tunable bichromatic lattices for ... - department of physics

University of California Santa Barbara

Doctoral Thesis

Tunable Bichromatic Lattices forUltracold Quantum Simulation

Author:

Alan Long

Supervisor:

Dr. David Weld

A thesis submitted in fulfillment of the requirements

for the degree of Bachelors of Science

in the

Weld Lab

Department of Physics

June 2015

Declaration of Authorship

I, Alan Long, declare that this thesis titled, ’Tunable Bichromatic Lattices for Ultracold

Quantum Simulation’ and the work presented in it are my own. I confirm that:

� This work was done wholly or mainly while in candidature for a research degree

at this University.

� Where any part of this thesis has previously been submitted for a degree or any

other qualification at this University or any other institution, this has been clearly

stated.

� Where I have consulted the published work of others, this is always clearly at-

tributed.

� Where I have quoted from the work of others, the source is always given. With

the exception of such quotations, this thesis is entirely my own work.

� I have acknowledged all main sources of help.

� Where the thesis is based on work done by myself jointly with others, I have made

clear exactly what was done by others and what I have contributed myself.

Signed:

Date:

i

“Thanks to my solid academic training, today I can write hundreds of words on virtually

any topic without possessing a shred of information, which is how I got a good job in

journalism.”

Dave Barry

UNIVERSITY OF CALIFORNIA SANTA BARBARA

Abstract

Faculty Name

Department of Physics

Bachelors of Science

Tunable Bichromatic Lattices for Ultracold Quantum Simulation

by Alan Long

We have created and implemented novel methods of quantum simulation. Through uti-

lization of the AC Stark shift, we have developed optical lattices that simulate both

periodic and quasi-periodic ionic lattices. The lattices are highly and precisely control-

lable in situ with minimal size, complexity, and cost.

Acknowledgements

I would like to thank Ruwan Seneratne,Zach Geiger, Kurt Fujiwara, Dr. Boris Shraiman,

Dr. Vyachaslav Lebedev, and my adviser Dr. David Weld for their help and support.

iv

Contents

Declaration of Authorship i

Abstract iii

Acknowledgements iv

Contents v

List of Figures vi

Symbols vii

1 Implementation 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Brief Overview of Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Brief Overview of Experiment . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Monochromatic Lattices 5

2.1 Physics of optical trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Physics of Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Bichromatic Lattices 12

3.1 Mathematical Basis of Bicromaticity . . . . . . . . . . . . . . . . . . . . . 13

4 Implementation 19

4.1 Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2 Apparatus Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.3 Measurement of Lattice Parameters . . . . . . . . . . . . . . . . . . . . . 27

4.3.1 Beam Profiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.3.2 Phase measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Lattice Characterization 30

6 Conclusion 34

7 Bibliography 35

v

List of Figures

1.1 A classically induced dipole. . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Atoms are attracted to minima for blue detuned light and maxima forred detuned light. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 A diagram of two Gaussian beams overlapping at and angle 2φ. Bothbeams propagate from the top of the page to bottom. . . . . . . . . . . . 7

2.3 An example of an optical lattice taken from our camera. . . . . . . . . . . 9

2.4 The same optical lattice. Height corresponds to intensity in arbitraryunits. The x and y axes are in µm. . . . . . . . . . . . . . . . . . . . . . . 10

2.5 A side view of the lattice in Figure 2.3. . . . . . . . . . . . . . . . . . . . . 11

3.1 A bichromatic lattice created with our setup. Lattice C is a superpositionof lattices A and B. Note concurrent maxima and A and B correspond toa maximum in C and similarly with minima. . . . . . . . . . . . . . . . . 12

3.2 An idealization of a bichromatic lattice. The blue is the bichromaticlattice resultant from the red and green lattices. ν = 3

5 . . . . . . . . . . . 13

3.3 The purple is a superposition of the red and blue. . . . . . . . . . . . . . . 14

3.4 The graph of possible energy eigenvalues as opposed to ν, called the Hof-stadter butterfly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.1 A beamsplitter pair, used to create two parallel beams of a controllablespacing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2 An example of a BS pair in our setup. . . . . . . . . . . . . . . . . . . . . 20

4.3 A diagram of the phase modulation apparatus. . . . . . . . . . . . . . . . 22

4.4 The phase modulation in out setup. . . . . . . . . . . . . . . . . . . . . . 22

4.5 A diagram of the main setup. Not to scale. Part numbers refer to thisdiagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.6 A Solidworks model of the setup. This is a slightly different version madefor free space coupling. Beyond the placement of mirrors and lack of fibersit is identical. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.7 The interferometry apparatus used for phase measurement. . . . . . . . . 28

4.8 The interferometry apparatus as seen from above. . . . . . . . . . . . . . . 29

4.9 The interferometry apparatus at an angle. . . . . . . . . . . . . . . . . . . 29

5.1 Spatial frequency as a function of beam spacing. . . . . . . . . . . . . . . 30

5.2 Lattice slice measured using the razor blade technique. . . . . . . . . . . . 31

5.3 Interferometry measurements and expected values. . . . . . . . . . . . . . 32

5.4 Overnight phase drift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

vi

Symbols

E electric field N C−1

p dipole C m

U potential J

I intensity W m−2

α dipole proportionality constant unitless

ω angular frequency of light s−1

ω0 angular frequency of optical transition s−1

ν spatial frequency of lattice m

λ wavelength of light m

φ angle of intersection of beams rad

ψ lattice phase rad

Υ envelope function unitless

vii

Chapter 1

Implementation

1.1 Motivation

In the world of condensed matter, crystals are a subject of much importance. Crystals

are seen everywhere and utilized constantly, from piezoelectric drivers to a simple clock.

Crystals are also used as idealizations of solids. The notion of a solid as balls on springs

in a pattern is in many ways the fundamental assumption behind condensed matter.

A crystal can be defined by a pattern, a length scale, and an atom or atoms that form

it. Regrettably all three of these are not easy to control. It is of course impossible

to change the materials making a crystal. The structure is essentially static as it is a

result of the materials in question. Finally, the length scale, e.g. the spacing between

adjacent atoms, can be controlled slightly with thermal expansion and contraction, but

what change may be found is very small and requires large temperature changes.

Because crystals are so useful, we want to discover new crystals with various properties.

However, due to the above reasons, it is difficult to actually create a specified crystal.

It then would seen that some way of crystal simulation would be needed, which must

have certain properties: It must retain the “pattern of balls on springs” behavior, that is

to say the potential must be spatially periodic and approximately a harmonic oscillator

around lattice sites, and it must be able to be controlled, that is to say the pattern,

length scales, and atoms used must all be able to be arbitrarily changed. So before

anything with the actual structure is done, some way of trapping arbitrary atoms must

be found.

1

Chapter 1. Introduction 2

Figure 1.1: A classically induced dipole.

1.2 Brief Overview of Lattices

The immediate answer one might think is to use an electric field. However there are

some problems. First of all, the atoms are neutral, so the best one could hope for is

dipole effects. Secondly, if the field is constant with time, the fact that ∇2V = 0, as we

are working in free space, means that there could not be a minimum or maximum of

the potential except at the boundaries, which is obviously not where we want the atoms

to go. So something must be done to remedy this and actually trap the atoms at a point.

This may be done by utilizing the AC Stark shift. This effect occurs when a dipole is

induced on the atom by an electric field, and then the dipole interacts with that field.

When an atom is in an electric field, the field will pull the electrons in one direction and

the proton in the other as they have opposite charges. This induces a dipole on the atom

proportional to the field. The energy of a dipole p in an electric field E is U = p·E. IF E

is constant, when p will be constant and the atom will simply be in a constant potential.

However, if the field is oscillating, the dipole will as well. In this case U = E · p is no

longer trivially constant, and forms a varying potential. This is the AC Stark shift. So

then, in order to trap an atom, an oscillating field must be used. Luckily there is an

oscillating field that is easy to use and control, light. Light is an electromagnetic wave

and as such will induce an AC Stark shift on the atom. The resultant potential energy

will depend on the intensity of the light, U = AI for a constant A depending on the

atom and light used. This simplifies the trapping problem greatly, instead of needing

to make a periodic potential field, one simply needs to make a periodic intensity. This

may be done through interference.


When two beams of light meet they interfere as they are waves. In the simplest case of

two plane waves of the form E = cos(~k · ~x+ ωt) interacting, the resultant intensity is

I = I0 cos2(πx

ν)

Where ν = 2π|k1−k2| is the spatial frequency. As the potential is proportional to the

intensity then

V ∝ cos2(πx

ν)

This is of great use to us. It would appear to satisfy both the periodic requirement

and the harmonic oscillator requirement, as V will be approximately parabolic at the

extrema. Furthermore, it is not reliant on the atom (it turns out the proportionality

constant of the intensity and the potential is, but that only changes the strength not

the function) so it may be used for any atom.

All that is left is the arbitrary pattern, which it does admittedly fail at, and the arbitrary

length scale. The length scale in this equation is ν which, it turns out, is related to the

angle the beams intersect at. So being able to control this angle would give control of

ν, called the spatial frequency.

In addition, though this method cannot make a truly arbitrary pattern, it can do more

than a simple cos2. One might imagine many beams being overlapped forming other

interesting patterns. These beams might all have the same wavelength so they would all

interfere, or pairs of different wavelength so they would not interfere (at a frequency at

atoms could feel), or some combination of the two. Through this many interesting and

physically relevant patterns may be seen. The most simple of which, and the topic of our

research, is two non-interfering patterns overlapping with an irrational periodicity ratio,

i.e. ν1ν2

/∈ Q. This will cause the resultant pattern to be quasiperiodic, structured but

non-repeating. Atoms trapped in such a potential would approximate a quasicrystal, a

quasiperiodic crystal.

So this looks like an appealing method. It satisfied 4.5 of the requirements, it is fairly

straight forward, and it does not require anything unreasonable, simply a good light

source. Because of this, we have used these interference patterns, called optical lattices,

in our research.

These lattices may be used to simulate condensed matter systems. By trapping atoms

in the lattices we can realize many models for ionic lattices, such as the tight binding


model and the Hubbard model. Through this we can get a deeper understanding of the

behavior of these systems normally not accessible trough computer simulations.

1.3 Brief Overview of Experiment

It is our goal to simulate such systems. Of particular interest to us are 1 dimensional

quasicrystals realized using quasiperiodic lattices. In order to create this, we use Li

atoms cooled to a Bose Einstein condensate (BEC) in high vacuum. We trap the BEC

in a quasiperiodic lattice with the help of additional traps. We then use the control

ability of the lattice parameters to probe the effects of certain configurations and pa-

rameters. We create this lattice using two pairs of focused laser light, creating two

lattices the superposition of which is quasiperiodic. We believe this set up will realize a

quasiperiodicity in atoms in a controllable manner and greatly further our knowledge of

quasicrystals.

Chapter 2

Monochromatic Lattices

2.1 Physics of optical trapping

In order to work with atoms, there needs to be a way of controlling them. One can

obviously not bolt an atom to a breadboard. What we need is some sort of potential,

and subsequently some sort of force, to keep the atoms where we want them. It seems

like the obvious solution would be to use some field force, which does end up being the

right answer, but with some difficulties. There is only one such force we are able to

manipulate with any ease, the electromagnetic force, so that is an obvious choice of tool

when compared to say, gravity. However we face the immediate issue of the fact we are

using neutral atoms. This means we can, at best, get dipole effects. So, looking back

on what we need. We need to induce dipoles on atoms, we need to use those dipoles to

put the atoms in a potential, and that potential needs to be controllable and confine the

atoms to a region. How might one do this, you may ask. Well it’s in the next section.

Let there be an oscillating electric field. Using complex notation,

~E = EEeiωt

Where E is the direction of polarization and E is the complex magnitude. Any dipole

induced by this field must be proportional to the field. So

~p = α~E

p = αE

for some complex α, called the polarizability. For any induced dipole the potential is

U = −1

2〈pE〉

5

Chapter 2. Monochromatic Lattices 6

U = −1

2〈α(Eeiωt)2〉

U = −1

2Re(α)|E|2

So then we get non trivial potential, which is what we wanted. So, an oscillating electric

field is the answer to our problem. It should be apparent that light fits the bill. In this

case the potential becomes

U =1

2ǫ0cRe(α)I

where I is the intensity of the light. However we still do not know what α is. By looking

at power absorption and scattering rates, it turns out that α can be found, and that it

depends on the atom being trapped and the the transitions in the atom. We can simplify

this by utilizing Lorentz’s classical oscillator model. Specifically, the constant is

α = 6πǫ0c3 Γ

ω40 − ω2ω2

0 − iω3Γ

Where ω is the frequency of the light, ω0 is the frequency associated with the opti-

cal transition and also the eigenfrequency of the classical oscillator model, and Γ is a

damping constant due to dipole radiation. It is

Γ =e2ω2

0

6πǫ0mec3

If we assume that Γ is small, i.e. the dipole radiation is negligible, the expression may

be reduced. This is a valid approximation as the coefficient in front of the ω20 in the

expression is approximately 6× 10−24s. So making this approximation, α becomes real,

and may be simply plugged into the expression for U to get

U =3πc2Γ

ω20(ω

2 − ω20)I

U =e2

2cǫ0me(ω2 − ω20)I

Note the sign of the expression. All constants and I are positive, so the sign is deter-

mined by the difference in squares of the frequencies. If ω > ω0 the potential will be

greater the greater the intensity, meaning atoms are repelled from areas of high inten-

sity. Conversely, ω < ω0 means that a high intensity corresponds to a low potential, and

atoms are attracted to high intensity. These are referred to as blue detuned and red

detuned respectfully. The effect that they would have on atoms can be seen in Figure

2.1


Figure 2.1: Atoms are attracted to minima for blue detuned light and maxima forred detuned light.

For the blue detuned case the atoms tend to the intensity minima and in the red detuned

case they tend to the maxima.

2.2 Physics of Lattices

Let’s leave the math of trapping for a bit and look at interference. It turns out that

interference patterns end up being quite useful, as we’ll see in a moment. Suppose that

there are two Gaussian beams overlapping at their waists as in Figure 2.2.

We’ll make some assumptions and definitions about the beams. First, assume the

Figure 2.2: A diagram of two Gaussian beams overlapping at and angle 2φ. Bothbeams propagate from the top of the page to bottom.


beams propagate from the top of the page to the bottom. Let the beams intersect at an

angle 2φ as shown. Let the beams have the same intensity, and the same wavelength λ.

Let the beams differ in phase by 2ψ. Assume that the beams are both linearly polarized

into the page. Finally assume that the beams are Gaussian and that they are uniform in

the region where they overlap, that is we need not consider the beams spreading. This

seems like a lot, but the only real approximation we made was that the beams were

uniform at their waists, which is true up to first order. The rest is just definitions or

things that can be controlled in an experiment.

Moving forward, these beams will interfere and form a standing wave. By symmetry the

pattern must be constant, bar the envelope function, in the y or z directions. So then

we can consider the x direction alone. Both beams will be of the form

E = E0er2

ω20±(ikl+ψ

2)+iωt

Where w0 is the waist, k is the wave vector, and r, l are cylindrical coordinates. If we

only concern ourselves with the plane perpendicular to their combined wave vectors, i.e.

the x− z plane, the equation becomes

E = E0e−x2cos(φ)2+z2

ω20±i(kxsin(φ)−ψ)+iωt

Adding the two gives

E = E0e−x2cos(φ)2+z2

ω20+iωt (

eikxsin(φ)−ψ + e−ikxsin(φ)+ψ)

E = 2E0ex2cos(φ)2+z2

ω20+iωt

cos

(

2π sin(φ)

λx− ψ

)

Squaring the norm to get intensity gives

I = 4E20e

2x2cos(φ)2+z2

ω20 cos2(

2π sin(φ)

λx− ψ

)

I = 4E20e

2x2cos(φ)2+z2

ω20 cos2(

2π sin(φ)

λx− ψ

)

So, from equation relating intensity and potential above

U =12πc2Γ

ω20(ω

2 − ω20)E2

0e2x2cos(φ)2+z2

ω20 cos2(

2π sin(φ)

λx− ψ

)


For ease’s sake lets define

U0 =12πc2Γ

ω20(ω

2 − ω20)E2

0

Υ = e2x2cos(φ)2+z2

ω20

ν =λ

2 sin(φ)

Then the equation is

U = U0Υcos2(πx

ν− ψ)

Which is much cleaner. We see that the equation has three main parts. First a scaling

constant U0 related to the set up and atom, then an envelope function Υ, and finally a

periodic pattern, the cos2 term, with frequency ν and phase −ψ. Figure 2.3 is a picture

Figure 2.3: An example of an optical lattice taken from our camera.

of an actual lattice created in lab. Figure 2.4 a visualization of data gathered using our

set up, with height corresponding to potential (with arbitrary units) and the x and y

units in µm. Note the Gaussian envelope and the periodic nature. We call this pattern

an optical lattice.

Figure 2.5 shows a side view of the lattice above, with U0 and ν labeled. For a single

1-dimensional lattice such as this, these are the most important parameters. In our

experiment we try to make the lattice as wide as possible so there will be a large section

where Υ ≈ 1, as then the lattice is completely periodic. In this case, the phase of the

lattice does not matter as it is symmetric under translation. This leaves well depth and

spatial frequency as the two most important values that define a lattice.


Figure 2.4: The same optical lattice. Height corresponds to intensity in arbitraryunits. The x and y axes are in µm.

So, at least in one dimension and assuming Υ ≈ 1, we have atoms being affected by a

periodic potential. This should bring to mind crystals. Crystals are, at their simplest,

periodic patterns of atoms. Atoms trapped in the lattice would tend to the maxima of

the lattice, assuming red detuning. We will assume that the lattices are red detuned

unless otherwise noted. Blue detuning would lead to identical situations, but with atoms

at minima. This would create a pattern of equally spaced atoms, essentially equivalent

to a crystal with lattice constant ν. The analogy becomes even better. To second order,

cos2(kx) = 1 − k2x2 at a maximum. Which means that the potential will be approx-

imately quadratic at the lattice sites, as one might expect. This is impotent however,

because it means that the quantum system is approximately a harmonic oscillator. It is

very common in condensed matter to model atoms in an ionic lattice as in a harmonic

oscillator at each lattice site. When in an optical lattice, atoms will experience the same

effects, modeling the behavior of an ionic lattice. This correspondence is the reason for

the name ”optical lattice” and why terminology of the two may be interchanged. Optical

lattices then simulate the behavior of crystals both on a large scale, where the lattice

structure is most important, and a small scale, where the behavior of atoms at a given

lattice site is most important.


Figure 2.5: A side view of the lattice in Figure 2.3.

This is incredibly useful. It is very rare that a complicated system can be completely

solved. Most often, it must be solved numerically. In many cases, specifically cases with

strongly interacting particles, the computations needed to model the system evolving

with time become prohibitively large. Because of this, computer simulations are capped

at a certain point. If one were to have a lattice with a Hamiltonian similar to that of

an ionic lattice, it would evolve in time similarly to the actual crystal. Through this it

is possible to model the effects on an ionic lattice over a much greater time span than

with computers.

This method may be used for many models. If the well is sufficiently deep, tunneling will

be small. This means that interactions between lattice sites will also be small. Because

of this electrons will stay close to a particular lattice site and thus with a particular atom.

This is a realization of the tight binding model. It can be shown that the Hubbard model

can also be realized using lattice.

Chapter 3

Bichromatic Lattices

Figure 3.1: A bichromatic lattice created with our setup. Lattice C is a superpositionof lattices A and B. Note concurrent maxima and A and B correspond to a maximum

in C and similarly with minima.

A very interesting and useful implementation of lattices is creating superpositions of

them, forming superlattices. This can create many interesting patterns and effects.

This simplest case of this is a bichromatic lattice, the superposition of two lattice with

the same direction but different spatial frequencies. It is called a bichomatic lattice due

to the fact that is could be formed with two different colored beam pairs interfering, but

it is easier to create it with same color beams. This can be done by having the pairs

intersect at the same spot but at different angles, which is the method we have used in

our experiments.

12

Chapter 3. Bichromatic Lattices 13

However,how exactly could this be done? If you had four beams crossing each would

interfere with every other one, making 6 lattices, not just the two wanted. This can

be solved in two ways. First, the polarizations between pairs may be made orthogonal.

This would mean that a beam would only meaningfully interfere with the other beam of

its pair, The other method would to have the wavelengths of the beams slightly detunes

from one another. Though the beams would interfere, the interference would oscillate

so rapidly that the atoms would not be affected by it.

3.1 Mathematical Basis of Bicromaticity

So the lattices form a pattern, but what kind of pattern. For this section we will assume

that the Gaussian envelope is wide enough that Υ ≈ 1 for a sufficiently large portion of

both lattices, i.e. we don’t have to care about it. Also, for ease’s sake, assume that both

lattices have a amplitude of 1 and the periodicity of the wider lattice is 1, whatever

effects there are should be independent of absolute scale, so this can be done. Then

the pattern will simply be the two lattices added, letting the smaller lattice have a

periodicity 0 < ν < 1

V = cos2(πx) + cos2(πx

ν

)

The resultant pattern cross section would look something like Figure 3.2.

Figure 3.2: An idealization of a bichromatic lattice. The blue is the bichromaticlattice resultant from the red and green lattices. ν = 3

5.

To better understand exactly what we are looking at, let us first assume that ν ∈ Q. The

picture above shows such a case, specifically when ν = 35 . Note that the pattern seems

to repeat itself. As it turns out this is a property of all rational bichromatic lattices. Let

us try to find where this pattern repeats. Assuming that the phase of both lattices is


0, they will both have a maximum when x = 0 so then we simply need to find the next

dual maximum. cos2 has its maxima when its argument is nπ where n is any integer.

So wider lattice will have a maximum every integer. So it is simply finding an integer

n s.t. νn ∈ N. As ν ∈ Q, ν = pqfor some p, q ∈ N where p and q are mutually prime.

So then if nqp

∈ N then p|nq but as q is mutually prime to p, p|n. So then the smallest

integer s.t. the pattern returns to its maximum is p. We can observe this in Figure 3.2.

I said earlier that the picture was of the case where ν = 35 and in fact, as you can see,

there is a maximum where x = 3 and this is the first maximum. So then the case where

ν is rational is fairly straightforward.

However, the irrational case is, luckily, far more interesting. Though the above work

will obviously not work directly for this case, it can be used for approximations. An

irrational number may be approximated using rational numbers, by continued fractions

or decimal expansion or any other such way. So then, as we consider shorter lattices

with spatial frequencies increasingly close to the irrational value we want, the resultant

bichromatic lattice should get closer and closer to the one we want. However, as the ra-

tional approximation nears the irrational number, the numerators get increasingly large,

meaning that the limit of the repetition distance as the periodicity ratio becomes irra-

Figure 3.3: The purple is a superposition of the red and blue.

tional goes to infinity. So then two lattices with irrational spatial frequencies will never

have their pattern repeat. This means that the pattern is no longer periodic, and is

now quasiperiodic. That is to say it is ordered, but not strictly periodic. This is very

interesting because the atoms loading into the trap will form into this pattern. So, like

the atoms in the, periodic, monochromatic lattice formed a crystal, these atoms would

form a quasicrystal. Quasicrystals are the subject of much research and are still being

found. Having the ability to simulate not just quasicrystals but custom order quasicrys-

tals would be a massive boon for our knowledge on a subject. Luckily this is what

these quasiperiodic lattices offer. Of particular interest to us are lattice where the ratio


between the lattices is the golden ratio, φ = 1+√5

2 , as we will show shortly.

We now know quasiperiodicity is created by irrational ν and any irrational ν will make

a quasi periodic lattice. One might now ask, does the choice of ν matter? It would be

very convenient if ν could be set to any irrational number and have the same effects.

This is regrettably not the case. Consider ν = 12 +

√2

10000 . This is irrational and will

make quasiperiodic lattice, however it will be very close to the lattice created by ν = 12 .

Quantitatively, the peaks will be .05 away, a good estimate for ”significant”, from each

other at x ≈ 35.5. Meaning that within any given 35 long interval, the quasiperiodic

lattice looks the same as the rational lattice, but with some phase. This means that

any quasiperiodic effects will not be evident on any scales of that size or smaller. These

effects are the reason we want these superlattices so a lattice like this would not allow

for any experimentation to be done. This is an example of a ratio we do not want, a

“bad choice” of ν.

So then, how do we get good choices. Let’s take a step back and think about an irra-

tional lattice mathematically. As rational lattices are much easier to work with, it would

be beneficial to find rational approximations. Though there are many ways to go about

this, it turns out that the method of continued fractions is both the easiest and the most

helpful.

Any real number x may be expressed as a continued fraction of the form

x = a0 +1

a1 +1

a2+1

...+ 1an

if x is rational and

x = a0 +1

a1 +1

a2+1

...

if x is irrational, with a0 ∈ Z and ai ∈ N when i > 0. By truncating at some an, one

can create a rational approximation of an irrational number x. That is to say, is x is

defined as above in the irrational case, the approximation, call it xn would look like the

rational case above.

So then, if ν is some irrational ratio of spatial frequencies, it may be expressed as a

continued fraction. As ν is defined to be 0 < ν < 1, disregarding the case where ν = 1

as it is trivial, we know that a0 = 0 for any ratio. Let us similarly define νn as the

continued fraction truncated at the nth entry. νn ∈ Q so νn = pqfor some p, q ∈ N. To


find what these are, we can use induction. Consider just looking at the partition of ai

where i > j for some j. This part will be

aj +1

· · ·+ 1an

This will obviously be rational, let it befnjgnj

, where the superscripts denote the fact we

are considering νn, not showing powers. Note that the fractional part of the expression

will be1

aj+1 +1

···+ 1an

=gnj+1

fnj+1

So we have

aj +gnj+1

fnj+1

=fnjgnj

So

fnj = ajfnj+1 + gnj+1

gnj = fnj+1

Looking at j = n we see fnn = an and gnn = 1. This process can be done to find fn1 and

gn1 . Putting these into the original equation gives

νn =1fn1gn1

=fn2fn1

So then a lattice of this ratio would have a repetition distance of fn2 (if n = 1 we define

f12 = 1).

Now we have a method of approximating irrational lattices and characterizing those

approximations, and so we can start to answer the very aforementioned question “what

is a ‘good” lattice?”. What went wrong with the bad example above was that distance

between repetition for some successive approximations was very large, making there be

a large gap in necessary length scale for the lattice to noticeably change. So what we

want is for any given successive difference to be as small as possible. This can be done

using strong induction. First the base case, n = 2. So we are trying to minimize f22 −f12 ,

however this is just a2 − 1, and the minimal natural number is 1 so a2 = 1. Now for

induction, assume ai = 1 for 1 < i ≤ n. We are again trying to approximate fn+12 − fn2 .

However as fn2 is set by our inductive hypothesis, this is the same as minimizing fn+12 .

As ai = 1 when 1 < j < n+ 1,

fn+1j = fn+1

j+1 + fn+1j+2


. With fn+1n+1 = an and fn+1

n+2 = 1. This has a closed form solution of

fn+1n−m = F (m+ 1)an+1 + F (m)

where F (m) is themth Fibonacci number. This is have a minimum again when an+1 = 1

so that must be the case. So by induction ai = 1 when i > 1. The case where ai = 1

for every i ≥ 0 is the continued fraction of φ, the golden ratio. So ν = 1a1+φ−1 . When

a1 = 1 this is just ν = φ−1.

So now we have a target periodicity ratio so that the successive distances of repetition

are minimized. Not unsurprisingly it minimizes the ais. In fact, it is generally true that

numbers with one or more large ais will be “bad”. For example, 12 +

√2

10000 from above

has a1 = a2 = 1 but a3 = 1767. So then there is a massive jump from the second ap-

proximation resolution distance to the third. which is the jump from ν ≈ 12 to a better

approximation, which is what we saw earlier.

This is one reason why the golden ratio is so interesting and why the work above actually

matters. The golden ratio will have its large n approximations be used the earliest,

making its quasiperiodic behavior “appear” on the shortest scale. Because we have

limited scale, this means such a lattice will be the most quasiperiodic possible. This

lattice is so interesting it has its own name, a Fibonacci lattice. It is so called because

approximations of the golden ratio are the ratio of successive Fibonacci numbers, which

can actually be seen above

fn+1j = fn+1

j+1 + fn+1j+2

This is simply the definition of the Fibonacci sequence, as its starts with 1,1 as proved.

As said before, quasiperiodic lattices can be used to realize quasicrystals. There is

much research currently being done on quasicrystals. There are hundreds of known

quasicrystals, which in the grand scheme of things is actually quite small. Quasicrystals

were only discovered in 1982, a discovery which would later win the 2011 Nobel prize.

In our specif case, one dimensional quasicrystals, the ratio of the spatial frequencies and

the energy eigenvalue have the following relationship

This is referred to as the Hofstadter Butterfly as it was discovered by Douglas Hofs-

tadter in 1976 and he doesn’t know what butterflies look like. The Butterfly has many

interesting properties, it is highly symmetric and a fractal.


Figure 3.4: The graph of possible energy eigenvalues as opposed to ν, called theHofstadter butterfly.

Chapter 4

Implementation

4.1 Controllability

Figure 4.1: A beamsplitter pair, used to create two parallel beams of a controllablespacing.

In order to create the lattice, we utilize a method originally created by Foote, a beam

splitter pair. Figure 4.1 shows a diagram of the prototypical pair. Looking first at

the red beam, the beam is sent into the pair from the right. It first passes through

a λ/2 waveplate. The waveplate is rotated such that an equal amount of power will

be retroreflected and transmitted by the right beamsplitter. The beam then hits the

cube, half is reflected out to a lens, which we will return to, and half is transmitted.

As the transmitted beam is polarized such that it passes through the first beamsplitter,

19

Chapter 4. Implementation 20

Figure 4.2: An example of a BS pair in our setup.

it will pass through the second entirely. the beam then passes through a λ/4 wave-

plate, is reflected off a mirror, and passes though the waveplate again. The waveplate

is rotated such that it causes the polarization of the beam to be rotated 90◦. As it

is now orthogonal to how it was originally, and that was entirely transmitted, it will

now be entirely reflected by the left beamsplitter. It then joins the other reflected beam

on the lens. The lens focuses the beams to a point, where they interfere creating a lattice.

This set-up has many advantages. First, the beams are inherently parallel. Because the

beamsplitters will form essentially two mirrors orthogonal to one another, the beams will

be parallel, even if the beam does not come in perfectly normal to the cube face. This

both makes set up easier, as it takes away a degree of freedom that would otherwise

need to be exact, and it allows for the outgoing beams to be directed to the left or

right slightly. In addition, the set up is compact. In our final apparatus, we had two of

these pairs, along with all other necessary optics, fitted to a 10 by 12 inch breadboard.

Furthermore, it makes power control easy. Because the two outgoing beams are a set

fraction of the incoming beam, the outgoing beams will always have the same relative

power, and to change their power is simply a matter of changing the incoming beam.

This is immensely useful as beams of different powers will not create a good lattice, as the

wells will no be as deep as they could be due to the part of the stronger beam not canceled

by the weaker. This both allows for better lattices, and makes power modulation, for


example a noise eater, easy to adapt to the set up. Finally, the set up makes predicting

the periodicity very easy. In order to know the periodicity, you of course need to know

at what angle the beams interfere. This is somewhat difficult to measure in practice.

However, with this set up, one simply needs to know the spacing between beams, which

is easily measured, and the focal length of the lens, which is known, to figure out the

angle using simple trigonometry. If the beam comes in x distance from the cube faces

the periodicity is

ν =λ

2 sin(

tan−1(

xf

))

where f is the focal length. This is, at least in comparison to the other expressions we’ve

used, very simple. Through this it is easy to get a quick approximation of the periodicity.

Of course, this is not a very exact method, and much better ways of measuring periodicity

will be talked about at length, but this is a very good check to make sure you’re at least

in the ballpark of your intended value.

In addition to these, this set-up makes the lattice very easy to tune. To see this consider

the green beam. In practice the two beams being shown in would be the same color,

but they are different here for visualization purposes. If the red beam is some distance

x above the from of the pair, the green beam is x + ∆x away. The green beam goes

through the apparatus exactly as the red beam, and similarly comes out in two halves

into the lens. However, It will be the same distance from the center line as it was from

the cube faces, x+∆x. Because of this, it will be at some angle φ+∆φ. As the angle

has changed, the lattice will have a different periodicity, with the new value being

ν +∆ν =λ

2 sin(

tan−1(

x+∆xf

))

So by simply changing the position of the beam (or by changing the position of the

cubes), the periodicity may be easily controlled. This is very helpful because beam/cube

position can be controlled using movable stages, which are readily accessible, accurate,

and programmable. This is a simple, elegant solution to the problem of period control.

Of course, period is not the only thing of interest. As stated above there is also trap

depth and phase. Trap depth is very easy to control, just increase the power of the

beams, but phase modulation requires more work. Phase modulation of the lattice is

achieved through phase modulation of one of the beams, in this case the retroreflected

one. The movement of the mirror will change the path length, thus changing the phase.

Specifically, is the mirror is moved some distance ∆x, the path length will change by

2∆x, because the incident beam and the reflected beam are both changed by ∆x. This

will result in a phase change of the beam by ψ = 2∆xλ

.

This will result in a change in phase of the lattice. This is due to the fact that the


Figure 4.3: A diagram of the phase modulation apparatus.

Figure 4.4: The phase modulation in out setup.

locations of the maxima of the beam has now changed, so the places where both beams

are at their maxima, i.e. the maxima of the lattice, will also spatially change. If you

were looking at the lattice as the mirror was slowly moved, you would see the maxima

appear to translate across the screen, but have the Gaussian envelope stay in its place.


Figure 4.5: A diagram of the main setup. Not to scale. Part numbers refer to thisdiagram.

4.2 Apparatus Overview

The beam is emitted by the laser. It is then sent through a fiber optic cable two two

parallel and identical setups. The light it then recollimated with two fiber collimators

(1,3). It then is reflected off of mirrors (2,3) and then passes through beam samplers

(6,8) splitting off a small fraction of the beam onto photodiodes (5,7). The signal from

the photodiode is then sent to a noise eater which modulates the power of the corre-

sponding beam, correcting for any power fluctuations. There is a parallel identical set

up that creates another beam. This beam is also modulated in a similar way using a

noise eater

Both beams will pass through beamsplitter pairs (9-18), creating two beam pairs. For

both of these beamsplitter pairs, the retroreflective mirrors (14,17) is attached to a

piezoelectric crystal (15,18). This allows for the optical path length of the retroreflected

beam coming from the beamsplitter pair to have its optical path length changed thus


changing the phase of the lattice.

Finally the beams are recombined in a large beamsplitter (22), using a large mirror (19).

The both pairs must pass through two λ/2 waveplates (20,21) so that they the vast ma-

jority of the power is directed right. A small portion of the beams will be transmitted

or reflected down into the phase measurement apparatus (24). The four beams then are

focused by a lens (23) to a point. At this point a bichromatic superlattice will be formed.

Now for a part by part breakdown. First the recollimators (1,3). The recollimator is

used, as one might expect, to recollimate the beam coming from the fiber. The light will

come out of the fiber in many directions and the recollimator uses a lens to focus the

beam back to a normal Gaussian one. It is important for the recollimator to collimate

the beam to the proper width. Rather counter-intuitively the smaller the beam going

into a lens the larger it will be at the focus. This is due to how a Gaussian beam works,

as the smaller the beam waist the faster the beam will expand. This remains true for

a focused beam, as it is still Gaussian. So a larger size on the lens would mean the

beam expanded from the waist faster meaning a smaller waist. It is preferable to have

a large waist as that allows for more lattice sites. Because of this, it is best to have a

recollimator which outputs a small beam.

Next the power modulation. A small portion of the beam is picked off with a beam

sampler (6,8). This is sent to photodiodes (5,7). The reason for this is so that the

power of each beam may be controlled. There are many things that can cause power

fluctuations, movement in the fiber, changes in power consumption at a point upstream

from our setup, polarization changes. We want our lattices to be a stable as possible,

which includes having a stable power, and thus stable well depths. To this end, use

the beam sampler to get a known fraction of the beam. We have built in lab devices

including a photodiode and a PID to collect this beam and measure its power. It then

takes this and outputs an electrical signal which we use as a feedback with an AOM to

keep the power constant.

Looking at the upper beam as the lower beam is identical, it then passes through the first

beamsplitter pair (7-10). This acts just as the pair discussed earlier. It produces two

parallel beams as the normal pair does. However, there are some differences. Firstly the

beamsplitters (12) themselves are translated as opposed to the beam being translated.

This is because, as will be explained shortly, the retroreflective mirror (14) is very small,

7 mm. This is smaller than the range of positions we want from the beam so we must

move the pair itself instead. The second thing of note is that the retroreflective mirror


Figure 4.6: A Solidworks model of the setup. This is a slightly different versionmade for free space coupling. Beyond the placement of mirrors and lack of fibers it is

identical.

(14) is attached to a piezo (15). This allows for the mirror to be translated on a scale

smaller than the wavelength thus changing the optical path length of the retroreflected

beam, and thus change the phase of the lattice. This will be discussed much more in

depth later. The bottom apparatus is identical. The piezo itself is 5 mm on each side,

meaning we cannot attach a large mirror. This causes our choice in mirror size, 7mm.

The beams are combined on the large beamsplitter (22). The beamsplitter is a polarizing

one . This causes the beam pairs to be of orthogonal polarizations. It is very important

to have this as it allows the beam pairs to not interfere. If they were of the same polar-

ization, not only would the pairs interfere, any two beams would, including ones from

different pairs. This would end up forming a superposition of six lattices instead of two.

Though this might lead to some interesting physics, it is not our goal. The waveplates’

(20,21) purpose if to rotate the polarization of the lower pair by nearly 90◦ as they were

just reflected out of polarizing beamsplitters and must now pass through one, and to

rotate the top two beams very slightly. This causes the majority of the beams to be put

to the right and a small fraction to the bottom. These bottom beams are sent into an

interferometer (24). This is used to measure their phases and control the phase of the


lattice.

Finally the beams are focused by a lens (23). The focal length of the lens is important

as it determines how changes to the beam spacing effects the periodicity. It is also

important, along with the radius of the lens, as they form a lower bound to the spatial

frequencies achievable. In principle, the beam spacing may be set to 0. In practice this

is not the case as the beams have width and will hit the edges of the beamsplitters,

however the spacing may still be made quite small. At 0 spacing, the spatial frequency

is

ν =λ

2 sin(tan−1( 0f))

=λ

0= ∞

So ideally the spatial frequency is unbounded from above ideally. In practice we can not

get the beams arbitrarily close, but as we move the beams closer we quickly approach a

periodicity larger than our spot size, making the issue moot. However it is bound from

below. The largest beam spacing possible is the width of the lens. This is rather obvious

as if the beam spacing was more than that the beams would not hit the lens and just

pass by. As a larger spacing means a smaller periodicity, an upper bound to the spacing

should correspond to a lower bound of the spatial frequency. So, if r is the radius of the

lens, the minimum achievable spatial frequency is

ν =λ

2 sin(tan−1( rf))

It is preferable to have this as small as possible, allowing for the largest range of spatial

frequencies. This will happen with a large r and a small f . Because of this bigger lenses

with shorted focal lengths are preferable, and used in the set up, as opposed to smaller

longer focal length lenses.

As stated earlier the spatial frequencies of the lattices are changed by changing the

relative position of the beam and the beamsplitters. In both cases, the beamsplitter pairs

are placed on moving micrometer stages. The stages used are 9066-X from Newport.

Both stages are moved using mechanical actuators, 8301NF. The actuators are computer

controlled. Using this, we are able to attain minimum movement of 30 nm, corresponding

to a change of ¡4 pm of the spatial frequency.


4.3 Measurement of Lattice Parameters

4.3.1 Beam Profiling

After making a lattice, there is a salient, and surprisingly non-trivial, question: is it the

lattice I want? This is a problem that it turns out is actually quite difficult to solve.

The cause of this is the nature of lattices. They are simply interference patterns, so you

cant actually see them directly just by looking down on one. Whatever is seeing the

lattice would have to look at it inside the lattice to actually know anything about its

structure. There seems to be a simple solution to this problem, put a camera there. A

camera CCD would be able to pick up the light and therefore simply measure the lattice

directly. For wide lattices, this method works quite well, all pictures in this thesis were

taken this way. However this method has drawbacks with scale. It is difficult to get a

camera with small enough pixels to image a small lattice properly. The camera we used

had pixels that were 8.3× 8.3 µm. As we want to image lattices smaller than this, such

a camera would not work. In fact, We have found it impossible to find a camera with

sufficiently small pixel size.

There are ways around this though, we have attempted to use a razor blade to profile the

lattice, as one might a Gaussian beam. We attached the razor to a piezo stage and moved

across the site of the lattice. The beams that created the lattice are then refocused onto

a photo-diode. The signal from the photo-diode is the total power at any given time,

meaning its derivative is the amount of power cover by the razor in a small amount of

time, i.e. the power in the small amount of space covered in that time. This means we

can take the signal from the photo-diode, take its derivative numerically, and use that

as a characterization of the lattice. This has the benefit of being able to be done a scale

dependent on the minimal incremental movement of the motor. This can be brought

down to ∼ 100 nm. As will be discussed in the results section, we ran into many issues

with this method. However, we believe that this method shows promise and deserves

further research. As explained earlier, the phase of the lattice is modulated by changing

the position of the mirror on the order of 10−2λ. This is done by using a piezoelectric

driver. A small mirror is attached to the piezo. The piezo it then used to move the

mirror by putting a voltage across it. The voltage necessary to move half a wavelength

is such that it is very easy to control. For our use λ2 ≈ .5µm which corresponds to a

voltage across the piezo of ≈ 30V. 30V is easy to achieve and allows for phase changes

with better than 1% precision.


4.3.2 Phase measurement

In order to change the phase of the lattices, we need to know what the phases of the

lattices are. To do this, we form an interferometer. We take the low power beams from

Figure 4.7: The interferometry apparatus used for phase measurement.

the combining PBS and send them to a non-polarizing cube (1). Half of the beam will

pass through, be reflected off a mirror (2), and then be partially reflected to the left by

the cube. The other half will be reflected right and pass through a lens (3). A mirror (4)

is situated at the focal length of the lens, and reflects the beams back through the lens.

The lens bends the beams back to parallel, but the beams have now switched places,

as the mirror and lens essentially form a 1-1 telescope. The reversed beams then are

partially transmitted to the left through the cube. The bottom two pairs of beams are

sent into a beam dump (5). The top two pairs is sent to two photodiodes (6,7). This

forms an interferometer between the two beams in each pair. This way we can easily

measure the relative phase. This allows us to both better modulate the phase as we

can see what we are doing, and correct for phase noise. There is much phase noise, both

on the short term and long term. We can send the signal from the photodoide though

a noise eater and form a feedback loop that controls the piezo voltage. Through this we

can easily and readily both modulate and stabilize the phase of the lattices.


Figure 4.8: The interferometry apparatus as seen from above.

Figure 4.9: The interferometry apparatus at an angle.

Chapter 5

Lattice Characterization

Figure 5.1: Spatial frequency as a function of beam spacing.

One of the first measurements we preformed using our setup was seeing if our period

modulation was working as we expected it to. Using a micrometer,we changed the

spacing between beams for a 1-D lattice and measured the resultant lattice’s period

based on that. We did this by taking pictures of the lattice directly using a camera and

then extracting data on the lattice using fitting algorithms. The data was plotted and

fit to our expected curve, as in figure 14. As a check of accuracy, the wavelength of

the laser was left as a variable and the data fitted with it as the parameter. Our laser

is very tightly locked and serves as an excellent reference point. Our laser is tuned to

670.9 nm, and out measured value was 668.9 nm. This is a difference of .3%. Our data

matches up excellently with the theory and bolsters our confidence in the setup.

30

Chapter 5. Lattice Characterization 31

Figure 5.2: Lattice slice measured using the razor blade technique.

This is data from our razor blade measurements. As said in section 4, this was an ef-

fort to have better lattice characterization on much smaller length scales due to CCDs

being unable to resolve such small patterns. As one can see, there is a distinct shape

of a lattice that can be seen and analyzed. To that extent the method was partially

successful. However there are many apparent issues. It is obviously very noisy. It has

been smoothed twice to attain the black line. This noise is caused by there being small

amounts of noise in the original error function-like signal, in blue. This noise is almost

negligible in comparison to the signal, however because we are taking the derivative and

it changes on a much shorter timescale than the signal we want, the derivatives end up

being roughly similar magnitudes. Because of this the signal must be smoothed by a

large margin for usable data to be gained. In addition, one may notice that the x axis is

time not distance. This is due to difficulty calibrating the stage used. This is a hardware

issue entirely, but an important and difficult one.

Our phase measurement was very successful. It can be seen in figure 16. The blue data

points are the photo-diode signal taken a normalized by its maximum. The green is the

expected behavior from the piezo’s data sheet. As you can see the curve is obeyed very

well up to around the 12V mark. After this point the behavior becomes much less like

what we expect. We do not know why this change occurs, however we have two full

periods to work with before this point, so there is no actual need for us to involve those

voltages.


Figure 5.3: Interferometry measurements and expected values.

We used the interferometer to measure the phase drift of the lattice overnight. The

results are in figure 17. Data was begun to be taken at around 11pm. The sharp drop

Figure 5.4: Overnight phase drift.

off at then end of hour 2 is the last grad student leaving the lab. Before this one can

see the noise and chaotic behavior of the phase when there are people working in the

lab. The reason there is such a steep decline is that the lights were switched off. The

lights act as the background as we are using a photo-diode, so the drop off is the room

becoming dark. The phase then changes linearly. The photo-diode is measuring the sin2

of the phase, so the periodic signal corresponds to a linear phase change. This is due to

thermal effects. The temperature in the ab steadily decreases when people are not in it

and the lights are off. This causes the breadboard to contract, moving the retroreflective

mirror closer to the BS pair. This causes a phase shift due to a changing path length.


The effect is small, the distance moved is about 7µm over 6 hours, but its enough to

effect the phase. In this case, we have a phase shift of around a cycle an hour. This

is easily manageable and only requires occasional adjustments. The change in small

time scale noise can also be see. As the room and the building calm down though the

night, the noise noticeably decreases. We do not know what happened at hour 8, we are

looking into it. Around hour 9, i.e. 8 am, people start arriving and the noise returns.

midway through hour 9 someone returns to lab and changes things withe the laser, the

sudden lack of a signal, and then turns on the lights, the jump in signal.

Chapter 6

Conclusion

In all we have successfully created lattices able to simulate complicated condensed mat-

ter systems. We have created lattices that are highly controllable, capable of tuning

their spatial frequency. We have also stabilized and modulated multiple lattices simul-

taneously in situ. Using our set up we have created novel methods of controlling the

phase of lattices and modulating them using a feedback loop. By superimposing these

lattices we have realized a quasiperiodic system. We hope to implement these lattices

with an actual BEC in the near future to utilize their potential.

34

Chapter 7

Bibliography

1. R. Grimm et al: Advances in Atomic Molecular and Optical Physics 42, 39 (1987).

2. S. Al Assam et al: PRA 82, 021604(R) (2010).

3. R. A. Williams et al: Opt. Express 16, 16977 (2008).

4. T. C. Li et al: Opt. Express 16, 5465 (2008).

5. Fallani et al: Opt. Express 13, 4303 (2005).

6. Peil et al: PRA 67, 051603(R) (2003).

7. Michael Lohse: Large-Spacing Optical Lattices for Many-Body Physics with De-

generate Quantum Gases, 2012.

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