University of Amsterdam

van der Waals-Zeeman instituut

Raman laser system for coherentmanipulation of cold atoms

With a focus on the phase lock electronics

July 11, 2012

Author:M. Dinkgreve

6192882

Supervisor:Dr. Q.A. Beaufils

Dr. R.J.C. SpreeuwDr. N.J. van Druten

Abstract

This report presents a Raman laser system to drive atoms between the two hyper-fine levels of the ground state of 87Rb. Therefore two external cavity diode lasers areplaced in a phase lock loop (PLL). The lasers are tuned to emit light at a wavelengthof 780,241 nm and have a frequency difference of 6,834682 GHz, corresponding tothe atomic transition between the F=1 and F=2 levels of the 5S1/2 state of 87Rb.Saturated absorption spectroscopy is used to successfully visualize the D2 lines ofRubidium. One of the lasers is locked on an atomic transition by polarization spec-troscopy. The PLL is used to stabilize the relative frequency and phase between thetwo lasers. The coherence of the system is tested in an electromagnetically inducedtransparency experiment on Rubidium in a vapor cell. This shows some coherence,but using the phase-feedback loop above the frequency-feedback loop, does not im-prove the coherence of the system. The bandwidth of the PLL is too low to obtaina stable phase lock.

Report on a 12EC Bachelor Project in Physics and Astrophysics,executed in the period 02-04-2012 until 11-07-2012

Contents

1 Introduction 2

2 Theory 32.1 Energy levels of an atom [6] . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Multi-level system [6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Raman Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Rubidium [6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 External cavity diode lasers 113.1 Setup inside the laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Working with the lasers [6] . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 Individual line width of the lasers . . . . . . . . . . . . . . . . . . . . . . . 13

4 Spectroscopy on a Rubidium cell 144.1 Doppler-free Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Frequency lock one laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5 Raman laser system 195.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.2.1 Optical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.2.2 Electronic setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.3.1 Response time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.3.2 Error signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.3.3 Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6 Electromagnetically induced transparency (EIT) 256.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

7 Conclusions 28

8 Populaire samenvatting 29

1

1 Introduction

The research about quantum information processing and quantum computing has becomemore popular in the last few years. Coherent control of quantum systems is an importantpart of this.

In this thesis a Raman laser system is presented. This setup is invented to excitecold atoms on a microchip [13]. Therefore a stable two-photon transition between thetwo hyperfine states of the ground state of Rubidium 87 is needed. The actual frequencydifference between the two qubit states is in the microwave range (6,834682GHz), whichcorresponds to a wavelength of a few centimeters. This is too large to excite the atomson a microchip, the wavelength is larger than the distance between the atoms on the chip.Therefore two lasers with higher frequencies, in the near infrared, are used with a frequencydifference of 6,834682GHz. This coherent process is called a Raman transition, where anatom is brought from one state to another, via a virtual level.

To do this two independent external cavity diode lasers are used in a phase lockedloop (PLL). The lasers will be frequency locked and phase locked to each other. So it isimportant to have a very stable frequency difference between the lasers and even moreimportant is the stability of the relative phase. In order to test the performance of thephase lock and the coherence of the system an EIT experiment is done.

In chapter 1, a theoretical analysis of the three level Raman transitions is shown. Inchapter 2, external cavity diode lasers, used for the experiments are described. Chapter3 describes the different kinds of spectroscopy that are used to frequency lock one laser.Chapter 4 presents the setup for a Raman laser system and the results of the locking.Finally in chapter 5 an EIT experiment is described in order to test the coherence of thesystem.

2

2 Theory

2.1 Energy levels of an atom [6]

An atom has different energy levels. These are eigenstates of the atom, with discreteenergies. The energy of the state depends on the configuration of the electrons in thedifferent electron shells. These energy levels split up and shift on a smaller scale, by differenteffects. Fine structure, the most significant effect, is the result of spin-orbit coupling andthe relativistic correction. On a smaller scale, hyperfine structure in the atomic spectrumis a result of the properties of the atomic nucleus. The property of main focus here is theangular momentum of the atomic nucleus, which interacts with the electrons [14].

The total electronic angular momentum operator, characterizing spin-orbit interaction,is: J = L+S, with L the orbital angular momentum operator and S the spin operator. Thefine structure level is indicated by the spectroscopic notation: n2S+1LJ . Since the hyperfineinteraction is three orders of magnitude smaller than the spin-orbit interaction, the nuclearangular momentum I couples directly to the total electronic angular momentum, insteadof coupling both to the orbital angular momentum and the spin (L and S). So the totalatomic angular momentum F is given by: F = J + I. The quantum numbers F and mF

characterize the hyperfine splitting. For a given F , in the absence of a magnetic field, thestates are (2F + 1)-fold degenerate [14].

Atomic transitions [6]

An atom can go from one state to another by interaction with light. There are three pos-sible mechanisms: stimulated absorption, stimulated emission and spontaneous emission.Stimulated absorption is the transition to a higher energy level by absorption of a photonwith a frequency corresponding to the energy difference between the levels (E = hω). Bystimulated emission, one photon with the right frequency is absorbed and two are emitted.With spontaneous emission, the transition to a state with a lower energy is initiated byzero-point radiation, so without any applied electromagnetic field.

2.2 Multi-level system [6]

An atom with its different energy levels can be considered as a multi-level system. Here, atime-dependent theory for a multi-level system is developed, which will produce formulasfor the probabilities of atomic transitions.

3

A multilevel system has n states which are eigenstates of the unperturbed Hamiltonian,H0:

H0ψ = Enψ (1)

These eigenstates are orthonormal:

< ψm|ψn > (2)

At t=0 the perturbation is turned on:

H = H0 +H ′(t) (3)

Since the eigenstates constitute a complete set, the wave function can be written as alinear combination of them:

Ψ(t) =∑n

cn(t)ψne−iEnt/h (4)

This wave function satisfies the time-dependent Schrodinger equation:

HΨ = ih∂Ψ

∂t(5)

To obtain cn(t), equation (3) and (4) are filled in to equation (5):

∑n

cn(t)H0ψne−iEnt/h +

∑n

cn(t)H ′(t)ψne−iEnt/h =

ih∑n

∂cn(t)

∂tψne

−iEnt/h +∑n

cn(t)Enψne−iEnt/h

∑n

cn(t)H ′(t)ψne−iEnt/h = ih

∑n

∂cn(t)

∂tψne

−iEnt/h

Taking the inner product with ψm and exploiting the orthogonality of the wave functionsgives:

∑n

cn(t) < ψm |H ′(t)|ψn > e−iEnt/h = ih∑n

∂cn(t)

∂t< ψm|ψn > e−iEnt/h

∑n

cn(t)H ′mne−iEnt/h = ih

∂cm(t)

∂te−iEmt/h

− ih

∑n

cn(t)H ′mnei(Em−En)t/h =

∂cm(t)

∂t(6)

with

H ′mn ≡< ψm |H ′(t)|ψn > (7)

4

In the case of a laser system, the perturbation is an electromagnetic wave. The primaryresponse of the atoms is to the electric part of the wave. If the spatial variation of theelectromagnetic field is ignored (because the wavelength is much bigger than the atom)and it is polarized along the z-direction, propagating in the y-direction, then the field thatinfluences the atom can be written:

−→E = E0cos(ωt)k (8)

The energy of a charge in a electric field is given by −q∫ −→E · d~r, from which follows:

H ′ = −qE0zcos(ωt)

H ′mn = −~pE0cos(ωt)

with the electric dipole moment:

~p = q < ψm |z|ψn > (9)

Inserting this in equation 6 yields:

∂cm(t)

∂t=i

h~pE0

∑n

cn(t)cos(ωt)ei(Em−En)t/h (10)

If one only looks at the two states N and M, equation 10 gives:

∂cM(t)

∂t=i

h~pE0cos(ωt)e

i(EM−EN )t/hCN (11)

=i

2h~pE0

(ei

(ω+

EM−ENh

)t+ e

−i(ω−EM−EN

h

)t)CN (12)

∂cN(t)

∂t=i

h~pE0cos(ωt)e

−i(EM−EN )t/hCM (13)

=i

2h~pE0

(ei

(ω−EM−EN

h

)t+ e

−i(ω+

EM−ENh

)t)CM (14)

When the system is close to resonance: EM−EN

h− ω EM−EN

h+ ω, the rotating wave

approximation can be used. This gives:

∂cM(t)

∂t=

i

2h~pE0e

−i(ω−EM−EN

h

)tCN (15)

∂cN(t)

∂t=

i

2h~pE0e

i

(ω−EM−EN

h

)tCM (16)

Combining these equations yields:

5

∂2cM(t)

∂t2+ i

(ω − EM − EN

h

)∂cM(t)

∂t+

∣∣∣∣∣~pE0

2h

∣∣∣∣∣2

CM(t) = 0 (17)

When it is assumed that the system starts in the state ψN : cN(0) = 1 and cM(0) = 0,the solution to this equation is [5]:

cM(t) =|~p|E0

h

√(EM−EN

h− ω

)2+|~p|2E2

0

h2

sin

√√√√(EM − EN

h− ω

)2

+

(|~p|E0

h

)2t

2

(18)

The probability of a transition from state N to M equals |cM(t)|2:

PN→M(t) =|~p|2E2

0

h2((

EM−EN

h− ω

)2+|~p|2E2

0

h2

)sin2

√√√√(EM − EN

h− ω

)2

+

(|~p|E0

h

)2t

2

(19)

2.3 Raman Transitions

Rabi frequency and detuning of the laser light

In a two level quantum system there are two different states where the atoms can be. Whenthere is an electric field applied, or in this case a laser beam, with the right frequency ofthe atomic transition the atoms can be excited from the φ0 state to the φ1 state. Theatoms go in the excited state φ1, but after a certain time they will fall back to the φ0 state,by spontaneous emission. The life time of the excited state gives the natural linewidth dueto the uncertainty principle of Heisenberg:

∆E ·∆t ≥ h

2(20)

Where the uncertainty in energy ∆E is proportional to the lifetime ∆t.The probability to find the atoms in state φ0 or φ1 is |c0(t)|2 and |c1(t)|2, respectively.

And normalization of the wave function requires that |c0(t)|2 + |c1(t)|2 = 1. In time, theprobability is going to oscillate between the two states. This oscillation has a specificfrequency called the Rabi frequency, which is defined as:

Ωi,j ≡~di,j · ~E0

h(21)

where ~d is again the transition dipole moment an ~E0 the (vector) electric field amplitude.This frequency is associated with the strength of the coupling between the electric field

6

and the transition, so the polarization of the light and the dipole moment. When theseare parallel the interaction is strongest.

The Rabi frequency depends on the chosen detuning. The generalized Rabi frequencyis defined as:

Ω′ =√

Ω2 + δ2 (22)

At zero detuning, the system is at resonance. The probability oscillates between the twostates φ0 and φ1, Rabi-oscillations (Ω′ = Ω). A given pi-pulse (π/Ω) can now bring allthe atoms from state φ0 to φ1. To prevent resonant excitations to the excited state thedetuning must be larger then the natural linewidth of the excited state. If the detuningis not zero the system will oscillate at the generalized Rabi frequency and not equallybetween the two states anymore. The probability of the atoms to be in φ1 is lower. Thiswill change the Rabi frequency. For example, when the detuning is 1Ω, this leads to aoscillation between a superposition between the two states. So this will give a higher Rabifrequency, see figure 1.

Figure 1: The Probability of finding the atoms in the φ1 state [9].

Stimulated Raman transitions

When looking at a multilevel system every state has its own detuning. In this thesis onlya three level Raman system is used. This three level system is commonly known as a threelevel Λ configuration. There are three levels at which the atoms can be φ1, φ2 and φ3. Thereis no direct coupling between φ1 and φ2 so therefore the atoms will be excited from φ1 toφ3 (absorption) and from φ3 to φ2 (stimulated emission). This coherent process is called aRaman transition, see figure 2. With Raman transitions an atom is brought from one stateto another, via an intermediate level. To do this, two laser fields are applied. One laserhas a frequency of ωL1 (between φ1 and φ3) with a single-photon detuning ∆ = (ω13−ωL1)and the other at ωL2 (between φ2 and φ3) with a Raman detuning δ = (ω23− ωL2−∆). Ifthe detuning ∆ is too small the atoms might actually populate the higher level φ3 insteadof going to the final state φ2 immediately. The φ3 level is only a virtual level, so if thedetuning is larger than the natural linewidth the atoms will never go there.

7

Figure 2: Energy levels of the three level Raman system.

As discussed before the probability between two transitions in a multilevel system isgiven by (19). For a three level Raman system this formula will be the same, but now wehave n = 1, 2, 3. So formula (9) becomes:

~d = q < φ3|z|φn > (n = 1, 2) (23)

Where q = −e if an atomic nuclear charge is considered. Then the Rabi frequency is:

Ω =−eE0

h< φ3|z|φn > (24)

With a detuning (δ) of:

δ =Em − En

h± ωL (25)

Now the equation for the probability for one state to another in a multi-level system, interms of detuning and Rabi frequency, becomes:

PN→M(t) =Ω2

Ω2 + δ2sin2

(√Ω2 + δ2

2t

)(26)

Here√

Ω2 + δ2 is just the generalized Rabi frequency as in (22). At resonance, δ = 0, theprobability is just oscillating at the normal Rabi frequency.

PN→M(t) = sin2(

Ωt

2

)(27)

For a three level system with two transitions and two detunings, δ and ∆, as shown infigure 2, the generalized Rabi frequency is defined as:

Ω′ =

√(|Ω1|2

∆− |Ω2|2

∆− δ) + 4

|Ω1|2|Ω2|2∆2

(28)

8

Figure 3: Probability of finding an atom in the φ2 state, with Ω1 = Ω2 and ∆ = 10Ω1 andδ = 0.2Ω1[9].

Here Ω1 is the Rabi frequency of the first transition, Ω2 the Rabi frequency of the secondtransition, ∆ the detuning on φ3 and δ the detuning on φ2 [8]. The first term betweenbrackets contains the light shift |Ω1|2/|∆ and |Ω1|2/|∆, and the detuning δ. Figure 3 showsthe probability depending on the detuning δ.

9

2.4 Rubidium [6]

In many experiments concerning cold atoms, Rubidium atoms are used. Rubidium is analkali metal, with atomic number 37. It is hydrogenlike and has a relatively large abundanceon the earth. The only natural isotopes are: 85Rb (72%) and 87Rb (28%).

Figure 4: The fine and hyperfine structure of the low levels of 87Rb in absence of a magnetic field[14].

The structure of the Rubidium atom is determined by the effects described earlier.There are several transitions that have frequencies in the visible range. Two groups ofthem are called the D1 and D2 lines: transitions from a 5S1/2-level to a 5P1/2-level andfrom a 5S1/2-level to a 5P3/2-level, respectively. The D1 lines interact with photons withwavelengths close to 795nm and the D2 lines with wavelengths close to 780nm [11, 12].The focus in this thesis lies on the D2 lines. By the selection rules, the only transitionsthat can take place are the ones: F → F ′ = F − 1, F → F ′ = F and F → F ′ = F + 1.

10

3 External cavity diode lasers

3.1 Setup inside the laser

The lasers used for the experiments are external cavity diode lasers (ECDLs). Each laserconsists of a semi conductor laser diode with an external cavity used to reduce its linewidthand to control the frequency. The laser diode with a collimating lens and a diffractiongrating, are shown in figure 5. Furthermore the grating is mounted on a piezoelectrictransducer (PZT), which allows for electrical tuning of the frequency. The whole laserdiode and grating setup is mounted on a Peltier element, which stabilizes the temperature.This again is mounted on a large metal base. An AD590 temperature sensor is used tomeasure the temperature.

The diode is of the type Sanyo DL-7140-201W with a wavelength of 782 ±2nm and isplaced in a laser and collimator holder. The collimating lens is for adjusting the laser beamby screwing the lens closer or further away from the diode. If the diode is placed at thefocal point of the lens the light will overall have the same beam size. This can be tested bychecking if the beam size stays the same over a distance of a few meters. Then the lightfrom the diode is going to the grating. This grating diffracts the light in all directions. Thezero-order light is the output beam for the laser and the first order diffracted light givesoptical feedback to the diode. The diffraction angle corresponds to different wavelengths.So changing the angle of the grating will roughly change the wavelength of the laser by amaximum of 2nm. The more the frequency is pushed away from the original frequency ofthe diode, the harder it is to hold the frequency stable. For the experiments a wavelengthof 780nm is required, so the diode is pushed to it’s limits.

Figure 5: Top view of the external cavity diode laser, showing: (1) the laser diode, (2) collimatinglens, (3) diffraction grating, (4) Piezoelectric transducer.

For adjusting the feedback to the diode also the vertical angle of the grating is impor-tant. If this is misaligned there will not be an actual laser beam. To optimize the feedbackto the laser the threshold current must be as low as possible. The threshold value is theminimum current at which the laser starts lasing. On a piece of paper you can see the lightflashing. This can be seen even better on an oscilloscope. The threshold must be as low aspossible by changing both the horizontal and vertical angle of the grating. This will alsoimprove the power of the beam. After the threshold current is reached, the power of thelaser beam behaves linearly with respect to the current.

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3.2 Working with the lasers [6]

When the laser is installed, a more precise tuning of the wavelength of the emitted lightis needed. Next to turning the grating, there are three ways to adjust the wavelength:the temperature controller, the current controller and the electrical signal that goes tothe piezoelectric transducer. The interplay between the different factors determines thefrequency emitted by the laser.

Temperature

As stated above, the temperature around the laser is stabilized by a Peltier element. Thisis controlled by a Thorlabs TED200 temperature controller, which gets feedback from thetemperature sensor in the laser. Generally, the frequency increases when the temperatureis increased. However, this process can be highly nonlinear. It is close to linear when thetemperature is adjusted by a small amount, with the current in a large enough, stablerange. For stable control of the temperature, the temperature must be set no more thanfive degrees above and below room temperature.

Current

The amount of current injected to the laser can be adjusted by a Thorlabs LDC202 currentcontroller. Increasing the current generally leads to a lower frequency and increases theemitted power linearly. There is hysteresis in this process. Close attention should bepaid to the maximum current that the diode can bear, in this case 140 mA. The outputpower can be increased - without changing the frequency - by increasing the current anddecreasing the temperature simultaneously.

PZT

The length of the external cavity can be changed by sending an electrical signal to the piezo-electric transducer (PZT). In the experiments described in this thesis, the PZT is connectedto a function generator which generates a triangle shaped wave with a frequency of 30 Hz.Hereby the laser is scanned over a range of frequencies during the experiments. The PZTis also connected to a lock unit built by the UvA electronics department. Therewith thefrequency at which the scan starts (i.e. the offset) and the length of the scan range (i.e.the level) can be adjusted.

Single mode and coherence control

Figure 6 shows how different factors influence the lasing frequency. The mode with themaximum gain is emitted. Changing the temperature shifts all profiles asynchronously.Changing the current changes the external an internal mode, thus moves profile 2 and 3,but with different speeds. Therefore, it can happen that two lasing modes can becomeequally favorable. Then, the laser will emit both frequencies, thus operates multi modal.

12

While scanning the PZT, profiles 3 and 4 change. Therefore, a different single mode canbecome favorable. Then a mode hop occurs, i.e. the lasers starts lasing at a differentfrequency [2]. It is desirable to find a temperature at which the amount of mode-hops inone scan range is the lowest.

To improve the mode hop-free tuning range, profile 2 must be shifted with profiles 3 and4. This can be done by modulating the current by applying a signal to it which is similarto the signal that scans the PZT [2]. The current controller must be connected to the samefunction generator as the PZT and the signal must be inverted, for the two signals to bealways out of phase by exactly π. The connection between the function generator and thecurrent goes via a lock unit, to be able to tune the size and change the polarization of thesignal. This procedure improved the mode hop-free tuning range from∼ 4 GHz to∼ 9 GHz.

Figure 6: Factors that determine the frequency emitted by the laser. 1: laser diode gain medium,2: internal resonator mode structure, 3: external resonator mode structure, 4: grating profile,[2].

3.3 Individual line width of the lasers

The line width of a laser is a measure of the fluctuation in emitted frequency. For the laserdiodes used in the experiments, the line width in free-running mode (i.e. without opticalfeedback) will be around 40 MHz. This is too high for the experiments in this project.The use of an external cavity should decrease the line width. For a laser mode to interfereconstructively, the wavelength of the light should not only fit an integral number of timesin the internal cavity, but also in the external cavity. The line width can be estimatedby tuning the lasers to be multi modal. Then, the two modes emitted by the laser willinterfere, resulting in a beat signal. This beat can be measured with a photo diode andevaluated on a spectrum analyzer. The width of the beat is a measure for the line widthof the laser. The individual line width of the EDCL’s used is estimated to be 250-300 kHz.

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4 Spectroscopy on a Rubidium cell

Working with lasers in atomic physics experiments, it is important to measure and controlthe frequency of the laser. It is very useful to have a known physical reference to lock thelaser frequency on. In this chapter is described how a spectrum of the Rubidium D2 linesis visualized and how this is used to lock the laser.

4.1 Doppler-free Spectroscopy

One way to show the Rubidium spectrum is by a simple absorption spectroscopy. A setupfor this would be to send a laser beam through a vapor cell and measure the intensity of thetransmitted light in a photo diode. When the frequency of the laser is modulated over arange of 780,24nm (corresponding to the D2 lines of rubidium), the light will be absorbedat (or near) an atomic transition. Looking at the transmitted light on an oscilloscope,results in a dip at every atomic transition frequency, showing the spectrum. However,these spectral lines (dips) are not infinitely thin, they are broadened. This is significantlycause by Doppler broadening. The atoms in the vapor cell have different velocities. Atomsmoving towards or away from the laser light will see a higher (red shifted), respectivelylower (blue shifted) frequency. This will cause a Gaussian shaped absorption profile in thetransmission spectrum. Close transitions cannot be distinguished by this broadening.

Figure 7: Setup for a saturation spectroscopy containing a ECDL, a λ/2 wave plate, a polarizingbeam splitter (PBS), a Rubidium cell and a photo diode (PD).

A way to eliminate the Doppler broadening is by saturated absorption spectroscopy.Two counter-propagating beams, from one laser, go through the vapor cell (with 85Rb and87Rb). The optical setup is shown in figure 7. One of the beams has a much higher powerand is called the pump beam. This one will not go into the photo diode, but will only serveto excite the atoms. The weaker beam, called the probe beam, is measured in the photodiode. The idea of saturated absorption spectroscopy is that the pump beam will diminishthe absorption of the weaker probe beam at the transitions. This only works for atomswith a velocity close to zero, because only these atoms will see the two laser beams at thesame frequency. Therefore the Doppler broadening is eliminated by this technique, whichis also called a Doppler free spectroscopy [3]. Figure 8 shows the transmission spectra withand without pump beam.

14

Figure 8: A part of the Rubidium spectrum measured by saturation spectroscopy: a) withoutpump beam and b) with pump beam.

4.2 Frequency lock one laser

To stabilize and control the frequency of the laser polarization spectroscopy is used [6, 10].By this technique the spectrum obtained by saturated absorption spectroscopy will bedifferentiated. This results in sharp lines going through zero at the transition frequencies.See figure 9. For locking the laser home made feedback electronics are placed in a feedbackloop. A block diagram of the feedback electronics is shown in figure 10. The photo diodesignal is used to generate a feedback signal to the locking electronics. The frequencywhere the signal is crossing 0 Volt will be the locking frequency. The electronics will givefeedback to the PZT, so the frequency is pushed back to the 0 Volt signal. To lock thelaser, the Locking/Dither switch is first set to Dither. The scanning range is made smallerby lowering the level and the offset is used to scan over the right transition frequency.When the signal is crossing 0 Volt at the preferred frequency, the Locking/Dither switch isturned to Locking. The gain is to adjust the intensity of the feedback signal. If the gain isto low the frequency can drift away. When it is too high, the PZT is correcting too muchand the frequency of the laser will oscillate. It is best to turn the gain as high as possible,just before the laser starts oscillating. Using polarization spectroscopy will improve thislocking, because of the steep lines going through zero it is less likely the laser will jump toother transitions.

15

Figure 9: A part of the Rubidium spectrum corresponding to the 87Rb: F = 2→ F ′ = 1, 2, 3 (onthe left) and 85Rb: F = 3 → F ′ = 2, 3, 4 (on the right) measured by a) saturation spectroscopyand b) polarization spectroscopy.

Figure 10: Block diagram of the feedback electronics used to lock the laser [9].

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4.3 Results

Figure 11 shows the D2 line spectrum of Rubidium, scanned over a range of 7,5 GHz.In order to characterize the different frequencies in the obtained spectrum the frequencydifferences between the different peaks were compared to the hyper fine energies from [ref].The transitions and their corresponding energies are listed in Table 1.

Figure 11: Absorption spectrum of the Rubidium D2 lines. Corresponding to a) 87Rb: F =2 → F ′ = 1, 2, 3 b) 85Rb: F = 3 → F ′ = 2, 3, 4 c) 85Rb: F = 2 → F ′ = 1, 2, 3 and d) 87Rb:F = 1→ F ′ = 0, 1, 2.

Transition Literature [11, 12] ExperimentF = 2→ F ′ = 1, 87Rb 384227,7 GHz 384227,8 GHzF = 2→ F ′ = 2, 87Rb 384227,9 GHz 384227,9 GHzF = 2→ F ′ = 3, 87Rb 384228,1 GHz 384228.0 GHzF = 3→ F ′ = 2, 85Rb 384229,1 GHz 384229.0 GHzF = 3→ F ′ = 3, 85Rb 384229,1 GHz 384229.1 GHzF = 3→ F ′ = 4, 85Rb 384229,3 GHz 384229.2 GHzF = 2→ F ′ = 1, 85Rb 384232,1 GHz not measuredF = 2→ F ′ = 2, 85Rb 384232,1 GHz 384232.1 GHzF = 2→ F ′ = 3, 85Rb 384232,2 GHz not measuredF = 1→ F ′ = 0, 87Rb 384234,5 GHz 384234.2 GHzF = 1→ F ′ = 1, 87Rb 384234,5 GHz 384234.3 GHzF = 1→ F ′ = 2, 87Rb 384234,7 GHz 384234.4 GHz

Table 1: Measured frequencies for the obsurved transitions.

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The Doppler peaks are fitted with a Gaussian Amplitude function:

y = y0 + Ae−(x−xc)2

2ω2 , (29)

Where ω is the full width at half maximum (FWHM), A the amplitude, x the frequencyand xc the frequency at the center of the peak. This is used to show the improvement ofpreciseness in frequency of the measurements, by involving the pump beam in saturatedabsorption spectroscopy. Figure 12 shows the Doppler broadened absorption profile with-out the pump beam, resulting in a FWHM of 0,21335 GHz. In figure 13, the absorptionprofile with pump beam is presented, resulting in a FWHM of 0,0173 GHz. This representsan improvement of an order of magnitude.

The frequency lock of one laser by polarization spectroscopy is established. A lock ona closed atomic transition can be stable for a day. For any other transitions the lock isstable for about 1 hour. The bandwidth of the locking is 3 MHz.

Figure 12: Doppler broadened absorption profile of the transition: 87Rb, F = 2 → F ′ = 1, 2, 3with a Gaussian Amplitude fit.

Figure 13: Doppler-free absorption profile showing the transitions: 87Rb, F = 2 → F ′ = 1, 2, 3.On the right zoomed in on one peak with a Gaussian Amplitude fit.

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5 Raman laser system

5.1 Introduction

In section 2 is shown that the three-level Raman system can drive a photon from onequantum state to another via an intermediate state. In this experiment a Raman lasersystem is used to drive 87Rb atoms from the F = 1 state to the F = 2 state levels (52S1/2).Therefore two independent lasers are used with a frequency difference of 6,834682 GHz,corresponding to the transition frequency between F = 1 and F = 2. For the Ramantransitions the intermediate state is chosen to be the excited state (52P3/2) of the D2 linesof Rubidium. The transitions and the different states are shown in figure 14. In theexperiments is chosen to lock the master laser (ML) on one of the excited states of 85Rb.In this way there is a detuning of about 2GHz for the system.

Figure 14: Raman transitions between the ground state and excited state of the Rubidium D2lines. The master laser (ML) is tuned on the 85Rb excited states and the slave laser (SL) has afrequency difference of 6,8 GHz plus a Raman detuning.

Finally the two lasers must be frequency and phase locked to each other. Then thelasers will have a stable frequency difference and have the same phase. If the phase lockis stable this will correspond to a coherent system that can drive atoms between the twoground states of 87Rb.

5.2 Experimental setup

In this setup there are two independent lasers in a phase locked-loop (PLL). The two lasersare combined in a polarizing beam splitter cube (PBS) and are going into a fast photo diode(FPD). This FPD creates an interference signal, the beat note, at the frequency differenceof the two lasers. The master laser can be locked on some reference (an atomic transitionof the 85Rb excited state) or left unlocked if the absolute frequency is not important.

The beat note signal is compared with a reference by phase lock electronics. Thiselectronics provide two error signals, a slow and a fast signal, to feedback to the slave laser(SL). So the SL will be frequency and phase locked on the ML.

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5.2.1 Optical setup

Figure 15 shows the optical setup for the PLL. Both lasers are going through an opticalisolator (OI). This prevents the laser light from reflecting back into the laser diode. Thenboth lasers are split off in a PBS. A λ/2 wave plate can adjust the polarization, so thesplitting of the beams can be controlled. The two laser beams are mixed in a PBS. The lightnow has two different orthogonal polarizations. To create an interference signal anotherPBS is placed, but rotated 45 degrees, so 50% of the signal remaines and has the samepolarization. Finally the beams go through a lens into the fast photo diode (FPD). Only afraction of the light is going to the FPD. This is 1, 5mW for each laser. Most of the poweris available for experiments. The ML can be frequency locked by polarization spectroscopy,see previous section.

Figure 15: Optical setup, showing a Phase locked loop. OI=optical isolator,λ/2 =half wave plate,λ/4 =quarter wave plate, PBS=polorized beam splitter, RPBS= 45o rotated PBS, PD=photodiode, FPD=fast photo diode

5.2.2 Electronic setup

Figure 16 shows a block diagram of the electronic part of the PLL. The beat note signalfrom the fast photo diode is first down mixed and then compared to a reference by thephase lock electronics. This will give two error signals, a slow and a fast signal, that givefeedback to the SL.

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Figure 16: Block diagram of the electronic part of the PLL. On the left: down mixing the signals.On the right: Phase lock electronics.

Down mixing

The beat note signal from the FPD is around 6,8GHz. This is too fast for the phaselock electronics, so the signal is first down mixed with a microwave reference signal, on aMini-Circuit ZX05− 14+ frequency mixer. The intermediate frequency (IF) of the downmixing is νIF = |νbeat − νMWref |. This frequency should be around 5 to 80 MHz. Afterdown mixing the IF signal goes to the phase lock electronics where it will be compared toa second reference, applied at the same frequency, νIF .

Phase lock electronics

The phase lock device used for this experiment is based on the analog devices AD9901integrated circuit [1]. This consists of a phase/frequency discriminator that can act in twodifferent modes: as a linear phase detector or a frequency discriminator. When the beatnote and reference frequency are very close or the same, only the phase detector is active.On the other hand, when the frequencies are too different the frequency discriminator takesover. When the frequencies are corrected, the phase detector is active again.

Figure 17: Block diagram of the analog Divices AD9901 digital phase/frequency discriminator.

Figure 17 shows a block diagram of the main components of the phase lock electronics.There are 4 ’D’ flip-flops and an exclusive-OR gate (XOR). The two input signals, from

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the beat note (oscillator input) and the reference (reference input), are pulse trains. Theflip-flops convert this signals at every rising edge, so the output signal is exclusive a blockfunction. By this the frequencies of the signals are divided by two. The outputs of theflip-flops are compared in the XOR, this gives an high signal when they are different anda low signal when they are the same, this is also a block function. When the duty cycle ofthe output XOR is 50% the wave signals are phase locked (figure 18a). This correspondsto a phase lock at φ = π. If one of the input signals is converted the phase lock is atφ = 0. When the signals are out of phase the duty cycle is higher or lower then 50%(figure 18b). If the two input signals are out of phase (constant high or low duty cycle) thefrequency discriminator takes over, until they are in phase again (duty cycle is around 50%).Because of environment vibrations there will always be some small frequency fluctuations.Therefore the phase lock electronics keep changing between the two modes, phase detectionand frequency discriminator. This way the two lasers can be phase/frequency locked for along time (several hours).

Figure 18: Illustration of the input/output relationship at lock. a) Input signals in phase. b)Input signals out of phase.

Feedback to the laser

Finally the phase lock device has two outputs, a slow and a fast error signal. The slow onegives feedback to the piezoelectric transducer (PZT) to adjust the frequency, so the laserswill be frequency locked. The fast error signal gives feedback by modulating the current,this compensates for the phase so the lasers can be phase locked. To modulate the currentof the SL, a current modulator as shown in figure 19 is implanted between the currentcontroller and the laser diode. The fast error signal is connected to the RF input. Thiscurrent modulator is based on a phase lead loop.

5.3 Results

5.3.1 Response time

In order to characterize the phase lock the response time the fast output of the phase lockdevice is measured. To do this a reference block function of 50 to 200 kHz is put into

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Figure 19: Schematics of the current modulation electronics consisting a 50Ω resistance, a phaselead network of a parallel 1kΩ resistance and a 330 nF capacitor. And a 1µF AC couplingcapacitor [9].

the RF of the current modulator (instead of the fast error signal) while the beat note isstill running in frequency lock. The output of the fast error signal is now compared to thereference. The response time of the whole ciruit is 1,0 to 1,7µs. This variation is due tothe different reference frequencies. Fgure 20 shows a delay time of 1,7µs for a referencefrequency of 50kHz. This is divided into the delay time of the phase lock electronics(200ns) and the delay of the rest of the PLL (1,5µs), the rest of the PLL contains the wiresand other electronics. The response time of the phase lock electronics corresponds to abandwidth of 5 MHz. For a good phase lock the bandwidth of the phase lock electronicsshould be a several times larger then the bandwidth of the beat note so the lock will notgo away due to environment fluctuations.

Figure 20: Oscilloscoop signals showing the response time of the whole circuit (1,5µs) and theactual phase lock electronics delay time (200ns), measured with a reference of 50kHz.

5.3.2 Error signals

Also the slow and fast error signals with and without phase lock are compared. The resultsare shown in figure 21. When the slow and fast error signals are connected (give feedback) tothe SL, the slow signal is lower then when the fast signal is not connected. This implicatesthat the fast signal is taking over. This shows that the phase lock electronics work well.

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The fast signal is also lower when connected, but this is partly due to the resistance inthe current modulation. Important is that the signal is total random without given anyfeedback and becomes structured once the laser is given feedback.

Figure 21: Error signals of the slow (purple) and fast (blue) output. a) With frequency lock. b)With phase lock.

5.3.3 Bandwidth

The bandwidth of the beat note is 1,5 to 2,0 MHz, see figure 22. When the Phase lock isapplied the bandwidth is a lot wider because the laser starts oscillating. The fast outputsignal that goes to the current modulator gets quickly too high, so the beat note signalstarts oscillating. To overcome this an attenuator, of 4 dBm, is implanted in-between theoutput and the current modulator. Also a narrow peak as in [7] is not observed. This canbe due to the low resolution of the spectrum analyzer that is used, which has a averagingtime of 5ms. The peak that is expected is about 10 Hz so is impossible to see with thisspectrum analyzer. However, an actual phase lock is not achieved.

Figure 22: Down mixed beat note signal.

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6 Electromagnetically induced transparency (EIT)

In order to test the performance of the phase lock and the coherence of the laser system anEIT experiment is performed. EIT is a classical experiment base on a quantum mechanicaleffect about the interference between the transition probabilities of a three level system,showing a transparency in the absorption/transmission spectrum when a coherent systemis at resonance [4]. The amount of transparency gives an indication of how well the systemcan be in a coherent superposition of the ground states of 87Rb.

Figure 23: Atomic configuration for EIT experiments.

6.1 Experimental setup

For the EIT measurements the two laser beams, with different polarizations, go co-propagatingthrough a rubidium cell. After the cell the two beams are split up by a beam splitting cubeand the probe beam is measured in a photo diode (figure 24). The pump beam is obtainedfrom the ML and is turned to the F = 2→ F ′ = 2 transition in the D2 line of 87Rb. Thiswill serve as the control field of the EIT configuration. The probe field is obtained fromthe SL and is turned to the F = 1→ F ′ = 2 with different detuning, corresponding to theΛ configuration shown in figure 23. In order to measure the transmission due to EIT, theprobe field is scanned over a range of 900 MHz. The pump beam has a power of 16mWand the probe of 150µW. The temperature of the cell is at 21oC.

Figure 24: Optical setup for EIT measurements in 87Rb.

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6.2 Results

Figure 25a shows the EIT measurements: with and without phase lock, and with nocoupling (without pump). The relevant trough is around zero detuning. There are alsotwo troughs at negative detuning, these are corresponding to the lower transitions of theexcited state (52P3/2). The transparency peak, in the relevant trough, is at a detuning of16MHz instead of at zero detuning. This can be due to the lock of the ML, which has afrequency of νML = 384227, 90± 0, 05GHz. So the shift of 16MHz is within this margin.

Figure 25b shows a measurement of the relevant transparency peak with more datapoints. At phase lock the EIT peak is expected to be larger then without phase lock,because the system is more coherent when the lasers are in phase. However, it seems thateven with a frequency lock only, the system is coherent enough to show a good EIT peak.So the phase lock does not improve the coherence of the system. With phase lock the peakis even lower. This can be due to the experimental setup of the EIT. A higher temperaturewill increase the density within the cell, this results in a better chance of exciting the atoms(hitting the atoms with the laser beam). Also the overlap of the two beams in the cell canbe optimized more. On the other hand, when these results are not due to the EIT setup,this means that the phase lock does not work well enough to improve the coherence of thesystem.

Unfortunately there was not enough time within this Bachelor project to figure thatout. Testing the Raman laser system on cold atoms on a chip would be a better option tocheck the phase lock.

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a)

b)

Figure 25: EIT measurements showing a) the transmission spectrum with and without phaselock, and with no coupling. b) More precise measurement of the transparency peak with andwithout phase lock.

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7 Conclusions

The aim of this project was to establish a Raman laser system for coherent manipulationof atoms between the two hyperfine levels of the 87Rb ground state. However a properphase lock of the two lasers is not accomplished. With saturated absorption spectroscopythe D2 lines of Rubidium are visualized. A mode-hop free scanning range of 9 GHz isachieved. This provides to scan over the whole spectrum of the D2 lines of 85Rb and 87Rbat once. The locking of one laser, by polarization spectroscopy, is held stable for one hourto one day, depending on the atomic transition. The linewidth of the locking is 3MHz,which is normal for this kind of setup. The individual bandwidth of the lasers is around300 kHz. The bandwidth of the relative frequency lock of the two lasers, the beat note,is measured at 1,5 to 2,0 MHz. For phase locking the lasers the noise bandwidth of thePLL must be a several times larger than the bandwidth of the beat note. The lockingbandwidth of the phase lock electronics is measured at 5 MHz, this should be enough forphase locking. However the delay of the rest of the PLL is 1,5µs, this is too large. Becausedifferent delay times were measured for different frequencies of the reference, the currentmodulation electronics seem to be the main problem. But also the delay of the wires andthe other objects in the setup of the PLL should be further improved to accomplish betterresults.

In order to test the coherence of the system an electromagnetically induced transparencyexperiment is performed. This shows some transparency when the lasers are frequencylocked only. However when the phase lock is applied (the fast error signal is giving feedbackto the laser), there is less transparency observed. This means that the phase lock is notimproving the coherence of the system, it is even worst. This is because of the large delaytime of the PLL. Thereby the phase lock feedback is correcting too slow for improving therelative phase between the lasers.

For further experiments the delay time of the PLL must be decreased. This can be doneby improving the current modulation electronics and using shorter cables in the setup. Alsofurther investigation in understanding the phase lock electronics and replacing the filterinside the phase lock device should be considered. Finally, the system can be tested onan EIT experiment again. If the system is coherent enough, this system will be able tomanipulate cold atoms on a microchip.

Thanks

I would like to thank Quentin Beaufils for his daily guidance and patient help and feed-back. My thanks also goes to Shayla Jansen, for the inspiring teamwork on parts of theexperiment. Finally I would like to thank Robert Spreeuw and Klaasjan van Druten forsharing their experiences and for their overall guidance during this project. I had a greattime at the Quantum Gasses and Quantum Information group and certainly learned a lotabout experimental physics.

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8 Populaire samenvatting

In dit Bachelorproject wordt een opzet voor een Raman laser systeem beschreven. Dit kanworden gebruikt om atomen op een microchip zo te manipuleren dat ze bepaalde eigen-schappen gaan vertonen. Dit draagt bij aan het onderzoek naar kwantum computers.

In een atoom bewegen de elektronen in verschillende banen (schillen) om de atoomkernheen. Ze kunnen naar verschillende banen springen, zo bepalen ze de energie van hetatoom. Een atoom kan hierdoor verschillende energieniveaus hebben, in de kwantumfysicanoemt men dat energie eigentoestanden. De laagste energietoestand is de grondtoestand,hier zitten de atomen het liefst in en wordt aangeduid met |0 >. Door met een laser opatomen te schijnen gaan de elektronen sneller bewegen en gaat het atoom naar een hogerenergieniveau |1 >. Dit gebeurt alleen als de laser precies de juiste energie heeft, die cor-respondeert met de juiste frequency. De elektronen moeten namelijk precies genoeg energiehebben (de overgangsenergie) om naar een andere baan te gaan, ze kunnen niet tussen debanen in zitten. De energie van de laser kan zo ingesteld worden dat de elektronen precieseen baan kunnen opschuiven. Het atoom gaat dan van de grondtoestand naar de eersteaangeslagen toestand |1 > (dit is een hoger energieniveau). De elektronen blijven een tijdjein deze andere baan zitten (aangeslagen toestand), maar vervallen op den duur weer terugnaar hun oude baan (de grondtoestand). De tijd die het atoom in de aangeslagen toestandblijft zitten noemen we de levensduur van de toestand. Met behulp van een laser krijgje een verdeling van atomen met een bepaalde kans dat ze in de |0 > of |1 > toestandzitten. Op een gegeven moment kan een atoom zich in een superpositie bevinden van detwee toestanden. Superpositie betekend dat een atoom op enig moment zowel in de eneals in de andere toestand kan zijn. Dit wordt gebruikt voor het verwerken van informatiebij kwantum computers.

In dit experiment willen we atomen tussen twee bepaalde energietoestanden laten varieren.Het energieverschil tussen de twee toestanden is zo klein dat we een laser met een grotegolflengte van meerdere centimeters nodig zouden hebben. Omdat atomen slechts op enkelenanometers van elkaar vandaan zitten heeft een laser met zo’n grote golflengte geen enkeleffect op de atomen. Daarom gebruiken we twee lasers met heel kleine golflengtes, dit cor-respondeert met hoge frequenties (f=v/λ). De twee lasers hebben dus een hoge frequentiemet een klein frequentie verschil, dit verschil moet precies de overgangsenergie/overgangs-frequentie zijn. De atomen worden door de ene laser van de |0 > toestand naar een veelhogere toestand gebracht en worden dan weer door de andere laser van dat hoge niveaunaar de |1 > toestand geduwd. Dit noemen we Raman overgangen, de atomen gaan viaeen omweg naar het juiste energieniveau. Voor dit experiment is het ook belangrijk dathet verschil tussen de lasers stabiel is, maar ze moeten daarbij ook in fase lopen. Dit zorgtervoor dat dit een samenhangend geheel wordt, een coherent systeem. Het is gelukt om detwee lasers een stabiel frequentieverschil te geven, maar niet gelukt om ze in fase te latenlopen. Dat is best lastig omdat de lasers niet precies dezelfde frequentie hebben, maar eenklein verschil (de overgangsenergie). Daardoor verandert het faseverschil de steeds tijd enmoet er constant gecorrigeerd wordt.

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[3] Unknown Author. Saturated absorption spectroscopy, 2010.

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[7] A.M. Marino and C.R. Stroud. Phase-locked laser system for use in atomic coherenceexperiments. Review of Scientific Instruments, 2008.

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[10] C.P. Pearman, C.S. Adams, S.G. Cox, P.F. Griffin, D.A. Smith, and I.G. Hughes.Polarization spectroscopy of a closed atomic transition: applications to laser frequencylocking. Journal of Physics B, 2002.

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[14] J.T.M. Walraven. Atomic Physics: Lecture course. 2008.

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