Raman laser system and radio frequencyexperiments for driving two-photon

transitions in 87Rb

Caspar OckeloenAugust 26, 2008

Abstract

This report presents two approaches for driving two-photon transitions between hyper-fine states in the 87Rb ground state. These hyperfine states are intended for use as qubitstates in quantum information processing with neutral atoms. The first approach uses mi-crowave and radio frequency fields, and experiments on microwave single- and two-photontransitions showing the magnetically dependent Zeeman splitting are presented. The sec-ond approach uses two near infrared laser light fields, and a Raman laser setup using twoexternal cavity diode lasers with relative frequency and phase locking is presented. A phaselock is not yet possible because the combined line width of the lasers (around 6 MHz) iscomparable tot he locking bandwidth of the phase locked loop, but a robust relative fre-quency lock with a large capturing range is achieved. A theoretical analysis of two-photontransitions is also included.

Bachelors Project in physicsUniversiteit van Amsterdam

Contents

Introduction 2

1 Theoretical background 31.1 The theory of Raman transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Atom in a radiation field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Two-level Rabi system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.3 Two-photon Raman transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Diode lasers and Doppler-free spectroscopy 82.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 External cavity diode lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.2 Doppler-free saturated absorption spectroscopy . . . . . . . . . . . . . . . . . . 92.2.3 Locking the lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.1 Laser power characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.2 Doppler-free saturated absorption spectroscopy . . . . . . . . . . . . . . . . . . 12

3 Zeeman level spectroscopy and two photon interactions 143.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2.1 Zeeman level spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.2 Two-photon interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3.1 Zeeman level spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3.2 Two photon interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Raman laser system 224.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1.1 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2.1 Optical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2.2 Electronic setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.3.1 Response times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.3.2 Phase locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Conclusions 29

Populaire samenvatting 30

1

Introduction

Quantum information processing on a magnetic lattice atomchip

Quantum information processing requires an array of qubits, quantum systems with two isolated, long-lived states, which can be manipulated individually and undergo specific interactions between at leasttwo qubits. Many different quantum systems are considered for use as qubits. In [1], a two-dimensionallattice of microtraps, loaded with clouds of ultracold 87Rubidium atoms, on a permanent magneticatom chip is realized. This system can potentially be used for quantum information processing, usingthe F = 1,mF = −1 and F = 2,mF = 1 states of the 87Rb ground state as qubit states.

To drive transitions between these qubit states, a two-photon transition has to be used due to aselection rule. In principle, two microwave fields with a combined frequency of 6.834 682 GHz can beapplied. This, however, corresponds to wavelengths of at least 2 cm, making it difficult to couple thestates in a single microtrap, which is only 20µm away from it neighbor.

Raman laser

An alternative method of driving a transition between the states mentioned above, is by using a two-photon Raman laser pulse. This Raman pulse consists of a light field with two frequencies (say, ω1

and ω2), with a frequency difference of (in this case) 6.834 682 GHz. An atom in the F = 1,mF = −1state can then be driven, via a higher (virtual) state to F = 2,mF = 1, and vice versa.

By choosing the two frequencies close to (but not at) an optical transition, lasers can be used togenerate the Raman pulse and the pulse can be focussed on to a single microtrap.

This report

In this report, both the microwave and Raman laser approach to drive the mentioned qubit transitionare investigated. In chapter 1, a theoretical analysis of the two-level transitions is shown. In chapter2, the diode lasers used for building the Raman laser are discussed, and some basic applicationsare investigated. Chapter 3 presents experiments done to investigate the energy levels of the 87Rbground state, and an experiment where the qubit transition is driven by a microwave frequency andradio frequency field. Finally, chapter 4 discusses a setup for a Raman laser system, and the resultsachieved so far.

2

1 Theoretical background

1.1 The theory of Raman transitions

In this section, a derivation of the behavior of Raman transitions will be given. First, a one-photonabsorption will be described by treating the atom as a two-level system, following [2]. Then it will beshown, following [4], that under certain conditions a two-photon stimulated Raman transition can bedescribed in a way similar to the one-photon absorption.

1.1.1 Atom in a radiation field

Consider an atom with one electron in a radiation field. The time-dependend Schrodinger equation is

ih∂Ψ(~r, t)∂t

= HΨ(~r, t), (1.1)

where ~r is the coordinate of the electron.The total Hamiltonian H of the atom can be written as

H(~r, t) = H0(~r) +H ′(~r, t), (1.2)

where H0 is the time-independent Hamiltonian of the atom with no field present and H ′ is the inter-action with the radiation field.

The functions φn(~r) are the eigenfunctions of H0 and their eigenvalues En are known:

H0φn(~r) = Enφn(~r)En = hωn (1.3)

The solution Ψ(~r, t) of the time-dependent Schrodinger equation, can be expanded in terms of thecomplete set φn(~r) and the time-dependent factors ck(t).

Ψ(~r, t) =∑k

ck(t)φk(~r)e−iωkt. (1.4)

Using this expansion, and noting that 〈φj |φk〉 = δjk and 〈φj |H0|φk〉 = Ejδjk, the Schrodingerequation can be written as

ih∂

∂t

∑k

ckφke−iωkt = (H0 +H ′)

∑k

ckφke−iωkt

∫φ∗j ih

∂

∂t

∑k

ckφke−iωktd3~r =

∫φ∗j (H0 +H ′)

∑k

ckφke−iωktd3~r

ih∂

∂t

∑k

ck〈φj |φk〉e−iωkt =∑k

ck〈φj |H0 +H ′|φk〉e−iωkt

ih∂

∂tcje−iωjt =

∑k

ck (〈φj |H0|φk〉+ 〈φj |H ′|φk〉) e−iωkt

ih

(∂cj∂t− iωjcj

)e−iωjt = cj〈φj |Ho|φj〉e−iωjt +

∑k

ck〈φj |H ′|φk〉e−iωkt

(ih∂cj∂t

+ Ejcj

)e−iωjt = cjEje

−iωjt +∑k

ck〈φj |H ′|φk〉e−iωkt

3

ih∂cj∂t

e−iωjt =∑k

ck〈φj |H ′|φk〉e−iωkt

ih∂cj∂t

=∑k

ckH′jke

i(ωj−ωk)t, (1.5)

where H ′jk(t) ≡ 〈φj |H ′(t)|φk〉.

1.1.2 Two-level Rabi system

In the case of absorption of laser light, the radiation field can be regarded as having a single frequencyωL. We choose this frequency close to a specific atomic resonance between two states, say φ0 and φ1,with frequency ωa ≡ (ω1 − ω0). The laser has a detuning δ ≡ (ωL − ωa). We can then treat the atomas a two-level system, taking only the two states 0 and 1 into account. This truncates the sum in (1.5)to only the first two terms.

We consider the case where H ′ has only the non-diagonal matrix elements H ′01 and H ′10 nonzero.Note that H ′10 = H ′∗01 because H ′ is Hermitian.

Equation (1.5) is now reduced to only the following two equations:

ih∂c0∂t

= c1H′01e−iωat

ih∂c1∂t

= c0H′10e

iωat. (1.6)

The interaction between the electric field ~E(~r, t) and the electric dipole moment of the atom ~d = e~ris given by

H ′(t) = − ~E(~r, t) · ~d = −e ~E(~r, t) · ~r. (1.7)

Using a plane wave traveling in the positive z direction, with amplitude E0 and (unit) polarizationvector ε,

~E(~r, t) = E0ε cos(kz − ωLt), (1.8)

the interaction Hamiltonian becomes

H ′(t) = −eE0r cos(kz − ωLt), (1.9)

where the dipole moment is assumed parallel to the polarization vector.The matrix element H ′10 is

H ′10(t) = 〈ψ1|H ′(t)|ψ0〉

= −eE0

∫ψ1r cos(kz − ωLt)ψ2d

3~r. (1.10)

The wave functions φn(~r) are almost entirely contained within a region much smaller than the opticalwavelength, so we assume ~E(~r, t) to be constant (in space) over this region and take the cosine out ofthe integral. This is known as the electric dipole approximation.

H ′10(t) = −eE0

∫ψ1rψ2d

3r cos(kz − ωLt)

= −eE0〈φ1|r|φ0〉 cos(kz − ωLt)= hΩ cos(kz − ωLt), (1.11)

where Ω is the Rabi frequency, defined by

Ω ≡ −eE0

h〈φ1|r|φ0〉. (1.12)

To solve the two equations (1.6) we make the rotating wave approximation (RWA). This is doneby assuming that the detuning δ ≡ (ωL − ωa) is small compared to ωL + ωa. The RWA can bemade by replacing the cosine in (1.11) with only 1

2ei(kz−ωLt) (see [3] for an argumentation using

4

pertubation theory). The two equations (1.6) can then be uncoupled by differentiating the first oneand substituting, to find

Ω2

4c0 − iδ

∂c0∂t

+∂2c0∂t2

= 0

Ω2

4c1 + iδ

∂c1∂t

+∂2c1∂t2

= 0. (1.13)

With initial conditions c0(0) = 1 and c1(0) = 0 (the atom is initially in the state φ0), the solutionsare (see [2])

c0(t) =

cosΩ′t2− i δ

Ω′sin

Ω′t2

eiδt/2

c1(t) = −i ΩΩ′

sinΩ′t2e−iδt/2, (1.14)

where

Ω′ ≡√

Ω2 + δ2 (1.15)

is the generalized Rabi frequency at detuning δ.To conclude, the probability to find the atom in the state φ0 or φ1 (|c0(t)|2 and |c1(t)|2, respectively)

oscillates at the generalized Rabi frequency Ω′. For zero detuning (δ = 0) the probability |c1(t)|2oscillates between 0 and 1 at Ω′ = Ω. Thus, a light pulse of length π/Ω (known as a π-pulse) will bringthe atom from state φ0 to φ1. As the detuning increases, the generalized Rabi frequency increases, andthe amplitude of the probability-oscillation decreases (measuring the state of the atom after applyinga π-pulse will not always yield φ1). Figure 1.1 shows the probability |c1(t)|2 for several detunings.

0 1 2 30 , 0

0 , 2

0 , 4

0 , 6

0 , 8

1 , 0

δ=2Ω

δ=ΩProba

bility

T i m e (π/Ω)

δ=0

Fig. 1.1: Probability |c1(t)|2 of finding an atom in the state φ1, shown for δ = 0 (zero detuning), δ = Ω andδ = 2Ω.

1.1.3 Two-photon Raman transitions

Now, a three level atom is considered, with levels φ1, φ2 and φ3 as shown in figure 1.2a. Thisconfiguration is known as a three level Λ configuration. Two laser fields are applied. The first laser istuned at ω1L and couples φ1 and φ3 with a detuning ∆ = (ω13−ω1L), where ω13 is the atomic resonancebetween φ1 and φ3. The second laser, at ω2L, couples φ2 and φ3, with a detuning (∆+δ) = (ω23−ω2L),where ω23 is the resonance between φ2 and φ3. There is no direct coupling between φ1 and φ2.

As in the previous section, the Hamiltonian H is written as the sum of an internal HamiltonianH0 and an interaction Hamiltonian H ′. Only H ′ is considered, taking φn (n = 1, 2, 3) as solutions forH0. The light field of the two lasers is written as:

~E(~r, t) =12E1εe

i(k1Lz−ω1Lt) +12E2εe

i(k2Lz−ω2Lt)+c.c. (1.16)

5

a φ3

φ2

φ1

∆

δ

ω1L

ω2L

b

0 2 0 4 00 , 0

0 , 2

0 , 4

0 , 6

0 , 8

1 , 0

δ=0.2Ω1

δ=0.1Ω1

Proba

bility

T i m e

δ=0

Fig. 1.2: (a) Energy levels of the three level Raman system. (b)Probability |c2(t)|2 of finding an atom in thestate φ2, with Ω1 = Ω2 and ∆ = 10Ω1, shown for δ = 0, δ = 0.1Ω1 and δ = 0.2Ω1.

and, again making the RWA, the Hamiltonian becomes:

H ′ =

0 0 hΩ1e−i(k1Lz−ω1Lt)

0 0 hΩ2e−i(k2Lz−ω2Lt)

hΩ∗1ei(k1Lz−ω1Lt) hΩ∗2e

i(k2Lz−ω2Lt) 0

, (1.17)

with the Rabi frequencies

Ωn = −eEn2h〈ψ3|r|ψn〉 (n = 1, 2) (1.18)

The sum in the Schrodinger equation (1.5) is truncated to three terms, corresponding to the threelevels. The remaining equations are:

∂c1∂t

= ic3Ω1e−ik1ze−i∆t

∂c2∂t

= ic3Ω2e−ik2ze−i(∆+δ)t

∂c3∂t

= ic1Ω∗1eik1zei∆t + ic2Ω∗1e

ik2zei(∆+δ)t (1.19)

A large detuning ∆ is required to suppress spontaneous emission. With the conditions

∆ >> |Ω1|, |Ω2|, δ, (1.20)

we can assume that c1 and c2 oscillate slowly compared to ∆. The equation for ∂c3/∂t can then beintegrated directly, ignoring the time dependence in c1 and c2:

c3 = −ic1Ω∗1∆eik1z+i∆t

−ic2Ω∗2

∆ + δeik1z+i(∆+δ)t, (1.21)

and substituted into (1.19), reducing the problem to a two-level problem:

∂c1∂t

= ic1|Ω1|2

∆+ ic2

Ω1Ω∗2∆ + δ

ei(k2−k1)zeiδt

∂c2∂t

= ic1Ω∗1Ω2

∆ei(k1−k2)zeiδt + ic2

|Ω2|2

∆ + δ. (1.22)

Here, the remaining spatial oscillations (ei(k2−k1)z and ei(k1−k2)z) have a large wavelength comparedto the atomic scale, and can be left out.

6

The time dependence of the right hand side of both these equations can be canceled with thesubstitutions (effectively transforming to a rotating frame)

γ1 = c1e−iδt/2

γ2 = c2eiδt/2. (1.23)

The equations (1.22) then become (written in matrix form)

∂

∂t

(γ1

γ2

)= −iHrot

(γ1

γ2

),

where

Hrot =

(− |Ω1|2

∆ + δ2 −Ω1Ω∗2

∆

−Ω∗1Ω2∆ − |Ω2|2

∆ − δ2

). (1.24)

The general solution to this equation is(γ1(t)γ2(t)

)= e−iHrott

(γ1(0)γ2(0)

), (1.25)

or, in terms of c1 and c2(c1(t)c2(t)

)=(eiδt/2 0

0 e−iδt/2

)e−iHrott

(c1(0)c2(0)

). (1.26)

With the initial values c1(0) = 1 and c2(0) = 0, this yields

c1(t) = ei

(|Ω1|

2

∆ +|Ω2|

2

∆ +δ

)t2

(cos

ωt

2+i

ω

(|Ω1|2

∆− |Ω2|2

∆− δ)

sinωt

2

),

c2(t) = ei

(|Ω1|

2

∆ +|Ω2|

2

∆ −δ)

t2

(i

ω

2Ω∗1Ω2

∆sin

ωt

2

), (1.27)

where

ω =

√(|Ω1|2

∆− |Ω2|2

∆− δ)2

+ 4|Ω1|2|Ω2|2

∆2. (1.28)

The frequency ω is now the new Rabi frequency for the two-photon interaction. The first term inbrackets contains the light shifts Ω1|2/∆ and Ω2|2/∆, and the detuning δ. If δ is chosen such that thelight shifts are compensated and the first term is zero, the probability |c2(t)|2 oscillates between 0 and1. Figure 1.2b shows some examples of this probability.

7

2 Diode lasers and Doppler-freespectroscopy

2.1 Introduction

All experiments presented in this report, use diode lasers and common optical components. Thischapter describes the lasers used, a spectroscopy experiment and a setup to electronically lock thelaser frequencies to a known physical reference, in this case a Rubidium transition.

2.2 Experimental setup

2.2.1 External cavity diode lasers

The two lasers used in all experiments, are known as external cavity diode lasers (ECDLs). EachECDL consists of a laser diode, a collimating lens, and a diffraction grating, as shown in figure 2.1.The zero-order diffracted light from the grating is the output beam of the ECDL, but the first orderis aligned to go back into the laser diode. This provides optical feedback to the laser, effectivelyextending the laser cavity up to the diffraction grating. Adjusting the angle of the grating allows thelaser wavelength to be tuned.

The diffraction grating is mounted on a piezoelectric transducer (PZT), which allows for electricaltuning of the wavelength, by changing the length of the external cavity.

The laser diode and grating setup is mounted on a Peltier element, which in turn is mountedon a large metal base. The Peltier element is used to stabilize the temperature of the diode. Thetemperature is measured with an AD590 temperature sensor, and the Peltier element is controlled bya Thorlabs TED200 temperature controller. In all experiments, the temperature was set to 17.5 C.

The laser diodes used are of unknown make and type, but were specified to run at a wavelength of790 nm and provide up to 100 mW of optical power. The diodes are controlled by a Thorlabs LDC202current controller, running in constant current mode. Laser diodes require a threshold current Ith tostart lasing, and have a linear current/power characteristic for currents above Ith.

When the grating is properly aligned, the threshold current is lowered, because around threshold,more photons are in the laser diode. This property can be used to align the grating. The laser diodecurrent is modulated around threshold, and the intensity of the laser is measured with an optical power

12

34

Fig. 2.1: Top view of the external cavity diode laser, showing (1) laser diode, (2) collimating lens, (3) diffractiongrating, and (4) piezoelectric transducer.

8

meter (with analog output) or a photodiode, and shown on an oscilloscope. The modulation showsthe threshold in the time domain of the oscilloscope, and the grating alignment can be optimized forlow threshold current. This procedure was used to align the lasers, and reach stable lasing modes ata wavelength of 795 nm, corresponding to the Rubidium D1 line.

The current/power characteristic of both laser diodes was measured with an optical power meter,and the wavelength with a wavemeter. These measurements were done on the free running diodes(with only the diode and collimating lens), and on the ECDL setup with diffraction grating, alignedto have the lasers run at a wavelength of 795 nm.

To verify the wavelength measured with the wavemeter, the laser frequencies were modulated withthe PZT, and directed through a rubidium vapor cell. The grating angle was then adjusted untilfluorescence was observed from the vapor cell with an infrared viewer, indicating the laser frequencywas at an atomic transition.

2.2.2 Doppler-free saturated absorption spectroscopy

To measure and control the frequency of a laser, it is useful to have a known physical reference. Inthis experiment, the lasers are used to record a spectrum of the D1 line of rubidium gas (a naturalmixture of the isotopes 85Rb and 87Rb), to investigate the scan range (how far the laser frequency canbe modulated without mode hops) of the lasers, and confirm the measured wavelength.

The spectroscopy setup is shown in figure 2.2. The technique used is called saturated absorptionspectroscopy, and will be discussed here.

Absorption spectroscopy

A simple spectroscopy setup would be to pass a laser beam through a vapor cell, and record theintensity of the transmitted laser light with a photodiode. When the laser frequency is modulated,laser light will be absorbed by the vapor when it is at (or near) an atomic transition, decreasing theamount of transmitted light. The photodiode signal, when displayed on an oscilloscope, will then showdips at the atomic transition frequencies, thus creating a spectrum.

In this kind of experiment, the width of the spectral lines (dips) is determined dominantly byDoppler broadening [5]. Doppler broadening occurs because the atoms in the vapor cell have a randomvelocity, and ”see” the laser light Doppler shifted depending on their velocity parallel to the laser beam.Atoms moving towards the laser light, ”see” a higher frequency light, and therefore absorb light whenthe laser frequency is lower than the atomic transition. Conversely, atoms moving away from the laserlight, absorb light when the laser frequency is higher than the atomic transition. The result is linebroadening, creating lines with a Gaussian shape known as the Doppler profile, as shown in 2.3a.

Saturated absorption spectroscopy

Saturated absorption offers a way to eliminate the Doppler broadening [5]. Instead of one laser beam,two counter-propagating beams, from the same laser, are sent through the vapor cell. In practice thereis a small angle between the beams, as shown in figure 2.2. One of the two beams is much strongerthan the other, and called the pump beam. The intensity of the weak beam, called the probe beam,is again measured with a photodiode.

Rb cell

Laser

PBS λ/2

PD

Fig. 2.2: Setup used for saturated absorption spectroscopy. PBS = polarizing beam splitter cube, λ/2 = halfwave plate, PD = Photodiode.

9

a b

ω

Fig. 2.3: Schematic of probe laser light transmitted through a Rb vapor cell, as the laser frequency ω scansacross a transition, (a) without saturation and (b) with saturating pump beam.

When the laser frequency is scanned across an atomic transition, the photodiode signal again showsthe Doppler profile. Atoms moving towards the probe beam still absorb probe beam light when thelaser frequency is below the transition. These atoms move away from the pump beam, absorbingpump beam light when the laser frequency is above the transition (note, however, that the pumpbeam transmitted light is not monitored). So, these atoms are either resonant with the probe beam orwith the pump beam, but not with both at the same time. The same (but opposite) argument holdsfor atoms moving away from the probe and towards the pump.

However, atoms with zero velocity parallel to the beams, see the same frequency in both pumpand probe beams. When the laser frequency (and therefore both beams) are resonant with these zero-velocity atoms, the atoms can absorb light from either beam. Since the pump beam is much strongerthan the probe beam, more atoms will absorb pump light than probe light. The pump beam (partly)saturates the transition, creating a transparency for the probe beam. The recorded spectrum thereforeshows the Doppler profile, as stated, but with a narrower peak in it, as shown in figure 2.3b. Thus, amuch smaller line width is achieved, showing the so-called sub-Doppler structure of the spectrum.

The saturated absorption spectroscopy experiment is set up as shown in figure 2.2. A polarizingbeam splitter cube allows some light through, and reflects some at a 90 angle, based on the polarizationof the incoming light. Light polarized horizontally (figure 2.2 is a top view) goes through, light polarizedvertically reflects. A half wave plate is used to adjust the linear polarization of the laser light, creatinga strong pump beam and a weak probe beam. The probe beam is further attenuated with a neutraldensity filter, before entering the vapor cell. A diaphragm is used before the photodiode, to avoidreflected light from the pump beam from hitting the photo diode.

Scan range

The laser frequency can be scanned by applying a voltage across the PZT, and thereby changing thelength of the external cavity. However, when changing the length too far, a different lasing mode canbecome more efficient, in which case the laser will jump to that mode. This is called a mode hop.When scanning the lasers with the PZT only, a scan range (the frequency range that can be scannedwithout encountering mode hops) in the order of 1 GHz was observed.

The current supplied to the diode influences which lasing mode is preferable, and adjusting thelaser current while scanning the PZT can greatly improve the scan range. In general, if the PZT isset to higher frequencies (by shortening the external cavity), a lower diode current is required to avoidmode hops. To achieve this while scanning, an inverting amplifier was built, as shown in figure 2.4.

The LDC202 diode current controller allows modulation if supplied with a 0-10V modulation signal.The amplifier has to invert the scanning signal, amplify (or attenuate) it by an adjustable factor andadd a positive constant voltage to it. The amplifier built consist of a series of two inverting amplifiers,

-+

10k

10k

-+

22k

0-22k

-+

22k

22k

22k0-22k

Vin

-

Vin

+

Vout

-15 V

Fig. 2.4: Amplifier used for modulating the diode current while scanning the PZT. The amplifier consists ofthree stages. First, the signal is inverted. Then, the signal is inverted again with adjustable gain between 0and 1. Finally, an adjustable constant (negative) voltage is added and the resulted signal is inverted again.The first inverting stage can be skipped, providing a non-inverting input.

10

the second one of which has an adjustable gain between 0 and 1, and an inverting summing amplifier,adding an adjustable voltage.

Using this amplifier to scan both PZT and diode current simultaneously, the whole Rb D1 linecould be scanned with both lasers, creating a scan range of 9 GHz. See section 2.3.2 for the resultingspectra.

2.2.3 Locking the lasers

The spectroscopy setup described above, was used to lock the lasers frequency to a Rb atomic transi-tion, using home built feedback electronics. The feedback loop is described in this section.

A block diagram of the feedback electronics is shown in figure 2.5. The photodiode signal from thespectroscopy setup (figure 2.2) is added to a negative offset, chosen such that the slope of a Dopplerprofile or a sub-Doppler peak crosses 0 Volt. The frequency where this 0 Volt crossing occurs, willbe the locking frequency. The signal is then fed into a pre gain amplifier, which can be used for fastcurrent feedback (not used in this experiment), and into the main amplifier, after optionally beingattenuated.

The output of the amplifier is integrated, and sent to a high Voltage amplifier (not shown) to drivethe 150 V piezoelectric transducer.

Photodiode Input (“Lock IN”)

Attenuator 0 / 20 / 40 dB

Amplifier

“Lock in amplifier”

Gain adjust “Gain out”

Oscilloscope

Integrator

Integrated lock output (“Lock out”)

Piezo driver

“Pre gain out”

Optional current feedback

“Locking / Dither” switch

Offset

Scan controls (“Dither”)

Offset Level

“Dither in”

Function generator

Fig. 2.5: Block diagram of the feedback electronics used to lock the lasers. The spectroscopy photodiode signalis added to a negative offset, amplified and integrated to provide an error signal for the grating PZT. The scancontrols are for finding the desired atomic transition. For reference, the names used on the actual device aregiven in quotation marks.

Procedure to lock a laser

To lock a laser, the Locking/Dither switch is first set to Dither. As dither input a triangle wave issupplied, and the photodiode signal (from Gain out) is viewed on an oscilloscope (triggered on thedither input signal). This shows the spectrum, and by adjusting the dither offset and level, one canzoom in to the desired transition to lock to. Once only one transition is visible, and the locking offsetis adjusted for the desired lock frequency, the Locking/Dither switch is set to Locking, and the laserwill be locked.

11

When the laser is locked, the oscilloscope mentioned shows the feedback signal. The lock inamplifier gain can then be adjusted to make sure the laser frequency doesn’t drift away (too low gain)or oscillate (too high gain).

2.3 Results

2.3.1 Laser power characteristics

The optical output power of both diodes is shown in figure 2.6, both free running (without feedback,just the diode and collimating lens) and with grating feedback (aligned to achieve lowest thresholdcurrent and run at the Rubidium D1 line (795 nm)).

On both lasers, the threshold current when the diode is free running is about 25 mA. The feedbacklowers the threshold current to 18 mA on both lasers. The slope of power/current characteristic islowered by the feedback. When free running, both diodes output more then 95mW at a current of120 mA. In laser 2, the feedback reduces the maximum power to 68 mW. In laser 1 (with feedback),the power/current characteristic becomes nonlinear for currents above 90 mA, and the highest outputpower achieved is 44 mW. This nonlinearity is likely due to problems with the feedback, and canprobably be solved by changing the alignment. For the experiments described in this report, however,the achieved power is high enough.

0 4 0 8 0 1 2 00

4 0

8 0

L a s e r 1 f r e e r u n n i n gL a s e r 2 f r e e r u n n i n gL a s e r 1 w i t h f e e d b a c kL a s e r 2 w i t h f e e d b a c k

Outpu

tpow

er(m

W)

C u r r e n t ( m A )

Fig. 2.6: Measured optical output power of both laser diodes as function of the current through the diode.Open symbols represent the power without optical feedback (”free running”), closed symbols represent thepower with grating feedback.

2.3.2 Doppler-free saturated absorption spectroscopy

Figure 2.7 shows the D1 line spectra recorded with both lasers, by scanning the Piezo and laser currentas described in section 2.2.2 The sub Doppler peaks were identified by comparing (uncalibrated)frequency differences between different peaks with hyperfine level energies from [6]. The frequencyscale was calibrated (for each laser separately) by fitting the four 87Rb peaks to the literature data.Zero on the frequency scale corresponds to the D1 line center as used in the cited literature. Table 2.1shows the identification and relative frequencies of the peaks recorded with laser 2.

Cross-over peaks

Peaks 4 and 7 in figure 2.3 do not correspond to an atomic resonance, but have frequencies exactly inbetween their two neighboring peaks. These are known as cross-over peaks, and are a side effect of

12

saturated absorption spectroscopy.When the laser frequency is at a cross-over peak, say peak 4 in the figure, there is a velocity group

moving towards the pump beam for which the pump beam is Doppler shifted to be at resonance witha higher frequency transition (peak 5 in this example). So, this velocity group is saturated by thepump beam. However, this same velocity group is moving away from the probe beam at such velocitythat the probe beam is Doppler shifted to be resonant with a lower frequency transition (peak 3 inthis example). Since this velocity group is saturated, the probe beam is not (or less) absorbed by itand the cross-over peak appears in the spectrum.

The same argument holds for a velocity group moving away from the pump beam, making thecross-over peak stronger.

- 4 0 0 0 - 2 0 0 0 0 2 0 0 0 4 0 0 0

L a s e r 2

Trans

missi

on(ar

bitrar

yunit

s)

D e t u n i n g ( M H z )

L a s e r 1

12

3 4

5

67 8

91 0

Fig. 2.7: Spectrum of the Rubidium D1 line, recorded with both lasers. The horizontal scale was adjusted byfitting the 87Rb sub Doppler peaks to data from [6]. The two spectra have different and shifted vertical scales.Numbers in the bottom spectrum refer to table 2.1.

Peak Transition Measured frequency Literature1 87Rb F = 2→ F′ = 1 -3100 MHz -3073 MHz2 87Rb F = 2→ F′ = 2 -2229 MHz -2257 MHz3 85Rb F = 2→ F′ = 1 -1481 MHz4 Crossover peak between peak 3 and 5 -1298 MHz5 85Rb F = 2→ F′ = 2 -1103 MHz6 85Rb F = 1→ F′ = 1 not measured7 Crossover peak between peak 6 and 8 -1755 MHz8 85Rb F = 1→ F′ = 2 1950 MHz9 87Rb F = 1→ F′ = 1 3785 MHz 3761 MHz10 87Rb F = 1→ F′ = 2 4554 MHz 4578 MHz

Table 2.1: Measured frequencies of the observed transitions, compared to literature. F and F’ refer tohyperfine sublevels in the 52S1/2 and 52P1/2 levels, respectively. Peak numbers and data refer to theright part of figure 2.6. Literature values from [6].

13

3 Zeeman level spectroscopy and twophoton interactions

3.1 Introduction

In this chapter, several experiments involving the Zeeman sub levels of the ground state of Rubidiumare presented.

3.2 Experimental setup

3.2.1 Zeeman level spectroscopy

The ground state (52S1/2) of 87Rubidium is split into two hyperfine levels, F=1 and F=2, with anenergy difference corresponding to 6.8347 GHz. The F=1 level is split into three levels (mF = −1, 0, 1)under the influence of a magnetic field with a Zeeman splitting of (to first order) -0.70 MHz/G, andthe F=2 level is split into five levels (mF = −2,−1, 0, 1, 2) with Zeeman splitting +0.70 MHz/G [6].For a single-photon transition, mF is only allowed to change by ∆mF = −1, 0, 1, as shown in figure3.1b.

In the experiment described in this section, a spectrum of the allowed transitions between theZeeman sub levels of the ground state hyperfine levels is recorded. The setup is shown in figure 3.2.At the heart of the experiment is a vapor cell containing Rubidium gas and buffer gas. To raise thevapor pressure, the cell is heated using a 100W halogen lamp.

Laser light (carried through an optical fiber) is shone through the vapor cell and focussed on aphotodiode. The laser frequency is locked to the (52S1/2,F = 2→ 52P1/2,F = 1) transition, using the

Ener

gy

Ener

gy

52S1/2

52P1/2

F = 1

F = 2

6.8 GHz

F = 1

F = 2

6.8 GHz

795 nm

mF= -1 0 1

mF= -2 -1 0 1 2

a b

Fig. 3.1: Energy levels of 87Rb. (a) Ground state hyperfine splitting and first excited state. Laser light (solidarrow) pumps atoms from F=2 into the excited state. Decay (dashed arrows) allows atoms to go back toF=1 or F=2. Circles on a level represent atoms in that state. (b) Zeeman sub levels of the ground statehyperfine levels. Lines represent allowed single-photon transitions. The dashed and dotted lines are two pairsof degenerate transitions.

14

PD

F L L FC

1

2

2

3 3

Rb cell

Fig. 3.2: Setup for spectroscopy of the Zeeman sub levels, showing (1) microwave coupler, (2) static magneticfield coils and (3) radio frequency coils. FC = fiber coupler, L = lens, F = color filter, PD = photodiode.

setup described in section 2.2.3. In the vapor cell, atoms in the ground state F=2 level, are pumped bythe laser to the excited state. The atoms can decay back to either the F=1 or F=2 ground state level,but atoms ending up in F=2 are quickly pumped back into the excited state. Effectively, therefore,the laser pumps atoms from the F=2 into the F=1 ground state level.

A microwave source generates microwaves at 6.8347 GHz. These are amplified to about 1W maxi-mum, and coupled to the vapor cell with a microwave coupler. The microwave frequency is modulatedby a triangle signal over a bandwidth of 10MHz. When the microwave frequency corresponds to atransition energy, the microwaves couple the F=1 and F=2 ground state levels. This stimulates atomsfrom the F=1 state to the F=2 state, repopulating the F=2 state. The pump laser is then used todetect this: the atoms in F=2 absorb laser light while being pumped to F=1 once again, resultingin less laser light transmission and a lower signal at the photodiode. So, the photodiode signal is ameasure for wether the microwaves are resonant with a transition.

To study the magnetic field dependence of the setup, coils are added to produce a static magneticfield in de vapor cell

As mentioned before, the (first order) Zeeman splitting of the F=1 ground state level is -0.70MHz/G, and the splitting of the F=2 level +0.70 MHz/G. Because of this, the (F = 1,mF = −1)→(F = 2,mF = 0) transition has the same energy as the (F = 1,mF = 0) → (F = 2,mF = −1)transition. The pair (F = 1,mF = 0) → (F = 2,mF = 1) and (F = 1,mF = 1) → (F = 2,mF = 0)is also degenerate. The two degenerate pairs are shown in figure 3.1b as dashed and dotted lines,respectively.

Taking these degeneracies into account, and the fact that there are 9 allowed transitions, 7 peakscan be expected in the spectrum. The center peak, (F = 1,mF = 0)→ (F = 2,mF = 0), is expectedto be independent of magnetic fields. .

3.2.2 Two-photon interactions

In the following experiment, two radio frequency coils are added to the experiment with 10 windingseach, connected in series, as shown in figure 3.2. The coils are used to apply a radio frequency (RF)field, parallel to the laser field. The frequency of the RF field is varied from 100 to 1500 kHz.

The radio frequency field, when used in conjunction with the microwave (MW) field, allow fortwo-photon transitions between Zeeman sub levels of the F=1 and the F=2 ground state hyperfinelevels. Because 2 photons are involved, the quantum number mF can change by up to two (instead ofone).

Figure 3.3 shows the Zeeman sub levels, as before, but now with a virtual level created by the RFfield. The effect of the RF field can be thought of as the addition of such virtual levels above andbelow every real level. In a two-photon interaction, the MW field couples the initial state to one of thevirtual levels, and the RF field couples the virtual level to the final state. As an example, the figureshows a possible transition from mF = −1 to mF = 1.

A spectrum is recorded by choosing a specific RF frequency and scanning the MW frequency.With two-photon transitions, transitions with the same MW frequencies as in the previous (1 photon)experiment can be expected, as well as transitions at exactly the RF frequency higher and lower. Thespectrum should thus be as before, but with two sidebands (separated by the RF frequency) on everypeak.

15

Ener

gy

F = 1

F = 2

mF= -1 0 1

mF= -2 -1 0 1 2

Mic

row

ave

Radio frequency

Fig. 3.3: Zeeman sub levels of the hyperfine ground state levels of 87Rb with an example of a two-photoninteraction. The dashed line represents a virtual level.

3.3 Results

3.3.1 Zeeman level spectroscopy

Figure 3.4a shows a spectrum recorded with no applied magnetic field. The splitting observed iscaused by the random magnetic field present in the laboratory. The splitting between the differenttransitions is 0.48 MHz, corresponding to a magnetic field of 0.7 Gauss (the earth magnetic field inthe Netherlands is about 0.5 Gauss).

As expected, the spectrum shows 7 equally spaced peaks. Comparing to figure 3.1b, the peaks canbe identified as (from left to right):

1 F = 1,mF = 1 → F = 2,mF = 22 F = 1,mF = 1 → F = 2,mF = 13 F = 1,mF = 1 → F = 2,mF = 0 and F = 1,mF = 0 → F = 2,mF = 14 F = 1,mF = 0 → F = 2,mF = 05 F = 1,mF = 0 → F = 2,mF = −1 and F = 1,mF = −1 → F = 2,mF = 06 F = 1,mF = −1 → F = 2,mF = −17 F = 1,mF = −1 → F = 2,mF = −2

The F = 1,mF = 0 → F = 2,mF = 0 transition (”0-0 transition”) is commonly used as frequencyreference in commercial atomic clocks. To achieve high frequency stability, the line width of thistransition has to be as small as possible. By carefully selecting the buffer gas pressure and adjustingother parameters, a FWHM line width in the order of 100 Hz can be achieved [8].

The line width in figure 3.4a is about 100 kHz. By adjusting the vapor cell temperature (andtherefore gas pressure) and the microwave power, to reduce pressure broadening, the line width wasdecreased to about 20kHz, as shown in figure 3.4b. The microwave frequency scan speed was alsoreduced, to reduce the asymmetry visible in both spectra. In figure 3.4b, there is still a visibleasymmetry, significantly influencing the line width. The origin of this asymmetry is unknown.

Magnetic field dependence

Spectra were recorded at various applied magnetic fields, in a direction perpendicular to the laserbeam. Multiple gaussian curves were fitted to each spectrum, to obtain the transition frequencies asfunction of magnetic field. The result is shown in figure 3.5. Error bars in the figure represent thepeak width of each peak.

16

a

- 2 , 0 - 1 , 5 - 1 , 0 - 0 , 5 0 , 0 0 , 5 1 , 0 1 , 5 2 , 00 , 9 0

0 , 9 2

0 , 9 4

0 , 9 6

0 , 9 8

1 , 0 0

Relat

ivetra

nsmi

ssion

R e l a t i v e f r e q u e n c y ( M H z )

b

- 1 5 0 - 1 0 0 - 5 0 0 5 0 1 0 0 1 5 0- 0 , 4

- 0 , 3

- 0 , 2

- 0 , 1

0 , 0

Trans

missi

on

R e l a t i v e f r e q u e n c y ( k H z )

Fig. 3.4: Microwave spectra of the 87Rb ground state hyperfine sub levels. (a) Spectrum showing all sevenresonances (horizontal scale offset at an arbitrary frequency). (b) Narrowest line width (about 20 kHz FWHM)observed on the F = 1,mF = 0→ F = 2,mF = 0 transition. The microwave frequency scales are relative to thecenter transition.

The magnetic field scale was not known or measured, but calculated from fitting the data points,assuming the field to be weak enough to consider only linear Zeeman shifting. In the fit, an arbitrary(in strength an direction) laboratory field was taken as a fit parameter. The fit function used, shownas lines in the figure, was

f = n√

(sI + ox)2 + o2yz,

where f is the transition frequency (MHz), n = (−3,−2, · · · , 3) determines which transition isconsidered, s is the slope of the linear Zeeman splitting (in MHz per Ampere of current through thecoils), I the current (A), ox is the offset (MHz) accounting for an arbitrary field present parallel tothe direction of the applied field, and oyz is the offset (MHz) accounting for an arbitrary field in

the directions perpendicular to the direction of the applied field.√o2x + o2

yz/(0.70MHz/G) yields alaboratory field of 0.72 Gauss.

The residuals shown in 3.5 get larger with larger field strength. This does not imply that theassumption that the measurement stays in the linear Zeeman splitting regime is unjustified, since thepeak width also gets larger at larger field strength. One explanation for this is that the magneticfield is not homogeneous in the entire vapour cell. Furthermore, the parameters ox and oyz in thefit function are only important for low field strength. So, at low field strength there are more fittingparameters and a better fit can be expected.

Magnetometry

Experiments similar to the experiment described here, are developed for use as magnetometer. Oneexample is given in [9]. This experiment can also be used as magnetometer. Measuring the position ofany two peaks in the recorded spectra can be used to measure the magnetic field. Given the measuredpeak width of 20 kHz, and measuring the outer two peaks, would yield a sensitivity of 0.01 Gauss.This sensitivity can be improved, because all seven peaks can be measured, and the peak positionscan be determined more accurately than the peak width.

3.3.2 Two photon interactions

Figure 3.6 shows spectra recorded with an RF field present while scanning the MW frequency. Theside bands on every peak are visible as expected. A static magnetic field is applied to clearly separatethe side bands from adjacent peaks.

It was noticed that, for some values of the magnetic field and RF frequency, even more sidebands(separated from the peak by 2 or 3 times the RF frequency) can be seen. Figure 3.7 shows a clear

17

- 1 5 - 1 0 - 5 0 5 1 0 1 5

- 3 0

- 2 0

- 1 0

0

1 0

2 0

3 0

0 , 0

0 , 2

0 , 4

0 , 6

Relat

ivefre

quen

cy(M

Hz)

A p p l i e d m a g n e t i c f i e l d ( G )

Fig. 3.5: Bottom: magnetic field dependence of the Zeeman splitting. Lines represent a fitted function. Errorbars represent individual peak widths. Top: residuals.

3

18

- 5 0 50 , 8 8

0 , 9 2

0 , 9 6

1 , 0 0

Relat

ivetra

nsmi

ssion

R e l a t i v e m i c r o w a v e f r e q u e n c y ( M H z )

5 0 0 k H z

2 . 3 M H z

Fig. 3.6: Spectrum with both microwave (MW) and radio frequency (RF) fields applied. The MW frequencyis scanned and shown on the horizontal axis. RF=500kHz, and a magnetic field of 3.3 G is applied. Theexpected side bands are visible.

- 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5

0 , 9 6

0 , 9 8

1 , 0 0 1 . 3 M H z1 . 3 M H z

Relat

ivetra

nsmi

ssion

R e l a t i v e m i c r o w a v e f r e q u e n c y ( M H z )

1 . 3 M H z1 . 3 M H z

3 . 1 M H z

Fig. 3.7: Spectrum with RF=1.3MHz and a magnetic field of 4.4 G, showing 3 pairs of side bands on eachpeak. This indicates 2-, 3- and 4-photon transitions.

example of this, with the magnetic field at 3.3G and the RF frequency at 1.3MHz. This can beinterpreted as 3-photon and even 4-photon processes, involving one MW photon and 2 or 3 RF photons.The more photons involved in the transition, the lower one expects the probability of that transitionoccurring. Indeed, the side bands attributed to 3-photon processes are smaller than the first sidebands, and the 4-photon side bands are yet smaller.

Figure 3.8 shows peak positions (found by taking the locally lowest data point in the recordedspectrum) from spectra recorded at different RF frequencies. The magnetic field strength was 3.3G. Lines representing the expected positions for 1-, 2- and 3-photon processes are drawn, making itpossible to attribute most of the peaks to these processes. 4-photon processes, and possibly morephoton processes, should account for the remaining peaks.

19

Magnetic field dependence

Figure 3.9 shows peak positions from spectra recorded with different magnetic field strength. Peaksfrom 3- and 4-photon transitions were omitted. The upper and lower blue line in the figure, which canboth be associated with the F = 1,mF = −1→ F = 2,mF = 1 and F = 1,mF = 1→ F = 2,mF = −1two-photon transitions, are independent of magnetic field.

0 5 0 0 1 0 0 0 1 5 0 0

2

4

6

8

1 0

1 2

1 4

1 6

1 8

Relat

ivemi

crowa

vefre

quen

cy(M

Hz)

R a d i o f r e q u e n c y ( K H z )

Fig. 3.8: Peak positions from spectra with different RF frequencies at a fixed magnetic field of 3 G. Diamondsrepresent all detected peaks. Lines represent expected values for up to three-photon processes.

20

1 , 6 1 , 8 2 , 0 2 , 2 2 , 4 2 , 6 2 , 8 3 , 02

4

6

8

1 0

1 2

1 4

1 6

1 8

Relat

ivemi

crowa

vefre

quen

cy(M

Hz)

M a g n e t i c f i e l d ( G )

Fig. 3.9: Peak positions from spectra with different magnetic field strength, with a fixed RF frequency of600kHz. Thick lines represent one-photon transitions, thin lines represent 2-photon transitions. Higher ordertransitions were omitted.

21

4 Raman laser system

4.1 Introduction

As shown in section 1.1, two-photon transitions can be used to drive an atom from one level to anothervia a third, intermediate level. In the previous chapter, experiments with two-photon transitions usingradio frequency and microwave photons were shown. In this chapter, a setup for a Raman laser systemwill be discussed. The Raman laser system is intended to drive 87Rb atoms from the F = 1,mF = −1to the F = 2,mF = 1 ground state (5S1/2) levels and back, using an intermediate level in the firstexcited state (5P1/2). Figure 4.1 shows the levels and corresponding frequencies.

Ener

gy

52S1/2

52P1/2

F = 1, mF = -1

F' = 1F' = 2

F = 2, mF = 1

6.8 GHz

795 nm ω2Lω1L

Fig. 4.1: Partial level scheme of the 87Rb, showing the levels used in the Raman transition. ω1L and ω2L arethe desired frequencies of laser 1 and laser 2. ωRaman = ω1L − ω2L.

4.1.1 Requirements

To drive the aforementioned Raman transition, two laser light fields with wavelengths of 795nm,corresponding to the D1 transition, must be applied, with a frequency difference of 6.834 682 610GHz, plus or minus a chosen detuning δ. The qubit states are observed to have coherence timesexceeding 1 second in a microchip trap [10], so the frequency difference must be stable to at least 1Hz: the phase of a mixed signal from the to light fields (for instance, by generating a beat note, seebelow) must remain in phase with a reference signal for at least one second. Some fluctuation of therelative phase between beat note and reference, on a shorter timescale, is allowed.

Independent control of the two laser light fields, for example by adjusting the intensity and/orfrequency of one of the two using an acousto-optic modulator (AOM), is desired.

4.2 Experimental setup

There are several possible approaches to achieve the required laser light fields. Some of these relyon deriving the two fields directly from one laser, including directly modulating the laser current[11] or modulating the laser beam with an electro-optic modulator (EOM) or severel AOMs [12].These methods have limitations in the difference frequency that can be achieved and in tunability andindependent control of the beams [13].

22

Rb cell

Laser 1

Laser 2

FPD

FC

PD

FM

λ/2

A

A

ND

λ/2 GPOI

λ/2

λ/2

PBS

RPBS

GPOI

PBS

Fig. 4.2: Optical part of the Raman laser setup. λ/2 = half wave plate, OI = optical isolator, GP = glassplate, PBS = polarizing beam splitter cube, RPBS = PBS rotated 45, FM = flick mirror, ND = neutraldensity filter, A = aperture, PD = photodiode, FPD = fast photodiode, FC = fiber coupler.

In this setup, we use two independent lasers in a phase locked-loop (PLL). The basic principleof this is as follows. The two lasers, laser 1 and laser 2, are combined on a beam splitter cube andshone on a fast photodiode. This creates an interference signal, or beat note, on the photodiode,resulting in an electrical signal at exactly the difference frequency of the two lasers. This signal(beat signal) is compared to a reference signal by phase lock electronics. The phase lock electronicsprovide an error signal to feed back to laser 2. If the beat and reference frequency are identical, theerror signal should be proportional to the relative phase φ between beat and error signal, to keep therelative phase constant (usually 0) and thereby actually phase lock the two signals. If the beat andreference frequency are different, the error signal should be proportional, or at least have the sign of,the relative frequency between beat and reference signal. In this setup, laser 1 can be either locked tosome reference (for example an atomic transition, as in section 2.2.3), or left unlocked if the absolutefrequency is not important.

4.2.1 Optical setup

Figure 4.2 shows the optical setup used in this experiment. The two lasers each go through an opticalisolator. Then, a fraction (about 10%) of each beam is split off by a glass plate, to go to a saturatedabsorption spectroscopy setup (see chapter 2). Only one spectroscopy setup is used, and a flick mirroris used to choose which laser goes to the spectroscopy. The spectroscopy is not used to lock any of thelasers, but only to provide a reference when adjusting both lasers and setting their initial frequencies.

The laser light that passes through the glass plates, is mixed on a polarizing beam splitter cube(PBS). Half wave plates are used to adjust the fraction of light going to the fast photodiode (FPD).This is about 3 mW in total, 1.5 mW from either laser. The light out of the PBS had two orthogonalpolarizations. To create an interference signal, another PBS, but rotated 45, is used, discarding 50%of the signal and letting 50% through to the photodiode. The FPD is a New Focus 1577-A 12GHzphotoreceiver.

23

MixerZX05-14+

LO

RF IF

MW reference IF reference

Beat noteinput

Phase lock electronics

Laser 2Piezo driver

SlowLPF

FastLPF

Laser 2Current mod

AD9901

REF

VCO

OUT

Fig. 4.3: Block diagram of the electronic part of the phase-locked loop. LPF = low pass filter.

4.2.2 Electronic setup

Figure 4.3 shows a block diagram of the electronic part of the phase-locked loop. The signal from thefast photodiode is first downmixed to an intermediate frequency, and then compared to a reference byphase lock electronics. This results in an error signal, that is fed back to laser 2.

Down mixing

The beat note frequency from the FPD, νbeat, should then be around 6.8 GHz, too fast for directusage in our phase lock electronics. Therefore, it is first downmixed with a microwave reference signal,frequency νMWref , on a Mini-Circuits ZX05-14+ frequency mixer. The intermediate frequency (IF)of the mixer is νIF = |νbeat − νMWref |. It is impossible to distinguish between νbeat > νMWref andνbeat < νMWref , by looking at νIF . To overcome this, νMWref is chosen detuned from the desiredlaser difference frequency νRaman, by an amount νIFref between 5 and 80 MHz. To phase lock thetwo lasers, the phase lock electronics now should compare the mixer IF to a supplied second referenceat νIFref .

Phase lock electronics

The phase lock electronics used in this setup is based on the analog devices AD9901 integrated circuit.This is a digital phase/frequency discriminator, featuring a linear error signal in a relative phase rangeφ = 0 · · · 2π, phase locking at φ = π (but inverting one of the input signals results in lock at φ = 0).

Figure 4.4 shows a block diagram of the AD9901, and an example of the expected output signalwhen presented with two digital input signals (in this case, the signal, or (voltage controlled) oscillator,is faster than the reference). This example is without any feedback, just to show the operation of theAD9901.

The first stage in the AD9901 consists of a flip-flop for each input, configured to invert its outputon every rising edge of the input signal. This divides the input frequencies by two, and gives a cleansignal regardless of the duty cycles of the input signals. The outputs from both flip-flops are mixedon a XOR gate, to produce the unfiltered phase lock error signal, as shown from point 1 to point 4 infigure 4.4b. At point 1, φ is close to zero and the output of the XOR gate is zero most of the time.Low-pass-filtering this signal, would result in a signal near the logical 0 voltage. However, since theSignal is faster than the Reference, φ will increase in time. At point 2, φ = π, and the output of theXOR gate is half the time low and half the time high. After low-pass-filtering, this would create anerror signal halfway between the logical 0 and logical 1 voltages. At point 3, φ is getting closer to 2π,and the filtered error signal would be close to the logical 1 voltage. The dashed line in figure 4.4bshows the filtered output signal, behaving linearly over this described range.

The second stage shown in figure 4.4a is the frequency discriminator, and comes into action at point4 in figure 4.4b. At this point, the Reference has a phase lead of φ = 2π, and the logic described abovewould yield an incorrect error signal. The two frequency discriminator flip-flops and NAND/ANDgates suppress the XOR gate output, and maintain a constant high output signal. At point 5, φ = 4π,and the AD9901 goes back into phase detection mode. However, as the AD9901 cycles between phasedetection and frequency detection modes, the error signal is longer high than low resulting in a suitablesignal for slow (Piezo) feedback to correct the frequency.

24

a

b

5 4 3 2

Err

or S

igna

l

Time 1

Sig

nal

Flip

Flo

p

Ref

eren

ceFl

ip F

lop

Sig

nal

Ref

eren

ce

Fig. 4.4: Analog Devices AD9901 digital phase/frequency discriminator. (a) Block diagram. (b) Sample inputand output signals.

25

In the electronics used in this experiment, as shown in figure 4.3, the output of the AD9901 isused for two outputs, a slow and a fast output. The slow output is designed for frequency locking,using Piezo feedback, and the fast output (up to several tens of MHz) is designed for phase lockingthe lasers, using current feedback.

Current feedback

To allow for fast feedback, a direct laser current modulation input was implemented in laser 2. Figure4.5 shows the schematics of this input, which is based on a design in reference [14]. It consists of a50Ω input resistor, a resistor/capacitor network to generate a phase lead, and a 1 µF AC couplingcapacitor.

RF input

50 Ω

1 kΩ

330 nF1 µF Laser diode

Currentcontroller

phase lead

Fig. 4.5: Schematic of fast current feedback electronics used in the experiment.

4.3 Results

4.3.1 Response times

To characterize the potential robustness of the phase locked loop, the response times of the opticaland electronic system were measured.

Figure 4.6 shows the response of the downmixed beat note signal to a square pulse input on thecurrent modulation of laser 2. It shows a response time of about 50 ns, with little dispersion. Thisresponse time includes any delay in the current modulation, the laser diode itself, optics, the fastphotodiode, mixer and cables.

The only other delay in the phase locked loop comes from the phase lock electronics. Looking atthe phase lock data (figure 4.7), this delay was estimated between 50 and 100 ns.

The total delay of the PLL, between 100 and 150 ns, corresponds to a locking bandwidth of about6 to 10 MHz.

4.3.2 Phase locking

With the setup described, an actual phase lock between the two lasers was not achieved. Figure 4.7shows reference, beat note and error signals in a situation where the two lasers are frequency lockedto each other, but not phase locked. In the figure, the beat note signal is slower than the referencesignal, and the Piezo feedback is correcting for that. On a longer timescale, this results in a robustfrequency lock.

The error signals shown in figure 4.7 clearly correspond to the expected low pass filtered outputsignals of the AD9901. The numbers 1 through 4 correspond to the numbers in figure 4.4b.

The capturing range of the frequency lock is, in theory, twice the IF reference frequency in onedirection (the downmixed beat note signal frequency can cross zero but not minus the IF referencefrequency, for the feedback to be of correct sign), and limited only by the bandwidth of the phase lockelectronics in the other direction. If laser 2 is near a mode hop, this can also limit the capturing range.In practice, a capturing range of 100MHz or more was often observed.

Part c of figure 4.7 shows a Fourier transform of the (downmixed) beat note signal, on a 10µstimescale. The line width of the beat note signal on this timescale is about 6 MHz, comparable to thelocking bandwidth of the PLL. This explains why a phase lock is not achieved.

An upper limit on the line width of the beat note signal can also be measured using the capturingrange. Frequency locking was still possible with the IF frequency at 5 MHz, but not robustly at 4 MHz

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0 2 0 0 4 0 0 6 0 0- 0 . 4 5- 0 . 4 0- 0 3 5- 0 3 0- 0 2 5- 0 2 0

- 0 . 0 0 7

0 . 0 0 0

0 . 0 0 7

3 0

4 0

5 0

6 0

7 0

Frequ

ency

(MHz

)Be

atno

te(V)

Input

(V)

T i m e ( n s )

I n p u tB e a t n o t e f r e q u e n c y

Fig. 4.6: Beat note response of a square input pulse to the current modulation input of laser 2. Top: downmixedbeat signal. Bottom: input signal (line) and instantaneous frequency of the beat note signal (squares), showingthe response time. The instantaneous frequency is calculated from the distances between adjacent peaks.

or lower, because then the probability of relative frequency to cross -4 MHz and thus the feedback tobe of incorrect sign became real. So, the line width of the beat note signal must be less than 10 MHz.

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-0.020.00

0.02-16

-14-12

-10-8

-9.8

-9.6

-9.4

-9.2

-9.0

-8.8

-0.020.00

0.02

6080

100

-90-60-30 -0.3

-0.2

-0.1

0.00.10.20.3

Errorsignalamplitude

b

AmplitudeAmplitudea

c

Slowe

rrors

ignal

Fast

error

signa

l4

32

Refer

ence

Signa

l

Time(

micro

seco

nds)

1

FFTMagnitude

Frequ

ency

(MHz

)

Fig. 4.7: Raman laser data. (a) IF reference and downmixed beat note signal, together with the slow and fasterror signal from the phase lock electronics. (b) Zoom on the black rectangle in (a). (c) Fourier transform ofthe beat note signal in (a), showing the combined line width of the two lasers.

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Conclusions

This bachelors project aimed at building a Raman laser system consisting of two lasers in a phaselocked loop. In this process, working with extended cavity diode lasers and various experimentaltechniques had to be learned. This resulted in several useful experiments.

Results

As a first introduction to diode lasers and optical experiments, the laser diodes were characterized andused to record spectra of the Rubidium D1 line. These spectra are in good agreement with literature,and the diode lasers proved to have good long term stability, i.e. show only small frequency drift overseveral hours.

As a second experiment, single-photon and two-photon transitions between the ground state levelsin a Rubidium cell with buffer gas were observed. This resulted in a simple, self-calibrating exper-iment showing Zeeman splitting and multi-photon transitions. This experiment is also effectively amagnetometer, sensitive to at least 0.01 Gauss.

The final experiment was building the Raman laser system. Using electronics based on a digitalphase/frequency detector, a reliable relative frequency lock between the two lasers, with a line width of6 MHz and a large capturing range (in the order of 100 MHz or more) was achieved. This experimentalso characterizes the line widths of the individual lasers to be around 4 MHz. The combined linewidth of the lasers is comparable to the estimated locking bandwidth of the PLL, and indeed provedtoo large for an actual phase lock.

Further improvements

To achieve a robust and flexible phase locked Raman laser system, the setup described here can beuseful, if the diode lasers are replaced and/or optimized to achieve smaller individual line widths.One way to achieve this can be to use lower power laser diodes and amplify the Raman laser beamwith a tapered amplifier. Reduction of the line width could also be achievable by better isolating thesetup from vibrations, investigating different diode laser mounts and carefully optimizing electroniccomponents.

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Populaire samenvatting

In dit bachelorprojectverslag wordt het ontwikkelen van een Raman lasersysteem beschreven, metenkele andere experimenten die in dit proces zijn gebruikt.

Overgangen in atomen

Afhankelijk van de baan die de elektronen van een atoom om de atoomkern hebben, kan een atoomzich in verschillende toestanden bevinden. Deze toestanden hebben over het algemeen verschillendeenergieen. Stel dat een atoom in een of andere toestand 0 is, en wordt beschenen met licht (foto-nen) waarvan de energie (E = hf , met h de constante van Planck en f de frequentie van het licht)overeenkomt met het energieverschil tussen toestand 0 en toestand 1, dan kan het atoom een fotonopnemen en overgaan naar toestand 1. Evenzo kan het atoom weer overgaan naar toestand 0, en daar-bij weer een foton uitzenden. Laserlicht is bijzonder geschikt om dit soort overgangen in atomen voorelkaar te krijgen, omdat de bundel goed te richten is, en het licht bij veel lasers precies een frequentieheeft. Een laserbundel is echter niet te maken van licht met een hele lage frequentie (bijvoorbeeldmicrogolven) omdat dit licht een lange golflengte heeft (in dit geval enkele cm). Een algemene eigen-schap van alle golfverschijnselen, dus ook licht, is dat je het niet beter kunt bundelen of focusserendan ongeveer de golflengte zelf.

Het doel van het Raman lasersysteem is om een microgolfovergang te kunnen stimuleren, corre-sponderend met licht met een golflengte van ongeveer 4 cm, met een bundel die focusseerbaar is op eengebied van 20 µm. Om dit te bereiken worden twee laserbundels gebruikt, beide met een golflengte vanongeveer 795nm, maar met een klein energieverschil (dus frequentieverschil) dat precies overeenkomtmet de gewenste microgolfovergang. De eerste bundel brengt het atoom van toestand 0 naar een toes-tand met veel hogere energie, en de tweede bundel brengt hem weer omlaag, maar dan naar toestand1. In de quantummechanica zijn dit geen twee losse processen, maar een enkel proces, dat bekendstaat als een Raman-overgang.

Quantumprocessor

Het Raman laser systeem is een tool om in een ontwerp voor een quantumprocessor te gebruiken [1].In een computer wordt informatie opgeslagen in bits, die 0 of 1 kunnen zijn. In een quantumprocessorwordt informatie opgeslagen in qubits, quantumsystemen die 0, 1 of een combinatie van 0 en 1 kunnenzijn. In dit geval zijn de qubits atomen, met zoals hiervoor genoemd toestanden 0 en 1 (en nog veleandere die geen rol spelen in het experiment). De atomen worden gevangen in een rooster, met eenheel klein wolkje atomen elke 20µm. De Ramanlaser kan dan gebruikt worden om 1 wolkje van detoestand 0 naar 1, terug naar 0, of naar een bepaalde combinatie (superpositie) van de twee te brengen.

Resultaten en conclusies

Om het frequentieverschil tussen de twee laserbundels precies goed te krijgen en te houden, wordt defrequentie van de tweede laser steeds aangepast aan de hand van de toevallige veranderingen van deeerste en de tweede. De toevallige veranderingen zijn echter te groot om snel te kunnen corrigeren.Hierdoor blijft het frequentieverschil wel ongeveer goed, maar om de gewenste nauwkeurigheid vanenkele Hz te bereiken, zijn verbeteringen aan de stabiliteit van de lasers nodig.

De andere experimenten in dit verslag laten de eigenschappen van verschillende soorten overgangenin rubidiumatomen zien.

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References

[1] S. Whitlock, R. Gerritsma, T. Fernholz, and R. J. C. Spreeuw, ”Two-dimensional array of micro-traps with atomic shift register on a chip,”, arXiv preprint arXiv:0803.2151v1 (2008)

[2] H. J. Metcalf and P van der Straten, ”Laser Cooling and Trapping,” p. 3-7, Springer Verlag,New-York, 1999

[3] D. J. Griffiths, ”Introduction to Quantum Mechanics,” 2nd edition, Prentice Hall, 2004

[4] K. Moler, D. S. Weiss, M. Kasevich, S. Chu, ”Theoretical analysis of velocity-selective Ramantransitions,” Physical Review A 45, 342 (1992).

[5] New Focus Application Note 7, ”FM Spectroscopy with Tunable Diode Lasers,”http://www.newfocus.com/products/documents/literature/apnote7.pdf

[6] D. Steck, ”Rubidium 87 D Line Data,” http://steck.us/alkalidata/, version 1.6 (2003).

[7] J. Camparo, ”The rubidium atomic clock and basic research,” Physics Today 60 issue 11, 33 (2007)

[8] J. Camparo, J. Coffer and J. Townsend, ”Laser-pumped atomic clock exploiting pressure-broadenedoptical transitions,” Journal of the Optical Society of America B 22, 512 (2005).

[9] P. D. D. Schwindt, S. Knappe, V. Shah, L. Hollberg and J. Kitching, ”Chip-scale atomic magne-tometer,” Applied Physics Letters 85, 6409 (2004).

[10] P. Treutlein, P. Hommelhoff, T. Steinmetz, T. W. Hansch and J. Reichel, ”Coherence in MicrochipTraps,” Physical Review Letters 92, 203005 (2004).

[11] C. J. Myatt, N. R. Newbury and C. E. Wieman, ”Simplified atom trap by using direct microwavemodulation of a diode laser,” Optics Letters 18, 649 (1993)

[12] F. B. J. Buchkremer, R. Dumke, Ch. Buggle, G. Birkl, and W. Ertmer, ”Low-cost setup forgeneration of 3 GHz frequency difference phase-locked laser light,” Review of Scientific Instruments71, 3306 (2000)

[13] A. M. Marino and C. R. Stroud, Jr., ”Phase-locked laser system for use in atomic coherenceexperiments,” Review of Scientific Instruments 79, 013104 (2008)

[14] L. Cacciapuoti, M. de Angelis, M. Fattori, G. Lamporesi, T. Petelski, M. Prevedilli, J. Stuhlerand G. M. Tino, ”Analog+digital phase and frequency detector for phase locking of diode lasers,”Review of Scientific Instruments 76, 053111 (2005)

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