Electromagnetically Induced Transparency: A Thesis Presented to … · 2016-05-19 ·...

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 thesisand my career at Reed could never have been completed:

– Lucas Illing, whose guidance and seemingly bottomless knowledge got me througheven the most difficult of thesis crises;

– Jay Ewing, for the technical knowhow, for the avid reassurance, and for a sleekapparatus;

– John Essick, Joel Franklin, Darrell Schroeter, and David Griffiths, who werealways available to mull over confusing topics and usually solve all of my prob-lems;

– 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 processand also throughout the course of my life;

– Zubenelgenubi Scott and Indy Liu, for being the best officemates in the wholesubbasement (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 rightamount #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 methrough the final stretch; and,

– Emilia and Ines Furman and Arrow and Vega Henson, who remind me everydaythat the universe is a beautiful place.

Thank you to all.

Table of Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 1: Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Derivation of the EIT Hamiltonian . . . . . . . . . . . . . . . . . . . 3

1.1.1 Bare-Atom Hamiltonian . . . . . . . . . . . . . . . . . . . . . 41.1.2 Applied Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.3 Rotating Wave Approximation . . . . . . . . . . . . . . . . . . 61.1.4 Time and Phase Independent Frame . . . . . . . . . . . . . . 7

1.2 Dressed State Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.1 Dressed States . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.2 The Dark State and The Bright State . . . . . . . . . . . . . . 101.2.3 Higher-Level Systems . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Absorption in EIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3.1 Density Operator . . . . . . . . . . . . . . . . . . . . . . . . . 141.3.2 Von Neumann Equation . . . . . . . . . . . . . . . . . . . . . 151.3.3 Von Neumann Equation in EIT . . . . . . . . . . . . . . . . . 161.3.4 Complex Susceptibility And Its Implications . . . . . . . . . . 18

Chapter 2: Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.1 Rubidium and Zeeman Splitting . . . . . . . . . . . . . . . . . . . . . 232.2 Preliminary Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.1 Diode Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2.2 Saturated Absorption Spectroscopy . . . . . . . . . . . . . . . 25

2.3 Primary EIT Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3.1 Components of Polarization . . . . . . . . . . . . . . . . . . . 272.3.2 Zeeman Apparatus . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4 Procedural Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Chapter 3: Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.2 Possible Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.1 F = 1→ F ′ = 0 Transition of the D2 Line . . . . . . . . . . . 373.2.2 D1 Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2.3 Laser Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2.4 Higher Resolution Detection and Real-Time Scanning . . . . . 38

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

List of Figures

0.1 A typical absorption spectrum, and an EIT absorption spectrum. . . 20.2 Λ-type structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1 All three level structures capable of demonstrating EIT . . . . . . . . 41.2 Λ-type level structure with applied field energies (~ωp and ~ωc) and

their energy detunings from resonance (~∆p and ~∆c). . . . . . . . . 51.3 Energy diagram of a six-level system . . . . . . . . . . . . . . . . . . 111.4 Linear susceptibility plotted in arbitrary units as a function of normal-

ized probe frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1 The F = 1→ F ′ = 1 transitions . . . . . . . . . . . . . . . . . . . . . 242.2 Schematic of saturated absorption setup. . . . . . . . . . . . . . . . . 262.3 Hyperfine spectra of 87Rb. . . . . . . . . . . . . . . . . . . . . . . . . 272.4 Schematic of the primary EIT setup . . . . . . . . . . . . . . . . . . . 282.5 Section and 3D rendering of the Zeeman apparatus . . . . . . . . . . 312.6 Magnetic field attenuated by mu metal shielding inside the Zeeman

apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.7 Magnetic field driven by the solenoid inside the Zeeman apparatus . . 33

3.1 Experimental EIT absorption of 87Rb . . . . . . . . . . . . . . . . . . 363.2 Simulation of absorption and dispersion as the decoherence parameter

γ2 varies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Abstract

Electromagnetically induced transparency (EIT) — a quantum phenomenon whereina material undergoes optical and dispersive modifications under the influence of reso-nant electromagnetic fields — was observed in warm rubidium vapor using the Zeemanmethod. Specifically, the absorption of a probe laser through the atomic medium ina magnetic field was measured, and a transparency window was observed around atypically absorbent atomic resonance, consistent with EIT.

Introduction

Electromagnetically induced transparency (EIT) is a quantum phenomenon thatmakes use of resonant electromagnetic fields to alter the optical properties of anatomic medium. Under typical conditions — that is, in the presence of a single near-resonant field — atoms will absorb energy from the surrounding field and excite intohigher energy states. The absorption spectrum, as a function of field frequency, fol-lows a Lorentzian curve peaked around the natural frequency of the resonant atomictransition, 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 whichthe medium exhibits high transparency, as shown in Fig. 0.1b. Hence, we get thename electromagnetically induced transparency. In addition, the dispersion propertiesare also affected, giving rise to many useful applications such as effective nonlinearfrequency conversion, slow light and nonlinear mixing.

EIT was first theorized in 1990 by Harris et al. [1] as a variation on coherentpopulation trapping. The following year, EIT was observed experimentally by Bolleret al. [2] in strontium vapor and by Field et al. [3] in lead vapor. Since then EIT hasbeen an extremely prevalent topic of research in physics. In 1996, Jain et al. [4] showedits applications in nonlinear mixing, wherein light beams of a certain wavelength areconverted to another wavelength with high efficiency. In 1998, Ling et al. [5] createdan electromagnetically induced diffraction grating from a gaseous atomic medium.In 1999, Budker et al. [6] used EIT to control the group velocity of light travelingthrough 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 ofoptical 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 andthe remaining field would be obscured. This gives rise to even more applications suchas optical switching [7].

The quantum mechanical mechanisms underlying EIT are quite complicated andnot easily intuited. One simple way to think of EIT is as a result of destructive inter-ference of transition probabilities. Given the three-level structure in Fig. 0.2, thereexist two paths between state |1〉 and state |3〉 (excluding longer paths): |1〉 → |3〉and |1〉 → |3〉 → |2〉 → |3〉. If the probabilities of each of these transitions occur-ring in an atom have opposing phases, destructive interference could occur and resultin no transitions from |1〉 to |3〉 [8]. This would, in turn, lead to a reduction inthe absorption of the field resonant with that transition. More rigorous derivations of

2 Introduction

HaL

D

Α

HbL

Dp

Α

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.

È1\

È3\

È2\

Figure 0.2: Λ-type structure.

three-level systems are discussed in Chapter 1, as well as a novel derivation describingthe six-level system used in this experiment.

This experiment explores specifically Zeeman EIT, where the transparency windowis seen not as a function of the incoming light but as a function of the strength of amagnetic 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 theresonant frequencies of the transitions. In essence, instead of changing the frequencyof light to match the atomic resonance — as in standard EIT — we will change thelevel splittings such that the resonant frequency matches that of the incoming light.

Specifics about how Zeeman EIT is observed in this experiment are discussed inChapter 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 fullyquantum mechanical derivation is not strictly necessary. This chapter will make asemiclassical derivation of the Hamiltonian for EIT, which treats atoms as quantummechanical objects that interact with classical vector fields. This derivation followsclosely the works of Weatherall [9] and Erickson [10]. We will then interpret theHamiltonian in two ways. First, we will take a look at the “dressed” state basis ofthe 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 ofsuch a system. The semiclassical approach is sufficient to see both the mechanismsbehind EIT and its effects. The main tradeoff of this derivation is that, while it ismuch simpler than the fully quantum mechanical derivation, it does not account forspontaneous emission of the atoms. Instead, we will add necessary phenomenologicaldecay 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 con-figuration 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 ex-periment exhibit a combination of Λ-type and V-type energy level structures. SinceΛ-atoms have a very low decay rate from the |2〉 state, the effect of EIT is muchstronger than in the other configurations [10]. For this reason, we will focus on thetheory as it applies to Λ-structure; however, the following derivations generalize toany of these energy structures, and further, to Zeeman EIT.

Let |1〉, |2〉, and |3〉 be the names given to the ground state, the metastable state,and the excited state, respectively, of the atom without any fields applied; theirenergies will be defined in terms of their corresponding frequencies as

En = ~ωn, (1.1)

for n = 1, 2, 3. While |1〉 and |2〉 can both excite into |3〉, we assume that the transitionbetween |1〉 and |2〉 is dipole forbidden. Therefore, the resonant frequencies between

4 Chapter 1. Theory

È1\

È3\

È2\

È1\

È3\

È2\

È1\

È2\

È3\

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, (1.2a)

ω23 = ω3 − ω2. (1.2b)

1.1.1 Bare-Atom Hamiltonian

Before any external field is applied, the energy eigenvalues and their eigenstates aredenoted ~ωn and |n〉, respectively. We can enforce completeness and orthogonalitysuch that ∑

n

|n〉〈n| = 1, (1.3)

and〈n|m〉 = δnm. (1.4)

The bare-atom Hamiltonian H0 can therefore be expressed in this basis as a matrixwith elements

H0 =

~ω1 0 00 ~ω2 00 0 ~ω3

. (1.5)

Without any external stimulation, this system will tend to settle in the lowest energy|1〉 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 otherstates are relatively unoccupied and can be neglected. The control field is tuned tothe resonant frequency ωc ≈ ω23 with amplitude Ec, and the probe field is tuned tothe other resonance ωp ≈ ω13 with amplitude Ep. A energy diagram of this scheme isshown in Fig. 1.2. This yields an electric field

E = Ep cos (ωpt− kp · r) + Ec cos (ωct− kc · r) , (1.6)

1.1. Derivation of the EIT Hamiltonian 5

È1\

È3\

È2\

ÑΩ13

ÑΩ12

ÑΩp

ÑΩc

ÑDp ÑDc

Figure 1.2: Λ-type level structure with applied field energies (~ωp and ~ωc) and theirenergy detunings from resonance (~∆p and ~∆c).

where the magnitude of the wavevector k = 2πλ

for the wavelength λ of the fields.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 (ωpt) + Ec cos (ωct) . (1.7)

The fields will perturb the Hamiltonian from that of the bare atom such that

H = H0 +H1, (1.8)

where H1 is the interaction term between the fields and the atom and can be writtenas

H1 = −qE · d. (1.9)

This assumes that the atom behaves similarly to an electric dipole with charge q andseparation vector d. This is a particularly good approximation for hydrogenic atoms— like rubidium — with only one electron in their outer shell. The dipole will quicklyalign with the fields, so we can instead write

H1 = −qEd. (1.10)

Now define the dipole moment operator ℘ ≡ qd and its elements ℘mn ≡ 〈n|℘|m〉.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 |1〉 → |2〉 transition to bedipole forbidden. Additionally, the diagonal terms must go to zero because of thespherical symmetry of the wavefunction [9]. Therefore, the matrix representation ofthe Hamiltonian simplifies further to

H1 = −E

0 0 ℘13

0 0 ℘23

℘31 ℘32 0

. (1.12)

6 Chapter 1. Theory

1.1.3 Rotating Wave Approximation

As it is, the interaction term of the Hamiltonian looks simple, but thus far, the electricfield has not played its part. In order to clearly see its effect, it is useful to transforminto the interaction picture of the unperturbed system. To do that, we use a unitarymatrix U defined as the time evolution operator, such that

U(t) = eiH0t/~ =

eiω1t 0 00 eiω2t 00 0 eiω3t

. (1.13)

Applying this transformation to H1 yields

UH1U† = −E

0 0 ℘13e−iω13t

0 0 ℘23e−iω23t

℘31eiω13t ℘32e

iω23t 0

. (1.14)

As seen from Eq. 1.7, the electric field is a sum of cosines. These can be writteninstead as exponentials:

E =Ep2

(eiωpt + e−iωpt

)+Ec2

(eiωct + e−iωct

). (1.15)

Plugging this into the transformed Hamiltonian, the nonzero elements become

(UH1U†)13 = −℘13

(Ep2

(eiωpt + e−iωpt

)+Ec2

(eiωct + e−iωct

))e−iω13t, (1.16a)

(UH1U†)23 = −℘23

(Ep2

(eiωpt + e−iωpt

)+Ec2

(eiωct + e−iωct

))e−iω23t, (1.16b)

(UH1U†)31 = −℘31

(Ep2

(eiωpt + e−iωpt

)+Ec2

(eiωct + e−iωct

))eiω13t, (1.16c)

(UH1U†)32 = −℘32

(Ep2

(eiωpt + e−iωpt

)+Ec2

(eiωct + e−iωct

))eiω23t. (1.16d)

In the rotating wave approximation, it is assumed that the rapidly oscillating termswill average out quickly and therefore can be ignored. Thus, exponential terms withlarge imaginary arguments drop out. Since ωc ≈ ω23 and ωp ≈ ω13, there will be oneexponential term in each matrix element that oscillates slowly enough to survive thisapproximation. Namely,

(UH1U†)13 = −1

2Ep℘13e

i(ωp−ω13)t, (1.17a)

(UH1U†)23 = −1

2Ec℘23e

i(ωc−ω23)t, (1.17b)

(UH1U†)31 = −1

2Ep℘31e

i(ω13−ωp)t, (1.17c)

(UH1U†)32 = −1

2Ec℘32e

i(ω23−ωc)t. (1.17d)

1.1. Derivation of the EIT Hamiltonian 7

Transforming back to the Schrodinger picture, the interaction Hamiltonian becomes

H1 = U †(UH1U

†)U= −1

2

0 0 Ep℘13eiωpt

0 0 Ec℘23eiωct

Ep℘31e−iωpt Ec℘32e

−iωct 0

. (1.18)

The electric dipole operators can be represented in terms of their magnitudes andphases such that

℘13 = ℘∗31 = |℘13|eiφp , (1.19a)

℘23 = ℘∗32 = |℘23|eiφc . (1.19b)

The Rabi frequencies of this system are defined to be

Ωp =Ep|℘13|

~, (1.20a)

Ωc =Ec|℘23|

~. (1.20b)

Plugging these definitions into the interaction Hamiltonian and summing it with thebare-atom Hamiltonian, the full Hamiltonian in the presence of applied fields becomes

H =~2

2ω1 0 −Ωpei(ωpt+φp)

0 2ω2 −Ωcei(ωct+φc)

−Ωpe−i(ωpt+φp) −Ωce

−i(ωct+φc) 2ω3

. (1.21)

1.1.4 Time and Phase Independent Frame

The Hamiltonian in Eq. 1.21 still depends on phases and time, which muddies theinterpretation. To clearly see how EIT will come from such a Hamiltonian, we musttransform into a new basis — the corotating basis. The unitary matrix that willachieve this transformation is defined as

U(t) =

e−i(ωpt+φp) 0 00 e−i(ωct+φc) 00 0 1

. (1.22)

Note that the full Hamiltonian has a different set of eigenstates than the bare atom.Define |n′〉 to be the eigenstates ofH. There are also analogous eigenstates |n′〉 = U |n′〉of the corotating Hamiltonian H. For the corotating eigenstates to satisfy Schrodinger’s

8 Chapter 1. Theory

equation, we must have

H|n′〉 = i~∂

∂t|n′〉

= i~∂

∂t

(U |n′〉

)= i~

(∂U

∂t|n′〉+ U

∂|n′〉∂t

)

= i~

(∂U

∂t|n′〉+

1

i~UH|n′〉

)

=

(i~∂U

∂tU † + UHU †

)U |n′〉

H|n′〉 =

(i~∂U

∂tU † + UHU †

)|n′〉. (1.23)

Therefore, the transformation of the Hamiltonian into the corotating frame is

H = i~∂U

∂tU † + UHU † (1.24)

=~2

2ωp 0 00 2ωc 00 0 0

+~2

2ω1 0 −Ωp

0 2ω2 −Ωc

−Ωp −Ωc 2ω3

H =

~2

2(ω1 + ωp) 0 −Ωp

0 2(ω2 + ωc) −Ωc

−Ωp −Ωc 2ω3

. (1.25)

By noting that the physical interpretation of a Hamiltonian is unaltered by the addi-tion of a scalar multiple of the identity, the Hamiltonian can be coerced into an evenmore interpretable form. That is, adding −~(ω1 + ωp)1 yields

H =~2

0 0 −Ωp

0 2(ω2 + ωc − ω1 − ωp) −Ωc

−Ωp −Ωc 2(ω3 − ω1 − ωp)

. (1.26)

Finally, defining the laser detunings from resonance as ∆p ≡ ω13−ωp = ω3−ω1−ωpand ∆c ≡ ω23 − ωc = ω3 − ω2 − ωc, the EIT Hamiltonian takes on its standard form:

H =~2

0 0 −Ωp

0 2(∆p −∆c) −Ωc

−Ωp −Ωc 2∆p

. (1.27)

1.2. Dressed State Picture 9

1.2 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 approximatethat ∆p ≈ ∆c ≡ ∆. This is reasonable because these detunings are small comparedto other relevant frequencies in the experiment.

Solving the characteristic equation

det∣∣∣H − λ1∣∣∣ = 0, (1.28)

yields the following eigenvalues:

λ0 = 0, (1.29a)

λ− ≡ ~ω− =~2

(∆−

√∆2 + Ω2

p + Ω2c

), (1.29b)

λ+ ≡ ~ω+ =~2

(∆ +

√∆2 + Ω2

p + Ω2c

). (1.29c)

The corresponding eigenstates of the “dressed” system, expressed as linear combina-tions of the bare-atom states, are

|0〉 =Ωc|1〉 − Ωp|2〉√

Ω2p + Ω2

c

, (1.30a)

|−〉 = − Ωp|1〉+ Ωc|2〉∆−

√∆2 + Ω2

p + Ω2c

+ |3〉, (1.30b)

|+〉 =Ωp|1〉+ Ωc|2〉

∆ +√

∆2 + Ω2p + Ω2

c

− |3〉. (1.30c)

Note that an atom in state |0〉 cannot excite into state |3〉 because |0〉 has no com-ponent of the |3〉 state. For this reason, we call |0〉 the “dark” state.

When the fields are close to resonance — that is, ∆ ≈ 0 — the upper dressedstates can be normalized and approximated by

|−〉 =1√2

(Ωp|1〉+ Ωc|2〉√

Ω2p + Ω2

c

+ |3〉

), (1.31a)

|+〉 =1√2

(Ωp|1〉+ Ωc|2〉√

Ω2p + Ω2

c

− |3〉

). (1.31b)

10 Chapter 1. Theory

1.2.2 The Dark State and The Bright State

If we assume that the amplitude of the probe beam is small compared to that of thecontrol, then Ωp Ωc. Using this approximation and normalizing, the dark state |0〉becomes

|0〉 ≈ |1〉. (1.32)

Under these conditions, the dark state is approximately the same as the ground stateof the bare atom. Therefore, the dark state is not only decoupled from the othertwo, but additionally, it is a stationary state of both the bare-atom Hamiltonian andthe dressed system. This means that, due to their time-independence, atoms in theground state will remain there and never excite. Thus, the probe beam will remainunabsorbed.

It would seem that atoms that start out in some admixture of the upper stateswould remain excited. However, this semiclassical treatment of a quantum phe-nomenon neglects the possibility of spontaneous emission from the upper states intothe ground state. In actuality, these states will decay into the ground state and thenbe 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 thenature of the non-dark states |+〉 and |−〉. Using Eqs. 1.31a and 1.31b, we may definetwo linear combinations of the states

|a〉 ≡ |+〉 − |−〉√2

= |3〉, (1.33a)

|b〉 ≡ |+〉+ |−〉√2

=Ωp|1〉+ Ωc|2〉√

Ω2p + Ω2

c

. (1.33b)

|a〉 and |b〉 are not stationary states. In fact, the strong control field acts to oscillateatoms back and forth between these two states. For this reason we shall call |b〉 thebright state; it accounts for all transitions to and from the excited state |a〉. Usingthe same weak probe field approximation as before, we can write

|b〉 ≈ |2〉. (1.34)

Therefore, in the presence of both electric fields, atoms in state |2〉 are still able toabsorb energy from the control field and excite into state |3〉. However, |1〉 decouplesfrom 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-levelsystems — specifically Λ-type energy structures. However, in practice, few systemsconsist of just three energy levels. In fact, the atomic levels relevant to this experimentconsist of six levels, as shown in Fig. 1.3.

Most direct methods of calculating absorption — like the methods discussed inthe following section — would be much more difficult for systems with more thanthree levels. The dressed state analysis, however, is more or less the same. For this


È1\È2\

È3\

È4\È5\

È6\

Figure 1.3: Energy diagram of a six-level system. Solid arrows indicate strong controlfields while dashed arrows represent weak probe fields.

reason, it is often more convenient to verify that a dark state exists for a given systembefore going through any computation-heavy analysis.

Consider the six-level system described in Fig. 1.3. In this scenario, there are twocontrol beams, both with energy ~ωc ≈ ~ω24 ≈ ~ω35 — shown as solid arrows — andtwo 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 00 ~ω2 0 0 0 00 0 ~ω3 0 0 00 0 0 ~ω4 0 00 0 0 0 ~ω5 00 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

2ω1 0 0 0 −Ωcei(ωct+φc) 0

0 2ω2 0 −Ωpei(ωpt+φp) 0 −Ωcei(ωct+φc)

0 0 2ω3 0 −Ωpei(ωpt+φp) 0

0 −Ωpe−i(ωpt+φp) 0 2ω4 0 0

−Ωce−i(ωct+φc) 0 −Ωpe−i(ωpt+φp) 0 2ω5 0

0 −Ωce−i(ωct+φc) 0 0 0 2ω6

.

(1.36)Now, we wish to transform the Hamiltonian into a frame that is time and phase


independent using the unitary matrix

U(t) =

e−i(ωpt+φp) 0 0 0 0 0

0 1 0 0 0 00 0 e−i(ωct+φc) 0 0 00 0 0 ei(ωct+φc) 0 00 0 0 0 1 00 0 0 0 0 ei(ωpt+φp)

. (1.37)

Plugging into Eq. 1.24 yields the corotating Hamiltonian

H =~2

2 (ω1 + ωp) 0 0 0 −Ωc 0

0 2ω2 0 −Ωp 0 −Ωc

0 0 2 (ω3 + ωc) 0 −Ωp 00 −Ωp 0 2 (ω4 − ωc) 0 0−Ωc 0 −Ωp 0 2ω5 0

0 −Ωc 0 0 0 2 (ω6 − ωp)

. (1.38)

We can now assume the relative energy spacings of each cluster of three levels arethe same. This implies that we can express all six frequencies in terms of ω2 and therelevant spacings such that

ω1 = ω2 − δ, ω4 = ω2 + ω0 − δ, (1.39a)

ω2 = ω2, ω5 = ω2 + ω0, (1.39b)

ω3 = ω2 + δ, ω6 = ω2 + ω0 + δ, (1.39c)

where ~δ is the spacing between adjacent states within a single cluster, and ~ω0 isthe energy gap between the two clusters. This yields two resonant frequencies

ω15 = ω26 = ω0 + δ, (1.40a)

ω24 = ω35 = ω0 − δ (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. Thisyields

H =~2

0 0 0 0 −Ωc 00 2(δ−ωp) 0 −Ωp 0 −Ωc0 0 2(2δ+ωc−ωp) 0 −Ωp 00 −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, (1.42a)

∆c = ω24 − ωc = ω0 − δ − ωc, (1.42b)


the Hamiltonian becomes

H =~2

0 0 0 0 −Ωc 00 2(∆p−ω0) 0 −Ωp 0 −Ωc0 0 2(∆p+∆c) 0 −Ωp 00 −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 diagonalterms simplify, and the Hamiltonian of interest becomes

H =~2

0 0 0 0 −Ωc 00 −2ω0 0 −Ωp 0 −Ωc

0 0 0 0 −Ωp 00 −Ωp 0 −2ω0 0 0−Ωc 0 −Ωp 0 0 0

0 −Ωc 0 0 0 −2ω0

, (1.44)

with the following eigenstates:

|0Λ〉 =Ωc|1〉 − Ωp|3〉√

Ω2p + Ω2

c

, (1.45a)

|0V〉 =Ωp|4〉 − Ωc|6〉√

Ω2p + Ω2

c

, (1.45b)

|a〉 =

√Ω2p + Ω2

c |2〉+ Ωc|4〉+ Ωp|6〉√2Ω2

p + 2Ω2c

, (1.45c)

|b〉 =

√Ω2p + Ω2

c |2〉 − Ωc|4〉 − Ωp|6〉√2Ω2

p + 2Ω2c

, (1.45d)

|c〉 =

√Ω2p + Ω2

c |1〉+ Ωc|3〉+ Ωp|5〉√2Ω2

p + 2Ω2c

, (1.45e)

|d〉 =

√Ω2p + Ω2

c |1〉 − Ωc|3〉 − Ωp|5〉√2Ω2

p + 2Ω2c

. (1.45f)

As expected, we see that one of the eigenstates — the one that is convenientlynamed |0Λ〉 — looks almost exactly like the dark state in the three-level system. Asin the Λ-type scheme, when the probe beam amplitude is small, i.e. Ωp Ωc, we find

|0Λ〉 ≈ |1〉. (1.46)

This shows that the states |1〉, |3〉, and |5〉 make up a mini-Λ scheme inside the greatersix-level system and behave similarly to the normal Λ-type structure.

Additionally, there is another state of a similar form. That is, under the sameapproximation,

|0V〉 ≈ |6〉. (1.47)

This indicates that there are in fact two dark states of the system: one of the Λ schemeand the other of the V scheme made up of states |2〉, |4〉, and |6〉. It is important to


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, theywill tend to populate the Λ scheme states because there is a greater probability thatan atom will decay into |1〉 or |3〉 than into |2〉. Therefore, the EIT effects due to theΛ 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 offrequencies — an appropriate assumption when using the Zeeman method to showEIT, as in this experiment — both associated transparency windows will occur con-currently, 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, butit is not as useful in making physical predictions about the medium. Since thisexperiment deals not with a single atom, but with a whole ensemble of atoms, thenext calculation will relate ideas of the previous sections to measurable macroscopicquantities. In particular, we will calculate the electric susceptibility χ of the mediumin the presence of fields. The susceptibility is of interest because its real and imaginaryparts determine the two primary effects of EIT: its real part determines the dispersionof 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 = ε0χE, (1.48)

where P is the dielectric polarization and ε0 is the permittivity of free space. Thepolarization can also be defined as

P = N〈℘〉, (1.49)

where N is the total number of atoms. Therefore, Eqs. 1.48 and 1.49 can be equated,yielding

ε0χE = N〈℘〉. (1.50)

Almost all of the quantities above are macroscopic and measurable, so in order tocalculate the susceptibility, we only need to find the expectation value of the electricdipole operator.

1.3.1 Density Operator

For pure quantum states, an expectation value is defined in terms of the wavefunction|Ψ〉, such that for some operator Λ,

〈Λ〉 = 〈Ψ|Λ|Ψ〉. (1.51)

However, it is often difficult to find the wavefunction of one atom, much less of theentire 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 stateis 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 |n〉and follow a distribution Pn, then the expectation value becomes

〈Λ〉 =∑n

Pn〈n|Λ|n〉. (1.52)

For this reason, it is simpler to define expectation values in terms of the densityoperator

ρ =∑n

Pn|n〉〈n|, (1.53)

where Pn is the probability of an atom being in state |n〉. To see how this works,recall the completeness and orthogonality relations in Eqs. 1.3 and 1.4, and compute,

〈Λ〉 =∑n

Pn〈n|Λ|n〉

=∑mn

Pn〈n|Λ|m〉〈m|n〉

=∑mn

〈m|Pn|n〉〈n|Λ|m〉

=∑m

〈m|ρΛ|m〉

〈Λ〉 = Tr(ρΛ). (1.54)

This definition of an expectation value makes no reference to the specific state of theatoms. Instead, it makes uses of statistical distributions within the system as definedin the density operator.

1.3.2 Von Neumann Equation

Since Schrodinger’s equation is defined in terms of the wavefunction, a different rela-tion exists in terms of the density operator; this relation is called the von Neumannequation.

In order to derive the von Neumann equation, recall that Schrodinger’s equationand its adjoint are expressed by

d

dt|ψ〉 = − i

~H|ψ〉, d

dt〈ψ| = i

~〈ψ|H. (1.55)


Computing the time derivative of the density operator, we find

ρ =∑n

Pn (|n〉〈n|+ |n〉〈n|)

= − i~∑n

Pn (H|n〉〈n| − |n〉〈n|H)

= − i~

(Hρ− ρH)

= − i~

[H, ρ] . (1.56)

Again, our model has omitted the possibility of spontaneous emission from states|2〉 and |3〉. To count for this, we include an additional term

ρ = − i~

[H, ρ]− 1

2Γ, ρ , (1.57)

where the straight brackets represent a commutator and the curly brackets representan anti-commutator. Here, Γ is defined in terms of the decay rate γn from a state |n〉as

Γ =

0 0 00 γ2 00 0 γ3

, (1.58)

and γ1 = 0 because the atom cannot decay out of the ground state.The additional decay term only accounts for atoms leaving states |2〉 and |3〉. It

does not specify where they go. In reality, the majority of these atoms may decay intostate |1〉, 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 asdecay rates of atoms leaving the three-level system completely.

In coordinate form, von Neumann’s equation becomes

ρij = −∑k

[i

~(Hikρkj − ρikHkj) +

1

2(Γikρkj + ρikΓkj)

]. (1.59)

1.3.3 Von Neumann Equation in EIT

Using von Neumann’s equation, written in component form, and the three-level EITHamiltonian H, from Eq. 1.27 — expressed in the corotating basis — the densityoperator ρ can be calculated in the same basis. The diagonal matrix elements of ˙ρcan be written in terms of ρ as

˙ρ11 =iΩp

2(ρ31 − ρ13) , (1.60a)

˙ρ22 = −γ2ρ22 +iΩc

2(ρ32 − ρ23) , (1.60b)

˙ρ33 = −γ3ρ33 −iΩp

2(ρ31 − ρ13)− iΩp

2(ρ32 − ρ23) . (1.60c)


Additionally, the off-diagonal terms are

˙ρ12 = ˙ρ∗21 =(−γ2

2+ i(∆p −∆c)

)ρ12 −

iΩc

2ρ13 +

iΩp

2ρ32, (1.61a)

˙ρ13 = ˙ρ∗31 =(−γ3

2+ i∆p

)ρ13 +

iΩp

2(ρ33 − ρ11)− iΩc

2ρ12, (1.61b)

˙ρ23 = ˙ρ∗32 =

(−1

2(γ2 + γ3) + i∆c

)ρ23 +

iΩc

2(ρ33 − ρ22)− iΩp

2ρ12. (1.61c)

We will now assume that the majority of the atoms occupy the ground state. This is agood approximation both because atoms are constantly being pumped by the controlbeam from |2〉 into |1〉, and also because spontaneously emitted atoms continuouslyrepopulate the ground state. Therefore, since the diagonal terms ρnn can be thoughtof as the fraction of atoms in the population of |n〉, we see that ρ11 ≈ 1 and ρ22 ≈ρ33 ≈ 01. This yields

˙ρ12 = ˙ρ∗21 =(−γ2

2+ i(∆p −∆c)

)ρ12 −

iΩc

2ρ13 +

iΩp

2ρ32, (1.62a)

˙ρ13 = ˙ρ∗31 =(−γ3

2+ i∆p

)ρ13 −

iΩp

2− iΩc

2ρ12, (1.62b)

˙ρ23 = ˙ρ∗32 =

(−γ2 + γ3

2+ i∆c

)ρ23 −

iΩp

2ρ12. (1.62c)

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 ˙ρ23 = 0), ρ23 is linearly proportionalto Ωp, so then the iΩp

2ρ32 term in Eq. 1.62a can be dropped. Thus, we have

˙ρ12 = ˙ρ∗21 =(−γ2

2+ i(∆p −∆c)

)ρ12 −

iΩc

2ρ13, (1.63a)

˙ρ13 = ˙ρ∗31 =(−γ3

2+ i∆p

)ρ13 −

iΩp

2− iΩc

2ρ12. (1.63b)

These are now in the form of two coupled differential equations. To solve, they canbe expressed as a single matrix equation, such that(

˙ρ12

˙ρ13

)=

(−γ2

2+ i(∆p −∆c) − iΩc

2

− iΩc2

−γ32

+ i∆p

)(ρ12

ρ13

)+

(0

− iΩp2

)(1.64)

Now define

X =

(ρ12

ρ13

), (1.65a)

M =

(−γ2

2+ i(∆p −∆c) − iΩc

2

− iΩc2

−γ32

+ i∆p

), (1.65b)

A =

(0

− iΩp2

), (1.65c)

1Obviously states |2〉 and |3〉 are not empty, but their populations are so much smaller than thatof |1〉 that they can be neglected.


and Eq. 1.64 becomes simply,

X = MX + A. (1.66)

The steady state solution to this matrix equation — assuming X = 0 — is

X = −M−1A. (1.67)

This yields solutions for the matrix elements of the density operator,

ρ12 =ΩcΩp

2iγ3 (∆c −∆p) + γ2 (γ3 − 2i∆p) + 4∆c∆p − 4∆2p + Ω2

c

, (1.68a)

ρ13 =i (γ2 + 2i (∆c −∆p)) Ωp

2iγ3 (∆c −∆p) + γ2 (γ3 − 2i∆p) + 4∆c∆p − 4∆2p + Ω2

c

. (1.68b)

These matrix elements are, however, still in the corotating basis. In order to transformback to the Schrodinger picture, we would have to compute

ρ = U †ρU

=

ρ11 ρ12ei(ωp+φp−ωc−φc)t ρ13e

i(ωp+φp)t

ρ21e−i(ωp+φp−ωc−φc)t ρ22 ρ23e

i(ωc+φc)t

ρ31e−i(ωp+φp)t ρ32e

−i(ωc+φc)t ρ33

(1.69)

1.3.4 Complex Susceptibility And Its Implications

We now have a matrix expression for the density operator in the corotating frame ρ.To ease the amount of computation, we will first show that ρ13 is the only matrixelement of the density operator in the Schrodinger picture needed to calculate theelectric susceptibility χ. Recall that polarization can be expressed as

P = N〈℘〉= NTr (ρ℘)

= NTr

ρ11 ρ12 ρ13

ρ21 ρ22 ρ23

ρ31 ρ32 ρ33

0 0 ℘13

0 0 ℘23

℘31 ℘32 0

= NTr

ρ13℘31 ρ13℘32 ρ11℘13 + ρ12℘23

ρ23℘31 ρ23℘32 ρ21℘13 + ρ22℘23

ρ33℘31 ρ33℘32 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 densityelements as

P = N(ρ13℘31e

i(ωp+φp)t + ρ23℘32ei(ωc+φc)t + ρ31℘13e

−i(ωp+φp)t + ρ32℘23e−i(ωc+φc)t

).

(1.71)


The polarization can also be expressed in terms of the electric field as defined inEq. 1.15 such that

P = ε0χE

=ε0χ(ωp)Ep

2

(eiωpt + e−iωpt

)+ε0χ(ωc)Ec

2

(eiωct + e−iωct

). (1.72)

Notice that since χ depends on frequency, there appear two different susceptibilitiesin the equation above. We are particularly interested in χ(ωp) ≡ χp because thedesired effect is transmission of the probe beam. To isolate this parameter, we matchexponentials in Eqs. 1.71 and 1.72 — specifically, those that contain a factor of eiωpt.This yields

ε0χpEp2

= Nρ13℘31eiφpt. (1.73)

Recalling that ℘31 = |℘13|e−iφpt and solving for χp gives

χp =2N |℘13|ε0Ep

ρ13. (1.74)

Finally, using Eq. 1.68b we can write a full expression for the complex electric sus-ceptibility as

χp =2N |℘13|ε0Ep

i (γ2 + 2i (∆c −∆p)) Ωp

2iγ3 (∆c −∆p) + γ2 (γ3 − 2i∆p) + 4∆c∆p − 4∆2p + Ω2

c

. (1.75)

The real and imaginary parts can be written

Re(χp) = −2N |℘13|ε0Ep

2(γ2

2∆p + (∆c −∆p)(4 (∆c −∆p) ∆p + Ω2

c

))Ωp(

γ22 + 4 (∆c −∆p)

2) (γ2

3 + 4∆2p

)+ 2 (γ2γ3 + 4 (∆c −∆p) ∆c) Ω2

c + Ω4c

,

(1.76a)

Im(χp) =2N |℘13|ε0Ep

(γ2

2γ3 + 4γ3 (∆c −∆p)2 + γ2Ω2

c

)Ωp(

γ22 + 4 (∆c −∆p)

2) (γ2

3 + 4∆2p

)+ 2 (γ2γ3 + 4 (∆c −∆p) ∆c) Ω2

c + Ω4c

.

(1.76b)

We recall that the real part of the linear susceptibility is proportional to the dis-persion 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 dras-tically when the probe field is on resonance. This window of transparency is preciselywhat is meant by EIT. With only one or the other applied field, we would measurea typical absorption spectrum, but with both, we see a cancelation, such that theprobe beam remains unabsorbed. Additionally, the dispersion becomes much steepernear resonance. Though this experiment will not dwell much on the dispersion, it isworth noting that such a dispersion relation in a zone of low absorption is conduciveof complex optical phenomena such as slow light [6].


HaL

DpΓ3

Χp

HbL

DpΓ3

Χp

Figure 1.4: Linear susceptibility plotted in arbitrary units as a function of normalizedprobe frequency. The solid line shows the imaginary part of χp (which is proportionalto the absorption), and the dashed line shows the real part of χp (which is proportionalto the dispersion). Probe frequency is normalized with respect to γ3, which allows usto plot in units of γ3. For (a) we set ∆c = 0, Ωc = 0, and γ2 = 104γ3. For (b) we set∆c = 0, Ωc = γ3, and γ2 = 104γ3.


As one last check, if we set Ωc = 0 — corresponding to the case when the controlbeam is switched off — we get

Re(χp) = −2N |℘13|ε0Ep

2Ωp∆p

γ3 + 4∆2p

, (1.77a)

Im(χp) =2N |℘13|ε0Ep

Ωpγ3

γ3 + 4∆2p

. (1.77b)

As shown in the plots of Eq. 1.77 in Fig. 1.4a, the absorption and dispersion revertback 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 mag-netic field, effectively establishing the existence of EIT. The theory discussed in theprevious chapter was derived for standard EIT — which makes use of two lasers. Thetheory also generalizes to Zeeman EIT. The major distinction is that the multiple-level system is created by applying a magnetic field across atoms with degenerateenergy levels. This splits the levels creating a suitable energy scheme for EIT. Dueto this distinction, it is possible to use only one laser and scan the magnitude of themagnetic field surrounding the atoms to acquire data. Details on how this is possiblewill 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 energystructure and the availability of applicable optical equipment. Additionally, sincerubidium is ubiquitous in the study of EIT and saturated absorption spectroscopy, itmakes for easy comparison to past research.

Specifically, the D2 transition in the 87Rb isotope is used. This transition isbetween the ground state |5s 2S1/2〉 and the excited state |5s 2P3/2〉 and is resonantat a wavelength of 780.24 nm. Within these states there exist several hyperfine levelsdenoted by their quantum numbers F for the ground state and F ′ for the excitedstate. The hyperfine levels used in this experiment are F = 1 and F ′ = 1.

These levels were chosen because of the prominence of the peak associated withtheir atomic transition on a hyperfine absorption spectrum and for the simplicity oftheir Zeeman splitting. The Zeeman effect occurs when states of the same energybut distinct angular momenta are introduced to a magnetic field. Here, angularmomentum is denoted by the quantum numbers mF and m′F for the ground andexcited state, respectively. Each state (F = 1 and F ′ = 1) can have three momentamF = 0,±1, so they each split into three distinct levels. The resulting six-levelsystem is shown in Fig. 2.1. In theory, it may seem simpler to use the F ′ = 0level because this excited state does not split due to the Zeeman effect. However, inpractice, its associated absorption line is too small to detect using the equipment in

24 Chapter 2. Experiment

mF=-1mF=0

mF=1

mF'=-1mF'=0

mF'=1

F¢ 1

F 1

Figure 2.1: The F = 1 → F ′ = 1 transitions with the Zeeman split levels. Solidarrows 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 momen-tum by one quantum number. That is, σ+ will excite mF = −1→ m′F = 0. Similarly,a right-circularly polarized σ− beam will take mF = 1→ m′F = 0.

At the same time, the mF = 0 state will also excite into m′F = ±1 states forming aV-type structure. This combination of Λ and V schemes is the same energy structureshown to have EIT dark states in Section 1.2.3. Luckily, EIT is seen for both of theseconfigurations when the Zeeman splitting is minimal, or when the applied magneticfield is close to zero. This means their transparency windows will occur concurrentlyon the absorption curve. Due to spontaneous decay from the upper levels, atoms willtend to settle in the Λ states, so the contribution of the Λ scheme to the EIT signalwill be much greater that that of the V scheme. Therefore, we still expect absorptionsimilar to that of the three-level system.

In terms of the states of the three-level Λ system discussed in Chapter 1, themF = ±1 states act as |1〉 and |2〉 exchangeably, depending on the sign of the appliedmagnetic field1. The m′F = 0 state acts as |3〉.

2.2 Preliminary Setup

The setup consists of a diode laser which is parked at the resonant frequency usingsaturated absorption spectroscopy. The beam is then properly polarized and shonethrough a cell of rubidium surrounded by magnetic field coils. The transmitted beamis then detected by a power meter, and an absorption curve is obtained.

1When 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 25

2.2.1 Diode Laser

The 780 nm diode laser was a product of a past thesis at Reed College [14]. It hasa Littrow configuration in which an external cavity is formed between the rear facetof the diode and a diffraction grating. The grating is angled such that the first-orderdiffraction is reflected back into the diode to form a feedback loop. Zeroth-orderreflection exits the cavity as the lasing beam. The lasing wavelength is determinedby the length of the external cavity.

The laser is driven by a ILX Lightwave LDX-3525 precision current source andcooled by a Peltier cooler configured with a ILX Lightwave LDT-5910B PID precisionthermo-electric temperature controller. The frequency of the laser can be fine tunedby a piezoelectric crystal which changes the cavity length by small amounts. A voltageis applied across the piezo crystal by a Thorlabs MDT694A piezo controller, which isdriven by an Agilent 33210A arbitrary waveform signal generator.

This allows the laser to be scanned across a range of wavelengths according to theoutput voltage of the signal generator. The piezo crystal can scan the laser frequencyover a range of 1-2 GHz without mode jumping.

The exiting laser light enters a Conoptics Model 713A optical isolator and is splitby a 95-5 beamsplitter. The transmitted beam is much stronger, and it continues intothe primary EIT setup. The weaker reflected beam is directed toward the saturatedabsorption spectroscopy setup described in the following section.

2.2.2 Saturated Absorption Spectroscopy

Doppler-free saturated absorption spectroscopy is a standard experimental methodused to park the laser onto a particular frequency resonant with a hyperfine tran-sition [15]. We will use this technique to tune the laser to the F = 1 → F ′ = 1transition in 87Rb.

Without saturated absorption spectroscopy, this would prove to be a difficulttask; since the atoms in a cell of gas will have some spread of velocities accordingto a Maxwell-Boltzmann distribution, each velocity group will “appear” to have anatomic 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 spectroscopyis a technique used to resolve the hyperfine peaks by counterpropagating a pump andprobe 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 facetof the plate and again off the back. These are the probe2 beams which pass througha rubidium cell before being detected by photodiodes. Another much stronger pumpbeam is transmitted through the glass plate. The pump is redirected to counterprop-agate with one of the probe beams as they pass through the medium.

Due to the opposing Doppler shifts, the counterpropagating beams cannot be si-

2It is important to note that the pump and probe beams in the saturated absorption spectroscopysetup are distinct from those in the primary EIT setup and those mentioned in the theory sections.


CurrentController

Temp.Controller

SignalGenerator

PiezoController

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 littleabsorption. The laser is scanned in frequency by sending a positive triangle waveformthrough the piezo controller. Two spectra result: one from the unsaturated probebeam — a typical Doppler-broadened absorption spectrum — and another from thesaturated beam — consisting of narrow peaks at the hyperfine resonance lines. Thedifference of these two signals yields a resolvable hyperfine spectrum. Examples ofsuch spectra are shown in Fig. 2.3.

To park the laser onto the F = 1→ F ′ = 1 peak, the frequency scanning range isdecreased in amplitude while observing the detector signal. This effectively narrowsthe spectrum onto the desired peak. Then, the scan is stopped altogether. The piezovoltage can also be manually adjusted to ensure the laser is stabilized at the rightfrequency.

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 aboutone run of measurements. In future experiments, it may be necessary to use feedbackcontrol electronics to stabilize the laser for longer amounts of time [17].

2.3. Primary EIT Setup 27

F'=2F'=1

0.2 0.4 0.6 0.8Vscan HVL

1

2

3

4

5

6

Vspectrum HVLF'=3F'=2

0.1 0.2 0.3 0.4 0.5 0.6 0.7Vscan HVL

5

10

15

Vspectrum HVL

Figure 2.3: Hyperfine spectra of 87Rb. F = 1→ F ′ transitions (left) and F = 2→ F ′

transitions (right). These measurements are consistent with the results of MacAdamset al. [16].

2.3 Primary EIT Setup

The majority of the light emitted by the laser — now parked on the hyperfine line ofinterest — 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, thesingle beam will be composed of two components of orthogonal polarizations. Thesecomponents will act as independent control and probe beams, though they make upthe same beam. For this reason, it is easiest to discuss how optical elements affecteach polarization component individually.

As shown in Fig. 2.4, the light first passes through a half-wave plate. Based on theangle of the half-wave plate, the resulting light will have some specific combinationof linear polarizations, which will later become the control and probe components ofthe beam. The angle of the half-wave plate is chosen such that the ratio of intensitiesof “control” polarization to the “probe” polarization is 9:1. The beam then passesthrough a quarter-wave plate, which converts the linearly polarized components tocircular polarizations. This produces the σ+ and the σ− needed to excite the atomicmedium. After the beam passes through the rubidium cell, it is converted back tolinear polarization by another quarter-wave plate, and the linear components are splitby a polarizing beamsplitter. The power of the resulting probe beam is measured bya ThorLabs PM100D optical power meter.

To see that this works explicitly, we will use Jones calculus [18] to calculate therelevant polarizations at the lettered points in Fig. 2.4. Assume that the polarizationoutput by the laser is the y-direction, where the y-direction points upwards perpen-dicular to the optics table.3 Therefore, the initial Jones vector at point A and the

3The starting polarization may very well be some admixture of x- and y-polarizations, but thiscan be corrected for by properly setting the angle of the half-wave plate.


SASSetup

Λ2

Λ4

ZeemanApparatus

Rb Cell

Λ4 PBS BD

OpticalIsolator

TunableDiode Laser

PowerMeter

ConstantVoltageSource

A

B

C

D EÄÄÄÄÄÄÄÄÄ

Figure 2.4: Schematic of the primary EIT setup. Abbreviations used in this figureare: 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

PA =

(01

), JHWP =

(cos 2θ sin 2θsin 2θ − cos 2θ

), (2.1)

respectively, where θ is the angle of the half-wave plate’s fast axis with respect to they-direction. After the half-wave plate, the polarization at point B will be

PB = JHWPPA

=

(cos 2θ sin 2θsin 2θ − cos 2θ

)(01

)=

(sin 2θ− cos 2θ

). (2.2)

Clearly, θ can be chosen to create any control-to-probe ratio desired. In this ex-periment, only the 9:1 ratio will be studied, but changing this ratio leads to other


interesting observable effects, such as coherent population trapping [19] and electro-magnetically induced absorption [20].

The Jones matrix for a quarter-wave plate with its fast axis 45 from the y-direction is

JQWP =1√2

(1 ii 1

). (2.3)

Therefore, the polarization at point C will be

PC = JQWPPB

=1√2

(1 ii 1

)(sin 2θ− cos 2θ

)=

1√2

(sin 2θ − i cos 2θi sin 2θ − cos 2θ

)=

1√2

[sin 2θ

(1i

)− i cos 2θ

(1−i

)]= σ+ sin 2θ − σ−i cos 2θ, (2.4)

where the left-circular and the right-circular polarizations are defined to be

σ+ ≡1√2

(1i

), σ− ≡

1√2

(1−i

), (2.5)

respectively. Thus, the polarization at point C has a component of each circularpolarization.

When a beam reflects off a mirror, left-circularly polarized light becomes right-circularly polarized and vice versa. Therefore, the polarization at point D is

PD = σ− sin 2θ − σ+i cos 2θ (2.6)

=1√2

(sin 2θ − i cos 2θ−i sin 2θ + cos 2θ

)(2.7)

Finally, the beam goes through another quarter-wave plate at 45. The resultingpolarization at point E will be

PE = JQWPPD

=1√2

(1 ii 1

)1√2

(cos 2θ + i sin 2θ−i cos 2θ − sin 2θ

)=

(sin 2θcos 2θ

). (2.8)

The beam has now returned to a composition of linearly polarized components suchthat 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 andwhich to the probe beam: σ+ or σ−. In fact, it does not matter all that much. InChapter 1, we assumed that the probe field was resonant with the transition from


the lowest energy state and the control was resonant with that of the higher energyground state. However, as we saw in Section 2.1, when the magnetic field around therubidium cell is swept from positive to negative values, the two ground states tradeplaces. 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 twoquarter-wave plates and the rubidium cell were removed from the setup. This allowedthe 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. Thiscorresponds to having only y-polarized light. 11.39 mW of power was transmittedthrough the beamsplitter, and only 243 µW was reflected.

Then, one of the quarter-wave plates was put in place and adjusted until thetransmitted and reflected beams were as close to equal as possible. This alignment ofthe quarter-wave plate converts linear polarization to circular. The transmitted beamhad power 5.40 mW and the reflected beam had 5.79 mW. The first quarter-wave platewas removed and replaced by the remaining quarter-wave plate, and this process wasrepeated. This time, the transmitted power was 5.98 mW, and the reflected powerwas 5.40 mW.

Finally, after all of the wave plates were returned to their spots in the opticalsetup, the half-wave plate was readjusted such that the ratio of the power of thebeams transmitted and reflected through the polarizing beamsplitter was 9:1. Theirtrue 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 cellis placed within a Zeeman apparatus. A section and 3D rendering of the apparatusare displayed in Fig. 2.5. The cell is enclosed in an acrylic tube wrapped in a solenoidof copper wire. The total resistance of the coil is 14.62 Ω. For the experiment, a100 Ω external resistor was added to attain a finer absorption curve. The wire exitsthe apparatus at its end caps and is driven by a constant voltage source to induce amagnetic field within the solenoid along the apparatus’ axis. The solenoid is aboutthree times longer than the cell, while their radii are comparable, so near the centerof the apparatus, the induced magnetic field can be approximated as constant. Thiswill be shown more explicitly later in the section.

Outside of the solenoid are three cylindrical layers of mu metal. Mu metal is amaterial with a very high magnetic permeability for the purpose of shielding the cellfrom exterior magnetic fields. The shielding we used had a relative permeability4

of 80000 [21]. The geometry of the shielding — three concentric cylinders — waschosen because inserting gaps, in fact, makes the shielding much more effective. Thisconfiguration was also relatively simple to construct.

The effectiveness of the shielding was determined by measuring the magneticfield at various positions within the apparatus without current passing through the

4Relative permeability is a dimensionless quantity defined as µR = µ/µ0 where µ is the absolutepermeability and µ0 is the permeability of free space.


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 z HcmL

1

2

3

4

5

6

x HcmL

Figure 2.5: Section (top) and 3D rendering (bottom) of the Zeeman apparatus. Im-ages by Jay Ewing.


2 4 6 8 10 12z HcmL

-20

-10

10

20

30

40

50

BA HmGL

2 4 6 8 10 12z HcmL

50

100

150

200

250

BT HmGL

0.2 0.4 0.6 0.8 1.0 1.2 1.4r HcmL

-10

10

20

30

40

50

BA HmGL

0.2 0.4 0.6 0.8 1.0 1.2 1.4r HcmL

50

100

150

200

250

BT HmGL

Figure 2.6: Magnetic field attenuated by mu metal shielding inside the Zeeman ap-paratus as a function of cylindrical coordinates r and z. The left two plots show theaxial field BA while the right two show the transverse field BT . The dashed line is av-erage value of the magnetic field strength in the solenoid. The solid lines correspondto the magnetic field outside the mu metal shielding.

solenoid. Axial (along the axis of the apparatus) and transverse fields — BA andBT , respectively — were measured as functions of the cylindrical coordinates r andz, with the origin chosen to be at the center of the apparatus. The results are plottedin Fig. 2.6. For reference, each graph also includes a solid line that corresponds tothe field outside the shielding.

In each case, the field is attenuated by a factor of about 5. The magnitude of themagnetic field of the Earth in our lab is on the order of 520 mG; assuming this is themain source of exterior fields, this mu metal shielding reduces the exterior field suchthat it is easily canceled by the applied field.

If the solenoid were infinitely long and perfectly aligned, the axial magnetic fieldinduced by flowing current would be

BA = µ0nI (2.9)

inside the solenoid, where n is the density of coils in the solenoid that the current Iflows through, and µ0 is the permeability of free space. The transverse field wouldbe zero. Of course, the solenoid is actually finite in length and may not be perfectlyaligned, so the field will vary within the apparatus. The field may also pick up anonzero transverse component. However, within a range close to the center of the


2 4 6 8 10z HcmL

6800

7200

7400

7600

7800

8000

8200

BA HmGL

2 4 6 8 10z HcmL

550

600

650

700

BT HmGL

0.2 0.4 0.6 0.8 1.0 1.2 1.4r HcmL

8040

8060

8070

8080

8090

BA HmGL

0.2 0.4 0.6 0.8 1.0 1.2 1.4r HcmL

520

540

560

580

BT HmGL

Figure 2.7: Magnetic field driven by a solenoid inside the Zeeman apparatus (withthe mu metal shielding in place). Field strength is plotted as a function of cylindricalcoordinates r and z with 300 mA flowing through the solenoid. The left two plotsshow the axial field BA while the right two show the transverse field BT . The dashedline in the upper right plot shows the uniform magnetic field inside an infinitely longsolenoid.

solenoid — where the cell sits — the magnetic field can be approximated by that ofan infinitely long solenoid.

The uniformity of the magnetic field induced by the current was also measuredas 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 areplotted 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 belittle 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 approximationto 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 parallelto the axial direction.


2.4 Procedural Overview

Most of the procedure has already been explained in the previous section’s discussionof the setup. This section will concisely summarize the steps taken in this experimentto obtain data.

First, the quarter-wave plates were adjusted so they convert linear polarizationsto circular and vice versa. Then, the half-wave plate was adjusted so that the ratioof powers of the control beam component of polarization and that of the probe beamwas 9:1.

Next, the current powering the laser was increased until reaching a suitable lasingpower, 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 positivevoltage triangle waveform. The range of frequencies in the scan was adjusted makingsmall changes in temperature and current in the laser, until infrared fluorescence inthe rubidium cell was observed indicating near-resonance.

Then, the saturated absorption spectrum — obtained in the difference of signalsobserved by the photodiodes — was viewed on an oscilloscope. The F = 1→ F ′ = 1peak was identified on the spectrum, and, by adjusting the DC offset on the piezocontroller, the scanning range was shifted such that the peak was near the low-frequency end of the spectrum. Then, the amplitude of the triangle waveform wasdecreased, effectively narrowing the frequency range and zooming in on the transitionpeak. This process was continued until the spectrum displayed just the peak at thelow-end end of the spectrum, and then the scan was stopped by turning off the outputof the signal generator.

Finally, the magnetic field in the Zeeman apparatus was turned on by running aconstant voltage across the solenoid. The transmittance of the probe component ofthe beam was then recorded as a function of the voltage across the solenoid. Sincethe resistance of the solenoid is known, its current can easily be determined from thevoltage 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 ′ = 1 hyperfine transition of the D2

line in 87Rb in the presence of a magnetic field. The resulting data is discussed in thischapter, 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 vari-ety of voltages applied to the solenoid. Solenoid voltage is not, however, a preferableor intuitive independent variable in this analysis. Therefore, voltage data was con-verted to current using Ohm’s law. The induced magnetic field was then calculatedfrom the current; since the field was not perfectly uniform within the cell, the axialfield of an infinite solenoid from Eq. 2.9 was used. Five runs of data were taken thisway, and the power measurements were averaged across all the runs. Each data pointwas subtracted from the transmission power through the medium at a frequency farfrom 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 trans-parency window induced by EIT. We do not see the decrease in absorption far fromresonance because the magnetic fields in this experiment were not large enough tomove 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 exper-iment. 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-levelΛ scheme. The scheme in our experiment, created from the F = 1 → F ′ = 1 transi-tions, was a six-level system with both Λ-type and V-type structure. Therefore, whilethe dressed state analysis is sufficient in predicting that EIT exists for the six-levelsystem, there are some subtle differences.

For example, we assumed that, in the three-level system, the decoherence termγ2 was very small. When γ2 becomes large, the transparency window gets shallowerand shallower, as shown in Fig. 3.2. What would the analogous term in the six-

1The 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

-10 -5 0 5 10

6.0

6.5

7.0

7.5

8.0

-300 -200 -100 0 100 200 300

I HmAL

PHΜ

WL

B HmGL

Figure 3.1: Experimental EIT absorption of 87Rb. The horizontal axis has two scales,one for current and one for magnetic field strength.

DpΓ3

ImH ΧpL

DpΓ3

ReH ΧpL

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 representhigher 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 theexcited state of the Λ scheme, then there are four decay rates we need to take intoconsideration: that of the metastable state of the Λ scheme and also of all three statesin the V scheme. Therefore, it seems there may be a much higher chance of decay outof the system, which would greatly reduce the magnitude of the transparency signal.

3.2 Possible Improvements

3.2.1 F = 1→ F ′ = 0 Transition of the D2 Line

The ideal energy scheme for Zeeman EIT would be made from a F = 1 → F ′ = 0hyperfine transition. The ground state would split into three levels while the excitedstate would not split at all. The Λ scheme similar to the one used in this experimentwould form but without the two extra excited states. When circularly polarized fieldswere applied, the mF = 0 state would not have anywhere to transition, thereforedecoupling it from the system and leaving a true three-level Λ structure. This wouldgreatly reduce the γ2 term, and the theory of three-level systems would be moreapplicable. Additionally, unwanted excitations into higher energy levels would begreatly reduced with this transition.

However, the F = 1→ F ′ = 0 hyperfine peak of the D2 line in 87Rb is very small.With the existing experimental setup, as well as time constraints, it was too difficultto park the laser onto the resonant frequency. However, by properly optimizingthe alignment of the saturated absorption setup, one could significantly improve theresolution of the hyperfine spectrum. In this case, it may be possible to utilize theF = 1→ F ′ = 0 transition to observe EIT.

3.2.2 D1 Line

The D1 line in 87Rb is much more frequently used in Zeeman EIT. D1 is resonant ata wavelength 794.98 nm, which is too high for the diode laser used in this experimentto 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 itsenergetic structure, it may be necessary repump atoms back into the ground state ofthe 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 frequencywould have to be readjusted using the saturated absorption setup. The time ofstability could be greatly increased by frequency locking the laser. This would involvesetting up an electronic feedback loop so that when the laser started to drift, it wouldbe able to automatically restabilize. This would increase the number of data pointsthat could be taken in a single run and minimize the the extraneous absorptive effectscaused 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 precisionjust high enough to see a clear EIT signal. However, to get such a result, we had toaverage 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 fieldstrength.

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 thesolenoid using a signal generator would allow us to see the signal on an oscilloscope —much like the way the saturated absorption spectra are viewed.

2For example, if the D1 line was used, and it exhibited a stronger EIT signal.3The voltage would have to be varied slowly enough that Ohm’s law still accurately predicted

the current in the solenoid.

Conclusion

This thesis aimed to show theoretically and experimentally the effects of electromag-netically induced transparency. We first showed two derivations of EIT in three-levelsystems: direct computation of the linearized electric susceptibility and the dressedstate analysis. We also used the dressed state picture to show that a six-level systemexhibits EIT.

We then introduced the Zeeman method for observing EIT. We showed thatthrough clever polarization, a Zeeman setup requires only a single beam to elicitEIT. We then used this setup to measure an absorption curve through warm rubid-ium vapor. The existence of a clear transparency window centered at the transitionresonance is evidence of EIT for an atomic medium in a magnetic field.

References

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[12] G. T. Purves, Absorption And Dispersion In Atomic Vapours: Applications ToInterferometery, Ph.d. thesis, Durham University (2006).

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42 References

[16] K. B. MacAdam, A. Steinbach, and C. Weiman, Am. J. Phys. 60, 12 (1992).

[17] F. G. Atkinson, Diode Laser Stabilization, Undergraduate thesis, Reed College(1996).

[18] E. Hecht, Optics, 4th ed. (Pearson Education, Inc., San Francisco, CA, 2002).

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[20] B. W. Worth, Experiments On Zeeman-Based Electromagnetically Induced Trans-parency And Optical Sensing In Turbid Media, Masters thesis, Miami University(2013).

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custommagneticshielding.magneticshield.com/Asset/mu-2.pdf.

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