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Magnetic Field Effects on the 1083 nm Atomic Line of Helium. Optical Pumping of Helium and

a r X i v :p h y s i c s /0203040v 2 [p h y s i c s .a t o m -p h ] 24 J u n 2002

EPJ manuscript No.

(will be inserted by the editor)

Magnetic Field E?ects on the 1083nm Atomic Line of Helium

Optical Pumping of Helium and Optical Polarisation Measurement in High Magnetic Field

E.Courtade 1,

F.Marion 1,P.-J.Nacher 1a ,

G.Tastevin 1,K.Kiersnowski 2,and T.Dohnalik 2

1Laboratoire Kastler Brossel b ,24rue Lhomond,75231Paris cedex 05,France.

Marian Smoluchowski Institute of Physics,Jagiellonian University,ul.Reymonta 4,30-059Krak′o w,Poland.

February 2,2008

Abstract.The structure of the excited 23S and 23P triplet states of 3He and 4He in an applied magnetic

?eld B is studied using di?erent approximations of the atomic Hamiltonian.All optical transitions (line positions and intensities)of the 1083nm 23S-23P transition are computed as a function of B .The e?ect of metastability exchange collisions between atoms in the ground state and in the 23S metastable state is studied,and rate equations are derived,for the populations these states in the general case of an isotopic mixture in an arbitrary ?eld B .It is shown that the usual spin-temperature description remains valid.A simple optical pumping model based on these rate equations is used to study the B -dependence of the population couplings which result from the exchange collisions.Simple spectroscopy measurements are performed using a single-frequency laser diode on the 1083nm transition.The accuracy of frequency scans and of measurements of transition intensities is studied.Systematic experimental veri?cations are made for B =0to 1.5T.Optical pumping e?ects resulting from hyper?ne decoupling in high ?eld are observed to be in good agreement with the predictions of the simple model.Based on adequately chosen absorption measurements at 1083nm,a general optical method to measure the nuclear polarisation of the atoms in the ground state in an arbitrary ?eld is described.It is demonstrated at B ～0.1T,a ?eld for which the usual optical methods could not operate.

PACS.32.60.+i Zeeman and Stark e?ects –32.70.-n Intensities and shapes of atomic spectral lines –32.80.Bx Level crossing and optical pumping

1Introduction

Highly polarised 3He is used for several applications in various domains,for instance to prepare polarised targets for nuclear physics experiments [1],to obtain spin ?lters for cold neutrons [2,3],or to perform magnetic resonance imaging (MRI)of air spaces in human lungs [4,5].A very e?cient and widely used polarisation method relies on op-tical pumping of the 23S metastable state of helium with 1083nm resonant light [6,7].Transfer of nuclear polarisa-tion to atoms in the ground state is ensured by metastabil-ity exchange collisions.Optical Pumping (OP)is usually performed in low magnetic ?eld (up to a few mT),required only to prevent fast relaxation of the optically prepared orientation.OP can provide high nuclear polarisation,up to 80%[8],but e?ciently operates only at low pressure (of order 1mbar)[9].E?cient production of large amounts of polarised gas is a key issue for most applications,which often require a dense gas.For instance,polarised gas must

2 E.Courtade et al.:The 1083nm Helium Transition in High Magnetic Field

in the structures of the di?erent excited levels of helium.In order to populate the 23S metastable state and per-form OP,a plasma discharge is sustained in the helium gas.In the various atomic and molecular excited states which are populated in the plasma,hyper?ne interaction transfers nuclear orientation to electronic spin and orbital orientations.The electronic angular momentum is in turn converted into circular polarisation of the emitted light or otherwise lost during collisions.This process is actu-ally put to use in the standard optical

[8,17]in which the circular polarisation of a spectral line emitted by the plasma is nuclear polarisation is inferred.The an applied magnetic ?eld unfortunately sitivity of the standard optical detection 10mT,and hence a di?erent measurement be used in high ?elds.This transfer of an adverse e?ect in OP situations by of nuclear polarisation in the gas.The of an applied ?eld reduces this polarisation thus signi?cantly improve the OP ations of limited e?ciency,such as low high pressures.At low temperature (4.2metastability exchange cross section sets a neck and strongly limits the e?ciency of OP ?eld increase from 1to 40mT was observed increase in nuclear polarisation from 17%to situation [21].At high gas pressures (above the proportion of metastable atoms is creation of metastable molecular species is factors which tend to reduce the e?ciency not surprising that a signi?cant by suppressing relaxation channels in high A systematic investigation of various vant for OP in non-standard conditions (high high pressure)has been made,and results elsewhere [22,23].As mentioned earlier,an surement method of nuclear polarisation in must be developed.It is based on ments of a probe beam,and requires a of magnetic ?eld e?ects on the 1083nm In this article we report on a detailed study an applied magnetic ?eld on the structure of 23P atomic levels,both in 3He and 4He.In cal section we ?rst present results obtained e?ective Hamiltonian and discuss the puted line positions and intensities at ?elds,then discuss the e?ect of an applied metastability exchange collisions between and the 11S 0ground state level of helium.mental section we present results of line positions and intensities up to 1.5T,optical measurement technique of the nuclear of 3He.

Let us ?nally mention that these principally motivated by OP developments,results could be of interest to design or ments performed on helium atoms in the 23S in an applied ?eld,as long as metrological required.This includes laser cooling and Zeeman slowing [24]of a metastable atom beam,magnetic trapping and evaporative cooling which recently allowed two groups to obtain Bose Einstein condensation with metastable 4He atoms [25,26],and similar experiments which may probe the in?uence of Fermi statistics with ultracold metastable 3

He atoms.

E.Courtade et al.:The 1083nm Helium Transition in High Magnetic Field 3

analyse the results of ?ne-structure intervals measured us-ing

microwave transitions in an applied magnetic ?eld in 4

He [33]and 3He [34].

All this work has produced a wealth of data which may now be used to compute the level structures and transi-tion probabilities with a very high accuracy up to high magnetic ?elds (several Tesla),in spite of a small per-sisting disagreement between theory and experiments for the g -factor of the Zeeman e?ect in the 23P state [32,35].However,such accurate computation can be very time-consuming,especially for 3He in which ?ne-structure,hy-per?ne and Zeeman interaction terms of comparable im-portance have to be considered.A simpli?ed approach was proposed by the Yale group [36],based on the use of an e?ective Hamiltonian.The hyper?ne mixing of the 23P and singlet 21P states is the only one considered in this e?ective Hamiltonian,and its parameters have been ex-perimentally determined [34].A theoretical calculation of the 3He 23P structure with an accuracy of order 1MHz later con?rmed the validity of this phenomenological ap-proach [37].In the present article,we propose a further simpli?cation of the e?ective Hamiltonian introduced by the Yale group,in which couplings to the singlet states are not explicitly considered.Instead we implicitly take into account these coupling terms,at least in part,by using the 23P splittings measured in zero ?eld to set the eigenvalues of the ?ne-structure matrices.

Results obtained using this simpli?ed e?ective Hamil-tonian will be compared in section 2.1.3to those of the one including the couplings to the 21P levels,with 3or 6addi-tional magnetic sublevels,depending on the isotope.The e?ect of additional terms in a more elaborate form of the Zeeman Hamiltonian [32]than the linear approximation which we use will also be evaluated.2.1A simple e?ective Hamiltonian 2.1.1Notations

The 11

S ground level of helium is a singlet spin state (S =0),with no orbital angular momentum (L =0),and hence has no total electronic angular momentum (J =0).In 3He,the nucleus thus carries the only angular momen-tum I ,giving rise to the two magnetic sublevels m I =±1/2.Their relative populations,(1±M )/2,de?ne the nuclear polarisation M of the ground state.

The level structure of 4He excited states is determined from the ?ne-structure term and the Zeeman term in the Hamiltonian.The ?ne-structure term H fs is easily ex-pressed in the total angular momentum representation (J )using parameters given in tables 4and 5in the Appendix.For the Zeeman term in the applied magnetic ?eld B we shall use the simple linear form :

H (4)

Z =μB (g ′L L +g ′

S S )·B.(1)

The values of Bohr magneton μB ,and of the orbital and

spin g -factors g ′L

and g ′S are given in table 6in the Ap-pendix.The situation is quite simple for the 23S state,for

which the 3sublevels |m S (with m S =±1,0)are eigen-states at any ?eld.For the 23P state the Zeeman term

couples sublevels of di?erent J (which however have the same m J )and is more easily expressed in the decoupled (L,S )representation.We note |m L ,m S the vectors of the decoupled basis used in the (L,S )representation and |J ;m J those of the coupled basis,which are the zero-?eld 9eigenstates of the 23P state.We call Z 1to Z 9(by increas-ing value of the energy,see ?gure 2)the 9sublevels of the 23P state in arbitrary ?eld.To simplify notations used in the following,we similarly call Y 1to Y 3the 3sublevels of the 23S state.

Fig.2.Left :magnetic sublevels of 4He in the 23S (bottom)and 23P (top)levels in very low magnetic ?eld.J is thus a good quantum number and m J is used to di?erentiate the atomic states.When a small ?eld is applied,the sublevel degeneracies are removed and energies increase for increasing values of m J (i.e.from Z 1to Z 9).Right :magnetic sublevels of 3He in very low magnetic ?eld.F and m F are now good quantum numbers to di?erentiate the atomic states.Only the F =3/2P x and P y states involve a strong mixing of states of di?erent J values [7].Energies increase for increasing values of m F for all sublevels except in the 23P 0state (i.e.from A 1to A 6and B 1to B 16).

The transition probabilities of all components of the 1083nm line are evaluated using the properties of the elec-tric dipole transition operator.For a monochromatic laser light with frequency ω/2π,polarisation vector e λ,and in-tensity I las (expressed in W/m 2),the photon absorption

4 E.Courtade et al.:The 1083nm Helium Transition in High Magnetic Field

rate for a transition from Y i to Z j is given by :

m e ωΓ′

I las

(Γ′/2)

3mc 2

f,(3)

from which Γ=1.022×107

s ?1

is obtained.Atomic colli-sions contribute to Γ′

by a pressure-dependent amount of

order 108s ?1/mbar [40].The transition matrix elements

(4)

ij (e λ)are evaluated in the (L,S )representation for each of the three light polarisation states λ=x ±iy (circularly polarised light σ±propagating along the z -axis set by the ?eld B )and λ=z (transverse light with πpolarisation)using the selection rules given in table 7in the Appendix.The transformation operator P 4given in table 10in the Appendix is used to change between the (L,S )and the (J )representations in the 23P state of 4He and thus write a simple 9×9matrix expression for the Hamiltonian in the (L,S )representation,H 9:

H 9=P ?1

4H 9fs P 4+H 9Z ,

(4)

where H 9fs is the diagonal matrix of table 4in the Ap-pendix and H 9Z the diagonal matrix written from equa-tion 1.

The level structure of 3He excited states is determined from the total Hamiltonian,including hyper?ne interac-tions.The ?ne-structure term H fs is expressed in the to-tal angular momentum representation (J )using slightly di?erent parameters (table 5in the Appendix)due to the small mass dependence of the ?ne-structure intervals.For the Zeeman term we also take into account the nuclear contribution:

H (3)

Z =μB (g ′L L +g ′S S +g I I )·B.(5)

The values of all g -factors for 3He are listed in table 6in the Appendix.For the hyper?ne term,we consider the simple contact interaction between the nuclear and elec-tronic spins and a correction term H cor

hfs :

H hfs =A L I ·S +H cor

hfs .

(6)

The matrix elements of the main contact interaction term in the decoupled (S,I )representation and the values of the constants A S and A P for the 23S and 23P states are given in table 8in the Appendix.The correction term only exists for the 2P state [36,37].When restricted to the triplet levels,it only depends on 2parameters which are given with the matrix elements in table 11in the Appendix.As in the case of 4He,we compute level structure in the decoupled representation,now (L,S,I ).The vectors of the decoupled bases are noted |m S :m I for the 23S state,and |m L ,m S :m I for the 23P state.For the 23S state,the 6×6matrix expression H 6of the Hamiltonian is :

H 6=H 6Z +F 6,

(7)

where H 6Z is the diagonal matrix written from equation 5and F 6the matrix representation of H hfs (table 8in the Appendix).For the 23P state,the 18×18matrix H 18is

constructed from H 9fs ,F 6,H cor

hfs and H (3)Z :

m ′L ,m ′S :m ′

I |H 18|m L ,m S :m I =

δ(m ′I ,m I ) m ′L ,m ′

S |H 9fs |m L ,m S

+δ(m ′L ,m L ) m ′S :m ′

I |F 6|m S :m I +

m ′L ,m ′S :m ′I H cor

hfs +H (3)Z m L ,m S :m I .

The mass-dependent constants in H 9fs and F 6are set to

the values given for 3He in the Appendix.

Notations similar to those of the zero-?eld calculation of reference [7]are used.The 6sublevels of the 23S state are called A 1to A 6,the 18levels of the 23P state B 1to B 18by increasing values of the energy (see ?gure 2)1.An alternative notation system,given in table 12in the Appendix,is also used for the 6sublevels of the 23S state when an explicit reference to the total angular momentum projection m F is desired.Transition probabilities from A i to B j due to monochromatic light are given by a formula similar to equation 2:

m e ωΓ′

I las

(Γ′/2)

There is a di?erence with notations used in reference [7]:

within each level of given F,labelling of state names was in-creased for convenience from largest to lower values of m F .In an applied ?eld,this would correspond to an energy decrease inside each of the F levels (except the P 0,F =1/2level).This is the reason for the new labelling convention,which is more convenient in an applied ?eld.

E.Courtade et al.:The1083nm Helium Transition in High Magnetic Field5

In the following,the dependence of the transition ma-trix elements on the polarisation vector eλwill not be explicitly written.Given the selection rules described in the Appendix,the polarisation vector corresponding to a non-zero transition matrix element can be unambiguously determined.

All optical transition energies ω(4)ij and ωij will be referenced to ω57(0),the energy of the C1transition in null magnetic?eld,which connects levels A5and A6to levels B7to B10.Energy di?erences will be noted?,for instance:

?(4)ij/ =ω(4)ij?ω57(0),?ij/ =ωij?ω57(0).(10) They are determined from computed energy splittings in the23S and23P states of each isotope.The isotope shift contribution is adjusted to

obtain the precisely measured C9-D2interval(810.599MHz,[29]).

2.1.2Numerical Results

Numerical computation of level structure and absorption spectra in an applied?eld B is performed in3steps.Matri-ces H9(for4He),H6and H18(for3He)are computed?rst.

A standard matrix diagonalisation package(the double-precision versions of the jacobi and eigsrt routines[41])is then used to compute and sort all eigenvalues(energies) and eigenvectors(components of the atomic states in the

decoupled bases).All transition matrix elements T(4)

ij and

T ij are?nally evaluated,and results are output to?les for further use or graphic display.Actual computation time is insigni?cant(e.g.20ms per value of B on a PC using a compiled Fortran program[42]),so that we chose not to use the matrix symmetries to reduce the matrix sizes and the computational load,in contrast with references [36,37].We thus directly obtain all transitions of the1083 nm line:19transitions for4He(6forσ+and forσ?,7 forπlight polarisation),and70transitions for3He(22for σ+and forσ?,26forπlight polarisation).These num-bers are reduced for B=0to18for4He(the Y2→Z7, i.e.|0 →|1;0 πtransition has a null probability)and to 64for3He(the F=1/2→F=5/2transitions,from A5or A6to any of B1to B6are forbidden by selection rules). Due to level degeneracy,the usual D0,D1and D2lines of

4He,and C1to C9lines of3He[7]are obtained for B=0 (?gure3,and table9in the Appendix).

An applied magnetic?eld removes level degeneracies (Zeeman splittings appear)and modi?es the atomic states and hence the optical transition probabilities.Examples of level Zeeman energy shifts are shown in?gure4for the23S state and the highest-lying23P levels(originating from the23P0state at low?eld).In the23S state,the two Zeeman splittings are simply proportional to B(28 GHz/T)for4He,as inferred from equation1.For3He, signi?cant deviations from linearity occur for A2,A3and A5above a?eld of order0.1T,for which Zeeman shifts are signi?cant compared to the hyper?ne splitting.Level crossing of A4and A5(and hence interchange of names Fig.3.Top:Components of the1083nm line in4He(left) and3He(right)for B=0.D J is the usual name of the tran-sition to the23P J level.C1to C9refer to the transitions by increasing energy.C0is the additional lowest energy compo-nent corresponding to a forbidden transition in zero?eld(see text).Bottom:positions of all the spectral lines resulting from level splittings and isotope shift.The total frequency span from C1to D0is70.442GHz.

of the two states)occurs at0.1619T.A high-?eld decou-pled regime is almost reached for1.5T(?gure4,bottom).

Analytical expression for state energies and components

on the decoupled{|m S:m I }basis are given in the Ap-pendix(equations A10to A17).In the23P state,the sit-

uation is more complex due to?ne-structure interactions

and to the larger number of levels.In particular no level

remains una?ected by B in contrast with the situation for Y2.As a result,πtransitions in4He from Y2to Z3or Z9have non-zero Zeeman frequency shifts,which are pro-portional to B2at low and moderate?eld although they originate from the linear Zeeman term in the Hamiltonian (equation1).Even the23P0states,for which level mixing is weakest due to the large?ne-structure gap,experience signi?cant?eld e?ects(?gure4,top and top insert).The splitting between B17and B18(2.468GHz/T at low?eld) is only weakly a?ected by the nuclear term in equation5 (±16.2MHz/T),and mostly results from level mixing and electronic Zeeman e?ects.

Examples of optical transition intensities and Zeeman frequency shifts are shown in?gure5for4He(πlight polarisation).Shifts of transition frequencies are clearly visible on the projection onto the base plane,the usual Zeeman diagram(in which no line crossing occurs).Large changes are also induced by the applied?eld on the transi-tion probabilities.In particular,the forbidden D1transition (Y2→Z7)has a matrix element which linearly increases

at low?eld(T(4)

=3.3×|B|),and approaches1in high?eld (curve labelled b in?gure5).The two other strong lines

6 E.Courtade et al.:The1083nm Helium Transition in High Magnetic Field

Fig.4.Level energies E S and E P of the23S and23P states are computed as a function of the applied?eld B.They result from the?ne-structure and Zeeman energy contributions for 4He(dashed lines),with the additional hyper?ne contributions

for3He(solid lines).The inserts are close-ups of the low-?eld regions.In the23P state(top),only the highest lying levels are shown for simplicity.

at high?eld(2curves labelled a superimposed in?gure

5) originate from the low-?eld Y1→Z2and Y3→Z8tran-sitions of D2and D1lines.All other line intensities tend to0at high?eld,in a way consistent with the sum rules of equation9.

The splittings of the C8and C9lines of3He at mod-erate?eld are displayed in?gure6.Circularly polarised light,which is used in OP experiments to deposit angular momentum in the gas,is more e?ciently absorbed forσ?polarisation when a?eld is applied.This can tentatively be related to results of our OP experiments:at B=0.1T, the most e?cient pumping line is actually found to be C9,σ?.OP results at B=0.6T reported in reference[15] also indicate thatσ?polarisation is more e?cient,in spite of?uorescence measurements suggesting an imperfect po-larisation of the light.One must however note that OP e?ciency does not only depend on light absorption,and that level structures and metastability exchange collisions (which will be discussed below)also play a key role.Let us ?nally mention that the forbidden C0transition becomes Fig.5.3D plot of the transition matrix elements forπlight polarisation in4He,T(4)

(π),(vertical axis)and of transition energy di?erence?(with respect to C1)as a function of B.To highlight the changes of the transition intensities,the symbol

sizes(symbol areas)are proportional to T(4)

.a and b are the strongest lines in high?eld(see text).The projection onto the base plane is the usual Zeeman diagram.

allowed in an applied?eld,but that the transition prob-abilities remain very weak and vanish again at high?eld.

Fig.6.Plot of the Zeeman line shifts for the C8and C9lines of3He.The symbol types refer to di?erent light polarisation states.The symbol sizes(areas)are proportional to the corre-sponding transition matrix elements T ij.

E.Courtade et al.:The1083nm Helium Transition in High Magnetic Field7

2.1.3Discussion

To assess the consequences of the main approximation in

our calculations,namely the restriction of the Hamilto-

nian to the triplet23P con?guration,we have made a sys-

tematic comparison to the results of the full calculation

including singlet-triplet mixing.

In the case of4He,exact energies are of course obtained

for B=0,and level energy di?erences do not exceed1kHz

for B=2T.This very good agreement results from the

fact that the lowest order contribution of the singlet ad-

mixture in the23P

state to the Zeeman e?ect is a third

order perturbation[30,43].For similar reasons,computed

transition probabilities are also almost identical,and our

simple calculation is perfectly adequate for most purposes

if the proper parameters of table5in the Appendix are

used.

For3He we have compared the results of the full cal-

culation involving all24sublevels in the2P states as de-

scribed by the Yale group[36]and two forms of the sim-

pli?ed calculation restricted to the18sublevels of the23P

state.The simpler form only contains the contact interac-

tion term A P I·S in the hyper?ne Hamiltonian of equa-

tion6.The more complete calculations makes use of the

full hyper?ne interaction,with the correction term H cor

hfs

introducing two additional parameters,d and e.

The simpler1-parameter calculation is similar to the

zero-?eld calculation of reference[7](with updated values

of all energy parameters),and provides a limited accuracy

as shown in?gure7.Errors on the computed energies of

Fig.7.Errors in calculated energies for the18sublevels of the

3He23P state resulting from the sole use of the contact term

A P I·S in the hyper?ne Hamiltonian are plotted for di?erent

magnetic?eld https://www.doczj.com/doc/843948348.html,rge changes of the errors occur

close to level anticrossings(e.g.m F=1/2B15and B17levels

close to1.18T,see?gure4).

the levels of several tens of MHz cannot be signi?cantly re-

duced by adjusting A P to a more appropriate value.This

is a direct evidence that the correction term H cor

hfs

unam-

biguously modi?es the energy level diagram,with clear

?eld-dependent signatures at moderate?eld and around

1T.Errors on the transition matrix elements up to a few

10?3also result from neclecting H cor

hfs

.Using this simple1-

parameter form of the hyper?ne Hamiltonian should thus

be avoided when a better accuracy is desired.

The3-parameter calculation has also been compared

to the full calculation,and a slight adjustment of param-

eters A P,d and e with respect to the corresponding pa-

rameters C,D/2and E/5of the full calculation(which

also includes the o?-diagonal parameter C′[36]),now pro-

vides a more satisfactory agreement for the level energies,

shown in?gure8.Transitions matrix elements T ij are also

Fig.8.Errors in calculated energies for the18sublevels of the

3He23P state resulting from the use of the full3-parameter

hyper?ne Hamiltonian are plotted for di?erent magnetic?eld

intensities.As in?gure7,large changes of the errors occur

close to the B15-B17level anticrossing around1.18T.

obtained with good accuracy(1.3×10?4r.m.s.di?erence

on average for all transitions and?elds,with a maximum

di?erence of5×10?4).The values of the3parameters A P,

d and

e di?er by-0.019%,3.4%and4.6%from the corre-

sponding parameters in the full calculation.This adjust-

ment may e?ectively represent part of the o?-diagonal ef-

fects in the singlet admixture,and allows to decrease the

energy errors,especially below1T(by a factor2-3).

The other important approximation in our calculations

is related to the simpli?cations in the Zeeman Hamiltonian

written in equations1and5.The?rst neglected term is

the non-diagonal contribution proportional to the small

parameter g x in equation A1.The non-zero elements of

this additional contribution are of orderμB Bg x/5,i.e.1-

2MHz/T in frequency units.The other neglected term

results from the quadratic Zeeman e?ect[30,32,43],which

introduces matrix elements of order(μB B)2/R∞(R∞=

3.29×106GHz stands for the Rydberg constant in fre-

quency units),i.e.60kHz/T2.For most applications,and

in particular for the experimental situations considered in

the following,corrections introduced by these additional

8 E.Courtade et al.:The1083nm Helium Transition in High Magnetic Field terms in the Zeeman Hamiltonian would indeed be negli-

gible,not larger for instance than the errors in?gure8.

2.2Metastability exchange

So far we have only addressed the e?ects of an applied

magnetic?eld on the level energies and structures of the

23S and23P states of helium,and the consequences for the

1083nm optical transition.In this section we now consider

the e?ect of the applied?eld on the angular momentum

transfer between atoms during the so-called metastability

exchange(ME)collisions,which are spin-exchange binary

collisions between an atom in the ground state and one

in the metastable23S state.They play an important role

in all experimental situations where the atoms in the23S

state are a minority species in a plasma(e.g.with a num-

ber density1010–1011cm?3compared to2.6×1016for the

total density of atoms at1mbar in typical OP conditions).

These collisions may be neglected only for experiments on

atomic beams or trapped atoms for which radiative forces

are used to separate the metastable atoms from the much

denser gas of atoms in the ground state.

To describe the statistical properties of a mixture of

ground state and23S state atoms we shall use a standard

density operator formalism and extend the treatment in-

troduced by Partridge and Series[44]for ME collisions,

later improved[45,46]and used for an OP model[7],to

the case of an arbitrary magnetic?eld.Our?rst goal is

to show that the“spin temperature”concept[47,48,7]re-

mains valid,and that the population distribution in the

23S state is fully determined(in the absence of OP and of

relaxation)by the nuclear polarisation M of the ground

state.This is the result on which the optical detection

method of section3.3relies.The second goal is to provide

a formalism allowing to study the consequences of hyper-

?ne decoupling on spin transfer during ME collisions.

2.2.1Derivation of rate equations

We shall consider situations in which no resonance is driven

between the two magnetic sublevels of the ground state of

3He(the ground state of4He has no structure)nor be-

tween sublevels of the23S state.We can thus a priori

assume that the density operatorsρg(in the ground state

of3He),ρ3(in the23S state of3He)andρ4(in the23S

state of4He)are statistical mixtures of eigenstates of the

Hamiltonian.The corresponding density matrices are thus

diagonal and contain only populations(no coherences).

With obvious notations:

ρg=1+M

2|? ?|(11)

ρ3= 61a i|A i A i|(12)

ρ4= 31y i|Y i Y i|,(13) where a i and y i are relative populations( a i= y i=1).

An important feature of ME in helium is that in prac-tice no depolarisation occurs during the collisions,due to the fact that all involved angular momenta are spins[46]. Three kinds of ME collisions occur in an isotopic mixture, depending on the nature of the colliding atoms(3He or 4He)and on their state(ground state:3,4He,23S state: 3,4He?).

1.Following a ME collision between3He atoms:

3He+3He?→3He?+3He

ρgρ3ρ′3ρ′g

the nuclear and electronic angular momenta are re-combined in such a way that the density operators just after collisionρ′g andρ′3are given by:

ρ′g=Tr eρ3(14)

ρ′3=ρg?Tr nρ3(15) where Tr e,Tr n are trace operators over the electronic and nuclear variables respectively[45].

2.Following the collision between a ground state4He

atom and a23S state3He atom:

4He+3He?→4He?+3He

ρ3ρ′′4ρ′′g

the density operators just after collision are:

ρ′′g=Tr eρ3(16)

ρ′′4=Tr nρ3.(17) 3.Following the collision between a23S state4He atom

and a ground state3He atom:

3He+4He?→3He?+4He

ρgρ4ρ′′′3

the density operator of the outgoing23S state3He atom is:

ρ′′′3=ρg?ρ4.(18) The partial trace operations in equations14to17 do not introduce any coherence term in density matri-ces.In contrast,the tensor products in equations15and 18introduce several o?-diagonal terms.These are driving terms which may lead to the development of coherences in the density operators,but we shall show in the fol-lowing that they can usually be neglected.Since m F is a good quantum number,o?-diagonal terms in the ten-sor products only appear between2states of equal m F. With the state names de?ned in the Appendix(table12), these o?-diagonal terms are proportional to the operators A h+ A l+ and A l+ A h+ (for m F=1/2), A h? A l? and A l? A h? (for m F=-1/2).These four operators contain a similar sinθ±cosθ±factor,whereθ+andθ?are?eld-dependent level mixing parameters(see equations A14to A17in the Appendix).They have a time evolution char-acterised by a fast precessing phase,exp(i?E±t/ ),de-pending on the frequency splittings?E±/ of the eigen-states of equal m F.

E.Courtade et al.:The 1083nm Helium Transition in High Magnetic Field 9

The time evolution of the density operators due to ME collisions is obtained from a detailed balance of the de-parture and arrival processes for the 3kinds of collisions.This is

similar to

the method introduced by Dupont-Roc et al .[45]to derive a rate equation for ρg and ρ3in pure 3

He at low magnetic ?eld.It is extended here to isotopic mixtures,and the e?ect of an arbitrary magnetic ?eld is discussed.

The contribution of ME collisions to a rate equation for the density operator ρ3of the 23S state 3He is obtained computing an ensemble average in the gas of the terms arising from collisions with a 3He atom (collision type 1,?rst line of the rate equation)and from collisions involving di?erent isotopes (types 2and 3,second line): d

σv 33,34.For each time-dependent o?-diagonal

coherence,the average of the fast precessing phase involves

a weighting factor exp(??t

/τ)to account for the distribu-tion of times ?t elapsed after a collision: exp i?E ±?t

/ =1dt

n 3ρ3

=n 3N 3σv 34

?n 3N 4ρ3+n 4N 3

Πi ρg ?ρ4Πi

(21)

in which we note Πi =|A i A i |the projector on the eigen-state A i .The main di?erences with the result derived in [45]are the additional term (second line)in equation 21,which re-?ects the e?ect of 3He ?4He collisions,and the replacement of the projectors on the F substates by the more general projectors Πi on the eigenstates in the applied ?eld B .Considering the ?eld dependence of the frequency split-tings ?E ±/ ,the condition on the ME collision rate 1/τis only slightly more stringent at 0.08T that for B =0,but can be considerably relaxed at high magnetic ?eld.The linear increase of ?E ±and the 1/B decrease of sin θ±(see ?gure 31)in the common factor of all coherences pro-vide a B 2increase of the acceptable collision rate,and hence of the operating pressure,for which coherences can be neglected in all density matrices,and equations 11to 13are valid.

The other rate equations are directly derived from equa-tions 14,16and 17.Since there is no tensor product,hence no coherence source term in the relevant arrival terms,ex-plicit ensemble averages are not required and the contri-bution of ME is: d

σv 33(?ρg +Tr e ρ3)

n 4ρ4

=?n 4N 3

σv 34Tr n ρ3.(23)

A trace operation performed on equations 21and 23shows

that

dt n 3

ME ,(24)as expected since the three ME processes preserve the to-tal number of atoms in the 23S excited state.2.2.2Steady-state solutions

Total rate equations can be obtained by adding relaxation

and OP terms to the ME terms of equations 21and 23.They can be used to compute steady-state density ma-trices for the 23S state,while the nuclear polarisation M (and hence ρg )may have a very slow rate of change (com-pared to the ME collision rate 1/τ).In these steady-state situations,the contribution of ME collisions to the total rate equations for the 23S state just compensates the OP and relaxation contributions.Since the latter are traceless terms (OP and relaxation only operate population trans-fers between sublevels of a given isotope),by taking the trace of equation 21or 23one ?nds that the 23S state and ground state number densities have the same isotopic ratio R :

R =n 4/n 3=N 4/N 3.(25)

This results from having implicitly assumed in section 2.2.1that all kinds of collisions have the same ME cross sec-

10 E.Courtade et al.:The1083nm Helium Transition in High Magnetic Field tion2.In fact destruction and excitation of of the23S

state in the gas may di?erently a?ect3He and4He atoms,

e.g.due to the di?erent di?usion times to the cell walls.

However in practice this has little e?ect compared to the very frequent ME collisions,which impose the condition of equation25for number densities.

To obtain simpler expressions for the rate equations, we introduce the parameters:

1/τe=N3σv34(26)

1/T e=n3σv34(27)

μ=σv34(28) The rates1/T e and1/τe correspond to the total ME col-lision rate for a3He atom in the ground state and in the 23S state respectively;in steady state,equation25can be used to show thatτe/T e=n3/N3=n4/N4.Since the ther-mal velocity distributions simply scale with the reduced masses of colliding atoms,the value of the dimensionless parameterμcan be evaluated from the energy dependence ofσ.From reference[18]one can estimateμ?1.07at room temperature.Rewriting equation26as:

1/τe=(N3+N4)

1+R

,(29)

the ME rate in an isotopic mixture thus di?ers from that in pure3He by at most7%(at large R)for a given total density.

Using these notations and the isotopic ratio R(equa-tion25),the contributions of ME to rate equations be-come:

d T e(?ρg+Tr eρ3)(30) dτe1

ρ3 ME=1R+μ Πi×

×(μρg?Tr nρ3+Rρg?ρ4)Πi}.(32)

In a way similar to that of reference[7]we can trans-form equations30to32into an equivalent set of coupled rate equations for the nuclear polarisation M and for the

M ME=1

y i ME=1R+μ ?y i+6 k=1G4ik a k (34) dτe ?a i+

E.Courtade et al.:The 1083nm Helium Transition in High Magnetic Field 11

2.2.4OP e?ects on populations

The general problem of computing all the atomic popu-lations in arbitrary conditions is not considered in this work.It has been addressed

in the low ?eld limit using speci?c models in pure 3He [7]and in isotopic mixtures [50].Here we extend it to arbitrary magnetic ?elds,but only consider the particular situation where there is no ground state nuclear polarisation (M =0).It can be met in absorption spectroscopy experiments,in which a weak probing light beam may induce a deviation from the uni-form population distribution imposed by M =0(in?nite spin temperature).We shall assume in the following that the OP process results from a depopulation mechanism in which population changes are created only by excitation of atoms from selected 23S sublevels and not by sponta-neous emission from the 23P state (where populations are randomised by fast relaxation processes).In 3He,this de-population mechanism has been checked to dominate for pressures above ～1mbar [6,7].

Under such conditions,and further assuming that Zee-man splittings are large enough for a single sublevel in the 23S state to be pumped (A p or Y p ,depending on the probed isotope),the OP contribution to the time evolu-tion of populations has a very simple form.Given the ME contribution of equation 34or 35,the total rate equation for the pumped isotope becomes:

d dt y i ME +y p

?δY (i,p )

(41)

or d dt a i ME +a p 6?δA (i,p )

,(42)

where 1/τp is the pumping rate (equation 2or 8)and

δA (i,p )=1(resp.δY (i,p )=1)if the level A i (resp.Y i )is the pumped level,0otherwise.In steady state,the relative populations can be obtained from the kernel of the 9×9matrix representation of the set of rate equations 34and 42,or 35and 41.

When a 3He line is probed,the 4He relative popula-tions y i are fully imposed by ME processes only (equa-tion 34),and can be subsituted in equation 42.The rela-tive populations a i can then be obtained as the kernel of a 6×6matrix which depends on the pumped level index p ,on the reduced pumping rate τe /τp and on the magnetic ?eld B,but not on the isotopic ratio R (as discussed at the end of the Appendix).Results showing the decrease of the pumped population a p ,and hence of the absorp-tion of the incident light,are displayed in ?gure 9as a function of B .A strong absorption decrease is found at high B,due to larger population changes induces by OP when ME less e?ciently transfers the angular momentum to the ground state (let us recall that we assume M =0,so that the ground state is a reservoir of in?nite spin tem-perature).Population changes are found to be weaker for

the two states A l +and A h

?,which both have |0:± as the main component in high ?eld (curves b and b ′in ?gure 9).

When a 4He line is probed,the substitution and re-duction performed above cannot be made.The full set of

Fig.9.OP on a 3He level :the computed decrease of the relative population a p of the pumped level of 3He below its equilibrium value (1/6)is plotted as a function of B for two values of the reduced pumping rate τe /τp :10?2(solid lines)

and 10?3(dashed lines).For the 2populations a p =a l +or a h

?,the results fall on the same curve (labelled b or b ′depending on τe /τp ).For the 4other populations,a signi?cantly larger decrease is found (groups of curves a and a ′).The insert is an expanded view of the low-?eld region (with a linear B -scale).

populations turns out to vary with the isotopic ratio R.However,the OP e?ects are found to depend mostly on the product Rτe /τp ,as shown in ?gure 10in the case of the state Y 3(m S =1).This in?uence of the isotopic com-position extends over a wide ?eld range,which justi?es the use of isotopic mixtures to reduce the bias resulting from OP e?ects in our systematic quantitative measurements of 4He line intensities.Figure 11displays a comparison of computed OP e?ects for the 3sublevels of the 4He 23S states.Results for the populations y 1and y 3are quite sim-ilar,except at large R.Weaker OP e?ects are found for the population y 2,for which m S =0.This feature is similar to the weaker OP e?ects found in 3He for the two states which have m S =0,m I =±1/2as main angular momenta components in high ?eld (see ?gure 9).

3Experimental

In this part we present experimental results of laser ab-sorption experiments probing the ?ne,hyper?ne and Zee-man splittings of the 1083nm transition in 3He and 4He.Accurate microwave measurements [27,33,34]and high pre-cision laser spectroscopy experiments [28,29,51,52]are usu-ally performed on helium atomic beams.In contrast,we have made unsophisticated absorption measurements us-ing a single free-running laser diode on helium gas in cells at room temperature,under the usual operating condi-tions for nuclear polarisation of 3He by OP.An important consequence of these conditions is the Doppler broadening

12 E.Courtade et al.:The1083nm Helium Transition in High Magnetic Field

Fig.10.OP on a4He level:the decrease of the4He population y3below its equilibrium value(1/3)is plotted as a function of B.The population has been computed for two values of the reduced pumping rateτe/τp:10?2(solid lines)and10?3 (dashed lines),and di?erent values of R(ranging from100to 10?3).The population decrease mostly depends on the product Rτe/τp(10?1,10?2,...10?5for curves a,b,...e).

Fig.11.OP on a4He level:the computed decrease of4He populations y1,y2and y3due to OP e?ects is plotted as a function of B.The reduced pumping rateτe/τp=10?2is that of the solid lines in?gure10.Four values of the isotopic ratio R are used(Rτe/τp=10?2to10?4for curves a to d).

of all absorption lines.Each transition frequencyωij/2π(in the rest frames of the atoms)is replaced by a Gaussian distribution of width?:

?=(ωij/2π)

ln2,are respec-tively1.98and1.72GHz).The narrow Lorentzian res-onant absorption pro?les(with widthsΓ′/2)in equations

2and8are thus strongly modi?ed,and broader Voigt

pro?les are obtained for isolated lines in standard absorp-

tion experiments.ForΓ′??,these are almost Gaus-sian pro?les of width?.In the following,we?rst report

on Doppler-free absorption measurements from which a

good determination of frequency splittings is obtained, then on integrated absorption intensities in di?erent mag-netic?elds.We?nally demonstrate that absorption mea-surements may provide a convenient determination of the nuclear polarisation M.

3.1Line positions:saturated absorption measurements 3.1.1Experimental setup

The experiment arrangement is sketched in?gure12.The Fabry Perot interferometer

Fig.12.Main elements of a saturated absorption experiment.

A magnetic?eld

B is applied either along the pump and probe beams,or perpendicular to the beams.Polarising elements (Pol.)consist of linear polarisers and/or quarter-wave plates depending on the desired polarisations of the pump and probe beams.

helium gas samples are enclosed in sealed cylindrical Pyrex glass cells,5cm in diameter and5cm in length.Cells are ?lled after a careful cleaning procedure:bakeout at700K under high vacuum for several days,followed by100W RF (27MHz)or microwave(2.4GHz)discharges in helium with frequent gas changes until only helium lines are de-tected in the plasma?uorescence.Most results presented here have been recorded in cells?lled with0.53mbar of 3He or helium mixture(25%3He,75%4He).Di?erent gas samples(from0.2to30mbar,from10%to100%3He) have also been used to study the e?ect of pressure and isotopic composition on the observed signals.A weak RF discharge(<1W at3MHz)is used to populate the23S state in the cell during absorption measurements.The RF excitation is obtained using a pair of external electrodes. Aligning the RF electric?eld with B provides a higher density of23S states and better OP results in high mag-netic?eld.

Most data have been acquired using a specially de-signed air-core resistive magnet(100mm bore diameter,

E.Courtade et al.:The 1083nm Helium Transition in High Magnetic Field 13

with transverse optical access for 20mm beams).In spite of a reduced footprint (30×20cm 2,axis height 15cm)which conveniently permits installation on an optical ta-ble,this magnet provides a fair homogeneity.The com-puted relative inhomogeneity is <10?3over a 5cm long cylindrical volume 1cm in diameter (typical volume probed by the light beams in spectroscopy measurements),and <3×10?3over the total cell volume.This usually induces low enough magnetic relaxation of nuclear polarisation in OP experiments.The ?eld-to-current ratio was calibrated using optically detected NMR resonance of 3He.Experi-ments for B up to 0.12T were performed with this coil.Several saturated absorption experiments have been re-peated and extended up to 0.22T using a standard elec-tromagnet in the Institute of Physics in Krakow.

The laser source is a 50mW laser diode (model 6702-H1formerly manufactured by Spectra Diode Laborato-ries).Its output is collimated into a quasi-parallel beam (typically 2×6mm in size)using an anti-re?ection coated lens (f =8mm).A wedge-shaped plate is used to split the beam into a main pump beam (92%of the total in-tensity),a reference beam sent through a

confocal Fabry Perot interferometer,and a probe beam (?gure 12).A small aperture limiting the probe beam diameter is used to only probe atoms lying in the central region of the pump beam.A small angle is set between the counterpropagating beams.This angle and su?cient optical isolation from the interferometer are required to avoid any feedback of light onto the laser diode,so that the emitted laser frequency only depends on internal parameters of the laser.The po-larisations of pump and probe beams are adjusted using combinations of polarising cubes,1/2-wave,1/4-wave re-tarding plates.For all the measurements of the present work,the same polarisation was used for pump and probe beams.However valuable information on collisional pro-cesses can be obtained using di?erent polarisation and/or di?erent frequencies for the two beams.

The probe beam absorption is measured using a modu-lation technique.The RF discharge intensity is modulated at a frequency f RF low enough for the density of the ab-sorbing atoms 23S to synchronously vary (f RF ～100Hz).The signal from the photodiode monitoring the transmit-ted probe beam is analysed using a lock-in ampli?er.The amplitude of the probe modulation thus measured,and the average value of the transmitted probe intensity are both sampled (at 20Hz),digitised,and stored.Absorp-tion spectra are obtained from the ratios of the probe modulation by the probe average intensity.This proce-dure strongly reduces e?ects of laser intensity changes and of optical thickness of the gas on the measured absorp-tions [22].

With its DBR (Distributed Bragg Re?ector)technol-ogy,this laser diode combines a good monochromaticity and easy frequency tuning.It could be further narrowed and stabilised using external selective elements,and thus used for high-resolution spectroscopy of the 1083nm line of helium [51].Here we have simply used a free running diode,with a linewidth dominated by the Schalow-Townes broadening factor of order 2-3MHz FWHM [53].The laser

frequency depends on temperature and current with typi-cal sensitivities 20GHz/K and 0.5GHz/mA respectively.To reduce temperature drifts,the diode is enclosed in a temperature regulated 1-dm https://www.doczj.com/doc/843948348.html,ing a custom made diode temperature and current controller,we obtain an overall temperature stability better than a mK,and a current stability of order a μA.Frequency drifts over sev-eral minutes and low frequency jitter are thus limited to 10MHz at most.

The laser frequency is adjusted,locked or swept by temperature control using the on-chip Peltier cell and tem-perature sensor.The small deviation from linearity of the frequency response was systematically measured during the frequency sweeps by recording the peaks transmitted by the 150MHz FSR interferometer.Figure 13displays an example of such a measurement:for a control voltage V (in Volts)the frequency o?set (in GHz)is 52.5×(1+0.048V )V .The linear term in this conversion factor is usually taken from more accurate zero-?eld line splitting measurements (see below),but this small non-linear correction is used to improve the determination of frequency scales for ex-tended sweeps.

Fig.13.Insert :fraction of recording of the light intensity transmitted by the Fabry Perot interferometer during a tem-perature controlled laser frequency scan (1V of control voltage approximately induces changes of 2.5K in laser temperature and 50GHz in emitted frequency).Main plot :residues of a linear ?t (open triangles)or parabolic ?t (solid circles)of all transmitted peak positions.

3.1.2Zero-?eld measurements

Results presented in this section have in fact been ob-tained in the earth ?eld in which Zeeman energy changes are of order 1MHz or less and thus too small to a?ect the results given the accuracy of our measurements.A small permanent magnet is placed near the cell to strongly re-duce the ?eld homogeneity and thus prevent any nuclear polarisation to build up when the pump beam is absorbed.

14 E.Courtade et al.:The1083nm Helium Transition in High Magnetic Field An example of frequency sweep over the whole3He ab-

sorption spectrum is given in?gure14(3He pressure:0.53

mbar).The frequency scale is determined using the ac-

Fig.14.Zero-?eld saturated absorption spectrum of3He.The

upper traces(a)represent the probe absorption signals when

the pump beam is applied(solid line)or blocked(dotted line).

Traces b and c(an expanded section of trace b)are the di?er-

ence of probe absorptions with and without the pump beam,

with a×4vertical scale factor.The sections of the spectrum

comprising the lines C2to C5provide a much larger signal and

are displayed with a reduced amplitude.Frequency sweep rate

is160MHz/s,a value ensuring su?cient signal quality and

manageable laser frequency drifts.Transition lines are labelled

C1to C9,crossover resonances between C i and C j are labelled

χi?j.The vertical bars on the axes represent the computed

line positions and intensities.

curately known value3A S/2=6.7397GHz of the C8-C9

splitting.The saturated absorption spectrum(solid line in

?gure14a)combines the usual broad absorption lines and

several narrow dips.The dips reveal a reduced absorption

of the probe by atoms interacting with the pump beam.

These are atoms with negligible velocity along the beam

direction(Doppler-free resonances)or atoms with a veloc-

ity such that the Doppler shift is just half of the splitting of

two transitions from or to a common level(crossover reso-

nances).An absorption signal recorded without the pump

beam(dotted line in?gure14a)only shows the broad fea-

tures,which can be?t by a Doppler Gaussian pro?le for

isolated lines(e.g.C8or C9).It is used to obtain the nar-

row resonances in?gure14b by signal substraction.The

widthδof the narrow resonances is50MHz FWHM in

the conditions of?gure14.It can be attributed to several

combined broadening processes,including pump satura-

tion e?ects and signal acquisition?ltering.

Broader line features,the“pedestals”on which nar-

row resonances stand,result from the combined e?ects of

collisions in the gas.Fully elastic velocity changing col-

lisions tend to impose the same population distribution

in all velocity classes,and thus to identically decrease

the probe absorption over the whole velocity pro?le.In

contrast,ME collisions involving a3He atom in a gas

where the nuclear polarisation is M=0induce a net loss

of angular momentum3(a uniform population distribu-

tion tends to be imposed by the in?nite spin tempera-

ture,see section2.2.3).In steady-state,the un-pumped

velocity classes will thus acquire only part of the pop-

ulation changes imposed in the pumped velocity class.

The relative amplitude of the resulting broad pedestal

results from the competition of collisional transfer and

collisional loss of orientation.One consequence is that

it is almost pressure-independent,as was experimentally

checked.Another consequence is that it strongly depends

on4He concentration in isotopic mixtures,as illustrated

in?gure15.The much larger fractional amplitude of the

Fig.15.Zero-?eld saturated absorption spectrum of the D0

line in a25%3He-75%4He mixture(same total pressure,

0.53mbar,as for?gure14).The lower trace also displays the

squared absorption amplitude(open circles),with an arbitrary

weight adjusted to?t the saturated absorption pedestal(see

text).

pedestal results from the weaker depolarising e?ect of the

ME collisions:only25%of the exchange collisions occur

in this case with a depolarised3He atom and thus con-

E.Courtade et al.:The1083nm Helium Transition in High Magnetic Field15 tribute to bleach the pump-induced population changes,

while all collisions induce velocity changes.In?gure15

is also displayed the squared absorption pro?le,which

closely matches the shape of the pedestal.This is a general

feature also observed in3He spectra,which results from

the linear response to the rather low pump power.Assum-

ing that the relative population changes are proportional

to the absorbed pump power,frequency detuning reduces

by the same Doppler pro?le factor both these population

changes and the probed total population.The resulting

squared Doppler pro?le is still Gaussian,but with reduced

√

width?/

16 E.Courtade et al.:The1083nm Helium Transition in High Magnetic Field

Fig.17.a:Plot of frequency shifts on the3He C8and C9 lines in an applied magnetic?eld measured by a saturated ab-sorption technique.Solid lines:computed Zeeman shifts(same as in?gure6).Up and down triangles:data forσ+andσ?po-larisations of light beams.Solid and open symbols:measure-ment using the air-core and iron yoke magnets respectively. Crosses:result of corrections applied the open symbol data set(see text).b:Di?erences between experimental and com-puted values for the measurements in the iron yoke magnet are plotted as a function of B.Open symbols:raw data(no correction);?lled symbols:data with?eld corrections only; crosses:data with?eld and frequency corrections(see text). attributed to a global frequency o?set from scan to scan. Estimating the o?set of each scan from the average of the3 recorded line positions,the remaining di?erences are sig-ni?cantly reduced(9MHz r.m.s.forσ+andσ?).The accuracy of these line position measurements is thus com-parable to that of the zero-?eld measurements(?gure16).

For the measurements in the iron yoke electromagnet (up to0.25T),a similar frequency o?set adjustment is not su?cient to signi?cantly reduce the di?erences be-tween measured and computed line positions(?gure17b). For instance,they amount to130MHz r.m.s.for theσ+ measurements(open up triangles).Moreover,splittings within a given scan consistently di?er from the computed ones.Assuming that the magnetic?eld is not exactly pro-portional to the applied current due to the presence of a soft iron yoke in the magnet,we use the largest measured splitting on the C9line(the transitions originating from the sublevels m F=±3/2)to determine the actual value of the magnetic?eld during each scan.Solid symbols in?g-ure17b are computed di?erences for these actual values of the magnetic?eld.Crosses(?gures17a and17b)corre-spond to fully corrected data,allowing also for frequency o?set adjusments.The?nal remaining di?erences(22MHz r.m.s.)are still larger than in the air-core magnet,yet this agreement is su?cient to support our?eld correction and o?set adjustment procedures in view of the line intensity measurements described in the following section.

3.2Line intensities

3.2.1Experimental setup

The experimental arrangement is either the one sketched in?gure12,in which the pump beam is blocked and only a weak probe beam(usually0.1-1mW/cm2)is transmitted through the cell,or a simpli?ed version of the setup when no saturated absorption measurement is performed.

The magnetic?eld is obtained by di?erent means de-pending on its intensity.Data at moderate?eld(up to 0.22T)data have been acquired using the resistive mag-nets described in section3.1.1.Higher?eld measurements have been performed in the bore and fringe?eld of a1.5T MRI superconducting magnet[54].With the laser source and all the electronics remaining in a low-?eld region sev-eral meters away from the magnet bore,cells were succes-sively placed at?ve locations on the magnet axis with?eld values ranging from B=6mT to1.5T.The1.5T value (in the bore)is accurately known from the routinely mea-sured1H NMR frequency(63.830MHz,hence1.4992T). The lower?eld intensities,6mT and0.4T,have been mea-sured using a Hall probe with a nominal accuracy of1%. The intermediate values,0.95T and1.33T,are deduced from the measured Zeeman splittings with a similar accu-racy of1%.Due to the steep magnetic?eld decrease near the edge of the magnet bore,large gradients cause a sig-ni?cant inhomogeneous broadening of the absorption lines in some situations discussed in section3.2.3.For these high?eld measurements,the Fabry Perot interferometer was not implemented and no accurate on-site check of the laser frequency scale was performed.Instead,the usual corrections determined during the saturated absorption measurements(see section3.1.1and?gure13)are used for the analysis of the recorded absorption spectra.

A systematic study of the e?ect of experimental con-ditions such as gas pressure,RF discharge intensity,and probe beam power and intensity,has been performed both in low?eld(the earth?eld)and at0.08T.The main char-acteristic features of the recorded spectra have been found to be quite insensitive to these parameters.Absorption spectra have been analysed in detail to accurately deduce the relative populations of di?erent sublevels(from the ra-tios of line intensities)and the average atomic density of atoms in the23S state in the probe beam(from its absorp-tion).A careful analysis of lineshapes and linewidths re-veals three di?erent systematic e?ects which are discussed in the next three sections.

3.2.2E?ect of imperfect3He isotopic purity

Evidence of a systematic e?ect on absorption pro?les was observed in experiments with nominally pure3He gas.A

E.Courtade et al.:The 1083nm Helium Transition in High Magnetic Field 17

typical recording of the C 8and C 9absorption lines for B =0

and residuals

to di?erent Gaussian ?ts are shown in ?gure 18.A straightforward ?t by two independent

Fig.19.Absorption measurements on the C 8and C 9lines performed in a ?eld B =0.111T in nominally pure 3He at 0.53mbar for circular polarisations (upper graph :σ+,lower graph :σ?).Traces a correspond to the raw absorption signals (solid lines)and computed pro?les (dotted lines,see text).Traces b are di?erences (residue plots)between absorption signals,corrected for light polarisation defects,and computed pro?les.Traces c are the computed pro?les for pure 4He,scaled down in order to correspond to an isotopic ratio R =0.4%.Expanded vertical scales (×5,right hand side axes)are used for traces b and c .

sorption signals are similar to the computed ones (dot-

18 E.Courtade et al.:The1083nm Helium Transition in High Magnetic Field ted lines),but two signi?cant di?erences appear.First,

a small line is visible for instance in the upper graph

aroundε/h=30GHz,where noσ+transition should be

observed.This is due to an imperfect circular polarisation

of the probe beam in the cell,resulting in part from stress-

induced circular dichroism in the cell windows(measured

to be of order0.2%).The amount of light with the wrong

polarisation is determined for each recording,and each sig-

nal is corrected by substracting the appropriate fraction

of the signal with the other circular polarisation(0.79%

and0.18%corrections forσ+andσ?recordings in?g-

ure19)4.Second,the corrected signals may then tenta-

tively be?t by the sum of three Gaussian pro?les centred

on the frequencies independently determined in a satu-

rated absorption experiment(see?gure17).But,as is the

case in zero?eld,this procedure does not provide satisfac-

tory results for the amplitudes and linewidths of the split

C9lines(corresponding to the transitions originating from

any of the lowest levels of the23S state,A1to A4,to the

highest levels of the23P state,B17and B18).In?gure19,

traces b display the di?erences between the corrected sig-

nals and the computed sums of three Gaussian pro?les

with a common linewidth and with amplitudes propor-

tional to the nominal transition probabilities in this?eld.

Two global amplitudes and the linewidth(?=1.221GHz)

are adjusted to?t the two C8transitions,and the resulting

di?erences are best accounted for assuming a proportion

R=0.4%of4He in the gas(traces c in?gure19).In order

to better reproduce the experimental recordings,the am-

plitudes of the split C9lines are?nally allowed to vary.

Fits with excellent reducedχ2and no visible feature in

the residues are obtained and the obtained ratios of in-

tensities are given in table1for this typical recording.

The same kind of discrepancy as in zero?eld is obtained

polarisation

computed

0.37570.18803

T4,17T2,17

0.313110.16111

C8/C90.833400.85683

ratio0.851980.86360

deviation+21%?3.6%

https://www.doczj.com/doc/843948348.html,puted line intensities T ij for B=0.1111T and

their ratios are compared to the experimental line amplitude

ratios of?gure19.

for the strong components of the transitions;it happens

to be much worse for the weakest line in some recordings,

including that of?gure19,for reasons which are not un-

derstood.

In an attempt to test whether the remaining line in-

tensity disagreements with the computed values may re-

E.Courtade et al.:The 1083nm Helium Transition in High Magnetic Field 19

respect to the computed ones is usually small (of order ±3%)after correcting for the presence of some 4He in our 3

He cells and for the observed light polarisation imperfec-tions.This di?erence does not arise from signal-to-noise limitations and its origin is not known.This may set a practical limit to the accuracy with which an absolute measurement of population ratios can be made using this experimental technique.Similar discrepancies are also ob-served at higher ?elds,situations for which the additional systematic e?ects described in the following sections must ?rst be discussed.

3.2.3E?ects of magnetic ?eld gradients

As previously mentioned (section

3.2.1),absorption ex-periments above

0.25T have been performed in the fringe ?eld of a 1.5T magnet.The exact ?eld map of this magnet was not measured,but the ?eld decreases from 1.33T to 0.95T in only 25cm,so that ?eld gradients along the axis may exceed 15mT/cm at some locations.In contrast,?eld variations in transverse planes are much smaller in the vicinity of the ?eld axis.As a result,a negligible broad-ening of absorption lines is induced by the ?eld gradient for perpendicular probe beams (σor πpolarisation for a linear polarisation perpendicular or parallel to B ).For probe beams propagating along the ?eld axis (σpolarisa-tion for a linear polarisation,σ+or σ?polarisation for a circular polarisation),the range of ?elds δB applied to the probed atoms is the product of the ?eld gradient by the cell length.This ?eld spread δB induces spreads δ?ij /h of optical transition frequencies which are proportional to δB and to the derivatives d?ij /dB.Each spread δ?ij /h thus depends on the probed transition as illustrated in ?gure 21for B =0.95T.

https://www.doczj.com/doc/843948348.html,puted ?eld-derivatives of the transitions energies ?ij /h are plotted for all 3He and 4He circular polarisation tran-sitions as a function of ?ij /h for B =0.95T.Inhomogeneous line broadenings proportional to these values are induced by a ?eld gradient.

The resulting absorption pro?les are given by the con-volution of the Doppler velocity distribution and of the fre-

quency spread functions,computed from the density dis-tribution of metastable atoms,the ?eld gradient at the cell location and the derivatives d?ij /dB.Assuming for sim-plicity a uniform density of atoms in the 23S state,com-puted broadened absorption pro?les only depend on the dimensionless ratio δ?ij /h?of the transition frequency spread to the Doppler width.Figures 22a and 22b display examples of computed pro?les (absorption as a function of this reduced detuning).For increasing frequency spreads,

of c ?to δ?a of

20 E.Courtade et al.:The1083nm Helium Transition in High Magnetic Field

Fig.23.Part of an absorption spectrum recorded at B=0.95T in a3He-4He mixture at2.13mbar(isotopic ratio R=3)using a longitudinal probe beam(open symbols).The solid curve is the spectrum computed for this R,assuming thermal Doppler widths for the di?erent lines.Its amplitude is scaled to repro-duce that of the47GHz line.Experimental broadening of the absorption lines is directly found on the101GHz line,and suggested by the residual absorption between the main lines (non-zero signal at50and55GHz),which is2-3times larger than computed.The insert is an expanded portion of the spec-trum,in which several3He and4He lines overlap(the computed absorption of pure4He is displayed as a dotted line).Experi-mental broadening of the absorption lines also clearly appears for this part of the spectrum.

nal(symbols)roughly correspond to the pro?le computed assuming the nominal Doppler widths?and neglecting frequency spreads(solid line).Clear amplitude di?erences are however observed,in particular for the most intense line.They can be attributed to OP

e?ects,and will be discussed in detail in the next section(3.2.4).Moreover, signi?cant linewidth di?erences appear,for instance on the isolated most shifted4He line(?tted apparent linewidth ?app=1.42GHz,30%larger than?4).Most of the other lines actually result from the overlap of several atomic transitions,and a straightforward Gaussian?t is not rele-vant.For example,the computed total absorption around ?/h=72GHz(insert in?gure23,solid line)results from the two4He transitions at66.7and72.1GHz(dotted line) and the two3He transitions at65.2and71.1GHz.

To better analyse the e?ect of the probe beam direc-tion on the shape and widths of di?erent lines,we plot in?gure24the absorption data using the same method as in?gure22b.The open symbols correspond to three of the lines in?gure23,with a probe beam along the di-rection?eld,and the solid symbols to lines from a similar recording with a perpendicular probe beam.Apart from the asymmetries on the71GHz lines which result from the additional3He line on the low-frequency side of the absorption pro?le(which is thus not retained in the anal-ysis),satisfactory linear?ts reveal Gaussian-like pro?les and provide the values of the corresponding linewidths. The experiments performed with a transverse probe pro-vide data with the steepest slopes and hence the smallest B

of ij

symbols:data of?gure23,longitudinal probe beam.Solid symbols:data for a transverse probe beam.The straight lines represent Gaussian absorption pro?les,and the values of their slopes are used to extract the apparent linewidths?app.The smallest linewidths,consistent with the4He Doppler width,are obtained for a transverse probe(?4=1.13and1.105GHz for the47and72GHz lines).Larger apparent linewidths are ob-tained for a longitudinal probe(?app=1.27,1.31and1.42GHz for the47,72and101GHz lines).

linewidths,while those using a longitudinal probe lead to larger apparent linewidths as https://www.doczj.com/doc/843948348.html,ing the convo-lution results of?gure22c,the frequency spreadsδ?ij/h for these lines are computed to be1.4,1.65and1.85GHz respectively.From the derivatives d?ij/dB for these transi-tions(?gure21),one computes that a?eld spreadδB=52mT would produce frequency spreads of1.08,1.65and1.85 GHz,corresponding to the observed broadenings for the 72and101GHz lines.The additional apparent broaden-ing for the47GHz experimental line may be attributed to OP e?ects for this rather intense line(see the next sec-tion3.2.4).For the estimated value of the?eld gradient (15mT/cm),this52mT?eld spread would be obtained for an e?ective cell length of3.5cm.This is quite satis-factory since the density of atoms in the metastable23S state vanishes at the cell walls,so that most of the probed atoms lie within a distance shorter than the actual gap between the end windows of our5cm-long cells(external length).

To summarise the results obtained in this section,the expected e?ects of strong?eld gradients on absorption lines have indeed been observed.They introduce a sys-tematic broadening and amplitude decrease of absorption signals which can be signi?cant(up to30%changes in our experiments),and must be considered in quantitative analyses.For isolated lines,area measurements are not af-fected by the transition frequency spreads.However,most lines usually overlap in a recorded absorption spectrum (especially in an isotopic mixture),and the quantitative

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MLC 580C N ,5131/04.02Phone:+49 7622/695-5Fax:+49 7622/695-602 e-mail:info@ac-magnetic.de https://www.doczj.com/doc/843948348.html,

Magnetic Control Systems Sdn.Bhd.No.16, Jalan Kartunis U1/47Temasya Ind.Park, Section U140150 Shah Alam, Selangor Darul Ehsan, Malaysia Phone:(+60) 3 / 55691718eMail: info@https://www.doczj.com/doc/843948348.html,.my Magnetic Control Systems (Shanghai) Co. Ltd.999 Ning-qiao Road, Bldg. 2W/1F Pudong New Area Shanghai 201206, China Phone:(+86) 21/ 58 341717eMail: magnetic@https://www.doczj.com/doc/843948348.html, Magnetic Automation Pty. Ltd.19 Beverage Drive Tullamarine, Victoria 3043, Australia Phone:(+61) 3 / 93 30 10 33eMail: info@https://www.doczj.com/doc/843948348.html, Magnetic Automation Corp.3160 Murrell Road Rockledge, FL 32955, USA Phone:(+1) 321/ 635 85 85eMail: info@https://www.doczj.com/doc/843948348.html, Magnetic Autocontrol Pvt.Ltd.Calve Chateau, 2B, IInd Floor Kilpauk 322 Poonamallee High Road IND Chennai, 600010 / India Phone:(+91) 44 6400 443eMail: magneticsales@https://www.doczj.com/doc/843948348.html,

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