ABSTRACT

RADIATION TRAPPING IN OPTICAL MOLASSES

By Soo Y. Kim

Using three orthogonal pairs of counterpropagating red-detuned lasers and a magnetic field gradient, we have constructed a trap of 85Rb atoms at temperatures reduced to the order of 100 microKelvins and densities of 109 atoms/cm3. When the atoms interact with the light, they are absorbing and spontaneously emitting photons. The emitted photons from atoms can be absorbed by neighboring atoms. This phenomenon, called radiation trapping, limits the density and temperature of the trap and also destroys the atomic coherence. We can non- invasively measure the effects of radiation trapping by measuring the intensity correlation function of the light radiated from the trap. We have found that theoretically, the effects of radiation trapping can be detected in dilute atomic samples with an on-resonance optical depth as low as 0.1, more than an order of magnitude less than seen before. Experimental progress is reported.

RADIATION TRAPPING IN OPTICAL MOLASSES

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 Soo Y. Kim Miami University Oxford, Ohio 2003

Advisor ______Samir Bali

Reader ______Perry Rice

Reader______S. Douglas Marcum

Table of Contents

1. Introduction and Motivation 1 2. Laser Cooling and Trapping 3 2.1 Laser Cooling 3 2.2 Magnetic Field Gradient Trapping 5 3. Radiation Trapping 7 3.1 First-order and Second-order Coherence 7 3.2 Optical Bloch Equations with Radiation Trapping for MovingAtoms 9 3.3 Second-order Coherence for Moving Atoms with Radiation Trapping 10 4. The Apparatus 15 4.1 Atomic System 15 4.2 Vacuum Chamber 18 4.3 Frequency Tuned Laser Diode 19 4.3.1 External Cavity Diode Laser 19 4.3.2 Saturated Absorption 25 4.3.3 Locking Mechanisms 29 4.4 The Trapping Setup 31 4.5 Magnetic Coils 35 4.5.1 Magnetic Field Gradient Coils 35 4.5.2 Nulling Coils 37 4.5.3 Switching Circuit 38 5. Characteristics Measurements 40 5.1 Number Measurement 40 5.2 Loading Curve 41 5.3 Density Measurements 43 5.4 Temperature Measurement 45 6. Intensity Correlations 47 7. Conclusion and Future Goals 50 References 51 Appendices 53 Appendix A: Parts and Suppliers 53 Appendix B: Circuits 56

*note that references are in brackets [ ], and parts/supplies are superscript. ii

Acknowledgements

I would like to thank my advisor, Dr. Samir Bali, for taking me into his lab and giving me the opportunity to experience building a full lab from nothing but an empty optics table. He has truly been an integral part of my graduate career at Miami University. I would also like to thank the people I have closely worked with in lab for the past three years: Matthew Beeler, Ronald Stites, Brian Pollock, Charlie LaPlante, Laura Feeney, and Kristy Kallback-Rose. Without them, the lab would not be where it is today. I would also like to thank our collaborating professors, Dr. Perry Rice and Dr. S. Douglas Marcum for all their help and for being the readers for this thesis.

Financial support for this project was provided by the College of Arts and Science, the Office for the Advancement of Scholarship and Teaching (OAST), and Miami University. Also, supports from the Cottrell Foundation (Research Corporation), the Petroleum Research Fund of the American Chemical Society are appreciated.

A paper entitled “Sensitive Detection of Radiation Trapping in Cold-Atom Clouds” has been accepted for publication in Physical Review A from work done for this thesis.

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Dedicated to the future Jedis

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1. Introduction and Motivation

Cold atoms have been a hot area of research in recent years. It started in 1933 when R. Frisch at the University of Hamburg showed that light can exert a force on atoms [1]. But it took almost 40 years (1970) for A. Ashkin at Bell Laboratories to show trapping by using a focused laser beam, or optical tweezers, to trap small latex spheres immersed in a medium [2]. He proposed that atoms could be trapped in the same way with resonant light. Five years later, in 1975, the idea of laser cooling was independently presented by two groups: Hänsch and Schawlow at Stanford University [3] and Wineland and Dehmelt at the University of Washington [4]. Both looked at the use of three orthogonal pairs of counterpropagating beams to slow down atoms. The beams are tuned below the resonant frequency of the atoms, so that when the atoms approach the beams, some portion of them are Doppler shifted into resonance and interact with the laser. Any beams that are propagating in the direction of the motion of the atom would be Doppler shifted out of resonance, and not affect the atom. Thus the atoms are slowed down and cooled, given that average speed is related to temperature. Then ten years later, in 1985, S. Chu et al at AT&T Bell Laboratories, created an optical molasses with laser beams [5]. In 1987, again at AT&T Bell Laboratories, Raab et al developed the magneto-optical trap (MOT) using a magnetic field gradient to cool and trap atoms [6]. Finally in 1997, S. Chu, C. Cohen-Tannoudji, and W. Phillips received the Nobel Prize for the atom trap. Also in 2001, E. Cornell, W. Ketterle, and W. Phillips were awarded the Nobel Prize for creating Bose-Einstein condensates (BEC). Cold atoms have been used to create atomic clocks [7], further BEC research [8,9,10] and many other applications.

We use the MOT to observe the effects of radiation trapping. Atoms trapped in a MOT are continuously interacting with laser light and absorbing and emitting photons. In a dense atom trap, the likelihood that an atom could absorb the emitted photons from another atom increases. The absorption of the photon pushes the two atoms apart, increasing the temperature and decreasing the density of the trap. This also decreases the coherence of the trap due to the random phase and polarization of the spontaneously emitted photon [11,12]. This feature, called radiation trapping, was first seen in cold atom clouds by Walker et al

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[13,14] in 1990. They found that for traps of 1010~1011 atoms/cm3 and an optical depth of 3, radiation trapping became evident. In 1996, Bali et al [15] studied the intensity correlations of scattered light from laser-cooled atoms and encountered preliminary effects of radiation trapping, but further study was not done on the effects. In 2001, Matsko, Novikova, Scully, and Welch [16] noticed that in their electromagnetically induced transparency (EIT) experiments, radiation trapping effects are present. To explain them, they developed a simple theory for radiation trapping. However, this applies specifically to EIT experiments, which involve three-level atoms, usually hot. They neglect Doppler broadening and direct interactions with the laser. We present a theory using intensity correlations to express the effects of radiation trapping, including Doppler broadening and direct interactions, for an optical molasses. We find that radiation trapping is prevalent in traps with a density of 109 and an optical depth of 0.1, which is more than one order of magnitude less than seen before [13,14] and can be measured with a non-invasive technique.

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2. Laser Cooling and Trapping

To trap atoms, first they must be slowed down to very low velocities. This can be achieved by having resonant counterpropagating laser beams bombard an atom in three orthogonal directions. Since for a monatomic ideal gas, temperature is related to kinetic energy, 3 1 k T = mv 2 , (2.1) 2 B 2 the reduction of velocity leads to a reduction of temperature and therefore slow atoms are said to be cold. The situation of the cooled atoms and six cooling laser beams is called an optical molasses. However, the lasers do not exert a position-dependent force upon the atoms. A magnetic field gradient is necessary to provide the position-dependent force to create an atom trap.

2.1 Laser Cooling At room temperature, gas atoms are moving with average speeds around half a kilometer per second. The kinetic energies of the atoms can be described by a Maxwell-Boltzmann distribution. The atoms in the low velocity tail of the distribution can be slowed down to speeds of centimeters per second using photons. The atom can be imagined as a big bowling ball and the little photons of light as ping pong balls. Conservation of momentum shows that one ping pong ball may be insufficient to significantly affect the bowling ball, but with enough ping pong balls, the bowling ball can be slowed down. Thus, a gas atom can be slowed down to low velocities with a massive quantity of photons, which a laser can supply. For example, a 1mW beam of a 780 nm laser provides 3.9×1015 photons/sec. Although the bowling ball and ping pong ball model provides a nice picture, it does not accurately depict what actually occurs. When a photon encounters an atom, the energy from the photon excites an electron in a ground state of the atom to an excited state. Therefore, only photons with the correct energy, or matching frequency ωo, will be absorbed by the atom. However, since the atom is moving at a velocity v, the atom perceives the frequency of the photon ω to have been Doppler C G G shifted by k ⋅ v , where k is the wave propagation vector of the light. Since the momentum

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G of a photon is >k , this multiplied with the average photon absorption rate of an atom gives the average force [17] on the atom, G G I I > γ o Fcool = k G G , (2.1.1) 2 2 1 + I I o + [2(δ + k ⋅ v) γ ] where γ is the decay rate of the atom, I is the intensity of the laser light, and δ is the laser detuning, δ = ωo −ω . We will assume, without any loss in generality, that the laser is red- detuned, i.e. I. The recoil due to emission is not taken into account because the direction of emission from the atom is random and the average of this recoil force would sum to zero. In 1D, the net force exerted by counterpropagating laser beams on an atom moving in the positive direction is G G >kγ I ≈ 1 1 ’ F = ∆ − ÷, (2.1.2) cool ∆ 2 2 ÷ 2 I o «1 + I I o + [2(δ + kv) γ ] 1 + I I o + [2(δ − kv) γ ] ◊ assuming that the two light fields do not interfere and act on the atom independently. In the approximation that kv << γ , δ and using the binomial series expansion, the force simplifies

to G I 2 G > 2 ( δ γ ) Fcool = −4 k v (2.1.3) 2 2 I o [1 + I I o + (2δ γ ) ] for three pairs of orthogonal counterpropagating beams. As the equation shows, the force is directed opposite of velocity and therefore clearly is a damping force. If the laser is red-detuned, the atom is Doppler shifted into resonance with light propagating opposite its direction of motion and shifted further out of resonance with light propagating in its direction of motion. Consequently, the atom mostly interacts with the opposing beam and is slowed down. For the case of counterpropagating blue-detuned beams, δ < 0 , the atom would be further accelerated. Therefore, to slow an atom in all three spatial dimensions, six counterpropagating red-detuned lasers in three orthogonal directions are used. However, as one can see, the lasers only exert a damping force. The atoms are slowed wherever they are, but there is no preferred spot where the atoms want to gather. Below, we see how a magnetic field gradient is used to provide a position-dependent force.

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2.2 Magnetic Field Gradient Trapping G G G The electron spin and angular momentum, S and L , and nuclear spin I , produce a net G magnetic moment of an atom, characterized by the total angular momentum F , where G H G G F = S + L + I . When an external magnetic field is introduced, the energy levels of the atom split, known as the Zeeman effect. The energy shift of a level is dependent on mF , the projection of the total angular momentum of the atom on the magnetic field axis [18], > mF e B ∆ E = , (2.2.1) 2me c where e is the electron charge, me is the electron mass, and B is the external magnetic field, given by

2 » ÿ µ NIR 1 1 B o … Ÿ x = 3 − 3 , (2.2.2) 2 … 2 2 2 2 2 2 Ÿ (R + (A + x) ) (R + (A − x) ) ⁄ where R is the radius of the coil, A is the half the distance between the two coils, NI is the number of ampturns, and x is the distance away from the midpoint between the coils. The resonant frequencies of the levels shift by

mF eB ∆υ = . (2.2.3) 4πme c For simplicity, we will consider an atom with a ground state of F = 0 and an excited state of F′ = 1. In the presence of a magnetic field, the ground state is unperturbed, but the excited state splits to three levels, mF = 1,0,−1. For a magnetic field gradient, the energy levels split as shown in Fig. 2.2.1. For a laser- cooled atom at a distance z from the center, it experiences a force towards the center of the trap from Eq. 2.1.1 (using v = 0 for simplicity), G G >kγ I ≈ 1 1 ’ F = ∆ − ÷ . (2.2.4) trap ∆ 2 2 ÷ 2 I o «1 + I I o + [2(δ − ∆υ ) γ ] 1 + I I o + [2(δ + ∆υ ) γ ] ◊

Similarly to the cooling force, we use the approximation that ∆ v << γ , δ and the binomial expansion to see that the trapping force simplifies to G G > I (2δ γ ) Ftrap = 4 k ∆ v . (2.2.5) I 2 2 o [1+ I I o + (2δ γ ) ]

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The position dependence of the force is not easily seen in Eq. 2.2.5 because it is hidden

in the magnetic field in ∆v. Thus, the polarized laser light and magnetic field gradient provide the cooling and position-dependent forces that result in an atomic magneto- optical trap.

E mf=1 |e> (F’ = 1) 0 -1 σ+ σ- ω B |g> (F = 0) x” x’ x x = 0 B = 0

Fig. 2.2.1. The excited energy level of an atom splits with respect to the B-field. At point x’, the laser light - ω is closer to resonance with the m f = −1 level, which is pumped by σ -polarized light, and as a result,

pushed toward x = 0 . Likewise, at point x”, the laser light is closer to resonance with the m f = +1 level, which is pumped by σ+-polarized light, and pushed again towards x = 0 .

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3. Radiation Trapping

As an atom interacts with the laser light, it absorbs and emits photons. As will be explained in chapter 6, these photons exit the trap and can be detected for measurements on the trap or can be just lost in space. However, as the density in a trap increases, the photon mean free path decreases and it is more probable that an emitted photon could be absorbed by a neighboring atom instead of just leaving the trap. This phenomenon, called radiation trapping, prevents the trap from getting denser, raises the temperature of the trap, and reduces coherence. As the photon is emitted from the atom, the atom recoils in the opposite direction of the movement of the photon. When the photon is absorbed by another atom, this atom recoils in the direction of the photon movement. Therefore, the two atoms are pushed away from each other, decreasing the density of the trap. The recoil introduces motion to the atoms, increasing temperature. Because the polarization and phase of the emitted photon is random, the coherence of the trap decreases when atoms start absorbing these incoherent photons. By looking at the density and temperature limits and the decrease in coherence, radiation trapping can be detected in the trap.

3.1 First-order and Second-order Coherence The density and temperature can be directly measured, as is shown in Ch. 5, but the decrease in coherence is more difficult to observe. We can look at the first order coherence g(1)(τ) of the trap, which is defined [19] as

¿Eˆ * (t)Eˆ(t +τ )– g (1) (τ ) = , (3.1.1) ¿Eˆ * (t)Eˆ(t)– which looks at the electric field Eˆ at different times. As one can easily see, for τ = 0 , g (1) (0) = 1. According to the Weiner-Khintchine theorem, this is related to the Fourier transform of the normalized frequency spectrum [19],

1 ∞ F(ω) = — g (1) (τ )e −iωτ dτ . (3.1.2) 2π −∞ But rather than measuring the field, we find it simpler to measure the intensity of the radiating photons. The intensity correlation, or second-order coherence g(2)(t), is defined as

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¿Iˆ(t)Iˆ(t +τ )– g (2) (τ ) = . (3.1.3) ¿Iˆ(t)– 2 where I(t) is the intensity from the radiating atoms at a time t and I(t+τ) is the intensity of the radiating atoms at a time delay τ later. For chaotic light from a large number of spatially coherent sample, we have the relation of the second-order correlation to the first order correlation g(1)(τ), [19] g (2) (τ ) = 1+ | g (1) (τ ) |2 . (3.1.4)

Knowing that g (1) (0) = 1, we see that g (2) (0) = 2 . This equation does not work for a coherent light source, such as a laser beam, where g (2) (0) = 1 as can be seen from the definition of g(2)(τ). Since radiating atoms emit their radiate independently of each other, the radiation acts as a chaotic light source. Since g(2)(τ) only observes photons radiated from the atoms, it is a non-invasive measurement. There is not just one radiating frequency but a range of frequencies because of Doppler- broadening caused by the range of velocities of the atoms. The general relation is [19]

(2) 2 2 g (τ ) =1 + exp(−∆ D τ ) , (3.1.5)

where ∆D is the Doppler-broadened full width at half-maximum height in the frequency

1 2k BT ln 2 ∆ D = = 2ωo 2 . (3.1.6) τ c Mc

ωo is the resonant frequency of the atom, M is the mass of the atom, T is the temperature, kB is the Boltzmann constant, and τc is the coherence time. As one can see in Eq. 3.1.6, the temperature of the trap can be found from g(2)(τ). Bali, et al. found that for clouds under ~105 atoms, with densities where radiation trapping should be insignificant, the temperature measurements from the intensity correlations (from Eq. 3.1.6) agreed with temperature measurements obtained by use of the time-of-flight (TOF) method [17]. These two measurements are compared because the TOF is a simple and very direct approach to measuring the temperature of the atoms, while the intensity correlations have many parameters that could affect the inferred temperature value. But, for clouds of more than ~105 atoms, it was found that the two temperature measurements no longer agree. For example, for 3×106 atoms (detuning ∆=−1.85Γ), the correlation method of measuring 8

temperature suggested 120 µK but the TOF method measured 50 µK. Bali, et al. speculated that perhaps radiation trapping could have caused this difference due to the increase in spectral width via radiation trapping. However, no attempt was made to include radiation trapping in the theory.

3.2 Optical Bloch Equations with Radiation Trapping for Moving Atoms We have constructed optical Bloch equations (OBE) for a two-level atom excited by a laser, including spontaneous emission and radiation trapping. First, we see that the number of “collisions” a photon encounters while moving through the cloud of density n and length l, is nlσ , where σ is the photon cross-section for absorption. This is also known as the optical depth [20]. Since in our case, the number of collisions is <<1, we can treat it as a probability that a photon is absorbed by an atom. If we multiply this by the rate of atoms decaying from the excited state, γρee, where γ is the decay rate and
ee is the population in the excited state, we obtain the rate of absorption, R, of scattered photons by the atoms. R can also be

described by the probability of available radiated photons, as γnth [16], where we use “ nth ” to represent the probability of absorption of a fluorescent photon. Thus, by setting the two equations for R equal, we can obtain the formula for nth

nth = nlσρ ee . (3.2.1) The probability of the atom of being in excited state is found by [17]

1 I / I o ρ ee = 2 , (3.2.2) 2 1 + I I o + (2δ / γ )

where I is the intensity of all six trapping beams, δ is the detuning of the laser and Io is the saturation intensity which is 1.64 mW/cm2 for 85Rb. For a trap with a density around 108-109 I atoms/cm3, l ≈1 mm, = 2 , σ=2.91×10-13 m2, γ = (26.63 ns) –1 [17], and δ is one linewidth, I o we find the optical depth to range from 0.01~0.2, and nth from about 0.001~0.02. This treatment is only valid for small optical depths and nth <<1. Our starting point is the OBEs for a two-level atom. Including spontaneous emission and radiation trapping, they are

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# # i * ρ ee = −ρ gg = −γ ′ρ ee + nthγ + (Ωρ ge − Ω ρ eg ) (3.2.3) 2

# # * ≈ γ ′ ’ iΩ ρ eg = ρ ge = −∆ − iδ ÷ρ eg − (ρ ee − ρ gg ). (3.2.4) « 2 ◊ 2 where we define γ ′ ≡ (2nth +1)γ and γ is the population decay rate of the excited state

3 2 ωeg µ γ = . (3.2.5) > 3 3πε 0 c

Here, ωeg is the resonant frequency of the atom and µ is the dipole moment [17]. Knowing # # that ρ ee + ρ gg = 1, we see that ρ ee + ρ gg = 0 . ρee and ρgg are the atomic populations in the excited and ground states, respectively, and ρ eg and ρ ge relate to the coherence of the atomic dipole [17]. Thus, ρ eg and ρ ge decrease due to the incoherent process of radiation trapping.

ρee decreases and ρ gg increases from the spontaneous emission process. Ω is the Rabi

frequency, defined as G E Ω = e(r ⋅ εˆ) o , (3.2.6) eg > G where Eo is the amplitude of the electric driving field and er eg is the induced dipole moment. Since the atoms are moving, we need to account for the velocity of the atoms into the Rabi frequency. For a moving atom with velocity v, we find that the OBEs become

G G G G # # i ik ⋅vt * ik ⋅vt ρ ee (t) = −ρ gg (t) = −γ ′ρ ee + nthγ + (Ωe ρ ge − Ω e ρ eg ) (3.2.7) 2 ′ G G # # * ≈ γ ’ iΩ ik ⋅vt ρ eg (t) = ρ ge (t) = −∆ − iδ ÷ρ eg − e (ρ ee − ρ gg ). (3.2.8) « 2 ◊ 2

3.3 Second-order Coherence for Moving Atoms with Radiation Trapping If we treat the field radiated by an atomic dipole as an operator, we have G G ˆ G E(r, t) = K(r )σˆ + (t) , (3.3.1) where σˆ ± (t) are defined as usual as the atomic raising and lowering operators [20,21] and

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G G G G G ω 2 ≈ d d ⋅ r r ’ K(r ) ∆ ( ) ÷ , (3.3.2) = 2 ∆ − 3 ÷ 4πε o c « r r ◊ G G is the usual spatial dipole pattern at point r radiated by an electric dipole d at the origin, oscillating at ω, The spatial dependence cancels in Eq. 3.1.1 and we have ¿σˆ (t)σˆ (t +τ )– g (1) (τ ) = + − ss , (3.3.3) ¿σˆ + (t)σˆ − (t)– ss where the subscript ss denotes the steady-state. ρeg and ρge can be expressed as the

iωt −iωt expectation values of the raising and lowering operators, ρ eg ≡ σˆ − e and ρ ge ≡ σˆ + e [20]. However, as Eq. 3.3.3 shows, we need to calculate the two-time correlation function of the operators. By use of the quantum regression theorem, we can use the single-time expectation values to find the two-time correlation function for g(2)(τ) [21,22]. The quantum % % % regression theorem states that if there is an operatorO , and O(t +τ ) = ƒ a j (τ ) O j (t) , then j` % % % % Oi (t)O(t +τ ) = ƒ a j (τ ) Oi (t)O j (t) . Or in words, the fluctuation (two-time correlation) j

regresses in the same manner [i.e. using the same factors aj] as the mean. The quantum regression theorem can be used in certain conditions described in Refs. 21 and 22. In our % case, by solving the OBEs in Eqs. 3.2.7 and 3.2.8, we can write an equation for σ − (t +τ ) in % % terms of σ + (t) and σ − (t) . Then the quantum regression theorem allows us to pre- % multiply the terms of the equation by σ + (t) to obtain

G G iωt −(γ ′ 2+iδ )τ iΩ * −ik ⋅v (t+τ ) σˆ + (t)σˆ − (t +τ ) e = ρee (t)e + ρeg (t)e 2(γ ′ 2 + iδ ′) (3.3.4) (1− e −(γ ′ 2+iδ ′)τ ) G G where δ ′ ≡ δ − k ⋅ v , assuming weak excitation. For the steady state solution, where t → ∞ ,

ss ss we simply replace ρ ee (t) and ρeg (t) by their steady state solutions. ρee and ρeg can be found

ss 1 I I o by making Eqs. 3.2.3 and 3.2.4 equal to zero. We find that ρee = 2 and 2 1+ I I o + (2δ γ ′)

ss iΩ I I o ρeg = , where s = [17]. We find that Eq. 3.3.4 then becomes 2(γ ′ 2 − iδ )(1 + s) 1+ (2δ γ )2

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γ 2 Ω G G γ − iω τ γ ′ ˆ t ˆ t n e − γ ′τ 2 e eg e − ik ⋅vτ e − iωτ σ + ( )σ − ( + τ ) ss = th + 2 . (3.3.5) γ ′ 4δ ′2 + γ ′2 + 2 Ω

We also take the average over the Maxwell-Boltzmann distribution of velocities,

≈ m 2 ’ 2 2 eikvτ = — eikvτ ∆ e(−mv 2kBT ) ÷dv = e−k τ kBT 2m to get v ∆ ÷ « 2πk BT ◊

γ − iω τ 2 2 ˆ t ˆ t n e − γ ′τ 2e eg ss e − k τ k B T 2m e − iωt σ + ( )σ − ( +τ ) ss = [ th + ρ ee,0 ] (3.3.6) γ ′ G ss 2 2 2 2 G where ρ ee,0 = Ω (4δ ′ + γ ′ + 2 Ω ). For this average, we replace k ⋅ v with –kv since the dominant force comes from the laser beams that counter the motion of the atom. Also, we assume that the detuning isδ ′ ≈ δ , since kv is small. For molasses at a temperature of 50
K, k is 8×106 m-1 and v is 0.1 m/s, where as the detuning, at one linewidth is 5.98 MHz. Using the two approximations of detuning and direction in Eq. 3.3.3, we can find that

2 2 −γ ′τ / 2 −iω aτ ss −k B Tk τ / 2m −iwτ (1) (nth e e + ρ ee,0 e e ) g (τ ) = ss . (3.3.7) nth + ρ ee,0 With Eq. 3.1.3, we obtain our final form,

(2) 1 g (τ ) = 1+ ss nth + ρee,0 . (3.3.8) 2 2 2 2 2 ss − k B Tk τ / m 2 − γ ′τ ss −γ ′τ / 2 − k B Tk τ / 2m × [ρ ee,0 e + nth e + 2nth ρ ee,0e e cos δτ ] 2 Since nth is <<1, the nth term is negligible. We only looked at the atom in one position, but if we average over atomic positions over the whole cloud (assuming the cloud to be much more than one wavelength in size), the cosine term may cancel owing to the random atomic

ss locations [23]. The only term that is left in the equation then, is the dominant ρee,0 term,

which does have nth in it. We plot Eq. 3.3.8 for three different values of nth in Fig. 3.3.1. As can be seen, as nth increases, the coherence decreases. The curve with nth=0.01, optical depth 9 3 of 0.15, corresponds to a trap with a density of 1×10 atoms/cm and the curve with nth=0.02, optical depth of 0.3, corresponds to a trap with a density of 2×109 atoms/cm3. From this, we expect to see effects of radiation trapping at traps with low optical densities, more than an order of magnitude less than in Refs. 13,14. Figures 3.3.2 and 3.3.3 plot the three terms separately.

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(2) Fig. 3.3.1. Theoretical plot of g (τ) for three different values of nth. The inset shows the effect of the cosine 2 and nth terms.

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(2) ss Fig. 3.3.2. Theoretical plot of g (τ ) with only the dominant term with ρee,0 for three different values

of nth.

2 Fig. 3.3.3. The cosine and nth terms for a value of nth=0.02. As one can see, these two terms have a much smaller effect than the dominant term shown in Fig. 3.3.2.

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4. The Apparatus

By using an external cavity, we can tune the frequency of a laser diode to match the resonant frequency of atomic rubidium. The tuned laser light is red-detuned and sent in three orthogonal directions into a rubidium-filled vacuum chamber. With the addition of a magnetic field gradient, we can cool atoms and create a magneto-optical trap (MOT).

4.1 Atomic System The alkali metals are popular candidates for laser cooling and trapping because the excitation frequency of the transition between the ground state and the first excited state is in the visible region [17]. Because it is easy to obtain inexpensive infrared laser diodes, we choose rubidium, whose frequency is in the infrared. There are two natural isotopes, 85Rb and 87Rb, but we choose to trap 85Rb because it is more abundant (72%). In the ground state, the valence electron in atomic 85Rb lies in the 5s state, where 5 is the principal quantum number which describes the energy level and size, and “s” describes the G G orbital angular momentum of the electron L , which in this case, L = 0 . When the atom G interacts with the light, the electron jumps to the excited state, 5p, where L = 1. The G G interaction between the spin of the electron S and the angular momentum L causes a splitting, called the fine splitting, of the 5p state, into 5p1/2 and 5p3/2. The half-integer G G G G number is the total angular momentum J of the electron, where J = L + S . Because for the G 5s state L = 0 , the level does not split and is simply 5s1/2.

If the interaction of the angular momentum of the electron and the nuclear spin is considered, an even smaller splitting, called the hyperfine splitting, of the energy levels G occurs. For 85Rb, the magnitude of the nuclear spin I , is 5/2. Thus, the different levels are G G G G described by F = J + I , the total atomic angular momentum, where F ranges from G G G G J − I to J + I .

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The trapping transition desired is the one from the 5s1/2 level to the 5p3/2 state. When excited, an atom obeys the dipole transition selection rule, where ∆F = 0,±1. We choose to keep the trapping laser frequency at the F=3  F’=4 transition. However, as can be seen in Fig. 4.1.1 and by the selection rule, the atom can also be off-resonantly excited to the F’= 2 and 3 levels as well. Then when the atom decays from either the F’= 2 or 3 state, it may fall back into the F=2 level, which will not be in resonance with the trapping laser and within a few cycles, each lasting the upper state lifetime of 27 ns, the atom quickly becomes unaffected by the laser. To prevent this, a repumping laser is added, which is tuned to one of the F=2  F’=2 or 3 transitions. Then when the atom decays again, it can fall into either of the two ground states which are both being excited. The atomic spectrum of the two transitions can be seen in Fig. 4.1.2. Thus, the atom will continue to stay in resonance with the trapping laser, as well as the repumping laser. Both lasers are also slightly red-detuned to account for the Doppler effect of the moving atoms, as is explained in Ch. 2.

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Fig. 4.1.1. The Hyperfine Levels of Rubidum

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F=2 F’ F=1 F’

85Rb 87Rb

F=3 F’ F=2 F’

Fig. 4.1.2. The two transitions for both isotopes of rubidium.

4.2 Vacuum Chamber The chamber is a steel sphere with 7 pairs of windows protruding from the body. Three are in orthogonal positions, used for the trapping beams to enter and leave the chamber. Three different pump systems are used to bring down and maintain the pressure within the chamber to approximately 2×10-9 Torr. The first pump is the roughing pump, which mechanically pumps the chamber from atmospheric pressure to the milliTorr range. The second is a turbo pump, which mechanically pumps the pressure down to near 10-7 Torr. The final pump is the ion pump, which ionizes the gas and magnetically pulls out ions, allowing us to reach 10-9 Torr. After the chamber is pumped down to low pressures, rubidium is introduced through a leak valve attached to the vacuum chamber. The valve attaches to a short flexible vacuum tube which contains a glass vial of rubidium. After breaking the vial, the tube is heated and the

18

valve opened to allow the rubidium to enter the chamber. After many days of filling and heating, the rubidium had saturated the walls of the chamber enough so that we did not have to continuously fill the chamber to get a decent sized trap volume.

4.3 Frequency Tuned Laser Diode 4.3.1 External Cavity Diode Laser To use laser light to trap atoms, the atoms must be able to interact with the laser light, which happens when the laser is on resonance with the atom. Because there are no laser diodes made specifically to match the frequency of rubidium, we create an external cavity around a laser diode to tune it to the exact frequency we desire. The external cavity setup is set on an aluminum base plate, Fig. 4.3.1.1, mounted on Sorbothane.

Fig. 4.3.1.1. Baseplate

19

The Sorbothane absorbs and reduces any vibrations that could destabilize the external cavity system. This is all placed within an aluminum box to reduce any sudden thermal fluctuations and air currents from the outside environment since we also temperature tune the laser to assist in frequency tuning. The walls are held together with screws and the bottom of the box protrudes on two sides so that we can secure the box to the table with clamps. However, the laser light must exit the box so holes are made in the box. Also, we need to be able to adjust mounts in the cavity system, so holes are made in the walls of the box that allow in hex-key screwdrivers. To minimize any external influences, tape is placed over the adjustment holes and opened only when needed. The laser light exit holes are left open to prevent any reduction in power. Inside the box, we have the external cavity laser system. A laser diode is placed in aluminum housing, Fig. 4.3.1.2, so that the longer axis of the elliptical beam is parallel to the table.

1.125

0.750 0.219 0.563 0.172 0.375

4-40 Drill and Tap Ø0.235

7

.

0 0 .

3 0.020 2 5

Ø0.175 Ø0.221 0.875 0.187 0.375 0.040 0.125

0.040

8-32 Drill and Tap

Fig. 4.3.1.2. Laser mount and Teflon spacer

A driver1 provides current to the laser diode, which has a safety feature that allows us to set a current limit so that we do not destroy the diode. A Teflon spacer, Figure 4.3.1.2, is set between the housing and the aluminum base plate to thermally isolate the laser diode mount from the rest of the setup.

20

Ridges are made on the bottom of the Teflon spacer and diode mount so that there is a better grip onto the surface below. Nylon screws secure the diode mount to the base plate to prevent any heat conduction. A thermoelectric heater/cooler2 is secured to the housing with adhesive3 to thermally stabilize the laser diode. By applying a current across the leads, we can see that the side with the leads is the active side. A heat sink is attached to the inactive side of the heater/cooler to rapidly dissipate the heat. A thermistor4 is semi-secured with heat sink paste in a hole drilled as close as possible to the laser diode to monitor the temperature. The heater/cooler and thermistor allow us to set and control the temperature of the laser diode with a controller5. The controller has a safety aspect that sets limits on the current so that we do not destroy the heater/cooler by trying to overwork it. After the laser diode, a collimating lens6 is critically placed to collimate the diverging beam, Fig. 4.3.1.3.

Fig. 4.3.1.3. Collimating lens mount

The lens is achromatic, with a wavelength range of 600-1020 nm, where our wavelength of 780 nm falls clearly within the range. The center of the lens should be at the same height as the center of the laser diode to ensure the beam is at constant height. A micrometer screw finely adjusts the spacing between the lens and laser diode. The lens should be placed so that the laser beam has a constant size and shape over a distance of a few meters. Next, a beamsplitter7 is glued onto a 1 cm3 aluminum cube and secured onto the baseplate by a screw. It is used to transmit half of the collimated light and reflect the other half, which will later be used as the main trapping beam. 21

The last element in the external cavity of the laser diode is a diffraction grating8, glued onto the side of a mount9. Placing a small amount on the corners of the grating should secure it in place (but still allow us the possibility to remove it with some ease if we need to). The transmitted beam from the beamsplitter cube above hits the diffraction grating, which disperses different wavelengths into different angles, except for the zero order. We can adjust the diffraction grating to a certain angle by adjusting the grating mount and have the desired wavelength go back through the beamsplitter into the laser diode. This allows us to selectively increase the power of the desired wavelength in our laser diode so that that wavelength will predominantly constitute the trapping beam. Two piezo-electric disks10 soldered back to back are placed behind the diffraction grating to allow us to even more finely adjust the angle of the grating. These piezos are ones normally found in buzzers. Plastic spacers cut from transparency sheets are placed between the piezo and the mount so that there is no electrical conduction between the two. Grease is applied on the plastic that is in contact with the ball of the micrometer screw so that the screw turns smoothly. By applying a current from a function generator, we can cause the piezo to flex and thus move the diffraction grating by very small amounts. Because the disks protrude below the mount, a deep groove has to be made in the baseplate so that the diffraction grating mount can move. The final setup is shown in Fig. 4.3.1.4. In order for us to have frequency tuning, we have to ensure that the light from the diffraction grating must be aligned so that it goes back into the laser diode. When we look at the light reflected out of the beamsplitter, we see two spots. One spot is the beam reflected from the diffraction grating. By adjusting the angle of the diffraction grating mount, we can move one of the spots closer to the other which means the light is returning directly back into the laser diode. When near each other, the moving spot “jumps” into the stationary one and remains there even while the angle of the mount is still being adjusted. This situation exists only for a small range of angle adjustments, after which the two spots separate again. This short angular range represents the external cavity adjustments when we are tuning the frequency of the laser diode. For the trapping laser, we use a Sanyo laser diode11. This diode is on resonance when set at 25.6°-25.7° C with a current of 95-100 A, which is the typical operating current. The repumping laser is a Sharp laser diode12. This diode is on resonance at 26.1° C and has a

22

typical operating current of 68-73 A. Because everything drifts from day to day, we need to adjust the mounts by a method called threshold tuning so that we ensure maximum feedback. First, the current is brought down to the threshold current of the laser, or when the laser no longer lases. When the diffraction grating is horizontally and vertically aligned and the collimation optimized, the laser will begin to lase at currents lower than the threshold current. By bringing down the current in small increments, we can adjust the mount until the laser is barely lasing. This is the new threshold and the laser is optimized for wavelength selection. For example, the Sanyo laser threshold current is typically 30 mA, but with feedback, the new threshold current is 26 mA. For the Sharp, the typical threshold current is 50-60 mA, and the new threshold current is 40 mA. We have currently found a new high power, cheap Sharp laser diode13 that may be used in future applications.

23

Fig. 4.3.1.4. External Cavity Diode Laser

24

4.3.2 Saturated Absorption

To confirm that the laser diode is frequency tunable about the resonant frequency of rubidium, it is placed in a saturated absorption setup, Fig. 4.3.2.1.

Fig. 4.3.2.1. Saturated absorption setup.

25

A mirror is placed after the beamsplitter to steer the reflected beam out of the box. A linear polarizer is used to control the intensity of the beam to prevent power broadening, so we can get more distinct peaks.

Figure 4.3.2.2. a. Power broadened peaks. b. non-power broadened peaks. Then the beam encounters a thick glass plate14. About 4% of the light is reflected on each surface, giving us two weak reflected beams (separated by about 0.5 – 1 cm) and one strong transmitted beam. The two weak beams are reflected into a rubidium vapor cell15 and then reflected off a glass microscope slide into two photodiodes16. As the diffraction grating is ramped by the piezo, the laser is ramped through a small range of frequencies. When the beams are on resonance, they are absorbed by the rubidium atoms. Since the atoms are moving inside the cell, the different velocities Doppler shift them into and out of resonance with the beams. Thus, the signal into the photodiodes is a Doppler broadened absorption of the weak beams, as shown in Fig. 4.3.2.3.

26

Fig. 4.3.2.3. Doppler broadened signal. The strong transmitted beam is reflected off two mirrors17 and set to coincide with one of the two weak beams, making sure that it is not reflected into the photodiodes. The strong and the weak beam counter-propagate, so that atoms Doppler shifted into resonance with the weak beam are further Doppler shifted out of resonance with the strong beam. But for atoms moving with a velocity perpendicular to beam propagation, both beams are on resonance with the atom. When on resonance, the strong beam causes the atoms to be mostly in the upper state due to saturation. The weak beam is too weak to cause saturation and thus a Doppler-free saturation dip, also known as a Bennet hole, is burned into its absorption profile [24]. By subtracting the other weak beam’s Doppler broadened profile, a burned spectrum with no Doppler effects is obtained, as is demonstrated in Fig. 4.3.2.4.

27

Fig. 4.3.2.4. Removing Doppler effect in signal via saturated absorption. In reality, there are six Bennet holes, but for simplicity, only one is shown.

As the trapping laser is ramped, it is on resonance with the transitions from the F=3  F’=2, 3, 4 states. Hence we expect to see three peaks, not six, corresponding to these transitions. The extra three peaks are called “cross-over peaks”. They appear because it is possible for an atom to be Doppler-shifted into resonance to a certain transition 1 (say F=3

 F’=3), such that ω = ω1 − δ , with the weak beam and simultaneously be Doppler-shifted into resonance to a different transition 2 (say F=3  F’=2), such that ω = ω2 + δ , with the counterpropagating strong beam, Fig. 4.3.2.5.

   Fig. 4.3.2.5. The atom is blue Doppler shifted into resonance with the weak beam, = 1- ,G andG red Doppler shifted into resonance with the strong beam, = 2+, where the detuningδ = k ⋅ v .

Again, the strong beam saturates, and a hole is burned in the Doppler spectrum at a frequency, (ω + ω ) which can easily be seen to be at the arithmetic mean of the two transitions, ω = 1 2 . 2

28

For the three transitions, there are three corresponding crossover peaks at the arithmetic means, thus resulting in six peaks in the spectrum, as is clearly seen in Fig. 4.3.2.6.

F’=3~4 F’=2~4

F’=4 F’=3 F’=2~3 F’=2

Figure 4.3.2.6. Trapping saturated absorption spectrum. The peaks with two numbers equal to F’ are the crossover peaks, whilst the atomic transitions have only one.

The stable Doppler-free saturated absorption spectrum confirms that our laser can be tuned to the trapping frequency of rubidium. By reducing the ramp on the piezo, we can selectively choose a frequency that we want to use. Knowing that we want to trap on a frequency a few linewidths red-detuned from the F=3  F’=4 transition, we can lock onto that specific frequency. A locking circuit and an AOM are used to lock onto the desired trapping frequency.

4.3.3 Locking Mechanisms Using a locking circuit [25], we can lock onto the side of any desired peak of the spectrum, but not at an arbitrary position in the hyperfine spectrum. This is a problem because to trap moving atoms, the trapping laser must be red-detuned, typically between 1 – 4 linewidths. Thus, an Acousto-Optical Modulator19 (AOM), consisting of an RF wave propagating through a crystal, is used. Applying an RF voltage on the piezoelectric transducer inside the

AOM, an acoustic wave is sent into the medium (PbMoO4). The acoustic wave perturbs the refractive index of the medium, which results in an optical diffraction grating. When the beam enters the wave front at the Bragg angle, the intensity of the first order is maximized. 29

The first order frequency is also shifted because of the sound wave. It can be thought of as a Doppler shift as the wave encounters a surface moving at the velocity of the sound wave [26]. Our AOM can shift the frequency by 60 – 100 MHz. The crossover peak, F=3  F’=3~4, next to the trapping transition, F=3  F’=4, is 60.3 MHz away, and the next crossover peak, F=3  F’=2~4, is 92 MHz away, Fig. 4.3.3.1.

Fig. 4.3.3.1. The six peaks are schematically drawn separately for clarity with their corresponding detunings, with respect to the trapping peak.

Because we want to red-detune the laser and the first crossover peak is at the minimum modulation of the AOM, we find it convenient to lock onto the F=3  F’=2~4 transition. Then referring to the AOM data sheet, we can adjust the voltage so that we are the detuned from resonance by a desired amount.

30

4.4 The Trapping Setup The trapping laser light must undergo a few changes before it is ready to be used to trap atoms. There are four major elements in the trapping beam shaping setup: the anamorphic prism pair19, the optical isolator20, the AOM, and the telescope. As mentioned earlier, the light from the laser diode is elliptical. Thus, an anamorphic prism pair is used to circularize the beam by reducing one side of the beam while unchanging the other. Figure 4.4.1 shows how the prism pairs are aligned. For different ellipticities, the angles of the prism pairs differ as the figure shows.

Fig. 4.4.1. The alignment of the anamorphic prism pair. The elliptical beam enters in the right and leaves circular from the left. For a 2:1 ratio, the values for A, B, C, D are 21.2°, 6.0°, 5.1mm, and 5.3 mm, respectively. For a 3:1 ratio, the values are 30.4°, 0.1°, 6.4 mm, and 6.4 mm, respectively.

The power of the beam, measured at an aperture set after the anamorphic prism pair, is found to decrease by 10%. Next, the circular beam is sent into an optical isolator. The optical isolator is used to prevent backreflections from feeding back into the laser system and destabilizing the laser. The isolator contains a Faraday rotator, which consists of a magneto-optically active cylindrical crystal mounted inside a concentric permanent ring magnet, and two polarizers on the two ends. The plane of polarization of the input polarizer is set parallel to the linear polarization of the laser beam. Next, the Faraday isolator rotates the polarization of the light by 45° in a direction determined by only the direction of the co-axial magnetic field, regardless of the propagation direction of the beam [26]. The plane of polarization of the

31

output polarizer is then set parallel to the rotated laser beam. The power of the beam, measured at an aperture set after the isolator, is found to decrease by 15%. If light reflects back into the optical isolator, the polarization of the beam encounters an additional rotation of 45° in the same direction as before and will be perpendicular to the input polarizer. Consequently, no light continues back into the laser system. Next, the light is focused into the AOM, described earlier, so that we can modulate the frequency and precisely detune the light by 1-4 linewidths. A lens is used to focus the light into the diffraction grating in the crystal formed by the RF-generated acoustic wave. The beam size needs to stay constant through the length of the crystal so that it encounters the same number of slits of the diffraction grating evenly. This can be done by making the confocal parameter, or the axial distance for which the beam radius lies within a factor 2 of its minimum, to be much greater than the length of the crystal inside the AOM. The

confocal parameter 2zo, can be found by combining the following equations λ Wo = , 4.4.1 πθ o and

2Wo 2zo = , 4.4.2 θ o where Wo is the beam waist radius, λ is the wavelength of the beam, and θo is the angle of the

−1 ≈W ’ light [4]. θo is found by tan ∆ ÷ where W is the initial beam size radius and f is the focal « f ◊ length of the lens. For a 3mm diameter size beam and a lens with a focal length of 20 cm, the confocal parameter is found to be only 0.88 cm, whereas a lens with a focal length of 30 cm has a confocal parameter of 2 cm. Since the crystal in our AOM is 0.8cm along the beam path, a lens with at least a focal length of 30 cm must be used. To conserve tabletop space, a 30 cm lens is used instead of a lens with a greater focal length. Another 30 cm lens is set focal length away after the AOM to bring the beam back to its original size. Also, a λ/221 plate is placed before the AOM to ensure that the polarization of the beam is horizontal, which is preferred by the AOM, i.e. diffraction efficiency is maximized. An aperture is used to select and transmit only the first order beam for the trap. After optimization of power, the beam is reduced by 75% through the AOM. Because the beam is so small in diameter, a 32

telescope is used to magnify the beam. A 20cm lens is placed focal length after a microscope objective of 10X magnification. Using an aperture, we cut the beam to a diameter of 15mm. Finally, the total power of the beam has been reduced to 25% of the total initial power out of the laser. In our case, we get a trapping beam of 7-8 mW from an initial laser beam of 30 mW. A λ/2 plate is placed before a polarizing beamsplitter22 so that we can adjust the beam that 20% of the light goes into the x-direction and 80% of the light goes into the y and z- directions. A 50/50 beamsplitter23 splits the stronger beam equally between the y and z- directions. Finally, before the three orthogonal beams enter the vacuum chamber, they are circularly polarized by λ/4 plates24, shown in Fig. 4.4.2. σ - σ -

σ+ σ -

x σ+ σ+

z y

Fig. 4.4.2. The polarization of the light is determined by the gradient caused by the magnetic coils.

As can been seen in the figure, the light in the y and z-directions are polarized opposite of the light in the x-direction. This is because the gradient in the y and z-directions are in the G G opposite direction of the x-direction. Maxwell’s equations give us ∇ ⋅ B = 0 and that shows ∂B ≈ ∂B ∂B ’ us that = −∆ + ÷ , which explains the flip in sign of the gradient in the y and z- ∂x « ∂y ∂z ◊ directions.

33

B σ+ σ -

x

B σ - σ+

y, z

Fig. 4.4.3. The gradient is opposite in the x-direction as the y and z-directions. Thus, the polarizations of the incoming laser beams must also be different according to the field.

By aligning the beams to a mark on a plastic disk, we center each beam so that the trap will form in the center of the chamber. On the other side of the chamber, another λ/4 plate, in an arbitrary orientation, and a mirror are placed to reflect the beams straight back into the chamber with opposite polarizations. The retro-reflection alignment is done by aligning the reflection back onto the mirrors that were before the chamber. Thus, the trap has six orthogonal circularly polarized beams coinciding at the center of the chamber. The repumping laser setup is not as complex as the trapping laser. An anamorphic prism pair circularizes the beam like the trapping beam. We do not have an optical isolator, but since there are not many components in the setup, each lens and mirror are adjusted so that the backreflections are misaligned as to not enter the laser system. Two lenses are used to increase the bean size, modulated by an aperture. The beam is sent into the same polarizing beamsplitter as the trapping beam to ensure that they coincide. The proportional intensity is not crucial for the repumping as with the trapping since it only needs to be in the chamber, so polarizations and powers are not adjusted.

34

4.5 Magnetic Coils 4.5.1 Magnetic Field Gradient Coils The lasers act as velocity-damping forces in all six directions and cool the atoms. However, since they do not exert position-dependent forces, they are not enough to create an atom trap. We rely on a magnetic gradient to gather the atoms in one central spot where the magnetic field is zero. By putting two coils in an anti-Helmholtz configuration as in Fig. 4.5.1.1, we can create the gradient needed. The coils are parallel to each other, but have currents running in opposite directions. Although the Helmholtz configuration has the distance between the coils the same as the radius, due to spatial restrictions, the radius is smaller. This will give us a nonlinear gradient, but for a centimeter area in the center of the chamber where the trap resides, the gradient is close to being linear. The gradient perpendicular to the axis of the coils is half the magnitude of the gradient along the axis.

R x

A

Fig. 4.5.1.1. The magnetic gradient coils are set up in an anti-Helmholtz configuration.

We found the magnetic field for a point x in Eq. 2.2.2. To find the gradient, the derivative of the field is taken and we obtain Eq. 4.5.1.1.

2 » ÿ NIR ∂B 3µ0 … 2A + 2x 2A − 2x Ÿ = − 5 + 5 (4.5.1.1) ∂x 4 … 2 2 2 2 2 2 Ÿ (R + (A + x) ) (R + (A − x) ) ⁄ 35

If we take the first two terms of the Taylor expansion of Eq. 2.2.2., B(x) = B(0)+ B′(0)x , we

2 see that we get B(x) = 0 − (3µo NIR A)x and is indeed linear, giving us a constant gradient for small x. Since we have multi-layer coils, we take the midpoint of the layers for the radius and obtain 4.06 cm for R. The center of the chamber is located about 7.62 cm from the center of the coils. Because of space limit, we can fit 150 turns with 10 amps25 (1500 ampturns) on our flanges and calculate a gradient of around 15 G/cm at the center of the trap, as seen in Fig. 4.5.1.2.

Magnetic Field Gradient

25

20 ) s s u a g

( 15

t n e i d a r

G 10

d l e i F 5

0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 distance (cm)

Fig. 4.5.1.2. The magnetic field gradient in a centimeter long spot at the center of the trap.

As one can see from Eq. 4.5.1.1, the closer together the coils are, the greater the gradient will be. In order to fit it between the other protruding flanges and be as close to the chamber as possible, the coils were wound in a graded fashion. The coils are wound by a lathe on a Teflon form, Fig. 4.5.1.3. Teflon is used because it can tolerate high temperatures, up to around 200°C, and the coils reach a little over 100°C. Teflon also is an insulator, so it prevents any possible leakage of current from the coils to the chamber or ground and also prevents any eddy currents from occurring when the coils are rapidly shut off. 36

0.5”

0.03”

2.8” 4.5”

1.76”

Fig. 4.5.1.3. The Teflon form for the magnetic coils. The big piece on the side was made so that the form could be held in the lathe.

4.5.2 Nulling Coils The atoms will gather where the magnetic field is zero. But if there are extraneous magnetic fields from outside influences such as the Earth’s magnetic field, there are “heating” forces on the atoms. To cancel the stray fields in all three orthogonal directions, there are three sets of Helmholtz coils on each axis. Unlike the gradient coils, these are not switched on and off since they are for background noise effects. The coils are powered by a Hewlett Packard 6629A DC power supply. The HP power supply has four independent channels which allow us to apply a different current to each pair of coils. Each coil consists of 30 turns on each axis. We find that for molasses, we need 0.8 A in the x-axis, 0.35 A in the y-axis, and 0.5 A in the z-axis. However, when the gradient coils are turned off and the atoms fall away, they seem to be affected by an external magnetic field. No settings of the nulling coils seem to cancel that external magnetic field and also keep the field at the center of the chamber 0 G. Originally the coils were wound right onto the flanges, which did not put them in the true Helmholtz configuration where the radius is equal to the distance between the two coils. However, we

37

find that the field is not constant, changing by 76 mG only 1 cm from the center of the chamber as shown in Fig. 4.5.2.1.

dc coils field

2.13 2.03 1.93 )

s 1.83 s

u 1.73

a 4.06 cm radius g

( 1.63 15.24 cm radius d

l 1.53 e i

f 1.43 1.33 1.23 1.13 -2 -1 0 1 2 distance from the center (cm)

Fig. 4.5.2.1. The field due to the DC coils for the non-Helmholtz configuration, 4.06 cm radius, and the Helmholtz configuration, 15.24 cm radius. The Helmholtz configuration stays constant whereas the non- Helmholtz configuration changes by 280 mG at ±2 cm.

Thus new Helmholtz nulling coils with the radius of 15.24 cm are to be made. We hope to see cancellations of all extraneous magnetic fields.

4.5.3 Switching Circuit The radiation trapping experiments are done on an optical molasses, which is an optical atom cloud with no magnetic forces. An optical molasses is colder than a MOT because the magnetic field creates additional splittings that complicate the system and increase energy inside the atoms. Thus, to get optical molasses for certain measurements, the gradient coils must be switched off within a few milliseconds. However, due to Lenz’ law, induced Eddy currents prevent the currents from being shut off in such a short time period. A modified RL circuit designed and constructed by Miami University’s instrumentation lab is used (see appendix B). The coils are found to have an inductance of 2 mH, and the resistor in place is 50 ohms, so the time constant is found to be 40 µs in Eq. 4.5.3.1 L τ = (4.5.3.1) R 38

where L is the inductance and R is the resistor. If we switch the coils on and off at a frequency of 100 Hz, the power dissipated in the resistor is 10 W, as shown in Eq. 4.5.3.2, 1 P = fLI 2 (4.5.3.2) 2 where f is the switching frequency. The circuit is versatile to handle configurations with currents ranging from 1 to 10 A and coils with inductances from 2 to 80 mH by changing one resistor. With the gradient and the frequency-detuned stable laser system, we are able to obtain a MOT in the vacuum chamber. The switching circuit allows us to have optical molasses from which we can observe radiation trapping using intensity correlations, which is explained in the next chapter.

39

5. Characteristic Measurements

Now that the atom trap has been attained, it has to be characterized. There are four measurements used to characterize the trap are the following: number of atoms, loading curve, density, and temperature.

5.1 Number Measurement The number measurement is a non-invasive method of estimating the number of atoms in the trap. It is done by simply placing a lens26 and a detector27 outside a window of the chamber, 9.25 cm from the center of the chamber, where the trap resides. The lens is chosen to have a focal length of 5 cm to create a 1:1 image of the trap onto a photodetector placed after the lens. Although the scattered light gathered by the lens is weak, it is sufficient for one to use an IR viewer to position the image onto a photodiode. Then using the voltage reading, the number of atoms can be determined. First, the photon scattering rate of an atom in a laser field is determined by Eq. 2.1.1,

γ I I o γ p = 2 , (5.1.1) 2 1+ I I o + [2δ γ ] where I is the laser intensity, Io is the saturation intensity of the atom, γ is the decay rate which is 3.7×107 cycles/sec for Rb, and δ is the detuning of the laser. Assuming that the laser is at saturation intensity and the detuning is one linewidth γ, the scattering rate is found to be 3.1×106 cycles/sec. The power of light radiated from an atom can be > determined by multiplying the energy from a photon ωo by the number of photons that are scattered, or the scattering rate > Prad = ωoγ p , (5.1.2)

For a Rb atom, the power radiated is 7.8×10-13 W/atom. Using geometry, the solid angle Ω of the trap that is being transmitted through the window can be calculated by dividing the area of the window by the squared distance from the trap to window. By dividing by 4π, the percentage of the total radiation can be obtained. In our case, the area of our window is 10.6 cm2, and 9.3 cm from the trap, which allows us to observe 1% of the total radiation from the trap. Therefore, if only 1% of the radiation from the

40

trap is seen, we expect to see about 7.8×10-15 W/atom. Knowing the response of the photodiode, which is 0.51 A/W in our case, we can find that we will measure 4.0×10-15 A/atom. Since an oscilloscope measures voltage, not current, a current-voltage converter is used to measure the signal from the photodiode. We use the highest gain, which is 1×108, so each atom gives 4.0×10-7 V/atom. Finally, we find that the number of atoms can be found by

Vsig N atoms = . (5.1.3) Vatom For a trap with laser intensity 6.8 mW at the 15mm aperture, gradient of about 6 G/cm , we get a signal of around 2 V, which correlates to 5×106 atoms in our trap. The tuning voltage on the AOM is found to be near 14.5 V (refer to AOM datasheet), corresponding to about half a linewidth blue-detuned, but this is not accurate because we do not know exactly on which frequency the laser is locked. We expect that the beams are actually 1~2 linewidths red-detuned.

5.2 Loading Curve Using the same setup as the number measurement, the loading curve of the trap can be recorded on an oscilloscope. The loading curve can be used to find the density of the trap or the pressure of the chamber. If we define an atom loading rate L and an elastic collision loss rate γe, the rate of which atoms are knocked out of the trap by background gas molecules colliding into them, the overall atom trap rate can be determined: dN = L − γ N . (5.2.1) dt e A solution to the differential equation is as follows L N(t) = (1− e −γ et ). (5.2.2) γ e At the steady state, when t = ∞ , the solution condenses to simply L N ss = , (5.2.3) γ e where Nss is the number of atoms in the trap when it reaches steady state. For early times, the number of atoms in the trap is very low, so the rate of loading/loss is

41

dependent solely on L. Thus, knowing the number of atoms from the number measurement and the slope of the loading curve near t = 0 , the elastic collision rate can be found. For a trap of 6.2×105 atoms, we get a loading curve shown in Fig. 5.2.1.

Loading Curve

0.3 y = 0.15x

0.25

0.2 ) V (

0.15 y t i s n e

t 0.1 n I

0.05

0

-0.05 0 2 4 6 8 10 Time (s)

Fig. 5.2.1 The loading curve for a trap with trapping laser beams at 7.3mW at our typical detuning of about 2.5 linewidths.

For early times, the slope, or L, is 0.15 V/sec which correlates to 3.8×105 atoms/sec, giving us a collision rate γe of 0.6 collision/sec. Or an exponential curve can be directly fitted to find the collision rate. The elastic collision rate can also be described as [17]

γ e = nσ a v , (5.2.4) where n is the number density, σa is the atom-atom collision cross-section, and v is the average velocity. The velocity is found using Eq. 2.1, which is 340 m/s at room temperature. For Rb atoms, the cross-sectional area for collisions is 2.5×10-17 m2 [27]. Using room temperature for the temperature of the gas atoms, the number density can be calculated. The ideal gas law relates the number density to pressure,

42

P n = , (5.2.5) k BT

-9 where kB is the Boltzmann constant. Since our pressures are typically around 2×10 Torr, and the density is about 6.4×1013 atoms/m3, we calculate a collision rate of 0.6 collision/sec, agreeing with our previous calculation.

5.3 Density Measurements As mentioned above, the density of the atom trap can also be measured directly. A very weak on-resonant probe beam is sent through the center of the trap and onto a photodiode on the other side of the chamber, making sure that the beam does not saturate the photodiode. A weak probe is used to prevent power broadening [19]. The probe beam is ramped in and out of resonance by varying the current. The intensity of the probe beam on the photodiode similarly fluctuates as the light is absorbed by the trap when on resonance and passes through unchanged when off resonance. Knowing that the absorption rate of the light through atoms is dI = −nσI , (5.3.1) dz where σ is the cross section of absorption of an atom, and n is the density [17]. This differential equation can be solved easily,

−nlσ I(l) = I o e , (5.3.2) where l is the length of the trap. Since this is the amount absorbed, subtracting it from Io gives the intensity of the probe after it passes through the cloud. If we denote the I − I relative absorption o as A, we get I o

A =1 − e −nlσ . (5.3.3) If we solve Eq. 5.3.3 for the number density, we find ln(1− A) n = − (5.3.4) lσ We use the intensity of the probe beam that passes through the trap when off-resonance as Io to account for any scatter from the windows of the chamber. Then when the probe is on resonance, the intensity of the probe I decreases due to absorption.

43

In Figs. 5.3.1 and 5.3.2, the peaks are the three transitions of the trapping spectra of F=3F’=2, 3, 4.

Number Density Measurement

0

-1

-2

-3 ) V (

y -4 t i s

n Io e -5 I t n I -6

-7

-8

-9 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Time (s)

Fig. 5.3.1. The off-resonance intensity of the probe is 8.4V, which is Io.

44

Density Measurement

0.6

0.5

0.4 ) V

( 0.3

e g a t l

o 0.2 v

0.1

0 0 0.01 0.02 0.03 0.04 0.05 0.06 -0.1 time (s)

Fig. 5.3.2. Because the absorption is much less than the intensity of the probe beam, we had to look at the peaks at a finer scale. The tallest peak is our trapping peak, 504 mV, which is Io-I.

The peaks in Figs. 5.3.1 and 5.3.2 are measured for a trap of 6×106 atoms. Using Eq. 5.3.1, for an absorption cross-sectional area of 290.7×10-11 cm2 [17], and a trap length of 1mm, we find that the density of this trap is 2×108 atoms/cm3.

5.4 Temperature Measurement The peaks from absorption, however, are wider than they should be because of Doppler broadening. Since Doppler broadening is related to temperature by Eq. 3.1.5, the temperature can be calculated by the broadening difference. The broadening in this case is 21.3 MHz, so the temperature for this trap is found to be 340 mK. Though we made the probe beam very weak, there is still some power broadening. Because of that, the temperature found is not the exact temperature of the trap. Therefore, another temperature method is also used.

45

To find the temperature of atoms, the time-of-flight, or TOF, method is utilized. This method completely destroys the trap so is made after all measurements are taken. An on-resonance probe “sheet”, 4mm x 8 mm, is placed 1cm under the trap. When the magnetic field gradient is shut off, the atoms fall through the sheet by the effects of gravity. The lens and photodiode from the number measurement are moved to catch the light scattered from the atoms passing through the probe sheet. Using the time it takes from the gradient to shut off to when the atoms fall through the probe sheet, and the acceleration due to gravity, the velocity of the atoms can be easily determined with simple kinematics 1 ∆x = v t + gt 2 . (5.4.1) o 2 The velocity of the atoms can be related to temperature using the ideal gas law, as mentioned in earlier sections. For a trap at 100
K and a probe distance of 1 cm, we expect to see a time of 66 ms. To prevent any miscellaneous scatter inside the vacuum chamber, the trapping and repumper laser beams are shut off with the use of a mechanical shutter28 in the setup path. We are still attempting to implement a TOF measurement.

46

6. Intensity Correlations

As explained in chapter 3, intensity correlations can be used to observe radiation trapping forces in an atom trap. Figure 6.1 shows a brief overall view of the setup for the intensity correlation measurement.

Lens f=5cm Optic cable dia=100
m

detector I(t) Photon counting electronics

computer

Fig. 6.1. Schematic diagram of the apparatus.

Because the window that is used is angled upwards from the chamber, an optical rail29 is used to align all the components of the setup along the axis of the window. Like the number measurement, a lens with a focal length of 5cm is used to gather radiation from the trap. Immediately after the lens, is an iris30, which we can close to 1 mm. The iris is used to focus the radiation into an optic cable, but it also used for spatial coherence. In Ch. 3, we assumed a spatially coherent sample, but that is not realistic. Equation 3.1.4 actually is g (2) (τ ) = 1+ S | g (1) (τ ) |2 , (6.1) where S is the spatial coherence. To form a simple picture of spatial coherence, let us consider a beam of constant intensity and two points on a plane perpendicular to the direction of the beam propagation. If the beam is spatially coherent, the phase difference measured at the two points will be constant at any time. Even if the phase is randomly fluctuating, the phase difference between the two points will remain constant. If the beam is not spatially coherent, the phase difference between the two points will change [20]. A point source produces a spatially coherent beam, but the signal from a point source is weak. Thus, in 47

order to facilitate our measurement taking, we look at a finite area of the trap. Consequently, this reduces our spatial coherence, which is 0.25 in our case. For a more in-depth discussion on spatial coherence, refer to Ref. 28. The light is focused into a 100µm diameter optic cable31. The cable is set into a 6-axis kinematic mount32 to ensure that light from the lens is centered into the cable. However, since single photons are counted, the system needs to be free from background light. A lens tube33 covers the 10cm span from the iris to the optic cable with an IR filter34 is placed in the tube. Also, the lens tube is shrouded in a black cloth to cover any leaks in the setup. The cable takes the light to a single-photon counting module35 on the table. Every time it detects a photon, a 5V TTL pulse is outputted. The detector is also shrouded in black cloth. The approximate number of photons being sent to the detector is monitored on a rough scale just so that the diode is not oversaturated. The output is also sent to a multichannel scaler36 (MCS) which registers the incoming pulses and sorts them into user defined time bins. The user has to define the number and size of bins and number of passes to take. The MCS can sort the data into a maximum of approximately 16,500 bins with no lost counts between the bins. We set the number of passes to be 1,100 ns as the size of the bin, and the maximum number of allowed bins. Thus, the MCS would count pulses in 100 ns intervals and place them in each corresponding bin. Then the software program outputs a plot of the number of pulses vs. time in intervals of 100 ns. This plot was taken into a program [28], which calculated g(2)(τ) and plotted it vs. τ. Figure 6.2 is an early attempt of taking intensity correlations on an optical molasses and a MOT. As one can see, the data does not start at 2, but near 1.25, due to the spatial coherence mentioned earlier. Also, for later times, the data does not fall to 1, as one would expect. A measurement was done on laser light and white light to test the apparatus. We found that the correlation measurements for both are 1, as expected. The noise in the data makes it difficult to see the effects of radiation trapping that we want to measure. We expect the molasses data to fall faster than the MOT data, but it is hard to ascertain from our plot. We hope that with a better locking technique and nulling coils, we may be able to obtain better measurements.

48

Fig. 6.2 We have taken and plotted g(2)() vs.  for an optical molasses and a MOT.

49

7. Conclusion and Future Goals

As shown in the last chapter, we have attempted to take intensity correlation measurements on optical molasses and MOT. However, we find that we have many noise issues that impair the data. It is hard to discern the effects of radiation trapping due to noise in the data that we propose are from the laser systems. Even with the naked eye, we can see the trap fluctuating in intensity. So we are currently working on incorporating a lock-in amplifier and a better locking system to more stably lock onto the trapping frequency.

Currently we are also working on making true Helmholtz coils for the dc nulling coils. As mentioned in chapter 4, we suspect that the reason we have not been able to obtain TOF measurements are due to stray magnetic fields not being exactly canceled by our previous nulling coils. Instead of falling down through the probe beam, the atoms are perhaps experiencing forces away from the probe. We expect that the installation of the new nulling coils will allow us to make the TOF temperature measurements.

The next step after radiation trapping in molasses is to make an optical lattice. We want to look at the motion atoms exhibit in an optical lattice. Because the atoms are always absorbing and emitting photons, one expects them to exhibit a Brownian random walk. However, when the depth of the potential wells of the optical lattice is decreased, the atoms may fly out of the well, traveling freely for some time before falling into a well again. This action can be described as Levy motion, a type of non-Brownian random walk. Levy motion is determined by sudden long range changes in motion, called Levy flights, which can characterize the motion of the atom when it escapes the well. Using the intensity correlation method, we intend to observe these atomic motions.

50

References

[1] R. Frisch, Z Phys. 86, 42 (1933).

[2] A. Ashkin, Phys. Rev. Lett. 24, 156 (1970).

[3] T.W. Hänsch and A.L. Schawlow, Opt. Commun. 13, 68 (1975).

[4] D. Wineland and H. Dehmelt, Bull. Am. Phys. Soc. 20, 637 (1975).

[5] S. Chu, L. Hollberg, J.E. Bjorkholm, A. Cable, and A. Ashkin, Phys. Rev. Lett. 55, 48 (1985).

[6] E.L. Raab, M. Prentiss, A. Cable, S. Chu, and D.E. Pritchard, Phys. Rev. Lett. 59, 2631 (1987).

[7] http://www.boulder.nist.gov/timefreq/cesium/fountain.htm.

[8] M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, E.A. Cornell, Science, 269, 198 (1995).

[9] K.B. Davis, M.O. Mewes, M.R. Andrews, M.J. Van Druten, D.S. Durfee, D.M. Kurn, W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995).

[10] C.C. Bradley, C.A. Sackett, R.G. Hulet, Phys. Rev. Lett. 78, 985 (1997).

[11] M. Fleischhauer, Europhys. Lett. 45, 659 (1999).

[12] G. Ankerhold, M. Schiffer, D. Mutschall, T. Scholz, and W. Lange, Phys. Rev. A 48, R4031 (1993).

[13] T. Walker, D. Sesko, and C. Wieman, Phys Rev. Lett. 64, 408 (1990).

[14] D. Sesko, T. Walker, and C. Wieman, J. Opt. Soc. Am. B, 8, 946 (1991).

[15] S. Bali, D. Hoffman, J. Siman, and T. Walker, Phys. Rev. A 53, 3469 (1996).

[16] A. Matsko, I. Novikova, M. Scully, and G. Welch, Phys. Rev. Lett. 87, 133601 (2001).

[17] Metcalf, Harold J. and Peter van der Straten. Laser Cooling and Trapping. Springer, New York (1999).

[18] Shankar, R., Principles of Quantum Mechanics, (Plenum Press, New York, 1994), p.398.

51

[19] R. Loudon, The Quantum Theory of Light (Oxford University Press, Oxford, 1983).

[20] P. W. Milonni and J. H. Eberly, Lasers, Wiley, 1988.

[21] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge, 1995.

[22] M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge 1997.

[23] R. Kaiser, private communication.

[24] W. Demtröder, Laser Spectroscopy. Springer, Berlin (1996).

[25] K.B. MacAdam, A. Steinbach and C. Wieman, Am. J. Phys. 60, 1098-1111, (1992).

[26] B.E.A. Saleh, and M.C. Teich. Fundamentals of Photonics. John Wiley & Sons, Inc., New York (1991).

[27] S. Bali, K.M. O’Hara, M.E. Gehm, S.R. Granade, and J.E. Thomas, Phys. Rev. A, 60, R29-R32 (1999).

[28] M. Beeler, Honors Thesis, Miami University, 2003.

52

Appendix A: Parts and Suppliers:

1. Ultra Low Noise Current Source, LDX 3620, ILX Lightwave, 800.459.9459, www.ilxlightwave.com.

2. Thermoelectric Cooler, 1”x ½”, CPO.8-63-06L, MELCOR, 1040 Spruce St., Trenton, NJ 08648, 609.393.4178, www.melcor.com.

3. Clear Thermal Adhesive. Minco Products, Inc., 7300 Commerce Lane, Minneapolis, MN 55432, 763.571.3121, www.minco.com.

4. 10 kΩ NTC Leaded Thermistor, 2322 640 55103 (BC1489), Digikey, 701 Brooks Avenue South, Thief River Falls, MN 56701, 800.DIGIKEY, www.digikey.com.

5. Temperature Controller, LDT 5910B, ILX Lightwave, 800.459.9459, www.ilxlightwave.com.

6. Collimating Lens, 1403.108.022, f= 5mm, numerical aperture 0.5, Rodenstock/LINOS Photonics, 459 Fortune Blvd, Milford, MA 01757, 800.334.5678, www.rodenstockoptics.com.

7. Non-Polarizing Beamsplitter, 1 x 1 cm, 03 BSL053, Melles Griot, 2051 Palomar Airport Road, 200, Carlsbad, CA 92009, 800.835.2626, www.mellesgriot.com.

8. Gold Diffraction Grating, 12.7 x 12.7 mm, 1200 lines/mm, NT43247, Edmund Industrial Optics, 101 E. Gloucester Pike, Barrington, NJ 08007, 800.363.1992, www.edmundoptics.com.

9. Kinematic Mirror Mount, model 9809, hex key adjustments, Thorlabs, 435 Route 206 North, Newton, NJ 07860, 973.300.3600, www.thorlabs.com.

10. Piezoelectric Ceramic Buzzer Element, Feedback Type, diameter 20 mm/15mm, 102- 1129-ND (CEB-20FD64), Digikey (address as above).

11. Sanyo 70 mW Laser Diode, 5.6 mm package, single mode, DL7140-201, Thorlabs (address as above).

12. Sharp 30 mW Laser Diode, 9 mm package, LT024MD0, not available.

13. Sharp 120 mW Laser Diode, 5.6 mm package, single mode, GH0781JA2C (425-1809- ND), Digikey, (address as above).

14. BK7 Plane Window, diameter 1”, thickness 0.375”, PW-1037-C, CVI Laser Corporation, 200 Dorado Place SE, Albuquerque, NM 87123, 800.296.9541, www.cvilaser.com.

53

15. Rubidium Vapor Cell, 780/795 nm, 25mm x 76 mm, 2010-RB-03, Newport Corporation, 1791 Deere Ave., Irvine, CA 92606, 800.222.6440, www.newport.com.

16. Photodetector, 1.1 x 1.1 mm, S1336-18BK, Hamamatsu Corporation, 250 Wood Avenue, Middlesex, NJ 08846, 800.524.0504, www.usa.hamamatsu.com.

17. Fused Silica Broadband Mirror, diameter 1”, 750-1100 nm, BB1-E03, Thorlabs, (address as above). note: all mirrors in setup are either gold mirrors or these mirrors.

18. Acousto-Optical Modulator, 1250C-2-804B, and Deflector Driver, D332B-805, Isomet, 5263 Port Royal Road, Springfield, VA 21151, 703.321.8301, www.isomet.com.

19. Unmounted Anamorphic Prism Pair, 06 GPU 001, Melles Griot, (address as above).

20. Optical Isolator, IO-5-NIR-LP, Optics for Research, 315 Bloomfield Ave., PO Box 82, Caldwell, NJ 07006, 973.228.4480, www.opticsforresearch.com.

21. Half-wave plate, QWPM-780-08-2-R10, CVI Laser Corp., (address as above).

22. Polarizing Beamsplitter, 25.4 mm, 03PBS057, Melles Griot, (address as above).

23. 50/50 Beamsplitter, 25.4 mm, 03PSD052, Melles Griot, (address as above).

24. Quarter-wave plate, QWPM-780-08-4-R10, CVI Laser Corp., (address as above).

25. 60V/10A Power Supply, DLM60-10M9G, Sorenson, 9250 Brown Deer Road, San Diego, CA 92121, 800.525.2024, www.sorensen.com.

26. Double-Convex Lens, diameter 50mm, A45-905, Edmund Industrial Optics, (address as above).

27. Photodetector, 10 x 10 mm, S1337-1010BR, Hamamatsu Corporation, (address as above).

28. Electronic Shutter, 76992, Shutter Controller, 76995, Oriel Instruments, 150 Long Beach Boulevard, Stratford, CT 06497, 203.377.8282, www.oriel.com.

29. Optical Rail, 12”, RLA1200, Thorlabs, (address as above).

30. SM2 Series Iris Diaphragm, SM2D25, Thorlabs, (address as above).

31. Optic Cable, diameter 100
m, FC connections on both ends, 1 meter, n.a. 0.29, multimode, F-MLD-C, Newport Corporation, (address as above).

32. 6-Axis Kinematic Mount, K6X, Thorlabs, (address as above).

54

33. SM1 and SM2 Lens Tubes, SM1LXX and SM2LXX. SM1 to SM2 Adapter, SM1A2, Thorlabs, (address as above).

34. Color Glass Filter, diameter 25.4, Cut-on frequency 715 nm, FSR-RG715, Newport Corporation, (address as above).

35. Single Photon Counting Module, SPCM-AQR-14-FC, Pacer Components (Perkin- Elmer/EG&G Canada), Unit 4 Horseshoe Park, Pangbourne, Reading, Berkshire RG8 7JW, UK, 0118.9845280, www.pacer.co.uk.

36. Turbo MCS, T914, Ortec, 801 South Illinois Ave., Oak Ridge, TN 37831, 800.251.9750, www.ortec-online.com.

55

Appendix B:

*note: to reduce noise, the following alterations have been made on each circuit.

1. All power supplies to circuits are buffered to power ground by 4700
F capacitors. 2. Power ground is connected to circuit ground. Power ground is also connected to chassis ground through a 0.1
F capacitor and a 10 Ω resistor set in parallel. 3. All chip power supplies are buffered to ground by 0.1 pF capacitors.

Circuit diagrams:

1. Old Current to Voltage Converter [25]

56

2. New Current to Voltage Converter, courtesy of Univ. of Wisc-Mad

57

+5 V

1 8 Current Supply 0.1 TTL 2 7 Microfarads 2 kohms OP275 1.2 kohms 3 6 300 Ohms 4 5

Switching Power box module Debounced Switch +5 V

TTL 1 kohm To Circuit

1 kohm

+5 V

3. Old Switching Circuit, with the coils in either parallel or series connection, RURU10060 diode, HGTG30N60C3 FET, V271HA32/38Z20 varistor, 74L500 NAND gate chip.

58

4. a. Magnetic Field Switching Circuit (1)

59

b. Magnetic Field Switching Circuit (2)

60

c. Magnetic Field Switching Circuit (3)

61

d. Magnetic Field Switching Circuit (4)

62

5. Previous Locking Circuit [25]

6. The ±6.95V Design

63

7. New Locking Circuit, courtesy of Univ. of Wisc-Mad 64