ABSTRACT

LIGHT SHIFT MEASUREMENTS OF COLD RUBIDIUM ATOMS USING RAMAN PUMP-PROBE SPECTROSCOPY Nathan Jon Souther

We have measured light shifts, also known as the A.C. Stark effect, in cold Rubidium atoms using pump-probe spectroscopy. The measurement was made both for atoms in a magneto optical trap (MOT) and for atoms that were in an optical molasses. We show that while the measured light shifts agree with theory for optical molasses there are additional Zeeman shifts in the MOT that the theory does not account for. To the best of our knowledge, this is the first time a careful systematic measurement has been performed in cold atoms of light shift as a function of intensity. LIGHT SHIFT MEASUREMENTS OF COLD RUBIDIUM ATOMS USING RAMAN PUMP-PROBE SPECTROSCOPY 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 Nathan Jon Souther Miami University Oxford, Ohio 2009

Advisor Samir Bali

Reader Bur¸cinBayram

Reader James Clemens

Reader Perry Rice TABLE OF CONTENTS

List of Figures ...... iv

Dedication ...... vi

Acknowledgments ...... vii

CHAPTER PAGE

1 Background and Motivation ...... 1

1.1 Organization of this Thesis ...... 3

2 Theory and Principles of Light Shift Measurement ...... 5

2.1 Magneto Optical Trap ...... 5 2.2 Doppler Cooling ...... 6 2.3 Trapping the Cooled Atoms in One Spot ...... 7 2.4 Sub-Doppler Cooling ...... 8 2.4.1 Motion Induced Atomic Orientation Cooling . . . . . 8 2.4.2 Sisyphus Cooling ...... 11 2.5 Light Shift for a Two Level Atom ...... 12 2.6 Simplified Light Shift Model for a Multi-Level Atom . . . . . 19 2.7 Expected Signal of Light Shift Measurement ...... 20

3 The Trapping and Repumper Systems ...... 26

3.1 The External Cavity Diode Laser ...... 26 3.2 Anamorphic Prism Pair ...... 28 3.3 Faraday Rotator ...... 29 3.4 Creating Six Trapping Beams ...... 33 3.5 Repumper Laser ...... 34

4 Locking the Lasers ...... 36

4.1 Saturated Absorption Set Up ...... 36 4.2 Lowering Threshold ...... 38

ii 4.3 Acousto Optical Modulator ...... 40 4.4 Locking the Trap Beam ...... 41

5 Frequency Scannable Probe ...... 47

5.1 Spectrum Analyzer ...... 52

6 The Vacuum System and Magnetic Field Considerations ...... 57

6.1 Vacuum System ...... 57 6.2 Canceling the Earth’s Magnetic Field ...... 58 6.3 Applying Magnetic Gradient ...... 58

7 Experimental Procedure ...... 63

7.1 Preparation ...... 63 7.2 Data Collection, MOT ...... 64 7.3 Data Collection, Molasses ...... 66

8 Results and Discussion ...... 68

8.1 The Theoretical Prediction ...... 68 8.2 Molasses and MOT Data ...... 70

9 Conclusions and Future Outlook ...... 71

9.1 Conclusions ...... 71 9.2 Future Outlook ...... 71 9.2.1 Independent Pump-Probe ...... 72 9.3 Optical Fiber ...... 73 9.4 Independent Pump-Probe Measurements ...... 73 Bibliography ...... 78

iii LIST OF FIGURES

FIGURE PAGE

2.1 Laser Cooling ...... 6 2.2 Atom Trapping ...... 8 2.3 Polarization Gradient for σ+ and σ− polarized Light ...... 9 2.4 Clebsch-Gordan Coefficients for J = 1 to J 0 = 2 Atomic System . . . 10 2.5 Motion Induced Population Imbalance ...... 11 2.6 Polarization Gradient and Energy Light Shift for π-polarized Light . 13 2.7 Clebsch-Gordan Coefficients for J=1/2 to J’=3/2 Atomic System . . 14 2.8 Two Level Atom ...... 15 2.9 Ground State Light Shift ...... 18 2.10 Clebsch-Gordan Coefficients for the F=3 to F0= 4 Atomic System . . 23 2.11 Light Shifts in the F=3 Ground State Atomic Levels ...... 24 2.12 Cause of Raman Peaks ...... 25 3.1 Trapping Beam Optical Layout ...... 27 3.2 Anamorphic Prism Pair Setup ...... 29 3.3 Anamorphic Prism Pair Angles ...... 30 3.4 Anamorphic Prism Pair Displacements ...... 31 4.1 Saturated Absorption Spectroscopy Setup ...... 37 4.2 Saturated Spectrum Hyperfine Ground State Pictures ...... 43 4.3 BNC Connections with Laser Locking Box ...... 44 4.4 Circuit for Creating Variable Fixed Offset for AOM ...... 45

iv 4.5 Locking Points for Different Detunings of Trap Laser ...... 46 5.1 Optical Layout for the Pump-Probe ...... 48 5.2 Circuit for Creating Variable Scanning Offset for AOM ...... 51 5.3 Double-pass AO Setup for Probe ...... 52 5.4 Single Pass Frequency Shift ...... 55 5.5 Double Pass Frequency Shift ...... 56 6.1 Getter Pin Diagram ...... 59 6.2 Magnetic Field Turn Off ...... 61 6.3 Magnetic Field Turn On ...... 62 7.1 Typical Light Shift Measurement ...... 65 7.2 Molasses Data Taking Timing ...... 67 8.1 MOT and Molasses Data ...... 69 9.1 Independent Pump Probe Setup ...... 72 9.2 Multi Mode vs. Single Mode Fibers ...... 74 9.3 Circuit Diagram for Pump Shutter Control ...... 75 9.4 Pump Shutter Opening ...... 76 9.5 Spectrum for Independent Pump-Probe ...... 77 9.6 Trap Shutter Closing ...... 77

v To Drea

vi ACKNOWLEDGMENTS

I would first like to thank my advisor Samir along with undergraduates Peter Harnish and Richard Wagner for all their work along side of me, including those 30 hour days in the lab. I would not have been able to accomplish what I have without their help. I am also indebted to Michael Eldridge and Mark Fisher. If it was not for Mike’s helpfulness, expertise, and availability work in the lab would have been a lot slower. He was always able to make or help me make the pieces that I needed in the lab. Mark’s helpfulness and ability to have random electronic components stored away on his shelves has proved invaluable and saved us a lot of time. I’m thankful for all the department faculty and staff that taught me and helped me in numerous ways, along with my fellow graduate students who provided me with entertainment, and especially those who provided used of their vehicles. Finally, I would like to thank my family for all the support they have given me over the years and Andrea for putting up with me being 700 miles away for two years.

vii CHAPTER 1 BACKGROUND AND MOTIVATION

A magneto optical trap (MOT) has become a standard tool in recent years, allowing for cold atomic samples in the micro Kelvin temperature range. Doppler broadening and collision rates in the MOT are greatly reduced. These properties make cold atoms an attractive environment to conduct fundamental experiments in atomic and optical physics. Since this setting has become so popular it is important to fully understand it. The use of pump-probe spectroscopy allows for diagnostic measurements of the MOT while still in operation. Due to the low powers used in the probe, the measure- ments are nondestructive allowing for continuous measurement of the behavior of the atom in the trap and the parameters of the trap. Pump-probe spectroscopy can also be used right after the atoms are released from the trap as is the case with optical molasses which are formed by turning off the magnetic field gradient in the MOT. Through a careful study of the MOT or molasseses new information is gained about the dynamics of the cold atoms. One particular area of interest is the exploration of Raman transitions that occur between the light shifted Zeeman sublevels of the trapped atoms. By exploring the correlation between the light shift and radiation pressure among the trapped atoms one group [1] has suggested using negative pressures to induce an optical implosion of trapped atoms. Grison et al. [2] was the first group to experimentally show, using a probe beam, the existence of population differences in

1 Zeeman ground state atomic sublevels. Another group [3] has studied the polarization dependence of the probe when performing spectroscopy of a MOT. The differences they found are signatures of optical anisotropy in the cold atoms. Our own motivation to study Raman spectroscopy of the light shifted Zeeman sublevels of the cold trapped atoms is threefold. First, since there are many schools with working MOTs, there is clearly a peda- gogical interest to demonstrate spectroscopy on a sub-MHz scale. Almost all MOT setups use saturated absorption techniques to lock the trapping lasers. Saturated Absorption offers a great introduction to atomic spectroscopy at the few MHz level to the advance undergraduate and graduate student. We show here that with the addition of a weak probe beam that can be scanned in frequency, a $30 photo diode detector, and a standard oscilloscope one can easily perform atomic spectroscopy with 0.01-0.1 MHz resolution. Second, upon scouring the literature we have found, to our surprise, no evidence of a systematic measurement of light shifts in cold atoms as a function of laser intensity. Several interesting questions arise: Many atomic groups use the MOT as a target for a variety of spectroscopic mea- surements [4], without feeling a need to instead use molasses (obtained by turning off the magnetic field gradient in the MOT). Molasses are colder and denser than the MOT, however the disadvantage is that molasses are transient, with the atoms diffusing away in a few hundred milliseconds. Hence, the question arises: While the MOT is a more convenient test-bed than molasses, is it possible at all to reliably measure light shifts in the MOT? How badly are these measurements affected by the presence of the magnetic field gradient? The traditional theoretical treatment of the light shift is based on a completely coherent light-atom interaction where spontaneous emission can be taken to be ab- sent. How well does this coherent theory compare with measurements performed on cold atoms trapped by near-resonant light?

2 Third, our immediate goal is to build an optical lattice for the purpose of studying anomalous atomic transport that has been predicted in near-resonant lattices. We wish to use pump-probe spectroscopy as our main diagnostic tool to detect and char- acterize the lattice in the near future. The measurement of the light shift performed in this thesis serves as a good means for us to acquire expertise in the techniques associated with pump-probe spectroscopy. Here, we wish to use pump-probe spec- troscopy as a tool to examine the dynamics of cold trapped atoms. In this method, two beams of light are made incident on the atoms: one, the pump, is held at a fixed frequency while the other, the probe, is scanned in frequency around the pump frequency. By looking at the frequency-dependent absorption and amplification of the probe beam one can gather information about the population distribution of the atoms among the various Zeeman sub-levels, and also information about the shifts in these Zeeman levels. 1.1 Organization of this Thesis

In Chapter 2 the theory of light shifts is explained, we give background theory as well as a description of how we predict the light shifts we intend to measure. A substantial part of the thesis is also used to describe the MOT, which along with the pump-probe spectroscopy setup, forms the heart of our experiments. In Chapter 2 the principles of how atoms are cooled and trapped in the MOT are described, while Chapters 3-7 gives details on how these ideas are actually implemented in the lab. Chapter 3 describes the paths trap and repumper beams and the optics used to create the MOT. In Chapter 4 there is a description of the probe beam which is used to measure the light shifts. Chapter 5 explains how the lasers are locked. Chapters 6 and 7 explain the tools used to trap the atoms in the vacuum chamber. In Chapter 8 the results of our pump-probe spectroscopy experiments are displayed along with interpretation. Chapter 9 explains future outlook of the these experiments and where

3 to go next. It focuses on using another laser to create an independent pump-probe pair.

4 CHAPTER 2 THEORY AND PRINCIPLES OF LIGHT SHIFT MEASUREMENT

In this chapter we begin by presenting the principles of operation of a MOT, namely, how Doppler cooling and sub-Doppler cooling work. Next we present a theoretical calculation of the light shift for a simple two level atom. Finally we show how one can set up a simple theoretical model to estimate the light shifts we expect to measure for a real multi level atom. 2.1 Magneto Optical Trap

In order to collect atoms a magneto optical trap (MOT) is used. The basic idea is that optical beams are used to cool, i.e. slow, the atoms and a magnetic field is used to collect these slowed atoms at one specific location. The laser consists of a red detuned beam that is split into three beams, each retro reflected to form six beams counter-propagating in three orthogonal directions, all of which meet in the center of a vacuum chamber. Since the laser light is red detuned Doppler cooling occurs. The Doppler cooling slows the atoms down and creates an optical molasses. The cooling process is described in the next section. After the atoms are cooled a position dependent magnetic field from an anti-Helmholtz coil is used to collect the cooled atoms in the center of the chamber. The collection of atoms is described in Section 2.3.

5 2.2 Doppler Cooling

To understand how Doppler cooling works, the simple case of an atom moving along the z-axis in a one dimensional trap is examined. In a one dimensional trap two lasers beams, both of which are red detuned, are set up to be counter-propagating along the z-axis (Figure 2.1).

ω-Δ ω-Δ

Figure 2.1: The two counter-propagating beams are both red detuned causing the atom to be more in resonance with the beam it is traveling towards.

Since the atom is moving the frequency of the light becomes Doppler shifted and the atom sees a different frequency of light. The frequency of the light that the atom is traveling towards is shifted up proportional to the atom’s speed. Conversely the beam that the atom is traveling away from is shifted down to a lower frequency. Because the trapping beams are red detuned the atom is more likely to interact with a beam if it sees the beam as being frequency shifted up towards resonance. Therefore the atom is more likely to interact with the beam traveling opposite to the direction of the atom’s velocity. When the atom interacts with a photon it absorbs the photon and gets a momentum kick in the direction the photon was traveling. The atom, now in the excited state, must emit a photon and return to the ground state by either the process of stimulated emission or spontaneous emission. If the atom emits the photon by stimulated emission it will emit in the same direction as the incoming photon and thus, after this absorption-emission cycle, the atom is neither slowed

6 down nor sped up. However, if the atom returns to the ground state by the process of spontaneous emission the emitted photon has an equally likely chance to be emitted in any direction. The momentum kick given by spontaneous emission, when averaged over many photon recoils, yields zero. Thus, there is a net momentum loss from the absorption of photons in the direction opposite that of the momentum of the atom. From this momentum kick the atoms are slowed down, creating an optical molasses. This description can easily be expanded to three dimensions by including beams in both the x and y directions as well. 2.3 Trapping the Cooled Atoms in One Spot

The next step is to collect the cooled atoms in optical molasses all in one spot to build up a trapped ball. For this purpose, a magnetic gradient is used (see Section 6.3). The magnetic gradient produces zero magnetic field at the center of the chamber and increases the magnetic field with distance from the center. Due to the Zeeman effect the energy levels of each atom are shifted in proportion to the magnetic field. Shown below is a toy model of a Jg = 0 → Je = 1 atom, where J is the total (spin + orbital) angular momentum of the electron. (When we use J we are assuming the nuclear spin is zero. The symbol F is used when we need to include a non zero nuclear spin.) As shown in Figure 2.2 an atom positioned on the positive side of z axis (where z = 0 is defined as B = 0 and is in the center of the chamber) will be more likely to be excited to the mJ = −1 state since it is closer to resonance with the laser. The only way that the atom can be excited to the mJ = −1 state is if it absorbs a σ− photon. The σ− photons are all coming from the z > 0 side and thus the atom will receive a momentum kick towards z = 0. Since the magnetic field is reversed on the other side of z = 0 the atom is more likely to be excited by a σ+ and be pushed towards the center in the positive z direction.

7 E

1 (J = 1) 0

-1 hν laser σ+ σ- B

(J = 0) z

The atoms collect at one point (where B = 0 ).

Figure 2.2: The magnetic field gradient that causes the Zeeman splittings makes it preferential for the atoms to get pushed to the center.

2.4 Sub-Doppler Cooling

It is noted that the cold atoms achieve temperatures below the Doppler cooling limit. For example, the Doppler limit for the Rubidium atom is 148 µK. However, in the lab we attain temperatures of 50 µK or less. This leads to the realization that sub-Doppler cooling schemes are present. These cooling processes are discussed in Dalibard et al [5]. Below I will discuss the two different processes of sub-Doppler cooling: motion induced atomic orientation cooling, and Sisyphus cooling.

2.4.1 Motion Induced Atomic Orientation Cooling

In our MOT the trapping beams consist of σ+ and σ− counter-propagating beams. The interference of these polarizations create a linear polarization perpendicular to the axis of the beams that rotates in space, see Figure 2.3.

Using Jg = 1 → Je = 2 toy model of the atom, we can examine the process of

8 σ+ σ-

Figure 2.3: The polarization gradient for two counter-propagating beams of orthog- onally circularly polarized light. The result is a linear polarization that rotates in space.

motion induced atomic orientation cooling. First consider a stationary atom. Looking at the Clebsch-Gordan coefficients shown in Figure 2.4 we can find the probability of this stationary atom to be in a particular state. Since the net polarization seen by a stationary atom at any location is linear, the atom undergoes only linear excitations.

2 The probability of a transition from |1, 0i to |2, 0i is 3 whereas the probability of 1 going from |1, ±1i to |2, ±1i is 2 . It is seen that the atoms are more likely to end up in the |1, 0i state then the |1, ±1i states because,

!2 !2 r2 r1 1 |1, 0i → |1, ±1i : = (2.1) 3 6 9 !2 !2 r1 r1 1 |1, ±1i → |1, 0i : = (2.2) 2 2 4 As explained in Section 2.5, Equations (2.1) and (2.2) mean that the light shift is

9 greater in the |1, 0i state than the |1, ±1i states by a factor of 4 . Thus for a stationary 4 atom the probability of being in |1, +1i or |1, −1i is equal to 17 , and the probability 9 of being in |1, 0i is equal to 17 . Now if we consider an atom that is moving along the z-axis in the helix-like linearly polarized light field (Figure 2.3), then in the frame of the atom the polarization is

9 -2 -1 0 1 2 J'=2 σ − 1

π

σ + 1 J=1

-1 0 1

Figure 2.4: The Clebsch-Gordan coefficients for all the Zeeman sublevels of a J = 1 to J 0 = 2 transition. The top row gives the coefficients for left circular excitation, the middle for linear excitation and the bottom row for right circular excitation. For 0 example, the Clebsch-Gordan coefficients for the |J = 1, mJ = 1i ↔ |J = 2, mJ0 = 0i q 1 0 q 1 0 transition is 6 , and that the |J = 1, mJ = 0i ↔ |J = 2, mJ = 1i transition is 2 .

linear and rotating. To account for this rotation a switch to a rotating frame is made. This switch places the atom in a linear polarization just like the stationary atom, however there is an inertial field created from the rotating frame. The atom now “sees” a fictitious “magnetic field”-like term which creates a “Zeeman”-like shift given below that is dependent on the atom’s velocity:

∆E z ∝ −m“B” (2.3) ~ where “B” is a fictitious “magnetic field” of magnitude kv and our m is the magnetic quantum number. Equation (2.3) suggests that for an atom moving toward z < 0 (i.e. v < 0) the ground state sublevel with m=+1 gets a slight additional shift (upward closer to resonance with the excited state) while the m=-1 sublevel gets shifted an extra bit downward, see Figure 2.5. A slight population imbalance between the |1, +1i and |1, −1i states is created, with such atoms (v < 0) being in |1, +1i with probability >

4 4 17 and in |1, −1i with probability < 17 . Therefore these atoms (v < 0) preferentially

10 σ+ σ- σ+ σ- +1 -1 -1 +1 0 0

v < 0 v > 0

Figure 2.5: Motion induced population imbalance in |1, +1i and |1, −1i leads to sub-Doppler cooling.

absorb from the σ+ beam (6 times more likely than from the σ− beam, from Figure 2.4) and get pushed in a direction opposite to their motion, and are thus cooled. Similarly, atoms moving toward z > 0(v > 0) preferentially occupy the |1, −1i state and preferentially absorb from the σ− beam (6 times more likely then σ+) thus getting further cooled. This is the cooling mechanism undergone by the atoms in our MOT.

2.4.2 Sisyphus Cooling

Sisyphus cooling is another form of sub-Doppler cooling that arises from an en- tirely different mechanism. This type of cooling does not occur in our MOT, we only provide a brief description here for completion. Sisyphus cooling occurs in a light field created by orthogonally polarized counter-propagating linear light. When or- thogonal linear polarizations interfere the result is a light field of spatially oscillating

11 polarization, where the period of the oscillation is dependent on the wavelength of the light. Figure 2.6 shows polarization dependence on position of the light field. If we consider a toy model of a stationary atom with a Jg = 1/2 to Je = 3/2 transition (the simplest model necessary to understand Sisyphus cooling) (Figure 2.7), we see

+ 1 1 that if the atom were to see only σ polarization it would be pumped to the 2 , + 2 1 1 state. In this case the 2 , + 2 state has a light shift 3 times of the light shift in the 1 1 − 2 , − 2 state. The opposite is true for σ polarization. For linear polarization it becomes equally probable for the atom to be in either state and thus the light shifts are equal. This gives rise to oscillating energy levels in which are shown in Figure 2.6B. To understand the cooling we look at an atom in motion. Whichever ground state a moving atom happens to be in whenever it nears the top of the hill, the atom gets optically pumped to the other ground state, i.e. to the bottom. The atom keeps moving along and it loses energy climbing to the top of the next energy level before it is optically pumped back down again. In this way the atom keeps losing energy and further cools. 2.5 Light Shift for a Two Level Atom

The following is a calculation for the light shift of atomic energy levels of a two level atom in an electromagnetic field. This light shift is also known as the a.c. Stark shift. We start with the Schr¨odingerequation, ∂ i |Ψi = H |Ψi (2.4) ~∂t where |Ψi is the wave function and H is the Hamiltonian operator for the “light + atom” system:

H = H0 + V (2.5) where H0 is the total energy operator for the bare atom (no light). V is the light-atom interaction energy operator and is given by:

V = −e~r · E~ (r, t) (2.6)

12 a)

z

b)

Figure 2.6: a) The polarization gradient for the interference of two orthogonally polarized linear light beams. b) Energy level shifts of the ground state.

13 3 1 1 3 − − + + 3 2 2 2 2 J'= 1 2 σ − 1 3

π 2 2 3 3

1 σ + 1 3 1 J= 2 − 1 + 1 2 2

Figure 2.7: The Clebsch-Gordan coefficients for a J=1/2 to J’=3/2 transition. The top row gives the coefficients for left circular excitation, the middle for linear excita- tion and the bottom row for right circular excitation.

However since the wavelength of the light is much larger than the size of the atom the long wavelength approximation can be used and the r dependence on E~ can be ignored. Therefore, Vˆ = −e~r · E~ (t) (2.7) where −e~r is the induced dipole moment. We consider a 2 level atom with the bare eigenstates |gi and |ei (Figure 2.8)

Here Ee is the energy of the excited state, Eg is the energy of the ground state and ~ωeg is the difference in energy between the two states. For simplicity we set

Eg = 0 and thus Ee = ~ωeg. We have

H0 |gi = Eg |gi and H0 |ei = Ee |ei , (2.8) and since |gi and |ei form an orthonormal basis we also have,

hg |gi = he |ei = 1 and hg |ei = he |gi = 0 (2.9)

14 e

g

Figure 2.8: The energy levels of a simple two level model

Therefore, any arbitrary wave function can be expanded in terms of these orthonormal basis vectors,

|Φ(t)i = ag(t) |gi + ae(t) |ei (2.10)

Plugging this into the Schr¨odingerequation (2.4) we get,

~ (H0 − e~r · E(t))(ag(t) |gi + ae(t) |ei) = i~(˙ag(t) |gi +a ˙ e(t) |ei)

~ ~ agEg |gi + aeEe |ei + ag(−e~r · E) |gi + ae(−e~r · E) |ei = i~a˙ g |gi + i~a˙ e |ei) (2.11)

Projecting onto |gi and |ei), respectively, we find,

~ i~a˙ g = agEg + ae hg| (−e~r · E) |ei (2.12) ~ i~a˙ e = aeEe + ag he| (−e~r · E) |gi (2.13)

Assuming the excitation to be a plane wave, we have

~ E(t) =E ˆ 0 cos(ωLt) 1 = Eˆ e−iωLt + c.c. (2.14) 2 0 where  is the laser polarization and ωL is the frequency of the laser. The matrix

15 elements in Equations (2.12) and (2.13) can be simplified, using Equation (2.14). We find

he| (−e~r · E~ ) |gi = he| (−e~r) |gi · E~ 1 = − er · Eˆ (e−iωLt + eiωLt) (2.15) 2 eg 0 1 hg| (−e~r) · E~ |ei = − er · Eˆ (e−iωLt + eiωLt) (2.16) 2 ge 0

Defining the Rabi frequency χ as:

er · Eˆ χ = eg 0 (2.17) ~ and its complex conjugate er · Eˆ χ∗ = ge 0 (2.18) ~ we find upon substituting equations (2.15)-(2.18) in equations (2.12) and (2.13),

1 ia˙ = − a χ∗(e−iωLt + eiωLt) (2.19) g 2 e 1 ia˙ = a ω − a χ(e−iωLt + eiωLt) (2.20) e e eg 2 e

For convenience we make the substitutions,

ag(t) = cg(t) (2.21)

−iωLt ae(t) = ce(t)e (2.22) where ce(t) is the observable time dependent effect that survives the averaging over many optical cycles. Plugging Equations (2.21) and (2.22) into Equations (2.19)- (2.20) we obtain:

1 ic˙ = − χ∗c (e−2iωLt + 1) (2.23) g 2 e

∂ ia˙ = i (c (t)e−iωLt) e ∂t e 1 ic˙ = (ω − ω )c − χc (1 + e2iωLt) (2.24) e eg L e 2 g

16 Ignoring the terms oscillating at twice the optical frequency (this is called the

Rotating Wave Approximation, or RWA), and defining the laser detuning ∆ ≡ (ωeg −

ωL) we obtain thec ˙-equations:

χ∗ ic˙ = − c (2.25) g 2 e χ ic˙ = ∆c − c (2.26) e e 2 g

In order to conveniently solve these equations we may choose to write the atomic wave function, in the rotating wave approximation, as |ΦRWAi and express it as a column matrix as follows,   cg |ΦRWAi = cg |gi + ce |ei =   (2.27) ce

Noting that the Schr¨odingerequation can be cast as

∂ H |Φ i = i |Φ i (2.28) RWA RWA ~∂t RWA we see that equations (2.25) and (2.26), when expressed in the form

   −χ∗    i~c˙g 0 ~ cg = 2 (2.29)    −χ    i~c˙e 2 ~ ~∆ ce are merely a restatement of the Schr¨odinger equation in the RWA. In order to solve for the eigenvalues (denoted by λ below) of HRWA, we set the determinant equal to zero:

−χ∗ 0 − λ 2 ~ = 0 (2.30) −χ 2 ~ ~∆ − λ thus obtaining the eigenvalues  q  λ = ~ ∆ ± ∆2 + |χ|2 (2.31) ± 2

Equation (2.31) is saying that the ground state |gi has gotten shifted, owing to  q  ~ 2 2 the interaction of the atom with light, from 0 to 2 ∆ − ∆ + |χ| . This shift is

17 -0.003329638

g g

a b c

Figure 2.9: The shift in the ground state energy level of a two level atom. This |χ| is the light shift. a) The unshifted energy. b) When ∆ << 1 the energy shift is |χ| quadratic in |χ|. c) When ∆ >> 1 the energy shift is linear in |χ|. (see Equation 2.32)

called the light shift. The excited state has a symmetric light shift in the opposite direction. In the discussion that follows we focus exclusively on the light shifts in the ground state, not the excited state. This is because, for cold atom experiments, the excited state fractions are typically small, just a few percent, meaning that an overwhelming majority of the atoms spend most of their time in the ground state and are thus affected by spatial and temporal variations in the ground state light shift. The only time the excited states come into play is when the (red-detuned) excitation makes the atom undergo a transition, and even then the atom barely spends any time in

18 the excited state, typically hopping down to the ground state within a spontaneous emission life time (27 ns for Rb). 2.6 Simplified Light Shift Model for a Multi-Level Atom

If we turn our attention now from a two level model to a real multilevel atom we need to take into account the relative strength of the different transitions. In our case, the transitions of interest are the F = 3 → F 0 = 4 transitions in 85Rb. A diagram of the Clebsch-Gordan coefficients for the F = 3 → F 0 = 4 transition is shown in Figure 2.10. Including the probability of making different atomic transitions scales the light shift by the square of the Clebsch-Gordan coefficient. Thus, the light shift for any one of the ground states mF (−3 ≤ mF ≤ 3) is given by:

 q  ~ 2 2 2 (δLS)m = ∆ − ∆ + |χ| Cm ,m 0 (2.32) F 2 F F where CmF ,mF 0 is the Clebsch-Gordan coefficient for the transition between the ground state mF and the excited state mF 0 . In order to get Equation (2.32) in a form we can compare with measurements we use the relation of the Rabi frequency to the laser intensity I:

I 2 |χ|2 ≡ 2 (2.33) ISAT Γ where Γ is the natural line width and ISAT is the saturation intensity for all F = 3 → F 0 = 4 transitions in 85Rb. We find the light shift to be, in units of frequency,

s ! δ Γ ∆ ∆2 I LS = − + C 2 (2.34) 2 mF mF 0 ~ 2 Γ Γ 2ISAT We consider the case of a stationary multi-level atom for simplicity. This means that, from Figure 2.3 and as argued previously in Section 2.4.1, the atom sees just a linear polarization. Thus, in order to estimate the light shifts of the F = 3 ground state levels in Figure 2.11, we need to use in Equation (2.34) the Clebsch-Gordan coefficients corresponding to linearly polarized excitation. Because the light shift is

19 proportional to the square of the Clebsch-Gordan coefficient (see Equation (2.32), or Equation (2.34)) we can indicate the relative light-shifts of the F = 3 ground state sublevels as shown in Figure 2.11. The magnitude of the light shift for the |3, 0i

2 2 ground level is greatest with |C00| taking a value of 7 = 0.29, and the light shift for 2 1 the |3, ±3i ground levels is least with |C±3±3| taking a value of 8 = 0.125. We note from Figure 2.11 that the light shifts for the |3, ±2i, |3, ±1i and |3, 0i ground states are all rather similar. For simplicity we have decided to approximate 2 these close-lying states as one, with a single value of CmF mF 0 that is taken to be 0.25 (the average of 0.21, 0.21, 0.27, 0.27, and 0.29). In this case the number of ground states is effectively reduced to three and the Rb system begins to resemble the vastly easier to analyze J = 1 → J = 2 toy model shown in Figure 2.4. 2.7 Expected Signal of Light Shift Measurement

In an idealized pump-probe measurement of the light shift, there is a) a strong pump beam that creates the light shifts, and b) a weak probe beam that extracts information about the atomic population distribution without affecting the environ- ment created for the atoms by the pump. The pump is fixed in frequency, while the probe beam is frequency-scanned around the pump. In our case, the six σ+σ− MOT beams act as the pump beams and the probe is a weak linearly polarized external beam that is almost collinear with one of the MOT beams. The whole point of Section 2.6 above is that the expected signal for a J = 1 → J = 2 toy atom would not look too different from that for the 85Rb F = 3 → F = 4 atom. Consider a J = 1 → J = 2 atom subjected to a strong pump beam, with light shifts as indicated in Figure 2.12 the |1, 0i ground sublevel is shifted the most

2 2 2 with |C0,0| taking a value of 3 while for the |1, ±1i levels |C±1,±1| takes the value 1 2 (see Figure 2.7). We have already shown in Equations (2.1) and (2.2) and the 9 discussion thereafter that most of the atomic population ( 17 th) resides in the |1, 0i 4 ground sublevel, while the |1, ±1i sublevels have 17 th each.

20 Lets consider the action on this toy atom of a pump with fixed frequency ωpump and a probe with variable frequency ωprobe, both of which are red-detuned from resonance with the excited state. Just as in the case of the sub-Doppler cooling with σ± beams, we switch to the reference frame of the helically rotating linear polarization, so that the pump beam has linear polarization. However, the probe now has π, σ+ and σ− components. We show below that, in conjunction with the π-polarized pump, the σ+and σ− components give rise to Raman transitions between the light-shifted levels in Figure 2.12. The π-polarized component of the probe does not play any role.

When ωprobe < ωpump , ωpump is closer to resonance with the excited state and thus is far more likely than ωprobe to pump the atoms in the |1, 0i state to the excited state

(to the |2, 0i state because ωpump is π-polarized) and then, by stimulated emission back into the pump, to the |1, 0i state again. However, as ωprobe approaches the + − value ωpump − δLS the excited atoms are increasingly stimulated by the σ and σ components of the probe to emit into the probe beam and drop from |2, 0i to the |1, ±1i ground states (see Figure 2.12 (a)). This absorption from the pump and stimulated emission into the probe causes a gain in the transmitted probe power.

Note that the probe gain peaks when ωprobe = ωpump − δLS.

On the other hand, as ωprobe becomes larger than ωpump , it is now ωprobe that is closer to resonance with the excited state and thus far more likely than ωpump to excite the atoms sitting in the |1, 0i state to the excited state (to the |2, ±1i states) and then, by stimulated emission back into the probe, to the |1, 0i state again. However, as shown in Figure 2.12 (b), when ωprobe approaches the value ωpump +δLS the excited atoms are increasingly stimulated by the pump to emit into the pump beam and drop from |2, ±1i to the |1, 0i ground states. This absorption from the probe and stimulated emission into the pump causes a loss in the transmitted probe power. Note that the probe loss peaks when ωprobe = ωpump + δLS.

In between, when ωprobe = ωpump , neither ωprobe nor ωpump are efficiently tuned to cause stimulated emission from the excited state to the |1, ±1i states. As shown in

21 Figure 2.12 (c), both the pump and the probe are equally likely to excite the atom to the excited state ( |2, 0i for pump and |2, ±1i for probe), and by stimulated emission back into themselves, back into the |1, 0i state. Of course, if a photon is absorbed from a beam and then emitted back into it by stimulated emission, the beam shows neither gain nor loss. Thus, we expect to see in the probe transmission spectrum a gain “peak” centered at ωprobe = ωpump − δLS and a loss “dip” centered at ωprobe = ωpump + δLS , as shown in Figure 2.12 (d). Measuring the frequency separation between the centers of the peak and the dip yields twice the light shift. Thus the light shift is determined. Note that we expect the size of the dip to be larger than the peak-gain because the pump is typically an order of magnitude stronger in intensity than the probe. Thus an atom sitting in the excited state in Figure 2.12 (b) is more likely to emit a photon by stimulated emission into the pump and drop to the |1, ±1i states than emit by stimulated emission back into the probe and drop down to |1, 0i. This gives rise to a large loss of photons from the probe into the pump. On the other hand, an atom sitting in the excited state in Figure 2.12 (a) is less likely to emit a photon by stimulated emission into the probe and drop to the |1, ±1i states than emit by stimulated emission back into the pump and drop to |1, ±1i. This gives rise to a small gain of photons from the pump into the probe.

22 Figure 2.10: The Clebsch-Gordan coefficients for a F=3 to F0=4 transition. The top row gives the coefficients for left circular excitation, the middle for linear excitation and the bottom row for right circular excitation.

23 F'=4 J'=2

.125 .125 -3 3 -1 1 .21 .21 F=3 J=1 .27 .29 .27 -2 2 -1 0 1 0

Figure 2.11: The light shift differences in the F=3 atomic states. The square of the Clebsch-Gordan coefficient is shown above each state. The circled states are all approximately the same compared to the |3, ±3i states, thus they can be averaged and approximated as one state.

24 (a) (b) (c)

unshifted

-1 +1 -1 +1 -1 +1 0 0 0

Transmission Probe Intensity

Figure 2.12: Top: Three cases where the which give rise to the Raman signals. (a) Probe gain (b) Probe loss (c) Probe gain = Probe loss. The pump beam leads to π-polarized transitions and is shown with dashed arrows. The probe beam leads to σ+σ− transitions and is shown with solid arrows. Below: Depiction of how a probe transmission signal would look.

25 CHAPTER 3 THE TRAPPING AND REPUMPER SYSTEMS

In order to create the magneto optical trap two lasers are used, a trap laser and a repumper laser. The trap is the main laser that cools the atoms in the chamber. To get the beams ready to trap they must first go through a series of optics. In this chapter I will explain the path of the trap and repumper beams through each component. The diagram of the trap beam path, shown below in Figure 3.1, will be referenced throughout the chapter. 3.1 The External Cavity Diode Laser

The laser is an external cavity diode laser (ECDL). The diode laser we use is a single mode diode laser from Sharp (Model # GH0781JA2C). The current from the laser diode is controlled by ILX Lightwave LDX-3620 Ultra Low Noise Current Source. This current source has a stability of 10ppm over a 30 minute time period. The laser diode is mounted in an aluminum mount that has a hole for a thermistor near the laser. On top of the mount is an attached thermo electric cooler and a heat sink that can be used to control the temperature of the laser. This thermo electric cooler is run by ILX Lightwave LDT-5910B temperature controller which has temperature stability control of 0.005oC. The laser beam is first collimated by a lens with a 5 mm focal length. This is best done by sending the beam out a few meters and adjusting the lens until the far spot is the same size as it is near the lens. The whole laser setup is fastened on a mount with a fine pitch screw for adjusting the separation

26 H1 PBS2 NPBS APP P1 L2 I1 H3 H2 PBS1 L1 ECDL1 Faraday X-Axis Rotator Trap Beam

AO “-1”

To Saturated Absorption Z-Axis Y-Axis Trap Beam Trap Beam

Figure 3.1: The diagram for the trapping beam.

of the lens from the laser. Directly after the collimating lens a non polarizing beam splitter splits the laser into two paths. The reflected light creates the main beam used for atom trapping and the transmitted beam is incident on a diffraction grating. This diffraction is the frequency tuning element in the ECDL. The diffraction grating is aligned to send the -1 order beam back into the laser. The beam travels back through the beam splitter creating another beam out the other port of the beam splitter to be used for frequency locking the laser via saturated absorption. The diffraction grating is on a mount that allows for control over the angle at which the light hits it. This control allows for changes in the frequency of the first order light sent back to the laser. While the mount gives fine control, still finer control is needed. A piezo electric transducer is placed in between the horizontal adjust screw and the front plate of this mount. This piezo can be adjusted, by applying a voltage to it. By ramping the piezo the frequency of the laser can be scanned. The mount that the laser setup is on is placed on a rubber matting (Sorbothane) to help isolate the laser system from table

27 vibrations. All of this is encased inside an aluminum box to help isolate the laser from airflow, temperature changes, and stray light from other sources. The aluminum box has small holes to allow for the adjusting of the diffraction gratings horizontal and vertical tilts, as well as holes to allow the output of the two beams coming from the laser, one for trapping, the other for frequency locking as already mentioned. 3.2 Anamorphic Prism Pair

The trap beam coming out from ECDL1 is first shaped to a circular beam by an anamorphic prism pair, APP as indicated in Figure 3.1. The anamorphic prism pair is a set of two prisms that shape a beam of light. Since the light from the diode laser comes out in an elliptical beam, the prisms are used to shape the light into a circular beam. The elliptical beam is circularized by compressing or expanding the beam along one of the axes of the beam profile. The angle at which the beam is incident on the first prism surface is chosen to be Brewster’s angle; this minimizes loss through the prism pair. The polarization must also be set to satisfy Brewster’s condition. This is done by simply rotating a half-wave plate (H1) placed before the prism pair and measuring the output power on the other end of the prisms. The angle of the second prism is chosen to change the magnification along the horizontal axis, thus it’s placement depends on the ellipticity of the incoming beam. The anamorphic prisms are mounted 1.5 mm apart from each other. This distance does not affect the magnification. However, it will affect how far the beam is displaced from the incident line. The angles and displacements are given in tabular form in the Melles Griot catalog [6] In order to decide the orientation of the prism pair for magnifications that are not specified in the table the data was graphed in Figures

3.3 and 3.4. Figure 3.2 shows the layout of the prism pair. Angles α1 and α2 can be changed and the distances ‘e’ and ‘d’ are changed to align the second prism to the beam. It is also noted that this was done to expand beyond the given parameters

28 1.5

A

E D

B α 1 α 2

Figure 3.2: Orientation of the anamorphic prism pair the angles and distance from center line are altered to obtain different magnifications.

of 2 to 6 times magnification. However, this ended up not working very well as the deflections became too large to fit on the prism pair. The prism pair for the trap beam is set to magnify -1.88 times along the horizontal axis. 3.3 Faraday Rotator

Next the circularized beam is sent through a Faraday Rotator, as shown in Figure 3.1, in order to cut out any back reflections. An optical Faraday rotator is a device that acts like a one-way valve for light. The light is sent through one way and any retro

29 50

40

30 ) g e d ( 20 e l g n A 10

0 2 2.5 3 3.5 4 4.5 5 5.5 6

-10 Magnification

Figure 3.3: This graph shows the angles that should be used in order to create a specified magnification. The solid line is α1 and the dotted line is α2 on the APP diagram Figure 3.2.

30 8

7.5

) 7 m m (

n o

i 6.5 t a r e ep

S 6

5.5

5 2 2.5 3 3.5 4 4.5 5 5.5 6 Magnification

Figure 3.4: This graph shows the displacements that should be used in order to create a specified magnification. The solid line is e and the dotted line is d on the APP diagram Figure 3.2.

31 reflections are kept from coming back and causing interference with the laser. Thus the device acts as an “optical isolator” for the laser, isolating the laser from unwanted feedback. The device works by using a pair of polarizers and a magnetic field. After passing through the first (input) polarizer the light is linearly polarized. Next the light passes through a rare-earth doped optical crystal across which a collinear magnetic field is applied. The polarization of the light is rotated by 45o due to the Faraday effect and allowed to pass through the second (output) polarizer. Any retro light will pass back through the output polarizer first and rotate by another 45o in the magnetic field such that it finds itself at 90o with respect to the first polarizer’s transmission axis. Two different types of Faraday rotators are used in our experiment, model IO-5- NIR-LP and IO-2.5D-780-PBS from Optics for Research. For the trap beam we use Model IO-5-NIR-LP which has a larger active aperture with a diameter of 4.9 mm and allows the user to achieve 45o Faraday rotation in the optical crystal for a wide variety of wavelengths by tuning the length of the crystal across which the collinear magnetic field is applied. Furthermore, the polarizers can be independently rotated to increase efficiency. In order to maximize the transmission through model IO-5-NIR-LP we follow these steps: 1) The beam is first aligned through with the output polarizer removed. The input polarizer is rotated for maximum transmission. 2) The Faraday Rotator is then flipped around so the laser enters through the “output” end. The output polarizer is added and rotated to minimize transmission, this will be around 45o with respect to the first polarizer. 3) The Faraday Rotator is now flipped back to the correct orientation, and the length the crystal is in the magnetic field adjusted to maximize transmission. Steps 1-3 may need to be reiterated two or three times before both “maximum transmission” and “maximum isolation” are achieved. Typically we can achieve 86% transmition with -32 dB isolation.

32 3.4 Creating Six Trapping Beams

After the Faraday Rotator, a pinhole is in place for alignment and to block the rejected light from the Rotator. Next, a half-wave plate (H2) and a polarizing cube beam splitter (PBS1) allows for the diverting of some light to the probe beam (see section 5) and the rest continues onward as the trap. Maximum transmission through the beam splitter is 98 percent. Two mirrors are used to align the beam through a telescope set up formed by the lens system L1-L2 in Figure 3.1. This telescope set up expands the beam to a large size and a portion is selected with an adjustable iris (I1) to be used as the trap beam. The portion we select is of fairly uniform intensity across its cross section. The beam is expanded by being put through a 22:1 microscope objective (L1) and then collimated with a large 20 cm focal length lens (L2) placed 20 cm away. The iris is then placed right behind the lens. When the iris has a 15mm diameter a typical power is 5.85 mW. The beams are then divided into three beams via two cube beam splitters the first being polarizing and the second being a 50/50 non-polarizing cube beam splitter (PBS2 and NPBS respectively in Figure 3.1). A half-wave plate (H3) is used to set the polarization to divert 20 percent of the power, the remaining 80 percent gets split by the next beam splitter into two beams each of 40 percent of the original power. The 20 percent beam is sent up through the large windows of the chamber and the the 40 percent beams are but through the smaller windows making an X shape. Each of the beams is sent through a quarter-wave plate just before entering the chamber so that the light becomes circularly polarized. The orientation of the quarter-wave plates is critical [7]. After the chamber the beams go through another quarter wave plate and then are incident on a mirror that retroreflects the light. Thus, the retroreflected beam is circularly polarized again but in the opposite direction. The retro beam traverses all the optics again and is aligned so the majority of the beam just misses a pinhole

33 (P1) directly after the Faraday Rotator (see Figure 3.1). This small mis-alignment is used to eliminate feedback into the laser while still keeping the beams overlapping in the chamber. In order to align the beams through the chamber plastic caps are placed over the small windows with dots in the center of the caps. When the beam is centered on both caps the beam also travels through the center of the chamber. For the beam that goes though the large window we use Plexiglas shields, made by Michael Eldridge the Physics instrument maker, to align the beam. These shields are semi circles that fit over the large window and have periodic markings on them. The purpose of the markings is to later enable the introduction of Lattice laser beams into the chamber at a variety of predetermined angles such that they pass through the center. The trap beam can be aligned to the center of each of the shields and thus through the center of the chamber. 3.5 Repumper Laser

The repumper beam setup is very similar to the trap beam setup. The beam first goes through an anamorphic prism pair to become circularized. The beam then is directed through a Faraday Rotator of (Model IO-2.5D-780-PBS). Since the power of the repumper is less crucial then of the trap the smaller Rotator is used. Typical transmission through this Rotator is 70 percent. The repumper is then sent on a path using five mirrors before going to a telescoping set up to be expanded, collimated and introduced into the chamber. The reason five mirrors are used is the repumper and trap beam need to spatially overlap at a point where a shutter is introduced for timing purposes for future experiments with an optical lattice. For the telescope set up a 22:1 microscope objective is used to expand the beam and a large 95 mm lens is used to collimate the beam. Again an iris is used to select a bright uniform region of the beam. The profile of the repumper is far less crucial

34 than the trap laser and as such a much larger portion of the beam is used. When the iris is set to 15 mm a typical power of the repumper is 2.4 mW. The repumper is then directed with two mirrors to the polarizing beam splitter and combined with the trap beam at PBS2 in Figure 3.1. It does not matter what percent of the repumper ends up in what beam just that the light get to the atoms.

35 CHAPTER 4 LOCKING THE LASERS

In order to cool atoms the lasers need to be locked to high precision, within a fraction of an atomic line width (6 MHz for Rb). To accomplish this, saturated absorption and a locking circuit are used. In this chapter I explain how the lasers are locked. 4.1 Saturated Absorption Set Up

Saturated absorption is a spectroscopic technique that allows analysis of the laser frequency using atoms at room temperature. A laser beam passing through a va- por cell filled with Rb gas is absorbed when the laser’s frequency is resonant with the Rb atoms. This absorption can be viewed on an oscilloscope. However the fre- quencies are Doppler broadened due to the fact that the atoms are moving, thus the frequencies are in poor resolution. To fix this problem another much stronger beam is directed counter propagating with the first weaker beam, which saturates the absorption of atoms traveling perpendicular to both beams. Those perpendicular propagating atoms are the only ones that are in resonance with both beams. Since the atoms are moving perpendicular to the beams the light sees them as stationary. The atoms do not absorb any of the weak resonant beam because the stronger beam saturates their absorption, thus there are dips in the absorption spectrum of the weak beam. The saturated absorption setup (see Figure 4.1) uses the beam from the back of ECDL1. A thick glass plate (G1) is used to split the beam into two weak reflected

36 For trapping (see Figure 3.1)

ECDL1 M2 G1 G2

Vapor Cell

Scope

I-V Converter M1

Figure 4.1: The Setup for Saturated Absorption.

beams, each four percent of the original beam, and a strong transmitted beam. The weak beams are aligned through the vapor cell and then directed, using a thin glass plate (G2), in to two photo detectors. The photo detectors are wired in subtraction so the difference between the beams is what appears in the output to the oscilloscope. Just looking at one beam, if the laser is frequency ramping around resonance with the Rb atoms in the vapor cell the Doppler broadened peaks are seen. The large peak represents 85Rb, which is 72% abundant, and the smaller peak is 87Rb, 28% abundant. When the light from both beams is directed into the detectors equally the Doppler peaks are canceled out and a flat line is visible on the oscilloscope.

37 The strong beam is then, with two mirrors (M1 and M2), directed through the va- por cell from the other side and aligned such that it is completely counter propagating with one of the weak beams. The absorption in that beam is now saturated and as such there are dips in the absorption signal where the laser frequency is the resonance frequency of the Rb atoms. Since these dips occur only in one of the weak beams the subtracted signal shows only these dips, which are the hyperfine peaks. In 85Rb the peaks correspond to the transitions F = 3 → F 0 = 2, 3, 4 and F = 2 → F 0 = 1, 2, 3 while in 87Rb the peaks correspond to the transitions F = 2 → F 0 = 1, 2, 3 and F = 1 → F 0 = 0, 1, 2. There are also extra peaks in the spectrum, these peaks are the crossover peaks. The crossover peaks are “fake” peaks that occur at the arithmetic mean between the real hyperfine peaks. Because of the Doppler effect the atoms in the combined (strong and weak) beam see a laser frequency of ω − ∆ on the strong and ω + ∆ on the counter-propagating weak beem. ∆ being the Doppler shift. Thus, if the atom has more than one hyperfine resonance frequencies, say ω1 and ω2,then there is a frequency where the atom can be in resonance with both the strong and weak beam besides ω = ω1 and ω = ω2. This occurs if ω − ∆ = ω1 and ω + ∆ = ω2, i.e. by

ω1+ω2 adding the zero equations, we find this occurs at ω = 2 . Thus, here the ∆ = 0

ω1+ω2 case creates the “real” peaks and the ω = 2 correspond to a “fake” peak. This “fake” peak appears much bigger than the “real” peaks since different velocity classes may contribute to the signal, not just the one velocity class moving perpendicularly to both the strong and weak beams in the “real” peak case. 4.2 Lowering Threshold

Since the lasers tunability comes from the fact that the 1st order light reflected off a diffraction grating is directed back into the laser it is crucial that the set up be aligned optimally. In order to check that the set up is aligned the threshold of the laser is lowered. To lower the threshold the current is turned down until the output

38 light displays a dramatic drop in intensity, this usually occurs at a current somewhere around 30 mA but is unique for each laser. Once this threshold is found it can be lowered. By adjusting the vertical and horizontal screws on the mount that holds the diffraction grating the output light will change in intensity. The goal is to make the current at which the threshold occurs to be as low as possible. The best way to achieve the lowest threshold is to lower the current just below threshold and then adjust the vertical and horizontal screws until the highest intensity is achieved, then repeating the process. The process is repeated until changing the screws can not produce a higher intensity of the output beam. It is easier to find the threshold if the ramp to the laser is turned all the way down or unplugged. The collimation screw can also be adjusted to gain a lower threshold. Including this screw just adds another iteration to the process, all three are adjusted to achieve the lowest threshold. Once the threshold is lowered the current is increased to a desirable level for operating the lasers. This level depends on the laser and on the temperature but is around 100 mA. With the ramp back to maximum the horizontal screw is turned very slowly clockwise while looking in the vapor cell with an IR viewer. The screw should be turned until the resonance flash is visible in the vapor cell. This is easier to see with the room lights off. Once a faint resonance is seen the horizontal screw can be adjusted to obtain maximum brightness. To gain finer control attention is turn to the oscilloscope and the current is adjusted until the resonance spectrum are seen, recall Section 4.1. If the threshold has been properly adjusted both the 85Rb F = 3 → F 0 and 87Rb F = 2 → F 0 families of peaks should be visible in one scan without any mode-hops. In order to lock the lasers to a particular frequency, a homebuilt servoloop is used. A home built function generator is used to produce a triangle wave that supplies the ramp for the laser as well as the triggering for the oscilloscope. In our case the same function generator is used to ramp all three lasers. A “locking circuit” takes the

39 amplified saturated absorption signal and controls the laser’s piezo to hold the laser at a particular frequency. Use of the locking circuit to lock is further explained in Section 4.4. Below is a diagram of how the circuit devices are connected (Figure 4.3). 4.3 Acousto Optical Modulator

The trapping beam is unique among the lasers in our set up in that before the saturated absorption takes place the beam is first passed through an AOM. The AOM shifts the frequency of the beam in order to lock the Saturated Absorption beam on a easily lockable peak, while the trapping beam frequency is then actually near the actual resonance. The Acousto Optical Modulator, or AOM, diffracts light with sound waves using the acousto optic effect. That is, a periodic mechanical strain is introduced into the crystal by an acoustic wave, which the laser passes through. This strain in the crystal sets up a periodic variation of the index of refraction, which in turn creates a diffraction pattern. This induced diffraction grating causes the laser light to experience Bragg diffraction, enabling frequency shifting of the light. The acoustic wave is set up in the crystal by an RF-wave driver; the driver uses an oscillating electrical voltage to vibrate a piezoelectric transducer. The diffraction can be related by the following equation:

mλ sin θ = (4.1) 2Λ where θ is the angle of deflection, m is the order of diffraction (...-2,-1,0,1,2...), λ is the wavelength of the laser, and Λ is the wavelength of the modulating sound. When aligning a laser through the AOM the incident angle can be altered to maximize transmission in the desired order. In our case this is either the +1 order or the -1 order. One should be able to achieve transmission of around 75 percent in the desired order. By changing the voltage applied to the driver the frequency and angle of the laser

40 light is changed. The AOM can change the frequency between 60 and 100 MHz. The different drivers have specified voltages for the corresponding change in frequency. In order to control how much the frequency changes when the light passes through the AOMs, we built circuits to supply variable voltages to the “VT ” terminal of the AOM drivers. We have three AOMs all from ISOMET, for two of them the crystal is model # 1205C-2-804B and the Driver is model # D322B-805, the third AOM more recently purchased has the same crystal but the driver is new and is model # 620C-80. The two different models have different requirements. Two identical AOM units are used for the pump and the probe. One needs to have a ramping voltage added and the other just needs a fixed voltage. The third AOM has different voltage needs. All three AOM voltage delivery circuits are designed to use the 15 Volts that is readily available in our lab. The AOM that is used to offset the locking frequency in the trap laser saturated absorption setup has different specifications then the other two as it is a newer model. Instead of needing 5 to 15 Volts the AOM uses 0 to 10 Volts. This poses difficulty when designing from a 15 volt input since the input impedance needs to be less than 50 Ω. Thus a 741 op-amp is used with a voltage divider to one input and the other being attached to ground. For our resisters selected the output voltage goes between 3.9 and 9.3 Volts, this provides ample range to work with. 4.4 Locking the Trap Beam

Other then the AOM the saturated absorption setup is the same as described in Section 4.1. The output of the differencing photo diodes in Figure 4.1 is first amplified via a current voltage converter, then se to the locking circuit. The locking circuit has three knobs and two switches, the knobs are: “horizontal”, “vertical”, and “gain/ramp” and the switches are: “invert” and “lock”. The BNC connections are “saturated absorption in”, “saturated absorption out”, and “ramp in”. When resonance can be seen in the vapor cell, the current to the laser is finely adjusted

41 until the saturated absorption spectrum from “saturated absorption in” is seen. Once found, the ramp knob is used to zoom in at points a) b) or c) on the red detuned side of crossover peak between the F = 3 → F 0 = 4 and F = 3 → F 0 = 2 transitions, see Figure 4.5. This side is kept in center of the scope by using the “horizontal knob” of the locking circuit and the “offset knob” of the current to voltage amplifier. The “saturated absorption out” signal must also be kept centered by adjusting the “vertical knob” on the locking circuit. Once zoomed in until the signal is a flat line the “lock” switch is flipped and the “gain/ramp knob” is increased until just before the onset of feedback oscillations. If the line stays in the center of the oscilloscope the beam is locked. If the line drifts up or down immediately after locking, the “invert” switch maybe set for the wrong slope, flipping it and locking again may work. In order to see how far away from resonance the trapping beam is, the oscilloscope must first be calibrated. From the spectrum on the oscilloscope the distance between two peaks are measured. Since the peaks occur at known frequencies a ratio of oscilloscope distance to frequency is set up, in our case it is 92 MHz = .0198 ms. Thus if we chose to lock at point b) in Figure 4.5 which is 97.1 MHz red detuned from the trapping peak, (which is shifted by a frequency of 86 MHz upward courtesy of the AOM) the trap beam is only 11.1 MHz from the trapping peak and thus 1.85 Γ red detuned (where Γ = 6 MHz, the natural line width of Rubidium). Locking near the top of the peak at point c) allows for a trap laser detuning of -1.23Γ and locking near the base at a) is -2.47Γ. The repumper has a similar locking circuit to the trap laser. The repumper is locked on the left side of the F=2 85Rb spectrum (Figure 4.2 a)). This is easiest to find by first locating the stronger F = 3 → F 0 (Figure 4.2 c)) and 87Rb F=2 → F’ (d)) spectra and turning down the current slowly until you find the 85Rb F = 2 → F 0 spectrum (a)).

42 a) b)

d) c)

Figure 4.2: Pictures of the different saturated absorption spectra a) 85Rb F = 2 → F 0 = 1, 2, 3 and is the repumping transition. b) 87Rb F = 1 → F 0 = 0, 1, 2. c) 85Rb F = 3 → F 0 = 2, 3, 4 and is the trapping transition. d) 87Rb F = 2 → F 0 = 1, 2, 3. It is noted that these pictures are not on the same scale, the peaks in b) and d) are really farther separated than those in a) and c).

43

Figure 4.3: This shows the connections made to and from the locking circuit.

44 +15V

R 3

V- -15V 10kΩ V out + +15V V+ 741

R 4 R 1 R 2

Figure 4.4: Circuit diagram for delivering the voltage to the AOM that offsets the locking frequency. R1 and R2 = 18 kΩ, R3 = 6.5 kΩ, R4=1kΩ, Vout = 3.9 to 9.3 Volts.

45 85 Trap laser

0.02

0.01

0 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 a) -0.01

b) -0.02

c) -0.03 -0.04

Figure 4.5: The saturated spectrum of the trapping transition F = 3 → F 0 = 4, 85Rb for the trap laser.

46 CHAPTER 5 FREQUENCY SCANNABLE PROBE

In this chapter I describe how the probe beams are set up to be frequency tunable around the frequency of the pump beams. In order for the pump-probe spectroscopy to happen, both the pump and probe beam need to be derived from the same laser. In our case the “pump beam” is formed by the six MOT trapping beams and the probe beam is a single beam that is derived from the trap laser. The probe needs to be scanned in frequency around the pump frequency which is kept fixed. An AOM is used to scan the probe. However since the AOM imparts an offset frequency shift the probe must be directed through two AOMs: the first AOM detunes the probe away from the pump by a fixed frequency and the second tunes it back but ramps the frequency thus scanning around the pump. The location of the probe must also be spatially held fixed at the site of the cold atom cloud so both AOMs need to be set up in a double pass system. A polarizing beam splitter (PBS1 in Figure 5.1) is used to derive some light out of the MOT trap beam and form the probe beam. Since the probe beam needs to be very weak only 6 percent of the light needs to be diverted. Using two mirrors the probe is directed into the first double-pass AOM set up. The beam first goes through a half-wave plate (H3) and then polarizing beam splitter (PBS2). The half wave plate is set to maximize the light bent through the beam splitter towards the AOM, the maximum transmission is about 93 percent. 300 mm lenses (L1 and L2 in Figure 5.3) are used to focus the beam down to a smaller size to maximize transmission through

47 Scope Amp

H1APP H2 PBS1 Trapping/Pump ECLD1 Faraday Beam Rotator Trapped Atoms H3 AO “-1” Trapping/Pump Subtraction PBS2 AO1 “-1” Beam Beam To Saturated Q1 Absorption H4 Q2 Scanning Probe PBS3 AO2 “+1” Beam

H5 P1 L3 H6 PBS4

Figure 5.1: Layout of the beam paths for pump-probe spectroscopy of the MOT and molasses.

the small aperture of the AOM. Care is taken however not to focus the beam too small or the beam will not interact with enough spatial periods of the diffraction grating formed in the crystal to be properly detuned. The waist of our beam is calculated to be 0.15 mm at the AOM. Positioning the AOM so that the -1 order is maximum, about 82 percent is trans- mitted in that order. The beam then goes through another 300 mm lens to become collimated and a quarter-wave plate (Q1 Figure 5.1) and then is incident on a mirror. It passes back through the setup along the same path, aligned with pinholes. The mirror is adjusted to allow for maximum transmission back through the AOM in the -1 order. Since the beam traverses the AOM twice it experiences two frequency shifts both in the same direction. After the beam is collimated again it passes through the polarizing beam splitter, since the beam passed twice through a quarter-wave plate the polarization is rotated allowing the beam to take a new path as it passes

48 back through the beam splitter (PBS2). The quarter-wave plate (Q1) is turned to maximize this transmission. This transmission is around 62 percent of the light in the single AOM pass. Using two more mirrors the beam is directed into a second AOM double pass set up (Scanning AO). After a half-wave plate (H4) and a polarizing beam splitter (PBS3) the max transmission is around 89 percent. Again the AOM is adjusted for maximum transmission however this time it is in the +1 order. Maximum transmission of around 71 percent is achieved. After being sent back through the AOM for a second pass with the quarter-wave platethere is a transmission of 66 percent. This AOM has a modulating voltage applied to it so the beams frequency and position (after the first pass) is scanning. To control the power and polarization of the probe, it is sent through a half-wave plate (H5) and a polarizer (P1). Thus we can vary the power with the half-wave plate and vary both the polarization and power with the polarizer. The probe beam is sized to a 2 mm diameter with a pinhole and then put through a 750 mm lens (L3). This lens focuses the probe on the ball of atoms. At the atoms the probe has a diameter of 0.37 mm, calculated by the formula: Dπw f = (5.1) 4λ where f is the focal length of the lens, D is the diameter of the incident beam, w is the diameter of the beam at the focal point, and λ is the wavelength of the beam. After the lens the probe is sent through a half-wave plate (H6) and a polariz- ing beam splitter (PBS4). The beam splitter creates two beams, the probe aligned through the ball of atoms and a subtraction beam which is aligned close to the atoms but not through them. The half-wave plate (H6) is set to make both beams equal intensity so they will subtract each other (a 50/50 nonpolarizing beam splitter was originally used in place of PBS4, however, it was found to not be able to split the beam exactly evenly so the added control, given by the half-wave plate, was needed). After going through the chamber the beams are directed to a pair of photo diodes tied

49 in subtraction (these are the same design as the detectors for the saturated absorption setups). In order to align the beams the probe intensity is turned up enough to see an effect on the ball, this is done by diverting more power from the trap to the probe. The last mirror before the chamber is adjusted to send the beam right through the atoms, a “tail” of atoms being blown away by the strong resonant laser beam should be visible on the screen and can be centered on the ball. Since the camera is not at the same angle as the beam it is helpful to look in to the chamber directly from behind the probe beam and a hole should be visible in the atom ball making it look like a donut. Once this is accomplished the probe power is turned back down so it is below the point where the atom ball seems affected. The subtraction beam is then blocked and the probe is optimized into one of the detectors by maximizing the output on the oscilloscope using the mirror after the chamber. The probe beam is then blocked and the subtraction beam is similarly optimized into the other detector. With the beams blocked and no trapped atoms the signal is set to zero using the offset on a current to voltage converter. Next, unblocking the beams the signal is set to zero by adjusting the half-wave plate (H6) before the probe and subtraction beam separation. This assures that the beams are both equal intensity at the detectors and as such any technical noise from the beam should be canceled out. The AOM that is used for a fixed frequency offset to the probe beam has the easiest circuit. It is simply a voltage divider with one of the resistors being a potentiometer. The resistor values are chosen such that a voltage range of 4.28 to 14.73 volts is achieved for the output. R1 = 39 Ω and R2 = 50 kΩ POT plus a 488 Ω resistor. For the AOM used ramp the probe beam through different frequencies a ramping voltage must be applied. In order for a ramp to be applied a 741 op-amp is used (see Figure 5.2). A voltage divider with a potentiometer, similar to what was used for the other voltage delivery circuit for the probe, provides one input while the other input is a ramping voltage from a function generator. It is important to note that in this

50 R 4 R 3

R 5 V- +15V + R 1 GND V+ V in -15V 741

V out R 2

Figure 5.2: Circuit diagram for delivering the tuning voltage to the AOM that scans the probe frequency. R1 and R2 = 13 kΩ, R3 = 10 kΩ POT, R4=66.5 kΩ, R5 = 22 kΩ Vin comes from a function generator.

configuration the voltage from the voltage divider gets doubled and the input voltage is subtracted from there. Thus the voltage that goes to the AOM is:   POT + R5 Vout = 2V0 − Vin (5.2) R4 + POT + R5 where Vout is the voltage to the AOM, V0 is the supply voltage for the voltage divider,

Vin is the voltage from the function generator, POT is the potentiometers resistance, and the Rs are various resistors.

51 PBS L1 +1 AO 0 -1 H1 L2 Q1

Figure 5.3: Diagram of a double-pass AOM set up.

A ramping frequency can be created by applying a ramping voltage. This ramp- ing however, also ramps the angle of deflection. In order to keep the beam almost stationary in space the beam can be directed back into the laser in what is called a “double-pass configuration” (see Figure 5.3). This allows for larger changes in fre- quency, since the beam is frequency shifted twice, but keeps the position of the beam after the double-pass the same. 5.1 Spectrum Analyzer

A spectrum analyzer is a device used to measure the spectral features of an input light. Thus, it can be used to measure the frequency difference between two laser beams. The spectrum analyzer is composed of a Fabry-Perot interferometer that is ramped, along with a detector and an amplifier. Our device is encased inside an aluminum box which is in a Plexiglas box in order to cut down on mechanical and

52 thermal noise. The output of the amplified signal can be viewed on an oscilloscope. The oscilloscope shows peaks corresponding to the wavelength of the incoming light. Similar to the optical fiber, the alignment into the interferometer is crucial. If the alignment is not optimal the output signal is broadened and it becomes difficult to see what wavelength the light is at. In order to optimize the alignment the interferometer is held in a four-axis mount. The interferometer consists of two slightly transparent mirrors, multiple reflections occur in between the mirrors and the transmission is then measured by a detector. Depending on whether the separation distance between the mirrors is an even or odd number of wavelengths of the laser beam the transmission either has constructive or destructive interference. This separation distance is finely controlled by the ramping voltage being applied to a piezoelectric transducer attached to one of the mirrors. The interference output is measured by a detector and the signal can be displayed by an oscilloscope. In order to quantify the performance of the Fabry-Perot interferometer a dimensionless value known as the finesse is used. The finess is a function of the two mirrors. The finesse is also defined as the Free Spectral Range (FSR) over the spectral resolution (∆ν), and is related to the number of times the beam reflects back and forth between the mirrors before leaking out. The more the beams reflect in between the two mirrors the more complete the interference process is which leads to higher resolution measurements. The FSR is the range which is scanned over. For our Spectrum Analyzer, model SA-200-9, the manufacturer’s manual states that the FSR = 2 GHz and the finesse for our wavelength (λ < 1000nm) is 200, thus the minimum resolvable bandwidth is 10 MHz. In order for the Spectrum Analyzer to be useful it needs to first be calibrated. By measuring the period on the oscilloscope over which the peaks repeat we know that this value corresponds to the FSR of 2 GHz. Thus the scaling on the oscilloscope can be converted accordingly. Using the Spectrum Analyzer we measured beams both before and after passing through an AOM. The ramp was set so that the signal repeated on the oscilloscope

53 as seen in figure 5.4. The distance between the repeated peaks is the FSR (labeled in Figures 5.4 and 5.5 and was measured to be 46.8 ms. The separation between the two different peaks is the frequency separation between the beam before and after is was passed through the AOM. This separation was measured to be 2.01 ms. Scaling these values with the known fact that the FSR is 2 GHz, gives a frequency difference of 85.9 MHz. This was measured when the non-ramping probe AOM (AO1 in Figure 5.1) had a tuning voltage of 11.71 volts which corresponds to a specified shift of 86 MHz agreeing with what was measured. Using the peaks the minimum resolvable bandwidth can also be checked the FWHM of a peak was measured to be 600 µs this scales to give a minimum resolvable bandwidth of 25.6 MHz. The double-pass AO was also tested as seen in Figure 5.5. For this situation the separation was measure to be 3.74 ms and the 2 GHz scaled to 44.4 ms giving a double-pass frequency shift of 170.5 MHz.

54 1.4

1.2

1

0.8

0.6

0.4

0.2 FSR

0 -0.03 -0.02 -0.01 0 0.01 0.02

Figure 5.4: Spectrum Analyzer output showing the frequency difference between the probe beam before and after going through a single pass of an AOM. The FSR is 2 GHz making the detuning, ∆=85.9 MHz.

55 1

0.5

0 -0.02 -0.01 0 0.01 0.02 0.03

-0.5

-1

-1.5 FSR

-2

Figure 5.5: Spectrum Analyzer output showing the frequency difference between the probe beam before and after going through a double pass of an AOM. The FSR is 2 GHz making the detuning, ∆=170.5 MHz

56 CHAPTER 6 THE VACUUM SYSTEM AND MAGNETIC FIELD CONSIDERATIONS

Once the beams are prepared and directed into the vacuum chamber a few more steps are needed to obtain the atoms. In this chapter I will explain the systems we use to create a vacuume and the magnect fields we need to trap cold atoms. 6.1 Vacuum System

The vacuum system consists of the vacuum chamber and a set of three pumps; a mechanical roughing pump, turbo pump, and ion pump. The vacuum is used to limit the collisions between the trapped atoms and any other atoms or ions in the area. Our vacuum system is capable of reaching pressures less than 10−10 Torr. The roughing pump we use is a Varian SD-40 double vane mechanical roughing pump with a NW 16 kwik flange, capable of pumping from atmospheric pressure down to 10−2 Torr. The turbo pump we use is a Varian V-60 turbo pump with a NW kwik flange input and 4.5” conflat flange output. The turbo pump brings the pressure down to 10−6 Torr. The final pump we use is a 200 L/s Physical Electronics ion pump. This pump brings the pressure down to the 10−10 Torr. This final pressure is displayed on a Digital Multi-Pump Controller. The Vacuum chamber we use is a 26-port Extended Octagon chamber from Kim- ball Physics (MCF800-E020080.16). The inner size of the chamber is about 2700 cm3. The chamber is connected on top of a six-way cross. The distance to the small

57 windows is 110 mm (center to outer surface) with an additional 11 mm for the flange and bolt thickness. The distance to the large windows is 106 mm (center to outer surface) with an additional 13.84 mm for the flange and bolt thickness. The Vacuum Chamber is filled with rubidium atoms via a set of rubidium getters from SAES. By applying a current through the getters rubidium gas is released into the chamber. Within the chamber there are eight getters. The getters are accessed by an eight pin connector on the outside of the chamber, although only 3 pairs of pins are used. Two pairs of pins have four getters attached and one pair has two getters attached. Previously the two getter pair was being used, however, the getters seemed to run out since upwards of 6 A were needed to be applied in order to fill to the desired amount. The switch was made to pins two and three where the optimum current setting is around 2.4 Amps. Figure 6.1 shows a pin diagram of the getters. 6.2 Canceling the Earth’s Magnetic Field

The magneto optical trap is sensitive to magnetic fields; as such the earth’s mag- netic field must be canceled over the region of the trap. In order to cancel out the field six coils are set up around the chamber in a cube orientation where each side is 64 cm long. Each of the coils is made up of a parallel ribbon cable and wrapped around a cube side 12 times. The coils have a resistance of about 140 ohms. Each of the six coils is hooked up to it’s own power supply, this allows for magnetic field control in each direction. Five of the coils are driven by a MPJA HY1802D power supply and the other coil is driven by a Elenco XP-603 (this is connected to the −z coil). 6.3 Applying Magnetic Gradient

A magnetic gradient is used to collect the cooled atoms into the center of the vacuum chamber. In order to set up this magnetic gradient two coils are used in an anti-Helmholtz configuration. 10 AWG enamel coated wire is used to make the coils, each coil is made of 108 turns with an inner and outer diameter of 16.5 cm and 21.5

58 7 1

6 2

5 3 4

Figure 6.1: Shows the orientation of the getter pins, where up is towards the top of the chamber. Pins 4 and 5 are the used up two getter pair and we are currently using pins 2 and 3 which have four getters. Pins 6 and 7 also have four getters attached but are not being used. There is no connections on pin 1 or the center pin.

59 cm respectively. The resistance of the wire is 0.35 ohms. The two coils are wired in series with each other, a Sorenson DLM 20-30 is used to apply 10 A to the coils. The two coils are set up on either end of the chamber parallel to each other. The current in each coil runs in different directions so the magnetic field set up by each coil points in opposite directions. The magnetic field gradient is controlled by a current control circuit built by Lynn Johnson in the Instrumentation Laboratory. It can be turned off and on by both a push button trigger and by sending a TTL logic pulse to the turn off switch. Once the switch is triggered the current through the coils are turned off quickly. Our current is turned off in 2.5 ms and turns on in 50 ms. The turn off time of the current through the coils is shown in Figure 6.3. This was measured by connecting a single loop of wire around one of the coils and measuring the induced emf produced in that wire.

60 -6 -4 -2 0 2 4 6

Time (ms)

Figure 6.2: This shows the current to the coils that produce the magnetic field gradient turning off in 2.5 ms.

61 -30 -20 -10 0 10 20 30

Time (ms)

Figure 6.3: This shows the current to the coils that produce the magnetic field gradient turning on in 50 ms.

62 CHAPTER 7 EXPERIMENTAL PROCEDURE

7.1 Preparation

To get ready for the data taking process all beams must be optimally aligned. This requires checking the power outputs of the beams and, if they are below standard operating powers, optimizing the power through each optic. The trap beam is aligned so each beam is going through the center of the chamber, and the retro reflections are aligned so the beams can be seen on pinhole P1 after the optical Faraday rotator. The repumper is aligned so that it overlaps the trap beams. Once the beams are aligned in the chamber the repumper and trap are frequency locked as described in Section4.4 The trap beam is locked towards the base of the spectrum at point a) in Figure 4.5 to achieve a red detuning of 2.5 line widths away from resonance. With the ball of atoms visible on the television screen the retro-reflecting mirrors are tweaked to compact the ball. By killing the magnetic gradient the dc magnetic field canceling coils are tweaked to produce the coldest molasses. When set correctly the molasses should puff out and very slowly fall downward. The size of the atomic cloud was measured to be 2.3 mm in diameter. This measurement was made by using the camera and measuring the image of the ball on the television screen to be 4.5 cm. By turning the camera away from the chamber and placing a ruler the same distance away from the camera as the center of the chamber is (12.1 cm), the television screen appears to be 11 mm. The screen height of the

63 television is 21.5 cm, thus 11mm in real size is equivalent to 21.5 cm of magnified view. Thus, 11 mm 4.5 cm(mag) = 2.3 mm (7.1) 21.5 cm(mag) Once the MOT is optimized the probe is aligned through the ball. This is most easily done by turning up the probe power until the atoms visibly being pushed. The probe is aligned to go through the center of the ball. A mirror placed after the chamber is adjusted to maximize the probe into a detector. The same after-chamber alignment is done with the probe subtraction beam. The power of the subtraction beam is turned up to make sure that it is indeed harmlessly passing alongside of the trapped atoms and is not affecting the ball in any way. Typical pump powers measured after H3 in 3.1 ranged between 3.3 to 6 mW. Powers in the probe were kept low compared to the pump and typically ranged from 3 to 0.2 µW measured just before the chamber. These corespond to intensities at the

mW ball of 3 to 5.5 cm2 for the pump and for the probe. 7.2 Data Collection, MOT

When taking data the trapping beam’s power is measured after the beam is ex- panded and before it is split into three separate beams. This power is used to calculate the pump intensity as explained in detail in Section 8.1. Since the probe beam is de- rived from the trap beam changing the trap/pump power also changes the probe power. The probe power is then set by turning the half-wave plate (H5 in Figure 5.1) before the polarizer (P1). Since our optics are not ideal the polarization of the probe beam is also changed, this alters the proportion of power that goes to the subtraction beam through the polarizing beam splitter (PBS4). In order to correct for this the half-wave plate (H6) before the beam splitter is turned until both the probe beam and subtraction beam are of equal intensities. The intensity is then measured right before the beam enters the chamber. If the intensity is too high, such that it is in

64 y ensit Int robe P

1MHz

Probe Detuning

Figure 7.1: A typical light shift data run, the light shift is measured by taking half the distance between the peak and trough of the signal.

danger of affecting the atoms, or too low, such that it does not produce a readable signal, the power of the probe beam is adjusted again by using H6. Once the signal is seen on the oscilloscope the signal can be optimized by tweaking the half-wave plate (H6) before the beam splitter. To do this, the probe is blocked and the null signal is centered on the oscilloscope with the current-to-voltage converter’s offset. Then the probe is unblocked and re centered on the oscilloscope by changing H6. This process ensures that the probe and probe subtraction beam are the same intensity and helps to cancel out technical noise from the laser. With the signal on the scope ten averages were taken and the light shift was measured. The pump intensity was changed by adjusting the half-wave plate (H2 in Figure

65 5.1) before the beam splitter (PBS1) that separates the probe from the pump. The process was then repeated for multiple pump intensities. 7.3 Data Collection, Molasses

To take data in optical molasses instead of a MOT the magnetic field must be turned off. We turn off the magnetic fields and then ramp the probe frequency, then turn the magnetic field back on and build up the ball so the number of atoms builds up again. We follow the timed sequence of events shown in Figure 7.2 for data collection. A function generator creates a 1 Hertz square wave that triggers our timing cir- cuits. The first circuit takes the input pulse and shortens the pulse width to 30 ms. This shortened pulse is connected to the stop trigger for the magnetic gradient, thus the gradient is off for 30 ms of every second. Another timing circuit is used to create a much shorter pulse width of 2.94 ms, this pulse is used to trigger a second function generator. The triggered function generator crates a 150 Hz, 1 volt triangle wave, this is used as our ramp for scanning the probe acousto optical modulator. The triangle wave which is sent to the probe is also used to trigger the oscilloscope. Our data are all contained in the 3.33 ms of the downward slope of the probe ramp. The reason the data taking is delayed is the magnetic field takes 2.5 ms to completely turn off and we wait another 2.04 ms before taking data thus giving time for the molasses to form. Data is taken the same way as outlined above in Section 7.2. However the timing is such that one scan is taken every second. Ten averages were taken and the light shift was measured.

66 30 ms Master Trigger Signal

Gradient B-field Trigger

Gradient B-field

Ramp Trigger

Probe Ramp

t=0 Molasses Measuring Trap Forming of Light Shift Reset 2.04 ms 3.3 ms 2.5 ms

Figure 7.2: This shows the timing scheme used when taking data in the optical molasses.

67 CHAPTER 8 RESULTS AND DISCUSSION

In this chapter we present, and discus, the results that we have obtained for pump- probe spectroscopy of atoms in the MOT and in optical molasses. Figure 8.1 shows the main experimental result of this thesis - a first systematic measurement of the light shift of cold atoms as a function of laser intensity. It is evident that the data for molasses fits the theory much better then the data for atoms in the MOT. 8.1 The Theoretical Prediction

We start our analysis by first describing how we arrived at the theoretical predic- tion indicated by the solid line in Figure 8.1. The theory graphed in Figure 8.1 is Equation (2.34), reproduced below:

s ! δ Γ ∆ ∆2 I LS = − + C 2 2 mF ,mF 0 ~ 2 Γ Γ 2ISAT

∆ where Γ is the natural line width of Rb: 5.98 MHz, Γ is the trap detuning which is set at -2.5 (with error of ±0.25, which leads to an error bar in the theoretical prediction

0 as shown), ISAT is the saturation intensity for the F = 3 → F = 4 transition of Rb: 1.64 mW , and as discussed in Section 2.6 we take C 2 to be 0.25. We now need cm2 mF ,mF 0 to find the intensity. We do this by experimental measurement, as described below. First we measure the power and intensity in the trapping laser just before it is split into x, y, and z beams at PBS2 in Figure 3.1. However, owing to a myriad of optics that guide this light into the vacuum chamber, only 83 % of this light remains

68 400 MOT Data

Molasses Data 350 Theory

300

250 t (kHz) f 200 Shi

ht 150 ig L 100

50

0 2.5 3 3.5 4 4.5 5 5.5 6 Pump Intensity (mW/cm2)

Figure 8.1: Measurement of ground state light shifts in the MOT and molasses versus the intensity of the trapping beams. A linear fit was set to each data set: kHz MOT data had a slope of 29.27, Molasses of 21.42, and the Theory of 20.77 mW/cm2

just before the chamber. We determined this by measuring the power in each of the three beams just before entering the chamber. However, this is still not the actual intensity seen by the atoms, for the light has to travel through the chamber windows, and background vapor to get to the trapped ball of atoms. We measured the power in each beam emerging on the other side of the chamber and found that the trapping beams had lost an additional 14%-in other words an additional 7% from the entry point until the midpoint of the chamber. This means that the intensity of the pump beam when it first presents itself to the cold atom cloud is down by a factor 0.83(.93) = 0.77 from the intensity measured just before PBS2. This pump intensity is, in principle, “doubled” because the MOT beams are retroreflected back through the chamber. However, this is not true because the beam after passing through the cold atom ball traverses the far half of the chamber and window (losing 7%), is retroreflected by the mirror and quarter-wave plate combination (2% loss),

69 then traverses again through the window and half the chamber (losing 7%) before it gets to make a second pass through the atoms. Thus the total pump intensity (from the MOT beams) seen by the atoms is 0.77 + 0.77 (0.93)2 (0.98) = 1.4, not 2, times the intensity measured before PBS2. The error bar in the theory plot stems from uncertainty in our measured value of the MOT detuning ∆/Γ by ±0.25, owing to uncertainty in our knowledge of where exactly the trap laser is locked on the side of the saturated absorption peak (see Figure 4.5). In otherwords when we lock at the point (a) to obtain ∆/Γ = -2.5, the locaton of point (a) has a uncertainty of ±0.25Γ. In Figure 8.1 we have chosen to show the error bars for just the two farthest points. 8.2 Molasses and MOT Data

The slopes of the data are in good agreement especially the molasses data. Thus, our faith is reaffirmed in the simplified theoretical model in Section 2.6 we have used to describe the F = 3 → F 0 = 4 Rb atom. While the MOT slope is still in agreement with the theory, the values for the light shift are consistently higher. The only difference between the MOT and the molasses is the magnetic field gradient: The gradient is still on in the MOT, while it is switched off in molasses. We make sure that we commence making measurements in molasses only after residual eddy fields, that are setup upon rapidly turning off the magnetic field gradient, have completely died out (as mentioned in Section 6.3 this time is 2.5 ms). The presence of the magnetic gradient must cause extra Zeeman shifts that our light shift theory does not account for. The gradient in our case is 12 Gauss/cm, which means that across a 2.3 mm trap, the range of magnetic fields seen at the extremities of the cloud is ±1.4 Gauss. For Rb, the Zeeman shift is 0.5 MHz/Gauss between every adjacent Zeeman sublevel.

70 CHAPTER 9 CONCLUSIONS AND FUTURE OUTLOOK

9.1 Conclusions

We have made a systematic measurement of the ground state light shift for cold 85Rb atoms and have found a significant differences between the measurements for atoms in a magneto trap (MOT) as compared to atoms in optical molasses. The measured light shift for atoms in molasses agree better with the theoretically predicted values than the MOT does. The data follows the same trend as the prediction but the measured values are consistently higher we believe this discrepancy is because of additional Zeeman shifts. The shifts if the MOT are significantly higher due to the magnetic gradient. 9.2 Future Outlook

An independent pump-probe laser has also been set up. As mentioned in Chapter 1, a major motivation for us to study the method of pump probe spectroscopy in cold atoms is to use the techniques to detect and characterize an optical lattice. In this case, the MOT laser is turned off while the lattice is kept on and an independent laser is used for the creation of pump and probe beams for pump-probe spectroscopy of the optical lattice. Using this additional laser we will be able to probe the atoms with the trapping beams turned off. This will allow for further exploration of the light shifts in a different environment such as in a lattice.

71 9.2.1 Independent Pump-Probe

Scope Amp

APP H1 PBS Trapping ECDL1 Faraday Beam Rotator

AO “-1” L4 H6 PBS Pump Trapped Beam Atoms To Saturated Absorption Subtraction Single Mode Beam L1 L2 H2 PBS L3 Optical Fiber Faraday ECDL2 Rotator Probe H3 Beam

PBS AO “-1” Q1 To Saturated H4 Absorption Q2 Scanning H5 PBS AO “+1”

PBS

Figure 9.1: Diagram of setup with a independent laser for the pump and probe beams.

To create pump and probe beams that are independent from the trapping beams another laser is used. This laser is designed the same as the others and it has a typical power of 25 mW for the main beam out of the box. The laser beam profile from this laser has an ellipticity of 1.6:1 and as such it is not vary effective to use a set of anamorphic prism pairs to circularize the beam. The beam is shaped only by irises along it’s path. Two mirror are used to align the beam into the optical Faraday rotator. Typical transmission achieved through the OFR is 75 percent. A half-wave plate (H2) and polarizing beam splitter is used to separate the pump beam from the probe beam. The half wave plate is used to send

72 the minimum amount of light to the probe this is typically 7 percent. The probe is then sent over to the same double AOM set up as the probe beam that was derived from the trap beam. The beam is then directed in to a fiber optic cable described in 9.3. Transmission through the fiber varied from day to day as high as 50 percent have been achieved but other times the transmission was only 30 percent. 9.3 Optical Fiber

A single mode optical fiber (model # SM8005-6125 from ThorLabs) is used to create a clean Gaussian TEM00 beam. The fiber is mounted on the fixed part of a three-way translation stage. A microscope objective is mounted on the movable part of the translation stage and is used to focus the laser into the fiber optic. In order to archive maximum transmission through the fiber the laser has to be focused directly into the fiber, this means the objective has to be not only in the precise location horizontally and vertically but also there is a strong dependence on the distance of the objective from the fiber. The laser also must be collinear with the transmission axes of the objective and the fiber. Two mirrors for aligning the beam into the fiber are used to help achieve this. It is important to stress the use of a single mode fiber, though a multi mode fiber allows for more power transmittance and is easier to align the beam profile is not Gaussian. This difference can easily be seen by expanding the beams after they have passed through the fiber, see Figure 9.2. 9.4 Independent Pump-Probe Measurements

For this measurement the MOT trapping beam intensity is again varied, however since our pump and probe beams are derived from another laser they both stay at the same intensity. It is also a simpler theoretically since there is now only one pump beam being retro reflected compared to the three beams in the MOT. The pump beam is also more collinear with the probe, which again makes it a simpler case. A shutter is used to keep the pump beam from pushing the atoms at all times.

73 Figure 9.2: Images of the output of both a multi mode fiber (left) and a single mode fiber (right). It is easily seen that the single mode fiber produces a more Gaussian beam profile.

74 BNC

C

+15V

E

+5V V in

Figure 9.3: The circuit diagram for sending 15 volts to the pump shutter using TTL logic signals.

This shutter is triggered from the magnetic field. The magnetic field triggering pulse is sent to the shutter control circuit, see figure 9.3. The circuit input TTL logic signal is sent to a silicon NPN power transistor (model # 2N3055), which gates 15 volts to open the shutter. The shutter has a delay time of 4.08 ms and a time to open of 1.92 ms, thus a time from triggering to allowing the full beam to pass through is 6 ms (Figure 9.4). The time delay allows for time for the atoms to equilibrate until the pump is turned on for data taking. The probe beam is also turned off for all times when it is not ramping, this is done by using the sync of the second function generator and putting the TTL pulse through an inverter and sending it to the turn on/off switch for the non ramping AOM. When the AOM is not trigger all the light is in the zeroth order and is blocked by a pinhole, when the trigger is on the -1st order light goes through the pinhole and continuous to through the ramping AOM and is used as the probe beam. Since the pump and probe beams are on only during the data taking period they have minimum affect on the atoms, thus creating more accurate data. Since there is currently no AOM on the pump beam the only way to achieve different detunings is to lock the laser at different points. Looking at figure 9.5 we

75 -15 -10 -5 0 5 10 15 Time (ms) Figure 9.4: Shows the light beam power through the pump shutter and the signal that tells the shutter when to open.

can see that locking on various heights of the peak gives different detunings. Locking at a) is 9.25 Γ, b) is 8.44 Γ, c) 7.2 Γ, and d) is 5 Γ away from resonance. We also tried using a shutter to block the MOT trapping and repumper beams, however we found that the shutter is not fast enough (Figure 9.6) and the cold atoms are all gone by the time we can take data. This problem could be solved with another AOM used in the MOT beams, it would allow for faster blocking of the beam.

76 85 Pump-Probe Spectra

0.1 a) 0.08

0.06

0.04 b) 0.02

0 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 -0.02 c) d) -0.04

-0.06

-0.08

Figure 9.5: The spectrum of the 85 Rb for the independent pump-probe laser.

-3 -2 -1 0 1 2 3 Time (ms) Figure 9.6: Shows the light beam power through the trap shutter and the signal that tells the shutter when to close.

77 BIBLIOGRAPHY

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