Double Electromagnetically Induced Transparency

8/14/2019 Double Electromagnetically Induced Transparency

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UNIVERSITY OF CALGARY

Double Electromagnetically Induced Transparency

With Application to Nonlinear Optics at Low Light Levels

by

Andrew John MacRae

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF MASTER OF SCIENCE

DEPARTMENT OF PHYSICS AND ASTRONOMY

CALGARY, ALBERTA

July, 2008

c Andrew John MacRae 2008


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UNIVERSITY OF CALGARY

FACULTY OF GRADUATE STUDIES

The undersigned certify that they have read, and recommend to the Faculty of GraduateStudies for acceptance, a thesis entitled Science submitted by Andrew John MacRaein partial fulfillment of the requirements for the degree of MASTER OF SCIENCE.

Chairman, Dr. Alexander LvovskyDepartment of Physics and Astronomy

Chairman, Dr. Karl-Peter MarzlinDepartment of Physics and Astronomy

Chairman, Dr. Wolfgang TittelDepartment of Physics and Astronomy

Chairman, Dr. Sergey MoiseevKazan Physical-Technical Institute,Russian Academy of Sciences

Date

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Abstract

Electromagnetically Induced Transparency (EIT) has become an invaluable resource for

numerous applications in quantum information theory, fundamental physics, nonlinear

optics and precision metrology. This thesis is concerned with an extension to EIT: double-

EIT, in which two separate optical fields experience EIT due to a coherence between a

common ground state. Double EIT has appeared as the basis for a number of theoretical

proposals but has yet to be demonstrated and analyzed experimentally.

In this thesis, a double-EIT system is experimentally implemented and analyzed,

using a hot vapor of Rubidium-87 atoms. The interplay between the two signal fields isstudied and properties are found to emerge which may prove useful for application. An

application of double EIT which in principle allows large nonlinear interactions between

light pulses as low as several photons per atomic cross section is discussed. Progress

towards a physical demonstration of this effect is reported and prospects for further work

in this direction are then described.

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Table of Contents

Approval Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Underlying Physical Principles . . . . . . . . . . . . . . . . . . . . . . . . 32.1 Electromagnetically Induced Transparency . . . . . . . . . . . . . . . . . 3

2.1.1 Basic Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 Absorption and Dispersion Profiles . . . . . . . . . . . . . . . . . 52.1.3 Slow Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.4 The Storage of Light . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.5 Theory vs. Experiment . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Rubidium-87: The Physical System . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Atomic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.2 Interaction with a Magnetic Field: Zeeman Splitting . . . . . . . 162.2.3 The AC-Stark Shift . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Underlying Experimental Principles . . . . . . . . . . . . . . . . . . . . . 213.1 Stable Diode Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Laser Phase Lock Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3 Scanning and Pulsing the Light: Acousto-Optical Modulators . . . . . . 253.4 Housing the Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.5 Detecting the Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 Double Electromagnetically Induced Transparency . . . . . . . . . . . . . 324.1 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.1.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 334.1.2 CW DEIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.1.3 Simultaneous Slowdown of Co-Propagating Pulses . . . . . . . . . 384.1.4 Simultaneous Storage . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 A Theoretical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3 Further Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 Giant Optical Nonlinearities With DEIT . . . . . . . . . . . . . . . . . . 485.1 EIT-Based Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.2 Physical Implementation in 87Rb . . . . . . . . . . . . . . . . . . . . . . 51

5.3 Experimental Observations and Progress . . . . . . . . . . . . . . . . . . 525.4 Prospects for Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

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List of Figures

2.1 Atomic level scheme for EIT . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Absorption and dispersion profiles associated with EIT. . . . . . . . . . . 7

2.3 Experimental plot of the slow light effect . . . . . . . . . . . . . . . . . . 82.4 EIT absorption in the frequency domain - pulse distortion occurs as aresult of asymmetric absorption. . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Storage of light using EIT. . . . . . . . . . . . . . . . . . . . . . . . . . . 112.6 Experimental EIT dispersion and absorption profiles. . . . . . . . . . . . 132.7 Overview of the atomic system used in the following experiments. Exper-

iments were performed on the D1 transition at 794.979nm. . . . . . . . . 152.8 Splitting of the degenerate Zeeman sublevels with applied field. . . . . . 19

3.1 Diode lasers used in the experiments. . . . . . . . . . . . . . . . . . . . . 223.2 Block diagram of the optical phase-locked loop used in the experiments . 24

3.3 Beat frrequency between locked lasers . . . . . . . . . . . . . . . . . . . . 263.4 AOM driver block diagram. . . . . . . . . . . . . . . . . . . . . . . . . . 273.5 Delay generator designed for the experiments . . . . . . . . . . . . . . . . 283.6 Phase detection scheme for the experiment . . . . . . . . . . . . . . . . . 30

4.1 Atomic scheme used in experiment. Observation of simultaneous EITtransparency dips. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2 Experimental setup used to realize double EIT. . . . . . . . . . . . . . . 354.3 Zeeman EIT for various transitions. . . . . . . . . . . . . . . . . . . . . . 364.4 Cross-talk between EIT signal fields. . . . . . . . . . . . . . . . . . . . . 374.5 Simultaneous slow light with no atomic preparation. . . . . . . . . . . . . 39

4.6 Simultaneous group velocity of the pulses as a function of pump power.Matched Pulses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.7 Simultaneous storage and retrieval of two signal pulses using a single pumpfield. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.8 Storage run optimized for the hyperfine field. . . . . . . . . . . . . . . . . 434.9 The theoretical tripod scheme analyzed in the text. . . . . . . . . . . . . 444.10 Theoretical prediction and experimental observation of optical pumping

effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.1 N-type atomic level scheme proposed by Schmidt and Imamoglu. Off-resonant coupling to |s creates a Stark shift leading to a large shift in the

phase of the signal. The inset shows a typical non-EIT XPM scheme forwhich the coefficient is orders of magnitude less. . . . . . . . . . . . . . 49

5.2 Atomic system used to explore large XPM. . . . . . . . . . . . . . . . . . 515.3 Splitting of EIT lines with applied magnetic field . . . . . . . . . . . . . 535.4 Deviation of 2-photon resonance of the Stark field EIT as a result of the

nonlinear Zeeman shift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.5 The phase of the hyperfine pulse with and without the Stark Field . . . . 56

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Chapter 1

Introduction

In recent years, advances in the theoretical understanding of quantum optics, has lead to

the prospect of using quantum properties of light for the construction of quantum com-

puters [1], the encryption of sensitive information [2], and ultra-sensitive measurements

of fundamental parameters[3]. In parallel with the theoretical progress, new experimen-

tal techniques and technologies have made the implementation of these ideas feasable.

Amoung these advances is the Electromagnetically Induced Transparency (EIT) effect[4]. EIT allows for the ultra-slow propagation of a light pulse with minimal loss [5],

and the temporary storage of classical [6] or non-classical light[7]. As this progress has

continued in classical and quantum optical technology, several proposals seek to further

exploit the EIT effect to control the relative properties of separate light pulses using mul-

tiple EIT systems[8, 9]. In particualr, double EIT which involves the EIT efect on two

separate signal fields, has been proposed to realize large optical nonlinearities between

single photons [10], all optical light buffering [11], and quantum logic gates for use in a

quantum computer [12].

One of the benefits of using light as a carrier of quantum information is its ability to

travel (as quickly as nature will allow) with minimal unwanted interaction with the envi-

ronment. However, given the robustness of light against cross-talk with outside sources

and its tendancy to propagate, it is difficult to store and manipulate this information

soley with another light channel. In order to overcome this challenge, nonlinear inter-

actions requiring light levels as low as several photons per pulse, have been proposed

in which an ensemble of atoms mediate the effect of one light pulse on another. In the

following chapters, a scheme is explored which shows promise towards the realization of

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such an effect.

Chapter 2 outlines the theoretical concepts addressed in the remainder of the thesis.

The formalism of EIT is first briefly presented and its use in creating slow and stored

light is described. A description of the atomic system used to perform the experiments is

then given and the relevent features of the interaction of the atomic sytem with external

fields is outlined.

In chapter 3, the technical details of the experiment are layed out. Since the majority

of the work of an experimental project lies in developing techniques and technology to

measure physical parameters, this chapter outlines the bulk of the effort to obtain the

results described in the remaining portion of the thesis.The main results of the thesis are presented in chapter four. The observation of

double EIT is first reported and a full study of the interplay between the fields is given.

The slow light dynamics are explored and with the use of preparatory pulses, the group

velocity of the slowed light is manipulated and mathced. The simultaneous storage of

two seperate light pulses is described. A simple model of a four-level double EIT scheme

is analyzed and shown to display the behaviour examined in the experiment.

Finally, chapter five discusses the possible use of this system to exhibit a previously

proposed [9] EIT-based nonlinear scheme. Progress towards this goal is reported and

further prospects are explored.


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Chapter 2

Underlying Physical Principles

The purpose of this chapter is to give a theoretical description of the physical effects

relevant to the work presented in this thesis. A full description of EIT itself could

fill a large chapter in a textbook and still be lacking in completeness, however such a

description certainly does not belong here. The purpose of this chapter is therefore not

to reinvent the wheel, but rather to provide a picture of the effects specific to this

thesis. For a full descpription, refer to [13].This chapter begins with a description of the cornerstone of this work: electromag-

netically induced transparency, including a comparison of what is normally presented

in textbooks, with what is observed in a typical experiment. Next a description of the

atomic system used is given, including its interaction with electromagnetic fields.

2.1 Electromagnetically Induced Transparency

The primary physical principle underlying the research described in this thesis is the

Electromagnetically Induced Transparency (EIT) effect. In EIT, the optical response

of an atomic medium is modified by a coherence induced via applied light fields. The

physical basis of this effect is quantum interference between excitation pathways.

2.1.1 Basic Formalism

The EIT effect can be understood in terms of the phenomenon of a dark state, first used to

describe coherent population trapping [14]. To address the concept of a dark state, con-

sider and atom in a lambda-type scheme which is coupled to light fields with Rabi frequen-

cies p and s (Fig. 2.1). Assuming the dipole and rotating wave approximations[15],

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Figure 2.1: Atomic level scheme used to demonstrate CPT and EIT.

the full Hamiltonian for this system in the absence of decoherence is given by

H =h

2

0 0 1

0 2(1 2) 2

1 2 22

. (2.1)

where i is the detuning from state |i. The eigenstates can be shown to be:

|+ = sinsin |1 + cos |e + cossin |2 (2.2)

| = sincos |1 sin |e + coscos |2 (2.3)

|d = sin |1 cos |2 (2.4)

where = tan1(s/p), and = tan1 2p +

2s/

/2 are known as the mixing

angles. Of particular interest is the dark state |d, a coherent superposition of the

ground states which contains no |e term, and therefore does not couple to the excited

state. In other words, if the atoms are prepared in this state, they will be transparent to

a light field resonant to the |d |e transition.


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The idea behind EIT is to efficiently drive the atomic system into this dark state using

a perturbative signal field for s and a strong coupling field for p. Under these condi-

tions, the majority of the population is in the signal ground state (11 1), cos 1,

and sin 0 so that |d |1, i.e. the system population is driven into a dark state

from which the weak signal field couples. Under these conditions, the signal field experi-

ences transparency. The quantum mechanical basis for this transparency is a destructive

interference of the probablity of excitation pathways: when the atoms are in this dark

state superposition, the probabilty amplitude of the pump exciting an atom from |1 is

equal and opposite to that of the signal exciting from state |2.

2.1.2 Absorption and Dispersion Profiles

The atomic response to an optical field is described by the susceptibility . For the

present discussion, we are interested in the linear susceptibility. However a key feature of

EIT is that the vanishing linear susceptibility which eliminates absorption is accompanied

by a significant higher order term (3) [13] - a fact that will be exploited when discussing

the large Kerr effect in chapter 5.

In order to obtain an expression for , the master equation for the density matrix is

solved in the steady-state ( = 0.) The master equation includes spontaneous emission

e1,e2 from |e to |1 and |2 as well as ground-state dephasing, 12 on |1 and |2. Defin-

ing the super-operators: LA [, A], and Lrelax(ab) ab/2 (2 |b a| |a b| |a a| |a

Solving the master equation in the steady state then requires that we solve the algebraic

equation:

0 =1

ihLH+ LSE(e1) + LSE(e2) + LSE(11) + LSE(22), (2.5)

from which we determine the density matrix .

In order to relate the susceptibility to the density matrix elements in 2.5, first note


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that for a linear dielectric such as a dilute gas, we can write the atomic polarization as:

P = 0E. (2.6)

Furthermore, the atomic polarization is defined as the mean dipole moment per unit

volume so that, in the density matrix formalism;

P =N

Vd = Tr

d

(2.7)

where N is the number of relevant atoms, V is the effective volume, = N/V is the

atomic density, and d is the dipole operator. For a given transition from state |i to |j,

the dipole operator is given by dij = ji |j i| where ji is the electronic dipole element for

the |i |j transition. Using this in equation 2.7 and dividing out the Rabi frequency

(h E) gives:

= |ij|

2

0hij + c.c.. (2.8)

For a single atom, equation 2.8 would suffice. For an optical medium such as an

atomic vapor, the transmission is described by the amplitude transfer function [13]:

T(, z) = eikz()/2 (2.9)

where k = /2 is the wave-number of the transmitted light. From this, it is apparent

that the imaginary portion of gives rise to absorption, whereas the real component leads

to dispersion.

Inserting the off-diagonal density matrix from equation 2.5 solved to first order, and

identifying terms which rotate at exp[ipt] gives the rather cumbersome equation:


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Figure 2.2: Imaginary and real components of the susceptibility in equation 2.8 cor-responding to the absorption and dispersion profiles respectively. The steep dispersionleads to slow light (see next section) while the vanishing absorption allows for the losslesstransmission of this light.

=|13|

2

0h

4

|c|2 4

4221 + i

8231 + 221

|c|

2 + 2131

|c|2 + (31 + 2i)(31 + 2i)2

. (2.10)

Figure 2.2 displays the real and imaginary components of susceptibility corresponding

to dispersion and absorption profiles respectively in the absence of decoherence. Note

that with no decoherence, perfect transparency is always attained for = 0. The presence

of decoherence lessens the effect of transparency which can be partially compensated for

by driving the atoms harder with a stronger pump field. Note as well that the FWHM of

the EIT signal is proportional to the strength of the pump field |p| [16]. This provides

a simple method for manipulating bandwidth for the EIT system, which is important for

the applications discussed in the following subsections.

2.1.3 Slow Light

The steep dispersion curve associated with EIT (Fig. 2.2) has a profound impact for

pulsed light, which by nature is spread out in frequency. Specifically, since the group


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0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [s]

N

ormilizedSignalatDetector[a.u

]

Pulse with no atoms

Slowed pulse

Figure 2.3: Slow light effect on the hyperfine channel. The square trace show the originalpulse obtained in the absence of the atomic gas, the distorted pulse is measured after

placing the atoms in the beam path. The distortion is a result of the narrowness of theEIT window in this configuration ( 100kHz.) The delay of the pulse corresponds to agroup velocity of 33km/s and a pulse compression of 104 (300m 3cm.)

velocity of a pulse of light is defined as

vg d

dk= c [n() + (dn/d)]1 , (2.11)

the steep dispersion curve leads to a large dn/d in the denominator, leading to ultra-slow

light pulses. Group velocities on the order of 10m/s [5] have been obtained, a fact which

is worth a second thought - one can ride a bike faster than a pulse of light progagates in

an EIT medium!

Figure 2.3 displays the slow light observed in our experiment in which a 1 s pulse is


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delayed as is passes through a 10cm cell, corresponding to a group velocity of 33km/s.

A often quoted character of slow light with EIT is that whereas the group velocity

(vg dn/d) 1, the group velocity dispersion d2n/d2 is vanishing. This means

that in principle, there is no distortion of the waveform as it propagates through the

EIT medium. A quick glance at Fig. 2.3 shows that this is not the case in the above

experiment. The reason for this lies in a fundamental tradeoff off an EIT system: one

would like the for the pulse to fit spatially in the cell, requiring a large group velocity

reduction and a short pulse. However the group velocity of the pulse increases with the

width of the EIT window and the spectral bandwidth of a pulse is inversely proportional

to the temporal length. Therefore, compressing the pulse spatially increases the pulsespectrally and vice versa, so a happy medium of must be found. In the case of the pulse

shown above, the large group velocity reduction occurred at the expense of the absorption

of the side bands (see Fig. 2.4.)

2.1.4 The Storage of Light

As a pulse of light propagates through an EIT medium, it becomes compressed by a

factor of c/vg. During this propagation, the interaction of the light with the atomic en-

semble forms a spin wave which propagates at velocity vg . The light-spin subspace form a

quasi-particle known as a dark-state polariton [17]. If the pump field is switched off adi-

abatically, the polariton becomes a pure spin wave (between the two EIT ground states),

which maintains the coherent properties of the light pulse, but does not propagate. If the

field is switched back on adiabatically, the dark-state polariton accelerates once again,

and the original light pulse is recovered from the atoms. In this way, light may be stored

in atoms and storage times of over 1 ms have been reported[6], as well as the storage of

quantum light [7, 18, 19]. It should be noted however, that the retrieval efficiency of a

stored light pulse is limited due to decoherence of the atoms, and storage efficiencies are


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Figure 2.4: Transmission through a narrow EIT medium results in pulse distortion.Inset: Fourier transform of the square pulses used in the experiments is a sinc function.Transmission through the Lorentzian EIT profile with FWHM much less than the the sincsideband leads to their attenuation. The retrieved pulse is no longer a top hat functionin the time domain.


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Figure 2.5: Typical light storage run in the experiment. Dotted curve shows the switchingsequence of the control field.

typically no higher than 25%. Figure 2.5 shows a typical storage run obtained in the lab.

2.1.5 Theory vs. Experiment

In the literature, it is common to see the EIT dispersion curves plotted as in figure 2.2

which are for a single atom in the absence of decoherence. In practice however, several

forms of decoherence are always present. Furthermore, in a hot atomic vapor, the EIT

effect occurres over an ensemble of atoms which are Doppler broadened. Additionally,

the condition of one-photon detuning = 0 is rarely met precisely. Each of these effects

conspire to give absorption and dispersion profiles which differ markedly from that of a

decoherence-free single atom.

With the presence of decoherence, the ground state coherence which causes EIT is nolonger pure, and total transparency is never obtained at two-photon resonance. By using

a sufficiently strong pump however, the transparence at resonance can be improved, but

at the the expense of a less steep dispersion curve. If decoherence rates become too large,


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the dark state can not be induced at all, and the EIT effect will not occur [13].

In a hot atomic vapor, the random motion of the atoms causes a distribution of

Doppler shifts, so that each atom has a velocity-dependant resonant frequency with

respect to the incoming light. Since the probe and pump beams in a typical EIT setup

are co-propagating, they see the same velocity distribution in the atoms and therefore

see the same relative (two-photon) detuning. Although the two photon resonance is

independent of the Doppler broadening, the one photon detuning is not. With =

0, noticeable asymmetry is present in the EIT linewidth. The absorption profile is a

convolution of the profile of a single atom with a Boltzmann distribution having width

=

8kT/mc

2

0, where 0 is the resonant wavelength of the atomic transition. Thishas the effect of significantly narrowing the EIT linewidth since the FWHM of the EIT

line now depends on the inverse Doppler linewidth WD rather than the inverse natural

linewidth [16]. Since WD/ 100, the EIT linewidth is considerably narrower in a

Doppler Broadened system. Figure 2.6 displays the absorption and dispersion profiles

present in the laboratory.

2.2 Rubidium-87: The Physical System

The experiments described in this work involve the interaction of light fields with a

dilute gas of 87Rb atoms. Rubidium is an Alkali metal having a single valence electron

orbiting a filled shell. Such Hydrogenic atoms have the favorable property that they are

affected by a (spin-independent) central potential V(r) and as a result, may be described

in the well known Hydrogen atom formalism [20]. In addition to simplicity of analysis,87Rb has transitions in the neighborhood of 800nm, a wavelength accessible by many

commercially available diode lasers. The particular atomic transitions addressed in this

thesis are sublevels of the 5S 5P transition which are due to fields internal to the


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1000 800 600 400 200 0 200 400 600 800 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

TwoPhoton Detuning [kHz]

Signalfromd

etector[a.u.]

Phase of Signal Field

Absorption of Signal Field

Figure 2.6: Phase and transmission of signal field under EIT is monitored while scanningover 2-photon resonance.


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atom. These levels experience additional energy shifts due to the presence of static

and time-varying electromagnetic fields. The following subsection describes the precise

atomic level addressed by the light fields used in the experiments.

2.2.1 Atomic Structure

Due to the coupling of the electronic orbital momentum L with the electronic spin S

(so called L S coupling), this single spectral line exhibits fine structure and is split

into several lines. The number of distinct energies due to the fine structure splitting is

determined by the standard addition of angular momentum [21]. Using J = L + S as

our total angular momentum:

|L S| J L + S. (2.12)

For the S-shell, L = 0 and since S = 1/2 for the electron, there is only one possible

value for J, namely J = 1/2. For the P-shell (L = 1), J = 1/2 or J = 3/2 are each valid

and so the 5S 5P transition is split into a fine structure doublet: the D1 transition

(52S1/2 52P1/2), and the D2 transition (5

2S1/2 52P3/2). Since these transitions are

separated by over 1THz (the D1 transition occurs at 794.979nm and the D2 at 780.241nm)

they are often treated separately. The remainder of this chapter will deal with the D1

transition.

Just as the interaction of the orbital momentum with the electronic spin angular

momentum leads to fine-structure within the lines, the interaction of the nuclear spin I

with the electronic spin yields an additional energy shifts leading to hyperfine structure.

Just as before, total angular momentum F = J + I is used as in equation 2.12, giving

|J I| F J + I. For Rubidium, I = 3/2 and since, for both shells on the D1

transition. J = 1/2, we have the possibility to use F = 1 or F = 2. Note that for the

D2 transition, we can also have J = 3/2, yielding F = 0, F = 1, F = 2, or F = 3 for


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Figure 2.7: Overview of the atomic system used in the following experiments. Experi-ments were performed on the D1 transition at 794.979nm.

the P-shell. This is one advantage of using the D1 transition in an experiment; there are

simply less levels to worry about!

To get a value for the hyperfine splitting, one uses first order perturbation theory

with the hydrogenic Hamiltonian experiencing a perturbation in the form of a magnetic

field due to the dipole moment of the nucleus. The first order correction gives:

hf =52Lj

2[F(F + 1) I(I + 1) J(J + 1)] =

4F(F + 1) 18

452Lj (2.13)

where 52Lj is the magnetic dipole constant for the particular state. Numerically,

52S1/2/h = 3.417GHz and 52P1/2/h = 408.3MHz [22]. Plugging in these values, we find

that the S-shell of the D1 transition is split into two lines separated by 6.834GHz and

the P-shell is split into two lines 816.7MHz apart. The structure described above is

summarized in figure 2.7.


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2.2.2 Interaction with a Magnetic Field: Zeeman Splitting

The previous subsection described the 5S 5P transition which actually contains fine

structure sub-levels, each of which in turn, contain hyperfine sublevels. To go one step

further down the rabbit hole[23], each of the hyperfine levels F contains 2F + 1 mag-

netic sublevels labeled by their magnetic number mF. The degeneracy of the fine and

hyperfine sub-levels is lifted by magnetic fields dynamically generated within the atom.

The magnetic sublevels become non-degenerate in the presence of an external magnetic

field B. In the experiments described in this thesis, the atoms are placed in a uniform,

time independent magnetic field aligned with the beam path which is chosen to be the

axis of quantization: B(t) = Bz. The effective Hamiltonian describing the interaction of

the atoms with this magnetic field is:

HB =Bh

(gSS + gLL + gII) B =BB

h(gSSz + gLLz + gIIz) (2.14)

where B = h(1.3996)MHz/G is the Bohr magneton, and gS,gJ, and gI are the spin,

orbital, and nuclear g-factors respectively. If the splitting due to the magnetic field is

small due to the fine structure splitting ( 1THz here), then J is a good quantum number

and we can write

HB =BB

h(gJJz + gIIz) (2.15)

where gJ is the Lande g-factor (gJ(52S1/2) = 2.0023 2, and gJ(5

2P1/2) = 0.6666

2/3.) The calculation of Zeeman splitting in the weak or strong field regimes is a simple

excersise in perturbation theory. For the weak case, the splitting is small compared to

the hyperfine splitting and so 2.15 can be written as HB =BBh

(gFFz). First order

perturbation theory then gives the correction: Eweak =gFBBh Fz =

gFBBh mF, i.e.

the splitting is linear in magnetic number mF. On the other hand if the field is strong


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so that the hyperfine splitting is small compared to the induced splitting, then Jz is

kept and the splitting is (neglecting the nuclear spin - a correction of less that 0.1%)

linear in mJ: Estrong =gFBBh

mJ. Note that this result is somewhat counterintuitive:

for weak fields, the lines are split into 2([2] + 1) + 2([1] + 1) = 8 distinct lines ( F = 2

has mF = 2, 1, 0, 1, 2, F = 2 has mF = 1, 0, 1) but as the strength of this field

increases, these energy levels collect into two distinct levels (mJ = 1/2 or +1/2.) This

transition occurs in the intermediate field where neither the external nor internal field

dominates. Naturally, the calculation of energy splitting requires more effort than in the

perturbative cases. Unfortunately, it is this intermediate field regime which is required

for the experiments described in chapter 5 and therefore, the calculation must be carriedout.

The approach will be to diagonalize the Hamiltonian including the hyperfine and

Zeeman interaction in order to determine the energy splitting. The full Hamiltonian is:

Hhfs+ HB = 52LjI J+gJBB

hJz = 52Lj (Ix Jx + Iy Jy + Iz Jz) +

gJBB

hJz (2.16)

where the additional offset due to the nuclear spin has been neglected, since gI/gJ

1. The product of the J and I operators is understood to be the tensor product, so

that Jz 14 Jz, where 14 is the 4-dimensional identity matrix. Using the Pauli spin

matrices for J and their spin 3/2 analogs for I, and working in the (Jz, mJ, Iz, mI) basis,

we have (with 52Lj and = BgI/h for brevity):


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18

34 +

2B 0 0 0 0 0 0 0

0 34

+ 2

B

32

0 0 0 0 0

0 324 +

2B 0 0 0 0 0

0 0 0 4

2

B 0 0 0

0 0 0 4 +2 B 0 0 0

0 0 0 0 0 4 2B

32 0

0 0 0 0 032

34

+ 2

B 0

0 0 0 0 0 0 0 34 +2B

(2.17)

Since this matrix is block diagonal, its eigenvalues are straightforward:

E1,2 =3

4

2B

E3,4 = 1

4

a 2

(2)2 + (B)2

E5,6 = 1

4

2

(2)2 2B + (B)2

E7,8 = 14

2

(2)2 + 2B + (B)2

.

Figure 2.8 displays the energy lines as a function of applied field.

2.2.3 The AC-Stark Shift

The final relevant physical effect which will be discussed in this chapter is the AC Stark

shift. Consider a two level atom coupled off resonantly to a light field. The bare-state

Hamiltonian gives energy eigenvalues corresponding to the energy difference between the

atomic levels. Turning on an off-resonant coupling field between the levels introduces

transition terms to the Hamiltonian. Since we have now modified the Hamiltonian, it

is reasonable to expect that the energy eigenvalues have changed, in turn changing the


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Figure 2.8: Splitting of the degenerate Zeeman sublevels with applied field. Note that inthe weak field approximation, the splitting is linear in mf and in the strong field regime,linear in mj .)


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20

spacing between the two levels - an effect known as the AC-Stark shift.

A full analysis of this effect is given in [24], but a simple expression for the magnitude

of the energy shift E can be arrived at by considering a far off-resonant field so that the

transition probability is small, and using second-order perturbation theory. Recall that

the second order correction to the energy of a system with Hamiltonian experiencing a

small perturbation His given by:

E(2) =i=j

|i| H |j|2

Ei Ej(2.18)

where Ek terms correspond to the unperturbed eigen-energies. In the case of off

resonant coupling, the perturbative Hamiltonian is given by h12/2 |1 2|+c.c.. Denoting

E2 E1 h, and inserting into 2.18 gives the AC-Stark shift:

EStark =h ||2

. (2.19)


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Chapter 3

Underlying Experimental Principles

In this chapter, the primary tools required to perform the experiments are described.

Since a large part of the work performed during the past 18 months involved the design,

construction, and optimization of the techniques and devices necessary to accurately

measure the physical effects of interest, this thesis would not be complete without a

summary of the main aspects of the experimental implementation.

3.1 Stable Diode Lasers

A quantum optics experiment such as described in this thesis would not be possible

without a high quality light source. It should provide a single mode collimated beam, be

tunable over a wide range of frequencies, while at the same time have a narrow range of

fluctuations from its central frequency. At the same time, since funding is unfortunately

never an inexhaustible resource, it should not be prohibitively expensive. In order to

meet each of these requirements, external-cavity diode lasers (ECDL) by on a design

described in [25] were constructed for use in this experiment.

The basic components of the ECDL are displayed in figure 3.1. The laser diode

is formed by a PN-junction semiconductor which emits light over its relatively wide

( 1THz) gain profile whose center frequency at a given temperature depends on the

injection current. The surfaces of the diode form a Fabry-Perot cavity, which has acomb of resonant frequencies giving a number of spectral modes, each having short term

linewidth on the order of tens of MHz. The purpose of the external cavity is to select a

single mode and further reduce the bandwidth by two orders of magnitude. The external

21


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Figure 3.1: (a) Sketch of the laser diode system used in this experiment: External cavityis formed between diode and first order reflection from grating. Output frequency is set

by holding temperature constant and modulating both the external cavity length andthe injection current.(b) Photograph of one of the ECDLs in the lab.

cavity consists of a reflective, blazed diffraction grating: the first order couples back into

the laser diode forming a cavity, and the zeroth order is reflected to an out-coupling

mirror. The light which leaves the laser is at this point a stable and narrow-band source

suitable for use in an experiment.

In order to tune the output frequency of the laser, one may change either the tem-

perature of the diode, the injection current, or the length of the external cavity. Since a

change in temperature is a considerably slow process, the temperature which corresponds

to the neighborhood of the desired output frequency is determined ahead of time and

then held constant by active feedback using a Peltier TEC and a temperature sensor

for continuous measurement. For finer corrections, the length of the external cavity is

adjusted by a piezoelectric transducer which is attached to the grating mount. Using

this technique, the output frequencies can be scanned over a range of several GHz at a

bandwidth of several kHz. If this scan is made to be too large, the laser may hop modes

and will have to be re-calibrated to obtain single mode operation. Finally, for higher

bandwidth corrections, the injection current to laser diode may be modulated, allowing


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23

for a correction bandwidth of up to 1MHz, over a limited frequency range.

Putting the system together is a simple task but the initial alignment can be a hum-

bling experience. However, once this stage is complete, high quality operation is main-

tained with little further work. Self made diode lasers have a tendency to take on a

personality of their own and require a good deal of getting used to; so much so that our

lasers have acquired their own names: Hubert and Alvar.

3.2 Laser Phase Lock Loop

When a number of light fields coherently interact with an atomic ensemble, it is impera-

tive that each of these light sources are relatively in-phase. If each of these fields originate

from the same coherent source, then this condition is automatically satisfied. However,

if the required fields differ too greatly in frequency, they need to come from a separate

source. Even if each source is itself coherent, the relative phase between the two fields

may fluctuate so rapidly as to render their coherent interaction impossible. This problem

may be circumvented by locking the two sources together in phase by using a feedback

loop. This section describes the Laser Phase-Lock Loop (PLL) used to lock the ECDLs

in this experiment. The PLL constructed in this experiment was based on the design

by J. Appel, an earlier student in the research group. Full details will be described in a

publication currently under preparation.

A slave laser operating at frequency 1 is locked to a master laser at 2 when their

relative phase changes at fixed rate, specifically, the beat frequency 21. This is

especially important for experiments in EIT, where it is the relative (2-photon) detuningwhich is the crucial parameter. In order to monitor this beat frequency, the two signals

are interfered on a beam splitter and observed with a photodiode. Letting E1(t) = E1ei1t

and E2(t) = E2ei2t be the light fields in question, the signal observed at the photodiode


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Figure 3.2: Block diagram for the operation of the Optical PLL. The optical phase ismeasured as the AC-component of the interference between the lasers. A stable frequency

source provides the reference. The two signals are divided to a common frequency andpassed into an Analog Devices ADF 4107 PLL chip producing an error signal. The FETand external cavity feedback signals are then separately outcoupled

is proportional to |E1(t) + E2(t)|2, or

I = I0 + I1cos (t + ) (3.1)

where I0

= |E1|2 + |E

2|2 and I

1= 2|E

1E2|, and = Arg[E

1E22

]. In other words,

is measured directly as the AC component at the photo-detector. Since the hyperfine

levels in 87Rb are separated by 6.8GHz, a photodetector with at least 7GHz bandwidth

is required. Originally this was accomplished by using a New Focus 1577A diode, but the

by placing an additional filter at the input of teh PLL circuit, the Hamamatsu G-4176

proved to provide the same results with more than an order of magnitude reduction in

price.

The basic operation of the laser PLL is sketched in figure 3.2. The beat frequency

is placed at one end of the phase-frequency detector (PFD), and a stable reference is at

the other. The PFD operates as follows: the input signal is divided down by an integer

factor N, while the the reference is divided by an integer R. N and R are chosen so that


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25

/N = ref/R. The frequency divided signals are then locked in phase by comparing

them at a charge pump with a small but finite dead time: if the divided reference signal

switches high before the divided optical signal, a positive voltage results. Similarly, if the

divided optical signal switches before the reference, a negative voltage is produced. After

a brief programable dead time, the charge pump is reset and begins listening again. This

forms the error signal for the feedback loop: if the signals begin to change too quickly

or slowly, a correcting signal is applied to the ECDL. The error signal is split up into

low and high frequency components which are sent to the ECDL as piezoelectric and

injection current modulation respectively. When the the relative optical phase occurs at

Nref/R, the error signal is zero, and the lasers are phase locked. Using this technique,the relative phase fluctuations between the two lasers can be reduced to a few tens of

Hz. Without such a phase lock, EIT using two separate sources would not be possible.

Figure 3.3 displays the beat frequency spectrum between the two lasers with the PLL

applied.

3.3 Scanning and Pulsing the Light: Acousto-Optical Modulators

While the injection current and piezoelectric-controlled grating allow the frequency of the

laser to be roughly set, they do not suffice to fine tune and scan the frequency on the kHz

level. Furthermore one often requires short ( 1s) pulses with rise time of 100ns or less,

which is not possible by using the laser controller alone (the transient behavior present

when switching on a diode laser is on the order of several seconds.) In order to achieve the

required frequency and amplitude modulation one can use an acousto-optical modulator(AOM). An AOM is a crystal with a piezo-electric modulator attached to one side.

Applying a periodic signal to the piezo with frequency sends traveling waves across

the crystal creating a moving, periodic phase-grating. Light at frequency 0 incident on


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26

4 3 2 1 0 1 2 3 460

50

40

30

20

10

0

ff0

[MHz]

BandPower[dB]

Figure 3.3: With the use of an OPLL, the beat frequency between the optical beams isfixed to less than one part in 106. Here the lasers each at frequencies of 377THz, areset at a fixed difference of f0 = 6.75470GHz. The phase noise is effectively pushed awayfrom the carrier band. Note that the logarithmic scale used since on a linear scale, thesideband noise is unidentifiable as compared to the carrier band.


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Figure 3.4: Block diagram of the electronic driver used for the AOMs in the lab. Sincestable 10Mhz sources are abundundan, but higher frequency are modulations are requiredfor the AOMs, this driver multiplies the 10MHz input by a programable integer factor,the signal is then amplified to 30dBm and fed directly into the AOM. A high speed switchallows the signal to be turned on for a brief period, allowing the creation of pulses.

the crystal is then Bragg-scattered via a photon-phonon interaction creating spatially

separated sidebands at frequencies = 0 . Typically, only one of the spatially

separated sidebands is selected with an iris and then out-coupled to the experiment. By

modulating the frequency or amplitude of the signal passed to the AOM piezo, amplitude

frequency or amplitude modulation is performed on the light. Using this technique,

steady frequency scans of anywhere from 500Hz to 100MHz are possible, depending

on the bandwidth of the particular AOM. By sending a pulsed signal at the desired

frequency to the AOM, a pulse of light will emerge from the device which has a duration

limited only by the time required for the compressional wave to traverse the crystal:

min = Lcrystal/vsound 100ns.

In order to create the signal to be sent to the AOM, a switchable high-frequency

driver was designed to operate the 80MHz and 200MHz center frequency AOMs used

in this experiment. The block diagram of this driver is shown in figure 3.4. A 10MHz

stable reference input is multiplied by a programmable integer and send to a fast (4.5ns

rise time) switch. This allows a stable input to always be present while using a different

source to create the switching signal.

In order to precisely control the arrival time of the light pulses in the experiments, the


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Figure 3.5: Using two pairs of monostable multivibrators, one triggered on the risingedge and the other on the falling, a signal input pulse triggers two pulses of programablewidth, whose delay can be individually set. This proved to be an individual resource formatching the arrival time of the pulses to the atoms.

switching signals to the diver must be able to be independently controllable. To this

end, a triggerable delay generator was constructed which, when triggered produced an

output pulse of programmable width after a programmable delay into both of its output

channels. This delay generator is summarized in figure 3.5.

3.4 Housing the Atoms

The experiments described in this thesis were performed in Rb vapor at a temperature

ranging from 35oC to 70oC contained in a 10cm long, 2.5cm diameter glass tube. The

primary advantage of using an atomic vapor over a collection of cold/trapped atoms is

the minimal amount of preparation needed to perform experiments on the atoms. This

simplicity comes at the cost of higher decoherence rates and Doppler broadened atomic


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29

transitions as compared to a cold, trapped atom system such as a magneto-optical trap.

The vapor cell is housed in a large steel oven with a heater and temperature controller

keeping the system at a fixed temperature. In order avoid random Zeeman shifts of the

atoms energy levels, several layers of metal shielding surround the atomic cell in the

oven. Two large solenoids surround the oven as well allowing for the application of a

reasonably stable, uniform magnetic field along the axis of laser propagation.

3.5 Detecting the Light

It is safe to say that every experimental result presented in this thesis relied on the

detection of light; a few words should then be said about the method used to accomplish

this task. Since macroscopic amounts of light were used (1W of 795nm light corresponds

to a photon flux of 1012s1), simple Si photodetectors could be used in place of single

photon detectors or photo-multiplier tubes. In such a detector, the photovoltaic effect

transforms the energy from an incident light beam into an electrical signal. This is

accomplished through a small energy gap between the valence and conduction bands of

the detector: when the incident light has enough energy to transfer an electron from the

valence to conduction band, the accumulated charge results in a flow of current. Since

there is always electronic noise present in an electronic circuit, there is a minimum light

level detectable by a photo-detector, below which, the current due to the incident light

is indecipherable from the current due to noise.

There are two ways of looking at a signal coming into a detector: one can plot the

amplitude of the signal as a function of time (time-domain detection), or one can plot theinstantaneous amplitude of a given frequency range (frequency domain detection.) Figure

2.3 is an example of a time domain measurement and Fig. 3.3 shows a frequency domain

measurement. Since the typical frequency of an optical field is 1014Hz which is orders


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Figure 3.6: The phase detection system used in the experiment. The same frequencysource which drives the AOM is compared to the optical beat frequency between the mod-ulated and unmodulated light signals. Each signal is compared at a AD8302 RF/Phasedetector and the dc-output which correscorrespondse relative phase between the signalsis observed on a scope with a resolution of 17.5mrad/mV.

of magnitude faster than possible with electronics, the spectrum of a light signal cannot

be measured directly. To circumvent this problem, one can observe the beat frequency

spectrum of the optical signal of interest mixed with a local oscillator; a technique known

as heterodyne detection.

In addition to the amplitude of an optical signal, the phase is also of interest. For

example, one may be interested in the phase accumulated while passing through a non-

linear medium as compared to the case where this medium is not present. In chapter

five, a experiment which requires the measurement of the phase of short (1s) pulse of

light is described. A significant challenge to overcome was the determination of the op-

tical phase given only a brief pulse. In order to overcome this obstacle, a fast electronic

phase detector was designed and constructed capable of detecting the phase difference

between two signals oscillating at up to several GHz. The schematic for the phase de-

tection system developed is shown in figure 3.6. The heart of the system is a Analog

Devices AD8302 phase comparator chip which provides an output voltage proportional


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to the relative phase of its two inputs, with phase accuracy of 1 0/mV. Since the rise

time for a full swing of the output is 400ns, it is fast enough to accurately detect the

phase of pulses with duration on the order of 1s. The 200MHz phase reference is sent

to the phase detector as well as an AOM driving the signal of interest. The signal is

then mixed on a fast photodetector with a local oscillator taken from the signal before

passage through the AOM. The resultant 200MHz beat signal is then phase stable with

the reference which provides the input for the phase detector. The output channel then

gives a direct measure of the optical phase, modulo 2.

3.6 Summary

In this chapter, the primary components of the experiments described in the next two

chapters were described in detail. Specifically, an external cavity diode laser was con-

structed and a working laser was put into operation in order to provide the light sources.

A laser PLL was constructed and configured to lock these two lasers together at 6 .831GHz,

with sub-Hz precision. AOM drivers operating at 80MHz and 200MHz were designed

and constructed in order to scan and switch the light signals independently in order to

fine-tune the arrival of simultaneously created pulses, a delay generator was designed.

Finally a phase detector capable of detecting the optical phase of pulses as short as 1s

was designed, manufactured, and put into operation.


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Chapter 4

Double Electromagnetically Induced Transparency

At the beginning of this project, when the idea of demonstrating a EIT based nonlinearity

for pulsed light was first discussed, perhaps the biggest obstacle was the creation of a

double-EIT system - a task which had yet to be experimentally realized. Specifically,

it was not known whether the coherence present in one signal field would detrimentally

affect the other. Furthermore, if we were able to obtain the simultaneous slowdown of

the co-propagating pulses, would we be to manipulate the system in some way in order tomatch the group velocities of these pulses so that they may be of use in the nonlinearities

experiment. The first task was therefore to attempt to experimentally realize a double-

EIT scheme and use it to produce matched pulses. This endeavor turned out to be not

only challenging, but interesting in its own right.

As further motivation for studying double EIT, note that while EIT itself has become

a cornerstone of many methods for controlling optical fields, having applications ranging

from precision interferometry to all-optical buffers in classical and quantum information

networks, as well as forming the basis of a quantum memory for quantum computation.

However, several theoretical proposals have suggested Double EIT as a vehicle for extend-

ing the utility of EIT schemes by creating transparency conditions for two signal fields

simultaneously [9, 26, 12]. This provides the possibility for coherent control [27, 15, 8] and

nonlinear interaction between weak optical fields [9, 26]. Double EIT allows propagation

of the two signal fields with minimal loss, and increases the interaction time between

pulses due to group velocity reduction, making double EIT a promising candidate for

numerous applications in quantum computation and communication.

In this chapter, a double EIT system is thoroughly investigated including the inter-

32


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dependency of the two signal fields in both continuous-wave (CW) and pulsed cases. In

the CW case the effect of optical pumping due to the second signal field is investigated.

It is shown that the delay induced in the pulses can be adjusted and matched through

preparation of the atomic states. In addition to quantum computational gates, this tech-

nique could find application in quantum communication protocols that use frequency

multiplexed information channels, such as simultaneous quantum memory, simple qubit

operations and correction of time delays [11]. Finally, a theoretical model exploring the

double EIT system in the following experiments is investigated, and many of the key

experimental observations are reproduced. Many details of this chapter may be found in

[28].

4.1 Experimental Results

The double EIT system explored in this work is displayed in figure 4.1, operating on the

D1 transition of 87Rb (see chapter2.) The pump field is right-circularly polarized, and

couples the F = 2 F = 2 transition. The first signal field is of left circular polarization

and couples the same transition, except addresses a different Zeeman sublevel. For this

reason, we name this field, the Zeeman field. Similarly, the hyperfine field is of right

circular polarization and couples out of the second hyperfine ground level F = 1

F = 2. These particular transitions were chosen to form the tripod scheme due to their

applicability to a nonlinear optics scheme (see Chapter 5.) It should be noted that not

all combinations of atomic transitions give the same quality EIT. Figure 4.3 shows the

Zeeman EIT for various transitions.

4.1.1 Experimental Setup

The experimental setup is displayed in Fig. 4.2. The experiments were performed in a

12-cm atomic cell containing the 87Rb gas and 10 torr Ne buffer gas maintained at 45C.


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Figure 4.1: Left: the double EIT scheme is of tripod configuration formed from theZeeman sublevels of the hyperfine levels of the 87Rb D1 transition. The +-polarized

pump couples F=2 F=2, the + hyperfine signal field couples F=1 F=2, andthe Zeeman signal field is -polarized and couples F=2 F=2. Right: Double EITtransparency peaks were observed by scanning the pump-field detuning, simultaneouslyscanning 2-photon detuning for each signal field. Note: signal fields are separated infrequency here by 6.863 GHz

The fields were produced by two self-made external cavity diode lasers. As mentioned in

chapter 3, in order that the fields to act coherently on a single set of atoms, the lasers

were phase locked at 6.834 GHz. One of the lasers provided the pump and Zeeman fields,

while the other generated the hyperfine field and the local oscillator for its detection. The

pump and each of the signal fields passed through AOMs allowing them to be scanned

in frequency, or switched on or off independently. The beams were then spatially mode

matched and passed through a quarter-wave plate to convert from linear to circular

polarization. Typical laser beam power in these experiments amounted to 2 mW 6

mW for the pump, and 1W150W for the Zeeman and hyperfine fields. The beam

diameters in the cell were about 750m. After passage through the atomic medium,

another quarter-wave plate and a polarizing beam splitter (PBS) separated the fields

into two paths for detection; one with the Zeeman field, and the other containing the

hyperfine and pump fields.


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Figure 4.2: Experimental setup used to realize double EIT.

The Zeeman field was measured directly with a photodiode. However, the detection

of the hyperfine field proved to be more challenging - since the weak hyperfine field was

in the same spatial and polarization mode as the strong pump, it could not be spatially

separated and had be measured in the presence of the much stronger pump. To this

end heterodyne detection was employed, with a local oscillator differing in frequency by

200MHz, taken from before the hyperfine beam passes through the AOM. The resultant

beat frequency was observed with a spectrum analyzer in a time-resolved, zero frequency

span setting with a resolution of 200 ns.

4.1.2 CW DEIT

The first logical step in exploring a double EIT system was to look for the simultaneous

appearance of the EIT transparency dip on each signal field. This was accomplished by

scanning the frequency of the pump field, effectively scanning the two-photon detuning

of the Zeeman and hyperfine EIT systems simultaneously. Simultaneous transparency

peaks occurred as 2-photon resonance was met between each signal field and the pump


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1000 800 600 400 200 0 200 400 600 800 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2Photon Detuning [kHz]

Transmission

F=1>F=1

F=2>F=1

F=2>F=2

Figure 4.3: EIT in the Zeeman configuration for various transitions. Note the transitionF = 1 F = 2 is not present since no Zeeman EIT was found on this transition

as seen in figure 4.1. It was noted that the transparency at resonance was significantly

greater for the hyperfine EIT than for the Zeeman. This can be explained by the fact

that the Zeeman field couples to an excited level which is not part of a Lambda system

(mF = 1, see figure 4.1), whereas the hyperfine field does not, leading to absorption for

any atoms in the mF = 1 ground state. Additionally the decoherence rate between the

Zeeman ground state and pump is in general not equal to that of the hyperfine ground

state and pump and can in principle be much higher.

Once double EIT was observed, this first question to explore was the effect of one EIT

system on the other: could the tripod system be treated independently as two separate

EIT systems or is there some interplay between the two signals? To answer this, a single


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Figure 4.4: The effect of one signal field on the other in the DEIT system. In (a),the absorption spectrum of the hyperfine field, fixed at 50 W, is measured for various

powers of the Zeeman EIT signal. In (b), the Zeeman signal is of constant intensity andthe hyperfine power is varied.

EIT line was observed with the other field turned off at first. The second field, set to a

relatively low power was then switched on and its power was increased while monitoring

the EIT profile of the first field. The results of this experiment are displayed in figure 4.4.

The hyperfine field was set to 50W and the profile was observed for various Zeeman field

powers. It became immediately apparent that even for weak Zeeman fields (< 10W,)

there was a significant effect on the hyperfine EIT. Specifically, the EIT contrast (defined

as the ratio of absorption at 2-photon resonance divided by the absorption well off of

2-photon resonance) was considerable enhanced. Increasing the Zeeman signal strength

furthered the contrast enhancement without noticeably affecting the transparency at

two-photon resonance. Finally when the Zeeman field became much stronger than the

hyperfine field (> 150W) the two-photon resonant transparency began to decrease, due

to the decoherence from the non-perturbative presence of the Zeeman field which acted

as an additional pump [16]. The experiment was then repeated for the Zeeman field,

with the hyperfine field slowly increasing in intensity and a similar effect was noted (Fig.

4.4.)


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The above effect can be explained by optical pumping from one ground state to the

other. From the atomic level scheme (Fig. 4.1), we see that when the hyperfine field is

absent, the atomic population will collect in the F = 1 ground state. Turning on the

hyperfine field will now serve to repopulate the F = 2 ground state, specifically the m = 2

sublevel, increasing the effective atomic density of atoms experiencing EIT. Similarly,

with only the hyperfine field present, the |F = 2, m = 2 ground state will accumulate,

and turning on the Zeeman field will have the effect of increasing the number of atoms

in the F = 1 ground state. Thus the relative contrasts can be adjusted by selecting

appropriate powers for each field. The ability to manipulate the EIT lineshape proved

extremely useful when dealing with slow light as described in the next subsection.

4.1.3 Simultaneous Slowdown of Co-Propagating Pulses

Once DEIT had been observed in continuous wave (CW) regime, the next step was to

observe the slow light effect simultaneously on two pulses. The majority of the proposals

involving DEIT mention at the beginning of this chapter involve co-propagating pulses,

each traveling with identical group velocity. By sending a 1s pulse from the Zeeman

and hyperfine fields simultaneously through the EIT medium, each experienced a reduced

group velocity. However a significant problem was that while the slow light effect was

present for each field, there was a severe group velocity mismatch between the pulses (Fig.

4.5.) Furthermore, without the ability to control the relative group velocity between the

pulses, the simultaneous slowdown of two separate pulses observed in this experiment

would be of very limited applicability to the aforementioned proposals.

This initial limitation can be overcome by noting that the group velocity of a light

pulse is ultimately determined by the EIT lineshape - that is, the real component of the

linear susceptibility. In the previous section, it was found that by having a second field

present in the EIT system, the lineshape could be manipulated by changing the strength


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9 10 11 12 13 14 15 16 17 180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [s]

NormalizedSignalatDetector[au]

Zeeman Pulse: No Atoms

Zeeman Slowed Pulse

Hyperfine Pulse: No Atoms

Hyperfine Slowed Pulse

vg

= 298 km/sv

g= 33.0 km/s

Figure 4.5: Without atomic preparation pulses, the group velocities of the simultaneousslowed pulses are severely mismatched.


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of that field. Note that in the previous section, it was actually the imaginary component

of the susceptibility that was being manipulated, but the real component is related to

the imaginary through the Kronig-Kramers dispersion relations [29]:

[] =2

P0

[]

2 20d. (4.1)

To exploit the change in the dispersion and therefore the slow-light dynamics, a

preparatory pulse was sent in the hyperfine chanel and then switched off momentarily

before sending the simultaneous pulses through the atoms. By adjusting the strength and

duration of the preparatory pulse, the amount of optical pumping from one ground state

to another was increased or decreased. Since the group velocity of a pulse depends on the

atomic density in the relevant transition [5], optical pumping out of the hyperfine ground

state into the Zeeman ground state has the effect of increasing the group velocity of the

hyperfine pulse while reducing that of the Zeeman. Since the group velocity reduction was

initially less dramatic for the Zeeman pulse (Fig. 4.5,) the hyperfine channel preparatory

pulse had the effect of correcting for this initial discrepancy. Figure 4.6(a) shows the

results obtained by using a 500s hyperfine initial pulse with varying power followedby simultaneous 2s pulses in each signal field. In the neighborhood of Pprep = 55W,

the group velocities of the pulses are found to be matched at 155km/s. The solid lines

are produced from a numerical simulation of a tripod DEIT system (see next section.)

Figure 4.6(b) shows the signals at the detector for the 55W data point. Note that,

as mentioned before, the transparency at two photon resonance for the Zeeman EIT

is considerably less than that of the hyperfine EIT. Consequently, a transmitted slow

Zeeman pulse suffers considerably more absorption than the hyperfine pulse. For this

reason, the Zeeman pulse is scaled by a factor of 10 in the figure.


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Figure 4.6: (a) The group velocity of each pulse is shown as a function of preparatorypulse power. Note that the curves intersect at which point, the pulses are matched. (b)Shows the traces for the Pprep = 55W point.

4.1.4 Simultaneous Storage

After exploring the simultaneous slowdown of pulses with matched group velocities, then

next step was to switch off the control field during the propagation of these pulses in

order to store the pulses simultaneously and, (hopefully) retrieve the pulses a time later

by switching the control field back on. Figure 4.7 displays the results of a typical storage

run with a storage time of 10s. Storage times of 100s were routinely observed, butwith decreasing retrieval efficiency.

One of the main limitations of simultaneous storage is that the pulse compression

factor vg/c may not be small enough to have the pulse fit spatially in the cell. A 2s

pulse in free space has spatial length ct = 600m. For our 12cm cell, this requires

vg/c = 2 104 corresponding to vg = 20km/s, which is an order of magnitude lower

than obtained in the experimental setup. When a pulse does not fit entirely into the cell,

only the portion of the pulse which is within the cell at the instant that the pump field

is switched off is stored, the rest of which escapes. Thus aside from the loss of efficiency

due to decoherence in the atoms, another practical limitation is the pulse compression. It

should be noted that the hyperfine group velocity was capable of group velocity reductions


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Figure 4.7: Simultaneous storage and retrieval of two signal pulses using a single pumpfield.


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0 2 4 6 8 10 12 140

0.05

0.1

0.15

0.2

0.25

0.3

Time[s]

Signalatdetector[a.u]

Figure 4.8: Storage run optimized for the hyperfine field.

corresponding to spatial compression well within a pulse length. In this case, the retrieval

efficiency was only limitted by atomic decoherence (Fig. 4.8.)

4.2 A Theoretical Model

In order to gain understanding of our observations, a theoretical model of a tripod atomic

system was evaluated for which the susceptibility numerically determined. Although the

double EIT system studied in the previous section consists of 16 levels, forming several

tripod EIT systems, these tripod systems are degenerate and the properties of the system

as a whole can be effectively modelled by examining a single tripod system. The model

consists of a common excited state |e and three ground states |z, |h, |p, coupled by

two weak signal fields z, h, and a pump p respectively. The decoherence mechanisms

included dephasing (decay of the off-diagonal matrix elements) ij acting within each

pair of ground states, and population exchange G between levels |h and |z (Fig. 4.9).

In the rotating wave approximation, the light-atom Hamiltonian is given by:


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Figure 4.9: The theoretical tripod scheme analyzed in the text.

0 h p z

h h 0 0

p 0 p 0

z 0 0 z

. (4.2)

where i is the detuning of the light field with respect to the |i |e atomic transition.

The decoherence terms may be modeled by introducing the relaxation matrix:

ee

h + p + z

1

2ehh

1

2epp

1

2heh eeb + G

zz hh + ppph

1

2phph bdG

1

2hzhz

1

2pep

1

2hpph eep pp

ph + pz

1

2pzpz

1

2zez Gzh

1

2zhhz

1

2pz aaG (dd bb) + ppph

. (4.3)

By solving the Liouville equation using eqs. 4.2 and the decoherence matrix above, the

steady-state density matrix was obtained (see equation 2.5.)

The main purpose of the theoretical analysis of this system was to investigate the

effect of the preparatory pulse on the slow-light dynamics of the system. A nontrivial


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assumption in this model is that the steady-state approach is valid. Recall that in the

experiment there is a long (500s) pulse, which is switched off for off 5s before

sending the twin pulses. In order to verify the validity of the steady state approximation

the relaxation time of the system was measured. Specifically, with the preparatory pulse

on, the other field sees a higher atomic density than it does when the field is off. For a

finite time after the preparatory is switched off, the atomic system remains unchanged

before it begins to decay to the no prep. field state. This time was measured to be on

the order of relax 100s. Since relax off, the steady-state approach is justified.

The group velocities of the signal pulses were calculated using the linear response

theory with the initial state given by the steady state density matrix. The susceptibilityof the atomic gas for the two signal fields was evaluated numerically for varying strengths

of the preparation field. The parameters of the system were set to fit the group velocity

behavior to that observed experimentally. A good fit was obtained with a Doppler width

of 500 MHz, z = h = p = 6 MHz, hz = hp 5 kHz. zp 40 kHz, and G = 50 Hz.

The results of this analysis using the above parameters is displayed as the solid line in

figure 4.6.

In the CW picture, our theoretical model has also been able to qualitatively reproduce

the enhancement of the transparency contrast for one signal field when increasing the

strength of the other. Figure 4.10 shows the experimental and theoretical plots of the

CW EIT linewidth with increasing field strength on the opposite EIT signal.

It was found that a crucial parameter in obtaining a good fit was the decoherence

between the two signal ground states |h and |z. As hz was increased, the optical

pumping effect became much more significant. With small hz, enhanced transparency on

2-photon resonance in the CW regime emerged due to the presence of an additional dark

state formed by energy levels |h and |z. Experimentally this enhancement was verified

by manipulating the detuning of the hyperfine pulse and observing a small transparency


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Figure 4.10: Comparison of the optical pumping effect on the EIT contrast in the exper-iment to that predicted by the theoretical model

dip, in addition to the transparency due to the pump field, corresponding to this dark

state.

Also interesting is the role of the population exchange rate G. As noted in [16],

the fraction of this mechanism in ground state decoherence is small compared to pure

dephasing ij . This finding is confirmed by the present experiment, yet the population

exchange mechanism cannot be completely neglected. If we set G = 0, the populations

of states |h and |z would depend only on the ratio between h and z, but not on theirabsolute magnitude. The nonlinear effect of the signal fields on each other could then be

observed at arbitrarily low field strength, which is not in agreement with experimental

observation. This discrepancy can be addressed by setting a small, but nonzero value of

G, which governs the ground state populations at low signal intensities. Intuitively, the

non-zero population exchange serves to continuously repopulate the ground state which

allows for a more significant optical pumping effect dependant on the strength of the

signal field which couples out of this level.


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4.3 Further Prospects

The ability to manipulate the relative group velocities of light pulses could prove promis-

ing as an all-optical light buffer for use in classical optical signal processing. Since the

pulses are present at separate frequencies, a frequency multiplexing scheme could be im-

plemented in which light at separate frequencies are separated, independently addressed,

and rejoined.

The preparatory pulsing which was used to equalize the group velocities of the pulses

was used in the Zeeman channel and was found to further enhance the slow light effect

on the hyperfine pulse by nearly a factor of 10. Although not of interest for creating

matched light pulses, the Zeeman field could be used to set a programmable delay in

an optical pulse. This itself could find application as an optical light buffer acting on a

single channel.

The ability to simultaneous store light pulses at separate frequencies shows that dou-

ble EIT can be used as a broadband memory - acting independently on two pulses of

light. There are several limitations to this scheme, namely that the memory would only

work for discrete frequencies and the second pulse could not enter the atoms while a pulse

is being stored (since no pump field would lead quickly to absorption.) Nevertheless, the

ability to independently store and retrieve light pulses at separate frequencies could find

application in quantum memory.


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Chapter 5

Giant Optical Nonlinearities With DEIT

One of the key features of EIT is that while vanishing linear absorption and unity index

of refraction are present at two-photon resonance due to destructive interference of exci-

tation pathways, the third order susceptibility is at a maximum as a result of constructive

interference [22]. This allows for the creation of large optical nonlinearities such as cross-

phase modulation (XPM). In this chapter, progress is reported towards implementing a

large XPM scheme using double EIT.

5.1 EIT-Based Nonlinearities

The key attribute of an EIT system allowing for large cross-Kerr interaction is the steep

dispersion about two-photon resonance - the same mechanism responsible for slow light

(Fig. 2.2.) Since the optical phase shift introduced by an atomic medium is determined

by the real component of the susceptibility or equivalently, the index of refraction, a slight

change in the two photon detuning will correspond to a large shift in phase. Specifically,

the optical phase shift induced by a medium of length L on a transmitted pulse is given

by:

(n) = L

cn(). (5.1)

Suppose then that the presence of an additional field changes in some way, the two photon

resonance condition of the EIT system. Ifnf and ni are the indices of refraction with and

without the additional field causing the shift in two-photon resonance, the XPM phase

shift is defined as XPM = (nf) (ni). Thus, an EIT medium has the capability to

48


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Figure 5.1: N-type atomic level scheme proposed by Schmidt and Imamoglu. Off-resonantcoupling to |s creates a Stark shift leading to a large shift in the phase of the signal.The inset shows a typical non-EIT XPM scheme for which the coefficient is orders ofmagnitude less.

induce a large XPM between two fields, so long as one field can affect the detuning of

the other.

The first proposal to exploit this nonlinear effect came about in 1996 by Schmidt

and Imamoglu [30]. The idea is to start with a standard EIT system, but to couple an

additional field, off resonantly from the EIT signal ground state to an additional state

|s, forming an N-type scheme (Fig 5.1.) The |g |e |p lambda system produces the

standard steep dispersion associated with EIT, but the now ground state |p experiences

an AC-Stark shift as a result of the off-resonant coupling to |x. This shift in energy in

turn shifts the 2-photon resonance of the EIT system, creating a large phase shift.

Compared to a conventional cross-phase modulation scheme involving three atomic

levels not experiencing EIT (see figure 5.1 inset,) the N-type scheme produces a nonlinear

coefficient which is many orders of magnitude greater. Specifically, the ratio of Re[(3)]

nonlinearities from the EIT to non-EIT case is 3.3 109. The magnitude of this nonlin-

earity allows for the possibility of nonlinear interactions at the single photon level. The


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ability to create a conditional -phase shift between photons would be of tremendous

importance to the quantum information community since it would allow for the creation

of a quantum controlled-phase gate [12]; a universal quantum phase gate [1].

A major limitation of this scheme is that the generated nonlinear phase shift is pro-

portional to the interaction length Lint, of the signal and axillary fields. For short pulses,

this becomes a severe restriction on the maximum possible phase shift, due to the group

velocity mismatch between the two pulses. The Stark pulse travels at the speed of light c,

whereas the signal pulse experiences the pronounced group velocity reduction associated

with EIT. As a result, the axillary pulse quickly escapes the slowed signal. Quantita-

tively, if the signal and Stark pulses have temporal length and are initially matched,the pulses will be mismatched after the peak of the Stark pulse has travelled a distance

corresponding to the spatial length l = vsigsig, of the signal pulse: int = l/c (the

distance travelled by slowed pulse is neglected since vsig/c 1.) Thus the effective length

over which the pulses can interact is given by:

l = vsigcint = vsig

c2

csig. (5.2)

In general, for propagation through an atomic cell of length L, the interaction length is

Lint =min[L, l] [13]. For a typical EIT group velocity of 30km/s and 1s pulse lengths,

this corresponds to Lint = l = 3m. At this level, the maximal allowable phase shift is

less than 0.1rad for single photon pulses, assuming perfect transparency [13]. Thus, the

above scheme is rather limited for use as a quantum phase gate since the magnitude of

the phase shift is much less than .


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Figure 5.2: (a) The large XPM scheme proposed to implement experimentally. The|p |z |h provides the double EIT tripod scheme gives the sow light effect to bothsignal fields. |h also couples off-resonantly to |s creating large cross phase modulation.It is imperitive that the strength of the magnetic field is such that the Zeeman shift isin the nonlinear, intermediate field regime in order to break the |X |s |p EIT scheme.(b) Illustrates the need for a magnetic field.

5.2 Physical Implementation in 87Rb

The above difficulty may be overcome if both the signal and Stark fields have matched

group velocities. In this case, l and Lint is limited only by the cell length.

There

Double Electromagnetically Induced Transparency

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