Single photons in an imperfect array of beam-splitters: Interplay between percolation, backscattering and transient localization

2.9 MB·

by  C. M. Chandrashekar

Single photons in an imperfect array of beam-splitters: Interplay between percolation,
backscattering and transient localization
C. M. Chandrashekar,1,∗ S. Melville,2 and Th. Busch1
1Quantum Systems Unit, Okinawa Institute of Science and Technology Graduate University, Okinawa, Japan
2The Queen’s College, University of Oxford, United Kingdom
Photons in optical networks can be used in multi-path interferometry and various quantum infor-
mation processing and communication protocols. Large networks, however, are often not free from
defects, which can appear randomly between the lattice sites and are caused either by production
faultsordeliberateintroduction. Inthisworkwepresentnumericalsimulationsofthebehaviourof
asinglephotoninjectedintoaregularlatticeofbeam-splittingcomponentsinthepresenceofdefects
that cause perfect backward reflections. We find that the photon dynamics is quickly dominated
bythebackscatteringprocesses,andasmallfractionofreflectorsinthepathsofthebeam-splitting
arraystronglyaffectsthepercolationprobabilityofthephoton. Wecarefullyexaminesuchsystems
4
andshowaninterestinginterplaybetweentheprobabilitiesofpercolation,backscatteringandtem-
1
porary localization. We also discuss the sensitivity of these probabilities to lattice size, timescale,
0
injection point, fraction of reflectors and boundary conditions.
2

r
p
I. INTRODUCTION there is a non-trivial tradeoff between the probabilities
A
for localization, percolation, and backscattering.
1 Our presentation is organised as follows. In sectionII
Recent developments in experimental techniques have
we define the dynamics of a photon in a completely con-
allowed the realisation and study of many complex pho-
] nected array of beam-splitters and in sectionIII we sim-
h tonicsystemssuchasmultipath,multiphotoninterferom-
ulate the dynamics in the presence of a number of reflec-
p eters that exhibit high fidelity quantum interference[1–
tors between adjacent beam-splitters. We then calculate
-
t 6]. This stems from, and also stimulates, a great deal of the probabilities of percolation, backscattering and tem-
n
interestinusingphotonsasinformationcarriersforvari-
a porarylocalizationandconcludewithadiscussionofthe
ousquantuminformationprocessingandcommunication
u results in sectionIV.
q protocols[7–13]. However, buildingthelargeopticalnet-
[ works for photon propagation required by some of these

protocols is not an easy task and imperfections in the
3 II. PHOTON PROPAGATION IN A REGULAR
coupling between different sections of a network can ap-
v ARRAY OF BEAM-SPLITTERS
pear. It is therefore important to discuss and simulate
0
0 simple toy models of single photon propagation in an ir-
A photon incident on a beam-splitter can be written
6 regulararrayofbeam-splitters,inordertoachieveabet-
as the Fock state |n ,n ,n ,n (cid:105). For a single photon,
1 terunderstandingofhowproposedlargeopticalnetworks a b c d
1. might behave in practice. na + nb + nc + nd = 1 with each n being an integer
and the indices a,b,c,d specifying the four beam-splitter
0
Here we present a numerical study of the behaviour of arms. Infigure1(a)weshowaschematicofaphotonim-
4
a single photon injected into a regular lattice of beam pinging on a beam-splitter and indicate the correspond-
1
: splitting components (modelling the network), in which ing transmitting and reflecting paths. In figure1(b) we
v weallowforperfectreflectionstooccurbetweenacertain definethefourarmsofthebeam-splitterasa,b,c,andd
i
X fractionofthelatticesites(modellingthesystemdefects, and indicate the corresponding Fock states for a photon
oranintentionalfeatureofthenetwork). Thoughthethe travelling in one of the associated modes. This allows to
r
a presence of the reflectors introduces irregular paths for define annihilation operators aˆ,ˆb,cˆ,dˆ, such that
photon propagation, the operation at each lattice site is
considered to be an ideal lossless beam-splitter, where aˆ|1000(cid:105)=|0000(cid:105), aˆ†|0000(cid:105)=|1000(cid:105) (1)
the input and output operators are related by a uni-
[aˆ,aˆ†]=1; [aˆ,ˆb]=[aˆ,cˆ]=[aˆ,dˆ]=0, (2)
tarytransformation. Wefindthatthephotonisconfined
withinalatticeofsizeN×N overtimescalesproportional
to N, but that these vary considerably with factors such andanalogouslyfortheotherthreeoperators(ˆb,cˆ,dˆ)cor-
astheinjectionpointandtheboundaryconditionsofthe responding to the remaining three indices (nb,nc,nd).
lattice, which we choose as either reflective or absorp- Thus the action of a beam-splitter on a photon may be
tive. This allows for temporary localization of the pho- regarded as the action of the effective Hamiltonian
ton within the lattice network and, as time progresses,
1 (cid:16) (cid:17) 1 (cid:16) (cid:17)
H = √ aˆ†−iˆb† aˆ+ √ ˆb†−iaˆ† ˆb
2 2
(3)
1 (cid:16) (cid:17) 1 (cid:16) (cid:17)
+ √ cˆ†−idˆ† cˆ+ √ dˆ†−icˆ† dˆ,
∗Electronicaddress: [email protected] 2 2
2
FIG. 1: (a) Schematic of a beam-splitter (BS) with the out-
putpaths(reflectedandtransmitted)indicatedforoneofthe
possibleinputstates. (b)Allpossiblephotonmodesoutgoing
from the beam-splitter. In the percolation direction the in-
put state |1000(cid:105) leads to the output states |1000(cid:105) and |0100(cid:105) FIG.2: Schematicofanarrayofbeam-splittersarrangedina
and the input state |0100(cid:105) leads to |0100(cid:105) and |1000(cid:105). In the squarelatticewithdetectors(D)atallpossibleoutputports,
backscattering direction the input state |0010(cid:105) leads to the which register the photon once it has moved through the ar-
output states |0010(cid:105) and |0001(cid:105) and the input state |0001(cid:105) ray. The small graph at the right hand side indicates the
leads to |0001(cid:105) and |0010(cid:105) possible paths for a photon entering in |1000(cid:105).
wherethefactorofiaccountsforaphaseshiftofπduring
beam-splittersasunity,thetotalprobabilityforthepho-
reflection.
tontoreachanedgeofalatticeofsizeN×N isP(t)=0
We will now consider an array of beam-splitters, each
for t≤N, P(t)=1 for t>2N and 0≤P(t)≤1 for any
positioned at the vertices of a square lattice and la-
time N <t<2N.
belled by (x,y) (see figure2). Initially, a single pho-
ton is injected at (x,y) = (1,1) in state |1000(cid:105) and
we can describe its dynamics using the product basis
|n ,n ,n ,n (cid:105)⊗H , where H is the position Hilbert III. PHOTON PROPAGATION IN AN ARRAY
a b c d x,y x,y
OF BEAM-SPLITTERS WITH BACKWARD
space. Therefore, the initial state at the injection point
REFLECTORS
as shown in figure2 will be given by
|Ψ(t=0)(cid:105)=|1000(cid:105)⊗|x=1,y =1(cid:105). (4) Backward reflection and loss of photons between the
lattice site are two of the most fundamental processes
The action of the beam-splitting operator, which acts that can affect the forward propagation of a photon in
only on the Fock state |na,nb,nc,nd(cid:105) and leaves the po- an array of beam-splitting components. In this section
sition states unchanged, will be H (equation(3)), and we will discuss the additional effects that appear when a
the evolution of the position state is given by the shift certainnumberofbackwardreflectorsareintroducedinto
operation the path. While the results are specific to the setup, the
treatmentwepresentcanserveasageneralframeworkfor
(cid:88)
S = |1000(cid:105)(cid:104)1000|⊗|x+1,y(cid:105)(cid:104)x,y| other forms of irregularities in the path of the photons.
(x,y) In figure3 we show the effect a reflector, positioned
+|0100(cid:105)(cid:104)0100|⊗|x,y+1(cid:105)(cid:104)x,y| (5) between two beam-splitters, has on the path of a photon
and in figure4 a schematic of an array of beam-splitters
+|0010(cid:105)(cid:104)0010|⊗|x−1,y(cid:105)(cid:104)x,y|
interspersed with a number of reflectors is given. In or-
+|0001(cid:105)(cid:104)0001|⊗|x,y−1(cid:105)(cid:104)x,y|.
der to model the effect of perfect reflection at the beam-
splitters, we consider the initial state at the injection
Hence the successive action of H and S on the product
point to be given by equation(4). Note that for sym-
state|n ,n ,n ,n (cid:105)⊗|x,y(cid:105)advancesthesystemonetime
a b c d metry reasons the results obtained below also hold for
step, andaftertstepsthestateofthephotonisgivenby
a photon initially entering in mode |0100(cid:105). For all com-
|Ψ(t)(cid:105)=[S(H ⊗1)]t|Ψ(t=0)(cid:105). (6) pletely connected vertices the Hamiltonian H is given in
equation(3) and can be written as,
In this regular evolution the photon will never be scat-
tered into the modes |0010(cid:105) and |0001(cid:105) therefore it can  1 −i 0 0 aˆ†
only exit at the upper and right-hand side edges of the H = √1 (cid:0)aˆ ˆb cˆ dˆ(cid:1)−i 1 0 0 ˆb† . (7)
lattice. We call this forward propagation. If we define 2  0 0 1 −icˆ†
the time required for the photon to travel between two 0 0 −i 1 dˆ†
3
operator is then
(cid:88)
S = |1000(cid:105)(cid:104)1000|⊗|x+(1−k ),y(cid:105)(cid:104)x,y|
a
(x,y)
+|0100(cid:105)(cid:104)0100|⊗|x,y+(1−k )(cid:105)(cid:104)x,y|
b
+|0010(cid:105)(cid:104)0010|⊗|x−(1−k ),y(cid:105)(cid:104)x,y|
c
+|0001(cid:105)(cid:104)0001|⊗|x,y−(1−k )(cid:105)(cid:104)x,y|, (10)
d
andthesystemevolvesaccordingtothemodifiedequiva-
lentofequation(6). Thisensuresthataphotonin,forex-
ample, the|0100(cid:105)modewillscatterintothe|0001(cid:105)mode
and acquire a phase shift of π when hitting a reflector.
FIG.3: Schematicoftwoneighbouringbeam-splitterswitha
This photon will also be unaffected by S, so that it en-
reflector(R)intheconnectingpath. Theinitialoutputstate
counters the same beam-splitter a second time at the
fromthebluebeam-splitter(left-handside)is|0100(cid:105)andthe
one from the red beam-splitter (right-hand side) is |0001(cid:105). subsequent time step. The distribution of reflectors in
Thus ka(x,y)=1, kc(x+1,y)=1. the lattice is given by a consistent set of ki(x,y) such
that k (x,y)=k (x+1,y), etc.
a c
During this evolution the reflections can lead to
backscattering of the photon (i.e. scattering into the
modes|0010(cid:105)and|0001(cid:105)),whichopensthepossibilityfor
the photon to exit along the lattice edges on the left and
the bottom. Additionally, sufficiently nearby groups of
such reflectors can lead to temporary localization of the
photon in the lattice. Therefore, in addition to the per-
colationprobability,thesystemischaracterisedbyprob-
abilities for backscattering and localization. Assuming
an arrangement of detectors as shown in figure4, perco-
lation corresponds to the photon exiting the lattice from
eitheroftheedges(x , y)or(x, y ), backscattering
max max
corresponds to exiting the lattice from the edges along
(1, y) and (x, 1), and localization corresponds to tem-
porary confinement within the lattice for times t ≥ 2N.
Sinceallpossiblephotonpathsarereversible,localization
is of course only transient. For the initial state given in
FIG.4: Schematicofthearrayofbeam-splittersinasquare
equation(4), i.e. injecting a single photon at one of the
lattice with impurities given by perfect reflectors. Photon
detectors along (x = 1,y) and (x,y = 1) will register the cornersofthelattice,weshowinfigure5theprobabilities
backscattering of the photon due to the presence of the re- ofpercolation,backscatteringandtemporarylocalization
flectors. Thesmallgraphsattherighthandsideindicatethe asafunctionofthefractionofconnectionsbetweenadja-
possible paths for a photon at each vertex. cent beam-splitters that are not disturbed by a reflector.
These probabilities are obtained after averaging over a
large number of realizations.
When a reflector is present in an arm between two ver- The probabilities for lattices of different sizes N ×N,
tices the general Hamiltonian can be written in the form where N = 50,100,200 and 400, at time t = 2N are
shown in figure5(a). One can note that the probability
aˆ† forbackscatteringdominatesuntilthefractionofconnec-
H = √1 (cid:0)aˆ ˆb cˆ dˆ(cid:1)Rˆb† , (8) tionsbetweentheadjacentbeam-splittersisclosetounity
2 cˆ† andonecanthinkofthefractionatwhichafiniteproba-
dˆ† bilityforpercolationappearsastheanaloguetotheclas-
sical percolation threshold[15–17]. This behaviour can
where R is given by be easily understood by realising that encountering a re-
flector once leads to scattering into the modes that lead
 1−k −i(1−k ) −ik k  to backscattering, and encountering a second reflector is
a b a b
−i(1−k ) 1−k k −ik necessary to scatter into the percolation modes again.
R= a b a b 
 −ik k 1−k −i(1−k ) Since the injection point is located at the corner of the
c d c d
k −ik −i(1−k ) 1−k networkfurthestawayfromanydetectorsforpercolation,
c d c d
(9) reflectionearlyonduringthepropagationprocessleadto
with k = 0 if the nth arm is open and k = 1 if the the domination of the backscattering probability. When
n n
nth arm contains a reflector. The corresponding shift the fraction of connections is closer to unity, but before
4
FIG. 6: Schematic of an array of beam-splitters in a square
latticeinterspersedwithasmallnumberofperfectlyreflecting
surfaces and reflecting boundaries. A photon backscattered
along the injection side of the lattice is fed back to the lat-
tice due to reflectors placed along these sides, except at the
injection point.
The interplay between backscattering, localization
and percolation can be changed by introducing reflect-
ing edges in the backscattering direction and allowing
backscattered photons to only exit at the injection point
(x = 1,y = 1) (see figure6). Unsurprisingly one can
see from figure7(a), where we show the probabilities for
different lattices sizes, that at t = 2N backscattering is
reduced and instead an increase in temporary localiza-
tion is observed compared to the situation when reflect-
FIG.5: Probabilityofphotonpercolation,backscatteringand
ing edges are absent (see figure5(a)). Backscattering is
temporarylocalizationasafunctionofthefractionofconnec-
tions between adjacent beam-splitters. (a) Probabilities for still significant though, since photons scattered early on
latticesofdifferentsizes,N×N,whereN =50,100,200and inthepercolationprocesshaveahighprobabilitytoexit
400 are shown at time t=2N. (b) Probabilities for a lattice through the entry beam-splitter and this probability is
of size N = 100 for different times. Strong backscattering is further increased by the coherent backscattering[14]. In
clearly visible until the fraction of connections between the figure7(b) we show the probabilities for different times,
adjacent beam-splitters is close to unity. and find that the probability for localization monotoni-
cally decreases while both, the backscattering and per-
colation probability rise. One can again note that the
dependence on the lattice size (the number of beam-
the steep increase in percolation probability dominates, splitters) has only a weak influence on the probabilities.
temporary confinement of the photon within the lattice Comparing both cases above one can note that for
can be seen. This indicates that, while a large num- the identical initial condition given by equation(4), the
ber of reflectors leads to quick expulsion of the photon asymptotic behaviour is identical: as the fraction of con-
along the sides with (1,y) and (x,1), a decreasing num- nections goes to unity the percolation probability goes
ber allows for geometries in which the photon bounces to one, whereas for a fraction of connections around 0.5,
aroundinsidethelatticeforalongtime. Alargefraction backscattering has a probability of one. While differ-
of good connections between the beam-splitting compo- ent initial conditions exhibit qualitatively the same in-
nents is therefore required for the photon to percolate terplay between transient localization and percolation
across an array of beam-splitters. From figure5(a), one (withstrongdependenceondetectorplacementandweak
can also note that the lattice size (the number of beam- dependence on lattice size), the general asymptotic be-
splitters)hasonlyaweakinfluenceontheseprobabilities. haviour will change. An example of this is shown in
The probabilities for a lattice with N =100 for different figure8 for the situation without reflecting boundary
timesareshowninfigure5(b)andonecanseetheproba- conditions and where we have chosen |ψ(t = 0)(cid:105) =
bility of temporary localization decreasing with time, as |1000(cid:105)⊗|1x ,1y (cid:105). Forconsistencywewillagainde-
2 max 2 max
expected. finepercolationasexitingthelatticeinthemodes|1000(cid:105)
5
FIG.8: Probabilityofphotonpercolation,backscatteringand
temporarylocalizationasafunctionoffractionofconnections
betweentheadjacentbeam-splittersintheabsenceofreflect-
ing boundaries and with the photon incident at the center of
thelatticearray. TheprobabilitiesforalatticewithN =100
for different times are shown.
IV. DISCUSSION AND CONCLUSION
In this work we have modelled a large optical network
consisting of a regular array of beam-splitters, and con-
sidered the effects stemming from randomly introduced
reflective defects. The presence of these defects has a
significant influence on the transport properties of the
system - with the percolation probability for a photon
decaying rapidly even for only a small percentage of de-
FIG.7: Probabilityofphotonpercolation,backscatteringand
fective paths (∼10%). We have also found the existence
temporarylocalizationasafunctionoffractionofconnections
betweentheadjacentbeam-splittersforthesituationwherea ofatransient‘localised’state,whichconfinesthephoton
backscattered photon is fed back to the lattice at the edges. within the lattice over finite timescales.
(a) Probabilities for lattices of different sizes, N ×N, where In region of small percentages of defects, an inter-
N = 50,100,200 and 400 are shown at time t = 2N. (b) esting interplay between the three possible scenarios
Probabilities for a lattice with N =100 for different times. takes place: the photon percolates forward, the pho-
ton backscatters, or the photon remains within the lat-
tice. These relative probabilities are fairly insensitive to
changes in the lattice size, but vary significantly if the
or |0100(cid:105) and backscattering as having encountered an distribution of detectors around the lattice is altered (by
odd number of reflectors before leaving the lattice in the replacing some detectors with reflectors, feeding those
modes |0010(cid:105) or |0001(cid:105). In figure8 we show the resulting photons back into the lattice). With fewer detectors
probabilitiesandonecannotethatthepercolationprob- around the lattice edges, the localization probability is
ability is almost same as the backscattering probability finite over a much longer timescales, before giving way
until the fraction of connection gets closer to unity (0.8 to both, backscattering and percolation. If the injection
for t = 1600 when N = 100). After that the backscat- point is near a particular lattice edge, a large probabil-
tering probability decreases to zero and the percolation ity for the photon to exit the lattice via this edge exists
probability rises to one, as the photon can transverse (backscattering processes dominate), and if the injection
the upper right quarter of the network most of the times pointisfarfromalatticeedge,long-livedlocalizationcan
withoutencounteringareflector. Fractionofconnections be seen.
smaller than 0.5 results in localization with probability The implicationfor large optical networks isthat even
one. Unlike the transient localization for the model with smallfractionsofreflectivedefectswillsignificantlyalter
theinjectionpointatoneofthecornersofthelattice,the thepathtakenbythephotonthroughthesystem. There-
localization for injection at the middle of the lattice is a fore, quantum communication systems using optical net-
permanent localization, due to the absence of a detector works will be very sensitive to defects and require addi-
close to the injection point. tional strategies to combat imperfections. These could,
6
for example, consist of the suitable use of additional re- Acknowledgments
flectors to feed stray photon amplitudes back into the
system. The study of multipath interferometer or large
optical networks are therefore very valuable to identify
the percentages of defective components a system can SM would like to thank OIST Graduate University
tolerate and to test ideas to correct them in oder to ob- for the support for summer internship during which this
tain reliable devices. work was carried out.
[1] A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, and J. P. J. Mosley, E. Andersson, I. Jex, and Ch. Silberhorn,
L.O’Brien,Silica-on-siliconwaveguidequantumcircuits, Photons Walking the Line: A Quantum Walk with Ad-
Science 320, 646 (2008). justable Coin Operations, Phys. Rev. Lett. 104, 050502
[2] Graham D. Marshall, Alberto Politi, Jonathan C. F. (2010).
Matthews, Peter Dekker, Martin Ams, Michael J. With- [10] A. Regensburger, C. Bersch, B. Hinrichs, G. On-
ford, and Jeremy L. O’Brien, Laser written waveguide ishchukov, A. Schreiber, C. Silberhorn, and Ulf Peschel,
photonic quantum circuits, Opt. Express 170, 12546 - Photonpropagationinadiscretefibernetwork: Aninter-
12554 (2009). playofcoherenceandlossesPhys.Rev.Lett.107,233902
[3] Alberto Peruzzo, Mirko Lobino, Jonathan C. F. (2011).
Matthews, Nobuyuki Matsuda, Alberto Politi, Kon- [11] MatthewA.Broome,AlessandroFedrizzi,SalehRahimi-
stantinos Poulios, Xiao-Qi Zhou, Yoav Lahini, Nur Is- Keshari, Justin Dove, Scott Aaronson, Timothy Ralph,
mail, Kerstin Wo¨rhoff, Yaron Bromberg, Yaron Silber- Andrew G. White, Photonic Boson Sampling in a Tun-
berg, Mark G. Thompson, Jeremy L. O’Brien, Quantum able Circuit, Science 339, 794-798 (2013).
Walks of Correlated Photons, Science 329, 1500 (2010). [12] Justin B. Spring, Benjamin J. Metcalf, Peter C.
[4] Anthony Laing, Alberto Peruzzo, Alberto Politi, Maria Humphreys, W. Steven Kolthammer, Xian-Min Jin,
Rodas Verde, Matthaeus Halder, Timothy C. Ralph, MarcoBarbieri,AnimeshDatta,NicholasThomas-Peter,
MarkG.Thompson,JeremyL.O’Brien,High-fidelity op- Nathan K. Langford, Dmytro Kundys, James C. Gates,
eration of quantum photonic circuits, Appl. Phys. Lett. BrianJ.Smith,PeterG.R.Smith,IanA.Walmsley,Bo-
97, 211109 (2010). son Sampling on a Photonic Chip, Science 339, 798-801
[5] A. Peruzzo, A. Laing, A. Politi, T. Rudolph, and J. L. (2013).
O’Brien, Multimode quantum interference of photons in [13] M. Tillmann, B. Daki´c, R. Heilmann, S. Nolte, A. Sza-
multiport integrated devices, Nature Communications 2 meit,andP.Walther,ExperimentalBosonSampling,Na-
224 (2011). ture Photonics 7, 540-544 (2013).
[6] Linda Sansoni, Fabio Sciarrino, Giuseppe Vallone, [14] E. Akkermans, P.E. Wolf, and R. Maynard, Coherent
Paolo Mataloni, Andrea Crespi, Roberta Ramponi, Backscattering of Light by Disordered Media: Analysis
andRobertoOsellame, Two-Particle Bosonic-Fermionic of the Peak Line Shape, Phys. Rev. Lett. 56, 14711474
Quantum Walk via Integrated Photonics, Phys. Rev. (1986).
Lett. 108, 010502 (2012). [15] S.Kirkpatrick,Rev.Mod.Phys.45,574-588(1973);B.
[7] Y. Yamamoto, C. Santori, G. Solomon, J. Vuckovic, D. Bolloba´s,etal.,Percolation,CambridgeUniversityPress
Fattal, E. Waks, and E. Diamanti, Single photons for (2006).
quantuminformationsystems,ProgressinInformatics1, [16] D.StaufferandA.Aharony,IntroductiontoPercolation
pp.5-37 (2005). Theory (2nd ed.), CRC Press (1994).
[8] A. Politi, J. C. F. Matthews, and J. L. O’Brien, Shor’s [17] C. M. Chandrashekar, Th. Busch, Quantum Percolation
quantum factoring algorithm on a photonic chip,Science and transition point for directed discrete-time quantum
3250, 1221 (2009). walk, arXiv:1303.7013.
[9] A. Schreiber, K. N. Cassemiro, V. Potocek, A. Ga´bris,