32nd URSI GASS, Montreal, 19–26 August 2017

Single and multiple atomic dipole radiators

Aiyin Y. Liu(1), Tian Xia(1), Lingling Meng(1) and Weng C. Chew(1)

(1) Department of Electrical and Computer Engineering, University of Illinois at Urbana Champaign, Illinois, USA

Abstract

The radiation from dipole atoms situated in a general loss-less electromagnetic environment is studied in a fully quan-tized model by direct factorization of the Hamiltonian. Thefactorized Hamiltonian feature coupled excitations of theatoms and fields.

1 Introduction

The interaction between atoms and quantized electromag-netic fields, or photons, have long attracted much attention[1]. Both theorists and experimentalists have long marveledat the rich physics hidden in these simple models [2–9]. Re-cent advances in artificial atoms, such as superconductingquantum circuits [10] or semiconductor quantum dots [11],seem to have paved the way toward manipulating manyatoms and photons for applications in quantum informationprocessing or quantum computing [12]. Quantum coher-ence and the cooperative effects among the atoms as well asbetween atoms and photons are essential to such systems.Rigorous modeling of these effects calls for incorporatingmore complex and realistic electromagnetic environmentsin the model [13]. Computational electromagnetics (CEM)have proved useful for such a task in many a different fieldin the past.

In this paper, our recent work on utilizing CEM, in par-ticular the dyadic Green’s function (DGF), in studying thecooperative effects between atoms and photons is summa-rized [9, 13–15]. Section 2 introduces the dipole Hamilto-nian. Section 3 outlines the procedure of direct factoriza-tion to solve the quantum dynamics using the DGF. Section4 presents some preliminary results and discussions.

2 The dipole interaction Hamiltonian

The dipole coupling Hamiltonian for a collection of atomscoupled to a lossless electromagnetic system is [5–7, 9]:

H = ∑i

ωi b†i bi +∑

kωk a†

k ak +∑i,k

[gi,k b†

i ak +g∗i,k a†k bi

](1)

The first two terms in (1) are the free atom and free fieldcontributions, respectively. If the atoms are independentharmonic oscillators, then the atom creation/annihilation

operators b†j and bi obey the same type of commutation re-

lations as their field counterparts [3].

[bi, b†j ] = δi, j [ak, a

†k′ ] = δk,k′ (2)

The third term in (1) represents the dipole interaction,whose strength is given by [3, 6, 9]:

gi,k =

√ωk

2εEk(ri) ·di (3)

Fock states of the total system takes the form [9]:

|Ψ〉= ∏i

b†nii√ni!|gi〉⊗∏

k

a†nkk√nk!|0〉 (4)

where the n’s are excitation numbers of the atoms or fieldmodes. The rotating wave approximation (RWA) [3] is ap-plied in (1). Hence, dynamics governed by (1) only coupleFock states with the same total excitation number [8, 9].When said number is 1, projection of (1) onto the corre-sponding Fock states in (4) gives the Hamiltonian in thesingle excitation subspace [9].

H1 = ∑i

ωi|ei〉〈ei|+∑k

ωk|1k〉〈1k|

+∑i,k

[gi,k |ei〉〈1k|+g∗i,k |1k〉〈ei|

](5)

This Hamiltonian applies to the case where the atoms areindependent two-level systems and is the starting point ofmany discussions in the literature [2, 6, 7]. Assuming theelectromagnetic environment contain M modes and there isa total of Q atoms, we readily write down an explicit matrixconstruction of (5) if the modes of the EM environment isknown.

H1 =

[HA,1 gg† HF,1

](6)

This is a (Q+M)× (Q+M) Hermitian matrix. Numeri-cal diagonalization of (6) will in fact reveal all the eigen-energies of (1), not restricted to the single excitation sub-space. However, this is not practical for general EM envi-ronments. We next turn to a Green’s function based method.

3 Dressed atom fields and dressed states

The factorized Hamiltonian of (1) has the form [16]:

H = ∑ν

ων c†νcν (7)

The operators cν and c†ν

have expansion [2, 4, 9, 16]:

cν = ∑i

Ai(ν) bi +∑k

F(ν ,k) ak (8)

c†ν= ∑

iA∗i (ν) b†

i +∑k

F∗(ν ,k) a†k (9)

These are coupled oscillations of the atoms and fields,which we force to obey:

[cν , c†ν ′ ] = δν ,ν ′ (10)

Equation (10) imposes on the expansion coefficients:

∑i

Ai(ν)A∗i (ν′)+∑

kF(ν ,k)F∗(ν ′,k) = δν ,ν ′ (11)

Using (10) and (7) we have:

[cν , H] = ων cν (12)

These are therefore the real excitations of the system. Writ-ten in terms of the expansion coefficients, (12) further im-poses that:

Ai(ν) [ων −ωi] = ∑k

F(ν ,k)g∗i,k (13)

F(ν ,k) [ων −ωk] = ∑i

Ai(ν)gi,k (14)

Solving (11) with (13) and (14) will supply the coefficientsin (8) as well as the new eigen-energies in (7).

If the original field spectrum has degeneracy W > Q at aparticular frequency ω , there may be field excitations un-coupled to the atoms. The coefficients of such excitationscan be determined from the atomic coupling constants. Thenumber of such excitations at ω is W −Q′, where Q′ givesthe number of atoms with linearly independent couplingconstants in this subset of degenerate modes [6, 9].

Excitations coupled to the atom are solved as an impliciteigenvalue problem, in which the roots of a matrix determi-nant supply the eigen-energies.

det[M1(ων)

]= 0, M2(ων) ·A = ων A (15)

The matrix element of M1(ων ) is:

M1 i, j = G∗i, j(ων)−δi, j (ων −ωi) (16)

where:

Gi, j(ων) = ∑k

gi,k g∗j,kων −ωk

(17)

The vector A in the second part of (15) determines theatomic expansion coefficients for a particular excitationwith ων up to an arbitrary scaling factor. We use (11) tonormalize them against a matrix M3(ων).

A† ·M3(ων) ·A = 1 (18)

M3 i, j = δi, j−∂ωνG∗i, j(ων) (19)

From the order of the determinant in (15) it can be shownthat the total number of solutions, including excitationscoupled and uncoupled to the atoms, shall be W +Q. Whenthe electromagnetic spectrum features a continuum (or sev-eral continua), such accounting is no longer possible. Thedenominator in (17) needs to be treated as a distribution[2, 4, 9, 16]:

1ων −ωk

→ P.V.1

ων −ωk+Z(ων)δ (ων −ωk) (20)

where Z(ων) is to be determined by the first part of (15),in place of ων . It will be convenient to define the principalvalue and singular parts of Gi, j(ων):

∆i, j(ω) = ∑k

P.V.gi,k g∗j,kω−ωk

(21)

Γi, j(ω) = ∑k

gi,k g∗j,k δ (ω−ωk) (22)

These are the cooperative radiative shift and decay rates ofthe atoms, respectively. Equations (16) and (18) become:

M2 i, j = ∆∗i, j(ων)+Z(ων)Γ

∗i, j(ων)+δi, j ωi (23)

A† ·Γ∗ ·A =1

π2 +Z2(ων)(24)

Finally, the important relation between the cooperative de-cay rate and dyadic Green’s function allows us to attain thegoal of applying this calculation to a general electromag-netic environment using computational electromagnetics.

Γi, j(ω) =k2

ε πdi · Im

[G(ri,rj;ω)

]·d j (25)

The procedure outlined above presents the essential stepsin the factorization, the details of these calculations will bepublished elsewhere [17].

4 Bound states

In the single atom case, it is shown that a discrete energybelow the electromagnetic continuum exists if [9]:

limω→Ω

+1

ρ(ω)|g(ω)|2 > 0 (26)

Here Ω1 is the cutoff frequency of the electromagnetic con-tinuum, ρ(ω) is the density of states and g(ω) is the cou-pling constants translated to the energy variable. This realexcitation of the system is partly an excited atom and partlya photon packet bound around the atom [8, 9, 18]. In themultiple atom case, existence of these bound states are notimmediately obvious when we inspect the determinant in(15). However, under weak inter-atomic coupling, we ex-pect the bound states around each atom to engage in tight-binding interaction. There will be in general Q such boundstates below the continuum. As the coupling strength in-creases, the shifting of these energies represents a very in-teresting problem, possibly leading to bound states insidethe continuum. These will be demonstrated with numeri-cal examples, which are reserved for a separate publication[17].

Figure 1. Plot of the survival probability (log scale) ofthe excited atom over time showing the effect of boundstate on the spontaneous emission. The interference fromthe atom-photon bound state forbids the excited atom fromcompletely decaying, which is indicated by a non-zero tailin the long time. Varying the transition energy of the freeatom, ωe, as well as the coupling strength between the atomand fields effect the value of the tail [9].

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