| d+ − d− |0i go to zero as circularly polarized light cannot drive that transition. It should also be noted that d± will only act on |±1i. This means our equation becomes

Ex hf| Hd·E |ii = √ (C+1 h0| d+ |+1i − C−1 h0| d− |−1i) (2.11) 2

Using the Wigner-Eckart Theorem it can be shown that [9]

h0| d+ |+1i = h0| d− |−1i (2.12) which leads to

Ex hf| Hd·E |ii = √ h0| d+ |+1i (C+1 − C−1) (2.13) 2

12 So we see that

hf| Hd·E |ii ∝ C+1 − C−1 (2.14)

The Case of Y-Polarized Light

Through a similar method as x-polarization we start with E = Eyyˆ which leads the −iE Hamiltonian of this system to be written as H = √ y (d + d ) which working through d·E 2 + − similar steps we see that

−iEy hf| Hd·E |ii = √ (C+1 h0| d+ |+1i − C−1 h0| d− |−1i) (2.15) 2

and after application of the Wigner-Eckart Theorem

hf| Hd·E |ii ∝ C+1 + C−1 (2.16)

Consequences

Now we can look at what should be happening in the simpliﬁed model of our laboratory

system. Our initial state should be |C+1| = |C−1|. This assumption is valid because the vapor cell is at room temperature so thermal eﬀects should cause the system to be in its most probable state, with population equally distributed among the ground states. For the

case of x-polarization and C+1 = C−1 Eqn. 2.14 leads to

hf| Hd·E |ii = 0 (2.17)

This is what we refer to as a dark state. The eﬀect of the light on the system is that you are never able to make the transition to the excited state. The opposite is true for

C+1 = −C−1 for this polarization where these probabilites add and we get an absorbing bright state. However, this absorbing state is destroyed by spontaneous emission. This allows the system another chance of going into the dark state. After a few optical cycles the

13 ensemble of atoms will fall into a dark state after a few optical cycles. For y-polarization the populations add to each other to create the opposite for the same starting populations.

For y-polarized light we see an absorbing bright state for C+1 = C−1 and a dark state for

C+1 = −C−1.

14 Chapter 3

Experimental Setup and Procedure

This chapter will contain all the the experimental details to construct and operate a system a laser system for our EIT and EIA experiments. All necessary experimental details for recreating our data should be included in the appropriate sections.

3.1 External Cavity Diode Laser System

Our experiment, the same as many other atomic physics experiments, needs a tunable and frequency stable light source as they deal with very closely spaced energy levels of atomic transitions. These laser systems are standard throughout the community [10] due to their price and tunability. Due to the high price of comercially available laser systems many experimental groups build their own laser systems for a fraction of the commercial price. The laser diodes are inexpensive ranging in price from about $30 to $45 per laser diode purchased from Thorlabs (model number: L785P090). Including the controlling electronics and other optics the entire diode system can be built for around $3,000. Following the design of [11] we have implemented an external cavity diode laser (ECDL) system with the basic components shown in Fig. 3.1.

3.1.1 Construction of the ECDL

An aluminium case is custom built to house the laser for partial protection from temperature variations in the laboratory. Other than a moderate level of temperature control

15 Figure 3.1: The physical ECDL system with labeled components. The beam paths are shown in red and blue. Red correspond to the beam coming out out of the diode and the experimental beam. The light blue corresponds to the beam reflected from the diffraction grating that goes to the saturated absorption setup. this case also allows the laser system to be moved from one location to another with only minor adjustments having to be made afterward. On the baseplate of this housing a piece of Sorbothane is placed to mount the laser system on. Sorbothane is a rubber like material that greatly decreases vibrational noise. Any mechanical energy that causes deformation in the material is dispersed as heat. A custom built baseplate is placed on top of the Sorbothane upon which we are able to mount the remaining pieces including the beam splittler, laser diode mount, collimating lens mount and diffraction grating mount. On our custom built baseplate we first attach the laser diode mount. This is accomplished by placing a teflon strip between the laser diode mount and the baseplate and connecting

16 Figure 3.2: The basic design of the ECDL system. Two orders of the diffraction grating reflection are shown and labeled by their order (0 and 1). The red beam is the beam incident upon the diffraction grating while the green beam is the -1 order copropagating beam. them with teflon screws to electrically and thermally isolate the laser mount and baseplate from each other. As many of our fine adjustments require us to touch around the base plate it is imperative that we do not discharge any additional charges we might have accumulated through the laser diode. Any transient current through the diode can cause complete failure and death of the diode. The laser diode is then mounted snugly ensuring first that the person installing the diode is electrically grounded. Once the diode is turned on and the current turned to about 50mA the grounded person needs to fix the ellipticity of the beam. The rectangular output facet of the laser diode means that only certain modes are allowed to propagate inside of the diode which helps control the polarization of the output beam. The elliptical beam should be oriented such that the major axis of the ellipse is horizontal or parallel to the plane of the baseplate or optics bench. The polarization of the beam is orthogonal to the major axis of the ellipse or parallel to the minor axis. To view the beam a card or piece of paper is used a few inches from the diverging beam such that the ellipticity of the beam is obvious. The orientation change has to be done while grounded and having slightly loosened the screws

17 holding the laser diode into its’ mount and physically turning the diode. Once the diode is oriented correctly the diode mount screws must be retightened. As seen in Fig. 3.2 the next piece of our laser system is the collimating lens to correct the divergent beam emitted from the laser diode. A collimating lens is placed immediately after the diode to collect the divergent light and focus it at an infinite distance. For lab purposes this infinite distance is typically a few meters. The process of collimation is explained in Sec. 3.1.2. The light then goes to a 50/50 nonpolarizing beamsplitter which evenly splits the laser power. Half of the power is transmitted through the beamsplitter and the other half is reflected off of the center of the beamsplitter. The beamsplitter is mounted on a small piece of aluminum with double sided tape. The aluminum piece is screwed into the baseplate and the beamsplitter and its mount are rotated such that it is both a snug fit with the screw and that the face of the beamsplitter is perpindicular to the direction of laser propagation. Attaching the diffraction grating to its’ mount is the next step. One must ensure that the arrow drawn on the top or bottom of the diffraction grating is pointing in the direction you want the +1 diffracted order to go. If the grating is parallel to the output transmission surface of the beamsplitter the arrow should point in the direction of the experimental beam. Once you know what orientation you need to mount the grating it should be superglued to a Thorlabs mount with standard off-the-shelf superfine pitched screws. It should be glued on that mount so that the -1 order goes back into the laser diode. On the reverse side of this Thorlabs mount two piezo electric transducers(PZT) are placed back to back between the fine pitchscrew behind the diffraction grating and the backplate of the mount. The center of the crystal should be as close to the point where the screw contacts the grating mount as seen in Fig. 3.3. These PZTs allow for fine control over the angular placement of the diffraction grating. We are also able to apply a triangle signal to the PZTs and scan the angle of the diffraction grating at a known rate, thus scanning the frequency the laser. For more information on obtaining feedback see Sec. 3.1.3.

18 Figure 3.3: Shown is the position of the PZTs mounted behind the grating. The ﬁne adjustment screws are also labeled. The third screw should not be adjusted.

The final piece that is attached to the baseplate is a steering mirror. This mirror can be directly placed on the baseplate with a small line of glue or it can be affixed onto the front plate of the Thorlabs mount, next to the diffraction grating. Placing the mirror on the baseplate means that small grating adjustments do not change the SAS beam position as much. This is the prefered position when creating a new laser system but can be more tricky to initially position the mirror correctly. If the mirror is instead attached to the diffraction grating mount small changes in output beam position will continuously be made as you change the position of the grating. This can be helpful once you are already on feedback and have found resonance. Knowing the exact spot the output beam needs to be for resonance can make adjusting to resonance after it has drifted away easier as you can see the changes in position in the saturated absorption setup. The actual ECDL setup is shown in Fig. 3.1. Attached on top of the laser diode mount is a thermoelectric cooler(TEC) and a heatsink. The TEC is attached with a piece of heat tape cut to cover the most surface area between both the TEC and laser diode mount and the TEC and heatsink. After affixing the TEC and heatsink for temperature control of the

19 laser system we must also be able to monitor the temperature. For this a small hole was drilled into the laser diode mount close to where the laser is inserted. Into this hole we apply a small amount of thermally conductive heat paste. Then we insert a 10kΩ thermistor so that we can monitor the temperature of the laser diode electronically and apply a current in the proper direction to the TEC to control the temperature of the laser diode to less than 0.1◦C. These pieces are also shown in Fig. 3.4.

Figure 3.4: A view of the laser system from behind the laser diode. Shown are the pertinant components (thermistor and TEC) for temperature control of the laser diode. The diﬀraction grating and mirror are shown as well.

The temperature of the diode needs to be controlled to prevent frequency drift. Our temperature controlling electronics are able to tell when any drift in our temperature occurs and apply a current to the TEC to compensate. TECs work using the peltier effect which causes the different sides of the TEC to have different temperatures based on the direction of current applied. The temperature must be set before looking for resonance features otherwise the frequency output of the laser will be continuously drifting around. To do this initial temperature setting we enclose the laser system in its box by carefully running the wires through the hole and securing them down. We then turn the current on the laser system to a value we expect to be running the system at, typically 95mA. The system is allowed

20 to equilibrate with this current going through the diode for about 15-20minutes. After that time we set the temperature that we will lock at to the current temperature of the system. This allows us to precisely control the temperature while at the same time minimizing the work that needs to be done by the TEC module. The following subsections will be a walk through some of the more intricate parts of constructing an ECDL system for an atomic physics experiment.

3.1.2 Collimation

The light coming out of the laser diode is both highly divergent and broadband. To conduct our experiments we need a beam that is of consistent intensity over the interaction cross section in our vapor cell. Any divergence of our beam will cause a decrease in laser intensity over the beam path. To correct for this divergence we must first collimate the light from the diode. We do this using a aspherical lens placed immediately after the laser diode to collect the light. This lens must be large enough to collect all of the light coming out of the diode and placed such that the focal length of the lens is at the exit facet of the laser diode. To ensure the position of the lens is correct we must place it into the system and look at the profile of the light passing through. To ensure the beam is nondivergent over the length of the experiment we use a mirror to reflect the experimental beam across the laboratory and look at the profile of the beam along the length. If the beam profile changes we loosen the screws holding the collimating mount and move it forward or backward along the track created by the screws. If the beam is focusing at some point move the aspherical lens forward and backwards to make this point be as far away from the exit facet of the diode as possible before locking the lens mount onto the baseplate. Finer adjustments can be made with the collimating screw connected to the baseplate. Once the beam is focused as far out as you are able to see it you need to make a small turn of the fine adjustment screw. Turning this

21 screw about 1/8 of one turn counter-clockwise should move the focus further away ﬁnishing the collimation of the beam.

3.1.3 Feedback

The natural bandwidth of the laser diode is much larger than necessary for atomic physics experiments. To fix this we must provide the diode with a seed frequency that will then be amplified by the diode. The frequency selection element we use is a diffraction grating. A large bandwidth input beam is seperated out based on frequency into different orders as you would expect with a diffracting element. Coming off of the grating are three orders. The 0th order is a simple reflection while parts of the band are reflected at slightly different angles. The -1 order and the 0th order are shown in Fig. 3.2. We position the diffraction grating mount by hand so the -1 order is going back through the beamsplitter and collimating lens into the laser diode. The bandwidth of the diffracted piece going back into the diode is much less than the original beam. This extra piece acts as a seed for the diode to amplify that frequency band instead of the natural output. A trick for getting this feedback mechanism started is if you take a small strip of paper with a hole in it and place it after the beamsplitter so the main beam goes through the hole you should be able to adjust the mount so the -1 order goes back through that hole. You can see this -1 order moving as you adjust the mount if you look closely at the output reflections of the grating. It helps if you have seen this -1 order before you start making adjustments and then watch as it moves toward the hole in the paper. This is of course a coarse adjustment for feedback but is necessary to make the next step as easy as possible. If we look at the output that goes to the SAS side (labeled in Fig. 3.2) far enough away, about 2m, we should see a solitary beam that moves as expected with the horizontal and vertical superfine pitched screws. As these screws are adjusted at some point a second beam will pop up close to the moving beam, this signifies you are close to feedback. If you

22 move the inital beam towards this secondary beam they will visibly jump together. It can be difficult to find this point. A useful tip to do this is to start at some point and sweep the horizontal a few full turns across. If you do not see the jump in that sweep move the beam vertically up or down by a little more than the beam diameter and sweep horizontally the other direction, repeat until feedback is found. This method will ensure that you find the feedback position quickly. After they jump together you should be able to move the ”horizontal adjust” a large amount without the beam shape moving at all. At this point the laser system is ”in feedback.” In feedback mode tuning the ”horizontal adjust” changes the frequency of the laser. A quick way to check if a laser is on feed back if you look at the experimental beam output and block the diffraction grating the beam output should drop in intensity by a large amount near threshold (see 3.1.4).

3.1.4 Threshold

Threshold is defined as the point the diode begins to lase. At low current values the diode emits like an infrared LED emitting noncoherent light. At a certain current value the diode undergoes something akin to a phase transition between an LED and a laser. The intensity of the light at this point jumps by a large and noticeable amount when seen with an infrared viewer. Perfecting the feedback will lower the theshold current for this transition. To find the threshold we look at the experimental beam and slowly turn the current down until we see the transition. Raw threshold, or the lowest possible threshold for the diode, is typically at a current of 28-30mA. If the current is more than 1mA above this minimum the veritcal adjustment, the more sensitive screw, is made first. A small adjustment of the vertical adjustment screw should cause the system to come more into feedback or the output to have greater intensity. This is continued by slowly dropping the current and making small adjustments in the vertical or horizontal screws to minimize the current at which this transition occurs.

23 One other fact that should be noted about threshold is that the threshold current value in the resonance position of the horizontal screw is slightly higher than the raw threshold current value. This diﬀerence is because the position of the diﬀraction grating where feedback is at a maximum corresponds to the natural wavelength of the diode (typically 785nm in our case).

3.1.5 Resonance

Finding resonance for the first time with a laser system is done by first finding feedback and threshold. After that a ±5V triangle wave is applied to the PZT at a frequency of about 10Hz. This triangle wave ramp of the PZT means the grating is very slightly changing angle which changes the frequency of the laser and thus sweeping through a frequency range based on the voltage applied to the PZT. This ramping frequency allows us to look for the resonant frequency for the rubidium in a vapor cell placed in the laser path. From the raw threshold position of the horizontal screw the resonant position is generally around half of a turn clockwise. However this is obviously not a hard and fast rule. If the current is turned to 90-95mA and the horizontal screw is swept slowly and carefully from one end of feedback to the other one should find resonance at a given current value. gives you the best opportunity to find resonance at a given current value. If a flash heralding resonance in the vapor cell is not seen, the current needs to be changed. The quickest and easiest way to do this is to change the current value by 1mA at a time and sweep through the feedback range. Once a flicker is seen in the bulb it should be maximised for brightness with small changes of the current value and horizontal position. To know what transition is being excited by the laser we use a method known as saturated absorption spectroscopy.

24 3.2 Saturated Absorption Spectroscopy

One of the simplest things that can be viewed with our laser systems is the doppler free saturated absorption setup. This setup allows for the resolution of hyperﬁne structure of rubidium vapor using a very simple optical setup; as in [11]. A sample spectrum without using saturated absorption spectroscopy is shown in Fig. 3.6a.

Figure 3.5: Shown is a sample saturated absorption setup used in all of our atomic physics experiments. The relevent beam lines are shown. The redlines are the output beam of laser and the strong beam going through the cell. The yellow lines are the weak beams reﬂected oﬀ of the thick glass plate and into photodiodes. Note that the strong red beam overlabs and is counter propagating to one of the weak yellow beams.

A saturated absorption set up as seen in Fig. 3.5 consist of two weak probe beams and a strong pump beam that is counter-propagating and overlapped with one of the weak beams. Any atoms moving perpindicularly to the direction of beam propagation will be simultaneously in resonance with all of the beams as they do not see a doppler shifted frequency. The atoms that are interacting with the weak and strong beam pair should be saturated by the strong beam. This saturation with the strong beam means the atoms

25 no longer absorb the on resonant photons and causes Doppler-free ”holes” into the Doppler broadened transmission profile of the overlapped beam at the hyperfine transition frequencies as in Fig. 3.6b. The other beam does not have these ”holes” in its’ profile so we are able to subtract these profiles and the result is a spectrum that shows the hyperfine transitions.

(a) Doppler broadened spectrum. (b) Doppler free ”holes”.

Figure 3.6: Without saturated absorption spectroscopy the transmitted profile of the laser is Doppler broadened as shown in 3.6a. The transitions being shown are the F = 3 to F 0 in 85Rb and F = 2 to F 0 in 87Rb. Note that no hyperfine features are discernible due to this Doppler broadening that is caused by the motion of the atoms in the vapor cell. 3.6b shows the Doppler free ”holes”. These ”holes” are a result of the atomic vapor being saturated by a counter propagating strong beam and not interacting with the weak probe beam. 3.6c shows the Doppler free hyperfine spectrum that is the result of subtracting the two weak probe signals.

26 3.3 Optical Setup

Seen in Fig. 3.7 is the apparatus that was used to obtain our EIT and EIA data. The function and purpose of the individual pieces of the apparatus will be described below. As mentioned briefly in Sec. 3.1.1 the beam coming out of the laser diode is elliptical in shape. The first step in our optical setup is to shape our beam. The elliptical beam is put through a half wave plate and anamorphic prism pair. The initial half wave plate (H1) corrects t he polarization of the beam for maximum power through the prism pair by minimizing losses due to reflection. The prism pair is used to magnify one axis of the ellipse while maintaining the size of the beam along the other axis. This is the reason it is imperative the ellipse is set horizontally along the major axis during the construction of the laser system. The prism pair set up is shown in Fig. 3.8.

Figure 3.7: Our setup to measure EIT and EIA. The beam is shaped and transmitted through the vapor cell and the transmittance is measured using a photodiode.

27 Figure 3.8: The geometry of an anamorphic prism pair as shown in the CVI Melles Griot catalog [12].

The manufacturer of the anamorphic prism pairs gives geometry dimensions down to a magnification of two. However, our newer laser diodes have an ellipticity (major axis:minor axis) of less than 2:1. To compensate for this we have extrapolated the manufacturer’s data so that our anamorphic prism pairs will compensate for laser diode ellipticity between 1 and 2. The seperation and angles are needed for a given magnifaction are shown in Fig. A.1 and Table A.1. After the beam has been initially shaped it is put through a half wave plate and polarizer combination (H1 and P). This allows the experimenter the ability to control the power going to the remaining parts of the system. When the subtraction beam is about the same strength as the probe beam going through the cell this combination allows you to lower or raise the power to both beams equally. After this combination is another half wave plate (H2). This wave plate controls the ratio of powers between the experimental and subtraction beam due to the next element, the polarizing beam splitter (PBS). Polarizing beam splitters allow one polarization of light to transmit through unchanged while the orthogonal polarization is reflected. Changing the direction of the beam’s polarization therefore changes how much power is transmitted and how much is reflected into the

28 different output beams. The reflected beam is used as a subtraction beam. This beam goes directly into one of the subtraction photodiodes and acts as a DC offset to the experimental signal. The transmitted beam is sent through another half wave plate (H3) set with its’ fast axis 45◦ to the output polarization of the beamsplitter. Setting the wave plate here flips the polarization by 90◦, making it vertically polarized. This polarization is needed when we changed the transverse magnetic field of our system and look at EIT and EIA spectra. Polarization and power control issues have now been addressed so we must deal with the beam profile. Even upon almost circularizing a beam using an anamorphic prism pair the profile will probably not be perfectly gaussian. We use a microscope objective lens (L1) to magnify our beam and a secondary lens (L2) to collimate this light. The collimation process is very similar to Sec. 3.1.2 in principle but with less fine control. The second lens should be moved into position where its’ focal point is at the focal point of the microscope objective lens. Once this beam is collimated an iris is inserted into the system and opened to a known diameter. This diameter becomes our experimental beam diameter and was set to 3.0mm. This value allowed a large intensity range and a good signal to noise ratio. Beyond just setting the beam diameter the iris is used to select a section of our beam that has uniform intensity. The beam can be oddly shaped with a variable intensity profile due to imperfections in adjustments. We want the atoms to see a uniform field over the vapor cell so we select a part of the beam that is constant intensity to put through the iris. A beam diameter over a few centimeters does not allow a uniform profile to be found. This does of course depend on how much magnification the microscope objective lens causes. For example a 20x objective will allow a beam diameter that can be up to about 1cm whereas a 4x objective is limited to 3 − 4cm constant intensity diameter. This beam is then incident upon the center of a rubidium vapor cell and the light is collected in the back by focusing it onto the other subtraction photodiode. The beam passes through the center of the cell because the magnetic field is best controlled and most uniform at the center of the solenoid.

29 3.4 Magnetic Field Cancellation and Control

In this section the issue of magnetic fields within our vapor cell will be investigated. Any stray magnetic field due to the earth or other objects in the laboratory could cause unexpected behavior in our experiment. These fields are canceled so that we can more precisely control the field inside of the vapor cell. The field inside of the vapor cell is controlled using a solenoid wrapped around the cell.

3.4.1 Helmholtz Coils

The Helmholtz coils used to cancel the earth’s magnetic field are shown in Fig. 3.9. These coils use computer ribbon cable instead of thicker copper wire to cancel out the field in each direction. This ribbon cable doesnt allow as much current to flow through the wire but the magnetic field due to a solenoid (the large coils) is proportional to both the current and number of turns of wire. With less total current we are able to create a magnetic field equivalent to a larger current copper wire at a much lower cost. The magnetic field along the length of our vapor cell or in thez ˆ-direction should follow [13].

2 8µ0Ia 1 Bz = [ q + π 2 d 2 2 d 2 (a + (z − 2 ) ) 2a + (z − 2 ) 1 q ] 2 d 2 2 d 2 (a + (z + 2 ) ) 2a + (z + 2 ) where ’a’ is the distance from the center of the coil to the edge and ’d’ is the separation between the two connected Helmholtz coils. This leads to less than a 2.5% diﬀerence in the magnetic ﬁeld over the length of our vapor cell. These theoretical plots match, in shape, the experimental values included in AppendixB

30 Figure 3.9: To cancel the earth’s magnetic field we use the helmholtz coil arrangement shown here. Each coil has independent current control to create a static magnetic field that is the same magnitude and oppostie direction of earth’s field.

specifically Fig. B.3 in form. It is not important that the theory and experimental plots are identical as they use different absolute values. The magnetic fields are controlled independent of each other by applying currents to individual coil pairs. The currents in these coils are controlled by seperate individual current supplies. The current needed to create a field of about 1G beyond canceling the earth’s field through the center of our cell is about 1A. An issue that arises when using these current supplies directly is that the total resistance of the individual stranded wire is very low because the ribbons are connected in parallel. This means that a small change in the voltage supplied by the current supplies means a large increase in current. The magnetic field is dependent on the current so the finer control on the current in the system means a finer control of the magnetic field. To accomplish this we must insert additional resistance into each current supply circuit due to the small resistance in the coils to ensure the power dissipated through

31 (a) The magnetic ﬁeld as a function of distance from the center of a helmholtz coil along thez ˆ-direction. The geometry of the coil matches that of our z coil. The ﬁeld is relatively constant over a 10cm range arround the origin.

(b) Fractional difference of the magnetic field as a function of distance from the center of a helmholtz coil is shown. Over the length of our vapor cell (7.2cm) the magnetic field should vary less than 2.5%.

Figure 3.10: Theoretical plots of magnetic ﬁeld uniformity due to a Helmholtz coil.

32 the additional resistors is not to great. The power dissapated by a resistor is dictated by Joule’s law, P = I2R, which leads to Joule heating in the resistor. Passing a current of 1A though a large value resistor would burn most resistors out. To accomplish this we place two 8Ω resistors that are made to be able to dissapate 25W of power in series with the coils. These additional resistors are placed on a large aluminum heat sink to efficiently dissipate the heat at high current values. Before inserting the vapor cell into the Helmholtz coil array the fields must be zeroed. To do this a Sypris 7030 Gauss/Tesla meter with a 3-axis Hall sensor probe is inserted into the Helmholtz coil system. The probe should be inserted so that the sensor end is in the center of the array where the center of the vapr cell will be. This probe is outfitted with 3 separate Hall sensors that need to be aligned in accordance with the coordinate system shown in Fig. 3.9. Once the probe is inserted in the center of the Helmholtz coil system the B-field cancellation follows. This is done by adjusting the current in one of the coils and watching how it effects thex ˆ,y ˆ, andz ˆ magnetic field strengths. Small adjustments should be made to the probe such that changing the current connected to one set of coils changes the filed only in that direction. Any change in the magnitudes of the other field strengths should be nominal in comparison to the field being changed. To zero the field at the center of the coil array the currents must be adjusted to cancel the earth’s field. To zero the sensor system the shielding cover provided with the pobe is placed over the probe. All three directions of the sensor are zeroed simultaneously by pressing and holding the ’Enter’ button and then pressing the only ’Zero’ button lit up. After a few seconds the Hall sensors will be zeroed. Once the sielding is removed each coil current is adjusted such that the probe reading is 0G in each direction. Between every measurement or every two to three minutes the sensor system must be rezeroed to account for electrical drift of the sensor. For use in the experiment current values that give a certain transverse magnetic field strengths are also needed. This is done using the same Hall probe and changing the current

33 to control the field in thex ˆ direction. Current values through the coils are taken at 0.1G increments from 0G to 1G. This transverse field effects both the size and width of the EIT and EIA features dicussed in Chapter4.

3.4.2 Magnetic Field Ramp Control

The longitudinal magnetic ﬁeld is controlled using a solenoid of copper wire wrapped around the vapor cell shown in Fig. 3.11. The magnetic ﬁeld at the center of the solenoid follows

µ0NI Bcenter = √ (3.1) L2 + 4R2

where µ0 is the magnetic permeability of free space, N is the number of turns in the solenoid, I is the current in the solenoid, L is the length of the solenoid, and R is the radius. With a 0.99A maximum current being supplied to the 121 turn coil the maximum ﬁeld in the center of the vapor cell is 22.7G. The current is oscillated at 3.45Hz between ±22.7G using a triangle waveform so that the current and magnetic ﬁeld change linearly.

Figure 3.11: Creating a longitudinal magnetic ﬁeld: The rubidium vapor cell is wrapped with copper wire in the shape of a solenoid. This copper coil has 121 turns over the length of the vapor cell. This is located inside the Helmholtz coils and can be seen Fig. 3.9.

The copper wire is held on to the vapor cell by end mounted teﬂon clamps. The teﬂon

34 prevents eddy currents caused by the ramping magnetic ﬁeld that would cause our ﬁeld readings inside the vapor cell to become unreliable. The single linearly polarized beam is passed through the center of the vapor cell.

3.5 The Atomic System

85 87 The atomic transitions we are exciting are the D2 lines of Rb and Rb, the energy level diagram for these transitions can be seen in Fig. 3.14 and Fig. 3.15 respectively. These ﬁgures show all of our transitions. The transitions we see EIT on are from the lower of the

2 0 0 two 5 S1/2 to the set of levels F = F or F = F ± 1 . EIA is observed on the upper ground state transitions.

(a) 85Rb lower ground state transitions (b) 87Rb lower ground state transitions

Figure 3.12: Saturated absorption spectra shapes for each set of hyperﬁne transitions from the upper ground states of the D2 lines. These result in EIT features. The horizontal (frequency) scales of the graphs are not the same.

While reducing the frequency ramp signal to reduce the laser output bandwidth the frequency must be tuned by changing the angle of the diﬀraction grating with the voltage

35 to the piezoelectric transducers. If the ramp signal is reduced to zero the frequency should be tuned to the minimum noise position in the EIT or EIA spectrum. Only one low noise point occurs while tuning through the hyperﬁne peaks.

(a) 85Rb upper ground state transitions. (b) 87Rb upper ground state transitions.

Figure 3.13: Saturated absorption spectra for each set of hyperﬁne transitions from the lower ground states of the Rb D2 lines. These result in EIA features. The horizontal (frequency) scales of the graphs are not the same.

36 85 Figure 3.14: Rb D2 line data. Level spacings are included for all transitions. EIT is seen on the transition from F = 2 to F 0 and EIA on F = 3 to F 0. Included in [14]

37 87 Figure 3.15: Rb D2 line data. Level spacings are included for all transitions. EIT is seen on the transition from F = 1 to F 0 and EIA on F = 2 to F 0. Included in [15]

38 Chapter 4

Results and Discussion

In this chapter example of EIT and EIA spectra are shown for each of the four hyperfine transition systems in Sec. 3.5. There are two transition systems for 85Rb: F = 3 → F 0 = 2, 3, 4 and F = 2 → F 0 = 1, 2, 3 and two transition systems for 87Rb: F = 2 → F 0 = 1, 2, 3 and F = 1 → F 0 = 0, 1, 2. Data taken on the response of these spectra to varying magnetic fields are presented as well. The frequency detuning shown in the figures of this chapter comes from the longitudinal magnetic field scan. Using an oscilloscope to measure the spectrum we must convert the time scale to a frequency scale along the x-axis. This is done by calculating the amount of magnetic field sweep over a given time scale. For the coil shown in Fig. 3.11 using a 1A current creates a 22.7G magnetic field. Taking into account the frequency of the coil driver providing the current the magnetic field changes at a rate of 0.311G/ms. Fig. 4.1 shows the oscillator strengths for the various hyperfine transitions on which we have observed EIT and EIA signals. From this figure we can estimate what type of spectrum to expect for a given hyperfine transition system. The largest oscillator strength within a given hyperfine transition system should determin whether we observe EIT or EIA, as explained previously in Sec. 1.2.

39 Figure 4.1: The relative oscillator strengths for the hyperﬁne transition systems used in our experiment. Adapted from [7]. The lines follow the same rules as before in Fig. 1.4.

4.1 EIT on the 85Rb F = 2 → F 0 = 1, 2, 3 transition sys-

tem

We have observed EIT in the lower ground state hyperfine transition system F = 2 → F 0 = 1, 2, 3 of 85Rb. The effect of intensity and transverse magnetic field on EIT features have been studeied. The transverse magnetic field applied to the cell in this chapter is always perpendicular to beam propagation and is in the same direction as the polarization of the laser field. The lower ground state transition corresponds to Fig. 4.1c. From this figure it is obvious we expect EIT on these transitions because the bold and solid lines are about the same length as the dashed line resulting in a very strong EIT signal. This is mirrored when we look at our EIT signal in Fig. 4.2a. Even with a low intensity we have a strong and very clear

40 signal. The size of this signal increases in both height and width as a function of transverse magnetic ﬁeld and intensity as seen in Fig. 4.3. All of this data show a minimum EIT and EIA feature width occuring at a non-zero transverse magnetic ﬁeld. This provides further motivation for the development of the theory to explain this subtle feature.

(a) (b)

Figure 4.2: Sample 85Rb EIT spectra taken at low intensities(0.5mW/cm2), low transverse magnetic field (0G), high intensities (1.5mW/cm2), and high transverse magnetic field (1G). A and δ are studied as a function of transverse magnetic field and intensity in Fig. 4.3 .

41 (a) (b)

Figure 4.3: 85Rb EIT feature height and width as functions of intensity and transverse magnetic ﬁeld. The width is compared to the natural linewidth of 6MHz. Representative error bars are shown on one point.

4.2 EIA on the 85Rb F = 3 → F 0 = 2, 3, 4 transition sys-

tem

The probability distributions for the 85Rb F = 3 → F 0 = 4 transition in Fig. 4.1b shows a clear dominance over the other hyperﬁne transitions. By this we expect an EIA signal. Representative spectra are shown in Fig. 4.4.

42 (a) (b)

Figure 4.4: 85Rb EIA spectra taken at low intensities( 0.5mW/cm2), low transverse magnetic ﬁeld (0G), high intensities (1.5mW/cm2), and high transverse magnetic ﬁeld (1G).

43 (a) (b)

Figure 4.5: 85Rb EIA feature height and width as functions of intensity and transverse magnetic ﬁeld. The width is compared to the natural linewidth of 6MHz. Representative error bars are shown on one point.

We see that the dependencies of the spectral features of EIA are similar in shape to those of the EIT transition in 85Rb.

44 4.3 EIT on the 87Rb F = 1 → F 0 = 0, 1, 2 transition sys-

tem

Transitions on 87Rb show typically smaller features for two reasons. Our vapor cell contains a natural isotope mixture of rubidium (72% 87Rb and 28% 85Rb). Having less atoms to interact with should create a smaller signal. The other reason goes back to the oscillation strengths. Having weaker transitions also means the signal strength should be lower.

(a) (b)

Figure 4.6: 87Rb EIT spectra taken at low intensities( 0.5mW/cm2), low transverse magnetic ﬁeld (0G), high intensities (1.5mW/cm2), and high transverse magnetic ﬁeld (1G).

45 (a) (b)

Figure 4.7: 87Rb EIT feature height and width as functions of intensity and transverse magnetic ﬁeld. The width is compared to the natural linewidth of 6MHz. Representative error bars are shown on one point.

4.4 EIA on the 87Rb F = 2 → F 0 = 1, 2, 3 transition sys-

tem

The prominence of the effect of a transverse magnetic field and intensity on our features is best demonstrated by Fig. 4.8. When either the transverse field or intensity is changed the feature gets much larger both in height and width. The signal to noise ratio is greatly

46 improved using a higher intensity beam. The transverse ﬁeld does not appear to improve this ratio.

(a) (b)

Figure 4.8: 87Rb EIA spectra taken at low intensities( 0.5mW/cm2), low transverse magnetic ﬁeld (0G), high intensities (1.5mW/cm2), and high transverse magnetic ﬁeld (1G).

47 (a) (b)

Figure 4.9: 85Rb EIT feature height and width as functions of intensity and transverse magnetic ﬁeld. The width is compared to the natural linewidth of 6MHz. Representative error bars are shown on one point.

48 Chapter 5

Conclusions and Future Outlook

5.1 Conclusion

We have observed EIT and EIA in both 85Rb and 85Rb. How these spectra vary based on both magnetic ﬁelds and intensities of our probe beam has been investigated. We have successfully implemented a Zeeman tuned method for EIT and EIA using a single linearly polarized laser beam. These signals are observed with good signal to noise ratio over the range of intensities and transverse magnetic ﬁelds shown.

5.2 Future Outlook

Our goal is to extend our exploration of EIT and EIA phenomena in more depth on both isotopes of Rubidium. We plan to use a combination of Zeeman tuned EIT and two linear, orthogonally polarized laser beams to look at other coherent eﬀects in our system. A phenomena called sign reversal has been obtained using 87Rb as seen in [16]. We have preliminary data that shows we are seeing a similar result in 85Rb on the F = 2 → F 0 = 1, 2, 3 transition system. When increasing the coupling beam power the EIT feature begins to spread out when the coupling beam power becomes close to the probe beam power. As the coupling beam power begins to exceed the probe power by a large amount the EIT signal turns into an EIA signal as seen in Fig. 5.1. We believe this is due to the coupling beam closing some of the open transitions in the F = 2 → F 0 = 1, 2, 3 transition system or the

49 F = 1 → F 0 = 0, 1, 2 transition system in [16]. If more transitions are closed the EIA signal becomes stronger as it becomes less eﬀected by the spontaneous emission pathways destroying the signal. Some basic experimental evidence for this hypothesis is that the EIA signals do not undergo sign reversal when a coupling beam is introduced.

Figure 5.1: Evidence for sign reversal using 85Rb. The probe beam intensity is 214µW/cm2. The coupling beam powers equal to a) 0µW/cm2, b) 224µW/cm2, c) 319µW/cm2, d) 553µW/cm2.

For further theoretical development we plan to use the Atomic Density Matrix (ADM) Mathematica package developed my the Rochester group at the University of California, Berkeley [17]. This package is capable of creating and solving the optical Bloch equations and Liouville equation for multilevel atomic systems such as ours. The ADM package has been used to some extent with the EIT system in 87Rb by [16].

50 Appendix A

Anamorphic Prism Pair Data

The data and graphs from extrapolation of manufacturer’s data on the magniﬁcation of an anamorphic prism pair and included in this appendix.

Figure A.1: Graphs containing information (as seen in Table A.1) on the seperation and angles necessary for given magniﬁcation.

51 Table A.1: Anamorphic prism pair geometry setting values for a range of magniﬁcations. Angles (α1 and α2) are in degrees and displacement lengths (d and e) are in mm.

Magniﬁcation α1 α2 Displacement e Displacement d 1 7 15.6 3 3.7 1.2 10.4 13.2 3.5 4.1 1.4 13.6 11 4 4.4 1.6 16.5 9.1 4.4 4.7 1.8 19.1 7.3 4.8 5 2 21.5 5.7 5.1 5.3 2.2 23.7 4.3 5.4 5.6 2.4 25.6 3.1 5.7 5.8 2.6 27.4 2 6 6 2.8 29 1 6.2 6.2 3 30.4 0.2 6.4 6.4 3.2 31.6 -0.6 6.6 6.5 3.4 32.7 -1.2 6.7 6.7 3.6 33.7 -1.7 6.9 6.8 3.8 34.6 -2.2 7 6.9 4 35.3 -2.6 7.1 7 4.2 36 -2.9 7.2 7.1 4.4 36.6 -3.2 7.3 7.2 4.6 37.2 -3.4 7.4 7.3 4.8 37.7 -3.6 7.5 7.3 5 38.1 -3.8 7.6 7.4 5.2 38.6 -4 7.6 7.5 5.4 39.1 -4.2 7.7 7.5 5.6 39.5 -4.4 7.8 7.6 5.8 40 -4.7 7.8 7.6 6 40.6 -5 7.9 7.7 6.2 41.2 -5.3 8 7.7 6.4 41.8 -5.7 8.1 7.8 6.6 42.6 -6.2 8.2 7.9 6.8 43.4 -6.8 8.3 8 7 44.4 -7.4 8.5 8

52 Appendix B

Magnetic Field Uniformity

The theoretical data shown in Sec. 3.4.1 agrees with this experimental data. The theory data used arbitrary values for determining the strength of the ﬁeld along the Bz direction over the length scale shown below. All of this data was taken by Iris Zhang [9]. The vapor cell is 24.6mm in diameter and 71.8mm in length. This means that the magnetic ﬁeld along thex ˆ- andy ˆ-directions must be uniform over about 12mm on either side of the origin and thez ˆ-direction should be uniform over the length of the cell.

Figure B.1: Magnetic ﬁeld over a range of x positions blown up to show detail.

Thez ˆ-direction of the magnetic ﬁeld in Fig. B.3 is uniform over a large range. This matches with the theoretical plots from Sec. 3.4.1.

53 Figure B.2: Magnetic ﬁeld over a range of y positions, blown up to show detail.

Figure B.3: Magnetic ﬁeld over a range of z positions.

54 Appendix C

Coil Driver Schematic

The coil driver schematic for the driver used to provide current to the solenoid around the vapor cell. The driver was designed and built by the Miami University Instrumentation Lab.

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