Download Electromagnetically Induced Transparency: The

Electromagnetically Induced Transparency:
The Zeeman Method
A Thesis
Presented to
The Division of Mathematics and Natural Sciences
Reed College
In Partial Fulfillment
of the Requirements for the Degree
Bachelor of Arts
W. Ace Furman
May 2016
Approved for the Division
(Physics)
Lucas Illing
Acknowledgements
Before I begin, I would like to mention a handful of people without whom this thesis
and my career at Reed could never have been completed:
– Lucas Illing, whose guidance and seemingly bottomless knowledge got me through
even the most difficult of thesis crises;
– Jay Ewing, for the technical knowhow, for the avid reassurance, and for a sleek
apparatus;
– John Essick, Joel Franklin, Darrell Schroeter, and David Griffiths, who were
always available to mull over confusing topics and usually solve all of my problems;
– Bob Ormond, firstly, for all the anecdotes and, secondly, for the electronics;
– Johnny Powell and Noah Muldavin, for reminding me to hang loose;
– Gary and Frankie Furman, who ensured my sanity during the thesis process
and also throughout the course of my life;
– Zubenelgenubi Scott and Indy Liu, for being the best officemates in the whole
subbasement (Dominion anyone?);
– Alex Deich and Naomi Gendler, for teaching me what friends are for;
– Colleen Werkheiser, without whom I would not have known where to even start;
– Seth Gross, for making Mondays the best day of the week, which, by the way,
is no trivial matter;
– Taylor Holdaway and Kai Addae, for helping me procrastinate just the right
amount #Slurpees;
– Spencer Fussy and Carly Goldblatt, for accepting me the way I am;
– Jack Taylor, for undisclosed reasons
;
– Munyo Frey, whose continuous encouragement and loving outlook pulled me
through the final stretch; and,
– Emilia and Inés Furman and Arrow and Vega Henson, who remind me everyday
that the universe is a beautiful place.
Thank you to all.
Table of Contents
Introduction
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1
Chapter 1: Theory . . . . . . . . . . . . . . . . . . . . .
1.1 Derivation of the EIT Hamiltonian . . . . . . . . .
1.1.1 Bare-Atom Hamiltonian . . . . . . . . . . .
1.1.2 Applied Fields . . . . . . . . . . . . . . . . .
1.1.3 Rotating Wave Approximation . . . . . . . .
1.1.4 Time and Phase Independent Frame . . . .
1.2 Dressed State Picture . . . . . . . . . . . . . . . . .
1.2.1 Dressed States . . . . . . . . . . . . . . . . .
1.2.2 The Dark State and The Bright State . . . .
1.2.3 Higher-Level Systems . . . . . . . . . . . . .
1.3 Absorption in EIT . . . . . . . . . . . . . . . . . .
1.3.1 Density Operator . . . . . . . . . . . . . . .
1.3.2 Von Neumann Equation . . . . . . . . . . .
1.3.3 Von Neumann Equation in EIT . . . . . . .
1.3.4 Complex Susceptibility And Its Implications
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3
3
4
4
6
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9
9
10
10
14
14
15
16
18
Chapter 2: Experiment . . . . . . . . . . . . . .
2.1 Rubidium and Zeeman Splitting . . . . . .
2.2 Preliminary Setup . . . . . . . . . . . . . .
2.2.1 Diode Laser . . . . . . . . . . . . .
2.2.2 Saturated Absorption Spectroscopy
2.3 Primary EIT Setup . . . . . . . . . . . . .
2.3.1 Components of Polarization . . . .
2.3.2 Zeeman Apparatus . . . . . . . . .
2.4 Procedural Overview . . . . . . . . . . . .
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23
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24
25
25
27
27
30
34
Chapter 3: Results . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Possible Improvements . . . . . . . . . . . . . . . . . . . . .
3.2.1 F = 1 → F 0 = 0 Transition of the D2 Line . . . . . .
3.2.2 D1 Line . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Laser Locking . . . . . . . . . . . . . . . . . . . . . .
3.2.4 Higher Resolution Detection and Real-Time Scanning
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35
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38
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Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
List of Figures
0.1
0.2
A typical absorption spectrum, and an EIT absorption spectrum. . .
Λ-type structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2
1.1
1.2
All three level structures capable of demonstrating EIT . . . . . . . .
Λ-type level structure with applied field energies (~ωp and ~ωc ) and
their energy detunings from resonance (~∆p and ~∆c ). . . . . . . . .
Energy diagram of a six-level system . . . . . . . . . . . . . . . . . .
Linear susceptibility plotted in arbitrary units as a function of normalized probe frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3
1.4
2.1
2.2
2.3
2.4
2.5
2.6
2.7
3.1
3.2
The F = 1 → F 0 = 1 transitions . . . . . . . . . . . . . . . . . . . . .
Schematic of saturated absorption setup. . . . . . . . . . . . . . . . .
Hyperfine spectra of 87 Rb. . . . . . . . . . . . . . . . . . . . . . . . .
Schematic of the primary EIT setup . . . . . . . . . . . . . . . . . . .
Section and 3D rendering of the Zeeman apparatus . . . . . . . . . .
Magnetic field attenuated by mu metal shielding inside the Zeeman
apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Magnetic field driven by the solenoid inside the Zeeman apparatus . .
Experimental EIT absorption of 87 Rb . . . . . . . . . . . . . . . . . .
Simulation of absorption and dispersion as the decoherence parameter
γ2 varies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
11
20
24
26
27
28
31
32
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36
Abstract
Electromagnetically induced transparency (EIT) — a quantum phenomenon wherein
a material undergoes optical and dispersive modifications under the influence of resonant electromagnetic fields — was observed in warm rubidium vapor using the Zeeman
method. Specifically, the absorption of a probe laser through the atomic medium in
a magnetic field was measured, and a transparency window was observed around a
typically absorbent atomic resonance, consistent with EIT.
Introduction
Electromagnetically induced transparency (EIT) is a quantum phenomenon that
makes use of resonant electromagnetic fields to alter the optical properties of an
atomic medium. Under typical conditions — that is, in the presence of a single nearresonant field — atoms will absorb energy from the surrounding field and excite into
higher energy states. The absorption spectrum, as a function of field frequency, follows a Lorentzian curve peaked around the natural frequency of the resonant atomic
transition, as shown in Fig. 0.1a.
In EIT, two fields are introduced, each resonant with a distinct atomic transition.
This alters the absorption spectrum to include a window of frequencies, for which
the medium exhibits high transparency, as shown in Fig. 0.1b. Hence, we get the
name electromagnetically induced transparency. In addition, the dispersion properties
are also affected, giving rise to many useful applications such as effective nonlinear
frequency conversion, slow light and nonlinear mixing.
EIT was first theorized in 1990 by Harris et al. [1] as a variation on coherent
population trapping. The following year, EIT was observed experimentally by Boller
et al. [2] in strontium vapor and by Field et al. [3] in lead vapor. Since then EIT has
been an extremely prevalent topic of research in physics. In 1996, Jain et al. [4] showed
its applications in nonlinear mixing, wherein light beams of a certain wavelength are
converted to another wavelength with high efficiency. In 1998, Ling et al. [5] created
an electromagnetically induced diffraction grating from a gaseous atomic medium.
In 1999, Budker et al. [6] used EIT to control the group velocity of light traveling
through rubidium vapor. They slowed group velocities of light to 8 m/s.
Since two fields must be present to see EIT, it is one of the few examples of
optical setups in which light signals interact with other light signals. For example,
say one field were to be turned off; the medium would become absorbent again and
the remaining field would be obscured. This gives rise to even more applications such
as optical switching [7].
The quantum mechanical mechanisms underlying EIT are quite complicated and
not easily intuited. One simple way to think of EIT is as a result of destructive interference of transition probabilities. Given the three-level structure in Fig. 0.2, there
exist two paths between state |1i and state |3i (excluding longer paths): |1i → |3i
and |1i → |3i → |2i → |3i. If the probabilities of each of these transitions occurring in an atom have opposing phases, destructive interference could occur and result
in no transitions from |1i to |3i [8]. This would, in turn, lead to a reduction in
the absorption of the field resonant with that transition. More rigorous derivations of
2
Introduction
Α
Α
HaL
HbL
Dp
D
Figure 0.1: (a) A typical absorption spectrum, and (b) an EIT absorption spectrum.
In both cases, ∆ is the laser detuning from atomic resonance. The subscript p in (b)
signifies that the probe laser is being scanned.
È3\
È2\
È1\
Figure 0.2: Λ-type structure.
three-level systems are discussed in Chapter 1, as well as a novel derivation describing
the six-level system used in this experiment.
This experiment explores specifically Zeeman EIT, where the transparency window
is seen not as a function of the incoming light but as a function of the strength of a
magnetic field applied across the atomic medium. This is a result of the Zeeman effect,
which allows us to split degenerate energy levels in the atoms, effectively changing the
resonant frequencies of the transitions. In essence, instead of changing the frequency
of light to match the atomic resonance — as in standard EIT — we will change the
level splittings such that the resonant frequency matches that of the incoming light.
Specifics about how Zeeman EIT is observed in this experiment are discussed in
Chapter 2. The results and analysis of the experiment are shown in Chapter 3.
Chapter 1
Theory
EIT is a quantum phenomenon that cannot be predicted classically. However, a fully
quantum mechanical derivation is not strictly necessary. This chapter will make a
semiclassical derivation of the Hamiltonian for EIT, which treats atoms as quantum
mechanical objects that interact with classical vector fields. This derivation follows
closely the works of Weatherall [9] and Erickson [10]. We will then interpret the
Hamiltonian in two ways. First, we will take a look at the “dressed” state basis of
the EIT Hamiltonian to see how a dark state is induced by the applied fields. Then,
we will use the Hamiltonian to explicitly calculate the absorption and dispersion of
such a system. The semiclassical approach is sufficient to see both the mechanisms
behind EIT and its effects. The main tradeoff of this derivation is that, while it is
much simpler than the fully quantum mechanical derivation, it does not account for
spontaneous emission of the atoms. Instead, we will add necessary phenomenological
decay terms by hand in Section 1.3.2.
1.1
Derivation of the EIT Hamiltonian
EIT is only seen in atoms with specific energetic properties. Three-level energy configuration must match one of the following: Λ-type, V-type, or ladder (cascade) type,
all shown in Fig. 1.1. The transitions in rubidium that will be utilized in this experiment exhibit a combination of Λ-type and V-type energy level structures. Since
Λ-atoms have a very low decay rate from the |2i state, the effect of EIT is much
stronger than in the other configurations [10]. For this reason, we will focus on the
theory as it applies to Λ-structure; however, the following derivations generalize to
any of these energy structures, and further, to Zeeman EIT.
Let |1i, |2i, and |3i be the names given to the ground state, the metastable state,
and the excited state, respectively, of the atom without any fields applied; their
energies will be defined in terms of their corresponding frequencies as
En = ~ωn ,
(1.1)
for n = 1, 2, 3. While |1i and |2i can both excite into |3i, we assume that the transition
between |1i and |2i is dipole forbidden. Therefore, the resonant frequencies between
4
Chapter 1. Theory
È3\
È3\
È3\
È2\
È2\
È2\
È1\
È1\
È1\
Figure 1.1: All three level structures capable of demonstrating EIT, namely, Λ-type
(left), V-type (middle), and ladder type (right).
relevant energy levels are
ω13 = ω3 − ω1 ,
ω23 = ω3 − ω2 .
1.1.1
(1.2a)
(1.2b)
Bare-Atom Hamiltonian
Before any external field is applied, the energy eigenvalues and their eigenstates are
denoted ~ωn and |ni, respectively. We can enforce completeness and orthogonality
such that
X
|nihn| = 1,
(1.3)
n
and
hn|mi = δnm .
(1.4)
The bare-atom Hamiltonian H0 can therefore be expressed in this basis as a matrix
with elements


~ω1 0
0
H0 =  0 ~ω2 0  .
(1.5)
0
0 ~ω3
Without any external stimulation, this system will tend to settle in the lowest energy
|1i state due to spontaneous emission from higher energy states.
1.1.2
Applied Fields
Now consider that two fields are applied so that electrons populate the upper states.
We can assume that, other than the three states in the Λ-configuration, all other
states are relatively unoccupied and can be neglected. The control field is tuned to
the resonant frequency ωc ≈ ω23 with amplitude Ec , and the probe field is tuned to
the other resonance ωp ≈ ω13 with amplitude Ep . A energy diagram of this scheme is
shown in Fig. 1.2. This yields an electric field
E = Ep cos (ωp t − kp · r) + Ec cos (ωc t − kc · r) ,
(1.6)
1.1. Derivation of the EIT Hamiltonian
5
È3\
ÑD p
ÑDc
ÑΩ12
ÑΩ p
ÑΩc
ÑΩ13
È2\
È1\
Figure 1.2: Λ-type level structure with applied field energies (~ωp and ~ωc ) and their
energy detunings from resonance (~∆p and ~∆c ).
where the magnitude of the wavevector k = 2π
for the wavelength λ of the fields.
λ
Note that for wavelengths much larger than the diameter of an atom, we have λ r;
thus, k · r is small, so it can be dropped, leaving
E = Ep cos (ωp t) + Ec cos (ωc t) .
(1.7)
The fields will perturb the Hamiltonian from that of the bare atom such that
H = H0 + H 1 ,
(1.8)
where H1 is the interaction term between the fields and the atom and can be written
as
H1 = −qE · d.
(1.9)
This assumes that the atom behaves similarly to an electric dipole with charge q and
separation vector d. This is a particularly good approximation for hydrogenic atoms
— like rubidium — with only one electron in their outer shell. The dipole will quickly
align with the fields, so we can instead write
H1 = −qEd.
(1.10)
Now define the dipole moment operator ℘ ≡ qd and its elements ℘mn ≡ hn|℘|mi.
With these definitions, the interaction Hamiltonian becomes
H1 = −℘E.
(1.11)
Note that many of the matrix elements of this Hamiltonian are zero. For example,
the dipole elements ℘12 and ℘21 must be zero for the |1i → |2i transition to be
dipole forbidden. Additionally, the diagonal terms must go to zero because of the
spherical symmetry of the wavefunction [9]. Therefore, the matrix representation of
the Hamiltonian simplifies further to


0
0 ℘13
0 ℘23  .
H1 = −E  0
(1.12)
℘31 ℘32 0
6
1.1.3
Chapter 1. Theory
Rotating Wave Approximation
As it is, the interaction term of the Hamiltonian looks simple, but thus far, the electric
field has not played its part. In order to clearly see its effect, it is useful to transform
into the interaction picture of the unperturbed system. To do that, we use a unitary
matrix U defined as the time evolution operator, such that
 iω t

e 1
0
0
0 .
U (t) = eiH0 t/~ =  0 eiω2 t
(1.13)
iω3 t
0
0 e
Applying this transformation to H1 yields


0
0
℘13 e−iω13 t
0
0
℘23 e−iω23 t  .
U H1 U † = −E 
℘31 eiω13 t ℘32 eiω23 t
0
(1.14)
As seen from Eq. 1.7, the electric field is a sum of cosines. These can be written
instead as exponentials:
E=
Ec iωc t
Ep iωp t
e
+ e−iωp t +
e
+ e−iωc t .
2
2
(1.15)
Plugging this into the transformed Hamiltonian, the nonzero elements become
Ec iωc t
−iω13 t
Ep iωp t
−iωp t
−iωc t
†
e
+e
+
e
+e
(U H1 U )13 = −℘13
e
,
(1.16a)
2
2
Ec iωc t
−iω23 t
Ep iωp t
†
−iωp t
−iωc t
(U H1 U )23 = −℘23
e
+e
e
+e
+
e
,
(1.16b)
2
2
Ec iωc t
iω13 t
Ep iωp t
−iωp t
−iωc t
†
e
+e
e
+e
+
e
,
(1.16c)
(U H1 U )31 = −℘31
2
2
Ec iωc t
iω23 t
Ep iωp t
−iωp t
−iωc t
†
(U H1 U )32 = −℘32
e
+e
+
e
+e
e
.
(1.16d)
2
2
In the rotating wave approximation, it is assumed that the rapidly oscillating terms
will average out quickly and therefore can be ignored. Thus, exponential terms with
large imaginary arguments drop out. Since ωc ≈ ω23 and ωp ≈ ω13 , there will be one
exponential term in each matrix element that oscillates slowly enough to survive this
approximation. Namely,
1
(U H1 U † )13 = − Ep ℘13 ei(ωp −ω13 )t ,
2
1
†
(U H1 U )23 = − Ec ℘23 ei(ωc −ω23 )t ,
2
1
†
(U H1 U )31 = − Ep ℘31 ei(ω13 −ωp )t ,
2
1
†
(U H1 U )32 = − Ec ℘32 ei(ω23 −ωc )t .
2
(1.17a)
(1.17b)
(1.17c)
(1.17d)
1.1. Derivation of the EIT Hamiltonian
7
Transforming back to the Schrödinger picture, the interaction Hamiltonian becomes
H1 = U † U H1 U † U


0
0
Ep ℘13 eiωp t
1
0
0
Ec ℘23 eiωc t  .
=− 
2
−iωp t
−iωc t
Ep ℘31 e
Ec ℘32 e
0
(1.18)
The electric dipole operators can be represented in terms of their magnitudes and
phases such that
℘13 = ℘∗31 = |℘13 |eiφp ,
℘23 =
℘∗32
= |℘23 |e
iφc
.
(1.19a)
(1.19b)
The Rabi frequencies of this system are defined to be
Ep |℘13 |
,
~
Ec |℘23 |
Ωc =
.
~
Ωp =
(1.20a)
(1.20b)
Plugging these definitions into the interaction Hamiltonian and summing it with the
bare-atom Hamiltonian, the full Hamiltonian in the presence of applied fields becomes

2ω1
0
−Ωp ei(ωp t+φp )
~
H= 
0
2ω2
−Ωc ei(ωc t+φc )  .
2
−i(ωp t+φp )
−i(ωc t+φc )
−Ωp e
−Ωc e
2ω3

1.1.4
(1.21)
Time and Phase Independent Frame
The Hamiltonian in Eq. 1.21 still depends on phases and time, which muddies the
interpretation. To clearly see how EIT will come from such a Hamiltonian, we must
transform into a new basis — the corotating basis. The unitary matrix that will
achieve this transformation is defined as

e−i(ωp t+φp )
0
0
Ũ (t) = 
0
e−i(ωc t+φc ) 0  .
0
0
1

(1.22)
Note that the full Hamiltonian has a different set of eigenstates than the bare atom.
Define |n0 i to be the eigenstates of H. There are also analogous eigenstates |ñ0 i = Ũ |n0 i
of the corotating Hamiltonian H̃. For the corotating eigenstates to satisfy Schrödinger’s
8
Chapter 1. Theory
equation, we must have
∂ 0
|ñ i
∂t
∂ 0
= i~
Ũ |n i
∂t
!
∂|n0 i
∂ Ũ 0
|n i + Ũ
= i~
∂t
∂t
H̃|ñ0 i = i~
!
∂ Ũ 0
1
= i~
|n i + Ũ H|n0 i
∂t
i~
!
∂ Ũ †
Ũ + Ũ HŨ † Ũ |n0 i
= i~
∂t
!
∂ Ũ †
0
†
H̃|ñ i = i~
Ũ + Ũ HŨ |ñ0 i.
∂t
(1.23)
Therefore, the transformation of the Hamiltonian into the corotating frame is
∂ Ũ †
Ũ + Ũ HŨ †
∂t




2ωp 0 0
2ω1
0
−Ωp
~
~
2ω2 −Ωc 
=  0 2ωc 0  +  0
2
2
0
0 0
−Ωp −Ωc 2ω3


2(ω1 + ωp )
0
−Ωp
~
0
2(ω2 + ωc ) −Ωc  .
H̃ = 
2
−Ωp
−Ωc
2ω3
H̃ = i~
(1.24)
(1.25)
By noting that the physical interpretation of a Hamiltonian is unaltered by the addition of a scalar multiple of the identity, the Hamiltonian can be coerced into an even
more interpretable form. That is, adding −~(ω1 + ωp )1 yields


0
0
−Ωp
~
.
2(ω2 + ωc − ω1 − ωp )
−Ωc
H̃ =  0
(1.26)
2
−Ωp
−Ωc
2(ω3 − ω1 − ωp )
Finally, defining the laser detunings from resonance as ∆p ≡ ω13 − ωp = ω3 − ω1 − ωp
and ∆c ≡ ω23 − ωc = ω3 − ω2 − ωc , the EIT Hamiltonian takes on its standard form:


0
0
−Ωp
~
2(∆p − ∆c ) −Ωc  .
H̃ =  0
(1.27)
2
−Ωp
−Ωc
2∆p
1.2. Dressed State Picture
1.2
9
Dressed State Picture
The following dressed state analysis draws heavily on the work of Fleischhauer et al.
[11], Purves [12], and Marangos [13].
1.2.1
Dressed States
In order to analyze the eigensystem of the EIT Hamiltonian, we will approximate
that ∆p ≈ ∆c ≡ ∆. This is reasonable because these detunings are small compared
to other relevant frequencies in the experiment.
Solving the characteristic equation
det H̃ − λ1 = 0,
(1.28)
yields the following eigenvalues:
λ0 = 0,
(1.29a)
~
q
∆ − ∆2 + Ω2p + Ω2c ,
2
q
~
λ+ ≡ ~ω+ =
∆ + ∆2 + Ω2p + Ω2c .
2
λ− ≡ ~ω− =
(1.29b)
(1.29c)
The corresponding eigenstates of the “dressed” system, expressed as linear combinations of the bare-atom states, are
Ωc |1i − Ωp |2i
p 2
,
Ωp + Ω2c
Ωp |1i + Ωc |2i
p
|−i = −
+ |3i,
∆ − ∆2 + Ω2p + Ω2c
Ωp |1i + Ωc |2i
p
− |3i.
|+i =
∆ + ∆2 + Ω2p + Ω2c
|0i =
(1.30a)
(1.30b)
(1.30c)
Note that an atom in state |0i cannot excite into state |3i because |0i has no component of the |3i state. For this reason, we call |0i the “dark” state.
When the fields are close to resonance — that is, ∆ ≈ 0 — the upper dressed
states can be normalized and approximated by
1
|−i = √
2
1
|+i = √
2
!
Ωp |1i + Ωc |2i
p 2
+ |3i ,
Ωp + Ω2c
!
Ωp |1i + Ωc |2i
p 2
− |3i .
Ωp + Ω2c
(1.31a)
(1.31b)
10
1.2.2
Chapter 1. Theory
The Dark State and The Bright State
If we assume that the amplitude of the probe beam is small compared to that of the
control, then Ωp Ωc . Using this approximation and normalizing, the dark state |0i
becomes
|0i ≈ |1i.
(1.32)
Under these conditions, the dark state is approximately the same as the ground state
of the bare atom. Therefore, the dark state is not only decoupled from the other
two, but additionally, it is a stationary state of both the bare-atom Hamiltonian and
the dressed system. This means that, due to their time-independence, atoms in the
ground state will remain there and never excite. Thus, the probe beam will remain
unabsorbed.
It would seem that atoms that start out in some admixture of the upper states
would remain excited. However, this semiclassical treatment of a quantum phenomenon neglects the possibility of spontaneous emission from the upper states into
the ground state. In actuality, these states will decay into the ground state and then
be trapped there. Spontaneous emission will be discussed further in Section 1.3.2.
As one final note about dressed states, we wish to gain some intuition into the
nature of the non-dark states |+i and |−i. Using Eqs. 1.31a and 1.31b, we may define
two linear combinations of the states
|+i − |−i
√
= |3i,
2
|+i + |−i
Ωp |1i + Ωc |2i
√
|bi ≡
= p 2
.
Ωp + Ω2c
2
|ai ≡
(1.33a)
(1.33b)
|ai and |bi are not stationary states. In fact, the strong control field acts to oscillate
atoms back and forth between these two states. For this reason we shall call |bi the
bright state; it accounts for all transitions to and from the excited state |ai. Using
the same weak probe field approximation as before, we can write
|bi ≈ |2i.
(1.34)
Therefore, in the presence of both electric fields, atoms in state |2i are still able to
absorb energy from the control field and excite into state |3i. However, |1i decouples
from the excited state completely, and we see no absorption of the probe beam.
1.2.3
Higher-Level Systems
The majority of the theory included in this discussion of EIT applies to three-level
systems — specifically Λ-type energy structures. However, in practice, few systems
consist of just three energy levels. In fact, the atomic levels relevant to this experiment
consist of six levels, as shown in Fig. 1.3.
Most direct methods of calculating absorption — like the methods discussed in
the following section — would be much more difficult for systems with more than
three levels. The dressed state analysis, however, is more or less the same. For this
1.2. Dressed State Picture
11
È6\
È5\
È4\
È3\
È2\
È1\
Figure 1.3: Energy diagram of a six-level system. Solid arrows indicate strong control
fields while dashed arrows represent weak probe fields.
reason, it is often more convenient to verify that a dark state exists for a given system
before going through any computation-heavy analysis.
Consider the six-level system described in Fig. 1.3. In this scenario, there are two
control beams, both with energy ~ωc ≈ ~ω24 ≈ ~ω35 — shown as solid arrows — and
two probe beams, both with energy ~ωp ≈ ~ω15 ≈ ~ω26 — shown as dashed arrows.
We again start with a bare-atom Hamiltonian that looks like




H0 = 



~ω1 0
0
0
0
0
0 ~ω2 0
0
0
0
0
0 ~ω3 0
0
0
0
0
0 ~ω4 0
0
0
0
0
0 ~ω5 0
0
0
0
0
0 ~ω6








(1.35)
After taking the rotating wave approximation, we can express the full Hamiltonian —
analogous to Eq. 1.21 — as

~

H= 
2
0
2ω2
0
0
0
0
−Ωp e−i(ωp t+φp )
2ω3
0
0
2ω4
0
0
0
0
2ω5
0
0
2ω6
0
−Ωc e−i(ωc t+φc )
0
−Ωp e−i(ωp t+φp )
0
−Ωc e−i(ωc t+φc )
0
0
−Ωc ei(ωc t+φc )
0
0
−Ωc ei(ωc t+φc )
−Ωp ei(ωp t+φp )
0
2ω1
0
i(ωp t+φp )
−Ωp e



.

(1.36)
Now, we wish to transform the Hamiltonian into a frame that is time and phase
12
Chapter 1. Theory
independent using the unitary matrix




Ũ (t) = 



e−i(ωp t+φp )
0
0
0
0
0
Plugging into Eq. 1.24

2 (ω1 + ωp )

0

~
0
H̃ = 

0
2

−Ωc
0
0
0
0
1
0
0
−i(ωc t+φc )
0 e
0
0
0
ei(ωc t+φc )
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0 ei(ωp t+φp )




.



(1.37)
yields the corotating Hamiltonian
0
0
0
−Ωc
0
2ω2
0
−Ωp
0
−Ωc
0
2 (ω3 + ωc )
0
−Ωp
0
−Ωp
0
2 (ω4 − ωc )
0
0
0
−Ωp
0
2ω5
0
−Ωc
0
0
0
2 (ω6 − ωp )




 . (1.38)



We can now assume the relative energy spacings of each cluster of three levels are
the same. This implies that we can express all six frequencies in terms of ω2 and the
relevant spacings such that
ω1 = ω2 − δ,
ω2 = ω2 ,
ω3 = ω2 + δ,
ω4 = ω2 + ω0 − δ,
ω5 = ω2 + ω 0 ,
ω6 = ω2 + ω0 + δ,
(1.39a)
(1.39b)
(1.39c)
where ~δ is the spacing between adjacent states within a single cluster, and ~ω0 is
the energy gap between the two clusters. This yields two resonant frequencies
ω15 = ω26 = ω0 + δ,
ω24 = ω35 = ω0 − δ
(1.40a)
(1.40b)
Since adding a scalar multiple of the identity preserves the spectrum of the matrix,
we may add −~ (ω1 + ωp ) 1 and express the frequencies in terms of ω0 and δ. This
yields
 0

0
0
0
−Ω
0
c
~
H̃ = 
2
0 2(δ−ωp )
0
−Ωp
0
−Ωc

0
0
2(2δ+ωc −ωp )
0
−Ωp
0
.
0
−Ωp
0
2(ω0 −ωc −ωp )
0
0

−Ωc
0
−Ωp
0
2(ω0 +δ−ωp )
0
0
−Ωc
0
0
0
2(ω0 +2δ−2ωp )
(1.41)
In terms of the detunings
∆p = ω15 − ωp = ω0 + δ − ωp ,
∆c = ω24 − ωc = ω0 − δ − ωc ,
(1.42a)
(1.42b)
1.2. Dressed State Picture
13
the Hamiltonian becomes
 0
H̃ =
~
2

0
0
0
−Ωc
0
0 2(∆p −ω0 )
0
−Ωp
0
−Ωc
 0

0
2(∆p +∆c )
0
−Ωp
0
 0
.
−Ωp
0
2(∆p +∆c −ω0 ) 0
0
−Ωc
0
−Ωp
0
2∆p
0
0
−Ωc
0
0
0 2(2∆p −ω0 )
(1.43)
In the case when the control and probe laser are both on resonance, the diagonal
terms simplify, and the Hamiltonian of interest becomes


0
0
0
0
−Ωc
0
 0
−2ω0
0
−Ωp
0
−Ωc 



~
0
0
0
0
−Ω
0
p
,
(1.44)
H̃ = 

−Ωp
0
−2ω0
0
0 
2 0

 −Ωc
0
−Ωp
0
0
0 
0
−Ωc
0
0
0
−2ω0
with the following eigenstates:
|0Λ i =
|0V i =
|ai =
|bi =
|ci =
|di =
Ωc |1i − Ωp |3i
p 2
,
Ωp + Ω2c
Ωp |4i − Ωc |6i
p 2
,
Ωp + Ω2c
p 2
Ωp + Ω2c |2i + Ωc |4i + Ωp |6i
p 2
,
2Ωp + 2Ω2c
p 2
Ωp + Ω2c |2i − Ωc |4i − Ωp |6i
p 2
,
2Ωp + 2Ω2c
p 2
Ωp + Ω2c |1i + Ωc |3i + Ωp |5i
p 2
,
2Ωp + 2Ω2c
p 2
Ωp + Ω2c |1i − Ωc |3i − Ωp |5i
p 2
.
2Ωp + 2Ω2c
(1.45a)
(1.45b)
(1.45c)
(1.45d)
(1.45e)
(1.45f)
As expected, we see that one of the eigenstates — the one that is conveniently
named |0Λ i — looks almost exactly like the dark state in the three-level system. As
in the Λ-type scheme, when the probe beam amplitude is small, i.e. Ωp Ωc , we find
|0Λ i ≈ |1i.
(1.46)
This shows that the states |1i, |3i, and |5i make up a mini-Λ scheme inside the greater
six-level system and behave similarly to the normal Λ-type structure.
Additionally, there is another state of a similar form. That is, under the same
approximation,
|0V i ≈ |6i.
(1.47)
This indicates that there are in fact two dark states of the system: one of the Λ scheme
and the other of the V scheme made up of states |2i, |4i, and |6i. It is important to
14
Chapter 1. Theory
note that, due to spontaneous emission from the upper states into the lower states,
atoms will transition between Λ scheme states and V scheme states; however, they
will tend to populate the Λ scheme states because there is a greater probability that
an atom will decay into |1i or |3i than into |2i. Therefore, the EIT effects due to the
Λ scheme will be much stronger than those of the V scheme.
Assuming that both the Λ and V schemes become resonant at the same set of
frequencies — an appropriate assumption when using the Zeeman method to show
EIT, as in this experiment — both associated transparency windows will occur concurrently, and appear as one trough in the absorption spectrum.
1.3
Absorption in EIT
The dressed state approach is helpful in showing the mechanism behind EIT, but
it is not as useful in making physical predictions about the medium. Since this
experiment deals not with a single atom, but with a whole ensemble of atoms, the
next calculation will relate ideas of the previous sections to measurable macroscopic
quantities. In particular, we will calculate the electric susceptibility χ of the medium
in the presence of fields. The susceptibility is of interest because its real and imaginary
parts determine the two primary effects of EIT: its real part determines the dispersion
of the medium while its imaginary part governs the absorption. From this calculation,
both of these properties will be predicted as functions of probe frequency.
The linearized susceptibility is defined by the relation
P = 0 χE,
(1.48)
where P is the dielectric polarization and 0 is the permittivity of free space. The
polarization can also be defined as
P = N h℘i,
(1.49)
where N is the total number of atoms. Therefore, Eqs. 1.48 and 1.49 can be equated,
yielding
0 χE = N h℘i.
(1.50)
Almost all of the quantities above are macroscopic and measurable, so in order to
calculate the susceptibility, we only need to find the expectation value of the electric
dipole operator.
1.3.1
Density Operator
For pure quantum states, an expectation value is defined in terms of the wavefunction
|Ψi, such that for some operator Λ,
hΛi = hΨ|Λ|Ψi.
(1.51)
However, it is often difficult to find the wavefunction of one atom, much less of the
entire system of atoms. Additionally, an atom may exist in a mixed state. While a
1.3. Absorption in EIT
15
pure state may be determined by a superposition of stationary states, a mixed state
is a statistical mixture of pure quantum states that do not interfere within a system.
If for example, the states of the individual atoms in the system are expressed as |ni
and follow a distribution Pn , then the expectation value becomes
hΛi =
X
Pn hn|Λ|ni.
(1.52)
n
For this reason, it is simpler to define expectation values in terms of the density
operator
X
ρ=
Pn |nihn|,
(1.53)
n
where Pn is the probability of an atom being in state |ni. To see how this works,
recall the completeness and orthogonality relations in Eqs. 1.3 and 1.4, and compute,
hΛi =
X
=
X
Pn hn|Λ|ni
n
Pn hn|Λ|mihm|ni
mn
=
X
hm|Pn |nihn|Λ|mi
mn
=
X
hm|ρΛ|mi
m
hΛi = Tr(ρΛ).
(1.54)
This definition of an expectation value makes no reference to the specific state of the
atoms. Instead, it makes uses of statistical distributions within the system as defined
in the density operator.
1.3.2
Von Neumann Equation
Since Schrödinger’s equation is defined in terms of the wavefunction, a different relation exists in terms of the density operator; this relation is called the von Neumann
equation.
In order to derive the von Neumann equation, recall that Schrödinger’s equation
and its adjoint are expressed by
d
i
|ψi = − H|ψi,
dt
~
d
i
hψ| = hψ|H.
dt
~
(1.55)
16
Chapter 1. Theory
Computing the time derivative of the density operator, we find
X
ρ̇ =
Pn (|ṅihn| + |nihṅ|)
n
=−
iX
Pn (H|nihn| − |nihn|H)
~ n
i
= − (Hρ − ρH)
~
i
= − [H, ρ] .
~
(1.56)
Again, our model has omitted the possibility of spontaneous emission from states
|2i and |3i. To count for this, we include an additional term
1
i
ρ̇ = − [H, ρ] − {Γ, ρ} ,
~
2
(1.57)
where the straight brackets represent a commutator and the curly brackets represent
an anti-commutator. Here, Γ is defined in terms of the decay rate γn from a state |ni
as


0 0 0
Γ =  0 γ2 0  ,
(1.58)
0 0 γ3
and γ1 = 0 because the atom cannot decay out of the ground state.
The additional decay term only accounts for atoms leaving states |2i and |3i. It
does not specify where they go. In reality, the majority of these atoms may decay into
state |1i, but since there are already a large number of atoms in the ground state,
the addition of the decayed atoms can be neglected. Instead, we treat γ2 and γ3 as
decay rates of atoms leaving the three-level system completely.
In coordinate form, von Neumann’s equation becomes
Xi
1
(Hik ρkj − ρik Hkj ) + (Γik ρkj + ρik Γkj ) .
(1.59)
ρ̇ij = −
~
2
k
1.3.3
Von Neumann Equation in EIT
Using von Neumann’s equation, written in component form, and the three-level EIT
Hamiltonian H̃, from Eq. 1.27 — expressed in the corotating basis — the density
operator ρ̃ can be calculated in the same basis. The diagonal matrix elements of ρ̃˙
can be written in terms of ρ̃ as
iΩp
ρ̃˙ 11 =
(ρ̃31 − ρ̃13 ) ,
2
iΩc
(ρ̃32 − ρ̃23 ) ,
ρ̃˙ 22 = −γ2 ρ̃22 +
2
iΩp
iΩp
ρ̃˙ 33 = −γ3 ρ̃33 −
(ρ̃31 − ρ̃13 ) −
(ρ̃32 − ρ̃23 ) .
2
2
(1.60a)
(1.60b)
(1.60c)
1.3. Absorption in EIT
17
Additionally, the off-diagonal terms are
˙ρ̃12 = ρ̃˙ ∗21 = − γ2 + i(∆p − ∆c ) ρ̃12 − iΩc ρ̃13 + iΩp ρ̃32 ,
2
2
2
γ
iΩ
iΩ
3
p
c
(ρ̃33 − ρ̃11 ) −
ρ̃12 ,
ρ̃˙ 13 = ρ̃˙ ∗31 = − + i∆p ρ̃13 +
2
2
2
1
iΩc
iΩp
ρ̃˙ 23 = ρ̃˙ ∗32 = − (γ2 + γ3 ) + i∆c ρ̃23 +
(ρ̃33 − ρ̃22 ) −
ρ̃12 .
2
2
2
(1.61a)
(1.61b)
(1.61c)
We will now assume that the majority of the atoms occupy the ground state. This is a
good approximation both because atoms are constantly being pumped by the control
beam from |2i into |1i, and also because spontaneously emitted atoms continuously
repopulate the ground state. Therefore, since the diagonal terms ρnn can be thought
of as the fraction of atoms in the population of |ni, we see that ρ11 ≈ 1 and ρ22 ≈
ρ33 ≈ 01 . This yields
γ
iΩc
iΩp
2
ρ̃13 +
ρ̃32 ,
(1.62a)
ρ̃˙ 12 = ρ̃˙ ∗21 = − + i(∆p − ∆c ) ρ̃12 −
2
2
2
γ
iΩp iΩc
3
ρ̃˙ 13 = ρ̃˙ ∗31 = − + i∆p ρ̃13 −
−
ρ̃12 ,
(1.62b)
2
2
2
γ2 + γ3
iΩp
ρ̃˙ 23 = ρ̃˙ ∗32 = −
+ i∆c ρ̃23 −
ρ̃12 .
(1.62c)
2
2
Now, the weak probe approximation can be used to drop all terms proportional to Ω2p .
In the steady state solutions for Eq. 1.62c (where ρ̃˙ 23 = 0), ρ̃23 is linearly proportional
to Ωp , so then the iΩ2p ρ̃32 term in Eq. 1.62a can be dropped. Thus, we have
˙ρ̃12 = ρ̃˙ ∗21 = − γ2 + i(∆p − ∆c ) ρ̃12 − iΩc ρ̃13 ,
(1.63a)
2
2
γ
iΩp iΩc
3
ρ̃˙ 13 = ρ̃˙ ∗31 = − + i∆p ρ̃13 −
−
ρ̃12 .
(1.63b)
2
2
2
These are now in the form of two coupled differential equations. To solve, they can
be expressed as a single matrix equation, such that
γ2
0
ρ̃˙ 12
− 2 + i(∆p − ∆c )
− iΩ2 c
ρ̃12
+
(1.64)
=
ρ̃13
− iΩ2 c
− γ23 + i∆p
− iΩ2p
ρ̃˙ 13
Now define
X=
ρ̃12
ρ̃13
,
− γ22 + i(∆p − ∆c )
− iΩ2 c
M=
γ3
iΩc
−
− 2 + i∆p
2
0
A=
,
− iΩ2p
1
(1.65a)
,
(1.65b)
(1.65c)
Obviously states |2i and |3i are not empty, but their populations are so much smaller than that
of |1i that they can be neglected.
18
Chapter 1. Theory
and Eq. 1.64 becomes simply,
Ẋ = MX + A.
(1.66)
The steady state solution to this matrix equation — assuming Ẋ = 0 — is
X = −M−1 A.
(1.67)
This yields solutions for the matrix elements of the density operator,
Ωc Ωp
,
2iγ3 (∆c − ∆p ) + γ2 (γ3 − 2i∆p ) + 4∆c ∆p − 4∆2p + Ω2c
i (γ2 + 2i (∆c − ∆p )) Ωp
=
.
2iγ3 (∆c − ∆p ) + γ2 (γ3 − 2i∆p ) + 4∆c ∆p − 4∆2p + Ω2c
ρ̃12 =
(1.68a)
ρ̃13
(1.68b)
These matrix elements are, however, still in the corotating basis. In order to transform
back to the Schrödinger picture, we would have to compute
ρ = Ũ † ρ̃Ũ

ρ̃11
=  ρ̃21 e−i(ωp +φp −ωc −φc )t
ρ̃31 e−i(ωp +φp )t
1.3.4

ρ̃12 ei(ωp +φp −ωc −φc )t ρ̃13 ei(ωp +φp )t
ρ̃22
ρ̃23 ei(ωc +φc )t 
ρ̃32 e−i(ωc +φc )t
ρ̃33
(1.69)
Complex Susceptibility And Its Implications
We now have a matrix expression for the density operator in the corotating frame ρ̃.
To ease the amount of computation, we will first show that ρ13 is the only matrix
element of the density operator in the Schrödinger picture needed to calculate the
electric susceptibility χ. Recall that polarization can be expressed as
P = N h℘i
= N Tr (ρ℘)

ρ11 ρ12 ρ13


ρ21 ρ22 ρ23
= N Tr
ρ31 ρ32 ρ33

ρ13 ℘31 ρ13 ℘32

= N Tr ρ23 ℘31 ρ23 ℘32
ρ33 ℘31 ρ33 ℘32


0
0 ℘13
 0
0 ℘23 
℘31 ℘32 0

ρ11 ℘13 + ρ12 ℘23
ρ21 ℘13 + ρ22 ℘23 
ρ31 ℘13 + ρ32 ℘23
= N (ρ13 ℘31 + ρ23 ℘32 + ρ31 ℘13 + ρ32 ℘23 ) .
(1.70)
Using Eq. 1.69, the polarization can be expressed in terms of the rotated density
elements as
P = N ρ̃13 ℘31 ei(ωp +φp )t + ρ̃23 ℘32 ei(ωc +φc )t + ρ̃31 ℘13 e−i(ωp +φp )t + ρ̃32 ℘23 e−i(ωc +φc )t .
(1.71)
1.3. Absorption in EIT
19
The polarization can also be expressed in terms of the electric field as defined in
Eq. 1.15 such that
P = 0 χE
0 χ(ωc )Ec iωc t
0 χ(ωp )Ep iωp t
e
+ e−iωp t +
e
+ e−iωc t .
=
2
2
(1.72)
Notice that since χ depends on frequency, there appear two different susceptibilities
in the equation above. We are particularly interested in χ(ωp ) ≡ χp because the
desired effect is transmission of the probe beam. To isolate this parameter, we match
exponentials in Eqs. 1.71 and 1.72 — specifically, those that contain a factor of eiωp t .
This yields
0 χp Ep
= N ρ̃13 ℘31 eiφp t .
(1.73)
2
Recalling that ℘31 = |℘13 |e−iφp t and solving for χp gives
χp =
2N |℘13 |
ρ̃13 .
0 Ep
(1.74)
Finally, using Eq. 1.68b we can write a full expression for the complex electric susceptibility as
χp =
i (γ2 + 2i (∆c − ∆p )) Ωp
2N |℘13 |
.
0 Ep 2iγ3 (∆c − ∆p ) + γ2 (γ3 − 2i∆p ) + 4∆c ∆p − 4∆2p + Ω2c
(1.75)
The real and imaginary parts can be written
2 γ22 ∆p + (∆c − ∆p ) 4 (∆c − ∆p ) ∆p + Ω2c Ωp
2N |℘13 |
Re(χp ) = −
,
0 E p
γ22 + 4 (∆c − ∆p )2 γ32 + 4∆2p + 2 (γ2 γ3 + 4 (∆c − ∆p ) ∆c ) Ω2c + Ω4c
(1.76a)
γ22 γ3 + 4γ3 (∆c − ∆p )2 + γ2 Ω2c Ωp
2N |℘13 |
Im(χp ) =
.
0 E p
γ22 + 4 (∆c − ∆p )2 γ32 + 4∆2p + 2 (γ2 γ3 + 4 (∆c − ∆p ) ∆c ) Ω2c + Ω4c
(1.76b)
We recall that the real part of the linear susceptibility is proportional to the dispersion of the medium; likewise, the imaginary part is proportional to the absorption.
Referring to the plot of Eq. 1.76 in Fig. 1.4b, we see that the absorption falls off drastically when the probe field is on resonance. This window of transparency is precisely
what is meant by EIT. With only one or the other applied field, we would measure
a typical absorption spectrum, but with both, we see a cancelation, such that the
probe beam remains unabsorbed. Additionally, the dispersion becomes much steeper
near resonance. Though this experiment will not dwell much on the dispersion, it is
worth noting that such a dispersion relation in a zone of low absorption is conducive
of complex optical phenomena such as slow light [6].
20
Chapter 1. Theory
Χp
HaL
D p Γ3
Χp
HbL
D p Γ3
Figure 1.4: Linear susceptibility plotted in arbitrary units as a function of normalized
probe frequency. The solid line shows the imaginary part of χp (which is proportional
to the absorption), and the dashed line shows the real part of χp (which is proportional
to the dispersion). Probe frequency is normalized with respect to γ3 , which allows us
to plot in units of γ3 . For (a) we set ∆c = 0, Ωc = 0, and γ2 = 104 γ3 . For (b) we set
∆c = 0, Ωc = γ3 , and γ2 = 104 γ3 .
1.3. Absorption in EIT
21
As one last check, if we set Ωc = 0 — corresponding to the case when the control
beam is switched off — we get
2N |℘13 | 2Ωp ∆p
,
0 Ep γ3 + 4∆2p
2N |℘13 | Ωp γ3
Im(χp ) =
.
0 Ep γ3 + 4∆2p
Re(χp ) = −
(1.77a)
(1.77b)
As shown in the plots of Eq. 1.77 in Fig. 1.4a, the absorption and dispersion revert
back to typical trends as when EIT is not present, as expected.
Chapter 2
Experiment
This experiment aims to measure an absorption curve of rubidium atoms in a magnetic field, effectively establishing the existence of EIT. The theory discussed in the
previous chapter was derived for standard EIT — which makes use of two lasers. The
theory also generalizes to Zeeman EIT. The major distinction is that the multiplelevel system is created by applying a magnetic field across atoms with degenerate
energy levels. This splits the levels creating a suitable energy scheme for EIT. Due
to this distinction, it is possible to use only one laser and scan the magnitude of the
magnetic field surrounding the atoms to acquire data. Details on how this is possible
will be discussed in this chapter.
2.1
Rubidium and Zeeman Splitting
Rubidium gas was chosen to be the atomic medium in this study for its energy
structure and the availability of applicable optical equipment. Additionally, since
rubidium is ubiquitous in the study of EIT and saturated absorption spectroscopy, it
makes for easy comparison to past research.
Specifically, the D2 transition in the 87 Rb isotope is used. This transition is
between the ground state |5s 2 S1/2 i and the excited state |5s 2 P3/2 i and is resonant
at a wavelength of 780.24 nm. Within these states there exist several hyperfine levels
denoted by their quantum numbers F for the ground state and F 0 for the excited
state. The hyperfine levels used in this experiment are F = 1 and F 0 = 1.
These levels were chosen because of the prominence of the peak associated with
their atomic transition on a hyperfine absorption spectrum and for the simplicity of
their Zeeman splitting. The Zeeman effect occurs when states of the same energy
but distinct angular momenta are introduced to a magnetic field. Here, angular
momentum is denoted by the quantum numbers mF and m0F for the ground and
excited state, respectively. Each state (F = 1 and F 0 = 1) can have three momenta
mF = 0, ±1, so they each split into three distinct levels. The resulting six-level
system is shown in Fig. 2.1. In theory, it may seem simpler to use the F 0 = 0
level because this excited state does not split due to the Zeeman effect. However, in
practice, its associated absorption line is too small to detect using the equipment in
24
Chapter 2. Experiment
F¢ ‡ 1 mF '=-1
F‡1
mF =-1
mF '=0
mF =0
mF '=1
mF =1
Figure 2.1: The F = 1 → F 0 = 1 transitions with the Zeeman split levels. Solid
arrows represent transitions induced by σ+ and dashed lines represent that of σ− .
this experiment.
In order to construct a Λ-type level structure, two circularly polarized fields are
applied. A left-circularly polarized σ+ beam will increase an atom’s angular momentum by one quantum number. That is, σ+ will excite mF = −1 → m0F = 0. Similarly,
a right-circularly polarized σ− beam will take mF = 1 → m0F = 0.
At the same time, the mF = 0 state will also excite into m0F = ±1 states forming a
V-type structure. This combination of Λ and V schemes is the same energy structure
shown to have EIT dark states in Section 1.2.3. Luckily, EIT is seen for both of these
configurations when the Zeeman splitting is minimal, or when the applied magnetic
field is close to zero. This means their transparency windows will occur concurrently
on the absorption curve. Due to spontaneous decay from the upper levels, atoms will
tend to settle in the Λ states, so the contribution of the Λ scheme to the EIT signal
will be much greater that that of the V scheme. Therefore, we still expect absorption
similar to that of the three-level system.
In terms of the states of the three-level Λ system discussed in Chapter 1, the
mF = ±1 states act as |1i and |2i exchangeably, depending on the sign of the applied
magnetic field1 . The m0F = 0 state acts as |3i.
2.2
Preliminary Setup
The setup consists of a diode laser which is parked at the resonant frequency using
saturated absorption spectroscopy. The beam is then properly polarized and shone
through a cell of rubidium surrounded by magnetic field coils. The transmitted beam
is then detected by a power meter, and an absorption curve is obtained.
1
When the magnetic field is positive, the mF = 1 state will have higher energy than mF = −1;
when the magnetic field is negative, the energy splitting is reversed.
2.2. Preliminary Setup
2.2.1
25
Diode Laser
The 780 nm diode laser was a product of a past thesis at Reed College [14]. It has
a Littrow configuration in which an external cavity is formed between the rear facet
of the diode and a diffraction grating. The grating is angled such that the first-order
diffraction is reflected back into the diode to form a feedback loop. Zeroth-order
reflection exits the cavity as the lasing beam. The lasing wavelength is determined
by the length of the external cavity.
The laser is driven by a ILX Lightwave LDX-3525 precision current source and
cooled by a Peltier cooler configured with a ILX Lightwave LDT-5910B PID precision
thermo-electric temperature controller. The frequency of the laser can be fine tuned
by a piezoelectric crystal which changes the cavity length by small amounts. A voltage
is applied across the piezo crystal by a Thorlabs MDT694A piezo controller, which is
driven by an Agilent 33210A arbitrary waveform signal generator.
This allows the laser to be scanned across a range of wavelengths according to the
output voltage of the signal generator. The piezo crystal can scan the laser frequency
over a range of 1-2 GHz without mode jumping.
The exiting laser light enters a Conoptics Model 713A optical isolator and is split
by a 95-5 beamsplitter. The transmitted beam is much stronger, and it continues into
the primary EIT setup. The weaker reflected beam is directed toward the saturated
absorption spectroscopy setup described in the following section.
2.2.2
Saturated Absorption Spectroscopy
Doppler-free saturated absorption spectroscopy is a standard experimental method
used to park the laser onto a particular frequency resonant with a hyperfine transition [15]. We will use this technique to tune the laser to the F = 1 → F 0 = 1
transition in 87 Rb.
Without saturated absorption spectroscopy, this would prove to be a difficult
task; since the atoms in a cell of gas will have some spread of velocities according
to a Maxwell-Boltzmann distribution, each velocity group will “appear” to have an
atomic resonance frequency that is Doppler-shifted by some amount. For this reason,
the hyperfine peaks will be Doppler-broadened and, in a typical absorption spectrum,
smoothed into one conglomerate absorption peak. Saturated absorption spectroscopy
is a technique used to resolve the hyperfine peaks by counterpropagating a pump and
probe beam and saturating the atoms in their excited state.
A schematic of the saturated absorption spectroscopy setup is shown in Fig. 2.2.
The beam from the laser is reflected twice off a glass plate — once off the front facet
of the plate and again off the back. These are the probe2 beams which pass through
a rubidium cell before being detected by photodiodes. Another much stronger pump
beam is transmitted through the glass plate. The pump is redirected to counterpropagate with one of the probe beams as they pass through the medium.
Due to the opposing Doppler shifts, the counterpropagating beams cannot be si2
It is important to note that the pump and probe beams in the saturated absorption spectroscopy
setup are distinct from those in the primary EIT setup and those mentioned in the theory sections.
26
Chapter 2. Experiment
Signal
Generator
Temp.
Controller
Piezo
Controller
Current
Controller
Figure 2.2: Schematic of saturated absorption setup.
multaneously resonant except with the zero-velocity group of atoms. In this scenario,
the pump beam will excite the vast majority of the atoms into an excited state —
“saturating” that transition. This allows the probe beam to pass through with little
absorption. The laser is scanned in frequency by sending a positive triangle waveform
through the piezo controller. Two spectra result: one from the unsaturated probe
beam — a typical Doppler-broadened absorption spectrum — and another from the
saturated beam — consisting of narrow peaks at the hyperfine resonance lines. The
difference of these two signals yields a resolvable hyperfine spectrum. Examples of
such spectra are shown in Fig. 2.3.
To park the laser onto the F = 1 → F 0 = 1 peak, the frequency scanning range is
decreased in amplitude while observing the detector signal. This effectively narrows
the spectrum onto the desired peak. Then, the scan is stopped altogether. The piezo
voltage can also be manually adjusted to ensure the laser is stabilized at the right
frequency.
Since we do not actively lock the laser frequency, the signal will drift. However,
the laser stays at resonance for about 3-5 minutes, which is long enough to take about
one run of measurements. In future experiments, it may be necessary to use feedback
control electronics to stabilize the laser for longer amounts of time [17].
2.3. Primary EIT Setup
27
Vspectrum HVL
Vspectrum HVL
6
F'=1
F'=2
F'=2
F'=3
15
5
4
10
3
2
5
1
0.2
0.4
0.6
0.8
Vscan HVL
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Vscan HVL
Figure 2.3: Hyperfine spectra of 87 Rb. F = 1 → F 0 transitions (left) and F = 2 → F 0
transitions (right). These measurements are consistent with the results of MacAdams
et al. [16].
2.3
Primary EIT Setup
The majority of the light emitted by the laser — now parked on the hyperfine line of
interest — passes through the primary setup where EIT is observed.
2.3.1
Components of Polarization
The novel aspect of this setup is that only one laser is used. To achieve this, the
single beam will be composed of two components of orthogonal polarizations. These
components will act as independent control and probe beams, though they make up
the same beam. For this reason, it is easiest to discuss how optical elements affect
each polarization component individually.
As shown in Fig. 2.4, the light first passes through a half-wave plate. Based on the
angle of the half-wave plate, the resulting light will have some specific combination
of linear polarizations, which will later become the control and probe components of
the beam. The angle of the half-wave plate is chosen such that the ratio of intensities
of “control” polarization to the “probe” polarization is 9:1. The beam then passes
through a quarter-wave plate, which converts the linearly polarized components to
circular polarizations. This produces the σ+ and the σ− needed to excite the atomic
medium. After the beam passes through the rubidium cell, it is converted back to
linear polarization by another quarter-wave plate, and the linear components are split
by a polarizing beamsplitter. The power of the resulting probe beam is measured by
a ThorLabs PM100D optical power meter.
To see that this works explicitly, we will use Jones calculus [18] to calculate the
relevant polarizations at the lettered points in Fig. 2.4. Assume that the polarization
output by the laser is the y-direction, where the y-direction points upwards perpendicular to the optics table.3 Therefore, the initial Jones vector at point A and the
3
The starting polarization may very well be some admixture of x- and y-polarizations, but this
can be corrected for by properly setting the angle of the half-wave plate.
28
Chapter 2. Experiment
Zeeman
Apparatus
ÄÄÄÄÄÄÄÄÄ
D
PBS
ː4
Rb Cell
BD
E
ŸŸŸŸŸŸŸŸŸ
C
ː4
Constant
Voltage
Source
B
Power
Meter
ː2
A
SAS
Setup
Optical
Isolator
Tunable
Diode Laser
Figure 2.4: Schematic of the primary EIT setup. Abbreviations used in this figure
are: λ/2 - half-wave plate; λ/4 - quarter-wave plate; PBS - polarizing beamsplitter;
BD - beam dump. Points A-E are included solely for ease of reference.
Jones matrix associated with the half-wave plate are
0
cos 2θ sin 2θ
PA =
,
JHWP =
,
1
sin 2θ − cos 2θ
(2.1)
respectively, where θ is the angle of the half-wave plate’s fast axis with respect to the
y-direction. After the half-wave plate, the polarization at point B will be
PB = JHWP PA
cos 2θ sin 2θ
0
=
sin 2θ − cos 2θ
1
sin 2θ
=
.
− cos 2θ
(2.2)
Clearly, θ can be chosen to create any control-to-probe ratio desired. In this experiment, only the 9:1 ratio will be studied, but changing this ratio leads to other
2.3. Primary EIT Setup
29
interesting observable effects, such as coherent population trapping [19] and electromagnetically induced absorption [20].
The Jones matrix for a quarter-wave plate with its fast axis 45◦ from the ydirection is
1
1 i
.
(2.3)
JQWP = √
2 i 1
Therefore, the polarization at point C will be
PC = JQWP PB
1
1 i
sin 2θ
=√
− cos 2θ
2 i 1
1
sin 2θ − i cos 2θ
=√
2 i sin 2θ − cos 2θ
1
1
1
= √ sin 2θ
− i cos 2θ
i
−i
2
= σ+ sin 2θ − σ− i cos 2θ,
where the left-circular and the right-circular polarizations are defined to be
1
1
1
1
σ+ ≡ √
,
σ− ≡ √
,
2 i
2 −i
(2.4)
(2.5)
respectively. Thus, the polarization at point C has a component of each circular
polarization.
When a beam reflects off a mirror, left-circularly polarized light becomes rightcircularly polarized and vice versa. Therefore, the polarization at point D is
PD = σ− sin 2θ − σ+ i cos 2θ
1
sin 2θ − i cos 2θ
=√
2 −i sin 2θ + cos 2θ
(2.6)
(2.7)
Finally, the beam goes through another quarter-wave plate at 45◦ . The resulting
polarization at point E will be
PE = JQWP PD
1
1
1 i
cos 2θ + i sin 2θ
√
=√
2 i 1
2 −i cos 2θ − sin 2θ
sin 2θ
=
.
cos 2θ
(2.8)
The beam has now returned to a composition of linearly polarized components such
that the polarizing beamsplitter can pick off the probe component to be measured.
It is unclear at this point which polarization corresponds to the control beam and
which to the probe beam: σ+ or σ− . In fact, it does not matter all that much. In
Chapter 1, we assumed that the probe field was resonant with the transition from
30
Chapter 2. Experiment
the lowest energy state and the control was resonant with that of the higher energy
ground state. However, as we saw in Section 2.1, when the magnetic field around the
rubidium cell is swept from positive to negative values, the two ground states trade
places. For this reason, either choice will yield similar results.
It took a series of steps to set the wave plates to the correct angles. First, the two
quarter-wave plates and the rubidium cell were removed from the setup. This allowed
the beam to pass through only the half-wave plate and the polarizing beamsplitter.
The half-wave plate was adjusted until the transmitted beam was maximized. This
corresponds to having only y-polarized light. 11.39 mW of power was transmitted
through the beamsplitter, and only 243 µW was reflected.
Then, one of the quarter-wave plates was put in place and adjusted until the
transmitted and reflected beams were as close to equal as possible. This alignment of
the quarter-wave plate converts linear polarization to circular. The transmitted beam
had power 5.40 mW and the reflected beam had 5.79 mW. The first quarter-wave plate
was removed and replaced by the remaining quarter-wave plate, and this process was
repeated. This time, the transmitted power was 5.98 mW, and the reflected power
was 5.40 mW.
Finally, after all of the wave plates were returned to their spots in the optical
setup, the half-wave plate was readjusted such that the ratio of the power of the
beams transmitted and reflected through the polarizing beamsplitter was 9:1. Their
true powers were 9.92 mW and 1.171 mW.
2.3.2
Zeeman Apparatus
EIT actually occurs within the rubidium cell. In order to induce this effect, the cell
is placed within a Zeeman apparatus. A section and 3D rendering of the apparatus
are displayed in Fig. 2.5. The cell is enclosed in an acrylic tube wrapped in a solenoid
of copper wire. The total resistance of the coil is 14.62 Ω. For the experiment, a
100 Ω external resistor was added to attain a finer absorption curve. The wire exits
the apparatus at its end caps and is driven by a constant voltage source to induce a
magnetic field within the solenoid along the apparatus’ axis. The solenoid is about
three times longer than the cell, while their radii are comparable, so near the center
of the apparatus, the induced magnetic field can be approximated as constant. This
will be shown more explicitly later in the section.
Outside of the solenoid are three cylindrical layers of mu metal. Mu metal is a
material with a very high magnetic permeability for the purpose of shielding the cell
from exterior magnetic fields. The shielding we used had a relative permeability4
of 80000 [21]. The geometry of the shielding — three concentric cylinders — was
chosen because inserting gaps, in fact, makes the shielding much more effective. This
configuration was also relatively simple to construct.
The effectiveness of the shielding was determined by measuring the magnetic
field at various positions within the apparatus without current passing through the
4
Relative permeability is a dimensionless quantity defined as µR = µ/µ0 where µ is the absolute
permeability and µ0 is the permeability of free space.
2.3. Primary EIT Setup
31
1
2
3
4
5
6
x HcmL
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15
z HcmL
Figure 2.5: Section (top) and 3D rendering (bottom) of the Zeeman apparatus. Images by Jay Ewing.
32
Chapter 2. Experiment
BT HmGL
BA HmGL
50
250
40
200
30
150
20
10
100
2
4
6
8
10
12
z HcmL
50
-10
2
-20
4
6
8
10
12
z HcmL
BT HmGL
BA HmGL
50
250
40
200
30
150
20
100
10
0.2
0.4
0.6
0.8
1.0
1.2
1.4
r HcmL 50
-10
0.2
0.4
0.6
0.8
1.0
1.2
1.4
r HcmL
Figure 2.6: Magnetic field attenuated by mu metal shielding inside the Zeeman apparatus as a function of cylindrical coordinates r and z. The left two plots show the
axial field BA while the right two show the transverse field BT . The dashed line is average value of the magnetic field strength in the solenoid. The solid lines correspond
to the magnetic field outside the mu metal shielding.
solenoid. Axial (along the axis of the apparatus) and transverse fields — BA and
BT , respectively — were measured as functions of the cylindrical coordinates r and
z, with the origin chosen to be at the center of the apparatus. The results are plotted
in Fig. 2.6. For reference, each graph also includes a solid line that corresponds to
the field outside the shielding.
In each case, the field is attenuated by a factor of about 5. The magnitude of the
magnetic field of the Earth in our lab is on the order of 520 mG; assuming this is the
main source of exterior fields, this mu metal shielding reduces the exterior field such
that it is easily canceled by the applied field.
If the solenoid were infinitely long and perfectly aligned, the axial magnetic field
induced by flowing current would be
BA = µ0 nI
(2.9)
inside the solenoid, where n is the density of coils in the solenoid that the current I
flows through, and µ0 is the permeability of free space. The transverse field would
be zero. Of course, the solenoid is actually finite in length and may not be perfectly
aligned, so the field will vary within the apparatus. The field may also pick up a
nonzero transverse component. However, within a range close to the center of the
2.3. Primary EIT Setup
33
BT HmGL
BA HmGL
8200
700
8000
7800
650
7600
600
7400
7200
550
2
6800
4
6
8
10
z HcmL
2
BA HmGL
8090
BT HmGL
8080
580
4
6
8
10
z HcmL
8070
560
8060
0.2
8040
0.4
0.6
0.8
1.0
1.2
1.4
r HcmL 540
0.2
0.4
0.6
0.8
1.0
1.2
1.4
r HcmL
520
Figure 2.7: Magnetic field driven by a solenoid inside the Zeeman apparatus (with
the mu metal shielding in place). Field strength is plotted as a function of cylindrical
coordinates r and z with 300 mA flowing through the solenoid. The left two plots
show the axial field BA while the right two show the transverse field BT . The dashed
line in the upper right plot shows the uniform magnetic field inside an infinitely long
solenoid.
solenoid — where the cell sits — the magnetic field can be approximated by that of
an infinitely long solenoid.
The uniformity of the magnetic field induced by the current was also measured
as a function of position. This time, the solenoid was carrying a current of 300 mA.
Again, axial and transverse fields were measured as functions of x and z, and are
plotted in Fig. 2.7. Each measurement has an approximate error of ∆B = 20 mG.
The plots tend to plateau as z goes to zero. Additionally, there seems to be
little change in field strength as we move in the r-direction. The field is not, however,
uniform within the error in the measurement. Nevertheless, it is a good approximation
to consider the field to be constant across the rubidium cell.
Also, the transverse magnetic field is fairly large compared to the expected value.
This is most likely because the magnetic field probe was not perfectly aligned parallel
to the axial direction.
34
2.4
Chapter 2. Experiment
Procedural Overview
Most of the procedure has already been explained in the previous section’s discussion
of the setup. This section will concisely summarize the steps taken in this experiment
to obtain data.
First, the quarter-wave plates were adjusted so they convert linear polarizations
to circular and vice versa. Then, the half-wave plate was adjusted so that the ratio
of powers of the control beam component of polarization and that of the probe beam
was 9:1.
Next, the current powering the laser was increased until reaching a suitable lasing
power, and the saturated absorption setup was used to park the laser. To do this,
the frequency of the laser was scanned by driving the piezo crystal with a positive
voltage triangle waveform. The range of frequencies in the scan was adjusted making
small changes in temperature and current in the laser, until infrared fluorescence in
the rubidium cell was observed indicating near-resonance.
Then, the saturated absorption spectrum — obtained in the difference of signals
observed by the photodiodes — was viewed on an oscilloscope. The F = 1 → F 0 = 1
peak was identified on the spectrum, and, by adjusting the DC offset on the piezo
controller, the scanning range was shifted such that the peak was near the lowfrequency end of the spectrum. Then, the amplitude of the triangle waveform was
decreased, effectively narrowing the frequency range and zooming in on the transition
peak. This process was continued until the spectrum displayed just the peak at the
low-end end of the spectrum, and then the scan was stopped by turning off the output
of the signal generator.
Finally, the magnetic field in the Zeeman apparatus was turned on by running a
constant voltage across the solenoid. The transmittance of the probe component of
the beam was then recorded as a function of the voltage across the solenoid. Since
the resistance of the solenoid is known, its current can easily be determined from the
voltage using Ohm’s law, and from that, the induced magnetic field can be calculated,
yielding an absorption curve as a function of magnetic field strength.
Chapter 3
Results
An EIT signal was observed for the F = 1 → F 0 = 1 hyperfine transition of the D2
line in 87 Rb in the presence of a magnetic field. The resulting data is discussed in this
chapter, as well as improvements to the experiment that could yield clearer results.
3.1
Data Analysis
Probe beam power transmitted through the atomic medium was measured for a variety of voltages applied to the solenoid. Solenoid voltage is not, however, a preferable
or intuitive independent variable in this analysis. Therefore, voltage data was converted to current using Ohm’s law. The induced magnetic field was then calculated
from the current; since the field was not perfectly uniform within the cell, the axial
field of an infinite solenoid from Eq. 2.9 was used. Five runs of data were taken this
way, and the power measurements were averaged across all the runs. Each data point
was subtracted from the transmission power through the medium at a frequency far
from resonance. The resulting absorption data is shown in Fig. 3.1.
As expected, there is a reduction in absorption near B = 01 due to the transparency window induced by EIT. We do not see the decrease in absorption far from
resonance because the magnetic fields in this experiment were not large enough to
move far off resonance. Also, the decrease in absorption is only on the order of 2 µW.
It was almost too small a signal for the precision of the equipment used in this experiment. This is also a relatively small change in comparison to the proposed theory.
However, the absorption spectrum derived in Section 1.3 was for a pure three-level
Λ scheme. The scheme in our experiment, created from the F = 1 → F 0 = 1 transitions, was a six-level system with both Λ-type and V-type structure. Therefore, while
the dressed state analysis is sufficient in predicting that EIT exists for the six-level
system, there are some subtle differences.
For example, we assumed that, in the three-level system, the decoherence term
γ2 was very small. When γ2 becomes large, the transparency window gets shallower
and shallower, as shown in Fig. 3.2. What would the analogous term in the six1
The transparency window is not centered perfectly at B = 0, but it is well within the error of
±10 mG due to Earth’s magnetic field attenuated by the mu metal, as shown in Fig. 2.6.
36
Chapter 3. Results
-300
-200
-100
B HmGL
0
100
200
300
8.0
P HΜWL
7.5
7.0
6.5
6.0
-10
-5
0
I HmAL
5
10
Figure 3.1: Experimental EIT absorption of 87 Rb. The horizontal axis has two scales,
one for current and one for magnetic field strength.
ImH Χ p L
ReH Χ p L
D p Γ3
D p Γ3
Figure 3.2: Simulation of absorption and dispersion as the decoherence parameter γ2
varies. Light purple corresponds to low values of γ2 and the darker lines represent
higher values of γ2 . The values were varied from 0 to 1.4γ3 .
3.2. Possible Improvements
37
level system be? If we think of γ2 as the rate of decay from states that are not the
excited state of the Λ scheme, then there are four decay rates we need to take into
consideration: that of the metastable state of the Λ scheme and also of all three states
in the V scheme. Therefore, it seems there may be a much higher chance of decay out
of the system, which would greatly reduce the magnitude of the transparency signal.
3.2
3.2.1
Possible Improvements
F = 1 → F 0 = 0 Transition of the D2 Line
The ideal energy scheme for Zeeman EIT would be made from a F = 1 → F 0 = 0
hyperfine transition. The ground state would split into three levels while the excited
state would not split at all. The Λ scheme similar to the one used in this experiment
would form but without the two extra excited states. When circularly polarized fields
were applied, the mF = 0 state would not have anywhere to transition, therefore
decoupling it from the system and leaving a true three-level Λ structure. This would
greatly reduce the γ2 term, and the theory of three-level systems would be more
applicable. Additionally, unwanted excitations into higher energy levels would be
greatly reduced with this transition.
However, the F = 1 → F 0 = 0 hyperfine peak of the D2 line in 87 Rb is very small.
With the existing experimental setup, as well as time constraints, it was too difficult
to park the laser onto the resonant frequency. However, by properly optimizing
the alignment of the saturated absorption setup, one could significantly improve the
resolution of the hyperfine spectrum. In this case, it may be possible to utilize the
F = 1 → F 0 = 0 transition to observe EIT.
3.2.2
D1 Line
The D1 line in 87 Rb is much more frequently used in Zeeman EIT. D1 is resonant at
a wavelength 794.98 nm, which is too high for the diode laser used in this experiment
to reach. With a new laser diode, however, seeing D1 spectral peaks would be simple.
The D1 line would make comparison to past work much easier; however, it due to its
energetic structure, it may be necessary repump atoms back into the ground state of
the appropriate transition.
3.2.3
Laser Locking
One major source of error in this experiment was due to the laser’s drift in frequency.
The frequency was stable for up to about 5 minutes, after which time the frequency
would have to be readjusted using the saturated absorption setup. The time of
stability could be greatly increased by frequency locking the laser. This would involve
setting up an electronic feedback loop so that when the laser started to drift, it would
be able to automatically restabilize. This would increase the number of data points
that could be taken in a single run and minimize the the extraneous absorptive effects
caused by unwanted change in frequency.
38
Chapter 3. Results
3.2.4
Higher Resolution Detection and Real-Time Scanning
The power meter used to detect the transmission through the medium had precision
just high enough to see a clear EIT signal. However, to get such a result, we had to
average over many runs. A single run would look very choppy and under-resolved.
For this reason, it was difficult to make use of real-time scanning of the magnetic field
strength.
If either the EIT signal were stronger2 or the power meter had higher resolution,
real-time scanning would be more realizable. Scanning the voltage slowly3 across the
solenoid using a signal generator would allow us to see the signal on an oscilloscope —
much like the way the saturated absorption spectra are viewed.
2
For example, if the D1 line was used, and it exhibited a stronger EIT signal.
The voltage would have to be varied slowly enough that Ohm’s law still accurately predicted
the current in the solenoid.
3
Conclusion
This thesis aimed to show theoretically and experimentally the effects of electromagnetically induced transparency. We first showed two derivations of EIT in three-level
systems: direct computation of the linearized electric susceptibility and the dressed
state analysis. We also used the dressed state picture to show that a six-level system
exhibits EIT.
We then introduced the Zeeman method for observing EIT. We showed that
through clever polarization, a Zeeman setup requires only a single beam to elicit
EIT. We then used this setup to measure an absorption curve through warm rubidium vapor. The existence of a clear transparency window centered at the transition
resonance is evidence of EIT for an atomic medium in a magnetic field.
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