Table Of ContentLehmann-Symanzik-Zimmermann Reduction Approach to
Multi-Photon Scattering in Coupled-Resonator Arrays
T. Shi and C. P. Sun
Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing, 100190, China
Wepresentaquantumfield theoretical approach basedon theLehmann-Symanzik-Zimmermann
reduction for the multi-photon scattering process in a nano-architecture consisting of the coupled-
resonator arrays (CRA), which are also coupled to some artificial atoms as a controlling quantum
node. By making use of this approach, we find the bound states of single photon for an elemen-
0
tary unit, the T-type CRA, and explicitly obtain its multi-photon scattering S-matrix in various
1
situations. Wealso usethismethodtocalculate themulti-photonS-matrices forthemorecomplex
0
quantumnetwork constructed with main T-typeCRA’s, such as a H-typeCRA waveguide.
2
n
PACSnumbers: 03.65.Nk,42.50.-p,11.55.-m,72.10.Fk
a
J
8 I. INTRODUCTION the waveguide with nonlinear dispersion relation. It can
2
be linearized in the high frequency regime in which pho-
ton momentum k π/2a with lattice spacing a, to
] In order to realize all-optical quantum devices1,2, it is ∼ ±
h approximate the linear dispersion relation for the con-
very crucial to explore the physical mechanism for the
p ventional waveguide.
- single photon generation, transport and one shot detec-
t tion etal.. Thus, we need a comprehensive understand- In the recent study4, the single photon transmission
n
a ing of the fundamental processes of coherent photonic andreflectioncoefficientswerecalculatedfortheincident
u scatteringinthe solidstatebasedconfinedsystems,such photonwithanyenergyinthe T-typestructure(a quan-
q as the photonic crystal with artificial band gaps. Es- tum node coupled to a CRA, see Fig. 1), which demon-
[ sentially, by coupling the system with an extra two-level strate the novel lineshapes beyond the Breit-Wigner20
2 system (TLS) to form a hybrid system, for controllably andtheFanolineshapes16. Thiskindofinvestigationwas
v transportofphotonsthebasicelementisaquantumnode carried out only for the case with single photon. Also,
9 or quantum switch3. It was abstracted as the so-called there are only quite a few researches on the two photon
7 photon transistor most recently1,4. For quantum infor- case, for which we mention an elegant theoretical ap-
2 mation processing, such quantum node can coherently proach for two photon scattering8,9 based on the Bethe-
1 controlthequantumstatetransferinsomequantumnet- ansatz21–27. Moreover, we have to say that it sounds
.
9 work5–7. Actually, to manipulate the coherent transport very difficult to prepare the system only with one pho-
0 of photons, the quantum node is tunable so that it can ton or two photons exactly, thus these previous studies
8 behave either as a perfect mirror totally reflecting pho- need to be improvedfor multi-photon processes oriented
0
tons,orasanidealtransparentmediumallowingphotons by practical application.
:
v to pass throughly. Theoretically, the quantum node for Actually, the study for multi-photon transport is very
i
X single-photon in all-optical architectures was extensively importantto realize a practicalall-opticaldevices. How-
studied in one dimensional waveguide by making use of ever, these subtle approaches for single and two photon
r
a the standard scattering approaches2,8–13. cases mentioned above4,8,9,19, such as the discrete coor-
dinate scattering theory and the Bethe-ansatz technique
The photonic quantum node is usually modelled as a
localized TLS14, which can be implemented as an arti- with fixed scatterers,are not feasibly generalized for the
realistic multi-photon processes, even for two-photon or
ficial atom, coupled to the photons transported in the
coupled-resonator arrays (CRA)15. The atomic parame- three-photonprocesses. Therefore,ourpresentsystemat-
icalapproachbasedonquantumfieldtheoryissignificant
ters,e.g.,theenergylevelspacing,aretunableto control
since it is obviously feasible and intrinsically natural for
the propagationof photons. Recently, basedonthis the-
the generalizationto multi-photon scattering processes.
oretical model and its generalizations,the photon trans-
port in the CRA systems has been studied for different In this paper, we utilize the Lehmann-Symanzik-
purposes4,8,9,19. Here, we wouldlike to point outtwo re- Zimmermann(LSZ) reduction28 in quantum field theory
markable issues: (a) if only one photonallowedto trans- toinvestigatethe multi-photontransportinthe complex
port in the CRA with the artificial atom prepared at its CRA with some two- level scatterers. This method has
ground state, the hybrid system can be described by a beenusedtostudythesingleelectroninelasticscattering
simplemodelwithasinglestatecoupledtoacontinuum, inAndersonmodelandKondomodel29,30. Here,wedeal
which is referred to the Anderson-Fano-Lee model16–18 with the multi-photon scattering problem by studying
with single excitation; (b) the CRA can be regarded as the out-state of the scattered photons for an arbitrary
2
state of incident photons. In a middle stages, we cal-
culate the multi-photon scattering matrix (S-matrix) in
details. With the diagrammatic analysis, we find that
the basic element of the S-matrix is a connected trans-
fer matrix (T-matrix), which can be obtained from the
well-known LSZ reduction formula about the photonic
Green’s function. From the explicitly achieved expres-
sions of photonic out-states, we analysis in details quan-
tum statistical characters of photon transmission in the
situation with many photons. We find that, in the tight e
two level atom
binding CRA of T-type, there exist the single photon g
bound states. As a test, two photon transport in the T-
typewaveguideisre-considered,andourobtainedresults
T-Type CRA
accord with the recent woks8,9 using the Bethe-ansatz,
whichverifiesthe resultsbasedonthe LSZreductionap-
proach are valid. As the development, the three photon
scatteringisstudied,andtheoutgoingstatesofthethree
photon are given by this approach. Our present investi- FIG. 1: (Color online) The schematic for the complex CRA:
gationmainly based on these results, canbe regardedas TheT-typestructurewithCRAcoupledtoimpurity(maybea
a substantial development for its particular emphasis on two-levelscatterer)isthebasicelementtoconstructthemore
the multi-photon scattering. complicatedarchitectureofquantumnetwork. Theredcircles
Inpractice,theT-typephotonicelementwementioned denotethetwolevelimpurity,while thebluedotsdenotethe
photoniccoupled resonators.
aboveisthebasicblocktoconstituteacomplexquantum
networkcoherentlytransferringphotonsinacontrollable
fashion. Aslightlycomplicatedillustrationofsucharchi-
annihilation(creation) operator for the photonic single
tecture is the CRA waveguide of H-type. In this paper,
modewitheigen-frequencyω inthei-thcavity;V isthe
we also study two photon scattering processes in the H- 0
hybridization constant of the localized atom-photon in
type in details.
the 0-th site of CRA.
Thepaperisorganizedasfollows: inSec.IIandIII,we
Inthemomentumspace(k-space),theHamiltonian(1)
modelourhybridsystemformulti-photontransportand
is re-expressed as
presentthescatteringmatrixbasedontheLSZreduction
approach;in Sec. IV, we show that there exist the single
H = Ω e e + ε a†a
photon bound states in the tight binding T-type CRA; T | ih | k k k
in Sec. V, we study the multi-photon transport in the Xk
V
T-type waveguide; in Sec. VI, we discuss the two pho- + (a†σ−+H.c.), (2)
ton transport in the H-type waveguide; in Sec. VII, the √LXk k
results are summarized with some remarks.
with the photonic dispersion relation
II. SCATTERING MODEL FOR THE HYBRID εk =ω0 2Jcosk, (3)
−
SYSTEM
in CRA of length L. Here, we choose the cavity spac-
ing a = 1. In the high energy limits k π/2 and
A. Model setup ∼ ±
ω = πJ, the above dispersion relation is linearized as
0
ε k , which is the same as that in the conventional
k
In this subsection, we model the T-type CRA by the ∼ | |
waveguide. Thus, we can use CRA to simulate the con-
two-levelatomcoupledtophotonsinsideCRAillustrated
ventionalwaveguidein the high frequency limits. As the
in Fig. 1. The model Hamiltonian reads
basic element of the complex quantum network as illus-
tratedin Fig. 1, the two- levelatomplays the role of the
HT = Ω|eihe|+ [ω0a†iai−J(a†iai+1+H.c.)] photon transistor to control the photon transmission.
Xi It is obvious that, when confined within the single ex-
+V δ (a†σ−+h.c.), (1) citation subspace, our model is the same as the models
i0 i
Xi by Anderson, Fano and Lee. This observation motivates
us to consider how to use various approaches developed
where the operator σ− = g e denotes the flip from previously for the models by Anderson, Fano and Lee,
| ih |
the atomic ground state g to the excited state e to deal with the coherent processes of our system, espe-
| i | i
with the energy level spacing Ω. Here, J is the cially with multi-photons. To this end, we will compare
hopping constant characterizing the inter-cavity cou- our model (2) with the Anderson, Fano and Lee model
pling in the tight-binding approximation; a (a†) is the as follows.
i i
3
B. Relation to Anderson model with two heavy fermions (V, N) of masses m and
V
m . Here, the relativistic boson A with momentum
N k
By neglecting the Coulomb interaction, the Anderson k and rest mass µ possesses the dispersionrelation ωk =
model is the same as the Fano model, so we discuss the k2+µ2, and g is the three body coupling constant for
similarities and differences between the model (2) and tphe scattering of the three kinds of particles, V, N, and
Anderson model. The corresponding Hamiltonian reads A. T. D. Lee used this exact solvable model to study
the necessity of renormalization in quantum field theory
HA = εkc†kσckσ +εdfσ†fσ+HV +HU, (4) even without the perturbation expansion.
Xk,σ By comparing Lee model with the model (2), we find
where c (c† ) is the annihilation (creation)operatorof thatthe heavyfermionsV andN canbe regardedasthe
kσ kσ
excited state and ground state of the atom, respectively.
the conductive electron with dispersion relation ε and
k
spin σ. f (f†) is the annihilation(creation) operator of And the relativistic boson Ak can be considered as the
σ σ
photon in the model (2). There also exist some differ-
the impurity f-electron with energy ε and spin σ. The
d
ences between the Lee model andthe model (2). Firstly,
Coulombrepulsiveinteractionoftheimpurityf-electrons
isdescribedbytheHubbardtermH =Uf†f f†f . The inLeemodel,therenormalizationsoftheV-fermionmass
U ↑ ↑ ↓ ↓ m and coupling constant g should be taken into ac-
hybridization of a conductive electron with the localized V
count, and the Hamiltonian (6) is non-Hermitian. Thus
f-electron is depicted by the mixing term
we should investigate the unitarity of the S-matrix care-
H =V (c† f +H.c.). (5) fully31,32. Secondly,theHilbertspaceofheavyfermionis
V kσ σ
Xk,σ spanned by four basis states, i.e., |0i, ψV† |0i, ψN† |0i and
ψ†ψ† 0 . We conclude that the two models (2) and (6)
Anderson even used this model (4) to investigate the V N| i
are equivalent only in the single excitation case, i.e., one
localized magnetic state in the metal. By comparing the
relativistic boson A with one heavy fermion N or one
Hamiltonian (4) with the Hamiltonian (2), we find that k
heavy fermion V with no relativistic boson A in Lee
the atom and the photon in the model (2) can be con- k
model, and one photon with the atom prepared at the
sidered as the impurity f-electron and the conductive
ground state or no photon with the atom prepared at
electron in the Anderson model, respectively. But there
the excited state in the model (2).
stillexistsomedifferencesbetweenthetwomodels. First,
theHilbertspaceoftheatominthemodel(2)isspanned Infact, itisturnoutthatthemulti-particlescattering
by two basis states, e and g , but the Hilbert space of problems (such as N-θθ scattering)in the Andersonand
| i | i
the impurity f-electron is spanned by four basis states, Lee models were both successfully studied by the LSZ
0 , f† 0 , f† 0 andf†f† 0 . Secondly,inthe Anderson reduction approach29,30,33,34. In the following, we start
m| iode↑lt|hieco↓n|diuctivee↑lec↓t|roinsarefermions,whileinthe withthemodel(2)toinvestigatetheS-matrixofscatter-
model (2) photons are bosons. Thirdly, there is no Hub- ing photon in the hybrid CRA systems. For the further
bard term H in the model (2), while in the Anderson discussion,we firstgivea generalformalismto study the
U
modelifthereisonlyonef-electrontheCoulombinterac- multi-photon S-matrix in the complex CRA by the LSZ
tion vanishes, i.e., the Hubbard interaction H does not reduction formalism.
U
play any role. In conclusion, the two models are equiva-
lentonlyinthesingleexcitationcase,i.e.,oneconductive
electron with no f-electron or one impurity f-electron
with no conductive electron in the Anderson model, and
one photon with the atom prepared at the ground state III. LEHMANN-SYMANZIK-ZIMMERMANN
ornophotonwiththeatompreparedattheexcitedstate REDUCTION FOR PHOTON SCATTERING IN
GENERAL
inthemodel(2). Otherwise,inthemulti-excitationcase,
the two models (2) and (4) are obviously not equivalent.
In this section, we first briefly summarize the main
results of the LSZ reduction approach in quantum field
C. Relation to Lee model
theory, and then we use them to give a general formula
for multi-photon S-matrix in the T-type CRA. For in-
The Lee model18 with the Hamiltonian vestigating the multi-photon scattering, in the first sub-
H = m ψ†ψ +m ψ† ψ + ω A†A section, we first define the n-photon S-matrix and 2n-
L V V V N N N k k k point photonic Green’s function, whichare explicitly de-
Xk
scribed by the Feynman diagrams (Fig. 2), and utilized
1
+g (ψ†ψ A +H.c.), (6) the functional integral to obtain the 2n-point photonic
√2ω V V N k
Xk k Green’s function. In the second subsection, we use the
LSZ reduction approach to obtain a general form of the
describes a reaction process
n-photonS-matrixelementbyreducingtheexternallegs
V ⇄N +A , (7) (red curves in Fig. 2) of the Green’s function.
k
4
A. Main results of the LSZ reduction approach
Because the LSZ reduction approach relates to the
n n
photonicGreen’sfunction andS-matrix,weneedtogive
their definitions for discussing the LSZ reduction explic-
itly. The 2n-point photonic Green’s function
G (t′,...,t′ ;t ,...,t )
p1,...,pn;k1,...,kn 1 n 1 n
= Θ Ta (t′)...a (t′ )a† (t )...a† (t ) Θ , (8)
h | p1 1 pn n k1 1 kn n | i n n
is defined by the time ordering product T of the photon
n- 1 n- 1
creation(annihilation)operatora†(a )intheHeisenberg
k k
picture. Here, Θ denotes the ground state of the Hamil-
tonian H of the system.
The S-matrix element35,36
other disconnected diagrams
f i = f S i . (9)
outh | iin inh | | iin
of photons is defined through the overlap of incoming FIG.2: (Coloronline)TheFeynmandiagramsforLSZreduc-
state i andoutgoingscatteringstate f ,whichare tion: The red line denotes the free photon propagator. The
asymp|tiointically free, and may contain mu|ltiio-upthoton exci- grayshadecirclesdenotetheS-matrixelementsandtheblue
tation,i.e.,theyaremulti-photonstates. Inourcase,the shade circles denote the T-matrix elements. The Feynman
incoming state is diagram of S-matrix element is constructed by all kinds of
thedisconnecteddiagrams. Andeachofthethedisconnected
i =a† ...a† 0 . (10) diagrams contains some connected diagrams.
| iin k1 kn| i
In the present case, the S-matrix is given by
k p
+∞
S =T exp[ i dtH (t)], (11)
int
− Z
−∞
with the atom-photon hybridization Hamiltonian
k p k p
H =V(a†σ−+H.c.), (12)
int 0
in the interaction picture. (a) (b)
Using the diagrammatic analysis, we find that the
Feynman diagramof S-matrixelement is constructedby
summing up the contributions from all kinds of discon- FIG. 3: (Color online) The diagrammatic constructions of
the single photon S-matrix element: There exist two kinds
nected diagrams as shown in Fig. 2. Here, each of these
of disconnected diagrams (a) and (b). The red line denotes
disconnected diagram is made up of some connected di-
the free photon propagator. The gray shade circles denote
agrams which describe the T-matrix elements. As the the S-matrix elements and the blue shade circles denote the
specialcases,weconsiderthediagrammaticconstruction single photon T-matrix elements.
for the single photon and the two photon S-matrix ele-
ments by the connected T-matrix elements below. For
the single photon case, the S-matrix element of disconnected diagrams (a), (b) and (c) as shown in
Fig. 4. Obviously, the multi-photon S-matrix is totally
Sp;k =δkp+iTp;k, (13) determined by the connected T-matrix, so we only need
to find the photonic T-matrices.
is defined by the T-matrix element for single photon,
Fortunately, the intrinsic relation
where k and p are the momenta of the incoming and
n
outgoing photons. In this case, there exist two kinds of
iT = G [2πG−1(k )G−1(p )] , (15)
disconnected diagrams (a) and (b) as shown in Fig. 3. 2n 2n 0 r 0 r (cid:12)
For the two photon case, the S-matrix element rY=1 (cid:12)(cid:12)os
(cid:12)
between n-photon T-matrix element (cid:12)
S =S S +S S +iT , (14)
p1p2;k1k2 p1k1 p2k2 p2k1 p1k2 p1p2;k1,k2 T =T , (16)
2n p1,...,pn;k1,...,kn
isreducedbytheT-matrixelementoftwophotons,where
and photonic Green’s function
k andp (r =1,2)arethemomentaoftheincomingand
r r
outgoing photons. In this case, there exist three kinds G =G , (17)
2n p1,...,pn;k1,...,kn
5
k p where the generating functional
1 1
Z[η,η∗] = D[a a†]D[f f†]δ( f†f 1)
Z k k σ σ σ σ−
Xσ
k p
2 2 exp i[S+ dt (η∗a +η a†)] ,(20)
{ Z k k k k }
Xk
k p
1 1
is defined by the action S = dtL and the Lagrangian
R
L=if†∂ f +if†∂ f +i a†∂ a H . (21)
k2 p2 e t e g t g Xk k t k− T
(a)
In Eq. (20), we represent the two- level atom by the
k p k p
1 1 2 1 fermions fσ as
e e = f†f ,
k2 p2 k1 p2 |eihg| = fe†fe, (22)
(b) (c) | ih | e g
with the constraint
FIG. 4: (Color online) The diagrammatic constructions of f†f =1. (23)
σ σ
thetwo photon S-matrix element: There exist threekindsof σX=e,g
disconnected diagrams (a), (b) and (c). The red line denotes
the free photon propagator. The gray shade circles denote This constraint arises from the fact that the physical
the S-matrix elements and the blue shade circles denote the space of the atom spanned by e and g is two dimen-
two photon T-matrix elements. sions, while the physical space|oif the f|erimions spanned
by 0 , f† 0 , f† 0 and f†f† 0 is four dimensions. By
| i e | i g| i e g| i
integrating the photon field and the fermionic fields f
σ
in Eq. (20), the generating functional is obtained as
is given by the LSZ reduction formula, where
lnZ[η,η∗] = TrlnM[ξ,ξ∗]
i
G (k )= , (18) η (ω)2
0 r ω ε +i0+ i dωdk | k | , (24)
r− kr − Z ω ε +i0+
k
−
is the Green’s function of free photon and where the field variable is
G is the Fourier transformation of
Eqp1.,(..8.,)p.n;Hk1e,.r.e.,,knthe subscriptos denotes the onshell limit ξ(ω)=V dk ηk(ω) , (25)
ω εk. Finally,themulti-photonS-matrixelementsare Z 2πω−εk+i0+
→
determined by the photonic Green’s functions entirely.
and the matrix
In this paper, we use an elegant method, i.e., functional
integrals, to establish the desired relation (15). This M[ξ,ξ∗]= [ω−Ω+Σ(ω)]δωω′ ξ(ω−ω′) ,
method has been used to solve many quantum impurity (cid:18) ξ†(ω′ ω) (ω i0+)δωω′ (cid:19)
− −
problem29,30 in the condensed matter physics. Here, we (26)
show how it works for the multi-photon transmission in is defined by the self-energy
this hybrid system.
Γ
Σ(ω)=ReΣ(ω)+i ρ(ω), (27)
2
of the atom and Γ = V2. Here, the real part of the
B. General formula for S-matrix in the T-type self-energy is determined by the principal-value integral
CRA as
dk Γ
ReΣ(ω)= P , (28)
WeutilizethegeneratingfunctionalZ torepresentthe Z 2π ε ω
k
full time ordering Green’s function as −
and the imaginary part is proportional to the density of
state (DOS)
( 1)nδ2nlnZ[η ,η∗]
G = − k k ,
p1,...,pn;k1,...,kn δη∗ ...δη∗ δη ...δη (cid:12) 1
p1 pn k1 kn(cid:12)(cid:12)(cid:12)ηk=ηk∗=0(19) ρ(ω)=Xi |Di(ω)|, (29)
6
where D (ω) = ∂ ε and z is the real root of the
i k k|k=zi i
equationεzi =ω. Forthewaveguidetheself-energyΣ(ω) S-matrix element Eb(u)
is a constant. For the CRA the self-energy depends on
of the single photon
thefrequencyωandwehaveusedMarkovapproximation
to obtain Eq. (24). 2J
With the helps of Eqs. (15), (19) and (24), we can
achieve the multi-photon S-matrix by considering the w
0
Green’s function G2n. In the following, we use the LSZ 2J
approachtostudythemulti-photontransportinthecom-
plex CRA.
The exact Green function
of the two level atom E(d)
b
IV. BOUND STATES OF THE SINGLE
PHOTON TRANSPORT IN THE T-TYPE CRA
(a) (b)
In this section, we consider the single photon trans-
FIG. 5: (Color online) (a) The gray shade circle denotes the
port in the T-type CRA. This problem has been consid-
scattering matrix element of the single photon, and it is also
eredinRef.4,butthephotonboundstateshavenotbeen
the Green function of the two level atom. The poles of the
taken into account. There are two interesting physical
atom Green function determine the energies of the bound
phenomena about the photon bound states. (a) Firstly, states. (b) The band structure of T-type CRA is shown in
in the hybrid system the single photon bound states de- theright panel.
pict the states that the single photon is almost local-
ized in the cavity containing the two-level atom. In this
sense, the photon bound states can be used as the quan- Because the energies of bound states are not inside the
tum information storage of single photon. (b) Secondly, photonic energy band of the CRA, i.e., Eb ω0 > 2J,
| − |
as the basic element of single photon transistor, the T- the self-energy is
type CRA is usually coupled with other CRAs through
Γsign(E ω )
the two-level atom. These CRAs can be regarded as the Σ(E )= b− 0 . (33)
b
scatteringchannels. Itis provedthat19 the incidentpho- − (Eb ω0)2 4J2
− −
ton in another CRA, whose energy is resonance on the p
Then Eq. (32) becomes
bound state energy in the T-type CRA, will be totally
reflected. This phenomenon can be considered as the
Γsign(E ω )
one-dimension photonic Feshbach resonance, which can E Ω b− 0 =0. (34)
b
beusedtosimulatetheFeshbachresonanceintheatomic − − (Eb ω0)2 4J2
− −
and condensed matter physics. The above reasons moti- p
(d)
vate us to study the photon bound states in the T-type The above equation possesses two solutions (E and
b
CRA by using the LSZ approach. E(u)) for E : one solution E(d) is below the bottom of
b b b
The T-type CRA is described by the Hamiltonian (2). energy band, the other solution E(u) is above the top of
ThenEqs.(15),(19)and(24)immediatelygivethesingle b
energy band. The structure of the energy spectrum is
photon S-matrix element
shown in Fig 5b. The two bound states are
Sp;k =(1+rk)δkp+rkδ−kp, (30) B =[ ψ (x )a†+σ+] 0 , (35)
| ±i ± i i | i
Xi
with the reflection amplitude
with the wave-functions19 in the spatial representation
iΓ
r = − . (31)
k 2Jsink(εk−Ω)+iΓ ψ−(xi)= q(Eb(u(−)−)|ωx0|V)2−4J2e|x|lnκ−(Eb(u)), if Eb =Eb(u) ,
aHnedreR, weΣe(hωa)ve=u0tilfiozredthDeOscSatρt(eωr)ed=p2h/opto4nJi2n−th(ωe T−-tωy0p)2e ψ+(xi)= q(Eb(d)−Vω0)2−4J2e|x|lnκ+(Eb(d)), if Eb =Eb(d)
CRA. The result (30) is the same as that obtained in (36)
Ref.4. where
In the following, we study the single photon bound
E ω ω E
statesinthissystembyconsideringthepolesofS-matrix κ (E)= ( − 0)2 1 0− . (37)
(see Fig. 5a). The poles of S-matrix are the roots of the ± −r 2J − ± 2J
equation
It is obviously that lnκ (E) < 0 for the both cases
±
(u) (d)
Γ E = E and E = E ; the wave functions exponen-
b b
(Eb Ω)+i Σ(Eb)=0. (32) tially decay as x increases as shown in Eq. (36), which
− 2 | |
7
and
1.0
ΨHdLHxL He = Ω e e + ka† a
0.8 T | ih | k,e k,e
kX>0
efunction 00..46 +√V˜LXk>0(a†k,eσ−+H.c.), (41)
v
wa 0.2 where the operators
0.0
1
a = (a +a ),
-0.2 ΨHuLHxL k,e √2 k −k
-40 -20 0 20 40 1
a = (a a ), (42)
x k,o √2 k− −k
describe annihilations of the e-photon and o-photon.
FIG. 6: (Color online) The wave-functions of two photonic
bound state in the discrete coordinates. The norms of both Here, e-photon depicts the photon with even parity in
wave-functions exponentially decay as |x| increases. the momentum space, i.e., a−k,e = ak,e, and o-photon
depicts the photon with odd parity in the momentum
space, i.e., a = a . In Eq. (41), the effective cou-
−k,o k,o
insure that the photon is indeed localized to form the pling constant is V˜ −=√2V. For the o-photon, the Ho is
bound states. The wave-functions ψ (x ) are illustrated T
± i diagonalized in the bases a† 0 , so we only need to
in Fig. 6. { k,o| i}
find out the S-matrix for the the scattered e-photon. In
By short summary for this section, we verify the ex-
the following, we consider the S-matrix for the e-photon
istence of photon bound states in the T-type CRA by
according to the approach.
considering the poles of the S-matrix element. In the
next section, using the T-type CRA to simulate the T-
type waveguide, we study the multi-photon transport in
A. Single photon scattering
the T-type waveguide.
For the single e-photon case, Eqs. (19) and (24) give
V. SCATTERING MATRIX FOR PHOTONS IN the single photon Green’s function
THE T-TYPE WAVEGUIDE
i Γ
G(p;k)=[G(k) T G2(k)]δ , (43)
0 − 2πω α 0 pk
k
Inthissection,wefocusonthephotontransportinthe −
T-type waveguide. For the T-type waveguide, S-matrix where α = Ω iΓ /2 and Γ = V˜2. Here, we used
T T
−
element is firstly calculated out by the LSZ reduction. ReΣ(ω) = 0 and ρ(ω) = 1. Together with Eq. (15),
Secondly, by using the obtained S-matrix, we give the Eq. (43) gives the single e-photon T-matrix
out-stateforarbitraryincidentstateofphotons. Finally,
iΓ
byanalyzingthephotonout-stateinthespatialrepresen- iT =δ − T . (44)
p;k pk
k α
tation, we can obtain the quantum statistical properties
−
of the scattered photons, such as the photon bunching Next, we achieve the single e-photon S-matrix element
and anti-bunching. S = t δ by considering Eq. (13), where the trans-
p;k k pk
As shown in Sec. II, the waveguide is simulated by mission coefficient is
CRA in the limits k π/2 and ω = πJ, in which
∼ ± 0 k α∗
the dispersion relation of the photon is ε =v k = k . t = − . (45)
k g| | | | k k α
Here, we let the group velocity vg = 1 for convenient. −
Then the Hamiltonian This result (45) accords with that of Refs.8,9 based on
H = Ω e e + k a†a the Lippmann-Schwinger formalism.
T | ih | | | k k
Xk
V
+ (a†σ−+H.c.), (38) B. Two photon scattering
√L k
Xk
For the two e-photoncase,Eqs.(19) and(24) give the
describing the T-type waveguide becomes
two e-photon Green’s function as follows:
H(w) =He +Ho , (39)
T T T 2Γ2 G(k )G(k )G(p )G(p )
G = i T 0 1 0 2 0 1 0 2
with two parts p1,p2;k1,k2 (2π)3 (ω′ α)(ω′ α)
2− 1−
HTo = εka†k,oak,o, (40) (ω1+ω2−2α)δω1+ω2,ω1′+ω2′ . (46)
kX>0 × (ω1 α)(ω2 α)
− −
8
By taking the photon frequency ω and ω′ on shell, we
i i
obtain the T-matrix
Γ2 (k +k 2α)
iT = i T 1 2−
p1,p2;k1,k2 π (p α)(k α)
2 1
− −
δ
k1+k2,p1+p2 . (47)
×(p α)(k α)
1 2
− −
From the above equation, the two e-photon S-matrix el-
ement
(a)
S = iT
p1p2;k1k2 p1p2;k1,k2
+t t (δ δ +δ δ ). (48)
k1 k2 p1k1 p2k2 p2k1 p1k2
followsEq.(14)immediately. Iftwoincidentphotonsare
prepared in the state k ,k , the wave function8,9
1 2
| i
1 (b) (c)
x ,x out = S x ,x p ,p
h c | i 2 p1p2;k1k2h c | 1 2i
pX1p2
= eiExc 1 [t t cos(∆ x) FIG.7: (Coloronline)FeynmandiagramsforthreephotonS
2π k1 k2 k matrix: Thereexistthreekindsofdisconnecteddiagrams(a),
4Γ2ei(E−2Ω+iΓT)|x|/2 (b) and (c).
T ], (49)
−4∆2 (E 2Ω+iΓ )2
k− − T
δ(ω k )
of two outgoing photons in the spatial representation is F(2) = i i− Pi
obtained in terms of the two e-photon center of mass ωi,ωi′ XPQ (ω2′ −ωQ2)(ω3′ −ω3)(ω2′ −α)
coordinate x = (x + x )/2 and two photon relative
c 1 2 δ(ω′ p )
coordinate x = x x . Here, the total momentum i i− Qi , (52)
1 − 2 (ω αQ)(ω′ +ω′ ω α)
(energy) is E = k1 +k2 and the relative momentum is 3− 2 1− 2−
∆ = (k k )/2. When the photon momenta k and
k 1 2 1 and
−
k both satisfy the resonance condition k = k = Ω,
2 1 2
the envelop wave-function (49) exponentially decays as F(3) = iδ(ωi−kPi)
othuetgroeilnagtivtewocopohrdotinoantseaxttirnaccrteawsietsh. eTahcihsorethfleerctesfftehcattivtehlye ωi,ωi′ XPQ (ω2′ −ωQ2)(ω1′ −ω1)(ω1′ −α)
taondzefroor,mthae twwaovep-fhuontcotniobnoautnxd=sta0ted.ecIrfeaEse−s a2sΩ∆is keipnt- (ω αQ)(iωδ(ω+i′−ωpQiω)′ α). (53)
| k| 2− 2 3− 2−
creases, which implies the photons repulse against each
other effectively through interacting with the two- level Here, P = (P1,P2,P3) and Q = (Q1,Q2,Q3) are two
atom. The above results about two photon scattering different permutations of (1,2,3), and i = 1,2,3. The
accord with the results reported in Refs.8,9. three photon S-matrix element
S = i S T
p1,p2,p3;k1,k2,k3 pj;ki pλ,pβ;kγ,kδ
C. Three photon scattering i,jX=1,2,3γX,δ6=iλX,β6=j
+ S
For the three photon case,Eqs. (15) and (19) give the XPQi=Y1,2,3 pQi;kPi
connected T-matrix as
+iT , (54)
p1,p2,p3;k1,k2,k3
Γ3
iT =i T δ[ (k p )] F(a) , is constructed by summing up the contributions from
p1p2p3;k1k2k3 3(2π)2 i− i ki,pi
Xi a=X1,2,3 all disconnected Feynman diagrams as shown in Fig. 7.
(50) Here, γ,δ,λ,β 1,2,3 . Thesimilardiscussionscanbe
∈{ }
where the functions F(a) are defined as applicable to deal with the N-photonscattering process.
ωi,ωi′
Next, we consider the physical meaning of S-matrix.
δ(ω k )
F(1) = i i− Pi To this end, we first analysis the T-matrices, which
ωi,ωi′ XPQ (ω1′ −ωQ1)(ω3′ −ω3)(ω1−α) are |T2|2 = |Tp1p2;k1,k2|2 and |T3|2 = |Tp1,p2,p3;k1,k2,k3|2.
δ(ω′ p ) Here, T 2 describesthe two-photonbackgroundfluores-
(ω3′ −αQ)(iω1+i−ω2−Qiω1′ −α), (51) cence,|w2h|ich is explicitly discussed in Ref.8,9, and |T3|2
9
0
0
1 1 0.16
2 3 2
40 2
20 1
0 3 0 3
3 3
2 2
1 1
0 0 0
FIG.8: (Coloronline)Three-photonbackgroundfluorescence
forthetotalenergyofincidentphotonsE =k1+k2+k3 =3
and ΓT = 1, where the energy level spacing Ω is taken as
units: (a) the three photon are both on resonance with the
atom, i.e., k1 = k2 = k3 = Ω; (b) the energies of the three 0.07
photons are k1 =0.5, k2 =0.3, and k3 =2.2, respectively.
depicts the three-photonbackgroundfluorescence,which
is shown in Fig. 8. It is shown that when the three
photons are all on resonance with the atom, the three-
photon background fluorescence describing T3 2 is en- 0
| |
hanced largely.
The out-going state of three photons
out = S p ,p ,p , (55)
| i p1,p2,p3;k1,k2,k3| 1 2 3i
p1≤Xp2≤p3
isdeterminedbythe threephotonS-matrix,anditsspa-
tial representationthe wave-function reads as
FIG. 9: (Color online) The probability distribution of three
photon with one photon at origin, where the total energy of
hx1,x2,x3|outi incident photons E = k1 + k2 + k3 = 3, the energy level
= Sp1,6p(22,pπ3;)k31/,2k2,k3ei(p1x1+p2x2+p3x3). (56) sapreacbinogthΩoinsrteaskoennaansceunwiittsh,atnhdeaΓtTom=,1i:.e(.,a)k1th=ekth2r=eekp3h=otoΩn;
p1Xp2p3 (b) the energies of the three photons are k1 = 0.5, k2 = 1.5,
and k3 =1, respectively.
The contour maps of the probability distributions
x ,x ,x out 2 are numerically shown in Fig. 9. It
1 2 3
|h | i|
is illustrated in Fig. 9a that when the three photon are
all on resonance with the atom, the scattered photons multi-photon transport, the atom can induce the effec-
prefer the two photon bound state rather than the three tive interaction of photons. We can control the effective
photon bound state. That is, if two photons form the interaction by adjusting the energy level spacing of the
bound state, it is difficult to form three photon bound atom. Therefore,thecoherentmanipulationsofTLScan
state, namely, the two bounded photons repulse another result in a transitions from the repulsive case to attrac-
one effectively. When the three photon are not on res- tion of effective photon interactions.
onance, the probability distribution x ,x ,x out 2 is In addition, we point that the recent references.8,9 ob-
1 2 3
|h | i|
showninFig.9b. ItisillustratedinFig.9bthatitisalso tained the same results for two photons transport, but
difficult to realize the three photon bound state. If the our works are different from them: (1) Refs.8,9 only give
positionofonephotonisgiven,suchasx =0,othertwo the twophotoneigenstatesbyBethe-ansatzmethod, but
3
photons do not alwaysattract or repulse each other, but we find a general method37,38, i.e., the scattering Bethe
attracteachother at some points and repulse eachother ansatz technique (see Appendix), to derive the multi-
at other points, which are determined by the distance photon eigenstates; (2) though we can obtain the multi-
betweenthe twophotonandanotherphotonlocalizedat photoneigenstatesby the subtle scatteringBethe-ansatz
x = 0. It is follows the above discussion that, in the method,westillneedalotofcomplicatedcalculationsto
3
conceptual setup of the photon transistor, the two- level achieve the S-matrix by using the Lippmann-Schwinger
atom controls the coherent transport behaviors of single scattering theory. However, the LSZ approach can be
photon, such as the transmission and reflection. In the generalized to study the multi-photon scattering in the
10
In the high energy limits k π/2 and ω(s) =πJ ,
two level atom −→± 0 s
the dispersion relation of the photon is ε(s) v k with
k ∼ s| |
the group velocity v = 2J . In the case, the Hamilto-
e s s
nian (58) describes the photon transport in the H-type
g waveguide. For convenient, we use the operators
CRA-1
1
a(s) = (a +a ),
k,e √2 k,s −k,s
1
a(s) = (a a ), (59)
k,o √2 k,s− −k,s
CRA-2
to rewrite the Hamiltonian (58) as H = H(e) +H(o),
H H H
H-Type CRA where
H(o) = ε(s)a(s)†a(s), (60)
H k k,o k,o
k>0X,s=1,2
FIG. 10: (Color online) The schematic for the H-type CRA
for the o-photon and
is shown in this figure. The red circle denotes the two level
atom. The bluedots denote thecoupled resonators.
H(e) = Ω e e + ε(s)a(s)†a(s)
H | ih | k k,e k,e
k>0X,s=1,2
waveguide, such as the three photon transport in the
1
waveguide; (3) Except these, the LSZ approach is also + (V¯ a(s)†σ−+h.c.), (61)
√L s k,e
usedtodealwiththemulti-photonscatteringinthemore k>0X,s=1,2
complexCRA, suchasthe H-type CRA.Inthe nextsec-
tion, by using the H-type CRA to simulate the H-type for the e-photon with the effective coupling V¯ = √2V .
s s
waveguide, we investigate the multi-photon transport in Because Hamiltonian H(o) is diagonalized in the bases
H
the H-type waveguide. a(s)† 0 , so we only need to find the S-matrix for the
{ k,o | i}
e-photon.
VI. TWO PHOTON SCATTERING PROCESS IN
THE H-TYPE WAVEGUIDE
A. The LSZ reduction for e-photon scattering in
the H-type waveguide
In this section, we study the two photon scattering
process in the H-type waveguide. The conventional H-
For calculating the multi-photon S-matrix, we only
typewaveguideissimulatedbytheH-typeCRA(Fig.10)
consider the Green’s function and the S-matrix for the
in the high energy limits. The model Hamiltonian
e-photon in the H-type waveguide. In this case, the 2n-
H = Ω e e + δ (V a† σ−+h.c.) (57) point photonic Green’s function reads
H | ih | i0 s i,s
i,sX=1,2
Gj1,...,jn;i1,...,in (t′,...t′ ;t ,...t ) (62)
+ [ω0(s)a†i,sai,s−Js(a†i,sai+1,s+h.c.)], p1,...,pn;k1,...,kn 1 n 1 n
i,sX=1,2 = DTap(j11,)e(t′1)...ap(jnn,)e(t′n)ak(i11,)e†(t1)...a(kinn,)e†(tn)EH .
of the H-type CRA is defined by the hopping constant
J and the creation operator a† of the i-th single mode The S-matrix element is the overlap
s i,s
(s)
cavity with frequency ω in the CRA-s, where V is
0 s f i = f S i , (63)
the hybridization constant of localized atom-photon in outh | iin inh | | iin
the 0-th site of the CRA-s. Here, s denotes the CRA-1
of incoming wave i = a(i1)†...a(in)† 0 and outgoing
or the CRA-2 as shown in Fig. 10. In the k-space, the | iin k1,e kn,e | i
wave state f . As shown in the Sec. III, the basic
Hamiltonian (57) becomes | iout
part of the S-matrix is the T-matrix. The relation
H = Ω e e + ε(s)a† a
H | ih | k,Xs=1,2 k k,s k,s iT(H) = G(H) n [2πG−1(k )G−1(p )] , (64)
+√1Lk,Xs=1,2(Vsa†k,sσ−+h.c.), (58) 2n 2n rY=1 0ir r 0jr r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)os
between the 2n-point T-matrix element
(s) (s)
where the dispersion relation of photon is ε = ω
2J cosk. k 0 − T(H) =Tj1,...,jn;i1,...,in , (65)
s 2n p1,...,pn;k1,...,kn
