Interplay of coherence and interaction in light propagation through waveguide QEDlattices

Tarush Tiwari,1 Dibyendu Roy,2 and Rajeev Singh3

1School of Biochemical Engineering, Indian Institute of Technology (Banaras Hindu University), Varanasi 221005, India.2Raman Research Institute, Bangalore 560080, India.

3Department of Physics, Indian Institute of Technology (Banaras Hindu University), Varanasi 221005, India.(Dated: October 29, 2020)

We investigate the role of optical nonlinearity in light propagation through two different waveguideQED lattices, namely a chain of qubits with direct coupling between the nearest neighbors and achain of connected resonators to each of which a qubit is side-coupled. We apply an efficienttruncated Heisenberg-Langevin equations method to show loss of coherent light transmission withincreasing intensity in these lattices due to effective photon-photon interactions and related photonblockade mediated by nonlinearity in qubits. In contrast to the direct-coupled qubits, we find arevival in the coherent light transmission in the side-coupled qubits at relatively higher intensitiesdue to saturation of qubits by photons. We further study these lattices within the quasi-classicalapproximation, which fails for a broad set of parameters. We then devise a technique to modifythe quasi-classical analysis to give much better results. Lastly, we examine light propagation in aninhomogeneous lattice of side-coupled qubits and observe non-monotonicity in light transmissionwith increasing light intensity.

I. INTRODUCTION

The physics of mesoscopic collections of quantum par-ticles has been extensively explored in the past for in-vestigating quantum effects such as size confinements,interference, interactions, and band-topology [1]. Thequantization of conductances, persistent currents, andCoulomb blockade are well-known examples of quantumeffects in lower dimensional systems. Though these phe-nomena were particularly popular for mesoscopic electri-cal devices, recent efforts in quantum photonics have alsodemonstrated many interesting mesoscopic quantum ef-fects in various cavity [2], circuit [3] and waveguide QED[4, 5] set-ups. Some such recent discoveries are the pho-ton blockade and quantum nonlinearity for propagatingsingle photons through an elongated ensemble of lasercooled atoms in an optical-dipole trap [6, 7], a dissipativephase transition in a one-dimensional circuit QED lat-tice [8], and the cooperative Lamb shift in a mesoscopicarray of ions suspended in a Paul trap [9]. Motivatedby these fantastic developments in experiments, we herestudy nonlinear quantum transport of light through one-dimensional (1D) waveguide QED lattices. The waveg-uide QED lattices are characterized by either (a) an ab-sence of optical confinement (cavity) along the light prop-agation direction or (b) the coupling to and from thecavity(ies) dominating the internal system losses in theso-called overcoupled regime [4, 10]. These special fea-tures of the waveguide QED lattices separate our studyfrom other recently explored cavity and circuit QED lat-tices [8, 11–19].

The photonic systems have their own distinctive fea-tures different from electrical devices. Photons beingcharge-neutral do not interact with each other and canhave very long spatial and temporal coherence. Effec-tive interaction between photons can be realized through

their coupling to matter. Our present study employs suchlight-matter coupling to investigate the interplay of co-herence and interaction in photons’ quantum transportin a 1D mesoscopic array of qubits. These qubits canbe considered to be made of alkali atoms (e.g., rubidiumatoms in [6, 7]) or superconducting qubits (e.g., transmonqubit in [8]) or ions (e.g., Sr+ ions in [9]). The features ofthe above qubits and their interactions with light can bevery diverse. Nevertheless, we here consider some generalmodels of the light-matter coupling and the interactionsbetween qubits. We mainly choose two different modelsfor the 1D lattice of qubits; these are (a) a chain of qubitswith direct coupling between the nearest neighbors, and(b) a chain of connected resonators to each of which aqubit is side-coupled. While we do not have any confine-ment along the light-propagation for the direct-coupledqubits, we work in the overcoupled regime [4, 10] for thechain of connected resonators with side-coupled qubits.

The theoretical study of correlated quantum transportof light is a nontrivial problem. Many different theoret-ical approaches, including analytical [12, 15, 16, 20–35]and numerical [36–40] methods, have been explored inrecent years. One major challenge in applying these the-ories is the scaling of the system sizes from single or mul-tiple qubits to a mesoscopic collection of qubits. We heredevelop and implement a truncated Heisenberg-Langevinequations (THLE) approach and a quasi-classical analy-sis with complex interactions to scale up the system sizes.The Heisenberg-Langevin equations have been appliedin recent years for studying correlated light transmissionin a chain of two-level systems [40]. We here show thebenefits of truncation in the number of user variablesto efficiently capture the main features of the nonlinearquantum transport and scaling up the system sizes.

Next, we compare the THLE approach to a quasi-classical analysis (QCA) [41] recently used in the the-oretical analysis of experimental observation of dissipa-

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for remaining photonsNo Coherent Transmission,Tmax→ 0

High Effective Photon-Photon InteractionLow Coherent Transmission

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Qubits SaturateLow effective photon-photon interaction

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kjbbikeSide-coupled Qubits

Low Transmission

High Transmission

FIG. 1. Cartoons showing the arrangement of the direct andside-coupled qubits mediums through which the photons areincoming from the left and transmitted to the right. The bluecircles represent the qubits while the red ones represent theresonators. For each medium we show, in the parameter spaceof input intensity (Iin) and inter-site coupling (Jx), the regionsof high transmission with the color gradient. With increasingintensity we observe an initial increase in effective photon-photon interaction followed by the saturation of qubits at veryhigh intensity. The saturation of qubits has opposite effect ontransmission in the two mediums, causing a complete loss oftransmission in the direct-coupled system while a revival offull coherent transmission through the resonators for the side-coupled one.

tive phase transition in a 1D circuit QED lattice [8]. Thesystem in Fitzpatrick et al. [8] has relatively large qubitrelaxation as well as intrinsic photon loss from every siteof the medium. We are here interested in a waveguideQED system with negligible or no loss from the mid-dle sites of it as we wish to understand the interplayof coherence and interaction. We are surprised to findthat such a QCA badly fails in the absence of internallosses in bulk. We specifically notice that contrary to theresults from the THLE, there is always some frequencythat has perfect transmission within the QCA. It appearsthat the source of such discrepancy is due to failure ofQCA in capturing photons’ inelastic scattering and therelated effective interaction between photons generatedin our models. To revive the QCA in such situations,we propose to analytically continue (or modify) the in-teraction parameter to the complex plane. We develop asimple scheme to find the complex interaction parameterfor the modified QCA. We see the modified QCA workssignificantly well when it is compared to the THLE. Theadvantage of the modified QCA is that it is simpler toimplement numerically for much longer lattices than theTHLE.

We then apply the THLE and modified QCA to studythe interplay of coherence and interaction in light prop-agation through the two open waveguide QED latticemodels. At low intensities of the incident light, we findcoherent light propagation in both the models, which canbe investigated by any of THLE, QCA, and modified

QCA. The interaction dominates light propagation forincreasing intensities in an array of direct-coupled qubits.The transmission coefficient in this model falls with anincreasing power due to the photon-photon blockade,which is generated by the nonlinearity (interaction) atthe qubits. This regime can be adequately analyzed onlyby either THLE or modified QCA. For the side-coupledqubits, the effect of interaction on light propagation isnon-monotonic with increasing intensity of input lightfor the side-coupled qubits. The light transmission co-efficient at resonant condition first decays with growingpower, and then it again enhances with further increasein intensity when photons saturate the qubits. The restof the paper is organized in the following sections andappendices. The models, analysis, primary results, andoutlook are given in the next four sections. We also in-clude technical details of our research in the appendicesfor the interested readers.

II. MODELS

We consider a 1D medium either comprising of qubitswith direct coupling between the nearest neighbors or aresonator chain to which the qubits are side-coupled. Tomodel the transmission through the system, we couplethe ends of the medium to photon baths, where incidentlight from the left bath excites the qubit/resonator atone end and propagates through the system to reach theright bath at the other end (see Fig. 1). The entire systemincluding the baths is described by the Hamiltonian

H = HM +HLB +HRB +HLM +HRM , (1)

where HM represents the Hamiltonian of the mediumwhich, in this paper, can be the direct-coupled qubits(Hdirect) or resonator chain with side-coupled qubits(Hside). HLB (HRB) is the left (right) bath and HLM

(HRM ) is the coupling between the chain and the left(right) bath. The bath and the bath-medium couplingHamiltonians within the Markovian approximation are[40]

HLB =

∫ +∞

−∞dk ~ωka†LkaLk, (2)

HRB =

∫ +∞

−∞dk ~ωka†RkaRk, (3)

HLM =

∫ +∞

−∞dk ~gL(a†Lkχ1 + χ†1aLk), (4)

HRM =

∫ +∞

−∞dk ~gR(a†RkχN + χ†NaRk). (5)

The operators a†Lk and a†Rk are the creation operatorswith wave number k in the left and right side baths re-spectively. We set the ground state energy of the bath

to be zero and have [aLk′ , a†Lk] = [aRk′ , a

†Rk] = δ(k− k′

).

3

The bosonic operators χ1 and χN correspond to the leftand right most qubits/resonators of the medium, whichare coupled to the baths with strengths gL and gR re-spectively.

The baths are kept at a temperature of 0K to ensure acoherent light source. At time t0 the medium-bath cou-pling is switched on and incoming light interacts with theends of the chain. For a weak bath-medium coupling, wewrite HLM and HRM within the rotating wave approxi-mation where we drop the counter rotating terms.

A. Direct-coupled array of qubits

A simple Bose-Hubbard type model system that showscorrelated light propagation is a 1D array of qubits withdirect coupling between nearest neighbors [12, 14, 19]:

HM = Hdirect =

N∑i=1

~ωqib†i bi + ~Ub†i bi(b

†i bi − 1)

+

N−1∑i=1

2~Jx(b†i bi+1 + b†i+1bi). (6)

Here, b†i and bi are the bosonic creation and annihilationoperators at site i with ωqi being the qubit frequency

and [bi, b†j ] = δi,j . 2Jx is the coupling strength between

neighboring qubits. U determines the strength of on-siteinteraction, which is repulsive for positive U . In the limitU → ∞, the qubit maps to a two-level system. The on-site interaction at the qubits induces optical nonlinearityin the model by preventing multiple excitations by pho-tons, and is responsible for the inelastic light scatteringstudied in this work.

B. Side-coupled array of qubits

We also study a chain of connected resonators to eachof which a qubit is side-coupled. This model is similarto the system investigated in [8]. In contrast to [8], weconsider here no photon loss from the resonators to inves-tigate perfect coherence of the waveguide QED medium.The Hamiltonian of the model is

HM = Hside =

N∑i=1

(Hqi +Hr

i +Hrqi ) +

N−1∑i=1

Hhopi , (7)

where,

Hri = ~ωrif

†i fi, (8)

Hqi = ~ωqib

†i bi + ~Ub†i bi(b

†i bi − 1), (9)

Hrqi = ~gi(b†ifi + f†i bi), (10)

Hhopi = 2~Jx(f†i fi+1 + f†i+1fi). (11)

Similar to the directly coupled qubit chain, b†i and bi arethe creation and annihilation operators for the qubits

with qubit frequency ωqi and on-site interaction U . Theresonator chain is a one dimensional array of bosonic res-

onators with creation and annihilation operators f†i andfi, respectively, and resonator frequency ωri . Each res-onator in the chain is coupled to its nearest neighbors,with coupling strength 2Jx, and side-coupled with thequbit, with resonator-qubit coupling strength gi.

III. ANALYSIS OF THE MODELS

A. Integrating out the baths

The Heisenberg equations of the baths and resonatorscan be integrated out, and we only need to deal explicitlywith the Heisenberg equations of the qubits where thebath and resonator dynamics appear as parameters inthese equations. Integrating out the baths gives us thequantum Langevin equations (QLEs) for the qubits andresonators [40, 42]. Following Manasi and Roy [40], wefirst integrate out the baths. The Heisenberg equationsfor the photon operators of the bath, aLk and aRk areinhomogeneous linear differential equations. The initialcondition (at t = t0) will set the direction of the incominglight in the system. By solving the Heisenberg equationsfor the baths, we obtain the time evolution of the bathoperators [40] given by,

aLk(t) = Gk(t− t0)aLk(t0)− igL∫ t

t0

dt′Gk(t− t′)χ1(t′),

(12)

aRk(t) = Gk(t− t0)aRk(t0)− igR∫ t

t0

dt′Gk(t− t′)χN (t′),

(13)

where, Gk(τ) = θ(τ)e−ivgkτ is the retarded Green’s func-tion of the baths. The bath solutions, Eqs. 12-13, aresubstituted in the remaining equations of motion forthe qubits and give rise to the noise operators due tothe baths which are given as, ηL(t) =

∫∞−∞ dk Gk(t −

t0)gLaLk(t0) and ηR(t) =∫∞−∞ dk Gk(t − t0)gRaRk(t0).

These noises coming from the baths can be convertedto constant numbers by considering an appropriate ini-tial condition. Assuming that a coherent light is incidentfrom the left and the initial state (at t = t0) for thecoherent light source is |Ep, ωp〉,

aLk(t0)|Ep, ωp〉 = Epδ(vgk − ωp)|Ep, ωp〉, (14)

aRk(t0)|Ep, ωp〉 = 0, (15)

where Ep is the amplitude of the incoming light whileωp is its frequency. The initial condition converts thenoises from the bath into the Rabi frequency of the sys-tem, given by ΩL = gLEp/vg, and medium-bath couplingrates from the left and right baths given by, ΓL = πg2

L/vgand ΓR = πg2

R/vg respectively. The noise terms can now

4

be replaced in the QLEs for the qubits/resonators withthese constant terms. The Rabi frequency appears in thesystem of equations either as a constant source term or asprefactors to certain operators, while the bath-mediumcoupling rates only appear as the latter. The incomingphoton frequency (ωp) appears as time-dependent phasefactors in the QLEs which are absorbed by suitable re-definition of the operators.

B. Truncated Heisenberg-Langevin equations

As the qubits/resonators are bosonic, each of them canhave any number of excitations. These excited states

appear as powers of bi and b†i in the operators. To de-scribe the THLE, we need to consider every indepen-dent boson operator for each qubit/resonator for each

site; which are I, bi, b†i , b†i bi, b

2i , b†2i , b

†i b

2i , b†2i bi, b

†2i b

2i , ....

However, as there are infinite number of possible opera-tors, we must truncate the set of boson operators. If werestrict our system to have only one excitation, then onlythe first four of these operators would enter the equa-tions. In this case, for example, to obtain the set ofequations for two qubits we consider every operator in

the set I, b1, b†1, b†1b1 ⊗ I, b2, b

†2, b†2b2, which gives 16

independent operators. The operator I⊗I is ignored as ithas no dynamics and we are left with 15 operators. Thenumber of operators increases exponentially as number of

qubits (N) increases and are given by I, bi, b†i , b†i bi⊗N .

Including more excited states in our operator set willincrease the accuracy of the results, however, for rela-tively high on-site interaction (U), appropriate accuracycan be achieved by considering a limited number of ex-cited states. As more excited states per site (m) areincluded, the number of operators increases with a poly-nomial growth where the exponent is 2N , we denote mas maximum number of bosons per site. For example, ifm = 2, the number of operators for a chain of two qubits(N = 2) increases to 80 as the operator set becomes

I, bi, b†i , b†i bi, b

2i , b†2i , b

2i b†i , bib

†2i , b

2i b†2i ⊗2. Similarly, for a

chain with N qubits with m allowed excited states persite, we shall get (m+ 1)2N − 1 operators.

We illustrate the THLE analysis by writing the equa-tions of motion for the simplest system considered here,namely a single qubit. The expectation values of qubitoperators after integrating the baths from the equationsalong with its phase factor are denoted as,

Sjk = 〈b†j1 bk1〉e(−j+k)iωp(t−t0), (16)

where the phase factor has been chosen such that theRHS of the equations of motion have no explicit time de-pendence. A few examples can be, S01 = 〈b1〉eiωp(t−t0),

S22 = 〈b†21 b21〉. Applying this convention to our equa-

tions, and setting m = 2, we get the following set ofeight coupled differential equations:

dS01

dt= −(iδωq1 + ΓL + ΓR)S01 − 2iUS12 − iΩL, (17)

dS02

dt= −(δωq1 + 2ΓL + 2ΓR)S02 − 2iU(S02 + 2S13)

−2iΩLS01, (18)

dS11

dt= −2(ΓL + ΓR)S11 − iΩL(S10 − S01), (19)

dS12

dt= −(iδωq1 + 3ΓL + 3ΓR)S12 − 2iU(S12 + S23)

−iΩL(2S11 − S02), (20)

dS22

dt= 2(2ΓL + 2ΓR)S22 − iΩL(2S21 − 2S12). (21)

here, δωq1 = ωq1 − ωp and the remaining equations arethe hermitian conjugates of the equations 17, 18 and 20.In the truncation scheme with m = 2, we ignore all otherequations with higher values of j, k in Eq. 16. Note thatwe still get the operators we have ignored due to the trun-cation on the RHS of the above equation for example inEqs. 18-20 and their hermitian conjugates. We drop theseterms in the further analysis, and explicitly check the ac-curacy of the truncation by considering various values ofm. As mentioned before, increasing m increases accuracyof the results but also expands the set of equations.

The equations 17-21, and their hermitian conjugates,can be re-written in a matrix form with the vectorsS(t) = S01,S02,S10,S11,S12,S20,S21,S22T and Ω =−iΩL, 0, iΩL, 0, 0, 0, 0, 0T . The linear system of equa-tions can be represented very compactly as

dS(t)

dt= ZS(t) + Ω. (22)

Assuming the qubits in the ground state before irra-diation with photon, we solve the coupled set of equa-tions to get the steady-state results, which occurs whent t0 after some transient time. We obtain the long-

time (t→∞) steady-state solution by solving dS(t)dt = 0,

which gives the time independent solution of the equa-tions as S(t→∞) = −Z−1Ω.

C. Integrating out the resonators

To model the side-coupled system of qubits, we firstsolve the QLEs for the resonators coupled to the bathsand qubits, just as it was done in the case of baths. Fora single qubit-resonator pair, we denote the expectationvalues of resonator operators after integrating the bathssimilar to those of the qubits in Eq. 16 by:

Fjk(t) = 〈f†j1 fk1 〉e(−j+k)iωp(t−t0). (23)

5

With this notation, the QLE for the resonator variableturns into the following equation:

dF01(t)

dt= −(iδωr1 + Γ)F01(t)− igS01(t)− iΩL. (24)

Here, δωr1 = ωr1−ωp, Γ = ΓL+ΓR and S01(t) is the qubitoperator as mentioned in III B. Since the resonator has nonon-linearity (it is modeled as a harmonic oscillator), wenote that Eq. 24 is an inhomogeneous linear differentialequation which is not connected to any other variables ofthe resonator apart from F01. We now integrate Eq. 24to obtain a formal solution of the equation as:

F01(t) =F01(t0)GA(t− t0)− iΩL∫ t

t0

GA(t− t′)dt′

− ig∫ t

t0

GA(t− t′)S01(t′)dt′, (25)

where, GA(τ) = e−Aτθ(z) and A = iδωr1 + Γ. This isquite analogous to integrating out the baths, except thatthe last integral in Eq. 25 retains a memory of the qubits’time-evolution; thus, the Eq. 25 is non-Markovian in na-ture. For simplicity, we now make the Markovian ap-proximation, and we replace S01(t′) in Eq. 25 by S01(t).This simplifies the equation by neglecting any retarda-tion effect. We can now write at long-time (t → ∞)steady-state:

F01(t→∞) = −iΩLA− i g

AS01(t→∞). (26)

The Eq. 26 is then substituted in the differential equa-tions obtained after taking expectation of the QLEs ofthe qubits, which for m=2 at long time (t→∞) gives

dS01(t)

dt

∣∣∣∣t→∞

= −[iδωq1 +

g2

A

]S01 − 2iUS12 −

gΩLA

,

(27)

dS02(t)

dt

∣∣∣∣t→∞

= −2

[iδωq1 +

g2

A

]S02 − 2iU(S02 + 2S13)

−2gΩLAS01, (28)

dS11(t)

dt

∣∣∣∣t→∞

= −[g2

A+g2

A∗

]S11 − gΩL

[S10

A+S01

A∗

],

(29)

dS12(t)

dt

∣∣∣∣t→∞

= −[iδωq1 + 2

g2

A+g2

A∗

]S12

−2iU(S12 + 2S23)− gΩL

[2S11

A+S02

A∗

],

(30)

dS22(t)

dt

∣∣∣∣t→∞

= −2

[g2

A+g2

A∗

]S22 − 2gΩL

[S21

A+S12

A∗

],

(31)

and the rest of the equations are hermitian conjugates of27, 28, 30. The Eqs. 27-31 and its hermitian conjugatescan be re-written in the matrix form and solved to obtainthe steady-state solution in the same way as mentionedin III B.

For longer system lengths, similar to the system of sin-gle qubit-resonator pair, we obtain N linear inhomoge-neous QLEs for the resonators in an array of N resonator-qubit pairs. Also similar to the single site case, the abovementioned approximations can be applied to integratethe resonators’ QLEs and to obtain the long-time steady-state relations between F ’s and S’s. After integration,the steady-state relations between F ’s and S’s are inturn substituted in the differential equations obtainedafter taking expectation of the QLEs of the qubits. Asthe resonators can be integrated out, the numerical com-plexity of the problem is determined only by the qubits.

D. The quasi-classical analysis

The models studied here can also be investigated us-ing a quasi-classical system of equations, which are ob-

tained by the classical limit approximation (〈b†2j bj〉 =

|β|2βj), where the operators are replaced by their aver-ages (〈bj〉 = βj) [41]. This approximation limits the setof equations in their first moment. For the direct-coupledqubits system, we obtain

βj =− (iδωqj + Γj)βj − 2iU |βj |2βj− 2iJx(βj−1 + βj+1)− iΩLδ1,j , (32)

where, Γj = 0 for 2 ≤ j ≤ N −1, Γ1 = ΓL and ΓN = ΓR.If N = 1, Γ = ΓL+ΓR. We use open boundary conditionfor the medium, i.e., β0 = βN+1 = 0.

The quasi-classical equations can be derived for theside-coupled qubits in the same manner. For the res-onator chain, let 〈fj〉 = αj and hence for N resonator-qubit pairs in the side-coupled model we get 2N quasi-classical equations which are

αj =− (iδωrj + Γj)αj − 2iJx(αj−1 + αj+1),

− igjβj − iΩLδ1,j (33)

βj =− iδωqjβj − 2iU |βj |2βj − igjαj , (34)

where, Γj = 0 for 2 ≤ j ≤ N −1, Γ1 = ΓL and ΓN = ΓR.If N = 1, Γ = ΓL+ΓR. Again we use the open boundarycondition, i.e. α0 = αN+1 = 0.

These equations are solved using standard numeri-cal differential equation solvers to obtain the long-timesteady-state solutions. At very low intensities when theinteraction and non-linearity in the system are not signif-icant, the QCA has a perfect agreement with the THLEresults. However, with increasing intensity, the QCA failsto capture any inelastic scattering effects as predicted bythe THLE analysis. The peaks in transmission profilecalculated using QCA starts shifting towards higher fre-quencies as the intensity is increased. This drastic failure

6

of QCA is possibly a reflection of the fact that the approx-imation is missing the inelastic scattering of the photonsdue to interaction. If we consider the Eq. 32 carefully werealize that the interaction term is contributing as extraoccupation dependent contribution to the oscillation fre-quency instead of a term that can cause a decrease inoscillation amplitude. The blockade phenomena shouldhave contributed the latter. In particular, for the QCA,there is always some frequency that has perfect trans-mission. In the case where input intensity is very high,the shift can be so large that the peak may go outsidethe range of input frequencies considered and it can bemistakenly interpreted as very low transmission causedby interaction.

As we increase the system sizes further in the THLEanalysis, the number of THLEs grows exponentiallywhich tends to reach the computational limit even af-ter large truncation. In contrast to THLE analysis, thequasi-classical equations can be solved very easily asthere are only N equations involved. The quasi-classicalequations increase linearly with system size N and henceenables us to analyze inelastic scattering for hundreds ofqubits.

The non-linearity in the QCA due to on-site interac-tion term can cause more than one steady-states, e.g.,a multi-stability in the system which is absent in thepresent THLE analysis with one-time variables. It isalso possible to find stable oscillating solutions for thesequasi-classical equations. These non-linear features ofthe quasi-classical equations appear at high intensities,while the QCA begins to fail even at much lower inten-

sity. We also carried out a self-consistent mean-field anal-ysis of the system, (see Appendix C), but it also fails tocapture the inelastic scattering at higher intensities. In-terestingly, both the approximate analyses, namely QCAand self-consistent mean-field, give the same incorrect re-sults for a wide range of parameters.

E. Modified quasi-classical equations

A careful analysis of the quasi-classical equations re-veals that the above mentioned shift in the transmissionprofile is due to the on-site interaction term (U). Weobserve that we need to incorporate inelastic scatteringof light and related loss of photons in the QCA. To high-light the fact that this inelastic scattering and the relatedphoton loss are caused due to interaction, we modify theinteraction parameter U to become a complex number(Ueff). We choose its value to match the THLE analysisresults for the single qubit case (details of the derivationare given in Appendix A) which gives

Ueff =Γ (Syδωq1 − Γβrss)

2S2yΩL

−iΓ2

(Sy − βiss

)2S2

yΩL, (35)

where

βrss =|βss|Sx√S2x + S2

y

βiss =|βss|Sy√S2x + S2

y

, (36)

and

Sx = −ΩL((δωq1 + 2U)2δωq1 + 12UΩ2

L + 9Γ2δωq1)

(δωq1 + 2U)2(Γ2 + δω2

q1

)+ 4(δωq1 + 2U)UΩ2

L + 12UΩ2Lδωq1 + 9Γ4 + 9Γ2δω2

q1

(37)

Sy = −ΓΩL

((δωq1 + 2U)2 + 9Γ2

)(δωq1 + 2U)2

(Γ2 + δω2

q1

)+ 4(δωq1 + 2U)UΩ2

L + 12UΩ2Lδωq1 + 9Γ4 + 9Γ2δω2

q1

. (38)

where Γ = ΓL + ΓR and |βss| =√

(βrss)2 + (βiss)

2.

For a single qubit, we find almost perfect agreementbetween THLE and the modified QCA from its incep-tion, apart from small deviations due to truncation er-ror. Quite remarkably, we find that the modified equa-tions work well even for longer chains. The advantage ofusing quasi-classical equations lies in its simplicity andcomputational efficiency, which enables us to study thetransmission for much longer system sizes which are com-putationally more challenging.

The correction for the on-site interaction can also becarried out for the more realistic side-coupled system.Again, for the side-coupled system, we try to replicatethe THLE results using the QCA and obtain a similarexpression for Ueff (see Appendix B for details). For a

single qubit-resonator pair, the complex on-site interac-tion also nicely captures the inelastic scattering of theincident photons and related photon loss.

As Ueff was calculated using the single site case for boththe media, it does not depend on Jx. Nevertheless, wefind a good agreement between the THLE and correctedQCA for greater system sizes as long as relatively low val-ues of Jx (< ΓL = ΓR) are used. However, an increasein Jx causes disagreements between the THLE and mod-ified QCA. Albeit, the corrected QCA gives quite usefulresults for a good range of parameters in both the media.

For the direct-coupled qubit array, within the limitedsystem sizes that we could explore using the THLE anal-ysis, we find very little system size dependence of theresonant transmission. This suggests that the inelastic

7

scattering and photon loss caused by on-site interactionsseem to be dominated at the two ends. This is also con-sistent with earlier studies on two level atoms in absenceof nearest neighbor interactions [40]. However, we end upcausing too much of inelastic scattering and photon lossusing a homogeneous complex on-site interaction every-where in the modified QCA. In fact, the resonant trans-mission for a homogeneous complex on-site interactiondecreases with increasing system size and eventually fallsto zero. This mismatch can be surmounted by apply-ing complex on-site interaction only at the two ends ofthe medium while using its real part in the bulk of thechain. With this fix, we obtain system size independenttransmission upto a chain of 300 qubits.

F. Transmission coefficient for the medium

To obtain the output transmission from the right endof the medium, we calculate the solutions of the cou-pled differential equations in the THLE analysis for themedium. The intensity of incoming light from the leftside of the medium is Iin (= E2

p/2πv2g). For the direct-

coupled qubit medium, the transmission coefficient, attime t, for a medium of N qubits is given as

TTHLE(t) =2ΓRIin〈b†NbN 〉, (39)

where 〈b†NbN 〉 is the number operator expectation valueat the output qubit. We can similarly obtain the QCAtransmission coefficient by calculating the expectationvalue of the qubit operator βN from the QCA (Eq. 32).The QCA transmission coefficient is given as

TQCA(t) =2ΓRIin

β∗N (t)βN (t). (40)

For the side-coupled qubits, the output photons arecoming out from the resonator at the rightmost end andnot from the qubit as for the direct-coupled qubit system.Hence, for a medium of N qubit-resonators pairs, thetransmission coefficient is given as

TTHLE(t) =2ΓRIin〈f†NfN 〉, (41)

where 〈f†NfN 〉 is the expectation value of the number op-erator of the rightmost resonator. It is found by substi-tuting the steady-state solutions of the qubits’ variablesin the integrated output resonator’s equation at steady-state. For example, for a single qubit-resonator pair, tofind the resonator number operator (F11), we multiplyEq. 26 with its conjugate, to obtain the final relation as

F11 =Ω2L + gΩL(S01 + S10) + g2S11

AA∗. (42)

Longer system sizes can also be dealt in a similar man-ner. Similar to Eq. 40, we can obtain the transmission

coefficient for the side-coupled array of qubits using thequasi-classical equations (Eq. 34) as

TQCA(t) =2ΓRIin

α∗N (t)αN (t). (43)

IV. RESULTS

A. Transmission through direct-coupled array ofqubits

For the direct-coupled array of N qubits, we calculatethe steady-state solutions using both THLE and QCA.We first set vg = 1 and and measure various parametersin units of qubit frequency ωa. We assume all qubits intheir ground state before irradiation with the photons,i.e. S(t → t0) = 0 and βj(t → t0) = 0. The steady-state solutions of Eqs. 39 and 40 are denoted as S∞ andβ∞i , which we use to obtain the steady-state transmissioncoefficients: T ∞THLE and T ∞QCA. For the modified QCA wereplace the on-site interaction U with Ueff in Eq. 40 andobtain the transmission coefficient T ∞MQCA.

We show in Fig. 2 the steady-state transmission for asingle qubit and a pair of two directly coupled qubits us-ing THLE, QCA and modified QCA analyses. For bothsingle qubit and two direct-coupled qubit mediums, allthe transmission results are calculated with maximumsix bosons per qubit (m = 6) for the THLE analysis.For a single qubit, we show the transmission profiles, i.e.T ∞THLE, T ∞QCA and T ∞MQCA versus scaled frequency ωp/ωafor different input intensities (Iin) in Fig. 2(a,d,g). At lowinput intensity, Fig. 2(a), the photon transport throughthe medium is coherent and we observe that the transmis-sion profile for a single qubit has a single peak with maxi-mum possible transmission at resonance. The three anal-yses, namely THLE, QCA and modified QCA, agree com-pletely with each other as well. However, as we increasethe intensity, Fig. 2(d,g), the on-site interaction (U) atthe qubits creates effective interactions between photonsmediated by the qubits’ excitations. Such effective inter-actions or optical nonlinearities cause photon blockaderesulting in loss of coherent transport of photons, whichis observed as decreasing of maximum transmission inthe THLE transmission profile. Instead of showing a de-crease in maximum transmission, the QCA only showsa shift in the transmission profile Fig. 2(d,g), while themodified QCA agrees with the THLE results.

The decrease in maximum transmission showing lossof coherence due to photon blockade, at resonant inputfrequency ωp = ωa, is more clear in Fig. 2(j), where weshow lowering of transmission with increasing intensity.The reduction of transmission at very high light intensi-ties is due to the saturation of qubits by photons, whichcompletely blocks any transmission of light to the otherend. This results in no interaction between the remain-ing photons. The observed decrease in the QCA trans-mission is due to the shifting of the transmission profile,

8

0.0

0.5

1.0

T∞

(a)N= 1

T∞THLE

(b)N= 2 Jx = 0.01ωa

T∞QCA

(c)N= 2 Jx = 0.02ωa

T∞MQCA

0.0

0.5

1.0

T∞

(d) (e) (f)

0.92 1.00 1.08ωp/ωa

0.0

0.3

0.6

T∞

(g)

0.92 1.00 1.08ωp/ωa

(h)

0.92 1.00 1.08ωp/ωa

(i)

10-6 10-1Iinvg/ωa

0.0

0.5

1.0

T∞

(j)10-6 10-1

Iinvg/ωa

(k)10-6 10-1

Iinvg/ωa

(l)

10-6

10-4

10-2

I inv g/ωa

(m)T∞QCA(n)T∞QCA

(o)T∞QCA

10-6

10-4

10-2

I inv g/ωa

(p)T∞THLE(q)T∞THLE

(r)T∞THLE

0.92 1.00 1.08ωp/ωa

10-6

10-4

10-2

I inv g/ωa

(s)T∞MQCA

0.92 1.00 1.08ωp/ωa

(t)T∞MQCA

0.92 1.00 1.08ωp/ωa

(u)T∞MQCA

0.0

0.2

0.4

0.6

0.8

1.0

FIG. 2. Single and two direct-coupled qubits steady-statetransmission results comparing between the THLE (T ∞THLE),quasi-classical (T ∞QCA) and modified QCA (T ∞MQCA) analyses.First column shows transmission results for a single qubitwhile the other two columns show transmission results fortwo direct-coupled qubits at different values of Jx = 0.01ωa

and Jx = 0.02ωa. The first three rows show the effect ofincreasing the input intensity Iin (in terms of ωa/vg) of pho-tons as Iin = 1.12 × 10−6(a,b,c), Iin = 1.5 × 10−4(d,e,f) andIin = 0.01(g,h,i). The fourth row shows transmission withincreasing Iin at ωp = ωa(j,k) and ωp = 0.965ωa(l). Thelast three rows show variation in transmission profile for arange of Iin for the QCA, THLE and modified QCA anal-yses respectively. Other parameters are: ωq1 = ωq2 = ωa,ΓL = ΓR = 0.02ωa, U = 1.05ωa.

and as mentioned before there is always some frequencywith perfect transmission in this analysis, Fig. 2(i). Weshow the systemic variation of the entire transmissionprofile with intensities in Fig. 2(m,p,s) for QCA, THLEand modified QCA respectively. We observe that in allthe single qubit transmission results, the QCA fails tomatch the THLE analysis, while the modified QCA hasperfect agreement with THLE results for a large range ofIin. This is not surprising as the Ueff was derived from theTHLE of the single qubit model itself, and just signifiesthat the truncation errors are not significant.

In the second and third columns of Fig. 2, we presentthe results for two directly coupled qubits at differ-ent inter qubit coupling strengths (Jx). In the secondcolumn, which has low inter qubit coupling strength(2Jx = ΓL = ΓR), we observe a single peak in the lowintensity transmission profile, which is, in-fact due to theoverlapping of the two peaks appearing at resonance forthe two qubits in the system, Fig. 2(b). In the last col-umn, we increase Jx (Jx = ΓL = ΓR), which shifts theresonance peaks away from each other and we observethe two peaks separately, Fig. 2(c). Again, both QCAand modified QCA analyses are able to capture all thedetails of THLE analysis at low intensity with perfec-tion when the system shows coherent transport. Sim-ilar to the single qubit case, a higher intensity causesthe photon blockade phenomenon resulting in loss of co-herence and decreased photon transport in the medium,Fig. 2(e,f,h,i). In contrast to the single qubit case, pho-ton blockade at high Iin causes the transmission profilefeatures in the THLE analysis for the two qubit mediumto change, Fig. 2(i), as the two peaks appearing at low in-tensities have now disappeared. We observe that THLEshows strong disagreements with QCA, while a reason-able agreement with modified QCA at high intensities.

We want to stress that Ueff is calculated by consideringonly single qubit and no additional fitting has been done,and still the modified QCA has captured the transmis-sion reduction due to photon blockade fairly accurately.However, there is larger disagreement between these twoanalyses as Jx is increased. Even with a few limitations,the modified QCA is able to capture the photon block-ade effects predicted by the THLE analysis reasonablywell for a large set of parameters. Transmission fromthe three analyses as function of Iin for two qubit modelat different Jx is shown in Fig. 2(k,l), where the trans-mission decreases with increasing Iin. We see a slightlylarger disagreement between THLE and modified QCAtransmission at higher Jx. The overall variation in trans-mission profile of the QCA, THLE and modified QCAwith intensity can be seen by comparing the last threerows of Fig. 2, which shows the shift in the QCA trans-mission similar to the single qubit and the effectiveness ofthe modified QCA in matching the THLE transmissionresults.

We further show a comparison of the transmission pro-files for the three analyses for upto five directly coupledqubits in Fig. 3. The three columns in Fig. 3 correspond

9

0.0

0.5

1.0T∞

(a)

N= 3

T∞THLE

(b)

N= 4

T∞QCA

(c)

N= 5

T∞MQCA

0.0

0.5

1.0

T∞

(d) (e) (f)

0.0

0.3

0.6

T∞

(g) (h) (i)

0.92 1.00 1.08ωp/ωa

0.00

0.06

0.12

T∞

(j)

0.92 1.00 1.08ωp/ωa

(k)

0.92 1.00 1.08ωp/ωa

(l)

10-6 10-1Iinvg/ωa

0.0

0.5

1.0

T∞

(m)

10-6 10-1Iinvg/ωa

(n)

10-6 10-1Iinvg/ωa

(o)

FIG. 3. Longer direct-coupled system steady-state transmis-sion for the THLE (T ∞THLE), quasi-classical (T ∞QCA) and mod-ified QCA (T ∞MQCA) analyses. The three columns show trans-mission profiles for system sizes (N) of 3, 4, and 5 qubitsrespectively. In the first four rows we show transmissionprofiles with input intensity Iin(in terms of ωa/vg) increas-ing on going down the columns as Iin = 1.12 × 10−6(a,b,c),Iin = 1.5 × 10−4(d,e,f), Iin = 0.01(g,h,i) and Iin = 0.1(j,k,l).The last row shows transmission with intensity at input fre-quency ωp = ωa for N = 3, 5(m,o) and ωp = 0.978ωa

for N = 4(n). The remaining parameters are, ωqj = ωa,ΓL = ΓR = 0.02ωa, Jx = 0.02ωa, U = 1.05ωa.

to increasing system sizes of three, four and five qubitsfor the direct-coupled qubits medium. In the THLE anal-ysis, the three, four and five direct-coupled qubits arrayhas truncation at m = 5, 3 and 2 respectively. The trun-cation is larger, i.e. m decreases, for larger system sizesdue to the computational capacity being reached. How-ever, due to large on-site interaction U , we expect thetruncation errors to be negligible though we cannot checkthis explicitly for the largest system size. In the first fourrows, we show the transmission profile with increasing Iindown the column. At low Iin, Fig. 3(a,b,c), we observecoherent transmission with the number of peaks in theTHLE transmission profile increase as resonance pointsincreases with the system size. As the Iin is increased,similar to the single and two direct-coupled qubits casein Fig. 2, the THLE transmission profile shows the effectof photon blockade as maximum transmission decreaseswith Iin, Fig. 3(d,e,f). On further increase in Iin, thetransmission further decreases and the peaks disappear

in the THLE analysis, Fig. 3(g,h,i).At low Iin, both QCA and modified QCA matches

the THLE transmission profile. However, at high Iin,the QCA again fails to show any agreement with THLEresults while the modified QCA captures the loss oftransmission and shows much better agreement with theTHLE analysis. Instead of showing the decrease in trans-mission due to photon blockade as well as saturationof qubits by photons, the QCA transmission profile hasagain shifted with increasing intensity, which is similar tothe shifts in transmission profile as seen in Fig. 2(d,e,f).The modified QCA, while still being fairly accurate,does start to show some disagreement with THLE inFig. 3(g,h,i). In the last row, we show that the mod-ified QCA has been reasonably successful in predictingthe photon blockade over the simple QCA by comparingthe reduction of transmission with increasing Iin. Thesudden spike in the QCA transmission is due to the peaksencountered during the shifting of the transmission pro-file. There are some minor spikes in the modified QCAtransmission for the same reason as well.

B. Transmission through side-coupled chain ofqubits

In the case of side-coupled system of qubits with theresonators, for THLE we solve the system of qubits equa-tions, with resonators equation integrated out, to obtainthe transmission from the right end of the medium. Sim-ilar to the direct-coupled qubits case, we assume all thequbits and resonators to be in their ground state and setvg = 1. We then use Eq. 41, 42, 43 to obtain the transmis-sion coefficients T ∞THLE, T ∞QCA and T ∞MQCA for the THLE,QCA and modified QCA respectively. Unlike the direct-coupled case, we observe a reduction of coherent trans-port due to effective photon-photon interactions only atintermediate intensities for the side-coupled system. Atvery large intensities, the side-coupled system starts tobehave more like a resonator chain without any qubits.This is because the qubits get saturated by the photonsand are no longer able to create effective photon-photoninteractions blocking the photon transport through theresonator, which results in recovery of large coherenttransport at a high intensity. Thus the side-coupled sys-tem shows a stark difference in light transmission at ahigh intensity in comparison to the direct-coupled latticeinvestigated above.

We show the transmission results for the medium ofone qubit-resonator pair coupled to the baths in Fig. 4.The two columns in the figure correspond to two dif-ferent qubit-resonator coupling strengths (g), which areg = 0.02ωa on the left and g = 0.05ωa on the right. Thebasic features of the side-coupled qubits’ transmissionprofile at low intensity can be seen in Fig. 4(a,b), wherewe observe two transmission peaks with maximum coher-ent photon transport separated by a large valley with aminima in transmission at ωp = ωa. We have chosen the

10

0.92 1.00 1.080.0

0.5

1.0

T∞(a)

g= 0.02ωa

T∞THLE T∞QCA

0.92 1.00 1.08

(b)g= 0.05ωa

T∞MQCA

0.0

0.5

1.0

T∞

(c) (d)

0.0

0.5

1.0

T∞

(e) (f)

0.92 1.00 1.08ωp/ωa

0.0

0.5

1.0

T∞

(g)

0.92 1.00 1.08ωp/ωa

(h)

10-6 10-1Iinvg/ωa

0.5

1.0

T∞

(i)

10-6 10-1Iinvg/ωa

(j)

10-6

10-4

10-2

I inv g/ωa

(k)

T∞ QC

A

(l)

T∞ QC

A

10-6

10-4

10-2

I inv g/ωa

(m)

T∞ TH

LE

(n)

T∞ TH

LE

0.92 1.00 1.08ωp/ωa

10-6

10-4

10-2

I inv g/ωa

(o)

T∞ MQ

CA

0.92 1.00 1.08ωp/ωa

(p)

T∞ MQ

CA

0.0

0.2

0.4

0.6

0.8

1.0

FIG. 4. Single qubit side-coupled to a single res-onator transmission results showing comparison betweenthe THLE (T ∞THLE), quasi-classical (T ∞QCA) and modifiedQCA (T ∞MQCA) analyses. The first four rows show the ef-fect of increasing input intensity Iin (in terms of ωa/vg) asIin = 1.12× 10−6(a,b), Iin = 7.1× 10−4(c,d), Iin = 0.034(e,f)and Iin = 0.68(g,h) at two different resonator-qubit couplingstrengths of g = 0.02ωa and g = 0.05ωa in the two columns re-spectively. Change in transmission on increasing Iin is shownin (i,j) at, ωp = 0.98ωa(i) and ωp = 0.95ωa(j). Variation ofthe entire transmission profile with increasing Iin is shownin the last three rows for QCA, THLE and modified QCArespectively. The other parameters are: ωr1 = ωq1 = ωa,ΓL = ΓR = 0.02ωa and U = 1.05ωa.

resonator (ωr1) and qubit (ωq1) frequencies to be equalin Fig. 4, however, when resonator and qubit frequenciesare not equal, the minima in transmission occurs when-ever ωp is equal to the qubit frequency. The minima intransmission is seen when the qubit absorbs the incom-ing photons and we see a reduced photon transport. Theeffect of increasing g, at low intensity, can be seen inFig. 4(b) where the two peaks have now moved furtherapart and the dip between them has become broader,which suggests an increased influence of the qubit in themedium. At low intensity, both QCA and modified QCAagree to the THLE results perfectly as in the case ofdirect-coupled qubits described earlier.

In the first four rows, Iin gradually increases on go-ing down the columns. The THLE analysis shows thatthe increase in intensity initially causes disruption of co-herent photon transport and the transmission peaks tolower down, Fig. 4(c)-(h), which is similar to lowering oftransmission observed in the direct-coupled qubits sys-tem. The reduction in transmission is again due to theeffective photon-photon interactions at higher intensitiesmediated by the on-site interaction present in the qubits.Such effective interactions block the propagation of light.We observe that higher values of g in the second columncauses a greater loss of transmission as the influence ofthe qubits on generating the effective photon-photon in-teractions has increased, Fig. 4(e,f). We see that withincrease in Iin, the valley between the peaks rises gradu-ally until it supersedes the lowering of transmission. Atthis moment, the peaks disappear and we observe maxi-mum photon blockade and loss of transmission. At evenhigher intensity, Fig. 4(g,h), the rise of transmission atωp = ωa continues and the transmission profile starts tolook a bit like the one for a single resonator. This occursas the qubit gets saturated by photons and most incom-ing photons pass through the resonator and transmittedwith a little influence of the qubit. Thus, there are nolonger effective photon-photon interactions and relatedphoton blockade in transmission at very high intensities.

In the Fig. 4(c)-(f), we also observe that the QCAfails to match the transmission profile with increasingintensity. We observe an overall shift in the QCA trans-mission profile where the right half of the profile showsshifting as well as lowering of transmission and the leftpart moves towards the center. Eventually at high in-tensity, the QCA transmission profile becomes similar tothat of the resonator as the qubit saturates. The modi-fied QCA, however, shows perfect agreement with THLEresults for all Iin values which is, again, not surprising asthe correction (Ueff) was derived from the THLE for sin-gle resonator-qubit pair with enough truncation to cap-ture the photon blockade and saturation phenomena.

We show the variation of transmission with intensityin Fig. 4(i,j). We see that the QCA has matched the pho-ton blockade of transmission as predicted by the THLE tosome extent, but the modified QCA results have matchedthe THLE results perfectly. We must note that any low-ering of transmission in QCA is due to shifting of trans-

11

mission profile and not due to an effective photon-photoninteractions. These results shows that the QCA is ac-curate for some parameters in the side-coupled systembut not for all parameters, in particular when the co-herent transmission is low. On the other hand, for allparameters, the modified QCA has shown perfect agree-ment. This is further confirmed by the variation of thetransmission profile with the intensity as shown in thelast three rows of Fig. 4 for QCA, THLE and modifiedQCA respectively. We observe that the QCA transmis-sion profile is shifting towards right with increasing in-tensity, Fig. 4(k,l) and shows little agreement with theTHLE results at intermediate intensities. The modifiedQCA, however, has perfectly captured all the featurespredicted by the THLE analysis, Fig 4(m,n,o,p).

We show the transmission results for a medium of twoqubits coupled to a chain of two resonators at differentinter-resonator coupling strengths (Jx) and resonator-qubit coupling g in Fig. 5. Each column has constant Jx,where the first two columns have low Jx(= ΓL/2 = ΓR/2)and the last column has high Jx(= ΓL = ΓR). Eachcolumn also has a constant g with low g(= 0.02ωa) inthe first column and high g(= 0.05ωa) in the last twocolumns. At low intensity and Jx, the transmission pro-file from the THLE analysis shows coherent photon trans-port with two peaks separated by a valley with no photontransport at ωp = ωa, Fig. 5(a,b). Each individual peakis in-fact made of two peaks merged together. Increasingg causes the resonance peaks to move away from eachother, Fig. 5(a,b). The individual peak splits, similarto the direct-coupled model, as we increase Jx which re-sults into four peaks in the transmission profile with notransmission at ωp = ωa, Fig. 5(c).

The input intensity (Iin) is increased as we go downthe column in the first four rows. In all cases, from theTHLE analysis we observe disruption of coherent photontransport with increase in Iin due to effective photon-photon interaction along with gradual increase in trans-mission at ωp = ωa. At low Jx, photon-photon inter-action causes lowering of transmission peaks, similar tothe single resonator qubit pair case in Fig. 4, where thetransmission keeps decreasing till intermediate intensi-ties, Fig. 5(d,e,g,h). At even higher intensity, the qubitsstarts to get saturated by photons and a large coherentphoton transport is restored, Fig. 5(j,k). At higher Jx,we observe from the THLE analysis that out of four peaksthe ones near ωp = ωa lose transmission first as we in-crease Iin as shown in Fig. 5(f). With further increasein Iin, the effective photon-photon interactions and re-lated photon blockade increase, and their effect is seen inthe entire transmission profile, Fig. 5(i). At even higherIin, the qubits start to get saturated by photons withlarge coherent photon transport getting restored which,in the case of two resonators, shows two peaks as seenin Fig. 5(l). At very high intensities we essentially ob-serve transmission through a harmonic chain with someeffects from the qubits. In this limit the qubits can bethought of as additional baths, and the photon transport

is dominated by the resonator chain.

The QCA fails to match the transmission profile ofthe THLE analysis with increasing Iin and is unable tocapture the photon blockade phenomenon (as well as sat-uration of qubits) for most parameters. The QCA trans-mission profile instead shifts towards increasing ωp whencompared to the THLE results, Fig. 5(d,e,f). On theother hand, the modified QCA captures the loss of co-herent photon transport and other features predicted bythe THLE fairly accurately at low Jx. With increasedJx in the third column, the modified QCA is still ableto capture the features of the transmission profile butshows lesser agreement with THLE analysis in predict-ing the loss of coherent transmission.

The overall effect of Iin on transmission is shown inFig. 5(m,n,o), which shows loss of coherent transmis-sion peak due to effective photon-photon interactions forthe three analyses at various parameters. At low Jx,Fig. 5(m,n), loss of coherent transmission predicted byTHLE is captured fairly accurately by the modified QCA.The QCA does not capture the loss of transmission ac-curately in Fig. 5(m) but shows agreement at some inter-mediate intensities with the THLE analysis in Fig. 5(n).We must note that loss of transmission in QCA is due toshifting of transmission profile and not because of pho-ton blockade. At higher Jx, Fig. 5(o), modified QCAshows less accuracy in matching the lowering of transmis-sion predicted by THLE analysis while simple QCA haseven larger disagreement with THLE. We show the en-tire transmission profile with varying Iin in the last threerows of Fig. 5 for the QCA, THLE and modified QCA re-spectively. We can observe that for both values of Jx theQCA fails to match the features of THLE analysis resultsfor a large range of parameters. The modified QCA, how-ever, is able to capture most of the transmission featuresas predicted by THLE analysis. The agreement betweenthese two analysis is good at low Jx, however, we alsoobserve that the modified QCA becomes less accurate asJx is increased.

We show the effects of increasing system size in theside-coupled qubits medium in Fig 6 where the left andright columns are for three and four qubits respectively.The first four rows correspond to increasing Iin in everycolumn. Similar to Fig. 3, the THLE truncation is in-creased for two system sizes in each column with m = 4and m = 2 respectively and large U ensures the stabil-ity of transmission results. At low intensity, the side-coupled qubits medium shows coherent photon transportwith two major transmission peaks each having threesmaller peaks for N = 3 near maximum transmission,Fig. 6(a). Larger system sizes have more peaks, however,in Fig. 6(b) for a medium of four side-coupled qubits themiddle two peaks are merged and we can only observethree peaks distinctly in each major peak. Increasing res-onant peaks with system size can be observed distinctlyat higher Jx. When we increase Iin for each system size aswe go down the column, the THLE analysis shows loss ofcoherent photon transport as transmission decreases due

12

0.0

0.5

1.0

T∞

(a)Jx = 0.01ωa, g= 0.02ωa

T∞THLE

(b)Jx = 0.01ωa, g= 0.05ωa

T∞QCA

(c)Jx = 0.02ωa, g= 0.05ωa

T∞MQCA

0.0

0.5

1.0

T∞

(d) (e) (f)

0.0

0.5

1.0

T∞

(g) (h) (i)

0.92 1.00 1.08ωp/ωa

0.0

0.5

1.0

T∞

(j)

0.92 1.00 1.08ωp/ωa

(k)

0.92 1.00 1.08ωp/ωa

(l)

10-6 10-1Iinvg/ωa

0.0

0.5

1.0

T∞

(m)10-6 10-1

Iinvg/ωa

(n)10-6 10-1

Iinvg/ωa

(o)

10-6

10-4

10-2

I inv g/ωa

(p)

T∞ QC

A

(q) (r)

10-6

10-4

10-2

I inv g/ωa

(s)

T∞ TH

LE

(t) (u)

0.92 1.00 1.08ωp/ωa

10-6

10-4

10-2

I inv g/ωa

(v)

T∞ MQ

CA

0.92 1.00 1.08ωp/ωa

(w)

0.92 1.00 1.08ωp/ωa

(x)

0.0

0.2

0.4

0.6

0.8

1.0

FIG. 5. Two side-coupled qubits transmission results showingthe effects of inter-resonator coupling (Jx) as well as a com-parison between the THLE (T ∞THLE), quasi-classical (T ∞QCA)and modified QCA (T ∞MQCA) analyses. The three columnshave (Jx/ωa, g/ωa) = (0.01, 0.02), (0.01, 0.05) and (0.02, 0.05)respectively. The first four rows show the effect of in-creasing input intensity Iin (in terms of ωa/vg) as Iin =1.12 × 10−6(a,b,c), Iin = 7.1 × 10−4(d,e,f), Iin = 0.034(g,h,i)and Iin = 0.68(j,k,l). Transmission with increasing Iin isshown in (m,n,o) at ωp = 0.98ωa(m), ωp = 0.95ωa(n) andωp = 0.965ωa(o). Variation of the entire transmission pro-file with increasing Iin is shown in the last three rows eachfor QCA, THLE and modified QCA respectively. The otherparameters are: ωrj = ωqj = ωa, ΓL = ΓR = 0.02ωa andU = 1.05ωa.

0.0

0.5

1.0

T∞

N=3 (a)

T∞THLE T∞QCA

N=4 (b)

T∞MQCA

0.0

0.5

1.0

T∞

(c) (d)

0.0

0.5

1.0

T∞

(e) (f)

0.94 1.00 1.06ωp/ωa

0.0

0.5

1.0

T∞

(g)

0.94 1.00 1.06ωp/ωa

(h)

10-6 10-1Iinvg/ωa

0.0

0.5

1.0

T∞

(i)10-6 10-1

Iinvg/ωa

(j)

FIG. 6. Longer side-coupled system steady-state transmissionfor the THLE (T ∞THLE), quasi-classical (T ∞QCA) and modifiedQCA (T ∞MQCA) analyses. The two columns show results forsystem sizes (N) of 3 and 4 qubits respectively. The firstfour rows show transmission profiles with input intensity Iin(in terms of ωa/vg) increasing on going down the columns asIin = 1.12× 10−6(a,b), Iin = 7.1× 10−4(c,d), Iin = 0.034(e,f)and Iin = 0.135(g,h). The last row shows transmission withincreasing intensity for the corresponding system sizes at ωp =0.9872ωa(i) and ωp = 0.9896ωa(j). The remaining parametersare, ωrj = ωqj = ωa, ΓL = ΓR = 0.02ωa, Jx = 0.01ωa,g = 0.02ωa, U = 1.05ωa.

to the effective photon-photon interactions resulting intophoton blockade at intermediate Iin. Similar to Fig. 3,the peaks closer to the center show the loss of trans-mission first. At even higher intensity, Fig. 6(g,h), thequbits start to get saturated by photons, which resultsin a revival of coherent photon transport in the medium,and transmission profile becoming more dominated bythe resonator chain.

Both QCA and modified QCA show perfect agreementwith the THLE analysis at low Iin and capture all thefeatures for various parameters of the model, Fig. 6(a,b).However, the QCA fails to match the THLE transmissionprofile at intermediate Iin, Fig. 6(c,d), but have someagreement with THLE at very high Iin where qubit sat-uration has started to occur. At intermediate Iin, thetransmission profile of QCA shifts towards the resonantinput frequency and we observe lowering of transmissionpeaks in the right half of the transmission profile. Wewould like to note here that for some parameters the

13

0.0

0.5

1.0

T∞ TH

LE

(a)N= 3Homogeneous

(b)N= 4Inhomogeneous

0.0

0.3

0.6

T∞ TH

LE

(c) (d)

0.92 1.00 1.08ωp/ωa

0.0

0.5

1.0

T∞ TH

LE

(e)

0.92 1.00 1.08ωp/ωa

(f)

10-6 10-1Iinvg/ωa

0.0

0.5

1.0

T∞ TH

LE

(g)10-6 10-1

Iinvg/ωa

(h)

FIG. 7. Inhomogeneous side-coupled systems THLE trans-mission profile for system sizes N = 3, 4 in the left and rightcolumns respectively. The first three rows show transmissionprofiles with input intensity Iin (in terms of ωa/vg) increas-ing on going down the columns as Iin = 1.12 × 10−6(a,b),Iin = 9.4 × 10−3(c), Iin = 0.047(d) and Iin = 0.68(e,f).The last row shows transmission with Iin, for homogeneousand inhomogeneous qubits respectively, at ωp = 0.988ωa and0.992ωa for N = 3(g), and at ωp = 1.01ωa for N = 4(h). Forthe homogeneous medium, both system sizes have gj = 0.02ωa

and ωqj = ωa. For the inhomogeneous system N = 3 hasgj/ωa = (0.04, 0.01, 0.04), ωqj/ωa = (0.95, 1.0, 1.06) whileN = 4 has, gj/ωa = (0.05, 0.007, 0.02, 0.04), ωqj/ωa =(0.92, 1.0, 1.06, 1.08). The remaining parameters are, ωrj =ωa, ΓL = ΓR = 0.02ωa, Jx = 0.01ωa and U = 1.05ωa.

QCA starts showing oscillating behavior, in which casewe have shown the time average of the transmission. Themodified QCA, on the other hand, continues to showgood agreement as we increase the intensity till we reachintermediate intensities where the transmission loss dueto photon blockade is highest and where it also showssome deviation though not as much as QCA. At evenhigher Iin, with saturation of qubits, the modified QCAagain matches the THLE transmission profile with goodaccuracy.

We show the transmission from the three analyseswith increasing Iin at one of the intermediate peaks(ωp = 0.95ωa) in the last row of Fig. 6. We observe in-teresting non-monotonic behavior similar to Fig. 5(m,o)showing a decrease in coherent transmission due to ef-fective photon-photon interaction and then a revival ofcoherent transport through the resonator chain due tosaturation of qubits by photons. The QCA fails to pre-dict the loss of coherent transmission over a wide rangeof intensity, while the modified QCA has shown goodagreement with THLE analysis in predicting this loss.

Finally, in Fig. 7, we consider photon transport

through an inhomogeneous medium of side-coupledqubits, where we introduce inhomogeneity in qubit fre-quencies (δωqj ) and the qubit-resonator couplings (gj)[11, 18, 43]. For an inhomogeneous or disordered system,the coherent quantum behavior itself has the possibilityof showing reduced transmission due to Anderson local-ization at low intensity as seen in Fig. 7(a,b). As Iin isincreased there is an initial loss of quantum coherence dueto effective photon-photon interactions even for the inho-mogeneous system, 7(c,d). However, at high Iin, coherentphoton transport re-emerges as qubits get saturated byphotons and qubits have less effect on photons transitingthrough the resonators, Fig. 7(e,f). We observe that thetransmission profile for both homogeneous and inhomo-geneous qubits is very similar at high Iin as the resonatorsin the medium are all identical. The small difference athigh Iin is due to non-identical qubits being at differ-ent saturation levels. The non-monotonic loss and riseof transmission is observed in case of both homogeneousand inhomogeneous qubits as shown in Fig. 7(i,j). A veryinteresting effect is seen for the inhomogeneous system,namely, the effective photon-photon interactions end upresulting in a higher transmission than its coherent trans-port, which is lowered due to Anderson localization.

SUMMARY AND OUTLOOK

In this paper, we have explored transmission proper-ties of photons through lattices of direct and side-coupledqubits, where the qubits have on-site interaction. Withthe increase in intensity of incoming photons in a coher-ent state, both lattices develop effective photon-photoninteractions and related photon blockade mediated bythe on-site interaction at the qubits. For the direct-coupled qubits, at single photon limit the transmissionline-shape shows resonance peaks equal to the number ofqubits in the system. With an increase in intensity thephoton blockade occurs, which causes lowering of reso-nance transmission peaks until negligible transmission isobserved at a very high intensity when the qubits get sat-urated by photons. The side-coupled qubits, at low inten-sity, also show resonance peaks with maximum transmis-sion and a transmission minima at the qubit frequency.With an increase in intensity, the resonance peaks low-ers due to photon blockade along with rise in transmis-sion at the qubit frequency. At a very high intensity thequbits get saturated and their effects start to disappearresulting in a revival of coherent transport in the lat-tice of side-coupled qubits. A non-monotonic behaviorof transmission with increase in Iin is observed for bothhomogeneous and inhomogeneous side-coupled qubits.

We also performed a quasi-classical analysis for thetwo lattices, which completely fails to show any trans-mission loss due to photon blockade. For both lattices,the QCA shows agreement with THLE at single particlelimit but fails at higher intensities and shows a shift inthe QCA transmission profile. One of our main contri-

14

butions in this paper is to modify the QCA equationsby introducing a complex on-site interaction by match-ing the THLE and QCA transmissions for the single sitecase. The modified QCA is effective in capturing effectivephoton-photon interactions for a wide range of parame-ters even at larger system sizes. However, with increasein Jx, the effectiveness of modified QCA reduces. Wehope that further studies will provide more insight intoimproving this approach to work for an even wider set ofparameters.

The waveguide QED lattices explored in our work arecomplementary to an earlier experimental work in cir-cuit QED lattices [8], but we believe that the trans-port features observed by us are within the reach of re-cent experimental developments. Control over internallosses in the bulk of the lattice poses experimental chal-lenges, and the current theoretical study can readily beextended to incorporate some intrinsic losses from thequbits and resonators. It would also be useful to exam-ine the effects of inhomogeneity and optical nonlinearitysystematically and explore the possibility of many-bodylocalization [18, 43–46] in these systems. On the analysisside, there is room for further improvement with a moresystematic way of introducing the corrections such thatthe modified quasi-classical equations are applicable to amuch broader range of parameters. Perhaps some renor-malization procedure can also be developed that consid-ers one qubit at a time with more variables and the restof the system as renormalized parameters.

ACKNOWLEDGEMENTS

R.S. and D.R. acknowledge funding from the Depart-ment of Science and Technology, India, via the Ramanu-jan Fellowship. D.R. also acknowledges funding fromthe Ministry of Electronics & Information Technology(MeitY), India under grant for “Centre for Excellence inQuantum Technologies” with Ref. No. 4(7)/2020-ITEA.

Appendix A: Complex on-site interaction fordirect-coupled qubits

As mentioned in the main text, the quasi-classicalequations can be modified to match the THLE analy-sis by replacing the on-site interaction strength (U) witha complex on-site interaction (Ueff). To calculate Ueff forthe direct-coupled qubits, we consider a truncated sys-tem of five operators ST =

[S01 S01 S11 S12 S12

]and

calculate their THLE which are given by Eqs. 17, 19, 20and the conjugates of Eqs. 17, 20.

By solving Eqs. 19 and 20, we obtain the steady-statevalues S∞11 and S∞01 in terms of S∞01

S∞11 =iΩL

(S∞01 − S∞01

)2Γ

, (A1)

S∞12 =Ω2L

(S∞01 − S∞01

)Γ (i(δω + 2U) + 3Γ)

, (A2)

where, Γ = ΓL + ΓR. The steady-state solutions aresubstituted in Eq. 17 at steady-state which gives

S∞01 (−Γ− iδω)−2iUΩ2

L

(S∞01 − S∞01

)Γ (i(δω + 2U) + 3Γ)

− iΩL = 0, (A3)

The Eq. A3 can be re-written in real and imaginary partsof S∞01 by substituting S∞01 = Sx + iSy which gives twoequations for the two real variables. We solve these twoequations to obtain steady-state values of Sx and Sy asEqs. 37 and 38.

For comparison with the QCA at steady-state, thequasi-classical equation for the medium of single qubitis given as

− 2iUβ2ssβss − iΩL + βss (−Γ− iδω) = 0. (A4)

In order to match the transmissions for the THLE steady-state obtained above with the QCA steady-state, we ob-tain the magnitude of βss by demanding that the trans-mission obtained in the two calculations are the same,which gives

|βss|2 = S∞11 = −SyΩLΓ

. (A5)

Comparing transmission, however, does not give us thephase of βss but we choose it to be the same as that ofS∞01 which yields the Eq. 36. Finally, we re-define the Uas Ueff and re-arrange Eq. A4 to obtain

Ueff =−ΩL + βss (iΓ− δω)

2β2ssβss

. (A6)

With this choice of Ueff , the QCA steady-state analysisagrees with the THLE steady-state analysis for a singlesite by construction except for truncation errors.

Appendix B: Complex on-site interaction forside-coupled qubits

To calculate the complex on-site interaction (Ueff) forthe side-coupled qubits medium, we match the transmis-sion coefficients of the THLE and the QCA for a singleresonator-qubit pair. To calculate the steady-state solu-tions of the THLE, we consider a truncated set of theoperators ST =

[S01 S01 S11 S12 S12 S02 S02

]. The

THLEs for these operators are given by Eqs. 27-30 andtheir conjugates. In contrast to direct-coupled qubitscase, we require a larger operator set to calculate Ueff

in order to perfectly match the modified QCA resultswith those from the THLE analysis.

After solving the set of seven coupled THLEs, we ob-tain the expected steady state solution for S∞01 in termsof S∞11 as

S∞01 = E1 + E2S∞11 (B1)

15

and the expectation value of S∞11 is given as

S∞11 =−ΩL(A∗E∗1 +AE1)

g(A+A∗) + ΩL(AE2 +A∗E∗2 )(B2)

where,

E1 =−gΩL(iδωq1AA

∗ + 2g2A∗ + gA+ 2iUAA∗)(iδωq1A+ g2 + iUA)(iδωq1A+ g2)AA∗

(iδωq1A+ g2)(

(iδωq1AA∗ + 2g2A∗ + gA+ 2iUAA∗)(iδωq1A+ g2 + iUA)(iδωq1A+ g2) + 2iUg2Ω2

LA2)(B3)

E2 =4iUgΩL(iδωq1A+ g2 + iUA)A∗

(iδωq1AA∗ + 2g2A∗ + gA+ 2iUAA∗)(iδωq1A+ g2 + iUA)(iδωq1A+ g2)AA∗ + 2iUg2Ω2

LA

(B4)

where, A = iδωr1 + Γ and Γ = ΓL + ΓR. By integratingout the resonators we have obtained the expected valueF∞01 in Eq. 26

Finally, we match the steady state THLE operatorswith those from the QCA from Eqs. 33-34 by demandingα∞1 = F∞01 and β∞1 = S∞01 at steady-state. We replaceU with Ueff in the quasi-classical equations and solve forUeff which is given by

Ueff = −δωq1S∞01 + gF∞01

2|S∞01 |2S∞01

. (B5)

Appendix C: The self-consistent mean-field analysis

The propagation of photons through the medium canalso be studied using the mean-field approximation ap-plied on the on-site interaction term in the Hamiltonian.In the case of direct-coupled qubits, after applying themean-field approximation the Hamiltonian the mediumbecomes

Hmfdirect =~

N∑i=1

ωqib†i bi + Ub†i bi(〈b

†i bi〉 − 1)

+ ~N−1∑i=1

2Jx(b†i bi+1 + b†i+1bi). (C1)

The bath and bath coupling terms are also included inthe complete Hamiltonian of the medium. To obtain themean-field equations for the system, we derive a set ofN equations by using the Heisenberg equations for theoperators bj for each site given by

bj =− (iδωqj + Γj)bj − iU(〈b†jbj〉 − 1)bj

− 2iJx(bj−1 + bj+1)− iΩLδ1,j , (C2)

where, Γj = 0 for 2 ≤ j ≤ N −1, Γ1 = ΓL and ΓN = ΓR.If N = 1, Γ = ΓL + ΓR. We use open boundary con-dition for the medium, i.e. b0 = bN+1 = 0. Similarto the quasi-classical equations, the mean-field equationshas no dependence on other excited states and, therefore,N mean-field equations have all the necessary informa-tion to model the medium.

By comparing the Eq. C2 with the quasi-classical equa-tions, we observe that the on-site interaction term inmean-field equations has missed a factor of 2 and hasan additional -1 as well. Therefore, we put the factor of2 in the on-site interaction and remove -1 explicitly inthe numerical calculations.

Each of the mean-field equations has number opera-

tor (〈b†i bi〉) in the RHS. Therefore to solve these equa-tions at steady-state, we multiply each of the N equa-tions with its conjugate to derive a set of coupled equa-tions for number operators at each site. For example, weget the following self-consistent mean-field equations fora medium two direct-coupled qubits

〈b†1b1〉 =|δω2 + 2U〈b†2b2〉 − iΓR|2Ω2

L

|4J2x − (δω1 + 2U〈b†1b1〉 − iΓL)(δω2 + 2U〈b†2b2〉 − iΓR)|2

, (C3)

〈b†2b2〉 =4J2xΩ2

L

|4J2x − (δω1 + 2U〈b†1b1〉 − iΓL)(δω2 + 2U〈b†2b2〉 − iΓR)|2

. (C4)

In order to get numerical solutions of these equations we run a self-consistent loop over the set of equations

16

for a large number of iterations and calculate the outputtransmission using the formula

T ∞mf =2ΓRIin〈b†NbN 〉. (C5)

As mentioned in the main text, the self-consistentmean-field after making the corrections mentioned aboveagrees with QCA for a wide range of parameters.

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