Broadband photon-photon interactions mediated by cold atoms in a photonic crystalfiber

Marina Litinskaya,1 Edoardo Tignone,1 and Guido Pupillo1

1icFRC, IPCMS (UMR 7504) and ISIS (UMR 7006),Universite de Strasbourg and CNRS, 67000 Strasbourg, France.

(Dated: October 13, 2018)

We demonstrate theoretically that photon-photon attraction can be engineered in the continuumof scattering states for pairs of photons propagating in a hollow-core photonic crystal fiber filledwith cold atoms. The atoms are regularly spaced in an optical lattice configuration and the pho-tons are resonantly tuned to an internal atomic transition. We show that the hard-core repulsionresulting from saturation of the atomic transitions induces bunching in the photonic component ofthe collective atom-photon modes (polaritons). Bunching is obtained in a frequency range as largeas tens of GHz, and can be controlled by the inter-atomic separation. We provide a fully analyticalexplanation for this phenomenon by proving that correlations result from a mismatch of the quanti-zation volumes for atomic excitations and photons in the continuum. Even stronger correlations canbe observed for in-gap two-polariton bound states. Our theoretical results use parameters relevantfor current experiments with Rb atoms excited on the D2-line.

I. INTRODUCTION

There is growing interest in realising strongly interact-ing photons [1, 2] for applications in quantum informa-tion processing [3–10], quantum metrology [11–13] andmany-body physics [14–16]. Photonic non-linearities areoften induced by coupling photons to two-level emitters,as achieved in atomic and molecular setups with sin-gle emitters in a cavity configuration [17, 18]. Alter-natively, photonic non-linearities can result, e.g., fromthe anharmonicity of the multi-excitation spectra in theJaynes-Cummings Hamiltonian [19–23], or from dipolaror van-der-Waals interactions between atoms, such as inRydberg atoms under condition of electromagneticallyinduced transparency (EIT) [24–28]. Recent ground-breaking experiments [27, 28], in particular, have demon-strated attraction of photons caused by formation ofbound bipolariton states [29, 30].

In this work we propose the observation of photon-photon interactions and bunching in an ordered ensem-ble of two-level atoms confined to one-dimension (1D)and resonantly coupled to the transverse photons of acavity [Fig. 1(a)]. The interaction of atoms and light inthe strong exciton-photon coupling regime results in theformation of a doublet of polaritonic modes [Fig. 1(b)],corresponding to coherent superpositions of photonic andcollective atomic excitations (excitons). In our schemethe non-linearity results solely from the kinematic inter-action between these excitons: The latter amounts to ahard-core repulsion and is due to atomic saturability, asone atom can accommodate at most one excitation.

The existence of kinematic interaction in solids hasbeen known for decades [31], however, was always consid-ered as a very weak effect. In contrast, here we demon-strate that kinematic interaction in a cold atom setupmay lead to a pronounced bunching in the photonic com-ponent of the coupled polaritonic states. As opposedto narrow bound state resonances in the MHz frequency

range typical, e.g., of Rydberg atoms, this bunching ap-pears in the continuum of unbound two-polariton states,and can be observed in a broad GHz frequency rangefor parameters within the reach of current experimen-tal technologies.Via the exact solution for the subsystemconsisting of excitons uncoupled from light, we demon-strate that the bunching is the result of the mismatchof the quantization volumes for states with (excitons)and without (photons) hard core repulsion. Due to thebroadband nature of the effect, this type of non-linearityis expected to be comparatively resilient against decoher-ence. We conclude by discussing the occurrence of boundtwo-polariton states within spectral gaps.

The scheme we have in mind consists of two-level atomstrapped in a 1D optical lattice in the Mott insulator state(i.e. with one atom per lattice site), and confined insidea 1D resonant cavity. For concreteness, in this work weconsider the D2-line of Rb atoms placed into a hollow-core photonic crystal fibre – as in the experiment [32]with Sr. If the cavity losses are low and atoms are well-ordered, in this geometry one can work near the atomictransition without imposing the EIT condition to elim-inate absorption: Polaritonic states are coherent super-positions of photonic and collective atomic excitations,as opposed to incoherent absorption of individual atoms.Besides hollow-core fibers [8, 32–38], one can think ofother implementations of this scheme, as recent theoret-ical studies have investigated a variety of systems thatallow for coupling photons to an ensemble of two-levelemitters in a 1D configuration [39–45]. Solid-state re-alizations are also possible, e.g., using Si vacancies inphotonic crystals [46, 47].

II. MODEL

The setup consists of N atoms trapped on a lattice andcoupled to a cavity. Its Hamiltonian is

arX

iv:1

512.

0231

2v1

[qu

ant-

ph]

8 D

ec 2

015

2

H = E0

∑s

P †sPs + t∑s

(P †sPs+1 + P †sPs−1

)+∑qν

Ep(qν)b†(qν)b(qν)

+ g∑s,qν

[P †s b(qν)eiqs + Psb

†(qν)e−iqνs].

(1)

Here, Ps,P†s and b(qν),b†(qν) destroy and create

an atomic excitation at site s and a photon withthe wave vector qν along the cavity axis, respec-tively; Ep(qν) = c

√q2ν + q2

⊥ is the photon energy,qν = 2πν/(Na) (ν is an integer, a is the inter-particlespacing, c is the speed of light), and q⊥ is the trans-verse photon momentum [for the lowest mode of aperfect open cylindrical resonator of radius R, it isfound from the first zero of the function J0(q⊥R)];E0 is the atomic transition frequency, t ∝ d2/a3 isthe hopping energy for the atomic excitations in thenearest neighbor approximation, g = d

√2πE0/V is

the atom-light coupling constant, with d the transitiondipole moment and V = πR2Na the volume. Wenote that while the atomic part of equation (1) isreadily diagonalized by Frenkel exciton operators [31]P (qν) = 1√

N

∑n Pne

−iqνn describing extended wave

functions resulting from exciton hopping, the secondterm in the Hamiltonian is essentially negligible, sincethe bare exciton dispersion is much weaker than the po-lariton dispersion originating from light-matter coupling.

In the following, we solve the Schrodinger equation inthe two-particle subspace, where a wave function reads

|Ψ〉 =∑nm

{Anm√

2|bnbm〉+Bnm |bnPm〉+

Cnm√2|PnPm〉

}.

(2)

Here, bn =∑qνb(qν)eiqνn/

√N , while Anm, Bnm =

BSnm + BAnm and Cnm are the amplitudes for find-ing two relevant states (photons or atomic excitations)at sites n,m (the superscripts S,A stay for symmet-ric/antisymmetric; the amplitudes A and C are alwayssymmetric). We recast the Schrodinger equation as a setof equations for A,B and C, and solve it in terms of totaland relative wave vectors of two particles, Kν′ = qν1 +qν2and kν = (qν1 − qν2)/2, respectively, where only Kν′ is agood quantum number. Below we consider in detail thecase Kν′ = 0, and briefly discuss Kν′ 6= 0. For Kν′ = 0,qν2 = −qν1 , and kν = 2πν/(Na) with an integer indexν = (−N/2, N/2] (a detailed discussion of the quantumnumbers describing K and k for bosons on a lattice isprovided by Javanainen, J. et al. [48]). We obtain:

EρAρ(kν) = 2Ep(kν)Aρ(kν) +G√

2Bρ(kν),

EρBρ(kν) = [Ee(kν) + Ep(kν)]Bρ(kν)

+G√

2[Aρ(kν) + Cρ(kν)],

EρCρ(kν) = 2Ee(kν)Cρ(kν) +G√

2Bρ(kν) + Sρ,

(3)

(b)

k units of

Eun

itsof

G

(c)

210123

π

δ

p

e

kSC

E (k )

E (k )

E (k )LL

ν

ν

ν

ν

770

772

774

776

k units of 2ΠL

768

770

772

774

776

k units of 2ΠL

770

772

774

776

k units of 2ΠL

ETH

z

1 0.5 0 0.5 1

767.985

767.990

768.000

k units of Π1 0.5 0 0.5 1

767.985

767.995

768.000

k units of Π1 0.5 0 0.5 1

767.990

767.995

768.000

k units of Π1 0.5 0 0.5 1

767

767

767

767

k units of Π1 0.5 0 0.5 1

767.985

767.990

767.995

768.000

k units of Π1 0.5 0 0.5 1

767

767

767

767

0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6

0.0 0.2 0.4210121 0.50 0.5 1

768

LUE (k )ν

UUE (k )ν

ETH

z

767.995

767.990

767.985

767.9800.5 12

1012345

0

Eun

itsof

GE −

0

k of πν units

a = 2.6 mµUE (k )ν

LE (k )ν

FIG. 1: (a) Sketch of the two-level atomic ensemble em-bedded into a cylindrical cavity showing main parameters(see text). (b) Dispersion of lower and upper polaritons,EL(kν) and EU (kν), vs. those of uncoupled exciton and cav-ity photon, Ee(kν) and Ep(kν), with positive detuning δ =Ep(0)−E0. Shade marks the strong coupling region approx-imately restricted by kSC . (c) Non-interacting two-polaritonstates: ELL(kν) = 2EL(kν), ELU (kν) = EL(kν) + EU (kν),EUU (kν) = 2EU (kν), a = 2.66 µm, δ = 0. Note differentenergy scales for lower and upper halves of the plot. Insetshows three two-polariton branches plotted together at thesame energy scale.

where Ee(kν) = E0 + 2t cos akν is the exciton energy, ρlabels the two-polariton state and

Sρ = −G√

2

N

∑qν

Bρ(qν)− 4t

N

∑qν

Cρ(qν) cos aqν (4)

is a kν-independent term accounting for polariton-polariton scattering due to the kinematic interaction;B = BS and BA = 0 for Kν′ = 0; G = g

√N is the

collective atom-light coupling constant.For Sρ = 0, equations (3) describe non-interacting po-

laritons with dispersion determined by the condition

[E − 2EL(kν)][E − EL(kν)− EU (kν)]

× [E − 2EU (kν)] ≡ ∆(E, kν) = 0.(5)

Here, EL(kν) and EU (kν) are the energies of the lower(L) and upper (U) polaritons, respectively, with

EL,U (kν) =1

2

{Ee + Ep ∓

√(Ee − Ep)2 + 4G2

}. (6)

Figure 1(b) shows that the Brillouin zone can be roughlydivided into two distinct regions: The strong-couplingregion with kν < kSC near the atom-photon resonance[shaded region in Figs. 1(b) and (c)], and the region withkν > kSC , where polaritons essentially behave as un-coupled exciton and photon. The characteristic wave

3

vector kSC = 2√E0G/c~ is determined by the condi-

tion EL(kSC) = E0 with the parabolic approximationfor EL(kν).

For finite kinematic interaction Sρ 6= 0 and the solu-tions of equations (3) are wave packets of free-polaritonstates. Below we demonstrate that correlations betweenphotons arise as a result of constructive interferenceamong several components of these wave packets. Thiseffect is more prominent the larger the strong couplingregion. In the following, we explain it by first solvinganalytically the Schrodinger equation for bare excitonsuncoupled from photons, and then by describing theeffects of strong coupling of excitons to photons. Thelatter results in photonic bunching in a broad frequencyrange in the continuum. The existence of boundtwo-photon states within polaritonic gaps is discussedtowards the end of the work.

III. TWO-PHOTON CORRELATIONS

Equations (3) can be solved analytically. By using theequality

∑kνC(kν) ≡ 0, which follows from the kine-

matic interaction constraint∑kνC(kν) = C(n = 0) ≡ 0

(n is the relative distance between two excitations in thesite representation), we reduce the problem to three in-dependent equations of the form

xρ(kν) =1

N

∑kν′

xρ(kν′ ), (7)

with xρ(kν) representing the quantitiesAρ(kν)∆(Eρ, kν), Bρ(kν)∆(Eρ, kν)/[Eρ − 2Ep(kν)]and Cρ(kν)∆(Eρ, kν)/φ(Eρ, kν), where ∆(Eρ, kν)is defined in equation (5), and φ(E, kν) =[E − 2Ep(kν)][E − Ep(kν) − Ee(kν)] − 2G2. Theneach equation is solved by xρ(kν) = const(ρ). Introduc-ing the normalization constant

cρ =

(∑kν

[φ2(Eρ, kν) + 2G2(Eρ − 2Ep)2 + 4G4]

∆2(Eρ, kν)

)−1/2

,

(8)we finally write

Aρ(kν) =2G2cρ

∆(Eρ, kν),

Bρ(kν) =[G√

2(Eρ − 2Ep(kν))cρ]

∆(Eρ, kν),

Cρ(kν) =φ(Eρkν)cρ∆(Eρ, kν)

.

(9)

Figure 2(a) shows example results for the real-spaceFourier transform of the amplitudes (9) for the threestates from the bottom, middle and top of the two-polariton lower-lower- (LL-) band, whose energies are

(k )eff

(A(0) − A ) A

(A(0) − A )

X (L,α)//

0 N/2−N/2

3x

B(n)

C(n)A(n)

200xA(n)

20xB(n)5000xA(n)

200xB(n)C(n)

C(n)

n

(a)

(c)

2E0

2E - 2G0

E

(b)

0.0

0.2

−0.2

0.0

0.2

−0.2

0.0

0.2

−0.2

ν0 N/2N/4

L2E (k )

a = 5.3 µm

2 LE (k )ν

ρE

0

1

2

3

1 N/2ρ

ρ1

ρ2

ρ3

ρ3

ρ1

ρ2

FIG. 2: (a) Amplitudes X(n) − 〈X(n)〉, X = A (red), B(blue dashed) and C (green dotted), scaled by the factorsshown in each figure, for three two-polariton states markedby arrows in panel (b); n is the relative distance, N = 40, a =5.3 µm. (b) Exact two-polariton energies (cyan), comparedto the energies of non-interacting polaritons 2EL(kν) (bluesquares). Cyan points are plotted at the positions given bykeff(ρ). (c) Bunching strength as function of state number,absolute value (red circles) and scaled with the account ofphotonic amplitude of the state (blue squares).

indicated by arrows in Fig. 2(b). For small ρ (upperplot) all amplitudes resemble free wave states with asharp dip in the two-exciton amplitude C(n) at n = 0,which is a result of the hard-core constraint. Withthe increase of ρ within the LL-band, the amplitudesB(n) and C(n) demonstrate modulated oscillations,with the excitation probability increasing towards largerseparations. In contrast, the two-photon amplitude A(n)stops oscillating for larger ρ and displays a peak-likefeature centered at n = 0 (see lower plot): The two-photon amplitude thus demonstrates bunching in thepresence of repulsive kinematic interaction among atoms.

In order to clarify the behavior of the amplitudes, wefirst solve the Schrodinger equation

EµC(ex)µ (n) = (1− δn0)

{2E0C

(ex)µ (n) + 2t

×[C(ex)µ (n+ 1) + C(ex)

µ (n− 1)]} (10)

for two bare excitons interacting via the kinematic inter-action in the nearest neighbor approximation (i.e., in theabsence of coupling to photons). Remarkably, we findthat this latter equation is analytically solvable (see Ap-pendix A for details). Due to the exclusion of the stateC(ex)(n = 0) caused by the kinematic interaction, thestates are now described by a new set of wave vectors

4

{κµ}, with

κµ =2πµ

Na, (11)

and µ = [− (N−1)2 , (N−1)

2 ] half-integer. The set {κµ} haselements that lie exactly between those of the originalset {kν}. This suggests that the exciton-exciton kine-matic interaction, however weak, is a non-perturbative ef-

fect. The new two-exciton eigenenergies are then E(ex)µ =

2E0 + 4t cos aκµ, and their amplitudes read

C(ex)µ (n) = (1− δn0) sin aκµ|n|. (12)

For finite exciton-photon coupling and Eρ .2EL(kSC), the exciton-photon coupling prevails overexciton-exciton interactions. In this regime each eigen-state is to a good approximation described by a sin-gle kν , and the C-amplitudes behave approximately asCν(n) ∝ (1−δn0) cos ankν , i.e. as the symmetric part of aplane wave with 2ν nodes. For Eρ & 2EL(kSC), however,the exciton-exciton interaction prevails over light-mattercoupling, and the amplitudes C reach the exciton-likelimit, where they are well approximated by equation (12).In other words, with the increase of ρ, the two-polaritonquantum number ρ makes a smooth transition from ν-numbers to µ-numbers. We find that in all parameterregimes the two-polariton energy is well approximatedby the analytical expression Eρ = 2EL(keff(ρ)), where

keff(ρ) =2πρ∗Na

, ρ∗ = (ρ− 1)N/2− 1/2

N/2− 1, (13)

correspond to an effective wave vector keff and its la-bel ρ∗, respectively, interpolating between the two setsabove. This is shown in Fig. 2(b), where the exact numer-ical results for the energies from equations (3) (cyan dots)plotted as a function of the effective wave vectors keff per-fectly match the analytical estimates ELL(kν → keff(ρ)).

In the basis of the original wave vector set {kν},the gradual shift to the set {κµ} corresponds to theformation of wave packets. The resonant characterof the factor 1/∆(Eρ, kν) in equation (9) implies thatAρ(kν) is peaked at kν ∼ ±keff(ρ). However, thesecomponents dominate the shape of Aρ(kν) only forstates within the strong coupling region, i.e. whenkeff(ρ) < kSC . In contrast, for keff(ρ) > kSC , thecomponents Aρ(kν ∼ keff(ρ)) are strongly suppressed,as 1/∆(Eρ, kν) ∝ 1/E2

U (kν) ≈ 1/E2p(kν) decays fast

outside of the strong coupling region [see inset inFig. 1(c)]. Therefore, for larger ρ only low-kν states(with kν . kSC) are found to contribute to Aρ(kν); inother words, higher-kν states are too off-resonant toparticipate in the formation of the wave packets and, asa result, at the large-ρ amplitudes A(k) develop a singlemaximum around kν = 0. This cusp-like structure ofAρ(kν) results in the cusp-like shape of Aρ(n) in realspace [Fig. 2(a), middle and lower panels]. This explainsthe central result of this paper: The mismatch between

the quantum numbers describing the interacting (C) andnon-interacting (A − B) subsystems leads to systematictwo-photon bunching at n = 0. As Aρ(n) takes itsmaximum value at the same separation n = 0 for allρ states with keff(ρ) & kSC , we expect that it shouldnot be averaged out by a finite width of the excitingsource. In the Appendix B we quantify these argumentsand interpret the effect in terms of interference betweendifferent Aρ(kν)-components.

IV. CONTROLLING TWO-PHOTONCORRELATIONS

Correlations in the continuum of two-polariton statesshould be observable with current experimental technolo-gies with Rb or Sr atoms in a Mott insulator state withunit filling, placed in a hollow-core fiber, e.g., in the con-figuration of Okaba et al. [32]. For example, choosingthe radius of the fiber R = 0.299 µm, we bring the lowestcavity mode in resonance with the D2 transition in Rb(transition dipole d = 4.22 a.u.) at E0 = 384 THz. Wequantify the two-photon bunching by the figure of merit∆Aρ defined as

∆Aρ =|Aρ(n = 0)| − 〈Aρ〉

〈Aρ〉, 〈Aρ(n)〉 =

1

N

∑n

|Aρ(n)|

(14)if the difference is positive, and zero otherwise and plotit in Fig. 2(c) (red circles). The apparent decrease of ∆Afor large ρ is a result of the overall decrease of the pho-tonic wave function in the polaritonic state. To demon-strate that the bunching effect in fact increases with in-creasing ρ, in Fig. 2(c) we plot ∆Aρ/X

(L,α)(keff(ρ)) (blue

squares), where X(L,α)(keff(ρ)) is the two-photon compo-nent in the two-polariton wave function in the absence ofkinematic interaction, as defined in the Appendix B.

Figure 3(a) shows that ∆A(Eρ) changes by varyingthe lattice constant a from 532 nm to 5.32 µm. Coun-terintuitively, ∆A is found to increase for larger a, cor-responding to lower atomic density. This is explained bynoting that only states within the strong coupling region,which have both finite two-exciton amplitude (to interacteffectively) and finite two-photon amplitude (the observ-able), contribute to Aρ(n = 0). Therefore, photon bunch-ing is most pronounced when kSC is comparable to thesize of the Brillouin zone, which can be easily achievedin cold atom experiments. The latter condition requireslarger inter-atomic separations, as akSC/π ∝ a3/4. Thisalso explains why this type of bunching of continuumstates cannot be observed in solids: For electronic tran-sitions of ∼ 2 eV and typical lattice constants a ∼ 5 A,the relative size of the strong coupling region akSC/π ∼2a√ω0G/πc~ ∼ 10−4. On the contrary, Fig. 3(a) shows

that in cold atom systems considerable bunching can berealised in a continuous band of the order of several GHz.

In Figure 3(b) we examine ∆A vs energy for a few

5

∆A

K (units of π/a)

(c)

(b)

0−10 E−2E , GHz0

A

∆A

E−2E , GHz0

δ = Gδ = 0.5Gδ = 0δ = -0.5G

a = 5.3 µm, G = 5.3 GHz

∆A

(a)

0

1

2

3

a = 0.5 µm, G = 16.8 GHz

a = 2.6 µm, G = 7.5 GHz

0

0.5

0 N/2−N/2 n

A(n)B(n)C(n)

(d)

LL

LUgapstateE ρ

a = 50.3 µm

15 10 5 00

3

6

2 1 0 1 20.0

3.5

0.0

0.8

ρ1

ρ2

ρ1

ρ2

ρ

FIG. 3: (a) Figure of merit |∆A(n = 0)| as a function of thestate energy for Rb atoms, D2 line, for a = 532 nm, 2.66 µmand 5.32 µm. (b) |∆A(n = 0)| in the LL-band for a = 5.3 µmfor different values of detuning. (c) |∆A(n = 0)| as functionof K for two states, which at Kν

′ = 0 correspond to the statesmarked by ρ1 and ρ2 in panel (a); a = 5.3 µm. (d) Ampli-tudes A(n), B(n) and C(n) for the bipolariton gap state; a =50.3 µm. Inset displays the energies Eρ as a function of thestate number ρ, with the gap state shown in red.

values of the detuning δ = Ep(0)−E0 between the cavitymode Ep(0) and the excitonic resonance E0, by varying δbetween −G/2 and G. We find that large negative valuesof δ result in wider bunching frequency intervals due towider ELL bands.

Finally, it is of crucial importance that the bunchingsurvives for states with Kν′ 6= 0. The latter correspondto propagation of the center of mass of the two polari-tons, and can be directly observed in experiments. Theequations for the amplitudes at Kν′ 6= 0 are bulky andwill be published elsewhere. Here we only demonstratethe existence of bunching in a wide range of (kν1 , kν2)-pairs by plotting in Fig. 3(c) ∆A as a function of Kν′ fortwo states, selected so that at Kν = 0 they correspondto the full squares marked as ρ1 and ρ2 in Fig. 3(a).

V. GAP STATES

A gap in the polaritonic spectrum appears naturallywhen the detuning is positive. In contrast, for δ = 0and small a the two-polariton spectrum is non-gapped,as the energy for two lower polaritons ELL(π/a) equalsELL(π/a) = ELU (0) = 2E0, with ELU the energy of thelower-upper- (LU-) band. However if a is so large thatkSC ∼ π/a, then ELL(π/a) < 2E0 as the lower polari-ton does not reach the excitonic limit, and a small gap∆LU = ELU (kν = 0)−ELL(kν = π/a) opens between theLL- and LU- bands at zero detuning. In this regime onecan observe the formation of bunched states also within

the gapped region at the very bottom of the LU-band.Contrary to the discussion above, these are polariton-polariton bound states with very distinct wave functions,with A- and B-amplitudes peaked around n = 0 (the two-exciton C-component vanishes at n = 0 and therefore ismaximal for the separation |n| = 1, in accordance withthe hard core restriction) [see Fig. 3(d)]. For moderatea, the bound state merges with the LU-band. With theincrease of the lattice constant, at a ∼ 25 µm, the boundstate splits from the band and shifts into the gap. Fur-ther increase of a is accompanied by deeper penetrationinto the gap, dramatic increase of the photonic bunchingin this particular state, and suppression of the bunchingin the continuum.

The state above is a gap bipolariton forming underrepulsive kinematic interaction. This is similar to thekinematic biexciton appearing in organic crystals withtwo molecules in a unit cell [49]. However, there thekinematic biexciton overlaps with the continuum band,and can be easily destroyed by, e.g., disorder or cou-pling to phonons. In contrast, the kinematic bipolaritondescribed here is located in the gap and is thus stableagainst decoherence. Other types of bound states, bothbelow the LL-continuum and in the polariton gap, canform if the atoms interact via, e.g., dipole forces. Thiswould be analogous to, e.g., the gap bound states foundin atomic systems [50, 51] and Jaynes-Cummings-typemodels with repulsive interactions [52].

For a large enough the interaction of excitons withphotons from higher Brillouin zones becomes possible:An exciton with a wave vector kν is coupled not onlyto a photon with the same kν but also to photons withkν ± 2π/a, kν ± 4π/a, etc. For the setup described here,this would occur for a large value a & 50 µm. Whathappens in this regime will be the subject of a furtherinvestigation.

VI. DISCUSSION

In this article we have characterised the correlationsthat are generated for pairs of 1D photons propagat-ing in a hollow-core crystal fiber and coupled to an or-dered atomic ensemble. The correlations resemble two-photon attraction for states in the continuum of scatter-ing states. This two-photon bunching can be observed inGHz frequency window, which greatly exceeds MHz fre-quency intervals typical for the correlations induced by,e.g., three-level cold Rydberg atoms in the EIT regime,and can be controlled by tuning the inter-atomic spac-ing. Our results are valid as long as the polaritonic split-ting 2G exceeds strongly the sum of the excitonic andphotonic broadenings, which can be achieved by usinga high-quality cavity and by careful preparation of theatomic Mott insulator state. Other realizations of thisscheme are also possible, as long as broadenings can bekept smaller than the collective exciton-photon coupling.In addition to 1D hollow-core crystal fibers, promising

6

candidates are metallic nanowires [53] and nanophotonicwaveguides [15, 54–60].

The same effect as demonstrated here, and even in anexaggerated form, can exist for chains of two-level Ryd-berg atoms coupled to 1D cavity photons. We find thatthe formation of a large-radius Rydberg blockade sphereconsiderably enhances bunching, as the effect of thequantization volumes mismatch presented in our workbecomes more pronounced. In addition, the long-rangedynamical (dipole-dipole or van der Waals) interaction isfound to enrich the arising of non-linear effects. Theseresults will be subject of a forthcoming publication.

VII. ACKNOWLEDGEMENTS

We acknowledge support by the ERC-St Grant Cold-SIM (No. 307688), UdS via Labex NIE and IdEX, RYSQ.

Appendix A: Kinematic interaction for bare excitons

The Schrodinger equation for two bare excitons (G ≡0) on sites n1 and n2 interacting via kinematic interactionis

EC(ex)n1n2

= 2E0C(ex)n1n2

+ (1− δn1n2)

×∑s

(tn1sC

(ex)sn2

+ tn2sC(ex)n1s

)(A1)

with tij the long-range hopping energy, and the same-

site amplitude chosen as C(ex)nn = 0 [61]. Let n be n =

|n1 − n2|, and the index µ enumerate the eigenstates ofthis equation. We rewrite equation (A1) in the nearestneighbour approximation:

E(ex)µ C(ex)

µ (n) = (1− δn0){

2E0C(ex)µ (n) + 2t

× [C(ex)µ (n+ 1) + C(ex)

µ (n− 1)]}.

(A2)

One can verify that the normalized amplitudes whichsatisfy equation (A2) are

C(ex)µ (n) ≡ gn(µ) =

√2(1− δn0)√

Nsin |n|κµ (A3)

with wave vectors κµ introduced in equation (10) in themain text. The basis functions gn(µ) form an orthonor-mal set in both spaces, with the orthonormality condi-tions reflecting the permutation symmetry and the hard-core condition:∑

n gn(µ1)gn(µ2) = δ|µ1|,|µ2|,∑µ gn1

(µ)gn2(µ) = (1− δn10)δ|n1|,|n2|.

(A4)

The eigenenergies of two-exciton states are also de-scribed by the wave vectors κµ as

E(ex)µ = 2E0 + 4t cos aκµ. (A5)

These new wave vectors κµ = 2πµ/(Na) have a half-integer state index µ = [−(N − 1)/2, (N − 1)/2] and lieexactly between the positions of the standard wave vec-tors kν for non-interacting excitons. As a consequence,

the amplitudes C(ex)µ (kν) do not have poles, but rather

an enhanced components of kν ≈ κµ, as can be seen fromthe Fourier transform of equation (A3):

C(ex)µ (k) =

sin aκµ cos(akν) + (−1)µ sin akν sin(akνN/2)

cos akν − cos aκµ.

(A6)We conclude that the kinematic interaction is a weak,but absolutely non-perturbative effect for excitons. Letus now discuss the effect of the kinematic interaction forpolaritons.

Appendix B: Creation of the wave packets

Two-polariton states in the presence of kinematicinteraction can be viewed as composed of two sub-systems: the non-interacting subsystem (consisting ofphoton-photon and photon-exciton states) is describedby the quantum numbers kν = 2πν/(Na) with integer ν,whereas the interacting subsystem (consisting of exciton-exciton states) is described by κµ = 2πµ/(Na) with half-integer µ. In the following discussion we shall stress therole of the two wave vector sets, which will be reflected inthe adopted notations. The coupling between these twosubsystems is responsible for intermixing the correspond-ing wave vector sets {kν} and {κµ} and eventually leadsto the creation of the wave packets in the “original” wavevector set {kν}. In particular, at the lowest energies thecoupling of excitons to photons dominates over exciton-exciton interaction, and the corresponding polaritons arebetter described by kν . With the increase of the statenumber, instead, polaritons enter the exciton-like regimeand are better described by κµ.

We introduce the operator α†n, β†n and γ†n, which de-scribe, respectively, creation of two photons, one pho-ton and one exciton, and two excitons separated by adistance n. The two particle wave function takes theform |Ψ〉 =

∑s [A(s) |αs〉+B(s) |βs〉+ C(s) |γs〉], and

the Hamiltonian Heff = HAB + H(KI)C + H

(KI)AB−C is made

up of three terms:

HAB =∑n,m

[2Ep(n−m)α†nαm

+ (Ep(n−m) + Ee(n−m))β†nβm

]+G√

2∑n

[α†nβn + β†nαn

],

H(KI)C =

∑n,m

(1− δn0)2Ee(n−m)γ†nγm,

H(KI)AB−C = G

√2∑n

(1− δn0)[γ†nβn + β†nγn

],

(B1)

7

where the last one describes the coupling between the in-teracting (C) and non-interacting (AB) subsystems. Theresulting Schrodinger equation is identical to the Fouriertransform of equations (3) in the main text.

The first term HAB describes the subspace “photon-photon

⋃photon-exciton”, and is diagonalized by the

operators ξ(i)ν

†= X

(i,α)ν α†ν +X

(i,β)ν β†ν :

HAB =∑i=L,U

∑ν

E(p,i)ν ξ(i)

ν

†ξ(i)ν . (B2)

Here and below i = (U,L) is the index of the polaritonicbranch, ν = (−N/2, N/2] is the free-state wave index,and

E(p,i)ν = E(p)

ν +E

(e)ν + E

(p)ν ±

√(E

(p)ν − E(e)

ν

)2

+ 8G2

2(B3)

with E(p)ν ≡ Ep(kν), E

(e)ν ≡ Ee(kν). The energies

E(p,i={L,U})ν (with i = L corresponding to “−”, and

i = U to “+” in the right-hand side) are constructed assums of energies of one photon and one exciton-polaritonwith the coupling constant

√2G, taken at the same wave

vector kν . They naturally appear as solutions of the firsttwo lines of equations (3) in the main text with C ≡ 0.The photon-photon and photon-exciton amplitudes are

X(i,α)ν =

√√√√√√(E

(p,i)ν − E(p)

ν − E(e)ν

)2

2G2 +(E

(p,i)ν − E(p)

ν − E(e)ν

)2 ,

X(i,β)ν =

√1−

(X

(i,α)ν

)2

.

(B4)

The two-exciton part H(KI)C of the Hamiltonian is in-

stead diagonalized as

H(KI)C =

∑µ

E(ex)µ χ†µχµ, (B5)

with energy E(ex)µ defined in equation (A5), and

χ†µ =

N/2∑s=−N/2+1

gs(µ)γ†s . (B6)

Eventually, the interaction Hamiltonian HAB−C canbe rewritten in terms of ξ- and χ-operators as

H(KI)AB−C =

G

N

∑i=L,U

∑νµ

ΛνµX(i,β)ν

(χ†µξ

(i)ν + ξ(i)

ν

†χµ

)(B7)

with coupling coefficients Λνµ given by

Λνµ =1

2

[cot

π(ν + |µ|)N

− cotπ(ν − |µ|)

N

]. (B8)

The coefficients Λνµ intermix the wave vector sets kν andκµ.

The Schrodinger equation for the Hamiltonian Heff and

the wave function |Ψ〉 =∑iν p

(i)ν |ξ(i)

ν 〉+∑µ eµ |χµ〉 leads

to

(E − E(p,i)

ν

)p(i)ν =

GX(i,β)ν

N

∑µ

Λνµeµ,

(E − E(ex)

µ

)eµ =

G

N

∑i=L,U

∑ν

X(i,β)ν Λνµp

(i)ν .

(B9)

We can exclude the exciton-exciton amplitudes eµ fromequations (B9); in the absence of hopping (t ≡ 0) theresulting equation for piν reduces to

(E − E(p,i)

ν

)p(i)ν =

G2X(i,β)ν

2N(E − 2E0)

×∑

i′=L,U

∑ν′

Fνν′X(i′,β)ν′ p

(i′)ν′

(B10)

with the kernel

Fνν′ = N (δν,ν′ + δν,−ν′)− 2

N. (B11)

The first term in the right-hand side of this equation de-scribes the wave-vector-conserving scattering, while thesecond describes the formation of wave packets via scat-tering of non-interacting subsystem through the interact-ing one.

Within these notations, the amplitude for two photonsbeing separated by n lattice sites is

A(n) = 〈αn|Ψ〉 =1√N

∑i=L,U

∑ν

p(i)ν X(i,α)

ν e−2πiνnN

(B12)

so that A(0) =∑iν p

(i)ν X

(i,α)ν /

√N results from a collec-

tive effect of p-amplitudes that add up with a vanishingphase; large separation amplitudes are instead averagedout by the oscillating exponentials. The wider is the

distribution of p(i)ν , the larger A(0) is expected. Using

equations (B9) we get

A(0) =G

N√N

∑i=L,U

∑ν

X(i,α)ν X

(i,β)ν(

E − E(p,i)ν

)∑µ

Λνµeµ. (B13)

Due to the mismatch between quantum numbers ν and

µ the denominator (E − E(p,i)ν ) is not a real pole. How-

ever, it plays an important role in the establishing ofthe bunching, which occurs when E = Eρ is resonant

with the band of non-interacting states E(p,i)ν (B3); when

Eρ < min{E

(p,i)ν

}= E

(p,i=L)ν=0 the two-photon wave func-

tion looks unperturbed and exhibits plane-wave-like os-cillations. This criterium can be used as a good rule

8

of the thumb when deciding on whether a state witha given energy shows bunching or not. It looks like asif excitons talked to each other via virtual excitations– the eigenstates of the non-interacting subsystem. In-deed, the “real” elementary excitations are one-polaritonstates, while the energies (B3) do not have an indepen-dent physical meaning, except as a virtual scatteringchannel through which excitons interact.

Using the equality

C(s) = 〈γs|Ψ〉 =∑µ

eµgs(µ) (B14)

following from |Ψ〉 representation via A,B,C- and p, e-amplitudes, we find that 2eµ =

∑s gs(µ)C(s). For higher

ρ showing bunching, we can approximate C-amplitudesby bare exciton-exciton amplitudes (A3) times a normal-

ization coefficient X(γ)ρ , which accounts for the presence

of finite exciton-photon and photon-photon excitation inthe total wave function of ρ-th eigenstate. We then ob-

tain, using orthogonality of g-functions,

Aρ(0) ≈ GX(γ)ρ

2N32

∑i=L,U

∑ν

X(i,α)ν X

(i,β)ν

(Eρ − E(p,i)ν )

×[Λν(µ=ρ) + Λν(µ=−ρ)

].

(B15)

Equation (B15) shows that only those state contributeinto A(0), which simultaneously have non-vanishing pho-tonic and excitonic amplitudes, which is true only forthe strong coupling region. Therefore, the larger part ofthe Brilloiun zone the strong coupling region occupies,the stronger is the effect of bunching. This explains theincrease of the bunching efficiency with the increase ofthe lattice constant a: For larger a the Brillouin zone ofexciton is matched to a flatter part of the photon disper-sion, and the strong coupling region is larger. This scaleargument explains why in natural solids the kinematicinteraction is a negligible effect, while in atomic systemsit may lead to a qualitatively different behaviour.

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