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Vibrational exciton-mediated quantum state transfert: a simple model.
Vincent Pouthier∗
Institut UTINAM, Universit´e de Franche-Comt´e,
CNRS UMR 6213, 25030 Besanc¸on Cedex, France
(Dated: January 26, 2012)
A communication protocol is proposed in which quantum state transfer is mediated by a vibra-
tional exciton. We consider two distant molecular groups grafted on the sides of a lattice. These
groups behave as two quantum computers where the information in encoded and received. The
lattice playstherole of acommunication channelalong which theexciton propagates and interacts
withaphononbath. Specialattentionispaidfordescribingthesysteminvolvinganexcitondressed
by a single phonon mode. The Hamiltonian is thus solved exactly so that the relevance of the
2 perturbation theory is checked. Within the nonadiabatic weak-coupling limit, it is shown that the
1 systemsupportsthreequasi-degeneratestatesthatdefinetherelevantpathsfollowedbytheexciton
0 to tunnel between the computers. When the model parameters are judiciously chosen, construc-
2 tiveinterferences takeplace between these paths. Phonon-induceddecoherence is minimized and a
high-fidelityquantumstate transfer occurs overa broad temperature range.
n
a
PACSnumbers: 03.65.Yz,03.67.-a,63.22.-m,71.35.-y
J
5
2
I. INTRODUCTION qubit. Theexcitonpropertiesarethus describedbyare-
duceddensitymatrix(RDM)whosebehaviorisgoverned
]
h byageneralizedmasterequation(GME)50–52. Tounder-
Over short length scales, high-fidelity quantum-state
p stand quantum decoherence, we have studied excitonic
transfer (QST) from one region to another is a funda-
- coherences, i.e. the RDM matrix elements that measure
t mental task in quantum information processing1. It is
n the abilityofthe excitontobe inaqubitstate53–56. The
a required to ensure a perfect communication between the dynamics was described using a Fr¨ohlichmodel57 within
u different parts of a quantum computer (QC) or between
the nonadiabaticweak-couplinglimit, i.e. a commonsit-
q adjacent QCs. In that case, solid-state based system is
[ the ideal candidate for scalable computing for, at least, uation for vibrational excitons. It has been shown that
1 two main reasons2. First, in a lattice, the information is high-fidelityQST requiresthe use ofafinite-size system.
v encoded on elementary excitations that naturally propa- Indeed, in an infinite lattice53,54, the phonons behave
4 gate owing to inherent interactions between neighboring as a reservoir. The Markov limit is reached and stan-
8 sites. QST is thus spontaneously achieved without any dard GME approaches can be used. Dephasing limited-
1
external control. Second, no interfacing is needed be- coherentmotiontakesplacesothatthe coherenceslocal-
5
tweentheQCsandthecommunicationchannel(CC)that izepreventinganyefficientQST.Bycontrast,inafinite-
.
1 involves the same elementary excitations. Because the sizelattice,astrongnon-markovianregimeoccursresult-
0 CC depends onthe waythe informationis implemented, inginthebreakdownofGMEmethods55,56. Toovercome
2
different strategies have been elaborated, such as optical thisproblem,perturbationtheory(PT)hasbeenapplied
1
lattices3,arraysofquantumdots4,conductingpolymers5 successfully47–49. It has been shown that the phonons
:
v and spin networks6–16. However, it has been shown that evolve differently depending on whether the exciton oc-
Xi qubitsmaybe encodedonintramolecularvibrations17–23 cupies the vacuum or an excited state. Exciton-phonon
so that vibration-mediated QST is a promising alterna- entanglement takes place resulting in the decay of the
r
a tive for quantum information processing24,25. excitonic coherences. Nevertheless, the decoherence de-
Inthis paper, weproposea protocolin whicha molec- pends on the nature of the one-exciton state.
ular lattice plays the role of the CC, QST being me- In particular, for odd lattice sizes, the coherence in-
diated by high-frequency vibrational excitons. Indeed, volvingthestatelocatedatthebandcentersurvivesover
molecular lattices exhibit regularly distributed atomic an extremely long-time scale, even at high temperature.
subunits. Owingtodipole-dipoleinteractions,theenergy Unfortunately, this state is a stationary wave unable to
of a specific internal vibration delocalizes between these carryinformationbetweenthelatticesides. Nevertheless,
subunits giving rise to narrow-band excitons. Although we can take advantage of its robustness againstdecoher-
many properties of the excitons have been studied26–46, encetodefine aprotocolquitesimilartothatintroduced
their potential interest for QST has been suggested very by Yao et al. in spin chains16. The main idea is to con-
recently47–49. The main idea is to encode the informa- sider QST between two distant molecular groups that
tion on a vibrational qubit defined as a superimposition are grafted on the lattice sides. The structure must be
involving the vacuum and one-exciton states. designed so that a vibration of each molecular group is
Unfortunately, the exciton does not propagate freely resonantwiththerobuststateofthelatticewhileremain-
along the lattice. It interacts with a phonon bath that ing insensitive to the phonons. The grafted groups play
tends to destroy the coherent nature of any vibrational the role of the QCs whereas the lattice defines the CC.
2
YA (0) = a +b 0 YA (t) = a +beihL Owing to the confinement, one-excitonstates are N sta-
tionary waves with quantized wave vectors K = kπ/L,
F F k
S S with k =1,..,N and L=N +1, expressed as
w F F w
S S
N
x 2
ϕ = sin(K x)x . (2)
(0) (L) | ki rL k | i
w0 w0 w0 Xx=1
W W W W
The corresponding eigenenergies ω0 = ω +2Φcos(K )
k 0 k
M M M form N discrete energy levels centered on ω , i.e. the
(1) (x) (N) 0
energy of the so-called robust stationary wave ϕ .
L/2
| i
The phonons of the CC refer to the external motions
FIG. 1: Communication protocol between two distant com-
puters x = 0 and x = L. The information is carried by an of the lattice sites that behave as point masses M con-
exciton that propagates along a finite-sizelattice. nected via force constants W. They define N normal
modes with wave vectors Q = pπ/L and frequencies
p
Ω =Ω sin(Q /2),with p=1,..,N and Ω = 4W/M.
p c p c
Dipole-dipole interactions couple the QCs with the CC RestrictingourattentiontotheLFPMp=1,thpephonon
andallowtheexcitonpropagationsothatneithercontrol Hamiltonian is defined as H = Ωa†a, with Ω Ω , a†
B 1
≡
nor interfacing is required. and a being standard boson operators. In the phonon
In such a confined system, strong non-markovian ef- Hilbertspace B,theeigenstatesarethusthewell-known
fects will occur so that the dynamics will be addressed numberstatesEn . Asdetailedpreviously47,the exciton-
within the effective Hamiltonian concept provided by phonon interac|tiion V = M(a† + a) favors the exciton
PT58,59. However, the fundamental question arises scattering from ϕk to ϕk±1 mediated by phonon ex-
| i | i
whether PT is relevant or not, depending on the model changes. The matrix elements of the M operator are
parameters. To answer that question, we shall restrict defined as
our attention to a system in which only the lowest fre-
quency phonon mode (LFPM) is considered, this mode Mkk′ =η(δk,k′+1+δk,k′−1), (3)
beingresponsibleforthelargestperturbationoftheexci-
ton dynamics47,48. Due to its simplicity, this model can where η = [(EBΩ/L)(1 (Ω/Ωc)2)]1/2, EB being the
−
be solved exactly so that the PT performance can been small polaron binding energy expressed in terms of the
checked easily. coupling strength χ as EB =χ2/W.
The QCs are formed by two molecular groups x = 0
The paper is organized as follows. In Sec. II, the
and x = L whose internal dynamics is described by a
exciton-phononHamiltonianisdescribedandQSTisfor-
two-levelsystem (Fig. 1). Let 0 (resp. L ) be the first
mulated in terms of the excitonic coherences. Then, PT
| i | i
excited state of the 0th (resp. Lth) moleculargroupand
is applied to evaluate the system eigenproperties and
ω thecorrespondingBohrfrequency. Thevacuumstate
to determine an approximate expression of the exciton S
describesthe twogroupsintheirgroundstate. Lo-
RDM. In Sec. III, a numerical analysis is performed in qc
|⊘ i
cated far enough from each other, these groups do not
which PT is compared with exact calculations. The re-
interact. Their internal dynamics is governed by the
sults are discussed and interpreted in Sec. IV.
Hamiltonian H =ω (0 0 + L L).
qc S
| ih | | ih |
The QCs, assumed to be sufficiently far from the lat-
tice, are insensitive to the phonons. By contrast, owing
II. THEORETICAL BACKGROUND
to dipole-dipole interactions, they interact with the lat-
tice exciton. This interaction originates in a vibrational
A. Model Hamiltonian energy transfer between the group 0 (resp. L) and the
lattice site x=1 (resp. x=N) as
We consider a CC formed by a 1D molecular lattice
W =Φ (0 1 + 1 0 + L N + N L), (4)
that contains an odd number of sites N (Fig. 1). Each S | ih | | ih | | ih | | ih |
site x=1,...,N is occupied by an atomic subunit whose
where Φ = ǫΦ, ǫ being the strength of the coupling
internaldynamicsisdescribedbyatwo-levelsystem. Let S
between the QCs and the CC.
x be the first excited state of the xth two-level system
| i Within this model, the exciton dynamics is governed
and ω the corresponding Bohr frequency. The vacuum
0 by the Hamiltonian H =H +H +W. It acts in the
state describes all the two-level systems in their A cc qc
ground|⊘sctcaite. In the CC, the exciton Hamiltonian is de- Hilbert space EA whose dimension is N +3. EA is gen-
erated by the vacuum = and by N +2
fined in terms of the hopping constant Φ as (~=1) |⊘i |⊘cci⊗|⊘qci
one-exciton eigenstates ψ , associated to the eigenfre-
µ
| i
quencies ω , with µ=0,...,L. The exciton Hamiltonian
N N−1 µ
Hcc =xX=1ω0|xihx|+ xX=1Φ(|x+1ihx|+|xihx+1|). (1) imsitnheusthreesweroitntee-nexacsitHonAs=tatPesLµ,=w0eωµfi|rψstµifihψxµω|.STtoo dωe0tesro-
3
energy w +2F The exciton-phonon Hamiltonian is thus written as
j1 0 y0 H = H0+V where H0 = HA+HB is the unperturbed
Hamiltonian. It acts in the Hilbert space =
A B
E E ⊗E
that is partitioned into independent subspaces as =
j E
L2-1 E0 ⊕ E1. In the zero-exciton subspace E0, V = 0 so
that the unperturbed states are eigenstates of H. They
j w w+ y correspond to tensor products |⊘i⊗|ni that describe n
0 L2 L 0 w yy+-o ftruerebpedhosntoantess. I|Ψn0µt,hneio=ne|-ψexµcii⊗ton|nsiubresfpearcteoEf1r,etehpehuonnpoenrs-
j - accompanied by an exciton in state ψµ . Because V
L2+1 turns on, they are no longer eigenstat|es oif H. The ex-
acteigenstates Ψ ,witheigenenergiesE ,correspondto
i i
| i
entangled exciton-phonon states.
The coupling V favors exciton scattering from ψ to
j y | µi
N w -2F L ψµ′ through phonon exchanges. The allowed transi-
0 | i
tions are specified by the selection rules ψµ M ψµ′ =
h | | i 6
0 (Eq.(3)). Because ψ interacts with the phonons
FIG.2: Exciton energy spectrumfor ǫ=0(left side) andfor µ
| i
ǫ6=0 (right side). through its dependence with respect to the stationary
waves,these rules are quite similar to those that charac-
terize the CC (ψ is coupled with ψ ). However,
µ µ±1
| i | i
that a resonance occurs between the two localized states two main differences occur. First, because ψo only de-
| i
0 and L and the robust stationary wave ϕ (Fig. pends on the localized states, it remains insensitive to
L/2
|2)i. The|n,iǫ is assumed to be sufficiently sm|all sio that thephononbath. Second,asplittingoccursforthetran-
off-resonantinteractionsbetweenthelocalizedstatesand sitions involving ψ± . Indeed, because ψL/2±2 reduces
| i | i
theremainingstationarywavescanbe disregarded. This to ϕL/2±1 , it interacts with ψ+ and ψ− , both states
| i | i | i
condition is satisfied provided that ǫ π 2/L. As a depending on ϕL/2 . Nevertheless, within the nonadia-
≪ | i
result,the one-excitoneigenstates|ψµiare spplitinto two baticlimit,i.e. for4Φ<Ωcandǫ≪π 2/L,theallowed
groups (Fig. 2). transitions do not conserve the energyp. There is no res-
The first group contains N 1 states that reduce onance between the coupled unperturbed states so that
−
to the stationary waves ϕk , with k = L/2. For second order PT can be applied to treat V in the weak
| i 6
(µωµ==0,ω.µ0..+,L1)/w2h−er2e,asthfoeyr µar=e Lde/fi2n+ed2,a..s.,|Lψ,µtih=ey |rϕedµ+uc1ei cPoTupislinvaglildimaittt(eEmBpe≪ratuΦr)e.TApsrodveitdaeidledthpatreLvi<ouLsl∗y4w7,i4t8h,
to |ψµi=|ϕµ−1i (ωµ =ωµ0−1). The secondgroupresults L∗ ≈ 0.2Ω2c/EBkBT. When L > L∗, quasi-resonances
fromthehybridizationbetweenthedegeneratestates 0 , take place between the unperturbed coupled states and
| i
ϕ and L . Indeed, 0 (resp. L ) interacts with PT breaks down.
L/2
| i | i | i | i
ϕ throughthecouplingconstantg =ǫΦ 2/L(resp.
L/2
|g′ =gisin(Nπ/2)). Whenǫ=0,thesecouplinpgsraisethe
6 B. Excitonic coherences and QST
degeneracy so that the eigenstates are defined as
1 1 ∆N Withoutanyperturbation,theCCandthe QCsarein
ψ = 0 ϕ + L
| ±i 2| i± √2| L/2i 2 | i thermal equilibrium at temperature T. Assuming that
ψo = 1 0 ∆N L , (5) sωt0at≫e.kBByTc,oanltlrtahste,ttwhoe-plehvoenlosnysstfeomrms aarethienrmthaelirbagtrhoudned-
| i √2| i− √2| i
scribedby the density matrixρ =exp( βH )/Z , Z
B B B B
−
where ∆N = sin(Nπ/2) and with the convention µ = being the phonon partition function (β = 1/kBT). In
L/2 1 and µ = L/2 o. The corresponding thatcase,oneassumesthatthe internalvibrationsinter-
eigen∓frequ≡en±cies are expressed≡as actwithanexternalsourcesothattheexcitonisinitially
preparedin a state ψ = . This step is supposed to
A
| i6 |⊘i
ω = ω 2ǫΦ/√L be rather fast when comparedwith the typicaltime evo-
± 0
± lutionofthephonons. Thefullsystemisthusbroughtin
ω = ω . (6)
o 0 a configuration out of equilibrium and its initial density
matrix is defined as ρ(0)= ψ ψ ρ .
ψ± and ψo define three quasi-degenerate states that | Aih A|⊗ B
| i | i To study QST between the QCs, ψ defines a qubit
involve a superimposition of the localized states, either A
| i
implemented on the molecular group x=0 as
symmetric or antisymmetric. Note that ψ is exactly
o
| i
located at the band center and it does not depend on
ψ =α +β 0 , (7)
A
the stationary wave ϕ . This is no longer the case | i |⊘i | i
L/2
for ψ and ψ tha|t lieijust above and just below the where α2 + β 2 = 1. Our aim is thus to measure the
+ −
| i | i | | | |
band center, respectively. ability of the system to freely evolve in time so that this
4
initial qubit is copied on the second QC x=L (Fig. 1). transformation U = exp(S) that generates a new point
Whatever its duration, QST must be realized with the ofviewinwhichtheeffectiveHamiltonianHˆ =UHU† is
largestfidelitydespitethecouplingwiththephonons. To block-diagonal in the unperturbed basis. The generator
define this fidelity measure, different objects have been S isexpandedasaTaylorseriesinthecouplingV sothat
introduced60,oneofthemostwidelyusedbeingcertainly Hˆ becomes diagonalwith respect to the phononnumber
the so-called average Schumacher’s fidelity6. Here, we states, only. Up to second order, it is written as
restrict our attention to the excitonic coherences. In-
deed, the exciton properties are encoded in the RDM Hˆ =H +δH +(Ω+δΩ)a†a (9)
A A
σ(t) = Tr [ρ(t)], where Tr is a partial trace over the
B B
phonon degrees of freedom. The coherences are thus the where δH and δΩ are operators in whose matrix
A A
off-diagonalmatrixelementsσx⊘(t). Theyprovideinfor- elementsaredefinedas(intheunperturEbedbasis ψµ )
mation about the ability of the xth two-level system to {| i}
develop a superimposition between its ground state and
L
1 M M M M
its first excited state a time t. δH = µ2µ µµ1 + µ1µ µµ2
When the initial qubit is implemented on the 0th QC, Aµ1µ2 2µX=0ωµ1 −ωµ−Ω ωµ2 −ωµ−Ω
the coherence σ (0) is turned on at time t=0. There-
0⊘ L
fore, the ability of the Lth QC to develop a superim- δΩ = 1 Mµ2µMµµ1 + Mµ1µMµµ2
position involving and L at time t is given by µ1µ2 2 ω ω Ω ω ω Ω
|⊘i | i µX=0 µ1 − µ− µ2 − µ−
σ (t)=G (t)σ (0) with
L⊘ L0 0⊘
L
1 M M M M
G (t)= LTr ρ eiHBte−iHt 0 . (8) + µ2µ µµ1 + µ1µ µµ2 (10)
L0 h | B(cid:2) B (cid:3)| i 2µX=0ωµ1 −ωµ+Ω ωµ2 −ωµ+Ω
The effective exciton propagator G (t) generalizes the
L0
conceptoftransitionamplitude. Ityieldstheprobability δH is the correction of the exciton Hamiltonian ow-
A
amplitude to observe the exciton in L at time t given ing to the coupling with the phonons. It results from
| i
that it was in 0 at t = 0. Its effective nature results the spontaneous emission of a phonon during which the
| i
fromthefactthattheexcitoninteractswiththephonons exciton realizes a transition from ψ to ψ . How-
during its transition. ever, in the nonadiabatic limit, th|e µe1niergy|isµinot con-
The effective propagator is the central object of the served during the transition. The emitted phonon is im-
present study. The condition GL0(t) = 1 reveals that mediately reabsorbed and the exciton realizes a second
| |
theLthQCreachesastateattimetthatisequivalentto transition from ψ to ψ . In other words the exci-
the initial state, to a phase factor. Note that this condi- ton does no long|erµipropa|gµa2tei freely but it is dressed by
tion is exactly the QST fidelity when the coupling with a virtual phonon cloud. This dressing renormalizes the
the phonons is disregarded6. Consequently, depending excitonicenergiesω byanamountδω =δH . Inad-
µ µ Aµµ
onthevalueofthemodelparameters,studyingthemax- dition, it induces effective interactions δH between
Aµ1µ2
imum value of GL0(t) provides key information about distinct excitonic states that can no longer be neglected
| |
the fidelity of the QST. for quasi-degenerate states.
Similarly, δΩ defines the correction of the phonon fre-
quency. It has two origins. First, the phonon can be
C. Perturbation theory
absorbed giving rise to excitonic transitions. Because
this process does not conserve the energy, the phonon is
In its operatorial formulation58, standard PT involves immediately re-emitted. Second, the phonon can induce
a unitary transformation that provides a new point of thestimulatedemissionofasecondphononduringwhich
view in which the exciton-phonon dynamics is described the exciton realizes transitions. But, as previously, the
by an effective Hamiltonian. The key point is that this emitted phonon is immediately reabsorbed. Both mech-
Hamiltonian is diagonalin the unperturbed basis. Quite anisms are virtual processesindicating that the phonons
powerful to treat finite-size lattices47,48, this approach are dressed by virtual excitonic transitions.
breaks down in the present situation because the unper- To diagonalizeHˆ for eachphonon number, we use the
turbedHamiltonianexhibitsquasi-degeneratestates. For
fact that δΩ is smaller than δH within the weak cou-
A
small ǫ values, Ψ0 and Ψ0 have almost the same
| +,ni | −,ni pling limit. Consequently, let χν be the eigenstates of
energy. Although they do not directly interact through | i
H + δH and ωˆ the corresponding eigenfrequencies
A A ν
V, they are coupled with the same unperturbed states.
(ν = 0,...,L). Up to second order in V, one introduces
Consequently, effective couplings occur between quasi-
δΩ = χ δΩχ as the correction of the phonon fre-
ν ν ν
degenerate states resulting in errors in the calculations h | | i
quency induced by the exciton that occupies the state
of the corrected energies. χ . Within these notations, Hˆ is rewritten as
ν
To overcome this problem, quasi-degenerate PT is | i
applied59 (Appendix A). To proceed, we take advan-
L
tage of the fact that the effective couplings conserve the Hˆ ωˆ χ χ +Hˆ(ν) χ χ , (11)
phonon number. Therefore, our procedure involves a ≈νX=0 ν| νih ν| B ⊗| νih ν|
5
whereHˆ(ν) =(Ω+δΩ )a†a isthe Hamiltonianthatgov-
B ν
ernsthephonondynamicswhentheexcitonisinthestate
χ .
ν
| i
In the new point of view, the exciton-phonon dynam-
ics is thus governedby the effective Hamiltonian Hˆ that
is diagonal in the basis χ n . Its eigenvalues de-
ν
| i⊗| i
fine the system eigenfrequencies up to second order in
V as E = ωˆ +n(Ω+δΩ ). Hˆ does no longer refer
ν,n ν ν
toindependentexcitationsbutitcharacterizesentangled
exciton-phononstates. A state χ describes an exciton
ν
| i
dressed by a virtual phonon cloud whereas the number
state n describesnphononsclothedbyvirtualexcitonic
| i
transitions. This entanglement is clearly highlighted in
the starting point of view in which the eigenstates do no
longer factorize as Ψ =U† χ n .
ν,n ν
| i | i⊗| i
Asdetailedpreviously48,PTisparticularlysuitablefor
deriving an approximate expression for G (t). To pro-
L0 FIG.3: Ei−Eν,n vsEi for ǫ=0.01, χ=10pN andL=10.
ceed, we first introduce U and diagonalize H in Eq.(8).
Then, we define the effective density matrix
1
ρ(ν)(t)= exp (βΩ+iδΩ t)a†a , (12)
B Z(ν)(t) − ν
B (cid:2) (cid:3)
where Z(ν)(t) = (1 exp[ (βΩ+iδΩ t)])−1. Strictly
B − − ν
speaking, ρ(ν)(t) is not a density matrix since it yields
B
complex values for the phonon population. However,
it is isomorphic to ρ with the correspondence βΩ
B
→
βΩ+iδΩ t and it provides averages equivalent to ther-
ν
mal averages. Consequently, after simple algebraic ma-
nipulations, G (t) is rewritten as48
L0
L Z(ν)(t)
G (t) = B exp[ iωˆ t] (13)
L0 ν
Z − ×
νX=0 B FIG.4: Ei (thinx)andEν,n (fulllines) vsη forǫ=0.01and
Tr ρ(ν)(t) LU†(t)χ χ U (0)0 . L=10.
Bh B h | ν | νih ν| ν | ii
where Uν(t) = eiHˆB(ν)tUe−iHˆB(ν)t. Expanding U in a Tay- better is the agreement. Of course, ∆E increases with
lorserieswithrespecttoV,onefinallyobtainsthesecond E indicating that the PT accuracy decreases with the
i
order expression of GL0(t), as detailed in Appendix B. phononnumber n because V scales as √n. For instance,
for E 100Φ (n 50), ∆E is approximately equal to
i
10−3Φ≈whereasitr≈eaches5 10−3ΦforE 200Φ. Nev-
i
III. NUMERICAL RESULTS ertheless, we have verified×that ∆E is sma≈ller than the
energy level spacing indicating that PT remains valid,
In this section, numerical calculations are carried out even for quite large energies. As shown in the inset,
to show the relevance of PT for describing QST. To pro- the curve ∆E vs E behaves almost periodically, with
i
ceed, the previous formalism is applied to amide-I vi- a period approximately equal to Ω, indicating that the
brations in α-helices, a system for which the parameters PT accuracy depends on the nature of the unperturbed
are well-known26–36 : ω = 1660 cm−1, W = 15 Nm−1, states. PT is exact for the unperturbed states that in-
0
M =1.8 10−25kg,Ω =96.86cm−1 andΦ=7.8cm−1. volve the exciton state insensitive to the phonons. By
c
×
To avoid PT breakdown, the size is set to L = 10 and contrast,unperturbedstateslocatedintheneighborhood
the exciton-phonon coupling strength is fixed to χ = 10 of quasi-degenerate states are the less well-corrected.
pN. Special attention will be paid for characterizing the As shown in Fig. 4, ∆E slightly increases with χ
influence ofthe couplingǫ betweenthe QCsandthe CC. so that PT remains valid in the intermediate coupling
The difference ∆E =E E betweenexactandap- regime. Itreproducestheinfluenceoftheexciton-phonon
i ν,n
−
proximate energies is shown in Fig. 3. For ǫ=0.01, PT coupling on the energy levels up to χ=20 pN (η =1.41
provides a very good estimate of the energy spectrum cm−1). In particular, PT accounts for the energy level
overa broadenergyscale. The smallerthe energyis,the crossingprocessthataffectsthedressedstates. Owingto
6
o
FIG.6: (Coloronline) ωˆµ−ωµ−δωµ vsǫforχ=10pNand
L=10.
FIG. 5: (Color online) δωµ vs ǫ for χ=10 pN and L=10. w
+
w
the coupling V, unperturbed states hybridize giving rise +
toenergyshiftandsplittingcharacteristicofanti-crossing
phenomena. Someenergylevelsrepeleachotherwhereas
other energy levels get closer. These levels describing
w
exact uncoupled eigenstates, energy level crossing takes -
place.
w
Theǫdependenceoftheexcitonenergycorrectionδω -
µ
is shown in Fig. 5. Owing to the dressing by virtual
phonons, the exciton energy is redshifted and δω de-
µ
creases linearly with E . However, this behavior de-
B
pends onthe excitonstate. Forthe state ψ insensitive
o
| i
to the phonons, δωo = 0. By contrast, the stationary FIG.7: (Coloronline)ǫdependenceofωˆ± (fulllines)andω±
waves µ = o, experience an energy correction approx- (dashed lines) for χ=10 pN and L=10.
6 ±
imately equal to 0.2E . This shift reduces to 0.1E
B B
− −
for the states ψ and ψ , whose energies define the
0 L
| i | i
band edges, and for the quasi-degenerate states ψ . states mainly correspond to superimpositions involving
±
| i
Moreover, Fig. 5 reveals that the energy correction of ψ and ψ .
+ −
| i | i
the stationary waves is almost ǫ independent. This is The ǫ dependence of the energy ωˆ of the quasi-
±
no longer the case for the quasi-degenerate states be- degenerate dressed state χ is shown in Fig. 7. For
±
| i
cause δω (resp. δω ) decreases (resp. increases) lin- quite large ǫ values, ωˆ (resp. ωˆ ) is slightly red-shifted
+ − + −
early with ǫ (not well distinguishable in Fig. 5). Note when compared with ω (resp. ω ). In that case,
+ −
that δω =δω =0.108E when ǫ=0. ωˆ follows the ǫ dependence of the corrected bare en-
+ − B ±
Fig. 6 shows the ǫ dependence of the difference be- ergy ω± + δω± and χ+ (resp. χ− ) basically cor-
| i | i
tween the energy ωˆµ of a dressed state |χµi and the cor- responds to |ψ+i (resp. |ψ−i). As ǫ decreases down
rectedenergy ω +δω ofa barestate ψ . For the sta- to zero, ωˆ decreases and it reaches ω . By contrast,
µ µ µ + 0
| i
tionarywavesµ=o, ,acorrespondenceoccursbetween ωˆ− increases and it converges to a value located below
6 ±
bothenergiessuggestingthatthedressingmainlyinduces the band center. The energy difference characterizes an
an energy renormalization without significantly modify- anti-crossing phenomena so that the energy levels of the
ing the state. In fact, we have verified that χ ψ dressed states repel each other. This is the signature
µ µ
| i ≈ | i
for µ = . Note that the correspondence is exact for of the hybridization between ψ+ and ψ− that results
6 ± | i | i
the state insensitive to the phonons so that one defines in the formation of two quasi-degenerate dressed states
a dressed state |χoi ≡ |ψoi (ν = L/2 ≡ o). By contrast, |χ±i≈(|ψ+i±|ψ−i)/√2.
the correspondence disappears for the quasi-degenerate The ǫ dependence of the phonon frequency correction
statesandthe smallerǫis,thelargeristhe energydiffer- δΩ is shown in Fig. 8. The phonon frequency is ei-
ν
ence. A strong hybridization occurs between ψ that ther red-shifted or blue-shifted depending on the nature
±
| i
results in the formation of two quasi-degenerate dressed of the exciton. A redshift is induced when the exciton
states. Denoted χ (ν = L/2 1 ), these dressed occupiesadressedstate χ withν =0,...,L/2 1. The
± ν
| i ∓ ≡ ± | i −
7
1.0
e=0.020
0.8
)| 0.6
(t0
L
G 0.4
|
(-)
(o) 0.2
(+)
0.0
1.0
e=0.013
0.8
)| 0.6
(t0
L
G 0.4
|
0.2
FIG. 8: (Color online) δΩν vsǫ for χ=10 pN and L=10. 0.0
0 2000 4000 6000 8000 10000
Ft
closertothebandedgetheexcitonenergyislocated,the
larger is the shift. By contrast, an exciton that occupies FIG. 9: (Color online) Time evolution of |GL0(t)| for χ=10
a state χ with ν = L/2+ 1,...,L yields a blueshift pN,L=10andT =300K.Exactcalculations(dashedlines)
ν
| i and PT (full lines).
of the phonon frequency. As previously, excitonic states
near the band edge favor the largest phonon frequency
shift. Note that in the state χ , the exciton produces
o
no phonon frequency shift. A|s sihown in Fig. 8, when T=100 K
the exciton occupies a state isomorphic to a stationary
wave, the phonon frequency shift is ǫ independent. By
contrast,a strong ǫ dependence occurs when the exciton
is in a quasi-degeneratedressedstate χ . In that case,
±
| i
δΩ < 0 increases when ǫ decreases whereas δΩ > 0
+ − T=200 K
decreases when ǫ decreases. For nonvanishing ǫ values,
it turns out that δΩ δΩ . However, when ǫ tends
− +
≈ −
to zero, δΩ converges to zero whereas δΩ reaches a
+ −
T=300 K
quite small nonvanishing value. In other words, when
ǫ 0, the frequency shift is negligible when the phonon
→
isaccompaniedbyanexcitonineither χ , χ or χ .
o + −
| i | i | i
The key ingredients entering PT being characterized,
letusnowstudytheeffectiveexcitonpropagator GL0(t) FIG.10: (Coloronline)Max[|GL0(t)|]vsǫforχ=10pNand
| |
whose time evolution is displayed in Fig. 9. The fig- L=10. Exactcalculations(dashedlines)andPT(fulllines).
ure shows that PT perfectly agrees with exact calcu-
lations over a long-time scale. Initially equal to zero,
G (t) first increases with time. Then, at time T , By contrast, when ǫ = 0.013, G (t) exhibits a single
| L0 | M | L0 |
it reaches a maximum value G quite close to unity. absolute maximum whose behavior remains quite stable
M
Finally, it develops damped oscillations that fluctuate whenǫvaries. ItoccursatT =699.85Φ−1anditsvalue
M
around 1/2. These oscillations support a high-frequency GM =0.97 is very close to unity.
small-amplitude modulation whose behavior depends on The ǫ dependence of the maximum value G =
M
ǫ. Whenǫ=0.020, G (t) exhibitsadoublemaximum. Max[G (t)] is illustrated in Fig. 10 for different tem-
L0 L0
Att=248.47Φ−1, G| (t) |firstreachesalocalmaximum peratu|res. Th|e figure shows that PT provides results in
L0
| |
whose value is equal to 0.83. Then, the absolute maxi- a quite goodagreementwith exact calculations. A small
mum G = 0.88 occurs at T = 495.48Φ−1. Such a discrepancy occurs at high temperature and for small ǫ
M M
temporalstructureis verysensitiveto ǫ. Whenǫslightly values, only. When ǫ decreases from 0.05, the curve G
M
varies around 0.020, the absolute maximum jumps be- vsǫexhibits aseriesofminima andmaximawhosevalue
tween the two values of the double maximum so that depends on the temperature. A local minimum corre-
T exhibits a discontinuous character with respect to ǫ. sponds to a singularity. The curve exhibits a kind of
M
8
T∗ 40Kforǫ=0.01whereasT∗ 130Kforǫ=0.02.
≈ ≈
Finally, for ǫ=0.013, the curve G (ǫ) lies in the neigh-
M
borhood of its absolute maximum. Consequently, G
M
slightly decreases with the temperature. It varies from
(almost) unity for T = 10 K to 0.97 for T = 300 K. It
scales as G 1 (T/T )2 with T 1510 K so that
M 0 0
≈ − ≈
the lost information remains smaller than 3 %.
IV. DISCUSSION
According to the numerical results, PT is a powerful
toolfordescribingtheexciton-phonondynamics. Within
the nonadiabatic weak-coupling limit, it accurately ac-
counts for the spectral properties of the system over a
broadenergyscale. Moreover,PTisparticularlysuitable
FIG. 11: (Color online) Max[|GL0(t)|] vs T for χ = 10 pN
forcharacterizingthetimeevolutionoftheeffectiveexci-
and L = 10. The calculations have been carried out using
PT. ton propagatorover a long-time scale, even at high tem-
perature. In that context, both exact and approximate
calculations have revealed the potential powerfulness of
the proposed communication protocol. The key point
cuspintheneighborhoodofwhichthetime T (ǫ)isdis-
M
is that its efficiency strongly depends on the coupling
continuous. By contrast, a local maximum corresponds
ǫ between the QCs and the CC. When ǫ is judiciously
toawelldefinedpointwherethefirstderivativeofG (ǫ)
M
chosen, it turns out that the maximum value of G (t)
vanishes. Closetoalocalmaximum,T (ǫ)remainscon- L0
M | |
is quite close to unity. A high-fidelity QST occurs over
tinuous. Whatever the temperature, it turns out that
a broad temperature range, the lost information during
the absolute minimum of G (ǫ) occurs for ǫ 0.021.
M
≈ the transfer remaining smaller than 3 %. By contrast,
Its value slightly decreases with the temperature and it
specificǫvaluesinduceaholeinthe transmissioncurves.
variesfrom0.86forT =100Kto0.84forT =300K.By
The maximum value of the effective exciton propagator
contrast,theabsolutemaximumtakesplaceforǫ 0.013
≈ deviates from unity resulting in the impoverishment of
T. Equalto0.99forT =100K,itdecreasesto0.97for
∀ the transfered information. The lost information drasti-
T =300K. Note that if one defines P =100 (1 G )
M
× − cally increases and it can represent almost 15 % of the
as the percentage of the lost information, one obtains
initially implemented infromation at high temperature.
P 3%athightemperature. Whenǫtendstozero,the
≈ Of course, these results depend on both the lattice size
maximaandtheminimaarebecomingmorefrequentbut
and the exciton-phonon coupling strength whose the in-
less pronounced. G tends to a temperature dependent
M
crease reduces the fidelity of the transfer.
value approximately equal to 0.99 and 0.95 for T = 100
Tointerpretthese results,PTcanbe usedforderiving
and 300 K, respectively. We have verified that the curve
a simplified expression of G (t). Indeed, our numeri-
G vsǫdependsonboththelatticesizeandtheexciton- L0
M
cal studies have revealed that the so-called diagonal ap-
phononcoupling strength. These parametersmodify the
proximation works quite-well. Consequently, the exact
positionandthevalueoftheminimaandofthemaxima.
exciton-phonon eigenstates Ψ basically reduce to the
For instance, for ǫ = 0.01 and T = 300 K, one obtains i
| i
effective Hamiltonian eigenstates χ n . The trans-
G = 0.93 for L = 10 and χ = 10 pN, G = 0.89 for ν
M M | i⊗| i
formationU inEq.(13)behavesasthe unitoperatorand
L = 20 and χ = 10 pN and G = 0.73 for L = 10 and
M
theexciton-phononentanglementmainlyresultsfromthe
χ = 20 pN. Increasing either L or χ reduces the QST
modificationoftheexcitonstates ψ χ ,therenor-
fidelity. µ ν
| i→| i
malization of the exciton energies ω ωˆ and the cor-
Finally, the temperature dependence of GM is illus- rectionofthe phononfrequency Ω µΩ→+δνΩ . G (t) is
trated in Fig. 11 for different ǫ values. The calcula- → ν L0
thus rewritten as
tions havebeen carriedoutusing PT that is particularly
suitable for the considered ǫ values. When ǫ = 0.021, L
the curve GM(ǫ) lies in the neighborhood of its absolute GL0(t)≈ Fν(t)e−iωˆνthL|χνihχν|0i, (14)
minimum. Consequently,overthetemperaturerangedis- νX=0
played in the figure, G takes the smallest values. Al-
M where F (t)=Z(ν)(t)/Z is the decoherence factor as
most uniform, it is approximately equal to 0.86 0.01. ν B B
±
The corresponding lost information is thus about 14 %. 1 e−βΩ
For ǫ = 0.01 and 0.02, a critical temperature T∗ dis- Fν(t)= 1 e−−βΩ−iδΩνt. (15)
criminates between two regimes. When T < T∗, GM −
rapidly decreases with the temperature. By contrast, In Eq.(14), G (t) is the sum of the probability ampli-
L0
when T > T∗, smooth variations take place. Note that tudes associated to the different paths that the exciton
9
can follow to tunnel between the QCs. A given path de- ǫ as δω = ∞ ( 1)rE ǫr. The positive coefficients
± − r=0 − r
fines a transition through the dressed state χν . The Er are definedPas
| i
corresponding amplitude involves the weight of the lo-
calized states in the dressed state, a phase factor that η2 E¯r E¯r
E = + , (16)
accountsforthefreeevolutionofthedressedexcitonand r 2 (cid:18)(Ω+∆ω)r+1 (Ω ∆ω)r+1(cid:19)
−
thedecoherencefactor. Thislatercontributionoriginates
in the exciton-phonon interaction. Indeed, when Eq.(8) where E¯ = 2Φ/√L and ∆ω = ω0 ω0 . In the
| L/2 − L/2±1|
is developed in the phonon number state basis, GL0(t) nonadiabatic limit, E0 EB/L so that δω+ (resp.
≈ −
canbeinterpretedasfollows. Thesystembeingprepared δω ) decreases (resp. increases) linearly with ǫ form
−
in the factorized state |0i⊗|ni, GL0(t) is the probabil- −EB/L, in a quite good agreement with the results dis-
ity amplitude to observethe system in a factorized state played in Fig. 5. Note that δω is similar to the shift
±
L exp( inΩt)n . Itthus describesanexcitonictran- experienced by the band edge states because the weight
| i⊗ − | i
sition during whichthe phonons evolvefreely. When the of ϕ in ψ is equal to 1/√2.
L/2 ±
| i | i
exciton occupies the state χν , the phonon frequency is In addition to energy renormalization, the exciton-
| i
modified. The probability amplitude that the phonons phonon interaction yields an effective coupling v =
+−
evolve freely reduces to a phase factor exp(−inδΩνt). −(δω++δω−)/2between|ψ+iand|ψ−i(Eq.(10)). Quite
Of course, such a phase factor does not affect the exci- similar in magnitude to the energy shifts, this coupling
tonic coherence when the phonons are initially in a pure cannolongerbeneglectedresultinginastronghybridiza-
state. However, at finite temperature, the phonons are tion between the quasi-degenerate states. These states
described by a statistical mixture of number states. A behaveasatwo-levelsystemindependentontheremain-
thermal average is required and it yields a sum over dif- ing states whose eigenstates define two quasi-degenerate
ferent phase factors which interfere with the other. The dressed states as
interferences lead to a decay of the excitonic coherence
encoded in the decoherence factor. χ+ +cos(θ)ψ+ +sin(θ)ψ−
| i ≈ | i | i
At this step, Eq.(14) can be simplified because only χ− sin(θ)ψ+ +cos(θ)ψ− . (17)
| i ≈ − | i | i
thedressedstatesthatinvolvethelocalizedstates 0 and
L contributesignificantlytotheexcitonpropagat|ori. In- The corresponding eigenenergies are expressed as
| i
deed,thebarestates ψ thatcorrespondtothestation-
arywavesoftheCC(|µµ=io, )arenotdegenerated. The ωˆ =ω + δω++δω− 1√∆, (18)
6 ± ± 0 2 ± 2
dressingbyvirtualphononsmainlyinducesenergyrenor-
malization without significantly modifying their nature. where
Therefore, the system exhibits N 1 dressed states iso-
morphic to these stationary waves−, i.e. χ ψ and ∆ = (ω +δω ω δω )2+4v2
| µi ≈ | µi + +− −− − +−
ωaˆrµe≈almωµos+tδinωdµefpoernµde6=ntoo,n±.thBeelcoacuasliezethdessteadterse,ssthedeisrtcaotnes- cos(2θ) = ω++δω+−ω−−δω−. (19)
√∆
tributiontoG (t)canbedisregarded. Notethat ψ is
L0 µ
| i
coupled with zµ neighboring states through phonon ex- The hybridization is encoded in the θ parameter whose
changes. In the nonadiabatic limit (ωµ ωµ±1 Ω), value depends on both the exciton-phonon coupling
| − | ≪
Eq.(10) yields δωµ ≈ −zµEB/L, as observed in Fig. 5. strength, measured by δω±, and the energy difference
Because z1 = zL = 1, the shift experienced by the band δ = ω+ ω− 2E¯ǫ. Two asymptotic situations oc-
edge states µ = 1 and µ = L is two times smaller than cur. For−quite≡large ǫ values, δω δ so that the
±
the shift experienced by the remaining stationary waves influence of the quasi-degeneracy| is n|e≪gligible. The hy-
for which zµ = 2. In a market contrast, the bare state bridization is weak and the dressed states reduce to the
|inψgoiddroeesssendotstiantteeraχcotwithψtoheopnhloynionnvso.lvTehsethcoerlroecspaloiznedd- bTahreebsatareteesn,eir.ge.ies|χex±pier≈ien|cψe±aisaimndilaωˆr±red≈shωif±t a+ppδrωo±x-.
| i ≡ | i
states 0 and L (see Eq.(5)) and the decoherence fac- imately equal to E /L, as displayed in Fig. 7 for
tidoeraFlop(|at)tihrefdoruct|ehseiteoxucintiotny.icCtorannsesiqtuioenntblye,tw|χeoeindtehfienQesCasn. ǫth≈e 0h.y0b5r.idBizyactioonntr−iassts,tBrfoonrgs.maOllnǫevoabltuaeisn,s|δθω±|≫π/4δ aanndd
≈
The fundamental point concerns the quasi-degenerate χ (ψ ψ )/√2. The energy ωˆ reaches the
± + − +
states ψ that are profoundly perturbed by the dress- |banid≈cent|erain±d|itscialesasωˆ ω +(E¯ E )2ǫ2/2E .
± + 0 1 0
ingme|chaniism. Asforthestationarywaves,theexciton- Bycontrast,ωˆ ω 2E 2E≈ǫ2 (E¯ E−)2ǫ2/2E lies
− 0 0 2 1 0
≈ − − − −
phononinteractioninducesaredshiftδω ofeachenergy closetoω 2E /L,inagoodagreementwiththeresults
± 0 B
−
ω . Thisinteractionoriginatesinthedependenceof ψ displayed in Fig. 7. Note that in that case χ reduces
± ± +
| i | i
withrespecttotherobuststationarywave ϕ . There- toasuperimpositionofthe localizedstateswhereas χ
L/2 −
| i | i
fore,eachquasi-degeneratestateiscoupledwithtwosta- tends to the robust stationary wave.
tionary waves through phonon exchanges. In that case, BycombiningEqs.(5)and(17), itis easyto showthat
it is easy to extract δω from Eq.(10). In doing so, the the quasi-degenerate dressed states depend on the local-
±
shifts can be expanded as a Taylorseries with respectto ized states. They thus define two relevant paths for the
10
excitonic transition between the QCs. However, when
the exciton occupies χ , it induces a shift δΩ of the
± ±
| i
phonon frequency. Quantum decoherence is no longer
negligible and the decoherence factors is expressed as
e−in¯δΩ±t
F (t) , (20)
±
≈ 1+4∆n¯2sin2(δΩ t/2)
±
q
where n¯ = [exp(βΩ) 1]−1 and ∆n¯2 = n¯(n¯ +1). Note
−
that Eq.(20) has been obtained from Eq.(15) by simpli-
fying the time dependence of the argument of the deco-
herence factor. By inserting Eq.(17) into Eq.(10), the
phonon frequency shift is written as
∞
δΩ 2cos(2θ) E ǫ2r+1. (21)
± 2r+1
≈∓
Xr=0
In a quite good agreement with the results displayed in
Fig. 8, Eq.(21) reveals that when the exciton occupies
the state χ (resp. χ ), it induces a redshift (resp.
+ −
| i | i
blueshift) of the phonon frequency. Moreover, as ob-
served in Fig. 8 when ǫ is not too small, one obtains
δΩ = δΩ . However, when ǫ tends to zero, Eq.(21)
− +
yields δ−Ω 2E (E¯ E )ǫ2/E . Such a behavior
± 1 1 0
≈ ∓ −
slightly differs from our numerical results. As shown in
Fig. 8, δΩ vanishes when ǫ tends to zero whereas δΩ
+ −
reachesarathersmallvalue. Thisdiscrepancyoriginates
inthe factthat the exactdressedstates χ slightly de-
±
| i
pendonthe stationarywaves ψ , withµ=o, . When
µ
| i 6 ±
ǫtendstozero,suchadependenceismorepronouncedfor
χ . These additional components, that are not taken
−
| i
into accountin Eq.(17), yield a nonvanishingδΩ value.
−
In that context, it turns out that χ , χ and χ
define the main paths followed by th|eoeixc|ito+nito tu|nn−eil FIG. 12: (a) GM(ǫ) vs ǫ for χ=10 pN, L=10 and T =300
K using PT. (b) α=Ws/Wf vs ǫ. (c) TM/Tf vsǫ.
between the QCs. Consequently, G (t) is rewritten as
L0
1
G (t) ∆ exp( iω t) + (22) interferences between the different amplitudes will occur
L0 N 0
≈ − − (cid:20) 2 resulting in a characteristic time evolution of G (t).
L0
| |
1 As time evolves, G (t) will reach a maximum whose
− 4|F+(t)|exp(−iW+t)[1+sin(2θ)] value GM will dep|enLd0on|the model parameters through
1 the interference pattern.
F (t) exp(+iW t)[1 sin(2θ)] ,
− − To understand this phenomenon intuitively, let us fo-
− 4| | − (cid:21)
cus our attention on the influence of the coupling ǫ be-
where W = (ωˆ ω +n¯δΩ ) defines a positive fre- tween the QCs and the CC. Provided that ǫ is not too
± ± 0 ±
± −
quency relative to the band center ω . According to small,thenumericalresultssuggestthatδΩ δΩ >
0 − +
≈−
Eq.(22), G (t) is the sum of the three transition am- 0. The modulus of the decoherence factor is thus al-
L0
plitudes, τ = 1/2 and τ = F (t) exp( iW t)[1 most independent on the nature of the exciton so that
o ± ± ±
−| | ∓ ±
sin(2θ)]/4, connected to the three relevant paths. Be- F (t) F(t) ν = . Moreover, it turns out that the
ν
| | ≈ ∀ ±
cause χ is insensitive to the phonons, the probability influence ofθ remainsnegligible,i.e. sin(2θ) 0. There-
o
| i ≈
amplitudeτ istheweight1/2ofthelocalizedstates. By fore, after straightforward algebraic manipulations, one
o
contrast, the probability amplitude τ that the exciton obtains
±
tunnels through χ± involves a phase factor whose time 1
evolution is gove|rnedi by T± = π/W±. This phase fac- |GL0(t)|2 ≈ 4[1 + F2(t)cos2(Wst) (23)
tor is weighted by both the decoherence factor and the
2F(t)cos(W t)cos(W t)],
s f
weight of the localized states. Consequently, depending −
on the value of the model parameters, W , θ and F (t) where W =(W W )/2 and W =(W +W )/2. In
± ± s − + f − +
−
will take particular values so that each probability am- the righthandside of Eq.(23), the firsttermis the prob-
plitude will develop a specific time evolution. Quantum ability τ 2 thattheexcitonrealizesatransitionthrough
o
| |
