Report of the internship at IISc

Vivishek Sudhir

18th June – 20th July, 2010

Resume

Name: Vivishek Sudhir

Date of Birth: 28th June, 1988

Academic Qualification: B. Tech. (Electrical & Electronics)

Current Affiliation: Graduate StudentCenter for Quantum ControlDepartment of PhysicsImperial College, London

Permanent Address: ‘Krishnangie’, Puthur RoadKoppam JunctionPalakkad 678001, Kerala

E-mail: vivishek.sudhir@gmail.com

Division: Atomic & Optical Physics LabDepartment of PhysicsIndian Institute of Science, Bangalore

Supervisor: Prof. Vasant Natarjan

1

Acknowledgement

At the outset, I would like to thank Prof. Vasant Natarajan for giving me this opportunity to workin his group. Without his welcoming attitude, I would not have learned so much nor would I haveexperienced such a deep passion for experimental physics. I would also like to identify the strongsupport that he has extended to me and for patiently listening to all my chatter.

Without the friendly support of the students in the lab, I would not have got to speed with much ofthe stringent demands of experimentation. In particular, I would like to identify the helping hands ofKanhaiya, Ketan, Alok and Ranjitha.

Last, but never the least, I would like to thank Raghuveer for all the help he has done, from procuringthe copper plates and screws to organizing the wonderful trip to Hoggenekal.

Thank you all!

Vivishek Sudhir

2

Contents

Resume 1

Acknowledgment 2

1 Zeeman tuned Doppler cooling 41.1 Theory of Doppler cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Zeeman tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Design and test of Zeeman slower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1 Axial magnetic field and current coils . . . . . . . . . . . . . . . . . . . . . . . 61.3 Adaptive Zeeman tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Study of Hyperfine structure 122.1 Theory of hyperfine interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.1 First and second order hyperfine corrections . . . . . . . . . . . . . . . . . . . . 13

Bibliography 15

3

Chapter 1

Zeeman tuned Doppler cooling

Cooling of few-body atomic/molecular systems has become an essential tool in modern atomic physics,particularly in the field of condensate physics. In this context, “cooling” refers to the attainment of afew-body ensemble which is as close to its theoretical ground state as possible. This means that theensemble necessarily has its thermal rotational and vibrational modes frozen. Thus, the individualentities of the ensemble are slowed down to a velocity as close to rest as possible (remember that theaverage velocity and temperature of an ensemble are related, hence the usage of the term “cooling”).

For systems which possess an internal structure constituted by bound states, the preferred methodof slowing is by photon recoil. In this method, a laser beam of suitable energy is focussed against theoncoming matter beam; the bound particles are excited by photon absorption and in the subsequentspontaneous decay of the excited state, a photon is emitted, which provides a recoil momentum inthe direction opposite to the beam velocity, thus slowing it down.

One of the most efficient and advantageous cooling method is the so called Doppler cooling tech-nique. Using this method, one can achieve temperatures as low as a few hundred micro Kelvins; itis theoretically impossible to cool below this Doppler limit. Other methods like Sisyphus cooling orevaporative cooling are used on a Doppler cooled system to take it to lower temperatures.

1.1 Theory of Doppler cooling

In most experiments in atomic physics, two distinct levels of an atom are chosen to be resonant witha laser beam shooting photon in the direction opposing the direction of motion of the atomic beam.Then the photon recoil force slows the atoms as they move in the optical “drag” of the radiation.

To understand Doppler cooling, it is necessary to first understand how a two-level atom interactswith the radiation field. Let ρ be the density matrix of the two-level atom, i.e.,

ρ =(ρee ρegρge ρgg

)=(cec∗e cec

∗g

cgc∗e cgc

∗g

),

so that we have the two implicit relations, ρee + ρgg = 1 and ρ∗eg = ρge; thus, the density matrixhas only three independent real parameters. In the interaction picture, the Hamiltonian for an atominteracting with a classical radiation field is 1,

H = −eE · r, (1.1)

where, E = E0eeikz−ωf t is the electric field.

When an atom and a radiation field interact with each other, there is a possibility that the excitedatomic state may spontaneously decay or that the ground state may spontaneously get excited; these

1as pointed out by C. Cohen-Tannoudji etal. [1], there are some serious covariance issues in the form of the interactionHamiltonian (1.1) when the radiation field is also quantized, but for the present semi-classical treatment, this is welljustified [2]

4

effects are included in the dynamical equations by explicitly adding a term Γρ into the Liouvilleequation for the density matrix, which then reads,

dρ

dt=

1i~

[H, ρ] + Γρ.

Written out in full, the Liouville equation takes the form,

d

dt

ρggρeeρgeρeg

=

0 γ −iΩ/2 iΩ∗/20 −γ iΩ/2 −iΩ∗/2

−iΩ∗/2 iΩ∗/2 − (iδ + γ/2) 0iΩ/2 −iΩ/2 0 − (−iδ + γ/2)

ρggρeeρgeρeg

, (1.2)

where ρge = ρgee−iδt, Ω = − eE0

~ 〈e|r|g〉 is the Rabi frequency, δ = ωf −ωa is the detuning and γ is thespontaneous transition rate. In arriving at the above form of the dynamical equation, we have madeuse of the dipole and rotating wave approximations [3].

Introducing new real variables, u = ρee, v = <e(ρge) and w = =m(ρge), the above equations canbe recast into the form,

d

dt

uvw

=

−γ 0 Ω0 −γ/2 −δ−Ω δ −γ/2

uvw

+

00

Ω/2

. (1.3)

These are the optical Bloch equations.In steady state, the upper level population ρee is given by,

ρee =Ω2/4

δ2 + Ω2/2 + γ2/4,

so that the scattering force is given by,

F = ~kγΩ2/4

δ2 + Ω2/2 + γ2/4, (1.4)

which we have obtained by multiplying the photon momentum by the upper level decay rate andthe upper level population. It is easily noted that this photon recoil force has its maximum value of~kγ/2 when the detuning δ = 0 i.e., for maximum recoil force, the atom and light should be in prefectresonance.

But now, since the atom is moving through the light beam, the frequency of the laser as seen by theatom will be Doppler shifted by an amount −kV , where V (t) is the velocity of the atom. So, if, initiallythe atom and laser are in resonance, as it slows down by photon recoil, it moves below resonance andthe subsequent number of photons it absorbs will fall down gradually and further cooling becomesimpossible. In an older technique, initially, the laser is tuned slightly below the atomic transition sothat as the atom starts to experience the Doppler effect, the laser comes in resonance and cooling canbe effectively affected.

1.1.1 Zeeman tuning

To circumvent the problem of the laser falling out of resonance due to Doppler shifting, one usesthe technique of Zeeman tuning. Zeeman tuning is a far more effective method of maintaining zerodetuning than, say red shifting the laser initially, which still only gives a brief window of perfectresonance.

The idea here is to apply a magnetic field on the atom in such a fashion that the variation of thedetuning due to Doppler shift is exactly balanced by the Zeeman shift of the atomic levels induced bythis magnetic field. The objective is to establish a magnetic field such that the Zeeman shift of theatomic level nullifies the Doppler shift, thus maintaining perfect resonance throughout the stoppingdistance.

5

If L is the desired stopping distance, then the velocity profile of the atom across this distance isgiven by,

V (z) = V0

(1− z

L

)1/2,

where we have assumed constant deceleration. For the Zeeman effect to exactly counter the Dopplershift, the condition is,

ωa +µBB(z)

~= ωf + kV (z).

The above two equations give the required magnetic field profile viz.,

B(z) = B0 +Bi

(1− z

L

)1/2, (1.5)

where µBB0 = −~δ and µBBi = ~kV0.

1.2 Design and test of Zeeman slower

The particular atom that we aim to cool here is Ytterbium, a lanthanide with atomic number 70.The ultimate aim of trapping Yb is to perform sensitive studies on its electron dipole moments andfor that, one requires an isotope whose nucleus is a fermion. Thus, we choose the isotope with massnumber 171, which has a nuclear spin of 1

2 . The ground state electronic configuration of this isotopeis [Xe]4f146s0. Some of the excited energy states useful for cooling and trapping is shown in Fig. 1.1.

Figure 1.1: Low-lying electronic energy levels of Yb17170

The transition 1P1 → 1S0 is the one used for Zeeman tuned Doppler cooling. However, this transitionis not radiatively closed, so that the population in 1S0 decays to the 3P0, 3P1 and 3P2 states. The 3P1

population decays back to the 1S0 state via the emission of a 556nm photon, closing the decay cycle.But the states 3P0 and 3P2 are meta-stable, so that they do not decay back and so the populationexcited into them are lost forever in the context of laser cooling. Thus, the relative stability of thesestates limit the time available for laser cooling, so that there is a maximum distance within which theatom has to be cooled or it cannot be cooled any further.

The fact that the thermal Yb source gives off Yb atoms at a most-probable velocity of 350m/sand that we require to slow them down to about 20m/s most-probable and the fact that our Zeemanslower had to be within 40cm length, fixed the required magnetic field constants B0 and Bi.

1.2.1 Axial magnetic field and current coils

Now the required magnetic field profile on the axis of the Zeeman slower flight tube is given by (1.5),which is a semi-parabola opening in the −z direction. The way to achieve this profile is by using alarge number of current carrying coils wound over the flight tube, and energized by passing currentover it. Consider such a coil confined in the xy−plane as shown in Fig. 1.2.

6

Figure 1.2: Current carrying loop in xy−plane [4]

Using standard spherical coordinates, the current density J has components only in the φ direction,given by,

Jφ = I sin θ′δ(cos θ′

) δ (r′ − a)a

,

so that the vectorial current density is given by,

J = −Jφ sinφ′i + Jφ cosφ′j.

Then by the Biot-Savart law,

A(r) =µ0

4π

∫J(r′)|r− r′|

d3r′,

is the magnetic vector potential. For our problem, since there is an axial symmetry, only the Aφ com-ponent survives, and is given by the integral (after the preliminary integration over delta functions),

Aφ(r, θ) =µ0Ia

4π

∫ 2π

0

cosφ′ dφ′

(a2 + r2 − 2ar sin θ cosφ′)1/2.

This integral can be evaluated in closed form using elliptic functions, to give the result,

Aφ(r, θ) =µ0

4π4Ia√

a2 + r2 + 2ar sin θ

[(2− k2)K(k)− 2E(k)

k2

], (1.6)

where K(x) and E(x) are complete elliptic integrals of the first kind and the argument k is given by,

k2 =4ar sin θ

a2 + r2 + 2ar sin θ.

In an actual experiment, the atomic beam travels, neglecting the beam width itself, through theaxis (θ → 0, r → z) and it is the axial magnetic field Bz(z) that needs to fit the profile (1.5).Evaluating the axial component of the magnetic field using (1.6), we get,

Bz(z) =µ0

2Ia2

(z2 + a2)3/2. (1.7)

The hypothesis is that the profile (1.5) can be obtained by winding current coils carrying the samecurrent, but geometrically arranged in layers arranged one on top of the other, with varying numberof turns in each layer.

7

We assume that the coil arrangement consists of N layers with the nth layer consisting of amaximum of Mn coil turns. We also assume that the conductor has a radius r and carries a uniformsteady current I throughout all the turns in all the layers. Then, the magnetic field produced by themth turn of coil in the nth layer is given by,

Bmn(z) =µ0I

2r

(n+ 1

2

)2[(n+ 1

2

)2 +(z −m− 1

2

)2]3/2,

so that the magnetic field produced by the totality of all the coil turns in all the layers is given by,

Bcoil(z) =µ0I

2r

N∑n=1

Mn∑m=1

(n+ 1

2

)2[(n+ 1

2

)2 +(z −m− 1

2

)2]3/2. (1.8)

Now the “error” in the axial field profile is given by the function,

e(z) = |B(z)−Bcoil(z)|2,

where B(z) is the theoretically required profile from (1.5). The problem of finding the required coilconfiguration can now be formulated as the mathematical problem of minimizing e(z) with respect tothe variables N,M1,M2...MN, which is well-posed and well known. It can be solved in a straightforward manner by employing the MMSE (minimum mean square error) algorithm, which is wellstudied in other contexts [5].

We designed the coil configuration in two separate segments, the reverse and the forward. This wasnecessitated because the field strengths we required were on the higher side, upto a maximum of 300Gauss. The reverse windings were made with a 0.9mm thick cooled copper wire and the current usedwas 60A. The forward part was wound using a high-capacity cooled copper wire of 1.2mm diameterand a current of 127A was passed through it. The exact coil configurations for both the windings aregiven in Table 1.2.1.

Coil layer Reverse coil turns Forward coil turnsLayer 1 9 turns 4 turnsLayer 2 9 turns 4 turnsLayer 3 14 turns 4 turnsLayer 3 14 turns 4 turnsLayer 4 2 turns 4 turnsLayer 5 5 turns 4 turnsLayer 6 5 turns 4 turnsLayer 7 5 turns 6 turnsLayer 8 5 turns –

Table 1.1: Coil configuration of the forward and reverse segments of the Zeeman tuner

The Zeeman tuner was built acording to the above specifications and the axial magnetic fieldwas measured before the equipment was deployed on the actual experimental setup. Fig. 1.3 showsthe experimentally measured axial magnetic field strength and its near perfect concordance with thetheoretically desired profile.

The main laser system is the one used to effect cooling by coupling the 1P1 → 1S0 tansition. ASpectraPhys MilleniaXs laser operating at 532nm is fed to a Coherent 899-21 Ti:Sapphire ring laserto produce a continuously tunable output from 700 − 800nm. The Ti:Sa is frequency stabilized bylocking it to a Febry-Perot reference cavity at 797.6nm. This is then frequency doubled using a LaserAnalytic System - WaveTrain doubler. The final output is 398.8nm at just above 120mW.

The final RF magneto-optical trapping cavity consists of two pairs of coils in anti-Helmholtzconfiguration, made of hollow copper tubes through which chilled water is circulated to keep themcool. The complete arrangement, commonly called the Ioffe-Pritchard MOT is shown in Fig. 1.4.

8

Figure 1.3: Theoretically desired and experimentally obtained axial magnetic field strength insideZeeman tuner

Figure 1.4: Schematic of a Ioffe-Pritchard trap configuration

1.3 Adaptive Zeeman tuning

The analysis of §1.1 assumes that the deceleration of the atom via photon recoil is constant i.e., thatthe recoil force is a constant over time. This assumption has its origins in the expression (1.4), whichwas calculated assuming that the upper level population has reached steady state. It is of course truethat after a brief transient, the population reaches a steady state value; but it is a fact that during thetransient period, the laser is significantly off resonance since this is the time when the atomic cloudhas the highest velocity and hence experiences the maximum Doppler shift. Thus, during this period,the atom does not really undergo much slowing and so the initial segment of the Zeeman slower issimply a flight tube for all practical purposes.

The developments presented in this section, aims at rectifying this ineffectiveness and forms one ofthe novel pieces of work done during the internship period. It is quite obvious that when the atom ismoving through a Doppler shifted laser field, the optical Bloch equations (1.3), are to be supplementedwith another equation for the linear velocity of the ensemble, a freedom of degree which is very muchcoupled with the internal atomic degrees of freedom.

To obatin this final equation, one notes that the detuning δ is a time varying parameter, unlike

9

what it is made to look in (1.3); in particular, we have

dδ

dt=

d

dt(ωf − ωa − k〈v〉)

= −kd〈v〉dt

=k

m〈∂H∂z〉,

where in going to the last line, we have used the Ehrenfest Theorem. Using the dipole and rotatingwave approximations, one can express the above equation after taking the necessary expectation valuesas,

dδ

dt=

~km

(∂Ω∂z

ρge +∂Ω∗

∂zρeg

).

Using the fact that the Rabi frequency Ω is real, the above equation is equally well an equation forthe linear velocity V (t) in terms of the atomic coherences. Thus, we arrive at the full coupled opticalBloch equations,

d

dt

uvwV

=

−γ 0 Ω 00 −γ/2 kV − δ0 0−Ω δ0 − kV −γ/2 00 2s 0 0

uvwV

+

00

Ω/20

, (1.9)

where s ≡ ~m∂Ω∂z . It is to be noted that the above equations form a set of nonlinear coupled ODE’s

and is thus not possible to find closed form analytical solutions for them; it becomes necessary tosolve them numerically.

Instead of doing that, a realtime simulator was programmed using Simulink to simulate the dy-namics predicted by the coupled optical Bloch equations (1.9). The state diagram for the equationsin Simulink is shown in Fig. 1.5.

Figure 1.5: Simulink state diagram for the coupled optical Bloch equation simulator

10

The system was simulated for a realistic scenario of atoms starting out at 100m/s, but beingcontrolled by a cooling laser of very little power, namely of the order of a few pico-watts. In thestandard Zeeman tuned Doppler cooling situation of previous sections, the claculations predict thatthe atoms would be cooled to the Doppler limit in about 20secs. But in this case, where even theinitial transient region is utilized for resonant cooling, the results of the simulation Fig. 1.6 show thatthe atoms are cooled to the Doppler limit in about 4secs.

Figure 1.6: Results of a typical simulation of the coupled optical Bloch equations (1.9) in Simulink.The top three plots show the variables u(t), v(t), w(t), while the bottom most plot shows V (t), theatomic velocity.

Thus, it is seen that utilization of the initial transient dynamics can markedly improve the resultof laser cooling, especially for the case if light atoms [6]. The other use is for the development of arealtime automatic Zeeman tuner for a Doppler cooling system. In this case, instead of creating aspace varying magnetic profile for Zeeman tuning, one can use realtime simulations of the coupledoptical Blocj equations (1.9), to arrive at the time dependent magnetic field required for perfectresonant coupling of the laser and atom. Once this is known, it is easily possible to use a “gated”current to develop the required field. This technique would eliminate the need for a cumbersome coilsystem for the Zeeman tuner. These are developments for the future.

11

Chapter 2

Study of Hyperfine structure

In the typical study of the atomic spectra, the nucleus is considered a point entity, which does notexert any other influence on the electron(s) other than the Coulombic central force; this state ofaffairs is modeled, in the relativistic case, by a Hamiltonian which is the sum of the Dirac-CoulombHamiltonian and the configuration interaction term,

HDC = cα ·

(∑i

pi

)+ βmc2 −

∑i

Ze2

ri+∑j

Ze2

|ri − rj |

, (2.1)

where, the α, β are anti-commuting Dirac matrices and the sums are over the electrons. In theabove expression, the last double sum represents the configuration interaction (CI) terms, while theremaining terms represent the Dirac-Coulomb terms.

The above Hamiltonian has undergone several revisions [7], each bringing in a new term to accountfor various fine structure effects, including the fine splitting, the Lamb shift and the Zeeman-Starkshifts. All these effects have been due to the presence of electromagnetic fields that affect the electronstructure; with the former two originating from vacuum polarization while the later originating fromexternally applied fields.

On the other hand, the hyperfine splitting is a fine structure effect caused due to the finite size ofthe nucleus. In particular, it is caused by the overlap of the electronic wavefunctions with that of thenucleus. As the spherically symmetric component of the electron-nucleus interaction is already takeninto account in (2.1), hyperfine splitting is mainly caused by direction dependent coupling betweenthe electrons and the nucleus.

2.1 Theory of hyperfine interaction

The fact that the nucleus has a finite size means that its charge distribution is may exert a directiondependent force on the electronic cloud; of course, this is typified by the multipole moment expansionof the nuclear electromagnetic potentials [8],

φ(r) =∑k

1rk+1

(∑µ

(−1)µCk,µ(r)Qk,−µ

)

A(r) = −i∑k

1rk+1

√k + 1k

(∑µ

(−1)µCk,µ(r)Mk,−µ

),

(2.2)

where, Ck,µ and Ck,µ are the normalized scalar and vector spherical harmonic respectively; while,Qk,−µ andMk,−µ are the components of the nuclear electric and magnetic 2k–pole moment operatorsrespectively.

In this abstract model, we still approximate the nucleus by a collection of multipoles, rather thanelucidate its exact potential surfaces using a nuclear structure calculation.

12

The hyperfine interaction Hamiltonian is given by,

Hhfs(r) =∑i

e (α ·A(ri)− φ(ri)) , (2.3)

where the sum is over the electron coordinates. The above interaction Hamiltonian is the minimallycoupled Dirac Hamiltonian with the nuclear momentum set to zero, the Born-Oppenheimer approxi-mation, which basically assumes that the nucleus is massive enough to be considered stationary.

Substituting the multipole expansions (2.2) in the hyperfine Hamiltonian (2.3), gives,

Hhfs =∑k,µ

(−1)µTk,µTk,µ, (2.4)

where the kth rank spherical tensor Tk,µ acts on the electronic states and Tk,µ acts on the nuclearspace. We make the identification Tk,µ ≡Mk,µ for odd k and Tk,µ ≡ Qk,µ for even k. Explicitly, theelectronic tensors are given by,

Tk,µ =∑i

tk,µ(ri),

where,

tk,µ(r) =

− 1rk+1Ck,µ(r) even k

− irk+1

√k+1k α ·Ck,µ(r) odd k

(2.5)

2.1.1 First and second order hyperfine corrections

In the presence of multipolar fields typical of the hyperfine interaction, the total electronic angularmomentum J is no longer a conserved quantity; it is the “total” angular momentum F = I + J thatis conserved. Here, I is the nuclear angular momentum. Thus, the proper basis set of states are theones formed by taking the tensor product of the electronic and nuclear states, viz.,

|γIJFMF 〉 =∑

MJ ,MI

CFMFJMJ ;IMI

|γJMJ〉 ⊗ |IMI〉, (2.6)

where the expansion coefficients are the usual Clebsch–Gordan coefficients and γ represents the setof all the other electronic quantum numbers. For fixed values of I and J , we have the selection rule,|J − I| ≤ F ≤ J + I.

Using perturbation theory, the first order corrections in the electronic energies due to the hyperfineinteraction can be shown to be given by the diagonal elements of Hhfs in the coupled basis, i.e.,

W(1)F = 〈γIJFMF |Hhfs|γIJFMF 〉. (2.7)

Similarly, the second order correction is given by,

W(2)F =

∑γ′J ′

|〈γ′IJ ′FMF |Hhfs|γIJFMF 〉|2

Eγ′J ′ − EγJ, (2.8)

where the sum excludes the case γ′ = γ, J ′ = J .Before we actually set out to evaluate the above expression, it is useful to introduce a set of c-

number parameters that characterize the hyperfine interaction and which have a direct bearing withexperimental data. These hyperfine structure constants, are usually denoted by A,B,C,D... [9] andare defined in terms of the diagonal products of the “stretched” matrix elements of the hyperfinespherical tensors T and T . For any tensor operator Ok,µ, its “stretched” matrix element is defined asthe expectation value of its zeroth component in the stretched state |I,MI = I〉, i.e.,

〈Ok〉I ≡ 〈I,MI = I|Ok,µ=0|I,MI = I〉.

13

In particular, the magnetic dipole, electric quadrupole, magnetic octupole and electric hexadecapolemoments of the nucleus are defined in terms of these c-number stretched matrix elements via,

µ = 〈M1〉IQ = 2〈Q2〉I

Ω = −〈M3〉IΠ = 〈Q4〉I .

Now, the hyperfine structure constants A,B,C,D are defined as [9],

A =1IJ〈T1〉I 〈T1〉J =

µ

IJ〈T1〉J

B = 4 〈T2〉I 〈T2〉J = 2Q 〈T2〉JC = 〈T3〉I 〈T3〉J = −Ω 〈T3〉JD = 〈T4〉I 〈T4〉J = Π 〈T4〉J .

Now, the first order correction (2.7) can be expressed in the form [10],

W(1)F =

∑k

Xk(I, J, F ) 〈Tk〉I 〈Tk〉J , (2.9)

where the coefficients Xk are given by,

Xk(I, J, F ) =(−1)I+J+F

F J Ik I J

(J k J−J 0 J

) (I k I−I 0 I

) .

In the above expression,(j1 j2 j3m1 m2 m3

)stands for the Wigner 3–j symbol while

j1 j2 j3j4 j5 j6

stands

for the Wigner 6–j symbol. The second order correction (2.8), can also be explicitly expressed as [11],

W(2)F =

∑γ′J ′

(−1)J−J′

EγJ − Eγ′J ′

∑k,l

I J FJ ′ I k

I J FJ ′ I l

〈I|Tk|I〉〈I|Tl|I〉〈γJ |Tk|γ′J ′〉〈γ′J ′|Tl|γJ〉.

(2.10)Before the expressions (2.9) (2.10) can be used in an actual calculation, it is useful to know

the explicit expressions for the electronic spherical tensor operator’s expectation over the relativisticelectronic Dirac wavefunctions. In the “low” energy regime (where QED effects are negligible) ofrelativistic atomic theory, the typical electronic wavefunction is given by the bispinor,

|nκm〉 =1r

(iPnκ(r)Ωκm(r)Qnκ(r)Ω−κm(r)

),

where P,Q are the radial parts of the wavefunctions obtained as eigenstates of the Hamiltonian (2.1)and Ω are spherical spinors [8], which are eigenstates of the operator K = 1 − σ · L with eigenvalueκ = (l − j)(2j + 1). Then we have the following important result [10],

〈n′κ′|tk|nκ〉 =

−〈κ′|Ck|κ〉

∫∞0

drrk+1 (Pn′κ′Pnκ +Qn′κ′Qnκ) even k (electric)

〈κ′|Ck|−κ〉(κ′+κk

) ∫∞0

drrk+1 (Pn′κ′Qnκ +Qn′κ′Pnκ) odd k (magnetic),

where the selection rules: |j − j′| ≤ k ≤ j + j′ and l + l′ even, are to be observed.

14

Bibliography

[1] C. Cohen-Tannoudji, J. D. Roc, and G. Grynberg, Photons and Atoms (John-Wiley and Sons,1997).

[2] C. Cohen-Tannoudji, in Fundamental Systems in Quantum Optics, edited by J. Dalibard, J. M.Raimond, and J. Zinn-Justin, Ecole de Physique des Houches (Elsevier Science, 1990), vol. SessionLIII.

[3] H. J. Metcalf and P. van der Straaten, Laser cooling and trapping (Springer, 2002).

[4] J. D. Jackson, Classical Electrodynamics (John-Wiley and Sons, 2009), 3rd ed.

[5] S. Haykin, Adaptive Signal Processing (Tata Mc-Graw Hill, 2004).

[6] G. B. A. etal (ALPHA Collaboration), Phys. Rev. Lett. 105, 013003 (2010).

[7] B. Brandsen and C. Joachain, Physics of Atoms and Molecules (John-Wiley and Sons, 1990).

[8] W. R. Johnson, Lectures on Atomic Physics (Springer, 2007).

[9] W. Dankworth, J. Ferch, and H. Gebauer, Z. Phys. 267, 229 (1974).

[10] K. Beloy, A. Derevianko, and W. R. Johnson, Phys. Rev. A 77, 012512 (2008).

[11] K. Beloy and A. Derevianko, Phys. Rev. A 78, 032519 (2008).

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