Optical Pumping and Magnetic Resonance James Dragan Lab Partner: Stefan Evans Physics Department, Stony Brook University, Stony Brook, NY 11794. (Dated: October 4, 2013) We optically pump electrons in Rb 85 to the 5 2 S 1/2 , F=2, m f =+2 state and in Rb 87 to the 5 2 S1/2, F=3, m f =+3 using the D 1 line λ = 795 nm, emitted by a Rb Lamp, which becomes σ + polarized due to a linear polarizer and a quarter-wave plate. We verify the pumping using an RF signal and stimulated emission to populate the next lower m f level and observe the transmitted light read by a photodiode. We show that this transition is achieved in both isotopes. Using the absorption frequency we determine the Earths Magnetic field B Earth , along the quantization axis z. The presence of this field splits the degenerate states in the hyperfine structure in absence of an external B field, by using theoretical values for g F . We go onto determining the effects of power broadening the absorption dip. Lastly we measure Land˜ eg-factor, g F in both isotopes from measuring an applied B field from Maxwells coils and the corresponding resonance frequency. 1. INTRODUCTION Optical Pumping is a method first developed by Alfred Kastler 1 (who was awarded the Nobel Prize in 1966), that has become a widely used technique in experimental physics ever since. Optical pumping is a process in which electromagnetic radiation is used to pump electrons into a well- defined quantum state. The process by which this happens is dependent on the atomic structure of the sample and the properties of the radiation. The utilization of polarization and the selection rules for m leads to being able to pump the electrons into a dark state where there are no magnetic sublevels, m f , to excite to. This can be used even when there is an m f level to excite to and then the experimenter has a well-defined two-level system. The first thing one must account for is the atomic structure of the atomic sample.
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Optical Pumping and Magnetic Resonance
James Dragan
Lab Partner: Stefan Evans
Physics Department, Stony Brook University, Stony Brook, NY 11794.
(Dated: October 4, 2013)
We optically pump electrons in Rb85 to the 52S1/2 , F=2, mf=+2 state and in Rb87 to the
52S1/2, F=3, mf=+3 using the D1 line λ = 795 nm, emitted by a Rb Lamp, which becomes
σ+ polarized due to a linear polarizer and a quarter-wave plate. We verify the pumping
using an RF signal and stimulated emission to populate the next lower mf level and observe
the transmitted light read by a photodiode. We show that this transition is achieved in both
isotopes. Using the absorption frequency we determine the Earths Magnetic field BEarth,
along the quantization axis z. The presence of this field splits the degenerate states in the
hyperfine structure in absence of an external B field, by using theoretical values for gF . We
go onto determining the effects of power broadening the absorption dip. Lastly we measure
Landeg-factor, gF in both isotopes from measuring an applied B field from Maxwells coils
and the corresponding resonance frequency.
1. INTRODUCTION
Optical Pumping is a method first developed by Alfred Kastler1 (who was awarded the Nobel
Prize in 1966), that has become a widely used technique in experimental physics ever since. Optical
pumping is a process in which electromagnetic radiation is used to pump electrons into a well-
defined quantum state. The process by which this happens is dependent on the atomic structure
of the sample and the properties of the radiation. The utilization of polarization and the selection
rules for m leads to being able to pump the electrons into a dark state where there are no magnetic
sublevels, mf , to excite to. This can be used even when there is an mf level to excite to and then
the experimenter has a well-defined two-level system. The first thing one must account for is the
atomic structure of the atomic sample.
2
1.1. Fine Structure
When an electron undergoes orbital motion, there is an associated orbital magnetic moment
defined as ~µ = I ~A where I = −e(ω/2π) and ~A = π~r2. Rewriting ω in terms of the angular
momentum L we find that
|~µ| = −e~2m
l =−e~2m
√L(L+ 1) , (1)
where√L(L+ 1) are the eigenvalues of the L operator and e~
2m = µB = 9.27 × 10−24JT−1 is the
Bohr magneton.
In the presence of an external field, the magnetic moment will undergo a precession due to
the cross product of the two terms, resulting n a torque vector. Defining the fields axis along the
quantization axis ~Lz one finds that the frequency of precession is the Larmor frequency:
ωL = γB0 . (2)
Here γ = µB/~ is the gyromagnetic ratio. As stated we have defined the quantization axis to
be ~lz. It is important to notice that |Lz| < |~L|. This means that the magnetic moment will never
completely align with the field and thus it will always precess as described above.
Our next step is to account for the spin of the electron by which S=1/2 and ms = -1/2, 1/2. It
is shown that the resutling spin magnetic moment is3
~µs = −gse
2m~S . (3)
The Lande g-factor for spin, gs, was theorized by Dirac to be 2, and then shown through
Quantum Electrodynamics to be equal to 2.0023. As an aside, the motivation to measure gF in
this experiment is due to the fact that this number is disputed and gF depends upon this value.
Next we must change our reference frame to the electron, e−, which sees a nucleus precessing
around itself. Because the nucleus is charged, for reasons discussed in later sections, the rotating
charge produces a magnetic field. Using the Biot-Savart Law we find
~Bl =Ze2µ0
4πmr3~l . (4)
3
The electrons spin magnetic moment interacts with this field through the relation
HFS =Ze2µ0
8πm2r2(S · L) (5)
where (S · L) is given by the relation
J2 = L2 + 2L · S + S2 . (6)
Rearranging the terms, we find
L · S =1
2(J2 + L2 + S2) =
~2
2[J(J + 1)− L(L+ 1)− S(S + 1)] . (7)
The full expression for the fine structure correction to the Hamiltonian is given by3
HFS =Egα
2
5
(1
J + 1/2
). (8)
If the atom had no angular momentum from the nucleus then this would hold enough information
to fully describe the energy levels. In the case of this experiment we must take into account the
spin of the nucleus.
1.2. Hyperfine Structure
Due to the spin of the nucleus, I = 3/2 in 87Rb and I = 5/2 in 85Rb, I and J couple to give our
grand angular momentum quantum number F = |~F | = |~I + ~J |. Looking at the nucleus’ magnetic
moment we see
~µI = +gIµN~~I , (9)
where gI is the Lande g-factor for the nucleus’ spin, |~I| = ~√I(I + 1), and µN =
e~2M
=µB
1836.
The hyperfine perturbation to the Hamiltonian is given as follows:
HHFS = −~µI · ~BJ (10)
4
where ~BJ is given by
~BJ =~J√
J(J + 1). (11)
Thus
HHFS =gIµN~
Bj1√
J(J + 1)(~I · ~J) . (12)
We solve for (~I · ~J) in the same procedure as Eq (1.6) by defining the total atomic angular momentum
number ~F = ~I + ~J . Where |~F | = ~√F (F + 1) and F has values F = |I − J |, .., |I + J | of integer
steps. For each hyperfine level, there are 2F+1 magnetic sublevels, mF . Solving for (~I · ~J) we find
(I · J) =1
2[F (F + 1)− I(I + 1)− J(J + 1)] (13)
This gives us our full Hamiltonian for the hyperfine structure:
HHFS =gIµNBJ
2√J(J + 1)
[F (F + 1)− I(I + 1)− J(J + 1)] . (14)
This is the full form for our hyperfine structure which we can now use to map out the energy
levels of Rubidium.
1.3. Energy Levels of 85Rb and 87Rb
In this experiment we use naturally occurring Rubidium which comes in two isotopes, 85Rb
(72% abundance and nuclear spin quantum number I=5/2) and 87Rb (28% abundance and nuclear
spin quantum number I=3/2)2. Alkali Atoms are defined by having a positively charged core
with a single valence electron, in our case occupying the 5s orbital. The electron shell of orbitals
[(1s2)(2s2)(2p6)(3s2)(3p6)(4s2)(3d10)(4p6)] are all filled making this Rb+ core spherically symmetric
with a total angular momentum of Lc=0, spin orbital angular momentum of Sc=0. The LS-
coupled angular momentum quantum number ~Jc=|Jc|=| ~Lc + ~Sc|=0 where J is defined quantum
mechanically to take values of increasing integers between |L− S| to |L+ S|.
Because the ion core does not contribute any total angular momentum, all the momentum
quantum numbers come from the valence electrons. Considering the first electronic ground state,
5
with ionization energy 4.177eV or 296.81nm, in Rubidium (5s)2S1/2 we see that n=5, l=0, S=1/2,
L=0 and therefore J = 1/2. Since J takes on only one value there is no fine structure in the
(5s)2S1/2 state. For the first excited state, (5p)2P ,however there is fine structure splitting. We
find that n=5, l =1, S = 1/2, L=1 and therefore J = 1/2, 3/2 which gives two fine structure levels
of (5p)2P1/2 and (5p)2P3/2. The spin orbit coupling energy term is given by Eq. (8).
Using this equation we find that the (5p)2P1/2 is lower than the (5p)2P3/2 state. The energy
to couple (5s)2S1/2 to (5p)2P1/2 is given as λ = 795 nm which is referred to as the D1 line. The
energy to couple (5s)2S1/2 to (5p)2P3/2 is given as λ = 780 nm which is referred to as D2 line2,3.
In this experiment, both the D1 and D2 line are produced by the Rb Lamp, but we filter out the
D2 line so that only the D1 line is incident on the atoms. It should also be noted that the lifetime
of these excited states are extremely small, ≈ 10−8s which is instantaneous with respect to how
fast the photodiode can detect changes in the input power.
If we account for the nucleus, and its spin quantum number then we find even finer splittings
in energy spectra. Looking at the 85Rb isotope, I = 5/2, we find splittings in both fine structures.
In the ground state 52S1/2 the degeneracy is split into two hyperfine levels F = 2, F =3. For the
52P1/2 state we find F = 2, 3 whereas for the 52P3/2 state F = 1, 2, 3, 4. Looking at the 87Rb
isotope with I = 3/2 we find that the ground state 52S1/2 is split to F = 1 and F = 2. In the two
excited states we find that 52P1/2 is split into F = 1 and F = 2, while the 52P3/2 state has F = 0,
1, 2, 3.
As stated for each hyperfine F level, there are 2F + 1 mF sublevels if the degeneracy is lifted.
For F = 1 there are 3 mF levels corresponding to mF = -1, 0, +1. For F = 2, mF = -2, -1, 0, +1,
+2. For F = 3, mF = -3, -2, -1, 0, +1, +2, +3 and for F = 4, mF = -4, -3, -2, -1, 0, +1, +2, +3,
+4. A diagram of the energy structure for each isotope in the 52S1/2 and 52P1/2 state is shown.
These are the corresponding ground state and excited state excited from the D1 line.
6
D1 Line 795 nm
F = 2
F = 1
F = 1
F = 2
.8 GHz
6.8 GHz
-2 -1 0 +1 +2
-2 -1 0 +1 +2
-1 0 +1
-1 0 +1
D1 Line 795 nm
F = 3
F = 2
F = 2
F = 3
-3 -2 -1 0 +1 +2 +3
-3 -2 -1 0 +1 +2 +3
-2 -1 0 +1 +2
-2 -1 0 +1 +2 361.58 MHz
3.035 GHz
FIG. 1: The energy diagram for the D1 line for 85Rb and 87Rb is shown along with the corresponding energy spacing.
We find that in the presence of no external field, the hyperfine level F is degenerate. If a field is applied, then there
is an energy spacing between mF porportional to the strength of the field. The number of these levels correspond to
2F+1.
In this experiment we use frequencies in the kHz range to make transitions between mF levels
once the degeneracy is split. It is clear from the diagram above that we know transitions are made
in the same hyperfine F level based on the large energy separation between F levels. We now have
enough information to present the concept of optical pumping.
1.4. Optical Pumping
In this experiment the incident light on the Rb cell is σ+ polarized. Due to the selection rules
we find that no transition can occur unless ∆m = 0,±1 6. These solutions correspond to the
three distinct polarization types which are π, σ− and σ+ with π referring to linear polarization,
σ− referring to left-hand circularly polarized and σ+ referring to right-hand circularly polarized
7
light with respect to the direction of propagation. This experiment utilizes σ+ polarized light to
send atoms to the highest mF level, which becomes a dark state, as it has nowhere to excite to
after (since we block the D2 line). After a transition is made to an excited state, m′F = mF + 1,
it will decay rapidly and spontaneously. It is important to recall that the lifetime is on the order
of ×10−8s. When spontaneous decay occurs, the direction of emission is uniform in all directions
and polarization of the emitted light is arbitary, in that it can make a transition of ∆mF = 0,±1.
This is indicated by the stripped line in the figure below.
-2 -1 0 +1 +2
F = 2
F = 1
F = 1
F = 2
FIG. 2: Here the effects of σ+ polarized light incident on an atom are shown. Since the light has angular momentum
the selection rules tell us that for σ+ light, ∆m = 1 in a transition. We see this results in electrons going to the right.
This diagram shows optical pumping to the F = 2, mF = +2 level in 87Rb. The same case is true in 85Rb except we
optically pump to F = 3, mF = +3. Once the electron has undergone stimulated absorption, it will excite to a higher
energy level. When it undergoes spontaneous emission the polarization of the light is random and therefore it can
make any of the three transitions indicated by a dashed line. After many lifetimes of absorption and decay/emission
we find all the electrons in the F = 2, mF = +2 state. This is the dark state. In the case of σ− light we can pump
electrons to the left, to the F = 2, mF = −2 in this case.
8
If the electron undergoes stimulated emission then the polarization it encounters will be σ−
sending it back to the original mF level it started at. It quickly becomes clear that to effectively
optically pump, the incident light on the atoms must be present over at least ten lifetimes. In
our case, if we take the lifetime of the excited state to be 10−8 then in a full second we have
≈ 108 cycles of absorption and decay or emission. Therefore, practically after only one second, the
chances of all the electrons being optically pumped is very high. The efficiency is limited due to
other factors that are noted in the Procedure section. At time t = 0 we assume that all the mF
states are equally populated due to thermal excitations, kbT ≈ 1012s >> GHz. It must be noted
that a static external field must be present to split the degeneracy of the mf levels, otherwise we
could not optically pump. Experimentally the magnetic field from the Earth, Bearth, is enough to
split the degeneracy. Now let’s look more in depth at how static fields perturbe the system.
1.5. Interaction with Static External Fields
This experiment utilizes a static magnetic field to split the degeneracy of the mF levels. A
similar phenomena occurs when there is a static electric field present but for the purposes of this
experiment those such effects will not be discussed. As mentioned, each hyperfine level consists of
2F+1 degenerate sublevels, mF . In the presence of an external magnetic field the degeneracy is
broken.
HB =µB~
(gsSz + gLLz + gIIz) ·Bz (15)
Eq. (15) is the Hamiltonian[4][5] describing the interaction with the magnetic field along the
atomic quantization axis. We see that the B field interacts with the magnetic dipole moments of
the electron spin, electron orbit, and the nuclear spin. gS , gL, gI are the electron spin, electron
orbital, and nuclear ”g-factors”. Here, gL = 1 −me/mnucleus , which we approximate to 1. The
exact value of gs has been measured to high precision to be 2.00231930436153(53)[7] which we will
approximate to 2 in this lab. To calculate gJ we look at the magnetic moment associated with J:
(~µS)J = [(~µL)J + (~µS)J ] · J|J |
(16)
where
9
(~µL)J =µB2
~L · ~J|J |
(~µS)J = −2µB2
~S · ~J|J |
. (17)
Solving for ~µJ we find,
~µJ = −µB2
[3
2+S(S + 1)− L(L+ 1)
J(J + 1)
]~J , (18)
which gives us[3],
gJ '3
2+S(S + 1)− L(L+ 1)
J(J + 1). (19)
The g-factor to consider is gF . Again we look at the magnetic moment associated with ~F . Solving
in the same fashion as above we find,
~µF = −gJF (F + 1) + J(J + 1)− I(I + 1)
2F (F + 1)
µB~~F , (20)
which gives us a value of gF where we neglect the nuclear term because it is a correction of .1%.
We repeat this approximation below for the final form the Hamiltonian for the same reason.[3][4][5]
gF ' gj[1 +
J(J + 1)− I(I + 1)
F (F + 1)
]. (21)
Writing the Hamiltonian for the hyperfine structure’s interaction with an external magnetic field