Characterization of a magneto-optical trap of Rubidium atoms and building the repump laser

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Characterization of a magneto-optical trap of

Rubidium atoms and building the repump laser

Jelle van Urk

10646760

Report Bachelor Project Physics and Astronomy

Date of submission: August 14, 2017

University of Amsterdam Faculty of Science

Size 15 EC

Conducted between May 5, 2017 and August 14, 2017 Institute Van der Waals-Zeeman Institute Supervisor Dr. R.J.C. Spreeuw

Daily supervisors Carla Sanna and David Davtyan Second Assessor Dr. R. Gerritsma

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Abstract

In this report the magneto-optical trap of 87Rb atoms is characterized. Through absorption imaging with a camera the size, optical density and number of atoms are determined. The obtained image of the MOT is fitted as a Gaussian. From there the standard deviations in x and y directions are determined and by knowing the pixel size, the size of the MOT is calculated. The MOT has an area of 0.833 mm2 and the number of atoms is 1.31×106. The fluorescence is measured with a photodiode, which led to 3.56×106 atoms in the cloud. In addition, the lifetime and build up time for the MOT is measured under different circumstances. The set up for the repump laser is also discussed.

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1 Introduction

Classical computers are getting to their limit of how much they can calculate and simulate. Therefore the development of quantum simulators and quantum chips is nowadays a widely examined part in physics. Instead of the classical bit, the qubit is developed. Because of the quantum mechanical behaviour, they can be 0, 1 or a superposition of the two. This gives more possibilities with a single qubit than a bit (0 and 1) in a classical computer, which makes these quantum computers faster. To better understand this quantum behaviour, experiments in quantum simulation are done. Quantum simulators manipulate simple quantum systems to get a better understanding of more difficult systems which are almost impossible to simulate with a supercomputer. One way of realizing systems for quantum computation or quantum simulation is by creating atom chips. In these chips small clouds of ultracold atoms are loaded into magnetic traps.

The first step in such experiments is to create cold atoms, in this case 87_Rb.

They are produced with the use of a magneto-optical trap (MOT), which can cool the cloud of atoms to a few microkelvin. A MOT uses a combination of a cooling and repump laser and a quadrupole magnetic field to trap the atoms in a vacuum chamber. These lasers need a well-defined frequency, before they are able to interact with the atoms. In this report a comprehensive description of the repump laser is given. The cooling laser and the vacuum chamber were already there at the start of this experiment. After the alignment of the cooling laser was fixed, the MOT was characterized, with the use of two different techniques: absorption imaging with a camera and measuring fluorescence with a photodiode.

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2 Magneto-optical trap

2.1 Theory

A magneto-optical trap (MOT) is a trap that consist of a combination of laser beams and a magnetic field gradient, made for cooling and trapping neutral atoms. The procedure to make a MOT can be sketched as follows. Two identical coils with opposite current sign (anti-Helmholtz configuration) produce the quadrupole magnetic field of the trap. The opposite currents generate a magnetic field which is zero in the center and increases from the center to the edge of the trap, as shown in figure 1.

Figure 1: A quadrupole magnetic trap. The anti-Helmholtz configuration produce a magnetic field, which is zero in the center and increases in all directions. The presence of the magnetic field causes the magnetic energy levels MJ of the

excited atoms to Zeeman shift. MJ = 1 is shifted up for B > 0 and shifted down

for B < 0. For MJ = −1 the opposite happens. This shift is bigger as the atom

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Figure 2: A magneto-optical trap in one dimension. The magnetic field, which is zero in the center and increases in both directions, produce a Zeeman shift for the excited atoms. The laser with opposite circular polarization are red detuned. For an atom on the right side of the trap, the ∆M = −1 is closer to resonance with the laser and is therefore excited by the σ− laser beam and kicked towards the center.

An atom can be excited by three different polarizations of light: linearly polar-ized light (π-polarization) or circular polarpolar-ized light (σ+- or the opposite σ− -polarization). An atom starts in the ground state with J = 0 and MJ = 0.

σ+_{-polarized light excites this atom to J = 1 and ∆M}

J = 1, while σ−-polarized

light excites it to ∆MJ = −1.

Now two counterpropagating laser beams with opposite circular polarization are sent through the trap. The two lasers are red detuned. It depends on the position of the atom in the trap to which state it will be excited. For an atom located on the right side of the trap the MJ = −1 is shifted down and it is excited by the

σ−-polarized light incident from the right. Therefore the atom is kicked towards the center. The opposite happens for atoms on the left side of the trap. There the atom is excited to the MJ = 1 state by the laser with σ+-polarization [4], but

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2.2 Laser system

The laser system for the MOT consists of two lasers, one for cooling and one for repumping of the atoms. They excite the atoms via the D2 line located at a wave-length 780,24 nm, see Appendix A [6]. To prevent the lasers from drifting, they are locked to the correct transition frequency. To find the transition, saturated absorption spectroscopy is used (section 3.2.1).

To slow the atoms down, the cooling laser is used to perform Doppler cooling. The photons in the laser beam interact with the87Rb atoms in the vacuum cham-ber. If the photons frequency is close to the transition frequency of the atom, they will be absorbed, resulting in an excitation of the atom to a higher energy level. The cooling laser is locked to the |5S1

2, F = 2i to |5P 3

2, F = (1, 3)i crossover

transition. An AOM (section 3.1) shifts the laser up by 200 MHz. This results in a red detuning of 13 MHz to the |5S1

2, F = 2i to |5P 3

2, F = 3i transition to achieve

the correct cooling frequency [8]. The atoms move with finite velocity through the chamber. This means that atoms counterpropagating with the beam see the photons blue shifted and atoms copropagating with the beam see the photons red shifted. To counteract this Doppler effect the light of the laser is red detuned by 13 MHz. An atom moving to the right gets excited by a counterpropagating laser. Then the atoms decay through spontaneous decay to their previous ground state (|5S1

2, F = 2i). On average the change in momentum through the photon

emission is zero, because the atom emits a photon in a random direction. The change in momentum through absorption is always to the opposite direction of the velocity, so therefore the atom gets slowed down.

Because the laser is red detuned, there is a possibility that the cooling laser excites atoms to the |5P3

2, F = 2i state instead of the |5P 3

2, F = 3i state. From there

the atoms can decay to the |5S1

2, F = 1i ground state and leave the cooling cycle.

This decay would lead to the atoms escaping the trap. Therefore a repump laser is necessary to prevent from losing these atoms. The repump laser is resonant to

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cooling process.

3 Repump laser

3.1 Acousto-optic modulator

Like all the lasers in the experiment, the repump laser is locked to a well defined frequency in order to excite the Rubidium atoms to the correct energy state. To achieve it, acousto-optic modulators (AOM’s) are used to shift the laser frequency. An AOM uses sound waves of a certain frequency travelling through a crystal to diffract an incoming light beam (see figure 3).

Figure 3: Bragg diffraction of an incoming beam with wavelength λ inside a crystal. Sound waves in the crystal with wavelength Λ diffract the beam. Λ is added to or subtracted from λ to create the different diffraction orders.

The angle of the diffracted beam depends on the wavelength of the incoming light and the sound wave, which is emitted by the AOM. The diffraction of the light is described by Bragg’s law:

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In this equation λ is the wavelength of the light, Λ is the wavelength of the sound wave, θ is the angle of diffraction and m is an integer, which gives the order of diffraction [3]. This diffraction occurs when photons from the light beam interact and scatter off from the phonons from the sound wave. The sign of m depends on the fact that some photons absorb a phonon (+) and other photons create a phonon (-), and thus lose energy, due to the sound wave of the AOM. The change in frequency can be calculated using the following equation [3]:

∆f = mEphonon

h , (2)

where ∆f is the frequency change of the photons and Ephonon = hF , so ∆f = mF ,

with F the frequency of the sound wave.

3.2 Experimental work

In the three picture in this chapter the setup for repump laser is shown. The red arrows give the trajectory of the laser. The setup is divided into two distinct parts. The first part changes the beam’s shape, size and polarization to optimize the coupling and the frequency, so that it can be used for repumping87Rb atoms to the correct energy state. The second part, enclosed by the orange square, con-tains the vapor cell and the photodiode detector, to perform saturated absorption spectroscopy to measure the exact transitions of 87_{Rb. This technique will be}

described in detail in section 3.2.1.

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Figure 4: The first box of the repump laser setup. The DL100 diode provides a high intensity laser. It is divided in two parts: the spectroscopy part (orange square) and the part which gives the laser the right properties for the experiment. The laser produces linear polarized light with a power of 67mW. The output beam of the laser diode is elliptical, which is not useful for coupling the beam into a fiber. To reshape the beam, a pair of prisms is placed directly after the diode in the following configuration, (see figure 5) (2).

Figure 5: Anamorphic expansion prism configuration. The incoming elliptical beam is changed in one direction, so the beams shape is changed to a circular one.

The prisms are placed in such a way, that the beam shape is only changed in one direction. This configuration is called anamorphic expansion prism configuration [7]. After the prisms, the circularly shaped beam passes through an isolator (3). The isolator prevents any reflected light to hit the diode, which could destabilize the laser through optical feedback. After the isolater, the beam passes through

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a half waveplate (4) and a polarizing beam splitter (PBS) (5). The half wave-plate rotates the linear polarization, which can therefore change the amount of S-polarized light (perpendicular to the plane of incidence) and P-polarized light (parallel to the plane of incidence) in the beam. The PBS reflects S-polarized light with an angle of 90◦. P-polarized light isn’t affected and propagates in the original direction. The amount of light which is transmitted/reflected depends on the orientation of the half waveplate. So this combination is used to distribute the power between the experiment and the saturated absorption spectroscopy setup (with the orange square).

The transmitted light continues its way to the spectroscopy setup (see next sec-tion). The reflected beam hits a mirror (6) and reaches a telescope (7). The telescope is a Keplerian beam expander (figure 6).

Figure 6: Keplerian beam expander. Two lenses of respectively f1 = 50 mm and

f2 = 100 mm are used to magnify the beam with M = f_f2₁.

The magnification M of the beam depends on the focal lengths of the two lenses: M = f2

f1

. (3)

In this setup f1 = 50mm and f2 = 100mm, which gives a magnification factor of

M = 2. The enlargement of the beamsize helps the coupling of the beam into the fiber at the laser coupler (10). The lens inside of the coupler focuses the beam in order to match the fiber mode. The following equation gives a simple relation of the beam diameter D, the diameter d of the spot inside the fiber, the focal length

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f of the lens inside the input coupler and the wavelength λ of the laser beam: d = 4

π λf

D. (4)

To optimize the coupling of the beam into the fiber, it is necessary to get a d = 4.6 × 10−6m. Knowing that λ = 780 × 10−9m and f = 6.2 × 10−3m, it is calculated with equation (4) that D = 1.3×10−3m. The diameter of the beam was measured by taking a picture of the beam and fitting the data with a Gaussian beam. The pictures of the beam before and after the telescope are shown in figure 7. The pictures were taken with a camera from Thorlabs.

Before telescope After telescope

Figure 7: Two pictures of the beam. The picture on the left side is the beam before the telescope, the picture on the right side is the magnified beam after the telescope.

The beam intensity is Gaussian distributed. Therefore the data can be fitted with the following equation:

I(x) = A + I0e

−(x−µ)2

2σ2 . (5)

µ is the position on the camera where the peak intensity I0 of the beam is located.

σ is the standard deviation of the Gaussian distribution. Figure 8 shows the fit for the beam after the telescope.

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Figure 8: A Gaussian fit of the intensity of the beam after the telescope. 4σ is the width of the beam at 95% of the total intensity of the beam, which is used to estimate the beamsize. I0 is the peak intensity in the center.

In the fit 4σ corresponds to the width of the beam at 95% of the total intensity. To estimate the width of the beam, 2σ is multiplied by 2. With this procedure, it is determined that the diameter of the beam before the telescope is approximately 0.65 × 10−3m. Therefore the Keplerian telescope (7) is used to magnify the beam to D = 1.3 × 10−3m. The PBS (8) is used to clean up the polarization and the half waveplate (9) is used to give the beam the correct linear polarization, which is necessary to optimize the coupling to the fiber.

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Figure 9: The second box where the AOM is located. The frequency is changed by 80MHz before the laser is coupled in the fiber for the experiment.

From the output coupler (1) the beam passes through a mechanical shutter (2). After the two mirrors (3), the beam is sent through the crystal in the AOM (4). The repump laser is locked 80MHz below the |5S1

2, F = 1i to |5P 3

2, F = 2i

transition. The AOM provides the frequency shift needed to make the lasers frequency resonant to this transition. To get rid of the unnecessary diffraction orders, an aperture is used (5). It only transmits the first order diffracted beam, which has the correct frequency. Before the laser reaches another input coupler (8), it goes through PBS (6) and waveplate (7). The PBS and half waveplate have the same purpose as in box 1.

After every step the laser loses power. To find the MOT in this experiment, the power of the repump beam that enters the vacuum chamber has to be at least 7.5 mW. Table 1 gives the power and transmission of the laser during the setup.

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After: Power (mW) Transmission (%) Laser 67.0 100 Prisms 65.5 97.8 Isolator 57.5 87.8 PBS (5) 52.5 91.3 PBS (8) 49.0 93.3 Fiber coupling (10) 22.6 46.1 AOM 16.5 73 PBS (6) 13.4 81.2 Fiber coupling (8) 9.1 67.9

Table 1: This table shows the loss in power of the repump laser during the setup. The amount of transmission is shown in percentage after every step.

3.2.1 Saturated absorption spectroscopy

As mentioned above it is important to lock the laser to the right transition fre-quency of the 87_{Rb atoms. To find the transitions, saturated absorption}

spec-troscopy is performed. The setup is shown in figure 10.

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prop-The transmitted part of the beam in the PBS (5) first passes through a tele-scope (1). The telescope magnifies the beam size to optimize the spectroscopy by increasing the amount of interactions in the vapour cell. The light is still P-polarized, so it is transmitted through the PBS (2). The beam propagates through the vapour cell (3). The quarter waveplate (4) changes the polarization to σ-polarization before the beam is reflected by the mirror (5). After the sec-ond propagation through the waveplate, the light is π-polarized again, but now orthogonal to the incoming beam (S-polarization). The beam is now reflected by the PBS and is sent through a lens (6), which focuses the beam in the photodiode (7) to detect the optical signal and convert it into an electric signal.

Figure 11 shows the principle of saturated absorption spectroscopy. Two beams, both on resonance with the atoms transition frequency, propagate in opposite direction through the vapour cell filled with 87_{Rb atoms. The two beams are}

coherent. The incident beam works as a pump beam. By reflecting the incident beam with a mirror it returns through the vapour cell and acts as a probe beam. The probe beam is directed to a photodiode, which gives the spectroscopy signal.

Figure 11: The incoming beam is used as a pump beam and propagates through the vapour cell. The reflected beam then acts as a probe beam in cell. The atoms with zero velocity are already excited by the pump beam. Therefore the photodiode measures a peak in the absorption spectrum of the probe beam. The pump beam is on resonance with the atomic transition frequency. It propa-gates through the vapour cell and excites atoms in a range around this frequency. This results in a broad absorption hole in the spectrum instead of a narrow one,

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also called Doppler broadening [1]. The reflected beam moves in the other di-rection through the atom cloud acting as a probe beam. The probe gives same absorption hole as the pump gives, only with a narrow peak at exactly the tran-sition frequency. This happens because the beams counterpropagate through the cell. An atom that moves to the right through the cell is blue shifted for the pump beam and red shifted for the probe beam. Therefore this atom sees two different frequencies. The beams only have the same frequency for atoms with zero veloc-ity. The pump laser has already excited these atoms and they will therefore not absorb photons from the probe beam anymore. This results in a sharp peak in the absorption spectrum at exactly the atomic transition frequency.

4 Characterization of the MOT

To optimize the MOT, it is important to characterize the atomic cloud. For the characterization two techniques were used. The first is measuring the fluorescence of the MOT with a photodiode and the second technique is absorption imaging with a complementary metal–oxide–semiconductor (CMOS) camera. In this re-search the following properties of the cloud were determined: lifetime, build up time, optical density, the number of atoms and the size of the cloud.

4.1 Fluorescence measurement

For the lifetime, build up time and the number of atoms the photodiode is used. In figure 12 the setup for this measurements is shown. Part of the fluorescence of the MOT hits the lens and it is focused on the photodiode.

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Figure 12: The schematic setup the fluorescence measurement. Some of the fluo-rescence of the MOT is focused on the photodiode by the lens (f = 9cm).

The photodiode measures the fluorescence of the MOT and converts the light to an electrical signal. An oscilloscope displays this signal in a voltage over time diagram. The signal consists not only of the fluorescence of the MOT, but also of the light refracted from the chip and the chip mounting. To get the total voltage of the MOT itself a lifetime measurement is taken. The dispenser is switched off and the oscilloscope shows the decrease in fluorescence due to atoms leaving the trap. The data of the measurement is fitted with the standard function for an exponential decay:

V (t) = V0+ V1e−

t−t0

τ , (6)

For the build up time almost the same equation is used: V (t) = V0 + V1(1 − e−

t−t0

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where V0 is the voltage without the MOT, t0 the starting point of the

measure-ment and τ the mean lifetime/build up time of the cloud. V1 is the difference in

voltage and therefore the voltage induced by the MOT. Figure 13 shows a lifetime measurement.

Figure 13: Lifetime measurement with the photodiode. The dispenser is switched off at t ≈ 3 s and the photodiode measures a decrease of fluorescence due to atoms leaving the trap. The data is fitted and used to calculate the mean lifetime of the MOT. The red line is the fit and the blue dots are the data points.

In this case V1 = 5.0 × 10−3V. The mean lifetime, which is the average time an

atom spends in the trap, is the fit parameter of this measurement and is fitted at τ = 7.9 ± 0.1 s. The opposite measurement is taken to get the build up time for the MOT. When all the atoms have left the trap, the dispenser is switched on to measure the increase in fluorescence.

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Figure 14: Build up measurement. The photodiode measures the increase of fluorescence when the dispenser is switched on. The red line is the fit and the blue dots are the data points.

The oscilloscope can’t measure longer than 600 seconds. For this reason the mea-surement stops before the MOT is finished building up. For the fit to be accurate, it needs a well defined start and end. Because there is no end in this measure-ment, it is not clear where the starting point is located. The starting point for the fit is put in manually and lies between 34 s and 36 s. The mean build up time for the MOT is τ = 477.7 ± 13.1 s for the start at 34 s and τ = 492.5 ± 13.8 s for the start at 36 s. This gives an extra error of approximately 10%. The large number for the build up time is probably because it takes time to warm up the dispenser. In addition, it takes time to build up background pressure inside the vacuum chamber.

Besides the dispenser measurements, three other build up measurements were done. These measurements were taken by switching on the magnetic field, re-pump and cooling laser respectively, to see the difference in build up times. The dispenser is on during the these measurements, so the background pressure of the Rubidium atoms in the chamber was kept constant. The mean build up time for

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the MOT by switching on the magnetic field is τ = 2.07 ± 0.04 s (figure 15).

Figure 15: Build up measurement by switching on the magnetic field. The red line is the fit and the blue dots are the data points.

The next figure is the build up time for the MOT after switching on the repump laser. The mean build up time is approximately the same as for the magnetic field, τ = 2.1 ± 0.1 s.

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Figure 16: Build up measurement. The red line is the fit and the blue dots are the data points.

For the cooling laser (figure 17) there is a large jump at the beginning. This is because the photodiode measures a lot of light from the cooling laser which is scattered from the chip mounting. So it first jumps to the value with no MOT before it starts to build up. The mean build up time for this measurement is τ = 1.39 ± 0.04 s. The error of the fit would suggest that the fit for the last measurement is better than for the others. However, it is clear to see that this is not true. Therefore the goodness of fit is evaluated for these three measurement: the R-squared value. This is a statistical measure that checks how close the data is to the fitted line. R-squared lies always between 0 and 1. The closer this number is to one, the better the fit is. For these three experiments the R-squared values are 0.9899, 0.9772 and 0.9642 respectively. This indicates that the fit for the cooling laser is indeed worse than the other two.

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Figure 17: Build up measurement by switching on the cooling laser. At the start there is a big jump in the signal, because most of the light measured by the photodiode comes from scattered light from the cooling laser. The red line is the fit and the blue dots are the data points.

As mentioned in section 3.1, atoms in the MOT emit photons resonant to the atoms transition and decay to a lower energy state through spontaneous emission. They do that by emitting a photon in a random direction. These photons are detected by the photodiode. The rate by which the atom cloud emits photons, is called the scattering rate ΓSC. ΓSC is described by the following equation [5]:

ΓSC = Γ 2 I/Isat 1 + I/Isat+ 4∆2/Γ2 . (8)

Γ is the natural linewidth of the excited state (2π × 6, 06 MHz) [6], ∆ is the detuning of the cooling laser (2π × 13 MHz). The total intensity of the laser is 80 mW, which is divided in four different beams with a diameter of 2.8 cm. I is the total peak intensity of the four beams, (26.74mW_cm2) and Isat is the saturation

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intensity (1.67mW_cm2) [6]:

Isat =

2π2

~Γc

3λ3 , (9)

with λ the wavelength of the D2 transition line. For a more elaborate explanation of the scattering rate, see ref [5] and chapter two in ref [4]. Finally this gives a scattering rate of ΓSC = 2π × 1.37 MHz.

The amount of light that hits the lens is calculated by the ratio κ, the fractional solid angle:

κ = πr

2

4πd2, (10)

where πr2 is the area of the lens, with r the radius of the lens (1.25 cm). 4πd2 is the total area of the sphere with d the distance from the MOT to the lens (18 cm). The power of the MOT is given by the following equation [5]:

PMOT= κΓSCN E. (11)

Here E = hc_λ = 2.55 × 10−19J is the energy of one photon emitted by an atom in the cloud at a wavelength of 780.24 nm. N is the number of atoms trapped inside the MOT. The power of MOT, that is measured by the photodiode is:

PPD =

V

GS(λ) (12)

where V is the output voltage and G and S are the total impedance (in Ω) of the oscilloscope and the sensitivity (in _WA) of the photodiode respectively. The sensitivity of the photodiode depends on the wavelength of the light and is given by the manufacturer. For this wavelength the sensitivity S = 0.5_WA. The used oscilloscope (Rigol DS1074) has a total impedance of G = 106Ω. The number of atoms in the MOT can be estimated by equating PMOT and PPD:

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N = V κSΓSCEG

. (13)

With all the numbers, this equation gives an estimation for the total number of atoms, N = 3.56 × 106 _atoms.

4.2 Absorption imaging

Another widely used technique for characterizing a MOT is absorption imaging. For this technique a third laser is used, the probe laser, which is on resonance with the |5S1

2, F = 2i to |5P 3

2, F = 3i transition. The schematic setup for the

absorption imaging is shown below.

Figure 18: Schematic setup for the absorption imaging. The probe has a diameter bigger than the MOT. Therefore the beam is locally absorbed, which leads to a dark spot on the picture. The MOT is focused on a CMOS camera. The incident beam is demagnified by the telescope to match the size of the chip. The telescope

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The beam, which has a diameter bigger than the MOT, is sent directly through the MOT and after the vacuum chamber the light is captured by a 10 bit CMOS camera from IDS. The camera has a resolution of 1280 × 1080 pixel with a pixel size of 5.3 µm × 5.3 µm. Equation 3 gives the demagnification of the beam through the telescope, M = 1₂. The incoming beam has a Gaussian shape with peak intensity I0 = 0.17mW_cm2 and frequency ω. Due to the absorption of photons by the

atoms in the cloud, the beam intensity is locally decreased. The final intensity that is captured by the camera is given by the following equation [2]:

I(ω) = I0e

−ODΓ2

4(ω−ω2₀)+Γ2_. ₍₁₄₎

ω0 is the resonant frequency of the used transition. As mentioned above, the

probe laser is on resonance with the transition, so ω = ω0. This reduces I(ω) to:

Iout = Iine−OD. (15)

OD is the optical density of the cloud:

OD = ln Iin Iout

. (16)

The optical density is also given by OD = N lσ0. Here n is the number of atoms

per unit volume of the MOT and l is the thickness. σ0 = 3λ

2

2π = 2.91 × 10 −9_cm2

is the maximum cross-section of a Rubidium atom [1]. The measurement begins with taking three images: one with the MOT, one without the MOT and one completely dark. The pictures are taken automatically. Every cycle takes around 25 s. The measurement starts with switching on the cooling and repump laser and the coils. The dispenser is turned on with a current of 7.5A for 15 s per cycle to build up the MOT. After that the dispenser, cooling laser and the coils are switched off. The probe laser is turned on after one ms for 50 µs and then the first picture is taken with the camera. This is done before the atoms leave the

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trap. Therefore the MOT absorbs photons from the probe laser and a dark spot is seen on the picture: an absorption image. Five seconds later, with no MOT, the second picture is taken. Another five seconds later an image with no lasers on is taken, to finish the cycle and complete the set of three pictures, see figure 19.

MOT No MOT Dark

Figure 19: Three pictures taken by the camera. The left is the picture with the MOT. In the red ellips there is a dark spot with respect to the same picture with no MOT in the middle. On the right there is a picture with no laser, which is subtracted from the other two.

To get the correct intensity, the dark image is subtracted from both the image with the MOT as for the image without the MOT. To get the optical density with this data, Eq.(16) is extended to:

OD = ln Iin− Idark Iout− Idark

. (17)

The intensity per pixel is converted to an electrical signal. This data from the camera is then processed, which leads to the following images:

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Figure 20: The data images. On the left there is the actual absorption image of the MOT. The image on the right is the optical density of the MOT, which is calculated with equation 17.

Furthermore, the density of atoms in the cloud is given by ρ(x, y, z). The atoms reduce the beams intensity according to Beer’s law [9]:

dI

dz = −ρσI. (18)

In this equation σ = σ0

(2∆_Γ)2₊₁.The detuning is zero and therefore σ is just σ0. The

solution of this equation gives:

I(z) = I0e−nσ0, (19)

where n = R∞

−∞dzρ(x, y, z) is the column density of atoms [cm

−2_{] of the cloud,}

perpendicular the xy plane. The column of this cloud has an area of one pixel on the camera. Equation 15 and 19 are used to calculate the total number of atoms inside the cloud. Both equations are divided by I0 and then differentiated

with respect to the lasers frequency ω. The two equations for lnI(ω)_I

0

are then equalized to get an equation for the optical density of the atoms per column per

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pixel:

−OD = −nσ =⇒ n = OD σ0

. (20)

Under the assumption that the optical density and ρ(x, y, z) are constant within each pixel, the total number of atoms can be calculated:

N = X

all pixels

npixelApixelm2, (21)

which leads to:

N = Apixelm 2 σ0 X all pixels ODpixel. (22)

Here m = 2 is the demagnification of the pixel area due to the telescope. The data for the optical density is fitted as a Gaussian. From there the standard deviations in both x and y direction are determined: σx = 37.7 pixels and σy = 59.8 pixels.

The area of the MOT is then 0.833 mm2_{. Knowing the size of the cloud, the}

optical density for every pixel in the cloud is summed up. This gives an estimated value for the number of atoms in the cloud with absorption imaging, 1.31 × 106 atoms.

5 Discussion/Conclusion

For the experiment to work properly, a minimum power of 7.5 mW is needed for the repump laser. Eventually this is exceeded by approximately 1.5 mW. The initial laser power 67 mW is reduced by a total of 86.5%. This is due to coupling efficiencies of 53.3% and 32.1%. This loss could be lower if the whole setup was build in one box instead of two. In this way the laser has to be coupled into a fiber only once, which would reduce the loss.

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ing differ by a factor 3: 3.56 × 106 _{atoms for the photodiode and 1.31 × 10}6 _atoms

for the absorption imaging with a size of 0.833 mm2_{. It would be logical if these}

numbers were the other way round, because for the measurement with the fluores-cence it is difficult to focus the MOT properly on the photodiode. The oscilloscope only shows a voltage over time diagram and therefore the signal can be from both the MOT as for any scattered light inside and outside the vacuum chamber. With the camera the MOT is seen on the picture and so it is a lot easier to focus the MOT properly. One explanation for this difference is the cross section σ0. The

cross section used to calculate the number of atoms through absorption imaging is based on the two-level atom system. The Rubidium atoms have more than two levels. The cross section is probably lower for such a system, which would increase the number of atoms. Another possible explanation for this difference is the absence of the error. Because of the difficult focusing of the MOT on the photodiode, it is possible that the error of this measurement is large with respect to the absorption imaging. For further research it would be useful to take time for this calculation.

The lifetime and build up time is also determined under different circumstances. The mean lifetime and mean build up time of the cloud, when the trap is main-tained and the dispenser is turned off/on, are τ = 7.9 ± 0.1 sec and τ = 492.9 ± 13.8 sec respectively. There is a big difference in build up time when the dispenser is on and the lasers and magnetic field are used to determine the mean build up time. For the measurement with the repump and cooling laser the mean build up time is τ = 2.2 ± 0.1 sec and for the cooling laser τ = 1.39 ± 0.04 sec. This differ-ence is probably due to the fact that the dispenser is on during the experiment and there is no loss in background pressure.

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6 Populaire samenvatting

Experimentalisten proberen tegenwoordig steeds moeilijkere systemen te beschri-jven. Systemen die zelfs met een supercomputer niet gesimuleerd kunnen worden. Daarom wordt er veel onderzoek gedaan naar het ontwikkelen van quantumcom-puters die wel de rekenkracht hebben om dit uit te voeren. In tegenstelling tot de klassieke bit (0 of 1), kan een qubit 0, 1 of een combinatie van de twee zijn. Om dit quantummechanische gedrag beter te begrijpen worden quantum simulatoren ontwikkeld. Zo worden atoom chips gemaakt, door super koude atomen te laden in magnetische vallen op de chip.

Dit proces begint met het vangen van koude atomen in een magneto-optical trap (MOT). Een MOT maakt gebruik van een combinatie van een sterk magneetveld en lasers. Als het licht van de laser niet de juiste eigenschappen heeft, zullen de atomen niets van de fotonen merken. Door het licht de juiste eigenschappen te geven, zoals golflengte, frequentie en polarisatie, kunnen fotonen in de laser Rubidium atomen exciteren naar een hogere energietoestand. De absorptie van fotonen gebeurt in tegengestelde richting van de voortbeweging van de atomen, waardoor ze afremmen/afkoelen. De opbouw van een van deze lasers is besproken in dit verslag. Om het uiteindelijke experiment voor quantumsimulatie te opti-maliseren is het belangrijk om de eigenschappen van de MOT te weten, zoals de grootte van het wolkje atomen en het aantal atomen dat zich erin bevindt. Twee technieken om dit te bepalen zullen ook in dit verslag te vinden zijn.

7 Acknowlegdement

For the past three months I’ve working in a great team to do my bachelor project. A special thanks to Arthur La Rooij for telling me there was no bachelor student working in this group. He contacted Robert Spreeuw for me, to ask if there was room in his lab for me. Therefore I also want to thank Robert for giving me the

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and I’m really curious in the progress for the coming months. In particular I want to thank Carla Sanna and David Davtyan to be my daily supervisors. From start to end they were there to help me with everything I needed.

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References

[1] Christopher J. Foot, Atomic Physics, 2005, Department of Physics, Unversity Press Oxford

[2] Kathrin Luksch, Measurement of the Number of Atoms in a Magneto-Optical Trap Using Absorption Imaging, 2012, National University of Singapore [3] D.J. McCarron, A Guide to Acousto-Optic Modulators, 2007, Durham

Uni-veristy

[4] Harold J. Metcalf and Peter van der Straten, Laser Cooling and Trapping, 1999, Springer

[5] Matthew J. Pritchard, Manipulation of ultracold atoms using magnetic and optical fields, 2006, University Durham

[6] Daniel A. Steck, Rubidium 87 D Line Data, 2001, Los Alamos National Laboratory

[7] Orazio Svelto, Principles of Lasers, 2010, Spinger

[8] Atreju Tauschinsky, Rydberg Atoms on a Chip and in a Cell, 2013, University of Amsterdam, PhD Thesis

[9] Gordon McDonald, Detecting Atomic Shot Noise On Ultra-cold Atom Clouds, 2017, University of Sydney

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A

87

Characterization of a magneto-optical trap of Rubidium atoms and building the repump laser