2S 3S two-photon transition frequency measurement of ...ywliu/thesis/chiahsiang.pdf · Abstract 2S...

2S → 3S two-photon transition frequencymeasurement of atomic lithium

Chia-Hsiang HsuNational Tsing Hua University

August 21, 2005

Abstract

2S → 3S two-photon transition frequency

measurement of atomic lithiumMaster s dissertation

Chia-Hsiang HsuNational Tsing Hua University, Taiwan 2005

Lithium 2 2S1/2 → 3 2S1/2 transition has been observed by high-precision laser spec-troscopy using two-photon Doppler-free excitation and fluorescence detection. Thetwo-photon excitation is performed in a weakly collimated atomic beam using atitanium-sapphire (TIS) ring laser at 735 nm with 450 mW laser power. Four tran-sition lines, including isotopes 6Li and 7Li, were observed and the linewidth is 10MHz and SNR of the strongest line is ∼ 100. Absolute frequencies of all hyperfinecomponents have been measured to an uncertainty of 280 kHz using optical femtosec-ond comb. The resulting frequency of 7Li : F = 2 − 2 is 407808975.87(13) MHz andisotope shift is 11454.95(51) MHz. There are discrepancies with recent work [1], andthis may be due to laser instability in our system. The results are compared withthe theoretical works, including relativistic and QED energy contributions, and theaccuracy is improved by an order of magnitude. Combined with theory, the squarednuclear radii difference between 6Li and 7Li is 1.2(3) fm2.

Contents

1 Introduction 1

1.1 Lithium and fundamental atomic physics . . . . . . . . . . . . . . . . 1

1.1.1 Precision physics of simple atoms . . . . . . . . . . . . . . . . 1

1.1.2 Interests in lithium property . . . . . . . . . . . . . . . . . . . 2

1.2 Atomic theory calculations . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Nonrelativistic wave functions [2] . . . . . . . . . . . . . . . . 4

1.2.2 Relativistic correction . . . . . . . . . . . . . . . . . . . . . . 6

1.2.3 Quantum electrodynamic corrections . . . . . . . . . . . . . . 7

1.2.4 Nuclear size correction . . . . . . . . . . . . . . . . . . . . . . 8

1.2.5 Transition energy and isotope shifts . . . . . . . . . . . . . . . 9

1.3 Doppler-free two-photon transition . . . . . . . . . . . . . . . . . . . 10

1.3.1 Multiphotonic transitions . . . . . . . . . . . . . . . . . . . . 11

1.3.2 Two-photon transition probability . . . . . . . . . . . . . . . . 11

1.3.3 Two-photon absorption lineshape . . . . . . . . . . . . . . . . 13

1.3.4 Light shifts (ac Stark effect) . . . . . . . . . . . . . . . . . . . 15

1.4 Optical femtosecond comb based on Mode-locked Ti:sapphire laser . . 16

1.4.1 Mode-locked Ti:sapphire laser . . . . . . . . . . . . . . . . . . 17

1.4.2 Supercontinuum generation . . . . . . . . . . . . . . . . . . . 18

2 Experiment 21

2.1 Atomic structure of lithium . . . . . . . . . . . . . . . . . . . . . . . 21

2.1.1 Lithium property . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1.2 Energy level diagram . . . . . . . . . . . . . . . . . . . . . . . 23

I

CONTENTS II

2.2 Review of 2S → 3S two-photon spectroscopy . . . . . . . . . . . . . . 23

2.3 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.1 Laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.2 Atomic lithium beam . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.3 Fluorescence detection . . . . . . . . . . . . . . . . . . . . . . 25

2.4 Absolute frequency measurement . . . . . . . . . . . . . . . . . . . . 26

2.4.1 Femtosecond comb system . . . . . . . . . . . . . . . . . . . . 26

2.4.2 Repetition rate and offset frequency stabilization . . . . . . . 26

2.4.3 Femtosecond comb test . . . . . . . . . . . . . . . . . . . . . . 27

2.4.4 Beat measurement . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Results and discussions 33

3.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 Systematic effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2.1 Doppler background . . . . . . . . . . . . . . . . . . . . . . . 36

3.2.2 Light shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2.3 Second-order Doppler shift . . . . . . . . . . . . . . . . . . . . 38

3.3 Hyperfine constant, isotope shift, and nuclear size . . . . . . . . . . . 39

3.4 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 Conclusion 43

List of Figures

1.1 Schematic diagram of theoretical calculation process. . . . . . . . . . 8

1.2 Two-photon diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3 Energy level diagram of two-photon transition. . . . . . . . . . . . . . 13

1.4 Lineshape simulation of two-photon transition. . . . . . . . . . . . . . 15

1.5 Schematic experimental arrangement of laser excitation spectroscopy

with reduced Doppler width in a collimated atomic beam. . . . . . . 16

1.6 Ultrashort pulse train emitted by a mode locked laser and the corre-

sponding spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.7 Frequency spectrum of comb lines and the unknown laser frequency. . 20

2.1 Lithium vapor pressures (torr) versus temperatures (C) and the fitting

curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Partial energy level diagram for lithium (6Li and 7Li). . . . . . . . . . 28

2.3 Hyperfine structure of 6Li and 7Li. . . . . . . . . . . . . . . . . . . . 29

2.4 Experimental setup using 735 nm TIS laser. . . . . . . . . . . . . . . 30

2.5 The spectrum after the photonic crystal fiber caught by the portable

spectrometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.6 The experimental setup of the femtosecond comb system. . . . . . . . 32

3.1 Two-photon spectrum of 6Li. . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Two-photon spectrum of 7Li. . . . . . . . . . . . . . . . . . . . . . . . 35

3.3 The beat signal recorded in spectrum analyzer. . . . . . . . . . . . . 36

3.4 The histogram of 7Li 2S1/2 → 3S1/2 (F = 2→ F = 2) transition. . . . 37

III

LIST OF FIGURES IV

3.5 The histogram of 7Li 2S1/2 → 3S1/2 (F = 1→ F = 1) transition. . . . 38

3.6 The histogram of 6Li 2S1/2 → 3S1/2 transition. . . . . . . . . . . . . . 39

3.7 Line profile simulation for this experiment. . . . . . . . . . . . . . . . 40

List of Tables

2.1 Some physical properties of lithium at 20C, 1 atm. . . . . . . . . . . 21

2.2 Lithium vapor pressures (torr) versus temperatures (C). . . . . . . . 22

2.3 Decay channels of lithium. . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 2S → 3S two-photon transition of 7Li and 6Li measured by Bushaw et

al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1 Measured frequency comparison with ref [1]. . . . . . . . . . . . . . . 40

3.2 Hyperfine structure constants, isotope shift, and transition energies of

the 6,7Li two-photon transition 2 2S1/2 → 3 2S1/2. . . . . . . . . . . . . 41

3.3 Values for the squared difference in nuclear radii. . . . . . . . . . . . 42

V

Chapter 1

Introduction

1.1 Lithium and fundamental atomic physics

1.1.1 Precision physics of simple atoms

Simple atoms are a basic physical object and their simplicity has been a challenge

for theory and experiment for some time. A simple atom should be explained with

a physically transparent and simple theory and studied with experiments based on

simple ideas. At present-day the field is still attractive to physicists because of the

clear physical nature of different phenomena and a possibility to perform both precise

calculations and measurements.

After a century of work in the physics of simple atoms, the list of such atoms

available for study is quite long [3]:

• hydrogen, and its isotopes: deuterium and tritium;

• exotic atoms: positronium, muonium, muonic atoms, pionic atoms, antiprotonic

atoms, etc. ;

• antihydrogen;

• helium atom;

• hydrogen-like and helium-like ions;

1

CHAPTER 1. Introduction 2

• the Rydberg states, etc.

Based on optical precision spectroscopy of the simple atoms, the related studies of

applications include:

• precision determination of fundamental constants;

• precision test of QED and bound state QED;

• study of proton structure and other particle properties;

• study of nuclear structure;

• search for variation of fundamental constants;

• search for violation of fundamental symmetries;

• new frequency standards etc.

Precision physics of simple atoms offers the opportunity of interdisciplinary ex-

change between atomic spetroscopy, nuclear and particle physics and quantum field

theory. Fundamental physical constants are universal and their values are needed

for different problems of physics and metrology. The determination of constants is a

necessary and important part of most of the so-called precision test of the QED and

bound state QED and that makes the precision physics of simple atoms an important

field of a general interest.

1.1.2 Interests in lithium property

Hydrogen and other two-body systems have long been regarded as the ‘fundamental’

systems of atomic physics because the Schrodinger (or Dirac) equation can be solved

exactly to give a lowest-order description of the system. The results to be reviewd

here will show that helium (and other three-body systems) now stand on the same

footing with hydrogen in that solutions to the Schrodinger equation that are essentially

exact for all practical purposes are readily obtained. The mian idea of high precision

variational calculation is expanding the wavefunction in Hylleraas coordinates.


Lithium is one of the fundamental few-body problems of atomic physics. The

method for the calculation of accurate nonrelativistic wave functions and energies can

in priciple be applied to more complex atoms and molecules. The principal difficulties

are that the number of terms required in the basis set to reach a given level of accuracy

grows extremely rapidly with the number of particles, and the correlated integrals

become much more difficult to evaluate. In the case of lithium (and Li-like ions) have

results of spectroscopic accuracy been obtained [4].

Data on nuclear charge radii is of fundamental importance for nuclear physics.

Among the unstable isotopes, 11Li is of particular interest since it consists of a 9Li

core surrounded by a “halo” of two loosely bound neutrons [5]. Halo nuclei are mainly

characterized by a small binding energy for one or more of their nucleons, and they

have mainly been seen in ground states of light nuclei. The best known one-neutron

halo nucleus is 11Be and among the two-neutron halo nuclei the best established ones

are 11Li and 6He. Furthermore, 8B is a well-explored candidate for a one-proton halo

and 8He is thought to be a four-neutron halo.

The 2S − 3S transition will be used for studies on the Li isotopes because of

the high-resolution achievable with Doppler-free two-photon excitation. Therefore, a

precise isotope shift (IS) measurement for this transition is needed to verify that all

contributions have been calculated correctly and to sufficient accuracy.

1.2 Atomic theory calculations

This section gives a brief survey of the variational method used to solve the nonrel-

ativistic problem, and the principal effects that must be taken into account in order

to estimate the relativistic and QED corrections. All the contributions to the energy

and their orders of magnitude are expanded with parameters α, the fine structure

constant, and µ/M , the ratio of the reduced electron mass to the nuclear mass for Li.

The lithium low-lying energy level and isotope shift for 2 2PJ → 2 2S and 3 2S → 2 2S

transition energies in lithium are calculated variationally in Hylleraas coordinates,

including nonrelativistic, relativistic, and QED terms, and the lowest-order finite nu-


clear size correction, by Drake et al [2][4][6][7][8][9][10][11].

1.2.1 Nonrelativistic wave functions [2]

Perturbation theory

After rescaling distances according to r → (m/µ)r, the Hamiltonian for a three-

electron atomic system is

H = H0 + λH ′, (1.1)

with

H0 = −1

2

3∑i=1

∇2i − Z

3∑i=1

1

ri+

3∑i>j

1

rij, (1.2)

and the mass polarization term

H ′ =3∑i>j

∇i · ∇j, (1.3)

in unit of 2RM , where RM = (1 − µ/M)R∞, µ = mM/(m + M) is the electron

reduced mass, and λ = −µ/M , which can be treated as a perturbation parameter.

The Schrodinger equation

HΨ = EΨ (1.4)

can be solved perturbatively by expanding Ψ and E according to

Ψ = Ψ0 + λΨ1 + · · · ,

E = ε0 + λε1 + λ2ε2 + · · · . (in unit of 2RM)

(1.5)

Finite nuclear mass effects and the mass polarization operator are taken into account

up to second order by perturbation theory and the explicit mass dependence of E is

E = ε0 + λ(ε0 + ε1) + λ2(ε1 + ε2) +O(λ3) in unit of 2R∞. (1.6)


Variational calculations

The presence of the 1/rij Coulomb repulsion term makes the Schrodinger equation

nonseparable, so both Ψ0 and Ψ1 were solved variationally in multiple basis sets in

Hylleraas coordinates containing terms of the form [12]

Ψ = AΩ∑µ

Cµφµ (1.7)

φ(r1, r2, r3) = rj11 rj22 r

j33 r

j12

12 rj23

23 rj31

31 e−αr1−βr2−γr3

×YLM(l1l2)l12,l3(r1, r2, r3)χ1 , (1.8)

where A is the antisymmetrizer and Cµ the variationally determined expansion co-

efficients. YLM(l1l2)l12,l3is a vector-coupled product of spherical harmonics for the three

electrons to form a state of total andgular momentum L, and χ1 is a spin function

with spin angular momentum 1/2. α, β, and γ are additional nonlinear scale factor

that can be separately adjusted to optimize the energy.

The pricipal computational step (process solving the eigenfunction Ψ and eigen-

value E) is to diagonalize the H matrix in the non-orthogonal Hylleraas basis sets.

This is equivalent to satisfying the variational condition

δ

∫Ψ∗(H − E)Ψdτ = 0. (1.9)

All terms of powers j1, j2, · · · , j31 are nominally included such that

j1 + j2 + j3 + j12 + j23 + j31 ≤ Ω (1.10)

and the convergence of the eigenvalues is studied as Ω is progressively increased.

Asymptotic expansion method [6]

The evaluation of correlated three-electron integrals arising from Eq. 1.9 involving

correlations among all three particle pairs r12, r23, and r31 is complicated. There is

no longer a simple, close-form expression, as in the calculation for helium. To reach


high precision it requires larger basis set and the radial integrals converge very slowly

with extremely time consuming. There are some accuracy considerations:

• the radial integrals,

• individual matrix elements,

• solution of the secular equation,

• resulting lowest eigenvalues.

With the expansion for the interelectron coordinate r12 derived by Perkins [13],

rj12 =

L1∑q=0

Pq(cos θ12)

L2∑

k=0

Cjqkrq+2k< rj−q−2k

> ,

an asymptotic-expansion method reduces the time required to calculate even the

most troublesome integrals and also preserves accuracy and numerical stability. The

method has recently been successfully applied to the low-lying states of lithium. It

may be also applied to the more sigular integrals which arise in the calculation of

relativistic coorections from the Breit interaction.

1.2.2 Relativistic correction

The lowest-order relativistic corrections of O(α2) come from the Breit-Pauli inter-

action, and the relativistic recoil terms are finite nuclear mass corrections to these.

The anomalous magnetic momentum terms are taken into account as corrections of

O(α2) to the basic Breit-Pauli interaction. The lowest-order relativistic corrections of

O(α2) and the spin-dependent anomalous magnetic moment corrections of O(α3) can

be written in the form (in atomic units) [14]:

∆Erel = 〈Ψ |Hrel|Ψ〉J , (1.11)


where Ψ is a nonrelativistic wave function, and Hrel is defined by

Hrel = B1 +B2 +B3e +B3Z +B5

+Zπα2

2

3∑i=1

δ(ri)− πα2

3∑i>j

(1 +8

3si · sj)δ(rij)

+m

M(∆2 + ∆3Z)

+γ(2B3Z +4

3B3e +

2

3B

(1)3e + 2B5 +

m

M∆3z) (1.12)

Bi are the Breit-Pauli interaction terms [15], the terms proportional to m/M are the

nuclear relativistic recoil corrections derived by Stone [16], and the terms propor-

tional to γ, Euler’s constant, are the anomalous magnetic momentum corrections. In

addition, finite-nuclear-mass corrections of order O(α2/M) a.u. come from the mass

scaling of these terms, cross terms with the mass polarization operator.

1.2.3 Quantum electrodynamic corrections

The QED correction can be written in the form according to the formulation of

McKenzie and Drake:

EQED = EL,1 + EL,2 + EHO, (1.13)

where EL,1 and EL,2 represent the lowest-order α3 Ry electron-nucleus and electron-

electron terms, respectly, and EHO represents higher-order terms of O(α4) Ry and

higher. EL,1 is given by the Kabir-Salpeter formula [17]:

EL,1 =4

3α3Z

∑i

〈δ(ri)〉[ln(α)−2 − β(nLS) +

19

30

], (1.14)

where β(nLS) = lnk0(nLS) is the Bethe lagarithm for a state with principal, angular

momentum, and spin quantum numbers n, L, and S, respectively, and EL,2 accounts

for the Araki-Sucher terms [18]:

EL,2(nLS) =

(14

3lnα +

164

15

)∑i>j

〈δ(rij)〉 − 14

3Q, (1.15)


where

Q =1

4π

∑i>j

lima→0

⟨r−3ij (a) + 4π(γ + lna)δ(rij)

⟩. (1.16)

The principal difficulty is the calculation of the so-called Bethe logarithm, which

determines the dominant part of the electron self-energy. The dominant contribution

comes from a sum over inner shell excitations to intermediate state lying high in the

photoionization continuum. In addition, high accuracy is required since the first few

significant figures in the Bethe logarithm are state independent, and so cancal from

the physically relevant transition frequencies. Recently a novel finite basis set method

[11] is used to calculate the Bethe logarithm for the ground 2 2S1/2 and excited 3 2S1/2

states of lithium. The basis sets are constructed to span a huge range of distance scales

within a single calculation, leading to well-converged values for the Bethe logarithm.

1.2.4 Nuclear size correction

Finally, the last correction to be included is that due to finite nuclear size. It is given

in lowest-order by

∆Enuc =2πZr2

rms

3〈δ(ri)〉 , (1.17)

where rrms = Rrms/aBohr, Rrms is the root-mean-square radius of the nuclear charge

distribution, and aBohr is the Bohr radius. A mass scaling factor of (µ/m)3 is included

in the definition of 〈δ(ri)〉.

H rel

H QED

E NR

E rel

E QED

&

& variational

method

perturbation

theory

asymptotic

expansion

method

expectation

value

E total

(mass dependent)

isotope shift

H NR

=H 0 + H

NR

Figure 1.1: Schematic diagram of theoretical calculation process.


1.2.5 Transition energy and isotope shifts

All the above relativistic, QED, and nuclear size contributions are taken into account

as corrections to the total energy. Expectation values for this terms are calculated

from nonrelativistic wavefunctions expressed in Hylleraas coordinates and solved vari-

ationally in Eq. 1.1. The perturbing effect of mass polarization on the expectation

valus of the operators can be obtained using

Ψ = Ψ0 + λ(Ψ1 − 〈Ψ1|Ψ0〉Ψ0) + · · · , (1.18)

where the first two terms of the right-hand side are orthogonal to each other. Thus

for an operator A

〈Ψ |A|Ψ〉 = a0 + λa1 + · · · ,

a0 = 〈Ψ0 |A|Ψ0〉 ,

a1 = 2 〈Ψ0 |A|Ψ1〉 − 2 〈Ψ0|Ψ1〉〈Ψ0 |A|Ψ0〉(1.19)

in unit (µ/m)n 2R∞, where −n is the degree of homogeneity of operator A in three-

electron coordinates space such that A(βr) = β−nA(r). Using (µ/m)n = (1 + λ)n ≈1 + nλ, one has the explicit mass-dependent formula

〈Ψ |A|Ψ〉 = a0 + λ(na0 + a1) +O(λ2) in unit of 2R∞. (1.20)

Combining all the coefficients of µ/M in the calculations gives the 2 2S → 3 2S tran-

sition:

f2 2S−3 2S = −0.133 767 15(64) (µ/M)

+0.123 648 10(29) (µ/M)2

−0.666 646 3(55) r2rms + 1.980 2(19) r2

rms(µ/M), (1.21)

for all Li isotopes in units of 2R∞, and 2 2S → 2 2P1/2,3/2 transitions are also given in

[2].


Determination of nuclear radii

The formula gives the Li isotope shifts and can be provided to determine the charge

radii of each Li isotopes. The principal is that, if all mass-dependent contribution to

the IS can be calculated with sufficient accuracy, the residual discrepancy between

experiment and computation is caused by differences in nuclear charge radii. To a

first approximation, the QED terms are independent of the nuclear mass, and so they

largely cancel from the calculated isotope shift. The significance of this method is

therefore that the nuclear radius can be determined independently of QED uncertain-

ties. The nuclear radius squared of an arbitrary isotope ALi relative to 6Li is expressed

in the form

R2rms(

ALi) = R2rms(

6Li) +EA

meas − EA0

C, (1.22)

EAmeas is the measured isotope shift for ALi relative to 6Li, and EA

0 contains all the

calculated contributions to the isotope shift except for the nuclear size contributions.

The constant C:

C =2πZ

3

[〈δ(ri)〉i − 〈δ(ri)〉f

]

= −1.5661 MHz/fm2 for 2 2S → 3 2S (1.23)

depends on the transition i→ f in question, but it is nearly independent of the mass

number A.

The recent results of theoretical calculation by Drake et al. for 2 2S1/2 → 2 2S1/2

transition frequency of 7Li is 27206.0926(9) cm−1 [11], and the isotope shift value of

EA0 for 7Li− 6Li is 11453.00(6) MHz [10].

1.3 Doppler-free two-photon transition

The limitation of accurate measurements in the optical frequency of a gas sample is

mianly due to the thermal velocities of the particles. The Doppler effect causes a

frequency shift experienced by a moving particle. Thus the spectrum appears to be a

broad Gaussian lineshape, rather than the narrow Lorenzian lineshape caused by the


finite lifetime of the upper level. In two-photon transitions, the Doppler broadening

is eliminated by using two counterpropagating beams.

1.3.1 Multiphotonic transitions

Like one photon E1 transitions, multiphoton absorptions may also occur in optical

transitions. The first evidence of two-photon absorption in the optical range between

discrete levels of atoms was performed by Abella(1962) [19].

Consider an atom of velocity V interacting with several plane waves with each

wavetor ki, The first-order Doppler shift for each wave is V · ki. If the atom absorbs

n photons k1, k2, . . . , kn, the n-photon transition is shifted by a quanty equal to

n∑i=1

ki · V , (1.24)

and if∑ki = 0 the n-photon absorption will not be Doppler-shifted . Suppose that a

two-photon transition can occur between the levels Eg and Ee of an atom in a standing

electromagnetic wave of angular frequency ω. If the atom absorbs one photon from

each wave, the conservation of energy implies (Fig. 1.2)

Ee − Eg = ~(ω + kV ) + ~(ω − kV ) = 2~ω. (1.25)

which has no velocity dependence of atoms. It means that, at resonance, all the atoms

can absorbe two photons irrespective of their velocities. Theorectically, the width of

this resonance is of the same order of magnitude as the natural width Γe (inverse of the

excited state lift time). In order to apply this effect to high-resolution spectroscopy,

the laser linewidth should be smaller than the natural width of the transition.

1.3.2 Two-photon transition probability

The two-photon transition probability may be calculated using perturbation theory to

second order. The probability of exciting an atom from ground state g to an excited


Atom velocity = v

w w

Figure 1.2: Two-photon diagram.

state e is equal to [20]

P 2ge(δω) =

∣∣∣∣∣∑i

〈e |H1| i〉〈i |H2| g〉+ 〈e |H2| i〉〈i |H1| g〉∆ωi

∣∣∣∣∣

2Γe

4δω2 + 14Γ2e

+

∣∣∣∣∣∑i

〈e |H1| i〉〈i |H1| g〉∆ωi

∣∣∣∣∣

2Γe

4(δω − kV )2 + 14Γ2e

+

∣∣∣∣∣∑i

〈e |H2| i〉〈i |H2| g〉∆ωi

∣∣∣∣∣

2Γe

4(δω + kV )2 + 14Γ2e

. (1.26)

The notation used is as follow (illustrated in Fig. 1.3):

• H1 and H2 are the electric dipole interaction Hamiltonian of the atom with

incident and reflected waves,

• δω = ω − ω0 is the difference between the laser frequency ω and the resonance

frequency ω0 = (Ee − Eg)/2~,

• ∆ωi = ω−(Ei−Eg)/~ is the energy defect of the one-photon transition for each

intermediate state i,

The first term includes two processes: the atom absorbs a photon of the incident wave

followed of the reflected wave, or the reflected wave is first absorbed. The second and

third terms indicate the absorption of two photons of the incident waves or reflected

waves, and are dependent on atom velocities. The electric dipole interaction Hamil-

tonian is proportional to the elctric field, so the two-photon transition probability is

proportional to I2, square of the electric intensity. The probability is approximately


E r

E g

Two-photon process

One-photon process

E e

Energy detuning 0

w

0 w

w

Figure 1.3: Energy level diagram of two-photon transition, where ~∆ω is energy de-tuning, ω0 is the transition angular frequency, Er is a real imtermediate state betweenground state Eg and excited state Ee.

proportional to the inverse square of the energy detuning. In order to obtain a sig-

nificant transition probability, it is needed a real intermediate state near the middle

of the transition and with sufficient laser power. One of the limitations is that two-

photon transition is only allowe in E1 forbidden transitions, like S → S, S → D or

P → P . The states g and e have the same parity, opposite to the intermediate states

i.

1.3.3 Two-photon absorption lineshape

For two opposite light beams with the same polarization, frequency and intensity, the

two-photon transition probability from Eq. 1.26 can be written:

P 2ge(δω) =

∣∣∣∣∣∑i

〈e |H| i〉〈i |H| g〉∆ωi

∣∣∣∣∣

2

×[

4Γe4δω2 + 1

4Γ2e

+Γe

4(δω − kV )2 + 14Γ2e

+Γe

4(δω + kV )2 + 14Γ2e

]. (1.27)


Due to two possible processes for an atom absorbing two photons from two counter-

propagating beams, the probability appears as four times of a two-photon absorption

in a travelling wave. At resonance all the atoms contribute to transition intensity,

whereas at off-resonance there would be one atom group of velocities absorbing pho-

tons propagating in the same direction due to Doppler effect. Thus integrate the

transition probability with Maxwellian velocity distribution of atoms, and the two-

photon absorption lineshape appears as the superposition of one broadened Doppler

background (Gaussian) and a much narrower Lorentzian profile (see Fig. 1.4).

The ratio of the areas of Lorentzian and Gaussing profile is equal to 2. If Γe and

∆νD are the natural and Doppler widths, the intensity of the narrow Lorenztian curve

appears to be of the order of ∆νD/Γe lager than the intensity of the Gaussian curve.

In most cases the Gaussian Doppler curve will appear as a very small background.

The explicit expression of the transition probability at resonance is:

P (2)ge (res) =

1

Γe

(3

π

r0λgrλre~c

)2(P

S

)2ωgrωre∆ω2

r

fgrfre

× |〈Je1me − q|Jrmr〉〈Jr1mr − q|Jgmg〉|2 (1.28)

where r0 = e2/4πε0mc2 = 2.8×10−13 cm is the classical radius of the electron; fgr, fre

are the oscillator strengths of these one-photon transitions; P and S are respectively

the power and the cross section of the laser beam. The index q of the Clebsch-Gordan

coefficients charaterizes the polarizations of the light (q = +1, 0, −1 respectively for

the polarizations σ+, π, σ−).

Atomic beam method

On the contrary to cell gases, atomic beam means that the atoms are confined to

move in one direction (Fig. 1.5). If the laser beam is perpendicular to the atomic

beam, only the transverse velocity components contribute to Doppler frequency shift

V · k = Vtk. A well collimated atomic beam leads to narrow transverse velocity

distributions and so reduces the Doppler broadening effect. Approximately for the

atomic beam the transverse velocity distribution is Maxwellian and the Doppler width


Natural Linewidth

Doppler width

Frequency

Inte

nsity

Figure 1.4: Lineshape simulation of two-photon transition.

for beam diverging angle θ is:

∆νD′ ≈ ∆νD sin θ, (1.29)

and ∆νD =2ν0

cυp√

ln2 =2ν0

c

√2kT

mln2 (1.30)

where ∆νD is the Doppler width for a volume of gas, and υp is most probable speed

of atom. This technique also provides a collision-free environment but has the disad-

vantage of low number density interacting with the light source, on the contrary to

cell gases.

1.3.4 Light shifts (ac Stark effect)

It is important to consider whether there might be any systematic factors which might

shift the resonance. The light intensity which is required to induce the two-photon

transition might lead to light shifts [21], which can be interpreted as an ac-Stark effect.

The light shifts of ground and excited states caused by the photon energy detuned


oven

b

d laser

PMT

A

slit

Figure 1.5: Schematic experimental arrangement of laser excitation spectroscopy withreduced Doppler width in a collimated atomic beam.

from Er − Eg real transition is given by [20]:

δωg = 2〈g |H| r〉〈r |H| g〉

∆ωr(1.31)

δωe = 2〈e |H| r〉〈r |H| e〉

∆ωr. (1.32)

The factor of 2 corresponds to the two travelling waves (the value of the shift is twice

the value in a single travelling wave). The shift of the two-photon transition is equal

to (δωe − δωg) and is proportional to the inverse of the energy detuning.

1.4 Optical femtosecond comb based on Mode-locked

Ti:sapphire laser

Optical femtosecond comb is a revolutionary technique for optical frequency metrol-

ogy. Mode-locked lasers generate ultrashort optical pulses by establishing a fixed

relationalship across a broad spectrum [22]. The special feature of the laser is the

frequency spectrum consisting of a large number of exactly equidistant modes, like a

“frequency ruler” to measure any unknown frequency within an optical region. The

entire spectrum can be described by only two numbers: the frequency mode spacing

and an offset frequency which characterizes the absolute position of the comb with

respect to the frequency zero point. The absolute frequency of any coherent optical

source can be determined by detecting its beat note frequency with the one of the


comb mode.

1.4.1 Mode-locked Ti:sapphire laser

Optical frequency comb (OFC) generator is based on Kerr-lens mode-locked femtosec-

ond lasers [23][24]. The pulses are established by interference of successive longitudinal

modes oscillating simultaneously in the laser cavity. In general, The frequency compo-

nents are random in phase and not equal spacing. Mode-locking by fixing the relative

phase of all lasing modes can be achieved by knocking the cavity mirror. A single

pulse that got amplified is composed of frequency modes with phase locked together

and has a peak-peak period:

τrep =2L

c(1.33)

where L is cavity length. It is equivalent to that this short pulse bounces in cavity and

continuously gets amplified with a period of a round trip time, τrep. The repetition

rate of the mode-locked laser is equal to the mode spacing relative to the cavity:

frep =1

τrep=

c

2L. (1.34)

In general with such ultrashort pulses, the relative phase between peaks of the pulse

envolope and the underlying electric-field carrier wave is not constant from pulse to

pulse because the group and phase velocities differ inside the laser cavity. The carrier

wave (moves with phase velocity) shifts by ∆φ after each round trip with respect

to the pulse envelope (moves with group velocity) (Fig. 1.6). As a result, the comb

frequencies are integer multiples of the repetion rate added an offset frequency:

f0 =∆φfrep

2π, (1.35)

which is due to the difference between the group and phase velocity.


1.4.2 Supercontinuum generation

The spectrum of mode-locked laser typically can not cover an octave, i.e. the highest

frequencies are a factor of 2 larger than the lowest frequencies. Recently the introduc-

tion of high nonlinear photonic crystal fiber (PCF) has enable efficient generation of

coherent supercontinuum radiation [25]. The highly ninlinear photonic crystal fibers

guide light in a small solid silica core, surrounded by a micro-structured cladding

formed by a periodic arrangement of air holes in silica. The optical properties of the

core closely resemble those of a rod of glass suspended in air, resulting in strong con-

finement of light and a large nonlinear coefficient. Ultrafast laser pulses are passed

through the highly nonlinear photonic crystal fiber to generate a supercontinuum

spanning more than one octave. With the progresses of the femtosecond lasers and

photonic crystal fibers, the frequency comb from visible to near infrared region is well

established.

Finally, with stabilized repetition rate (frep) and offset frequency (f0), absolute

frequency of each comb line is known. Thus, the absolute frequency of a laser can be

determined by the beat frequency between the comb and the laser (Fig. 1.7):

f = nfrep ± f0 ± fbeat. (1.36)


fr

fn=nf

r+f

0

f

f0

Frequency domain

I(f)

0

Time domain

T = 1/frep

∆φ

t

E(t)

Figure 1.6: Ultrashort pulse train emitted by a mode locked laser and the correspond-ing spectrum.


frequency

Intensity

Beat frequency

Unknown laser frequency

Figure 1.7: Frequency spectrum of comb lines and the unknown laser frequency. Thefrequency difference between the laser and nearby comb line is beat frequency.

Chapter 2

Experiment

The 2 2S1/2 → 3 2S1/2 transition is used for studies on the isotopes 6Li and 7Li because

of the high-resolution achievable with Doppler-free two-photon excitation.

2.1 Atomic structure of lithium

2.1.1 Lithium property

Lithium is the lightest of all metals, with a density only about half that of water. It is

silvery in appearance, much like Na and K, other members of the alkali metal series. It

reacts with water, but not as vigorously as sodium, and the flame is a dazzling white

when it burns strongly. The physical properties of lithium is partly listed below:

Atomic number 3Group alkali metalAtomic mass average 6.914Density(@ 300K) 0.534 g/ccMelting point 180.7 CBoiling point 1342 CMolar volume 13 cm3/mole

Table 2.1: Some physical properties of lithium at 20C, 1 atm.

The fitting function of the pressure-temperature is:

log10P = 2.34− 20.20e(−T/347.95) (2.1)

21

CHAPTER 2. Experiment 22

Vapor pressure (torr) 10−8 10−7 10−6 10−5 10−4

Temperature (C) 235 268 306 350 404

Vapor pressure (torr) 10−3 10−2 10−2 1Temperature (C) 467 537 627 747

Table 2.2: Lithium vapor pressures (torr) versus temperatures (C).

200 300 400 500 600 700 800

-8

-6

-4

-2

0

Li va

po

r p

ressu

re lo

g[P

] (t

orr

)

Temperature ( o C)

Figure 2.1: Lithium vapor pressures (torr) versus temperatures (C) and the fittingcurve.

where P is vapor pressure in torr of lithium and T is the temperature in unit of C.

There are two stable naturally occurring lithium isotopes , 7Li and 6Li. The abun-

dance of 7Li and 6Li is 92.5% and 7.5%, respectively. The short-lived radioactive

isotopes with half-life T1/2 = 838 ms (8Li) and 178.3 ms (9Li) can be produced syn-

thetically by dircting high energy 12C beam onto a tungsten target [26]. The study

of 11Li, the most prominent neutron halo nucleus, is of great current interest because

this isotope consists of a 9Li core with a “halo” of two loosely bound neutrons orbiting

the nucleus [5].


2.1.2 Energy level diagram

The partial energy level diagram of lithium is shown in Fig. 2.2. The nuclear spins are

1 and 3/2 for 6Li and 7Li, respectively. The 2S1/2 states of 6Li and 7Li each split into

two hyperfine substates as shown in Fig. 2.3. The ground-state hyperfine intervals are

known to extremely high accuracy [27]. The most recent and accurate measurements

with the atomic beam magnetic resonance method yield for the separation between the

6Li(2S1/2) F = 1/2 and F = 3/2 states values of 228.205259(3) MHz, and 7Li(2S1/2)

F = 1 and F = 2 states values of 803.5040866(10) MHz.

Since selection rules limit two-photon transitions from an S state to ∆F = 0, there

are four observable transitions between the 2 2S1/2 and 3 2S1/2 levels , and these are

marked as A, B, C, and D for 7Li : F = 2 − 2, 7Li : F = 1− 1, 6Li : F = 3/2− 3/2,

and 6Li : F = 1/2− 1/2, respectively (see Fig. 2.3). ∆f7, ∆f6, and ∆f76 are denoted

by B-A, D-C, and A-C, respectively.

Decay routes A value (1/s)

3S1/2 → 2P3/2 2.178× 107

3S1/2 → 2P1/2 1.092× 107

2P3/2 → 2S1/2 3.721× 107

2P1/2 → 2S1/2 3.721× 107

Table 2.3: Decay channels of lithium.

In this two-photon transition, there are two decay channels for the upper level (see

Table. 2.3). The lifetime of upper 3S1/2 level state derived from A values is 29.8 ns

and the natural linewidth (FWHM) is:

(2.178× 107 + 1.092× 107)s−1

2π× 1

2= 2.6 MHz. (2.2)

The factor 1/2 is due to that the exciting frequency is half of the two-photon transition.

2.2 Review of 2S → 3S two-photon spectroscopy

Bushaw et al. [1] have studied the 2S → 3S transition of 6,7Li by high-precision laser

spectroscopy using two-photon Doppler-free excitation and photoionization detection.


Two-photon excitation is performed in a weakly collimated (' 5 FWHM) atomic

beam with a titanium-sapphire (Ti:sapphire, TIS) ring laser, followed by ionization

with a portion of the 514.5 nm light from its Ar ion pump laser. The TIS frequency is

stabilized by offset locking to a single-mode HeNe laser [28], using an evacuated and

temperature-stabilized scanning confocal interferometer as a transfer oscillator. The

measurement results are summarized below:

frequency (MHz)f(7Li : F = 2) 815 617 954(3)

∆f76 11 280.687(26)∆f7 617.291(22)∆f6 175.311(23)

Table 2.4: 2S → 3S two-photon transition of 7Li and 6Li measured by Bushaw et al.∆f7, ∆f76, ∆f6 are relative frequencies corresponding to B-A, A-C, D-C (see Fig. 2.3).

2.3 Experimental setup

2.3.1 Laser system

The experimental setup is shown in Fig. 2.4. In this experiment, the CW (continuous

wave) single-frequency ring Ti:Sapphire (TIS) laser (model TIS-SF-07, Tekhnoscan)

is used as exciting light source. Ths TIS laser is pumped by a diode-pumped 532

nm laser (Verdi-V6, Coherent). In optimized condition, the TIS laser generates a

maximum power of 600 mW at 735 nm with 5W pumped.

In two-photon excitation the laser light is totally retro-reflected, so a Faraday

optical isolator is used to prevent optical feedback. A small portion of laser light

was reflected out for laser diagnosis. A wavemeter (home-made) is used to determine

the laser frequency with accuracy to 2 GHz. Parts of the reflected light are sent

to a reference cavity (home-made, F.S.R = 300 MHz) and a scanning Fabry-Perot

(home-made, F.S.R = 1 GHz) to determine the scanning range of the laser and for

monitoring the laser mode, respectively. Most part of the reflected light about 20

mW is coupled into a fiber, sending to the femtosecond comb system with about 3


mW output for absolute frequency measurement.

2.3.2 Atomic lithium beam

Lithium metal is loaded in a stainless-steel oven. This oven is wound by a resistive

heater (resistance ≈ 25 Ω, heating length = 2 m) and two thermal couples are attached

onto the front and rear side of the oven for monitoring the oven temperature. To

avoid oxidation in air the lithium was being loaded under the condition of argon air

filled. The lithium oven is heated up to ∼500C ,corresponding to a vapor pressure

∼ 3.5 × 10−5 torr of lithium and the most probable speed 1356 m/s and a number

density 8×1021 m−3, in a vacuum chamber (∼ 10−6torr). The producing vapor effuses

through a small hole (diameter = 2 mm) drilled in front side of the heating oven. The

lithium source is a weakly collimated atom beam with diverging angle θ ∼ 2.4.

2.3.3 Fluorescence detection

Because the transitions were observed to be quite narrow (FWHM ≈ 10 MHz), the

laser frequency is set manually to the peak of each line. The laser frequency is read

from the wavemeter with uncertainty of 2 GHz. The 735 nm two-photon exciting laser

is directed into the vacuum chamber through a window. The laser is focused with

a lens (focal length = 50 cm) such that it intersects with the lithium beam at right

angles and retro-reflected by a concave mirror (R = 25cm). The focused laser beam

diameter is 100 µm corresponding to a Rayleigh range 1 cm. The focusing points of

the focused and that of the retro-reflected coincide within the lithium beam region

to optimize the two-photon transition. This implies that the retro-reflected beam is

the same as the incident beam anywhere. Due to reflection of the lens the laser light

may be resonant between the lens and the concave mirror, so the lens is tilted with a

small angle.

When two-photon excitation occurs, the fluorescence decay at 670 nm, correspond-

ing to the decay path 2P → 2S, is observed. The fluorescence from the laser/atom

interaction region is collected by a lens (f = 2 cm), and detected by the photomul-


tiplier tube (PMT, R446 Hamamatsu). A 670.8-nm interference filter (bandwidth 3

nm FWHM) is placed in front of the PMT to limit spurious counts from scattered

laser light and background radiation, as shown in Fig. 2.4. The PMT is housed in a

gray PVC tube to reduce the background noise and providing a rigid mount.

The laser light is amplitude modulated (modulation frequency 2500Hz) by a chop-

per. The fluorescence detected by the PMT is demodulated using a lock-in amplifier

(SCITEC Instrum). The demodulated signal and the transmission of the reference

cavity are recorded by a computer using a data-acqusition (DAQ) card with a Lab-

View program.

2.4 Absolute frequency measurement

2.4.1 Femtosecond comb system

The optical femtosecond comb (OFC) in this experiment is setup in the professor Shy’s

laboratory. The femtosecond comb laser source is Kerr-lens mode-lock Ti:sapphire

laser (GigaJet 20) and has a repetition rate of 1 GHz and pulse width of about 30

fs. A commercial photonic crystal fiber was used to expand the spectrum to contain

an octave. The octave spanning spectrum output from the photonic crystal fiber is

shown in Fig. 2.5.

2.4.2 Repetition rate and offset frequency stabilization

To lock the repetition rate frequency at a specific frequency is controlling the cavity

length with a piezoelectric transducer (PZT). The repetition frequency of the fem-

tosecond laser is phase-locked to a 1 GHz signal synthesizer from a stable microwave

source referenced on the rubidium frequency standard (Fig. 2.6). By mixing the sig-

nals of repetion frequency and from synthesizer, the frequency-difference signal is fed

into a phase-locked loop (PLL). The error signal by the PLL is amplified to drive the

PZT of the ring cavity. To obtain the offset frequency of the femtosecond laser, part

of the laser after spectrum broadening is frequency-doubled using nonlinear optical


crystal LBO. Coupling of the original and doubling waves and measuring the hytero-

dyne beat between these lines yields a different frequency, which is just the offset

frequency:

δ = 2fn − f2n = 2(nfrep + f0)− (2nfrep + f0) = f0. (2.3)

The offset frequency is also phase-locked to a synthesizer and fed into a digital PLL for

controlling the pump power with an acousto-optic modulator (AOM). The accuracy

of the optical femtosecond comb is ∼ 10−11 level.

2.4.3 Femtosecond comb test

The OFC was checked by measuring the well-known I2 transition at 780 nm (R(26)0-

14:a10 component [29]). The light source TIS laser (in Prof. Shy’s Lab), which is a

commercial system with internal locking to a reference cavity, is more stable than the

TIS laser in Li measurement. The measurement was performed in frequemcy scan

mode, and using the same data-acquisition program as in this Li experiment. The

resulting frequency of I2 shows an accuracy of less than 200 kHz. This shows the OFC

and the scanning method can reach an accuracy of 200 kHz.

2.4.4 Beat measurement

In the lithium two-photon transitions part of the TIS 735 nm laser light (about 20

mW) is coupled into a fiber, sending to the femtosecond comb system with about 3

mW output for absolute frequency measurement. A 730 nm interference bandpass

filter (bandwidth 10 nm FWHM) is used to filter out the unnecessary femtosecond

comb lines and avoid saturation of the photodiode.


I=1

Li(6)

2P 1/2

2P 3/2

Li(7)

I=3/2

F=2

F=2

F=1

F=1

F=1/2

F=1/2

F=3/2

F=3/2

3S 1/2

2S 1/2

670nm

812nm

735nm x2

Figure 2.2: Partial energy level diagram for lithium (6Li and 7Li). (not scaled)


A

B

C

D

7 Li 6 Li

F=2

F=2

F=1

F=1

F=3/2

F=3/2

F=1/2

F=1/2

3S 1/2

2S 1/2

228.205259(3) MHz 803.5040866(10) MHz

3/4A 3S

5/4A 3S

1/2A 3S

A 3S

Figure 2.3: Hyperfine structure of 6Li and 7Li. ∆f7, ∆f76, ∆f6 are relative frequenciescorresponding to B-A, A-C, D-C. (not scaled)


Ti:sapphire laser

735 nm

Femtosecond comb

system

chopper

controller Lock-in

Amplifier

Wavemeter

fiber

Isolator

Reference cavity PD

chopper

f=50cm

R=25cm

vacuum chamber

Li atomic beam

PMT

Data Aquisition

Figure 2.4: Experimental setup using 735 nm TIS laser. (PMT: photomultiplier)


200 400 600 800 1000 1200

2000

2200

2400

2600

2800

3000

3200

735nm

Inte

nsity (

ab

. u

nit)

Wavelength (nm)

Figure 2.5: The spectrum after the photonic crystal fiber caught by the portablespectrometer.


R b

c l

o c k

(

1 0 M H z )

T u

n a

b l e

b a

n d

p a s s

f i l

t e r

A M

P

R F

S y n

t h e

s i z e

r

L o

o p

F i l

t e r

1 6

: 1

H V

R F

S y n

t h e

s i z e

r A

O M

d r i

v e r

T I S

l a s e

r @

7 3

5 n

m

A M

P

P r e

s c a l

a r

P h

a s e - f

r e q

u e n

c y

d e t e

c t o

r

L o

o p

F i l

t e r

Millennia V

P Z

T

O C

P D

A P

D

T u

n a b

l e

b a n

d p

a s s

f i l t

e r

P C

F

L B

O

P B

S

P B

S

P B

S

U n

i v e

r s a l

C o

u n

t e r

R b

c l o

c k

( 1

0 M

H z

) A

P D

G i g

a J e t

2 0

A O

M

R b c

l o c

k (

1 0

M H

z )

Figure 2.6: The experimental setup of the femtosecond comb system. (PD: photodetector, PZT: piezoelectric transducer, OC: output coupler, AOM: acoustic-opticmodulator, HV: high voltage amplifier, PCF: photonic crystal fiber, λ/2: half-waveplate, PBS: polarizing beam splitter, APD: avalanche photo detector)

Chapter 3

Results and discussions

3.1 Results

The four transition lines of lithium 2S → 3S two-photon transition, including isotopes

7Li and 6Li, are studied in this work. The fluorescence signals recorded with reference

cavity transmission and laser scan are shown in Fig. 3.1 and Fig. 3.2. The signal-to-

noise ratio (SNR) of the largest signal (7Li : F = 2 → 2) is ∼ 100. Due to that the

quantities contained in natural lithium metal of 7Li and 6Li are respectively 92.5%

and 7.5%, the signal of isotope 6Li is by a factor 1/12 smaller than of 7Li.

A typical beat frequency spectrum is shown in Fig. 3.3. The intensity of beat

note is 20∼30 dBm above the noise level. The beat signal was amplified and the

frequency beat note was counted using an universal counter (53132A HP) which is

also referenced to the rubidium clock. A tunable bandpass filter is used to reduce the

interference from the 1 GHz beat signal (the repetition rate of the comb). According

to the counted beat frequency combined with the repetition rate and offset frequency,

the absolute frequency of the 735 nm TIS laser was determined. The fluorescence

signals of several different scans are integrated as a histogram for ecah transition.

The errorbar of the histogram is given by the average deviation:

σ = 〈|yi − 〈y〉|〉 , (3.1)

33

CHAPTER 3. Results and discussions 34

Figure 3.1: Two-photon spectrum of 6Li. The F.S.R of the reference cavity is 300MHz.

of the averaged signal of the same laser frequency with binning 0.5 MHz. Fig. 3.4,

Fig. 3.5 and Fig. 3.6 show the result spectrums of the four hyperfine components in this

two-photon transition. The spectrum data is fitted to a Voigt function of a program

using ROOT platform (from CERN). The Voigt fitting function is a convolution of

Gaussian and Lorentzian functions:

Lorentz(ν) =2

π

σL4(ν − ν0)2 + σ2

L

, (3.2)

Gauss(ν) =1√

2π∆exp(−(ν − ν0)2

2∆2), (3.3)


Figure 3.2: Two-photon spectrum of 7Li. The F.S.R of the reference cavity is 300MHz.

where ν0 is the spectra centre frequency. The Lorentzian linewidth is σL and the

Doopler width is σD = 8√

ln2 ∆. The fitting result shows that the peak has a

full width 10 MHz of half-maximum (FWHM) with 4.5 MHz Lorentzian component,

slightly larger than the 2.6 MHz expected from the 29.8 ns lifetime of the 3S state.

The linewidth of a Doppler-free transition is also contributed to the transit time

broadening. With the thermal velocity of 1356 m/s of lithium passing through the

focused laser beam waist of 100 µm, there is a finite interaction time 73 ns of the

lithium atom with the light field, corresponding to a transit time broadening 2.1

MHz. The fitting results give the absolute transition frequencies of each line listed as

Table. 3.1 (a).


540 560 580 600 620 640 660 680 700 720 740

-85

-80

-75

-70

-65

-60

-55

-50

-45

Inte

nsity (

dB

m)

Beat note frequency (MHz)

Figure 3.3: The beat signal recorded in spectrum analyzer.

The absolute frequencies of the four hyperfine components in this two-photon

transition are measured to an uncertainty of < 300 kHz. The transition interval

frequencies are listed in Table. 3.1 (b).

3.2 Systematic effect

In the spectra of two-photon transition, the measured transition frequency or level

energies may be shifted by some systematic effects discussed below:

3.2.1 Doppler background

There are four lines in this two-photon transition. The lineshape distortion due to

tails of neighboring transitions results in a frequency shift of the line center. The

lineshape is simulated by adding a Lorentzian function with a Gaussian function as


AEntries 100Mean 976RMS 8.264

/ ndf 2χ 13 / 95Amplitude_a 2.5± 91.4 Center 0.1± 975.9 Gw_a 0.319± 2.808 Lw_a 0.52± 4.55 offset 0.0188± 0.3405

Frequency-407808000 (MHz)955 960 965 970 975 980 985 990 995 1000

Sig

nal

(ab

. nu

it)

1

2

3

4

5

6

7

8

AEntries 100Mean 976RMS 8.264

/ ndf 2χ 13 / 95Amplitude_a 2.5± 91.4 Center 0.1± 975.9 Gw_a 0.319± 2.808 Lw_a 0.52± 4.55 offset 0.0188± 0.3405

Li(7):F=2->2

Figure 3.4: The histogram of 7Li 2S1/2 → 3S1/2 (F = 2 → F = 2) transition. Thecenter frequency is 407808975.87(13) MHz. The linewidth (FWHM) is 10 MHz with4.5 MHz Lorentzian component (natural linewidth 2.6 MHz, transit time broadening= 2.1 MHz) .

Doppler background:

2A

π

σL4(ν − ν0)2 + σ2

L

+B√2π∆

exp(−(ν − ν0)2

2∆2) (3.4)

where A and B represent the area of Lorentzian and Gaussian profile, respectively.

The Doppler width can be estimated from Eq. 1.30. For lithium gas at 500C with

the most probable velocity υp (1356 m/s) the Doppler width is 3 GHz. For lithium

atomic beam in this experiment the beam diverging angle θ ∼ 2.4 corresponds to a

reduced Doppler width 130 MHz. By setting Lorentzian width 5 MHz and area ratio

B/A=1/2, the simulation is shown in Fig. 3.7. The peak value ratio of the Loretzian

and Gaussian profile is 35. Typically the nearest transition frequency interval is 88

MHz and the result shift of peak center due to neighboring line profile is within 1

kHz. In fact the tails and Doppler background is covered under noise level in this

experiment.


BEntries 100

Mean 282.8

RMS 8.486

/ ndf 2χ 19.88 / 95Amplitude_a 1.71± 54.29

Center 0.2± 282.1

Gw_a 0.373± 2.591 Lw_a 0.554± 4.904

offset 0.0121± 0.2078

Frequency-407809000 (MHz)265 270 275 280 285 290 295 300 305 310

Sig

nal

(ab

. nu

it)

0

1

2

3

4

5 BEntries 100

Mean 282.8

RMS 8.486

/ ndf 2χ 19.88 / 95Amplitude_a 1.71± 54.29

Center 0.2± 282.1

Gw_a 0.373± 2.591 Lw_a 0.554± 4.904

offset 0.0121± 0.2078

Li(7):F=1->1

Figure 3.5: The histogram of 7Li 2S1/2 → 3S1/2 (F = 1 → F = 1) transition. Thecenter frequency is 407809282.12(16) MHz.

3.2.2 Light shift

For the lithium 2S → 3S two-photon transition, the energy detuning ∆ωi = ω −(Ei − Eg)/~ is about 40 THz in frequency, in contrast to laser frequency 407.8 THz.

With laser power 450 mW the light shift estimated from Eq. 1.32 in this experiment

is within 10 kHz.

3.2.3 Second-order Doppler shift

In two-photon transition with two counterpropagating light beams, the first-order

Doppler shift is eliminated. To second-order Doppler shift [30]:

δν(2) = ν0υ2

2c2, (3.5)

which is dependent on the speed υ of the atoms but independent on the direction.

With the most probable speed 1356 m/s in this experiment, the second-order Doppler

shift is 4.2 kHz.


CDEntries 400

Mean 368.4

RMS 49.29

/ ndf 2χ 97.09 / 391

Amplitude_a 0.362± 7.804

Center_a 0.3± 334.3

Gw_a 0.823± 2.233

Lw_a 1.034± 7.333

Amplitude_b 0.223± 3.312

Center_b 0.3± 421.2

Gw_b 0.702± 2.784

Lw_b 1.466± 5.009

offset 0.00180± 0.04089

Frequency-407803000 (MHz)300 320 340 360 380 400 420 440 460 480

Sig

nal

(ab

. nu

it)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

CDEntries 400

Mean 368.4

RMS 49.29

/ ndf 2χ 97.09 / 391

Amplitude_a 0.362± 7.804

Center_a 0.3± 334.3

Gw_a 0.823± 2.233

Lw_a 1.034± 7.333

Amplitude_b 0.223± 3.312

Center_b 0.3± 421.2

Gw_b 0.702± 2.784

Lw_b 1.466± 5.009

offset 0.00180± 0.04089

Li(6):F=3/2->3/2, F=1/2->1/2

Figure 3.6: The histogram of 6Li 2S1/2 → 3S1/2 (the left is F = 3/2→ F = 3/2 andthe right is F = 1/2→ F = 1/2) transition. The center frequency is 407803334.27(28)and 407803421.17(26) MHz, respectively.

3.3 Hyperfine constant, isotope shift, and nuclear

size

The A3S magnetic-dipole hyperfine constants are derived from the observed splittings

as

A3S(nLi) = (∆fgs −∆fn)/(I + 1/2), (3.6)

where I is the nuclear spin and the 2S ground state hfs splittings ∆fgs are precisely

known from atomic beam magnetic resonance measurements [27]. Similarly, the iso-

tope shift of the hfs center of gravity is

IS = ∆f76 +3

4(A2S − A3S)7Li − 1

2(A2S − A3S)6Li. (3.7)

Resulting values for A3S and IS are given in Table. 3.2 with uncertainties obtained

from normal error propagation through Eqs. 3.6 and 3.7.

The IS can be compared with the most recent theoretical value [2] to determine


(a)

Peak Transition Laser frequency (MHz) [1] ∆fA 7Li : F = 2− 2 407808975.87(13) 407808977.0(15) -1.1B 7Li : F = 1− 1 407809282.12(16) 407809285.6(15) -3.5C 6Li : F = 3/2− 3/2 407803334.27(28) 407803336.7(15) -2.4D 6Li : F = 1/2− 1/2 407803421.17(26) 407803424.3(15) -3.1

(b)

frequency (MHz) [1] ∆fB-A 306.25(21) 306.646(11) -2.4A-C 5641.60(31) 5640.344(13) -1.3D-C 86.90(38) 87.656(12) -0.8

Table 3.1: (a) Lithium 2S → 3S two-photon transition frequencies and comparisonwith ref [1]. (b) Transition interval frequencies and comparison with ref [1].

50 100 150 200

0.1

0.2

0.3

Figure 3.7: Line profile simulation for this experiment. The area ratio of Lorentzianand Gaussian profile is 2. The linewidth is set to 5 and 130 MHz, respectively.

the nuclear size between isotopes, 6Li and 7Li. All mass-dependent contribution to the

IS was calculated in theory with sufficient accuracy, then from Eq. 1.22 the difference

in the square of the nuclear charge radii is given:

R2rms(

6Li)− R2rms(

7Li) = 1.2± 0.3 fm2. (3.8)

Tere are some disagrement among the various spectroscopic determinations as shown

in Table. 3.3.


6Li 7Li Ref.

A3S, MHz 34(13) 95(10) [31]94.68(22) [32]

35.263(15) 93.106(11) [1]93.09 [8]*93.084 [33]*

36.27(51) 95.50(21) ThisIS, MHz 11435(20) [31]

11453.734(30) [1]11454.24(5)(39) [10]*11454.95(51) This

E, cm−1 27205.7129(10) 27206.0952(10) [34]27205.71214(10) 27206.09420(10) [1]

27206.0924(39) [10]*27205.711975(20) 27206.094071(10) This

Table 3.2: Hyperfine structure constants, isotope shift, and transition energies of the6,7Li two-photon transition 2 2S1/2 → 3 2S1/2. Theoretical calculations indicated by *.The second uncertainty is from uncertainty in nuclear charge radii.

3.4 Discussions

There are some discrepancies (-1.1 ∼ -3.5 MHz) in transition frequency compared

with [1]. The optical femtosecond comb with frequency scanning method can reach

an accuracy of 200 kHz, as described in section 2.4.3. The dicrepancies may be due

to the laser instability in our system. In laser scan mode, the laser freqency does

not vary smoothly as monotonic increasing or decreasing, but fluctuates during the

frequency scan. Typically the frequency fluctuation is about 1 MHz in 100 ms which

is observed using the scanning Fabry-Perot cavity or the beat spectrum. This leads

to a counting error in the beat measurement with counter (gate time = 100 ms).

The 2S−3S transition energy is corrected to the hfs center of gravity and given in

cm−1 in Table. 3.2 for comparison with prior values. The results are in good agreement

with the theoretical value [10] and the accuracy is improved by an order of magnitude.

The result for IS is in agreement with the theoretical value, but the discrepancy with

Ref. [1] is still large.


Method IS (MHz) ∆R2 Ref.Electron scattering 0.79(25) [35]

Li(2S − 3S) IS 11456.734(30) 0.47(5) [1]11454.95(51) 1.2(3) This

Li(2S − 2P1/2) IS 10534.26(13) 0.84(8) [36]10533.13(15) 0.38(7) [37]10532.9(6) 0.29(25) [38]10533.160(68) 0.39(4) [1]

Li(2S − 2P3/2) IS 10533.59(19) 0.41(8) [36]10534.93(15) 0.96(7) [37]10533.3(5) 0.29(21) [38]

Li+(2 3S1 − 2 3PJ) IS 0.73(5) [39]

Table 3.3: Values for the squared difference in nuclear radii ∆R2 = R2(6Li)−R2(7Li)in unit of fm2.

Chapter 4

Conclusion

Absolute frequencies of lithium 2 2S1/2 → 3 2S1/2 two-photon transition have been

measured to an uncertainty < 300 kHz using optical femtosecond comb, including

isotopes 6Li and 7Li. The two-photon excitation is performed in a weakly collimated

atomic beam with a Ti:sapphire ring laser at 735 nm (450 mW). Four transition lines

are observed and the signal-to-noise ratio (SNR) of the strongest line (7Li : F = 2−2)

is ∼ 100. The results are in good agreement with the theoretical value and the

accuracy is improved by an order of magnitude (Table. 3.2).

From the transition frequencies, the magnetic-dipole hyperfine constants of 3S1/2

state and isotope shift are also deduced (Table. 3.2). Comparing with the theoretical

calculation the nuclear size difference between 6Li and 7Li can also deduced, but there

are some disagreement with other previous works (Table. 3.3). The discrepancy may

be due to the laser instability.

Future work

In the future, possible works for improvements in this experiment are listed below:

• The signal of isotope 6Li should be improved.

• Stabilization and external scan of the TIS laser by side of fringe lock to a cavity.

• An enhance cavity method may be used to improve the signal with higher SNR.

43

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