Stimulated Raman Transitions Between Hyperfine Ground States of Magnetically Trapped Rubidium-87 Atoms

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Stimulated Raman Transitions

Between Hyperfine Ground States of

Magnetically Trapped Rubidium-87

Atoms

Tony Hubert

Thesis presented for partial fulfillment of the degree of

Master of Science (MSc.) in physics

Supervisors: Dr. R.J.C. Spreeuw and Julian Naber

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Instructies

LOGO

BEELDMERK

WOORDMERK

Institute of Physics

University of Amsterdam

The Netherlands

April 2015

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Abstract

In this thesis we present an approach for coherent manipulation of the hyper-fine ground states of magnetically trapped 87Rb atoms through two-photon stimulated Raman transitions. We describe the theory behind stimulated Raman transitions in which light shifts of the hyperfine ground states man-ifest themselves naturally, and calculate the specific laser intensity ratio for which the differential light shift vanishes. We also show how for the spe-cific |F = 1, mF= −1i −→ |F = 2, mF= 1i transition, the two possible Ra-man paths through the D2 line interfere destructively, causing a significant supression of the Raman Rabi frequency. A phase-locked laser system is de-scribed in detail and experimental results are given. Measurements of the differential light shifts are in perfect agreement with theory and we elucidate the importance of minimizing them in order to increase coherence. We also discuss several Raman line broadening effects, including Doppler broadening and power broadening. Preliminary attempts at observing Raman Rabi os-cillations stress the importance of laser intensity stability and confinement of atoms within the laser beam’s interaction region.

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Acknowledgements

I would like to thank everyone involved in aiding me in this project. Partic-ularly Robert Spreeuw for giving me the opportunity and Julian Naber for guiding me through it.

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Chapter 1 Introduction

Coherent control of two-state quantum systems has been subject to many recent studies, both theoretical and experimental. Applications range from atomic clocks and magnetometers to the more popularized advances in quan-tum information processing. The subject discussed in this thesis concerns mostly the latter, in which a two-state quantum system is commonly re-ferred to as a quantum bit or q-bit. The value of the q-bit is determined by the state the two-level system: either |0i, |1i or a superposition of both.

Among the proposed physical systems and physical phenomena that could possibly serve as a platform for quantum bits are superconducting circuits [14], nuclear magnetic resonance [12], trapped ions [11] and trapped neutral atoms. The main advantages of using neutral atoms are their relatively weak interaction with the environment and the variety of available techniques and tools that can be used to manipulate both their external and internal degrees of freedom.

Particularly suitable neutral atoms are alkali metals such as rubidium (Rb) and caesium (Cs). These atoms, having two long-lived hyperfine ground states at an energy splitting of a few GHz and their first excited state being hundreds of THz away, can effectively be treated as two-level systems under the right circumstances.

Recent studies have proposed a quantum information platform based on two-dimensional on-chip lattices of magnetic microtraps for ultracold 87Rb atoms [20][22]. The idea behind a lattice of microtraps is that every individ-ual trapped atomic cloud could act as a single q-bit. Interactions between neighbouring microtraps are then facilitated by strong dipole-dipole inter-actions of highly excited Rydberg states, potentially allowing for Rydberg-blockade based entanglement and ultimately quantum gates [16][19].

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6

0 100 200 300 400 500 600 700 x (µm) y (µm) 0 20 40 60 80 100 120 140 0 50 100 150 10 20 30

FIG. 7. Absorption imaging of the loaded magnetic lattice.

The rubidium atoms are loaded simultaneously into hexagonal

(left side) and square (right side) lattices. A magnified view of

the center region with the crossover between both geometries

is shown in the upper part.

V. CONCLUSION AND OUTLOOK

We have described the design and construction of an

advanced atom chip based on a lattice of permanent

mag-netic microtraps for neutral atoms. Considerations which

we have taken into account include: the electrical

cur-rent load and thermal load of the preliminary magnetic

trap based on current conducting wires, the possibility for

rapid chip exchange, the mechanical stability of the chip,

compactness and space limitations, optical accessibility

and the requirement to image with a resolution in the

micrometer range, and the needs of ultra-high vacuum.

Following the construction and initial testing, the new

atom chip has been placed in vacuum and finally tested

by loading rubidium atoms into the magnetic microtraps

after a sequence of compression and cooling stages.

Having atoms in a 10 µm spacing lattice on an atom

chip will allow us to access the Rydberg dipole-dipole

interaction regime required for quantum information

ex-periments. At the same time we find that we are reaching

some limits in the current protocol for new atom chip

fab-rication. To reach smaller dimensions, we will need new

materials and new methods of fabrication which include

magnetic films with smaller grain size. In addition it will

be necessary to switch from UV optical lithography to

e-beam lithography. This will then enable fabrication of

magnetic lattices on the 100 nm length scale, at which

analog quantum simulation can be implemented. The

technological challenges and solutions of miniaturization

into the nanometer regime are examined in detail

sepa-rately

39

. The ultimate limits on permanent magnet atom

chips will be determined to a large extent by advances in

the fields of magnetic materials and magnetic storage.

Several exciting directions for the development of atom

chips based on permanent magnetic films lie ahead.

Tech-nological advances such as multi-layering can be

ex-ploited to enable single site addressability with large

par-allelism. An atom chip of multi-layers will allow the

in-ricated from a low conductance material with reduced

Johnson noise such as ITO. This electrode layer will

en-able single site addressability by having tiny local electric

fields act on the atom, taking it in and out of resonance

with RF, microwave, or light radiation. The ability to

ad-dress a single atom within a large-scale two-dimensional

atomic register enables the demonstration of algorithms

which have so far been out of reach for both neutral atom

and ion systems, and is part of the foundation of digital

quantum simulation.

VI. ACKNOWLEDGEMENTS

The authors would like to thank Anne de Visser for

help in magnetizing the chip, Mattijs Bakker and the

Technology Centre of the UvA Faculty of Science for

their support. RF gratefully acknowledges support by

the Miller Institute for Basic Research in Science,

Uni-versity of California at Berkeley. This work is part of

the research program of the Foundation for Fundamental

Research on Matter (FOM), which is part of the

Nether-lands Organization for Scientific Research (NWO). The

UvA authors acknowledge support from the EU Marie

Curie ITN COHERENCE Network.

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Figure 1.1: Absorption imaging of a magnetic microtrap lattice loaded with ultracold 87Rb atoms. The atoms can be loaded simultaneously into hexago-nal (left side) and square (right side) lattices. The lattice period is 10µm, al-lowing for inter-microtrap dipole-dipole Rydberg interactions of the trapped atomic clouds. [20]

Coherent manipulation of the hyperfine ground states of87Rb atoms has been successfully demonstrated using a combination of radiofrequency- and microwaves [21]. However, their relatively long wavelengths limit the spa-tial resolution of this approach. It would be impossible to adress a single microtrap.

A different approach might be to coherently manipulate the87Rb ground states through stimulated Raman transitions. Using a combination of two near infrared (∼ 780 nm) lasers, we can drive the same ground state transi-tion, but with a spatial resolution high enough to potentially excite individual microtraps.

Coherent stimulated Raman transitions between hyperfine ground states of alkali atoms have been proposed before [10], and have recently been suc-cessfully demonstrated in optically trapped single cesium atoms [2] and rubid-ium atoms [1]. Experimental reports on similar transitions in atomic clouds remain scarse, although there have been made attempts on room temperature rubidium vapour [21].

In this work we describe our, arguably successful, attempts at the experi-mental realization of coherent stimulated Raman transitions between hyper-fine ground states of magnetically trapped 87Rb atoms.

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We shall begin by briefly introducing the general theory of stimulated Raman transitions as well as describing the more specific case of 87Rb. Next, we explain the setup that was designed to drive Raman experiment. We conclude with our experimental results, conclusion and outlook on future improvement and challenges.

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Chapter 2 Theory

2.1 Rubidium-87 Level Structure

Rubidium-87 (87Rb) is one of two naturally occuring isotopes of Rubidium, the other being 85Rb. Its specific characteristics and energy level structure make it a popular atom for quantum and atom optics experiments. We begin by briefly outlining the 87Rb fine- and hyperfine energy structure in both zero-field and an externally applied magnetic field.

2.1.1 (Zero-Field) Fine and Hyperfine Structure

87_{Rb is an Alkali atom with one electron in the outermost 5S shell. The} transition to the 5P excited state is split into two D-line components; the D1 line (52S_1/2 → 52_P

1/2) and the D2 line (52S1/2 → 52P3/2). Of these two com-ponents the latter has been of much more relevance to experimental atom physics due to the fact that its cycling transition can be exploited for the cooling and trapping of 87Rb. There is also a strong supply of relatively af-fordable and available laser diodes that can operate at the D2line wavelength (780nm).

The 52S_1/2, 52P_1/2 and 52P_3/2 states are split into multiple hyperfine F states, where F = I + J is the total angular momentum of the atom, I being the nuclear angular momentum and J = L + S the sum of the electron’s angular momentum L and spin S. F = |F| takes on values |I + J | ≥ F ≥ |I − J| in steps on one integer (or one quantum of angular momentum ¯h). The entire D1 and D2 fine and hyperfine level structure is shown in figure 2.1.

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5S 5P (nL) 52_S 1/2 52P_1/2 52P3/2 (n2S+1LJ) F=3 F=2 F=1 F=0 F=2 F=1 F=2 F=1 D2780.24 nm D1794.98 nm ∆HFS= 2π × 6.834 673 617 GHz δ2,3= 2π × 267 MHz δ1,2= 2π × 157 MHz δ0,1= 2π × 72 MHz 2π × 812 MHz

Figure 2.1: Zero-field fine and hyperfine energy level structure of 87Rb.

2.1.2 Zeeman-Splitting

Each of the hyperfine (F ) energy levels contains 2F + 1 magnetic sublevels (mF). These sublevels are degenerate in the absence of any externally ap-plied magnetic fields. However, when such fields are apap-plied, the degeneracy is broken. This effect is known as Zeeman-splitting and the Hamiltonian de-scribing the atomic interaction with a magnetic field B along the z-direction is given by HZ = µB ¯ h (gSS + gLL + gII) · B = µB ¯ h (gSSz+ gLLz+ gIIz) Bz, (2.1)

where µB = e¯h/2me is the Bohr magneton and gS, gLand gI are respectively the electron spin, nuclear spin and electron orbital g-factors. They have all been measured experimentally with great accuracy.

In the low-field limit where the hyperfine-splitting always dominates the Zeeman interaction (∆EZ _∆Ehfs_{), the total atom angular momentum} F = J + L + S is always conserved and the Zeeman interaction Hamiltonian can be written in the coupled representation[17]:

HZ = 1 ¯

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in which the hyperfine Land´e factor gF is approximately gF ' 1 +

F (F + 1) + J (J + 1) − I(I + 1)

2F (F + 1) . (2.3)

In this low-field, coupled representation, we may assume that the Zeeman interaction perturbs the hyperfine eigenstates. To first order, the perturba-tion then gives rise to an energy shift on top of the hyperfine-structure and the Zeeman interaction effectively lifts the degeneracy in mF:

∆E_|F,mZ

Fi = µBgFmFBz, (2.4)

in which mF can take all integer values limited to the interval F ≥ mF ≥ −F . We now calculate the Zeeman splitting of the hyperfine (F ) levels of the 52S1/2 ground state of 87Rb. For the 52S1/2 state we have J = 1/2 and I = 3/2, so that gF becomes gF '      −1 2 for F = 1 1 2 for F = 2. (2.5)

The resulting Zeeman shifts are shown in figure 2.2.

0 5 10 15 20 −10 −5 0 5 10 Magnetic Field (G) ∆ E Z (MHz) 52_S 1/2, F = 1 mF= −1 mF= 0 mF= 1 0 5 10 15 20 −20 −10 0 10 20 Magnetic Field (G) ∆ E Z (MHz) 52_S 1/2, F = 2 mF= 2 mF= 1 mF= 0 mF= −1 mF= −2

Figure 2.2: Zeeman splitting of the 52S_1/2 ground states in the low field regime.

2.1.3 Hyperfine Ground States As Quantum Bits

In the low-field regime the Zeeman sub-levels |F = 1, mF= −1i and |F = 2, mF= 1i can be considered to be good candidates for the role of quantum-bits (|0i, |1i) in quantum information experiments for the following reasons:

1. Both of these states are low-field seekers, making them trappable in the local minima of magnetic potential fields.

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2. Beyond first order approximation, the slopes of the Zeeman shifts of |F = 1, mF= −1i and |F = 2, mF= 1i depend on the magnetic field, and to obtain a more exact value for the energy difference between the states one should include a second order Zeeman shift. At B = Bm= 3.228 917 G these two slopes are exactly equal. So when a cloud of atoms is trapped in a magnetic potential with the trap bottom at Bm, the energy shift between the two states remains nearly constant along the atomic cloud and the effects of field fluctuations is minimized. Bm is often termed as the magic field. The hyperfine splitting of the |F = 1, mF= −1i and |F = 2, mF= 1i ground states including a second order Zeeman shift at BM is ∆B_HFSm = 6.834 678 114 GHz.

3. It is possible to coherently drive transitions between these two states, so that we can manipulate quantum bits to be in |0i, |1i or an entangled superposition of the two: e.g. √1

2 (|0i + |1i).

2.2 Stimulated Raman Transitions

In order to realize coherent transitions between the two hyperfine87Rb ground states |0i = |F = 1, mF= −1i and |1i = |F = 2, mF= 1i, for which |∆mF| = 2, one needs to drive a two-photon transition. One way to do this is by using a combination of microwave (MW) and radio-frequency (RF) fields in a lad-der transitions, as shown in figure 2.3. The transition between |0i and |1i is driven through an intermediate state (|F = 2, mF= 0i) to which the MW field is coupled with a fixed detuning to prevent unwanted resonant scatter-ing. The RF field simultaneously couples this intermediate state to the final state |1i. Although a RF+MW transition is relatively easy to establish, the long wavelengths involved result in a very poor spatial resolution. Also, the maximum practically atainable two-photon Rabi frequency (or atomic cou-pling to the field) is much lower than could potentially be had for a minimally divergent, high intensity laser beam.

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F=1 F=2 ∆HFS= 2π × 6.8 GHz -2 -1 0 1 2 mF ωMW, ΩMW ωRF, ΩRF

Figure 2.3: Two-photon excitation scheme for the |F = 1, mF= −1i → |F = 2, mF= 1i transition in the 52S1/2 87Rb ground state using a combi-nation of microwave (MW) and radiofrequency (RF) fields.

A much higher spatial resolution can be obtained through stimulated Raman transitions by employing lasers in a Λ-configuration (figure 2.4). In such a configuration, two atom states are coherently coupled to each other through an energetically higher lying intermediate state. In the following sections the dynamics of stimulated Raman transitions will be explained, starting with a general theory of the matter and followed up by a more detailed description for the specific case of 87Rb.

2.2.1 General Theory of Stimulated Raman Transitions

We consider a simple three level Raman system as illustrated in figure 2.4. The aim of such a stimulated Raman system is to coherently drive transi-tions between two states (|0i and |1i) through an energetically higher lying intermediate state (|2i) by employing two coupled lasers.

One laser, which we call the pump laser, couples the state |0i to the intermediate state |2i with frequency ωP. Another laser, called the Stokes laser, simultaneously couples state |1i to the same intermediate state with ωS.

To avoid driving population into the intermediate state we detune the two lasers from resonance by ∆ (Note: in figure 2.4, ∆ is taken to be negative). The detuning from the two-photon resonance frequency is denoted by δ, which we take to be much smaller than ∆.

The system’s dynamics can be obtained by solving the time-dependent (TD) Schr¨odinger equation for the system’s Hamiltonian. For simplicity, we take the energy of the ground state |0i to be zero. In the rotating wave

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€

ω

P, ΩP € ωS, ΩS € 0 € 1 € 2 € Γ € δ € Δ € ΔE Pump Laser Stokes Laser

Figure 2.4: Basic scheme of a stimulated Raman system. States |0i and |1i are coupled to each other through an intermediate level |2i. This two-photon transition is being driven by two separate lasers, each coupling a separate state to the intermediate level. Both lasers are detuned from |2i by ∆ to minimize unwanted one-photon transitions. The two-photon detuning is denoted δ.

approximation (RWA) [18], the Hamiltonian for this system in the ordered basis {|0i , |1i , |2i} can be written as

ˆ H = h¯ 2   0 0 ΩP 0 −2δ ΩS ΩP ΩS −2∆  , (2.6)

In which the off-diagonal terms represent the coupling between states due to the pump and Stokes lasers with respective Rabi-frequencies ΩP and ΩS. The Rabi frequency of a laser coupled to a transition is determined by the laser-field amplitude at the atom E(r0) and the expectation value of the dipole-transition operator ˆD:

ΩL_g,e = −1 ¯

hhe| ˆD · EL(r0) |gi , (2.7) where |gi and |ei are the ground and excited state respectively and the symbol L is used to differentiate between lasers (EP for the pump and ES for the Stokes laser)[18].

The time dependent state Ψ(t) of the three-level atom can be expressed as a superposition of its unperturbed eigenstates (|0i, |1i and |2i) weighed

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by their respective time dependent probability amplitudes Ci(t):

Ψ(t) = C0(t) |0i + C1(t) |1i + C2(t) |2i . (2.8) Solving the TD Schr¨odinger equation

ˆ

HΨ(t) = i¯hd

dtΨ(t) (2.9)

gives us a set of three coupled equations for the probability amplitudes Ci(t):

i ˙C0(t) = 1 2ΩPC2(t), i ˙C1(t) = 1 2ΩSC2(t) − δC1(t), i ˙C2(t) = 1 2(ΩPC0(t) + ΩSC1(t)) − ∆C2(t). (2.10)

Assuming that the detuning ∆ is much greater than the Rabi-frequencies, we can adiabatically eliminate any population in state |2i. That is, the population in |2i will undergo much faster oscillations than |0i and |1i so that we can assume ˙C2(t) to average out to zero over a large number of cycles. By doing this, we can reduce our system to an effective two-level scheme: i ˙C₀(t) = ΩP 4∆ (ΩPC0(t) + ΩSC1(t)) , i ˙C1(t) = ΩS 4∆ (ΩPC0(t) + ΩSC1(t)) − δC1(t), (2.11)

for which the effective Hamiltonian is ˆ Heff = ¯ h 4 Ω2_P/∆ ΩPΩS/∆ ΩPΩS/∆ Ω2S/∆ − 4δ . (2.12)

Here, the diagonal terms give rise to an energy shift Λ of |0i and |1i respec-tively, shifting the energy levels up or down depending on the sign of the laser detuning: Λ₀ = ¯hΩ 2 P 4∆, Λ1 = ¯h Ω2_S 4∆. (2.13)

Because these energy shifts are caused by the coupling of light, they are known as light shifts (also referred to as AC Stark shifts). The effective differential light shift δΛ of the transition frequency between the two states is then given by

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and the effective two-photon Raman detuning becomes

δeff = δ − δΛ (2.15)

Note: Remember that we have assumed ∆ ΩP, ΩS. For a smaller detun-ing the approximation fails and this expression for the light-shifts does not apply anymore.

Again, the off-diagonal terms represent the coupling between the two states, for which we now define the two-photon Rabi frequency as

ΩR = ΩPΩS

2∆ . (2.16)

Solving the Schr¨odinger equation for Heff under the assumption that the initial population is in state |0i we arrive at the time-dependent population densities ρi(t): ρ0(t) = |C0(t)|2 = 1 + Ω2_R 2Ω2₀ [cos Ω0t − 1] , ρ1(t) = |C1(t)|2 = Ω2_R 2Ω2₀ [1 − cos Ω0t] , (2.17)

where we defined the generalized Rabi frequency as

Ω0 = q

Ω2_R+ δ_eff2 . (2.18) Equation 2.17 describes an oscillating probability of finding atoms in states |0i or |1i, known as Rabi oscillations. As a complete population in-version only occurs for δ_eff = 0, it is of foremost importance that the lasers in a Raman laser system are capable of operating at exact two-photon Ra-man resonance with minimal fluctuations in frequency. This can be readily achieved by employing a phase-locked laser setup, which will be thoroughly described later in this thesis. Also, since δΛ depends on the ratio of laser in-tensities, precise control and stability of laser power is essential. An unstable laser intensity ratio will result in fluctuations of the two-photon resonance frequency, effectively causing a decoherence of the Raman transitions.

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0 2 4 6 8 10 12 14 16 18 0 0.2 0.4 0.6 0.8 1 Time (seconds) ρ1 (t ) δ_eff= 0 δ_eff= 1ΩR δ_eff= 2ΩR

Figure 2.5: Rabi oscillations of the population density ρ1(t) for ΩR = 1 rad s−1 and different δeff. Complete population inversion only occurs at δeff = 0. For larger detuning the maximum transition probability decreases, while the generalized Rabi frequency Ω0 increases.

Cross Coupling

One thing we have not explicitly considered is cross coupling of the lasers (see figure 2.6). Let us assume that the lasers are on Raman resonance (δ = 0). Although the Stokes laser is meant to couple state |1i to |2i with detuning ∆, it also unintentionally couples |0i to |2i with detuning ∆ − ∆E. The same argument can be made for the pump beam, which inadvertently couples |1i to |2i with detuning ∆ + ∆E.

In practice, as long as ∆E is larger than ∆, the cross coupling does not lead to single- or two-photon transitions, and therefore should not affect the two-photon Rabi frequency. However, it does add considerably to the light shifts. Taking this cross coupling into account, the more complete expression for the light shift of a ground state becomes:

Λg = ΛPg + Λ S g, (2.19) so that: Λ₀ = ¯h 4 " (ΩP_0,2)2 ∆ + (ΩS_0,2)2 ∆ − ∆E # , (2.20) Λ₁ = ¯h 4 " (ΩS_1,2)2 ∆ + (ΩP_1,2)2 ∆ + ∆E # . (2.21)

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€ ωP, Ω0,2 P € ωS, Ω0,2 S € 0 € 1 € 2 € Γ € Δ € ΔE Pump Laser Stokes Laser € Δ + ΔE (a) € ωP, Ω0,2 P € ωS, Ω0,2 S € 0 € 1 € 2 € Γ € Δ € ΔE Pump Laser Stokes Laser € Δ − ΔE (b)

Figure 2.6: Cross coupling of the pump and Stokes lasers in a stimulated Raman system. Although not on purpose, each individual laser effectively couples both ground states to the intermediate excited state with a different detuning. While the Stokes laser is intended to couple the ground state |1i to |2i with detuning ∆, it also effectively couples |0i to |2i with detuning ∆ + ∆E (2.6a). The opposite can be said of the pump laser (2.6b).

2.2.2 Stimulated Raman Transitions in

87

Rb

The previous description of a stimulated Raman system is valid for a three-level atom. However, as we have shown before,87Rb is not a three-level atom. Both the ground state and D2 excited state are split into multiple hyperfine levels F, which in turn have different sublevels mF. To obtain the complete expressions for the light shifts Λg and the two-photon Rabi-frequency ΩR we must take into account that every sublevel couples to the light fields with a different Rabi-frequency and detuning.

A complete picture of all relevant sublevels and their respective detuning relative to both lasers is given in figure 2.7. We will only concern ourselves with the D2 excitation line as we are far enough away from the D1 line to completely neglect its contributions.

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52_S 1/2 52_P 3/2 D2 line 780 nm ∆0 ∆₁ ∆2 ∆₃ Pump Laser Stokes Laser F=1 F=2 ∆HFS 2π × 6.834 GHz F=0 F=1 F=2 F=3 -3 -2 -1 0 1 2 3 m_F δ2,3 δ1,2 δ0,1

Figure 2.7: Complete level structure of the D2 line in 87Rb, including the detunings of all intermediate hyperfine (F) states with respect to the two Raman (pump and Stokes) lasers. Hyperfine energy splittings of the excited states are denoted δ0,1, δ1,2and δ2,3. The |F = 1, mF= −1i and |F = 2, mF= 1i ground states (colored red) are separated by the hyperfine splitting ∆HFS= 2π × 6.834GHz.

The Rabi frequency Ω (given by equation 2.17) with which an atomic level couples to the light field is characterized by the matrix elements of the dipole operator ˆD: hFg, mg| ˆD |Fe, mei. In order to calculate these matrix elements, we use the Wigner-Eckart theorem to factor out the mF dependency and write the elements as a product of a reduced matrix element and a Clebsch-Gordan coefficient [17]:

hFg, mg| ˆD |Fe, mei = DFg,FehFe, me, 1, q | Fgmgi . (2.22)

The reduced matrix element DFg,Fe is independent of magnetic sublevels and

can be expressed in units of D2,3, which is the matrix element of the closed transition of the 87Rb D2 line (F = 2 ←→ F = 3):

DFg,Fe = D2,3dFg,Fe. (2.23) In which D2,3 = s ¯ hΓ30λ 3 D2 8π2 = 3.58 × 10 −29 _{C m,} _(2.24)

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coupling strength dFg,Fe is given by dFg,Fe = q (2Fg+ 1)(2Je+ 1)(−1)Fg+I+Je+1 Fg Fe 1 Je Jg I 6j , (2.25)

where {}6j is the 6j-Racah symbol.

The second factor in equation 2.22 is the Clebsch-Gordan coefficient and describes the coupling between different sublevels through the absorbtion or emission of a photon with spherical polarization q. For which q is labeled 0, 1 or -1 for π, σ+ or σ− respectively. In the cartesian basis {ˆx, ˆy, ˆz} this is defined as: q =      (ˆx − iˆy)/√2 , q = −1 ˆ z , q = 0 −(ˆx + iˆy)/√2 , q = 1. (2.26)

For tables of the 87Rb D2 hyperfine dipole matrix elements I refer to [17]. Using the formalism above we can now write the Rabi frequency for the coupling between specific magnetic sublevels as

ΩFg,mg,Fe,me = Γ

r I 2Isat

dFg,FehFe, me, 1, q | Fgmgi, (2.27)

in which I is the laser intensity and the saturation intensity Isat is defined as

Isat = ¯ h2Γ2

D2,3

= 1.669 33 (35) mW/cm2. (2.28)

Multi-level Light Shifts

As both Raman lasers couple to multiple sublevels with a different detuning, they both produce multiple light shifts. To obtain an expression for the light shift of a ground state, we have to sum over all intermediate sublevels. Using equation 2.27 for the sub-level specific Rabi frequency we obtain the following expression for the light shift of a ground state caused by a laser L with intensity IL: ΛL_F_g_,m_g = Γ 2 8 IL Isat X Fe,me,q d2_F_g_,F_ehFe, me, 1, q | Fgmgi2 ∆L Fg,Fe , (2.29)

Let us consider a Raman laser field with σ++ σ− polarization parallel to the quantization (B-field) axis (as is the case in our experiment), the contribu-tion to the light shift due to π-transicontribu-tions are zero. In this case, the only

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sublevels that do contribute to the light shifts of the |F = 1, mF= −1i and |F = 2, mF= 1i ground states are given in figure 2.8.

52_S 1/2 52P3/2 F=1 F=2 F=0 F=1 F=2 F=3 -3 -2 -1 0 1 2 3 mF σ− σ+ σ− σ+

Figure 2.8: All contributions from the D₂line for σ++σ−polarized light that add to the light shifts of the |1, −1i and |2, 1i ground states (coloured red). The blue and green arrows indicate the coupling of intermediate sublevels to |1, −1i and |2, 1i respectively.

Using equation 2.29 we can then calculate the exact light shifts of both ground states, where we also take into account the contribution of the pre-viously described cross coupling:

Λ_1,−1 = ΛP_1,−1+ ΛS_1,−1 = Γ 2 192 " IP Isat 7 ∆2 + 5 ∆1 + 4 ∆0 + IS Isat 7 ∆2− ∆HFS + 5 ∆1− ∆HFS + 4 ∆0− ∆HFS # , (2.30) and Λ2,1 = ΛP2,1+ ΛS2,1 = Γ 2 960 " IS Isat 52 ∆₃ + 25 ∆₂ + 3 ∆₁ + IP Isat 52 ∆3+ ∆HFS + 25 ∆2+ ∆HFS + 3 ∆1+ ∆HFS # . (2.31)

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Here, IP and IS are the pump and Stokes laser intensities respectively, ∆_HFS = 2π × 6.834 GHz is the ground state hyperfine splitting and ∆_0,1,2,3 denotes the laser detuning with respect to hyperfine levels F = 0, 1, 2, 3 re-spectively as shown in figure 2.7.

Figure 2.9 shows the differential light shift, δ_Λ = Λ_2,1 − Λ_1,−1, of the |F = 1, mF= −1i and |F = 2, mF= 1i hyperfine ground states as a function of the intensity ratio IS/IP of the probe (IP) and Stokes (IS) lasers. It shows that for a fixed detuning ∆, there is a ratio R₀ for which the differential light shifts are exactly zero, independent of the total laser intensity.

0 0.2 0.4 0.6 0.8 1 −50 0 50 IP/IS δΛ (kHz) IS = 0.5 mW/mm2 IS = 1.0 mW/mm2 IS = 1.5 mW/mm2 IS = 2.0 mW/mm2 ∆3 = −2π × 2713 MHz IS IP = R0

Figure 2.9: Differential light shift δΛ of the |F = 1, mF= −1i and |F = 2, mF= 1i hyperfine 87Rb ground states versus laser intensities. IS and IP are the Stokes and pump laser intensities respectively. This calculation is specific for a detuning ∆3 = −2π × 2713 MHz. Notice that there is a specific laser intensity ratio, R0, for which δΛ = 0. R0 is independent of total laser intensity and depends on the detuning ∆.

Because a differential light shift δΛ leads to an effective Raman detuning of the lasers (equation 2.15), it is necessary to correct for this in an experi-ment. One straightforward way of doing this would be to fine-tune the laser frequencies to adjust for the induced detuning, which can be done relatively easy and with great accuracy in a phase locked laser setup. However, as the light shifts depend on the laser intensities, which are spatially distributed in a Gaussian manner, this approach fails to fulfill the resonance condition for

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−3.0 −2.5 −2.0 −1.5 −1.0 −0.5 1.8 2 2.2 2.4 2.6 2.8 ∆3/2π (GHz) R0 (I S /I P ) ∆3< 0 (a) 0.5 1.0 1.5 2.0 2.5 3.0 0.3 0.35 0.4 0.45 0.5 0.55 ∆3/2π (GHz) R0 (I S /I P ) ∆3> 0 (b)

Figure 2.10: Zero differential light-shift intensity ratio R₀ as a function of∆₃ for both (a) red and (b) blue detuning.

all atoms struck by the laser beams. Not only does this decrease the max-imum population transfer, it also gives rise to unwanted light shift induced effects such as spectral line broadening and damping of the Rabi oscillations. Some of these effects will be discussed in the Results chapter.

Another solution is to fix both laser powers to the intensity ratio R0 for which δΛ = 0. This way, all atoms struck by the two laser beams experi-ence no differential light shift, provided that their Gaussian intensity profiles overlap perfectly (this can be achieved by coupling both lasers in and out of the same fiber). Calculating R0 amounts to equating the differential light shift to zero and solving for the laser intensities:

δΛ = Λ2,1− Λ1,−1 = 0 −→ IS IP

= R0. (2.32)

An analytical expression for R0 can be written in terms of the laser detunings ∆ with respect to the 52P3/2 excited hyperfine states (see figure 2.7):

R0 = −_∆ 52 3+∆HFS + 35 ∆2 − 25 ∆2+∆HFS + 25 ∆1 − 3 ∆1+∆HFS + 20 ∆0 52 ∆3 + 25 ∆2 − 35 ∆2−∆HFS + 3 ∆1 − 25 ∆1−∆HFS − 20 ∆0−∆HFS (2.33)

Figure 2.10 shows R0 as a function of ∆3.

Damping of Two-Photon Rabi-Frequency

The presence of multiple hyperfine intermediate states in 87Rb also results in multiple contributions two the two-photon Rabi frequency. There are two possible paths that contribute to the |1, −1i → |2, 1i Raman transition:

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either through the |1, 0i or |2, 0i hyperfine 52P3/2 sublevel. Both paths are shown in figure 2.11. Including both contributions, the complete expression for the two-photon Raman Rabi frequency then becomes

ΩR= Ω|1,−1i,|1,0iΩ|2,1i,|1,0i 2∆1 +Ω|1,−1i,|2,0iΩ|2,1i,|2,0i 2∆2 . (2.34) 52S1/2 52P3/2 σ+ σ− F=1 F=2 F=0 F=1 F=2 F=3 -3 -2 -1 0 1 2 3 mF

Figure 2.11: There are two possible stimulated Raman paths for the |1, −2i −→ |2, 1i transition. Either through the |1, 0i or |2, 0i hyperfine 52P3/2 sublevel.

Unfortunately, the transition strengths of these paths are exactly equal in magnitude, but opposite in sign, causing both Raman paths to destructively interfere:

h2, 1| ˆD |1, 0i h1, 0| ˆD |1, −1i = − h2, 1| ˆD |2, 0i h2, 0| ˆD |1, −1i , (2.35) so that

Ω|1,−1i,|1,0iΩ|2,1i,|1,0i+ Ω|1,−1i,|2,0iΩ|2,1i,|2,0i= 0. (2.36) Using this, we can then rewrite the two-photon Raman Rabi frequency as

ΩR = 1 2Ω|1,−1i,|1,0iΩ|2,1i,|1,0i 1 ∆1 − 1 ∆2 , (2.37)

or more explicitly as a product of a ”one-path” Raman Rabi frequency and a damping factor D(∆₁):

ΩR =

Ω|1,−1i,|1,0iΩ|2,1i,|1,0i

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where D(∆1) = δ_1,2B δB 1,2− ∆1 , (2.39)

and δ_1,2B is the hyperfine splitting of between the F = 1 and F = 2 52P3/2 hyperfine states including the zeeman shifts for an applied magnetic field flux B (see figure 2.7). For a detuning of ∆3 = −2π × 2713 MHz, and δ_1,2B=0 = 2π × 156.9 MHz we have D ≈ 0.05, causing a severe suppression of the Raman Rabi frequency.

Spontaneous Raman Scattering

Despite the large detuning from the intermediate 52S3/2 hyperfine states, the Raman lasers may still off-resonantly scatter from them.

Every time an atom spontaneously scatters from an intermediate state into one of the ground state sublevels, it is lost in the coherent Rabi oscillation of the atom cloud. This atom may then again couple with the laser fields and be driven into excitation, but it’s individual Rabi oscillation will now be out of phase with the rest of the atoms, effectively dampening the entire cloud’s average Rabi oscillations. We call this a decoherence effect.

To evaluate the decoherence effect of this spontaneous Raman scattering we begin by writing the scattering rate Rsc_F_i_,m_i_,F_f_,m_f for scattering events from an initial ground state |Fi, mii to a final ground state |Ff, mfi induced by a laser field coupling to multiple intermediate excited sublevels |Fe, mei. In the low saturation limit, where ∆ Ω, we can write Rsc

Fi,mi,Ff,mf as the absolute

square of the sum of all possible scattering paths:

Rsc_F i,mi,Ff,mf ' Γ 8 X Fe,me ΩFi,mi,Fe,me ∆Fg,Fe Asc Ff,mf,Fe,me 2 , (2.40)

for which we have defined the scattering amplitude to be

Asc_F_f_,m_f_,F_e_,m_e = dFf,FehFf, mf, 1, q | Femei . (2.41)

Calculating the time dependent population densities of the ground state sublevels then comes down to solving the master equation in Lindblad form:

d

dtρ = − i ¯

h[Heff, ρ] + L(ρ), (2.42) where ρ = |Ψi hΨ| is the system’s density matrix, H_eff is the multi-sublevel equivalent of equation 2.12 including all ground state F = 1 and F = 2 Zee-man shifts normalized with respect to |1, −1i and |2, 1i respectively (the nor-malization ensures that we remain on Raman resonance for |1, −1i −→ |2, 1i

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independent of the Zeeman shift).L(ρ) is called the Lindblad superoperator, which is obtained as follows,

L(ρ) = 1 2 X j= Fimi Ffmf 2cjρc † j − c † jcjρ − ρc † jcj , (2.43) and cFimi Ffmf = (Rsc_F_i_m_i Ffmf )1/2|Ff, mfi hFi, mi| . (2.44)

A numerical solution of equation 2.42 at Raman resonance (δeff = 0) and zero differential light shift (δΛ = 0 and I_I_PS = R0) is plotted for ρ|1,−1i and ρ_|2,1iin figure 2.12. It predicts a strong scattering induced decoherence effect, that dampens the two-photon Rabi amplitude by more than half after two cycles. 0 1,000 2,000 3,000 4,000 5,000 6,000 0 0.2 0.4 0.6 0.8 1 Time (µs) ρ|F ,m F i ρ_|1,−1i ρ|2,1i ρ|2,2i B = 3.23G I = 1.0 mW/mm2 Ω0 = 544 Hz ∆₃ = −2π × 2713 MHz

Figure 2.12: Resonant Raman Rabi oscillations of the |1, −1i, |2, 1i and |2, 2i magnetically trappable ground states. We assumed Raman resonance and zero differential light shift of the |1, −1i and |2, 1i ground states (δeff = 0). The overall detuning with respect to F = 3 is ∆3 = −2π × 2713 MHz and we included first order Zeeman shifts of the 52S_1/2 hyperfine mF sublevels at a magnetic field of B = 3.23 Gauss. For a combined laser intensity of IS+ IP = 1.0 mW, we find a Rabi frequency of 544 Hz. The oscillations are damped by spontaneous Raman scattering, effectively causing a decoherence of the system.

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Generally, the effect of scattering can be greatly reduced by simply choos-ing a large detunchoos-ing ∆, as the scatterchoos-ing rate Rsc is suppressed much faster (Rsc _{∝ 1/∆}2_{) than the two-photon Rabi frequency (Ω}

R ∝ 1/∆). However, this does not apply to our specific case, in which we also have to take into ac-count the previously described damping of ΩR (equation 2.37). What makes this damping particularly problematic is that for increasingly large detuning it causes ΩR to also scale as ∝ 1/∆2. Effectively, this means that Rsc and ΩR are similarly suppressed, making it less advantageous to use very large ∆.

It should be noted that in these calculations, we haven’t accounted for other decoherence effects such as trap-losses, field fluctuations and collisional effects.

Intensity Ratio Fluctuations

Fluctuations in the power of the individual Raman lasers may cause fluctu-ating differential light shifts, therefore increasing the generalized Rabi fre-quency (equation 2.18). To elucidate the effect of unstable individual laser powers we have plotted the Rabi dynamics (solutions to equation 2.42) for fluctuating intensity ratios near R0 in figure 4.3.

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0 200 400 600 800 1,000 0 0.2 0.4 0.6 0.8 Pulse Length (µs) P opulation Densit y ρ|2,1i ρ|2,2i (a) IS/IP = R0 0 200 400 600 800 1,000 0 0.2 0.4 0.6 0.8 Pulse Length (µs) P opulation Densit y ρ|2,1i ρ|2,2i (b) IS/IP = 1.01 · R0 0 200 400 600 800 1,000 0 0.2 0.4 0.6 0.8 Pulse Length (µs) P opulation Densit y ρ|2,1i ρ|2,2i (c) IS/IP = 1.02 · R0 0 200 400 600 800 1,000 0 0.2 0.4 0.6 0.8 Pulse Length (µs) P opulation Densit y ρ|2,1i ρ|2,2i (d) IS/IP = 1.03 · R0

Figure 2.13: Simulations of F = 2 Rabi oscillations including spontaneous Raman scattering for different Raman laser intensity ratios. The calculations show that for minimal deviations from the optimal intensity ratio R0, the light shift induced Raman detuning significantly affects the generalized Rabi frequency and maximum population transfer. The calculation is based on an arbitrary Stokes laser intensity of IS = 3.25 mW/mm2 and detuning ∆3 = 2π × 2.7 GHz.

Clearly, the effect of power fluctuations is quite significant. A 2% increase of power in the Stokes laser almost doubles the generalized Rabi frequency and more than halves the maximum population transfer.

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Chapter 3 Experimental Setup

We now turn to describe the experimental setup employed in our pursuit to realize coherent stimulated Raman transitions. We shall begin by discussing the Raman laser system in detail, followed by a brief explaination of the cooling, trapping and optical pumping mechanisms used to prepare the87Rb atoms for the experiment. We conclude this technical section by describing our method of imaging and (52S1/2, F = 2) population detection.

3.1 Raman Laser System

The Raman laser system can be divided into four parts (figure 3.1), all serving their particular purpose:

1. Raman lasers. The two lasers responsible for a two-photon stimu-lated Raman transition are arranged in a master-slave configuration. That is, the slave laser (SL) is set up to follow the master laser (ML) with a frequency difference exactly equal to the ground state hyperfine splitting ∆_HFS = 6.834 GHz (see figure 2.7).

2. Phase lock. In order to stabilize the frequency difference between the master and slave laser, we employ a phase locking mechanism. This mechanism locks the slave laser’s frequency and phase relative to the master laser through an electronical feedback loop. The frequency difference can then be fine-tuned with high precision on a computer. 3. Polarization spectroscopy and master laser lock. The absolute

master laser frequency is locked on a 87Rb transition with a fixed de-tuning ∆ using Doppler-free polarization spectroscopy.

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4. Pulse mechanism. Before both lasers are coupled into an optical fibre leading to the experiment, they are coupled into an acousto-optic modulator (AOM), allowing us to accurately pulse the laser light with variable pulse lengths.

Pulse Mechanism & Experiment Phase Lock Mechanism Polarization Spectroscopy & ML Lock Slave Laser (SL) Master Laser (ML) € λ 2 € λ 2 € λ 2 € λ 2 € λ 2 PBS PBS PBS PBS 50:50

Figure 3.1: Overview of the Raman laser system. Both lasers are split up into multiple beams leading to the various parts of the system. A combination of polarizing beam splitters (PBS) and have-wave plater (λ/2) allow us to carefully tune the amount of light into each section.

3.1.1 Raman Lasers

At the heart of our stimulated Raman laser system lie two commerically avail-able DL 100 narrow linewidth tunavail-able external cavity diode lasers (ECDL’s) from manufacturer TOPTICA fitted with Axcel Photonics laser diodes. These lasers can be set up to operate at a center wavelength of 780 nm with a typ-ical course tuning range of about 3 nm and mode-hop free tuning range of 20 GHz. The typical maximum output power for our specific laser diodes is about 150 mW and the typical linewidth lies between 100 KHz and 1 MHz. The DL 100 provides two means for fast tuning of the lasing frequency: 1. by directly controlling the laser diode driving current.

2. by controlling the voltage over the piezo actuated grating angle. It is important to note that they differ dramatically in bandwidth. Slower electronic feedback up to a few tens of kHz is often fed to the grating piezo,

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while the current may be modulated at frequencies in the order of a few tens of MHz. In practice, this means that for a simple frequency lock the piezo grating often suffices. For much faster locking mechanisms (such as phase locks) it is often necessary to modulate the current also.

3.1.2 Phase Lock

In order to stabilize the frequency difference of the master and slave lasers and maximize their phase coherence (that is, to minimize their relative phase noise), we employ a phase locking mechanism. The electronic system is de-signed to lock the beat note signal produced by both lasers to a reference RF oscillator. The error signal produced by comparing the beat note with the reference signal can be fed back into proportional-integral-derivative (PID) controllers which then actively adjust the slave laser frequency, thereby sta-bilizing and narrowing the beat note and thus fixing the lasers’ relative fre-quency and phase. A simplified schematic of this phase locked loop (PLL) is depicted in figure 3.2. Spectrum Analyzer Amplifier Splitter Slave Laser (SL) SL Piezo Piezo Amplifier PID SL FET Mixer PBS Rotated 45° Fast PD Amplifier MW 6.834 GHz + 25 MHz ML + SL Beam Error Signal Beat Signal ~ 6.8 GHz Beat Signal ~ 25 MHz Lens PC Frequency Detector mFalc Phase Detector + PID RF 25 MHz

Figure 3.2: A simplified schematic of the phase locking electronics. See the text for a detailed explanation.

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The beat note signal of the ML and SL is obtained through a fast photode-tector (FPD). We use a New Focus amplified free-space photoreceiver that converts optical signals up to 12 GHz. Before the two overlapped beams are collected on the FPD, they pass through a 45◦ rotated polarizing beam splitter (PBS) to ensure equal linear polarizations. Since the FPD mea-sures the intensity (∝ |E|2) of the combined laser field, its signal equals the squared sum of both individual laser amplitudes. It therefore comprises of two components: an extremely fast component equal to the sum of both laser frequencies and a slower component equal to the difference of the laser frequencies. The former is in the order of hundreds of THz and is left com-pletely undetectable by the FPD. The latter is termed the beat note and should be approximately 6.8 GHz (if we tune our lasers properly).

The detected beat note signal is then amplified and mixed down to a secondary beat signal at a frequency more suited to our feedback electronics. More specifically, the primary beat signal is multiplicatively mixed with a microwave (MW) oscillator set exactly equal to the Zeeman shifted 87Rb hyperfine splitting of the |F = 1, mF= −1i and |F = 2, mF= 1i ground states ∆Bm

HFS = 6.834 678 114 GHz plus an additional 25 MHz. This way, if the secondary beat signal is to be locked exactly at 25 MHz, we know that the primary beat signal must be at exactly 6.834 678 114 GHz. Fine-tuning the exact primary beat frequency can be done by simply tuning the MW oscillator through the connected laboratory control computer (PC). The MW oscillator is referenced to an external 10 MHz rubidium atomic clock.

We split part of the secondary beat signal to a spectrum analyzer, so that we can monitor it in real time.

The remaining signal is amplified and fed into a phase- and frequency detector, which both compare the secondary beat note to a referenced 25 MHz RF oscillator and produce an error signal accordingly. For frequency detection we use the frequency discrimination function of an Analog Devices AD9901 digital phase/frequency discriminator, which outputs a ”slow” error signal in the order of a few kHz. This frequency error signal is then sent to a PID controller which controls the slave laser’s piezo actuated grating. For phase detection we employ a commercially available Toptica mFALC module, which is a combined analog phase detector and PID controller. The mFalc produces a ”fast” phase error signal in the order of a few MHz and directly controls the slave laser diode current through the SL’s build in field effect transistor (FET).

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20 22 24 26 28 30 −20 −40 −60 −80 Beat Frequency (MHz) P o w er (dBm) Frequency Locked Phase Locked

Figure 3.3: Beat note signal obtained from the spectrum analyzer, under the effect of both the frequency lock and phase lock.

The procedure to obtaining a successful phase lock is as follows:

1. We first manually tune the SL to be approximately 6.8 GHz away from the ML, and make sure the SL is free from any occuring mode hops. 2. Next, we activate the SL frequency feedback loop by engaging the

grat-ing piezo controllgrat-ing PID. This locks the primary beat note to be cen-tered exactly at 6.834 GHz (and thus the secondary beat to exactly 25 MHz). This frequency lock serves only as a support to the eventual phase lock. Its role is to stabilize the beat note for the phase detector to use and retrieve the beat in case the lock is lost for some reason. 3. Finally, the phase lock loop is activated by employing the mFALC phase

detector/PID. The fast current feedback of the mFALC significantly reduces the ML and SL’s relative phase noise and therefore greatly reduces the beat note’s linewidth.

Figure 3.3 shows the secondary beat note signal, obtained from the spec-trum analyzer, under the effect of both the frequency lock and phase lock. We observe a great reduction of the beat note’s linewidth: from approximately 3 MHz for the frequency lock alone, to less than 100 Hz (the spectrum an-alyzer’s resolution limit) with the phase lock engaged. We also observe a significant increase of beat note signal of up to 40 dBm relative to the back-ground.

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3.1.3 Polarization Spectroscopy and Master Laser Lock

We use a polarization spectroscopy based electronical feedback mechanism to lock the master laser’s frequency to an atomic Rubidium D2transition. Polar-ization spectroscopy is used to create a frequency error signal that can be fed back to the ML’s piezo actuated grating through a PID controlled feedback loop. The advantage of this technique over alternative locking mechanisms is that it does not rely on the modulation of the laser frequency, thus preventing the addition of any unwanted noise on the laser light.

Polarization Spectroscopy

Polarization spectroscopy is a high resolution spectroscopic method that can be used to probe polarization dependent optical properties of a measured atomic medium. We use a combination of two counter-propagating laser beams to pump and probe a atomic vapour cell containing both 87Rb and 85_Rb.

The pump beam has a relatively high intensity and is circularly polarized (either σ+ or σ−). It serves to optically pump the atomic ensemble into a collective stretched state. That is, the circularly polarized light will only induce ∆mF = 1 or ∆mF = −1 transitions (for σ+ or σ− respectively) and therefore pump the atomic population into the highest, respectively lowest, mF hyperfine magnetic sublevels. The result is a non-uniform population of different magnetic sublevels.

The linearly polarized, low intensity probe beam will observe this imposed anisotropy of the atomic medium as a birefringence. As a linearly polarized beam can be decomposed into a linear combination of σ+ and σ−, the probe beam will then experience a decrease in absorbtion for one orthogonal po-larization component and an increase for the other. This effect is strongest for closed transitions (such as F = 2 ←→ F = 3 in 87Rb), as their excited states never decay out of the pumping cycle (F=3 can only decay back into F=2). Retrieving both individual polarization components from the probe beam can be done by simply filtering them through a polarizing beam splitter (PBS).

The counter-propagating alingment of the beams allows Doppler-free ab-sorbtion measurements, as used in standard saturated abab-sorbtion spectroscopy. For a more elaborate description of polarization spectroscopy and saturated absorbtion spectroscopy I refer to [3] and [15] respectively.

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Master Laser (ML) ML Piezo / Grating Piezo Amplifier PID Controller

€

λ 2 PBS

€

λ 4 PBS Differential Amplifier Vapour Cell 85_{Rb +}87_Rb Pump Beam Probe Beam PD1 PD2 ML Beam Error Signal

Figure 3.4: A simplified schematic of the polarization spectroscopy based master laser (ML) lock.

Polarization Spectroscopy Based Laser Lock

A simplified schematic of the polarization spectroscopy based master laser (ML) locking mechanism is given in figure 3.4. The linearly (π) polarized ML beam is split up into a pump- and probe beam using a polarizing beam splitter (PBS), where the pump/probe intensity ratio may be adjusted using a half-wave plate (λ/2). The pump beam’s polarization is then made circular (either σ+ or σ−) using a quarter-wave plate (λ/4). Both beams are then directed through the87Rb+85Rb atomic vapour cell in a counter-propagating alignment.

Using another PBS, the π polarized probe beam is then split into its two orthogonal polarization components (σ+ and σ−), which are individually collected by two photodiodes (PD1 and PD2). Both photodiode signals are substracted from each other and the difference is amplified using a differential amplifier, resulting in a dispersive error signal (see figure 3.5) around an atomic transition resonance frequency. This error signal is fed back to the ML’s piezo actuated grating through a PID loop, thus effectively locking the laser frequency on an atomic transition.

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Laser Frequency (a.u.) Signal (a.u.) PD 1 PD 2 F0_{=2 c.o. 4} F0_{=3 c.o. 4} F0₌₄ F = 3 → F0

Laser Frequency (a.u.)

Signal

(a.u.)

F0_{=2 c.o. 4}

F0_{=3 c.o. 4}

F0₌₄

Figure 3.5: Saturated polarization spectroscopy of atomic 85Rb transitions. The left figure shows the individual photodiode signals (PD1, PD2) of the two orthogonal polarizition components (σ+, σ−) of the pump beam. The figure on the right shows the dispersive error signal obtained by substracting both photodiode signals. Closed transitions and their repective cross-overs (c.o.) with other transitions are strongest. Each atomic transition creates its own lockable error signal.

If we now want to lock the master laser on a 87Rb transition with a fixed detuning ∆, we can simply lock on a nearby 85Rb transition instead. For example, the 85Rb F = 2 −→ F = 3 closed transition has its resonance frequency -2526 MHz away from the 87Rb F == 1 −→ F = 2 transition. So locking the ML on this particular 85Rb transition will be effectively equal to locking to the87Rb transition with a fixed detuning of ∆ = 2π × −2526 MHz. A disadvantage of this locking method is that we are quite limited in our accessible range of detunings. It should also be noted that with this method, we are effectively locking on the side of a transitional line, instead of on its center. Depending on the sign of error signal’s slope, this results in a slightly lower or higher locked detuning than expected.

Also, as the magnetic sublevels may undergo Zeeman splitting, this lock-ing mechanism is very sensitive to external magnetic fields. For a typical background magnetic field of 0.5 Gauss (the approximate average magnetic field at the earth’s surface), the zeeman shifts of both the 87Rb and 85Rb magnetic sublevels will only be in the order of 2π × 1 MHz. In our magnetic trap, the magnetic field at the trap bottom is approximately 3.2 Gauss, and the maximum zeeman shifts will be closer to 2π × 10 MHz. For the sake of simplicity, when we refer to a detunig ∆ in this thesis, it will be with respect to the mF = 0 sublevels, which do not undergo a Zeeman shift in the low-field limit.

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3.1.4 Pulse Mechanism

Before the laser beams are coupled into an optical fibre leading to the exper-iment, they pass through an acousto-optic modulator, allowing us to accu-rately pulse the laser light in rectangular pulses of variable duration.

An acousto-optic modulator (AOM) is a device which can be used for controlling the power, frequency or spatial direction of a laser beam. It is based on the acousto-optic effect, i.e. the modification of the refractive index by an oscillating mechanical pressure. The key element of an AOM is a transparent crystal attached to a piezoelectric transducer used to excite a acoustic wave through the crystal. The laser light then experiences Brillouin scattering at the generated periodic refractive index and interference occurs similar to Bragg diffraction. The frequency of the 1st order diffracted light is Doppler shifted by an amount equal to the frequency of the acoustic wave. By only using the 1st order diffracted laser beam and driving the transducer with square shaped RF pulses, we can effectively produce precisely tuned laser pulses. 1st_Order Diffracted Beam Transmitted Beam ( 0th_{Order )} Acoustic Wave Transducer Absorber € θ

Figure 3.6: The basic working principle of an acousto-optic modula-tor (AOM). A piezoelectric transducer sends acoustic waves through a transparant crystal. An absorber prevents standing waves from forming in-side the crystal. The incoming laser beam then scatters off the acoustically generated periodic refractive index, resulting in a diffracted beam at a diffrac-tion angle θ. By switching on and off the transducer, we can effectively pulse the 1st order diffracted laser beam.

We use a commercially available Isomet 1205C-2 AOM driven by an 80 MHz RF oscillator at 1.2 Watts. It is designed to operate at wavelengths between 633-830 nm with a maximum 1storder deflection efficiency of ≥ 85% at a maximum beam diameter of 2.0 mm. Narrowing down the beam diam-eter results in a slightly lower deflection efficiency, but can also significantly shorten the pulse’s rise time τ . To obtain steep edged rectangular shaped

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pulses we opt for a minimal rise time of approximately 25 ns, at a (I = 1/e2) beam diameter of 100-150 µm and a maximum deflection efficiency of about 60%.

Figure 3.7 shows a schematic of the pulse mechanism. The overlapped master laser (ML) and slave laser (SL) beams are focused down through the AOM using a telescope, after which only the 1st order diffracted beam is coupled into an optical fibre leading to the experiment. Driving the AOM is an 80 MHz RF oscillator which is coupled to a TTL (transistor-transistor logic) driven electronical switch. The switch allows us to turn on and off the RF signal at an arbitrary pulse length, thereby effectively pulsing the fibre-coupled laser beam.

Amplifier To Experiment

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λ 2 PBS For Polarization Filtering Lens AOM Beam Dump 1st ML + SL Beam

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λ 2 Optical Fibre Switch RF 80 MHz TTL Signal PC 0th

Figure 3.7: A simplified schematic of the AOM based laser pulsing mecha-nism.

To account for the latency between the TTL pulse and the actual laser pulse we measure the laser pulse on a photodiode (PD) at the point of interest (e.g. at the end of the optical fibre), and display both signals (TTL + PD) on an oscilloscope. Figure 3.8 shows a measurement for a 200 ns pulse after the optical fibre leading to the experiment. The pulse delay includes the latency of all involved electronics and optics.

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.5 1 1.5 2 2.5 Time (µs) Signal (a.u.) TTL Pulse PD Signal Delay ≈ 0.7 µs τ ≈ 25 ns

Figure 3.8: Oscilloscope measurement of the TTL signal and the correspond-ing laser pulse measured on a photodiode (PD). We observe a laser pulse delay of approximately 0.7 µs relative to the TTL pulse. The laser pulse rise time τ is approximated to be 25 ns. The slower rising drift of the pulse’s maximum signal intensity is due to the thermalization of the AOM crystal, which takes approximately 0.4µ s to settle.

A often discussed problematic feature of the AOM is the crystal’s ther-malization [6], which, from the moment the transducer is turned on, takes approximately 0.4 µs before it stabilizes. This thermalization effect leads to beam pointing of the diffracted beam, i.e. it leads to a temporarily drifting diffraction angle θ. This beam pointing effect causes the fibre coupling effi-ciency to slowly drift until the thermalization settles. For longer pulses, in the order of tens of µs, this effect can be largely neglected. In general, the effect may be minimized by tuning the fibre coupling to the specific pulse length used.

3.2 Sample preparation and Laser Injection

3.2.1 Sample preparation

We initially load 7 × 107 87Rb atoms into a vacuum chamber using an ul-trapure evaporative 87Rb dispenser. The atomic cloud is laser cooled using a laser slightly detuned from the F = 2 ←→ F = 3 cycling transition and trapped in a magneto optical trap (MOT) [9]. The initial cloud is then cooled to ' 50 µK using optical molasses [22]. A repump laser is used to

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pump any atoms that have decayed into the F = 1 state back into the MOT cycle.

After the optical molasses, we switch off the repump lasers with the MOT lasers still on until almost all the atoms have decayed back into the F = 1 ground state. The atoms are then loaded into a Ioffe-Pritchard type magnetic trap, where it is further cooled to ' 10 µK using RF evaporation [4][5]. In the magnetic trap, only atoms in the |F = 1, mF= −1i state remain, as the other magnetic sublevels are untrappable (see figure 2.2). Finally, the magnetic trap potential’s bottom is set at the magic field value of Bm= 3.2 G. At this point the trap contains ' 106 atoms.

3.2.2 Laser Injection

Once the atoms are prepared in the |F = 1, mF= −1i ground state, we turn on the Raman lasers and inject the laser light into the vacuum chamber and through the atomic cloud.

Straight from the optical fibre, the light is arbitrarily linearly polarized. To ensure a correct polarization with respect to the quantization axis (which lies parallel to the bias magnetic field at the magnetic trap bottom), the laser beam goes through a half-wave plate (λ/2) and a polarizing beam splitter (PBS). The PBS is positioned so that the reflected beam’s linear polarization is perpendicular to the trap bottom’s magnetic field direction, i.e. the Raman light’s polarization is σ++σ−with respect to the quantization axis. The half-wave plate is used to maximize intensity of the reflected beam.

The Raman laser beam is directed through the centre of the atomic cloud, so that the position of the Gaussian laser beam’s peak intensity coincides with cloud’s highest atom number density.

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Optical Fibre

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λ 2 PBS From Raman Lasers

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!

B

Magnetic Trap Potential Atomic Cloud Mirror

Figure 3.9: Before the Raman lasers are sent through the trapped atomic cloud, the beams’ polarizations are adjusted to be linearly polarized perpen-dicular to the cloud’s quantization axis. The quantization axis is defined by the bias magnetic field direction ˜B at the bottom of the magnetic trap potential. The polarizing beam splitter (PBS) only reflects light that has a linear polarization perpendicular to ˜B. A half-wave plate λ/2 is used to maximize the intensity of the reflected beam.

3.3 Imaging

Information about the atomic cloud, and the |F = 2, mF= 1i ground state population in particular, is obtained through absorbtion imaging. We mea-sure the attenuation of laser light passing through the atomic cloud by imag-ing the cloud’s shadow on a CCD camera.

For the imaging of atoms in the F = 2 ground state we probe the cloud’s absorbance using laser beam resonant to the F = 2 −→ F = 3 transition (3.11) and linearly polarized with respect to the quantization axis. A 90:10 (transmition : reflection) beam splitter first reflects part of the beam through the atoms after which it is reflected back into the cloud by a mirror mounted inside the vacuum chamber. Part of the back-reflected beam then travels through the beam splitter and is focused onto the CCD camera. See [13] for a more detailed explanation of absorption imaging. To prevent imaging of atoms in the non-trappable |F = 2, mF= (0, −1, −2)i ground states we wait until all of these atoms have been lost from the trapped cloud before taking an image (typically a few ms).

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Atomic Cloud 90:10 Beam Splitter Mirror Probe Laser Beam CCD Camera

Figure 3.10: Simplified drawing of the imaging setup. The cloud’s absorbance is probed by a laser beam resonant to the F = 2 −→ F = 3 transition.

The law of Bouguer-Lambert-Beer relates the attenuation of light trav-elling a distance l through a column of material (the atomic cloud) to the scattering cross-section σ and number density n inside the column of the attenuating species (the atoms) in the material. It states that the intensity of light decreases exponentially with distance inside the material:

If = I0e−σnl, (3.1)

where I₀ and If are the intensity of light before and after attenuation respec-tively.

The optical depth (O.D.) is defined as the product of the scattering cross-section and column density:

O.D. ≡ σn (3.2)

We may then express the optical depth in terms of the transmittance: O.D. = − ln If

I0

. (3.3)

In order to retrieve the optical depth from the probe beam, we take three images: a light image of the un-attenuated probe beam, taken in the absence of atoms; a shadow image of the attenuated probe beam as it has passed through the atomic cloud; and finally a dark image, in the absence of both the atoms and probe beam. The dark image is substracted from the other two images to correct for any unwanted background light and the optical depth per pixel can then be calculated as follows:

O.D. = − ln Shadow − Dark Light − Dark

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An example of an imaged atomic cloud with population in the F = 2 ground state is given in figure 3.11.

(a) Light (b) Dark

(c) Shadow (d) Optical Depth (O.D.)

Figure 3.11: Absorption imaging of a magnetically trapped atomic cloud with population in the F = 2 ground state. The optical depth (d) is obtained through equation 3.3.

Stimulated Raman Transitions Between Hyperfine Ground States of Magnetically Trapped Rubidium-87 Atoms