Decoherence and disorder in quantum walks: From ballistic spread to localization

1.8 MB·English

by  A. Schreiber

Decoherence and disorder in quantum walks: From ballistic spread to localization
A. Schreiber,1, K. N. Cassemiro,1 V. Potoˇcek,2 A. G´abris,2, I. Jex,2 and Ch. Silberhorn3,4
∗ †
1Max Planck Institute for the Science of Light, Gu¨nther-Scharowsky-Str. 1 / Bau 24, 91058 Erlangen, Germany.
2Department of Physics, FNSPE, Czech Technical University in Prague, Bˇrehov´a 7, 115 19 Praha, Czech Republic.
3Max Planck Institute for the Science of Light, Gu¨nther-Scharowsky-str. 1 / Bau 24, 91058 Erlangen, Germany.
4 University of Paderborn, Applied Physics, Warburger Str. 100, 33098 Paderborn, Germany.
(Dated: January 14, 2011)
Weinvestigatetheimpactofdecoherenceandstaticdisorderonthedynamicsofquantumparticles
moving in a periodic lattice. Our experiment relies on the photonic implementation of a one-
dimensional quantumwalk. The purequantum evolution is characterized by a ballistic spread of a
photon’swavepacketalong28steps. Byapplyingcontrolledtime-dependentoperationswesimulate
1 threedifferentenvironmentalinfluencesonthesystem,resultinginafastballisticspread,adiffusive
1
classical walk and the first Anderson localization in a discrete quantum walk architecture.
0
2
PACSnumbers: 03.65.Yz,05.40.Fb,71.23.-k,71.55Jv,03.67.Ac

n
a
J Randomwalksdescribetheprobabilisticevolutionofa However,althoughtheoretically predicted inthe context

3 classical particle in a structured space resulting in a dif- of quantum walks [23–25], the effect has never been ob-
1 fusive transport. In contrast, endowing the walker with served in a discrete quantum walk scenario.

quantum mechanical properties typically leads to a bal- Furthermore, it is interesting to note that the energy
]
h listic spread of the particle’s wave function [1]. The co- transport in photosynthetic light-harvesting systems is
p herent nature of quantum walks has been theoretically influenced by both, static and dynamic disorders, and
- explored, providing interesting results for a wide range it is precisely the interplay between the two effects that
t
n ofapplications. Theystatenotonlyauniversalplatform lead to the highly efficient transfer in those molecular
a
forquantumcomputing[2]butalsoconstituteapowerful complexes [4, 5]. Thus, in order to simulate a realistic
u
q tool for modelling biological systems [3–5], thus hinting influence of the environment, we go further in our stud-
[ towards the mechanism of energy transfer in photosyn- ies by investigating the effect of dynamical noise, which

thesis. Quantum walks of single particles on a line have typicallyinducesdecoherence[26,27]. Utilizing the abil-
1
beenexperimentallyrealizedinseveralsystems,e.g. with ity to easily tune the conditions for the quantum walk,
v
8 trapped atoms [6] and ions [7, 8]; energy levels in NMR we demonstrate here the diverse dynamics of quantum
3 schemes [9, 10]; photons in waveguide structures [11], a particles propagating in these different systems.
6 beam splitter array [12], and in a fiber loop configura- Inourexperimentwerealizethequantumwalkofpho-
2
tion [13]. Although these experiments opened up a new tons by employing a linear optical network. The evolu-
.
1 route to higher dimensional quantum systems, more so- tion of the particle’s wave function |ψ(x)i is given by
0
phisticated quantum walks need to be implemented to
1
pursue the realmof realapplications. A firststep in this |ψ(x)i→γ |ψ(x)i+ β |ψ(k)i, (1)
1 x x,k
: direction has been recently reported [14], in which two Xk=x
v 6
particlesexecute asimultaneouswalkanddisplayintrin-
i with the position dependent amplitudes γ and β de-
X sic quantum correlations. x x,k
termining the probability of the particle to stay at the
r One of the most important requirements for realizing discrete position x or evolve to the adjacent sites k, re-
a
quantumwalk-basedprotocolsistheabilitytocontrolthe
spectively.
dynamics of the walk, that is to access and manipulate Westudytheexpansionoftheparticle’swavepacketin
thewalker’sstateinapositiondependentway[15,16]. In
four different scenarios. (i) Firstof allwe implement the
thispaperwepresentthefirstexperimentalrealizationof
quantum walk in a homogeneous lattice, showing that it
quantum walks with tunable dynamics. We investigate presents an evolution that is free from decoherence. (ii)
the evolution of quantum particles moving in a discrete
Next, we introduce static disorder by manipulating the
environment presenting static and dynamic disorders.
lattice parametersγ andβ ,thus observingAnderson
x x,k
As predicted by Anderson in 1958 [17], static disorder localization. We then examine two scenarios leading to
leads to an absence of diffusion and the wave function decoherence,whichessentiallydiffer inthe time scalesof
of the particle becomes localized, which, e.g. would ren- the occurring dynamic perturbations. (iii) In this case a
der a conductor to behave as an insulator. Anderson dynamic randomizationof the lattice parameters γ and
x
localization has been experimentally investigated in dif- β simulatesthe evolutionofaparticleinteractingwith
x,k
ferent physical scenarios, e.g. employing photons mov- a fast fluctuating environment. The resulting dephasing
ing in semiconductor powders [18] and photonic lattices suppresses the underlying interference effects and hence
[19, 20], or even via Bose-Einstein condensates [21, 22]. causes the particle to evolve just like in a classical ran-
2
dom walk [6, 8]. (iv) In the last scenario we simulate
a slowly changing homogeneous environment. While γ
x
andβ arestableduringasinglerealization,aslowdrift
x,k
leadstodifferentconditionsforsubsequentparticles,thus
affecting results obtained in an ensemble measurement.
In a discrete quantum walk the position of a parti-
cle evolves according to its internal coin state |ci. For
our photonic implementation we use the linear horizon-
tal |Hi = (1,0)T and vertical |Vi = (0,1)T polarization
of light. The state of the photon after N steps of the
walk is found by applying the unitary transformation
U = N SˆCˆ to the initial state |ψ(x) i = |x i⊗|c i.
n=1 n 0 0 0
The cQoin operation Cˆ (x) manipulates the polarization
n
of the photon in dependence on the position x and the
stepnumber n. Inthe basis{|Hi,|Vi}the coinoperator FIG. 1: (a) Schematic setup. The coin state is manipu-
lated via half-wave plate (HWP) and electro-optic modula-
is given in matrix form by
tor (EOM). The two polarizing beam splitters (PBS) allow
eiφH(x) 0 cos(2θ) sin(2θ) toimplement thestep operation. APDs: Avalanchephotodi-
C(x)= , (2) odes;(b)Probabilitydistributionafter28stepsofasymmet-
(cid:18) 0 eiφV(x) (cid:19)(cid:18) sin(2θ) −cos(2θ)(cid:19)
ricHadamardwalkwithinitialcircularpolarization. Stacked
bars: Adapted theory splitted into the two coin states |Vi
withthediagonalmatrixrepresentingaphaseshiftφ (x)
H (blue, bottom) and |Hi (red, top). Gray dots show experi-
for horizontal and φ (x) for vertical polarizations, while
V mental data for vertical polarization, black dots the sum of
the secondmatrix correspondsto a polarizationrotation
bothpolarizations. Errorbarscorrespondtostatisticalerrors.
of 2θ. The step operation Sˆ shifts the position x of the
photonby +1 ifthe polarizationis horizontalandby −1
if it is vertical.
stepoperationis realizedinthe time domainviatwo po-
Following Eq.(1), the evolution of the wave function
larizing beam splitters (PBS) and a fiber delay line, in
with the step number n is given by
which horizontally polarized light follows a longer path
|ψ(x) i=γ |ψ(x) i+β (|ψ(x+2) i+|ψ(x−2) i). (Fig. 1(a)). The resulting temporal difference of 5.9 ns
n+2 x n x,x 2 n n
± (3) between both polarization components corresponds to a
Note that the transition coefficients γ and β are step in the spatial domain of x±1. After a full evolu-
x x,x 2
fullysetbythecoinoperationsC (x)andC (x±). By tion the photon wave packet is distributed over several
n+1 n+2
changing the parameters φ(x) and θ in a controlled discretespatialpositionsor,equivalently,overrespective
H/V
waywecanalterthecoefficientsandhencecreatediverse timewindows. Fordetectionthephotongetscoupledout
typesofphysicalconditionsforaquantumwalkscenario. of the loop by a beam splitter with a probability of 12%
A simple measure to quantify the spread of the wave per step. We employ two avalanche photodiodes (APD)
function in the different systems is provided by the vari- tomeasurethephoton’stimeandpolarizationproperties,
ance σ2 of the final spatial distribution. While the de- which gives information about the number of steps, the
coherence free quantum walk presents a ballistic spread, specific position of the photon, as well as its coin state.
withσ2 ∝n2,theclassicalrandomwalkisdiffusive,char- Theprobabilitythataphotonundergoesafullroundtrip
acterizedbyσ2 =n. Incontrasttoboth,inaonedimen- without getting lost or detected is given by ηsetup =0.55
sionalsystemwithstatic disorderthe wavepacketshows (0.22) without (with) the EOM and the detection effi-
exponential localization after a short initial expansion. ciencyisηdet =0.06perstep. Todeterminethestatistical
The stagnation of the wave packet spread is thus evi- distribution of one specific step we detected more than
denced by a constant variance. 104eventsinanoverallmeasurementtimeofmaximally1
The functional principle of our experimental setup is h, limited by the setup stability. This guaranteedan ab-
sketchedinFig. 1(a)andisdiscussedindetailin[13]. We solutestatisticalerrorofthe assessedprobabilityateach
generatetheinputphotonswithapulseddiodelaserwith position of less than 0.01. An average photon number
acentralwavelengthof805nm,apulsewidthof88psand per pulse at the detected step of less than hni < 0.003
arepetitionrateof110kHz. Theinitialpolarizationstate ensured a negligible probability of multi-photon events
of the photons is preparedwith retardationplates. Each P(n>1)/P(n=1)<0.02.
coinoperationconsistsofapolarizationrotation,whichis (i)Homogeneouslattice.—Inthefirstofourfourquan-
realizedwithahalf-waveplate(HWP),andasubsequent tum walk scenarios we investigate a homogeneous envi-
phaseshiftimplementedbyafastswitchingelectro-optic ronment, thus testing the intrinsic coherence properties
modulator (EOM), as described in Eq.(2). The proper- of the setup. The spatial distribution after 28 steps can
ties of the EOM impose that φ (x)/φ (x) ≈ 3.5. The be seen in Fig. 1(b). We used the initial state |ψ i =
V H 0
3
|0i⊗ 1 (|Hi+i|Vi) and the Hadamard coin (θ = π/8)
√2
ateachposition. Thefinalstateclearlyshowsthecharac-
teristic shape of a fully coherent quantum walk: the two
pronounced side peaks and the low probability around
the initial position. Moreover, the polarization analysis
confirms the expected dependence of the particle’s final
position on its coin state. An adapted theory includ-
ingonly smallimperfections ofthe coinparameterθ, the
initial coin state and differential losses between the two
polarizationsfullyexplainsthefinalspatialandpolariza-
tion distribution. The quality of the result can be quan-
tifiedbythe distanced(P ,P )= 1 |P (x)−P (x)|
m th 2 x m th
between the measured P and the tPheoretical P prob-
m th
ability distributions. It ranges between 0 for identical
distributions and 1 for a complete mismatch. The dis-
FIG.2: Measuredprobabilitydistribution(front)andrespec-
tanceofthemeasuredwalktotheadaptedquantumthe-
tive theory (back, gray bars) of 11 steps of a quantum walk
ory is d(P ,P ) = 0.052±0.015. For comparison we
m qw (θ =π/8) with static disorder (a), dynamic disorder (b) and
calculatedthedistanceto thefully decoherent(classical) in a decoherence free environment (Inset (c)). The insets in
scenario, obtaining d(P ,P ) = 0.661±0.015. Hence, (a) and (b) show the measured distribution in semilog scale
m cl
our result confirms an almost decoherence free evolution with linear (a) and parabolic fit (b). (c) Transition of the
after 28 steps. variance from ballistic quantum walk to diffusive/ localized
evolutionduetodynamic(redsquares)andstatic(greendots)
(ii) Static disorder.— We implemented the evolution
disorderwithincreasingdisorderstrengthΦmax;dashedlines:
of a particle in an environment with static disorder us-
theorywithoutadaptionforexperimentalimperfections. The
ing a quantum walk with variable coin operation. To red solid line marks the variance of a classical random walk.
create a static disorder a coin operation is required, (Vertical error is smaller than the dotsize). (d) Relative fre-
which is position and not step dependent. In our sys- quency f(|φV|) of the applied phases φV for the signal with
tem this is realized by a controlled phase shift φ (x), intervalΦmax =(1.02±0.05)π. Thedashedlineindicatesthe
H/V
uniform distribution.
such that the photon acquires the same phase any in-
stance it appears at position x. To generate a random
static phase pattern we applied a periodic noise signal
to the EOM. The periodicity of the signal was carefully noise signal, thus eliminating position dependent phase
adjusted to ensure that the applied phase shift opera- correlations. Decoherence appears as a consequence of
tion is strictly position dependent. Using different phase the dynamically varying phase suffered by the quantum
patterns at subsequent runs allows to average over var- particle during the evolution. As a result, the photon
ious disorders, as considered in the model of Anderson. undergoes a classical random walk, revealing a binomial
The strength of disorder is determined by the maximal probability distribution (Fig. 2(b)). In contrast to the
applied phase shift Φ , which defines the uniform in- previouscase,thespatialprofileofthewavepacketshows
max
terval φ (x) ∈ [−Φ ,Φ ], from which the phases a parabolic shape in the semilog scale (inset, Fig. 2(b)).
V max max
are chosen. The probability distribution after eleven AstepwiseincreaseofthedisorderstrengthΦ nicely
max
steps is shown in Fig. 2(a). We used the initial state demonstratesthecontrolledtransitionofthesystemfrom
|ψ i = |0i⊗|Hi, θ = π/8 and a high disorder strength the ballistic evolution (decoherence free quantum walk)
0
(Φ = (1.14±0.05)π). In contrast to the decoherence towardsthe diffusive evolution/localizationin a scenario
max
free quantum walk (Φ =0, inset of Fig. 2(c)), in the with dynamic/static disorder (Fig. 2(c)). For this pur-
max
disordered scenario the expansion of the wave packet is pose we characterize the resulting expansion profile by
highly suppressed. We observe a strictly enhanced ar- its variance σ2. Without decoherence (Φ = 0) the
max
rival probability around the initial position, which also ballistically spreading wave packet shows a large expan-
displays the predicted exponential decay. This striking sion induced by quantum interference after eleven steps.
signature of Anderson localization is emphasized by lin- The slightly higher variance in comparisonto the theory
ear fits in the semilog scaled plot (inset of Fig. 2(a)). can be explained by small polarization rotations in the
Our results are in agreement with a theoretical model EOM. In a system with dynamic disorder, decoherence
determined by a Monte Carlo simulation of 104 different reduces the expansion of the wave packet to the level of
phase patterns compatible with our experiment. Com- a diffusive classical particle. In contrast, static disorder
pared to (i), the number of steps is reduced due to the leads to a stagnation of the spread and hence an even
additional losses introduced by the EOM. smaller variance. Our results clearly demonstrate how
(iii)Fastfluctuations.—Togenerateasystemwithdy- the amount and kind of disorder influence the expansion
namic disorder we detuned the temporal length of the of the particle’s wave packet.
4
Finally, the geometry of the setup allows to observe
easilythe wavepacket’sevolutionstepbystepinallfour
scenarios(Fig. 3(b)). Forcases (i) and (iv) we observea
ballisticspread,withanevenfasterexpansioninasystem
with slow fluctuations. The evolution with fast dynamic
disorder (iii) is clearly diffusive. Lastly, under the con-
dition of static disorder (ii) the variance saturates after
few steps andthe dynamics is dominatedbythe effectof
Anderson localization. For comparison, we show in Fig.
3(c) a theoretical plot for the evolution of the variance
over fifty steps. The parameters used in simulation and
experiment are equivalent to the experimental settings
used for Figs. 1(b), 2(a-b) and 3(a).
In conclusion, we presented how disorder and fluctua-
FIG.3: (a) Averagedprobability distribution inaslow deco- tionsinaperiodiclatticecaninfluencethe evolutionofa
herence scenario with different coin angles θ∈[0,π/4]: Mea-
traversingparticle. Weobservedafastballisticspreadfor
surement (orange, front) and theory (gray, back); (b)+(c)
slowly changing lattice parameters, a diffusive spread in
Trendofthevariancewiththenumberofsteps: Measurement
thecaseofdynamicaldisorderandAndersonlocalization
up to 12 steps (b) and simulation up to 50 steps (c). While
photonsundergoingthedecoherencefreequantumwalk(blue forlatticeswithstaticdisorder. Furthermore,weshowed
triangles) and the evolution with slow dynamic disorder (or- the controlled transition between the different regimes.
ange dots) show a ballistic behavior, in a classical random Thehighflexibilityandcontrolallowsnotonlythestudy
walk (red squares) they move diffusively, and, finally, in the offurtherdecoherencephenomenainquantumwalksbut
case of static disorder (green diamonds) they stagnate.
also to simulate specific physical scenarios of interest for
the solidstate andbiophysicscommunity. Moreover,the
possibility to manipulate quantum walks with time de-
The agreement between theory and measurement in pendent coin operations is a fundamental step towards
the completely dephased scenario (Fig. 2(b)) confirms a the realization of quantum walk-based protocols.
sufficient randomness of the applied noise signal. Fur- We thank P.P. Rohde for helpful discussions. We
thermore, an independent interferometric measurement acknowledge financial support from the German Israel
revealedtherelativefrequencyoftheusedphasesf(|φV|), Foundation (Project 970/2007). K.N.C. and I.J. ac-
as can be seen in Fig. 2(d) with Φmax = (1.02±0.05)π. knowledge support from the Alexander von Humboldt
However, small imperfections of the EOM lead to a dis- Foundation; V.P., A.G. and I.J. from MSMT LC06002
tributionslightlyoffuniformity,whichalsoinfluencesthe and MSM 6840770039.
measured variances shown in Fig. 2(c).
(iv)Slowfluctuations.—Asthefourthscenariowesim-
ulated fluctuations in a homogeneous system, but with
parametersthat changein a time scalemuch largerthan
∗ Electronic address: [email protected]
thefulldurationofasinglequantumwalk. Althoughthe
† Secondary address: Research Institute for Solid State
individual evolution is not affected under these circum-
Physics and Optics,Hungarian Academy of Sciences, H-
stances, an ensemble measurement of subsequent walks
1525 Budapest, P. O.Box 49, Hungary.
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