• No results found

Dynamics and applications of excited cold atoms

N/A
N/A
Protected

Academic year: 2021

Share "Dynamics and applications of excited cold atoms"

Copied!
115
0
0

Bezig met laden.... (Bekijk nu de volledige tekst)

Hele tekst

(1)

Dynamics and applications of excited cold atoms

Citation for published version (APA):

Claessens, B. J. (2006). Dynamics and applications of excited cold atoms. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR609972

DOI:

10.6100/IR609972

Document status and date: Published: 01/01/2006

Document Version:

Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.

• The final author version and the galley proof are versions of the publication after peer review.

• The final published version features the final layout of the paper including the volume, issue and page numbers.

Link to publication

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain

• You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:

www.tue.nl/taverne Take down policy

If you believe that this document breaches copyright please contact us at: openaccess@tue.nl

(2)

Excited Cold Atoms

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de

Technische Universiteit Eindhoven, op gezag van de

Rector Magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie aangewezen door het College voor

Promoties in het openbaar te verdedigen

op donderdag 6 juli 2006 om 16.00 uur

door

Bert Jan Claessens

geboren te Maaseik, Belgi¨e

(3)

en

prof.dr. M.J. van der Wiel Copromotor:

dr.ir. E.J.D. Vredenbregt

Druk: Universiteitsdrukkerij Technische Universiteit Eindhoven Ontwerp Omslag: Jan-Willem Luiten © Nickel/Mills/Ledroit.

CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN Claessens, Bert Jan

Dynamics and Applications of Excited Cold Atoms/ by Bert Jan Claessens. Eindhoven : Technische Universiteit Eindhoven, 2006. -Proefschrift.

ISBN-10:90-386-2531-6 ISBN-13:978-90-386-2531-7 NUR 926

Trefw.: laserkoeling / Magneto-Optische val / deeltjesversnellers / ultra koud plasma

Subject Headings: laser cooling / Magneto-Optical Trap / particle accelerators / ultracold plasma

(4)
(5)

Eindhoven University of Technology P.O. Box 513

5600 MB Eindhoven The Netherlands

Cover Requiem chevalier vampire © Nickel/Mills/Ledroit.

The work described in this thesis has been carried out at the Physics Department of the Eindhoven University of Technology, and is part of the research program of the ‘Stichting voor Fundamenteel Onderzoek der Materie’ (FOM), which is financially supported by the ‘Nederlandse Organisatie voor Wetenschappelijk Onderzoek’ (NWO).

(6)

1 Introduction 1

1.1 Cold atoms . . . 1

1.2 Laser cooling and trapping . . . 2

1.2.1 Laser cooling . . . 2

1.2.2 Magneto-Optical Trap . . . 4

1.2.3 Rubidium . . . 5

1.3 Ultracold plasmas and cold rydberg atoms . . . 6

1.4 Cold electron bunches . . . 8

1.4.1 Brightness . . . 8

1.4.2 Self-fields . . . 10

1.5 This thesis . . . 12

Bibliography . . . 13

2 Accurate measurement of the photoionization cross section of the (2p)5(3p) 3D 3 state of neon 15 2.1 Introduction . . . 16

2.2 Experimental setup . . . 17

2.3 Measurement of the photoionization cross section . . . 18

2.3.1 Linear loss rate . . . 18

2.3.2 Loss rate Vs UV intensity . . . 20

2.3.3 Excited state fraction . . . 21

2.4 Sources of uncertainty and conclusions . . . 24

2.5 Acknowledgements . . . 25

Bibliography . . . 25

3 Dipole-Dipole interactions in a frozen Rydberg gas 27 3.1 Introduction . . . 28

3.2 Dipole-Dipole interactions . . . 28

3.3 Setup . . . 31

3.3.1 Magneto-Optical Trap and detection . . . 31

3.3.2 Pulsed laser system . . . 32

(7)

3.6 Line broadening due to dipole-dipole interactions . . . 39

3.6.1 Introduction . . . 39

3.6.2 Line shape model . . . 40

3.6.3 Experiment . . . 42

3.6.4 Results . . . 42

3.7 Ramsey experiments . . . 45

3.8 Discussion and conclusions . . . 47

Bibliography . . . 50

4 Ultracold electron source 51 Bibliography . . . 57

5 First generation pulsed electron source setup 59 5.1 Introduction . . . 59

5.2 Loading a Magneto-Optical Trap . . . 59

5.3 2-D MOT . . . 62 5.4 Laser setup . . . 64 5.5 Vacuum chamber . . . 65 5.6 Trapping beams . . . 67 5.7 Magnetic fields . . . 69 5.8 MOT diagnostics . . . 70 5.9 Excitation . . . 71 5.10 Plasma diagnostics . . . 72

5.11 Second generation accelerator . . . 72

5.12 Conclusions . . . 73

Bibliography . . . 75

6 Experimental results 77 6.1 Introduction . . . 77

6.2 MOT characterization . . . 77

6.2.1 Number of trapped atoms . . . 77

6.2.2 Lifetime and loading rate . . . 78

6.2.3 Temperature and density . . . 80

6.3 Electron and ion beam generation . . . 80

6.3.1 Experimental routine . . . 81 6.3.2 Electron signal . . . 82 6.3.3 Ion signal . . . 83 6.3.4 Plasma fraction . . . 84 6.3.5 Total charge . . . 85 6.3.6 Spatial distributions . . . 87

(8)

7 Prospects 95 Bibliography . . . 97 Summary 99 Samenvatting 101 Dankwoord 103 cv 105

(9)
(10)

Introduction

1.1 Cold atoms

In 1987, Raab et al. [1] demonstrated the Magneto-Optical trap (MOT) for the first time. In this device, atoms are collected and trapped in a single point-like volume in space and their temperature is greatly reduced by the combination of six laser beams and a quadrupole magnetic field. The MOT provided the experimentalist with a new and practical means to cool and trap dilute samples of neutral atoms [2, 3]. In an MOT, parameters such as atom number, density and temperature can be accurately controlled and can be measured with relative ease. On top of this an MOT is a very pure system where only one atomic isotope is trapped in a single well-defined state. Furthermore, because of the low temperature in combination with the relatively low density, the corresponding phase-space density is high with little interaction between the trapped atoms. All these properties caused the MOT to initiate a cascade of experiments that revolutionized the world of atomic physics. Probably the most dramatic impact came from the the realization of a Bose Einstein Condensate (BEC) in a magnetostatic trap [4] that was loaded with atoms from an MOT. A BEC was produced for the first time in 1995 by Cornell et al. [5] providing exceptional experimen-tal control over a mesoscopic quantum system. The field of degenerate quantum gasses flourished, leading to tantalizing experiments ranging from Bardeen-Cooper-Schrieffer pair-ing [6] to quantum noise correlation measurements [7].

All this work has been rewarded with not one, but two Nobel prizes within the time span of four years. In 1997, Chu, Cohen-Tannoudji and Phillips received the Nobel prize for the development of methods to cool and trap atoms with laser light. The second Nobel prize was awarded to Cornell, Ketterle and Wieman in 2001 for the demonstration of a BEC in dilute gases of alkali atoms, and for early fundamental studies of their properties.

In 1999, Killian et al. succeeded in creating the first Ultracold Plasma (UCP) [8] from an MOT. In their pioneering work, atoms in an MOT were photo-ionized with a pulsed laser, creating a plasma with very low electron (≈ 10 K) and ion (≈ 1 K) temperatures. These experiments opened the door to laboratory study of a strongly coupled plasma, i.e. a plasma where the potential energy of the ions and the electrons is larger than their

(11)

ki-netic energy. A year after the work performed by Killian et al., Robinson et al. also [9] succeeded in creating a UCP from a gas of cold Rydberg atoms excited from an MOT. Until now, most of the work in this field has been devoted to the fundamental properties and the dynamics of a UCP [10, 11]. This thesis on the other hand, deals with using a UCP or gas of cold Rydberg atoms as a source of a very high brightness electron beam [12]. The essential idea behind this is that the low electron temperature leads to a low electron beam divergence and energy spread and therefore to an electron beam with a very high brightness. |eñ |gñ w+kv w-kv on resonance F=- va A C p B p- kh hk

Figure 1.1: Left: (A) Impression of laser cooling showing an atom with momentum p

counter-propagated by a light field. (B) The atom absorbs the photon and the corresponding momentum. The resulting momentum of the atom is p − ~k. (C) The atom decays to the ground state and emits a photon in a random direction. The result of many of these cycles is a momentum transfer in the opposite direction of the photon. The momenta transferred by all spontaneously emitted photons add up to zero. Right: The Doppler shift (kv) shifts the laser (ω) counterpropagating with respect to the atom closer to resonance. This leads to more absorption of the light counter-propagating the atom. The result is that the atom experiences a damping force.

1.2 Laser cooling and trapping

1.2.1 Laser cooling

The operation of an MOT is based on laser cooling [2]. The physics behind it is that an atom in a laser light field with wavelength λ experiences a force due to the momentum transfer caused by the absorption and emission of photons with momentum ~k, with k the wavevector of the laser, |k| = 2π/λ. When a two-level atom in its ground state |gi moving in a light field absorbs a photon (Fig. 1.1B), it is excited to the excited state |ei and its momentum changes by ~k in the direction of the light field. After a typical time (τ = 1/Γ with Γ the natural linewidth of the transition) the atom decays to the ground state again and emits a photon in a random direction, in which it also gets a momentum kick

(12)

(Fig. 1.1C). After many of these cycles, the recoils of the spontaneously emitted photons add up to zero. The momentum kicks of the absorbed photons on the other hand add up in the direction of the light field. For a simple two-level transition, this results in a force F due to one single-frequency laser beam given by [2]:

F = ~kΓ 2

s

1 + s + (2δ/Γ)2. (1.1)

Here, s = I/I0 is the saturation parameter, I the intensity of the laser, I0 the saturation

intensity and δ = ω − ω0 the detuning of the laser frequency ω compared to the atomic

resonance frequency ω0.

When the atom is moving with a velocity v, it experiences a Doppler shift δD = −k·v. As a

result, when an atom is moving in standing light wave consisting of two counter-propagating light fields which are red detuned (δ < 0), the Doppler shift will cause an imbalance in the forces of the two counter-propagating beams. In the reference frame of the moving atom, the Doppler effect will shift the counter-propagating light field closer to resonance, from which the atom will consequently feel a larger force. This is illustrated in the right part of Fig. 1.1. The resulting force on the atom from the light field is approximately linear with the velocity, i.e. F ∼= −αv (see Fig. 1.1). The atom experiences a damping force, as though it were moving in a (optical) molasses [13] , this leads to cooling of the atoms.

s

-s

+

M

J

0

1

-1

0

-1

1

M

J

w

B

B

E

z

J=0

J=1

Figure 1.2: Illustration of the principle of a Magneto-Optical Trap (J=0 to J = 1 transition).

The atoms are illuminated by counter-propagating laser beams of circularly-polarized light at the center of a quadrupole magnetic field. The circularly polarized light field in addition to the position dependent Zeeman shift causes a position dependent force in addition to the frictional force illustrated in Fig. 1.1.

(13)

1.2.2 Magneto-Optical Trap

A three-dimensional optical molasses results in a force that only has a velocity dependence. In order to create a trap, one also needs a spatially dependent force. In an MOT this is provided by adding a magnetic field gradient to an optical molasses and by carefully choosing the polarization of the light beams. An MOT consists of three orthogonal pairs of red-detuned, counter-propagating, circularly-polarized laser beams intersecting at the center of a magnetic quadrupole field, generated by a pair of anti-Helmholtz coils [2, 3]. The trapping principle of the MOT is explained in Fig. 1.2 for a J = 0 (ground state) to

J = 1 (excited state) transition.

The counterpropagating laser beams have opposite circularly polarization, i.e. σ+ and

σ− light. The magnetic field induces a linear Zeeman shift near the center of the trap.

As such the degeneracy of the magnetic sub-levels (MJ = 0, ±1) of the excited state is

lifted. On the right side of the center, the MJ = −1 state is tuned closer to resonance

and correspondingly on the left side the MJ = +1 state. As a consequence, on the right

side the atom interacts with σ− light, while on the left side the atom interacts with the

σ+ light (selection rules state that σ± light drives ∆M

J = ±1 transitions). Thus, if the

polarization of the counterpropagating beams is set correctly, the atoms are driven to the center of the trap and as such the force becomes spatially dependent in addition to the velocity dependence coming from the polarization-independent molasses force.

In good approximation, the motion of an atom trapped in an MOT can be described by an over-damped harmonic oscillator [3]. Near the center of the MOT the force on a atom with velocity v at position z can be written as:

FMOT = −αv − αβk z. (1.2)

The quantity α is the damping coefficient, given by:

α = −~k2 I I0

2δ/Γ

(1 + (2δ/Γ)2)2, (1.3)

while β incorporates the effect of the magnetic field and is given by:

β = gµB

~

dB

dz . (1.4)

Here g is the Land´e factor, µBthe Bohr magneton and dB/dz the gradient of the magnetic

field in the z-direction (dB/dz = −2dB/dr). As such the motion of a particle entering the trapping region with a velocity below a certain capture velocity vc can be approximated

by that of a damped harmonic oscillator. Under typical operation conditions the atom undergoes over-damped simple harmonic motion to the center of the trap.

A standard MOT contains about 109 atoms at a density of about 1010cm−3 and a

tempera-ture of about 300 µK [14]. The largest MOT reported so far, to our knowledge, in terms of both density and number of atoms was reported in 1993 by Ketterle et al.. They succeeded in creating an MOT containing 1010 atoms at a density of 1012 cm−3 and a temperature of

(14)

      

F

1

3

2

4

4

 

3

2

        ! " #$

5P

% &'

5S

( )*

Figure 1.3: Level diagram for 85Rb of the hyperfine levels 5S

1/2 and 5P3/2, indicating the trapping laser transition and the repumper transition.

1.2.3 Rubidium

Because of the relative ease with which they can be trapped and the availability of low cost diode lasers to drive the laser cooling transition, alkali atoms are often used for trapping and cooling. In our experiments we mainly use rubidium (85Rb). The optical transitions

used for trapping and cooling are shown in Fig. 1.3. Laser cooling and trapping is done from the ground 5S(J = 1/2) to the fine-structure state 5P(J = 3/2). Since 85Rb also has

a nuclear spin (I = 5/2) both the S and the P state have a hyperfine-structure (F = I + J). Trapping and cooling is done with a ”trapping laser” operated at the closed hyperfine transition 5S1/2,F =3 → 5P3/2,F=4 at a wavelength of 780 nm. Light from this laser also

off-resonantly excites the 5P3/2,F=3 state, from which the atoms can decay back to the

5S1/2,F=2 hyperfine level of the ground state. Once the atoms are in this state, they can no

longer be exited by the trapping light. Because of this, one also has to apply a ”repumping

laser ”, i.e., a laser that pumps the atoms that fall back to the 5S1/2,F=2 state to the

5P3/2,F=3 state, from where they can fall back to the 5S1/2,F=3 state. Table 1.1 gives a

(15)

Table 1.1: Some important characteristic quantities for 85Rb and the laser cycling transition

5S1/2(F=3)↔5P3/2(F=4).

Quantity Symbol Value

Atomic mass m 85 a.m.u.=1.41×10−25 kg

Wavelength λ 780.24 nm (in vacuum)

Natural linewidth Γ 5.98 MHz

Lifetime of 5P3/2(F=4) τ = 1/(2πΓ) 26.63 ns

Saturation intensity I0 1.64 mW cm−2

Recoil velocity vrec = ~k/m 0.602 cm s−1

Doppler limit, velocity vD = (~Γ/2m)1/2 11.85 cm s−1

Doppler limit, temperature TD = ~Γ/2kB 142.41 µK

D

E=kT

                ! "# $% &'( )*+ ,-. /

Figure 1.4: (A) Creation of a UCP by exciting a Rydberg level that evolves spontaneously to

a UCP under the right conditions. (B) Creation of a UCP by photo-ionizing the atoms from an MOT to just above the ionization threshold, the excess energy ∆E is mainly transferred to the electrons as kinetic energy.

1.3 Ultracold plasmas and cold rydberg atoms

A conventional ”cold” and neutral plasma has a temperature of about 10000 K. This is the minimum temperature at which a reasonable fraction (1 %) of the electrons has an energy in excess of 5 eV, which is the lowest possible energy an electron can have to ionize an atom [17]. As mentioned in Section 1.1, the first experimental realization of an Ultracold Plasma (UCP), performed at the National Institute for Standards and Technology (NIST), was reported by Killian et al. [8]. The temperature in a UCP is on the order of 1 K for the ions and 10 K for the electrons [10, 18], orders of magnitude less than in a conventional plasma. This plasma was produced by photoionization of a cloud of laser cooled Xe atoms just above the ionization threshold (Fig. 1.4B). The excess energy, ∆E, i.e., the difference

(16)

Position

PotentialEnergy

A B

Ionisation

Figure 1.5: Impression of correlation heating, (A) initially the atoms in an MOT have no

spatial correlation and little energy spread. (B) The atoms are photo-ionized and the electrons and ions are not in the minima of the created potential energy landscape. This potential energy is subsequently converted to kinetic energy.

between the photon energy and the ionization threshold, is transferred to the electrons and the ions. Due to the large mass ratio, most of it goes to the electrons. As a result, the initial electron temperature can be adjusted by changing the laser wavelength and is limited by the bandwidth of the laser. In the experiment performed at NIST, the initial electron temperature could be varied between 0.1 and 1000 K and the initial ion temperature could be as low as 10 µK. The number of atoms ionized could be adjusted by changing the energy of the ionization laser. In this way 105 ions could be produced at a peak density of about

109 cm−3.

Conventionally, an ionized gas is considered to be a plasma if λD, the Debye screening

length, is smaller than the size of the sample:

λD =

r

²0kBT

e2n , (1.5)

where e is the elementary charge, ²0 is the electric permittivity of free space, kB the

Boltz-mann constant, n the electron density and T the electron temperature. In the NIST experiment, the Debye screening length was about 500 nm, which is small compared to the size of the sample which was about 300 µm. Therefore the system created can indeed be regarded as a plasma.

The properties of such a UCP are such that the plasma parameters offer practical exper-imental access to the regime of strongly coupled plasmas. These are plasmas where the thermal energy is less then the Coulomb interaction energy. For a neutral plasma with electron and ion temperatures Te and Ti, this ratio is characterized by the electron and ion

(17)

coupling parameters Γe and Γi: Γe = e 2 4π²0akBTe , Γi = ΓeTe Ti e−λDa . (1.6)

The exponential term comes from the shielding of ion-ion interactions by electrons, a = (4πn/3)−1/3 is the Wigner-Seitz radius. For densities of about 109 cm−3 and temperatures

mentioned above, the coupling parameters can be as high as Γe = 10 and Γi = 1000

respectively, i.e., deep in the strongly coupled regime (Γ >> 1) where one expects ordering effects such as Coulomb crystallization.

On the basis of the results of molecular dynamics simulations [19–21], however, it was soon realized that immediately after creation of the plasma, a rapid intrinsic heating effect occurs. Because there is hardly any spatial correlation between the atoms in an MOT, the plasma is initially also completely uncorrelated. As such the electrons and ions of a UCP are initially not at a potential energy minimum. The subsequent conversion of potential energy into kinetic energy rapidly heats both the electrons and ions. This is depicted in Fig. 1.5. Each subsystem (ions and electrons) heats up until its corresponding coupling parameter is approximately 1 on a time scale of the inverse plasma frequency 1/ωp =

p

0/ne2, with m the mass of the corresponding subsystem. For the electrons

this is in the ns timescale, for the ions this is in the µs timescale, at a density of 109 cm−3.

This, together with other heating mechanisms such as continuum lowering [22] (the electric field of the ions and electrons results in an effective lowering of the ionization treshold) and three body recombination [10] leads to an effective electron temperature of about 10 K. For the ions, effective temperatures of about 1 K have been measured by means of absorption imaging [11]. New techniques are being investigated to reduce the heating due to correlation heating. Suggested techniques are laser cooling [23, 24] of the ions, or adding correlation to the initial neutral atoms by placing them in a lattice [25] or by starting with a fermi degenerate gas [26].

Only a year after the first UCP produced by photoionization, Robinson et al. [9] reported the spontaneous evolution of a gas of cold Rydberg atoms into a UCP as mentioned in Section 1.1. This experiment was similar to the experiment performed at NIST, instead of tuning the laser just above the ionization threshold the laser was tuned just below the ionization threshold as depicted in Fig. 1.4B. Although the parameters of the plasma are much the same, the dynamics of the plasma formation are quite different. Chapter 3 gives a detailed description of how a cold gas of Rydberg atoms evolves into a UCP. The advantage of using a Rydberg gas to create a UCP over using photoionization is that using Rydberg atoms can offer more control over the plasma parameters.

1.4 Cold electron bunches

1.4.1 Brightness

Up until now, a UCP has mainly been a subject of research for its own sake, focussed on properties such as the electron and ion temperatures and understanding its dynamics.

(18)

A

W

z

x

Beam envelope

Electron trajectories

Figure 1.6: Geometry of a electron bunch at a focus. Indicated are the beam envelope and

the electron trajectories in a focus. A is the transverse area of the beam and Ω the solid angle. The goal of the ultracold electron bunches project at Eindhoven University of Technology on the other hand is to prove that a UCP or cold Rydberg gas has a enormous potential as a bright, pulsed electron source. The figure of merit for electron sources is the brightness [27], i.e. the current density per unit solid angle and per unit energy spread, as indicated in Fig. 1.6. The brightness summarizes the complex properties of an electron bunch and gives a good indication about its usefulness for potential applications.

Formally the fundamental characterization of a collisionless pulsed charged particle beam is given in terms of the Lorentz-invariant local phase space density distribution f (r, p), for electrons, p = γmev/

p

1 − v2/c2, with v the velocity, and me the electron mass, f (r, p) is

normalized to the number of particles N = R f (r, p)d3rd3p. As a result a good definition

for the Lorentz-invariant local 6D brightness is given by [28]:

B(r, p) ≡ em2

ec2f (r, p), (1.7)

with e the elementary charge, and c the speed of light.

For a particle bunch with a 6D gaussian phase space distribution the (peak) 6D brightness

B ≡ B(0, 0) at the center of the bunch can be written as: B = em2ec2 3 N σxσyσzσpxσpyσpz , (1.8) where σA is the 1/

e value of the quantity A. However in practice, the exact distribution

of a beam is not known, and typically it is the variance σA=

< A2 > − < A >2 that can

be measured, where <> indicates averaging over the distribution.

Furthermore, in a beam, degrees of freedom can be coupled. A linear lens for example, results in a linear coupling between x and px. If one would then use Eq. 1.8 as a definition

for the brightness, this would result in a gross underestimate. As a result, the normalized root-mean-square (rms) emittance is commonly used, for the x-direction e.g. the emittance is given by: ²x = 1 mec p < x2 >< p2 x > − < xpx >2. (1.9)

The emittance ² is a Lorentz-invariant measure for the focusability of the beam. Under the assumption that the x, y and z directions are decoupled, the following equation is a

(19)

practical estimate for the 6D brightness. B = 1 mec Ne (2π)3² x²y²z . (1.10)

In cases where the longitudinal energy spread is less critical however, one uses the transverse normalized brightness (B⊥) for pulsed electron bunches. This definition is mostly used.

Assuming that the distribution in the z − pz space is Gaussian, one defines for a beam

traveling in the z-direction:

B⊥= Ip 2² x²y , (1.11) where Ip = vzQ/

2πσz is the peak current. An important note here is that the

normal-ized transverse brightness is not Lorentz invariant, nevertheless for high energy electron bunches it is the figure of merit.

The relation between the normalized transverse brightness (B⊥) and the actual 6-D

bright-ness (B) is then:

B⊥= BσE

2π, (1.12)

with σE the longitudinal energy spread σE = vzσpz.

The normalized brightness scales as 1/T (in the case one can speak of a temperature). In the best performing sources (see Chapter 4) so far the limiting temperatures are of the order of 103-104 K, while for a UCP the electron temperature is of the order of 10 K.

This would result in a possible normalized brightness three orders of magnitude higher then current state of the art sources, all other circumstances being equal. This back of

the envelope calculation shows the huge promise of using a UCP as a electron source. A

detailed analysis of the potential of a UCP as a cold electron source is given in chapter 4.

1.4.2 Self-fields

Although the emittance of a bunch created from a plasma is extremely low immediately after creation, it must also be kept low. The emittance, as defined above, can be regarded as the rms surface area of the projection of a bunch on the x − px, y − py phase space

planes. The result is that the emittance corresponding to a curved projection is larger than the emittance corresponding to a projection with the same actual surface, but with a rectilinear projection. Figure 1.7 illustrates the phase space projections of a linear and a distorted bunch and the corresponding emittance, defined by the area of the ellipsoidal envelope of the projection.

The advantage of using this definition of emittance is that the practical quality of the beam is also included. In a typical accelerator, one uses optics that is as linear as possible, i.e. optics with little aberrations that can only rotate the projections of phase space. As a consequence, if one wants to focus the beam to a small spot, the non-linear part of the projection results in a larger spot size, reducing the brightness.

Probably the most detrimental effect for the emittance is curvature of phase space, due to the self-fields, resulting from the space-charge forces [28]. In typical electron accelerators

(20)

x

p

x

e

x

Figure 1.7: The projection on the x − px phase space plane is given for two bunches, as are the ellipsoids drawn around the projections defining the emittance. One has a linear projection corresponding to a low emittance and one has a curved projection corresponding to a large emittance. The surfaces of both projections are approximately equal.

these space-charge forces are the main limiting factor in the achievable brightness, i.e. the intrinsic brightness of the initial ionization process. Space-charge induced emittance growth is mainly important at low energies, because at high beam energies the space-charge forces are reduced due to relativistic effects.

To see this, consider a cylindrically symmetric bunch with velocity v and charge density distribution qn(r). The electric field in the radial direction is then derived from Gauss’s law and given by [29]:

Er = q ²0r Z r 0 n(r0)r0dr0. (1.13)

The azimuthal magnetic field is given by Ampere’s law:

= qvµ0 r Z r 0 n(r0)r0dr0, (1.14)

where µ0 is the permeability of free space. Combining Eq. 1.13 and Eq. 1.14 gives:

Fr= q(Er− vBθ) = qEr/γ2, (1.15)

where γ = [1 − (v/c)2]−1/2. Thus, the radial coulomb forces are suppressed by a factor γ2.

Including relativistic effects, the acceleration a of a electron moving with velocity v due to a force F is defined by [30]:

F = mγna + γ2a · v

c2 v

o

. (1.16)

Combining Eq. 1.15 and Eq. 1.16 shows that the acceleration due to space-charge forces is suppressed by a factor γ3 both in the radial and the longitudinal direction. The

(21)

practical accelerators, one wants to accelerate the bunch as fast as possible to relativistic velocities.

A completely new and fundamentally different way to reduce the effect of space charge forces on the emittance is by carefully shaping the radial distribution of the electron bunch. This is discussed in more detail in Chapter 4.

1.5 This thesis

This thesis presents work done in connection with the ultracold electron bunches (UCEB) project at Eindhoven University of Technology and is a collaboration of the groups Atomic physics and Quantum Technology and physics and applications of accelerators of the de-partment Applied Physics. The goal of this project is the production of high brightness electron bunches from a UCP or cold Rydberg gas.

In Chapter 2, we present a study of the effect of an ionizing laser on a sample of trapped atoms, and use this to obtain a precise measurement of the photoionization cross section. In this practical case we have used metastable neon in the3D

3 state, rather then Rb atoms.

This value is important for the practical production of a UCP of neon, since it determines the power needed to photoionize the atoms. A UCP of metastable neon is relevant for the production of a continuous ion beam of noble gas atoms, which is an extension of the UCEB project.

Chapter 3 presents work performed at the University of Virginia, devoted to the role of dipole-dipole interactions in a gas of cold Rydberg atoms. These dipole interactions can be used to accurately control the formation of UCP from a gas of cold Rydberg atoms. Po-tentially, dipole-dipole interactions can be used to reduce the initial electron temperature in a UCP. This could reduce the emittance of cold electron bunches even further.

In Chapter 4, we investigate the feasibility of a UCP as a source for an electron accelerator through simulations with the General Particle Tracer Code. We find that using a UCP as an electron source can result in an increase of brightness of over two orders of magnitude compared to conventional electron sources.

Chapter 5 gives a detailed discussion on the first setup designed and constructed for the fabrication of a UCP from rubidium atoms.

Chapter 6 presents the results of creating the first UCP from a cold Rydberg gas and imaging them on a phosphor screen. From these images we obtain an upper value for the initial emittance of our source.

Finally, in Chapter 7 we speculate on where the new field of cold atom charged particle sources might lead us.

(22)

Bibliography

[1] E. Raab, M. Prentiss, A. Cable, S. Chu, and D. Pritchard, Phys. Rev. Lett. 59, 2631 (1987). [2] H.J. Metcalf, P. van der Straten, Laser Cooling and Trapping (Springer, Berlin Heidelberg

New York, 1999).

[3] C.J. Foot, Atomic Physics (Oxford University Press, 2005). [4] W. H. Wing, Prog. Quant. Electr. 8, 181 (1984).

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

[6] M. W. Zwierlein, J. R. Abo-Shaeer, A. Schirotzek, C. H. Schunck and W. Ketterle, Nature 435, 1047 (2005).

[7] S. F¨olling, F. Gerbier, A. Widera, O. Mandel, T. Gericke and I. Bloch, Nature 434, 481 (2005).

[8] T.C. Killian, S. Kulin, S. D. Bergeson, L. A. Orozco, C. Orzel, and S. L. Rolston, Phys. Rev. Lett. 83, 4776 (1999).

[9] M. P. Robinson, B. Laburthe Tolra, Michael W. Noel, T. F. Gallagher and P. Pillet, Phys. Rev. Lett. 85, 4466 (2000).

[10] S. Kulin, T. C. Killian, S. D. Bergeson and S. L. Rolston, Phys. Rev. Lett. 85, 318 (2000). [11] C. E. Simien, Y. C. Chen, P. Gupta, S. Laha, Y. N. Martinez, P. G. Mickelson, S. B. Nagel,

and T. C. Killian, Phys. Rev. Lett. 92, 143001 (2004).

[12] B. J. Claessens, S. B. van der Geer, G. Taban, E. J. D. Vredenbregt, and O. J. Luiten, Phys. Rev. Lett. 95, 164801 (2005).

[13] S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, Phys. Rev. Lett. 57, 314 (1986). [14] E. W. Streed, A. P Chikkatur, T. L Gustavson, M. Boyd, Y. Torii, D. Schneble, G. K.

Campbell, D. E. Pritchard, W. Ketterle, cond-mat 0507348 (2005).

[15] W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, Phys. Rev. Lett. 70, 2253 (1993).

[16] C. Hawthorn, Ph.D. thesis (University of Melbourne, 2004).

[17] T. F. Gallagher, P. Pillet, M. P. Robinson, B. Laburthe-Tolra, and M. W. Noel, J. Opt. Soc. Am. B 20, 1091 (2002).

[18] J. L. Roberts, C. D. Fertig, M. J. Lim, and S. L. Rolston, Phys. Rev. Lett. 92, 253003 (2004).

[19] S. G. Kuzmin and T. M. O’Neil, Phys. Rev. Lett. 88, 065003 (2002). [20] F. Robicheaux and James D. Hanson, Phys. Rev. Lett. 88, 055002 (2002). [21] T. Pohl, T. Pattard, and J. M. Rost, Phys. Rev. A 70, 033416 (2004). [22] Y. Hahn, Phys. Lett. A 293, 266 (2002).

[23] T. C. Killian, Y. C. Chen, P. Gupta, S. Laha, Y. N. Martinez, P. G. Mickelson, S. B. Nagel, A. D. Saenz, and C. E. Simien, Plasma Phys. Control. Fusion 47, A 297 (2005).

(23)

[24] T. Pohl, T. Pattard, and J. M. Rost, Phys. Rev. Lett. 92, 155003 (2004) . [25] T. Pohl, T. Pattard, J.M. Rost, J. Phys. B: At. Mol. Opt. Phys. 37 L183 (2004). [26] M. S. Murillo Phys. Rev. Lett. 87, 115003 (2001) .

[27] For a recent overview, see P. Piot, in The physics and Applications of High Brightness

Electron Beams, edited by J. Rosenzweig, G. Travish, and L. Serafini (World Scientific,

Singapore, 2003), p. 127.

[28] S. B. van der Geer, M. J. de Loos, M. J. van der Wiel, and O. J. Luiten, to pe published. [29] F. Kiewit, Ph.D. thesis (Eindhoven University of Technology, 2003).

(24)

Accurate measurement of the photoionization cross

section of the (2p)

5

(3p)

3

D

3

state of neon

Abstract. We report a new measurement of the photoionization cross section for the (2p)5(3p) 3D

3 state of neon at the wavelengths of 351 and 364 nm a. These data were

obtained by monitoring the decay of the fluorescence of atoms trapped in a magneto-optical atom trap under the presence of a photoionizing laser, a technique developed by Dinneen et al [16]. We obtain absolute photoionization cross sections of 2.05 ± 0.25 × 10−18 cm2 at λ = 351 nm and 2.15 ± 0.25 × 10−18cm2 at λ = 364 nm, an improvement in accuracy of a factor of four over previously published values. These new values are not consistent with published theoretical data.

(25)

2.1 Introduction

The process of photoionization plays an important role in applied plasmas such as gas lasers or discharge lamps, and also as a technique for producing ionized matter for scientific study. Recently, for instance, it has become possible to study a virtually unexplored field in plasma physics experimentally through the production of a so-called ultracold plasma [1] by near-threshold photoionization of a sample of neutral atoms trapped in a magneto-optical trap (MOT). Such ultracold plasmas have ion temperatures around 1 mK and electron temperatures as low as 10 K at a density of about 109 cm−3, at which point

the electrostatic interaction energy between nearest neighbors becomes comparable to the kinetic energy. The plasma is then close to entering the strongly coupled regime where standard classical plasma physics assumptions may become invalid [2, 3].

Photoionization cross sections furthermore provide fundamental tests of atomic struc-ture calculations. [4–19]. Here, as is often the case, alkali-metal atoms and rare gas atoms are of particular importance as relatively simple test subjects. For ionization out of the ground state, absolute cross sections have been determined with great precision (uncer-tainty 1-3 %) over a wide range of photon energies of up to 4000 eV by using synchrotron radiation [5–7]. For excited states, ionization close to threshold has been studied using discharge and laser-excited atomic beams [8–13] and, only for alkali-metal atoms, also us-ing atom traps [16–19]. In the latter case, absolute values could be determined with an accuracy down to 10%.

The atom trap technique was pioneered by Dinneen et al. [16] and later applied by several other groups [17–19], with all experiments focusing exclusively on rubidium. Here we follow this lead to obtain absolute and precise measurements of the photoionization cross section of the excited (2p)5(3p) 3D

3 state of neon for two different ionization

wave-lengths. While much has been learnt about photoionization out of this and closely related states from, e.g., photo-electron spectra [10], electron angular distributions [11] and auto-ionization widths [12, 13, 15], so far the absolute value of the corresponding cross sections has only been known with a relative accuracy of 50 % [10, 11]. In this paper we present an independent and direct measurement based on studying the effect of an ionizing laser on the decay dynamics of laser-excited neon atoms trapped in a magneto-optical trap. The technique used here employs only relative measurements of atom numbers, which allowed the relative precision to be increased by a factor of four over the measurements of Siegel et

al. [11]. At the time, these authors necessarily had to rely on data on collisional ionization

processes to obtain an absolute value of their atom flux, which restricted the achieved accuracy.

This paper is organized as follows: Section 2.2 describes the experimental setup and the characteristics of the photoionization laser; Section 2.3 presents the experimental results and the corresponding analysis; Section 2.4 discusses the results and the corresponding uncertainties and compares our measured values with previous work.

(26)

Figure 2.1: Schematic drawing of the MOT and the UV laser beam. Depicted in the figure is

the telescope used for expanding the PI beam and the glass plate used for moving the PI beam with respect to the atom cloud.

2.2 Experimental setup

Metastable neon atoms are trapped in an MOT which is loaded from a bright atomic beam as described in [20, 21]. The atomic beam can produce approximately 2×1010 atoms per

second travelling at 100 m/s. These atoms are slowed down further in a second Zeeman slower and trapped in an MOT. The magnetic field gradient of the MOT is about 10 G/cm, and the detuning of the trapping laser was set at -0.5 Γ with Γ = (2π)8.2MHz the natural width of the atomic transition.

The laser light used for all laser cooling stages of both the atomic beam and the MOT was generated with a frequency stabilized Coherent 899 ring dye laser. This laser pro-duces 700 mW of laser light at a wavelength of 640.224 nm, resonant with the closed Ne(2p)5(3s)3P

2 →(2p)5(3p)3D3 optical transition. The laser light for the MOT beams was

spatially filtered through a fiber and expanded. The MOT beams have a Gaussian beam profile with a 1/√e intensity radius of 1.5±0.1 mm. The combined intensity of all MOT

beams was typically 16 mW/cm2.

The spatial profile of the fluorescence emitted by the MOT was imaged with a charge-coupled-device (CCD) camera and had a profile that could be fitted well with a Gaussian distribution. The 1/√e radius of the MOT fluorescence in the x (sx), and z (sz) (see

Fig. 2.1) direction was measured to be 80±20 µm. The intensity of the MOT beams was stabilized to better than 10−2 using an electronic controller connected to the modulation

(27)

two stabilized intensities within 400 µs.

The UV light for photoionization (PI beam) was produced with a Coherent Innova 90-5 argon-ion laser which we used in two different modes: (1) a dual wavelength mode or ”mixed mode”, in which the laser generates a maximum of 86.5 mW of 364 nm light and 53.5 mW of 351 nm light simultaneously, and (2) a ”non-mixed mode”in which the laser generates a maximum of 80 mW at 351 nm only.

Both these wavelengths generated by the laser are greater than 290 nm, the maximum wavelength for photoionization from the (2p)5(3s) 3P

2 state. This ensures that

photoion-ization only occurs from the (2p)5(3p)3D

3 state. The Gaussian PI beam was expanded to

a 1/√e intensity radius of 450±20 µm in the x-direction and 530±20 µm in the y-direction

at the position of the MOT (see Fig. 2.1). The beam was passed trough an anti-reflection coated glass plate, enabling small, controlled horizontal and vertical movements of the PI beam relative to the atom cloud, and then sent through a vacuum window into the trap under a small angle with the z-direction. The transmission of the vacuum window was measured to be 0.73 ± 0.02 at 351 nm and 0.80 ± 0.02 at 364 nm.

Relevant information was obtained via the measurement of the decay fluorescence using a Pulnix TM 1300 CCD camera. The camera measured total fluorescence power emitted by the trapped atoms at a rate of 10 frames/s. The fluorescence of the atoms could also be detected with an EMI 9862K photomultiplier tube (PMT). This was used to measure fast (≤ 400µs) changes in fluorescence power, resulting from a sudden change in MOT laser intensity. Both the CCD camera and the photomultiplier tube were only used for relative measurements so that calibration for absolute detection efficiency was not necessary.

A schematic representation of the MOT setup and the PI beam is given in Fig. 2.1.

2.3 Measurement of the photoionization cross section

2.3.1 Linear loss rate

The rate equation that describes the time evolution of the number of trapped atoms Nt in

the MOT with the presence of the PI beam is

dNt

dt = RL− NtBG+ ΓP I) − β

Z

n(r, t)2dr, (2.1)

where RLis the loading rate (determined by the beam flux and the capture efficiency of the

MOT) and ΓBG the decay rate due to density-independent losses such as background gas

collisions (the pressure in the trap chamber during operation is approximately 1.6×10−9

mbar). The rate constant β describes the density-dependent losses and n(r, t) is the atomic density distribution of the trapped atoms. Finally, ΓP I is the decay rate due to

photoion-ization from which the photoionphotoion-ization cross section can be determined as we show in subsection 2.3.2.

One aspect that makes photoionization experiments with metastable atoms different compared to similar experiments with alkali atoms is that the steady-state number in the

(28)

Fluorescence power (arb. units)

Time (s)

Figure 2.2: Fluorescence power from the atoms trapped in the MOT as a function of time

after loading is stopped, without (filled circles) and with (open circles) the presence of the PI beam. The solid lines are fits to Eq. (2.2).

MOT is mainly determined by two body losses [21]. A consequence of this is that it is very difficult to measure ΓP I by studying the effect of the photoionization laser on the

steady-state number of atoms in the MOT. Only its effect on the fill rate or on the decay rate of the MOT, and then only at low densities can be used. Here the dynamics are determined by linear losses, i.e., density independent losses. In this experiment we chose to study the decay dynamics when the atomic beam is switched off (RL = 0) since then the results do

not suffer from fluctuations in the beam flux.

In our trap the spatial distribution is to a good approximation independent of the number of atoms [21], i.e. n(r, t) = n(r)g(t) so that the solution to Eq. (2.1) (with RL= 0)

can be written as

Nt(t) =

Nt(0)exp(−tR)

1 + (βNt(0)/Vef f)[1 − exp(−tR)]

, (2.2)

where Vef f = (2π)3/2sxsysz is the effective trap volume [21] and R = ΓBG+ ΓP I the total

decay rate. Figure 2.2 shows two typical experimental decay curves fitted with Eq. (2.2), with and without the presence of the photoionizing laser. Fits to the data such as these enable R to be determined. For both curves the initial part of the decay is dominated by two-body losses. The effect of linear losses becomes visible at lower atom numbers, and therefore lower atomic densities.

(29)

(a)

(

)

R (s )

-1

R (s )

-1

Figure 2.3: The upper graph (a) shows the measured linear loss rate as a function of intensity

of the mixed-mode PI beam. The lower graph (b) shows the fitted loss rate as a function of intensity of the 351 nm-only PI beam. The solid lines are linear fits to the data.

2.3.2 Loss rate Vs UV intensity

The linear decay rate ΓP I due to photoionization by a monochromatic light source with

frequency ν can be written as

ΓP I =

IP If σ

, (2.3)

where IP I is the average photoionizing laser intensity incident on the 3D3 atoms in the

MOT, h is Planck’s constant and σ is the photoionization cross section at frequency ν [16]. The decay rate is proportional to the excited state fraction f , i.e., the fraction of atoms in the excited3D

3 state, since only these atoms can be ionized with the UV laser as discussed

in section 2.2. As the size of the atom cloud is much smaller than the diameter of the trapping beams, f is constant over the trapping region. Note that Eq. (2.3) is only valid when IP I is far from saturation; in the present experiment this is always the case. By

measuring R as a function of the average photoionizing laser intensity IP I, a value for f σ

(30)

The value of IP I is determined by the total UV laser power (PU V) and the spatial profile

of the photoionizing laser I(x, y), given by

I(x, y) = I0exp(− x2 2 x )exp(− y 2 2 y ), (2.4)

with σx and σy the 1/

e radii of the PI beam in the x and y directions and I0 =

PU V/(2πσxσy) the peak intensity. Furthermore, the spatial profile of the (3D3) atoms

in the MOT and the alignment of the MOT with respect to the UV laser have to be taken into account. As mentioned in section 2.2 we measured the spatial profiles of both the3D

3

atoms and the PI-laser by imaging them on a CCD camera and found that both spatial profiles were fitted well with a Gaussian distribution. Assuming that the MOT and the PI beam are well aligned with respect to one another (we will discuss alignment of the MOT and PI beam further in section 2.4), the average intensity can be found by averaging the spatial profile I(x, y) given by Eq. (2.4) over the normalized, transverse, spatial distribution ˜

n(x, y) of the trapped atoms, IP I =

Z Z

I(x, y)˜n(x, y) dxdy = q I0σxσy 2

x+ s2x)(σy2+ s2y)

, (2.5)

where (as before) sx and sy are the 1/

e radii in the x and y direction of the spatial profile

of the 3D

3 atoms. Since the size of the PI beam is much larger than the size of the 3D3

atom distribution, the value of IP I/I0 is rather insensitive to fluctuations in the size of the 3D

3 atom distribution.

Figure (2.3a) shows the measured linear loss rate as a function of intensity seen by the atoms when the laser was running in mixed mode (both 351 nm and 364 nm light). Each data point is a statistical average of at least five measurements. Figure (2.3b) shows the measured linear loss rate as a function of intensity seen by the atoms with the laser running in non-mixed mode (351 nm only). Once again, each data point is a statistical average of at least five separate measurements. The linear behavior of the data confirms the assumption of a loss rate constant that varies linearly with the laser intensity. The slope of these curves corresponds to the quantity f σ/hν = 0.79±0.05 cm2/J for the 351

nm-only beam, and f hσ/hνi = 0.83±0.02 cm2/J for the mixed mode UV beam, where h...i

indicates averaging according to the fractional power of the PI beam at the wavelengths of 364 and 351 nm.

2.3.3 Excited state fraction

To determine σ, the excited state fraction f must be ascertained. In order to overcome cumbersome calculations to determine the excited state fraction [22], we used an empirical expression developed by Townsend et al. [23],

f = 1

2

CS

(31)

Figure 2.4: Measured ratio of fluorescence powers P2/P1as a function of the ratio of intensities Im

2 /I1m. Every point is a statistical average over at least 8 measurements. The solid line is a fit to the data using Eq. 2.7. The dotted line represents Eq. 2.7 for A=0.

Here, S = Im/Im

0 , where I0m is the saturation intensity of the 3P2 3D3 transition, and

Im is the laser intensity of all MOT beams combined. The detuning of the MOT beams δ

is expressed in units of Γ, and for this experiment was set to δ = −0.5. The quantity C is a phenomenological factor which lies between the average of the squared Clebsch-Gordan coefficients of all involved transitions and 1. For the 3P

23D3 transition, the average of

the squared Clebsch-Gordan coefficients is 0.46.

The large uncertainty in C and in the effective intensity Im seen by the atoms, due

to laser beam imbalances and alignment uncertainties, makes it necessary to measure the excited state fraction. For this we adopted a modified version of the technique used by Townsend et al. [23]. We measured the power P1 of the fluorescence scattered by the atoms

at a certain trap-laser intensity Im

1 , which is proportional to the excited state fraction f1

at that intensity. We then suddenly changed the intensity and measured the power P2 of

the fluorescence emitted by the atoms with the new intensity Im

2 (Townsend et al. applied

a rapid change in detuning). The intensity was changed fast enough to make sure the loss in atom number and the movement of the atoms may be neglected.

Figure 2.4 shows the ratio P2/P1 as function of the ratio between the two intensities

Im

2 /I1m. Our data fits well with the following equation:

f2 f1 = I2m Im 1 1 + A 1 + A(Im 2 /I1m) , (2.7)

derived from Eq. (2.6). Here, A = CIm

1 /I0m(1 + 4δ2) is the only fit parameter. Having

(32)

state fraction at the MOT laser intensity used in the decay experiments (corresponding to

Im

2 /I1m = 1) to be 21.7±1.5 %. From Eq. (2.6) it follows that f = (1/2)A/(1 + A) so that

uncertainties in the value of the detuning have no influence on this determination.

From the fitted value of A we can extract an estimate for C by inserting the calculated effective intensity Im of the MOT beams, which serves as a consistency check. This results

in a value of 0.4±0.1 for the phenomenological constant C, which is indeed in the expected range. We note that this cannot be regarded as an actual measurement of C since which we did not independently determine the intensity of the trapping light experienced by the atoms.

R (s )

-1

R (s )

-1

Figure 2.5: Linear loss rate as a function of the relative displacement of the PI beam with

respect to the atom cloud in (a) the x-direction, and (b) the y-direction. The curves represent parabolic fits to the data.

(33)

2.4 Sources of uncertainty and conclusions

Combining the measurements of the linear loss rate f σ due to photoionization and the excited state fraction f yields a value of 2.05 ± 0.18 × 10−18 cm2 for σ at 351 nm, and a

value of 2.15 ± 0.16 × 10−18 cm2 at 364 nm. The value for the cross section at 351 nm was

determined directly from the measurements with the laser in non-mixed mode. The value obtained was then used to extract the cross section at 364 nm from the measurements with the laser in mixed mode. We note that these cross sections correspond to effectively unpolarized atoms, due to the presence of trapping beams with various polarizations. The uncertainties given here correspond only to the statistical standard deviations as deter-mined by the measurements of total loss rate R versus PI intensity IP I and the excited

state fraction f (which is the dominant source of the uncertainty).

In our measurements there were also several systematic sources of uncertainty; all of these are related to the precision with which the average intensity IP I is known, estimated

to be within 8%. This comes mainly from four contributions: (1) The uncertainty of the transmission through the vacuum window between the point where we measured the power and where the atoms are (2%); (2) The uncertainty in the power measured with our power meter (5%); (3) The uncertainty in determining the waists of both the UV laser and the atomic distribution in the MOT. This gives an uncertainty in the determined intensity of 3%; (4) The alignment of the UV laser with respect to the atom cloud. Special care was taken to make sure that the photoionization beam was aligned with respect to the MOT. Figure 2.5 shows the effective decay rate as a function of horizontal (a) and vertical (b) displacements of the PI beam with respect to the MOT, displaced via the rotatable glass plate (see Fig. (2.1)). Based on these measurements we conclude that the MOT was within 0.3 σ of the 1/√e intensity radius σ of the PI beam. This gives an additional uncertainty

of 5% for the average intensity.

Taking these various uncertainties into account by adding them quadratically, we con-clude that we measured the cross section for Ne3D

3 atoms to be 2.05 ± 0.25 × 10−18 cm2

at a wavelength of 351 nm and 2.15 ± 0.25 × 10−18 cm2 at a wavelength of 364 nm. The

relative accuracy obtained (≈12%) is quite comparable to the result of Dinneen et al. using rubidium and is dominated by the contribution from the excited state fraction.

When we compare our measurement with previous experimental work done at 351 nm by Siegel et al. [11] using an atomic beam, then we find that our measurements are in perfect agreement with their value but a factor of four more precise. These authors determined the photoionization cross section at 351 nm to be (2 ± 1) × 10−18 cm2 by reference to

data on collisional ionization on which their absolute flux of excited-state atoms could be calibrated. As the measurements here only rely on relative atom numbers as measured by the fluorescence yield, in principle they can be made arbitrarily more precise by further improvement of the statistical accuracy and elimination of systematical errors.

For neon there do not seem to exist any recent calculations of absolute cross sections for ionization out of the 3p-state, in contrast to the situation for the metastable (3s)3P

0,2

states [14]. To compare the experimental data we have to refer back to somewhat older theoretical values from Duzy and Hyman [24], from Chang [25] and from Chang and Kim

(34)

[26]. Duzy an Hyman used a central-field approximation for the potential experienced by the outer electron, including a core-polarizability that was adjusted semi-empirically to generate binding energies that would match experimental data of about 15 excited states. Fine-structure effects were not taken into account. The photoionization cross sections were subsequently calculated from the resulting wavefunction of the outer electron. In this way, a value of ≈ 5×10−18cm2 for the cross section at 364 nm was obtained. Chang [25] instead

applied a Hartree-Fock treatment which included the influence of many-body corrections to the cross section; these, however, turned out to be quite small (<10%) for ionization wavelengths greater than 340 nm. In a subsequent paper, therefore, Chang and Kim [26] used a single-configuration Hartree-Fock treatment and found σ ≈ 4.5 × 10−18 cm2 at

364 nm, with an about 10% smaller value at 351 nm. All these values are beyond the high 3σ-border of our data. This suggests that a refinement of the calculations may be in order, for which the current data may serve as a benchmark.

2.5 Acknowledgements

We would like to thank A. Kemper, R. Rumphorst, W. Kemper, H. van Doorn, V. Mo-gendorff, C. Hawthorn, L. van Moll, J. van de Ven and R. Gommers for technical and experimental assistance. We also acknowledge useful communications with H. Hotop, and thank Coherent Netherlands B.V. for the loan of the Innova 90 laser used in these exper-iments. This work was financially supported by the Australian Research Council and the Netherlands Foundation for Fundamental Research on Matter (FOM).

Bibliography

[1] T. C. Killian, S. Kulin, S. D. Bergeson, L. A. Orozco, C. Orzel, and S. L. Rolston, Phys. Rev. Lett. 83, 4776-4779 (1999)

[2] T. Pohl, T. Pattard, and J. M. Rost, Phys. Rev. Lett. 92, 155003 (2004) [3] F. Robicheaux and J. D. Hanson, Phys. Rev. Lett. 88, 055002 (2002)

[4] S. Alo¨ıse, P. OKeeffe, D. Cubaynes, M. Meyer, and A. N. Grum-Grzhimailo, Phys. Rev. Lett. 94, 223002 (2005)

[5] I. H. Suzuki and N. Saito, J. Electron Spectrosc. Relat. Phenom. 129, 71 (2003) [6] J.A.R. Samson , W.C. Stolte, J. Electron Spectrosc. Relat. Phenom. 123 265 (2002) [7] A.A. Sorokin, L.A. Shmaenok, S.V. Bobashev, B. M¨obus and G. Ulm, Phys. Rev. A 58,

2900 (1998).

[8] I.D. Petrov, V.L. Sukhorukov, E. Leber, and H. Hotop, Eur. Phys. J. D 10, 53 (2000) [9] R. Kau, I. D. Petrov, V. L. Sukhorukov and H Hotop, J. Phys. B: At. Mol. Opt. Phys. 29

5673 (1996)

[10] J. Ganz, B. Lewandowski, A. Siegel, W. Bussert, H. Waibel, M.-W. Ruf and H. Hotop, J. Phys. B: At. Mol. Phys. 15, L485-489 (1982).

(35)

[11] A. Siegel, J. Ganz, W. Bussert and H. Hotop, J. Phys. B: At. Mol. Phys. 16 2945 (1983) [12] J. Ganz, M. Raab, H. Hotop and J. Geiger, Phys. Rev. Lett. 53, 1547 (1984)

[13] D. Klar, K. Ueda, J. Ganz, K. Harth, W. Bussert, S. Baier, J.M. Weber, M.-W. Ruf and H. Hotop, J. Phys. B: At. Mol. Opt. Phys. 27, 4897 (1994)

[14] I.D. Petrov, V.L. Sukhorukov and H. Hotop, J. Phys. B: At. Mol. Opt. Phys. 32, 973 (1999). [15] I. D. Petrov, V. L. Sukhorukov and H. Hotop, J. Phys. B: At. Mol. Opt. Phys. 35, 323

(2002)

[16] T. P. Dinneen, C.D. Wallace, K.-Y.N. Tan, and P.L. Gould, Opt. Lett. 17, 1706 (1992). [17] J. R. Lowell, T. Northup, B. M. Patterson, T. Takekoshi, and R. J. Knize, Phys. Rev. A

66, 062704 (2002).

[18] D. N. Madsen, J. W. Thomsen, J. Phys. B: At. Mol. Phys. 35, 2173 (2002). [19] C. Gabbanini, S. Gozzini, A. Lucchesini, Opt. Comm. 141, 25 (1997).

[20] J.G.C. Tempelaars, R.J.W. Stas, P.G.M. Sebel, H.C.W. Beijerinck, and E.J.D. Vredenbregt, Eur. Phys. J. D 18,113 (2002).

[21] S.J.M. Kuppens, J.G.C. Tempelaars, V.P. Mogendorff, B.J. Claessens, H.C.W. Beijerinck, and E.J.D. Vredenbregt, Phys. Rev. A 65,023410 (2002).

[22] J. Javanainen, J. Opt. Soc. Am. B 10, 572 (1993).

[23] C. G. Townsend, N.H. Edwards, C.J. Cooper, K. P. Zetie, C. J. Foot, A. M. Steane, P. Szriftgiser, H. Perrin, and J. Dalibard, Phys. Rev. A 52, 1423 (1995).

[24] C. Duzy and H.A. Hyman, Phys. Rev. A 22, 1878 (1980). [25] T. N. Chang, J. Phys. B: At. Mol. Phys. 15, L81 (1982). [26] T. N. Chang and Y. S. Kim, Phys. Rev. A 26, 2728 (1982).

(36)

Dipole-Dipole interactions in a frozen Rydberg gas

Abstract. Here we studied the effects of dipole-dipole interactions in a gas of cold Rydberg atoms b. We show that dipole-dipole interactions affect the formation of an

Ultracold Plasma and directly and quantitatively show their existence by studying the broadening of microwave transitions. The measured widths scale proportional to the density and scale with the fourth power of n, the principal quantum number, as can be expected from dipole-dipole interactions. In a last set of experiments we show that Ramsey experiments can be used to measure interactions between Rydberg atoms. We measured the loss of contrast as a function of density which we attribute to the effect of dipole-dipole interactions.

bThe work described in this Chapter, to be submitted for publication, has been performed

(37)

3.1 Introduction

A Rydberg atom has at least one highly excited electron and is therefore mainly character-ized by the principal quantum number n of the excited electron (typically n ≥10) [1]. For a typical Rydberg atom, the excited electron effectively sees a core of one unit of positive charge. This results in properties very similar to a those of a hydrogen atom. When the atoms in a Magneto-Optical Trap (MOT) (see Chapter 5) are excited to Rydberg states a gas of cold Rydberg atoms is created with a temperature of ≈ 300 µK and a density of

≈ 109 cm−3. On a typical time scale of 1 µs, these atoms move only about 3 % of the

interatomic spacing, and as such the Rydberg gas is considered to be ”frozen” [2]. Re-cently these systems became of interest for mainly two reasons. For one, the exaggerated properties such as large dipole moments make the system ideal for studying many-body interactions. These can be of practical interest for quantum computing. For example, an excited Rydberg atom in a cloud of cold atoms may block excitation of other atoms. This would allow to create a single qubit from a cloud of atoms without the need to address individual atoms [3–5].

Second, recently it has been shown that dipole-dipole interactions in a cold Rydberg gas can trigger the evolution to an Ultracold Plasma (UCP). Two dipole-coupled Rydberg atoms can collide and ionize, triggering an avalanche ionization that results in a UCP [6]. In this Chapter we will report the results of an experimental study of the role of dipole-dipole interactions in a frozen Rydberg gas. First we give a brief description of dipole-dipole-dipole-dipole interactions after which we will discuss the main elements of the setup. In the next part we present an example measurement showing that the formation of a plasma can be triggered by driving dipole-dipole interactions. In the next part we show that the dipole-dipole in-teraction leads to line broadening of microwave transitions. In the last section we report preliminary results of a Ramsey experiment aimed at studying the effects of dipole-dipole interactions in an alternative way.

3.2 Dipole-Dipole interactions

The fundamental interactions between two neutral atoms are their electric and magnetic multipole interactions; the longest range of these is the dipole-dipole interaction. Classi-cally, two dipoles µ1 and µ2 interact through the dipole-dipole interaction V given by [7]:

V = µ1· µ2− 3(µ1· ˆR)(µ2· ˆR)

R3 , (3.1)

with ˆR the unit vector along the direction connecting the two dipoles and R the distance

between the two dipoles. This equation can be written in a scalar form, assuming parallel or anti-parallel orientation of µ1 and µ2 along the ˆz direction. This is for example the case

for transition dipoles induced by linearly polarized laser or microwave excitation. One then gets:

V = µ1µ2(1 − 3 cos2θ)

(38)

Here θ is the angle between ˆz and ˆR as indicated in Fig. 3.1a, where the dipoles are

as-sumed to be polarized along the z-axes. The angular dependency of Eq. 3.2 makes that the interaction can be either attractive or repulsive, depending on the angle θ. This is indicated in Fig. 3.1b.

In our experiments we typically work with Rydberg atoms in the n=40 state (the

transi-m 2 m 1 R q z

a

b

Figure 3.1: (a) Two dipoles, µ1and µ2are polarized along the z direction and separated by a distance R. The angle between the polarization of the dipole moments and the interatomic axes is referred to as θ. (b) Angular dependence of the dipole-dipole interaction. Three equipotential lines are depicted. The lobes on the top and on the bottom (with an angle θ ranging from 306◦ to +54◦ and from 126 to +216) correspond to an attractive interaction. The left and right lobe (with a angle ranging from 54◦to +126and from 216to +306) correspond to a repulsive interaction.

tion dipole moment µ for a Rydberg atom in the 35s state is approximately µ=1000 ea0[1]).

At a typical density of ρ =109cm−3, the average interatomic spacing < R >= (4ρπ/3)−1/3=

6 µm. The resulting interaction energy is of the order of 10 MHz. For a 85Rb Rydberg

atom the acceleration a, at this interaction strength (|F| = 3µ2/R4) is then of the order

25×103 m/s2. The motion induced by this acceleration results on a typical timescale of

about 1 µs in a displacement of about 15 nm, small compared to the average interatomic spacing (6 µm). In a typical sample, however, there are always pairs at a smaller inter-atomic spacing. For these pairs the motion becomes important and causes ionization. A pair of Rydberg atoms in the 40 s, p state, starting at a infinite separation and moving towards eachother on the attractive curve can ionize at an interatomic spacing of approxi-mately 2 µm. In Section 3.5 we will show that this ionization can trigger the formation of a UCP from a Rydberg gas.

In a quantum mechanical treatment, two atoms of opposite parity that are dipole con-nected (e.g. s and p atoms) can form two molecular states sp and ps which are degenerate at infinite interatomic spacing [2]. At finite separation, however, they are coupled by the dipole-dipole interaction V . In a very simplified approximation the Hamiltonian of this

Referenties

GERELATEERDE DOCUMENTEN

In het algemeen kan worden gesteld dat vooral de omstandigheden waaronder de rij taak moet worden uitgevoerd van belang zijn voor het mogelijke effect van reclame-uitingen op

‘Het Vlaams Welzijnsverbond staat voor boeiende uitdagingen in sectoren van zorg en ondersteuning van kwetsbare doelgroepen’, zegt Chantal Van Audenhove.. ‘Samen met het team

af te leiden dat de ontlading allesbehalve homogeen is, wat na- tuurlijk de nodige vraagtekens zet bij de diverse homogeniteits- eisen voor de twee optische diagnostieken. Indien

Bij de toetsing van het kniemodel san de resultsten van de knie- analyse-experimenten is kennie van de hierboven beschreven fak- toren van groot belang. Andersom zullen ze de

Er is veel artrose in de wervelkolom aangetroffen, er zijn twee individuen met geheelde botbreuken, twee kinderen zijn vermoedelijk aan tuberculose overleden, één

Het percentage planten met TRV symptomen (geplant als pit, 4ppp, schema 1) geteeld in besmette grond om primaire symptomen, (grijze kolommen) en het percentage zieke planten in

Hoewel de minerale samenstelling slechts een ruwe karakterisering van de beide soorten deeltjes is, geeft het wel aan dat ijzer- houdende colloïden die ontstaan als gevolg van

Deze kengetallen kunnen weliswaar door heel andere factoren beïnvloed worden dan gezondheid, maar ongewenste afwijkingen ten opzichte van de norm (zoals verstrekt