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Quantum information and the orbital angular momentum of light in a turbulent atmosphere
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Quantum information and the orbital angular momentum of light in a turbulent atmosphere
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Content
QuantumInformation and the Orbital
AngularMomentum of Light in a
TurbulentAtmosphere
A DISSERTATION PRESENTED
BY
JOSÉ RAÚL GONZÁLEZ ALONSO
TO
THE DEPARTMENT OF PHYSICS AND ASTRONOMY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
IN THE SUBJECT OF
PHYSICS
UNIVERSITY OF SOUTHERN CALIFORNIA
LOS ANGELES, CALIFORNIA
DECEMBER 2016
©2016 – JOSÉ RAÚL GONZÁLEZ ALONSO
ALL RIGHTS RESERVED.
Thesisadvisor: Professor ToddBrun JoséRaúl González Alonso
QuantumInformation and the Orbital Angular Momentum of Light
ina Turbulent Atmosphere
ABSTRACT
In this dissertation, we study the effects of a turbulent atmosphere on the quantum information
encodedin the orbital angular momentum of photons.
InthefirstandsecondchaptersweusethesinglescreenmodeltoinvestigatehowtheKolmogorov
turbulence affects Laguerre-Gauss states. In the first chapter we use the techniques of approximate
error correction, whereas in the second we use adaptive optics, to compensate the noise. We numeri-
callyexplore the effectiveness of either schemein eachrespective chapter.
Finally, in the third chapter we use an infinitesimal propagation equation to derive a discrete
Lindbladequationthatcanalsobeusedasanoisemodel. Weinvestigatetheeffectsofthedominant
Lindblad operators and discuss different techniques that could be used to counter their effects. We
alsodiscuss possible schemesto protect quantum information encoded across multiple photons.
iii
iv
Contents
0 INTRODUCTION 1
0.1 Quantum Error Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
0.2 Orbital Angular Momentum of Light and Atmospheric Turbulence . . . . . . . . . . . . 2
1 USING APPROXIMATE ERROR CORRECTION TO PROTECT OAM PHOTONS FROM
THE EFFECTS OF TURBULENCE 5
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 OAM States and Atmospheric Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Numerical Simulation: Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Numerical Simulations: Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5 Encoding and Protecting a Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 Encoding and Recovery Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 RECOVERING QUANTUM INFORMATION IN ORBITAL ANGULAR MOMENTUM OF
PHOTONS BY ADAPTIVE OPTICS 21
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Adaptive Optics and Zernike Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 CalculatingtheMatrixElementsoftheSuperoperatorRepresentationofAOandTur-
bulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Solving the integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3 PROTECTING ORBITAL ANGULAR MOMENTUM OF PHOTONS IN INFINITESIMAL
NOISY PROPAGATION 39
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Infinitesimal Propagation Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3 Lindblad Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Error Detection and Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.5 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4 CONCLUSIONS 59
REFERENCES 61
v
vi
Listingof figures
1.1 ExampleofthespectrumofRwhenmax
in
Æ3,max
out
Æ6,w
0
Æ0.01m,C
2
n
Æ1£10
¡14
m
¡2/3
,¸Æ1£10
¡6
m, zÆ500m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Contour plots of (a) j0
L
i and (b) j1
L
i before the noise process, where max
in
Æ 3,
max
out
Æ6, w
0
Æ0.01m, C
2
n
Æ1£10
¡14
m
¡2/3
,¸Æ1£10
¡6
m,and zÆ500m. . . . . . . 10
1.3 Contourplotsof(a)j0
L
iand(b)j1
L
iafterthenoiseprocess,wheremax
in
Æ3,max
out
Æ
6, w
0
Æ0.01m, C
2
n
Æ1£10
¡14
m
¡2/3
,¸Æ1£10
¡6
m, and zÆ500 m. . . . . . . . . . . . 11
1.4 Contour plots of (a) j0
L
i and (b) j1
L
i after the recovery process, where max
in
Æ 3,
max
out
Æ6, w
0
Æ0.01 m, C
2
n
Æ1£10
¡14
m
¡2/3
,¸Æ1£10
¡6
m,and zÆ500m. . . . . . . 12
1.5 Behavior of the channel fidelity as a function of w/r
0
for several values of max
in
and
max
out
. Inallcases, w
0
Æ0.01m, C
2
n
Æ1£10
¡14
m
¡2/3
,¸Æ1£10
¡6
m. . . . . . . . . . 17
2.1 Probability to observe the initial state after turbulence and adaptive optics corrections. 37
2.2 Probability to observe the state j3Å∆l,0i after the turbulence and adaptive optics
correction for initial statej3,0i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3 Probability to observe the state j3,∆pi after the turbulence and adaptive optics cor-
rection for initial statej3,0i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1 ExampleoftheoperatorC when¸Æ1.0£10
¡6
m,!
0
Æ0.01m,C
2
n
Æ1.0£10
¡14
m
¡2/3
,
t
z
Æ100,and LÆ4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2 ExampleoftheoperatorD when¸Æ1.0£10
¡6
m,!
0
Æ0.01m,C
2
n
Æ1.0£10
¡14
m
¡2/3
,
t
z
Æ100,and LÆ4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3 Magnitudes of the eigenvalues of the operator
e
D when ¸Æ1.0£10
¡6
m, !
0
Æ0.01 m,
C
2
n
Æ1.0£10
¡14
m
¡2/3
, t
z
Æ100,and LÆ4. . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.4 Example of the Lindblad operatorL
1
when ¸Æ1.0£10
¡6
m, !
0
Æ0.01 m, C
2
n
Æ1.0£
10
¡14
m
¡2/3
, t
z
Æ100,and LÆ4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.5 Example of the Lindblad operatorL
2
when ¸Æ1.0£10
¡6
m, !
0
Æ0.01 m, C
2
n
Æ1.0£
10
¡14
m
¡2/3
, t
z
Æ100,and LÆ4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.6 Example of the Lindblad operator
e
L
1
when ¸Æ1.0£10
¡6
m, !
0
Æ0.01 m, C
2
n
Æ1.0£
10
¡14
m
¡2/3
, t
z
Æ100, LÆ4,and nÆ3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.7 Example of the Lindblad operator
e
L
2
when ¸Æ1.0£10
¡6
m, !
0
Æ0.01 m, C
2
n
Æ1.0£
10
¡14
m
¡2/3
, t
z
Æ100, LÆ4,and nÆ3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.8 Trace of the state in Eq. (3.34) after evolving for t
z
when ¸Æ1.0£10
¡6
m, !
0
Æ0.01
m, C
2
n
Æ1.0£10
¡14
m
¡2/3
, and®Æ¯Æ
1
p
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
vii
3.9 Probability of a detectable error when using a state of the form given in Eq. (3.34)
after evolving for t
z
when ¸Æ1.0£10
¡6
m, !
0
Æ0.01 m, C
2
n
Æ1.0£10
¡14
m
¡2/3
, and
®Æ¯Æ
1
p
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.10 MinimumfidelitybetweenaninitialstateoftheformgiveninEq. (3.34)andthestate
afterevolving for t
z
when¸Æ1.0£10
¡6
m,!
0
Æ0.01 m, C
2
n
Æ1.0£10
¡14
m
¡2/3
. . . . . 56
viii
Listingof tables
1.1 The effect of the size of the input and output spaces on the channel fidelity when
w
0
Æ0.01m, C
2
n
Æ1£10
¡14
m
¡2/3
,and¸Æ1£10
¡6
m. . . . . . . . . . . . . . . . . . . . . 18
2.1 Angular integrals for Z
k
(
r
R
,µ
)
Z˜
k
(
r
R
,µ
)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Angular integrals for Z
k
(
r
R
,µ
)
Z˜
k
(
r
′
R
,µ
′
)
and Z
k
(
r
′
R
,µ
′
)
Z˜
k
(
r
R
,µ
)
. . . . . . . . . . . . . . 30
2.3 Angular integrals for Z
k
(
r
′
R
,µ
′
)
Z˜
k
(
r
′
R
,µ
′
)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
ix
A MIS PADRES
x
Acknowledgments
I OWE THANKS TO SO MANY PEOPLE, but perhaps above all, to my wonderful advisor Prof. Todd A.
BrunforhiscontinuoussupportinmyPh.D.studies. Itwouldhavebeenimpossibletocompletethis
dissertationwithout his sage advice,patient explanations,shared wisdom, and gentle criticism.
I am also grateful to Professors Lidar, Johson, Willner, Haas, Reichardt, and Zanardi for their
very useful suggestions and feedback in my qualifying exam and in the different chapters of this
dissertation.
IwouldalsoliketoextendmythankstoKung-ChuanHsu,ChrisCantwell,YicongZheng,Ching-
Yi Lai, Scout Kingery, Siddharth Muthukrishnan, Milad Marvian, and Lin Jing for their encourage-
ment,company,andsupportinthetryingtimesthatsometimesonecanencounteringraduateschool.
I am specially grateful to my board game pals and regular lunch companions Patrick Davidson, Jan
Florjanczyk,and Alex Leavittwithout whom I could not havebeen able to see this through.
Iwouldhavebeenmultipletimesatalosswereitnotfortheincrediblehelpinallmattersadmin-
istrative of Gerrielyn Ramos, Corine Wong, Anita Fung, Tim Boston, Betty Byers, Lisa Moeller, and
Christina Tasulis Williams. It is easy to take for granted such superb work, but without their help
mylife would havebeen immeasurably more difficult.
Finally,IwouldliketothankmylovelywifePaulinaPinedoforherunwaveringsupportandlove.
Mylife would lose muchof its color,beauty,and excitement without her.
xi
0
Introduction
0.1 QUANTUM ERROR CORRECTION
NATURE IS FUNDAMENTALLY QUANTUM MECHANICAL, and information is fundamentally physical.
If such is the case, why is it that we rarely, if ever, observe the weirdness of quantum mechanics in
our everyday lives? It is because of a process known as decoherence that destroys one key aspect of
quantum states: their ability to interfere, constructively or destructively. Preserving this ability is
keyfortheapplicationsthatharnessquantumstrangenesstoenablethingsthatwouldbeimpossible
usingclassicalphysics.
Howto protectinformation encodedin a quantumsystem isnot obvious. In fact,at firstglance,it
would appear that it is an impossible task. Unlike classical states, arbitrary quantum states cannot
be cloned or copied. In other words, it is not possible in a quantum system to apply a repetition
codelikeinaclassicalsystemtoprotectinformation. Moreover,thepossibleerrorstoprotectagainst
1
in the quantum mechanical case form a continuous set. Hence, it would appear that infinite preci-
sion is required to determine the error that needs to be corrected. Moreover, there is yet another
complication: measuring a quantum state generally destroys it. Therefore, it seems impossible to
measure which error has occurred without disrupting the state to be protected. Fortunately, none
ofthecomplicationsmentionedaboveisinsurmountable,andtheprotectionofquantuminformation
from errorsis possible.
With the discovery of the first quantum error correcting code, it was realized that information
couldberedundantlyencodedinaquantumsystemwithoutviolatingtheno-cloningtheorem. Rather
thancopyingthestate,itisembeddedinasubspaceofalargersystem,usingbasisstatesthatsatisfy
a set of parity checks or stabilizers. Moreover, it is not necessary to measure the full encoded state,
but just the error syndrome defined by the parity checks in order to detect whether an error had
occurred and the steps necessary to correct it. Finally, while the errors may form a continuum,
measuringtheparitycheckswillprojectthemontoadiscretesetoferrorsthatcanthenbecorrected.
Thus,itseems possible to protect quantum information from the deleterious effects of noise.
Interestingly,quantumerrorcorrectionisnotonlyappliedtoprotectinformationstored,ortrans-
mitted,butalsowhileitisundergoingadynamicchangeinacomputation. Remarkablyenough,itis
possible to show that arbitrarily good quantum computation can be done fault tolerantly, even if the
operations applied are faulty, as long as the error per gate is below a certain threshold. We are in-
terestedin applying the theory of errorcorrection to protecting quantum information encoded inthe
orbitalangularmomentumoflighttravelinginfreespaceinordertoprotectitfromthedecoherence
caused byatmospheric turbulence.
0.2 ORBITAL ANGULAR MOMENTUM OF LIGHT AND ATMOSPHERIC TURBULENCE
Unlike the case of a photon with polarization, the quantum description of the physical state of a
singlephotonwithorbitalangularmomentumrequiresaninfinitedimensionalHilbertspace. While,
in principle, this might appear as a terrible complication, it actually represents a great opportunity
sinceitmakesitpossibletoencodearbitrarilylargeamountsofinformationinonephoton. However,
there is a major downside: when sending one photon with OAM through free space for the purposes
2
oftransmittingquantuminformation,itisimportanttonotethatwhileittravelsthroughaturbulent
atmosphere,itwillundergorandomspatialaberrationscausedbyrefractiveindexfluctuationsthat,
at optical frequencies, will cause the photon’s state to decohere [1–5]. Therefore, if the objective is to
takeadvantageofthedesirablepropertiesofOAMforanyquantumapplicationitbecomesnecessary
to protect the state from such a loss of coherence. In order to do so, it is possible to use the theory
of quantum error correction [6–8] to protect the information encoded in a photon. However, it is
importanttofirstobtainanaccuratedescriptionofthedecoherenceprocessofthephotonintermsof
theformalismnormallyusedinthetheoryofquantumopensystems[9]tocharacterizethedifferent
quantumerrors that can affect the physical state of a photon with OAM.
The outline of this dissertation is as follows. In Ch. 1 we present how to use a single phase
model to study the effects of turbulence on Laguerre-Gauss states and how it is possible to use the
theoryofapproximateerrorcorrectingcodes[10–13]tomitigatetheeffectsofatmosphericturbulence
[14]. In Ch. 2 we still use a single phase screen model to study the turbulence effects, but now
we also investigate how the techniques of adaptive optics can be applied to the mitigation of the
turbulence effects on OAM states. Finally, in Ch. 3, we study the turbulence from the perspective
of an error model with multiple phase screens that leads to an infinitesimal propagation equation
analysis [15, 16]. We then derive a discrete Lindblad representation for this representation and
studythemostrelevanterroroperatorsandtheireffects. Wealsodiscusspossibleschemestoencode
quantuminformation across multiple photons.
3
4
1
Using Approximate ErrorCorrection to Protect
OAMPhotons from the Effects of Turbulence
1.1 INTRODUCTION
IT IS A WELL KNOWN FACT that in classical electromagnetic theory the total angular momentum of
a field can always be decomposed into the sum of two terms [17]. One of these terms is identifiable
as the orbital part while the other is identifiable as the polarization part of the angular momentum.
Upon expressing these quantities as quantum mechanical operators [18], it becomes clear that each
of them requires very different spaces to be diagonalized. In the case of the polarization angular
momentum, it is sufficient to have a two-dimensional Hilbert space. On the other hand, for the
orbitalcase,an infinite dimensional space is needed.
Photons have always been the information carriers of choice in quantum information, with many
5
protocols using polarization or time-bin degrees of freedom to encode quantum information [19, 20].
Exploitingthephoton’sorbitalangularmomentum(OAM)couldprovidedistinctiveadvantages. The
main one is an increased alphabet size for information transmission [21] This interesting property,
could be applied in quantum key distribution [22, 23] However, this potential can only be realized if
suitable quantum information can be encoded in the OAM photon states, and if it can be protected
from thedecohering effect of atmospheric turbulence [1, 24–29]
Tobetterunderstandthedecoherenceaphotonundergoeswhentravelingthroughtheatmosphere
we numerically simulated the evolution and extracted a Kraus operator sum decomposition for the
resulting error process. We then studied one scheme for encoding and correcting quantum informa-
tion.
The plan of this chapter is as follows. The theoretical framework for the turbulence model, and
the calculation of the Kraus operators, are presented in Secs. 1.2 and 1.3. In Sec. 1.4, we describe
theresultsofournumericalsimulations. InSec. 1.5,wediscusstheproblemofencodinginformation
when we cannot do perfect error correction. In Sec. 1.6 we present the result of encoding a qubit
in a photon with OAM and how the channel fidelity behaves with the distance traveled for a given
encoding. Finally,in Sec. 1.7 weconclude.
1.2 OAM STATES AND ATMOSPHERIC TURBULENCE
Consider a beam that initially has a spatial wave function corresponding to an eigenstate of OAM.
Write suchan eigenfunction in cylindrical coordinates as
hrjl
0
,p
0
iÆ
1
p
2¼
R
l
0
,p
0
(r,z)exp(il
0
µ), (1.1)
where r
2
Æ x
2
Åy
2
, µ Æ arctan
(
y
x
)
, and the beam proppagates in the z direction. The functions
R
l
0
,p
0
(r,z)areabasisfortheradialdependence,suchastheLaguerre-Gaussfunctions[24]. Through-
6
outthis chapterwe use
R
l
0
,p
0
(r,z) Æ
A
w(z)
(p
2r
w(z)
)
jl
0
j
L
jl
0
j
p
0
(
2r
2
w(z)
2
)
e
¡r
2
/w(z)
2
£e
¡ikr
2
/[2R(z)]
e
i(2p
0
Åjl
0
jÅ1)tan
¡1
(z/z
R
)
, (1.2)
where w(z)Æw
0
√
1Å(z/z
R
)
2
is the beam width, R(z)Æz[1Å(z
R
/z)
2
] is the radius of wave-front cur-
vature,and z
R
Æ
1
2
kw
2
0
is the Rayleighrange. Thequantity tan
¡1
(z/z
R
)is known as the Gouy phase.
We assume that the effects of the atmospheric turbulence can be represented by the action of an
operator
ˆ
T
φ
suchthat
hrj
ˆ
T
φ
jl
0
,p
0
iÆexp
(
iφ(r,µ)
)
hrjl
0
,p
0
i. (1.3)
Thatis,thespatiallyvaryingphasechangeφ(r,µ)representsthecumulativeeffectoffluctuationsin
therefractiveindexofair[30,31]Ingeneral,φ(r,z)willbearandomfunctiondrawnfromasuitable
ensemble. The state of the beam after the interaction with the environment is a superposition of
severalOAM eigenstates:
jl
0
,p
0
i
ˆ
T
φ
7!
1
∑
lÆ0
1
∑
pÆ0
®
l,p,l
0
,p
0
jl,pi. (1.4)
Thecoefficients
{
®
l,p,l
0
,p
0
}
are given by
®
l,p,l
0
,p
0
Æ
1
2¼
Ï
rdrdµR
l,p
(r,z)
£ exp{i[µ(l
0
¡l)¡φ(r,µ)]}R
l
0
,p
0
(r,z). (1.5)
Weareinterestedinstudyingtheeffectofturbulenceonthemostgeneralpossiblequantumstate.
By linearity, it is sufficient to consider initial operators of the form jl,pi
⟨
l
′
,p
′
¯
¯
. These operators
transformaccording to
jl,pi
⟨
l
′
,p
′
¯
¯
7!
ˆ
T
φ
jl,pi
⟨
l
′
,p
′
¯
¯ˆ
T
†
φ
. (1.6)
7
FromEqs. (1.4)–(1.6) it follows that
jl,pi
⟨
l
′
,p
′
¯
¯
7!
1
4¼
2
∑
˜
l,˜ p,
˜
l
′
,˜ p
′
∫∫∫∫
rdrdµr
′
dr
′
dµ
′
£ R˜
l,˜ p
(r,z)R
l,p
(r,z)exp
{
i
[
µ
(
l¡
˜
l
)
Åφ(r,µ)
]}
£ R˜
l
′
,˜ p
′(r
′
,z)R
l
′
,p
(r
′
,z)exp
{
¡i
[
µ
′
(
l
′
¡
˜
l
′
)
Åφ(r
′
,µ
′
)
]}
£
¯
¯˜
l, ˜ p
⟩⟨
˜
l
′
, ˜ p
′
¯
¯
. (1.7)
Since the variations in the atmosphere are random, we take their ensemble average. If one assumes
that the refractive index fluctuations in the atmosphere are a Gaussian random process with zero
mean,thenonecanusethathexp(ix)iÆexp
(
¡
1
2
⟨
x
2
⟩)
andEq. (1.7)toshowthatonaverageageneral
OAM stateundergoes the following transformation
jl,pi
⟨
l
′
,p
′
¯
¯
7!
1
2¼
∑
˜
l,˜ p,˜ p
′
Ñ
rdrd¹r
′
dr
′
R˜
l,˜ p
(r,z)
£ R
l,p
(r,z)R
l
′
Å
˜
l¡l,˜ p
′(r
′
,z)R
l
′
,p
′(r
′
,z)exp
[
i¹
(
l¡
˜
l
)]
£ exp
(
¡
D
φ
(¯
¯
r¡r
′
¯
¯
)
2
)
¯
¯˜
l, ˜ p
⟩⟨
l
′
Å
˜
l¡l, ˜ p
′
¯
¯
. (1.8)
In the equation aboveD
φ
is called the phase structure function of the aberrations. We assume that
this functionis rotationally invariant, that is,the aberrations are isotropic,so that
D
φ
(¯
¯
r¡r
′
¯
¯
)
ÆD
φ
(r,µ,r
′
,µ
′
)ÆD
φ
(r,µ¡µ
′
,r
′
,0).
This, along with the trivial change of variables ¹Æµ¡µ
′
, ºÆ
1
2
(µÅµ
′
) turns Eq. (1.7) into Eq. (1.8).
This phase structure function depends on the model of turbulence and the power spectral density of
the fluctuations[31, 32]. Forthe Kolmogorov turbulence theory,the functionD
φ
isgiven by [33, 34]
D
φ
(¯
¯
r¡r
′
¯
¯
)
Æ2
[
24
5
¡
(
6
5
)]
5/6
(
¯
¯
r¡r
′
¯
¯
r
0
)5/3
, (1.9)
8
where
r
0
Æ
(
16.6
¸
2
∫
L
dℓC
2
n
)
¡3/5
(1.10)
is the Fried parameter, which has dimensions of length [33] In (1.10), ¸ is the wavelength, L is the
propagationpath,ℓislengthalongthepropagationpath,andC
n
iscalledtheatmosphericrefractive
index structure constant. In spite of being called a constant, C
n
depends on altitude and may vary
alongthe path [33]Note that other models for turbulence maylead to different expressions [31]
By evaluating (1.8), we can extract the superoperator elements of a completely positive, trace-
preserving map
ˆ
T
φ
. This map represents the decoherence process of the OAM states. By analyzing
ˆ
T
φ
, we hope to find the dominant noise processes of this decoherence, and construct error-correction
procedures that protect against them. Both the integral and the analysis of the superoperator must,
ingeneral, be done numerically.
1.3 NUMERICAL SIMULATION: PROCEDURE
Eigenvalue Magnitude (a.u.)
Eigenvalue Number
0 1000 2000 3000
−2
0
2
4
6
8
10
12
Figure 1.1: Example of the spectrum of R when max
in
Æ3, max
out
Æ6, w
0
Æ0.01 m, C
2
n
Æ1£10
¡14
m
¡2/3
,
¸Æ1£10
¡6
m, zÆ500 m.
9
② ✲ ❛ ① ✐ s ❧ ❡ ♥ ❣ t ❤ ✭ ♠ ✮ ✁ ✂ ✄ ☎ ✆ ✝ ✞ ✟ ✠ ✡ ☛ ☞✌
☛ ✂ ✌
✍ ✵ ✿ ✵ ✷ ✵ ✵ ✿ ✵ ✷
✍ ✵ ✿ ✵ ✷
✍ ✵ ✿ ✵ ✶
✵
✵ ✿ ✵ ✶
✵ ✿ ✵ ✷
② ✲ ❛ ① ✐ s ❧ ❡ ♥ ❣ t ❤ ✭ ♠ ✮ ✁ ✂ ✄ ☎ ✆ ✝ ✞ ✟ ✠ ✡ ☛ ☞✌
☛ ❜ ✌
✍ ✵ ✿ ✵ ✷ ✵ ✵ ✿ ✵ ✷
✍ ✵ ✿ ✵ ✷
✍ ✵ ✿ ✵ ✶
✵
✵ ✿ ✵ ✶
✵ ✿ ✵ ✷
Figure 1.2: Contour plots of (a) j0
L
i and (b) j1
L
i before the noise process, where max
in
Æ 3, max
out
Æ 6,
w
0
Æ0.01 m, C
2
n
Æ1£10
¡14
m
¡2/3
, ¸Æ1£10
¡6
m, and zÆ500 m.
10
② ✲ ❛ ① ✐ s ❧ ❡ ♥ ❣ t ❤ ✭ ♠ ✮ ✁ ✂ ✄ ☎ ✆ ✝ ✞ ✟ ✠ ✡ ☛ ☞✌
☛ ✂ ✌
✍ ✵ ✿ ✵ ✷ ✵ ✵ ✿ ✵ ✷
✍ ✵ ✿ ✵ ✷
✍ ✵ ✿ ✵ ✶
✵
✵ ✿ ✵ ✶
✵ ✿ ✵ ✷
② ✲ ❛ ① ✐ s ❧ ❡ ♥ ❣ t ❤ ✭ ♠ ✮ ✁ ✂ ✄ ☎ ✆ ✝ ✞ ✟ ✠ ✡ ☛ ☞✌
☛ ❜ ✌
✍ ✵ ✿ ✵ ✷ ✵ ✵ ✿ ✵ ✷
✍ ✵ ✿ ✵ ✷
✍ ✵ ✿ ✵ ✶
✵
✵ ✿ ✵ ✶
✵ ✿ ✵ ✷
Figure 1.3: Contour plots of (a) j0
L
i and (b) j1
L
i after the noise process, where max
in
Æ3, max
out
Æ6, w
0
Æ
0.01 m, C
2
n
Æ1£10
¡14
m
¡2/3
, ¸Æ1£10
¡6
m, and zÆ500 m.
11
② ✲ ❛ ① ✐ s ❧ ❡ ♥ ❣ t ❤ ✭ ♠ ✮ ✁ ✂ ✄ ☎ ✆ ✝ ✞ ✟ ✠ ✡ ☛ ☞✌
☛ ✂ ✌
✍ ✵ ✿ ✵ ✷ ✵ ✵ ✿ ✵ ✷
✍ ✵ ✿ ✵ ✷
✍ ✵ ✿ ✵ ✶
✵
✵ ✿ ✵ ✶
✵ ✿ ✵ ✷
② ✲ ❛ ① ✐ s ❧ ❡ ♥ ❣ t ❤ ✭ ♠ ✮ ✁ ✂ ✄ ☎ ✆ ✝ ✞ ✟ ✠ ✡ ☛ ☞✌
☛ ❜ ✌
✍ ✵ ✿ ✵ ✷ ✵ ✵ ✿ ✵ ✷
✍ ✵ ✿ ✵ ✷
✍ ✵ ✿ ✵ ✶
✵
✵ ✿ ✵ ✶
✵ ✿ ✵ ✷
Figure 1.4: Contour plots of (a) j0
L
i and (b) j1
L
i after the recovery process, where max
in
Æ3, max
out
Æ6,
w
0
Æ0.01 m, C
2
n
Æ1£10
¡14
m
¡2/3
, ¸Æ1£10
¡6
m, and zÆ500 m.
12
The elements of the superoperator
ˆ
T
φ
can be represented as a matrix. Taking a cue from expres-
sion(1.7),the matrix elements ofT shouldsatisfy
(
ˆ
T
φ
½
)
(
˜
l,˜ p,
˜
l
′
,˜ p
′
)
Æ
∑
l,p,l
′
,p
′
T
(
˜
l,˜ p,
˜
l
′
,˜ p
′
),(l,p,l
′
,p
′
)
½
(l,p,l
′
,p
′
)
, (1.11)
where we represent density operators as vectors in an appropriately chosen Hilbert space. The ele-
mentsof the vector are
½
(l,p,l
′
,p
′
)
Æ
⟨
l,p
¯
¯
½
¯
¯
l
′
,p
′
⟩
. (1.12)
Withthis choice,the matrix elements ofT are
T
(
˜
l,˜ p,
˜
l
′
,˜ p
′
),(l,p,l
′
,p
′
)
Æ
8
>
>
<
>
>
:
0, if
˜
l
′
̸Æl
′
Å
˜
l¡l
Ð
in(1.8), if
˜
l
′
Æl
′
Å
˜
l¡l.
(1.13)
Asmentionedintheprevioussection,thematrixelementsofT,ingeneral,mustbeevaluatednu-
merically. Forthispurpose,wehaveusedtheintegrationalgorithmsin[35]and[36]asimplemented
byJohnson
*
, and the GNUScientific Library [37]
In principle, the states used for communication could be arbitrary superpositions of OAM states.
However, we will assume that the input states to the channel are restricted to a subspace of the
OAM Hilbert space with angular momentum quantum numbers below some maximum. This kind
of truncation of the Hilbert space also makes the analysis easier; but since the effects of turbulence
canchangetheOAMofaphoton,itcreatesthepossibilityofleakageerrors,wheretheinputphotons
are transformed out of the subspace. To minimize this problem, we will allow the output subspace
of the photons to be higher dimensional than the input subspace. That means that the matrix of the
superoperatorcan be rectangular.
In general, the matrix representation of the superoperator
ˆ
T
φ
is sparse, and when considering
different dimensions for the input and output spaces, rectangular. For the OAM states before the
noise process, we include states with orbital angular momentum projections l
in
such that jl
in
j·
max
in
. After the noise process, we include OAM states such that jl
out
j· max
out
. For the radial
*
Seehttp://ab-initio.mit.edu/wiki/index.php/Cubature
13
quantum number,we includestatesjl,pi suchthat 0·p·max
in
ormax
out
, respectively.
From the noise process represented by
ˆ
T
φ
, we can obtain a Kraus representation. To this end,
consider thematrixR definedby rearranging the elements ofT:
R
(
˜
l,˜ p,l,p),(
˜
l
′
,˜ p
′
,l
′
,p
′
)
ÆT
(
˜
l,˜ p,
˜
l
′
,˜ p
′
),(l,p,l
′
,p
′
)
. (1.14)
The matrixR is always square, regardless of the values of max
in
and max
out
we use to truncate the
input andoutputspaces. Withthis definition,
½
ˆ
T
φ
7!
∑
l,p,l
′
,p
′
˜
l,˜ p,
˜
l
′
,˜ p
′
T
(
˜
l,˜ p,
˜
l
′
,˜ p
′
),(l,p,l
′
,p
′
)
½
(l,p,l
′
,p
′
)
j
˜
l, ˜ pih
˜
l
′
, ˜ p
′
j
Æ
∑
l,p,l
′
,p
′
˜
l,˜ p,
˜
l
′
,˜ p
′
R
(
˜
l,˜ p,l,p),(
˜
l
′
,˜ p
′
,l
′
,p
′
)
ˆ
O
(
˜
l,˜ p)(l,p)
½
(
ˆ
O
(
˜
l
′
,˜ p
′
)(l
′
,p
′
)
)
†
,
where
ˆ
O
(
˜
l,˜ p)(l,p)
Æj
˜
l, ˜ pihl,pj. Wenow diagonalizeR:
R
(
˜
l,˜ p,l,p),(
˜
l
′
,˜ p
′
,l
′
,p
′
)
Æ
∑
k
¸
k
v
k
(
˜
l,˜ p,l,p)
(
v
k
(
˜
l
′
,˜ p
′
,l
′
,p
′
)
)
¤
, (1.15)
and use thisto obtain a Kraus decomposition
½
ˆ
T
φ
7!
∑
k
ˆ
A
k
½
ˆ
A
†
k
, (1.16)
with Krausoperators
ˆ
A
k
givenby
ˆ
A
k
Æ
∑
˜
l,˜ p,l,p
√
¸
k
v
k
(
˜
l,˜ p,l,p)
ˆ
O
(
˜
l,˜ p)(l,p)
. (1.17)
Thedecompositionin(1.16)willallowustoordertheKrausoperatorsbytheirrespectiveimportance,
thatis,bythesizeoftheireigenvalues,andwillalsogiveussomeintuitionaboutthedominanteffects
of thenoise process on our initial state.
14
1.4 NUMERICAL SIMULATIONS: EXAMPLES
Toillustrateourdiscussionsofar,wepresentsomeoftheresultsofthenumericalsimulationsneeded
toobtaintheelementsofTandR[seeEqs. (1.13)and(1.14)],andtheerroroperatorsdefinedbyEq.
(1.17).
First, we observe that the eigenspectrum of R is dominated by its few largest eigenvalues (see
Fig. 1.1). The first, and largest one, is for
ˆ
A
1
. Upon further inspection, we see that
ˆ
A
1
is close to the
identityoperator
ˆ
A
1
Æc
ˆ
Iň ", (1.18)
where∥ˆ "∥ is small. We note that some of the remaining nonzero eigenvalues are degenerate, corre-
spondingto a pair of error operators.
The actions of two error operators with the same eigenvalue parallel each other. One operator
raises the initial value of l by a given amount, while the other lowers it by the same amount. As the
eigenvalues become smaller and smaller, the values by which the pair of error operators raise and
lower l become larger and larger. Put differently, large changes in the value of OAM for a beam are
unlikely. ThequestionofhowtoencodeourinitialOAMstatestoprotectthemfromthenoiseprocess
willbe studied in the following sections.
1.5 ENCODING AND PROTECTING A QUBIT
In this section we explore the possibility of encoding a qubit using a suitable basis of OAM states to
protect it against turbulence. The basis states should be such that after encoding a qubit and then
performing a suitable noise recovery operation we maximize the quantum channel fidelity [38] of
the atmospheric turbulence channel. However, before choosing the encoding, we must know how to
implementthe error recovery map.
Since we are truncating both the input and output space we cannot expect to be able to do per-
fect quantum error correction: the error map is not trace preserving. However, we can do approxi-
mate quantum error correction (see [10–12]. The idea behind approximate quantum error correction
(AQEC for short) over a system with Hilbert space H and noise process E is to solve the triple
15
optimization problem
max
W
max
R
min
jÃi2H
0
F
(
jÃihÃj,W
¡1
◦R◦E◦W jÃihÃj
)
. (1.19)
Here,H
0
denotes the qudit space (a subspace ofH),E the error map,W the encoding map,R the
recovery map, and F the fidelity. One can fix an encoding map and then try to find the solutions to
the problem
max
R
min
jÃi2C
F
(
jÃihÃj,R◦EjÃihÃj
)
, (1.20)
whereC isthecodespace. Although(1.20)canbesolvedbyconvexoptimizationmethods[39],wecan
more easily use the transpose channel to construct a near optimal recovery map [12]. The recovery
map weuse is given by
R
P
Æ
∑
k
P
ˆ
A
†
k
E(P)
¡1/2
(¢)E(P)
¡1/2
ˆ
A
k
P. (1.21)
In Eq. (1.21), P is the projector onto the code space, andE(¢)Æ
∑
k
ˆ
A
k
(¢)
ˆ
A
†
k
is the error map. Finally,
the inverse ofE(P)(actually,a pseudoinverse) is taken over its support.
Now that we know how to implement the error recovery map for a particular choice of code, we
can write anexpression for the channelfidelityC:
C Æmax
P
1
d
2
∑
k,l
tr
°
°
°P
ˆ
A
†
k
E(P)
¡1/2
ˆ
A
l
°
°
°
2
, (1.22)
where P is the projector onto the code space, and d is the dimension of the space we are encoding.
Fora qubit, dÆ2.
1.6 ENCODING AND RECOVERY RESULTS
ToillustratetheperformanceoftheAQECapproach,wenowshowsomeexamplesofhowthechannel
fidelity varies with the path length of the path for a qubit (dÆ2) encoded using OAM. Ideally, we
shouldchooseourcodespacetomaximizethechannelfidelity. However,evenwiththesimplifications
of the AQEC, optimizing (1.22) has only been feasible for very small examples. In these examples,
the optimal encoding was usually one that maximized the distance between the OAM values of the
16
basis states. For this reason, we used the following encoding to investigate how robust OAM states
withAQEC are to atmospheric noise:
j0i
L
Æ
j¡max
in
,0iÅj¡max
in
,1i
p
2
, (1.23)
j1i
L
Æ
jmax
in
,max
in
¡1iÅjmax
in
,max
in
i
p
2
. (1.24)
In Figs. 1.2 - 1.4 we plot the basis states we have chosen for certain cases before and after the
noise process, and after the recovery. We can see that the recovery map visibly restores the states,
thoughnot perfectly,to their initial conditions.
Inoursimulationswealsoinvestigatedtheeffectofthedimensionsoftheinputandoutputspaces
on the channel fidelity. For this purpose, we used max
in
Æ1,2,3 for the input space, and max
out
Æ
max
in
,max
in
Å1,...,6fortheoutputspace. Theoutputspacewasalwaysatleastasbigastheinput
space. The effects of both the dimensions and the path length are summarized in Fig. 1.5 and in
Table 1.1.
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Channel Fidelity
w
r
0
1,2
1,3
1,4
1,5
1,6
2,3
2,4
2,5
2,6
3,4
3,5
3,6
Figure 1.5: Behavior of the channel fidelity as a function of w/r
0
for several values of max
in
and max
out
. In
all cases, w
0
Æ0.01 m, C
2
n
Æ1£10
¡14
m
¡2/3
, ¸Æ1£10
¡6
m.
17
Table 1.1: The effect of the size of the input and output spaces on the channel fidelity when w
0
Æ 0.01 m, C
2
n
Æ 1£10
¡14
m
¡2/3
, and ¸Æ
1£10
¡6
m.
max
in
max
out
w
r
0
Æ3.4046£10
¡3 w
r
0
Æ1.3561£10
¡2 w
r
0
Æ9.6954£10
¡2 w
r
0
Æ2.6639£10
¡1 w
r
0
Æ7.1673£10
¡1
1 1 0.9997 0.9965 0.8864 0.4749 0.1169
1 2 0.9999 0.9987 0.9511 0.6879 0.2528
1 3 0.9999 0.9988 0.9558 0.7399 0.3556
1 4 0.9999 0.9988 0.9569 0.7548 0.4240
1 5 0.9999 0.9988 0.9574 0.7603 0.4672
1 6 0.9999 0.9988 0.9577 0.7629 0.4942
2 2 0.9995 0.9954 0.8595 0.4659 0.1522
2 3 0.9999 0.9994 0.9709 0.7423 0.2927
2 4 1.0000 0.9997 0.9836 0.8399 0.4015
2 5 1.0000 0.9997 0.9862 0.8710 0.4815
2 6 1.0000 0.9998 0.9871 0.8824 0.5375
3 3 0.9994 0.9938 0.8258 0.4344 0.1700
3 4 0.9999 0.9992 0.9619 0.7110 0.3001
3 5 1.0000 0.9995 0.9805 0.8282 0.4044
3 6 1.0000 0.9996 0.9847 0.8754 0.4856
18
We expected that as we made the input space larger (higher values of max
in
) the values for the
channel fidelity would improve, as there would be more distance between the values of OAM in the
encodedstates. Weseethatingeneraltermsthisistrue,providedthattheoutputspaceisalsolarge
enough to reduce leakage errors due to truncation of the space (this is, for example, the reason why
thecase max
in
Æ2, max
out
Æ6has better channelfidelities than the case max
in
Æ3, max
out
Æ6).
Rather unsurprisingly, but disapointingly, we also see that the channel fidelity rapidly decays
as the ratio w/r
0
increases. This decay is so fast that, even with error correction, after w/r
0
Æ
2.6639£10
¡1
, the OAM states for a moderate level of turbulence C
2
n
Æ1£10
¡14
m
¡2/3
become unus-
ableforquantumcommunication. Thiscouldbeanartifactofourapproximations,orofasuboptimal
encoding or recovery. However, we believe it is more likely that OAM photons are too sensitive to
atmospheric turbulence to be useful for quantum communication without some much better method
of compensating for the noise. Perhaps coupling quantum error correction with other protection
methods, such as decoherence-free subspaces, or the use of adaptive optics, may prove fruitful in
increasing the communications range over which OAM photons can be used to transmit quantum
information. These are subjects for future research.
Ourresultsagreewiththeworkdone[26,32,40]and,fromanexperimentalperspective,fromthe
behavior observed in [41–43] However, unlike these works, we have derived a Kraus representation
for the atmospheric turbulence channel and studied the behavior of the channel fidelity, not the
channel capacity as is more commonly done. Also, we have studied the application of approximate
quantum error correction to the problem of protecting OAM photons from atmospheric turbulence,
somethingthat, as far as we know,has not been tried before.
1.7 CONCLUSIONS
In principle, OAM could be used to encode large amounts of quantum information per photon to be
transmitted through air. Through numerical simulations, we were able to extract a Kraus represen-
tation for the error process OAM photons undergo through a turbulent atmosphere. We described
theeffectsofturbulenceusingKolmogorov’smodel. ToprotectOAMphotonsformerrors,weapplied
the methods of approximate quantum error correction. Unfortunately, our numerical simulations
19
indicate that even with quantum error correction, the range over which we can use OAM for effec-
tive quantum communications is very limited. Perhaps it is possible to better protect OAM photons
from noise if we couple quantum error correction with other methods such as adaptive optics or
decoherence-free subspaces. These are the subject of ongoing research.
20
2
RecoveringQuantum Information in Orbital
AngularMomentum of Photons by Adaptive
Optics
2.1 INTRODUCTION
PHOTONS ARE VERY INTERESTINGcarriersforquantuminformationsincetheyarerelativelyeasyto
produceandtransmit. Mostcommonly,thepolarizationofphotons—atwodimensionalspace—isthe
choice for encoding qubits in free space, though dual-rail and time-bin encodings are also common
(particularly for transmission through optical fiber). However, higher dimensional spaces can be
obtained with the use of the orbital angular momentum (OAM) of photons [24, 44, 45], which could
potentially allow larger channel capacities and increased key generation rates for protocols such as
21
quantum keydistribution.
Unfortunately,OAMofphotonsishighlysensitivetoatmosphericturbulence[1,26,30,32,46,47],
and a method to protect quantum information encoded in the OAM of photons while they travel
in a turbulent atmosphere remains elusive [14]. If OAM photons are ever to be used in quantum
communications, it is imperative to properly understand the processes they undergo in a turbulent
atmosphere [1, 25, 26, 30–34, 46–57] and how tocompensate for them.
One possibility to mitigate the effects of turbulence is to use adaptive optics [58–60]. While this
possiblityhasbeenexploredforclassicalcommunication[61]itstillremainstobestudiedthoroughly
in the quantum case. In this paper, we explore the effects of atmospheric turbulence with adaptive
opticsonboththeradialandazimuthaldegreesoffreedom[62,63]ofOAMphotons. Forthispurpose,
we model atmospheric turbulence using the Kolmogorov spectrum, and the correction effects using
Zernike polynomials[58].
In Sec. 2.2 we expand the effects of turbulence in Zernike functions, and we use them in Sec. 2.3
to calculate the residual errors after the adaptive optics correction in a first order expansion. We
describetheseerrorsasasuperoperatoractingonthestateoftheinputphotons,andderiveintegral
expressionsforthematrixelementsofthissuperoperator. InSec.2.4weshowhowtocalculatethese
integrals,andin Sec. 2.5 weillustrate our discussion with numerical examples.
2.2 ADAPTIVE OPTICS AND ZERNIKE FUNCTIONS
As a photon propagates through a turbulent atmosphere, it encounters small fluctuations in the
density(andhencetheindexofrefraction)ofair. Thesefluctuationsvaryacrossthewavefrontofthe
photons. We model the cumulative effects of turbulence by a spatially-varying phase change φ(r,µ),
where r and µ are cylindrical coordinates across the wave front. (A third coordinate, z, represents
the distance along the beam, which is generally the distance from the transmitter to the receiver.)
The methodofadaptive optics is to estimate this phase and then compensate for its effects.
Thesemethodswereoriginallydevelopedforastronomy[60],buthavemorerecentlybeenapplied
to classical communication by OAM of photons through free space [61]. Adaptive optics works by
sendingabrightpulseoflightinastandardstatethroughthesamevolumeofairimmediatelybefore
22
the communication pulse. Because air moves slowly compared to light, the fluctuations encountered
bythisprobebeamwillbeveryclosetothatencounteredbythecommunicationpulse. Bymeasuring
howtheprobebeamisdistorted,thephasefunctionφ(r,µ)canestimated,andactiveopticalelements
canapply a compensating phase shift to cancel out the distortion.
How is this phase estimated? One approach is to expand φ(r,µ) in terms of an orthogonal set of
functions that are defined in the receiving aperture of a system. Such set is given by the Zernike
functions {Z
k
(r/R,µ)} [64], which are defined on a disk of radius R. In terms of these functions we
writethe phase as
φ(r,µ)Æ
1
∑
kÆ1
a
k
Z
k
(
r
R
,µ
)
. (2.1)
How do we define the functions {Z
k
(r/R,µ)}? We assume 0·r·R. For integers n,m such that n¸0,
n¸jmj and n¡jmj is even, we can define a joint index k using the ordering conventions in [58]. For
m̸Æ0and k even:
Z
k
(
r
R
,µ
)
Æ
1
R
√
2(nÅ1)
¼
P
jmj
n
(
r
R
)
cos(mφ). (2.2)
For m̸Æ0 and k odd:
Z
k
(
r
R
,µ
)
Æ
¡1
R
√
2(nÅ1)
¼
P
jmj
n
(
r
R
)
sin(mφ). (2.3)
Finally,for mÆ0:
Z
k
(
r
R
,µ
)
Æ
1
R
√
nÅ1
¼
P
0
n
(
r
R
)
. (2.4)
Thepolynomials P
jmj
n
aregiven by
P
jmj
n
(
r
R
)
Æ
n¡jmj
2
∑
sÆ0
(¡1)
s
(
n¡s
s
)(
n¡2s
n¡jmj
2
¡s
)
(
r
R
)
n¡2s
. (2.5)
The relationship between the integer parameters m and n and the joint index k is somewhat
complicated. Defining
TÆ
n(nÅ1)
2
, T
2
ÆT mod2, m
0
Æjmj mod2, m
1
Æjmj¡1 mod2,
23
the jointindex k as used in [58] is given by
kÆ1ÅTÅjmj¡(1¡±
jmj,0
)(H(m)[T
2
m
0
Å(1¡T
2
)m
1
]ÅH(¡m)[T
2
m
1
Å(1¡T
2
)m
0
]), (2.6)
where
H(x)Æ
8
>
>
<
>
>
:
0, if xÇ0,
1, if x¸0.
(2.7)
If mÈ0, then k is even, and if mÇ0, then k is odd. In the case where mÆ0, the parity of k can be
eitherevenorodd. Theindexkisconstructedinsuchawaythat,foragivenn,indiceswithasmaller
value of jmj are smaller. In what follows, to maintain clarity in switching back and forth between
theintegerparameters m and n andthejointindex k,wewilldenoteby n
k
and m
k
thetwointegers
corresponding to theparticular joint index k.
Withthedefinitions(2.2),(2.3),and(2.4),theZernikepolynomialsformanorthonormalsetonthe
disk ofradius R:
∫
R
0
∫
¼
¡¼
rdrdµZ
k
(
r
R
,µ
)
Z˜
k
(
r
R
,µ
)
Ʊ
k,
˜
k
. (2.8)
Since the Zernike functions form a complete orthonormal set on the disk of radius R, we can always
expand the phase change φ(r,µ) using them, as in Eq. (2.1). The main task in adaptive optics is
then to estimate the coefficients of the expansion a
k
up to a certain number of Zernike modes J,
and use this information to eliminate the aberrations due to the first J modes. In other words, an
experimental procedureyields a correction phaseφ
c
givenby
φ
c
(r,µ)Æ
J
∑
kÆ1
a
k
Z
k
(
r
R
,µ
)
(2.9)
24
whichis then subtracted fromφ(r,µ), leavinga residualphase [58]:
φ
A
(r,µ)Æφ(r,µ)¡φ
c
(r,µ)
Æ
1
∑
kÆJÅ1
a
k
Z
k
(
r
R
,µ
)
.
(2.10)
Generally, the coefficients a
k
are estimated using a procedure such as the Shack-Hartmann wave-
front sensing technique. However, because of the helical nature of the wavefronts of OAM states,
thismustbemodifiedtoincorporateabrightprobebeaminordertodothewavefrontestimation[61]
required in the adaptive optics correction procedure. The correction itself is done using fast active
optics.
2.3 CALCULATING THE MATRIX ELEMENTS OF THE SUPEROPERATOR REPRESENTATION
OF AO AND TURBULENCE
We are interested in the protecting the quantum information initially encoded in an eigenstate of
OAM from the effects of turbulence with the help of adaptive optics. We write the input state in
termsof basis vectorsjl
0
,p
0
i:
hrjl
0
,p
0
iÆ
1
p
2¼
R
l
0
,p
0
(r,z)exp(il
0
µ), (2.11)
wherer
2
Æx
2
Åy
2
,µÆarctan
(
y
x
)
. Weassumethestatepropagatesinthe zdirection. Inwhatfollows,
wewill use the Laguerre-Gauss functions R
l
0
,p
0
(r,z)[24]:
R
l
0
,p
0
(r,z)Æ
2A
l,p
w(z)
(
p
2r
w(z)
)
jl
0
j
L
jl
0
j
p
0
(
2r
2
w(z)
2
)
e
¡r
2
/w(z)
2
e
¡ikr
2
/[2R(z)]
e
i(2p
0
Åjl
0
jÅ1)arctan(z/z
R
)
, (2.12)
where w(z)Æw
0
√
1Å(z/z
R
)
2
is the beam width, R(z)Æz[1Å(z
R
/z)
2
] is the radius of wave-front cur-
vature, and z
R
Æ
1
2
kw
2
0
is the Rayleigh range. The quantity tan
¡1
(z/z
R
) is known as the Gouy phase,
25
and thenormalization constant A
l,p
is
A
l,p
Æ
√
p!
(pÅjlj)!
. (2.13)
The functions L
jlj
p
(x)are generalized Laguerre polynomials:
L
jlj
p
(x)Æ
p
∑
iÆ0
(¡1)
i
(
pÅjlj
p¡i
)
x
i
i!
. (2.14)
Similarly to our earlier work in [14], the effects of turbulence and adaptive optics corrections on a
basis state(2.11)is represented by an operator
ˆ
A
φ
A
suchthat
hrj
ˆ
A
φ
A
jl
0
,p
0
iÆexp
(
iφ
A
(r,µ)
)
hrjl
0
,p
0
i. (2.15)
However, this only describes the change of state for a particular realization of the noise. We must
make an ensemble average over φ
A
(r,µ) to find the superoperator representing the residual error
process aftertheadaptive optics.
The OAM eigenstates in Eq. (2.11) form a complete basis. Therefore, we can use them to expand
the state after the combined effects of the turbulent atmosphere and adaptive optics. In general,
after averagingover the noise the state will be mixed, so we represent it as a density matrix
½Æ
∑
l,l
′
,p,p
′
½
l,p;l
′
,p
′jl,pihl
′
,p
′
j.
By linearity, it suffices to know how the noise superoperator acts on outer products of the form
jl,pi
⟨
l
′
,p
′
¯
¯
:
jl,pi
⟨
l
′
,p
′
¯
¯
7!
1
4¼
2
∑
˜
l,˜ p,
˜
l
′
,˜ p
′
∫∫∫∫
rdrdµr
′
dr
′
dµ
′
R˜
l,˜ p
(r,z)R
l,p
(r,z)exp
[
i
[
µ
(
l¡
˜
l
)
Åφ
A
(r,µ)
]]
£R˜
l
′
,˜ p
′(r
′
,z)R
l
′
,p
(r
′
,z)exp
[
¡i
[
µ
′
(
l
′
¡
˜
l
′
)
Åφ
A
(r
′
,µ
′
)
]]¯
¯˜
l, ˜ p
⟩⟨
˜
l
′
, ˜ p
′
¯
¯
.
(2.16)
Since the atmospheric variations in the refraction index are random, we take the ensemble average
E[¢]of(2.16). Tocalculateit,weneedanexpressionforE
[
exp{i
(
φ
A
(r,µ)¡φ
A
(r
′
,µ
′
)
)
}
]
. Assumingthat
26
φ
A
is a Gaussian random variable with zero mean, it is straightforward to show that the ensemble
averagesimplifies to
E
[
exp{i
(
φ
A
(r,µ)¡φ
A
(r
′
,µ
′
)
)
}
]
Æexp
{
¡
1
2
E
[
(
φ
A
(r,µ)¡φ
A
(r
′
,µ
′
)
)
2
]
}
. (2.17)
Assuming that the residual effects of the uncorrected noise are small, we consider only terms up to
firstorder in the series expansion of the exponential:
exp
{
¡
1
2
E
[
(
φ
A
(r,µ)¡φ
A
(r
′
,µ
′
)
)
2
]
}
¼1¡
1
2
E
[
(
φ
A
(r,µ)¡φ
A
(r
′
,µ
′
)
)
2
]
. (2.18)
Expandingφ
A
interms of Zernike functions as in Eq. (2.10),we can write
E
[
(
φ
A
(r,µ)¡φ
A
(r
′
,µ
′
)
)
2
]
Æ
1
∑
k,
˜
kÆJÅ1
E
[
a
k
a˜
k
]
(
Z
k
(
r
R
,µ
)
¡Z
k
(
r
′
R
,µ
′
))(
Z˜
k
(
r
R
,µ
)
¡Z˜
k
(
r
′
R
,µ
′
))
. (2.19)
The covariance of the expansion coefficients can be calculated explicitly using the Fourier trans-
formsof the Zernike functions. If n
k
Ån˜
k
¸2, then E
[
a
k
a˜
k
]
is givenby
E
[
a
k
a˜
k
]
Æ(¡1)
1
2
(n˜
k
¡n
k
)
MR
2
(
R
r
0
)5
3
√
(n
k
Å1)(n˜
k
Å1)±
m
k
,m˜
k
I
n
k
,n˜
k
, (2.20)
where
MÆ
4
p
2
(
3
5
¡
(
6
5
))
5/6
¡
(
11
6
)
2
¼
11/3
¼0.04579117421711036, (2.21)
and
r
0
Æ
(
16.6
¸
2
∫
L
dℓC
2
n
)
¡3/5
(2.22)
is called the Fried parameter, which has dimensions of length [33]. (Because of the convention we
used in Eqs. (2.2–2.4), there is an extra factor of ¼R
2
compared to the expression in Ref. [58].) In
(2.22), ¸ is the wavelength, L is the propagation path, ℓ is a length element along the propagation
path, and C
2
n
is called the atmospheric refractive index structure constant and has units of L
¡
2
3
. In
spite of being called a constant, C
2
n
depends on altitude, pressure, temperature and may vary along
the path [31]. However, for the horizontal path free-space propagation, it may be approximated by a
27
constant [31]. Therefore,in whatfollows we approximate r
0
¼
(
16.6C
2
n
z/¸
2
)
¡3/5
.
Finally,we will usethat
I
n
k
,n˜
k
Æ
¼
11/3
¡
(
14
3
)
¡
(
1
2
(n
k
Ån˜
k
¡
14
3
Å3)
)
2¡
(
1
2
(n˜
k
¡n
k
Å
14
3
Å1)
)
¡
(
1
2
(n
k
¡n˜
k
Å
14
3
Å1)
)
¡
(
1
2
(n
k
Ån˜
k
Å
14
3
Å3)
). (2.23)
It is perhaps worth noting that the covariance of the expansion coefficients in Eq. (2.20) becomes
small as n
k
Ån˜
k
becomeslarge.
Puttingalloftheabovetogether,andthentakingtheensembleaverage,theexpressioninEq.(2.16)
is approximately
jl,pi
⟨
l
′
,p
′
¯
¯
7!jl,pi
⟨
l
′
,p
′
¯
¯
¡
1
8¼
2
∑
˜
l,˜ p,
˜
l
′
,˜ p
′
∑
k,
˜
k
∫∫∫∫
rdrdµr
′
dr
′
dµ
′
E
[
a
k
a˜
k
]
exp
[
i
[
µ
(
l¡
˜
l
)
¡µ
′
(
l
′
¡
˜
l
′
)]]
£
(
Z
k
(
r
R
,µ
)
¡Z
k
(
r
′
R
,µ
′
))(
Z˜
k
(
r
R
,µ
)
¡Z˜
k
(
r
′
R
,µ
′
))
£R˜
l,˜ p
(r,z)R
l,p
(r,z)R˜
l
′
,˜ p
′(r
′
,z)R
l
′
,p
(r
′
,z)
¯
¯˜
l, ˜ p
⟩⟨
˜
l
′
, ˜ p
′
¯
¯
.
(2.24)
After somework we can obtain exact expressions for all the integrals in Eq. (2.24).
2.4 SOLVING THE INTEGRALS
Whendoingtheintegrationofthezerothordertermoftheexpansion(i.e. wherenoZernikefunctions
appear),wesimplyusetheorthonormalityoftheLaguerre-Gaussfunctionstoconcludethatthisterm
reducesto±
l,
˜
l
±
p,˜ p
±
l
′
,
˜
l
′±
p
′
,˜ p
′. Calculatingthefirstorderterm(thatis,theonewithZernikefunctions)
of the expansionrequires a bit more work.
2.4.1 FIRST ORDER TERM OF THE EXPANSION: ANGULAR PART OF THE INTEGRATION
First,wenotethatsincethecovarianceoftheexpansioncoefficientsincludes±
m
k
,m˜
k
,thenkand
˜
k,as
wellasn
k
andn˜
k
,havethesameparityfornon-zeroterms. Hence,theangularpartoftheproductof
functions Z
k
and Z˜
k
willbehavethesamefunctionalform,andtheintegralsvanishunlessm
k
Æ ˜ m
k
.
Thus, there are only three possible cases for each of the angular forms of each product depending on
28
whether m
k
is negative,positive,or zero.
Allof these angular integrals take exactly the same form:
F
(
l,l
′
,
˜
l,
˜
l
′
,m
k
)
Æ
∫
¼
¡¼
∫
¼
¡¼
dµdµ
′
exp
[
i
[
µ
(
l¡
˜
l
)
¡µ
′
(
l
′
¡
˜
l
′
)]]
f(µ)g(µ
′
),
where the particular functions f(µ) and g(µ
′
) depend on which product of Zernike functions is being
integrated, and also on the value of m
k
. We have four possible products of Z
k
and Z˜
k
to analyze,
whichgiveusthreedistinctintegrals;andforeachcasetheresultisdifferentform
k
È0, m
k
Ç0,and
m
k
Æ0. Wesummarize theresults of all the angular integrals in the tables below.
Table 2.1: Angular integrals for Z
k
(
r
R
,µ
)
Z
˜
k
(
r
R
,µ
)
rangeof m
k
f(µ)g(µ
′
) result
m
k
Ç0 sin(m
k
µ)
2
F
1
(
l,l
′
,
˜
l,
˜
l
′
,m
k
)
Æ
8
>
>
<
>
>
:
2¼
2
if l
′
Æ
˜
l
′
and lÆ
˜
l
¡¼
2
if l
′
Æ
˜
l
′
and lÆ
˜
l§2m
k
0 otherwise
m
k
È0 cos(m
k
µ)
2
F
1
(
l,l
′
,
˜
l,
˜
l
′
,m
k
)
Æ
8
>
>
<
>
>
:
2¼
2
if l
′
Æ
˜
l
′
and lÆ
˜
l
¼
2
if l
′
Æ
˜
l
′
and lÆ
˜
l§2m
k
0 otherwise
m
k
Æ0 1 F
1
(
l,l
′
,
˜
l,
˜
l
′
,m
k
)
Æ
{
4¼
2
if l
′
Æ
˜
l
′
and lÆ
˜
l
0 otherwise
1. Thefirst case is Z
k
(
r
R
,µ
)
Z˜
k
(
r
R
,µ
)
. Theresults of the angular integrals are given in Table 2.1.
2. The second case is Z
k
(
r
R
,µ
)
Z˜
k
(
r
′
R
,µ
′
)
. The results of the angular integrals are given in Ta-
ble 2.2.
3. Thethirdcaseis Z
k
(
r
′
R
,µ
′
)
Z˜
k
(
r
R
,µ
)
,whoseintegralisequaltothatincase2becausem
k
Æm˜
k
.
Theresults of the angular integrals are also given in Table 2.2.
4. The fourth case is Z
k
(
r
′
R
,µ
′
)
Z˜
k
(
r
′
R
,µ
′
)
. The results of the angular integrals are given in Ta-
ble 2.3.
29
Table 2.2: Angular integrals for Z
k
(
r
R
,µ
)
Z
˜
k
(
r
′
R
,µ
′
)
and Z
k
(
r
′
R
,µ
′
)
Z
˜
k
(
r
R
,µ
)
rangeof m
k
f(µ)g(µ
′
) result
m
k
Ç0 sin(m
k
µ)sin(m
k
µ
′
) F
2,3
(
l,l
′
,
˜
l,
˜
l
′
,m
k
)
Æ
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
¼
2
if l
′
Æ
˜
l
′
Åm
k
and lÆ
˜
lÅm
k
¼
2
if l
′
Æ
˜
l
′
¡m
k
and lÆ
˜
l¡m
k
¡¼
2
if l
′
Æ
˜
l
′
Åm
k
and lÆ
˜
l¡m
k
¡¼
2
if l
′
Æ
˜
l
′
¡m
k
and lÆ
˜
lÅm
k
0 otherwise
m
k
È0 cos(m
k
µ)cos(m
k
µ
′
) F
2,3
(
l,l
′
,
˜
l,
˜
l
′
,m
k
)
Æ
{
¼
2
if l
′
Æ
˜
l
′
§m
k
and lÆ
˜
l∓m
k
0 otherwise
m
k
Æ0 1 F
2,3
(
l,l
′
,
˜
l,
˜
l
′
,m
k
)
Æ
{
4¼
2
if l
′
Æ
˜
l
′
and lÆ
˜
l
0 otherwise
2.4.2 FIRST ORDER TERM OF THE EXPANSION: RADIAL PART OF THE INTEGRATION
Now that we have completed the angular part of the integrals, we can proceed with the radial part.
The different cases are the same as we used above. For simplicity, we will denote the products of the
constants of normalization (and exponentials involving the Gouy phase) in Eqs. (2.2–2.4) and (2.12)
as
N
(
l,p,l
′
,p
′
,
˜
l, ˜ p,
˜
l
′
, ˜ p
′
,k,
˜
k
)
Æ
16ϵ
m
k
¼R
2
w(z)
4
A
l,p
A
l
′
,p
′A˜
l,˜ p
A˜
l
′
,˜ p
′
√
(n
k
Å1)
(
n˜
k
Å1
)
£e
[
iarctan
(
z
z
R
)
(2pÅjlj¡2p
′
¡jl
′
j¡2˜ p¡j
˜
ljÅ2˜ p
′
Åj
˜
l
′
j)
] (2.25)
where
ϵ
m
k
Æ
8
>
>
<
>
>
:
2, if m
k
̸Æ0,
1, if m
k
Æ0.
(2.26)
Intheintegralsthatfollow,weintegratetheproductoftherespectiveLaguerreandZernikepoly-
nomials term by term using the substitutions uÆ
2r
2
w(z)
2
and vÆ
2r
′2
w(z)
2
. Additionally, when possible
we have used the conditions that arise from the angular integrals (and the covariance of the expan-
30
Table 2.3: Angular integrals for Z
k
(
r
′
R
,µ
′
)
Z
˜
k
(
r
′
R
,µ
′
)
range of m
k
f(µ)g(µ
′
) result
m
k
Ç0 sin(m
k
µ
′
)
2
F
4
(
l,l
′
,
˜
l,
˜
l
′
,m
k
)
Æ
8
>
>
<
>
>
:
2¼
2
if l
′
Æ
˜
l
′
and lÆ
˜
l
¡¼
2
if l
′
Æ
˜
l
′
∓2m
k
and lÆ
˜
l
0 otherwise
m
k
È0 cos(m
k
µ
′
)
2
F
4
(
l,l
′
,
˜
l,
˜
l
′
,m
k
)
Æ
8
>
>
<
>
>
:
2¼
2
if l
′
Æ
˜
l
′
and lÆ
˜
l
¼
2
if l
′
Æ
˜
l
′
∓2m
k
and lÆ
˜
l
0 otherwise
m
k
Æ0 1 F
4
(
l,l
′
,
˜
l,
˜
l
′
,m
k
)
Æ
{
4¼
2
if l
′
Æ
˜
l
′
and lÆ
˜
l
0 otherwise
sion coefficients) relating the different indices (since the angular integrals vanish except for certain
combinations of indices). To get an exact expression, we integrate the product of all the polynomials
involvedterm by term using Eqs.(2.5)and (2.14).
Theintegrals all take the following form:
N
∫
R
0
dr
′
∫
R
0
drrr
′
L
jlj
p
(
2r
2
w(z)
2
)
L
j
˜
lj
˜ p
(
2r
2
w(z)
2
)
L
jl
′
j
p
′
(
2r
′
2
w(z)
2
)
L
jl
′
j
˜ p
′
(
2r
′
2
w(z)
2
)
f(r)g(r
′
)e
¡
2r
2
w(z)
2
¡
2r
′2
w(z)
2
, (2.27)
where f and g are functions that will depend on which product of Zernike functions is being inte-
grated.
1. Thefirstcaseis Z
k
(
r
R
,µ
)
Z˜
k
(
r
R
,µ
)
. Alloftheangularintegralsofthiscasevanishexceptwhen
l
′
Æ
˜
l
′
. The functions f and g in this case are
f(r)Æ
(
p
2r
w(z)
)
jljÅj
˜
lj
P
jm
k
j
n
k
(
r
R
)
P
jm
k
j
n˜
k
(
r
R
)
(2.28)
g(r
′
)Æ
(
p
2r
′
w(z)
)
2jl
′
j
(2.29)
31
andthe result of the integral is
N w(z)
4
16
(
p
′
Å
¯
¯
l
′
¯
¯
)
!
p
′
!
±
p
′
,˜ p
′G
1
(
l,p,
˜
l, ˜ p,k,
˜
k
)
, (2.30)
where
G
1
(
l,p,
˜
l, ˜ p,k,
˜
k
)
Æ
p
∑
jÆ0
˜ p
∑
˜ ȷÆ0
n
k
¡jm
k
j
2
∑
j
′
Æ0
n
˜
k
¡jm
k
j
2
∑
˜ ȷ
′
Æ0
(¡1)
(jÅj
′
Ř ȷŘ ȷ
′
)
j!˜ ȷ!
(
pÅjlj
p¡j
)(
˜ pÅ
¯
¯˜
l
¯
¯
˜ p¡ ˜ ȷ
)(
n
k
¡j
′
j
′
)(
n
k
¡2j
′
n
k
¡jm
k
j
2
¡j
′
)
£
(
n˜
k
¡ ˜ ȷ
′
˜ ȷ
′
)(
n˜
k
¡2˜ ȷ
′
n˜
k
¡jm
k
j
2
¡ ˜ ȷ
′
)(
p
2R
w(z)
)
(2j
′
Å2˜ ȷ
′
¡n
k
¡n˜
k
)
°
(
1
2
(
jljÅ
¯
¯˜
l
¯
¯
Ån
k
Ån˜
k
Å2
(
jÅ ˜ ȷ¡j
′
¡ ˜ ȷ
′
))
Å1,
2R
2
w(z)
2
)
.
(2.31)
InEq. (2.31),we haveused the lower incomplete gamma function
°(®,x)Æ
∫
x
0
dxe
¡t
t
®¡1
towrite the result of the term by term integration.
2. The second case is Z
k
(
r
R
,µ
)
Z˜
k
(
r
′
R
,µ
′
)
. In this case, the combinations of indices where the dif-
ferent angular integrals vanish depend on the values of m
k
, without a particular combination
commonto all values of m
k
. The functions f and g inthis case are
f(r)Æ
(
p
2r
w(z)
)
jljÅj
˜
lj
P
jm
k
j
n
k
(
r
R
)
(2.32)
g(r
′
)Æ
(
p
2r
′
w(z)
)
jl
′
jÅj
˜
l
′
j
P
jm
k
j
n˜
k
(
r
′
R
)
(2.33)
andafter a tedious but straightforwardcalculation, we find the result of the integral:
N w(z)
4
16
G
2
(
l,p,l
′
,p
′
,
˜
l, ˜ p,
˜
l
′
, ˜ p
′
,k,
˜
k
)
(2.34)
32
where
G
2
(
l,p,l
′
,p
′
,
˜
l, ˜ p,
˜
l
′
, ˜ p
′
,k,
˜
k
)
Æ
0
B
@
p
∑
jÆ0
˜ p
∑
˜ ȷÆ0
n
k
¡jm
k
j
2
∑
j
′
Æ0
(¡1)
(jŘ ȷÅj
′
)
j!˜ ȷ!
(
pÅjlj
p¡j
)(
˜ pÅ
¯
¯˜
l
¯
¯
˜ p¡ ˜ ȷ
)(
n
k
¡j
′
j
′
)(
n
k
¡2j
′
n
k
¡jm
k
j
2
¡j
′
)
£
(
p
2R
w(z)
)
(2j
′
¡n
k)
°
(
1
2
(
jljÅ
¯
¯˜
l
¯
¯
Ån
k
Å2
(
jÅ ˜ ȷ¡j
′
))
Å1,
2R
2
w(z)
2
)
1
A
£
0
B
@
p
′
∑
iÆ0
˜ p
′
∑
˜ ıÆ0
n
˜
k
¡jm
k
j
2
∑
i
′
Æ0
(¡1)
(iŘ ıÅi
′
)
i!˜ ı!
(
p
′
Å
¯
¯
l
′
¯
¯
p
′
¡i
)(
˜ p
′
Å
¯
¯˜
l
′
¯
¯
˜ p
′
¡˜ ı
)(
n˜
k
¡i
′
i
′
)(
n˜
k
¡2i
′
n˜
k
¡jm
k
j
2
¡i
′
)
£
(
p
2R
w(z)
)
(2i
′
¡n˜
k
)
°
(
1
2
(¯
¯
l
′
¯
¯
Å
¯
¯˜
l
′
¯
¯
Ån˜
k
Å2
(
iŘ ı¡i
′
))
Å1,
2R
2
w(z)
2
)
1
A
.
(2.35)
This solution also works for the Z
k
(
r
′
R
,µ
′
)
Z˜
k
(
r
R
,µ
)
terms, by using (2.35) and switching n
k
$
n˜
k
. In thiscase,we call the functionG
3
.
3. The third case is Z˜
k
(
r
′
R
,µ
′
)
Z˜
k
(
r
′
R
,µ
′
)
. In this case, the angular part of the integration (calcu-
latedabove) vanishes except when lÆ
˜
l,and the functions f and g are
f(r)Æ
(
p
2r
w(z)
)
2jlj
(2.36)
g(r
′
)Æ
(
p
2r
′
w(z)
)
jl
′
jÅj
˜
l
′
j
P
jm
k
j
n
k
(
r
′
R
)
P
jm
k
j
n˜
k
(
r
′
R
)
. (2.37)
Wecan follow the same procedure we used to obtain (2.31) to find theresult of the integral
N w(z)
4
16
(pÅjlj)!
p!
±
p,˜ p
G
4
(
l
′
,p
′
,
˜
l
′
, ˜ p
′
,k,
˜
k
)
(2.38)
33
where
G
4
(
l
′
,p
′
,
˜
l
′
, ˜ p
′
,k,
˜
k
)
Æ
p
′
∑
iÆ0
˜ p
′
∑
˜ ıÆ0
n
k
¡jm
k
j
2
∑
i
′
Æ0
n
˜
k
¡jm
k
j
2
∑
˜ ı
′
Æ0
(¡1)
(iÅi
′
Ř ıŘ ı
′
)
i!˜ ı!
(
p
′
Å
¯
¯
l
′
¯
¯
p
′
¡i
)(
˜ p
′
Å
¯
¯˜
l
′
¯
¯
˜ p
′
¡˜ ı
)(
n
k
¡i
′
i
′
)(
n
k
¡2i
′
n
k
¡jm
k
j
2
¡i
′
)
£
(
n˜
k
¡˜ ı
′
˜ ı
′
)(
n˜
k
¡2˜ ı
′
n˜
k
¡jm
k
j
2
¡˜ ı
′
)(
p
2R
w(z)
)
(2i
′
Å2˜ ı
′
¡n
k
¡n˜
k
)
°
(
1
2
(¯
¯
l
′
¯
¯
Å
¯
¯˜
l
′
¯
¯
Ån
k
Ån˜
k
Å2
(
iŘ ı¡i
′
¡˜ ı
′
))
Å1,
2R
2
w(z)
2
)
.
(2.39)
2.4.3 PUTTING IT ALL TOGETHER
Now that we have obtained the results for all of the integrals, we can finally write the result of the
first order expansion we did in (2.24). To this end, we use the following shorthand for the angular
and radial results of the integration. We take the definitions of F
1
from Table 2.1, F
2
and F
3
from
Table2.2,andF
4
fromTable2.3. FortheradialpartweusethedefinitionsofG
1,2,3,4
fromEqs.(2.31–
2.39). With all of this, we can write the combined effects of the turbulence and the adaptive optics
correction as
jl,pi
⟨
l
′
,p
′
¯
¯
7!jl,pi
⟨
l
′
,p
′
¯
¯
¡
1
8¼
2
∑
˜
l,˜ p,
˜
l
′
,˜ p
′
,k,
˜
k
C
(
±
p
′
,˜ p
′′
A
2
l
′
,p
′
F
1
G
1
¡F
2
G
2
¡F
3
G
3
Å
±
p,˜ p
A
2
l,p
F
4
G
4
)
¯
¯˜
l, ˜ p
⟩⟨
˜
l
′
, ˜ p
′
¯
¯
(2.40)
where
C ÆE
[
a
k
a˜
k
]
N
w(z)
4
16
Æ
ϵ
m
k
M
¼
A
l,p
A
l
′
,p
′A˜
l,˜ p
A˜
l
′
,˜ p
′(n
k
Å1)
(
n˜
k
Å1
)
(¡1)
1
2
(n˜
k
¡n
k
)
£I
n
k
,n˜
k
e
[
iarctan
(
z
z
R
)
(2pÅjlj¡2p
′
¡jl
′
j¡2˜ p¡j
˜
ljÅ2˜ p
′
Åj
˜
l
′
j)
]
±
m
k
,m˜
k
.
(2.41)
Theindicesk,
˜
karesuchthatk¸JÅ1,
˜
k¸JÅ1andn
k
Ån˜
k
¸2. Weuse J todenotethehighestorder
oftheZernikefunctionsusedinthecorrectionoftheturbulenceeffects(seeEqn.(2.10)). Additionally,
alltheOAMradialindicesarenon-negativeintegerswhereastheazimuthalindicescantakepositive
or negativeinteger values.
34
2.5 NUMERICAL EXAMPLE
WecanseefromEq. (2.40)that,uptofirstorder,themapthatrepresentstheeffectsoftheturbulence
and the adaptive optics, which we will denote by
ˆ
A
φ
A
, is a perturbation to the identity map. How
small this perturbation is, and how it behaves as we modify some of the parameters used in the
description of both the turbulence and the Zernike expansion, is something we will study below
using the same procedure we used previously in [14]. Since the map is linear (a superoperator) and
completely positive, we can represent it by a Choi matrix [65]. Diagonalizing the Choi matrix of this
representationallows us to find a set of Kraus operators for the map.
WecancharacterizetheperturbationbyusingthedimensionlessparametersR/w(z), w(z)/r
0
,and
z/z
R
, and the number J of modes used in the correction. Because we are doing a first order approx-
imation to the map that represents the effects of the turbulence and the adaptive optics corrections,
wewilllimitourselvestovaluesoftheparametersthatrepresentweakturbulenceandsmallresidual
errors.
In what follows, we will illustrate numerically (using the library in [37]) how changes in the
parameters affect the map given by Eq. (2.40). To facilitate the analysis, we will restrict the sizes
of the Hilbert spaces of the OAM states before and after the channel. Specifically, we will consider
inputstateswithazimuthalindicesthatsatisfyjl
in
j·3andradialindicessuchthat0·p
in
·6,and
outputstates with azimuthal indicesjl
out
j·6andradial indices 0·p
out
·6.
To ensure the validity of the approximation used to derive Eq. (2.40) we have considered cases
where the turbulence is weak by choosing a small value of C
2
n
. Moreover, we also chose parameters
in order to ensure that the scintillation (which we neglect) is weak. For the case of Gaussian beams,
thescintillation is weak when the Rytov variance [31]
¾
2
R
Æ1.637t
5
6
z
(
w(0)
r
0
)5
3
(2.42)
satisfiesthe condition [31]
¾
2
R
Ç
(
t
z
Å
1
t
z
)5
6
. (2.43)
35
For cases where no adaptive optics correction is applied (i.e. the number of modes corrected is
zero), we have used the numerical procedure from our previous work [14]. However, for all the cases
in which there are correction being applied, we use Eq. (2.40) to estimate the final state numerically
with thehelp of the excellent GNU Scientific Library [37].
Justasin[14],wefoundthatthespectrumoftheChoimatrixofthemap
ˆ
A
φ
A
isdominatedbyits
largesteigenvalue. Thiseigenvaluecorrespondstoanerroroperatorthatisclosetotheidentity. The
remaining nonzero eigenvalues mostly come in degenerate pairs. One of the error operators associ-
atedwiththesedegenerateeigenvalueslowerstheorbitalangularmomentumazimuthalnumberby
a given amount, while the other operator raises it by the same amount. As the amount by which the
OAMisraisedorloweredincreases,themagnitudeoftheeigenvaluedecreases(makingtheseerrors
weaker or less probable). These pairs of error operators can also raise or lower the radial number
p. The effects of these errors are more noticeable the larger the value of the azimuthal or radial
numbers become.
As an example of this dependence, consider a state that initially has lÆl
′
Æl
0
and pÆ p
′
Æ0. In
Fig. 2.1 we see that as the value of l
0
increases, the probability that the output state is the same
as the initial state decreases. However, as the number of corrected modes increases, the probability
that the input state is unchanged also increases, and the differences between these probabilities
for different values of l
0
become smaller. (Unsurprisingly, this is also true as the strength of the
turbulence decreases.)
We can also study the probability to measure neighboring modes to the initial state after the
effectsofthechannel. InFigs.2.2and2.3weseethatiftheinitialstateisj3,0i,thenasweincrease
the number of modes corrected (or decrease the turbulence strength), the probabilities to observe
neighboring states j3Å∆l,0i or j3,∆pi diminish, just as the probability to observe the initial state
increases. Interestingly, the probabilities to observe neighboring azimuthal modes are greater than
the probilitiesto observe neighboring radial modes.
36
Nu m b e r o f M o d e s C o r r e c t e d
0 10 15 20 30
P r ob ab i l i t y
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
l 0 = 0
l 0 = 1
l 0 = 2
l 0 = 3
(a)
R
w(z)
Æ9.2088,
w(z)
r
0
Æ0.2165, and
z
z
R
Æ0.4234
Nu m b e r of M o d e s C or r e c t e d
0 10 15 20 30
P r ob ab i l i t y
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
l 0 = 0
l 0 = 1
l 0 = 2
l 0 = 3
(b)
R
w(z)
Æ9.8596,
w(z)
r
0
Æ0.1167, and
z
z
R
Æ0.1693
Figure 2.1: Probability to observe the initial state after turbulence and adaptive optics corrections.
N u m b e r of M o d e s C o r r e c t e d
0 10 15 20 30
P r ob ab i l i t y
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
" l = ! 9
" l = ! 8
" l = ! 7
" l = ! 6
" l = ! 5
" l = ! 4
" l = ! 3
" l = ! 2
" l = ! 1
" l = 0
" l = 1
" l = 2
" l = 3
(a)
R
w(z)
Æ9.2088,
w(z)
r
0
Æ0.2165, and
z
z
R
Æ0.4234
Nu m b e r of M o d e s C or r e c t e d
0 10 15 20 30
P r ob ab i l i t y
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
" l = ! 9
" l = ! 8
" l = ! 7
" l = ! 6
" l = ! 5
" l = ! 4
" l = ! 3
" l = ! 2
" l = ! 1
" l = 0
" l = 1
" l = 2
" l = 3
(b)
R
w(z)
Æ9.8596,
w(z)
r
0
Æ0.1167, and
z
z
R
Æ0.1693
Figure 2.2: Probability to observe the state j3Å∆l,0i after the turbulence and adaptive optics correction for
initial state j3,0i.
37
Nu m b e r of M o d e s C or r e c t e d
0 10 15 20 30
P r ob ab i l i t y
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
" p = 0
" p = 1
" p = 2
" p = 3
" p = 4
" p = 5
" p = 6
(a)
R
w(z)
Æ9.2088,
w(z)
r
0
Æ0.2165, and
z
z
R
Æ0.4234
Nu m b e r o f M o d e s C o r r e c t e d
0 10 15 20 30
P r ob ab i l i t y
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
" p = 0
" p = 1
" p = 2
" p = 3
" p = 4
" p = 5
" p = 6
(b)
R
w(z)
Æ9.8596,
w(z)
r
0
Æ0.1167, and
z
z
R
Æ0.1693
Figure 2.3: Probability to observe the state j3,∆pi after the turbulence and adaptive optics correction for ini-
tial state j3,0i.
2.6 CONCLUSIONS
We have derived an approximate map for the effects of Kolmogorov atmospheric turbulence and
adaptive optics on orbital angular momentum states of a photon, assuming that the effects of the
uncorrected noise are small. Using this result, we numerically explored the dependence on some of
the dimensionless parameters used to characterize the noise process, and on the number of modes
correctedbyadaptiveoptics. Wehaveseenthatadaptiveopticsmaycompensateforsomeofthenoise
due to weak atmospheric turbulence. However, more research is required to extend this to the case
of strongturbulence,or to a case where only a few modes are corrected.
For the case we studied, however, the results seems quite reasonable: increasing the number of
modes corrected by adaptive optics increases the probability that the states are transmitted without
errors. Moreover, it is possible to find a Kraus map for the residual errors after adaptive optics
havebeen applied. An interesting possibility is tocombine adaptive optics (whichin a sensereduces
the noise in the channel) with quantum error correction (which could be used to protect quantum
information fromthe remaining weak errors). This is an important direction for future work.
38
3
Protecting OrbitalAngular Momentum of
Photons in InfinitesimalNoisy Propagation
3.1 INTRODUCTION
PHOTONS WITH ORBITAL ANGULAR MOMENTUM (OAM)[24,44,66,67]travelingthroughfreespace
hold the potential to become a useful system for high bandwidth applications in both classical and
quantum communications. However, to achieve this it will be necessary to protect the photons from
the pernicious effects caused by a turbulent atmosphere. Small fluctuations in the density of the
air lead to random changes in the index of refraction across the wave front of a propagating photon;
thesephaseshiftsdistortthestateofthephoton,causingshiftsinitsangularmomentumandradial
degrees of freedom. This distortion represents a strong source of noise for quantum communication
thatrequires robust error correction or suppression [14].
39
Inthequantumcase,understandingthisnoiserequiresprecisemodelingofthedecoherenceeffects
of such an atmosphere as well as an application of quantum error detection or correction schemes to
protect the quantum information encoded in eachphoton. To this end, we will be using the infinites-
imalpropagationequationofRoux[15,68]tomodelthenoiseprocessthataphotonundergoeswhile
traveling through the air, and will derive a Lindblad representation of this process with a discrete
set ofLindblad operators.
While in principle the orbital angular momentum states offer an infinite-dimensional Hilbert
space in which to store quantum information, in practice only a finite number of such states may
be practically prepared or manipulated. Larger quantum states, therefore, will have to be encoded
across multiple photons—particularly likely if the quantum information is encoded in a quantum
error-correcting code (QECC). However, multiple photons traveling together with a time separation
that is less than the characteristic time of the turbulence process will undergo the same random
phase shifts, as they propagate through the same volume of air with the same density fluctuations.
This means that the noise process for such a train of photons will be symmetric under permutations
of the photons. From the Lindblad representation for a single photon, we can therefore find the
representation fora train of closelyspaced photons.
We then briefly discuss possible methods for error correction with a codeword encoded in the an-
gularorbitalmomentaofatrainofphotons,whetheritispossibletoexploitthisexchangesymmetry
toboosttheresistancetonoise,andthepossibilityofcombiningerrorcorrectionwithadaptiveoptics.
Weendby giving a few numerical examples of our schemeand briefly discussing our results.
3.2 INFINITESIMAL PROPAGATION EQUATION
In [15, 68] the so-called infinitesimal propagation equation (IPE) for an OAM entangled biphoton
travelling in a turbulent atmosphere is presented. The formalism is then used in [16] to derive a
Lindblad-like equation for the evolution of a single photon with a continuous spectrum of Lindblad
operators. However, unlike the work in [16], we would like to obtain a discrete set of Lindblad
operatorsandinvestigatetheireffectsinordertogaininsightintotheerrorprocessanOAMphoton
experiences ina turbulent atmosphere and (hopefully) how to correct it.
40
We start with the IPE for a single OAM photon using the notation and results of Roux from
[15, 68]:
@
z
½
mn
(z)Æ¡iS
mu
(z)½
un
(z)Åi½
mv
(z)S
vn
(z)ÅL
mnuv
(z)½
uv
(z)¡L
T
½
mn
(z). (3.1)
Here, z represents the propagation distance of the photon along the beam path; this distance plays
therole of time in a usual master equation.
In Eq. (3.1), the indices m,n,u, and v each represent a collective index for both the radial and
orbital degrees of freedom of the OAM state represented by the density matrix ½. Repeated indices
implya summation. The operator that represents the free-space propagation is
S
x,y
(z)Æ
i
2k
∫
jKj
2
G
x
(K,z)G
¤
y
(K,z)
d
2
K
4¼
2
. (3.2)
Thedissipative part of the evolution is given by
L
mnuv
(z)Æk
2
∫
©
1
(K)W
m,u
(K,z)W
¤
n,v
(K,z)
d
2
K
4¼
2
. (3.3)
Finally,there is a divergent dissipative term that is given by
L
T
Æk
2
∫
©
1
(K)
d
2
K
4¼
2
. (3.4)
In Eqs. (3.2–(3.4)), the vector KÆ (k
x
,k
y
) is the two dimensional projection of the propagation
vectorkÆ(k
x
,k
y
,k
z
). The function G
x
(K,z) is the two-dimensional momentum space wave function
of the OAM basis. Therefore, because of the orthogonality of the OAM basis, the momentum space
wavefunctions satisfy
∫
G
x
(K,z)G
¤
y
(K,z)
d
2
K
4¼
2
Ʊ
x,y
. (3.5)
Theconvolution of the momentum space wavefunctions is denoted byW
x,y
(K,z):
W
x,y
(K,z)Æ
∫
G
x
(K
1
,z)G
¤
y
(K
1
¡K
,
z)
d
2
K
1
4¼
2
. (3.6)
In the IPE formalism it is possible in principle to use any power spectral density©
1
(K) for the
41
turbulencemodel. Forthecurrentchapter,ifweignoretheeffectoftheinnerscaleoftheturbulence,
we can start with the von Karman power spectral density [31] with the Fourier convention from
[15, 68]:
©
1
(K)Æ
20¼
2
C
2
n
9¡
(
1
3
)(
jKj
2
Å·
2
0
)
11/6
¼
8.186994C
2
n
(
jKj
2
Å·
2
0
)
11/6
. (3.7)
In Eq. (3.7), ·
2
0
is used for the outer scale of the turbulence. In our calculations, it allow us to
regularize the integrals in Eqs. (3.3) and (3.4) so that we can eventually take the large outer scale
limit ·
0
! 0. This is an important case because it allows us to study the Kolmogorov model of
turbulence.
3.2.1 GENERATING FUNCTIONS FOR THE INTEGRALS
To analyze the effects of turbulence via the IPE, we must write the integrals in Eqs. (3.2–3.4) in a
form that is more amenable to calculations. In [69], a generating function for the Laguerre-Gauss
modes is used to obtain generating functions for the different integrals in Eqs. (3.2–3.4). In what
follows,webriefly review these results.
The generating function forthe Laguerre-Gauss modes used in [15, 68]is
G Æ
1
∑
n,mÆ0
1
m!
L
m
n
(
2(u
2
Åv
2
)
1Åt
2
)[
w(1Åit)
1¡it
]
n
[(uÅiv)pÅ(u¡iv)q]
m
(1¡it)
1Åm
Æ
1
Ω(t,w)
exp
[
(uÅiv)p
Ω(t,w)
Å
(u¡iv)q
Ω(t,w)
¡
(1Åw)(u
2
Åv
2
)
Ω(t,w)
]
, (3.8)
whereΩ(t,w)Æ1¡w¡it¡iwt. Thenormalizedcoordinates u,v,and t aregivenby uÆx/!
0
, vÆy/!
0
and tÆz¸/¼!
2
0
. Intheseexpressions!
0
istheinitialbeamwaistand¸isthewavelengthofthebeam
of light. The parameters p, q, and w generate the Laguerre-Gauss modes by taking derivatives of
the generatingfunction according to the following rules:
M
LG
r,l
(u,v,t)Æ
8
>
>
>
>
>
<
>
>
>
>
>
:
N
[
1
r!
@
r
w
@
jlj
p
G
]
w,p,qÆ0
for lÈ0
N
[
1
r!
@
r
w
G
]
w,p,qÆ0
for lÆ0
N
[
1
r!
@
r
w
@
jlj
q
G
]
w,p,qÆ0
for lÇ0,
(3.9)
42
where r and l are the radial and azimuthal indices, respectively, andN is the normalization con-
stant:
N Æ
[
r!2
jljÅ1
¼(rÅjlj)!
]1/2
. (3.10)
The rule in Eq. (3.9) also applies when using the generating functions of the integrals for the
different terms in the IPE. However, for this we will need the Fourier transform of the generating
function,whichis
F{G}Æ
¼
1Åw
exp
[
i¼(aÅib)p
1Åw
Å
i¼(a¡ib)q
1Åw
¡
¼
2
(a
2
Åb
2
)Ω(t,w)
1Åw
]
. (3.11)
Here, a and b are related tothe wavevector components by k
x
Æ2¼a/!
0
and k
y
Æ2¼b/!
0
.
FREE-SPACE PROPAGATION TERM
Usingthegeneratingfunctionin(3.11),andthenexchangingtheorderoftheintegralandthederiva-
tives, we can obtain a generating function for Eq. (3.2). Furthermore, from the form of this function,
we find that the azimuthal indices involved must be equal, and that the radial indices can differ at
mostby one. In other words,the result of the integral in (3.2)simplifies to
S
m,n
(z)Æ
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
i(1ÅjljÅ2r)
2z
R
, l
m
Æl
n
Æl,r
m
Ær
n
Ær,
i(1ÅjljÅr)
1
2(1År)
1
2
2z
R
,
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
l
m
Æl
n
Æl,
jr
m
¡r
n
jÆ1,
rÆ
r
m
År
n
¡1
2
,
0, otherwise,
(3.12)
where the azimuthal indices are indicated by l
m
Æl
n
Æl and the radial indices are indicated by r
m
and r
n
.
43
DIVERGENT AND DISSIPATIVE TERMS
Withthe spectraldensity we chosein (3.7),the divergent dissipative term is
L
T
Æ
8¼
3
C
2
n
3·
10/6
0
¸
2
¡
(
1
3
)¼
30.86424C
2
n
·
10/6
0
¸
2
. (3.13)
However, to generate the function for the dissipative and divergent terms, we now need to use a
generating functionfor the radial indices of the modal correlation functions [15, 68]:
W
rG
(K,ϕ,z) Æ
exp(¡X)exp[i(l
m
¡l
n
)ϕ]E
jl
n
j
n
E
jl
m
j
m
(1¡w
m
w
n
)
[
r
n
!
(jl
n
jÅr
n
)!
]
1/2
[
r
m
!
(jl
m
jÅr
m
)!
]
1/2
£
M(l
m
,l
n
)
∑
sÆ0
jl
m
j!jl
n
j!(¡X)
¡s
(jl
m
j¡s)!(jl
n
j¡s)!s!
, (3.14)
where l
m
and l
n
are the azimuthal indices and r
m
and r
n
are their associated radial indices. More-
over [15, 68],
M(l
m
,l
n
)Æ
1
2
(jl
m
jÅjl
n
j¡jl
m
¡l
n
j), (3.15)
XÆ
K
2
³
m
³
¤
n
´
2
8¼
2
(1¡w
m
w
n
)
, (3.16)
E
m
Æ
i
p
2K³
m
´
4¼(1¡w
m
w
n
)
, (3.17)
E
n
Æ
i
p
2K³
¤
n
´
4¼(1¡w
m
w
n
)
,, (3.18)
where³
a
Æz
R
¡iz¡w
a
(z
R
Åiz),´Æ¸/!
0
,and we are using polar momentum space coordinates:
k
x
Åik
y
ÆKexp(iϕ),
Once again, if we first perform the integral of each term in the sum in each of the generating
functions for the convolution terms of Eq. (3.3), we can get a generating function for the radial part
of for L
mnuv
(z) in (3.3). Since we are interested in the Kolmogorov model for turbulence, we take
the limit ·
0
!0 in our expressions. This will produce terms that have divergences when mÆu and
44
nÆv, but fortunately, these divergences are canceled by those in L
T
. Therefore, after subtracting
thedivergenttermL
T
fortheappropriatecombinationsofindicesm,n,u,andv,andtakingthelimit
·
0
!0, we get a generating function for the radial part of Eq. (3.3) expressed as two sums, one for
eachof the convolutions in the integral. That is:
L
rG
mnuv
Æ
M(l
m
,l
u
)
∑
sÆ0
M(l
n
,l
v
)
∑
s
′
Æ0
(¡1)
sÅs
′
D(l
m
,r
m
,l
n
,r
n
,l
u
,r
u
,l
v
,r
v
,s,s
′
)
[
A(l
m
,r
m
,l
n
,r
n
,l
u
,r
u
,l
v
,r
v
)
¢Ω(t,w
m
)
jl
m
j¡s
(Ω
¤
(t,w
n
))
jl
n
j¡s
′
(Ω
¤
(t,w
u
))
jl
u
j¡s
Ω(t,w
v
)
jl
v
j¡s
′
¢(1¡w
m
w
u
)
¡jl
m
j¡jl
u
jÅs¡1
(1¡w
n
w
v
)
¡jl
n
j¡jl
v
jÅs
′
¡1
!
jl
m
jÅjl
n
jÅjl
u
jÅjl
v
j¡2(sÅs
′
)
0
¢¡
(
1
2
(jl
m
jÅjl
n
jÅjl
u
jÅjl
v
j)¡(sÅs
′
)¡
5
6
)
)
B(w
m
,w
n
,w
u
,w
v
,t,!
0
)
P(l
m
,l
n
,l
u
,l
v
,s,s
′
)
)
]
,
(3.19)
where
A(l
m
,r
m
,l
n
,r
n
,l
u
,r
u
,l
v
,r
v
)Æi
jl
m
jÅjl
u
j¡jl
n
j¡jl
v
j
5¼
3
C
2
n
√
r
m
!r
n
!r
u
!r
v
!jl
m
j!jl
n
j!jl
u
j!jl
v
j!, (3.20)
B(w
m
,w
n
,w
u
,w
v
,t,!
0
,s,s
′
)Æ¡
!
2
0
(w
m
w
u
¡1)(w
n
w
v
¡1)
£
(
t
2
(w
m
(w
n
(2w
u
w
v
Åw
u
Åw
v
)Åw
u
w
v
¡1)Åw
n
w
u
w
v
¡w
n
¡w
u
¡w
v
¡2)
Å2it(w
m
(w
n
w
u
¡w
n
w
v
¡w
u
w
v
Å1)Åw
n
w
u
w
v
¡w
n
¡w
u
Åw
v
)
Åw(2w
n
w
u
w
v
¡w
n
w
u
¡w
n
w
v
¡w
u
w
v
Å1)¡w
n
w
u
w
v
Åw
n
Åw
u
Åw
v
¡2),
(3.21)
P(l
m
,l
n
,l
u
,l
v
,s,s
′
)Æ¡
jl
m
j¡jl
n
j¡jl
u
j¡jl
v
j
2
ÅsÅs
′
Å
5
6
, (3.22)
D(l
m
,r
m
,l
n
,r
n
,l
u
,r
u
,l
v
,r
v
,s,s
′
)Æ9
p
2¸
2
¡
(
1
3
)
s!s
′
!
√
(jl
m
jÅr
m
)!(jl
n
jÅr
n
)!(jl
u
jÅr
u
)!(jl
v
jÅr
v
)!
£(jl
m
j¡s)!(jl
n
j¡s
′
)!(jl
u
j¡s)!(jl
v
j¡s
′
)!.
(3.23)
45
While these expressions appear very complicated, they are not particularly difficult to handle with
an appropriatecomputer program.
3.3 LINDBLAD EQUATION
Our goal in using generating functions for each of the terms in Eq. (3.1) is to rewrite it explicitly in
Lindblad form using a discrete set of Lindblad operators (unlike the continuous Lindblad operators
in[16]). ThisdiscretesetofLindbladoperatorsmaythenyieldabetterunderstandingoftheeffectof
atmospheric turbulence on OAM as an error process with a set of dominant error types (represented
by the dominant Lindblad operators). This, in turn, should be useful in designing QECCs to protect
quantum informationagainst these dominant errors.
Additionally, we can also use these Lindblad operators to build a simplified model for the noise
processonmultiplephotonswhenthetimeseparationbetweenphotonsislessthanthecharacteristic
time ofthe turbulence,and to design suitable encodings across multiple photons.
To achieve this, we want to rewrite Eq. (3.1) in the superoperator formalism in the following
manner:
@
z
col(½)Æcol(C(½)ÅD(½)), (3.24)
wherethenotation“col(½)”meanstowritethe N£N matrix½ asan N
2
£1columnvectorbystacking
thecolumnsof½ ontopofeachother. Alinearmapon½ thenbecomesan N
2
£N
2
matrixmultiplying
the vectorcol(½).
In Eq. (3.24), C(½)Æ¡i
[
H,½
]
represents the coherent evolution part of the Lindblad equation,
while D(½)Æ
∑
k
L
k
½L
†
k
¡
1
2
L
†
k
L
k
½¡
1
2
½L
†
k
L
k
represents the decoherent part of the evolution. Using
the identity[70]
col(AXB)ÆB
T
Acol(X),
we canrewrite these superoperators using their respective matrix representationsC andD:
C Æ¡i(IH¡HI), (3.25)
46
and
DÆ
∑
k
L
k
L
k
¡
1
2
IL
†
k
L
k
¡
1
2
L
†
k
L
k
I, (3.26)
soEq. (3.24)becomes a simple linear equation:
@
z
col(½)Æ(CÅD)col(½).
3.3.1 OBTAINING LINDBLAD OPERATORS FROM A SUPEROPERATOR REPRESENTATION
By an appropriate grouping of the indices in Eq. (3.1), one can numerically find N
2
£N
2
approxima-
tions to the matricesC andD for a given range of collective (i.e., both azimuthal and radial) OAM
indices. (SincetheHilbertspacesofboththeazimuthalandradialdegreesoffreedomareinprinciple
infinitedimensional, we approximate by truncating to a finite dimensional subspace.)
As we have done before, let us denote by u,v the collective OAM indices used to describe the
quantumstatebeforetheeffectsoftheturbulence,whiledenotingbym,nthecollectiveOAMindices
usedto describe the state after the effects of turbulence. The initial density matrix½ canbe written
½Æ
∑
u,v
½
(u,v)
juihvj.
Theseouter productsjuihvj aremapped into basis vectorsju,vi for½:
col(½)Æ
∑
u,v
½
(u,v)
ju,vi.
So the elements of the column vector col(½) are labeled by a pair of collective indices u,v, and the
elementsof the matricesC andD arelabeled by two pairs of indices (m,n) and (u,v).
We will use azimuthal and radial indices such that¡L·l
m
,l
n
,l
u
,l
v
·L and 0·r
m
,r
n
,r
u
,r
v
·L
with LÈ0. We choose L such that both pairs of collective indices (u,v) and (m,n) are less than or
equal to the cutoff N
2
. The cutoff of the indices might result in “leakage” errors of the state out of
thefinitedimensionalsubspace. Inprinciple,theeffectsofthisleakagecouldbedescribedbyadding
47
an extra term to Eq. (3.24) involving an anti-commutator with an (unknown) operator; but we will
ignore that term in the rest of this chapter, on the assumption that our cutoff is sufficiently higher
than theindices of the initial state that leakage errors are negligible.
Underalloftheaboveassumptions,wecanrewritethepartof(3.1)thatrepresentsthefree-space
propagation ofthe beam to be includedinC, as
(C)
(m,n),(u,v)
ÆS
u,m
±
v,n
¡S
v,n
±
m,u
. (3.27)
Forthe decoherentsuperoperator matrix we have
(D)
(m,n),(u,v)
Æ
1
r
m
!r
n
!r
u
!r
v
!
@
r
m
,r
n
,r
u
,r
v
w
m
,w
n
,w
u
,w
v
L
rG
m,n,u,v
¯
¯
¯
w
m
Æw
n
Æw
u
Æw
v
Æ0
. (3.28)
From this, and comparing with (3.25), and (3.26) we can extract a form for the Lindblad operators
usingtheproceduredescribedin[71]. Essentially,thisrequiresustoobtaintheeigenvectors{v
k
}and
eigenvalues {¸
k
} of
e
DÆP
I
Choi(D)P
I
, (3.29)
whereP
I
ÆII¡1/Ncol(I)col(I)
†
,andChoi(A)istheChoimatrixof A [65]. ALindbladoperatorL
k
is
constructed by taking the square root of the eigenvalue and reorganizing the eigenvectors to form a
matrix ofthe appropriate size:
√
¸
k
v
k
Æcol(L
k
).
With this procedure, we can extract the most important Lindblad operators, as measured by the
size of the eigenvalues ¸
k
used to construct them. As we will see below, if we order the eigenvalues
j¸
1
j¸j¸
2
j¸¢¢¢¸j¸
N
2j the magnitudes of the eigenvaluesj¸
k
j fall off rapidly with k. So the first few
Lindblad operatorsdominate the error process.
48
3.3.2 MULTIPLE CLOSELY-SPACED PHOTONS
As a photon propagates through a thin slice of turbulent air, its state is transformed by the random
fluctuations in the density (and hence in the index of refraction) across the wave front. If we knew
the precise values of the density fluctuations, we could describe the evolution of the state by some
unitarytransformation
½(z)!½(zÅ∆z)Æexp(¡i"H)½(z)exp(i"H)¼¡i"[H.½(z)]Å"
2
(
H½(z)H¡(1/2){H
2
,½(z)}
)
, (3.30)
where"isasmallparameterandHisarandomHamiltonianthatdescribestheunitaryevolutionof
thephoton state across the thin slice of air from z to zÅ∆z.
Of course, in practice we do not know H, and so we take an ensemble average of Eq. (3.30) over
all realizations ofH. Assuming that the ensemble average ofH vanishes (that is, mean zero noise),
the linear term in (3.30) goes away, but the quadratic terms do not, and this average transforms
Eq. (3.30) into a Lindblad equation. We see by inspection that the Lindblad operators {L
k
} must be
linear combinations of these possible HamiltoniansH. (If the mean is small but not exactly zero, it
ispossible for there also to be a Hamiltonian term in the Lindblad equation.)
Now suppose that a sequence of n photons all cross the slice of air from z to zÅ∆z within a time
that is short compared to the characteristic time of the turbulence. Then the state of each photon
experiencesexactlythesameunitarytransformationgiveninEq.(3.30),whichisthesameashaving
acollective Hamiltonian
e
H actingon all the photons of the form
e
HÆ
n
∑
iÆ1
(
i¡1
⊗
jÆ1
I
)
H
(
n
⊗
jÆiÅ1
I
)
. (3.31)
Theevolution of the n-photonstate then looks like
½(z)!½(zÅ∆z)Æexp
(
¡i"
e
H
)
½(z)exp
(
i"
e
H
)
¼¡i"[
e
H.½(z)]Å"
2
(
e
H½(z)
e
H¡(1/2){
e
H
2
,½(z)}
)
. (3.32)
Takingtheensembleaverageofthen-photonequation(3.32)willagainyieldaLindbladequation.
Furthermore, if {L
k
} for kÆ1,...,N
2
is the set of Lindblad operators for the one-photon case, we can
49
approximate theLindblad operators for the n-photoncaseby
e
L
k
Æ
n
∑
iÆ1
(
i¡1
⊗
jÆ1
I
)
L
k
(
n
⊗
jÆiÅ1
I
)
. (3.33)
We will discuss the properties of the Lindblad operators for the one- and n-photon cases in the fol-
lowing sections.
1 500 1000 1500 2025
1
500
1000
1500
2025
1 500 1000 1500 2025
1
500
1000
1500
2025
-3.6×10
-8
-1.8×10
-8
0
2.3×10
-8
4.5×10
-8
6.8×10
-8
9.0×10
-8
Figure 3.1: Example of the operatorC when ¸Æ1.0£10
¡6
m, !
0
Æ0.01 m, C
2
n
Æ1.0£10
¡14
m
¡2/3
, t
z
Æ100,
and LÆ4.
3.3.3 NUMERICAL EXAMPLES OF LINDBLAD OPERATORS
To get a better idea of how the matrix representation of the superoperatorsC andD look, we have
calculated some examples numerically. In Fig. 3.1, we show how the matrix for the coherent part
of the evolution in the IPE looks like. As expected from Eqs. (3.12) and (3.25), we obtain a purely
imaginary sparsematrix where all of the non-zero elements are concentrated around the diagonal.
On the other hand, as can be seen in Fig. 3.2 the decoherent part of the IPE evolution involves
50
1 500 1000 1500 2025
1
500
1000
1500
2025
1 500 1000 1500 2025
1
500
1000
1500
2025
-1.79
-1.30
-0.90
-0.40
0
0.48
(a) Real part ofD
1 500 1000 1500 2025
1
500
1000
1500
2025
1 500 1000 1500 2025
1
500
1000
1500
2025
-0.0062
-0.0042
-0.0021
0
0.0021
0.0042
0.0062
(b) Imaginary part ofD
Figure 3.2: Example of the operatorD when ¸Æ1.0£10
¡6
m, !
0
Æ0.01 m, C
2
n
Æ1.0£10
¡14
m
¡2/3
, t
z
Æ100,
and LÆ4.
0 50 100 150
10
-5
0.001
0.100
10
Eigenvalue number
Eigenvalue magnitude (AU)
Figure 3.3: Magnitudes of the eigenvalues of the operator
e
D when ¸Æ1.0£10
¡6
m, !
0
Æ0.01 m, C
2
n
Æ1.0£
10
¡14
m
¡2/3
, t
z
Æ100, and LÆ4.
51
elementsthatingeneralarecomplexnumbers. Whilethemagnitudeoftheseelementschangeswith
the propagation distance, the structure of the superoperator matrix remains the same. Also, we
can that the matrix elements representing transitions for either the azimuthal or radial degrees of
freedom becomesmaller in magnitude as the size of the shifts in these numbers becomes larger:
1 10 20 30 40 45
1
10
20
30
40
45
1 10 20 30 40 45
1
10
20
30
40
45
-1.1
-0.7
-0.4
0
0.4
0.7
1.1
(a) Real part of L
1
1 10 20 30 40 45
1
10
20
30
40
45
1 10 20 30 40 45
1
10
20
30
40
45
-0.022
-0.015
-0.007
0
0.007
0.014
(b) Imaginary part of L
1
Figure 3.4: Example of the Lindblad operator L
1
when ¸Æ1.0£10
¡6
m, !
0
Æ0.01 m, C
2
n
Æ1.0£10
¡14
m
¡2/3
,
t
z
Æ100, and LÆ4.
We can see in Fig. 3.3 that in the spectrum of
e
D the eigenvalues come in pairs. Each pair corre-
spondstoaset ofoperatorsthat raisesorlowersthe initialvaluesof l
m
and l
n
byasetamount, and
additionally changes the initial values of r
m
and r
n
. Moreover, since the spectrum is dominated by
the first two eigenvalues, this means that the first two Lindblad operators have an outsize influence
on thenoise process represented by the IPE.
ThetwomostdominantLindbladoperatorsaresparse,andrepresentashiftin l
m
,l
n
byoneunit,
accompanied by shifts in r
m
,r
n
that depend on the values of OAM. This can be seen from the blocks
aboveandbelowthediagonalinFigs.3.4and3.5. Italsoappearthattheeffectsoninitialstateswith
opposite signsof l
m
,l
n
aresimilar.
The properties mentioned above seem also to extend to the multiphoton case, as can be seen from
Figs.3.6 and 3.7.
52
1 10 20 30 40 45
1
10
20
30
40
45
1 10 20 30 40 45
1
10
20
30
40
45
-1.1
-0.7
-0.4
0
0.4
0.7
1.1
(a) Real part of L
2
1 10 20 30 40 45
1
10
20
30
40
45
1 10 20 30 40 45
1
10
20
30
40
45
-0.014
-0.007
0
0.007
0.015
0.022
(b) Imaginary part of L
2
Figure 3.5: Example of the Lindblad operator L
2
when ¸Æ1.0£10
¡6
m, !
0
Æ0.01 m, C
2
n
Æ1.0£10
¡14
m
¡2/3
,
t
z
Æ100, and LÆ4.
1 20000 40000 60000 80000 91125
1
20000
40000
60000
80000
91125
1 20000 40000 60000 80000 91125
1
20000
40000
60000
80000
91125
-0.0013
-0.0009
-0.0005
0
0.0004
0.0008
0.0011
0.0015
(a) Real part of
e
L
1
1 20000 40000 60000 80000 91125
1
20000
40000
60000
80000
91125
1 20000 40000 60000 80000 91125
1
20000
40000
60000
80000
91125
-0.000024
-0.000016
-8.×10
-6
0
9.×10
-6
0.000017
(b) Imaginary part of
e
L
1
Figure 3.6: Example of the Lindblad operator
e
L
1
when ¸Æ1.0£10
¡6
m, !
0
Æ0.01 m, C
2
n
Æ1.0£10
¡14
m
¡2/3
,
t
z
Æ100, LÆ4, and nÆ3.
53
1 20000 40000 60000 80000 91125
1
20000
40000
60000
80000
91125
1 20000 40000 60000 80000 91125
1
20000
40000
60000
80000
91125
-0.0013
-0.0009
-0.0005
0
0.0004
0.0008
0.0011
0.0015
(a) Real part of
e
L
2
1 20000 40000 60000 80000 91125
1
20000
40000
60000
80000
91125
1 20000 40000 60000 80000 91125
1
20000
40000
60000
80000
91125
-0.000017
-9.×10
-6
0
8.×10
-6
0.000016
0.000024
(b) Imaginary part of
e
L
2
Figure 3.7: Example of the Lindblad operator
e
L
2
when ¸Æ1.0£10
¡6
m, !
0
Æ0.01 m, C
2
n
Æ1.0£10
¡14
m
¡2/3
,
t
z
Æ100, LÆ4, and nÆ3.
3.4 ERROR DETECTION AND CORRECTION
3.4.1 AN ERROR-DETECTING CODE
Let us now consider one-photon states for which the value of the radial index is always zero. We
are going to use superposition of these states to build an error detecting code and investigate its
performance. Considerthe following states:
jÃ
n
iÆ®jn,0iůj¡n,0i (3.34)
where§naretheazimuthalquantumnumbersl and0istheradialquantumnumberr. Toillustrate
the effects of the truncation of the state space, we can calculate the trace of the state ½(t
z
) after
evolving for a dimensionless propagation distance of t
z
Æ z¸/¼!
2
0
, when the initial state is ½
n
(0)Æ
jÃ
n
ihÃ
n
j.
Since the Lindblad operators cause an initial state to scatter into neighboring modes, it is to be
expected that, because of the truncation of the space, the trace of the final state will decrease with
the propagation distance. We can see in Fig. 3.8 that this is precisely what happens. Moreover, we
54
n 1
n 2
n 3
n 4
n 5
n 6
0 10 20 30 40 50 60
0.0
0.2
0.4
0.6
0.8
1.0
t
z
Tr(ρ(t
z
))
Figure 3.8: Trace of the state in Eq. (3.34) after evolving for t
z
when ¸Æ1.0£10
¡6
m, !
0
Æ0.01 m, C
2
n
Æ
1.0£10
¡14
m
¡2/3
, and ®Æ¯Æ
1
p
2
.
n 1
n 2
n 3
n 4
n 5
n 6
0 10 20 30 40 50 60
0.0
0.2
0.4
0.6
0.8
1.0
t
z
P[error]
Figure 3.9: Probability of a detectable error when using a state of the form given in Eq. (3.34) after evolving
for t
z
when ¸Æ1.0£10
¡6
m, !
0
Æ0.01 m, C
2
n
Æ1.0£10
¡14
m
¡2/3
, and ®Æ¯Æ
1
p
2
.
55
can also see that this effect depends on the choice of initial state. As the OAM of the initial state
grows,theeffect of the dissipation becomes larger.
n 1
n 2
n 3
n 4
n 5
n 6
0 10 20 30 40 50 60
0.5
0.6
0.7
0.8
0.9
1.0
t
z
F
Figure 3.10: Minimum fidelity between an initial state of the form given in Eq. (3.34) and the state after
evolving for t
z
when ¸Æ1.0£10
¡6
m, !
0
Æ0.01 m, C
2
n
Æ1.0£10
¡14
m
¡2/3
.
Since the most relevant Lindblad operators by magnitude are those that shift the OAM value by
one unit, we can see that if we restrict our attention to these operators and use states of the form
given by (3.34) to build a code, then terms in the Lindblad equation that correspond to I, L
1
L
2
, or
L
2
L
1
correspond to no error, whereas terms that either change the initial value of r
m
,r
n
or take us
outofthetruncatedspacerepresentadetectableerror. Wecannumericallycalculatetheprobability
of such an error as a function of the propagation distance. The results of this calculation are seen in
Fig. (3.9). As can be seen, as the propagation distance increases, so does the probability of detecting
an error.
Finally, we can also consider how the fidelity between the initial and a renomalized final state
varies as a function of the propagation distance for different values of OAM of the initial state. For
each point in Fig. 3.10 we chose the values of ® and ¯ that minimize the fidelity. It is clear and
consistent with our expectations that as the propagation distance increases, the minimum fidelity
decreases.
56
3.4.2 A POSSIBLE SCHEME FOR ERROR CORRECTION IN OAM OF PHOTONS
This construction of an error-detecting code for OAM suggests one possible approach to error cor-
rection. This approach builds on standard QECCs by concatenating them with the type of error-
detectingcode described in the previous subsection.
The idea is quite simple. An [[n,k,d]] QECC encodes k logical qubits into n physical qubits and
hasminimumdistance d [7]. Suchacodecancorrectgeneralerrorsonupto
⌊
d¡1
2
⌋
qubits. However,
such codes can also correct up to d¡1 erasureerrors: that is, errors in which d¡1 qubits are erased
(that is, completely randomized or lost), but where it is known which qubits have been erased. This
meansthatifoneknowswhichqubitsinacodewordhaveerrors,butnotnecessarilywhattheerrors
are, then one could discard those qubits, replace them with new qubits in any state, and then carry
out the correction procedure for erasures. Note that knowing which qubits have errors makes a code
morepowerful, able to correct twice as many errors.
A natural scheme then presents itself. The physical qubits of the [[n,k,d]] are realized by n
closely-spaced OAM photons using the quantum error-detection code from the previous subsection.
Whenthesephotonsarereceived,onefirstmeasurestodetectwhetheranerrorhasoccurredoneach
of the n photons. Provided that no more than d¡1 photons have errors, they can be corrected as
erasureerrors.
It is quite possible to extend this scheme in a number of ways. Since the OAM space is infinite-
dimensional (in principle), one can encode qudits rather than qubits, and use a suitable qudit code
concatenated with a qudit quantum error-detecting code. It is also possible to store a qubit in the
polarization of each photon; these qubits are insensitive to the effects of turbulence. It is also quite
possible that much better QECCs than this can be designed for OAM of photons; but this straight-
forwardapproachshould work at least reasonably well if the noise is not too strong.
3.5 DISCUSSION AND CONCLUSIONS
We have seen that for the infinitesimal propagation equation derived in [15, 68] it is possible to
use the techniques from [71] to obtain a discrete Lindblad equation for the effects of Kolmogorov
turbulence on an OAM photon propagating through a turbulent atmosphere. A numerical analysis
57
of the Lindblad operators reveal that they come in pairs, and represent shifts in the OAM content of
a state. The dominant pair (as measured by the magnitude of the corresponding eigenvalue) shifts
the initial l
m
,l
n
byone and also can changethe values of r
m
,r
n
.
Based on this analysis of the dominant errors, we presented a simple quantum error-detecting
code,andshowedhowthiscouldbeconcatenatedwithan[[n,k,d]]QECCtogivenanerror-correction
procedure forquantum information encoded across the OAM of multiple photons.
We also found the form of the Lindblad operators when there are multiple photons propagating
with a time separation that is less than the characteristic time of the turbulence process. Interest-
ingly,theseLindbladoperators—whichcanbeinterpretedasdistincterrorprocesses—actcollectively
on the n photonsin a permutation-symmetric manner.
Thisraisesanintriguingquestioninquantumerrorcorrection. Giventhepermutationsymmetry
of the photon errors, is it possible to build a quantum error-correcting code for information encoded
across multiple photons that exploits the symmetry of the noise? For finite-dimensional spaces such
permutation symmetry might be expected to give rise to a decoherence-free subspace or noiseless
subsystem [72–74]. In an infinite-dimensional space—such as occurs in OAM—it seems that this
neednotbetrue. Butitseemslikelythatthesymmetrycouldstillbeexploitedbyaproperlydesigned
QECC.
Anotherintriguingpossibilityistocombinequantumerrorcorrectionwithamethodsuchasadap-
tive optics to achieve better performance in the face of turbulence [75]. Indeed, given the high noise
rates from atmospheric turbulence [14], it may be necessary to combine adaptive optics and quan-
tumerror-correctiontoallowpracticalquantumcommunicationatall. Thesetopics,however,arethe
subject ofongoing research.
58
4
Conclusions
In this dissertation, we studied the effects of a turbulent atmosphere on the quantum information
encodedin the orbital angular momentum of photons.
InitiallyinCh. 1weusedthesinglescreenmodeltoexplorehowtheKolmogorovturbulenceaffects
Laguerre-Gauss states. Using numerical simulations, we extracted an operator sum decomposition
of the noise process OAM photons undergo while traveling in a turbulent atmosphere. Additionally,
we investigated how to apply the methods of approximate quantum error correction to compensate
the noise the photons undergo. Sadly, our results indicate that even with these methods, the range
overwhichwe can use OAM for effective quantum communications is very limited.
Secondly in Ch. 2 we again used the same model and techniques as in Ch. 2 but we also incor-
porated the effects of adaptive optics corrections. While assuming that the effects of the uncorrected
noise are small, we obtained an expression to analyze the residual effects of incorporating adaptive
optics to compensate for some of the noise due to atmospheric turbulence. As could be expected, we
59
found that increasing the number of corrected modes increases the probability of transmitting the
states without error. However, more research is required to explore how to combine adaptive optics
with quantumerror correction.
Finally, in Ch. 3 we used an infinitesimal propagation equation originally derived by Roux [15,
68] to derive a discrete Lindblad equation that can also be used as a noise model. We numerically
explored the effects of the dominant Lindblad operators, and presented a simple quantum error-
detecting code. Additionally, we discussed how this could be concatenated with a quantum error
correcting codeto protect information encoded across the OAM of multiple photons.
60
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64
T
HIS THESIS WAS TYPESET using L
A
T
E
X,
originally developed by Leslie Lamport
and based onDonald Knuth’sT
E
X.
A template that can be used to format a PhD
dissertation with this look & feel has been
released under the permissive AGPL license,
and can be found online at github.com/asm-
products/Dissertate or from its lead author,
JordanSuchow,at suchow@post.harvard.edu.
65
Abstract (if available)
Abstract
In this dissertation, we study the effects of a turbulent atmosphere on the quantum information encoded in the orbital angular momentum of photons. ❧ In the first and second chapters we use the single screen model to investigate how the Kolmogorov turbulence affects Laguerre-Gauss states. In the first chapter we use the techniques of approximate error correction, whereas in the second we use adaptive optics, to compensate the noise. We numerically explore the effectiveness of either scheme in each respective chapter. ❧ Finally, in the third chapter we use an infinitesimal propagation equation to derive a discrete Lindblad equation that can also be used as a noise model. We investigate the effects of the dominant Lindblad operators and discuss different techniques that could be used to counter their effects. We also discuss possible schemes to protect quantum information encoded across multiple photons.
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University of Southern California Dissertations and Theses
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Asset Metadata
Creator
González Alonso, José Raúl
(author)
Core Title
Quantum information and the orbital angular momentum of light in a turbulent atmosphere
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Physics
Publication Date
10/18/2016
Defense Date
10/13/2016
Publisher
University of Southern California
(original),
University of Southern California. Libraries
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Tag
atmospheric turbulence,OAI-PMH Harvest,orbital angular momentum of light,quantum error correction,quantum information
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Language
English
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Advisor
Brun, Todd (
committee chair
), Haas, Stephan (
committee member
), Lidar, Daniel (
committee member
), Reichardt, Ben (
committee member
), Willner, Alan (
committee member
)
Creator Email
jrgonzal@usc.edu,yosefrajwulf@gmail.com
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c40-315010
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UC11214518
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etd-GonzalezAl-4883.pdf (filename),usctheses-c40-315010 (legacy record id)
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315010
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Dissertation
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González Alonso, José Raúl; Gonzalez Alonso, Jose Raul
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Tags
atmospheric turbulence
orbital angular momentum of light
quantum error correction
quantum information